diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzkksx" "b/data_all_eng_slimpj/shuffled/split2/finalzzkksx" new file mode 100644--- /dev/null +++ "b/data_all_eng_slimpj/shuffled/split2/finalzzkksx" @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\n\nThe morphological investigation of galaxies in the early Universe can provide important information on their\nformation and evolutionary processes.\nFor instance, a disturbed and multi-clump morphology, especially in the cold phase, may suggest\nthe presence of disc instabilities, and can give indication of feedback processes, as well as minor and\/or major merger events during the galaxy assembly \\citep[e.g.,][]{Tamburello:2015, Fiacconi:2016, Ceverino:2017, Pallottini:2017,Pallottini:2017a}.\nIn this context, the advent of facilities delivering high angular resolution observations has enabled us to probe the internal structure of galaxies in the distant Universe, revealing that clumpy morphologies are more common at higher redshift than at $z=0$ \\citep[e.g.][]{Forster-Schreiber:2006,Genzel:2008}. \nFor instance, the fraction of galaxies at $0.55.7$ \\cite{Jiang:2013} found that roughly half of the brightest galaxies ($M_{1500}$<-20.5 mag) are made of multiple components that may be merging.\nNear-infrared (NIR) high-resolution imaging have also revealed that irregular shapes with multi-clump morphology are prevalent in LAEs and LBGs at $z\\sim7$ within the epoch of reionization \\citep{Ouchi:2010, Sobral:2015, Matthee:2017, Bowler:2017}.\n\nAdditional identification of clumpy systems in the early epoch and a detailed characterisation of high-$z$ clumps or satellites is fundamental to constrain galaxy assembly. \nThe extended Atacama Large Millimetre\/submillimetre Array (ALMA) configurations enable us to reach high-angular\nresolution\nand exploit far-infrared (FIR) fine structure emission lines, such as [C{\\sc ii}]\\ at 158$\\mu$m, as\npowerful diagnostics to assess the morphology of primeval galaxies. [C{\\sc ii}]\\ is emitted primarily in the (mostly neutral) atomic and molecular gas associated with Photon\nDominated Regions (excited by the soft UV photons), but also in partly ionised regions and it is one of the\nprimary coolants of the ISM. Indeed, it is generally the strongest emission lines observed in the spectra of\ngalaxies. Since its first detection at high redshift \\citep{Maiolino:2005} this transition has then been\ndetected in large samples of distant galaxies. However, until recently, the [C{\\sc ii}]\\ emission was only detected\nin extreme environments, such as quasar host galaxies and SMGs, characterised by SFRs of several hundred\nsolar masses per year, not really representative of the bulk of the galaxy population at these epochs\n\\citep[e.g.][]{Maiolino:2009, Maiolino:2012, De-Breuck:2011, Wagg:2012, Gallerani:2012, Carilli:2013, Carniani:2013, Williams:2014, Riechers:2014, Yun:2015, Schreiber:2017, Trakhtenbrot:2017, Decarli:2017}.\nDetecting [C{\\sc ii}] \\ in ``normal'' galaxies has required the sensitivity delivered by ALMA.\nTo date, the [C{\\sc ii}]\\ line has been detected in several galaxies at $z>5$ and it is spatially resolved in most of these targets \\citep{Capak:2015,Willott:2015, Maiolino:2015, Knudsen:2016, Pentericci:2016, Bradac:2016, Smit:2017, Carniani:2017a,Matthee:2017}. \n\\cite{Smit:2017} recently presented [C{\\sc ii}]\\ observations of two galaxies at $z\\sim7$ characterised by a gradient of velocity consistent with a undisturbed rotating gas disk.\nHowever, most of the $z>5$ galaxies show extended and clumpy [C{\\sc ii}]\\ emission with velocities consistent with the systematic redshift of the galaxy ($|\\Delta$v$_{\\rm Ly\\alpha}|<500$~km\/s) but spatially offset relative to the rest-frame UV counterpart \\citep{Maiolino:2015, Willott:2015, Capak:2015,Carniani:2017a, Carniani:2017, Jones:2017}.\nIn many cases these offsets have been ignored or ascribed to astrometric uncertainties. However, based\non detailed astrometric analysis,\nit has been shown that most of these offsets are physical (a revised analysis will be given in this paper),\nhence they should be taken as an important signature of the evolutionary processes in the early phases of\ngalaxy formation.\nVarious scenarios have been proposed to explain the positional offsets between [C{\\sc ii}]\\ and star-forming regions such as\nstellar feedback clearing part of the ISM, gas accretion, wet mergers, dust obscuration and variations of the\nionisation parameter \\citep[e.g.][]{Vallini:2015, Katz:2017}. \n\\cite{Barisic:2017} and \\cite{Faisst:2017a} have recently found (rest-frame) UV faint companions whose locations is\nconsistent with the displaced [C{\\sc ii}]\\ emission, suggesting that the carbon line traces star-forming regions where the UV light is absorbed by dust.\n\nAs mentioned,\nmost previous studies have attempted to assess the nature of [C{\\sc ii}]\\ emission in primeval galaxies neglecting the positional offsets between the FIR line and rest-frame UV emission. \nThe goal of this paper is to assess the connection between [C{\\sc ii}]\\ and SFR in the early Universe by taking into\naccount the multi-clump morphology of galaxies at $z>5$ and by associating the components with their\nproper optical-UV counterparts (if detected). This is achieved by re-analysing ALMA [C{\\sc ii}]\\ observations of $z > 5$ star-forming galaxies, and by performing a detailed kinematical analysis of the [C{\\sc ii}]\\ line, in order to deblend the different components of the multi-clump systems. \nIn addition to previous ALMA observations, partly discussed in literature, we also make use of new ALMA data\ntargeting five $z\\sim6$ star-forming galaxies with SFR~$<20$~M$_{\\odot}$ yr$^{-1}$. In Section~\\ref{sec:sample} we detail the sample\nand the analysis of ALMA observations. The morphological analysis is presented in Section~\\ref{sec:Multi-component\nsystems}, while the relation between the [C{\\sc ii}]\\ and SFR is discussed in Section~\\ref{sec:dispersion}. In\nSection~\\ref{sec:lyaEW} we investigate the connection between [C{\\sc ii}]\\ luminosity and Ly$\\alpha$\\ strength in our sample.\nSection~\\ref{sec:spatially} focuses on the spatial extension of the [C{\\sc ii}]\\ and UV emission\nand the correlation between [C{\\sc ii}]\\ surface brightness and SFR surface density. We discuss the findings in Section~\\ref{sec:discussion}, while the conclusions of this work are reported in Section~\\ref{sec:conclusions}\n\nThroughout this paper we assume the following cosmological parameters: $H_0 = 67.8$ km s$^{-1}$ Mpc$^{-1}$ , $\\Omega_M = 0.308$, $\\Omega_\\Lambda$ = 0.685 \\citep{Planck:2016}\n\n\n\\section{Sample, observations and analysis}\\label{sec:sample}\n\n\\subsection{Archival data}\\label{sec:literature_sample}\n\n\\begin{table}\n \\centering\n \\caption{Overview of the $z>5$ star-forming galaxies observed with ALMA used in this paper, ordered by name.}\n \n \\addtolength{\\tabcolsep}{-3pt} \n \\begin{tabular}{lccccc }\n \\hline\n \\hline\n Target$^{(a)}$ & Ra$^{(b)}$ & Dec$^{(c)}$ & Ref.$^{(d)}$ & [C{\\sc ii}]$^{(e)}$ & Clumpy$^{(f)}$ \\\\\n \\hline\n \t\\multicolumn{6}{c}{\\emph{Literature Sample}} \\\\\n\t \\hline\nBDF3299 & 337.0511 & -35.1665 & 1,2 & \\cmark & \\cmark\\\\\nBDF512 & 336.9444 & -35.1188 & 1 & & \\\\\nCLM1 & 37.0124 & -4.2717 & 3 & \\cmark & \\\\\nCOSMOS13679 & 150.0990 & 2.3436 & 4 & \\cmark & \\\\\nCOSMOS24108 & 150.1972 & 2.4786 & 4 & \\cmark & \\cmark\\\\\nCOS-2987030247 & 150.1245 & 2.2173 & 5 & \\cmark & \\\\\nCOS-3018555981 & 150.1245 & 2.2666 & 5 & \\cmark & \\\\\nCR7 & 150.2417 & 1.8042 & 6 & \\cmark & \\cmark\\\\\nHimiko & 34.4898 & -5.1458 & 7,12 & \\cmark & \\cmark\\\\\nHZ8 & 150.0168 & 2.6266 & 8 & \\cmark & \\cmark \\\\\nHZ7 & 149.8769 & 2.1341 & 8 & \\cmark & \\\\\nHZ6 & 150.0896 & 2.5864 & 8 & \\cmark & \\cmark \\\\\nHZ4 & 149.6188 & 2.0518 & 8 & \\cmark & \\\\\nHZ3 & 150.0392 & 2.3371 & 8 & \\cmark & \\\\\nHZ9 & 149.9654 & 2.3783 & 8 & \\cmark & \\\\\nHZ10 & 150.2470 & 1.5554 & 8 & \\cmark & \\cmark \\\\\nHZ2 & 150.5170 & 1.9289 & 8 & \\cmark & \\cmark \\\\\nHZ1 & 149.9718 & 2.1181 & 8 & \\cmark & \\\\\nIOK-1 & 200.9492 & 27.4155 & 9 & & \\\\\nNTTDF6345 & 181.4039 & -7.7561 & 4 & \\cmark & \\\\\nSDF46975 & 200.9292 & 27.3414 & 1 & \\\\\nSXDF-NB1006-2 & 34.7357 & -5.3330 & 10 & & \\\\\nUDS16291 & 34.3561 & -5.1856 & 4 & \\cmark & \\\\\nWMH5 & 36.6126 & -4.8773 & 3,11 & \\cmark & \\cmark\\\\\n \\hline\n \t\\multicolumn{6}{|c|}{\\emph{Additional new data}} \\\\\n\t \\hline\nBDF2203 & 336.958 & -35.1472 & & \\cmark & \\\\\nGOODS3203 & 53.0928 & -27.8826 \\\\\nCOSMOS20521 & 150.1396 & 2.4269 & & & \\\\\nNTTDF2313 & 181.3804 & -7.6935 & & \\\\\nUDS4812 & 34.4768 & -5.2472 & & & \\\\\n\n \\hline\n\\end{tabular}\n\\\\\n\n\\begin{tablenotes}[flushleft]\n\\footnotesize\n\\item {\\bf Notes}.\n{\\bf (a)} Name of the source.\n{\\bf(b, c)} J2000 coordinates.\n{\\bf(d)} References in which ALMA observations are presented (\n[1] \\citealt {Maiolino:2015}; [2] \\citealt{Carniani:2017a};\n[3] \\citealt{Willott:2015}; [4] \\citealt{Pentericci:2016};\n[5] \\citealt{Smit:2017}; [6] \\citealt{Matthee:2017};\n[7] \\citealt{Ouchi:2013}; [8] \\citealt{Capak:2015};\n[9] \\citealt{Ota:2014}; [10] \\citealt{Ouchi:2013};\n[11] \\citealt{Jones:2017} [12] Carniani et al. 2017b)\n{\\bf(e)} Check mark, \\cmark, indicates that [C{\\sc ii}]\\ emission has been detected at the redshift of the galaxy.\n{\\bf(f)} Check mark, \\cmark, indicates that the galaxy has a clumpy morphology.\n\n\\end{tablenotes}\n\\label{tab:projects}\n\\end{table}\n\nThe sample is mainly drawn from the archive and literature by selecting only spectroscopically confirmed\nstar-forming galaxies at $z>5$ observed with ALMA in the [C{\\sc ii}]\\ line. \nWe limit our sample to those systems with SFR $\\lesssim\n100$ M$_{\\odot}$ yr$^{-1}$\\ since they are representative of the bulk of the galaxy population in the early Universe \\citep[e.g.][]{Robertson:2015, Carniani:2015}.\nThe list of selected sources is given in Table~\\ref{tab:projects}.\nThe sample does not include lensed systems \\citep{Knudsen:2016, Gonzalez-Lopez:2014, Bradac:2016, Schaerer:2015}\nsince magnification factor uncertainties may lead to large errors on SFR and [C{\\sc ii}]\\ luminosity estimates, as well\nas on the morphology analysis. \n\n\nFor the purpose of our investigation, which focuses on the nature and implications of the positional offsets\nbetween [C{\\sc ii}]\\ and UV emission, we have retrieved and re-analysed ALMA data revealing a [C{\\sc ii}]\\ detection at the systemic velocity of the galaxy (see Table~\\ref{tab:projects}). For these objects, ALMA observations have been calibrated following the prescriptions presented in previous works. \n\nIn addition to the rest-frame FIR images, we have also used Hubble Space Telescope ({\\it HST}) and Visible and Infrared Survey Telescope for Astronomy (VISTA) NIR observations (rest-frame UV at $z>5$). \nALMA, {\\it HST} and VISTA data have been aligned based on the location of serendipitous sources detected in both\nALMA continuum and NIR images by assuming that the millimetre emission of these sources is cospatial to\nthe near-infrared map. This is also supported by the fact that all foreground sources used for registering\nmillimetre and NIR images do not exhibit any multi-clump or merger-like morphologies indicating that\nastrometric offsets between the ALMA and NIR images are likely associated to astrometric calibrations. \nFor those observations revealing the presence of two (or more) serendipitous sources we have verified that the\nastrometric shift for each source is consistent with that estimated from the other source(s) in the same map.\nIn all cases we have checked that the estimated astrometric offset is consistent with those obtained by aligning NIR foreground sources and ALMA phase calibrators to their astrometric position from the GAIA Data Release 1 catalogue \\citep{Gaia-Collaboration:2016}. %\nFor those sources whose ALMA continuum map showing no serendipitous sources, we have matched the NIR foreground sources and ALMA phase calibrators to either GAIA Data Release~1 catalogue \\citep{Gaia-Collaboration:2016} or AllWISE catalogue \\citep{Cutri:2013}. \nWe note that in all ALMA datasets the locations of the various phase calibrators are in agreement within the\nerror with the GAIA and AllWISE catalogues, implying that, when a systemic (not physical)\noffset is seen, this is generally due to some small astrometric uncertainties in the optical-NIR\ndata.\nThese astrometric issues have been also discussed by \\cite{Dunlop:2016} who analysed ALMA images targeting the Hubble Ultra Deep Field.\nWe therefore applied the astrometric shifts, which span a range between 0.1\\arcsec\\ and 0.25\\arcsec, to the NIR images. \n\n\nWe note that additional systems with low SFRs have also been tentatively\ndetected in 14 [C{\\sc ii}]\\ line emitting candidates at $6 < z < 8$ \\citep{Aravena:2016a},\nwhich are not included in this analysis as they are not spectroscopically confirmed\nyet and $\\sim$60\\% of these objects are expected to be spurious .\n\n\n\n\n\\subsection{Additional new ALMA data}\n\\label{sec:additional}\n\n\nIn addition to the archival\/literature sample, we have also included new [C{\\sc ii}]\\ observations of five star-forming\ngalaxies at $z\\sim6$ with a SFR$\\sim$10 M$_{\\odot}$ yr$^{-1}$\\ observed with ALMA in Cycles 3 and 4 (P.I.\nPentericci). The five new sources, listed in Table~\\ref{tab:projects}, have been selected from a sample of\n$>120$ LBGs at $z\\sim6-7$. These have been spectroscopically confirmed thanks to recent ultra-deep spectroscopic\nobservations from the ESO Large Program CANDELSz7 \\citep[P.I. L. Pentericci;][]{DE-Barros:2017}. The proposed\nALMA programs aimed at observing [C{\\sc ii}]\\ emission in ten star-forming galaxies at $z>6$ with UV luminosities\nlower than -21 mag (UV SFR $<20$ M$_{\\odot}$ yr$^{-1}$) and spectroscopic redshift uncertainties $<$0.03, but only five\ngalaxies have been observed. The observations and data calibrations are presented in\nthe Appendix~\\ref{sec:appA}. We have registered NIR images to ALMA observations by matching the location of\nthe foreground, serendipitous continuum\nsources and ALMA calibrators to the position given by the GAIA Data Release 1 catalogue (Gaia Collaboration et al. 2016).\n\nWhile the continuum emission is not detected at the location of any of the five galaxies, we detect two and one serendipitous sources in the COSMOS20521 and BDF2313 continuum maps, respectively. \nThe positions of the serendipitous continuum sources is in agreement with the location of NIR foreground galaxies, thus confirming the astrometric shifts estimated from the catalogue.\n \n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.8\\columnwidth]{Figure\/FigureSpectrumBDF.pdf} %\n\\caption{ ALMA spectrum of BDF2203 showing a new [C{\\sc ii}]\\ detection. The velocity reference is set to the redshift defined by the Ly$\\alpha$.\nThe dotted grey lines shows the 1$\\sigma$ and -1$\\sigma$ per channel. The red line indicates the best fit 1D Gaussian line profile.} \n \\label{fig:appFig1}\n\\end{figure}\n\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.6\\columnwidth]{Figure\/FigBDF.pdf} %\n\\includegraphics[width=0.6\\columnwidth]{Figure\/FigBDFZoom.pdf} %\n\\caption{Top: [C{\\sc ii}]\\ flux map of BDF2203 obtained integrating the ALMA cube over the velocity range between -60 km\/s and 132 km\/s. Red solid contours are at levels of 2$\\sigma$,\n3$\\sigma$ and 4$\\sigma$, where $\\sigma=23$ mJy km\/s. Dashed black contours indicate negative values at -3 and -2 times the noise level in the same map.\nBottom: zoom of the central 2.5\\arcsec\\ region around BDF2203. [C{\\sc ii}]\\ flux map contours are over imposed on the NIR image. Red contours are at the same levels\nas in the top panel. The ALMA synthesised beam in the lower left corner in both panels. }\n \\label{fig:mapBdf}\n\\end{figure}\n\nThe [C{\\sc ii}]\\ emission is undetected in all but one of\nthese sources, that is, BDF2203.\nFigure~\\ref{fig:appFig1} shows the spectrum of the detected [C{\\sc ii}]\\ emission,\nwith a spectral rebinning of 40 km s$^{-1}$, while the spectra of the non-detections are shown in the Appendix (Figure~\\ref{fig:appFig2}).\n\nFor the four non-detections, we assume a full-width-at-half-maximum of\nFWHM = 100 km s$^{-1}$, which is consistent with [C{\\sc ii}]\\ line widths observed in other $z>6$ galaxies\n\\citep[e.g.][]{Pentericci:2016,Carniani:2017a, Carniani:2017}, and we infer 3$\\sigma$ upper limits on the [C{\\sc ii}]\\ luminosity (Table~\\ref{tab:appTab1}). \n\nThe redshift of the [C{\\sc ii}]\\ emission detected in BDF2203 is $z_{\\rm [CII]}=6.1224\\pm0.0005$, which is in agreement\nwith that inferred from Ly$\\alpha$ ($z_{\\rm Ly\\alpha}= 6.12\\pm0.03$). \nBy fitting a 1D Gaussian profile to the [C{\\sc ii}]\\ line we estimate a FWHM=$150\\pm50$ km\/s that is similar to those estimated in high-$z$ [C{\\sc ii}]-emitting galaxies \\citep[FWHM=50-250 km\/s;][]{Willott:2015, Pentericci:2016, Matthee:2017, Carniani:2017}.\nThe flux map of the [C{\\sc ii}]\\ line, extracted with a spectral width of 200 km\/s, is shown in Figure~\\ref{fig:mapBdf}.\nThe emission is detected in the map with a S\/N=5 and has an integrated flux density of $140\\pm35$ mJy km\/s, which\ncorresponds to L$_{\\rm[CII]}$$=(12.5\\pm2.5)\\times10^{7}$~L$_{\\odot}$\\ at z=6.12. By fitting a 2D gaussian profile to the flux map, we\nmeasure a size of ${\\rm(2.3\\pm0.5)\\arcsec\\times(1.2\\pm0.2)\\arcsec}$ with ${\\rm PA=88\\deg\\pm8\\deg}$ and a beam-deconvolved size of ${\\rm(1.4\\pm0.8)\\arcsec\\times(0.5\\pm0.4)\\arcsec}$ with a ${\\rm PA=80\\deg\\pm20\\deg}$. The [C{\\sc ii}]\\ emission is thus marginally resolved and has a diameter size of $\\sim5$~kpc. The [C{\\sc ii}]\\ flux map overlaps with the rest-frame UV emission that\nhas an extension of about 1 kpc. The centroids are separated by 0.25\\arcsec. However, given the ALMA beam size\n($1.90\\arcsec\\ \\times1.11\\arcsec$), such positional offset is consistent with the positional uncertainties $\\Delta\\theta\\approx\\frac{}{S\/N}\\sim0.2\\arcsec-0.4\\arcsec$. The low angular resolution and sensitivity of current observations are not sufficient to assess the morphology of the FIR line emission.\n\n\n\n\n\\subsection{SFR estimates and morphology analysis}\n\\label{sec:SFRestimates}\n\nThe continuum emission at rest-frame wavelengths around 158 $\\mu$m is associated to thermal emission from dust heated by\nthe UV emission of young stellar populations in galaxies.\nThe continuum emission is detected only in four sources within the selected sample. When compared with the typical UV-IR SED of galaxies, the\nweak rest-frame far-IR continuum indicates that these galaxies are on average characterised by low dust masses.\n\n\nThe spectral energy distribution (SED) of the thermal dust emission can be modelled with a greybody with dust\ntemperature T$_{\\rm d}$\\ and spectral index emissivity $\\beta$. However, one single photometric measurement is not sufficient\nto perform the SED fitting and constrain the two free parameters. We thus assume T$_{\\rm d}$\\ = 30K and $\\beta$ = 1.5, which are\nconsistent with those observed in local dwarf galaxies \\citep{Ota:2014}, to estimate the far-infrared (FIR) luminosity from the ALMA continuum observations. For those galaxies that are not detected in ALMA continuum images we infer a 3$\\sigma$ upper limit on ${\\rm L_{FIR} }$. \nWe note that the FIR emission strongly depends on the assumed dust temperature, yielding to a luminosity uncertainty of $\\Delta \\log({\\rm L_{FIR}}) = 0.6 $ \\citep{Faisst:2017a}.\n\nFor each galaxy we can infer the star-formation rate from both the UV (SFR$_{\\rm UV}$) and \n FIR (SFR$_{\\rm FIR}$) emission by adopting the calibrations presented in \\cite{Kennicutt:2012}.\nExcluding HZ9 and HZ10, all sources in the sample have ${\\rm SFR_{\\rm FIR}\/SFR_{\\rm UV} \\lessapprox 1}$ with an average value of ${\\rm SFR_{\\rm FIR}\/SFR_{\\rm UV}}\\approx0.6$ . Given that most of the FIR luminosity estimates are $3\\sigma$\nupper limits, the average SFR$_{\\rm FIR}$\/SFR$_{\\rm UV}$ is actually much lower than 0.6.\n\nGiven the small contribution of SFR$_{\\rm FIR}$ (either in the IR-detected sources or those with upper limits), we assume total ${\\rm SFR \\approx SFR_{\\rm UV}}$ with no dust correction for those galaxies without continuum detection. While we infer total ${\\rm SFR = SFR_{\\rm UV}+SFR_{\\rm FIR}}$ for those galaxies revealing continuum emission at 158 $\\mu$m.\n\n\n \nIn order to assess the multi-clump morphology of the [C{\\sc ii}]\\ emission in our sources, we perform a kinematical analysis on the retrieved ALMA cube, extracting channel\nmaps at different velocities relative to the redshift of the galaxies. This analysis enables us to disentangle the emission of complex systems having multiple\ncomponents at different velocities \\citep[e.g.][]{Carniani:2013, Riechers:2014}. We also estimate the size of the\nvarious UV and [C{\\sc ii}]\\ emission by fitting a 2D elliptical Gaussian profile to emission maps and by taking into account\nthe angular resolution of the observations.\n\nThe properties of the [C{\\sc ii}], FIR and UV emission, such as redshift, flux density, luminosity, SFR, and radius, are reported in Table~\\ref{tab:table1}\n\n\n\\section{Multi-components systems}\\label{sec:Multi-component systems}\n\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=2\\columnwidth]{Figure\/FigureStamps}\n %\n \\caption{Rest-frame UV images of the nine $z>5$ star-forming galaxies showing multi-clump morphology in [C{\\sc ii}]\\ and\/or rest-frame UV emission. The galaxies are ordered by redshift\n and each stamp is 3\\arcsec\\ on the side, with North to the top and East to the left. The red, blue, and green contours show the [C{\\sc ii}]\\ channel maps at different velocity intervals. \n The levels of the contours and channel maps velocities are indicated in the legends. The properties of each galaxy are reported in Table~\\ref{tab:table1}.}\n \\label{fig:fig1}\n\\end{figure*}\n\n\n\n\\cite{Carniani:2017a} discuss the origin of the positional and spectral offsets between optical emission and FIR lines observed in high-$z$ systems, suggesting that the observational properties can arise from distinct regions (or components) of galaxies. In order to understand the nature of these offsets, we have re-analysed ALMA [C{\\sc ii}]\\ observations for those galaxies showing [C{\\sc ii}]\\ detections and have performed a morphology analysis as discussed in Section~\\ref{sec:SFRestimates}.\nWe have found a clear multi-clump morphology in nine out of 21 $z>5$ galaxies having [C{\\sc ii}]\\ detections (Table~\\ref{tab:table1}). \nThe UV rest-frame images and [C{\\sc ii}]\\ maps are shown in Figure~\\ref{fig:fig1}, sorted by redshift.\nThe stamps are 3\\arcsec\\ across, or $\\sim$17.5~kpc at the average redshift of this sample ($= 6$), and show\nthe contours of the [C{\\sc ii}]\\ maps, obtained from our analysis, superimposed on \nthe rest-frame UV emission in grayscale. \n\nNew {\\it HST} near-IR images of HZ6, HZ8, and HZ10 have already been presented in \\cite{Barisic:2017} and \\cite{Faisst:2017a} revealing multi-component structures.\nThe location of the individual rest-frame UV clumps is consistent with the peak positions of the [C{\\sc ii}]\\ emission extracted at different velocities relative to the\nredshift of the brightest component (which is labelled ``a'' in all stamps). \nThe [C{\\sc ii}]\\ emission detected in all individual clumps has a level of significance\\footnote{$\\sigma$ is the rms of the channel map in which we detect the [C{\\sc ii}]\\ emission} higher than 5$\\sigma$ and it is spatially resolved (see Table~\\ref{tab:table1}). \nThe channel map analysis confirms that HZ8b and HZ10b (Hz8W and Hz10W in previous works) are at the same redshift of\nHZ8a and HZ10a, respectively. We note that the kinematic properties of HZ10 are consistent with the analysis reported by \\cite{Jones:2017a}, who claim that the velocity gradient observed in this galaxy matches a merger scenario rather than a rotating gas disk model.\n\n\nThe deeper {\\it HST} observations reveal also two faint companions close to the HZ2 galaxy within a projected distance of $\\sim$6~kpc.\nThese sources are not discussed in previous studies \\citep{Capak:2015,Barisic:2017}, since their redshifts were not\nfully spectroscopically confirmed by the [C{\\sc ii}]\\ line. \nHowever, a detailed kinematic investigation of the carbon line shows an extended emission with a morphology consistent with the rest-frame UV emission. \nAlthough the two faint companions are detected with a low level of significance ($\\sim3.5\\sigma$), the match between the three UV peaks and the [C{\\sc ii}]\\ emission supports the reliability of the ALMA detections. \nThe three sources, dubbed in this work as HZ2a, HZ2b and HZ2c, are at the same redshift and form a multi-component system similar to that observed in HZ6.\nThe angular resolution and the low sensitivity of current ALMA observations are not sufficient to spatially resolve the [C{\\sc ii}]\\ emission in HZ2b and HZ2c.\nHZ2a is instead spatially resolved, with a diameter of 2.5 kpc.