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+{"text":"\\section{Catalogue construction}\n\nWhilst semi-analytic models have proven useful in modelling and predicting numerous properties of the galaxy population (Kauffmann, White \\& Guiderdoni \\cite{kwg}; Cole et al \\cite{coleetal94}; Somerville \\& Primack \\cite{somervilleprimack} they provide only limited information on the spatial distribution of galaxies, and so it has been difficult to study in detail the clustering properties of galaxies using these models. Recently, semi-analytic models have been combined with N-body simulations to provide the required spatial information (Kauffmann, Nusser \\& Steinmetz \\cite{kns}; Governato et al \\cite{fabio}; Kauffmann et al \\cite{gketal}). We have used a similar technique to study the clustering of galaxies in CDM universes within well constrained semi-analytic models. The main difference between our technique and that of Kauffmann et al \\cite{gketal} is that whilst they extract the merging history of each dark matter halo directly from the simulation, we construct this history using the extended Press-Schechter theory. We find that this gives the same statistical results as the Kauffmann et al method and allows us to resolve merger trees to much smaller masses.\n\nTo construct a catalogue of galaxies containing spatial information we use the following procedure: (i) take the output from a dissipationless N-body simulation of dark matter and use a group finding algorithm (here we use the friends-of-friends algorithm with the standard linking length of $b=0.2$) to locate bound, virialised haloes of dark matter of 10 or more particles (such groups have been shown to be stable by Kauffmann et al \\cite{gketal}); (ii) determine the mass of each group, the position and velocity of its centre of mass and the positions and velocities of randomly selected particles within the group; (iii) for each group use a semi-analytic model of galaxy formation constrained to match the local B and K-band luminosity functions to determine the population of galaxies living within the dark matter halo; (iv) attach the central galaxy of the halo to the centre of mass of the group and attach any satellite galaxies to randomly selected particles within the halo so that galaxies trace mass within a given dark matter halo (which may not be exactly true in reality because of processes such as dynamical friction).\n\nThis results in a galaxy catalogue which can be analysed to determine the clustering properties of galaxies of any given luminosity, morphology, colour and so on. In fact by using this technique it is possible to produce catalogues of galaxies complete with spatial information (or alternatively redshifts, and angular coordinates) with any observationally motivated selection criteria. Furthermore, by identifying dark matter halos on the past lightcone of an observer a full kock galaxy redshift survey can be constructed. The luminosity functions determined from the galaxy catalogue are shown in Figure \\ref{fig1}. Although they become incomplete at the faint end because of the limited resolution of the N-body simulation there is good agreement between the bright ends and the observed luminosity functions. Since we only consider the clustering of galaxies for which our catalogue is complete the resolution limit is not important. We find however that an accurate match to the bright end of the luminosity function is important as it strongly constrains the resulting two-point correlation function. An example of the information produced by our model is given in Figure \\ref{fig2}, which shows a slice through a $\\Lambda$CDM N-body simulation upon which the positions of galaxies brighter than $M_{\\mathrm B} - 5 \\log h = -19.5$ (we define the Hubble constant to be ${\\mathrm H}_0 = 100 h$ km s$^{-1}$ Mpc$^{-1}$) have been overlaid as circles. It can be seen quite clearly that the galaxies trace the mass to some extent. To determine exactly how well galaxies trace the underlying dark matter we estimate the two-point correlation function of these galaxies.