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+{"text":"\\section{Introduction}\n\nOne of the most intriguing questions in modern astronomy is how the universe\nassembled into the structures we see today. A major \npart of this question is understanding how galaxies formed over\ncosmic time.  A popular and  rapidly developing method of tracing the \nevolution of galaxies is through examining galaxy morphologies and\nstructures over a range of redshifts (e.g. Conselice 2003; Conselice, \nRajgor \\& Myers 2008; Conselice et al. 2009; Lotz {et~al.$\\,$} 2008; Cassata\net al. 2010; Ricciardelli\net al. 2010; Weinzirl et al. 2011). While understanding how the morphologies of \ngalaxies evolve, and how matter in galaxies is structured, is fundamental, \nstructural studies of galaxies have thus far focused on measuring properties \nin one or more photometric bands, \nand using this as a tracer of\nevolution.  These types of structural analyses have always been measured in\nterms of relative luminosities, such that brighter parts of galaxies\ncontribute more towards their structure and morphology.  To understand \ngalaxies more fully however, it is important to study the evolution of \ngalaxy structure in terms of their stellar mass distribution, as this better\nreflects the underlying distribution of stars in a galaxy.\n\nMorphological studies have evolved from initial attempts to describe the \nrange of galaxy forms during the early-mid 20th century, towards modern \nefforts of linking the spatial distribution of a galaxy's stars to its \nformation history. In the move away from visual classifications, the goal \nhas been to quantify galaxy structure\/morphology in a way such that \nstructures can be measured automatically and in a reliable \nway. There are two broad approaches for automated classification of\ngalaxies - the parametric and non-parametric methods.\n\nIn the parametric approach, the light profiles of galaxies, as seen in the\nplane of the sky, are compared with predetermined analytic functions. For\nexample, single Sersic profiles are fit to an entire galaxy's light\nprofile.  Similarly, \nbulge-to-disc ($B\/D$) light ratios are computed by fitting\ntwo component profiles to a galaxy. These \nmethods are however  unable to parameterise directly any star formation or\nmerging activity which produces random or asymmetric structures within\ngalaxies.\nThe ``non-parametric'' structural approaches have no such presumed \nlight profiles implicit in their analyses (e.g., Shade et al. 1995; Abraham et al.\n1996; Conselice 1997, 2003; Lotz et al. 2004). This is a more \nnatural approach towards measuring galaxy structures, as the majority of \ngalaxies in the distant Universe have irregular structures that are not\nwell fit by parameterised forms\n(e.g. Conselice {et~al.$\\,$} 2005; Conselice, Rajgor \\& Myers 2008). Perhaps the \nmost successful and straightforward of the non-parametric\nsystems is the $CAS$ method, which uses a combination of concentration ($C$),\nasymmetry ($A$) and clumpiness ($S$) values to separate galaxy types \n(Conselice {et~al.$\\,$} 2000; Bershady {et~al.$\\,$} 2000;  Conselice 2003).\n\nBesides overall structure, the measurement of the size evolution of galaxies, as\nmeasured through half-light radii,\nalso has important implications for their formation histories. For example, \nmany studies such as Trujillo {et~al.$\\,$} (2007) and Buitrago et al. (2008) have\n measured galaxy sizes for systems at $z > 1$ and have found\na population of compact spheroid galaxies with number densities two orders of \nmagnitude higher than what we find in the local universe. The absence of a significant \nnumber of these small sized (as measured by half-light radii), high mass galaxies \nin the local Universe suggests that this population has evolved and has perhaps merged with \nother galaxies. However, these studies are measured based on the light originating from \nthese galaxies, and it is not clear whether sizes would change when measured \nusing the distribution of stellar mass rather than light.\n\nAll of these methods for measuring the resolved structures \nof high redshift galaxies depend on measurements made in one or more \nphotometric bands. As such, quantitative structures are  \ninfluenced by a combination of effects, including: regions of enhanced \nstar formation,  irregular dust distributions, differing ages and \nmetallicities of stellar populations and minor\/major mergers.\nThe morphology and structure of\na galaxy also depends strongly upon the rest-frame wavelength probed\n(e.g., Windhorst et al. 2002; Taylor-Mager et al. 2007).\nYoung stars, such as OB stars can dominate the appearances of galaxies\nat blue wavelengths, thereby giving a biased view of the underlying\nmass within the galaxy.  Longer wavelength observations improve the\nsituation, but at every wavelength, the light emitted is from a mixture\nof stars at a variety of ages and the dust content.    \n More fundamental is the structure  \nof the stellar mass distribution with a galaxy, as it is more of a direct tracer\nof the underlying potential. \nAlthough galaxy structure has been measured on rest-frame near-infrared and $I$-band \nimaging, which traces stellar mass to first order, we directly investigate the\nstellar mass images within this paper. \n\nWe use the method outlined in Lanyon-Foster, Conselice\n\\& Merrifield (2007; LCM07) to\nreconstruct stellar mass maps of galaxies within the GOODS fields at $z < 1$. \nWe then use this to investigate the evolution of the distribution of\ngalaxy stellar mass during the last half of the universe's history.\nWe use these stellar mass maps to directly measure \n$CAS$ parameters, and the sizes of these galaxies in stellar mass\nover this epoch. We test the assumptions that the $CAS$\nparameters and sizes \ncan be reliably measured in optical light by comparing the parameters \nfor these galaxies measured in  $the \nB_{435}$, $V_{606}$, $i_{775}$, and $z_{850}$ bands  and in stellar mass. \nWe finally investigate how these various wavelengths and stellar mass\nmaps can be used to classify galaxies by their formation modes, revealing \nhow these systems are assembling.\n\n\n\nThis paper is organised as follows: in \\S~\\ref{sec:data} and\n\\S~\\ref{sec:method} we describe the data, sample selection and method,\nincluding explanations of the  K-correction code we use and the $CAS$ \nanalysis. In \\S~\\ref{sec:results} we present our results, discussing the \nstellar mass maps themselves, the comparison between galaxy size and the \n$CAS$ parameters in $z_{850}$ and mass. We then explore the \nrelations between the structural parameters in $B_{435}$, $V_{606}$, \n$i_{775}$, $z_{850}$ and stellar mass. Finally, in \\S~\\ref{sec:conclusions}, \nwe discuss our conclusions and comment on future applications.  Throughout \nwe assume a standard cosmology of {\\it H}$_{0} = 70$ km s$^{-1}$ Mpc$^{-1}$, \nand $\\Omega_{\\rm m} = 1 - \\Omega_{\\Lambda}$ = 0.3.\n\n\\section{Data and Sample}\\label{sec:data}\n\n\\subsection{Data}\n\nThe primary source of our data consists of HST\/ACS imaging from the GOODS ACS \nimaging Treasury Program\\footnote{http:\/\/archive.stsci.edu\/prepds\/goods\/}. \nThe observations consist  of imaging in the $B_{435}$ (F435W), $V_{606}$ \n(F606W), $i_{775}$ (F775W) and $z_{850}$ (F850LP) pass-bands, covering the \nHubble Deep Field-North (HDF-N) area.    The central wavelengths of these\nfilters, and their full-width at half-maximum,\nare: F435W (4297, 1038 \\AA), F606W (5907, 2342 \\AA), \nF775W (7764, 1528 \\AA), F850L (9445, 1229 \\AA).\n The images have been reduced (using the ACS CALACS pipeline),\ncalibrated, stacked and mosaiced by the GOODS team (Giavalisco {et~al.$\\,$}\n2004). The field is provided  in many individual image sections, \nwith the HDF-N data divided into 17 sections, each of\n$8192\\times8192$ pixels.  The total area of the GOODS survey is roughly\n315 arcmin$^{2}$, and we utilise imaging which was drizzled  with\na pixel scale of 0.03 arcsec pixel$^{-1}$, giving an effective PSF\nsize of $\\sim 0.1$\\arcsec in the $z-$band.\n\n\nThe sample consists of galaxies selected in the GOODS-N field with $z \\leq\n1$ and a F814W magnitude of $< 24$. The sample was also\nrestricted to those galaxies with \ndata in all  four ACS photometric bands such that we can \nsubsequently obtain the most \naccurate K-corrections and stellar masses possible. The final sample \nconsists of 560 objects, which were individually cut out of the GOODS-N \nACS imaging, and handled as separate entities. This is described in detail \nin the  following section.\n\nPhotometric redshifts are available for the whole sample (Mobasher {et~al.$\\,$}\n2004), and spectroscopic redshifts are available for 404 out of 560 galaxies,\nas found by the Team Keck Redshift Survey (TKRS; Wirth {et~al.$\\,$}, 2004). When\nspectroscopic redshifts are available they are used, else the photometric\nvalues are substituted.  \n\n\n\\section{Method}\\label{sec:method}\n\n\\subsection{Stellar Mass Maps}\n\nTo create stellar mass maps of our galaxies, each pixel in each\ngalaxy image is treated individually\nthroughout the analysis. We convert fluxes from counts per pixel to apparent\nmagnitudes per square arc-second for each pixel in every image in all four\nphotometric bands, as well as calculating their associated errors. \nPixels with negative  fluxes\nwere assigned values four orders of magnitude smaller than the typical  flux\nso that even pixels with low signal-to-noise values can be mapped. This \nhowever means that our resulting values for the CAS parameters, especially\nasymmetry which uses the background light to do a correction, are potentially\ndifferent than when measured using optical light.\n\nBecause our galaxies are at a variety of redshifts, we have to carry out\nfitting to each SED for each pixel to calculate the stellar mass within\neach pixel.  \nK-corrections and stellar masses are calculated for each pixel of each image\nusing the ``K-Correct'' code of Blanton \\& Roweis (2007).  This is\nsimilar to previous methods outlined by Lanyon-Foster et al. (2007),\nBothun (1986), Abraham et al. (1999), and Welikala et al. (2008, 2011).\n\nThe code, K-Correct\\footnote{distributed at\nhttp:\/\/cosmo.nyu.edu\/blanton\/kcorrect\/}\nnaturally handles large datasets, and through its interpretation of the data\nin terms of physical stellar population models, outputs results in terms of\nstellar mass and star formation histories (SFHs) of each galaxy. The code\nallows the input of GOODS data and outputs a stellar mass for each object. The\ncode can also be modified so as to treat each pixel as a separate input object,\nwhich we do in this work.\n\nThe K-correction ($K_{QR}(z)$) between bandpass $R$, used to observe a galaxy\nwith apparent magnitude ($m_{R}$), at redshift $z$, and the desired \nbandpass $Q$ is defined as (e.g., Oke \\& Sandage 1968):\n\n\\begin{equation} \\label{eq:kcdef}\nm_R = M_Q + DM(z) + K_{QR}(z) - 5log(h)\n\\end{equation}\n\n\\noindent where\t\n\n\\begin{equation} \\label{eq:dmod}\nDM(z) = 25 + 5 log \\left[ \\frac{d_{L}}{h^{-1} Mpc} \\right]\n\\end{equation}\n\n\\noindent is the bolometric distance modulus calculated from the luminosity\ndistance, $d_L$, $M_Q$ is the absolute magnitude and $h$ = H$_{0}$\/100\nkm\\,s$^{-1}$\\,Mpc$^{-1}$. \n\nThe K-correct software contains model templates in electronic form\nand an implementation of the method to fit data to models. The code uses\nthe stellar population synthesis models of Bruzual \\& Charlot (2003) and\ncontains training sets of data from GALEX, SDSS, 2MASS, DEEP2 and GOODS.\nThe code finds the nonnegative linear combination of\n$N$ template star formation histories that best match the observations\nusing a minimum $\\chi^2$ comparison. With the entire set of galaxy\nobservations available, K-correct also fits for the $N$ template SFHs using\na nonnegative matrix factorisation algorithm.   K-correct\nnaturally handles data uncertainties, missing data, \nand deals with the complications of observing galaxy spectra\nphotometrically, using broadband filters of galaxies at varying redshifts.\n\nWe use a set of 485 spectral templates to fit to our galaxy pixel\nSEDs using Bruzual \\& Charlot (2003) models with \nthe Chabrier (2003) IMF and Padova (1994) isochrones. All of the six metallicities \navailable (mass fractions of elements heavier than He of $Z = 0.0001, 0.0004, 0.004, 0.008,\n0.02$ and $0.05$) are used.\n\nSome of the known areas of uncertainty in the stellar population models are in\nthe UV and IR regions. In the UV light from young or intermediate aged\nstellar populations can dominate the flux at\n$\\sim 1500 \\AA$.\nIn the near-IR, thermally pulsating asymptotic giant branch (TP-AGB) stars\ndominate the flux in some intermediate age populations (Maraston, 2005). We\ndiscuss this issue in terms of stellar masses in Conselice et al. (2007) who\nshow that the effects of TP-AGB stars do not kick in until at\nredshifts higher than the scope of this study ($z > 2$), and thus are not a\nconcern in the present work.\n\nResults are outputted from K-correct for each\ninput pixel, giving rest-frame fluxes at $NUV$, $U$, $B$, $V$, $R$, $I$ \nmagnitudes\nplus their associated errors, mass-to-light ratios in all of these bands and\nfinally the stellar mass in that pixel.  We then reconstruct the image of\nthe galaxy in stellar mass based on these calculations across the galaxy.  \nWe use a signal-to-noise limit of S\/N $= 3$ per pixel in stellar mass for all\noptical bands, for a reliable calculation.  We also investigate how the\nresults would change if pixels are summed together, finding similar results\nwith the exception of the asymmetry which tends to decrease at lower \nresolutions.\n\n\n\n\\begin{figure}\n\\includegraphics[angle=0, width=85mm, height=85mm]{lanyon.fig1.ps}\n\\caption{The number of galaxies in each of our sample classifications\nas described in \\S 3.2.   Galaxies were visually classified by\n  inspection of both $z_{850}$ images and segmentation maps.}\n\\label{fig:class}\n\\end{figure}\n\n\n\\subsection{Visual Classifications}\n\nWe have classified our entire sample visually, using the $z_{850}$ galaxy\nimages and the object segmentation maps. We have derived the classification\nsystem by applying the  most appropriate criteria to this particular\ndataset. The classification scheme is divided into nine categories, ranging\nfrom  compact objects (spherical with little or no apparent envelopes) to\nobviously merging systems (with evidence of tidal streams and  other merger\nattributes). Six objects in the catalogue were found to be unresolved and\nthese were removed from the sample.  We give a description of these types\nand how they were selected below.  We often refer to these galaxy types\nthroughout this paper, and Appendix B gives a description of these\ntypes in terms of our measured indices.  Figures 2-10 show examples of\nthese galaxy types.\n\n\\begin{figure}\n\\includegraphics[angle=0, width=85mm, height=130mm]{lanyon.fig4.eps}\n\\caption{Visual displays of the light (in $z_{850}$, left), stellar mass (middle) and the distribution of $M\/L$ ratios (right) for the compact elliptical (cE) galaxies.  The $M\/L$ map is scaled such that white corresponds to higher $M\/L$ values, and black to lower $M\/L$ values. This convention is used in the following  Figures ~\\ref{fig:massmap_pE} to ~\\ref{fig:massmap_M}.}\n\\label{fig:massmap_cE}\n\\end{figure}\n\n\n\n\\begin{figure}\n\\includegraphics[angle=0, width=85mm, height=130mm]{lanyon.fig5.eps}\n\\caption{Comparing the light (in $z_{850}$), stellar mass maps, and the \ndistribution of $M\/L$ ratios for examples of peculiar elliptical (pE) \ngalaxies. The scale and convention is the same as that used in \nFig. ~\\ref{fig:massmap_cE}.}\n\\label{fig:massmap_pE}\n\\end{figure}\n\n\n\\subsubsection{Compact Ellipticals}\n\nThe {\\em compact ellipticals}, denoted as ``cE'', were identified as compact \nobjects that varied smoothly across their radii when viewed morphologically. \nThese objects had little or no sign of an envelope usually \nassociated with early-type galaxies.\nThe compact Es are generally found at lower redshifts, at $z < 0.4$\nwith a few between $0.6 < z < 0.8$.  The distinguishing feature\nof these compact Es is their small sizes, with half-light radii\nof $< 1$ kpc.\n\n\n\\subsubsection{Ellipticals}\n\nThe elliptical galaxies are amongst the most popular and important\ngalaxy type in our study, as they are typically the most massive galaxies\nin the local universe.   One reason for this is that elliptical\ngalaxies, and massive galaxies in general, are the test-bed\nfor understanding theories of galaxy formation which often\nhave strong predictions for how the most massive galaxies\nshould form (e.g., Bertone \\& Conselice 2009). The ellipticals we see at high\nredshift are likely the progenitors of the massive\nellipticals found today.\n\nGalaxies with the familiar features of nearby {\\em elliptical galaxies} were \nclassed as such, and given the label ``E\".    28 E galaxies were identified, \nwhich at first glance have the appearance of early\ntypes, but on closer inspection and by varying the image contrast had some\npeculiar features, such as multiple nuclei or minor disturbances in the\ninternal structure. These objects were classified as \n{\\em peculiar ellipticals}, and are labelled as  pE (Conselice {et~al.$\\,$}  2007).\n\n\n\\subsubsection{Spirals\/Disks}\n\nObjects with a disc-plus-bulge structure\nwere classified as {\\em early-type spirals} (``eS\") if the bulge appears\nlarger than the disc component, and {\\em late-type spirals} (``lS\") if the \ndisc is larger than the central bulge. Edge-on disc galaxies are denoted  \nby ``EO\".\n\nThe spiral galaxies are amongst the most interesting for\nthis study given that they often contain two major stellar\npopulation types segregated spatially. Traditionally \nthis is seen as an older stellar population making up\nthe bulge, and the spiral arms consisting of\nyounger stellar populations.  \n\n\\subsubsection{Peculiars and Mergers}\n\nObjects that did not fit easily into any of the previously defined categories,\nbut whose segmentation map showed them to be singular, or unconnected to any\napparently nearby object in the image, were classified as {\\em peculiar},\n``Pec\". These are galaxies that could possibly have merged in the recent past.\n\nGalaxies whose morphology also ruled them out from any previously defined\nclass but whose segmentation map showed them to be connected to, or associated\nwith, at least one other object on the image, but without obvious merger\nsignatures, were classified as {\\em possible mergers} (``pM\"). Galaxies \nwith the pM\nrequirements and which showed obvious signs of merging, such as tidal tails,\nwere classified as {\\em mergers} (``M\").\n\nThis information is summarised for quick reference in Table\n\\ref{tab_classfcn}. The sample was independently classified four times. \nThe mode of the results\nwere taken to  be the true classification and where an indecisive split was\nproduced the lowest numerically valued type was chosen. Figure\n~\\ref{fig:class} shows a histogram of the sample classifications.\n\nWe note that there are a significant number of peculiar galaxies in our \nsample (e.g., Figure~1), much higher than what has been found at lower \nredshifts in previous work investigating galaxy morphology (e.g., Conselice \net al. 2005). \nThese are systems that at lower resolution would be classified as disk-like \ngalaxies in many cases. They are not necessarily merging systems, but are \nlikely normal galaxies in some type of formation, the modes of which are one\nof the features we investigate later in this paper.\n\nWe show examples of each of the above types  in Figures 2-10, with\nfive examples shown for each. These figures show the image of the galaxy\nin the $z_{850}$-band, the stellar mass maps, and the mass to light ratio for each\ngalaxy.\n\n\n\\begin{table}\n\\centering\n\\begin{tabular}{|l|l|}\n\\hline\nType   &   Description  \\\\\n\\hline\nE       &   Elliptical   \\\\\ncE      &   Compact E     \\\\\npE      &   Peculiar E   \\\\\neS      &   Early-Type Spiral \\\\\nlS      &   Late-Type Spiral  \\\\\nEO      &   Edge-on disc \\\\\nPec     &   Peculiar    \\\\\npM      &   Possible merger  \\\\\nM       &   Obvious merger  \\\\\n\n\\hline\n\n\\end{tabular}\n\\caption{Descriptions of the classification scheme.}\n\\label{tab_classfcn}\n\\end{table}\n\n\n\\begin{figure}\n\\includegraphics[angle=0, width=85mm, height=130mm]{lanyon.fig6.eps}\n\\caption{Comparing light, stellar mass maps and $M\/L$ for the elliptical (E) galaxies. The scale and convention is the same as that used in Fig. ~\\ref{fig:massmap_cE}.}\n\\label{fig:massmap_E}\n\\end{figure}\n\n\\begin{figure}\n\\includegraphics[angle=0, width=85mm, height=130mm]{lanyon.fig7.eps}\n\\caption{Comparing light, stellar mass maps and $M\/L$ for the early-type spiral galaxies (eS). The scale and convention is the same as that used in Fig. ~\\ref{fig:massmap_cE}.