\n\n\nAn additional star-forming galaxy with a complex morphology is WHM5 at $z=6.0695$, which appears to consist of multiple components in [C{\\sc ii}]: a compact source, seen also in dust emission, and an extended component at the location of the rest-frame UV emission \\citep{Willott:2015a, Jones:2017}. \nThe two components are separated by a projected distance of $\\sim3$ kpc and a velocity of $\\sim200$ km\/s. Both [C{\\sc ii}]\\\nemission components are spatially resolved ($\\sim$1.3 and $\\sim$2.8 kpc) in the ALMA observations with an angular resolution of 0.3\\arcsec\\ \\citep{Jones:2017}.\n\nAt higher redshift, the presence of multiple [C{\\sc ii}]\\ components has recently been reported in CR7 at $z$=6.6 by\n\\cite{Sobral:2015,Sobral:2017} and \\cite{Matthee:2015,Matthee:2017}. Three out of four detected [C{\\sc ii}]\\ clumps coincide\nwith the location of UV clumps. In two cases the [C{\\sc ii}]\\ emission is spatially resolved with a radius of $\\sim3-3.8$ kpc (see \\citealt{Matthee:2017} for details).\n\nAn additional multi-clump galaxy observed with ALMA is Himiko at $z\\sim6.695$ \\citep{Ouchi:2013}. The rest-frame UV image (see Figure~3 of \\citealt{Ouchi:2013})\nreveals that the galaxy comprises three sub-components with SFR spanning in range between 5 and 8 M$_{\\odot}$ yr$^{-1}$. The projected distance between the sources is of about 3-7 kpc.\nkpc. Out of the three sub-components have a Ly$\\alpha$\\ EW=68 \\AA\\, while the other two have EW less than 8 \\AA. Although early observations had reported non-detections\nof [C{\\sc ii}]\\ in this source, recently \\cite{Carniani:2017} have reported a clear detection with extended\/multi-clump morphology. The primary [C{\\sc ii}]\\ emission is\ncoincident with the Ly$\\alpha$ peak, while the UV clumps are much weaker in [C{\\sc ii}]\\ or even undetected. More generally the extended [C{\\sc ii}]\\ emission does not resembles the UV clumpy distribution.\n\nFor the remaining systems (COS24108 and BDF3299) showing a clear positional offsets between [C{\\sc ii}]\\ and UV emission in Figure~\\ref{fig:fig1}, we cannot speculate much more than what has been already done in previous works due to the lack of deeper ALMA and\/or {\\it HST} observations. \nAs discussed by \\citealt{Carniani:2017a}, these offsets are certainly associated with physically distinct sub-components, and the [C{\\sc ii}]\\ clumps with no\nUV counterpart may either be tracing star forming\nregions that are heavily obscured at UV wavelengths \\citep{Katz:2017}, or associated with accreting\/ejected gas.\n\n\n\n\n\\subsection{On the nature of the kpc-scale sub-components} \n\nThe deeper [C{\\sc ii}]\\ and rest-frame UV observations have unveiled the real multi-component nature of nine star-forming galaxies at $z>5$ further highlighting and\n(partly) explaining\npositional offsets between UV and [C{\\sc ii}]\\ emission in previous studies \\citep{Capak:2015,Maiolino:2015,Willott:2015,Pentericci:2016,Carniani:2017a}. These results\nsuggest that future evidence of displaced FIR line and UV emission should be not ignored since it generally reveals the\npresence of sub-components with different physical properties.\n\nSuch sub-components can be ascribed to either satellites in the process of accreting \\citep{Pallottini:2017a} or clumps ejected by past galactic outflows \\citep{Gallerani:2016}.\nHowever, in many cases both the SFR$_{\\rm UV}$ and the size of the various sub-components are comparable to those estimated for\nthe central galaxies, hence suggesting a\nmajor merger scenario in many cases.\nFuture ALMA observations with higher resolution and sensitivity are necessary to detect further sub-components in these and other systems and to perform a detailed dynamical analysis, which allow us to to estimate the dynamic mass and assess the nature of sub-components.\n\n\n\n\n\n\n\n\\section{The L$_{\\rm[CII]}$\\ - SFR relation at z=5--7}\\label{sec:dispersion}\n\nA tight relation between the [C{\\sc ii}]\\ luminosity and the global SFR is seen in local galaxy observations, at least\nwhen excluding extreme (ULIRG-like) cases \\citep{De-Looze:2014, Kapala:2015, Herrera-Camus:2015}.\nThis finding makes the [C{\\sc ii}]\\ line a promising tool to investigate the properties of early galaxies and to\ntrace their star formation. \nHowever, the behaviour of the [C{\\sc ii}]\\ line emission at $z>5$ seems to be more complex than observed in the local Universe. \nPrevious studies have shown that only a fraction of [C{\\sc ii}]\\ detections of early galaxies agree with the local relation, while most\nhigh-$z$ galaxies are broadly scattered, with claims that most of them are [C{\\sc ii}]-deficient relative to the local relation. \nHowever, most of previous high-$z$ studies classified multi-component systems as single objects in the L$_{\\rm[CII]}$-SF diagram.\nIf we associate each clump and\/or galaxy with its proper UV counterparts (or lack thereof), then the resulting location on the L$_{\\rm[CII]}$-SFR diagram changes significantly for these objects \n\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=1.6\\columnwidth]{Figure\/FigureLcii_SFR.pdf}\\\\\\ \\\\\n %\n \\caption{L$_{\\rm[CII]}$\\ as a function of SFR at $z=5-7$. Blue stars indicate the global SFR and [C{\\sc ii}]\\ luminosity for system having multiple sub-components,\n as discussed in Section~\\ref{sec:Multi-component systems}; in these cases the location of the individual sub-components are shown with yellow stars.\nRed circles show the location of the remaining high-$z$ galaxies (i.e. single systems) listed in Table~\\ref{tab:table1}, that do not exhibit any positional offsets and\/or disturbed morphology.\nIn the bottom-right of the figure we show an error bar that is representative for the whole sample.\nThe green line is the local relation for local star-forming galaxies \\citep{De-Looze:2014}.}\n \\label{fig:fig2}\n\\end{figure*}\n\nFigure~\\ref{fig:fig2} shows L$_{\\rm[CII]}$\\ as a function of SFR. The green line illustrates the local relation obtained\nby \\cite{De-Looze:2014} and its dispersion is given by the\nshaded area. Results for $z>5$ galaxies, as listed in Table~\\ref{tab:table1}, are shown with various symbols in\nFigure~\\ref{fig:fig2}. \nThe SFR estimation for the $z>5$ galaxies (and their sub-components) is discussed in Section~\\ref{sec:SFRestimates}.\nThe multiple-component objects (HZ2, HZ6, HZ8, HZ10, WHM5, Himiko, CR7, COS24108, and BDF3299) are split into several individual components with their own SFRs and L$_{\\rm[CII]}$. \nThe nine complex systems discussed in Section~\\ref{sec:Multi-component systems} are broken into 20 sub-components distributed on different\nregions of the L$_{\\rm[CII]}$-SFR plane. The location of the individual subcomponents is indicated with yellow stars, while the location of these\nsystems by integrating the whole [C{\\sc ii}]\\ and UV emission (i.e. ignoring that these are actually composed of different subsystems) is\nindicated with blue stars.\nThe four new [C{\\sc ii}]\\ non-detections presented in Section~\\ref{sec:additional} fall below the local relation, while the L$_{\\rm[CII]}$\\ for BDF2203 places this galaxy along the \\cite{De-Looze:2014} relation.\n\nOnce the association between [C{\\sc ii}]\\ emission and optical-UV counterparts is properly done,\nwe find that the resulting distribution occupies a large area of the L$_{\\rm[CII]}$-SFR plot with a large scatter both above and below the local relation.\nAbout 19 objects of the total sample are in agreement within 1 sigma with the local relation, but the remaining 24 systems have deviations, either above or below the relation, up to 3 sigma.\n\nIn order to quantify the L$_{\\rm[CII]}$-SFR offset of the high-$z$ sample from the relation found in the local population, we investigate the distribution of offsets relative to the local\nrelation. More specifically,\nfor each galaxy we calculate the offset from the relation as $\\Delta$Log([C{\\sc ii}])=Log([C{\\sc ii}])-Log([C{\\sc ii}]$_{\\rm expect-local}$), where Log([C{\\sc ii}]$_{\\rm expect-local}$) is the [C{\\sc ii}]\\ luminosity\nexpected from the local relation according to the SFR measured in the galaxy or sub-component.\nThe result of this distribution is shown in the left panel of Figure~\\ref{fig:fig2b}, while the right panel shows the distribution of the offsets. \nIn contrast with some previous claims based on fewer targets, the $\\Delta$Log([C{\\sc ii}]) distribution, which includes both detections and upper limits, does not exhibit any clear shift relative\nto the local relation (and whose distribution is shown with the dotted green histogram.\nThe number of objects below (23) and above (20) the L$_{\\rm[CII]}$-SFR relation are comparable. \n However, the dispersion of the\nis 0.48$\\pm$0.07, which is about two times larger than the uncertainty reported by \\cite{De-Looze:2014} for the local relation. Such larger dispersion may be associated to the presence of kpc-scale sub-components that are not common in the local Universe. However, we will discuss the possible origin of this dispersion in the next sections.\n\n\n\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=1\\columnwidth]{Figure\/FigureCIIdispersion.pdf}\n %\n\\caption{Deviations from the local L$_{\\rm[CII]}$-SFR relation as a function of SFR. The 1$\\sigma$ dispersion of the local relation is indicated with the shaded green region.\nThe distribution of the offsets is shown in the right panel. The black curve represents the Log([C{\\sc ii}])-Log([C{\\sc ii}]$_{\\rm expect-local}$) distribution from our sample, while the dotted\ngreen histogram is the distribution of the local relation. The dashed grey line shows a Gaussian fit to the high-$z$ distribution.}\n \\label{fig:fig2b}\n\\end{figure}\n\n\n\n\n\n\n\n\\section{The relation between Ly$\\alpha$\\ EW and [C{\\sc ii}]\\ emission}\\label{sec:lyaEW}\n\nIt is well known that Ly$\\alpha$\\ emission depends on the level of ionizing photons produced by star formation (or AGN activity) and radiative transfer effects in the ISM.\nModels and observations suggest that the Ly$\\alpha$\\ EW increases with decreasing metallicity and dust content \\citep{Raiter:2010, Song:2014}. \nSince the [C{\\sc ii}]\\ emission is sensitive on the ISM properties as well, and in particular the ISM heating through photoelectric ejection from dust grains,\nwe expect a relation between the [C{\\sc ii}]\\ luminosity and the Ly$\\alpha$\\ strength \\citep{Harikane:2017}.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=1\\columnwidth]{Figure\/FigureLcii_SFR_EW.pdf}\n %\n\\caption{ L$_{\\rm[CII]}$\\ as a function of SFR at $z=5-7$. \nStars and circles represent sub-components and individual galaxies, respectively.\nSymbols are colour-coded according to their Ly$\\alpha$\\ EWs (not corrected for the inter-galactic medium absorption), as indicated on the colour bar on the right.\nIn the bottom-right corner we show an error bar that is representative for the whole sample.\nThe green line are the local relation for local star-forming galaxies \\citep{De-Looze:2014} while its dispersion is indicated by the shaded green region.}\n \\label{fig:EW}\n\\end{figure}\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=1\\columnwidth]{Figure\/FigureDelta_EW.pdf}\n\\includegraphics[width=1\\columnwidth]{Figure\/FigureDelta_EWcorr.pdf}\n %\n\\caption{ Top panel: Offset from the local L$_{\\rm[CII]}$-SFR relation as a function of EW(Ly$\\alpha$). The right hand axis shows the [C{\\sc ii}]\/SFR ratio corresponding to the\ndeviations from the local L$_{\\rm[CII]}$-SFR relation.\nThe 1$\\sigma$ dispersion of the local relation is indicated by the shaded green region. Star and circles indicate sub-components and individual galaxies, respectively.\n The grey dashed relationship shows our linear fit (see text).\n Bottom panel: Same as the top panel but the EW(Ly$\\alpha$)$^{\\rm int}$ has been corrected for the IGM absorption by following \\citealt{Harikane:2017}.}\n \\label{fig:EW2}\n\\end{figure}\n\nIn Figure~\\ref{fig:EW} we show L$_{\\rm[CII]}$\\ as function of SFR by colour-coding the different symbols according to their Ly$\\alpha$\\ EWs (not corrected for the inter-galactic medium absorption). We include \nonly those sub-components and galaxies having a Ly$\\alpha$\\ EW measurement.\nThere is a weak tendency for galaxies with high EW(Ly$\\alpha$) to lie below the local L$_{\\rm[CII]}$-SFR relation, and vice-versa.\nThis is shown better in Figure~\\ref{fig:EW2} where the offset from the local L$_{\\rm[CII]}$-SFR relation\nis plotted as a function of EW(Ly$\\alpha$). Note that, since the slope of the local relation is one, plotting the deviation from the local relation\n($\\Delta\\log$([C{\\sc ii}])=log([C{\\sc ii}])-log([C{\\sc ii}]$_{\\rm expect-local}$)) is equivalent to plotting the ratio between [C{\\sc ii}]\\ luminosity and SFR, i.e. L$_{\\rm [CII]}$\/SFR, which is indeed given on the\nright hand axis of Figure~\\ref{fig:EW}.\nAlthough there is a large dispersion, there is a tentative indication that the offset from the local relation (hence L$_{\\rm [CII]}$\/SFR) anti-correlates with Ly$\\alpha$\\ EW. A linear\nfit gives\n$$\n{\\rm \\Delta \\log([C\\textsc{ii}]) = (0.55\\pm0.20)-(0.44\\pm0.15)\\log(EW(Ly\\alpha)) } \n$$\nand the dispersion around this best-fit is 0.25~dex.\n\nThe result does not change significantly if we attempt to correct the EW(Ly$\\alpha$) for IGM absorption, by following the prescription given by \\cite{Harikane:2017}.\nIn this case the relation is shown in the bottom panel of Figure~\\ref{fig:EW} and the resulting best-fit linear relation is\n$$\n{\\rm \\Delta \\log([C\\textsc{ii}]) = (0.56\\pm0.20)-(0.41\\pm0.14)\\log(EW(Ly\\alpha)^{int}) \\ \\ (0.30 {\\it dex})}\n$$\nand the dispersion around this best-fit is even larger, 0.3~dex.\n\n\\cite{Harikane:2017} find a relation between L$_{\\rm [CII]}$\/SFR and EW(Ly$\\alpha$) steeper than ours, though consistent within errors. The steeper relation found by\n\\cite{Harikane:2017} is probably associated with the fact that in their work they combine the global properties of galaxies and do not extract the subcomponents.\n\nSome anti-correlation between deviation from the local L$_{\\rm[CII]}$-SFR relation and EW(Ly$\\alpha$) (or, equivalently, between L$_{\\rm[CII]}$-SFR and EW(Ly$\\alpha$)$^{\\rm int}$) is expected\nfrom the dependence of these quantities from the metallicity, either directly\nor through the associated dust content, as already predicted by some models \\citep[e.g.][]{Vallini:2015,Pallottini:2017, Matthee:2017}.\nIndeed, lower metallicity implies lower amount of carbon available for cooling, but also less dust content. Indeed, since the heating of PDRs occurs primarily\nthrough photoelectric effect on dust grains, the lower is the dust content the lower is the heating efficiency of the gas in the PDR, hence the lower is the\nemissions of the [C{\\sc ii}]\\ cooling line. On the other hand, the lower the dust content the lower is the absorption of the Ly$\\alpha$ resonant line, hence the higher\nis the EW(Ly$\\alpha$).\n\n\n\\section{Spatially resolved L$_{\\rm[CII]}$--SFR relation}\\label{sec:spatially}\n\n\nIn the previous sections, we show that the L$_{\\rm[CII]}$-SFR relation at $z>5$ has a intrinsic dispersion larger than observed in the local Universe.\nSuch a large scatter suggests that the [C{\\sc ii}]\\ luminosity may not be good tracer of the SFR at least at early epochs. \n\nRecent spatially resolved studies have claimed that the [C{\\sc ii}]-SFR relation is better behaved in terms of SFR surface density and [C{\\sc ii}]\\ surface brightness than in global proprieties (L$_{\\rm[CII]}$\\ and SFR), since the surface brightness calibration is more closely related to the local UV field \\citep{Herrera-Camus:2015, Smith:2017}.\nIt is thus worth to analyse the relation $\\Sigma_{\\rm[CII]}$-$\\Sigma_{\\rm SFR}$\\ at $z>5$. \n\nIn this section we compare the spatial extension of the [C{\\sc ii}]\\ and UV emission and, then, we investigate the correlation between [C{\\sc ii}]\\ surface brightness and SFR surface density.\n\n\n\n\\subsection{Spatial extension of the [C{\\sc ii}]\\ emission}\\label{sec:spatialextension}\n\nFigure~\\ref{fig:fig_ext} shows the extension of the [C{\\sc ii}]\\ emission compared with the extension of star formation traced by the UV counterpart.\n[C{\\sc ii}]\\ emission is generally much more extended than the UV emission tracing unobscured star formation. This discrepancy may be partially\nassociated with observational effects. Indeed, while the high angular resolution of {\\it HST} enables to resolve small clumps, it may have low\nsensitivity to diffuse, extended emission. However, even by smoothing the UV images in the deepest observations available to us, we still\ndo not recover the extension observed in [C{\\sc ii}]. On the other hand, the resolution of the [C{\\sc ii}]\\ observations may, in some case, smear out\nclumps and result in an overall extended distribution. However, in many cases the ALMA observations achieve a resolution comparable, or even\nhigher, than {\\it HST} at UV rest-frame wavelengths and, despite this, we measure clearly larger [C{\\sc ii}]\\ sizes. Moreover, when high angular resolution\nobservation are used, these do reveal that a significant fraction of the [C{\\sc ii}]\\ flux is resolved out on large scales \\citep[see e.g. discussion\nin ][]{Carniani:2017a}. In conclusion, we believe that the different sizes between [C{\\sc ii}]\\ and UV emission are tracing truly different distribution\nof the [C{\\sc ii}]\\ and UV emission on different scales.\n\nThere could be various explanations for these differences. If the star formation associated with the [C{\\sc ii}]\\ emission\non large scale is heavily obscured, the UV light does not trace this component \\citep{Katz:2017}. This is certainly a possibility, although \none might expect that the bulk of the obscuration should affect the central region more heavily than the outer parts, hence one would\nexpect the opposite trend. In alternative, the extended component of [C{\\sc ii}]\\ may not be directly associated with star forming regions, but with\ncircumgalactic gas, either\nin accretion or ejected by the galaxy, and which is illuminated by the strong radiation field produced by the galaxy. \n\n\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=1\\columnwidth]{Figure\/Ruv_vs_Rcii.pdf}\\\\\\ \\\\\n %\n\\caption{Half-light radii of star formation regions, as measured from the (rest-frame) UV light, compared with the half-light radii of the associated [C{\\sc ii}]\\ emission. The dashed line indicates the 1:1 relation.}\n \\label{fig:fig_ext}\n\\end{figure}\n\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=1\\columnwidth]{Figure\/FigureECII_ESFR}\\\\\\ \\\\\n %\n\\caption{$\\Sigma_{\\rm[CII]}$ vs $\\Sigma_{\\rm SFR}$ \\ for $z>5$ star-forming galaxies detected both in [C{\\sc ii}]\\ and rest-frame UV. Symbols are as in Figure~\\ref{fig:fig2}. The grey line indicates the local\nrelation by \\citet{Herrera-Camus:2015}. The dashed line indicates the best linear fit on the $z>5$ sample.}\n \\label{fig:fig3}\n\\end{figure}\n\n\n\n\\subsection{Surface brightness}\\label{sec:surface brightness}\\label{sec:surface}\n\n\nOnce we have measured the extension of the [C{\\sc ii}]\\ emission and of the SFR regions, we can estimate the [C{\\sc ii}]\\ surface brightness and SFR surface density of each sub-component and of\nthe individual sources detected in [C{\\sc ii}]\\ and UV.\nFigure~\\ref{fig:fig3} shows the $\\Sigma_{\\rm[CII]}$-$\\Sigma_{\\rm SFR}$\\ relation, where we have included only those galaxies detected in both [C{\\sc ii}]\\ and UV emission. \nSystems that are not spatially resolved in [C{\\sc ii}]\\ (or UV) emission are indicated with lower limits. We also show the local relation by \\cite{Herrera-Camus:2015} and its dispersion. \nIn contrast to the L$_{\\rm[CII]}$-SFR diagram, there are no galaxies located significantly above the local relation, only a few galaxies\nare located on the local relation, and most galaxies spread largely below the local relation. This is primarily due to the\nlarge extension of the [C{\\sc ii}]\\ emission in these high redshift systems, as discussed in the previous section.\nBy fitting the $\\Sigma_{\\rm[CII]}$-$\\Sigma_{\\rm SFR}$\\ measurements for our sample we obtain the following relation:\n$$\n\\log(\\Sigma_{\\rm SFR}) = (0.63\\pm0.11)\\times(\\log(\\Sigma_{\\rm [CII]})-(25\\pm6)\n$$\nPart of the $\\Sigma_{\\rm[CII]}$\\ deficit may be ascribed to the metallicity of the gas.\nIndeed a similar deviations have been observed in local low-metallicity galaxies, in which the $\\Sigma_{\\rm[CII]}$-$\\Sigma_{\\rm SFR}$\\ calibration over-predicts the $\\Sigma_{\\rm[CII]}$\\\nby up to a factor of six.\nWe note that \\cite{Faisst:2017a} estimated a metallicity of 12+log(O\/H)$>$8.5 for HZ10 and HZ9, and a metallicity of 12+log(O\/H)$<$8.5 for HZ1, HZ2, and HZ4. \nThe former agree with the local $\\Sigma_{\\rm[CII]}$-$\\Sigma_{\\rm SFR}$\\ relation while HZ1, HZ2, and HZ4 have low $\\Sigma_{\\rm[CII]}$-$\\Sigma_{\\rm SFR}$\\ ratio.\nThis interpretation is also supported by the fact that all CR7 clumps, which have metallicities 12+log(O\/H)$<8.2$ (see discussion in \\citealt{Sobral:2017}) , are located well below the local relation. \nThe $\\Sigma_{\\rm[CII]}$\\ deficit is also akin to the simulations by \\cite{Pallottini:2017a} who investigated the physical properties of a simulated galaxy at $z=6$, with metallicity\n12+log(O\/H)=8.35. Such galaxy has a ${\\rm \\log(\\Sigma_{\\rm[CII]}\/erg \\ s^{-1} \\ kpc^{-2})=40.797}$ and ${\\rm \\log(\\Sigma_{\\rm SFR}\/M_\\odot yr^{-1})= 1.027}$, which places the mock\ngalaxy below the $\\Sigma_{\\rm[CII]}$-$\\Sigma_{\\rm SFR}$\\ relation found for local star-formation galaxies.\n\nHowever, the offset observed\nin high-$z$ galaxies is significantly larger (one order of magnitude or more) than the one observed in local low metallicity galaxies, so other effects\nare likely in place, which will be discussed in the next section.\nFinally, still within the context of the surface brightness and SFR surface density, we mention that, for local galaxies,\n\\cite{Smith:2017} found a dependence between the L$_{\\rm[CII]}$\/L$_{\\rm FIR}$ ratio and the SFR surface density, suggesting that the [C{\\sc ii}]\\ deficit increases strongly with increasing $\\Sigma_{\\rm SFR}$. Therefore the $\\Sigma_{\\rm SFR}$\\ dependence has a strong impact also on the \\mbox{L$_{\\rm[CII]}$-SFR} relation. \nIn our sample, we cannot verify this dependence as most galaxies have only an upper limit on the L$_{\\rm FIR}$. However, we can investigate the relation L$_{\\rm[CII]}$\/L$_{\\rm UV}$ and $\\Sigma_{\\rm SFR}$\\, since the contribution of SFR$_{\\rm FIR}$ to the total SFR is negligible (see discussion in Section~\\ref{sec:SFRestimates}).\nFigure~\\ref{fig:fig5} shows the observed L$_{\\rm[CII]}$\/L$_{\\rm UV}$ ratio spanning a range between 0.002\\% to 0.4\\% (over two orders of magnitude),\nwhile the SFR surface density spans the range between 1 and 30 ${\\rm M_{\\odot} \\ yr^{-1} \\ kpc^{-2}}$.\nThere is only a very weak correlation between\nL$_{\\rm[CII]}$\/L$_{\\rm UV}$ and $\\Sigma_{\\rm SFR}$, with very large dispersion (much larger than what observed locally for the\nL$_{\\rm[CII]}$\/L$_{\\rm FIR}$ and $\\Sigma_{\\rm SFR}$ relation). This indicates that in these high-$z$ systems the L$_{\\rm[CII]}$\/L$_{\\rm UV}$ line ratio (and L$_{\\rm[CII]}$\/SFR) does not strongly depend on the\nareal density with which galaxies form stars at $z>5$, and that other effects (metallicity, and other phenomena discussed in the next section) may contribute\nto the very large dispersion.\n\n\n\n\n\n\\section{Discussion on the [C{\\sc ii}]-SFR scaling relations at z=5--7}\\label{sec:discussion}\n\nClearly galaxies at $z>5$ behave differently, relative to their local counterparts, for what concerns the [C{\\sc ii}]\\ and SFR properties.\nSummarising the finding of the previous sections:\n1) both the \\mbox{L$_{\\rm[CII]}$-SFR} relation and the $\\Sigma_{\\rm[CII]}$--$\\Sigma_{\\rm SFR}$\\ relation have a scatter much larger than the local relations;\n2) contrary to some previous claims, the \\mbox{L$_{\\rm[CII]}$--SFR} relation is not offset relative to the local relation, while the $\\Sigma_{\\rm[CII]}$--$\\Sigma_{\\rm SFR}$\\ relation is clearly\noffset by showing much lower $\\Sigma_{\\rm[CII]}$ \\ relative to local relation;\n3) the extension of the [C{\\sc ii}]\\ emission is larger than the extent of the star formation traced by the UV emission. \n\n\nThe larger scatter observed in the SFR-L$_{\\rm[CII]}$\\ relation is certainly\nindicative of a broader range of properties spanned by such primeval galaxies relative to the local population.\nIndeed, \\mbox{high-$z$} cosmological simulations show that the [C{\\sc ii}]\\ emission strongly depends on the gas metallicity, ionisation parameter, and evolutionary stage of\nthe system and that all of these properties are expected to span a much broader range at high-$z$ with respect to local galaxies \\citep{Vallini:2015,Vallini:2016,Vallini:2017, Pallottini:2015, Pallottini:2017, Olsen:2017, Katz:2017}.