\n\n\\begin{figure}\n\\centering\n\\begin{tabular}{cc}\n\\psfig{file=benson-fig1a.ps,width=80mm} & \\psfig{file=benson-fig1b.ps,width=80mm}\n\\end{tabular}\n\\caption{The local B and K-band luminosity functions from our model compared to various observational determinations. Note the good agreement between model (solid line) and observations (symbols) at the bright end. Our galaxy catalogues become incomplete at the faint end of the luminosity functions (faintwards of $M_{\\mathrm B} - 5 \\log h \\approx -17.5$ in the B-band and $M_{\\mathrm K} - 5 \\log h \\approx -20.5$ in the K-band) due to the limited resolution of the N-body simulation. Our catalogues are constructed using only galaxies for which a complete sample is available within the model.}\n\\label{fig1}\n\\end{figure}\n\n\\begin{figure}\n\\centering\n\\hspace{0.in}\\psfig{file=benson-fig2.ps,width=100mm}\n\\caption{A slice through a $\\Lambda$CDM N-body simulation. The volume shown is 141 $\\times$ 141 $\\times$ 8 $h^{-1}$ Mpc. Dark matter is shown by the greyscale, with the darker areas being the most dense. Overlaid are the positions of all galaxies brighter than $M_{\\mathrm B} - 5 \\log h = -19.5$ indicated by circles.}\n\\label{fig2}\n\\end{figure}\n\n\\section{The two-point correlation function}\n\n\\begin{figure}[t]\n\\centering\n\\hspace{0.in}\\psfig{file=benson-fig3.ps,width=100mm}\n\\caption{The correlation function of galaxies brighter than $M_{\\mathrm B} - 5 \\log h = -19.5$ in a $\\Lambda$CDM cosmology. Points with errorbars show the APM galaxy correlation function of Baugh \\cite{APM}. The dotted line shows the dark matter correlation function whilst the solid line shows the correlation function of galaxies in our model (the dashed lines to either side indicate the Poisson errors on this correlation function).}\n\\label{fig3}\n\\end{figure}\n\nShown in Figure \\ref{fig3} is the two-point correlation function of galaxies brighter than $M_{\\mathrm B} - 5 \\log h = -19.5$ in our $\\Lambda$CDM model (which has $\\Omega _0 = 0.3$, $\\Lambda = 0.7$, $h = 0.7$ and $\\sigma _8 = 0.9$). This is compared to the correlation function of the underlying dark matter and to the observationally determined correlation function of galaxies in the APM survey from Baugh \\cite{APM}. Note that all of the curves shown here are real space correlation functions. The galaxy correlation function shows many interesting properties. Firstly it is biased with respect to the mass correlation function, and furthermore this bias is scale dependent. On small scales there is in fact an antibias, which is exactly what is needed to reconcile the theory with the observed galaxy correlation function. The observed and model galaxy correlation functions agree over a wide range of scales, both showing approximate power law behaviour. The scale-dependant bias seen in these models arises from a complex interplay of effects. On large scales the bias is due to the intrinsic bias of dark matter halos in a CDM universe as described by Mo \\& White \\cite{mowhite}. On smaller scales the bias is controlled by the way halos are populated with galaxies, specifically the variations in number of galaxies per halo for halos of a given mass. The Lagrangian radius exclusion of halos also affects the small scale bias. These issues are explored in greater detail by Benson et al \\cite{meetal}, who also study $\\Omega _0 = 1$ models and find that such models fail to reproduce the observed clustering of galaxies.