diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzmjyi" "b/data_all_eng_slimpj/shuffled/split2/finalzzmjyi" new file mode 100644--- /dev/null +++ "b/data_all_eng_slimpj/shuffled/split2/finalzzmjyi" @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\nDirect imaging has revealed more than a dozen planetary-mass companions around young stars \\citep[for a review see, e.g.,][]{bow16}. Intermediate resolution ($R$$\\gtrsim$1000--2000) infrared spectroscopy of these objects reveals a large number of spectroscopic features that can be compared with those of free-floating objects of similar mass and\/or temperature (e.g., brown dwarfs) as well as atmospheric models to constrain formation and early evolution scenarios. Owing to the mostly high contrast ratios and close separations to their host stars, however, only few planets have been studied spectroscopically at high signal-to-noise ratios (S\/N). A prime candidate for further investigation is the HD\\,106906\\ AB+b system. \n\n\\object{HD 106906} is a close binary star \\citep{lag17} at a distance of 102.8$\\pm$2.5\\,pc \\citep{gai16} in the Lower Centaurus Crux association \\citep[13$\\pm$2\\,Myr][]{pec12}. This binary star is known to harbor a circumstellar disk extending to $>$500\\,AU with a large inner hole and a co-moving low-mass companion at a projected separation of 7\\farcs1 \\citep[$\\sim$730\\,AU;][]{che05,bai14,kal15,lag16}. Previous estimates of the temperature and mass of the companion, based on low-resolution 1--2.5\\,\\mum\\ spectroscopy and 0.6--3.5\\,\\mum\\ photometry, placed it in the planetary-mass regime \\citep[\\Teff=1800\\,K, $M$=11$\\pm$2\\MJup;][]{bai14,wu16}. \n\nThe large distance of \\object{HD 106906 b} to its host has spawned discussion about its formation process. This close binary may either have formed like a star through gravitational collapse of a molecular cloud or it may have formed in the primary's disk. The latter may have happened in situ or closer to the star and scattered to its current position through interaction with the central binary and\/or other stars \\citep[e.g.,][]{rod17}. Alternatively, it may have formed around another star and was scattered \\emph{into} the system through a close encounter early in the history of the association \\citep{par12}.\n\nWe present here new high S\/N infrared spectroscopy of HD\\,106906\\,b. The observation and data reduction are described in Sect.~\\ref{sec:obs}, determinations of spectral type, effective temperature, luminosity, and mass are presented in Sect.~\\ref{sec:results}. We summarize the new findings and discuss their implications in Sect.~\\ref{sec:summary}.\n\n\n\\section{Observations and data reduction}\\label{sec:obs}\nObservations of HD\\,106906\\,b\\ were taken with the SINFONI integral field spectrograph on the Very Large Telescope between December 2014 and March 2015. The primary star HD\\,106906\\ ($V$=7.8\\,mag), 7\\farcs1 from the target, served as natural guide star for the adaptive optics system. We obtained spectra in $J$ (1.10--1.40\\,\\mum), $H$ (1.45--1.85\\,\\mum), and \\Ks\\ (1.95--2.45\\,\\mum) bands with a spatial pixel scale of 125\\,mas\\,$\\times$\\,250\\,mas, resulting in a field of view of 8\\arcsec$\\times$8\\arcsec\\ and a spectral resolution of $R$$\\approx$2000--4000. Object (O) and sky (S) observations followed an OSSOOSS\\dots\\ pattern with a $\\sim$1--2\\arcsec\\ offset between the two positions. The object frames were randomly offset with respect to each other within a radius of $\\sim$1\\arcsec, always keeping the bright primary outside the field of view. Reference stars with spectral types between B3 and B9 for the correction of telluric absorption were observed close in time to each science observation with the same instrumental setup and at similar airmass. A summary of the observation details is given in Table~\\ref{tab:obs}.\n\n\\begin{table*}\n \\caption{Observation summary\\label{tab:obs}}\n\\centering\\small\n\\begin{tabular}{lclccccccccc}\n\\hline\\hline\n &\n &\n &\n &\n &\n seeing &\n Strehl &\n FWHM\\tablefootmark{c} &\n Telluric &\n SpT &\n \\Teff\\tablefootmark{d} &\n airmass\\\\\n UT Date &\n Filter &\n $n_{\\rm exp}$$\\times$$t_{\\rm int}$ &\n $R\\!=\\!\\lambda\/\\Delta\\lambda$\\tablefootmark{a} &\n airmass &\n (arcsec) &\n ratio\\tablefootmark{b} &\n (arcsec) &\n ref. star &\n (ref. star) &\n (ref. star) &\n (ref. star)\\\\\n\\hline\n2014-12-31 & $H$ & 5$\\times$100\\,s & $\\sim$3000 & 1.32 & 1.1 & 10--16 & 0.35 & HIP\\,050038 & B5 & 15200 & 1.32 \\\\ \n2015-01-23 & $H$ & 2$\\times$100\\,s & $\\sim$3000 & 1.18 & 0.6 & $\\sim$38 & 0.32 & HIP\\,055480 & B8 & 11400 & 1.59 \\\\ \n2015-02-23 & $K$ & 4$\\times$300\\,s & $\\sim$4000 & 1.17 & 1.0 & 20--27 & 0.26 & HIP\\,059363 & B9 & 10500 & 1.19 \\\\\n2015-02-26 & $J$ & 3$\\times$300\\,s\\tablefootmark{e} & $\\sim$2000 & 1.21 & 1.2 & 5--15 & 0.46 & HIP\\,055938 & B3 & 19000 & 1.26 \\\\ \n2015-03-05 & $J$ & 6$\\times$300\\,s & $\\sim$2000 & 1.17 & 1.3 & 18--27 & 0.46 & HIP\\,057474 & B7 & 12500 & 1.24 \\\\ \n\\hline\n\\end{tabular}\n\\tablefoot{\n \\tablefoottext{a}{From the SINFONI manual.}\n \\tablefoottext{b}{According to SINFONI's Real Time Computer (RTC).}\n \\tablefoottext{c}{Measured at central wavelength.}\n \\tablefoottext{d}{Converted from spectral type using \\citet{cox_allens_2000}.}\n \\tablefoottext{e}{A total of four spectra were observed and extracted at this epoch, but one showed exceptionally high noise and was excluded from further analysis.}\n}\n\\end{table*}\n\nData reduction partly relies on the SINFONI pipeline (Ver.~2.7.0) in the esorex\\footnote{\\url{http:\/\/www.eso.org\/sci\/software\/cpl\/esorex.html}} environment. We used the standard settings and workflow until one integral field data cube was reconstructed per exposure. This involves dark subtraction, flat fielding, distortion correction, and wavelength calibration. The sky frame closest in time, reduced in the same way, was subtracted from each science frame. Since sky frames were taken with a small spatial offset relative to the science frames, most of these frames contain an image of the science target close to the edge of the detector. In order to avoid over subtraction, some sky frames had to be rejected if their point spread function (PSF) overlapped with the science target. Subsequently, we replaced any pixels flagged by the pipeline as bad by an interpolation of the nearest 6 good pixels in spectral direction using a custom \\emph{IDL} routine. The resulting integral field spectroscopic cubes have 64$\\times$64 pixels\\footnote{The SINFONI image plane is filled with 32$\\times$64 rectangular pixels. To create the cube, every spatial pixel was automatically split by the pipeline into two neighboring square pixels with identical pixel values.} along the spatial and 2202 pixels along the spectral direction. \n\nWe extracted 1D spectra from the 3D cubes with the procedure described in \\citet{dae13}, using apertures with radii identical to 0.8 times the full width at half maximum (FWHM) in each wavelength bin. The FWHM values were determined from Gaussian fits to the trace in the high spatial resolution direction of the cube. Larger apertures could not be used because of contamination of the sky frames with astrophysical sources close to the science target location. We investigated whether any systematic uncertainties are introduced by choosing a small extraction aperture. Using the standard star as a reference, we increased the aperture size from 0.8 to 4$\\times$FWHM and found variations of the extracted flux of $<$10\\% per pixel at all wavelengths. Because of the low S\/N of the target object in each individual wavelength bin, determination of the centroid and FWHM as a function of wavelength use a 2D Gaussian fitting to the bright telluric standard star in every slice of the cube. The extracted trace was shifted to the location of the target by fitting a 2D Gaussian profile to the target PSF after averaging 20--100 slices of the cube in spectral dimension.\n\nThe telluric standard stars were reduced and extracted in the same way as the science observations, but with a larger aperture (4$\\times$FWHM), and divided by a blackbody curve according to their effective temperature (Table~\\ref{tab:obs}). Hydrogen absorption lines were replaced by straight lines along the continuum before dividing the science spectra by the tellurics.\n\nFig.~\\ref{fig:im} shows spectral median images of one representative cube per filter. Extracted and averaged spectra are shown in Fig.~\\ref{fig:spectra}.\n\\begin{figure*}[tbh]\n\\includegraphics[width=0.32\\textwidth]{f1a.eps}\\hfill\n\\includegraphics[width=0.32\\textwidth]{f1b.eps}\\hfill\n\\includegraphics[width=0.32\\textwidth]{f1c.eps}\n\\caption{\\label{fig:im} Examples of individual, fully reduced, and sky-subtracted data cubes of HD\\,106906\\,b\\ in $J$, $H$, and $K$ band, median-collapsed in spectral direction. The bright region in the bottom right is created by the primary star just outside the field of view. Dark patches are visible where the sky image contains stars. An example aperture for the extraction of the spectra ($r$=0\\farcs26) is superimposed on HD\\,106906\\,b\\ on the $K$-band image in red. Panels show the full field of view of SINFONI (8$\\times$8\\arcsec); centering is subject to jitter offset. The color stretch is linear, the orientation of all panels as indicated in the \\Ks\\ image.}\n\\end{figure*}\n\\begin{figure*}[tbh]\n\\centering\n\\includegraphics[width=0.95\\textwidth]{f2a.eps}\\vspace{1ex}\n\\includegraphics[width=0.95\\textwidth]{f2b.eps}\\vspace{1ex}\n\\includegraphics[width=0.95\\textwidth]{f2c.eps}\n\\caption{\\label{fig:spectra}Fully reduced and averaged spectra in $J$ (top), $H$ (middle), and \\Ks\\ band (bottom) binned by a factor of 2. Green spectra show individual observing periods (most recent on top); black are averaged from all individual exposures per filter. Each spectrum has been normalized individually; offsets and flux scales are arbitrary. Regions highlighted in magenta may contain residual noise from the removal of absorption lines (mostly hydrogen) in the telluric standard. The wavelengths of some atomic and molecular absorption features of low-mass stellar objects are highlighted \\citep{cus05}. Regions of strong telluric absorption potentially causing residuals in the reduction are indicated by the black horizontal bars.}\n\\end{figure*}\nAs can be seen in the top two panels of Fig.~\\ref{fig:spectra}, we detect slight differences in the spectral slope of the continuum between the two observing epochs in $J$ and $H$ band, respectively. Normalized at the central wavelength in both filters, the amplitude of the variability at the band edges is $\\pm$$\\sim$10\\%. This may be a consequence of the small extraction apertures or of variable telluric water absorption during the observations or a combination thereof. Variable telluric absorption might apply in particular to our second epoch H-band observation where the largest airmass difference between the science and telluric standard target occurred ($\\Delta$airmass$\\sim$0.4). In addition, intrinsic spectral variability at this level has been previously observed for low-mass objects \\citep[e.g.,][]{apa13}. Since we find no significant difference for any of the extracted parameters in Sect.~\\ref{sec:results} when analyzing the individual epochs, we average all exposures for each filter band and perform all measurements on the combined spectra.\nUncertainties per spectral wavelength bin are calculated as the standard error of the mean between individual exposures of the same observing sequence. The measured S\/Ns are S\/N($J$)$\\approx$20\/pixel, S\/N($H$)$\\approx$20--50\/pix, and S\/N($K$)$\\approx$20--40\/pix.\n\n\\section{Analysis and results}\\label{sec:results}\n\n\\subsection{Spectral characteristics}\nThe spectra in Fig.~\\ref{fig:spectra} show the characteristic triangular shape of the H-band continuum, which is indicative of a young, low-mass, and low surface gravity object. Low gravity is quantitatively supported in the classification scheme by \\citet{all13}, which uses ``gravity scores'' based on spectral indices. Scores of 0, 1, and 2 identify objects with no, normal, and strong indications of low gravity, respectively. We measured FeH$_J$=1.12$^{+0.04}_{-0.03}$, K\\,I$_J$=1.06$\\pm$0.01, and H-cont=1.00$\\pm$0.02 resulting in gravity scores of 1, 1, and 2.\n\nWe see a large number of spectroscopic features that are likely caused by absorption through atoms and simple molecules in the atmosphere of HD\\,106906\\,b. The most prominent features are strong potassium lines between 1.15\\,\\mum\\ and 1.3\\,\\mum\\ as well as clearly detected carbon monoxide bands at $\\gtrsim$2.3\\,\\mum\\ \\citep{cus05}. These are common features of low-mass objects. Weaker, but significantly detected features include sodium, magnesium, and calcium. The rest of the spectrum appears to be dominated by weaker absorption lines that are partly blended. \n\nThe equivalent widths $W_\\lambda=\\int (F_\\lambda-F_\\mathrm{c})\/F_\\mathrm{c} \\,\\,\\mathrm{d}\\lambda$ of a selection of strong ($W_\\lambda>1$\\,\\AA) features are listed in Table~\\ref{tab:ews}. \n\\begin{table}\n \\caption{Equivalent widths of strong ($W_\\lambda>1$\\,\\AA) spectral features\\label{tab:ews}}\n\\centering\n\\begin{tabular}{ccr@{\\dots}lr@{\\,$\\pm$\\,}l}\n \\hline\\hline\n feature &\n $\\lambda_\\mathrm{c}$ (\\mum) &\n \\multicolumn{2}{c}{$\\lambda_{\\rm int}$ (\\mum)\\tablefootmark{a}} &\n \\multicolumn{2}{c}{$W_\\lambda$ (\\AA)} \\\\ \n \\hline\nNa\\,I & 1.138 & 1.1360 & 1.1420 & 7.57 & 0.60 \\\\\nK\\,I & 1.169 & 1.1670 & 1.1710 & 4.81 & 0.17 \\\\\nK\\,I & 1.177 & 1.1750 & 1.1805 & 5.81 & 0.39 \\\\\nK\\,I & 1.243 & 1.2415 & 1.2455 & 5.03 & 0.13 \\\\\nK\\,I & 1.253 & 1.2500 & 1.2550 & 4.26 & 0.26 \\\\\nK\\,I & 1.520 & 1.5150 & 1.5200 & 3.64 & 0.29 \\\\\n$^{12}$CO(6$-$3)& 1.618 & 1.6180 & 1.6230 & 2.63 & 0.16 \\\\\nFeH & 1.625 & 1.6240 & 1.6280 & 2.79 & 0.11 \\\\\nFeH? & 1.650 & 1.6480 & 1.6515 & 1.76 & 0.09 \\\\\nFeH? & 1.656 & 1.6540 & 1.6565 & 2.10 & 0.05 \\\\\nAlI & 2.110 & 2.1075 & 2.1165 & 4.11 & 0.40 \\\\\nNaI & 2.206 & 2.2057 & 2.2105 & 2.92 & 0.17 \\\\\nCaI & 2.26 & 2.2610 & 2.2670 & 2.69 & 0.29 \\\\\nMgI & 2.281 & 2.2800 & 2.2820 & 1.03 & 0.08 \\\\\n$^{12}$CO(2$-$0)& 2.294 & 2.2930 & 2.3100 & 18.6 & 1.1 \\\\\n$^{12}$CO(3$-$1)& 2.323 & 2.3220 & 2.3390 & 11.6 & 1.1 \\\\\n$^{12}$CO(4$-$2)& 2.353 & 2.3520 & 2.3690 & 15.3 & 1.7 \\\\\n\\hline\n\\end{tabular}\n\\tablefoot{\n \\tablefoottext{a}{Integration range to determine equivalent widths, taking into account line center shifts with respect to $\\lambda_\\mathrm{c}$. Line identfication and widths from \\citet{mey98,cus05,dae12,bon14}.}\n}\n\\end{table}\nThe continuum $F_\\mathrm{c}$ was measured with a quadratic polynomial fit to visually selected points along the continuum in regions devoid of strong absorption features. Polynomial fits to predefined sections of the spectrum, as used by \\citet{bon14}, did not lead to satisfactory fits of the continuum. Uncertainties of $W_\\lambda$ were obtained from the standard deviation of repeated measurements with different continuum fits following this recipe.\n\n\\subsection{Spectral type}\nWe determined the spectral type of HD\\,106906\\,b\\ with three different methods: visual spectral comparisons (Sect.~\\ref{sec:vis}), spectral indices (Sect.~\\ref{sec:indic}), and examination of the Na\\,I and K\\,I equivalent widths (Sect.~\\ref{sec:ews}). Based on the inferred spectral types of L1.5--L2, L0.9$\\pm$0.6, and L0$\\pm$2, respectively, we calculated a spectral type of L1.7$\\pm$0.3 (inverse variance-weighted average).\nThis value and its uncertainty is dominated by the visual inspection measurement. Given that we relied on the classifications by \\citet{all13} and \\citet{bon14}, which themselves are subject to systematic and random uncertainties of typically 0.5 to 1 subclasses, we adopted a conservative uncertainty of 1 subclass for our best estimate. In the commonly used scheme of half subclasses we obtain L1.5$\\pm$1.0, which is our best estimate for the spectral type of HD\\,106906\\,b. \n\n\\subsubsection{Visual inspection}\\label{sec:vis}\nWe compared our $JHK$ data to the spectral libraries from \\citet{all13} and \\citet{bon14}, which present and characterize spectra of young late-type objects of low gravity. \\citet{all13} examined 73 ultracool young (10--300\\,Myr) field dwarfs and classified these sources in a spectral type range of M5--L7. Their spectra have low ($R\\approx100$) and medium resolution ($R\\gtrsim750$--2000). \\citet{bon14} focused on a smaller sample of 15 young brown dwarfs mostly in the M6--L0 range. Their spectra were observed with the same instrument as used for our study (SINFONI) at equal spectral resolution in $J$ band (R$\\approx$2000) but slightly lower resolution in $H$ and $K$ ($R$$\\approx$1500).\n\nIn Fig.~\\ref{fig:visualinspection} we compare our observed spectra to the library spectra. Because our spectra were not observed simultaneously and we cannot calibrate the flux of the H-band spectrum owing to an unknown H-band magnitude, we cannot obtain a reliable relative flux calibration of the full $JHK$ spectrum. We thus compare each filter individually. \n\\begin{figure*}[tbh]\n \\centering\n\\includegraphics[width=0.5\\textheight]{f3a.eps}\\vspace{1ex}\n\\includegraphics[width=0.5\\textheight]{f3b.eps}\n\\caption{\\label{fig:visualinspection}Comparison of our spectra (black) with the libraries of young star spectra (red) by \\citet[top]{all13} and \\citet[bottom]{bon14}. The $J$, $H$, and $K$ spectra have been normalized and matched for each band individually.}\n\\end{figure*}\nFor each comparison, our target spectrum has been convolved with a Gaussian profile to match the spectral resolution of the respective library spectrum. The individual results are listed in Table~\\ref{tab:spectralindices}. Best fits range from M8 to L4. Based on the comparison from the \\citet{all13} library, we classify the companion as a L1.5--L2, while the \\citet{bon14} library suggests M8--L0. However, a caveat is that the latter library does not contain spectra with types L0.5--L3.5.\n\n\\begin{table}\n \\caption{Spectral type estimation\\label{tab:spectralindices}}\n\\centering\n\\begin{tabular}{lcc}\n\\hline\\hline\nData set\/Index & Ref. & Spectral type \\\\\n\\hline\n$J$ & 1 & L0.5\\,$\\pm$\\,1.5 \\\\\n$H$ & 1 & L2.5\\,$\\pm$\\,0.5 \\\\\n$K$ & 1 & L1.0\\,$\\pm$\\,1.0 \\\\\n\\multicolumn{2}{l}{{\\bf Weighted Average}\\tablefootmark{a}} & {\\bf L2.1\\,$\\pm$\\,0.4} \\\\\n\\hline\\\\[-1.5ex]\n$J$ & 2 & L0.0\\,$\\pm$\\,1.0 \\\\\n$H$ & 2 & L0.0\\,$\\pm$\\,4.0 \\\\\n$K$ & 2 & M9.5\\,$\\pm$\\,1.5 \\\\\n\\multicolumn{2}{l}{{\\bf Weighted Average}\\tablefootmark{a}} & {\\bf M9.9\\,$\\pm$\\,0.8} \\\\\n\\hline\nFeH & 3 & $>$L0.6\\tablefootmark{b} \\\\ \nH$_{2}$O-1 & 3 & L0.6\\,$\\pm$\\,1.1 \\\\ \nH$_{2}$O & 4 & L1.1\\,$\\pm$\\,0.8 \\\\ \nH$_{2}$O-2 & 3 & L1.0\\,$\\pm$\\,1.6 \\\\ \nH$_{2}$OD & 5 & M7.1\\,$\\pm$\\,0.8\\tablefootmark{b} \\\\ \n\\multicolumn{2}{l}{{\\bf Weighted Average}} & {\\bf L0.9\\,$\\pm$\\,0.6} \\\\\n\\hline\n\\end{tabular}\n\\tablefoot{Spectral types as estimated from visual inspection (top two sections) and spectral indices (bottom).\n \\tablefoottext{a}{Average for a spectral library, weighted by the inverse variance.}\n \\tablefoottext{b}{These measurements do not enter our final spectral type estimate because the FeH index does not cover the entire range of interest for HD\\,106906\\,b\\ \\citep[valid only for M3--L3;][]{sle04} and H$_2$OD is based on data that are strongly affected by residuals from the telluric correction \\citep[1.96--2.08\\,\\mum;][]{mcl03}.}\n }\n\\tablebib{(1) \\citealt{all13}; (4) \\citealt{bon14}; (3) \\citealt{sle04}; (4) \\citealt{all07}; (5) \\citealt{mcl03}.}\n\\end{table}\n\n\\subsubsection{Spectral indices}\\label{sec:indic}\nIn order to be consistent with previous studies, we adopted spectral indices that were used by \\citet{all13} and \\citet{bon14}. Specifically, we explored the H$_{2}$O index near 1.5\\,\\mum\\ \\citep{all07}, H$_{2}$O-1 ($\\sim$1.3\\,\\mum), and \\mbox{H$_{2}$O-2} ($\\sim$2.1\\,\\mum) indices \\citep{sle04} and the H$_{2}$OD index \\citep[$\\sim$2.0\\,\\mum,][]{mcl03}. Since the $J$ band contains many spectral features from various molecules, we also examined the $J$ band FeH spectral index \\citep{sle04}, near 1.2\\,\\mum. The uncertainties of the best inferred value of L2.5$\\pm$1.9, however, extend beyond the valid spectral type range of M3--L3. Thus, we did not include these measurements in our final spectral type estimate.\n\nTable~\\ref{tab:spectralindices} summarizes our results for each spectral index. Uncertainties were propagated from our measurement uncertainty and the quoted uncertainties of the polynomial fits in the original publications. \nCalculating the average of the spectral types determined using the H$_{2}$O, H$_{2}$O-1 and H$_{2}$O-2 indices and weighing the average by the inverse of the square of their uncertainty, we get a spectral type of L0.9$\\pm$0.6. \n\n\\subsubsection{Equivalent widths}\\label{sec:ews}\nOur spectra have a high S\/N and a relatively high resolution. We can use the additional information contained in our data compared to, for example, \\citet{all13} to improve our spectral type determination. We compare the equivalent widths of five $J$-band features of Na\\,I (1.138\\,\\mum) and K\\,I (1.169, 1.177, 1.243, and 1.253\\,\\mum; see Table~\\ref{tab:ews}) with those of other young objects as measured by \\citet{bon14} at the same spectral resolution. We find that the equivalent widths are broadly consistent with an L0$\\pm$2 object. However, the correlation of spectral type with the Na\\,I and K\\,I equivalent widths of young object spectra such as those of HD\\,106906\\,b\\ is weak and has a larger scatter compared to field objects, as is apparent from Fig.~11 in \\citet{bon14}. This diagnostic thus leads to less precise results than for field dwarfs. More independent determinations of precise equivalent widths and spectral types are needed to increase the usefulness of equivalent widths for spectral type determinations of young planetary-mass objects.\n\n\\subsection{Effective temperature and luminosity}\\label{sec:teff}\nTo obtain robust estimates of \\Teff\\ and \\Lbol, we use two strategies. The first determination is based on our best spectral type estimate and uses empirical correlations of \\Teff\\ and \\Lbol\\ with spectral type for young late-type objects. The second approach derives a luminosity of HD\\,106906\\,b\\ based on its brightness, empirically calibrated bolometric corrections, and the new Gaia distance, which is 10\\% larger than pre-Gaia estimates.\n\nOur analysis is based on the analysis of colors, luminosities, and effective temperatures of 152 young brown dwarfs and directly imaged planets by \\citet{fah16}. They have estimated \\Lbol\\ by integrating the available SEDs for their sample of objects between 0 and 1000 microns. Since the distances to these objects are known, \\citeauthor{fah16} have estimated radii using \\Lbol\\ and model isochrones. Then, based on radius and \\Lbol, they have used the Stefan-Boltzmann law to calculate \\Teff. We mostly use the results from the \\citeauthor{fah16} study as it supersedes the smaller young object sample by \\citet{fil15}. The latter study, however, presents an analysis of a comparison sample of 65 field stars as well as bolometric corrections for young low-mass objects that are used in Sect.~\\ref{sec:lum}.\n\n\\subsubsection{Effective temperature}\n\\citet{fah16} have provided polynomial fits to the \\Lbol\\ and \\Teff\\ correlations with spectral type for three categories of low-gravity objects. First, their ``YNG'' sample contains all spectra with indications of low gravity. Second, the ``YNG2'' contains YNG as a subset, but also includes directly imaged planetary-mass objects. Third, the ``GRP'' sample contains confirmed young moving group targets.\nFor our best estimate of the spectral type of HD\\,106906\\,b\\ (L1.5$\\pm$1.0), we obtain $T_{\\rm eff}^{\\rm YNG}$\\,=\\,1920\\,$\\pm$\\,210\\,K, $T_{\\rm eff}^{\\rm YNG2}$\\,=\\,1820\\,$\\pm$\\,240\\,K, and $T_{\\rm eff}^{\\rm GRP}$\\,=\\,1900\\,$\\pm$\\,240\\,K. Uncertainties were derived from both the spectral type uncertainty and the rms of the spectral type-\\Teff\\ relation by \\citet{fah16}. The temperatures are in good agreement with each other and with the \\citeauthor{bai14} value of \\Teff=1800$\\pm$100\\,K derived from evolutionary models. Since HD\\,106906\\,b\\ is best described by the YNG2 sample, we adopt \\Teff=1820\\,$\\pm$\\,240\\,K as our best estimate for its temperature.\n\nThis temperature is lower than what is obtained when assuming that HD\\,106906\\,b\\ is a field object ($T_{\\rm eff}^{\\rm field}$\\,=\\,2030\\,$\\pm$\\,180\\,K). It is also slightly lower than the field object value derived by \\citealt{bai14} of $T_{\\rm eff}^{\\rm field}$=$1950\\pm200$\\,K based on their later spectral type of L2.5$\\pm$1. This is because similar to young brown dwarfs, planets are redder than field dwarfs in the near-infrared bands \\citep[e.g.,][]{fah16}; this is likely explained by high-altitude clouds \\citep[e.g.,][]{bow10,cur11,mad11,mar12,ske12}. \n\n\\subsubsection{Luminosity\\label{sec:lum}}\nUsing the polynomial fits to the empirical luminosity-spectral type relations derived by \\citet{fah16}, we calculate values of $\\log\\left(L_\\mathrm{bol}^\\mathrm{YNG}\/L_\\odot\\right)$\\,=\\,$-$3.83\\,$\\pm$\\,0.35, $\\log\\left(L_\\mathrm{bol}^\\mathrm{YNG2}\/L_\\odot\\right)$\\,=\\,$-$3.64\\,$\\pm$\\,0.24, and $\\log\\left(L_\\mathrm{bol}^\\mathrm{GRP}\/L_\\odot\\right)$\\,=\\,$-$3.47\\,$\\pm$\\,0.31. Uncertainties, as estimated from a Monte Carlo simulation, take into account the spectral type uncertainty and the rms in $\\log L$ \\citep{fah16}. Again, we select YNG2 as the most appropriate value and listed the others to illustrate the range of answers depending on the underlying sample. \n\nFor an independent estimate of the luminosity we also use bolometric corrections of young stars for the $J$ band and \\Ks\\ band, respectively. These are available through polynomial fits to the YNG sample of \\citet{fil15}. With bolometric corrections for the HD\\,106906\\,b\\ $J$ and \\Ks\\ magnitudes of $BC_J$\\,=\\,1.54\\,$\\pm$\\,0.28\\,mag and $BC_{Ks}$\\,=\\,3.27\\,$\\pm$\\,0.13\\,mag, we derive luminosities of $\\log\\left(L_\\mathrm{bol}^{\\mathrm{YNG,BC}_{Ks}}\/L_\\odot\\right)$\\,=\\,$-$3.57\\,$\\pm$\\,0.05 and $\\log\\left(L_\\mathrm{bol}^{\\mathrm{YNG,BC}_J}\/L_\\odot\\right)$\\,=\\,$-$3.74\\,$\\pm$\\,0.11 calculated as\n\\begin{equation}\n \\log\\left(\\frac{L_\\mathrm{bol}}{L_\\odot}\\right)=0.4\\left(M_\\mathrm{bol,\\odot}-m+5\\log\\left(\\frac{d}{10\\,\\mathrm{pc}}\\right)-BC\\right)\\quad,\n\\end{equation}\nusing $M_\\mathrm{bol,\\odot}$=4.74\\,mag. Uncertainties were propagated using a Monte Carlo simulation based on the rms values reported by \\citet[$\\Delta$BC$_{Ks}$=0.126\\,mag, $\\Delta$BC$_{J}$=0.189\\,mag]{fil15} as well as the uncertainties of the distance modulus and our spectral type estimate.\nThe comparably faint luminosity derived from the $J$-band corrections is due to a red $J$$-$\\Ks\\ color, compared to atmospheric models and other targets of this spectral type, which was already observed by \\citet{wu16}. \n\nFrom the three estimates ($L_\\mathrm{bol}^\\mathrm{YNG2}$, $L_\\mathrm{bol}^{\\mathrm{YNG,BC}_J}$, and $L_\\mathrm{bol}^{\\mathrm{YNG,BC}_{Ks}}$) we derive $\\log\\left(L_\\mathrm{bol}\/L_\\odot\\right)$\\,=\\,$-$3.65\\,$\\pm$\\,0.08 as our best estimate of the bolometric luminosity of HD\\,106906\\,b\\ using a Monte Carlo simulation to propagate the uncertainties.\n\n\\subsection{Mass}\\label{sec:mass}\nWe used stellar evolution models to translate our inferred luminosity to mass.\nModel families include DUSTY \\citep{cha00}, BT-Settl \\citep{bar15}, and the Bern Exoplanet Tracks (BEX). The first two feature dusty atmospheres and are classified as ``hot start'' according to their high initial (i.e., post-formation) entropies. The BEX use initial entropies found in self-consistently coupled planet formation and evolution simulations \\citep{mor12}. The models were updated to employ Ames Cond atmospheric boundary conditions \\citep{all01} and include Deuterium burning \\citep{mol12}. The coupled formation and evolution calculations lead in population syntheses to planets with a range of post-formation entropies and deuterium abundances for more massive objects. This allows us to identify initial conditions for the evolution corresponding to the hottest, hot, warm, and cold post-formation states \\citep{mor17}.\n\n\nFig.~\\ref{fig:mass} shows HD\\,106906\\,b\\ with respect to the evolutionary tracks. \n\\begin{figure*}[tbh]\n \\centering\n \\setlength{\\unitlength}{\\textwidth}\n\\begin{picture}(1.0,1.2)\n \\put(0.00,0.80){\\includegraphics[width=\\columnwidth]{f4a.eps}}\n \\put(0.40,1.15){\\fontfamily{phv}\\selectfont DUSTY}\n \\put(0.50,0.80){\\includegraphics[width=\\columnwidth]{f4b.eps}}\n \\put(0.88,1.15){\\fontfamily{phv}\\selectfont BT-SETTL}\n \\put(0.00,0.40){\\includegraphics[width=\\columnwidth]{f4c.eps}}\n \\put(0.36,0.75){\\fontfamily{phv}\\selectfont BEX Hottest}\n \\put(0.50,0.40){\\includegraphics[width=\\columnwidth]{f4d.eps}}\n \\put(0.895,0.75){\\fontfamily{phv}\\selectfont BEX Hot}\n \\put(0.00,0.00){\\includegraphics[width=\\columnwidth]{f4e.eps}}\n \\put(0.38,0.35){\\fontfamily{phv}\\selectfont BEX Cold}\n\\end{picture}\n\\caption{\\label{fig:mass} HD\\,106906\\,b's luminosity and age (filled circle) compared to evolutionary tracks. References and extracted mass estimates are listed in Table~\\ref{tab:mass}. The locations of other young, low-mass companions are shown for comparison \\citep[open symbols, target names in the first panel;][]{cha17,bow16}.}\n\\end{figure*}\nWe adopt an age of 13$\\pm$2\\,Myr, assuming that there is no significant delay of the formation of HD\\,106906\\,b\\ compared to other members of Upper Scorpius and its host star. To estimate the mass of HD\\,106906\\,b, we linearly interpolate each grid at the age and luminosity of HD\\,106906\\,b. Mass estimates, listed in Table~\\ref{tab:mass}, range from $M$\\,=\\,12.5$\\pm$1.0\\,\\MJup\\ to $M$\\,=\\,14.2$^{+0.4}_{-0.9}$\\,\\MJup, depending on the initial entropy of the formation scenario, i.e., whether a hot star or cold start model was used, and on the atmospheric dust model. \nWhen assuming a 3\\,Myr formation timescale \\citep[cf.][]{for05} for HD\\,106906\\,b\\ (i.e., an effective age of 10$\\pm$2\\,Myr), the derived masses decrease by 0--0.7\\,\\MJup\\ (see Table~\\ref{tab:mass}).\n\n\\begin{table}\n \\caption{Masses from isochrone fits\\label{tab:mass}}\n\\centering\n\\begin{tabular}{lccc}\n\\hline\\hline\nModel & Ref. & $M$ (\\MJup) & $M_{t_{\\rm form}=3\\,{\\rm Myr}}$ (\\MJup) \\\\\n\\hline\nDUSTY & 1 & $12.3^{+0.9}_{-0.8}$ & $11.9^{+2.5}_{-0.9}$ \\\\\nBT-SETTL & 2 & $12.3^{+0.8}_{-0.7}$ & $11.9^{+1.7}_{-0.8}$ \\\\\nBEX Hottest & 3 & $12.8^{+1.1}_{-0.8}$ & $12.1^{+1.7}_{-1.1}$ \\\\\nBEX Hot & 3 & $13.1^{+0.8}_{-0.6}$ & $12.6^{+1.3}_{-0.9}$ \\\\\nBEX Cold\/Warm\\tablefootmark{a} & 3 & $14.0^{+0.2}_{-0.5}$ & $14.0^{+0.2}_{-0.4}$ \\\\\n\\hline\n\\end{tabular}\n\\tablefoot{\n \\tablefoottext{a}{The cold and warm evolutionary tracks of the BEX models predict identical luminosities for objects above $\\sim$10\\,\\MJup\\ \\citep{mor17}.}\n}\n\\tablebib{(1) \\citealt{cha00}, (2) \\citealt{bar15}, (3) \\citealt{mor17}.}\n\\end{table}\n\n\n\\subsection{Upper limits on accretion}\nWe do not detect any evidence for line emission, in particular not in the accretion-sensitive hydrogen features, i.e., Paschen-$\\beta$ (Pa$\\beta$, 1.282\\,\\mum) and Brackett-$\\gamma$ (Br$\\gamma$, 2.167\\,\\mum), or in the limit of the Brackett series ($H$ band). Accordingly, we do not see evidence for strong accretion activity onto HD\\,106906\\,b. Assuming magnetospheric accretion from an isolated circumplanetary disk onto HD\\,106906\\,b,\\ we infer a quantitative upper limit for the mass accretion rate based on the upper limit for the accretion luminosity $L_\\mathrm{acc}$ and empirical $L_\\mathrm{acc}$--$L_\\mathrm{line}$ relations derived for T~Tauri stars \\citep{alc14}.\n\nWe use the description by \\citet{gul98} to estimate a mass accretion rate from an accretion luminosity using\n\\begin{equation}\\label{eq:gul98}\n \\dot{M} = \\left(1-\\frac{R_{\\rm pl}}{R_\\mathrm{in}}\\right)^{-1}\\frac{L_\\mathrm{acc}\\,R_{\\rm pl}}{G\\,M_{\\rm pl}}\\quad.\n\\end{equation}\nFollowing \\citet{lov11}, we conservatively set the inner disk truncation radius $R_\\mathrm{in}$=2.7\\,$R_{\\rm pl}$. Larger radii, such as $R_\\mathrm{in}$=5\\,$R_{\\rm pl}$, which are commonly used for T Tauri stars, result in stronger constraints on the inferred upper limit of the mass accretion rate of up to a factor of 1.5. The mass and radius of the companion were set to $M_{\\rm pl}$=12.3\\,$\\pm$\\,0.8\\,\\MJup\\ and $R_{\\rm pl}$=1.6$\\,R_\\mathrm{Jup}$, which is consistent with our derived luminosity and temperature. Flux calibration of the spectra was achieved by normalizing the spectra, after multiplication with the spectral response curves of the respective filters, to the $J$ and \\Ks\\ magnitudes measured by \\citet{bai14}\\footnote{No $H$-band magnitudes have been published so far, so this technique could not be applied to our $H$-band spectrum.}. The measured continuum noise values in Pa$\\beta$ and Br$\\gamma$ of $\\Delta F {\\rm d}\\lambda$(Pa$\\beta$)\\,=\\,1.4$\\times$$10^{-18}$\\,erg\/s\/cm$^2$ and $\\Delta F {\\rm d}\\lambda$(Br$\\gamma$)\\,=\\,1.6$\\times$$10^{-18}$\\,erg\/s\/cm$^2$ translate to upper limits of the line luminosity $L_\\mathrm{line}$\\,=\\,$4\\pi F_\\mathrm{line}d^2$ of $\\log(L_{\\rm line}\/L_\\sun)$\\,$<$\\,$-8.78$ and $<$\\,$-8.95$, respectively (99\\% confidence). Using the $L_\\mathrm{acc}$--$L_\\mathrm{line}$ correlations listed \\citet{alc14} we infer $\\log(L_\\mathrm{acc}\/L_\\sun)$\\,<\\,$-5.63$ (Pa$\\beta$) and $\\log(L_\\mathrm{acc}\/L_\\sun)$\\,<\\,$-6.20$ (Br$\\gamma$). Using eq.~(\\ref{eq:gul98}) we infer upper limits for the mass accretion rate of $\\dot{M}<1.8\\times10^{-9}$\\,\\MJup{}yr$^{-1}$ (Pa$\\beta$) and $\\dot{M}<4.8\\times10^{-10}$\\,\\MJup{}yr$^{-1}$ (Br$\\gamma$).\n\nUncertainties and upper limits were calculated using a Monte Carlo simulation taking into account the uncertainties in measured flux, mass, distance, and the empirical $L_\\mathrm{acc}$--$L_\\mathrm{line}$ relations \\citep{alc14}. We conservatively account for systematic uncertainties such as the flux calibration and the subtraction of hydrogen absorption in the telluric standard star spectra by increasing the spectral noise by a factor of 2. All quoted upper limits are 99\\% confidence limits. \nSince the mass accretion rate as calculated from Br$\\gamma$ results in the more restrictive limit, we quote $\\dot{M}<4.8\\times10^{-10}$\\,\\MJup{}yr$^{-1}$ as our best estimate of the mass accretion rate for HD\\,106906\\,b.\n\nWe make a prediction for the expected accretion-induced line luminosity at the wavelength of H$\\alpha$ based on the derived line luminosity (using the more restrictive limit $\\log(L_\\mathrm{acc}\/L_\\sun)$\\,<\\,$-6.20$~). We use the $L_\\mathrm{acc}$--$L_\\mathrm{line}$ relation from \\citet{alc14} to derive $\\log(L_{\\rm H\\alpha}\/L_\\sun)$\\,$<$\\,$-6.69$ (99\\% confidence). \n\n\n\\section{Summary and discussion}\\label{sec:summary}\nWe present new high S\/N (20--50\/pix) intermediate-resolution VLT\/SINFONI 1--2.5\\,\\mum\\ integral field spectroscopy of HD\\,106906\\,b, which is a companion at the deuterium burning limit in the Upper Scorpius association.\nWe detect a number of spectral features indicative of a low mass for HD\\,106906\\,b, and also the H-band continuum shows the typical triangular shape of low gravity objects.\nComparison with spectral libraries of young and field objects as well as analyzing spectral features suggests a spectral type of L1.5$\\pm$1.0. Previous estimates based on low-resolution spectroscopy classified this object as either L2 (based on $H$-band spectroscopy) or L3 \\citep[$K$-band spectroscopy;][]{bai14}. \n\nWe use this new information and the recently published distance of HD\\,106906\\,b\\ \\citep[102.8$\\pm$2.5\\,pc]{gai16} to derive a luminosity of $\\log\\left(L_\\mathrm{bol}\/L_\\odot\\right)=-3.65\\pm0.08$ and an effective temperature of \\Teff\\,=\\,1820\\,$\\pm$\\,240\\,K; this recently published distance of this binary star is $\\sim$10\\% larger than the previous best estimate from the Hipparcos mission. Based on its luminosity and age compared to predictions from evolutionary models, we estimate a mass between $M$\\,=\\,12.3$^{+0.8}_{-0.7}$\\,\\MJup\\ (hot start) and $M$\\,=\\,14.0$^{+0.2}_{-0.5}$\\,\\MJup\\ (cold start) in the limit of zero formation time. Mass predictions are slightly lower if a formation timescale of 3\\,Myr is assumed of between $M$\\,=\\,$11.9^{+2.5}_{-0.9}$\\,\\MJup\\ and $M$\\,=\\,14.0$^{+0.2}_{-0.4}$\\,\\MJup.\n\nThe formation and early evolution of HD\\,106906\\,b\\ remain unclear. The low mass ratio of $q$$\\sim$0.004 with respect to the host system \\citep[$M_{\\rm HD106906AB}$\\,=\\,1.37\\,\\Msun+1.34\\,\\Msun;][]{lag17} suggests ``planet-like'' formation in a protoplanetary disk \\citep{pep14,reg16}. Formation through core accretion, however, is restricted to orbits much closer to the star than its current separation of $>$700\\,AU \\citep[e.g., $\\lesssim$35\\,AU in the simulations by][]{dod09}. Accordingly, subsequent migration would be required. Currently, however, no conclusive evidence for scattering with an internal planet or the central binary has been found \\citep[cf.][]{jil15}. An ejection scenario with subsequent braking through a stellar fly-by appears unlikely in the light of the simulations by \\citet{rod17}. While not strictly excluded, in situ formation through disk instability appears unlikely (though not excluded) because large disks $>$700\\,AU have been rarely observed around forming stars. Formation of HD\\,106906\\,b\\ in a star-like channel, i.e., through direct collapse of the parent cloud core into a triple system HD\\,106906{}AB+b, remains a possibility.\n\nThe degeneracy between the various formation scenarios could be broken when the orbit of HD\\,106906\\,b\\ around its host system becomes known. Its long orbital period of $>$3000\\,yr \\citep{jil15}, however, precludes precise direct measurement of the motion of HD\\,106906\\,b. Indirect information on the orbit of HD\\,106906\\,b, however, may be gathered from observations of the circumstellar disk whose eccentricity is thought to reflect interaction with the companion \\citep{nes17}. Alternatively, additional high precision spectroscopy in the atmospheric absorption bands (e.g., near 1.4\\,\\mum\\ and 2\\,\\mum) and at mid-IR wavelengths together with retrieval analysis \\citep[e.g.,][]{lin12} may be used to measure the chemical composition of the atmosphere of HD\\,106906\\,b, which is a function of its birth location. For example, an enhanced carbon-to-oxygen ratio suggests a formation in a disk close to the host star \\citep[e.g.,][]{obe11}. Direct collapse, in contrast, would produce a stellar abundance ratio. Future observation with high-precision spectroscopy instruments such as those on board the James Webb Space Telescope will help to break these degeneracies.\n\nWhile the presented spectra cover a number of hydrogen features (Paschen-$\\beta$, Brackett-$\\gamma,$ and higher), none of these features appears in emission. We thus exclude magnetospheric accretion onto HD\\,106906\\,b\\ at a rate of $\\dot{M}>4.8\\times10^{-10}$\\,\\MJup{}yr$^{-1}$ (99\\% confidence). With this measurement we can exclude accretion at a level several times below the accretion levels that have been observed for other targets at a similar mass and age as HD\\,106906\\,b. For example, \\citep{joe13} find a mass accretion rate of $\\dot{M}=8\\times10^{-9}$\\,\\MJup{}yr$^{-1}$ for OTS\\,44, a free-floating 12\\,\\MJup\\ member of the $\\sim$2\\,Myr-old Chamaeleon\\,I region. DH\\,Tau, an 11\\,\\MJup\\ companion at $\\sim$340\\,AU from its $\\sim$1--2\\,Myr-old parent star, is thought to accrete at a rate of $\\dot{M}=3.3\\times10^{-9}$\\,\\MJup{}yr$^{-1}$ \\citep{zho14}. For comparison with optical studies and as a reference for optical follow up, we predict an accretion-induced H$\\alpha$ line luminosity of $\\log(L_{\\rm H\\alpha}\/L_\\sun)$\\,$<$\\,$-6.69$. Previous observations of substellar companion candidates in H$\\alpha$ returned values of $\\log(L_{\\rm H\\alpha}\/L_\\sun)$\\,$=$\\,$-4.2$ \\citep[LkCa\\,15b;][]{sal15} and $\\log(L_{\\rm H\\alpha}\/L_\\sun)$\\,$=$\\,$-3.3$ \\citep[HD\\,142527B;][]{clo14}, again significantly larger than our limit for HD\\,106906\\,b. If HD\\,106906\\,b\\ were accreting, its rate must be significantly below that of these comparison objects. We thus conclude that HD\\,106906\\,b\\ features no significant amount of circumplanetary gas. However, the present data do not exclude the presence of a \\emph{gas-poor} disk around HD\\,106906\\,b, similar to what has been observed for the primary object HD\\,106906\\ \\citep{kal15,lag16}. In fact, \\citet{kal15} present weak evidence for the presence of circumplanetary dust around HD\\,106906\\,b\\ based on color excesses and a marginally spatially extended image of the source. \n\nOur new study makes HD\\,106906\\,b\\ one of the few low-\\Teff\\ companions for which high S\/N spectroscopy could be observed. As demonstrated in Fig.~\\ref{fig:models1}, most of the observed features in the spectrum must be real rather than due to measurement uncertainties\\footnote{We note that the selected model temperature is slightly below our best estimate of \\Teff=1820\\,K. This selection was made due to a bimodal $\\chi^2$ distribution of BT-SETTL model fits in \\logg--\\Teff\\ space which has local minima at \\Teff$\\approx$1600\\,K and \\Teff$\\sim$2600\\,K with a particularly poor representation of the K-band spectrum of HD\\,106906\\,b\\ by \\Teff$\\approx$1700--2100\\,K models. Detailed model fitting to explore reasons for this bimodality is beyond the scope of the current paper.}.\n\\begin{figure*}[tbh]\n \\centering\n \\setlength{\\unitlength}{\\textwidth}\n\\begin{picture}(1.0,0.7)\n \\put(0.00,0.00){\\includegraphics[width=1\\textwidth]{f5.eps}}\n \\put(0.06,0.65){\\fontfamily{phv}\\Large\\selectfont a)}\n \\put(0.06,0.48){\\fontfamily{phv}\\Large\\selectfont b)}\n \\put(0.06,0.3){\\fontfamily{phv}\\Large\\selectfont c)}\n \\put(0.06,0.135){\\fontfamily{phv}\\Large\\selectfont d)}\n\\end{picture}\n\\caption{\\label{fig:models1}{\\bf (a)} \\Teff=1600\\,K, \\logg=3.5 BT-SETTL model \\citep[green curve;][]{bar15}, smoothed to the resolution of our K-band spectrum HD\\,106906\\,b\\ (black) and scaled to the same average flux. Most features observed in the spectra are reproduced by the (virtually noise-free) model, supporting evidence that the measured features are caused by atmospheric absoprtion rather than noise. {\\bf (b)} Residuals are shown.\\ {\\bf (c)} and {\\bf (d)} show the same for a shorter wavelength range.}\n\\end{figure*}\nThese high fidelity observations were possible with current instrumentation because of the large angular separation of HD\\,106906\\,b\\ from its host star. Most other directly imaged systems feature much smaller separations, which reduce the achievable S\/N due to additional light from the primary. The current spectrum thus serves as a reference spectrum for future studies of young low-gravity objects in orbit around low-mass stars, in particular with regard to the higher sensitivity of the James Webb Space Telescope and the higher angular resolution of the upcoming generation of 30 m-class telescopes.\n\n\\begin{acknowledgements}\n We thank the anonymous referee for a thorough review and valuable suggestions for improvement.\n We thank Michael Line for valuable contributions in preparation of the observing time proposal. \n This work has been carried out within the frame of the National Centre for Competence in Research PlanetS supported by the Swiss National Science Foundation. S.D., S.P.Q., and G.-D.M.\\ acknowledge the financial support of the SNSF. C.M. and G.-D.M. acknowledge the support from the Swiss National Science Foundation under grant BSSGI0$\\_$155816 ``PlanetsInTime''. \n This work has made use of data from the European Space Agency (ESA) mission {\\it Gaia} (\\url{http:\/\/www.cosmos.esa.int\/gaia}), processed by the {\\it Gaia} Data Processing and Analysis Consortium (DPAC; \\url{http:\/\/www.cosmos.esa.int\/web\/gaia\/dpac\/consortium}). Funding for the DPAC has been provided by national institutions, in particular the institutions participating in the {\\it Gaia} Multilateral Agreement.\n\\end{acknowledgements}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section*{Introduction}\n\nIn this paper we investigate the extremal behavior of free resolutions of subschemes of complete intersections $X \\subseteq \\mathbb{P}^n$.\nOur motivating question is the following.\nLet $\\mathbf{d} = (d_1, \\ldots, d_c)$ be a degree sequence and $p(\\zeta)$ a Hilbert polynomial: \nare there uniform bounds on the syzygies of $Z\\subseteq X$, where $X\\subseteq \\mathbb{P}^n$ is a complete intersection of degrees $\\mathbf{d}$ and $Z\\subseteq X$ a closed subscheme with Hilbert polynomial $p(\\zeta)$?\n\nIn order to address this problem, we study Hilbert schemes of Clements-Lindstr\\\"om schemes, \ni.e. complete intersections $Y \\subseteq \\mathbb{P}^n$ defined by the vanishing of $x_1^{d_1}, \\ldots, x_{c}^{d_{c}}$.\nThey include $Y = \\mathbb{P}^n$ as special case, which is in fact interesting and non-trivial for most of our considerations.\nOur main contributions revolve around a new distinguished point on the Hilbert scheme $\\mathrm{Hilb}^{p(\\zeta)}(Y)$, called the \\emph{expansive point} (or subscheme, or ideal) and denoted by $\\mathrm{Exp}(p(\\zeta))$.\nWe adopt an abstract recursive approach in defining $\\mathrm{Exp}(p(\\zeta))$, based on seven axioms related to hyperplane sections, cf. Theorem \\ref{TheoremDefinitionExpansiveAxioms}.\nIn a sense, this gives rise to a theory of expansive ideals and Hilbert polynomials, which parallels the theory of lexicographic ideals and Hilbert functions.\n\nOur main result, Theorem \\ref{TheoremExtremalFiniteResolution}, states that $\\mathrm{Exp}(p(\\zeta))$ attains the largest possible number of $i$-th syzygies for a subscheme in $\\mathrm{Hilb}^{p(\\zeta)}(Y)$, for every homological degree $i$. \nNo such theorem exists for \\emph{graded} syzygies, since each Hilbert scheme has several maximal graded Betti tables. \nWe remark that considering expansive subschemes of Clements-Lindstr\\\"om schemes $Y$ for various degree sequences $\\mathbf{d}$, as opposed to just for $\\mathbb{P}^n$,\ncarries advantages.\nFirst, \nby taking the degree sequence into account, and restricting thus to a smaller Hilbert scheme,\none obtains sharper numerical bounds on Betti numbers.\nA similar point of view is adopted e.g. in \\cite{EiGrHa93}, where \n bounds on the number of points in intersections of quadric hypersurfaces are improved using the data of the degree sequence.\nMore importantly, \nour main result extends conjecturally to arbitrary complete intersections of $\\mathbb{P}^n$.\nIn fact we show that, under the validity of the Lex Plus Powers Conjecture, $\\mathrm{Exp}(p(\\zeta))$ yields uniform bounds for the syzygies of subschemes $Z\\in \\mathrm{Hilb}^{p(\\zeta)}(X)$ for \\emph{all} complete intersections $X\\subseteq \\mathbb{P}^n$ of degrees $\\mathbf{d}$,\nthus giving a complete answer to our motivating problem.\nSee Proposition \\ref{PropositionLPP} and Theorem \\ref{TheoremLPP}.\n \nWe apply the theory of expansive ideals also to infinite free resolutions over complete intersections, motivated by the recent progress in this area.\nOur second main result, Theorem \\ref{TheoremIfninite}, shows that, over a quadratic Clements-Lindstr\\\"om ring of characteristic 0, expansive ideals achieve extremal Betti numbers for the infinite free resolution.\nWe conjecture that this pattern holds for arbitrary degree sequences and base field.\n\nIn the case $Y=\\mathbb{P}^n$, Theorem \\ref{TheoremExtremalFiniteResolution} gives a new proof of \\cite[Theorem 1.1]{CaMu13}, \nwhich asserts the existence of a subscheme in $\\mathrm{Hilb}^{p(\\zeta)}(\\mathbb{P}^n)$ with extremal Betti numbers.\nThe authors remark in \\cite[Introduction]{CaMu13} that the proof, of combinatorial nature, is very long and complicated, and it would be desirable to have a better understanding of the structure and construction of such extremal subschemes.\nWe believe that, with the method developed in this work, we have found a satisfactory answer.\nIn fact, besides providing a short and more conceptual proof, \n the axioms of Theorem \\ref{TheoremDefinitionExpansiveAxioms} can be used to further illuminate the structure of expansive ideals.\nIn particular, we prove in Theorem \\ref{TheoremChainExpansive} that expansive ideals form descending chains of inclusions, \n starting with a saturated lex ideal, and each step of the chain is described explicitly. \n This fact serves as the basis for an efficient algorithm to compute $\\mathrm{Exp}(p(\\zeta))$.\nThe problem of finding an algorithm to determine a subscheme in $\\mathrm{Hilb}^{p(\\zeta)}(\\mathbb{P}^n)$ with maximal syzygies had been suggested also in \\cite[Section 5]{MoNa14}.\n\n\nBorel-fixed points have proved helpful in understanding the geometry of the Hilbert scheme, \ne.g. in questions of connectedness, smoothness, rationality, enumeration of components, and defining equations, see for instance\n\\cite{BeLeRo13,LeRo11,PeSt05,Ra19,Re95,ReSt97}.\nSeveral problems in this area remain open.\nOur work identifies a new distinguished Borel-fixed point, that is very different from the well-known lex point in many respects.\nWe hope that the notion of expansive point may lead to new perspectives or applications in the geometry of Hilbert schemes.\n \n\n\n\n\\section{Clements-Lindstr\\\"om rings}\\label{SectionPreliminaries}\n\nThis section serves the purpose of fixing the basic terminology for the paper.\n We introduce the rings that are central to this work, \n and some special classes of ideals. \n\nLet $\\mathbb{N}$ denote the set of nonnegative integers.\nThe symbol $\\Bbbk$ denotes an arbitrary field. \nAll rings considered in this work are Noetherian $\\mathbb{Z}$-graded $\\Bbbk$-algebras generated in degree 1, and all ideals and modules are graded;\nthese attributes are often assumed implicitly and omitted.\n\nIf $V$ is a $\\mathbb{Z}$-graded $\\Bbbk$-vector space, denote the $j$-th graded component by $[V]_j$.\nThe numerical function \n$\\mathrm{HF}(V) : \\mathbb{Z} \\rightarrow \\mathbb{N} \\cup \\{\\infty\\}$ defined by $\\mathrm{HF}(V,j) = \\dim_\\Bbbk[V]_j$ for all $j \\in \\mathbb{Z}$ is called the\n{\\bf Hilbert function} of $V$.\nIf there is a numerical polynomial $\\mathrm{HP}(V) \\in \\mathbb{Q}[\\zeta]$ such that \n $\\mathrm{HP}(V,j) = \\mathrm{HF}(V,j)$ for all $j \\gg 0$, then $\\mathrm{HP}(V) $ is called the {\\bf Hilbert polynomial} of $V$.\n\nThe maximal ideal of a ring $A$ is denoted by $\\mathfrak{m}_A$.\nAn ideal $I\\subseteq A$ is {\\bf saturated} if $I : \\mathfrak{m}_A = I$, \nequivalently, if $\\mathrm{depth}(A\/I)>0$;\nnotice that the unit ideal $I=A$ is saturated.\nThe saturation of $I \\subseteq A $ is defined as $I : \\mathfrak{m}_A^\\infty = \\cup_{t \\geq 0} I : \\mathfrak{m}_A^t$,\nand it is a saturated ideal with \n$\\mathrm{HP}(I : \\mathfrak{m}_A^\\infty) =\\mathrm{HP}(I)$.\n\nGiven a projective scheme $X= \\mathrm{Proj} A$ and a polynomial $p(\\zeta)\\in \\mathbb{Q}[\\zeta]$,\nthe {\\bf Hilbert scheme}, denote by $\\mathrm{Hilb}^{p(\\zeta)}(X)$, is the scheme parametrizing the closed subschemes $Z\\subseteq X$ with $\\mathrm{HP}(Z) =p(\\zeta)$.\nAs it is common in the literature, \nwe often identify a closed subscheme $Z \\subseteq X$ with its saturated ideal $I_Z \\subseteq A$ and with the point on the Hilbert scheme parametrizing it,\nand sometimes we extend attributes of one object to the other two.\nFor instance we may talk about strongly stable subschemes or lex points on the Hilbert scheme,\nand we adopt the following:\n\n\\begin{convention}\nIf $I\\subseteq A$ is an ideal, the expression ``$I\\in \\mathrm{Hilb}^{p(\\zeta)}(\\mathrm{Proj} A)$'' means that $I$ is saturated and $\\mathrm{HP}(A\/I) =p(\\zeta)$.\n\\end{convention}\n\nLet $A$ be a ring and $M$ a finite $A$-module.\nThe integers \n$\\beta_{i,j}^A(M) = \\dim_\\Bbbk[\\mathrm{Tor}_A^i(M,\\Bbbk)]_j$ and $\\beta_{i}^A(M) = \\dim_\\Bbbk\\mathrm{Tor}_A^i(M,\\Bbbk)$\nare the {\\bf graded Betti numbers} and the {\\bf (total) Betti numbers} of $M$, respectively.\n\n\\begin{convention}\\label{ConventionInfinity}\nWe will often use $\\mathbb{N}\\cup \\{\\infty\\}$ as index set and as range for exponents. \nWe adopt standard conventions on $\\infty$, namely that $m < \\infty $ and $\\infty-m = \\infty$ for all $m \\in \\mathbb{N}$.\nIf $r$ is an element in a ring then we set $r^\\infty := 0$.\nIf $d = \\infty$, the expression ``$\\ell < d$'' means ``$\\ell\\in \\mathbb{N}$''.\n\\end{convention}\n\n\n\\begin{definition}\nA \\textbf{Clements-Lindstr\\\"om ring} is a ring of the form \n$$\nA = \\frac{\\Bbbk [x_1, \\ldots, x_{m}]}{\\big(x_1^{d_1}, \\ldots, x_m^{d_{m}}\\big)}\n$$\nfor some sequence of integers $d_1 \\leq d_2 \\leq \\cdots \\leq d_{m}$ with $d_i \\in \\mathbb{N}\\cup \\{\\infty\\}$\nWe emphasize that $x_i^\\infty := 0$. \nThus, when $d_1 = \\infty$ the ring $A$ is simply a polynomial ring.\nOn the other hand, when $d_{m} < \\infty$ the ring $A$ is Artinian, \nand its only saturated ideal is the unit ideal.\n\\end{definition}\n\nFor the remainder of this section, let $A$ denote an arbitrary Clements-Lindstr\\\"om ring.\n\nAn ideal $I \\subseteq A$ is monomial if it is the image of a monomial ideal of $\\Bbbk [x_1, \\ldots, x_{m}]$. \nWe denote by $<_{\\mathrm{lex}}$ the lexicographic monomial order in $A$ induced by $x_1 > x_2 > \\cdots > x_{m}$.\nA monomial ideal $I \\subseteq A$ is \\textbf{lex} if $[I]_j$ is a vector space generated by an initial segment of monomials with respect to $<_{\\mathrm{lex}}$ for every $j$,\nequivalently, if $I$ is the image of a lex ideal of $\\Bbbk [x_1, \\ldots, x_{m}]$.\nThe saturation of a lex ideal is again lex.\nA theorem of Clements and Lindstr\\\"om, which includes classical results of Macaulay and Kruskal-Katona as special cases,\n states that lex ideals classify Hilbert functions in $A$:\n\n\\begin{prop}[\\cite{ClLi69}]\\label{PropositionCL}\nLet $A$ be a Clements-Linstr\\\"om ring and $I\\subseteq A$ an ideal.\nThere exists a unique lex ideal $L \\subseteq A$ such that $\\mathrm{HF}(L) = \\mathrm{HF}(I)$.\n\\end{prop}\n\nIf $\\mathcal{H}:\\mathbb{Z}\\rightarrow \\mathbb{N}$ is the Hilbert function of some ideal of $A$, we denote by ${\\mathrm{Lex}}(\\mathcal{H},A)$ the unique lex ideal $L\\subseteq A $ with $\\mathrm{HF}(L) = \\mathcal{H}$.\nIf $I \\subseteq A$ we define ${\\mathrm{Lex}}(I):={\\mathrm{Lex}}(\\mathrm{HF}(I), A)$. \n\nA monomial ideal $I \\subseteq A $ is {\\bf almost lex} if the last variable $x_{m}$ is a non-zerodivisor on $A\/I$ and\n $\\frac{I+(x_{m})}{(x_{m})}$ is a lex ideal of the Clements-Lindstr\\\"om ring $\\frac{A}{(x_{m})}$.\nIn particular, almost lex ideals are saturated.\nObserve that a lex ideal is not, in general, almost lex.\n\nA monomial ideal $I \\subseteq A$ is \\textbf{strongly stable} if for every nonzero monomial $\\mathbf{u} \\in I$ and $x_h $ dividing $\\mathbf{u}$,\nthen $\\frac{x_k \\mathbf{u}}{x_h}\\in I$ for all $k 0$, this is equivalent to the fact that $x_{m}$ does not divide any monomial minimal generator of $I$.\nThe saturation of a strongly stable ideal is again strongly stable.\nBoth lex ideals and almost lex ideals are strongly stable.\n\n\\begin{examples}\nLet $d_1=2, d_2 = 3, d_3=d_4 = \\infty$.\nThe associated Clements-Lindstr\\\"om ring is $A = \\frac{\\Bbbk[x_1, x_2, x_3, x_4]}{(x_1^2, x_2^3)}$.\nConsider the following ideals of $A$:\n\\begin{itemize}\n\\item $\\big(x_1x_2^2,\\, x_1x_2x_3 \\big)$ is both lex and almost lex;\n\\item $\\big(x_1x_2,\\, x_1x_3,\\, x_1x_4^2,\\, x_2^2x_3\\big)$ is lex but not almost lex, since it is not saturated;\n\\item $\\big( x_1 x_2,\\, x_1x_3,\\, x_2^2\\big)$ is almost lex but not lex, since $x_1x_4 >_{{\\mathrm{lex}}} x_2^2$;\n\\item $\\big( x_1 x_2,\\, x_2^2\\big)$ is strongly stable and saturated, but neither lex nor almost lex.\n\\end{itemize}\n\\end{examples}\n\n\\begin{remark}[The lex point]\\label{RemarkUniqueLex}\nFor every $p(\\zeta) \\in \\mathbb{Q}[\\zeta]$ such that $\\mathrm{Hilb}^{p(\\zeta)}(\\mathrm{Proj} A)\\ne \\emptyset$, \nthere is exactly one lex ideal in $\\mathrm{Hilb}^{p(\\zeta)}(\\mathrm{Proj} A)$.\nThe show existence,\ntake $I \\in \\mathrm{Hilb}^{p(\\zeta)}(\\mathrm{Proj} A)$ and let $L = {\\mathrm{Lex}}(I):\\mathfrak{m}_A^\\infty$.\nIt follows that $L$ is a saturated lex ideal of $A$ with $\\mathrm{HP}(I)=\\mathrm{HP}(L)$.\nIf $L,L'$ are two saturated lex ideals with $\\mathrm{HP}(L)=\\mathrm{HP}(L')$, \nthen $\\mathrm{HF}(L,d) = \\mathrm{HF}(L',d)$ for $d \\gg 0$, thus $[L]_d=[L']_d$.\nLet $K= \\big([L]_d\\big) \\subseteq A$, \nit follows that $L = L' = K : \\mathfrak{m}_A^\\infty$, proving uniqueness.\nIn the case of $\\mathrm{Hilb}^{p(\\zeta)}(\\mathbb{P}^n)$\nit is known that the lex point is smooth \\cite{ReSt97},\nhowever this is unknown for Clements-Lindstr\\\"om schemes \\cite{PeSt07}.\n\\end{remark}\n\nIf $\\mathrm{Hilb}^{p(\\zeta)}(\\mathrm{Proj} A)\\ne \\emptyset$, we denote by ${\\mathrm{Lex}}(p(\\zeta),A)$ the unique lex ideal in $\\mathrm{Hilb}^{p(\\zeta)}(\\mathrm{Proj} A)$.\nWe emphasize that the lex ideal of a Hilbert function and the lex ideal of a Hilbert polynomial are different concepts, and both are relevant for this work. \nThe notation ${\\mathrm{Lex}}(I)$ is reserved for the lex ideal with the same Hilbert function as the ideal $I$.\n\nBy Remark \\ref{RemarkUniqueLex} the set of monomial subschemes in $\\mathrm{Hilb}^{p(\\zeta)}(\\mathrm{Proj} A)$ is non-empty whenever $\\mathrm{Hilb}^{p(\\zeta)}(\\mathrm{Proj} A) \\ne \\emptyset$.\nOn the other hand, this set is always finite, \nas the next discussion shows.\n\n\\begin{remark}\\label{RemarkFinitelyManyForEachHP}\nThere are finitely many monomial subschemes in each $\\mathrm{Hilb}^{p(\\zeta)}(\\mathrm{Proj} A)$.\nTo see this, since the preimage in the polynomial ring of a saturated monomial ideal of $A$ is again saturated and monomial, \nit suffices to treat the case when $A$ is a polynomial ring. \nThere is a well-known upper bound, due to Gotzmann \\cite{Go78}, for the Castelnuovo-Mumford regularity of a saturated ideal $J\\subseteq A$ in terms of $\\mathrm{HP}(A\/J)$.\nThis implies the desired conclusion, since there are finitely many monomial ideals generated in bounded degrees.\n\\end{remark}\n\nWe remark that there are algorithms to produce all the strongly stable points or almost lex points of $\\mathrm{Hilb}^{p(\\zeta)}(\\mathbb{P}^n)$,\nsee for instance \\cite{AlLe18,CiLeMaRo11,MoNa14}. \nThese algorithms can be extended with minor modifications to the case of Clements-Lindstr\\\"om rings $A$.\n\n\n\n\n\\begin{example}\nLet $d_1=2, d_2=3$, $d_3=d_4=\\infty$ and consider \nthe associated Clements-Lindstr\\\"om ring $A = \\frac{\\Bbbk[x, y, z, w]}{(x^2, y^3)}$.\nFor $p(\\zeta) = 3\\zeta+5$ the strongly stable ideals in $\\mathrm{Hilb}^{p(\\zeta)}(\\mathrm{Proj} A)$ are\n \\begin{align*}\n & \\left(xy, xz^5 \\right)&\n&\\left( xy^2, xyz^2, xz^3\\right)&\n &\\left( xy^2, xyz, xz^4\\right)&\n \\\\\n & \\left( xy^2, xyz, y^2z^3\\right)&\n&\\left( xy^2, xyz^2, y^2z^2\\right)&\n& \\left( xy, y^2z^4\\right)&\n \\end{align*}\nThe ideals in the first row are almost lex, and ${\\mathrm{Lex}}(3\\zeta+5,A)=\\left(xy, xz^5 \\right)$.\n\\end{example}\n\n\n\\section{Decomposition of monomial ideals}\\label{SectionDecomposition}\n\nWe introduce an inductive decomposition of monomial ideals in Clements-Lindstr\\\"om rings.\nThis decomposition is particularly effective for strongly stable and (almost) lex ideals; \nit will play a fundamental role in the construction of expansive ideals in Section \\ref{SectionExpansivePoint}.\n\n\nFor the rest of the paper, we fix the following notation:\n\\begin{equation}\n\\label{DefinitionRings}\n\\begin{aligned}\n S & = \\Bbbk[x_1, x_2,\\ldots, x_{n}, x_{n+1}] \\qquad &R & = S \/ \\big(x_1^{d_1}, \\ldots, x_n^{d_n}\\big)\\\\ \n{\\overline{S}} &= \\Bbbk[x_1, x_2,\\ldots, x_{n-1}, x_{n+1}] \\qquad & {\\overline{R}} & = {\\overline{S}} \/ \\big(x_1^{d_1}, \\ldots, x_{n-1}^{d_{n-1}}\\big)\\\\\n{\\widetilde{S}} &= \\Bbbk[x_1, x_2,\\ldots, x_{n-1}, x_n] \\qquad &{\\widetilde{R}} &= {{\\widetilde{S}}}\/{\\big(x_1^{d_1}, \\ldots, x_n^{d_n}\\big)}\n\\end{aligned}\n\\end{equation}\nwhere $n \\in \\mathbb{N}$ and $d_1 \\leq d_2 \\leq \\cdots \\leq d_{n}$ is any sequence with $d_i \\in \\mathbb{N}\\cup \\{\\infty\\}$.\nIn other words, we will always set $d_{n+1} = \\infty$, \nso that $x_{n+1}^{d_{n+1}}=0$ and it will be omitted. \nIn this way, the Clements-Lindstr\\\"om rings $R$ and ${\\overline{R}}$ always have positive Krull dimension,\nwhereas the Clements-Lindstr\\\"om ring ${\\widetilde{R}}$ may be Artinian.\nThe rings ${\\overline{R}}$ and ${\\overline{S}}$ are defined only if $n>0$.\nWhen $n=0$ we have $R=S = \\Bbbk[x_1]$, and the only saturated ideals are the zero ideal and the unit ideal.\nObserve that ${\\overline{S}}$ and ${\\overline{R}}$ are algebra retracts of $S$ and $R$, respectively, and they may be regarded either as subrings or as factor rings;\nsimilarly for ${\\widetilde{S}}$ and ${\\widetilde{R}}$.\nBy abuse of notation, the symbols $x_i$ will be used to denote elements in different rings.