\nRecently, \\cite{Lagache:2017} investigated the expected dispersion of L$_{\\rm[CII]}$-SFR relation in the distant Universe by using a semi-analytical model of galaxy formation\nfor a large sample of simulated galaxies at $z>4$. They found a L$_{\\rm[CII]}$-SFR correlation with a large scatter of 0.4-0.8 dex, which is in agreement with our result. They claim that such large dispersion is associated to the combined effects of different gas contents, metallicities, and interstellar radiation fields in the simulated high-$z$ galaxies. \n\nAs we have shown, the mild anti-correlation with EW(Ly$\\alpha$), as well as the analysis of some individual galaxies for which the metallicity has been estimated,\ndoes suggest that the metallicity may play a role on the [C{\\sc ii}]\\ emission (either simply in terms of carbon abundance and\/or in terms of heating of the ISM through\nphotoelectric effect on dust grains, whose abundance scales with the metallicity). In particular, the lower metallicity of galaxies at $z>5$ \ncan explain some of the scatter towards low [C{\\sc ii}]\\ emission in both the L$_{\\rm[CII]}$-SFR relation and in the $\\Sigma_{\\rm[CII]}$-$\\Sigma_{\\rm SFR}$\\ relation. However, metallicity effects are\nunlikely to explain the scatter towards high [C{\\sc ii}]\\ emission in the L$_{\\rm[CII]}$-SFR relation. Moreover, for what concerns the $\\Sigma_{\\rm[CII]}$-$\\Sigma_{\\rm SFR}$\\ relation, the offset\nand large spread toward low $\\Sigma_{\\rm[CII]}$\\ is probably too large to be entirely ascribed to metallicity.\n\nLarge variations in ionisation parameter can also contribute to the spread in [C{\\sc ii}]\\ emission \\citep{Gracia-Carpio:2011, Katz:2017}. In particular, if [C{\\sc ii}]\\ in\nprimeval galaxies also traces circumgalactic gas in accretion and\/or expelled from the galaxy, and excited by the UV radiation of the central galaxy,\nthis would result into the observed larger [C{\\sc ii}]\\ sizes, hence lower $\\Sigma_{\\rm[CII]}$ \\ and lower ionisation parameter, and the latter would increase the total L$_{\\rm[CII]}$ \\ relative to the local\nrelation. On the other hand, young compact star forming, primeval galaxies would be characterised by higher ionisation parameter, which would reduce the [C{\\sc ii}]\\ emission,\nhence contributing to the spread towards low L$_{\\rm[CII]}$.\n\nAs mentioned in the previous sections, yet another possibility is that the UV emission associated with some of the [C{\\sc ii}]\\ clumps is heavily obscured. This would\nexplain the scatter above the L$_{\\rm[CII]}$-SFR relation, in the sense that for some of these systems the underlying SFR, as traced by the UV, is heavily underestimated. \nThis can be the case for a few galaxies. However, as we mentioned, most of these systems show no or weak continuum dust emission, indicative of low dust content.\n\nTo make further progress additional data at other wavelength will be, in the future, extremely valuable. In particular, JWST will enable to identify obscured stellar components\nas well as H$\\alpha$ emission associated with star formation. ALMA observations of other transitions, such has [OIII]88$\\mu$m (though observable only in some redshift ranges), has\nproved extremely useful to constrain these scenarios \\citep[e.g. ][]{Inoue:2016,Carniani:2017a}.\n\n\n\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=1\\columnwidth]{Figure\/FigureLcii_ESFR} %\n\\caption{L$_{\\rm[CII]}$\/L$_{\\rm UV}$ ratio as a function of the SFR surface density. Symbols are as in Figure~\\ref{fig:fig2}.}\n \\label{fig:fig5}\n\\end{figure}\n\n\n\n\\section{Conclusions}\\label{sec:conclusions}\n\nIn this work we have investigated the nature of the [C{\\sc ii}]\\ emission in star-forming galaxies at $z=5-7$. In particular, we have explored the positional offsets between UV and FIR line emissions, the\npresence of multiple components,\nand the implications on the [C{\\sc ii}]-SFR scaling relations in the distant Universe, once the correct association between [C{\\sc ii}]\\ and UV emission is properly taken into account.\nWe have performed our investigation in a sample of 29 $z>5$ ``normal'' star-forming galaxies (SFR $<$ 100 M$_{\\odot}$~yr$^{-1}$) observed with ALMA in the [C{\\sc ii}]\\ line. In addition to the re-analysis of archival objects,\nwe have also included new ALMA observations targeting five star-forming galaxies at $z\\sim6$ with SFR $\\sim$ 10 M$_{\\odot}$ yr$^{-1}$, resulting into a new detection. Our mains results are:\n\n\\begin{itemize}\n\n\n\n\\item The continuum emission around 158 $\\mu$m is not detected in most of the $z>5$ galaxies observed with ALMA, indicating a low dust content. By modelling the dust emission with a greybody spectrum\nwith dust temperature T$_{\\rm d}$ = 30 K and emissivity index $\\beta$ = 1.5, we have found that the SFR based on the FIR emission is, on average, lower than\nthe SFR measured from the UV emission by at least a factor 0.6, but probably much more (due to several upper limits).\n\n\\item By accurately registering ALMA and NIR images, and by kinematically discriminating multiple [C{\\sc ii}]\\ components, our analysis has revealed that the [C{\\sc ii}]\\ emission breaks into multiple\nsub-components in 9 out of the 21 galaxies having [C{\\sc ii}]\\ detections. In these nine targets we have observed the presence of 19 FIR-line emitting clumps. Only very few of these, if any, are associated\nwith the primary (brightest) UV counterpart, while the bulk of the [C{\\sc ii}]\\ is associated with fainter UV components.\nIn only three cases (COSMOS24108, Himiko, and BDF3299) the shallow NIR images have not enabled us to detect the UV counterparts associated with some of the [C{\\sc ii}]\\ clumps. \n\n\\item We have studied the relation between [C{\\sc ii}]\\ and SFR on the high-$z$ sample by taking into account the presence of these sub-components and the proper associations\nbetween [C{\\sc ii}]\\ and UV components. The distribution of $z>5$ galaxies on the L$_{\\rm[CII]}$-SFR diagram follows the local relation, but the dispersion is 1.8 times larger than that observed in nearby galaxies.\n\n\\item The deviation from the local L$_{\\rm[CII]}$-SFR relation shows a weak anti-correlation with EW(Ly$\\alpha$) though shallower and with larger dispersion than what found in other\nstudies that did not account for the multi-component nature of these systems.\n\n\\item Most of the objects in the high-$z$ sample are spatially resolved in [C{\\sc ii}]\\ and UV emission. \nThe extension of the [C{\\sc ii}]\\ emission is generally much larger than the extension of star forming regions traced by the UV emission.\n\n\\item In the $\\Sigma_{\\rm[CII]}$-$\\Sigma_{\\rm SFR}$\\ diagram $z>5$ galaxies are characterised by a large scatter with respect to local galaxies, and are mostly distributed below the local relation (i.e. fainter\n$\\Sigma_{\\rm[CII]}$\\ at a given $\\Sigma_{\\rm SFR}$). \n\n\\end{itemize}\n\n\nWe have suggested that a combination of different effects may be responsible for the different properties of high-$z$ galaxies in terms of [C{\\sc ii}]--SFR properties\nrelative to local galaxies. More specifically: 1) the low metallicity of high-$z$ galaxies may be responsible (also indirectly through the lower dust photoelectric heating) for part\nof the scatter towards lower [C{\\sc ii}]\\ emission relative to the local relations; 2) the presence of circumnuclear gas in accretion and\/or expelled from the galaxy may be responsible\nfor the larger size in [C{\\sc ii}]\\ relative to the SFR distribution and may also be responsible for the scatter of the L$_{\\rm[CII]}$-SFR distribution above the local relation\nas a consequence of lower ionisation parameter; 3) in compact young star forming regions the increased ionisation parameter and higher gas density may be responsible for\nthe suppression of [C{\\sc ii}]\\ for galaxies which are below the local relation; 4) dust obscuration may be responsible for both the different morphology between [C{\\sc ii}]\\ and UV emission\nand also for the scatter of sources above the local L$_{\\rm[CII]}$-SFR relation.\n\n\\section*{Acknowledgments}\n\nThis paper makes use of the following ALMA data: {\\small ADS\/JAO.ALMA\\#2012.1.00719.S, ADS\/JAO.ALMA\\#2012.A.00040.S, ADS\/JAO.ALMA\\#2013.A.00433.S ADS\/JAO.ALMA\\#2011.0.00115.S, ADS\/JAO.ALMA\\#2012.1.00033.S, ADS\/JAO.ALMA\\#2012.1.00523.S, ADS\/JAO.ALMA\\#2013.1.00815.S, ADS\/JAO.ALMA\\#2015.1.00834.S., ADS\/JAO.ALMA\\#2015.1.01105.S, and ADS\/JAO.ALMA\\#2016.1.01240.S}\n which can be retrieved from the ALMA data archive: https:\/\/almascience.eso.org\/ alma-data\/archive. ALMA is a partnership of ESO (representing its member states), NSF (USA) and NINS (Japan), together with NRC (Canada) and NSC and ASIAA (Taiwan), in cooperation with the Republic of Chile. The Joint ALMA Observatory is operated by ESO, AUI\/NRAO and NAOJ.\nWe are grateful to G. Jones to for providing the [C{\\sc ii}]\\ flux maps of WHM5.\nR.M. and S.C. acknowledge support by the Science and Technology Facilities Council (STFC). R.M. acknowledges ERC Advanced Grant 695671 ``QUENCH''. AF acknowledges support from the ERC Advanced Grant INTERSTELLAR H2020\/740120. \n\n\n\\setlength{\\labelwidth}{0pt}\n\\bibliographystyle{mn2e}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\n\\section{Introduction} \\label{sec1}\n\\mbox{} \\\\[-9mm]\n\nIn optical and acoustic radiation with boundary conditions given on a disk, the choice of basis functions to represent the boundary conditions is of great importance. Here one can consider the following requirements:\n\\bi{A.0}\n\\ITEM{A.} the basis functions should be appropriate given the physical background,\n\\ITEM{B.} the basis functions should be effective and accurate in their ability of representing functions vanishing outside the disk and arising in the particular physical context,\n\\ITEM{C.} the basis functions should have convenient analytic results for transformations that arise in a natural way in the given physical context.\n\\end{inspring}\nFor the case of optical diffraction with a circular pupil having both phase and amplitude non-uniformities, Zernike \\cite{ref1} introduced in 1934 his circle polynomials, denoted here as $Z_n^m(\\rho,\\vartheta)$ with radial variable $\\rho\\geq0$ and angular variable $\\vartheta$, vanishing for $\\rho>1$, which find nowadays wide-spread application in fields like optical engineering and lithography \\cite{ref2}--\\cite{ref7}, astronomy \\cite{ref8}--\\cite{ref10} and ophthalmology \\cite{ref11}--\\cite{ref13}. The Zernike circle polynomials are given for integer $n$ and $m$ such that $n-|m|$ is even and non-negative as\n\\begin{eqnarray} \\label{e1}\nZ_n^m(\\rho,\\vartheta) & = & R_n^{|m|}(\\rho)\\,e^{im\\vartheta}~,~~~~~~0\\leq\\rho<1\\,,~~0\\leq\\vartheta<2\\pi~, \\nonumber \\\\[2mm]\n& = & 0\\hspace*{2cm},~~~~~~\\rho>1~,\n\\end{eqnarray}\nwhere the radial polynomials $R_n^{|m|}$ are given by\n\\begin{equation} \\label{e2}\nR_n^{|m|}(\\rho)=\\rho^{|m|}\\,P_{\\frac{n-|m|}{2}}^{(0,|m|)}\\,(2\\rho^2-1)~,\n\\end{equation}\nwith\n$P_k^{(\\alpha,\\beta)}$ the general Jacobi polynomial as given in \\cite{ref14}, Ch.~22, \\cite{ref15}, Ch.~4 and \\cite{ref16}, Ch.~5, \\S4. The circle polynomials were investigated by Bhatia and Wolf \\cite{ref17} with respect to their appropriateness for use in optical diffraction theory, see issue A above, and were shown to arise more or less uniquely as orthogonal functions with polynomial radial dependence satisfying form invariance under rotations of the unit disk.\n\nThe orthogonality condition,\n\\begin{equation} \\label{e3}\n\\int\\limits_0^1\\int\\limits_0^{2\\pi}\\,Z_{n_1}^{m_1}(\\rho,\\vartheta)(Z_{n_2}^{m_2}(\\rho,\\vartheta))^{\\ast}\\,\\rho\\,d\\rho\\,d\\vartheta=\\frac{\\pi} {n+1}\\,\\delta_{m_1m_2}\\,\\delta_{n_1n_2}~,\n\\end{equation}\nwhere $n$ is either one of $n_1$ and $n_2$ at the right-hand side, together with the completeness, see \\cite{ref18}, App.~VII, end of Sec.~I, guarantees effective and accurate representation of square integrable functions on the unit disk in terms of their expansion coefficients with respect to the $Z_n^m$, see issue B above. The Zernike circle polynomials, notably those of azimuthal order $m=0$, were considered recently by Aarts and Janssen \\cite{ref19}--\\cite{ref22} for use in solving a variety of forward and inverse problems in acoustic radiation. This raised in the acoustic community \\cite{ref23} the question how the circle polynomials with $m=0$ compare to other sets of non-polynomial, radially symmetric, orthogonal functions on the disk. It was shown in \\cite{ref24} that the expansion coefficients, when using circle polynomials, properly reflect smoothness of the functions to be expanded in terms of decay and that they compare favourably in this respect with the expansion coefficients that occur when using orthogonal Bessel series expansions; the latter expansions are sometimes used both in the acoustic and the optical domain, see \\cite{ref25}--\\cite{refY}.\n\nThe present paper focuses on analytic properties, see issue C above, of basis functions, and in Sec.~\\ref{sec2} we present a number of such properties for the set of Zernike circle polynomials. In Sec.~\\ref{sec3} a generalization of the set of Zernike circle polynomials is introduced, viz.\\ the set of functions\n\\begin{eqnarray} \\label{e4}\nZ_n^{m,\\alpha}(\\rho,\\vartheta) & = & (1-\\rho^2)^{\\alpha}\\,\\rho^{|m|}\\,P_{\\frac{n-|m|}{2}}^{(\\alpha,|m|)} (2\\rho^2-1)\\,e^{im\\vartheta}~,~~~~~~0\\leq\\rho<1~, \\nonumber \\\\[2mm]\n& = & 0\\hspace*{6.1cm},~~~~~~\\rho>1~,\n\\end{eqnarray}\nwith $\\rho$, $\\vartheta$ and $n$, $m$ as before, see (\\ref{e1}), and parameter $\\alpha>{-}1$. These new Zernike-type functions arise sometimes more naturally in certain physical problems and can be more convenient when solving inverse problems in diffraction theory since the decay in the Fourier domain can be controlled by choosing the parameter $\\alpha$ appropriately. While issue A above should guide the choice of $\\alpha$, the matter of orthogonality and completeness is settled as in the classical case $\\alpha=0$. A more involved question is what becomes of the analytic properties, noted for the classical circle polynomials in Sec.~\\ref{sec2}, when $\\alpha\\neq0$. A major part of the present paper is concerned with answering this question, and this yields generalization of many of the results holding for the case $\\alpha=0$ and about which more specifics will be given at the end of Sec.~\\ref{sec3} where the basic properties of the generalized Zernike functions are presented.\n\n\\section{Analytic results for the classical circle polynomials} \\label{sec2}\n\\mbox{} \\\\[-9mm]\n\nThe Zernike circle polynomials were used by Nijboer in his 1942 thesis \\cite{ref27} for the computation of the point-spread functions in near best-focus planes pertaining to circular optical systems of low-to-medium numerical aperture. In that case, starting from a non-uniform pupil function\n\\begin{equation} \\label{e5}\nP(\\rho,\\vartheta)=A(\\rho,\\vartheta)\\,e^{i\\Phi(\\rho,\\vartheta)}~,~~~~~~0\\leq\\rho<1\\,,~~0\\leq\\vartheta<2\\pi~,\n\\end{equation}\nthe point-spread function $U(r,\\varphi\\,;\\,f)$ at defocus plane $f$ with polar coordinates $x+iy=r\\,e^{i\\varphi}$ is given in accordance with well established practices in Fourier optics as\n\\begin{equation} \\label{e6}\nU(r,\\varphi\\,;\\,f)=\\int\\limits_0^1\\int\\limits_0^{2\\pi}\\,e^{if\\rho^2}\\,e^{2\\pi i\\rho r\\cos(\\vartheta-\\varphi)}\\,P(\\rho,\\vartheta)\\,\\rho \\,d\\rho\\,d\\vartheta~.\n\\end{equation}\nIt was discovered by Zernike \\cite{ref1} that the Fourier transform of the circle polynomials,\n\\begin{equation} \\label{e7}\n({\\cal F}\\,Z_n^m)(r,\\varphi)=\\int\\limits_0^1\\int\\limits_0^{2\\pi}\\,e^{2\\pi i\\rho r\\cos(\\vartheta-\\varphi)}\\,Z_n^m(\\rho,\\vartheta)\\,\\rho\\,d\\rho\\,d\\vartheta\n\\end{equation}\nhas the closed-form result\n\\begin{equation} \\label{e8}\n({\\cal F}\\,Z_n^m)(r,\\varphi)=2\\pi\\,i^n\\,\\frac{J_{n+1}(2\\pi r)}{2\\pi r}\\,e^{im\\varphi}~,~~~~~~r\\geq0\\,,~~0\\leq\\varphi<2\\pi~,\n\\end{equation}\nwhere $J_{n+1}$ is the Bessel function of the first kind and order $n+1$. Thus, one expands the pupil function $P$ as\n\\begin{equation} \\label{e9}\nP(\\rho,\\vartheta)=\\sum_{n,m}\\,\\beta_n^m\\,Z_n^m(\\rho,\\vartheta)~,~~~~~~0\\leq\\rho<1\\,,~~0\\leq\\vartheta<2\\pi~,\n\\end{equation}\nand uses the result in (\\ref{e8}) to compute the point-spread function $U$ in (\\ref{e6}) in terms of the expansion coefficients $\\beta_n^m$. In the case of best focus, $f=0$, this gives a computation result for the point-spread function immediately. For small values of $f$, say $|f|\\leq1$, one expands the focal factor,\n\\begin{equation} \\label{e10}\ne^{if\\rho^2}=1+if\\rho^2-\\tfrac12\\,f^2\\rho^4-...\n\\end{equation}\nand spends some additional effort to write functions $\\rho^{2k}\\,Z_n^m(\\rho,\\vartheta)$ as a $k$-terms linear combination of circle polynomials with upper index $m$ to which (\\ref{e7}) applies. Also see \\cite{ref18}, Ch.~9, Secs.~2--4. These are the main features of the classical Nijboer-Zernike theory for computation of optical point-spread functions in the presence of aberrations.\n\nIn \\cite{ref28} Janssen has computed the point-spread function $U_n^m(r,\\varphi\\,;\\,f)$ pertaining to a single term $Z_n^m$ in the form\n\\begin{equation} \\label{e11}\nU_n^m(r,\\varphi\\,;\\,f)=2\\pi\\,i^{|m|}\\,V_n^{|m|}(r,f)\\,e^{im\\varphi}~,\n\\end{equation}\nwhere\n\\begin{eqnarray} \\label{e12}\nV_n^{|m|}(r,f) & = & \\int\\limits_0^1\\,e^{if\\rho^2}\\,R_n^{|m|}(\\rho)\\,J_{|m|}(2\\pi\\rho r)\\,\\rho\\,d\\rho~= \\nonumber \\\\[3.5mm]\n& = & e^{if}\\,\\sum_{l=0}^{\\infty}\\,\\Bigl(\\frac{{-}if}{\\pi r}\\Bigr)^l\\,\\sum_{j=0}^p\\, u_{lj}\\, \\frac{J_{|m|+l+2j+1}(2\\pi r)}{2\\pi r}\n\\end{eqnarray}\nwith explicitly given $u_{lj}$ ($p=\\frac12\\,(n-|m|)$). This result has led to what is called the extended Nijboer-Zernike (ENZ) theory of forward and inverse computation for optical aberrations. The theory in its present form allows computation of point-spread functions for general $f$ and for high-NA optical systems, including polarization and birefringence, as well as for multi-layer systems. See \\cite{ref29}--\\cite{ref33} and \\cite{ref34} for an overview. Furthermore, it provides a framework for estimating pupil functions $P$, in terms of expansion coefficients $\\beta$, from measured data $|U|^2$ of the intensity point-spread function in the focal region. The latter inverse problem has the basic assumption that $P$ deviates only mildly from being constant so that the term with $m=n=0$ in (\\ref{e9}) dominates the totality of all other terms. The theoretical intensity point-spread function can then, with modest error, be linearized around the leading term $|\\beta_0^0\\,U_0^0|^2$, and this leads, via a matching procedure with the measured data in the focal region, to a first estimate of the unknown coefficients. This procedure is made iterative by incorporating the totality of all deleted small cross-terms (involving the $\\beta_n^m$, $(m,n)\\neq(0,0)$, quadratically) in the matching procedure using the estimate of the $\\beta$'s from the previous steps. In practice one finds that pupil functions $P$ deviating from being constant by as much as 2.5 times the diffraction limit can be retrieved. See \\cite{ref31}, \\cite{ref32}, \\cite{ref34}, \\cite{ref35}, \\cite{ref36} for more details.\n\nThe result in (\\ref{e8}) is one evidence of computational appropriateness of the circle polynomials for forward and inverse problems, but there are others. Many of these are based on the basic NZ-result in (\\ref{e8}). In \\cite{ref37}--\\cite{ref38}, Cormack used this result to calculate the Radon transform ${\\cal R}_n^m$,\n\\begin{equation} \\label{e13}\n{\\cal R}_n^m(\\tau,\\psi)=\\int\\limits_{l(\\tau,\\psi)}\\,Z_n^m(\\nu,\\mu)\\,dl\n\\end{equation}\nof $Z_n^m$, with integration along the line $\\nu\\cos\\psi+\\mu\\sin\\psi=\\tau$ in the plane with $\\tau\\geq0$ and $\\psi\\in[0,2\\pi)$ and where $(\\nu,\\mu)=(\\rho\\cos\\vartheta,\\rho\\sin\\vartheta)$. The result is that\n\\begin{equation} \\label{e14}\n{\\cal R}_n^m(\\tau,\\psi)=\\frac{2}{n+1}\\,(1-\\tau^2)^{1\/2}\\,U_n(\\tau)\\,e^{im\\psi}~,~~~~~0\\leq\\tau\\leq1\\,,~~0\\leq\\psi<2\\pi\\,,\n\\end{equation}\nwhere $U_n$ is the Chebyshev polynomial of the second kind and degree $n$, \\cite{ref14}, Ch.~22. On this explicit form of the Radon transform of $Z_n^m$, Cormack based a method for estimating a function on the disk from its Radon transform by estimating its Zernike expansion coefficients through matching. (In 1979, Cormack was awarded the Nobel prize in medicine, together with Hounsfield, for their work in computerized tomography.) Cormack's result was used by Dirksen and Janssen \\cite{ref39} to find the integral representation\n\\begin{equation} \\label{e15}\nR_n^m(\\rho)=\\frac{1}{2\\pi}\\,\\int\\limits_0^{2\\pi}\\,U_n(\\rho\\cos\\vartheta)\\cos m\\vartheta\\,d\\vartheta\n\\end{equation}\n(integer $m,n\\geq0$) that displays, for any $\\rho\\geq0$, the value of the radial part at $\\rho$ in the form of the Fourier coefficient of a trigonometric polynomial. This formula (\\ref{e15}) can be discretized, error free when more that $m+n$ equidistant points in $[0,2\\pi]$ are used, and this yields a scheme of the DCT-type for computation of $R_n^m(\\rho)$.\n\nA further consequence of the basic NZ-result (\\ref{e8}) is the theory of shifted-and-scaled circle polynomials developed in \\cite{ref40}. For given $a\\geq0$, $b\\geq0$ with $a+b\\leq1$, there are developed explicit expressions for the coefficients $K_{nn'}^{mm'}(a,b)$ in the expansion\n\\begin{equation} \\label{e16}\nZ_n^m((a+b\\rho'\\cos\\vartheta',b\\rho'\\sin\\vartheta'))=\n\\sum_{n',m'}\\,K_{nn'}^{mm'}(a,b)\\,Z_{n'}^{m'}(\\rho'\\cos\\vartheta,\\rho'\\sin\\vartheta')~.\n\\end{equation}\nThis generalizes the result in \\cite{ref41},\n\\begin{equation} \\label{e17}\nR_n^m(\\varepsilon\\rho)=\\sum_{n'=m(2)n}\\,(R_n^{n'}(\\varepsilon)-R_n^{n'+2}(\\varepsilon))\\,R_{n'}^m(\\rho)~,\n\\end{equation}\non the Zernike expansion of scaled circle polynomials ($m\\geq 0$).\n\nThe basic NZ-result is also useful for the computation of various acoustic quantities that arise from a circular piston in a planar baffle. The complex amplitude $p({\\bf r},k)$ of the sound pressure at the field point ${\\bf r}=(x,y,z)$, $z\\geq0$, in front of the baffle plane $z=0$ due to a harmonic excitation $\\exp(i\\omega t)$ with wave number $k=\\omega\/c$, with $c$ the speed of sound, is given by Rayleigh's integral and King's integral as\n\\begin{eqnarray} \\label{e18}\np({\\bf r},k) & = & \\frac{i\\rho_0 ck}{2\\pi}\\,\\int\\limits_S\\,v(\\sigma)\\,\\frac{e^{-ikr'}}{r'}\\,dS~= \\nonumber \\\\[3.5mm]\n& = & i\\rho_0ck\\,\\int\\limits_0^{\\infty}\\,\\frac{e^{-z(u^2-k^2)^{1\/2}}}{(u^2-k^2)^{1\/2}}\\,J_0(wu)\\,V(u)\\,u\\,du~.\n\\end{eqnarray}\nHere $\\rho_0$ is the density of the medium, $v(\\sigma)$ is a non-uniform velocity profile assumed to depend on the radial variable $\\sigma=(\\nu^2+\\mu^2)^{1\/2}$ on the piston surface $S$ with center ${\\bf 0}$ and radius $a$, and $r'$ is the distance from the field point ${\\bf r}$ to the point $(\\nu,\\mu,0)$ on $S$. Furthermore, $w=(x^2+y^2)^{1\/2}$ is the distance from the field point ${\\bf r}$ to the $z$-axis, the root $(u^2-k^2)^{1\/2}$ has the value $i\\,\\sqrt{k^2-u^2}$ and $\\sqrt{u^2-k^2}$ for $0\\leq u\\leq k$ and $u\\geq k$, respectively with $\\sqrt{~~}$ non-negative in both cases, and $V(u)$ is the Hankel transform,\n\\begin{equation} \\label{e19}\nV(u)=\\int\\limits_0^a\\,J_0(u\\sigma)\\,v(\\sigma)\\,\\sigma\\,d\\sigma~,~~~~~~u\\geq0~,\n\\end{equation}\nof order 0 of $v(\\sigma)$.\n\nIn \\cite{ref19}, the on-axis pressure $p_{2l}({\\bf r}=(0,0,z),k)$ due to $v(\\sigma)=R_{2l}^0(\\sigma\/a)$, $0\\leq\\sigma\\leq a$, was shown from Rayleigh's integral and a special result on spherical Bessel functions to be given as\n\\begin{equation} \\label{e20}\np_{2l}({\\bf r}=(0,0,z),k)=\\tfrac12\\,\\rho_0\\,c(ka)^2({-}1)^l\\,j_l(kr_-)\\,h_l^{(2)}(kr_+)~,\n\\end{equation}\nwhere $j_l$ and $h_l^{(2)}$ are spherical Bessel functions, see \\cite{ref14}, Ch.~10 and in particular 10.1.45--46, and $r_{\\pm}=\\frac12\\,[(r^2+a^2)^{1\/2}\\pm r]$. This result was used in \\cite{ref19} for estimating a velocity profile from on-axis pressure data on the level of expansion coefficients with respect to radially symmetric circle polynomials.\n\nIn \\cite{ref20}, the King integral for the pressure is employed to express the pressure $p((1,0,0),k)$ at the edge, the reaction force $\\int\\limits_S\\,p\\,dS$ and the total radiated power $\\int\\limits_S\\,p(0)\\,v^{\\ast}(\\sigma)\\,dS$ in integral form. Expanding $v(\\sigma)$ into radially symmetric circle polynomials and using the basic NZ-result for an explicit expression of the Hankel transform $V(u)$ in (\\ref{e19}), this gives rise to integrals\n\\begin{equation} \\label{e21}\n\\int\\limits_0^{\\infty}\\,\\frac{J_m(au)\\,J_{n+1}(au)}{(u^2-k^2)^{1\/2}}\\,du~,~~~~~~\\int\\limits_0^{\\infty}\\, \\frac{J_{m+1}(au)\\,J_{n+1}(au)}{(u^2-k^2)^{1\/2}\\,u}\\,du\n\\end{equation}\nwith integer $m,n\\geq0$ of same parity. These integrals have been evaluated as a power series in $ka$ in \\cite{ref20}.\n\nIn \\cite{ref21}, the problem of sound radiation from a flexible spherical cap on a rigid sphere is considered. The scaling theory of Zernike circle polynomials, appropriately warped so as to account for the spherical geometry of the problem, is used to bring the standard solution of the Helmholtz equation with axially symmetric boundary data in a semi-analytic form per warped Zernike term. This gives rise to a coefficient-based solution of the inverse problem of estimating an axially symmetric velocity profile on the cap from measured pressure data in the space around the sphere.