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+{"text":"\\section{Introduction}\n\nThe observations presented in this paper are part of the \n{\\sl Hubble Space Telescope} Key Project on the Extragalactic Distance Scale,\na detailed description of which can be found in\n\\markcite{kfm1995}Kennicutt, Freedman \\& Mould (1995). \nThe main goal of the Key Project\nis to measure the Hubble Constant, H$_{0}$, to an accuracy of 10\\%, by\nusing Cepheids to calibrate several secondary distance indicators such as\nthe planetary nebula luminosity\nfunction, the Tully--Fisher relation, surface brightness fluctuations,\nand methods using supernovae.\nThese methods will then be used to\ndetermine the distances to more distant galaxies whereby the global\nHubble Constant can be measured.\nPublished results from other Key Project galaxies\ninclude M81 (\\markcite{free1994}Freedman {\\it et al.} 1994), \nM100 (\\markcite{lff1996}Ferrarese {\\it et al.} 1996),\nM101 (\\markcite{kelson1996}Kelson {\\it et al.} 1996), \n\\ngc{925} (\\markcite{silb1996}Silbermann {\\it et al.} 1996),\n\\ngc{3351} (\\markcite{grah1997}Graham {\\it et al.} 1997), \n\\ngc{3621} (\\markcite{raw1997}Rawson {\\it et al.} 1997),\n\\ngc{2090} (\\markcite{phel1998}Phelps {\\it et al.} 1998),\n\\ngc{4414} (\\markcite{turn1998}Turner {\\it et al.} 1998),\n\\ngc{7331} (\\markcite{hughes1998}Hughes {\\it et al.} 1998), and\n\\ngc{2541} (\\markcite{lff1998a}Ferrarese {\\it et al.} 1998).\n \nNGC~1365 ($\\alpha_{1950} = 3^{\\rm h} 31^{\\rm m}$, $\\delta_{1950} = -36\\arcdeg\n18\\arcmin$) is a large, symmetric, barred spiral galaxy with a measured\nheliocentric velocity of 1652 km~s$^{-1}$ \\markcite{st1981} \n(Sandage \\& Tammann 1981) located in the Fornax cluster of galaxies.\nIt is classified as an SBb(s)I galaxy by \\markcite{st1981}\nSandage \\& Tammann and as an SBs(b) galaxy by\n\\markcite{dev1991}de Vaucouleurs {\\it et al.} (1991). \nWork by Veron {\\it et al.} (1980) found that \\ngc{1365} contains a\nhidden Seyfert 1 nucleus. \\ngc{1365} has been extensively mapped in HI\nby Ondrechen \\& van der Hulst (1989) and more recently by\n\\markcite{jvm1995}J\\\"{o}rs\\\"{a}ter \\& van Moorsel (1995). In particular,\n\\markcite{jvm1995}J\\\"{o}rs\\\"{a}ter \\& van Moorsel found that the \ninner disk of \\ngc{1365} has a significantly different inclination\nangle (40\\arcdeg) compared to previous work using optical isophotes (55\\arcdeg)\nby Linblad (1978). Other inclination angles found in the literature are\n56\\arcdeg (Bartunov {\\it et al.} 1994), 61\\arcdeg (Schoniger \\& Sofue 1994,\nAaronson {\\it et al.} 1981),\n44\\arcdeg~$\\pm$ 5\\arcdeg~(Bureau, Mould, \\& Staveley-Smith 1996),\n46\\arcdeg $\\pm$ 8\\arcdeg~(Ondrechen \\& van der Hulst 1989)\nand 63\\arcdeg~(Tully 1988). This scatter is a result of \nthe warped nature of the \\ngc{1365} disk\n(\\markcite{jvm1995}J\\\"{o}rs\\\"{a}ter \\& van Moorsel 1995) combined with the \nvarious observational methods the authors used to determine the\nthe major and minor axis diameters.\nThe true inclination angle of \\ngc{1365} is not vital for our Cepheid\nwork but is critial for photometric and HI line-width corrections\nfor the Tully-Fisher (TF) relation. If one uses infrared ($H$ band) absolute \nmagnitudes the effect of using the various inclination angles between\n40\\arcdeg~to 63\\arcdeg~is minimal as the extinction correction in the infrared\nis small. \nHowever, the correction to the line-width, (sin $i$)$^{-1}$, is not small.\nThe corrected HI line width will decrease by\n40\\% if one uses 63\\arcdeg~instead of 40\\arcdeg. Detailed discussion of\nthe TF method is beyond the scope of this paper but we caution readers\nto accurately determine the inclination angle of \\ngc{1365} before using\nit as a TF calibrator.\n\nOverall, Fornax is a relatively compact cluster of about 350 galaxies\nwith a central, dense concentration of elliptical galaxies.\nThe center of the cluster is dominated by the large E0 galaxy\n\\ngc{1399}, with \\ngc{1365} lying $\\sim$0.5 Mpc from \\ngc{1399} as projected\non the sky. The heliocentric radial velocity of the Fornax cluster is \n1450 km~s$^{-1}$ with a dispersion\nof 330 km~s$^{-1}$ (\\markcite{hm1994}Held \\& Mould 1994). \nThe Fornax\ncluster has been the target of many studies involving secondary distance \nindicators, as reviewed by Bureau, Mould, \\& Staveley-Smith (1996). \nA Cepheid distance to the cluster will be an important step\nforward in calibrating the extragalactic distance scale.\n\nThis is the first of two papers on the Cepheid distance to \\ngc{1365}\nand the Fornax cluster. This paper describes the HST observations of\nCepheid variable stars in \\ngc{1365} and the distance derived to the\ngalaxy. A companion paper by \\markcite{mad1998a} Madore {\\it et al.} (1998a) \ndiscusses the\nimplications for the distance to the Fornax cluster and the calibration\nof the extragalactic distance scale.\nThe location of \\ngc{1365} within the Fornax cluster and the geometry\nof the local universe is discussed in \\markcite{mad1998b}Madore {\\it et al.}\n(1998b).\nWe note that the Key Project has recently observed two other galaxies within \nthe Fornax cluster, \\ngc{1425} (Mould {\\it et al.} 1998) and \\ngc{1326A} \n(Prosser {\\it et al.} 1998).\n\n\\section{Observations}\n\nNGC~1365 was imaged using the Hubble Space Telescope's Wide Field and\nPlanetary Camera 2 (WFPC2). A description of\nWFPC2 instrument is given in the HST WFPC2 Instrument Handbook\n(\\markcite{burr1994}Burrows {\\it et al.} 1994).  The camera\nconsists of four 800$\\times$800 pixel CCDs.\nChip 1 is the Planetary Camera with 0.046 arcsec pixels and an illuminated\n$\\sim 33\\times31$ arcsec field of view. The three other CCDs make up\nthe Wide Field Camera (Chips 2-4), each with 0.10 arcsec pixels and a\n$\\sim 1.25\\times1.25$ arcmin illuminated field of view. Each CCD has a readout\nnoise of about 7 $e^{-}$. A gain setting of 7 $e^{-}$\/ADU was used for all\nof the \\ngc{1365} observations.\n\nWFPC2 imaged the eastern part of \\ngc{1365},\nas seen in Figure 1, from 1995 August through September. As with all the \ngalaxies chosen for the Key Project,\nthe dates of observation were selected using a power-law time series\nto minimize period aliasing and maximize uniformity of phase coverage\nfor the expected range of Cepheid periods from 10 to 60 days\n(\\markcite{free1994}Freedman {\\it et al.} 1994).  Twelve epochs in\n$V$ (F555W) and four in $I$ (F814W) were obtained. Each \nepoch consisted of two exposures, taken on successive orbits,\nwith typical integration times of 2700 seconds. All observations\nwere made with the camera at an operating temperature of\n$-$88$\\arcdeg$ C.  Table 1 lists the image identification code,\nHeliocentric Julian Date of observation (mid-exposure), exposure time, \nand filter, of each observation. On 1995 August 28 the focus of HST\nwas changed, occurring between the 5th and 6th epoch of \\ngc{1365}\nobservations. The effect of this refocus is discussed in Section 3.1.\n\n\\section{Photometric Reductions}\n\nAll observations were preprocessed through the standard Space\nTelescope Science Institute (STScI) pipeline as described by\n\\markcite{holt1995b}Holtzman {\\it et al.}  (1995b).  The images were\ncalibrated with the most up-to-date version of the routine reference\nfiles provided by the Institute at the time the images were taken.\nOur post-pipeline processing included masking out the vignetted edges, \nbad columns, and bad pixels.\nThe images were then multiplied by a pixel area map to correct for the\nWFPC2 geometric distortion (\\markcite{hill1998}Hill {\\it et al.} 1998), \nmultiplied by 4, and converted to integer values. Then the ALLFRAME and\nDoPHOT photometry packages were used to obtain profile-fitting photometry\nof all the stars in the HST images.\nDetails of each method are described below.\n\n\\subsection{DAOPHOT II\/ALLFRAME Photometry}\n\n\nThe extraction of stellar photometry from CCD images using the\nALLFRAME (\\markcite{pbs1994}Stetson 1994) package first requires a \nrobust list of stars in the\nimages. The long exposures of \\ngc{1365} contain significant numbers of \ncosmic ray events which can be misidentified as stars by automated\nstar-finding programs. To solve this problem, images\nfor each chip were median averaged to produce a\nclean cosmic ray free image. DAOPHOT II and ALLSTAR\n(\\markcite{pbs1987}Stetson 1987, \\markcite{sdc1990}Stetson {\\it et al.} 1990, \nStetson 1991, 1992, 1994) were then used to identify the stars\nin the deep, cosmic ray free images for each chip.\nThe star lists were subsequently used by ALLFRAME to\nobtain profile-fitting photometry of the stars in the original\nimages. The point spread functions were derived from\npublic domain HST WFPC2 observations of the globular clusters\nPal 4, and \\ngc{2419}.\n\nTen to twenty fairly isolated stars in each WFPC2 chip were chosen to\ncorrect the profile-fitting ALLFRAME photometry out to the 0.5 arcsec\nsystem of \\markcite{holt1995a}Holtzman {\\it et al.} (1995a). These\nstars are listed in the appendix as secondary standards to our photometric\nreductions.\nTo remove the effects of nearby neighbors on the aperture photometry\nall the other stars were first subtracted from the images. Growth curves for\nthe isolated stars were constructed using DAOGROW \n(\\markcite{pbs1990}Stetson 1990). \nBest estimates\nof the aperture corrections were determined by running the program COLLECT\n(\\markcite{pbs1993}Stetson 1993), which uses the profile-fitting photometry\nand the growth-curve photometry to determine the best photometric correction \nout to the largest aperture, 0.5 arcsec in our case. \n\nAs with the previous Key Project galaxy \\ngc{925} \n(\\markcite{silb1996}Silbermann {\\it et al.} 1996),\nthe aperture corrections (ACs) for all the frames in each filter were averaged \nto produce mean ACs that were then applied to each frame to shift the\nCepheid photometry onto the \\markcite{holt1995a}Holtzman {\\it et al.} \n0.5 arcsec system.\nFor nonvariable stars, mean instrumental $V$ and $I$ magnitudes averaged over\nall epochs were calculated using DAOMASTER (\\markcite{pbs1993}Stetson 1993) \nand then shifted to \nthe 0.5 arcsec system using the mean ACs. However, for \\ngc{1365} we\nnoticed there were relatively large variations in the individual frame\nACs, typically on the order of $\\pm$ 0.07 mag, but for one case as large as\n$-$0.2 mag, from the mean values. \nThese variations are due to random photometric scatter in the small number of\nisolated, bright stars available to us for AC determination, and the\nrealization that on any given image, a few of these stars may be corrupted by\ncomic ray events or chip defects further reducing the number of usable AC\nstars for a particular image.\nThere was also a focus change between the\n5th and 6th epoch, which resulted in an obvious shift of about +0.1 mag in\nthe individual ACs for epochs obtained after the focus change.\nAs a result, we also calculated mean magnitudes and fit period-luminosity \nrelations for the Cepheids using photometry corrected by individual frame ACs.\nThe typical difference between\nmean $V$ and $I$ magnitudes for the Cepheids (mean ACs $-$ individual ACs) \nwas +0.01 mag, and for some\nof the Cepheids was as large as +0.02 mag.\nDue to the refocusing event and the relatively large variation in the \nindividual ACs we feel that the individual photometric measurements of the\nCepheids at each epoch are best represented using the individual ACs. The\neffect on the distance modulus to \\ngc{1365} is minimal, +0.01 mag.\nThroughout the rest of this paper we will\nuse the individual AC corrected Cepheid photometry.\n\nThe final form of the conversion equations for the ALLFRAME photometry is: \n\n\\begin{equation}\nM = m + 2.5\\log t - C_{1}(V-I) + C_{2}(V-I)^{2} + AC + ZP\n\\end{equation}\n\n\\noindent \nwhere M the standard magnitude, m is the \ninstrumental magnitude, $t$ is the exposure time, $C_{1}$ and $C_{2}$ are\nthe color coefficients, AC is the aperture \ncorrection (different for each frame for the Cepheids), and ZP is the zero \npoint.\nSince we have shifted our photometry to the 0.5 arcsec system of\n\\markcite{holt1995a}Holtzman {\\it et al.} (1995a) we use their color\ncoefficients from their Table 7: $C_{1} = -0.052$ for $V$ and $-$0.063 for $I$,\nand $C_{2} = 0.027$ for $V$ and 0.025 for $I$.\nThe ZP term includes the \\markcite{hill1998}Hill {\\it et al.} (1998) long \nexposure zero point, \nthe ALLFRAME zero point of $-$25.0 mag,  and a correction for multiplying the \nimages by four before converting them to integers (2.5log(4.0)).  For $V$ band \nthe ZP terms are $-$0.984 $\\pm$ 0.02, $-$0.973 $\\pm$ 0.01, \n$-$0.965 $\\pm$ 0.01, and $-$0.989 $\\pm$ 0.02 for Chips 1$-$4 respectively,\nand for\n$I$ band are $-$1.879 $\\pm$ 0.04, $-$1.838 $\\pm$ 0.02, $-$1.857 $\\pm$ 0.02, \nand $-$1.886 $\\pm$ 0.01 for Chips 1$-$4 respectively.\nFor each epoch, an initial guess of each star's color, $V-I$ = 0.0, was used\nand the $V$ and $I$ equations were then solved iteratively. \n\nFor the nonvariable stars, an average magnitude over all epochs was calculated\nusing DAOMASTER, which corrects for frame-by-frame differences due to \nvarying exposure times and focus, and the equations above were used to \ncalculate the final standard $V$ and $I$ magnitudes using mean aperture \ncorrections for each chip and filter combination. \n\n\\subsection{DoPHOT Photometry}\n\nAs a double-blind check on our reduction procedures, we\nseparately reduced the \\ngc{1365} data using a variation \nof DoPHOT (\\markcite{schec1993}Schechter et al. 1993) described\nby \\markcite{saha1996}Saha {\\it et al.} (1996).\nThe DoPHOT reductions followed the procedure described by\n\\markcite{saha1996}Saha {\\it et al.},\n\\markcite{lff1996} Ferrarese {\\it et al.} (1996) and \n\\markcite{silb1996}Silbermann {\\it et al.} (1996). Here we \nonly mention aspects of the reductions that were unique to NGC 1365.\nThe DoPHOT reduction procedure identifies cosmic rays when\ncombining the single epoch exposures, prior to running DoPHOT\n(\\markcite{saha1994}Saha {\\it et al.} 1994).  \nDue to the unusually long exposure times in\nthe \\ngc{1365} images, however, the number of pixels affected by cosmic\nrays was so large that significant\nnumbers of residual cosmic ray artifacts remained in the combined\nimages.  To overcome this problem, the combined images for each epoch\nwere further combined in pairs, to create a single clean master image.\nThe master image was then compared with each original image to identify\nand flag the cosmic rays.  The process was then repeated, but with\nthe pixels affected by cosmic ray events in both single epoch image pairs\nflagged.  This process was iterated several times to ensure both a\nclean master image, and that tips of bright stars were not incorrectly\nidentified as cosmic rays and removed. These master images were then used to\ngenerate a coordinate list for the DoPHOT photometry.\n\nCalibration of the DoPHOT photometry was carried out as follows. \nAperture corrections were determined from the \\ngc{1365} frames\nand applied to the raw magnitudes.\nThe magnitudes were then corrected to an exposure\ntime of 1 second, and a zero point calibration was applied to\nbring them to the 0.5 arcsec system of \\markcite{holt1995a}Holtzman \n{\\it et al.} (1995a).  These zero point corrections are as\ngiven in \\markcite{holt1995a}Holtzman {\\it et al.}, but with a \nsmall correction applied to account for differences in star and sky \napertures (\\markcite{hill1998}Hill {\\it et al.} 1998).\nThe prescription of \\markcite{holt1995a}Holtzman {\\it et al.}\nwas then used to convert the instrumental magnitudes\nto standard Johnson $V$ and Cousins $I$ magnitudes. \n\n\\subsection{Comparison of DAOPHOT and DoPHOT Photometric Systems}\n\nThe independent data reductions using ALLFRAME and DoPHOT provide a\nrobust external test for the accuracy of the profile-fitting photometry of\nthese crowded fields.  \nFigure 2 shows the comparison between the DoPHOT and ALLFRAME final photometry.\nAs expected we see increased scatter as one goes to fainter magnitudes.\nWe also see that for some of the filter and chip combinations (i.e. Chip 1 $V$\nand $I$ bands, and Chip 2 $V$ band) there are small scale errors, on the order\nof 0.01 mag mag$^{-1}$. The nature of these scale errors is still not \ncompletely\nclear at this time, but it is most likely due to small differences in the sky \ndetermination, which translate into correlations between the photometric\nerror and the star magnitudes. Artificial star simulations (discussed below) \nshow that DoPHOT is\nsomewhat more robust than ALLFRAME in separating close companions, which will\ntherefore be measured brighter by ALLFRAME than DoPHOT. This explains the \nlarger number of outlyers found with positive DoPHOT$-$ALLFRAME residuals for \nsome filter\/chip combinations.\n\n\nThe horizontal line in each panel in Figure 2 marks the average difference\nbetween DoPHOT and ALLFRAME,\nusing stars brighter than 25 mag and removing wildly discrepant stars.\nThe differences are listed in Table 2. The errors are the rms of the means.\nIn general the differences are on the order of\n$\\pm$ 0.1 mag, which is slightly larger than DoPHOT\/ALLFRAME comparisons\nfor other Key Project galaxies ($\\sim 0.07$ mag or less for\nM101, \\ngc{925}, \\ngc{3351}, \\ngc{2090}, and \\ngc{3621}). \nAn extensive set of simulations was carried out adding artificial stars to the\nNGC 1365 frames, with the intent of understanding the nature of the observed\ndifferences. The main result of these observations, discussed in detail in a\nsubsequent paper (\\markcite{lff1998b}Ferrarese {\\it et al.} 1998b), \nis that the dominant cause of\nuncertainty lies in the aperture corrections, while errors in sky \ndeterminations\nand ability to resolve close companions play only a second order role. The\npaucity of bright isolated stars in crowded fields makes the determination of\naperture corrections problematic at best. As an extreme example, the ALLFRAME \naperture corrections derived for the first and second exposure of the first \n$I$ epoch differ by 0.3 mag for the PC. \nSince focus\/jitter changes between consecutive orbit exposures are irrelevant, \n0.3 mag can be taken as a reasonable\nestimate of the uncertainty in the aperture corrections for that particular\nchip\/filter\/epoch combination (more typical variations in the aperture \ncorrections are $\\pm$ 0.07 mag for both the DoPHOT and ALLFRAME datasets). \nSimilar considerations hold for the DoPHOT\naperture corrections. \nThe DoPHOT$-$ALLFRAME differences observed for the bright\nNGC 1365 stars are therefore found to be not significant when compared to the\nuncertainty in the aperture corrections, and simply reflect the limit to which\nphotometric reduction can be pushed in these rather extreme fields. \n\n \nWe made a similar DoPHOT-ALLFRAME comparison\nfor the 34 Cepheids used to fit the period-luminosity relations (see Section 6).\nThe results of those comparisons are shown in Figure 3, and listed at the\nbottom of Table 2. The DoPHOT-ALLFRAME differences for the Cepheids show more \nscatter, as expected, since they are $\\sim$2 mag fainter than the brighter\nnonvariable stars in Figure 2, but overall, the Cepheids mirror the nonvariable\nstar DoPHOT-ALLFRAME differences.\n\n\\section{Identification of Variable Stars}\n\nTwo methods were used to search for variable stars using the ALLFRAME \ndataset. In both cases, the search for variables was done using only \nthe $V$ photometry. The $I$ photometry was used to help confirm variability and \nto determine $V-I$ colors. \nThe first method was a search for stars with unusually high dispersion in \ntheir mean $V$ magnitudes. A few candidate variables were found this way. \nThe more fruitful method employed a variation on the correlated variability \ntest suggested by \\markcite{ws1993}Welch \\& Stetson (1993).  \nFirst, the average magnitude and standard deviation \nover all epochs was calculated for each star.  Any magnitude\nmore than 2 standard deviations from the average was discarded. This removed \nmany of the cosmic ray events. As another filter, if the magnitude \ndifference between two single epoch observations was greater than 2.75 mag\nthe epoch was thrown out. Note that cosmic ray events that lead to\nreasonably measured magnitudes (i.e. about the same as the average magnitude)\nslip through these filters. \nFor each pair of observations in an epoch, the difference between\neach observation and the average magnitude is then calculated. The two\ndifferences are multiplied together and summed over all epochs. \nThe net result is that true variables will consistently have both\nsingle epoch observations brighter or fainter than the average magnitude,\nincreasing the sum over all epochs. Random high or low observations will\ntend to scatter around the average but not systematically within a single\nepoch, so nonvariable stars will have lower sum values. \nTypical values of the sum were $\\ge$ 1.0 for the variable candidates while\nthe nonvariable stars scattered around $\\le$ 0.2, with increased scatter as\none went to the faintest magnitudes. Note that we discarded bad data only to\nderive a variability index.\n\nAfter obtaining a set of variable candidates from\nthe above procedure, the photometry for each candidate was plotted against \ndate of observation. We were then able to note an approximate period for\nthe candidate as well as any observations affected by cosmic-ray events.\nAt the faint end, where we expected more contamination by nonvariable stars, \nwe were able to exclude obvious nonvariables immediately. For candidate \nvariables that passed this stage, any cosmic-ray event or otherwise\ncorrupted observations were removed in anticipation of determining a period \nof variation. Periods for the candidate variables were found using a\nphase-dispersion minimization routine as described by\n\\markcite{stell1978}Stellingwerf (1978).  The resulting light curves\nwere checked by eye to verify the best period for each candidate.\nErrors\non the periods were determined by examining changes in the light curve as \nvarious periods were used. When the light curve became visibly degraded\n(i.e. photometry points out of phase) an upper\/lower limit to the period\ncould safely be assigned.\nThese errors are subjective but provide the reader with a guide to how\nwell the light curves are sampled. As a final step, the local environment\nof each candidate was inspected to check for severe crowding.\n\nThe search for variables in the DoPHOT reductions followed closely the\nprocedure described in \\markcite{sh1990}Saha \\& Hoessel (1990) and in\n\\markcite{lff1996}Ferrarese {\\it et al.} (1996).  Candidates that\nwere classed as having a $\\geq$ 99\\% confidence of being variables\n(based on a reduced chi-squared test) were then checked for\nperiodicity using a variant of the method of \\markcite{lk1965}Lafler\n\\& Kinman (1965).  The number of spurious variables was minimized by\nrequiring that the reduced chi-squared statistic be greater than 2.0\nwhen the minimum and maximum values were removed from the calculation.\nThe light curves of each variable candidate were then inspected\nindividually and any alternate minima in the phase dispersion relation\nwere checked to see which produced the best Cepheid light\ncurve. Generally the minimum in the phase dispersion plot produced the\nbest light curve. The image of each Cepheid candidate was also\ninspected at a number of epochs. Those falling in severely crowded\nregions or in areas dominated by CCD defects were also excluded.\n\nThe two lists of candidate variables, one from the ALLFRAME photometry\nand one from the DoPHOT photometry, were then compared. Any candidates found\nin only one dataset were located in the other dataset and checked for\nvariability. \nCandidates that appeared to be real variables in {\\it both} the\nALLFRAME and DoPHOT datasets make up our final sample of   \n52 Cepheids in \\ngc{1365}. \nFinder charts for the 52 Cepheids are shown in Figures 4 and 5.\nThe Cepheid astrometry, periods, and period errors are given in Table 3.\nThe variables have been labeled V1 through V52 in order of descending period.\nColumn 1 in Table 3 identifies the Cepheids. Column 2 lists the CCD chip the\nCepheid is on. Chip 1 is the Planetary Camera and Chips 2-4 are the\nWide Field Camera chips. Columns 3 and 4 list the pixel position of\neach Cepheid, as found on image u2s70202t (see Table 1). Columns\n5 and 6 give the right ascension and declination in J2000 coordinates\nfor each Cepheid.  Column 7 lists the Cepheid period in days. Column 8\ngives the period error in days and column 9 lists the logarithm of the\nperiod. In addition to these 52 variables, \nthere are several stars that appear to be definitely variable but for \none reason or another did not make it into our list of definite Cepheids.\nReasons include uncertain periods, questionable environment (very crowded)\nor variability seen in either the ALLFRAME or DoPHOT dataset but not both.\nPositions, mean $V$ and $I$ magnitudes, and possible periods for these stars \nare given in Table A2 of the appendix.\n\n\\section{Variable Light Curves and Parameters}\n  \nTo construct the ALLFRAME light curves, magnitudes obtained from images \ntaken within a single epoch were averaged, with the resulting mean magnitude \nthen plotted. Light curves for the Cepheids are shown in Figure 6 and the\nALLFRAME photometry is listed \nin Tables 4 and 5. The error bars in Figure 6 are \naverages of the two single exposure errors as reported by ALLFRAME that make up \neach epoch. In cases where one of the two single exposure magnitudes is\nbad (i.e. cosmic ray event), the magnitude and error are from the surviving \nmeasurement. As with previous Key Project galaxies, ALLFRAME \noverestimates the error for a given photometric measurement in these \nundersampled WFPC2 images. This effect is discussed in the analysis of WFPC2 \ndata from M101 by \\markcite{pbs1998a}Stetson {\\it et al.} (1998a).\nBriefly, when we\ncompare photometric measurements within a given epoch for \\ngc{1365}\nwe find that the difference between them is significantly less than\nthe errors quoted by ALLFRAME. Typical real photometric differences\nare $\\pm$ 0.06 mag, at the magnitude level of the Cepheids,\nfor both $V$ and $I$, while ALLFRAME\nreports errors significantly larger than $\\pm$ 0.10 mag.\n\nMean $V$ magnitudes for the Cepheids were determined two different ways.\nFirst, as in other papers in this series, since the observations were \npreselected to evenly sample a\ntypical 10$-$60 day Cepheid light curve \n(\\markcite{free1994}Freedman {\\it et al.} 1994), \nunweighted intensity averaged\nmean magnitudes were calculated.  Second, phase-weighted mean intensity \nmagnitudes $<m>$ were also calculated using\n\n\\begin{equation}\n<m> = -2.5\\log[\\sum_{i}^{N} 0.5(\\phi_{i+1} - \n\\phi_{i-1})10^{-0.4m_{i}}]\n\\end{equation}\n\n\\noindent \nwhere $\\phi$ is the phase, and the sum is over the entire light\ncycle. The average difference between the unweighted and\nphase-weighted intensity averaged mean $V$ magnitudes is only $-$0.027\n$\\pm$ 0.002 mag for the 52 Cepheids.  This difference is quite small,\nas expected, since most of the Cepheids have nearly uniformly sampled\nlight curves.\n\nWith only four $I$ observations, total mean $I$ magnitudes were calculated\nas follows. Using the $V$ and $I$ magnitudes at the four $I$ epochs,\naverage $V$ and average $I$ magnitudes were calculated. Then, the\ndifference between the four-epoch $V$ average ($<V>_{4}$) and 12-epoch\n$V$ mean magnitude ($<V>_{12}$) was calculated for each Cepheid.\nSince the amplitude of Cepheids in $V$ is almost exactly twice the\namplitude in $I$ (V:I = 1.00:0.51, \\markcite{free1988} Freedman 1988),\nthe four-epoch $I$ magnitude was corrected to obtain the full\n12-epoch $I$ magnitude, as follows:\n\n\\begin{equation}\n<I>_{12}~=~<I>_{4} + 0.51(<V>_{12} - <V>_{4}).\n\\end{equation}\n\n\\noindent \nCosmic-ray corrupted data were removed before determining mean\nmagnitudes.  Figure 7 shows the $I$ vs\n($V-I$) color-magnitude diagram for the HST field of \\ngc{1365},\nhighlighting the 52 Cepheids. The Cepheids fall neatly between\nthe well-defined blue plume and weak red plume of supergiants.\n\nTable 6 lists derived ALLFRAME photometric parameters for the Cepheids.\nColumn 1 identifies the Cepheids. \nColumns 2-5 list the intensity-average and phase-weighted mean $V$\nmagnitudes and errors.\nColumns 6-9 list the intensity-average and phase-weighted mean $I$\nmagnitudes and errors. \nAll of the errors listed in Table 6 are mean magnitude dispersions.\nThey reflect the uncertainty due to the star's variability \n(the amplitude of variation)\nand are not derived from the overestimated ALLFRAME errors.\nColumn 10 and 11 lists the intensity-averaged and phase-weighted $V-I$ \ncolor of each Cepheid. \nA symbol in column 12 \nindicates the Cepheid was used in the Cepheid period-luminosity relation\nfit to determine the distance to \\ngc{1365} based on the criteria listed\nin the next section. \n\n\\section{Period-Luminosity Relations and the Distance to \\ngc{1365}}\n\nStandard period-luminosity (PL) relations for the LMC Cepheids are\nadopted from \\markcite{mf1991} Madore \\& Freedman (1991) which\nassume a true LMC distance modulus of 18.50 mag and total\nline-of-sight LMC Cepheid reddening of E($B-V$) = 0.10 mag:\n\n\\begin{equation}\nM_{V} = -2.76\\log {\\rm P} - 1.40\n~~~{\\rm and}~~~ M_{I} = -3.06\\log {\\rm P} - 1.81 .\n\\end{equation}\n\nTo determine the \ndistance to \\ngc{1365} a subset of the 52 Cepheids were chosen based on the \nfollowing\ncriteria. The period of the Cepheid had to be between 10 and 47 days.\nThe lower limit of 10 days is common for all of the Key Project galaxies\nand was chosen to avoid first overtone pulsators which have periods less \nthan $\\sim$ 10 days (Madore \\& Freedman 1991). The longest period of 47 days \nwas estimated from our observing window of 49 days and our actual sequence\nof observations throughout that window. \nNone of our Cepheids have periods under\n10 days but 5 Cepheids have periods $> 47$ days, so they are\nexcluded from the fit.\nEach variable also had to have a Cepheid-like light curve and\nthe same period, to within 10\\%, in the ALLFRAME and DoPHOT datasets.\nNext, to exclude Cepheids that were too crowded, each Cepheid had to \ncontribute more than 50\\% of\nthe light within a 2 pixel box surrounding it.\nOur last criterion was that each Cepheid \nhad to have a typical Cepheid-like color, 0.5 $\\leq V-I \\leq$ 1.5.\nThere are a total of 34 Cepheids that satisfied all of the above criteria and\nthey are indicated in column 11 of Table 6. These 34 Cepheids were used to\nfit the PL relations and determine the distance to \\ngc{1365}.\n\nIn order to avoid incompleteness bias in fitting a slope to the\n\\ngc{1365} data, only the zero point of the regression was fitted,\nwith the slope of the fit fixed to the LMC values.\nThe phase-weighted $V$ and $I$ PL relations are shown in Figure 8. The filled \ncircles are the 34 Cepheids used to fit the PL relations, while the open \ncircles are the other Cepheids. The solid line in each\nfigure represents the best fit to each dataset. The dashed lines drawn\nat $\\pm$ 0.54 mag for the $V$ PL relation and $\\pm$ 0.36 mag for the\n$I$ PL relation represent 2-sigma deviations from the mean PL\nrelations. In the absence of significant differential reddening the intrinsic\nwidth of the Cepheid instability strip is expected to place the\n\\ngc{1365} Cepheids within these limits. \nWe note that the full sample of 52 Cepheids are well contained within\nthe $V$ and $I$ instability strips in Figure 8.\nFrom the PL fits, the apparent distance moduli to \\ngc{1365}\nare $\\mu_{V}$ = 31.70 $\\pm$ 0.05 mag\\, and $\\mu_{I}$ = 31.54 $\\pm$ 0.06 mag\\,,\nwhere the errors are\ncalculated from the observed scatter in the \\ngc{1365} PL data\nthemselves, appropriately reduced by the sample size of Cepheids.\n\nThe observed difference in the apparent distance moduli for \\ngc{1365}\ngives $\\mu_{V} - \\mu_{I}$ = $E(\\hbox{\\it V--I)}$ = 0.16 $\\pm$ 0.08 mag\\,. \nThe Key Project has adopted a reddening law of $R_{V} = A_{V}\/E(V-I) = 2.45$\nwhich is consistent with the work of Dean, Warren \\& Cousins (1978),\n\\markcite{card1989}Cardelli {\\it et al.} (1989) and Stanek (1996).\nWe therefore obtain A$_{V}$ = 0.40\\,~ for this region of \\ngc{1365}.\nThe true distance modulus to \\ngc{1365} is then\n31.31 $\\pm$ 0.08 mag\\,, corresponding to a\nlinear distance of 18.3 $\\pm$ 0.7 Mpc\\, (internal errors only). \nTo test how robust our calculated distance to \\ngc{1365} was,\nwe also calculated the distance modulus using intensity averaged mean\nmagnitudes for the 34 Cepheids.\nThe resulting true distance modulus, 31.34 mag, is just slightly larger \nthan our result using phase-weighted mean magnitudes. \nAs another test we used the full compliment of \n52 Cepheids to fit the PL relations, using phase-weighted mean magnitudes,\nand obtained a distance modulus of 31.35 mag. As expected we do see a\nslight variation in the distance modulus depending on which mean magnitudes\nare used or the sample size of Cepheids but the net result is that all of\nthese values are contained within our 31.31 $\\pm$ 0.08 mag\\, distance modulus and error.\n\nThe distance to \\ngc{1365} was derived independently using the Cepheid\nparameters derived from the DoPHOT reductions. \nThe resulting apparent distance moduli\nare $\\mu_{V}$ = 31.64 $\\pm$ 0.07 mag\\, and  $\\mu_{I}$ = 31.49 $\\pm$ 0.07 mag\\,,\nwith a true modulus $\\mu_{0}$ = 31.26 $\\pm$ 0.10 mag\\,.  These differ from\nthe ALLFRAME moduli by $-$0.06, $-$0.05, and $-$0.05 mag,\nrespectively. Despite the differences in DoPHOT and ALLFRAME photometry\n(Figures 2 and 3) the true distance moduli agree very well.\nTo better understand why this is so\nwe fit PL relations separately\nfor each WFPC2 chip, as seen in Table 7. The scatter in individual chip\napparent distance moduli is quite small for the ALLFRAME photometry and\nsomewhat more scattered for the DoPHOT photometry. \nAlso, DoPHOT consistently measures a smaller \nreddening compared\nto ALLFRAME for each chip and filter, except for Chip 2. Since over one-third\nof the Cepheids used to fit the PL relations are in Chip 2, this reduces the\noverall discrepancy in reddening when combining the Cepheids in all four chips.\nAs a test, we fit PL relations excluding the Chip 2 Cepheids. \nThe resulting true\ndistance moduli are then 31.31 mag (no change) for ALLFRAME and 31.37 mag \n(change of +0.11 mag) for DoPHOT. The net effect of this comparison is\nthat the ALLFRAME and DoPHOT true distance moduli still agree within\ntheir errors. Table 7 summarizes all of these PL fit tests. Column 1 in Table 7\nindicates if the dataset used was the phase-weighted or intensity averaged\nmean $V$ and $I$ magnitudes. Column 2 lists which chip subset was used (1,2,3\nor 4) or all 4 chips (1-4). Column 3 gives the number of Cepheids used in\nthe fit. Columns 4 and 5 give the apparent distance moduli. Column 6 gives\nthe extinction in magnitudes. Column 7 lists the true distance moduli for\neach PL fit test. The ALLFRAME results are on top, and the corresponding\nDoPHOT results are at the bottom.