\n\nSince $R = {\\widetilde{R}}[x_{n+1}]$ there is a tight relation between invariants of ideals of $R$ and ${\\widetilde{R}}$;\nwe summarize the main formulas in the following remark.\n\n\n\\begin{remark}\\label{RemarkModuloXNplus1}\nLet $I\\subseteq R$ be a saturated ideal such that $I:x_{n+1}=I$.\nDenote by ${\\widetilde{I}} = \\frac{I+(x_{n+1})}{(x_{n+1})} \\subseteq {\\widetilde{R}}$ the image of $I$ in ${\\widetilde{R}}$.\n Then $\\mathrm{HF}({\\widetilde{I}}, d) = \\mathrm{HF}(I,d)-\\mathrm{HF}(I,d-1)$ for $d\\in \\mathbb{Z}$ and $\\mathrm{HP}({\\widetilde{I}}, \\zeta) = \\mathrm{HF}(I,\\zeta)-\\mathrm{HF}(I,\\zeta-1)$.\nFurthermore, for all $i,j$ we have\n\\begin{align*}\n\\beta_{i,j}^{\\widetilde{S}}({\\widetilde{R}}\/{\\widetilde{I}}) &= \\beta_{i,j}^S(R\/I)\n&\\beta_{i,j}^S({\\widetilde{R}}\/{\\widetilde{I}}) &= \\beta_{i,j}^S(R\/I) + \\beta_{i-1,j-1}^S(R\/I)\\\\\n\\beta_{i,j}^{\\widetilde{R}}({\\widetilde{R}}\/{\\widetilde{I}}) &= \\beta_{i,j}^R(R\/I)\n&\\beta_{i,j}^R({\\widetilde{R}}\/{\\widetilde{I}}) &= \\beta_{i,j}^R(R\/I) + \\beta_{i-1,j-1}^R(R\/I).\n\\end{align*}\nIf $I$ is monomial or strongly stable, then so is ${\\widetilde{I}}$.\nConversely, given any strongly stable $K\\subseteq {\\widetilde{R}}$, the extension $ K R \\subseteq R$ of $K$ to $R$ is a saturated strongly stable ideal\nwhose image in ${\\widetilde{R}}$ is $K$.\n\\end{remark}\n\n\nFor a monomial ideal $I \\subseteq R$ there exist uniquely determined monomial ideals $I_\\ell \\subseteq {\\overline{R}}$ such that\nthe following decomposition of ${\\overline{R}}$-modules holds\n\\begin{equation}\\label{EqDecomposition}\nI = \\bigoplus_{\\ell=0}^{d_{n}-1} I_\\ell x_{n}^\\ell.\n\\end{equation}\nThe set of components $\\{I_\\ell\\}$ is finite if $d_n < \\infty$, infinite otherwise.\nThroughout the paper, the notation $I_\\ell$ will always refer to this decomposition; \nit should not be confused with graded components, denoted instead by $[I]_j$.\n\n\n\\begin{example}\nLet $R = {\\Bbbk[x_1,x_2, x_3, x_4]}\/{\\big(x_1^3,x_2^3,x_3^4\\big)}$,\nso that ${\\overline{R}} = {\\Bbbk[x_1,x_2, x_4]}\/{\\big(x_1^3,x_2^3\\big)}$.\nThe components of the ideal $I = \\big( x_1^2,\\, x_1x_2^2x_3,\\, x_1x_2x_3^2,\\,x_1x_3^3,\\, x_2^2x_3^2\\big) \\subseteq R$ are the ${\\overline{R}}$-ideals\n$$\nI_0 = \\big(x_1^2\\big), \\quad I_1 = \\big( x_1^2, x_1x_2^2\\big), \\quad I_2 = \\big( x_1^2, x_1x_2, x_2^2\\big), \\quad I_3 =\\big( x_1, x_2^2\\big).\n$$\n\\end{example}\n\nWe are going to record some elementary properties of the decomposition \\eqref{EqDecomposition}.\nFirst, we define a partial order $\\preceq$ among univariate polynomials with rational coefficients.\n\n\\begin{definition}\\label{DeinitionPartialOrderPolynomials}\nLet $p(\\zeta),q(\\zeta) \\in \\mathbb{Q}[\\zeta]$.\nWe set $p(\\zeta)\\preceq q(\\zeta)$ if $q(\\zeta) - p(\\zeta)$ is a non-negative constant polynomial,\ni.e., if the coefficients of positive degree coincide in $p(\\zeta)$ and $q(\\zeta)$ and the constant terms satisfy $p(0) \\leq q(0)$.\n\\end{definition}\n\nRecall that the constant term of a Hilbert polynomial is always an integer, carrying the same information as the arithmetic genus.\n\n\\begin{prop}\\label{PropositionFiniteLengthRank}\nLet $I\\subseteq J \\subseteq R$ be saturated strongly stable ideals.\nThe quotient $J\/I$ is a free module over $\\Bbbk[x_{n+1}]$ via restriction of scalars.\nThe following conditions are equivalent\n\\begin{itemize}\n\\item[(i)] $\\mathrm{HP}(I) \\preceq \\mathrm{HP}(J)$ \\vspace*{0.1cm}\n\\item[(ii)]$\\mathrm{rank}_{\\Bbbk[x_{n+1}]} \\left(J\/I \\right)<\\infty $ \\vspace*{0.1cm}\n\\item[(iii)]$\\dim_\\Bbbk \\big({{\\widetilde{J}}}\/{{\\widetilde{I}}}\\big)< \\infty$\n\\end{itemize}\nand if these conditions are satisfied then \n$\n\\mathrm{HP}(J) -\\mathrm{HP}(I) = \\mathrm{rank}_{\\Bbbk[x_{n+1}]} \\left({J}\/{I} \\right) = \\dim_\\Bbbk \\big({{\\widetilde{J}}}\/{{\\widetilde{I}}}\\big).\n$\n\\end{prop}\n\n\\begin{proof}\nLet $M = J\/I$ and $\\widetilde{M} = {\\widetilde{J}}\/{\\widetilde{I}}$, with notation as in Remark \\ref{RemarkModuloXNplus1}.\nThen $M \\cong \\widetilde{M} \\otimes_{\\widetilde{R}} {\\widetilde{R}}[x_{n+1}] \\cong \\widetilde{M} \\otimes_\\Bbbk \\Bbbk[x_{n+1}]$, implying the first statement and the equality\n $\\mathrm{rank}_{\\Bbbk[x_{n+1}]} \\left({J}\/{I} \\right) = \\dim_\\Bbbk \\big({{\\widetilde{J}}}\/{{\\widetilde{I}}}\\big)$.\nSince $I \\subseteq J$, we have $\\mathrm{HP}(I) \\preceq \\mathrm{HP}(J)$ if and only if $\\mathrm{HP}(M)=\\mathrm{HP}(J) -\\mathrm{HP}(I) $ is a constant, \nequivalently $M$ has Krull dimension at most 1, equivalently $M$ is a finite $\\Bbbk[x_{n+1}]$-module.\nFinally, the rank of a finite free $\\Bbbk[x_{n+1}]$-module is equal to its Hilbert polynomial.\n\\end{proof}\n\nIn the next proposition we list basic properties of the decomposition \\eqref{EqDecomposition}.\n\n\\begin{prop}\\label{PropositionElementaryPropertiesDecomposition}\nLet $R$ be a Clements-Lindstr\\\"om ring\nand $I \\subseteq R $ a monomial ideal.\n\\begin{itemize}\n\\item[(1)] The sequence $\\{ I_\\ell\\}$ is a non-decreasing chain of ideals of ${\\overline{R}}$.\n\\item[(2)] If $d_{n}= \\infty$ the sequence $\\{ I_\\ell\\}$ is eventually constant, \nand the limit is equal to the ideal\n$$\nI_\\infty := \\frac{I : (x_{n})^\\infty + (x_{n})}{(x_{n})} \\subseteq {\\overline{R}}.\n$$\n\\item[(3)] $I$ is strongly stable if and only if $I_\\ell$ is strongly stable for all $\\ell$ and $(x_1, \\ldots, x_{n-1}) I_\\ell \\subseteq I_{\\ell-1}$ for all $\\ell\\geq 1$.\n\\item[(4)] If $I$ is strongly stable, then $I$ is saturated if and only if $I_\\ell$ is saturated for every $\\ell$.\n\\item[(5)] $\\mathrm{HP}\\big((x_1, \\ldots, x_{n})I\\big) \\preceq \\mathrm{HP}\\big(I)$.\n\\end{itemize}\nNow assume that $I$ is strongly stable and saturated.\n\\begin{itemize}\n\\item[(6)] $I_{\\ell+1}\/I_{\\ell}$ is a finite free $\\Bbbk[x_{n+1}]$-module of rank equal to the integer $\\mathrm{HP}(I_{\\ell+1}) -\\mathrm{HP}(I_{\\ell}) $.\n\\item[(7)] If $d_n = \\infty$ then $\\mathrm{HP}(I_\\infty,\\zeta) = \\mathrm{HP}(I,\\zeta) - \\mathrm{HP}(I,\\zeta-1)$.\n\\item[(8)] $\\mathrm{HP}(I_\\ell,\\zeta) - \\mathrm{HP}(I,\\zeta)+\\mathrm{HP}(I,\\zeta-1)$ is a constant polynomial.\nIn particular, the coefficient of $\\zeta^h$ in $\\mathrm{HP}(I_{\\ell})$, where $h>0$, is independent of $\\ell$ and uniquely determined by $\\mathrm{HP}(I)$.\n\\item[(9)] $\\mathrm{HP}(I_{\\ell_1}) \\preceq \\mathrm{HP}(I_{\\ell_2})$ for all $\\ell_1 \\leq \\ell_2 < d_{n}$.\n\\item[(10)] There exists a saturated strongly stable ideal $J\\subseteq I$ such that $\\mathrm{HP}(I)-\\mathrm{HP}(J) = 1$.\n\\end{itemize}\n\\end{prop}\n\n\\begin{proof}\n(1) follows immediately from \\eqref{EqDecomposition} since the ideal $I$ is closed under multiplication by $x_n$.\n\nAssume $d_n = \\infty$. \nSince ${\\overline{R}}$ is Noetherian, the non-decreasing sequence $\\{ I_\\ell\\}$ stabilizes.\nChoose $\\ell_0\\in \\mathbb{N}$ so that $I_\\ell = I_{\\ell_0}$ for all $\\ell \\geq \\ell_0$,\nand consider the ideal $J = \\bigoplus_{\\ell= 0}^{d_{n}-1} I_{\\ell_0} x_{n}^\\ell \\subseteq R$.\nThen we have $J : x_n = J$, $I \\subseteq J$, and $x_n^{\\ell_0}J \\subseteq I$.\nIt follows that $J = I: (x_n)^\\infty$, and thus $I_\\infty = I_{\\ell_0} = \\frac{J+(x_n)}{(x_n)} = \\frac{I : (x_{n})^\\infty + (x_{n})}{(x_{n})} \\subseteq {\\overline{R}}$, proving (2). \n\n(3) holds by definition of strongly stable ideal.\n\n(4) follows since a strongly stable ideal is saturated if and only if the last variable $x_{n+1}$ does not divide any of its monomial minimal generators,\nunder our assumption that $\\dim R >0$.\n\n(5) holds because the $R$-module $I\/(x_1,\\ldots, x_n)I$ has Krull dimension at most 1,\nhence its Hilbert polynomial is a non-negative constant.\n\n(6) follows from (4) and Proposition \\ref{PropositionFiniteLengthRank}, since by (3) $I_{\\ell+1}\/I_{\\ell}$ is \nannihilated by $(x_1, \\ldots, x_{n-1})$ and thus it is a finite $\\Bbbk[x_{n+1}]$-module.\n\n With notation as in the proof of (2), let $J = I : (x_n)^\\infty = \\bigoplus_{\\ell= 0}^{d_{n}-1} I_{\\ell_0} x_{n}^\\ell \\subseteq R$.\nSince $x_n$ is a non-zerodivisor on $R\/J$ and $I_\\infty = \\frac{J+(x_n)}{(x_n)} $, \nwe have $\\mathrm{HP}(I_\\infty, \\zeta) = \\mathrm{HP}(J,\\zeta) - \\mathrm{HP}(J,\\zeta-1)$.\nHowever, from (6) we see that $J\/I$ has Krull dimension at most 1, so that $\\mathrm{HP}(J) -\\mathrm{HP}(I)$ is a constant polynomial, and (7) follows.\n\nIf $d_n < \\infty$ then $R$ has Krull dimension 1,\n the coefficient of $\\zeta^h$ in $\\mathrm{HP}(I_{\\ell})$ and $\\mathrm{HP}(I)$ is 0 for $h>0$, thus (8) holds in this case.\n Now assume $d_n = \\infty$.\nBy (6) we have that $\\mathrm{HP}(I_{\\ell}) - \\mathrm{HP}(I_{\\ell-1})\\in \\mathbb{N}$,\n hence the coefficient of $\\zeta^h$ in $\\mathrm{HP}(I_{\\ell})$ is independent of $\\ell$ for $h>0$.\n Therefore, the claim (8) reduces to the case of the component $I_\\infty$, and follows thus from (7).\n \n(9) follows immediately from (1) and (6).\n \nLet $\\mathbf{u}_1, \\ldots, \\mathbf{u}_t$ be the minimal monomial generators of $I$ ordered decreasingly in $<_{\\mathrm{lex}}$.\nThen the ideal $J = (\\mathbf{u}_1, \\ldots, \\mathbf{u}_{t-1},x_1 \\mathbf{u}_t, \\ldots, x_{n} \\mathbf{u}_t)$ satisfies (10).\n\\end{proof}\n\n\n\n\n\\section{The expansive point in the Hilbert scheme}\\label{SectionExpansivePoint}\n\n\nThis section represents the core of the paper: here \nwe introduce the expansive point on the Hilbert scheme of a Clements-Lindstr\\\"om scheme.\nWe develop a machine to deal with expansive ideals both from an abstract and computational perspective.\nThe reader may choose to skip the proof of Theorem \\ref{TheoremDefinitionExpansiveAxioms} -- the rest of the paper relies on the axioms \\hyperref[A1]{(A1)} through \\hyperref[A7]{(A7)}, but does not use the proof.\nRecall our standing notation \\eqref{DefinitionRings} on Clements-Lindstr\\\"om rings.\n\n\\begin{thm}\\label{TheoremDefinitionExpansiveAxioms}\nLet $R$ be a Clements-Lindstr\\\"om ring.\nFor every polynomial\n $p(\\zeta) \\in \\mathbb{Q}[\\zeta]$ such that $\\mathrm{Hilb}^{p(\\zeta)}(\\mathrm{Proj} R) \\ne \\emptyset$\n there exists a unique ideal $\\mathrm{Exp}(p,R) \\in \\mathrm{Hilb}^{p(\\zeta)}(\\mathrm{Proj} R)$, \n called the {\\bf expansive ideal} with Hilbert polynomial $p(\\zeta)$,\nsuch that the following axioms are satisfied:\n\\begin{enumerate}\n\\item[(A1)\\label{A1}] $ \\mathrm{Exp}(p,R)$ is strongly stable;\n\n\\item[(A2)\\label{A2}] the components $ \\mathrm{Exp}(p,R)_\\ell \\subseteq {\\overline{R}}$ are expansive for all $\\ell$;\n\n\\item[(A3)\\label{A3}] given two polynomials $p(\\zeta) \\preceq p'(\\zeta) $ we have $\\mathrm{Exp}(p',R) \\subseteq \\mathrm{Exp}(p,R)$;\n\n\\item[(A4)\\label{A4}] $(x_1, \\ldots, x_{n}) \\mathrm{Exp}(p,R) $ is expansive;\n\n\\item[(A5)\\label{A5}] if $q(\\zeta) = \\mathrm{HP}\\big({\\overline{R}}\/\\mathrm{Exp}(p,R)_k\\big)-1$ is such that \n$\\mathrm{Hilb}^{q(\\zeta)}\\big(\\mathrm{Proj} {\\overline{R}}\\big) \\ne \\emptyset$, for some $k < d_n$, then for all $h < k$ we have\n$\n\\mathrm{Exp}(p,R)_h \\subseteq (x_1, \\ldots, x_{n-1})^{k-h} \\mathrm{Exp}\\big(q(\\zeta),{\\overline{R}}\\big);\n$\n\n\\item[(A6)\\label{A6}] if $J\\in \\mathrm{Hilb}^{p(\\zeta)}(\\mathrm{Proj} R)$ is strongly stable then for every $0\\leq \\rho \\leq d_n-1$ we have\n$$\n\\sum_{\\ell = 0}^\\rho \\mathrm{HP}\\big(\\mathrm{Exp}(p,R)_\\ell\\big) \\preceq \\sum_{\\ell = 0}^\\rho \\mathrm{HP}(J_\\ell) ;\n$$\n\n\\item[(A7)\\label{A7}] if $J\\in \\mathrm{Hilb}^{p(\\zeta)}(\\mathrm{Proj} R)$ is strongly stable then \n$\\mathrm{HP}\\big( (x_1, \\ldots, x_{n}) \\mathrm{Exp}(p,R) \\big)\\preceq\\mathrm{HP}\\big((x_1, \\ldots, x_{n}) J\\big).$\n\n\n\\end{enumerate}\n\\end{thm}\n\n\\noindent\nWe define $\\mathrm{Exp}(I) = \\mathrm{Exp}(p(\\zeta), R)$ \nif $I \\in \\mathrm{Hilb}^{p(\\zeta)}(\\mathrm{Proj} R)$.\n\n\n\n\\begin{proof}[Proof of Theorem \\ref{TheoremDefinitionExpansiveAxioms}]\nWe prove the theorem by induction on the number of variables $n+1$.\nFor the base of the induction $n=0$ there is nothing to prove, so we assume $n>0$.\nFix a polynomial $p(\\zeta)$ such that $\\mathrm{Hilb}^{p(\\zeta)}(\\mathrm{Proj} R) \\ne \\emptyset$.\nIt follows by Proposition \\ref{PropositionElementaryPropertiesDecomposition} (8) \nthat for any two strongly stable $I,J \\in \\mathrm{Hilb}^{p(\\zeta)}(\\mathrm{Proj} R) $ and $0\\leq h,k k$ be maximal such that $E_k = E_\\chi$.\n\nWe exhibit a strongly stable $J \\in \\mathrm{Hilb}^{p(\\zeta)}(\\mathrm{Proj} R)$ generating a contradiction. Set \n\\begin{equation}\\label{ConstructionJContradiction}\nJ = \\bigoplus_{\\ell \\ne \\eta,\\chi} E_\\ell x_n^{\\ell} \\oplus \\mathrm{Exp}\\big(\\mathrm{HP}\\big({\\overline{R}}\/E_\\eta\\big)+1,{\\overline{R}}\\big) x_n^\\eta \\oplus \\mathrm{Exp}\\big(\\mathrm{HP}\\big({\\overline{R}}\/E_\\chi\\big)-1,{\\overline{R}}\\big) x_n^\\chi \\subseteq R.\n\\end{equation}\nNote that both $\\mathrm{Exp}\\big(\\mathrm{HP}\\big({\\overline{R}}\/E_\\eta\\big)+1,{\\overline{R}}\\big)$ and $\\mathrm{Exp}\\big(\\mathrm{HP}\\big({\\overline{R}}\/E_\\chi\\big)-1,{\\overline{R}}\\big)$\nexist by induction, since the corresponding Hilbert schemes are nonempty.\nFor the former, by Proposition \\ref{PropositionElementaryPropertiesDecomposition} (10) there exists some ideal $\\mathcal{J} \\subseteq E_h $ with \n$\\mathrm{HP}\\big({\\overline{R}}\/\\mathcal{J}\\big) = \\mathrm{HP}\\big({\\overline{R}}\/E_h\\big)+1$.\nFor the latter, the Hilbert scheme of $q(\\zeta) $ is nonempty by assumption.\nWe observe that $J$ is an ideal of $R$, as its components form a non-decreasing sequence:\nby \\hyperref[A3]{(A3)}, since we already know that $E$ is an ideal, it only remains to check the two inclusions\n$ E_{\\eta-1} \\subseteq\\mathrm{Exp}\\big(\\mathrm{HP}\\big({\\overline{R}}\/E_\\eta\\big)+1,{\\overline{R}}\\big)$ and $ \\mathrm{Exp}\\big(\\mathrm{HP}\\big({\\overline{R}}\/E_\\chi\\big)-1,{\\overline{R}}\\big)\\subseteq E_{\\chi+1}$,\nbut they follow by the choice of $\\eta, \\chi$.\nClearly $J$ is monomial, and it is saturated since $x_{n+1}$ does not divide its minimal generators.\n\nWe prove that $J$ is strongly stable.\nBy Proposition \\ref{PropositionElementaryPropertiesDecomposition} (3)\nit suffices to show that $(x_1, \\ldots, x_{n-1})J_\\ell\\subseteq J_{\\ell-1}$ for each $\\ell$,\nsince all $J_\\ell$ are strongly stable.\nUsing Proposition \\ref{PropositionElementaryPropertiesDecomposition} (5) and (8), we see that the $\\mathrm{HP}\\big({\\overline{R}}\/(x_1, \\ldots, x_{n-1})J_\\ell\\big)-\\mathrm{HP}\\big({\\overline{R}}\/J_{\\ell-1}\\big) $ is an integer.\nBoth $(x_1, \\ldots, x_{n-1})J_\\ell$ and $ J_{\\ell-1}$ are expansive,\nhence it suffices to show \n$ \\mathrm{HP}\\big({\\overline{R}}\/(x_1, \\ldots, x_{n-1})J_\\ell\\big)- \\mathrm{HP}\\big({\\overline{R}}\/J_{\\ell-1}\\big)\\geq 0 $.\nThere are three cases that do not follow from $E$ being strongly stable.\nIf $\\ell = \\chi = \\eta+1$,\nnecessarily $\\chi = k$ and $\\eta =h$,\nhence $F = (x_1, \\ldots, x_{n-1})^{k-h} \\mathrm{Exp}\\big( q(\\zeta),{\\overline{R}}\\big)\\subsetneq E_h$\nbecomes $(x_1, \\ldots, x_{n-1}) J_\\chi \\subsetneq E_{\\eta}$,\nso $\\mathrm{HP}(E_\\eta) - \\mathrm{HP}\\big((x_1, \\ldots, x_{n-1})J_\\chi\\big)>0$ and \n$ \\mathrm{HP}\\big({\\overline{R}}\/(x_1, \\ldots, x_{n-1})J_\\ell\\big)- \\mathrm{HP}\\big({\\overline{R}}\/J_{\\ell-1}\\big)\\geq 0 $.\nIf $\\ell = \\eta+1 < \\chi$,\nwe must prove $ \\mathrm{HP}\\big({\\overline{R}}\/(x_1, \\ldots, x_{n-1})J_{\\eta+1}\\big)- \\mathrm{HP}\\big({\\overline{R}}\/J_{\\eta}\\big)\\geq 0$,\ni.e. $\\mathrm{HP}\\big({\\overline{R}}\/(x_1, \\ldots, x_{n-1})E_{\\eta+1}\\big) -\\mathrm{HP}\\big({\\overline{R}}\/E_\\eta\\big)-1 \\geq 0$, \nequivalently, $(x_1, \\ldots, x_{n-1})E_{\\eta+1} \\subsetneq E_\\eta$.\nBut if this were false, then $(x_1, \\ldots, x_{n-1})E_{\\eta+1} = E_\\eta$, since $E$ is strongly stable.\nIn particular $E_{\\eta+1} \\ne E_\\eta$, forcing $\\eta = h$,\nand the pair $h+1,k$ would contradict the choice of $h,k$.\nFinally, \nif $\\ell = \\chi > \\eta+1$,\nwe must show \n$ \\mathrm{HP}\\big({\\overline{R}}\/(x_1, \\ldots, x_{n-1})J_\\chi\\big)- \\mathrm{HP}\\big({\\overline{R}}\/J_{\\chi-1}\\big)\\geq 0 $.\nIf this were false,\nthen $E_{\\chi-1} \\subsetneq (x_1, \\ldots, x_{n-1})E_\\chi$,\nforcing $\\chi = k$,\nand $\\mathrm{Exp}\\big(\\mathrm{HP}\\big({\\overline{R}}\/E_{k-1}\\big)+1,{\\overline{R}}) \\subseteq (x_1, \\ldots, x_{n-1})\\mathrm{Exp}\\big(q(\\zeta)+1,{\\overline{R}}\\big)$.\nWe obtain\n$(x_1, \\ldots, x_{n-1})^{k-h-1}\\mathrm{Exp}\\big(\\mathrm{HP}\\big({\\overline{R}}\/E_{k-1}\\big)+1,{\\overline{R}}) \\subseteq (x_1, \\ldots, x_{n-1})^{k-h}\\mathrm{Exp}\\big(q(\\zeta)+1,{\\overline{R}}\\big) = F\\subsetneq E_h$,\nand the pair $h,k-1$ contradicts the choice of $h,k$.\nThus $J$ is strongly stable.\n\nFrom \\eqref{ConstructionJContradiction} it follows immediately that\n$\\sum_{\\ell=0}^\\rho \\mathrm{HP}(J_\\ell) \\preceq \\sum_{\\ell=0}^\\rho \\mathrm{HP}(E_\\ell)$ for every $\\rho < d_n$\nand $\\sum_{\\ell=0}^\\eta \\mathrm{HP}(J_\\ell) \\prec \\sum_{\\ell=0}^\\eta \\mathrm{HP}(E_\\ell)$.\nThis yields a contradiction to the choice of $I$, proving \\hyperref[A5]{(A5)}.\n\n\n\nIn order to verify \\hyperref[A3]{(A3)}, we prove a stronger statement:\n\\begin{enumerate}\n\\item[$(\\dagger)$]\nif $E,E'\\subseteq R$ are saturated monomial ideals satisfying \\hyperref[A1]{(A1)}, \\hyperref[A2]{(A2)}, \\hyperref[A5]{(A5)}, and\nsuch that $\\mathrm{HP}(E')-\\mathrm{HP}(E)$ is an integer, then $E \\subseteq E'$ or $E' \\subseteq E$.\n\\end{enumerate}\nSince $\\mathrm{HP}(E')-\\mathrm{HP}(E)\\in \\mathbb{Z}$, by \nProposition \\ref{PropositionElementaryPropertiesDecomposition} (8) $\\mathrm{HP}(E'_h)-\\mathrm{HP}(E_k)\\in \\mathbb{Z}$ for every $h,k$.\nSuppose $E \\ne E'$, and let $h$ be the least index such that $E_h \\ne E'_h$. \nBy axiom \\hyperref[A3]{(A3)} either $E_h \\subsetneq E'_h$ or $E'_h \\subsetneq E_h$;\n assume, for instance, that $E_h \\subsetneq E'_h$.\nAssume by contradiction that there exists $h < k < d_n$ with $E_k \\not\\subseteq E'_k$.\nUsing \\hyperref[A3]{(A3)} again we find that $E'_k \\subsetneq E_k$ and $\\mathrm{HP}(E'_k)\\prec \\mathrm{HP}(E_k)$,\nso $\\mathrm{HP}\\big({\\overline{R}}\/E_k\\big)\\preceq \\mathrm{HP}\\big({\\overline{R}}\/E'_k\\big)-1 $ and \n$\n\\mathrm{Exp}\\big(\\mathrm{HP}\\big({\\overline{R}}\/E'_k)-1, {\\overline{R}}\\big) \\subseteq E_k.\n$\nSince $E$ is strongly stable, we have $(x_1, \\ldots, x_{n-1})^{k-h}E_k \\subseteq E_h$.\nBy \\hyperref[A5]{(A5)} we get $E'_h \\subseteq (x_1, \\ldots, x_{n-1})^{k-h} \\mathrm{Exp}\\big(\\mathrm{HP}\\big({\\overline{R}}\/E'_k)-1, {\\overline{R}}\\big)$.\nWe derive the contradiction\n $$\n E_h \\subsetneq E'_h\\subseteq (x_1, \\ldots, x_{n-1})^{k-h}\\mathrm{Exp}\\big(\\mathrm{HP}\\big({\\overline{R}}\/E'_k)-1, {\\overline{R}}\\big) \\subseteq (x_1, \\ldots, x_{n-1})^{k-h}E_k \\subseteq E_h.\n $$\nThis proves the claim $(\\dagger)$, which implies the axiom \\hyperref[A3]{(A3)} and also the uniqueness of the expansive ideal for each Hilbert polynomial.\n\n\nNext, we show \\hyperref[A6]{(A6)}.\nWe must prove that \n$\n\\big\\{ \\sum_{\\ell=0}^\\rho \\mathrm{HP}(E_\\ell) \\big\\}_{\\rho=0}^{d_n-1}\n$\nis the unique minimal sequence, with respect to componentwise comparison by $\\preceq$, among all strongly stable ideals $J\\in \\mathrm{Hilb}^{p(\\zeta)}(\\mathrm{Proj} R)$.\nLet $J\\in \\mathrm{Hilb}^{p(\\zeta)}(\\mathrm{Proj} R)$ be strongly stable\nsuch that \n$\n\\big\\{ \\sum_{\\ell=0}^\\rho \\mathrm{HP}(J_\\ell) \\big\\}_{\\rho=0}^{d_n-1}\n$\n is also minimal with respect to componentwise comparison. \nAs in \\eqref{EqConstructionExpansive}\nwe define\n$$ \nE' = \\bigoplus_{\\ell1$.\nLet $M' = (x_1, \\ldots, x_n) M$ and $M'' = M\/M'$.\nBoth $M'$ and $M''$ are finite $\\mathbb{Z}^{n+1}$-graded $R$-modules.\nAs $\\Bbbk[x_{n+1}]$-modules via restriction of scalars,\n $M'$ is free of rank less than $c$,\nwhereas $M''$ is also free, by multidegree reasons, and it satisfies (ii).\nThe conclusions follow by induction on $c$ from the exact sequence $0\\rightarrow M' \\rightarrow M \\rightarrow M'' \\rightarrow 0$.\n\\end{proof}\n\n\n\\begin{thm}\\label{TheoremExtremalFiniteResolution}\nLet $S = \\Bbbk[x_1, \\ldots, x_{n+1}]$ be a polynomial ring and $R= S\/(x_1^{d_1}, \\ldots, x_n^{d_n})$ a Clements-Lindstr\\\"om ring, where $2 \\leq d_1 \\leq \\cdots \\leq d_n \\leq \\infty$.\nFor each polynomial $p(\\zeta)$ we have\n$$\n\\beta_{i}^S\\big(R\/I\\big) \\leq \\beta_{i}^S\\big(R\/\\mathrm{Exp}(p(\\zeta))\\big) \n$$\nfor all $I\\in\\mathrm{Hilb}^{p(\\zeta)}(\\mathrm{Proj} R)$ and all $i \\geq 0$.\n\\end{thm}\n\n\\begin{proof}\nWe proceed by induction on $n$ and the case $n=0$ is trivial, so let $n>0$.\nBy Lemma \\ref{LemmaAlmostLexMerminMurai}, we may assume without loss of generality that $I$ is a strongly stable monomial ideal.\nLet ${\\mathcal{I}}, {\\mathcal{E}}$ denote the preimages of $I, \\mathrm{Exp}(p) \\subseteq R$ in the polynomial ring $S$.\n We have decompositions \n\\begin{align}\\label{EqDecompositionIandE}\n{\\mathcal{I}} &= \\bigoplus_{\\ell=0}^\\infty {\\mathcal{I}}_\\ell x_n^\\ell, &{\\mathcal{E}} &= \\bigoplus_{\\ell=0}^\\infty {\\mathcal{E}}_\\ell x_n^\\ell\n\\end{align}\nwhere ${\\mathcal{I}}_\\ell, {\\mathcal{E}}_\\ell$ are ideals of ${\\overline{S}}$. \nSpecifically, ${\\mathcal{I}}_\\ell \\subseteq {\\overline{S}}$ is the preimage of $I_\\ell \\subseteq {\\overline{R}}$ if $\\ell < d_n$, and ${\\mathcal{I}}_\\ell = {\\overline{S}} $ if $d_n \\leq \\ell < \\infty$; \nlikewise for ${\\mathcal{E}}_\\ell$.\nSince $R\/I \\cong S\/{\\mathcal{I}}$ and $R\/\\mathrm{Exp}(p) \\cong S\/{\\mathcal{E}}$,\nwe must prove that $\\beta_{i}^S({\\mathcal{I}}) \\leq \\beta_{i}^S({\\mathcal{E}})$ for all $i$.\nThe variable $x_n$ is a non-zerodivisor on $S, {\\mathcal{I}}, {\\mathcal{E}},$ so it suffices to \n prove $\\beta_{i}^{\\overline{S}}({\\mathcal{I}}\/x_n{\\mathcal{I}}) \\leq \\beta_{i}^{\\overline{S}}({\\mathcal{E}}\/x_n{\\mathcal{E}})$ for all $i$.\n \n \n Let ${\\mathcal{J}}\\subseteq {\\overline{S}}$ denote the preimage of $\\mathrm{Exp}(I_0) \\subseteq {\\overline{R}}$.\n Since ${\\overline{S}}\/{\\mathcal{I}}_0 \\cong R\/I_0$ and ${\\overline{S}}\/{\\mathcal{J}} \\cong {\\overline{R}} \/ \\mathrm{Exp}(I_0)$, \n by induction we have $ \\beta_i^{\\overline{S}}({\\mathcal{I}}_0) \\leq \\beta^{\\overline{S}}({\\mathcal{J}})$ for every $i \\geq 0$.\n By Corollary \\ref{CorollaryHyperplaneSections} we have $\\mathrm{HP}\\big(\\mathrm{Exp}(p)+(x_n)\\big ) \\preceq\\mathrm{HP}\\big(I+(x_n)\\big )$.\n Note that $I_0 \\cong \\frac{I+(x_n)}{(x_n)} $ and $\\mathrm{Exp}(p)_0 \\cong \\frac{\\mathrm{Exp}(p)+(x_n)}{(x_n)} $, so \n $\\mathrm{HP}(\\mathrm{Exp}(p)_0 ) \\preceq\\mathrm{HP}(I_0)$. \n By \\hyperref[A3]{(A3)} we conclude that $\\mathrm{Exp}(p)_0 \\subseteq \\mathrm{Exp}(I_0)$, and hence ${\\mathcal{E}}_0 \\subseteq {\\mathcal{J}}$.\nBy Proposition \\ref{PropositionFiniteLengthRank}\nthe quotient ${\\mathcal{J}}\/{\\mathcal{E}}_0 \\cong \\mathrm{Exp}(I_0)\/ \\mathrm{Exp}(p)_0$ is a free $\\Bbbk[x_{n+1}]$-module over of rank \n $c_0 = \\mathrm{HP}\\big( \\mathrm{Exp}(I_0)\/ \\mathrm{Exp}(p)_0 \\big)\n= \\mathrm{HP}\\big(I+(x_n)\\big )-\\mathrm{HP}\\big(\\mathrm{Exp}(p)+(x_n)\\big )$. \nUsing the short exact sequence $ 0\\rightarrow {\\mathcal{E}}_0 \\rightarrow {\\mathcal{J}} \\rightarrow {\\mathcal{J}}\/{\\mathcal{E}}_0 \\rightarrow 0$ and Lemma \\ref{LemmaUpperBoundBettiFreeModule} (i) we obtain\n\\begin{equation}\\label{EqFirstEstimateFiniteResolution}\n \\beta_i^{\\overline{S}}({\\mathcal{I}}_0) \\leq \n\\beta_i^{\\overline{S}} ( {\\mathcal{J}}) \\leq \\beta_i^{\\overline{S}} ( {\\mathcal{E}}_0) + \\beta_i^{\\overline{S}} ( {\\mathcal{J}}\/{\\mathcal{E}}_0) \\leq \\beta_i^{\\overline{S}} ( {\\mathcal{E}}_0) + c_0\\beta_i^{\\overline{S}}\\big(\\Bbbk[x_{n+1}]\\big).\n\\end{equation}\n\nSuppose first that $d_n = \\infty$. \nFrom \\eqref{EqDecompositionIandE} we deduce\n decompositions of ${\\overline{S}}$-modules \n \\begin{equation}\\label{EqDecompositionQuotientsIandE}\n\\frac{{\\mathcal{I}}}{x_n{\\mathcal{I}}}\\cong {\\mathcal{I}}_0 \\oplus \\bigoplus_{\\ell=0}^\\infty \\frac{{\\mathcal{I}}_\\ell}{{\\mathcal{I}}_{\\ell-1}} \\cong {\\mathcal{I}}_0 \\oplus \\bigoplus_{\\ell=0}^\\infty \\frac{I_\\ell}{I_{\\ell-1}},\n \\qquad\n \\frac{{\\mathcal{E}}}{x_n{\\mathcal{E}}}\\cong {\\mathcal{E}}_0\\oplus \\bigoplus_{\\ell=0}^\\infty \\frac{{\\mathcal{E}}_\\ell}{{\\mathcal{E}}_{\\ell-1}} \\cong {\\mathcal{E}}_0\\oplus \\bigoplus_{\\ell=0}^\\infty \\frac{\\mathrm{Exp}(p)_\\ell}{\\mathrm{Exp}(p)_{\\ell-1}}. \n \\end{equation}\nBy Proposition \\ref{PropositionElementaryPropertiesDecomposition} (6) the ${\\overline{S}}$-modules $\\bigoplus_{\\ell=0}^\\infty \\frac{I_\\ell}{I_{\\ell-1}}$ and $\\bigoplus_{\\ell=0}^\\infty \\frac{\\mathrm{Exp}(p)_\\ell}{\\mathrm{Exp}(p)_{\\ell-1}}$ are free $\\Bbbk[x_{n+1}]$-modules of rank $ c_1 = \\mathrm{HP}(I_\\infty)- \\mathrm{HP}(I_0)\\in \\mathbb{N}$ and $c_2 = \\mathrm{HP}(\\mathrm{Exp}(p)_\\infty)- \\mathrm{HP}(\\mathrm{Exp}(p)_0)\\in \\mathbb{N}$, respectively.\nMoreover, by \\ref{PropositionElementaryPropertiesDecomposition} (3) these modules are annihilated by $(x_1, \\ldots, x_{n-1})\\subseteq {\\overline{S}}$.\nUsing Lemma \\ref{LemmaUpperBoundBettiFreeModule} (ii) and \ncombining with \\eqref{EqFirstEstimateFiniteResolution} we obtain\n\\begin{align*}\n\\beta_{i}^{\\overline{S}}({\\mathcal{I}}\/x_n{\\mathcal{I}}) & = \\beta_{i}^{\\overline{S}}({\\mathcal{I}}_0) + \\beta_{i}^{\\overline{S}}\\left( \\bigoplus_{\\ell=0}^\\infty \\frac{{\\mathcal{I}}_\\ell}{{\\mathcal{I}}_{\\ell-1}}\\right) = \\beta_{i}^{\\overline{S}}({\\mathcal{I}}_0) + c_1\\beta_{i}^{\\overline{S}}\\big(\\Bbbk[x_{n+1}]\\big)\\\\\n& \\leq \\beta_i^{\\overline{S}} ( {\\mathcal{E}}_0) + (c_0+ c_1)\\beta_{i}^{\\overline{S}}\\big(\\Bbbk[x_{n+1}]\\big).\n\\end{align*}\nWe claim that $ \\beta_i^{\\overline{S}} ( {\\mathcal{E}}\/x_n{\\mathcal{E}}) = \\beta_i^{\\overline{S}} ( {\\mathcal{E}}_0) + (c_0+ c_1)\\beta_{i}^{\\overline{S}}\\big(\\Bbbk[x_{n+1}]\\big)$, concluding the proof in this case. \nThis follows from \\eqref{EqDecompositionQuotientsIandE} and Lemma \\ref{LemmaUpperBoundBettiFreeModule} (ii) once we show that $\\bigoplus_{\\ell=0}^\\infty \\frac{\\mathrm{Exp}(p)_\\ell}{\\mathrm{Exp}(p)_{\\ell-1}}$ has rank $c_0+c_1$ as $\\Bbbk[x_{n+1}]$-module,\nthat is, $c_2 = c_0+c_1$.\nBut this is true by definition of $c_0$, $c_1$, $c_2$, additivity of Hilbert polynomials, and the fact that \n$\\mathrm{HP}(I_\\infty) = \\mathrm{HP}(\\mathrm{Exp}(p)_\\infty)$ by Proposition \\ref{PropositionElementaryPropertiesDecomposition} (7).\n\nNow suppose that $d_n < \\infty$.\nThe decompositions of ${\\overline{S}}$-modules obtained from \\eqref{EqDecompositionIandE} become \n \\begin{equation}\\label{EqSecondDecompositionQuotientsIandE}\n\\frac{{\\mathcal{I}}}{x_n{\\mathcal{I}}}\\cong{\\mathcal{I}}_0 \\oplus \\bigoplus_{\\ell=0}^{d_n-1} \\frac{I_\\ell}{I_{\\ell-1}} \\oplus \\frac{{\\overline{R}}}{I_{d_n-1}},\n \\qquad\n \\frac{{\\mathcal{E}}}{x_n{\\mathcal{E}}}\\cong {\\mathcal{E}}_0\\oplus \\bigoplus_{\\ell=0}^{d_n-1} \\frac{\\mathrm{Exp}(p)_\\ell}{\\mathrm{Exp}(p)_{\\ell-1}} \\oplus \\frac{{\\overline{R}}}{\\mathrm{Exp}(p)_{d_n-1}}.\n \\end{equation}\nOur goal is to estimate $\\beta_{i}^{\\overline{S}}({\\overline{R}}\/I_{d_n-1})$.\nBy induction\n$\\beta_{i}^{\\overline{S}}({\\overline{R}}\/I_{d_n-1}) \\leq \\beta_{i}^{\\overline{S}}\\big({\\overline{R}}\/\\mathrm{Exp}(I_{d_n-1})\\big)\n$ \nfor all $i\\geq 0$.\nNote that both $R, {\\overline{R}}$ have Krull dimension 1, since $d_n <\\infty$,\nhence all Hilbert polynomials of ideals are constant.\nBy additivity of $\\mathrm{HP}(-)$ we have the formulas\n\\begin{align*}\n&\\mathrm{HP}(I) = \\sum_{\\ell=0}^{d_n-1} \\mathrm{HP}(I_\\ell)\n&& \\mathrm{HP}\\big(I+(x_n^{d_n-1})\\big) = \\sum_{\\ell=0}^{d_n-2} \\mathrm{HP}(I_\\ell) + \\mathrm{HP}({\\overline{R}})\\\\\n&\\mathrm{HP}(\\mathrm{Exp}(p)) = \\sum_{\\ell=0}^{d_n-1} \\mathrm{HP}(\\mathrm{Exp}(p)_\\ell) \\quad\n&&\\mathrm{HP}\\big(\\mathrm{Exp}(p)+(x_n^{d_n-1})\\big) = \\sum_{\\ell=0}^{d_n-2} \\mathrm{HP}(\\mathrm{Exp}(p)_\\ell) + \\mathrm{HP}({\\overline{R}}).\n\\end{align*}\nBy Corollary \\ref{CorollaryHyperplaneSections} we have $\\mathrm{HP}\\big(\\mathrm{Exp}(p)+(x_n^{d_n-1})) \\preceq \\mathrm{HP}\\big(I+(x_n^{d_n-1}))$.\nSince $\\mathrm{HP}(I) = \\mathrm{HP}(\\mathrm{Exp}(p))$, the formulas above imply that $\\mathrm{HP}(I_{d_n-1}) \\preceq \\mathrm{HP}(\\mathrm{Exp}(p)_{d_n-1})$, and \nusing \\hyperref[A3]{(A3)} we deduce $\\mathrm{Exp}(I_{d_n-1}) \\subseteq \\mathrm{Exp}(p)_{d_n-1}$.\nFrom the short exact sequence \n$$\n0 \\rightarrow \\frac{\\mathrm{Exp}(p)_{d_n-1}}{\\mathrm{Exp}(I_{d_n-1})} \\rightarrow \\frac{{\\overline{R}}}{\\mathrm{Exp}(I_{d_n-1})} \\rightarrow \\frac{{\\overline{R}}}{\\mathrm{Exp}(p)_{d_n-1}} \\rightarrow 0\n$$\nwe obtain\n\\begin{equation}\\label{EqEstimateLastPart}\n\\beta_{i}^{\\overline{S}}({\\overline{R}}\/I_{d_n-1}) \\leq \\beta_{i}^{\\overline{S}}\\big({\\overline{R}}\/\\mathrm{Exp}(I_{d_n-1})\\big) \\leq \\beta_{i}^{\\overline{S}}\\left(\\frac{\\mathrm{Exp}(p)_{d_n-1}}{\\mathrm{Exp}(I_{d_n-1})}\\right) + \\beta_{i}^{\\overline{S}}\\left(\\frac{{\\overline{R}}}{\\mathrm{Exp}(p)_{d_n-1}}\\right).\n\\end{equation}\n\nFinally, we are going to use \\eqref{EqSecondDecompositionQuotientsIandE} to give an upper bound for $\\beta_i^{\\overline{S}}({\\mathcal{I}}\/x_n{\\mathcal{I}})$.\nAs before, \nthe ${\\overline{S}}$-modules \n$\\bigoplus_{\\ell=0}^{d_n-1} \\frac{I_\\ell}{I_{\\ell-1}}$ and $\\bigoplus_{\\ell=0}^{d_n-1} \\frac{\\mathrm{Exp}(p)_\\ell}{\\mathrm{Exp}(p)_{\\ell-1}}$ are annihilated by $(x_1, \\ldots, x_{n-1})\\subseteq {\\overline{S}}$,\nand by Proposition \\ref{PropositionElementaryPropertiesDecomposition} (6) they are free $\\Bbbk[x_{n+1}]$-modules of ranks \n $ c'_1= \\mathrm{HP}(I_{d_n-1})- \\mathrm{HP}(I_0)$ and $c'_2 = \\mathrm{HP}\\big(\\mathrm{Exp}(p)_{d_n-1}\\big)- \\mathrm{HP}\\big(\\mathrm{Exp}(p)_0\\big)$, respectively.\nBy Proposition \\ref{PropositionFiniteLengthRank},\nthe module $\\frac{\\mathrm{Exp}(p)_{d_n-1}}{\\mathrm{Exp}(I_{d_n-1})}$ is also free over $\\Bbbk[x_{n+1}]$, of rank $c_3 = \\mathrm{HP}\\big(\\mathrm{Exp}(p)_{d_n-1}\\big)- \\mathrm{HP}\\big(\\mathrm{Exp}(I_{d_n-1}\\big)$.\nCombining the decomposition \\eqref{EqSecondDecompositionQuotientsIandE} and the bounds \\eqref{EqFirstEstimateFiniteResolution}, \\eqref{EqEstimateLastPart}, and using Lemma \\ref{LemmaUpperBoundBettiFreeModule} (i) we find\n\\begin{align*}\n&\\quad\\beta_{i}^{\\overline{S}}({\\mathcal{I}}\/x_n{\\mathcal{I}}) = \\beta_{i}^{\\overline{S}}({\\mathcal{I}}_0) + \\beta_{i}^{\\overline{S}}\\left( \\bigoplus_{\\ell=0}^{d_n-1} \\frac{I_\\ell}{I_{\\ell-1}}\\right) + \\beta_{i}^{\\overline{S}}\\left(\\frac{{\\overline{R}}}{I_{d_n-1}}\\right) \\\\\n\\leq &\n\\left[ \\beta_i^{\\overline{S}} ( {\\mathcal{E}}_0) + c_0\\beta_i^{\\overline{S}}\\big(\\Bbbk[x_{n+1}]\\big)\\right] \n+ c'_1\\beta_i^{\\overline{S}}\\big(\\Bbbk[x_{n+1}]\\big) + \\left[ c_3\\beta_i^{\\overline{S}}\\big(\\Bbbk[x_{n+1}]\\big) + \\beta_{i}^{\\overline{S}}\\left(\\frac{{\\overline{R}}}{\\mathrm{Exp}(p)_{d_n-1}}\\right)\\right]\\\\\n= & \\beta_i^{\\overline{S}} ( {\\mathcal{E}}_0) + (c_0+c'_1+c_3)\\beta_i^{\\overline{S}}\\big(\\Bbbk[x_{n+1}]\\big) + \\beta_{i}^{\\overline{S}}\\left(\\frac{{\\overline{R}}}{\\mathrm{Exp}(p)_{d_n-1}}\\right).\n\\end{align*}\nThe expression in the last line is equal to \n$ \\beta_i^{\\overline{S}} ( {\\mathcal{E}}\/x_n{\\mathcal{E}})$ \nbecause of \\eqref{EqSecondDecompositionQuotientsIandE}, Lemma \\ref{LemmaUpperBoundBettiFreeModule} (ii),\nand the fact that $c'_2 = c_0+c'_1+c_3$.\nThis concludes the proof.\n\\end{proof}\n\nWe remark that, in the case of $\\mathbb{P}^n$, \nthe existence of a point in $\\mathrm{Hilb}^{p(\\zeta)}(\\mathbb{P}^n)$ satisfying the conclusion of Theorem \\ref{TheoremExtremalFiniteResolution}\nwas proved in \\cite[Theorem 1.1]{CaMu13}, cf. the Introduction.\n\n\n\\begin{remark}\nThe numerical bounds on the Betti numbers provided by Theorem \\ref{TheoremExtremalFiniteResolution} do not depend on $\\Bbbk$,\nas it follows from the combinatorial formula of \\cite[Proposition 2.1]{Mu08}.\n\\end{remark}\n\n\n\n\nGiven integers $d_1 \\leq d_2 \\leq \\cdots \\leq d_n \\leq \\infty$,\nwe say that an ideal is a complete intersection of degree sequence $d_1, \\ldots, d_n$ if\nit is generated by a regular sequence $f_1, \\ldots, f_c$ with $d_i = \\deg(f_i)$ for every $i \\leq c:=\\max\\{j \\, : d_j < \\infty\\}$.\nWe emphasize that a complete intersection of degree sequence $d_1, \\ldots, d_n$ may have codimension $c \\leq n$.\n\n\nA remarkable consequence of Theorem \\ref{TheoremExtremalFiniteResolution} is the fact that, conjecturally, \nthe expansive subscheme $\\mathrm{Exp}(p(\\zeta),R)\\in \\mathrm{Hilb}^{p(\\zeta)}(\\mathrm{Proj} R)$\n has the largest number of syzygies \n among all subschemes $Z\\in \\mathrm{Hilb}^{p(\\zeta)}(X)$ of \\emph{any} complete intersection $X \\subseteq \\mathbb{P}^n$\n of degree sequence $d_1, \\ldots, d_n$.\n To justify this claim, \nwe recall the statements of two famous conjectures on complete intersections.\nFor our purposes, it is convenient to state them in terms of ideals of ${\\widetilde{S}} = \\Bbbk[x_1, \\ldots, x_n]$.\n \n \n\n\\begin{itemize}\n\\item \\emph{Eisenbud-Green-Harris Conjecture}:\nIf $I\\subseteq {\\widetilde{S}}$ contains a regular sequence of degree sequence $d_1, \\ldots, d_n$, \nthen there exists a lex ideal $L\\subseteq {\\widetilde{S}}$ with $\\mathrm{HF}(I) = \\mathrm{HF} \\big( L + (x_1^{d_1},\\ldots, x_n^{d_n})\\big)$.\n\n\\item \\emph{Lex-plus-powers Conjecture}: \nIf $I\\subseteq {\\widetilde{S}}$ contains a regular sequence of degree sequence $d_1, \\ldots, d_n$ and if\nthere exists a lex ideal $L\\subseteq {\\widetilde{S}}$ with $\\mathrm{HF}(I) = \\mathrm{HF} \\big( L + (x_1^{d_1},\\ldots, x_n^{d_n})\\big)$,\nthen $\\beta_{i,j}^{\\widetilde{S}}(I) \\leq \\beta_{i,j}^{\\widetilde{S}}( L + (x_1^{d_1},\\ldots, x_n^{d_n})\\big) $\nfor all $i,j$.\n\\end{itemize}\n\nThe first conjecture was proposed in \\cite{EiGrHa93}, \nwhereas the second one is attributed to Charalambous and Evans in \\cite{FrRi07}.\nDespite the apparently independent statements, it is known that the Lex-plus-powers Conjecture implies the Eisenbud-Green-Harris Conjecture.\n\n\\begin{prop}\\label{PropositionLPP}\nLet $X \\subseteq \\mathbb{P}^n$ be a complete intersection \nwith degree sequence $d_1 \\leq d_2 \\leq \\cdots \\leq d_n \\leq \\infty$.\nIf the Lex-plus-powers conjecture is true, then \n$\n\\beta_{i}^S\\big(S\/I_Z\\big) \\leq \\beta_{i}^S\\big(R\/\\mathrm{Exp}(p)\\big) \n$\nfor every closed subscheme $Z\\in\\mathrm{Hilb}^{p(\\zeta)}(X)$ and all $i \\geq 0$.\n\\end{prop}\n\\begin{proof}\nAs in Lemma \\ref{LemmaAlmostLexMerminMurai},\nwe may assume that $x_{n+1}$ is a non-zerodivisor on $S\/I_Z$, and we \nconsider ${\\widetilde{I}} = \\frac{I_Z+(x_{n+1})}{(x_{n+1})}\\subseteq {\\widetilde{S}}$.\nBy assumption the Lex-plus-powers Conjecture and the Eisenbud-Green-Harris Conjecture hold,\ntherefore there exists a lex ideal ${\\widetilde{L}} \\subseteq {\\widetilde{R}}$ such that $\\mathrm{HF}({\\widetilde{S}}\/{\\widetilde{I}}) =\\mathrm{HF}({\\widetilde{R}}\/{\\widetilde{L}}) $\nand $\\beta_{i,j}^{\\widetilde{S}}({\\widetilde{S}}\/{\\widetilde{I}}) \\leq \\beta_{i,j}^{\\widetilde{S}}({\\widetilde{R}}\/{\\widetilde{L}}) $ for all $i,j\\geq 0$.\nThe extension $L= {\\widetilde{L}} R \\subseteq R$ is an almost lex ideal, and \nusing Remark \\ref{RemarkModuloXNplus1} we deduce $\\mathrm{HF}(L)=\\mathrm{HF}(I_Z)$ and \n$\\beta_{i,j}^S(S\/I_Z) \\leq \\beta_{i,j}^S(R\/L) $ for all $i, j\\geq 0$. \nBy Theorem \\ref{TheoremExtremalFiniteResolution} we have \n$\\beta_{i}^S(R\/L) \\leq \\beta_{i}^S(R\/\\mathrm{Exp}(p)) $ for all $i\\geq 0$, and this concludes the proof.\n\\end{proof}\n\n\n\nIn particular,\nby \\cite[Main Theorem]{CaSa18}\n we obtain the following result.\n\n\\begin{thm}\\label{TheoremLPP}\nAssume $\\mathrm{char}(\\Bbbk) = 0$. \nLet $X \\subseteq \\mathbb{P}^n_\\Bbbk$ be a complete intersection \nwith degree sequence \nsuch that $d_j > \\sum_{h=1}^{j-1}(d_h-1)$ for all $j\\geq 3$.\nThen \n$\n\\beta_{i}^S\\big(S\/I_Z\\big) \\leq \\beta_{i}^S\\big(R\/\\mathrm{Exp}(p)\\big) \n$\nfor every closed subscheme $Z\\in\\mathrm{Hilb}^{p(\\zeta)}(X)$ and all $i \\geq 0$.\n\\end{thm}\n\n\n\n\n\\section{Infinite free resolutions}\\label{SectionInfinite}\n\n\nIn this section we investigate bounds for the Betti numbers of infinite free resolutions over a Clements-Lindstr\\\"om ring.\nMinimal free resolutions over complete intersections have attracted much attention in the past few years, \nand significant progress has been achieved in the description of their asymptotic behavior,\nsee e.g.\n\\cite{BuWa15,EiPe16,EiPeSc19}.\nWe conjecture that expansive subschemes exhibit extremal infinite free resolutions, and prove this conjecture for quadratic Clements-Linstr\\\"om rings in characteristic zero.\nWe also deduce the extremality of the deviations of expansive subschemes of $\\mathbb{P}^n$, and in particular the extremality of the Poincar\\'e series.\n\nWe begin by proposing the following problem.\n\n\\begin{conj}\\label{ConjectureInfiniteResolution}\nLet $R$ be a Clements-Lindstr\\\"om ring. \nWe have\n$\\beta_{i}^{R}(I) \\leq \\beta_{i}^{R}(\\mathrm{Exp}(p)) $ \nfor every $I\\in \\mathrm{Hilb}^{p(\\zeta)}(\\mathrm{Proj} R)$ and every $i \\geq 0$.\n\\end{conj}\n\nWhen the ground field has characteristic zero,\n\\cite[Theorem 1.4]{MuPe12} reduces the problem to almost lex ideals, \n proceeding as in Lemma \\ref{LemmaAlmostLexMerminMurai}.\n\n\n\\begin{lemma}\\label{LemmaReductionStableInfiniteResolution}\nAssume that $\\mathrm{char}(\\Bbbk)=0$.\nFor every $I\\in \\mathrm{Hilb}^{p(\\zeta)}(\\mathrm{Proj} R)$ there exists an almost lex $J\\in \\mathrm{Hilb}^{p(\\zeta)}(\\mathrm{Proj} R)$\nsuch that $\\mathrm{HF}(I)=\\mathrm{HF}(J)$ and $\\beta_{i,j}^R(I)\\leq \\beta_{i,j}^R(J)$ for all $i,j$.\n\\end{lemma}\n\nThe following theorem is the main result of this section.\nThe proof employs a construction from \\cite{ArAvHe00,EiPoYu03,GaHiPe02}.\n\n\\begin{thm}\\label{TheoremIfninite}\nAssume that $\\mathrm{char}(\\Bbbk)=0$.\nLet $R$ be a Clements-Lindstr\\\"om ring with $d_j\\in \\{ 2, \\infty\\}$ for every $j$.\nWe have\n$\n\\beta_{i}^R\\big(I\\big) \\leq \\beta_{i}^R\\big(\\mathrm{Exp}(p)\\big) \n$\nfor every $I\\in\\mathrm{Hilb}^{p(\\zeta)}(\\mathrm{Proj} R)$ and every $i \\geq 0$.\n\\end{thm}\n\n\\begin{proof}\nWe proceed by induction on $n$, and the case $n=0$ is trivial, so assume $n>0$.\nBy Lemma \\ref{LemmaReductionStableInfiniteResolution} we may assume that $I$ is strongly stable.\nIn addition to the notation established in \\eqref{DefinitionRings},\nin this proof we consider the ``intermediate'' ring \n$$\nT = \\frac{S}{(x_1^{d_1}, \\ldots, x_{n-1}^{d_{n-1}})}\n$$\nso that $R=T\/(x_n^{d_n})$.\nBy assumption, we either have $d_n = \\infty$, in which case $T=R$,\nor $d_n =2$.\n\n\nConsider the ideal ${\\mathcal{I}}\\subseteq T$ generated by the monomials of $T$ corresponding to the minimal generators of $I\\subseteq R$.\nNotice that ${\\mathcal{I}}$ is smaller than the preimage of $I$ in $T$ if $d_n=2$, whereas ${\\mathcal{I}}=I$ if $d_n=\\infty$.\nSince $x_{n}$ is a non-zerodivisor on $T$ and ${\\mathcal{I}}$, and $T\/(x_{n}) \\cong {\\overline{R}}$, \nwe have $\\beta_{i,j}^{T}({\\mathcal{I}}) = \\beta_{i,j}^{{\\overline{R}}}({\\mathcal{I}}\/x_{n}{\\mathcal{I}})$.\nWe have a decomposition of ${\\overline{R}}$-modules\n$$\n\\frac{{\\mathcal{I}}}{x_{n}{\\mathcal{I}}} = I_0 \\oplus \\bigoplus_{\\ell =1}^{d_n-1} \\frac{I_\\ell}{I_{\\ell-1}}.\n$$\nBy induction $\\beta_i^{\\overline{R}}(I_0)\\leq\\beta_i^{\\overline{R}}(\\mathrm{Exp}(I_0))$.\nIn the proof of Theorem \\ref{TheoremExtremalFiniteResolution} \nwe established that $\\mathrm{Exp}(p)_0 \\subseteq \\mathrm{Exp}(I_0)$, \nand that \n$\\frac{\\mathrm{Exp}(I_0)}{\\mathrm{Exp}(p)_0}$\nis a free $\\Bbbk[x_{n+1}]$-module via restriction of scalars of rank $c_0 = \\mathrm{HP}(I_0)-\\mathrm{HP}(\\mathrm{Exp}(p)_0)$.\nBy Lemma \\ref{LemmaUpperBoundBettiFreeModule} (i) we obtain\n\\begin{equation}\\label{InequalityBetti0InfiniteResolution}\n\\beta_i^{\\overline{R}}(I_0)\\leq\\beta_i^{\\overline{R}}(\\mathrm{Exp}(I_0)) \\leq \\beta_i^{\\overline{R}}(\\mathrm{Exp}(p)_0) + c_0 \\beta_i^{{\\overline{R}}}(\\Bbbk[x_{n+1}]).\n\\end{equation}\nWe also saw, using Propositions \\ref{PropositionFiniteLengthRank} and \\ref{PropositionElementaryPropertiesDecomposition},\nthat the ${\\overline{R}}$-module $\\oplus_{\\ell =1}^{d_n-1} \\frac{I_\\ell}{I_{\\ell-1}}$ \nis annihilated by $(x_1, \\ldots, x_{n-1})$, and is a free $\\Bbbk[x_{n+1}]$-module of rank $c_1 =\\mathrm{HP}(I_{d_n-1}) - \\mathrm{HP}(I_0)$.\nBy Lemma \\ref{LemmaUpperBoundBettiFreeModule}~(ii) \n\\begin{equation}\\label{EqBettiNumbersTandOvR}\n\\beta_i^{T}({\\mathcal{I}}) = \\beta_i^{{\\overline{R}}}(I_0) + c_1\\beta_i^{{\\overline{R}}}(\\Bbbk[x_{n+1}]).\n\\end{equation}\n\n\nIf $ d_n = \\infty$ then the formula \\eqref{EqBettiNumbersTandOvR} becomes \n$\n\\beta_i^{R}(I) = \\beta_i^{{\\overline{R}}}(I_0) + c_1 \\beta_i^{{\\overline{R}}}(\\Bbbk[x_{n+1}]),\n$\nand likewise we obtain\n$\n\\beta_i^{R}(\\mathrm{Exp}(p)) = \\beta_i^{{\\overline{R}}}(\\mathrm{Exp}(p)_0) + c_2 \\beta_i^{{\\overline{R}}}(\\Bbbk[x_{n+1}])\n$\nwhere $c_2 = \\mathrm{HP}(\\mathrm{Exp}(p)_\\infty)- \\mathrm{HP}(\\mathrm{Exp}(p)_0).$\nBy Proposition \\ref{PropositionElementaryPropertiesDecomposition} (7) we have \n$\\mathrm{HP}(\\mathrm{Exp}(p)_\\infty)=\\mathrm{HP}(I_\\infty)$ and therefore $c_2=c_0+c_1$, and\ncombining with \\eqref{InequalityBetti0InfiniteResolution} we conclude that $\\beta_i^{R}(I) \\leq \n\\beta_i^{R}(\\mathrm{Exp}(p))$ as desired.\n\nNow assume that $d_n = 2$.\nWe regard $R,{\\overline{R}},$ and $T$ as $\\mathbb{Z}^{n+1}$-graded,\nbut we also consider the $\\mathbb{Z}$-grading induced by the variable $x_n$.\nIf $M$ is $\\mathbb{Z}^{n+1}$-graded $T$-module \nwe define $\\sigma(M)$ to be the vector space consisting of the graded components of $M$ with $x_n$-degrees $0$ or $1$.\nClearly, $\\sigma$ defines an exact functor from the category of $\\mathbb{Z}^{n+1}$-graded $T$-modules\nto the category of $\\mathbb{Z}^{n+1}$-graded $\\Bbbk$-vector spaces.\n\nLet ${\\mathbb{F}}$ be the minimal $\\mathbb{Z}^{n+1}$-graded free resolution of ${\\mathcal{I}}$ over $T$.\nThe $x_{n}$-twists in this resolution are all equal to 0 or 1:\nthis follows from the fact that ${\\mathbb{F}} \\otimes_T \\frac{T}{(x_{n})}$ is a minimal $\\mathbb{Z}^{n}$-graded free resolution of ${\\mathcal{I}}\/x_{n}{\\mathcal{I}}$ over ${\\overline{R}}$,\nand that ${\\mathcal{I}}\/x_{n}{\\mathcal{I}}$ is generated in $x_n$-degrees $0,1$.\nThe complex ${\\mathbb{E}}=\\sigma({\\mathbb{F}})$ is acyclic and minimal, in the sense that the image of its differential lies in $(x_1, \\ldots, x_{n+1}){\\mathbb{E}}$.\nEach direct summand in ${\\mathbb{F}}$ has the form $T(-\\delta_1, \\ldots, -\\delta_n, -\\delta_{n+1})$ with $\\delta_n \\in \\{0,1\\}$; \nthe corresponding summand in ${\\mathbb{E}}$ is a factor ring of $R = T\/(x_n^2)$, namely\n$$\n\\sigma\\big(T(-\\delta_1, \\ldots, -\\delta_n, -\\delta_{n+1})\\big) \\cong \\frac{R}{(x_n^{2-\\delta_n})}(-\\delta_1, \\ldots, -\\delta_n, -\\delta_{n+1}).\n$$\nThe cyclic $R$-module on the right hand side is free if and only if $\\delta_n = 0$.\nIn fact, ${\\mathbb{E}}$ is an acyclic minimal $\\mathbb{Z}^{n+1}$-graded complex of (not necessarily free) finitely generated $R$-modules.\nSince all the $x_n$-twists in ${\\mathbb{F}}$ are in $\\{0,1\\}$, every free summand of ${\\mathbb{F}}$ contributes with a non-zero summand in ${\\mathbb{E}}$.\nIn other words,\nthe numbers of generators in every homological degree $i$ is the same for ${\\mathbb{F}}$ and ${\\mathbb{E}}$, and this number is $\\beta_i^{T}({\\mathcal{I}})$.\nAmong the direct summands of ${\\mathbb{E}}$, the free modules are precisely those coming from copies of $T$ in ${\\mathbb{F}}$ with $x_n$-twist equal to 0.\nThese modules form themselves another complex ${\\mathbb{E}}'$, which is again minimal and acyclic, but it is even free.\nIn fact, ${\\mathbb{E}}'$ is the minimal free resolution of $I_0$ over ${\\overline{R}}$, since $I_0$ is the truncation of ${\\mathcal{I}}$ in $x_n$-degree $0$, and ${\\overline{R}}$ is the truncation of $T$ in $x_n$-degree 0. \nWe conclude that in homological degree $i$ in ${\\mathbb{E}}$ we have exactly $\\beta^{\\overline{R}}_i(I_0)$ free summands, i.e. copies of $R$.\n\nTo summarize,\n${\\mathbb{E}}$ is an acyclic minimal complex of $\\mathbb{Z}^{n+1}$-graded $R$-modules, \nit has $\\beta_i^{T}({\\mathcal{I}})$ generators in homological degree $i$,\nof which $\\beta^{\\overline{R}}_i(I_0)$ generate a free module $R$, \nwhereas the remaining ones generate a non-free module isomorphic to $R\/(x)$.\nThe number of non-free summands of ${\\mathbb{E}}$ in homological degree $i$ is therefore\n$\\beta_i^{T}({\\mathcal{I}})- \\beta^{\\overline{R}}_i(I_0) = c_1\\beta_i^{{\\overline{R}}}(\\Bbbk[x_{n+1}])$\nby \\eqref{EqBettiNumbersTandOvR}.\nNote also that the 0-homology of ${\\mathbb{E}}$ is $\\sigma(T\/{\\mathcal{I}}) =R\/I$.\n\n\nLet $E_j$ denote the module in homological degree $j$ in ${\\mathbb{E}}$.\nThe differentials of ${\\mathbb{E}}$ can be lifted to a complex of complexes, namely a double complex ${\\mathbb{D}}_I$ of $R$-modules where the \n$j$-th vertical complex is the minimal free resolution of $F_j$.\nBy construction, the double complex ${\\mathbb{D}}_I$ is free.\nFurthermore, it is minimal, \nand the total complex $\\mathrm{Tot}({\\mathbb{D}}_I)$ is a minimal $\\mathbb{Z}^{n+1}$-graded free resolution of $R\/I$ over $R$,\ncf. \n\\cite[Proposition 5.6]{EiPoYu03},\n\\cite[Theorem 1.3]{ArAvHe00},\nor\n\\cite[Theorem 2.10]{GaHiPe02}.\nRecall that the $R$-module $R\/(x)$ has an infinite minimal free resolution over $R$ with $\\beta_i^R(R\/(x))=1$ for every $i \\in \\mathbb{N}$.\nIt follows that in $\\mathbb{D}_I$, for each $i\\geq 0$, we have \n\\begin{itemize}\n\\item[$(\\ast)$]$\\beta^{\\overline{R}}_i(I_0)$ summands in homological bidegree $(i,0)$ arising from the free summands of ${\\mathbb{E}}$;\n\n\\item[$(\\ast \\ast)$] $c_1\\beta_i^{{\\overline{R}}}(\\Bbbk[x_{n+1}])$ summands in homological bidegree $(i,j)$ for all $j\\geq 0$,\narising from the non-free summands of ${\\mathbb{E}}$;\n\\end{itemize}\nwhere the first coordinate is horizontal and the second coordinate vertical.\nWe conclude that the Betti numbers of a saturated strongly stable $I\\subseteq R$ depend only on those of $I_0\\subseteq {\\overline{R}}$ and on the number \n$c_1 =\\mathrm{HP}(I_1) - \\mathrm{HP}(I_0)$. \n\nThe same construction for $\\mathrm{Exp}(p)$ yields a double complex ${\\mathbb{D}}_{\\mathrm{Exp}(p)}$.\nLet $c'_2 = \\mathrm{HP}(\\mathrm{Exp}(p)_1)- \\mathrm{HP}(\\mathrm{Exp}(p)_0)$.\nWe observed in the proof of Theorem \\ref{TheoremExtremalFiniteResolution} that $\\mathrm{HP}(I_{d_n-1}) \\preceq \\mathrm{HP}(\\mathrm{Exp}(p)_{d_n-1})$,\nthat is, $\\mathrm{HP}(I_{1}) \\preceq \\mathrm{HP}(\\mathrm{Exp}(p)_{1})$.\nWe deduce that $c'_2 \\geq c_0+c_1$.\nFinally, \nwe compare the contribution of the two types of summands $(\\ast)$ and $(\\ast\\ast)$ to the double complexes\n${\\mathbb{D}}_I$ and ${\\mathbb{D}}_{\\mathrm{Exp}(p)}$:\n\n\n\\begin{itemize}\n\\item[$(\\ast)$] \nFor every $i\\geq 0$,\nby \\eqref{InequalityBetti0InfiniteResolution}, \n${\\mathbb{D}}_I$ has at most $c_0\\beta_i^{{\\overline{R}}}(\\Bbbk[x_{n+1}])$ more summands in position $(i,0)$ than ${\\mathbb{D}}_{\\mathrm{Exp}(p)}$,\namong those arising from the free summands of ${\\mathbb{E}}$.\n\\item[$(\\ast\\ast)$] \nFor every $i,j\\geq 0$,\n$\\mathbb{D}_{\\mathrm{Exp}(p)}$ has at least $(c'_2-c_1)\\beta_i^{{\\overline{R}}}(\\Bbbk[x_{n+1}])$ more summands in position $(i,j)$ than\n$\\mathbb{D}_I$, \namong those arising from the non-free summands of ${\\mathbb{E}}$.\n\\end{itemize}\nThus $\\mathbb{D}_{\\mathrm{Exp}(p)}$ has at least as many copies of $R$ as $\\mathbb{D}_I$ in every position $(i,j)$.\nThis concludes the proof, \nsince \n$\\beta_i^R(I),\\beta_i^R(\\mathrm{Exp}(p))$ \nare the Betti numbers of $\\mathrm{Tot}(\\mathbb{D}_I),\\mathrm{Tot}(\\mathbb{D}_{\\mathrm{Exp}(p)})$ respectively.\n\\end{proof}\n\n\nIn the remainder of this section, we explore deviations and Poincar\\'e series of expansive subschemes. \nThe deviations of a ring $A$ are a sequence of integers $\\{ \\varepsilon_i(A)\\}_{i \\geq 1}$\nmeasuring several homological or cohomological data of $A$.\nExamples include: the generators of a Tate resolution of $A$ over a polynomial ring, as well as a Tate resolution of $\\Bbbk$ over $A$;\nthe ranks of the modules in a cotangent complex of $A$;\n the dimensions of the components of the homotopy Lie algebra $\\pi(A)$ of $A$.\nWe refer to \\cite[Sections 7 and 10]{Av98} for definitions and background.\n\n\n\\begin{lemma}\\label{LemmaNoVariable}\nLet $I \\in \\mathrm{Hilb}^{p(\\zeta)}(\\mathrm{Proj} R)$ be strongly stable. \nWe have the inclusion of vector spaces of linear forms\n$[\\mathrm{Exp}(p)]_1 \\subseteq [I]_1$.\n\\end{lemma}\n\\begin{proof}\nWe may assume $I \\ne R$.\nSince $\\mathrm{Exp}(p)$ is saturated and strongly stable, we have $[\\mathrm{Exp}(p)]_1 = \\langle x_1, \\ldots, x_m\\rangle_\\Bbbk$ for some $0\\leq m \\leq n$.\nIf $m=n$ then $\\mathrm{Exp}(p) = (x_1, \\ldots, x_n) \\subseteq R$ is the only strongly stable ideal in $\\mathrm{Hilb}^{p(\\zeta)}(\\mathrm{Proj} R)$,\nso $I= \\mathrm{Exp}(p)$.\nIf $m x_n > x_{n-1} > \\cdots > x_1$.\nThe usual lex order is denoted by $<_{{\\mathrm{lex}}}$.\nFurthermore, let $\\mathcal{G}(-)$ denote the set of minimal monomial generators of a monomial ideal,\nand $[\\mathcal{G}(-)]_d$ those of degree $d$.\n\n\\begin{thm}\\label{TheoremChainExpansive}\nLet $R$ be a Clements-Lindstr\\\"om ring and $p(\\zeta)$ such that $\\mathrm{Hilb}^{p(\\zeta)}(\\mathrm{Proj} R) \\ne \\emptyset$.\nLet $p'(\\zeta) = p(\\zeta) -p(\\zeta-1)$\nand $\\widetilde{L} = {\\mathrm{Lex}}(p'(\\zeta),{\\widetilde{R}})$.\n There exists a chain of almost lex ideals of $R$\n $$\n E^{(0)} \\supseteq E^{(1)} \\supseteq \\cdots \\supseteq E^{(c-1)} \\supseteq E^{(c)}\n $$\n such that $E^{(c)}=\\mathrm{Exp}(p,R)$, \n $E^{(0)} = \\widetilde{L} R \\subseteq R$ is the extension to $R$, \n and for each $k=0, \\ldots, c-1$ we have $\\frac{E^{(k)}}{E^{(k+1)}} = \\Bbbk[x_{n+1}] \\mathbf{u}^{(k)}$, where $\\mathbf{u}^{(k)}$\n is the following monomial of $E^{(k)}$\n$$\n\\mathbf{u}^{(k)} = \\min_{<_\\mathrm{opp}}\\left\\{\\min_{<_{\\mathrm{lex}}} \\left[\\mathcal{G}\\big(E^{(k)}\\big)\\right]_d\\, : \\, d \\in \\mathbb{N}\\right\\}.\n$$\n\\end{thm}\n\n\n\\begin{proof}\nDenote $E = \\mathrm{Exp}(p,R)$ and ${\\widetilde{E}} = \\frac{E+(x_{n+1})}{(x_{n+1})}\\subseteq {\\widetilde{R}}$.\nBy Proposition \\ref{PropositionExpansiveAlmostLex} ${\\widetilde{E}}$ is a lex ideal of ${\\widetilde{R}}$, \nand by Remark \\ref{RemarkModuloXNplus1} $\\mathrm{HP}({\\widetilde{E}},\\zeta) = \\mathrm{HP}(E,\\zeta) - \\mathrm{HP}(E,\\zeta-1)$.\nThe saturation ${\\widetilde{L}} = {\\widetilde{E}} : \\mathfrak{m}_{\\widetilde{R}}^\\infty \\subseteq {\\widetilde{R}}$ is a saturated lex ideal containing ${\\widetilde{E}}$ and with $\\mathrm{HP}({\\widetilde{L}})=\\mathrm{HP}({\\widetilde{E}})$,\nso ${\\widetilde{L}} = {\\mathrm{Lex}}(p'(\\zeta),{\\widetilde{R}})$.\n Let $E^{(0)} = \\widetilde{L} R \\subseteq R$, so\n $E\\subseteq E^{(0)} $ \n and $\\mathrm{HP}(E^{(0)})-\\mathrm{HP}(E)$ is the non-negative integer $c= \\dim_\\Bbbk({\\widetilde{L}}\/{\\widetilde{E}})$,\n cf. Proposition \\ref{PropositionFiniteLengthRank}.\n\nWe prove the theorem by induction on $c$.\nThe case $c=0$ is trivial, so assume $c>0$.\nDenoting $E' = \\mathrm{Exp}\\big(p(\\zeta)-1,R\\big)$ and ${\\widetilde{E}}' = \\frac{E'+(x_{n+1})}{(x_{n+1})}\\subseteq {\\widetilde{R}}$,\nwe have $E\\subseteq E'$ by axiom \\hyperref[A3]{(A3)}, \nand $\\mathrm{HP}(E'\/E) =1$.\nTaking images in ${\\widetilde{R}}$, the quotient ${\\widetilde{E}}'\/{\\widetilde{E}}$ is a 1-dimensional vector space generated by a monomial $\\mathbf{u}$ of ${\\widetilde{E}}'$,\nnecessarily $\\mathbf{u} \\in \\mathcal{G}(E')$.\nFurthermore, since ${\\widetilde{E}}$ is a lex ideal of ${\\widetilde{R}}$, \n$\\mathbf{u}$ must be the lowest monomial with respect to $<_{\\mathrm{lex}}$ in its graded component of ${\\widetilde{E}}'$,\nso\n$\\mathbf{u} =\\min_{<_{\\mathrm{lex}}} \\big[\\mathcal{G}\\big(E'\\big)\\big]_d$ for some $d$.\nSince ${\\widetilde{E}} : \\mathfrak{m}_{\\widetilde{R}}^\\infty = {\\widetilde{E}}' : \\mathfrak{m}_{\\widetilde{R}}^\\infty$ and the theorem holds for $E'$ by induction,\nit remains to show that \n$\\mathbf{u} = \\min_{<_\\mathrm{opp}}\\left\\{\\min_{<_{\\mathrm{lex}}} \\big[\\mathcal{G}\\big(E'\\big)\\big]_d\\, : \\, d \\in \\mathbb{N}\\right\\}$.\nIn other words, given\n $\\mathbf{v} =\\min_{<_{\\mathrm{lex}}} \\big[\\mathcal{G}\\big(E'\\big)\\big]_{d'}$ with $d \\ne d'$, we must show that $\\mathbf{v} >_\\mathrm{opp} \\mathbf{u}$.\n \nLet $I \\subseteq E'$ be the almost lex ideal such that \n${{\\widetilde{E}}'}\/{{\\widetilde{I}}}$ is the 1-dimensional vector space $\\langle \\mathbf{v}\\rangle$.\nNotice that $\\mathrm{HP}(I)=\\mathrm{HP}(E)$.\nLet \n$\n\\mathbf{u} = x_1^{u_1} \\cdots x_m^{u_m} \\cdots x_n^{u_n}$\nand\n$\\mathbf{v} = x_1^{v_1} \\cdots x_m^{v_m} \\cdots x_n^{v_n}\n$\nwhere $m= \\max\\{ i \\, : \\, u_i \\ne v_i\\}$.\nIn order to conclude the proof, we must show that $u_m > v_m$.\nIf $md$.\nSince $L'$ is lex, $\\mathbf{w}$ is a minimal generator, $x_{n+1}^{\\delta-d} \\mathbf{v}\\in L'$ is not a minimal generator, and the two monomials have the same degree, it follows that $\\mathbf{w} <_{{\\mathrm{lex}}} x_{n+1}^{\\delta-d} \\mathbf{v}$.\nHowever, $\\mathbf{w} \\in I$ but $x_{n+1}^{\\delta-d} \\mathbf{v}\\notin I$, so $I$ is not lex, as desired.\n\\end{remark}\n\nTheorem \\ref{TheoremChainExpansive} readily translates into an algorithm to compute $\\mathrm{Exp}(p(\\zeta)$ from $p(\\zeta)$, sketched below.\nFor the sake of completeness, we also include an algorithm to compute ${\\mathrm{Lex}}(p(\\zeta))$.\nThese algorithms have been implemented by the authors in Macaulay2 \\cite{M2}.\n\n\n\\begin{algorithm}[The expansive ideal of a Hilbert polynomial]\\label{AlgorithmExp}\nLet $R$ be a Clements-Lindstr\\\"om ring and $p(\\zeta) \\in \\mathbb{Q}[\\zeta]$ with $\\mathrm{Hilb}^{p(\\zeta)}(\\mathrm{Proj} R)\\ne \\emptyset$.\n\\begin{itemize}\n\\item If $p(\\zeta) =0$, return $ \\mathrm{Exp}\\big(p(\\zeta),R\\big):=R$.\n\\item If $p(\\zeta) \\ne0$, let $p'(\\zeta)=p(\\zeta)-p(\\zeta-1)$, $L^{(0)} := {\\mathrm{Lex}}\\big(p'(\\zeta),{\\widetilde{R}}\\big)R$,\n $c = p(\\zeta) - \\mathrm{HP}(R\/L^{(0)})$.\nFor each $k = 1, \\ldots, c$ let $\\mathbf{u}_1, \\ldots, \\mathbf{u}_t$ be the minimal generators of $L^{(k-1)}$\nso that $\\mathbf{u}_t = \\min_{<_\\mathrm{opp}}\\left\\{\\min_{<_{\\mathrm{lex}}}\\left[\\mathcal{G}\\big(H^{(k-1)}\\big)\\right]_d : d \\in \\mathbb{N}\\right\\}.$\nSet $L^{(k)} := ( \\mathbf{u}_1, \\ldots, \\mathbf{u}_{t-1}, x_1 \\mathbf{u}_t, x_2 \\mathbf{u}_t, \\ldots, x_n \\mathbf{u}_t )$.\n\\item Return $ \\mathrm{Exp}\\big(p(\\zeta),R\\big)=L^{(c)}$.\n\\end{itemize}\n\\end{algorithm}\n\n\\begin{algorithm}[The lex ideal of a Hilbert polynomial]\\label{AlgorithmLex}\nLet $R$ be a Clements-Lindstr\\\"om ring and $p(\\zeta) \\in \\mathbb{Q}[\\zeta]$ with $\\mathrm{Hilb}^{p(\\zeta)}(\\mathrm{Proj} R)\\ne \\emptyset$.\n\\begin{itemize}\n\\item If $p(\\zeta) =0$, return $ {\\mathrm{Lex}}\\big(p(\\zeta),R\\big):=R$.\n\\item If $p(\\zeta) \\ne0$, let $p'(\\zeta)=p(\\zeta)-p(\\zeta-1)$, $H^{(0)} := {\\mathrm{Lex}}\\big(p'(\\zeta),{\\widetilde{R}}\\big)R$,\n $c = p(\\zeta) - \\mathrm{HP}(R\/H^{(0)})$.\nFor each $k = 1, \\ldots, c,\\,$ let $ \\mathbf{w}_1, \\ldots, \\mathbf{w}_t$ be the minimal generators of $H^{(k-1)}$\nordered so that either $\\deg(\\mathbf{w}_i)<\\deg(\\mathbf{w}_{i+1})$ or $\\deg(\\mathbf{w}_i)=\\deg(\\mathbf{w}_{i+1})$ and $\\mathbf{w}_i>_{{\\mathrm{lex}}}\\mathbf{w}_{i+1}$.\nSet $H^{(k)} := ( \\mathbf{w}_1, \\ldots, \\mathbf{w}_{t-1}, x_1 \\mathbf{w}_t, x_2 \\mathbf{w}_t, \\ldots, x_n \\mathbf{w}_t )$.\n\\item Return $ {\\mathrm{Lex}}\\big(p(\\zeta),R\\big)=H^{(c)}$.\n\\end{itemize}\n\\end{algorithm}\n\nThe last result of this section shows that the lex point and the expansive point on $\\mathrm{Hilb}^{p(\\zeta)}(\\mathrm{Proj} R)$ are as different as they can be:\nthey are almost never equal, and if they are, then there is only one strongly stable point on the Hilbert scheme.\nIn the case of $\\mathbb{P}^n$ it follows that if $\\mathrm{Exp}(p(\\zeta),S) = {\\mathrm{Lex}}(p(\\zeta),S)$ then $\\mathrm{Hilb}^{p(\\zeta)}(\\mathbb{P}^n)$ is rational, irreducible, and smooth\n\\cite{LeRo11,ReSt97}.\n\nObserve that a saturated lex $L\\subseteq R$ is necessarily of the form\n\\begin{equation}\\label{EqDescriptionLex}\nL = \\big(x_1^{a_1+1}, x_1^{a_1}x_2^{a_2+1}, \\ldots, x_1^{a_1}\\cdots x_{r-1}^{a_{r-1}}x_r^{a_r+1}\\big)\n\\end{equation}\nfor some integers $a_i \\geq 0$, $r\\leq n$, and such that $x_1^{a_1}\\cdots x_{r-1}^{a_{r-1}}x_r^{a_r+1}\\ne 0$.\nNote that some of the other generators may be 0, if we have $a_i+1 = d_i$ for some $i$.\n\n\n\\begin{prop}\\label{PropositionLexEqualExp}\nLet $R$ be a Clements-Lindstr\\\"om ring and $p(\\zeta)\\in \\mathbb{Q}[\\zeta]$. \nWith notation as in Theorem \\ref{TheoremChainExpansive} and Remark \\ref{RemarkChainLex}\nwe have ${\\mathrm{Lex}}(p(\\zeta),R) = \\mathrm{Exp}(p(\\zeta),R) $ if and only if one of the following occurs\n\\begin{enumerate}\n\\item ${\\mathrm{Lex}}(p(\\zeta),R) = L^{(0)}$;\n\\item $L^{(0)} $ is generated in a single degree and ${\\mathrm{Lex}}(p(\\zeta),R) = L^{(1)}$;\n\\item $L^{(0)} $ is principal and ${\\mathrm{Lex}}(p(\\zeta),R) = L^{(2)}$;\n\\item $d_{n-1} <\\infty$ and ${\\mathrm{Lex}}(p(\\zeta),R) = \\big( x_1^{d_1-1}\\cdots x_{n-1}^{d_{n-1}} x_n^\\alpha \\big)$ for some $\\alpha \\in \\mathbb{N}$.\n\\end{enumerate}\nIn this case $\\mathrm{Hilb}^{p(\\zeta)}(\\mathrm{Proj} R)$ contains only one strongly stable point.\n\\end{prop}\n\\begin{proof}\nIt is easy to check, \nusing Theorem \\ref{TheoremChainExpansive} and Remark \\ref{RemarkChainLex}, that in each case (1), (2), (3), (4) we get \n$\\mathrm{Exp}(p(\\zeta),R) = E^{(c)} = L^{(c)} = {\\mathrm{Lex}}(p(\\zeta),R)$. \nAssume now that $L ={\\mathrm{Lex}}(p(\\zeta),R) = \\mathrm{Exp}(p(\\zeta),R)$, and its generators are as in \\eqref{EqDescriptionLex}.\nThen\n$$\nE^{(0)} = L^{(0)} = \\big(x_1^{a_1+1}, x_1^{a_1}x_2^{a_2+1}, \\ldots, x_1^{a_1}\\cdots x_{r-2}^{a_{r-2}+1}, x_1^{a_1}\\cdots x_{r-1}^{a_{r-1}}\\big).\n$$\nThis implies that the $L^{(k)}$'s are the only almost lex $I$ such that $L^{(0)} \\subseteq I \\subseteq L$, hence $E^{(k)}=L^{(k)}$ for all $k\\leq c$. \nIn particular, ${\\mathrm{Lex}}(p(\\zeta),R) = \\mathrm{Exp}(p(\\zeta),R)$ implies ${\\mathrm{Lex}}(p(\\zeta)-b,R) = \\mathrm{Exp}(p(\\zeta)-b,R)$ for every $b\\in \\mathbb{N}$ for which the Hilbert scheme is nonempty. \nObserve that the generators of a saturated lex ideal ordered as in \\eqref{EqDescriptionLex}\nare non-decreasing in degree, decreasing in $<_{{\\mathrm{lex}}}$, and increasing in $<_\\mathrm{opp}$.\nIt follows from Theorem \\ref{TheoremChainExpansive} and Remark \\ref{RemarkChainLex} that,\nwhenever ${\\mathrm{Lex}}(q(\\zeta),R) = \\mathrm{Exp}(q(\\zeta),R)$ for some $q(\\zeta)$, we have\n${\\mathrm{Lex}}(q(\\zeta)+1,R) = \\mathrm{Exp}(q(\\zeta)+1,R)$ if and only if ${\\mathrm{Lex}}(q(\\zeta))$ is generated in a single degree. \nFinally notice that if ${\\mathrm{Lex}}(q(\\zeta)+1,R)$ is generated in a single degree, then ${\\mathrm{Lex}}(q(\\zeta),R)$ is necessarily principal, \nand ${\\mathrm{Lex}}(q(\\zeta)+1)$ is principal only if ${\\mathrm{Lex}}(q(\\zeta),R) = \\big( x_1^{d_1-1}\\cdots x_{n-1}^{d_{n-1}} x_n^\\alpha \\big)$ for some $\\alpha \\in \\mathbb{N}$.\nThis forces one of (1), (2), (3), or (4) to occur.\n\n\nTo prove the last statement, let $I\\in \\mathrm{Hilb}^{p(\\zeta)}(\\mathrm{Proj} R)$ be strongly stable.\nNote that $L_0=\\mathrm{Exp}(p(\\zeta),R)_0 = {\\mathrm{Lex}}(p(\\zeta),R)_0\\subseteq {\\overline{R}}$ is both expansive and lex.\nLet $r(\\zeta) = \\mathrm{HP}({\\overline{R}}\/L_0)$.\nBy \\hyperref[A6]{(A6)} $\\mathrm{HP}(L_0) \\preceq \\mathrm{HP}(I_0)$, i.e.\n$\\mathrm{HP}({\\overline{R}}\/I_0)=r(\\zeta)-b$ for some $b\\in \\mathbb{N}$.\nAs observed above, this implies $\\mathrm{Exp}(r(\\zeta)-b,{\\overline{R}}) = {\\mathrm{Lex}}(r(\\zeta)-b,{\\overline{R}})$,\nthus by induction on $n$ we obtain $I_0=L_0$.\nIf case (1) holds,\nthen $L=L_0R$, as $x_n$ does not divide the generators of $L$.\nOn the other hand, $I_0R \\subseteq I$.\nWe have $\\mathrm{HP}(L_0) = \\mathrm{HP}(I_0)$ so $\\mathrm{HP}(L_0R) = \\mathrm{HP}(I_0R)$, and $\\mathrm{HP}(I) = \\mathrm{HP}(L)$, \nimplying $I = I_0R$ by Proposition \\ref{PropositionFiniteLengthRank}, so $I=L$ as desired.\nFor the other cases (2), (3), and (4), it suffices to observe that \n if ${\\mathrm{Lex}}(q(\\zeta),R)$ is generated in a single degree for some $q(\\zeta)$,\nthen $L(q(\\zeta)+1,R)$ is the only saturated strongly stable ideal $H \\subseteq {\\mathrm{Lex}}(q(\\zeta),R)$ with $\\mathrm{HP}(R\/H) = q(\\zeta) +1$.\n\\end{proof}\n\n\n\n\n\\section{Examples}\\label{SectionExamples}\n\nWe conclude the paper by \n exhibiting examples of expansive points in some Hilbert schemes, \n constructed by the methods of Section \\ref{SectionComputation},\nand numerical bounds on Betti numbers obtained by the results of Section \\ref{SectionSyzygies}.\n\nWe begin with the analysis of expansive subschemes of dimension 0.\nIt follows by axiom \\hyperref[A4]{(A4)}\nthat the 0-dimensional subschemes defined by $(x_1, \\ldots, x_n)^\\delta$ are expansive for every $\\delta\\geq 0$.