diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzlvnf" "b/data_all_eng_slimpj/shuffled/split2/finalzzlvnf" new file mode 100644--- /dev/null +++ "b/data_all_eng_slimpj/shuffled/split2/finalzzlvnf" @@ -0,0 +1,5 @@ +{"text":"\\section{INTRODUCTION}\n\nEarly-type galaxies have been studied in great detail in rich clusters\nup to redshift one and above (see Stanford, Eisenhard \\&\nDickinson, 1998), with the basic\nresult that passive evolution in luminosity and no evolution in mass is\nruling them to at least $z\\simeq 1$. Old ages and an early coeval epoch of\nformation are then implied by these observations. It is still unclear, \nhowever, how much the rich-cluster environment is representative\nof the general population. Indeed, contradictory results have been\nreported about the evolutionary properties of early-type field galaxies.\n\nThe analysis by Im et al. (1996) of a sample of elliptical galaxies in the HST\nMedium Deep Survey seems to indicate that these galaxies share similar\nproperties with the cluster objects, a result based however on a sample of\n376 galaxies with only 24 spectroscopic redshifts and photometric \nredshifts based on (V--I) colors. Similarly, Lilly et al. (1996) find that the luminosity function of red galaxies in the Canada-France\nRedshift Survey (CFRS) does not exhibit significant changes in the redshift\ninterval $0.2\\leq z \\leq1$. These observations imply an early\nepoch of formation for the elliptical galaxies.\n\nThese conclusions have been questioned by Kauffmann, Charlot, \\& White (1996)\n(see also Baugh, Cole, \\& Frenk, 1996), who claim evidence for a decrease \nin the mass \ndensity already at $z\\leq 1$ for early-type galaxies in the CFRS, \nthese too selected from V--I colors.\n\nAn exceedingly deep and clean view of the field galaxy populations to high\nredshifts is provided by a long integration of HST in the so-called {\\it\nHubble Deep Field} (HDF, Williams et al., 1996). Though having provided\nrelevant constraints on galaxy formation and evolution and about the\nepoch of production of metals (e.g. Madau et al.\\\n1996; Connolly et al.\\ 1997; Sawicki, Lin, \\& Yee, 1997), \nthe interpretation of these data alone is made difficult by the\nrelatively short selection wavelengths (essentially those\nof the U, B, V, I bands), which\nimply strong evolutionary and K-corrections as a function of redshift\n(Giavalisco, Steidel, \\& Macchetto, 1996). \nThese may be particularly severe for\nthe early-type galaxies, because of their quickly evolving optical\nspectra and extreme K-corrections (e.g. Maoz, 1997).\n\nAnother unsolved problem specifically affecting short wavelength observations\nconcerns the possible effects of dust present\nin the line-of-sight to the object: even small amounts may seriously affect\nspectra corresponding to rest-frame far-UV wavelengths.\nVarious analyses attempting to solve this issue have compared the observed\nUV spectra with templates of young dusty galaxies (e.g. Meurer et al.\\ 1997).\nIt is evident, however,\nthat even small variations in the dust properties and in the assumed spectral\ntemplates imply widely discrepant predictions for the extinction corrections\n(published estimates range from less than 1 to more than 3 magnitudes of \nextinction in high-z galaxies). Any inferences\nabout the history of star-formation are correspondingly uncertain. \n\nWe try in this paper an alternative approach to the past history of galaxies\nbased on a thorough photometric and morphological analysis of a complete\nK-band selected sample of elliptical and S0 galaxies in the HDF.\nThe selection in the K infrared band helps to overcome most of\nthe above problems of optical selection, in particular minimizes the effects\nof K- and evolutionary corrections and of extinction by any residual\nintervening dust. Another crucial advantage of using near-infrared data\nis that the integrated fluxes at these wavelengths are contributed by\nall (low-mass) stars dominating the baryonic content of a galaxy.\n\nVarious deep integrations have been performed in the HDF at near-IR\n(Cowie et al.\\ 1996; Dickinson et al. 1997) and mid-IR wavelengths\n(Rowan-Robinson et al. 1997; see also \nAussel et al. 1998), which, combined with\nthe extreme quality of the morphological information and the very good\nspectroscopic coverage, make this area unique, in particular for the\ninvestigation of early-type galaxies outside rich clusters.\n\nWe concentrate here on a (morphologically selected)\nsample of 35 early-type galaxies. Our interest for this sub-population rests\non its homogeneous morphological properties, indicative of a\npurely stellar emission (hence requiring a relatively simple\nmodelling), and on the expected old ages (by comparison with their rich cluster\nanalogues). Altogether, this class of objects is expected to inform about\nthe high-redshift side of the star-formation history, which has perhaps been \nonly partly sampled by direct optical selection via the {\\it drop-out} \ntechnique. In our approach, \nthe inferred evolutionary history of stellar populations is essentially\nunaffected by the mentioned problems related to dust extinction.\n\nSection 2 of this paper provides details about the selection of the sample,\nthe adopted procedures in the photometric and morphological analyses, and\nstatistical tests of completeness carefully dealing with \nobservational limits on the total flux and surface brightness. \nIn Section 3 we analyze physical properties of galaxies as\ninferred from the morphology and from fits to the broad-band UV-optical-IR\nspectra. Section 4 addresses the statistical properties of the sample\n(counts, redshift distributions, identification statistics), and compare them\nwith those of local samples of early-type galaxies. \nWe discuss, in particular, evidence that the redshift distribution\nbreaks above $z\\gtrsim 1.3$.\nVarious interpretations of these results are discussed in Section 5, \nwhere the effects of repeated merging events, probably in the presence \nof a dust-polluted medium, are considered. \nThis information is used in Section 6 to constrain the history of star \nformation of stellar populations, and to compare it with other published \nestimates.\nOur main results are finally summarized in Section 7.\n\nWe adopt $H_0=50\\ Km\/s\/Mpc$ throughout the paper. The analysis is made for\ntwo values of the cosmological deceleration parameter, $q_0=0.5$ and 0.15, assuming\nzero cosmological constant $\\Lambda$. Note that the effects of increased time-scales \nand volumes of an open universe might be also obtained in a closure world model\nwith a non-zero $\\Lambda$. \n\n\n\\section{SAMPLE SELECTION AND PHOTOMETRY}\n\nDickinson et al. (1997) obtained deep near-infrared images of the $HDF$ \nwith the IRIM camera mounted at the KPNO-4m Telescope. The\ncamera employs a 256x256 NICMOS-3 array with $0''.16$\/pixel, but the\nreleased images are geometrically transformed and rebinned (with\nappropriate pixel weighting) into a 1024x1024 format. \nIRIM has observed the same area in the J, H and K filters, for a\ntotal of 12, 11.5 and 23 hours, respectively. Formal\n5-$\\sigma$ limiting magnitudes for the HDF\/IRIM images, computed\nfrom the measured sky noise within a $2''$ diameter circular aperture,\nare $23^m.45$ at J, $22^m.29$ at H, and $21^m.92$ at K, whereas the\nimage quality is $\\sim 1''.0$ $FWHM$.\n\nOur galaxy sample has been extracted from the HDF\/IRIM K--band image \nthrough a preliminar selection based on\nthe automatic photometry provided by SExtractor (Bertin and Arnouts 1996).\nIt is flux--limited in the K--band and it includes only galaxies whose\nmorphology strongly suggests an {\\it early--type} classification. The measure of\nmagnitude is based on fixed 2.5$''$ aperture photometry (which best accounts for\nthe seeing of the K image) and applying a stellar correction for the outer part.\n\n\n\\subsection{Monte-Carlo tests of completeness}\n\nIn order to set the appropriate value of the magnitude limit $K_L$ for\ninclusion in our sample, we have produced a synthetic frame containing\na 10$\\times$10 grid of toy galaxies with $r^{1\/4}$ luminosity profiles,\nhaving total $K$ magnitudes and effective radii $r_e$ spanning the\nranges $19^m.00\\div 21^m.25$ (step $0^m.25$) and\n$0^{\\prime\\prime}.08\\div 0^{\\prime\\prime}.80$ (step\n$0^{\\prime\\prime}.08$), respectively. These limits were chosen to\nprovide a full characterization of the performances of SExtractor\n(i.e. identification capability and magnitude estimate of galaxies) in\nthe critical range of $K$ magnitudes and for values of the true\neffective radius typical of the $HDF$ ellipticals (Fasano and\nFilippi~1998, hereafter FF98; see also Fasano et al.~1998, hereafter\nFA98).\n\nWe are particularly interested in evaluating the probability of\ndetection, as well as possible biases in the magnitude estimation, as\na function of the magnitude itself and of $r_e$ (i.e. of the `true'\naverage surface brightness $<\\mu_e^K>$, and, after all, of the\nredshift). We have convolved this synthetic frame using a $PSF$\nderived by multi--gaussian fitting of the few stars included in the\n$K$--band image. Then, a bootstrapping procedure has been carried out\nin order to produce 10 different images mimicking the 'true' image\nnoise. These images have been processed with SExtractor to get,\nfor each toy galaxy, ten different SExtractor estimates of the total \n$K$ magnitude, and then the average magnitude $$, the standard\ndeviation $\\sigma_{K_{SEx}}$ (or the difference $\\Delta K =\n-K_{true}$).\n\nFrom these simulations of galaxies having $r^{1\/4}$ luminosity profiles\nwe conclude the following. {\\it a)} For the\nwhole range of tested magnitudes and radii, the fraction of detections\nis equal to unity up to $\\simeq 21^m.4$ and $<\\mu_e^K>\\simeq\n22^m.0$. {\\it b)} \nThe magnitudes estimated by SExtractor are systematically fainter\nthan the true magnitudes, the bias mainly depending on the\ngalaxy surface brightness.\n\nFigure 1a shows that, if we consider only galaxies with\n$\\lesssim 20^m.7$ ($K_{lim}$ hereafter), the standard\ndeviation of the SExtractor magnitude estimates is less than $0^m.05$\n($\\sigma_{max}$ hereafter). Figure 1b reports $\\Delta K$ as a\nfunction of $<\\mu_e^K(SEx)>$ for the sub--sample of toy galaxies with\n$\\lesssim K_{lim}$. It shows that $\\Delta K$ systematically\nincreases at increasing $<\\mu_e^K(SEx)>$, approaching the maximum value\n$\\Delta K_{max}\\sim 0^m.5$ for $<\\mu_e^K(SEx)>\\simeq 22^m.0$. The solid\nline in Figure 1b is a polynomial fit to the data:\n\n\\begin{equation}\n\\log(\\Delta K) = -0.830 + 0.147\\times<\\mu_*^K> + 0.019\\times<\\mu_*^K>^2\n\\end{equation}\n\n\\noindent where $<\\mu_*^K> = <\\mu_e^K(SEx)> - 19$.\n\nThe plots in Figure 1 provide a straightforward indication of the\nstrategy to be used in order to obtain a complete flux limited sample\nof early--type galaxies from the $IRIM$ $K$--band image using\nSExtractor. The procedure consists of the following steps: 1) to\nproduce a catalog of objects with $K_{SEx}\\lesssim K_{lim}=20^m.7$;\n2) to check the morphology of each object on the WFPC2 frames,\nremoving stars and late--type (or irregular) galaxies from the sample;\n3) to exploit the high resolution and depth of the $V_{606}$ and\/or\n$I_{814}$ images to derive the effective radii of the galaxies and use\nthese values of $r_e$ to compute $<\\mu_e^K>$, assuming a roughly\nconstant color profile; 4) to apply to the $K_{SEx}$ magnitudes the\nstatistical corrections given in the previous equation; 5) to include\nin the final sample only galaxies with corrected magnitude less than\nor equal to $K_L = K_{lim} - \\Delta K_{max} - \\sigma_{max} = 20^m.15$.\n\n\\subsection{The morphological filter}\n\nThe most delicate steps in the previous scheme are those concerning\nthe morphological analysis and the evaluation of the effective radii\n(points 2 and 3 of the previous list). They are widely discussed in\nFF98 and FA98 and we refer to these papers for details. Here we wish\nto summarize the 'flow--chart' of the morphological filtering for the\npresent sample. The SExtractor automatic photometry of the $IRIM$\n$K$--band image produced a preliminar sample of 109 objects with\n$K_{SEx}\\leq 20^m.7$ (first point of the previous\nselection scheme). All these objects were examined with the {\\it\nIMEXAM--IRAF} tool to produce a first (conservative) screening against\nstars or late-type and irregular galaxies, resulting in a temporary\nlist of 47 {\\it early--type} candidates. \nOur quantitative morphological classification filter assumes a dominant\n$r^{-1\/4}$ profile for the bulk of the light distribution in the galaxy.\n\nFor 29 galaxies in this list\nthe surface photometry in the ST--$V_{606}$ band is available in FF98\nand the total AB magnitudes in the four $WFPC2-HDF$ bands are given in\nFA98, together with the equivalent effective radii derived after\ndeconvolution of the luminosity profiles. Four more galaxies included\nin both our and FF98 samples, were not analyzed by FA98. For these\ngalaxies we computed optical AB magnitudes and effective radii according \nto the prescriptions given in FA98. The remaining 14 galaxies in the\ntemporary list are not in the FF98 sample, since in the $V_{606}$ \nband they did not satisfy the selection criteria adopted in order to \nsecure a reliable morphological analysis. We performed the detailed\nsurface photometry of these galaxies in the $I_{814}$ band (where they \nshow a better S\/N ratio), producing luminosity and geometric profiles \nof each galaxy. From this analysis, 3 objects were recognized to\nbe '{\\it disk-dominated}' galaxies (likely $Sa$), whereas two more objects \nshowed peculiar or unclassifiable profiles. These galaxies were excluded\nfrom the temporary list. \n\nThe remaining 9 galaxies with surface photometry \nin the $I_{814}$ band were analyzed following the same procedure described \nin FA98 to derive the total AB magnitudes in the optical bands, \nas well as the '{\\it true}' (deconvolved) equivalent effective radii.\nFigure 2 shows the luminosity profiles of these galaxies derived from \nthe surface photometry in the $I_{814}$ band.\n\n\\subsection{The final sample}\n\nAltogether, of the 47 candidate E\/S0 galaxies with $K_{SEx}\\leq 20.7$, 33 were\nalready classified as E\/S0 by FA98 and FF98, while 9 are confirmed as E\/S0\nby the present paper, for a total of 42 objects in the incomplete sample at\n$K_{SEx}\\leq 20.7$. To ensure a highly reliable completeness limit, following\nsteps 4 and 5 of the selection scheme discussed in Sect. 2.1, \nwe used equation (1) to correct the $K_{SEx}$ magnitudes of the 42\ngalaxies in the temporary sample. In this way we \nobtain our final sample of 35 galaxies with corrected magnitude $K\\leq 20^m.15$ over the\nHDF (3 WF + 1 Planetary Camera) area of 5.7 square arcmin. Some basic data \non the sample are listed in Table 1. \n\nThe $H$ and $J$ magnitudes listed in the Table have been obtained\nrunning SExtractor on the corresponding $IRIM$ images and\naccounting for the expected biases with the same procedure adopted for\nthe $K$ magnitudes (simulations of toy galaxies, convolution with proper\n$PSF$, noise bootstrapping and correlation between bias and average\nsurface brightness). The corresponding equations are:\n\n\\begin{equation}\n\\log(\\Delta H) = -0.903 + 0.288\\times<\\mu_*^H> - 0.0158\\times<\\mu_*^H>^2\n\\end{equation}\n\n\\begin{equation}\n\\log(\\Delta J) = -1.014 + 0.162\\times<\\mu_*^J> + 0.0140\\times<\\mu_*^J>^2\n\\end{equation}\n\n\\noindent \nwhere $<\\mu_*^H> = <\\mu_e^H(SEx)> - 20$ and $<\\mu_*^J> = <\\mu_e^J(SEx)> \n- 19$.\n\nFor 15 galaxies in Table 1 spectroscopic redshifts are available in the\nliterature or in the WEB (see notes to Table 1), while for the remaining \n20 objects a\nreliable estimate of $z$ is obtained from fits of the optical-IR\nbroad-band spectra (see Sect. 3.2).\n\nThe last column in Table 1 gives some information on the surface photometry.\nIn particular, the symbol (814 Ph.) indicates galaxies whose surface \nphotometry has been obtained in the $I_{814}$ band, while the symbols \n(T1), (T2) and (T3) indicate galaxies which, according to FF98, belong \nto the `{\\it Normal}', '{\\it Flat}' and '{\\it Merger}' class, respectively. \nThese attributes refers to the classification by FF98 \n(see also Fasano et al. 1996),\nbased on the luminosity profiles, of {\\it early--type} galaxies \nin the $HDF$: {\\it 1)} the '{\\it Normal}' class, for which\nthe de~Vaucouleurs law is closely followed down to the innermost isophote \nnot significantly affected by the $PSF$; {\\it 2)} the '{\\it Flat}' class, \ncharacterized by an inward flattening of the luminosity \nprofiles (with respect to the de~Vaucouleurs' law) which cannot be \nascribed to the effect of the $PSF$ ; {\\it 3)} the '{\\it Merger}' class, \nin which isophotal contours show the existence of complex \ninner structures (two or more nuclei) embedded inside a common envelope, \nwhich roughly obeys the $r^{1\/4}$ law.\n\nThe completeness of the sample has been evaluated from numerical simulations \nas described in Sect. 2.1. \nWe have not used the classical $V\/V_{max}$ test to this purpose,\nsince it is equally sensitive to departures from spatial homogeneity\nthan to completeness. The question of the distribution of objects in the\nspace-time will be addressed in Sect. 4 below.\n\n\n\n\n\n\\section{ANALYSIS OF THE BROAD-BAND SPECTRA AND OF THE SURFACE BRIGHTNESS\nDISTRIBUTIONS} \n\n\nThe surface photometry of the images of our sample galaxies has shown that \nfor the large majority of them \nthere are no morphological signatures of the presence of dust,\ne.g. obscured lanes across the galaxy, asymmetric brightness distributions\nor profiles deviating from the $r^{-1\/4}$ over the bulk of the galaxy\n(see however Witt, Thronson, \\& Capuano, 1992, for a cautionary remark about the\nlatter point).\nOnly three galaxies belong to the morphological class T3, where on-going\nstar-formation, possibly enshrouded by \ndust, is likely to have a role. Four more objects \nbelong to the class T2, for which any dust, whenever present, should most probably \nbe confined to the inner core of the galaxy, hence should not affect the \nglobal emission. Five out of 7 of the morphologically peculiar objects have blue (class [a], as discussed in Sect. 3.2) spectra.\n\nAn obvious way to test for the presence of dust would be to look for its\nre-radiation at infrared wavelengths. The Infrared Space Observatory has\nobserved the HDF in two broad-band filters centered at 6.7 and 15 $\\mu m$.\nWhile the former, for redshifted objects, is dominated by photospheric \nemission of old stars, the latter is contributed by very hot dust and\ntransiently heated molecules or very small dust particles present in the ISM.\nMann et al. (1997) and Goldschmidt et al. (1997) report catalogues of sources\ndown to rather conservative flux limits. At 15 $\\mu m$ only one ISO source is\nin common with our list (ID 2-251-0), a galaxy with a point-source evidencing\nin the U and B, probably an AGN. Though a conclusion has to wait for a \nrefined analysis of the ISO observations, aimed at deeper flux limits \n(Aussel et al. 1998, Desert et al. 1988), \nthe available information seems to confirm \nthat there is little room for dust emission by our E\/S0 galaxies. \n\nThe present analysis will take advantage of this simplified behavior for the \nbulk of our objects,\nby allowing modelling of galaxy spectra as the integrated emission of purely \nstellar populations. The inclusion of the additional effect of dust extinction,\nwhenever present, would have substantially weakened our conclusions, since\ndegenerate sets of solutions could have been possible, implying\nuncertain estimates of the photometric redshift and of the galaxy age.