MegaMorph -- multi-wavelength measurement of galaxy structure: Sérsic profile fits to galaxies near and far
Marina Vika, Steven P. Bamford, Boris Haeussler, Alex L. Rojas, Andrea Borch, Robert C. Nichol
MMon. Not. R. Astron. Soc. , 1–29 (2012) Printed 7 June 2018 (MN L A TEX style file v2.2)
MegaMorph – multi-wavelength measurement of galaxy structure:S´ersic profile fits to galaxies near and far
Marina Vika, Steven P. Bamford, Boris H¨außler, , Alex L. Rojas, Andrea Borch, Robert C. Nichol Carnegie Mellon University in Qatar, Education City, PO Box 24866, Doha, Qatar School of Physics and Astronomy, The University of Nottingham, University Park, Nottingham, NG7 2RD, UK Institute of Cosmology and Gravitation (ICG), University of Portsmouth, Dennis Sciama Building, Burnaby Road Portsmouth, PO1 3FX, UK
Accepted ... Received ...; in original form ...
ABSTRACT
We demonstrate a new multi-wavelength technique for two-dimensional parametric modellingof galaxy surface-brightness profiles, which we have incorporated into the widely used soft-ware
GALFIT . Our new method, named
GALFITM , extends
GALFIT
GALFITM is evaluated by fitting elliptical S´ersic profiles to ugriz imag-ing data for 4026 galaxies, comprising the original SDSS imaging for 163 low redshift( v < ∼ km s − ) galaxies and 3863 artificially redshifted ( . < ∼ z < ∼ . ) images of thesame galaxies.Comparing results from single-band and multi-band techniques, we show that GALFITM significantly improves the extraction of information, particularly from bands with low signal-to-noise ratio (e.g., u and z SDSS bands) when combined with higher signal-to-noise images.We also study systematic trends in the recovered parameters, particularly S´ersic index, thatappear when one performs measurements of the same galaxies at successively higher redshifts.We argue that it is vital that studies investigating the evolution of galaxy structure are carefulto avoid or correct for these biases.The resulting multi-band photometric structural parameters for our sample of 163 galax-ies are provided. We demonstrate the importance of considering multi-band measurements byshowing that the S´ersic indices of spiral galaxies increase to redder wavelengths, as expectedfor composite bulge-disk systems. Finally, for the ellipticals in our sample, which should bewell-represented by single-S´ersic models, we compare our measured parameters to those fromprevious studies.
Key words: galaxies: photometry — galaxies: fundamental parameters — galaxies: structure— methods: data analysis — techniques: image processing
The radiation output of typical galaxies is dominated by starlight,with most energy emerging at optical–near-infrared wavelengths(e.g., Driver et al. 2012). From integrated photometry alone, al-though preferably combined with distance information, e.g., cos-mological redshifts, we can learn a great deal about galaxy stel-lar populations (e.g., Walcher et al. 2011). Measurements of lumi-nosity at a number of wavelengths may be used to infer physicalquantities, such as stellar mass, age and metallicity (e.g. Pforr et al.2012). Including near-infrared (NIR) data is particularly useful forbreaking the degeneracy between age and metallicity (Worthey1994). However, the integrated stellar population is only one as-pect of a galaxy’s character. By studying how the stellar population is spatially distributed, e.g. via surface photometry, we may gainfurther insights into a galaxy’s history. In particular, the stellar dis-tribution provides information on the dynamical state of a galaxy,without requiring observationally expensive resolved spectroscopy(e.g., Krajnovi´c et al. 2011).The most basic spatially resolved measurement is simply thesize of a galaxy. Just measuring this can place strong constraintson models of galaxy formation (e.g., Mao, Mo, & White 1998),although defining and estimating a galaxy’s size is by no meanstrivial. More sophisticated descriptions of a galaxy’s surface bright-ness distribution include visual morphology (e.g., de Vaucouleurset al. 1991; Lintott et al. 2011), concentration (e.g., Shimasaku et al.2001), asymmetry and clumpyness (e.g., Conselice 2003), multi- c (cid:13) a r X i v : . [ a s t r o - ph . C O ] J u l Vika et al.
Gaussian expansion (Cappellari 2002), S´ersic models (Sersic 1968;Graham & Driver 2005) and their extensions (Andrae et al. 2011),bulge-disk models (e.g., Simard et al. 2002), and more sophisti-cated multi-component models (e.g., Peng et al. 2010).Studying spatially resolved surface brightness distributions isobviously easier and more precise for relatively nearby galaxies.Indeed, for some it is possible to determine such distributions fromcounts of individual stars (e.g., Ibata et al. 2009). Even withoutgoing to this extreme, it is possible to discern a great level of de-tail: thin disks, thick disks, bulges, bars, rings, spiral arms, nu-clei, cores and streams (e.g., Yoachim & Dalcanton 2006). Eachof these features may be associated with different stellar popula-tions (e.g., Seigar & James 1998). Clearly, characterising the struc-ture of galaxies to this level is a extremely complex task. Even forvery nearby galaxies, studies of early-type galaxy kinematics haveshown that visual morphologies and photometric structural param-eters can be misleading (Emsellem et al. 2011).Once we move beyond our local neighbourhood fine struc-tural distinctions become more difficult, and only broad featuresmay be discerned. Furthermore, for large samples there is a needfor robust automated measurements, which favours simpler ap-proaches. In this context, integrated properties, such as total lumi-nosity and colour, provide an important starting point (e.g., Blantonet al. 2001; Baldry et al. 2004), but additional physical insight is af-forded by even simple size and structure information (e.g., Blantonet al. 2003; Kauffmann et al. 2003; Shen et al. 2003). Such studiesdemonstrate a clear division of the galaxy population in terms ofspheroid- and disk-dominated systems, which have different rela-tionships with mass, colour and environment (e.g., Guo et al. 2009).The fundamentally different nature of spheroid and disk compo-nents highlights the need to separate these structures, and considertheir individual properties, whenever possible (MacArthur et al.2003; Allen et al. 2006; Cameron et al. 2009; Simard et al. 2011). Itis even becoming clear that there exist different types of spheroids,formed via distinct processes, which may be distinguished by struc-tural measurements (e.g., Kormendy & Kennicutt 2004; Athanas-soula 2005; Fisher & Drory 2010; Fabricius et al. 2012).At still higher redshifts, decomposing galaxy components be-comes difficult. However, single-component parameters, such aseffective radius and S´ersic index, can still be used to explore vari-ations in the structures of galaxies. One active area of study con-cerns the evolution of early-type galaxy sizes (Hopkins et al. 2010).Members of the massive early-type galaxy population at z ∼ appear to be substantially more compact than those today (e.g.,Buitrago et al. 2008; Trujillo et al. 2009; Cassata et al. 2010).This suggests, when combined with expectations from hierarchi-cal merging, that some of these early-formed objects must increasetheir sizes dramatically, without significantly changing their stellarmass (although see Mancini et al. 2010 and Trujillo et al. 2012).Structural measurements have also been used to quantify evolutionin the morphological composition of the galaxy population, findingthat massive galaxies were considerably more disky at z ∼ thanthey are in the local Universe (van der Wel et al. 2011; Bruce et al.2012; Buitrago et al. 2013).Studies of galaxy structure typically require robust estimatesof stellar mass and recent star formation, in order to study trendsand ensure that comparisons are made between similar popula-tions of objects. These quantities, in turn, require accurate measure-ments of galaxy colour, which are compared with representativestellar population models to deduce the desired physical estimates.These models necessarily make simplifying assumptions regard-ing star-formation history, metallicity, and dust extinction (Walcher et al. 2011). They are also usually constrained using global colours,discarding any information on spatial variations due to multiplecomponents. However, the fact that colour typically varies radi-ally within a galaxy indicates the presence of multiple star forma-tion histories that are, to some degree, localised. Incorporating thisknowledge into our estimates of stellar mass and star formationhistory – for example, by allowing different structural componentsof a galaxy to have different colours – can potentially make theseestimates more robust and meaningful.In addition to the variation of stellar populations betweengalaxy components, one should consider population gradientswithin individual components. This is especially true in the caseof low-quality imaging, where decomposing each galaxy into morethan one component becomes difficult, or for fundamentally single-component systems, e.g. elliptical galaxies (Suh et al. 2010). Ap-proaches include extracting radial colour profiles using annularphotometry (e.g., Bell & de Jong 2000; Weinmann et al. 2009),or estimating physical parameters pixel-by-pixel from colour maps(e.g., Welikala et al. 2008; Lanyon-Foster et al. 2012). These tech-niques have been used very effectively, although they are primarilysuitable for data with reasonably high spatial resolution, as they donot account for the point spread function (PSF). This means thatgalaxies with smaller angular sizes will be systematically biased toflatter gradients. For some studies one can choose to consider onlylarge and/or nearby galaxies, and thereby avoid resolution issues.However, in order to obtain sufficiently large samples for environ-mental studies, or to ensure completeness as a function of size, onemust necessarily push to the resolution limits of current surveys.For galaxies which can be reasonably well described bysmooth, axisymmetric models, assuming such a model allows oneto account for the effect of the PSF. Assuming an appropriatemodel also enables more accurate surface brightness profile mea-surements, particularly for low signal-to-noise data. However, mea-suring colours using model-based methods typically assumes thateach component (the entire galaxy in the case of single-componentmodels) has a homogeneous colour. Determining structural param-eters independently in each band results in colours that are hard tointerpret, as the fluxes can correspond to models with very differ-ent shapes in each band. Usually, therefore, structural parametersare first determined on one image, and these are subsequently ap-plied to images in other wavebands with total flux as the only freeparameter (e.g., Lackner & Gunn 2012). La Barbera & de Carvalho(2009) take a hybrid approach, performing independent profile fitsto each waveband, and then performing elliptical annular photome-try on the resulting models to determine PSF-corrected colour gra-dients (utilised to great effect in La Barbera et al. 2010). Whilesome techniques allow their models to be fit to two images simul-taneously (e.g., Simard et al. 2002), until very recently (Bamford etal., in prep.) no solutions permitted the use of an arbitrary numberof images at different wavelengths.Prompted by all of the above issues, we embarked upon aproject, named ‘MegaMorph’, to investigate ways of improving ourability to extract physically-meaningful structural information fromgalaxy images (Bamford et al. 2011). In this paper we present re-sults obtained by applying our new technique for measuring galaxyprofiles in multi-wavelength imaging. The development of thistechnique was motivated by a desire to preserve the concept ofdistinct structural components, while allowing for realistic colourgradients within them. It was also driven by a determination tomake more effective use of the wealth of spatially-resolved multi-wavelength imaging available from modern surveys. To achieveour goals we have extended standard two-dimensional profile fit- c (cid:13) , 1–29 egaMorph II ting techniques to include wavelength-dependent models for whichthe degree of variation with wavelength is controllable. We choseto implement these techniques by modifying GALFIT
GALFIT as GALFITM .This paper is one of a series that will explore the advan-tages of our multi-wavelength approach to measuring galaxy struc-tural properties. In Bamford et al. (in prep.; hereafter Paper I) wepresent this new tool in more detail, describing the new features anddemonstrating its use through some specific examples. In H¨außleret al. (2012) (Paper III) we test our new method on large datasets,by automating both the preparation of the data and the fitting pro-cess itself. For this purpose we adapt
GALAPAGOS (Barden et al.2012) to use
GALFITM , and quantify its performance by applyingit to data from the GAMA survey (Driver et al. 2011) and to simu-lations. The resulting measurements of GAMA galaxies, in partic-ular the variation of structural parameters with wavelength, will bestudied further in Vulcani et al. (Paper IV; in prep.). In the presentpaper, as well as Papers III and IV, we perform and utilise the re-sults of single-S´ersic profile fitting. Ultimately, however, we expectthe greatest benefits of multi-band fitting when dealing with multi-component models. A further set of papers will present the resultsof bulge-disk decompositions on the same data.This paper investigates the performance of
GALFITM for fit-ting single-S´ersic profiles to galaxy images with a wide range ofresolution and signal-to-noise. This is achieved by analysing SDSSimages of large, nearby galaxies, as well as versions of these imagesthat have been convolved and resampled to simulate their appear-ance at a range of redshifts. These results are used to study trendsin statistical scatter and potential systematic biases in measured pa-rameters as the galaxies are considered at increasing distance. Asa further test, the structural parameters of the elliptical galaxies inour sample, as obtained on the original images, are compared tothose obtained by independent studies.In Section 2 we present the criteria for selecting our parentgalaxy sample and the methodology used to construct the artifi-cially redshifted sample. In Section 3 we briefly introduce our newtechnique and utilise it to fit wavelength-dependent profiles to u , g , r , i , z images for both original and artificially redshifted galaxysamples. We present a comparison of the results from our multi-band fitting method to those obtained by fitting each band inde-pendently. In Section 4 we discuss these comparisons and their im-plications, and compare our multi-band structural parameters fora subsample of elliptical galaxies with parameters derived by pre-vious studies. We conclude and discuss future work in Section 5.Throughout the paper, we adopt a standard cosmological modelwith Ω Λ = 0 . , Ω m = 0 . and H = 70 km s − Mpc − . Our primary aim in this paper is to evaluate the benefits of mod-elling galaxy structure by simultaneously fitting imaging data ata variety of wavelengths. For this purpose we select galaxies thathave parametric surface brightness profile measurements by previ-ous studies (using single-band data), and which are located within the footprint of the Sloan Digital Sky Survey (SDSS; Abazajianet al. 2009). We compile a sample of 168 galaxies with images in u , g , r , i , z passbands, which have profile fits presented in one ofthe following papers: Pohlen & Trujillo (2006), M¨ollenhoff (2004),Moriondo et al. (1998) and Caon et al. (1993).The galaxies in our sample are typically larger than a singleSDSS frame. The production of images for these galaxies is accom-plished with the help of M ONTAGE (Jacob et al. 2010), which per-forms the transformations, rebinning and background adjustmentnecessary to combine the individual frames into a single mosaic.We ensure that each mosaic contains the entire galaxy as well assufficient sky area for an accurate background estimation. In theright-hand panels of Figures 5, 6 and 7 we show three example ofthis imaging data for galaxies NGC4570, NGC4274 and NGC4431.The M
ONTAGE
SDSS tool scales each image to a constantphotometric zeropoint ( mag arcsec − s − , corresponding to . mag s − with an exposure time of . s, or . mag).In order for GALFIT to construct a reasonable sigma image (giv-ing an estimate of the noise on each pixel, and hence affecting therelative weighting of pixels and final parameter uncertainties), theimages produced by M
ONTAGE were rescaled to approximate elec-tron counts. This was done by assuming values of the zeropoint andgain for each band that were obtained by averaging over the survey.Five out of 168 galaxies are excluded from the further analy-sis. NGC0988 and UGC04684 are removed because of flux con-tamination due to nearby saturated stars which hinder accuratephotometry. We also exclude the blue-shifted galaxies NGC3031,NGC4406 and NGC4419 as they cannot be appropriately red-shifted by
FERENGI (see Section 2.2). The final sample consists of163 low redshift ( v < ∼ km s − ) galaxies of mixed morphology(from ellipticals to late-type spirals). We do not attempt to constructa complete or representative sample of galaxies, but a wide varietyof galaxies are present in the sample. The global properties of thesample are tabulated in Table 1. We wish to assess the accuracy and reliability of galaxy profile fit-ting over wide ranges of spatial resolution and signal-to-noise, and,in particular, compare the performance of single- and multi-bandfitting techniques. In order to do so, we require a sample of galaxieswith known properties, or at least properties measured on very highquality imaging, which have been observed with a variety of resolu-tions and signal-to-noise levels. Samples of real observations withsuch a wide variety of imaging qualities are very limited, and wouldmake for a highly heterogeneous dataset. In principle, completelysimulated images could be used. However, it is difficult to producesimulations which reproduce the detailed and varied appearance ofreal galaxies, with spiral arms, bars, star-formation regions and dis-turbances. We utilise simulations based on smooth profiles in PaperIII, but in this paper we wish to test our fitting technique on morerealistic images.We therefore choose to construct a sample of artificially red-shifted images. These are produced using
FERENGI (Barden et al.2008). For each of the 163 nearby galaxies we create a set of arti-ficially redshifted u , g , r , i , z images. The artificial redshifting ap-plies cosmological changes in angular size, surface brightness and,optionally, spectral energy distribution, to simulate the observationof a given galaxy at a greater distance (for details see Barden et al.2008). FERENGI can also account for changes in instrumentation,such as pixel scale, PSF, filters and zeropoint. However, in our case c (cid:13) , 1–29 Vika et al.
Figure 1.
Artificially redshifted r -band images of galaxy NGC2742 at red-shifts . , . , . and . , together with their masks and outputs ofour multi-band fitting procedure. The redshifted input images are shown inthe upper panels. The corresponding GALFITM best-fitting model and itsresiduals (image − model) are shown in the middle panels. The lower pan-els display the masks that were used during the fit. Note that the resolutionand flux changes with redshift, images are displayed at constant physicalsize with an adaptive intensity scale. we use real SDSS imaging to simulate SDSS observations, so thesedetails remain unchanged.While FERENGI can account for the k-correction, i.e. thechange in flux due to the change in the restframe wavelength sam-pled by a given observational filter for galaxies at different red-shifts, we choose to disable this feature. In this way the redshiftedgalaxies should retain a constant absolute magnitude in each bandas a function of redshift. This is less physically realistic, but al-lows us to test the ability of
GALFIT to measure magnitudes as thenoise increases and resolution decreases in a more stringent man-ner, without the uncertainties that would be introduced by bandpassshifting.As galaxies become fainter and less resolved with increasingredshift, the fitting process becomes more challenging. Faint com-ponents that are visible at low redshifts may no longer be apparentat greater distances. We therefore anticipate greater uncertaintieson the recovered structural parameters and potentially systematicbiases.The galaxies are artificially redshifted to values in the range . < z < . with a step of . . However, if the true redshiftof a galaxy is higher than . the production of the artificial im-ages starts at the next higher redshift, so that the artificial galaxiesare always more distant than in the original imaging. In addition,we cease the redshifting process when the resulting image wouldbe smaller than 21 pixels on a side, as at this point the galaxy iscomparable in size to the PSF. Smaller galaxies, therefore, do nothave artificial images all the way out to z = 0 . .The top panels of Fig. 1 illustrate 4 of the 25 artificially red-shifted r -band images constructed for the galaxy NGC2742. No- tice the reduction in resolution (a result of the decreasing angularsize with respect to both the PSF and pixel scale) and diminishingsignal-to-noise with increasing redshift.One limitation of our method of artificially redshifting realgalaxies is that in some cases the galaxies have neighbouring ob-jects. FERENGI treats all objects in the image in the same way, andso these neighbours are also artificially redshifted. In the case oftrue galaxy groups this is realistic, and provides a way of assessingthe impact of such systems on structural measurements. However,in the case of stars, as well as foreground or background galax-ies, the resulting image is not entirely realistic. Furthermore, asour original images are large, they include a relatively high numberof stars. In this respect, the artificially-redshifted images thereforecontain more stars than would be expected for real observations ofhigher-redshift galaxies. On the other hand, the stars are faded bythe same amount as the galaxy, which counteracts this effect. In theend we expect our artificial images to present a reasonably realisticdistribution of neighbouring objects.At high redshifts we notice blending occurring: neighbouringgalaxies or stars tend to be merged by the worsening resolution.This process distorts the shape of the target object, resulting inchanges to the structural parameter measurements. Such changesin shape are also accompanied, in some cases, by small changes ofthe magnitude (mostly increasing of the flux of the target galaxy).For instance, in Fig. 2 at z = 0 . we can see, around galaxyNGC4387, three additional sources that are well separated from thecentral object of interest. At z = 0 . two of the three sources have‘blended’ with the main galaxy and finally, at redshift z = 0 . , allfour objects have blended into one extended source. GALFIT
3, and hence
GALFITM , can make use of mask images dur-ing the fitting process. These masks indicate the areas of each im-age that are contaminated with light from sources that are not in-tended to be modelled by
GALFIT . The masks are additional FITSfiles created by using SE
XTRACTOR (Bertin & Arnouts 1996) seg-mentation maps. Pixels flagged in the mask image are not includedin the fitting process. Initially, we create separate masks for each ofthe five bands, and then take the union of these to produce a singlemerged mask. We perform a detailed visual check on every maskto ensure that they do not include the main galaxy or parts thereof.When a mask was found to cover part of the galaxy flux the maskimage was corrected. The final merged mask is used for all wave-lengths in the
GALFIT fitting process. This ensures that the fittedarea is identical in each band and for fits with both
GALFIT
GALFITM . Differences between bands and methods are thereforenot attributable to variations in masking. The masks are generatedindependently for each redshift, in order to mimic the deblendingissues faced by real observations. The lower panels of Fig. 1 showfour masks at different redshifts for the galaxy NGC2742.The estimation of the background level is important for a suc-cessful fit (e.g., H¨aussler et al. 2007). For all galaxies, the sky back-ground value is measured before running
GALFIT . To estimate thesky for both the real images and artificial
FERENGI images we usea similar method to that employed by
GALAPAGOS . This methodapplies elliptical annuli centred on the galaxy to measure the sur-face brightness as a function of radius. The sky value is selected atthe point where the surface brightness gradient is robustly and con-servatively judged to be flat to the accuracy permitted by the skynoise. The procedure is described in more detail in Barden et al.(2012). In the case of the original images we have also employed a c (cid:13) , 1–29 egaMorph II Figure 2.
Artificially redshifted images of galaxy NGC4387 at redshifts . , . and . , together with models fit using our multi-band tech-nique. This Figure illustrates the artificial blending of nearby objects due tothe artificial redshifting done with FERENGI . second approach for estimating the sky values. For this we simplycalculate the resistant (sigma-clipped) mean of the large M ONTAGE mosaics. Each M
ONTAGE image has at least 60 per cent of its areafree of bright objects, which enables an accurate sky determination.The sky background values are kept fixed during the fitting process.The galaxy light distribution, especially in the inner region,is distorted by the effect of atmospheric seeing, which needs to becorrected for (Trujillo et al. 2001).
GALFIT achieves this by con-volving the S´ersic function with the PSF in the fitting process.PSFs for our images are provided by SDSS, however for simplicitywe have used a single typical PSF for each passband throughoutthis work. For all our highly-resolved real images, small variationsin the PSF will make minimal difference to the obtained parame-ters. For our artificially-redshifted images,
FERENGI produces out-put images which match the provided PSF. The same PSF is alsosuppled to
GALFIT , so the two are entirely consistent.
GALFIT and
GALFITM
We have adapted
GALFIT version 3.0.2 for the requirements of thisproject, with the consent of the original developer, C. Peng. To dif-ferentiate our modified version from the standard release we referto it as
GALFITM . For reference, all the work in this paper uses
GALFITM version 0.1.2.1. The code will be publicly released inthe near future. Development is continuing, primarily to improveease-of-use and incorporate the additional features mentioned be-low. However, the general performance of the technique is expectedto remain as presented in this paper.
GALFIT fits a variety of 2D analytic functions to galaxy im-ages. Multiple components may be added, for example a detailedmodel might include a bulge, disc, bar, nucleus and the sky back-ground. Using the Levenberg-Marquardt (LM) algorithm,
GAL - FIT minimises the χ residual between a galaxy image and thePSF-convolved model by modifying the free parameters. The finalmodel is accepted when χ reaches a minimum. The calculation of χ is weighted by a sigma map, which may either be provided orinternally created. The reader can refer to Peng et al. (2002, 2010)for a detailed description of GALFIT .The standard version of
GALFIT
GALFITM , can accept an arbitrary numberof (pixel-registered) images of the same region of sky at differentwavelengths (e.g., we use images in five wavelength bands for thisstudy, while Paper III uses nine bands). To these images
GALFITM fits a single, wavelength-dependent, model. As for
GALFIT
3, thismodel may comprise one or more component functions, each with anumber of parameters, e.g. centre position ( x c , y c ), magnitude ( m ),effective radius ( r e ), S´ersic index ( n ), axial ratio ( b/a ) and positionangle ( θ ) for a single-S´ersic function. To extend these componentfunctions to multi-wavelength, their free parameters are replacedwith functions of wavelength. These are chosen to be Chebyshevpolynomials, as they possess several convenient properties (see Pa-per I and III for more details). In the fitting algorithm, the standardparameters are thus replaced by the coefficients of these polynomi-als. For each standard parameter the user can select the maximumpolynomial order for which the coefficients are free to vary in thefitting process. For example, some parameters may be set to havea specific constant value or vary with wavelength in a manner thatis fixed, e.g., one might fix n = 1 to model an exponential diskcomponent. Other parameters may be allowed to vary as a constantfunction of wavelength, e.g. one might like to allow the centre x and y coordinates to vary, but require that they are the same inevery band. Still other parameters may be permitted to vary withwavelength as linear, quadratic, or higher-order functions, e.g., onecould choose to allow r e to vary quadratically with wavelength, inorder to account for colour gradients. As one allows higher ordercoefficients to be fit, the function becomes more flexible, such thatwhen the number of free coefficients for a standard parameter is thesame as the number of input images, that parameter can effectivelyvary independently between bands.The implementation of this technique has been achieved in ageneral manner, so all of the standard GALFIT component functionsare available, and all of the standard parameters of those functionsare treated identically. For full details please refer to Paper I. Inthis work the polynomials, describing how the model parametersvary between images, are functions of wavelength. However, notethat the value corresponding to each band is user-definable. It istherefore straightforward to use alternatives, e.g. the logarithm ofwavelength, as the variable for these polynomials.With
GALFITM , one is faced with a decision of how much free-dom to give each parameter to vary with wavelength. This can beadvised by expectations from other studies, physical intuition, andlimitations imposed by the signal-to-noise and resolution of the im-ages. A useful approach is to consider the variations seen when theparameters are fit to each image independently. For example, inthe top left panel of Fig. 3 we plot the variation of effective radiuswith wavelength, from single-band fits, for eleven example galax-ies. Six of these galaxies show no trend, one (red line) shows a lin-ear increase, and four of the largest galaxies exhibit a more complexvariation of r e with wavelength (see Section 4.3.1 for further dis-cussion of this issue). Considering S´ersic index, plotted in a similarfashion in the top left panel of Fig. 4, trends of increasing n withwavelength are evident for the majority of objects. GALFIT incorporates constraints, which impose limits on thevariation of parameters during the fitting process. Some of these arehard-coded (such as ensuring that sizes cannot become negative) c (cid:13) , 1–29 Vika et al. r e ff ( k p c ) l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l Montage (single−band) l l l l ll l l ll l l l ll ll l l ll l l l ll l l l ll l l l ll l l l ll l l l ll l l l l
Ferengi z=0.01(single−band) r e ff ( k p c ) l l l l ll l l ll l l l ll l l ll l l ll l l l ll l l l ll l l l ll l l l ll l l l ll l l l l Ferengi z=0.06(single−band) r e ff ( k p c ) l l l l ll l l l ll l l l ll l l ll l l l ll l l l ll l l ll l l l ll l l l ll l l l ll l l l l Ferengi z=0.12(single−band) r e ff ( k p c ) l l l l ll l l l ll l l l ll l l l ll l l l ll l l l ll l l l ll l l l ll l l l ll l l l ll l l l l u g r i zMontage (multi−band) r e ff ( k p c ) l l l l ll l l l ll l l l ll l l l ll l l l ll l l l ll l l l ll l l l ll l l l ll l l l ll l l l l u g r i zFerengi z=0.01(multi−band) r e ff ( k p c ) l l l l ll l l l ll l l l ll l l l ll l l l ll l l l ll l l l ll l l l ll l l l ll l l l ll l l l l u g r i zFerengi z=0.06(multi−band) r e ff ( k p c ) l l l l ll l l l ll l l l ll l l l ll l l l ll l l l ll l l l ll l l l ll l l l ll l l l ll l l l l u g r i zFerengi z=0.12(multi−band) Figure 3.
Example fitting results for eleven galaxies: 1-NGC853, 2-NGC2775, 3-NGC3521, 4-NGC3992, 5-NGC4274, 6-NGC4431, 7-NGC4434, 8-NGC4496A,9-NGC4570, 10-NGC4623, 11-NGC4900. We show recovered effective radius for both single-band (top row) and multi-band (bottom row)fitting methods. The leftmost column shows results from fitting the original images, while the remaining columns show results from fitting images artificiallyredshifted to redshifts 0.01, 0.06 and 0.12. Multi-band fitting results have a smooth dependence on wavelength as, by design, r e is only permitted to varylinearly with wavelength. A typical one sigma error is given for the largest and smallest galaxy for the M ONTAGE images. The errors have estimated inSection 3.3. The eleven galaxies were chosen to show a variety of parameters, while ensuring some large galaxies were included. Note that in the panels withredshifted images there are a few cases where the effective radius, mostly in u -band, takes values above the upper limit of the plot, e.g., for NGC3992 at z = 0 . the r e ,u = 35 kpc. while others may be optionally specified by the user. Constraintsare useful to improve the reliability and efficiency early in the fit-ting process, by excluding regions of parameter space which arebelieved to be unphysical or eliminated by other considerations.However, if the minimisation routine repeatedly encounters con-straints, this is an indication that a good model fit to the data cannotbe achieved. In this case at least some of the resulting parameterswill typically lie very close to a constraint boundary. These param-eters for such a fit are likely to be seriously biased, and hence it issensible to discard them from further analysis.The extension to multi-band fitting required the implementa-tion of constraints to be significantly modified in GALFITM . If aproposed parameter step would violate a constraint,
GALFIT
GALFITM , constraint violations are avoided by simply notchanging the offending parameter. For instance, if an individualstep would violate a constraint, then the coefficients correspondingto the affected parameter are not changed on that step. The subse-quent step may not violate constraints, in which case the parameterwill continue to converge toward its optimum value. However, inthe standard implementation of GALFITs LM algorithm, if a stepsuccessfully reduced the χ , then the algorithm is ‘encouraged’ andthe next step will be larger. In some cases this can lead to one or twoparameters repeatedly attempting to violate constraints and hencebecoming ‘frozen’ for the duration of the fit. To mitigate this, wefound that periodically substantially reducing the global step sizewould often lead to ‘frozen’ parameters being able to take a smallstep towards their constraint boundary and hence ‘thawing’. Tests(see comparison in Figure 9) show that this approach works well,and GALFIT
GALFITM typically return very similar results, c (cid:13) , 1–29 egaMorph II l l l l ll l l l ll l l l ll l l l ll l l l ll l l l ll l l l ll l l l ll l l l ll l l l ll l l l l Montage (single−band) n l l l l ll l l l ll l l l ll l l l ll l l l ll l l l ll l l l ll l l l ll l l l ll l l l ll l l l l Ferengi z=0.01(single−band) n l l l l ll l l l ll l l l ll l l l ll l l l ll l l l ll l l l ll l l l ll l l l ll l l l ll l l l l Ferengi z=0.06(single−band) n l l l ll l l l ll l l l ll l l l ll l l l ll l l l ll l l ll l l l ll l l l ll l l l ll l l l l Ferengi z=0.12(single−band) l l l l ll l l l ll l l l ll l l l ll l l l ll l l l ll l l l ll l l l ll l l l ll l l l ll l l l l u g r i zMontage (multi−band) n l l l l ll l l l ll l l l ll l l l ll l l l ll l l l ll l l l ll l l l ll l l l ll l l l ll l l l l u g r i zFerengi z=0.01(multi−band) n l l l l ll l l l ll l l l ll l l l ll l l l ll l l l ll l l l ll l l l ll l l l ll l l l ll l l l l u g r i zFerengi z=0.06(multi−band) n l l l l ll l l l ll l l l ll l l l ll l l l ll l l l ll l l l ll l l l ll l l l ll l l l ll l l l l u g r i zFerengi z=0.12(multi−band) Figure 4.