}\n\\label{fig:massmap_eS}\n\\end{figure}\n\n\n\\begin{figure}\n\\includegraphics[angle=0, width=85mm, height=130mm]{lanyon.fig8.eps}\n\\caption{Comparing light, stellar mass maps and $M\/L$ for the late-type spiral galaxies (lS). The scale and convention is the same as that used in Fig. ~\\ref{fig:massmap_cE}.}\n\\label{fig:massmap_lS}\n\\end{figure}\n\n\\begin{figure}\n\\includegraphics[angle=0, width=85mm, height=130mm]{lanyon.fig9.eps}\n\\caption{Comparing light, stellar mass maps and $M\/L$ for the edge-on disk galaxy (EO) galaxies. The scale and convention is the same as that used in Fig. ~\\ref{fig:massmap_cE}.}\n\\label{fig:massmap_EO}\n\\end{figure}\n\n\n\\begin{figure}\n\\includegraphics[angle=0, width=85mm, height=130mm]{lanyon.fig10.eps}\n\\caption{Comparing light, stellar mass maps and $M\/L$ for the peculiar (Pec) galaxies. The scale and convention is the same as that used in Fig. ~\\ref{fig:massmap_cE}.}\n\\label{fig:massmap_Pec}\n\\end{figure}\n\n\\begin{figure}\n\\includegraphics[angle=0, width=85mm, height=130mm]{lanyon.fig11.eps}\n\\caption{Comparing light, stellar mass maps and $M\/L$ for the possible-merging (pM) galaxies. The scale and convention is the same as that used in Fig. ~\\ref{fig:massmap_cE}.}\n\\label{fig:massmap_pM}\n\\end{figure}\n\n\\begin{figure}\n\\includegraphics[angle=0, width=85mm, height=130mm]{lanyon.fig12.eps}\n\\caption{Comparing light, stellar mass maps and $M\/L$ for the merging (M) galaxies. The scale and convention is the same as that used in Fig. ~\\ref{fig:massmap_cE}.}\n\\label{fig:massmap_M}\n\\end{figure}\n\n\\subsection{CAS Analysis}\\label{sec:analysis}\n\nWe use the concentration, asymmetry, clumpiness ($CAS$) parameters to\nquantitatively measure the structures of our sample, in all available bands,\n$BViz$, and on the stellar mass maps.  We also measure the Gini and M$_{20}$ \nparameters, forming an extensive non-parametric method for measuring the\nstructures and morphologies of galaxies in resolved CCD images (e.g.,\nConselice et al. 2000a; Bershady et al. 2000; Conselice et al. 2002; Lotz\net al. 2004;\nConselice 2003; Lotz et al. 2008). The premise for using these parameters is\nto tap into the light distributions of galaxies, which reveal their past and\npresent formation modes (Conselice 2003). The regions into which the\ntraditional Hubble types fall in CAS parameter space is well understand\nfrom local galaxy comparisons. For\nexample, selecting objects with $A > 0.35$ finds systems that are\nhighly disturbed, and nearly all are major galaxy mergers (e.g., Conselice et\nal. 2000b; Conselice 2003; Hernandez-Toledo et al. 2005; Conselice 2006a).\nA more detailed\nanalysis of this problem is provided in Appendix A for optical light.\n\n\n\\begin{figure*}\n\\includegraphics[angle=0, width=148mm, height=138mm]{masscas.eps}\n\\caption{The change in the asymmetries and concentrations measured on our\nstellar mass maps as a function of\nredshift due simply to distance effects.  Show at the top is the change in the asymmetry\nparameters for a mixture of nearby galaxies of various types: ellipticals, spirals and\nirregulars, while the bottom two panels are for the concentration index.   The left hand\nside for both shows the change for the entire sample we simulation with the red dots and\nerrorbars showing the average change and 1 $\\sigma$ variation of that change.  The right\nhand side shows these simulations divided up between ellipticals (black solid line), \nspiral galaxies (blue dotted line) and irregulars (red dashed line.)}\n\\label{fig:sim}\n\\end{figure*}\n\n\n\\begin{figure}\n\\includegraphics[angle=0, width=90mm, height=90mm]{lanyon.fig2.ps}\n\\caption{The colour-magnitude relation for our sample of galaxies.\nDisplayed is the ($V_{606} - z_{850}$) colour vs. the magnitude of\nthe galaxy as observed in the $z_{850}$ band.  Different galaxy\ntypes are displayed, including: compact ellipticals (cE), peculiar\nellipticals (pE), ellipticals (E), early-type spirals (eS), late-type\nspirals (lS), edge-on disk galaxies (EO), peculiar galaxies (Pec),\npossible-merger galaxies (pM) and merging systems (M).  This key is used\n  throughout the remainder of this paper.}\n\\label{fig:CMD}\n\\end{figure}\n\n\nWe measure the structural parameters for the GOODS sample using the method of\nConselice {et~al.$\\,$} (2008), with slight adjustments made for the stellar mass \nmaps, to enable the code to handle the large values of stellar masses per \npixel, rather than flux.  The radius of each individual galaxy within the \npostage stamp image\/mass map is measured on the stellar mass map, and we \ndefine all our indices within the Petrosian radii (e.g., Petrosian 1976; \nBershady {et~al.$\\,$} 2000; Conselice 2003).  the Petrosian radius has been \nfound to be a better are more reliable radius (and reproducible) than the\nisophotal radius (Petrosian 1976).  \nThe limits and relationship to other radii is described in detail in\nterms of total light in a galaxy by Graham et al. (2005).\n\nCircular apertures are used for measuring our \nPetrosian radii and quantitative parameter estimation. \nThe Petrosian radius used to measure our parameters is defined by,\n\n$$R_{\\rm Petr} = 1.5 \\times r(\\eta = 0.2),$$\n\n\\noindent where $r(\\eta = 0.2)$ is the radius where the surface brightness \n(or stellar mass per unit area) is\n20 percent of the surface brightness (or stellar mass per unit area)\nwithin that radius (Bershady et al. 2000).  Note that this is a distance\nindependent measurement, given that surface brightness dimming effects\nboth measurements of surface brightness in the same way.  \n\nAccounting for background light and noise is extremely important when\nmeasuring structural parameters, especially for faint galaxies, and this \nmust also be dealt with for the stellar mass maps. The measured \nparameters in the $B_{435}$, $V_{606}$, $i_{775}$ and $z_{850}$ bands and \nstellar mass images are corrected as described\nin Conselice {et~al.$\\,$} (2008), by considering a background area close to the\nobject, and the segmentation map of the object. Using a background close to\nthe galaxy itself, any problems introduced by objects imaged on a large mosaic,\nwith a non-uniform weight map, and the object itself being faint compared to\nthe background, are alleviated. We review below how the\n$CAS$ parameters are measured. These are described in more detail\nin Bershady et al. (2000), Conselice et al. (2000), Conselice (2003) and Lotz\net al. (2008).\n\n\n\n\n\\subsubsection{Asymmetry}\n\nWe measure the asymmetry of a galaxy by taking the original galaxy image (or\nstellar mass map), rotating it by 180 degrees about its centre, and then \nsubtracting the two images (Conselice 1997), with corrections for background \nand radius (see Conselice {et~al.$\\,$} 2000a for details). The centre of rotation \nis found using an iterative process, which locates the minimum asymmetry. \nA correction is made within the stellar mass maps for this parameter, \nas it was found that uncertainties in the mass-to-light ratio adversely \naffected the asymmetry measurement. We found that approximately 15 percent \nof the stellar mass in the asymmetry calculation is left\nover from random fluctuations in $M\/L$.  This was compensated for including\nan addition correction for this uncertainty through a slight additional\nsubtraction to the asymmetry signal.  This was done by scaling the \nasymmetric residuals of the rotated and subtracted image by an increase of\n15\\%, that is only 85\\% of the residual is considered.\n\nWe measure the background\nlight in the same way as for the light measures of asymmetry, although because\nwe are dealing with a conversion to stellar mass, which is not trivially\ndone for the background, we have some background values which are lower\nthan zero.  To deal with this, we placed all the negative pixels in the\nstellar mass map background to zero.  \n\nThe equation for calculating asymmetry is:\n\n\\begin{equation}\nA = {\\rm min} \\left(\\frac{\\Sigma|I_{0}-I_{180}|}{\\Sigma|I_{0}|}\\right) - {\\rm\n  min} \\left(\\frac{\\Sigma|B_{0}-B_{180}|}{\\Sigma|I_{0}|}\\right)\n\\label{eq:asym}\n\\end{equation}\n\n\\noindent Where $I_{0}$ denotes the original image pixels, $I_{180}$ is the\nimage after rotating by 180$^{\\circ}\\,$. The background subtraction is made using\nlight (or mass) from a blank sky area, $B_{0}$, and is minimised using the same process\nas for the object itself. The higher the\nvalues of $A$, the higher the degree of asymmetry the galaxy possesses, the\nmost extreme cases usually corresponding to merger candidates (Conselice 2003).\n\n\n\n\\subsubsection{Concentration}\n\nThe concentration parameter measures the the intensity of light (or stellar \nmass) contained within a pre-defined central region, compared to a larger \nregion towards the edge of the visible galaxy. Concentration is most often \ndefined as the ratio of the flux contained within circular radii possessing \n20 percent and 80 percent ($r_{20}$, $r_{80}$) of the total galaxy flux,\n\n\\begin{equation}\nC = 5 \\times {\\rm log} \\left(\\frac{r_{80}}{r_{20}}\\right).\n\\end{equation}\n\n\\noindent A higher value of $C$ corresponds to an object where more light is\ncontained within the central region. This measurement has been shown to\ncorrelate with the halo, and total stellar masses, of nearby \ngalaxies (e.g. Bershady {et~al.$\\,$} 2000; Conselice 2003).\n\n\\begin{figure}\n\\centering\n\\includegraphics[angle=0, width=90mm, height=90mm]{lanyon.fig3.ps}\n\\caption{Colour in observed ($V_{606} - z_{850}$) vs. Redshift ($z$) \nfor our sample. The points displayed are the same as in Fig. ~\\ref{fig:CMD}.}\n\\label{fig:colz}\n\\end{figure}\n\n\\subsubsection{Clumpiness}\n\nClumpiness ($S$) is related to asymmetry, in that it is used to measure the\namount of light (or mass) in a galaxy that exists in discrete, clumpy\ndistributions. In this definition, a smooth galaxy contains light at low\nspatial frequencies, such as elliptical galaxies, whereas clumpy systems have\nmost of their light contained in high spatial frequencies. Star forming\ngalaxies tend to be clumpy in structure, with high $S$ values. We measure\nclumpiness via:\n\n\\begin{equation} \nS = 10 \\times \\left[\\left(\\frac{\\Sigma\n(I_{x,y}-I^{\\sigma}_{x,y})}{\\Sigma I_{x,y} }\\right) - \\left(\\frac{\\Sigma\n(B_{x,y}-B^{\\sigma}_{x,y})}{\\Sigma I_{x,y}}\\right) \\right],\n\\end{equation}\n\n\\noindent where, the original image $I_{x,y}$ is blurred to produce a\nsecondary image,  $I^{\\sigma}_{x,y}$. The secondary image is subtracted from\nthe original image to create a residual map, showing only the high frequency\nstructures contained within the galaxy (Conselice 2003). The residuals are\nquantified by normalising, using the total light in the original galaxy image,\nand then subtracting the normalised residual sky. The smoothing kernel,\n$\\sigma$, used is determined from the radius of the galaxy, and has the value\n$\\sigma = 0.2 \\cdot 1.5 \\times r(\\eta = 0.2)$ (Conselice 2003). The centres of\nthe galaxies (roughly the inner 1\/10th the Petrosian radius) are removed \nduring this procedure.  This parameter is furthermore the most difficult\nto measure and is best measured on galaxies imaged with a high S\/N ratio, and\nare only of limited use in this paper.  Also, the Clumpiness is highly\nsensitive to resolution or seeing, although the ACS imaging we use at these\nredshifts is within the range where values can be properly compared to \nnearby calibration data sets (Conselice 2003).\n\n\n\n\n\n\\subsubsection{Gini}\n\nThe Gini coefficient ($G$) is defined through the Lorentz curve of \nthe light distribution and does\nnot depend on a predefined central position, thus distinguishing it from the\nconcentration parameter. If $G$ is zero, the galaxy has a uniform surface\nbrightness, whereas if a single pixel contains all the flux, $G$ is unity. To\ncalculate $G$ efficiently, the pixel flux (or stellar mass) values, \n$f_{i}$, are\nsorted into increasing order and the following formula is used:\n\n\\begin{equation}\nG = \\frac{1}{\\vert \\bar{f} \\vert n\\left(n-1\\right)} \\sum_{i}^n \\left(2i - n -\n1\\right) \\vert f_{i} \\vert\n\\label{eq:gini}\n\\end{equation}\n\n\\noindent where $n$ is the total number of pixels in the galaxy (Lotz {et~al.$\\,$}\n2004, 2008).\n\n\\subsubsection{M$_{20}$}\n\nM$_{20}$ is anti-correlated with concentration such that galaxies with high\nconcentration have low M$_{20}$ values. M$_{20}$ is a direct tracer of the\nbrightest regions in a galaxy, but does not require a pre-determined central\ncoordinate, rather this is calculated to be the location of minimal\nM$_{20}$. M$_{20}$ is more sensitive to merger signatures than concentration\nand is normalised by the total moment of the galaxy. It is defined as\n\n\\begin{equation}\nM_{20} \\equiv {\\rm log} 10 \\left(\\frac{\\sum_{i} M_{i}}{M_{tot}}\\right), {\\rm while} \\sum_{i} f_{i} < 0.2f_{tot}\n\\label{Eq:M20}\n\\end{equation}\n\n\\begin{equation}\nM_{tot} = \\sum_{i}^n M_{i} = \\sum_{i}^n f_{i}[(x_{i} - x_{c})^2 + (y_{i} - y_{c})^2]\n\\label{Eq:Mtot}\n\\end{equation}\n\n\\noindent where $x_{c}$, $y_{c}$ is the galaxy centre (Lotz {et~al.$\\,$} 2004).\n\n\n\n\\subsection{Simulations of CAS parameters}\n\nOne issue that we must address within this paper is how well we are measuring structure on\nthe stellar mass\nmaps for our sample of galaxies, and importantly how the measurement of stellar mass structure\nwould change with redshift due only to cosmological effects. This is critical if we are\nto make any inferences of the evolution of galaxies in terms of their stellar mass distribution \nover time.  This issue\nhas been addressed before for visual images of galaxies in Conselice (2003) and in Beauvais \n\\& Bothun (1999) for velocity fields in spiral galaxies. Here we address it for the stellar\nmass maps.\n\nWe simulate how our stellar mass map creation process, and CAS measurements would vary as a function of\nredshift due to a decrease in the signal to noise and resolution. We carry out this process\nin the same way we have for the actual galaxies in the GOODS fields that we analyse in\nthis paper. We take a sample of 82\nnearby galaxies of various types - ellipticals, spirals, and irregulars, convert their\nflux values to stellar mass.  We also go through the entire process we do on GOODS, including\nsetting equal to zero the background values with negative stellar masses.  \n\nThis stellar mass conversion is done after the galaxy is redshifted, so that we are mimicking as\nmuch as possible how the real observations and measurements are done.  The simulation itself\nis done by simulating each image at some redshift z$_1$ to how it would appear at z$_2$, with z$_2 >$ z$_1$.\nIn our case z$_{1} \\sim 0$.  When carrying out these simulations of placing lower redshift galaxies\nto high redshifts we calculate first the rebinning factor, b, which is the reduction in apparent size of \na galaxy's image when viewed at higher redshift.  The other major factor is  the relative amounts of \nflux from the sky and galaxy, and the noise produced from the galaxy, sky, dark current, and imaging instrument\n(e.g., read noise) from ACS.   The process and details for how these simulations are done can be found\nin Conselice (2003).  \n \nIn summary, the surface brightness of the simulated image must be reduced such that the equation,\n\n\\begin{equation}\n4\\pi \\alpha_{z_1} N_{z_1} p_{z_1} (1+z_{1}) = 4\\pi \\alpha_{z_2} N_{z_2} p_{z_2} \n(1+z_{2}) \\frac{\\Delta \\lambda_{z_1}}{\\Delta \\lambda_{z_2}},\n\\end{equation}\n\n\\noindent holds, whereby the galaxy as observed in one filter, and is simulated in\nanother with central rest-frame\nwavelengths of $\\lambda_{z_1}$ and $\\lambda_{z_2}$ and widths \n$\\Delta \\lambda_{z_1}$ and $\\Delta \\lambda_{z_1}$.  In the above equation\n$N_{z}$ is the total number of pixels within the galaxy at $z$ and p$_{z}$ is the average ADU counts \nper pixel.   The calibration constant\n$\\alpha_{z}$ is in units of erg s$^{-1}$ cm$^{-2}$ \\AA$^{-1}$ ADU$^{-1}$.\n\nA sky background, and noise from this backgrounds is added to these images by \n$B_{z} \\times t_{z}$, where $B_{z}$ is background flux in units of\nin units of ADU s$^{-1}$. For ACS simulations we take these values from the measured background\nbased on GOODS imaging (Giavalisco et al. 2004) checked to be consistent with\nthe values from the  ACS handbook.  Other noise effects are then added, including \nread-noise scaled for the number of read-outs, dark current and photon noise from the background.  \nOur resulting images are then smoothed by the ACS PSF as generated by Tiny-Tim (Kriss et al. 2001),\nalthough using PSFs measured from stars give the exact same results. \n\nFrom these simulated images at redshifts z = 0.5,1 and 2, we then reconstruct the stellar mass\nimages in the same way as we do for our original galaxies, and then measure the CAS parameters on\nthese stellar mass images.  The results of this are shown in Figure~\\ref{fig:sim} for the concentration\nand asymmetry parameters.  The M$_{20}$ and Gini parameters however have similar behaviours,\nas do the galaxy half-light radii.\nWe find that the clumpiness indices have a similar average difference, but with a much larger\nscatter.  \n  \nWhat we find overall is that the trend with redshift is such that the concentration and asymmetry \nparameters on average do not change significantly when measured in stellar mass (Figure ~\\ref{fig:sim}).  \nWe show in the right panel of Figure ~\\ref{fig:sim} the change in C and A when divided into\nearly\/late\/peculiar types.  Again, on average there is not a significant change with redshift,\nalthough individual galaxies clearly can have significant differences at higher redshift.  \n\nWe also investigate how our parameters change at each redshift between the stellar mass\nand the visual images after the simulation.  We find very little difference except that at the highest redshifts\nof our simulations, at $z = 2$, we find that the average change in the asymmetry parameter\nchanges $\\delta A = (A_{\\rm opt} - A_{\\rm mass}) \\sim -0.1$, such that the stellar mass image\nis more asymmetric than the optical light one.   \nWe also investigate the same quantities that we use in later figures,\nfinding that twice the difference in the asymmetries of the optical and stellar mass\nimage, divided by the sum of these, increases steadily until it reaches a values of\n$-1.1$ at $z = 2$ similar to what we see later in this paper for the actual values.  However\nat $z = 1$ we find that this change is less and closer to $-0.1$.  \n\n\n\n\n\\section{Analysis}\\label{sec:results}\n\nThe following sections describes the analysis of our sample, \nand what we can learn from examining their\nstructures in stellar mass maps, and how this evolves\nover time.  We first examine the\ndistribution of visual classifications for our sample and\ntheir global colours.  We later describe the stellar mass maps we\nconstruct for these galaxies, and then finally present a \nstructural analysis of these systems based on the distribution\nof their stellar mass.\n\n\n\\subsection{Global Properties of the Sample}\n\nIn Figure ~\\ref{fig:CMD} we show the observed colour magnitude diagram for our\nsample.  Each morphological class is plotted as a different symbol; solid red  \ncircles represent the cE's, open red circles show the pE galaxies and the  \nE galaxies are represented by red stars. Solid blue squares show the eS  \nand open blue squares the lS systems. Edge-on disk galaxies are represented by black \ndiagonal crosses and solid green triangles show the Pec systems. \npM and M galaxies are represented by open green triangles and open \ngreen stars respectively. This key is used throughout the rest of the \npaper in the various figures. \n\n\nBased on this, we find that our classified early-type galaxies are on average \nredder and brighter than the peculiars and pM\/M systems, while the spirals \ntend to be bluer across a range of brightnesses, but the differences are not \nso clear as would be expected for a sample at low-$z$ only.\n This colour-magnitude diagram does not show such an obvious morphological \nsequence as is seen for purely Hubble type galaxies in the nearby universe\n(e.g., Conselice 2006b). \n\nThe relation between $(V_{606} - z_{850})$ colour and redshift is shown in\nFigure ~\\ref{fig:colz}. The symbols are the same as those in Fig.\n~\\ref{fig:CMD}. As can be seen, there is a general trend for all types to \nbecome redder with increasing redshift, which is an expected feature of redshift. \nThe early types are reddest across the whole redshift range, followed by the \neS and lS galaxies, with the Pec\/pM\/M systems the bluest across all \nredshifts. The compact ellipticals are blue and \nfaint. Note however that some pM and eS galaxies are quite red.  \nWhat this reveals is that there is a decoupled relation between the \ncolour and morphologies of galaxies not seen in nearby galaxies where\nthis correlation is strong. A specific morphological type as measured\nvisually at high-z cannot be used to predict the colour of the galaxy.