\n\nFinally, returning to baffled-piston radiation, in \\cite{ref22} the impulse response $h({\\bf r},t)$, $t\\geq0$, at a field point ${\\bf r}=(x,y,z)$ in front of the baffle is considered. This $h({\\bf r},t)$ is obtained as a Fourier inversion integral with respect to wave number $k$ of the velocity potential $\\varphi({\\bf r},k)=(i\\rho_0ck)^{-1}\\,p({\\bf r},k)$, which can be evaluated, via Rayleigh's integral, as an integral of the velocity profile $v$ over all points in the baffle plane at a common distance $(c^2t^2-z^2)^{1\/2}$ from the field point. For the case that $v(\\sigma)=R_{2l}^0(\\sigma\/a)$, $0\\leq\\sigma\\leq a$, the latter integral can be evaluated explicitly using the addition theorem for Legendre polynomials. This explicit result can be used to compute impulse responses for general $v$ by expanding such a $v$ into radially symmetric circle functions. And, of course, an inverse problem, with a coefficient-based solution, can again be formulated.\n\n\\section{Generalized Zernike circle functions} \\label{sec3}\n\\mbox{} \\\\[-9mm]\n\nThe Zernike circle polynomials, see (\\ref{e1}), all have modulus 1 at the edge $\\rho=1$ of the unit disk while in both Optics and Acoustics it frequently occurs that the non-uniformity behaves differently towards the edge of the disk.\n\nIn optical design, it is often desirable to have a pupil function $P$ whose point-spread function $U$, see (\\ref{e6}), decays relatively fast outside the focal region where the specifications of the designer are to be met. The Zernike circle polynomials are discontinuous at the edge $\\rho=1$, and the corresponding point-spread functions $U_n^m(r,\\varphi\\,;\\,f)$ have poor decay, like $r^{-3\/2}$ as $r\\rightarrow\\infty$.\n\nIn the ENZ point-spread functions, see Sec.~\\ref{sec2}, the key assumption is that the pupil function's deviation from being constant is not large. When $|P|$ gets small at the edge, which happens, for instance, when the source is a pinhole with a positive diameter while the objective lens has a large NA, this basic assumption is not met. In such a case, aberration retrieval with the ENZ method may become cumbersome, even in its iterative version.\n\nIn the theory of acoustic radiation from a baffled, planar piston with radially symmetric boundary conditions, Streng \\cite{ref42} follows the approach of Bouwkamp \\cite{ref43} in solving the Helmholtz equation for this case and postulates normalized pressure functions on the disk of the form\n\\begin{equation} \\label{e22}\n\\sum_{n=0}^{\\infty}\\,a_n(1-\\rho^2)^{n+1\/2}~,~~~~~~0\\leq\\rho<1~.\n\\end{equation}\nIn the light of the later developments of this paper, it is interesting to note here that Bouwkamp himself prefers expansions that involve the functions\n\\begin{equation} \\label{e23}\n(1-\\rho^2)^{1\/2}\\,P_l^{(1\/2,0)}(2\\rho^2-1)=({-}1)^l\\,P_{2l+1}((1-\\rho^2)^{1\/2})~,~~~~~~0\\leq\\rho<1~.\n\\end{equation}\nSimilarly, Mellow \\cite{ref44} has velocity profiles on the disk of the form (normalized to the unit disk)\n\\begin{equation} \\label{e24}\nv(\\rho)=\\sum_{n=0}^{\\infty}\\,b_n(1-\\rho^2)^{n-1\/2}~,~~~~~~0\\leq\\rho<1~.\n\\end{equation}\nIn \\cite{ref44} and \\cite{ref45}--\\cite{ref46}, the postulates (\\ref{e22}), (\\ref{e24}) are used to solve design problems in acoustic radiation from a membrane in a circular disk in terms of the expansion coefficients $a_n$, $b_n$. In principle, these design problems could also be solved when radially symmetric circle polynomials instead of $(1-\\rho^2)^{n\\pm1\/2}$ were used as trial functions, but this would become cumbersome, the functions $(1-\\rho^2)^{\\pm1\/2}$ themselves already having a poorly convergent expansion with respect to the $R_{2l}^0(\\rho)$.\n\nIn the three instances just discussed, it would be quite helpful when the system of Zernike circle polynomials would be replaced by systems whose members exhibit an appropriate behaviour at the edge of the disk. These new systems should satisfy the requirements A, B and C mentioned in the beginning of Sec.~\\ref{sec1}. In this paper, the set of functions\n\\begin{eqnarray} \\label{e25}\nZ_n^{m,\\alpha}(\\rho,\\vartheta) & \\!\\!= & \\!\\!(1-\\rho^2)^{\\alpha}\\,\\rho^{|m|}\\,P_{\\frac{n-|m|}{2}}^{(\\alpha,|m|)}(2\\rho^2-1)\\, e^{im\\vartheta}~,~~~0\\leq\\rho<1\\,,~0\\leq\\vartheta<2\\pi \\nonumber \\\\[3mm]\n& \\!\\!= & \\!\\!0\\hspace*{6.1cm},~~~\\rho>1~,\n\\end{eqnarray}\nis considered for this purpose. In (\\ref{e25}), the parameter $\\alpha>{-}1$, and $n$ and $m$ are integers such that $n-|m|$ is even and non-negative, and the $P_k^{(\\alpha,\\beta)}$ are Jacobi polynomials as before. Observe that the radial parts\n\\begin{equation} \\label{e26}\nR_n^{|m|,\\alpha}(\\rho)=(1-\\rho^2)^{\\alpha}\\, \\rho^{|m|}\\, P_{\\frac{n-|m|}{2}}^{(\\alpha,|m|)}(2\\rho^2-1)~,~~~~~~0\\leq\\rho<1~,\n\\end{equation}\ndepend non-polynomially on $\\rho$, unless $\\alpha=0,1,...\\,$. The radial parts occur essentially in Tango \\cite{ref47}, Sec.~1, but there the factor $(1-\\rho^2)^{\\alpha}$ is replaced by $(1-\\rho^2)^{\\alpha\/2}$ with somewhat different restrictions on $\\alpha$ and $|m|$. There is also a relation with the disk polynomials as they occur, for instance, in the work of Koornwinder \\cite{ref48}:\n\\begin{equation} \\label{e27}\nZ_n^{m,\\alpha}(\\rho,\\vartheta)=(1-\\rho^2)^{\\alpha}\\,D_{\\frac{n+m}{2}\\,,\\,\\frac{n-m}{2}}^{\\alpha}(\\rho\\,e^{i\\vartheta})~,\n\\end{equation}\nwith\n\\begin{equation} \\label{28}\nD_{k,l}^{\\alpha}(\\rho\\,e^{i\\vartheta})=\\rho^{|k-l|}\\,P_{\\min(k,l)}^{(\\alpha,|k-l|)}(2\\rho^2-1)\\,e^{i(k-l)\\vartheta}\n\\end{equation}\nfor $k,l=0,1,...\\,$. Thus, the disk polynomials omit the factor $(1-\\rho^2)^{\\alpha}$ altogether.\n\nFurther definitions used in this paper are\n\\begin{equation} \\label{e29}\np=\\frac{n-|m|}{2}~,~~~~~~q=\\frac{n+|m|}{2}\n\\end{equation}\nfor integers $n$ and $m$ such that $n-|m|$ is even and non-negative, and the generalized Pochhammer symbol\n\\begin{equation} \\label{e30}\n(x)_y=\\frac{\\Gamma(x+y)}{\\Gamma(x)}~.\n\\end{equation}\n\nBy orthogonality of the $P_k^{(\\alpha,\\beta)}(x)$ with respect to the weight function $(1-x)^{\\alpha}(1+x)^{\\beta}$ on $[{-}1,1]$, there holds\n\\begin{eqnarray} \\label{e31}\n& \\mbox{} & \\frac{1}{\\pi}\\,\\int\\limits_0^1\\int\\limits_0^{2\\pi}\\,(1-\\rho^2)^{-\\alpha}\\,Z_{n_1}^{m_1,\\alpha}(\\rho,\\vartheta) (Z_{n_2}^{m_2,\\alpha}(\\rho,\\vartheta))^{\\ast}\\,\\rho\\,d\\rho\\,d\\vartheta~= \\nonumber \\\\[3.5mm]\n& & =~\\frac{(p+1)_{\\alpha}}{(p+|m|+1)_{\\alpha}}~\\frac{\\delta_{m_1m_2}\\,\\delta_{n_1n_2}}{n_1+\\alpha+1}~.\n\\end{eqnarray}\n\nWe note that for any integer $N=0,1,...$\n\\begin{equation} \\label{e32}\n\\rho^{|m|}\\,P_{\\frac{n-|m|}{2}}^{(\\alpha,|m|)}(2\\rho^2-1)\\,e^{im\\vartheta}\n\\end{equation}\nwith $n=0,1,...,N$ and $m={-}n,{-}n+2,...,n$ are $\\frac12\\,(N+1)(N+2)$ linearly independent functions of the form $\\sum_{i,j}\\,a_{ij}\\nu^i\\mu^j$ ($\\nu=\\rho\\cos\\vartheta$, $\\mu=\\rho\\sin\\vartheta$) where the summation is over integer $i,j\\geq0$ with $i+j\\leq N$. Therefore, by Weierstrass theorem the functions in (\\ref{e32}) are complete.\n\nThis paper focuses on establishing versions for the generalized Zernike functions of the analytic results that were presented for the classical circle polynomials in Sec.~\\ref{sec2}. Thus, in Sec.~\\ref{sec4}, the Fourier transform of $Z_n^{m,\\alpha}$ is computed in terms of Bessel functions and a Weber-Schafheitlin representation of the radial functions $R_n^{m,\\alpha}$ is given. In Sec.~\\ref{sec5}, the Radon transform of $Z_n^{m,\\alpha}$ is computed in terms of the Gegenbauer polynomials $C_n^{\\alpha+1}$, and a representation of the radial functions as Fourier coefficients of the periodic function $C_n^{\\alpha+1}(\\rho\\cos\\vartheta)$, $0\\leq\\vartheta<2\\pi$, for a fixed $\\rho\\in[0,1)$ is given. Thus a computation scheme of the DCT-type for the radial functions arises. Then, in Sec.~\\ref{sec6}, a scaling result is given. This result is somewhat more awkward than in the classical case since the radial functions $R_n^{m,\\alpha}(\\rho)$ have restrictions on their behaviour as $\\rho\\uparrow1$ while the scaled radial functions $R_n^{m,\\alpha}(\\varepsilon\\rho)$, to be considered with $0<\\varepsilon<1$, do not. Next, in Sec.~\\ref{sec7}, the new Zernike functions are expanded in terms of the classical circle polynomials, with an explicit expression for the expansion coefficients. This makes it possible to transfer all forward computation results from the ENZ theory and from the acoustic Nijboer-Zernike (ANZ) theory for the classical case to the new setting in a semi-analytic form. This is useful for those cases that a closed form or a simple semi-analytic form is not available or awkward to find in the new setting. In Secs.~\\ref{sec8}--\\ref{sec10}, the focus is on generalizing the results from the ANZ theory, as developed in \\cite{ref19}--\\cite{ref22} to the more general setting. This gives rise in Sec.~\\ref{sec8} to a power series representation of the basic integrals that occur when various acoustic quantities are computed from King's integral for the sound pressure (in the case of baffled-piston radiation). An inverse problem, in which the velocity profile is estimated in terms of its expansion coefficients from near-field measurements via Weyl's formula, is considered in Sec.~\\ref{sec9}. Finally, in Sec.~\\ref{sec10}, the trial functions as used by Streng and Mellow, following Bouwkamp's solution of the diffraction problem for a circular aperture, are compared with the $Z_{2l}^{0,\\alpha={\\pm}1\/2}$. In \\cite{ref46}, Mellow and K\\\"arkk\\\"ainen consider radiation from a disk with concentric rings, and for this an indefinite integral involving the radial functions $R_{2l}^{0,{\\pm}1\/2}(\\rho)$ is required. This integral is computed in closed form.\n\n\\section{Fourier transform of generalized Zernike circle functions} \\label{sec4}\n\\mbox{} \\\\[-9mm]\n\nIn this section, the 2D Fourier transform of $Z_n^{m,\\alpha}$ is computed. It is convenient to write here\n\\begin{equation} \\label{e33}\nZ_n^{m,\\alpha}(\\nu,\\mu)\\equiv Z_n^{m,\\alpha}(\\rho,\\vartheta)~,\n\\end{equation}\nwhere $\\nu+i\\mu=\\rho\\,e^{i\\vartheta}$ with $\\rho\\geq0$ and $0\\leq\\vartheta<2\\pi$. \\\\ \\\\\n{\\bf Theorem 4.1.}~~For $\\alpha>{-}1$ and integer $n$, $m$ such that $p=\\frac12\\,(n-|m|)$ is a non-negative integer, there holds\n\\begin{eqnarray} \\label{e34}\n& \\mbox{} & \\int\\limits\\!\\!\\int\\limits\\,e^{2\\pi i\\nu x+2\\pi i\\mu y}\\,Z_n^{m,\\alpha}(\\nu,\\mu)\\,d\\nu\\,d\\mu~= \\nonumber \\\\[3.5mm]\n& & =~2\\pi\\,i^n\\,2^{\\alpha}(p+1)_{\\alpha}\\,\\frac{J_{n+\\alpha+1}(2\\pi r)}{(2\\pi r)^{\\alpha+1}}\\,e^{im\\varphi}~,\n\\end{eqnarray}\nwhere $x+iy=r\\,e^{i\\varphi}$ with $r\\geq0$ and $0\\leq\\varphi<2\\pi$. Furthermore,\n\\begin{equation} \\label{e35}\n\\displaystyle \\int\\limits_0^1\\,R_n^{|m|,\\alpha}(\\rho)\\,J_{|m|}(2\\pi\\rho r)\\,\\rho\\,d\\rho=({-}1)^p\\,2^{\\alpha}(p+1)_{\\alpha}\\, \\frac{J_{n+\\alpha+1}(2\\pi r)}{(2\\pi r)^{\\alpha+1}}~,\n\\end{equation}\nand\n\\begin{equation} \\label{e36}\nR_n^{|m|,\\alpha}(\\rho)=({-}1)^p\\,2^{\\alpha}(p+1)_{\\alpha}\\,\\displaystyle \\int\\limits_0^{\\infty}\\, \\frac{J_{n+\\alpha+1}(t)\\,J_{|m|}(\\rho t)}{t^{\\alpha}}\\,dt~,~~~~~0\\leq\\rho<1~.\n\\end{equation}\n{\\bf Proof.}~~First consider the case that $m\\geq0$. It follows from \\cite{ref14}, 11.4.33 (Weber-Schafheitlin integral), with\n\\begin{equation} \\label{e37}\n\\mu=n+p+1=m+2p+\\alpha+1\\,,~~a=1\\,,~~\\nu=m\\,,~~b=\\rho\\in[0,1)\\,,~~\\lambda=\\alpha\n\\end{equation}\nthat\n\\begin{eqnarray} \\label{e38}\n& \\mbox{} & \\int\\limits_0^{\\infty}\\,\\frac{J_{n+\\alpha+1}(t)\\,J_m(\\rho t)}{t^{\\alpha}}\\,dt~= \\nonumber \\\\[3mm]\n& & =~\\frac{\\rho^m\\,\\Gamma(m+p+1)}{2^{\\alpha}\\, \\Gamma(m+1)\\,\\Gamma(p+\\alpha+1)}\\,F({-}p-\\alpha,m+p+1\\,;\\,m+1\\,;\\,\\rho^2)~. \\nonumber \\\\\n& & \\mbox{}\n\\end{eqnarray}\nNext from \\cite{ref14}, 15.3.3 with\n\\begin{equation} \\label{e39}\na=m+1+p+\\alpha\\,,~~~b={-}p\\,,~~~c=m+1\\,,~~~z=\\rho^2\n\\end{equation}\nit follows that\n\\begin{eqnarray} \\label{e40}\n& \\mbox{} & F({-}p-\\alpha,m+p+1\\,;\\,m+1\\,;\\,\\rho^2)~= \\nonumber \\\\[3mm]\n& & =~(1-\\rho^2)^{\\alpha}\\,F({-}p,m+1+\\alpha+p\\,;\\,m+1\\,;\\,\\rho^2)~= \\nonumber \\\\[3mm]\n& & =~(1-\\rho^2)^{\\alpha}\\,\\frac{p!}{(m+1)_p}\\,P_p^{(m,\\alpha)}(1-2\\rho^2)~,\n\\end{eqnarray}\nwhere in the last step \\cite{ref14}, 15.4.6 with $n=p$, $\\alpha=m$, $\\beta=\\alpha$ and $z=\\rho^2$ has been used. Therefore, for general integer $m$, $|m|\\leq n$,\n\\begin{eqnarray} \\label{e41}\n& \\mbox{} & \\int\\limits_0^{\\infty}\\,\\frac{J_{n+\\alpha+1}(t)\\,J_{|m|}(\\rho t)}{t^{\\alpha}}\\,dt~= \\nonumber \\\\[3.5mm]\n& & =~\\frac{({-}1)^p}{2^{\\alpha}(p+1)_{\\alpha}}\\,\\rho^{|m|}(1-\\rho^2)^{\\alpha}\\,P_p^{(\\alpha,|m|)}(2\\rho^2-1) = \\frac{({-}1)^p}{2^{\\alpha}(p+1)_{\\alpha}}\\,R_n^{|m|,\\alpha}(\\rho)~, \\nonumber \\\\\n& & \\mbox{}\n\\end{eqnarray}\nwhere $P_k^{(\\alpha,\\beta)}({-}x)=({-}1)^k\\,P_k^{(\\beta,\\alpha)}({-}x)$ and the definition (\\ref{e26}) of $R_n^{|m|,\\alpha}(\\rho)$ have been used. This establishes (\\ref{e36}).\n\nNext,\n\\begin{eqnarray} \\label{e42}\n& \\mbox{} & \\int\\limits_0^{\\infty}\\int\\limits_0^{2\\pi}\\,e^{-2\\pi i\\nu x-2\\pi i\\mu y}\\,\\frac{J_{n+\\alpha+1}(2\\pi r)} {(2\\pi r)^{\\alpha+1}}\\,e^{im\\varphi}\\,r\\,dr\\,d\\varphi~= \\nonumber \\\\[3.5mm]\n& & =~2\\pi\\,i^m\\,\\int\\limits_0^{\\infty}\\,J_m({-}2\\pi r\\rho)\\,\\frac{J_{n+\\alpha+1}(2\\pi r)}{(2\\pi r)^{\\alpha+1}}\\,r\\,dr\\,e^{im\\vartheta}~= \\nonumber \\\\[3.5mm]\n& & =~\\frac{({-}1)^{|m|}}{2\\pi}\\,\\int\\limits_0^{\\infty}\\,\\frac{J_{|m|}(t)\\,J_{n+\\alpha+1}(t)}{t^{\\alpha}}\\,dt\\, e^{im\\vartheta}~,\n\\end{eqnarray}\nwhere, subsequently, use is made of\n\\begin{equation} \\label{e43}\n\\nu x+\\mu y=\\rho\\,r\\cos(\\vartheta-\\varphi)~,\n\\end{equation}\n\\begin{equation} \\label{e44}\n\\frac{1}{2\\pi}\\,\\int\\limits_0^{2\\pi}\\,e^{-it\\cos\\vartheta}\\,e^{im\\vartheta}\\,d\\vartheta=i^m\\,J_m({-}t)=({-}i)^{|m|}\\,J_{|m|}(t)~,\n\\end{equation}\nsee \\cite{ref14}, Sec.~9.1, and where the substitution $t=2\\pi r$ has been used in the last step in (\\ref{e42}). Therefore, using (\\ref{e41}), the definitions (\\ref{e25})--(\\ref{e26}) and $n=|m|+2p$, it follows that\n\\begin{eqnarray} \\label{45}\n& \\mbox{} & \\int\\limits\\!\\!\\int\\limits\\,e^{-2\\pi i\\mu x-2\\pi i\\mu y}\\,\\frac{J_{n+\\alpha+1}(2\\pi r)}{(2\\pi r)^{\\alpha+1}}\\,r\\,dr\\,d\\varphi~= \\nonumber \\\\[3mm]\n& & =~\\frac{({-}i)^n}{2\\pi}~\\frac{1}{2^{\\alpha}(p+1)_{\\alpha}}\\,Z_n^{m,\\alpha}(\\rho,\\vartheta)~.\n\\end{eqnarray}\nThen (\\ref{e34}) follows by 2D Fourier inversion. Now also (\\ref{e35}) follows using the definitions (\\ref{e25})--(\\ref{e26}) in (\\ref{e34}) and proceeding as in (\\ref{e42})--(\\ref{e44}). \\\\ \\\\\n{\\bf Notes.} \\\\[-7mm]\n\\bi{1.0}\n\\ITEM{1.} The result in (\\ref{e34}) generalizes the case $\\alpha=0$ in (\\ref{e7}) to general $\\alpha>{-}1$.\n\\ITEM{2.} The result in (\\ref{e35}) gives the Hankel transform of order $|m|$ of the radial part $R_n^{|m|,\\alpha}$, and (\\ref{e36}) is what one gets by inverse Hankel transformation of order $|m|$. These two results are reminiscent of, but clearly different from, the results in (\\ref{e10}) and (\\ref{e11}) of \\cite{ref47}.\n\\ITEM{3.} By the asymptotics of the Bessel functions, see \\cite{ref14}, 9.2.1, it is seen that the Fourier transform of $Z_n^{m,\\alpha}$ decays as $r^{-\\alpha-3\/2}$ as $r\\rightarrow\\infty$.\n\\end{inspring}\n\n\\section{Radon transform of generalized Zernike circle functions; integral representation and DCT-formula for radial parts} \\label{sec5}\n\\mbox{} \\\\[-9mm]\n\nIn this section, the Radon transform of the generalized circle functions is expressed in terms of Gegenbauer polynomials. Furthermore, an integral representation involving these Gegenbauer polynomials for the radial parts is proved, and a method of the DCT-type for computation of the radial parts is shown to follow from this integral representation.\n\nThe Radon transform of $Z_n^{m,\\alpha}$ is given by\n\\begin{equation} \\label{e46}\n({\\cal R}\\,Z_n^{m,\\alpha})(\\tau,\\psi)=\\int\\limits_{l(\\tau,\\psi)}\\,Z_n^{m,\\alpha}(\\nu,\\mu)\\,dl\n=\\int\\limits\\,R_n^{|m|,\\alpha}(\\rho(t))\\,e^{im\\vartheta(t)}\\,dt~,\n\\end{equation}\nwith $l(\\tau,\\psi)$ and $\\rho$, $\\vartheta$ given in Fig.~1 for $\\tau\\geq0$ and $0\\leq\\psi<2\\pi$. Thus\n\\begin{equation} \\label{e47}\n\\rho(t)=(\\tau^2+t^2)^{1\/2}\\,,~~\\vartheta(t)=\\psi+\\arctan(t\/\\tau)=\\psi+{\\rm sgn}(t)\\arccos(\\tau\/\\rho(t))~.\n\\end{equation}\n\\begin{figure}[htb!]\n\\begin{center}\n\\includegraphics[width=0.70\\linewidth]{Report2011Fig1.eps}\n\\caption{Line of integration $l(\\tau,\\psi)$ and integration point $P=Q+t({-}\\sin\\psi,\\cos\\psi)$ with $-\\infty{-}1$ and integers $n$, $m$ such that $p=\\frac12\\,(n-|m|)$ is a non-negative integer that\n\\begin{equation} \\label{e49}\n({\\cal R}\\,Z_n^{m,\\alpha})(\\tau,\\psi)= \\frac{(p+1)_{\\alpha}}{(n+1)_{2\\alpha}}~\\frac{2^{2\\alpha+1}\\,\\Gamma(\\alpha+1)} {n+2\\alpha+1}\\,(1-\\tau^2)^{\\alpha+1\/2}\\,C_n^{\\alpha+1}(\\tau)\\,e^{im\\psi}\n\\end{equation}\nwhen $0\\leq\\tau\\leq1$ and $({\\cal R}\\,Z_n^{m,\\alpha})(\\tau,\\psi)=0$ for $\\tau>1$. Here $C_n^{\\alpha+1}$ is the Gegenbauer (or ultraspherical) polynomial corresponding to the weight function $(1-x^2)^{\\alpha+1\/2}$, $-11~,\n\\end{eqnarray}\nand so, from (\\ref{e48})\n\\begin{eqnarray} \\label{e51}\n({\\cal R}\\,Z_n^{m,\\alpha})(\\tau,\\psi) & = & e^{im\\psi}\\,\\int\\limits_0^1\\,R_n^{m,\\alpha}(\\rho)\\,\\left( i^m\\,\\int\\limits_{-\\infty}^{\\infty}\\,e^{-i\\tau t\/\\rho}\\,J_m(t)\\,dt\\right)\\,d\\rho~= \\nonumber \\\\[3.5mm]\n& = & e^{im\\psi}\\,i^m\\,\\int\\limits_{-\\infty}^{\\infty}\\,e^{-i\\tau s}\\,\\left(\\int\\limits_0^1\\,R_n^{m,\\alpha}(\\rho)\\,J_m(\\rho s) \\,\\rho\\,d\\rho\\right)\\,ds~= \\nonumber \\\\[3.5mm]\n& = & e^{im\\psi}\\,i^n\\,2^{\\alpha}(p+1)_{\\alpha}\\,\\int\\limits_{-\\infty}^{\\infty}\\,e^{-i\\tau s}\\, \\frac{J_{n+\\alpha+1}(s)} {s^{\\alpha+1}}\\,ds~,\n\\end{eqnarray}\nwhere (\\ref{e35}) has been used, together with $i^m({-}1)^p=i^n$.\n\nBy \\cite{ref49}, 1.12, item (10), there holds for even $n$ ($a=1$, $2n$ instead of $n$, $\\nu=\\alpha+1$)\n\\begin{eqnarray} \\label{e52}\n& \\mbox{} & 2\\,\\int\\limits_0^{\\infty}\\cos\\tau s\\,\\frac{J_{n+\\alpha+1}(s)}{s^{\\alpha+1}}\\,ds~= \\nonumber \\\\[3.5mm]\n& & =~({-}1)^{\\frac12 n}\\,\\frac{2^{\\alpha+1}\\,n!\\,\\Gamma(\\alpha+1)}{\\Gamma(n+2\\alpha+2)}\\, (1-\\tau^2)^{\\alpha+1\/2} \\,C_n^{\\alpha+1}(\\tau)~,~~~~~~0\\leq\\tau<1~, \\nonumber \\\\\n& \\mbox{} &\n\\end{eqnarray}\nwhile the integral in (\\ref{e52}) equals 0 when $\\tau>1$. By \\cite{ref49}, 2.12, item (10), there holds for odd $n$ ($a=1$, $2n+1$ instead of $n$, $\\nu=\\alpha+1$)\n\\begin{eqnarray} \\label{e53}\n& \\mbox{} & -2i\\,\\int\\limits_0^{\\infty}\\sin\\tau s\\,\\frac{J_{n+\\alpha+1}(s)}{s^{\\alpha+1}}\\,ds~= \\nonumber \\\\[3.5mm]\n& & =~{-}i({-}1)^{\\frac{n-1}{2}}\\,\\frac{2^{\\alpha+1}\\,n!\\,\\Gamma(\\alpha+1)}{\\Gamma(n+2\\alpha+2)}\\, (1-\\tau^2)^{\\alpha+1\/2}\\,C_n^{\\alpha+1}(\\tau)~,~~~~~~0\\leq\\tau<1~, \\nonumber \\\\\n& \\mbox{} &\n\\end{eqnarray}\nwhile the integral in (\\ref{e53}) equals 0 when $\\tau>1$. Using (\\ref{e52}) and (\\ref{e53}) in (\\ref{e51}), it follows upon some administration with $i^n$ and $({-}1)^{\\frac12 n}$, $({-}1)^{\\frac12 (n-1)}$ that for $0\\leq\\tau<1$\n\\begin{equation} \\label{e54}\n({\\cal R}\\,Z_n^{m,\\alpha})(\\tau,\\psi)=(p+1)_{\\alpha}\\, \\frac{2^{2\\alpha+1}\\,n!\\,\\Gamma(\\alpha+1)} {\\Gamma(n+2\\alpha+2)}\\,(1-\\tau^2)^{\\alpha+1\/2}\\,C_n^{\\alpha+1}(\\tau)\\,e^{im\\psi}~,\n\\end{equation}\nwhile $({\\cal R}\\,Z_n^{m,\\alpha})(\\tau,\\psi)=0$ when $\\tau>1$. The result follows now upon some further administration with Pochhammer symbols. \\\\ \\\\\n{\\bf Theorem 5.2.}~~There holds for $\\alpha>{-}1$ and integer $n$ and $m$ such that $n-|m|$ is even and non-negative\n\\begin{equation} \\label{e55}\n\\Bigl(\\!\\begin{array}{c} q+\\alpha \\\\ q\\end{array}\\!\\Bigr)(1-\\rho^2)^{-\\alpha}\\,R_n^{|m|,\\alpha}(\\rho)=\\frac{1}{2\\pi}\\,\\int\\limits_0^{2\\pi} \\,C_n^{\\alpha+1}(\\rho\\cos\\vartheta)\\,e^{-im\\vartheta}\\,d\\vartheta~,\n\\end{equation}\nwhere $p=\\frac12\\,(n-|m|)$, $q=\\frac12\\,(n+|m|)$. \\\\[2mm]\n{\\bf Proof.}~~There holds, taking $|\\nu|=\\tau$ and $\\psi=0$ or $\\pi$ according as $\\nu \\geq 0$ or $\\nu < 0$ in (\\ref{e49}), with separate consideration of even $m$ and odd $m$ in the case $\\nu < 0$,\n\\begin{equation} \\label{e56}\n\\int\\limits_{-\\sqrt{1-\\nu^2}}^{\\sqrt{1-\\nu^2}}\\,Z_n^{m,\\alpha}(\\nu,\\mu)\\,d\\mu=K_{nm}(1-\\nu^2)^{\\alpha+1\/2}\\,C_n^{\\alpha+1}(\\nu)~,\n\\end{equation}\nwhere\n\\begin{equation} \\label{e57}\nK_{nm}=\\frac{(p+1)_{\\alpha}}{(n+1)_{2\\alpha}}~\\frac{2^{2\\alpha+1}\\,\\Gamma(\\alpha+1)}{n+2\\alpha+1}~.\n\\end{equation}\nExpand\n\\begin{equation} \\label{e58}\nC_n^{\\alpha+1}(\\nu)=\\sum_{n',m'}\\,\\beta_{nn'}^{mm'}(1-\\rho^2)^{-\\alpha}\\,Z_{n'}^{m',\\alpha}(\\nu,\\mu)\n\\end{equation}\nas a function depending only on $\\nu$ with $\\nu^2+\\mu^2\\leq1$. By orthogonality, see (\\ref{e31}), there holds\n\\begin{equation} \\label{e59}\n\\beta_{nn'}^{mm'}=L_{n'm'}^{-1}\\,\\int\\limits\\hspace*{-6mm}\\int\\limits_{\\nu^2+\\mu^2\\leq1}\\,C_n^{\\alpha+1}(\\nu)\\, Z_{n'}^{m',\\alpha} (\\nu,\\mu)\\,d\\nu\\,d\\mu~,\n\\end{equation}\nwhere\n\\begin{equation} \\label{e60}\nL_{n'm'}=\\frac{\\pi}{n'+\\alpha+1}~\\frac{(p'+1)_{\\alpha}}{(p'+|m'|+1)_{\\alpha}}~.\n\\end{equation}\nUsing (\\ref{e56}), it is seen that\n\\begin{eqnarray} \\label{e61}\n\\beta_{nn'}^{mm'} & = & L_{n'm'}^{-1}\\,\\int\\limits_{-1}^1\\,C_n^{\\alpha+1}(\\nu)\\left(\\int\\limits_{-\\sqrt{1-\\nu^2}}^{\\sqrt{1-\\nu^2}}\\, Z_{n'}^{m',\\alpha}(\\nu,\\mu)\\,d\\mu\\right)\\,d\\nu~= \\nonumber \\\\[3.5mm]\n& = & L_{n'm'}^{-1}\\,K_{n'm'}\\,\\int\\limits_{-1}^1\\,(1-\\nu^2)^{\\alpha+1\/2}\\,C_n^{\\alpha+1}(\\nu)\\,C_{n'}^{\\alpha+1}(\\nu)\\,d\\nu~= \\nonumber \\\\[3.5mm]\n& = & L_{n'm'}^{-1}\\,K_{n'm'}\\,M_n\\,\\delta_{nn'}~,\n\\end{eqnarray}\nwhere $M_n$ follows from orthogonality of the $C^{\\alpha+1}$ as, see \\cite{ref16}, (7.8),\n\\begin{equation} \\label{e62}\nM_n=\\frac{\\pi\\cdot2^{-2\\alpha-1}\\cdot\\Gamma(n+2\\alpha+2)}{n!\\,(n+\\alpha+1)\\,\\Gamma^2(\\alpha+1)}~.\n\\end{equation}\nWith $\\nu=\\rho\\cos\\vartheta$ it then follows that\n\\begin{eqnarray} \\label{e63}\nC_n^{\\alpha+1}(\\rho\\cos\\vartheta) & = & \\sum_{m'}\\,\\beta_{nn}^{mm'}(1-\\rho^2)^{-\\alpha}\\,Z_n^{m',\\alpha}(\\rho,\\vartheta)~= \\nonumber \\\\[3.5mm]\n& = & \\sum_{m'}\\,\\beta_{nn}^{mm'}\\,\\rho^{|m'|}\\,P_{\\frac{n-|m'|}{2}}^{(\\alpha,|m'|)}(2\\rho^2-1)\\,e^{im\\vartheta}~.\n\\end{eqnarray}\nTherefore, see (\\ref{e26}),\n\\begin{equation} \\label{e64}\n\\beta_{nn'}^{mm'}\\,\\rho^{|m'|}\\,P_{\\frac{n-|m'|}{2}}^{(\\alpha,|m'|)}(2\\rho^2-1)=\\frac{1}{2\\pi}\\,\\int\\limits_0^{2\\pi}\\, C_n^{\\alpha+1}(\\rho\\cos\\vartheta)\\,e^{-im'\\vartheta}\\,d\\vartheta\n\\end{equation}\nfor integer $m'$ such that $p'':=\\frac12\\,(n-|m'|)$ is a non-negative integer. An explicit computation from (\\ref{e57}), (\\ref{e60}), (\\ref{e61}) yields now\n\\begin{eqnarray} \\label{e65}\n\\beta_{nn}^{mm'} & = & L_{nm'}^{-1}\\,K_{nm'}\\,M_n~= \\nonumber \\\\[3.5mm]\n& = & \\Bigl(\\frac{\\pi}{n+\\alpha+1}~\\frac{(p''+1)_{\\alpha}}{(p''+|m'|+1)_{\\alpha}}\\Bigr)^{-1}\\cdot\n\\frac{(p''+1)_{\\alpha}}{(n+1)_{2\\alpha}}~\\frac{2^{2\\alpha+1}\\,\\Gamma(\\alpha+1)}{n+2\\alpha+1}~\\cdot \\nonumber \\\\[3.5mm]\n& & \\cdot~\n\\frac{\\pi\\cdot2^{-2\\alpha-1}\\,\\Gamma(n+2\\alpha+2)}{n!\\,(n+\\alpha+1)\\,\\Gamma^2(\\alpha+1)}~= \\nonumber \\\\[3.5mm]\n& = & \\frac{(p''+|m'|+1)_{\\alpha}}{\\Gamma(\\alpha+1)}=\\frac{\\Gamma(q''+\\alpha+1)}{\\Gamma(q''+1)\\,\\Gamma(\\alpha+1)} =\\Bigl(\\!\\begin{array}{c} q''+\\alpha \\\\ q \\end{array}\\!\\Bigr)~,\n\\end{eqnarray}\nin which $q''=p''+|m'|=\\frac12\\,(n+|m'|)$. Now replace $m'$ by $m$ in (\\ref{e64}) and (\\ref{e65}), and (\\ref{e55}) results. \\\\ \\\\\n{\\bf Notes.} \\\\[-7mm]\n\\bi{1.0}\n\\ITEM{1.} Theorem 5.1 generalizes the case $\\alpha=0$ in (\\ref{e14}) to general $\\alpha>{-}1$. A further generalization, to orthogonal functions on spheres of general dimension $N$ instead of disks, is provided in \\cite{refX}, Theorem 3.1. The proof of Theorem~5.1 as given here follows rather closely the approach of Cormack in \\cite{ref37}--\\cite{ref38} which differs from the approach used in \\cite{refX}.\n\\ITEM{2.} Theorem 5.2 generalizes the case $\\alpha=0$ of the integral representation of the $R_n^{|m|}$ in \\cite{ref39}, (A.10) to the case of general $\\alpha>{-}1$. Furthermore, for a fixed $\\rho\\in(0,1)$, the formula (\\ref{e55}) can be discretized to\n\\begin{equation} \\label{e66}\n\\Bigl(\\!\\begin{array}{c} q+\\alpha \\\\ q \\end{array}\\!\\Bigr)(1-\\rho^2)^{-\\alpha}\\,R_n^{|m|,\\alpha}(\\rho)=\n\\frac1N\\,\\sum_{k=0}^N\\,C_n^{\\alpha+1}\\Bigl(\\rho\\cos\\frac{2\\pi k}{N}\\Bigr)\\,e^{-2\\pi i\\frac{k}{N}}\n\\end{equation}\nwhen $N$ is an integer $>\\:n+|m|$. This yields a method of the DFT-type to compute the radial parts fast and reliably.\n\\ITEM{3.