\nMore generally, we can characterize all of them explicitly.\n\n\n\\begin{example}[0-dimensional subschemes]\\label{Ex0Dimension}\nLet $c\\in \\mathbb{N}$, then $\\mathrm{Exp}(c)$ is the unique almost lex ideal of $\\mathrm{Hilb}^c(\\mathrm{Proj} R)$ generated in at most two consecutive degrees. \nEquivalently,\n$\\mathrm{Exp}(c) = (x_1, \\ldots, x_n)^\\delta + I$\nwhere $\\delta = \\min\\{ d \\, : \\mathrm{HP}\\big(R\/(x_1, \\ldots, x_n)^d\\big) \\geq c\\}$ and \n$I\\subseteq R$ is an almost lex ideal generated in degree $\\delta -1$.\nNote that $I$ is necessarily generated by the first $\\mathrm{HP}\\big(R\/(x_1, \\ldots, x_n)^\\delta\\big) - c$ monomials of $[{\\widetilde{R}}]_{\\delta-1}$ in the lex order.\nThis statement follows by induction on $c$, using the chain of ideals in Theorem \\ref{TheoremChainExpansive}.\nIn the special case of $\\mathrm{Hilb}^c(\\mathbb{P}^n)$, we recover the main result of \\cite{Va94}.\n\\end{example}\n\n\\begin{example}\nThe simplest known reducible Hilbert scheme of points is $\\mathrm{Hilb}^8(\\mathbb{P}^4)$, see\n\\cite{CaErVeVi09}.\nIt is the union of two irreducible components of dimension 32 and 23. \nThe expansive subscheme is \n$E = (x_1, \\ldots, x_4)^3+ (x_1^2, x_1x_2, x_2^2, x_1x_3, x_2x_3, x_3^2, x_1x_4)$,\nand it lies in the intersection of the two components. \nTo verify this , consider the vector space \n $W = [{\\widetilde{S}}\/{\\widetilde{E}}]_2 = \\langle x_2x_4, x_3x_4, x_4^2\\rangle_\\Bbbk$\nand the bilinear form \n$B : \\big([{\\widetilde{S}}]_1\\otimes W \\big)^{\\otimes 2} \\rightarrow \\bigwedge^3W \\cong \\Bbbk$\ngiven by $B(\\ell_1 \\otimes q_1 , \\ell_2 \\otimes q_2) = \\ell_1\\ell_2 \\wedge q_1 \\wedge q_2$.\nThen $B$ is degenerate, since\n$B(x_2\\otimes x_2x_4 , \\ell_2\\otimes q_2 ) = x_2 \\ell_2 \\wedge x_2x_4 \\wedge q_2 = 0$ \nfor every $\\ell_2, q_2$, \nso the conclusion follows from \n \\cite[Theorem 1.3]{CaErVeVi09}.\n\\end{example}\n\n\n\\begin{example}\nWe exhibit three situations, found in \\cite{Ra19}, \nwhere the lex point and the expansive point are the only two strongly stable points of the Hilbert scheme.\n\n(1) Let $p(\\zeta) = { \\zeta + n-2 \\choose n-2} +\\zeta +1$.\nThen $\\mathrm{Hilb}^{p(\\zeta)}(\\mathbb{P}^n)$ is the union of two irreducible components $\\mathcal{H}, \\mathcal{H}'$, \nwhose general points are respectively a line and an $(n-2)$-plane in general position, and a line intersecting an $(n-2)$-plane union an isolated point \\cite[Section 3]{Ra19}.\nThe lex point ${\\mathrm{Lex}}(p(\\zeta)) = (x_1, x_2^2, x_2x_3, \\ldots, x_2x_{n-2}, x_2 x_{n-1}^2, x_2x_{n-1}x_n)$ lies in the interior of $\\mathcal{H}'$.\nThe expansive point \n$ \\mathrm{Exp}(p(\\zeta)) = (x_1^2, x_1x_2, \\ldots, x_1x_n, x_2^2, x_2x_3, \\ldots, x_2x_{n-1}) $\nlies in the intersection $\\mathcal{H}\\cap \\mathcal{H}'$.\n\n(2) Let $p(\\zeta) = { \\zeta + n-2 \\choose n-2} +2$.\nThen $\\mathrm{Hilb}^{p(\\zeta)}(\\mathbb{P}^n)$ is irreducible, and its general point parametrizes an $(n-2)$-plane and 2 isolated points \\cite[Section 5.1]{Ra19}.\nWe have \n${\\mathrm{Lex}}(p(\\zeta)) = (x_1, x_2^2, x_2x_3, \\ldots, x_2x_{n-1}, x_2x_n^2)$ and\n$ \\mathrm{Exp}(p(\\zeta)) = (x_1, x_2)(x_1, \\ldots, x_{n})$.\nThe $\\mathrm{GL}(n+1)$-orbit of $ \\mathrm{Exp}(p(\\zeta))$ is the singular locus of $\\mathrm{Hilb}^{p(\\zeta)}(\\mathbb{P}^n)$.\n\n(3) Let $p(\\zeta) = {\\zeta +n \\choose n} - {\\zeta +n-d \\choose n} +3$.\nThen $\\mathrm{Hilb}^{p(\\zeta)}(\\mathbb{P}^n)$ is smooth, and its general point parametrizes a hypersurface of degree $d$ with 3 isolated points \\cite[Section 5.2]{Ra19}.\nWe have ${\\mathrm{Lex}}(p(\\zeta)) = x_1^d(x_1, x_2, \\ldots, x_{n-1}, x_n^3)$ and\n$ \\mathrm{Exp}(p(\\zeta)) = x_1^d (x_1, x_2, \\ldots, x_{n-2}, x_{n-1}^2,x_{n-1}x_n,x_{n}^2)$.\n\\end{example}\n\n\n\\begin{example}[Twisted cubics]\nThe Hilbert scheme $ \\mathrm{Hilb}^{3\\zeta+1}(\\mathbb{P}^3)$ is described in \\cite{PiSc85}. \nIt is the union of two rational smooth irreducible components $\\mathcal{H}, \\mathcal{H}'$, \nwhose general point parametrizes respectively a twisted cubic and a plane cubic union a point in $\\mathbb{P}^3$.\nThere are three strongly stable points in $ \\mathrm{Hilb}^{3\\zeta+1}(\\mathbb{P}^3)$.\nThe point $(x^2,xy,y^2)$ lies in the interior of $\\mathcal{H}$,\nand it is the generic initial ideal of the twisted cubic with respect to $<_{{\\mathrm{lex}}}$.\nThe point ${\\mathrm{Lex}}(3\\zeta+1) =( x, y^4, y^3z )$ lies in the interior of $\\mathcal{H}'$.\nFinally, $\\mathrm{Exp}(3\\zeta+1)=(x^2, xy, xz, y^3)$\n lies in the intersection $\\mathcal{H}\\cap \\mathcal{H}'$\nand gives the most degenerate curve in this Hilbert scheme, namely a line tripled in the plane with a spatial embedded point. \nThe universal deformation space of $\\mathrm{Exp}(3\\zeta+1)$ is studied in \\cite[Lemma 6]{PiSc85}\nto deduce the rationality of $\\mathcal{H}, \\mathcal{H}',$ and $ \\mathcal{H}\\cap\\mathcal{H}'$.\n\\end{example}\n\n\n\n\\begin{example}\nLet $ R = \\Bbbk[x_1, x_2, x_3, x_4, x_5]\/(x_1^2, x_2^3, x_3^3)$. \nThe upper bounds on the syzygies of $I \\in \\mathrm{Hilb}^{7\\zeta}(\\mathrm{Proj} R)$ are \n$$\n\\beta_1^S(R\/I) \\leq 7,\\quad\n\\beta_2^S(R\/I) \\leq 13, \\quad\n\\beta_3^S(R\/I) \\leq 9,\\quad\n\\beta_4^S(R\/I) \\leq 2.\n$$\nIf we ignore the data of the degree sequence and \nregard $\\mathrm{Proj} R\/I$ as a subscheme in \n$\\mathrm{Hilb}^{7\\zeta}(\\mathbb{P}^4)$,\nwe obtain the coarser bounds \n$$\n\\beta_1^S(R\/I) \\leq 19,\\quad\n\\beta_2^S(R\/I) \\leq 42, \\quad\n\\beta_3^S(R\/I) \\leq 33,\\quad\n\\beta_4^S(R\/I) \\leq 9.\n$$\n\\end{example}\n\n\n\n\\begin{example}\nLet $\\mathcal{S}\\subseteq \\mathbb{P}^4$ be a complete intersection of a quadric and a cubic hypersurface.\nUsing Theorem \\ref{TheoremLPP} we find that a 1-dimensional subscheme $\\mathcal{C}\\in \\mathrm{Hilb}^{5\\zeta+10}(\\mathcal{S})$ has syzygies bounded by \n$\n\\beta_0^S(I_\\mathcal{C}) \\leq 17,\n\\beta_1^S(I_\\mathcal{C}) \\leq 39, \n\\beta_2^S(I_\\mathcal{C}) \\leq 32,\n\\beta_3^S(I_\\mathcal{C}) \\leq 9.\n$\n\\end{example}\n\n\n\n\n\\begin{example}\nAn elliptic quartic $\\mathcal{C}\\subseteq \\mathbb{P}_{\\mathbb{C}}^3$ is the complete intersection of 2 quadric surfaces. \nFor any 0-dimensional subscheme $\\mathcal{Z}\\subseteq \\mathcal{C}$ we claim that the following bounds hold\n$$\n\\beta_0^S(I_\\mathcal{Z}) \\leq 6,\\,\n\\beta_1^S(I_\\mathcal{Z}) \\leq 9, \\,\n\\beta_2^S(I_\\mathcal{Z}) \\leq 4.\n$$\nTo see this, let $R = \\mathbb{C}[x,y,z,w]\/(x^2,y^2)$.\nIt follows from Example \\ref{Ex0Dimension} that\n$E=\\mathrm{Exp}(\\mathrm{HP}(S\/I_\\mathcal{Z}), R)$ is either one of \n$(x,y,z), (x,y,z^2), (x,yz,z^2)$ or it has the form \n$(xyz^\\alpha, xz^{\\alpha+1+\\delta_1}, yz^{\\alpha+1+\\delta_2}, z^{\\alpha+2+\\delta_3})$\nfor some integers $\\alpha\\in \\mathbb{N}$ and $0\\leq \\delta_1 \\leq \\delta_2 \\leq \\delta_3\\leq 1$.\nThe claim follows now from Theorem \\ref{TheoremLPP} computing a resolution of $R\/E$.\nObserve that results that do not take degree sequences into account (such as those of \\cite{CaMu13} or \\cite{Va94})\ndo not yield any bounds in this example, \nsince the Betti numbers of arbitrary 0-dimensional subschemes $\\mathcal{Z}\\subseteq \\mathbb{P}^n$ are obviously unbounded. \n\\end{example}\n\n\n\n\\subsection*{Acknowledgments}\nThe second author would like to thank Ritvik Ramkumar for some helpful conversations.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\n\n\n\\subsection{Setup}\n\n\nWe first evaluate our method on ImageNet. We sample a set of 1000 images from the ImageNet validation set that are initially classified correctly to avoid artificially inflating the success rate. Since the predicted probability is available for every class, we minimize the probability of the correct class as adversarial loss in untargeted attacks, and maximize the probability of the target class in targeted attacks. We sample a target class uniformly at random for all targeted attacks.\n\nNext, we evaluate SimBA in the real-world setting of attacking the Google Cloud Vision API. Due to the extreme budget required by baselines that might cost up to \\$150 per image\\footnote{The Google API charges \\$1.50 for 1000 image queries.}, here we only compare to LFBA, which we found to be the most query efficient baseline.\n\nIn our experiments, we limit SimBA and SimBA-DCT to at most $T=10,000$ iterations for untargeted attacks and to $T=30,000$ for targeted attacks. For SimBA-DCT, we keep the first $1\/8$th of all frequencies, and add an additional $1\/32$nd of the frequencies whenever we exhaust available frequencies without succeeding. For both methods, we use a fixed step size of $\\epsilon = 0.2$.\n\n\\subsection{ImageNet results}\n\\label{sec:imagenet}\n\n\\paragraph{Success rate comparison (\\autoref{fig:methods_comparison}).}\nWe demonstrate the query efficiency of our method in comparison to the QL attack -- arguably the state-of-the-art black-box attack method to date -- by plotting the average success rate against the number of queries. Figure \\ref{fig:methods_comparison} shows the comparison for both untargeted and targeted attacks. The dotted lines show progression of average $L_2$-norm throughout optimization. Both SimBA and SimBA-DCT achieve dramatically faster increase in success rate in both untargeted and targeted scenarios. The average $L_2$-norm for both methods are also significantly lower.\n\n\\begin{figure*}[t!]\n\\centering\n\\includegraphics[width=0.49\\textwidth]{fig\/queries_untargeted2.pdf}\n\\includegraphics[width=0.49\\textwidth]{fig\/queries_targeted2.pdf}\n\\vspace{-1ex}\n\\caption{Histogram of number of queries required until a successful attack (over 1000 target images). SimBA-DCT is highly right skewed, suggesting that only a handful of images require more than a small number of queries. The \\emph{median} number of queries required by SimBA-DCT for untargeted attack is only 582. However, limiting to the low frequency basis results in SimBA-DCT failing to find a successful adversarial image after $60,000$ queries, whereas SimBA can achieve $100\\%$ success rate consistently.\n\\label{fig:queries_histogram}}\n\\end{figure*}\n\n\\begin{table*}[ht!]\n\\centering\n\\resizebox{0.49\\textwidth}{!}{\n\\begin{tabular}{cccc}\n\\multicolumn{4}{c}{\\large \\hspace{4ex} \\textbf{Untargeted}} \\\\\n\\hline\n\\textbf{Attack} & \\textbf{Average queries} & \\textbf{Average $L_2$} & \\textbf{Success rate} \\\\\n\\hline\n\\multicolumn{4}{c}{\\hspace{4ex} Label-only} \\\\\n\\hline\nBoundary attack & 123,407 & 5.98 & $100\\%$ \\\\\nOpt-attack & 71,100 & 6.98 & $100\\%$ \\\\\nLFBA & 30,000 & 6.34 & $100\\%$ \\\\\n\\hline\n\\multicolumn{4}{c}{\\hspace{4ex} Score-based} \\\\\n\\hline\nQL-attack & 28,174 & 8.27 & $85.4\\%$ \\\\\nBandits-TD & 5,251 & 5.00 & $80.5\\%$ \\\\\n\\textbf{SimBA} & 1,665 & 3.98 & $98.6\\%$ \\\\\n\\textbf{SimBA-DCT} & {\\bf 1,283} & 3.06 & $97.8\\%$ \\\\\n\\hline\n\\end{tabular}\n}\n\\resizebox{0.49\\textwidth}{!}{\n\\begin{tabular}{cccc}\n\\multicolumn{4}{c}{\\large \\hspace{3ex} \\textbf{Targeted}} \\\\\n\\hline\n\\textbf{Attack} & \\textbf{Average queries} & \\textbf{Average $L_2$} & \\textbf{Success rate} \\\\\n\\hline\n\\multicolumn{4}{c}{\\hspace{4ex} Score-based} \\\\\n\\hline\nQL-attack & 20,614 & 11.39 & $98.7\\%$ \\\\\nAutoZOOM & 13,525 & 26.74 & $100\\%$ \\\\\n\\textbf{SimBA} & {\\bf 7,899} & 9.53 & $100\\%$ \\\\\n\\textbf{SimBA-DCT} & 8,824 & 7.04 & $96.5\\%$ \\\\\n\\hline\n\\end{tabular}\n}\n\\vspace{1ex}\n\\caption{Average query count for untargeted (left) and targeted (right) attacks on ImageNet. Methods are evaluated on three different metrics: average number of queries until success (lower is better), average perturbation $L_2$-norm (lower is better), and success rate (higher is better). Both SimBA and SimBA-DCT achieve close to $100\\%$ success rate, similar to other methods in comparison, but require significantly fewer model queries while achieving \\emph{lower} average $L_2$ distortion.\n\\label{table:method_comparison_imagenet}}\n\\end{table*}\n\n\\paragraph{Query distributions (\\autoref{fig:queries_histogram}).} In \\autoref{fig:queries_histogram} we plot the histogram of model queries made by both SimBA and SimBA-DCT over 1000 random images. Notice that the distributions are highly right skewed so the median query count is a much more representative aggregate statistic than average query count. These median counts for SimBA and SimBA-DCT are only 944 and 582, respectively. In the targeted case, SimBA-DCT can construct an adversarial perturbation within only $4,854$ median queries but failed to do so after $60,000$ queries for approximately $2.5\\%$ of the images. In contrast, SimBA achieves a success rate of $100\\%$ with a median query count of $7,038$.\n\nThis result shows a fundamental trade-off when selecting the orthonormal basis $Q$. Restricting to only the low frequency DCT basis vectors for SimBA-DCT results in faster average rate of descent for most images, but may fail to admit an adversarial perturbation for some images. This phenomenon has been observed by \\citet{guo2018low} for optimization-based white-box attacks as well. Finding the right spectrum to operate in on a per-image basis may be key to further improving the query efficiency and success rate of black-box attack algorithms. We leave this promising direction for future work.\n\n\n\\paragraph{Aggregate statistics (\\autoref{table:method_comparison_imagenet}).}\n\\autoref{table:method_comparison_imagenet} computes aggregate statistics of model queries, success rate, and perturbation $L_2$-norm across different attack algorithms. We reproduce the result for LFBA, QL-attack and Bandits-TD using default hyperparameters, and present numbers reported by the original authors' papers for Boundary Attack\\footnote{Result reproduced by \\citet{cheng2018query}}, Opt-attack, and AutoZOOM. The target model is a pretrained ResNet-50 \\citep{he2016residual} network, with the exception of AutoZOOM, which used an Inception v3 \\citep{szegedy2016rethinking} network. Some of the attacks operate under the harder label-only setting (i.e., only the predicted label is observed), which may impact their query efficiency due to observation of partial information. Nevertheless, we include these methods in the table for completeness.\n\nThe three columns in the table show all the relevant metrics for evaluating a black-box attack. Ideally, an attack should succeed often, construct perturbation with low $L_2$ norm, and do so within very few queries. It is possible to artificially reduce the number of model queries by lowering success rate and\/or increasing perturbation norm. To ensure fair comparison, we enforce our methods achieve close to $100\\%$ success rate and compare the other two metrics. Note that the success rate for boundary attack and LFBA are always $100\\%$ since both methods begin with very large perturbations to guarantee misclassification and gradually reduce the perturbation norm.\n\nBoth SimBA and SimBA-DCT have \\emph{significantly lower} average $L_2$-norm than all baseline methods. For untargeted attack, our methods require 3-4x fewer queries (at $1,665$ and $1,232$, respectively) compared to the strongest baseline method -- Bandits-TD -- which only achieves a $80\\%$ success rate. For targeted attack (right table), the evaluated methods are much more comparable, but both SimBA and SimBA-DCT still require significantly fewer queries than QL-attack and AutoZOOM.\n\n\\begin{figure}[t]\n\\vspace{1ex}\n\\centering\n\\includegraphics[width=\\columnwidth]{fig\/models_comparison_both2.pdf}\n\\caption{Comparison of success rate versus number of model queries across different network architectures for untargeted SimBA (solid line) and SimBA-DCT (dashed line) attacks. Both methods can successfully construct adversarial perturbations within $20,000$ queries with high probability. DenseNet is the most vulnerable against both attacks, admitting a success rate of almost $100\\%$ after only 6,000 queries for SimBA and 4000 queries for SimBA-DCT. Inception v3 is much more difficult to attack for both methods.\n\\label{fig:network_comparison}}\n\\vspace{1ex}\n\\end{figure}\n\n\\paragraph{Evaluating different networks (\\autoref{fig:network_comparison}).}\nTo verify that our attack is robust against different model architectures, we evaluate SimBA and SimBA-DCT additionally against DenseNet-121 \\citep{huang2016densely} and Inception v3 \\citep{szegedy2016rethinking} networks. Figure \\ref{fig:network_comparison} shows success rate across the number of model queries for an untargeted attack against the three different network architectures. ResNet-50 and DenseNet-121 exhibit a similar degree of vulnerability against our attacks. However, Inception v3 is noticeably more difficult to attack, requiring more than $10,000$ queries to successfully attack with some images. Nevertheless, both methods can successfully construct adversarial perturbations against all models with high probability.\n\n\n\n\n\n\\begin{figure}[ht!]\n\\centering\\includegraphics[width=1.08\\columnwidth]{fig\/image_samples_combined.pdf}\n\\caption{Randomly selected images before and after adversarial perturbation by SimBA, SimBA-DCT and QL attack. The constructed perturbation is imperceptible for all three methods, but the perturbation $L_2$-norms for SimBA and SimBA-DCT are significantly lower than that of QL attack across all images. Our methods are capable of constructing an adversarial example in comparable or fewer queries than QL attack -- as few as 36 queries in some cases! Zoom in for detail. \\label{fig:image_samples}}\n\\vspace{-2ex}\n\\end{figure}\n\n\\begin{figure}[t]\n \\centering\n \\vspace{1ex}\n \\includegraphics[width=0.8\\columnwidth]{fig\/GCV_comparison2.pdf}\n \\vspace{-1ex}\n \\caption{Plot of success rate across number of model queries for Google Cloud Vision attack. SimBA is able to achieve close to $70\\%$ success rate after only 5000 queries, while the success rate for LFBA has only reached $25\\%$. \\label{fig:gcv_succ_plot}}\n \\vspace{-3ex}\n\\end{figure}\n\n\\begin{figure*}[t]\n \\centering\n \\includegraphics[width=\\textwidth]{fig\/GCV_screenshot.pdf}\n \\caption{Screenshot of Google Cloud Vision labeling results on a sample image before and after adversarial perturbation. The original image contains a set of camera instruments. The adversarial image successfully replaced the top concepts with guns and weapons. See supplementary material for additional samples. \\label{fig:gcv_screenshot}}\n \\vspace{-2ex}\n\\end{figure*}\n\n\n\\paragraph{Qualitative results (\\autoref{fig:image_samples}).}\nFor qualitative evaluation of our method, we present several randomly selected images before and after adversarial perturbation by untargeted attack. For comparison, we attack the same set of images using QL attack. Figure \\ref{fig:image_samples} shows the clean and perturbed images along with the perturbation $L_2$-norm and number of queries. While all attacks are highly successful at changing the label, the norms of adversarial perturbations constructed by SimBA and SimBA-DCT are much smaller than that of QL attack. Both methods requires consistently fewer queries than QL attack for almost all images. In fact, SimBA-DCT was able to find an adversarial image in as few as 36 model queries! Notice that the perturbation produced by SimBA contains sparse but sharp differences, constituting a low $L_0$-norm attack. SimBA-DCT produces perturbations that are sparse in frequency space, and the resulting change in pixel space is spread out across all pixels.\n\n\\subsection{Google Cloud Vision attack}\n\\label{sec:gcv}\n\nTo demonstrate the efficacy of our attack against real world systems, we attack the Google Cloud Vision API, an online machine learning service that provides labels for arbitrary input images. For a given image, the API returns a list of top concepts contained in the image and their associated probabilities. Since the full list of probabilities associated with every label is unavailable, we define an untargeted attack that aims to remove the top 3 concepts in the original. We use the maximum of the original top 3 concepts' returned probabilities as the adversarial loss and use SimBA to minimize this loss. Figure \\ref{fig:gcv_screenshot} shows a sample image before and after the attack. The original image (left) contains concepts related to camera instruments. SimBA successfully replaced the top concepts with weapon-related objects with imperceptible change to the original image. Additional samples are included in the supplementary material.\n\nSince our attack can be executed efficiently, we evaluate its effectiveness over an aggregate of 50 random images. For the LFBA baseline, we define an attack as successful if the produced perturbation has an $L_2$-norm of at most the highest $L_2$-norm in a successful run of our attack. Figure \\ref{fig:gcv_succ_plot} shows the average success rate of both attacks across number of queries. SimBA achieves a final success rate of $70\\%$ after only 5000 API calls, while LFBA is able to succeed only $25\\%$ of the time under the same query budge. To the best of our knowledge, this is the first adversarial attack result on Google Cloud Vision that has a high reported success rate within very limited number of queries.\n\n\n\\section{Introduction}\n\\input{introduction.tex}\n\n\\section{Background}\n\\input{background.tex}\n\n\\section{A Simple Black-box Attack}\n\\input{method.tex}\n\n\\section{Experimental Evaluation}\n\\input{experiment.tex}\n\n\\section{Related Work}\n\\input{related.tex}\n\n\\section{Discussion and Conclusion}\n\\input{conclusion.tex}\n\n\n\n\\subsection{Hyper-parameters}\n\n\n\\paragraph{Cartesian basis.}\nA natural first choice for the set of orthogonal search directions $Q$ is the standard basis $Q = I$, which corresponds to performing our algorithm directly in pixel space. Essentially each iteration we are increasing or decreasing one color of a single randomly chosen pixel. Attacking in this basis corresponds to an $L_0$-attack, where the adversary aims to change as few pixels as possible.\n\n\\paragraph{Discrete cosine basis.} Recent work has discovered that random noise in low frequency space is more likely to be adversarial \\citep{guo2018low}. To exploit this fact, we follow \\citet{guo2018low} and propose to exploit the \\emph{discrete cosine transform} (DCT). The discrete cosine transform is an orthonormal transformation that maps signals in a 2D image space $\\mathbb{R}^{d \\times d}$ to frequency coefficients corresponding to magnitudes of cosine wave functions. In what follows, we will refer to the set of orthonormal frequencies extracted by the DCT as $Q_{\\dct}$. While the full set of directions $Q_{\\dct}$ contains $d \\times d$ frequencies, we keep only a fraction $r$ of the lowest frequency directions in order to make the adversarial perturbation in the low frequency space.\n\n\\paragraph{General basis.} In general, we believe that our attack can be used with any orthonormal basis, provided that the basis vectors can be sampled efficiently. This is especially challenging for high resolution datasets such as ImageNet since each orthonormal basis vector has dimensionality $d \\times d$. Iterative sampling methods such as Gram-Schmidt process cannot be used due to linear memory cost in the number of sampled vectors. Thus, we choose to evaluate our attack using only the standard basis vectors and DCT basis vectors for their efficiency and natural suitability to images.\n\n\\paragraph{Learning rate $\\epsilon$.}\nGiven any set of search directions $Q$, some directions may decrease $p_h(y\\mid\\mathbf{x})$ more than others. Further, it is possible for the output probability $p_{h}(y\\mid\\mathbf{x} + \\epsilon \\mathbf{q})$ to be non-monotonic in $\\epsilon$.\nIn \\autoref{fig:eps_vs_prob}, we plot the relative decrease in probability as a function of $\\epsilon$ for randomly sampled search directions in both pixel space and the DCT space. The probabilities correspond to prediction on an ImageNet validation sample by a ResNet-50 model. This figure highlights an illuminating result:\nthe probability $p_h(y\\mid\\mathbf{x}\\pm\\epsilon\\mathbf{q})$ \\emph{decreases monotonically} in $\\epsilon$ with surprising consistency (across random images and vectors $\\mathbf{q}$)!\nAlthough some directions eventually increase the true class probability, the expected change in this probability is negative with a relatively steep slope.\nThis means that our algorithm is not overly sensitive to the choice of $\\epsilon$ and the iterates will decrease the true class probability quickly. The figure also shows that search in the DCT space tends to lead to steeper descent directions than pixel space.\nAs we show in the next section, we can tightly bound the final $L_2$-norm of the perturbation given a choice of $\\epsilon$ and maximum number of steps $T$, so the choice of $\\epsilon$ depends \\emph{primarily on budget considerations} with respect to $\\|\\delta\\|_2$.\n\n\\paragraph{Budget considerations.}\nBy exploiting the orthonormality of the basis $Q$ we can bound the norm of $\\delta$ tightly. Each iteration a basis vector is either added, subtracted, or discarded (if neither direction yields a reduction of the output probability.)\nLet $\\alpha_i \\in \\{-\\epsilon, 0, \\epsilon\\}$ denote the sign of the search direction chosen at step $t$, so\n\\begin{equation*}\n \\delta_{t + 1} = \\delta_{t} + \\alpha_t\\mathbf{q}_{t}.\n\\end{equation*}\nWe can recursively expand $\\delta_{t + 1} = \\delta_{t } + \\alpha_t\\mathbf{q}_{t}$. In general, the\nfinal perturbation $\\delta_{T}$ after $T$ steps can be written as a sum of these individual search directions:\n\\begin{equation*}\n \\delta_{T} = \\sum_{t=1}^{T} \\alpha_t \\mathbf{q}_{t}.\n\\end{equation*}\nSince the directions $\\mathbf{q}_{t}$ are orthogonal, $\\mathbf{q}_{t}^{\\top}\\mathbf{q}_{t'} = 0$ for any $t \\neq t'$. We can therefore compute the $L_2$-norm of the adversarial perturbation:\n\\begin{align*}\n \\Vert \\delta_{T} \\Vert_2^2 = \\left\\| \\sum_{t=1}^{T} \\alpha_t \\mathbf{q}_{t} \\right\\|_2^2\n = \\sum_{t=1}^{T} \\Vert \\alpha_i \\mathbf{q}_{t} \\Vert_2^2\n\t\t\t\t\t\t\t\t\t\t\t\t\t &= \\sum_{t=1}^{T} \\alpha_t^2 \\Vert \\mathbf{q}_{t} \\Vert_2^2 \\\\\n &\\leq T\\epsilon^2.\n\\end{align*}\nHere, the second equality follows from the orthogonality of $\\mathbf{q}_{t}$ and $\\mathbf{q}_{t'}$, and the last inequality is tight if all queries result in a step of either $\\epsilon$ or $-\\epsilon$. Thus the adversarial perturbation has $L_2$-norm at most $\\sqrt{T} \\epsilon$ after $T$ iterations. This result holds for any orthonormal basis (e.g. $Q_{\\dct}$).\n\nOur analysis highlights an important trade-off: for query-limited scenarios, we may reduce the number of iterations by setting $\\epsilon$ higher, incurring higher perturbation $L_2$-norm. If a low norm solution is more desirable, reducing $\\epsilon$ will allow quadratically more queries at the same $L_2$-norm. A more thorough theoretical analysis of this trade-off could improve query efficiency.\n\n\n\\section{Experiment on CIFAR-10}\n\nIn this section, we evaluate SimBA and SimBA-DCT on a ResNet-50 model trained on CIFAR-10. Both attacks remain very efficient on this new dataset without any hyperparameter tuning.\n\n\\autoref{fig:queries_histogram_cifar} shows the distribution of queries required for a successful targeted attack to a random target label. In contrast to the experiment on ImageNet, the use of low frequency DCT basis is less effective due to the reduced image dimensionality. Both SimBA and SimBA-DCT perform similarly, with SimBA-DCT having a slightly heavier tail.\n\n\\autoref{table:method_comparison_cifar} shows aggregate statistics for the attack on CIFAR-10. Both methods achieve a success rate of $100\\%$ when limited to a maximum of $10,000$ queries. The actual required queries is much fewer, with both methods averaging to approximately $300$ queries, matching the median. SimBA-DCT has a slightly worse performance compared to SimBA due its query distribution having a slightly heavier tail. Nevertheless, the average query count is in line with state-of-the-art attacks on CIFAR-10. For instance, AutoZOOM achieves a mean query count of $259$ with an average $L_2$-norm of 3.53.\n\n\\begin{figure}[ht!]\n\\centering\n\\includegraphics[width=0.5\\textwidth]{fig\/queries_targeted_cifar.pdf}\n\\caption{Histogram of number of queries required until a successful targeted attack on CIFAR-10 (over 1000 target images).\n\\label{fig:queries_histogram_cifar}}\n\\end{figure}\n\n\\begin{table}[ht!]\n\\centering\n\\begin{tabular}{ccccc}\n\\hline\n\\textbf{Attack} & \\textbf{Average queries} & \\textbf{Median queries} & \\textbf{Average $L_2$} & \\textbf{Success rate} \\\\\n\\hline\n\\textbf{SimBA} & 322 & 297 & 2.04 & $100\\%$ \\\\\n\\textbf{SimBA-DCT} & 353 & 312 & 2.21 & $100\\%$ \\\\\n\\hline\n\\end{tabular}\n\\vspace{1ex}\n\\caption{Average query count for SimBA and SimBA-DCT on CIFAR-10.\n\\label{table:method_comparison_cifar}}\n\\end{table}\n\n\\section{Additional image samples for attack on Google Cloud Vision}\n\nTo demonstrate the generality of our evaluation of the Google Cloud Vision attack, we show 10 additional random images before and after perturbation by SimBA. In all cases, we successfully remove the top 3 original labels.\n\n\\begin{figure}[h!]\n \\vspace{4ex}\n \\centering\n \\begin{subfigure}[]{0.9\\textwidth}\n \\includegraphics[width=\\textwidth]{fig\/example1_reduced.pdf}\n \\end{subfigure}\n \\begin{subfigure}[]{0.9\\textwidth}\n \\includegraphics[width=\\textwidth]{fig\/example2_reduced.pdf}\n \\end{subfigure}\n \\begin{subfigure}[]{.9\\textwidth}\n \\includegraphics[width=\\textwidth]{fig\/example3_reduced.pdf}\n \\end{subfigure}\n \\begin{subfigure}[]{.9\\textwidth}\n \\includegraphics[width=\\textwidth]{fig\/example4_reduced.pdf}\n \\end{subfigure}\n\\end{figure}\n\n\\begin{figure}[h!]\n \\ContinuedFloat\n \\centering\n \\begin{subfigure}[]{.9\\textwidth}\n \\includegraphics[width=\\textwidth]{fig\/example5_reduced.pdf}\n \\end{subfigure}\n \\begin{subfigure}[]{.9\\textwidth}\n \\includegraphics[width=\\textwidth]{fig\/example6_reduced.pdf}\n \\end{subfigure}\n \\begin{subfigure}[]{.9\\textwidth}\n \\includegraphics[width=\\textwidth]{fig\/example7_reduced.pdf}\n \\end{subfigure}\n \\begin{subfigure}[]{.9\\textwidth}\n \\includegraphics[width=\\textwidth]{fig\/example8_reduced.pdf}\n \\end{subfigure}\n\\end{figure}\n\n\\begin{figure}[h!]\n \\ContinuedFloat\n \\centering\n \\begin{subfigure}[]{.9\\textwidth}\n \\includegraphics[width=\\textwidth]{fig\/example9_reduced.pdf}\n \\end{subfigure}\n \\begin{subfigure}[]{.9\\textwidth}\n \\includegraphics[width=\\textwidth]{fig\/example10_reduced.pdf}\n \\end{subfigure}\n \\caption{Additional adversarial images on Google Cloud Vision.}\n\\end{figure}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\\label{introduction}\n\nThe lack of precise control over standard condensed matter systems has hindered\nthe development of experiments that can probe systematically the effects\nof strong correlations. However, the large degree of control in atomic systems, has\nmade them powerful tools for studying many condensed matter phenomena, \nand in particular novel superfluid phases~\\cite{bourdel1,regal2,chin,partridge1,kinast2,zwierlein3}.\nFor instance, a current research frontier is the study of fermion mixtures with population \nimbalance~\\cite{mit,rice,mit-2,rice-2}. \nSince the population of each component as well as their interaction strength\nare experimentally tunable, these knobs enabled the\nstudy of the BCS to BEC evolution in population imbalanced two-component fermion\nsuperfluids~\\cite{mit,rice,mit-2,rice-2}.\nIn contrast with the crossover physics found in the population balanced \ncase~\\cite{leggett,nsr,carlos}, these experiments have demonstrated the \nexistence of phase transitions between normal and superfluid phases, \nas well as phase separation between superfluid (paired) and normal (excess) \nfermions as a function of \npopulation imbalance~\\cite{liu,bedaque,sedrakian,castorina,carlson,pao,sheehy}.\n\nMotivated by these recent experiments, there has been intense theoretical\ninterest in understanding the phase diagram of population\nimbalanced mixtures~\\cite{torma,pieri,yi,silva,haque,iskin-mixture,lobo,liu-mixture,mizushima}.\nSo far, an accurate description of such mixtures is only available \nin the weak fermion attraction limit, and it is yet to be \ndeveloped for intermediate fermion attraction around unitarity, or\nfor the strong attraction limit. Some progress has been made \nin the strong attraction limit, where Fermi-Fermi mixtures were described\nas a weakly interacting Bose-Fermi mixture~\\cite{pieri, iskin-mixture, iskin-mixture2, taylor},\nhowever, the effective boson-fermion and boson-boson scattering parameters were\nobtained only in the Born approximation.\nStrictly speaking, the Bose-Fermi description is valid only in the strong attraction limit, \nbut may also provide semi-quantitative understanding of the phase diagram close to unitarity. \nThus, the main goal of this manuscript is to analyze the boson-fermion\nand boson-boson scattering parameters beyond the Born approximation\nfor arbitrary mass ratio of fermions, and use the effective Bose-Fermi mixture description\nto generate improved phase diagrams and density profiles of Fermi-Fermi \nmixtures with equal or unequal masses in the strong \nattraction limit beyond the Born and mean-field \napproximations~\\cite{iskin-mixture,pao-mixture,iskin-mixture2,duan-mixture,parish}. \n\nThe main results of this manuscript are as follows.\nFirst, we analyze three- and four-fermion scattering \nprocesses and obtain the exact boson-fermion and boson-boson scattering lengths \nas a function of mass anisotropy. Second, we use the exact boson-fermion\nand boson-boson scattering parameters to construct the phase\ndiagram for Fermi-Fermi mixtures in the strong attraction limit.\nIn addition to the normal (N) and uniform superfluid (U) phases,\nwe find two different non-uniform phase separated (PS) states: \n(1) phase separation between pure unpaired (excess) \nand pure paired fermions (molecular bosons), and\n(2) phase separation between pure excess fermions \nand a mixture of excess fermions and molecular bosons, \ndepending on the fermion-fermion scattering parameter.\nThe phase boundaries are very sensitive to the masses of the fermions, and also to\nthe boson-fermion and boson-boson interactions. \nFor equal mass mixtures, our results for the phase boundary between the PS(2) and \nthe uniform (U) phase improves on previous saddle-point \n(mean-field) results, and the quantitative changes are substantial, but not\ndramatic.\nHowever, there is a dramatic increase in quantitative differences between \nmean-field and the present results for unequal mass mixtures as the mass ratio\ndeviates from one. In particular, these differences are more pronounced when \nheavier fermions are in excess indicating the importance of taking into account\nscattering processes beyond the Born approximation.\nFurthermore, we discuss the effects of the trapping potential on the density \nprofiles of condensed and non-condensed molecular bosons, as well as excess \nfermions at zero and finite temperatures. Lastly, we discuss the implications \nof our findings to possible experiments involving Fermi-Fermi mixtures \nwith equal or unequal masses and equal or unequal populations.\n\nThe remainder of this manuscript is organized as follows.\nIn Sec.~\\ref{sec:hamiltonian}, we discuss briefly the Hamiltonian for Fermi-Fermi \nmixtures with equal or unequal masses and emphasize \nthat the system reduces effectively to a Bose-Fermi mixture \nof molecular bosons and excess fermions in the strong \nattraction limit.\nIn Sec.~\\ref{sec:a_BF}, we analyze the exact boson-fermion and boson-boson \nscattering lengths as a function of mass anisotropy, which are used to \ncalculate the resulting phase diagram of Fermi-Fermi mixtures in \nthe strong attraction limit. \nIn Sec.~\\ref{sec:BFmixture}, we discuss the stability of the effective\nBose-Fermi mixture of molecular bosons and excess fermions in three, two and one dimensions.\nWe analyze the stability of population imbalanced Fermi-Fermi mixtures \nin the strong attraction limit in Sec.~\\ref{sec:FFmixture}, and \nwe construct the phase diagrams of these systems in Sec.~\\ref{sec:phase.diagram}.\nIn Sec.~\\ref{sec:trap}, we discuss the effects of harmonic traps on the density \nprofiles of condensed and non-condensed molecular bosons, and excess fermions \nat zero and finite temperatures, and show their experimental signatures.\nLastly, we present a summary of our conclusions in Sec.~\\ref{sec:conclusions}.\n\n\n\n\n\n\n\n\n\n\n\\section{Hamiltonian}\n\\label{sec:hamiltonian}\n\nIn order to calculate the correct phase diagrams of Fermi-Fermi mixtures\nin the strong attraction limit, it is necessary to obtain first the correct\nscattering parameters between two Bose molecules (paired fermions), \nand also between a Bose molecule and an unpaired fermion. To achieve this task,\nwe begin by describing the Hamiltonian density for a mixture of fermions \n(in units of $\\hbar = k_B = 1$) as \n\\begin{eqnarray}\n\\label{eqn:hamiltonian}\nH (x) &=& \\sum_\\sigma \\bar \\psi_{\\sigma} (x) \\left[ - \\frac {\\nabla^2}{ 2m_{\\sigma} } - \\mu_{\\sigma} \\right] \n\\psi_{\\sigma} \\nonumber \\\\\n&-& g \\bar \\psi_{\\uparrow} (x) \\bar \\psi_{\\downarrow} (x) \\psi_{\\downarrow} (x) \\psi_{\\uparrow} (x),\n\\end{eqnarray}\nwhere $\\bar \\psi_{\\sigma} (x)$ is the field corresponding to the creation\nof a fermion with pseudospin index $\\sigma$, at position and time $x \\equiv ({\\bf r}, \\tau)$.\nHere, $g > 0$ is the strength of the attractive fermion-fermion interaction \nand $\\sigma$ identifies two types of ($\\uparrow$ and $\\downarrow$) \nfermions. This notation allows the analysis of a mixture of fermions with \nequal or unequal masses, as well as equal or unequal chemical potentials.\nThroughout the manuscript, we assume that the lighter fermions are always $\\uparrow$ and\nthat the heavier fermions are always $\\downarrow$, as intuitively suggested by the direction of the arrows,\nand that the chemical potentials $\\mu_{\\sigma}$ fix the population (density) $n_{\\sigma}$ \nof each type of fermion independently.\n\nThe contact interaction Hamiltonian given in Eq.~(\\ref{eqn:hamiltonian})\ngeneralizes the equal mass and equal chemical potential Hamiltonian that is used to study \nthe BCS to BEC evolution within the functional integral formalism~\\cite{carlos}.\nThe functional integral formulation~\\cite{carlos,iskin-mixture,iskin-mixture2,taylor} captures \nsome essential features of the evolution from BCS to BEC superfluidity for Fermi mixtures \nwith equal or unequal masses as well as with equal or unequal populations. \nHowever, truly quantitative results are currently possible only in the BCS limit, \nwhere the theory is simple, but the temperatures required to reach the BCS regime \nare very low and hard to be achieved experimentally.\nIn the unitarity regime, experiments can be performed and the phase diagram \ncan be explored since the critical temperature for superfluidity is attainable,\nbut an accurate theoretical description of this regime is still lacking.\nWhile in the BEC limit, not only the temperature required to reach the BEC regime is \nexperimentally achievable, but also the theory becomes simple since the Fermi-Fermi mixtures \ncan be described effectively by a weakly interacting mixture of molecular bosons and \nexcess fermions~\\cite{pieri,iskin-mixture,iskin-mixture2,taylor}.\nHowever, the initial proposals of such effective Bose-Fermi mixtures~\\cite{pieri,iskin-mixture}\ncan provide only semi-quantitative results for comparison with experiments in the\nBEC regime since the scattering parameters for two molecular bosons and a molecular boson \nand an excess fermion are obtained only in the Born approximation.\nIn order to overcome this defficiency, we discuss next the boson-fermion \nand boson-boson scattering parameters beyond the Born approximation for arbitrary mass ratio. \nThe correct scattering parameters will be used in Sec.~\\ref{sec:phase.diagram} \nto construct phase diagrams and density profiles for \nquantitative comparisons with experiments in the BEC regime.\n\n\n\n\n\n\n\n\n\n\\section{Boson-Fermion and Boson-Boson Scattering Lengths}\n\\label{sec:a_BF}\n\nMixtures of two types of fermions in the strong attraction limit can be described \nby effective Bose-Fermi models~\\cite{pieri,iskin-mixture,iskin-mixture2,taylor}, \nwhere fermion pairs behave as molecular bosons and interact weakly with\neach other and with excess unpaired fermions.\nScattering lengths between two molecular bosons ($a_{BB}$), and between\na molecular boson and an excess fermion ($a_{BF})$ were calculated\nin the Born approximation using many body techniques for \nequal~\\cite{carlos, pieri} and unequal masses~\\cite{iskin-mixture, iskin-mixture2}.\nHowever, these results do not agree with calculations using \nfew body techniques~\\cite{petrov,petrov-abf},\nbecause it is necessary to go beyond the Born approximation.\n\nIn the case of fermion mixtures with equal masses, while the Born approximation \nin many body theory leads to $a_{BB} = 2a_F$~\\cite{carlos} and \n$a_{BF} = 8a_F\/3$~\\cite{pieri, iskin-mixture}, the results from few body techniques are \n$a_{BB} \\approx 0.60a_F$~\\cite{petrov} and $a_{BF} \\approx 1.18a_F$~\\cite{skorniakov}.\nHowever, a diagrammatic approach beyond the Born approximation~\\cite{brodsky, levinsen} \nfor equal mass fermions recovers the few body results. \nIn this section, we generalize these diagrammatic \napproaches and analyze the boson-fermion and boson-boson scattering parameters \nfor two types of fermions with unequal masses in order \nto make quantitative predictions for experiments \ninvolving Fermi-Fermi mixtures in the strong attraction limit. \nHere, we show that the diagrammatic many body approach \nfor unequal mass fermions produce results consistent with three and\nfour body techniques that were recently used to obtain $a_{BF}$ and $a_{BB}$ as a function of \nmass ratio~\\cite{petrov, petrov-abf}. \nWe analyze the technical aspects of the boson-fermion scattering parameter for unequal\nmass fermions in some detail, since they are much easier to present, \nwhile we do not discuss in great detail the technical aspects of the boson-boson scattering\nparameter for unequal masses, as they are extremely cumbersome.\nDetailed descriptions of the boson-fermion and boson-boson scattering parameters for equal\nmass fermions can be found in the literature~\\cite{brodsky, levinsen}.\n\n\\begin{figure} [htb]\n\\psfrag{a}{\\LARGE $-\\mathbf{k}, w_F$}\n\\psfrag{b}{\\LARGE $\\mathbf{k}, w_B + \\epsilon_b$}\n\\psfrag{c}{\\LARGE $\\mathbf{k}+\\mathbf{p}, w_B - w_F + \\epsilon_b + p_0$}\n\\psfrag{d}{\\LARGE $-\\mathbf{q}, w_F - q_0$}\n\\psfrag{i}{\\LARGE $\\mathbf{q}, w_B + \\epsilon_b + q_0$}\n\\psfrag{f}{\\LARGE $\\mathbf{q}+\\mathbf{p}, w_B - w_F + \\epsilon_b + p_0 + q_0$}\n\\psfrag{g}{\\LARGE $-\\mathbf{p}, w_F - p_0$}\n\\psfrag{h}{\\LARGE $\\mathbf{p}, w_B + \\epsilon_b + p_0$}\n\\centerline{\\scalebox{0.4}{\\includegraphics{t-matrix.eps}}}\n\\caption{\\label{fig:t-matrix}\nDiagrammatic representation of the integral equation for the\nboson-fermion scattering T-matrix $T_\\mathbf{k}^{BF}(\\mathbf{p},p_0)$.\nHere, we use the notation $(-\\uparrow) \\equiv \\downarrow$ and vice versa.\n}\n\\end{figure}\n\nWe begin our analysis by describing a zero temperature $(T = 0)$ diagrammatic representation for \nthe boson-fermion scattering T-matrix $T_\\mathbf{k}^{BF}(\\mathbf{p},p_0)$ \nas shown in Fig.~\\ref{fig:t-matrix}, where $w_F = k^2\/(2m_F)$ and $w_B = k^2\/(2m_B)$ \nare the kinetic energies for the excess fermions and molecular bosons, respectively, \nand $\\epsilon_b = - 1\/(m_{\\uparrow \\downarrow} a_F^2) < 0$\nis the binding energy of the molecular bosons.\nHere,\n\\begin{eqnarray}\nm_B &=& m_\\uparrow + m_\\downarrow \\\\\nm_{\\uparrow\\downarrow} &=& \\frac{2m_\\uparrow m_\\downarrow}{m_\\uparrow + m_\\downarrow}\n\\end{eqnarray}\nare masses of the molecular bosons and twice the reduced mass of \nthe $\\uparrow$ and $\\downarrow$ fermions, respectively.\nIn this figure, single lines represent retarded free fermion propagators\n\\begin{equation}\nG_{0,\\sigma}(\\mathbf{k}, w) = \\frac{1}{w - w_\\sigma + \\mu_\\sigma + i0^+},\n\\end{equation}\nwhere $w_\\sigma = k^2\/(2m_\\sigma)$ is the energy and $\\mu_\\sigma$ is the\nchemical potential of the $\\sigma$-type fermions.\nSimilarly double lines represent the retarded molecular boson propagators \n\\begin{equation}\n\\label{eqn:bose-propagator}\nD_0(\\mathbf{k}, w) = \\frac{4\\pi\/m_{\\uparrow\\downarrow}^{3\/2}}\n{|\\epsilon_b|^{1\/2} - (w_B - w - \\mu_\\uparrow - \\mu_\\downarrow - i0^+)^{1\/2}}\n\\end{equation}\nobtained from the expansion of the effective action~\\cite{iskin-mixture,iskin-mixture2},\nand a corresponding RPA re-summation of the fermion polarization bubbles leading to\n$\nD_0(\\mathbf{k}, w) = - g\/[1 + g \\Gamma (\\mathbf{k}, w)]\n$\nwhere the fermion polarization bubble is \n$\n\\Gamma (\\mathbf{k}, w) = \\sum_{{\\bf q}, q_0} G_{0,\\sigma}(\\mathbf{k} + \\mathbf{q}, w + q_0 ) \nG_{0,\\sigma}(-\\mathbf{q}, -q_0 ).\n$\nIntegration over the internal momentum ${\\bf q}$ and frequency $q_0$ leads to \n$\n\\Gamma (\\mathbf{k}, w) = \n\\Gamma (0,0) + [m_{\\uparrow\\downarrow}^{3\/2} \/ (4 \\pi)]\n\\left( w_B - w - \\mu_{\\uparrow} - \\mu_{\\downarrow} - i0^{+} \\right)^{1\/2}\n$\nwhich in combination with the definition of the fermion-fermion scattering \nlength $a_F = m_{\\uparrow\\downarrow} T^{FF} (0,0)\/ 4\\pi$, and\nthe fermion-fermion T-matrix \n$\nT^{FF} (0,0) = - g \/ [1 + g \\Gamma (\\mathbf{0}, 0)]\n$\nlead to the final result described in Eq.~(\\ref{eqn:bose-propagator}).\n\nOn the right hand side of Fig.~\\ref{fig:t-matrix}, the first diagram \nrepresents a fermion exchange process, and all other (infinitely many) \npossible processes are included in the second diagram.\nIn all diagrams, we choose $\\uparrow$ ($\\downarrow$) to label\nlighter (heavier) fermions such that lighter (heavier) fermions\nare in excess when $F \\equiv \\uparrow$ $(F \\equiv \\downarrow)$. \nThis choice spans all possible mass ratios.\nIn the following, we set $\\mu_\\sigma = 0$ since all of the calculations are \nperformed for three fermions scattering in vacuum.\nThe T-matrix $T_\\mathbf{k}^{BF}(\\mathbf{p},p_0)$ \nsatisfies the following integral equation\n\\begin{eqnarray}\n&& T_\\mathbf{k}^{BF}(\\mathbf{p},p_0) = \n- G_{0,-F}(\\mathbf{k}+\\mathbf{p}, w_B - w_F + \\epsilon_b + p_0) \\nonumber \\\\\n&& - \\sum_{\\mathbf{q}, q_0} D_0(\\mathbf{q}, w_B + \\epsilon_b + q_0) \nG_{0,F}(-\\mathbf{q}, w_F - q_0) \\times \\\\\n&& T_\\mathbf{k}^{BF}(\\mathbf{q},q_0) G_{0,-F}(\\mathbf{p} + \\mathbf{q}, w_B - w_F + \\epsilon_b + p_0 + q_0) \\nonumber,\n\\label{eqn:t-matrix}\n\\end{eqnarray}\nwhere we used $(-\\uparrow) \\equiv \\downarrow$ and vice versa.\nOn the right hand side, we can sum over frequency $q_0$ by closing the \nintegration contour in the upper half-plane, where $T_\\mathbf{k}^{BF}(\\mathbf{q},q_0)$ \nand $D_0(\\mathbf{q}, w_B + \\epsilon_b + q_0)$ are analytic functions of $q_0$.\nSince this integration sets $q_0 = k^2\/(2m_F) - q^2\/(2m_F)$, we set \n$p_0 = k^2\/(2m_F) - p^2\/(2m_F)$ in order to have the same frequency dependence\nfor the T-matrix on both sides~\\cite{levinsen}. \nSince we are interested in the zero-range low energy s-wave scattering,\nwe average out the angular dependences of $\\mathbf{k}$ and $\\mathbf{p}$.\nWhen $k \\to 0$, the generalized integral equation defined in Eq.~(\\ref{eqn:t-matrix})\ncan be expressed in terms of the boson-fermion scattering function $a_{k \\to 0} (p)$ as\n\\begin{eqnarray}\n&& \\frac{m_{\\uparrow\\downarrow} a_0^{BF}(p)\/m_{BF}} \n{a_F^{-1} + (m_{\\uparrow\\downarrow} p^2\/m_{BF} + a_F^{-2})^{1\/2}} \n= \\frac{1}{p^2 + a_F^{-2}} - \\frac{m_B}{2\\pi m_F} \\nonumber \\\\\n&& \\int_0^\\infty \\frac{dq}{qp} \n\\ln\\left( \\frac{q^2 + 2m_F qp\/m_B + p^2 + a_F^{-2}} {q^2 - 2m_F qp\/m_B + p^2 + a_F^{-2}} \\right) \na_0^{BF}(q).\n\\label{eqn:ieqn}\n\\end{eqnarray}\nHere, we used the definition of the boson-fermion scattering length \n\\begin{equation}\na_k^{BF}(p) = \\frac{m_{BF}}{m_{\\uparrow\\downarrow}^{3\/2}}\n\\left[|\\epsilon_b|^{1\/2} + \\left(\\frac{p^2-k^2}{m_{BF}} - \\epsilon_b\\right)^{1\/2}\\right] \nT_k^{BF}(p),\n\\label{eqn:a-scattering}\n\\end{equation}\nwith its full momentum dependence,\nwhere $m_{BF}$ is twice the reduced mass of an excess fermion \nand a molecular boson given by\n\\begin{equation}\nm_{BF} = \\frac{2m_B m_F}{m_B + m_F}.\n\\end{equation}\nThe integral equation shown in Eq.~(\\ref{eqn:ieqn}) as well\nas the scattering length expression shown in Eq.~(\\ref{eqn:a-scattering}) \nreduce to the results for the equal masses~\\cite{brodsky, levinsen} when \n$m_\\uparrow = m_\\downarrow = m$.\nNotice that only the fermion exchange process is taken into account in \nthe Born approximation, and that neglecting the second term on the right hand side \nof Eq.~(\\ref{eqn:ieqn}) leads to\n$\na_0^{BF}(0) = 2(m_{BF}\/m_{\\uparrow\\downarrow})a_F\n$\nwhich is consistent with our previous results~\\cite{iskin-mixture, iskin-mixture2}. \nHowever, we need to include both terms in order to find the exact boson-fermion scattering length.\n\n\\begin{figure} [htb]\n\\centerline{\\scalebox{0.5}{\\includegraphics{BFscattering-mr.eps}}}\n\\centerline{\\scalebox{0.5}{\\includegraphics{BBscattering-mr.eps}}}\n\\caption{\\label{fig:a_BF-mr}\nThe boson-fermion scattering length $a_0^{BF}(0)\/a_F$ versus mass anisotropy \n$m_\\uparrow\/m_\\downarrow$ is shown in (a) when lighter $\\uparrow$-type (hollow circles) or \nheavier (solid circles) $\\downarrow$-type fermions are in excess.\nThe boson-boson scattering length $a^{BB}(0)\/a_F$ versus mass anisotropy \n$m_\\uparrow\/m_\\downarrow$ is shown in (b). In addition, the scattering lengths in the \nBorn approximation are shown as solid lines.\n}\n\\end{figure}\n\nNext, we solve numerically the integral equation given in Eq.~(\\ref{eqn:ieqn}),\nand obtain $a_0^{BF}(p)$ as a function of the mass anisotropy $m_\\uparrow\/m_\\downarrow$.\nThe exact solutions and the Born approximation values of\n$a_0^{BF}(0)$ are shown in Fig.~\\ref{fig:a_BF-mr}(a).\nWhen $m_\\uparrow = m_\\downarrow$, we find $a_0^{BF}(0) \\approx 1.179a_F$, which \nis consistent with results previously found for equal masses~\\cite{skorniakov, petrov, petrov-abf, brodsky, levinsen, taylor}.\nNotice that $a_0^{BF}(0)$ decreases (increases) from this value with increasing mass \nanisotropy when the lighter (heavier) fermions are in excess.\nIn addition, in the limit of $m_\\uparrow\/m_\\downarrow \\to 0$, while\n$a_0^{BF}(0) \\to a_F$ when the lighter fermions are in excess, \n$a_0^{BF}(0)$ grows rapidly when the heavier fermions are in excess.\nNotice also that the Born approximation values for $a_0^{BF}(0)$ are not in agreement\nwith the exact values for any mass anisotropy, but the general qualitative trends \nare captured by the Born approximation as can be seen from Fig.~\\ref{fig:a_BF-mr}(a).\n\nIn addition, we present results for $a^{BB}(0)$\nas a function of mass anisotropy $m_\\uparrow\/m_\\downarrow$ in Fig.~\\ref{fig:a_BF-mr}(b).\nThe exact results of the boson-boson scattering parameters for unequal mass fermions \ncan be obtained either by extending the diagrammatic approach\nfor equal mass fermions~\\cite{brodsky, levinsen}\nor by using few body techniques~\\cite{petrov}. The Born approximation values\n$\na^{BB}(0) = (m_B\/m_{\\uparrow\\downarrow}) a_F\n$\nare found in Ref.~\\cite{iskin-mixture, iskin-mixture2}.\nSince the technical details to calculate the boson-boson scattering parameters \nare quite cumbersome, and not particularly illuminating, here we just mention \nthat the results found in the literature~\\cite{petrov} can also be \nobtained diagramatically for the unequal mass case.\nAs shown in Fig.~\\ref{fig:a_BF-mr}(b), $a^{BB}(0)$ grows very slowly as\nthe mass ratio $m_{\\uparrow}\/m_{\\downarrow}$ decreases, in constrast with\nthe much more rapid growth of the Born approximation values of $a^{BB}(0)$.\nAs expected, the Born approximation values of $a^{BB}(0)$ \nare not in agreement with the exact values for any mass anisotropy.\n\n\n\\begin{table} [htb]\n\\begin{tabular}{|c|c|c|c|c|c|}\n\\hline\n$\\uparrow$ & $\\downarrow$ & $m_\\uparrow\/m_\\downarrow$ & $a_{BB}\/a_F$ & $a_{B\\uparrow}\/a_F$ & $a_{B\\downarrow}\/a_F$ \\\\\n\\hline\n$^{6}$Li & $^{6}$Li & 1.000 & 0.593 & 1.179 & 1.179 \\\\\n$^{6}$Li & $^{40}$K & 0.150 & 0.695 & 1.010 & 1.984 \\\\\n$^{6}$Li & $^{87}$Sr & 0.068 & 1.123 & 1.003 & 2.512 \\\\\n$^{6}$Li & $^{171}$Yb & 0.035 & - & 1.001 & 3.023 \\\\\n$^{40}$K & $^{40}$K & 1.000 & 0.593 & 1.179 & 1.179 \\\\\n$^{40}$K & $^{87}$Sr & 0.460 & 0.597 & 1.064 & 1.411 \\\\\n$^{40}$K & $^{171}$Yb & 0.234 & 0.629 & 1.022 & 1.723 \\\\\n$^{87}$Sr & $^{87}$Sr & 1.000 & 0.593 & 1.179 & 1.179 \\\\\n$^{87}$Sr & $^{171}$Yb & 0.508 & 0.599 & 1.073 & 1.374 \\\\\n$^{171}$Yb & $^{171}$Yb & 1.000 & 0.593 & 1.179 & 1.179 \\\\\n\\hline\n\\end{tabular}\n\\caption{\\label{table:a} Exact boson-boson ($a_{BB}$) and boson-fermion ($a_{BF}$) \nscattering lengths for a list of two-species Fermi-Fermi mixtures.\nHere, $a_{B \\uparrow}$ ($a_{B \\downarrow}$) corresponds to excess-type \nof $\\uparrow$ ($\\downarrow$) fermions.\n}\n\\end{table}\n\nSeveral atomic gases of fermions ($^{6}$Li, $^{40}$K, $^{87}$Sr~\\cite{innsbruck}, \nand $^{171}$Yb~\\cite{fukuhara}) are being currently investigated,\nand experimental methods for studying mixtures of two types of fermions \nare being developed in several groups. Thus, anticipating future experiments \ninvolving mixtures of two types of fermions, \nwe show in Table~\\ref{table:a} the boson-fermion and boson-boson scattering lengths\nfor a few mixtures.\n\nHere, we would like to make two comments. First, it is quite remarkable that\nthe diagramatic many body approach recovers the few body results for boson-fermion and\nboson-boson scattering lengths for arbitrary mass ratio $m_{\\uparrow}\/m_{\\downarrow}$.\nThe diagramatic many body approach is performed in momentum space, while the few body \napproach is performed in real space. The two methods are equivalent because all of the possible\nscattering processes are taken into account exactly in the diagramatic many body approach \nat $T = 0$ for three or four fermions. However, the calculation of the \nscattering parameters for three or four fermions in the presence of many others \n(arbitrary number of particles) at zero or finite temperatures requires \na full many body approach, which is inevitably approximate and more difficult to implement.\nSecond, our analysis does not include the effects related to \nEfimov (three body bound) states,\nwhich may become important when analyzing the scattering parameters as a function\nof mass ratio $m_{\\uparrow}\/m_{\\downarrow}$~\\cite{petrov-abf}. In particular, the mixtures of \n$^6$Li and $^{87}$Sr or $^6$Li and $^{171}$Yb have mass ratios of \n$m_{Li}\/m_{Sr} \\approx 0.068$ and $m_{Li}\/m_{Yb} \\approx 0.035$, which are below the critical\nratio $m_{\\uparrow}\/m_{\\downarrow} \\approx 0.073$ for the emergence of Efimov states. \n\nHaving presented the boson-fermion and boson-boson scattering lengths\nfor arbitrary mass ratio $m_{\\uparrow}\/m_{\\downarrow}$, we discuss next the\nresulting phase diagrams for Fermi-Fermi mixtures in the strong attraction\nlimit, where the system can be effectively described by a\nBose-Fermi mixture~\\cite{pieri,iskin-mixture,iskin-mixture2,taylor}\nof molecular bosons and excess unpaired fermions.\n\n\n\n\n\n\n\n\n\\section{Bose-Fermi Mixtures at Zero Temperature}\n\\label{sec:BFmixture}\n\nIn this section we use the effective Bose-Fermi mixture \ndescription~\\cite{pieri,iskin-mixture,taylor}\nto analyze the phase diagram of population imbalanced \nFermi-Fermi mixtures in the strong attraction limit. \nWe describe first the general stability conditions \nfor Bose-Fermi mixtures at zero temperature, and use \nthis connection to discuss the stability and phase \ndiagrams of Fermi-Fermi mixtures in the \nstrong attraction limit.\n\n\n\n\n\n\n\n\n\n\n\\subsection{Bose-Fermi Mixtures}\n\\label{sec:atomic-BFmixture}\n\nThe ground state of Bose-Fermi mixtures can be described by the\nfree energy~\\cite{viverit, iskin-mixture2}\n\\begin{equation}\n{\\cal E} = {\\cal E}_B + {\\cal E}_F + \\frac{U_{BB} n_B^2}{2} + U_{BF} n_F n_B \n- \\mu_F n_F - \\mu_{B} n_B,\n\\label{eqn:energy}\n\\end{equation}\nwhich characterizes the center-of-mass degrees of freedom for a mixture\nof single-hyperfine-state bosons and fermions. \nHere $\\mu_F$ and $n_F$ ($\\mu_B$ and $n_B$) are the density and chemical \npotential of fermions (bosons), $\\epsilon_F$ is the Fermi energy \nof the fermions, and $U_{BB}$ and $U_{BF}$ are the repulsive \nboson-boson and boson-fermion interaction strengths.\nThe density of single-hyperfine-state fermions in three dimensions is given by\n$\nn_F = (1\/V) \\sum_\\mathbf{k}^{|\\mathbf{k}| < k_F} 1 = k_F^3\/(6\\pi^2),\n$\nwhere $k_F$ is the Fermi momentum and $V$ is the volume. \nThe first term in Eq.~(\\ref{eqn:energy}) is the total kinetic energy of \nbosons, which is assumed to be much smaller than all other energies, and is neglected.\nThis assumption is very good since essentially all bosons are condensed in\nthe ${\\bf k} = {\\bf 0}$ state, when the boson-boson and boson-fermion \ninteractions are weak.\nThe second term in Eq.~(\\ref{eqn:energy}) is the total kinetic energy of \nfermions, which in three dimensions is given by\n$\n{\\cal E}_F = (1\/V) \\sum_\\mathbf{k}^{|\\mathbf{k}| < k_F} \\epsilon_{\\mathbf{k},F} \n= 3 \\epsilon_F n_F \/ 5,\n$\nwhere $\\epsilon_{\\mathbf{k},F} = |\\mathbf{k}|^2\/(2m_F)$.\n\nFrom the free energy given in Eq.~(\\ref{eqn:energy}), we obtain the fermion\nand boson chemical potentials using the condition \n$\\partial {\\cal E} \/\\partial n_{i} = 0$ with $i = F, B$, leading to\n\\begin{eqnarray}\n\\mu_F &=& \\epsilon_F + U_{BF} n_B, \\\\\n\\mu_B &=& U_{BB} n_B + U_{BF} n_F.\n\\end{eqnarray}\nThen, we use the positive definiteness of the Bose-Fermi compressibility matrix \n$\\kappa_{i,j} = \\partial \\mu_i\/\\partial n_j$,\n\\begin{eqnarray}\n\\frac{\\partial \\mu_F}{\\partial n_F} \\frac{\\partial \\mu_B}{\\partial n_B} - \n\\frac{\\partial \\mu_F}{\\partial n_B} \\frac{\\partial \\mu_B}{\\partial n_F} > 0,\n\\end{eqnarray}\nand find that bosons and fermions phase separate when the condition\n\\begin{equation}\nn_F \\ge \\frac{4\\pi^4}{3m_F^3} \\frac{U_{BB}^3}{U_{BF}^6}, \\hspace{3mm} \\textrm{(3D)}\n\\label{eqn:stab.3D}\n\\end{equation}\nis satisfied in three-dimensional systems~\\cite{viverit, iskin-mixture2}.\nTherefore, the stability of uniform superfluidity puts an upper \nlimit on the density of fermions in three-dimensions.\n\nFollowing a similar approach in lower dimensions, where\n$n_F = k_F^2\/(4\\pi)$ and ${\\cal E}_F = \\epsilon_F n_F \/ 2$ in two dimensions,\nand $n_F = k_F\/\\pi$ and ${\\cal E}_F = \\epsilon_F n_F \/ 3$ in one dimension,\nwe find that the bosons and the fermions phase separate when the conditions\n\\begin{eqnarray}\n1 &\\le& \\frac{m_F}{2\\pi} \\frac{U_{BF}^2}{U_{BB}}, \\hspace{3mm} \\textrm{(2D)} \n\\label{eqn:stab.2D} \\\\\nn_F &\\le& \\frac{m_F}{\\pi^2} \\frac{U_{BF}^2}{U_{BB}}, \\hspace{3mm} \\textrm{(1D)}\n\\label{eqn:stab.1D}\n\\end{eqnarray}\nare satisfied, respectively, for two- and one-dimensional systems.\nNotice that, the stability of uniform superfluidity puts a lower limit\nin one dimension, which is in sharp contrast with the three-dimensional result.\nFurthermore, the stability condition in two dimensions does not depend\nexplicitly on the density of fermions (see also Ref.~\\cite{tempere}). \nHowever, the results in lower dimensions have to be used with caution, \nsince quantum fluctuations are more pronounced, and may affect these stability conditions. \n\nFor an atomic Bose-Fermi mixture, we can also describe analytically a finer\nstructure of phases. There are four possible phases~\\cite{viverit}:\n(I) PS(1) where there is phase separation between pure fermions and pure bosons;\n(II) PS(2) where there is phase separation between pure fermions,\nand a mixture of fermions and bosons;\n(III) PS(3) where there is phase separation between pure bosons,\nand a mixture of fermions and bosons; and\n(IV) PS(4) where there is phase separation between two different mixtures \nof fermions and bosons.\n\nFor a three-dimensional weakly interacting Bose-Fermi mixture, we follow Ref.~\\cite{viverit} \nand find that there are only two stable phases within the phase separation region:\n(I) PS(1) where there is phase separation between pure fermions and pure bosons, and \n(II) PS(2) where there is phase separation between pure fermions,\nand a mixture of fermions and bosons.\nWe obtain analytically the condition\n\\begin{equation}\nn_F \\ge \\frac{1125 \\pi^4}{128 m_F^4} \\frac{U_{BB}^3}{U_{BF}^6} \n- \\frac{5}{4} \\frac{U_{BB}}{U_{BF}} n_B, \\hspace{3mm} \\textrm{(3D)}\n\\label{eqn:stab2.3D}\n\\end{equation}\nfor the transition from the PS(2) to the PS(1) phase~\\cite{iskin-mixture2}.\n\nIn lower dimensions, we find that the structure of the phase diagram is quite different.\nIn two dimensions, the phase separated region consists only of \nPS(1) where there is phase separation between pure fermions and pure bosons.\nWhile, for a one-dimensional weakly interacting Bose-Fermi mixture, \nthe phase separated region consists also of two regions:\n(I) PS(1) where there is phase separation between pure fermions and pure bosons,\nand (III) PS(3) where there is phase separation between pure bosons,\nand a mixture of fermions and bosons. \nWe obtain analytically the condition\n\\begin{equation}\nn_F \\le \\frac{3 m_F}{2\\pi^2} \\frac{U_{BF}^2}{U_{BB}} \n- \\frac{U_{BB}}{U_{BF}} n_B, \\hspace{3mm} \\textrm{(1D)}\n\\label{eqn:stab2.1D}\n\\end{equation}\nfor the transition from the PS(3) to the PS(1) phase.