Example fitting results for S´ersic index. The figure layout, galaxies and colour coding is the same as Fig. 3. even when constraints are important (see Section 4.1 and Paper III).In some cases the parameter continues to migrate to the constraintboundary, so the final fit is treated as unusable.In order to store the increased outputs in an efficient manner,the FITS file produced by
GALFIT has been extended for
GALFITM .In addition to the original, model and residual images for all bands,this file contains the PSFs and several new tables. These tables con-tain all the input details and results of the fit, as well as some addi-tional information, including the number of iterations, timings anda record of when constraints were encountered during the fit.
We obtain the structural parameters of our galaxies by fitting two-dimensional elliptical single-S´ersic models using both
GALFIT
GALFITM . The radial surface brightness profile is given by theS´ersic function (Sersic 1968): I ( r ) = I e exp (cid:26) − b n (cid:20)(cid:16) rr e (cid:17) /n − (cid:21)(cid:27) (1)where r e is the effective or half-light radius, I e is the intensity atthe effective radius, n is the S´ersic index and b n is a function of n (Graham & Driver 2005). The term r e describes the physicalsize of the galaxy and n gives an indication of the concentration of the light distribution. When n is equal to the values . , , , theS´ersic profile is equivalent to a Gaussian, exponential and de Vau-couleurs (1959) profile, respectively. Most galaxies are consideredto be primarily two component systems, comprising a disk withan exponential ( n = 1 ) profile, and a bulge, typically well repre-sented by a S´ersic function with n > . At low resolution, a singleS´ersic function can do a reasonable job of representing the overallprofile of such a two-component system. In this case, the resultingparameters reflect a combination of both the disk and the bulge.On the other hand, elliptical galaxies are generally regarded as be-ing single component systems, well described by a S´ersic profile(although, this picture too breaks down when considered in detail).We run three types of fit, each of which is performed on the u , g , r , i , z band images for all of our original and redshifted galax-ies. For the first ( S3 ) we fit each band individually with GALFIT SM ) we repeat the same fitsusing GALFITM (again single-band fitting), and finally ( MM ) we fiteach galaxy with GALFITM using all the five bands simultaneously(multi-band fitting).In the multi-band fitting we choose to allow magnitudes tovary completely freely between bands. In practice, this means thatwe set the wavelength dependence of magnitude to be described bya quartic polynomial, i.e. with as many coefficients as the numberof bands. We allow full freedom as we wish to avoid any potential c (cid:13) , 1–29 Vika et al. − − − − z M ( m ag ) lllllllllllllllllllllllll points: SMlines: S3 l z M ( m ag ) lllllllllllllllllllllllll points: MM l z s i z e s [, ]/ lllllllllllllllllllllllll points: SMlines: S3 l z cs i z e s [, ]/ e ff e c t i v e r ad i u s ( k p c ) lllllllllllllllllllllllll points: MM l . . . . . z n lllllllllllllllllllllllll l z n lllllllllllllllllllllllll l z b / a lllllllllllllllllllllllll l z b / a b / a . . . . . . lllllllllllllllllllllllll l − z PA lllllllllllllllllllllllll l z PA lllllllllllllllllllllllll l z c h i − s qua r e lllllllllllllllllllllllll l l l l l l l l l l l l l l l l l l l l l l l l l l z c h i − s qua r e c h i − s qua r e . . . . ll Bandszirguall l z N o . o f It e r . lllllllllllllllllllllllll l l l l l l l l l l l l l l l l l l l l l l l l l l z N o . o f It e r . l z C P U t i m e ( s e c ) lllllllllllllllllllllllll l l l l l l l l l l l l l l l l l l l l l l l l l l z C P U t i m e ( s e c ) C P U t i m e ( s e c ) l Figure 5.
Left: A series of plots presenting the variation of recovered parameters as a function of redshift for galaxy NGC4570. First and third columns showthe results of single-band fits (points:
GALFITM , lines:
GALFIT
3) while the second and the fourth columns show results of our multi-band fits. The points atredshift zero in each panel give the values for the original galaxy image, while the rest of the points represent the artificially redshifted images. The dashedhorizontal lines in the S´ersic index panels indicate soft constraint applied on S´ersic index. Note that, for example, in case of multi-band fitting the smoothnessof the u -band magnitudes is a result of the other profile parameters being naturally constrained by the higher signal-to-noise bands, rather than being due toany externally imposed constraints. Right: The five u, g, r, i, z SDSS images, created with M
ONTAGE , for the same galaxy. biases on the recovered magnitudes, and hence colours, which mayresult from assuming a lower-order polynomial dependence.For effective radius and S´ersic index we elect to allow linearvariations with wavelength. In contrast to magnitude, we expectphysical variations in these parameters to be smaller and smoother,particularly when compared to their measurement uncertainties andovert the limited optical wavelength range considered in this paper( ∼ – nm). This decision is also supported by the outcomesof single-band fitting. Figures 3 and 4 show the fitting results, ef-fective radius and S´ersic index respectively, for eleven examplegalaxies fitted in our original (left column) and artificially red-shifted images (remaining columns). The single-band results (toprow) for most of our galaxies are consistent with constant or lineartrends with wavelength for both n and r e . These trends are well-represented by the multi-band fits (bottom row). Note that n , inparticular, consistently shows significant trends with wavelength.Allowing only constant n in the multi-band fits, or imposing struc-tural parameters measured in one band to measure fluxes in theother bands, would fail to account for this behaviour. Note that dueto low signal-to-noise ratios some recovered parameters in Fig. 3are out of the plotting area. We choose to not expand the scale ofthe figure in order to include these individual values but to focus onthe reliable values instead. Finally, we assume that the shapes of our galaxies do notchange within the range of wavelength defined by SDSS bands, sowe specify the galaxy center, the axis ratio and the position angleto be constant with wavelength. We have performed an inspectionof plots similar to Figs. 3 and 4 for these parameters, and find nosignificant evidence to contradict this decision. These parametersare still free to vary during the fit, just without any variation withwavelength.In all the three runs ( S3 , SM , MM ) we use the same initial pa-rameters for galaxy center ( x c , y c ) , magnitude ( m ) , effective ra-dius ( r e ), S´ersic index ( n ) , axis ratio ( b/a ) , position angle ( θ ) andsky background value (although different values are used for eachgalaxy image, as described below).First, we fit the sample of original galaxy images, produceddirectly from SDSS data using M ONTAGE . The initial magnitudevalues for the minimisation are taken from the SDSS aperture mag-nitudes , the center ( x, y ) initial values where set to the middle ofeach image , while for the rest of the parameters we use constantinitial values for all the galaxies in all the bands ( r e = 10 pixels, http://cas.sdss.org/astro/en/tools/crossid The M
ONTAGE images are centred on each galaxy.c (cid:13) , 1–29 egaMorph II − − − − z M ( m ag ) lllllllllllllllllllllllll points: SMlines: S3 l z M ( m ag ) lllllllllllllllllllllllll points: MM l z s i z e s [, ]/ llllllllllllllllllll llll points: SMlines: S3 l z cs i z e s [, ]/ e ff e c t i v e r ad i u s ( k p c ) lllllllllllllllllllllllll points: MM l . . . . . z n lllllllllllllllllllllllll l z n lllllllllllllllllllllllll l z b / a lllllllllllllllllllllllll l z b / a b / a . . . . . . lllllllllllllllllllllllll l − z PA lllllllllllllllllllllllll l z PA lllllllllllllllllllllllll l z c h i − s qua r e lllllllllllllllllllllllll l l l l l l l l l l l l l l l l l l l l l l l l l l z c h i − s qua r e c h i − s qua r e . . . . ll Bandszirguall l z N o . o f It e r . lllllllllllllllllllllllll l l l l l l l l l l l l l l l l l l l l l l l l l l z N o . o f It e r . l z C P U t i m e ( s e c ) lllllllllllllllllllllllll l l l l l l l l l l l l l l l l l l l l l l l l l l z C P U t i m e ( s e c ) C P U t i m e ( s e c ) l Figure 6.
A series of plots presenting the variation of recovered parameters as a function of redshift for galaxy NGC4274. The layout is the same as Fig. 5. n = 1 . , b/a = 0 . , θ = 0 ). The initial effective radius value waschosen to be smaller than all the measured effective radii, relying on GALFIT ’s ability to accurately determine the size of a well-resolvedgalaxy independent of the initial value. All the parameters are al-lowed to vary during the fitting process, with the exception of thesky background, which is kept fixed. We additionally apply a softconstraint on S´ersic index, requiring n to vary within the range . to , and on the center, such that x and y may vary by no morethan the square root of their initial values. These constraints are notimportant for the high quality original images, but we include themto be consistent with the fits to the artificially redshifted images.Once we have established accurate fits for the original images,we repeat the same three runs of the fitting process, as describedabove ( S3 , SM , MM ), on the artificially redshifted images. The out-put parameter values from the GALFITM multi-band fits ( MM ) tothe original images (given in Table 1) are used as initial values forfitting the lowest redshift artificial images. The output parametersfrom these fits were then used to derive initial values for all thehigher redshift images. The initial values for S´ersic index, axis ra-tio and position angle were the same at all redshifts as, in princi-ple, the geometry should remain constant with redshift. Apparentmagnitude and effective radius were cosmologically adjusted, suchthat the initial absolute values of these quantities are constant withredshift. In order to ensure we do not introduce wavelength trendsthrough the initial parameters, we use the same initial values of ef-fective radius, S´ersic index, axis ratio and position angle for all thebands. These are therefore initially constant with wavelength, even if they are subsequently allowed to vary with wavelength during thefitting process. The common initial value is determined by takingthe median of the values for each band.For each galaxy we visually inspect all recovered parametersfor both original and artificial images. In Figs. 5, 6 and 7 we presenta summary of the results for three example galaxies together withthe five u, g, r, i, z original images. Similar plots are available forall the 163 nearby galaxies. In these figures, the first and the thirdcolumns show the results of the single-band fits (points: SM , lines: S3 ) while the second and the fourth columns show the results ofthe multi-bands fits ( MM ). The first row of panels shows the abso-lute magnitude ( M ) and effective radius, where the fit was success-ful, both for original and artificially-redshifted images. The sec-ond row shows the S´ersic index and axis ratio, while the third rowshows presents the position angle and the minimised chi-square.Finally, the last row shows the number of iterations needed to fiteach galaxy and the cpu time taken by the fitting process. For sim-plicity, in these figures, absolute magnitude and effective radius aredetermined assuming distances simply derived from the observedredshift and adopted cosmology. The values shown in these figuresare therefore different to those shown in Section 4.3.1 and Table 2,where accurate distance estimates are used.Figure 5 presents the recovered structural parameters for thegalaxy NGC4570. The first thing to mention is that, ignoring u -band for a moment, the trends with artificial redshift are pleasinglyflat for both single- and multi-band fits. We also note that the re-sults for the single-band approach agree very well between GAL - c (cid:13) , 1–29 Vika et al. − − − − z M ( m ag ) lllllllllllllllllllll points: SMlines: S3 l z M ( m ag ) lllllllllllllllllllll points: MM l z s i z e s [, ]/ lllllllllllllllllllll points: SMlines: S3 l z cs i z e s [, ]/ e ff e c t i v e r ad i u s ( k p c ) lllllllllllllllllllll points: MM l . . . . . z n lllllllllllllllllllll l z n lllllllllllllllllllll l z b / a lllllllllllllllllllll l z b / a b / a . . . . . . lllllllllllllllllllll l − z PA lllllllllllllllllllll l z PA lllllllllllllllllllll l z c h i − s qua r e lllllllllllllllllllll l l l l l l l l l l l l l l l l l l l l l l z c h i − s qua r e c h i − s qua r e . . . . ll Bandszirguall l z N o . o f It e r . lllllllllllllllllllll l l l l l l l l l l l l l l l l l l l l l l z N o . o f It e r . l z C P U t i m e ( s e c ) lllllllllllllllllllll l l l l l l l l l l l l l l l l l l l l l l z C P U t i m e ( s e c ) C P U t i m e ( s e c ) l Figure 7.
A series of plots presenting the variation of recovered parameters as a function of redshift for galaxy NGC4431. The layout is the same as Figs. 5 and6. In this example both single- and multi-band fits fail after redshift ∼ . . In this case we believe that is due to our inability to construct an accurate maskbeyond this redshift, i.e. SExtractor no longer deblends two neighbouring objects. The output model therefore changes shape and magnitude with redshift. FIT S3 , lines in columns 1 and 3) and GALFITM ( SM , points incolumns 1 and 3). The results for single- and multi-band ( MM , pointsin columns 2 and 4) techniques also generally agree very well. If wenow consider the u -band, the lowest signal-to-noise band, we seethat this too agrees between all fit types to z ∼ . . However, forthe single-band methods, beyond z ∼ . , first n begins to showan increasing scatter, then r e and b/a , and finally M . Other bandsstart to show significant increases in scatter for z > ∼ . , particu-larly for n . In contrast, the multi-band fitting results remain reliablefor all bands to higher redshifts. Significantly increased scatter doesnot set in for any parameters until z > ∼ . .The more reliable behaviour of our multi-band method is a re-sult of their structural parameters being constrained by more data,with many of the values required to display a degree of consistencybetween bands. Remember that there is no restriction on the varia-tion of magnitude with wavelength, but effective radius and S´ersicindex may only vary linearly, while axis ratio and position angle areconstant with wavelength. The behaviours seen in Fig. 5 are typicalfor many of the galaxies in our sample.We show two more examples, for NGC4274 in Fig. 6 andNGC4431 in Fig. 7. NGC4274 is another case where the individual u -band fits show a large scatter, while the multi-band results remainsensible, and consistent with those obtained for the original image.Again, our multi-band approach also reduces the scatter in the otherbands, enabling more sensible results to be recovered for high arti- ficial redshifts, where the data quality is very poor. NGC4431 is amuch less luminous galaxy, for which at z > ∼ . both single- andmulti-band fitting have high scatter and the structure of the galaxycannot be extracted. Prior to this, all methods produce reasonableresults. For NGC4431, and similar cases where single- and multi-band fitting processes failed to give consistent results with redshift,we investigated the reason by examining the galaxy images, inputmasks, output models and residuals (via figures similar to Fig. 1).In all cases we found that the divergencies in the derived param-eters were due to the main galaxy becoming blended with nearbyobjects or being undetectable in multiple bands.In general, multi-band fitting improves the stability of the re-sults, particularly for the lower signal-to-noise bands, increasingthe distance out to which meaningful multi-wavelength structuralinformation can be recovered for a galaxy of a given luminosity.However, it cannot help in cases where the galaxy becomes unre-solved or undetected in multiple bands (i.e., more than two of thefive used in this work). The parameter that typically shows the mostimprovement is effective radius, followed by S´ersic index and thenmagnitude. Even though the multi-band method improves the re-liability of the parameters, it is fairly common for galaxies in oursample to show systematic trends in some parameters with artificialredshift. These are usually present to a similar degree with all fittingmethods. We believe that these are mostly a result of the decreasingimage quality. However, some of these trends may be attributable c (cid:13) , 1–29 egaMorph II ll l ll lll l l llll ll ll ll ll lll ll l ll lll l l ll ll l lll l l ll ll lll l l l lll l lll ll llll lll l llll ll l l l ll ll lll ll l l l lll l l ll ll ll ll ll ll ll llll ll ll l llll llll ll ll ll llll l ll l lll l lll l ll ll llll lll l − . . . m ring.sky[M] m [ M ] - m r i ng . sky [ M ] l l ll l lll l l llll ll ll ll lllll ll l ll llll l ll ll l lll ll ll ll ll l ll l llll lll ll llll lll l l lll ll l l l ll ll lll ll l l ll ll l l ll ll ll ll ll ll ll llll ll ll l llll ll ll ll ll ll llll l ll l lll llll l ll ll llll lll ll l ll l lll ll llll l l ll ll lllll ll l ll llll l ll ll l lll ll llll ll l ll l llll lll ll llll lll l l lll ll l l l ll ll lll ll l l ll ll l l ll ll ll ll ll ll ll llll l l ll l llll ll ll ll ll ll llll l ll l lll lllll ll ll ll ll lll ll l ll l lll ll llll l l ll ll lllll ll l ll llll l ll ll l lll ll llll ll l ll l llll lll ll llll lll l l lll ll l l l ll ll lll ll l l ll ll l l ll ll ll ll ll ll ll llll l l ll lllll ll ll ll ll ll llll l ll l l ll lllll ll ll ll ll lll ll l ll l lll ll llll l l llll lllll ll l ll lllll ll ll l ll l ll llll ll l ll l llll lll ll ll ll lll l l lll ll l l l ll ll lll ll l l ll ll l l ll ll ll ll ll ll ll llll l l ll lllll ll ll ll ll ll llll l ll ll ll lllll ll ll ll ll lll llll llll llll l lll ll llll l l l ll lll l lll lll ll lll l ll ll l l ll lll ll l l llll ll ll l l ll l lll ll ll l ll lll l lll ll lllll l ll llll llll ll ll lll l l lllll lll l l ll ll ll ll ll ll ll lll ll l l ll ll lll ll lll lll ll l . . . r eff , ring.sky[M] r e ff[ M ] r e ff , r i ng . sky [ M ] lll llll llll l lll ll llll l l l ll lll l lll lll ll lll l ll ll l l ll lll ll l l llll ll ll l l ll l lll ll ll lll l ll l ll l ll lllll l ll llll llll ll ll lll l l lllll lll l l ll ll ll ll ll ll ll lll ll l l ll ll lll ll lll lll ll llll llll llll l lll ll llll l l l ll llll lll lll ll ll l l ll ll l l ll lll ll l l llll ll ll l l ll l lll l l ll lll l ll l ll l ll lllll l ll llll llll ll ll lll l l ll lll lll l l ll ll ll ll ll ll ll lll ll l l ll ll lll ll lll lll ll llll llll l lll l lll ll llll l l l ll ll lllll lll ll ll l l ll ll l l ll lll ll l l ll ll ll ll l l ll l lll l l ll lll l ll l ll l ll lllll l ll llll llll ll ll lll l l ll lll lll l l ll ll ll ll ll ll ll lll lll l ll ll lll ll lll lll ll llll llll l lll l lll ll llll l l l ll ll lllll lll ll ll l l ll ll ll ll lll ll l l ll ll ll ll l l ll llll l l ll lll l ll l ll l ll lllll l ll llll ll ll ll ll lll l l ll lll lll l l ll l l ll ll ll ll ll lll lll l ll ll l ll ll lll lll ll l lllll Bandszirgu llll ll ll ll l ll lll l ll llll l ll ll ll l l lll l ll lll ll ll l l lll l l l lll ll ll ll l llll l ll l lll lll l llll ll ll lll l lllll llll l l llll ll ll ll ll ll lll l ll ll l ll lll lll ll ll ll llll l ll l ll lll ll ll l ll ll ll l . . . . n ring.sky[M] n [ M ] n r i ng . sky [ M ] llll ll ll lll ll lll l ll lll ll ll ll ll ll lll l ll lll ll ll l l lll l l l lll ll ll ll l llll l lll lll lll l llll ll ll lll l lllll ll ll l l llll ll ll ll ll lllll l ll ll l ll lll l ll ll ll l l llll l ll l ll ll lll ll l ll ll ll l lll l ll ll lll ll lll l l lll l ll ll ll ll ll llll ll lll ll ll l l lll ll l ll l l l ll ll l llll l lll lll lll l llll ll ll lll l lllll ll ll l l llll ll ll ll ll lllll l ll lll ll lll l ll ll ll l l llll l ll l ll ll ll l ll l ll ll ll l lll lll ll lll ll ll l l l lll l ll ll ll ll ll llll ll lll ll ll l l lll lll ll ll l ll ll lllll l lll lll lll lllll ll ll lll l lllll ll ll l l llll ll ll ll ll l llll l ll lll ll llll ll ll ll l l lllll ll l ll ll ll l ll l ll ll ll l lll lll ll lll l l ll l l l lll l ll ll ll ll ll ll ll ll lll ll ll l l lll lll ll ll ll l ll lllll l l ll lll lll lllll ll ll l lll lllll ll ll l l ll ll ll ll ll ll l llll l ll lll ll llll ll ll ll l l lllll ll l ll ll ll lll l ll ll lll Figure 8.
Plots showing the difference in parameters from multi-band fitting (MM) when measured using two different sky values, as described in § to the inability of the S´ersic function to represent the profile ofsome galaxies, as well as deblending and sky measurement issues.The observed systematic biases with redshift do not obey a consis-tent pattern that we have been able to discern, with the exceptionof S´ersic index. This commonly decreases as the galaxy becomesartificially redshifted to greater distances. This issue is discussedfurther in Section 4.2.In order to demonstrate the capabilities of the MM approachfor more accurately measuring the physical properties of galaxies,we chose to limit some parameters freedom to vary with wave-length, with reference to the trends seen in the results of our single-band fits. As we will see in the results below, these assumptionsallow us to more accurately recover parameters in noisy images.However, it is important to recognise that, while these constraintsincrease the stability of the fits, there is also the risk of introducingsystematic biases in cases where the true wavelength dependenceof the profile does not correspond to that assumed. Low signal-to-noise images would be most susceptible to such systematics as theparameters would be mostly constrained by the higher signal-to-noise data. For instance, if the light profiles of galaxies in the (noisy) u -band are significantly different to that in gri , restricting the effec-tive radius to linear variations with wavelength would impose thestructure determined in gri on u . As well as introducing a bias on r e ,u , this may also result in a systematic error in magnitude, de-spite it being allowed to vary freely. On the other hand, if we hadtried to fit the low signal-to-noise u -band image independently, themuch-reduced information regarding the center and profile of thegalaxy would result in large statistical uncertainties, which wouldpropagate into uncertainties in the magnitude.Our method produces measurements for the parameters in allbands. However, for low signal-to-noise images there will be a par-tial interpolation or extrapolation from the higher signal-to-noisedata. What signal there is in the low signal-to-noise bands will havesome influence on the fit. Any systematic bias will therefore be lessthan would have resulted from an extrapolation based only on thehigh signal-to-noise bands. Such systematics may be reduced bygiving the model more freedom to vary with wavelength, at the costof increased statistical uncertainties on the resulting parameters inthe low signal-to-noise bands. A further consideration is that withlow signal-to-noise data there is a danger of overfitting, for exam- c (cid:13) , 1–29 Vika et al. ple, the S´ersic profile adapting to fit a noise peak rather than thegalaxy. Even when fitting the bands independently, the possibilitystill remains for significant systematic errors.A compromise must therefore be made by any user of ourMM method. One can allow more freedom for parameters to varywith wavelength, giving (perhaps) less potential for systematics bi-ases, but resulting in higher statistical uncertainties; or vice versa.The flexibility of our multi-band fitting approach allows the user tobalance systematic and statistical uncertainties by using indepen-dent observational results, physical insight and knowledge of theirdataset to guide their assumptions for how galaxy parameters varywith wavelength.
GALFIT internally computes statistical errors from the covariancematrix constructed by the Levenberg-Marquardt algorithm. Theseerrors would be statically correct if the only source of uncertaintywere Poisson noise. However, in reality they are substantially un-derestimated, as they do not take into account the parameter de-pendencies on (external) sky estimation, PSF accuracy, model mis-match and correlated noise. For example, variations within the ac-ceptable range of sky values may result in systematic parameter un-certainties much greater than the formal
GALFIT errors (Vika et al.2012).We determine the sky value in the original galaxies using twodifferent methods, in order to obtain a rough indication of howthis uncertainty affects the structural parameters. Figure 8 showsthe parameter differences for the two sky measurements, whichare described in Section 2.3. The purpose of this comparison isnot to fully study the parameter dependency on sky, as this hasbeen done by various other studies (e.g., H¨aussler et al. 2007), butrather to provide an indicative uncertainty value for our measure-ments. The two different sky estimations lead to uncertainties perband ( u , g , r , i , z ) for m ( ± . , ± . , ± . , ± . , ± . ), r e ( ± %, ± %, ± %, ± %, ± % ) and n ( ± %, ± %, ± %, ± %, ± %). These uncertainties refer only to the orig-inal (M ONTAGE -derived) images using multi-band fitting and notthe redshifted images. They are reproduced at the top of Table 1.As the purpose of this paper is focus on fitting single-S´ersicfunctions, we do not attempt to fit more complex models, or derivethe parameters of bulges and disks separately. This will be the sub-ject of a subsequent paper. Despite their simplicity, single-S´ersicmodels are often capable of giving a good overall representationof galaxy surface brightness profiles, even in the case of multi-component systems. In such cases the values for S´ersic index andeffective radius will reflect the superposition of the components.For example, an n ∼ bulge together with an exponential diskof similar brightness, will usually result in an intermediate n forthe single-S´ersic fit, e.g., n ∼ . . On the other hand, ellipticalgalaxies are often considered to be single-component systems, wellrepresented by a single-S´ersic at all radii. GALFIT assumes that the intrinsic profile being fit is perfectlyrepresented by the assumed model. Mismatches between the realprofile and the model, e.g., as in the case of galaxies with a bulgeand disk, produce an additional contribution to the uncertaintiesin model parameters, beyond those estimated by
GALFIT . Further-more, model mismatch may make model parameters more sensitiveto systematic biases as a function of signal-to-noise and/or resolu-tion, as mismatches become less significant as image quality de-creases. This issue is discussed further in the following section.