} The integral representation in (\\ref{e55}) is an excellent starting point to derive the asymptotics of the radial parts, by stationary phase methods, etc., when $n,|m|\\rightarrow\\infty$ such that $n\/|m|\\rightarrow\\kappa\\in(0,1)$, with $\\alpha$ fixed and $\\rho$ bounded away from 0 and 1. See \\cite{ref50}, ENZ document, Sec.~7, item~4.\n\\end{inspring}\n\n\\section{Scaling theory for generalized Zernike circle functions} \\label{sec6}\n\\mbox{} \\\\[-9mm]\n\nScaling theory for generalized Zernike circle functions is compromised by the occurrence of the factor $(1-\\rho^2)^{\\alpha}$. In the first place, one has to restrict the scaling parameter $\\varepsilon$ in $Z_n^{m,\\alpha}(\\varepsilon\\rho,\\vartheta)$ to the range $0\\leq\\varepsilon<1$. While this restriction is quite natural, the scaling results, such as (\\ref{e17}), for the case that $\\alpha=0$ allows unrestricted values of $\\varepsilon$ due to polynomial form of the radial parts. Furthermore, the value of $Z_n^{m,\\alpha}(\\varepsilon\\rho,\\vartheta)$ at $\\rho=1$ (with $0\\leq\\varepsilon<1$) is in general finite and unequal to 0. This implies that the only natural candidate among the systems $(Z_{n'}^{m',\\alpha'} (\\rho,\\vartheta))_{n',m'}$ as expansion set for $Z_n^{m,\\alpha}(\\varepsilon\\rho,\\vartheta)$ is the case that $\\alpha'=0$. Finally, an extension to shift-and-scaling theory, as in \\cite{ref40}, seems also more cumbersome. Nevertheless, for the radial part there is the following result. \\\\ \\\\\n{\\bf Theorem 6.1.}~~Let $\\alpha>{-}1$, $0\\leq\\varepsilon<1$, and let $n$, $m$ be non-negative integers such that $p=\\frac12\\,(n-m)$ is a non-negative integer. Then\n\\begin{equation} \\label{e67}\nR_n^{m,\\alpha}(\\varepsilon\\rho)=\\sum_{n'=m,m+2,...}\\,C_{nn'}^{m,\\alpha}(\\varepsilon)\\,R_{n'}^m(\\rho)~,~~~~~~0\\leq\\rho<1~,\n\\end{equation}\nin which the $C$'s can be expressed as the sum of two hypergeometric functions ${}_2F_1$. Furthermore,\n\\begin{eqnarray} \\label{e68}\n& \\mbox{} & C_{nn'}^{m,\\alpha}(\\varepsilon)=\\frac{(p+1)_{\\alpha}}{(p'+1)_{\\alpha}}\\,\\Bigl(R_n^{n',\\alpha}(\\varepsilon)-\\frac{p'+\\alpha}{p'}\\, R_n^{n'+2,\\alpha}(\\varepsilon)\\Bigr)~, \\nonumber \\\\[3mm]\n& & \\hspace*{5.2cm}n'=m,m+2,...,n-2~,\n\\end{eqnarray}\nwhere $p'=\\frac12\\,(n-n')$. Finally, when $\\alpha$ is a non-negative integer, it holds that\n\\begin{equation} \\label{e69}\nC_{nn'}^{m,\\alpha}(\\varepsilon)=0~,~~~~~~n'=n+2\\alpha+2,n+2\\alpha+4,...~.\n\\end{equation}\n{\\bf Proof.}~~By the orthogonality condition (\\ref{e31}) for the case $\\alpha=0$, there holds for $n'=m,m+2,...$\n\\begin{equation} \\label{e70}\nC_{nn'}^{m,\\alpha}(\\varepsilon)=2(n'+1)\\,\\int\\limits_0^1\\,R_n^{m,\\alpha}(\\varepsilon\\rho)\\,R_{n'}^m(\\rho)\\,\\rho\\,d\\rho~.\n\\end{equation}\nNow, by (\\ref{e36}) and, subsequently (\\ref{e35}) with $\\alpha=0$, $2\\pi r=\\varepsilon t$,\n\\begin{eqnarray} \\label{e71}\n& \\mbox{} & \\int\\limits_0^1\\,R_n^{m,\\alpha}(\\varepsilon\\rho)\\,R_{n'}^m(\\rho)\\,\\rho\\,d\\rho~= \\nonumber \\\\[3.5mm]\n& & =~({-}1)^p\\,2^{\\alpha}(p+1)_{\\alpha}\\,\\int\\limits_0^1\\,\\left(\\int\\limits_0^{\\infty}\\, \\frac{J_{n+\\alpha+1}(t)\\,J_m(\\varepsilon\\rho t)} {t^{\\alpha}}\\,dt\\right)\\,R_{n'}^m(\\rho)\\,\\rho\\,d\\rho~= \\nonumber \\\\[3.5mm]\n& & =~({-}1)^p\\,2^{\\alpha}(p+1)_{\\alpha}\\,\\int\\limits_0^{\\infty}\\,\\frac{J_{n+\\alpha+1}(t)}{t^{\\alpha}}\\, \\left(\\int\\limits_0^1\\,R_{n'}^m(\\rho)\\,J_m(\\varepsilon\\rho t)\\,\\rho\\,d\\rho\\right)\\,dt~= \\nonumber \\\\[3.5mm]\n& & =~({-}1)^p\\,2^{\\alpha}(p+1)_{\\alpha}({-}1)^{(n'-m)\/2}\\,\\int\\limits_0^{\\infty}\\,\\frac{J_{n+\\alpha+1}(t)\\, J_{n'+1}(\\varepsilon t)} {t^{\\alpha}\\cdot\\varepsilon t}\\,dt~.\n\\end{eqnarray}\nUsing \\cite{ref14}, first item in 9.1.27,\n\\begin{equation} \\label{e72}\n\\frac{J_{n'+1}(z)}{z}=\\frac{1}{2(n'+1)}\\,(J_{n'}(z)+J_{n'+2}(z))~,\n\\end{equation}\nit is then found that\n\\begin{equation} \\label{e73}\nC_{nn'}^{m,\\alpha}(\\varepsilon)=({-}1)^{(n+n'-2m)\/2}\\,2^{\\alpha}(p+1)_{\\alpha}(I_{nn'}(\\alpha,\\varepsilon)+I_{n,n'+2}(\\alpha, \\varepsilon))~,\n\\end{equation}\nwhere\n\\begin{equation} \\label{e74}\nI_{nn''}(\\alpha,\\varepsilon)=\\int\\limits_0^{\\infty}\\,\\frac{J_{n+\\alpha+1}(t)\\,J_{n''}(\\varepsilon t)}{t^{\\alpha}}\\,dt~.\n\\end{equation}\nNow, for $n''=m,m+2,...,n\\,$, there holds by (\\ref{e35})\n\\begin{equation} \\label{e75}\nI_{nn''}(\\alpha,\\varepsilon)=\\frac{({-}1)^{p''}}{2^{\\alpha}(p''+1)_{\\alpha}}\\,R_n^{n'',\\alpha}(\\varepsilon)~,\n\\end{equation}\nwhere $p''=\\frac12\\,(n-n'')$. For $n''=n+2,n+4,...\\,$, there holds by (\\ref{e38})\n\\begin{equation} \\label{e76}\nI_{nn''}(\\alpha,\\varepsilon)=\\frac{\\varepsilon^m\\,\\Gamma(n''+p''+1)}{2^{\\alpha}\\,\\Gamma(n''+1)\\,\\Gamma(p''+\\alpha+1)} \\,F({-}p''-\\alpha,n''+p''+1\\,;\\,n''+1\\,;\\,\\varepsilon^2)~,\n\\end{equation}\nwhere again $p''=\\frac12\\,(n-n'')$. When $\\alpha$ is a non-negative integer, (\\ref{e76}) vanishes when $p''+\\alpha$ is a negative integer, i.e., when\n\\begin{equation} \\label{e77}\nn''=n+2\\alpha+2,n+2\\alpha+4,...~.\n\\end{equation}\nWhen $\\alpha$ is not an integer, infinitely many of the $I_{nn''}$ must be expected to be non-vanishing.\n\nFrom (\\ref{e73}) and (\\ref{e75})it is found for $n'=m,m+2,...,n-2$ that\n\\begin{eqnarray} \\label{e78}\nC_{nn'}^{m,\\alpha} & = & ({-}1)^{(n+n'-2m)\/2}\\,2^{\\alpha}(p+1)_{\\alpha}~\\cdot \\nonumber \\\\[3.5mm]\n& & \\cdot~\\left[\\frac{({-}1)^{(n-n')\/2}}{2^{\\alpha}\\Bigl(\\dfrac{n-n'}{2}+1\\Bigr)_{\\alpha}}\\, R_n^{n',\\alpha}(\\varepsilon) + \\frac{({-}1)^{(n-n'-2)\/2}}{2^{\\alpha}\\Bigl(\\dfrac{n-n'}{2}\\Bigr)_{\\alpha}}\\,R_n^{n'+2,\\alpha}(\\varepsilon)\\right]~= \\nonumber \\\\[3.5mm]\n& = & \\frac{(p+1)_{\\alpha}}{(p'+1)_{\\alpha}}\\,\\Bigl(R_n^{n',\\alpha}(\\varepsilon)-\\frac{p'+\\alpha}{p'}\\, R_n^{n'+2,\\alpha} (\\varepsilon)\\Bigr)~,\n\\end{eqnarray}\nwhere $p'=\\frac12\\,(n-n')$ and the proof is complete.\n\n\\section{Forward computation schemes for generalized Zernike circle polynomials from ordinary ENZ and ANZ} \\label{sec7}\n\\mbox{} \\\\[-9mm]\n\nIn this section, the generalized Zernike circle functions $Z_n^{m,\\alpha}$ are expanded with respect to the system $(Z_{n'}^{m'})_{m',n'}$ of classical circle polynomials. The azimuthal dependence is in all cases through the factor $\\exp(im\\vartheta)$, and this allows restriction of the attention to the radial parts only. \\\\ \\\\\n{\\bf Theorem 7.1.}~~Let $\\alpha>{-}1$ and let $n$, $m$ be non-negative integers such that $p=\\frac12\\,(n-m)$ is a non-negative integer. Then\n\\begin{equation} \\label{e79}\nR_n^{m,\\alpha}(\\rho)=\\sum_{k=0}^{\\infty}\\,C_k\\,R_{m+2k}^m(\\rho)~,~~~~~~0\\leq\\rho<1~,\n\\end{equation}\nwhere $C_k=0$ for $k=0,1,...,p-1$ and\n\\begin{eqnarray} \\label{e80}\nC_k & = & ({-}1)^{k-p}\\,\\frac{m+2k+1}{m+k+p+\\alpha+1}\\,\\Bigl(\\!\\begin{array}{c} m+p+k \\\\ p \\end{array}\\!\\Bigr) \\Bigl(\\!\\begin{array}{c} \\alpha \\\\ k-p \\end{array}\\!\\Bigr)\\,\/ \\nonumber \\\\[3.5mm]\n& & \/\\,\\Bigl(\\!\\begin{array}{c} m+k+p+\\alpha \\\\ m+k \\end{array}\\!\\Bigr)~= \\nonumber \\\\[3.5mm]\n& = & \\frac{m+2k+1}{m+k+p+\\alpha+1}~\\frac{({-}\\alpha)_{k-p}}{(k-p)!}~\\frac{(p+1)_{\\alpha}}{(m+k+p+1)_{\\alpha}}\n\\end{eqnarray}\nwhen $k=p,p+1,...~$. \\\\[2mm]\n{\\bf Proof.}~~By the orthogonality condition (\\ref{e31}) with $\\alpha=0$, there holds\n\\begin{equation} \\label{e81}\nC_k=2(m+2k+1)\\,\\int\\limits_0^1\\,R_n^{m,\\alpha}(\\rho)\\,R_{m+2k}^m(\\rho)\\,\\rho\\,d\\rho~.\n\\end{equation}\nUsing the definition (\\ref{e26}) of $R_{n=m+2p}^{m,\\alpha}$ and $R_{m+2k}^m$, and using in the integral in (\\ref{e81}) the substitution\n\\begin{equation} \\label{e82}\nx=2\\rho^2-1\\in[{-}1,1]\\,,~~\\rho^2=\\tfrac12\\,(1+x)\\,,~~1-\\rho^2=\\tfrac12\\,(1-x)\\,,~~\\rho\\,d\\rho=\\tfrac14\\,dx~,\n\\end{equation}\nthis becomes\n\\begin{eqnarray} \\label{e83}\n& \\mbox{} & \\hspace*{-5mm}C_k~= \\nonumber \\\\[3mm]\n& & \\hspace*{-5mm}=~2(m+2k+1)\\,\\int\\limits_0^1\\,\\rho^{2m}(1-\\rho^2)^{\\alpha}\\,P_p^{(\\alpha,m)}(2\\rho^2-1)\\,P_k^{(0,m)}(2\\rho^2-1)\\, \\rho\\,d\\rho~= \\nonumber \\\\[3.5mm]\n& & \\hspace*{-5mm}=~\\frac{m+2k+1}{2^{m+\\alpha+1}}\\,\\int\\limits_{-1}^1\\,(1-x)^{\\alpha}\\,(1+x)^m\\,P_p^{(\\alpha,m)}(x)\\,P_k^{(0,m)}(x)\\,dx~.\n\\end{eqnarray}\nBy Rodriguez' formula, see \\cite{ref15}, (4.3.1) or \\cite{ref16}, p.~161, there holds\n\\begin{equation} \\label{e84}\n(1-x)^{\\alpha}\\,(1+x)^m\\,P_p^{(\\alpha,m)}(x)=\\frac{({-}1)^p}{2^p\\,p!}\\,\\Bigl(\\frac{d}{dx}\\Bigr)^p\\, [(1-x)^{\\alpha+p}(1+x)^{m+p}]~.\n\\end{equation}\nThen, by $p$ partial integrations from (\\ref{e83}) and (\\ref{e84}),\n\\begin{equation} \\label{e85}\nC_k=\\frac{m+2k+1}{2^{m+p+\\alpha+1}\\,p!}\\,\\int\\limits_{-1}^1\\,(1-x)^{p+\\alpha}(1+x)^{p+m}\\Bigl(\\frac{d}{dx}\\Bigr)^p\\,P_k^{(0,m)} (x)\\,dx~.\n\\end{equation}\nNext, by using, see \\cite{ref15}, (4.2.17) or \\cite{ref16}, (4.14),\n\\begin{equation} \\label{e86}\n\\frac{d}{dx}\\,P_n^{(\\alpha,\\beta)}(x)=\\tfrac12\\,(n+\\alpha+\\beta+1)\\,P_{n-1}^{(\\alpha+1,\\beta+1)}(x)\n\\end{equation}\nrepeatedly, it is found that for $k\\geq p$\n\\begin{equation} \\label{e87}\n\\Bigl(\\frac{d}{dx}\\Bigr)^p\\,P_k^{(0,m)}(x)=\\frac{1}{2^p}~\\frac{(k+m+p)!}{(k+m)!}\\,P_{k-p}^{(p,m+p)}(x)~,\n\\end{equation}\nwhile this vanishes for $k{-}1$, and let $n$, $m$ be non-negative integers such that $n-m$ is even and non-negative. Then the through-focus point-spread function $U_n^{m,\\alpha}(r,\\varphi\\,;\\,f)$ corresponding to $Z_n^{m,\\alpha}(\\rho,\\vartheta)$, see (\\ref{e6}), is given by\n\\begin{equation} \\label{e94}\nU_n^{m,\\alpha}(r,\\varphi\\,;\\,f)=2\\pi\\,i^m\\,e^{im\\varphi}\\,\\sum_{k=0}^{\\infty}\\,C_k\\,V_{m+2k}^m(r,f)\n\\end{equation}\nwith $V_{m+2k}^m$ given in semi-analytic form in (\\ref{e12}) and $C_k$ given in (\\ref{e80}). \\\\[2mm]\n{\\bf Proof.}~~Just insert the expansion (\\ref{e79}) into the integral\n\\begin{equation} \\label{e95}\n\\int\\limits_0^1\\int\\limits_0^{2\\pi}\\,e^{if\\rho^2}\\,e^{2\\pi i\\rho r\\cos(\\vartheta-\\varphi)}\\,Z_n^{m,\\alpha}(\\rho,\\vartheta)\\,\\rho\\,d\\rho\\,d\\vartheta\n\\end{equation}\nfor $U_n^{m,\\alpha}$ and use (\\ref{e11}). \\\\ \\\\\n{\\bf Notes.} \\\\[-7mm]\n\\bi{1.0}\n\\ITEM{1.} In a similar fashion, the on-axis pressure $p_{2i}^{0,\\alpha}((0,0,z),k)$ due to the radially symmetric velocity profile $v(\\sigma)=Z_{2i}^{0,\\alpha}(\\sigma\/a,\\vartheta)$, see (\\ref{e18}), can be obtained from the on-axis pressures $p_{2j}((0,0,z),k)$ due to $Z_{2j}^0(\\sigma\/a,0)$, given in (\\ref{e20}), and the coefficients $C_j$, case $m=0$, in (\\ref{e80}).\n\\ITEM{2.} The availability of the through-focus point-spread functions $U_n^{m,\\alpha}$ per Theorem~7.2 makes it possible to do aberration retrieval, with pupil functions expanded in generalized circle functions, in the same spirit as this is done in the ordinary ENZ theory, see Sec.~\\ref{sec2}. Similarly, radially symmetric velocity profiles, expanded into radially symmetric generalized circle functions, can be retrieved with the same approach that is used in the ordinary ANZ theory, see \\cite{ref19}, Sec.~V.\n\\end{inspring}\n\n\\section{Acoustic quantities for baffled-piston radiation from King's integral with generalized circle functions as velocity profiles} \\label{sec8}\n\\mbox{} \\\\[-9mm]\n\nIn this section various acoustic quantities that arise from baffled-piston radiation with a velocity profile that is expanded into generalized Zernike circle functions are computed in semi-analytic form. The starting point is King's integral, second integral expression in (\\ref{e18}), in which $V(u)$ is the Hankel transform (\\ref{e19}) of order 0 of $v(\\sigma)$. Having expanded $v(\\sigma)$ in radially symmetric circle functions $Z_{2j}^{0,\\alpha}=R_{2j}^{0,\\alpha}$, the Hankel transforms\n\\begin{eqnarray} \\label{96}\nV_{2j}^{0,\\alpha}(u) & = & \\int\\limits_0^a\\,R_{2j}^{0,\\alpha}(\\sigma\/a)\\,J_0(u\\sigma)\\,\\sigma\\,d\\sigma~= \\nonumber \\\\[3.5mm]\n& = & ({-}1)^j\\,a^2\\,2^{\\alpha}(j+1)_{\\alpha}\\,\\frac{J_{2j+\\alpha+1}(au)}{(au)^{\\alpha+1}}~,~~~~~~j=0,1,...~,\n\\end{eqnarray}\nsee (\\ref{e35}) arise.\n\nThe acoustic quantities considered are \\\\\n-- pressure $p((a,0,0)\\,;\\,k)$ at edge of the radiator, \\\\[2mm]\n-- reaction force $F=\\int\\limits_S\\,p\\,dS$ on the radiator, \\\\\n-- the power output $P=\\int\\limits_S\\,p\\,v^{\\ast}\\,dS$ of the radiator.\n\nBy taking $z=0$, $w=a$ in King's integral, it is seen that\n\\begin{equation} \\label{e97}\np_{{\\rm edge}}=p((a,0,0)\\,;\\,k)=i\\rho_0ck\\,\\int\\limits_0^{\\infty}\\,\\frac{J_0(au)\\,V(u)}{(u^2-k^2)^{1\/2}}\\,u\\,du~.\n\\end{equation}\nIt was shown in \\cite{ref20}, Sec.~III from the integral result\n\\begin{equation} \\label{e98}\n\\int\\limits_0^a\\,J_0(\\sigma u)\\,\\sigma\\,d\\sigma=au^{-1}J_1(au)\n\\end{equation}\nby taking $z=0$ in King's integral and integrating over $w$, $0\\leq w{-}1$ and let $n$, $m$ be non-negative integers such that $n-m$ is non-negative and even. Then\n\\begin{eqnarray} \\label{e103}\n& \\mbox{} & i\\,\\int\\limits_0^{\\infty}\\,\\frac{J_{m+\\beta}(au)\\,J_{n+\\gamma+1}(au)}{(u^2-k^2)^{1\/2}\\,u^{\\beta+\\gamma}}\\,du~= \\nonumber \\\\[3.5mm]\n& & =~\\frac{-({-}1)^p\\,a^{2\\varepsilon}}{2ka}\\,\\sum_{l=1}^{\\infty}\\,\\frac{(\\tfrac12\\,l+\\tfrac12)_{\\varepsilon}} {(\\tfrac12\\,l+\\varepsilon)_{\\delta}}~\\frac{({-}\\tfrac12\\,l+1-\\beta)_p({-}\\tfrac12\\,l+1)_q({-}ika)^l}{\\Gamma(\\tfrac12\\,l+p+\\gamma+1) \\,\\Gamma(\\tfrac12\\,l+q+2\\varepsilon+1)}~, \\nonumber \\\\[1mm]\n& \\mbox{} &\n\\end{eqnarray}\nwhere\n\\begin{equation} \\label{e104}\n\\varepsilon=\\tfrac12\\,(\\beta+\\gamma)\\,,~~\\delta=\\tfrac12\\,(\\beta-\\gamma)\\,,~~p=\\tfrac12\\,(n-m)\\,,~~q=\\tfrac12\\,(n+m)~.\n\\end{equation}\n{\\bf Proof.}~~The plan of the proof is entirely the same as that of the proofs of the results in \\cite{ref20}, Appendix~A, and so only the main steps with key intermediate results are given.\n\nWith $(u^2-k^2)^{1\/2}$ as defined below (\\ref{e18}) there holds\n\\begin{eqnarray} \\label{e105}\n& \\mbox{} & i\\,\\int\\limits_0^{\\infty}\\,\\frac{J_{m+\\beta}(au)\\,J_{n+\\gamma+1}(au)}{(u^2-k^2)^{1\/2}\\,u^{\\beta+\\gamma}}\\,du~= \\nonumber \\\\[3.5mm]\n& & =~\\int\\limits_0^k\\,\\frac{J_{m+\\beta}(au)\\,J_{n+\\gamma+1}(au)}{u^{\\beta+\\gamma}\\,\\sqrt{k^2-u^2}}\\,du+i\\,\\int\\limits_k^{\\infty}\\, \\frac{J_{m+\\beta}(au)\\,J_{n+\\gamma+1}(au)}{u^{\\beta+\\gamma}\\,\\sqrt{u^2-k^2}}\\,du~= \\nonumber \\\\[3.5mm]\n& & =~I_1+I_2~.\n\\end{eqnarray}\nAs to the integral $I_1$ in (\\ref{e105}), the product of the two Bessel functions is written as an integral, see \\cite{ref52}, beginning of \\S13.61, and \\cite{ref20}, (A8), from $-\\infty i$ to $\\infty i$, the order of integration is reversed and it is used that\n\\begin{equation} \\label{e106}\n\\int\\limits_0^k\\,\\frac{u^{2q+2s+1}}{\\sqrt{k^2-u^2}}\\,du=k^{2q+2s+1}\\,\\frac{\\Gamma(q+1+s)\\,\\Gamma(\\tfrac12)} {\\Gamma(q+\\tfrac32+s)}\n\\end{equation}\nto obtain\n\\begin{eqnarray} \\label{e107}\n& \\mbox{} & \\hspace*{-7mm}I_1=\\frac{\\tfrac12\\,\\Gamma(\\tfrac12)}{2\\pi i}\\,(\\tfrac12\\,a)^{2\\varepsilon}\\,(\\tfrac12\\,ka)^{2q}~\\cdot \\nonumber \\\\[3.5mm]\n& & \\hspace*{-7mm}\\cdot~\\int\\limits_{-\\infty i}^{\\infty i}\\,\\frac{\\Gamma({-}s)\\,\\Gamma(2q+2\\varepsilon+2s+2)\\,\\Gamma(q+1+s)(\\tfrac12\\,ka)^{2s+1}} {\\Gamma(m+\\beta+s+1)\\,\\Gamma(n+\\gamma+s+2)\\,\\Gamma(2q+2\\varepsilon+s+2)\\,\\Gamma(q+s+\\tfrac32)}\\,ds~. \\nonumber \\\\\n& \\mbox{} &\n\\end{eqnarray}\nThe choice of the integration contour is such that it has all poles of $\\Gamma({-}s)$ on its right and all poles of $\\Gamma(2q+2\\varepsilon+2s+2)$ on its left (this is possible since $q\\geq0$ and $\\beta+\\gamma>{-}2$). Closing the contour to the right, thereby enclosing all poles of $\\Gamma({-}s)$ at $s=j=0,1,...$ with residues $({-}1)^{j+1}\/j!$, and using the duplication formula of the $\\Gamma$-function to write\n\\begin{equation} \\label{e108}\n\\Gamma(2q+2\\varepsilon+2j+2)=\\frac{2^{2q+2\\varepsilon+2j+1}}{\\Gamma(\\tfrac12)}\\,\\Gamma(q+\\varepsilon+j+1)\\,\\Gamma(q+\\varepsilon+j+\\tfrac32)~,\n\\end{equation}\nit follows that\n\\begin{eqnarray} \\label{e109}\nI_1 & = & \\tfrac12\\,a^{2\\varepsilon}\\,\\sum_{j=0}^{\\infty}\\,({-}1)^j\\,\\frac{\\Gamma(q+j+1)}{j!}~\\frac{\\Gamma(q+\\varepsilon+j+\\tfrac32)} {\\Gamma(q+j+\\tfrac32)}~\\frac{\\Gamma(q+\\varepsilon+j)}{\\Gamma(m+\\beta+j+1)}~\\cdot \\nonumber \\\\[3.5mm]\n& & \\cdot~\\frac{(ka)^{2(q+j)+1}}{\\Gamma(n+\\gamma+j+2)\\,\\Gamma(2q+2\\varepsilon+j+2)}~.\n\\end{eqnarray}\nReplacing $j+q+1$ by $j=q+1,q+2,...\\,$, it is found after some administration with Pochhammer symbols (such as $(x-q)_q=({-}1)^q(1-x)_q$) that\n\\begin{equation} \\label{e110}\nI_1=\\frac{-({-}1)^p\\,a^{2\\varepsilon}}{2ka}\\,\\sum_{j=q+1}^{\\infty}\\,\\frac{(j+\\tfrac12)_{\\varepsilon}}{(j+\\varepsilon)_{\\delta}}~\\frac{({-}j+1)_q ({-}j+1-\\beta)_p({-}ika)^{2j}}{\\Gamma(j+p+\\gamma+1)\\,\\Gamma(j+q+2\\varepsilon+1)}~.\n\\end{equation}\nHere it may be observed that $({-}j+1)_q=0$ for $j=1,...,q\\,$, so that the summation in (\\ref{e110}) could start at $j=1$ as well.\n\nAs to the integral $I_2$ in (\\ref{e105}), again the integral representation for the product of two Bessel functions is used, the integration order is reversed, and it is used that\n\\begin{equation} \\label{e111}\n\\int\\limits_k^{\\infty}\\,\\frac{u^{2q+2s+1}}{\\sqrt{u^2-k^2}}\\,du=\\tfrac12\\,k^{2q+2s+1}\\,\\frac{\\Gamma(\\tfrac12)\\, \\Gamma({-}q-s-\\tfrac12)}{\\Gamma({-}q-s)}~.\n\\end{equation}\nThis yields\n\\begin{eqnarray} \\label{e112}\n& \\mbox{} & \\hspace*{-7mm}I_2=\\tfrac12\\,i\\,\\Gamma(\\tfrac12)(\\tfrac12\\,a)^{2\\varepsilon}~\\cdot \\nonumber \\\\[3.5mm]\n& & \\hspace*{-7mm}\\cdot\\,\\int\\limits_{-\\infty i}^{\\infty i}\\,\\frac{\\Gamma({-}s)}{\\Gamma({-}q-s)}~\\frac{\\Gamma(2q+2\\varepsilon+2s+2) \\,\\Gamma({-}q-s-\\tfrac12)(\\tfrac12\\,ka)^{2s+2q+1}} {\\Gamma(m+\\beta+s+1)\\,\\Gamma(n+\\gamma+s+2)\\,\\Gamma(2q+2\\varepsilon+s+2)}\\, ds~. \\nonumber \\\\\n& \\mbox{} &\n\\end{eqnarray}\nThe factor $\\Gamma({-}s)\/\\Gamma({-}q-s)$ is a polynomial since $q=0,1,...\\,$, and so the integrand has its poles at $s=j-q-\\tfrac12\\,$, $j=0,1,...\\,$, and at $s={-}r-q-\\varepsilon-1$, $r=0,1,...\\,$. Since $\\varepsilon=(\\beta+\\gamma)\/2>{-}1$, the integration contour can be chosen such that all poles $j-q-\\tfrac12\\,$, with residues $({-}1)^{j+1}\/j!$, lie to the right of it while all poles ${-}r-q-\\varepsilon-1$ lie to the left of it. Closing the contour to the right, and using the duplication formula again, to write\n\\begin{equation} \\label{e113}\n\\Gamma(2j+2\\varepsilon+1)=\\frac{2^{2j+2\\varepsilon}}{\\Gamma(\\tfrac12)}\\,\\Gamma(j+\\varepsilon+\\tfrac12)\\,\\Gamma(j+\\varepsilon+1)~,\n\\end{equation}\nit follows that\n\\begin{eqnarray} \\label{e114}\nI_2 & = & \\tfrac12\\,i\\,a^{2\\varepsilon}\\,\\sum_{j=0}^{\\infty}\\,({-}1)^j\\,\\frac{\\Gamma(j+1+\\varepsilon)}{\\Gamma(j+1)}~\\frac{\\Gamma({-}j+\\tfrac12+q)} {\\Gamma({-}j+\\tfrac12)}~\\frac{\\Gamma(j+\\varepsilon+\\tfrac12)}{\\Gamma(j+\\beta+\\tfrac12-p)}~\\cdot \\nonumber \\\\[3.5mm]\n& & \\cdot~\\frac{(ka)^{2j}}{\\Gamma(j+\\gamma+\\tfrac32+p)\\,\\Gamma(j+2\\varepsilon+\\tfrac32+q)}~.\n\\end{eqnarray}\nThen some administration with Pochhammer symbols (such as $\\Gamma(x)\/\\Gamma(x-p)=({-}1)^p(1-x)_p$) yields\n\\begin{equation} \\label{e115}\nI_2=\\frac{-({-}1)^p\\,a^{2\\varepsilon}}{2ka}\\,\\sum_{j=0}^{\\infty}\\,\\frac{(j+1)_{\\varepsilon}} {(j+\\varepsilon+\\tfrac12)_{\\delta}}~\\frac{({-}j+\\tfrac12)_q({-}j-\\beta+\\tfrac12)_p({-}ika)^{2j+1}} {\\Gamma(j+p+\\gamma+\\tfrac32)\\,\\Gamma(j+q+2\\varepsilon+\\tfrac32)}~.\n\\end{equation}\n\nThe result in (\\ref{e103}) is now obtained by adding $I_1$ in (\\ref{e110}), with summation starting at $j=1$, and $I_2$ in (\\ref{e115}), while observing that the terms $j$ in (\\ref{e110}) yield the terms in (\\ref{e103}) with even $l=2j$, $j=1,2,...\\,$, and that the terms $j$ in (\\ref{e115}) yield the terms in (\\ref{e103}) with odd $l=2j+1$, $j=0,1,...\\,$.\n\n\\section{Estimating generalized velocity profiles in baffled-piston radiation from near-field pressure data via Weyl's formula} \\label{sec9}\n\\mbox{} \\\\[-9mm]\n\nIn this section a brief sketch is given of how one can estimate, in the setting of baffled-piston radiation, a not necessarily radially symmetric velocity profile from near-field pressure data. The starting point is the Rayleigh integral for the pressure, first integral expression in (\\ref{e18}), that is written in normalized form as\n\\begin{equation} \\label{e116}\np(\\nu,\\mu\\,;\\,\\zeta)=\\int\\limits_{~S}\\hspace*{-2mm}\\int\\limits\\,v(\\nu',\\mu')\\,\\frac{e^{ikar'}}{r'}\\,d\\nu'\\,d\\mu'~,\n\\end{equation}\nwhere\n\\begin{equation} \\label{e117}\nr'=((\\nu-\\nu')^2+(\\mu-\\mu')^2+\\zeta^2)^{1\/2}\n\\end{equation}\nis the distance from the field point $(\\nu,\\mu,\\zeta)$, with $\\zeta\\geq0$, to the point $(\\nu',\\mu',0)$ on the radiating surface $S$, for which we take the unit disk. For a fixed value of $\\zeta>0$, the equation (\\ref{e116}) can be written as\n\\begin{equation} \\label{e118}\np(\\nu,\\mu\\,;\\,\\zeta)=(v\\,{\\ast\\ast}\\,W({\\cdot},{\\cdot}\\,;\\,\\zeta))(\\nu,\\mu)~,\n\\end{equation}\nwhere $\\ast\\ast$ denotes $2D$ convolution and\n\\begin{equation} \\label{e119}\nW(\\nu,\\mu\\,;\\,\\zeta)=\\frac{\\exp[ika(\\zeta^2+\\nu^2+\\mu^2)^{1\/2}]}{(\\zeta^2+\\nu^2+\\mu^2)^{1\/2}}~.\n\\end{equation}\nUsing the Fourier transform ${\\cal F}$, defined as\n\\begin{equation} \\label{e120}\n({\\cal F}q)(x,y)=\\int\\limits_{-\\infty}^{\\infty}\\int\\limits_{-\\infty}^{\\infty}\\,e^{2\\pi i\\nu x+2\\pi i\\mu y}\\,q(\\nu,\\mu)\\,d\\nu\\,d\\mu~,~~~~~~x,y\\in{\\Bbb R}~,\n\\end{equation}\nthe formula (\\ref{e118}) can be written as\n\\begin{equation} \\label{e121}\n{\\cal F}\\,[p({\\cdot},{\\cdot}\\,;\\,\\zeta)]={\\cal F}v\\cdot{\\cal F}\\,[W({\\cdot},{\\cdot}\\,;\\,\\zeta)]~.\n\\end{equation}\nBy Weyl's result on the representation of spherical waves, see \\cite{ref18}, Sec.~13.2.1, there holds\n\\begin{equation} \\label{e122}\n{\\cal F}\\,[W({\\cdot},{\\cdot}\\,;\\,\\zeta)](x,y)=2\\pi i\\, \\frac{\\exp\\,[i\\zeta((ka)^2-(2\\pi x)^2-(2\\pi y)^2)^{1\/2}]} {((ka)^2-(2\\pi x)^2-(2\\pi y)^2)^{1\/2}}~,\n\\end{equation}\nwith the same definition of the square root as the one that was used in connection with King's integral in (\\ref{e18}).\n\nNow assume that the unknown velocity profile vanishes outside the unit disk and that it has a $(1-\\nu^2-\\mu^2)^{\\alpha}$-behaviour at the edge of the unit disk, where $\\alpha>{-}1$. Then $v$ has an expansion\n\\begin{equation} \\label{e123}\nv(\\nu,\\mu)=\\sum_{n,m}\\,C_n^{m,\\alpha}\\,Z_n^{m,\\alpha}(\\nu,\\mu)\n\\end{equation}\nin generalized Zernike circle functions, with Fourier transform\n\\begin{equation} \\label{e124}\n{\\cal F}v=\\sum_{n,m}\\,C_n^{m,\\alpha}\\,{\\cal F}\\,Z_n^{m,\\alpha}~,\n\\end{equation}\nin which ${\\cal F}\\,Z_n^{m,\\alpha}$ is given explicitly in Sec.~\\ref{sec4}. Thus, when the pressure $p$ is measured in the near-field plane $(\\nu,\\mu\\,;\\,\\zeta)$ with $\\zeta$ fixed, one can estimate $v$ on the level of its expansion coefficients $C_n^{m,\\alpha}$ by adopting a matching approach in (\\ref{e121}), using Weyl's result in (\\ref{e122}) and the result of Sec.~\\ref{sec4} in (\\ref{e124}). \\\\ \\\\\n\n\\section{Comparison with trial functions as used in acoustic design by Mellow and K\\\"arkk\\\"ainen} \\label{sec10}\n\\mbox{} \\\\[-9mm]\n\nMellow and K\\\"arkk\\\"ainen are concerned with design problems in acoustic radiation from a resilient disk (radius $a$) in an infinite or finite baffle ($z=0$), see \\cite{ref45}--\\cite{ref46}. The front and rear pressure distributions $p_+$ and $p_-$, $p_-={-}p_+$, are assumed to be radially symmetric and to have the form\n\\begin{equation} \\label{e125}\n\\sum_{l=0}^{\\infty}\\,a_l(1-(\\sigma\/a)^2)^{l+1\/2}\n\\end{equation}\non the disk in accordance with the choice of trial functions used by Streng \\cite{ref42} which is based on the work of Bouwkamp \\cite{ref43}. In the case that the normal gradient of the pressure at $z=0$ is considered, as is done by Mellow in \\cite{ref44}, an expansion of the form\n\\begin{equation} \\label{e126}\n\\sum_{l=0}^{\\infty}\\,b_l(1-(\\sigma\/a)^2)^{l-1\/2}\n\\end{equation}\non the disk has to be considered.\n\nIn the design problem considered in \\cite{ref45}--\\cite{ref46}, the coefficients $a_l$ in (\\ref{e125}) are to be found such that the pressure gradient $\\frac{\\partial p}{\\partial z}\\,(w,z=0{+})$ equals a desired function $\\Phi(w)$ of the distance $w$ of a point in the baffle plane to the origin. In the design problem considered in \\cite{ref44}, the coefficients $b_l$ in (\\ref{e126}) are to be found such that $p(w,z=0{+})$ equals a desired function $\\Psi(w)$.