\nNotice that the structure of the PS(3) phase in one-dimension is \nvery different from the structure of the PS(2) phase in three dimensions.\nAgain, the results in lower dimensions have to be used with caution, \nsince the quantum fluctuations are more pronounced, and may affect these stability conditions. \n\nNext, we concentrate only on the three-dimensional case, and use\nthe stability conditions found above as well as the interaction (scattering)\nparameters obtained in Sec.~\\ref{sec:a_BF} to analyze the phase\ndiagrams of Fermi-Fermi mixtures in the strong attraction limit.\n\n\n\n\n\n\n\n\n\n\\subsection{Fermi-Fermi Mixtures \\\\ in the Strong Attraction Limit}\n\\label{sec:FFmixture}\n\nTo make an analogy between Bose-Fermi mixtures and population\nimbalanced Fermi-Fermi mixtures in the strong attraction limit, \nwe identify $F \\equiv \\{\\uparrow$ or $\\downarrow\\}$ as the excess fermions.\nThis identification leads to the density of excess fermions ($n_E$) and molecular\nbosons ($n_B$) given by\n\\begin{eqnarray}\nn_E &=& n_F - n_{-F} = |n_\\uparrow - n_\\downarrow|, \\\\\nn_B &=& \\frac{n - n_E}{2} = n_{-F},\n\\end{eqnarray}\nrespectively, where $n = n_\\uparrow + n_\\downarrow$ is the total density of \n$\\uparrow$- and $\\downarrow$-type fermions.\nHere, we use $(-\\uparrow) \\equiv \\downarrow$ and vice versa.\nFor instance, if $F \\equiv \\uparrow$ fermions are in excess, the density\nof excess fermions and molecular bosons are $n_E = n_\\uparrow - n_\\downarrow$ and \n$n_B = (n-n_E)\/2 = n_\\downarrow$, respectively, such that all $\\downarrow$-type\nfermions are paired with some of the $\\uparrow$-type fermions to form molecular bosons,\nbut there are $\\uparrow$-type fermions left unpaired.\nIt is important to emphasize that the internal degrees of freedom \n(electronic, vibrational, and rotational) of molecular bosons are\nnot explicitly considered here, in the same spirit of the description of atomic bosons \npresented in Sec.~\\ref{sec:atomic-BFmixture}, where the electronic degrees of freedom were\nalso not explicitly considered. \n\nFor three dimensions, we define the boson-boson and boson-fermion interaction strengths \n\\begin{eqnarray}\nU_{BB} &=& \\frac{4\\pi a_{BB}}{m_B} = \\frac{4\\pi \\gamma_B}{m_B} a_F, \n\\label{eqn:U_BB} \\\\\nU_{BF} &=& \\frac{4\\pi a_{BF}}{m_{BF}} = \\frac{4\\pi \\beta_F}{m_{BF}} a_F,\n\\label{eqn:U_BF}\n\\end{eqnarray}\nwhere $a_F$, $a_{BB} = \\gamma_B a_F$ and $a_{BF} = \\beta_F a_F$ are the \nfermion-fermion, boson-boson and boson-fermion scattering lengths.\nHere, $\\gamma_B = a_{BB}\/a_F$ and $\\beta_F = a_{BF}\/a_F$ are constants, \nwhich are found in Sec.~\\ref{sec:a_BF}\nas shown in Fig.~\\ref{fig:a_BF-mr} and Table~\\ref{table:a}.\nIn addition, we define the population imbalance parameter\n\\begin{equation}\nP = \\frac{N_\\uparrow - N_\\downarrow}{N_\\uparrow + N_\\downarrow} \n= \\frac{n_\\uparrow - n_\\downarrow}{n_\\uparrow + n_\\downarrow},\n\\end{equation}\nsuch that $|P| = n_E\/n$, and $n = K_F^3\/(3\\pi^2)$, where $N_\\sigma$ is the\nnumber of $\\sigma$-type fermions and $K_F$ is the Fermi \nmomentum corresponding to the total density of fermions defined\nby $K_F^3 = (k_{F,\\downarrow}^3 + k_{F,\\uparrow}^3)\/2$.\n\nUsing these definitions, the phase separation condition Eq.~(\\ref{eqn:stab.3D}) becomes\n\\begin{equation}\n|P| \\ge \\frac{\\pi^3 \\gamma_B^3 m_{BF}^6}{16 \\beta_F^6 m_B^3 m_F^3} \\lambda^3,\n\\label{eqn:psc}\n\\end{equation}\nwhere $\\lambda = 1\/(K_F a_F)$ is the scattering parameter.\nSimilarly, the condition given in Eq.~(\\ref{eqn:stab2.3D}) becomes\n\\begin{equation}\n|P| \\left(1 - \\frac{5 \\gamma_B m_{BF}}{8 \\beta_F m_B} \\right) \n\\ge \\frac{3375 \\pi^3 \\gamma_B^3 m_{BF}^6}{8192 \\beta_F^6 m_B^3 m_F^3}\\lambda^3 \n- \\frac{5\\gamma_B m_{BF}}{8 \\beta_F m_B},\n\\label{eqn:psc2}\n\\end{equation}\nfor the transition from the PS(2) to the PS(1) phase.\n\nWe would like to emphasize that these analytic expressions given in Eqs.~(\\ref{eqn:psc}) \nand~(\\ref{eqn:psc2}) are valid only in the strong attraction limit \nwhen $1\/(K_F a_F) \\gg 1$, but \nthey may still give semi-quantitative results for $1\/(K_F a_F) \\gtrsim 1$.\nClose to the unitarity, the Bose-Fermi description of Fermi-Fermi mixtures \nin terms of molecular bosons and excess fermions is not reliable, since the binding energy \nof molecular bosons is small and the interactions between molecular bosons and excess fermions \nor between two molecular bosons may be sufficient to cause dissociation of the\nmolecules into directly scattering fermions. \nHowever, there may be an intermediate regime between unitarity and the \nstrict BEC limit where we can describe Fermi-Fermi mixtures in terms\nof a mixture of molecular bosons and excess fermions such that\nthe molecular bosons can dissociate due to boson-boson or boson-fermion interactions,\nbut be in chemical equilibrium with excess fermions. \nWhen dissociation of molecular bosons is included, the system is no longer\na binary mixture of molecular bosons and excess fermions, but a \nternary mixture of molecular bosons, dissociated bosons ($\\uparrow \\downarrow \\rightleftharpoons \\uparrow + \\downarrow$), \nand excess fermions, or effectively a ternary mixture of molecular bosons, \nand $\\uparrow$- and $\\downarrow$-type fermions. \nIn the case of ternary mixtures, there can be a large number of phase separated regimes. \nIf we confine our discussion to the equillibrium of a maximum of two phases \nof this ternary mixture then several other situations can be encountered. \nFor example, when $\\downarrow$-type fermions are in excess, a possible sequence of phases\nfor fixed population imbalance $P$ and increasing scattering parameter $1\/(K_F a_F)$ is:\n(1) Normal phase (N) of partially polarized fermions $\\to$ \n(2) mixture of molecular bosons\nand $\\uparrow$-type fermions phase separated from partially polarized normal \nfermions $\\to$ \n(3) molecular bosons phase separated from excess $\\downarrow$-type\nfermions $\\to$ \n(4) mixture of molecular bosons and $\\downarrow$-type excess fermions\nphase separated from $\\downarrow$-type excess fermions $\\to$ \n(5) coexistence of molecular bosons and $\\downarrow$-type excess fermions.\n\nTherefore, as long as Fermi-Fermi mixtures can be regarded as a binary mixture\nof non-dissociated molecular bosons and excess fermions, \nthe expressions given in Eqs.~(\\ref{eqn:psc}) and~(\\ref{eqn:psc2}) \nmay be used as a guide for the boundaries between phase \nseparated (non-uniform) and the mixed (uniform) phases for any mixture of fermions. \nIn particular, Eqs.~(\\ref{eqn:psc}) and~(\\ref{eqn:psc2}) may serve as estimators for the phase boundaries\nof equal or unequal mass Fermi-Fermi mixtures with population imbalance, \nas discussed next. \n\n\n\n\n\n\n\n\n\n\n\\subsection{Phase Diagrams of Fermi-Fermi Mixtures in the Strong Attraction Limit}\n\\label{sec:phase.diagram}\n\nAmong many possibilities of Fermi-Fermi mixtures (see Table~\\ref{table:a}), we focus our\nanalysis on population imbalanced mixtures of \n$^{6}$Li or $^{40}$K atoms where $m_\\uparrow = m_\\downarrow$, and\n$^{6}$Li and $^{40}$K atoms where $m_\\uparrow = 0.15 m_\\downarrow$.\n\nIn Fig.~\\ref{fig:BEC}, we show phase diagrams of population imbalance $P$\nand scattering parameter $1\/(K_F a_F)$ for equal mass mixtures \nwhen $m_\\uparrow = m_\\downarrow$, and for unequal mass mixtures \nwhen $m_\\uparrow = 0.15 m_\\downarrow$.\nIn these diagrams, we choose $\\uparrow$ ($\\downarrow$) to label\nlighter (heavier) fermions such that lighter (heavier) fermions\nare in excess when $P > 0$ $(P < 0)$. \nAlthough these diagrams are strictly valid in the strong attraction limit\nwhen $1\/(K_F a_F) \\gg 1$, they are qualitatively correct \nwhen $1\/(K_F a_F) \\gtrsim 1$ or as long as the molecular bosons are\nnot dissociated. In the later case, the system may be approximately \ndescribed as a ternary mixture of molecular bosons,\n$\\uparrow$- and $\\downarrow$-type fermions and many other phases are \npossible, as discussed in Sec.~\\ref{sec:FFmixture}.\n\n\n\\begin{figure} [htb]\n\\centerline{\\scalebox{0.5}{\\includegraphics{BFphase-v-mr.1.eps} }}\n\\centerline{\\scalebox{0.5}{\\includegraphics{BFphase-v-mr.0.15.eps} }}\n\\caption{\\label{fig:BEC}\nPhase diagram of population imbalance $P = (n_\\uparrow - n_\\downarrow)\/(n_\\uparrow + n_\\downarrow)$ \nversus scattering parameter $1\/(K_F a_F)$ for \n(a) equal masses when $m_\\uparrow = m_\\downarrow$, and\n(b) unequal masses when $m_\\uparrow = 0.15 m_\\downarrow$\nin the BEC side.\nWe show normal (N), uniform superfluid (U),\nand phase separated non-uniform superfluid phases PS(1) and PS(2).\n}\n\\end{figure}\n\nIn these figures, we show the following phases: \n(i) the normal (N) phase corresponding to balanced $(P = 0$) or \nimbalanced $(P \\ne 0)$ mixture of unpaired $\\uparrow$- or $\\downarrow$-type fermions;\n(ii) uniform superfluid (U) phase where paired\n(molecular bosons) and unpaired fermions coexist; \nand (iii) phase separated (PS) non-uniform superfluid phases.\nThe PS(1) region labels phase separation between pure excess fermions and \nsuperfluid molecular bosons, while the PS(2) region labels phase separation between \npure excess fermions, and a mixture of excess fermions and superfluid molecular bosons.\nThe phase boundary between U and PS(2) phases is determined from Eq.~(\\ref{eqn:psc}), \nand the phase boundary between PS(2) and PS(1) phases is determined \nfrom Eq.~(\\ref{eqn:psc2}).\nFor a fixed mass anisotropy, when $|P|$ is large, \nwe find phase transitions from PS(1) $\\to$ PS(2) $\\to$ U as \nthe interaction strength $1\/(K_F a_F)$ increases.\nHowever, when $|P|$ is very small, we find a phase transition directly from the \nPS(1) to the U phase as $1\/(K_F a_F)$ increases.\n\nWe would like to remark in passing that the phase diagrams for \nmixtures of $^6$Li and $^{87}$Sr or $^6$Li and $^{171}$Yb with mass ratios of \n$m_{Li}\/m_{Sr} \\approx 0.068$ and $m_{Li}\/m_{Yb} \\approx 0.035$, which are below the critical\nratio $m_{\\uparrow}\/m_{\\downarrow} \\approx 0.073$ for the emergence \nof Efimov (three body bound) states are much richer, since phase separation and\ncoexistence phases involving Efimov states (trimers), molecular bosons\nand excess fermions are also present.\n\nIt is also important to emphasize that since we use the exact boson-boson and \nboson-fermion scattering lengths, our phase diagrams in the strong \nattraction limit already include fluctuation corrections beyond the Born approximation. \nFor comparison, the corresponding phase diagrams within the Born approximation \nare described in Fig.~\\ref{fig:BEC-born}, where the phase boundaries in the\npopulation imbalance $P$ versus scattering parameter $1\/(K_F a_F)$ plane are shown\nfor equal $(m_\\uparrow = m_\\downarrow)$ and unequal $(m_\\uparrow = 0.15 m_\\downarrow)$\nmass mixtures. A direct comparison of Figs.~\\ref{fig:BEC} and~\\ref{fig:BEC-born}\nshows that the results beyond the Born approximation are quantitatively \ndifferent from the saddle-point results~\\cite{pao, sheehy} in the equal mass case.\nThese quantitative differences become significantly large for unequal \nmass mixtures when heavier fermions are \nin excess~\\cite{iskin-mixture,iskin-mixture2,pao-mixture,parish} \ndue to the large sensitivity of the exact scattering parameters on \nthe mass ratio $m_{\\uparrow}\/ m_{\\downarrow}$ as shown\nin Fig.~\\ref{fig:a_BF-mr} and Table~\\ref{table:a}.\nHowever, the same phases are present in both cases, indicating that the \nBorn approximation captures the basic qualitative features, but fails\nto produce the phase boundaries quantitatively.\n\nLastly, we would like to point out the presence of several triple points\nin the phase diagrams shown in Figs.~\\ref{fig:BEC} and~\\ref{fig:BEC-born}. \nAlong the $\\vert P \\vert = 1$ lines, we find several triple points as $1\/(K_F a_F)$ increases\nwhere the fully polarized normal phases ($P = \\pm 1$) merge with (i) the partially polarized normal (N)\nand the PS(1) phase; or with (ii) the PS(1) and PS(2) phases; or with (iii) \nthe PS(2) and U phases. Furthermore, there is also an additional triple point that\noccurs for small $\\vert P \\vert$ (iv) where the phases PS(1), PS(2) and U meet.\nThe precise locations of these triple points can be obtained for any mass ratio \nand scattering parameter from Eqs.~(\\ref{eqn:psc})\nand~(\\ref{eqn:psc2}) using the equal sign $(=)$ condition.\nThe triple point for case (ii) can be obtained by setting $\\vert P \\vert = 1$ in \nEq.~(\\ref{eqn:psc2}), and the triple point for case (iii) \ncan be obtained by setting $\\vert P \\vert = 1$ in Eq.~(\\ref{eqn:psc}). \nFinally, the triple point for case (iv) can be obtained\nby using the equal sign $(=)$ condition of Eqs.~(\\ref{eqn:psc}) and~(\\ref{eqn:psc2}) and by\nsolving the two equations simultaneously.\n\n\\begin{figure} [htb]\n\\centerline{\\scalebox{0.5}{\\includegraphics{BFphase-v-mr.1-born.eps} }}\n\\centerline{\\scalebox{0.5}{\\includegraphics{BFphase-v-mr.0.15-born.eps} }}\n\\caption{\\label{fig:BEC-born}\nPhase diagram of population imbalance $P = (n_\\uparrow - n_\\downarrow)\/(n_\\uparrow + n_\\downarrow)$ \nversus scattering parameter $1\/(K_F a_F)$ in the Born approximation for \n(a) equal masses when $m_\\uparrow = m_\\downarrow$, and\n(b) unequal masses when $m_\\uparrow = 0.15 m_\\downarrow$\nin the BEC side.\nWe show normal (N), uniform superfluid (U),\nand phase separated non-uniform superfluid phases PS(1) and PS(2).\n}\n\\end{figure}\n\nHaving analyzed the phase diagrams for non-trapped continuous systems, we discuss\nnext the effects of the trapping potential which \nare necessary to understand experiments involving ultracold \nFermi-Fermi mixtures.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Trapped Bose-Fermi Mixtures at Zero and Finite Temperatures}\n\\label{sec:trap}\n\nIn this section, we use again the simpler description of the effective Bose-Fermi mixture\nto describe trapped Fermi-Fermi mixtures in the strong attraction limit.\nFor this purpose, we present first the theory of trapped Bose-Fermi mixtures \nat zero and finite temperatures, and then discuss the density profiles \nof trapped Fermi-Fermi mixtures in the strong attraction limit using the\nrelation between the two systems described in Sec.~\\ref{sec:BFmixture}.\n\n\n\n\n\n\n\n\n\n\\subsection{Bose-Fermi mixtures}\n\\label{sec:trap.atomic}\n\nThe Hamiltonian density for a Bose-Fermi mixture in an external potential can be \nwritten as\n\\begin{eqnarray}\n\\label{eqn:bose-fermi-hamiltonian}\nH ( {\\bf r} ) & = & K_F( {\\bf r} ) + V_F( {\\bf r} ){\\hat n_F} ( {\\bf r} ) + K_B( {\\bf r} ) \n+ V_B( {\\bf r} ) {\\hat n_B} ( {\\bf r} ) \\nonumber \\\\\n& + & U_{BF} {\\hat n_B} ( {\\bf r} ) {\\hat n_F} ( {\\bf r} ) \n+ (U_{BB}\/2) {\\hat n_B} ( {\\bf r} ) {\\hat n_B}( {\\bf r} ), \n\\end{eqnarray}\nwhere $ K_i( {\\bf r} ) = \n\\psi_{i}^{\\dagger} ( {\\bf r} ) [-\\nabla^2\/(2m_i) - \\mu_i] \\psi_{i} ( {\\bf r} )$ \nrepresents the kinetic and chemical potential terms for fermions $(i \\equiv F)$ \nor bosons $(i \\equiv B)$ in a single hyperfine state. \nHere, ${\\hat n_i}({\\bf r}) = \\psi_{i}^{\\dagger}({\\bf r}) \\psi_i ({\\bf r})$\nrepresent the local density operators, and $U_{BF}$ and $U_{BB}$ represent the \nboson-fermion and boson-boson interaction. The single-hyperfine-state fermions\nare non-interacting, but obey the Pauli exclusion principle. \n\nFor simplicity, we approximate the trapping potential by an isotropic harmonic function\nwhere the potential energy is \n\\begin{eqnarray}\nV_{i} ( {\\bf r} ) = \\frac{1}{2} \\alpha_{i} r^2.\n\\end{eqnarray}\nHere, $\\alpha_{i} = m_{i} w_{i}^2$ is proportional to the trapping \nfrequency of bosons $(i \\equiv B)$ or fermions $(i \\equiv F)$, which is typically different \nfor each kind of atom. Since the potential and the interactions are isotropic,\nthe effective potentials and densities depend only on $r = \\vert {\\bf r} \\vert$.\n\nIn the presence of such trapping potentials, the bosons and fermions \nfeel the effective potentials \n\\begin{eqnarray}\nV_{B,{\\rm eff}} ( r ) &=& V_B ( r ) + 2U_{BB} n_B(r) + U_{BF} n_F(r),\n\\label{eqn:V_B} \\\\\nV_{F,{\\rm eff}} ( r ) &=& V_F (r) + U_{BF} n_B(r),\n\\label{eqn:V_F}\n\\end{eqnarray}\nrespectively, where $n_F (r)$ is the local density of fermions, and \n\\begin{equation}\nn_B(r) = n_C(r) + n_{NC}(r).\n\\end{equation}\nis the total local density of bosons.\nHere, $n_C (r)$ and $n_{NC} (r)$ are the density of condensed and non-condensed \nbosons, respectively.\n\nThe number of condensed bosons is determined from the Gross-Pitaevskii \nequation leading to\n\\begin{equation}\nn_C (r) = \\frac{\\mu_B - V_B (r) - 2U_{BB} n_{NC}(r) - U_{BF} n_F(r)} {U_{BB}}\n\\label{eqn:n_Cr}\n\\end{equation}\nwithin the Thomas-Fermi approximation (TFA), where the kinetic energy\nof the bosons is neglected.\nThis relation is valid when the condition \n$ \\mu_B - V_B (r) - 2U_{BB} n_{NC} (r) - U_{BF} n_F(r) \\ge 0$\nis satisfied, otherwise, $ n_C(r) = 0 $. \n\nIn our analysis, we treat both the non-condensed bosons and the fermions \nas ideal gases~\\cite{amoruso} in effective potentials \ngiven in Eqs.~(\\ref{eqn:V_B}) and~(\\ref{eqn:V_F}).\nIn addition, we assume that non-condensed bosons are in thermal equilibrium\nwith condensed bosons at the same chemical potential $\\mu_B$.\nWithin these approximations, the density of non-condensed bosons and \nfermions are given by\n\\begin{eqnarray}\nn_{NC}(r) &=& \\frac{1}{V}\\sum_{\\mathbf{k}} b[\\epsilon_{\\mathbf{k},B} - \\mu_B + V_{B,{\\rm eff}}(r)], \n\\label{eqn:n_NCr} \\\\\nn_F(r) &=& \\frac{1}{V}\\sum_{\\mathbf{k}} f[\\epsilon_{\\mathbf{k},F} - \\mu_F + V_{F, {\\rm eff}}(r)],\n\\label{eqn:n_Fr}\n\\end{eqnarray}\nwhere $b(x) = 1\/[\\exp(x\/T) - 1]$ is the Bose,\nand $f(x) = 1\/[\\exp(x\/T) + 1]$ is the Fermi distribution.\nHere, $\\epsilon_{\\mathbf{k},i} = |\\mathbf{k}|^2\/(2m_{i})$ is the kinetic energy\nof bosons $(i \\equiv B)$ or fermions $(i \\equiv F)$.\n\nNotice that, at zero temperature, all bosons condense \nsuch that $n_{NC}(r) = 0$, and $n_B(r) = n_C(r)$, leading to \n\\begin{eqnarray}\nn_B(r) &=& \\frac{\\mu_B - V_B(r) - U_{BF} n_F(r)} {U_{BB}}, \\\\\nn_F(r) &=& \\frac{\\{2m_F [\\mu_F - V_F(r) - U_{BF} n_B(r) ]\\}^{3\/2} }{6\\pi^2}, \n\\end{eqnarray}\nfor the densities of bosons and fermions, respectively~\\cite{iskin-mixture2}. \nThe first expression is valid when the condition \n$\\mu_B - V_B(r) - U_{BF} n_F(r) \\ge 0$ is satisfied, \notherwise, $n_B(r) = 0$.\nThe second expression is valid when the condition \n$\\mu_F - V_F(r) - U_{BF} n_B(r) \\ge 0$ is satisfied, \notherwise, $n_F(r) = 0$.\n\nThe chemical potentials of bosons and fermions are determined from \nfixing the total number of bosons and the number of fermions, \nindependently, as follows\n\\begin{eqnarray}\nN_B &=& \\int d^3r n_B(r), \n\\label{eqn:N_B}\\\\\nN_F &=& \\int d^3r n_F(r),\n\\label{eqn:N_F}\n\\end{eqnarray}\nwhere the integration is over all space. Therefore, in order to find the density \nprofiles for condensed and non-condensed bosons, as wells as for fermions, \nwe need to solve Eqs.~(\\ref{eqn:N_B}) and~(\\ref{eqn:N_F}) \nfor $\\mu_B$ and $\\mu_F$ self-consistently.\nNext, we discuss the density profiles of Fermi-Fermi mixtures \nin the strong attraction limit using the effective Bose-Fermi description presented.\n\n\n\n\n\n\n\n\n\n\\subsection{Fermi-Fermi Mixtures \\\\ in the Strong Attraction Limit}\n\\label{sec:trap.FF}\n\nTo make the connection between Bose-Fermi mixtures and population\nimbalanced Fermi-Fermi mixtures in the strong attraction limit,\nwe identify $F \\equiv \\{\\uparrow$ or $\\downarrow\\}$ as the excess fermions.\nThis identification leads to the density of excess fermions ($n_E$) and molecular\nbosons ($n_B$) given by\n\\begin{eqnarray}\nn_E(r) &=& n_F(r) - n_{-F}(r) = |n_\\uparrow(r) - n_\\downarrow(r)|, \\\\\nn_B(r) &=& \\frac{n(r) - n_E(r)}{2} = n_{-F}(r),\n\\end{eqnarray}\nrespectively, where $n(r) = n_\\uparrow(r) + n_\\downarrow(r)$ is the total density of \n$\\uparrow$- and $\\downarrow$-type fermions.\nHere, we use $(-\\uparrow) \\equiv \\downarrow$ and vice versa.\nFor instance, if $F \\equiv \\uparrow$ fermions are in excess, the density\nof excess fermions and molecular bosons are $n_E (r) = n_\\uparrow (r) - n_\\downarrow (r)$ \nand $n_B(r) = [n (r) - n_E (r)]\/2 = n_\\downarrow(r)$, respectively, such that \nall $\\downarrow$-type fermions are paired with some of the $\\uparrow$-type fermions \nto form molecular bosons, but there are $\\uparrow$-type fermions left unpaired.\nIn addition, we identify $\\alpha_B = \\alpha_\\uparrow + \\alpha_\\downarrow$, where\n$\\alpha_\\sigma = m_\\sigma w_\\sigma^2$ is different for different atoms.\n\n\\begin{figure} [htb]\n\\centerline{\\scalebox{0.49}{\\includegraphics{BFmixture-mr.1-p.0.5-v.2-zero-a.eps} }}\n\\centerline{\\scalebox{0.49}{\\includegraphics{BFmixture-mr.1-p.0.5-v.2-T.0.2-a.eps} }}\n\\centerline{\\scalebox{0.49}{\\includegraphics{BFmixture-mr.1-p.0.5-v.2-T.0.35-a.eps} }}\n\\centerline{\\scalebox{0.49}{\\includegraphics{BFmixture-mr.1-p.0.5-v.2-T.0.41-a.eps} }}\n\\caption{\\label{fig:ncond}\nDensity (in units of $K_F^3$) of condensed ($n_C$) and non-condensed ($n_{NC}$) \nmolecular bosons, and excess ($n_E$) fermions versus trap radius $r\/R_{TF}$\nis shown for \n(a) $T = 0$,\n(b) $T = 0.2 E_F$,\n(c) $T = 0.35 E_F$, and\n(d) $T = 0.41 E_F$.\nHere, $m_\\uparrow = m_\\downarrow$, the population imbalance parameter is $P = 0.5$ \nand the scattering length parameter is $1\/(K_F a_F) = 2$.\n}\n\\end{figure}\n\nAs an example, we look at the equal mass case $m_F = m_{\\uparrow} = m_{\\downarrow} = m$\nand $m_B = m_{\\uparrow} + m_{\\downarrow} = 2m$, and solve the self-consistent relations\nEqs.~(\\ref{eqn:N_B}) and~(\\ref{eqn:N_F}) for a two-hyperfine-state mixture of $^6$Li only\nor $^{40}$K only. In our numerical analysis, we choose \n$\\alpha_F = \\alpha_\\uparrow = \\alpha_\\downarrow = \\alpha$\nand $\\alpha_B = 2\\alpha$, scattering parameter $1\/(K_F a_F) = 2$ \nand population imbalance parameter $P = 0.5$ such that $n_\\uparrow = 3n_\\downarrow$. \nWe define a characteristic energy $E_F = K_F^2\/(2m)$ in terms of the wavevector $K_F$ \nand the fermion mass $m$ and assume it to be the known Fermi energy of a gas of non\ninteracting fermions in the trapping potential $V_F(r)$. \nWe scale the radial distance with the effective Thomas-Fermi (TF) radius $R_{TF}$ \ndefined by $E_F = \\alpha R_{TF}^2\/2$. With this identification, the total number of trapped fermions \n$N = (E_F\/w_F)^3\/3$, with $w_F = \\sqrt{\\alpha\/m}$, \ncan be rewritten as $N = K_F^3 R_{TF}^3\/24$ in terms of $K_F$ and $R_{TF}$. \n\nSince we are interested also in the temperature dependence of the density\nprofiles, we recall that the critical temperature for Bose-Einstein condensation \nof a non-interacting harmonically trapped Bose gas is $T_{BEC} = w_B [ N_B \/\\zeta (3) ]^{1\/3}$, \nwhere $w_B = \\sqrt{\\alpha_B\/ m_B}$. In our Fermi-Fermi mixture the number of bosons is\n$N_B = N (1 - \\vert P \\vert )\/2$ expressed in terms of the total\nnumber of fermions $N = N_\\uparrow + N_\\downarrow$ and population imbalance \n$P = (N_\\uparrow - N_\\downarrow)\/N$.\nUsing the expression for $N_B$, the expression of $N$ in terms of $E_F$ and $w_F$,\nand that $w_B = w_F$ for equal masses, we find \n\\begin{equation}\nT_{BEC} = \\left[ \\frac{1-|P|}{6 \\zeta(3)} \\right]^\\frac{1}{3} E_F \n\\approx 0.518 (1-|P|)^{1\/3} E_F,\n\\label{eqn:tbec}\n\\end{equation}\nfor the critical BEC temperature, which is valid when $1\/(K_F a_F) \\to \\infty$. \nHere, $\\zeta(x)$ is the Zeta function and $\\zeta(3) \\approx 1.202$.\nTherefore, for $P = 0.5$, we obtain $T_{BEC} \\approx 0.41 E_F$.\nNotice that $T_{BEC}$ for population imbalance $P$ given in Eq.~(\\ref{eqn:tbec}) is a generalization \nof the results for equal populations~\\cite{griffin, perali}.\n\n\nIn Fig.~\\ref{fig:ncond}, we show the density (in units of $K_F^3$) of condensed \n($n_C$) and non-condensed ($n_{NC}$) molecular bosons, and excess ($n_E$) \nfermions as a function trap radius $r\/R_{TF}$ for four temperatures:\n(a) $T = 0$,\n(b) $T = 0.2 E_F$,\n(c) $T = 0.35 E_F$, and\n(d) $T = 0.41 E_F$.\nAt zero temperature $(T = 0)$, as shown in Fig.~\\ref{fig:ncond}(a), we find that \nall of the molecular bosons are condensed, and that they are concentrated \nclose to the center of the trap.\nIn contrast, the majority of excess fermions are pushed away from the center \ntowards the edges of the trap due to the repulsive boson-fermion interaction and\nthe high concentration of condensed molecular \nbosons. Therefore, there is a clear indication of phase separation between \nmolecular bosons and excess fermions.\nWhen the temperature is increased to $T = 0.2E_F$\nshown in Fig.~\\ref{fig:ncond}(b), some of the molecular bosons are not \ncondensed. These non-condensed molecular bosons are also pushed away\nfrom the center towards the edges of the trap just like the excess fermions.\nFurther increase in temperature increases (decreases) the number of \nnon-condensed (condensed) molecular bosons as can be seen in Fig.~\\ref{fig:ncond}(c).\nFor temperatures close to $T_{BEC}$ and above, all of the molecular bosons \nbecome non-condensed as shown in Fig.~\\ref{fig:ncond}(d) \nhaving a Gaussian-like density distribution.\nSimilarly, the excess fermions also have Gaussian-like density distribution \nfor temperatures at $T_{BEC}$ and above due to the absence of the condensate. \n\nTherefore, at zero temperature, we find that the trapping potential tends to \nfavor phase separation into a PS(1)-rich phase where regions of almost pure fermions \nand almost pure bosons are separated. However, at finite temperatures, the system\ndevelops a PS(2)-rich phase where regions of almost pure fermions and of almost \nfully mixed bosons and fermions are separated. The region of coexistence of\nbosons and fermions can be further broken down into a domain of coexisting\nexcess fermions with condensed and non-condensed bosons, and into a domain of \ncoexisting excess fermions and non-condensed bosons as can be seen in \nFig.~\\ref{fig:ncond}(d). Again, if the molecular bosons are allowed to dissociate,\nthe system can be described by a ternary \nmixture as discussed in Sec.~\\ref{sec:FFmixture} and\nthe phase diagram can be even richer, especially closer to the unitarity.\n\n\\begin{figure} [htb]\n\\centerline{\\scalebox{0.49}{\\includegraphics{BFmixture-mr.1-p.0.5-v.2-zero-b.eps} }}\n\\centerline{\\scalebox{0.49}{\\includegraphics{BFmixture-mr.1-p.0.5-v.2-T.0.2-b.eps} }}\n\\centerline{\\scalebox{0.49}{\\includegraphics{BFmixture-mr.1-p.0.5-v.2-T.0.35-b.eps} }}\n\\centerline{\\scalebox{0.49}{\\includegraphics{BFmixture-mr.1-p.0.5-v.2-T.0.41-b.eps} }}\n\\caption{\\label{fig:nupdown}\nDensity (in units of $K_F^3$) of $\\uparrow$- ($n_\\uparrow$) and $\\downarrow$-type\n($n_\\downarrow$) fermions versus trap radius $r\/R_{TF}$\nis shown for \n(a) $T = 0$,\n(b) $T = 0.2 E_F$,\n(c) $T = 0.35 E_F$, and\n(d) $T = 0.41 E_F$.\nHere, $m_\\uparrow = m_\\downarrow$, population imbalance parameter is $P = 0.5$ \nand scattering length parameter is $1\/(K_F a_F) = 2$.\n}\n\\end{figure}\n\nIn Fig.~\\ref{fig:nupdown}, we show the density (in units of $K_F^3$)\nof $\\uparrow$- ($n_\\uparrow$) and $\\downarrow$-type ($n_\\downarrow$) fermions \nas a function of radius $r\/R_{TF}$ for four temperatures:\n(a) $T = 0$,\n(b) $T = 0.2 E_F$,\n(c) $T = 0.35 E_F$, and\n(d) $T = 0.41 E_F$.\nAt zero temperature, as shown in Fig.~\\ref{fig:nupdown}(a), we find that the density \nof $\\uparrow$- and $\\downarrow$-type fermions are similar close to the center of the \ntrapping potential, while some of the excess-type fermions are close to the edges.\nWhen the temperature increases to $T = 0.2E_F$\nshown in Fig.~\\ref{fig:nupdown}(b) or to $T = 0.35E_F$\nshown in Fig.~\\ref{fig:nupdown}(c), the density of $\\uparrow$- and $\\downarrow$-type\nfermions become different at the center of the trap. In addition, \nboth $\\uparrow$- and $\\downarrow$-type fermions exist towards the\nedges. At temperatures close to $T_{BEC}$ and above, the density profiles \nof $\\uparrow$- and $\\downarrow$-type fermions have the standard shapes of weakly interacting\ntrapped Fermi gases.\n\nIn this section, we have shown that the effective Bose-Fermi description of Fermi-Fermi mixtures\nis applicable in the strong attraction limit, and thus can provide good quantitative\ncomparisons to experiments in the same regime, since the\nexact boson-fermion and boson-boson scattering parameters\nwere used to obtain the phase diagrams and density profiles.\nThus, we think that it is particularly important to perform experiments\nfor different population imbalances in the strong attraction limit, \nwhere the theory is simple. The situation is somewhat more complicated \nnear unitarity where quantitative comparisons between theory and experiment \nare more difficult. Furthermore, there are also some differences between the experimental studies\nof MIT~\\cite{mit, mit-2} and Rice~\\cite{rice,rice-2} groups performed near unitarity,\nsince the shapes of their traps and the number of trapped atoms are quite different.\n\nHaving concluded the analysis of the effects of trapping potentials on \nFermi-Fermi mixtures in the strong attraction limit, next we give a summary of our conclusions.\n\n\n\n\n\n\\section{Conclusions}\n\\label{sec:conclusions}\n\nIn summary, we used the effective Bose-Fermi mixture description to obtain\nthe phase diagrams of Fermi-Fermi mixtures with equal or \nunequal masses and equal or unequal populations in the strong attraction limit.\nFor this purpose, we analyzed first the exact boson-fermion \nand boson-boson scattering lengths as a function of mass anisotropy, \nand then we constructed the phase diagrams of Fermi-Fermi mixtures\nin the BEC regime.\n\nWe showed that three-dimensional non-trapped fermion mixtures with population imbalance\nexhibit phase separation in addition to the normal polarized mixture of fermions \nand uniform mixture of superfluid and excess fermions. \nIn the BEC regime, we found two different non-uniform phase separated states: \nPS(1), where there is phase separation between pure unpaired (excess) \nand pure paired fermions (molecular bosons); and\nPS(2), where there is phase separation between pure excess fermions \nand a mixture of excess fermions and molecular bosons.\nFor equal mass mixtures, our results for the phase boundaries\nare quantitatively different from previous saddle-point results,\nand these quantitative differences become more pronounced \nfor unequal mass mixtures when heavier fermions are in excess \nindicating the importance of taking into account scattering processes \nbeyond the Born approximation. \n\nWe also discussed the effects of trapping potentials on the density profiles of \ncondensed and non-condensed molecular bosons, and excess fermions at zero and finite \ntemperatures. At zero temperature, we found that almost all of the condensed\nbosons are at the center of the trap, while the excess fermions are pushed to the edges\ndue to the repulsive boson-fermion interactions. At finite temperatures, we found that\nnon-condensed pairs and excess fermions are created at the center of the trap \nat the expense of an overall reduction of condensed bosons. \nFinally, at temperatures above the BEC temperature, the number of condensed bosons vanish,\nand the system becomes a mixture of weakly interacting non-condensed bosons\nand excess fermions. Finally, we discussed that our findings can provide \ngood quantitative comparisons to experiments performed in the same \nregime of validity of the theory (BEC regime), since the boson-fermion \nand boson-boson scattering parameters that enter our calculations are exact \nin the dilute limit.\n\nLastly, we think that it is important to perform experiments with Fermi-Fermi mixtures\nin the strong attraction limit (BEC regime) where the theoretical description is\nsimple. In this limit, additional superfluid and normal phases and richer density \nprofiles proposed here can be observed, and directly compared with the theory.\n\n\nWe thank NSF (DMR-0304380) for support.\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\\label{sec:introduction}\n\nTo tackle the pressing challenge of climate change \\cite{bai2018six} and the urgent call by the United Nations to reduce the adverse environmental impact of cities \\cite{assembly2015sustainable}, there is an increasing effort to study the impact of human mobility on urban well-being dimensions, such as traffic congestion, air pollution, and carbon dioxide emissions.\nAn element that enriches the complexity of this challenge is the widespread diffusion of GPS navigation apps such as TomTom, Google Maps, and Waze, which use routing algorithms, heuristics and AI to suggest the best path to reach a user's desired destination. \nAlthough undoubtedly useful, particularly when exploring an unfamiliar city,\nGPS navigation apps may also cause several issues in the urban environment: since they are typically optimised to keep an individual's trip as short as possible, they do not care about collective effects on the city, such as whether the traffic can be absorbed by the streets, compromises safety or creates more pollution \\cite{debaets2014route, macfarlane2019when, siuhi2016opportunities}. \nMany documented cases show that GPS navigation apps may create chaos: for example, they sometimes divert heavy traffic through side roads in nearby towns, with such an impact on the residents' life that experts stated that these apps are a new agent claiming a ``right to the city\"~\\cite{leonia, fisher2022algorithms}.\n\nBeyond the anecdotal, preliminary research show that the impact of navigation apps on the urban environment is mixed \\cite{ericsson2006optimizing, samaras2016quantification}. \nOn the one hand, navigation apps may provide benefits in mitigating carbon dioxide emissions \\cite{arora2021quantifying}; on the other hand, they may increase the population exposure to pollution in densely-populated areas \\cite{perezprada2017managing}. \nOverall, existing studies are sporadic and yield contradictory results, leading to a picture of the navigation apps' impact on the urban environment that is mainly unclear and incomplete. \nIn particular, the literature still lacks a rigorous framework to assess and compare the impact of navigation apps on urban well-being.\n\nIn this paper, we design a simulation framework -- TraffiCO$_2$ -- to assess the impact of GPS navigation apps on urban well-being in terms of carbon dioxide (CO$_2$) emissions.\nThe framework uses GPS data to reconstruct a city's mobility demand, and navigation apps' APIs to obtain routing suggestions (paths on the road network) for a set of vehicles' origin and destination.\nThen, TraffiCO$_2$ relies on a traffic simulator that considers all aspects of vehicular mobility (e.g., jams, queues at traffic lights, roads capacity), to generate trajectories that describe when each vehicle visits each road in its path.\nFinally, the framework use an emission model to estimate CO$_2$ emissions for each segment in the road network.\n\nWe apply TraffiCO$_2$ to the city of Milan (Italy) to study how emissions changes as we increase the fraction of vehicles that follow navigation apps' routing.\nWe find that settings in which either all vehicles or none of them follow a navigation apps' routing suggestion lead to the highest amount of CO$_2$ emissions. \nThese settings also correspond to the most uneven distribution of the CO$_2$ emissions across the roads, with a few roads suffering the greater quantity of emissions.\nIn contrast, a scenario where just a fraction of the vehicles (around 50\\%) follows navigation apps' suggestion and the remaining part follow a randomly perturbed fastest path, reduces the vehicles' overall emissions and distributes them more evenly on the road network. \nWe also find that the navigation apps' routing affects the spatial distribution of emissions, making Milan's external ring road more polluted while unloading the internal roads from the emissions.\nNotably, adding perturbation to the vehicles' path is beneficial, as it reduces the overall emissions, distributes them more evenly, and reduces the vehicles' average travel time.\n\nOur simulation framework is a useful tool to assess and compare routing strategies, helping drivers, institutions, and policymakers understand the impact of navigation apps on the urban environment. \nMoreover, TraffiCO$_2$ is a first step towards designing and testing next-generation routing principles that may increase urban well-being while satisfying individual needs. \n\n\n\n\n\n\n\n\\subsection*{Open Source}\nWe provide the implementation of TraffiCO$_2$, the code and the link to the data to reproduce our study at \\url{https:\/\/bit.ly\/traffico2_gh}.\n\n\\section{Related Work}\n\\subsection*{Environmental impact of vehicular traffic}\nThe environmental impact generated by vehicles (e.g., air pollution) is becoming increasingly evident in urban environments.\nExisting methods to quantify vehicles' emissions range between two extremes. On the one hand, some approaches rely on measurements performed on small samples of vehicles with high spatio-temporal resolutions, such as those coming from particulate sensors~\\cite{desouza2020} or portable emissions measurement systems (PEMS)~\\cite{chong2020,lujan2018}. \nThese sensors measure emissions in real-world driving conditions, producing accurate estimates but hardly generalisable patterns due to the limited sample size. For example, two studies~\\cite{lujan2018,chong2020} analyse emissions from PEMS of one and three vehicles, finding that the highest emissions are associated with the urban part of the route, flat roads, and low speed.\n\nOn the other hand, some studies cover a region's entire fleet, such as those using odometer readings from annual safety inspections.\nThese data describe each vehicle's age, fuel type, engine volume, and mileage, used in macroscopic models to estimate annual emissions.\nTwo studies~\\cite{chatterton2015use,diao2014vehicle} use odometer readings to compute mean annual emissions for UK postcode areas and explore the built-environment effects (e.g., work accessibility) on the vehicles' annual miles travelled in Boston.\nUnfortunately, odometer readings miss critical information such as instantaneous speed and acceleration~\\cite{kancharla2018incorporating,choudhary2016urban,ferreira2015impact,zheng2017influence}, making it challenging to track emissions over time and map them to suburban areas.\n\nSomewhere between these two extremes lie works that use GPS traces, which describe human mobility in great detail~\\cite{luca2020deep,barbosa2018human} and can cover a representative fraction of the vehicle fleet~\\cite{pappalardo2013understanding, bohm2021improving}. \nThese data allow computing instantaneous speed and acceleration, which can be used within microscopic models to obtain emissions estimates in high spatio-temporal resolution.\nSeveral studies use GPS traces to analyse the vehicles' emissions at different spatio-temporal scales~\\cite{bohm2021improving,nyhan2016,liu2019}, investigate the relationship between emissions and the urban environment~\\cite{reznik2018}, vehicle miles travelled and fuel consumption~\\cite{wang2014using}, or trip rates and travel mode choice~\\cite{cervero1997travel}. \nOther studies concentrate on congestion-related emissions~\\cite{gately2017urban} or braking~\\cite{chen2020mining}, emissions associated with ride-hailing~\\cite{sui2019gps} and bus stops' positioning~\\cite{yu2020mobile}, the impact of urban policies~\\cite{rahman2017tribute}, methods for emission modelling~\\cite{zhu2020high,aziz2018novel}, and air quality monitoring~\\cite{desouza2020}.\n\n\\subsection*{Microscopic traffic simulation} \nMicroscopic traffic simulators are crucial to simulate vehicular traffic given a mobility demand.\nA notable example is SUMO (Simulation of Urban MObility), an open-source, portable, and multi-modal tool designed to handle traffic simulations on road networks~\\cite{Microscopic2018Lopez, alazzawi2018simulating, krajzewicz2012recent}. \nSUMO allows controlling several aspects of traffic, from fuel consumption to vehicle emissions and routing strategies.\nSeveral works use SUMO to study the impact of vehicular traffic on the urban environment.\nAlazzawi et al.~\\cite{alazzawi2018simulating} simulate the introduction of a fleet of automated shared vehicles into Milan, finding that a fleet of 9500 of these vehicles helps mitigate traffic congestion and emissions.\nMalik et al.~\\cite{malik2019evaluation} propose a traffic system that re-routes an emergency vehicle in case of traffic jams to reduce travel time and pollution. \nKrajzewicz et al.~\\cite{krajzewicz2005simulation} use optical information systems to optimise traffic lights over junctions better than traditional approaches.\nZubillaga et al.~\\cite{zubillaga2014measuring} compare the traditional traffic-light coordination (green-wave method) with a self-organising method that adapts to traffic demands, showing how the latter is way better. \n\n\\subsection*{Urban routing and impact of navigation systems}\nPeople's natural routing choices may significantly deviate from the optimal route.\nThe origin of these sub-optimal human routes may lie in several factors, e.g., the environment in which one grew up \\cite{coutrot2022entropy}, the subjective perception of space \\cite{norman2005perception}, the presence of landmarks \\cite{foo2005do}, and even the usage of an electric vehicle \\cite{jensen2020route}.\n\nZhu et al.~\\cite{zhu2015do} find that only $34\\%$ of trips, mostly very short and very long ones, follow the shortest time path.\nLima et al.~\\cite{lima2016understanding} find that $53\\%$ of the drivers' routes are not optimal and that most drivers use a few preferred routes for their journeys.\nXu et al.~\\cite{xu2021understanding} find similar results using location-based services data, which are less accurate but more pervasive than vehicular trajectory data.\nBongiorno et al.~\\cite{bongiorno2021vector} find that people increasingly deviate from the shortest path as the distance between origin and destination increases. \nSimilarly, Manley et al.~\\cite{manley2015shortest} show that people's route choice results from multiple decisions made at each anchor of the route.\nGiven the evident uncertainty in people's routing strategies, the impact of navigation apps' routing suggestions on the urban environment is not negligible~\\cite{macfarlane2019when}.\nFor this reason, some effort has been put into developing ``environmentally-friendly'' navigation apps that minimise fuel consumption instead of travel time~\\cite{barth2007environmentally}.\nStudies on the effects of this eco-routing, either as fuel savings~\\cite{ericsson2006optimizing} or system-wide impact~\\cite{ahn2013network,samaras2016quantification}, find that the urban impact of navigation apps is mixed: green navigation apps can reduce CO$_2$ emissions but increase the population exposure to nitrogen oxides~\\cite{perezprada2017managing}. \nFollowing the same research line, Mehrvarz et al.~\\cite{mehrvarz2020optimal} find that the fastest route suggested by navigation apps may not optimise fuel consumption.\nIn contrast, Arora et al.~\\cite{arora2021quantifying} show that following Google Maps' routing strategy saves $1.7\\%$ of CO$_2$ emissions and $6.5\\%$ travel time. \n\n\\subsection*{Position of our work}\nExisting studies are sporadic and yield contradictory results, and the impact of navigation apps on urban well-being is mainly unclear and incomplete.\nMoreover, the existing literature lacks a rigorous framework to assess and compare the various navigation apps and routing strategies.\nWe fill this gap by proposing TraffiCO$_2$, a simulation framework to estimate the adverse impact of different routing strategies on total CO$_2$ emissions and their distribution of a city's road network.\nThus, our work extends the branch of literature on sustainable urban mobility by providing a rigorous experimental framework to disentangle the impact of routing criteria on the urban environment. \n\n\\begin{figure*}[htb!]\n \\centering\n \\includegraphics[width=\\linewidth]{imgs\/Navigators4.pdf}\n \\caption{\nSchema of the TraffiCO$_2$ simulation framework.\n(a) The city is split into squared tiles using scikit-mobility \\cite{scikitmob}. \n(b.1) Real data are used to estimate mobility flows (OD matrix) within the city. (b.2-b.3) A trip is created by selecting at random an origin-destination pair from the OD matrix and two edges on the road network.\n(b.5) Some routing algorithm is used to convert each trip into a path on the road network. \n(c) Steps b.1-b.5 are repeated $N$ times ($N$ = number of vehicles) to obtain a multiset of routed paths.\n(d) A traffic simulator (SUMO) is used to simulate the urban traffic generated by the routed paths (e).}\n \\label{fig:schema}\n\\end{figure*}\n\n\\section{Simulation Framework}\n\\label{sec:simulation_framework}\nWe design a simulation framework -- TraffiCO$_2$ -- to generate a realistic urban traffic considering different routing strategies (Figure~\\ref{fig:schema}).\nFirst, we use real data to generate a \\textit{mobility demand} describing trips (origin-destination pairs) in an urban environment.\nSecond, we transform each trip into a path on the road network using some routing algorithm, obtaining a multiset of \\textit{routed paths}.\nThird, we use an agent-based model (SUMO) that, considering realistic aspects of vehicular mobility (e.g., jams, traffic lights, slowdowns), simulates an \\textit{urban traffic} based on the multiset of routed paths.\nFinally, we compute the vehicles' travel time and the CO$_2$ emissions on each road from the trajectories generated by SUMO.\n\n\n\\subsection{Road Network} \n\\label{sec:road_network}\nIt describes the road infrastructure where the vehicles move during the simulation. \nThe road network is a directed graph $G=(V, E)$, where $V$ is the set of nodes representing road intersections and $E$ is the set of edges representing roads. \nBoth nodes and edges may have attributes, such as: traffic lights, number of lanes, road speed limit and type (e.g., motorway, secondary road).\nThese attributes are used by SUMO to simulate realistic aspects of vehicular mobility.\n\n\n\\subsection{Mobility Demand}\n\\label{sec:mobility_demand}\nThe \\emph{mobility demand} $D = \\{T_1, \\dots, T_N\\}$ is a multiset of $N$ trips (one per each vehicle) within an urban environment. \nA single trip $T_v=(o,d)$ for a vehicle $v$ is defined by its origin location $o$ and destination location $d$.\nTo compute $D$, we first divide the urban environment into squared tiles of a given side (Figure \\ref{fig:schema}a). Second, we use real mobility data to estimate the flows between the tiles, thus obtaining an origin-destination matrix $M$ where an element $m_{o, d}\\in M$ describes the number of vehicles' trips that start in tile $o$ and end in tile $d$ (Figure \\ref{fig:schema}b.1). \n\nThen, we iterate $N$ times the following procedure. A vehicle's $v$ trip is a pair $T_v=(e_o, e_d)$ generated by selecting at random a matrix element $m_{o,d} \\in M$ with a probability $p_{o, d} \\propto m_{o, d}$ and uniformly at random two edges $e_o, e_d \\in E$ within tiles $o$ and $d$, respectively (Figure \\ref{fig:schema}b.2, b.3). \n\n\\subsection{Paths generation}\n\\label{sec:paths_generation}\nWe translate the $N$ trips in $D$ into $N$ paths obtaining a multiset $\\overline{D}$ of \\emph{routed paths} within the urban environment. \nEach path $P_v(e_o,e_d, R) {=} (e_o, \\dots, e_d)$ of a vehicle $v$\nis a sequence of edges on the road network connecting $e_o$ and $e_d$ (Figure \\ref{fig:schema}b.5), obtained by some routing algorithm $R$. \nWhen a vehicle's path is generated by a routing algorithm $R$, we say that the vehicle is $R$-routed.\nIn $\\overline{D} = {\\{P_1, \\dots, P_N\\}}$, the routed paths (Figure \\ref{fig:schema}c) are generated independently by (different) routing algorithms. \n\n\\subsection{Traffic Simulation}\n\\label{sec:traffic_simulation}\nWe simulate the vehicular traffic generated by the routed paths in $\\overline{D}$ using SUMO (Simulation of Urban MObility) \\cite{Microscopic2018Lopez, alazzawi2018simulating, krajzewicz2012recent} (Figure~\\ref{fig:schema}d). \nSUMO explicitly models each vehicle's physics and dynamics, including their routes through the road network, allowing us to simulate vehicular traffic realistically, including traffic jams, queues at traffic lights, and slowdowns due to heavy traffic.\nSUMO outputs an \\emph{urban traffic} (Figure \\ref{fig:schema}e), i.e., a multiset $S(\\overline{D}, \\mathcal{T}) = \\{\\overline{P_1}, \\dots, \\overline{P_N}\\}$ where: \n\\begin{itemize}\n\\item $\\mathcal{T} = (t^{(1)}, \\dots, t^{(N)})$ is the sequence of departure times of the $N$ paths in $\\overline{D}$, where $t^{(i)} \\in \\mathcal{T}$ is a timestamp chosen uniformly at random from the simulation interval (e.g., 1 hour); \n\\item a vehicle $v$'s trajectory $\\overline{P_v} \\in S(\\overline{D},\\mathcal{T})$ is defined as: $$\\overline{P_v} = ((e_o, t_1^{(v)}), \\dots, (e_d, t_m^{(v)}))$$ where $m$ is the length of path $\\overline{P_v}$.\n\n\\end{itemize}\n\n\\subsection{Vehicle Emissions}\n\\label{sec:vehicle_emissions}\nWe assess the impact of the urban traffic $S(\\overline{D}, \\mathcal{T})$ using a model that estimates the vehicles' emissions based on their trajectories $\\overline{P_1}, \\dots, \\overline{P_N}$. \nSpecifically, we use the HBEFA3 emission model based on the Handbook of Emission Factors for Road Transport (HBEFA) database~\\cite{infras2013handbook}. \nThe HBEFA3-based model estimates the vehicle's instantaneous CO$_2$ emissions relying on the following function, which is linked to the power the vehicle's engine produces in each trajectory point $j$ to overcome the driving resistance force~\\cite{krajzewicz2015second}:\n$$\\mathcal{E}(j) = c_0 + c_1sa + c_2sa^2 + c_3s + c_4s^2 + c_5s^3$$\nwhere $s$ and $a$ are the vehicle's speed and acceleration in point $j$, respectively, and $c_0,\\dots,c_5$ are parameters changing\nper emission type and vehicle taken from the HBEFA database.\n\nWe compute the amount of CO$_2$ emissions on each edge $e \\in E$ by summing all the emissions corresponding to any vehicle $v$'s trajectory point that fall on $e$, i.e., $\\mathcal{E}(e) = \\sum_v \\sum_{j \\in \\overline{P_v}} \\mathcal{E}(j)$.\nFinally, we construct a weighted road network $\\overline{G} = (V, \\overline{E})$ where each edge $\\overline{e} \\in \\overline{E}$ is associated with the attribute $\\mathcal{E}(e)$ describing the amount of CO$_2$ emissions on it.\n\n\n\\section{Experimental Setup}\n\\label{sec:experimental_setup}\nWe apply TraffiCO$_2$ to a 45 km$^2$ area in the city centre of Milan, Italy, for which we have GPS data describing 17k private vehicles travelling between April 2nd and 8th, 2007 (114k GPS points).\nWe split Milan into squared tiles with 1 km side, detect the tiles where each vehicle starts and stops \\cite{Ramaswamy2004Project, scikitmob}, and compute the origin-destination matrix $M$ of vehicles' flows.\n\nWe download Milan's road network $G=(V, E)$ from OpenStreetMap ($|V|=5551$ nodes and $|E|=36,945$ edges) and preprocess it to fix incorrect information regarding turns, intersections, road interruptions, the number of lanes per road, and other inaccuracies that characterise these data \\cite{Argota2022Getting}. \n\nBased on the matrix $M$ and the road network $G$, we compute the mobility demand $D$ with $N=15,000$ vehicles using the procedure described in Section \\ref{sec:mobility_demand}.\nWe choose $N=15,000$ because it minimises the difference between the distribution of travel time of real trajectories and simulated ones, a common way to assess the realism of a simulated urban traffic \\cite{Argota2022Getting} (see Appendix \\ref{sec:traf-cal} for detail). \n\nTo translate the mobility demand $D$ into the multiset of routed paths $\\overline{D}$, we consider two routing algorithms: OpenStreetMap (OSM) and TomTom (TT). \nOSM is a public voluntary geographic information system, which provides APIs\\footnote{\\href{https:\/\/openrouteservice.org\/dev\/\\#\/api-docs}{https:\/\/openrouteservice.org\/dev\/\\#\/api-docs}} to generate paths between locations.\nTT is a commercial navigation system service that provides APIs\\footnote{\\href{https:\/\/developer.tomtom.com\/routing-api\/documentation\/product-information\/introduction}{https:\/\/developer.tomtom.com\/routing-api}} for routes generation.\nFor OSM and TT, we use a routing principle that suggests a path between two locations as a trade-off between travel time and distance.\nNote that OSM and TT implement this principle differently, i.e., they may provide different paths for the same origin-destination pair.\n\nWe obtain the path of vehicles that do not follow navigation apps' suggestions using Duarouter (DR), a routing algorithm provided by SUMO\\footnote{\\href{https:\/\/sumo.dlr.de\/docs\/duarouter.html}{https:\/\/sumo.dlr.de\/docs\/duarouter.html}} that suggests the fastest path (i.e., shortest travel time) between two edges on the road network.\nThe fastest path may be perturbed using a randomisation parameter $w \\in [1, +\\infty)$, where $w=1$ means no randomisation (i.e., the fastest path). The higher $w$, the more randomly perturbed the fastest path is.\nTherefore, DR with $w > 1$ allows us to model the driving behaviour of vehicles that do not strictly follow navigation apps' suggestions and, simultaneously, to model the imperfection and non-rationality of human drivers \\cite{Seele2012Cognitive}. \nIndeed, individuals get distracted when driving (e.g., take wrong turns), and they lack complete knowledge of the city's traffic, the road network, and the best path to reach a destination. \nIn our experiments, we use $w=5$ because it is the value that minimises the difference between the distribution of travel time of real trajectories and simulated ones (see Appendix \\ref{sec:traf-cal}).\nFigure \\ref{fig:trajectories} shows four examples of paths generated by OSM, TT, DR with $w=1$ and DR with $w=5$ between the same origin-destination pair (trip).\n\n\\begin{figure}[htb!]\n \\centering\n \\includegraphics[width=0.9\\linewidth]{imgs\/plot_trajs.pdf}\n \\caption{Examples of routed paths between an origin and destination pair according to OSM (blue), TT (red), DR with $w=5$ (orange), and DR with $w=1$ (black).}\n \\label{fig:trajectories}\n\\end{figure}\n\n\nGiven a routing algorithm $R \\in \\{\\text{OSM}, \\text{TT}\\}$, we study its impact on the urban environment generating 11 multisets of routed paths $\\overline{D}^{(R)}_0, \\dots, \\overline{D}^{(R)}_{10}$. \nIn each multiset $\\overline{D}^{(R)}_i$ ($i=0, \\dots, 10$), $(i \\cdot 10)$\\% of the paths (chosen uniformly at random among the $N$ paths) are $R$-routed ($R {\\in} \\{\\text{OSM}, \\text{TT}\\}$) and the remaining paths are routed by DR with $w=5$. \nFor example, for $i=5$ and $R = \\text{TT}$, $\\overline{D}_5^{(R)}$ contains 50\\% of the paths routed by TT and the remaining vehicles routed by DR with $w=5$.\nSimilarly, $i=7$ means that 70\\% of the vehicles are TT-routed and 30\\% are DR-routed ($w=5$).\n\nTo make experiments more robust, for each $i = 0, \\dots, 10$, we generate $\\overline{D}_i^{(R)}$ ten times, each one with a different choice of $R$-routed vehicles that are chosen uniformly at random.\nFinally, we generate the urban traffic $S(\\overline{D}_{i}^{(R)}, \\mathcal{T})$ for each multiset of routed paths $\\overline{D}_{i}^{(R)}$, where departure times in $\\mathcal{T}$ are chosen uniformly at random between 0 and 3600 seconds, i.e., we simulate one hour of traffic in Milan. \nThen, for each urban traffic $S(\\overline{D}_{i}^{(R)}, \\mathcal{T})$, we compute the CO$_2$ emissions for each trajectory and aggregate them at the edge level, obtaining the weighted road network $\\overline{G}_i^{(R)}$.\nFrom $\\overline{G}_i^{(R)}$, we obtain the overall CO$_2$ emissions in Milan $\\mathcal{E}_i^{(R)} = \\sum_{e\\in E} \\mathcal{E}(e)$, where $\\mathcal{E}(e)$ is the amount of emissions on edge $e$.\n\n\\begin{figure}[htb!]\n \\centering\n \\subfigure[\\large OSM]{\n \\includegraphics[width=0.9\\linewidth]{imgs\/plot_ccdf_OSM__AVG_across_repetitions__INSET_ZOOM__V2.png}}\n \\subfigure[\\large TT]{\\includegraphics[width=0.9\\linewidth]{imgs\/plot_ccdf_TT__AVG_across_repetitions__INSET_ZOOM__V2.png}}\n \\caption{\n The Complementary Cumulative Distribution Functions (CCDFs) of the CO$_2$ (in mg) emitted on the roads (averaged across 10 repetitions) by the vehicles of the generated urban traffic $S(\\overline{D}_{i}^{(R)}, \\mathcal{T})$, for OSM (a) and TT (b). \n Colours represent routed paths with increasing percentage of $R$-routed vehicles.\n The inset plot zooms on the distributions' tail.}\n \\label{fig:ccdfs}\n\\end{figure}\n\n\n\n\\section{Results}\n\\label{sec:results}\nWe study how the distribution of the CO$_2$ emissions across Milan's roads changes varying the percentage of $R$-routed vehicles, i.e., varying $i=0, \\dots,10$ of $\\overline{D}^{(R)}_i$, $R \\in \\{\\text{OSM}, \\text{TT}\\}$.\nIn general, emissions distribute across roads in a heterogeneous way: a few grossly polluted roads coexist with roads with significantly fewer emissions (Figure \\ref{fig:ccdfs}).\nIndeed, these distributions are associated with a Gini index $g_{\\text{\\tiny OSM}} \\in [0.864, 0.876]$ and $g_{\\text{\\tiny TT}} \\in [0.860, 0.868]$ (Figure~\\ref{fig:gini&totco2}a) and are well approximated by a truncated power-law with the exponent $\\alpha_{\\text{\\tiny OSM}} \\in [1.76,1.89]$ and $\\alpha_{\\text{\\tiny TT}} \\in [1.76,2.00]$ (Figure~\\ref{fig:alpha}). \nSee Appendix \\ref{sec:fitting_distributions} for details on the curve fitting.\nIn particular, the distributions are the least uneven when 50\\% and 70\\% of the vehicles are OSM-routed and TT-routed, respectively (Figure~\\ref{fig:ccdfs}).\n\nThe analysis of how the total CO$_2$ emissions ($\\mathcal{E}$) varies with the percentage of $R$-routed vehicles also reveals a clear pattern: when either all vehicles are $R$-routed or none of them, the overall emissions are maximised (Figure \\ref{fig:gini&totco2}b).\nIn contrast, the overall emissions are minimised when just half of the vehicles are R-routed.\nIn other words, $\\mathcal{E}_5^{(R)} < \\mathcal{E}_0^{(R)}$ and $\\mathcal{E}_5^{(R)} < \\mathcal{E}_{10}^{(R)}$.\nNote that the total emissions when all vehicles are TT-routed are much heavier than when all are OSM-routed (Figure \\ref{fig:gini&totco2}b).\nThis result suggests that TT's routing algorithm recommends paths that generate a smaller adverse impact on urban well-being than OSM.\n\nThe fraction of $R$-routed vehicles also influence the spatial distribution of emissions in the city.\nIn particular, comparing the two scenarios that maximise the total emissions (0\\% and 100\\% of $R$-routed vehicles) with the scenario that minimises them (50\\% of $R$-routed vehicles) helps understand where vehicles are being routed, consequently revealing emissions hot spots.\nIn Figure \\ref{fig:map_diff}a, we show the difference between the per-road emissions (normalized by the road length) when none of the vehicle is OSM-routed and 50\\% of them are (i.e., $\\mathcal{E}_0^{\\text{\\tiny (OSM)}}(e) - \\mathcal{E}_5^{\\text{\\tiny (OSM)}}(e)$, $\\forall e \\in E$).\nSimilarly, Figure \\ref{fig:map_diff}b shows the normalized emissions difference when all vehicles are OSM-routed and 50\\% of them are (i.e., $\\mathcal{E}_{10}^{\\text{\\tiny(OSM)}}(e) - \\mathcal{E}_5^{\\text{\\tiny (OSM)}}(e)$, $\\forall e \\in E$).\nWe find that when 100\\% of vehicles are OSM-routed, the emissions are more concentrated towards Milan's ring road (Figure \\ref{fig:map_diff}b).\nIn contrast, when none of them is OSM-routed, the emissions are more concentrated towards the city centre (Figure \\ref{fig:map_diff}a).\nWe find similar results for TT (Figure \\ref{fig:map_diff_TT}).\n\n\\begin{figure*}[htb!]\n \\centering\n \\subfigure[\\large Gini index]{\n \\includegraphics[width=0.8\\columnwidth]{imgs\/plot_errorbars_gini__V2.pdf}}\n \\subfigure[\\large Total CO$_2$]{\\includegraphics[width=0.8\\columnwidth]{imgs\/plot_errorbars_AVG_total_co2__V2.pdf}}\n \\caption{\n Gini index of the CO$_2$ distribution (a) and total CO$_2$ emissions (b) varying the percentage of $R$-routed vehicles, for OSM (blue) and TT (red).\n \n \n In the error bars, points indicate the average Gini index (a) and the total CO$_2$ (b) over ten simulations with different choices of $R$-routed vehicles chosen uniformly at random.\n Vertical bars indicate the standard deviation. \n }\n \\label{fig:gini&totco2}\n\\end{figure*}\n\n\\begin{figure*}[htb!]\n \\centering\n \\subfigure[]{\n \\includegraphics[width=0.85\\columnwidth]{imgs\/Milan_map_co2_diff_norm_by_len__osm_0_osm_50.pdf}}\n \\subfigure[]{\\includegraphics[width=0.85\\columnwidth]{imgs\/Milan_map_co2_diff_norm_by_len__osm_100_osm_50.pdf}}\n \\caption{\n The difference in the total CO$_2$ emitted on each road (in mg per meter of road) when: (a) none of the vehicles is OSM-routed and 50\\% of them are\n ($\\mathcal{E}_0^{\\text{\\tiny (OSM)}}(e) - \\mathcal{E}_5^{\\text{\\tiny (OSM)}}(e)$, $\\forall e \\in E$); (b) all vehicles are OSM-routed and 50\\% of them are ($\\mathcal{E}_{10}^{\\text{\\tiny (OSM)}}(e) - \\mathcal{E}_5^{\\text{\\tiny (OSM)}}(e)$, $\\forall e \\in E$).