GALFITM versus
GALFIT Having applied various modifications to the original
GALFIT
3, wefirst wish to ascertain that
GALFITM continues to return the same (orat least similar) results when applied to the same data with an iden-tical configuration. We therefore compare the single-band
GALFIT S3 ) outputs with those from single-band GALFITM ( SM ) and, in-vestigate the cause of any differences.In Fig. 9, we compare the derived parameters from the single-band fits for the artificially redshifted sample only. We plot the dif-ference between the GALFIT n , r e , m ) and GALFITM parameters ( n [M] , r [M]e , m [M] ) as a function of the ‘true’ apparent r -band magnitude. This magnitude ( m [M] r, ) is determined by cos-mologically transforming the value obtained from the single-bandfit to the most local artificially-redshifted image. Using this magni-tude, rather than the value measured for each image, avoids scatteron the recovered parameters from entering the horizontal axis ofthis plot. Each column shows a different band, from u on the left,to z on the right. We include all the parameters even when they arein close proximity to a constraint. However, we indicate these unre-liable fits by plotting their points in grey. For simplicity we do notshow the parameters derived from the original M ONTAGE images.As can be seen from Fig. 9, for the majority of the objects bothmethods return the same results, thus we get a straight horizontalline at zero or one. We find that the fraction of galaxies for whichthe magnitude difference is less that . mag is 58 per cent, whilethis fraction increases to 71 per cent for ∆ m < . mag. Mostof the more deviant points we find are consistent within ∆ m < . mag up to m r ∼ mag for u - and z -bands and m r ∼ magfor the rest of the bands. As galaxies get fainter we notice a gradualincrease of the scatter, which becomes large after m r > . Inthe second row of panels, for r e , we also notice that there is somefraction of galaxies with ratio very close to zero. This is causedby cases where the recovered GALFIT . pixels, whereas for GALFITM these values are avoided.The main reason for differences in the recovered parametersbetween
GALFIT
GALFITM is due to the handling of con-straints between the different codes, although there may also becontributions from a number of more minor modifications (see Sec-tion 3.1 and Paper I). From inspection of Fig. 9 we note that much,but not all, of the scatter in these plots is due to objects that havetheir parameters derived from fits which end up on constraints.Thus, the scatter in this plot, and the significant differences between
GALFITM and
GALFIT
GALFIT
3, and which are probably sensi-tive to the initial conditions, in addition to the details of the fittingroutine.Finally, note that there is a cloud of points, particularly notice-able in the u -band, which is systematically offset to fainter magni-tude, smaller size and lower S´ersic index. This contains the major-ity of objects which disagree between S3 and SM , but for which theparameters do not end on a constraint boundary, and corresponds toa variety of objects and redshifts. We note that this feature is muchmore prevalent in low signal-to-noise images, where the χ surfaceis likely to be complicated, with multiple minima, but we are notyet entirely certain of the reasons for its appearance. However, allof the fits in the cloud have encountered constraints during the fit-ting process, and we strongly suspect that differences in the wayconstraints are handled by GALFITM and
GALFIT c (cid:13) , 1–29 egaMorph II l l l l l llllllllllllllllllll l l l l l lllllllllllllllllll l l l l l llllllllllllllllllll l l l l l lllllllllllllllllll l l l l l llllllllllll l l l l l llll l l l l l ll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l lllllllllllllllllll l l l l l lllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l lllllllllllll l l l l l llllllllllllllllllll l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l lllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllll l l l l l llllllllllllllllllll l l l l l lllllllllllllllllll l l l l l lllllllllllllllllll l l l l l llllllllllllllllllll l 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llllllllllllllllllll l l l l l ll ll lllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l ll l ll l l l l lllll ll ll l l l l lll llllllllllll l l l l l lllllll lll lllll l l l l l lllllll lllll lllll l l l l l lllllllllllllllllll l l ll l l l l l l l ll llll l l l l l llllllllllllll lll l l l l l l lll lllll llll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l lllllllllllllllllll l l l l l llllllllllllllllllll l l l ll ll lllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l lllllllllllllllllll l l l l l lllllllllllllllll l l l l l llllllllllllllllllll l l l l llll l l l l l lll ll lllllllllll l l l l l lllll l l l l l lll llllllllllllll l l l l l llllllllllllllllllll l l l l l ll ll lll l l l l l llllll l l l l l llllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllll l l l l l llllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l lllllllllllllllllll l l l l l llllllllll l l l l l lllllll llllllllll l l l l l llllllllllllllllllll l l l l l lllllllllllllllllll l l l l l lllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l lllllllllllll llll l l l l l lllllllllllllllllll l l l l l llllllllll lllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l lllllllllllllllll l l l l l lllllllllllllllllll l l l l l lllllllllllllll lll l l l l l llllllllllll ll l l l l l lllll lllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l lllllllllllllllllll l l l l l lllllll lllll lllll l l l l l lllllllllllllllllll l l l l l lllllllllllllllllll l l l l l llllllllllllllllllll l l l l l lllllllllllllllllll l l l l l llllllllllllllllllll l l l l l lllllllllllllllllll l l l l l llllllllllllllllll l l l l l llllll llllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l lllllllllllllllllll l l l l l lllllllllllllllll l l l l l lllllllllllllllllll l l l l l lllllllllllllllllll l l l l l lllllll lllllllllll
12 16 20 . . . . . nn [ M ] ll lllllllllll llllll llllllllllllllll l lllll l ll l l lll lllllll llllllllllll l lllll lll l l llll llllllllllll lllllllllllll lllll l llllll lllllll lll l llllll l lllllll l l l l l ll llllllll ll ll lll ll lllllllllllll lllllll ll ll lll llll llllll lll lll llllllllllll llll lllllllllll llllllllllll lll lllllll lllllll llllllllll llllllllllll llllllllllllllll lll lllllllllll l lllllllllllllllllllll ll ll lll l lllllllllllllll l llllllll l l l lll ll l l lllll llllllll lllllllllllllllll llll ll lllllllllllll lllll llll lll l llllllllll l l ll lll l llllllllll ll l llllllllllllllll llll l l l l ll lllllll lll l l l l ll l ll l lll l ll l lllllllllll llllll lll lllll l l ll lllllllllll l ll l l l lllll lll lll lllll llll l lll llll l ll llllll ll l l llll lll l l l l llllll llll lllllllll ll l l lll l llllllllllllllllll lllll lll l l lll l llll l lll lll ll ll lll lllllll lllllllllll lll l lll lllllllllllllllll ll llll l l l l l llllllllllllllllllll l lllllllllllll l llllllllll llll lllllllllll lllllllllllllll ll llllllllllllllll llll llllll lll l ll lllllll l llllllllllllllllllll ll l ll lll lllllllllllllll l l l lllllllllllllllllllllllll ll l lllll ll l llllllllll llll ll l llll l lll ll lll llllllllllllllllllllllllllll llllllllllllll lllll lllll l llllll l l lllllllllllll l l l l lllllllllllllllllllllllllll ll llllllllllll l l lllllllllll lllllllllllll l llllllllllllll lllll lllllllllllllll lllllllllll lllllllllllllllllll llllllllllll llllll lllllllll l l l l llllllllllllllllll l l l l l lllll ll l l l l l lll llllllll l l l l l lll l l l l l ll l ll l ll l l l l l llllll lll lll l l l l l lllll lll l l l l l l ll llll lll l l l l l ll l lll l l l l l lllllllllll l l l l l lllll l l l ll l l l l l llllllllllllllllllll l l l l l llllllllllll llll l l l l l lllllllllllllll l l l l l llll ll lll ll l l l l ll l l l l l lllll l ll l ll l l l l llllllllllllllllllll l l l l l lllll l llll l llllll ll l l l l l lll lllll lll l l l l ll lll l l l l l lllllllllllllllllll l l l l l llllllllllllllllll l l l l l l ll lllll l l l l l llllllllllllll llll l l l l l llll ll l ll l l l l l lllllll ll lll l l l l l llll l ll l l l l l llll l lll l l l l l lll llll l l l l l llllllllll llll l l l l l lll l lll llll l l l l l lllllll l lll l l l l l llllll l ll l l l l l lllll llllllll l l l l l llllllllllllllllllll l l l l l lllllllllllllll l l l l ll l l l l l l l l ll l l l l l lllllllllllllllllll l l l l l lllllllllllllll l l l l l l ll ll ll l l l l l lllllllllll ll l l l l l llllllllllllllllll l l l l l lllllllllllll l l l l l lllllllllllllllll l l l l l llll l l ll l l l l l llllllllllll llll l l l l l llllllllllllllllll l l l l l llllllllllll l l llll l l l l l llllllllllll l l l l l ll lll ll l l l l l lllllll l l l l l ll ll llllll l l l l l lllllllllllll ll l l l l l lll llll l ll l l l l l llll ll lllllll l l l l l llllllllllllllllllll l l l l l lll l lllll l l l l l llllllllllllllllll l l l l l lllll lllll llll l l l l l lllllll lllllll l l l l l lllllllllllllllll l l l l l lllllllllll lllllll l l l l l lllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllll ll l l l l ll l l l l l lllllll ll l l l llllllllllllllllllll l l l l l lllll lllllllllllll l l ll l l l l l llllll ll llllllll l l l l l lllllllllllllllllll ll l l l l ll l l l ll ll l l l l ll l l l l l lll l l l l l llllllllllllllllllll l l l l l ll l l l l l lllllllllll lll l l l l ll l l l l l llllllllllllllllllll l l l l l ll lllllllllllllllll l l l l l llllll llllll llll l l l l l llllllllllllllll l l l l l lll l l l l l l l llll lll l lllllllllllllllllll l l l l l llllllllllllll l l l l l ll l lll l l l l l lllllllllll lllll l ll l l l l l ll ll l l l l l ll l l l l l lll ll lll l l l l l lllllllllllllllll l l l l l ll l l l l ll l l l l l lll l l llllll l l l l l llllllllllllllllll l l l l l ll l l lll l l l l l llllllllllllllllllll l l l l l llll lllll l l l l l llllll l lll l l l l l l ll l l l l l lllllllllllll ll l l l l l llllllllllllllllllll l l l l l ll l l l l l lllllllllllll ll l l l l l l ll ll l l l l l llllllllll l l l l l lllll ll l l l l l lllllllllllll lll l l l l l ll llllll l l l l l lllll ll l l l l l llllllllll llll l l l l l llll ll l l ll l l l l l lllllllllllllllll l l l l l lllllllll ll l l l l l lll ll l l ll l l l l l ll ll l l ll l l l l l lllllllllllllllllll l l ll l l l l l l l ll lllllllll l l l l l lll ll l l l l l lllllllllllllllllll ll l l l l l lllll l lll l l l l l lll l lll l ll l l l l l llllllllllll ll l l l l l llll l ll l l l l l llllllllll lllll l l l l l lllll llllllllll l l l l l lllllll l llllll l l l lllllll lllllllllll l l l l l lll l l l l l llll l lll l l l l l lll l ll l l l l l ll ll l l l l l llll ll ll l lll l l l l lll ll l lll ll ll l l l l l lll ll l l l l l l lll ll l l l l l l l l l ll l l l l l lll ll l l l l l l l ll ll l l l l l llllllll llllll l l l l l lll ll ll l l l l l lllll llll l l l l l l lll l l l l l lllllll ll l l l l l lll l ll l l l l l l ll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l llllllll ll l l l l l l lll l l l l l ll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllll l l l l l l lllllllllllllllllll l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllll ll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllll l l l l l l lllllllllllllllllll l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllll l llllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllll l l l l l l llllllllllllll l l l l l l llllllllllllll l l l l l l ll lllllllllllllll l l l l l l llllll llllllll l l l l l l llllllllllllllllll l l ll l l l l l l llllll lllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lll l l l l l l lllllllllllllllllll l l l l l l llll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllll l l l l l l llllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll
12 16 20 ll lllllllllllll lllllllll llllllllllllllll l llllll ll l l l lll l l lll lllllll llllllllllll l llllll lll l llllllllllllllllllllll l lllllllllllll lllll l llllllll l lllllll llll l lllllll l lllllllll l l l l l ll llllllll llllll lll ll llllllllllllll lll llll ll ll lll llllllllllll lll lllll llllllllllllll lllllllllllllllll llllllllllll lll llll ll ll lllllll llllllllll llllllllllll lllllllllllllllllll lll l lllllllllll l lllllllllllllllllllll lllllll l ll l lllllllllllllll l lllllllll l l l l lllll l llllllll llllllll lllllllllllllllllllllllll ll lllllllllllll lllllllllll llll l llllllllll l l ll lll l llllllllll ll lllllllll lllllllllllllllll l l l l llllll llllllll l ll l l l l l llllll l l llllllllllll l lllllll lllll ll llllllllllllll l l ll lllllllll l ll l lll l llllll llll lllllllllllllllll l llll llll l l ll l ll llllll ll l l llll lll l l l l lllllll llll llllllllllll ll l l lllll llllllllllllll llllllll lllll lll l ll llll l lllll l llll llll ll ll lll llllllll llllllllllll lll l lll l lllllllllllllllll l l llll l l l l l l lllllllllllllllllll l llllllllllllll l lllllllllllll llll llllllllllll llllllllllllllll ll l llllllllllllllllllllll l lllllllll l l lllllllll l l lllllllllllllllllll ll l ll lll lllllllllllllll l l l l lllllllllllllllllllll llllll lll l l llll lll llll l l llllllllllll llll ll l ll ll l lll ll l ll llllllllllllllllllllllllllll l llllllllllllll lllllllllllll l lllllll l l llllllllllllll l l l l ll llllllllllllllllllllllllll ll llllllllllll l l ll llllllllll lllllllllllllll l llllllllllllllll lllll lllllllllllllll llllllllllll lllllllllllllllllllll lllllllllllll llllllllllllllll l l l l l lllllllllllllllll l l l l l l llll ll l l l l l l ll llllllll l l l l l lll l l l l l ll l ll l ll l l l l l l lllll lll lll l l l l l l llll lll l l l l l l ll llll lll l l l l l l l l lll l l l l l l llllllllll l l l l l l llll l l l ll l l l l l l lllllllllllllllllll l l l l l l lllllllllll llll l l l l l l llllllllllllll l l l l l l lll 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l l l l l llllllllllll ll l l l l l l ll llll l ll l l l l l l llllll llllllll l l l l l l lllllllllllllllllll l l l l l l ll l lll ll l l l l l l lllllllllllllllll l l l l l l llll llllllllll l l l l l l llllll lllllll l l l l l l llllllllllllll ll l l l l l l llllllllll lllllll l l l l l l llllllllllllll ll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllll ll ll l l l l l ll l l l l l l llllll ll l l l l lllllllllllllllllll l l l l l l llll lllllllllllll l l lll l l l l l l lllll l l llllllll l l l l l l llllllllllllllllll l ll l l l l ll l l l l l ll l l l l l ll l l l l l lll l l l l l l lllllllllllllllllll l l l l ll ll l l l l l l lllllll llllllll l l l l l ll l l l l l l lllllllllllllllllll l l l l l l l lllllllllllllllll l l l l l l lllll llllll llll l l l l l l lllllllllllllll l l l lll lll lll l l l l l l l ll ll lll l l llllllllllllllllll l l l l l l lllllllllllll l l l l l l l l lll l l l l l l llllllllll lllll l l ll l l l l l l l ll l l l l l ll l l l l l l llllll lll l l l l l l llllllllllllllll l l l l l ll l l l l ll l l l l l l ll l l llllll l l l l l l lllllllllllllllll l l l l l l l l l ll ll l l l l l l lllllllllllllllllll l l l l l l lll lllll l l l l l l lllll l lll l l l l l l l ll l l l l l l llllllllllll ll l l l l l l lllllllllllllllllll l l l l l ll l l l l l l lllllll lllll ll l l l l l l ll ll l l l l l l lllllllll l l l l l lllll ll l l l l l l llllllllllll lll l l l l l l l llllll l l l l l l llll ll l l l l l l lllllllll llll l l l l l l lll ll l l ll l l l l l l llllllllllllllll l l l l l l llllllll ll l l l l l l ll ll l l ll l l l l l l l ll l l l l ll l l l l l l llllllllllllllllll l l ll l l l l l l l ll lllllllll l l l l l l ll ll l l l l l l llllllllllllllllll ll l l l l l l llll l lll l l l l l l ll l ll l llll l l l l l l llllllll lll l ll l l l l l l lll l l ll l l l l l l lllllllll lllll l l l l l lllll llllllllll l l l l l l llllll l llllll l l l l llllll lllllllllll l l l l l lll l l l l l l lll l lll l l l l l l ll l ll l l l l l l l ll l l l l l l lll ll ll l lll l l l l lll ll l l ll ll ll l l l l l l ll ll l l l l l l lll ll l l l l l l l l l ll l l l l l l ll ll l l l l l l l ll ll l l l l l l lllllll llllll l l l l l l ll ll ll l l l l l l llll llll l l l l l l ll ll l l l l l l llllll l ll l l l l l l ll l ll l l l l l l ll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllll l l l l l l lll l l l l l ll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllll ll l l l l l l lllllllllllllllllll l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllll ll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllll l l l l l l lllllllllllllllllll l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllll llllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllll l l l l l l llllllllllllll l l l l l l llllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllll l l l l l l llllllllllllllllll l ll l l l l l l llllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lll l l l l l l llllllllllllllllll l l l l l l llll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllll l l l l l l llllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l llllllllllllll ll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll
12 16 20 m r , (mag) ll lllllllllllll lllllllll lllllllllllllllll l lllllllll l l l lll l l lll lllllll llllllllllll l llllll lll l llllllllllllllllllllll l lllllllllllll lllll l llllllll l lllllll ll l l llllll l lllllllll l l l l l ll llllllll llllll lll ll llllllllllllll lll llll ll ll lll llllllllllll lll lllll llllllllllllll lllllllllllllllll llllllllllll lll llll ll ll llllll llllllllll llllllllllll lllllllllllllllllll lll l llllllllllll l lllllllllllllllllllll lllllll l ll l lllllllllllllll l lllllllll l l l l lllll l llllllll llllllll lllllllllllllllllllllllll ll lllllllllllll lllllllllll llll l llllllllll l l ll lll l llllllllll ll lllllllll lllllllllllllllll l l l l llll llllllll l ll l l l l l llllll l l llllllllllll l lllllll lllll llllllllllllllllll l l lllllllllllll ll l lll llllllll llll lllllllllllllllll l llll llll l l ll l ll llllll ll l l llll lll l l l l lllllll llll llllllllllll ll l l lllll lllllllllllll llllllll lllll lll l ll llll l lllll l llll llll ll ll lll llllllll llllllllllll lll l lll l lllllllllllllllll l l llll l l l l l l lllllllllllllllllll l llllllllllllll l llllllllllll llll llllllllllll llllllllllllllll ll l llllllllllllllllllllll l lllllllll l l lllllllll l l lllllllllllllllllll ll l ll lll lllllllllllllll l l l l lllllllllllllllllllll llllll lll l l llll lll llll l l lllllll lll llll ll l ll ll l lll ll l ll llllllllllllllllllllllllllll l lllllllllllll lllllllllllll l lllllll l l llllllllllllll l l l l ll llllllllllllllllllllllllll ll llllllllllll l l ll llllllllll lllllllllllllll l llllllllllllllll lllll lllllllllllllll llllllllllll lllllllllllllllllllll lllllllllllll llllllllllllllll l l l l l lllllllllllllllll l l l l l l llll ll l l l l l l ll llllllll l l l l l lll l l l l l ll l ll l ll l l l l l l lllll lll lll l l l l l l llll lll l l l l l l ll llll lll l l l l l l l l lll l l l l l l llllllllll l l l l l l llll l l l ll l l l l l l lllllllllllllllllll l l l l l l lllllllllll llll l l l l l l llllllllllllll l l l l l l lll ll lll ll l l l l ll l l l l l l llll l l l l lll l l l l l lllllllllllllllllll l l l l l l llll l llll l llllll l ll l l l l l l ll lllll lll l l l l l l lll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllll l l l l l l ll lllll l l l l l l lllllllllllll llll l l l l l l lll ll l ll l l l l l l llllll ll lll l l l l l l lll l ll l l l l l l lll l lll l l l l l l ll llll l l l l l l lllllllll llll l l l l l l ll l lll llll l l l l l l llllll l lll l l l l l l lllll l ll l l l l l l llll llllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllll l l l l ll l l l l l l ll l ll l l l l l l llllllllllllllllll l l l l l l llllllllllllll l l l l l l ll ll ll l l l l l l llllllllll ll l l l l l l lllllllllllllllll l l l l l l lllllll lllll l l l l l l llllllllllllllll l l l l l l lll l l ll l l l l l l lllllllllll llll l l l l l l lllllllllllllllll l l l l l l lllllllllll l l lllll l l l l l l llllllll lll l l l l l l l lll l ll l l l l l l llllll l l l l l l l ll llllll l l l l l l llllllllllll ll l l l l l l ll llll l ll l l l l l l llllll llllllll l l l l l l lllllllllllllllllll l l l l l l ll l lll ll l l l l l l lllllllllllllllll l l l l l l llll llllllllll l l l l l l llllll lllllll l l l l l l llllllllllllll ll l l l l l l llllllllll lllllll l l l l l l llllllllllllll ll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllll ll ll l l l l l ll l l l l l l llllll ll l l l l lllllllllllllllllll l l l l l l llll lllllllllllll l l lll l l l l l l lllll l l llllllll l l l l l l llllllllllllllllll l ll l l l l ll l l l l l ll l l l l l ll l l l l l lll l l l l l l lllllllllllllllllll l l l l ll ll l l l l l l lllllll llllllll l l l l l ll l l l l l l lllllllllllllllllll l l l l l l l lllllllllllllllll l l l l l l lllll llllll llll l l l l l l lllllllllllllll l l l lll lll lll l l l l l l l ll ll lll l l llllllllllllllllll l l l l l l lllllllllllll l l l l l l l l lll l l l l l l llllllllll lllll l l ll l l l l l l l ll l l l l l ll l l l l l l llllll lll l l l l l l llllllllllllllll l l l l l ll l l l l ll l l l l l l ll l l llllll l l l l l l lllllllllllllllll l l l l l l l l l ll ll l l l l l l lllllllllllllllllll l l l l l l lll lllll l l l l l l lllll l lll l l l l l l l ll l l l l l l llllllllllll ll l l l l l l lllllllllllllllllll l l l l l ll l l l l l l lllllll lllll ll l l l l l l ll ll l l l l l l lllllllll l l l l l lllll ll l l l l l l llllllllllll lll l l l l l l l llllll l l l l l l llll ll l l l l l l lllllllll llll l l l l l l lll ll l l ll l l l l l l llllllllllllllll l l l l l l llllllll ll l l l l l l ll ll l l ll l l l l l l l ll l l l l ll l l l l l l llllllllllllllllll l l ll l l l l l l l ll lllllllll l l l l l l ll ll l l l l l l llllllllllllllllll ll l l l l l l llll l lll l l l l l l ll l ll l llll l l l l l l llllllll lll l ll l l l l l l lll l ll l l l l l l lllllllll lllll l l l l l lllll llllllllll l l l l l l llllll l llllll l l l l llllll lllllllllll l l l l l lll l l l l l l lll l lll l l l l l l ll l ll l l l l l l l ll l l l l l l lll ll ll l lll l l l l lll ll l l ll ll ll l l l l l l ll ll l l l l l l lll ll l l l l l l l l l ll l l l l l l ll ll l l l l l l l ll ll l l l l l l lllllll llllll l l l l l l ll ll ll l l l l l l llll llll l l l l l l ll ll l l l l l l llllll l ll l l l l l l ll l ll l l l l l l ll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllll l l l l l l lll l l l l ll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllll llll l l l l l l lllllllllllllllllll l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllll ll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllll l l l l l l lllllllllllllllllll l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllll lllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllll l l l l l l llllllllllllll l l l l l l lllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllll l l l l l l llllllllllllllllll l l l ll l l l l l l llllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllll lllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lll l l l l l l lllllllllllllllllll l l l l l l llll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllll l l l l l l llllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllll l l l l l l llllllll lllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l llllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll
12 16 20 ll lllllllllllll lllllllll lllllllllllllllll l llllllllll l l l lll l lll lllllll llllllllllll l llllll lll l llllllllllllllllllllll l lllllllllllll lllll l llllllll l llllll lll l lllllll l lllllllll l l l l l ll llllllll llllll lll ll llllllllllllll lll llll ll ll lll llllllllllll lll lllll llllllllllllll lllllll lllllllll llllllllllll lll llll ll ll lllllll llllllllll llllllllllll lllllllllllllllllll lll l llllllllllll l lllllllllllllllllllll lllllll l ll l lllllllllllllll l lllllllll l l l l lllll l llllllll llllllll lllllllllllllllllllllllll ll lllllllllllll lllllllllll llll l lllllllll l l ll lll l llllllllll ll lllllllll lllllllllllllllll l l l l llllllllllllllll l ll l l l l l llllll l l llllllllllll l lllllll llllll llllllllllllllllll l l llllllllllllll l l ll l lll llllllll llll lllllllllllllllll l llll llll l l ll l ll llllll ll l l llll lll l l l l lllllll llll llllllllllll ll l l lllll llllllllllllll llllllll llll lll l ll lll l lllll l llll llll ll ll lll llllllll llllllllllll lll l lll l lllllllllllllllll l l llll l l l l l l lllllllllllllllllll l llllllllllllll l ll lllllllll llll llllllllllll llllllllllllllll ll l llllllllllllllllllllll l lllllllll l l lllllllll l l lllllllllllllllllll ll l ll lll lllllllllllllll l l l l lllllllllllllllllllll llllll lll l l llll lll llll l l lllllllll llll ll l ll ll l lll ll l ll llllllllllllllllllllllllllll l llllllllllllll lllllllllllll l lllllll l l llllllllllllll l l l l ll llllllllllllllllllllllllll ll llllllllllll l l ll llllllllll lllllllllllllll l lllllllllllllll lllll lllllllllllllll llllllllllll lllllllllllllllllllll lllllllllllll llllllllllllllll l l l l l lllllllllllllllll l l l l l l llll ll l l l l l l ll llllllll l l l l l lll l l l l l ll l ll l ll l l l l l l lllll lll lll l l l l l l llll lll l l l l l l ll llll lll l l l l l l l l lll l l l l l l llllllllll l l l l l l llll l l l ll l l l l l l lllllllllllllllllll l l l l l l lllllllllll llll l l l l l l lllllllll lllll l l l l l l lll ll lll ll l l l l ll l l l l l l llll l l l l lll l l l l l lllllllllllllllllll l l l l l l llll l llll l llllll l ll l l l l l l ll lllll lll l l l l l l lll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllll l l l l l l ll lllll l l l l l l lllllllllllll llll l l l l l l lll ll l ll l l l l l l llllll ll lll l l l l l l lll l ll l l l l l l lll l lll l l l l l l ll llll l l l l l l lllllllll llll l l l l l l ll l lll llll l l l l l l llllll l lll l l l l l l lllll l ll l l l l l l llll llllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllll l l l l ll l l l l l l ll l ll l l l l l l llllllllllllllllll l l l l l l llllllllllllll l l l l l l ll ll ll l l l l l l llllllllll ll l l l l l l lllllllllllllllll l l l l l l lllllll lllll l l l l l l llllllllllllllll l l l l l l lll l l ll l l l l l l lllllllllll llll l l l l l l lllllllllllllllll l l l l l l lllllllllll l l lllll l l l l l l llllllll lll l l l l l l l lll l ll l l l l l l llllll l l l l l l l ll llllll l l l l l l llllllllllll ll l l l l l l ll llll l ll l l l l l l llllll llllllll l l l l l l lllllllllllllllllll l l l l l l ll l lll ll l l l l l l lllllllllllllllll l l l l l l llll llllllllll l l l l l l llllll lllllll l l l l l l llllllllllllll ll l l l l l l llllllllll lllllll l l l l l l llllllllllllll ll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllll ll ll l l l l l ll l l l l l l llllll ll l l l l lllllllllllllllllll l l l l l l llll lllllllllllll l l lll l l l l l l lllll l l llllllll l l l l l l llllllllllllllllll l ll l l l l ll l l l l l ll l l l l l ll l l l l l lll l l l l l l lllllllllllllllllll l l l l ll ll l l l l l l lllllll llllllll l l l l l ll l l l l l l lllllllllllllllllll l l l l l l l lllllllllllllllll l l l l l l lllll llllll llll l l l l l l lllllllllllllll l l l lll lll lll l l l l l l l ll ll lll l l llllllllllllllllll l l l l l l lllllllllllll l l l l l l l l lll l l l l l l llllllllll lllll l l ll l l l l l l l ll l l l l l ll l l l l l l llllll lll l l l l l l llllllllllllllll l l l l l ll l l l l ll l l l l l l ll l l llllll l l l l l l lllllllllllllllll l l l l l l l l l ll ll l l l l l l lllllllllllllllllll l l l l l l lll lllll l l l l l l lllll l lll l l l l l l l ll l l l l l l llllllllllll ll l l l l l l lllllllllllllllllll l l l l l ll l l l l l l lllllll lllll ll l l l l l l ll ll l l l l l l lllllllll l l l l l lllll ll l l l l l l llllllllllll lll l l l l l l l llllll l l l l l l llll ll l l l l l l lllllllll llll l l l l l l lll ll l l ll l l l l l l llllllllllllllll l l l l l l llllllll ll l l l l l l ll ll l l ll l l l l l l l ll l l l l ll l l l l l l llllllllllllllllll l l ll l l l l l l l ll lllllllll l l l l l l ll ll l l l l l l llllllllllllllllll ll l l l l l l llll l lll l l l l l l ll l ll l llll l l l l l l llllllll lll l ll l l l l l l lll l l ll l l l l l l lllllllll lllll l l l l l lllll llllllllll l l l l l l llllll l llllll l l l l llllll lllllllllll l l l l l lll l l l l l l lll l lll l l l l l l ll l ll l l l l l l l ll l l l l l l lll ll ll l lll l l l l lll ll l l ll ll ll l l l l l l ll ll l l l l l l lll ll l l l l l l l l l ll l l l l l l ll ll l l l l l l l ll ll l l l l l l lllllll llllll l l l l l l ll ll ll l l l l l l llll llll l l l l l l ll ll l l l l l l llllll l ll l l l l l l ll l ll l l l l l l ll l l l l l l lllllllllllllllllll l l l l l l llllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllll l l l l l l lll l l l l l ll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llll lllllll l l l l l l lllllllllllllllllll l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllll llllll lll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllll l l l l l l lllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllll ll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l llllll l l l l l l lllllllllllllllllll l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllll lllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllll l l l l l l lllllllllllll l l l l l l lllllllllll l l l l l l llllllllllllllll l l l l l l lll l lllllllllll l l l l l l llllllllllllllllll l l l ll l l l l l l lllllllllll l l l l l l lllllllllllllllllll l l l l l l lllll lllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l ll l l l l l l llllllll lllll l l l l l l llll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllll ll l l l l l l llllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllll l l l l l l lllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllll l l l l l l lllllll lllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lll lllllllllllll l l l l l l llll lllllllllllll l l l l l l llllllllllll lll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l llllllllllll llll l l l l l l lllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllll l l l l l l llllllllllllll ll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllll lll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllll
12 16 20 ll llllllllll llllllll lllllllllllllllll l llllllll l l l lll l l lll lllllll llllllllll l llllll lll l lllllllllllllllllll l lllllllllllll lllll l llllllll l llll ll llll l lllllll l lllllllll l l l l l ll llllllll lllll lll ll llllllllllllll lll llll ll ll lll l lllll lll lll lllll llllllllllllll lllllllllllllllll lllllllllll lll llll ll ll llllll llllllllll llllllllllll lllllllllllllllll lll l llllllllllll l lllllllllllllllllllll lllllll l ll l lllllllllllllll l lllllll l l l l lllll l lllllll llllllll lllllllllllllllllllllllll ll lllllllllllll llll llllll llll l lllllllll l l ll lll l llllllllll ll l lllllll lllllllllllllllll l l l l llllllllllll lll l l l l l lllll l l llllllll lll l lllllllll lllllllllllllll l l l llllllllllll l l ll l lll lllll llll llll lllllllllll l llll llll l l ll l ll lll ll l l llll lll l l l l lllllll llll llllllllllll ll l l llll lll lllll llllllll llll lll l ll l ll l lllll l llll llll ll ll lll llllllll llllllllllll ll l lll l lllllllllllllll ll llll l l l l l l llllllllllllllllll l llllllllllllll l l lllllllllll llll llllllllllll llllllllllllllll ll l llllllllllllllllllllll l lllllllll l l lllll l l lllllllllllllllllll ll l ll lll llllllllllllll l l l l lllllllllllllllllllll llllll lll l lllll lll llll l l llllllllll llll ll l ll ll l lll ll l ll lllllllllllllllll lllllll l lllllllllllll llllll lll l lllllll l l llllllllllllll l l l l ll lllllllllllllllllll llll ll llllllllll l l lllllllllll lllllllllllll l llllllll lllll lllll lllllllllllllll ll llllllll llllllllllllllll llll lllllllllllll llllllllllllllll l l l l l lllllllllllllllll l l l l l l lllll l l l l l l ll llllllll l l l l l lll l l l l l ll l ll l ll l l l l l l lllll lll lll l l l l l l llll lll l l l l l l ll llll lll l l l l l l l l lll l l l l l l llllllllll l l l l l l llll l l ll l l l l l l lllllllllllllllllll l l l l l l lllllllllll llll l l l l l l llllllllllllll l l l l l l lll ll lll ll l l l l ll l l l l l l llll l ll l lll l l l l l lllllllllllllllllll l l l l l l llll l llll l llllll l ll l l l l l l ll lllll lll l l l l l l lll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllll l l l l l l ll lllll l l l l l l lllllllllllll llll l l l l l l lll ll l ll l l l l l l llllll ll lll l l l l l l lll l ll l l l l l l lll l lll l l l l l l ll llll l l l l l l lllllllll llll l l l l l l ll l lll llll l l l l l l llllll l lll l l l l l l lllll l ll l l l l l l llll llllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllll l l l l ll l l l l l l ll l ll l l l l l l llllllllllllllllll l l l l l l llllllllllllll l l l l l l ll ll ll l l l l l l llllllllll ll l l l l l l lllllllllllllllll l l l l l l lllllll lllll l l l l l l llllllllllllllll l l l l l l lll l l ll l l l l l l lllllllllll llll l l l l l l lllllllllllllllll l l l l l l lllllllllll l l lllll l l l l l l lllllllllll l l l l l l l lll l ll l l l l l l llllll l l l l l l l ll llllll l l l l l l llllllllllll ll l l l l l l ll llll l ll l l l l l l llllll lllllll l l l l l l lllllllllllllllllll l l l l l l ll l lllll l l l l l l lllllllllllllllll l l l l l l llll llllllllll l l l l l l llllll lllllll l l l l l l llllllllllllll ll l l l l l l llllllllll lllllll l l l l l l llllllllllllll ll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllll ll ll l l l l l ll l l l l l l llllll ll l l l l lllllllllllllllllll l l l l l l llll lllllllllllll l l lll l l l l l l lllll l l llllllll l l l l l l llllllllllllllllll l ll l l l l ll l l l l l ll l l l l l ll l l l l l ll l l l l l l lllllllllllllllllll l l l l ll ll l l l l l l lllllll llllllll l l l l l ll l l l l l l lllllllllllllllllll l l l l l l l lllllllllllllllll l l l l l l lllll llllll llll l l l l l l lllllllllllllll l l l lll lll ll l l l l l l l ll ll lll l l llllllllllllllllll l l l l l l lllllllllllll l l l l l l l l lll l l l l l l llllllllll lllll l l ll l l l l l l l ll l l l l l ll l l l l l l llllll lll l l l l l l llllllllllllllll l l l l l ll l l l l ll l l l l l l ll l l llllll l l l l l l lllllllllllllllll l l l l l l l l l ll ll l l l l l l lllllllllllllllllll l l l l l l lll lllll l l l l l l lllll l lll l l l l l l l ll l l l l l l llllllllllll ll l l l l l l lllllllllllllllllll l l l l l ll l l l l l l lllllll lllll ll l l l l l l ll ll l l l l l l lllllllll l l l l l lllll ll l l l l l l llllllllllll lll l l l l l l l llllll l l l l l l llll ll l l l l l l lllllllll llll l l l l l l lll ll l l ll l l l l l l llllllllllllllll l l l l l l llllllll ll l l l l l l ll ll l l ll l l l l l l l ll l ll ll l l l l l l llllllllllllllllll l l ll l l l l l l l ll lllllllll l l l l l l ll ll l l l l l l llllllllllllllllll ll l l l l l l llll l lll l l l l l l ll l ll l llll l l l l l l lllllllllll l ll l l l l l l lll l l ll l l l l l l lllllllll lllll l l l l l lllll llllllllll l l l l l l llllll l llllll l l l l llllll lllllllllll l l l l l lll l l l l l l lll l lll l l l l l l ll l ll l l l l l l l ll l l l l l l lll ll ll l lll l l l l lll ll l l ll ll ll l l l l l l lll l l l l l l lll ll l l l l l l l l l ll l l l l l l ll ll l l l l l l l ll ll l l l l l l lllllll llllll l l l l l l ll ll ll l l l l l l llll llll l l l l l l ll ll l l l l l l llllll l ll l l l l l l ll l ll l l l l l l
Figure 9.