\n\nThe pressure $p(w,z)$, $z\\geq0$ can be expressed in terms of the boundary data $p_+$, $p_-$ via the dipole version of King's integral. Similarly, via the common version of King's integral, the pressure $p(w,z)$, $z\\geq0$, can be expressed in terms of the normal gradient $\\frac{\\partial p}{\\partial z}\\,(w,0)$ of $p$ at $z=0$. Inserting the series expansion (\\ref{e125}) and (\\ref{e126}) into the appropriate version of King's integral, the integrals\n\\begin{equation} \\label{e127}\n\\int\\limits_0^{\\infty}\\,\\Bigl(\\frac{1}{\\mu}\\Bigr)^{l\\pm 1\/2}\\,J_0(w\\mu)\\,J_{l\\pm 1\/2+1}(a\\mu)\\,\\sigma^{\\pm 1}\\,d\\mu\n\\end{equation}\narise where $\\sigma={-}i(\\mu^2-k^2)^{1\/2}$ and where the $\\pm$ follows the sign choice in the exponent $l\\pm 1\/2$ of $(1-(\\sigma\/a)^2)$ in (\\ref{e125}) and (\\ref{e126}). \nTo obtain (\\ref{e127}), an explicit result, due to Sonine, for the Hankel transform of order 0 of the functions $(1-(\\sigma\/a)^2)^{l\\pm 1\/2}$ has been used. The integrals in (\\ref{e127}) are evaluated in the form of a double power series in $ka$ and $w\/a$ in \\cite{ref44}--\\cite{ref46}. Thus, having the pressure available in this semi-analytic form, comprising the coefficients $a_l$ or $b_l$, one can evaluate $\\frac{\\partial p}{\\partial z}\\,(w,z=0{+})$ and $p(w,z=0{+})$ and find the coefficients by requiring a best match\n with the desired function $\\Phi(w)$ or $\\Psi(w)$.\n\n In the approach of the present paper, the starting point would be an expansion of the form\n \\begin{equation} \\label{e128}\n \\sum_{l=0}^{\\infty}\\,c_l\\,R_{2l}^{0,{\\pm} 1\/2}(\\sigma\/a)\n \\end{equation}\n of the pressure ($+$-sign) or pressure gradient ($-$-sign) on the disk. Following the approach in \\cite{ref44}--\\cite{ref46}, using either form of King's integral, this gives rise to the integrals\n \\begin{equation} \\label{e129}\n \\int\\limits_0^{\\infty}\\,\\Bigl(\\frac{1}{\\mu}\\Bigr)^{\\pm 1\/2}\\,J_0(w\\mu)\\,J_{2l\\pm1\/2+1}(a\\mu)\\,\\sigma^{\\pm 1}\\,d\\mu~,\n \\end{equation}\n where now the result of Sec.~\\ref{sec4} on the Hankel transform of $R_{2l}^{0,{\\pm}1\/2}$ has been used. The integral in (\\ref{e129}) is of the same type as the one in (\\ref{e127}) and can be evaluated by the method given in \\cite{ref44}--\\cite{ref46}.\n\n\\subsection{Numerical considerations} \\label{subsec10.1}\n\\mbox{} \\\\[-9mm]\n\nIn either approach, it is required to find coefficients such that a best match occurs between the semi-analytically computed pressure gradient or pressure at $z=0{+}$, comprising the coefficients, and the desired functions $\\Phi$ or $\\Psi$. For any $L=1,2,...\\,$, the linear span of the function systems\n\\begin{equation} \\label{e130}\n\\{(1-(\\sigma\/a)^2)^{l\\pm 1\/2}\\,|\\,l=0,1,...,L-1\\}\n\\end{equation}\nand\n\\begin{equation} \\label{e131}\n\\{R_{2l}^{0,{\\pm}1\/2}(\\sigma\/a)\\,|\\,l=0,1,...,L-1\\}\n\\end{equation}\nis the same. So matching using the first $L$ functions in (\\ref{e125}), (\\ref{e126}) yields the same result for the best matching pressure gradient or pressure at $z=0{+}$ as matching using the first $L$ functions in (\\ref{e128}), in theory. For small values of $L$, one finds numerically practically the same result when either system in (\\ref{e130}), (\\ref{e131}) is used. In the case that large values $L$ of the number of coefficients to be matched are required, the approach based on (\\ref{e125}), (\\ref{e126}) is expected to experience numerical problems while the one based on (\\ref{e129}) is likely not to have such problems. This is due to the fact that the functions in (\\ref{e130}) are nearly linearly dependent while the ones in (\\ref{e131}), due to orthogonality, are not, and this is expected to remain so after the linear transformation associated with either version of King's integral. Furthermore, it is to be expected that the semi-analytic forms, used in the matching procedure, that arise from any of the terms $(1-(\\sigma\/a)^2)^{l\\pm1\/2}$ must be used with much higher truncation levels than those that arise from the terms $R_{2l}^{0,{\\pm}1\/2}(\\sigma\/a)$.\n\nAll this can be illustrated by comparing the expansion coefficients of a $(1-(\\sigma\/a)^2)^{k\\pm1\/2}$ with respect to the system in (\\ref{e131}) with those of an $R_{2p}^{0,{\\pm}1\/2}(\\sigma\/a)$ with respect to the system in (\\ref{e130}). There is the following general result. \\\\ \\\\\n{\\bf Theorem 10.1.}~~For $m=0,1,...\\,$, $\\alpha>{-}1$ and $k,p=0,1,...$ there holds\n\\begin{equation} \\label{e132}\n\\rho^m(1-\\rho^2)^{k+\\alpha}\\,e^{im\\vartheta}=\\sum_{l=0}^k\\,D_{m+2l,k}^{m,\\alpha}\\,Z_{m+2l}^{m,\\alpha}(\\rho,\\vartheta)~,\n\\end{equation}\n\\begin{equation} \\label{e133}\nZ_{m+2p}^{m,\\alpha}(\\rho,\\vartheta)=\\sum_{r=0}^p\\,E_{r,m+2p}^{m,\\alpha}\\,\\rho^m(1-\\rho^2)^{r+\\alpha}\\,e^{im\\alpha}~,\n\\end{equation}\nwhere\n\\begin{eqnarray} \\label{e134}\n& \\mbox{} & D_{m+2l,k}^{m,\\alpha}=\\frac{(\\alpha+1)_k}{(m+\\alpha+1)_k}~\\frac{m+2l+\\alpha+1}{m+k+l+\\alpha+1}~\\frac{({-}k)_l(m+\\alpha+1)_l} {(\\alpha+1)_l(m+k+\\alpha+1)_l}~, \\nonumber \\\\[2mm]\n& \\mbox{} &\n\\end{eqnarray}\n\\begin{equation} \\label{e135}\nE_{r,m+2p}^{m,\\alpha}=\\frac{(\\alpha+1)_p}{(1)_p}~\\frac{({-}p)_r(m+p+\\alpha+1)_r}{(\\alpha+1)_r(1)_r}~.\n\\end{equation}\n\\mbox{} \\\\\n{\\bf Proof.}~~The proof of (\\ref{e132}), (\\ref{e134}) is quite similar to the one of Theorem~7.1, and so only the main steps and key intermediate results are given. By orthogonality, see (\\ref{e31}), and the substitution in (\\ref{e82}), there holds\n\\begin{equation} \\label{e136}\nD_{m+2l,k}^{m,\\alpha}=\\frac{m+2l+\\alpha+1}{2^{m+k+\\alpha+1}}~\\frac{(l+m+1)_{\\alpha}}{(l+1)_{\\alpha}}\\,\\int\\limits_{-1}^1\\, (1-x)^{k+\\alpha}(1+x)^m\\,P_l^{(\\alpha,m)}(x)\\,dx~.\n\\end{equation}\nNext, Rodriguez' formula, (see (\\ref{e84})), is used, and $l$ partial integrations are performed. There results for $k\\geq l$\n\\begin{eqnarray} \\label{e137}\n& \\mbox{} & \\frac{1}{2^{m+k+\\alpha+1}}\\,\\int\\limits_{-1}^1\\,(1-x)^{k+\\alpha}(1+x)^m\\,P_l^{(\\alpha,m)}(x)\\,dx~= \\nonumber \\\\[3.5mm]\n& & =~\\frac{({-}1)^l}{2^{m+k+l+\\alpha+1}}~\\frac{\\Gamma(k+l)}{l!\\,\\Gamma(k+1-l)}\\,\\int\\limits_{-1}^1\\, (1-x)^{k+\\beta}(1+x)^{l+m}\\,dx~,\n\\end{eqnarray}\nwhile this vanishes for $k 1$ such that if \\textsc{MinSU}{} is $\\rho$-approximable in pseudo-polynomial time then $\\PNP$. \n\n\\paragraph{Notation and definitions.} \n\nFor every finite set $S$, $\\card{S}$ denotes the cardinality of~$S$.\nFor all sets $A$ and $S$, $S^A$ denotes the set of all families of elements of $S$ indexed by~$A$.\n\nThe ring of rational integers is denoted~$\\mathbb{Z}$.\nFor every integer $n \\ge 0$,\n$\\seg{1}{n}$ denotes the set of all $k \\in \\mathbb{Z}$ such that $1 \\le k \\le n$.\n\nA(n undirected) \\emph{graph} is a pair $G = (V, E)$, where $V$ is a finite set and $E$ is a set of $2$-element subsets of~$V$: \nthe elements of $V$ are the \\emph{vertices} of $G$,\nthe elements of $E$ are the \\emph{edges} of $G$,\nand for each edge $e \\in E$, the elements of $e$ are the \\emph{extremities} of~$e$.\n\nLet \\textsc{Min} be a minimization problem.\nThe \\emph{decision problem associated with} \\textsc{Min} is: \ngiven an instance $I$ of \\textsc{Min} and an integer $k \\ge 0$, \ndecide whether there exists a solution of \\textsc{Min} on $I$ with measure at most~$k$.\n\n\n\n\\section{Membership} \\label{sec:msu-easy}\n\nFor each instance $\\inda{X}$ of \\textsc{MinSU}, \nthe set of all feasible solutions of \\textsc{MinSU}{} on $\\inda{X}$ equals $\\mathbb{Z}^A$, which is infinite.\nTherefore, \n\\textsc{MinSU}{} is not an \\emph{$\\mathrm{NP}$-optimization problem} \\cite{ausiello03complexity}, \nand thus the membership of \\textup{SU}{} in $\\mathrm{NP}$ is not completely trivial.\n\nLet $G = (V, E)$ be a graph.\nA \\emph{disconnection} of $G$ is \na pair $(B, C)$ such that \n$B \\ne \\emptyset$, \n$C \\ne \\emptyset$, \n$B \\cap C = \\emptyset$,\n$V = B \\cup C$,\nand \nfor every $(b, c) \\in B \\times C$, $\\{ b, c \\} \\notin E$.\nA graph is called \\emph{disconnected} if it admits a disconnection.\nA graph that is not disconnected is called \\emph{connected}.\n\nLet $\\inda{Y}$ be an indexed family of sets.\nThe \\emph{intersection graph} of $\\inda{Y}$ is defined as follows: \nits vertex set equals $A$ and \nfor all $b$, $c \\in A$ with $b \\ne c$, \n$\\{ b, c \\}$ is one of its edges if, and only if, $Y_b \\cap Y_c \\ne \\emptyset$.\n\n\\begin{lemma} \\label{lem:discon-opt}\nLet $\\inda{Y}$ be an instance of \\textsc{MinSU}.\nIf the intersection graph of $\\inda{Y}$ is disconnected \nthen \nthere exists $\\inda{u} \\in \\mathbb{Z}^A$ such that\n\\begin{equation} \\label{eq:X+t=A}\n\\card{\\bigcup_{a \\in A} (Y_a + u_a )}\n< \n\\card{\\bigcup_{a \\in A} Y_a} \\, . \n\\end{equation}\n\\end{lemma}\n\n\\begin{proof}\nFor each subset $B \\subseteq A$, put $Y_{B} = \\bigcup_{b \\in B} Y_b$.\nLet $(B, C)$ be a disconnection of the intersection graph of $\\inda{Y}$.\nLet \n$r \\in Y_B$ \nand \n$s \\in Y_C$\nbe fixed.\nSet \n$u_b = - r$ for every~$b \\in B$ \nand \n$u_c = - s$ for every~$c \\in C$.\nOn the one hand, we have \n$Y_B \\cap Y_C = \\emptyset$\nand \n$Y_A = Y_B \\cup Y_C$,\nso \n\\begin{equation} \\label{eq:A=BC}\n\\card{Y_A } = \\card{Y_B} + \\card{Y_C} \\, .\n\\end{equation}\nOn the other hand, we have \n$$\n\\bigcup_{a \\in A} (Y_a + u_a ) = (Y_B - r) \\cup (Y_C - s) \n$$\nand \n$$\n(Y_B - r) \\cap (Y_C - s) \\ne \\emptyset \n$$\nbecause $0 \\in (Y_B - r) \\cap (Y_C - s)$; \nit follows \n\\begin{equation} \\label{eq:A+t=BC}\n \\card{\\bigcup_{a \\in A} (Y_a + u_a ) }\n < \n \\card{Y_B}\n +\n \\card{Y_C} \\, .\n\\end{equation}\nIt now suffices to combine Equations~\\eqref{eq:A=BC} and \\eqref{eq:A+t=BC} to obtain Equation~\\eqref{eq:X+t=A}.\n\\qed\n\\end{proof}\n\nLemma~\\ref{lem:discon-opt} can be restated as follows:\n\n\\begin{lemma} \\label{lem:opt-con}\nLet $\\inda{X}$ be an instance of \\textsc{MinSU}.\nFor any optimum solution $\\inda{t}$ of \\textsc{MinSU}{} on $\\inda{X}$,\nthe intersection graph of $\\left( X_a + t_a \\right)_{a \\in A}$ is connected.\n\\end{lemma}\n\n\\begin{proof}\nLet $\\inda{t} \\in \\mathbb{Z}^A$ be such that the intersection graph of $\\left( X_a + t_a \\right)_{a \\in A}$ is disconnected.\nSet $Y_a = X_a + t_a$ for each $a \\in A$.\nBy Lemma~\\ref{lem:discon-opt}, \nthere exists $\\left( u_a \\right)_{a \\in A} \\in \\mathbb{Z}^A$ such that Equation~\\eqref{eq:X+t=A} holds.\nIt follows that $\\left( t_a + u_a \\right)_{a \\in A}$ is a better solution of \\textsc{MinSU}{} on $\\inda{X}$ than $\\inda{t}$.\n\\qed\n\\end{proof}\n\n\n\\begin{definition} \\label{def:SGpi}\nLet $G = (V, E)$ be a graph.\nPut $$\\tilde E = \\left\\{ (a, b) \\in V \\times V : \\{ a, b \\} \\in E \\right\\}\\, .$$\nAn \\emph{antisymmetric edge-weight function} on $G$ is \na function $\\varpi$ from $\\tilde E$ to $\\mathbb{Z}$ such that \n$\\varpi(b, c) = - \\varpi(c, b)$\nfor every $(b, c) \\in \\tilde E$.\nFor each antisymmetric edge-weight function $\\varpi$ on $G$, \ndefine $S(G, \\varpi)$ as the set of all $\\left( t_a \\right)_{a \\in V} \\in \\mathbb{Z}^V$ such that \n$t_b - t_c = \\varpi(b, c)$ \nfor all $(b, c) \\in \\tilde E$.\n\\end{definition}\n\n\nLet us comment Definition~\\ref{def:SGpi}.\nThe function $\\varpi$ assigns both a magnitude and an orientation to each edge of~$G$:\nfor all $a$, $b \\in V$ such that $\\{ a, b \\} \\in E$, \nthe magnitude of $\\{ a, b \\}$ is the absolute value of $\\varpi(a, b)$\nand \nthe orientation of $\\{ a, b \\}$ \nis determined by the sign of $\\varpi(a, b)$.\nIt is clear that for every $\\left( t_a \\right)_{a \\in V} \\in S(G, \\varpi)$ and every $u \\in \\mathbb{Z}$, \n$ \\left( t_a + u \\right)_{a \\in V} \\in S(G, \\varpi)$.\nIf $G$ is connected then \neither $S(G, \\varpi)$ is empty or there exists $\\left( t_a \\right)_{a \\in V} \\in \\mathbb{Z}^V$ such that \n$S(G, \\varpi) = \\left\\{ \\left( t_a + u \\right)_{a \\in V} : u \\in \\mathbb{Z} \\right\\}$.\nIf $G$ is connected and $S(G, \\varpi) \\ne \\emptyset$\nthen \nfor any $(b, u) \\in V \\times \\mathbb{Z}$,\nthe unique element $\\left( t_a \\right)_{a \\in V} \\in S(G, \\varpi)$ that satisfies $t_b = u$ is computable from $G$, $\\varpi$, $b$, and $u$ in polynomial time.\nA \\emph{closed walk} in $G$ is a finite sequence \n$(a_0, a_1, a_2, \\dotsc, a_k)$ \nsuch that \n$a_0 = a_k$\nand \n$\\{ a_{i - 1}, a_i \\} \\in E$ for every $i \\in \\seg{1}{k}$;\nthe \\emph{weight} of $(a_0, a_1, a_2, \\dotsc, a_k)$ under $\\varpi$ is defined as \n$\\varpi(a_0, a_1) + \\varpi(a_1, a_2) + \\dotsb + \\varpi(a_{k - 1}, a_k)$.\nA (simple) \\emph{cycle} in $G$ is a closed walk $(a_0, a_1, a_2, \\dotsc, a_k)$ in $G$ such that \nfor all $i$, $j \\in \\seg{1}{k}$, \n$a_i = a_j$ implies $i = j$. \nThe following three conditions are equivalent:\n\\begin{enumerate}\n \\item The set $S(G, \\varpi)$ is non-empty. \n \\item The weight under $\\varpi$ of every closed walk in $G$ equals~$0$.\n\\item The weight under $\\varpi$ of every cycle in $G$ equals~$0$.\n\\end{enumerate}\nThe second and third conditions can be thought as abstract forms of Kirchhoff's voltage law.\n\nA \\emph{tree} is a connected graph with one fewer edges than vertices, or equivalently, an acyclic connected graph.\nAn arbitrary graph $G = (V, E)$ is connected \nif, and only if, \nthere exists a subset $E' \\subseteq E$ such that $(V, E')$ is a tree \n($(V, E')$ is then called a \\emph{spanning tree} of~$G$). \n\n\\begin{lemma} \\label{lem:tree-Xa}\nLet $\\inda{X}$ be an instance of \\textsc{MinSU}. \nThere exist a tree $H$ with vertex set $A$ and an antisymmetric edge-weight function $\\varpi$ on $H$ that satisfy the following two conditions:\n\\begin{enumerate}\n \\item \\label{cond:range-varpi}\nEvery integer in the range of $\\varpi$ can be written as the difference of two elements of $\\bigcup_{a \\in A} X_a$.\n \\item \\label{cond:SUpi-opt}\nEvery element of $S(H, \\varpi)$ is an optimum solution of \\textsc{MinSU}{} on $\\inda{X}$.\n\\end{enumerate}\n\\end{lemma}\n\n\\begin{proof}\nLet $\\inda{t}$ be an optimum solution of \\textsc{MinSU}{} on $\\inda{X}$.\nLet $H$ be a spanning tree of the intersection graph of $\\left( X_a + t_a \\right)_{a \\in A}$: such a tree exists \nby Lemma~\\ref{lem:opt-con}.\nLet $\\varpi$ be the antisymmetric edge-weight function on $H$ defined by:\nfor all $b$, $c \\in A$ such that $\\{ b, c \\}$ is an edge of $H$, \n$\\varpi(b, c) = t_b - t_c$. \n\nFor all $b$, $c \\in A$, \n such that $\\{ b, c \\}$ is an edge of the intersection graph of $\\left( X_a + t_a \\right)_{a \\in A}$,\n$(X_b + t_b) \\cap (X_c + t_c)$ is non-empty, \nand thus \n$t_b - t_c$ belongs to $X_c - X_b$.\nTherefore, the first condition holds.\nNow, remark that $S(H, \\varpi) = \\left\\{ \\left( t_a + u \\right)_{a \\in A} : u \\in \\mathbb{Z} \\right\\}$,\nso the second condition holds.\n\\qed\n\\end{proof}\n\n\n\\begin{theorem} \\label{th:SU-inNP}\n\\textup{SU}{} belongs to $\\mathrm{NP}$.\n\\end{theorem}\n\n\\begin{proof}\nLet $\\left(\\inda{X}, k \\right)$ be an arbitrary instance of \\textup{SU}.\nWe propose the following (non-deterministic) algorithm to decide whether $\\left(\\inda{X}, k \\right)$ is a yes-instance of \\textup{SU}:\n\\begin{itemize}\n \\item \nGuess a tree $H$ with vertex set $A$ and an antisymmetric edge-weight function $\\varpi$ on $H$ such that the first condition of Lemma~\\ref{lem:tree-Xa} holds.\n\\item \nCompute an element $\\inda{t} \\in S(H, \\varpi)$.\n\\item \nCheck whether the cardinality of $\\bigcup_{a \\in A} (X_a + t_a)$ is at most~$k$.\n\\end{itemize}\nBy Lemma~\\ref{lem:tree-Xa}, the algorithm is correct.\nMoreover, \nthe bit-length of the guess (\\emph{i.e}, the ordered pair $(H, \\varpi)$) \nis polynomial in \nthe bit-length of the input (\\emph{i.e}, the instance $\\inda{X}$), \nso the algorithm can be implemented in non-deterministic polynomial time.\n\\qed\n\\end{proof}\n\nLet $m$ be a positive integer and let $X$ be a subset of $\\mathbb{Z}$ such that $X = -X$.\nOn each given $m$-edge graph,\nthere are exactly $\\card{X}^m$ distinct antisymmetric edge-weight functions whose ranges are subsets of~$X$. \n\nLet $n$ be a positive integer and let $\\mathcal{T}_n$ denote the set of all trees with vertex set $\\seg{1}{n}$.\n\\emph{Cayley's formula} ensures $\\card{\\mathcal{T}_n} = n^{n - 2}$ \\cite{harary73graphical}. \nMoreover, every tree can be reconstructed in polynomial time from its \\emph{Pr\\\"ufer code} \\cite{harary73graphical},\nso $\\mathcal{T}_n$ is enumerable in $O \\left( n^{O(n)} \\right)$ time.\n\n\\begin{theorem}\nThere exists an algorithm that, \nfor each instance $\\inda{X}$ of \\textsc{MinSU}{} given as input,\nreturns an optimum solution of \\textsc{MinSU}{} on $\\inda{X}$ in \n$O \\left( N^{O(\\card{A})} \\right)$ \ntime, where $N$ denotes the bit-length of $\\inda{X}$.\n\\end{theorem}\n\n\\begin{proof}\nPut $U = \\bigcup_{a \\in A} X_a$.\nLet $\\mathcal{H}$ denote the set of all ordered pairs of the form $(H, \\varpi)$, \nwhere \n$H$ is a tree with vertex set $A$ \nand \n$\\varpi$ is an antisymmetric edge-weight function on $H$ whose range is a subset of $U - U$.\nWe propose the following algorithm to solve \\textsc{MinSU}{} on $\\inda{X}$:\n\\begin{itemize}\n \\item For each $(H, \\varpi) \\in \\mathcal{H}$, compute an element of $S(H, \\varpi)$.\n \\item Return a best solution of \\textsc{MinSU}{} on $\\inda{X}$ among those computed at the previous step.\n\\end{itemize}\nBy Lemma~\\ref{lem:tree-Xa}, the algorithm returns an optimum solution of \\textsc{MinSU}{} on $\\inda{X}$.\nMoreover, remark that $\\card{\\mathcal{H}} = \\card{A}^{\\card{A} - 2} \\card{U - U}^{\\card{A} - 1}$\nand that $\\mathcal{H}$ is enumerable in $O \\left( N^{O(\\card{A})} \\right)$ time.\nTherefore, the algorithm can be implemented to run in $O \\left( N^{O(\\card{A})} \\right)$ time.\n\\qed\n\\end{proof}\n\n\n\n\\section{Hardness} \\label{sec:msu-hard}\n\nThe aim of this section is prove the hardness results for \\textsc{MinSU}.\n\nLet $G = (V, E)$ be a graph. \nA \\emph{vertex cover} of $G$ is a subset $C \\subseteq V$ such that $C \\cap e \\ne \\emptyset$ for every $e \\in E$: \na vertex cover is a subset of vertices that contains at least one extremity of each edge.\n\n\\defpb\n{\\textsc{Minimum} \\textsc{Vertex Cover}{} (\\textsc{MinVC})}\n{a graph~$G$.}\n{a vertex cover $C$ of~$G$.}\n{the cardinality of~$C$.}\nThe decision problem associated with \\textsc{MinVC}{} is named \\textsc{Vertex Cover}{} (\\textup{VC}).\nIt is well-known that \\textup{VC}{} is $\\mathrm{NP}$-complete \\cite{garey79computers}.\n\n\nTo prove that \\textup{SU}{} is (strongly) $\\mathrm{NP}$-complete, \nwe show that \\textup{VC}{} Karp-reduces to (a suitable restriction of) \\textup{SU}.\nThe following gadget plays a crucial role in our reduction as well as in other reductions \nthat can be found in the literature \\cite{nicolas08hardness,michael10complexity}:\n\n\\begin{definition}\nFor each integer $n \\ge 1$, \ndefine $R_n = \\left\\{ (i - 1) n^2 + i^2 : i \\in \\seg{1}{n} \\right\\}$.\n\\end{definition}\n \nA \\emph{Golomb ruler} \\cite{gardner83wheels,sidon32satz,babcock53intermodulation} is a finite subset $R \\subseteq \\mathbb{Z}$ that satisfies the following three equivalent conditions:\n\\begin{itemize}\n \\item For every $t \\in \\mathbb{Z}$, $t \\ne 0$ implies $\\card{R \\cap (R + t)} \\le 1$.\n \\item For every integer $d > 0$, there exists at most one $(r, s) \\in R \\times R$ such that $r - s = d$.\n \\item For all $r_1$, $r_2$, $s_1$, $s_2 \\in R$, $r_1 + r_2 = s_1 + s_2$ implies $\\{ r_1, r_2 \\} = \\{ s_1, s_2 \\}$.\n\\end{itemize}\nActually, only the first condition is referred to in what follows.\nAmong other convenient properties our gadget sets are Golomb rulers:\n\n\\begin{lemma} \\label{lem:golomb}\nLet $n$ be a positive integer.\nThe following four properties hold.\n\\begin{enumerate}\n\\item \\label{ppty:subset} The least element of $R_n$ is $1$ and the greatest element of $R_n$ is~$n^3$.\n\\item \\label{ppty:card} The cardinality of $R_n$ equals~$n$.\n\\item \\label{ppty:distn} \nThe distance between any two elements of $R_n$ is at least $n^2 + 3$.\n\\item \\label{ppty:Golomb} $R_n$ is a Golomb ruler.\n\\end{enumerate}\n\\end{lemma}\n\n\\begin{proof}\nProperties~\\ref{ppty:subset} and~\\ref{ppty:card} are clear.\nProofs of Property~\\ref{ppty:Golomb} can be found in \\cite{nicolas08hardness,michael10complexity}.\nFinally, \nremark that for every $i \\ge 1$, \nwe have \n$$\n\\left( i n^2 + {(i + 1)}^2 \\right)\n-\n \\left( (i - 1) n^2 + i^2 \\right) \n =\n n^2 + 2i + 1 \n \\ge \n n^2 + 3 \\, .\n$$\nHence, the distance the distance between any two consecutive elements of $R_n$ is at least \n$n^2 + 3$, and thus Property~\\ref{ppty:distn} holds.\n\\qed\n\\end{proof}\n\n\n\\begin{theorem} \\label{th:MSU-NPC}\n\\textup{SU}{} is strongly $\\mathrm{NP}$-hard. \n\\end{theorem}\n\n\\begin{proof}\nPut $f(x) = \\left( \\tfrac{1}{4} x + 2 \\right)^3 + \\tfrac{1}{2}x - 4$.\nLet \\textsc{Aux}{} denote the restriction of \\textup{SU}{} to those instances $\\left( \\inda{X}, k \\right)$ \nsuch that \nthe absolute value of every integer in \n$\\left\\{ k \\right\\} \\cup \\bigcup_{a \\in A} X_a$ is at most $f\\left( \\max_{a \\in A} \\card{X_a} \\right)$.\nWe prove that \\textsc{Aux}{} is $\\mathrm{NP}$-hard which implies the theorem.\nMore precisely, we show that \\textup{VC}{} Karp-reduces to \\textsc{Aux}.\n\n\n\\paragraph{Presentation of the reduction.}\n\nLet $I$ be an arbitrary instance of \\textup{VC}.\nThe reduction maps $I$ to an instance $J$ of \\textup{SU}{} that is defined as follows.\nLet $G$, $V$, $E$, and $k$ be such that $I = (G, k)$ and $G = (V, E)$.\nLet $n$ denote the cardinality of $V$.\nWithout loss of generality, \nwe may assume \n$V = \\seg{1}{n}$ \nand \n$k < n$ because $I$ is a yes-instance of \\textup{VC}{} whenever $k \\ge n$. \nLet $\\left( y_e \\right)_{e \\in E}$, $\\left( z_e \\right)_{e \\in E} \\in V^E$ be such that $e = \\left\\{ y_e, z_e \\right\\}$ for every $e \\in E$.\nSet \n \\begin{gather*}\n A = \\{ \\emptyset \\} \\cup E \\, , \\\\\n s = {(n + 4)}^3\\,, \\\\\n R = R_{n + 4}\\,, \\\\\n X_\\emptyset = (V - s - n) \\cup (R - s) \\cup (R + n) \\cup (V + s + n)\\,, \\\\\n X_e = \\left\\{ z_e - n \\right\\} \\cup R \\cup \\left\\{ y_e + s \\right\\} \n\\end{gather*}\nfor each $e \\in E$, and \n\\begin{gather*}\nJ = \\left( \\inda{X}, \\card{X_\\emptyset} + k \\right) \\, . \n\\end{gather*}\nClearly, $J$ is computable from $I$ in polynomial time.\n\n\\paragraph{An instance of \\textsc{Aux}.}\n Let us prove that $J$ is in fact an instance of \\textsc{Aux}.\nWith the help of Lemmas~\\ref{lem:golomb}.\\ref{ppty:subset} and~\\ref{lem:golomb}.\\ref{ppty:card},\nit is easy to see \nthat $1 - s - n$ is the least element of $\\bigcup_{a \\in A} X_a$,\nthat $s + 2n$ is the greatest element of $\\bigcup_{a \\in A} X_a$, and \nthat the cardinality of $X_\\emptyset$ equals $4n + 8$. \nThe latter property implies $\\card{X_\\emptyset} + k < 5 n + 8$.\nHence, \nthe absolute value of every integer in \n$\\left\\{ \\card{X_\\emptyset} + k \\right\\} \\cup \\bigcup_{a \\in A} X_a$ \nis at most $s + 2n$.\nNow, remark that $s + 2n = f(\\card{X_\\emptyset}) \\le f\\left( \\max_{a \\in A} \\card{X_a} \\right)$. \n\n\\paragraph{Correctness of the reduction.}\nIt remains to prove that \n$I$ is a yes-instance of \\textup{VC}{} \nif, and only if,\n$J$ is a yes-instance of \\textup{SU}.\n\n\\begin{lemma} \\label{lem:Xens}\nFor every $e \\in E$,\nit holds true that \n\\begin{enumerate}\n \\item \\label{ppty:s} $(X_e - s) \\setminus X_\\emptyset = \\left\\{ y_e \\right\\}$ and that\n \\item \\label{ppty:n} $(X_e + n) \\setminus X_\\emptyset = \\left\\{ z_e \\right\\}$.\n\\end{enumerate}\n\\end{lemma}\n\n\\begin{proof}\n We only prove Property~\\ref{ppty:s} because Property~\\ref{ppty:n} can be proven in the same way. \nPut $Y = \\left\\{ z_e - s - n \\right\\} \\cup (R - s)$.\nIt is clear that \n$X_e - s = Y \\cup \\left\\{ y_e \\right\\}$ \nand \n$Y \\subseteq X_\\emptyset$. \nTherefore, we have \n\\begin{equation} \\label{eq:Xe-Y-y}\n X_e - s = \\left\\{ y_e \\right\\} \\setminus X_\\emptyset \\, .\n\\end{equation}\nMoreover, it follows from Lemma~\\ref{lem:golomb}.\\ref{ppty:subset} that \n\\begin{itemize}\n \\item the greatest element of ${(V - s - n)} \\cup {(R - s)}$ equals $0$ and that\n \\item the least element of ${(R + n)} \\cup {(V + s + n)}$ equals $n + 1$.\n\\end{itemize}\nTherefore, $X_\\emptyset$ does not contain any element of $\\seg{1}{n}$.\nIn particular, $y_e$ does not belong to $X_\\emptyset$.\nCombining the latter fact with Equation~\\eqref{eq:Xe-Y-y}, \nwe obtain Property~\\ref{ppty:s}.\n\\qed\n\\end{proof}\n\n\n\\begin{lemma} \\label{lem:tpms}\nFor every $t \\in \\mathbb{Z}$, \n$\\card{(R + t) \\setminus X_\\emptyset} < n$ implies $t \\in \\left\\{ - s, + n \\right\\}$. \n\\end{lemma}\n\n\\begin{proof}\nLet us first bound from above the cardinality of $(R + t) \\cap X_\\emptyset$.\nFor each $\\tau \\in \\mathbb{Z}$, put\n$\nP_\\tau = (R + t) \\cap (V + \\tau)\n$\nand \n$\nQ_\\tau = (R + t) \\cap (R + \\tau).\n$\nFirst, \nit follows from Lemma~\\ref{lem:golomb}.\\ref{ppty:distn} that \n$\\card{P_\\tau} \\le 1$.\nSecond, \n$\\tau \\ne t$ implies $\\card{Q_\\tau} \\le 1$ by Lemma~\\ref{lem:golomb}.\\ref{ppty:Golomb}.\nAnd third, \nit holds that\n$$(R + t) \\cap X_\\emptyset = P_{- s - n} \\cup Q_{-s} \\cup Q_n \\cup P_{s + n} \\, .$$\nNow, assume $t \\notin \\{ - s, + n \\}$.\nFrom the preceding three facts, \nwe deduce that\n$$\n\\card{(R + t) \\cap X_\\emptyset} \n\\le \n\\card{P_{-s - n}} + \n\\card{Q_{-s}} + \n\\card{Q_n} + \n\\card{P_{s + n}} \n\\le \n4 \\,. \n$$\n(In fact, \nit is not hard to see that $\\card{(R + t) \\cap X_\\emptyset} \\le 2$ holds: \n$t > +n$ implies $P_{- s - n} = Q_{-s} = \\emptyset$, \n$- s < t < +n$ implies $P_{- s - n} = P_{s + n} = \\emptyset$, and \n$t < - s$ implies $Q_n = P_{s + n} = \\emptyset$.)\nSince $\\card{R + t} = n + 4$ by Lemma~\\ref{lem:golomb}.\\ref{ppty:card}, \nwe finally get \n$$\n\\card{(R + t) \\setminus X_\\emptyset} = n + 4 - \\card{(R + t) \\cap X_\\emptyset } \\ge n \\, .