\n\n\\subsection{Stellar Mass and Mass to Light Ratio Maps}\n\nBy calculating stellar mass to light ratios and the stellar mass within\neach pixel of each galaxy image in our sample, we reconstruct the image of\neach galaxy in terms of stellar mass and mass-to-light\nratio. We do this for each galaxy, using the method described in LCM07, \nand present five representative examples for each classification in Figures \n~\\ref{fig:massmap_cE} to ~\\ref{fig:massmap_M}. This figures present galaxy\nimages in $z_{850}$ (left), stellar mass (centre) and $(M\/L)_B$ (right), which\nwe discuss below. The images are grey-scaled such that the darkest pixels\nrepresent those brightest in $z_{850}$, and those that are most\nmassive. In the mass-to-light image, the whitest pixels are those with the\nhighest ($M\/L$), or redder in colour.  We explain below how these maps\nappear for our various types.\n\n\\subsubsection{Compact Ellipticals}\n\nFigure ~\\ref{fig:massmap_cE} shows five examples of the compact elliptical\n(cE) population in the sample. These galaxies appear similar in their \nstellar mass maps as they do in their $z_{850}$ images at first\nglance.  The  \nmass-to-light maps are not as uniform, and there are important quantitative\ndifferences between the stellar mass and light images. \nFor example, the galaxy 18246 appears to have a relatively \nhigher ($M\/L$) in its core compared to its outer parts, whilst galaxy\n25035 appears to have a low ($M\/L$) in its centre, suggesting that it has a \nbluer core, \nlikely due to star formation.  Galaxy 36503 contains a high ($M\/L$) \nring surrounding a blue core.\n\n\\subsubsection{Peculiar Ellipticals}\n\nFive examples of the peculiar elliptical (pE) galaxies are displayed in\nFig. ~\\ref{fig:massmap_pE}. These galaxies appear  more diffuse in stellar \nmass than in $z_{850}$, in part due to the loss in contrast in the stellar \nmass maps, but also due to their blue inner colours \n(Fig. ~\\ref{fig:massmap_pE}). As can be seen in the ($M\/L$) ratio maps of \nthese galaxies, they have blue cores, and the effect is most pronounced in \n22932 and 27429.  These galaxies also have blue overall \ncolours compared to pure E galaxies (Fig. ~\\ref{fig:colz}).   These peculiar \nellipticals have been seen and studied before in papers such as Conselice \net al. (2007).  They generally have a high asymmetry, and are found amongst \nthe most massive galaxies in the universe at $z \\sim 1$.  As can be seen in \nthe M\/L maps for these systems (Fig. ~\\ref{fig:massmap_pE}), these galaxies \nalso have a diversity in how young and old stellar populations, including\ndust, are distributed in these systems compared to normal elliptical galaxies.\n\n\\subsubsection{Ellipticals}\n\nFigure ~\\ref{fig:massmap_E} presents five examples of early-types from our\nsample. These galaxies appear more diffuse and more distributed spatially\nin their stellar \nmass than in the $z_{850}$ band, especially at large radii. These galaxies \nhave larger ($M\/L$) ratios in their centres, except for 36419, which is blue, \nand has a more complicated structure in ($M\/L$) than the others. Overall, \nthe E galaxies morphologically  have very similar structures in stellar mass \nand $z_{850}$.\n\n\\subsubsection{Spirals}\n\nFigures ~\\ref{fig:massmap_eS} and ~\\ref{fig:massmap_lS} show the $z_{850}$,\nstellar mass and ($M\/L$) maps for the early-type, and late-type spirals. \nAs found in LCM07,  structures within discs, including prominent spiral \narms, are often (but not always) smoothed out in stellar mass. This \nis true for early types as  well as late-type spirals, as can be seen for\nexample in\ngalaxy 25465 (Fig. ~\\ref{fig:massmap_eS}). Inspection of the ($M\/L$) maps \nreveal that the discs of many spirals are bluer than their bulges, as expected.\n\nThe mass-to-light ratio maps of the edge on galaxies \n(Fig. ~\\ref{fig:massmap_EO}) are mostly homogeneous in structure. However, \nthe maps of 19280, 39312 and 49722 all show some patches of low $M\/L$. \nIndeed, 19280 appears to be blue across the whole image, which would \nimply that the dust content of this galaxy is low, which is unusual for \nedge-on galaxies.\n\n\\subsubsection{Peculiars \\& Mergers}\n\nFigure ~\\ref{fig:massmap_Pec} shows the F814W images, stellar mass maps and M\/L \nratio maps for the Peculiar galaxies within our sample. The Peculiar \ngalaxies 22690 and 25584 (Fig. ~\\ref{fig:massmap_Pec}) appear to contain\n objects near the primary galaxy, but in both cases it is the larger galaxies on the\nright that is the target.  Although 22690 looks disturbed in $z_{850}$, \nthe effect of smoothing in the stellar mass map is also seen here. \nThis indicates that disturbed regions of the galaxy are due to star \nforming regions. The $z_{850}$ image of\n22690 does not have the regular morphology of a nearby spiral galaxy, although\nthe stellar mass map has a structure one would expect of such a \ngalaxy, having a central bulge surrounded by a smooth disc. The ($M\/L$) \nmap also shows a red central region surrounded by bluer pixels.\n\nThe mass-to-light maps of the Peculiars generally show that they are blue and\nalso difficult to see in stellar mass, due to their low mass to light ratio. \nThis suggests that star formation\nitself might be difficult to trace in stellar mass. However, the stellar mass \nin the pM galaxy\n18917 (Fig. ~\\ref{fig:massmap_pM}) traces the light in the galaxy, despite\nhaving a low ($M\/L$).  Note also that the peculiars are often peculiar spirals,\nin the sense that they look like nearly normal spirals, but with some\npeculiar features. Often these peculiars are quite small as well, and many of \nthem are very likely spirals in some type of formation.\n\n\nThe merging galaxy system 21839 (Fig. ~\\ref{fig:massmap_M}) is similar in \nstellar mass to its $z_{850}$ image, but possesses a more intricate structure \nin ($M\/L$). The mass-to-light map shows the smaller merging object to be blue,\nwhereas the brighter galaxy appears more red but with blue regions in its core.\n \nThe system 29800 (Fig. ~\\ref{fig:massmap_M}), whilst clearly appearing to\nbe a\nmerger between two galaxies of approximately equal brightness in $z_{850}$,\nwould likely be classified as a disc galaxy in stellar mass. The bottom of \nthe two bright nuclei has disappeared completely in the stellar mass image, \nalthough it is visible as a low ($M\/L$) patch in the mass-to-light map.\n\n\n\\subsection{Comparison Between Galaxy Properties in Mass and Light}\n\nIn this section we compare how the distribution of stellar mass in a galaxy\ncompares to the distribution of light as seen in the $z_{850}$ ACS imaging.\nOne of our goals is to determine how appropriate studies in $z_{850}$ and \nsimilar red bands are for measuring the mass content and structure of a galaxy, \nand how stellar mass quantitative morphologies differ from those measured in light. \n\nIn this section we investigate the relationship between galaxy properties \nin stellar mass and $z_{850}$-band light. For galaxy size (as  \nmeasured by the half-light radius) and the $CAS$ parameters we  \nplot the normalised difference between $z_{850}$ and stellar \nmass versus the mean value of $z_{850}$ and stellar mass as a representative  \nfigure to determine how these various quantities change.\n\n\n\\subsubsection{Galaxy Half-light and Half-mass Radii}\n\nOne of the basic features of a galaxy is its size, which in this paper we\nquantitatively mean the half-light radii.  Galaxy half-light radii are \nfound to strongly evolve with time, such that galaxies at higher redshifts have\na more compact structure (e.g., Ferguson et al. 2004; Trujillo et al. 2007; \nBuitrago et al. 2008; Carrasco et al. 2010).\nHowever, every size measurement has been carried out through measurements\nof light, and it is desirable to determine how the half-mass radii of galaxies measured\nin stellar mass maps compares with half-light radii measured using light.\n\nAs such, we have calculated half-light (or half-mass) radii ($Re$), in kpc, \nfor our sample in both stellar mass and $z_{850}$ light, and investigate whether \nhalf-light\/mass radii are comparable. \nWe define the normalised size difference as \n\n$$2[R_{\\rm e}(z_{850})-R_{\\rm e}(M_{*})]\/[R_{\\rm e}(z_{850})+R_{\\rm e}(M_{*})],$$  \n\n\\noindent and the average size as \n\n$$\\frac{1}{2}[R_{\\rm e}(z_{850})+R_{\\rm e}(M_{*})].$$ \n\n\\noindent We use these two calculations so as to avoid\nbiasing the analysis when the values become very\nsmall in either the stellar mass image, or in the $z_{850}$ band.\nWe plot the relation between these values in  \nFigure ~\\ref{fig:size_cf}. The horizontal dashed line marks the position\nof equal size ($Re(z_{850}) = Re(M_{*})$). The symbols are the same as in \nFigure~\\ref{fig:CMD}, and the average dispersion for the sample is\nrepresented by the black square points on the right of the plot. The larger\nblack square points in the figure show the average values and the\nmeasured dispersion for\nthe ranges of $\\frac{1}{2}\\left( Re(z_{850}) + Re(M_{*}) \\right)$ between \nzero and one kpc, one and two kpc, up to six kpc. This convention is \nused for each of the parameter comparisons in \nSections ~\\ref{sec:conc_cf} to ~\\ref{sec:M20_cf}.\n\nWe note that there is a steady progression for increasing average galaxy size\nwith morphology,  from the compact ellipticals, early types\nand early spirals to the late-type spirals, with the peculiars and edge-on\ngalaxies being more randomly distributed. The points are scattered about the\nline of equality such that nearly half (42 percent) of the galaxies have \n$Re(z_{850}) > Re(M_{*})$. There is, therefore, a slight tendency for \nsizes in masses to be higher, but this does not vary significantly with \naverage galaxy size and type (see Table~2).\n\nThere is no clear tendency for any particular morphological type to be larger\nin either the $z_{850}$ or stellar mass image, which suggests that there \nis no bias introduced \nby using measurements in $z_{850}$ band data for size measurements \n(Trujillo {et~al.$\\,$}  2007; Buitrago et al. 2008).  Table ~\\ref{tab:size_cf}.  \nshows that the average difference between the sizes in the \nstellar mass maps and the $z_{850}$-band image is essentially zero, \ndemonstrating that the measurements of half-radii in light does not differ \nsignificantly from the measurements in the distribution of stellar mass.\n\nWe also examine the normalised difference in size\nagainst ($V_{606}$ - $z_{850}$) colour to test whether there is a tendency for\nbluer galaxies to have larger radii in stellar mass than in light. This\nis what we might expect to find if star formation dominated the light near the\ncentre of the galaxy. While, on average, colour does not change with size\ndifference, we find that extreme half-radii differences between stellar\nmass and $z_{850}$ are mostly in blue systems.\n\n\n\\begin{figure}\n\\centering\n\\includegraphics[angle=0, width=90mm, height=90mm]{lanyon.fig13.ps}\n\\caption{Normalised effective-radius difference vs. mean \neffective radius of the sample in $z_{850}$\n  and stellar mass. The error bar to the right of the plot shows the average\n  dispersion for the whole sample. The large black squares denote the average\n  values in equally spaced bins of mean size, with error bars showing the\ndispersions of these values.  This convention\n  is used in all of the parameter comparison plots.  The symbols plotted\nhere are the same as in Figure~12.}\n\\label{fig:size_cf}\n\\end{figure}\n\n\n\n\\begin{table}\n\\centering\n\\begin{tabular}{c  r  r}\n\\hline \\hline\n\\\\\nType & Mean Size & Normalised Size Difference \\\\\n       &   (kpc)                               &         \\\\                \n\\\\\n\\hline\ncE   &   $0.6 \\pm(0.3)$  &  $0.14 \\pm(0.24)$   \\\\\npE   &   $1.8 \\pm(0.2)$   &  $-0.11 \\pm(0.11)$     \\\\\nE    &   $1.8 \\pm(0.3)$   &  $0.05 \\pm(0.11)$   \\\\\neS   &   $2.6 \\pm(0.4)$   &  $0.00 \\pm(0.12)$     \\\\\nlS   &   $3.7 \\pm(0.6)$   &  $-0.10 \\pm(0.12)$     \\\\\nEO   &   $2.7 \\pm(0.5)$  &   $0.02 \\pm(0.13)$    \\\\\nPec  &   $2.6 \\pm(0.5)$   &  $-0.09 \\pm(0.13)$     \\\\\npM   &   $2.7 \\pm(0.6)$   &  $-0.07 \\pm(0.13)$     \\\\\nM    &   $3.0 \\pm(0.6)$  &   $-0.08 \\pm(0.03)$     \\\\ \n\\hline\n\\end{tabular}\n\\caption{Comparisons between mean $z_{850}$-stellar mass half-mass\nradius ($R_{\\rm e}$) and normalised size difference in $z_{850}$ and stellar \nmass, organised by morphological type.} \n\\label{tab:size_cf}\n\\end{table}\n\n\n\\subsubsection{Concentration}\\label{sec:conc_cf}\n\nAnalogous to the galaxy size comparison in the previous section, we compare\nthe concentration parameter in $z_{850}$ and stellar mass images, as plotted \nin Figure~\\ref{fig:conc_cf}.  The\ndashed line shows $C(z_{850}) = C(M_{*})$ and the black square points\nrepresent the average values in equally spaced bins of average concentration.\n\nThe average concentration, in both $z_{850}$ and stellar mass, is generally \nhigh for the early types, and lower in the late-types and peculiars, \nwith the early-type spirals being spread between low and high values. \nThe compact elliptical galaxies have a low average concentration, due \nto the light being spread evenly across a galaxy's pixels.\n\nThe average trend shows that the ratio $\\frac{R_{80}}{R_{20}}$ is slightly\nlower in the stellar mass maps for the early types compared to the $z_{850}$\nband concentration. We find that in general R$_{80}$ is lower\nand R$_{20}$ is higher, compared to those in the $z_{850}$-band\nimage, although the effect is mostly due to a smaller R$_{80}$ although\nthis effects is quite small.  It is possible that \nstellar mass is less centrally concentrated towards the centre of \nthe galaxy in the early-types (raising\nR$_{20}$), and\/or the stellar mass is more diffuse at larger radii than\nthe distribution of light (lowering\nR$_{80}$). It is also the case that for many galaxies\nmore star formation is seen in the outer regions of the galaxies \nin the $z_{850}$-band, but these pixels are less dominant in stellar mass, \nraising the value of R$_{80}$ creating higher values of $C$ as seen.\n\n\n\n\\begin{figure}\n\\centering\n\\includegraphics[angle=0, width=90mm, height=90mm]{lanyon.fig15.ps}\n\\caption{Comparison between the concentration parameter in the $z_{850}$\nband  and within\n stellar mass maps. The error bar to the right of the plot shows the average\n  dispersion for the whole sample. The large black squares denote the average\n  values in equally spaced bins of mean concentration, plus dispersions.  The\nsymbols used here are the same as in Figure~12.}\n\\label{fig:conc_cf}\n\\end{figure}\n\n\n\\subsubsection{Asymmetry}\\label{sec:asym_cf}\n\n\nBefore we discuss the comparison of our stellar mass map asymmetry values\nto the $z_{850}$-band asymmetries, we note a few things.  \nFirst, the lowest asymmetry values for the sample are negative \nin all bands, mainly due to the background correction which for very\nsymmetric galaxies will sometimes be larger than the asymmetry in the galaxy\nitself.   This has the effect, \nas seen in Figure ~\\ref{fig:asym_cf}, of skewing the low average asymmetry \nvalues differences for the smallest stellar mass measured asymmetries. \nAs can be seen in Figure~16, except for these galaxies with low asymmetry \nvalues, there is a clear tendency for galaxies to be more asymmetric in \nstellar mass than in $z_{850}$, this being the case for a large\nfraction of the non-early type sample. The stellar mass maps also have a \ngreater spread in asymmetry values than $z_{850}$, and this can be more \nclearly seen in \\S ~\\ref{sec:AC}.\n\n However, we also note that the blue regions of galaxies are \nnot so well traced by the stellar mass\nmaps, and often led to difficulties with the image contrast, making the galaxy\nfeatures harder to see, as in the case of the late-spiral 19535\n(Fig. ~\\ref{fig:massmap_lS}).  Further galaxies, such as 38722 (Figure~6), \nare clearly more asymmetric and lopsided when viewed in the stellar mass \nband.  In this case, it appears that one spiral arm remains while the other \ndisappears.\n\nThere are several reasons why the asymmetry value in the \nstellar mass maps is higher than in the images.  First, \nwhen calculating $A$, noise tends to be magnified due to the process of\nsubtracting images.   The simulations discussed in \\S 3.4 show that\nthere is a tendency for the average galaxies to become more asymmetric\nin stellar mass measurements than in light.  This can explain part of\nthis, but we find that the average difference of around $-1$ in the\nrelative difference is too high to be accounted for solely by these\nredshift effects.  We discuss some of the reasons for why the\nasymmetries will be higher in the stellar mass maps than in light.\n\nWhen calculating stellar masses for each pixel we assume\nthat the $M\/L$ ratio is approximately constant in the surrounding \npixels. This should be the case for early type galaxies,\nespecially, due to their uniform stellar populations. To test how much this\nvariance in $M\/L$ could be affecting the $A$ values, we have measured\nthe $M\/L$ pixel variance in a typical early type galaxy (30976,\nFig. ~\\ref{fig:massmap_E}). The mean $M\/L$ for a pixel in this galaxy is\n$(M\/L)_{B} = 1.29$ with a typical standard deviation in the surrounding pixels\nof $\\sigma_{M\/L} = 0.27$. Although a correction has been applied to minimise\nthis effect (see \\S 3.3.1), it is not enough to account for\nthe high asymmetry signal in the spiral galaxy stellar mass maps, and thus \nthis asymmetry is likely a real effect due to the nature of the calculation\nof this parameter, and especially the background, as described in \\S 3.3.1.\n\n\\begin{figure}\n\\centering\n\\includegraphics[angle=0, width=90mm, height=90mm]{lanyon.fig16.ps}\n\\caption{Comparison between the asymmetry parameter in $z_{850}$ and stellar mass. The \npoints and conventions are the same as in Figure~12.  Note that the asymmetry\nin the stellar mass is lower than the $z-$band for only the early types. This\nis due to the method of measuring the asymmetry, where the sky background\nis handled in a different way than for imaging, resulting in higher values\nfor later type galaxies (\\S 4.3.3). }\n\\label{fig:asym_cf}\n\\end{figure}\n\n\nTo further understand why late-type spiral galaxies are more asymmetric \nin stellar mass than $z_{850}$, we have examined these galaxies in detail \nand present four examples of the late-type sample in both \n$z_{850}$ (Fig. ~\\ref{fig:spirals_z}), and stellar mass maps.  \nTwo of these galaxies have low asymmetries\nin both stellar mass and $z_{850}$ (25751 and 21448), and galaxy 21448 in \nparticular, demonstrates the smoothing of spiral arms, which we observed \npreviously in nearby galaxies (LCM08). However, two of these galaxies have \nhigher asymmetries in stellar mass than in $z_{850}$ (19535 and 34946). \nObject 19535, especially, shows little relation in stellar mass to the \n$z_{850}$ image. The star forming knots and central bulge, which can be \nseen clearly in $z_{850}$, translate to large scale asymmetries in stellar \nmass. There are several possible explanations for this effect. The star \nforming regions in 19535 could truly be more massive per pixel than the \nsurrounding disk, due to the high masses and densities of gas and dust \nrequired to form such massive star forming regions. This would lead to \nsuch regions not being smoothed out in stellar mass, causing higher $A(M_{*})$\nvalues. It is also possible that what we have interpreted as star forming\nknots in the disk in some cases could be minor merger signatures, that is, \nlow-mass galaxies falling into the object itself.\n\n\nThere is also a trend with morphological-type with increasing asymmetry \nsuch that early types have low asymmetry and late-types higher asymmetry. \nOur findings confirm previous studies who found similar trends \n(e.g. Conselice {et~al.$\\,$}\\,  2000). We have also investigated  the relation \nbetween asymmetry in stellar mass and $(V_{606}-z_{850})$\ncolour, and examine this after splitting into two\nredshift bands: $z < 0.6$ and $0.6 \\leq z < 1$. There is a clear trend for\ngalaxies with  a higher degree of asymmetry to be blue, with\n$\\langle(V_{606}-z_{850})\\vert_{A(M_{*})<0.35}\\rangle = 1.01(\\pm0.20)$ and\n$\\langle(V_{606}-z_{850})\\vert_{A(M_{*})\\geq0.35}\\rangle = 1.22(\\pm0.27)$.  \nThis trend has also been \nfound for  asymmetries measured in light (Conselice {et~al.$\\,$} 2003). \n\n\n\n\\begin{figure*}\n\\centering\n\\includegraphics[angle=0, width=188mm, height=95mm]{newfig17.eps}\n\\caption{Figure showing the $z_{850}$-band imaging (left) and\nstellar mass maps (right) for four late-type spiral \ngalaxies. Each galaxy is \nlabelled with its ID number and $A(z_{850})$ value. Each image is \n4.5 arcseconds on each side. \nShown at the top of each image is the ID and the asymmetry parameter\nmeasured on the stellar mass maps to be compared with the same\nvalue computed in the $z_{850}$ band.