Comparison of
GALFIT S3 ) fits versus GALFITM single-band ( SM ) fits. This plot shows the recovered S´ersic model parameters asa function of the r -band apparent magnitude as cosmologically scaled from the SM fit to the lowest-redshift artificial image for each galaxy (using thismagnitude, rather than the value measured for each image, avoids scatter on the recovered parameters from entering the horizontal axis of this plot). Upperpanels: difference in magnitude; middle panels: ratio of effective radii; lower panels: ratio of S´ersic indices. Only parameters for the artificially-redshiftedimages are included in this figure. The grey points indicate that at least one fit ( S3 or SM ) ended in close proximity to a constraint, and for which the recoveredparameters are hence deemed to be unreliable. We believe that the approach taken by
GALFITM is superior to thatof
GALFIT
3, particularly in low signal-to-noise cases.With noisy data it is quite common for the first few minimi-sation steps to be large and encounter constraints. In such cases
GALFIT
GALFITM will continue from the previous accept-able value of the offending parameter. The two approaches maytherefore converge to different minima. However, the one found byGALFITM is more likely to be in a ‘reasonable’ part of parameterspace. Alternatively, in both
GALFITM and
GALFIT χ -space. Asdescribed earlier, GALFITM attempts to avoid this by periodicallytaking a very small step, which may lead to systematic differencesin its results versus
GALFIT
GALFITM does indeed produce better results than
GALFIT χ values for SM and S3 . These are generallyconsistent, but in cases where the two methods return very differ- ent parameters the SM fit usually has a lower χ , and is therefore abetter representation of the data. Having checked that applying
GALFITM to single-band data ( SM )is sufficiently consistent with GALFIT S3 ), we now move on tocompare the performance of single- versus multi-band methods. Inparticular, we wish to investigate how multi-band fitting modifiesthe uncertainties on the measured parameters, and check whetherit introduces any systematic deviations compared with the single-band results. In Fig. 10, we compare the parameters from single-band fits using GALFITM ( SM ; n [M] , r [M]e , m [M] ) with those frommulti-band fits ( MM ; n [M , all] , r [M , all]e , m [M , all] , where ‘all’ indicatesthat all five u , g , r , i , z bands have been used in the fit). This figureplots the difference in the magnitudes, and the ratios of effectiveradius and S´ersic index, as a function of the apparent magnitude, inan equivalent manner to Fig. 9.As expected, both algorithms are consistent in extracting pa-rameters in the g , r and i bands, while in the shallower bands ( z c (cid:13) , 1–29 Vika et al. l l l l l llllllllllllllllllll l l l l l lllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l lllllll lllll l l l l l llll l l l l l ll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l lllllllllllll l l l l l llllllllllllllllllll l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l lllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l lllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l lllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllll ll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l lllllllllllllllllll l l l l l lllllll l l l l l llllllllllllllllllll l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l lllllll lllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l lllllll l l l l l lllllllllllllll l l l l l llllllllllllllll l l l l l llllllllllllllllllll l l l l l lllllllllllllllllll l l l l l llllllllllllllllllll l l ll l l l l l llllllllllll l l l l l llllllllllllllllllll l l l l l lllllllllllllllllll l l l l l llllllllllllllllllll l 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lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l llllllll lllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllll l l l l l l llllllllllll llll
12 16 20 m r , (mag) ll lllllllllllll lllllllll l llllllll llllllll l llllllllll l l l ll l l lll lllllll llllllllllll l llllll lll l llllllllllllllllllllll l lllllllllllll lllll l llllllll l lllllll l ll l l llllll l lllllllll l l l l l ll llllllll llllll lll ll llllllllllllll lll llll ll ll lll llllllllllll lll lllll llllllllllllll llllllllllllllll l llllllllllll lll llll ll ll lllllll llllllllll llllllllllll lllllllllllllllllll lll l llllllllllll l lllllllllllllllllllll lllllll l ll l lllllllllllllll l llllllll l l l l lllll l llllllll llllllll lllllllllllllllllllllllll ll lllllllllllll lllllllllll llll l l lllllllll l l ll lll l llllllllll ll ll lllllll lllllllllllllllll l l l l llllllllllllllll l ll l l l l l l lllll l l llllllll llll l lllllll llllll llllllllllllllllll l l lllllllllllllll l l ll l l ll llll llll llll lllllllllllllllll l llll llll l l l l l ll llllll ll l l llll lll l l l l lllllll llll llllllllllll ll l l l llll llllllllllllll llllllll lllll lll l ll llll l lllll l lll l llll ll ll lll llllllll llllllllllll lll l lll l lllllllllllllllll l l llll l l l l l l l llllllllllllllllll l llllllllllllll l lllllllllllll llll llllllllllll llllllllllllllll ll l llllllllllllllllllllll l lllllllll l l l llllllll l l l llllllllllllllllll ll l ll lll lllllllllllllll l l l l lllllllllllllllllllll llllll lll l l llll lll llll l l lllllllllll llll ll l ll ll l lll ll l ll lllllllllllllllllll lllllllll l llllllllllllll lllllllllllll l lllllll l l llllllllllllll l l l l ll llllllllllllllllllllllllll ll llllllllllll l l ll llllllllll lllllllllllllll l llll llllllllll ll lllll ll llllllllllll llllll lllll lllllllllllllllllllll lllllllllll lllllllllll lll l l l l l lllllllllllllllll l l l l l l llll ll l l l l l l ll llllllll l l l l l lll l l l l l ll l ll l ll l l l l l l lllll ll l lll l l l l l l llll lll l l l l l l ll llll lll l l l l l l l l lll l l l l l l llllllllll l l l l l l l lll l l l ll l l l l l l lllllllllllllllllll l l l l l l lllllllllll llll l l l l l l lllllllll lllll l l l l l l l ll ll lll ll l l l l ll l l l l l l llll l l l l lll l l l l l lllllllllllllllllll l l l l l l llll l ll ll l llllll l ll l l l l l l ll lllll lll l l l l l l lll l l l l l l l lllllllllllllllll l l l l l l lllllllllllllllll l l l l l l ll lllll l l l l l l lllllllllllll llll l l l l l l l ll ll l ll l l l l l l llllll ll lll l l l l l l lll l ll l l l l l l lll l lll l l l l l l ll llll l l l l l l l llllllll llll l l l l l l ll l lll llll l l l l l l l lllll l lll l l l l l l lllll l ll l l l l l l llll llllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllll l l l l ll l l l l l l ll l ll l l l l l l llllllllllllllllll l l l l l l llllllllllllll l l l l l l ll ll ll l l l l l l llllllllll ll l l l l l l lllllllllllllllll l l l l l l lllllll lllll l l l l l l llllllllllllllll l l l l l l lll l l ll l l l l l l lllllllllll llll l l l l l l lllllllllllllllll l l l l l l lllllllllll l l lllll l l l l l l llllllll lll l l l l l l l lll l ll l l l l l l llllll l l l l l l l ll llllll l l l l l l llllllllllll ll l l l l l l ll llll l ll l l l l l l llllll llllllll l l l l l l lllllllllllllllllll l l l l l l ll l lll ll l l l l l l lllllllllllllllll l l l l l l llll llllllllll l l l l l l llllll lllllll l l l l l l llllllllllllll ll l l l l l l llllllllll lllllll l l l l l l llllllllllllll ll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllll ll ll l l l l l ll l l l l l l l lllll ll l l l l lllllllllllllllllll l l l l l l llll lllllllllllll l l lll l l l l l l lllll l l llllllll l l l l l l llllllllllllllllll l ll l l l l ll l l l l l ll l l l l l ll l l l l l lll l l l l l l lllllllllllllllllll l l l l ll ll l l l l l l lllllll llllllll l l l l l ll l l l l l l l llllllllllllllllll l l l l l l l lllllllllllllllll l l l l l l lllll llllll llll l l l l l l l llllllllllllll l l l lll lll lll l l l l l l l ll ll lll l l llllllllllllllllll l l l l l l l llllllllllll l l l l l l l l lll l l l l l l llllllllll lllll l l ll l l l l l l l ll l l l l l ll l l l l l l llllll lll l l l l l l l lllllllllllllll l l l l l ll l l l l ll l l l l l l ll l l llllll l l l l l l lllllllllllllllll l l l l l l l l l ll ll l l l l l l lllllllllllllllllll l l l l l l lll lllll l l l l l l lllll l lll l l l l l l l ll l l l l l l llllllllllll ll l l l l l l lllllllllllllllllll l l l l l ll l l l l l l lllllll l llll ll l l l l l l ll ll l l l l l l lllllllll l l l l l lllll ll l l l l l l l llllllll lll lll l l l l l l l llllll l l l l l l llll ll l l l l l l lllllllll llll l l l l l l lll ll l l ll l l l l l l llllllllllllllll l l l l l l llllllll ll l l l l l l ll ll l l ll l l l l l l l ll l l l l ll l l l l l l llllllllllllllllll l l ll l l l l l l l ll ll lllllll l l l l l l ll ll l l l l l l llllllllllllllllll ll l l l l l l l lll l lll l l l l l l ll l ll l llll l l l l l l llllllll lll l ll l l l l l l lll l l ll l l l l l l lllllllll lllll l l l l l lllll llllllllll l l l l l l llllll l llllll l l l l llllll lllllllllll l l l l l lll l l l l l l lll l lll l l l l l l l l l ll l l l l l l l ll l l l l l l lll ll ll l lll l l l l lll ll l l l l ll ll l l l l l l ll ll l l l l l l lll ll l l l l l l l l l ll l l l l l l ll ll l l l l l l l ll ll l l l l l l lllllll llllll l l l l l l l l ll ll l l l l l l llll llll l l l l l l ll ll l l l l l l llllll l ll l l l l l l ll l ll l l l l l l ll l l l l l l l llllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllll l l l l l l ll l l l l ll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllll llll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllll llllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l l lllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllll lllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l lllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllll l l l l l l l llllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllll ll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllll l l l l l l lllllllllllllllllll l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllll lllllllllll l l l l l l l llllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l llllll l l l l l l llllllllllllll l l l l l l llllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l ll l l l l l l l lllllllllll l l l l l l lllllllllllllllllll l l l l l l l lllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllll lllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lll l l l l l l lllllllllllllllllll l l l l l l l lll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllll ll l l l l l l l lllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lll l l l l l l lllllllllllllllllll l l l l l l lllllllll l ll l l l l l l lllllllllllllllllll l l l l l l lllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l llllllll lllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllll lllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll
12 16 20 ll lllllllllllll lllllllll lllllllll llllllll l llllllllll l l l ll l lll lllllll llllllllllll l llllll lll l lllllll lllllllllllll l lllllllllllll lllll l llllllll l lllll l llll l l llllll l lllllllll l l l l l ll llllllll llllll lll ll llllllllllllll lll l lll ll ll lll llllllllllll lll lllll llllllllllllll lllll llllllllll l llllllllllll lll llll ll ll lllllll llllllllll llllllllllll llllllllllllllllll lll l llllllllllll l lllllllllllllllllllll lllllll l ll l lllllllllllllll l lllllllll l l l l lllll l llllllll llllllll lllllllll lllllllllllllll ll lllllllllllll lllllllllll llll l l lllllllll l l ll lll l llllllllll ll ll lllllll lllllllllllllllll l l l l llllllllllllllll l ll l l l l l llllll l l llllllll llll l lllllll lllll llllllllllllllllll l l lllllllllllllll l l ll l l ll llll llll llll lllllllllllllllll l llll llll l l ll l ll llllll ll l l llll lll l l l l lllllll llll llllllllllll ll l l lllll llllllllllllll l lllllll lllll lll l ll l ll l lllll l lll l llll ll ll lll llllllll llllllllllll lll l lll l lllllllllllllllll l l llll l l l l l l lllllllll l ll l llllllllllllll l lllllllllllll llll llllllllllll llllllllllllllll ll l llllllllllllllllllllll l lllllllll l l l llllllll l l l llllllllllllllllll ll l ll lll lllllllllllllll l l l l l llllllllllllllllllll llllll lll l l llll lll llll l l l lllllllll llll ll l ll ll l lll ll l ll lllllllllllllllllll lllllllll l llllllllllllll lllllllllllll l lllllll l l llllllllllllll l l l l ll llllllllllllllllllllllllll ll llllllllllll l l ll llllllllll lllllllllllllll l llll llllllllll ll lllll ll lllllllllllll llllllllllll lllllllllllllllllllll lllllllllllll llllllllllllllll l l l l l l llllllllllllllll l l l l l l llll ll l l l l l l ll llllllll l l l l l lll l l l l l ll l ll l ll l l l l l l lllll ll l lll l l l l l l llll lll l l l l l l ll llll lll l l l l l l l l lll l l l l l l llllllllll l l l l l l llll l l l ll l l l l l l lllllllllllllllllll l l l l l l lllllllllll llll l l l l l l lllllllll lllll l l l l l l lll ll lll ll l l l l ll l l l l l l llll l l l l lll l l l l l l llllllllllllllllll l l l l l l l lll l ll ll l llllll l ll l l l l l l l l lllll lll l l l l l l lll l l l l l l llllllllllllllllll l l l l l l l llllllllllllllll l l l l l l ll lllll l l l l l l lllllllllllll llll l l l l l l lll ll l ll l l l l l l llllll ll lll l l l l l l lll l ll l l l l l l lll l ll l l l l l l l l llll l l l l l l lllllllll llll l l l l l l ll l lll llll l l l l l l llllll l lll l l l l l l l llll l ll l l l l l l llll llllllll l l l l l l l llllllllllllllllll l l l l l l llllllllllllll l l l l ll l l l l l l ll l ll l l l l l l l lllllllllllllllll l l l l l l l lllllllllllll l l l l l l ll ll ll l l l l l l llllllllll ll l l l l l l lllllllllllllllll l l l l l l l llllll lllll l l l l l l l lllllllllllllll l l l l l l l ll l l ll l l l l l l lllllllllll llll l l l l l l l llllllllllllllll l l l l l l l llllllllll l l lllll l l l l l l llllllll lll l l l l l l l lll l ll l l l l l l llllll l l l l l l l ll llllll l l l l l l llllllllllll ll l l l l l l ll llll l ll l l l l l l llllll llllllll l l l l l l lllllllllllllllllll l l l l l l ll l lll ll l l l l l l l llllllllllllllll l l l l l l llll llllllllll l l l l l l l lllll lllllll l l l l l l l lllllllllllll ll l l l l l l l lllllllll lllllll l l l l l l llllllllllllll ll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllll ll ll l l l l l ll l l l l l l llllll ll l l l l l llllllllllllllllll l l l l l l llll lllllllllllll l l lll l l l l l l l llll l l llllllll l l l l l l l lllllllllllllllll l ll l l l l ll l l l l l ll l l l l l ll l l l l l lll l l l l l l lllllllllllllllllll l l l l ll ll l l l l l l lllllll llllllll l l l l l ll l l l l l l lllllllllllllllllll l l l l l l l lllllllllllllllll l l l l l l l llll llllll llll l l l l l l lllllllllllllll l l l lll lll lll l l l l l l l ll ll lll l l llllllllllllllllll l l l l l l lllllllllllll l l l l l l l l lll l l l l l l llllllllll lllll l l ll l l l l l l l ll l l l l l ll l l l l l l llllll lll l l l l l l llllllllllllllll l l l l l ll l l l l ll l l l l l l ll l l llllll l l l l l l l llllllllllllllll l l l l l l l l l ll ll l l l l l l lllllllllllllllllll l l l l l l lll lllll l l l l l l lllll l lll l l l l l l l ll l l l l l l llllllllllll ll l l l l l l lllllllllllllllllll l l l l l ll l l l l l l lllllll l llll ll l l l l l l ll ll l l l l l l lllllllll l l l l l lllll ll l l l l l l lllllllll lll lll l l l l l l l llllll l l l l l l llll ll l l l l l l lllllllll llll l l l l l l lll ll l l ll l l l l l l l lllllllllllllll l l l l l l llllllll ll l l l l l l ll ll l l ll l l l l l l l ll l l l l ll l l l l l l llllllllllllllllll l l ll l l l l l l l ll ll lllllll l l l l l l ll ll l l l l l l llllllllllllllllll ll l l l l l l llll l lll l l l l l l ll l ll l llll l l l l l l llllllll lll l ll l l l l l l lll l l ll l l l l l l lllllllll lllll l l l l l lllll llllllllll l l l l l l llllll l llllll l l l l llllll lllllllllll l l l l l lll l l l l l l lll l lll l l l l l l ll l ll l l l l l l l ll l l l l l l l ll ll ll l lll l l l l l ll ll l l ll ll ll l l l l l l l l ll l l l l l l lll ll l l l l l l l l l ll l l l l l l ll ll l l l l l l l ll ll l l l l l l lllllll llllll l l l l l l ll ll ll l l l l l l llll lll l l l l l l ll ll l l l l l l llllll l ll l l l l l l ll l ll l l l l l l ll l l l l l l lllllllllllllllllll l l l l l l llllllll lllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllll l l l l l l llllllllllll l l l l l l ll l l l l l ll l l l l l l lllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l llllllllllllll lll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllll l l l l l l lllll l ll l l l l l l lllllllllllllllllll l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l llllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l llllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllll l l l l l l llllllllll llllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllll lll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllll lllllllllll l l l l l l lllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllll lllllllll l l l l l l llllllll llllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllll lll ll l l l l l l llllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllll ll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllll l l l l l l lllllllllllllllll l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllll llllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllll l l l l l l llllllllllllll l l l l l l lllllllllllllll l l l l l l lll llllllll llll l l l l l l lllllll ll lllll l l l l l l lllllllllllllllllll l l ll l l l l l l llllll lllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lll l l l l l l lllllllllllllllllll l l l l l l llll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lll lll l l l l l l llllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l llllllllllllll l l l l l l lllllll llllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllll l l l l l lllll ll llll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllll lll l l l l l l lllllllllllllllllll l l l l l l llllllll lllllllll l l l l l l lllllllllllllllllll l l l l l l lllllll lllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllll ll lll ll l l l l l l llllllllllllllllll l l l l l l llllllllllllllll l l l l l l lll l lll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllll ll l l l l l l lllllllllllllll ll l l l l l l lllllllllllllllllll l l l l l l lllllll lll llll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllll l l l l l l lllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllll l l l l l l lllllllllllllllll
12 16 20 ll llll llllllll llllllll llllllllll lllll l llllllllll l l l ll l l lll lllllll lllllllllll l lllllll l lllllllllllllllllllll l lllll llllll lllll l llllll l lllll ll l lllllll l l lllllll l l l l l llllllll llllll ll ll llllllllllllll lll llll ll ll lll lllll llllll lll lll llllllllllllll lllll lllllllll llllllllllll lll llll ll lllllll llllllllll llllllllllll lllllll lllllllllll lll l llllllllllll l llllllllllllllll lll lllllll ll l l lllllllllll l lllllllll l l l l lllll l llllllll llllllll lllll lllllllllllllll ll l lllllllllll llllllllll llll l lllll ll l l ll lll l lllllllll ll lllllllll llllllllllllllll l l l l lllllllllllllll l ll l l l l l llllll l l llllllllllll l lllllll llllll lll lllllll llll l l lll ll lllll l l ll l lll l llllll llll lllllllllllllllll l llll llll l l ll l ll llllll ll l l llll lll l l l l lllllll llll lllllllllll ll l l lllll llllllllllllll llllllll llll lll l ll lll l lllll l llll lll ll ll lll llllllll lllllllllll lll l lll l llllllllllllll l l llll l l l l l l lllllllllllllllllll l llllllllllllll ll lllll llll llllllllll llllllllllllllll ll l lllllllllllllllll llll l lllllllll l l llllllll l l lllllll lllllllll ll l ll lll llllllllllllll l l l l lllllllllllllllllllll llllll lll l l llll lll llll l l llllllllll llll ll l ll ll l lll ll l ll lllllllll llllllllllllllllll l lll lllll llllllllllll l ll lll l lll l l l l ll llllllllllllllllllllllllll ll llllllll ll l l ll llllll ll lllllllllllllll lll lll lllllll lllll llllllllllll lllllllllll llllllllllllllllllll lllllllllll lllllllllllllll l l l l l lllllllllllllllll l l l l l l llll ll l l l l l l ll llllllll l l l l l lll l l l l l ll l ll l ll l l l l l l lllll llll l l l l l l llll lll l l l l l l ll llll lll l l l l l l l l lll l l l l l l llllllllll l l l l l l llll l l l ll l l l l l l lllllllllllllllllll l l l l l l lllllllllll llll l l l l l l llllllllllllll l l l l l l lll ll llll l l l l ll l l l l l l llll l l l l lll l l l l l lllllllllllllllllll l l l l l l llll l llll llllll l ll l l l l l l ll lllll lll l l l l l l lll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllll l l l l l l ll lllll l l l l l l lllllllllllll llll l l l l l l lll ll l ll l l l l l l llllll ll ll l l l l l l lll l ll l l l l l l lll l lll l l l l l l ll ll l l l l l l lllllllll llll l l l l l l ll l lll lll l l l l l l llllll l lll l l l l l l lllll l ll l l l l l l llll llllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllll l l l l ll l l l l l l ll l ll l l l l l l lllllllllllllll ll l l l l l l llllllllllllll l l l l l l ll ll ll l l l l l l llllllllll ll l l l l l l lllllllllllllllll l l l l l l lllllll lllll l l l l l l llllllll lllllll l l l l l l lll l l ll l l l l l l lllllllllll llll l l l l l l lllllllllllllllll l l l l l l lllllllllll l l lllll l l l l l l llllllll lll l l l l l l l lll l ll l l l l l l llllll l l l l l l l ll llllll l l l l l l llllllllllll ll l l l l l l ll llll l ll l l l l l l llllll llllllll l l l l l l lllllllllllllllllll l l l l l l ll l lll ll l l l l l l lllllllllllllllll l l l l l l llll llllllllll l l l l l l llllll lllllll l l l l l l llllllllllllll ll l l l l l l llllllllll lllllll l l l l l l llllllllllllll ll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllll ll ll l l l l l ll l l l l l l lllllll l l l l lllllllllllllllllll l l l l l l llll lllllllllllll l l lll l l l l l l lllll l l llllllll l l l l l l llllllllllllllllll l ll l l l l ll l l l l l ll l l l l l ll l l l l l lll l l l l l l lllllllllllllllllll l l l l ll ll l l l l l l lllllll llllllll l l l l l ll l l l l l l lllllllllllllllllll l l l l l l l lllllllllllllllll l l l l l l lllll llllll llll l l l l l l lllllllllllllll l l l lll lll lll l l l l l l l ll ll ll l l llllllllllllllllll l l l l l l lllllllllllll l l l l l l l l lll l l l l l l llllllllll lllll l l ll l l l l l l l ll l l l l l ll l l l l l l llllll lll l l l l l l llllllllllllllll l l l l ll l l l l ll l l l l l l ll l l llllll l l l l l l lllllllllllllllll l l l l l l l l l ll ll l l l l l l lllllllllllllllllll l l l l l l lll lllll l l l l l l lllll l lll l l l l l l ll l l l l l l llllllllllll ll l l l l l l lllllllllllllllllll l l l l l ll l l l l l l lllllll lllll ll l l l l l l ll ll l l l l l l lllllllll l l l l l lllll l l l l l l llllllllllll lll l l l l l l l llllll l l l l l l llll ll l l l l l l lllllllll llll l l l l l l lll ll l l ll l l l l l l llllllllllllllll l l l l l l lllllllll l l l l l l ll ll l l ll l l l l l l l ll l l l ll l l l l l l llllllllllllllllll l l ll l l l l l l l ll lllllllll l l l l l l ll ll l l l l l l llllllllllllllllll ll l l l l l l llll l lll l l l l l l ll l ll l llll l l l l l l llllllll lll l ll l l l l l l lll l l ll l l l l l l lllllllll lllll l l l l l lllll llllllllll l l l l l l llllll l llllll l l l l llllll lllllllllll l l l l l lll l l l l l l lll l lll l l l l l l ll ll l l l l l l l ll l l l l l l lll ll ll l ll l l l l lll ll l l ll ll ll l l l l l l ll ll l l l l l l lll ll l l l l l l l l l ll l l l l l l ll ll l l l l l l l lll l l l l l l lllllll llllll l l l l l l ll ll ll l l l l l l llll lll l l l l l l ll ll l l l l l l llllll l ll l l l l l l ll l ll l l l l l l ll
Figure 10.