\n$$ \n\\qed \n\\end{proof}\n\n\n\\paragraph{(If).} \n\nAssume that $I$ is a yes-instance of \\textup{VC}.\nThen, there exists a vertex cover $C$ of $G$ with $\\card{C} \\le k$.\nPut \n$F = \\left\\{ e \\in E : y_e \\in C \\right\\}$.\nSet\n$t_\\emptyset = 0$, \n$t_e = -s$ for each $e \\in F$, and \n$t_e = +n$ for each $e \\in E \\setminus F$.\nOn the one hand, \nit holds that \n\\begin{equation} \\label{eq:Xa-ta-X0}\n\\card{\\bigcup_{a \\in A} (X_a + t_a) } = \\card{X_\\emptyset} + \\card{\\bigcup_{e \\in E} (X_e + t_e ) \\setminus X_\\emptyset} \n\\end{equation}\nbecause $t_\\emptyset = 0$.\nOn the other hand,\nit follows from Lemma~\\ref{lem:Xens} that \n\\begin{equation} \\label{eq:Xe-te-ye-ze}\n\\bigcup_{e \\in E} (X_e + t_e) \\setminus X_\\emptyset \n= \n\\left\\{ y_e : e \\in F \\right\\} \\cup \\left\\{ z_e : e \\in E \\setminus F \\right\\} \\, .\n\\end{equation}\nSince the right-hand side of Equation~\\eqref{eq:Xe-te-ye-ze} is a subset of~$C$, \nwe have\n\\begin{equation} \\label{eq:Xe-te-X0-k}\n\\card{\\bigcup_{e \\in E} (X_e + t_e) \\setminus X_\\emptyset } \\le k \\, .\n\\end{equation}\nWe then get \n\\begin{equation} \\label{eq:J-yes}\n \\card{\\bigcup_{a \\in A} (X_a + t_a) } \\le \\card{X_\\emptyset} + k \n\\end{equation}\nby combining Equations~\\eqref{eq:Xa-ta-X0} and~\\eqref{eq:Xe-te-X0-k}.\nHence, $J$ is a yes-instance of \\textup{SU}.\n\n\\paragraph{(Only if).} \nAssume that $J$ is a yes-instance of \\textup{SU}.\nThen, there exists $\\inda{t} \\in \\mathbb{Z}^A$ such that Equation~\\eqref{eq:J-yes} holds.\nReplacing \n$\\inda{t}$ with $\\left( t_a - t_\\emptyset \\right)_{a \\in A}$ \nleaves the cardinality of $\\bigcup_{a \\in A} (X_a + t_a)$ unchanged; \ntherefore, \nwe may assume that $t_\\emptyset = 0$;\nin particular, Equation~\\eqref{eq:Xa-ta-X0} holds.\n\nPut \n$$\nC = \\bigcup_{e \\in E} (X_e + t_e) \\setminus X_\\emptyset \\, .\n$$\nCombining Equations~\\eqref{eq:Xa-ta-X0} and~\\eqref{eq:J-yes}, \nwe obtain Equation~\\eqref{eq:Xe-te-X0-k}, or equivalently, $\\card{C} \\le k$.\nNow, let us prove that $C$ is a vertex cover of~$G$.\nConsider an arbitrary edge~$e \\in E$.\nSince we have\n$$\n(R + t_e) \\setminus X_\\emptyset \n \\subseteq \n(X_e + t_e) \\setminus X_\\emptyset \\subseteq C \\, ,\n$$ \nit follows from Lemma~\\ref{lem:tpms} that $t_e \\in \\{ -s, +n \\}$.\nConsequently, Lemma~\\ref{lem:Xens} ensures that some extremity of $e$ belongs to $(X_e + t_e) \\setminus X_\\emptyset$,\nand this extremity is \\emph{a fortiori} in~$C$.\nHence, $I$ is a yes-instance of \\textup{VC}.\n\\qed\n\\end{proof}\n\n\nA graph $G = (V, E)$ is called \\emph{cubic} if for every vertex $v \\in V$, \nthe degree of $v$ in $G$ \n(\\emph{i.e.}, the cardinality of $\\left\\{ w \\in V : \\{ v, w \\} \\in E \\right\\}$) \nequals~$3$. \nLet \\textsc{MinVC}$3$ denote the restriction of \\textsc{MinVC}{} to cubic graphs.\n\\textsc{MinVC}$3$ is $\\mathrm{APX}$-complete under L-reduction \\cite{alimonti00APX};\nmoreover, if \\textsc{MinVC}$3$ is $\\frac{100}{99}$-approximable in polynomial time then $\\PNP$ \\cite{chlebik06complexity}.\n\nTo prove that \\textsc{MinSU}{} is ``strongly'' $\\mathrm{APX}$-hard,\nwhich is a better result than Theorem~\\ref{th:MSU-NPC},\nwe show that \\textsc{MinVC}$3$ L-reduces to a suitable restriction of \\textsc{MinSU}.\nIn fact, we simply adapt the proof of Theorem~\\ref{th:MSU-NPC}.\n \n\n\\begin{theorem} \\label{th:MSU-APX}\nThere exists a real constant $\\rho > 1$ such that \nif \\textsc{MinSU}{} is $\\rho$-approximable in pseudo-polynomial time then $\\PNP$.\n\\end{theorem}\n\n\\begin{proof}\nLet $f$ be as in the proof of Theorem~\\ref{th:MSU-NPC} \nand \nlet \\textsc{MinAux}{} denote the restriction of \\textsc{MinSU}{} to those instances $\\inda{X}$ such that \nthe absolute value of every integer in $\\bigcup_{a \\in A} X_a$ is at most $f\\left( \\max_{a \\in A} \\card{X_a} \\right)$.\nWe prove that \\textsc{MinAux}{} is $\\mathrm{APX}$-hard, \nwhich implies the theorem \nbecause \nevery pseudo-polynomial-time approximation algorithm for \\textsc{MinSU}{} \nis \n a polynomial-time approximation algorithm for \\textsc{MinAux}.\nMore precisely, we show that \n\\textsc{MinVC}$3$ {L-reduces} \\cite{ausiello03complexity} to \\textsc{MinAux}.\nWe use the notation of the proof of Theorem~\\ref{th:MSU-NPC}. \n\n\n\\paragraph{From a graph to an instance of \\textsc{Aux}.}\n\nLet $\\tau$ denote the minimum cardinality of a vertex cover of~$G$.\nLet $\\upsilon$ denote the minimum cardinality of $\\bigcup_{a \\in A} (X_a + t_a)$ over all $\\inda{t} \\in \\mathbb{Z}^A$.\nClearly, \n$\\inda{X}$ is computable from $G$ in polynomial time ($\\inda{X}$ is independent of $k$),\n$\\inda{X}$ is an instance of \\textsc{MinAux}, and \n$\\upsilon = \\card{X_\\emptyset} + \\tau = 4n + 8 + \\tau$.\n\nNow, assume that $G$ is cubic and $n \\ge 24$.\nThe first assumption implies $3 \\tau \\ge \\card{E} \\ge n$.\nIt follows \n$$\n4n + 8 \n= \\left( 4 + \\frac{8}{n} \\right) n \n\\le \\left( 12 + \\frac{24}{n} \\right) \\tau\n\\le 13\\tau \\, ,\n$$\nand thus $\\upsilon \\le 14 \\tau$.\n\n\n\\paragraph{From a solution of \\textsc{Aux}{} to a vertex cover.}\n\nLet $\\inda{t} \\in \\mathbb{Z}^A$. \nPut \n$$k = \\card{\\bigcup_{a \\in A} (X_a + t_a)} - \\card{X_\\emptyset}\\,.$$\nThere exists a vertex cover $C$ of $G$ \nthat satisfies \n$\\card{C} \\le k$, \n or equivalently, \n$$\\card{C} - \\tau \\le \\card{\\bigcup_{a \\in A} (X_a + t_a)} - \\upsilon \\, . $$\nMoreover, such a vertex cover is computable from $G$ and $\\inda{t}$ in polynomial time:\n\\begin{itemize}\n \\item if $k \\ge n$ then set $C = V$ and \n \\item if $k < n$ then set $C = \\bigcup_{e \\in E} (X_e + t_e - t_\\emptyset) \\setminus X_\\emptyset$.\n\\end{itemize}\n\n\n\\paragraph{Conclusion.}\n\nLet $\\varepsilon$ be a positive real number.\nIf \\textsc{MinSU}{} is ${(1 + \\varepsilon)}$-approximable in pseudo-polynomial time \nthen \\textsc{MinVC}$3$ is ${(1 + 14\\varepsilon)}$-approximable in polynomial time.\nTherefore, if \\textsc{MinSU}{} is $\\frac{1387}{1386}$-approximable in pseudo-polynomial then $\\PNP$.\n\\qed\n\\end{proof}\n\n \nAn immediate corollary of Theorem~\\ref{th:MSU-APX} is that \\textsc{MinSU}{} does not\nadmit any (pseudo-)polynomial time approximation scheme.\n\n\n\n\\section{Open questions}\n\nThe following three questions remain open:\nDoes there exist a constant $\\rho > 1$ such that \n\\textsc{MinSU}{} is $\\rho$-approximable in (pseudo-)polynomial time?\nIs \\textup{SU}{} \\emph{fixed-parameter tractable} \\cite{flum06parameterized} with respect to parameter $\\card{A}$?\nIs \\textup{SU}{} solvable in polynomial time for bounded $\\max_{a \\in A} \\card{X_a}$?\n\n\n\n\n\n\n\\bibliographystyle{abbrv}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\\IEEEPARstart{W}{ith} the rapid standardization process of 5G\nnetworks~\\cite{busari20185g}, millimeter waves (mm-waves) have garnered great attention worldwide from industry, academia, and government.\nA major issue is to better understand the propagation characteristics of mm-wave signals.\nMany mm-wave channel measurement campaigns have recently been carried out in urban and suburban environments~\\cite{rappaport2013millimeter, rappaport2017small, zhang201728ghzicc, zhang2018improving}. \n{\n However, very limited effort has been put into validating and improving current channel models in overcoming vegetation blockage. This is a key element of sensor data collection in forestry and agriculture~\\cite{wu2017propagation} for preventing cost incurred by under\/over-deployment of the sensors and improving their communication performance.}\nIn~\\cite{papazian1992wideband}, a constant excess path loss of around 25~dB was observed at 28.8~GHz through a pecan orchard for paths with roughly 8 to 20 trees. \nMore recent works~\\cite{zhang2018improving, rappaport201573} reported low attenuation values per unit foliage depth of 0.07~dB\/m at 28~GHz and of 0.4~dB\/m with 3~dB deviation at 73~GHz, respectively.\n{\n In~\\cite{shaik2016millimeter}, attenuation with a dual-slope structure was observed for out-of-leaf measurements at 15~GHz, 28~GHz and 38~GHz in forest environments.}\nMoreover, even though a variety of modeling approaches have been considered, most of them ignore site-specific geographic features~\\cite{zhang2018improving}.\nA comprehensive analysis for attenuation in vegetation is required to validate those observations\nand make improvements\nto mm-wave propagation modeling.\n\n{We explore this gap by investigating mm-wave propagation at 28~GHz through a coniferous forest in Boulder, Colorado, where we recorded a total of 1415 basic transmission loss measurements.}\nA comprehensive model comparison is provided to elucidate the pros and cons of different modeling approaches for predicting signal attenuation through vegetation. Novel site-specific models with consistently better performance than existing models are developed. \n\n\\section{Measurement Setup}\n\nThe measurement system in our previous work~\\cite{zhang201728ghzicc} was utilized. The receiver (RX), with a chip rate of 399.95 megachips per second, was installed in a backpack and powered by a lithium-ion polymer battery for portability purposes. As illustrated in \\Cref{fig_measurement_overviews}\\subref{fig_rx_tracks}, the transmitter (TX) was set up at the edge of the forest, while the RX was moved in the coniferous forest to continuously record the signal along with the GPS location information. Basic transmission losses were computed accordingly. The TX antenna was adjusted before each signal recording activity to point to the middle area of the track to be covered. Beam alignment was achieved at the RX side using a compass.\nWe also obtained satellite images from Google Maps and LiDAR data from the United States Geological Survey (USGS). Tree locations were manually labeled accordingly. Foliage regions were automatically extracted by comparing the LiDAR data with USGS terrain elevation data. \nThese site-specific geographic features of the forest are illustrated in \\Cref{fig_measurement_overviews}\\subref{fig_tree_locaitons_zoomed_in}. \n{\n Boulder is a semi-arid environment with low humidity and minimal rainfall. Measurements were performed on a warm spring day under mostly sunny conditions.\n}\n\n\n\\begin{figure}[t]\n\t\\captionsetup[subfigure]{justification=centering}\n\t\\centering\n\t\\subfloat[RX tracks illustrated with basic transmission loss results\\vspace{-1.5mm}]{\n {\\parbox{3in}{\\centering\\includegraphics[height=2in]{.\/eps\/zhang_WCL2018-1396_fig1.pdf}}}\n\t\t\\label{fig_rx_tracks}\n }\n \\hfil \n\t\\subfloat[Site-specific information available for the measurement site]{\n\t\t{\\parbox{3in}{\\centering\\includegraphics[height=2in,trim={0.01in 0.19in 0 0.19in},clip]{.\/eps\/zhang_WCL2018-1396_fig2.pdf}}}\n\t\t\\label{fig_tree_locaitons_zoomed_in}\n\t}\n\t\\caption{\n\t Overview of the measurement campaign. \n \t(a) The TX was installed at the edge of a coniferous forest. The RX followed 10 different tracks.\n One basic transmission loss result was computed for each second of the recorded signal to match the GPS data. (b) We have zoomed into the dotted-square area in (a) to better illustrate the site-specific features. \n {\n Overlaid on top of the satellite image are\n LiDAR data,\n foliage regions, and\n trunk locations, respectively.\n \n }}\n\t\\label{fig_measurement_overviews}\n \\vspace{-1mm}\n\\end{figure}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Foliage Analysis for the Coniferous Forest}\n\nWe compared three empirical foliage analysis models: the partition-dependent attenuation factor (AF) model~\\cite{durgin1998measurements}, the ITU-R obstruction by woodland model~\\cite{itu2016vegetation}, and Weissberger's model~\\cite{weissberger1982initial}.\nTo tune these models,\nfour parameters were computed for each measurement location: the distance between the TX and the RX, the number of tree trunks within the first Fresnel zone, the foliage depth along the line-of-sight (LoS) path, and the foliage area within the first Fresnel zone. These computations were performed in a three-dimensional (3D) reference system using Universal Transverse Mercator coordinates $(x, y)$ and altitude. Based on these results, site-specific models were introduced to improve path loss predictions.\n\n\nAll channel models considered here generate excess attenuation values on top of a site-general channel model.\nWe use the free-space path loss (FSPL) model as the baseline generic model. The path loss $PL$ in dB at the RX location $s$ is then composed of two parts:\n\\begin{equation}\n\t\\nonumber PL(s) = FSPL\\left[ d(s) \\right] + EPL(s) \\,\\text{,} \n\\end{equation}\nwhere $FSPL\\left[ d(s) \\right]$ is the FSPL in dB at a RX-to-TX distance of $d$ at $s$, and $EPL(s)$ is the excess path loss in dB at $s$.\n\n\\subsection{The AF Propagation Model~\\cite{durgin1998measurements}}\n\nThe partition-dependent AF propagation model takes advantage of site-specific information by assuming that each instance of one type of obstacle along the LoS path will incur a constant excess path loss.\nIn our case, we counted the number of trees, $N(s)$, along the LoS path to $s$ and added a constant excess path loss in dB, $L_0$, for each of the trees, as follows:\n\\begin{equation}\n\t\\nonumber EPL(s) = N(s) \\cdot L_0 \\,\\text{.}\n\\end{equation}\n\nConsidering the forest size and the number of RX locations involved, it is extremely difficult and time-consuming to count $N$ at each $s$ on-site. In our work, we simplified the trees, making them vertical lines rather than estimating the cylinder of each tree. Then, the number of trees within the first Fresnel zone for each $s$ was estimated based on manually labeled trunk locations\nand used as the number of obstacle trees. \n\n\\newcommand{2.7in}{2.7in}\n\\newcommand{-2mm}{-2mm}\n\\begin{figure}[t]\n\t\\centering\n\t\\includegraphics[width=2.7in\/560*640]{.\/eps\/zhang_WCL2018-1396_fig3.pdf}\n \\vspace*{-2mm}\n\t\\caption{The AF propagation model degenerates to a constant-loss-per-tree model in our case. Its predictions fit the shape of the measurement results but have a poor overall accuracy. \n\n }\n\t\\label{model_comparison_partition_dependent}\n \\vspace{-1mm}\n\\end{figure}\n\n\\Cref{model_comparison_partition_dependent} shows the predictions obtained from \nthe AF model. The unknown constant $L_0$ was fit according to the measurement data, resulting in a value of 6.47~dB per tree.\nAs can be seen, the AF model closely follows the shape formed by the measurement results. However, it suffers in predicting the correct amount of excess path loss in general. This is expected because \n{\n we have only considered the trunk locations for counting trees, but their physical sizes\n \n also play a critical role in attenuating the signal.}\nThe root mean squared error (RMSE) for the AF model compared with the measurements is 27.96~dB, achieving a 11.47~dB improvement over the FSPL model but still significantly worse than those for the other two empirical models discussed below.\n{\n \n \n We observe that it may be possible to improve the AF model by classifying trees into different size categories and assigning each category a loss value. \n However, in our case, trees grew in clusters, making it extremely challenging to distinguish individual canopies and to properly classify trees.}\n\n\n\n\\subsection{ITU-R Obstruction by Woodland Model~\\cite{itu2016vegetation}}\n\nThe ITU-R obstruction by woodland model assumes one terminal (the TX or the RX) is located within woodland or similar extensive vegetation, which fits well our measurement scenario. Instead of the number of trees, the ITU model uses the length of the path within the woodland in meters, $d_w(s)$, \n{\nwhich is the distance from the woodland edge to the terminal in the woodland,}\nto estimate the excess path loss:\n\\begin{equation}\n\\label{equ_itu}\n\tEPL(s) = A_m \\left[1-\\text{exp}(-d_w(s)\\cdot\\gamma \/ A_m)\\right] \\text{,}\n\\end{equation}\nwhere $\\gamma\\approx 6$~dB\/m is the typical specific attenuation for very short vegetative paths at 28~GHz, and $A_m$ is the maximum attenuation in dB.\nThe most distinguishing feature of this model is the upper limit $A_m$ imposed on the excess path loss.\n\nSince the TX was installed approximately 15~m away from the forest, this offset has been taken away from the 3D RX-to-TX distance to estimate $d_w(s)$, with the negative values clipped to zero.\nAlso, $A_m$ is yet to be determined in \\cite{itu2016vegetation} for 28~GHz signals, so we fitted it to our measurement results to obtain the best possible performance, which yielded $A_m \\approx 34.5$~dB. The resulting predictions are plotted in \\Cref{fig_itu_results}. The ITU model exhibits the best fit among the empirical models considered, with an overall RMSE of 20.08~dB. \nHowever, it clearly overestimates the path loss for locations with $d_w$ smaller than 30~m. At those locations, the LoS path may be clear or blocked by only a couple of trees, differing from a typical woodland blockage scenario. \nOn the other hand, the ITU model underestimates the path loss for large $d_w$. \n\n\\begin{figure}[t]\n\t\\centering\n\t\\includegraphics[width=2.7in]{.\/eps\/zhang_WCL2018-1396_fig4.pdf}\n \\vspace*{-2mm}\n\t\\caption{Predictions from the ITU obstruction by woodland model. As a comparison, the predictions from one site-specific model, which is covered in \\Cref{section_site_specific_models}, are also shown. The site-specific model follows the measurements better than the ITU model at the lower and higher ends of $d_w$.}\n\t\\label{fig_itu_results}\n \n\\end{figure}\n\n\\subsection{Weissberger's Model~\\cite{weissberger1982initial}}\n\n\nWeissberger's model, or Weissberger's modified exponential decay (WMED) model, can be formulated as follows:\n\\newcommand{-0.8em}{-0.8em}\n\\begin{equation}\n\t\\nonumber EPL(s) = {\n\t\t\\begin{cases}\n\t\t\t0.45\\,f_c^{{0.284}}\\,d_f(s)\t\t\t &\\hspace{-0.8em}{\\mbox{, if }}0$14 differently from those with less foliage blockage.\n\n\n\n\nWe have taken an image processing approach to \n{\n automatically}\nobtain the site-specific foliage depth, $d_f(s)$, \n{\n which is the sum total distance for the intersections of the direct path and the foliage regions.}\nBoth the LiDAR data and the terrain elevation data from the USGS were rasterized onto the same set of reference location points. The foliage regions were then extracted by thresholding their difference,\nresulting in the foliage regions illustrated in \\Cref{fig_measurement_overviews}\\subref{fig_tree_locaitons_zoomed_in}. Along the LoS path, the ratio of the number of foliage region pixels over the total number of pixels was calculated and multiplied with the corresponding 3D RX-to-TX distance to get the foliage depth for each RX location. \n\n\\Cref{fig_foliage_depth_based_models}\\subref{fig_weiss_results} compares the predictions from the WMED model with the measurement results. \nOverall, the WMED model gives a reasonably good RMSE value of 22.19~dB. \n\n\n\n\n\n\n\n\n\\newcommand{\\modelPerformFigWidth-1mm} %{\\modelPerformFigWidth}{2.7in-1mm}\n\\begin{figure}[t]\n\t\\centering\n\t\\subfloat[Results from Weissberger's model\\vspace{-1.5mm}]{\n\t\t\\includegraphics[width=\\modelPerformFigWidth-1mm} %{\\modelPerformFigWidth]{.\/eps\/zhang_WCL2018-1396_fig5.pdf}\n\t\t\\label{fig_weiss_results}\n\t}\n\t\\hfil\n\t\\subfloat[Results from site-specific models]{\n\t\t\\includegraphics[width=\\modelPerformFigWidth-1mm} %{\\modelPerformFigWidth]{.\/eps\/zhang_WCL2018-1396_fig6.pdf}\n\t\t\\label{fig_site_spec_results}\n\t}\n\t\\caption{{\n\tPredictions from foliage-depth-based models. Results from \\textit{model C}, which makes predictions based on the foliage area in the first Fresnel zon\n\t, are also plotted in (b) for comparison.\n \n \n \n \n \n }}\n\t\\label{fig_foliage_depth_based_models}\n \\vspace{-1mm}\n\\end{figure}\n\n\\subsection{Site-specific Models}\n\\label{section_site_specific_models}\n\nUsing high-precision publicly available geographic information, existing channel models can be tuned with well-estimated site-specific parameters.\nAs a result, simple but powerful site-specific models can be constructed as alternatives. We refer to these as \"site-specific\" models because their performance depends heavily on \nthe accuracy of\nthe parameters evaluated for each site.\n\n\n\nBy combining the idea of \n{\n evaluating the blockage condition individually for each $s$ from the AF model} \nand the two-slope modeling approach in the WMED model, we constructed \\textit{model A-I}:\n\\renewcommand{-0.8em}{-0.9em}\n\\begin{equation}\n\t\\nonumber EPL(s) = {\n\t\t\\begin{cases}\n\t\t\td_f(s)\\cdot L_1 \\,\\,\\text{,}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,{\\mbox{if }}0\\leqslant\\,&\\hspace{-0.8em}d_f(s)\\leqslant D_f\\\\\n D_f L_1 + \\left[ d_f(s)-D_f \\right] L_2\t\\,\\,{\\mbox{, if }}&\\hspace{-0.8em}d_f(s)>D_f\n\t\t\\end{cases}\n }\n\\end{equation}\nwhere $d_f(s)$ is the foliage depth in meters at $s$, $L_1$ and $L_2$ are two constants for adjusting the extra loss in dB caused by each meter of foliage, and $D_f$ is the boundary determining when $L_2$ will take effect. The upper bound from the ITU model can be imposed by setting $L_2=0$ to form \\textit{model A-II}:\n\\renewcommand{-0.8em}{-0.8em}\n\\begin{equation}\n\t\\nonumber EPL(s) = {\n\t\t\\begin{cases}\n\t\t\td_f(s)\\cdot L_1 \t&\\hspace{-0.8em}{\\mbox{, if }}0\\leqslant d_f(s)\\leqslant D_f\\\\\n D_f \\cdot L_1 \t&\\hspace{-0.8em}{\\mbox{, if }}d_f(s)>D_f\n\t\t\\end{cases}\n }\n\\end{equation}\n\nWe also reused the ITU model in \\Cref{equ_itu} with site-specific foliage depth to form \\textit{model B}. That is, $d_f(s)$ is used instead of $d_w(s)$, and parameters $A_m$ and $\\gamma$ are set according to the measurements. \n\nFor a fair performance comparison for these three models, we used the WMED boundary $D_f = 14$ for \\textit{model A-I} to leave only two adjustable parameters. After fitting these models to our data, we found $L_1 \\approx 2.39$~dB\/m and $L_2 \\approx 0.12$~dB\/m for \\textit{model A-I}, $L_1 \\approx 2.09$~dB\/m and $D_f \\approx 17.87$~m for \\textit{model A-II}, along with $A_m \\approx 38.04$~dB and $\\gamma \\approx 4.47$~dB\/m for \\textit{model B}. \nThe resulting predictions are plotted in \\Cref{fig_foliage_depth_based_models}\\subref{fig_site_spec_results}. The corresponding RMSE\nvalues are summarized in \\Cref{tab_overall_perf}, together with those for the traditional models as references. Note that the site-specific models perform very similarly, and each unit of foliage depth tends to contribute less to the excess loss as foliage depth grows. \\textit{Model A-I} does not limit the excess loss as the other two site-specific models do, but it performs slightly better than \\textit{model A-II} in terms of RMSE. Overall, \\textit{model B} performs the best, but computationally, it is more demanding because of its exponential form. \n\nWe can further push the best RMSE performance to 19.18~dB with \\textit{Model C}\n\\begin{equation}\n\t\\centering\n\t\\nonumber EPL(s) = {\n\t\t\\begin{cases}\n 0 \\,\\,\\text{,}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,&{\\mbox{if }a_f(s)=0} \\\\\n\t\t\ta_f(s)\\cdot L_1 + L_0 \\,\\,\\text{,}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,&{\\mbox{if }}0A_f\n\t\t\\end{cases}\n }\n\\end{equation}\n{where foliage area $a_f(s)$ is the sum total area for the intersections between the first Fresnel zone at RX location $s$ and the foliage regions;} \n$L_0$ (dB), $L_1$ (dB\/m$^2$), and $L_2$ (dB\/m$^2$) are constants adjusting the excess loss contribution; and $A_f$ is the boundary determining when the foliage is deep enough for $L_2$ to take effect. According to our measurement results, we have $L_0 \\approx 19.14$~dB, $L_1 \\approx 2.09$~dB\/m$^2$, $L_2 \\approx 0.06$~dB\/m$^2$, and $A_f \\approx 18.02$~m$^2$. This model has a sudden jump at the origin. Its prediction results are also shown in \\Cref{fig_foliage_depth_based_models}\\subref{fig_site_spec_results} for reference.\n\n\\begin{figure}[t]\n\t\\centering\n\t\\subfloat[Regional RMSE improvement over the ITU model\\vspace{-1.5mm}]{\n\t\t\\includegraphics[width=\\modelPerformFigWidth-1mm} %{\\modelPerformFigWidth]{.\/eps\/zhang_WCL2018-1396_fig7.pdf}\n\t\t\\label{fig_regional_perf_itu} \n\t}\n\t\\hfil\n\t\\subfloat[Regional RMSE improvement over the WMED model]{\n\t\t\\includegraphics[width=\\modelPerformFigWidth-1mm} %{\\modelPerformFigWidth]{.\/eps\/zhang_WCL2018-1396_fig8.pdf}\n\t\t\\label{fig_regional_perf_wmed}\n\t}\n\t\\caption{Regional performance improvement for site-specific models using a window size of 10~m. (a) Compared with the ITU model, site-specific models work significantly better for locations close to the TX and reasonably better for those far away. However, models \\textit{A-I}, \\textit{A-II}, and \\textit{B} suffer a severe performance degradation for $d_w$$\\in[35, 60]$~m, which is less of an issue for \\textit{model C}. (b) Compared with the WMED model, site-specific models again work reasonably better for extreme cases. A performance deterioration is observed at a foliage depth of around 80~m, where models \\textit{A-I} and \\textit{C} are less influenced.