\n\nTo check for any effects due to incompleteness at the faintest magnitudes\nwe split the final sample of 34 Cepheids into two sets of 17 Cepheids,\na bright and a faint sample, sorted using the $V$ phase-weighted ALLFRAME\nphotometry. The true distance modulus is\n31.36 $\\pm$ 0.11 mag for the bright sample and 31.28 $\\pm$ 0.12 mag for\nthe faint sample. The slight changes in distance moduli seen by excluding\nthe brightest or faintest Cepheids are not significant compared to the\nerrors and we conclude that we\nare not affected by incompleteness at the faintest magnitudes.\n\n\\subsection{Error Budget}\n\nTable 8 presents the error budget for the distance to \\ngc{1365}. \nThe errors are classified as either random or systematic based on how we\nwill use \n\\ngc{1365} to determine the Hubble Constant. For example, the LMC distance\nmodulus uncertainty and PL relation zero point uncertainties\nare systematic errors because the Key Project is using the same LMC distance\nmodulus and Cepheid PL relation slopes for all of our\ngalaxies. We now discuss each source of error in Table 8 in detail.\n\nThe Key Project has adopted an LMC distance modulus of 18.50 $\\pm$ 0.10 mag.\nThe review of published distances to the LMC, via various techniques,\nby \\markcite{west1997}Westerlund (1997) (his Table 2.8) indicates that the \ndistance modulus\nto the LMC is still uncertain at the 0.10 mag level. \nMore recently, \nGould \\& Uza (1998), using observations of a light echo from\nSupernova 1987A, suggest an LMC distance modulus no larger than \n18.37 $\\pm$ 0.04 mag \nor 18.44 $\\pm$ 0.05 mag for a circular or elliptical ring respectively. \nPanagia {\\it et al.} (1998) used HST observations of the ring\naround SN 1987A to obtain an LMC distance modulus of 18.58 $\\pm$ 0.05 mag.\nWork by Wood, Arnold, \\& Sebo (1997) using models of the LMC Cepheid HV 905 \nproduced an LMC distance modulus of 18.51 $\\pm$ 0.05 mag. Recent results from \nthe MACHO Project (Alcock {\\it et al.} 1997)\nusing double mode RR Lyrae stars give an LMC distance modulus of \n18.48 $\\pm$ 0.19 mag.\nIn light of these recent results we feel that our adopted LMC distance modulus\nand error are still valid.\n\n\\markcite{mf1991}Madore \\& Freedman (1991) fit Cepheid PL\nrelations\nto 32 LMC Cepheids using {\\it BVRI} photoelectric photometry. The dispersions\nfor a single point about the PL relations are $\\pm$ 0.27 mag for $V$ and \n$\\pm$ 0.18 mag for $I$. The PL zero point errors are obtained \nby dividing by the square root of the number of Cepheids used in the fit,\ngiving us \n$\\pm$ 0.05 mag for $V$ and $\\pm$ 0.03 for $I$, items (2) and (3) in Table 8.\n\nThe LMC distance modulus and PL relation zero point uncertainties\nare sources of systematic error within the Key Project, since we use\nthe Madore \\& Freedman equations for all of the galaxies, and all galaxy\ndeterminations will change systematically as improvements to the zero point\nbecome available. \nCombining items (1), (2), and (3) in Table 8 in quadrature we obtain a PL \nrelation uncertainty of $\\pm$ 0.12 mag.\n\n\\markcite{hill1998}Hill {\\it et al.} (1998) estimated our WFPC2 zero point \nuncertainties to be \n$\\pm$ 0.02 mag for both $V$ and $I$ by comparing ground-based and HST \nobservations of M100. In addition we must include the uncertainties in the \naperture corrections.  \nWhile there is one case where two aperture corrections within an epoch\ndiffered by 0.3 mag (Chip 1 $I$ band, first epoch), the scatter about the mean \nALLFRAME aperture corrections is \n$\\pm$ 0.07 mag for $V$ and $\\pm$ 0.06 mag for $I$.\nWe will take these to be the errors in zero point due to the aperture\ncorrections. We then combine the \\markcite{hill1998}Hill {\\it et al.} errors \nand the aperture\ncorrection errors in quadrature to obtain WFPC2 zero point errors of \n$\\pm$ 0.07 mag in $V$ and $\\pm$ 0.06 mag in $I$, items (4) and (5) in Table 8. \nSince the photometric errors in the two bands are uncorrelated\nwe combine items (4) and (5) in quadrature,\nbut we must weight these errors, $\\sigma_{V}$ and $\\sigma_{I}$, by the \ndiffering effects of reddening, as given by\n$[(1-R)^{2}(\\sigma_{V})^{2} + R^{2}(\\sigma_{I})^{2}]^{1\/2}$.\nAs stated previously, we have adopted a reddening law of\n$R_{V}= A_{V}\/E(V-I)$ = 2.45. Our photometric contribution to the error in\nthe distance modulus is therefore $\\pm$ 0.18 mag.\n\nEarly work by the Key Project discovered what appeared to be a long versus\nshort exposure zero point offset, such that stars in exposures \nlonger than approximately\n1000 seconds were systematically brighter, by approximately 0.05 mag, than\nstars in exposures of less than 1000 seconds duration. The effect is now\nthought to be a charge transfer efficiency effect in the WFPC2 CCDs. This\neffect is discussed in detail in \\markcite{hill1998}Hill {\\it et al.} (1998) \nand also in\n\\markcite{wh1997}Whitmore \\& Heyer (1997). For the Key Project\nwe have used the long exposure zero point for our photometric calibration.\nAt this time, an offset of +0.05 mag for both $V$ and $I$ is thought to be the \nbest estimate to correct for this effect. \nAdditional work on the zero points is currently\nbeing undertaken by \\markcite{pbs1998b}Stetson {\\it et al.} (1998b) \nbased on an \nextensive set of ground based and HST data.\nThis effect is included in Table 8 as a source of systematic error in the\ndistance to \\ngc{1365}.\n\nThere is an additional distance modulus error introduced from fitting the\nCepheid PL relations. \nFrom Section 5, these errors are $\\pm$ 0.05 mag\nin $V$ and $\\pm$ 0.06 mag in $I$, and  include photometric errors,\ndifferential reddening and the apparent width of the instability strip in\nboth $V$ and $I$. These errors are combined in quadrature in item (R2)\nin Table 8.\n\nThere is concern that the Cepheid PL relation may have a metallicity\ndependence. The Cepheids in the calibrating LMC are thought to be\nrelatively metal poor compared Fornax and Virgo cluster galaxies\n(Kennicutt {\\it et al.} 1998),\nso a metallicity dependence in the Cepheid PL relation\nwould be a source of systematic error in our distance determination.\nKennicutt {\\it et al.} measured\na marginal metallicity dependence in one of the Key Project\ngalaxies, M101, leading to a\nshift in distance modulus of \n$\\delta(m-M)_0\/\\delta[O\/H] = -0.24 \\pm 0.16$ mag\/dex.\nWe can use this relation to estimate the systematic error in distance modulus \nto \\ngc{1365} due to a metallicity dependence on the Cepheid PL relations.\nZaritsky, Kennicutt \\& Huchra (1994)\nfound the oxygen abundance in \\ngc{1365} to be $12 + {\\rm log}(O\/H) = 9.0$\nwhile the measured oxygen\nabundance in the LMC on this scale is $12 + {\\rm log}(O\/H) = 8.50$ \n(Kennicutt {\\it et al.} 1998).\nThe 0.5 dex difference in oxygen abundance would, if applied, \nincrease the true distance modulus by +0.12 mag or a 6\\% \nincrease in the distance to \\ngc{1365}.\n\nThe total uncertainty in distance modulus to \\ngc{1365} due to random\nerrors is obtained by combining the distance modulus and PL fit uncertainties\nin quadrature:\n$[0.18^{2} + 0.08^{2}]^{1\/2}$ = $\\pm$ 0.20 mag. \n\n\nThe total systematic error in the distance to \\ngc{1365} is obtained by\ncombining the systematic errors, the LMC Cepheid PL relation, a possible \nmetallicity dependence, and the long versus short exposure zero point,\nin quadrature:\n$[0.12^{2} + 0.12^{2}+0.05^{2}]^{1\/2}$ = $\\pm$ 0.18 mag.\nThus our final distance modulus to \\ngc{1365} is \n31.31 $\\pm$ 0.20 (random) $\\pm$ 0.18 (systematic).\nTh implications of a Cepheid distance to \\ngc{1365} and the Fornax\ncluster are discussed in the companion paper Madore {\\it et al.} (1998).\n\n\\section{Conclusion}\nWe have used the HST WFPC2 instrument to obtain 12 epochs in $V$ and\n4 epochs in $I$ of the eastern part of \\ngc{1365} located in the Fornax Cluster.\nThe images were reduced separately using the ALLFRAME and DoPHOT \nphotometry packages. The raw photometry was transformed to standard\nJohnson $V$ and Cousins $I$ photometry via Hill {\\it et al.} (1998)\nzero points, Holtzman {\\it et al.} (1995a) color terms, and \naperture corrections\nderived from the \\ngc{1365} images. The $\\lesssim$ 0.1 mag difference\nbetween the final ALLFRAME and DoPHOT photometry is most likely due to\nthe relatively large uncertainties in both the ALLFRAME and DoPHOT\naperture corrections, which show a scatter of $\\pm$ 0.07 mag. \n\nA total of 52 Cepheids\nwere discovered in \\ngc{1365}, ranging in period from 14 to 60 days. \nA subset of 34 Cepheids was chosen based on period, light curve\nshape, color and relative crowding to fit Cepheid period-luminosity relations\nusing fixed slopes from \n\\markcite{mf1991} Madore \\& Freedman (1991). A distance modulus of\n31.31 $\\pm$ 0.20 (random) $\\pm$ 0.18 (systematic) mag, corresponding to \n18.3 $\\pm$ 1.7 (random) $\\pm$ 1.6 (systematic) Mpc is obtained from the \nALLFRAME photometry. A distance modulus of 31.26 $\\pm$ 0.10 mag\\, (internal errors only) \nwas obtained from the DoPHOT photometry.\nThe average reddening in this region of \\ngc{1365} was measured to be\n$E(\\hbox{\\it V--I)}$ = 0.16 $\\pm$ 0.08 mag\\, using ALLFRAME and 0.15 $\\pm$ 0.10 mag \nusing DoPHOT.\nThe apparent agreement in distance between ALLFRAME and DoPHOT despite the\nobserved $\\lesssim$ 0.1 mag photometric differences is due to DoPHOT measuring\na lower reddening to \\ngc{1365} in WFPC2 Chips 1, 3 and 4 and a higher\nreddening than ALLFRAME in Chip 2. Over one-third of the Cepheids in the\nPL fit sample are found in Chip 2, reducing the {\\it mean} difference in\nreddening and distance modulus between DoPHOT and ALLFRAME.\nThe companion paper by Madore {\\it et al.} (1998) discusses in detail\nthe implications of a Cepheid distance to \\ngc{1365}.\n\n\\acknowledgments\nThe work presented in this paper is based on observations made by the\nNASA\/ESA Hubble Space Telescope, obtained by the Space Telescope\nScience Institute, which is operated by AURA, Inc. under NASA contract\nNo. 5-26555.  We gratefully acknowledge the support of the NASA and\nSTScI support staff, with special thanks our program coordinator, Doug\nVan Orsow.  Support for this work was provided by NASA through grant\nGO-2227-87A from STScI. \nThis paper is based partially on data obtained at the Las Campanas\nObservatory 2.5m telescope in Chile, owned and operated by the Carnegie\nInstitution of Washington.\nSMGH and PBS are grateful to NATO for travel support\nvia a Collaborative Research Grant (960178).\nLF acknowledges support by NASA through Hubble Fellowship grant\nHF-01081.01-96A awarded by the Space Telescope Science Institute,\nwhich is operated by AURA, Inc., for NASA under contract NSA 5-26555.\nThe research described in this paper was partially carried out by the Jet\nPropulsion Laboratory, California Institute of Technology, under a\ncontract with the National Aeronautics and Space Administration.  This\nresearch has made use of the NASA\/IPAC Extragalactic Database (NED)\nwhich is operated by the Jet Propulsion Laboratory, California\nInstitute of Technology, under a contract with the National\nAeronautics and Space Administration. \n\n\n\n\n\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\vspace{-2mm}\n\nInherent uncertainties derived from different root causes have realized as serious hurdles to find effective solutions for real world problems. Critical safety concerns have been brought due to lack of considering diverse causes of uncertainties, resulting in high risk due to misinterpretation of uncertainties (e.g., misdetection or misclassification of an object by an autonomous vehicle).  Graph neural networks (GNNs)~\\cite{kipf2017semi, velickovic2018graph} have received tremendous attention in the data science community. Despite their superior performance in semi-supervised node classification and regression, they didn't consider various types of uncertainties in the their decision process.  Predictive uncertainty estimation~\\cite{kendall2017uncertainties} using Bayesian NNs (BNNs) has been explored for classification prediction and regression in the computer vision applications, based on aleatoric uncertainty (AU) and epistemic uncertainty (EU). AU refers to data uncertainty from statistical randomness (e.g., inherent noises in observations) while EU indicates model uncertainty due to limited knowledge (e.g., ignorance) in collected data.  In the belief or evidence theory domain, Subjective Logic (SL)~\\cite{josang2018uncertainty} considered vacuity (or a lack of evidence or ignorance) as uncertainty in a subjective opinion. Recently other uncertainty types, such as dissonance, consonance, vagueness, and monosonance~\\cite{josang2018uncertainty}, have been discussed based on SL to measure them based on their different root causes. \n\n\\vspace{-1mm}\nWe first considered multidimensional uncertainty types in both deep learning (DL) and belief and evidence theory domains for node-level classification, misclassification detection, and out-of-distribution (OOD) detection tasks.  By leveraging the learning capability of GNNs and considering multidimensional uncertainties, we propose a uncertainty-aware estimation framework by quantifying different uncertainty types associated with the predicted class probabilities.  \nIn this work, we made the following {\\bf key contributions}:\n\\vspace{-2mm}\n\\begin{itemize}[leftmargin=*, noitemsep]\n\\item \\textbf{A multi-source uncertainty framework for GNNs}. Our proposed framework first provides the estimation of various types of uncertainty from both DL and evidence\/belief theory domains, such as dissonance (derived from conflicting evidence) and vacuity (derived from lack of evidence).  In addition, we designed a Graph-based Kernel Dirichlet distribution Estimation (GKDE) method to reduce errors in quantifying predictive uncertainties.\n\\item \\textbf{Theoretical analysis}:  Our work is the first that provides a theoretical analysis about the relationships between different types of uncertainties considered in this work.  We demonstrate via a theoretical analysis that an OOD node may have a high predictive uncertainty under GKDE.\n\\item \\textbf{Comprehensive experiments for validating the performance of our proposed framework}: Based on the six real graph datasets, we compared the performance of our proposed framework with that of other competitive counterparts. We found that the dissonance-based detection yielded the best results in misclassification detection while vacuity-based detection best performed in OOD detection. \n\\end{itemize}\n\\vspace{-2mm}\nNote that we use the term `predictive uncertainty' in order to mean uncertainty estimated to solve prediction problems.\n\n\\vspace{-2mm}\n\\section{Related Work} \\label{sec:related-work}\n\\vspace{-2mm}\nDL research has mainly considered {\\it aleatoric} uncertainty (AU) and {\\it epistemic} uncertainty (EU) using BNNs for computer vision applications.  AU consists of homoscedastic uncertainty (i.e., constant errors for different inputs) and heteroscedastic uncertainty (i.e., different errors for different inputs)~\\cite{gal2016uncertainty}.  A Bayesian DL framework was presented to simultaneously estimate both AU and EU in regression (e.g., depth regression) and classification (e.g., semantic segmentation) tasks~\\cite{kendall2017uncertainties}.  Later, {\\em distributional uncertainty} was defined based on distributional mismatch between testing and training data distributions~\\cite{malinin2018predictive}.  {\\em Dropout variational inference}~\\cite{gal2016dropout} was used for an approximate inference in BNNs using epistemic uncertainty, similar to \\textit{DropEdge}~\\cite{rong2019dropedge}.  Other algorithms have considered overall uncertainty in node classification~\\cite{eswaran2017power, liu2020uncertainty, zhang2019bayesian}. However, no prior work has considered uncertainty decomposition in GNNs. \n\nIn the belief (or evidence) theory domain, uncertainty reasoning has been substantially explored, such as Fuzzy Logic~\\cite{de1995intelligent}, Dempster-Shafer Theory (DST)~\\cite{sentz2002combination}, or Subjective Logic (SL)~\\cite{josang2016subjective}.  Belief theory focuses on reasoning inherent uncertainty in information caused by unreliable, incomplete, deceptive, or conflicting evidence.  SL considered predictive uncertainty in subjective opinions in terms of {\\em vacuity} (i.e., a lack of evidence) and {\\em vagueness} (i.e., failing in discriminating a belief state)~\\cite{josang2016subjective}. Recently, other uncertainty types have been studied, such as {\\em dissonance} caused by conflicting evidence\\cite{josang2018uncertainty}.\nIn the deep NNs, \\cite{sensoy2018evidential} proposed evidential deep learning (EDL) model, using SL to train a deterministic NN for supervised classification in computer vision based on the sum of squared loss.  However, EDL didn't consider a general method of estimating multidimensional uncertainty or graph structure. \n\\vspace{-2mm}\n\\section{Multidimensional Uncertainty and Subjective Logic}\n\\vspace{-2mm}\n\nThis section provides the overview of SL and discusses SL-based multiple types of uncertainties, called {\\em evidential uncertainty}, with the measures of \\textit{vacuity} and \\textit{dissonance}.  In addition, we give a brief overview of {\\em probabilistic uncertainty}, discussing the measures of \\textit{aleatoric} and \\textit{epistemic}.\n\n\\vspace{-2mm}\n\\subsection{Subjective Logic}\\label{SL}\n\\vspace{-2mm}\nSL offers the formulation of a subjective opinion based on both probabilistic logic (PL)~\\cite{nilsson1986probabilistic} and belief theory (BT)~\\cite{shafer1976mathematical} with two unique extensions.  First, SL explicitly represents uncertainty by introducing vacuity of evidence (or uncertainty mass) in its opinion representation.  This addresses the limitations of PL by modeling a lack of confidence in probabilities.  Second, SL extends the traditional BT by incorporating base rates as the prior probabilities in Bayesian theory.  The Bayesian nature of SL allows it to use second-order uncertainty to express and reason the uncertainty mass, where second-order uncertainty is represented by a probability density function (PDF) over first-order probabilities~\\cite{josang2016subjective}.  For multi-class problems, we use a multinomial distribution (i.e., first-order uncertainty) to model class probabilities and a Dirichlet PDF (i.e., second-order uncertainty) to model the distribution of class probabilities. Second-order uncertainty enriches uncertainty representation with evidence information, playing a key role in \ndistinguish OOD from conflicting prediction as detailed later. \n\n\\vspace{-1mm}\nOpinions are the arguments in SL. In the multi-class setting, the multinomial opinion of a random variable $y$ in domain $\\mathbb{Y}=\\{1,...,K\\}$ is given by a triplet as: \n\\vspace{-1mm}\n\\begin{align}\n\\label{eq:so}\n    \\omega =  ({\\bm{b}},u,{\\bm{a}}), \\text{ with }  \\sum\\nolimits_{k=1}^K b_k + u =1,\n\\end{align}\nwhere ${\\bm{b}} = (b_1, \\ldots,b_K)^T ,u$, and ${\\bm{a}}=(a_1, \\ldots, a_K)^T$ denote the belief mass distribution over $\\mathbb{Y}$, uncertainty mass representing vacuity of evidence, and base rate distribution over $\\mathbb{Y}$, respectively, and $\\forall k, a_k\\ge 0, b_k\\ge 0, u\\ge0$. The probability that $y$ is assigned to the $k$-class is given by $ P(y = k)= b_k + a_ku$,\nwhich combines the belief mass with the uncertain mass using the base rates. In the multi-class setting, $a_k$ can be regarded as the prior preference over the $k$-th class. When no specific preference is given, we assign all the base rates as $1\/K$.\n\\vspace{-1mm}\n\\subsection{Evidential Uncertainty}\n\\vspace{-1mm}\nIn this section, we explain how the second order uncertainty (evidential uncertainty) is derived from the first order uncertainty as a Dirichlet PDF.  Given a set of random variables\n $\\textbf{p}=(p_1,...,p_K)^T$, where $\\textbf{p}$ is distributed on a simplex of dimensionality $K-1$, a conditional distribution $P(y = k|\\textbf{p})=p_k$ can be represented by the marginal distribution, $P(y)=\\int P(y|\\textbf{p})p(\\textbf{p})d\\textbf{p}$. We define $p(\\textbf{p})$ as a Dirichlet PDF over $\\textbf{p}$: $\\text{Dir}(\\textbf{p}|\\boldsymbol{\\alpha})$, where $\\boldsymbol{\\alpha}=(\\alpha_1, \\ldots, \\alpha_K)^T$ is a $K$-dimensional strength vector, with $\\alpha_k\\ge0$ denoting the effective number of observations of the $k$-th class. SL explicitly introduces uncertain evidence through a weight $W$ representing non-informative evidence and redefines the strength parameter as: \n    $\\alpha_k=e_k+a_kW, \\text{ with } e_k \\ge 0, \\forall k \\in \\mathbb{Y}$,\nwhere $e_k$ is the amount of evidence (or the number of observations) to support the $k$-th class and $W$ is usually set to $K$, i.e., the number of classes. Given the new definition of the strength parameter, the expectation of the class probabilities $\\textbf{p}=(p_1, \\ldots, p_K)^T$ is given by: \n\\begin{align}\n    \\mathbb{E}[p_k]=\\frac{\\alpha_k}{\\sum_{j=1}^K \\alpha_j}=\\frac{e_k+a_kW}{\\sum_{j=1}^K e_j + W}, \n\\end{align}\nwhere $a_k=1\/K$. By marginalizing out $\\textbf{p}$, we can derive an evidence-based expression of belief mass and uncertainty mass: \n\\begin{align}\n\\label{eq:belief}\n    b_k= \\frac{e_k}{S} \\quad \\forall k \\in \\mathbb{Y}, \\quad u_v= \\frac{W}{S}, \\text{ with } S = \\sum_{k=1}^K \\alpha_k. \n\\end{align}\nSL categorizes uncertainty into two primary sources~\\cite{josang2016subjective}: (1) basic belief uncertainty derived from single belief masses, and (2) intra-belief uncertainty based on the relationships between different belief masses. These two sources of uncertainty can be boiled down to {\\em \\textbf{vacuity}} and {\\em \\textbf{dissonance}}, respectively, that correspond to vacuous belief and contradicting beliefs.  In particular, vacuity of an opinion $\\omega$ is captured by uncertainty mass $u_v$ in~\\eqref{eq:belief} while dissonance of an opinion~\\cite{josang2018uncertainty} is formulated by:\n\\begin{align}\n\\label{eq:dis}\n    diss(\\omega)=\\sum_{k=1}^{K}\\Big(\\frac{b_k \\sum_{j\\neq k}b_j \\text{Bal}(b_j,b_k)}{\\sum_{j\\neq k}b_j}\\Big), \\text{Bal}(b_j,b_k)=\n  \\begin{cases} \n  1-\\frac{|b_j-b_k|}{b_j+b_k} & \\text{if $b_i b_j \\neq 0$}\\\\\n  0 & \\text{if $\\min(b_i,b_j)=0$}, \n  \\end{cases}\n\\end{align}\nwhere $\\text{Bal}(b_j,b_k)$ is the relative mass balance function between two belief masses. The belief dissonance of an opinion is measured based on how much belief supports individual classes. Consider a binary classification example with a binomial opinion given by $(b_1,b_2,u,{\\bm{a}}) = (0.49, 0.49, 0.02, {\\bm{a}})$. Based on \\eqref{eq:dis}, it has a dissonance value of $0.98$. In this case, although the vacuity is close to zero, a high dissonance indicates that one cannot make a clear decision because both two classes have the same amount of supporting both beliefs, which reveals strong conflict within the opinion.\n\n\\vspace{-1mm}\n\\subsection{Probabilistic Uncertainty}   \n\\label{sect:multi-dim uncertainty}\n\\vspace{-1mm}\nFor classification, the estimation of the probabilistic uncertainty relies on the design of an appropriate Bayesian DL model with parameters $\\bm{\\theta}$. Given input $x$ and dataset $\\mathcal{G}$, we estimate a class probability by $P(y|x) = \\int P(y|x;\\bm{\\theta}) P(\\bm{\\theta}|\\mathcal{G}) d\\bm{\\theta}$, and obtain \\textbf{\\textit{epistemic uncertainty}} estimated by mutual information~\\cite{depeweg2018decomposition, malinin2018predictive}:\n\\begin{eqnarray}\n\\footnotesize\n\\vspace{-2mm}\n\\underbrace{I(y, \\bm{\\theta}|x, \\mathcal{G})}_{\\text{\\textbf{\\textit{Epistemic}}}} =\\underbrace{\\mathcal{H}\\big[ \\mathbb{E}_{P(\\bm{\\theta}|\\mathcal{G})}[P(y|x;\\bm{\\theta})] \\big]}_{\\text{\\textbf{\\textit{Entropy}}}} -  \\underbrace{\\mathbb{E}_{P(\\bm{\\theta}|\\mathcal{G})}\\big[\\mathcal{H}[P(y|x;\\bm{\\theta})] \\big]}_{\\text{\\textbf{\\textit{Aleatoric}}}}, \n\\vspace{-2mm} \\label{eq:epistemic}\n\\end{eqnarray}\nwhere $\\mathcal{H}(\\cdot)$ is Shannon's entropy of a probability distribution. The first term indicates {\\bf \\textit{entropy}} that represents the total uncertainty while the second term is {\\bf \\textit{aleatoric}} that indicates data uncertainty.  By computing the difference between entropy and aleatoric uncertainties, we obtain epistemic uncertainty, which refers to uncertainty from model parameters. \n\n\\section{Relationships Between Multiple Uncertainties}\n\\begin{wrapfigure}{R}{0.45\\textwidth}\n  \\vspace{-5mm} \n  \\centering\n  \\includegraphics[width=0.42\\textwidth]{fig\/fig_1.png}\n  \\vspace{-2mm}\n  \\caption{\\footnotesize{Multiple uncertainties of different prediction. Let ${\\bf u}=[u_v, u_{diss}, u_{alea}, u_{epis}, u_{en}]$.\\vspace{-5mm}}}\\label{fig:example}\n  \\vspace{-1mm}\n\\end{wrapfigure}\nWe use the shorthand notations $u_{v}$, $u_{diss}$, $u_{alea}$, $u_{epis}$, and $u_{en}$ to represent vacuity, dissonance, aleatoric, epistemic, and entropy, respectively.  \n\nTo interpret multiple types of uncertainty, we show three prediction scenarios of 3-class classification in Figure~\\ref{fig:example}, in each of which the strength parameters $\\alpha = [\\alpha_1, \\alpha_2, \\alpha_3]$ are known.  To make a prediction with high confidence, the subjective multinomial opinion, following a Dirichlet distribution, will yield a sharp distribution on one corner of the simplex (see Figure~\\ref{fig:example} (a)). For a prediction with conflicting evidence, called a conflicting prediction (CP), the multinomial opinion should yield a central distribution, representing confidence to predict a flat categorical distribution over class labels (see Figure~\\ref{fig:example} (b)).  For an OOD scenario with $\\alpha=[1, 1, 1]$, the multinomial opinion would yield a flat distribution over the simplex (Figure~\\ref{fig:example} (c)), indicating high uncertainty due to the lack of evidence.  The first technical contribution of this work is as follows.\n\n\\begin{restatable}{theorem}{primetheorem}\nWe consider a simplified scenario, where a multinomial random variable $y$ follows a K-class categorical distribution: $y \\sim \\text{Cal}(\\textbf{p})$, the class probabilities $\\textbf{p}$ follow a Dirichlet distribution: $\\textbf{p}\\sim \\text{Dir}({\\bm \\alpha})$, and ${\\bm \\alpha}$ refer to the Dirichlet parameters. Given a total Dirichlet strength $S=\\sum_{i=1}^K \\alpha_i$, \nfor any opinion $\\omega$ on a multinomial random variable $y$, we have\n\\vspace{-2mm}\n\\begin{enumerate}\n\\item General relations on all prediction scenarios. \n    \n(a) $u_v+ u_{diss} \\le 1$; (b) $u_v > u_{epis}$.\n    \n   \n   \n   \n   \n    \\item Special relations on the OOD and the CP.\n    \\begin{enumerate}\n        \\item For an OOD sample with a uniform prediction (i.e., $\\alpha=[1, \\ldots, 1]$), we have \n    \\begin{eqnarray}\n    1= u_v = u_{en}>  u_{alea} > u_{epis} > u_{diss} = 0 \\nonumber\n    \\end{eqnarray}\n        \\item For an in-distribution sample with a conflicting prediction (i.e., $\\alpha=[\\alpha_1, \\ldots, \\alpha_K]$ with $\\alpha_1 = \\alpha_2 =\\cdots = \\alpha_K$, if $S \\rightarrow \\infty$),  we have \n    \\begin{eqnarray}\n    u_{en} = 1, \\lim_{S\\rightarrow \\infty} u_{diss} =\\lim_{S\\rightarrow \\infty} u_{alea} =1 , \\lim_{S\\rightarrow \\infty} u_{v} =\\lim_{S\\rightarrow \\infty} u_{epis} =0 \\nonumber\n    \\end{eqnarray}\n    \\text{with} $u_{en} > u_{alea}> u_{diss}> u_{v}>u_{epis} $.\n    \\end{enumerate}\n\\end{enumerate}\n\\label{theorem1}\n\\end{restatable}\n\n\\vspace{-2mm}\nThe proof of Theorem~\\ref{theorem1} can be found in Appendix A.1.  As demonstrated in Theorem~\\ref{theorem1} and Figure~\\ref{fig:example}, entropy cannot distinguish OOD (see Figure~\\ref{fig:example} (c)) and conflicting predictions (see Figure~\\ref{fig:example} (b)) because entropy is high for both cases. Similarly, neither aleatoric uncertainty nor epistemic uncertainty can distinguish OOD from conflicting predictions.  In both cases, aleatoric uncertainty is high while epistemic uncertainty is low.  On the other hand, vacuity and dissonance can clearly distinguish OOD from a conflicting prediction.  For example, OOD objects typically show high vacuity with low dissonance while conflicting predictions exhibit low vacuity with high dissonance.  This observation is confirmed through the empirical validation via our extensive experiments in terms of misclassification and OOD detection tasks.\n\\vspace{-2mm}\n\\section{Uncertainty-Aware Semi-Supervised Learning}\n\\vspace{-2mm}\nIn this section, we describe our proposed uncertainty framework based on semi-supervised node classification problem.  The overall description of the framework is shown in Figure~\\ref{fig:framework}. \n\\vspace{-2mm}\n\\subsection{Problem Definition} \\label{subsec:problem-definition}\n\\vspace{-1mm}\nGiven an input graph $\\mathcal{G} = (\\mathbb{V}, \\mathbb{E}, {\\bf r}, {\\bf y}_\\mathbb{L})$, where $\\mathbb{V} = \\{1, \\ldots, N \\}$ is a ground set of nodes, $\\mathbb{E} \\subseteq \\mathbb{V}\\times \\mathbb{V}$ is a ground set of edges, $\\textbf{r} = [\\textbf{r}_1, \\cdots, \\textbf{r}_N]^T \\in \\mathbb{R}^{N\\times d}$ is a node-level feature matrix, $\\textbf{r}_i\\in \\mathbb{R}^d$ is the feature vector of node $i$, $\\textbf{y}_{\\mathbb{L}}=\\{y_i \\mid i \\in \\mathbb{L}\\}$ are the labels of the training nodes $\\mathbb{L} \\subset \\mathbb{V}$, and $y_i \\in \\{1, \\ldots, K\\}$ is the class label of node $i$. {\\bf We aim to predict}: (1) the \\textbf{class probabilities} of the testing nodes: $\\textbf{p}_{\\mathbb{V} \\setminus \\mathbb{L}} = \\{\\textbf{p}_i \\in [0, 1]^K \\mid i \\in \\mathbb{V} \\setminus \\mathbb{L}\\}$; and (2) the \\textbf{associated multidimensional uncertainty estimates} introduced by different root causes: $\\mathbf{u}_{\\mathbb{V} \\setminus \\mathbb{L}} = \\{\\mathbf{u}_i \\in [0, 1]^m \\mid i \\in \\mathbb{V} \\setminus \\mathbb{L}\\}$, where $p_{i, k}$ is the probability that the class label $y_i = k$ and $m$ is the total number of\nuncertainty types. \n\n\\begin{figure*}[t!]\n  \\centering\n  \\vspace{-1mm}\n  \\includegraphics[width=0.75\\linewidth]{fig\/framework1.png}\n  \\vspace{-2mm}\n  \\scriptsize{\n \\caption{\\footnotesize \\textbf{Uncertainty Framework Overview.} Subjective Bayesian GNN (a) designed for estimating the different types of uncertainties (b).\n}\n  \\label{fig:framework}\n\\vspace{-4mm}\n }\n\\end{figure*}  \n\n\\vspace{-2mm}\n\\subsection{Proposed Uncertainty Framework} \\label{subsec:bay-dl}\n\\vspace{-2mm}\n\\textbf{Learning evidential uncertainty.}\nAs discussed in Section~\\ref{SL}, evidential uncertainty can be derived from multinomial opinions or equivalently Dirichlet distributions to model a probability distribution for the class probabilities. Therefore, we design a Subjective GNN (S-GNN) $f$ to form their multinomial opinions for the node-level Dirichlet distribution $\\text{Dir}(\\textbf{p}_i | {\\bm \\alpha}_i)$ of a given node $i$. Then, the conditional probability $P(\\textbf{p} |A, \\textbf{r}; \\bm{\\theta})$ can be obtained by:\n\\begin{eqnarray}\\small \nP(\\textbf{p} |A, \\textbf{r}; \\bm{\\theta} )=\\prod\\nolimits_{i=1}^N \\text{Dir}(\\textbf{p}_i|\\bm{\\alpha}_i), \\ \\bm{\\alpha}_i=f_i(A,\\textbf{r};\\bm{\\theta}), \\label{GCN_1}\n\\end{eqnarray}\nwhere $f_i$ is the output of S-GNN for node $i$, $\\bm{\\theta}$ is the model parameters, and $A$ is an adjacency matrix. The Dirichlet probability function $\\text{Dir}(\\textbf{p}_i | \\bm{\\alpha}_i)$ is defined by:\n\\begin{eqnarray}\\small \n    \\text{Dir}(\\textbf{p}_i | \\bm{\\alpha}_i)=\\frac{\\Gamma(S_i)}{\\prod_{k=1}^K \\Gamma(\\alpha_{ik})}\\prod\\nolimits_{k=1}^K p_{ik}^{\\alpha_{ik}-1}.\n\\end{eqnarray}\nNote that S-GNN is similar to classical GNN, except that we use an activation layer (e.g., \\textit{ReLU}) instead of the \\textit{softmax} layer (only outputs class probabilities).  This ensures that S-GNN would output non-negative values, which are taken as the parameters for the predicted Dirichlet distribution. \n\n\\textbf{Learning probabilistic uncertainty.}\nSince probabilistic uncertainty relies on a Bayesian framework, we proposed a Subjective Bayesian GNN (S-BGNN) that adapts S-GNN to a Bayesian framework, with the model parameters $\\bm{\\theta}$ following a prior distribution. The joint class probability of $\\textbf{y}$ can be estimated by: \n\\begin{eqnarray}\n\\vspace{-3mm}\n\\small \nP(\\textbf{y} |A, \\textbf{r}; \\mathcal{G}) &=& \\int \\int P(\\textbf{y} | \\textbf{p}) P(\\textbf{p} |A, \\textbf{r}; \\bm{\\theta} ) P(\\bm{\\theta} | \\mathcal{G}) d \\textbf{p} d\\bm{\\theta} \\nonumber \\\\\n&\\approx& \\frac{1}{M}\\sum_{m=1}^M  \\sum_{i=1}^N \\int P(\\textbf{y}_i | \\textbf{p}_i) P(\\textbf{p}_i | A, \\textbf{r};\\bm{\\theta}^{(m)} ) d \\textbf{p}_i, \\quad \\bm{\\theta}^{(m)} \\sim q( \\bm{\\theta}) \n\\label{Baye_model}\n\\vspace{-2mm}\n\\end{eqnarray}\nwhere $P(\\bm{\\theta} | \\mathcal{G})$ is the posterior, estimated via dropout inference, that provides an approximate solution of posterior $q(\\bm{\\theta})$ and taking samples from the posterior distribution of models~\\cite{gal2016dropout}.  Thanks to the benefit of dropout inference, training a DL model directly by minimizing the cross entropy (or square error) loss function can effectively minimize the KL-divergence between the approximated distribution and the full posterior (i.e., KL[$q(\\bm{\\theta})\\|P(\\theta|\\mathcal{G})$]) in variational inference~\\cite{gal2016dropout, kendall2015bayesian}. For interested readers, please refer to more detail in Appendix B.8.\n\nTherefore, training S-GNN with stochastic gradient descent enables learning of an approximated distribution of weights, which can provide good explainability of data and prevent overfitting.  We use a {\\em loss function} to compute its Bayes risk with respect to the sum of squares loss $\\|\\textbf{y}-\\textbf{p}\\|^2_2$ by:\n\\begin{eqnarray}\\small \n\\mathcal{L}(\\bm{\\theta}) =  \\sum\\nolimits_{i\\in \\mathbb{L}} \\int \\|\\textbf{y}_i-\\textbf{p}_i\\|^2_2 \\cdot P(\\textbf{p}_i |A, \\textbf{r}; \\bm{\\theta}) d \\textbf{p}_i \n=  \\sum\\nolimits_{i\\in \\mathbb{L}} \\sum\\nolimits_{k=1}^K \\big(y_{ik}-\\mathbb{E}[p_{ik}]\\big)^2 + \\text{Var}(p_{ik}),\n\\label{loss}\n\\end{eqnarray}\nwhere $\\textbf{y}_i$ is an one-hot vector encoding the ground-truth class with $y_{ij} = 1$ and $y_{ik} \\neq $ for all $k \\neq j$ and $j$ is a class label. Eq.~\\eqref{loss} aims to minimize the prediction error and variance, leading to maximizing the classification accuracy of each training node by removing excessive misleading evidence.\n\\vspace{-1mm}\n\\subsection{Graph-based Kernel Dirichlet distribution Estimation (GKDE)}\n\\begin{wrapfigure}{R}{0.4\\textwidth}\n  \\centering\n  \\includegraphics[width=0.4\\textwidth]{fig\/GKDE.png}\n \\caption{\\small{Illustration of GKDE. Estimate prior Dirichlet distribution $\\text{Dir}(\\hat{\\alpha})$ for node $j$ (red) based on training nodes (blue) and graph structure information.}}\n  \\label{fig:gkde}\n  \\vspace{-2mm}\n\\end{wrapfigure} \n\\vspace{-1mm}\nThe loss function in Eq.~\\eqref{loss} is designed to measure the sum of squared loss based on class labels of training nodes.  However, it does not directly measure the quality of the predicted node-level Dirichlet distributions.  To address this limitation, we proposed \\textit{Graph-based Kernel Dirichlet distribution Estimation} (GKDE) to better estimate node-level Dirichlet distributions by using graph structure information.  