\n\n\n\\subsection{The interpretative tool: a model for the evolutionary\nspectral energy distributions of the integrated starlight} \n\nThe UV-optical-near\/IR Spectral Energy Distributions (SED) for all\ngalaxies in the sample have been fitted with synthetic spectra based on\na model briefly summarized here. A general framework for this modelization \ncan be\nfound in Bressan, Chiosi, \\& Fagotto (1994), while a critical assessment of problems\nand limitations is discussed by Charlot, Worthey, \\& Bressan (1996).\n\nThe code follows two paths, one describing the chemical evolution \nof the galaxy's ISM, the other\nassociating to any galactic time a generation of stars with the\nappropriate metallicity, and adding up the contributions to the integrated\nlight of all previous stellar populations.\n\nThe chemical path adopts a Salpeter IMF with a lower limit $M_l= 0.15\\ \nM_\\odot$, and a Schmidt-type law for the\nStar Formation Rate (SFR): $\\Psi(t)=\\nu \\, M_{g}(t)^{k}$, where $\\nu$ is the\nSF efficiency. A further parameter is the time-scale $t_{infall}$ for the \ninfall of the primordial gas. The library of isochrones is \nbased on the Padova models, spanning a\nrange in metallicities from $Z=0.004$ to 0.05 (i.e. from 0.2 to 2.5 solar).\nThis range is appropriate for stellar populations in local early-type \ngalaxies, which have solar metallicity on average \n(e.g. Grillmair et al. 1996).\n\nThe isochrones are modified to account for the contribution by dust-embedded \nMira and OH stars during the AGB phase (Bressan, Granato \\& Silva 1998), when\ncircum-stellar dust reddens the optical emission and produces an IR bump\nat 5-20 $\\mu m$. This IR emission from AGB stars is important for\ngalactic ages of 0.1 to a few Gyrs.\n\nA number of evolutionary patterns for the time-dependent SFR $\\Psi(t)$\nhave been tried to reproduce the galaxy SEDs, to estimate the photometric\nredshifts for galaxies lacking the spectroscopic identification, and to\nfit the global statistical properties of the sample.\nHowever, we will mostly refer in the following to two paradigmatic evolution \ncases.\n\nThe first model (henceforth Model 1) reproduces a classical scheme for the formation\nof ellipticals (e.g. Larson 1974), i.e. a huge starburst on short\ntimescales, expected to occur at high redshifts.\nIn our approach the star-formation has a maximum at a galactic age of\n0.3 Gyr and continues at substantial rates up to 0.8 Gyr, after which it\nis assumed that the input of energy by stellar winds and supernovae\nproduced a sudden outflow of the ISM through a galactic wind, stopping\nthe bulk of the SF. The evolution at later epochs is mostly due to\npassive aging of already formed stellar populations. \nThis evolution pattern is achieved with the following choice of the\nparameters: $t_{infall}=0.1$ Gyr, $k=1$, $\\nu=2\\ Gyr^{-1}$. \nThe precise time dependence of the SF rate is reproduced in Figure 3 (dotted line).\nThe redshift for the onset of star-formation $z_F$ in the galaxy\nis a free parameter.\n\nOur second model (henceforth Model 2) \ngives up the concept that the stellar populations in field\nearly-type galaxies are almost coeval, and assumes instead that the\nstar-formation lasts for a significant fraction of the Hubble time.\nThis is obtained with the following parameters:\n$t_{infall}=1$ Gyr, $k=1$, $\\nu=1.3\\ Gyr^{-1}$. \nThe corresponding SF law has a broad peak at 1.4 Gyrs but goes on at\nsubstantial rates for a couple Gyrs more. For a value of the\nbaryonic mass of $M=10^{11}M_\\odot$ (as typical for our sample galaxies,\nsee below), this brings to an energetic unbalance in the ISM followed by\na galactic wind at 3.1 Gyrs (see continuous line in Fig. 3 \nfor the detailed evolution of the SF). \n\nThis adopted SF law is clearly an over-simplified\npicture of a process which has been likely more complex. In particular,\na protracted SF, as implied by the second model, is likely to have been occurred \nthrough a set of successive starbursts (e.g. due to mergers or strong dynamical interactions). Since we are dealing with the integrated emission of all\nstellar generations, there is virtually no difference, as for the broad-band\nspectral appearance in the after-burst phase, between a set\nof starbursts occurring over 3 Gyrs and a continuous SF during this period.\n\nTo provide these simplified schemes with more flexibility, we have allowed\na residual star-formation $\\Psi_0$ at a basal level to be added to the flux\nemitted by the passively evolving populations. \nThis allows to fit the spectra of the four bluest objects, and to improve\nthe fits at the shortest (U, B) wavelengths for additional galaxies.\nSince typical values for $\\Psi_0$ are much less than 1 $M_{\\odot}\/yr$\nand this residual SF does not contribute significantly to the mass and\nenergetics,\nthis parameter is used only in the spectral fits of Fig. 4 below and\nnever more in the subsequent analyses.\n\n$\\Psi_0$ is then a second free \nparameter. The baryonic mass $M$ and the photometric redshift (for cases\nwhere it is needed) are further parameters in the spectral fitting procedure.\n\n\n\\subsection{Fitting the galaxy broad-band spectra: evaluation of the\nphotometric redshifts and of the galactic ages} \n\n\\subsubsection{The photometric redshifts } \n\nFigure 4 is a collection of the observed spectra for the 35 sample galaxies. \nThe figure displays essentially two kinds of spectral behaviors: \nclass {\\it (a)} spectra, which are rather blue at all \nwavelengths, dominated by young stellar populations (10 objects\nout of 35); class {\\it (b)} spectra, with overall redder properties, \ndisplaying a typical two power-law behavior, with a break at $\\lambda\n\\simeq 0.4\\ \\mu m$ in the rest-frame spectrum in correspondence of the \nBalmer decrement (25 sources). \nNo objects, even those at the lowest redshifts, are found to\ndisplay very red colors, as would be expected for a very old stellar\npopulation dominating the spectrum. This result is consistent with \na statistical study by Zepf (1997) revealing a lack of very red objects in \nthe field.\n\nA crucial step in our analysis is the evaluation of redshifts\nfrom broad-band spectral fitting\nfor sources lacking a spectroscopic identification. The uncertainties \nrelated with this estimate have been discussed by many authors (see\nin particular Connolly et al. 1997; Hogg et al. 1998),\nwith the general outcome that the inclusion of near-infrared data\nin the analysis (added to the 4 HST bands) makes the redshift\nestimate quite more reliable.\n\nWith respect to these analyses we benefit here of various advantages.\nThe first one is due to the lack of evidence for dust in our objects,\nwhich breaks down at least one possible degeneracy in the parameter space\n(that is estimating a lower photometric z from a dust-reddened template).\nThe second one is that most of the observed spectra have rather \nhomogeneous and monotonic behaviors as typical of early-type galaxies,\nwith an easily discernible Balmer feature.\nA third potential advantage, as verified \"a posteriori\", \nis given by our relatively bright K-band selection, which\ntends to select objects only up to moderate redshifts ($z<1.5$, see Sect.\n4-5) and to exclude very high-z galaxies, whose redshift estimate would be\nquite more uncertain because of the strong evolutionary corrections in the \ntemplate spectra.\n\nMore specifically, \nfor class {\\it (b)} sources lacking spectroscopic redshifts (i.e. 15 galaxies\nin total)\nthe estimate of the redshift from spectral fitting is quite robust, thanks\nto a well-defined spectral break corresponding to the Balmer decrement,\ntypically occurring at $\\lambda \\sim$ 0.6 to 1 $\\mu m$. \nThe combined use of the\n3 near-IR and the 4 HST bands, with very small photometric errors, allows\nan accurate determination of the 4000 A break from a model-assisted spectral \ninterpolation.\nFor class {\\it (a)} spectra without spectroscopic redshifts (5 more galaxies),\nthere is still clear evidence for a 4000 A break, which is however less\nprecisely characterized in some instances. In these cases the photometric \nredshift may turn out to be significantly more uncertain.\n\nFigure 5 summarizes a comparison of our redshift estimate based on broad-band\nspectral fitting with the actual measurement from mid-resolution spectroscopy, for\nthe 15 galaxies in our sample with this information. The errorbars associated with\nthe photometric estimate are conservative $\\sim$95\\% confidence intervals based on\n$\\chi^2$ fitting using Model 2 as spectral template. In only one case \n(object 2-251-0) the photometric\nmeasure is significantly discrepant with respect to the spectroscopic one, but\nthis happens for the galaxy containing a nuclear point-like source, \nwhich affects the match of HST and near-IR photometry.\nFig. 5 shows that our process exploiting 7-band spectral data is \noverall quite reliable and that no systematic effects are present.\nOn the other hand, it is clear that the 15 galaxies with optical spectroscopy \nare not randomply sampled from our source list, as shown in particular by Fig. \n7 below: they tend to be lower-redshift, bluer objects, with SF protracted\nto recent cosmic epochs. Such spectral behaviour has encouraged a spectroscopic\nfollow-up, while the redder higher-z ones will require a substantial dedicated\nfuture effort.\n\nAltogether, we estimate that the errors associated with redshifts\nevaluated from class {\\it (b)} spectra should not typically exceed $10\\%$\nin $z$,\nwhile a conservative estimate may be closer to $20\\%$ for class {\\it (a)}\nspectra (see also Connolly et al. and Hogg et al. for similar conclusions).\nNone of our results will be significantly affected by these uncertainties.\n\n\n\\subsubsection{Evaluation of the galactic ages } \n\nFor all sample galaxies, but for object 4-727-0, we found acceptable fits \nwith Model 2. These solutions were found \nby optimizing the 3 (4) free parameters discussed in the previous Section.\nThe spectral solutions illustrated in Fig. 4 (whose parameters are reported\nin Table 2) refer to the case $q_0=0.15$. Consistent solutions were also \nfound for Model 2 in a closure world model \n(see again Table 2 for the corresponding best-fitting parameters).\n\nThe failure of Model 1 to reproduce the data is illustrated in Figure 6,\nwhich compares the broad-band spectrum of a typical galaxy in the sample\nwith two \npredictions of the model at varying $z_F$. The continuous curve is\nthe spectrum predicted by Model 1 with $z_F=5$ and $q_0=0.15$. With this choice \nof the parameters, this corresponds to a stellar population beginning \nformation at z=5 and ending it at z=3.6 (hence observed 5.5 Gyrs after the \nend of the SF in the source frame): the prediction is clearly far too \nred both in the optical and even in the near-IR. The dashed curve\ncorresponds to the same model with $z_F=1.3$, and the additional contribution\nof ongoing SF by 0.2 $M_{\\odot}\/yr$ to better reproduce the U band flux.\nIf the overall fit is better, there is still a significant excess flux\nin the observed B band with respect to the model spectrum. We have found this\nexcess, as well as sometimes one in the V band, to be a general\ncharacteristic of the observed spectra: \neven for the reddest galaxies in the sample it seemed difficult to reproduce the spectra with stellar populations\nformed during a relatively brief time interval (as in Model 1), even including the additional contribution of ongoing star-formation.\n\nThe better performance of Model 2 is interpreted by us as indicating a rather\nprotracted star formation activity within each galaxy (typically 3 Gyrs in the \nmodel), rather than a single short-lived starburst (even one occurring at low\nredshift). A qualitatively similar result is achieved \nby FA98 analyzing a sample of HDF early-type galaxies \nselected in the $V_{606}$ band.\n\nThe second conclusion concerns the epoch when the stellar \nformation has taken place in the typical sample galaxy. Figure 7 is a plot of\nthe redshift corresponding to the peak of the star-formation versus the\nmass in baryons, according to our best-fits obtained with Model 2. \nPanels (a) and (b) refer to the two solutions with $q_0=0.15$ and $q_0=0.5$.\nThere are clearly two zones of avoidance in the figure: low-mass objects\n($M<2\\ 10^{10}\\ M_{\\odot}$) have peak SF confined to $z\\lesssim 1$, while \nthe high-mass ones \nappear to form stars mostly at $z\\gtrsim 1$. This is partly an artifact\nof the magnitude limits (in the former case) and of the small sampled\nvolume combined with the low space density of very massive galaxies \n(in the latter). While only a careful examination of all selection effects\nwill allow to obtain unbiased estimates of the cosmic SF history\nfrom this dataset (see Sect. 6 below), {\\it it is clear from Fig. 7 that the \nredshift interval of\nz=1 to 2 (1 to 3 for $q_0=0.5$) corresponds to a very active phase of SF\nfor the sample galaxies}.\n\nTo translate this into a constraint on the age for the typical\nearly-type galaxy in the field, we report in Figure 8 the rest-frame V--K\nand B--J colors as a function of redshift, compared with the predictions of \nsingle stellar populations with solar metal abundances. The rest-frame B--J\ncolors are computed by interpolating the galaxy observed spectra using the \nbest-fit models of Table 2, while the V--K's require substantial extrapolation\nat the longer wavelengths, hence are to be taken with some care.\nThe figure shows that the typical ages range from 1.5 to 3 Gyrs, rather\nindependently from the redshift (if any, there is quite a marginal tendency \nfor the higher-z galaxies to display bluer colors).\nFor comparison, the average colors of local galaxies are V--K$\\simeq$ \n3.2--3.3, quite significantly redder. \nThis difference is probably enhanced by a bias induced by the flux limit \nin the K-selected sample, emphasizing relatively bluer objects observed\nclose to their main event of star-formation. Again this will be subject of\nfurther inspection in the next chapters.\n\nAnother check of this is reported in Fig. 6, where the broad-band\nspectrum of the typical galaxy is compared with that of the local\nearly-type M32 (F. Bertola, private communication). The present\ninterpretation of M32's blue light is that it may be dominated by the\nHR turn-off population\nof a $\\sim 4$ Gyrs old stellar population of about solar\nmetallicity (O'Connell 1986), whose bright RGB and AGB stars have been\nrecently resolved (Freedman 1992, Grillmair et al. 1996). Older ages\nmight fit only for a metal-deficient dominant population (Renzini \\& Buzzoni\n1986). Our typical galaxy is bluer than M32 and, assuming a solar \nmetallicity, it turns out to be younger than 4 Gyrs.\n\nThe previous considerations have impinged upon a well-known problem when \ndating stellar populations from broad-band spectral analyses, i.e. the \ndegeneracy between age and metallicity. An estimate of the age needs \na guess on the metallicity.\nAs already discussed, the very regular morphologies and the lack of evidence\nfor the presence of dust in the large majority of our sample galaxies\nsuggest that they have already mostly completed their bulk of the SF \nand have metallicities comparable to those of the \nlocal counterparts (i.e. solar on average, see Carollo \\& Danziger 1994). \nIf true, then a conclusion seems unavoidable, that their typical age is close to \n1.5 to 2 Gyrs on average, at the observed redshift. \n\nFigure 9 illustrates the effects of drastic changes in the metallic \ncontent of the stellar populations on the rest-frame colors. \nThe two horizontal lines in both panels bracket\nthe typical colors for the sample galaxies. \nLarge age differences are found as a function of the metallicity for \na given color and for old stellar populations. \nOn the other hand, {\\it since our observed colors are rather blue, \nthe uncertainty due to the unknown metallicity is moderate in absolute terms}.\nFor example, with reference to Fig. 9b, the typical observed color\nB--J$\\simeq 2.5$ would correspond to an age of 1 Gyr for 2 times solar,\nto 2 Gyr for a 0.4 solar and 3 Gyr for a 0.2 solar metallicity.\nThe conclusion is that, {\\it unless we accept that the observed galaxies \nare very significantly metal-deficient (which would be difficult to \nreconcile with observations of local objects), our age determinations should\nnot be seriously in error}.\n\n\n\\subsection{Constraints from the size vs. surface brightness distributions\n} \n\nFurther significant constraints onto the nature and evolutionary status\nof our sample galaxies may be gained from a detailed analysis of their\nmorphological properties.\n\nThe study of the galaxy spectral shapes in the previous Sect. has \nindicated that intermediate (rather than old) ages, with large spreads, \nare typical. If so, then one would expect quite appreciable luminosity \nevolution due to the -- mostly passive -- aging of the stellar populations\nfrom the redshift of the observation to the present time.\nA way to check it, as discussed in FA98, would be to compare\nthe sizes and the average surface brightness (within the effective radius)\nof distant galaxies with those of the local ones. \nTable 1 (columns 6 and 15) reports these data for all galaxies in our sample.\n\nFigure 10 compares the same data with the mean locus ((thick continuos \nline) of the\nrelation between the effective radius $r_e$ and the corresponding average \nsurface brightness in the V band $<\\mu_e>_V$ (the Kormendy relation). \nIn this figure the effective radii are expressed in Kpc,\nwhile the surface brightness has been corrected (K- and evolutionary\ncorrections) according to Model 1.\n\nThere is a clear offset in Fig. 10, by $\\sim 1.5$ magnitudes, with respect to\nthe locus representing local galaxies (see for more details on the latter\nFA98 and Jorgensen et al., 1995): \nthe surface brightness of distant galaxies is\nmore luminous on average than implied by Model 1 -- one in which, we remind,\nthe bulk of SF was completed by $z=3.5$.\n\nFigure 11 shows that much better consistency is achieved with Model 2,\nboth in the V (panel [a]) and in the K bands (panel [b]).\nThe local Kormendy relation in the band plotted in Fig. 11b is taken from \nPahre et al. (1995). \nIn this case the evolution in luminosity between the redshift of the\nobservation and the present time is stronger (stars are younger on average),\nwith local and distant galaxies consistently tracing the same population,\nas expected.\n\n\n\\section{STATISTICAL PROPERTIES OF THE COMPLETE SAMPLE: THE REDSHIFT \nDISTRIBUTION}\n\nThough small, our K-selected sample has been tested with great care\nin Sect. 2 for completeness and reliability in source selection,\nand it is then suitable for a detailed statistical analysis.