\n Red roads indicate a positive difference; blue ones indicate a negative difference.\n \n \n }\n \\label{fig:map_diff}\n\\end{figure*}\n\n\n\n\\begin{figure*}[htb!]\n \\centering\n \\subfigure[\\large OSM - Gini]{\n\t\\includegraphics[width=0.3\\textwidth]{imgs\/plot_errorbars_w_analysis_gini_OSM.pdf}}\n\t\\subfigure[\\large OSM - CO$_2$]{\n\t\\includegraphics[width=0.3\\textwidth]{imgs\/plot_errorbars_w_analysis_totalco2_OSM.pdf}}\n\t\\subfigure[\\large OSM - travel time]{\n\t\\includegraphics[width=0.3\\textwidth]{imgs\/plot_errorbars_w_analysis_avgtraveltime_OSM.pdf}}\n\t\\subfigure[\\large TT - Gini]{\n\t\\includegraphics[width=0.3\\textwidth]{imgs\/plot_errorbars_w_analysis_gini_TT.pdf}}\n\t\\subfigure[\\large TT - CO$_2$]{\n\t\\includegraphics[width=0.3\\textwidth]{imgs\/plot_errorbars_w_analysis_totalco2_TT.pdf}}\n\t\\subfigure[\\large TT - travel time]{\n\t\\includegraphics[width=0.3\\textwidth]{imgs\/plot_errorbars_w_analysis_avgtraveltime_TT.pdf}}\n \\caption{\n Gini index of CO$_2$ distribution (a,d), total CO$_2$ emissions (b,e), average travel time per vehicle (c,f), for $w = 1, \\dots, 15$, varying the percentage of $R$-routed vehicles, for OSM (a,b,c) and TT (d,e,f).\n \n In the error bars, points indicate the average Gini index (a,d), total CO$_2$ emissions (b,e), average travel time (c,f) over ten simulations with different choices of $R$-routed vehicles chosen uniformly at random.\n Vertical bars indicate the standard deviation.}\n \\label{fig:w_osm_tt}\n\\end{figure*}\n\n\n\\paragraph{Impact of randomization}\nWe investigate the impact of DR's randomisation parameter $w$ on the total emissions, their distribution over the road network, and the vehicles' average travel time.\nFor this purpose, we repeat the simulation of the urban traffic varying $w = 1, \\dots, 15$. \nWe remind that the higher $w$, the more randomly perturbed DR's fastest path is (see Appendix \\ref{sec:perturbation}).\n\nFirst, changing $w$ does not affect considerably the shape of the emission distributions: the Gini index still ranges in $g_{\\text{\\tiny OSM}} \\in [0.860, 0.879]$ and $g_{\\text{\\tiny TT}} \\in [0.858, 0.879]$ (Figure \\ref{fig:w_osm_tt}a, d) and the truncated power-law exponent in $\\alpha_{\\text{\\tiny OSM}} \\in [1.71,1.94]$ and $\\alpha_{\\text{\\tiny TT}} \\in [1.71,2.08]$.\nSecond, the higher $w$, the more even the distributions (Figure \\ref{fig:w_osm_tt}a, d), the lower the emissions (Figure \\ref{fig:w_osm_tt}b, e), and the shorter the average travel time (Figure \\ref{fig:w_osm_tt}c, f).\nThird, the total emissions are minimised when 40-60\\% of the vehicles are $R$-routed, and so are the average travel time and the distributions' Gini index (Figure \\ref{fig:w_osm_tt}).\n\n\n\n\n\n\n\n\n\\section{Discussion}\nOur study provide several interesting results. \nWe discuss and interpret them in the following paragraphs.\n\n\\textbf{Navigation apps do impact on emissions.} \nIn general, CO$_2$ emissions are unevenly distributed across the city roads, and this unevenness is exacerbated when all vehicles or none of them are $R$-routed. \nIn contrast, the total CO$_2$ emissions as well as their heterogeneity are minimised when around 50-70\\% of the vehicles are $R$-routed. \nThese results clearly show that navigation apps do have a non-negligible adverse impact on the urban environment.\n\n\\textbf{Path perturbation is beneficial.}\nWe also find that path perturbation (through DR's $w$ parameter) is beneficial: the more we randomise the vehicles' paths, the shorter their travel time and the lower the amount of emissions in the city.\nThis may be because the vehicles' perturbed paths are more ``diverse'' (i.e., they spread over more roads), thus reducing congestion and consequently the amount of emissions and travel time.\nThe role of path randomisation is interesting and deserves further investigation.\n\n\n\\textbf{Trips' spatial distribution matters.} \nUrban planners and policymakers may be interested in investigating the impact of navigation apps on the spatial distribution of vehicles' emissions. \nOur study shows that, in Milan, the more vehicles are $R$-routed, the less emissions concentrate in the city center and the more in the external ring road.\nPresumably, this may be because the navigation apps \nroute the vehicles preferably on the city's arterial roads (such as the ring road) to keep the individual path as fast as possible.\n\n\\textbf{Different navigation apps, different impact.} \nOur TraffiCO$_2$ simulation framework is a useful tool to compare the impact of different navigation apps on urban well-being. In our study, we find that TomTom is better than OpenStreetMap in minimising the emissions distribution's inequality, and the city's total emissions and average travel time. \nAlthough TomTom's algorithm is not public, we may suppose that TomTom's heuristics are more sophisticated, e.g., they use more or higher-quality information than OpenStreetMap, such as traffic data, detailed information about typical road speed, capacity, and length.\n\n\\textbf{A novel what-if analysis tool.} \nTraffiCO$_2$ is a simulation framework to compare the impact of different routing strategies on urban well-being in terms of amount of emissions and their spatial and statistical distribution. \nOur tool allows us to go beyond state-of-the-art studies, which investigate the reduction of emissions for a fixed fraction of vehicles routed by a navigation app ~\\cite{arora2021quantifying}.\n\n\n\n\\section{Conclusion}\n\\label{sec:conclusion}\nScenarios where either all vehicles or none of them follow a navigation app's suggestion lead to the highest CO$_2$ emissions and the most uneven distribution of emissions per road.\nWe show that when just a fraction of the vehicles (around half of them) follows the navigation app's suggestions, such an adverse impact is minimised.\nThis minimisation also holds when introducing more randomness in non-R-routed paths, which leads to a reduction of the vehicles' average travel time, overall CO$_2$ emissions, and inequality in the distribution of CO$_2$ emissions across the road network.\n\nWe plan to improve and extend this study in several directions. \nFirst and foremost, we plan to extend the set of navigation apps considered (e.g., to Google Maps) and to study the environmental impact of a fleet of vehicles that use various navigation apps.\nIn this work, we focus on light-duty vehicles only and assume that all vehicles carry the same engine type. \nIncluding heavy-duty vehicles (e.g., buses, trucks) and considering different vehicle engine ages and types (e.g. diesel, LPG, petrol) would provide a more complete mosaic of the emissions on the road network.\nWe also look forward to apply our framework to various cities, to investigate how the impact of navigation apps varies with city size, shape, road network, and other characteristics.\nMoreover, we could study the impact of navigation apps in terms of other pollutants, such as nitrogen oxides, ozone, particulate matter, and volatile organic compounds, using emissions models similar to those used in this paper.\n\nIn the meantime, our work is a first step towards designing next-generation routing algorithms that, as our results suggest, should consider some degree of path randomisation to increase urban well-being while still satisfying individual needs.\n\n\n\n\n\n\\begin{acks}\nThis work has been partially supported by EU project H2020 Humane-AI-net G.A. 952026, EU project H2020 SoBigData++ G.A. 871042, ERC-2018-ADG G.A. 834756 \"XAI: Science and technology for the eXplanation of AI decision making\", and by the CHIST-ERA grant CHIST-ERA-19-XAI-010, by MUR (grant No. not yet available), FWF (grant No. I 5205), EPSRC (grant No. EP\/V055712\/1), NCN (grant No. 2020\/02\/Y\/ST6\/00064), ETAg (grant No. SLTAT21096), BNSF (grant No. K$\\Pi$-06-AOO2\/5). \n\nWe thank Davide Nicola, Pietro Iemmello, Ren\u00e9 Ferretti, Dante Milonga and Maurizio Turone for the inspiration. \nWe thank Daniele Fadda for his precious\nsupport in data visualization.\n\\end{acks}\n\n\\bibliographystyle{ACM-Reference-Format}\n\n\\section{Introduction}\nACM's consolidated article template, introduced in 2017, provides a\nconsistent \\LaTeX\\ style for use across ACM publications, and\nincorporates accessibility and metadata-extraction functionality\nnecessary for future Digital Library endeavors. Numerous ACM and\nSIG-specific \\LaTeX\\ templates have been examined, and their unique\nfeatures incorporated into this single new template.\n\nIf you are new to publishing with ACM, this document is a valuable\nguide to the process of preparing your work for publication. If you\nhave published with ACM before, this document provides insight and\ninstruction into more recent changes to the article template.\n\nThe ``\\verb|acmart|'' document class can be used to prepare articles\nfor any ACM publication --- conference or journal, and for any stage\nof publication, from review to final ``camera-ready'' copy, to the\nauthor's own version, with {\\itshape very} few changes to the source.\n\n\\section{Template Overview}\nAs noted in the introduction, the ``\\verb|acmart|'' document class can\nbe used to prepare many different kinds of documentation --- a\ndouble-blind initial submission of a full-length technical paper, a\ntwo-page SIGGRAPH Emerging Technologies abstract, a ``camera-ready''\njournal article, a SIGCHI Extended Abstract, and more --- all by\nselecting the appropriate {\\itshape template style} and {\\itshape\n template parameters}.\n\nThis document will explain the major features of the document\nclass. For further information, the {\\itshape \\LaTeX\\ User's Guide} is\navailable from\n\\url{https:\/\/www.acm.org\/publications\/proceedings-template}.\n\n\\subsection{Template Styles}\n\nThe primary parameter given to the ``\\verb|acmart|'' document class is\nthe {\\itshape template style} which corresponds to the kind of publication\nor SIG publishing the work. This parameter is enclosed in square\nbrackets and is a part of the {\\verb|documentclass|} command:\n\\begin{verbatim}\n \\documentclass[STYLE]{acmart}\n\\end{verbatim}\n\nJournals use one of three template styles. All but three ACM journals\nuse the {\\verb|acmsmall|} template style:\n\\begin{itemize}\n\\item {\\texttt{acmsmall}}: The default journal template style.\n\\item {\\texttt{acmlarge}}: Used by JOCCH and TAP.\n\\item {\\texttt{acmtog}}: Used by TOG.\n\\end{itemize}\n\nThe majority of conference proceedings documentation will use the {\\verb|acmconf|} template style.\n\\begin{itemize}\n\\item {\\texttt{acmconf}}: The default proceedings template style.\n\\item{\\texttt{sigchi}}: Used for SIGCHI conference articles.\n\\item{\\texttt{sigchi-a}}: Used for SIGCHI ``Extended Abstract'' articles.\n\\item{\\texttt{sigplan}}: Used for SIGPLAN conference articles.\n\\end{itemize}\n\n\\subsection{Template Parameters}\n\nIn addition to specifying the {\\itshape template style} to be used in\nformatting your work, there are a number of {\\itshape template parameters}\nwhich modify some part of the applied template style. A complete list\nof these parameters can be found in the {\\itshape \\LaTeX\\ User's Guide.}\n\nFrequently-used parameters, or combinations of parameters, include:\n\\begin{itemize}\n\\item {\\texttt{anonymous,review}}: Suitable for a ``double-blind''\n conference submission. Anonymizes the work and includes line\n numbers. Use with the \\texttt{\\acmSubmissionID} command to print the\n submission's unique ID on each page of the work.\n\\item{\\texttt{authorversion}}: Produces a version of the work suitable\n for posting by the author.\n\\item{\\texttt{screen}}: Produces colored hyperlinks.\n\\end{itemize}\n\nThis document uses the following string as the first command in the\nsource file:\n\\begin{verbatim}\n\\documentclass[sigconf]{acmart}\n\\end{verbatim}\n\n\\section{Modifications}\n\nModifying the template --- including but not limited to: adjusting\nmargins, typeface sizes, line spacing, paragraph and list definitions,\nand the use of the \\verb|\\vspace| command to manually adjust the\nvertical spacing between elements of your work --- is not allowed.\n\n{\\bfseries Your document will be returned to you for revision if\n modifications are discovered.}\n\n\\section{Typefaces}\n\nThe ``\\verb|acmart|'' document class requires the use of the\n``Libertine'' typeface family. Your \\TeX\\ installation should include\nthis set of packages. Please do not substitute other typefaces. The\n``\\verb|lmodern|'' and ``\\verb|ltimes|'' packages should not be used,\nas they will override the built-in typeface families.\n\n\\section{Title Information}\n\nThe title of your work should use capital letters appropriately -\n\\url{https:\/\/capitalizemytitle.com\/} has useful rules for\ncapitalization. Use the {\\verb|title|} command to define the title of\nyour work. If your work has a subtitle, define it with the\n{\\verb|subtitle|} command. Do not insert line breaks in your title.\n\nIf your title is lengthy, you must define a short version to be used\nin the page headers, to prevent overlapping text. The \\verb|title|\ncommand has a ``short title'' parameter:\n\\begin{verbatim}\n \\title[short title]{full title}\n\\end{verbatim}\n\n\\section{Authors and Affiliations}\n\nEach author must be defined separately for accurate metadata\nidentification. As an exception, multiple authors may share one\naffiliation. Authors' names should not be abbreviated; use full first\nnames wherever possible. Include authors' e-mail addresses whenever\npossible.\n\nGrouping authors' names or e-mail addresses, or providing an ``e-mail\nalias,'' as shown below, is not acceptable:\n\\begin{verbatim}\n \\author{Brooke Aster, David Mehldau}\n \\email{dave,judy,steve@university.edu}\n \\email{firstname.lastname@phillips.org}\n\\end{verbatim}\n\nThe \\verb|authornote| and \\verb|authornotemark| commands allow a note\nto apply to multiple authors --- for example, if the first two authors\nof an article contributed equally to the work.\n\nIf your author list is lengthy, you must define a shortened version of\nthe list of authors to be used in the page headers, to prevent\noverlapping text. The following command should be placed just after\nthe last \\verb|\\author{}| definition:\n\\begin{verbatim}\n \\renewcommand{\\shortauthors}{McCartney, et al.}\n\\end{verbatim}\nOmitting this command will force the use of a concatenated list of all\nof the authors' names, which may result in overlapping text in the\npage headers.\n\nThe article template's documentation, available at\n\\url{https:\/\/www.acm.org\/publications\/proceedings-template}, has a\ncomplete explanation of these commands and tips for their effective\nuse.\n\nNote that authors' addresses are mandatory for journal articles.\n\n\\section{Rights Information}\n\nAuthors of any work published by ACM will need to complete a rights\nform. Depending on the kind of work, and the rights management choice\nmade by the author, this may be copyright transfer, permission,\nlicense, or an OA (open access) agreement.\n\nRegardless of the rights management choice, the author will receive a\ncopy of the completed rights form once it has been submitted. This\nform contains \\LaTeX\\ commands that must be copied into the source\ndocument. When the document source is compiled, these commands and\ntheir parameters add formatted text to several areas of the final\ndocument:\n\\begin{itemize}\n\\item the ``ACM Reference Format'' text on the first page.\n\\item the ``rights management'' text on the first page.\n\\item the conference information in the page header(s).\n\\end{itemize}\n\nRights information is unique to the work; if you are preparing several\nworks for an event, make sure to use the correct set of commands with\neach of the works.\n\nThe ACM Reference Format text is required for all articles over one\npage in length, and is optional for one-page articles (abstracts).\n\n\\section{CCS Concepts and User-Defined Keywords}\n\nTwo elements of the ``acmart'' document class provide powerful\ntaxonomic tools for you to help readers find your work in an online\nsearch.\n\nThe ACM Computing Classification System ---\n\\url{https:\/\/www.acm.org\/publications\/class-2012} --- is a set of\nclassifiers and concepts that describe the computing\ndiscipline. Authors can select entries from this classification\nsystem, via \\url{https:\/\/dl.acm.org\/ccs\/ccs.cfm}, and generate the\ncommands to be included in the \\LaTeX\\ source.\n\nUser-defined keywords are a comma-separated list of words and phrases\nof the authors' choosing, providing a more flexible way of describing\nthe research being presented.\n\nCCS concepts and user-defined keywords are required for for all\narticles over two pages in length, and are optional for one- and\ntwo-page articles (or abstracts).\n\n\\section{Sectioning Commands}\n\nYour work should use standard \\LaTeX\\ sectioning commands:\n\\verb|section|, \\verb|subsection|, \\verb|subsubsection|, and\n\\verb|paragraph|. They should be numbered; do not remove the numbering\nfrom the commands.\n\nSimulating a sectioning command by setting the first word or words of\na paragraph in boldface or italicized text is {\\bfseries not allowed.}\n\n\\section{Tables}\n\nThe ``\\verb|acmart|'' document class includes the ``\\verb|booktabs|''\npackage --- \\url{https:\/\/ctan.org\/pkg\/booktabs} --- for preparing\nhigh-quality tables.\n\nTable captions are placed {\\itshape above} the table.\n\nBecause tables cannot be split across pages, the best placement for\nthem is typically the top of the page nearest their initial cite. To\nensure this proper ``floating'' placement of tables, use the\nenvironment \\textbf{table} to enclose the table's contents and the\ntable caption. The contents of the table itself must go in the\n\\textbf{tabular} environment, to be aligned properly in rows and\ncolumns, with the desired horizontal and vertical rules. Again,\ndetailed instructions on \\textbf{tabular} material are found in the\n\\textit{\\LaTeX\\ User's Guide}.\n\nImmediately following this sentence is the point at which\nTable~\\ref{tab:freq} is included in the input file; compare the\nplacement of the table here with the table in the printed output of\nthis document.\n\n\\begin{table}\n \\caption{Frequency of Special Characters}\n \\label{tab:freq}\n \\begin{tabular}{ccl}\n \\toprule\n Non-English or Math&Frequency&Comments\\\\\n \\midrule\n \\O & 1 in 1,000& For Swedish names\\\\\n $\\pi$ & 1 in 5& Common in math\\\\\n \\$ & 4 in 5 & Used in business\\\\\n $\\Psi^2_1$ & 1 in 40,000& Unexplained usage\\\\\n \\bottomrule\n\\end{tabular}\n\\end{table}\n\nTo set a wider table, which takes up the whole width of the page's\nlive area, use the environment \\textbf{table*} to enclose the table's\ncontents and the table caption. As with a single-column table, this\nwide table will ``float'' to a location deemed more\ndesirable. Immediately following this sentence is the point at which\nTable~\\ref{tab:commands} is included in the input file; again, it is\ninstructive to compare the placement of the table here with the table\nin the printed output of this document.\n\n\\begin{table*}\n \\caption{Some Typical Commands}\n \\label{tab:commands}\n \\begin{tabular}{ccl}\n \\toprule\n Command &A Number & Comments\\\\\n \\midrule\n \\texttt{{\\char'134}author} & 100& Author \\\\\n \\texttt{{\\char'134}table}& 300 & For tables\\\\\n \\texttt{{\\char'134}table*}& 400& For wider tables\\\\\n \\bottomrule\n \\end{tabular}\n\\end{table*}\n\nAlways use midrule to separate table header rows from data rows, and\nuse it only for this purpose. This enables assistive technologies to\nrecognise table headers and support their users in navigating tables\nmore easily.\n\n\\section{Math Equations}\nYou may want to display math equations in three distinct styles:\ninline, numbered or non-numbered display. Each of the three are\ndiscussed in the next sections.\n\n\\subsection{Inline (In-text) Equations}\nA formula that appears in the running text is called an inline or\nin-text formula. It is produced by the \\textbf{math} environment,\nwhich can be invoked with the usual\n\\texttt{{\\char'134}begin\\,\\ldots{\\char'134}end} construction or with\nthe short form \\texttt{\\$\\,\\ldots\\$}. You can use any of the symbols\nand structures, from $\\alpha$ to $\\omega$, available in\n\\LaTeX~\\cite{Lamport:LaTeX}; this section will simply show a few\nexamples of in-text equations in context. Notice how this equation:\n\\begin{math}\n \\lim_{n\\rightarrow \\infty}x=0\n\\end{math},\nset here in in-line math style, looks slightly different when\nset in display style. (See next section).\n\n\\subsection{Display Equations}\nA numbered display equation---one set off by vertical space from the\ntext and centered horizontally---is produced by the \\textbf{equation}\nenvironment. An unnumbered display equation is produced by the\n\\textbf{displaymath} environment.\n\nAgain, in either environment, you can use any of the symbols and\nstructures available in \\LaTeX\\@; this section will just give a couple\nof examples of display equations in context. First, consider the\nequation, shown as an inline equation above:\n\\begin{equation}\n \\lim_{n\\rightarrow \\infty}x=0\n\\end{equation}\nNotice how it is formatted somewhat differently in\nthe \\textbf{displaymath}\nenvironment. Now, we'll enter an unnumbered equation:\n\\begin{displaymath}\n \\sum_{i=0}^{\\infty} x + 1\n\\end{displaymath}\nand follow it with another numbered equation:\n\\begin{equation}\n \\sum_{i=0}^{\\infty}x_i=\\int_{0}^{\\pi+2} f\n\\end{equation}\njust to demonstrate \\LaTeX's able handling of numbering.\n\n\\section{Figures}\n\nThe ``\\verb|figure|'' environment should be used for figures. One or\nmore images can be placed within a figure. If your figure contains\nthird-party material, you must clearly identify it as such, as shown\nin the example below.\n\\begin{figure}[h]\n \\centering\n \\includegraphics[width=\\linewidth]{sample-franklin}\n \\caption{1907 Franklin Model D roadster. Photograph by Harris \\&\n Ewing, Inc. [Public domain], via Wikimedia\n Commons. (\\url{https:\/\/goo.gl\/VLCRBB}).}\n \\Description{A woman and a girl in white dresses sit in an open car.}\n\\end{figure}\n\nYour figures should contain a caption which describes the figure to\nthe reader.\n\nFigure captions are placed {\\itshape below} the figure.\n\nEvery figure should also have a figure description unless it is purely\ndecorative. These descriptions convey what's in the image to someone\nwho cannot see it. They are also used by search engine crawlers for\nindexing images, and when images cannot be loaded.\n\nA figure description must be unformatted plain text less than 2000\ncharacters long (including spaces). {\\bfseries Figure descriptions\n should not repeat the figure caption \u2013 their purpose is to capture\n important information that is not already provided in the caption or\n the main text of the paper.} For figures that convey important and\ncomplex new information, a short text description may not be\nadequate. More complex alternative descriptions can be placed in an\nappendix and referenced in a short figure description. For example,\nprovide a data table capturing the information in a bar chart, or a\nstructured list representing a graph. For additional information\nregarding how best to write figure descriptions and why doing this is\nso important, please see\n\\url{https:\/\/www.acm.org\/publications\/taps\/describing-figures\/}.\n\n\\subsection{The ``Teaser Figure''}\n\nA ``teaser figure'' is an image, or set of images in one figure, that\nare placed after all author and affiliation information, and before\nthe body of the article, spanning the page. If you wish to have such a\nfigure in your article, place the command immediately before the\n\\verb|\\maketitle| command:\n\\begin{verbatim}\n \\begin{teaserfigure}\n \\includegraphics[width=\\textwidth]{sampleteaser}\n \\caption{figure caption}\n \\Description{figure description}\n \\end{teaserfigure}\n\\end{verbatim}\n\n\\section{Citations and Bibliographies}\n\nThe use of \\BibTeX\\ for the preparation and formatting of one's\nreferences is strongly recommended. Authors' names should be complete\n--- use full first names (``Donald E. Knuth'') not initials\n(``D. E. Knuth'') --- and the salient identifying features of a\nreference should be included: title, year, volume, number, pages,\narticle DOI, etc.\n\nThe bibliography is included in your source document with these two\ncommands, placed just before the \\verb|\\end{document}| command:\n\\begin{verbatim}\n \\bibliographystyle{ACM-Reference-Format}\n \n\\section{Introduction}\nACM's consolidated article template, introduced in 2017, provides a\nconsistent \\LaTeX\\ style for use across ACM publications, and\nincorporates accessibility and metadata-extraction functionality\nnecessary for future Digital Library endeavors. Numerous ACM and\nSIG-specific \\LaTeX\\ templates have been examined, and their unique\nfeatures incorporated into this single new template.\n\nIf you are new to publishing with ACM, this document is a valuable\nguide to the process of preparing your work for publication. If you\nhave published with ACM before, this document provides insight and\ninstruction into more recent changes to the article template.\n\nThe ``\\verb|acmart|'' document class can be used to prepare articles\nfor any ACM publication --- conference or journal, and for any stage\nof publication, from review to final ``camera-ready'' copy, to the\nauthor's own version, with {\\itshape very} few changes to the source.\n\n\\section{Template Overview}\nAs noted in the introduction, the ``\\verb|acmart|'' document class can\nbe used to prepare many different kinds of documentation --- a\ndouble-blind initial submission of a full-length technical paper, a\ntwo-page SIGGRAPH Emerging Technologies abstract, a ``camera-ready''\njournal article, a SIGCHI Extended Abstract, and more --- all by\nselecting the appropriate {\\itshape template style} and {\\itshape\n template parameters}.\n\nThis document will explain the major features of the document\nclass. For further information, the {\\itshape \\LaTeX\\ User's Guide} is\navailable from\n\\url{https:\/\/www.acm.org\/publications\/proceedings-template}.\n\n\\subsection{Template Styles}\n\nThe primary parameter given to the ``\\verb|acmart|'' document class is\nthe {\\itshape template style} which corresponds to the kind of publication\nor SIG publishing the work. This parameter is enclosed in square\nbrackets and is a part of the {\\verb|documentclass|} command:\n\\begin{verbatim}\n \\documentclass[STYLE]{acmart}\n\\end{verbatim}\n\nJournals use one of three template styles. All but three ACM journals\nuse the {\\verb|acmsmall|} template style:\n\\begin{itemize}\n\\item {\\texttt{acmsmall}}: The default journal template style.\n\\item {\\texttt{acmlarge}}: Used by JOCCH and TAP.\n\\item {\\texttt{acmtog}}: Used by TOG.\n\\end{itemize}\n\nThe majority of conference proceedings documentation will use the {\\verb|acmconf|} template style.\n\\begin{itemize}\n\\item {\\texttt{acmconf}}: The default proceedings template style.\n\\item{\\texttt{sigchi}}: Used for SIGCHI conference articles.\n\\item{\\texttt{sigchi-a}}: Used for SIGCHI ``Extended Abstract'' articles.\n\\item{\\texttt{sigplan}}: Used for SIGPLAN conference articles.\n\\end{itemize}\n\n\\subsection{Template Parameters}\n\nIn addition to specifying the {\\itshape template style} to be used in\nformatting your work, there are a number of {\\itshape template parameters}\nwhich modify some part of the applied template style. A complete list\nof these parameters can be found in the {\\itshape \\LaTeX\\ User's Guide.}\n\nFrequently-used parameters, or combinations of parameters, include:\n\\begin{itemize}\n\\item {\\texttt{anonymous,review}}: Suitable for a ``double-blind''\n conference submission. Anonymizes the work and includes line\n numbers. Use with the \\texttt{\\acmSubmissionID} command to print the\n submission's unique ID on each page of the work.\n\\item{\\texttt{authorversion}}: Produces a version of the work suitable\n for posting by the author.\n\\item{\\texttt{screen}}: Produces colored hyperlinks.\n\\end{itemize}\n\nThis document uses the following string as the first command in the\nsource file:\n\\begin{verbatim}\n\\documentclass[acmlarge]{acmart}\n\\end{verbatim}\n\n\\section{Modifications}\n\nModifying the template --- including but not limited to: adjusting\nmargins, typeface sizes, line spacing, paragraph and list definitions,\nand the use of the \\verb|\\vspace| command to manually adjust the\nvertical spacing between elements of your work --- is not allowed.\n\n{\\bfseries Your document will be returned to you for revision if\n modifications are discovered.}\n\n\\section{Typefaces}\n\nThe ``\\verb|acmart|'' document class requires the use of the\n``Libertine'' typeface family. Your \\TeX\\ installation should include\nthis set of packages. Please do not substitute other typefaces. The\n``\\verb|lmodern|'' and ``\\verb|ltimes|'' packages should not be used,\nas they will override the built-in typeface families.\n\n\\section{Title Information}\n\nThe title of your work should use capital letters appropriately -\n\\url{https:\/\/capitalizemytitle.com\/} has useful rules for\ncapitalization. Use the {\\verb|title|} command to define the title of\nyour work. If your work has a subtitle, define it with the\n{\\verb|subtitle|} command. Do not insert line breaks in your title.\n\nIf your title is lengthy, you must define a short version to be used\nin the page headers, to prevent overlapping text. The \\verb|title|\ncommand has a ``short title'' parameter:\n\\begin{verbatim}\n \\title[short title]{full title}\n\\end{verbatim}\n\n\\section{Authors and Affiliations}\n\nEach author must be defined separately for accurate metadata\nidentification. As an exception, multiple authors may share one\naffiliation. Authors' names should not be abbreviated; use full first\nnames wherever possible. Include authors' e-mail addresses whenever\npossible.\n\nGrouping authors' names or e-mail addresses, or providing an ``e-mail\nalias,'' as shown below, is not acceptable:\n\\begin{verbatim}\n \\author{Brooke Aster, David Mehldau}\n \\email{dave,judy,steve@university.edu}\n \\email{firstname.lastname@phillips.org}\n\\end{verbatim}\n\nThe \\verb|authornote| and \\verb|authornotemark| commands allow a note\nto apply to multiple authors --- for example, if the first two authors\nof an article contributed equally to the work.\n\nIf your author list is lengthy, you must define a shortened version of\nthe list of authors to be used in the page headers, to prevent\noverlapping text. The following command should be placed just after\nthe last \\verb|\\author{}| definition:\n\\begin{verbatim}\n \\renewcommand{\\shortauthors}{McCartney, et al.}\n\\end{verbatim}\nOmitting this command will force the use of a concatenated list of all\nof the authors' names, which may result in overlapping text in the\npage headers.\n\nThe article template's documentation, available at\n\\url{https:\/\/www.acm.org\/publications\/proceedings-template}, has a\ncomplete explanation of these commands and tips for their effective\nuse.\n\nNote that authors' addresses are mandatory for journal articles.\n\n\\section{Rights Information}\n\nAuthors of any work published by ACM will need to complete a rights\nform. Depending on the kind of work, and the rights management choice\nmade by the author, this may be copyright transfer, permission,\nlicense, or an OA (open access) agreement.\n\nRegardless of the rights management choice, the author will receive a\ncopy of the completed rights form once it has been submitted. This\nform contains \\LaTeX\\ commands that must be copied into the source\ndocument. When the document source is compiled, these commands and\ntheir parameters add formatted text to several areas of the final\ndocument:\n\\begin{itemize}\n\\item the ``ACM Reference Format'' text on the first page.\n\\item the ``rights management'' text on the first page.\n\\item the conference information in the page header(s).\n\\end{itemize}\n\nRights information is unique to the work; if you are preparing several\nworks for an event, make sure to use the correct set of commands with\neach of the works.\n\nThe ACM Reference Format text is required for all articles over one\npage in length, and is optional for one-page articles (abstracts).\n\n\\section{CCS Concepts and User-Defined Keywords}\n\nTwo elements of the ``acmart'' document class provide powerful\ntaxonomic tools for you to help readers find your work in an online\nsearch.\n\nThe ACM Computing Classification System ---\n\\url{https:\/\/www.acm.org\/publications\/class-2012} --- is a set of\nclassifiers and concepts that describe the computing\ndiscipline. Authors can select entries from this classification\nsystem, via \\url{https:\/\/dl.acm.org\/ccs\/ccs.cfm}, and generate the\ncommands to be included in the \\LaTeX\\ source.\n\nUser-defined keywords are a comma-separated list of words and phrases\nof the authors' choosing, providing a more flexible way of describing\nthe research being presented.\n\nCCS concepts and user-defined keywords are required for for all\narticles over two pages in length, and are optional for one- and\ntwo-page articles (or abstracts).\n\n\\section{Sectioning Commands}\n\nYour work should use standard \\LaTeX\\ sectioning commands:\n\\verb|section|, \\verb|subsection|, \\verb|subsubsection|, and\n\\verb|paragraph|. They should be numbered; do not remove the numbering\nfrom the commands.\n\nSimulating a sectioning command by setting the first word or words of\na paragraph in boldface or italicized text is {\\bfseries not allowed.}\n\n\\section{Tables}\n\nThe ``\\verb|acmart|'' document class includes the ``\\verb|booktabs|''\npackage --- \\url{https:\/\/ctan.org\/pkg\/booktabs} --- for preparing\nhigh-quality tables.\n\nTable captions are placed {\\itshape above} the table.\n\nBecause tables cannot be split across pages, the best placement for\nthem is typically the top of the page nearest their initial cite. To\nensure this proper ``floating'' placement of tables, use the\nenvironment \\textbf{table} to enclose the table's contents and the\ntable caption. The contents of the table itself must go in the\n\\textbf{tabular} environment, to be aligned properly in rows and\ncolumns, with the desired horizontal and vertical rules. Again,\ndetailed instructions on \\textbf{tabular} material are found in the\n\\textit{\\LaTeX\\ User's Guide}.\n\nImmediately following this sentence is the point at which\nTable~\\ref{tab:freq} is included in the input file; compare the\nplacement of the table here with the table in the printed output of\nthis document.\n\n\\begin{table}\n \\caption{Frequency of Special Characters}\n \\label{tab:freq}\n \\begin{tabular}{ccl}\n \\toprule\n Non-English or Math&Frequency&Comments\\\\\n \\midrule\n \\O & 1 in 1,000& For Swedish names\\\\\n $\\pi$ & 1 in 5& Common in math\\\\\n \\$ & 4 in 5 & Used in business\\\\\n $\\Psi^2_1$ & 1 in 40,000& Unexplained usage\\\\\n \\bottomrule\n\\end{tabular}\n\\end{table}\n\nTo set a wider table, which takes up the whole width of the page's\nlive area, use the environment \\textbf{table*} to enclose the table's\ncontents and the table caption. As with a single-column table, this\nwide table will ``float'' to a location deemed more\ndesirable. Immediately following this sentence is the point at which\nTable~\\ref{tab:commands} is included in the input file; again, it is\ninstructive to compare the placement of the table here with the table\nin the printed output of this document.\n\n\\begin{table*}\n \\caption{Some Typical Commands}\n \\label{tab:commands}\n \\begin{tabular}{ccl}\n \\toprule\n Command &A Number & Comments\\\\\n \\midrule\n \\texttt{{\\char'134}author} & 100& Author \\\\\n \\texttt{{\\char'134}table}& 300 & For tables\\\\\n \\texttt{{\\char'134}table*}& 400& For wider tables\\\\\n \\bottomrule\n \\end{tabular}\n\\end{table*}\n\nAlways use midrule to separate table header rows from data rows, and\nuse it only for this purpose. This enables assistive technologies to\nrecognise table headers and support their users in navigating tables\nmore easily.\n\n\\section{Math Equations}\nYou may want to display math equations in three distinct styles:\ninline, numbered or non-numbered display. Each of the three are\ndiscussed in the next sections.\n\n\\subsection{Inline (In-text) Equations}\nA formula that appears in the running text is called an inline or\nin-text formula. It is produced by the \\textbf{math} environment,\nwhich can be invoked with the usual\n\\texttt{{\\char'134}begin\\,\\ldots{\\char'134}end} construction or with\nthe short form \\texttt{\\$\\,\\ldots\\$}. You can use any of the symbols\nand structures, from $\\alpha$ to $\\omega$, available in\n\\LaTeX~\\cite{Lamport:LaTeX}; this section will simply show a few\nexamples of in-text equations in context. Notice how this equation:\n\\begin{math}\n \\lim_{n\\rightarrow \\infty}x=0\n\\end{math},\nset here in in-line math style, looks slightly different when\nset in display style. (See next section).\n\n\\subsection{Display Equations}\nA numbered display equation---one set off by vertical space from the\ntext and centered horizontally---is produced by the \\textbf{equation}\nenvironment. An unnumbered display equation is produced by the\n\\textbf{displaymath} environment.\n\nAgain, in either environment, you can use any of the symbols and\nstructures available in \\LaTeX\\@; this section will just give a couple\nof examples of display equations in context. First, consider the\nequation, shown as an inline equation above:\n\\begin{equation}\n \\lim_{n\\rightarrow \\infty}x=0\n\\end{equation}\nNotice how it is formatted somewhat differently in\nthe \\textbf{displaymath}\nenvironment. Now, we'll enter an unnumbered equation:\n\\begin{displaymath}\n \\sum_{i=0}^{\\infty} x + 1\n\\end{displaymath}\nand follow it with another numbered equation:\n\\begin{equation}\n \\sum_{i=0}^{\\infty}x_i=\\int_{0}^{\\pi+2} f\n\\end{equation}\njust to demonstrate \\LaTeX's able handling of numbering.\n\n\\section{Figures}\n\nThe ``\\verb|figure|'' environment should be used for figures. One or\nmore images can be placed within a figure. If your figure contains\nthird-party material, you must clearly identify it as such, as shown\nin the example below.\n\\begin{figure}[h]\n \\centering\n \\includegraphics[width=\\linewidth]{sample-franklin}\n \\caption{1907 Franklin Model D roadster. Photograph by Harris \\&\n Ewing, Inc. [Public domain], via Wikimedia\n Commons. (\\url{https:\/\/goo.gl\/VLCRBB}).}\n \\Description{A woman and a girl in white dresses sit in an open car.}\n\\end{figure}\n\nYour figures should contain a caption which describes the figure to\nthe reader.\n\nFigure captions are placed {\\itshape below} the figure.\n\nEvery figure should also have a figure description unless it is purely\ndecorative. These descriptions convey what's in the image to someone\nwho cannot see it. They are also used by search engine crawlers for\nindexing images, and when images cannot be loaded.\n\nA figure description must be unformatted plain text less than 2000\ncharacters long (including spaces). {\\bfseries Figure descriptions\n should not repeat the figure caption \u2013 their purpose is to capture\n important information that is not already provided in the caption or\n the main text of the paper.} For figures that convey important and\ncomplex new information, a short text description may not be\nadequate. More complex alternative descriptions can be placed in an\nappendix and referenced in a short figure description. For example,\nprovide a data table capturing the information in a bar chart, or a\nstructured list representing a graph. For additional information\nregarding how best to write figure descriptions and why doing this is\nso important, please see\n\\url{https:\/\/www.acm.org\/publications\/taps\/describing-figures\/}.\n\n\\subsection{The ``Teaser Figure''}\n\nA ``teaser figure'' is an image, or set of images in one figure, that\nare placed after all author and affiliation information, and before\nthe body of the article, spanning the page. If you wish to have such a\nfigure in your article, place the command immediately before the\n\\verb|\\maketitle| command:\n\\begin{verbatim}\n \\begin{teaserfigure}\n \\includegraphics[width=\\textwidth]{sampleteaser}\n \\caption{figure caption}\n \\Description{figure description}\n \\end{teaserfigure}\n\\end{verbatim}\n\n\\section{Citations and Bibliographies}\n\nThe use of \\BibTeX\\ for the preparation and formatting of one's\nreferences is strongly recommended. Authors' names should be complete\n--- use full first names (``Donald E. Knuth'') not initials\n(``D. E. Knuth'') --- and the salient identifying features of a\nreference should be included: title, year, volume, number, pages,\narticle DOI, etc.\n\nThe bibliography is included in your source document with these two\ncommands, placed just before the \\verb|\\end{document}| command:\n\\begin{verbatim}\n \\bibliographystyle{ACM-Reference-Format}\n \n\\section{Introduction}\nACM's consolidated article template, introduced in 2017, provides a\nconsistent \\LaTeX\\ style for use across ACM publications, and\nincorporates accessibility and metadata-extraction functionality\nnecessary for future Digital Library endeavors. Numerous ACM and\nSIG-specific \\LaTeX\\ templates have been examined, and their unique\nfeatures incorporated into this single new template.\n\nIf you are new to publishing with ACM, this document is a valuable\nguide to the process of preparing your work for publication. If you\nhave published with ACM before, this document provides insight and\ninstruction into more recent changes to the article template.\n\nThe ``\\verb|acmart|'' document class can be used to prepare articles\nfor any ACM publication --- conference or journal, and for any stage\nof publication, from review to final ``camera-ready'' copy, to the\nauthor's own version, with {\\itshape very} few changes to the source.\n\n\\section{Template Overview}\nAs noted in the introduction, the ``\\verb|acmart|'' document class can\nbe used to prepare many different kinds of documentation --- a\ndouble-blind initial submission of a full-length technical paper, a\ntwo-page SIGGRAPH Emerging Technologies abstract, a ``camera-ready''\njournal article, a SIGCHI Extended Abstract, and more --- all by\nselecting the appropriate {\\itshape template style} and {\\itshape\n template parameters}.\n\nThis document will explain the major features of the document\nclass. For further information, the {\\itshape \\LaTeX\\ User's Guide} is\navailable from\n\\url{https:\/\/www.acm.org\/publications\/proceedings-template}.\n\n\\subsection{Template Styles}\n\nThe primary parameter given to the ``\\verb|acmart|'' document class is\nthe {\\itshape template style} which corresponds to the kind of publication\nor SIG publishing the work. This parameter is enclosed in square\nbrackets and is a part of the {\\verb|documentclass|} command:\n\\begin{verbatim}\n \\documentclass[STYLE]{acmart}\n\\end{verbatim}\n\nJournals use one of three template styles. All but three ACM journals\nuse the {\\verb|acmsmall|} template style:\n\\begin{itemize}\n\\item {\\texttt{acmsmall}}: The default journal template style.\n\\item {\\texttt{acmlarge}}: Used by JOCCH and TAP.\n\\item {\\texttt{acmtog}}: Used by TOG.\n\\end{itemize}\n\nThe majority of conference proceedings documentation will use the {\\verb|acmconf|} template style.\n\\begin{itemize}\n\\item {\\texttt{acmconf}}: The default proceedings template style.\n\\item{\\texttt{sigchi}}: Used for SIGCHI conference articles.\n\\item{\\texttt{sigchi-a}}: Used for SIGCHI ``Extended Abstract'' articles.\n\\item{\\texttt{sigplan}}: Used for SIGPLAN conference articles.\n\\end{itemize}\n\n\\subsection{Template Parameters}\n\nIn addition to specifying the {\\itshape template style} to be used in\nformatting your work, there are a number of {\\itshape template parameters}\nwhich modify some part of the applied template style. A complete list\nof these parameters can be found in the {\\itshape \\LaTeX\\ User's Guide.}\n\nFrequently-used parameters, or combinations of parameters, include:\n\\begin{itemize}\n\\item {\\texttt{anonymous,review}}: Suitable for a ``double-blind''\n conference submission. Anonymizes the work and includes line\n numbers. Use with the \\texttt{\\acmSubmissionID} command to print the\n submission's unique ID on each page of the work.\n\\item{\\texttt{authorversion}}: Produces a version of the work suitable\n for posting by the author.\n\\item{\\texttt{screen}}: Produces colored hyperlinks.\n\\end{itemize}\n\nThis document uses the following string as the first command in the\nsource file:\n\\begin{verbatim}\n\\end{verbatim}\n\n\\section{Modifications}\n\nModifying the template --- including but not limited to: adjusting\nmargins, typeface sizes, line spacing, paragraph and list definitions,\nand the use of the \\verb|\\vspace| command to manually adjust the\nvertical spacing between elements of your work --- is not allowed.\n\n{\\bfseries Your document will be returned to you for revision if\n modifications are discovered.}\n\n\\section{Typefaces}\n\nThe ``\\verb|acmart|'' document class requires the use of the\n``Libertine'' typeface family. Your \\TeX\\ installation should include\nthis set of packages. Please do not substitute other typefaces. The\n``\\verb|lmodern|'' and ``\\verb|ltimes|'' packages should not be used,\nas they will override the built-in typeface families.\n\n\\section{Title Information}\n\nThe title of your work should use capital letters appropriately -\n\\url{https:\/\/capitalizemytitle.com\/} has useful rules for\ncapitalization. Use the {\\verb|title|} command to define the title of\nyour work. If your work has a subtitle, define it with the\n{\\verb|subtitle|} command. Do not insert line breaks in your title.\n\nIf your title is lengthy, you must define a short version to be used\nin the page headers, to prevent overlapping text. The \\verb|title|\ncommand has a ``short title'' parameter:\n\\begin{verbatim}\n \\title[short title]{full title}\n\\end{verbatim}\n\n\\section{Authors and Affiliations}\n\nEach author must be defined separately for accurate metadata\nidentification. As an exception, multiple authors may share one\naffiliation. Authors' names should not be abbreviated; use full first\nnames wherever possible. Include authors' e-mail addresses whenever\npossible.\n\nGrouping authors' names or e-mail addresses, or providing an ``e-mail\nalias,'' as shown below, is not acceptable:\n\\begin{verbatim}\n \\author{Brooke Aster, David Mehldau}\n \\email{dave,judy,steve@university.edu}\n \\email{firstname.lastname@phillips.org}\n\\end{verbatim}\n\nThe \\verb|authornote| and \\verb|authornotemark| commands allow a note\nto apply to multiple authors --- for example, if the first two authors\nof an article contributed equally to the work.\n\nIf your author list is lengthy, you must define a shortened version of\nthe list of authors to be used in the page headers, to prevent\noverlapping text. The following command should be placed just after\nthe last \\verb|\\author{}| definition:\n\\begin{verbatim}\n \\renewcommand{\\shortauthors}{McCartney, et al.}\n\\end{verbatim}\nOmitting this command will force the use of a concatenated list of all\nof the authors' names, which may result in overlapping text in the\npage headers.\n\nThe article template's documentation, available at\n\\url{https:\/\/www.acm.org\/publications\/proceedings-template}, has a\ncomplete explanation of these commands and tips for their effective\nuse.\n\nNote that authors' addresses are mandatory for journal articles.\n\n\\section{Rights Information}\n\nAuthors of any work published by ACM will need to complete a rights\nform. Depending on the kind of work, and the rights management choice\nmade by the author, this may be copyright transfer, permission,\nlicense, or an OA (open access) agreement.\n\nRegardless of the rights management choice, the author will receive a\ncopy of the completed rights form once it has been submitted. This\nform contains \\LaTeX\\ commands that must be copied into the source\ndocument. When the document source is compiled, these commands and\ntheir parameters add formatted text to several areas of the final\ndocument:\n\\begin{itemize}\n\\item the ``ACM Reference Format'' text on the first page.\n\\item the ``rights management'' text on the first page.\n\\item the conference information in the page header(s).\n\\end{itemize}\n\nRights information is unique to the work; if you are preparing several\nworks for an event, make sure to use the correct set of commands with\neach of the works.\n\nThe ACM Reference Format text is required for all articles over one\npage in length, and is optional for one-page articles (abstracts).\n\n\\section{CCS Concepts and User-Defined Keywords}\n\nTwo elements of the ``acmart'' document class provide powerful\ntaxonomic tools for you to help readers find your work in an online\nsearch.\n\nThe ACM Computing Classification System ---\n\\url{https:\/\/www.acm.org\/publications\/class-2012} --- is a set of\nclassifiers and concepts that describe the computing\ndiscipline. Authors can select entries from this classification\nsystem, via \\url{https:\/\/dl.acm.org\/ccs\/ccs.cfm}, and generate the\ncommands to be included in the \\LaTeX\\ source.\n\nUser-defined keywords are a comma-separated list of words and phrases\nof the authors' choosing, providing a more flexible way of describing\nthe research being presented.\n\nCCS concepts and user-defined keywords are required for for all\narticles over two pages in length, and are optional for one- and\ntwo-page articles (or abstracts).\n\n\\section{Sectioning Commands}\n\nYour work should use standard \\LaTeX\\ sectioning commands:\n\\verb|section|, \\verb|subsection|, \\verb|subsubsection|, and\n\\verb|paragraph|. They should be numbered; do not remove the numbering\nfrom the commands.\n\nSimulating a sectioning command by setting the first word or words of\na paragraph in boldface or italicized text is {\\bfseries not allowed.}\n\n\\section{Tables}\n\nThe ``\\verb|acmart|'' document class includes the ``\\verb|booktabs|''\npackage --- \\url{https:\/\/ctan.org\/pkg\/booktabs} --- for preparing\nhigh-quality tables.\n\nTable captions are placed {\\itshape above} the table.\n\nBecause tables cannot be split across pages, the best placement for\nthem is typically the top of the page nearest their initial cite. To\nensure this proper ``floating'' placement of tables, use the\nenvironment \\textbf{table} to enclose the table's contents and the\ntable caption. The contents of the table itself must go in the\n\\textbf{tabular} environment, to be aligned properly in rows and\ncolumns, with the desired horizontal and vertical rules. Again,\ndetailed instructions on \\textbf{tabular} material are found in the\n\\textit{\\LaTeX\\ User's Guide}.\n\nImmediately following this sentence is the point at which\nTable~\\ref{tab:freq} is included in the input file; compare the\nplacement of the table here with the table in the printed output of\nthis document.\n\n\\begin{table}\n \\caption{Frequency of Special Characters}\n \\label{tab:freq}\n \\begin{tabular}{ccl}\n \\toprule\n Non-English or Math&Frequency&Comments\\\\\n \\midrule\n \\O & 1 in 1,000& For Swedish names\\\\\n $\\pi$ & 1 in 5& Common in math\\\\\n \\$ & 4 in 5 & Used in business\\\\\n $\\Psi^2_1$ & 1 in 40,000& Unexplained usage\\\\\n \\bottomrule\n\\end{tabular}\n\\end{table}\n\nTo set a wider table, which takes up the whole width of the page's\nlive area, use the environment \\textbf{table*} to enclose the table's\ncontents and the table caption. As with a single-column table, this\nwide table will ``float'' to a location deemed more\ndesirable. Immediately following this sentence is the point at which\nTable~\\ref{tab:commands} is included in the input file; again, it is\ninstructive to compare the placement of the table here with the table\nin the printed output of this document.\n\n\\begin{table*}\n \\caption{Some Typical Commands}\n \\label{tab:commands}\n \\begin{tabular}{ccl}\n \\toprule\n Command &A Number & Comments\\\\\n \\midrule\n \\texttt{{\\char'134}author} & 100& Author \\\\\n \\texttt{{\\char'134}table}& 300 & For tables\\\\\n \\texttt{{\\char'134}table*}& 400& For wider tables\\\\\n \\bottomrule\n \\end{tabular}\n\\end{table*}\n\nAlways use midrule to separate table header rows from data rows, and\nuse it only for this purpose. This enables assistive technologies to\nrecognise table headers and support their users in navigating tables\nmore easily.\n\n\\section{Math Equations}\nYou may want to display math equations in three distinct styles:\ninline, numbered or non-numbered display. Each of the three are\ndiscussed in the next sections.\n\n\\subsection{Inline (In-text) Equations}\nA formula that appears in the running text is called an inline or\nin-text formula. It is produced by the \\textbf{math} environment,\nwhich can be invoked with the usual\n\\texttt{{\\char'134}begin\\,\\ldots{\\char'134}end} construction or with\nthe short form \\texttt{\\$\\,\\ldots\\$}. You can use any of the symbols\nand structures, from $\\alpha$ to $\\omega$, available in\n\\LaTeX~\\cite{Lamport:LaTeX}; this section will simply show a few\nexamples of in-text equations in context. Notice how this equation:\n\\begin{math}\n \\lim_{n\\rightarrow \\infty}x=0\n\\end{math},\nset here in in-line math style, looks slightly different when\nset in display style. (See next section).\n\n\\subsection{Display Equations}\nA numbered display equation---one set off by vertical space from the\ntext and centered horizontally---is produced by the \\textbf{equation}\nenvironment. An unnumbered display equation is produced by the\n\\textbf{displaymath} environment.\n\nAgain, in either environment, you can use any of the symbols and\nstructures available in \\LaTeX\\@; this section will just give a couple\nof examples of display equations in context. First, consider the\nequation, shown as an inline equation above:\n\\begin{equation}\n \\lim_{n\\rightarrow \\infty}x=0\n\\end{equation}\nNotice how it is formatted somewhat differently in\nthe \\textbf{displaymath}\nenvironment. Now, we'll enter an unnumbered equation:\n\\begin{displaymath}\n \\sum_{i=0}^{\\infty} x + 1\n\\end{displaymath}\nand follow it with another numbered equation:\n\\begin{equation}\n \\sum_{i=0}^{\\infty}x_i=\\int_{0}^{\\pi+2} f\n\\end{equation}\njust to demonstrate \\LaTeX's able handling of numbering.\n\n\\section{Figures}\n\nThe ``\\verb|figure|'' environment should be used for figures. One or\nmore images can be placed within a figure. If your figure contains\nthird-party material, you must clearly identify it as such, as shown\nin the example below.\n\\begin{figure}[h]\n \\centering\n \\includegraphics[width=\\linewidth]{sample-franklin}\n \\caption{1907 Franklin Model D roadster. Photograph by Harris \\&\n Ewing, Inc. [Public domain], via Wikimedia\n Commons. (\\url{https:\/\/goo.gl\/VLCRBB}).}\n \\Description{A woman and a girl in white dresses sit in an open car.}\n\\end{figure}\n\nYour figures should contain a caption which describes the figure to\nthe reader.\n\nFigure captions are placed {\\itshape below} the figure.\n\nEvery figure should also have a figure description unless it is purely\ndecorative. These descriptions convey what's in the image to someone\nwho cannot see it. They are also used by search engine crawlers for\nindexing images, and when images cannot be loaded.\n\nA figure description must be unformatted plain text less than 2000\ncharacters long (including spaces). {\\bfseries Figure descriptions\n should not repeat the figure caption \u2013 their purpose is to capture\n important information that is not already provided in the caption or\n the main text of the paper.} For figures that convey important and\ncomplex new information, a short text description may not be\nadequate. More complex alternative descriptions can be placed in an\nappendix and referenced in a short figure description. For example,\nprovide a data table capturing the information in a bar chart, or a\nstructured list representing a graph. For additional information\nregarding how best to write figure descriptions and why doing this is\nso important, please see\n\\url{https:\/\/www.acm.org\/publications\/taps\/describing-figures\/}.\n\n\\subsection{The ``Teaser Figure''}\n\nA ``teaser figure'' is an image, or set of images in one figure, that\nare placed after all author and affiliation information, and before\nthe body of the article, spanning the page. If you wish to have such a\nfigure in your article, place the command immediately before the\n\\verb|\\maketitle| command:\n\\begin{verbatim}\n \\begin{teaserfigure}\n \\includegraphics[width=\\textwidth]{sampleteaser}\n \\caption{figure caption}\n \\Description{figure description}\n \\end{teaserfigure}\n\\end{verbatim}\n\n\\section{Citations and Bibliographies}\n\nThe use of \\BibTeX\\ for the preparation and formatting of one's\nreferences is strongly recommended. Authors' names should be complete\n--- use full first names (``Donald E. Knuth'') not initials\n(``D. E. Knuth'') --- and the salient identifying features of a\nreference should be included: title, year, volume, number, pages,\narticle DOI, etc.\n\n\nUsing the BibLaTeX system, the bibliography is included in your source\ndocument with the following command, placed just before the \\verb|\\end{document}| command:\n\\begin{verbatim}\n \\printbibliography\n\\end{verbatim}\n\nThe command \\verb|\\addbibresource{bibfile}| declares the \\BibTeX\\ file to use\nin the {\\bfseries preamble} (before the command\n``\\verb|\\begin{document}|'') of your \\LaTeX\\ source\nwhere ``\\verb|bibfile|'' is the name, \\emph{with} the ``\\verb|.bib|'' suffix.\nNotice that \\verb|\\addbibresource| takes only one argument: to declare multiple files,\nuse multiple instances of the command.\n\nCitations and references are numbered by default. A small number of\nACM publications have citations and references formatted in the\n``author year'' style; for these exceptions, please pass the option \\verb|style=acmauthoryear|\nto the \\verb|biblatex| package loaded in the {\\bfseries preamble} (before the command\n``\\verb|\\begin{document}|'') of your \\LaTeX\\ source.\n\n\n Some examples. A paginated journal article \\cite{Abril07}, an\n enumerated journal article \\cite{Cohen07}, a reference to an entire\n issue \\cite{JCohen96}, a monograph (whole book) \\cite{Kosiur01}, a\n monograph\/whole book in a series (see 2a in spec. document)\n \\cite{Harel79}, a divisible-book such as an anthology or compilation\n \\cite{Editor00} followed by the same example, however we only output\n the series if the volume number is given \\cite{Editor00a} (so\n Editor00a's series should NOT be present since it has no vol. no.),\n a chapter in a divisible book \\cite{Spector90}, a chapter in a\n divisible book in a series \\cite{Douglass98}, a multi-volume work as\n book \\cite{Knuth97}, a couple of articles in a proceedings (of a\n conference, symposium, workshop for example) (paginated proceedings\n article) \\cite{Andler79, Hagerup1993}, a proceedings article with\n all possible elements \\cite{Smith10}, an example of an enumerated\n proceedings article \\cite{VanGundy07}, an informally published work\n \\cite{Harel78}, a couple of preprints \\cite{Bornmann2019,\n AnzarootPBM14}, a doctoral dissertation \\cite{Clarkson85}, a\n master's thesis: \\cite{anisi03}, an online document \/ world wide web\n resource \\cite{Thornburg01, Ablamowicz07, Poker06}, a video game\n (Case 1) \\cite{Obama08} and (Case 2) \\cite{Novak03} and \\cite{Lee05}\n and (Case 3) a patent \\cite{JoeScientist001}, work accepted for\n publication \\cite{rous08}, 'YYYYb'-test for prolific author\n \\cite{SaeediMEJ10} and \\cite{SaeediJETC10}. Other cites might\n contain 'duplicate' DOI and URLs (some SIAM articles)\n \\cite{Kirschmer:2010:AEI:1958016.1958018}. Boris \/ Barbara Beeton:\n multi-volume works as books \\cite{MR781536} and \\cite{MR781537}. A\n couple of citations with DOIs:\n \\cite{2004:ITE:1009386.1010128,Kirschmer:2010:AEI:1958016.1958018}. Online\n citations: \\cite{TUGInstmem, Thornburg01, CTANacmart}.\n Data Artifacts: \\cite{UMassCitations}.\n Software project: ~\\cite{cgal,delebecque:hal-02090402}. Software Version: ~\\cite{gf-tag-sound-repo,}. Software Module: ~\\cite{cgal:lp-gi-20a}. Code fragment: ~\\cite{simplemapper}.\n\n\\section{Acknowledgments}\n\nIdentification of funding sources and other support, and thanks to\nindividuals and groups that assisted in the research and the\npreparation of the work should be included in an acknowledgment\nsection, which is placed just before the reference section in your\ndocument.\n\nThis section has a special environment:\n\\begin{verbatim}\n \\begin{acks}\n ...\n \\end{acks}\n\\end{verbatim}\nso that the information contained therein can be more easily collected\nduring the article metadata extraction phase, and to ensure\nconsistency in the spelling of the section heading.\n\nAuthors should not prepare this section as a numbered or unnumbered {\\verb|\\section|}; please use the ``{\\verb|acks|}'' environment.\n\n\\section{Appendices}\n\nIf your work needs an appendix, add it before the\n``\\verb|\\end{document}|'' command at the conclusion of your source\ndocument.\n\nStart the appendix with the ``\\verb|appendix|'' command:\n\\begin{verbatim}\n \n\\section{Introduction}\nACM's consolidated article template, introduced in 2017, provides a\nconsistent \\LaTeX\\ style for use across ACM publications, and\nincorporates accessibility and metadata-extraction functionality\nnecessary for future Digital Library endeavors. Numerous ACM and\nSIG-specific \\LaTeX\\ templates have been examined, and their unique\nfeatures incorporated into this single new template.