The key idea of the GKDE is to estimate prior Dirichlet distribution parameters for each node based on the class labels of training nodes (see Figure~\\ref{fig:gkde}). Then, we use the estimated prior Dirichlet distribution in the training process to learn the following patterns: (i) nodes with a high vacuity will be shown far from training nodes; and (ii) nodes with a high dissonance will be shown near the boundaries of classes.\n\n\\vspace{-1mm}\nBased on SL, let each training node represent one evidence for its class label. Denote the contribution of evidence estimation for node $j$ from training node $i$ by $\\mathbf{h}(y_i,d_{ij}) =[h_1, \\ldots, h_k, \\ldots, h_K] \\in[0, 1]^K$, where $h_k(y_i,d_{ij})$ is obtained by: \n\\vspace{-1mm}\n\\begin{eqnarray}\nh_k(y_i,d_{ij}) = \\begin{cases}0 & y_i \\neq k \\\\ g(d_{ij}) & y_i = k,  \\end{cases}\n\\vspace{-2mm}\n\\end{eqnarray}\n $g(d_{ij}) = \\frac{1}{\\sigma \\sqrt{2\\pi}}\\exp({-\\frac{{d^2_{ij}}}{2\\sigma^2}})$ is the Gaussian kernel function used to estimate the distribution effect between nodes $i$ and $j$, and $d_{ij}$ means the \\textbf{node-level distance} (\\textbf{a shortest path between nodes $i$ and $j$}), and $\\sigma$ is the bandwidth parameter. The prior evidence is estimated based GKDE: $\\hat{\\bm{e}}_j = \\sum_{i\\in \\mathbb{L}} \\mathbf{h}(y_i,d_{ij})$, where $\\mathbb{L}$ is a set of training nodes and the prior Dirichlet distribution $\\hat{\\bm{\\alpha}}_j = \\hat{\\bm{e}}_j +\\bf 1$.  \nDuring the training process, we minimize the KL-divergence between model predictions of Dirichlet distribution and prior distribution: $\\min \\text{KL}[\\text{Dir}(\\bm{\\alpha}) \\| \\text{Dir}(\\hat{\\bm{\\alpha}})]$.\nThis process can prioritize the extent of data relevance based on the estimated evidential uncertainty, which is proven effective based on the proposition below.\n\n\\begin{restatable}{proposition}{primeproposition}\nGiven $L$ training nodes, for any testing nodes $i$ and $j$, let ${\\bm d}_i = [d_{i1}, \\ldots, d_{iL}]$ be the vector of graph distances from nodes $i$ to training nodes and ${\\bm d}_j = [d_{j1}, \\ldots, d_{jL}]$ be the graph distances from nodes $j$ to training nodes, where $d_{il}$ is the node-level distance between nodes $i$ and $l$. If for all $l\\in \\{1, \\ldots, L\\}$, $d_{il} \\ge d_{jl}$, then we have\n\\begin{eqnarray}\n\\hat{u}_{v_i} \\ge \\hat{u}_{v_j}, \\nonumber\n\\end{eqnarray}\nwhere $ \\hat{u}_{v_i}$ and $\\hat{u}_{v_j}$ refer to vacuity uncertainties of nodes $i$ and $j$ estimated based on GKDE.\n\\label{theorem: vacuity}\n\\vspace{-1mm}\n\\end{restatable}\nThe proof for this proposition can be found in Appendix A.2. The above proposition shows that if a testing node is too far from training nodes, the vacuity will increase, implying that an OOD node is expected to have a high vacuity.\n\nIn addition, we designed a simple iterative knowledge distillation method~\\cite{hinton2015distilling} (i.e., Teacher Network) to refine the node-level classification probabilities. The key idea is to train our proposed model (Student) to imitate the outputs of a pre-train a vanilla GNN (Teacher) by adding a regularization term of KL-divergence. This leads to solving the following optimization problem: \n\\begin{eqnarray}\\small \n\\vspace{-1mm}\n\\min\\nolimits_{\\bm{\\theta}}  \\mathcal{L}(\\bm{\\theta}) + \\lambda_1 \\text{KL}[\\text{Dir}({\\bm \\alpha}) \\| \\text{Dir}(\\hat{\\bm \\alpha})] + \\lambda_2  \\text{KL}[P(\\textbf{y} \\mid A,\\textbf{r};\\mathcal{G}) \\parallel P(\\textbf{y}|\\hat{\\textbf{p}})], \n\\label{joint loss}\n\\vspace{-1mm}\n\\end{eqnarray}\nwhere $\\hat{\\textbf{p}}$ is the vanilla GNN's (Teacher) output and $\\lambda_1$ and $\\lambda_2$ are trade-off parameters. \n\n\\section{Experiments} \\label{sec:exp-results-analysis}\n\\vspace{-1mm}\nIn this section, we conduct experiments on the tasks of misclassification and OOD detections to answer the following questions for semi-supervised node classification:\n\n\\vspace{-1mm}\n\\noindent {\\bf Q1. Misclassification Detection:} What type of uncertainty is the most promising indicator of high confidence in node classification predictions? \n\n\\vspace{-1mm}\n\\noindent {\\bf Q2. OOD Detection:}  What type of uncertainty is a key indicator of accurate detection of OOD nodes? \n\n\\vspace{-1mm}\n\\noindent {\\bf Q3. GKDE with Uncertainty Estimates:}  How can GKDE help enhance prediction tasks with what types of uncertainty estimates?\n\n\\vspace{-1mm}\nThrough extensive experiments, we found the following answers for the above questions:\n\n\\vspace{-1mm}\n\\noindent {\\bf A1.} Dissonance (i.e., uncertainty due to conflicting evidence) is more effective than other uncertainty estimates in misclassification detection. \n\\vspace{-1mm}\n\n\\noindent {\\bf A2.}  Vacuity (i.e., uncertainty due to lack of confidence) is more effective than other uncertainty estimates in OOD detection.\n\n\\vspace{-1mm}\n\\noindent {\\bf A3.}  GKDE can indeed help improve the estimation quality of node-level Dirichlet distributions, resulting in a higher OOD detection.\n\n\\vspace{-2mm}\n\\subsection{Experiment Setup} \n\\vspace{-2mm}\n\\textbf{Datasets}: We used six datasets, including three citation network datasets~\\cite{sen2008collective} (i.e., Cora, Citeseer, Pubmed) and three new datasets~\\cite{shchur2018pitfalls} (i.e., Coauthor Physics, Amazon Computer, and Amazon Photo). We summarized the description and experimental setup of the used datasets in Appendix B.2\\footnote{The source code and datasets are accessible at \\href{https:\/\/github.com\/zxj32\/uncertainty-GNN}{\\color{magenta}{https:\/\/github.com\/zxj32\/uncertainty-GNN}}}.  \n\n\\vspace{-1mm}\n\\textbf{Comparing Schemes}: \nWe conducted the extensive comparative performance analysis based on our proposed models and several state-of-the-art competitive counterparts. We implemented all models based on the most popular GNN model, GCN~\\cite{kipf2017semi}.  We compared our model (S-BGCN-T-K) against: (1) Softmax-based GCN~\\cite{kipf2017semi} with uncertainty measured based on entropy; and (2) Drop-GCN that adapts the Monte-Carlo Dropout~\\cite{gal2016dropout, ryu2019uncertainty} into the GCN model to learn probabilistic uncertainty; (3) EDL-GCN that adapts the EDL model~\\cite{sensoy2018evidential} with GCN to estimate evidential uncertainty; (4) DPN-GCN that adapts the DPN~\\cite{malinin2018predictive} method with GCN to estimate probabilistic uncertainty. We evaluated the performance of all models considered using the area under the ROC (AUROC) curve and area under the Precision-Recall (AUPR) curve in both experiments~\\cite{hendrycks2016baseline}.\n\n\\vspace{-1mm}\n\\subsection{Results}\n\\vspace{-1mm}\n\\noindent {\\bf Misclassification Detection.} The misclassification detection experiment involves detecting whether a given prediction is incorrect using an uncertainty estimate.  Table~\\ref{AUPR:uncertainty} shows that S-BGCN-T-K outperforms all baseline models under the AUROC and AUPR for misclassification detection. The outperformance of dissonance-based detection is fairly impressive. This confirms that low dissonance (a small amount of conflicting evidence) is the key to maximize the accuracy of node classification prediction.  We observe the following performance order: ${\\tt Dissonance} > {\\tt Entropy} \\approx {\\tt Aleatoric} > {\\tt Vacuity} \\approx {\\tt Epistemic}$, which is aligned with our conjecture: higher dissonance with conflicting prediction leads to higher misclassification detection. We also conducted experiments on additional three datasets and observed similar trends of the results, as demonstrated in Appendix C.\n\\vspace{-1mm}\n\\begin{table*}[th!]\n\\scriptsize\n\\caption{AUROC and AUPR for the Misclassification Detection.}\n\\vspace{-2mm}\n\\centering\n\\begin{tabular}{c||c|ccccc|ccccc|c}\n\\hline\n\\multirow{2}{*}{Data} & \\multirow{2}{*}{Model} & \\multicolumn{5}{c|}{AUROC} & \\multicolumn{5}{c|}{AUPR} & \\multirow{2}{*}{Acc} \\\\  \n                  &                   & Va.\\tnote{*}& Dis. & Al. & Ep. & En. & Va. & Dis. &  Al. & Ep. &En. & \\\\ \\hline\n\\multirow{5}{*}{Cora} & S-BGCN-T-K & 70.6 & \\textbf{82.4} & 75.3 & 68.8 & 77.7&  90.3  & \\textbf{95.4} & 92.4 & 87.8 &93.4 & \\textbf{82.0} \\\\ \n                  &  EDL-GCN &  70.2  &  81.5  & -   &  - &  76.9  &90.0 &  94.6  &  -  & - &  93.6 & 81.5 \\\\    \n                  &    DPN-GCN &  -  &  -  & 78.3   &  75.5 &  77.3  & -  &  -  &  92.4  & 92.0 &92.4 & 80.8 \\\\  \n                  & Drop-GCN &  -  &  -  & 73.9   &  66.7  & 76.9  & -  &  -  &  92.7  & 90.0 &93.6 & 81.3 \\\\ \n                  &GCN & -  &  -  &  -  &  -   & 79.6&  -  &  -  &  -  &  -  & 94.1 & 81.5 \\\\ \n                  \\hline\n\\multirow{5}{*}{Citeseer} &  S-BGCN-T-K & 65.4& \\textbf{74.0}& 67.2& 60.7&  70.0& 79.8 & \\textbf{85.6} & 82.2 & 75.2 &83.5& \\textbf{71.0}  \\\\ \n                   &  EDL-GCN &  64.9  &  73.6  & -   &  - &  69.6  &79.2 &  84.6 &  -  & - &  82.9 & 70.2 \\\\    \n                  &    DPN-GCN &  -  &  -  & 66.0   &  64.9 &  65.5  & -  &  -  & 78.7  & 77.6 &78.1 & 68.1 \\\\  \n                  &                  Drop-GCN &   - &  -  & 66.4 & 60.8 &   69.8 &   -  & -& 82.3 & 77.8  &83.7 & 70.9  \\\\ \n                  &                  GCN &  -  &  -  &  -  &  -  & 71.4 &  -  &  -  &  -  & -   &83.2 & 70.3 \\\\ \\hline\n\\multirow{5}{*}{Pubmed} &  S-BGCN-T-K & 64.1 & \\textbf{73.3} & 69.3 & 64.2 &  70.7& 85.6 & \\textbf{90.8} & 88.8& 86.1   &89.2 & \\textbf{79.3} \\\\\n                  &  EDL-GCN &  62.6  &  69.0  & -   &  - &  67.2  &84.6 &  88.9 &  -  & - &  81.7 & 79.0 \\\\    \n                  &    DPN-GCN &  -  &  -  & 72.7   &  69.2 &  72.5  & -  &  -  &  87.8  & 86.8 &87.7 & 77.1 \\\\  \n                  &                  Drop-GCN &  -  &  -  &  67.3 & 66.1 &   67.2&  -  &  -& 88.6 & 85.6   &89.0 & 79.0  \\\\ \n                  &                  GCN &  -  &  -  &   - & -  & 68.5&  -  &   - & -   &  - &89.2 & 79.0 \\\\ \\hline\n\\end{tabular}\n\\begin{tablenotes}\\scriptsize\n\\centering\n\\item[*] Va.: Vacuity, Dis.: Dissonance, Al.: Aleatoric, Ep.: Epistemic, En.: Entropy \n\\end{tablenotes}\n\\vspace{2mm}\n\\label{AUPR:uncertainty}\n\\vspace{-5mm}\n\\end{table*}\n\n\n\n\n\\begin{table*}[th!]\n\\scriptsize\n\\caption{AUROC and AUPR for the OOD Detection.}\n\\vspace{-2mm}\n\\centering\n\\begin{tabular}{c||c|ccccc|ccccc}\n\\hline\n\\multirow{2}{*}{Data} & \\multirow{2}{*}{Model} & \\multicolumn{5}{c|}{AUROC} & \\multicolumn{5}{c}{AUPR} \\\\  \n                  &                   & Va.\\tnote{*}& Dis. & Al. & Ep.  &En. & Va. & Dis. &  Al. & Ep. & En. \\\\ \\hline\n\\multirow{4}{*}{Cora} & S-BGCN-T-K & \\textbf{87.6} & 75.5 & 85.5 & 70.8  & 84.8&  \\textbf{78.4} & 49.0 & 75.3 & 44.5 &  73.1  \\\\ \n                  &    EDL-GCN &            84.5 & 81.0 & & -& 83.3&    74.2 & 53.2 & - & -&71.4  \\\\ \n                  &       DPN-GCN &  -  &  -  & 77.3   &  78.9 &  78.3  & -  &  -  &  58.5  & 62.8 &  63.0  \\\\ \n                  &       Drop-GCN &  -  &  -  & 81.9   &  70.5 &  80.9  & -  &  -  &  69.7  & 44.2 &  67.2  \\\\  \n                  &                  GCN & -  &  -  &  -  &  -  &  80.7&  -  &  -  &  -  &  -  & 66.9 \\\\ \\hline\n\\multirow{4}{*}{Citeseer} &  S-BGCN-T-K & \\textbf{84.8} &55.2&78.4 & 55.1 &  74.0& \\textbf{86.8} & 54.1 & 80.8 & 55.8 & 74.0  \\\\ \n                  &  EDL-GCN &                 78.4 &59.4&- & - &  69.1&         79.8 & 57.3 & - & - & 69.0  \\\\ \n                  & DPN-GCN &   - &  -  & 68.3 & 72.2 &  69.5 &   -  & -& 68.5 & 72.1   & 70.3  \\\\ \n                  &                  Drop-GCN &   - &  -  & 72.3 & 61.4 &  70.6 &   -  & -& 73.5 & 60.8   & 70.0  \\\\ \n                  &                  GCN &  -  &  -  &  -  &  -  &  70.8 &  -  &  -  &  -  & -   & 70.2  \\\\ \\hline\n\\multirow{4}{*}{Pubmed} &  S-BGCN-T-K & \\textbf{74.6} &67.9& 71.8 & 59.2 & 72.2& \\textbf{69.6} & 52.9 & 63.6& 44.0   &56.5  \\\\\n                  &    EDL-GCN & 71.5 &68.2& - & - &  70.5& 65.3 & 53.1 & -& -   & 55.0  \\\\\n                   & DPN-GCN &  -  &  -  &  63.5 & 63.7 &   63.5&  -  &  -& 50.7 & 53.9   & 51.1  \\\\ \n                  &                  Drop-GCN &  -  &  -  &  68.7 & 60.8 &   66.7&  -  &  -& 59.7 & 46.7   & 54.8  \\\\ \n                  &                  GCN &  -  &  -  &   - & -   &  68.3&  -  &   -   &  -  & -&55.3 \\\\ \\hline\n\\multirow{4}{*}{Amazon Photo} &  S-BGCN-T-K & \\textbf{93.4} & 76.4& 91.4& 32.2  & 91.4&         \\textbf{ 94.8} & 68.0 & 92.3& 42.3   & 92.5  \\\\\n                  &     EDL-GCN & 63.4 & 78.1& - & - & 79.2&          66.2 & 74.8 & -&-   & 81.2  \\\\\n                  & DPN-GCN &  -  &  -  &  83.6 &  83.6 &   83.6&  -  &  -& 82.6 & 82.4 &   82.5 \\\\ \n                  &                  Drop-GCN &  -  &  -  &  84.5 & 58.7 &   84.3&  -  &  -& 87.0 & 57.7   &86.9 \\\\ \n                  &                  GCN &  -  &  -  &   - & -   &   84.4&  -  &   -   &  -  & -&87.0 \\\\ \\hline\n\\multirow{4}{*}{Amazon Computer} &  S-BGCN-T-K & \\textbf{82.3} & 76.6& 80.9& 55.4  & 80.9& \\textbf{70.5} & 52.8 & 60.9& 35.9 & 60.6  \\\\\n                  &    EDL-GCN & 53.2 & 70.1& - & - &  70.0&    33.2 & 43.9 & -& - & 45.7  \\\\\n                   &                  DPN-GCN &  -  &  -  &  77.6 & 77.7 & 77.7 &  -  &  -& 50.8 & 51.2 & 51.0 \\\\ \n                  &                  Drop-GCN &  -  &  -  &  74.4 & 70.5 &  74.3&  -  &  -& 50.0 & 46.7   &  49.8 \\\\ \n                  &                  GCN &  -  &  -  &   - & -   &   74.0&    - & -   &  -  & -&48.7 \\\\ \\hline\n\\multirow{4}{*}{Coauthor Physics} &  S-BGCN-T-K & \\textbf{91.3} & 87.6& 89.7& 61.8 & 89.8& \\textbf{72.2} & 56.6 & 68.1& 25.9 & 67.9  \\\\\n                  &     EDL-GCN & 88.2 & 85.8& - & -  & 87.6&    67.1 & 51.2 & -&- & 62.1  \\\\\n                   &                  DPN-GCN &  -  &  -  & 85.5 &85.6 &   85.5&  -  &  -& 59.8 & 60.2 &   59.8 \\\\ \n                  &                  Drop-GCN &  -  &  -  &  89.2 & 78.4 &   89.3&  -  &  -& 66.6 & 37.1   &66.5 \\\\ \n                  &                  GCN &  -  &  -  &   -    &  - & 89.1&  -  &  -   &  -  & -&64.0 \\\\ \\hline\n\\end{tabular}\n\\begin{tablenotes}\\scriptsize\n\\centering\n\\item[*] Va.: Vacuity, Dis.: Dissonance, Al.: Aleatoric, Ep.: Epistemic, En.: Entropy \n\\end{tablenotes}\n\\vspace{2mm}\n\\label{Table: AUROC_AUPR:ood}\n\\vspace{-7mm}\n\\end{table*}\n\n\\vspace{-0.3mm}\n\\noindent {\\bf OOD Detection.} This experiment involves detecting whether an input example is out-of-distribution (OOD) given an estimate of uncertainty. For semi-supervised node classification, we randomly selected one to four categories as OOD categories and trained the models based on training nodes of the other categories. Due to the space constraint, the experimental setup for the OOD detection is detailed in Appendix B.3. \n\n\\vspace{-1mm}\nIn Table~\\ref{Table: AUROC_AUPR:ood}, across six network datasets, our vacuity-based detection significantly outperformed the other competitive methods, exceeding the performance of the epistemic uncertainty and other type of uncertainties. This demonstrates that vacuity-based model is more effective than other uncertainty estimates-based counterparts in increasing OOD detection. We observed the following performance order: ${\\tt Vacuity} > {\\tt Entropy} \\approx {\\tt Aleatoric} > {\\tt Epistemic} \\approx {\\tt Dissonance}$, which is consistent with the theoretical results as shown in Theorem~\\ref{theorem1}.\n\n\\vspace{-1mm}\n\\noindent {\\bf Ablation Study.}\nWe conducted additional experiments (see Table~\\ref{Ablation}) in order to demonstrate the contributions of the key technical components, including GKDE, Teacher Network, and subjective Bayesian framework.  The key findings obtained from this experiment are: (1) GKDE can enhance the OOD detection (i.e., 30\\% increase with vacuity), which is consistent with our theoretical proof about the outperformance of GKDE in uncertainty estimation, i.e., OOD nodes have a higher vacuity than other nodes; and (2) the Teacher Network can further improve the node classification accuracy.\n\\vspace{-2mm}\n\\subsection{Why is Epistemic Uncertainty Less Effective than Vacuity?}\n\\vspace{-2mm}\nAlthough epistemic uncertainty is known to be effective to improve OOD detection~\\cite{gal2016dropout, kendall2017uncertainties} in computer vision applications, our results demonstrate it is less effective than our vacuity-based approach.  The first potential reason is that epistemic uncertainty is always smaller than vacuity (From Theorem~\\ref{theorem1}), which potentially indicates that epistemic may capture less information related to OOD. Another potential reason is that the previous success of epistemic uncertainty for OOD detection is limited to supervised learning in computer vision applications, but its  effectiveness for OOD detection was not sufficiently validated in semi-supervised learning tasks.  Recall that epistemic uncertainty (i.e., model uncertainty) is calculated based on mutual information (see Eq.~\\eqref{eq:epistemic}).  In a semi-supervised setting, the features of unlabeled nodes are also fed to a model for training process to provide the model with a high confidence on its output.  For example, the model output $P(\\textbf{y}|A, \\textbf{r};\\theta)$ would not change too much even with differently sampled parameters $\\bm{\\theta}$, i.e., $P(\\textbf{y}|A, \\textbf{r};\\theta^{(i)})\\approx P(\\textbf{y}|A, \\textbf{r};\\theta^{(j)})$, which result in a low epistemic uncertainty.  We also designed a semi-supervised learning experiment for image classification and observed a consistent pattern with the results demonstrated in Appendix C.6. \n\n\\begin{table*}[th!]\n\\scriptsize\n\\caption{Ablation study of our proposed models: (1) {\\tt S-GCN}: Subjective GCN with vacuity and dissonance estimation; (2) {\\tt S-BGCN}: S-GCN with Bayesian framework; (3) {\\tt S-BGCN-T}: S-BGCN with a Teacher Network; (4) {\\tt S-BGCN-T-K}: S-BGCN-T with GKDE to improve uncertainty estimation.}\n\\vspace{1mm}\n\\centering\n\\vspace{-3mm}\n\\begin{tabular}{c||c|ccccc|ccccc|c}\n\\hline\n\\multirow{2}{*}{Data} & \\multirow{2}{*}{Model} & \\multicolumn{5}{c|}{AUROC (Misclassification Detection)} & \\multicolumn{5}{c|}{AUPR (Misclassification Detection)} & \\multirow{2}{*}{Acc} \\\\\n                  &                   & Va.\\tnote{*}& Dis. & Al. & Ep. & En. & Va. & Dis. &  Al. & Ep. &En. & \\\\ \\hline\n\\multirow{4}{*}{Cora} & S-BGCN-T-K & 70.6 & 82.4 & 75.3 & 68.8 & 77.7&  90.3  & \\textbf{95.4} & 92.4 & 87.8 &93.4 & 82.0 \\\\ \n & S-BGCN-T &  70.8  &  \\textbf{82.5}  &75.3  & 68.9 & 77.8  &90.4  &  \\textbf{95.4}  &  92.6  & 88.0 &93.4 & \\textbf{82.2} \\\\\n                  & S-BGCN &  69.8  &  81.4  & 73.9   &  66.7  & 76.9  & 89.4  &  94.3  &  92.3  & 88.0 &93.1 & 81.2 \\\\  \n                   & S-GCN &  70.2  &  81.5  & -   & -  & 76.9  & 90.0  &  94.6  &  -  & - &93.6 & 81.5 \\\\\n                  \\hline\n & & \\multicolumn{5}{c|}{AUROC (OOD Detection)} & \\multicolumn{5}{c|}{AUPR (OOD Detection)} &  \\\\ \\hline \n\\multirow{4}{*}{Amazon Photo} &  S-BGCN-T-K & \\textbf{93.4} & 76.4& 91.4& 32.2  & 91.4&         \\textbf{ 94.8} & 68.0 & 92.3& 42.3   & 92.5 &-  \\\\\n                  &     S-BGCN-T & 64.0 & 77.5& 79.9 & 52.6 & 79.8&          67.0 & 75.3 & 82.0& 53.7   & 81.9&-  \\\\\n                  & S-BGCN & 63.0 & 76.6& 79.8 & 52.7 & 79.7&          66.5 & 75.1 & 82.1& 53.9   & 81.7&- \\\\ \n                  & S-GCN & 64.0 & 77.1& - & - & 79.6&          67.0 & 74.9 & -& -   & 81.6&- \\\\ \\hline\n\\end{tabular}\n\\begin{tablenotes}\\scriptsize\n\\centering\n\\item[*] Va.: Vacuity, Dis.: Dissonance, Al.: Aleatoric, Ep.: Epistemic, En.: Entropy \n\\end{tablenotes}\n\\vspace{2mm}\n\\label{Ablation}\n\\vspace{-5mm}\n\\end{table*}\n\n\\vspace{-2mm}\n\\section{Conclusion} \\label{sec:conclusion}\n\\vspace{-2mm}\nIn this work, we proposed a multi-source uncertainty framework of GNNs for semi-supervised node classification. Our proposed framework provides an effective way of predicting node classification and out-of-distribution detection considering multiple types of uncertainty. We leveraged various types of uncertainty estimates from both DL and evidence\/belief theory domains. Through our extensive experiments, we found that dissonance-based detection yielded the best performance on misclassification detection while vacuity-based detection performed the best for OOD detection, compared to other competitive counterparts.  In particular, it was noticeable that applying GKDE and the Teacher network further enhanced the accuracy in node classification and uncertainty estimates.\n\n\\section*{Acknowledgments}\nWe would like to thank Yuzhe Ou for providing proof suggestions. \nThis work is supported by the National Science Foundation\n(NSF) under Grant No \\#1815696 and \\#1750911.\n\n\n\\section*{Broader Impact}\nIn this paper, we propose a uncertainty-aware semi-supervised learning framework of GNN for predicting multi-dimensional uncertainties for the task of semi-supervised node classification. Our proposed framework can be applied to a wide range of applications, including computer vision, natural language processing, recommendation systems, traffic prediction, generative models and many more~\\cite{zhou2018graph}. Our proposed framework can be applied to predict multiple uncertainties of different roots for GNNs in these applications, improving the understanding of individual decisions, as well as the underlying models. While there will be important impacts resulting from the use of GNNs in general, our focus in this work is on investigating the impact of using our method to predict multi-source uncertainties for such systems. The additional benefits of this method include improvement of safety and transparency in decision-critical applications to avoid overconfident prediction, which can easily lead to misclassification. \n\nWe see promising research opportunities that can adopt our uncertainty framework, such as investigating whether this uncertainty framework can further enhance misclassification detection or OOD detection. To mitigate the risk from different types of uncertainties, we encourage future research to understand the impacts of this proposed uncertainty framework to solve other real world problems.\n\n\\newpage\n\\small\n\\medskip\n\n\\bibliographystyle{abbrv}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nMany of the advances of quantum computation based on superconducting qubits rely on the ability to readout the qubit state by measuring microwave photons leaking out of a superconducting resonator \\cite{wallraff-schoelkopf-nature2004}. Thanks to the development of near-quantum-limited Josephson parametric amplifiers (JPAs) \\cite{beltran-lehnert-apl2007,bergeal-devoret-nature2010,zhou-esteve-prb2014,eichler-wallraff-prl2014}, high-fidelity single-shot qubit readout is now possible \\cite{mallet-esteve-nphys2009,walter-wallraff-prappl2017}. These amplifiers are, moreover, finding use in a wide range of applications, from measuring quantum features in the radiation emitted by mesoscopic conductors \\cite{zakka-portier-prl2010,gasse-reulet-prl2013,stehlik-petta-prappl2015,westig-portier-prl2017,simoneau-reulet-prb2017}, to the detection of small ensembles of electronic spins \\cite{bienfait-bertet-prx2017}, and even to the search for dark matter \\cite{brubaker-carosi-prl2017}. JPAs are also versatile sources of single- and two-mode squeezed states \\cite{beltran-lehnert-nphys2008,eichler-wallraff-prl2014}, which have been used to confirm decade old predictions in quantum optics \\cite{murch-siddiqui-nature2013,toyli-siddiqi-prx2016}, and to improve electron-spin resonance spectroscopy \\cite{bienfait-bertet-prx2017}. On the theoretical side, squeezed states were proposed as a resource to improve qubit readout and to perform high-fidelity gates \\cite{didier-blais-clerk-prl2015,puri-blais-prl2016,royer-blais-quantum2017}, or as a basis for continuous variable quantum computing \\cite{braunstein-loock-rmp2005,grimsmo-blais-npjqi2017}.\n\nCurrent JPAs are able to amplify signals to more than 20 dB, and to squeeze vacuum fluctuations by $7$ dB ($12$ dB) in single- (two-) mode experiments \\cite{boutin-blais-prappl2017,eichler-wallraff-prl2014}. However, in this devices, the amplification bandwidth is limited to hundreds of megahertz \\cite{mutus-martinis-apl2014,roy-vijay-apl2015,westig-klapwijk-arxiv2017}. At the price of increasing device fabrication complexity, much larger amplification bandwidth, over $\\sim 3$ GHz, has been demonstrated with the recently developed Josephson traveling wave parametric amplifier \\cite{macklin-siddiqi-science2015}. The development of a simpler quantum-limited microwave amplifier, generating far-separated two-mode squeezed states and capable of amplifying signals over gigahertz bandwidths, is still needed to further advance quantum information processing science. It would also be an important tool to better characterize the radiation emitted by mesoscopic conductors, for which there is an increasing body of interesting predictions \\cite{beenakker-schomerus-prl2004,armour-rimberg-prl2013,leppakangas-johansson-prl2015,mendes-mora-njp2015,mendes-mora-prb2016}. Here, we propose such a simple broadband parametric amplifier, consisting of a single dc- and ac-voltage biased superconductor-insulator-superconductor (SIS) junction. The device can be operated in both phase-sensitive and phase-preserving modes and can be used for near-quantum-limited amplification and two-mode squeezing in few GHz bandwidth.\n\nThe proposed setup, illustrated in Fig.~\\ref{fig1}(a), operates as an amplifier in reflection mode. Parametric amplification is possible by taking advantage of the strong non-linearity of the transport characteristics of the junction. To this end, we consider a dc-voltage bias $V$ smaller than twice the superconducting gap $\\Delta$. This sets the junction as an open circuit, and the conduction of quasiparticles is enabled by applying a sinusoidal ac-voltage $V_\\text{ac}(t) =V_\\text{ac}\\cos(2\\omega_0 t)$, with $\\omega_0$ the measurement center frequency. This voltage combination gives rise to modulations of the admittance of the junction $Y_n(\\omega)$, with $n$ the $n$-th harmonic of the pump. As illustrated in Fig.~\\ref{fig1}(b), the finite-frequency admittance for dc-voltage approaching $(2\\Delta-\\hbar\\omega_0)\/e$ is dominated by a large non-linear susceptance Im$Y_1(\\omega_0)$ \\cite{barone-paterno-book,lee-apl1982}. On the other hand, the dissipative term, $\\textrm{Re}Y_0(\\omega_0)$, is only weakly affected by the ac-voltage and it is dominant for $eV > 2\\Delta-\\hbar\\omega_0$. Taking advantage of dc-voltage control, we demonstrate that parametric amplification and squeezing emerges when $Y_1(\\omega_0)$ is much larger than Re$Y_0(\\omega_0)$. As we show below, $Y_1(\\omega_0)$ is related to the coherent conversion process of $2\\hbar\\omega_0$ quanta of energy from the pump to two photons of frequency $\\omega_0$, while Re$Y_0(\\omega_0)$ is related to dissipative effects due to the tunneling of quasiparticles. It is surprising that even though SIS junctions are routinely used as high-frequency microwave quantum-limited mixers \\cite{trucker-feldman-rmp1985}, exploiting a very similar principle, their operation as parametric amplifiers have been mostly disregarded \\cite{lee-apl1982,devyatov-zorin-jap1986}. Here, we use the input-output formalism \\cite{yurke-denker-pra1984}, together with photon-assisted tunneling theory \\cite{tien1963} to compute the parametric amplification and squeezing properties of an ac- and dc-biased SIS junction \\cite{trucker-feldman-rmp1985}. The resulting Heisenberg-Langevin equations \\cite{gardiner-collet-pra1985} are numerically solved, allowing us to explore parametric amplification far from the small detuning limit considered previously \\cite{lee-apl1982,devyatov-zorin-jap1986}. For an Aluminum junction ($\\Delta = 180\\ \\mu$eV) pumped at $2\\omega_0 = 8$ GHz and operated at temperatures $T \\ll \\Delta\/k_B$, with $k_B$ the Boltzmann constant, we find that when operated in the phase-sensitive mode this device can produce more than 30 dB of gain and -20 dB of squeezing. On the other hand, operated in the phase preserving-mode, it achieves 15 dB of gain with an added noise close to the quantum limit and -14 dB of two-mode squeezing, in both cases over a large bandwidth exceeding 4 GHz. Moreover, the gain and the squeezing bandwidths are easily tuned by the dc-voltage and ac-pumping, allowing for instance to achieve higher gain in a reduced yet sizable bandwidth.\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[width=0.45\\textwidth]{fig1.pdf}\n\\caption{(a) Electrical scheme of the device: a SIS junction is dc-voltage biased close to the onset of quasiparticle transport $eV \\sim 2\\Delta$, while it is ac pumped with a single tone $V_\\text{ac} \\cos(2\\omega_0 t)$. Current conservation at the coupling mode ($x=0$) allows to relate the transmission line outgoing field $a_\\text{out}[\\omega]$ with its incoming field $a_\\text{in}[\\omega]$ and the current flowing through the junction $\\hat{I}_\\text{J}[\\omega]$. (b) Non-linear admittance as a function of the dc-voltage for an Aluminum junction ($\\Delta = 180\\ \\mu$eV) and $eV_\\text{ac} = 0.135 \\times 2\\hbar\\omega_0$. On one hand, as shown in Sec.~\\ref{model}, dissipation emerges from the absorption and emission of photons, which is illustrated by Re$Y_0(\\omega)$. Notably, Re$Y_0(\\omega)$ is only slightly perturbed by such small ac pumping. However, the pumping gives rise to a parametric interaction characterized by a sizable non-linear admittance $Y_1(\\omega_0)$. For dc-voltages in the range $2(\\Delta-\\hbar\\omega_0)<eV<2\\Delta-\\hbar\\omega_0$, the non-linear admittance $Y_1(\\omega_0)$ dominates over the dissipative response Re$Y_0(\\omega)$ giving rise to parametric amplification and squeezing. \\label{fig1}}\n\\end{center}\n\\end{figure}\n\nThis article is organized as follows: In Sec.~\\ref{model} we describe the input-output formalism used to characterize the device. Section~\\ref{resultsA} presents the results for an ideal SIS junction for which the transport response rises steeply for $eV=2\\Delta+n\\hbar\\omega_0$. The effects of the low-frequency noise, which captures most non-ideal effects, on the amplifier is described in Sec.~\\ref{resultsB}. Final remarks are presented in Sec.~\\ref{conclusions}.\n\\section{Model \\label{model}}\n\nWe consider a SIS junction in parallel with its capacitance $C$ and connected to a transmission line (TL), see Fig.~\\ref{fig1}(a). The Hamiltonian of the device is $\\hat{H} = \\hat{H}_\\text{qp} + \\hat{H}_\\text{ee} + \\hat{H}_\\text{t}$, where $\\hat{H}_\\text{qp}$ describes the quasiparticles in the superconductors forming the junction, $\\hat{H}_\\text{ee}$ is the electromagnetic environment Hamiltonian, which includes the TL and the capacitor $C$ of the junction. The last term is the tunneling Hamiltonian dressed by the environment and takes the form $\\hat{H}_\\text{t} = \\hat{\\mathcal{T}}(t) \\exp[i e\\hat{\\Phi}(t)\/\\hbar] + \\text{h.c.}$ with \\cite{devoret-urbina-prl1990}\n\\begin{equation}\n \\hat{\\mathcal{T}}(t) = \\sum_{l,r}t_{lr}\\hat{c}_l^\\dagger\\hat{c}_r e^{i [eV t\/\\hbar + \\varphi_{ac}(t)]},\n\\end{equation}\nthe tunneling operator transferring a quasiparticle from the right ($r$) to the left ($l$) side of the junction. In this expression, $\\hat{c}_{r(l)}$ is the annihilation operator of a quasiparticle in the right (left) superconductor and $t_{lr} \\equiv t_{rl}$ the tunneling probability amplitude. The phase $\\varphi_\\text{ac}(t) = (eV_\\text{ac}\/2\\hbar\\omega_0)\\sin(2\\omega_0 t)$ takes into account the ac-voltage. Finally, $\\hat{\\Phi}(t) \\equiv \\hat{\\Phi}(t,x=0)$ is the TL flux at the position of the junction. In the interaction picture with respect to $\\hat{H}_\\text{t}$, this flux is written in terms of the incoming $\\an{in} [\\omega]$ and outgoing $\\an{out} [\\omega]$ fields as \\cite{yurke-denker-pra1984}\n\\begin{align} \\label{phase}\n\\hat{\\Phi}(t,x) &= \\sqrt{\\frac{\\hbar Z_0}{4\\pi}} \\int_{0}^{\\infty} \\frac{d\\omega}{\\sqrt{\\omega}} \\left[ \\an{in}[\\omega]  e^{-i(\\omega t +k_\\omega x) } \\right.  \\nonumber \\\\\n&\\left.  + \\an{out}[\\omega]  e^{-i(\\omega t- k_\\omega x)}  + \\text{ h.c.} \\right],\n\\end{align} \nwith $k_\\omega = \\omega \\sqrt{L_0 C_0}$ the wave number and $Z_0 = \\sqrt{L_0\/C_0}$ the TL impedance defined in terms of its inductance $L_0$ and capacitance $C_0$ per unity of length. The incoming (outgoing) field obeys the commutation relation $[\\an{in (out)}[\\omega],\\ad{in (out)}[\\omega^\\prime]] = \\delta(\\omega - \\omega^\\prime)$.\n\nTo characterize the radiation emitted by the junction, we use the input-output formalism adapted to circuits coupled to quantum conductors \\cite{leppakangas-johansson-prl2015,grimsmo-blais-prl2106,mora-portier-prb2017}. To obtain the input-output boundary condition, the first step is to derive the Heisenberg equations of motions for both TL charge and flux at the position of the junction ($x=0$). The resulting equations are combined to express current conservation\n\\begin{equation}\nC \\ddot{\\hat{\\Phi}}(t,x=0) - \\frac{1}{L_0} \\frac{\\partial \\hat{\\Phi}(t,x)}{\\partial x} \\bigg|_{x=0} = \\hat{I}_\\text{J}(t),\n\\end{equation}\nwhich connects the outgoing field to the incoming field and the current of the junction $\\hat{I}_\\text{J}(t)$. Using Eq.~\\eqref{phase}, the above equation can be expressed as an input-output relation\n\\begin{equation} \\label{inout0}\n \\an {out}[\\omega] = \\frac{1 + i Z_0 C\\omega}{1 - i Z_0 C\\omega} \\an {in}[\\omega] + i \\sqrt{\\frac{Z_0}{\\pi \\hbar \\omega}} \\frac{\\hat{I}_\\text{J}[\\omega]}{1 - i Z_0 C\\omega}, \n\\end{equation}\nwhere $\\hat{I}_\\text{J}[\\omega]$ is the Fourier transform of $\\hat{I}_\\text{J}(t)$. The current $\\hat{I}_\\text{J}[\\omega]$ depends not only on the dc- and ac-voltages, but also on the TL voltage, $\\hat{V}_\\text{TL}(t) = \\dot{\\hat{\\Phi}}(t)$, implying that $\\hat{I}_\\text{J}[\\omega]$ and $\\an {in}[\\omega]$ do not commute. To circumvent the non-commutation between $\\hat{I}_\\text{J}[\\omega]$ and $\\an {in}[\\omega]$, we take advantage of the weak TL-junction coupling $\\sqrt{\\pi Z_0\/R_K} \\ll 1$ (for a typical $Z_0=50 \\Omega$ TL impedance, with $R_K \\simeq 25.8$ k$\\Omega$ the quantum of resistance), and compute $\\hat{I}_\\text{J}[\\omega]$ to second-order in the TL-junction coupling \\cite{mendes-mora-njp2015,grimsmo-blais-prl2106}. In the time-domain and for weak coupling to the environment, the current operator takes the form\n\\begin{equation}\n\\hat{I}_\\text{J}(t) =\\hat{I}_\\text{qp}(t) + \\frac{e^2}{\\hbar^2}\\hat{H}_\\text{qp}^\\text{t}(t)\\hat{\\Phi}(t),  \n\\end{equation}\nwhere $\\hat{I}_\\text{qp}(t) = i(e\/\\hbar) (\\hat{\\mathcal{T}}^\\dagger(t) - \\text{h.c.})$ and $\\hat{H}_\\text{qp}^\\text{t}(t) = \\hat{\\mathcal{T}}(t) + \\text{h.c.