\nIn consideration of the high-quality morphological and photometric data\navailable, this study may provide an accurate characterization of the \ndistribution of elliptical\/S0 galaxies in the space-time. \n\nWe consider here two statistical observables, the counts as a function of\nmorphological type and of limiting K magnitude, and the redshift distribution \nD(z).\n\nFigure 12 is a collection of galaxy counts in the K and HK bands, splitted \ninto two morphological components: the early-type galaxies (open squares), \nselected according to the criteria described in Section 2 (see Table 1), \nand late-type systems -- including spirals, irregulars, and starbursts \n(open triangles). \nThe K band counts in the left-hand panel\n are derived from our sample discussed in \nSect. 2 down to $K=20.1$ . The counts in the H+K band in the right-hand panel \nare derived from a sample by Cowie et al. (1996), including complete \nmorphological information for HK brighter than 21.5.\n\n\nThe number counts for early- and late-type galaxies display rather different \nslopes at the faint end, with the early-types converging very fast at\n$K>19$ and the late-types showing steadily increasing counts.\nSimilar results about the morphological number counts are reported by \nDriver et al. (1988), who find in particular a deficit of E\/S0 at\n$I_{814}>22$ compared with passively evolving models.\n\nThe distribution of the -- either spectroscopic or photometrically estimated -- \nredshifts for the 35 sample galaxies is reproduced as a thick line in Figure\n13. The observational distribution shows monotonic increasing values\nfrom $z$=0 to $z\\simeq 1.3$, with a marked peak at $z\\sim 1.1$. \n\nA remarkable feature in the distribution is \na sudden disappearance of objects at $z\\gtrsim 1.3$.\nThis absence may look surprising at first sight, taking into account that\nthe dominant stellar populations \nfor objects observed at $z\\sim 1$ are 1 to 3 Gyr old, whose luminosity\nhas then to increase, if any, at $z>1$.\n\nLet us first discuss the significance of this apparent redshift cutoff, in the \nlight of all selection effects operating in the sample. The first obvious\nconcern is our ability to identify in the K-band image\nextended emissions corresponding to high redshift elliptical\/S0's when going \nfrom the typical observed redshift $z\\sim 1$ to larger cosmic distances.\n\nFigure 14 illustrates the effect of increasing redshift on the average surface \nbrightness and the effective radius, for two evolutionary paths\n(both adopting $q_0=0.15$). \nThe top line corresponds to a typical galaxy in our sample having \n$<\\mu_e^K>=18$, $r_e=2$ Kpc (cfr. the distribution of values scaled to \nzero-redshift in Fig. 11b), and observed at $z$ ranging from 0 to 2.5.\nThe scaling with $z$ of the surface brightness has been calculated taking into\naccount all cosmological, K- and evolutionary effects. \nTo be conservative, these were computed according\nto Model 1 with a low formation redshift $z_F=3$. It turns out \nthat the brightening of the stellar populations, while approaching the SF \nphase at $z>1.5$, counter-balances any cosmological and K-correction\ndimming. In this case there is no significant dependence of the surface\nbrightness on redshift. A limiting situation is provided by the bottom curve\nin Fig. 14, representing the lowest surface brightness galaxy in the sample\nand luminosity evolution after Model 1 with a high formation redshift $z_F=6$. \nIn this case the brightness dimming effects dominate and there is an appreciable\nexcursion in $<\\mu_e^K>$ as a function of $z$. \nIn spite of this, in this case as well as\nin all other more favorable ones, the faint object would be still be\ndetectable up to z=2.5\nabove the completeness limit, estimated by simulations in Sect. 2 to be \n$<\\mu_e^K>=22$.\nNote that the same conclusions hold for any more actively evolving models\n(e.g. Model 2).\n\nAltogether, the K-band image is sensitive enough to allow easy detection of \neven moderately or non evolving galaxies up to at least $z\\sim 2.5$. \nThen the \nturnover in D(z) apparently occurring at much lower $z$ cannot be due\nto limitations in detecting faint extended structures.\n\nWe investigate in the following how the other fundamental selection condition, that \nin the total flux ($K<20.15$), operates for different \ncosmological and evolutionary models.\nTo perform this exercise, a reliable local luminosity function (LLF)\nof galaxies in the K band is required. Our adopted LLF is taken from\nGardner et al. (1997), which significantly updates previous\ndeterminations. The separate contributions of early- (E\/S0) and late-type \n(Sp\/Ir) galaxies have been calculated using the optical LLFs \nsplitted into various morphological types by Franceschini et al.\n(1988) and translated to the K band with type-dependent B-K colours.\n\nThe K-band luminosity function of galaxies has been combined with \nvarious evolutionary models to predict sample statistics,\nunder the simplifying assumption of a galaxy mass function\nconstant with cosmic time.\n\nWe find that the shape of the LLF estimated by Gardner et al. \ncan be more naturally\nreconciled with the total faint K-band counts of Fig. 12 in an open universe with\n$\\Lambda=0$, \nwhile a closure one would require some \"ad hoc\" evolutionary prescriptions. \nThis is because the latter has not enough volume at $z\\sim 1$ to fill in the \ncounts, given the space density of galaxies implied by the K LLF.\nThen Model 1 for E\/S0 galaxies (see Sect. 2.2), \nsupplemented with a moderate evolution for Sp\/Ir galaxies as in Mazzei\net al. (1992), could in principle provide a fair fit to the total\ncounts for $q_0=0.15$.\n\nBut a further crucial constraint is set by the observed $z$-distribution\nD(z) of early-type galaxies in the K-HDF (Fig. 13). From $z=0$ to $z=1.2$, \nD(z) is consistent with a (mostly passive) \nevolution in an open universe (with $q_0$=0.1-0.2) with zero cosmological \nconstant. Again, a closure world \nmodel with $\\Lambda=0$ would require a much larger comoving \nluminosity density of massive E\/S0s at $z$=1 than locally.\n\nHence, there is no evidence, up to this epoch, of an evolution of the baryon\nmass function. This is difficult to reconcile with a decrease of the mass \nand luminosity due\nto progressive disappearance of big galaxies in favor of smaller mass \nunits, as implied by some specifications of the hierarchical clustering \nscheme (e.g. Baugh, Cole, \\& Frenk, 1996). \nThe case for a strong negative evolution of the early-type population\n(by a factor 2-3 less in number density at $z$=1; e.g. \nKauffmann, Charlot \\& White 1996)\ndoes not seem supported by our analysis of the HDF.\n\nAbove z=1.2, however, both Models 1 and 2 (whichever formation redshift $z_F$ \nis assumed) have difficulties to\nreproduce the redshift distribution reported in Fig. 13 and the identification\nstatistics in Figure 12. \nIn particular it is not able to explain the large number of sources \nat $z=1$ followed by the rapid convergence above: the\nprediction would be of a much more gentle distribution, with less a\npronounced peak and a large tail of galaxies (including typically half of\nsample objects) observable above $z=1.2$. \nLower values of $q_0$ would allow better fits of the observed D(z) up \nto $z=1.3$, but would also worsen the mis-fit at higher $z$.\n\nA similar effect is observed in the\ncounts as a function of the morphological class (Fig. 12). Here again\nsources identified as E\/S0s show steep counts to K=19 and a sudden\nconvergence thereafter, while Model 1 would predict a much less rapid\nchange in slope. These problems remain forcing in various ways the \nmodel parameters. \n\n\n\\section{DUST EXTINCTION AND MERGING EFFECTS DURING A PROTRACTED SF PHASE}\n\nWe briefly discuss here possible explanations of the results obtained \nin Sections 3 and 4.\nWe look for solutions explaining the global\nstatistical properties of the sample (see Sect. 4) as well as the \nvariety of the single galactic spectra (Sect. 3).\n\n\\subsection{Merging}\n\nIn the previous Sect. we have found that\nat redshifts larger than 1.3 early-type galaxies suddenly disappear \nfrom the K-selected sample, while we would expect to observe them to much\nhigher $z$, given the predicted luminosity enhancement when approaching\nthe most intense phase of star formation. \nA comparison of Figures 7 and 13 suggests that there may be a relation \nbetween the redshift interval where such a disappearance occurs and the one\nbracketing the major episodes of star formation, \nas evaluated from the analysis of the stellar populations in distant galaxies.\nIt is clear from Fig. 7 that the interval $11$, is then a very appealing interpretation\nfor our results.\n\nAnother effect of merging would be to lower the flux detectability during\nthe SF phase, due to the mass function rapidly evolving at $z>1$.\nWe did not attempt to quantify the possible effects of merging on the\nvisibility of the early evolution phases of E\/S0 galaxies. We defer for an\nexercise of this kind to dedicated treatments (e.g. Baugh, Cole, \\& Frenk \n1996).\n\nAnother plausible reason of flux leakage during SF is related to\nthe extinction by dust in the medium where stars are forming. This dust\nis very likely present, left over by preexisting generations of massive \nstars. We spend the rest of this Section\nin a more quantitative evaluation of the possible effects of dust.\nAn argument strongly suggestive of its occurrence will be given in Section 6.\n\n\n\\subsection{Dust effects during a prolonged SF phase in field ellipticals}\n\nDust is observed in significant, or even large, amounts in a wide variety\nof objects at any redshifts. The most distant astrophysical sites\nexplored so far, the quasars at redshift 4 to 5, have shown to contain,\nat that early epoch, dust amounts comparable with those of the most\nmassive galaxies today (Omont et al. 1997; Andreani, Franceschini, \\& Granato, 1997).\nBut also distant radio galaxies, Lyman \"drop-outs\",\nand damped $Ly\\alpha$ galaxies (Pettini et al. 1997)\nhave invariably shown the presence of dust.\nIn several galaxies, among the infrared selected, IRAS has found \nthat the dust re-processes a dominant fraction of the stellar UV radiation\ninto the infrared (the ultra- and hyper-luminous IR galaxies, see e.g. Sanders \nand Mirabel, 1996; Rowan-Robinson 1997).\n\nIndeed, a single short-lived starburst may produce enough metals via\nmassive stellar outflows during the AGB and supernova phase. Recent ISO\nobservations (Lagage et al. 1996) have proven that dust grains are synthesized\nalmost simultaneously with the metals, as soon as they are made available \nto the medium.\n\nA common interpretation of ultra-luminous IR galaxies is that, during a\nmerger or a close interaction, even moderate amounts of dust are spread\naround to make an extended dusty core in the starburst. In these conditions\nit is very likely that any major merger happens in a dust enshrouded medium,\nseveral examples of which have been found at both low- (e.g. Arp 220,\nNGC 6090, M82, see Silva et al. 1998) and high-redshifts (e.g. IRAS F10214;\nHR10; see also Ivison et al. 1998).\n\nWe provide here a simplified description of the effects of dust during the\nstar-formation phase and test it on the statistical properties of our\nsample as discussed in the previous Section.\n\nThe simplification we adopt is to assume, instead of a discrete set of \nsuccessive merging-driven starbursts as would be a realistic physical\nsituation, a more continuous process of star-formation with a temporal\nevolution following that of Model 2 in Figure 3 and occurring in a \ndust-enriched medium (we remind that in Model 2 the SF has a peak \nat 1.4 Gyr after beginning and keeps on until 3.1 Gyrs. We do not consider the \neffects of dust in Model 1, as it would only remove the $z>3$ sources and because\nit is in any case inconsistent, see Sect. 3.2.2). \nDust associated with the residual ISM \nextinguishes the light emitted by high redshift objects.\nThe idea is that the SF efficiency in field galaxies with deep dark-matter \npotentials is not so high to produce a galactic wind on\nshort timescales after the onset of SF. Hence an ISM is present in the \ngalaxy for an appreciable fraction of the Hubble time, during which the ISM \nis progressively enriched of metals and dust. While the gas fraction\ndiminishes as stars are continuously formed, \nits metallicity increases and keeps the dust optical depth\nroughly constant with time.\n\nWe have then complemented the Model 2 of Sect. 3.1 to account for\nthe effect of extinction and re-radiation by dust in star-forming\nmolecular regions and more diffused in the galaxy body during the active\nphase of SF using a model by Silva et al. (1988). \nThe model is precisely as described in Sect. 3.1 for the whole passively\nevolving phase after the end of SF.\n\nA detailed description of the code is given in Silva et al.\nHere we summarize its basic features.\nThe amount of dust in the galaxy is assumed proportional to the residual gas\nfraction and chemical abundance of C and Si, while the\nrelative amount of molecular to diffuse gas ($M_m\/M_g$) is a model parameter.\nThe molecular gas is divided into spherical clouds of assigned mass\n($5\\ 10^{5}\\; \\rm{M}_{\\odot}$) and radius ($16\\ pc$). \nEach generation of stars born within the cloud progressively\nescapes it on timescales of several tens of Myrs. The two parameters \nregulating this process (see eq. [8] in Silva et al. 1998) are the age \n$t_0$ of \nthe oldest stars still embedded and the escape timescale $\\alpha$.\nThe emerging spectrum is obtained by\nsolving the radiative transfer through the cloud.\nBefore escaping the galaxy, the light arising from young stars\/molecular\ncomplexes, as well as from older stellar generations, further interacts\nwith dust present in the diffuse gas component (the latter dominates the \nglobal galactic extinction at late epochs, when the SF rate is low).\nThe dimming of starlight and consequent dust emission are\ncomputed by describing the galaxy as a spherically symmetric system\nsubdivided into volume elements, with radial dependencies of stars and gas \ndescribed by a King profile with a core radius $r_c=200\\ pc$,\nas reasonable for early-type systems. \n\nIn the general framework for the time evolution of the SF rate \ngiven by Model 2 (see Sect. 3.1 and the continuous line in Fig. 3), \nwe found a self-consistent \nsolution for the dust extinction and emission with the following \nparameter values: baryonic mass $M=10^{11}M_\\odot$, (typical for our sample\ngalaxies), $M_m\/M_g=0.3$, $t_0=0.1\\ Gyr$ and $\\alpha=100\\ Gyr^{-1}$. After \n3.1 Gyr,\nthe balance of gas thermal energy vs. input from supernovae breaks, and\nformally most of the small residual gas is then lost by the galaxy. \nAfter this event, a very low-level SF activity may keep on, due to either\npartial efficiency of the wind, or to stellar re-cycling.\nThe average stellar metallicity of the remnant is roughly solar.\n\nThis scheme naturally accounts for the evidence previously discussed \nin Sect. 3 of a substantial spread in the ages of distant field E\/S0,\nshowing a combination of massive\namounts of old stellar populations and younger stars.\n\nThe same scheme, when convolved with a K-band LLF, \nsuccessfully accounts for the statistical \ndistributions in Figs. 12 and 13, in addition to other galaxy counts in the\nK-band. Now the cutoffs of both D(z) at $z>1.3$ and of the counts for E\/S0s \nat $K>19$ are reproduced as the effect of dust extinction during the SF phase.\n\nWe stress again that, though not physically unplausible, our treatment of\nthe SF as a continuous process during a substantial fraction of the\nHubble time at high-z is likely an over-semplification of a more complex\nprocess characterized by a discrete set of SF episodes. However, there is \nvirtually no difference between the two cases as far as the \"observable\"\nlater, passively evolving, phase is considered.\n\n\n\\section{THE STAR-FORMATION HISTORY OF EARLY-TYPE FIELD GALAXIES}\n\n\nThis Section is devoted to a quantitative evaluation \nof the star-formation history of early-type galaxies in the field. \nThough it is not certainly the first time this is attempted,\nour approach is substantially different and complementary, and brings some\nsignificant advantages with respect to previous efforts\n(see e.g. Lilly et al. Madau et al. 1996; Connolly et al. 1997; Madau, \nPozzetti, \\& Dickinson 1998).\n\nMost of the previous analyses have concentrated on very large \n(e.g. thousands of objects for analyses dealing with the HDF itself or the \nCFRS) samples of galaxies without morphological differentiation and with \nsometimes very limited fractions (typically $<10\\%$ if we exclude CFRS) of\nspectroscopic identifications, directly\nobserved throughout the whole redshift range sampled by the very\nsensitive optical images.\n\nIn addition to the admittedly uncertain redshift estimation via the Lyman\ndrop-out technique, a recognized difficulty inherent in these \"direct\"\nevaluations of the cosmological star-formation rate is due to the\nessentially unknown effect of dust. Given the poor knowledge of dust\nproperties, an extinction correction is subject to tremendous uncertainties\neven for high-z galaxies with good optical spectra, which are in any\ncase a minority.\n\nOur present approach attempts to by-pass all these kinds of problems by\nconfining the analysis to a relatively small sample (35 objects)\nwith extremely accurate information per object.\nSuch an information concerns high-quality photometric data over a large\nwavelength interval ($0.3<\\lambda <2.2\\ \\mu m$), a good fraction (43\\%)\nof spectroscopically confirmed redshifts, a very accurate morphological\ninformation allowing to select a sample (that of early-types)\nwith likely homogeneous properties of star-formation.\nThe sample galaxies have been processed through a grid of synthetic\ngalaxy spectra, with the essential aim to date the stellar populations\npresent in the galaxy at the time of observation.\n\nThe selection waveband, the K band, has been chosen not only to minimize\nthe selection effects, due to the evolutionary and K-corrections, but\nalso to allow testing a wide range of stellar ages and masses, not\npossible if only optical data were considered.\nThe basic uncertain factor when dating stellar populations, i.e. \nthe metallicity of stars, was discussed in Sect. 3.2: our\nconclusion was that, given the relatively blue colors of galaxies\nat the time of observation, our results on the ages cannot be\ndrastically in error unless we consider highly non- (either super- or\nsub-) solar metallicities. In particular our results should be correct\nif our objects, having mostly completed their star formation, are \ncharacterized by the same solar metallicities which are typically observed \nin local galaxies.\nThe results of this best-fitting process are summarized in Table 2.\n\nOur task here is to account for all reasonable selection effects operating\nin the sample, to correct for incompleteness (that is for the part of the LF\nunobserved because of the flux limit) and to build up a \"population\" \nhistory from those of the single galaxies.