\n\nIf you are new to publishing with ACM, this document is a valuable\nguide to the process of preparing your work for publication. If you\nhave published with ACM before, this document provides insight and\ninstruction into more recent changes to the article template.\n\nThe ``\\verb|acmart|'' document class can be used to prepare articles\nfor any ACM publication --- conference or journal, and for any stage\nof publication, from review to final ``camera-ready'' copy, to the\nauthor's own version, with {\\itshape very} few changes to the source.\n\n\\section{Template Overview}\nAs noted in the introduction, the ``\\verb|acmart|'' document class can\nbe used to prepare many different kinds of documentation --- a\ndouble-blind initial submission of a full-length technical paper, a\ntwo-page SIGGRAPH Emerging Technologies abstract, a ``camera-ready''\njournal article, a SIGCHI Extended Abstract, and more --- all by\nselecting the appropriate {\\itshape template style} and {\\itshape\n template parameters}.\n\nThis document will explain the major features of the document\nclass. For further information, the {\\itshape \\LaTeX\\ User's Guide} is\navailable from\n\\url{https:\/\/www.acm.org\/publications\/proceedings-template}.\n\n\\subsection{Template Styles}\n\nThe primary parameter given to the ``\\verb|acmart|'' document class is\nthe {\\itshape template style} which corresponds to the kind of publication\nor SIG publishing the work. This parameter is enclosed in square\nbrackets and is a part of the {\\verb|documentclass|} command:\n\\begin{verbatim}\n \\documentclass[STYLE]{acmart}\n\\end{verbatim}\n\nJournals use one of three template styles. All but three ACM journals\nuse the {\\verb|acmsmall|} template style:\n\\begin{itemize}\n\\item {\\texttt{acmsmall}}: The default journal template style.\n\\item {\\texttt{acmlarge}}: Used by JOCCH and TAP.\n\\item {\\texttt{acmtog}}: Used by TOG.\n\\end{itemize}\n\nThe majority of conference proceedings documentation will use the {\\verb|acmconf|} template style.\n\\begin{itemize}\n\\item {\\texttt{acmconf}}: The default proceedings template style.\n\\item{\\texttt{sigchi}}: Used for SIGCHI conference articles.\n\\item{\\texttt{sigchi-a}}: Used for SIGCHI ``Extended Abstract'' articles.\n\\item{\\texttt{sigplan}}: Used for SIGPLAN conference articles.\n\\end{itemize}\n\n\\subsection{Template Parameters}\n\nIn addition to specifying the {\\itshape template style} to be used in\nformatting your work, there are a number of {\\itshape template parameters}\nwhich modify some part of the applied template style. A complete list\nof these parameters can be found in the {\\itshape \\LaTeX\\ User's Guide.}\n\nFrequently-used parameters, or combinations of parameters, include:\n\\begin{itemize}\n\\item {\\texttt{anonymous,review}}: Suitable for a ``double-blind''\n conference submission. Anonymizes the work and includes line\n numbers. Use with the \\texttt{\\acmSubmissionID} command to print the\n submission's unique ID on each page of the work.\n\\item{\\texttt{authorversion}}: Produces a version of the work suitable\n for posting by the author.\n\\item{\\texttt{screen}}: Produces colored hyperlinks.\n\\end{itemize}\n\nThis document uses the following string as the first command in the\nsource file:\n\\begin{verbatim}\n\\documentclass[acmsmall]{acmart}\n\\end{verbatim}\n\n\\section{Modifications}\n\nModifying the template --- including but not limited to: adjusting\nmargins, typeface sizes, line spacing, paragraph and list definitions,\nand the use of the \\verb|\\vspace| command to manually adjust the\nvertical spacing between elements of your work --- is not allowed.\n\n{\\bfseries Your document will be returned to you for revision if\n modifications are discovered.}\n\n\\section{Typefaces}\n\nThe ``\\verb|acmart|'' document class requires the use of the\n``Libertine'' typeface family. Your \\TeX\\ installation should include\nthis set of packages. Please do not substitute other typefaces. The\n``\\verb|lmodern|'' and ``\\verb|ltimes|'' packages should not be used,\nas they will override the built-in typeface families.\n\n\\section{Title Information}\n\nThe title of your work should use capital letters appropriately -\n\\url{https:\/\/capitalizemytitle.com\/} has useful rules for\ncapitalization. Use the {\\verb|title|} command to define the title of\nyour work. If your work has a subtitle, define it with the\n{\\verb|subtitle|} command. Do not insert line breaks in your title.\n\nIf your title is lengthy, you must define a short version to be used\nin the page headers, to prevent overlapping text. The \\verb|title|\ncommand has a ``short title'' parameter:\n\\begin{verbatim}\n \\title[short title]{full title}\n\\end{verbatim}\n\n\\section{Authors and Affiliations}\n\nEach author must be defined separately for accurate metadata\nidentification. As an exception, multiple authors may share one\naffiliation. Authors' names should not be abbreviated; use full first\nnames wherever possible. Include authors' e-mail addresses whenever\npossible.\n\nGrouping authors' names or e-mail addresses, or providing an ``e-mail\nalias,'' as shown below, is not acceptable:\n\\begin{verbatim}\n \\author{Brooke Aster, David Mehldau}\n \\email{dave,judy,steve@university.edu}\n \\email{firstname.lastname@phillips.org}\n\\end{verbatim}\n\nThe \\verb|authornote| and \\verb|authornotemark| commands allow a note\nto apply to multiple authors --- for example, if the first two authors\nof an article contributed equally to the work.\n\nIf your author list is lengthy, you must define a shortened version of\nthe list of authors to be used in the page headers, to prevent\noverlapping text. The following command should be placed just after\nthe last \\verb|\\author{}| definition:\n\\begin{verbatim}\n \\renewcommand{\\shortauthors}{McCartney, et al.}\n\\end{verbatim}\nOmitting this command will force the use of a concatenated list of all\nof the authors' names, which may result in overlapping text in the\npage headers.\n\nThe article template's documentation, available at\n\\url{https:\/\/www.acm.org\/publications\/proceedings-template}, has a\ncomplete explanation of these commands and tips for their effective\nuse.\n\nNote that authors' addresses are mandatory for journal articles.\n\n\\section{Rights Information}\n\nAuthors of any work published by ACM will need to complete a rights\nform. Depending on the kind of work, and the rights management choice\nmade by the author, this may be copyright transfer, permission,\nlicense, or an OA (open access) agreement.\n\nRegardless of the rights management choice, the author will receive a\ncopy of the completed rights form once it has been submitted. This\nform contains \\LaTeX\\ commands that must be copied into the source\ndocument. When the document source is compiled, these commands and\ntheir parameters add formatted text to several areas of the final\ndocument:\n\\begin{itemize}\n\\item the ``ACM Reference Format'' text on the first page.\n\\item the ``rights management'' text on the first page.\n\\item the conference information in the page header(s).\n\\end{itemize}\n\nRights information is unique to the work; if you are preparing several\nworks for an event, make sure to use the correct set of commands with\neach of the works.\n\nThe ACM Reference Format text is required for all articles over one\npage in length, and is optional for one-page articles (abstracts).\n\n\\section{CCS Concepts and User-Defined Keywords}\n\nTwo elements of the ``acmart'' document class provide powerful\ntaxonomic tools for you to help readers find your work in an online\nsearch.\n\nThe ACM Computing Classification System ---\n\\url{https:\/\/www.acm.org\/publications\/class-2012} --- is a set of\nclassifiers and concepts that describe the computing\ndiscipline. Authors can select entries from this classification\nsystem, via \\url{https:\/\/dl.acm.org\/ccs\/ccs.cfm}, and generate the\ncommands to be included in the \\LaTeX\\ source.\n\nUser-defined keywords are a comma-separated list of words and phrases\nof the authors' choosing, providing a more flexible way of describing\nthe research being presented.\n\nCCS concepts and user-defined keywords are required for for all\narticles over two pages in length, and are optional for one- and\ntwo-page articles (or abstracts).\n\n\\section{Sectioning Commands}\n\nYour work should use standard \\LaTeX\\ sectioning commands:\n\\verb|section|, \\verb|subsection|, \\verb|subsubsection|, and\n\\verb|paragraph|. They should be numbered; do not remove the numbering\nfrom the commands.\n\nSimulating a sectioning command by setting the first word or words of\na paragraph in boldface or italicized text is {\\bfseries not allowed.}\n\n\\section{Tables}\n\nThe ``\\verb|acmart|'' document class includes the ``\\verb|booktabs|''\npackage --- \\url{https:\/\/ctan.org\/pkg\/booktabs} --- for preparing\nhigh-quality tables.\n\nTable captions are placed {\\itshape above} the table.\n\nBecause tables cannot be split across pages, the best placement for\nthem is typically the top of the page nearest their initial cite. To\nensure this proper ``floating'' placement of tables, use the\nenvironment \\textbf{table} to enclose the table's contents and the\ntable caption. The contents of the table itself must go in the\n\\textbf{tabular} environment, to be aligned properly in rows and\ncolumns, with the desired horizontal and vertical rules. Again,\ndetailed instructions on \\textbf{tabular} material are found in the\n\\textit{\\LaTeX\\ User's Guide}.\n\nImmediately following this sentence is the point at which\nTable~\\ref{tab:freq} is included in the input file; compare the\nplacement of the table here with the table in the printed output of\nthis document.\n\n\\begin{table}\n \\caption{Frequency of Special Characters}\n \\label{tab:freq}\n \\begin{tabular}{ccl}\n \\toprule\n Non-English or Math&Frequency&Comments\\\\\n \\midrule\n \\O & 1 in 1,000& For Swedish names\\\\\n $\\pi$ & 1 in 5& Common in math\\\\\n \\$ & 4 in 5 & Used in business\\\\\n $\\Psi^2_1$ & 1 in 40,000& Unexplained usage\\\\\n \\bottomrule\n\\end{tabular}\n\\end{table}\n\nTo set a wider table, which takes up the whole width of the page's\nlive area, use the environment \\textbf{table*} to enclose the table's\ncontents and the table caption. As with a single-column table, this\nwide table will ``float'' to a location deemed more\ndesirable. Immediately following this sentence is the point at which\nTable~\\ref{tab:commands} is included in the input file; again, it is\ninstructive to compare the placement of the table here with the table\nin the printed output of this document.\n\n\\begin{table*}\n \\caption{Some Typical Commands}\n \\label{tab:commands}\n \\begin{tabular}{ccl}\n \\toprule\n Command &A Number & Comments\\\\\n \\midrule\n \\texttt{{\\char'134}author} & 100& Author \\\\\n \\texttt{{\\char'134}table}& 300 & For tables\\\\\n \\texttt{{\\char'134}table*}& 400& For wider tables\\\\\n \\bottomrule\n \\end{tabular}\n\\end{table*}\n\nAlways use midrule to separate table header rows from data rows, and\nuse it only for this purpose. This enables assistive technologies to\nrecognise table headers and support their users in navigating tables\nmore easily.\n\n\\section{Math Equations}\nYou may want to display math equations in three distinct styles:\ninline, numbered or non-numbered display. Each of the three are\ndiscussed in the next sections.\n\n\\subsection{Inline (In-text) Equations}\nA formula that appears in the running text is called an inline or\nin-text formula. It is produced by the \\textbf{math} environment,\nwhich can be invoked with the usual\n\\texttt{{\\char'134}begin\\,\\ldots{\\char'134}end} construction or with\nthe short form \\texttt{\\$\\,\\ldots\\$}. You can use any of the symbols\nand structures, from $\\alpha$ to $\\omega$, available in\n\\LaTeX~\\cite{Lamport:LaTeX}; this section will simply show a few\nexamples of in-text equations in context. Notice how this equation:\n\\begin{math}\n \\lim_{n\\rightarrow \\infty}x=0\n\\end{math},\nset here in in-line math style, looks slightly different when\nset in display style. (See next section).\n\n\\subsection{Display Equations}\nA numbered display equation---one set off by vertical space from the\ntext and centered horizontally---is produced by the \\textbf{equation}\nenvironment. An unnumbered display equation is produced by the\n\\textbf{displaymath} environment.\n\nAgain, in either environment, you can use any of the symbols and\nstructures available in \\LaTeX\\@; this section will just give a couple\nof examples of display equations in context. First, consider the\nequation, shown as an inline equation above:\n\\begin{equation}\n \\lim_{n\\rightarrow \\infty}x=0\n\\end{equation}\nNotice how it is formatted somewhat differently in\nthe \\textbf{displaymath}\nenvironment. Now, we'll enter an unnumbered equation:\n\\begin{displaymath}\n \\sum_{i=0}^{\\infty} x + 1\n\\end{displaymath}\nand follow it with another numbered equation:\n\\begin{equation}\n \\sum_{i=0}^{\\infty}x_i=\\int_{0}^{\\pi+2} f\n\\end{equation}\njust to demonstrate \\LaTeX's able handling of numbering.\n\n\\section{Figures}\n\nThe ``\\verb|figure|'' environment should be used for figures. One or\nmore images can be placed within a figure. If your figure contains\nthird-party material, you must clearly identify it as such, as shown\nin the example below.\n\\begin{figure}[h]\n \\centering\n \\includegraphics[width=\\linewidth]{sample-franklin}\n \\caption{1907 Franklin Model D roadster. Photograph by Harris \\&\n Ewing, Inc. [Public domain], via Wikimedia\n Commons. (\\url{https:\/\/goo.gl\/VLCRBB}).}\n \\Description{A woman and a girl in white dresses sit in an open car.}\n\\end{figure}\n\nYour figures should contain a caption which describes the figure to\nthe reader.\n\nFigure captions are placed {\\itshape below} the figure.\n\nEvery figure should also have a figure description unless it is purely\ndecorative. These descriptions convey what's in the image to someone\nwho cannot see it. They are also used by search engine crawlers for\nindexing images, and when images cannot be loaded.\n\nA figure description must be unformatted plain text less than 2000\ncharacters long (including spaces). {\\bfseries Figure descriptions\n should not repeat the figure caption \u2013 their purpose is to capture\n important information that is not already provided in the caption or\n the main text of the paper.} For figures that convey important and\ncomplex new information, a short text description may not be\nadequate. More complex alternative descriptions can be placed in an\nappendix and referenced in a short figure description. For example,\nprovide a data table capturing the information in a bar chart, or a\nstructured list representing a graph. For additional information\nregarding how best to write figure descriptions and why doing this is\nso important, please see\n\\url{https:\/\/www.acm.org\/publications\/taps\/describing-figures\/}.\n\n\\subsection{The ``Teaser Figure''}\n\nA ``teaser figure'' is an image, or set of images in one figure, that\nare placed after all author and affiliation information, and before\nthe body of the article, spanning the page. If you wish to have such a\nfigure in your article, place the command immediately before the\n\\verb|\\maketitle| command:\n\\begin{verbatim}\n \\begin{teaserfigure}\n \\includegraphics[width=\\textwidth]{sampleteaser}\n \\caption{figure caption}\n \\Description{figure description}\n \\end{teaserfigure}\n\\end{verbatim}\n\n\\section{Citations and Bibliographies}\n\nThe use of \\BibTeX\\ for the preparation and formatting of one's\nreferences is strongly recommended. Authors' names should be complete\n--- use full first names (``Donald E. Knuth'') not initials\n(``D. E. Knuth'') --- and the salient identifying features of a\nreference should be included: title, year, volume, number, pages,\narticle DOI, etc.\n\nThe bibliography is included in your source document with these two\ncommands, placed just before the \\verb|\\end{document}| command:\n\\begin{verbatim}\n \\bibliographystyle{ACM-Reference-Format}\n \n\\section{Introduction}\nACM's consolidated article template, introduced in 2017, provides a\nconsistent \\LaTeX\\ style for use across ACM publications, and\nincorporates accessibility and metadata-extraction functionality\nnecessary for future Digital Library endeavors. Numerous ACM and\nSIG-specific \\LaTeX\\ templates have been examined, and their unique\nfeatures incorporated into this single new template.\n\nIf you are new to publishing with ACM, this document is a valuable\nguide to the process of preparing your work for publication. If you\nhave published with ACM before, this document provides insight and\ninstruction into more recent changes to the article template.\n\nThe ``\\verb|acmart|'' document class can be used to prepare articles\nfor any ACM publication --- conference or journal, and for any stage\nof publication, from review to final ``camera-ready'' copy, to the\nauthor's own version, with {\\itshape very} few changes to the source.\n\n\\section{Template Overview}\nAs noted in the introduction, the ``\\verb|acmart|'' document class can\nbe used to prepare many different kinds of documentation --- a\ndouble-blind initial submission of a full-length technical paper, a\ntwo-page SIGGRAPH Emerging Technologies abstract, a ``camera-ready''\njournal article, a SIGCHI Extended Abstract, and more --- all by\nselecting the appropriate {\\itshape template style} and {\\itshape\n template parameters}.\n\nThis document will explain the major features of the document\nclass. For further information, the {\\itshape \\LaTeX\\ User's Guide} is\navailable from\n\\url{https:\/\/www.acm.org\/publications\/proceedings-template}.\n\n\\subsection{Template Styles}\n\nThe primary parameter given to the ``\\verb|acmart|'' document class is\nthe {\\itshape template style} which corresponds to the kind of publication\nor SIG publishing the work. This parameter is enclosed in square\nbrackets and is a part of the {\\verb|documentclass|} command:\n\\begin{verbatim}\n \\documentclass[STYLE]{acmart}\n\\end{verbatim}\n\nJournals use one of three template styles. All but three ACM journals\nuse the {\\verb|acmsmall|} template style:\n\\begin{itemize}\n\\item {\\texttt{acmsmall}}: The default journal template style.\n\\item {\\texttt{acmlarge}}: Used by JOCCH and TAP.\n\\item {\\texttt{acmtog}}: Used by TOG.\n\\end{itemize}\n\nThe majority of conference proceedings documentation will use the {\\verb|acmconf|} template style.\n\\begin{itemize}\n\\item {\\texttt{acmconf}}: The default proceedings template style.\n\\item{\\texttt{sigchi}}: Used for SIGCHI conference articles.\n\\item{\\texttt{sigchi-a}}: Used for SIGCHI ``Extended Abstract'' articles.\n\\item{\\texttt{sigplan}}: Used for SIGPLAN conference articles.\n\\end{itemize}\n\n\\subsection{Template Parameters}\n\nIn addition to specifying the {\\itshape template style} to be used in\nformatting your work, there are a number of {\\itshape template parameters}\nwhich modify some part of the applied template style. A complete list\nof these parameters can be found in the {\\itshape \\LaTeX\\ User's Guide.}\n\nFrequently-used parameters, or combinations of parameters, include:\n\\begin{itemize}\n\\item {\\texttt{anonymous,review}}: Suitable for a ``double-blind''\n conference submission. Anonymizes the work and includes line\n numbers. Use with the \\texttt{\\acmSubmissionID} command to print the\n submission's unique ID on each page of the work.\n\\item{\\texttt{authorversion}}: Produces a version of the work suitable\n for posting by the author.\n\\item{\\texttt{screen}}: Produces colored hyperlinks.\n\\end{itemize}\n\nThis document uses the following string as the first command in the\nsource file:\n\\begin{verbatim}\n\\documentclass[acmsmall,screen,review]{acmart}\n\\end{verbatim}\n\n\\section{Modifications}\n\nModifying the template --- including but not limited to: adjusting\nmargins, typeface sizes, line spacing, paragraph and list definitions,\nand the use of the \\verb|\\vspace| command to manually adjust the\nvertical spacing between elements of your work --- is not allowed.\n\n{\\bfseries Your document will be returned to you for revision if\n modifications are discovered.}\n\n\\section{Typefaces}\n\nThe ``\\verb|acmart|'' document class requires the use of the\n``Libertine'' typeface family. Your \\TeX\\ installation should include\nthis set of packages. Please do not substitute other typefaces. The\n``\\verb|lmodern|'' and ``\\verb|ltimes|'' packages should not be used,\nas they will override the built-in typeface families.\n\n\\section{Title Information}\n\nThe title of your work should use capital letters appropriately -\n\\url{https:\/\/capitalizemytitle.com\/} has useful rules for\ncapitalization. Use the {\\verb|title|} command to define the title of\nyour work. If your work has a subtitle, define it with the\n{\\verb|subtitle|} command. Do not insert line breaks in your title.\n\nIf your title is lengthy, you must define a short version to be used\nin the page headers, to prevent overlapping text. The \\verb|title|\ncommand has a ``short title'' parameter:\n\\begin{verbatim}\n \\title[short title]{full title}\n\\end{verbatim}\n\n\\section{Authors and Affiliations}\n\nEach author must be defined separately for accurate metadata\nidentification. As an exception, multiple authors may share one\naffiliation. Authors' names should not be abbreviated; use full first\nnames wherever possible. Include authors' e-mail addresses whenever\npossible.\n\nGrouping authors' names or e-mail addresses, or providing an ``e-mail\nalias,'' as shown below, is not acceptable:\n\\begin{verbatim}\n \\author{Brooke Aster, David Mehldau}\n \\email{dave,judy,steve@university.edu}\n \\email{firstname.lastname@phillips.org}\n\\end{verbatim}\n\nThe \\verb|authornote| and \\verb|authornotemark| commands allow a note\nto apply to multiple authors --- for example, if the first two authors\nof an article contributed equally to the work.\n\nIf your author list is lengthy, you must define a shortened version of\nthe list of authors to be used in the page headers, to prevent\noverlapping text. The following command should be placed just after\nthe last \\verb|\\author{}| definition:\n\\begin{verbatim}\n \\renewcommand{\\shortauthors}{McCartney, et al.}\n\\end{verbatim}\nOmitting this command will force the use of a concatenated list of all\nof the authors' names, which may result in overlapping text in the\npage headers.\n\nThe article template's documentation, available at\n\\url{https:\/\/www.acm.org\/publications\/proceedings-template}, has a\ncomplete explanation of these commands and tips for their effective\nuse.\n\nNote that authors' addresses are mandatory for journal articles.\n\n\\section{Rights Information}\n\nAuthors of any work published by ACM will need to complete a rights\nform. Depending on the kind of work, and the rights management choice\nmade by the author, this may be copyright transfer, permission,\nlicense, or an OA (open access) agreement.\n\nRegardless of the rights management choice, the author will receive a\ncopy of the completed rights form once it has been submitted. This\nform contains \\LaTeX\\ commands that must be copied into the source\ndocument. When the document source is compiled, these commands and\ntheir parameters add formatted text to several areas of the final\ndocument:\n\\begin{itemize}\n\\item the ``ACM Reference Format'' text on the first page.\n\\item the ``rights management'' text on the first page.\n\\item the conference information in the page header(s).\n\\end{itemize}\n\nRights information is unique to the work; if you are preparing several\nworks for an event, make sure to use the correct set of commands with\neach of the works.\n\nThe ACM Reference Format text is required for all articles over one\npage in length, and is optional for one-page articles (abstracts).\n\n\\section{CCS Concepts and User-Defined Keywords}\n\nTwo elements of the ``acmart'' document class provide powerful\ntaxonomic tools for you to help readers find your work in an online\nsearch.\n\nThe ACM Computing Classification System ---\n\\url{https:\/\/www.acm.org\/publications\/class-2012} --- is a set of\nclassifiers and concepts that describe the computing\ndiscipline. Authors can select entries from this classification\nsystem, via \\url{https:\/\/dl.acm.org\/ccs\/ccs.cfm}, and generate the\ncommands to be included in the \\LaTeX\\ source.\n\nUser-defined keywords are a comma-separated list of words and phrases\nof the authors' choosing, providing a more flexible way of describing\nthe research being presented.\n\nCCS concepts and user-defined keywords are required for for all\narticles over two pages in length, and are optional for one- and\ntwo-page articles (or abstracts).\n\n\\section{Sectioning Commands}\n\nYour work should use standard \\LaTeX\\ sectioning commands:\n\\verb|section|, \\verb|subsection|, \\verb|subsubsection|, and\n\\verb|paragraph|. They should be numbered; do not remove the numbering\nfrom the commands.\n\nSimulating a sectioning command by setting the first word or words of\na paragraph in boldface or italicized text is {\\bfseries not allowed.}\n\n\\section{Tables}\n\nThe ``\\verb|acmart|'' document class includes the ``\\verb|booktabs|''\npackage --- \\url{https:\/\/ctan.org\/pkg\/booktabs} --- for preparing\nhigh-quality tables.\n\nTable captions are placed {\\itshape above} the table.\n\nBecause tables cannot be split across pages, the best placement for\nthem is typically the top of the page nearest their initial cite. To\nensure this proper ``floating'' placement of tables, use the\nenvironment \\textbf{table} to enclose the table's contents and the\ntable caption. The contents of the table itself must go in the\n\\textbf{tabular} environment, to be aligned properly in rows and\ncolumns, with the desired horizontal and vertical rules. Again,\ndetailed instructions on \\textbf{tabular} material are found in the\n\\textit{\\LaTeX\\ User's Guide}.\n\nImmediately following this sentence is the point at which\nTable~\\ref{tab:freq} is included in the input file; compare the\nplacement of the table here with the table in the printed output of\nthis document.\n\n\\begin{table}\n \\caption{Frequency of Special Characters}\n \\label{tab:freq}\n \\begin{tabular}{ccl}\n \\toprule\n Non-English or Math&Frequency&Comments\\\\\n \\midrule\n \\O & 1 in 1,000& For Swedish names\\\\\n $\\pi$ & 1 in 5& Common in math\\\\\n \\$ & 4 in 5 & Used in business\\\\\n $\\Psi^2_1$ & 1 in 40,000& Unexplained usage\\\\\n \\bottomrule\n\\end{tabular}\n\\end{table}\n\nTo set a wider table, which takes up the whole width of the page's\nlive area, use the environment \\textbf{table*} to enclose the table's\ncontents and the table caption. As with a single-column table, this\nwide table will ``float'' to a location deemed more\ndesirable. Immediately following this sentence is the point at which\nTable~\\ref{tab:commands} is included in the input file; again, it is\ninstructive to compare the placement of the table here with the table\nin the printed output of this document.\n\n\\begin{table*}\n \\caption{Some Typical Commands}\n \\label{tab:commands}\n \\begin{tabular}{ccl}\n \\toprule\n Command &A Number & Comments\\\\\n \\midrule\n \\texttt{{\\char'134}author} & 100& Author \\\\\n \\texttt{{\\char'134}table}& 300 & For tables\\\\\n \\texttt{{\\char'134}table*}& 400& For wider tables\\\\\n \\bottomrule\n \\end{tabular}\n\\end{table*}\n\nAlways use midrule to separate table header rows from data rows, and\nuse it only for this purpose. This enables assistive technologies to\nrecognise table headers and support their users in navigating tables\nmore easily.\n\n\\section{Math Equations}\nYou may want to display math equations in three distinct styles:\ninline, numbered or non-numbered display. Each of the three are\ndiscussed in the next sections.\n\n\\subsection{Inline (In-text) Equations}\nA formula that appears in the running text is called an inline or\nin-text formula. It is produced by the \\textbf{math} environment,\nwhich can be invoked with the usual\n\\texttt{{\\char'134}begin\\,\\ldots{\\char'134}end} construction or with\nthe short form \\texttt{\\$\\,\\ldots\\$}. You can use any of the symbols\nand structures, from $\\alpha$ to $\\omega$, available in\n\\LaTeX~\\cite{Lamport:LaTeX}; this section will simply show a few\nexamples of in-text equations in context. Notice how this equation:\n\\begin{math}\n \\lim_{n\\rightarrow \\infty}x=0\n\\end{math},\nset here in in-line math style, looks slightly different when\nset in display style. (See next section).\n\n\\subsection{Display Equations}\nA numbered display equation---one set off by vertical space from the\ntext and centered horizontally---is produced by the \\textbf{equation}\nenvironment. An unnumbered display equation is produced by the\n\\textbf{displaymath} environment.\n\nAgain, in either environment, you can use any of the symbols and\nstructures available in \\LaTeX\\@; this section will just give a couple\nof examples of display equations in context. First, consider the\nequation, shown as an inline equation above:\n\\begin{equation}\n \\lim_{n\\rightarrow \\infty}x=0\n\\end{equation}\nNotice how it is formatted somewhat differently in\nthe \\textbf{displaymath}\nenvironment. Now, we'll enter an unnumbered equation:\n\\begin{displaymath}\n \\sum_{i=0}^{\\infty} x + 1\n\\end{displaymath}\nand follow it with another numbered equation:\n\\begin{equation}\n \\sum_{i=0}^{\\infty}x_i=\\int_{0}^{\\pi+2} f\n\\end{equation}\njust to demonstrate \\LaTeX's able handling of numbering.\n\n\\section{Figures}\n\nThe ``\\verb|figure|'' environment should be used for figures. One or\nmore images can be placed within a figure. If your figure contains\nthird-party material, you must clearly identify it as such, as shown\nin the example below.\n\\begin{figure}[h]\n \\centering\n \\includegraphics[width=\\linewidth]{sample-franklin}\n \\caption{1907 Franklin Model D roadster. Photograph by Harris \\&\n Ewing, Inc. [Public domain], via Wikimedia\n Commons. (\\url{https:\/\/goo.gl\/VLCRBB}).}\n \\Description{A woman and a girl in white dresses sit in an open car.}\n\\end{figure}\n\nYour figures should contain a caption which describes the figure to\nthe reader.\n\nFigure captions are placed {\\itshape below} the figure.\n\nEvery figure should also have a figure description unless it is purely\ndecorative. These descriptions convey what's in the image to someone\nwho cannot see it. They are also used by search engine crawlers for\nindexing images, and when images cannot be loaded.\n\nA figure description must be unformatted plain text less than 2000\ncharacters long (including spaces). {\\bfseries Figure descriptions\n should not repeat the figure caption \u2013 their purpose is to capture\n important information that is not already provided in the caption or\n the main text of the paper.} For figures that convey important and\ncomplex new information, a short text description may not be\nadequate. More complex alternative descriptions can be placed in an\nappendix and referenced in a short figure description. For example,\nprovide a data table capturing the information in a bar chart, or a\nstructured list representing a graph. For additional information\nregarding how best to write figure descriptions and why doing this is\nso important, please see\n\\url{https:\/\/www.acm.org\/publications\/taps\/describing-figures\/}.\n\n\\subsection{The ``Teaser Figure''}\n\nA ``teaser figure'' is an image, or set of images in one figure, that\nare placed after all author and affiliation information, and before\nthe body of the article, spanning the page. If you wish to have such a\nfigure in your article, place the command immediately before the\n\\verb|\\maketitle| command:\n\\begin{verbatim}\n \\begin{teaserfigure}\n \\includegraphics[width=\\textwidth]{sampleteaser}\n \\caption{figure caption}\n \\Description{figure description}\n \\end{teaserfigure}\n\\end{verbatim}\n\n\\section{Citations and Bibliographies}\n\nThe use of \\BibTeX\\ for the preparation and formatting of one's\nreferences is strongly recommended. Authors' names should be complete\n--- use full first names (``Donald E. Knuth'') not initials\n(``D. E. Knuth'') --- and the salient identifying features of a\nreference should be included: title, year, volume, number, pages,\narticle DOI, etc.\n\nThe bibliography is included in your source document with these two\ncommands, placed just before the \\verb|\\end{document}| command:\n\\begin{verbatim}\n \\bibliographystyle{ACM-Reference-Format}\n \n\\section{Introduction}\nACM's consolidated article template, introduced in 2017, provides a\nconsistent \\LaTeX\\ style for use across ACM publications, and\nincorporates accessibility and metadata-extraction functionality\nnecessary for future Digital Library endeavors. Numerous ACM and\nSIG-specific \\LaTeX\\ templates have been examined, and their unique\nfeatures incorporated into this single new template.\n\nIf you are new to publishing with ACM, this document is a valuable\nguide to the process of preparing your work for publication. If you\nhave published with ACM before, this document provides insight and\ninstruction into more recent changes to the article template.\n\nThe ``\\verb|acmart|'' document class can be used to prepare articles\nfor any ACM publication --- conference or journal, and for any stage\nof publication, from review to final ``camera-ready'' copy, to the\nauthor's own version, with {\\itshape very} few changes to the source.\n\n\\section{Template Overview}\nAs noted in the introduction, the ``\\verb|acmart|'' document class can\nbe used to prepare many different kinds of documentation --- a\ndouble-blind initial submission of a full-length technical paper, a\ntwo-page SIGGRAPH Emerging Technologies abstract, a ``camera-ready''\njournal article, a SIGCHI Extended Abstract, and more --- all by\nselecting the appropriate {\\itshape template style} and {\\itshape\n template parameters}.\n\nThis document will explain the major features of the document\nclass. For further information, the {\\itshape \\LaTeX\\ User's Guide} is\navailable from\n\\url{https:\/\/www.acm.org\/publications\/proceedings-template}.\n\n\\subsection{Template Styles}\n\nThe primary parameter given to the ``\\verb|acmart|'' document class is\nthe {\\itshape template style} which corresponds to the kind of publication\nor SIG publishing the work. This parameter is enclosed in square\nbrackets and is a part of the {\\verb|documentclass|} command:\n\\begin{verbatim}\n \\documentclass[STYLE]{acmart}\n\\end{verbatim}\n\nJournals use one of three template styles. All but three ACM journals\nuse the {\\verb|acmsmall|} template style:\n\\begin{itemize}\n\\item {\\texttt{acmsmall}}: The default journal template style.\n\\item {\\texttt{acmlarge}}: Used by JOCCH and TAP.\n\\item {\\texttt{acmtog}}: Used by TOG.\n\\end{itemize}\n\nThe majority of conference proceedings documentation will use the {\\verb|acmconf|} template style.\n\\begin{itemize}\n\\item {\\texttt{acmconf}}: The default proceedings template style.\n\\item{\\texttt{sigchi}}: Used for SIGCHI conference articles.\n\\item{\\texttt{sigchi-a}}: Used for SIGCHI ``Extended Abstract'' articles.\n\\item{\\texttt{sigplan}}: Used for SIGPLAN conference articles.\n\\end{itemize}\n\n\\subsection{Template Parameters}\n\nIn addition to specifying the {\\itshape template style} to be used in\nformatting your work, there are a number of {\\itshape template parameters}\nwhich modify some part of the applied template style. A complete list\nof these parameters can be found in the {\\itshape \\LaTeX\\ User's Guide.}\n\nFrequently-used parameters, or combinations of parameters, include:\n\\begin{itemize}\n\\item {\\texttt{anonymous,review}}: Suitable for a ``double-blind''\n conference submission. Anonymizes the work and includes line\n numbers. Use with the \\texttt{\\acmSubmissionID} command to print the\n submission's unique ID on each page of the work.\n\\item{\\texttt{authorversion}}: Produces a version of the work suitable\n for posting by the author.\n\\item{\\texttt{screen}}: Produces colored hyperlinks.\n\\end{itemize}\n\nThis document uses the following string as the first command in the\nsource file:\n\\begin{verbatim}\n\\documentclass[acmsmall]{acmart}\n\\end{verbatim}\n\n\\section{Modifications}\n\nModifying the template --- including but not limited to: adjusting\nmargins, typeface sizes, line spacing, paragraph and list definitions,\nand the use of the \\verb|\\vspace| command to manually adjust the\nvertical spacing between elements of your work --- is not allowed.\n\n{\\bfseries Your document will be returned to you for revision if\n modifications are discovered.}\n\n\\section{Typefaces}\n\nThe ``\\verb|acmart|'' document class requires the use of the\n``Libertine'' typeface family. Your \\TeX\\ installation should include\nthis set of packages. Please do not substitute other typefaces. The\n``\\verb|lmodern|'' and ``\\verb|ltimes|'' packages should not be used,\nas they will override the built-in typeface families.\n\n\\section{Title Information}\n\nThe title of your work should use capital letters appropriately -\n\\url{https:\/\/capitalizemytitle.com\/} has useful rules for\ncapitalization. Use the {\\verb|title|} command to define the title of\nyour work. If your work has a subtitle, define it with the\n{\\verb|subtitle|} command. Do not insert line breaks in your title.\n\nIf your title is lengthy, you must define a short version to be used\nin the page headers, to prevent overlapping text. The \\verb|title|\ncommand has a ``short title'' parameter:\n\\begin{verbatim}\n \\title[short title]{full title}\n\\end{verbatim}\n\n\\section{Authors and Affiliations}\n\nEach author must be defined separately for accurate metadata\nidentification. As an exception, multiple authors may share one\naffiliation. Authors' names should not be abbreviated; use full first\nnames wherever possible. Include authors' e-mail addresses whenever\npossible.\n\nGrouping authors' names or e-mail addresses, or providing an ``e-mail\nalias,'' as shown below, is not acceptable:\n\\begin{verbatim}\n \\author{Brooke Aster, David Mehldau}\n \\email{dave,judy,steve@university.edu}\n \\email{firstname.lastname@phillips.org}\n\\end{verbatim}\n\nThe \\verb|authornote| and \\verb|authornotemark| commands allow a note\nto apply to multiple authors --- for example, if the first two authors\nof an article contributed equally to the work.\n\nIf your author list is lengthy, you must define a shortened version of\nthe list of authors to be used in the page headers, to prevent\noverlapping text. The following command should be placed just after\nthe last \\verb|\\author{}| definition:\n\\begin{verbatim}\n \\renewcommand{\\shortauthors}{McCartney, et al.}\n\\end{verbatim}\nOmitting this command will force the use of a concatenated list of all\nof the authors' names, which may result in overlapping text in the\npage headers.\n\nThe article template's documentation, available at\n\\url{https:\/\/www.acm.org\/publications\/proceedings-template}, has a\ncomplete explanation of these commands and tips for their effective\nuse.\n\nNote that authors' addresses are mandatory for journal articles.\n\n\\section{Rights Information}\n\nAuthors of any work published by ACM will need to complete a rights\nform. Depending on the kind of work, and the rights management choice\nmade by the author, this may be copyright transfer, permission,\nlicense, or an OA (open access) agreement.\n\nRegardless of the rights management choice, the author will receive a\ncopy of the completed rights form once it has been submitted. This\nform contains \\LaTeX\\ commands that must be copied into the source\ndocument. When the document source is compiled, these commands and\ntheir parameters add formatted text to several areas of the final\ndocument:\n\\begin{itemize}\n\\item the ``ACM Reference Format'' text on the first page.\n\\item the ``rights management'' text on the first page.\n\\item the conference information in the page header(s).\n\\end{itemize}\n\nRights information is unique to the work; if you are preparing several\nworks for an event, make sure to use the correct set of commands with\neach of the works.\n\nThe ACM Reference Format text is required for all articles over one\npage in length, and is optional for one-page articles (abstracts).\n\n\\section{CCS Concepts and User-Defined Keywords}\n\nTwo elements of the ``acmart'' document class provide powerful\ntaxonomic tools for you to help readers find your work in an online\nsearch.\n\nThe ACM Computing Classification System ---\n\\url{https:\/\/www.acm.org\/publications\/class-2012} --- is a set of\nclassifiers and concepts that describe the computing\ndiscipline. Authors can select entries from this classification\nsystem, via \\url{https:\/\/dl.acm.org\/ccs\/ccs.cfm}, and generate the\ncommands to be included in the \\LaTeX\\ source.\n\nUser-defined keywords are a comma-separated list of words and phrases\nof the authors' choosing, providing a more flexible way of describing\nthe research being presented.\n\nCCS concepts and user-defined keywords are required for for all\narticles over two pages in length, and are optional for one- and\ntwo-page articles (or abstracts).\n\n\\section{Sectioning Commands}\n\nYour work should use standard \\LaTeX\\ sectioning commands:\n\\verb|section|, \\verb|subsection|, \\verb|subsubsection|, and\n\\verb|paragraph|. They should be numbered; do not remove the numbering\nfrom the commands.\n\nSimulating a sectioning command by setting the first word or words of\na paragraph in boldface or italicized text is {\\bfseries not allowed.}\n\n\\section{Tables}\n\nThe ``\\verb|acmart|'' document class includes the ``\\verb|booktabs|''\npackage --- \\url{https:\/\/ctan.org\/pkg\/booktabs} --- for preparing\nhigh-quality tables.\n\nTable captions are placed {\\itshape above} the table.\n\nBecause tables cannot be split across pages, the best placement for\nthem is typically the top of the page nearest their initial cite. To\nensure this proper ``floating'' placement of tables, use the\nenvironment \\textbf{table} to enclose the table's contents and the\ntable caption. The contents of the table itself must go in the\n\\textbf{tabular} environment, to be aligned properly in rows and\ncolumns, with the desired horizontal and vertical rules. Again,\ndetailed instructions on \\textbf{tabular} material are found in the\n\\textit{\\LaTeX\\ User's Guide}.\n\nImmediately following this sentence is the point at which\nTable~\\ref{tab:freq} is included in the input file; compare the\nplacement of the table here with the table in the printed output of\nthis document.\n\n\\begin{table}\n \\caption{Frequency of Special Characters}\n \\label{tab:freq}\n \\begin{tabular}{ccl}\n \\toprule\n Non-English or Math&Frequency&Comments\\\\\n \\midrule\n \\O & 1 in 1,000& For Swedish names\\\\\n $\\pi$ & 1 in 5& Common in math\\\\\n \\$ & 4 in 5 & Used in business\\\\\n $\\Psi^2_1$ & 1 in 40,000& Unexplained usage\\\\\n \\bottomrule\n\\end{tabular}\n\\end{table}\n\nTo set a wider table, which takes up the whole width of the page's\nlive area, use the environment \\textbf{table*} to enclose the table's\ncontents and the table caption. As with a single-column table, this\nwide table will ``float'' to a location deemed more\ndesirable. Immediately following this sentence is the point at which\nTable~\\ref{tab:commands} is included in the input file; again, it is\ninstructive to compare the placement of the table here with the table\nin the printed output of this document.\n\n\\begin{table*}\n \\caption{Some Typical Commands}\n \\label{tab:commands}\n \\begin{tabular}{ccl}\n \\toprule\n Command &A Number & Comments\\\\\n \\midrule\n \\texttt{{\\char'134}author} & 100& Author \\\\\n \\texttt{{\\char'134}table}& 300 & For tables\\\\\n \\texttt{{\\char'134}table*}& 400& For wider tables\\\\\n \\bottomrule\n \\end{tabular}\n\\end{table*}\n\nAlways use midrule to separate table header rows from data rows, and\nuse it only for this purpose. This enables assistive technologies to\nrecognise table headers and support their users in navigating tables\nmore easily.\n\n\\section{Math Equations}\nYou may want to display math equations in three distinct styles:\ninline, numbered or non-numbered display. Each of the three are\ndiscussed in the next sections.\n\n\\subsection{Inline (In-text) Equations}\nA formula that appears in the running text is called an inline or\nin-text formula. It is produced by the \\textbf{math} environment,\nwhich can be invoked with the usual\n\\texttt{{\\char'134}begin\\,\\ldots{\\char'134}end} construction or with\nthe short form \\texttt{\\$\\,\\ldots\\$}. You can use any of the symbols\nand structures, from $\\alpha$ to $\\omega$, available in\n\\LaTeX~\\cite{Lamport:LaTeX}; this section will simply show a few\nexamples of in-text equations in context. Notice how this equation:\n\\begin{math}\n \\lim_{n\\rightarrow \\infty}x=0\n\\end{math},\nset here in in-line math style, looks slightly different when\nset in display style. (See next section).\n\n\\subsection{Display Equations}\nA numbered display equation---one set off by vertical space from the\ntext and centered horizontally---is produced by the \\textbf{equation}\nenvironment. An unnumbered display equation is produced by the\n\\textbf{displaymath} environment.\n\nAgain, in either environment, you can use any of the symbols and\nstructures available in \\LaTeX\\@; this section will just give a couple\nof examples of display equations in context. First, consider the\nequation, shown as an inline equation above:\n\\begin{equation}\n \\lim_{n\\rightarrow \\infty}x=0\n\\end{equation}\nNotice how it is formatted somewhat differently in\nthe \\textbf{displaymath}\nenvironment. Now, we'll enter an unnumbered equation:\n\\begin{displaymath}\n \\sum_{i=0}^{\\infty} x + 1\n\\end{displaymath}\nand follow it with another numbered equation:\n\\begin{equation}\n \\sum_{i=0}^{\\infty}x_i=\\int_{0}^{\\pi+2} f\n\\end{equation}\njust to demonstrate \\LaTeX's able handling of numbering.\n\n\\section{Figures}\n\nThe ``\\verb|figure|'' environment should be used for figures. One or\nmore images can be placed within a figure. If your figure contains\nthird-party material, you must clearly identify it as such, as shown\nin the example below.\n\\begin{figure}[h]\n \\centering\n \\includegraphics[width=\\linewidth]{sample-franklin}\n \\caption{1907 Franklin Model D roadster. Photograph by Harris \\&\n Ewing, Inc. [Public domain], via Wikimedia\n Commons. (\\url{https:\/\/goo.gl\/VLCRBB}).}\n \\Description{A woman and a girl in white dresses sit in an open car.}\n\\end{figure}\n\nYour figures should contain a caption which describes the figure to\nthe reader.\n\nFigure captions are placed {\\itshape below} the figure.\n\nEvery figure should also have a figure description unless it is purely\ndecorative. These descriptions convey what's in the image to someone\nwho cannot see it. They are also used by search engine crawlers for\nindexing images, and when images cannot be loaded.\n\nA figure description must be unformatted plain text less than 2000\ncharacters long (including spaces). {\\bfseries Figure descriptions\n should not repeat the figure caption \u2013 their purpose is to capture\n important information that is not already provided in the caption or\n the main text of the paper.} For figures that convey important and\ncomplex new information, a short text description may not be\nadequate. More complex alternative descriptions can be placed in an\nappendix and referenced in a short figure description. For example,\nprovide a data table capturing the information in a bar chart, or a\nstructured list representing a graph. For additional information\nregarding how best to write figure descriptions and why doing this is\nso important, please see\n\\url{https:\/\/www.acm.org\/publications\/taps\/describing-figures\/}.\n\n\\subsection{The ``Teaser Figure''}\n\nA ``teaser figure'' is an image, or set of images in one figure, that\nare placed after all author and affiliation information, and before\nthe body of the article, spanning the page. If you wish to have such a\nfigure in your article, place the command immediately before the\n\\verb|\\maketitle| command:\n\\begin{verbatim}\n \\begin{teaserfigure}\n \\includegraphics[width=\\textwidth]{sampleteaser}\n \\caption{figure caption}\n \\Description{figure description}\n \\end{teaserfigure}\n\\end{verbatim}\n\n\\section{Citations and Bibliographies}\n\nThe use of \\BibTeX\\ for the preparation and formatting of one's\nreferences is strongly recommended. Authors' names should be complete\n--- use full first names (``Donald E. Knuth'') not initials\n(``D. E. Knuth'') --- and the salient identifying features of a\nreference should be included: title, year, volume, number, pages,\narticle DOI, etc.\n\nThe bibliography is included in your source document with these two\ncommands, placed just before the \\verb|\\end{document}| command:\n\\begin{verbatim}\n \\bibliographystyle{ACM-Reference-Format}\n \n\\section{Introduction}\nACM's consolidated article template, introduced in 2017, provides a\nconsistent \\LaTeX\\ style for use across ACM publications, and\nincorporates accessibility and metadata-extraction functionality\nnecessary for future Digital Library endeavors. Numerous ACM and\nSIG-specific \\LaTeX\\ templates have been examined, and their unique\nfeatures incorporated into this single new template.\n\nIf you are new to publishing with ACM, this document is a valuable\nguide to the process of preparing your work for publication. If you\nhave published with ACM before, this document provides insight and\ninstruction into more recent changes to the article template.\n\nThe ``\\verb|acmart|'' document class can be used to prepare articles\nfor any ACM publication --- conference or journal, and for any stage\nof publication, from review to final ``camera-ready'' copy, to the\nauthor's own version, with {\\itshape very} few changes to the source.\n\n\\section{Template Overview}\nAs noted in the introduction, the ``\\verb|acmart|'' document class can\nbe used to prepare many different kinds of documentation --- a\ndouble-blind initial submission of a full-length technical paper, a\ntwo-page SIGGRAPH Emerging Technologies abstract, a ``camera-ready''\njournal article, a SIGCHI Extended Abstract, and more --- all by\nselecting the appropriate {\\itshape template style} and {\\itshape\n template parameters}.\n\nThis document will explain the major features of the document\nclass. For further information, the {\\itshape \\LaTeX\\ User's Guide} is\navailable from\n\\url{https:\/\/www.acm.org\/publications\/proceedings-template}.\n\n\\subsection{Template Styles}\n\nThe primary parameter given to the ``\\verb|acmart|'' document class is\nthe {\\itshape template style} which corresponds to the kind of publication\nor SIG publishing the work. This parameter is enclosed in square\nbrackets and is a part of the {\\verb|documentclass|} command:\n\\begin{verbatim}\n \\documentclass[STYLE]{acmart}\n\\end{verbatim}\n\nJournals use one of three template styles. All but three ACM journals\nuse the {\\verb|acmsmall|} template style:\n\\begin{itemize}\n\\item {\\texttt{acmsmall}}: The default journal template style.\n\\item {\\texttt{acmlarge}}: Used by JOCCH and TAP.\n\\item {\\texttt{acmtog}}: Used by TOG.\n\\end{itemize}\n\nThe majority of conference proceedings documentation will use the {\\verb|acmconf|} template style.\n\\begin{itemize}\n\\item {\\texttt{acmconf}}: The default proceedings template style.\n\\item{\\texttt{sigchi}}: Used for SIGCHI conference articles.\n\\item{\\texttt{sigchi-a}}: Used for SIGCHI ``Extended Abstract'' articles.\n\\item{\\texttt{sigplan}}: Used for SIGPLAN conference articles.\n\\end{itemize}\n\n\\subsection{Template Parameters}\n\nIn addition to specifying the {\\itshape template style} to be used in\nformatting your work, there are a number of {\\itshape template parameters}\nwhich modify some part of the applied template style. A complete list\nof these parameters can be found in the {\\itshape \\LaTeX\\ User's Guide.}\n\nFrequently-used parameters, or combinations of parameters, include:\n\\begin{itemize}\n\\item {\\texttt{anonymous,review}}: Suitable for a ``double-blind''\n conference submission. Anonymizes the work and includes line\n numbers. Use with the \\texttt{\\acmSubmissionID} command to print the\n submission's unique ID on each page of the work.\n\\item{\\texttt{authorversion}}: Produces a version of the work suitable\n for posting by the author.\n\\item{\\texttt{screen}}: Produces colored hyperlinks.\n\\end{itemize}\n\nThis document uses the following string as the first command in the\nsource file:\n\\begin{verbatim}\n\\documentclass[acmtog]{acmart}\n\\end{verbatim}\n\n\\section{Modifications}\n\nModifying the template --- including but not limited to: adjusting\nmargins, typeface sizes, line spacing, paragraph and list definitions,\nand the use of the \\verb|\\vspace| command to manually adjust the\nvertical spacing between elements of your work --- is not allowed.\n\n{\\bfseries Your document will be returned to you for revision if\n modifications are discovered.}\n\n\\section{Typefaces}\n\nThe ``\\verb|acmart|'' document class requires the use of the\n``Libertine'' typeface family. Your \\TeX\\ installation should include\nthis set of packages. Please do not substitute other typefaces. The\n``\\verb|lmodern|'' and ``\\verb|ltimes|'' packages should not be used,\nas they will override the built-in typeface families.\n\n\\section{Title Information}\n\nThe title of your work should use capital letters appropriately -\n\\url{https:\/\/capitalizemytitle.com\/} has useful rules for\ncapitalization. Use the {\\verb|title|} command to define the title of\nyour work. If your work has a subtitle, define it with the\n{\\verb|subtitle|} command. Do not insert line breaks in your title.\n\nIf your title is lengthy, you must define a short version to be used\nin the page headers, to prevent overlapping text. The \\verb|title|\ncommand has a ``short title'' parameter:\n\\begin{verbatim}\n \\title[short title]{full title}\n\\end{verbatim}\n\n\\section{Authors and Affiliations}\n\nEach author must be defined separately for accurate metadata\nidentification. As an exception, multiple authors may share one\naffiliation. Authors' names should not be abbreviated; use full first\nnames wherever possible. Include authors' e-mail addresses whenever\npossible.\n\nGrouping authors' names or e-mail addresses, or providing an ``e-mail\nalias,'' as shown below, is not acceptable:\n\\begin{verbatim}\n \\author{Brooke Aster, David Mehldau}\n \\email{dave,judy,steve@university.edu}\n \\email{firstname.lastname@phillips.org}\n\\end{verbatim}\n\nThe \\verb|authornote| and \\verb|authornotemark| commands allow a note\nto apply to multiple authors --- for example, if the first two authors\nof an article contributed equally to the work.\n\nIf your author list is lengthy, you must define a shortened version of\nthe list of authors to be used in the page headers, to prevent\noverlapping text. The following command should be placed just after\nthe last \\verb|\\author{}| definition:\n\\begin{verbatim}\n \\renewcommand{\\shortauthors}{McCartney, et al.}\n\\end{verbatim}\nOmitting this command will force the use of a concatenated list of all\nof the authors' names, which may result in overlapping text in the\npage headers.\n\nThe article template's documentation, available at\n\\url{https:\/\/www.acm.org\/publications\/proceedings-template}, has a\ncomplete explanation of these commands and tips for their effective\nuse.\n\nNote that authors' addresses are mandatory for journal articles.\n\n\\section{Rights Information}\n\nAuthors of any work published by ACM will need to complete a rights\nform. Depending on the kind of work, and the rights management choice\nmade by the author, this may be copyright transfer, permission,\nlicense, or an OA (open access) agreement.\n\nRegardless of the rights management choice, the author will receive a\ncopy of the completed rights form once it has been submitted. This\nform contains \\LaTeX\\ commands that must be copied into the source\ndocument. When the document source is compiled, these commands and\ntheir parameters add formatted text to several areas of the final\ndocument:\n\\begin{itemize}\n\\item the ``ACM Reference Format'' text on the first page.\n\\item the ``rights management'' text on the first page.\n\\item the conference information in the page header(s).\n\\end{itemize}\n\nRights information is unique to the work; if you are preparing several\nworks for an event, make sure to use the correct set of commands with\neach of the works.\n\nThe ACM Reference Format text is required for all articles over one\npage in length, and is optional for one-page articles (abstracts).\n\n\\section{CCS Concepts and User-Defined Keywords}\n\nTwo elements of the ``acmart'' document class provide powerful\ntaxonomic tools for you to help readers find your work in an online\nsearch.\n\nThe ACM Computing Classification System ---\n\\url{https:\/\/www.acm.org\/publications\/class-2012} --- is a set of\nclassifiers and concepts that describe the computing\ndiscipline. Authors can select entries from this classification\nsystem, via \\url{https:\/\/dl.acm.org\/ccs\/ccs.cfm}, and generate the\ncommands to be included in the \\LaTeX\\ source.\n\nUser-defined keywords are a comma-separated list of words and phrases\nof the authors' choosing, providing a more flexible way of describing\nthe research being presented.\n\nCCS concepts and user-defined keywords are required for for all\narticles over two pages in length, and are optional for one- and\ntwo-page articles (or abstracts).\n\n\\section{Sectioning Commands}\n\nYour work should use standard \\LaTeX\\ sectioning commands:\n\\verb|section|, \\verb|subsection|, \\verb|subsubsection|, and\n\\verb|paragraph|. They should be numbered; do not remove the numbering\nfrom the commands.\n\nSimulating a sectioning command by setting the first word or words of\na paragraph in boldface or italicized text is {\\bfseries not allowed.}\n\n\\section{Tables}\n\nThe ``\\verb|acmart|'' document class includes the ``\\verb|booktabs|''\npackage --- \\url{https:\/\/ctan.org\/pkg\/booktabs} --- for preparing\nhigh-quality tables.\n\nTable captions are placed {\\itshape above} the table.\n\nBecause tables cannot be split across pages, the best placement for\nthem is typically the top of the page nearest their initial cite. To\nensure this proper ``floating'' placement of tables, use the\nenvironment \\textbf{table} to enclose the table's contents and the\ntable caption. The contents of the table itself must go in the\n\\textbf{tabular} environment, to be aligned properly in rows and\ncolumns, with the desired horizontal and vertical rules. Again,\ndetailed instructions on \\textbf{tabular} material are found in the\n\\textit{\\LaTeX\\ User's Guide}.\n\nImmediately following this sentence is the point at which\nTable~\\ref{tab:freq} is included in the input file; compare the\nplacement of the table here with the table in the printed output of\nthis document.\n\n\\begin{table}\n \\caption{Frequency of Special Characters}\n \\label{tab:freq}\n \\begin{tabular}{ccl}\n \\toprule\n Non-English or Math&Frequency&Comments\\\\\n \\midrule\n \\O & 1 in 1,000& For Swedish names\\\\\n $\\pi$ & 1 in 5& Common in math\\\\\n \\$ & 4 in 5 & Used in business\\\\\n $\\Psi^2_1$ & 1 in 40,000& Unexplained usage\\\\\n \\bottomrule\n\\end{tabular}\n\\end{table}\n\nTo set a wider table, which takes up the whole width of the page's\nlive area, use the environment \\textbf{table*} to enclose the table's\ncontents and the table caption. As with a single-column table, this\nwide table will ``float'' to a location deemed more\ndesirable. Immediately following this sentence is the point at which\nTable~\\ref{tab:commands} is included in the input file; again, it is\ninstructive to compare the placement of the table here with the table\nin the printed output of this document.\n\n\\begin{table*}\n \\caption{Some Typical Commands}\n \\label{tab:commands}\n \\begin{tabular}{ccl}\n \\toprule\n Command &A Number & Comments\\\\\n \\midrule\n \\texttt{{\\char'134}author} & 100& Author \\\\\n \\texttt{{\\char'134}table}& 300 & For tables\\\\\n \\texttt{{\\char'134}table*}& 400& For wider tables\\\\\n \\bottomrule\n \\end{tabular}\n\\end{table*}\n\nAlways use midrule to separate table header rows from data rows, and\nuse it only for this purpose. This enables assistive technologies to\nrecognise table headers and support their users in navigating tables\nmore easily.\n\n\\section{Math Equations}\nYou may want to display math equations in three distinct styles:\ninline, numbered or non-numbered display. Each of the three are\ndiscussed in the next sections.\n\n\\subsection{Inline (In-text) Equations}\nA formula that appears in the running text is called an inline or\nin-text formula. It is produced by the \\textbf{math} environment,\nwhich can be invoked with the usual\n\\texttt{{\\char'134}begin\\,\\ldots{\\char'134}end} construction or with\nthe short form \\texttt{\\$\\,\\ldots\\$}. You can use any of the symbols\nand structures, from $\\alpha$ to $\\omega$, available in\n\\LaTeX~\\cite{Lamport:LaTeX}; this section will simply show a few\nexamples of in-text equations in context. Notice how this equation:\n\\begin{math}\n \\lim_{n\\rightarrow \\infty}x=0\n\\end{math},\nset here in in-line math style, looks slightly different when\nset in display style. (See next section).