}$ are respectively the quasiparticle current and tunneling operators in the absence of TL voltage. The final step is to time-evolve $\\hat{I}_\\text{J}(t)$. Using linear response theory, the quasiparticles operators are time-evolved, in the interaction picture, with the interaction Hamiltonian $H_\\text{int}(t)= -\\hat{I}_\\text{qp}(t)\\hat{\\Phi}(t)$. Once more, we take advantage of the weak TL-junction coupling to expand the time-evolution operator to first-order. We further neglect quantum fluctuations of the current operator, and average over the quasiparticle operators in the terms proportional to $\\hat{\\Phi}(t)$. Under these approximations, the current operator is written in frequency space as\n\\begin{equation} \\label{current}\n\\hat{I}_\\text{J}[\\omega] = \\hat{I}_\\text{qp}[\\omega] + \\sum_n Y_{n}(\\omega- 2n \\omega_0) \\hat{V}_\\text{TL}[\\omega - 2n\\omega_0],\n\\end{equation}\nwith $\\hat{I}_\\text{qp}[\\omega]$ and $\\hat{V}_\\text{TL}[\\omega]$ the Fourier transforms of $\\hat{I}_\\text{qp}(t)$ and $\\hat{V}_\\text{TL}(t)$, respectively. The generalized admittance \n\\begin{align}\nY_n(\\omega) &= -\\frac{i}{\\hbar} \\int \\frac{d\\omega_1}{2\\pi}\\frac{S_n(\\omega)-S_n(2n\\omega_0-\\omega)}{(\\omega_1 +\\omega + i0^+)(\\omega_1 + i0^+)},\n\\end{align}\nrelates the current response at frequency $\\omega$ to the TL-voltage dynamics at frequency $\\omega - 2n\\omega_0$ \\cite{mora-portier-prb2017}. The admittance is defined in terms of the photon-assisted current-current correlator $\\langle \\hat{I}_\\text{qp}(\\omega) \\hat{I}_\\text{qp}(\\omega^\\prime) \\rangle = 2\\pi \\sum_n S_n(\\omega) \\delta(\\omega+\\omega^\\prime - 2n\\omega_0)$ where \n\\begin{align} \\label{noise-PE}\nS_n(\\omega) & =\\frac{1}{2} \\sum_{n_1=-\\infty}^{\\infty} J_{n_1}(\\rho) [J_{n_1+n}(\\rho) S_\\text{eq}(\\hbar\\omega+eV+2n_1\\hbar\\omega_0) \\nonumber \\\\\n&+ J_{n_1-n}(\\rho) S_\\text{eq}(\\hbar\\omega-eV-2n_1\\hbar\\omega_0)]\n\\end{align}\nis non-symmetrized photon-assisted current noise \\cite{mendes-mora-njp2015,grimsmo-blais-prl2106,mora-portier-prb2017}. It is defined in terms of the Bessel function $J_n(\\rho = eV_\\text{ac}\/2\\hbar\\omega_0)$ and of the equilibrium current noise $S_\\text{eq}(\\omega) = 2e \\langle \\hat{I}_{\\text{qp},\\omega} \\rangle \/[1- \\exp{(-\\hbar \\omega\/k_\\text{B} T)}]$, with $\\langle \\hat{I}_{\\text{qp},\\omega} \\rangle$ is the SIS junction quasiparticle current \\cite{ingold-nazarov-book1992}.\n\nEquation~\\eqref{current} describes the linear response of the junction due to TL-voltage fluctuations. In the absence of ac-voltage, the linear response is strictly local in frequency with $Y_{n\\neq 0}(\\omega) = 0$. On the other hand, the addition of the ac-bias enables the response of $Y_{n\\neq 0}(\\omega)\\neq 0$ due to TL-voltage fluctuations at frequencies beating with the harmonics of the pump frequency $2n \\omega_0$. It is important to mention that $Y_{n\\neq 0}(\\omega)$ is only different of zero for non-linear junctions \\cite{mora-portier-prb2017}. Consequently, in this device, parametric interaction is due to the combination of both photon-assisted transport and non-linearity. For instance, the non-linearity converts a single pump photon of frequency $2\\omega_0$ into two photons of frequency $\\omega_0$ \\cite{clerk-schoelkopf-rmp2010}. In the SIS amplifier, this process is given by Im$Y_1(\\omega)$, which describes the coherent conversion of $2\\hbar\\omega_0$ quantum of energy from the pump to two photons of frequency $\\omega_0$ \\cite{mendes-mora-njp2015}. As the junction absorbs photons, thereby introducing losses, the SIS amplifier has to be operated in the dc-voltage range in which parametric interaction, which is related to $Y_1(\\omega_0)$, dominates over dissipative effects $\\propto \\mathrm{Re}Y_0(\\omega_0)$. Fig.~\\ref{fig1}(b) illustrates the dc-voltage dependence of $Y_1(\\omega_0)$ and Re$Y_0(\\omega_0)$ for a weak pump amplitude $eV_\\text{ac}=0.135 \\times 2\\hbar\\omega_0$. We observe that $Y_1(\\omega_0)$ present strong discontinuities at $eV = 2\\Delta+n\\hbar\\omega_0$, which is a characteristic of photon-assisted transport and non-linearity, while Re$Y_0(\\omega_0)$ is weakly affected by the ac-voltage. Moreover, $Y_1(\\omega_0)$ is larger than Re$Y_0(\\omega_0)$ for $eV < 2\\Delta-\\hbar\\omega_0$, which sets dc-voltage operational point of the device. Finally, it is important to emphasize that $\\hat{I}_\\text{qp}[\\omega]$ and $Y_{n}(\\omega- 2n \\omega_0)$ depend only on the quasiparticles dynamics determined by $H_\\text{qp} + H_\\text{qp}^\\text{t}(t)$, and on the intrinsic characteristics of the junction, e.g., the superconducting gap $\\Delta$ and normal state resistance $R_T$, bias conditions, and temperature.\n\nFinally, replacing Eq.~\\eqref{current} into Eq.~\\eqref{inout0}, we obtain the expression for the outgoing field in terms of the incoming field and of the quasiparticles current operators\n\\begin{align} \\label{inout1}\n\\an{out}[\\omega+\\omega_0] & = r_\\omega\\an{in}[\\omega+\\omega_0] + \\gamma_\\omega\\ad{in}[\\omega_0-\\omega]\\nonumber \\\\\n&+ \\alpha_\\omega \\hat{I}_\\text{qp} [\\omega+\\omega_0] + \\beta_\\omega \\hat{I}_\\text{qp} [\\omega-\\omega_0],\n\\end{align}\nwhere $r_\\omega = \\Delta_\\omega^{-1}[P_{-,\\omega+\\omega_0}P_{+,\\omega_0-\\omega}^* + A_\\omega]$ is the reflection coefficient, with $P_{\\pm,\\omega} = \\kappa_{\\pm,\\omega} \\mp i \\omega$ and $\\kappa_{\\pm,\\omega}=(Z_0^{-1}\\pm \\text{Re}Y_{0,\\omega})\/C$ defining the total ($+$) and effective ($-$) photon decay rate of the device. Here, terms proportional to $Z_0^{-1}$ and $\\text{Re}Y_{0,\\omega}$ are the internal TL decay rate at which the junction absorb and emits photons, respectively. The dependence of the reflection coefficient with parametric term is given by $A_\\omega = Y_{1,\\omega-\\omega_0}Y_{1,-\\omega-\\omega_0}^*\/C^2$. Finally, $\\Delta_\\omega =P_{+,\\omega+\\omega_0}P_{+,\\omega_0-\\omega}^* - A_\\omega$. Note that, we neglected a frequency shift contribution proportional to $\\text{Im}Y_{0,\\omega}$. In the second term of Eq.~\\eqref{inout1}, we have introduced $\\gamma_\\omega = 2(Z_0 C \\Delta_\\omega )^{-1} \\sqrt{(\\omega_0 - \\omega)\/(\\omega + \\omega_0)} Y_{1,\\omega-\\omega_0}$. Moreover, the coefficients of $\\hat{I}_\\text{qp}[\\omega\\pm\\omega_0]$ are $\\alpha_\\omega =  \\mu_\\omega  \\Delta_\\omega^{-1} P_{+,\\omega_0-\\omega}^*$, and $\\beta_\\omega = \\mu_\\omega^*  \\Delta_\\omega^{-1} Y_{1,\\omega-\\omega_0}\/C$, with $\\mu_\\omega= i\/C\\sqrt{\\pi Z_0 \\hbar(\\omega+\\omega_0)}$. \n\nEquation~\\eqref{inout1} is a central result of this paper and show that the dc- and ac-voltage biased SIS junction acts as an amplifier operating in reflection mode. More specifically, for a fixed frequency $\\omega \\neq 0$, the operators $\\an{in}[\\omega+\\omega_0]$ and $\\ad{in}[\\omega_0-\\omega]$ commute and can be seen as the input signal and the idler mode operators of a phase-preserving amplifier \\cite{caves-prd1982,clerk-schoelkopf-rmp2010}. In this mode, the gain is simply given by $G(\\omega) = |r_\\omega|^2$ and the three last terms of Eq.~\\eqref{inout1} correspond to added noise, which limits the amplifier quantum efficiency. The added noise is \\cite{caves-prd1982,boutin-blais-prappl2017}\n\\begin{align} \\label{addednoise1}\n\\mathcal{A}(\\omega) &= \\frac{1}{2} \\left( 1 -\\frac{1}{G(\\omega)} +  \\frac{4\\pi}{G(\\omega)} \\left[ |\\alpha_\\omega|^2 S_0(\\omega+\\omega_0) \\nonumber \\right. \\right. \\\\\n&\\left. \\left. + |\\beta_\\omega|^2 S_0(\\omega-\\omega_0) + 2\\text{Re} [ \\alpha_\\omega\\beta_\\omega^*] S_1(\\omega+\\omega_0) \\right] \\right),\n\\end{align} \nwhere the first two terms describe the noise added of an ideal phase-preserving amplifier. The remaining terms are noise generated by the tunneling of quasiparticles, and they originate from absorption and emission processes of one and two quantum of energy $\\hbar\\omega_0$ by the junction. The above equation shows that, in the high-gain limit, the SIS amplifier is near quantum-limited even in the presence of this quasiparticle noise. Thus, to mitigate the effects of the quasiparticle noise, the operational voltages of the device are such that these noise terms [three last terms of Eq.~\\eqref{addednoise1}] are much smaller than the parametric interaction. As illustrated in Fig.~\\ref{fig1}(b), the optimal dc-voltage lies in the interval $2(\\Delta- \\hbar\\omega_0) < eV < 2\\Delta - \\hbar\\omega_0$, where absorption and emission processes by the junction, $\\propto \\text{Re}Y_0(\\omega)$, are weak and parametric interaction, $\\propto Y_1(\\omega)$, is dominant. \n\nFor $\\omega = 0$, $\\an{in}[\\omega_0]$ and $\\ad{in}[\\omega_0]$ do not commute, and the first two terms of Eq.~\\eqref{inout1} rather characterize an ideal phase-sensitive amplifier \\cite{caves-prd1982,clerk-schoelkopf-rmp2010}. In this operational mode, the gain $G_0$ is defined as a combination of $r_0$, $\\gamma_0$ and the phases of the input and output fields \\cite{boutin-blais-prappl2017}. The added noise is given by a linear combination of the two last terms of Eq.~\\eqref{inout1} and it also depends on the input and output field phases \\cite{boutin-blais-prappl2017}.\n\nTo better characterize the radiation emitted by the SIS amplifier, we also compute the output power spectrum $S_\\theta(\\omega)$, defined in terms of $\\langle \\Delta \\hat{X}_{\\theta}[\\omega]  \\Delta \\hat{X}_{\\theta}[\\omega^\\prime] \\rangle = S_\\theta(\\omega) \\delta(\\omega+\\omega^\\prime)$, with $\\hat{X}_{\\theta}[\\omega] = e^{-i \\theta}\\hat{a}_\\text{out}[\\omega+\\omega_0] + e^{i \\theta}\\hat{a}_\\text{out}^{\\dagger}[\\omega_0 -\\omega]$ the output field quadrature with fluctuations $\\Delta \\hat{X}_{\\theta}[\\omega] = \\hat{X}_{\\theta}[\\omega] - \\langle \\hat{X}_{\\theta}[\\omega]  \\rangle$ and $\\theta$ the phase of the output field. Similarly to amplification, the emitted radiation can be also characterized by single-mode $S_\\theta(\\omega=0) < 1$ or two-mode $S_\\theta(\\omega \\neq 0) < 1$ squeezed state.\n\nIt is worth mentioning that to derive Eq.~\\eqref{inout1} we neglected conversion processes where photons of frequency $|\\omega| > 2\\omega_0$ are up- or down-converted to $\\omega$. This approximation is justified by the fact that the $R_TC$ time of the junction acts as a high-frequency cutoff. For the results presented in the next section, we take $\\omega_{RC} = 1\/R_T C = 2\\pi \\times 30$ GHz as a fixed parameter of the junction, which is obtained for a normal state resistance $R_T = 50 \\Omega$ and $C=100$ fF. In addition, a low-pass filter can be used to filter all frequencies above $2\\omega_0$. Since we are interested in signals of frequency $\\omega_0 \\ll \\Delta\/\\hbar$ amplified by operating the junction close to the onset of quasi-particle transport $eV \\approx 2\\Delta$, we also ignored current and noise terms originating from the tunneling of Cooper pairs whose Josephson frequency $\\sim 4\\Delta\/\\hbar \\gg 2\\omega_0$ would be efficiently filtered as well.\n\n\\section{Results \\label{results}}\n\nWe first present results for an ideal SIS junction, with transport response rising steeply for voltages $eV = 2\\Delta + n\\hbar \\omega_0$ as illustrated in Fig.~\\ref{fig1}(b). We then investigate how gain and squeezing properties are affected by low-frequency noise, which are smoothing out the transport response of the junction and, consequently, diminishing the strength of the parametric interaction.\n\n\\subsection{Ideal SIS junction \\label{resultsA}}\n\nBefore turning to the full amplification and squeezing frequency dependences, we first present results for $\\omega = 0$ which corresponds to the phase-sensitive mode. Figs.~\\ref{fig2}(a) and (b) illustrate, respectively, the phase-sensitive gain and single-mode squeezing as a function of $\\Delta\/\\hbar\\omega_0$, when optimizing $V$ and $V_\\text{ac}$ to maximize single-mode squeezing. Here, $\\Delta$ varies while $\\omega_0\/2\\pi$ is kept fixed and equal to 4 GHz. Already at small $\\Delta\/\\hbar\\omega_0 \\simeq 2$ we observe 15 dB of amplification and $-10$ dB of squeezing. As $\\Delta\/\\hbar\\omega_0$ increases, the strength of the nonlinearities giving rise to parametric interaction increases, thus enhancing gain and squeezing as illustrated in Fig.~\\ref{fig2}. This can be understood by the fact that $S_n(\\omega)$ and, consequently, $Y_n(\\omega)$ are proportional to $\\Delta\/R_T$ in the limit of $\\Delta\/\\hbar\\omega_0 \\rightarrow \\infty$. Thus, increasing $\\Delta$ leads to an increase of the parametric interaction strength $\\propto Y_1(\\omega)$. For $\\Delta\/\\hbar\\omega_0 = 30$, amplification and squeezing reach $43$ dB and $-25$ dB, respectively. In Fig.~\\ref{fig2}, the dashed-line corresponds to an Aluminum junction. As expected, the optimal dc-voltage is $eV^\\text{opt} \\lesssim 2\\Delta - \\hbar\\omega_0$ for all values of $\\Delta\/\\hbar\\omega_0$ (not shown). At this dc-voltage, the susceptance Im$Y_1$ is larger than the dissipative contributions $\\propto$ Re$Y_0$, thus making the parametric interaction the dominant process over dissipation, as illustrated in Fig.~\\ref{fig1}(b). It is interesting to note that $V^\\text{opt}$ is also the optimal operational point of SIS mixers \\cite{trucker-feldman-rmp1985}. On the other hand, the optimal ac-voltage $V_\\text{ac}^\\text{opt}$ decreases as $\\Delta\/\\hbar\\omega_0$ increases.\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[width=0.45\\textwidth]{fig2.pdf}\n\\caption{(a) Phase-sensitive gain and (b) single-mode squeezing as a function of $\\Delta\/\\hbar\\omega_0$ for $\\omega_0\/2\\pi = 4$ GHz. The dashed line marks the values for an Aluminum superconducting junction. For each value of $\\Delta\/\\hbar\\omega_0$ there are optimal ac- and dc-voltages that maximize single-mode squeezing. The optimal dc-voltage is always equal to $eV^\\text{opt} \\lesssim 2\\Delta - \\hbar \\omega_0$, while the optimal ac-voltage amplitude diminishes as $\\Delta\/\\hbar\\omega_0$ increases (not shown here). \\label{fig2}} \n\\end{center}\n\\end{figure}\n\nFor the remainder of this article, we consider an Aluminum SIS junction with $\\Delta\/\\hbar\\omega_0\\approx 10.9$ for $\\omega_0\/2\\pi = 4$ GHz. In the phase-sensitive mode ($\\omega=0$), gain and squeezing are respectively $\\sim 34$ dB and $\\sim -20$ dB at the optimal operational point ($eV^\\text{opt} \\approx 20.762 \\times \\hbar\\omega_0$ and $eV_\\text{ac}^\\text{opt} \\approx 0.06 \\times 2\\hbar\\omega_0$). Moreover, the noise added by tunneling of quasiparticles is of the order of $10^{-4}$ added photons, meaning that the amplifier operates near the quantum limit.\n\nWe now investigate the frequency dependence of phase-preserving gain and two-mode squeezing, and how they are controlled by dc- and ac-voltage amplitudes. First, we find that for the optimal voltages, $V^\\text{opt}$ and $V_\\text{ac}^\\text{opt}$, both gain and squeezing vanish at finite detuning ($\\omega\\neq 0$). This is explained by the fact that, at finite detuning, the absorption of single photons is the dominant process already for $eV = 2\\Delta - \\hbar(\\omega+\\omega_0) < V^\\text{opt}$, while parametric interaction is dominant for $eV \\lesssim 2\\Delta - \\hbar(\\omega+\\omega_0)$. The response of the junction at finite detuning $\\omega$ is similar to the one presented in Fig.~\\ref{fig1}(b). To achieve parametric amplification at finite detuning, we set dc-voltage to a value smaller than $V^\\text{opt}$. In this case, there is a frequency range [$e(V-V^\\text{opt})\/\\hbar,e(V^\\text{opt}-V)\/\\hbar$] where the parametric interaction is the dominant mechanism, leading to phase-preserving amplification and two-mode squeezing at finite detuning. This frequency range defines the operational bandwidth of the device.\n\nFigure~\\ref{fig3}(a) illustrates the phase-preserving gain while panel (b) shows two-mode squeezing as a function of the detuning $(\\omega-\\omega_0)\/\\omega_0$ for three different values of $V$ and for fixed $eV_\\text{ac} = 0.1 \\times 2\\hbar \\omega_0$. These results show striking features. The first is that the bandwidth is controlled by the dc-voltage and, as anticipated, equal to $2 e(V^\\text{opt}- V)\/\\hbar$ for $2(\\Delta- \\hbar\\omega_0) < eV < 2\\Delta - \\hbar\\omega_0$. As expected, the largest bandwidth occurs for $eV \\approx 2\\Delta - 2\\hbar\\omega_0$. Both gain and squeezing spectra are flat over a 4 GHz band, with more than 10 dB of phase-preserving gain and -14 dB of squeezing. Moreover, the broadband squeezing spectrum characterizes the generation of far-separated two-mode squeezed states.\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[width=0.45\\textwidth]{fig3.pdf}\n\\caption{Aluminum junction ($\\Delta\/\\hbar\\omega_0 \\approx 10.9$): phase-preserving gain (a) and two-mode squeezing (b) as a function of the frequency detuning from the pump frequency $\\omega_0\/2\\pi = 4$ GHz. We consider a fixed ac-voltage amplitude $V_\\text{ac} = 0.1 \\times 2\\hbar\\omega_0$ for three different values of dc-voltages, $V \\approx 2\\Delta-1.2 \\hbar \\omega_0$ (cyan line), $V \\approx 2\\Delta-1.5 \\hbar \\omega_0$ (dark-blue line), and $V \\approx 2\\Delta- 2 \\hbar \\omega_0$ (black line). At $\\omega=\\omega_0$, the amplifier is operated in the phase-sensitive mode. \\label{fig3}}\n\\end{center}\n\\end{figure}\n\nAs mentioned before, the tunneling of quasiparticles also generates noise preventing the amplifier from operating at the quantum-limit. To further characterize the SIS junction parametric amplifier, we fix $eV \\approx 2(\\Delta- \\hbar\\omega_0)$ and investigate the impact of the ac-voltage amplitude on the gain and added noise. Figs.~\\ref{fig4}(a) and (b) illustrate the gain and added noise for three different values of $V_\\text{ac}$, respectively. The gain is observed to raise with increasing ac-voltage amplitude, while adding noise and decreasing the bandwidth. More importantly, gain can be significantly increased while the added noise remains of the same order. For instance, increasing $V_\\text{ac}$ from $0.11 \\times 2\\hbar \\omega_0$ to $0.135 \\times 2\\hbar \\omega_0$, the gain jumps from 14 dB to approximately 30 dB, while the added noise remains practically the same. The bandwidth is reduced with the increased gain, nonetheless it is of the same order of the measured bandwidth of the Josephson traveling wave parametric amplifier \\cite{macklin-siddiqi-science2015}. The increase of both gain and added noise is due to the enhancement of photon-assisted tunneling of quasiparticles, which controls both parametric amplification and added noise [three last terms of Eq.~\\eqref{addednoise1}]. However, even with the increase of the added noise with $V_\\text{ac}$, the SIS amplifier operates near the high-gain quantum limit of an ideal phase-preserving amplifier [dashed line in Fig.~\\ref{fig4}(b)] \\cite{clerk-schoelkopf-rmp2010}.\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[width=0.45\\textwidth]{fig4.pdf}\n\\caption{ Aluminum junction ($\\Delta\/\\hbar\\omega_0 \\approx 10.9$): Gain spectrum (a) and added noise power spectral density (b) as a function of the frequency detuning from $\\omega_0\/2\\pi = 4$ GHz for $V\\approx 2\\Delta-2\\hbar\\omega_0$ and three different values of ac-voltage amplitude, $V_\\text{ac} \\approx 0.08 \\times 2\\hbar \\omega_0$ (cyan line), $V_\\text{ac} \\approx 0.11 \\times 2\\hbar \\omega_0$ (blue line), and $V_\\text{ac} \\approx 0.135 \\times 2\\hbar \\omega_0$ (black line). The dashed line marks the added noise for an ideal phase-preserving amplifier. In this case, half-photon is added during the amplification process. As the ac-voltage amplitude increases the gain increases and the bandwidth is reduced. However, the added noise still near the quantum limit for all values of $V_\\text{ac}$. \\label{fig4}}\n\\end{center}\n\\end{figure}\n\nFinally, a higher degree of amplification and squeezing can be obtained by fabricating a junction with normal state resistance smaller than 50 $\\Omega$. However, due to the impedance mismatch between the junction and the TL, the bandwidth goes to zero as $R_\\text{T}$ diminishes. This can be understood by the fact that internal reflections of the microwave signal, caused by the large impedance mismatch, can interfere destructively, thereby destroying the coherent coupling between the signal ($\\an{in}[\\omega+\\omega_0]$) and the idler ($\\ad{in}[\\omega_0 - \\omega]$) frequency modes. Also, as shown in Fig.~\\ref{fig2}, a SIS junction with large superconducting gap leads to higher gain and squeezing. For instance, a Niobium junction ($\\Delta\/h \\sim 750$ GHz) would, in principle, generate higher gain and squeezing than an Aluminum junction. \n\nFinally, a higher degree of amplification and squeezing can be obtained by fabricating a junction with normal state resistance smaller than 50 $\\Omega$. However, due to the impedance mismatch between the junction and the TL, the bandwidth and gain goes to zero as $R_\\text{T}$ diminishes. Also, as shown in Fig.~\\ref{fig2}, a SIS junction with large superconducting gap leads to higher gain and squeezing. For instance, a Niobium junction ($\\Delta\/h \\sim 750$ GHz) would, in principle, generate higher gain and squeezing than an Aluminum junction. Moreover, the operational bandwidth could be extended to the THz range with NbSi or NbN junctions.\n\n\\subsection{Effects of low-frequency noise \\label{resultsB}}\n\nThe results presented in the previous section were obtained considering an ideal SIS junction in the low-temperature limit $k_\\text{B} T \\ll 2\\Delta$ for which the transport response rises steeply for $eV=2\\Delta+n\\hbar\\omega_0$ \\cite{barone-paterno-book}. However, this is an idealized situation and, in practice, temperature or low-frequency noise can smoothen the transport response. Here, we consider the effects of low-frequency noise on gain and squeezing properties of the SIS amplifier. These effects are included by assuming that the junction interact with a low-frequency electromagnetic environment \\cite{hofheinz-esteve-prl2011} and that the transport properties are described by the $P(E)$-theory \\cite{ingold-nazarov-book1992}. Indeed, this approach has been shown to quantitatively explain the finite Dynes tunneling density of states, usually observed below the dc-transport gap in a normal-insulator-superconductor junctions and the corresponding smoothing of the BCS coherence peak \\cite{pekola-tsai-prl2010,barone-paterno-book}. In this approach, the low-frequency noise modifies the equilibrium noise current noise to \n\\begin{equation} \\label{noiseeq-pe}\nS_\\text{eq}^\\text{eff}(\\omega) = \\int_{-\\infty}^{\\infty} S_\\text{eq}(\\hbar\\omega - E) P(E)  dE,\n\\end{equation}\nwhere $P(E)$ is the probability density of a tunneling electron emitting energy and takes the form for low-frequency noise \\cite{ingold-nazarov-book1992}\n\\begin{equation}\nP(E) = \\frac{1}{\\sqrt{4\\pi E_c  k_\\text{B} T}}\\exp \\left( - \\frac{(E-E_c)^2}{4  E_c   k_\\text{B} T} \\right) ,\n\\end{equation}\nwith $E_c = e^2\/2C_{bt}$ is the capacitor charging energy. This model is justified by the fact that the low noise biasing scheme consists of a resistive voltage divider followed by large capacitive filtering \\cite{hofheinz-esteve-prl2011,chen-rimberg-prb2014}. The low-frequency impedance is thus the parallel combination of a resistance with a large capacitance $C_\\text{bt}$. This filtering scheme reduces the bandwidth over which low-frequency voltage noise can develop to the kHz range \\cite{chen-rimberg-prb2014}, making the low-frequency voltage fluctuations fully classical. With this model, the theory developed in Sec.~\\ref{model} remains the same except for the replacement of $S_\\text{eq}(\\omega)$ by $S_\\text{eq}^\\text{eff}(\\omega)$ in Eq.~\\eqref{noise-PE}.\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[width=0.45\\textwidth]{fig5.pdf}\n\\caption{Aluminum junction ($\\Delta\/\\hbar\\omega_0 \\approx 10.9$): Gain (a) and squeezing (b) as a function of frequency detuning in the presence (solid lines) and in the absence (dashed line) of low-frequency electromagnetic environment. The two solid lines illustrate an efficient ($C_\\text{bt}\\sim 1$~nF - blue line) and inefficient ($C_\\text{bt}\\sim 10$~pF - cyan line) low-frequency noise filtering scheme. As expected, for an efficient filtering scheme, both gain and squeezing are quantitatively equal to the result obtained in absence of low-frequency electromagnetic environment. \\label{fig5}}\n\\end{center}\n\\end{figure}\n\nFigure~\\ref{fig5} illustrates the resulting effect on both (a) gain and (b) squeezing for $C_\\text{bt}\\sim 1$~nF (dark-blue line) and $C_\\text{bt}\\sim 10$~pF (light-blue line) corresponding to an efficient and inefficient low-frequency noise filtering scheme, respectively. These results are compared with the ideal case (dashed-line), where the junction does not interact with the low-frequency environment. For an Aluminum junction ($\\Delta\/\\hbar\\omega_0 \\approx 10.9$) and $T=15$ mK, $C_\\text{bt}\\sim 1$~nF corresponds to rms voltage fluctuations of $\\sqrt{k_\\text{B}T\/2C_\\text{bt}} \\sim 11$~nV (dark-blue line), the gain and squeezing are only weakly affected by the low-frequency environment which is efficiently filtered. The effect of low-frequency noise on $S_\\text{eq}^\\text{eff}(\\omega)$ is illustrated in the inset of Fig.~\\ref{fig5}. For $C_\\text{bt} \\sim 1$~nF, $S_\\text{eq}^\\text{eff}(\\omega)$ is almost indistinguishable from the ideal case (dashed line), a signature that the low-frequency noise is efficiently filtered. On the other hand, under less efficient filtering, $C_{bt} \\sim 10$~pF corresponding to $\\sim 0.1~\\mu$V rms voltage fluctuations (light-blue line), the equilibrium current noise rises smoothly and its behavior near $2\\Delta\/\\hbar$ deviates from the ideal case (see inset). In this case, low-frequency noise diminishes both gain and squeezing amplitudes (light-blue line). However, the bandwidth is only weakly modified.\n\n\\section{Conclusions \\label{conclusions}}\n\nWe have proposed a near-quantum-limited broadband amplifier and squeezer based on the photon-assisted tunneling of quasiparticles in a SIS junction. This device can function as a phase-sensitive or phase-preserving amplifier. The gain can be tuned by the dc- and ac-voltages amplitude to reach 30 dB over 2 GHz or 14 dB over 4 GHz. This device is also a source of two-mode squeezing with bandwidth over more than 4 GHz and $-14$ dB of squeezing. Moreover, gain and two-mode squeezing can be fine -tuned in-situ by changing the pumping tone frequency and dc-voltage bias. Therefore, the design and fabrication simplicity of this SIS amplifier, together with its operational mode flexibility, makes it a versatile near-quantum-limited microwave amplifier and squeezer, which can be easily integrated in many quantum microwaves experiments. \n\n\\section*{Acknowledgments}\n\nUCM thank S. Boutin, A. L. Grimsmo and M. Westig for fruitful discussion, and the Quantronics group for hospitality. UCM and AB were supported from Canada First Research Excellence Fund and NSERC. BR was supported by Canada Excellence Research Chairs, Government of Canada, Natural Sciences and Engineering Research Council of Canada, Qu\\'ebec MEIE, Qu\\'ebec FRQNT via INTRIQ, Universit\\'e de Sherbrooke via EPIQ, and Canada Foundation for Innovation. The research at CEA-Saclay has received funding from the European Research Council under the European Union's Programme for Research and Innovation (Horizon 2020)\/ERC Grant Agreement No. 639039, and support from the ANR AnPhoTeQ research contract.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nWe introduce the rosette harmonic mappings, analogous to the $n$-cusped\nhypocycloid mappings. For each integer $n$ where $n\\geq3,$ we obtain a family\nof mappings, that in many instances have only cyclic rather than dihedral\nsymmetry. The mappings have $n$ cusps, or in some cases $n$ nodes rather than\ncusps. It is interesting to consider one particular mapping for each $n$ in\nwhich the boundary of the mapping is continuous, \\ but with arcs of constancy.\nOur main goal is to establish the univalence of the rosette harmonic mappings.\nAdditionally we describe the location and orientation of cusps and nodes. We\nalso define a fundamental set from which the full graph of a rosette mapping\ncan be reconstructed, which is useful for computational efficiency.\n\nWe begin by establishing some notation and standard terminology associated\nwith planar harmonic mappings. A \\textbf{harmonic mapping} $f$ is a complex\nvalued univalent harmonic function defined on a region in the complex plane $%\n\\mathbb{C}\n$. Harmonic mappings can be arrived at in a variety of ways, for example by\nadding different harmonic functions together, or by using the Poisson integral\nformula, and more recently by using the shear construction, first described in\n\\cite{ClunieSheilSmall}. Univalence is not guaranteed however, except for in\nthe latter approach. For any harmonic mapping $f,$ we write $f=h+\\bar{g}$\nwhere $h$ and $g$ are analytic, and call $h$ and $g$ the \\textbf{analytic} and\n\\textbf{co-analytic} parts of $f$, respectively. The decomposition is unique\nup to the constant terms of $h$ and $g,$ and $h+\\bar{g}$ is known as the\n\\textbf{canonical decomposition }of $f$\\textbf{. \\ }Our mappings are defined\non $U,$ the open unit disk in the complex plane. The Jacobian $J_{f}$ of $f$\nis given by $J_{f}\\left(  z\\right)  =\\left\\vert f_{z}\\left(  z\\right)\n\\right\\vert ^{2}-\\left\\vert f_{\\bar{z}}\\left(  z\\right)  \\right\\vert\n^{2}=\\left\\vert h^{\\prime}\\left(  z\\right)  \\right\\vert ^{2}-\\left\\vert\ng^{\\prime}\\left(  z\\right)  \\right\\vert ^{2}$. We say that $f$ is\n\\textbf{sense-preserving} if $J_{f}\\left(  z\\right)  >0$ in $U.$ A theorem of\nL\\v{e}wy \\cite{Lewy} states that a harmonic function $f$ is locally one-to-one\nif $J_{f}$ is non-vanishing in $U$. Thus $f$ is locally one-to-one and\nsense-preserving if and only if $\\left\\vert g\\right\\vert <\\left\\vert\nh\\right\\vert $ and thus, there exists a meromorphic function known as the\n\\textbf{analytic dilatation} of $f,$ given by $\\omega_{f}\\left(  z\\right)\n=g^{\\prime}\/h^{\\prime}.$ Note that the analytic dilatation is related to the\n\\textbf{complex dilatation }$\\mu_{f}=\\bar{g}^{\\prime}\/h^{\\prime}$ from the\ntheory of quasiconformal mappings. We refer here to $\\omega_{f}\\left(\nz\\right)  $ simply as the \\textbf{dilatation} of $f $ - for more information\nsee \\cite{PeterHMBook} and \\cite{ECA}.\n\nThe rosette harmonic mappings of Definition \\ref{defnharmonic} are\nmodifications of a simple harmonic mapping known as the hypocycloid harmonic\nmapping, with image under the unit disk bounded by a $n$-cusped hypocycloid.\nThe symbol $\\partial A$ here denotes the topological boundary of the set $A$.\n\n\\begin{example}\n\\label{hypo}Let $n\\in%\n\\mathbb{N}\n,$ $n\\geq3.$ The \\textbf{hypocycloid harmonic mapping} is defined on $U$ by\n$f_{hyp}\\left(  z\\right)  =z+\\frac{1}{n-1}\\bar{z}^{n-1}$ with analytic and\nco-analytic parts $z$ and $\\frac{1}{n-1}z^{n-1}$. The dilatation is\n$\\omega_{f}\\left(  z\\right)  =z^{n-2}$ and $J_{f}\\left(  z\\right)\n=1^{2}-\\left\\vert \\bar{z}^{n-2}\\right\\vert ^{2}=1-\\left\\vert z^{2n-4}%\n\\right\\vert .$ Clearly $J_{f}\\left(  z\\right)  >0$ in $U$ so $f$ is locally\none to one$.$ It is also univalent on $U$ (see for instance \\cite{PeterHMBook}\nor \\cite{ECA}). Upon extension to $\\bar{U},$ we can consider the boundary\ncurve $f\\left(  e^{it}\\right)  ,$ which for $n=4\\ $is the familiar astroid\ncurve from calculus. We consider the boundary extension $f_{hyp}\\left(\ne^{it}\\right)  =e^{it}+\\frac{1}{n-1}e^{-i\\left(  n-1\\right)  t}$ which has\nsingular points (where the derivative is $0$ ) precisely when $t=2k\\pi\/n$ ,\n$k=1,2,...,n$. The left of Figure \\ref{rosette0} shows the image of $\\bar{U}$\nunder $f_{hyp} $ for $n=6$, where $f_{hyp}$ maps the unit circle $\\partial U$\nonto a $6$-cusped hypocycloid.\n\\end{example}\n\n\n\\begin{figure}[h]%\n\\centering\n\\includegraphics[\nheight=2.3514in,\nwidth=5.028in\n]%\n{rosette0.eps}%\n\\caption{Images of a regular polar grid in $U$ under the 6-cusped hypocycloid\n(left) and a 6-cusped rosette mapping (right) defined by $f_{0}\\left(\nz\\right)  =z\\,_{2}F_{1}\\left(  \\frac{1}{2},\\frac{1}{12};\\frac{13}{12}%\n;z^{12}\\right)  +\\frac{1}{5}\\bar{z}^{5}\\,\\overline{_{2}F_{1}\\left(  \\frac\n{1}{2},\\frac{5}{12};\\frac{17}{12};z^{12}\\right)  }$}%\n\\label{rosette0}%\n\\end{figure}\n\n\nThe rosette harmonic mappings introduced here can be viewed as modifications\nof the hypocycloid mappings, and are formulated by incorporating Gauss\nhypergeometric $_{2}F_{1}\\,\\ $factors into the analytic and co-analytic parts.\nThe rosette mappings $f_{\\beta}$ will be defined in Section 3, but an example\nof a 6-cusped rosette mapping appears on the right of Figure \\ref{rosette0}.\nIn comparison with cusps of the 6-cusped hypocycloid, the rosette has cusps\nthat are more \"pointy\". Figure \\ref{five} indicates further examples of\nrosette mappings for $n=6$ in which the images of the unit disk may have\nrotational but not reflectional symmetry. The process by which we obtain\nrosette mappings with essentially different features is by rotating the\nanalytic and co-analytic parts relative to one another.\n\nThis article is organized as follows: In Section 2, the properties of the\n$_{2}F_{1}$ \\linebreak hypergeometric functions utilized in the definition of\nthe rosette harmonic mappings are described. In Section 3, the rosette\nharmonic mappings are defined and their rotational and reflective symmetries\nare explored. In Section 4 we describe the boundary and arcs and the nodes and\ncusps seen there. In particular, we highlight a remarkable situation in which\nthe analytic and co-analytic parts \"cancel\" on $\\partial U,$ leading to arcs\nof constancy in the extension to the boundary. This has significant\nconsequences for the associated minimal surface that lifts from this map, as\ndescribed in \\cite{AbdullahMcDougall}. In Section 5 we use the argument\nprinciple for harmonic functions to show that the rosette harmonic mappings\nare in fact univalent. We also describe a computationally efficient way to\nconstruct the image of the unit disk by using rotations of a smaller\n\"fundamental set\".\n\n\\section{Hypergeometric functions\\label{hyper}}\n\nWe begin by defining two Gauss hypergeometric $_{2}F_{1}$ functions.\n\n\\begin{definition}\n\\label{hyperDef}Let $U$ denote the unit disk. For $n\\geq2$ and $z\\in\\bar{U},$\nconsider the Gauss $_{2}F_{1}$ hypergeometric functions\n\\begin{align*}\nH_{n}\\left(  z\\right)   &  =\\,_{2}F_{1}\\left(  \\frac{1}{2},\\frac{1}%\n{2n},1+\\frac{1}{2n},z\\right) \\\\\nG_{n}\\left(  z\\right)   &  =\\,_{2}F_{1}\\left(  \\frac{1}{2},\\frac{1}{2}%\n-\\frac{1}{2n},\\frac{3}{2}-\\frac{1}{2n},z\\right)\n\\end{align*}\nfor $n\\geq2.$ Note that when $n=2$ that $G_{2}=H_{2}$.\n\\end{definition}\n\nBy the Corollary in \\cite{MerkesScott}, $H_{n}$ and $G_{n}$ map the unit disk\nonto a convex region. Since $H_{n}\\left(  0\\right)  =G_{n}\\left(  0\\right)\n=1,$ the convex regions $H_{n}\\left(  U\\right)  $ and $G_{n}\\left(  U\\right)\n$ contain $1$. Thus $H_{n}$ and $G_{n}$ can be considered to be perturbations\nof the constant function of unit value$.$ In Figure \\ref{pert} where $n=6$,\none can see the distortion from unity in $H_{6}\\left(  z\\right)  $ and\n$G_{6}\\left(  z\\right)  .$ The reflectional symmetry in the real axis is also\napparent. These and further properties of the hypergeometric functions $H_{n}\n$ and $G_{n}$ are stated in Proposition \\ref{hyperProps}.%\n\n\\begin{figure}[h]%\n\\centering\n\\includegraphics[\nheight=1.3699in,\nwidth=3.9297in\n]%\n{pert.eps}%\n\\caption{Images of a polar grid in $U$ under $H_{6}$ and $G_{6}.