\n\nThis derivation of the SF history, though subject to the mentioned\nmodellistic uncertainties, should be minimally or not affected at all by the\ndust extinction problem, as it deals with stellar populations during\na relatively late dust-free phase.\n\nOur approach is otherwise the same as attempted by Lilly et al. (1996) and\nMadau et al. (1996), among others.\nWe consider the contribution of all galaxies in our sampled volume\nto the star-formation rate per unit comoving volume. The time-dependent\nstar-formation rate per any single galaxy is given by that of the \nbest-fitting model\n(i.e. Model 2 in the vast majority of galaxies, Model 1 for a \ncouple of them), and by the formation redshift\n$z_F$, everything scaled according to the best-fit baryonic mass $M$.\n\nThe contribution of each galaxy to the global comoving SF rate has been\nestimated by dividing the time-dependent SF by the maximum\ncomoving volume $V_{max}$ within which the object would still be visible\nabove the sample flux limit.\nWe defer to Lilly et al. (1995, 1996) for a detailed treatment of how \n$V_{max}$ is computed.\n\nA further correction factor to apply to the comoving SF rate takes into \naccount the portion of the\nluminosity function lost by the flux-limited sample within any\nredshift interval (at any $z$ only galaxies brighter than K=20.1\ncan be detected). The correction is simply computed as the \nratio of the total luminosity density to the luminosity-weighted integral \nof the LF above the luminosity corresponding to the flux limit at that $z$. \nThis correction for completeness is not\nvery large (of the order of a few tens percent), because our adopted\nK-band LLF is relatively flat at the faint end (see Connolly et al., 1997,\nfor further details on this point).\n\nThe global rate of SF $\\Psi (z)$, which is the mass in stars formed per\nyear and unit comoving volume (expressed in solar masses per year and \nper cubic Mpc), is then the summed contribution by all \ngalaxies in our sample. Figure 15 reports different estimates of \n$\\Psi (z)$ based on the two assumptions of $q_0=0.15$ and $q_0=0.5$. \nThe latter, in particular, is compared with estimates\nby Lilly et al. (1996), Madau et al. (1996) and Connolly et al. (1997),\nwhile a similar match is not possible for our best guess solution\nwith $q_0=0.15$. \n\nThe calculation of $V_{max}$ and of the completeness correction depend\non the assumed rate of luminosity evolution.\nThe shaded regions in Fig. 15 mark two boundaries estimated with two \ndifferent evolutionary \ncorrections for luminosity evolution, to provide an idea of\nthe uncertainties. \nThe lower limit is computed from Model 1 (moderate evolution at $z<3.5$, no\nz-cutoff). As such, it provides a strict lower boundary to $\\Psi (z)$.\nFor the upper curve, $V_{max}$ is computed after Model 2, i.e. assuming\nstronger luminosity evolution and a cutoff in the available volume at\n$z\\simeq 1.5$. This upper curve corresponds to our best-guess $\\Psi (z)$.\nUnfortunately, a precise evaluation of the uncertainties is not\npossible at this stage.\n\nFigures 16 and 17 transform the information on the global SF rate\n$\\Psi (z)$ of Fig. 15 into one on the actual mass density in stars \nsynthesized within any redshift interval (in solar masses\nper unit comoving volume and per redshift interval). \nThis is simply the time integral of the function $\\Psi [z]$, and better \nquantifies when the various stellar populations are actually formed.\nFigs. 16 and 17 correspond to the usual two choices for $q_0$, with \n$V_{max}$ computed from Model 2.\nIn each figure this information is also differentiated into various \ngalactic mass intervals.\n\nAs it is apparent, there is a tendency for low-mass galaxies to have \na star-formation activity protracted to lower redshifts. Indeed, several \nmoderate to low-mass galaxies show young ages at low redshifts (see Fig. 7).\nHowever, although we have been as careful as possible in correcting for all\nvarious selection and incompleteness effects, we cannot be conclusive about this,\nin particular because of the limited size of our sample. Deeper and richer\nsamples are needed to settle the issue.\n\nThe dashed lines in the various panels of Figs. 16 and 17 provide the\nintegral distributions (reported on a linear vertical scale ranging from 0 to\n100\\% in each panel)\nof the stellar mass synthesized as a function of $z$. For example, for $q_0=0.15$,\nthe dashed line in the top panel of Fig. 16 shows that in massive \ngalaxies the SF is already over at $z=1$, while the third panel \nindicates that, for $M<5\\ 10^{10}\\ M_\\odot$, 40\\% of the stars are generated \nat $z<1$.\n\nIt is evident from Figs. 15 to 17 that the two cases considered, of an\nEinstein-de Sitter or an open universe, entail somehow different\nsolutions for the evolutionary star-formation rate. In the closure world model\nwith $\\Lambda=0$\nthere is a limited amount of time between the redshift of the observation\n(typically $z\\sim 1$) and the Big-Bang: this implies relatively higher\nvalues of $z_F$ and a function $\\Psi (z)$ keeping flat to the highest $z$.\nAn open universe, instead, with more cosmic time available, entails typically\nlower values of $z_F$ and a $\\Psi (z)$ peaked at slightly lower redshifts.\n\nThis reflects on the mass fractions of stars generated at the various\nredshifts. For $q_0=0.15$, 80\\% of the mass in stars is formed between\nz=1 and z=3, while the remaining 20\\% is made at higher or lower $z$.\n For $q_0=0.5$, the same 80\\% of stars are formed between z=1.2 and z=4.2.\nThe median redshifts in the stellar formation process turn out to be\nclose to z=1.8 and z=2.5 in the two cases respectively.\n\nThe SF rate $\\Psi (z)$ for our early-type field galaxy population\nis compared in Fig. 15 with the average rate\n(shaded horizontal regions) expected for early-type galaxies in rich clusters,\nassumed that these synthesize between z=5 and 1.5 all metals observed \nin the intra-cluster medium (Mushotzky \\& Loewenstein 1997). \nThe comparison is mediated by a crucial\nassumption we made on the stellar IMF to have a Salpeter's form. \nAn IMF more weighted in favor of massive stars would decrease the \nSF requirement for cluster ellipticals. \nThere is in any case an indication that field ellipticals\nbehave as less efficient metal producers than the rich cluster counterparts.\nIt is quite conceivable that the large number of dynamical interactions\nunderwent by galaxies in the denser cluster environment might have triggered\na more intense SF during the early Gyrs of the cluster lifetime, with a\nglobally more efficient metal yield.\n\nThis evidence also matches a likely difference in the ages of cluster \nversus field ellipticals: while the properties of cluster galaxies imply\nthat the SF was completed already before z=2 (Stanford et al. 1998), \ntypically 50\\% of stars in the field are produced at more recent epochs.\n\nFinally, it is interesting to compare in Figure 15 (lower panel) the \nindications on the SF history $\\Psi (z)$, as inferred from direct \ninspection of high-redshift galaxies (see data points in the figure), \nwith our predictions based on \nmodelling of the stellar populations. The latter refer to only a sub-class \nof galaxies, those\nclassified as early-types. Yet, this prediction already exceeds the global\namount inferred by high-z \"drop-outs\".\nThere is a suggestion, in particular, that for $z>1.5$ the direct estimate\nbased on optical-UV measurements misses some light, if a sub-population alone\nalready exceeds the global estimated amount. This may be taken as an evidence\nthat dust indeed played a role in at least partly obscuring the young\nstars during the active phase of star formation. If moderately \ntrue for the average\ngas-rich spiral, this might have been particularly relevant during the\nstarburst events bringing to the formation of early-type galaxies.\n\n\n\\section{CONCLUSIONS AND PERSPECTIVES}\n\nEarly-type galaxies are commonly believed to contain among the oldest stellar \nsystems in the universe. Analyses of galaxies in rich clusters,\nwhich found weakly evolving spectra and non-evolving dynamical conditions\nup to z=1 and slightly above, have confirmed this intuition.\n\nIn spite of a morphological similarity which may be taken as indicative of a \ncommon origin, no conclusive results were achieved so far about the early-type \ngalaxies in the field. Our contribution here has been devoted to exploit\na near-IR sample \nof 35 distant early-types in the HDF with $K<20.15$, with optimal \nmorphological information and spectro-photometric coverage, to study\nproperties of distant early-type galaxies outside rich clusters. \nNear-IR observations provide a view of galaxy evolution minimally biased\nby the effects of the evolutionary K-correction, dust extinction and\nchanges in the $M\/L$ ratio due to the aging of stellar populations. \nThe basic conclusions of our analysis may be summarized as follows.\n\n\n\\begin{itemize}\n\n\n\\item \nThe broad-band spectral energy distributions of the sample galaxies, \ntogether with the assumption of a Salpeter's initial mass function\nwithin $M_l=0.15$ and $M_u=100\\ M_\\odot$, allow us to date their dominant \nstellar populations. The majority of\nbright early-type galaxies in this field are found at redshifts $z\n\\lesssim 1.3$ to display colors indicative of a fairly wide range of\nages (typically 1.5 to 3 Gyrs).\nThis evidence adds to that of a substantially broad distribution of the\nformation redshifts $z_F$ from galaxy to galaxy in the sample. So, there is\nno coeval event of star-formation for early-types in this field, \nprobably at variance with what happens for cluster galaxies.\nThere is a tendency for lower-mass systems to be typically younger than the\nmassive counterparts, but a firm conclusion has to wait for deeper and richer \nsamples.\nA spread in the ages and a somewhat protracted SF activity in field, as \nopposed to cluster, ellipticals may be consistent with their observed\nnarrow-band spectral indices ($H_\\beta$, $Mg2$, etc.; see\nBressan, Chiosi \\& Tantalo, 1996).\n\nA protracted SF activity in field ellipticals may also bear some relationship\nwith their observed\npreferentially \"disky\" morphology (Shioya and Taniguchi, 1993).\nUnfortunately, our sample is too faint to allow testing this. Of the 15\ngalaxies (out of 35) bright enough to have the $c_4$ \nmorphological parameter measured, 11 show rather clearly a \"disky\" morphology, \nand the other 4 a \"boxy\" shape. \n\n\n\\item\nThe basic uncertainty when dating stellar populations -- the metallicity of \nstars -- has been the subject of a careful discussion. Given the relatively \nblue colors of the detected galaxies at the typical redshift of one, \nthe age uncertainty cannot be very large, as it would when considering\nold populations in local galaxies. Our result should then be fairly\nrobust.\n\n\\item\nBecause of the different cosmological timescales, the redshift-dependent\nstar-formation history inferred from these data depends to some extent on the\nassumed value for the cosmological deceleration parameter. \nWe find that the major episodes of star-formation\nbuilding up 80\\% of the mass in stars for typical $M^{\\star}$ galaxies \nhave taken place during the wide\nredshift interval between $z=1$ and $z=4$ for $q_0$=0.5, which becomes $z=1$ \nto $z=3$ for $q_0$=0.15.\nLower-mass ($M<5\\ 10^{10}\\ M_{\\odot}$) systems tend to have their bulk of SF \nprotracted to lower redshifts (down to almost the present time).\n\n\\item\nThe previous items dealt with the ages of the dominant stellar populations\nin distant field galaxies. The evolution history for the dynamical \nassembly of these stars into a galactic body might\nhave been entirely different. However, our estimated galactic\nmasses, for a Salpeter IMF, are found in the range from a few $\\sim 10^{9}\\\nM_\\odot$ to a few $10^{11}\\ M_\\odot$ already at $z\\simeq 1$. So the\nmassive end of the E\/S0 population appears to be mostly in place by that \ncosmic epoch, with space densities, masses and luminosities consistent with\nthose of the local field population. \nIt is to be investigated how this result compares with published findings\n(see e.g. Kauffmann \\& Charlot 1998)\nof a strong decrease of the comoving mass density of early-type\ngalaxies already by $z\\simeq 1$. We suggest that a question to keep under\nscrutiny in these analyses concerns the color classification, as we find,\nafter a detailed morphological and photometric analysis, that these objects \nusually display blue young populations mixed with old red stars.\n\n\\item\nThe present sample is characterized by a remarkable absence of objects at \n$z>1.3$, which should be \ndetectable during the luminous star-formation phase expected to\nhappen at these redshifts. This conclusion seems robust:\nif optical spectroscopy has difficulties to enter this redshift\ndomain, our analysis, using broad-band galaxy SEDs when spectroscopic\nredshifts are not available, has no obvious biases.\nThe uncertainty in the photometric estimate of $z$ for this kind\nof spectra is small. We discuss solutions for this sudden disappearance\nin terms of: \n{\\it a)} merging events, triggering the SF, which imply strongly perturbed \nmorphologies and which may prevent selecting them by our\nmorphological classification filter, and {\\it b)} a dust-polluted\nISM obscuring the (either continuous or episodic) events of star-formation, \nafter which gas consumption (or a galactic wind) cleans up the galaxy.\nWe conclude that the likely solution is a combination thereof, \ni.e. a set of dust-enshrouded merging-driven starbursts occurred during\nthe first few Gyrs of the galaxy's lifetime.\n\n\\item\nA comparison between the observed SF rates with the level needed to synthesize\nthe metals in the ICM indicates that field ellipticals could have been \nslightly less efficient\nmetal producers than the cluster galaxies. This difference adds to the\none in the ages and age spread.\n\n\\item\nThe comparison in Fig. 15b of our estimated SF rate $\\Psi (z)$ \nat $z>1.5$ with those inferred\nfrom the optical colors of high-redshift galaxies (via the Lyman \"drop-out\" \nor other photometric techniques) provides a {\\it direct indication that a \nfraction of light\nemitted during the starburst episodes (in particular those concerning the\nformation of early-type galaxies) has been lost, probably obscured by dust}. \nThe recently detected IR\/sub-millimetric\nextragalactic background (Puget et al. 1996; Hauser et al. \n1997) may be a trace\nof this phenomenon. Our results on the cosmological SF rate reported \nin Fig. 15b (note that our best-guess coincides with the upper limit\nof the shaded region) are very close to the level predicted by Burigana\net al. (1997) to reproduce the spectral intensity of the cosmic \ninfrared background.\n\nIt will obviously be essential to have a direct test of such a \ndust-extinguished SF, but this will not be easy until powerful\ndedicated instrumentation will be available. \nThe difficulty is illustrated in Figure 18, \nshowing possible spectra corresponding to various ages of\na massive ($M=3\\ 10^{11}\\ M_\\odot$) galaxy at $z=2$, according to Model 2, \nduring the starburst and post-starburst phases. \nAs indicated by the figure, various planned missions\n(in launch-time order, SIRTF, ESA's FIRST,\nNASA's NGST, and some ground-based observatories in\nexceptionally dry sites, like the South Pole) could discover \nintense activity of star-formation at $z=1.5$ to 3 or 4.\nAlso interesting tests of these ideas could be soon achieved with SCUBA \non JCMT, and perhaps with ISO.\nIn any case, observations at long wavelengths will be needed to completely\ncharacterize the early evolutionary phases of (spheroidal) galaxies.\n\n\\item\nWhile our main conclusions are moderately dependent on the\nassumed value of $q_0$, an open universe or one with non-zero $\\Lambda$ are \nfavored in our analysis by \nthe match of the K-band local luminosity functions with the observed \nnumbers of faint distant galaxies. \n\n\\item\nThere are two main sources of uncertainty in our analysis. \nThe redshift distribution in Figure 12 (and in particular the\npronounced peak at $z=1.1$) may be somehow affected\nby the possible presence of a background cluster or group at $z\\sim 1$.\nA few to several of the galaxies with photometric redshifts in the interval \n$10$, will be denoted by $ {\\Omega_T} $. The time\ninterval of the cylindrical domain $ {\\Omega_T} $, that is the interval $(0,T)$ in\nthe case of $ {\\Omega_T} = \\Omega\\times (0,T)$, will be denoted by $I( {\\Omega_T} )$.\nBy $\\Ball{x}{r}$ I will mean an $n$ dimensional ball as a subset of $\\Real^n_x$ with center at $x$ and\nof radius $r$. $Q(x,t,r)$ will denote the cylinder\n$$\nQ(x,t,r):=\\Ball{x}{r}\\times (t-r^2,t).\n$$\nI will write $\\textrm{B}_r$ or $\\textrm{B}$ instead of\n$\\Ball{x}{r}$ when $x$ or $r$ are clear from the context. Similarly,\nsometimes I will write $Q_r$ or $Q$ instead of $Q(x,t,r)$. For a\nLebesgue measurable set $S$ by $|S|$ I will mean Lebesgue measure of\n$S$. Finally $V( {\\Omega_T} ; \\Real^n)$ will denote the closure of $C^1( {\\Omega_T} ; \\Real^n)$\nfunctions under the norm\n\\begin{equation*}\n\\norm{v}_{V( {\\Omega_T} )}^2 = \\sup_{t\\in I( {\\Omega_T} )} \\int_{\\Omega}|v(x,t)|^2\\,\\textrm{d}x +\n\\iint_{ {\\Omega_T} } |\\mathbf{\\nabla} v(x,t)|^2\\,\\textrm{d}x\\,\\textrm{d}t.\n\\end{equation*}\n\n\\section{Partial regularity of nonlinear systems}\nIn this section we recall partial regularity results for weak\nsolutions of quasi-linear parabolic systems of the form\n\\begin{equation}\\label{eq:nonlinear_system}\nu^i_t - (A^{\\alpha\\beta}_{ij}(x, \\mathbf{u}) u^j_{x_\\beta})_{x_\\alpha}=0 \\quad\n\\forall i \\in \\set{1\\ldots N},\n\\end{equation}\nwhere coefficients $A^{\\alpha\\beta}_{ij}$ satisfy the following condition of\n\\textbf{strong ellipticity}:\n\\begin{equation}\\label{eq:strong_ellipticity}\nA^{\\alpha\\beta}_{ij}\\xi^i_\\alpha \\xi^j_\\beta \\ge \\lambda|\\xi|^2\n\\textrm{ for all }\\xi\\in \\mathds{R} ^{nN}, \\textrm{ for some }\\lambda>0.\n\\end{equation}\nBy a weak solution in this case I mean a function $\\mathbf{u}\\in\nV( {\\Omega_T} ; \\Real^N)$ that satisfies\n$$\n\\iint_ {\\Omega_T} - u^iv^i_t + A^{\\alpha\\beta}_{ij} u^j_{x_\\beta}\nv^i_{x_\\alpha}\\,\\textrm{d}x\\textrm{d}t = 0, \\quad \\textrm{for all }\\mathbf{v}\\in \\Hkz{1}( {\\Omega_T} ;\\Real^N).\n$$\nIt is well known that weak solutions of systems of this type possess partial\nregularity under some appropriate continuity conditions on\n$A^{\\alpha\\beta}_{ij}$. The important result in this area is the following\nlocal regularity result due to Giaquinta and Struwe:\n~\\cite{GiaquintaStruwe82}\n\\begin{theorem}[Local regularity condition]\\label{thm:local_regularity}\nSuppose coefficients $A^{\\alpha\\beta}_{ij}$ satisfy condition\n\\eqref{eq:strong_ellipticity}, are continuous and bounded. Also\nsuppose $\\mathbf{u}\\in V( {\\Omega_T} )$ is a weak solution of\n\\eqref{eq:nonlinear_system}. Then, if for some $(x_0, t_0)\\in {\\Omega_T} $\n\\begin{equation}\\label{eq:local_regularity_condition}\n\\liminf_{R\\rightarrow 0}\\frac{1}{R^n}\\iint_{Q(x_0, t_0 ,R)} |\\mathbf{\\nabla} \\mathbf{u}|^2 \\,\\textrm{d}x\\textrm{d}t= 0,\n\\end{equation}\nthen $\\mathbf{u}$ is H\\\"older continuous in the neighborhood of $(x_0, t_0)$.