Comparison between
GALFITM single-band ( SM ) fits versus GALFITM multi-band ( MM ) fits, plotted in a similar manner to Fig. 9. We show therecovered S´ersic model parameters as a function of the r -band apparent magnitude as cosmologicaly scaled from the MM fit to the lowest-redshift artificialimage for each galaxy. Columns show results for u , g , r , i , z bands, respectively. Upper panels: difference in magnitude; middle panels: ratio of effectiveradii; lower panels: ratio of S´ersic indices. Values of m [M] − m [M , all] > , r [M]e /r [M , all]e > and n [M] /n [M , all] > , indicate that the magnitude isfainter, effective radius larger and S´ersic index greater, respectively, when measured in the single-band image compared to that obtained using the multi-bandapproach. The solid lines indicate the median, while dashed lines are the 16th- and 84th-percentiles of the distribution, all determined within one magnitudebins. Only parameters for the artificially-redshifted images are included in this figure. The grey points indicate that at least one fit ( SM or MM ) ended in closeproximity to a constraint, and for which the recovered parameters are hence deemed to be unreliable. and especially u ) we see a weaker correspondence between the twomethods. There are no discernible systematic differences betweenthe SM and MM parameters except for the S´ersic index values inthe u and z bands. The systematic differences in the u - and z -bandS´ersic index values will be investigated with the subsequent figures,and discussed further below.Although most galaxies behave similarly and constitute themain body of points, there are occasional exceptions. For exam-ple, an outlying ‘line’ of points can be seen in the magnitude plotsabove ( r -band) and below ( g and i -band) the main sequence ofpoints. This line is caused by one galaxy, NGC 4374, an ellipticalgalaxy known to host a strong active galactic nucleus (Cˆot´e et al.2006). Visually inspecting our NGC 4374 redshifted images revealsa prominent, unresolved nucleus. This extra central flux appears toproduce inconsistent results for SM and MM ; fits using both meth-ods have issues. This serves to highlight that in cases where theassumed profile is a bad match to reality, no method will produce satisfactory results. A more appropriate way to fit this galaxy wouldbe to add a central PSF component in the model.The scatter in this plot is a simple function of object magni-tude; the fainter an object, the greater the scatter between the SM and MM parameters. We quantify this scatter in a robust manner, ashalf of the range between the 16th- and 84th-percentiles (which isequivalent to the standard deviation for a Gaussian distribution), inorder to avoid being overly influenced by a relatively small numberof severe outliers. We also exclude all fits that ended up in closeproximity to any of the constraints in either the single- or multi-band fits. At m [M , all] r, = 18 the typical scatter per band ( u , g , r , i , z ) for m is ( ± . , ± . , ± . , ± . , ± . , for r e is( ± %, ± %, ± %, ± %, ± % ) and for n is ( ± %, ± %, ± %, ± %, ± %).The scatter is worst in the case of the S´ersic index, mainly be-cause this parameter is most sensitive to changes in the details inthe fitting. Magnitudes are integrated quantities, and are thereforegenerally the easiest galaxy properties to extract from imaging. De- c (cid:13) , 1–29 egaMorph II l l l l l llllllllllllllllllll l l l l l lllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l lllllll llll l l l l l llll l l l l l ll l l l l l lllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l lllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l lllllllllllllllllll l l l l l llllllllllllllllllll l l l l l l llll ll l l l l l llllllllllllllllllll l l l l l lllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l lllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l lllllllllllllll l l l l l ll ll lll ll l l l l l llllll lllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllll l l l l l llllllllllllllllllll l l l l llllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllll llll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l lll l llllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l lllllllllllllll l l l l l lllllll l l l l l llllllllllllllllllll l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l lllllll lllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l lllllll l l l l l lllllllllllllll l l l l l llllllllllllllll l l l l l llllllllllllllllllll l l l l l lllllllllllllllllll l l l l l llllllllllllllll l ll l l l l l ll lllllllll l l l l l llllllllllllllllllll l l l l l lllllllllllllllllll l l l l l lllllll ll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l l ll ll l l l l l llllllllllllllllllll l l l l l l l ll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llll l l l l l lllllllllllllllllll l l l l l lllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l lllllllll l l l l l ll ll l l l l l llllllllllllllllllll l l l l l llllllllllllllllll l l l l l lllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l lllllllllllllll l l l l l llllllllllllllllllll l l l l l l llllll l ll l l llll l l l l l llll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l lllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l lllllllllllllllllll l l l l l llllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l lllllllllllllllllll − . − . . . . mag0[−sumr, 3] M [ M ] - M z m i n [ M ] l l l lllllll lllllllll lll llll l ll l l l l l ll l llllllll llll l ll ll lllllll lll ll lllll ll l lll l l l l l l llll llll ll llllll lllllllll ll l llllll lll l ll llll ll lll llllll l l l l ll l l llll lllllllllll ll lll lll ll lll ll lllllll llllll l l lllll llll lll llllll llllll l l l l l lll l ll ll llllllllll l l l l l lll l l lll llll l l l ll l l l l l l l lll llll lll l ll ll l l llll ll l lllllll llllllll ll l l llllll l l llllllllllllll llll lllllll l l lllll llllllllll l l l l lllll llll llllllllll l ll llll l l l l l llllllllllllllll l lll llllll ll l llllllllllll l l l l l lllllll ll l l lllll l l l ll lllllll lll l l l l l l l l l lll lll l llllllllllll l l llllll l l l l ll l lll ll l llll ll ll lll lllll lllll ll llllll l l l l l l llllll l ll l l lllll l ll lllllllllllllll l l l l l llllll l lllllll ll lll llll lllllll l llll lllll lllllllll l lll l l l l ll l l l l ll llllllll lll lllllll lll l llll lll llll lllllllll llllllll ll l llllllllllllllllll ll l ll l l llll ll llllllll lllllllll l l l l l llllllllllllllllllll l l l l l l llll ll l l l l l ll ll lll ll l l l l l llllll lllllllllllll l l l l l llllllllllllllllllll l l l l llllllllll l l l l l llllllllllll llll l l l l l lll l llllllllll l l l l l llllllllllllllllllll l l l l l lllllllllllllll l l l llllllllllllllllllll l l l l l llllllllllllllll l ll l l l l l lllllll ll l l l l l l l ll l l l l l lllll l l l l l ll ll l l l l l l llllll l ll l l llll l l l l l lllllllll u l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l llllllllllll l l l l l l lll l l l l l ll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l llllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l l llllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllll ll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllll l l l l l l lllllllllllllllllll l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllll lllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllll l l l l l l llllllllllllll l l l l l l lllllllllllllll l l l l l l l llllllllllllllllll l l l l l l llllllllllllllllll l l l l l l llllllllllllllllll l l l ll l l l l l l llllllllllll l l l l l l l llllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllll lllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lll l l l l l l lllllllllllllllllll l l l l l l llll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllll l l l l l l llllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l llllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll mag0[−sumr, 3] dab s [ − s u m r , ] ll lll ll llll lllll lll lllllll l l lll lllllll l l llllll llll l ll llllllllllllll l lll lllll llllllllllllll l l llllllll llllll ll l llllll llll llllllll l l l llllll llll lll l g l l l l l l l llllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllll l l l l l l lll l l l l l ll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllll l l l l l l lllllllllllllllllll l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l lllllllllllllll l l l l l l l llllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllll ll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l llllll l l l l l l l llllllllllllllllll l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllll lllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllll l l l l l l l lllllllllllll l l l l l l lllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l llllllllllllllllll l l l ll l l l l l l l lllllllllll l l l l l l l llllllllllllllllll l l l l l l llllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l l llllllll lllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l ll l l l l l l lllllllllllllllllll l l l l l l l lll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllll l l l l l l llllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l llllllllllllllllll mag0[−sumr, 3] dab s [ − s u m r , ] r l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllll l l l l l l lll l l l l l ll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllll l l l l l l lllllllllllllllllll l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllll ll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l lllll l l l l l l lllllllllllllllllll l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllll lllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l lllll l l l l l l l lllllllllllll l l l l l l lllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l l lllllllllllllllll l l l ll l l l l l l llllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllll lllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l ll l l l l l l lllllllllllllllllll l l l l l l llll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllll l l l l l l llllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l ll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l llllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l lllllllllllllllll mag0[−sumr, 3] dab s [ − s u m r , ] lll llll lll l lll lll ll lll l ll lllllll l l l l l l l lllllllll lllllll ll llll ll lll l l ll lll ll lll l ll llllllllllllll llll llllllll lll llllll llll lll l l l l l l l lllllllllllllllll i l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllll l l l l l l lll l l l l l ll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllll l l l l l l lllllllllllllllllll l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllll ll l l l l l l lllllllllllllllllll l l l l ll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllll l l l l l l lllllllllllllllllll l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllll lllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllll l l l l l l llllllllllllll l l l l l l lllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l llllllllllllllllll l l l ll l l l l l l llllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lll l l l l l l lllllllllllllllllll l l l l l l llll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllll l l l l l l llllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllll ll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllll mag0[−sumr, 3] dab s [ − s u m r , ] ll ll l lllllll l lll lll ll lllll ll ll l llllll ll l ll lll llll llllll ll l llll ll llll lll l llllll lllll lll lllllllll l l l l l l lllll l l ll l llll ll llll l l llllllll lllllllll llll llll lllllll llll llll ll lllll l lll lllll l llll l llllllllllllllllll l lll ll llll l ll lllll l l l lll l ll lll l ll ll llllllll ll lllllllll ll l l lllllll l lllllllllllllllll l llllll lllll lll llll ll llll ll ll lllll l lllll lllllllllllll l l l l ll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll z l l l l l lllllllllllllllllll l l l l l llllllllllllllllll l l l l l lllllllllllllllllll l l l l l llllllllllllllllllll l l l l l lllllllllllll l l l l l llll l l l l ll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllll lllllllll l l l l l ll l lllllllllllll l l l l l lllll llllllllllll l l l l l lllllllllllllll ll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l lllll llll lllllll l l l l l llllllllllllllllllll l l l l l ll llllllllll l l l l l llllllllllllllllllll l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l lllllllllllllllllll l l l l l lllllllllllllllll l l l l l lllllllllllllllllll l l l l l llllllllllllllllll l l l l l lllllllllllllllllll l l l l l llllll lllll llllll l l l l l lllllllllllllllllll l l l l l lll llll llllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllll l l l l l lll llllllllllllllll l l l l l lllllll llllll ll l l l l l llllllllllllllllllll l l l l l lllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l l llllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l lllllllllllllllllll l l l l l llllllllllllllllllll l l l l l lllllll llllll l l l l l llllllll lllllllll l l l l l lllllllllllllllllll l l l l l llllllllllllllllll l l l l l llllllllllllllllllll l l l l l lllllllllllllllllll l l l l l lllllllllllllllllll l l l l l lllllllllllllllllll l l l l l llllllllllllllllllll l l l l l lllllllllllllllll l l l l l llllllllllllllllllll l l l l l lllllll llllllllllll l l l l l lllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l lllllllllllllllllll l l l l l llllllllllllllllllll l l l l l l llll l llllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l lllllllllllllllll l l l l l lllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l lllllllllllllllllll l l l l l lll lll l l l l l lllllllllllllllllll l l l ll l lllllllllllll l l l l l lll l llll l l l l l lllll l lll ll l l l l l l lllll llll l l l l l llllllllllllllllllll l l l l lll l l l l l l lll ll l l l l l lllllll llllllll l l l l l lll llllllllllll l l l l l llllllllllllllll l l l l l lllllllllllllllllll l l l ll l l l l l llllllllllll l l l l l l llllllllllllllll l l l l l lllll llllllllllll l l l l l llllllllllllllllllll l l l l l l lllllllllllllllll l l l l l llllllllllllllllllll l l l l l lllllllllllllll l l l l l lllllllllllllllllll l l l l l lllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llll l l l l l llllllllllllllllllll l l l l l ll ll l l l l l lllllllllllllllllll l l l l l lllllllllll l lll l l l l llllllll l l l l l lllllll l l l l l lllllllllllllllllll l l l l l lllllllllllllllllll l l l l l llll lllllllllllllll l l l l l llllllllllllllll l l l l l llllllllllllllllllll l l l l l lll lllll l l l l l lllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l ll l l l l l lllll llllll llll l l l l l lllllllllllll ll l l l l l lllllllllllllllllll l l l ll lllll l l l l l llllllllllllllll l l l l l lllllllllllllllllll l l l l l llllll l llllllllll l l l l l lll lllllllllllllll l l l l l llllllllllllllllll l l l l l lllllllllllllll ll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l lllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l lllllllllllllllllll l l l l l lllllllllllllllllll l l l l l lllllllllllllllllll l l l l l lllll lllllllllllll l l l l l llllll llll llllll l l l l l lllllllllllllllllll l l l l l lllllllllll lllllll l l l l l llllllllllllllllllll l l l lllllll llllllllllll l l l l l lllllllllll lll l l l l l lllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llll lllll ll l l l l l llllllllllllllllllll l l l l l lllllllllllllllllll l l l l l lllllllllllllllllll l l l l l lllllllllllllllllll l l l l l llllllllllllllllllll l l l l l lllllllllllllllllll l l l l l lllllllllllllllll l l l l l lllllllllllllllllll l l l l l llll l llllllllllll l l l l l lllllllllllllllllll l l l l l lllllllllllllllllll l l l l l lllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llll lllllllllllll . . . . . mag0[−sumr, 3] r e [ M ] r e , z m i n [ M ] l l ll llll lllllllllllll llll ll l l l l l ll l llll lll llll llll ll l l l l lll l l l l l l llll ll ll llllllll llllll l l lllll l l l l ll l l ll llll ll lll ll ll l lllllll llllll l l llll llllll llllll lllll l l l l l lll l ll l llllll l l l l l l l l ll lll l l l ll l l l l l l l lll llll ll l l ll l l ll l l ll ll ll l l l lllll ll llllll lll ll l l l lllll l lllllllll l lll l l l l l lllllllllllllllllll l l llll llll l ll llllllll l l l l l lllllllllll llll l l l ll l l l ll ll lllll lllllllllll l l l l l lllllllllllllllllll l llll lll l ll lll l lllll l l l l ll ll l ll l lllll ll l l l l l llll l l l l l l llllll l ll l l l llll l llll ll l l l ll l l llll llll l llll llll lllllllll l ll l l l l ll llllllllll l l llllll ll l llll lll llll llllllll llllllllllllll ll l llll llllll llllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l lllllll llllllllllll l l l l l llllllllllllllllllll l l l l l lllllllllllllllll l l l l l llllllllllllllllllll l l l l l lllllllllllllllllll l l l ll l lllllllllllll l l l l l lllllllllllllllllll l l l ll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l ll ll l l l l l lllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l ll llll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllll lllll l l l l l lll l l l l l ll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllll l l l l l l l llllllllllll lll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllll ll l l l l l l llllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l llllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllll llllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l l llllllllllllll l l l l l l l llllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllll ll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l llllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lll ll l l l l l l lllllllllllllllllll l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllll llllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l llllll l l l l l l lllllll l ll l l l l l l lllllllllllllll l l l l l l l lll l lllll l l l l l l llllllllllllllllll l l l l l l llllllllllllllllll l l ll l l l l l l llllllllllll l l l l l l l llllllllllllllllll l l l l l l llll lllllll l l l l l l lllllllllllllllllll l l l l l l l lllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllll lllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllll lll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lll l l l l l l lllll llllll l l l l l l llll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllll l l l l l l llllll l l l l l l l llllllllllllllllll l l l l l l llllllllllllll lll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l llllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllll ll l l l l l l lllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l ll l ll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllll llllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l lllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll mag0[−sumr, 3] r . r a t i o [ − s u m r , ] ll lllllll l ll llllll l l lll lll l l lll l ll llll ll llll ll l l llll l lllll lll lll l l l l l l l l llllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllll llll l l l l l l lll l l l l l ll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllll l ll l l l l l l lllllllllllllllllll l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l lllllllllllllll l l l l l l l llllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllll llll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l lllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllll ll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l l lllllll lllllllll l l l l l l llllll l l l l l l l llllllllllllllllll l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllll lllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllll l l l l l l l llllll ll l l l l l l lllllllllllllll l l l l l l lllllllllllllllll l l l l l l llllllllllllllllll l l l l l l llllllllllllllllll l l l ll l l l l l l l lllllllllll l l l l l l l llllllllllllllllll l l l l l l llllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l l llllllll lllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l ll l l l l l l lllllllllllllllllll l l l l l l l lll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllll l l l l l l lll ll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllll l l l l l l lllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llll lllll ll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllll lllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllll llllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllll l l l l l l llllllllllll ll mag0[−sumr, 3] r . r a t i o [ − s u m r , ] l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllll ll lll l l l l l ll l l l l l ll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllll l llll l l l l l l lllllllllllllllllll l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllll llllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllll ll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l lll l l l l l l lllllllllllllllllll l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllll llllllllll l l l l l l llllllll l l l l l l lllllllllllllllllll l l l l l l l lllll l l l l l l l llllllll ll l l l l l l lllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l l lllllllllllllllll l l ll l l l l l l lllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllll llllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllll lllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l ll l l l l l l lllllllllllllllllll l l l l l l llll l l l l l l llllll l lllllllll l l l l l l lllllllllllllllllll l l l l l l lllll ll l l l l l l llll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l l lllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l ll l l l l l l lllllllllllllllllll l l l l l l llllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllll l lll l l l l l l l llllllllllllllllll l l l l l l llllllllllllllllll l l l l l l l lllllll lllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llll llll lllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllll lllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllll l mag0[−sumr, 3] r . r a t i o [ − s u m r , ] ll llll lllll l l llllll l l l l l l llllll ll llll lll l l ll ll lll ll lll l l llll llll ll l l l l l l lllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llll lllllllll l l l l l l lllllll llll l l l l l l lll l l l l l ll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllll lll llll l l l l l l lllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllll l l l l l l lllllllllllllllllll l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllll lllllllllllll l l l l l l llllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllll llll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllll llll l l l l l l lllllllllllllllllll l l l l l l llll ll llllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllll l l l l l l llllllll llllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllll ll l l l l l l llllllllllllll ll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l ll lll l l l l l l llllllllllllllllll l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllll lllllllllll l l l l l l lllll llllllllllll l l l l l l lllllllllllllllllll l l l l l l ll lll l l l l l l llll llllllll l l l l l l llllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllll l l l l l l llllllllllllllllll l l ll l l l l l l lllll llllll l l l l l l llll lllllllllllll l l l l l l llllllllll llll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllll l l l l l l llllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lll l l l l l l lllllllllllllllllll l l l l l l llllllll lllllllll l l l l l l lllllllllllllllllll l l l l l l lll l l l l l l lllllllllllllllllll l l l l l l llll l l l l l l lllllll ll llllll l l l l l l lllllllllllllllllll l l l l l l llllllll l l l l l l llllll l l l l l l llllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l llllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllll l l l l l l lllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllll llllllllllll l l l l l l llllllllllllllllll l l l l l l llllllll l llllll l l l l l l lllllllllllllllllll l l l l l l lllllll lllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllll llllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllll lllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l lllllllllllllllllll l l l l l l llllllllllllll lll l l l l l l lllllllllllllllllll l l l l l l l ll l llllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllll llll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lll lllllllllllllll l l l l l l lll llllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l llllllllllllllllll l l l l l l llllllllll llllll mag0[−sumr, 3] r . r a t i o [ − s u m r , ] ll lllll l ll l ll l ll l lllll ll l l lll llllll ll l lll ll llll lll llll l l l l l lll l llll l l ll l ll ll llll l lllllll lllllllll llll llll llllll lll ll ll l lll lll ll l ll llll l ll ll l l lll l lll l ll ll llllll llllll l l llllllll llllll llllll lllll lllll lll llll lllll ll lll llll llll lllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l llllllllllllllllllll l l l l l lllllllllllllllllll l l l l l llllllllllllllllllll l l l l l lll llllllllllllll l l l l l lllllllllllll l l l llll l l l l l ll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllll lllllllll l l l l l ll l llllllllllllll l l l l l llllllllllllllllllll l l l l l lllllllllllllll ll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l lllll llll llllll l l l l l llllllllllllllll l l l l l ll llllllllll l l l l l llllllllllllllllllll l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l lll llllllllllllllll l l l l l llllllllllllllllllll l l l l l lllllllllllllllllll l l l l l llllllllllllllllllll l l l l l lllllllllllllllllll l l l l l lllllllllllllllll l l l l l llllllllllllllllll l l l l l llllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllll lllllllllllll l l l l l lllllllllllllll l l l l l l l llllllll l l l l l llllllllllllllllllll l l l l l lllllllllllllllllll l l l l l lll lllllllllllllll l l l l l lllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l lllllllllllllllllll l l l l l llllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllll l l l l l llllllllllllllllllll l l l l l lllllllllllllllllll l l l l l llllllllllllllllllll l l l l l lllllllllllllllllll l l l l l llllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l lllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllll ll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l lllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l lllllll l l l l l llllllllllllllllllll l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l lllllll lllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l lllllll l l l l l ll l lllllllll l l l l lllllllllllllll l l l l l llllllllllllllllllll l l l l l lllllllllllllllll l l l l l lllllllllllllllllll l l l ll l l l l l lllllllllllll l l l l l llllllllllllllllllll l l l l l lllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l 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l llllllllllllllllll l l l l l lllllllllllllllllll l l l l l llllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l lllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l lllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l lllllllllllllllllll l l l l l llll ll lllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllll ll l l l llllllllllllllllllll l l l l l llllllllllll llll l l l l l lllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllll lllllllllllll l l l l l llll lllllllllllll l l l l l llll l lllllllll l l l l l lllllllllllllllllll l l l l l llllllllllllllll l l l l l llllllllllllllllll l l l l l lllllllllllllllllll l l l l l lllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llll l llllllllll l l l l l llllllllllllllllllll l l l l l llllll llllllllll l l l l l lllllllllllllll lll 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12 16 20 . . . . . n [ M ] n z m i n [ M ] l l l lllllll lllllllllllllllll l ll l l l l l ll l llll lll ll ll lllll ll ll l l l l lll l l l l l l llll ll ll llllllllllllll l ll llll ll lll lllllll l l l l ll l l lll llllllllll ll ll lll ll lll ll lllllll l llll l l lllll llll ll llllll llllll l l l l l lll l ll ll llllllllll l l l l l lll l l lll llll l l l ll l l l l l l l llll llll lll l ll ll l l llll ll l lllllll llllllll ll l l llllll l l llllllllllllll llll llllll l l llll llllllll l l l lllll llll llllllllll l ll lll l l l l l lllllllllllllllllll l l llll llllll ll l llllllllllll l l l l l lllllllllll llll l l lllll l l l ll lllllll lllll lllllllllll l l l l l lllllllllllllllllll l lll ll l llllllllllll l l lllll l l l l ll ll l lll ll l lllll ll ll lll lllll lllll lllll l l l l l l llllll l ll l l l llll l ll ll ll lll l l l l l llllll l llllll lllll l ll llll lllllllll l lll l l l l ll l llllllllll l l llllllll ll l lll lll l llll llllll llllllll l ll lll llllll ll l l lll 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l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllll ll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l ll llllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l llllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllll lllll l l l l l l llllllllll lll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l l llllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllll ll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllll l l l l l l lllllllllllllllllll l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllll lllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllll l l l l l l lllllll l ll l l l l l l lllllllllllllll l l l l l l l llllllllllllllllll l l l l l l llllllllllllllllll l l l l l l llllllllllllllllll l l ll l l l l l l llllllllllll l l l l l l l llllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllll lllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lll l l l l l l lllll lllllllll l l l l l l llll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllll l l l l l l llllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l llllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllll ll l l l l l l lllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l lllllllllllllllll l l l l l l l lllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll
12 16 20 n . r a t i o [ − s u m r , ] lll llll ll llllll l l ll lllllll l l llllll llll l llllllllllll l ll lllll llllllllllllll l lllll llllll ll l ll llll llllllll l l l lllll lll lll l l l l l l l l llllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllll llll l l l l l l ll l l l l l ll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllll ll l l l l l l lllllllllllllllllll l l l l l lllllllllllllllllll l l l l l l l lllllllllllll lll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l lllllllllllllll l l l l l l l llllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l lllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllll ll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l llllll l l l l l l l llllllllllllllllll l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllll lllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllll l l l l l l l llllll lllll l l l l l l lllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l llllllllllllllllll l l l ll l l l l l l l lllllllllll l l l l l l l llllllllllllllllll l l l l l l llllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l l llllllll lllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l ll l l l l l l llllll lllllllllll l l l l l l l lll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllll l l l l l l llllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllll l l l l l l lllllllllllllllll l l l l l l lllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllll lllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllll l l l l l l llllllllllll lll
12 16 20m r , (mag) n . r a t i o [ − s u m r , ] l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllll l l l l l ll l l l l l ll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllll llll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllll l llll l l l l l l lllllllllllllllllll l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllll lll lllll l l l l l l lllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllll llllllll l l l l l l lllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllll ll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l lllll l l l l l l lllllllllllllllllll l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllll llllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l lllll l l l l l l l lllllllll ll l l l l l l lllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l l lllllllllllllllll l l ll l l l l l l llllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllll lllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l ll l l l l l l llllllllllllll lll l l l l l l llll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllll ll l l l l l l llllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllll l l l l l l lllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l ll l l l l l l lllllllllllllllllll l l l l l l llllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l llllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l lllllllllllllllll
12 16 20 n . r a t i o [ − s u m r , ] lll llll lllll l ll llllll l l l l l l l lllllllll lllllll ll llll lll l l lllll lll ll lll ll ll llllllll lll llllll llll lll l l l l l l l lllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llll lllllllll l l l l l l llllllllllll l l l l l ll l l l l l ll l l l l l l lllllllllllllllll l l l l l l llllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllll lllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllll lllll l l l l l l lllllllllllllllllll l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllll llllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l llllllll lllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllll ll l l l l l l lllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllll l l l l l l lllllllllllllllllll l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllll llllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllll l l l l l l lll lllllllll l l l l l l lllll lllllllll l l l l l l lllllllllllllllllll l l l l l l lllllll lllllllll l l l l l l llllllllllllllllll l l l ll l l l l l l llllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lll l l l l l l lllllllllllllllllll l l l l l l llll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lll llll l l l l l l llllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l llllllllllllll l l l l l l lllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllll l llllll l l l l l l lllllllllllllllllll l l l l l l lllllll lllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllll l l l l l l lll l llllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllll llllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllll l l l l l l lllllllllllllllll
12 16 20 n . r a t i o [ − s u m r , ] ll llllllll llllll lll ll llllll l ll lll lll lllll ll l llll ll llll lll lllll llll lll lllllll l l l l l lll l lllll l l ll l llll ll llll l l llllllll lllllllll lllllll lllllll lll llll ll lllll l lll lllll l llll l llllllllllllllllll l lll ll llll l ll lllll l lll l ll lll l ll ll llllll ll llllll l l lllllllllll l lllllllllllllllll l ll lllllll ll llll lllll ll lll l lllll lll llllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll Figure 11.