$}%\n\\label{pert}%\n\\end{figure}\n\n\n\\begin{proposition}\n\\label{hyperProps} Let $n\\geq3.$ \\newline(i) The Taylor coefficients $c_{m}$\nof $H_{n}\\left(  z\\right)  \\,$and $d_{m}$ of $G_{n}\\left(  z\\right)  $ are\ngiven by the formula\n\\[\nc_{m}=A_{m}\\frac{1}{2mn+1}\\text{ and }d_{m}=A_{m}\\frac{n-1}{n\\left(\n2m+1\\right)  -1},\\text{ }m\\geq0,\n\\]\nwhere $A_{0}=1,$ and $A_{m}=\\left(\n\\begin{array}\n[c]{c}%\n2m-1\\\\\nm-1\n\\end{array}\n\\right)  \/2^{2m-1},$ for $m\\geq1.$\\newline(ii) The hypergeometric functions\n$H_{n}$ and $G_{n}$ have reflectional symmetry\n\\[\nH_{n}\\left(  \\bar{z}\\right)  =\\overline{H_{n}\\left(  z\\right)  }\\text{ and\n}G_{n}\\left(  \\bar{z}\\right)  =\\overline{G_{n}\\left(  z\\right)  }.\n\\]\n(iii) Both $H_{n}\\ $and $G_{n}$ are univalent mappings of the open unit disk\nonto a bounded convex region in the right half plane. Moreover, both series\nconverge absolutely on the closed unit disk.\\newline(iv) Define $K_{n}%\n=H_{n}\\left(  1\\right)  $. Both $H_{n}\\ $and $G_{n}$ are positive for\n$x\\in\\left[  -1,1\\right]  ,$ with the interval $\\left[  H_{n}\\left(\n-1\\right)  ,H_{n}\\left(  1\\right)  \\right]  $ strictly contained in $\\left[\nG_{n}\\left(  -1\\right)  ,G_{n}\\left(  1\\right)  \\right]  ,$ and the values at\n$1$ are as follows:\n\\begin{align}\nH_{n}\\left(  1\\right)   &  =K_{n}=\\sqrt{\\pi}\\frac{\\Gamma\\left(  1+\\frac{1}%\n{2n}\\right)  }{\\Gamma\\left(  \\frac{1}{2}+\\frac{1}{2n}\\right)  },\\label{Kn}\\\\\nG_{n}\\left(  1\\right)   &  =\\left(  n-1\\right)  \\tan\\left(  \\frac{\\pi}%\n{2n}\\right)  K_{n}=\\sqrt{\\pi}\\frac{\\Gamma\\left(  \\frac{3}{2}-\\frac{1}%\n{2n}\\right)  }{\\Gamma\\left(  1-\\frac{1}{2n}\\right)  }.\\nonumber\n\\end{align}\nMoreover,\n\\[\n5\/6<G_{n}\\left(  -1\\right)  <H_{n}\\left(  -1\\right)  <1<H_{n}\\left(  1\\right)\n<G_{n}\\left(  1\\right)  <2.\n\\]\n\n\\end{proposition}\n\n\\begin{proof}\nWe first note the form of the Taylor coefficients $c_{m}$ of $H_{n},$ and\n$d_{m}$ of $G_{n},$ where $n\\geq2,$ and $m\\geq0.$ By the definition of\nhypergeometric series, and using the Pochhammer rising factorial symbol, we\nobtain\n\\[\nc_{m}=\\frac{\\left(  \\frac{1}{2}\\right)  _{m}\\left(  \\frac{1}{2n}\\right)  _{m}%\n}{m!\\left(  1+\\frac{1}{2n}\\right)  _{m}}=A_{m}\\frac{\\left(  \\frac{1}%\n{2n}\\right)  \\left(  \\frac{1}{2n}+1\\right)  \\left(  \\frac{1}{2n}+2\\right)\n\\ldots\\left(  \\frac{1}{2n}+m-1\\right)  }{\\left(  \\frac{1}{2n}+1\\right)\n\\left(  \\frac{1}{2n}+2\\right)  \\ldots\\left(  \\frac{1}{2n}+m-1\\right)  \\left(\n\\frac{1}{2n}+m\\right)  }%\n\\]\nwhere we define $A_{m}=\\left(  \\frac{1}{2}\\right)  _{m}\/m!$ . All but two\nfactors cancel, leaving\n\\[\nc_{m}=A_{m}\\frac{\\left(  \\frac{1}{2n}\\right)  }{\\left(  \\frac{1}{2n}+m\\right)\n}=A_{m}\\frac{1}{2mn+1}.\n\\]\nA more familiar formula for $A_{m}$ in terms of binomial coefficients is\nobtained from\n\\[\nA_{m}=\\frac{1\\cdot3\\cdot5\\ldots\\left(  2m-1\\right)  }{2^{m}m!}=\\frac{\\left(\n2m-1\\right)  !}{\\left(  m-1\\right)  !2^{m-1}2^{m}m!}=\\left(\n\\begin{array}\n[c]{c}%\n2m-1\\\\\nm-1\n\\end{array}\n\\right)  \/2^{2m-1}.\n\\]\nThe mth Taylor coefficient of $G_{n}$ is\n\\[\nd_{m}=\\frac{\\left(  \\frac{1}{2}\\right)  _{m}\\left(  \\frac{1}{2}-\\frac{1}%\n{2n}\\right)  _{m}}{m!\\left(  \\frac{3}{2}-\\frac{1}{2n}\\right)  _{m}}=A_{m}%\n\\frac{\\left(  \\frac{1}{2}-\\frac{1}{2n}\\right)  _{m}}{\\left(  \\frac{3}{2}%\n-\\frac{1}{2n}\\right)  _{m}}.\n\\]\nAgain, upon expanding the Pochhammer symbols and simplifying we obtain\n\\[\nd_{m}=A_{m}\\frac{\\left(  \\frac{1}{2}-\\frac{1}{2n}\\right)  }{\\left(\n\\frac{2m+1}{2}-\\frac{1}{2n}\\right)  }=A_{m}\\frac{n-1}{n\\left(  2m+1\\right)\n-1},\n\\]\nestablishing (i). Clearly these are positive term series$,$ so the stated\nsymmetries involving conjugation in (ii) hold. We now prove (iv). Because the\ncoefficients $c_{m}$ and $d_{m}$ are positive for all $m=0,1,2,...,$ the real\nline maps to the real line under both $H_{n}$ and $G_{n}.$ Moreover, for $x>0$\nthe function values are positive, and increasing with $x$. Clearly\n$H_{n}\\left(  1\\right)  >1>H_{n}\\left(  -1\\right)  ,$ because the $c_{m}$ are\npositive with $c_{0}=1,$ and for $x<0,$ we obtain an alternating series. For\n$G_{n},$ we see that $d_{0}-d_{1}<G_{n}\\left(  -1\\right)  <d_{0}-d_{1}+d_{2}.$\nComputing the lower bound, we obtain $d_{0}=1,$ and $d_{1}=A_{1}\\frac{1}%\n{2n+1}=\\frac{1}{4n+2},$ so\n\\[\nG_{n}\\left(  -1\\right)  >d_{0}-d_{1}=1-\\frac{1}{2}\\frac{n-1}{3n-1}>1-\\frac\n{1}{6}=5\/6.\n\\]\nTo see that $G_{n}\\left(  -1\\right)  <H_{n}\\left(  -1\\right)  ,$ we again use\nthe alternating series test and show that the lower bound $c_{0}-c_{1}$ for\n$H_{n}\\left(  -1\\right)  $ is larger than the upper bound $d_{0}-d_{1}+d_{2}$\nfor $G_{n}\\left(  -1\\right)  .$ We show that the equivalent inequality\n$c_{1}+d_{2}\\leq d_{1}$ holds$.$ Note that $A_{1}=\\frac{1}{2}$ and\n$A_{2}=\\frac{3}{8},$ so the inequality becomes\n\\[\n\\frac{1}{2}\\frac{1}{2n+1}+\\frac{3}{8}\\frac{n-1}{5n-1}\\leq\\frac{1}{2}\\frac\n{n-1}{3n-1}%\n\\]\nwhich after some algebra is equivalent to $\\left(  n-3\\right)  \\left(\n22n^{2}-7n+1\\right)  \\geq0.$ The latter is true for $n\\geq3,$ with equality at\n$n=3.$ Thus the inequalities $G_{n}\\left(  -1\\right)  <H_{n}\\left(  -1\\right)\n<1$ hold for $n\\geq3$. Finally, coefficient by coefficient, the following\nequivalent inequalities are equivalent to $c_{m}<d_{m}$ for $m=1,2,3,...$\n\\begin{align*}\nA_{m}\\frac{1}{2mn+1}  &  <A_{m}\\frac{n-1}{n\\left(  2m+1\\right)  -1}\\\\\n0  &  <2mn\\left(  n-2\\right)  ,\n\\end{align*}\nand this last inequality is clearly true for all $m=1,2,...$ when $n>2$. Thus\n$H_{n}\\left(  1\\right)  <G_{n}\\left(  1\\right)  .$ Finally we use Theorem 18\nin \\S 32$\\ $of \\cite{Rainville} to compute\n\\[\nK_{n}=H_{n}\\left(  1\\right)  =\\sqrt{\\pi}\\frac{\\Gamma\\left(  1+\\frac{1}%\n{2n}\\right)  }{\\Gamma\\left(  \\frac{1}{2}+\\frac{1}{2n}\\right)  }\\text{ and\n}G_{n}\\left(  1\\right)  =\\sqrt{\\pi}\\frac{\\Gamma\\left(  \\frac{3}{2}-\\frac\n{1}{2n}\\right)  }{\\Gamma\\left(  1-\\frac{1}{2n}\\right)  }%\n\\]\nNote that\n\\[\nG_{n}\\left(  1\\right)  \/H_{n}\\left(  1\\right)  =\\frac{\\Gamma\\left(  \\frac\n{3}{2}-\\frac{1}{2n}\\right)  }{\\Gamma\\left(  1-\\frac{1}{2n}\\right)  }%\n\\frac{\\Gamma\\left(  \\frac{1}{2}+\\frac{1}{2n}\\right)  }{\\Gamma\\left(\n1+\\frac{1}{2n}\\right)  }=\\frac{\\Gamma\\left(  \\frac{1}{2}+\\frac{1}{2n}\\right)\n\\Gamma\\left(  \\frac{3}{2}-\\frac{1}{2n}\\right)  }{\\Gamma\\left(  1+\\frac{1}%\n{2n}\\right)  \\Gamma\\left(  1-\\frac{1}{2n}\\right)  }.\n\\]\nFrom the well known identities for the gamma function, $\\Gamma\\left(\nz+1\\right)  =z\\Gamma\\left(  z\\right)  ,$ and $\\Gamma\\left(  z\\right)\n\\Gamma\\left(  1-z\\right)  =\\pi\/\\sin\\left(  \\pi z\\right)  ,$ we obtain\n\\[\n\\Gamma\\left(  1+z\\right)  \\Gamma\\left(  1-z\\right)  =\\left(  \\pi z\\right)\n\/\\sin\\left(  \\pi z\\right)  .\n\\]\nTherefore\n\\[\nG_{n}\\left(  1\\right)  \/H_{n}\\left(  1\\right)  =\\frac{\\pi\\left(  \\frac{1}%\n{2}-\\frac{1}{2n}\\right)  }{\\sin\\left(  \\pi\\left(  \\frac{1}{2}-\\frac{1}%\n{2n}\\right)  \\right)  }\\frac{\\sin\\left(  \\pi\\frac{1}{2n}\\right)  }{\\pi\\frac\n{1}{2n}}=\\left(  n-1\\right)  \\frac{\\sin\\left(  \\pi\/2n\\right)  }{\\cos\\left(\n\\pi\/2n\\right)  }=\\left(  n-1\\right)  \\tan\\left(  \\pi\/2n\\right)  .\n\\]\n\n\nWe finish by proving (iii). The radius of convergence of any Gauss\nhypergeometric series is 1. However the convergence on the closed disk is\nguaranteed for $_{2}F_{1}\\left(  a,b,c,z\\right)  $ whenever $\\operatorname{Re}%\n\\left(  c-a-b\\right)  >0$ (Section 29 of \\cite{Rainville}); here\n$\\operatorname{Re}\\left(  c-a-b\\right)  =1\/2\\,\\,$for both $G_{n}$ and $H_{n}$.\nBecause both $G_{n}$ and $H_{n}$ satisfy (ii) of the Corollary in\n\\cite{MerkesScott}, $_{2}F_{1}\\left(  a,b,c,z\\right)  $ maps the open disk\nunivalently onto a convex region. Because of positive Taylor coefficients, we\nhave $G_{n}\\left(  -1\\right)  <\\operatorname{Re}G_{n}\\left(  z\\right)\n<G_{n}\\left(  1\\right)  $ and $H_{n}\\left(  -1\\right)  <\\operatorname{Re}%\nH_{n}\\left(  z\\right)  <H_{n}\\left(  1\\right)  .$ Thus the images\n$H_{n}\\left(  U\\right)  $ and $G_{n}\\left(  U\\right)  $ are convex sets within\nthe lines $\\operatorname{Re}z=5\/6$ and $\\operatorname{Re}z=2.$\n\\end{proof}\n\nExamples of the hypergeometric functions $H_{n}$ and $G_{n}$ are calculated in\n\\cite{AbdullahMcDougall}, through applying the shear construction of\n\\cite{ClunieSheilSmall} to a particular conformal mapping onto a regular\n12-gon. This shear was in fact the origin of the first rosette mapping. The\nfunctions $H_{n}$ and $G_{n}$ appear in antiderivatives associated with\nseveral related\\ radical expressions in \\cite{AbdullahMcDougall}, and are\nrestated here.\n\n\\begin{proposition}\n\\label{hypergeometric} Let $n\\geq2,\\,$and let $z\\in U.$ Then\n\\begin{align}\n\\int_{0}^{z}\\frac{1}{\\sqrt{1-\\zeta^{2n}}}d\\zeta &  =zH_{n}\\left(\nz^{2n}\\right) \\label{hypint1}\\\\\n\\int_{0}^{z}\\frac{\\zeta^{n-2}}{\\sqrt{1-\\zeta^{2n}}}d\\zeta &  =\\frac{z^{n-1}%\n}{n-1}G_{n}\\left(  z^{2n}\\right)  . \\label{hypint2}%\n\\end{align}\n\n\\end{proposition}\n\n\\begin{proof}\nThe integrations are carried out using the Gauss hypergeometric formula, as\nindicated in \\cite{AbdullahMcDougall}. The integrals are restated here in\nterms of the notation introduced in Definition \\ref{hyperDef}.\n\\end{proof}\n\n\\section{Rosette Harmonic Mappings}\n\nThe analytic antiderivatives of Proposition \\ref{hypergeometric} are defined\nbelow as $h_{n}\\left(  z\\right)  $ and $g_{n}\\left(  z\\right)  .$ These become\n(rotations aside) the analytic and co-analytic parts of the rosette mappings\nin Definition \\ref{defnharmonic}. We note the similarity with the analytic and\nco-analytic parts of the hypocycloid, modified here with hypergeometric factors.\n\n\\begin{definition}\n\\label{defnhg}For $n\\geq2$ and $n\\in%\n\\mathbb{N}\n$, define the functions\n\\[\nh_{n}\\left(  z\\right)  =z\\,H_{n}\\left(  z^{2n}\\right)  ,\\text{ \\ }g_{n}\\left(\nz\\right)  =\\frac{z^{n-1}}{n-1}G_{n}\\left(  z^{2n}\\right)  ,\\text{ }z\\in\\bar{U}%\n\\]\nwhere $H_{n}$ and $G_{n}$ are the hypergeometric $_{2}F_{1}$ functions of\nDefinition \\ref{hyperDef}.\n\\end{definition}\n\n\\begin{remark}\nIn the current context, we consider only $n\\geq3$ for rosette harmonic\nmappings, but we note that when $n=2,$ $h_{n}\\left(  z\\right)  =g_{n}\\left(\nz\\right)  $ becomes the standard Schwarz-Christoffel mapping of the unit disk\nonto a square.\n\\end{remark}\n\nThe function $h_{n}$ can easily be shown to be starlike and therefore\nunivalent on the unit disk. However $g_{n}$ is not univalent for $n\\geq3$. The\nimage $h_{6}\\left(  \\bar{U}\\right)  $ is shown on the left of Figure\n\\ref{canon6}, and a portion of the graph of $\\overline{g_{6}}\\left(  \\bar\n{U}\\right)  $ is shown on the right, with $z$ restricted to $\\left\\{  z\\in\n\\bar{U}:\\arg z\\in\\left(  0,\\left(  5\/2\\right)  \\pi\/n\\right)  \\right\\}  $.%\n\n\\begin{figure}[h]%\n\\centering\n\\includegraphics[\nheight=2.3817in,\nwidth=4.5074in\n]%\n{canon6.eps}%\n\\caption{Image of $\\bar{U}$ under $h_{6}$ (left)$,$ and a sector of $\\bar{U}$\nunder $\\overline{g_{6}}$ (right)$.$ Note that the image of $\\overline{g_{6}}$\nis relatively enlarged$.$ }%\n\\label{canon6}%\n\\end{figure}\n\n\n\\begin{proposition}\n\\label{analco}Let $n\\geq3$ and let $\\zeta$ be a primitive $2nth$ root of\nunity$.$ Then $h_{n}\\left(  z\\right)  $ and $g_{n}\\left(  z\\right)  $ are\ncontinuous on $\\bar{U}$ and analytic on $\\overline{U}\\backslash\\left\\{\n\\zeta^{j}:j=1,2,...,2n\\right\\}  .$ Additionally the following hold.\n\\newline(i) Reflectional symmetries $h_{n}\\left(  \\bar{z}\\right)\n=\\overline{h_{n}\\left(  z\\right)  }$ and $g_{n}\\left(  \\bar{z}\\right)\n=\\overline{g_{n}\\left(  z\\right)  }$ are present.\\newline(ii) The functions\n$h_{n}\\left(  e^{it}\\right)  $ and $g_{n}\\left(  e^{it}\\right)  $ have\nmagnitudes $\\left\\vert H_{n}\\left(  e^{i2nt}\\right)  \\right\\vert $ and\n\\newline$\\frac{1}{n-1}\\left\\vert G_{n}\\left(  e^{i2nt}\\right)  \\right\\vert ,$\nand these magnitudes are each $\\left(  \\pi\/n\\right)  $-periodic as functions\nof $t.$ Moreover, convergence at $H\\left(  1\\right)  $ and $G\\left(  1\\right)\n$ is not uniform$.$\\newline(iii) Both $h_{n}$ and $g_{n}$ exhibit $2n$-fold\nrotational symmetry: for \\ $j\\in%\n\\mathbb{Z}\n,$\n\\begin{equation}\nh_{n}\\left(  e^{ij\\pi\/n}z\\right)  =e^{ij\\pi\/n}h_{n}\\left(  z\\right)  \\text{\nand }g_{n}\\left(  e^{ij\\pi\/n}z\\right)  =\\left(  -1\\right)  ^{j}e^{-ij\\pi\n\/n}g_{n}\\left(  z\\right)  . \\label{ga}%\n\\end{equation}\n(iv) The derivatives of $h_{n}\\left(  z\\right)  $ and $g_{n}\\left(  z\\right)\n$ are given by\n\\begin{equation}\nh_{n}^{\\prime}\\left(  z\\right)  =\\frac{1}{\\sqrt{1-z^{2n}}}\\text{ and }%\ng_{n}^{\\prime}\\left(  z\\right)  =\\frac{z^{n-2}}{\\sqrt{1-z^{2n}}}.\n\\label{derivatives}%\n\\end{equation}\n\n\\end{proposition}\n\n\\begin{proof}\nThe stated symmetries in (i) follow from the reflections $H_{n}\\left(  \\bar\n{z}^{2n}\\right)  =\\overline{H_{n}\\left(  z^{2n}\\right)  }$ and $G_{n}\\left(\n\\bar{z}^{2n}\\right)  =\\overline{G_{n}\\left(  z^{2n}\\right)  }$ of Proposition\n\\ref{hyperProps} (ii). For (ii) we easily compute $\\left\\vert h_{n}\\left(\ne^{it}\\right)  \\right\\vert =\\left\\vert e^{it}H_{n}\\left(  e^{i2nt}\\right)\n\\right\\vert $ and $\\left\\vert g_{n}\\left(  e^{it}\\right)  \\right\\vert\n=\\left\\vert e^{i\\left(  n-1\\right)  t}G_{n}\\left(  e^{i2nt}\\right)\n\\right\\vert \/\\left(  n-1\\right)  $ to see that $h_{n}\\left(  e^{it}\\right)  $\nand $g_{n}\\left(  e^{it}\\right)  $ have the stated magnitudes. These\nmagnitudes have period $\\pi\/n$ as functions of $t,$ since each of\n$H_{n}\\left(  e^{it}\\right)  $ and $G_{n}\\left(  e^{it}\\right)  $ have period\n$2\\pi$ as functions of $t.$ For (iii), first note that $\\left(  e^{ij\\pi\n\/n}z\\right)  ^{2n}=z^{2n}$ and $\\left(  e^{ij\\pi\/n}\\right)  ^{n-1}=e^{ij\\pi\n}e^{-ij\\pi\/n}=\\left(  -1\\right)  ^{j}e^{-ij\\pi\/n}$ for any integer $j.$ Thus\n\\[\nh_{n}\\left(  e^{ij\\pi\/n}z\\right)  =e^{ij\\pi\/n}zH_{n}\\left(  \\left(\ne^{ij\\pi\/n}z\\right)  ^{2n}\\right)  =e^{ij\\pi\/n}h_{n}\\left(  z\\right)  ,\\text{\nand}%\n\\]%\n\\[\ng_{n}\\left(  e^{ij\\pi\/n}z\\right)  =\\frac{1}{n-1}\\left(  e^{ij\\pi\/n}z\\right)\n^{n-1}G_{n}\\left(  \\left(  e^{ij\\pi\/n}z\\right)  ^{2n}\\right)  =e^{ij\\left(\n\\pi-\\pi\/n\\right)  }g_{n}\\left(  z\\right)  .\n\\]\n\\newline For (iv), the derivatives are immediate from (\\ref{hypint1}) and\n(\\ref{hypint2}). The stated region of analyticity of the functions $h_{n}$ and\n$g_{n}$ is due to the fact that their derivatives can be analytically\ncontinued across the boundary, except at the 2nth roots of unity where\nderivatives do not exist. Moreover the convergence of $H\\left(  1\\right)  $\nand $G\\left(  1\\right)  $ is not uniform, since this would imply analyticity\nat $\\zeta$.\n\\end{proof}\n\n\\begin{corollary}\n\\label{rayshg}Let $z=re^{ij\\pi\/n},$ and $j\\in%\n\\mathbb{Z}\n.$ Then\n\\[\nh_{n}\\left(  re^{ij\\pi\/n}\\right)  =e^{ij\\pi\/n}h_{n}\\left(  r\\right)  \\text{\nand }\\overline{g_{n}\\left(  re^{ij\\pi\/n}\\right)  }=\\left(  -1\\right)\n^{j}e^{ij\\pi\/n}g_{n}\\left(  r\\right)\n\\]\nand on these rays, $\\arg\\left(  h_{n}\\left(  z\\right)  \\right)  =\\arg z,$ and\n$\\arg\\left(  \\overline{g_{n}\\left(  z\\right)  }\\right)  =\\arg\\left(  \\frac\n{1}{n-1}\\bar{z}^{n-1}\\right)  .$\n\\end{corollary}\n\n\\begin{proof}\nBy Proposition \\ref{hyperProps} (iv), $H_{n}$ and $G_{n}$ map reals to\npositive reals. Thus $h_{n}$ and $g_{n}$ map positive reals to positive reals.\nNow let $z=re^{ij\\pi\/n}$ for $r>0$ in formula (\\ref{ga}).\n\\end{proof}\n\nWe now define the rosette harmonic mappings.\n\n\\begin{definition}\n\\label{defnharmonic}Let $n\\in%\n\\mathbb{N}\n$ with $n\\geq3,$ and let $h_{n}$ and $g_{n}$ be defined as in Definition\n\\ref{defnhg}. For each $\\beta\\in%\n\\mathbb{R}\n,$ define the \\textbf{rosette harmonic mapping}%\n\\[\nf_{\\beta}\\left(  z\\right)  =e^{i\\beta\/2}h_{n}\\left(  z\\right)  +e^{-i\\beta\n\/2}\\overline{g_{n}\\left(  z\\right)  },\\text{ }z\\in\\bar{U}.\n\\]\nThen $f_{\\beta}\\left(  z\\right)  $ is harmonic on $U,$ and continuous on\n$\\bar{U}.$ Denote the dilatation of $f_{\\beta}\\left(  z\\right)  $ by\n$\\omega\\left(  z\\right)  $ on $\\bar{U}\\backslash\\left\\{  \\zeta^{j}%\n:j=1,2,...,2n\\right\\}  ,$ where $\\zeta$ is a primitive 2nth root of unity.\n\\end{definition}\n\n\\begin{remark}\nFor simplicity of the notation, we do not notate the value of $n,$ which is\napparent from context, and fixed as a constant in our discussions$.$ The\ndilatation is independent of $\\beta$ and we simplify our notation to $\\omega$\ninstead of using $\\omega_{f_{\\beta}}.$\n\\end{remark}\n\nThe derivatives of the analytic and co-analytic parts (for $\\beta=0$) in\nequation (\\ref{derivatives}) are the same as the corresponding derivatives for\nthe hypocycloid, except for the radical factor. This factor affects the\nargument of the summands in the canonical decomposition, but Corollary\n\\ref{rayshg}\\ shows that along rays $\\left\\{  re^{ij\\pi\/n}:r>0\\right\\}  $\nwhere $j\\in%\n\\mathbb{Z}\n,$ that $h_{n}$ and $\\overline{g_{n}}$ are collinear. Moreover Corollary\n\\ref{rayshg} shows that on these same rays, that $h_{n}$ and $\\bar{g}_{n}$\nhave the same arguments as their analytic and anti-analytic counterparts in\nthe hypocycloid mapping. Thus $f_{0}\\left(  z\\right)  $ and $f_{hyp}\\left(\nz\\right)  $ are collinear along these radial lines (as indicated in Figure\n\\ref{rosette0}). Figure \\ref{five} shows images $f_{\\beta}\\left(  U\\right)  $\nas $\\beta$ varies$,$ and suggests that different harmonic mappings $f_{\\beta}$\nare obtained for the four different values of $\\beta$ there. This contrasts\nwith the hypocycloid mapping, where rotating the analytic and anti-analytic\nparts $z$ and $\\frac{1}{n-1}\\bar{z}^{n-1}$ relative to one another does not\nyield a graph that is essentially different.\n\nOne may ask what other maps could be obtained through adding different\nrotations of $h_{n}$ and $\\overline{g_{n}}$. The next proposition shows that\nif we consider arbitrary rotations of the analytic and co-analytic parts, by\n$\\theta$ and $\\tilde{\\theta}$ say, then the result will in fact be a rotation\nof a rosette harmonic mapping.\n\n\\begin{proposition}\nLet $\\theta$ and $\\tilde{\\theta}$ be arbitrary real angles. Then $e^{i\\theta\n}h_{n}\\left(  z\\right)  +\\overline{e^{i\\tilde{\\theta}}g_{n}\\left(  z\\right)\n}$ is a rotation $e^{i\\gamma}f_{\\beta}$ for some real angles $\\gamma$ and\n$\\beta.$\n\\end{proposition}\n\n\\begin{proof}\nNote that the harmonic function $e^{i\\theta}h_{n}\\left(  z\\right)\n+\\overline{e^{i\\tilde{\\theta}}g_{n}\\left(  z\\right)  }$ can be rewritten%\n\\begin{align*}\n&  e^{-i\\tilde{\\theta}}\\left(  e^{i\\theta}e^{i\\tilde{\\theta}}h_{n}\\left(\nz\\right)  +\\overline{g_{n}\\left(  z\\right)  }\\right) \\\\\n&  =e^{-i\\tilde{\\theta}}e^{i\\left(  \\theta+\\tilde{\\theta}\\right)  \/2}\\left(\ne^{i\\left(  \\theta+\\tilde{\\theta}\\right)  \/2}h_{n}\\left(  z\\right)\n+e^{-i\\left(  \\theta+\\tilde{\\theta}\\right)  \/2}\\overline{g_{n}\\left(\nz\\right)  }\\right) \\\\\n&  =e^{i\\left(  \\theta-\\tilde{\\theta}\\right)  \/2}f_{\\left(  \\theta\n+\\tilde{\\theta}\\right)  \/2}\\left(  z\\right)\n\\end{align*}\nThus $e^{i\\theta}h_{n}\\left(  z\\right)  +\\overline{e^{i\\tilde{\\theta}}%\ng_{n}\\left(  z\\right)  }$ can be obtained by a rotation by $\\gamma=\\left(\n\\theta-\\tilde{\\theta}\\right)  \/2$ of the map $f_{\\beta}$ where $\\beta=\\left(\n\\theta+\\tilde{\\theta}\\right)  \/2.$\n\\end{proof}\n\nThus the family of mappings $\\left\\{  f_{\\beta}:\\beta\\in%\n\\mathbb{R}\n\\right\\}  $ represents all of the different mappings, up to rotation, that\narise from arbitrary rotations of $h_{n}$ and $g_{n}$. We show in Proposition\n\\ref{parametrize} that $f_{\\beta+\\pi}$ is essentially the same mapping as\n$f_{\\beta}.$\n\n\\begin{proposition}\n\\label{parametrize}(i) For any \\ $\\beta\\in%\n\\mathbb{R}\n,$ the functions $f_{\\beta}$ and $f_{\\beta+\\pi}$ are related by\n\\begin{equation}\nf_{\\beta}\\left(  z\\right)  =e^{-i\\left(  \\frac{\\pi}{2}+\\frac{\\pi}{n}\\right)\n}f_{\\beta+\\pi}\\left(  e^{i\\frac{\\pi}{n}}z\\right)  . \\label{rotbet}%\n\\end{equation}\nThus the image $f_{\\beta+\\pi}\\left(  U\\right)  $ is equal to a rotation of the\nimage $f_{\\beta}\\left(  U\\right)  $.\\newline(ii) If $\\beta<0,$ then the image\nof any point under $f_{\\beta}$ is a reflection in the real axis of the same\npoint under $f_{-\\beta},$ where $-\\beta>0.$ \\ Specifically, $f_{\\beta}\\left(\n\\bar{z}\\right)  =\\overline{f_{-\\beta}\\left(  z\\right)  }.$\n\\end{proposition}\n\n\\begin{proof}\n(i) We compute, using $j=1$ in equation (\\ref{ga}),%\n\\[\nf_{\\pi+\\beta}\\left(  e^{i\\frac{\\pi}{n}}z\\right)  =e^{i\\left(  \\pi\n\/2+\\beta\/2\\right)  }e^{i\\pi\/n}h_{n}\\left(  z\\right)  +e^{-i\\left(  \\pi\n\/2+\\beta\/2\\right)  }\\overline{\\left(  -1\\right)  e^{-i\\pi\/n}g_{n}\\left(\nz\\right)  }.\n\\]\nMultiplying by $e^{-i\\left(  \\frac{\\pi}{2}+\\frac{\\pi}{n}\\right)  },$ and\nnoting that $-e^{-i\\frac{\\pi}{2}}=e^{i\\frac{\\pi}{2}},$ $\\ $%\n\\[\ne^{-i\\left(  \\frac{\\pi}{2}+\\frac{\\pi}{n}\\right)  }f_{\\pi+\\beta}\\left(\ne^{i\\frac{\\pi}{n}}z\\right)  =e^{i\\beta\/2}h_{n}\\left(  z\\right)  +e^{-i\\beta\n\/2}g_{n}\\left(  z\\right)  =f_{\\beta}\\left(  z\\right)  .\n\\]\n(ii) Once again, by computation:\n\\[\nf_{\\beta}\\left(  \\bar{z}\\right)  =e^{i\\frac{\\beta}{2}}h_{n}\\left(  \\bar\n{z}\\right)  +e^{-i\\frac{\\beta}{2}}\\overline{g_{n}\\left(  \\bar{z}\\right)\n}=e^{i\\frac{\\beta}{2}}\\overline{h_{n}\\left(  z\\right)  }+e^{-i\\frac{\\beta}{2}%\n}g_{n}\\left(  z\\right)\n\\]\nwhile\n\\[\nf_{-\\beta}\\left(  z\\right)  =e^{-i\\frac{\\beta}{2}}h_{n}\\left(  z\\right)\n+e^{i\\frac{\\beta}{2}}\\overline{g_{n}\\left(  z\\right)  }.\n\\]\nThe last two expressions are conjugates of one another. $\\blacksquare$\n\\end{proof}\n\n\\begin{corollary}\n\\label{transit}Let $\\tilde{\\beta}\\in%\n\\mathbb{R}\n,$ and let $\\tilde{\\beta}=\\beta+l\\pi$ for some $l\\in%\n\\mathbb{Z}\n.$ Then%\n\\begin{equation}\nf_{\\beta+l\\pi}\\left(  z\\right)  =e^{il\\left(  \\pi\/n+\\pi\/2\\right)  }f_{\\beta\n}\\left(  e^{-il\\pi\/n}z\\right)  . \\label{trans}%\n\\end{equation}\n\n\\end{corollary}\n\n\\begin{proof}\nSolving equation (\\ref{rotbet}) we have $f_{\\beta+\\pi}\\left(  z\\right)\n=e^{i\\left(  \\pi\/2+\\pi\/n\\right)  }f_{\\beta}\\left(  e^{-i\\pi\/n}z\\right)  .$\nEquation (\\ref{trans}) is obtained by repeated application of this equation.\n\\end{proof}\n\nProposition \\ref{parametrize} (i) shows that up to rotations (pre and post\ncomposed), all rosette mappings are represented in the set $\\left\\{  f_{\\beta\n}:\\beta\\in(-\\pi\/2,\\pi\/2]\\right\\}  .$ We will see in Section 4 that these\nrosette mappings are all distinct from one another in that no rosette mapping\nin the set can be obtained by rotations from another. Proposition\n\\ref{parametrize} (ii) allows us to consider just $\\left\\{  f_{\\beta}:\\beta\n\\in\\left[  0,\\pi\/2\\right]  \\right\\}  $ to obtain all rosette mappings, up to\nrotations and reflection$.$\n\\begin{figure}[h]%\n\\centering\n\\includegraphics[\nheight=4.3811in,\nwidth=4.3811in\n]%\n{five.eps}%\n\\caption{Images of $U$ under $f_{\\beta}$ with $n=5.$ The darker \"radial curve\"\nindicates the image of $\\left[  0,1\\right]  .$ The graphs $f_{\\pi\/3}$ and\n$f_{-\\pi\/3}$ illustrate Proposition \\ref{parametrize} (ii). The images\n$f_{-\\pi\/3}\\left(  U\\right)  $ and $f_{\\pi\/3}\\left(  U\\right)  $ have cyclic\nsymmetry while $f_{0}\\left(  U\\right)  $ and $f_{\\pi\/2}\\left(  U\\right)  $\nexhibit dihedral symmetry, illustrating Theorem \\ref{symmetry}. For $f_{\\pi\n\/2}$ the line of symmetry $\\arg z=\\pi\/4-\\pi\/\\left(  2n\\right)  $ is\nindicated.}%\n\\label{five}%\n\\end{figure}\n\n\nFigure \\ref{five} illustrates that $f_{-\\pi\/3}\\left(  U\\right)  $ and\n$f_{\\pi\/3}\\left(  U\\right)  $ are reflections of one another. The example\ngraphs in Figure \\ref{five} also demonstrate the rotational and reflectional\nsymmetries apparent within any particular graph $f_{\\beta}\\left(  U\\right)  ,$\nas stated in the following theorem$.$\n\n\\begin{theorem}\n\\label{symmetry}Let $n\\in%\n\\mathbb{N}\n$ and $n\\geq3.$ \\newline(i) The harmonic functions $f_{\\beta}\\left(  z\\right)\n$, $\\beta\\in%\n\\mathbb{R}\n$ have dilatation $\\omega\\left(  z\\right)  =z^{n-2}$ for $z\\in U.$\\newline(ii)\nThe rosette mapping $f_{\\beta}\\left(  z\\right)  $, $\\beta\\in%\n\\mathbb{R}\n$ has n-fold rotational symmetry, that is\n\\begin{equation}\nf_{\\beta}\\left(  e^{i2k\\pi\/n}z\\right)  =e^{i2k\\pi\/n}f_{\\beta}\\left(  z\\right)\n,\\text{ where }z\\in\\bar{U},\\text{ }k\\in%\n\\mathbb{Z}\n. \\label{RS}%\n\\end{equation}\n\\newline(iii) If $\\beta$ is an integer multiple of $\\pi\/2,$ then the image\n$f_{\\beta}\\left(  U\\right)  $ has reflectional symmetry$.$ For $\\beta=0$ and\n$\\beta=\\pi\/2$ the reflections are%\n\\begin{equation}\nf_{0}\\left(  \\overline{z}\\right)  =\\overline{f_{0}\\left(  z\\right)  }\\text{\nand }e^{i\\eta}f_{\\pi\/2}\\left(  e^{i\\gamma}\\overline{z}\\right)  =\\overline\n{e^{i\\eta}f_{\\pi\/2}\\left(  e^{i\\gamma}z\\right)  }, \\label{reflect}%\n\\end{equation}\nwhere $\\eta=\\pi\/\\left(  2n\\right)  -\\pi\/4$ and $\\gamma=-\\pi\/\\left(  2n\\right)\n.$ Thus if $z$ and $z^{\\prime}$ are reflections in $\\arg z=\\gamma,$ then\n$f_{\\pi\/2}\\left(  z\\right)  $ and $f_{\\pi\/2}\\left(  z^{\\prime}\\right)  $ are\nreflections in $\\arg z=\\eta$. \\newline\n\\end{theorem}\n\n\\begin{proof}\n(i) We compute $\\omega\\left(  z\\right)  =$ $g_{n}^{\\prime}\\left(  z\\right)\n\/h_{n}^{\\prime}\\left(  z\\right)  $ from the derivative expressions in\n(\\ref{derivatives}), noting that the constant $e^{i\\beta\/2},$ and the\nradicals, cancel leaving $z^{n-2}.$ \\newline(ii) From equation (\\ref{ga}) with\n$j=2k$ we have $h_{n}\\left(  e^{i2k\\pi\/n}z\\right)  =e^{i2k\\pi\/n}h_{n}\\left(\nz\\right)  $ and $\\overline{g_{n}\\left(  e^{i2k\\pi\/n}z\\right)  }=e^{i2k\\pi\n\/n}\\overline{g_{n}\\left(  z\\right)  }$. Thus%\n\\[\nf_{\\beta}\\left(  e^{i2k\\pi\/n}z\\right)  =e^{i2k\\pi\/n}\\left(  e^{i\\beta\/2}%\nh_{n}\\left(  z\\right)  +e^{-i\\beta\/2}\\overline{g_{n}\\left(  z\\right)\n}\\right)  =e^{i2k\\pi\/n}f_{\\beta}\\left(  z\\right)  \\text{.}%\n\\]\n\\allowbreak(iii) From Proposition \\ref{analco}, $f_{0}(\\overline{z}%\n)=h_{n}(\\bar{z})+\\overline{g_{n}\\left(  \\bar{z}\\right)  }=\\overline{h_{n}%\n(z)}+g_{n}\\left(  z\\right)  =\\overline{f_{0}\\left(  z\\right)  }.$ For\n$f_{\\pi\/2},$ consider the expressions $f_{\\pi\/2}\\left(  e^{i\\gamma}%\n\\overline{z}\\right)  $ and $f_{\\pi\/2}\\left(  e^{i\\gamma}z\\right)  $:\\newline%\n\\begin{align*}\nf_{\\pi\/2}\\left(  e^{i\\gamma}\\overline{z}\\right)   &  =e^{i\\pi\/4}h_{n}\\left(\ne^{-i\\pi\/\\left(  2n\\right)  }\\bar{z}\\right)  +e^{-i\\pi\/4}\\overline\n{g_{n}\\left(  e^{-i\\pi\/\\left(  2n\\right)  }\\bar{z}\\right)  }\\\\\n&  =e^{i\\pi\/4}\\overline{h_{n}\\left(  e^{i\\pi\/\\left(  2n\\right)  }z\\right)\n}+e^{-i\\pi\/4}g_{n}\\left(  e^{i\\pi\/\\left(  2n\\right)  }z\\right) \\\\\n&  =e^{i\\pi\/4}\\overline{e^{i\\pi\/\\left(  2n\\right)  }zH\\left(  -z^{2n}\\right)\n}+e^{-i\\pi\/4}ie^{-i\\pi\/\\left(  2n\\right)  }\\frac{z^{n-1}G\\left(\n-z^{2n}\\right)  }{n-1}\\\\\n&  =e^{i\\left(  \\pi\/4-\\pi\/\\left(  2n\\right)  \\right)  }\\left(  \\overline\n{zH\\left(  -z^{2n}\\right)  }+\\frac{z^{n-1}G\\left(  -z^{2n}\\right)  }%\n{n-1}\\right)  .\n\\end{align*}\nNote that $\\left(  e^{i\\pi\/\\left(  2n\\right)  }\\right)  ^{n-1}=ie^{-i\\pi\n\/\\left(  2n\\right)  }$ and $\\left(  e^{-i\\pi\/\\left(  2n\\right)  }z\\right)\n^{2n}=-z^{2n},$ as used in the calculation above$.$ We also use $\\left(\ne^{-i\\pi\/\\left(  2n\\right)  }\\right)  ^{n-1}=-ie^{i\\pi\/\\left(  2n\\right)  }%\n\\ $below:\\newline%\n\\begin{align*}\nf_{\\pi\/2}\\left(  e^{i\\gamma}z\\right)   &  =e^{i\\pi\/4}h_{n}\\left(\ne^{-i\\frac{\\pi}{2n}}z\\right)  +e^{-i\\pi\/4}\\overline{g_{n}\\left(\ne^{-i\\frac{\\pi}{2n}}z\\right)  }\\\\\n&  =e^{i\\pi\/4}\\left(  e^{-i\\frac{\\pi}{2n}}z\\right)  H\\left(  e^{-i\\pi}%\nz^{2n}\\right)  +e^{-i\\pi\/4}\\overline{\\left(  -i\\right)  e^{i\\frac{\\pi}{2n}%\n}\\frac{z^{n-1}}{n-1}G\\left(  -z^{2n}\\right)  }\\\\\n&  =e^{i\\left(  \\pi\/4-\\pi\/\\left(  2n\\right)  \\right)  }\\left(  zH\\left(\n-z^{2n}\\right)  +\\overline{\\frac{z^{n-1}}{n-1}G\\left(  -z^{2n}\\right)\n}\\right)  .\n\\end{align*}\nMultiplying each of $f_{\\pi\/2}\\left(  e^{i\\gamma}\\overline{z}\\right)  $ and\n$f_{\\pi\/2}\\left(  e^{i\\gamma}z\\right)  $ by $e^{i\\eta}=e^{i\\left(  \\pi\n\/4-\\pi\/\\left(  2n\\right)  \\right)  },$ we see that $e^{i\\eta}$ $f_{\\pi\n\/2}\\left(  e^{i\\gamma}\\overline{z}\\right)  $ and $e^{i\\eta}f_{\\pi\/2}\\left(\ne^{i\\gamma}z\\right)  $ are conjugates of one another. $\\ $Thus the stated\nequations in (iii) hold.\n\\end{proof}\n\nIn Section 4 we use features of the boundary $\\partial f_{\\beta}\\left(\nU\\right)  $ that allow us to demonstrate the precise symmetry group for each\ngraph $f_{\\beta}\\left(  U\\right)  $, $\\beta\\in%\n\\mathbb{R}\n$ (see Corollary \\ref{noreflect}). Theorem \\ref{symmetry} also shows that for\na given $n\\geq3$ the rosette mappings $f_{\\beta}$ and $n$-cusped hypocycloid\nall have the same dilatation. The equal dilatations result in similarities in\nthe tangents of the rosette and hypocycloid boundary curves, to be discussed\nfurther in Section 4.\n\nWe finish this section by laying out geometric features that are specific to\nrosette mappings for $\\beta$ in the interval $\\beta\\in(-\\pi\/2,\\pi\/2].$ These\nfacts in combination with equation (\\ref{trans}) will allow us to extend our\nconclusions for any $\\beta\\in%\n\\mathbb{R}\n.$\n\n\\begin{lemma}\n\\label{convexconcave} Let $n\\in%\n\\mathbb{N}\n,$ $n\\geq3,$ and recall $K_{n}=\\sqrt{\\pi}\\,\\Gamma\\left(  1+\\frac{1}%\n{2n}\\right)  \/\\Gamma\\left(  \\frac{1}{2}+\\frac{1}{2n}\\right)  .$ \\newline(i)\nFor $\\beta\\in(-\\pi\/2,\\pi\/2],$ we have polar forms for $f_{\\beta}\\left(\n1\\right)  $ and $f_{\\beta}\\left(  e^{i\\pi\/n}\\right)  $ given by magnitudes\n$\\left\\vert f_{\\beta}\\left(  1\\right)  \\right\\vert $ and $\\left\\vert f_{\\beta\n}\\left(  e^{i\\pi\/n}\\right)  \\right\\vert $ which are, respectively,\n\\begin{equation}\nK_{n}\\sqrt{\\sec^{2}\\left(  \\frac{\\pi}{2n}\\right)  +2\\tan\\left(  \\frac{\\pi}%\n{2n}\\right)  \\cos\\beta}\\text{ and }K_{n}\\sqrt{\\sec^{2}\\left(  \\frac{\\pi}%\n{2n}\\right)  -2\\tan\\left(  \\frac{\\pi}{2n}\\right)  \\cos\\beta}, \\label{cnmag}%\n\\end{equation}\nand by the arguments $\\psi=\\arg\\left(  f_{\\beta}\\left(  1\\right)  \\right)  $\nand $\\pi\/n+\\psi^{\\prime}=\\arg\\left(  f_{\\beta}\\left(  e^{i\\pi\/n}\\right)\n\\right)  ,$ where%\n\\begin{equation}\n\\tan\\psi=\\frac{1-\\tan\\left(  \\frac{\\pi}{2n}\\right)  }{1+\\tan\\left(  \\frac{\\pi\n}{2n}\\right)  }\\tan\\left(  \\frac{\\beta}{2}\\right)  \\text{ and }\\tan\n\\psi^{\\prime}=\\frac{1+\\tan\\left(  \\frac{\\pi}{2n}\\right)  }{1-\\tan\\left(\n\\frac{\\pi}{2n}\\right)  }\\tan\\left(  \\frac{\\beta}{2}\\right)  . \\label{cnarg}%\n\\end{equation}\nIf $\\beta=0$ then both $\\psi$ and $\\psi^{\\prime}$ are zero. If $\\beta=\\pi\/2,$\nthese angles reduce to \\linebreak$\\psi=\\pi\/4-\\pi\/\\left(  2n\\right)  $ and\n$\\psi^{\\prime}=\\pi\/4+\\pi\/\\left(  2n\\right)  .$\\newline(ii) If $\\beta\\in\n(0,\\pi\/2],$ then the curves $f_{\\beta}\\left(  r\\right)  $ and $f_{\\beta\n}\\left(  re^{i\\pi\/n}\\right)  $ have strictly increasing magnitude. Moreover,\n$\\arg\\frac{\\partial}{\\partial r}f_{\\beta}\\left(  r\\right)  $ decreases\nstrictly with $r$, and $\\arg\\frac{\\partial}{\\partial r}f_{\\beta}\\left(\nre^{i\\pi\/n}\\right)  ,$ increases strictly with $r$. We also have the following\narguments for the the tangents of these curves at the origin\n\\[\n\\lim_{r\\rightarrow0^{+}}\\arg\\frac{\\partial}{\\partial r}f_{\\beta}\\left(\nr\\right)  =\\beta\/2\\text{ and }\\lim_{r\\rightarrow0^{+}}\\arg\\frac{\\partial\n}{\\partial r}f_{\\beta}\\left(  re^{i\\pi\/n}\\right)  =\\beta\/2+\\pi\/n,\n\\]\nand at the boundary\n\\[\n\\lim_{r\\rightarrow1^{-}}\\arg\\frac{\\partial}{\\partial r}f_{\\beta}\\left(\nr\\right)  =0\\text{ and }\\lim_{r\\rightarrow1^{-}}\\arg\\frac{\\partial}{\\partial\nr}f_{\\beta}\\left(  re^{i\\pi\/n}\\right)  =\\pi\/2+\\pi\/n.\n\\]\n\\newline(iii) For $\\beta=0$ the curves $f_{0}\\left(  r\\right)  $ and\n$f_{0}\\left(  re^{i\\pi\/n}\\right)  $ are straight lines - the images have\nconstant argument of $0$ and $\\pi\/n,$ respectively.\n\\end{lemma}\n\n\\begin{proof}\nTo prove (i), we compute\n\\begin{align*}\nf_{\\beta}\\left(  r\\right)   &  =e^{i\\beta\/2}h_{n}\\left(  r\\right)\n+e^{-i\\beta\/2}g_{n}\\left(  r\\right) \\\\\n&  =\\cos\\left(  \\beta\/2\\right)  \\left(  h_{n}\\left(  r\\right)  +g_{n}\\left(\nr\\right)  \\right)  +i\\sin\\left(  \\beta\/2\\right)  \\left(  h_{n}\\left(\nr\\right)  -g_{n}\\left(  r\\right)  \\right)  ,\n\\end{align*}\nand, upon recalling from Corollary \\ref{rayshg} that $\\overline{g_{n}\\left(\nre^{i\\pi\/n}\\right)  }=e^{-i\\pi\/n}g_{n}\\left(  r\\right)  ,$\n\\begin{align*}\nf_{\\beta}\\left(  re^{i\\pi\/n}\\right)   &  =e^{i\\beta\/2}h_{n}\\left(  re^{i\\pi\n\/n}\\right)  +e^{-i\\beta\/2}\\overline{e^{-i\\pi\/n}g_{n}\\left(  r\\right)  }\\\\\n&  =e^{i\\pi\/n}\\left(  \\cos\\left(  \\beta\/2\\right)  \\left(  h_{n}\\left(\nr\\right)  -g_{n}\\left(  r\\right)  \\right)  +i\\sin\\left(  \\beta\/2\\right)\n\\left(  h_{n}\\left(  r\\right)  +g_{n}\\left(  r\\right)  \\right)  \\right)  .\n\\end{align*}\nWe recall the formulae $h_{n}\\left(  1\\right)  =H_{n}\\left(  1\\right)  =K_{n}$\nand $g_{n}\\left(  1\\right)  =G_{n}\\left(  1\\right)  \/\\left(  n-1\\right)\n=\\tan\\left(  \\pi\/\\left(  2n\\right)  \\right)  K_{n}$ from Proposition\n\\ref{hyperProps}. Using continuity of $f_{\\beta}$ on $\\bar{U}$, we take limits\nas $r\\rightarrow1^{-}$ to get%\n\\begin{align*}\nf_{\\beta}\\left(  1\\right)   &  =K_{n}\\left(  \\cos\\left(  \\beta\/2\\right)\n\\left(  1+\\tan\\left(  \\frac{\\pi}{2n}\\right)  \\right)  +i\\sin\\left(\n\\beta\/2\\right)  \\left(  1-\\tan\\left(  \\frac{\\pi}{2n}\\right)  \\right)  \\right)\n\\text{ and}\\\\\nf_{\\beta}\\left(  e^{i\\pi\/n}\\right)   &  =e^{i\\pi\/n}K_{n}\\left(  \\cos\\left(\n\\beta\/2\\right)  \\left(  1-\\tan\\left(  \\frac{\\pi}{2n}\\right)  \\right)\n+i\\sin\\left(  \\beta\/2\\right)  \\left(  1+\\tan\\left(  \\frac{\\pi}{2n}\\right)\n\\right)  \\right)  .\n\\end{align*}\nMoreover, writing $\\psi=\\arg f_{\\beta}\\left(  1\\right)  ,$ and $\\pi\n\/n+\\psi^{\\prime}=\\arg f_{\\beta}\\left(  e^{i\\pi\/n}\\right)  ,$ we take ratios of\nimaginary to real parts of $f_{\\beta}\\left(  1\\right)  $ and $f_{\\beta}\\left(\ne^{i\\pi\/n}\\right)  $ to easily obtain the stated formulae for $\\tan\\psi$ and\n$\\tan\\psi^{\\prime}.