}\n\\label{fig:spirals_z}\n\\end{figure*}\n\n\n\n\n\n\n\n\n\n\n\\begin{figure}\n\\centering\n\\includegraphics[angle=0, width=90mm, height=90mm]{lanyon.fig21.ps}\n\\caption{Comparison between the Gini parameter in $z_{850}$ and stellar mass. \nThe points and conventions are the same as in Figure~12.}\n\\label{fig:gini_cf}\n\\end{figure}\n\n\n\n\\begin{figure}\n\\centering\n\\includegraphics[angle=0, width=90mm, height=90mm]{lanyon.fig22.ps}\n\\caption{Comparison between the M$_{20}$ parameter in $z_{850}$ and stellar mass. The points and conventions are the same as in Figure~12.   Because of the inverse nature of M$_{20}$, galaxies with positive differences on\nthe y-axis are those with more `concentrated' light profiles than in the\nstellar mass maps.}\n\\label{fig:m20_cf}\n\\end{figure}\n\n\n\n\\subsubsection{Clumpiness}\n\nThe clumpiness values in stellar mass maps have a higher degree of \nuncertainty, as discussed in Section 3.3.3.\nOverall, we find a\ntrend for clumpiness to be greater in stellar mass than $z_{850}$, and this is the \ncase for 91 percent of the sample. However for larger values of average $S$, \nthere is a trend such that $S(M_{*}) \\rightarrow S(z_{850})$. \n\n\\begin{figure*}\n\\centering\n\\includegraphics[angle=0, width=148mm, height=148mm]{lanyon.fig23.eps}\n\\caption{Asymmetry vs. concentration in the $B_{435}$, $V_{606}$, $i_{775}$\n  and $z_{850}$ bands. Solid lines show the classification\ncriteria of Conselice (2003). \nThe point\n  in the bottom left segment of the plot shows the average error bar for the\n  whole sample. }\n\\label{fig:AC_bviz}\n\\end{figure*}\n\n\n\\begin{figure}\n\\centering\n\\includegraphics[angle=0, width=90mm, height=90mm]{lanyon.fig24.ps}\n\\caption{The asymmetry-concentration plane for the stellar mass maps.  The \nplotting  convention is the same as for Figure~12.}\n\\label{fig:AC_mass}\n\\end{figure}\n\n\nThe clumpiness parameter is similar to asymmetry, but picks out small \nscale features, such as compact star clusters, as opposed to large scale \nasymmetries. It is, therefore, not surprising that uncertainties in $M\/L$ \ncause a large number of the sample to have $S(M_{*}) > S(_{850})$, as \nvariations in $M\/L$ from pixel to pixel filter through in the \ncalculation of $S(M_{*})$. Such variations affect $A(M_{*})$ and $S(M_{*})$ \nmore than \nthe other parameters, as these calculations involve the subtraction of \nimages, which amplifies the $M\/L$ variations and large\nvariations of level within the background.\n\n\n\n\n\\subsubsection{Gini Index}\n\nFigure ~\\ref{fig:gini_cf} shows the comparison between the values of the Gini\nparameter in stellar mass and the $z_{850}$ band. \nAlthough the early-types are more strongly clustered around equality, \n$G(M_{*}) = G(z_{850})$, there is a clear trend for the sample to have \nhigher $G$ in stellar mass, with $G(M_{*}) > G(z_{850})$ in non-early types\nwithin our sample. This comparison of $G(z_{850})$ and $G(M_{*})$ \ndisplays a greater separation between types. The late-type spirals \nshow an approximately even \nspread in Gini values, from ~0.45 to ~0.93, but all of these have a higher \ndifference between $z_{850}$ and stellar mass than the early-types\nand compact ellipticals. \n\nFigure ~\\ref{fig:gini_cf} reveals that the Gini index is higher within the stellar \nmass maps than in \nthe $z_{850}$ band, except for early-type galaxies, indicating \nthat most of the \nstellar mass in later morphological types is contained within fewer pixels\nwithin the stellar mass image. \nThis is a significant difference between the early and late-types in our \nsample.  The reason for this is that the bright blue\nregions of these galaxies vanish as the M\/L ratio is inversely proportional\nto L.  This creates effectively a mass distribution with\na larger fraction of the mass contained within fewer pixels.\nFor example the bulges become more prominent for the late-type disks while the\narms vanish.  The bulges in these systems become more prominent and this\nwill rise the value of the Gini index.\n\nWhat can often been seen in stellar mass maps  of late-types and \nmergers\/peculiars is that these bright outer regions vanish, \nleaving only the central part of the galaxy  including most objects in \nFigure~6 which loose their outer parts\nand appear almost as early-types.  Figure~18 also shows that the \nGini index for the early-types in \nstellar mass and light are similar, likely because of the \nsmall variation in the M\/L ratios of the various pixels. \n\n\\subsubsection{M$_{20}$}\\label{sec:M20_cf}\n\nWe show the differences between the M$_{20}$ index in the stellar mass\nmaps and the $z_{850}$ band in Figure ~\\ref{fig:m20_cf}.\nDue to the reversed parity in the M$_{20}$ values, points which lie above the\ndashed line in Figure ~\\ref{fig:m20_cf} have M$_{20}(M_{*}) >$\nM$_{20}(z_{850})$, that is, the stellar mass is more more diffuse than the\nlight for most systems.   This would change however if we used\na smaller radius than the standard definition.  Overall, M$_{20}(M_{*})$ is higher\n(less negative) than M$_{20}(z_{850})$ for most of the sample.\n\nThe M$_{20}$ parameter traces the brightest 20 percent of the flux in the \ngalaxy, or the greatest 20 percent of the stellar mass. This parameter is  \nheavily weighted by the spatial distribution of the most luminous, or most \nmassive, pixels, but is normalised to remove any dependence on galaxy size \nand total flux\/stellar mass. M$_{20}$, like $G$, is also not dependent on a \nfixed, pre-determined centre.\n\nTherefore, if stellar mass exactly follows light within a galaxy we would \nexpect that\nM$_{20}(M_{*})$ = M$_{20}(z_{850})$. Figure ~\\ref{fig:m20_cf} shows that, while\nthis is approximately the case for early-type galaxies and early spirals, the\nsame does not apply for peculiars, late-type spirals and edge-on discs. The\ncompact ellipticals especially differ from this trend, \nhaving M$_{20}(M_{*})$ $<$ M$_{20}$($z_{850}$).  This is certainly due to \nthe fact that these compact ellipticals have blue cores which render their \nstellar mass maps more compact in the centre than for galaxies with \nredder cores.\n\n\n\\subsection{Stellar Mass Structure}\n\n\\subsubsection{Concentration vs. Asymmetry} \\label{sec:AC}\n\nWe examine how galaxies fall in the classic \nconcentration-asymmetry plane (e.g., Conselice et al. 2000) to determine \nif different galaxy types, as determine visually can be better separated \nat different wavelengths in this parameter space.\n\nWe plot the relation between asymmetry and concentration in light for the\nsample in Fig.~\\ref{fig:AC_bviz} and in stellar mass in \nFig. ~\\ref{fig:AC_mass}.  Over-plotted are the classification  criteria \nof Conselice (2003) and Bershady et al. (2000). Galaxies with \n$A>0.35$ are classified as mergers, \nwhile  those which lie above the top line are early-types. \nGalaxies to  the left of the middle line in Figure~20 \\& 21, \nare labelled \nmid-types and those to right of this line are classified as  late-types. \n\n\n\n\n\n\nThese criteria for identifying mergers and \nlate-types appear  more appropriate for our sample in the $i_{775}$ and \n$z_{850}$ bands, while galaxies  are more mixed in these regions in stellar \nmass, having many more\ngalaxies with high asymmetries, and spirals are not easily distinguished from\nmerging systems.  However, the uniqueness of the early-type region is more\nobvious in stellar mass\nmaps, with most E  galaxies in the $i_{775}$ and $z_{850}$ band plots \nfalling into the mid-type region (Fig. ~\\ref{fig:AC_mass}).\n\n\n\n\\begin{figure*}\n\\centering\n\\includegraphics[angle=0, width=148mm, height=148mm]{lanyon.fig25.eps}\n\\caption{Gini-M$_{20}$ for the $B_{435}$, $V_{606}$, $i_{775}$ and $z_{850}$\nbands. The solid lines mark the classification criteria of Lotz {et~al.$\\,$}  (2004;\nEquations ~\\ref{eq:lotz_mergerline} and  ~\\ref{eq:lotz_normalline}).}\n\\label{fig:gm20_bviz}\n\\end{figure*}\n\nWe find that for our sample, $z_{850}$ is the best band in which to measure\nthese parameters for separating morphological types from each other. \nIt can also be seen in Fig. ~\\ref{fig:AC_bviz}  that the $A$-$C$ \nrelation breaks down when viewed in bluer bands, as also shown for nearby \ngalaxies in Taylor-Mager et al. (2007).  We find that the scatter in $A$-$C$ increases for all\ngalaxy types towards bluer wavelengths, the effect is more pronounced for \nlate-type morphologies.\n\n\n\\begin{figure}\n\\centering\n\\includegraphics[angle=0, width=90mm, height=160mm]{lanyon.fig26.eps}\n\\caption{Gini-M$_{20}$ for $z_{850}$ (top) and the stellar mass maps \n(bottom). In $z_{850}$ (top) the dashed lines are the classification \ncriteria of Lotz {et~al.$\\,$}  (2004; Equations ~\\ref{eq:lotz_mergerline} and\n  ~\\ref{eq:lotz_normalline}) while the solid lines mark our revised criteria\n  in $z_{850}$ (Equations ~\\ref{eq:me_mergerline} and\n  ~\\ref{eq:me_normalline}). In stellar mass (bottom) the Lotz {et~al.$\\,$}\\,  \ncriteria are\n  marked by the dashed lines, our $z_{850}$ criteria are shown by the\n  dot-dashed lines and the solid line shows our separation of early and late\n  types\/mergers, in stellar mass.}\n\\label{fig:gm20_mass}\n\\end{figure}\n\n\n\\begin{figure}\n\\centering\n\\includegraphics[angle=0, width=90mm, height=90mm]{lanyon.fig20.ps}\n\\caption{Asymmetry in stellar mass maps vs. redshift. The solid line shows the \nfit to the E galaxies (Eq. ~\\ref{eqn:E_Am_z}), which is approximately \nconstant across redshifts ($z$).  The dashed line illustrates the fit to \nthe late-type spiral galaxies (Eq. ~\\ref{eqn:lS_Am_z}). The late-type \nspiral galaxies show a larger $A(M_{*})$ with increasingly higher redshifts.}\n\\label{fig:Am_z}\n\\end{figure}\n\n\nFigure ~\\ref{fig:AC_mass} shows the $A$-$C$ relation for the stellar mass \nmaps. There is a greater range in asymmetry values in stellar mass and the \ncriterion suggested by Conselice (2003) for merging systems, of $A > 0.35$, \neven includes a few E and some eS galaxies. There is, however, still a clear \nseparation in types between early and peculiar type systems, but the spirals \n(especially lS galaxies) significantly overlap between the two. This makes \nit difficult to distinguish between morphological types in stellar \nmass within $A-C$.  However, given the fact that many of these spirals appear\nto have some kind of merger or formation mode, the stellar mass structure\nis superior for finding galaxies in active evolution.\n\n\n\\subsubsection{Gini vs. M$_{20}$}\n\nWe plot the relation between $G$ and M$_{20}$ for light and stellar mass in \nFigures ~\\ref{fig:gm20_bviz} and ~\\ref{fig:gm20_mass}. The solid lines in Fig\n~\\ref{fig:gm20_bviz} are taken from Lotz {et~al.$\\,$} (2008), who used these\nparameters to\nclassify their sample of normal galaxies and ULIRGs. Equation\n~\\ref{eq:lotz_mergerline} describes the line above which Lotz {et~al.$\\,$} classify\ntheir sample as mergers, and Eq. ~\\ref{eq:lotz_normalline} describes the line\nseparating early and late-type galaxies. \n\n\\begin{equation}\nG = -0.14 \\cdot M_{20} + 0.33\n\\label{eq:lotz_mergerline}\n\\end{equation}\n\n\\begin{equation}\nG = 0.14 \\cdot M_{20} + 0.80\n\\label{eq:lotz_normalline}\n\\end{equation}\n\nWe again see that the cleanest, in terms of separating morphological types, optical\nrelation is in $z_{850}$, although our sample is not best represented by the\nLotz {et~al.$\\,$}  classifications in Gini-M$_{20}$ space, with many early-type\ngalaxies falling into the merging region of the plot. This is partially due to\nthe fact that $G$ and M$_{20}$ are calculated differently here then in Lotz\n{et~al.$\\,$}, as described in \\S3.3 and Lisker {et~al.$\\,$} (2008).  However, we cannot\nrule out that the classifications for this sample may be intrinsically\ndifferent, and it is worrying  that all of the edge-on galaxies fall into the\nmerging region of $G$-M$_{20}$.    We also use a similar\nprocedure to look at the same rest-frame wavelength as Lotz\net al. does. \n\nWe modify the Lotz et al. (2008) relations for $z_{850}$ (top) and \nstellar mass (bottom) and show this in Fig. ~\\ref{fig:gm20_mass}. The \nsolid lines in the $z_{850}$\nplot are our revisions to the Lotz {et~al.$\\,$} relations, based on our sample\n(Equations ~\\ref{eq:me_mergerline} for mergers and ~\\ref{eq:me_normalline}\nfor normal systems), \n\n\n\\begin{equation}\nG(z_{850}) = -0.14 \\cdot M_{20}(z_{850}) + 0.38\n\\label{eq:me_mergerline}\n\\end{equation}\n\n\\begin{equation}\nG(z_{850}) = 0.14 \\cdot M_{20}(z_{850}) + 0.74\n\\label{eq:me_normalline}\n\\end{equation}\n\n\\noindent while the dashed lines are the original Lotz {et~al.$\\,$} relations. In\nthe bottom panel of Fig. ~\\ref{fig:gm20_mass}, the solid line shows our\nrevision of the merger defining line for the stellar mass maps,\n(Eq. ~\\ref{eq:mass_mergerline}), \n\n\\begin{equation}\nG(M_{*}) = -0.14 \\cdot M_{20}(M_{*}) + 0.45,\n\\label{eq:mass_mergerline}\n\\end{equation}\n\n\\noindent while here our relations for $z_{850}$ and\nthose of Lotz {et~al.$\\,$} are shown as dot-dashed and dashed, respectively.\n\nAs with the relation for $A$($M_{*}$)-$C$($M_{*}$), the \n$G$($M_{*}$)-M$_{20}(M_{*})$ relation shows\nthe same general trend as its $z_{850}$ counterpart, but with a larger spread\nin values. The spirals, again, are not as neatly defined as in $z_{850}$, with\nthe eS galaxies separated from the early-types and the late-type\nspirals and edge-on disk galaxies\noccupying the merging region. However, there is a clear separation in\nthe stellar mass Gini vs. $M_{20}$ which does not exist for the optical\nlight.  Several of the pE galaxies lie in the merger\nregion, even as defined by the higher stellar mass line\n(Eq. ~\\ref{eq:mass_mergerline}). This has also been noted by Conselice {et~al.$\\,$}\n(2007), and is likely due to these objects having multiple nuclei.\n\nThe $G$($M_{*}$)-M$_{20}(M_{*})$ relation shows a clear early\/late-type \nsplit, defined by Eq. ~\\ref{eq:mass_mergerline}, as the Sb\/Sc\/Irr region, as \ndescribed by\nLotz {et~al.$\\,$} is not appropriate to apply to our sample in\nstellar mass.  We discuss other correlations: asymmetry vs. clumpiness,\nconcentration vs. M$_{20}$, size vs. concentration, and asymmetry\nvs. M$_{20}$ in the appendix A.  We evaluate in this appendix\nother scaling relationship\nbetween these non-parametric relationships, including asymmetry-clumpiness,\nconcentration and M$_{20}$, size and concentration, and asymmetry and\nM$_{20}$.\n\n\n\\subsection{The Evolution of Galaxy Stellar Mass Structure with Redshift}\n\n\nIn this section we investigate the changes in our galaxy stellar mass\nmaps with redshifts. We take a quantitative approach here and find\nthat at $z < 1$ there is little change in the stellar mass\nstructure with redshift, with the exception of the asymmetries of\nthese galaxies which decreases for some galaxy types at $z < 1$.\n\nWe plot $A(M_{*})$ against redshift (Figure\n~\\ref{fig:Am_z}) where we can see a general trend for\ngalaxies to become more asymmetric in stellar mass at\nhigher redshifts.   \nWe find that not all galaxy types follow this pattern\nhowever, with the most notable exception being the\nellipticals.  As shown in Fig. ~\\ref{fig:Am_z}\nthere is no trend for the ellipticals \nto become more asymmetric in stellar mass at higher\nredshifts.\n\nWe note that while the Es remain at approximately the same\nlow asymmetries across the range in redshift, there is a trend for\nspiral galaxies to become increasingly asymmetric at higher redshifts. We\nmeasure this trend by fitting both the ellipticals (solid line) and lS (dashed line)\ngalaxies, plotted in Figure ~\\ref{fig:Am_z} and quantified in Equations\n~\\ref{eqn:E_Am_z} and ~\\ref{eqn:lS_Am_z}, below.\n\n\\begin{equation}\nA_{E}(M_{*}) = 0.049(\\pm0.079) \\cdot z + 0.021(\\pm0.056)\n\\label{eqn:E_Am_z}\n\\end{equation}\n\n\\noindent is the best fit for the ellipticals and for the late-type spirals,\nthe best fit is,\n\n\\begin{equation}\nA_{lS}(M_{*}) = 0.327(\\pm0.093) \\cdot z + 0.175(\\pm0.064).\n\\label{eqn:lS_Am_z}\n\\end{equation}\n\n\\noindent Equation ~\\ref{eqn:E_Am_z} shows that the early-type \ngalaxies retain\napproximately constant asymmetries across $0 < z < 1$. Equation\n~\\ref{eqn:lS_Am_z} indicates that there is a relation for late-type spirals,\nsuch that $A(M_{*})$ increases with redshift.\n\n\n\n\nWe have already discussed in \\S 4.3.3 how the late-type spiral galaxies\nhave stellar mass asymmetries larger than their asymmetries in\noptical light.   Since the late-type spirals are galaxies\nwhich have arms\/disks that are larger\/brighter than\nthe bulge component, these galaxies are therefore\ndominated in terms of their light by their spiral\nstructures and arms.  These galaxies show the\nmost diversity and contract between spatial distributions\nwithin their M\/L and stellar mass maps \n(e.g., Fig. ~\\ref{fig:massmap_lS}). \n\n\n\n\nWe note that not all of the late-type spirals have high asymmetries\nin the stellar mass maps, some are as asymmetric or less than in the\n$z_{850}$-band (Fig. ~\\ref{fig:asym_cf}).  Furthermore, we find that\nin general the disk galaxies with higher stellar mass map \nasymmetries have a bluer\ncolour, which is the case at both high\nand low redshifts.   In\nmany of these cases, the structures are lopsided and\/or contain\nouter features that remain after normalising by the M\/L map.\nThis interpretation of stellar mass being more localised than the light\nis consistent with a higher Gini index in the stellar mass\nbands than in the light.\n\nSome of these asymmetric spirals, as well as many of the peculiars\nresemble the so-called ``clump\nclusters'' and clumpy spirals found by Elmegreen {et~al.$\\,$} (2005, 2007), \nespecially in stellar mass maps.  Their low concentration in mass\nis also similar to the Luminous Diffuse Objects (LDOs) found\nat similar redshifts (Conselice et al. 2004).  \nIndeed, there are some late-type spirals galaxies that we would\nhave classified differently, perhaps as Pec or pM systems, had we performed\nthe classification in stellar mass, rather than in $z_{850}$. \nThe clumps in galaxies\nfound by Elmegreen {et~al.$\\,$} are also massive, with typical values in the region\n$\\sim10^{8} - 10^{9}M_{\\odot}$ (Elmegreen \\& Elmegreen 2005), and reside in\nyoung disks at high redshift. It is likely that some of our highly asymmetric\nlate-type spiral galaxies are similar, or in the same class as these clumpy \nspirals.  These features may also be the result of minor merger events.  \n\n\\section{Conclusions}\\label{sec:conclusions}\n\n\nWe conduct a pixel by pixel study of the structures of 560 \ngalaxies found in the \nGOODS-N field at $z < 1$ within stellar mass maps and Hubble\nSpace Telescope ACS $BViz$ wavebands.\nWe measure stellar masses for each pixel of each galaxy image from our\n$BViz$ images by fitting to stellar population models. We use these \nvalues to construct stellar mass maps for our sample and compare morphologies \nin $BViz$ and stellar mass. Our major findings and results include:\n\nI. We construct stellar mass maps and mass-to-light ratio maps for each \ngalaxy and we present examples of each morphological type, in the \n$z_{850}$ band, stellar mass maps, and $(M\/L)_B$ in \nFigures ~\\ref{fig:massmap_cE} to ~\\ref{fig:massmap_M}. We\nfind that some compact elliptical (cE) galaxies have blue cores, and more\ncomplicated internal structures than in either $z_{850}$ or stellar mass would \nreveal on their own.  Many of the peculiar\nellipticals (pE) also have blue cores and we assert that these objects may\nhave merged in their recent histories.\nThe early and late-type spirals display more varied patterns. We find that for\nsome of these galaxies, structures seen in light (e.g. spiral arms) are\nsmoothed out in stellar mass. However, this effect does not hold true\nfor all galaxies, nor for all features seen in light.\nStellar mass and ($M\/L$) maps of the `peculiar' galaxies are complicated and \nvary across the sample. Although it is more difficult to make out features in \nstellar mass for galaxies that have very blue colours, \nstructures not seen in light are revealed. \n\n\nII. We compare the half-light\nradius, $R_{\\rm e}$, in $z_{850}$ with half-mass radii for our sample\nas a function of morphological type. We find no systematic tendency \nfor any particular morphological type to have larger $R_{\\rm e}$ in stellar mass \nthan in the  $z_{850}$ band, and\nthus conclude that there is no bias introduced by measuring galaxy sizes in\n$z_{850}$.\n\nIII. We find a clear tendency for many galaxies to be more asymmetric\nin stellar mass than in $z_{850}$. We also find that morphology correlates\nwith asymmetries, with early-types having low $A(M_{*})$, and late-types \nhigher values of $A(M_{*})$. We find a relation between colour and \n$A(M_{*})$ such that bluer galaxies have higher stellar mass asymmetry\ndifferences between optical light and stellar mass maps. \nThe late-type \nspiral galaxies in the sample have higher $A(M_{*})$ than \nwould be expected from their asymmetries in $z_{850}$. We discuss possible \ncauses of this effect, including regions of enhanced star formation also \npossessing higher stellar masses, and the evolution of spirals. We note that these highly asymmetric\nspirals resemble the clumpy disks of Elmegreen {et~al.