Parameter deviations for single-band
GALFITM fits ( SM ) relative to the lowest redshift ( z min ) fit, plotted as a function of the r -band apparentmagnitude, as cosmologically scaled from the SM fit to the lowest-redshift artificial image for each galaxy. Values of M [M] − M [M]zmin > , r [M]e /r [M]e , zmin > and n [M] /n [M]zmin > , indicate that the magnitude is fainter, effective radius larger and S´ersic index greater, respectively, when measured in the higher-redshiftimage compared to that obtained from the z min image. The black points denote the recovered parameters for the 3863 artificially-redshifted galaxies. Theblue points denote the galaxies that r e ,x, zmin [M] /r [M]e ,r, zmin > where x represents the band in consideration. The number of ‘blue’ outliers is 464, 0, 25 and100 for the u, g, i, z bands, respectively. The red points are additional outliers selected with the r [M]e ,x /r [M]e ,r > or r [M]e ,x /r [M]e ,r < . criteria (see text formore details). The number of ‘red’ outliers not included in the ‘blue’ sample are 596, 144, 99, 323. The solid red lines indicate the median, while dashed redlines are the 16th- and 84th-percentiles of the distribution, all determined within one magnitude bins. The magnitudes and effective radii obtained from theredshifted images have been cosmologically converted to their absolute values at the true redshift of each galaxy for the purpose of these comparisons. termining the effective radius may be thought of as finding a radiusfor which the integrated flux within and without are equal. The be-haviour of the profile within each of those two regions does notmatter, and there is no ambiguity in the definition for any arbitrarymonotonic profile. However, determining S´ersic index requires fur-ther information, particularly regarding the behaviour of the profileat its peak and in its tail. Higher S´ersic indices imply a greater shiftof flux from around the half-light radius and into both the centreand outskirts of the profile, but these pieces of information need notbe consistent for an arbitrary monotonic profile. The S´ersic index istherefore more difficult to measure accurately and is more suscep-tible to changes in signal-to-noise and resolution. Furthermore, aswe have seen in Figs. 3 and 4, S´ersic index is typically a strongerfunction of wavelength than effective radius. It may be expectedto show more significant changes between single- and multi-bandmethods. The origin of the scatter in Fig. 10 is obviously a result of thesingle- and multi-band fitting methods measuring different values.However, from this figure it is not clear if one or both methodsintroduce the scatter, i.e, whether one of the methods is superior.To further investigate the advantages of multi-band versus single-band fitting, we test the stability of the structural parameters as thegalaxy becomes fainter and less resolved, as a result of the artificialredshifting. We have already shown examples, in Figs. 4, 5 and 6,for individual galaxies where the increased stability of multi-bandfitting is apparent. Now, in Figs. 11, 12 and 13 we demonstratethat this improvement is true for the whole sample of artificially-redshifted galaxies.In these figures, we show the deviations of the fit parame-ters relative to the values obtained from the image with the low-est artificial-redshift ( z min ). We choose to compare to this image,rather than the original, as the extreme change in resolution means c (cid:13) , 1–29 Vika et al. l l l l l llllllllllllllllllll l l l l l lllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l lllllllllllll l l l l l llll l l l l l ll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l lllllllllllll l l l l l llllllllllllllllllll l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l lllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l lllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l lllllllllllll lllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l lllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l lllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllll ll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l lllllll l l l l l llllllllllllllllllll l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l ll llll lllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l lllllll l l l l l llllllllllllll l l l l l llllllllllllllll l l l l l llllllllllllllllllll l l l l l lllllllllllllllllll l l l l l llllllllllllllllllll l ll l l l l l ll llllllllll l l l l l llllllllllllllllllll l l l l l lllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllll l l l l l llllllllllllllllllll l l l l l lllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llll l l l l l llllllllllllllllllll l l l l l lllll l l l l l lllllll llllllllllll l l l l l lllllllllllllllllll l l l l l lllllllll l l l l l lllllll l l l l l llllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllll llllllll l l l l l lllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l lllllllllllllllllll l l l l l lllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l lllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l lllllllllllllllllll − . − . . . . cmag0[−sumcr, 3] M [ M , a ll ] - M z m i n [ M , a ll ] ll llllll ll l llll llll ll lllll lll lll ll l l llllll u l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllll l l l l l l lll l l l l l ll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllll l l l l l l lllllllllllllllllll l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllll ll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l lllll l l l l l l lllllllllllllllllll l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllll lllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllll l l l l l l llllllllllllll l l l l l l lllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l lllllllllllllllll l l l l l l l llllllllllllllllll ll l l l l l l llllllllllll l l l l l l l llllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllll lllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l l ll l l l l l l l llllllllllllllllll l l l l l l llll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllll l l l l l l llllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l l lllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l l ll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l lllllllllllllllll cmag0[−sumcr, 3] c dab s [ − s u m c r , ] g l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l l lllllllllll l l l l l l l ll l l l l l ll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l llllllllllll l l l l l l lllllllllllllllllll l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllll ll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l llllll l l l l l l l llllllllllllllllll l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllll lllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l lllll l l l l l l llllllllllllll l l l l l l l llllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll ll l l l l l l l lllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l l llllllll lllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l ll l l l l l l lllllllllllllllllll l l l l l l llll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l llllllll l l l l l l llllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l llllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll cmag0[−sumcr, 3] c dab s [ − s u m c r , ] r l l l l l l l llllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllll l l l l l l lll l l l l l ll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l l lllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l lllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllll l l l l l l l llllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllll ll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllll l l l l l l lllllllllllllllllll l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllll lllllllllll l l l l l l l llllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l llllll l l l l l l llllllllllllll l l l l l l lllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll ll l l l l l l l lllllllllll l l l l l l lllllllllllllllllll l l l l l l l lllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllll lllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lll l l l l l l lllllllllllllllllll l l l l l l l lll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllll l l l l l l l lllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll cmag0[−sumcr, 3] c dab s [ − s u m c r , ] i l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllll l l l l l l lll l l l l l ll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllll l l l l l l lllllllllllllllllll l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllll ll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllll l l l l l l lllllllllllllllllll l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllll lllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllll l l l l l l llllllllllllll l l l l l l lllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l ll l l l l l l llllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lll l l l l l l lllllllllllllllllll l l l l l l llll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllll l l l l l l llllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllll cmag0[−sumcr, 3] c dab s [ − s u m c r , ] l lllllllll lll llll l l llll llll lllll ll llll l z l l l l l llllllllllllllllllll l l l l l lllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllll lllll l l l l l llll l l l l l ll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l lllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllll l l l l l llllllllllllllllllll l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l lllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l lllllllllllllllllll l l l l l lllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l lllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l lllllll l l l l l llllllllllllllllllll l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l lllllll llllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l lllllll l l l l l lllllllllll ll l l l l l llllllllllllllll l l l l l llllllllllllllllllll l l l l l lllllllllllllllllll l l l l l llllllllllllllllllll l l ll l l l l l lllllllllllll l l l l l llllllllllllllllllll l l l l l lllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l lllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llll l l l l l llllllllllllllllllll l l l l l lllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l lllllllll l l l l l lllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l lllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllll l l l l l llllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l lllll l lll llll l l l l l llllllllllllllllllll l l l l l lllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llll lllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l lllllllllllllllllll . . . . . cmag0[−sumcr, 3] r e [ M , a ll ] r e , z m i n [ M , a ll ] ll llllll ll l l llll llll ll lllll lll lll ll l l llllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllll lllll l l l l l l lll l l l l l ll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllll llll l l l l l l lllllllllllllllllll l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l lllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllll ll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l lllll l l l l l l lllllllllllllllllll l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllll lllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllll l l l l l l lllllllll ll l l l l l l lllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l lllllllllllllllll l l l l l l l llllllllllllllllll l l ll l l l l l l llllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllll lllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l l ll l l l l l l l llllllllllllllllll l l l l l l llll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllll l l l l l l llllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l l lllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l l ll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l l llllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l lll l l ll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllll cmag0[−sumcr, 3] c r . r a t i o [ − s u m c r , ] l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l l lllll lllll l l l l l l l ll l l l l l ll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllll llll l l l l l l lllllllllllllllllll l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllll ll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l llllll l l l l l l l llllllllllllllllll l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllll lllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l lllll l l l l l l lllllllll lll l l l l l l l llllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l ll l l l l l l l lllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l l llllllll lllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l ll l l l l l l lllllllllllllllllll l l l l l l llll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l llllllll l l l l l l llllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllll l lll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllll lllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllll cmag0[−sumcr, 3] c r . r a t i o [ − s u m c r , ] l l l l l l l llllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllll l l l l l l lll l l l l l ll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l l lllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l lllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllll l l l l l l l llllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllll ll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllll l l l l l l lllllllllllllllllll l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllll lllllllllll l l l l l l l llllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l llllll l l l l l l lllllllll lll l l l l l l lllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l ll l l l l l l l lllllllllll l l l l l l lllllllllllllllllll l l l l l l l lllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllll lllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lll l l l l l l lllllllllllllllllll l l l l l l l lll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllll l l l l l l l lllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllll l lll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllll llllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllll cmag0[−sumcr, 3] c r . r a t i o [ − s u m c r , ] l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllll l l l l l l lll l l l l ll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllll l l l l l l lllllllllllllllllll l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllll ll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllll l l l l l l lllllllllllllllllll l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllll lllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllll l l l l l l lllllllllll l l l l l l lllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l llllllllllllllllllll l l l l l l llllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lll l l l l l l lllllllllllllllllll l l l l l l llll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllll l l l l l l lll ll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllll lll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllll cmag0[−sumcr, 3] c r . r a t i o [ − s u m c r , ] l lllllllll lll llllll l l llll llll lllll ll ll llll ll l l l l llllllllllllllllllll l l l l l lllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllll lllll l l l l l l ll l l l l l ll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l lllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l lllllllllllll l l l l l llllllllllllllllllll l l l l llllllllllllllllllll l l l l l lllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l lllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l lllllllllllllllllll l l l l l llll llllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l lllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l lllllll l l l l l llllllllllllllllllll l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l lllllll lllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l lllllll l l l l l llllllllllllll l l l l l llllllllllllllll l l l l l llllllllllllllllllll l l l l l lllllllllllllllllll l l l l l llllllllllllllllllll l l ll l l l l l lllllllllllll l l l l l llllllllllllllllllll l l l l l lllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l lllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llll l l l l l lllllllllllllllllll l l l l l lllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l lllllllll l l l l l lllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l lllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llll l l l l l llllllllllllllllllll l l l l l lllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l lllllllllllllllllll l l l l l llllllllllllllllllll l l l l l 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12 16 20 . . . . . n [ M , a ll ] n z m i n [ M , a ll ] ll llll ll l l llll llll ll lllll lll lll ll l l llllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllll l l l l l l lll l l l l l ll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllll l l l l l l lllllllllllllllllll l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l lllllllllllllllll l l l l l l lllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllll ll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l lllll l l l l l l lllllllllllllllllll l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllll lllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllll l l l l l l llllllllllllll l l l l l l lllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l lllllllllllllllll l l l l l l l llllllllllllllllll l l ll l l l l l l llllllllllll l l l l l l l llllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllll lllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l l ll l l l l l l l llllllllllllllllll l l l l l l llll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllll l l l l l l llllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l l lllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l l ll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lll lllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l lllllllllllllllll
12 16 20 c n . r a t i o [ − s u m c r , ] l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l l lllllllllll l l l l l l l ll l l l l l ll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l llllllllllll l l l l l l lllllllllllllllllll l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l llllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllll ll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l llllll l l l l l l l llllllllllllllllll l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllll lllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l lllll l l l l l l llllllllllllll l l l l l l l llllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l ll l l l l l l l lllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l l llllllll lllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l ll l l l l l l lllllllllllllllllll l l l l l l llll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l llllllll l l l l l l llllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lll l l l l l l lllllllllllllllllll l l l l l l l lllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lll lllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll
12 16 20m r , (mag) c n . r a t i o [ − s u m c r , ] l l l l l l l llllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllll l l l l l l lll l l l l l ll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l l lllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l lllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l lllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllll l l l l l l l llllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllll ll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllll l l l l l l lllllllllllllllllll l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllll lllllllllll l l l l l l l llllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l llllll l l l l l l lllllllll llll l l l l l l lllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l ll l l l l l l l lllllllllll l l l l l l lllllllllllllllllll l l l l l l l lllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllll lllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lll l l l l l l lllllllllllllllllll l l l l l l l lll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllll l l l l l l l lllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllll l l l l l l lllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll
12 16 20 c n . r a t i o [ − s u m c r , ] l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllll l l l l l l lll l l l l l ll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllll l l l l l l lllllllllllllllllll l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllll ll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllll l l l l l l lllllllllllllllllll l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllll lllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllll l l l l l l lllllllllllll l l l l l l lllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l ll l l l l l l llllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lll l l l l l l lllllllllllllllllll l l l l l l llll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllll l l l l l l llllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllll l l l l l l lllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l 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12 16 20 c n . r a t i o [ − s u m c r , ] l lllllllll lll llllll l l llll llll lllll ll ll llll l Figure 12.
In a similar manner to Fig. 11, this figure shows parameter deviations for multi-band
GALFITM fits ( MM ) relative to the lowest redshift ( z min ) fit,plotted as a function of the r -band apparent magnitude, as cosmologically scaled from the MM fit to the lowest-redshift artificial image for each galaxy. Theblack points denote the recovered parameters for the 3863 artificially-redshifted galaxies. The red points have been selected in the same way as Fig. 11, butusing MM data in this case. The number of the red outliers is 43, 0, 0 and 43 for u, g, i, z bands respectively. We do not find any blue outliers using MM data. Thesolid green lines indicate the median, while dashed green lines are the 16th- and 84th-percentiles of the distribution, all determined within one magnitude bins.To more easily compare to the SM fits, we include the lines from Fig. 11 in red. For consistency, we calculate the green lines using exactly the same galaxiesas were used in determining the red lines. The magnitudes and effective radii obtained from the redshifted images have been cosmologically converted to theirabsolute values at the true redshift of each galaxy for the purpose of these comparisons. that some of the nearest galaxies show offsets between parame-ters from the original and z min images. We wish to avoid includingthis additional scatter in these plots, and instead choose to focus onchanges over redshift ranges more typical for galaxy surveys (i.e. z > . ). Obviously, the apparent magnitude and angular sizeof galaxies will change as we measure them in successively moredistant artificially-redshifted images. To avoid this known cosmo-logical trend from complicating our interpretation, in these figureswe use absolute magnitude and physical size, in kpc, determinedusing the same cosmology used by FERENGI to create the artificialimages.Firstly, in Fig. 11 we examine the ability of the single-bandmethod ( SM ) to recover the redshifted parameters. The first rowof panels shows the offsets in absolute magnitude measured in allthe artificially-redshifted images, relative to the value obtained forthe lowest artificial redshift ( z min ) for each galaxy. Similarly, inthe second row we show the ratio of the effective radius relative tothat for the z min image, and finally, the third row shows the same for S´ersic index. The data in each panel is plotted against r -bandapparent magnitude ( m [M] r, ), cosmologically scaled from the valuefor the SM fit to the z min image for each galaxy, as also used inFig. 9. The columns show the parameter deviations for each of the u , g , r , i , z bands. We overplot the median and the 16th- and 84th-percentiles as a function of magnitude to guide the eye and helpidentify systematic trends, while reducing the influence of severeoutliers. In determining these lines, we exclude all fits that endedup in close proximity to any of the constraints in either the single-or multi-band fits (these points are not shown). For reference, thefraction of galaxies for which the single-band fit failed or ended onconstraints is 20%, 6%, 4%, 5%, 13% for u , g , r , i , z , respectively.Coloured outlying points, selected as described below, are also ex-cluded when determining the trend lines, such that they representthe behaviour of the main bulk of fits.We investigated the origin of the outlying points in these plots,most obvious in the r e , u -band panel in Fig. 11. The ‘lines’ ofpoints offset from the main distribution are primarily due to appar- c (cid:13) , 1–29 egaMorph II l l l l l llllllllllllllllllll l l l l l lllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l lllllllllllll l l l l l llll l l l l l ll l l l l l lllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l lllllllllllll l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l lllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l lllllllllllll lllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l lllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l lllllllllllllll l ll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllll ll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll lll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l lllllllllllllllllll l lll l l l l l lllllll l l l l l llllllllllllllllllll l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l ll llll lllllllllll l l l l l lllllllllllllllllll l l l l l llllllllllllllllllll l l l l l lllllll l l l l l lllllllllllllll l l l l l llllllllllllllll l l l l l llllllllllllllllllll l l l l l lllllllllllllllllll l l l l l ll llllllllll l l l l l lllllllllllllllllll l l l l l lllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllll l l l l l llllllllllllllllllll l l l l l lllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llll l l l l l llllllllllllllllllll l l l l l lllll l l l l l lllllll llllllllllll l l l l l lllllllllllllllllll l l l l l lllllllll l l l l l llllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllll llllllll l l l l l lllllllllllllllllll l l l l l llll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l lllllllllllllllllll l l l l l lllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l lllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l lllllllllllllllllll − . − . . . . cmag0[−sumcr, 3] M [ M , a ll ] - M z m i n [ M ] l l l lllllll llllllllllllll llll l ll l l l l l ll l llllllll llll l ll ll lllllll lll ll lllll llllll ll llllll lll llll ll l llllll lll l ll llll ll lll llllll l ll l l llll lllllllllll ll ll lll lll ll lll ll lllllll l lll lll llllll llllll l l l l l lll l ll ll llllllllll lll llll l l l l l ll lll l ll ll l l llll ll l lllllll llllllll l l ll llllll l l lllllllllllll llll lllllll l l lllll llllllllll l l l l lllll llll llllllllll l ll lllll llllllll ll l llllllllllll l l lllll l l l l llllll lllll llllllllllll llll lll l lllllllllll l l lllllll ll l lllll ll ll lll lllll lllll ll llllll l ll lllllllllllllll l l l l l llllll l lllllll ll lll llll lllllll l llll lllll lllllllll l lll l l l l ll l ll l l ll llllllll lll lllllll lll l llll lll llll lllllllll llllllll ll l llllllllllllllllll ll l ll l l lll ll llllllll lllllllll l l l l l lllllllllllllllllll l ll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l lllllllllllllllllll l lll l l l l l lllllllllllllllllll l lll l l l llllllllllllllllllll l l l l l lllll l l l l l lllllllll u l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllll l l l l l l lll l l l l l ll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllll l l l l l l lllllllllllllllllll l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllll ll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l lllll l l l l l l lllllllllllllllllll l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllll lllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllll l l l l l l llllllllllllll l l l l l l lllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l lllllllllllllllll l l l l l l l llllllllllllllllll l l ll l l l l l l llllllllllll l l l l l l l llllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllll lllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l l ll l l l l l l l llllllllllllllllll l l l l l l llll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllll l l l l l l llllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l l lllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l l ll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l lllllllllllllllll cmag0[−sumcr, 3] dab s _ mm _ s m [ − s u m c r , ] ll lll ll llll lllll lll lllllll l l lll lllllll l l llllll llll l ll llllllllllllll l ll l lllll llllllllllllll l l llllllll llllll ll l llllll llll llllllll l l l llllll llll lll l g l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l l lllllllllll l l l l l l l ll l l l l l ll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l llllllllllll l l l l l l lllllllllllllllllll l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllll ll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l llllll l l l l l l l llllllllllllllllll l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllll lllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l lllll l l l l l l llllllllllllll l l l l l l l llllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l ll l l l l l l l lllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l l llllllll lllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l ll l l l l l l lllllllllllllllllll l l l l l l llll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l llllllll l l l l l l llllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l llllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll cmag0[−sumcr, 3] dab s _ mm _ s m [ − s u m c r , ] r l l l l l l l llllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllll l l l l l l lll l l l l l ll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l l lllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l lllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllll l l l l l l l llllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllll ll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllll l l l l l l lllllllllllllllllll l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllll lllllllllll l l l l l l l llllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l llllll l l l l l l llllllllllllll l l l l l l lllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l ll l l l l l l l lllllllllll l l l l l l lllllllllllllllllll l l l l l l l lllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllll lllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lll l l l l l l lllllllllllllllllll l l l l l l l lll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllll l l l l l l l lllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll cmag0[−sumcr, 3] dab s _ mm _ s m [ − s u m c r , ] lll llll lll l lll lll ll lll l ll lllllll l l l l l l l lllllllll lllllll ll llll ll lll l l ll lll ll lll l ll llllllllllllll llll llllllll lll llllll llll lll l l l l l l l lllllllllllllllll i l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllll l l l l l l lll l l l l l ll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllll l l l l l l lllllllllllllllllll l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllll ll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllll l l l l l l lllllllllllllllllll l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllll lllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllll l l l l l l llllllllllllll l l l l l l lllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l ll l l l l l l llllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lll l l l l l l lllllllllllllllllll l l l l l l llll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllll l l l l l l llllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllll cmag0[−sumcr, 3] dab s _ mm _ s m [ − s u m c r , ] ll ll l llllllll l lll lll ll lllll ll ll l lllll ll l ll lll llll llllll ll l llll ll llll lll l llllll lllll lll lllllllll lllll l l ll l llll ll ll l l llllllll lllllllll llll llll lllllll llll l lll ll lllll l lll lllll l llll l llllllllllllllllll l lll ll llll l ll lllll l l l lll l ll lll l ll ll llllllll ll lllllllll ll l l llllllll l lllllllllllllllll l ll llllll lllll lll llll ll llll ll ll lllll l lllll llllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll z l l l l l llllllllllllllllllll l l l l l lllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllll lllll l l l l l llll l l l l l ll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l lllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllll l l l l l llllllllllllllllllll l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l lllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l lllllllllllllllllll l l l l l lllllllllllllllllll l l l l l llllllllllllllllllll l l l l l 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l l llll l ll lll l l l l l llllllllllllllllllll l l l l l lllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llll lllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l lllllllllllllllllll . . . . . cmag0[−sumcr, 3] r e [ M , a ll ] r e , z m i n [ M ] l l l lllllll llllllll llll llll l ll l l l l l ll l llll lllll llll l lll lllllll lll ll lllll ll l l l llll l l l l 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l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllll llll l l l l l l lllllllllllllllllll l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l lllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllll ll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l lllll l l l l l l lllllllllllllllllll l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllll lllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllll l l l l l l lllllllll ll l l l l l l lllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l lllllllllllllllll l l l l l l l llllllllllllllllll ll l l l l l l llllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllll lllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l l ll l l l l l l l llllllllllllllllll l l l l l l llll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllll l l l l l l llllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l l lllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l l ll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllll l l l l l l l llllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l lll llll l ll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllll cmag0[−sumcr, 3] r _ mm _ s m [ − s u m c r , ] llll ll llll l lll lll llllll l l lll lllllll l l llllll llll l ll llllllllllll l ll lllll lllllllllllll l l llllllll llllll ll l llll llll lll l l l llllll llll ll l l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l l lllll lllll l l l l l l l ll l l l l l ll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllll llll l l l l l l lllllllllllllllllll l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllll ll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l llllll l l l l l l l llllllllllllllllll l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllll lllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l lllll l l l l l l lllllllll lll l l l l l l l llllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll ll l l l l l l l lllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l l llllllll lllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l ll l l l l l l lllllllllllllllllll l l l l l l llll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l llllllll l l l l l l llllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllll l lll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllll lllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllll cmag0[−sumcr, 3] r _ mm _ s m [ − s u m c r , ] l l l l l l l llllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllll l l l l l l lll l l l l l ll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l l lllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l lllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllll l l l l l l l llllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllll ll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllll l l l l l l lllllllllllllllllll l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllll lllllllllll l l l l l l l llllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l llllll l l l l l l lllllllll lll l l l l l l lllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll ll l l l l l l l lllllllllll l l l l l l lllllllllllllllllll l l l l l l l lllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllll lllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lll l l l l l l lllllllllllllllllll l l l l l l l lll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllll l l l l l l l lllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllll l lll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllll llllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllll cmag0[−sumcr, 3] r _ mm _ s m [ − s u m c r , ] lll llllll l lll lll ll lll l ll lllllll l l l l l l l lllllllll lllllll ll llll ll ll l l llll ll lll l ll lllllllllllll llll l ll lll lllll llll lll l l l l l l l lllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllll l l l l l l lll l l l l ll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllll l l l l l l lllllllllllllllllll l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllll ll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllll l l l l l l lllllllllllllllllll l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllll lllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllll l l l l l l lllllllllll l l l l l l lllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l ll l l l l l l llllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lll l l l l l l lllllllllllllllllll l l l l l l llll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllll l l l ll ll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllll lll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllll lllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllll cmag0[−sumcr, 3] r _ mm _ s m [ − s u m c r , ] ll ll l llllllll l lll lll ll lllll lll ll l llllll ll l ll lll llll llllll ll l llll ll llll lll l llllll lllll lll lllllllll l l l l l lll l l lllll l l ll l llll ll llll l l llllllll lllllllll llll llll lllllll llll lll ll lllll l lll lllll l llll l llllllllllllllllll l lll ll llllllll l l l lll l ll ll l ll ll llllllll ll lllllllll ll l l llll llll l lllllllllllllllll l llllll lllll lll llll ll llll ll ll lllll l lllll lllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l llllllllllllllllllll l l l l l lllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllll lllll l l l l l llll l l l l l ll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l lllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l lllllllllllll l l l l l llllllllllllllllllll l l l l llllllllllllllllllll l l l l l lllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l lllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllll l l l l l llll lllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l lllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l lllllll l l l l l llllllllllllllllllll l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l lllllll lllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllll l l l l l llllllllllllll l l l l l llllllllllllllll l l l l l llllllllllllllllllll l l l l l lllllllllllllllllll l l l l l llllllllllllllllllll l l ll l l l l l lllllllllllll l l l l l llllllllllllllllllll l l l l l lllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l lllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llll l l l l l llllllllllllllllll l l l l l lllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l lllllllll l l l l l lllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l lllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llll l l l l l llllllllllllllllllll l l l l l lllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l lllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l lllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llll lllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l lllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllll
12 16 20 . . . . . n [ M , a ll ] n z m i n [ M ] l l l lllllll llllllll llll llll l ll l l l l l ll l llll lllll llll l lll lllllll lll ll lllll ll l l l llll l l l l l l llll llll ll llllll llllllll ll lll lll l ll llll ll lll lllllll l l l l ll l l llll lllllllllll ll ll lll lll ll lll ll lllllll llllllll l l lllll llll lll llllll llllll l l l l l lll l ll ll llllllllll l l l l l lll l l lll llll l l l ll l l l l l l l llll llll lll l ll ll l l llll ll l lllllll llllllll ll l ll llllll l l llllllllllllll llll lllllll l l llll lllllllll l l l l lllll llll llllllllll l lll llll l l l l l llllllllllllllllllll l l lll llllllll ll l llllllllllll l l l l l lllllllllll llll l l lllll l l l ll lllllll lllll lllllllllllll l l l l l lllllllllllllllllll l llll lll l llllllllllll l l llllll l l l l ll ll l lll ll l lllll ll ll lll lllll lllll ll llllll l l l l l l llllll l ll l l l llllll l ll llllllll llllll l l l l l llllll l lllllll ll lll llll lllllll l lll lllll lllllllll l lll l l l l ll l lllllllllll l l ll llllllll lll lllllll lll l llll lll llll llllllllllllllll ll l llllllllllllllllll ll l ll l l llll lllllllll llllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l ll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l lllll l l l l l lllllll l l l l l llllllllllllllllllll l l l l l llllllllllllllllllll l l l l l lllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllll l l l l l l lll l l l l l ll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllll l l l l l l lllllllllllllllllll l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l lllllllllllllllll l l l l l l lllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllll ll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l lllll l l l l l l lllllllllllllllllll l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllll lllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllll l l l l l l llllllllllllll l l l l l l lllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l lllllllllllllllll l l l l l l l llllllllllllllllll l ll l l l l l l llllllllllll l l l l l l l llllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllll lllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l l ll l l l l l l l llllllllllllllllll l l l l l l llll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllll l l l l l l llllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l l lllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l l ll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lll lllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l lllllllllllllllll
12 16 20 n_ mm _ s m [ − s u m c r , ] ll lll ll llll lllll lll lllll l l lll lllllll l l llllll llll l ll llllllllllllll l ll l lllll llllllllllllll l l llllllll llllll ll l llllll llll llllllll l l l llllll llll lll l l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l l lllllllllll l l l l l l l ll l l l l l ll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l llllllllllll l l l l l l lllllllllllllllllll l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l llllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllll ll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l llllll l l l l l l l llllllllllllllllll l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllll lllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l lllll l l l l l l llllllllllllll l l l l l l l llllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll ll l l l l l l l lllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l l llllllll lllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l ll l l l l l l lllllllllllllllllll l l l l l l llll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l llllllll l l l l l l llllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lll l l l l l l lllllllllllllllllll l l l l l l l lllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lll lllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll
12 16 20 m r , (mag) n_ mm _ s m [ − s u m c r , ] l l l l l l l llllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllll l l l l l l lll l l l l l ll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l l lllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l lllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l lllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllll l l l l l l l llllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllll ll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllll l l l l l l lllllllllllllllllll l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllll lllllllllll l l l l l l l llllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l llllll l l l l l l lllllllll llll l l l l l l lllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll ll l l l l l l l lllllllllll l l l l l l lllllllllllllllllll l l l l l l l lllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllll lllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lll l l l l l l lllllllllllllllllll l l l l l l l lll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllll l l l l l l l lllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllll l l l l l l lllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l 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12 16 20 n_ mm _ s m [ − s u m c r , ] lll llllll l lll lll ll lll l ll lllllll l l l l l l l lllllllll lllllll ll llll ll lll l l llll ll lll l ll lllllllllll llll llllllll lll llllll llll lll l l l l l l l lllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllll l l l l l l lll l l l l l ll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllll l l l l l l lllllllllllllllllll l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllll ll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllll l l l l l l lllllllllllllllllll l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllll lllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllll l l l l l l lllllllllllll l l l l l l lllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll ll l l l l l l llllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lll l l l l l l lllllllllllllllllll l l l l l l llll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllll l l l l l l llllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllll l l l l l l lllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l 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12 16 20 n_ mm _ s m [ − s u m c r , ] ll ll l llllllll l lll lll ll lllll lll ll l llllll ll l ll lll llll llllll ll l llll ll llll lll l llllll lllll lll lllllllll l l l l l lll l l lllll l l ll l llll ll llll l l llllllll lllllllll llll llll lllllll llll lll ll lllll l lll lllll l llll l llllllllllllllllll l lll ll llll l ll lllll l l l lll l lll l ll ll llllllll ll lllllll ll l l lllllllllll l lllllllllllllllll l ll llllll lllll lll llll ll llll ll ll lllll l lllll llllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l lllllllllllllllllll l l l l l l llllllllllllllllll Figure 13.
In a similar manner to Fig. 11, this figure shows parameter deviations for multi-band
GALFITM fits ( MM ) relative to the lowest redshift ( z min ) SM fit, plotted as a function of the r -band apparent magnitude, as cosmologically scaled from the MM fit to the lowest-redshift artificial image for each galaxy.The black points denote the recovered parameters for the 3863 artificially-redshifted galaxies. The red and blue points have been selected in the manner wayas in Fig 11, based only on the SM results, since this method is responsible for their outlying values. The solid cyan lines indicate the median, while dashedcyan lines are the 16th- and 84th-percentiles of the distribution, all determined within one magnitude bins. To more easily compare to the SM fits, we includethe lines from Fig. 11 in red. For consistency, we calculate the green lines using exactly the same galaxies as were used in determining the red lines. Themagnitudes and effective radii obtained from the redshifted images have been cosmologically converted to their absolute values at the true redshift of eachgalaxy for the purpose of these comparisons. ent inconsistencies in the lowest -redshift ( z min ) fits. In these casesthe effective radius is unreasonably large in the z min fit, but be-comes more reasonable at higher redshifts, leading to a systematicoffset in r [M]e /r [M]e , zmin . To illustrate this we identify cases wherethe z min fits have an effective radius, in the band under consider-ation, that is more than three times larger than the effective radiusin the r -band. These galaxies are denoted by blue points. Varia-tion in effective radius by more than a factor of three over opticalwavelengths is very unlikely to be physical, e.g. see Fig. 3. Theexcessively large effective radii in these z min fits tend to be accom-panied by high S´ersic indices and overly-bright magnitudes. Thisbehaviour is discussed further below.Where the z min fits are consistent between bands, further out-liers may also be identified by considering cases where instead thefits at higher redshift become inconsistent. To illustrate this, we de-note with red points those fits where the effective radius in a specificband ( u, g, i, z ) differs from the equivalent r -band value by more than a factor of three. These points do not display a consistent sys-tematic behaviour in S´ersic index and magnitude.We select both types of outlier using the SM results in Fig. 11and 13 and MM results in Fig. 12. In this way we identify caseswhere either the SM or MM method fails to fit a reasonable model,under the assumption that the r -band fit results are robust.In Fig. 12 we show a similar plot to Fig. 11, but now for theresults of our multi-band ( MM ) method. The median and (robustly-determined) scatter are shown by green lines. The multi-band ap-proach dramatically reduces, by a factor of 4-40, the fraction offits which fail or produce results in close proximity to constraints.These fractions are 5%, 0.1%, 0.1%, 0.1% and 2% for u , g , r , i , z , respectively. We also find dramatically fewer ‘red’ outliers, andno ‘blue’ outliers at all. To allow a fair comparison, we thereforedetermine the green lines only using those galaxies with both suc-cessful SM and MM fits and excluding the galaxies that are outliersboth in SM and MM results. To facilitate the comparison we includein Fig. 12 the SM median and scatter lines from Fig. 11 in red. c (cid:13) , 1–29 Vika et al.