$ We also readily obtain\n\\begin{align*}\n\\left\\vert f_{\\beta}\\left(  1\\right)  \\right\\vert ^{2}  &  =K_{n}^{2}\\left(\n1+2\\tan\\left(  \\frac{\\pi}{2n}\\right)  \\cos\\beta+\\tan^{2}\\left(  \\frac{\\pi}%\n{2n}\\right)  \\right)  \\text{ and }\\\\\n\\left\\vert f_{\\beta}\\left(  e^{i\\pi\/n}\\right)  \\right\\vert ^{2}  &  =K_{n}%\n^{2}\\left(  1-2\\tan\\left(  \\frac{\\pi}{2n}\\right)  \\cos\\beta+\\tan^{2}\\left(\n\\frac{\\pi}{2n}\\right)  \\right)\n\\end{align*}\nWe can rewrite $1+\\tan^{2}\\left(  \\pi\/\\left(  2n\\right)  \\right)  =\\sec\n^{2}\\left(  \\pi\/\\left(  2n\\right)  \\right)  $ to obtain the magnitudes stated\nin (i)$.$ For (ii), taking derivatives of the formulae first derived for\n$f_{\\beta}\\left(  r\\right)  $ and $f_{\\beta}\\left(  re^{i\\pi\/n}\\right)  ,$ we\nobtain\n\\begin{align*}\n\\frac{\\partial}{\\partial r}f_{\\beta}\\left(  r\\right)   &  =\\cos\\left(\n\\beta\/2\\right)  \\left(  \\frac{1+r^{n-2}}{\\sqrt{1-r^{2n}}}\\right)\n+i\\sin\\left(  \\beta\/2\\right)  \\left(  \\frac{1-r^{n-2}}{\\sqrt{1-r^{2n}}%\n}\\right)  \\text{ and }\\\\\n\\frac{\\partial}{\\partial r}f_{\\beta}\\left(  re^{i\\pi\/n}\\right)   &\n=e^{i\\pi\/n}\\left(  \\cos\\left(  \\beta\/2\\right)  \\left(  \\frac{1-r^{n-2}}%\n{\\sqrt{1-r^{2n}}}\\right)  +i\\sin\\left(  \\beta\/2\\right)  \\left(  \\frac\n{1+r^{n-2}}{\\sqrt{1-r^{2n}}}\\right)  \\right)  .\n\\end{align*}\n\n\nThe arguments of these derivatives, namely $\\arg\\frac{\\partial}{\\partial\nr}f_{\\beta}\\left(  r\\right)  $ and $\\arg\\frac{\\partial}{\\partial r}f_{\\beta\n}\\left(  re^{i\\pi\/n}\\right)  $ are respectively\\\n\\[\n\\arctan\\left(  \\tan\\left(  \\beta\/2\\right)  \\frac{1-r^{n-2}}{1+r^{n-2}}\\right)\n\\text{ and }\\pi\/n+\\arctan\\left(  \\tan\\left(  \\beta\/2\\right)  \\frac{1+r^{n-2}%\n}{1-r^{n-2}}\\right)  ,\n\\]\nwhence the stated monotonicity of $\\arg\\frac{\\partial}{\\partial r}f_{\\beta\n}\\left(  r\\right)  $ and $\\arg\\frac{\\partial}{\\partial r}f_{\\beta}\\left(\nre^{i\\pi\/n}\\right)  $ in (ii).\\linebreak The limits in (ii) are now easily\ncomputed. Note that as $r\\rightarrow1^{-},$ the ratio $\\frac{1+r^{n-2}%\n}{1-r^{n-2}}$ becomes infinite, and $\\arctan\\left(  \\tan\\left(  \\beta\n\/2\\right)  \\frac{1+r^{n-2}}{1-r^{n-2}}\\right)  $ approaches $\\pi\/2.\\ $We can\nalso calculate the magnitudes $\\left\\vert \\frac{\\partial}{\\partial r}f_{\\beta\n}\\left(  r\\right)  \\right\\vert ^{2}$ and $\\left\\vert \\frac{\\partial}{\\partial\nr}f_{\\beta}\\left(  re^{i\\pi\/n}\\right)  \\right\\vert ^{2},$ respectively, as\n$\\frac{1\\pm2r^{n-2}\\cos\\beta+r^{2n-4}}{1-r^{2n}}>\\frac{\\left(  1-r^{n-2}%\n\\right)  ^{2}}{1-r^{2n}}>0.$ Thus both curves $f_{\\beta}\\left(  r\\right)  $\nand $f_{\\beta}\\left(  re^{i\\pi\/n}\\right)  $ have increasing magnitude as $r$\nincreases. Finally to prove (iii), where $\\beta=0,$ we have \\newline%\n$f_{0}\\left(  r\\right)  =\\left(  1\/\\sqrt{2}\\right)  \\left(  h_{n}\\left(\nr\\right)  +g_{n}\\left(  r\\right)  \\right)  >0$. Moreover, $\\arg f_{0}\\left(\nre^{i\\pi\/n}\\right)  =\\frac{\\pi}{n}+\\arctan\\left(  0\\right)  =\\frac{\\pi}{n},$\nso $f_{0}\\left(  re^{i\\frac{\\pi}{n}}\\right)  $ maps to a ray emanating from\nthe origin with argument $\\pi\/n.$\n\\end{proof}\n\n\\section{Cusps and Nodes}\n\nWe now examine the boundary curves for the rosette harmonic mappings, which\nallows us to describe the cusps and other features that are apparent in the\ngraphs$.$ The boundary curve is also the key in our approach in Section 5 to\nproving univalence of the rosette harmonic mappings.\n\nFor a fixed $n\\geq3$, a rosette harmonic mapping $f_{\\beta}$ of Definition\n\\ref{defnharmonic} extends analytically to $\\partial U$, except at isolated\nvalues on $\\partial U$. Indeed recall that $h_{n}$ and $g_{n}$ of Definition\n\\ref{defnhg} are analytic except at the $2n$th roots of unity (Proposition\n\\ref{analco}). In contrast, the $n$-cusped hypocycloid $f_{hyp}$ of Example\n1.1 extends continuously to $\\bar{U}$ but with just $n$ values in $\\partial U\n$ where the boundary function is not regular.\n\nNevertheless, Figures \\ref{rosette0}\\ and \\ref{five} show rosette mappings\n$f_{\\beta},$ where exactly $n$ cusps are apparent. A striking similarity\nbetween the rosette and hypocycloid mappings is the common dilatation\n$\\omega\\left(  z\\right)  =z^{n-2}.$ We note the following consequence for a\nharmonic function $f$ on $U.$ Where $\\alpha\\left(  t\\right)  =f\\left(\ne^{it}\\right)  $ exists with a continuous derivative $\\alpha^{\\prime}\\left(\nt\\right)  $, Corollary 2.2b of \\cite{HenSchob} implies that $\\operatorname{Im}%\n\\left(  \\sqrt{\\omega_{f}\\left(  e^{it}\\right)  }\\alpha^{\\prime}\\left(\nt\\right)  \\right)  =0$ (see also Section 7.4 of \\cite{PeterHMBook}). Thus on\nintervals where $\\alpha^{\\prime}\\left(  t\\right)  $ is continuous and\nnon-zero, and for dilatation $\\omega_{f}\\left(  z\\right)  =z^{n-2}$, we have\n\\begin{equation}\n\\arg\\alpha^{\\prime}\\left(  t\\right)  \\equiv-\\arg\\sqrt{e^{i\\left(  n-2\\right)\nt}}\\equiv-\\left(  n\/2-1\\right)  t\\text{ }(\\operatorname{mod}\\pi).\n\\label{HSarg}%\n\\end{equation}\nWe will find for rosette mappings $f_{\\beta}$ of Definition \\ref{defnharmonic}\nthat have cusps, $n$ of the $2n$ singular points are \"removable\" in a sense to\nbe described. Furthermore, for $n\\geq3$ and consistent with (\\ref{HSarg}), the\nformula\n\\begin{equation}\n\\fbox{$\\arg\\alpha^\\prime\\left(  t\\right)  =k\\pi-\\left(  \\frac{n}{2}-1\\right)\nt$} \\label{compass}%\n\\end{equation}\nholds (except at possibly one point) on each of the $n$ intervals $\\left(\n\\left(  2k-2\\right)  \\pi\/n,2k\\pi\/n\\right)  $, $\\ k=1,2,...,n,$ both when\n$\\alpha$ is the boundary function of an $n$-cusped hypocycloid \\textit{and\n}when $\\alpha$ is the boundary function of a rosette (provided it has cusps).\nFormula (\\ref{compass}) will not be valid for example for a rosette mapping\n$f_{\\pi\/2}$, where there are arcs for which the boundary function is\nconstant$.$ To proceed we first give a definition of cusp, node, and singular\npoint of a curve.\n\n\\begin{definition}\n\\label{cusp}An \\textbf{isolated singular point} on a curve $\\alpha\\left(\nt\\right)  $ is a point $\\alpha\\left(  t_{0}\\right)  $ at which either\n$\\alpha^{\\prime}\\left(  t_{0}\\right)  =0$ or $\\alpha^{\\prime}\\left(\nt_{0}\\right)  $ is not defined, but for which $\\alpha^{\\prime}\\left(\nt\\right)  $ is defined and non-zero in a neighborhood of $t_{0}$. Define the\nquantities\n\\[\n\\arg\\alpha^{\\prime}\\left(  t_{0}\\right)  ^{-}=\\lim_{t\\nearrow t_{0}^{-}}%\n\\arg\\alpha^{\\prime}\\left(  t\\right)  \\text{ and }\\arg\\alpha^{\\prime}\\left(\nt_{0}\\right)  ^{+}=\\lim_{t\\searrow t_{0}^{+}}\\arg\\alpha^{\\prime}\\left(\nt\\right)\n\\]\nwhere they exist. An isolated singular point for which $\\arg\\alpha^{\\prime\n}\\left(  t_{0}\\right)  ^{+}$and $\\arg\\alpha^{\\prime}\\left(  t_{0}\\right)  ^{-}\n$ differ by $\\pi$ is defined to be a \\textbf{cusp.} The line $L$ through the\ncusp $\\alpha\\left(  t_{0}\\right)  $ containing points with argument equal to\n$\\arg\\alpha^{\\prime}\\left(  t_{0}\\right)  ^{+}$ (or $\\arg\\alpha^{\\prime\n}\\left(  t_{0}\\right)  ^{-}$) is called the \\textbf{axis of the cusp}, or\nsimply the \\textbf{axis}$.$ Define a \\textbf{node} to be an isolated singular\npoint on the curve at which $\\arg\\alpha^{\\prime}\\left(  t_{0}\\right)\n^{+}-\\arg\\alpha^{\\prime}\\left(  t_{0}\\right)  ^{-}=\\theta\\not \\equiv\n\\pi\\,\\left(  \\operatorname{mod}2\\pi\\right)  .$ In this case, $\\theta$ is the\n\\textbf{exterior angle} of the node. The interior angle at the node is then\n$\\pi-\\theta.$ If the exterior angle is $0,$ then we call the node a\n\\textbf{removable node}. A node is described as a \\textbf{corner} in some sources.\n\\end{definition}\n\nBoth $h_{n}$ and $g_{n}$ have nodes with exterior (and interior) angle\n$\\pi\/2,$ as seen in Figure \\ref{canon6}: since $\\overline{g_{n}}$ is a\nreflection of $g_{n}$, the nodes of $\\overline{g_{6}}$ in Figure \\ref{canon6}\nappear with exterior angle $-\\pi\/2$ rather than $\\pi\/2$. The lower right image\nin Figure \\ref{five} indicates a rosette mapping with nodes rather than cusps,\nbut the remaining images in Figure \\ref{five} show examples with cusps. The\nfollowing lemma provides a convenient way to invoke equation (\\ref{compass})\nand draw conclusions about cusps of the boundary extension.\n\n\\begin{lemma}\n\\label{cusps}Let $n\\geq3,$ and $f\\left(  z\\right)  $ be a harmonic mapping on\n$U$, with continuous extension to $\\bar{U},$ so that $\\alpha\\left(  t\\right)\n=f\\left(  e^{it}\\right)  $ is defined on $\\partial U$. Let $k=1,2,...,n.$\n\\newline(i)\\ Suppose that $\\alpha^{\\prime}$ is defined and non-zero except at\n$t=2k\\pi\/n,$ and that $\\alpha$ satisfies (\\ref{compass}) on each interval\n$\\left(  \\left(  2k-2\\right)  \\pi\/n,2k\\pi\/n\\right)  .$ Then for each $k,$\n$\\alpha\\left(  2k\\pi\/n\\right)  $ is a cusp$,$ and the cusp axis has argument\n$2k\\pi\/n.$\\newline(ii)\\ Suppose that $f$ has $n$-fold rotational symmetry\n$f\\left(  e^{i2k\\pi\/n}z\\right)  =e^{i2k\\pi\/n}f\\left(  z\\right)  ,$ and\nthat\\ (\\ref{compass}) holds for $k=1$ on the interval $\\left(  0,2\\pi\n\/n\\right)  .$ Then $\\alpha$ satisfies (\\ref{compass}) on each interval\n$\\left(  \\left(  2k-2\\right)  \\pi\/n,2k\\pi\/n\\right)  $ and the conclusions of\n(i) hold.\n\\end{lemma}\n\n\\begin{proof}\nFor (i), we evaluate limits at $t=\\frac{2k\\pi}{n}$ as follows using\n(\\ref{compass}):\n\\begin{align}\n\\arg\\alpha^{\\prime}\\left(  2k\\pi\/n\\right)  ^{-}  &  =\\lim_{t\\nearrow\n2k\\pi\/n^{-}}k\\pi-\\left(  \\frac{n}{2}-1\\right)  t=\\frac{2k\\pi}{n},\\text{\nand}\\label{cuspslopes}\\\\\n\\arg\\alpha^{\\prime}\\left(  2k\\pi\/n\\right)  ^{+}  &  =\\lim_{t\\searrow\n2k\\pi\/n^{+}}\\left(  k+1\\right)  \\pi-\\left(  \\frac{n}{2}-1\\right)  t=\\pi\n+\\frac{2k\\pi}{n}.\\nonumber\n\\end{align}\nThus $\\arg\\alpha^{\\prime}\\left(  2\\pi\/n\\right)  ^{+}$ and $\\arg\\alpha^{\\prime\n}\\left(  2\\pi\/n\\right)  ^{-}$ differ by $\\pi,$ so $\\alpha\\left(\n2k\\pi\/n\\right)  $ is a cusp. We also see the axis has argument $2k\\pi\/n$ (or\nequivalently $\\pi+2k\\pi\/n$). For (ii), we use the fact that $\\alpha\\left(\nt+2k\\pi\/n\\right)  =e^{i2k\\pi\/n}\\alpha\\left(  t\\right)  $ to conclude\n\\begin{equation}\n\\arg\\alpha\\left(  t+2k\\pi\/n\\right)  =2k\\pi\/n+\\arg\\alpha\\left(  t\\right)  .\n\\label{badd}%\n\\end{equation}\nGiven that (\\ref{compass}) holds for $k=1,$ we can extend it to each interval\n$\\left(  \\left(  2k-2\\right)  \\pi\/n,2k\\pi\/n\\right)  ,$ $k=2,3,...,n,$ using\n(\\ref{badd}), and so the conclusions of (i)\\ hold also.\n\\end{proof}\n\n\\begin{remark}\n\\label{extend}Lemma \\ref{cusps} remains valid (with appropriate adjustments to\nthe interval on which (\\ref{compass}) holds), even if for finitely many points\nin $\\left(  \\left(  2k-2\\right)  \\pi\/n,2k\\pi\/n\\right)  ,$ $\\arg\\alpha^{\\prime\n}\\left(  t\\right)  $ does not exist. To compute the limits (\\ref{cuspslopes})\nwe only need equation (\\ref{compass}) to hold in a neighborhood of the\nendpoints $2k\\pi\/n$.\n\\end{remark}\n\nThe following proposition surely appears in the literature, but for\ncompleteness, we use Lemma \\ref{cusps} to demonstrate the properties of the\nhypocycloid cusps.\n\n\\begin{proposition}\n\\label{hypocycloid}Let $n\\geq3,$ and let $f_{hyp}\\left(  z\\right)  =z+\\frac\n{1}{n-1}\\bar{z}^{n-1}$ be the hypocycloid harmonic mapping, and let\n$\\alpha_{hyp}\\left(  t\\right)  =f_{hyp}\\left(  e^{it}\\right)  .$ For\n$k=1,2,...,n,$ formula (\\ref{compass}) holds with $\\alpha=\\alpha_{hyp}$ for\nall $t\\in\\left(  \\left(  2k-2\\right)  \\pi\/n,2k\\pi\/n\\right)  .$ Moreover,\n$\\alpha_{hyp}$ has precisely $n$ cusps $\\alpha_{hyp}\\left(  2k\\pi\/n\\right)\n=\\frac{n}{n-1}e^{i2k\\pi\/n};$ the cusp axis has argument $2k\\pi\/n.$ In\ntraversing from one cusp to the next, $\\alpha_{hyp}$ has total curvature\n$\\pi-2\\pi\/n.$\n\\end{proposition}\n\n\\begin{proof}\nWe have $\\alpha_{hyp}\\left(  t\\right)  =h\\left(  e^{it}\\right)  +\\bar\n{g}\\left(  e^{it}\\right)  $ where $h\\left(  e^{it}\\right)  =e^{it}$ and\n$g\\left(  e^{it}\\right)  =\\frac{1}{n-1}e^{i\\left(  n-1\\right)  t}.$ We already\nnoted the singular points at $2k\\pi\/n$ in Example 1.1. With $z=e^{it}$ we\napply the chain rule, obtaining $\\frac{d}{dt}h\\left(  e^{it}\\right)  =ie^{it}$\nand $\\frac{d}{dt}g\\left(  e^{it}\\right)  =ie^{i\\left(  n-1\\right)  t}.$ The\nmagnitudes are both $1,$ so $\\alpha_{hyp}^{\\prime}\\left(  t\\right)  $ will\nhave argument equal to the mean of $\\arg\\frac{d}{dt}h\\left(  e^{it}\\right)  $\nand $\\arg\\frac{d}{dt}g\\left(  e^{-it}\\right)  ,$ when (for example) we choose\nbranches of the arguments that lie within $\\pi$ of one another. To this end,\nwe take\n\\begin{equation}\n\\arg\\frac{d}{dt}h\\left(  e^{it}\\right)  =\\pi\/2+t\\text{ and }\\arg\\frac{d}%\n{dt}g\\left(  e^{-it}\\right)  =3\\pi\/2-\\left(  n-1\\right)  t \\label{hypderivs}%\n\\end{equation}\non the interval $\\left(  0,2\\pi\/n\\right)  .$ Thus the mean is $\\left(\n2\\pi-\\left(  n\/2-1\\right)  t\\right)  \/2,$ which is equation (\\ref{compass})\nfor $k=1 $. We also have $f_{hyp}\\left(  e^{i2k\\pi\/n}z\\right)  =e^{i2k\\pi\n\/n}z+\\frac{1}{n-1}\\left(  e^{-i2k\\pi\/n}\\bar{z}\\right)  ^{n-1}.$ We factor out\nthe $e^{i2k\\pi\/n},$ using $\\left(  e^{-i2k\\pi\/n}\\right)  ^{n-1}=e^{-i2k\\pi\n\/n},$ and obtain $e^{i2k\\pi\/n}f_{hyp}\\left(  z\\right)  ,$ showing that\n$f_{hyp}$ has rotational symmetry$.$ By Lemma \\ref{cusps} (ii), $\\alpha\n_{hyp}\\left(  2k\\pi\/n\\right)  $ is a cusp and the axis has argument $2k\\pi\/n$\nfor each $k=1,2,...,n.$ Using $z=1$ above, we also obtain $f_{hyp}\\left(\ne^{i2k\\pi\/n}\\right)  =e^{i2k\\pi\/n}+\\frac{1}{n-1}\\left(  e^{i2k\\pi\/n}\\right)\n=\\frac{n}{n-1}e^{i2k\\pi\/n}$. The total curvature of $\\alpha_{hyp}$ is measured\nwith the change in argument of the unit tangent, or equivalently the change in\n$\\arg\\alpha_{hyp}^{\\prime}.$ Since this is monotonic and linear in $t$\n(equation (\\ref{compass})), the total change in $\\arg\\alpha_{hyp}^{\\prime} $\nover any of the given intervals is equal to $\\left(  n\/2-1\\right)  $ times the\ninterval length $2\\pi\/n$, and so we obtain $\\pi-2\\pi\/n$.\n\\end{proof}\n\nWe compute formulae for the derivatives $\\left(  d\/dt\\right)  h_{n}\\left(\ne^{it}\\right)  $ and $\\left(  d\/dt\\right)  g_{n}\\left(  e^{it}\\right)  .$ In\ncontrast with the hypocycloid, the arguments $\\left(  d\/dt\\right)\nh_{n}\\left(  e^{it}\\right)  $ and $\\left(  d\/dt\\right)  \\overline{g_{n}\\left(\ne^{it}\\right)  } $ differ by a constant angle of $\\pm\\pi\/2.$\n\n\\begin{proposition}\n\\label{linearangles}Let $n\\geq3,$ and let $\\zeta$ be a primitive 2nth root of\nunity, and consider $h_{n}$ and $g_{n}$ defined in Definition \\ref{defnhg},\nanalytic on $\\bar{U}\\backslash\\left\\{  \\zeta^{j}:j=1,2,...,2n\\right\\}  .$ Let\n$j=1,2,...,2n.$\\newline i) On each interval $\\left(  \\left(  j-1\\right)\n\\pi\/n,j\\pi\/n\\right)  ,\\ $derivatives $d\/dt$ of both $h_{n}\\left(\ne^{it}\\right)  $ and $g_{n}\\left(  e^{it}\\right)  $ have magnitude\n$1\/\\left\\vert \\sqrt{1-e^{i2nt}}\\right\\vert ,$ and the arguments are linear\nmonotonic functions, expressible as%\n\\begin{align}\n\\arg\\left(  \\frac{d}{dt}h_{n}\\left(  e^{it}\\right)  \\right)   &\n=3\\pi\/4-\\left(  n\/2-1\\right)  t+\\left(  j-1\\right)  \\pi\/2,\\text{\nand}\\label{argghh}\\\\\n\\arg\\left(  \\frac{d}{dt}g_{n}\\left(  e^{it}\\right)  \\right)   &\n=3\\pi\/4+\\left(  n\/2-1\\right)  t+\\left(  j-1\\right)  \\pi\/2. \\label{argghg}%\n\\end{align}\n\\newline(ii) The functions $h_{n}\\left(  e^{it}\\right)  $ and $g_{n}\\left(\ne^{it}\\right)  $ each have $2n$ singular points $h_{n}\\left(  e^{ij\\pi\n\/n}\\right)  $ and $g_{n}\\left(  e^{ij\\pi\/n}\\right)  ,$ which are each nodes\nwith exterior (and interior) angle $\\pi\/2.$\\newline(iii) The difference in the\narguments $\\arg\\frac{d}{dt}h_{n}\\left(  e^{it}\\right)  -\\arg\\frac{d}%\n{dt}\\overline{g_{n}}\\left(  e^{it}\\right)  $ is constant on $\\left(  \\left(\nj-1\\right)  \\pi\/n,j\\pi\/n\\right)  ,$ and is alternately $+\\pi\/2$ when $j$ is\neven, and $-\\pi\/2$ when $j$ is odd$.$\n\\end{proposition}\n\n\\begin{proof}\n(i) Recall the derivatives $\\frac{dh}{dz}=\\frac{1}{\\sqrt{1-z^{2n}}}$ and\n$\\frac{dg}{dz}=\\frac{z^{n-2}}{\\sqrt{1-z^{2n}}}$ of Proposition \\ref{analco}\n(iii). \\ We have $\\frac{d}{dt}h_{n}\\left(  e^{it}\\right)  =\\frac{ie^{it}%\n}{\\sqrt{1-e^{i2nt}}}$ and $\\frac{d}{dt}g_{n}\\left(  e^{it}\\right)\n=\\frac{ie^{i\\left(  n-1\\right)  t}}{\\sqrt{1-e^{i2nt}}}.$ Each derivative has\nmagnitude $1\/\\left\\vert \\sqrt{1-e^{i2nt}}\\right\\vert ,$ and singular points\noccur at the 2nth roots of unity, where $e^{i2nt}=1.$ We compute\n\\begin{align*}\n\\arg\\frac{d}{dt}h_{n}\\left(  e^{it}\\right)   &  =\\arg ie^{it}-\\frac{1}{2}%\n\\arg\\left(  1-e^{i2nt}\\right) \\\\\n&  =\\pi\/2+t+\\frac{1}{2}\\arctan\\frac{\\sin\\left(  2nt\\right)  }{1-\\cos\\left(\n2nt\\right)  }.\n\\end{align*}\nThe latter term reduces to $\\pi\/4-\\left(  n\/2\\right)  t,$ which can be seen\nfor example using the half angle formula for cotangent; $\\operatorname{arccot}%\n\\frac{\\sin\\left(  2nt\\right)  }{1-\\cos\\left(  2nt\\right)  }=nt$ and $\\arctan\nX=\\pi\/2-\\operatorname{arccot}X.$ Thus on $\\left(  0,\\pi\/n\\right)  ,$%\n\\[\n\\arg\\frac{d}{dt}h_{n}\\left(  e^{it}\\right)  =3\\pi\/4-\\left(  n\/2-1\\right)  t.\n\\]\nAt $t=\\pi\/\\left(  2n\\right)  ,$ $\\sqrt{1-e^{i2nt}}$ becomes real and\n$\\arg\\frac{d}{dt}h_{n}\\left(  e^{i\\pi\/2n}\\right)  =\\arg\\left(  ie^{i\\pi\n\/2n}\\right)  =\\pi\/2+\\pi\/\\left(  2n\\right)  ,$ which is consistent with our\nformula on $\\left(  0,\\pi\/n\\right)  $; this choice of branch of $\\arctan$\ngives $\\arg\\frac{d}{dt}h_{n}\\left(  e^{it}\\right)  $ an \"initial\" value\n$3\\pi\/4$ (the limit as $t\\searrow0^{+}$) and is evidently consistent with the\nargument at $t=e^{i\\pi\/2n}$ (see also Figure \\ref{canon6}). From the\nrotational symmetry equation (\\ref{ga}) for $h_{n}$%\n\\[\n\\arg\\frac{d}{dt}h_{n}\\left(  e^{i\\left(  t+j\\pi\/n\\right)  }\\right)  =\\arg\n\\frac{d}{dt}h_{n}\\left(  e^{it}\\right)  +j\\pi\/n.\n\\]\nAdding $\\left(  j-1\\right)  \\pi\/2$ extends our formula for $\\arg\\frac{d}%\n{dt}h_{n}\\left(  e^{it}\\right)  $ from $\\left(  0,\\pi\/n\\right)  $ to the\ninterval $\\left(  \\left(  j-1\\right)  \\pi\/n,j\\pi\/n\\right)  ,$ giving\n(\\ref{argghh}). Similarly on $\\left(  0,\\pi\/n\\right)  $ we obtain\n\\[\n\\arg\\frac{d}{dt}g_{n}\\left(  e^{it}\\right)  =\\pi\/2+\\left(  n-1\\right)\nt+\\frac{1}{2}\\arctan\\frac{\\sin\\left(  2nt\\right)  }{1-\\cos\\left(  2nt\\right)\n}=3\\pi\/4+\\left(  n\/2-1\\right)  t,\n\\]\na branch of the argument for which $\\arg\\frac{d}{dt}g_{n}\\left(  e^{i\\pi\n\/2n}\\right)  =\\pi-\\pi\/\\left(  2n\\right)  $ as expected$,$ so again with\ninitial value $3\\pi\/4$ on $\\left(  0,\\pi\/n\\right)  .$ From the rotational\nsymmetry equation (\\ref{ga}) for $g_{n}$ we have $g_{n}\\left(  e^{ij\\pi\n\/n}z\\right)  =e^{ij\\left(  \\pi-\\pi\/n\\right)  }g_{n}\\left(  z\\right)  ,$ so\n\\[\n\\arg\\frac{d}{dt}g_{n}\\left(  e^{i\\left(  t+j\\pi\/n\\right)  }\\right)  =\\arg\n\\frac{d}{dt}g_{n}\\left(  e^{it}\\right)  +j\\left(  \\pi-\\pi\/n\\right)  .\n\\]\nWe extend our formula for $\\arg\\frac{d}{dt}g_{n}\\left(  e^{it}\\right)  $ for\n$t\\in$ $\\left(  \\left(  j-1\\right)  \\pi\/n,j\\pi\/n\\right)  $ as before, adding\n$\\left(  j-1\\right)  \\pi\/2$, leading to equation (\\ref{argghg}). For (ii) we\nnote that as we pass from the interval $\\left(  \\left(  j-1\\right)  \\pi\n\/n,j\\pi\/n\\right)  $ to $\\left(  j\\pi\/n,\\left(  j+1\\right)  \\pi\/n\\right)  ,$\nboth $\\arg\\frac{d}{dt}h_{n}\\left(  e^{it}\\right)  $ and $\\arg\\frac{d}{dt}%\ng_{n}\\left(  e^{it}\\right)  $ increase by $\\pi\/2$ at $j\\pi\/n.$ Thus\n$h_{n}\\left(  e^{ij\\pi\/n}\\right)  $ and $g_{n}\\left(  e^{ij\\pi\/n}\\right)  $\neach are nodes with exterior angle $\\pi\/2.$ To prove (iii) we compute the\ndifference in $\\arg\\frac{d}{dt}h_{n}\\left(  e^{it}\\right)  $ and $\\arg\\frac\n{d}{dt}\\overline{g_{n}}\\left(  e^{it}\\right)  $ as $\\arg\\frac{d}{dt}%\nh_{n}\\left(  e^{it}\\right)  +\\arg\\frac{d}{dt}g_{n}\\left(  e^{it}\\right)  ,$ so\nfor \\newline$t\\in\\left(  \\left(  j-1\\right)  \\pi\/n,j\\pi\/n\\right)  ,$\n\\begin{equation}\n\\arg\\frac{d}{dt}h_{n}\\left(  e^{it}\\right)  -\\arg\\frac{d}{dt}\\overline{g_{n}%\n}\\left(  e^{it}\\right)  =3\\pi\/2+\\left(  j-1\\right)  \\pi=\\left(  2j+1\\right)\n\\pi\/2. \\label{hgdiff}%\n\\end{equation}\n\n\\end{proof}\n\n\\begin{remark}\nThe rosette mappings are distinguished from the hypocycloid in that $\\arg$\n$\\frac{d}{dt}h_{n}\\left(  e^{it}\\right)  $ and $\\arg\\frac{d}{dt}%\n\\overline{g_{n}\\left(  e^{it}\\right)  }$ are decreasing in lockstep. As a\nresult the curves $h_{n}\\left(  e^{it}\\right)  $ and $\\overline{g_{n}}\\left(\ne^{it}\\right)  ,$ and ultimately $g_{n}\\left(  e^{it}\\right)  $ must be rigid\nmotions of one another, as illustrated in Figure \\ref{canon6}, and Figure\n\\ref{transmirror}, and proved in Corollary \\ref{rigid}. For the hypocycloid,\nthe arguments of the derivatives of the analytic and anti-analytic parts\n(\\ref{hypderivs}) are also linear, but with non-equal slopes with differing sign.\n\\end{remark}\n\n\\begin{corollary}\n\\label{rigid}The graph $h_{n}\\left(  e^{it}\\right)  $ on an interval $\\left(\n\\left(  j-1\\right)  \\pi\/n,j\\pi\/n\\right)  $ and the graph of $g_{n}\\left(\ne^{it}\\right)  $ on an interval $\\left(  \\left(  j^{\\prime}-1\\right)\n\\pi\/n,j^{\\prime}\\pi\/n\\right)  $ are identical, up to a translation and\nrotation$,$ where $j,j^{\\prime}=1,2,...,2n.$ Moreover the two curves have\nopposite orientation.\n\\end{corollary}\n\n\\begin{proof}\nThe tangents $\\frac{d}{dt}h_{n}\\left(  e^{it}\\right)  $ and $\\frac{d}%\n{dt}\\overline{g_{n}}\\left(  e^{it}\\right)  $ have equal magnitudes on\n\\linebreak$\\left(  \\left(  j-1\\right)  \\pi\/n,j\\pi\/n\\right)  $, where their\narguments differ by a constant. Thus $h_{n}\\left(  e^{it}\\right)  $ and\n$\\overline{g_{n}\\left(  e^{it}\\right)  }$ have equal arclength and curvature,\nand so are equal up to a translation and rotation by the fundamental theorem\nof plane curves. Both $h_{n}\\left(  e^{it}\\right)  $ and $g_{n}\\left(\ne^{it}\\right)  $ have rotational symmetry (Proposition \\ref{analco}) so the\nprevious statement is true even when $h_{n}\\left(  e^{it}\\right)  $ and\n$g_{n}\\left(  e^{it}\\right)  $ are defined on different arcs. Proposition\n\\ref{analco} also shows that the curve $h_{n}\\left(  e^{int}\\right)  $ also\nhas reflectional symmetry, so the graph of $h_{n}$ has symmetry group\n$D_{2n}.$ Thus $\\overline{h_{n}\\left(  e^{it}\\right)  }$ is also a rotation of\n$h_{n}\\left(  e^{it}\\right)  $ on any interval $\\left(  \\left(  j-1\\right)\n\\pi\/n,j\\pi\/n\\right)  ,$ where the pair has opposite orientation. We conclude\n$\\overline{h_{n}\\left(  e^{it}\\right)  }$ and $\\overline{g_{n}\\left(\ne^{it}\\right)  }$ are rigid motions of one another with opposite orientation,\nand the Corollary follows upon conjugation.\n\\end{proof}\n\nProposition \\ref{linearangles} allows us to compute the derivative of the\nboundary function of a rosette harmonic mapping. The cosine rule for triangles\nis useful for adding numbers of the same magnitude, and we recall its\napplication in the following remark.\n\n\\begin{remark}\n\\label{cosine} For $X>0,$ the cosine rule yields\n\\[\n\\left\\vert Xe^{i\\theta_{1}}+Xe^{i\\theta_{2}}\\right\\vert =X\\left\\vert\ne^{i\\theta_{1}}+e^{i\\theta_{2}}\\right\\vert =\\sqrt{2}X\n\\sqrt{ \n1+\\cos \\left\\vert \\theta_{1}-\\theta_{2}\\right\\vert \n} .\n\\]\n\n\\end{remark}\n\n\\begin{theorem}\n\\label{fbdry}For $n\\geq3$ and $\\beta\\in(-\\pi\/2,\\pi\/2],$ and let $f_{\\beta}$ be\na rosette harmonic mapping defined in Definition \\ref{defnharmonic}. Consider\nthe boundary curve $\\alpha_{\\beta}\\left(  t\\right)  =f_{\\beta}\\left(\ne^{it}\\right)  ,$ $t\\in\\partial U.$ The derivative $\\alpha_{\\beta}^{\\prime\n}\\left(  t\\right)  $ exists and is continuous on $\\partial U, $ except at the\n$2n$ multiples of $\\pi\/n.$ Let $k=1,2,...,n.$ \\newline(i) For $\\left\\vert\n\\beta\\right\\vert <\\pi\/2,$ $\\alpha_{\\beta}$ satisfies (\\ref{compass}) on\n$\\left(  \\left(  2k-2\\right)  \\pi\/n,2k\\pi\/n\\right)  ,$ except at $t=\\left(\n2k-1\\right)  \\pi\/n$ where $\\alpha_{\\beta}^{\\prime}\\left(  t\\right)  $ is\nundefined. Moreover the magnitude of $\\alpha_{\\beta}^{\\prime}\\left(  t\\right)\n$, which is strictly non-zero, is%\n\\begin{equation}\n\\left\\vert \\alpha_{\\beta}^{\\prime}\\left(  t\\right)  \\right\\vert =\\sqrt{2}%\n\\sqrt{1\\pm\\sin\\left(  \\beta\\right)  }\/\\left\\vert \\sqrt{1-e^{i2nt}}\\right\\vert\n. \\label{mag}%\n\\end{equation}\nHere the $\\sin$ term is subtracted on the first half of the interval $\\left(\n\\left(  2k-2\\right)  \\pi\/n,2k\\pi\/n\\right)  ,$ and added on the second\nhalf.\\newline(ii) When $\\beta=\\pi\/2,$ on the first half of the interval\n$\\left(  \\left(  2k-2\\right)  \\pi\/n,2k\\pi\/n\\right)  ,$ $\\alpha_{\\pi\/2}$\nsatisfies (\\ref{compass})$,$ while $\\alpha_{\\pi\/2}^{\\prime}\\left(  t\\right)  $\nis strictly non-zero there with $\\left\\vert \\alpha_{\\pi\/2}^{\\prime}\\left(\nt\\right)  \\right\\vert =\\sqrt{2}\/\\left\\vert \\sqrt{1-e^{2int}}\\right\\vert $. On\nthe second half of the interval $\\left(  \\left(  2k-2\\right)  \\pi\n\/n,2k\\pi\/n\\right)  ,$ $\\alpha_{\\pi\/2}^{\\prime}\\left(  t\\right)  =0$ and\n$\\alpha_{\\pi\/2}\\left(  t\\right)  $ is constant.\n\\end{theorem}\n\n\\begin{proof}\nNote that the summands $\\frac{d}{dt}e^{i\\beta\/2}h_{n}\\left(  e^{it}\\right)  $\nand $\\frac{d}{dt}e^{-i\\beta\/2}g_{n}\\left(  e^{-it}\\right)  $ of $\\alpha\n_{\\beta}^{\\prime}\\left(  t\\right)  =\\frac{d}{dt}\\arg f_{\\beta}\\left(\ne^{it}\\right)  $ have the same magnitude, namely $1\/\\left\\vert \\sqrt\n{1-e^{2int}}\\right\\vert $. \\ From equation (\\ref{hgdiff}), the angle between\n$\\frac{d}{dt}h_{n}\\left(  e^{it}\\right)  $ and $\\frac{d}{dt}\\overline{g_{n}%\n}\\left(  e^{it}\\right)  $ is the constant $\\left(  -1\\right)  ^{j}\\pi\/2$ on\neach interval $\\left(  \\left(  j-1\\right)  \\pi\/n,j\\pi\/n\\right)  ,$ and the\npresence of $\\beta$ changes this difference to $\\beta+\\left(  -1\\right)\n^{j}\\pi\/2.$ Thus from the cosine rule (see Remark \\ref{cosine}) we obtain the\nmagnitude $\\left\\vert \\alpha_{\\beta}^{\\prime}\\left(  t\\right)  \\right\\vert\n=\\frac{\\sqrt{2}\\sqrt{1+\\cos\\left(  \\beta+\\left(  -1\\right)  ^{j}\\pi\/2\\right)\n}}{\\left\\vert \\sqrt{1-e^{i2nt}}\\right\\vert }=\\frac{\\sqrt{2}\\sqrt{1+\\left(\n-1\\right)  ^{j}\\sin\\left(  \\beta\\right)  }}{\\left\\vert \\sqrt{1-e^{i2nt}%\n}\\right\\vert },$ for $t\\neq j\\pi\/n.$ This proves equation (\\ref{mag}). Note\nthat since $\\left\\vert \\beta\\right\\vert <\\pi\/2$, $\\alpha_{\\beta}^{\\prime\n}\\left(  t\\right)  \\neq0.$ We turn to the argument of $\\alpha_{\\beta}^{\\prime\n}.$ We can utilize arithmetic means involving (\\ref{argghh}) and\n(\\ref{argghg}), \\ or simply make use of (\\ref{HSarg}). For either approach,\nthe initial argument $\\arg\\alpha_{\\beta}^{\\prime}\\left(  0\\right)  ^{+}$ must\nbe determined as either $\\pi$ or $0.$ The initial values of $\\arg\\frac{d}%\n{dt}h_{n}\\left(  e^{it}\\right)  $ and $\\arg\\frac{d}{dt}\\overline{g_{n}}\\left(\ne^{it}\\right)  $ as $t\\searrow0^{+}$ are $3\\pi\/4$ and $-3\\pi\/4$\nrespectively$.$ Therefore the initial angle $\\arg\\left(  \\alpha_{0}^{\\prime\n}\\left(  0\\right)  ^{+}\\right)  $ is $\\pi$ rather than $0,$ and $\\arg\n\\alpha_{0}^{\\prime}\\left(  t\\right)  =\\pi-\\left(  n\/2-1\\right)  t.$ This\nformula for $\\arg\\alpha_{0}^{\\prime}\\left(  t\\right)  $ holds throughout\n$\\left(  0,\\pi\/n\\right)  ,$ since $\\alpha_{\\beta}\\left(  t\\right)  $ is\ncontinuous$.$ The initial angles $\\frac{d}{dt}h_{n}\\left(  e^{it}\\right)  $\nand $\\frac{d}{dt}\\overline{g_{n}}\\left(  e^{it}\\right)  $ as $t\\searrow0^{+}$\non $\\left(  \\pi\/n,2\\pi\/n\\right)  $ become $3\\pi\/4+\\pi\/n$ and $\\pi\/4+\\pi\/n$\nrespectively (using Proposition \\ref{linearangles}, or rotational symmetry).\nThe initial value of $\\arg\\alpha_{0}^{\\prime}\\left(  t\\right)  $ on $\\left(\n\\pi\/n,2\\pi\/n\\right)  $ is therefore $\\pi\/2+\\pi\/n,$ consistent with\n(\\ref{compass}). Thus the formula $\\arg\\alpha_{0}^{\\prime}\\left(  t\\right)\n=\\pi-\\left(  n\/2-1\\right)  t$ holds throughout $\\left(  0,2\\pi\/n\\right)  $,\nwhere defined. For \\linebreak$\\left\\vert \\beta\\right\\vert <\\pi\/2,$ the means\nof $\\arg\\frac{d}{dt}e^{i\\beta\/2}h_{n}\\left(  e^{it}\\right)  $ and $\\arg\n\\frac{d}{dt}e^{-i\\beta\/2}\\overline{g_{n}}\\left(  e^{it}\\right)  $ are the same\nas for $\\beta=0$: the initial angles $\\arg\\frac{d}{dt}e^{i\\beta\/2}h_{n}\\left(\ne^{it}\\right)  $ and $\\arg\\frac{d}{dt}e^{-i\\beta\/2}\\overline{g_{n}}\\left(\ne^{it}\\right)  $ as $t\\searrow0^{+}$ on $\\left(  0,\\pi\/n\\right)  $ are\nrespectively $3\\pi\/4+\\beta\/2$ (third quadrant) and $-3\\pi\/4-\\beta\/2$ (fourth\nquadrant)$.$ Thus the initial angle $\\arg\\left(  \\alpha_{\\beta}^{\\prime\n}\\left(  0\\right)  ^{+}\\right)  $ maintains the value $\\pi$ (rather than\n$0$)$.$ Similarly the initial angles $\\arg\\frac{d}{dt}e^{i\\beta\/2}h_{n}\\left(\ne^{it}\\right)  $ and $\\arg\\frac{d}{dt}e^{-i\\beta\/2}\\overline{g_{n}}\\left(\ne^{it}\\right)  \/2$ as $t\\searrow0^{+}$ on $\\left(  \\pi\/n,2\\pi\/n\\right)  $ are\nrespectively $3\\pi\/4+\\pi\/n+\\beta\/2$ and $\\pi\/4+\\pi\/n-\\beta\/2.$ Thus the\ninitial angle $\\arg\\left(  \\alpha_{\\beta}^{\\prime}\\left(  \\pi\/n\\right)\n^{+}\\right)  $ also remains fixed as $\\pi\/2+\\pi\/n.$ We conclude that the\nequation for $\\arg\\alpha_{0}^{\\prime}\\left(  t\\right)  $ is valid for\n$\\arg\\alpha_{\\beta}^{\\prime}\\left(  t\\right)  $ on $\\left(  0,2\\pi\/n\\right)  $\n(note this would \\textit{not} be the case for $\\left\\vert \\beta\\right\\vert\n\\in\\left(  \\pi\/2,\\pi\\right)  $). Thus equation (\\ref{compass}) holds for\n$k=1,$ with $\\alpha_{\\beta}$ in place of $\\alpha,$ except at $t=\\pi\/n$ where\n$\\arg\\alpha_{\\beta}^{\\prime}\\left(  t\\right)  $ is not defined$.$ By Theorem\n\\ref{symmetry} (ii), $f_{\\beta}\\ $has $n$-fold rotational symmetry needed to\ninvoke Lemma \\ref{cusps} (ii), and in view of Remark \\ref{extend}, formula\n(\\ref{compass}) holds on each interval $\\left(  \\left(  2k-2\\right)\n\\pi\/n,2k\\pi\/n\\right)  ,$ for $t\\neq\\left(  2k-1\\right)  \\pi\/n.$ This proves (i).\n\nWe now consider (ii), with $\\beta=\\pi\/2$. From Proposition \\ref{linearangles}\n(iii), the angle between $\\frac{d}{dt}e^{i\\pi\/4}h_{n}\\left(  e^{it}\\right)  $\nand $\\arg\\frac{d}{dt}e^{-i\\pi\/4}\\overline{g_{n}}\\left(  e^{it}\\right)  $ is\n$\\beta+\\left(  -1\\right)  ^{j}\\pi\/2$ (mod $2\\pi$), which is either $0$ or\n$\\pi.$ For even $j,$ we see that $\\frac{d}{dt}e^{i\\pi\/4}h_{n}\\left(\ne^{it}\\right)  $ and $\\frac{d}{dt}e^{-i\\pi\/4}\\overline{g_{n}}\\left(\ne^{it}\\right)  $ cancel, since their arguments differ by $\\pi.$ Then\n$\\alpha_{\\beta}^{\\prime}\\left(  t\\right)  $ $=0,$ and $\\alpha_{\\beta}$ is\nconstant on $\\left(  \\left(  j-1\\right)  \\pi\/n,j\\pi\/n\\right)  .$ For odd $j,$\nthe two summands have the same argument, so $\\arg\\alpha_{\\pi\/2}^{\\prime\n}\\left(  t\\right)  =\\arg\\frac{d}{dt}e^{i\\pi\/4}h_{n}\\left(  e^{it}\\right)  .$\nAdding $\\pi\/4$ to formula (\\ref{argghh}), we obtain equation $\\arg\\alpha\n_{\\pi\/2}^{\\prime}\\left(  t\\right)  =\\pi-\\left(  \\frac{n}{2}-1\\right)\nt+\\left(  j-1\\right)  \\pi\/2$ on $\\left(  \\left(  j-1\\right)  \\pi\n\/n,j\\pi\/n\\right)  .$ But this subinterval is the \"first half\" of the interval\n$\\left(  \\left(  2k-2\\right)  \\pi\/n,2k\\pi\/n\\right)  $ where $j=2k-1,$ so\nreplacing $j$ in our formula for $\\arg\\alpha_{\\pi\/2}^{\\prime}\\left(  t\\right)\n$ we obtain $\\pi-\\left(  \\frac{n}{2}-1\\right)  t+\\left(  2k-2\\right)  \\pi\/2$,\nwhich is (\\ref{compass}). Moreover the non-zero magnitude of $\\alpha_{\\pi\n\/2}^{\\prime}\\left(  t\\right)  $ is $\\left\\vert \\alpha_{\\pi\/2}^{\\prime}\\left(\nt\\right)  \\right\\vert =\\sqrt{2}\/\\left\\vert \\sqrt{1-e^{2int}}\\right\\vert .$\n\\end{proof}\n\n\\begin{corollary}\n\\label{cuspdir}For $\\beta\\in\\left(  -\\pi\/2,\\pi\/2\\right)  $, let $\\alpha\n_{\\beta}\\left(  t\\right)  =f_{\\beta}\\left(  e^{it}\\right)  ,$ and\n$k=1,2,...,n. $ Then the singular points\\ of $\\alpha_{\\beta}$ occurring at\nmultiples of $\\pi\/n\\ $are alternately cusps, and removable nodes. At\n$t=\\left(  2k-1\\right)  \\pi\/n$, the discontinuity in $\\arg\\alpha_{\\beta\n}^{\\prime}$ is removable, and $\\alpha_{\\beta}\\left(  \\left(  2k-1\\right)\n\\pi\/n\\right)  $ is a removable node. In traversing from the cusp\n$\\alpha_{\\beta}\\left(  \\left(  2k-2\\right)  \\pi\/n\\right)  $ to the cusp\n$\\alpha_{\\beta}\\left(  2k\\pi\/n\\right)  ,$ $\\alpha_{\\beta}$ has total curvature\n$\\pi-2\\pi\/n.$ The total curvature over the first half of the interval, is\nequal to the total curvature over the second half of the interval $\\left(\n\\left(  2k-2\\right)  \\pi\/n,2k\\pi\/n\\right)  ,$ and is $\\pi\/2-\\pi\/n$.\n\\end{corollary}\n\n\\begin{proof}\nAs noted in the proof of (i) above, Lemma \\ref{cusps} still applies with\n(\\ref{compass}) holding on $\\left(  \\left(  2k-2\\right)  \\pi\/n,2k\\pi\/n\\right)\n$ except at the center $t=\\left(  2k-1\\right)  \\pi\/n$ of the interval$.$ We\ntherefore use (\\ref{compass}) on the punctured interval to evaluate the\nlimits\n\\begin{equation}\n\\arg\\alpha_{\\beta}^{\\prime}\\left(  \\left(  2k-1\\right)  \\pi\/n\\right)\n^{-}=\\arg\\alpha_{\\beta}^{\\prime}\\left(  \\left(  2k-1\\right)  \\pi\/n\\right)\n^{+}=\\pi\/2+\\left(  2k-1\\right)  \\pi\/n, \\label{nodedir}%\n\\end{equation}\nso the exterior angle is $0$ and $\\alpha_{\\beta}\\left(  \\left(  2k-1\\right)\n\\pi\/n\\right)  $ is a removable node. We note that the discontinuity in\n$\\arg\\alpha_{\\beta}^{\\prime}$ at $\\left(  2k-1\\right)  \\pi\/n$ is removable.\nAgain by Lemma \\ref{cusps}, $\\alpha_{\\beta}\\left(  2k\\pi\/n\\right)  $ is a cusp\nand the axis has argument $2k\\pi\/n.$ The equation (\\ref{compass}) is monotonic\nin $t$, so the total change in $\\arg\\alpha_{\\beta}^{\\prime}$ is equal to\n$\\pi-2\\pi\/n$ on the interval $\\left(  \\left(  2k-2\\right)  \\pi\/n,2k\\pi\n\/n\\right)  .$ Since (\\ref{compass}) is linear, half of this change, namely\n$\\pi\/2-\\pi\/n,$ occurs on each half of the interval $\\left(  \\left(\n2k-2\\right)  \\pi\/n,2k\\pi\/n\\right)  .$\n\\end{proof}\n\n\n\\begin{figure}[h]%\n\\centering\n\\includegraphics[\nheight=2.226in,\nwidth=4.3742in\n]%\n{fiveL2.eps}%\n\\caption{For $n=5$ with $\\beta=\\pi\/4$ (left) and $\\beta=2\\pi\/5$ (right), the\nnodes and cusps are indicated by a dots, and interlace with one another by\nargument. The argument of each node and cusp increases with $\\beta$ by\nequation (\\ref{cnarg}). Node and cusp locations are as described $\\left\\vert\n\\beta\\right\\vert <\\pi\/2$ in Theorem \\ref{cuspsnodes}. }%\n\\label{cuspnodepic5}%\n\\end{figure}\nBy Corollary \\ref{cuspdir}, the rosette harmonic mappings $f_{\\beta}$ for\n$\\left\\vert \\beta\\right\\vert <\\pi\/2$ have $n$-cusps, just as for the\n$n$-cusped hypocycloid mappings. Moreover with $n$ fixed, corresponding cusps\nfor different mappings have cusp axes that are parallel. This can be seen in\nFigure \\ref{cuspnodepic5} and in Figure \\ref{five} where cusps $\\alpha_{\\beta\n}\\left(  0\\right)  $ have axes parallel to the real axis$.