\n\\end{theorem}\nCondition \\eqref{eq:local_regularity_condition} is the basis for the proofs of\neverywhere regularity that we will discuss in the rest of this paper. We will, however, need one more result due to Giaquinta and Struwe ~\\cite{GiaquintaStruwe82}.\n\\begin{lemma}[$\\Lp{p}$ estimate]\\label{lem:lp_estimate}\nLet $\\mathbf{u}$ be a weak solution of the system \\eqref{eq:nonlinear_system}. Then there exists an exponent $p>2$ such that $|\\mathbf{\\nabla} \\mathbf{u}|\\in \\Lp{p}_{loc}( {\\Omega_T} )$; moreover for all $Q_R\\subset Q_{4R} \\subset {\\Omega_T} $ we have\n\\begin{equation}\\label{eq:Lp_estimate}\n\\left(\\Xiint{\\rule{1.2em}{0.5pt}}_{Q_R} |\\mathbf{\\nabla} \\mathbf{u}|^p\\,\\textrm{d}x\\textrm{d}t\\right)^{1\/p}\\leq C \\left(\\Xiint{\\rule{1.2em}{0.5pt}}_{Q_{4R}} |\\mathbf{\\nabla} \\mathbf{u}|^2\\,\\textrm{d}x\\textrm{d}t\\right)^{1\/2}.\n\\end{equation}\n\\end{lemma}\n\n\\section{Generalized diffusion equations}\nIn this section I discuss the type of parabolic systems I will refer to as\n\\textbf{diffusion system}. Let $\\Phi:\\Real^N\\rightarrow \\mathds{R} $ be a strictly convex, twice continuously differentiable function with\n\\begin{equation}\\label{eq:strict_convexity}\n\\lambda |\\xi|^2 \\le \\Phi_{z_i z_j} \\xi^i \\xi^j \\leq \\Lambda |\\xi|^2.\n\\end{equation}\nThen we say that $\\mathbf{u}$ is a weak solution of\n\\begin{equation}\\label{eq:gen_diff_eq}\n\\mathbf{u}_t - \\Delta\\left[\\mathbf{\\nabla}\\Phi(\\mathbf{u})\\right] = 0\n\\end{equation}\nif $\\mathbf{u}\\in V( {\\Omega_T} ; \\Real^N)$ and for\nall $\\mathbf{w} \\in \\Hkz{1}( {\\Omega_T} ; \\Real^N)$\n\\begin{equation}\n\\iint_ {\\Omega_T} -u^iw^i_t + (\\Phi_{z_i}(\\mathbf{u}))_{x_\\alpha}\nw^i_{x_\\alpha} \\,\\textrm{d}x\\textrm{d}t = 0.\n\\end{equation}\nThis is a standard quasi-linear elliptic system of the type in\n(\\ref{eq:nonlinear_system}), since we can rewrite it as\n\\begin{equation}\nu^i_t + \\left(\\Phi_{z_i z_j}(\\mathbf{u})u^j_{x_\\alpha}\\right)_{x_\\alpha} = 0\n\\quad \\textrm{for all }i\\in \\set{1\\ldots N}.\n\\end{equation}\nIf in addition $\\mathbf{u}$ is bounded I say $\\mathbf{u}$ is bounded weak solution.\n\nAs I have mentioned before this equation is a generalization of scalar nonlinear diffusion equation. It has several nice properties. First, its flow is a contraction in $\\Hk{-1}(\\Omega)$ and its solutions are unique. Second, weak solutions of \\eqref{eq:gen_diff_eq} are in $\\Hk{1}_{loc}(0,T; \\Lp{2}_{loc}(\\Omega))$, that is their weak derivatives in time are in $\\Lp{2}_{loc}( {\\Omega_T} )$. Furthermore, if the solution is bounded, the gradient in $x$ is actually in $\\Lp{4}_{loc}( {\\Omega_T} )$.\n\nFirst I show that flow is a contraction in $\\Hk{-1}(\\Omega)$.\n\\begin{theorem}[Uniqueness]\\label{thm:flow_contraction}\nLet $u_0, u_1\\in V( {\\Omega_T} ; \\Real^N)$ be two weak solutions of \\eqref{eq:gen_diff_eq} with the same boundary conditions, that is $u_0(\\cdot,t)\\equiv u_1(\\cdot,t)$ on $\\partial \\Omega$ for almost all $t\\in [0,T]$. Denote by $i:\\Lp{2}(\\Omega) \\rightarrow \\Hk{-1}(\\Omega)$ the natural embedding of square integrable functions in $\\Hk{-1}$ defined by\n$$\ni(f)(\\phi) := \\int_\\Omega f\\phi \\,\\textrm{d}x.\n$$\nThen we have for $T \\geq t_1 \\geq t_0 \\geq 0$\n\\begin{equation}\\label{eq:contraction}\n\\norm{i(u_0(t_1)-u_1(t_1))}_{\\Hk{-1}(\\Omega)}\\leq e^{\\lambda(t_1-t_0)}\\norm{i(u_0(t_0)-u_1(t_0))}_{\\Hk{-1}(\\Omega)},\n\\end{equation}\nwhere $\\lambda$ as in \\eqref{eq:strict_convexity}.\n\\end{theorem}\n\\begin{proof}\nLet us denote by $f_h$ the Steklov average of $f$ defined as\n$$\nf_h(x,t):=\\int_t^{t+h} f(x,s)\\,\\textrm{d}s.\n$$\nAlso for simplicity we will write $\\mathbf{v}_k$ for $\\mathbf{\\nabla}\\Phi(\\mathbf{u}_k)$ with $k=1,2$. It is not very hard to show that $(\\mathbf{u}_k)_h$ and $(\\mathbf{v}_k)_h$ weakly satisfy\n\\begin{equation}\\label{eq:milified_eq}\n((\\mathbf{u}_k)_h)_t - \\Delta((\\mathbf{v}_k)_h)=0.\n\\end{equation}\nLet us denote the solution of\n$$\n\\Delta w = f, \\quad w\\equiv 0\\textrm{ on }\\Omega\n$$\nby $\\Delta^{-1}f$. Then the $\\Hk{-1}(\\Omega)$ norm of $i(f)$ is given by\n$$\n\\norm{i(f)}_{\\Hk{-1}(\\Omega)}^2 = \\int_\\Omega |\\mathbf{\\nabla} (\\Delta^{-1} f)|^2\\,\\textrm{d}x.\n$$\nFor simplicity for an $f\\in \\Lp{2}(\\Omega)$ let us write $\\norm{f}_{\\Hk{-1}}$ instead of $\\norm{i(f)}_{\\Hk{-1}(\\Omega)}$. Now fix $h$. For $t_0, t_1 \\in (h, T-h)$ we compute\n\\begin{equation}\\label{eq:uniquenes_mollify}\n\\left(e^{2\\lambda t}\\norm{(\\mathbf{u}_1)_h - (\\mathbf{u}_0)_h}^2_{\\Hk{-1}}\\right)\\Big|_{t_0}^{t_1} = I_1 + I_2,\n\\end{equation}\nwhere\n\\begin{equation*}\nI_1 = \\int_{t_0}^{t_1}2\\lambda e^{2\\lambda t}\\norm{(\\mathbf{u}_1)_h - (\\mathbf{u}_0)_h}^2_{\\Hk{-1}}\\Big|_{t}dt,\n\\end{equation*}\nand\n\\begin{equation*}\nI_2 = \\int_{t_0}^{t_1} \\int_\\Omega 2 e^{2\\lambda t}\\mathbf{\\nabla}\\Delta^{-1}( (\\mathbf{u}_1)_h - (\\mathbf{u}_0)_h)\\cdot \\mathbf{\\nabla}\\Delta^{-1}( (\\mathbf{u}_1)_h - (\\mathbf{u}_0)_h)_t \\,\\textrm{d}x\\textrm{d}t.\n\\end{equation*}\nUsing the equation and the fact that $\\mathbf{u}_0$ and $\\mathbf{u}_1$ have the same trace, for $I_2$ we obtain\n\\begin{equation*}\nI_2 = -\\int_{t_0}^{t_1} \\int_\\Omega 2 e^{2\\lambda t} ( (\\mathbf{u}_1)_h - (\\mathbf{u}_0)_h)\\cdot ( (\\mathbf{v}_1)_h - (\\mathbf{v}_0)_h)\\,\\textrm{d}x\\textrm{d}t\n\\end{equation*}\nTaking the limit of both sides of the equation \\eqref{eq:uniquenes_mollify} as $h\\rightarrow 0$ we get\n\\begin{equation}\n\\left(e^{2\\lambda t}\\norm{(\\mathbf{u}_1 - \\mathbf{u}_0}^2_{\\Hk{-1}}\\right)\\Big|_{t_0}^{t_1} = I_3 + I_4,\n\\end{equation}\nwhere\n\\begin{equation*}\nI_3 = \\int_{t_0}^{t_1}2\\lambda e^{2\\lambda t}\\norm{\\mathbf{u}_1 - \\mathbf{u}_0}^2_{\\Hk{-1}}\\Big|_{t}dt,\n\\end{equation*}\nand\n\\begin{eqnarray*}\nI_4 & = & -\\int_{t_0}^{t_1} \\int_\\Omega 2 e^{2\\lambda t} ( \\mathbf{u}_1 - \\mathbf{u}_0)\\cdot ( \\mathbf{v}_1 - \\mathbf{v}_0)\\,\\textrm{d}x\\textrm{d}t\\\\\n& \\leq & -\\int_{t_0}^{t_1} \\int_\\Omega 2\\lambda e^{2\\lambda t} |\\mathbf{u}_1 - \\mathbf{u}_0|^2\\,\\textrm{d}x\\textrm{d}t\\\\\n& \\leq & -\\int_{t_0}^{t_1} 2 \\lambda e^{2\\lambda t} \\norm{\\mathbf{u}_1 - \\mathbf{u}_0}^2_{\\Hk{-1}}\\Big|_{t}dt.\n\\end{eqnarray*}\nThus we see that $I_3+I_4 \\leq 0$ and we establish the result for $T > t_1 \\geq t_0 > 0$. We establish the Theorem by taking the limit as $t_0\\rightarrow 0$ and $t_1\\rightarrow T$.\n\\qed\\end{proof}\nLet us now derive the $\\Hk{2}$ estimates for the solutions of \\eqref{eq:gen_diff_eq}. In general, quasi-linear parabolic systems do not have $\\Hk{2}$ estimates and the fact that solutions of generalized diffusion equations do have them is the consequence of the special structure that equation \\eqref{eq:gen_diff_eq} possesses.\n\\begin{theorem}[$\\Hk{2}$ estimates]\\label{H2_estimate}\nLet $u\\in V( {\\Omega_T} ; \\Real^N)$ be a weak solution of \\eqref{eq:gen_diff_eq}. Then $u_t\\in \\Lp{2}_{loc}( {\\Omega_T} )$ and for $Q(x,t,r)\\subsetneq Q(x,t,R) \\subset {\\Omega_T} $ we have the following estimates\n\\begin{equation}\\label{eq:H2_estimate}\n\\iint_{Q(x,t,r)} |\\mathbf{u}_t|^2\\,\\textrm{d}x\\textrm{d}t\n\\leq \\frac{C}{(R-r)^2} \\iint_{Q(x,t,R)} |\\mathbf{\\nabla}\\mathbf{u}|^2 \\,\\textrm{d}x\\textrm{d}t,\n\\end{equation}\n\\begin{equation}\\label{eq:Hessian_estimate}\n\\iint_{Q(x,t,r)} |\\mathbf{\\nabla}^2 (\\mathbf{\\nabla}_z\\Phi(\\mathbf{u}))|^2\\,\\textrm{d}x\\textrm{d}t \\leq \\frac{C}{(R-r)^2} \\iint_{Q(x,t,R)} |\\mathbf{\\nabla}\\mathbf{u}|^2 \\,\\textrm{d}x\\textrm{d}t.\n\\end{equation}\n\\end{theorem}\n\\begin{proof}\nAs before we will write $\\mathbf{v}$ for $\\mathbf{\\nabla}\\Phi(\\mathbf{u})$ and $f_h$ for Steklov average of $f$. Denote by $\\xi\\in C^\\infty_0(\\Omega)$ a smooth bump function supported in $\\Ball{x}{R}\\subset\\Omega$, which is identically one on $\\Ball{x}{r}\\subset\\Omega$ with $\\norm{\\mathbf{\\nabla}\\xi}_\\infty \\leq C\/(R-r)$. Also denote by $\\eta\\in C^\\infty_0((t-R^2,t])$ a function that is identically one on $[t-r^2,t]$ and supported in $[t-R^2,t]$ with $|\\eta'|\\leq C\/(R^2-r^2)$. Then multiplying equation\n$$\n(\\mathbf{u}_h)_t - \\Delta (\\mathbf{v}_h) = 0\n$$\nby $(\\mathbf{v}_h)_t\\xi^2\\eta$ we obtain\n$$\n\\int_{t-R^2}^t \\int_\\Omega (\\mathbf{u}_h)_t(\\mathbf{v}_h)_t\\xi^2\\eta + \\mathbf{\\nabla} (\\mathbf{v}_h)\\mathbf{\\nabla}(\\mathbf{v}_h)_t\\xi^2\\eta\n+ \\mathbf{\\nabla} (\\mathbf{v}_h)(\\mathbf{v}_h)_t 2\\xi\\mathbf{\\nabla}\\xi\\eta \\,\\textrm{d}x\\textrm{d}t = 0.\n$$\nUsing strict convexity on the first term, integrating the second term in time by parts and using H\\\"older inequality twice we obtain\n\\begin{eqnarray*}\n\\int_{t-R^2}^t \\int_\\Omega \\frac{\\lambda}{2}|(\\mathbf{u}_h)_t|^2\\xi^2\\eta \\,\\textrm{d}x\\textrm{d}t & + & \\int_\\Omega |\\mathbf{\\nabla}(\\mathbf{v}_h)|^2\\xi^2\\eta \\Big|_t \\,\\textrm{d}x \\\\\n&\\leq & C (\\norm{\\eta'}_\\infty + \\norm{\\mathbf{\\nabla}\\xi}_\\infty^2)\\int_{t-R^2}^t \\int_\\Omega |\\mathbf{\\nabla}(\\mathbf{v}_h)|^2\\xi^2\\,\\textrm{d}x\n\\end{eqnarray*}\nThus taking a limit as $h\\rightarrow 0$ we deduce that $u_t\\in L^2_{loc}(\\Omega)$ and derive estimate \\eqref{eq:H2_estimate} as claimed.\nTo prove \\eqref{eq:Hessian_estimate}, we use $\\Hk{2}$ estimates for the Laplacian to get the following for a fixed $t$:\n$$\n\\int_{\\Ball{x}{r}\\times\\set{t}} |\\mathbf{\\nabla}^2 \\mathbf{v}|^2\\,\\textrm{d}x \\leq \\frac{C}{(R-r)^2} \\int_{\\Ball{x}{R}\\times\\set{t}} |\\mathbf{\\nabla}\\mathbf{v}|^2 \\,\\textrm{d}x+ \\int_{\\Ball{x}{R}\\times\\set{t}} |u_t|^2\\,\\textrm{d}x.\n$$\nIntegrating in time and using estimate \\eqref{eq:H2_estimate} we get\n\\begin{equation*}\n\\iint_{Q(x,t,r)} |\\mathbf{\\nabla}^2 \\mathbf{v}|^2\\,\\textrm{d}x\\textrm{d}t \\leq \\frac{C}{(R-r)^2} \\iint_{Q(x,t,R)} |\\mathbf{\\nabla}\\mathbf{v}|^2 \\,\\textrm{d}x\\textrm{d}t.\n\\end{equation*}\n\\qed\\end{proof}\n\nIn addition to $\\Hk{2}$ estimates for weak solutions, bounded weak solutions of equation \\eqref{eq:gen_diff_eq} are actually in $\\Lp{4}_{loc}$. This is rather unusual since most quasi-linear equations do not have $\\Lp{4}$ estimates.\n\\begin{theorem}[$\\Lp{4}$ estimate for bounded solutions]\\label{thm:lp_bound_est}\nLet $\\mathbf{u}$ be a weak bounded solution of the equation \\eqref{eq:gen_diff_eq}. Then $\\mathbf{\\nabla} \\mathbf{u}$ is locally in $\\Lp{4}$ and for $Q(x,t,r)\\subsetneq Q(x,t,R)\\subset {\\Omega_T} $ we have the following estimate:\n\\begin{equation}\\label{eq:L4_estimate}\n\\iint_{Q(x,t,r)} |\\mathbf{\\nabla}\\mathbf{u}|^4\\,\\textrm{d}x\\textrm{d}t \\leq\n\\frac{C\\norm{\\mathbf{u}}_\\infty^2}{(R-r)^2}\\iint_{Q(x,t,R)} |\\mathbf{\\nabla}\\mathbf{u}|^2\\,\\textrm{d}x\\textrm{d}t.\n\\end{equation}\n\\end{theorem}\n\\begin{proof}\nLet us again denote $\\mathbf{\\nabla} \\Phi(\\mathbf{u})$ by $\\mathbf{v}$ and let $\\tau: \\mathds{R} \\rightarrow \\mathds{R} $ be a smooth increasing function that is linear on $(-\\infty, 1]$ and constant on $[2, \\infty)$. For some large enough constant $C_1$, $C_1\\tau(z)\\geq z (\\tau'(z))^2$. Define $\\tau_\\epsilon$ as $\\tau_\\epsilon(x):= \\tau(\\epsilon x)\/\\epsilon$. Notice that $\\norm{\\tau_\\epsilon'}_\\infty \\leq C_0$ and $C_1\\tau_\\epsilon(z)\\geq z (\\tau_\\epsilon'(z))^2$ with constants independent of $\\epsilon$. Letting $\\xi$ be a smooth bump function as in the proof of Theorem above, multiply the equation \\eqref{eq:gen_diff_eq} by $\\mathbf{v}\\tau_\\epsilon(|\\mathbf{\\nabla}\\mathbf{v}|^2)\\xi^2$ and integrate by parts to obtain\n\\begin{multline*}\n\\iint_{Q(x,t,R)} u^i_tv^i\\tau_\\epsilon(|\\mathbf{\\nabla}\\mathbf{v}|^2)\\xi^2 + |\\mathbf{\\nabla}\\mathbf{v}|^2\\tau_\\epsilon(|\\mathbf{\\nabla}\\mathbf{v}|^2)\\xi^2 \\\\\n+ 2v^iv^i_{x_\\alpha}v^j_{x_\\beta}\\tau_\\epsilon'(|\\mathbf{\\nabla}\\mathbf{v}|^2)v^j_{x_\\alpha x_\\beta}\\xi^2 + 2v^iv^i_{x_\\alpha}\\tau_\\epsilon(|\\mathbf{\\nabla} \\mathbf{v}|^2)\\xi\\xi_{x_\\alpha}\\,\\textrm{d}x\\textrm{d}t=0.\n\\end{multline*}\nHence we compute\n\\begin{eqnarray*}\n\\iint_{Q(x,t,R)} |\\mathbf{\\nabla}\\mathbf{v}|^2\\tau_\\epsilon(|\\mathbf{\\nabla}\\mathbf{v}|^2)\\xi^2\\,\\textrm{d}x\\textrm{d}t \\leq \\iint_{Q(x,t,R)} C_2\\norm{\\mathbf{v}}_\\infty^2|\\mathbf{u}_t|^2\\xi^2 + \\frac{1}{2}\\tau^2_\\epsilon(|\\mathbf{\\nabla}\\mathbf{v}|^2)\\xi^2\\\\\n+ C_3\\norm{\\mathbf{\\nabla}\\xi}^2_\\infty \\norm{\\mathbf{v}}_\\infty^2 |\\mathbf{\\nabla}\\mathbf{v}|^2\n+ C_1\\norm{\\mathbf{v}}^2_\\infty|\\mathbf{\\nabla}^2\\mathbf{v}|^2 \\xi^2 + \\frac{1}{4C_1}|\\mathbf{\\nabla}\\mathbf{v}|^4 \\tau_\\epsilon'(|\\mathbf{\\nabla}\\mathbf{v}|^2)^2\\xi^2 \\,\\textrm{d}x\\textrm{d}t.\n\\end{eqnarray*}\nSimplifying and using estimates \\eqref{eq:H2_estimate}, \\eqref{eq:Hessian_estimate} and $C_1\\tau_\\epsilon(z)\\geq z (\\tau_\\epsilon'(z))^2$ we deduce\n$$\n\\iint_{Q(x,t,r)} |\\mathbf{\\nabla}\\mathbf{v}|^2\\tau_\\epsilon(|\\mathbf{\\nabla}\\mathbf{v}|^2)\\,\\textrm{d}x\\textrm{d}t \\leq\n\\frac{C\\norm{\\mathbf{u}}_\\infty^2}{(R-r)^2}\\iint_{Q(x,t,R)} |\\mathbf{\\nabla}\\mathbf{u}|^2\\,\\textrm{d}x\\textrm{d}t.\n$$\nTaking a limit as $\\epsilon\\rightarrow 0$ we deduce by monotone convergence theorem that $\\mathbf{\\nabla}\\mathbf{u}$ is locally in $\\Lp{4}$ and the estimate as claimed in the statement of the Theorem.\n\\qed\\end{proof}\n\\begin{remark}\nThe fact that the gradient is actually locally in $\\Lp{4}$ implies that the singular set of a bounded solution has parabolic Hausdorff dimension smaller than $n-2$. As you may recall from Theorem \\ref{thm:local_regularity} the singular set is contained in the set\n$$\n\\set{(x,t)\\in {\\Omega_T} \\ \\Big| \\ \\liminf_{R\\rightarrow 0}\\frac{1}{R^n}\\iint_{Q(x, t ,R)} |\\mathbf{\\nabla} \\mathbf{u}|^2\\,\\textrm{d}x\\textrm{d}t > 0 }.\n$$\nNow using the H\\\"older inequality and the fact that $\\mathbf{u}$ locally in $\\Lp{4}$ we compute\n\\begin{eqnarray*}\n\\frac{1}{R^n}\\iint_{Q(x,t,R)} |\\mathbf{\\nabla}\\mathbf{u}|^2 \\,\\textrm{d}x\\textrm{d}t & \\leq &\n\\frac{1}{R^n}\\left( \\iint_{Q(x,t,R)} |\\mathbf{\\nabla}\\mathbf{u}|^4\\,\\textrm{d}x\\textrm{d}t\\right)^{1\/2}|Q(x,t,R)|^{1\/2}\\\\\n&\\leq& C R^{\\frac{2-n}{2}}\\left(\\iint_{Q(x,t,R)} |\\mathbf{\\nabla}\\mathbf{u}|^4 \\,\\textrm{d}x\\textrm{d}t\\right)^{1\/2}\\\\\n&\\leq& C \\left(R^{2-n}\\iint_{Q(x,t,R)} |\\mathbf{\\nabla}\\mathbf{u}|^4 \\,\\textrm{d}x\\textrm{d}t\\right)^{1\/2}.\n\\end{eqnarray*}\nHence the singular set must be contained in the set\n$$\n\\set{(x,t)\\in {\\Omega_T} \\ \\Big| \\ \\liminf_{R\\rightarrow 0}\\frac{1}{R^{n-2}}\\iint_{Q(x, t ,R)} |\\mathbf{\\nabla} \\mathbf{u}|^4 \\,\\textrm{d}x\\textrm{d}t > 0 },\n$$\nwhich has $n-2$ parabolic Hausdorff measure zero.\n\\end{remark}\n\n\n\\section{The key lemma}\nIn this section I discuss the parabolic version of the lemma that seems to be\nkey in the proof of everywhere regularity of some elliptic systems. Not\nsurprisingly, it will turn out that the parabolic lemma is crucial to proving\neverywhere regularity of solutions to some types of parabolic systems. The\nelliptic lemma, to which I refer, is well known and the proof of it can be found\nin ~\\cite{Giaquinta83} in Chapter 7 as part of Theorem 1.1.\n\nBefore we proceed with the discussion of this elliptic lemma let us recall that\ncoefficients $a_{\\alpha\\beta}$ are called \\textbf{strictly elliptic} if there\nexists $\\lambda >0$ such that\n$$\na_{\\alpha\\beta}\\xi^\\alpha\\xi^\\beta \\ge \\lambda |\\xi|^2\\quad\\textrm{for all }\\xi\\in \\Real^n.\n$$\n\\begin{lemma}\\label{lem:elliptic_key_lemma}\nSuppose coefficients $a_{\\alpha\\beta}(x)$ are strictly elliptic, bounded and measurable. Let $u\\in V(\\Omega)$, $f\\in \\Lp{1}(\\Omega)$ be nonnegative functions satisfying\n\\begin{equation}\\label{eq:elliptic_key_lemma_key_pdi}\n- (a_{\\alpha\\beta}u_{x_\\beta})_{x_\\alpha} + f \\le 0\n\\end{equation}\non $\\Omega$. For any $x_0\\in\\Omega$ for which\n$\\Ball{x_0}{R_0}\\subset \\Omega$ for some $R_0$, we have the\nfollowing:\n\\begin{equation}\\label{eq:elliptic_key_lemma_Morrey_decay}\n\\liminf_{R\\rightarrow 0} \\frac{1}{R^{n-2}}\\int_{\\Ball{x_0}{R}} f\\,\\textrm{d}x = 0.\n\\end{equation}\n\\end{lemma}\n\nThe proof of this lemma is rather simple and follows easily from the elliptic\nHarnack inequality. Because of the peculiar geometry of the parabolic\nHarnack inequality, the elliptic proof does not translate directly into the parabolic case. Instead, by adopting proof of elliptic lemma to parabolic equations, one is able to control $f$ on the cylinders whose top centers are slightly shifted back in time. In fact it is not true that $f$ can be controlled on the cylinders whose top centers are not fixed, without additional assumptions on $f$. It turns out, however, that the assumption that one needs to impose to prove the parabolic version of the lemma is satisfied in applications to everywhere regularity of parabolic systems. What we need to assume is that for some $\\alpha > 1$ the $\\Lp{\\alpha}$ norm of $f$ on a cylinder is controlled by the $\\Lp{1}$ norm of $f$, perhaps on a larger cylinder.\n\n\\begin{lemma}[Key Lemma]\\label{lem:key_lemma}\nSuppose coefficients $a_{\\alpha\\beta}(x,t)$ are strictly\nelliptic, bounded and measurable.\nLet $u\\in V( {\\Omega_T} )$, $f\\in \\Lp{1}( {\\Omega_T} )$ be nonnegative\nfunctions weakly satisfying\n\\begin{equation}\\label{eq:key_lemma_key_pdi}\nu_t - (a_{\\alpha\\beta}u_{x_\\beta})_{x_\\alpha} + f \\le 0\n\\end{equation}\non $ {\\Omega_T} $.\nFurther suppose that for some $\\alpha >1$ our $f$ satisfies the following:\n\\begin{equation}\\label{eq:higher_Lp}\n\\left(\\Xiint{\\rule{1.2em}{0.5pt}}_{Q_R} f^\\alpha \\,\\textrm{d}x\\textrm{d}t\\right)^{1\/\\alpha}\\leq C \\ \\Xiint{\\rule{1.2em}{0.5pt}}_{Q_{4R}} f \\,\\textrm{d}x\\textrm{d}t,\n\\end{equation}\nfor all $Q_R\\subset Q_{4R}\\subset {\\Omega_T} $. Then for any $(x_0, t_0)\\in {\\Omega_T} $ for which\n$$\n\\textrm{B}_{R_0}(x_0)\\times (t_0 - R_0^2, t_0+R_0^2)\\subset {\\Omega_T} , \\textrm{ for some } R_0,\n$$\nwe have the following:\n\\begin{equation}\\label{eq:key_lemma_Morrey_decay}\n\\liminf_{R\\rightarrow 0} \\frac{1}{R^{n}}\\iint_{Q(x_0,t_0,R)} f \\,\\textrm{d}x\\textrm{d}t = 0.\n\\end{equation}\n\\end{lemma}\n\\begin{proof}\nFix $0 <\\sigma \\le 1\/4$. Set\n$$\nR_i:= \\sigma^i R_0, \\quad Q_i:= Q(x_0, t_0, R_i), \\quad M_i:= \\sup_{Q_i} u\n$$\nand\n$$\nQ'_i:= Q(x_0,t_0 - 4\\sigma^2 R_i^2, (1-8\\sigma^2)^{1\/2}R_i),\n$$\n$$\nQ''_i:= Q(x_0,t_0 - 2\\sigma^2 R_i^2, (1-4\\sigma^2)^{1\/2}R_i).\n$$\nWe divide the proof in three steps. In the first step we show that for any $\\sigma \\in (0, 1\/4]$ we have\n$$\n\\lim_{i\\rightarrow \\infty}\\frac{1}{R_i^n}\\iint_{Q'_i} f \\,\\textrm{d}x\\textrm{d}t = 0.\n$$\nOnce we have done that, we show that we can control $f$ on $Q_i\\backslash Q'_i$ with the help of the assumption on $f$. Finally we will put it all together to conclude the lemma. \\\\\n\\noindent \\emph{Step 1}. \nFix $i$, and set $z := M_i - u$. We see that $z \\geq 0$ on $Q_i$ and $z$ satisfies\n\\begin{equation}\\label{eq:kl_z}\nz_t - (a_{\\alpha\\beta} z_{x_\\beta})_{x_\\alpha} \\ge f.\n\\end{equation}\nIn particular, due to parabolic Harnack inequality (see Theorem 6.24 in ~\\cite{Lieberman96})\n\\begin{equation}\\label{eq:kl_harnack_z}\n\\Xiint{\\rule{1.2em}{0.5pt}}_{Q''_i} z\\,\\textrm{d}x\\textrm{d}t \\leq C \\inf_{Q_{i+1}} z.