For the MM results, shown by the green lines in Fig. 12, at m [M , all] r, = 18 the typical scatter per band ( u , g , r , i , z ) for m is ( ± . , ± . , ± . , ± . , ± . , for r e is ( ± %, ± %, ± %, ± %, ± % ) and for n is ( ± %, ± %, ± %, ± %, ± %).In contrast, for the single-band SM results, shown by the redlines in Figs. 11 and 12, at m [M , all] r, = 18 the typical scatter perband ( u , g , r , i , z ) for m is ( ± . , ± . , ± . , ± . , ± . ,for r e is ( ± %, ± %, ± %, ± %, ± % ) and for n is( ± %, ± %, ± %, ± %, ± %). Recall that both the blueand red outliers were excluded before we determined these scatters.In the upper row of panels in Fig. 12, we can compare therecovery of magnitude for SM and MM fits as the galaxies becomemore distant and hence smaller and fainter. Both SM and MM re-sults show similar systematic trends. These are absent in the bluerbands, but in the red there appears to be a small bias toward re-covering fainter absolute magnitudes for objects viewed at largerdistances (and hence with fainter apparent magnitudes). The sys-tematic trends are discussed further below. There is a hint that thebias in the z -band is slightly smaller for MM . The two methods showsimilar scatter in g , r and i , but for the lower signal-to-noise bands, z and especially u , the MM results show significantly smaller scatter.In the middle row, we compare the results for effective radius.Again, we see that in the higher signal-to-noise bands the meth-ods return almost identical results. However, the MM method signif-icantly reduces the scatter in z , and makes a dramatic improvementin u -band. Part of the reason for this reduction in scatter is a resultof allowing only limited variations in this parameter with wave-length. By design, the r e in the different bands can only vary asa linear function of wavelength. Therefore the u -band size is con-strained using information from all the bands. Recall that, in Sec-tion 3.2, we chose the linear variation with the aim of improvingthe fitting results, by applying the physical assumption that thesesystems should only show smooth variations with wavelength, ob-serving that single-band ( SM ) parameters are consistent with linearvariation within the uncertainties (see top left panels of Figs. 3 and4). Note that there are outlying lines of points below the mainbody of points in all of the effective radius panels in both Figs. 11and 12. These points have not be selected as outliers (at least in g , r , i ) by either the ‘blue’ or ‘red’ criteria, because their effectiveradii in each band are reasonably consistent with the r -band val-ues. We have found that these points are the result of three galaxies(NGC 3631, NGC 4321 and NGC 4725), which share the charac-teristic that fits to their lower artificially-redshifted images mea-sure substantially higher effective radius than is measured on eitherthe original (M ONTAGE ) images, or the higher-redshift (
FERENGI )images. All methods ( S3 , SM and MM ) result in similar behaviourfor these galaxies. These are all approximately face-on spirals withsmall, prominent bulges and extended faint disks. Not unsurpris-ingly, it may be that this distinctively two-component morphologyis more susceptible to systematic variations in S´ersic fit parametersat intermediate resolution. Preliminary results from bulge-disk fitsconfirm this impression.Finally, the bottom row of Fig. 12, presents the results forS´ersic index. The comparison between SM and MM is very similar tothat for effective radius. S´ersic indices recovered by our multi-band( MM ) method, with n varying linearly with wavelength, show re-duced scatter in z and, in particular, u bands. The systematic trendsseen for r e and n versus m [M , all] r, are also significantly reduced in z and u when using the MM method. Overall, the comparison of SM versus MM fitting shows that forthe high signal-to-noise bands ( g , r and i ) the improvement in ex-tracting parameters is minor, while for low signal-to-noise bands( u and z ) there is a clear reduction of both the systematic and sta-tistical uncertainties for fainter galaxies. Another important advan-tage of MM versus SM is that the multi-band approach increases thenumber of galaxies that are successfully fitted. The improvement issmall for samples of large bright galaxies but can be very significantwhen we move to fainter samples. This advantage of multi-band fit-ting is more evident in Paper III, were we deal with thousands offaint galaxies, and show that this approach has the ability to obtainmeaningful measurements even in extreme cases where the galaxyis not visible in some of the bands.In Figs. 11 and 12 we notice a systematic effect wherebyboth SM and MM methods return systematically fainter magnitudes,smaller effective radii and lower S´ersic indices when going tofainter apparent magnitudes (i.e., for galaxies artificially-redshiftedto greater distances). A significant source of systematic uncertaintyin both figures originates from the sky determination. In principle,for the artificially-redshifted images we know the fundamental skylevel, as it has been explicitly simulated by FERENGI . However, inpractice there are faint, unresolved, and hence unmasked, objects inthe images, in addition to the galaxy of interest. In order to accountfor the contribution of these objects we need to measure the appar-ent sky value. At higher artificial redshifts, neighbouring objectsbecome fainter (due to surface brightness dimming), cover a largerarea of the image (due to the worsening angular resolution) and areless likely to be detected by SE
XTRACTOR and hence masked. Thecontribution of such sources to the sky therefore varies, and so asky value must be independently determined for each artificially-redshifted image.During the
FERENGI artificial-redshifting process, the angularsize of the original image, and hence the number of pixels that sam-ple it, are reduced. At z = 0 . the artificial images have decreasedto about per cent of the size (and so . per cent of the area) ofthe first redshifted image (i.e. with z = 0 . ). Furthermore, thePSF convolution acts to increase the size of the galaxy relative tothe image. For high artificial redshifts, the resulting small imagesdo not provide enough free sky pixels for a secure sky determina-tion and may lead to an overestimation of the sky value. This wouldresult in systematic trends in m , r e and n in the directions seen.However, we suspect that this is not the only cause of the observedtrends for three reasons. Firstly, the trends, particularly in the caseof S´ersic index, can be seen to set in at relatively bright magnitudes,and hence low redshifts, where all images still contain a large frac-tion of sky pixels. Secondly, for single-band fitting, the strength ofthe trend increases for bands with lower signal-to-noise, suggestingthat the systematics are dependent on noise, and not just resolution.Thirdly, the trends are significantly improved by multi-band fitting,even though the same sky is used for both methods.A plausible alternative cause of these trends is systematic mis-matches between the true galaxy profiles and the S´ersic model.Many of our galaxies are spirals, and are hence better describedby two-components, a bulge and a disk, than they are by asingle-S´ersic function. Even elliptical galaxies are not always well-represented by S´ersic profiles, particularly in their centres, as dis-cussed further in Section 4.3.1. One consequence of the bulgesin spiral galaxies is that they display excess light in their centres,when compared to a single-S´ersic profile fit at larger radii. This ex-cess will be resolved, and be apparent with high signal-to-noise, forlarge galaxies at low redshift. It would therefore be expected that afit to the full image would attempt to fit this feature, resulting in a c (cid:13) , 1–29 egaMorph II relatively large S´ersic index. For example, for an exponential diskwith a bright, well resolved n ∼ bulge we might obtain a single-S´ersic profile with n ∼ . At higher redshift, the same galaxieswould be less resolved and the noise relatively higher, lowering thesignificance of the central bulge. The fit will therefore be more con-strained than previously by flux at larger radii. A lower S´ersic indexfor the single-component fit would be a natural outcome, perhaps n ∼ in our above example. Due to the covariance between n , r e and m , shifts to lower n would be accompanied by smaller r e and fainter m as is seen (Guo et al. 2009; Yoon et al. 2011). Thisis an important issue that should be considered carefully by anystudy investigating the evolution of galaxy structural parameterswith redshift.With increasing redshift, neighbouring objects often becomeblended with the main galaxy. This effect is present in both realhigher-redshift data, and in the artificially-redshifted images. Asmentioned in Section 2.2, a downside of the redshifting process isthat FERENGI treats both galaxies and stars in the same way. To-gether with the high frequency of stars in the original images, thismeans that blending with similarly bright neighbours probably oc-curs more often in our images than it would in real data. Mask-ing is not always effective, due to the difficult task of identifyingand separating these blended neighbours. In such cases, the recov-ered parameters, for both in single- and multi-band fits, may be dis-torted. This highlights how important it is to carefully mask and fitneighbouring objects, particularly when fitting faint galaxies. It isalso important to recognise that inseparable blends will always in-troduce a contamination when attempting to measure galaxy struc-tural parameters in low resolution data. These effects will introduceadditional scatter in the recovered parameters, primarily at faint ap-parent magnitudes.Masking issues should affect both SM and MM methods equally,while systematic trends are present for both, but somewhat reducedfor MM . The differences in the scatter between these two methodscompares the performance of the two methods. From this we haveshown that our multi-band technique reduces the statistical uncer-tainties associated with lower signal-to-noise and lower physicalresolution images and enables the structural parameters of galaxiesto be more accurately measured, when compared with fitting eachband independently.Now, after studying the performance of the SM and MM meth-ods separately in Figs. 11 and 12, we can better understand the be-haviour seen in Fig. 10. First we note that the scatter in Fig. 10, par-ticularly for the g , r and i bands, is significantly smaller than thatin Figs. 11 and 12. This must be because galaxies experience simi-lar systematic trends with both SM and MM ; Fig. 10 is just showingresidual differences between the two methods. The same argumentexplains the systematic decline of the S´ersic index ratio in Fig. 10.Both methods measure smaller values of n as the galaxies becomemore distant, but the systematic effect is greater for SM than MM (Fig. 12; red line versus green line, respectively). This differenceresults in the moderately trend in the purple line in Fig. 10.We have shown that MM is more stable and robust to redshift ef-fects, in terms of the fraction of successful fits, reduced systematicbiases and smaller statistical scatter. However, we have not fullytested whether the MM results are more accurate, i.e. whether its re-sults are closer to the true values than SM . Recall that the assump-tion of a particular dependence of n and r e on wavelength (linear inthis case) may introduce systematic biases if it is overly restrictive.To investigate this we assume that the lowest redshift ( z min ) single-band ( SM ) fit accurately represents the structure of each galaxy (orat least is less affected by systematic effects than MM ). We will test the accuracy of the MM fits by looking for any systematic bias be-tween the MM results and the z min SM results.In Fig. 13 we present a hybrid of Figs. 11 and 12, showingthe parameter differences between our multi-band ( MM ) fits to eachartificially-redshifted image and the single-band ( SM ) fits to thelowest redshift image. In a similar manner to the previous figures,we determine the median and scatter lines using only those imageswith successful fits using both SM and MM methods, and excludingthe ‘blue’ and ‘red’ outliers, as defined in Fig. 11. We plot the me-dian and scatter in blue. In order to aid the comparison we includethe red lines from Fig. 11. In the upper row, we can compare the re-covered magnitudes, in the middle row we compare the recoveredeffective radius and in the lower panels the S´ersic indexes, as thegalaxies become more distant. As seen before, there is little changein behaviour for g , r and i , but there are clear differences for thelower signal-to-noise bands, z and especially u .For distant, faint galaxies the scatter of the MM results is sim-ilar, or slightly better, than SM , and the biases are significantly im-proved. For such galaxies, the benefits of the improved precisionand robustness of the multi-band approach outweigh any systematicinaccuracies. For nearby, bright galaxies there remains a significantscatter, which is associated with differences between SM and MM fitsthat persist even for reasonable quality imaging. At high signal-to-noise, one might suspect that SM results are more accurate and MM results biased. However, despite the scatter, there is little evidencefor a systematic difference between the two methods at the brightend. Furthermore, by comparing Figs. 11 and 12, it is clear that thevast majority of outliers in Fig. 13 are due to inconsistencies in the SM fits (blue and red points), even for good quality imaging. Alsorecall that the number of fits which encounter constraints, and arehence unusable, is many times greater for SM . We incline to trustthe MM results over the SM results as the latter are more likely toreturn fits with errors. Therefore, we conclude that the benefits ofimproved robustness in the multi-band approach largely outweighconcerns about systematic biases. Nevertheless, we advise that fu-ture studies should consider the scientific consequences of system-atic biases resulting from assuming a restrictive wavelength depen-dence for n and r e , and if necessary increase the flexibility of thisdependence. This is extremely easy to do in GALFITM , by increas-ing the order of the relevant polynomial when setting up the fit.
ONTAGE images
In Fig. 14 we plot the distribution of S´ersic indices for the originalimages, using n values obtained from our multi-band ( MM ) single-S´ersic fits. The black dotted lines show the distribution of ellipticaland lenticular galaxies. The red lines show early-type spirals, whileblue lines show late-type spirals, both including barred galaxies.Finally, green dashed lines show pure disk galaxies and irregulars.This plot illustrates how S´ersic index changes with both morpho-logical type and wavelength. For the E/S0 sample we can see thatthe majority of early-type galaxies have S´ersic indices between 3and 5, and show only minor variations with wavelength. It is inter-esting to note a hint of a bimodality, with n peaked at values aroundboth 4 and 2. We have investigated this and find that there is a signthat this is a result of ellipticals inhabiting the high- n peak with S0stending to have lower- n . However, low number statistics prevent usfrom pursuing this further.Early-type spirals show a wide range of n values from ∼ . to ∼ . . The morphological classification of spiral galaxies isusually based up on both the tightness of their spirals and their c (cid:13) , 1–29 Vika et al. n u D en s i t y . . . . n g D en s i t y . . . . n r D en s i t y . . . . n i D en s i t y . . . . n z D en s i t y . . . . −−−− E/S0 (34)Sa/Sb/SBa/SBb (39)Sbc/Sc/Sd/SBc−d (79)Sdm/Sm/Irr (11) Figure 14.
Distribution of S´ersic indices obtained from MM fits to the original imaging for our sample of 163 galaxies. Each histogram shows the distributionfor a specific range of morphologies. Each histogram has been normalised so the total area (integral) is equal to one. Each panel plots the results for a differentband. bulge-to-disk ratio. These are generally considered to be corre-lated, such that earlier-type spirals are associated with both tighter-winding arms and higher bulge-to-disk ratios. We would thereforeexpect that early-type spiral galaxies would have more prominentbulges, compared to late-type spirals, and hence higher S´ersic in-dices. However, as we can see from Fig. 14, early-type spirals haveequal large and small n values and this distribution shows no ob-vious changes with wavelength. It appears that visually classifiedearly-type spirals can possess a wide range of n , and hence bulge-to-disk ratio.Next we consider late-type spiral galaxies. We see that latespirals (Sbc – Sd) are concentrated around n = 1 , with a smalltail to higher n , as expected if they are typically dominated by anexponential disk. Bulgeless disks (Sdm/Sm) and irregular galaxiesare even more concentrated around n ∼ . Furthermore, for bothsamples of late-type galaxies, as we move from blue to red wave-lengths we see the distribution shift consistently to higher S´ersicindices, from < ∼ in u -band to ∼ . in z -band).That effect can be attributed to a combination of two phe-nomena. First, the small bulge or pseudobulge that these galaxies contain will become more prominent at longer wavelengths, due toit typically having a redder colour than the disk. Second, at bluerwavelengths the light emission is more attenuated by dust, whichmay act to flatten the profile at shorter wavelengths (Pastrav et al.2013). All 26 elliptical galaxies in our sample also have S´ersic profile pa-rameters measured by the classic work of Caon et al. (1993, here-after C93). More recently, Chen et al. (2010, hereafter C10) havealso measured these parameters for 19 galaxies in our sample usingimaging from SDSS. In this section we investigate the reliability ofour method by comparing our derived parameters with those mea-sured by C93 and C10. For this purpose, we use only the structuralparameters obtained from multi-band fits to the original M ONTAGE as classified in Caon et al. (1993) and/or the NASA/IPAC ExtragalacticDatabase (NED) c (cid:13) , 1–29 egaMorph II ll l lll l lll ll ll lll ll ll ll l ll −16 −17 −18 −19 −20 −21 −22 −23 − − M B ( mag ) M B - M B v ega [ M , a ll ] lll llll l l ll lll lll lll ll ll ll ll −16 −17 −18 −19 −20 −21 −22 −23 − − M g * ( mag ) M g *- M g [ M , a ll ] ll llll l ll l lll l ll ll l ll lll ll ll ll lll r B ( arcsec ) r g [ M , a ll ] ( a r cs e c ) ll l llll ll ll l ll l ll lll ll ll ll ll r g * ( arcsec ) r g [ M , a ll ] ( a r cs e c ) l l llll l ll l lll l lll ll ll lll ll ll ll l ll n B n g [ M , a ll ] ll lll lll l ll l ll l ll l ll ll ll ll ll n g * n g [ M , a ll ] l lll lll Figure 15.
Comparison between structural parameters derived in this work and those obtained by C93 for 26 common elliptical galaxies (left column) and byC10 for 19 common galaxies (right column). The red points highlight galaxies with r e, B > arcsec. Error bars in r g and n g show the estimated uncertaintiesof 11 per cent in each. images. In order to perform a sensible comparison with C93, our g -band AB magnitudes have been converted to B -band values and tothe Vega zeropoint system. We adopt the transformations providedby Blanton & Roweis (2007): M B = M g + 0 . . M g − M r ) − . , (2) M B, Vega = M B, AB + 0 . . (3)In Table 2 we list the absolute magnitude from our multi-bandfit, after converting to B -band Vega magnitudes, the galaxy dis-tance, the galactic extinction in g -band from NED (Schlafly &Finkbeiner 2011), C93’s B -band absolute magnitude ( M ∗ B ), effec-tive radius ( r ∗ e ,B ) and their equivalent profile S´ersic index ( n ∗ eq ,B ).As C93’s magnitudes and effective radii are presented in differ-ent format to that produced by GALFIT , we have derived M ∗ B and r ∗ e ,B from equation (9) of C93 using b n = 1 . n − . , c n = (2 . b n ) / ln(10) (Graham & Driver 2005) and the distancesfound in Table 2. The parameters measured by C10 can be easilyextracted directly from their paper and compared to our withoutany conversions, and so are not included in Table 2. All the abso- lute magnitude values presented in this section have been calculatedassuming the distances found in Table 2.In Fig. 15 we show the comparison between the parametersderived in this work and those from both C93 and C10. C93 werethe first to introduce S´ersic index as the third free parameter of theS´ersic function, until then it was common to keep it fixed at a valueof 4. In the left panels of Fig. 15 we compare our multi-band resultsthose of C93. We see that the two sets of derived properties are inreasonable agreement. However, there is a systematic trend, suchthat large galaxies, with r e ,B > ∼ arcsec, are measured to havesmaller sizes and fainter magnitudes using our method, and viceversa for smaller galaxies. In the right panels of Fig. 15 we comparewith C10. These measurements show much better agreement withours, independent of the galaxy size. However, in both lower panelsof Fig. 15 we note that where both C93 or C10 find n ∗ eq ,B > ∼ ourindices remain around n ∼ .That C93 measure slightly different values compared to boththis study and C10 is not unexpected. Both this study and C10 arebased on recent imaging data from SDSS and apply modern fit-ting techniques, whereas C93 was one of the first papers to apply c (cid:13) , 1–29 Vika et al. this fitting method, using lower quality images. They are thereforemore likely to suffer from systematic uncertainties, especially forthe most extended galaxies in their sample. Measuring the effectiveradius and S´ersic index of large elliptical galaxies is a not as easyas it might seem. Their low surface brightness haloes can be veryextended and are difficult to determine (Kormendy et al. 2009). AsH¨aussler et al. (2007) have shown, a galaxy with n = 4 is muchharder to fit than a galaxy with n = 1 , primarily due to uncertain-ties in the sky estimate. Small overestimations of the backgroundeffectively clip the outer faint halo and result in an underestimationof effective radius, magnitude and S´ersic index.Furthermore, large elliptical galaxies are known to show lightprofiles that diverge from exact S´ersic functions. It has been shownthat bright ( M B < − . ) elliptical galaxies have cores in theircentral regions, which deviate from a S´ersic model in the sensethat the surface brightness is lower than would be expected fromextrapolating the outer S´ersic profile inwards (e.g., Ferrarese et al.1994; Graham 2011). Usually, detailed fitting of these objects eitherexcludes the inner part of the galaxy, uses an additional functionto model the central light distribution (e.g. Lauer et al. 1995; Fer-rarese et al. 2006) or adopts a core-S´ersic function (Trujillo et al.2004). For instance, both studies that we compare with (C93 andC10) have excluded the inner few arcsec of each galaxy light pro-file from the fitting process. The decision of how much of the innerregion should be excluded from the fit typically requires an indi-vidual visual inspection of the one dimensional surface brightnessdistribution of each galaxy. This process is beyond of the aim ofthis paper. Furthermore, we prefer to keep the overall fitting pro-cess homogeneous and automated, such that our results can informthe interpretation of studies involving much larger samples. Thecore-S´ersic function could possibly provide better fits to the ellip-ticals in our original images but would be difficult to constrain incase of our artificially-redshifted sample. An added complicationis that, at present, the core-S´ersic function is not implemented in GALFIT .To further investigate the characteristics of these galaxies, andspecifically the discrepancies we see compared to C93 and C10 athigh- n , in Fig. 16 we plot the ∆ Dg parameter for 19 of our el-liptical galaxies, as derived by Glass et al. (2011). This parameteridentifies whether galaxies have a flux deficit or excess in their in-ner regions. Negative ∆ Dg values are indicative of a core structurein the center of a galaxy (see Glass et al. (2011) for more details).We highlight in red the same large galaxies as in Fig. 15. It is clearthat the galaxies that show the largest deviation from C93, in leftpanel in Fig. 15, display negative values in Fig. 16. Therefore, thegalaxies with central flux deficits are those that show different n and r e , between the different studies. Similar investigation regard-ing galaxy size discrepancies can be found in C10 their figure 11.Central deficits and excesses in elliptical galaxies only affect the in-ner few percent of r e , so these deviations from the S´ersic functionare only resolved for the larger galaxies, which typically show cen-tral deficits. In the case of smaller galaxies, the excess region is notresolved in the SDSS imaging and thus does not significantly im-pact upon our S´ersic function fits. Finally, note that both C93 andC10 use one-dimensional fitting methods, in contrast to our two-dimensional approach, so the difference between our S´ersic valuesmay partly result from this (Vika et al. 2012).At higher redshifts, e.g. in our artificially-redshifted images,we would expect that small deviations from S´ersic profiles in thecentral regions would rapidly become less important. We have seenin Section 4.2 that the ‘central excesses’ associated with bulges indisk galaxies result in a trend such that the recovered S´ersic index Figure 16.
The ∆ Dg parameter quantifies the light deficit (core galaxies,negative values) or light excess (cuspy or power law galaxies, positive val-ues) that appear in the central region of early type galaxies. We plot herevalues of ∆ Dg , obtained from Glass et al. (2011) for 19 of our 26 ellipti-cal galaxies, versus absolute g -band magnitude. The galaxies shown in redpoints are the same as in Fig. 15, i.e., those with r e, B > arcsec. decreases with increasing redshift. Ellipticals with a central deficitshould show the opposite behaviour. Indeed, upon examination wefound that all seven galaxies with negative ∆ Dg values show anincrease in their measured S´ersic index with redshift or stay flat.This contrasts with the remaining ellipticals – which mostly showflat, or occasionally decreasing, trends – and spirals, which oftendisplay systematic decreases in n with redshift (see Section 4.2).This supports the picture that when we resolve, and attempt to fit,the centres of cored elliptical galaxies we recover smaller S´ersic in-dices, while as galaxies become less resolved with increasing red-shift, the S´ersic function is less affected by the central behaviourand the fitting process returns higher n values. In this paper we have presented a modified version of
GALFIT
GALFITM , which can constrain wavelength-dependent mod-els of galaxy surface-brightness profiles using multi-band imagingdata. We assess the performance of this new feature by applying itto fit elliptical single-S´ersic profile models to SDSS ugriz imag-ing for 4026 galaxies (163 at low redshift and 3863 artificially red-shifted to greater distances).We show that
GALFITM produces very similar results to
GAL - FIT
GALFITM when fit-ting single-band images individually to those obtained by fittingmultiple bands simultaneously. We find that these two methods per-form similarly for obtaining structural parameters in the g , r and i c (cid:13) , 1–29 egaMorph II bands, but that our multi-band approach shows greater differencesin the lower signal-to-noise z and u bands.In order to demonstrate that the multi-band method producesmore robust results, we study the stability of the recovered struc-tural parameters as the galaxies become more distant, and hencethe images become noisier and less well resolved. We show thatthe GALFITM multi-band technique results in significant reductionsin both the statistical and systematic uncertainties on structural pa-rameter measurements. Additionally, the multi-band technique sig-nificantly increases the fraction of galaxies that return successfulfitting results. These improvements are most noticeable for lowquality images, but in all the tests we performed multi-band fittingis found to improve upon, or at least do no worse than, single-bandresults in terms of accuracy and robustness.In datasets with a variety of signal-to-noise levels, our ap-proach involves a degree of interpolation or extrapolation of param-eters constrained in the high signal-to-noise images to help deter-mine parameter values for the lower signal-to-noise images. Thereis thus a balance to be struck between reducing statistical uncer-tainties and avoiding systematic errors. Our method provides theflexibility to adapt the assumptions for the wavelength dependenceof the model, depending on the galaxy population being studied,the nature of the dataset and the aim of the analysis.Using the multi-band fitting method we measure and presentsingle-S´ersic structural parameters for the 163 galaxies in our sam-ple, as measured on the full-quality SDSS images. We comparethe distributions of S´ersic index for subsamples with different vi-sual morphologies, finding an anticipated trend for spiral galaxiesto show higher S´ersic indices in redder bands. Focussing on el-liptical galaxies, we compare our multi-band measurements withstructural parameters obtained by two independent studies. Over-all, we find good agreement between our multi-band fitting resultsand single-band fitting results from the literature, and are able tooffer explanations for the small discrepancies that exist.We conclude that using multi-band fitting improves oversingle-band fitting for the extraction of structural parameters, par-ticularly for datasets containing multi-band images with a mix-ture of signal-to-noise and resolution. Another advantage of thismethod, which has not been explored in this paper, but will beutilised in forthcoming work, is that it naturally produces robustcolour gradients. The multi-wavelength approach produces a morephysically consistent model, which we anticipate will enable us tomake more meaningful inferences than would otherwise be possi-ble.