$\\,$} (2007), and Conselice et al. (2004) \nand are experiencing either minor merging activity, or bulge formation\nthrough accretion of disk material.\n\nIV. We find that the Gini index in stellar mass is higher than in the\nz-band ($G(M_{*}) > G(z_{850})$) for all galaxies except the early-types,\nindicating that most of the stellar mass in later morphological types is \ncontained within fewer pixels. Late-type and peculiar morphologies show a \ntrend for M$_{20}(M_{*})$ $>$ M$_{20}(z_{850})$, suggesting that the \nbrightest $20$ percent of the brightest pixels is not necessarily where \nthe greatest $20$ percent of the stellar mass is located.\n\nV. We investigate the relations between several combinations of the\n$CAS$ parameters in $BViz$ and stellar mass, and compare our \nresults to previous morphological studies of this type (e.g. Conselice, \nRajgor \\& Myers 2008; Lotz {et~al.$\\,$}  2004; 2008). We find that $z_{850}$ is \nthe most appropriate photometric band to utilise for a $z < 1$ sample of \ngalaxies with all morphologies, although stellar mass maps are better at\ndistinguishing active galaxies from passive ones and is more a physical\nmeasure of structure.\n\n\nWe furthermore compare our sample classifications in $G$-M$_{20}$ to those of Lotz {et~al.$\\,$}\n(2008) and find that the Lotz {et~al.$\\,$} criteria do not best describe our\nsample. We revise the Lotz {et~al.$\\,$} (2008) criteria to best fit our sample\nin the $z_{850}$-band and within the stellar mass maps, and find that \nearly-types, late-types and mergers can be separated. However, the edge-on\ndisk galaxies remain problematic and cannot be distinguished from mergers in\n$G(z_{850})$-M$_{20}(z_{850})$. We find that $G(M_{*})$-M$_{20}(M_{*})$ \ncan be used to broadly separate early from late-type galaxies, but the \ncriteria cannot distinguish between late-type\/edge-on disks \nfrom peculiar\/merger systems.\n\nVI. We find a relationship between $R_{\\rm e}$ and $C$ (see appendix A) \nfor early-type galaxies in both\n$z_{850}$ and stellar mass. In each case we find that galaxies with higher\nconcentrations have larger radii, and this relation is steeper in $z_{850}$\nthan stellar mass. We also investigate asymmetry versus M$_{20}$ and find that\nthese parameters display a similar relation to $A-C$ in $i_{775}$ and\n$z_{850}$. In stellar mass, $A(M_{*})$-M$_{20}(M_{*})$ shows a tighter relation \nthan $A(M_{*})-C(M_{*})$, with a clear separation between early and late-type\nsystems. Thus, we conclude that, for all parameters, late-type\nspiral galaxies overlap with with Pec\/pM\/M systems which may be\nultimately taken from the same subset of galaxies. Structural studies in \nstellar mass do however track both minor and major merging events, and\ncan be used to find galaxies in active galaxy evolution modes.\n\nWe thank the GOODS team for making their data public, and STFC for a studentship towards supporting this work\nand the Leverhulme Trust for support.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nHighly frustrated magnets are a fascinating area of research with\nmany challenges and surprises \\cite{SRFB04,Diep05}. One exotic\ncase is the spin-1\/2 Heisenberg antiferromagnet on the kagom\\'e\nlattice, where in particular an unusually large number of low-lying\nsinglets was observed numerically \\cite{lech97,waldt98}. An\neffective Hamiltonian approach to the strongly trimerized kagom\\'e lattice\n\\cite{sub95} was then successful in explaining\nthe unusual properties of the homogeneous lattice \\cite{mila98}.\n\nInterest in this situation has been renewed recently due to the\nsuggestion that strongly trimerized kagom\\'e lattices can be\nrealized by fermionic quantum gases in optical lattices\n\\cite{SBCEFL04,FehrmannThesis}. For two fermions per triangle,\none obtains the aforementioned effective Hamiltonian on a\ntriangular lattice \\cite{sub95}, but without the original\nmagnetic degrees of freedom. This effective Hamiltonian\nalso describes the spin-1\/2 Heisenberg antiferromagnet on the\ntrimerized kagom\\'e lattice at one third of the saturation magnetization\n\\cite{CGHHPRSS}. Furthermore, this model shares some features\nwith pure orbital models on the square lattice \\cite{NuFr05,DBM05},\nbut it is substantially more frustrated than these models.\n\nNumerical studies of the effective quantum Hamiltonian\n\\cite{DEHFSL05,DFEBSL05,CEHPSprep}\nprovide evidence for an ordered groundstate and a possible\nfinite-temperature ordering transition. Since the Hamiltonian only has\ndiscrete symmetries, one expects indeed a finite-temperature\nphase transition if the groundstate is ordered.\nFurthermore, quantum fluctuations should be unimportant for the\ngeneric properties of such a phase transition, which motivates\nus to study the classical counterpart of the model at finite temperatures.\n\n\\section{Model and symmetries}\n\n\\begin{figure}[tb]\n\\begin{center}\n\\includegraphics[width=0.35\\columnwidth]{lat-ham.eps}\n\\end{center}\n\\caption{Part of the triangular lattice. Arrows indicate the direction\nof the unit vectors $\\vec{e}_{i;\\langle i,j\\rangle}$ entering the\nHamiltonian (\\ref{eqH}).\n\\label{fig:LatHam}\n}\n\\end{figure}\n\nIn this paper we study a Hamiltonian which is given in terms\nof spins $\\vec{S}_i$ by\n\\begin{equation}\nH = J \\, \\sum_{\\langle i,j\\rangle}\n \\left(2 \\, \\vec{e}_{i;\\langle i,j\\rangle} \\cdot \\vec{S}_i\\right) \\,\n \\left(2 \\, \\vec{e}_{j;\\langle i,j\\rangle} \\cdot \\vec{S}_j\\right) \\, .\n\\label{eqH}\n\\end{equation}\nThe sum runs over the bonds of nearest neighbours $\\langle i,j\\rangle$\nof a triangular lattice with $N$ sites.\nFor each bond, only certain projections of the spins $\\vec{S}_i$\non unit vectors $\\vec{e}_{i;\\langle i,j\\rangle}$ enter the interaction.\nThese directions are\nsketched in Fig.\\ \\ref{fig:LatHam}. Note that these directions depend\nboth on the bond $\\langle i,j\\rangle$ and the corresponding end $i$ or $j$\nsuch that three different projections of\neach spin $\\vec{S}_i$ enter the interaction with its six nearest neighbours.\n\nIn the derivation from the trimerized kagom\\'e lattice \\cite{sub95,SBCEFL04,Z05},\nthe $\\vec{S}_i$ are pseudo-spin operators acting on the two\nchiralities on each triangle and should therefore be considered as quantum\nspin-1\/2 operators. Here we will treat the $\\vec{S}_i$ as classical\nunit vectors. Since only the $x$- and $y$-components enter the\nHamiltonian (\\ref{eqH}), one may take the $\\vec{S}_i$ as `planar' two-component\nvectors. On the other hand, in the quantum case commutation relations\ndictate the presence of the $z$-component as well, such that taking the\n$\\vec{S}_i$ as `spherical' three-component vectors is another natural choice\n\\cite{CEHPSprep}. Here we will compare both choices and thus assess qualitatively\nthe effect of omitting the $z$-components.\n\nThe internal symmetries of the Hamiltonian (\\ref{eqH}) constitute the\ndihedral group $D_6$, {\\it i.e.}, the symmetry group of a regular hexagon.\nSome of its elements consist of a simultaneous transformation of the spins\n$\\vec{S}_i$ and the lattice \\cite{CEHPSprep}. Since these symmetries are\nonly discrete, a finite-temperature phase transition is allowed above an\nordered groundstate.\n\nThe derivation from a spin model \\cite{sub95,Z05} yields a positive $J>0$,\nwhile it may also be possible to realize $J<0$ \\cite{FehrmannThesis} in\na Fermi gas in an optical lattice \\cite{SBCEFL04}. In the following we will\nfirst discuss the case of a negative exchange constant $J<0$ and then\nturn to the case of a positive exchange constant $J>0$. The second case\nis more interesting, but also turns out to be more difficult to handle.\n\n\\section{Negative exchange constant}\n\nFor $J<0$ there is a one-parameter family of ordered groundstates,\n$\\vec{S}_i = (\\cos(\\phi_i),\\,\\sin(\\phi_i),\\,0)$ with $\\phi_a = \\theta$,\n$\\phi_b = \\theta+2\\,\\pi\/3$ and $\\phi_c = \\theta-2\\,\\pi\/3$ on the three\nsublattices $a$, $b$, and $c$, respectively ($120^{\\circ}$ N\\'eel order).\nThe energy of these states, $E^{J<0} = 6\\,J\\,N$, is independent of\n$\\theta$. Computing the free energy $\\mathcal{F}^{J<0}(\\theta)$ by\nincluding the effect of Gaussian fluctuations we find that\n$\\mathcal{F}^{J<0}(\\theta)$ has minima at $\\theta = (2\\,n+1)\\,\\pi\/6$,\n$n = 0,\\,1,\\, \\dots,\\,5$ \\cite{CEHPSprep}. This implies that the above\n$120^{\\circ}$ N\\'eel structures lock in at these angles.\n\n\\begin{figure}[tb]\n\\begin{center}\n\\includegraphics[width=\\columnwidth]{LTneg.eps}\n\\end{center}\n\\caption{MC results for $J<0$, $T = 10^{-3}\\,\\abs{J}$,\nand planar spins. (a) Snapshot of a configuration\non a $12\\times 12$ lattice.\nPeriodic boundary conditions are imposed at the edges.\n(b) Histogram of angles $\\phi_i$, averaged over 1000\nconfigurations.\n\\label{fig:CFGneg}\n}\n\\end{figure}\n\nWe have performed Monte-Carlo (MC) simulations for $J<0$ using a standard\nsingle-spin flip Metropolis algorithm \\cite{LB00}.\nA snapshot of a low-temperature configuration on an\n$N=12\\times 12$ lattice is shown in Fig.\\ \\ref{fig:CFGneg}(a).\nThe $120^{\\circ}$ ordering is clearly seen in such snapshots.\nFig.\\ \\ref{fig:CFGneg}(b) shows histograms of the angles $\\phi_i$.\nOne observes that with increasing lattice size $N$, pronounced maxima\nemerge in the probability $P(\\phi_i)$ to observe an angle $\\phi_i$\nat the predicted lock-in values $\\theta = (2\\,n+1)\\,\\pi\/6$.\n\n\\begin{figure}[tb]\n\\begin{center}\n\\includegraphics[width=0.65 \\columnwidth]{CnegN.eps}\n\\end{center}\n\\caption{MC results for the specific heat $C$ for $J<0$.\nFull lines are for planar spins; dashed lines for spherical spins.\nIncreasing line widths denote increasing system sizes $N$.\nError bars are negligible on the scale of the figure.\n\\label{fig:Cneg}\n}\n\\end{figure}\n\nThermodynamic quantities have been computed by averaging over at least 100\nindependent MC simulations. Each simulation was started at high\ntemperatures and slowly cooled to lower temperatures in order to minimize\nequilibration times. A first quantity, namely the specific heat $C$,\nis shown in Fig.\\ \\ref{fig:Cneg}. There is a maximum in $C$ at\n$T \\approx 1.8\\,\\abs{J}$ for spherical spins, and for planar spins at a\nhigher temperature $T\\approx 2.2\\,\\abs{J}$. The fact that the value of\n$C$ around the maximum increases with $N$ indicates a phase transition.\nFor $T \\to 0$, the equipartition theorem predicts a contribution $1\/2$\nto the specific heat per transverse degree of freedom. Indeed, the\nlow-temperature results are very close to $C\/N = 1\/2$ and $1$ for planar\nand spherical spins, respectively.\n\n\\begin{figure}[tb]\n\\begin{center}\n\\includegraphics[width=0.65 \\columnwidth]{Mneg.eps}\n\\end{center}\n\\caption{MC results for the square of the sublattice\nmagnetization $m_s^2$ for $J<0$.\nFull lines are for planar spins; dashed lines for spherical spins.\nIncreasing line widths denote increasing $N$.\nError bars are negligible on the scale of the figure.\n\\label{fig:Mneg}\n}\n\\end{figure}\n\nIn order to quantify the expected\norder, we introduce the sublattice order parameter\n\\begin{equation}\n\\vec{M}_s = {3 \\over N} \\sum_{i \\in {\\cal L}} \\vec{S}_i \\, ,\n\\label{defMsubl}\n\\end{equation}\nwhere the sum runs over one of the three sublattices ${\\cal L}$\nof the triangular lattice. Fig.\\ \\ref{fig:Mneg} plots the square\nof this sublattice order parameter\n\\begin{equation}\nm_s^2 = \\left\\langle \\vec{M}_s^2 \\right\\rangle \\, ,\n\\label{defMs}\n\\end{equation}\nwhich is a scalar quantity and expected to be non-zero in a\nthree-sublattice ordered state. Note, however, that the order\nparameter (\\ref{defMsubl}) is insensitive to the mutual orientation\nof spins on the three sublattices.\nIn Fig.\\ \\ref{fig:Mneg} we observe that $m_s^2$ converges to\nzero with $N \\to \\infty$ at high temperatures, while a non-zero\nvalue persists at lower temperatures, consistent with a\nphase transition around $T\\approx 2\\,\\abs{J}$, as already\nindicated by the specific heat. Furthermore, the sublattice order\nagain points to a higher transition temperature for planar spins than for\nspherical spins. It should also be noted that there are noticeable\nquantitative differences between the values of $m_s^2$ for\nplanar and spherical spins over the entire temperature range.\nJust for $T \\to 0$ both planar and spherical spins yield $m_s^2 \\to 1$,\nas expected for a perfectly ordered groundstate.\n\nIn order to determine the transition temperature $T_c$ more accurately,\nwe use the `Binder cumulant' \\cite{Binder81,LB00} associated to\nthe order parameter (\\ref{defMsubl}) via\n\\begin{equation}\nU_4 = 1+ A - A \\,\n{\\left\\langle \\vec{M}_s^4 \\right\\rangle \\over\n\\left\\langle \\vec{M}_s^2 \\right\\rangle^2} \\quad\n{\\rm with} \\quad\nA = \\cases{1 & for planar spins, \\cr\n{3 \\over 2} & for spherical spins. }\n\\label{defBinder}\n\\end{equation}\nThe constants in (\\ref{defBinder}) are chosen such that $U_4=0$ for a\nGaussian distribution of the order parameter $P(\\vec{M}_s) \\propto\n\\exp\\left(-c\\,\\vec{M}_s^2\\right)$. Such a distribution is expected at high\ntemperatures, leading to $U_4 \\to 0$ for $T \\gg \\abs{J}$. Conversely, a\nperfectly ordered state yields $\\left\\langle \\vec{M}_s^4 \\right\\rangle =\n\\left\\langle \\vec{M}_s^2 \\right\\rangle^2$ such that with the prefactors as\nin (\\ref{defBinder}) we find $U_4 = 1$. Hence, for an ordered state we\nexpect $U_4 \\approx 1$ for $T < T_c$.\n\n\\begin{figure}[tb]\n\\begin{center}\n\\includegraphics[width=0.93 \\columnwidth]{UnegN.eps}\n\\end{center}\n\\caption{Main panel:\nMC results for the Binder cumulant for $J<0$.\nFull lines are for planar spins; dashed lines for spherical spins.\nIncreasing line widths denote increasing $N$.\nError bars are negligible on the scale of the figure.\nInset: Binder cumulant for planar spins close to the\ncritical temperature.\n\\label{fig:Uneg}\n}\n\\end{figure}\n\nFig.\\ \\ref{fig:Uneg} shows our results for the Binder cumulant, as defined\nin (\\ref{defBinder}). The transition temperature $T_c$ can be estimated\nfrom the crossings of the Binder cumulants at different sizes $N$\n\\cite{Binder81,LB00}, which are shown for planar spins in the inset of\nFig.\\ \\ref{fig:Uneg}. Since we have not been aiming at high precision, we\ncannot perform a finite-size extrapolation. Nevertheless, we obtain a\nrough estimate\n\\begin{equation}\n{T_c^{\\rm planar}} \\approx 2 \\, {\\abs{J}}\\, ,\n\\label{TcNegPlanar}\n\\end{equation}\nwith an error on the order of a few percent. The corresponding\nvalue for spherical spins is estimated as\n${T_c^{\\rm spherical}} \\approx 1.57 \\, {\\abs{J}}$ \\cite{CEHPSprep}.\nWe therefore conclude that out-of-plane fluctuations reduce $T_c$\nby about $20\\%$.\n\n\\section{Positive exchange constant}\n\nAs in the case $J<0$, there is a one-parameter family of $120^{\\circ}$\nN\\'eel ordered groundstates with energy $-3\\,J\\,N$. They differ from the\nstates found for $J<0$ by an interchange of the spin directions on the\n$b$- and $c$-sublattices. The contribution of Gaussian fluctuations around\nthese states yields a free energy $\\mathcal{F}^{J>0}(\\theta)$ which has\nminima at $\\theta = \\pi \\,n\/3$, $n = 0,\\,1,\\, \\dots,\\,5$. Hence the N\\'eel\nstructure locks in at these angles for $J>0$. Surprisingly, an inspection\nof all states of finite cells of the lattice, in which the mutual angles\nbetween pairs of spins are multiples of $2\\,\\pi\/3$ \\cite{CEHPSprep},\nreveals that there is a macroscopic number of groundstates in addition to\nthe $120^{\\circ}$ N\\'eel states, {\\it i.e.}, the number of ground states\ngrows exponentially with $N$. The $120^{\\circ}$ N\\'{e}el state has $N\/3$\nsoft modes, all other groundstates have fewer soft modes \\cite{CEHPSprep}.\nAt low but finite temperatures the $120^{\\circ}$ N\\'eel state will\ntherefore be selected by a thermal order-by-disorder mechanism \\cite{OBD}.\n\n\\begin{figure}[tb]\n\\begin{center}\n\\includegraphics[width=0.5 \\columnwidth]{planarcfg_10_0.eps}\n\\hspace*{-0.12 \\columnwidth}\n\\includegraphics[width=0.5 \\columnwidth]{planarcfg_10_3.eps}\n\\end{center}\n\\caption{Snapshots of configurations generated by the exchange\nmethod at $T \\approx 10^{-2} \\, J$\non a $12\\times 12$ lattice for planar spins and $J>0$.\nPeriodic boundary conditions are imposed at the edges.\n\\label{fig:CFGpos}\n}\n\\end{figure}\n\nThe large number of groundstates which are separated by an energy barrier\nrenders it extremely difficult to thermalize a simple MC simulation at\nsufficiently low temperatures. We have therefore performed exchange MC\nsimulations (also known as `parallel tempering') \\cite{HN96,eM98} for\n$J>0$, using a parallel implementation based on MPI. 96 replicas were\ndistributed over the temperature range $T\/J = 0.01, \\ldots, 0.7$ in a\nmanner to ensure an acceptance rate for exchange moves of at least 70\\%.\nStatistical analysis was performed by binning the time series at each\ntemperature. Fig.~\\ref{fig:CFGpos} shows snapshots of two low-temperature\nconfigurations with $N=12\\times 12$ generated by the exchange method with\nplanar spins. In contrast to the case $J<0$, it is difficult to decide on\nthe basis of such snapshots if any order arises for $J>0$: there are\ndefinitely large fluctuations including domain walls in the system. These\nfluctuations are in fact a necessary ingredient of the order-by-disorder\nmechanism. A careful quantitative analysis is therefore clearly needed.\n\n\\begin{figure}[tb]\n\\begin{center}\n\\includegraphics[width=0.65 \\columnwidth]{Cpos.eps}\n\\end{center}\n\\caption{Exchange MC results for the specific heat $C$ for $J>0$.\n(a) Full lines are for planar spins; dashed lines for spherical spins.\nIncreasing line widths denote increasing system sizes $N$.\nError bars are on the order of the width of the lines in this panel.\nThe other panels show the\nlow-temperature parts of the results for spherical spins (b),\nand planar spins (c).\nThe system sizes in panels (b) and (c) range from $N=6\\times6$\n(bottom) to $N=18\\times18$ (panel (b), top),\nand $N=12\\times12$ (panel (c), top), respectively.\n\\label{fig:Cpos}\n}\n\\end{figure}\n\nWe start with the specific heat $C$, shown in Fig.\\ \\ref{fig:Cpos}. There\nis a broad maximum at $T \\approx 0.25\\,J$ for planar spins and $T \\approx\n0.3\\,J$ for spherical spins (see Fig.\\ \\ref{fig:Cpos}(a)). However, this\nmaximum does not correspond to a phase transition, as one can infer from\nthe small finite-size effects. There is a second small peak in $C$ at a\nlower temperature $T \\approx 0.02 \\, J$, see panels (b) and (c) of Fig.\\\n\\ref{fig:Cpos}. Since this peak increases with $N$, it is consistent with\na phase transition. Note that at the lowest temperatures $C\/N$ is clearly\nsmaller than $1\/2$ and $1$ for planar and spherical spins, respectively.\nIndeed, in-plane fluctuations around the $120^\\circ$ state yield one\nbranch of soft modes, which we expect to contribute only $N\/12$ to $C$\nrather than $N\/6$, as the other two branches. Thus, we expect $C\/N = 5\/12\n= 0.41666\\dots$ for planar spins and $C\/N = 11\/12 = 0.91666\\dots$ for\nspherical spins in the limit $T \\to 0$. Our MC results tend in this\ndirection, but we have not reached sufficiently low temperatures to fully\nverify this prediction. Lastly, we note that the difference between $C\/N$\nfor planar and spherical spins is consistent with $1\/2$ within error bars\nonly for $T \\lesssim 0.02\\, J$.\n\n\\begin{figure}[tb]\n\\begin{center}\n\\includegraphics[width=0.65 \\columnwidth]{Mpos.eps}\n\\end{center}\n\\caption{Exchange MC results for the square of the sublattice\nmagnetization $m_s^2$ for $J>0$.\nFull lines are for planar spins; dashed lines for spherical spins.\nIncreasing line widths denote increasing $N$.\nError bars are negligible on the scale of the figure.\n\\label{fig:Mpos}\n}\n\\end{figure}\n\nNext, in Fig.\\ \\ref{fig:Mpos} we show the square of the sublattice\nmagnetization (\\ref{defMs}) for $J>0$. First, we observe that it remains\nsmall for almost all temperatures and shoots up just at the left side of\nFig.\\ \\ref{fig:Mpos} where we measure a maximal value of $m_s^2 \\approx\n0.4$ for $T=J\/100$ on the $N=6\\times 6$ lattice. This is consistent with a\nphase transition into a three-sublattice ordered state in the vicinity of\nthe low-temperature peak in the specific heat $C$. In marked difference\nwith the case $J<0$ (see Fig.\\ \\ref{fig:Mneg}), here we observe only small\ndifferences between planar and spherical spins (at least for the sizes\nwhere we have data for planar spins, {\\it i.e.}, $N=6\\times6$, $9\\times9$\nand $12\\times12$).