$ The parallelism of\naxes follows from the identical unit tangent values of the boundary\nextensions, which also explains the total curvature of $\\pi-2\\pi\/n$ from one\ncusp to the next described both in Proposition \\ref{hypocycloid} and Corollary\n\\ref{cuspdir}.\n\nThe last graph $f_{\\pi\/2}\\left(  U\\right)  $ in Figure \\ref{five} does not\nhave cusps, but nodes with an acute interior angle, which we now examine.\n\n\\begin{corollary}\n\\label{nodes}Let $\\alpha_{\\pi\/2}\\left(  t\\right)  =f_{\\pi\/2}\\left(\ne^{it}\\right)  ,$ and let $k=1,2,...,n.$ On the first half of the interval\n$\\left(  \\left(  2k-2\\right)  \\pi\/n,2k\\pi\/n\\right)  ,\\ $the total curvature of\n$\\alpha_{\\pi\/2}$ is $\\pi\/2-\\pi\/n,$ while $\\alpha_{\\pi\/2}$ is constant\notherwise$.$ There is a piecewise smooth parametrization $\\tilde{\\alpha}%\n_{\\pi\/2},$ with the same graph as $\\alpha_{\\pi\/2}$ over $\\partial U$, and just\n$n$ singular points at $\\tilde{\\alpha}_{\\pi\/2}\\left(  2k\\pi\/n\\right)\n=\\alpha_{\\pi\/2}\\left(  2k\\pi\/n\\right)  $ which are nodes of $\\tilde{\\alpha\n}_{\\pi\/2}$ with interior angle $\\pi\/2-\\pi\/n.$\n\\end{corollary}\n\n\\begin{proof}\nSince $\\alpha_{\\pi\/2}^{\\prime}\\left(  t\\right)  =0$ on $\\left(  \\left(\n2k-1\\right)  \\pi\/n,2k\\pi\/n\\right)  ,$ the singularities are not isolated, and\nmoreover these intervals are arcs of constancy for the boundary function of\n$f_{\\pi\/2}$. We define $\\tilde{\\alpha}_{\\pi\/2},$ defined piecewise on\n$[0,2\\pi)$ by%\n\\begin{equation}\n\\tilde{\\alpha}_{\\pi\/2}\\left(  t\\right)  =\\alpha_{\\pi\/2}\\left(  \\left(\nk-1\\right)  \\pi\/n+t\/2\\right)  ,\\text{ }t\\in\\lbrack\\left(  2k-2\\right)\n\\pi\/n,2k\\pi\/n). \\label{halfspeed}%\n\\end{equation}\n\n\nThen on each interval in (\\ref{halfspeed}) the curve $\\tilde{\\alpha}_{\\pi\/2} $\nhas the values\\newline$\\alpha_{\\pi\/2}\\left(  \\left[  \\left(  2k-2\\right)\n\\pi\/n,\\left(  2k-1\\right)  \\pi\/n\\right]  \\right)  $. Moreover, $\\tilde{\\alpha\n}_{\\pi\/2}\\left(  2k\\pi\/n\\right)  =\\alpha_{\\pi\/2}\\left(  2k\\pi\/n\\right)  .$\nThus $\\tilde{\\alpha}_{\\pi\/2}$ is also continuous, with the same image as\n$\\alpha_{\\pi\/2}$ on $\\left[  0,2\\pi\\right]  .$ We compute\n\\begin{align*}\n\\lim_{t\\rightarrow2k\\pi\/n^{-}}\\arg\\tilde{\\alpha}_{\\pi\/2}^{\\prime}\\left(\nt\\right)   &  =\\lim_{t\\rightarrow2k\\pi\/n^{-}}\\arg\\alpha_{\\pi\/2}^{\\prime\n}\\left(  \\left(  k-1\\right)  \\pi\/n+t\/2\\right) \\\\\n&  =\\lim_{t^{\\prime}\\rightarrow\\left(  2k-1\\right)  \\pi\/n}2k\\pi\/2-\\left(\n\\frac{n}{2}-1\\right)  t^{\\prime}=k\\pi-\\left(  \\frac{n}{2}-1\\right)  \\left(\n2k-1\\right)  \\frac{\\pi}{n}.\n\\end{align*}\nand using the formula for $\\arg\\tilde{\\alpha}_{\\pi\/2}^{\\prime}$ on $\\left(\n2k\\pi\/n,2\\left(  k+1\\right)  \\pi\/n\\right)  $ obtain\n\\begin{align*}\n\\lim_{t\\rightarrow2k\\pi\/n^{+}}\\arg\\tilde{\\alpha}_{\\pi\/2}^{\\prime}\\left(\nt\\right)   &  =\\lim_{t\\rightarrow2k\\pi\/n^{+}}\\arg\\alpha_{\\pi\/2}^{\\prime\n}\\left(  k\\pi\/n+t\/2\\right) \\\\\n&  =\\lim_{t^{\\prime}\\rightarrow2k\\pi\/n}\\left(  2k+2\\right)  \\pi\/2-\\left(\n\\frac{n}{2}-1\\right)  t^{\\prime}\\\\\n&  =\\left(  k+1\\right)  \\pi-\\left(  \\frac{n}{2}-1\\right)  2k\\frac{\\pi}{n}.\n\\end{align*}\nThe difference $\\arg\\tilde{\\alpha}_{\\pi\/2}^{\\prime}\\left(  2k\\pi\/n\\right)\n^{+}-\\arg\\tilde{\\alpha}_{\\pi\/2}^{\\prime}\\left(  2k\\pi\/n\\right)  ^{-}$ is thus\n$\\pi-\\left(  \\frac{n}{2}-1\\right)  \\pi\/n=\\pi\/2+\\pi\/n,$ so we have a node with\ninterior angle $\\pi\/2-\\pi\/n.$\n\\end{proof}\n\n\\begin{remark}\n\\label{node}The node $\\tilde{\\alpha}_{\\pi\/2}\\left(  2k\\pi\/n\\right)  $ can be\nwritten $e^{i2k\\pi\/n}\\alpha_{\\pi\/2}\\left(  t\\right)  $ for any \\newline%\n$t\\in\\left[  \\left(  2k-1\\right)  \\pi\/n,2k\\pi\/n\\right]  ,$ where $\\alpha\n_{\\pi\/2}$ is constant$.$ We write the nodes of $\\tilde{\\alpha}_{\\pi\/2}$ as\n\\newline$e^{i2k\\pi\/n}f_{\\pi\/2}\\left(  1\\right)  $ when convenient$,$ and refer\nto nodes of $\\tilde{\\alpha}_{\\pi\/2}$ as the nodes of $f_{\\pi\/2}.$\n\\end{remark}\n\n\\begin{example}\nWhen $\\beta=\\pi\/2$ and with $n=5,$ then on intervals $\\left(  0,\\pi\/5\\right)\n,$ $\\left(  2\\pi\/5,3\\pi\/5\\right)  ,$ ... the boundary arcs $e^{i\\pi\/4}%\nh_{5}\\left(  e^{it}\\right)  $ and $e^{-i\\pi\/4}\\overline{g_{5}}\\left(\ne^{-it}\\right)  $ are translates of one another, and on intervals $\\left(\n\\pi\/5,2\\pi\/5\\right)  ,$ $\\left(  3\\pi\/5,5\\pi\/5\\right)  ,$ ..., the boundary\narcs are mirrors of one another. Figure \\ref{transmirror} (right) shows the\narc of constancy $\\left(  \\pi\/5,2\\pi\/5\\right)  $ on which $\\ e^{i\\pi\/4}%\nh_{5}\\left(  e^{it}\\right)  $ and $e^{-i\\pi\/4}\\overline{g_{5}}\\left(\ne^{-it}\\right)  $ are mirror images, and where $f_{\\pi\/2}\\left(\ne^{it}\\right)  $ is equal to the node $e^{i2\\pi\/5}f_{\\pi\/2}\\left(  1\\right)  $\n(indicated with a larger dot in the second quadrant).\n\\begin{figure}[ptb]%\n\\centering\n\\includegraphics[\nheight=2.4059in,\nwidth=5.0548in\n]%\n{transmir5.eps}%\n\\caption{Images of sectors in $U$ of $e^{i\\pi\/4}h_{5}$ and $e^{-i\\pi\n\/4}\\overline{g_{5}}$ with $\\arg z\\in\\left(  0,\\pi\/5\\right)  $ shaded (left)\nand $\\arg z\\in\\left(  \\pi\/5,2\\pi\/5\\right)  $ shaded (right), for $n=5.$ The\nbounding curves (thickened)$,$ are translates (left) and reflections (right).\nA tangent is indicated for $h_{5}\\left(  e^{it}\\right)  $ and $\\overline\n{g_{5}\\left(  e^{it}\\right)  }$ in each case, along with a portion of\n$f_{\\pi\/2}\\left(  e^{it}\\right)  $, which is constant (right) on $\\left(\n\\pi\/5,2\\pi\/5\\right)  .$}%\n\\label{transmirror}%\n\\end{figure}\n\n\\end{example}\n\nWe now complete our description of the symmetries within the graphs of the\nrosette mappings.\n\n\\begin{corollary}\n\\label{noreflect}Let $\\beta\\in%\n\\mathbb{R}\n$ and $n\\geq3.$ If $\\beta$ is not a multiple of $\\pi\/2,$ then the image set\n$f_{\\beta}\\left(  U\\right)  $ does not have reflectional symmetry. In this\ncase, $f_{\\beta}\\left(  U\\right)  $ has symmetry group $C_{n}.$ Otherwise,\n$\\beta$ is a multiple of $\\pi\/2,$ and $f_{\\beta}\\left(  U\\right)  $ has\nsymmetry group $D_{n}.$\n\\end{corollary}\n\n\\begin{proof}\nAll rosette mappings have at least $n$ fold rotational symmetry, by\nProposition \\ref{symmetry} (ii). Since $f_{\\beta}$ has either exactly $n$\ncusps, or exactly $n$ non-removable nodes, $f_{\\beta}\\left(  U\\right)  $\ncannot have a higher order of symmetry. For $\\left\\vert \\beta\\right\\vert\n\\in\\left(  0,\\pi\/2\\right)  ,$ the axis of the cusp through $f_{\\beta}\\left(\n1\\right)  $ is parallel to the real axis, while the radial ray through $0$ and\n$f_{\\beta}\\left(  1\\right)  $ has argument $\\psi=\\arg f_{\\beta}\\left(\n1\\right)  ,$ distinct for each $\\beta$ in $\\left(  -\\pi\/2,\\pi\/2\\right)  $ by\nequation (\\ref{cnarg})$.$ Moreover as noted in Lemma \\ref{convexconcave},\n$\\psi$ is acute$,$ and has the same sign as $\\beta.$ Thus if $\\left\\vert\n\\beta\\right\\vert \\in\\left(  0,\\pi\/2\\right)  ,$ then any reflection of\n$f_{\\beta}\\left(  U\\right)  $ results changing the sign of the angle between\nthe reflected cusp axis and the reflected radial ray, resulting in a distinct\nreflected image set. Thus the symmetry group of $f_{\\beta}\\left(  U\\right)  $\nis $C_{n}$ for $\\left\\vert \\beta\\right\\vert \\in\\left(  0,\\pi\/2\\right)  .$ This\nfact extends by formula (\\ref{trans}) to any real $\\beta$ that is not a\nmultiple of $\\pi\/2.$ If $\\beta=0$ or $\\beta=\\pi\/2$ we already established that\n$f_{\\beta}\\left(  U\\right)  $ has reflectional symmetry. We conclude that the\nsets $f_{0}\\left(  U\\right)  $ and $f_{\\pi\/2}\\left(  U\\right)  $ have dihedral\nsymmetry group $D_{n}$. If $\\beta=l\\pi\/2 $ for some $l\\in%\n\\mathbb{Z}\n$ then by formula (\\ref{trans}), $f_{\\beta}\\left(  U\\right)  $ is a rotation\nof either $f_{\\pi\/2}\\left(  U\\right)  $ or $f_{0}\\left(  U\\right)  ,$ and thus\nhas rotational symmetry also.\n\\end{proof}\n\n\\begin{corollary}\n\\label{differ}Let $n\\geq3.$ For distinct $\\beta$ and $\\beta^{\\prime}$ in the\ninterval $(-\\pi\/2,\\pi\/2],$ the image sets $f_{\\beta}\\left(  U\\right)  $ and\n$f_{\\beta^{\\prime}}\\left(  U\\right)  $ are not scalings or rotations of one\nanother. Moreover with the parameter $\\beta$ within the set $\\left[\n0,\\pi\/2\\right]  ,$ all images of the unit disk under a rosette harmonic\nmapping are obtained, up to rotation and reflection.\n\\end{corollary}\n\n\\begin{proof}\nThe proof of Corollary \\ref{noreflect} shows that if $\\beta\\in\\left(\n-\\pi\/2,\\pi\/2\\right)  ,$ the angle between the cusp axis through $f_{\\beta\n}\\left(  1\\right)  $ and the radial line from $0$ to $f_{\\beta}\\left(\n1\\right)  $ intersect at an angle that is distinct for each choice of $\\beta.\n$ Thus $f_{\\beta}\\left(  U\\right)  $ is different from any rotation or scaling\nof $f_{\\beta^{\\prime}}\\left(  U\\right)  $ for any other $\\beta^{\\prime}%\n\\in\\left(  -\\pi\/2,\\pi\/2\\right)  .$ Moreover $f_{\\pi\/2}$ is the only $f_{\\beta\n}$ without cusps for $\\beta\\in(-\\pi\/2,\\pi\/2].$ To prove the second statement,\nlet $\\tilde{\\beta}\\in%\n\\mathbb{R}\n\\backslash(-\\pi\/2,\\pi\/2].$ Upon reducing $\\tilde{\\beta}$ modulo $\\pi,$ we\nobtain an equivalent $\\beta\\equiv\\tilde{\\beta}$ $(\\operatorname{mod}$ $\\pi)$\nwhere $\\beta\\in(-\\pi\/2,\\pi\/2].$ By possibly repeated application of\n(\\ref{rotbet}) in Proposition \\ref{parametrize} (i), $f_{\\tilde{\\beta}}\\left(\nz\\right)  $ is obtained as a pre- and post- composition of a rotation with\n$f_{\\beta}\\left(  z\\right)  .$ Thus it is sufficient to consider $\\beta\n\\in(-\\pi\/2,\\pi\/2].$ If $\\beta<0,$ then by Proposition \\ref{parametrize} (ii),\n$f_{\\beta}\\left(  z\\right)  $ is a reflection in the real axis of $f_{-\\beta\n}\\left(  z\\right)  $ where $-\\beta\\in\\left[  0,\\pi\/2\\right]  .$\n\\end{proof}\n\nWe turn to describing features of the graphs $f_{\\beta}\\left(  U\\right)  $ for\n$\\beta$ restricted to $(-\\pi\/2,\\pi\/2].$ We may use equation (\\ref{trans}) to\nidentify cusps and nodes for $\\beta$ outside this interval, but we make the\ncautionary observation that while for $\\beta\\in\\left(  -\\pi\/2,\\pi\/2\\right)  $\n$f_{\\beta}\\left(  1\\right)  $ is a cusp and $f_{\\beta}\\left(  e^{i\\pi\n\/n}\\right)  $ is a node, adding $\\pi$ to $\\beta$ results in $f_{\\beta+\\pi\n}\\left(  1\\right)  $ being a node and $f_{\\beta+\\pi}\\left(  e^{i\\pi\/n}\\right)\n$ being a cusp. Quantities such as the magnitude of a cusp, the magnitude of a\nnode, and the distribution of angle between cusps neighboring nodes however$,$\nare independent of the interval to which $\\beta$ is restricted$.$ These\nquantities are derived in Theorem \\ref{cuspsnodes}, with the help of Lemma\n\\ref{convexconcave}.\n\nTheorem \\ref{cuspsnodes} shows that for fixed $n,$ the image $f_{0}\\left(\nU\\right)  $ has maximal diameter, and $diam\\left(  f_{\\left\\vert\n\\beta\\right\\vert }\\left(  U\\right)  \\right)  $ decreases with $\\left\\vert\n\\beta\\right\\vert \\in\\left(  0,\\pi\/2\\right)  .\\,$This is illustrated in Figure\n\\ref{five}, where all four graphs are plotted on the same scale. We also see\nthat for $\\beta=0,$ the cusps and nodes are equally spaced with angular\nseparation $\\pi\/n,$ but this becomes increasingly unbalanced as $\\left\\vert\n\\beta\\right\\vert $ increases to $\\pi\/2.$ As $\\beta\\nearrow\\pi\/2^{-},$ each\nnode approaches the subsequent\\footnote{Subsequent cusp (or node) here\nindicates the cusp (or node) with the next largest argument.} cusp, as shown\nin Theorem \\ref{cuspsnodes} (ii).\n\n\\begin{theorem}\n\\label{cuspsnodes}Let $n\\geq3,$ $\\beta\\in(-\\pi\/2,\\pi\/2],$ and let\n$\\alpha_{\\beta}\\left(  t\\right)  =f_{\\beta}\\left(  e^{it}\\right)  $ where\n$f_{\\beta}$ is a rosette harmonic mapping$.$ Recall the constant $K_{n}$\ndefined in Lemma \\ref{convexconcave}$.$ \\newline(i)$\\ $If $\\beta\\in\\left(\n-\\pi\/2,\\pi\/2\\right)  $ then the magnitude of the cusps decreases strictly if\n$\\left\\vert \\beta\\right\\vert \\in\\lbrack0,\\pi\/2),$ from a maximum $K_{n}%\n\\sec\\left(  \\frac{\\pi}{2n}\\right)  ,$ and with infimum $K_{n}\\left(\n1+\\tan\\left(  \\frac{\\pi}{2n}\\right)  \\right)  .$ The removable nodes have\nmagnitudes that increase strictly if $\\left\\vert \\beta\\right\\vert \\in\n\\lbrack0,\\pi\/2),$ from the minimum $K_{n}\\left(  1+\\tan\\left(  \\frac{\\pi}%\n{2n}\\right)  \\right)  $ and with supremum $K_{n}\\sec\\left(  \\frac{\\pi}%\n{2n}\\right)  .$\\newline(ii) If $\\beta\\in\\left(  -\\pi\/2,\\pi\/2\\right)  $ then\nthe angle between a cusp neighboring node is\n\\begin{equation}\n\\pi\/n\\pm\\arctan\\left(  \\frac{2\\tan\\left(  \\frac{\\pi}{2n}\\right)  \\sin\\left(\n\\beta\\right)  }{1-\\tan^{2}\\left(  \\frac{\\pi}{2n}\\right)  }\\right)  ,\n\\label{sep}%\n\\end{equation}\nwhere we use the positive sign when the node has the more positive argument.\nFor $\\beta=0,$ the cusps and neighboring nodes are separated by equal angles\nof $\\pi\/n$. The difference in arguments of a cusp and subsequent node\nincreases from $0$ to $2\\pi\/n$ with $\\beta\\in\\left(  -\\pi\/2,\\pi\/2\\right)  $.\n\\newline(iii) For $\\beta=\\pi\/2,$ the $n$ nodes of the rosette mapping\n$f_{\\pi\/2}$ have magnitude \\linebreak$K_{n}\\sec\\frac{\\pi}{2n},$ and for the\nnode $f_{\\pi\/2}\\left(  1\\right)  $ we have $\\arg\\left(  f_{\\pi\/2}\\left(\n1\\right)  \\right)  =\\pi\/4-\\pi\/\\left(  2n\\right)  $.\n\\end{theorem}\n\n\\begin{proof}\nWe begin with (i). Due to the $\\cos\\beta$ term in (\\ref{cnmag}), the cusp\n$\\left\\vert f_{\\beta}\\left(  1\\right)  \\right\\vert $ is decreasing with\n$\\left\\vert \\beta\\right\\vert $ and the node $\\left\\vert f_{\\beta}\\left(\ne^{i\\pi\/n}\\right)  \\right\\vert $ is increasing with $\\left\\vert \\beta\n\\right\\vert .$ Note that\n\\[\n\\sec^{2}\\left(  \\frac{\\pi}{2n}\\right)  \\pm2\\tan\\left(  \\frac{\\pi}{2n}\\right)\n\\cos\\beta=1+\\tan^{2}\\left(  \\frac{\\pi}{2n}\\right)  \\pm2\\tan\\left(  \\frac{\\pi\n}{2n}\\right)  =\\left(  1\\pm\\tan\\left(  \\frac{\\pi}{2n}\\right)  \\right)  ^{2}.\n\\]\nIn view of this equation, the maximum of $\\left\\vert f_{\\beta}\\left(\n1\\right)  \\right\\vert $ is $K_{n}\\left(  1+\\tan\\left(  \\frac{\\pi}{2n}\\right)\n\\right)  $ when $\\beta=0,$ and $\\left\\vert f_{\\beta}\\left(  1\\right)\n\\right\\vert $ approaches $K_{n}\\sec\\left(  \\frac{\\pi}{2n}\\right)  $ as $\\beta$\napproaches $\\pi\/2.$ Similarly, the minimum of $\\left\\vert f_{\\beta}\\left(\ne^{i\\pi\/n}\\right)  \\right\\vert $ is $K_{n}\\left(  1-\\tan\\left(  \\frac{\\pi}%\n{2n}\\right)  \\right)  $ when $\\beta=0,$ and approaches $K_{n}\\left(\n\\sec\\left(  \\frac{\\pi}{2n}\\right)  \\right)  $ as an upper bound as $\\left\\vert\n\\beta\\right\\vert $ approaches $\\pi\/2,$ proving (i)$.$ For (ii) formula\n(\\ref{cnarg}) shows us $\\arg f_{\\beta}\\left(  1\\right)  $ and $\\arg f_{\\beta\n}\\left(  e^{i\\pi\/n}\\right)  $ are increasing with $\\beta$ on $\\left(\n-\\pi\/2,\\pi\/2\\right)  .$ Clearly $\\arg\\left(  f_{0}\\left(  1\\right)  \\right)\n=0$ and $\\arg f_{0}\\left(  e^{i\\pi\/n}\\right)  =\\pi\/n.$ Additionally we have\n$\\arg\\left(  f_{\\beta}\\left(  1\\right)  \\right)  <\\pi\/n<\\arg f_{\\beta}\\left(\ne^{i\\pi\/n}\\right)  \\,\\ $for $\\beta\\in\\left(  0,\\pi\/2\\right)  .$ We compute the\nangle between this cusp and node to be $\\arg f_{\\beta}\\left(  e^{i\\pi\n\/n}\\right)  -\\arg f_{\\beta}\\left(  1\\right)  =\\pi\/n+\\psi^{\\prime}-\\psi,$ and\nwe use the arctangent formula $\\arctan\\psi^{\\prime}-\\arctan\\psi=\\arctan\\left(\n\\frac{\\psi^{\\prime}-\\psi}{1+\\psi^{\\prime}\\psi}\\right)  $ where $\\psi^{\\prime}$\nand $\\psi$ are as defined in equation (\\ref{cnarg}). Thus after a short\ncalculation we obtain formula (\\ref{sep}). This difference increases from\n$\\pi\/n$ and approaches $2\\pi\/n$ as $\\beta$ increases through the interval\n$[0,\\pi\/2)$. Now suppose that $\\beta\\in\\left(  -\\pi\/2,0\\right)  .$ The\nreflection of the node $f_{\\beta}\\left(  e^{i\\pi\/n}\\right)  $ in the real axis\nis the node $f_{-\\beta}\\left(  e^{-i\\pi\/n}\\right)  $ of $f_{-\\beta}$ and the\nreflection of $f_{\\beta}\\left(  1\\right)  $ is the cusp $f_{-\\beta}\\left(\n1\\right)  .$ From rotational symmetry, the angular difference between $\\arg\nf_{-\\beta}\\left(  1\\right)  $ and $\\arg f_{-\\beta}\\left(  e^{-i\\pi\/n}\\right)\n$ is the same as the that of $\\arg f_{-\\beta}\\left(  e^{i2\\pi\/n}\\right)  $ and\n$\\arg f_{-\\beta}\\left(  e^{i\\pi\/n}\\right)  .$ Again using rotational symmetry,\nthis is $2\\pi\/n-\\left(  \\psi-\\psi^{\\prime}\\right)  =\\pi\/n-\\arctan\\left(\n\\frac{\\psi^{\\prime}-\\psi}{1+\\psi^{\\prime}\\psi}\\right)  =\\pi\/n-\\arctan\\left(\n\\frac{2\\tan\\left(  \\frac{\\pi}{2n}\\right)  \\sin\\left(  -\\beta\\right)  }%\n{1-\\tan^{2}\\left(  \\frac{\\pi}{2n}\\right)  }\\right)  =\\pi\/n+\\arctan\\left(\n\\frac{2\\tan\\left(  \\frac{\\pi}{2n}\\right)  \\sin\\left(  \\beta\\right)  }%\n{1-\\tan^{2}\\left(  \\frac{\\pi}{2n}\\right)  }\\right)  $. For (iii), with\n$\\beta=\\pi\/2, $ we noted in Remark \\ref{node} after Corollary \\ref{nodes} that\nthe nodes can be expressed as rotations of $f_{\\pi\/2}\\left(  1\\right)  ,$ for\nwhich we observed the stated quantities in Lemma \\ref{convexconcave} (i)$.$\n\\end{proof}\n\nThe next example illustrates the separation of cusps and nodes as it varies\nwith $\\beta,$ as described in Theorem \\ref{cuspsnodes} (see also Figure\n\\ref{cuspnodepic5})$.$ The relative proximity of a node and neighboring cusp\ncoupled with equal total curvature of the boundary curve between any node and\ncusp (Corollary \\ref{cuspdir}) gives rise to the appearance of a cresting wave\nat the cusp.\n\n\\begin{example}\n\\label{cuspnody}For $n=5,$ the arguments of the nodes and cusps of $f_{0}$ are\nequally spaced by $\\pi\/n=\\pi\/5.$ By Lemma \\ref{convexconcave}, as $\\beta\n\\in\\lbrack0,\\pi\/2)$ increases, the separation of a cusp and subsequent node\ngrows from $\\pi\/5=36^{\\circ}$ towards $2\\pi\/5=72^{\\circ}. $ For $\\beta=\\pi\/4$\nthis separation is $\\pi\/5+\\arctan\\sqrt{5\/2-\\sqrt{5}}\\approx63^{\\circ},$ while\nfor $\\beta=2\\pi\/5$ it grows to $\\pi\/5+\\arctan\\sqrt{5\/4-\\sqrt{5}\/4}%\n\\approx71^{\\circ}$ (see Figure \\ref{cuspnodepic5}). For a specific cusp or\nnode $f_{\\beta}\\left(  e^{ij_{0}\\pi\/n}\\right)  ,$ $\\arg f_{\\beta}\\left(\ne^{ij_{0}\\pi\/n}\\right)  $ increases with $\\beta$ (see proof of Theorem\n\\ref{cuspsnodes} (ii))$,\\ $which is also illustrated in Figure\n\\ref{cuspnodepic5} as $\\beta$ increases from $\\pi\/4$ to $2\\pi\/5.$ By Corollary\n\\ref{cuspdir}, the total curvature of the boundary of $f_{\\beta}$ between\nneighboring cusps is $\\pi-2\\pi\/n=3\\pi\/5=108^{\\circ}.$ The total curvature of\nthe boundary from a cusp to a neighboring node is half of this, namely\n$54^{\\circ}$. Figure \\ref{cuspnodepic5} also indicates the phenomenon\ndescribed in Theorem \\ref{cuspsnodes}, that while both nodes and cusps \"rotate\ncounterclockwise\" as $\\beta\\in\\left(  0,\\pi\/2\\right)  $ increases, the nodes\ndo so at a greater rate..\n\\end{example}\n\nFinally we point out that the argument of the boundary curve fails to be\nstrictly increasing on the whole of $\\partial U,$ for $\\beta$ with $\\left\\vert\n\\beta\\right\\vert \\in\\left(  0,\\pi\/2\\right)  .$\n\n\\begin{corollary}\n\\label{parallel}Let $n\\geq3$, $\\beta\\in\\left(  -\\pi\/2,\\pi\/2\\right)  $ and\n$\\alpha_{\\beta}$ be the boundary function of the rosette harmonic mapping\n$f_{\\beta}.$ Then for each $k=1,2,...,n,$ $\\arg\\alpha_{\\beta}\\left(  \\left(\n2k-2\\right)  \\pi\/n\\right)  ,\\ $\\newline$\\arg\\alpha_{\\beta}\\left(  \\left(\n2k-1\\right)  \\pi\/n\\right)  ,$ and $\\arg\\alpha_{\\beta}\\left(  2k\\pi\/n\\right)  $\noccur in increasing order$,$ but there exists an interval on which\n$\\arg\\left(  \\alpha_{\\beta}\\left(  t\\right)  \\right)  $ is decreasing.\n\\end{corollary}\n\n\\begin{proof}\nTheorem \\ref{cuspsnodes} (ii) shows that the angle $\\gamma$ between\\ a cusp\nand subsequent node is given by formula (\\ref{sep}) which belongs to $\\left(\n0,2\\pi\/n\\right)  $. Because the cusps are separated by angle $2\\pi\/n,$ the\nangle from a cusp to a subsequent node is then $2\\pi\/n-\\gamma\\in\\left(\n0,2\\pi\/n\\right)  .$ Finally with $\\beta\\in\\left(  0,\\pi\/2\\right)  ,$ suppose\nthat $\\arg\\alpha_{\\beta}\\left(  -\\epsilon\\right)  \\leq\\psi=\\arg\\alpha_{\\beta\n}\\left(  0\\right)  $ for some $-\\epsilon\\in\\left(  -\\pi\/n,0\\right)  ,$ but for\nwhich $\\arg\\alpha_{\\beta}^{\\prime}\\left(  -\\epsilon\\right)  <\\tan\\psi.$ Such\nan $\\epsilon$ exists because $\\alpha_{\\beta}$ satisfies (\\ref{cuspslopes})\nwith $k=0,$ so $\\arg\\alpha_{\\beta}^{\\prime}\\left(  0\\right)  ^{-}=0.$ Since\n$\\arg\\alpha_{\\beta}^{\\prime}\\left(  t\\right)  $ is decreasing, we have\n$\\arg\\alpha_{\\beta}\\left(  t\\right)  <\\psi$ for all $t\\in\\left(\n-\\epsilon\/2,0\\right)  ,$ and so $\\alpha_{\\beta}\\left(  t\\right)  $ lies in a\nhalf-plane formed by a line parallel to to $\\arg z=\\psi,$ but with smaller\nimaginary part, and this half plane does not contain $\\alpha_{\\beta}\\left(\n0\\right)  .$ This contradicts continuity of $\\alpha_{\\beta}\\left(  t\\right)\n\\rightarrow\\alpha_{\\beta}\\left(  0\\right)  $ as $t\\nearrow0^{-}.$ Thus there\nis an interval $\\left(  -\\epsilon,0\\right)  $ on which $\\arg\\alpha_{\\beta\n}\\left(  t\\right)  >\\psi.$ On this interval, $\\arg\\alpha_{\\beta}\\left(\nt\\right)  $ decreases to $\\psi,$ and so rotates negatively relative to the\norigin. A similar argument holds when $\\beta\\in\\left(  -\\pi\/2,0\\right)  .$\n\\end{proof}\n\n\\section{Univalence and Fundamental Sets}\n\nOur approach to proving the univalence of $f_{\\beta}$ is to use the argument\nprinciple for harmonic functions. We note that various proofs of univalence\nfor rosette mappings $f_{\\beta}$ are possible. The following theorem describes\nthe winding number of the boundary curve $\\alpha\\left(  t\\right)  =f_{\\beta\n}\\left(  e^{it}\\right)  ,$ so that we can apply the argument principle.\n\n\\begin{lemma}\n\\label{unibdry}For fixed $\\beta\\in\\lbrack0,\\pi\/2),$ the boundary\n$\\alpha_{\\beta}\\left(  t\\right)  =f_{\\beta}\\left(  e^{it}\\right)  ,$\n$t\\in\\partial U$ is a simple, positively oriented closed curve. While\n$\\alpha_{\\pi\/2}$ has arcs of constancy, the parametrization $\\tilde{\\alpha\n}_{\\pi\/2}$ of equation (\\ref{halfspeed}) is a simple, positively oriented\nclosed curve on $\\partial U.$\n\\end{lemma}\n\n\\begin{proof}\nLet $\\beta\\in\\left(  0,\\pi\/2\\right)  .$ We give a separate argument for\n$\\beta=0$ and for $\\beta=\\pi\/2.$ We first show that $\\alpha_{\\beta}$ is one to\none when restricted to $\\left(  0,2\\pi\/n\\right)  ,$ and that the boundary\ncurve portion $\\alpha_{\\beta}\\left(  \\left(  0,2\\pi\/n\\right)  \\right)  $ lies\nin a sector $S.$ We then show that the portion of the graph of $\\alpha_{\\beta\n}$ restricted to the interval $\\left(  2k\\pi\/n,2\\left(  k+1\\right)\n\\pi\/n\\right)  $ is contained within the set $e^{i2k\\pi\/n}S.$ The sets\n$\\left\\{  e^{i2k\\pi\/n}S:k=0,1,...,n-1\\right\\}  $ are then seen to be pairwise\ndisjoint, so that the curve $\\alpha_{\\beta}$ has no self intersections on\n$[0,2\\pi),$ and we conclude $\\alpha_{\\beta}$ is a simple closed curve on\n$\\partial U.$\n\nDefine $L_{k}$ to be the axis of the cusp $\\alpha_{\\beta}\\left(  2k\\pi\\right)\n,$ so with argument equal to $2k\\pi\/n,$ $k=0,1,..,n-1.$ Recall from Theorem\n(\\ref{fbdry}) that $\\alpha_{\\beta}$ satisfies (\\ref{compass}) (which holds\nexcept at $t=\\left(  2k+1\\right)  \\pi\/n$). We begin with $\\alpha_{\\beta}$ on\nthe interval $\\left(  0,2\\pi\/n\\right)  .$ Let $p$ be the intersection of\n$L_{0}$ (parallel to the real axis) and $L_{1}.$ These non-collinear lines\nform the boundary of four unbounded open connected sectors, and we define $S$\nto be the sector with vertex $p$ \\ for which the (distinct) cusps\n$\\alpha_{\\beta}\\left(  0\\right)  \\in L_{0}$ and $\\alpha_{\\beta}\\left(\n2\\pi\/n\\right)  \\in L_{1}$ belong to $\\partial S.$ By equation (\\ref{compass}%\n)$,$ $\\arg\\alpha_{\\beta}^{\\prime}\\left(  t\\right)  $ decreasing and\n$\\arg\\alpha_{\\beta}^{\\prime}\\left(  0\\right)  ^{+}=\\pi.$ Thus the curve\n$\\alpha_{\\beta}$ lies \"above\" $L_{0}$. If $\\alpha_{\\beta}$ were to cross\n$L_{1}$ at some $t_{1}\\in\\left(  0,2\\pi\/n\\right)  ,$ then $\\arg\\alpha_{\\beta\n}^{\\prime}\\left(  t_{1}^{\\prime}\\right)  <2\\pi\/n$ for some $t_{1}^{\\prime}%\n\\in\\left(  t_{1},2\\pi\/n\\right)  $ in order for $\\arg\\alpha_{\\beta}^{\\prime\n}\\left(  2\\pi\/n\\right)  ^{-}=2\\pi\/n$ in equation (\\ref{cuspslopes}) to hold.\nHowever $\\arg\\alpha_{\\beta}^{\\prime}\\left(  \\left(  0,2\\pi\/n\\right)  \\right)\n\\subset\\left(  2\\pi\/n,\\pi\\right)  $ so no such intersection can occur, and we\nconclude that $\\alpha_{\\beta}$ restricted to $\\left(  0,2\\pi\/n\\right)  $ lies\nwithin the region $S.$ Moreover, because $\\arg\\alpha_{\\beta}^{\\prime}\\left(\nt\\right)  $ decreases strictly, with total change $\\pi-2\\pi\/n<\\pi,$\n$\\alpha_{\\beta}$ must be one to one on $\\left(  0,2\\pi\/n\\right)  .$ Now let\n$k=1,2,..,n,$ and consider the curve $\\alpha_{\\beta}$ restricted to $\\left(\n2k\\pi\/n,2\\left(  k+1\\right)  \\pi\/n\\right)  .$ By rotational symmetry of\n$f_{\\beta}$, $\\alpha_{\\beta}\\left(  t+2k\\pi\/n\\right)  =e^{i2k\\pi\/n}%\n\\alpha_{\\beta}\\left(  t\\right)  $, $\\alpha_{\\beta}$ is also one to one on\n$\\left(  2k\\pi\/n,2\\left(  k+1\\right)  \\pi\/n\\right)  $, and the graph\n$\\alpha_{\\beta}\\left(  \\left(  2k\\pi\/n,2\\left(  k+1\\right)  \\pi\/n\\right)\n\\right)  \\subseteq e^{i2k\\pi\/n}S.$ We complete the proof for $\\beta\\in\\left(\n0,\\pi\/2\\right)  $ by showing that the sets $\\left\\{  e^{i2k\\pi\/n}%\nS:k=0,1,...,n-1\\right\\}  $ are pairwise disjoint. Because of rotational\nsymmetry, the line $e^{i2k\\pi\/n}L_{0}$ passes through $e^{i2k\\pi\/n}%\n\\alpha_{\\beta}\\left(  0\\right)  =\\alpha_{\\beta}\\left(  2k\\pi\/n\\right)  ,$ a\ncusp, and $e^{i2k\\pi\/n}L_{0}$ has argument $2k\\pi\/n,$ so\\linebreak%\n\\ $e^{i2k\\pi\/n}L_{0}=L_{k}.$ Similarly $e^{i2k\\pi\/n}L_{1}=L_{k+1}.$ Thus\n$e^{i2k\\pi\/n}S$ is a sector with sides $L_{k}$ and $L_{k+1},$ and vertex at\ntheir point of intersection $e^{i2k\\pi\/n}p.$ The points $\\left\\{  e^{i2k\\pi\n\/n}p:k=0,..,n-1\\right\\}  $ form the vertices of a regular $n$-gon, that we\ndenote by $P.$ Then $P$ is centered on the origin, and we have\n$\\operatorname{Im}p>0,$ since the cusp $\\alpha_{\\beta}\\left(  0\\right)  $ has\nargument $\\psi=\\arg\\alpha_{\\beta}\\left(  0\\right)  \\in\\left(  0,\\pi\/4\\right)\n$ by Lemma \\ref{convexconcave} (ii). Note also that $L_{n}=L_{0},$ and\n$L_{n-1}=e^{-i2\\pi\/n}L_{0}.$ Thus $L_{0}$ and $L_{n-1}$ intersect at\n$e^{-i2\\pi\/n}p,$ with $\\operatorname{Im}\\left(  e^{-i2\\pi\/n}p\\right)\n=\\operatorname{Im}\\left(  p\\right)  .$ Therefore $p$ is in the second\nquadrant, and $e^{-i2\\pi\/n}p$ is in the first quadrant. We conclude that one\nbounding side of the sector $S$ is $\\left\\{  z\\in L_{0}:\\arg z\\leq\\arg\np\\right\\}  $. By rotational symmetry, the second side of $S$ is\\linebreak%\n\\ $\\left\\{  z\\in L_{1}:\\arg z\\leq2\\pi\/n+\\arg p\\right\\}  $. Thus the rays\n$\\left\\{  z\\in L_{k}:\\arg z\\leq2k\\pi\/n+\\arg p\\right\\}  $ that extend the sides\nof $P,$ each originating at $e^{i2k\\pi\/n}p$ and extending in the direction of\ndecreasing argument, form the boundaries of the sectors $e^{i2k\\pi\/n}S,$ where\n$k=0,1,...,n-1$ (see Figure 7 (left) where $S$ is shaded).%\n\n{\\includegraphics[\nheight=2.2364in,\nwidth=2.4561in\n]%\n{sectorsUni.eps}%\n}\n\\ \\\n{\\includegraphics[\nheight=2.1724in,\nwidth=2.1136in\n]%\n{sectorsPrime.eps}%\n}\n\\ \\ \\newline\\textsc{Figure 7.} {\\small The sector }$S${\\small \\ and regular\n}$n${\\small -gon }$P${\\small \\ (left). The set }$S^{\\prime}$ {\\small lies on\nthe non-zero side of the line }$L_{0},${\\small \\ and is bounded by }%\n$L_{0}^{\\prime}${\\small \\ and }$L_{1}^{\\prime}${\\small \\ (right).\\medskip}\n\nVarious approaches are possible to demonstrate that the sectors $e^{i2k\\pi\n\/n}S$ are disjoint. For instance, note that $\\arg p=\\pi\/2+\\pi\/n,$ and consider\nthe related sector\n\\[\nS^{\\prime}=\\left\\{  z\\in\\text{ext}\\left(  P\\right)  :\\arg z\\in\\left(\n\\pi\/2-\\pi\/n,\\pi\/2+\\pi\/n\\right)  \\right\\}  .\n\\]\nThe sets $e^{i2k\\pi\/n}S^{\\prime}$ are clearly disjoint for $k=0,1,...,n-1,$\nsince the arguments of points in the sets $e^{i2k\\pi\/n}S^{\\prime}$ for\ndistinct $k\\in\\left\\{  0,1,...,n-1\\right\\}  $ are in disjoint intervals (see\nthe right of Figure 7, where $S^{\\prime}$ is region that is shaded). Let\n$L_{k}^{\\prime}$ be the line through the origin and with argument\n$\\pi\/2+\\left(  2k-1\\right)  \\pi\/n,$ and let $E$ be the set $E=\\left\\{  z\\in\next\\left(  P\\right)  :\\arg z\\leq\\pi\/2-\\pi\/n,\\text{ }\\operatorname{Im}%\nz>\\operatorname{Im}p\\right\\}  ,$ bounded by $L_{0}$ and $L_{0}^{\\prime}$ and\n$P.$ We obtain $S$ from $S^{\\prime}$ by including the set $E,$ and excluding\nthe set $e^{i2k\\pi\/n}E$ (darker shaded region in the right of Figure 7) from\n$S^{\\prime}.$ Then $S=S^{\\prime}\\cup E\\backslash e^{i2\\pi\/n}E,$ and by\nrotational symmetry,\n\\[\ne^{i2k\\pi\/n}S=e^{i2k\\pi\/n}S^{\\prime}\\cup e^{i2k\\pi\/n}E\\backslash e^{i2\\left(\nk+1\\right)  \\pi\/n}E.\n\\]\nThus to obtain $e^{i2k\\pi\/n}S$ we have simply removed a sector with vertex\n$e^{i2\\left(  k-1\\right)  \\pi\/n},$ namely $e^{i2k\\pi\/n}E,$ from the set\n$e^{i2k\\pi\/n}S^{\\prime}$ and included its rotation $e^{i2\\left(  k+1\\right)\n\\pi\/n}E$ to obtain $e^{i2\\left(  k+1\\right)  \\pi\/n}S^{\\prime}.$ Because the\nsets $e^{i2k\\pi\/n}S^{\\prime}$ are disjoint, so are the sets $\\left\\{\ne^{i2k\\pi\/n}S:k=0,1,..,n-1\\right\\}  .$\n\nIf $\\beta=0$ then $p=0$ and the argument above does not apply, but a simple\nproof follows if we adapt the proof for $\\beta\\in\\left(  0,\\pi\/2\\right)  $ and\ndefine the sector $S$ to be $S=\\left\\{  z\\in%\n\\mathbb{C}\n:\\arg z\\in\\left(  0,2\\pi\/n\\right)  \\right\\}  ,$ in which case the sectors\n$\\left\\{  e^{2k\\pi i\/n}S:k=0,1,...,n-1\\right\\}  $ are clearly disjoint$.$\n\nIf $\\beta=\\pi\/2,$ then we adapt the proof so that $L_{k}$ is the line passing\nthrough node $\\alpha_{\\pi\/2}\\left(  2k\\pi\/n\\right)  $ with argument\n$\\alpha_{\\pi\/2}^{\\prime}\\left(  2k\\pi\/n\\right)  ^{+}$. The sectors then are\nbounded by the lines $L_{k},$ and the same same argument applies to show\n$\\tilde{\\alpha}_{\\pi\/2}$ is one to one on $S$ (note that $\\arg\\tilde{\\alpha\n}_{\\pi\/2}\\left(  \\left(  0,2\\pi\/n\\right)  \\right)  \\subset\\left(  \\pi\n\/2+\\pi\/\\left(  2n\\right)  ,\\pi\\right)  $ rather than $\\arg\\alpha_{\\beta\n}^{\\prime}\\left(  \\left(  0,2\\pi\/n\\right)  \\right)  \\subset\\left(  2\\pi\n\/n,\\pi\\right)  $).\n\\end{proof}\n\n\\begin{theorem}\n\\label{univalent}For any $n\\geq3$ in $%\n\\mathbb{N}\n,$ the harmonic functions $f_{\\beta}\\left(  z\\right)  $, $\\beta\\in%\n\\mathbb{R}\n$ defined in Definition \\ref{defnharmonic} are univalent, and so they are\nharmonic mappings.\n\\end{theorem}\n\n\\begin{proof}\nWe apply the argument principle for harmonic functions of\n\\cite{DurenHenLaugesen} to obtain univalence of $f_{\\beta}.$ We proved in\nLemma \\ref{unibdry} that for $\\beta\\in\\lbrack0,\\pi\/2),$ the boundary curve\n$\\alpha_{\\beta}\\left(  t\\right)  =f_{\\beta}\\left(  e^{it}\\right)  $ on\n$\\partial U$ is a simple closed curve about the origin$.$ Although\n$\\alpha_{\\pi\/2}$ has arcs of constancy, the winding number about each point\nenclosed by the curve $\\alpha_{\\beta}$ is still $1,$ so the argument principle\napplies for $\\beta=\\pi\/2$. Choose an arbitrary point $w_{0}$ enclosed by the\ncurve $\\alpha_{\\beta}.$ Then define $f=f_{\\beta}-w_{0},$ a harmonic function\ncontinuous in $\\bar{U}$, which does not have a zero on $\\partial U$. Moreover,\nsince $\\left\\vert \\omega_{f}\\right\\vert =\\left\\vert \\omega\\right\\vert <1$ in\n$U$, $f$ does not have any singular zeros in $U$. We see that $f\\left(\ne^{it}\\right)  =\\alpha_{\\beta}\\left(  t\\right)  -w_{0}$ has index 1 about the\norigin for $t\\in\\partial U$, so it follows from the argument principle that\n$f$ has exactly one zero in $U$. Thus $f_{\\beta}\\left(  z_{0}\\right)  =w_{0}$\nfor a unique $z_{0}\\in U.$ Since the choice of $z_{0}\\in U$ was arbitrary, we\nsee that $f_{\\beta}$ is onto the region enclosed by $\\alpha_{\\beta}.$ If\n$w_{1}$ is in the exterior of the region enclosed by the curve $\\alpha_{\\beta\n},$ then consider the function $\\tilde{f}=$ $f_{\\beta}-w_{1}.$ The harmonic\nfunction $\\tilde{f}$ satisfies the same hypotheses as did $f,$ but the winding\nnumber of $\\tilde{f}\\left(  e^{it}\\right)  $ about the origin is zero. Thus\nthere is no $z_{1}\\in U$ for which $f_{\\beta}\\left(  z_{1}\\right)  =w_{1}.$\nTherefore $f_{\\beta}\\left(  U\\right)  $ is contained in the interior of the\nregion bounded by $\\alpha_{\\beta}.$ If $\\beta\\in(-\\pi\/2,0),$ then $f_{\\beta}$\nis univalent since it is a reflection of $f_{-\\beta}$ where $-\\beta>0.$\nFinally if $\\beta\\not \\in (-\\pi\/2,\\pi\/2]$ then $f_{\\beta}$ $f_{\\beta}\\left(\nz\\right)  $ is a rotation of $f_{\\beta}\\left(  e^{-il\\pi\/n}z\\right)  $ for\nsome $l\\in%\n\\mathbb{Z}\n$ by Corollary \\ref{transit}, so is also univalent.\n\\end{proof}\n\n\\begin{remark}\nOn sectors $S$ of the closed unit disk for which $g_{n}\\left(  S\\right)  $ is\nconvex, we can show that $g_{n}$ is relatively more contractive than $h_{n},$\nin that for any two points $z_{0}$ and $z_{1}$ in the sector,\n\\[\n\\left\\vert h_{n}\\left(  z_{0}\\right)  -h_{n}\\left(  z_{1}\\right)  \\right\\vert\n\\geq\\left\\vert g_{n}\\left(  z_{0}\\right)  -g_{n}\\left(  z_{1}\\right)\n\\right\\vert\n\\]\nMoreover the inequality is strict when at least one point is not on $\\partial\nU.$ This fact can be used to show that $f_{\\beta}$ is one to one in the\nsectors $S$ of $\\bar{U}$ with argument in the range $[0,\\pi\/n)$, or with\nargument in the range $[\\pi\/n,2\\pi\/n).$ This leads to a direct proof of\nunivalence that does not rely on the argument principle.\n\\end{remark}\n\nWe now define a fundamental set, rotations of which make up the graph of the\nrosette harmonic mapping $f_{\\beta}$.\\ This set has an interesting\ndecomposition into two curvilinear triangles when $\\beta=0,$ or into a\ncurvilinear triangle and curvilinear bigon for $\\left\\vert \\beta\\right\\vert\n\\in(0,\\pi\/2]$. Moreover, the triangle is nowhere convex for $\\left\\vert\n\\beta\\right\\vert \\in(0,\\pi\/2],$ and when $\\beta=\\pi\/2$ the bigon has\nreflectional symmetry.\n\n\\setcounter{figure}{7}\n\n\\begin{definition}\n\\label{fundamental}For an interval $I$ with $\\left\\vert I\\right\\vert <2\\pi,$\ndefine the sector $S_{I}$ of the closed unit disk $\\bar{U}$ to be\n$S_{I}=\\left\\{  z\\in\\bar{U}:\\arg z\\in I,\\text{ }0\\leq r\\leq1\\right\\}  $. For\n$n\\in%\n\\mathbb{N}\n,$ $n\\geq3,$ $\\beta\\in(-\\pi\/2,\\pi\/2],$ let $f_{\\beta}$ be the rosette mapping\ndefined in Definition \\ref{defnharmonic}, and define the\\textbf{\\ fundamental\nset of the rosette mapping }$f_{\\beta}$ to be\n\\[\n\\mathcal{A}_{\\beta,n}=f_{\\beta}\\left(  S_{[0,2\\pi\/n)}\\right)  =\\left\\{\nf_{\\beta}\\left(  z\\right)  :0\\leq\\left\\vert z\\right\\vert \\leq1,\\text{ }%\n0\\leq\\arg z<2\\pi\/n\\right\\}  .