\n\\end{equation}\nLet $w$ solve backward time parabolic equation\n\\begin{equation}\\label{eq:kl_w}\n-w_t - (a_{\\alpha\\beta}w_{x_\\alpha})_{x_\\beta} = \\frac{1}{R^{2}_i}\\chi_{Q''_{i}}\n\\end{equation}\non $Q_i$ with $w\\equiv 0$ on the backward time parabolic\nboundary, that is\n$$\nw \\equiv 0 \\textrm{ on }(\\partial \\textrm{B}_{R_i}(x_0) \\times [t_0-R^2_{i}, t_0])\\cup\n(B_{R_i}(x_0)\\times \\set{t_0}).\n$$\nAt this point if $zw$ were actually differentiable in time, we\nwould multiply equation \\eqref{eq:kl_w} by $zw$ and\nintegrate by parts to obtain\n$$\n\\iint_{Q_i} -z\\left(\\frac{w^2}{2}\\right)_t + a_{\\alpha\\beta}\nz_\\beta\\left(\\frac{w^2}{2}\\right)_{x_\\alpha} + a_{\\alpha\\beta}\nw_{x_\\alpha}w_{x_\\beta}z \\,\\textrm{d}x\\textrm{d}t = \\frac{1}{R_i^2}\\iint_{Q''_{i}} zw\\,\\textrm{d}x\\textrm{d}t,\n$$\nand since $z$ satisfies equation \\eqref{eq:kl_z} we\nwould conclude that\n\\begin{equation}\\label{eq:key_lemma_sec_to_last}\n\\frac{1}{R_i^n}\\iint_{Q'_i} f\\left(\\frac{w^2}{2}\\right)\\,\\textrm{d}x\\textrm{d}t \\leq\n\\frac{1}{R_i^{n+2}}\\iint_{Q''_{i}} zw\\,\\textrm{d}x\\textrm{d}t.\n\\end{equation}\nIn general we cannot expect $zw$ to be differentiable in time.\nTo obtain equation \\eqref{eq:key_lemma_sec_to_last} rigorously\none would need to use Steklov average\n$$\n(zw)^h(x,t):=\\frac{1}{h}\\int_{t-h}^t z(x,\\tau)w(x,\\tau)\\,\\textrm{d} \\tau\n$$\nas a test function in \\eqref{eq:kl_w}. However, we will not do this\nhere, instead I refer the reader to Lemma 6.1 in\n~\\cite{Lieberman96}, where similar computation has been carried out.\n\nNow, since $w$ solves \\eqref{eq:kl_w}, by strong maximal\nprinciple $w \\ge \\theta >0$ on $Q'_i$ and also $w \\le C$\non $Q_i$, with bounds independent of $i$ (one can see this by\nscaling for example). Therefore, combining this observation with\ninequality \\eqref{eq:kl_harnack_z}, we obtain\n$$\n\\frac{1}{R_i^n}\\iint_{Q'_i} f \\,\\textrm{d}x\\textrm{d}t \\le C\\Xiint{\\rule{1.2em}{0.5pt}}_{Q''_{i}} z \\,\\textrm{d}x\\textrm{d}t \\le C \\inf_{Q_{i+1}} z = C (M_i - M_{i+1}).\n$$\nHowever, since\n$$\n\\sum_{i=0}^\\infty M_i - M_{i+1} \\leq \\sup_{Q_0} u,\n$$\nwe conclude that\n$$\n\\frac{1}{R_i^n}\\iint_{Q'_i} f \\,\\textrm{d}x\\textrm{d}t\\rightarrow 0 \\quad \\textrm{as}\\quad i \\rightarrow \\infty.\n$$\n\\\\\n\\noindent\\emph{ Step 2}.\nLet $A$ be some measurable set. We will show that for all $Q_R\\subset Q_{8R} \\subset Q_{R_0}$ and for all $\\epsilon$, there exists $\\delta$ such that if $|A\\cap Q_R|\\leq \\delta |Q_R|$, then\n$$\n\\frac{1}{R^n}\\iint_{A\\cap Q_R}f \\,\\textrm{d}x\\textrm{d}t \\leq \\epsilon.\n$$\nFirst, we can easily deduce by the argument similar to the one in part 1, that\n\\begin{equation}\\label{eq:kl_bound_f}\n\\frac{1}{(4R)^n}\\iint_{Q_{4R}}f \\,\\textrm{d}x\\textrm{d}t \\le C,\n\\end{equation}\nwhere constant is independent of $R$.\n\nWe now use \\eqref{eq:higher_Lp} to conclude that\n\\begin{eqnarray*}\n\\iint_{A\\cap Q_R} f\\,\\textrm{d}x\\textrm{d}t & \\le & \\left(\\frac{1}{|Q_R|}\\iint_{A\\cap Q_R} f^\\alpha \\,\\textrm{d}x\\textrm{d}t \\right)^{1\/\\alpha}|Q_R|^{1\/\\alpha}|A\\cap Q_R|^{1-1\/\\alpha} \\\\\n& = & C|Q_R|^{1\/\\alpha}|A\\cap Q_R|^{1- 1\/\\alpha} \\Xiint{\\rule{1.2em}{0.5pt}}_{Q_{4R}} f \\,\\textrm{d}x\\textrm{d}t \\\\\n& = & C\\left(\\frac{|A\\cap Q_R|}{|Q_R|}\\right)^{1-1\/\\alpha} \\iint_{Q_{4R}}f \\,\\textrm{d}x\\textrm{d}t \\leq C_1R^n\\delta^{1-1\/\\alpha}.\n\\end{eqnarray*}\nAbove, the last inequality follows by \\eqref{eq:kl_bound_f}.\n\n\\noindent \\emph{Step 3}.\\\\\nFinally we put everything together. Fix $\\epsilon >0$. First notice that by choosing $\\sigma$ small enough we can make\n$$\n|Q_i\\backslash Q'_i| \\leq \\delta |Q_i|,\n$$\nwhere $C_1\\delta^{1-1\/\\alpha} < \\epsilon \/2$. Then by first step we can find $i >2$ such that\n$$\n\\frac{1}{R_i^n}\\iint_{Q'_i} f \\,\\textrm{d}x\\textrm{d}t \\leq \\epsilon\/2.\n$$\nFinally, the above together with conclusion of second step gives us\n$$\n\\frac{1}{R_i^n}\\iint_{Q_i} f \\,\\textrm{d}x\\textrm{d}t = \\frac{1}{R_i^n}\\iint_{Q'_i} f \\,\\textrm{d}x\\textrm{d}t + \\frac{1}{R_i^n}\\iint_{Q_i\\backslash Q'_i} f \\,\\textrm{d}x\\textrm{d}t < \\epsilon.\n$$\n\\qed\\end{proof}\n\nNow we are in position to apply our key Lemma to deduce crucial importance of entropy in questions of everywhere regularity for parabolic systems.\n\\begin{theorem}[Entropy condition]\\label{thm:entropy_condition}\nLet $\\mathbf{u}$ be weak solution of \\eqref{eq:nonlinear_system}. Suppose there exists $\\phi\\in V( {\\Omega_T} )$ that together with $\\mathbf{u}$ weakly satisfy the following inequality:\n$$\n\\phi_t - (a_{\\alpha\\beta}\\phi_{x_\\beta})_{x_\\alpha} + \\lambda|\\mathbf{\\nabla} \\mathbf{u}|^2 \\leq 0,\n$$\nwhere $a_{\\alpha\\beta}$ are bounded and strictly elliptic. Then $\\mathbf{u}$ is everywhere H\\\"older continuous on the interior of $ {\\Omega_T} $.\n\\end{theorem}\n\\begin{proof}\nSince $\\mathbf{u}$ satisfies condition \\eqref{eq:Lp_estimate}, we see immediately that conditions of the key Lemma \\ref{lem:key_lemma} are satisfied. Therefore, we conclude that\n$$\n\\liminf_{R\\rightarrow 0}\\frac{1}{R^n}\\iint_{Q(X_0,R)} |\\mathbf{\\nabla} \\mathbf{u}|^2 \\,\\textrm{d}x\\textrm{d}t= 0.\n$$\nHowever, this is precisely the condition \\eqref{eq:local_regularity_condition} of Theorem \\ref{thm:local_regularity}. Hence we conclude that $\\mathbf{u}$ is everywhere H\\\"older continuous on the interior of $ {\\Omega_T} $.\n\\qed\\end{proof}\n\n\\section{Everywhere regularity of certain diffusion systems}\nIn this section we come back to the discussion of weak solutions of the diffusion system \\eqref{eq:gen_diff_eq}, imposing additional requirement that $\\Phi(z)$ is a function of only the norm of $z$. As we will see this allows us to conclude much more about solutions of this equation. In particular solutions are bounded if they are bounded initially and at the boundary. More importantly, solutions are actually everywhere H\\\"older continuous and thus smooth if $\\Phi$ is.\n\nFirst I will show that in the case when $\\Phi$ is a functions of only the norm, weak solutions of equation \\eqref{eq:gen_diff_eq} that are bounded initially and bounded at the boundary will remain bounded for all time.\n\\begin{theorem}[Boundedness]\\label{thm:boundedness}\nLet $\\mathbf{u}$ be a weak solution of the generalized diffusion equation \\eqref{eq:gen_diff_eq}. Also suppose that $\\norm{\\mathbf{u}(\\cdot,0)}_{\\Lp{\\infty}(\\Omega)}< \\infty$ and $\\norm{\\mathbf{u}(\\cdot,t)}_{\\Lp{\\infty}(\\partial\\Omega)} < \\infty$ for all $t\\in [0,T]$. Then $\\norm{\\mathbf{u}}_{\\Lp{\\infty}(\\Omega)} < \\infty$ and we have\n\\begin{equation}\\label{eq:sup_bound}\n\\norm{\\mathbf{u}(\\cdot, t)}_{\\Lp{\\infty}(\\Omega)}\\leq \\max\\set{\\norm{\\mathbf{u}(\\cdot,0)}_{\\Lp{\\infty}(\\Omega)}, \\sup_{s\\in [0,t]}\\norm{\\mathbf{u}(\\cdot,s)}_{\\Lp{\\infty}(\\partial\\Omega)} }.\n\\end{equation}\n\\end{theorem}\n\\begin{proof}\nSet $B$ to\n$$\nB = \\max\\set{\\norm{\\mathbf{u}(\\cdot,0)}_{\\Lp{\\infty}(\\Omega)}, \\sup_{s\\in [0,t]}\\norm{\\mathbf{u}(\\cdot,s)}_{\\Lp{\\infty}(\\partial\\Omega)}}.\n$$\nFix $\\epsilon > 0$ that is less than one. Let $\\gamma: \\mathds{R} _+\\rightarrow \\mathds{R} $ be a smooth convex function which is identically zero on $[0,B+\\epsilon]$, positive and increasing otherwise, and linear on $[B+1, \\infty)$. Also set $\\Gamma(z)$ to $\\gamma(|z|)$. Then we compute\n\\begin{eqnarray*}\n\\frac{d}{dt}\\int_\\Omega \\Gamma(\\mathbf{u}(x,t))\\,\\textrm{d}x & = & \\int_\\Omega \\Gamma_{z_i}u^i_t\\,\\textrm{d}x \\\\\n& = & \\int_\\Omega \\left(\\Gamma_{z_i}\\Phi_{z_i z_j} u^j_{x_\\alpha}\\right)_{x_\\alpha}\n- \\Gamma_{z_i z_k}\\Phi_{z_k z_j} u^i_{x_\\alpha}u^j_{x_\\alpha} \\,\\textrm{d}x \\\\\n& \\leq & \\int_{\\partial\\Omega} \\Gamma_{z_i}\\Phi_{z_i z_j} u^j_{x_\\alpha} \\nu_\\alpha \\,\\textrm{d}S \\\\\n& = & 0.\n\\end{eqnarray*}\nThe inequality is true because Hessians of two functions of only the norm commute and both $\\Gamma$ and $\\Phi$ are convex. The last equality is true because $\\gamma$ is identically zero on $[0,B+\\epsilon]$ and $|\\mathbf{u}|$ is less than or equal to $B$ on the boundary. Since $\\Gamma(\\mathbf{u})$ is positive and initially zero we conclude that $\\Gamma$ is zero up to time $t$ and thus $\\norm{\\mathbf{u}(\\cdot, t)}_\\infty \\leq B +\\epsilon$. Since the inequality is true for all $\\epsilon >0$ the Theorem follows.\n\\qed\\end{proof}\n\nThe next Lemma will show that if we suppose that $\\Phi(z)$ is a function only of the norm of $z$, then there exists an entropy that satisfies conditions of Theorem \\ref{thm:entropy_condition}.\n\\begin{lemma}\\label{lem:gen_diff_scalar}\nLet $\\mathbf{u}$ be a weak bounded solution of \\eqref{eq:gen_diff_eq} and suppose\n$\\Phi$ is of the form\n$$\n\\Phi(\\mathbf{u}) = \\phi(|\\mathbf{u}|).\n$$\nThen there is a continuously differentiable, strictly increasing function\n$\\gamma: \\mathds{R} \\rightarrow \\mathds{R} $ such that $\\phi=\\phi(|\\mathbf{u}|)$ weakly satisfies the following\ninequality:\n\\begin{equation}\n\\phi_t - \\Delta(\\gamma(\\phi)) + \\lambda^2 |\\mathbf{\\nabla}\\mathbf{u}|^2 \\leq 0.\n\\end{equation}\n\\end{lemma}\n\\begin{proof}\nFirst of all, without loss of generality we can suppose that $\\Phi(0)=0$.\nNotice that $\\phi$ satisfies the following equation weakly:\n$$\n\\phi_t - (\\Phi_{z_i}\\Phi_{z_i z_j}u^j_{x_\\alpha})_{x_\\alpha} +\n|\\mathbf{\\nabla}_x(\\mathbf{\\nabla}\\Phi(\\mathbf{u}))|^2=0.\n$$\nLooking at the quantity inside the divergence term we see that it is equal to\n$$\n\\Phi_{z_i}\\Phi_{z_i z_j}u^j_{x_\\alpha} =\n\\left(\\frac{1}{2}|\\mathbf{\\nabla}\\Phi(\\mathbf{u})|^2\\right)_{x_\\alpha}.\n$$\nSince $\\Phi(z) = \\phi(|z|)$, we observe that\n$$\n\\frac{1}{2}|\\mathbf{\\nabla} \\Phi(\\mathbf{u})|^2 = \\frac{1}{2} \\phi'(|\\mathbf{u}|)^2.\n$$\nLet $\\psi$ be the continuous inverse of $\\phi$. Set\n$$\n\\gamma(z) = \\int_0^z \\phi''(\\psi(t)) \\,\\textrm{d}t.\n$$\nThen multiplying $\\gamma'(\\phi(z))$ by $\\phi'(z)$ and integrating, we see that $\\gamma$ and $\\phi$ satisfy\n$$\n\\gamma(\\phi(z)) = \\frac{1}{2}\\phi'(z)^2.\n$$\nThis $\\gamma$ is continuously differentiable and strictly\nincreasing, since $\\phi$ is strictly convex. Therefore we\nconclude that $\\phi$ satisfies\n$$\n\\phi_t - \\Delta(\\gamma(\\phi))+|\\mathbf{\\nabla}_x(\\mathbf{\\nabla}\\Phi(\\mathbf{u}))|^2=0,\n$$\nand due to strict convexity the last term on the left hand side is greater or\nequal to $\\lambda^2 |\\mathbf{\\nabla} \\mathbf{u}|^2$.\n\\qed\\end{proof}\n\nNow we are in position to use the above Lemma \\ref{lem:gen_diff_scalar}\ntogether with Theorem \\ref{thm:entropy_condition} to deduce\n\\begin{theorem}\nWeak solutions of equation (\\ref{eq:gen_diff_eq}) are H\\\"older\ncontinuous in $ {\\Omega_T} $.\n\\end{theorem}\n\\begin{proof}\nLemma \\ref{lem:gen_diff_scalar} tells us, that for $\\mathbf{u}$ a weak bounded solution of \\eqref{eq:gen_diff_eq}, there exists $\\phi$ satisfying\n$$\n\\phi_t - (a\\phi_{x_\\alpha})_{x_\\alpha} + \\lambda^2 |\\mathbf{\\nabla}\\mathbf{u}|^2 \\leq 0,\n$$\nwhere $a(x,t) := \\gamma'(\\phi(x,t))$. However, this is precisely the condition of Theorem \\ref{thm:entropy_condition}. Therefore, we conclude the proof.\n\\qed\\end{proof}\n\n\\section{Strongly coupled parabolic systems}\nOne of the earliest nontrivial examples of quasi-linear\nparabolic systems whose solutions have interior everywhere\nregularity was due to Wiegner ~\\cite{Wiegner92}. These are the\nso-called strongly coupled parabolic systems of the following\nform:\n\\begin{equation}\\label{eq:strongly_coupled}\nu^i_t - (a_{\\alpha\\beta}u^i_{x_\\beta} +\nc^i_{\\alpha\\beta}H_{x_\\beta})_{x_\\alpha} = 0,\n\\end{equation}\nwhere\n\\begin{enumerate}\n\\item $H = H(\\mathbf{u})$ is a function of $\\mathbf{u}$;\n\\item $A^{ij}_{\\alpha\\beta}:= a_{\\alpha\\beta}\\delta^{ij} + c^i_{\\alpha\\beta}H_{z_j}$ and $a_{\\alpha\\beta}$\nare strictly elliptic in the sense that\n$$\n \\lambda |\\xi|^2\n\\le A^{ij}_{\\alpha\\beta}\\xi^i_\\alpha\\xi^j_\\beta,\\textrm{ and } \\lambda |\\zeta|^2 \\le a_{\\alpha\\beta} \\zeta^i \\zeta^j, \\quad \\textrm{for all }\\xi\\in \\mathds{R} ^{nN}, \\zeta\\in \\Real^n;\n$$\n\\item $H(z)$ is twice continuously differentiable and\n$\\lambda |\\zeta|^2 \\le H_{z_i z_j} \\zeta^i \\zeta^j;$\n\\item $a_{\\alpha \\beta}$ and $c^i_{\\alpha\\beta}$ are bounded.\n\\end{enumerate}\n\nAfter Wiegner, Dung ~\\cite{Dung99} also worked on these types of system. However, neither Wiegner's nor Dung's proofs of everywhere regularity for solutions of strongly coupled parabolic systems reduce to something analogous to the key Lemma \\ref{lem:key_lemma}. It is instructive to prove everywhere regularity of weak solutions to \\eqref{eq:strongly_coupled} using the key Lemma to illustrate an underlying similarity between strongly coupled parabolic systems and generalized diffusion equations \\eqref{eq:gen_diff_eq}. It appears that for both systems discussed in this paper the existence of an \\emph{entropy} is crucial for everywhere regularity of their solutions.\n\n\\begin{remark}\nWhen $\\Phi$ only depends on the norm of the gradient, diffusion system \\eqref{eq:gen_diff_eq} actually has the form of a strongly coupled system, except with possibly \\emph{non-convex} $H$. Indeed, if $\\Phi(z)= \\phi(|z|)$, then\n$$\n\\Phi_{z_i z_j}(z) = \\frac{\\phi'(|z|)}{|z|}\\delta_{ij} +\n\\frac{z_i}{|z|}\\left(\\phi''(|z|) - \\frac{\\phi'(|z|)}{|z|}\\right)\\frac{z_j}{|z|},\n$$\ntherefore, with\n$$\na_{\\alpha\\beta} = \\frac{\\phi'(|\\mathbf{u}|)}{|\\mathbf{u}|}\\delta_{\\alpha\\beta}, \\quad\nc^i_{\\alpha\\beta} =\\frac{u^i}{|\\mathbf{u}|}\\delta_{\\alpha\\beta},\n$$\nand\n$$\nH(z) = \\phi'(|z|) - \\int_0^{|z|} \\frac{\\phi'(s)}{s} \\,\\textrm{d}s,\n$$\nsystem \\eqref{eq:gen_diff_eq} has the form \\eqref{eq:strongly_coupled}.\n\\end{remark}\n\nWe can prove an interior everywhere regularity result for strongly coupled\nparabolic systems rather easily using Theorem \\ref{thm:entropy_condition}.\nAs in the previous section I will show existence of an entropy.\n\\begin{lemma}\nLet $\\mathbf{u} \\in V( {\\Omega_T} ; \\Real^N)$ be a weak bounded solution of\n(\\ref{eq:strongly_coupled}), then for some large enough $s$ there is a positive\nconstant $c$ such that $v := e^{sH}$ is a subsolution of the following\nequation:\n\\begin{equation}\\label{eq:strongly_coupled_scalar}\nv_t - (A_{\\alpha\\beta} v_{x_\\beta})_{x_\\alpha} + c|\\mathbf{\\nabla}\\mathbf{u}|^2 \\leq 0.\n\\end{equation}\n\\end{lemma}\n\\begin{proof}\nWe compute\n\\begin{eqnarray*}\n(e^{sH})_t &=& se^{sH}H_{z_i}(a_{\\alpha\\beta}u^i_{x_\\beta} +\nc^i_{\\alpha\\beta}H_{x_\\beta})_{x_\\alpha} \\\\\n& = & (A_{\\alpha\\beta}(e^{sH})_{x_\\beta})_{x_\\alpha} -\n(se^{sH}H_{z_i})_{x_\\alpha}(a_{\\alpha\\beta}u^i_{x_\\beta} +\nc^i_{\\alpha\\beta}H_{x_\\beta}).\n\\end{eqnarray*}\nThe last term on the right becomes\n\\begin{multline*}\n(se^{sH}H_{z_i})_{x_\\alpha}(a_{\\alpha\\beta}u^i_{x_\\beta}\n+c^i_{\\alpha\\beta}\nH_{x_\\beta}) = s^2 e^{sH}A_{\\alpha\\beta}H_{x_\\alpha} H_{x_\\beta} \\\\\n+ se^{sH}H_{z_i z_j}a_{\\alpha\\beta}u^i_{x_\\alpha}u^j_{x_\\beta} +\nse^{sH}H_{z_i z_j}c^i_{\\alpha\\beta}H_{x_\\beta}u^i_{x_\\alpha} \\\\\n\\ge se^{sH}\\left( \\lambda s|\\mathbf{\\nabla} H|^2 + \\lambda |\\mathbf{\\nabla} \\mathbf{u}|^2\n-C(\\epsilon)|\\mathbf{\\nabla} H|^2 - \\epsilon|\\mathbf{\\nabla} \\mathbf{u}|^2\\right) \\ge\n\\frac{\\lambda}{2} se^{sH}|\\mathbf{\\nabla} \\mathbf{u}|^2.\n\\end{multline*}\nThe last inequality follows by first making $\\epsilon$ small and then $s$ large. Since\n$\\mathbf{u}$ is bounded, we have $H$ is bounded from below. Therefore, for some\n$c$, $v$ satisfies equation (\\ref{eq:strongly_coupled_scalar}) as claimed.\n\\qed\\end{proof}\nAt this point we immediately conclude that conditions of Theorem \\ref{thm:entropy_condition} are satisfied. Therefore we have established\n\\begin{theorem}\nBounded weak solutions of (\\ref{eq:strongly_coupled}) are\nH\\\"older continuous in $ {\\Omega_T} $.\n\\end{theorem}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section*{Introduction}\n\nConsider the following coin-tossing game. To play the game, we will begin with a graph that has one active node with infinitely many inactive nodes on either side of it. In this paper, we will also refer to active nodes as occupied sites and inactive nodes as unoccupied sites. We will label the initial occupied site the origin. At the origin, there is a queue of particles that will perform random walks on the graph. This is seen in Figure \\ref{fig:begstate}, where occupied sites are black, unoccupied sites are white, the origin is labeled with ``$\\star$'' and the particles in the queue are indicated with ``$\\circ$''.\n\n\\begin{figure}\n\\[\n\\begin{tikzpicture}\n\\tikzstyle{wn}=[inner sep =3, fill=white, draw=black, circle];\n\\tikzstyle{bn}=[inner sep =3, fill=black, draw=black, circle];\n\\draw (0,0)--(1,0) node[wn] {} --(2,0) node[wn] {}--(3,0) node[wn] {}--(4,0) node[bn] {}--(5,0) node[wn] {}--(6,0) node[wn] {}--(7,0);\n\\draw (4,.2) node[above] {$\\circ$}; \\draw (4,.45) node[above] {$\\circ$}; \\draw (4,.7) node[above] {$\\circ$}; \n\\draw (4,1) node[above] {$\\vdots$};\n\\draw (4,-.2) node[below] {$\\star$};\n\\draw (0,0) node[left] {$\\cdots$};\n\\draw (7,0) node[right] {$\\cdots$};\n\\end{tikzpicture}\n\\]\n\\caption{The starting graph for our coin-tossing game.}\\label{fig:begstate}\n\\end{figure}\n\nWe imagine dropping the first particle in the queue onto the origin. Flipping heads will move the particle one position to the right and flipping tails will move it one position to the left. The game consists of flipping a coin repeatedly until the particle reaches an unoccupied site. Once an unoccupied site is reached, the particle settles there and occupies it. Now, the next particle in the queue is dropped onto the origin and the game begins again.\n\nThe illustration in Figure \\ref{fig:game} shows an example of the game being played for the first two particles in the queue. \nIn the example, we first flip heads. Thus, the first particle moves to the unoccupied site immediately to the right of the origin, settles there, and occupies it. Now, the graph has two occupied sites, each with infinitely many unoccupied sites on each side. Since a new site was occupied, the second particle in the queue is dropped onto the origin and the game begins again. With the next toss, there are two unoccupied sites that could become occupied: the site to the left of the origin or the site two to the right of the origin. Suppose the next coin toss is heads, followed by tails, then tails again. These three tosses would move the new particle to the occupied site, back to the origin, and then to the unoccupied site to the left of the origin. As a result, the particle settles at this site and occupies it. Next, the third next particle in the queue is dropped on the origin and the process continues.