\n\n\\begin{figure}[tb]\n\\begin{center}\n\\includegraphics[width=0.98 \\columnwidth]{UposN.eps}\n\\end{center}\n\\caption{Exchange MC results for the Binder cumulant for $J>0$.\n(a) Full lines are for planar spins; dashed lines for spherical spins.\nIncreasing line widths denote increasing $N$.\nError bars are at most on the order of the width of the lines.\n(b) Binder cumulant for planar spins close to the\ncritical temperature (\\ref{TcPosPlanar}). Lines of increasing slope\nare for increasing system sizes $N=6\\times6$, $9\\times 9$,\nand $12\\times12$.\n\\label{fig:Upos}\n}\n\\end{figure}\n\nFinally, Fig.\\ \\ref{fig:Upos} shows our results for the Binder cumulant,\nas defined in (\\ref{defBinder}). In contrast to the sublattice\nmagnetization $m_s^2$, the Binder cumulants for planar and spherical spins\nare close to each other only for $T \\lesssim 0.1\\,J$, provided we scale to\nthe same prefactor $A$ in the definition (\\ref{defBinder}). Again, our\nbest estimate for the transition temperature $T_c$ is obtained from the\ncrossings of the Binder cumulants at different sizes $N$ shown for planar\nspins in the inset of Fig.\\ \\ref{fig:Upos}. The only meaningful crossings\nare those of the $N=6\\times6$ with the $N=9\\times 9$ and $12\\times12$\ncurves, respectively. This leads to a rough estimate\n\\begin{equation}\n{T_c^{\\rm planar}} \\approx 0.0125 \\, {J}\\, .\n\\label{TcPosPlanar}\n\\end{equation}\nThis estimate is indistinguishable from the corresponding value for\nspherical spins \\cite{CEHPSprep}, as is expected since the spins lie\nessentially all in the $x$-$y$-plane for such low temperatures.\n\n\n\\section{Conclusions and outlook}\n\nIn this paper,\nwe have investigated finite-temperature properties of the classical\nversion of the Hamiltonian (\\ref{eqH}) on a triangular lattice.\n\nFor $J<0$, we have clear evidence for a low-temperature phase\nwith $120^\\circ$ N\\'eel order. The transition temperature is of\norder $\\abs{J}$ and can be determined with reasonable accuracy.\nPlanar and spherical spins yield qualitatively similar results,\nbut there are quantitative differences, in particular the\ntransition temperature (\\ref{TcNegPlanar}) is higher for planar spins.\nWe have performed further simulations for spherical spins \\cite{CEHPSprep}\nin order to determine critical properties. On the one hand, so far\nwe have no evidence for a discontinuity at $T_c$, on the other hand the\nassumption of a continuous phase transition yields very unusual\ncritical exponents \\cite{CEHPSprep} which violate the hyperscaling\nrelation. Thus, the most plausible scenario may be a\nweakly first-order transition.\n\nFor $J>0$, the groundstates are macroscopically degenerate \\cite{CEHPSprep}.\nA thermal order-by-disorder mechanism \\cite{OBD} predicts the\nselection of another $120^\\circ$ ordered state. The corresponding\nphase transition is just at the limits of detectability even with\nthe exchange MC method \\cite{HN96,eM98}: we find an extremely\nlow transition temperature (\\ref{TcPosPlanar})\nwhich is two orders of magnitude smaller than the overall energy scale $J$.\nIn this case, the spins lie essentially in the $x$-$y$-plane in the\nrelevant temperature region such that we obtain quantitatively extremely\nclose results for planar and spherical spins in the vicinity of $T_c$,\napart from a constant offset in the specific heat.\n\nThe features observed e.g.\\ in the specific heat for the classical variant\nresemble those found in the original quantum model\n\\cite{DEHFSL05,DFEBSL05,CEHPSprep}. Since only much smaller systems are\nnumerically accessible for the quantum model, the results for the classical\nvariant are an important tool for understanding the quantum case.\nIn particular, the finite-temperature phase transitions should be universal\nand may therefore be characterized in the classical model. However,\nhighly accurate data is needed for that purpose. Our result that spherical\nand planar spins exhibit the same qualitative features for $J<0$, and that\nfor $J>0$ the $z$-component is even quantitatively very small in the\nrelevant temperature range, may be useful in this context. Namely,\none may choose planar spins for further simulations, thus\nreducing the number of degrees of freedom to be updated.\n\n\\ack\n\nUseful discussions with F.\\ Mila are gratefully acknowledged. We are\nindebted to the CECPV, ULP Strasbourg for allocation of CPU time on an\nItanium 2 cluster. This work has been supported in part by the European\nScience Foundation through the Highly Frustrated Magnetism network.\nPresentation at HFM2006 is supported by the Deutsche\nForschungsgemeinschaft through SFB 602.\n\n\\section*{References}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Vortex state}\n \n The typical vortex magnetization configuration for an oblate hemispheroidal shell and a prolate hemispheroidal shell are depicted in Figures $\\ref{oblate_vortex}$ and $\\ref{prolate_vortex}$, respectively. While for the oblate hemispheroidal shell we can see clearly a core in the middle of the vortex, in the prolate hemispheroidal shell the core is spread all over the vortex.     \n \n \\begin{figure}\n\\begin{center}\n\\includegraphics [scale=0.16]{Aout45Bou25_Ain40Bin20-Vortex_Axes.eps}\n\\end{center}\n\\caption{The magnetization distribution in vortex state of an oblate hemispheroidal shell with $\\rm A_{in}=40$ nm, $\\rm C_{in}=20$ nm and thickness 5 nm. The color code represents the normalized z component of the magnetization, emphesizes the core in the center of the vortex.}   \\label{oblate_vortex}\n\\end{figure}\n\n\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics [scale=0.16]{Aout20Bou25_Ain15Bin20-Vortex.eps}\n\\end{center}\n\\caption{The magnetization distribution in vortex state of a prolate hemispheroidal shell with $\\rm A_{in}=15$ nm, $\\rm C_{in}=20$ nm and thickness 5 nm. The color code represents the normalized z component of the magnetization, shows that the core is not only in the center of the vortex.}   \\label{prolate_vortex}\n\\end{figure}\n\n\n\n\n\n Using the angular parametrisation for the normalised magnetisation \n \\begin{equation}\n\\label{m_eq}\n\\bold{m}=\\dfrac{\\bold{M}}{M}=(\\sin\\theta \\cos\\phi, \\sin\\theta\\sin\\phi, \\cos\\theta), \n\\end{equation}\n one can describe the vortex solution as follows:\n \n\\begin{equation}\n\\label{vortex_eq}\n\\cos\\theta=pf(r),\\qquad     \\phi=\\epsilon\\dfrac{\\pi}{2}+\\chi .\n\\end{equation}\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics [scale=0.5]{Vortex_core_thickness5.eps}\n\\end{center}\n\\caption{The normalized z component of the magnetization as a function of $r$ for hemispheroidal shells (Thickness=5 nm). (a) Symbols correspond to the micromagnetic simulations for semi-principal axis $\\rm A_{in}=40$ nm and different $\\rm C_{in}$.  (b) Symbols indicate the micromagnetic simulations results for semi-principal axis $\\rm A_{in}=15$ nm and three different $\\rm C_{in}$ values. Solid lines are the best ansatzes which fit to the simulations' results.}  \\label{profile_vortex5}\n\\end{figure}\n\n\n\n Here, $(r,\\chi,z)$ are the cylinder coordinates, $p=\\pm1$ is the vortex polarity which describes the vortex core magnetization (up or down) and $\\epsilon=\\pm1$ is the vortex chirality (clockwise or counterclockwise). The function $f(r)$ describes the out of surface structure of the vortex. In order to find a good fit to the oblate hemispheroidal shell, it is instructive to use an analogy with a vortex profile of the planar disk. There are known several models for describing the vortex in a disk shaped particles. For example the ansatz by Usov and Peschany \\cite{N.a1992,Guslienko2004} and the Kravchuk's ansatz \\cite{Kravchuk2007} which describes the vortex structure in disks and rings,\n\\begin{equation}\n\\label{Kravchuk_eq}\nf(r)=e^{-(\\dfrac{r}{\\xi})^2},\n\\end{equation}\nwhere the parameter $\\xi$ determines the radius of the vortex core. For the hemispherical shells Sheka et al. used a modified version of Usov's and Peschany's model \\cite{SHEKA2013} \n \\begin{equation}\n\\label{Usov_eq}\nf(r) = \\begin{cases} \\dfrac{r_c^2-r^2}{r_c^2+r^2} & \\text{if } r<r_c \\\\ 0 &\\text{if } r\\geq r_c   \\end{cases}.\n\\end{equation}\n\n  \n\n\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics [scale=0.5]{Vortex_core_thickness10.eps}\n\\end{center}\n\\caption{The normalized z component of the magnetization as a function of $r$ for hemispheroidal shells (Thickness=10 nm). (a) Symbols correspond to the micromagnetic simulations for semi-principal axis $\\rm A_{in}=40$ nm and different $\\rm C_{in}$.  (b) Symbols indicate the micromagnetic simulations results for semi-principal axis $\\rm A_{in}=15$ nm and three different different $\\rm C_{in}$ values. Solid lines are the best ansatzes which fit to the simulations' results.} \\label{profile_vortex10} \n\\end{figure}\n\n \nFigure $\\ref{profile_vortex5}$(a) shows the profile of the normalized z component of the magnetization as a function of $r$, for $\\rm A_{in}=40$ nm and shell thicknesses of 5 nm.  Kravchuk's ansatz (Eq. \\ref{Kravchuk_eq}) can be a good approximation only for $\\rm C_{in}\/\\rm A_{in} \\leq$ 0.75. The deviation from the Kravchuk's ansatz increases with increasing the aspect ratio ($\\rm C_{in}\/\\rm A_{in}$). The magnetization distributions for $1<\\rm \\frac{C_{in}}{A_{in}}<3$ do not fit at all to the Kravchuk's ansatz and do not correspond to Usov and Peschany's ansatz given by Eq. \\ref{Usov_eq}. Therefore, to describe $m_z$ vs $r$ in this range we propose the one parameter ansatz:\n\n\\begin{equation}\n\\label{like_Lor_eq}\nf(r)=\\frac{\\lambda^2}{\\lambda^2+r^2}.\n\\end{equation} \n     \nThe profile of the normalized z component of the magnetization as a function of $r$, for $\\rm A_{in}=15$ nm and shell thicknesses of 5 nm is plotted in Figure $\\ref{profile_vortex5}$(b). Equation \\ref{like_Lor_eq} can be a good approximation only for $\\rm C_{in}\/\\rm A_{in} \\leq$ 1.33. The magnetization distributions for $\\rm \\frac{C_{in}}{A_{in}}>2$ do not fit at all to the Kravchuk's ansatz or to Eq. \\ref{like_Lor_eq} and do not correspond to Usov and Peschany's ansatz given by Eq. \\ref{Usov_eq}. Therefore, to describe $m_z$ vs $r$ for $\\rm \\frac{C_{in}}{A_{in}}>2$ we suggest the two parameters ansatz:\n\n \\begin{equation}\n\\label{Schultz_eq}\nf(r)=1-(r\/R_c)^{\\alpha}.\n\\end{equation}\nThe fit is not good enough but it is the best simple ansatz.\n\nThe profile of the normalized z component of the magnetization as a function of $r$, for $\\rm A_{in}=40$ nm and shell thicknesses of 10 nm is presented in Figure $\\ref{profile_vortex10}$(a). The Kravchuk's ansatz is the best fit throughout the range ($\\rm C_{in}\/\\rm A_{in} \\leq$1.25).\n \nFigure $\\ref{profile_vortex10}$(b) depicts the profile of the normalized z component of the magnetization as a function of $r$, for $\\rm A_{in}=15$ nm and shell thicknesses of 10 nm. The best fit for $\\rm \\frac{C_{in}}{A_{in}} \\geq2.66$ is Eq. \\ref{Schultz_eq}  and for $0.33<\\rm \\frac{C_{in}}{A_{in}}<0.66$ is Eq. \\ref{like_Lor_eq}. Non of the above ansatz was good enough to describe $m_z$ vs $r$ for the middle size aspect ratios. \n\n \n\\section{Conclusions}\n\nWe present a detailed study of the ground state of magnetic nano hemispheroidal shells. In addition to the three magnetic ground states which exist in hemispherical magnetic shells we find another homogeneous ground state, the easy axis. This additional magnetic structure is the ground state only for small and elongated hemispheroidal shells. Like hemispherical shells \\citep{SHEKA2013}, as the dimensions increase there is more preference for vortex state than the onion state. The vortex profile cannot be described by only one ansatz. We need to choose the right ansatz for each hemispheroidal shell. Start with Kravchuk's ansatz for the wide hemispheroidal shells with low aspect ratio ($\\rm C_{in}\/\\rm A_{in}$), through Eq. \\ref{like_Lor_eq} and end with the ansatz which is given by Eq. \\ref{Schultz_eq} for the narrow hemispheroidal shells with large aspect ratio ($\\rm C_{in}\/\\rm A_{in}$).  \n \n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{PRELIMINARIES}\n\\subsection{Hamilton-Jacobi Reachability Analysis}\nLet $s \\le 0$ be time and $z \\in \\mathbb{R}^n$ be the state of an autonomous system. The evolution of the system state over time is described by a system of ordinary differential equations (ODE) below.\n\\begin{equation}\n\\label{eq:main_ode}\n    \\dot{z} = \\dv{z(s)}{s} = f(z(s), u(s), d(s))\n\\end{equation}\nwhere $u(\\cdot)$ and $d(\\cdot)$ denote the control and disturbance function respectively. The system dynamics $f$ are assumed to be uniformly continuous, bounded and Lipschitz continuous in $z$ for fixed $u$ and $d$. Given $u(\\cdot)$ and $d(\\cdot)$, there exists a unique trajectory that solves equation \\eqref{eq:main_ode}.\n\nThe trajectory or solution to equation \\eqref{eq:main_ode} is denoted as $\\zeta(s;z, t, u(\\cdot), d(\\cdot)): [t, 0] \\rightarrow \\mathbb{R}^n$ , which starts from state $z$ at time $t$ under control $u(\\cdot)$ and disturbances $d(\\cdot)$. $\\zeta$ satisfies \\eqref{eq:main_ode} almost everywhere with initial condition $\\zeta(t;z, t, u(\\cdot), d(\\cdot)) = z$.\n\nIn reachability analysis, we begin with a system dynamics described by an ODE and a target set that represents unsafe states\/obstacles \\cite{HJOverview}. We then solve a HJ PDE to obtain Backward Reachable Tube (BRT), defined as follows:\n\n\\begin{equation}\n\\begin{aligned}\n    \\bar{\\mathcal{A}} = \\{z: \\exists \\gamma \\in \\Gamma, \\forall u(\\cdot) \\in \\mathbb{U}, \\exists s \\in [t, 0], \\\\ \t\\zeta(s;z, t, u(\\cdot), d(\\cdot)) \\in \\mathcal{T} \\}\n\\end{aligned}\n\\end{equation}\n\nIn HJ reachability analysis, a target set $\\mathcal{T} \\subseteq  \\mathcal{R}^n$ is represented by the implicit surface function $V_{0}(z)$ as $\\mathcal{T} = \\{z: V_{0}(z) \\leq 0\\}$. The BRT is then the sub-level set of a value function $V(z, s)$ defined as below.\n\\begin{align}\n\\label{eq:eq3}\n    V(z, s) = \\min_{d(\\cdot)}\\max_{u(\\cdot) \\in \\mathbb{U}}\\min_{s\\in[t,0]}V_0(\\zeta(0;z, t, u(\\cdot), d(\\cdot)))\n\\end{align}\nWe assume disturbance is applied with non-anticipative strategies\\cite{bansal2017hamilton}. In a zero-sum differential game, the control input and disturbances have opposite objectives.\n\nThe value function $V(z, s)$ can be obtained as the viscosity solution of this HJ PDE: \n\\begin{equation}\n\\label{eq:HJ_variational_inequality}\n\\begin{gathered}\n    \\min\\{D_{s}V(z,s) + H(z, \\nabla V(z,s)), V(z,0) - V(z,s) \\} = 0\\\\ \n    V(z,0) = l(z), s \\in [t, 0] \\\\\n    H(z, \\nabla V(z, s)) =\\min_{\\gamma[u](\\cdot)}\\max_{u(\\cdot)} \\nabla V(z, s)^{T}f(z,u) \\end{gathered}\n\\end{equation}\n\nWe compute the HJ PDE until it converges. Numerical toolboxes based on level set methods such as \\cite{LsetToolbox1} are used to obtain a solution on a multi-dimensional grid for the above equation.\n\n\\subsection{Basic Numerical Solution}\nLet us store the value function on a multi-dimensional grid, with the numerical solution of the value function denoted as $V$. Let $N_{d}$ be the grid size on the $d$th axis ($1 \\le d \\le 4$). We also let $x_{d,i}$ denote the state of grid $i$ in dimension $d$.\nIn our approach throughout this paper, we will adopt the central differencing scheme for approximating derivatives in dimension $d$, which is defined as follows:\n\\begin{equation}\n\\label{eq:central_diff}\n\\begin{gathered}\n    D_{d}^{-}V(x_{d, i}) = \\dfrac{V(x_{d, i}) - V(x_{d, i-1})}{\\Delta x_{d}}, \\\\\n    D_{d}^{+}V(x_{d, i}) = \\dfrac{V(x_{d, i+1}) - V(x_{d, i})}{\\Delta x_{d}}, \\\\\n    D_{d}V(x_{d, i}) = \\dfrac{D_{d}^{+}V(x_{d, i}) + D_{d}^{-}V(x_{d, i})}{2} \\end{gathered}\n\\end{equation}\n\nThe two terms $D_{d}^{-}$ and $D_{d}^{+}$ are the left and right approximations respectively. Note that for grid points at each end of each dimension (i.e $i = N_d-1$, $i = 0$), \\eqref{eq:central_diff} is computed with extrapolated points.\nThe basic algorithm for solving \\eqref{eq:HJ_variational_inequality} on-grid for 4D systems is then described as follows: \n\n\\begin{algorithm}[h]\n\\caption{Value function solving procedures}\n\\label{HJalgorithm1}\n\\begin{algorithmic}[1]\n  \\State $V_{0}[N_1][N_2][N_3][N_4] \\leftarrow l(z)$\n    \\State \\texttt{\/\/Compute Hamiltonian term, and max, min deriv}\n  \\For{\\texttt{$i = 0 :N_1 - 1$; $j= 0:N_2 -1$; $k= 0:N_3 -1$; $l= 0:N_4 -1$}}   \n   \n   \n   \n        \\State \\texttt{Compute \\eqref{eq:central_diff} for } $1 \\leq d \\leq 4 $ \n       \n        \\State $ minDeriv \\leftarrow min(minDeriv, D_{d}V(x))$\n        \\State $ maxDeriv \\leftarrow max(maxDeriv, D_{d}V(x))$\n        \\State  $\\displaystyle u_{opt} \\leftarrow \\arg \\max_{u \\in \\mathbb{U}} \\nabla V(z, s)^{\\top}f(z,u)$  \n        \\State $\\dot{x} \\leftarrow f(z, u_{opt})$\n       \n        \\State $H_{i, j, k, l} \\leftarrow \\nabla V(z, s)^{\\top}\\dot{x}$\n        \n   \n   \n   \n  \\EndFor\n  \\State \\texttt{\/\/ Compute dissipation and add to H}\n  \\For{\\texttt{$i = 0 :N_1 - 1$; $j= 0:N_2 -1$; $k= 0:N_3 -1$; $l= 0:N_4 -1$}}\n  \\State $\\alpha_d(x) \\leftarrow \\max_{p \\in [minDeriv, maxDeriv]} \\abs{\\dfrac{\\partial H(x,p)}{\\partial p_d}}$\n  \\State $H_{i, j, k, l} \\leftarrow H_{i, j, k, l} - \\Sigma_{d=1}^{4} \\alpha_{d}(x)\\dfrac{D_{d}^{+}\\check(x) - D_{d}^{-}\\check(x) }{2}$\n  \\State $\\alpha_{d}^{max} \\leftarrow max(\\alpha_{d}^{max}, \\alpha_{d})$\n    \\EndFor\n\\State \\texttt{\/\/Compute stable integration time step}\n\\State $\\Delta t \\leftarrow (\\Sigma_{d=1}^{4}\\dfrac{\\abs{\\alpha_{d}^{max}}}{\\Delta x_{d}})^{-1}$\n  \\State $V_{t+1} \\leftarrow H\\Delta{t} + V_{t}$\n  \\State \\texttt{$V_{t+1} \\leftarrow min(V_0, V_{t+1})$}\n  \\State \\texttt{$\\epsilon \\leftarrow \\abs{V_{t+1} - V_{t}}$}\n  \\If{$\\epsilon < threshold$}\n    \\State \\texttt{$V_{t} \\leftarrow V_{t+1}$}\n    \\State \\texttt{Go to line 3}  \n  \\EndIf\n\\end{algorithmic}\n\\end{algorithm}\nThe above algorithm loops through the 4D array three times. In the first grid iteration, the Hamiltonian terms, maximum and minimum derivative is determined (lines 3-9). In the next grid iteration, the dissipation is computed and added to the Hamiltonian in order to make the computation stable. At the same time, the maximum alphas in all dimensions defined in line 13 are computed. These $\\alpha_d^{max}$ are used to determine the step bound $\\Delta t$. In the third grid iteration (line 19), each grid point is integrated for a length of $\\Delta t$.\n\\begin{figure}[H]\n    \\centering\n    \\includegraphics[width=.30\\textwidth]{mem_4dims.png}\n    \\caption{9 memory accesses (yellow + green colored grid) within each iteration for computing derivatives in all 4 directions as in line 3 (algorithm \\ref{HJalgorithm2})}\n    \\label{fig:my_label}\n\\end{figure}\nIn certain cases, $\\alpha_{d}(x)$ in line 13 is the same as computing the absolute value of $\\dot{x}$, which has been computed in line 8. In addition, in a lot of cases, $\\alpha_{d}^{max}$ stays the same across different time iterations. We also observed that $\\Delta t$ depends only on grid configuration and $\\alpha_{d}^{max}$. So instead of re-computing $\\Delta t$ every time and then loop through the 4D grid array again, we could pre-compute $\\Delta t$ and re-use it for all the time iterations. Combining these ideas together, throughout this paper, we will use the following algorithm with one grid looping, which is more computationally efficient:\n\\begin{algorithm}[h]\n\\caption{Value function solving procedures}\n\\label{HJalgorithm2}\n\\begin{algorithmic}[1]\n  \\State $V_{0}[N_1][N_2][N_3][N_4] \\leftarrow l(z)$\n  \\For{\\texttt{$i = 0 :N_1 - 1$; $j= 0:N_2 -1$; $k= 0:N_3 -1$; $l= 0:N_4 -1$}}\n        \\State \\texttt{Compute \\eqref{eq:central_diff} for } $1 \\leq d \\leq 4 $ \n        \n        \\State  $\\displaystyle u_{opt} \\leftarrow \\arg \\max_{u \\in \\mathbb{U}} \\nabla V(z, s)^{\\top}f(z,u)$  \n        \\State $\\dot{x} \\leftarrow f(z, u_{opt})$\n       \n        \\State $H_{i, j, k, l} \\leftarrow \\nabla V(z, s)^{\\top}\\dot{x}$\n        \\State $H_{i, j, k, l} \\leftarrow H_{i, j, k, l} - \\Sigma_{d=1}^{4} \\abs{\\dot{x_{d}}}\\dfrac{D_{d}^{+}(x) - D_{d}^{-}(x) }{2}$\n        \\State $V_{t+1,(i, j, k, l)} \\leftarrow H_{i, j, k ,l}\\Delta{t}_{precomputed} + V_{t, (i, j, k, l)}$\n        \\State \\texttt{$V_{t+1, (i, j, k, l)} \\leftarrow min(V_{0, (i, j, k, l)}, V_{t+1, (i, j, k, l)})$}\n  \\EndFor\n  \\State \\texttt{$\\epsilon \\leftarrow \\abs{V_{t+1} - V_{t}}$}\n  \\If{$\\epsilon < threshold$}\n    \\State \\texttt{$V_{t} \\leftarrow V_{t+1}$}\n    \\State \\texttt{Go to line 2}  \n  \\EndIf\n\\end{algorithmic}\n\\end{algorithm}\n\\subsection{Field Programmable Gate Arrays (FPGA)}\nFPGA are configurable intergrated circuits that are programmed for specific applications using hardware description language (HDL). \n\n\\begin{figure*}[h]\n  \\includegraphics[width=\\textwidth]{figures\/Overall-System.pdf}\n  \\caption{System overview on FPGA (Right). The initial value array is first transferred from DRAM to FPGA's on-chip memory. The memory buffer then distributes data to the 4 PEs to concurrently compute the new value function at 4 consecutive grid points. The output from PE is then written back to DRAM. Each fully pipelined PE outputs one grid point every clock cycle (Left). Inside the PE, there are hardware components that sequentially solve algorithm 2}\n    \\label{fig:overall_system}\n\\end{figure*}\n\nComputing platforms such as CPUs, GPUs, and FPGAs have a memory component and computing cores. Compute cores must request and receive all the necessary data from the memory component before proceeding with the computation. If the memory component cannot provide all data accesses the application requires to proceed at once, cores have to stall and wait, slowing down the computation.