ACKNOWLEDGMENTS
This publication was made possible by NPRP grant
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D., Katz N., 2011, MNRAS, 414, 1625 c (cid:13) , 1–29 egaMorph II T a b l e : G a l a xy S a m p l ea nd t h e fi t r e s u lt s a s d e r i v e d fr o m t h e m u lti - b a nd fi tti ng ; C o l u m n ( ) : G a l a xyn a m e ; C o l u m n ( ) : H ubb l e t yp e fr o m N E D ; C o l u m n ( - ) : a pp a r e n t m a gn it ud e s i n u , g , r , i , z p a ss b a nd s ; C o l u m n ( - ) : e ff ec ti v e r a d i u s ; C o l u m n ( - ) : S ´ e r s i c i nd e x . S ee S ec ti on3 f o r m o r e d e t a il s . N a m e T yp e m u m g m r m i m z r e , u r e , g r e , r r e , i r e , z n u n g n r n i n z ( m a g )( m a g )( m a g )( m a g )( m a g )( (cid:48)(cid:48) )( (cid:48)(cid:48) )( (cid:48)(cid:48) )( (cid:48)(cid:48) )( (cid:48)(cid:48) ) ( ± . )( ± . )( ± . )( ± . )( ± . )( ± % )( ± % )( ± % )( ± % )( ± % )( ± % )( ± % )( ± % )( ± % )( ± % ) I C S a . . . . . . . . . . . . . . . I C S B ( s ) b . . . . . . . . . . . . . . . I C S / Irr . . . . . . . . . . . . . . . I C S A B (r) c . . . . . . . . . . . . . . . I C E . . . . . . . . . . . . . . . I C S ( ) . . . . . . . . . . . . . . . I C E . . . . . . . . . . . . . . . NG C S A B ( s ) m . . . . . . . . . . . . . . . NG C S A B ( s ) c d . . . . . . . . . . . . . . . NG C S B (r s ) c . . . . . . . . . . . . . . . NG C S m p ec . . . . . . . . . . . . . . . NG C S A B (r s ) c . . . . . . . . . . . . . . . NG C S A B (r s ) c d . . . . . . . . . . . . . . . NG C ( R ) S A (r s ) b . . . . . . . . . . . . . . . NG C S A ( s ) c . . . . . . . . . . . . . . . NG C S A B (r s ) c . . . . . . . . . . . . . . . NG C S B (r s ) b . . . . . . . . . . . . . . . NG C S A ( s ) c d . . . . . . . . . . . . . . . NG C S B ( s ) b . . . . . . . . . . . . . . . NG C ( R ) S A (r) a . . . . . . . . . . . . . . . NG C S . . . . . . . . . . . . . . . NG C S A B (r s ) c . . . . . . . . . . . . . . . NG C S B (r) b . . . . . . . . . . . . . . . NG C S A ( s ) c . . . . . . . . . . . . . . . NG C S A (r) a b . . . . . . . . . . . . . . . NG C S A B (r s ) c . . . . . . . . . . . . . . . NG C S A (r) b . . . . . . . . . . . . . . . NG C S A ( s ) c . . . . . . . . . . . . . . . NG C S A B ( s ) c p ec . . . . . . . . . . . . . . . NG C S A B ( s ) c . . . . . . . . . . . . . . . NG C S A B d m . . . . . . . . . . . . . . . NG C S A B (r s ) b c . . . . . . . . . . . . . . . NG C S A B b c p ec . . . . . . . . . . . . . . . NG C S B (r s ) c . . . . . . . . . . . . . . . NG C S A ( s ) c d . . . . . . . . . . . . . . . NG C S A B (r s ) c . . . . . . . . . . . . . . . NG C S A B (r) c . . . . . . . . . . . . . . . NG C S B ( s ) c . . . . . . . . . . . . . . . NG C S A B (r s ) b c . . . . . . . . . . . . . . . NG C S B ( s ) b . . . . . . . . . . . . . . . NG C S d . . . . . . . . . . . . . . . NG C S A ( s ) . . . . . . . . . . . . . . . NG C S A ( s ) c . . . . . . . . . . . . . . . NG C S A (r) b c . . . . . . . . . . . . . . . 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D., Katz N., 2011, MNRAS, 414, 1625 c (cid:13) , 1–29 egaMorph II T a b l e : G a l a xy S a m p l ea nd t h e fi t r e s u lt s a s d e r i v e d fr o m t h e m u lti - b a nd fi tti ng ; C o l u m n ( ) : G a l a xyn a m e ; C o l u m n ( ) : H ubb l e t yp e fr o m N E D ; C o l u m n ( - ) : a pp a r e n t m a gn it ud e s i n u , g , r , i , z p a ss b a nd s ; C o l u m n ( - ) : e ff ec ti v e r a d i u s ; C o l u m n ( - ) : S ´ e r s i c i nd e x . S ee S ec ti on3 f o r m o r e d e t a il s . N a m e T yp e m u m g m r m i m z r e , u r e , g r e , r r e , i r e , z n u n g n r n i n z ( m a g )( m a g )( m a g )( m a g )( m a g )( (cid:48)(cid:48) )( (cid:48)(cid:48) )( (cid:48)(cid:48) )( (cid:48)(cid:48) )( (cid:48)(cid:48) ) ( ± . )( ± . )( ± . )( ± . )( ± . )( ± % )( ± % )( ± % )( ± % )( ± % )( ± % )( ± % )( ± % )( ± % )( ± % ) I C S a . . . . . . . . . . . . . . . I C S B ( s ) b . . . . . . . . . . . . . . . I C S / Irr . . . . . . . . . . . . . . . I C S A B (r) c . . . . . . . . . . . . . . . I C E . . . . . . . . . . . . . . . I C S ( ) . . . . . . . . . . . . . . . I C E . . . . . . . . . . . . . . . NG C S A B ( s ) m . . . . . . . . . . . . . . . NG C S A B ( s ) c d . . . . . . . . . . . . . . . NG C S B (r s ) c . . . . . . . . . . . . . . . NG C S m p ec . . . . . . . . . . . . . . . NG C S A B (r s ) c . . . . . . . . . . . . . . . NG C S A B (r s ) c d . . . . . . . . . . . . . . . NG C ( R ) S A (r s ) b . . . . . . . . . . . . . . . NG C S A ( s ) c . . . . . . . . . . . . . . . NG C S A B (r s ) c . . . . . . . . . . . . . . . NG C S B (r s ) b . . . . . . . . . . . . . . . NG C S A ( s ) c d . . . . . . . . . . . . . . . NG C S B ( s ) b . . . . . . . . . . . . . . . NG C ( R ) S A (r) a . . . . . . . . . . . . . . . NG C S . . . . . . . . . . . . . . . NG C S A B (r s ) c . . . . . . . . . . . . . . . NG C S B (r) b . . . . . . . . . . . . . . . NG C S A ( s ) c . . . . . . . . . . . . . . . NG C S A (r) a b . . . . . . . . . . . . . . . NG C S A B (r s ) c . . . . . . . . . . . . . . . NG C S A (r) b . . . . . . . . . . . . . . . NG C S A ( s ) c . . . . . . . . . . . . . . . NG C S A B ( s ) c p ec . . . . . . . . . . . . . . . NG C S A B ( s ) c . . . . . . . . . . . . . . . NG C S A B d m . . . . . . . . . . . . . . . NG C S A B (r s ) b c . . . . . . . . . . . . . . . NG C S A B b c p ec . . . . . . . . . . . . . . . NG C S B (r s ) c . . . . . . . . . . . . . . . NG C S A ( s ) c d . . . . . . . . . . . . . . . NG C S A B (r s ) c . . . . . . . . . . . . . . . NG C S A B (r) c . . . . . . . . . . . . . . . NG C S B ( s ) c . . . . . . . . . . . . . . . NG C S A B (r s ) b c . . . . . . . . . . . . . . . NG C S B ( s ) b . . . . . . . . . . . . . . . NG C S d . . . . . . . . . . . . . . . NG C S A ( s ) . . . . . . . . . . . . . . . NG C S A ( s ) c . . . . . . . . . . . . . . . NG C S A (r) b c . . . . . . . . . . . . . . . NG C S A B (r s ) b c . . . . . . . . . . . . . . . NG C S A B (r s ) c . . . . . . . . . . . . . . . NG C S A B (r s ) c . . . . . . . . . . . . . . . NG C S A ( s ) a b . . . . . . . . . . . . . . . NG C S A ( s ) c . . . . . . . . . . . . . . . NG C S A B (r) b . . . . . . . . . . . . . . . NG C S B (r s ) b c . . . . . . . . . . . . . . . NG C S A ( s ) b c . . . . . . . . . . . . . . . NG C S A (r s ) b c . . . . . . . . . . . . . . . c (cid:13)000 , 1–29 Vika et al. T a b l e c on ti nu e d fr o m p r e v i ou s p a g e N a m e T yp e m u m g m r m i m z r e , u r e , g r e , r r e , i r e , z n u n g n r n i n z ( m a g )( m a g )( m a g )( m a g )( m a g )( (cid:48)(cid:48) )( (cid:48)(cid:48) )( (cid:48)(cid:48) )( (cid:48)(cid:48) )( (cid:48)(cid:48) ) ( ± . )( ± . )( ± . )( ± . )( ± . )( ± % )( ± % )( ± % )( ± % )( ± % )( ± % )( ± % )( ± % )( ± % )( ± % ) NG C S A B ( s ) b . . . . . . . . . . . . . . . NG C ( R ) S A c . . . . . . . . . . . . . . . NG C B S A B ( s ) dp ec . . . . . . . . . . . . . . . NG C S B (r s ) d m . . . . . . . . . . . . . . . NG C S B (r) c . . . . . . . . . . . . . . . NG C E . . . . . . . . . . . . . . . NG C S B (r) b . . . . . . . . . . . . . . . NG C S A (r) + . . . . . . . . . . . . . . . NG C S A ( s ) c . . . . . . . . . . . . . . . NG C S B (r) . . . . . . . . . . . . . . . NG C E - . . . . . . . . . . . . . . . NG C S B / a . . . . . . . . . . . . . . . NG C S . . . . . . . . . . . . . . . NG C S B ( s ) c . . . . . . . . . . . . . . . NG C ( R ) S B (r) a b . . . . . . . . . . . . . . . NG C S + . . . . . . . . . . . . . . . NG C S A B (r s ) b c . . . . . . . . . . . . . . . NG C S A B ( s ) b c . . . . . . . . . . . . . . . NG C E . . . . . . . . . . . . . . . NG C S - . . . . . . . . . . . . . . . NG C S A . . . . . . . . . . . . . . . NG C E . . . . . . . . . . . . . . . NG C E . . . . . . . . . . . . . . . NG C S a . . . . . . . . . . . . . . . NG C E . . . . . . . . . . . . . . . NG C ( R ) S A ( s ) a . . . . . . . . . . . . . . . NG C E . . . . . . . . . . . . . . . NG C S / a . . . . . . . . . . . . . . . NG C S A (r) . . . . . . . . . . . . . . . NG C E / S ( ) . . . . . . . . . . . . . . . NG C E / d S . . . . . . . . . . . . . . . NG C S A ( s ) a b . . . . . . . . . . . . . . . NG C S ( ) . . . . . . . . . . . . . . . NG C E / E . . . . . . . . . . . . . . . NG C S A (r) . . . . . . . . . . . . . . . NG C E . . . . . . . . . . . . . . . NG C E / S . . . . . . . . . . . . . . . NG C E . . . . . . . . . . . . . . . NG C S ec . . . . . . . . . . . . . . . NG C S A (r) . . . . . . . . . . . . . . . NG C E . . . . . . . . . . . . . . . NG C S A B ( s ) c . . . . . . . . . . . . . . . NG C E / E p ec . . . . . . . . . . . . . . . NG C A S B (r s ) m . . . . . . . . . . . . . . . NG C A S B (r s ) d m . . . . . . . . . . . . . . . NG C S B ( s ) c d . . . . . . . . . . . . . . . NG C S B . . . . . . . . . . . . . . . NG C E . . . . . . . . . . . . . . . NG C E . . . . . . . . . . . . . . . NG C E . . . . . . . . . . . . . . . NG C S ( ) . . . . . . . . . . . . . . . NG C S ( ) . . . . . . . . . . . . . . . NG C E . . . . . . . . . . . . . . . NG C S B + . . . . . . . . . . . . . . . NG C E / S - . . . . . . . . . . . . . . . NG C S - . . . . . . . . . . . . . . . c (cid:13) , 1–29 egaMorph II T a b l e c on ti nu e d fr o m p r e v i ou s p a g e N a m e T yp e m u m g m r m i m z r e , u r e , g r e , r r e , i r e , z n u n g n r n i n z ( m a g )( m a g )( m a g )( m a g )( m a g )( (cid:48)(cid:48) )( (cid:48)(cid:48) )( (cid:48)(cid:48) )( (cid:48)(cid:48) )( (cid:48)(cid:48) ) ( ± . )( ± . )( ± . )( ± . )( ± . )( ± % )( ± % )( ± % )( ± % )( ± % )( ± % )( ± % )( ± % )( ± % )( ± % ) NG C E . . . . . . . . . . . . . . . NG C S A B (r s ) c d . . . . . . . . . . . . . . . NG C E . . . . . . . . . . . . . . . NG C S B ( s ) d . . . . . . . . . . . . . . . NG C S A ( s ) a b . . . . . . . . . . . . . . . NG C S A B (r) a b . . . . . . . . . . . . . . . NG C ( R ) S A (r) a b . . . . . . . . . . . . . . . NG C S A ( s ) a b . . . . . . . . . . . . . . . NG C S B (r s ) c . . . . . . . . . . . . . . . NG C S B ( s ) c d . . . . . . . . . . . . . . . NG C S A (r s ) b c . . . . . . . . . . . . . . . NG C S B ( s ) d m . . . . . . . . . . . . . . . NG C S B ( s ) bp ec . . . . . . . . . . . . . . . NG C S A B (r) c . . . . . . . . . . . . . . . NG C S B (r s ) c . . . . . . . . . . . . . . . NG C S A (r s ) b c p ec . . . . . . . . . . . . . . . NG C S A B (r) b . . . . . . . . . . . . . . . NG C S B ( s ) b . . . . . . . . . . . . . . . NG C S A ( s ) c . . . . . . . . . . . . . . . NG C S A B (r s ) c d . . . . . . . . . . . . . . . NG C S . . . . . . . . . . . . . . . NG C S A B (r s ) c . . . . . . . . . . . . . . . NG C S c dp ec . . . . . . . . . . . . . . . NG C S A ( s ) d . . . . . . . . . . . . . . . NG C S B (r s ) d . . . . . . . . . . . . . . . NG C S A B (r s ) b c p ec . . . . . . . . . . . . . . . NG C S A (r s ) c . . . . . . . . . . . . . . . NG C S A B (r s ) d . . . . . . . . . . . . . . . NG C S A B ( s ) b . . . . . . . . . . . . . . . NG C S B (r) b . . . . . . . . . . . . . . . NG C S A (r s ) b c . . . . . . . . . . . . . . . NG C ( R ) S A B (r s ) b . . . . . . . . . . . . . . . NG C S A ( s ) c d . . . . . . . . . . . . . . . NG C S . . . . . . . . . . . . . . . NG C S A ( s ) a . . . . . . . . . . . . . . . NG C S A B (r s ) d . . . . . . . . . . . . . . . UG C A S B ( s ) d . . . . . . . . . . . . . . . UG C A S c p . . . . . . . . . . . . . . . UG C S A B ( s ) c d . . . . . . . . . . . . . . . UG C S B c . . . . . . . . . . . . . . . UG C S d . . . . . . . . . . . . . . . UG C S B . . . . . . . . . . . . . . . UG C S . . . . . . . . . . . . . . . UG C S B ( s ) c d . . . . . . . . . . . . . . . UG C S B ( s ) d m . . . . . . . . . . . . . . . UG C S B ( s ) d . . . . . . . . . . . . . . . UG C S B ( s ) d m . . . . . . . . . . . . . . . UG C ( R ) S B b . . . . . . . . . . . . . . . UG C S A B (r s ) c . . . . . . . . . . . . . . . UG C S B ( s ) d . . . . . . . . . . . . . . . UG C S c . . . . . . . . . . . . . . . UG C S A B ( s ) c . . . . . . . . . . . . . . . UG C S c d . . . . . . . . . . . . . . . UG C S A B ( s ) m . . . . . . . . . . . . . . . c (cid:13)000
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D., Katz N., 2011, MNRAS, 414, 1625 c (cid:13) , 1–29 egaMorph II T a b l e : G a l a xy S a m p l ea nd t h e fi t r e s u lt s a s d e r i v e d fr o m t h e m u lti - b a nd fi tti ng ; C o l u m n ( ) : G a l a xyn a m e ; C o l u m n ( ) : H ubb l e t yp e fr o m N E D ; C o l u m n ( - ) : a pp a r e n t m a gn it ud e s i n u , g , r , i , z p a ss b a nd s ; C o l u m n ( - ) : e ff ec ti v e r a d i u s ; C o l u m n ( - ) : S ´ e r s i c i nd e x . S ee S ec ti on3 f o r m o r e d e t a il s . N a m e T yp e m u m g m r m i m z r e , u r e , g r e , r r e , i r e , z n u n g n r n i n z ( m a g )( m a g )( m a g )( m a g )( m a g )( (cid:48)(cid:48) )( (cid:48)(cid:48) )( (cid:48)(cid:48) )( (cid:48)(cid:48) )( (cid:48)(cid:48) ) ( ± . )( ± . )( ± . )( ± . )( ± . )( ± % )( ± % )( ± % )( ± % )( ± % )( ± % )( ± % )( ± % )( ± % )( ± % ) I C S a . . . . . . . . . . . . . . . I C S B ( s ) b . . . . . . . . . . . . . . . I C S / Irr . . . . . . . . . . . . . . . I C S A B (r) c . . . . . . . . . . . . . . . I C E . . . . . . . . . . . . . . . I C S ( ) . . . . . . . . . . . . . . . I C E . . . . . . . . . . . . . . . NG C S A B ( s ) m . . . . . . . . . . . . . . . NG C S A B ( s ) c d . . . . . . . . . . . . . . . NG C S B (r s ) c . . . . . . . . . . . . . . . NG C S m p ec . . . . . . . . . . . . . . . NG C S A B (r s ) c . . . . . . . . . . . . . . . NG C S A B (r s ) c d . . . . . . . . . . . . . . . NG C ( R ) S A (r s ) b . . . . . . . . . . . . . . . NG C S A ( s ) c . . . . . . . . . . . . . . . NG C S A B (r s ) c . . . . . . . . . . . . . . . NG C S B (r s ) b . . . . . . . . . . . . . . . NG C S A ( s ) c d . . . . . . . . . . . . . . . NG C S B ( s ) b . . . . . . . . . . . . . . . NG C ( R ) S A (r) a . . . . . . . . . . . . . . . NG C S . . . . . . . . . . . . . . . NG C S A B (r s ) c . . . . . . . . . . . . . . . NG C S B (r) b . . . . . . . . . . . . . . . NG C S A ( s ) c . . . . . . . . . . . . . . . NG C S A (r) a b . . . . . . . . . . . . . . . NG C S A B (r s ) c . . . . . . . . . . . . . . . NG C S A (r) b . . . . . . . . . . . . . . . NG C S A ( s ) c . . . . . . . . . . . . . . . NG C S A B ( s ) c p ec . . . . . . . . . . . . . . . NG C S A B ( s ) c . . . . . . . . . . . . . . . NG C S A B d m . . . . . . . . . . . . . . . NG C S A B (r s ) b c . . . . . . . . . . . . . . . NG C S A B b c p ec . . . . . . . . . . . . . . . NG C S B (r s ) c . . . . . . . . . . . . . . . NG C S A ( s ) c d . . . . . . . . . . . . . . . NG C S A B (r s ) c . . . . . . . . . . . . . . . NG C S A B (r) c . . . . . . . . . . . . . . . NG C S B ( s ) c . . . . . . . . . . . . . . . NG C S A B (r s ) b c . . . . . . . . . . . . . . . NG C S B ( s ) b . . . . . . . . . . . . . . . NG C S d . . . . . . . . . . . . . . . NG C S A ( s ) . . . . . . . . . . . . . . . NG C S A ( s ) c . . . . . . . . . . . . . . . NG C S A (r) b c . . . . . . . . . . . . . . . NG C S A B (r s ) b c . . . . . . . . . . . . . . . NG C S A B (r s ) c . . . . . . . . . . . . . . . NG C S A B (r s ) c . . . . . . . . . . . . . . . NG C S A ( s ) a b . . . . . . . . . . . . . . . NG C S A ( s ) c . . . . . . . . . . . . . . . NG C S A B (r) b . . . . . . . . . . . . . . . NG C S B (r s ) b c . . . . . . . . . . . . . . . NG C S A ( s ) b c . . . . . . . . . . . . . . . NG C S A (r s ) b c . . . . . . . . . . . . . . . c (cid:13)000 , 1–29 Vika et al. T a b l e c on ti nu e d fr o m p r e v i ou s p a g e N a m e T yp e m u m g m r m i m z r e , u r e , g r e , r r e , i r e , z n u n g n r n i n z ( m a g )( m a g )( m a g )( m a g )( m a g )( (cid:48)(cid:48) )( (cid:48)(cid:48) )( (cid:48)(cid:48) )( (cid:48)(cid:48) )( (cid:48)(cid:48) ) ( ± . )( ± . )( ± . )( ± . )( ± . )( ± % )( ± % )( ± % )( ± % )( ± % )( ± % )( ± % )( ± % )( ± % )( ± % ) NG C S A B ( s ) b . . . . . . . . . . . . . . . NG C ( R ) S A c . . . . . . . . . . . . . . . NG C B S A B ( s ) dp ec . . . . . . . . . . . . . . . NG C S B (r s ) d m . . . . . . . . . . . . . . . NG C S B (r) c . . . . . . . . . . . . . . . NG C E . . . . . . . . . . . . . . . NG C S B (r) b . . . . . . . . . . . . . . . NG C S A (r) + . . . . . . . . . . . . . . . NG C S A ( s ) c . . . . . . . . . . . . . . . NG C S B (r) . . . . . . . . . . . . . . . NG C E - . . . . . . . . . . . . . . . NG C S B / a . . . . . . . . . . . . . . . NG C S . . . . . . . . . . . . . . . NG C S B ( s ) c . . . . . . . . . . . . . . . NG C ( R ) S B (r) a b . . . . . . . . . . . . . . . NG C S + . . . . . . . . . . . . . . . NG C S A B (r s ) b c . . . . . . . . . . . . . . . NG C S A B ( s ) b c . . . . . . . . . . . . . . . NG C E . . . . . . . . . . . . . . . NG C S - . . . . . . . . . . . . . . . NG C S A . . . . . . . . . . . . . . . NG C E . . . . . . . . . . . . . . . NG C E . . . . . . . . . . . . . . . NG C S a . . . . . . . . . . . . . . . NG C E . . . . . . . . . . . . . . . NG C ( R ) S A ( s ) a . . . . . . . . . . . . . . . NG C E . . . . . . . . . . . . . . . NG C S / a . . . . . . . . . . . . . . . NG C S A (r) . . . . . . . . . . . . . . . NG C E / S ( ) . . . . . . . . . . . . . . . NG C E / d S . . . . . . . . . . . . . . . NG C S A ( s ) a b . . . . . . . . . . . . . . . NG C S ( ) . . . . . . . . . . . . . . . NG C E / E . . . . . . . . . . . . . . . NG C S A (r) . . . . . . . . . . . . . . . NG C E . . . . . . . . . . . . . . . NG C E / S . . . . . . . . . . . . . . . NG C E . . . . . . . . . . . . . . . NG C S ec . . . . . . . . . . . . . . . NG C S A (r) . . . . . . . . . . . . . . . NG C E . . . . . . . . . . . . . . . NG C S A B ( s ) c . . . . . . . . . . . . . . . NG C E / E p ec . . . . . . . . . . . . . . . NG C A S B (r s ) m . . . . . . . . . . . . . . . NG C A S B (r s ) d m . . . . . . . . . . . . . . . NG C S B ( s ) c d . . . . . . . . . . . . . . . NG C S B . . . . . . . . . . . . . . . NG C E . . . . . . . . . . . . . . . NG C E . . . . . . . . . . . . . . . NG C E . . . . . . . . . . . . . . . NG C S ( ) . . . . . . . . . . . . . . . NG C S ( ) . . . . . . . . . . . . . . . NG C E . . . . . . . . . . . . . . . NG C S B + . . . . . . . . . . . . . . . NG C E / S - . . . . . . . . . . . . . . . NG C S - . . . . . . . . . . . . . . . c (cid:13) , 1–29 egaMorph II T a b l e c on ti nu e d fr o m p r e v i ou s p a g e N a m e T yp e m u m g m r m i m z r e , u r e , g r e , r r e , i r e , z n u n g n r n i n z ( m a g )( m a g )( m a g )( m a g )( m a g )( (cid:48)(cid:48) )( (cid:48)(cid:48) )( (cid:48)(cid:48) )( (cid:48)(cid:48) )( (cid:48)(cid:48) ) ( ± . )( ± . )( ± . )( ± . )( ± . )( ± % )( ± % )( ± % )( ± % )( ± % )( ± % )( ± % )( ± % )( ± % )( ± % ) NG C E . . . . . . . . . . . . . . . NG C S A B (r s ) c d . . . . . . . . . . . . . . . NG C E . . . . . . . . . . . . . . . NG C S B ( s ) d . . . . . . . . . . . . . . . NG C S A ( s ) a b . . . . . . . . . . . . . . . NG C S A B (r) a b . . . . . . . . . . . . . . . NG C ( R ) S A (r) a b . . . . . . . . . . . . . . . NG C S A ( s ) a b . . . . . . . . . . . . . . . NG C S B (r s ) c . . . . . . . . . . . . . . . NG C S B ( s ) c d . . . . . . . . . . . . . . . NG C S A (r s ) b c . . . . . . . . . . . . . . . NG C S B ( s ) d m . . . . . . . . . . . . . . . NG C S B ( s ) bp ec . . . . . . . . . . . . . . . NG C S A B (r) c . . . . . . . . . . . . . . . NG C S B (r s ) c . . . . . . . . . . . . . . . NG C S A (r s ) b c p ec . . . . . . . . . . . . . . . NG C S A B (r) b . . . . . . . . . . . . . . . NG C S B ( s ) b . . . . . . . . . . . . . . . NG C S A ( s ) c . . . . . . . . . . . . . . . NG C S A B (r s ) c d . . . . . . . . . . . . . . . NG C S . . . . . . . . . . . . . . . NG C S A B (r s ) c . . . . . . . . . . . . . . . NG C S c dp ec . . . . . . . . . . . . . . . NG C S A ( s ) d . . . . . . . . . . . . . . . NG C S B (r s ) d . . . . . . . . . . . . . . . NG C S A B (r s ) b c p ec . . . . . . . . . . . . . . . NG C S A (r s ) c . . . . . . . . . . . . . . . NG C S A B (r s ) d . . . . . . . . . . . . . . . NG C S A B ( s ) b . . . . . . . . . . . . . . . NG C S B (r) b . . . . . . . . . . . . . . . NG C S A (r s ) b c . . . . . . . . . . . . . . . NG C ( R ) S A B (r s ) b . . . . . . . . . . . . . . . NG C S A ( s ) c d . . . . . . . . . . . . . . . NG C S . . . . . . . . . . . . . . . NG C S A ( s ) a . . . . . . . . . . . . . . . NG C S A B (r s ) d . . . . . . . . . . . . . . . UG C A S B ( s ) d . . . . . . . . . . . . . . . UG C A S c p . . . . . . . . . . . . . . . UG C S A B ( s ) c d . . . . . . . . . . . . . . . UG C S B c . . . . . . . . . . . . . . . UG C S d . . . . . . . . . . . . . . . UG C S B . . . . . . . . . . . . . . . UG C S . . . . . . . . . . . . . . . UG C S B ( s ) c d . . . . . . . . . . . . . . . UG C S B ( s ) d m . . . . . . . . . . . . . . . UG C S B ( s ) d . . . . . . . . . . . . . . . UG C S B ( s ) d m . . . . . . . . . . . . . . . UG C ( R ) S B b . . . . . . . . . . . . . . . UG C S A B (r s ) c . . . . . . . . . . . . . . . UG C S B ( s ) d . . . . . . . . . . . . . . . UG C S c . . . . . . . . . . . . . . . UG C S A B ( s ) c . . . . . . . . . . . . . . . UG C S c d . . . . . . . . . . . . . . . UG C S A B ( s ) m . . . . . . . . . . . . . . . c (cid:13)000 , 1–29 Vika et al. c (cid:13) , 1–29 egaMorph II Table 2.
Subsample of elliptical galaxies. The galaxy type and the galactic correction ( A g ) are from NED (and in brackets from C93). The absolutemagnitude M ∗ B , the effective radius ( r ∗ e ,B ) and the S´ersic index ( n ∗ eq ,B ) have been derived from one-dimensional fitting in C93. The M B is our g -bandmagnitude after conversion to the B -band and Vega system. The r e ,g and the n g are our g -band effective radius and S´ersic index. The distances havecome from Blakeslee et al. (2009) unless otherwise specified: a Tonry et al. (2001), b Jerjen et al. (2004). In the case of NGC4360 a mean Virgo distancehas been adopted.Galaxy VCC Type M ∗ B r ∗ e ,B n ∗ eq ,B distance A g M B r e ,g n g (mag) (arcsec) (Mpc) (mag) (mag) (arcsec)IC3468 VCC1422 E1 − .
89 21 .
08 2 .
21 15 . . − .
99 19 .
73 1 . IC3653 VCC1871 E3 − .
34 6 .
23 1 .
82 15 . . − .
35 7 .
23 1 . NGC4168 VCC49 E2 − .
23 17 .
44 2 .
53 30 . a . − .
43 31 .
07 2 . NGC4261 VCC345 E2-3 − .
76 124 .
53 7 .
65 31 . a . − .
44 45 .
52 4 . NGC4339 VCC648 E0 (S0) − .
60 27 .
57 3 .
50 16 . a . − .
75 26 .
26 3 . NGC4360 VCC722 E − .
48 18 .
92 3 .
33 16 . . − .
80 18 .
47 4 . NGC4365 VCC731 E3 − .
52 117 .
51 6 .
08 23 . . − .
42 74 .
72 4 . NGC4374 VCC763 E1 − .
69 190 .
85 8 .
47 18 . . − .
21 66 .
93 4 . NGC4387 VCC828 E5 − .
99 10 .
23 2 .
12 18 . . − .
12 13 .
31 2 . NGC4415 VCC929 S0/a (dE1) − .
02 18 .
59 1 .
85 14 . b . − .
23 19 .
46 1 . NGC4434 VCC1025 E0/S0(0) (E0) − .
74 14 .
45 4 .
19 22 . . − .
80 12 .
15 3 . NGC4436 VCC1036 dE6/dS0 (dS0) − .
81 16 .
07 2 .
40 15 . b . − .
84 21 .
15 2 . NGC4458 VCC1146 E0/E1 − .
03 20 .
03 2 .
84 16 . . − .
32 23 .
14 4 . NGC4464 VCC1178 E3 − .
07 6 .
52 2 .
61 13 . . − .
14 7 .
72 3 . NGC4472 VCC1226 E2/S0 (E2) − .
18 240 .
12 6 .
24 16 . . − .
00 121 .
94 4 . NGC4473 VCC1231 E5 − .
79 30 .
29 5 .
29 15 . . − .
86 37 .
03 4 . NGC4478 VCC1279 E2 − .
71 11 .
69 1 .
98 17 . . − .
80 12 .
84 1 . NGC4486 VCC1316 E0/E pec − .
76 170 .
91 6 .
51 16 . . − .
70 94 .
62 3 . NGC4551 VCC1630 E − .
84 12 .
71 1 .
89 16 . . − .
07 15 .
73 2 . NGC4552 VCC1632 E (S0(0)) − .
76 151 .
06 12 .
81 16 . . − .
43 44 .
72 4 . NGC4564 VCC1664 E6 − .
73 17 .
49 2 .
42 15 . . − .
98 24 .
38 3 . NGC4621 VCC1903 E5 − .
36 82 .
98 6 .
14 14 . . − .
46 72 .
64 5 . NGC4623 VCC1913 SB0+ ( E7) − .
30 16 .
68 1 .
66 13 . . − .
38 25 .
90 2 . NGC4636 VCC1939 E/S0-1 (E1) − .
73 137 .
61 2 .
86 14 . a . − .
91 105 .
57 3 . NGC4649 VCC1978 E2 (S02) − .
52 117 .
49 5 .
84 16 . . − .
49 74 .
56 3 . NGC4660 VCC2000 E5 − .
70 11 .
14 3 .
87 15 . . − .
71 13 .
26 3 . c (cid:13)000