}\n\t\\label{fig_regional_perfs}\n \\vspace{-1mm}\n\\end{figure}\n\n{\n The most important feature for these models is that \n they are fully automatic and thus can be applied in large-scale wireless communication networks. \n Site-specific information was fetched from Google and USGS servers. Foliage information was extracted, and channel modeling performed, by our automated algorithms.}\n \nAnother advantage of our site-specific models is their consistently good performance throughout the whole dataset. \nTo demonstrate this, regional RMSE improvements over the ITU and WMED models are evaluated in terms of $d_w$ and $d_f$, respectively, as summarized in \\Cref{fig_regional_perfs}. For our dataset, the ITU model works very well, as shown in \\Cref{tab_overall_perf}. However, according to \\Cref{fig_regional_perfs}\\subref{fig_regional_perf_itu}, the ITU model suffers from an RMSE degradation of as much as 20~dB compared with the site-specific models in the low-vegetation-coverage region ($d_w$$<$30~m). \nFor large $d_w$, this value is observed to be as much as 6~dB. A visual comparison for predictions from the ITU model and \\textit{model A-I} is provided in \\Cref{fig_itu_results}, where \\textit{model A-I} clearly works better for extreme cases at the low and high ends of $d_w$. Similar comparisons have been carried out for the WMED model in \\Cref{fig_foliage_depth_based_models}\\subref{fig_weiss_results} and \\Cref{fig_regional_perfs}\\subref{fig_regional_perf_wmed}. The WMED model slightly underestimates the path loss at RX locations with a small $d_f$ and overestimates it at \nlarge $d_f$. \n\n\\newcommand{1.1}{1.1}\n\\begin{table}[t]\n\t\\renewcommand{\\tabcolsep}{2.2pt\n\t\\renewcommand{\\arraystretch}{1.1}\n \\vspace{-1.5mm}\n\t\\caption{Overall Performance}\n\t\\label{tab_overall_perf}\n\t\\centering\n \\begin{tabular}{|>{\\bfseries}c||c|c|c|c|c|c|c|c|}\n \\hline\n & \\textbf{Baseline} & \\multicolumn{3}{c|}{\\textbf{Traditional}} & \\multicolumn{4}{c|}{\\textbf{Site-Specific}} \\\\ \\hline\n \\textbf{Model} & FSPL \t & AF & ITU & WMED & \\textit{A-I} & \\textit{A-II} & \\textit{B} & \\textit{C} \\\\ \\hline\n \\textbf{RMSE (dB)} & 39.43 \t & 27.96 & 20.08 & 22.19 & 19.96 & 20.02 & 19.93 & 19.18 \\\\ \\hline\n \n \\end{tabular}\n \\vspace{-1.5mm}\n\\end{table}\n\n\n\n\n\n\n\\section{Conclusion}\n\n{\n A comprehensive channel model comparison for attenuation through vegetation was conducted using measurements in a coniferous forest near Boulder, Colorado. Inspired by the results, we developed novel site-specific models for consistent improvement in prediction accuracy through shallow to deep vegetation blockages. They are fully automatic, easy to implement, and feasibly applicable to machine learning frameworks. \n}\n\n\n\\nocite{}\n\\bibliographystyle{IEEEtran}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\\label{sec:intro}\n\nLocalisation of a source of biological or chemical agent dispersing\nin the atmosphere is an important problem for national security\nand environmental monitoring applications \\cite{kendal}. Wind, as\nthe dominant transport mechanism in the atmosphere,\ncan generate strong turbulent motion, causing the released agent to\ndisperse as a plume whose spread increases with the downwind\ndistance \\cite{arya_98}. Assuming a constant release of the\ncontaminant, the problem involves estimation of source parameters:\nits location and intensity (release-rate). Two types of measurements\nare generally at disposal for source localisation: (i) the\nconcentration measurements at spatially distributed sensor\nlocations; (ii) the average wind speed and wind direction (typically\navailable from a nearby meteorological station).\n\nMany references are available on the topic of biochemical source\nlocalisation, assuming un-quantised (analog) concentration\nmeasurements. Standard solutions are based on optimisation\ntechniques, such as the nonlinear least squares \\cite{matthes_05} or\nsimulated annealing \\cite{thomson_07}. These methods are unreliable\ndue to local minima or poor convergence; in addition, they provide\nonly point estimates, without uncertainty intervals. The preferred\nalternative is the use of Bayesian techniques; they result in the\nposterior probability density function (PDF) of the source parameter\nvector, thereby providing an uncertainty measure to any point\nestimate derived from it. Most Bayesian methods for source\nestimation are based on Markov chain Monte Carlo (MCMC) technique,\nassuming either Gaussian or log-Gausiian likelihood function of\nmeasurements \\cite{keats_07,humphries_12,ortner,senocak_08}.\nRecently, a likelihood-free Bayesian method for source localisation\nwas proposed in \\cite{ristic_15}. \n\n\nBinary sensor networks have become widespread in environmental monitoring applications\nbecause binary sensors generate as little as one bit of information.\nSuch binary sensors allow inexpensive sensing with minimal communication\nrequirements \\cite{aslam_03}. In the context of binary sensor\nnetworks, an excellent overview of non-Bayesian chemical source\nlocalisation techniques is presented in \\cite{chen2008greedy}. Best\nachievable accuracy of source localisation using binary sensors has\nbeen discussed in \\cite{Ristic2014}.\n\nPrior work in using binary sensor data for biochemical source localisation assumes that the detection threshold\nof the sensor is known. It is a reasonable assumption for a commercial sensor whose sensitivity is specified (for example, in parts per\nmillion by volume (ppm$_v$) or grams per cubic meter) by the manufacturer and when the sensor is well calibrated. However, we consider at least two scenarios where the detection threshold of a binary sensor may not be accurately known. The first scenario is when a sensor's detection threshold goes off calibration due to environmental conditions such as temperature or humidity or ageing of the sensor. The second scenario is where the sensor is a human rather than a device. For example, imagine a person smelling a strong odour such as due to a gas leak or a decomposing animal carcass. When the person moves around, the smell will be detected in some locations but not in others, producing a binary measurement sequence without knowing the exact value of the threshold in ppm or g\/m$^3$. In this paper, we develop a Bayesian algorithm that carries out source parameter estimation based on such\n{\\em binary concentration measurements} where the sensor threshold is unknown. A Monte Carlo technique, importance sampling, is applied to calculate the posterior PDF approximately. The method is successfully\nvalidated using three experimental datasets obtained under different wind conditions.\n\n\n\n \n\n\\section{Models}\n\\label{s:2}\n\n\\subsection{Dispersion model}\nTo solve the source localisation problem described above, we propose a solution formulated in the Bayesian framework which relies on\ntwo mathematical models: the atmospheric dispersion model and the\nconcentration measurement model. A dispersion model mathematically describes the physical processes that govern the\natmospheric dispersion of the released agent within the plume. The\nprimary purpose of a dispersion model is to calculate the mean\nconcentration of emitted material at a given sensor location. A\nplethora of dispersion models are in use today\n\\cite{holmes_morawska_06} to account for specific weather\nconditions, terrain, source height, etc. In this paper, we adopt a\ntwo-dimensional dispersion model of ``particle encounters'' in a\nturbulent flow, described in \\cite{vergassola_07}. During a certain sensing\nperiod, each sensor experiences a Poisson distributed number of\n``encounters'' with released particles. The binary nature of\nmeasurements indicates that a sensor reading of binary ``1'' or a ``positive detection'' corresponds to\nthe number of such encounters exceeding a particular threshold.\n\n If a binary sensor with a particular threshold makes\npositive detections (binary ``1'') at some locations and zero detections (binary ``0'') at other locations due to a source of \na certain release rate, the measurements at these locations will be the same even if both the source release rate and the sensor detection\nthreshold were scaled up or down together by the same amount; it is the ratio between the source release rate and the sensor threshold that determines \nwhich sensor locations will have positive or zero readings. Therefore, when we estimate the source parameters using binary data from a sensor whose\ndetection threshold is unknown, it is not possible to estimate the absolute value of the source release rate; only the release rate normalised by the assumed sensor threshold can be estimated. Nevertheless, the source location, which is actually the parameter of main interest, can be estimated. Without loss of generality, in our \nexperiments, we assumed the sensor to output binary ``1'' if it encounters at least one particle during a sensing period and output a binary ``0'' otherwise. \n\nLet us assume that the biochemical source is located at $(x_0, y_0)$, with a normalised release rate of $Q_0$. The particles released from the source propagate with the isotropic diffusivity $D$, but can also be advected by wind. We assume the released particles to have an average lifetime of $\\tau$. While the wind speed is typically available from meteorological data from a nearby measuring station, we use this speed as the prior guess for a Bayesian estimate of the true, effective wind speed affecting the advection of particles. Accordingly, let us assume that the mean wind speed is $V$ and the mean wind direction coincides with the direction of the $x$ axis. We denote the PDF of the wind speed by $\\pi(V)$. A spherical sensor of\nsmall size $a$ at a location with coordinates $(x,y)$, non-coincidental with the source location $(x_0,y_0)$, will experience a series\nof encounters with the released particles. \n\n \nDenoting the parameter vector we wish to estimate, consisting of the source coordinates ($x_0$, $y_0$), normalised source release rate $Q_0$, and the wind speed $V$, by $\\mbox{\\boldmath$\\theta$} = [x_0\\;y_0\\;\nQ_0\\; V]^\\intercal$, the rate of particle encounters by the sensor at the $i$th location (where $i=1,\\dots,M$) with coordinates $(x_i,y_i)$ can be modelled\nas \\cite{vergassola_07}:\n\\begin{equation}\nR(x_i,y_i|\\mbox{\\boldmath$\\theta$}) =\n\\frac{Q_0}{\\ln\\left(\\frac{\\lambda}{a}\\right)}\\,\\exp\\left[{\\frac{(x_0-x_i)V}{2D}}\\right]\\cdot\nK_0\\left(\\frac{d_i(\\mbox{\\boldmath$\\theta$})}{\\lambda}\\right) \\label{e:disp}\n\\end{equation}\nwhere $D$, $\\tau$ and $a$ are known environmental and sensor parameters,\n\\begin{equation}\nd_i(\\mbox{\\boldmath$\\theta$}) = \\sqrt{(x_i-x_0)^2+(y_i-y_0)^2}\n\\end{equation}\nis the distance from the source to $i$th sensor location, $K_0$ is the modified\nBessel function of order zero, and\n\\begin{equation}\n\\lambda = \\sqrt{\\frac{D\\tau}{1+\\frac{V^2\\tau}{4D}}}.\n\\end{equation}\n\n\n\\subsection{Measurement model}\nThe stochastic process of sensor encounters with released particles\nis modelled by a Poisson distribution. The probability that \nsensor at location $(x_i,y_i)$ encounters\n$z\\in\\mathbb{Z}^+\\cup\\{0\\}$ particles ($z$ is a non-negative\ninteger) during a time interval $t_0$ is then:\n\\begin{equation}\n\\mathcal{P}(z; \\mu_i) = \\frac{(\\mu_i)^{z}}{z!}e^{-\\mu_i}\n\\label{e:likf}\n\\end{equation}\nwhere $\\mu_i = t_0\\cdot R(x_i,y_i|\\mbox{\\boldmath$\\theta$}) $ is the mean concentration\nat $(x_i,y_i)$. Equation (\\ref{e:likf}) represents the full\nspecification of the likelihood function of parameter vector $\\mbox{\\boldmath$\\theta$}$,\ngiven the sensor encounters $z$ counts at the $i$th position.\n\nHowever, because the actual sensor is binary, the measurement model is\n\\begin{equation}\nb_i = \\begin{cases} 1, & \\text{if } z =1,2,3,\\dots\\\\\n0, & \\text{if } z = 0.\n\\end{cases}\n\\end{equation}\nNote that $b_i$ is a Bernoulli random variable with the parameter\n\\begin{eqnarray} q_i(\\mbox{\\boldmath$\\theta$}) & = & \\text{Pr}\\{b_i=1\\}\\\\\n & = & \\sum_{z=1}^{\\infty} \\mathcal{P}(z;\\mu_i) \\\\\n & = & 1 - \\mathcal{P}(0;\\mu_i) \\\\\n & = & 1 - e^{-\\mu_i}.\n \\end{eqnarray}\nThe likelihood function for the sensor when it is at the $i$th location is then:\n\\begin{equation}\np(b_i|\\mbox{\\boldmath$\\theta$}) = [q_i(\\mbox{\\boldmath$\\theta$})]^{b_i} \\, [1-q_i(\\mbox{\\boldmath$\\theta$})]^{1-b_i}.\n\\end{equation}\nAssuming sensor measurements are conditionally independent, the likelihood\nfunction of the parameter vector $\\mbox{\\boldmath$\\theta$}$, given the binary\nmeasurement vector $\\mathbf{b} = [b_1,\\dots,b_M]^\\intercal$, is a product:\n\\begin{equation}\np(\\mathbf{b}|\\mbox{\\boldmath$\\theta$}) = \\prod_{i=1}^M [q_i(\\mbox{\\boldmath$\\theta$})]^{b_i} \\,\n[1-q_i(\\mbox{\\boldmath$\\theta$})]^{1-b_i}. \\label{e:lik}\n\\end{equation}\n\n\n\\section{Parameter estimation}\n\nThe estimation problem is formulated in the Bayesian framework. The goal\nis to compute the posterior PDF: the probability distribution of the\nparameter vector $\\mbox{\\boldmath$\\theta$}$ conditional on the measurement vector\n$\\mathbf{b}$. The posterior PDF provides a complete probabilistic\ndescription of the information contained in the measurements about\nthe parameter vector $\\mbox{\\boldmath$\\theta$}$. The basic elements required to compute the\nposterior distribution of are: (i) the prior distribution for the\nparameter vector $\\pi(\\mbox{\\boldmath$\\theta$})$ and (ii) the likelihood function\n$p(\\mathbf{b}|\\mbox{\\boldmath$\\theta$})$. Given these quantities, Bayes rule can be used to\nfind the posterior PDF as\n\\begin{equation}\np(\\mbox{\\boldmath$\\theta$}|\\mathbf{b}) = \\frac{p(\\mathbf{b}|\\mbox{\\boldmath$\\theta$})\\pi(\\mbox{\\boldmath$\\theta$})}{\\int p(\\mathbf{b}|\\mbox{\\boldmath$\\theta$})\\pi(\\mbox{\\boldmath$\\theta$})\nd\\mbox{\\boldmath$\\theta$}}.\n\\end{equation}\nThe prior distribution $\\pi(\\mbox{\\boldmath$\\theta$})$ is typically non-Gaussian. For\nexample, the source position can be restricted to polygon regions,\nwhile $Q_0$ and $V$ are strictly positive random variables.\nQuantities of interest related to $\\mbox{\\boldmath$\\theta$}$ (e.g., the posterior mean,\nvariance) can be computed from the posterior PDF.\n\n\\begin{figure*}[t!]\n\\centering\n\\begin{tabular}{cc}\n\\includegraphics[height=7.0cm]{data1_a.eps} &\n\\includegraphics[height=7.0cm]{data1_d.eps}\\\\\n\\includegraphics[height=5.3cm]{data1_bc.eps} &\n\\includegraphics[height=5.3cm]{data1_ef.eps}\n\\end{tabular}\n \\caption{\\footnotesize Estimation results for dataset 1: the left column is the prior PDF; the right column is the posterior\n PDF. Marginalised prior PDF $\\pi(x_0,y_0)$ in (a) and the marginalised posterior PDF $p(x_0,y_0|\\mathbf{b})$ in\n (d), approximated by random samples (indicated by scattered red dots).\nSensor locations (green squares are positive readings,\n blue circles are non-detections), as well as the building contours (black lines), also indicated in (a) and (d).\n Wind direction coincides with the $x$ axis. True source marked in (d) at $(-298.4,-342.6)$ (grey asterix). Figures (b) and (e) show the histograms corresponding to\n $\\pi(Q_0)$ and $p(Q_0|\\mathbf{b})$, respectively. Figures (c) and (f) show the histograms corresponding to $\\pi(V)$ and $p(V|\\mathbf{b})$, respectively.}\n \\label{f:1}\n\\end{figure*}\n\nOptimal Bayesian estimation is generally impossible because the\nposterior PDF cannot be found in closed-form; this is certainly the\ncase for the likelihood function specified in Sec.\\ref{s:2} and a\nnon-Gaussian prior $\\pi(\\mbox{\\boldmath$\\theta$})$. Hence we apply a Monte Carlo\napproximation technique of the optimal Bayesian estimation, known as\nimportance sampling \\cite{robert_casella}. This technique\napproximates the posterior PDF by a weighted random sample\n$\\{w_n,\\mbox{\\boldmath$\\theta$}_n\\}_{1\\leq n \\leq N}$, which is created as follows.\nFirst, a sample $\\{\\mbox{\\boldmath$\\theta$}_n\\}_{1\\leq n \\leq N}$ is drawn from an\nimportance distribution $\\varrho$, i.e., $\\mbox{\\boldmath$\\theta$}_n\\sim \\varrho(\\mbox{\\boldmath$\\theta$})$,\nfor $n=1,\\dots,N$. The unnormalised weights are then computed as:\n\\begin{equation}\n\\tilde{w}_n = \\frac{\\pi(\\mbox{\\boldmath$\\theta$}_n)}{\\varrho(\\mbox{\\boldmath$\\theta$}_n)}\\,p(\\mathbf{b}|\\mbox{\\boldmath$\\theta$}_n)\n\\end{equation}\nfor $n=1,\\dots,N$. Finally, the weights are normalised, i.e., $w_n =\n\\tilde{w}_n \/\\sum_{n=1}^N \\tilde{w}_n$, for $n=1,\\dots,N$. The\nchoice of importance distribution $\\varrho$ plays a significant role\nin the convergence of point estimators based on the approximation\n$\\{\\mbox{\\boldmath$\\theta$}_n\\}_{1\\leq n \\leq N}$. Ideally $\\varrho$ should resemble the\nposterior. Since the posterior is unknown, good importance\ndistributions are often designed iteratively (population Monte Carlo\n\\cite{robert_casella}, progressive correction \\cite{musso_et_al_00,\nmorelande_ristic_rad}). As $N\\rightarrow \\infty$, however, the\nchoice of $\\varrho$ is less relevant and even the prior $\\pi$ may be\nused as an (admittedly inefficient) importance distribution; this is\ndone in our implementation for convenience. Furthermore, once the weighted random\nsample $\\{w_n,\\mbox{\\boldmath$\\theta$}_n\\}_{1\\leq n \\leq N}$ is computed, resampling\n(with replacement) is carried out \\cite{pfbook} resulting in a\nsample with uniform weights. This last step was carried out mainly\nto improve the effect of visualisation of the posterior PDF (see\nfigures in the next section). In our implementation, the prior PDF\nis adopted as:\n\\begin{equation}\n\\pi(\\mbox{\\boldmath$\\theta$}) = \\pi(x_0,y_0)\\, \\pi(Q_0)\\, \\pi(V)\n\\end{equation}\nwhere: $\\pi(x_0,y_0)$ is a uniform distribution over designated\npolygon areas (e.g., buildings); $\\pi(Q_0)$ is a Gamma distribution\nwith shape $k$ and scale parameter $\\eta$; $\\pi(V)$ is a normal\ndistribution with mean $\\bar{V}$ and standard deviation $\\sigma_V$.\n\n\n\n\n\n\n\\section{Experimental results}\n\n\n\nThree experimental datasets collected using a single moving binary sensor with unknown detection threshold were used in the paper to demonstrate the algorithm performance(further details of the experiments cannot be revealed on security grounds). In all cases, algorithm parameters were\nadopted as follows: $a=0.2$ m, $D=1$ m$^2$\/s, $\\tau = 1000$ s,\n$t_0=1$ s, sample size $N=5000$, shape parameter $k=3$, scale\nparameter $\\eta=7$, standard deviation of wind speed $\\sigma_V =\n0.2$ m\/s. The wind conditions were different for the three datasets.\nThe mean wind direction was specified by angle $\\alpha$, measured\nanticlockwise from the $x$ axis. The values of wind parameters\n$(\\bar{V},\\alpha)$ were $(0.28 \\text{ m\/s}, 195^\\circ)$, $(0.12\n\\text{m\/s},-10^\\circ)$ and $(0.14 \\text{m\/s},150^\\circ)$ for the\nfirst, second and the third dataset, respectively.\n\n\n\nFig.\\ref{f:1} shows the results obtained using dataset 1: the left\ncolumn displays the prior PDF $\\pi(\\mbox{\\boldmath$\\theta$})$; the right column - the\nposterior PDF $p(\\mbox{\\boldmath$\\theta$}|\\mathbf{b})$. Figs. \\ref{f:1}.(a) and \\ref{f:1}.(d)\ndisplay the top down view of the area where the experiment was\ncarried out. The placement and the readings of the binary sensor\nare also marked (the number of sensor locations in dataset 1 is $M=45$).\nThe locations where the sensor reported positive readings (i.e., $b_i=1$), are marked by\ngreen squares; the remaining (non-detecting) sensor locations are marked by blue circles. Based on prior knowledge (e.g., intelligence reports), figures (a) and (d) also indicate the circular area where the source must be located (circle drawn in cyan colour). The center of this circle is the mean position of the sensor locations with\npositive readings; its radius is $150$m. Furthermore, figures (a)\nand (d) also display the countours of the buildings (black lines).\nBoth the marginal prior $\\pi(x_0,y_0)$ and the marginal posterior\nPDF $p(x_0,y_0|\\mathbf{b})$, are approximated by random samples marked by\nscattered red dots. Assuming the source must be inside one of the\nbuildings, the marginal prior PDF $\\pi(x_0,y_0)$ shown in figure (a)\nis a uniform PDF over the intersection of the cyan color bounded\ncircular area and the area covered by the buildings. The marginal\nposterior PDF $p(x_0,y_0|\\mathbf{b})$, shown in figure (d), concentrates in\nthe upper left corner of one of the buildings, thus \ndramatically reducing the initial uncertainty in the source location. The\ntrue source location is marked by an asterisk at $(-298.4,-342.6)$.\nFigs.\\ref{f:1}. (b) and (c) show the histograms of random samples\napproximating $\\pi(Q_0)$; and $\\pi(V)$, respectively. Figs.\n\\ref{f:1}.(e) and \\ref{f:1}.(f) display the histograms of random\nsamples approximating the marginal posteriors $p(Q_0|\\mathbf{b})$ and\n$\\pi(V|\\mathbf{b})$, respectively.\n\n\\begin{figure}[h!]\n\\centerline{\\includegraphics[height=7.0cm]{data2_a.eps}}\n\\centerline{\\includegraphics[height=5.3cm]{data2_bc.eps}}\n \\caption{\\footnotesize Estimation results obtained by processing dataset 2: (a) Marginalised posterior PDF $p(x_0,y_0|\\mathbf{b})$\n (scatter plot, red dots); sensor locations (green squares are positive readings,\n blue circles are non-detections); building contours (black lines);\n true source location at $(125.6,436.6)$ (grey asterix).\n Wind direction coincides with the $x$ axis. Figures (b) and (c) display the histograms of random samples approximating $p(Q_0|\\mathbf{b})$ and\n $p(V|\\mathbf{b})$, respectively. }\n \\label{f:2}\n\\end{figure}\n\\begin{figure}[h!]\n\\centerline{\\includegraphics[height=7.0cm]{data3_a.eps}}\n\\centerline{\\includegraphics[height=5.3cm]{data3_bc.eps}}\n \\caption{\\footnotesize Estimation results obtained by processing dataset 3: (a) Marginalised posterior PDF $p(x_0,y_0|\\mathbf{b})$\n (scatter plot, red dots); sensor locations (green squares are positive readings,\n blue circles are non-detections); building contours (black lines).\n True source location at $(31.2,-453.2)$ (grey asterix).\n Wind direction coincides with the $x$ axis. }\n \\label{f:3}\n\\end{figure}\n\n\nThe marginal posterior PDFs, obtained by processing dataset 2, are\nshown in Fig.\\ref{f:2}. The number of sensor measurement locations in this case\nwas $M=25$. The initial random sample $\\{\\mbox{\\boldmath$\\theta$}_n\\}_{1\\leq n \\leq N}$\nwas created in the same manner as for the case of dataset 1. From\nFig.\\ref{f:2}.(a) we can observe that the posterior PDF\n$p(x_0,y_0|\\mathbf{b})$ indicates fairly accurately the true source\nlocation marked by an asterisk at coordinates $(125.6,436.6)$.\n\n\n\nFinally, the marginal posterior PDFs, obtained by processing dataset\n3, are shown in Fig.\\ref{f:3}. The number of sensor locations in\nthis case was $M=27$. The initial random sample $\\{\\mbox{\\boldmath$\\theta$}_n\\}_{1\\leq n\n\\leq N}$ was created in the same manner as for the case of dataset\n1. From Fig.\\ref{f:3}.(a) we can observe that the support of the\nmarginal posterior PDF $p(x_0,y_0|\\mathbf{b})$ indeed contains the true\nsource location at coordinates $(31.2,-453.2)$. However, the\nconditions were such (the placement of sensor, the wind speed) that\nsome ambiguity in the source location remains. The resulting\nposterior PDF is bi-modal (two buildings contain scattered red\ndots), suggesting that the source must be located in one of them.\n\n\n\n\n\n\\section{Conclusions}\n\\label{s:6}\n\nThe paper proposed a simple Bayesian estimation algorithm for\nlocalisation of a continuous source of biochemical agent dispersing in the\natmosphere, using measurements collected at multiple locations by a single moving binary sensor whose detection threshold is unknown. The algorithm would also be applicable to a single snapshot of the measurements from a network of identical binary sensors. The sensor detection threshold may be unknown because it may have been drifted due to temperature, humidity, ageing, etc. Another possible scenario where the detection threshold of a binary sensor may be unknown is when a human, rather than a device, detects an odour at some locations but not others. In this scenario, the person can easily make the binary measurements of ``detection'' or ``non-detection'' without knowing the exact detection threshold in terms of ppm or g\/m$^3$ of the detected material. To enable source localisation in such scenarios, in our algorithm, we treat the source release rate, as well as the binary sensor threshold, as being unknown. Under these conditions, the algorithm can not estimate the absolute value of the source release rate and is only able to estimate the release rate normalised by the unknown sensor threshold. However, the algorithm can correctly estimate the location of the biochemical source. The performance of the\nalgorithm is demonstrated using three experimental datasets\ncollected in a semi-urban environment. In all three cases, the\nposterior density function included the true source location,\nthereby validating the proposed algorithm. Future work will consider\nintroducing the uncertainty in the mean wind direction and more\ndetailed dispersion models for urban environments.\n\n\\section*{References}\n\\bibliographystyle{elsarticle-num}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}