\n\\begin{figure*}[h]\n  \\includegraphics[width=\\textwidth, height=155.5pt]{figures\/PE_pipeline2.pdf}\n  \\caption{Pipelining schedule of a single PE. The PE's operation is an assembly line where multiple grid points could be processed at the same time at different stages. Each stage is physical hardware that computes specific parts of algorithm 2. At a particular stage and a particular cycle, the PE is busy computing a certain part of algorithm 2 for the grid point at the indices shown. Note that for simplicity, the indices shown here are for a single PE only.}\n  \\label{fig: pipelining_schedule}\n\\end{figure*}\nEfficient systems need to have both fast computing cores and fast data distributions from the memory. Depending on the application, the memory access and computing pattern will vary. General-purpose CPU\/GPU are often architected towards a reasonable performance for a wide variety of applications, but unoptimized for any particular application. FPGA chip, on the other hand, provides programmable digital circuits to design customized computing core and memory blocks. Thus, one can leverage knowledge about the details of the computing workload to design an efficient system accordingly with FGPA. With FPGA, one could control and achieve a higher degree of parallelism from the digital hardware level at the cost of programmability.\n\n\n\\subsection{Problem Description}\n\nA key observation of algorithm 2 is that each new grid point $V_{t+1}$ could be computed independently with each other within one time iteration and therefore, in parallel. We could then leverage a high degree of parallelism on FPGA by having many cores to update as many grid points concurrently as possible. \n\nHowever, before that, two challenges must be addressed. Firstly, memory blocks need to efficiently distribute data to compute cores. In order for a loop computation to proceed, each of these cores needs up to 9 data inputs (Fig 2.) and a memory design needs to satisfy this.  \nSecondly, a four-dimensional grid takes up tens of megabytes in memory and therefore cannot be fully fit to FPGA's on-chip memory for fast access.\n\n\nIn this paper, our goal is twofold. First, we will discuss our hardware design that can solve the above challenges and maximize parallel computation of algorithm 2 while efficiently making use of FPGA's on-chip memory. Next, we will show that this enables low latency of computation on FPGA which could be deployed in real-time systems.\n\n\\input{sections\/03-FPGA}\n\\input{sections\/04-Experiment}\n\n\\addtolength{\\textheight}{-12cm}  \n                                 \n                                 \n                                 \n                                 \n                                 \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\bibliographystyle{ieeetr}\n\n\n\\section{Introduction}\n\\label{sec:intro}\n\nAutonomous systems are becoming more prevalent in our lives. Examples of these systems include self-driving cars, unmanned aerial vehicles, rescue robots, etc. One key factor that will allow wider adoption of autonomous systems is the guaranteed safety of these systems. Despite tremendous progress in autonomous system research in areas such as motion planning, perception, and machine learning, deployment of these systems in environments that involve interactions with humans and other robots remains limited due to the potential danger these robotic systems can cause. Formal verification methods can help autonomous robots reach their untapped potential.\n \nHamilton-Jacobi (HJ) reachability analysis is a formal verification approach that provides guaranteed safety and goal satisfactions to autonomous systems under adversarial disturbances. There are many ways to do reachability analysis, solving the HJ PDE is one way to characterize sets of safe states and synthesizes optimal controllers, which involves calculating Backward Reachable Tube (BRT) that describes a set of states the system must stay out of in order to avoid obstacles. HJ reachability analysis has been successfully applied in practical applications such as aircraft safe landing \\cite{SafeLanding}, multi-vehicle path planning, multi-player reach avoid games \\cite{SafePlatoon}. \nThe appeal of this particular method is that it's very powerful in handling control and disturbances, nonlinear system dynamics, and flexible set representations.  \n\nThe main downside to HJ reachability is that it's solved on a multi-dimensional grid with the same number of dimensions as the number of state variables and scales exponentially with the number of dimensions. This prevents HJ formulation to be applied on real-time systems where safety is increasingly demanded. While 3D or smaller systems could be computed quickly with multi-core CPUs, practical systems that usually involve 4 to 5 system components can take several minutes to hours to compute. There have been researches that proposed decomposing high dimensional systems into smaller tractable sub-systems that can exactly compute \\cite{ExacDec} or overapproximate the BRT in certain cases \\cite{OverDec}. However, that challenge of applying HJ formulation on real-time systems remains, as some systems cannot be decomposed further than four dimensions, and over-approximation is introduced if projection methods are used. \n\nIn this paper, we expand the limit of the number of dimensions for which we could directly compute the BRT in \\emph{real time} through the use of FPGA. We would argue that customized hardware accelerators could complement well with those decomposition methods in making higher dimensional systems provably safe in real-time. As general-purpose computer no longer double its performance every two years due to the end of Moore's law, we have seen examples of successful hardware accelerations in other areas such as machine learning's training\/inference \\cite{TPU,Eyeriss, EIE}, robot's motion planning\\cite{DukePaper}.\n\n\nIn this paper, our contributions are as follows: \n\\begin{itemize}\n    \\item We prototype a customized hardware design on FPGA that accelerates HJ reachability analysis to 16x compared to state-of-the-art implementation and 142x compared to \\cite{LsetToolbox1} on 16-thread CPU for 4D system\n    \\item We demonstrate that the system could meet real-time requirement of guaranteeing safety in changing environments by re-computing BRT at 5Hz\n    \\item Demonstrate obstacle avoidance with a robot car driving in an environment in which new obstacles are introduced during run time at 5Hz.\n\\end{itemize}\n\n\n\\section{EXPERIMENT \\& RESULT}\n\\label{sec:exp}\n\\subsection{Experiment setup}\nIn this section, we demonstrate that our system can meet the real-time requirement through an obstacle avoidance demonstration in a changing environment.\n\nWe used a Tamiya TT02 model RC car\\cite{nvidiajetracer} controlled by an on-board Nvidia Jetson Nano microcontroller inside a $4$m $\\times$ $6$m room. We use the following extended Dubins car model for its dynamics:\n\\begin{equation}\n\\begin{split}\n\\label{eq:car_dynamics}\n    \\dot{x} &= v \\cos(\\theta)\\\\\n    \\dot{y} &= v \\sin(\\theta)\\\\\n    \\dot{v} &=a\\\\\n    \\dot{\\theta} &= v\\frac{\\tan(\\delta)}{L}\n\\end{split}\n\\end{equation}\n\n\\noindent where $a \\in \\left[-1.5, 1.5\\right]$,\n$\\delta \\in \\left[-\\frac{\\pi}{18}, \\frac{\\pi}{18}\\right]$,\nand $L = 0.3$m. The control inputs are the acceleration $a$ and the steering angle $\\delta$. We use a grid size of $60\\times60\\times20\\times36$ with resolutions of $0.1$m, $0.067$m, $0.2$m\/s and $0.17$ rad for $x$-position, $y$-position, speed and angle, respectively.\n\nInside the room, we use orange cones as obstacles and a motion capture system is used to accurately track the car's state and the position of the obstacles. We initialize the initial value function as follows:\n\\begin{equation}\n\\label{eq:init_V}\n    V_{0}(x, y, v, \\theta) = \\sqrt{(x-x_o)^2 + (y-y_o)^2} - R\n\\end{equation}\nwhere $x_o$ and $y_o$ are the obstacle's positions and R is the radius of the cone.\nObstacle's positions can be obtained from the motion capture system. Each of the cones has a radius of $0.08$m but is set as $0.75$m to account for the model mismatch between the car and the dynamics used.\n\nFor the experiment, we considered three different environments, with different cone placements, set up inside the room as shown in Fig. \\ref{fig:cones}. For each environment, a user manually controls the car and tries to steer into the cones.\n\\begin{equation}\n\\label{eq:closetoboundary}\n    V(x,y,v, \\theta) < 0.15\n\\end{equation}\nGiven the car's state, when \\eqref{eq:closetoboundary} is satisfied, the car is near the boundary of a BRT so optimal control computed from the value function is applied to safely avoid the cone. The optimal control is obtained from the value function as follows:\n\\begin{equation}\n\\label{eq:uopt_max}\n    u_{opt} =  \\arg \\max_{u \\in \\mathbb{U}} \\nabla V(x, y, v, \\theta , s)^{\\top}f(x, y, v, \\theta, u)\n\\end{equation}\n\n\\begin{figure}[h]\n\\centering\n\\begin{subfigure}[t]{.4\\textwidth}\n\\centering\n\\includegraphics[width=\\linewidth]{images\/car_center.png}\n\\caption{Environment 1}\n\\end{subfigure}\n\\hfill\n\\begin{subfigure}[t]{.4\\textwidth}\n\\centering\n\\includegraphics[width=\\linewidth]{images\/car_line_final.png}\n\\caption{Environment 2}\n\\end{subfigure}\n\\hfill\n\\begin{subfigure}[t]{.4\\textwidth}\n\\centering\n\\includegraphics[width=\\linewidth]{images\/car_apart_cross_final.png}\n\\caption{Environment 3}\n\\end{subfigure}\n\\caption{Different BRTs are used as the placement of cones change over time which limits where the RC car can be as it drives around the room.}\n\\label{fig:cones}\n\\end{figure}\n\n\nWe pre-compute the BRTs with a horizontal time of $0.5$s for three environments using optimized\\_dp\\cite{optimizedp} and demonstrate safety by correctly loading the value functions as the environment changes. \nWe choose to pre-compute the BRTs in order to emulate having an FPGA on-board without extra latency resulted from communication with a remote AWS instance.\nFor all environments, the maximum time step to make the integration stable is $0.007497$s. Initially, the room had a single cone but changed over time to different cone placements. \nThe BRT of a new environment could not be used until 200ms after the environment has changed, which is longer than the time taken to compute the same BRT on an FPGA.  A video of these experiments can be found at \\url{https:\/\/www.youtube.com\/playlist?list=PLUBop1d3Zm2vgPL4Hxtz8JufnIPmvrwlC}\n\n\\subsection{Hardware Correctness}\nWe use fixed-point data representations for hardware computation. In particular, we use 32 bits with 5 bits to represent the integer part (including sign) and 27 bits for the decimal part. With this choice, the precision of our computation is $2^{-27}=7.45\\times10^{-9}$ and the range of our computation is from $-16$ to $16$. The area we use for the experiment is $4$m $\\times$ $6$m, hence the largest absolute distance is the diagonal of $7.2$m. Therefore, the number of integer bits is enough to represent all possible values in the solution $V$, which has the physical interpretation of minimum distance to collision over time, given \\eqref{eq:eq3} and the choice of $V_0$ in \\eqref{eq:init_V}.\n\nWe choose to synthesize and implement our design on AWS F1 FPGA because of its flexibility and availability. To correctly input data to the FPGA, we first generate an initial value array based on the obstacles' positions and radius described by \\eqref{eq:init_V}. Then this value array is converted from floating-point to fixed-point number correctly based on the bit choice discussed above. Afterward, the value array is passed to the FPGA for the HJ PDE solving procedure to start.\n\nFor all three experiment, we verified the correctness of BRT generated by our hardware with the toolbox at \\cite{optimizedp} by comparing the maximum error between corresponding grid points. The toolbox uses 32-bit floating-point numbers. The numerical error resulting from the different representations is shown in table below for the three environments in table \\ref{table:Error}.\n\\begin{center}\n\\captionof{table}{ERROR COMPARISON} \n \\label{table:Error}\n \\begin{tabular}{||c c c c||} \n \\hline\n  & Env. 1 & Env. 2 & Env. 3 \\\\ [0.5ex] \n \\hline\\hline\n Error & $1.68\\times10^{-6}$ & $1.78\\times10^{-6}$& $1.37\\times10^{-6}$ \\\\ \n \\hline\n\\end{tabular}\n\\end{center}\nThese negligible errors are due to precision difference between fixed-point and floating point number. Even though the computation is repeated for many iterations, the maximum error does not grow dramatically over time. We believe that is because of the convergence property of BRT. As time grows, the rate of changes in the grid values also slows down leading to stable discrepancy between the correct floating point and fixed-point values.\n\n\\subsection{Computational speed and Resources Usage}\nTo measure the speed up for all three environments, we compare the computation time on AWS FPGA running at 250MHz against \\cite{optimizedp} and \\cite{LsetToolbox1} running on a 16-thread Intel(R) Core(TM) i9-9900K CPU at 3.60GHz. The latency here is the time it takes to compute the BRT. For FPGA, latency can be computed by multiplying the clock cycles with the clock period. The result is summarized in the table below.\n\n\n\\begin{center}\n\\captionof{table}{FPGA}\\label{table:fpga} \n \\begin{tabular}{||c c c c c||} \n \\hline\n    & \\ \\textbf{Clock cycles} & \\ \\textbf{Period} & \\textbf{Iterations} & \\textbf{Latency} \\\\ [0.4ex] \n \\hline\\hline\n Env. 1 & 44155209 & 4 ns & 67 & 0.176s\\\\ \n \\hline\n Env. 2 & 44155209 & 4 ns & 67 & 0.176s\\\\\n \\hline\n Env. 3 & 44155209 & 4 ns & 67 & 0.176s\\\\ [0.5ex]\n \\hline\n\\end{tabular}\n\\end{center}\n\n\\begin{center}\n\\captionof{table}{optimized\\_dp\\cite{optimizedp}} \\label{table:op_dp} \n \\begin{tabular}{||c c c c||} \n \\hline\n    & \\ \\textbf{Latency} & \\textbf{Iterations} & \\textbf{FGPA speed up} \\\\ [0.1ex] \n \\hline\\hline\n Env. 1 & 3.35 s & 67 & \\textbf{$\\times$18.9} \\\\ \n \\hline\n Env. 2 & 2.99 s & 67 & \\textbf{$\\times$17} \\\\\n \\hline\n Env. 3 & 3.42 s & 67 & \\textbf{$\\times$19.4}\\\\ [0.5ex]\n \\hline\n\\end{tabular}\n\\end{center}\n\n\\begin{center}\n \\captionof{table}{ToolboxLS\\cite{LsetToolbox1}} \\label{table:toolboxls} \n \\begin{tabular}{||c c c c||} \n \\hline\n    & \\ \\textbf{Latency} & \\textbf{Iterations} & \\textbf{FPGA speed up} \\\\ [0.1ex] \n \\hline\\hline\n Env. 1 & 25.11 s & 70 & \\textbf{$\\times$142} \\\\ \n \\hline\n Env. 2 & 25.14 s & 70 & \\textbf{$\\times$142} \\\\\n \\hline\n Env. 3 & 25.18 s & 70 & \\textbf{$\\times$142}\\\\ [0.5ex]\n \\hline\n\\end{tabular}\n\\end{center}\n \nIt can be observed that the latency of computation on FPGA is fixed and deterministic for all three environments while the latency on CPUs varies even though the computation remains the same. With the lower latency of $0.176$s, we are able to update the value function at a frequency of $5.68$Hz. The resources usage of our design for 4 PEs is shown in the table below. \n\\begin{center}\n \\begin{tabular}{||c c c c||} \n \\hline\n    & \\ \\textbf{LUT} & \\textbf{BRAM} & \\textbf{DSP} \\\\ [0.1ex] \n \\hline\\hline\n \\textbf{Used} & 26319 & 519 &  598\\\\ \n \\hline\n \\hline\n \\textbf{Available} & 111900 & 1680 & 5640\\\\ \n \\hline\n \\textbf{Utilization} & \\textbf{14.03\\%} & \\textbf{30.89\\%} & \\textbf{10.6\\%} \\\\[0.5ex]\n \\hline\n\\end{tabular}\n\\end{center}\nOn an FPGA, arithmetic operations on numbers are implemented using Digital Signal Processing (DSP) hardware or Look Up Table (LUT) that perform logical functions. Our design does not significantly consume most of the available resources and could be scaled up to a larger grid size.\n\n\\section{CONCLUSION}\nThis paper introduces a novel customized hardware design on FPGA that allows HJ reachability analysis to be computed $16$x faster than state-of-the-art implementation on a 16-thread CPU. Because of that, we are able to solve the HJ PDE at a frequency of 5Hz. The latency of our computation on FPGA is deterministic for all computation iterations, which is crucial for safety-critical systems. Our design approach presented here can be applied to system dynamics and potentially higher dimensional systems. \nFinally, we demonstrate that at 5Hz, a robot car can safely avoid obstacles and guarantee safety.\n\n\\section{Solving the HJ PDE Using FPGAs}\n\\label{sec:fpga}\n\nBefore going into details of the design, we will introduce some terminologies that will be relevant throughout the next section.\n\nIn digital systems, time is discretized into the unit of a \\emph{clock cycle}, which is the amount of time it takes for an operation such as computing, loading, and storing to proceed. Each clock cycle typically is a few nanoseconds.\nDynamic Random Access Memory (DRAM) is a type of memory that sits outside of the FPGA, which has higher memory capacity but takes a lot more clock cycles to access.\n\nOur custom hardware comprised two main components: on-chip memory buffer, and processing elements (PE) or computing cores (shown in Fig \\ref{fig:overall_system}). The memory buffer is on-chip storage, providing access to all the grid points a PE needs to compute a new value function. Each PE is a digital circuit that takes 9 grid points from the memory buffer to compute a new value function at a particular grid point according to algorithm 2 (line 3-10). In the following subsections, we will go into the details of each component.\n\\subsection{Indexed Processing Element (PE)}\n\nThe PE has the following target design objectives: \n(1) increase compute throughput (defined as the number of output generated per second) through pipelining, (2) reduce the computation time of each PE, (3) and ensure the correctness of result while minimizing data transfer between DRAM and FPGA.\n\nIn our design, we use 4 PEs (as shown in figure \\ref{fig:overall_system}). Each PE has an index $idx$ with $0 \\le idx \\le 3$ associated with it and computes the grid point $V_{t+1}(i, j, k, l + idx)$. At the beginning of the computation of algorithm 2, each PE takes as input a grid index $(i,j,k,l)$ and its 8 neighbours to start computing $V_{t+1}(i,j,k,l)$ according to algorithm 2 (line 2-10).\n\nTo increase computation throughput, each PE is fully pipelined. Similar to an assembly line, the PE operation is divided into multiple stages taking a few clock cycles to complete (Fig. \\ref{fig: pipelining_schedule}). Each stage within the pipeline is physical hardware that has a computation corresponding to one of the lines in algorithm 2 (line 3-10) for a particular index $i, j, k, l$. Every clock cycle, the result from previous stages will be loaded to the next stage, following the sequential order of algorithm 2. At any time during operations, the processing element is computing different intermediate components for multiple indices concurrently (explained in Fig.\\ref{fig: pipelining_schedule}).\n\nTo ensure that the computation is correct, inside each of the PE, there are indices counters to keep track of loop variable $i, j, k, l$, with the inner loop variable incrementing by one every clock cycle. These indices are used to correctly address the state vectors during system dynamics computation. To avoid accessing external DRAM we store these 4 state\/position vectors $x$ or any fixed non-linear functions such as $\\cos(\\cdot)$ and $\\sin(\\cdot)$ of these states as a lookup table stored in on-chip memory, as state vectors depend only on grid configuration and do not change with the environment. Each PE will have its own look-up table to avoid communications between PEs. Having this data on-chip will only require a few kilobytes of memory and no need to access DRAM throughout the computation.