\n\n\\begin{figure}\n\\[\n\\begin{tikzpicture}\n\\draw (0,0) node[scale=.75] (a) {\n\\begin{tikzpicture}\n\\draw (2.5,0) -- (3,0) node[inner sep =3, fill=white, draw=black, circle] {}--(4,0) node[inner sep =3, fill=black, draw=black, circle] {}--(5,0) node[inner sep =3, fill=white, draw=black, circle] {} -- (5.5,0);\n\\draw (4,-.2) node[below] {$\\star$}; \\draw (4,.2) node[above] {$\\circ$}; \\draw (2.5,0) node[left] {$\\dots$}; \\draw (5.5,0) node[right] {$\\dots$};\n\n\\end{tikzpicture}\n};\n\\draw (0,-2) node[scale=.75] (a2) {\n\\begin{tikzpicture}\n\\draw (2.5,0) -- (3,0) node[inner sep =3, fill=white, draw=black, circle] {}--(4,0) node[inner sep =3, fill=black, draw=black, circle] {}--(5,0) node[inner sep =3, fill=white, draw=black, circle] {} -- (5.5,0);\n\\draw (4,-.2) node[below] {$\\star$}; \\draw (5,.2) node[above] {$\\circ$}; \\draw (2.5,0) node[left] {$\\dots$}; \\draw (5.5,0) node[right] {$\\dots$};\n\\end{tikzpicture}\n};\n\\draw (5.5,-2) node[scale=.75] (b) {\n\\begin{tikzpicture}\n\\draw (2.5,0) -- (3,0) node[inner sep =3, fill=white, draw=black, circle] {}--(4,0) node[inner sep =3, fill=black, draw=black, circle] {}--(5,0) node[inner sep =3, fill=black, draw=black, circle] {}--(6,0) node[inner sep =3, fill=white, draw=black, circle] {} -- (6.5,0);\n\\draw (4,-.2) node[below] {$\\star$}; \\draw (4,.2) node[above] {$\\circ$}; \\draw (2.5,0) node[left] {$\\dots$}; \\draw (6.5,0) node[right] {$\\cdots$};\n\\end{tikzpicture}\n};\n\\draw (5.5,-4) node[scale=.75] (b2) {\n\\begin{tikzpicture}\n\\draw (2.5,0) -- (3,0) node[inner sep =3, fill=white, draw=black, circle] {}--(4,0) node[inner sep =3, fill=black, draw=black, circle] {}--(5,0) node[inner sep =3, fill=black, draw=black, circle] {}--(6,0) node[inner sep =3, fill=white, draw=black, circle] {} -- (6.5,0);\n\\draw (4,-.2) node[below] {$\\star$}; \\draw (5,.2) node[above] {$\\circ$}; \\draw (2.5,0) node[left] {$\\dots$}; \\draw (6.5,0) node[right] {$\\cdots$};\n\\end{tikzpicture}\n};\n\\draw (5.5,-6) node[scale=.75] (b3) {\n\\begin{tikzpicture}\n\\draw (2.5,0) -- (3,0) node[inner sep =3, fill=white, draw=black, circle] {}--(4,0) node[inner sep =3, fill=black, draw=black, circle] {}--(5,0) node[inner sep =3, fill=black, draw=black, circle] {}--(6,0) node[inner sep =3, fill=white, draw=black, circle] {} -- (6.5,0);\n\\draw (4,-.2) node[below] {$\\star$}; \\draw (4,.2) node[above] {$\\circ$}; \\draw (2.5,0) node[left] {$\\dots$}; \\draw (6.5,0) node[right] {$\\cdots$};\n\\end{tikzpicture}\n};\n\\draw (5.5,-8) node[scale=.75] (b4) {\n\\begin{tikzpicture}\n\\draw (2.5,0) -- (3,0) node[inner sep =3, fill=white, draw=black, circle] {}--(4,0) node[inner sep =3, fill=black, draw=black, circle] {}--(5,0) node[inner sep =3, fill=black, draw=black, circle] {}--(6,0) node[inner sep =3, fill=white, draw=black, circle] {} -- (6.5,0);\n\\draw (4,-.2) node[below] {$\\star$}; \\draw (3,.2) node[above] {$\\circ$}; \\draw (2.5,0) node[left] {$\\dots$}; \\draw (6.5,0) node[right] {$\\cdots$};\n\\end{tikzpicture}\n};\n\\draw (11.5,-8) node[scale=.75] (c) {\n\\begin{tikzpicture}\n\\draw (1.5,0) -- (2,0) node[inner sep =3, fill=white, draw=black, circle] {}--(3,0) node[inner sep =3, fill=black, draw=black, circle] {}--(4,0) node[inner sep =3, fill=black, draw=black, circle] {}--(5,0) node[inner sep =3, fill=black, draw=black, circle] {}--(6,0) node[inner sep =3, fill=white, draw=black, circle] {} -- (6.5,0);\n\\draw (4,-.2) node[below] {$\\star$}; \\draw (4,.2) node[above] {$\\circ$}; \\draw (1.5,0) node[left] {$\\dots$}; \\draw (6.5,0) node[right] {$\\cdots$};\n\\end{tikzpicture}\n};\n\\draw[->] (a) -- node[midway,right] {H} (a2);\n\\draw[->] (b) -- node[midway,right] {H} (b2);\n\\draw[->] (b2) -- node[midway,right] {T} (b3);\n\\draw[->] (b3) -- node[midway,right] {T} (b4);\n\\draw[->] (a2) -- node[midway,above,sloped] {new particle} (b);\n\\draw[->] (b4) -- node[midway,above,sloped] {new particle} (c);\n\\end{tikzpicture}\n\\] \n\\caption{Playing the coin-tossing game.}\\label{fig:game}\n\\end{figure}\n\nIn our game, each particle on its own is performing a simple one-dimensional random walk, which is a well-studied stochastic model. See \\cite{Sauer}. This game we have described, of sending particles to take random walks in succession, is known as Internal Diffusion Limited Aggregation (Internal DLA). This process was defined by Meakin and Deutch in \\cite{MeakinDeutch} to serve as a model for ``chemical etching.'' In a more mathematical work by Diaconis and Fulton \\cite{DiaconisFulton}, Internal DLA was studied using the language of Markov chains, with interesting algebraic connections. The limiting behavior of this model has been frequently studied in dimensions two and three (e.g., in a square grid, the limiting shape is a ball), but here we will consider the one-dimensional case. \n\nSome potential questions we can ask in the one-dimensional case for Internal DLA are:\n\\begin{enumerate}\n \\item What is the probability that, after releasing $n$ particles, $k$ of them settle to the right of the origin? \n \\item On average, how many coin tosses will it take for $n$ particles to settle? \n \\item Given a large queue of particles, if the coin is tossed $N$ times, what is the probability that there are $n$ occupied sites? \n \\item What if the particles do not perform a simple random walk and instead there is a probability assigned to each site? \n\\end{enumerate}\nIn this paper, we prove results for questions $(1)$ and $(2)$. Diaconis and Fulton addressed question $(1)$ and found the limiting distribution (as $n \\to \\infty$) is normal \\cite[Proposition 3.2]{DiaconisFulton}. We give a more precise characterization here. While we have investigated questions $(3)$ and $(4)$, we do not yet have any major results. Some comments on further pursuit of these questions are included at the end of the paper. \n\nIn Section 1, we consider question $(1)$ as a probability distribution for our coin-tossing game. That is, given that $n$ particles have left the origin and settled, we determine the probability that $k$ are to the right of the origin. We first used a Monte Carlo simulation to predict a distribution. Each simulation ran $100\\,000$ trials of the game and returned the minimum and maximum positions reached on the graph (assuming the origin is at position 0). Table \\ref{tab:Monte1} shows the observed probability of the game reaching maximum ending position $k$ for $1 \\leq n \\leq 7$.\n\n\\begin{table}\n\\[\n\\begin{array}{c | c c c c c c c | c}\nn \\backslash k & 0 & 1 & 2 & 3 & 4 & 5 & 6 & SSE\\\\\n\\hline\n1 & 1.00000 &&&&&&& 0.00000 \\\\\n2 & 0.50266 & 0.49734 &&&&&& 1.41512\\mathrm{e}{-10} \\\\\n3 & 0.16730 & 0.66561 & 0.16709 &&&&& 1.69687\\mathrm{e}{-10} \\\\\n4 & 0.04235 & 0.45694 & 0.45763 & 0.04308 &&&& 4.90051\\mathrm{e}{-10}\\\\\n5 & 0.00833 & 0.21704 & 0.54711 & 0.21906 & 0.00846 &&& 1.42356\\mathrm{e}{-10}\\\\\n6 & 0.00131 & 0.08109 & 0.41734 & 0.41990 & 0.07903 & 0.00133 && 8.36380\\mathrm{e}{-10}\\\\\n7 & 0.00018 & 0.02418 & 0.23675 & 0.47821 & 0.23671 & 0.02381 & 0.00016 & 1.82768\\mathrm{e}{-10}\n\\end{array}\n\\]\n\\caption{Monte Carlo results for reaching ending position $k$ for $1 \\leq n \\leq 7$. }\\label{tab:Monte1}\n\\end{table}\n\nWe know that when $n=1, k=0$ with probability $1$ because the only occupied site is the origin. Furthermore, we know that when $n=2,$ both $k=0$ and $k=1$ occur with probability $\\frac{1}{2}$ because $50\\%$ of the time we will flip heads and settle at the site to the right of the origin, and $50\\%$ of the time we will flip tails and settle at the site to the left of the origin. The data in Table \\ref{tab:Monte1} supports this fact. \n\nWhen $n=3$, our data suggests that there is a common denominator of $6$, and when $n=4$, our data suggests the common denominator may be $24$. For example, when $n=3$ and $k=0$, we see a probability of $0.16730$ which is very close to $0.1\\overline{666} = \\frac{1}{6}$. Similarly, when $n=4$ and $k=0$, we see $0.04235$ which is very close to $0.041\\overline{6} = \\frac{1}{24}$. If we consider these observations with our known probabilities for $n=1$ and $n=2$, we see that our denominators for $1 \\leq n \\leq 4$ may be $n!$. \n\nUpon recognizing this, we multiplied each row of Table \\ref{tab:Monte1} by a common denominator of $n!$ to try and find a nice distribution for the numerators. For example, multiplying row $5$ by $5!$ yields\n\\[\n[0.9996 \\quad 26.0448 \\quad 65.6532 \\quad 26.2872 \\quad 1.0152].\n\\]\nAfter multiplication by $n!$, we found that our data looked very close to the triangle of Eulerian numbers. See Table \\ref{table:eulnum}. Upon comparing our data in Table \\ref{tab:Monte1} to the corresponding Eulerian number divided by $n!$, we got a very close fit. The sum of squared errors, denoted $SSE$, for each $1 \\leq n \\leq 7$ is shown in the last column in Table \\ref{tab:Monte1}, comparing our data to the corresponding Eulerian number divided by $n!$. Thus, our Monte Carlo simulation suggests that our coin-tossing game follows an Eulerian probability distribution and we will prove this result in Theorem \\ref{thm:eulprob}.\\footnote{As communicated by Lionel Levine, it seems Jim Propp made this connection in unpublished work. In fact it was Levine, in a visit to DePaul University, who suggested we investigate this question in the first place.}\n\nRecall that for any $0 \\leq k \\leq n-1,$ the \\textit{Eulerian number $\\eulerian{n}{k}$} is the number of permutations of $\\{1,2,\\dots,n\\}$ with $k$ descents. See \\cite{Petersen}. The Eulerian numbers satisfy the recurrence\n\\begin{equation}\\label{eq:eulrec}\n\\eulerian{n}{k} = (n - k)\\eulerian{n - 1}{k - 1} + (k + 1)\\eulerian{n - 1}{k},\n\\end{equation}\nfor any $n \\geq 1$ and $0 \\leq k \\leq n - 1$, with boundary values $\\eulerian{n}{0} = \\eulerian{n}{n-1} = 1$ for all $n\\geq 0$. Table \\ref{table:eulnum} shows the triangle of Eulerian numbers for $1 \\leq n \\leq 6$. The Eulerian numbers are a finite combinatorial distribution and our coin-tossing game is a stochastic process. How these mathematical concepts are related is very puzzling at first glance. \n\\begin{table}\n\\begin{tabular}{c | c c c c c c}\n$n \\backslash k$ & 0 & 1 & 2 & 3 & 4 & 5\\\\\n\\hline\n1 & 1 \\\\\n2 & 1 & 1 \\\\\n3 & 1 & 4 & 1 \\\\\n4 & 1 & 11 & 11 & 1 \\\\\n5 & 1 & 26 & 66 & 26 & 1 \\\\\n6 & 1 & 57 & 302 & 302 & 57 & 1\n\\end{tabular}\n\\caption{Triangle of the Eulerian numbers for $1 \\leq n \\leq 6$.}\n\\label{table:eulnum}\n\\end{table}\n\nIn Section 2, we consider question $(2)$, the expected run time of the game. From our Monte Carlo simulations we recorded the average number of coin tosses needed for $n$ particles to settle. Let $E_n$ denote this number of expected tosses. The data from our simulation is shown in Table \\ref{tab:Monte2}. \n\n\\begin{table}\n\\[\n\\begin{array}{c | c ccccccccccccc}\nn& 1 & 2 & 3 & 4& 5& 6& 7& 8& 9 & 10 & 15 & 20 \\\\\n\\hline\nE_n & 0 & 1 & 3.00 & 6.66 & 12.49 & 20.98 & 32.64 & 47.96 & 67.54 & 91.68 & 300.24 & 699.79\n\\end{array}\n\\]\n\\caption{Monte Carlo results for $E_n$ for $1 \\leq n \\leq 10$, $n=15,$ and $n=20$.}\\label{tab:Monte2}\n\\end{table}\n\nUsing Lagrange interpolation, we find the number of tosses needed for the game to have $n$ occupied sites is roughly\n\\[\n\\frac{1}{12} n^3 + \\frac{1}{12} n^2.\n\\]\nWe will prove this result for expected run time in Theorem \\ref{thm:tosstosn}. \n\n\\section{Probability Distribution}\n\nTo begin, we define the possible states of the game and a two-parameter family of probabilities relating these states. \n\nFor any $n \\geq 1$ and $0 \\leq k \\leq n - 1$, let $s(n,k)$ denote the graph with $n$ occupied sites, $k$ of which are to the right of the origin. State $s(n,k)$ is shown in Figure \\ref{fig:snk}. We think of these graphs as possible states of our coin-tossing game. For example, \n\\[\ns(7,4) = \\begin{tikzpicture}[baseline=0]\n\\tikzstyle{wn}=[inner sep =3, fill=white, draw=black, circle];\n\\tikzstyle{bn}=[inner sep =3, fill=black, draw=black, circle];\n\\draw (0,0)--(1,0) node[wn] {} -- (2,0) node[bn] {}--(3,0) node[bn] {}--(4,0) node[bn] {}--(5,0) node[bn] {}--(6,0) node[bn] {}--(7,0) node[bn] {}--(8,0) node[bn] {}--(9,0) node[wn] {}--(10,0);\n\\draw (4,-.2) node[below] {$\\star$}; \\draw (0,0) node[left] {$\\cdots$}; \\draw (10,0) node[right] {$\\cdots$};\n\\end{tikzpicture}.\n\\]\n\n\\begin{figure}\n\\[\n\\begin{tikzpicture}[baseline=0]\n\\tikzstyle{wn}=[inner sep =3, fill=white, draw=black, circle];\n\\tikzstyle{bn}=[inner sep =3, fill=black, draw=black, circle];\n\\draw (0,0)--(1,0) node[wn] {} -- (2,0) node[bn] {}--(3,0) node[bn] {}--(4,0);\n\\draw (4.75,0)--(5.75,0) node[bn] {}--(6.75,0) node[bn] {}--(7.75,0);\n\\draw (8.5,0)--(9.5,0) node[bn] {}--(10.5,0) node[bn] {}--(11.5,0) node[wn] {}-- (12.5,0);\n\\draw (0,0) node[left] {$\\cdots$}; \\draw (4,0) node[right] {$\\cdots$}; \\draw (7.75,0) node[right] {$\\cdots$}; \\draw (12.5,0) node[right] {$\\cdots$};\n\\draw (5.75,-.2) node[below] {$\\star$};\n\\draw [decoration={brace,amplitude=6pt},decorate] (6.75,0.3) -- (10.5,0.3) node[midway,above=8pt] {$k$};\n\\draw [decoration={brace,amplitude=6pt},decorate] (2,0.3) -- (5.75,0.3) node[midway,above=8pt] {$n-k$};\n\\end{tikzpicture}\n\\]\n\\caption{State $s(n,k)$.}\\label{fig:snk}\n\\end{figure}\n\nThe graph $s(n,k)$ can only be reached from $s(n - 1,k - 1)$ or $s(n - 1,k)$ for any $n \\geq 1$ and $1 \\leq k \\leq n-2$. For example, state $s(4,2)$ can be reached only from state $s(3,1)$ or state $s(3,2)$. The boundary cases, states $s(n,0)$ and $s(n, n - 1)$, are exceptions: $s(n,0)$ can only be reached from $s(n-1,0)$ and $s(n,n-1)$ can only be reached from $s(n-1,n-2)$. The possible states of the game for $0 \\leq n \\leq 5$ are shown in Figure \\ref{fig:states}, with edges to indicate which states can be obtained from another.\n\\begin{figure}\n\\begin{center}\n\\[\n\\begin{tikzpicture}[scale=.7]\n \\node (oz) at (0,4) {$s(1,0)$}; \n \\node (tz) at (-2,2) {$s(2,0)$}; \\node (two) at (2,2) {$s(2,1)$};\n \\node (thz) at (-4,0) {$s(3,0)$}; \\node (tho) at (0,0) {$s(3,1)$}; \\node (thtw) at (4,0) {$s(3,2)$};\n \\node (fz) at (-6,-2) {$s(4,0)$}; \\node (fo) at (-2,-2) {$s(4,1)$}; \\node (ftw) at (2,-2) {$s(4,2)$}; \\node (fth) at (6,-2) {$s(4,3)$};\n \\node (ffz) at (-8,-4) {$s(5,0)$}; \\node (ffo) at (-4,-4) {$s(5,1)$}; \\node (fftw) at (0,-4) {$s(5,2)$}; \\node (ffth) at (4,-4) {$s(5,3)$}; \\node (fff) at (8,-4) {$s(5,4)$};\n \\draw (oz) -- (tz); \\draw (oz) -- (two); \n \\draw (tz) -- (thz); \\draw (tz) -- (tho); \\draw (two) -- (tho); \\draw (two) -- (thtw);\n \\draw (thz) -- (fz); \\draw (thz) -- (fo); \\draw (tho) -- (fo); \\draw (tho) -- (ftw); \\draw (thtw) -- (ftw); \\draw (thtw) -- (fth);\n \\draw (fz) -- (ffz); \\draw (fz) -- (ffo); \\draw (fo) -- (ffo); \\draw (fo) -- (fftw); \\draw (ftw) -- (fftw); \\draw (ftw) -- (ffth); \\draw (fth) -- (ffth); \\draw (fth) -- (fff);\n \\draw (ffz) node[below] {\\vdots}; \\draw (ffo) node[below] {\\vdots}; \\draw (fftw) node[below] {\\vdots}; \\draw (ffth) node[below] {\\vdots}; \\draw (fff) node[below] {\\vdots};\n\\end{tikzpicture}\n\\]\n\\end{center}\n\\caption{Possible states of the game for $0 \\leq n \\leq 5$.}\\label{fig:states}\n\\end{figure}\n\nWhile we know that the game will transition from one state to the next by moving down the lattice in Figure \\ref{fig:states}, we would like to know how likely each transition is. For example, we can ask how likely it is that the game will go from state $s(2,1)$ to state $s(3,1)$.\n\nFor any $n \\geq 1$ and $0 \\leq k \\leq n - 1$, let $p_{n,k}$ denote the probability of transitioning from $s(n - 1,k)$ to $s(n,k)$. Similarly, let $q_{n,k}$ denote the probability of transitioning from $s(n - 1,k - 1)$ to $s(n,k)$. We will call $p_{n,k}$ and $q_{n,k}$ transition probabilities. The transition probabilities in and out of state $s(n,k)$ are shown in Figure \\ref{fig:snktrans}. \n\nTo help us calculate the transition probabilities, we consider the classic Gambler's Ruin problem. \n\n\\subsection{Gambler's Ruin}\\label{gambruin}\n\nIn the Gambler's Ruin problem, we consider two gamblers: Gambler A and Gambler B. The gamblers are playing a game where they toss a coin. If the coin flip is tails, Gambler A takes a dollar from Gambler B. Similarly, if heads, Gambler B takes a dollar from Gambler A. Each player starts with some money, say $k$ dollars for Gambler A and $l$ dollars for Gambler B. The game ends when either Gambler A wins all $k+l$ dollars (Gambler B is ruined), or when Gambler B wins all $k+l$ dollars (Gambler A is ruined). The result says that the probability that Gambler A wins is $\\frac{k}{k+l}$ and the probability Gambler B wins is $\\frac{l}{k+l}$. See \\cite{GrinsteadSnell}.\n\nThis relates to our model in a straightforward way. Let's assume Gambler A has $4$ dollars and Gambler B has $3$ dollars. This corresponds to the following graph: \n\n\\[\n\\begin{tikzpicture}[baseline=0]\n\\tikzstyle{wn}=[inner sep =2.5, fill=white, draw=black, circle];\n\\tikzstyle{bn}=[inner sep =2.5, fill=black, draw=black, circle];\n\\draw (0,0)--(.75,0) node[wn] {} -- (1.5,0) node[bn] {}--(2.25,0) node[bn] {}--(3,0) node[bn] {}--(3.75,0) node[bn] {}--(4.5,0) node[bn] {}--(5.25,0) node[bn] {}--(6,0) node[wn] {}--(6.75,0);\n\\draw (3,-.2) node[below] {$\\star$}; \\draw (3,.2) node[above] {$\\circ$}; \\draw (0,0) node[left] {$\\cdots$}; \\draw (6.75,0) node[right] {$\\cdots$};\n\\draw [->] (.75,-1) node[below] {B is ruined} -- (.75,-.2); \\draw [->] (6,-1) node[below] {A is ruined} -- (6,-.2);\n\\draw [decoration={brace,amplitude=6pt},decorate] (1.5,0.15) -- (3,0.15) node[midway,above=12pt] {B};\n\\draw [decoration={brace,amplitude=6pt},decorate] (3,0.15) -- (5.25,0.15) node[midway,above=12pt] {A};\n\\end{tikzpicture}.\n\\]\nWe see that the location of the particle on the graph keeps track of the fortune for each player. That is, the money Gambler A has is equal to the number of active nodes to the right of and including the particle, while Gambler B's fortune equals the number of active nodes to the left of and including the particle. \n\nIf the coin toss is heads, Gambler B will take a dollar from Gambler A. So, continuing with our example, Gambler B will have $4$ dollars and Gambler A will have $3$ dollars. Thus, the game moves to the following state:\n\\[\n\\begin{tikzpicture}[baseline=0]\n\\tikzstyle{wn}=[inner sep =2.5, fill=white, draw=black, circle];\n\\tikzstyle{bn}=[inner sep =2.5, fill=black, draw=black, circle];\n\\draw (0,0)--(.75,0) node[wn] {} -- (1.5,0) node[bn] {}--(2.25,0) node[bn] {}--(3,0) node[bn] {}--(3.75,0) node[bn] {}--(4.5,0) node[bn] {}--(5.25,0) node[bn] {}--(6,0) node[wn] {}--(6.75,0);\n\\draw (3,-.2) node[below] {$\\star$}; \\draw (3.75,.2) node[above] {$\\circ$}; \\draw (0,0) node[left] {$\\cdots$}; \\draw (6.75,0) node[right] {$\\cdots$};\n\\draw [->] (.75,-1) node[below] {B is ruined} -- (.75,-.2); \\draw [->] (6,-1) node[below] {A is ruined} -- (6,-.2);\n\\draw [decoration={brace,amplitude=6pt},decorate] (1.5,0.15) -- (3.75,0.15) node[midway,above=12pt] {B};\n\\draw [decoration={brace,amplitude=6pt},decorate] (3.75,0.15) -- (5.25,0.15) node[midway,above=12pt] {A};\n\\end{tikzpicture}\n\\]\n\n\nIn general, state $s(n-1,k-1)$ corresponds to Gambler A having $k$ dollars and Gambler B having $l$ dollars such that $k+l=n$. Thus we see that Gambler A wins if the game goes to state $s(n,k-1)$ and Gambler B wins if the game goes to state $s(n,k)$. The Gambler's Ruin problem tells us that Gambler A will win with probability $\\frac{k}{n}$ and Gambler B will win with probability $\\frac{n-k}{n}$. Thus, we conclude for any $n \\geq 1$ and $1 \\leq k \\leq n - 1$, \n\\begin{equation}\\label{eq:pqtransprobs}\np_{n,k-1} = \\frac{k}{n} \\quad \\mbox{ and } \\quad q_{n,k} = \\frac{n-k}{n}.