\n\n\\subsection{Display Equations}\nA numbered display equation---one set off by vertical space from the\ntext and centered horizontally---is produced by the \\textbf{equation}\nenvironment. An unnumbered display equation is produced by the\n\\textbf{displaymath} environment.\n\nAgain, in either environment, you can use any of the symbols and\nstructures available in \\LaTeX\\@; this section will just give a couple\nof examples of display equations in context. First, consider the\nequation, shown as an inline equation above:\n\\begin{equation}\n \\lim_{n\\rightarrow \\infty}x=0\n\\end{equation}\nNotice how it is formatted somewhat differently in\nthe \\textbf{displaymath}\nenvironment. Now, we'll enter an unnumbered equation:\n\\begin{displaymath}\n \\sum_{i=0}^{\\infty} x + 1\n\\end{displaymath}\nand follow it with another numbered equation:\n\\begin{equation}\n \\sum_{i=0}^{\\infty}x_i=\\int_{0}^{\\pi+2} f\n\\end{equation}\njust to demonstrate \\LaTeX's able handling of numbering.\n\n\\section{Figures}\n\nThe ``\\verb|figure|'' environment should be used for figures. One or\nmore images can be placed within a figure. If your figure contains\nthird-party material, you must clearly identify it as such, as shown\nin the example below.\n\\begin{figure}[h]\n \\centering\n \\includegraphics[width=\\linewidth]{sample-franklin}\n \\caption{1907 Franklin Model D roadster. Photograph by Harris \\&\n Ewing, Inc. [Public domain], via Wikimedia\n Commons. (\\url{https:\/\/goo.gl\/VLCRBB}).}\n \\Description{A woman and a girl in white dresses sit in an open car.}\n\\end{figure}\n\nYour figures should contain a caption which describes the figure to\nthe reader.\n\nFigure captions are placed {\\itshape below} the figure.\n\nEvery figure should also have a figure description unless it is purely\ndecorative. These descriptions convey what's in the image to someone\nwho cannot see it. They are also used by search engine crawlers for\nindexing images, and when images cannot be loaded.\n\nA figure description must be unformatted plain text less than 2000\ncharacters long (including spaces). {\\bfseries Figure descriptions\n should not repeat the figure caption \u2013 their purpose is to capture\n important information that is not already provided in the caption or\n the main text of the paper.} For figures that convey important and\ncomplex new information, a short text description may not be\nadequate. More complex alternative descriptions can be placed in an\nappendix and referenced in a short figure description. For example,\nprovide a data table capturing the information in a bar chart, or a\nstructured list representing a graph. For additional information\nregarding how best to write figure descriptions and why doing this is\nso important, please see\n\\url{https:\/\/www.acm.org\/publications\/taps\/describing-figures\/}.\n\n\\subsection{The ``Teaser Figure''}\n\nA ``teaser figure'' is an image, or set of images in one figure, that\nare placed after all author and affiliation information, and before\nthe body of the article, spanning the page. If you wish to have such a\nfigure in your article, place the command immediately before the\n\\verb|\\maketitle| command:\n\\begin{verbatim}\n \\begin{teaserfigure}\n \\includegraphics[width=\\textwidth]{sampleteaser}\n \\caption{figure caption}\n \\Description{figure description}\n \\end{teaserfigure}\n\\end{verbatim}\n\n\\section{Citations and Bibliographies}\n\nThe use of \\BibTeX\\ for the preparation and formatting of one's\nreferences is strongly recommended. Authors' names should be complete\n--- use full first names (``Donald E. Knuth'') not initials\n(``D. E. Knuth'') --- and the salient identifying features of a\nreference should be included: title, year, volume, number, pages,\narticle DOI, etc.\n\nThe bibliography is included in your source document with these two\ncommands, placed just before the \\verb|\\end{document}| command:\n\\begin{verbatim}\n \\bibliographystyle{ACM-Reference-Format}\n \n\\section{Introduction}\nACM's consolidated article template, introduced in 2017, provides a\nconsistent \\LaTeX\\ style for use across ACM publications, and\nincorporates accessibility and metadata-extraction functionality\nnecessary for future Digital Library endeavors. Numerous ACM and\nSIG-specific \\LaTeX\\ templates have been examined, and their unique\nfeatures incorporated into this single new template.\n\nIf you are new to publishing with ACM, this document is a valuable\nguide to the process of preparing your work for publication. If you\nhave published with ACM before, this document provides insight and\ninstruction into more recent changes to the article template.\n\nThe ``\\verb|acmart|'' document class can be used to prepare articles\nfor any ACM publication --- conference or journal, and for any stage\nof publication, from review to final ``camera-ready'' copy, to the\nauthor's own version, with {\\itshape very} few changes to the source.\n\n\\section{Template Overview}\nAs noted in the introduction, the ``\\verb|acmart|'' document class can\nbe used to prepare many different kinds of documentation --- a\ndouble-blind initial submission of a full-length technical paper, a\ntwo-page SIGGRAPH Emerging Technologies abstract, a ``camera-ready''\njournal article, a SIGCHI Extended Abstract, and more --- all by\nselecting the appropriate {\\itshape template style} and {\\itshape\n template parameters}.\n\nThis document will explain the major features of the document\nclass. For further information, the {\\itshape \\LaTeX\\ User's Guide} is\navailable from\n\\url{https:\/\/www.acm.org\/publications\/proceedings-template}.\n\n\\subsection{Template Styles}\n\nThe primary parameter given to the ``\\verb|acmart|'' document class is\nthe {\\itshape template style} which corresponds to the kind of publication\nor SIG publishing the work. This parameter is enclosed in square\nbrackets and is a part of the {\\verb|documentclass|} command:\n\\begin{verbatim}\n \\documentclass[STYLE]{acmart}\n\\end{verbatim}\n\nJournals use one of three template styles. All but three ACM journals\nuse the {\\verb|acmsmall|} template style:\n\\begin{itemize}\n\\item {\\texttt{acmsmall}}: The default journal template style.\n\\item {\\texttt{acmlarge}}: Used by JOCCH and TAP.\n\\item {\\texttt{acmtog}}: Used by TOG.\n\\end{itemize}\n\nThe majority of conference proceedings documentation will use the {\\verb|acmconf|} template style.\n\\begin{itemize}\n\\item {\\texttt{acmconf}}: The default proceedings template style.\n\\item{\\texttt{sigchi}}: Used for SIGCHI conference articles.\n\\item{\\texttt{sigchi-a}}: Used for SIGCHI ``Extended Abstract'' articles.\n\\item{\\texttt{sigplan}}: Used for SIGPLAN conference articles.\n\\end{itemize}\n\n\\subsection{Template Parameters}\n\nIn addition to specifying the {\\itshape template style} to be used in\nformatting your work, there are a number of {\\itshape template parameters}\nwhich modify some part of the applied template style. A complete list\nof these parameters can be found in the {\\itshape \\LaTeX\\ User's Guide.}\n\nFrequently-used parameters, or combinations of parameters, include:\n\\begin{itemize}\n\\item {\\texttt{anonymous,review}}: Suitable for a ``double-blind''\n conference submission. Anonymizes the work and includes line\n numbers. Use with the \\texttt{\\acmSubmissionID} command to print the\n submission's unique ID on each page of the work.\n\\item{\\texttt{authorversion}}: Produces a version of the work suitable\n for posting by the author.\n\\item{\\texttt{screen}}: Produces colored hyperlinks.\n\\end{itemize}\n\nThis document uses the following string as the first command in the\nsource file:\n\\begin{verbatim}\n\\documentclass[sigconf,authordraft]{acmart}\n\\end{verbatim}\n\n\\section{Modifications}\n\nModifying the template --- including but not limited to: adjusting\nmargins, typeface sizes, line spacing, paragraph and list definitions,\nand the use of the \\verb|\\vspace| command to manually adjust the\nvertical spacing between elements of your work --- is not allowed.\n\n{\\bfseries Your document will be returned to you for revision if\n modifications are discovered.}\n\n\\section{Typefaces}\n\nThe ``\\verb|acmart|'' document class requires the use of the\n``Libertine'' typeface family. Your \\TeX\\ installation should include\nthis set of packages. Please do not substitute other typefaces. The\n``\\verb|lmodern|'' and ``\\verb|ltimes|'' packages should not be used,\nas they will override the built-in typeface families.\n\n\\section{Title Information}\n\nThe title of your work should use capital letters appropriately -\n\\url{https:\/\/capitalizemytitle.com\/} has useful rules for\ncapitalization. Use the {\\verb|title|} command to define the title of\nyour work. If your work has a subtitle, define it with the\n{\\verb|subtitle|} command. Do not insert line breaks in your title.\n\nIf your title is lengthy, you must define a short version to be used\nin the page headers, to prevent overlapping text. The \\verb|title|\ncommand has a ``short title'' parameter:\n\\begin{verbatim}\n \\title[short title]{full title}\n\\end{verbatim}\n\n\\section{Authors and Affiliations}\n\nEach author must be defined separately for accurate metadata\nidentification. As an exception, multiple authors may share one\naffiliation. Authors' names should not be abbreviated; use full first\nnames wherever possible. Include authors' e-mail addresses whenever\npossible.\n\nGrouping authors' names or e-mail addresses, or providing an ``e-mail\nalias,'' as shown below, is not acceptable:\n\\begin{verbatim}\n \\author{Brooke Aster, David Mehldau}\n \\email{dave,judy,steve@university.edu}\n \\email{firstname.lastname@phillips.org}\n\\end{verbatim}\n\nThe \\verb|authornote| and \\verb|authornotemark| commands allow a note\nto apply to multiple authors --- for example, if the first two authors\nof an article contributed equally to the work.\n\nIf your author list is lengthy, you must define a shortened version of\nthe list of authors to be used in the page headers, to prevent\noverlapping text. The following command should be placed just after\nthe last \\verb|\\author{}| definition:\n\\begin{verbatim}\n \\renewcommand{\\shortauthors}{McCartney, et al.}\n\\end{verbatim}\nOmitting this command will force the use of a concatenated list of all\nof the authors' names, which may result in overlapping text in the\npage headers.\n\nThe article template's documentation, available at\n\\url{https:\/\/www.acm.org\/publications\/proceedings-template}, has a\ncomplete explanation of these commands and tips for their effective\nuse.\n\nNote that authors' addresses are mandatory for journal articles.\n\n\\section{Rights Information}\n\nAuthors of any work published by ACM will need to complete a rights\nform. Depending on the kind of work, and the rights management choice\nmade by the author, this may be copyright transfer, permission,\nlicense, or an OA (open access) agreement.\n\nRegardless of the rights management choice, the author will receive a\ncopy of the completed rights form once it has been submitted. This\nform contains \\LaTeX\\ commands that must be copied into the source\ndocument. When the document source is compiled, these commands and\ntheir parameters add formatted text to several areas of the final\ndocument:\n\\begin{itemize}\n\\item the ``ACM Reference Format'' text on the first page.\n\\item the ``rights management'' text on the first page.\n\\item the conference information in the page header(s).\n\\end{itemize}\n\nRights information is unique to the work; if you are preparing several\nworks for an event, make sure to use the correct set of commands with\neach of the works.\n\nThe ACM Reference Format text is required for all articles over one\npage in length, and is optional for one-page articles (abstracts).\n\n\\section{CCS Concepts and User-Defined Keywords}\n\nTwo elements of the ``acmart'' document class provide powerful\ntaxonomic tools for you to help readers find your work in an online\nsearch.\n\nThe ACM Computing Classification System ---\n\\url{https:\/\/www.acm.org\/publications\/class-2012} --- is a set of\nclassifiers and concepts that describe the computing\ndiscipline. Authors can select entries from this classification\nsystem, via \\url{https:\/\/dl.acm.org\/ccs\/ccs.cfm}, and generate the\ncommands to be included in the \\LaTeX\\ source.\n\nUser-defined keywords are a comma-separated list of words and phrases\nof the authors' choosing, providing a more flexible way of describing\nthe research being presented.\n\nCCS concepts and user-defined keywords are required for for all\narticles over two pages in length, and are optional for one- and\ntwo-page articles (or abstracts).\n\n\\section{Sectioning Commands}\n\nYour work should use standard \\LaTeX\\ sectioning commands:\n\\verb|section|, \\verb|subsection|, \\verb|subsubsection|, and\n\\verb|paragraph|. They should be numbered; do not remove the numbering\nfrom the commands.\n\nSimulating a sectioning command by setting the first word or words of\na paragraph in boldface or italicized text is {\\bfseries not allowed.}\n\n\\section{Tables}\n\nThe ``\\verb|acmart|'' document class includes the ``\\verb|booktabs|''\npackage --- \\url{https:\/\/ctan.org\/pkg\/booktabs} --- for preparing\nhigh-quality tables.\n\nTable captions are placed {\\itshape above} the table.\n\nBecause tables cannot be split across pages, the best placement for\nthem is typically the top of the page nearest their initial cite. To\nensure this proper ``floating'' placement of tables, use the\nenvironment \\textbf{table} to enclose the table's contents and the\ntable caption. The contents of the table itself must go in the\n\\textbf{tabular} environment, to be aligned properly in rows and\ncolumns, with the desired horizontal and vertical rules. Again,\ndetailed instructions on \\textbf{tabular} material are found in the\n\\textit{\\LaTeX\\ User's Guide}.\n\nImmediately following this sentence is the point at which\nTable~\\ref{tab:freq} is included in the input file; compare the\nplacement of the table here with the table in the printed output of\nthis document.\n\n\\begin{table}\n \\caption{Frequency of Special Characters}\n \\label{tab:freq}\n \\begin{tabular}{ccl}\n \\toprule\n Non-English or Math&Frequency&Comments\\\\\n \\midrule\n \\O & 1 in 1,000& For Swedish names\\\\\n $\\pi$ & 1 in 5& Common in math\\\\\n \\$ & 4 in 5 & Used in business\\\\\n $\\Psi^2_1$ & 1 in 40,000& Unexplained usage\\\\\n \\bottomrule\n\\end{tabular}\n\\end{table}\n\nTo set a wider table, which takes up the whole width of the page's\nlive area, use the environment \\textbf{table*} to enclose the table's\ncontents and the table caption. As with a single-column table, this\nwide table will ``float'' to a location deemed more\ndesirable. Immediately following this sentence is the point at which\nTable~\\ref{tab:commands} is included in the input file; again, it is\ninstructive to compare the placement of the table here with the table\nin the printed output of this document.\n\n\\begin{table*}\n \\caption{Some Typical Commands}\n \\label{tab:commands}\n \\begin{tabular}{ccl}\n \\toprule\n Command &A Number & Comments\\\\\n \\midrule\n \\texttt{{\\char'134}author} & 100& Author \\\\\n \\texttt{{\\char'134}table}& 300 & For tables\\\\\n \\texttt{{\\char'134}table*}& 400& For wider tables\\\\\n \\bottomrule\n \\end{tabular}\n\\end{table*}\n\nAlways use midrule to separate table header rows from data rows, and\nuse it only for this purpose. This enables assistive technologies to\nrecognise table headers and support their users in navigating tables\nmore easily.\n\n\\section{Math Equations}\nYou may want to display math equations in three distinct styles:\ninline, numbered or non-numbered display. Each of the three are\ndiscussed in the next sections.\n\n\\subsection{Inline (In-text) Equations}\nA formula that appears in the running text is called an inline or\nin-text formula. It is produced by the \\textbf{math} environment,\nwhich can be invoked with the usual\n\\texttt{{\\char'134}begin\\,\\ldots{\\char'134}end} construction or with\nthe short form \\texttt{\\$\\,\\ldots\\$}. You can use any of the symbols\nand structures, from $\\alpha$ to $\\omega$, available in\n\\LaTeX~\\cite{Lamport:LaTeX}; this section will simply show a few\nexamples of in-text equations in context. Notice how this equation:\n\\begin{math}\n \\lim_{n\\rightarrow \\infty}x=0\n\\end{math},\nset here in in-line math style, looks slightly different when\nset in display style. (See next section).\n\n\\subsection{Display Equations}\nA numbered display equation---one set off by vertical space from the\ntext and centered horizontally---is produced by the \\textbf{equation}\nenvironment. An unnumbered display equation is produced by the\n\\textbf{displaymath} environment.\n\nAgain, in either environment, you can use any of the symbols and\nstructures available in \\LaTeX\\@; this section will just give a couple\nof examples of display equations in context. First, consider the\nequation, shown as an inline equation above:\n\\begin{equation}\n \\lim_{n\\rightarrow \\infty}x=0\n\\end{equation}\nNotice how it is formatted somewhat differently in\nthe \\textbf{displaymath}\nenvironment. Now, we'll enter an unnumbered equation:\n\\begin{displaymath}\n \\sum_{i=0}^{\\infty} x + 1\n\\end{displaymath}\nand follow it with another numbered equation:\n\\begin{equation}\n \\sum_{i=0}^{\\infty}x_i=\\int_{0}^{\\pi+2} f\n\\end{equation}\njust to demonstrate \\LaTeX's able handling of numbering.\n\n\\section{Figures}\n\nThe ``\\verb|figure|'' environment should be used for figures. One or\nmore images can be placed within a figure. If your figure contains\nthird-party material, you must clearly identify it as such, as shown\nin the example below.\n\\begin{figure}[h]\n \\centering\n \\includegraphics[width=\\linewidth]{sample-franklin}\n \\caption{1907 Franklin Model D roadster. Photograph by Harris \\&\n Ewing, Inc. [Public domain], via Wikimedia\n Commons. (\\url{https:\/\/goo.gl\/VLCRBB}).}\n \\Description{A woman and a girl in white dresses sit in an open car.}\n\\end{figure}\n\nYour figures should contain a caption which describes the figure to\nthe reader.\n\nFigure captions are placed {\\itshape below} the figure.\n\nEvery figure should also have a figure description unless it is purely\ndecorative. These descriptions convey what's in the image to someone\nwho cannot see it. They are also used by search engine crawlers for\nindexing images, and when images cannot be loaded.\n\nA figure description must be unformatted plain text less than 2000\ncharacters long (including spaces). {\\bfseries Figure descriptions\n should not repeat the figure caption \u2013 their purpose is to capture\n important information that is not already provided in the caption or\n the main text of the paper.} For figures that convey important and\ncomplex new information, a short text description may not be\nadequate. More complex alternative descriptions can be placed in an\nappendix and referenced in a short figure description. For example,\nprovide a data table capturing the information in a bar chart, or a\nstructured list representing a graph. For additional information\nregarding how best to write figure descriptions and why doing this is\nso important, please see\n\\url{https:\/\/www.acm.org\/publications\/taps\/describing-figures\/}.\n\n\\subsection{The ``Teaser Figure''}\n\nA ``teaser figure'' is an image, or set of images in one figure, that\nare placed after all author and affiliation information, and before\nthe body of the article, spanning the page. If you wish to have such a\nfigure in your article, place the command immediately before the\n\\verb|\\maketitle| command:\n\\begin{verbatim}\n \\begin{teaserfigure}\n \\includegraphics[width=\\textwidth]{sampleteaser}\n \\caption{figure caption}\n \\Description{figure description}\n \\end{teaserfigure}\n\\end{verbatim}\n\n\\section{Citations and Bibliographies}\n\nThe use of \\BibTeX\\ for the preparation and formatting of one's\nreferences is strongly recommended. Authors' names should be complete\n--- use full first names (``Donald E. Knuth'') not initials\n(``D. E. Knuth'') --- and the salient identifying features of a\nreference should be included: title, year, volume, number, pages,\narticle DOI, etc.\n\nThe bibliography is included in your source document with these two\ncommands, placed just before the \\verb|\\end{document}| command:\n\\begin{verbatim}\n \\bibliographystyle{ACM-Reference-Format}\n \n\\section{Introduction}\nACM's consolidated article template, introduced in 2017, provides a\nconsistent \\LaTeX\\ style for use across ACM publications, and\nincorporates accessibility and metadata-extraction functionality\nnecessary for future Digital Library endeavors. Numerous ACM and\nSIG-specific \\LaTeX\\ templates have been examined, and their unique\nfeatures incorporated into this single new template.\n\nIf you are new to publishing with ACM, this document is a valuable\nguide to the process of preparing your work for publication. If you\nhave published with ACM before, this document provides insight and\ninstruction into more recent changes to the article template.\n\nThe ``\\verb|acmart|'' document class can be used to prepare articles\nfor any ACM publication --- conference or journal, and for any stage\nof publication, from review to final ``camera-ready'' copy, to the\nauthor's own version, with {\\itshape very} few changes to the source.\n\n\\section{Template Overview}\nAs noted in the introduction, the ``\\verb|acmart|'' document class can\nbe used to prepare many different kinds of documentation --- a\ndouble-blind initial submission of a full-length technical paper, a\ntwo-page SIGGRAPH Emerging Technologies abstract, a ``camera-ready''\njournal article, a SIGCHI Extended Abstract, and more --- all by\nselecting the appropriate {\\itshape template style} and {\\itshape\n template parameters}.\n\nThis document will explain the major features of the document\nclass. For further information, the {\\itshape \\LaTeX\\ User's Guide} is\navailable from\n\\url{https:\/\/www.acm.org\/publications\/proceedings-template}.\n\n\\subsection{Template Styles}\n\nThe primary parameter given to the ``\\verb|acmart|'' document class is\nthe {\\itshape template style} which corresponds to the kind of publication\nor SIG publishing the work. This parameter is enclosed in square\nbrackets and is a part of the {\\verb|documentclass|} command:\n\\begin{verbatim}\n \\documentclass[STYLE]{acmart}\n\\end{verbatim}\n\nJournals use one of three template styles. All but three ACM journals\nuse the {\\verb|acmsmall|} template style:\n\\begin{itemize}\n\\item {\\texttt{acmsmall}}: The default journal template style.\n\\item {\\texttt{acmlarge}}: Used by JOCCH and TAP.\n\\item {\\texttt{acmtog}}: Used by TOG.\n\\end{itemize}\n\nThe majority of conference proceedings documentation will use the {\\verb|acmconf|} template style.\n\\begin{itemize}\n\\item {\\texttt{acmconf}}: The default proceedings template style.\n\\item{\\texttt{sigchi}}: Used for SIGCHI conference articles.\n\\item{\\texttt{sigchi-a}}: Used for SIGCHI ``Extended Abstract'' articles.\n\\item{\\texttt{sigplan}}: Used for SIGPLAN conference articles.\n\\end{itemize}\n\n\\subsection{Template Parameters}\n\nIn addition to specifying the {\\itshape template style} to be used in\nformatting your work, there are a number of {\\itshape template parameters}\nwhich modify some part of the applied template style. A complete list\nof these parameters can be found in the {\\itshape \\LaTeX\\ User's Guide.}\n\nFrequently-used parameters, or combinations of parameters, include:\n\\begin{itemize}\n\\item {\\texttt{anonymous,review}}: Suitable for a ``double-blind''\n conference submission. Anonymizes the work and includes line\n numbers. Use with the \\texttt{\\acmSubmissionID} command to print the\n submission's unique ID on each page of the work.\n\\item{\\texttt{authorversion}}: Produces a version of the work suitable\n for posting by the author.\n\\item{\\texttt{screen}}: Produces colored hyperlinks.\n\\end{itemize}\n\nThis document uses the following string as the first command in the\nsource file:\n\\begin{verbatim}\n\\documentclass[sigconf]{acmart}\n\\end{verbatim}\n\n\\section{Modifications}\n\nModifying the template --- including but not limited to: adjusting\nmargins, typeface sizes, line spacing, paragraph and list definitions,\nand the use of the \\verb|\\vspace| command to manually adjust the\nvertical spacing between elements of your work --- is not allowed.\n\n{\\bfseries Your document will be returned to you for revision if\n modifications are discovered.}\n\n\\section{Typefaces}\n\nThe ``\\verb|acmart|'' document class requires the use of the\n``Libertine'' typeface family. Your \\TeX\\ installation should include\nthis set of packages. Please do not substitute other typefaces. The\n``\\verb|lmodern|'' and ``\\verb|ltimes|'' packages should not be used,\nas they will override the built-in typeface families.\n\n\\section{Title Information}\n\nThe title of your work should use capital letters appropriately -\n\\url{https:\/\/capitalizemytitle.com\/} has useful rules for\ncapitalization. Use the {\\verb|title|} command to define the title of\nyour work. If your work has a subtitle, define it with the\n{\\verb|subtitle|} command. Do not insert line breaks in your title.\n\nIf your title is lengthy, you must define a short version to be used\nin the page headers, to prevent overlapping text. The \\verb|title|\ncommand has a ``short title'' parameter:\n\\begin{verbatim}\n \\title[short title]{full title}\n\\end{verbatim}\n\n\\section{Authors and Affiliations}\n\nEach author must be defined separately for accurate metadata\nidentification. As an exception, multiple authors may share one\naffiliation. Authors' names should not be abbreviated; use full first\nnames wherever possible. Include authors' e-mail addresses whenever\npossible.\n\nGrouping authors' names or e-mail addresses, or providing an ``e-mail\nalias,'' as shown below, is not acceptable:\n\\begin{verbatim}\n \\author{Brooke Aster, David Mehldau}\n \\email{dave,judy,steve@university.edu}\n \\email{firstname.lastname@phillips.org}\n\\end{verbatim}\n\nThe \\verb|authornote| and \\verb|authornotemark| commands allow a note\nto apply to multiple authors --- for example, if the first two authors\nof an article contributed equally to the work.\n\nIf your author list is lengthy, you must define a shortened version of\nthe list of authors to be used in the page headers, to prevent\noverlapping text. The following command should be placed just after\nthe last \\verb|\\author{}| definition:\n\\begin{verbatim}\n \\renewcommand{\\shortauthors}{McCartney, et al.}\n\\end{verbatim}\nOmitting this command will force the use of a concatenated list of all\nof the authors' names, which may result in overlapping text in the\npage headers.\n\nThe article template's documentation, available at\n\\url{https:\/\/www.acm.org\/publications\/proceedings-template}, has a\ncomplete explanation of these commands and tips for their effective\nuse.\n\nNote that authors' addresses are mandatory for journal articles.\n\n\\section{Rights Information}\n\nAuthors of any work published by ACM will need to complete a rights\nform. Depending on the kind of work, and the rights management choice\nmade by the author, this may be copyright transfer, permission,\nlicense, or an OA (open access) agreement.\n\nRegardless of the rights management choice, the author will receive a\ncopy of the completed rights form once it has been submitted. This\nform contains \\LaTeX\\ commands that must be copied into the source\ndocument. When the document source is compiled, these commands and\ntheir parameters add formatted text to several areas of the final\ndocument:\n\\begin{itemize}\n\\item the ``ACM Reference Format'' text on the first page.\n\\item the ``rights management'' text on the first page.\n\\item the conference information in the page header(s).\n\\end{itemize}\n\nRights information is unique to the work; if you are preparing several\nworks for an event, make sure to use the correct set of commands with\neach of the works.\n\nThe ACM Reference Format text is required for all articles over one\npage in length, and is optional for one-page articles (abstracts).\n\n\\section{CCS Concepts and User-Defined Keywords}\n\nTwo elements of the ``acmart'' document class provide powerful\ntaxonomic tools for you to help readers find your work in an online\nsearch.\n\nThe ACM Computing Classification System ---\n\\url{https:\/\/www.acm.org\/publications\/class-2012} --- is a set of\nclassifiers and concepts that describe the computing\ndiscipline. Authors can select entries from this classification\nsystem, via \\url{https:\/\/dl.acm.org\/ccs\/ccs.cfm}, and generate the\ncommands to be included in the \\LaTeX\\ source.\n\nUser-defined keywords are a comma-separated list of words and phrases\nof the authors' choosing, providing a more flexible way of describing\nthe research being presented.\n\nCCS concepts and user-defined keywords are required for for all\narticles over two pages in length, and are optional for one- and\ntwo-page articles (or abstracts).\n\n\\section{Sectioning Commands}\n\nYour work should use standard \\LaTeX\\ sectioning commands:\n\\verb|section|, \\verb|subsection|, \\verb|subsubsection|, and\n\\verb|paragraph|. They should be numbered; do not remove the numbering\nfrom the commands.\n\nSimulating a sectioning command by setting the first word or words of\na paragraph in boldface or italicized text is {\\bfseries not allowed.}\n\n\\section{Tables}\n\nThe ``\\verb|acmart|'' document class includes the ``\\verb|booktabs|''\npackage --- \\url{https:\/\/ctan.org\/pkg\/booktabs} --- for preparing\nhigh-quality tables.\n\nTable captions are placed {\\itshape above} the table.\n\nBecause tables cannot be split across pages, the best placement for\nthem is typically the top of the page nearest their initial cite. To\nensure this proper ``floating'' placement of tables, use the\nenvironment \\textbf{table} to enclose the table's contents and the\ntable caption. The contents of the table itself must go in the\n\\textbf{tabular} environment, to be aligned properly in rows and\ncolumns, with the desired horizontal and vertical rules. Again,\ndetailed instructions on \\textbf{tabular} material are found in the\n\\textit{\\LaTeX\\ User's Guide}.\n\nImmediately following this sentence is the point at which\nTable~\\ref{tab:freq} is included in the input file; compare the\nplacement of the table here with the table in the printed output of\nthis document.\n\n\\begin{table}\n \\caption{Frequency of Special Characters}\n \\label{tab:freq}\n \\begin{tabular}{ccl}\n \\toprule\n Non-English or Math&Frequency&Comments\\\\\n \\midrule\n \\O & 1 in 1,000& For Swedish names\\\\\n $\\pi$ & 1 in 5& Common in math\\\\\n \\$ & 4 in 5 & Used in business\\\\\n $\\Psi^2_1$ & 1 in 40,000& Unexplained usage\\\\\n \\bottomrule\n\\end{tabular}\n\\end{table}\n\nTo set a wider table, which takes up the whole width of the page's\nlive area, use the environment \\textbf{table*} to enclose the table's\ncontents and the table caption. As with a single-column table, this\nwide table will ``float'' to a location deemed more\ndesirable. Immediately following this sentence is the point at which\nTable~\\ref{tab:commands} is included in the input file; again, it is\ninstructive to compare the placement of the table here with the table\nin the printed output of this document.\n\n\\begin{table*}\n \\caption{Some Typical Commands}\n \\label{tab:commands}\n \\begin{tabular}{ccl}\n \\toprule\n Command &A Number & Comments\\\\\n \\midrule\n \\texttt{{\\char'134}author} & 100& Author \\\\\n \\texttt{{\\char'134}table}& 300 & For tables\\\\\n \\texttt{{\\char'134}table*}& 400& For wider tables\\\\\n \\bottomrule\n \\end{tabular}\n\\end{table*}\n\nAlways use midrule to separate table header rows from data rows, and\nuse it only for this purpose. This enables assistive technologies to\nrecognise table headers and support their users in navigating tables\nmore easily.\n\n\\section{Math Equations}\nYou may want to display math equations in three distinct styles:\ninline, numbered or non-numbered display. Each of the three are\ndiscussed in the next sections.\n\n\\subsection{Inline (In-text) Equations}\nA formula that appears in the running text is called an inline or\nin-text formula. It is produced by the \\textbf{math} environment,\nwhich can be invoked with the usual\n\\texttt{{\\char'134}begin\\,\\ldots{\\char'134}end} construction or with\nthe short form \\texttt{\\$\\,\\ldots\\$}. You can use any of the symbols\nand structures, from $\\alpha$ to $\\omega$, available in\n\\LaTeX~\\cite{Lamport:LaTeX}; this section will simply show a few\nexamples of in-text equations in context. Notice how this equation:\n\\begin{math}\n \\lim_{n\\rightarrow \\infty}x=0\n\\end{math},\nset here in in-line math style, looks slightly different when\nset in display style. (See next section).\n\n\\subsection{Display Equations}\nA numbered display equation---one set off by vertical space from the\ntext and centered horizontally---is produced by the \\textbf{equation}\nenvironment. An unnumbered display equation is produced by the\n\\textbf{displaymath} environment.\n\nAgain, in either environment, you can use any of the symbols and\nstructures available in \\LaTeX\\@; this section will just give a couple\nof examples of display equations in context. First, consider the\nequation, shown as an inline equation above:\n\\begin{equation}\n \\lim_{n\\rightarrow \\infty}x=0\n\\end{equation}\nNotice how it is formatted somewhat differently in\nthe \\textbf{displaymath}\nenvironment. Now, we'll enter an unnumbered equation:\n\\begin{displaymath}\n \\sum_{i=0}^{\\infty} x + 1\n\\end{displaymath}\nand follow it with another numbered equation:\n\\begin{equation}\n \\sum_{i=0}^{\\infty}x_i=\\int_{0}^{\\pi+2} f\n\\end{equation}\njust to demonstrate \\LaTeX's able handling of numbering.\n\n\\section{Figures}\n\nThe ``\\verb|figure|'' environment should be used for figures. One or\nmore images can be placed within a figure. If your figure contains\nthird-party material, you must clearly identify it as such, as shown\nin the example below.\n\\begin{figure}[h]\n \\centering\n \\includegraphics[width=\\linewidth]{sample-franklin}\n \\caption{1907 Franklin Model D roadster. Photograph by Harris \\&\n Ewing, Inc. [Public domain], via Wikimedia\n Commons. (\\url{https:\/\/goo.gl\/VLCRBB}).}\n \\Description{A woman and a girl in white dresses sit in an open car.}\n\\end{figure}\n\nYour figures should contain a caption which describes the figure to\nthe reader.\n\nFigure captions are placed {\\itshape below} the figure.\n\nEvery figure should also have a figure description unless it is purely\ndecorative. These descriptions convey what's in the image to someone\nwho cannot see it. They are also used by search engine crawlers for\nindexing images, and when images cannot be loaded.\n\nA figure description must be unformatted plain text less than 2000\ncharacters long (including spaces). {\\bfseries Figure descriptions\n should not repeat the figure caption \u2013 their purpose is to capture\n important information that is not already provided in the caption or\n the main text of the paper.} For figures that convey important and\ncomplex new information, a short text description may not be\nadequate. More complex alternative descriptions can be placed in an\nappendix and referenced in a short figure description. For example,\nprovide a data table capturing the information in a bar chart, or a\nstructured list representing a graph. For additional information\nregarding how best to write figure descriptions and why doing this is\nso important, please see\n\\url{https:\/\/www.acm.org\/publications\/taps\/describing-figures\/}.\n\n\\subsection{The ``Teaser Figure''}\n\nA ``teaser figure'' is an image, or set of images in one figure, that\nare placed after all author and affiliation information, and before\nthe body of the article, spanning the page. If you wish to have such a\nfigure in your article, place the command immediately before the\n\\verb|\\maketitle| command:\n\\begin{verbatim}\n \\begin{teaserfigure}\n \\includegraphics[width=\\textwidth]{sampleteaser}\n \\caption{figure caption}\n \\Description{figure description}\n \\end{teaserfigure}\n\\end{verbatim}\n\n\\section{Citations and Bibliographies}\n\nThe use of \\BibTeX\\ for the preparation and formatting of one's\nreferences is strongly recommended. Authors' names should be complete\n--- use full first names (``Donald E. Knuth'') not initials\n(``D. E. Knuth'') --- and the salient identifying features of a\nreference should be included: title, year, volume, number, pages,\narticle DOI, etc.\n\nThe bibliography is included in your source document with these two\ncommands, placed just before the \\verb|\\end{document}| command:\n\\begin{verbatim}\n \\bibliographystyle{ACM-Reference-Format}\n \n\\section{Introduction}\nACM's consolidated article template, introduced in 2017, provides a\nconsistent \\LaTeX\\ style for use across ACM publications, and\nincorporates accessibility and metadata-extraction functionality\nnecessary for future Digital Library endeavors. Numerous ACM and\nSIG-specific \\LaTeX\\ templates have been examined, and their unique\nfeatures incorporated into this single new template.\n\nIf you are new to publishing with ACM, this document is a valuable\nguide to the process of preparing your work for publication. If you\nhave published with ACM before, this document provides insight and\ninstruction into more recent changes to the article template.\n\nThe ``\\verb|acmart|'' document class can be used to prepare articles\nfor any ACM publication --- conference or journal, and for any stage\nof publication, from review to final ``camera-ready'' copy, to the\nauthor's own version, with {\\itshape very} few changes to the source.\n\n\\section{Template Overview}\nAs noted in the introduction, the ``\\verb|acmart|'' document class can\nbe used to prepare many different kinds of documentation --- a\ndouble-blind initial submission of a full-length technical paper, a\ntwo-page SIGGRAPH Emerging Technologies abstract, a ``camera-ready''\njournal article, a SIGCHI Extended Abstract, and more --- all by\nselecting the appropriate {\\itshape template style} and {\\itshape\n template parameters}.\n\nThis document will explain the major features of the document\nclass. For further information, the {\\itshape \\LaTeX\\ User's Guide} is\navailable from\n\\url{https:\/\/www.acm.org\/publications\/proceedings-template}.\n\n\\subsection{Template Styles}\n\nThe primary parameter given to the ``\\verb|acmart|'' document class is\nthe {\\itshape template style} which corresponds to the kind of publication\nor SIG publishing the work. This parameter is enclosed in square\nbrackets and is a part of the {\\verb|documentclass|} command:\n\\begin{verbatim}\n \\documentclass[STYLE]{acmart}\n\\end{verbatim}\n\nJournals use one of three template styles. All but three ACM journals\nuse the {\\verb|acmsmall|} template style:\n\\begin{itemize}\n\\item {\\texttt{acmsmall}}: The default journal template style.\n\\item {\\texttt{acmlarge}}: Used by JOCCH and TAP.\n\\item {\\texttt{acmtog}}: Used by TOG.\n\\end{itemize}\n\nThe majority of conference proceedings documentation will use the {\\verb|acmconf|} template style.\n\\begin{itemize}\n\\item {\\texttt{acmconf}}: The default proceedings template style.\n\\item{\\texttt{sigchi}}: Used for SIGCHI conference articles.\n\\item{\\texttt{sigchi-a}}: Used for SIGCHI ``Extended Abstract'' articles.\n\\item{\\texttt{sigplan}}: Used for SIGPLAN conference articles.\n\\end{itemize}\n\n\\subsection{Template Parameters}\n\nIn addition to specifying the {\\itshape template style} to be used in\nformatting your work, there are a number of {\\itshape template parameters}\nwhich modify some part of the applied template style. A complete list\nof these parameters can be found in the {\\itshape \\LaTeX\\ User's Guide.}\n\nFrequently-used parameters, or combinations of parameters, include:\n\\begin{itemize}\n\\item {\\texttt{anonymous,review}}: Suitable for a ``double-blind''\n conference submission. Anonymizes the work and includes line\n numbers. Use with the \\texttt{\\acmSubmissionID} command to print the\n submission's unique ID on each page of the work.\n\\item{\\texttt{authorversion}}: Produces a version of the work suitable\n for posting by the author.\n\\item{\\texttt{screen}}: Produces colored hyperlinks.\n\\end{itemize}\n\nThis document uses the following string as the first command in the\nsource file:\n\\begin{verbatim}\n\\documentclass[manuscript,screen,review]{acmart}\n\\end{verbatim}\n\n\\section{Modifications}\n\nModifying the template --- including but not limited to: adjusting\nmargins, typeface sizes, line spacing, paragraph and list definitions,\nand the use of the \\verb|\\vspace| command to manually adjust the\nvertical spacing between elements of your work --- is not allowed.\n\n{\\bfseries Your document will be returned to you for revision if\n modifications are discovered.}\n\n\\section{Typefaces}\n\nThe ``\\verb|acmart|'' document class requires the use of the\n``Libertine'' typeface family. Your \\TeX\\ installation should include\nthis set of packages. Please do not substitute other typefaces. The\n``\\verb|lmodern|'' and ``\\verb|ltimes|'' packages should not be used,\nas they will override the built-in typeface families.\n\n\\section{Title Information}\n\nThe title of your work should use capital letters appropriately -\n\\url{https:\/\/capitalizemytitle.com\/} has useful rules for\ncapitalization. Use the {\\verb|title|} command to define the title of\nyour work. If your work has a subtitle, define it with the\n{\\verb|subtitle|} command. Do not insert line breaks in your title.\n\nIf your title is lengthy, you must define a short version to be used\nin the page headers, to prevent overlapping text. The \\verb|title|\ncommand has a ``short title'' parameter:\n\\begin{verbatim}\n \\title[short title]{full title}\n\\end{verbatim}\n\n\\section{Authors and Affiliations}\n\nEach author must be defined separately for accurate metadata\nidentification. As an exception, multiple authors may share one\naffiliation. Authors' names should not be abbreviated; use full first\nnames wherever possible. Include authors' e-mail addresses whenever\npossible.\n\nGrouping authors' names or e-mail addresses, or providing an ``e-mail\nalias,'' as shown below, is not acceptable:\n\\begin{verbatim}\n \\author{Brooke Aster, David Mehldau}\n \\email{dave,judy,steve@university.edu}\n \\email{firstname.lastname@phillips.org}\n\\end{verbatim}\n\nThe \\verb|authornote| and \\verb|authornotemark| commands allow a note\nto apply to multiple authors --- for example, if the first two authors\nof an article contributed equally to the work.\n\nIf your author list is lengthy, you must define a shortened version of\nthe list of authors to be used in the page headers, to prevent\noverlapping text. The following command should be placed just after\nthe last \\verb|\\author{}| definition:\n\\begin{verbatim}\n \\renewcommand{\\shortauthors}{McCartney, et al.}\n\\end{verbatim}\nOmitting this command will force the use of a concatenated list of all\nof the authors' names, which may result in overlapping text in the\npage headers.\n\nThe article template's documentation, available at\n\\url{https:\/\/www.acm.org\/publications\/proceedings-template}, has a\ncomplete explanation of these commands and tips for their effective\nuse.\n\nNote that authors' addresses are mandatory for journal articles.\n\n\\section{Rights Information}\n\nAuthors of any work published by ACM will need to complete a rights\nform. Depending on the kind of work, and the rights management choice\nmade by the author, this may be copyright transfer, permission,\nlicense, or an OA (open access) agreement.\n\nRegardless of the rights management choice, the author will receive a\ncopy of the completed rights form once it has been submitted. This\nform contains \\LaTeX\\ commands that must be copied into the source\ndocument. When the document source is compiled, these commands and\ntheir parameters add formatted text to several areas of the final\ndocument:\n\\begin{itemize}\n\\item the ``ACM Reference Format'' text on the first page.\n\\item the ``rights management'' text on the first page.\n\\item the conference information in the page header(s).\n\\end{itemize}\n\nRights information is unique to the work; if you are preparing several\nworks for an event, make sure to use the correct set of commands with\neach of the works.\n\nThe ACM Reference Format text is required for all articles over one\npage in length, and is optional for one-page articles (abstracts).\n\n\\section{CCS Concepts and User-Defined Keywords}\n\nTwo elements of the ``acmart'' document class provide powerful\ntaxonomic tools for you to help readers find your work in an online\nsearch.\n\nThe ACM Computing Classification System ---\n\\url{https:\/\/www.acm.org\/publications\/class-2012} --- is a set of\nclassifiers and concepts that describe the computing\ndiscipline. Authors can select entries from this classification\nsystem, via \\url{https:\/\/dl.acm.org\/ccs\/ccs.cfm}, and generate the\ncommands to be included in the \\LaTeX\\ source.\n\nUser-defined keywords are a comma-separated list of words and phrases\nof the authors' choosing, providing a more flexible way of describing\nthe research being presented.\n\nCCS concepts and user-defined keywords are required for for all\narticles over two pages in length, and are optional for one- and\ntwo-page articles (or abstracts).\n\n\\section{Sectioning Commands}\n\nYour work should use standard \\LaTeX\\ sectioning commands:\n\\verb|section|, \\verb|subsection|, \\verb|subsubsection|, and\n\\verb|paragraph|. They should be numbered; do not remove the numbering\nfrom the commands.\n\nSimulating a sectioning command by setting the first word or words of\na paragraph in boldface or italicized text is {\\bfseries not allowed.}\n\n\\section{Tables}\n\nThe ``\\verb|acmart|'' document class includes the ``\\verb|booktabs|''\npackage --- \\url{https:\/\/ctan.org\/pkg\/booktabs} --- for preparing\nhigh-quality tables.\n\nTable captions are placed {\\itshape above} the table.\n\nBecause tables cannot be split across pages, the best placement for\nthem is typically the top of the page nearest their initial cite. To\nensure this proper ``floating'' placement of tables, use the\nenvironment \\textbf{table} to enclose the table's contents and the\ntable caption. The contents of the table itself must go in the\n\\textbf{tabular} environment, to be aligned properly in rows and\ncolumns, with the desired horizontal and vertical rules. Again,\ndetailed instructions on \\textbf{tabular} material are found in the\n\\textit{\\LaTeX\\ User's Guide}.\n\nImmediately following this sentence is the point at which\nTable~\\ref{tab:freq} is included in the input file; compare the\nplacement of the table here with the table in the printed output of\nthis document.\n\n\\begin{table}\n \\caption{Frequency of Special Characters}\n \\label{tab:freq}\n \\begin{tabular}{ccl}\n \\toprule\n Non-English or Math&Frequency&Comments\\\\\n \\midrule\n \\O & 1 in 1,000& For Swedish names\\\\\n $\\pi$ & 1 in 5& Common in math\\\\\n \\$ & 4 in 5 & Used in business\\\\\n $\\Psi^2_1$ & 1 in 40,000& Unexplained usage\\\\\n \\bottomrule\n\\end{tabular}\n\\end{table}\n\nTo set a wider table, which takes up the whole width of the page's\nlive area, use the environment \\textbf{table*} to enclose the table's\ncontents and the table caption. As with a single-column table, this\nwide table will ``float'' to a location deemed more\ndesirable. Immediately following this sentence is the point at which\nTable~\\ref{tab:commands} is included in the input file; again, it is\ninstructive to compare the placement of the table here with the table\nin the printed output of this document.\n\n\\begin{table*}\n \\caption{Some Typical Commands}\n \\label{tab:commands}\n \\begin{tabular}{ccl}\n \\toprule\n Command &A Number & Comments\\\\\n \\midrule\n \\texttt{{\\char'134}author} & 100& Author \\\\\n \\texttt{{\\char'134}table}& 300 & For tables\\\\\n \\texttt{{\\char'134}table*}& 400& For wider tables\\\\\n \\bottomrule\n \\end{tabular}\n\\end{table*}\n\nAlways use midrule to separate table header rows from data rows, and\nuse it only for this purpose. This enables assistive technologies to\nrecognise table headers and support their users in navigating tables\nmore easily.\n\n\\section{Math Equations}\nYou may want to display math equations in three distinct styles:\ninline, numbered or non-numbered display. Each of the three are\ndiscussed in the next sections.\n\n\\subsection{Inline (In-text) Equations}\nA formula that appears in the running text is called an inline or\nin-text formula. It is produced by the \\textbf{math} environment,\nwhich can be invoked with the usual\n\\texttt{{\\char'134}begin\\,\\ldots{\\char'134}end} construction or with\nthe short form \\texttt{\\$\\,\\ldots\\$}. You can use any of the symbols\nand structures, from $\\alpha$ to $\\omega$, available in\n\\LaTeX~\\cite{Lamport:LaTeX}; this section will simply show a few\nexamples of in-text equations in context. Notice how this equation:\n\\begin{math}\n \\lim_{n\\rightarrow \\infty}x=0\n\\end{math},\nset here in in-line math style, looks slightly different when\nset in display style. (See next section).\n\n\\subsection{Display Equations}\nA numbered display equation---one set off by vertical space from the\ntext and centered horizontally---is produced by the \\textbf{equation}\nenvironment. An unnumbered display equation is produced by the\n\\textbf{displaymath} environment.\n\nAgain, in either environment, you can use any of the symbols and\nstructures available in \\LaTeX\\@; this section will just give a couple\nof examples of display equations in context. First, consider the\nequation, shown as an inline equation above:\n\\begin{equation}\n \\lim_{n\\rightarrow \\infty}x=0\n\\end{equation}\nNotice how it is formatted somewhat differently in\nthe \\textbf{displaymath}\nenvironment. Now, we'll enter an unnumbered equation:\n\\begin{displaymath}\n \\sum_{i=0}^{\\infty} x + 1\n\\end{displaymath}\nand follow it with another numbered equation:\n\\begin{equation}\n \\sum_{i=0}^{\\infty}x_i=\\int_{0}^{\\pi+2} f\n\\end{equation}\njust to demonstrate \\LaTeX's able handling of numbering.\n\n\\section{Figures}\n\nThe ``\\verb|figure|'' environment should be used for figures. One or\nmore images can be placed within a figure. If your figure contains\nthird-party material, you must clearly identify it as such, as shown\nin the example below.\n\\begin{figure}[h]\n \\centering\n \\includegraphics[width=\\linewidth]{sample-franklin}\n \\caption{1907 Franklin Model D roadster. Photograph by Harris \\&\n Ewing, Inc. [Public domain], via Wikimedia\n Commons. (\\url{https:\/\/goo.gl\/VLCRBB}).}\n \\Description{A woman and a girl in white dresses sit in an open car.}\n\\end{figure}\n\nYour figures should contain a caption which describes the figure to\nthe reader.\n\nFigure captions are placed {\\itshape below} the figure.\n\nEvery figure should also have a figure description unless it is purely\ndecorative. These descriptions convey what's in the image to someone\nwho cannot see it. They are also used by search engine crawlers for\nindexing images, and when images cannot be loaded.\n\nA figure description must be unformatted plain text less than 2000\ncharacters long (including spaces). {\\bfseries Figure descriptions\n should not repeat the figure caption \u2013 their purpose is to capture\n important information that is not already provided in the caption or\n the main text of the paper.} For figures that convey important and\ncomplex new information, a short text description may not be\nadequate. More complex alternative descriptions can be placed in an\nappendix and referenced in a short figure description. For example,\nprovide a data table capturing the information in a bar chart, or a\nstructured list representing a graph. For additional information\nregarding how best to write figure descriptions and why doing this is\nso important, please see\n\\url{https:\/\/www.acm.org\/publications\/taps\/describing-figures\/}.\n\n\\subsection{The ``Teaser Figure''}\n\nA ``teaser figure'' is an image, or set of images in one figure, that\nare placed after all author and affiliation information, and before\nthe body of the article, spanning the page. If you wish to have such a\nfigure in your article, place the command immediately before the\n\\verb|\\maketitle| command:\n\\begin{verbatim}\n \\begin{teaserfigure}\n \\includegraphics[width=\\textwidth]{sampleteaser}\n \\caption{figure caption}\n \\Description{figure description}\n \\end{teaserfigure}\n\\end{verbatim}\n\n\\section{Citations and Bibliographies}\n\nThe use of \\BibTeX\\ for the preparation and formatting of one's\nreferences is strongly recommended. Authors' names should be complete\n--- use full first names (``Donald E. Knuth'') not initials\n(``D. E. Knuth'') --- and the salient identifying features of a\nreference should be included: title, year, volume, number, pages,\narticle DOI, etc.\n\nThe bibliography is included in your source document with these two\ncommands, placed just before the \\verb|\\end{document}| command:\n\\begin{verbatim}\n \\bibliographystyle{ACM-Reference-Format}\n \n\\section{Introduction}\nACM's consolidated article template, introduced in 2017, provides a\nconsistent \\LaTeX\\ style for use across ACM publications, and\nincorporates accessibility and metadata-extraction functionality\nnecessary for future Digital Library endeavors. Numerous ACM and\nSIG-specific \\LaTeX\\ templates have been examined, and their unique\nfeatures incorporated into this single new template.\n\nIf you are new to publishing with ACM, this document is a valuable\nguide to the process of preparing your work for publication. If you\nhave published with ACM before, this document provides insight and\ninstruction into more recent changes to the article template.\n\nThe ``\\verb|acmart|'' document class can be used to prepare articles\nfor any ACM publication --- conference or journal, and for any stage\nof publication, from review to final ``camera-ready'' copy, to the\nauthor's own version, with {\\itshape very} few changes to the source.\n\n\\section{Template Overview}\nAs noted in the introduction, the ``\\verb|acmart|'' document class can\nbe used to prepare many different kinds of documentation --- a\ndouble-blind initial submission of a full-length technical paper, a\ntwo-page SIGGRAPH Emerging Technologies abstract, a ``camera-ready''\njournal article, a SIGCHI Extended Abstract, and more --- all by\nselecting the appropriate {\\itshape template style} and {\\itshape\n template parameters}.\n\nThis document will explain the major features of the document\nclass. For further information, the {\\itshape \\LaTeX\\ User's Guide} is\navailable from\n\\url{https:\/\/www.acm.org\/publications\/proceedings-template}.\n\n\\subsection{Template Styles}\n\nThe primary parameter given to the ``\\verb|acmart|'' document class is\nthe {\\itshape template style} which corresponds to the kind of publication\nor SIG publishing the work. This parameter is enclosed in square\nbrackets and is a part of the {\\verb|documentclass|} command:\n\\begin{verbatim}\n \\documentclass[STYLE]{acmart}\n\\end{verbatim}\n\nJournals use one of three template styles. All but three ACM journals\nuse the {\\verb|acmsmall|} template style:\n\\begin{itemize}\n\\item {\\texttt{acmsmall}}: The default journal template style.\n\\item {\\texttt{acmlarge}}: Used by JOCCH and TAP.\n\\item {\\texttt{acmtog}}: Used by TOG.\n\\end{itemize}\n\nThe majority of conference proceedings documentation will use the {\\verb|acmconf|} template style.\n\\begin{itemize}\n\\item {\\texttt{acmconf}}: The default proceedings template style.\n\\item{\\texttt{sigchi}}: Used for SIGCHI conference articles.\n\\item{\\texttt{sigchi-a}}: Used for SIGCHI ``Extended Abstract'' articles.\n\\item{\\texttt{sigplan}}: Used for SIGPLAN conference articles.\n\\end{itemize}\n\n\\subsection{Template Parameters}\n\nIn addition to specifying the {\\itshape template style} to be used in\nformatting your work, there are a number of {\\itshape template parameters}\nwhich modify some part of the applied template style. A complete list\nof these parameters can be found in the {\\itshape \\LaTeX\\ User's Guide.}\n\nFrequently-used parameters, or combinations of parameters, include:\n\\begin{itemize}\n\\item {\\texttt{anonymous,review}}: Suitable for a ``double-blind''\n conference submission. Anonymizes the work and includes line\n numbers. Use with the \\texttt{\\acmSubmissionID} command to print the\n submission's unique ID on each page of the work.\n\\item{\\texttt{authorversion}}: Produces a version of the work suitable\n for posting by the author.\n\\item{\\texttt{screen}}: Produces colored hyperlinks.\n\\end{itemize}\n\nThis document uses the following string as the first command in the\nsource file:\n\\begin{verbatim}\n\\end{verbatim}\n\n\\section{Modifications}\n\nModifying the template --- including but not limited to: adjusting\nmargins, typeface sizes, line spacing, paragraph and list definitions,\nand the use of the \\verb|\\vspace| command to manually adjust the\nvertical spacing between elements of your work --- is not allowed.\n\n{\\bfseries Your document will be returned to you for revision if\n modifications are discovered.}\n\n\\section{Typefaces}\n\nThe ``\\verb|acmart|'' document class requires the use of the\n``Libertine'' typeface family. Your \\TeX\\ installation should include\nthis set of packages. Please do not substitute other typefaces. The\n``\\verb|lmodern|'' and ``\\verb|ltimes|'' packages should not be used,\nas they will override the built-in typeface families.\n\n\\section{Title Information}\n\nThe title of your work should use capital letters appropriately -\n\\url{https:\/\/capitalizemytitle.com\/} has useful rules for\ncapitalization. Use the {\\verb|title|} command to define the title of\nyour work. If your work has a subtitle, define it with the\n{\\verb|subtitle|} command. Do not insert line breaks in your title.\n\nIf your title is lengthy, you must define a short version to be used\nin the page headers, to prevent overlapping text. The \\verb|title|\ncommand has a ``short title'' parameter:\n\\begin{verbatim}\n \\title[short title]{full title}\n\\end{verbatim}\n\n\\section{Authors and Affiliations}\n\nEach author must be defined separately for accurate metadata\nidentification. As an exception, multiple authors may share one\naffiliation. Authors' names should not be abbreviated; use full first\nnames wherever possible. Include authors' e-mail addresses whenever\npossible.\n\nGrouping authors' names or e-mail addresses, or providing an ``e-mail\nalias,'' as shown below, is not acceptable:\n\\begin{verbatim}\n \\author{Brooke Aster, David Mehldau}\n \\email{dave,judy,steve@university.edu}\n \\email{firstname.lastname@phillips.org}\n\\end{verbatim}\n\nThe \\verb|authornote| and \\verb|authornotemark| commands allow a note\nto apply to multiple authors --- for example, if the first two authors\nof an article contributed equally to the work.\n\nIf your author list is lengthy, you must define a shortened version of\nthe list of authors to be used in the page headers, to prevent\noverlapping text. The following command should be placed just after\nthe last \\verb|\\author{}| definition:\n\\begin{verbatim}\n \\renewcommand{\\shortauthors}{McCartney, et al.}\n\\end{verbatim}\nOmitting this command will force the use of a concatenated list of all\nof the authors' names, which may result in overlapping text in the\npage headers.\n\nThe article template's documentation, available at\n\\url{https:\/\/www.acm.org\/publications\/proceedings-template}, has a\ncomplete explanation of these commands and tips for their effective\nuse.\n\nNote that authors' addresses are mandatory for journal articles.\n\n\\section{Rights Information}\n\nAuthors of any work published by ACM will need to complete a rights\nform. Depending on the kind of work, and the rights management choice\nmade by the author, this may be copyright transfer, permission,\nlicense, or an OA (open access) agreement.\n\nRegardless of the rights management choice, the author will receive a\ncopy of the completed rights form once it has been submitted. This\nform contains \\LaTeX\\ commands that must be copied into the source\ndocument. When the document source is compiled, these commands and\ntheir parameters add formatted text to several areas of the final\ndocument:\n\\begin{itemize}\n\\item the ``ACM Reference Format'' text on the first page.\n\\item the ``rights management'' text on the first page.\n\\item the conference information in the page header(s).\n\\end{itemize}\n\nRights information is unique to the work; if you are preparing several\nworks for an event, make sure to use the correct set of commands with\neach of the works.\n\nThe ACM Reference Format text is required for all articles over one\npage in length, and is optional for one-page articles (abstracts).\n\n\\section{CCS Concepts and User-Defined Keywords}\n\nTwo elements of the ``acmart'' document class provide powerful\ntaxonomic tools for you to help readers find your work in an online\nsearch.\n\nThe ACM Computing Classification System ---\n\\url{https:\/\/www.acm.org\/publications\/class-2012} --- is a set of\nclassifiers and concepts that describe the computing\ndiscipline. Authors can select entries from this classification\nsystem, via \\url{https:\/\/dl.acm.org\/ccs\/ccs.cfm}, and generate the\ncommands to be included in the \\LaTeX\\ source.