\n\\begin{figure*}[h]\n  \\centering\n  \\includegraphics[width=0.85\\textwidth, height=250pt]{figures\/Mem_Buf.pdf}\n  \\caption{Four lines of memory buffer supply all the grid data to the four PEs. Each of the rectangle blocks is a FIFO queue synthesized as Block RAM (BRAM). The overhead notation is the size of the FIFO queue with $N_1, N_2, N_3, N_4$ as the four grid dimensions. Note that the queue's size depends only on three dimensions. Every clock cycle, new grid points streamed from DRAM start entering each buffer line (left-hand side) and grid points at the end of the lines are discarded (right-hand side). }\n  \\label{fig: parallel_line}\n\\end{figure*}\n\\subsection{On-Chip Memory Buffer}\nThe memory buffer has the following key design objectives: (1) minimizing the amount of on-chip memory usage and external DRAM accesses while (2) concurrently providing 9 grid points to each PE every clock cycle.\n\nOne problem of working with a high-dimensional grid is that the whole grid can take up tens of megabytes and therefore cannot be fully fit to a state-of-the-art FPGA's on-chip memory. Instead of storing everything on-chip, in our design, grid points are streamed continuously from DRAM into an on-chip memory buffer (shown in Fig.\\ref{fig:overall_system}) and can be re-used many times for spatial derivatives computation in 4 dimensions before being thrown away. From the grid dimensions, we could compute the maximum reuse distance beyond which a grid point can be safely discarded as no longer needed. This maximum reuse distance is equal to the minimum size of on-chip memory buffer, which is dependent only on $N-1$ dimensions \\cite{soda} and can be fitted to an FPGA's on-chip memory. Our memory buffer structure is implemented as First In First Out (FIFO) queue data structure. Every clock cycle, a new grid point supplied from DRAM will start entering the FIFO queue while the grid point reaching at the end of the FIFO queue will be discarded (shown in Fig. \\ref{fig: parallel_line}).\n\nFPGA on-chip memory buffers are composed of standard Blocks of Random Access Memory (BRAM). Each BRAM has two-ports and at most two reads can be requested concurrently in the same clock cycle. If all 9 grid points (shown in Fig. \\ref{fig:my_label}) are stored in the same BRAM at the same time, a PE would then have to wait for 5 clock cycles before performing the computation. One way to increase the number of accesses per clock cycle is to duplicate the data in multiple BRAMs, but this would not work well for multidimensional arrays since these array copies easily exceed FPGA on-chip memory. A different technique would be \\emph{memory banking}, which is to partition the memory on-chip into multiple BRAM that could concurrently deliver data to the PE, allowing the PE to start to compute new value function for a grid point in one clock cycle. \n\nTo allow concurrent access for multiple PEs, we adopted the parallel memory buffer microarchitecture from \\cite{soda}. Corresponding to the number of PEs, our on-chip storage structure is made of 4 line buffers. Each of these line buffers is a sequence of BRAM connected acting in a queue fashion: a grid point moves towards the end of the line every clock cycle. The two endpoints of each BRAM (shown in Fig 4) provide tapping points that are connected as inputs to the PEs.\nThe number of PEs, therefore, is mainly limited by the DRAM bandwidth. \n\nWe also made modifications to the execution flow in \\cite{OptimalMicro} to accommodate for computing values function at the boundary. Once each of the buffer lines is half full, all the processing elements can start computing a new value function.    \n\n\\subsection{Fixed-Point Representation}\nComputing a new value function based on algorithm 2 involves multiple addition operations on floating-point numbers. At the hardware level, the addition of floating-point numbers is as computationally expensive as fixed-point multiplication, which would take up lots of resources and chip's area. Instead, we use fixed-point representations for our data to reduce the burden on the hardware. We will show in the next section that this has little impact on the correctness of the computation if the radix point is chosen carefully for the grid configuration.    ","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{intro}\n\nThe renormalization group procedure for effective particles (RGPEP) has been developed during the last years~\\cite{Glazek:2012qj,GlazekTrawinskiAdS,Gomez-Rocha:2015esa} as a non-perturbative tool for constructing bound-states in quantum chromodynamics (QCD)~\\citep{Wilsonetalweakcoupling}.\nIt introduces the concept of effective particles, which differ from the bare or canonical ones, by having size $s$, corresponding to the momentum scale $\\lambda=1\/s$. \nCreation and annihilation of effective particles in the Fock space are described by the action of effective particle operators, $a^\\dagger_s$ and $a_s$, on states built from the vacuum state $|0\\rangle$ using $a_s^\\dagger$; bare particle operators, $a^\\dagger_0$ and $a_0$, appearing in the canonical Hamiltonian,  create and annihilate pointlike particles (with size $s=0$).\n\nWe are interested in calculating the evolution of quark and gluon quantum states, describing their dynamics and studying their binding. In a single formulation, the sought effective Hamiltonian must provide a means for the constituent-like behavior of quarks and gluons in hadrons with the measured quantum numbers, and also an explanation for the short-distance phenomena of  weakly interacting pointlike partons.\n\nIn this work, we apply the RGPEP to  interacting gluons in the absence of quarks. We will demonstrate that the RGPEP passes the test of describing asymptotic freedom, which is a precondition for any approach aiming at using QCD, especially for tackling nonperturbative issues, such as the ones that emerge when one allows effective gluons to have masses~\\cite{Wilsonetalweakcoupling}.\n\nWe start from the regularized canonical Hamiltonian for quantum Yang-Mills field in the Fock space, obtained from the corresponding Lagrangian density. We use the RGPEP to  introduce effective particles and calculate a family of  effective Hamiltonians characterized by a scale or size parameter $s$. These Fock-space Hamiltonians depend on the effective-particle size parameter in an asymptotically free way: the coupling constant in a three-gluon interaction term vanishes with the inverse of $\\ln(1\/s)$.\n\nIn the following, we summarize the procedure, which is general and can be applied to any other quantum field theory.\nFor a more extended and detailed explanation, we refer the reader to Ref.~\\cite{Gomez-Rocha:2015esa}.\n\n\\section{Renormalization group procedure for effective particles }\n\\label{sec:1}\n\n\\subsection{Initial Hamiltonian}\n\\label{subsec:2:1}\n\nWe derive the canonical Hamiltonian for Yang-Mills theories from the Lagrangian density:\n\\begin{eqnarray}\n {\\cal L} = - {1 \\over 2} \\text{tr} \\, F^{\\mu \\nu}\nF_{\\mu \\nu} \\ ,\n\\end{eqnarray}\nwhere \n$F^{\\mu \\nu} = \\partial^\\mu A^\\nu \n- \\partial^\\nu A^\\mu + i g [A^\\mu, A^\\nu]$,  \n$A^\\mu = A^{a \\mu} t^a$, \n$ [t^a,t^b] = i f^{abc} t^c$, which leads to the energy-momentum tensor, \n\\begin{eqnarray}\n {\\cal T} ^{\\mu \\nu} & = &\n-F^{a \\mu \\alpha} \\partial^\\nu A^a_\\alpha + g^{\\mu \\nu} F^{a \\alpha\n\\beta} F^a_{\\alpha \\beta}\/4 \\ .\n\\end{eqnarray}\nWe choose the front-form (FF) of dynamics~\\citep{Dirac1949} which consists of setting the quantization surface on the hyperplane $x^+=x^0+x^3=0$. Using  the gauge $A^+=0$, the Lagrange equations lead to the condition\n\\begin{eqnarray}\nA^- & = & \n{ 1 \\over \\partial^+ } \\, 2 \\, \\partial^\\perp A^\\perp \n- { 2 \\over \\partial^{ + \\, 2} } \\ \nig \\, [ \\partial^+ A^\\perp, A^\\perp] \\ ,\n\\end{eqnarray}\nso that the only degrees of freedom are the fields $A^\\perp$.\n\nIntegration of $ {\\cal T} ^{+-}$ over the front $x^+=0$ leads to the FF energy of the constrained gluon field:\n\\begin{equation}\nP^- = {1 \\over 2}\\int dx^- d^2 x^\\perp  {\\cal T} ^{+-}\\, |_{x^+=0} \\quad .\n\\label{Hg}\n\\end{equation}\nThis operator contains a series of products of 2nd, 3rd or 4th powers of the field $A^\\mu$ or their derivatives. The energy momentum tensor $ {\\cal T} ^{+-}$ can be written as\n\\begin{eqnarray}\n {\\cal T} ^{+ -} = {\\cal H}_{A^2} + {\\cal H}_{A^3} + {\\cal H}_{A^4} + {\\cal\nH}_{[\\partial A A]^2} \\ ,\n\\end{eqnarray}\nwhere\n\\cite{Casher:1976ae,Thorn:1979gv,BrodskyLepage}\n\\begin{eqnarray}\n\\label{HA2}\n{\\cal H}_{A^2} & = & - {1\\over 2} A^{\\perp a } (\\partial^\\perp)^2 A^{\\perp a} \\  , \\\\\n\\label{HA3}\n{\\cal H}_{A^3} & = &  g \\, i\\partial_\\alpha A_\\beta^a [A^\\alpha,A^\\beta]^a  \\ , \\\\\n\\label{HA4}\n{\\cal H}_{A^4} & = &  - {1\\over 4} g^2 \\, [A_\\alpha,A_\\beta]^a[A^\\alpha,A^\\beta]^a  \\ , \\\\\n\\label{HA2A2}\n{\\cal H}_{[\\partial A A]^2} & = &  {1\\over 2}g^2 \\,\n[i\\partial^+A^\\perp,A^\\perp]^a {1 \\over (i\\partial^+)^2 }\n[i\\partial^+A^\\perp,A^\\perp]^a \\ .\n\\end{eqnarray}\nThe quantum canonical Hamiltonian is obtained by replacing the field $A^\\mu$ by\nthe quantum field operator\n\\begin{eqnarray}\n\\hat A^\\mu & = & \\sum_{\\sigma c} \\int [k] \\left[ t^c \\varepsilon^\\mu_{k\\sigma}\na_{k\\sigma c} e^{-ikx} + t^c \\varepsilon^{\\mu *}_{k\\sigma}\na^\\dagger_{k\\sigma c} e^{ikx}\\right]_{x^+=0} \\ ,\n\\end{eqnarray}\nwhere $[k] = \\theta(k^+)\ndk^+ d^2 k^\\perp\/(16\\pi^3 k^+)$,\nthe polarization four-vector is defined as $\\varepsilon^\\mu_{k\\sigma} \n= (\\varepsilon^+_{k\\sigma}=0, \\varepsilon^-_{k\\sigma} \n= 2k^\\perp \\varepsilon^\\perp_\\sigma\/k^+, \n\\varepsilon^\\perp_\\sigma)$ and the indices $\\sigma$ and $c$ denote spin and color quantum numbers, respectively. The creation and annihilation operators  satisfy the commutation relations\n\\begin{eqnarray}\n\\left[ a_{k\\sigma c}, a^\\dagger_{k'\\sigma' c'} \\right] \n& = & \nk^+\n\\tilde \\delta(k - k') \\,\\, \\delta^{\\sigma \\sigma'}\n\\, \\delta^{c c'} \\ , \n\\quad\n\\left[ a_{k\\sigma c}, a_{k'\\sigma' c'} \\right] \n\\ = \\\n\\left[ a^\\dagger_{k\\sigma c}, a^\\dagger_{k'\\sigma' c'} \\right] \n\\ = \\\n0 \\ ,\n\\end{eqnarray}\nwith $\\tilde \\delta(p) = 16 \\pi^3 \\delta(p^+) \\delta(p^1)\n\\delta(p^2)$.\n\n\nThe canonical Hamiltonian is divergent and  needs regularization. At every interaction term, every creation and annihilation operator in the canonical Hamiltonian is multiplied by a regulating factor\\footnote{Other regulating functions are available~\\cite{Gomez-Rocha:2015esa}. Finite dependence of the effective Hamiltonian on the small-x regularization may be thought to be related to the vacuum state problem, the phenomena of symmetry breaking and confinement~\\cite{Wilsonetalweakcoupling}.}\n\\begin{eqnarray}\nr_{\\Delta \\delta}(\\kappa^\\perp, x) \n& = & \n\\exp(-\\kappa^\\perp \/ \\Delta)\\, x^\\delta \\theta(x-\\epsilon) \\ ,\n\\end{eqnarray} \nwhere $x$ is the relative momentum fraction $x_{p\/P}=p^+\/P^+$, $\\kappa$ is the relative transverse momentum, $\\kappa_{p\/P}=p^\\perp-xP^\\perp$, and $P$ is the total momentum in the term one considers.\nThe regulating function prevents the interaction terms from acting if the change of transverse momentum between gluons were to exceed $\\Delta$, or if the change of longitudinal momentum fraction $x$ were to be smaller than  $\\delta$.\n\n\\subsection{Derivation of the effective Hamiltonian}\nThe RGPEP transforms bare, or point-like creation and annihilation operators into effective ones~\\citep{Glazek:2012qj}. Effective particle operators of size $s=t^{1\/4}$ are related to bare ones by certain unitary transformation\n\\begin{eqnarray}\n\\label{at}\na_t & = &  {\\cal U} _t \\, a_0 \\,  {\\cal U} _t^\\dagger  \\ .\n\\end{eqnarray}\nThe fact that the Hamiltonian operator cannot be affected by this change requires,\n\\begin{eqnarray}\n {\\cal H} _t(a_t) & = &  {\\cal H} _0(a_0)  \\ , \n\\end{eqnarray}\nwhich is equivalent to writing:\n\\begin{eqnarray}\n\\label{cHt}\n {\\cal H} _t(a_0) =  {\\cal U} _t^\\dagger  {\\cal H} _0(a_0) {\\cal U} _t  \\ .\n\\end{eqnarray}\nDifferentiating both sides of~(\\ref{cHt}) leads to the RGPEP equation:\n\\begin{eqnarray} \n\\label{ht1}\n {\\cal H} '_t(a_0) & = &\n[  {\\cal G} _t(a_0) ,  {\\cal H} _t(a_0) ] \\ ,\n\\end{eqnarray} \nwhere $ {\\cal G} _t = -  {\\cal U} _t^\\dagger  {\\cal U} '_t$,\nand therefore, \n$\n {\\cal U} _t \n= \nT \\exp{ \\left( - \\int_0^t d\\tau \\,  {\\cal G} _\\tau\n\\right) }$. $T$ denotes ordering in $\\tau$.\nThe RGPEP equation~(\\ref{ht1}) is the engine of this procedure. It governs the evolution of effective particles with the scale parameter $t$. It encodes the relation between pointlike quantum gluons appearing in the canonical Hamiltonian and the effective,\nor constituent ones referred to by effective phenomenological models describing bound states.\n\n We choose the generator to be the commutator $ {\\cal G} _t = [  {\\cal H} _f,  {\\cal H} _{Pt} ]$,\\footnote{Other generators are also allowed but may lead to more complicated expressions~\\cite{Glazek:2000dc}. Our choice is similar to Wegner's~\\cite{Wegner}.} where $ {\\cal H} _f$ is the non-interacting term of the Hamiltonian and $ {\\cal H} _{Pt}$ is defined in terms of $ {\\cal H} _t$ . \n\n$ {\\cal H} _t$ is a series of normal-ordered products of creation and annihilation operators,\n\\begin{eqnarray}\n\\label{Hstructure} \n {\\cal H} _t(a_0) =\n\\sum_{n=2}^\\infty \\, \n\\sum_{i_1, i_2, ..., i_n} \\, c_t(i_1,...,i_n) \\, \\, a^\\dagger_{0i_1}\n\\cdot \\cdot \\cdot a_{0i_n} \\, .\n\\end{eqnarray} \n$ {\\cal H} _{Pt}$ differs from $ {\\cal H} _t$ by \n the vertex total $+$-momentum factor,\n\\begin{eqnarray}\n\\label{HPstructure} \n {\\cal H} _{Pt}(a_0) & = &\n\\sum_{n=2}^\\infty \\, \n\\sum_{i_1, i_2, ..., i_n} \\, c_t(i_1,...,i_n) \\, \n\\left( {1 \\over\n2}\\sum_{k=1}^n p_{i_k}^+ \\right)^2 \\, \\, a^\\dagger_{0i_1}\n\\cdot \\cdot \\cdot a_{0i_n} \\, .\n\\end{eqnarray} \nThe initial condition for the differential equation~(\\ref{ht1}) is given by the regularized canonical Hamiltonian given in Section~\\ref{subsec:2:1} plus counterterms.\nMore precisely, the initial condition is given by the physical fact that at very small distances or very high energies, the regularized canonical Hamiltonian must be recovered and any regularization dependence must be removed. \n\nWe solve the RGPEP equation~(\\ref{ht1}) for the effective Hamiltonians using an expansion in powers of the coupling constant $g$  up to third order and we focus our studies on the structure of the three-gluon term~\\citep{Glazek:2012qj,Gomez-Rocha:2015esa}.\n\n\n\n\n\n\\section{The three-gluon vertex}\n\\label{sec:2}\n\nThe third-order effective Hamiltonian  expansion have the following structure:\n\\begin{eqnarray}\n\\label{Hpert}\nH_t & = & \nH_{11,0,t} + H_{11,g^2,t} + H_{21,g,t} + \nH_{12,g,t} + H_{31,g^2,t} + H_{13,g^2,t} + \nH_{22,g^2,t} + H_{21,g^3,t} + H_{12,g^3,t}  \\ .\n\\end{eqnarray}\nThe first and second subscripts indicate the number of creation and annihilation operators, respectively. The third subscript labels the order in powers of $g$. Finally, the last label indicates the dependence on the scale parameter $t$.\n\nThe initial condition at $t=0$ has the form:\n\\begin{eqnarray}\n\\label{Hper0}\nH_0 & = & \nH_{11,0,0} + H_{11,g^2,0} + H_{21,g,0} + \nH_{12,g,0} + H_{31,g^2,0} + H_{13,g^2,0} + \nH_{22,g^2,0} + H_{21,g^3,0} + H_{12,g^3,0}  \\ ,\n\\end{eqnarray} \nand consists of the regularized canonical Hamiltonian plus counterterms. The latter are calculated in such a way that $H_t$ remains finite when $\\Delta\\to\\infty$. It is not possible to remove the small-$x$ cutoff $\\delta$ at these point. However, this dependence will be of interest in higher-order calculations, since the small-$x$ phenomena are thought to be related to the vacuum-state behavior. The last step in the RGPEP is to replace bare creation and annihilation operators by effective ones.\n\nThe three-gluon vertex and the running of the Hamiltonian coupling are encoded in the sum of first- and third-order terms:\n\\begin{eqnarray}\nH_{(1+3),t} & = & (H_{21,g,t} + \nH_{12,g,t}) + ( H_{21,g^3,t} + H_{12,g^3,t} ) \\ .\n\\end{eqnarray}\nThe third-order solution requires the knowledge of the first and second-order solutions. The sum of all these contributions has the form\\footnote{The subscripts 1,2,3 refer to the gluon lines indicated in Fig.~\\ref{Fig-runingg}. So, e.g. $\\kappa_{12}^\\perp= x_{2\/3}\\kappa_{1\/3}^\\perp - x_{1\/3}\\kappa_{2\/3}^\\perp$ . }~\\citep{Gomez-Rocha:2015esa} (see Fig.~\\ref{Fig-runingg}):\n\\begin{eqnarray}\nH_{(1+3),t} & = &\n\\sum_{123}\\int[123] \\  \\tilde \\delta(k_1+k_2-k_3) \n\\ f_{12t} \\, \\left[ \\tilde Y_{21\\,t}(x_1,\\kappa_{12}^\\perp, \\sigma)a^\\dagger_{1t} a^\\dagger_{2t} a_{3t}  \\ + \\tilde Y_{12\\,t}(x_1,\\kappa_{12}^\\perp, \\sigma)\\ a_{3t}^\\dagger a_{2t} a_{1t} \\right]\\nonumber \\\\ \n\\end{eqnarray}\nwhere $f_{12 t}=e^{-(k_1+k_2)^4t}$ is a form factor and $\\tilde Y_{21\\,t}(x_1,\\kappa_{12}^\\perp, \\sigma)$ is the object of our study. \nWe define the Hamiltonian coupling constant $g_t$ as the coefficient in front of the canonical color, spin and momentum dependent factor $Y_{123}(x_1,\\kappa_{12}^\\perp,\\sigma)= i f^{c_1 c_2 c_3} [ \\varepsilon_1^*\\varepsilon_2^*\n\\cdot \\varepsilon_3\\kappa_{12}^\\perp - \\varepsilon_1^*\\varepsilon_3 \\cdot\n\\varepsilon_2^*\\kappa_{12}^\\perp {1\\over x_{2\/3}} - \\varepsilon_2^*\\varepsilon_3\n\\cdot \\varepsilon_1^*\\kappa_{12}^\\perp {1\\over x_{1\/3}} ]$ in the limit $\\kappa_{12}^\\perp\\to 0$, for some value of $x_1$ denoted by $x_0$.\nSo,\n\\begin{eqnarray}\n\\label{consider}\n\\lim_{\\kappa_{12}^\\perp \\to 0}\n\\tilde Y_t(x_1,\\kappa_{12}^\\perp, \\sigma) \n& = &\n\\lim_{\\kappa_{12}^\\perp \\to 0}\n\\left[\nc_t(x_1,\\kappa_{12}^\\perp) \nY_{123}(x_1,\\kappa_{12}^\\perp, \\sigma) \n+ \ng^3 \\tilde T_{3 \\,\\text{finite}}(x_1,\\kappa_{12}^\\perp, \\sigma) \n\\right]  .\n\\end{eqnarray}\nwhere $\\tilde T_{3 \\,\\text{finite}}(x_1,\\kappa_{12}^\\perp, \\sigma)$ is a finite part contained in the counterterm and does not contribute to the running coupling,\n\\begin{eqnarray}\n\\lim_{\\kappa_{12}^\\perp \\to 0}\nc_t(x_1,\\kappa_{12}^\\perp) \n& = &\ng + g^3 \\lim_{\\kappa_{12}^\\perp \\to 0}\n\\left[\n  c_{3t  }(x_1,\\kappa_{12}^\\perp) \n- \n  c_{3t_0}(x_1,\\kappa_{12}^\\perp) \n\\right] \\ . \n  \\label{limitc3}\n\\end{eqnarray}\nAnd assuming some value for $g_0$ at some small $t_0$, $g_{t_0}=g_0$,\n\\begin{eqnarray}\n\\label{gl1}\ng_t\n& \\equiv &  c_t(x_1) \n\\ = \\\n\\label{gl2}\ng_0 + g_0^3 \n\\left[\n  c_{3t  }(x_1) \n- \n  c_{3t_0}(x_1) \n\\right] \\ .\n\\end{eqnarray}\nWe introduce now the momentum scale parameter $\\lambda=t^{-1\/4}$. This yields,\n\\begin{eqnarray}\n\\label{gl}\ng_\\lambda & = &\ng_0 - { g_0^3 \\over 48 \\pi^2 }   N_c \\,   11 \\,\\ln\n{ \\lambda \\over \\lambda_0} \\ .\n\\end{eqnarray} \nDifferentiation of the latter with respect to $\\lambda$ leads to\n\\begin{eqnarray}\n\\lambda {d \\over d\\lambda} \\, g_\\lambda\n& = & \\beta_0 g_\\lambda^3 \\ , \\quad \\text{with}\\quad \\beta_0 \\, = \\, - { 11 N_c \\over 48\\pi^2 } \\ .\n\\end{eqnarray}\nThis result equals the asymptotic freedom result \nin Refs.~\\cite{Gross:1973id,Politzer:1973fx},\nwhen one identifies $\\lambda$ with the momentum \nscale of external gluon lines in Feynman diagrams.\nOur result also coincides with the expression obtained in~\\citep{Glazek:2000dc}, where an analogous calculation were performed using a different generator. \n\n\n\n\\begin{figure}[h]\n \\includegraphics[width=0.9\\textwidth]{runningg3rdorderNEW2}\n \\caption{Graphical representation of terms contributing to the effective three-gluon vertex (third-order expansion)~\\citep{Gomez-Rocha:2015esa}. Thin internal lines correspond to intermediate bare gluons and thick external lines  correspond to the creation and\nannihilation operators that appear in the three-gluon FF Hamiltonian interaction\nterm for effective gluons of size $s$.\nDashed lines with transverse bars represent the combined contributions of terms~(\\ref{HA4}) and~(\\ref{HA2A2}). The black dots indicate counterterms.}\n \\label{Fig-runingg}\n\\end{figure}\n\n\\section{Summary and conclusion}\n\nWe have applied the RGPEP to the quantum $SU(3)$ Yang-Mills theory and extracted the running coupling from the three-gluon-vertex  term in the third-order effective Hamiltonian.\nThe result turns out to be independent of the choice of the generator, as it coincides with the one obtained in an analogous calculation performed in~\\cite{Glazek:2000dc}, using a different generator. The present generator, however, leads to simpler equations than the older one, which is desired and needed for our forthcoming forth-order calculations, required for any attempt at description of physical systems using QCD~\\citep{Wilsonetalweakcoupling}. \nThe obtained running coupling is of the form that is familiar from other formalism and renormalization schemes and passes the test of producing asymptotic freedom, which any method aiming at solving QCD must past.\n\n\n\n\\begin{acknowledgements}\nPart of this work was supported by the Austrian Science Fund (FWF) under project No. P25121-N27. Fig.~\\ref{Fig-runingg} was produced with JaxoDraw~\\citep{Binosi:2003yf}. \n\\end{acknowledgements}\n\n\n \n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}