\n\\end{equation} \n\n\\begin{figure}\n\\[\n\\begin{tikzpicture}[scale=.7]\n \\node (nk) at (2,0) {$s(n,k)$};\n \\node (qnk) at (0,2) {}; \n \\node (pnk) at (4,2) {};\n \\node (pn1k) at (0,-2) {};\n \\node (qn1k1) at (4,-2) {};\n \\draw (qnk) -- node[left]{$q_{n,k}$} (nk); \\draw (pnk) -- node[right]{$p_{n,k}$} (nk); \\draw (pn1k) -- node[left]{$p_{n+1,k}$}(nk); \\draw (qn1k1) -- node[right]{$q_{n+1,k+1}$}(nk);\n \\end{tikzpicture}\n \\]\n \\caption{Transition probabilities for state $s(n,k)$.}\\label{fig:snktrans}\n \\end{figure}\n \n \\subsection{Eulerian distribution}\n\nWe use these equations for our transition probabilities to obtain our first main result, which is the probability of starting the game at $s(1,0)$ and arriving at state $s(n,k)$. Let $P(n,k)$ denote this probability. For example, $P(1,0) = 1,$ and since we move immediately to $s(2,0)$ or $s(2,1)$ after our first coin toss, we see $P(2,0) = \\frac{1}{2} = P(2,1)$. For $n \\geq 3,$ things are more interesting. \n\nFirst, we will consider the transition to the boundary states (when $k=0$ or $k=n-1$). We know that state $s(n,0)$ can only be reached from $s(n-1,0)$, so $P(n,0)$ will only depend on $P(n-1,0)$. Similarly, state $s(n,n-1)$ can only be reached from $s(n-1,n-2)$, so $P(n,n-1)$ depends only on $P(n-1,n-2)$. Using our formulas for the transition probabilities in Equation \\eqref{eq:pqtransprobs}, we have \n\\[\nP(n,0) = p_{n,0} P(n-1,0) \\quad \\mbox{and} \\quad P(n,n-1) = q_{n,n-1} P(n-1,n-2).\n\\]\nSince $p_{n,0}=q_{n,n-1}= \\frac{1}{n}$ and $P(1,0)=1$, we find \n\\begin{equation}\\label{eq:boundary}\nP(n,0) = \\frac{P(n-1,0)}{n} = \\frac{1}{n!} \\quad \\mbox{ and } \\quad P(n,n-1) = \\frac{P(n-1,n-2)}{n} = \\frac{1}{n!}.\n\\end{equation}\n\nNext, we will look at the non-boundary cases. Because $s(n,k)$ can only be reached from states $s(n-1,k-1)$ and $s(n-1,k)$, we know that $P(n,k)$ will only depend on $P(n-1,k-1)$ and $P(n-1,k)$. Using our formulas for the transition probabilities in Equation \\eqref{eq:pqtransprobs}, we find for $1 \\leq k \\leq n-2$, the following recurrence for $P(n,k)$:\n\\begin{align}\nP(n,k) &= q_{n,k} P(n-1,k-1) + p_{n,k} P(n-1,k), \\nonumber \\\\\n&= \\frac{n - k}{n} \\cdot P(n - 1,k - 1) + \\frac{k + 1}{n} \\cdot P(n - 1,k) \\label{eq:recursiveP}.\n\\end{align}\nThe recurrence relation of $P(n,k)$ in Equation \\eqref{eq:recursiveP} looks similar to that of the Eulerian number $\\eulerian{n}{k}$ from Equation \\eqref{eq:eulrec}. \n\nIf we assume \n\\[\nP(n-1,k) = \\frac{\\eulerian{n-1}{k}}{(n-1)!},\n\\] \nfor $1 \\leq k \\leq n-2$, then,\n\\begin{align*}\nP(n,k) &= q_{n,k}P(n-1,k-1) + p_{n,k}P(n-1,k) \\\\\n &= \\frac{n-k}{n} \\cdot \\frac{\\eulerian{n - 1}{k - 1}}{(n-1)!} + \\frac{k+1}{n} \\cdot \\frac{\\eulerian{n - 1}{k}}{(n-1)!},\\\\\n &=\\frac{(n - k)\\eulerian{n - 1}{k - 1} + (k + 1)\\eulerian{n - 1}{k}}{n!},\\\\\n &= \\frac{\\eulerian{n}{k}}{n!},\n\\end{align*}\nwhere the final equality follows from the recurrence for Eulerian numbers in Equation \\eqref{eq:recursiveP}.\nThis proves our first theorem.\n\n\\begin{thm}\\label{thm:eulprob}\nFor any $n \\geq 1$ and $0 \\leq k \\leq n - 1$,\n\\[\nP(n,k) = \\frac{\\eulerian{n}{k}}{n!}.\n\\]\n\\end{thm} \n\nOur first main result is that $P(n,k)$, the probability of arriving at state $s(n,k)$, is modeled by the Eulerian probability distribution. In other words, we found a nice result for the probability that after emptying a queue of $n$ particles, $k$ of them settle to the right of the origin. With question (1) answered, we now move onto question (2): the expected amount of time (i.e., the number of coin tosses) it takes for $n$ particles to settle. \n\n\\section{Expected run time}\n\nWe next turn to the expected run time of our coin-tossing game, i.e., we want to know how many tosses it will take for $n$ particles to settle. Recall from the introduction that $E_n$ denotes the expected number of tosses needed to reach a state with $n$ occupied sites. Notice that $E_1 = 0$, and $E_2 = 1$ because flipping heads or tails will cause a site to become occupied immediately. It gets more complicated once we start looking at cases for $n \\geq 3$. In the introduction, we gave an estimate for $E_n$ based on our Monte Carlo simulation. We now turn this into our next theorem.\n\n\\begin{thm}\\label{thm:tosstosn}\nFor any $n \\geq 1$, the expected number of tosses needed to reach a state with $n$ occupied sites is \n\\begin{equation}\\label{eq:actualtoss}\nE_n = \\frac{1}{12} n^3 + \\frac{1}{12} n^2.\n\\end{equation}\n\\end{thm}\n\nIf we compute the first differences of this sequence ($E_{n} - E_{n-1}$), we find the change in expected tosses. Let $\\Delta E_n$ denote the expected number of tosses it takes for the $n^{th}$ particle to perform a random walk and settle, given $n-1$ sites are already occupied. \nAssuming Equation \\eqref{eq:actualtoss}, we find\n\\[\n\\Delta E_n = \\frac{1}{12} n^3 + \\frac{1}{12} n^2 - \\left(\\frac{1}{12} (n-1)^3 + \\frac{1}{12} (n-1)^2\\right) = \\frac{1}{4}n^2 - \\frac{1}{12}n.\n\\]\nThus to obtain our run time result in Theorem \\ref{thm:tosstosn}, it suffices to show that the number of expected tosses for the $n^{th}$ particle, once dropped, to settle in an unoccupied site is\n\\begin{equation}\\label{eq:newtossquad}\n\\Delta E_n = \\frac{1}{4}n^2 - \\frac{1}{12}n.\n\\end{equation}\n\nBut by definition, the expected number of new steps is \n\\[\n\\Delta E_n = \\sum_{k=1}^{n-1}P(n-1,k-1) \\cdot E(n-1,k-1),\n\\]\nwhere $E(n-1,k-1)$ denotes the expected number of tosses to go from state $s(n-1,k-1)$ to $s(n,k)$ or $s(n,k-1)$. By Theorem \\ref{thm:eulprob}, we can write\n\\[\n \\Delta E_n = \\sum_{k=1}^{n-1} \\frac{\\eulerian{n-1}{k-1}}{(n-1)!} \\cdot E(n-1,k-1) = \\frac{1}{(n-1)!} \\sum_{k=1}^{n-1} \\eulerian{n-1}{k-1} E(n-1,k-1),\n\\]\ni.e., $\\Delta E_n$ is a weighted sum of Eulerian numbers. We will show $E(n-1,k-1)$ is simply $k(n-k)$, which will help us to evaluate the sum and verify \\eqref{eq:newtossquad}, and hence Theorem \\ref{thm:tosstosn}.\n\n\\subsection{Escape time for a random walk}\n\nConsider that the game is in state $s(n-1,k-1)$ pictured below, \n\\[\n\\begin{tikzpicture}\n\\tikzstyle{wn}=[inner sep =3, fill=white, draw=black, circle];\n\\tikzstyle{bn}=[inner sep =3, fill=black, draw=black, circle];\n\\draw (0,0)--(1,0) node[wn] {} --(2,0) node[bn] {}--(3,0) node[bn]{} -- (4,0);\n\\draw (4,0) node[right] {$\\cdots$};\n\\draw (4.75,0)--(5.75,0) node[bn] {} --(6.75,0) node[bn] {} -- (7.75,0) node[bn] {} -- (8.75,0); \n\\draw (8.75,0) node[right]{$\\cdots$};\n\\draw (9.5,0)--(10.5,0) node[bn] {}--(11.5,0) node[bn]{} -- (12.5,0) node[wn]{} -- (13.5,0);\n\\draw (0,0) node[left] {$\\cdots$};\n\\draw (13.5,0) node[right] {$\\cdots$};\n\\draw (6.75,-.2) node[below] {$\\star$}; \\draw (12.5,.2) node[above] {$k$}; \\draw (1,.2) node[above] {$l$};\n\\draw [decoration={brace,amplitude=6pt},decorate] (7.75,0.3) -- (11.5,0.3) node[midway,above=8pt] {$k-1$};\n\\draw [decoration={brace,amplitude=6pt},decorate] (2,0.3) -- (5.75,0.3) node[midway,above=8pt] {$l-1$};\n\\end{tikzpicture}\n\\]\nwhere $k+l = n$. We would like to know how to compute $E(n-1,k-1),$ i.e., how many coin tosses it will take for the game to reach either state $s(n,k)$ (activating on the right) or state $s(n,k-1)$ (activating on the left). To help us, recall that for the simple random walk, the expected time to reach the boundary of an interval $[-b,a]$ is $ab$. See \\cite{Sauer}. We translate this escape time result into the language of our game with the following claim. \n\\begin{cl}\\label{cl:esctime}\nFor any $k,l \\geq 1$ such that $k+l=n$, the expected number of tosses to go from state $s(n-1,k-1)$ to $s(n,k)$ or $s(n,k-1)$ is $E(n-1,k-1) = kl = k(n-k)$.\n\\end{cl}\n\nFor example, consider state $s(5,3)$ below.\n\\[\n\\begin{tikzpicture}\n\\tikzstyle{wn}=[inner sep =3, fill=white, draw=black, circle];\n\\tikzstyle{bn}=[inner sep =3, fill=black, draw=black, circle];\n\n\\draw (9,-2)--(9.5,-2) node[wn] {} -- (10.25,-2) node[bn] {} -- (11,-2) node[bn] {} -- (11.75,-2) node[bn] {} -- (12.5,-2) node[bn]{} -- (13.25,-2) node[bn] {} -- (14,-2) node[wn] {} -- (14.5,-2); \n\\draw (9,-2) node[left] {$\\cdots$}; \\draw (14.5,-2) node[right] {$\\cdots$};\n\\draw (11,-2.2) node[below] {$\\star$}; \\draw (9.5,-2.2) node[below] {\\scriptsize $l=2$}; \\draw (14,-2.2) node[below] {\\scriptsize $k=4$};\n\\draw (11.75,-2.65) node[below] {$s(5,3)$};\n\\end{tikzpicture}\n\\]\n\nAs shown, $l=2$ and $k=4$ in $s(5,3)$. We can calculate, using Claim \\ref{cl:esctime}, the expected number of tosses it takes to go from each state with $n=5$ to a state with $n=6$ occupied sites. Consider the chart below:\n\\[\n\\begin{centering}\n\\begin{tabular}{c | c | c | c} \nState & $k$ & $l$ & $E(5,k-1) = kl$ \\\\\n\\hline\n$s(5,0)$ & $1$ & $5$ & 5 \\\\\n$s(5,1)$ & $2$ & $4$ & 8 \\\\\n$s(5,2)$ & $3$ & $3$ & 9 \\\\\n$s(5,3)$ & $4$ & $2$ & 8 \\\\\n$s(5,4)$ & $5$ & $1$ & 5 \\\\\n\\end{tabular}\n\\end{centering}\n\\]\nIf we know there are five occupied sites (but not which 5), and we want to know how many tosses it would take for the sixth particle to settle, we calculate a weighted sum using our Eulerian probabilities. This is precisely our definition for $\\Delta E_6$. \n\\[\n\\Delta E_6 = \\left(\\frac{1}{120}\\right)(5) + \\left(\\frac{26}{120}\\right)(8) + \\left(\\frac{66}{120}\\right)(9) + \\left(\\frac{26}{120}\\right)(8) + \\left(\\frac{1}{120}\\right)(5) = \\frac{17}{2}. \n\\]\nThis agrees with Equation \\eqref{eq:newtossquad}, \n\\[ \n\\frac{1}{4} \\left(6^2\\right) - \\frac{1}{12} \\left(6\\right) = \\frac{17}{2}.\n\\]\n\nIn general, since $P(n,k) = \\frac{\\eulerian{n}{k}}{n!}$ by Theorem \\ref{thm:eulprob} and $E(n-1,k-1) = kl$ by Claim \\ref{cl:esctime}, we have the following Corollary of Theorem \\ref{thm:eulprob} and Claim \\ref{cl:esctime}.\n\n\\begin{cor}\\label{cor:deltaen}\nFor any $n \\geq 3$, the expected number of tosses for the $n^{th}$ particle to settle, given $n-1$ sites are occupied is \n\\begin{align*}\n\\Delta E_n &= \\sum_{k=1}^{n-1} \\frac{\\eulerian{n-1}{k-1}}{(n-1)!} \\cdot k(n-k) \\\\\n&= \\frac{1}{(n-1)!} \\sum_{k=1}^{n-1} \\eulerian{n-1}{k-1} \\cdot k(n-k).\n\\end{align*}\n\\end{cor}\n\nIt remains to see that this also equals the formula in Equation \\eqref{eq:newtossquad}. We will do so using generating function techniques.\n\n\\subsection{Eulerian polynomials and generating functions}\n\nThroughout this section we use the technique of \\emph{generating functions} which is explored in \\cite{Wilf}, and in particular \\cite{Petersen} for Eulerian numbers. The basic idea is to study power series whose coefficients are the numbers we are interested in, and then to use properties of these functions to deduce properties of the coefficients. \n\nTo this end, we define the $n^{th}$ bivariate Eulerian polynomial, denoted $A_{n}(s,t)$, as \n\\[\nA_{n}(s,t) = \\sum_{k=1}^{n} \\eulerian{n}{k-1} s^{k} t^{n+1-k}, \n\\]\ne.g., $A_5 (s,t) = st^5 + 26s^2t^4 + 66s^3t^3 + 11s^4t^2 + s^5t$. \n\nWe can get $\\Delta E_n$ from $A_{n-1} (s,t)$ as follows. First, we differentiate with respect to $s$ and $t$,\n\\begin{align*}\n\\frac{\\partial}{\\partial s} \\frac{\\partial}{\\partial t} \\left[A_{n-1}(s,t)\\right] &= \\frac{\\partial}{\\partial s} \\frac{\\partial}{\\partial t} \\left[\\sum_{k=1}^{n-1} \\eulerian{n-1}{k-1}s^kt^{n-k} \\right], \\\\\n&= \\sum_{k=1}^{n-1} \\eulerian{n-1}{k-1}k(n-k)s^{k-1}t^{n-k-1},\n\\end{align*}\nand when $s=1$ and $t=1$, this is \n\\begin{equation}\\label{eq:wgtsum}\n\\sum_{k=1}^{n-1} \\eulerian{n-1}{k-1}k(n-k) = (n-1)! \\cdot \\Delta E_n.\n\\end{equation}\n\nFor example, consider the case for $n=6$. We differentiate $A_5 (s,t)$ with respect to $s$ and $t$ and then set $s=t=1$,\n\\[\n\\frac{\\partial}{\\partial s} \\frac{\\partial}{\\partial t} \\left[A_{5}(s,t)\\right]_{s=t=1} = (1\\cdot 5) + 26(2 \\cdot 4) + 66(3\\cdot 3) + 26(4 \\cdot 2) + (5\\cdot 1).\n\\]\nNotice that this is precisely the weighted sum for $n=6$ from Equation \\eqref{eq:wgtsum},\n\\[\n\\sum_{k=0}^{4} \\eulerian{5}{k-1} k(6-k) = 5! \\cdot \\Delta E_6.\n\\]\n\nBefore we can use this idea to prove Equation \\eqref{eq:newtossquad} holds in general, we consider the exponential generating function for the Eulerian polynomial $A_n(s,t)$, denoted $F(s,t,z)$. By definition, we have \n\\begin{equation}\\label{eq:expgenf}\nF(s,t,z) = \\sum_{n \\geq 1}A_{n}(s,t) \\frac{z^n}{n!}.\n\\end{equation}\nThat is, expanding in $z$, we find \n\\[\nF(s,t,z) = stz + (s^2t + st^2 ) \\frac{z^2}{2} + (s^3t + 4s^2t^2 + st^3) \\frac{z^3}{3!} + \\cdots .\n\\]\n\nTo find a useful formula for $F(s,t,z),$ we work to the bivariate case from the univariate case, which is better known. \nNotice that \n\\begin{equation}\\label{eq:uni2bi}\nA_n (s,t) = \\sum_{k=1}^n \\eulerian{n}{k-1}s^kt^{n+1-k} = t^{n+1}\\sum_{k=1}^n \\eulerian{n}{k-1} \\left(\\frac{s}{t}\\right)^k= t^{n+1} A_n \\left(\\frac{s}{t}\\right),\n\\end{equation}\nwhere $A_n(x) = \\sum_{k=1}^{n} \\eulerian{n}{k-1} x^k$ is the single-variable Eulerian polynomial.\n\nThe exponential generating function for $A_n(x)$, denoted $G(x,z)$, is \n\\[\nG(x,z) = \\sum_{n\\geq 1} A_n (x) \\frac{z^n}{n!} = \\frac{x\\left(1-e^{z(x-1)}\\right)}{e^{z(x-1)} -x}.\n\\]\nSee \\cite{Petersen}. Thus, we see that \n\\begin{align}\nF(s,t,z) &= \\sum_{n \\geq 1} A_n (s,t) \\frac{z^n}{n!} \\nonumber \\\\\n&= t \\cdot \\sum_{n \\geq 1} A_n \\left(\\frac{s}{t}\\right) \\frac{(tz)^n}{n!} \\nonumber \\\\\n&= t \\cdot G\\left(\\frac{s}{t},tz\\right) \\nonumber \\\\\n&= \\frac{st\\left(1- e^{z(s-t)}\\right)}{te^{z(s-t)} -s} \\label{eq:fstz}. \n\\end{align}\n\nWith the function $F(s,t,z)$ in Equation \\eqref{eq:fstz}, we can calculate $\\Delta E_n$ simultaneously for all $n$ through the same steps of differentiation and substitution, i.e., \n\\begin{align*}\n\\frac{\\partial}{\\partial s} \\frac{\\partial}{\\partial t} [F(s,t,z)]_{s=t=1} &= \\sum_{n \\geq 1} \\left(\\frac{\\partial}{\\partial s} \\frac{\\partial}{\\partial t} [A_n(s,t)]_{s=t=1}\\right) \\frac{z^n}{n!}, \\\\\n&= \\sum_{n \\geq 1} n!\\cdot \\Delta E_{n+1} \\frac{z^n}{n!} = \\sum_{n \\geq 1} \\Delta E_{n+1} z^n.\n\\end{align*}\nTaking the deriviative at $s=t=1$ of the formula for $F(s,t,z)$ given in \\eqref{eq:fstz}, we find\n\\[\n\\frac{z^4 - 4z^3 + 6z^2 -6z}{6(z-1)^3} = \\sum_{n \\geq 1} \\Delta E_{n+1} \\cdot z^n. \n\\]\n\nNote that from Equation \\eqref{eq:newtossquad}, we would like to see\n\\begin{equation}\\label{eq:en1}\n\\Delta E_{n+1} = \\frac{1}{4} \\left(n+1\\right)^2 - \\frac{1}{12} \\left(n+1\\right) = \\frac{1}{4} n^2 + \\frac{5}{12} n + \\frac{1}{6}. \n\\end{equation}\n\nWe will now use generating function techniques to show that \n\\[\n\\frac{z^4 - 4z^3 + 6z^2 -6z}{6(z-1)^3} = \\sum_{n\\geq 1} \\left(\\frac{1}{4} n^2 + \\frac{5}{12} n + \\frac{1}{6}\\right) z^n.\n\\]\n \nBy differentiating the geometric series $1\/(1-z) = 1+z+z^2 + \\cdots$ twice (and dividing by $2$), we know that \n\\begin{equation}\\label{eq:genfunc}\n\\frac{1}{(1-z)^3} = \\sum_{n \\geq 0} \\binom{n+2}{2} z^n = \\binom{2}{2} + \\binom{3}{2}z + \\binom{4}{2} z^2 + \\cdots.\n\\end{equation}\n\nWe split our generating function for expectation into four pieces:\n\\[\n\\frac{z^4 - 4z^3 + 6z^2 -6z}{6(z-1)^3} = \\frac{6z}{6(1-z)^3} - \\frac{6z^2}{6(1-z)^3} + \\frac{4z^3}{6(1-z)^3} - \\frac{z^4}{6(1-z)^3},\n\\]\nand using the formula from Equation \\eqref{eq:genfunc}, we find \n\\begin{align*}\n\\frac{6z}{6(1-z)^3} &= z \\cdot \\sum_{n \\geq 0} \\binom{n+2}{2} z^n = \\sum_{n\\geq 1} \\binom{n+1}{2} z^n, \\\\\n-\\frac{6z^2}{6(1-z)^3} &=- z^2 \\cdot \\sum_{n \\geq 0} \\binom{n+2}{2} z^n = -\\sum_{n\\geq 2} \\binom{n}{2} z^n, \\\\\n\\frac{4z^3}{6(1-z)^3} &=\\frac{2z^3}{3} \\cdot \\sum_{n \\geq 0} \\binom{n+2}{2} z^n = \\sum_{n\\geq 3} \\frac{2}{3} \\binom{n-1}{2}z^n,\\\\\n- \\frac{z^4}{6(1-z)^3} &= - \\frac{z^4}{6} \\cdot \\sum_{n \\geq 0} \\binom{n+2}{2} z^n = \\sum_{n \\geq 4} \\frac{1}{6} \\binom{n-2}{2} z^n.\n\\end{align*}\n\nSumming these, we get\n\\begin{align*}\n \\frac{z^4 - 4z^3 + 6z^2 -6z}{6(z-1)^3} &= z + 2z^2 + \\frac{11}{3}z^3 \\\\\n & \\quad + \\sum_{n\\geq 4}\\left[\\binom{n+1}{2} -\\binom{n}{2} + \\frac{2}{3}\\binom{n-1}{2} + \\frac{1}{6}\\binom{n-2}{2} \\right] z^n.\n\\end{align*}\n\nUsing the binomial formula to simplify the general term for $n\\geq 4$, we find\n\\[\n\\binom{n+1}{2} - \\binom{n}{2} + \\frac{2}{3}\\binom{n-1}{2} - \\frac{1}{6}\\binom{n-1}{2} = \\frac{1}{4}n^2 + \\frac{5}{12}n + \\frac{1}{6},\n\\]\nwhich establishes Equation \\eqref{eq:en1}, and thus Theorem \\ref{thm:tosstosn}.\n\n\\section{Further remarks}\n\nWe also have Monte Carlo results about question (3), the probability that $n$ sites are occupied given a coin is tossed $N$ times (with a large queue of particles). From the data,\nit appears that the probabilities have a common denominator of $2^{n-1}$. After clearing denominators and rounding the values to the nearest integer, the results for $1\\leq n \\leq 7$ are as follows:\n\\[\n\\begin{array}{c | c c c c c}\nN \\backslash n & 2 & 3 & 4 & 5 \\\\\n\\hline\n1 & 1 \\\\\n2 & 1 & 1 \\\\\n3 & 1 & 3 \\\\\n4 & 1 & 4 & 3 \\\\\n5 & 1 & 9 & 6 \\\\\n6 & 1 & 10 & 18 & 3 \\\\\n7 & 1 & 21 & 32 & 10 \n\\end{array}\n\\]\nWe did not find this array in the On-Line Encyclopedia of Integer Sequences \\cite{Sloane}, though the first two columns have simple explanations. Also, we remark that the first nonzero entry in column $n$ occurs when $N = \\lceil \\frac{n}{2} \\rceil \\cdot \\lfloor \\frac{n}{2} \\rfloor$.\n \nWe finish with some thoughts and observations about question (4) about what happens when we no longer have a simple random walk. Using linear algebra, one can analyze the one-dimensional random walk for any fixed choice of probabilities of a particle moving left or right at a given site. The coin toss model we used is just the case where the probability is $\\frac{1}{2}$ at each site. We have experimented with other choices of probabilities, but without interesting results, except in the case of a biased coin toss. Here each site has the same fixed probability, but that probability is not necessarily $\\frac{1}{2}$. In this case we can use analysis similar to that in this paper to obtain similar results, which we summarize now. \n\nLet $p$ be the probability of a particle going to the right and $q$ the probability of going to the left, and let $\\rho=p\/q$. The biased Gambler's ruin gives transition probabilities of $[k]\/[n]$ where\n\\[\n [n] = \\frac{1-\\rho^n}{1-\\rho} = 1+\\rho+\\rho^2 + \\cdots + \\rho^{n-1}.\n\\]\nThen the same arguments we used in the fair coin case will show that \n\\begin{equation}\n P(n,k) = \\frac{ \\eulerian{n}{k}^{\\maj}}{[n]!},\n\\end{equation}\nwhere $[n]! = [n][n-1]\\cdots[2][1]$, and $\\eulerian{n}{k}^{\\maj}$ counts permutations of length $n$ with $k$ descents, weighted according to a power of $\\rho$ given by a statistic called \\emph{major index}. That is, \n\\[\n\\eulerian{n}{k}^{\\maj} = \\sum_{\\substack{w \\in S_n, \\\\ \\des(w) =k }} \\rho^{\\maj(w)}.\n\\]\nThis gives a refinement of the Eulerian distribution in terms of $\\rho$, as explained in Chapter 6 of \\cite{Petersen}. Studying questions like the limiting distribution and run time when $\\rho\\neq 1$ is the starting point for future work on this problem.\n\n\\section*{Acknowledgements}\n\nThe author is grateful for support from the Undergraduate Summer Research Program from the College of Science and Health at DePaul University, and the guidance of her advisor Kyle Petersen. Thanks also to Lionel Levine for suggesting the problem, and thanks to Jim Propp for comments on an earlier draft.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}