\n\nUser-defined keywords are a comma-separated list of words and phrases\nof the authors' choosing, providing a more flexible way of describing\nthe research being presented.\n\nCCS concepts and user-defined keywords are required for for all\narticles over two pages in length, and are optional for one- and\ntwo-page articles (or abstracts).\n\n\\section{Sectioning Commands}\n\nYour work should use standard \\LaTeX\\ sectioning commands:\n\\verb|section|, \\verb|subsection|, \\verb|subsubsection|, and\n\\verb|paragraph|. They should be numbered; do not remove the numbering\nfrom the commands.\n\nSimulating a sectioning command by setting the first word or words of\na paragraph in boldface or italicized text is {\\bfseries not allowed.}\n\n\\section{Tables}\n\nThe ``\\verb|acmart|'' document class includes the ``\\verb|booktabs|''\npackage --- \\url{https:\/\/ctan.org\/pkg\/booktabs} --- for preparing\nhigh-quality tables.\n\nTable captions are placed {\\itshape above} the table.\n\nBecause tables cannot be split across pages, the best placement for\nthem is typically the top of the page nearest their initial cite. To\nensure this proper ``floating'' placement of tables, use the\nenvironment \\textbf{table} to enclose the table's contents and the\ntable caption. The contents of the table itself must go in the\n\\textbf{tabular} environment, to be aligned properly in rows and\ncolumns, with the desired horizontal and vertical rules. Again,\ndetailed instructions on \\textbf{tabular} material are found in the\n\\textit{\\LaTeX\\ User's Guide}.\n\nImmediately following this sentence is the point at which\nTable~\\ref{tab:freq} is included in the input file; compare the\nplacement of the table here with the table in the printed output of\nthis document.\n\n\\begin{table}\n \\caption{Frequency of Special Characters}\n \\label{tab:freq}\n \\begin{tabular}{ccl}\n \\toprule\n Non-English or Math&Frequency&Comments\\\\\n \\midrule\n \\O & 1 in 1,000& For Swedish names\\\\\n $\\pi$ & 1 in 5& Common in math\\\\\n \\$ & 4 in 5 & Used in business\\\\\n $\\Psi^2_1$ & 1 in 40,000& Unexplained usage\\\\\n \\bottomrule\n\\end{tabular}\n\\end{table}\n\nTo set a wider table, which takes up the whole width of the page's\nlive area, use the environment \\textbf{table*} to enclose the table's\ncontents and the table caption. As with a single-column table, this\nwide table will ``float'' to a location deemed more\ndesirable. Immediately following this sentence is the point at which\nTable~\\ref{tab:commands} is included in the input file; again, it is\ninstructive to compare the placement of the table here with the table\nin the printed output of this document.\n\n\\begin{table*}\n \\caption{Some Typical Commands}\n \\label{tab:commands}\n \\begin{tabular}{ccl}\n \\toprule\n Command &A Number & Comments\\\\\n \\midrule\n \\texttt{{\\char'134}author} & 100& Author \\\\\n \\texttt{{\\char'134}table}& 300 & For tables\\\\\n \\texttt{{\\char'134}table*}& 400& For wider tables\\\\\n \\bottomrule\n \\end{tabular}\n\\end{table*}\n\nAlways use midrule to separate table header rows from data rows, and\nuse it only for this purpose. This enables assistive technologies to\nrecognise table headers and support their users in navigating tables\nmore easily.\n\n\\section{Math Equations}\nYou may want to display math equations in three distinct styles:\ninline, numbered or non-numbered display. Each of the three are\ndiscussed in the next sections.\n\n\\subsection{Inline (In-text) Equations}\nA formula that appears in the running text is called an inline or\nin-text formula. It is produced by the \\textbf{math} environment,\nwhich can be invoked with the usual\n\\texttt{{\\char'134}begin\\,\\ldots{\\char'134}end} construction or with\nthe short form \\texttt{\\$\\,\\ldots\\$}. You can use any of the symbols\nand structures, from $\\alpha$ to $\\omega$, available in\n\\LaTeX~\\cite{Lamport:LaTeX}; this section will simply show a few\nexamples of in-text equations in context. Notice how this equation:\n\\begin{math}\n \\lim_{n\\rightarrow \\infty}x=0\n\\end{math},\nset here in in-line math style, looks slightly different when\nset in display style. (See next section).\n\n\\subsection{Display Equations}\nA numbered display equation---one set off by vertical space from the\ntext and centered horizontally---is produced by the \\textbf{equation}\nenvironment. An unnumbered display equation is produced by the\n\\textbf{displaymath} environment.\n\nAgain, in either environment, you can use any of the symbols and\nstructures available in \\LaTeX\\@; this section will just give a couple\nof examples of display equations in context. First, consider the\nequation, shown as an inline equation above:\n\\begin{equation}\n \\lim_{n\\rightarrow \\infty}x=0\n\\end{equation}\nNotice how it is formatted somewhat differently in\nthe \\textbf{displaymath}\nenvironment. Now, we'll enter an unnumbered equation:\n\\begin{displaymath}\n \\sum_{i=0}^{\\infty} x + 1\n\\end{displaymath}\nand follow it with another numbered equation:\n\\begin{equation}\n \\sum_{i=0}^{\\infty}x_i=\\int_{0}^{\\pi+2} f\n\\end{equation}\njust to demonstrate \\LaTeX's able handling of numbering.\n\n\\section{Figures}\n\nThe ``\\verb|figure|'' environment should be used for figures. One or\nmore images can be placed within a figure. If your figure contains\nthird-party material, you must clearly identify it as such, as shown\nin the example below.\n\\begin{figure}[h]\n \\centering\n \\includegraphics[width=\\linewidth]{sample-franklin}\n \\caption{1907 Franklin Model D roadster. Photograph by Harris \\&\n Ewing, Inc. [Public domain], via Wikimedia\n Commons. (\\url{https:\/\/goo.gl\/VLCRBB}).}\n \\Description{A woman and a girl in white dresses sit in an open car.}\n\\end{figure}\n\nYour figures should contain a caption which describes the figure to\nthe reader.\n\nFigure captions are placed {\\itshape below} the figure.\n\nEvery figure should also have a figure description unless it is purely\ndecorative. These descriptions convey what's in the image to someone\nwho cannot see it. They are also used by search engine crawlers for\nindexing images, and when images cannot be loaded.\n\nA figure description must be unformatted plain text less than 2000\ncharacters long (including spaces). {\\bfseries Figure descriptions\n should not repeat the figure caption \u2013 their purpose is to capture\n important information that is not already provided in the caption or\n the main text of the paper.} For figures that convey important and\ncomplex new information, a short text description may not be\nadequate. More complex alternative descriptions can be placed in an\nappendix and referenced in a short figure description. For example,\nprovide a data table capturing the information in a bar chart, or a\nstructured list representing a graph. For additional information\nregarding how best to write figure descriptions and why doing this is\nso important, please see\n\\url{https:\/\/www.acm.org\/publications\/taps\/describing-figures\/}.\n\n\\subsection{The ``Teaser Figure''}\n\nA ``teaser figure'' is an image, or set of images in one figure, that\nare placed after all author and affiliation information, and before\nthe body of the article, spanning the page. If you wish to have such a\nfigure in your article, place the command immediately before the\n\\verb|\\maketitle| command:\n\\begin{verbatim}\n \\begin{teaserfigure}\n \\includegraphics[width=\\textwidth]{sampleteaser}\n \\caption{figure caption}\n \\Description{figure description}\n \\end{teaserfigure}\n\\end{verbatim}\n\n\\section{Citations and Bibliographies}\n\nThe use of \\BibTeX\\ for the preparation and formatting of one's\nreferences is strongly recommended. Authors' names should be complete\n--- use full first names (``Donald E. Knuth'') not initials\n(``D. E. Knuth'') --- and the salient identifying features of a\nreference should be included: title, year, volume, number, pages,\narticle DOI, etc.\n\n\nUsing the BibLaTeX system, the bibliography is included in your source\ndocument with the following command, placed just before the \\verb|\\end{document}| command:\n\\begin{verbatim}\n \\printbibliography\n\\end{verbatim}\n\nThe command \\verb|\\addbibresource{bibfile}| declares the \\BibTeX\\ file to use\nin the {\\bfseries preamble} (before the command\n``\\verb|\\begin{document}|'') of your \\LaTeX\\ source\nwhere ``\\verb|bibfile|'' is the name, \\emph{with} the ``\\verb|.bib|'' suffix.\nNotice that \\verb|\\addbibresource| takes only one argument: to declare multiple files,\nuse multiple instances of the command.\n\nCitations and references are numbered by default. A small number of\nACM publications have citations and references formatted in the\n``author year'' style; for these exceptions, please pass the option \\verb|style=acmauthoryear|\nto the \\verb|biblatex| package loaded in the {\\bfseries preamble} (before the command\n``\\verb|\\begin{document}|'') of your \\LaTeX\\ source.\n\n\n Some examples. A paginated journal article \\cite{Abril07}, an\n enumerated journal article \\cite{Cohen07}, a reference to an entire\n issue \\cite{JCohen96}, a monograph (whole book) \\cite{Kosiur01}, a\n monograph\/whole book in a series (see 2a in spec. document)\n \\cite{Harel79}, a divisible-book such as an anthology or compilation\n \\cite{Editor00} followed by the same example, however we only output\n the series if the volume number is given \\cite{Editor00a} (so\n Editor00a's series should NOT be present since it has no vol. no.),\n a chapter in a divisible book \\cite{Spector90}, a chapter in a\n divisible book in a series \\cite{Douglass98}, a multi-volume work as\n book \\cite{Knuth97}, a couple of articles in a proceedings (of a\n conference, symposium, workshop for example) (paginated proceedings\n article) \\cite{Andler79, Hagerup1993}, a proceedings article with\n all possible elements \\cite{Smith10}, an example of an enumerated\n proceedings article \\cite{VanGundy07}, an informally published work\n \\cite{Harel78}, a couple of preprints \\cite{Bornmann2019,\n AnzarootPBM14}, a doctoral dissertation \\cite{Clarkson85}, a\n master's thesis: \\cite{anisi03}, an online document \/ world wide web\n resource \\cite{Thornburg01, Ablamowicz07, Poker06}, a video game\n (Case 1) \\cite{Obama08} and (Case 2) \\cite{Novak03} and \\cite{Lee05}\n and (Case 3) a patent \\cite{JoeScientist001}, work accepted for\n publication \\cite{rous08}, 'YYYYb'-test for prolific author\n \\cite{SaeediMEJ10} and \\cite{SaeediJETC10}. Other cites might\n contain 'duplicate' DOI and URLs (some SIAM articles)\n \\cite{Kirschmer:2010:AEI:1958016.1958018}. Boris \/ Barbara Beeton:\n multi-volume works as books \\cite{MR781536} and \\cite{MR781537}. A\n couple of citations with DOIs:\n \\cite{2004:ITE:1009386.1010128,Kirschmer:2010:AEI:1958016.1958018}. Online\n citations: \\cite{TUGInstmem, Thornburg01, CTANacmart}.\n Data Artifacts: \\cite{UMassCitations}.\n Software project: ~\\cite{cgal,delebecque:hal-02090402}. Software Version: ~\\cite{gf-tag-sound-repo,}. Software Module: ~\\cite{cgal:lp-gi-20a}. Code fragment: ~\\cite{simplemapper}.\n\n\\section{Acknowledgments}\n\nIdentification of funding sources and other support, and thanks to\nindividuals and groups that assisted in the research and the\npreparation of the work should be included in an acknowledgment\nsection, which is placed just before the reference section in your\ndocument.\n\nThis section has a special environment:\n\\begin{verbatim}\n \\begin{acks}\n ...\n \\end{acks}\n\\end{verbatim}\nso that the information contained therein can be more easily collected\nduring the article metadata extraction phase, and to ensure\nconsistency in the spelling of the section heading.\n\nAuthors should not prepare this section as a numbered or unnumbered {\\verb|\\section|}; please use the ``{\\verb|acks|}'' environment.\n\n\\section{Appendices}\n\nIf your work needs an appendix, add it before the\n``\\verb|\\end{document}|'' command at the conclusion of your source\ndocument.\n\nStart the appendix with the ``\\verb|appendix|'' command:\n\\begin{verbatim}\n \n\\section{Introduction}\nACM's consolidated article template, introduced in 2017, provides a\nconsistent \\LaTeX\\ style for use across ACM publications, and\nincorporates accessibility and metadata-extraction functionality\nnecessary for future Digital Library endeavors. Numerous ACM and\nSIG-specific \\LaTeX\\ templates have been examined, and their unique\nfeatures incorporated into this single new template.\n\nIf you are new to publishing with ACM, this document is a valuable\nguide to the process of preparing your work for publication. If you\nhave published with ACM before, this document provides insight and\ninstruction into more recent changes to the article template.\n\nThe ``\\verb|acmart|'' document class can be used to prepare articles\nfor any ACM publication --- conference or journal, and for any stage\nof publication, from review to final ``camera-ready'' copy, to the\nauthor's own version, with {\\itshape very} few changes to the source.\n\n\\section{Template Overview}\nAs noted in the introduction, the ``\\verb|acmart|'' document class can\nbe used to prepare many different kinds of documentation --- a\ndouble-blind initial submission of a full-length technical paper, a\ntwo-page SIGGRAPH Emerging Technologies abstract, a ``camera-ready''\njournal article, a SIGCHI Extended Abstract, and more --- all by\nselecting the appropriate {\\itshape template style} and {\\itshape\n template parameters}.\n\nThis document will explain the major features of the document\nclass. For further information, the {\\itshape \\LaTeX\\ User's Guide} is\navailable from\n\\url{https:\/\/www.acm.org\/publications\/proceedings-template}.\n\n\\subsection{Template Styles}\n\nThe primary parameter given to the ``\\verb|acmart|'' document class is\nthe {\\itshape template style} which corresponds to the kind of publication\nor SIG publishing the work. This parameter is enclosed in square\nbrackets and is a part of the {\\verb|documentclass|} command:\n\\begin{verbatim}\n \\documentclass[STYLE]{acmart}\n\\end{verbatim}\n\nJournals use one of three template styles. All but three ACM journals\nuse the {\\verb|acmsmall|} template style:\n\\begin{itemize}\n\\item {\\texttt{acmsmall}}: The default journal template style.\n\\item {\\texttt{acmlarge}}: Used by JOCCH and TAP.\n\\item {\\texttt{acmtog}}: Used by TOG.\n\\end{itemize}\n\nThe majority of conference proceedings documentation will use the {\\verb|acmconf|} template style.\n\\begin{itemize}\n\\item {\\texttt{acmconf}}: The default proceedings template style.\n\\item{\\texttt{sigchi}}: Used for SIGCHI conference articles.\n\\item{\\texttt{sigchi-a}}: Used for SIGCHI ``Extended Abstract'' articles.\n\\item{\\texttt{sigplan}}: Used for SIGPLAN conference articles.\n\\end{itemize}\n\n\\subsection{Template Parameters}\n\nIn addition to specifying the {\\itshape template style} to be used in\nformatting your work, there are a number of {\\itshape template parameters}\nwhich modify some part of the applied template style. A complete list\nof these parameters can be found in the {\\itshape \\LaTeX\\ User's Guide.}\n\nFrequently-used parameters, or combinations of parameters, include:\n\\begin{itemize}\n\\item {\\texttt{anonymous,review}}: Suitable for a ``double-blind''\n conference submission. Anonymizes the work and includes line\n numbers. Use with the \\texttt{\\acmSubmissionID} command to print the\n submission's unique ID on each page of the work.\n\\item{\\texttt{authorversion}}: Produces a version of the work suitable\n for posting by the author.\n\\item{\\texttt{screen}}: Produces colored hyperlinks.\n\\end{itemize}\n\nThis document uses the following string as the first command in the\nsource file:\n\\begin{verbatim}\n\\documentclass[sigconf, language=french,\nlanguage=german, language=spanish, language=english]{acmart}\n\\end{verbatim}\n\n\\section{Modifications}\n\nModifying the template --- including but not limited to: adjusting\nmargins, typeface sizes, line spacing, paragraph and list definitions,\nand the use of the \\verb|\\vspace| command to manually adjust the\nvertical spacing between elements of your work --- is not allowed.\n\n{\\bfseries Your document will be returned to you for revision if\n modifications are discovered.}\n\n\\section{Typefaces}\n\nThe ``\\verb|acmart|'' document class requires the use of the\n``Libertine'' typeface family. Your \\TeX\\ installation should include\nthis set of packages. Please do not substitute other typefaces. The\n``\\verb|lmodern|'' and ``\\verb|ltimes|'' packages should not be used,\nas they will override the built-in typeface families.\n\n\\section{Title Information}\n\nThe title of your work should use capital letters appropriately -\n\\url{https:\/\/capitalizemytitle.com\/} has useful rules for\ncapitalization. Use the {\\verb|title|} command to define the title of\nyour work. If your work has a subtitle, define it with the\n{\\verb|subtitle|} command. Do not insert line breaks in your title.\n\nIf your title is lengthy, you must define a short version to be used\nin the page headers, to prevent overlapping text. The \\verb|title|\ncommand has a ``short title'' parameter:\n\\begin{verbatim}\n \\title[short title]{full title}\n\\end{verbatim}\n\n\\section{Authors and Affiliations}\n\nEach author must be defined separately for accurate metadata\nidentification. As an exception, multiple authors may share one\naffiliation. Authors' names should not be abbreviated; use full first\nnames wherever possible. Include authors' e-mail addresses whenever\npossible.\n\nGrouping authors' names or e-mail addresses, or providing an ``e-mail\nalias,'' as shown below, is not acceptable:\n\\begin{verbatim}\n \\author{Brooke Aster, David Mehldau}\n \\email{dave,judy,steve@university.edu}\n \\email{firstname.lastname@phillips.org}\n\\end{verbatim}\n\nThe \\verb|authornote| and \\verb|authornotemark| commands allow a note\nto apply to multiple authors --- for example, if the first two authors\nof an article contributed equally to the work.\n\nIf your author list is lengthy, you must define a shortened version of\nthe list of authors to be used in the page headers, to prevent\noverlapping text. The following command should be placed just after\nthe last \\verb|\\author{}| definition:\n\\begin{verbatim}\n \\renewcommand{\\shortauthors}{McCartney, et al.}\n\\end{verbatim}\nOmitting this command will force the use of a concatenated list of all\nof the authors' names, which may result in overlapping text in the\npage headers.\n\nThe article template's documentation, available at\n\\url{https:\/\/www.acm.org\/publications\/proceedings-template}, has a\ncomplete explanation of these commands and tips for their effective\nuse.\n\nNote that authors' addresses are mandatory for journal articles.\n\n\\section{Rights Information}\n\nAuthors of any work published by ACM will need to complete a rights\nform. Depending on the kind of work, and the rights management choice\nmade by the author, this may be copyright transfer, permission,\nlicense, or an OA (open access) agreement.\n\nRegardless of the rights management choice, the author will receive a\ncopy of the completed rights form once it has been submitted. This\nform contains \\LaTeX\\ commands that must be copied into the source\ndocument. When the document source is compiled, these commands and\ntheir parameters add formatted text to several areas of the final\ndocument:\n\\begin{itemize}\n\\item the ``ACM Reference Format'' text on the first page.\n\\item the ``rights management'' text on the first page.\n\\item the conference information in the page header(s).\n\\end{itemize}\n\nRights information is unique to the work; if you are preparing several\nworks for an event, make sure to use the correct set of commands with\neach of the works.\n\nThe ACM Reference Format text is required for all articles over one\npage in length, and is optional for one-page articles (abstracts).\n\n\\section{CCS Concepts and User-Defined Keywords}\n\nTwo elements of the ``acmart'' document class provide powerful\ntaxonomic tools for you to help readers find your work in an online\nsearch.\n\nThe ACM Computing Classification System ---\n\\url{https:\/\/www.acm.org\/publications\/class-2012} --- is a set of\nclassifiers and concepts that describe the computing\ndiscipline. Authors can select entries from this classification\nsystem, via \\url{https:\/\/dl.acm.org\/ccs\/ccs.cfm}, and generate the\ncommands to be included in the \\LaTeX\\ source.\n\nUser-defined keywords are a comma-separated list of words and phrases\nof the authors' choosing, providing a more flexible way of describing\nthe research being presented.\n\nCCS concepts and user-defined keywords are required for for all\narticles over two pages in length, and are optional for one- and\ntwo-page articles (or abstracts).\n\n\\section{Sectioning Commands}\n\nYour work should use standard \\LaTeX\\ sectioning commands:\n\\verb|section|, \\verb|subsection|, \\verb|subsubsection|, and\n\\verb|paragraph|. They should be numbered; do not remove the numbering\nfrom the commands.\n\nSimulating a sectioning command by setting the first word or words of\na paragraph in boldface or italicized text is {\\bfseries not allowed.}\n\n\\section{Tables}\n\nThe ``\\verb|acmart|'' document class includes the ``\\verb|booktabs|''\npackage --- \\url{https:\/\/ctan.org\/pkg\/booktabs} --- for preparing\nhigh-quality tables.\n\nTable captions are placed {\\itshape above} the table.\n\nBecause tables cannot be split across pages, the best placement for\nthem is typically the top of the page nearest their initial cite. To\nensure this proper ``floating'' placement of tables, use the\nenvironment \\textbf{table} to enclose the table's contents and the\ntable caption. The contents of the table itself must go in the\n\\textbf{tabular} environment, to be aligned properly in rows and\ncolumns, with the desired horizontal and vertical rules. Again,\ndetailed instructions on \\textbf{tabular} material are found in the\n\\textit{\\LaTeX\\ User's Guide}.\n\nImmediately following this sentence is the point at which\nTable~\\ref{tab:freq} is included in the input file; compare the\nplacement of the table here with the table in the printed output of\nthis document.\n\n\\begin{table}\n \\caption{Frequency of Special Characters}\n \\label{tab:freq}\n \\begin{tabular}{ccl}\n \\toprule\n Non-English or Math&Frequency&Comments\\\\\n \\midrule\n \\O & 1 in 1,000& For Swedish names\\\\\n $\\pi$ & 1 in 5& Common in math\\\\\n \\$ & 4 in 5 & Used in business\\\\\n $\\Psi^2_1$ & 1 in 40,000& Unexplained usage\\\\\n \\bottomrule\n\\end{tabular}\n\\end{table}\n\nTo set a wider table, which takes up the whole width of the page's\nlive area, use the environment \\textbf{table*} to enclose the table's\ncontents and the table caption. As with a single-column table, this\nwide table will ``float'' to a location deemed more\ndesirable. Immediately following this sentence is the point at which\nTable~\\ref{tab:commands} is included in the input file; again, it is\ninstructive to compare the placement of the table here with the table\nin the printed output of this document.\n\n\\begin{table*}\n \\caption{Some Typical Commands}\n \\label{tab:commands}\n \\begin{tabular}{ccl}\n \\toprule\n Command &A Number & Comments\\\\\n \\midrule\n \\texttt{{\\char'134}author} & 100& Author \\\\\n \\texttt{{\\char'134}table}& 300 & For tables\\\\\n \\texttt{{\\char'134}table*}& 400& For wider tables\\\\\n \\bottomrule\n \\end{tabular}\n\\end{table*}\n\nAlways use midrule to separate table header rows from data rows, and\nuse it only for this purpose. This enables assistive technologies to\nrecognise table headers and support their users in navigating tables\nmore easily.\n\n\\section{Math Equations}\nYou may want to display math equations in three distinct styles:\ninline, numbered or non-numbered display. Each of the three are\ndiscussed in the next sections.\n\n\\subsection{Inline (In-text) Equations}\nA formula that appears in the running text is called an inline or\nin-text formula. It is produced by the \\textbf{math} environment,\nwhich can be invoked with the usual\n\\texttt{{\\char'134}begin\\,\\ldots{\\char'134}end} construction or with\nthe short form \\texttt{\\$\\,\\ldots\\$}. You can use any of the symbols\nand structures, from $\\alpha$ to $\\omega$, available in\n\\LaTeX~\\cite{Lamport:LaTeX}; this section will simply show a few\nexamples of in-text equations in context. Notice how this equation:\n\\begin{math}\n \\lim_{n\\rightarrow \\infty}x=0\n\\end{math},\nset here in in-line math style, looks slightly different when\nset in display style. (See next section).\n\n\\subsection{Display Equations}\nA numbered display equation---one set off by vertical space from the\ntext and centered horizontally---is produced by the \\textbf{equation}\nenvironment. An unnumbered display equation is produced by the\n\\textbf{displaymath} environment.\n\nAgain, in either environment, you can use any of the symbols and\nstructures available in \\LaTeX\\@; this section will just give a couple\nof examples of display equations in context. First, consider the\nequation, shown as an inline equation above:\n\\begin{equation}\n \\lim_{n\\rightarrow \\infty}x=0\n\\end{equation}\nNotice how it is formatted somewhat differently in\nthe \\textbf{displaymath}\nenvironment. Now, we'll enter an unnumbered equation:\n\\begin{displaymath}\n \\sum_{i=0}^{\\infty} x + 1\n\\end{displaymath}\nand follow it with another numbered equation:\n\\begin{equation}\n \\sum_{i=0}^{\\infty}x_i=\\int_{0}^{\\pi+2} f\n\\end{equation}\njust to demonstrate \\LaTeX's able handling of numbering.\n\n\\section{Figures}\n\nThe ``\\verb|figure|'' environment should be used for figures. One or\nmore images can be placed within a figure. If your figure contains\nthird-party material, you must clearly identify it as such, as shown\nin the example below.\n\\begin{figure}[h]\n \\centering\n \\includegraphics[width=\\linewidth]{sample-franklin}\n \\caption{1907 Franklin Model D roadster. Photograph by Harris \\&\n Ewing, Inc. [Public domain], via Wikimedia\n Commons. (\\url{https:\/\/goo.gl\/VLCRBB}).}\n \\Description{A woman and a girl in white dresses sit in an open car.}\n\\end{figure}\n\nYour figures should contain a caption which describes the figure to\nthe reader.\n\nFigure captions are placed {\\itshape below} the figure.\n\nEvery figure should also have a figure description unless it is purely\ndecorative. These descriptions convey what's in the image to someone\nwho cannot see it. They are also used by search engine crawlers for\nindexing images, and when images cannot be loaded.\n\nA figure description must be unformatted plain text less than 2000\ncharacters long (including spaces). {\\bfseries Figure descriptions\n should not repeat the figure caption \u2013 their purpose is to capture\n important information that is not already provided in the caption or\n the main text of the paper.} For figures that convey important and\ncomplex new information, a short text description may not be\nadequate. More complex alternative descriptions can be placed in an\nappendix and referenced in a short figure description. For example,\nprovide a data table capturing the information in a bar chart, or a\nstructured list representing a graph. For additional information\nregarding how best to write figure descriptions and why doing this is\nso important, please see\n\\url{https:\/\/www.acm.org\/publications\/taps\/describing-figures\/}.\n\n\\subsection{The ``Teaser Figure''}\n\nA ``teaser figure'' is an image, or set of images in one figure, that\nare placed after all author and affiliation information, and before\nthe body of the article, spanning the page. If you wish to have such a\nfigure in your article, place the command immediately before the\n\\verb|\\maketitle| command:\n\\begin{verbatim}\n \\begin{teaserfigure}\n \\includegraphics[width=\\textwidth]{sampleteaser}\n \\caption{figure caption}\n \\Description{figure description}\n \\end{teaserfigure}\n\\end{verbatim}\n\n\\section{Citations and Bibliographies}\n\nThe use of \\BibTeX\\ for the preparation and formatting of one's\nreferences is strongly recommended. Authors' names should be complete\n--- use full first names (``Donald E. Knuth'') not initials\n(``D. E. Knuth'') --- and the salient identifying features of a\nreference should be included: title, year, volume, number, pages,\narticle DOI, etc.\n\nThe bibliography is included in your source document with these two\ncommands, placed just before the \\verb|\\end{document}| command:\n\\begin{verbatim}\n \\bibliographystyle{ACM-Reference-Format}\n \n\\section{Introduction}\nACM's consolidated article template, introduced in 2017, provides a\nconsistent \\LaTeX\\ style for use across ACM publications, and\nincorporates accessibility and metadata-extraction functionality\nnecessary for future Digital Library endeavors. Numerous ACM and\nSIG-specific \\LaTeX\\ templates have been examined, and their unique\nfeatures incorporated into this single new template.\n\nIf you are new to publishing with ACM, this document is a valuable\nguide to the process of preparing your work for publication. If you\nhave published with ACM before, this document provides insight and\ninstruction into more recent changes to the article template.\n\nThe ``\\verb|acmart|'' document class can be used to prepare articles\nfor any ACM publication --- conference or journal, and for any stage\nof publication, from review to final ``camera-ready'' copy, to the\nauthor's own version, with {\\itshape very} few changes to the source.\n\n\\section{Template Overview}\nAs noted in the introduction, the ``\\verb|acmart|'' document class can\nbe used to prepare many different kinds of documentation --- a\ndouble-blind initial submission of a full-length technical paper, a\ntwo-page SIGGRAPH Emerging Technologies abstract, a ``camera-ready''\njournal article, a SIGCHI Extended Abstract, and more --- all by\nselecting the appropriate {\\itshape template style} and {\\itshape\n template parameters}.\n\nThis document will explain the major features of the document\nclass. For further information, the {\\itshape \\LaTeX\\ User's Guide} is\navailable from\n\\url{https:\/\/www.acm.org\/publications\/proceedings-template}.\n\n\\subsection{Template Styles}\n\nThe primary parameter given to the ``\\verb|acmart|'' document class is\nthe {\\itshape template style} which corresponds to the kind of publication\nor SIG publishing the work. This parameter is enclosed in square\nbrackets and is a part of the {\\verb|documentclass|} command:\n\\begin{verbatim}\n \\documentclass[STYLE]{acmart}\n\\end{verbatim}\n\nJournals use one of three template styles. All but three ACM journals\nuse the {\\verb|acmsmall|} template style:\n\\begin{itemize}\n\\item {\\texttt{acmsmall}}: The default journal template style.\n\\item {\\texttt{acmlarge}}: Used by JOCCH and TAP.\n\\item {\\texttt{acmtog}}: Used by TOG.\n\\end{itemize}\n\nThe majority of conference proceedings documentation will use the {\\verb|acmconf|} template style.\n\\begin{itemize}\n\\item {\\texttt{acmconf}}: The default proceedings template style.\n\\item{\\texttt{sigchi}}: Used for SIGCHI conference articles.\n\\item{\\texttt{sigchi-a}}: Used for SIGCHI ``Extended Abstract'' articles.\n\\item{\\texttt{sigplan}}: Used for SIGPLAN conference articles.\n\\end{itemize}\n\n\\subsection{Template Parameters}\n\nIn addition to specifying the {\\itshape template style} to be used in\nformatting your work, there are a number of {\\itshape template parameters}\nwhich modify some part of the applied template style. A complete list\nof these parameters can be found in the {\\itshape \\LaTeX\\ User's Guide.}\n\nFrequently-used parameters, or combinations of parameters, include:\n\\begin{itemize}\n\\item {\\texttt{anonymous,review}}: Suitable for a ``double-blind''\n conference submission. Anonymizes the work and includes line\n numbers. Use with the \\texttt{\\acmSubmissionID} command to print the\n submission's unique ID on each page of the work.\n\\item{\\texttt{authorversion}}: Produces a version of the work suitable\n for posting by the author.\n\\item{\\texttt{screen}}: Produces colored hyperlinks.\n\\end{itemize}\n\nThis document uses the following string as the first command in the\nsource file:\n\\begin{verbatim}\n\\documentclass[sigplan,screen]{acmart}\n\\end{verbatim}\n\n\\section{Modifications}\n\nModifying the template --- including but not limited to: adjusting\nmargins, typeface sizes, line spacing, paragraph and list definitions,\nand the use of the \\verb|\\vspace| command to manually adjust the\nvertical spacing between elements of your work --- is not allowed.\n\n{\\bfseries Your document will be returned to you for revision if\n modifications are discovered.}\n\n\\section{Typefaces}\n\nThe ``\\verb|acmart|'' document class requires the use of the\n``Libertine'' typeface family. Your \\TeX\\ installation should include\nthis set of packages. Please do not substitute other typefaces. The\n``\\verb|lmodern|'' and ``\\verb|ltimes|'' packages should not be used,\nas they will override the built-in typeface families.\n\n\\section{Title Information}\n\nThe title of your work should use capital letters appropriately -\n\\url{https:\/\/capitalizemytitle.com\/} has useful rules for\ncapitalization. Use the {\\verb|title|} command to define the title of\nyour work. If your work has a subtitle, define it with the\n{\\verb|subtitle|} command. Do not insert line breaks in your title.\n\nIf your title is lengthy, you must define a short version to be used\nin the page headers, to prevent overlapping text. The \\verb|title|\ncommand has a ``short title'' parameter:\n\\begin{verbatim}\n \\title[short title]{full title}\n\\end{verbatim}\n\n\\section{Authors and Affiliations}\n\nEach author must be defined separately for accurate metadata\nidentification. As an exception, multiple authors may share one\naffiliation. Authors' names should not be abbreviated; use full first\nnames wherever possible. Include authors' e-mail addresses whenever\npossible.\n\nGrouping authors' names or e-mail addresses, or providing an ``e-mail\nalias,'' as shown below, is not acceptable:\n\\begin{verbatim}\n \\author{Brooke Aster, David Mehldau}\n \\email{dave,judy,steve@university.edu}\n \\email{firstname.lastname@phillips.org}\n\\end{verbatim}\n\nThe \\verb|authornote| and \\verb|authornotemark| commands allow a note\nto apply to multiple authors --- for example, if the first two authors\nof an article contributed equally to the work.\n\nIf your author list is lengthy, you must define a shortened version of\nthe list of authors to be used in the page headers, to prevent\noverlapping text. The following command should be placed just after\nthe last \\verb|\\author{}| definition:\n\\begin{verbatim}\n \\renewcommand{\\shortauthors}{McCartney, et al.}\n\\end{verbatim}\nOmitting this command will force the use of a concatenated list of all\nof the authors' names, which may result in overlapping text in the\npage headers.\n\nThe article template's documentation, available at\n\\url{https:\/\/www.acm.org\/publications\/proceedings-template}, has a\ncomplete explanation of these commands and tips for their effective\nuse.\n\nNote that authors' addresses are mandatory for journal articles.\n\n\\section{Rights Information}\n\nAuthors of any work published by ACM will need to complete a rights\nform. Depending on the kind of work, and the rights management choice\nmade by the author, this may be copyright transfer, permission,\nlicense, or an OA (open access) agreement.\n\nRegardless of the rights management choice, the author will receive a\ncopy of the completed rights form once it has been submitted. This\nform contains \\LaTeX\\ commands that must be copied into the source\ndocument. When the document source is compiled, these commands and\ntheir parameters add formatted text to several areas of the final\ndocument:\n\\begin{itemize}\n\\item the ``ACM Reference Format'' text on the first page.\n\\item the ``rights management'' text on the first page.\n\\item the conference information in the page header(s).\n\\end{itemize}\n\nRights information is unique to the work; if you are preparing several\nworks for an event, make sure to use the correct set of commands with\neach of the works.\n\nThe ACM Reference Format text is required for all articles over one\npage in length, and is optional for one-page articles (abstracts).\n\n\\section{CCS Concepts and User-Defined Keywords}\n\nTwo elements of the ``acmart'' document class provide powerful\ntaxonomic tools for you to help readers find your work in an online\nsearch.\n\nThe ACM Computing Classification System ---\n\\url{https:\/\/www.acm.org\/publications\/class-2012} --- is a set of\nclassifiers and concepts that describe the computing\ndiscipline. Authors can select entries from this classification\nsystem, via \\url{https:\/\/dl.acm.org\/ccs\/ccs.cfm}, and generate the\ncommands to be included in the \\LaTeX\\ source.\n\nUser-defined keywords are a comma-separated list of words and phrases\nof the authors' choosing, providing a more flexible way of describing\nthe research being presented.\n\nCCS concepts and user-defined keywords are required for for all\narticles over two pages in length, and are optional for one- and\ntwo-page articles (or abstracts).\n\n\\section{Sectioning Commands}\n\nYour work should use standard \\LaTeX\\ sectioning commands:\n\\verb|section|, \\verb|subsection|, \\verb|subsubsection|, and\n\\verb|paragraph|. They should be numbered; do not remove the numbering\nfrom the commands.\n\nSimulating a sectioning command by setting the first word or words of\na paragraph in boldface or italicized text is {\\bfseries not allowed.}\n\n\\section{Tables}\n\nThe ``\\verb|acmart|'' document class includes the ``\\verb|booktabs|''\npackage --- \\url{https:\/\/ctan.org\/pkg\/booktabs} --- for preparing\nhigh-quality tables.\n\nTable captions are placed {\\itshape above} the table.\n\nBecause tables cannot be split across pages, the best placement for\nthem is typically the top of the page nearest their initial cite. To\nensure this proper ``floating'' placement of tables, use the\nenvironment \\textbf{table} to enclose the table's contents and the\ntable caption. The contents of the table itself must go in the\n\\textbf{tabular} environment, to be aligned properly in rows and\ncolumns, with the desired horizontal and vertical rules. Again,\ndetailed instructions on \\textbf{tabular} material are found in the\n\\textit{\\LaTeX\\ User's Guide}.\n\nImmediately following this sentence is the point at which\nTable~\\ref{tab:freq} is included in the input file; compare the\nplacement of the table here with the table in the printed output of\nthis document.\n\n\\begin{table}\n \\caption{Frequency of Special Characters}\n \\label{tab:freq}\n \\begin{tabular}{ccl}\n \\toprule\n Non-English or Math&Frequency&Comments\\\\\n \\midrule\n \\O & 1 in 1,000& For Swedish names\\\\\n $\\pi$ & 1 in 5& Common in math\\\\\n \\$ & 4 in 5 & Used in business\\\\\n $\\Psi^2_1$ & 1 in 40,000& Unexplained usage\\\\\n \\bottomrule\n\\end{tabular}\n\\end{table}\n\nTo set a wider table, which takes up the whole width of the page's\nlive area, use the environment \\textbf{table*} to enclose the table's\ncontents and the table caption. As with a single-column table, this\nwide table will ``float'' to a location deemed more\ndesirable. Immediately following this sentence is the point at which\nTable~\\ref{tab:commands} is included in the input file; again, it is\ninstructive to compare the placement of the table here with the table\nin the printed output of this document.\n\n\\begin{table*}\n \\caption{Some Typical Commands}\n \\label{tab:commands}\n \\begin{tabular}{ccl}\n \\toprule\n Command &A Number & Comments\\\\\n \\midrule\n \\texttt{{\\char'134}author} & 100& Author \\\\\n \\texttt{{\\char'134}table}& 300 & For tables\\\\\n \\texttt{{\\char'134}table*}& 400& For wider tables\\\\\n \\bottomrule\n \\end{tabular}\n\\end{table*}\n\nAlways use midrule to separate table header rows from data rows, and\nuse it only for this purpose. This enables assistive technologies to\nrecognise table headers and support their users in navigating tables\nmore easily.\n\n\\section{Math Equations}\nYou may want to display math equations in three distinct styles:\ninline, numbered or non-numbered display. Each of the three are\ndiscussed in the next sections.\n\n\\subsection{Inline (In-text) Equations}\nA formula that appears in the running text is called an inline or\nin-text formula. It is produced by the \\textbf{math} environment,\nwhich can be invoked with the usual\n\\texttt{{\\char'134}begin\\,\\ldots{\\char'134}end} construction or with\nthe short form \\texttt{\\$\\,\\ldots\\$}. You can use any of the symbols\nand structures, from $\\alpha$ to $\\omega$, available in\n\\LaTeX~\\cite{Lamport:LaTeX}; this section will simply show a few\nexamples of in-text equations in context. Notice how this equation:\n\\begin{math}\n \\lim_{n\\rightarrow \\infty}x=0\n\\end{math},\nset here in in-line math style, looks slightly different when\nset in display style. (See next section).\n\n\\subsection{Display Equations}\nA numbered display equation---one set off by vertical space from the\ntext and centered horizontally---is produced by the \\textbf{equation}\nenvironment. An unnumbered display equation is produced by the\n\\textbf{displaymath} environment.\n\nAgain, in either environment, you can use any of the symbols and\nstructures available in \\LaTeX\\@; this section will just give a couple\nof examples of display equations in context. First, consider the\nequation, shown as an inline equation above:\n\\begin{equation}\n \\lim_{n\\rightarrow \\infty}x=0\n\\end{equation}\nNotice how it is formatted somewhat differently in\nthe \\textbf{displaymath}\nenvironment. Now, we'll enter an unnumbered equation:\n\\begin{displaymath}\n \\sum_{i=0}^{\\infty} x + 1\n\\end{displaymath}\nand follow it with another numbered equation:\n\\begin{equation}\n \\sum_{i=0}^{\\infty}x_i=\\int_{0}^{\\pi+2} f\n\\end{equation}\njust to demonstrate \\LaTeX's able handling of numbering.\n\n\\section{Figures}\n\nThe ``\\verb|figure|'' environment should be used for figures. One or\nmore images can be placed within a figure. If your figure contains\nthird-party material, you must clearly identify it as such, as shown\nin the example below.\n\\begin{figure}[h]\n \\centering\n \\includegraphics[width=\\linewidth]{sample-franklin}\n \\caption{1907 Franklin Model D roadster. Photograph by Harris \\&\n Ewing, Inc. [Public domain], via Wikimedia\n Commons. (\\url{https:\/\/goo.gl\/VLCRBB}).}\n \\Description{A woman and a girl in white dresses sit in an open car.}\n\\end{figure}\n\nYour figures should contain a caption which describes the figure to\nthe reader.\n\nFigure captions are placed {\\itshape below} the figure.\n\nEvery figure should also have a figure description unless it is purely\ndecorative. These descriptions convey what's in the image to someone\nwho cannot see it. They are also used by search engine crawlers for\nindexing images, and when images cannot be loaded.\n\nA figure description must be unformatted plain text less than 2000\ncharacters long (including spaces). {\\bfseries Figure descriptions\n should not repeat the figure caption \u2013 their purpose is to capture\n important information that is not already provided in the caption or\n the main text of the paper.} For figures that convey important and\ncomplex new information, a short text description may not be\nadequate. More complex alternative descriptions can be placed in an\nappendix and referenced in a short figure description. For example,\nprovide a data table capturing the information in a bar chart, or a\nstructured list representing a graph. For additional information\nregarding how best to write figure descriptions and why doing this is\nso important, please see\n\\url{https:\/\/www.acm.org\/publications\/taps\/describing-figures\/}.\n\n\\subsection{The ``Teaser Figure''}\n\nA ``teaser figure'' is an image, or set of images in one figure, that\nare placed after all author and affiliation information, and before\nthe body of the article, spanning the page. If you wish to have such a\nfigure in your article, place the command immediately before the\n\\verb|\\maketitle| command:\n\\begin{verbatim}\n \\begin{teaserfigure}\n \\includegraphics[width=\\textwidth]{sampleteaser}\n \\caption{figure caption}\n \\Description{figure description}\n \\end{teaserfigure}\n\\end{verbatim}\n\n\\section{Citations and Bibliographies}\n\nThe use of \\BibTeX\\ for the preparation and formatting of one's\nreferences is strongly recommended. Authors' names should be complete\n--- use full first names (``Donald E. Knuth'') not initials\n(``D. E. Knuth'') --- and the salient identifying features of a\nreference should be included: title, year, volume, number, pages,\narticle DOI, etc.\n\nThe bibliography is included in your source document with these two\ncommands, placed just before the \\verb|\\end{document}| command:\n\\begin{verbatim}\n \\bibliographystyle{ACM-Reference-Format}\n \n\\section{Introduction}\nACM's consolidated article template, introduced in 2017, provides a\nconsistent \\LaTeX\\ style for use across ACM publications, and\nincorporates accessibility and metadata-extraction functionality\nnecessary for future Digital Library endeavors. Numerous ACM and\nSIG-specific \\LaTeX\\ templates have been examined, and their unique\nfeatures incorporated into this single new template.\n\nIf you are new to publishing with ACM, this document is a valuable\nguide to the process of preparing your work for publication. If you\nhave published with ACM before, this document provides insight and\ninstruction into more recent changes to the article template.\n\nThe ``\\verb|acmart|'' document class can be used to prepare articles\nfor any ACM publication --- conference or journal, and for any stage\nof publication, from review to final ``camera-ready'' copy, to the\nauthor's own version, with {\\itshape very} few changes to the source.\n\n\\section{Template Overview}\nAs noted in the introduction, the ``\\verb|acmart|'' document class can\nbe used to prepare many different kinds of documentation --- a\ndouble-blind initial submission of a full-length technical paper, a\ntwo-page SIGGRAPH Emerging Technologies abstract, a ``camera-ready''\njournal article, a SIGCHI Extended Abstract, and more --- all by\nselecting the appropriate {\\itshape template style} and {\\itshape\n template parameters}.\n\nThis document will explain the major features of the document\nclass. For further information, the {\\itshape \\LaTeX\\ User's Guide} is\navailable from\n\\url{https:\/\/www.acm.org\/publications\/proceedings-template}.\n\n\\subsection{Template Styles}\n\nThe primary parameter given to the ``\\verb|acmart|'' document class is\nthe {\\itshape template style} which corresponds to the kind of publication\nor SIG publishing the work. This parameter is enclosed in square\nbrackets and is a part of the {\\verb|documentclass|} command:\n\\begin{verbatim}\n \\documentclass[STYLE]{acmart}\n\\end{verbatim}\n\nJournals use one of three template styles. All but three ACM journals\nuse the {\\verb|acmsmall|} template style:\n\\begin{itemize}\n\\item {\\texttt{acmsmall}}: The default journal template style.\n\\item {\\texttt{acmlarge}}: Used by JOCCH and TAP.\n\\item {\\texttt{acmtog}}: Used by TOG.\n\\end{itemize}\n\nThe majority of conference proceedings documentation will use the {\\verb|acmconf|} template style.\n\\begin{itemize}\n\\item {\\texttt{acmconf}}: The default proceedings template style.\n\\item{\\texttt{sigchi}}: Used for SIGCHI conference articles.\n\\item{\\texttt{sigchi-a}}: Used for SIGCHI ``Extended Abstract'' articles.\n\\item{\\texttt{sigplan}}: Used for SIGPLAN conference articles.\n\\end{itemize}\n\n\\subsection{Template Parameters}\n\nIn addition to specifying the {\\itshape template style} to be used in\nformatting your work, there are a number of {\\itshape template parameters}\nwhich modify some part of the applied template style. A complete list\nof these parameters can be found in the {\\itshape \\LaTeX\\ User's Guide.}\n\nFrequently-used parameters, or combinations of parameters, include:\n\\begin{itemize}\n\\item {\\texttt{anonymous,review}}: Suitable for a ``double-blind''\n conference submission. Anonymizes the work and includes line\n numbers. Use with the \\texttt{\\acmSubmissionID} command to print the\n submission's unique ID on each page of the work.\n\\item{\\texttt{authorversion}}: Produces a version of the work suitable\n for posting by the author.\n\\item{\\texttt{screen}}: Produces colored hyperlinks.\n\\end{itemize}\n\nThis document uses the following string as the first command in the\nsource file:\n\\begin{verbatim}\n\\documentclass[sigconf]{acmart}\n\\end{verbatim}\n\n\\section{Modifications}\n\nModifying the template --- including but not limited to: adjusting\nmargins, typeface sizes, line spacing, paragraph and list definitions,\nand the use of the \\verb|\\vspace| command to manually adjust the\nvertical spacing between elements of your work --- is not allowed.\n\n{\\bfseries Your document will be returned to you for revision if\n modifications are discovered.}\n\n\\section{Typefaces}\n\nThe ``\\verb|acmart|'' document class requires the use of the\n``Libertine'' typeface family. Your \\TeX\\ installation should include\nthis set of packages. Please do not substitute other typefaces. The\n``\\verb|lmodern|'' and ``\\verb|ltimes|'' packages should not be used,\nas they will override the built-in typeface families.\n\n\\section{Title Information}\n\nThe title of your work should use capital letters appropriately -\n\\url{https:\/\/capitalizemytitle.com\/} has useful rules for\ncapitalization. Use the {\\verb|title|} command to define the title of\nyour work. If your work has a subtitle, define it with the\n{\\verb|subtitle|} command. Do not insert line breaks in your title.\n\nIf your title is lengthy, you must define a short version to be used\nin the page headers, to prevent overlapping text. The \\verb|title|\ncommand has a ``short title'' parameter:\n\\begin{verbatim}\n \\title[short title]{full title}\n\\end{verbatim}\n\n\\section{Authors and Affiliations}\n\nEach author must be defined separately for accurate metadata\nidentification. As an exception, multiple authors may share one\naffiliation. Authors' names should not be abbreviated; use full first\nnames wherever possible. Include authors' e-mail addresses whenever\npossible.\n\nGrouping authors' names or e-mail addresses, or providing an ``e-mail\nalias,'' as shown below, is not acceptable:\n\\begin{verbatim}\n \\author{Brooke Aster, David Mehldau}\n \\email{dave,judy,steve@university.edu}\n \\email{firstname.lastname@phillips.org}\n\\end{verbatim}\n\nThe \\verb|authornote| and \\verb|authornotemark| commands allow a note\nto apply to multiple authors --- for example, if the first two authors\nof an article contributed equally to the work.\n\nIf your author list is lengthy, you must define a shortened version of\nthe list of authors to be used in the page headers, to prevent\noverlapping text. The following command should be placed just after\nthe last \\verb|\\author{}| definition:\n\\begin{verbatim}\n \\renewcommand{\\shortauthors}{McCartney, et al.}\n\\end{verbatim}\nOmitting this command will force the use of a concatenated list of all\nof the authors' names, which may result in overlapping text in the\npage headers.\n\nThe article template's documentation, available at\n\\url{https:\/\/www.acm.org\/publications\/proceedings-template}, has a\ncomplete explanation of these commands and tips for their effective\nuse.\n\nNote that authors' addresses are mandatory for journal articles.\n\n\\section{Rights Information}\n\nAuthors of any work published by ACM will need to complete a rights\nform. Depending on the kind of work, and the rights management choice\nmade by the author, this may be copyright transfer, permission,\nlicense, or an OA (open access) agreement.\n\nRegardless of the rights management choice, the author will receive a\ncopy of the completed rights form once it has been submitted. This\nform contains \\LaTeX\\ commands that must be copied into the source\ndocument. When the document source is compiled, these commands and\ntheir parameters add formatted text to several areas of the final\ndocument:\n\\begin{itemize}\n\\item the ``ACM Reference Format'' text on the first page.\n\\item the ``rights management'' text on the first page.\n\\item the conference information in the page header(s).\n\\end{itemize}\n\nRights information is unique to the work; if you are preparing several\nworks for an event, make sure to use the correct set of commands with\neach of the works.\n\nThe ACM Reference Format text is required for all articles over one\npage in length, and is optional for one-page articles (abstracts).\n\n\\section{CCS Concepts and User-Defined Keywords}\n\nTwo elements of the ``acmart'' document class provide powerful\ntaxonomic tools for you to help readers find your work in an online\nsearch.\n\nThe ACM Computing Classification System ---\n\\url{https:\/\/www.acm.org\/publications\/class-2012} --- is a set of\nclassifiers and concepts that describe the computing\ndiscipline. Authors can select entries from this classification\nsystem, via \\url{https:\/\/dl.acm.org\/ccs\/ccs.cfm}, and generate the\ncommands to be included in the \\LaTeX\\ source.\n\nUser-defined keywords are a comma-separated list of words and phrases\nof the authors' choosing, providing a more flexible way of describing\nthe research being presented.\n\nCCS concepts and user-defined keywords are required for for all\narticles over two pages in length, and are optional for one- and\ntwo-page articles (or abstracts).\n\n\\section{Sectioning Commands}\n\nYour work should use standard \\LaTeX\\ sectioning commands:\n\\verb|section|, \\verb|subsection|, \\verb|subsubsection|, and\n\\verb|paragraph|. They should be numbered; do not remove the numbering\nfrom the commands.\n\nSimulating a sectioning command by setting the first word or words of\na paragraph in boldface or italicized text is {\\bfseries not allowed.}\n\n\\section{Tables}\n\nThe ``\\verb|acmart|'' document class includes the ``\\verb|booktabs|''\npackage --- \\url{https:\/\/ctan.org\/pkg\/booktabs} --- for preparing\nhigh-quality tables.\n\nTable captions are placed {\\itshape above} the table.\n\nBecause tables cannot be split across pages, the best placement for\nthem is typically the top of the page nearest their initial cite. To\nensure this proper ``floating'' placement of tables, use the\nenvironment \\textbf{table} to enclose the table's contents and the\ntable caption. The contents of the table itself must go in the\n\\textbf{tabular} environment, to be aligned properly in rows and\ncolumns, with the desired horizontal and vertical rules. Again,\ndetailed instructions on \\textbf{tabular} material are found in the\n\\textit{\\LaTeX\\ User's Guide}.\n\nImmediately following this sentence is the point at which\nTable~\\ref{tab:freq} is included in the input file; compare the\nplacement of the table here with the table in the printed output of\nthis document.\n\n\\begin{table}\n \\caption{Frequency of Special Characters}\n \\label{tab:freq}\n \\begin{tabular}{ccl}\n \\toprule\n Non-English or Math&Frequency&Comments\\\\\n \\midrule\n \\O & 1 in 1,000& For Swedish names\\\\\n $\\pi$ & 1 in 5& Common in math\\\\\n \\$ & 4 in 5 & Used in business\\\\\n $\\Psi^2_1$ & 1 in 40,000& Unexplained usage\\\\\n \\bottomrule\n\\end{tabular}\n\\end{table}\n\nTo set a wider table, which takes up the whole width of the page's\nlive area, use the environment \\textbf{table*} to enclose the table's\ncontents and the table caption. As with a single-column table, this\nwide table will ``float'' to a location deemed more\ndesirable. Immediately following this sentence is the point at which\nTable~\\ref{tab:commands} is included in the input file; again, it is\ninstructive to compare the placement of the table here with the table\nin the printed output of this document.\n\n\\begin{table*}\n \\caption{Some Typical Commands}\n \\label{tab:commands}\n \\begin{tabular}{ccl}\n \\toprule\n Command &A Number & Comments\\\\\n \\midrule\n \\texttt{{\\char'134}author} & 100& Author \\\\\n \\texttt{{\\char'134}table}& 300 & For tables\\\\\n \\texttt{{\\char'134}table*}& 400& For wider tables\\\\\n \\bottomrule\n \\end{tabular}\n\\end{table*}\n\nAlways use midrule to separate table header rows from data rows, and\nuse it only for this purpose. This enables assistive technologies to\nrecognise table headers and support their users in navigating tables\nmore easily.\n\n\\section{Math Equations}\nYou may want to display math equations in three distinct styles:\ninline, numbered or non-numbered display. Each of the three are\ndiscussed in the next sections.\n\n\\subsection{Inline (In-text) Equations}\nA formula that appears in the running text is called an inline or\nin-text formula. It is produced by the \\textbf{math} environment,\nwhich can be invoked with the usual\n\\texttt{{\\char'134}begin\\,\\ldots{\\char'134}end} construction or with\nthe short form \\texttt{\\$\\,\\ldots\\$}. You can use any of the symbols\nand structures, from $\\alpha$ to $\\omega$, available in\n\\LaTeX~\\cite{Lamport:LaTeX}; this section will simply show a few\nexamples of in-text equations in context. Notice how this equation:\n\\begin{math}\n \\lim_{n\\rightarrow \\infty}x=0\n\\end{math},\nset here in in-line math style, looks slightly different when\nset in display style. (See next section).\n\n\\subsection{Display Equations}\nA numbered display equation---one set off by vertical space from the\ntext and centered horizontally---is produced by the \\textbf{equation}\nenvironment. An unnumbered display equation is produced by the\n\\textbf{displaymath} environment.\n\nAgain, in either environment, you can use any of the symbols and\nstructures available in \\LaTeX\\@; this section will just give a couple\nof examples of display equations in context. First, consider the\nequation, shown as an inline equation above:\n\\begin{equation}\n \\lim_{n\\rightarrow \\infty}x=0\n\\end{equation}\nNotice how it is formatted somewhat differently in\nthe \\textbf{displaymath}\nenvironment. Now, we'll enter an unnumbered equation:\n\\begin{displaymath}\n \\sum_{i=0}^{\\infty} x + 1\n\\end{displaymath}\nand follow it with another numbered equation:\n\\begin{equation}\n \\sum_{i=0}^{\\infty}x_i=\\int_{0}^{\\pi+2} f\n\\end{equation}\njust to demonstrate \\LaTeX's able handling of numbering.\n\n\\section{Figures}\n\nThe ``\\verb|figure|'' environment should be used for figures. One or\nmore images can be placed within a figure. If your figure contains\nthird-party material, you must clearly identify it as such, as shown\nin the example below.\n\\begin{figure}[h]\n \\centering\n \\includegraphics[width=\\linewidth]{sample-franklin}\n \\caption{1907 Franklin Model D roadster. Photograph by Harris \\&\n Ewing, Inc. [Public domain], via Wikimedia\n Commons. (\\url{https:\/\/goo.gl\/VLCRBB}).}\n \\Description{A woman and a girl in white dresses sit in an open car.}\n\\end{figure}\n\nYour figures should contain a caption which describes the figure to\nthe reader.\n\nFigure captions are placed {\\itshape below} the figure.\n\nEvery figure should also have a figure description unless it is purely\ndecorative. These descriptions convey what's in the image to someone\nwho cannot see it. They are also used by search engine crawlers for\nindexing images, and when images cannot be loaded.\n\nA figure description must be unformatted plain text less than 2000\ncharacters long (including spaces). {\\bfseries Figure descriptions\n should not repeat the figure caption \u2013 their purpose is to capture\n important information that is not already provided in the caption or\n the main text of the paper.} For figures that convey important and\ncomplex new information, a short text description may not be\nadequate. More complex alternative descriptions can be placed in an\nappendix and referenced in a short figure description. For example,\nprovide a data table capturing the information in a bar chart, or a\nstructured list representing a graph. For additional information\nregarding how best to write figure descriptions and why doing this is\nso important, please see\n\\url{https:\/\/www.acm.org\/publications\/taps\/describing-figures\/}.\n\n\\subsection{The ``Teaser Figure''}\n\nA ``teaser figure'' is an image, or set of images in one figure, that\nare placed after all author and affiliation information, and before\nthe body of the article, spanning the page. If you wish to have such a\nfigure in your article, place the command immediately before the\n\\verb|\\maketitle| command:\n\\begin{verbatim}\n \\begin{teaserfigure}\n \\includegraphics[width=\\textwidth]{sampleteaser}\n \\caption{figure caption}\n \\Description{figure description}\n \\end{teaserfigure}\n\\end{verbatim}\n\n\\section{Citations and Bibliographies}\n\nThe use of \\BibTeX\\ for the preparation and formatting of one's\nreferences is strongly recommended. Authors' names should be complete\n--- use full first names (``Donald E. Knuth'') not initials\n(``D. E. Knuth'') --- and the salient identifying features of a\nreference should be included: title, year, volume, number, pages,\narticle DOI, etc.\n\nThe bibliography is included in your source document with these two\ncommands, placed just before the \\verb|\\end{document}| command:\n\\begin{verbatim}\n \\bibliographystyle{ACM-Reference-Format}\n ","meta":{"redpajama_set_name":"RedPajamaArXiv"}}