\n\\]%\n\\begin{figure}[h]%\n\\centering\n\\includegraphics[\nheight=2.1793in,\nwidth=4.3517in\n]%\n{fund.eps}%\n\\caption{Fundamental sets $\\mathcal{A}_{\\pi\/5,5}$ (left) and $\\mathcal{A}%\n_{\\pi\/2,5}$ (right)}%\n\\label{fund}%\n\\end{figure}\n\n\\end{definition}\n\nThe univalence of $f_{\\beta}$ implies that the set $f_{\\beta}\\left(\nS_{[a,b)}\\right)  $ is bounded by the image of $f_{\\beta}\\left(  \\partial\nS_{[a,b)}\\right)  ,$ for $\\left\\vert b-a\\right\\vert <2\\pi.$ Thus the\nfundamental set $\\mathcal{A}_{\\beta,n}$ is bounded by $f_{\\beta}\\left(\n\\partial S_{[0,2\\pi\/n)}\\right)  .$ Definition \\ref{fundamental} and the\nfollowing proposition apply only to $\\beta\\in(-\\pi\/2,\\pi\/2],$ but we will use\nthe fundamental sets $\\mathcal{A}_{\\beta,n}$ with $\\beta$ from this restricted\nrange to express the graph of the rosette mapping $f_{\\tilde{\\beta}}\\left(\n\\bar{U}\\right)  ,$ for any $\\tilde{\\beta}\\in%\n\\mathbb{R}\n.$\n\nFor $\\beta\\in(-\\pi\/2,\\pi\/2]$ we note a further interesting decomposition in\nProposition \\ref{fundunion} of the set $\\mathcal{A}_{\\beta,n}$ for $\\beta\n\\neq0$ into a curvilinear triangle with sides that are nowhere convex, and a\ncurvilinear bigon, the latter having reflectional symmetry when $\\beta=\\pi\/2$\n(see Figure \\ref{fund}).\n\n\\begin{proposition}\n\\label{fundunion} Let $n\\in%\n\\mathbb{N}\n,$ $n\\geq3$ and $\\beta\\in(-\\pi\/2,\\pi\/2].$ Let $\\mathcal{A}_{\\beta,n}$ be the\nfundamental set defined in \\ref{fundamental} for the rosette mapping\n$f_{\\beta}$ of Definition \\ref{defnhg}. The set $\\mathcal{A}_{\\beta,n}$ is a\ncurvilinear triangle subtending an angle of $2\\pi\/n$ at the origin. Moreover,\n$\\mathcal{A}_{\\beta,n}$ is the disjoint union%\n\\[\n\\mathcal{A}_{\\beta,n}=f_{\\beta}\\left(  S_{[0,\\pi\/n)}\\right)  \\cup f_{\\beta\n}\\left(  S_{[\\pi\/n,2\\pi\/n)}\\right)  .\n\\]\n(i) For $\\beta\\in\\left(  0,\\pi\/2\\right)  $, the set $f_{\\beta}\\left(  \\partial\nS_{[0,\\pi\/n)}\\right)  $ is a curvilinear triangle with sides that are nowhere\nconvex$.$ The set $f_{\\beta}\\left(  \\partial S_{[\\pi\/n,2\\pi\/n)}\\right)  $ is a\ncurvilinear bigon, where one side is strictly convex, and the other side has a\nsingle inflection point. The angle subtended at the origin, both by the bigon\nand the triangle, is $\\pi\/n.$ The remaining angles in the bigon and triangle\nare $0.$ \\newline(ii)\\ If $\\beta=\\pi\/2$ then the statements in (i) hold,\nexcept the angle subtended by the bigon at the node of $f_{\\pi\/2}$ is\n$\\pi\/2-\\pi\/n.$ Additionally, the bigon $f_{\\pi\/2}\\left(  \\partial\nS_{[\\pi\/n,2\\pi\/n)}\\right)  $ is symmetric about the line through the vertices\nof the bigon$. $\\newline(iii) When $\\beta\\in\\left(  -\\pi\/2,0\\right)  $, the\nconclusions of (i) except $f_{\\beta}\\left(  \\partial S_{[\\pi\/n,2\\pi\n\/n)}\\right)  $ is the curvilinear triangle and $f_{\\beta}\\left(  \\partial\nS_{[0,\\pi\/n)}\\right)  $ is the curvilinear bigon.\\newline(iv)\\ When $\\beta=0,$\nthe sides of $f_{\\beta}\\left(  \\partial S_{[0,\\pi\/n)}\\right)  $ and $f_{\\beta\n}\\left(  \\partial S_{[\\pi\/n,2\\pi\/n)}\\right)  $ incident with the origin are\nline segments, and the curvilinear triangles $f_{\\beta}\\left(  \\partial\nS_{[0,\\pi\/n)}\\right)  $ and $f_{\\beta}\\left(  \\partial S_{[\\pi\/n,2\\pi\n\/n)}\\right)  $ are reflections of one another in the line containing their\ncommon side.\n\\end{proposition}\n\n\\begin{proof}\nWe let $\\alpha_{\\beta}\\left(  t\\right)  =f_{\\beta}\\left(  e^{it}\\right)  $ on\n$\\partial U.$ The proofs that follow combine facts about the image of\n$f_{\\beta}$ along radial lines of $U$ in Lemma \\ref{convexconcave}, and limits\nof $\\arg\\alpha_{\\beta}^{\\prime}$ at cusps and nodes. The image $\\mathcal{A}%\n_{\\beta,n}=f_{\\beta}\\left(  S_{[0,2\\pi\/n)}\\right)  $ is bounded by $f_{\\beta\n}\\left(  \\partial S_{[0,2\\pi\/n)}\\right)  .$ Since $f_{\\beta}\\left(  1\\right)\n$ and $f_{\\beta}\\left(  e^{i2\\pi\/n}\\right)  $ are distinct, $f_{\\beta}\\left(\n\\partial S_{[0,2\\pi\/n)}\\right)  $ is a curvilinear triangle (even when\n$\\beta=\\pi\/2$). At the origin, Lemma \\ref{convexconcave} (ii) states that\n$f_{\\beta}\\left(  r\\right)  $ and $f_{\\beta}\\left(  re^{i\\pi\/n}\\right)  $ have\ntangents with arguments $\\beta\/2$ and $\\beta\/2+\\pi\/n.$ The side $f_{\\beta\n}\\left(  re^{i2\\pi\/n}\\right)  $ is a rotation by $2\\pi\/n$ of $f_{\\beta}\\left(\nr\\right)  ,$ and so the argument of its tangent at $0$ is $\\beta\/2+2\\pi\/n.$\nThus the angle subtended by the vertex of $\\mathcal{A}_{\\beta,n}$ at the\norigin is $2\\pi\/n$. We now prove (i). Our observations about $\\frac{\\partial\n}{\\partial r}\\arg f_{\\beta}\\left(  r\\right)  ,$ $\\frac{\\partial}{\\partial\nr}\\arg f_{\\beta}\\left(  re^{i\\pi\/n}\\right)  ,$ and $\\frac{\\partial}{\\partial\nr}\\arg f_{\\beta}\\left(  re^{i2\\pi\/n}\\right)  $ at the origin show that the\nangle subtended by $f_{\\beta}\\left(  \\partial S_{[0,\\pi\/n)}\\right)  $ and by\n$f_{\\beta}\\left(  \\partial S_{[\\pi\/n,2\\pi\/n)}\\right)  $ at the origin is\n$\\pi\/n.$ At the boundary, we have by equation (\\ref{nodedir}) with $k=1$ in\nCorollary \\ref{cuspdir} that $\\arg\\alpha_{\\beta}^{\\prime}\\left(  t\\right)\n\\rightarrow\\pi\/2+\\pi\/n$ as $t\\rightarrow\\pi\/n^{+}.$ This matches\n$\\frac{\\partial}{\\partial r}\\arg f_{\\beta}\\left(  re^{i\\pi\/n}\\right)  $ at\n$f_{\\beta}\\left(  e^{i\\pi\/n}\\right)  =\\alpha_{\\beta}\\left(  \\pi\/n\\right)  $,\nby Lemma \\ref{convexconcave} (ii)$.$ Thus $f_{\\beta}\\left(  re^{i\\pi\n\/n}\\right)  $ and $\\alpha_{\\beta}\\left(  t\\right)  $ join to form a single\nsmooth curve, together forming one side of $f_{\\beta}\\left(  \\partial\nS_{[0,2\\pi\/n)}\\right)  ,$ whence the bigon. From Lemma \\ref{convexconcave}\n(ii), $\\frac{\\partial}{\\partial r}\\arg f_{\\beta}\\left(  re^{i\\pi\/n}\\right)  $\nis strictly increasing, yet $\\arg\\alpha_{\\beta}^{\\prime}\\left(  e^{it}\\right)\n$ is strictly decreasing by (\\ref{compass}). Thus $f_{\\beta}\\left(  e^{i\\pi\n\/n}\\right)  $ is an inflection point where curvature changes sign on the\nbigon. We also have $\\frac{\\partial}{\\partial r}\\arg f_{\\beta}\\left(\nr\\right)  $ is strictly decreasing, while $\\frac{\\partial}{\\partial r}\\arg\nf_{\\beta}\\left(  re^{i\\pi\/n}\\right)  $ is strictly increasing. Thus the sides\nof the triangle $f_{\\beta}\\left(  \\partial S_{[0,\\pi\/n)}\\right)  $ are nowhere\nconvex. Returning to the bigon, we now see that its second side $f_{\\beta\n}\\left(  re^{i2\\pi\/n}\\right)  $ is strictly convex, since $f_{\\beta}\\left(\nre^{i2\\pi\/n}\\right)  $ is a rotation about the origin of $f_{\\beta}\\left(\nr\\right)  $. We complete the proof of (i) by observing that when $\\beta\n\\in\\lbrack0,\\pi\/2),$ the vertices $f_{\\beta}\\left(  e^{i2\\pi\/n}\\right)  $ and\n$f_{\\beta}\\left(  1\\right)  $ are cusps of $\\alpha_{\\beta}, $ and the angle\nsubtended at $f_{\\beta}\\left(  e^{i2\\pi\/n}\\right)  $ and $f_{\\beta}\\left(\n1\\right)  $ is $0.$\n\nNow we turn to (ii), with $\\beta=\\pi\/2.$ We observed reflectional symmetry in\nTheorem \\ref{symmetry} (iii) of $f_{\\pi\/2}\\left(  re^{-i\\pi\/n}\\right)  $ and\n$f_{\\pi\/2}\\left(  r\\right)  $ about the axis through $0$ and the node\n$f_{\\pi\/2}\\left(  1\\right)  .$ Rotating by $2\\pi\/n$, the ray through $0$ and\nthe node $f_{\\pi\/2}\\left(  e^{i2\\pi\/n}\\right)  $ is also an axis of\nreflectional symmetry, in which the sides $f\\left(  re^{i\\pi\/n}\\right)  $ and\n$f\\left(  re^{i2\\pi\/n}\\right)  $ of the bigon are reflected. To see the\nsubtended angle at the boundary $\\alpha_{\\pi\/2}$, we use Lemma\n\\ref{convexconcave} (ii) as above to see the tangent of $f_{\\pi\/2}\\left(\nre^{i\\pi\/n}\\right)  $ approaches $\\pi\/2+\\pi\/n$ as $r\\rightarrow1^{-}.$ By the\nformula in the proof of Corollary \\ref{nodes} with $k=1$, $\\arg\\tilde{\\alpha\n}_{\\pi\/2}^{\\prime}\\left(  t\\right)  $ also approaches $\\pi\/2+\\pi\/n$ as\n$t\\rightarrow2\\pi\/n^{-}.$ We conclude that the side $f_{\\pi\/2}\\left(\nre^{i\\pi\/n}\\right)  $ of the bigon becomes tangent to the boundary\n$\\tilde{\\alpha}_{\\pi\/2}$ at the node $f_{\\pi\/2}\\left(  e^{i\\pi\/n}\\right)\n=f_{\\pi\/2}\\left(  e^{i2\\pi\/n}\\right)  $ as $t\\rightarrow2\\pi\/n^{-}.$ By\nreflectional symmetry, the second side of the bigon $f_{\\pi\/2}\\left(\nre^{i2\\pi\/n}\\right)  $ becomes tangent to the boundary $\\tilde{\\alpha}_{\\pi\n\/2}$ at as $t\\rightarrow2\\pi\/n^{+}.$ Thus the angle subtended in the bigon at\n$f_{\\pi\/2}\\left(  e^{i2\\pi\/n}\\right)  $ is the same as the interior angle of\nthe nodes of $\\tilde{\\alpha}_{\\pi\/2},$ namely $\\pi\/2-\\pi\/n$ by Corollary\n\\ref{nodes}$.$\n\nThe statements in (iii) follow readily using reflectional symmetry. For\n(iv)\\ we already noted in Lemma \\ref{convexconcave} (iii) that the images of\n$f_{0}\\left(  r\\right)  ,$ $f_{0}\\left(  re^{i\\pi\/n}\\right)  $ and\n$f_{0}\\left(  re^{i2\\pi\/n}\\right)  $ are linear. Reflectional symmetry was\nestablished in Corollary \\ref{noreflect}.\n\\end{proof}\n\nWe finish by decomposing the graph of an arbitrary rosette mapping$\\ $into a\ndisjoint union of $n$ rotations of a fundamental set of Definition\n\\ref{fundamental}.\n\n\\begin{corollary}\nLet $n\\geq3$ and $\\tilde{\\beta}\\in%\n\\mathbb{R}\n.$ The set $f_{\\tilde{\\beta}}\\left(  \\bar{U}\\right)  $ can be written as a\ndisjoint union of rotations of $\\mathcal{A}_{\\beta,n}.$ Specifically if\n$\\tilde{\\beta}=\\beta+l\\pi$ for $\\beta\\in(-\\pi\/2,\\pi\/2]$ then\n\\[\nf_{\\tilde{\\beta}}\\left(  \\bar{U}\\right)  =i^{l}%\n{\\displaystyle\\bigcup_{k=1}^{n}}\ne^{i\\left(  2k+l\\right)  \\pi\/n}\\mathcal{A}_{\\beta,n}.\n\\]\n\n\\end{corollary}\n\n\\begin{proof}\nBy Corollary \\ref{transit} we have $f_{\\tilde{\\beta}}\\left(  z\\right)\n=e^{il\\left(  \\pi\/n+\\pi\/2\\right)  }f_{\\beta}\\left(  e^{-il\\pi\/n}z\\right)  .$\nThus\n\\[\nf_{\\tilde{\\beta}}\\left(  \\bar{U}\\right)  =f_{\\tilde{\\beta}}\\left(  e^{il}%\n\\bar{U}\\right)  =e^{il\\left(  \\pi\/n+\\pi\/2\\right)  }f_{\\beta}\\left(  \\bar\n{U}\\right)  =i^{l}e^{il\\pi\/n}%\n{\\displaystyle\\bigcup_{k=1}^{n}}\ne^{i2k\\pi\/n}\\mathcal{A}_{\\beta,n}.\n\\]\n\n\\end{proof}\n\n\\bibliographystyle{amsalpha}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\partial{\\partial}\n\\def\\partial_i{\\partial_i}\n\\def\\frac{1}{2}{\\frac{1}{2}}\n\\defA^{+}_0{A^{+}_0}\n\\def\\psi_+{\\psi_+}\n\\def\\psi_-{\\psi_-}\n\\def\\psi_2{\\psi_2}\n\\def\\psi_1{\\psi_1}\n\\def\\psi^{\\dagger}_+{\\psi^{\\dagger}_+}\n\\def\\psi^{\\dagger}_-{\\psi^{\\dagger}_-}\n\\def\\overline{\\psi}{\\overline{\\psi}}\n\\def\\psi^{\\dag}{\\psi^{\\dag}}\n\\def\\chi^{\\dag}{\\chi^{\\dag}}\n\\def\\sla#1{#1\\!\\!\\!\/}\n\\defx^{+}{x^{+}}\n\\defx^{-}{x^{-}}\n\\defy^{-}{y^{-}}\n\\defx_\\perp{x_\\perp}\n\\defy_\\perp{y_\\perp}\n\\def\\underline{x}{\\underline{x}}\n\\def\\underline{y}{\\underline{y}}\n\\def\\underline{p}{\\underline{p}}\n\\def\\underline{n}{\\underline{n}}\n\\newcommand{\\newcommand}{\\newcommand}\n\\defp_{\\ulin}{p_{\\underline{n}}}\n\\newcommand{\\intl}{\\int\\limits_{-L}^{+L}\\!dx^-}\n\\newcommand{\\intv}{\\int\\limits_{V}^{}\\!d^3\\underline{x}}\n\\newcommand{\\intvy}{\\int\\limits_{V}^{}\\!d^3\\underline{y}}\n\\newcommand{\\intly}{\\int\\limits_{-L}^{+L}\\!{{dy^-}\\over\\!2}}\n\\newcommand{\\zmint}{\\int\\limits_{-L}^{+L}\\!{{dx^-}\\over{\\!2L}}}\n\\newcommand{\\intp}{\\int\\limits_{0}^{+\\infty}\\!{{dp^+}\\over\\!4\\pi}}\n\\def\\begin{equation}{\\begin{equation}}\n\\def\\end{equation}{\\end{equation}}\n\\def\\begin{eqnarray}{\\begin{eqnarray}}\n\\def\\end{eqnarray}{\\end{eqnarray}}\n\\begin{document}\n\\title{Higgs mechanism in a light front formulation}\n\\medskip\n\\author{ {\\sl L$\\!\\!$'ubom\\'{\\i}r Martinovi\\v c}\n\\footnote{E-mail: fyziluma@savba.sk}\\\\\nInstitute of Physics, Slovak Academy of Sciences\\\\  \nD\\'ubravsk\\'a cesta 9, 845 11 Bratislava, Slovakia\\\\\n\\phantom{.}\\\\\nand\\\\\n\\phantom{.}\\\\\n{\\sl Pierre Grang\\'e}\n\\footnote{E-mail: Pierre.GRANGE@LPTA.univ-montp2.fr}\\\\\nLaboratoire de Physique Th\\'eorique et Astroparticules\\\\ \nUniversit\\'e Montpellier II, Pl. E. Bataillon\\\\ \nMontpellier, F-34095 France}\n\\medskip\n\\maketitle\n\n\\begin{abstract}\nWe give a simple derivation of the Higgs phenomenon  \nin an abelian light front gauge theory. It is based on a finite-volume \nquantization with antiperiodic scalar fields and \nperiodic gauge field. An infinite set of degenerate vacua \nin the form of coherent states of the scalar field, that minimize  \nthe light front energy, is constructed. The corresponding effective \nHamiltonian describes a massive vector field whose third component is \ngenerated by the would-be Goldstone boson. This mechanism, \nunderstood here quantum mechanically in the form analogous to the spacelike \nquantization, is derived without gauge fixing as well as in the unitary \nand the light-cone gauge. \n\\end{abstract}\n\n\n\\vspace{5mm}\nSpontaneous symmetry breaking (SSB) of global as well as gauge symmetries \nhas not been fully understood in the light-front (LF) field theory, which \nis the formulation (most often hamiltonian) of relativistic dynamics \nthat uses the LF \nvariables $x^\\mu= (x^{+}, x^{-},x^1,x^2), x^\\pm=x^0 \\pm x^3$ and is quantized \non a surface of constant light-front time $x^{+}$ \\cite{Dir,LKS,Rohr}. \nConsequently, one deals with the LF field variables which satisfy field \nequations with a different structure than the equations of the conventional \nfield theory which parametrizes the spacetime by  \n$x^\\mu = (t,x^1,x^2,x^3)$ and is quantized at $t=0$. \nThe main reason for difficulties in obtaining a clear picture of SSB \nin the LF theory is the positivity of the spectrum of the LF momentum operator \n$P^+$.  Quite generally, the interacting-theory vacuum state coincides with \nthe free Fock vacuum if Fourier modes carrying $p^+=0$ (LF zero modes - ZM) can \nbe neglected. This simplifying feature is very useful in perturbative and \nbound-state calculations. However, it complicates the understanding of other \nnon-perturbative properties because it seems to prohibit any vacuum structure \nin LF theories and hence also the well established SSB pattern. Alternative \nschemes of the physics of broken phase have been given in the LF literature \n\\cite{lfssb,Yam,RT}. They are typically based on the operator scalar ZM \nwhich is present for periodic boundary conditions (BC) and which satisfies \nan equation of a constraint. The role of this  \nvariable in the phase transition of the $\\lambda \\phi^4(1+1)$ model was \nanalyzed by means of the Haag expansion in \\cite{SGW}.   \n\nThe Higgs mechanism in the LF formalism was studied on the tree level in the \ncontinuum formulation \\cite{Prem}. It was assumed that a scalar field contains \na c-number piece which gave a justification for performing a usual shift \nin the Lagrangian leading to the generation of the mass term for the gauge \nfield. A support for the above assumption comes from the fact that the solution \nof the zero-mode constraint of the real scalar field in the DLCQ analysis \ncontains such a constant non-operator part \\cite{Dave, LMSSB}. \n\nIn the present work, we study the SSB of an abelian symmetry in the Higgs \nmodel. Our approach is based on the discrete light-cone quantization method \n(DLCQ) considered as a hamiltonian analytical framework with large but finite \nnumber of Fourier modes to approximate quantum field \ntheory with its infinite number of degrees of freedom. A (regularized) unitary \noperator that shifts the scalar field by a constant will be used to transform  \nthe Fock space. The motivation for this step is a natural physical \nrequirement to find ground states in the broken phase which would correspond \nto a lower LF energy than the usual Fock vacuum. This is suggested already \nby considering minima of the classical LF potential energy. A procedure, \nequivalent to transforming the states, is to work with a transformed \nHamiltonian and calculate its matrix elements between the usual Fock states. \nIn this way one naturally arrives at the effective type of the Hamiltonian \nthat incorporates the usual pattern of the Higgs mechanism. \n\n  \nThe Lagrangian density of the abelian Higgs model that we wish to \nanalyze has the form  \n\\begin{equation}\n{\\cal L}=-\\frac{1}{4}F_{\\mu\\nu}F^{\\mu\\nu} + \\frac{1}{2} (D_\\mu\\phi)^\\dagger D^\\mu \\phi + \n\\frac{1}{2} \\mu^2 \\phi^\\dagger\\phi - \\frac{\\lambda}{4} (\\phi^\\dagger \\phi)^2,  \n\\label{lagr}\n\\end{equation}\nwhere $F_{\\mu\\nu}=\\partial_\\mu A_\\nu - \\partial_\\nu A_\\mu,~~D^\\mu\\phi=\\partial^\\mu\\phi \n+ieA^\\mu\\phi, \\mu^2 > 0$. The Lagrangian is invariant under two groups of \ntransformations: the global rotations of the complex scalar field $\\phi(x) \n\\rightarrow \\exp{\\big(-i\\beta\\big)}\\phi(x)$ and the local gauge transformations \n\\begin{equation}\n\\phi(x) \\rightarrow \\exp{\\big(-i\\omega(x)\\big)}\\phi(x),~~A^\\mu (x) \\rightarrow  \nA^\\mu (x) + \\partial^\\mu \\omega(x)\/e.\n\\label{GT}\n\\end{equation}\nIn terms of the LF variables, the Lagrangian (\\ref{lagr}) is \n\n\\begin{eqnarray}\n&&\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!{\\cal L}_{lf} = \n\\frac{1}{2} \\big(\\partial_+ A^+ - \\partial_- A^-\\big)^2 \n+ \\big(\\partial_+ A^i + \\frac{1}{2} \\partial_i A^-\\big)\\big(2\\partial_- A^i +\\partial_i \nA^+\\big) -  \n\\nonumber \\\\ \n&&\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\! - \\frac{1}{2} \\big(\\partial_1 A_2 - \\partial_2 A_1)^2 +  \n \\partial_+ \\phi^\\dagger\\partial_-\\phi + \\partial_-\\phi^\\dagger \\partial_+\\phi - \n\\frac{1}{2} \\partial_i\\phi^\\dagger\\partial_i\\phi -  \n\\frac{ie}{2}\\phi^\\dagger\\stackrel {\\leftrightarrow} {\\partial_+}\\phi A^+ - \\nonumber \\\\ \n&&\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\! - \\frac{ie}{2}\\phi^\\dagger\\delrlm\n\\phi A^- - \\frac{ie}{2}\\phi^\\dagger\\delrli\\phi A^i +  \n\\frac{e^2}{2}\\big(A^+A^- - A^iA^i\\big)\\phi^\\dagger\\phi + \\frac{\\mu^2}{2}\n\\phi^\\dagger\\phi - \\frac{\\lambda}{4}\\big(\\phi^\\dagger\\phi\\big)^2.\n\\label{lflagr}\n\\end{eqnarray} \nWriting $\\phi=\\sigma+i\\pi$, the conserved current corresponding to \nthe global symmetry is $J^\\mu (x)=-i\\phi^\\dagger (x) \\stackrel {\\leftrightarrow} {\\partial^\\gm} \\phi(x)=\n2\\sigma (x)\\stackrel {\\leftrightarrow} {\\partial^\\gm} \\pi (x)$. \n \nWe will work in a finite volume $V=8L^3$ with space coordinates restricted to \n$-L \\le x^{-},x^1,x^2 \\le L$. Our notation is $x^\\mu=(x^{+},\\underline{x}), \\underline{x}=\n(x^{-},x^1,x^2),~ p.x=\\frac{1}{2} p^-x^+ + \\frac{1}{2} p^+x^- - p^1x^1-p^2x^2$. The gauge \nfield will be chosen periodic in all three directions, while the scalar field \nwill be antiperiodic: $A^\\mu(x^{+},x^{-}=-L, x,y)=A^\\mu(x^{+},x^{-}=L,x,y), \n\\sigma(x^{+},x^{-}=-L,x,y)=-\\sigma(x^{+},x^{-}=L, x,y)$, and similarly in the \nperpendicular directions $x_\\perp \\equiv(x^1,x^2)$ \\cite{remark}.\nThe boundary conditions imply the discrete values \nof the three momentum labeled by a (half)integer and also lead  to the \npresence of global and proper zero modes of the gauge field \\cite{Alex}. The \nproper ZM are constrained variables that can modify the LF Hamiltonian. For \nsmall coupling the corrections may be evaluated perturbatively \\cite{AD}. \nWe shall however neglect the gauge-field ZM in the present discussion \nbecause they are not crucial for the phenomenon under study. The \nfields below refer then to the sector of normal Fourier modes.\n\nThe LF Hamiltonian, obtained in the canonical way from the \nLagrangian (\\ref{lflagr}), reads\n\\begin{eqnarray}\n&&\\!\\!\\!\\!\\!P^-=\\intv \\Big\\{F_{12}^2 + \\Pi^2_{A^+}+2\\Pi_{A^+}\\partial_- A^--\\Pi_{A^i}\n\\partial_i A^--  2e\\sigma\\delrlm\\pi A^-  \n- 2e\\sigma\\delrli \\pi A^i -  \\nonumber \\\\ \n&&\\!\\!\\!\\!\\! - e^2 A^2\\big(\\sigma^2+\\pi^2\\big)\n+ (\\partial_i\\sigma)^2 + (\\partial_i \\pi)^2 \n- \\mu^2\\big(\\sigma^2+\\pi^2\\big) + \n\\frac{\\lambda}{2}\\big(\\sigma^2+\\pi^2\\big)^2 \\Big\\}. \n\\label{Ham}\n\\end{eqnarray}\nHere $A^2=A^+A^- - A^iA^i, i=1,2$ and the canonical momenta are \n\\begin{eqnarray}\n&&\\!\\!\\!\\!\\!\\!\\Pi_{A^+}=\\partial_+ A^+-\\partial_- A^-,~\\Pi_{A^i} = \n2\\partial_- A^i + \\partial_i A^+ \n\\nonumber \\\\\n&&\\!\\!\\!\\!\\!\\!\\Pi_{A^-}=0,~\\Pi_\\sigma=2\\partial_- \\sigma-e\\pi A^+,\\Pi_\\pi=2\\partial_-\\pi+\ne\\sigma A^+.\n\\label{moms}\n\\end{eqnarray}\n\nAt $x^{+}=0$, we assume the usual LF commutation rules\n\\begin{eqnarray}\n&&\\big[\\sigma(x^{+},\\underline{x}),\\sigma(x^{+},\\underline{y})\\big] = \n\\frac{-i}{8}\\epsilon(x^{-}-y^{-})\\delta^2(x_\\perp-y_\\perp), \\nonumber \\\\\n&&\\big[\\pi(x^{+},\\underline{x}),\\pi(x^{+},\\underline{y})\\big] = \n\\frac{-i}{8}\\epsilon(x^{-}-y^{-})\\delta^2(x_\\perp-y_\\perp), \\nonumber \\\\\n&&\\big[A^+(x^{+},\\underline{x}),\\Pi_{A^+}(x^{+},\\underline{y})\\big]=\n\\frac{i}{2}\\delta^3(\\underline{x}-\\underline{y}) \\nonumber\\\\\n&&\\big[A^i(x^{+},\\underline{x}),\\Pi_{A^j}(x^{+},\\underline{y})\\big]= \n\\frac{i}{2}\\delta^{ij}\\delta^3(\\underline{x}-\\underline{y}).\n\\label{cr}\n\\end{eqnarray}\nThe mode expansions of the scalar fields are   \n\\begin{eqnarray}\n\\sigma(0,\\underline{x})=\\frac{1}{\\sqrt{V}}\\sum\\limits_{\\underline{n}}^{}\\frac{1}{\\sqrt{p^+_n}}\n\\big[a(p_{\\underline{n}})e^{-i p_{\\underline{n}} .\\underline{x}} + a^\\dagger(p_{\\underline{n}})e^{i p_{\\underline{n}} .\\underline{x}} \\big], \n\\nonumber \\\\\n\\pi(0,\\underline{x})=\\frac{1}{\\sqrt{V}}\\sum\\limits_{\\underline{n}}^{}\\frac{1}{\\sqrt{p^+_n}}\n\\big[c(p_{\\underline{n}})e^{-i p_{\\underline{n}}. \\underline{x}} + c^\\dagger(p_{\\underline{n}})e^{i p_{\\underline{n}}.\\underline{x}} \\big],\n\\label{fexp}\n\\end{eqnarray}\nwhere $p_{\\underline{n}}=(p_n^+,p_{n_1},p_{n_2}), p_n^+=\\frac{2\\pi}\n{L}n, n=1\/2,3\/2,\\dots$, and similarly for the perpendicular components.     \nThe global rotations are implemented by the unitary operators $V(\\beta)$ in terms of the charge $Q=\\intv J^+(x)$:\n\\begin{eqnarray}\n\\sigma(x) \\rightarrow V(\\beta)\\sigma(x)V^\\dagger(\\beta) = \\sigma(x)\\cos \\beta-\\pi(x)\\sin \\beta, \\nonumber \\\\\n\\pi(x) \\rightarrow V(\\beta)\\pi(x)V^\\dagger(\\beta) = \\sigma(x)\\sin \\beta+\\pi(x)\\cos \\beta.\n\\label{ccr}\n\\end{eqnarray}\nHere\n\\begin{equation}\nV(\\beta)=e^{i\\beta Q}\n\\label{uniq}\n\\end{equation}\nwith \n\\begin{equation}\nV(\\beta) = \\exp\\big\\{\\sum\\limits_{\\underline{n}}^{\\Lambda}\\Big(a^\\dagger(p_{\\underline{n}})\nc(p_{\\underline{n}}) - c^\\dagger(p_{\\underline{n}})a(p_{\\underline{n}})\\Big)\\big\\}. \n\\label{glim}\n\\end{equation}\nThe Hamiltonian (\\ref{Ham}) is invariant under $x^{+}$-independent gauge \ntransformations. They are implemented by the unitary operator\n\\begin{equation}\nU[\\omega(\\underline{x})]=\\exp \\Big\\{ i \\intv\\big[2\\Pi_{A^+}\\partial_- - \\Pi_{A^i}\\partial_i + \neJ^+\\big]\\omega(\\underline{x})\\Big\\} \\label{loim}\n\\end{equation}\nIndeed, we easily find \n\\begin{eqnarray}\n&&U[\\omega(\\underline{x})]\\phi(x)U^\\dagger[\\omega(\\underline{x})]=\\exp{\\big(-i\\omega(\\underline{x})\\big)}\n\\phi(x), \\nonumber \\\\\n&&U[\\omega(\\underline{x})]A^\\mu(x)U^\\dagger[\\omega(\\underline{x})]=\n\t\t\tA^\\mu(x) + e^{-1}\\partial^\\mu\\omega(\\underline{x}).\n\\label{gtimpl}\n\\end{eqnarray}\nConsider now the unitary operators \n\\begin{eqnarray}\nU_\\sigma(b)=\\exp\\Big\\{-2ib\\intv \\Pi_\\sigma(x)\\Big\\} \\nonumber \\\\\nU_\\pi(b)=\\exp\\Big\\{-2ib\\intv \\Pi_\\pi(x)\\Big\\}.\n\\label{shifto}\n\\end{eqnarray}\nThey shift the corresponding scalar field by a constant. To follow the usual \ntreatment, we will perform only shifts in the $\\sigma$-direction:\n\\begin{eqnarray}\nU_\\sigma(b) \\sigma(x) U^{-1}_\\sigma(b) = \\sigma(x) - 2ib \\intvy \\big[\\Pi_\\sigma(y),\\sigma(x)\\big] \n\\nonumber \\\\\n= \\sigma(x) - b \\epsilon_\\Lambda(L-x^{-}) \\epsilon_\\Lambda(L-x^1) \\epsilon_\\Lambda(L-x^2).\n\\label{shift}\n\\end{eqnarray}\nThe subscript $\\Lambda$ attached to the sign function $\\epsilon(\\underline{x})$ indicates that their \nFourier series is truncated at $\\Lambda$:\n\\begin{eqnarray}\n\\epsilon_\\Lambda(x^{-}) = \\frac{4i}{L} \\sum_{n=\\frac{1}{2}}^{\\Lambda} \\frac{1}{p_n^+} \\Big(e^{-ip_n^+ x^{-}} \n- e^{ip^+_n x^{-}}\\Big) \n\\label{sign}\n\\end{eqnarray}\nand analogously for the perpendicular components. The point is that one has to \ntake a large but finite number of field modes in all three space directions in order \nto have a well-defined operator $U_\\sigma(b)$. In practice, for $\\Lambda \\approx 10^3$ the \nsign functions are equal to unity to a very good approximation everywhere on the finite \ninterval $-L < x^{-},x^1,x^2 < L$ except for a very small neighborhood of the \nend-points. Therefore we \nwill not write these sign functions explicitly henceforth. \n\nBy means of the shift operator $U_\\sigma(b)$, we can define a set of states \n$\\vert b \\rangle = U_\\sigma(b)\\vert 0 \\rangle$ ($\\vert 0 \\rangle$ is the Fock \nvacuum). Minimizing the expectation value of the energy density $V^{-1}\\langle \nb \\vert P^- \\vert b \\rangle$, we easily find that the minimum of the LF energy, \nequal to $-\\frac{\\mu^4}{2\\lambda}$ is achieved for $b=\\frac{\\mu}{\\sqrt{\\lambda}} \\equiv \nv$. It is lower than the usual (vanishing) value of the LF energy in the \n``trivial'' vacuum $\\vert 0 \\rangle$. From Eq.(\\ref{shift}) we also have the \nproperty that the vacuum expectation value of the $\\sigma$-field is non-zero \nwhich is the indication of broken symmetry: \n\\begin{equation}\n\\langle v \\vert \\sigma(x) \\vert v \\rangle = \\langle 0 \\vert U^{-1}_\\sigma(v) \n\\sigma(x)U_\\sigma(v) \\vert 0 \\rangle = v.\n\\label{ssb}\n\\end{equation} \nHere, the sign functions multiplying the value $v$ are understood as in \nEq.(\\ref{shift}). The accompanying vacuum degeneracy is easily obtained by \nrotating our ``trial'' vacuum (chosen in the $\\sigma$-direction):\n\\begin{equation}\nV(\\beta)\\vert v \\rangle = V(\\beta)U_\\sigma(v)\\vert 0 \\rangle \\equiv \\vert v;\\beta \n\\rangle.\n\\label{fulv}\n\\end{equation}\nThus we have an infinite set of vacuum states corresponding to the above \nminimum of the LF energy and labeled by the angle $\\beta$.\n\nThe next step in the Hamiltonian formalism is to construct the space of states. \nA natural possibility would be to apply a string of creation operators \nof all fields to the new vacuum, chosen to be $\\vert v;0 \\rangle$, and \ncalculate the corresponding matrix elements of $P^-$. A simpler option is  \nto build a usual set of Fock states from the Fock vacuum $\\vert 0 \\rangle$ \nand transform all of them by $U_\\sigma(v)$. This type of states is known as \ndisplaced number states in quantum optics \\cite{qo}. In either case one can \neasily see that instead of the original Hamiltonian (\\ref{Ham}) one actually \nworks with the new ``effective'' LF Hamiltonian\n\\begin{equation}\n\\tilde{P}^- = U^{-1}_\\sigma(v) P^- U_\\sigma(v)\n\\label{Hef}\n\\end{equation}\nin which the $\\sigma$-field is shifted by the value $v$. This of course leads to \nthe structure known from the lagrangian formalism in the conventional field \ntheory \\cite{IZ}: the mass term of the gauge field of the form $e^2v^2A^2$ is \ngenerated, \nthe pion field becomes massless and the $\\sigma$-field acquires mass equal to \n$\\sqrt{2}\\mu$. The change in the Hamiltonian density shows this explicitely: \n\\begin{eqnarray}\n&&\\delta P^- = -\\frac{\\mu^4}{2\\lambda} + 3\\mu^2 \\sigma^2 + \\mu^2 \\pi^2 - e^2v^2A^2  \n- 2e^2v\\sigma A^2 +   \n\\nonumber\\\\\n&& + 2\\sqrt{\\lambda}\\mu \\sigma \\Big(\\sigma^2 + \\pi^2 \\Big)\n- 2ev\\Big(\\partial_- \\pi A^- + \\partial_i\\pi A^i\n\\Big). \n\\end{eqnarray}\nThe latter non-diagonal term and the kinetic term $(\\partial_i \\pi)^2$ can be \nremoved by introducing the new field $B$:\n\\begin{equation}\nB^{+}(x)=A^{+}+\\frac{2}{ev}\\partial_-\\pi,~ \nB^i(x) = A^i(x) -\\frac{1}{ev}\\partial_i \\pi(x), \n\\label{subst}\n\\end{equation}\nwhile $B^- = A^-$. In this way, the $\\pi$ field disappeared from the quadratic \npart of the Hamiltonian but it is still present in the interacting part. One \nmay suspect that it is actually a redundant degree of freedom because \nthe gauge freedom has not been removed.\n  \nIn a full analogy with the space-like treatment, a clearer \nphysical picture is obtained in the unitary gauge. Introducing the radial  \nand angular field variables: \n\\begin{equation}\n\\phi(x)=\\rho(x)e^{i\\Theta(x)\/v},\n\\label{radial}\n\\end{equation}\nthe LF Hamiltonian will take the form\n\\begin{eqnarray}\n&&\\!\\!\\!\\!\\!P^-_r=\\intv \\Big\\{\\Pi^2_{A^+} + 2\\Pi_{A^+}\\partial_- A^- - \n\\Pi_{A^i}\\partial_i A^- \n+ F_{12}^2 +(\\partial_i \\rho)^2 + \\nonumber \\\\\n&&\\!\\!\\!\\!\\! + \\rho^2(\\partial_i \\Theta\/v)^2  - 2e\\rho^2\\partial_- \nA^-\\Theta\/v  - 2e\\rho^2A^i \\partial_i \\Theta \/v -e^2\\rho^2 A^2 -\\mu^2\\rho^2 + \n\\frac{\\lambda}{2}\\rho^4 \\Big\\}. \n\\label{hamr}\n\\end{eqnarray}\nTo fix the gauge at the classical Lagrangian level, one observes that the gauge \ntransformations simply shift the angular field variable $\\Theta(x)$ by the \ngauge function $\\omega(x)$. Choosing $\\omega(x)=-\\Theta(x)\/v$, one has \n\\begin{equation}\n\\! \\phi(x) \\rightarrow \\rho(x),~A^\\mu (x) \\rightarrow B^\\mu (x) = A^\\mu (x) - \n\\frac{1}{ev}\\partial^\\mu \\Theta (x)\n\\label{newa}\n\\end{equation}\nwith the corresponding Lagrangian\n\\begin{equation}\n{\\cal L}_u = -\\frac{1}{4}G_{\\mu \\nu}G^{\\mu \\nu} + \\frac{1}{2} |\\partial_\\gm \\rho \n-ieB_\\mu \\rho|^2 + \\frac{1}{2}\\mu^2 \\rho^2 \n-\\frac{\\lambda}{4}\\rho^4.\n\\label{ulag}\n\\end{equation} \nTaking this gauge fixing over to the quantum theory, defined by the \ncommutation relation at $x^{+}=0$\n\\begin{equation}\n\\Big[\\rho(x^{+},\\underline{x}),\\rho(x^{+},\\underline{y})\\Big] = -\\frac{i}{8}\n\\epsilon(x^{-}-y^{-})\\delta^2(x_\\perp-y_\\perp),\n\\label{crr}\n\\end{equation}\nwe find the quantum LF Hamiltonian $P^-_u$ in the unitary gauge. It coincides \nwith the Hamiltonian (\\ref{hamr}) except for the missing $\\Theta$-terms and \nthe $B^\\mu$ replacing the $A^\\mu$ field. The equal-LF time algebra (\\ref{crr}) \nenables us to introduce the shift operator ($\\Pi_\\rho = 2\\partial_- \\rho)$\n\\begin{equation}\nU_\\rho(v)=\\exp\\Big\\{-2i v\\intv \\Pi_\\rho(x) \\Big\\}\n\\label{shifr}\n\\end{equation}\nwhich defines the ``effective'' LF Hamiltonian \n$\\tilde{P}^-_u = U^{-1}_\\rho (v) P^-_u U_\\rho(v)$ \ncorresponding to the unitary gauge:\n\\begin{eqnarray}\n&&\\!\\!\\!\\!\\!\\!\\tilde{P}^-_u = \\intv \\Big\\{\\Pi_{B^+}^2 + 2\\Pi_{B^+}\\partial_- B^- \n- \\Pi_{B^i}\\partial_i B^- + G_{12}^2 + ~~~~  \n\\nonumber \\\\ \n&&\\!\\!\\!\\!\\!\\!+ (\\partial_i\\rho)^2   \n-e^2(\\rho + v)^2B^2 - \\mu^2(\\rho + v)^2 + \\frac{\\lambda}{2}(\\rho + v)^4\\Big\\}. \n\\label{Hunit}\n\\end{eqnarray}\nFrom its form it is easy to find that it describes one massive scalar field \n$\\rho$ and a vector field with the mass $e^2v^2$. The massive vector field \nemerged as a combination of the the massless gauge field $A^\\mu$ and the \nscalar $\\Theta$ field. \n\nAnother possibility is to analyze the symmetry breaking in the light-cone \ngauge. This means that we set $A^+ = 0$ in the normal-mode sector. The \nstarting Hamiltonian and conjugate momenta are then the expressions \n(\\ref{Ham}),(\\ref{moms}) without the terms containing $A^+$. One proceeds \nas in the case without the gauge fixing, namely defines the \nshift operator $U_{\\sigma}(v)$ and constructs the infinite set of degenerate \n(approximative) vacuum states by applying the unitary operator $V(\\beta)$ \n(Eq.(\\ref{glim})) to the coherent-state vacuum $\\vert v \\rangle$. The \ncorresponding effective LF Hamiltonian is obtained by the transformation \n(\\ref{Hef}). One observes an important difference as compared \nwith the unitary-gauge treatment. It is related to the fact that the choice  \n$A^+ = 0$ eliminates the $A^+ A^-$ part of the vector field mass term \ngenerated by shifting the $\\sigma$ field in the $-e^2 A^2 (\\sigma^2 + \\pi^2)$ term \nin the Hamiltonian (\\ref{Ham}). Thus the massive vector field seems to have \nonly two components and this is not correct. The resolution of this difficulty \ncomes from the observation \\cite{Prem} that in the light-cone gauge the \nGauss' law becomes a constrained equation for the $A^-$ component of the \ngauge field: \n\\begin{equation}\n\\partial_-^2 A^-(x) + \\partial_- \\partial_i A^i(x) = e\\sigma(x) \\delrli \\pi(x). \n\\label{amcon}\n\\end{equation}\nThe shift of the $\\sigma$ field by means of the operator $U_\\sigma(v)$ generates \nan additional term of the form $ev\\partial_- \\pi(x)$ on the righ-hand side of this \nequation. Upon inserting the shifted constraint to the Hamiltonian, the latter \npiece leads to the new term $e^2v^2 \\pi^2$ ($i=1,2$): \n\\begin{equation}\n\\tilde{P}^-_{lc}= \\intv \\Big[F_{12}^2 + (\\partial_i A^i)^2 + (\\partial_i \\pi)^2 \n+ e^2v^2 \\big(\\pi^2 + A_i^2\\big) + \\dots \\Big].\n\\label{frep}\n\\end{equation}\nTo see that this Hamiltonian corresponds to a free massive vector meson field, \nit is useful to consider the gauge-invariant form of the scalar electrodynamics with a massive vector field \\cite{Sop}. It differs from the Lagrangian \nof the massless scalar QED by the term $\\frac{1}{2}(mA^\\mu-\\partial^\\mu B)^2$ which \nmakes the vector field massive. $B$ is a scalar field and $m$ a mass parameter. The usual formulation with the condition $\\partial_\\mu A^\\mu = 0$ corresponds \nto the gauge $B=0$. In the $A^+=0$ gauge we obtain  \n\\begin{equation}\nP^- = \\intv  \\Big[F_{12}^2 + \\big(\\partial_i A^i)^2 + (\\partial_i B)^2 + m^2 A_i^2 \n+m^2B^2\\Big],   \n\\label{lfmvbh}\n\\end{equation} \nplus the interaction terms. Comparing the two Hamiltonians, one can see that \nalso in the light-cone gauge picture of the LF Higgs mechanism the gauge field \nbecame massive possessing three components $(\\pi,A^1,A^2)$ with the mass \n$m=ev$. The mass term of the $\\sigma$ field is generated as in the previous case.  \n \nIn summary, we gave three versions of the Higgs phenomenon in the light front \nabelian Higgs model for different gauge choices. Our light front formulation \nwas based on the finite-volume quantization with antiperiodic boundary \nconditions for the scalar fields. Minimization of the LF energy led \nto the semiquantum description of the degenerate vacuum states. In this way, \nthe concept of the trivial LF vacuum containing no quanta was generalized to \na more complex vacuum state with the non-trivial structure. The overall picture of the spontaneous breaking of the (abelian) gauge symmetry was thus found to \nbe quite analogous to the conventional theory quantized on the space-like \nhypersurface, namely one scalar field and the gauge field become massive \n(the tree-level masses $e^2v^2$ and $\\sqrt{2}\\mu$, respectively) and there \nis no massless Goldstone boson in the particle spectrum.\n\nThis work was supported by the grant No. APVT 51-005704 of the Slovak Research  \nand Development Agency, by the French NATO fellowship and by \nIN2P3-CNRS. Hospitality of the LPTA Laboratory at the Montpellier University \nis also gratefully acknowledged by L.M..\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}