\n\n\\acknowledgements{}\n\nAJB's attendance at the X$^{\\mathrm th}$ Rencontres de Blois was funded in part by a grant from the European Commision.\n\n\n\\begin{bloisbib}\n\\bibitem{APM} Baugh C. M., 1996, \\mnras {280} {267}\n\\bibitem{meetal} Benson A. J., Cole S., Frenk C. S., Baugh C. M., Lacey C. G., 1998, {\\it in preparation}\n\\bibitem{coleetal94} Cole S., Lacey C. G., Baugh C. M., Frenk C. S., 1994, \\mnras {271} {781}\n\\bibitem{coleetal98} Cole S., Lacey C. G., Baugh C. M., Frenk C. S., 1998, {\\it in preparation}\n\\bibitem{colless} Colless M., Boyle B., 1997, {\\it astro-ph\/9710268}\n\\bibitem{gardner} Gardner J. P., Sharples R. M., Frenk C. S., Carrasco B. E., 1997, \\apj {480} {L99}\n\\bibitem{glazebrook} Glazebrook K. et al, 1995, {\\it astro-ph\/9503116}\n\\bibitem{fabio} Governato F. et al., 1998, \\nat {392} {359}\n\\bibitem{kwg} Kauffmann G., White S. D. M., Guiderdoni B., 1993, \\mnras {264} {201}\n\\bibitem{kns} Kauffmann G., Nusser A., Steinmetz M., 1997, \\mnras {286} {795}\n\\bibitem{gketal} Kauffmann G., Colberg J. M., Diaferio A., White S. D. M., 1998, {\\it astro-ph\/9805283}\n\\bibitem{loveday} Loveday J., Peterson B. A., Efstathiou G., Maddox S. J., 1992, \\apj {390} {338}\n\\bibitem{mowhite} Mo H. J., White S. D. M., 1996, \\mnras {282} {347}\n\\bibitem{arat} Ratcliffe A., Shanks T., Parker Q. A., Fong R., 1998, \\mnras {296} {173}\n\\bibitem{somervilleprimack} Somerville R., Primack J., 1998, {\\it astro-ph\/9802268}\n\\bibitem{szokoly} Szokoly G.P., Subbarao M.U., Conolly A.J., Mobasher B., 1998, {\\it astro-ph\/9801132}\n\\bibitem{zucca} Zucca E. et al, 1997, \\aa {326} {477}\n\\end{bloisbib}\n\\vfill\n\\end{document}\n\n\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:introduction}\n\n\nThe (original) Caldero-Chapoton map $X$ is an important object in\ncluster theory.  The arguments of $X$ are certain objects of a cluster category, and the values are the corresponding elements of a cluster algebra.  The map $X$ expresses\nthat the cluster category is a categorification of the\ncluster algebra, see \\cite{CC}, \\cite{CK}, \\cite{CK2}, \\cite{FK},\n\\cite{Palu}.  For example, Figure \\ref{fig:AR_quiver} shows the\nAuslander-Reiten (AR) quiver of $\\mathsf{C}( A_5 )$, the cluster category of\nDynkin type $A_5$, with a useful ``coordinate system''.  Figure\n\\ref{fig:frieze} shows the AR quiver again, with the values of $X$ on\nthe indecomposable objects of $\\mathsf{C}( A_5 )$.  The values are Laurent\npolynomials over $\\mathbb{Z}$; indeed, the cluster algebra consists of such\nLaurent polynomials.\n\\begin{figure}\n\\[\n  \\xymatrix @+1.8pc @!0 {\n    *+[blue]{\\{\\, 5,7 \\,\\}} \\ar[dr] && \\{\\, 6,8 \\,\\} \\ar[dr] && *+[blue]{\\{\\, 1,7 \\,\\}} \\ar[dr] && \\{\\, 2,8 \\,\\} \\ar[dr] && \\{\\, 1,3 \\,\\} \\ar@{.}[dd] \\\\\n    & \\{\\, 5,8 \\,\\} \\ar[dr] \\ar[ur] && \\{\\, 1,6 \\,\\} \\ar[dr] \\ar[ur] && *+[red]{\\{\\, 2,7 \\,\\}} \\ar[dr] \\ar[ur] && \\{\\, 3,8 \\,\\} \\ar[dr] \\ar[ur] \\\\\n    \\{\\, 4,8 \\,\\} \\ar[dr] \\ar[ur] \\ar@{.}[uu] \\ar@{.}[dd] && \\{\\, 1,5 \\,\\} \\ar[dr] \\ar[ur] && \\{\\, 2,6 \\,\\} \\ar[dr] \\ar[ur] && \\{\\, 3,7 \\,\\} \\ar[dr] \\ar[ur] && \\{\\, 4,8 \\,\\} \\ar@{.}[uu] \\ar@{.}[dd] \\\\\n    & \\{\\, 1,4 \\,\\} \\ar[dr] \\ar[ur] && *+[red]{\\{\\, 2,5 \\,\\}} \\ar[dr] \\ar[ur] && \\{\\, 3,6 \\,\\} \\ar[dr] \\ar[ur] && \\{\\, 4,7 \\,\\} \\ar[dr] \\ar[ur] \\\\\n    \\{\\, 1,3 \\,\\} \\ar[ur] && *+[blue]{\\{\\, 2,4 \\,\\}} \\ar[ur] && \\{\\, 3,5 \\,\\} \\ar[ur] && \\{\\, 4,6 \\,\\} \\ar[ur] && *+[blue]{\\{\\, 5,7 \\,\\}} \\\\\n                        }\n\\]\n\\caption{The Auslander-Reiten quiver of the cluster category $\\mathsf{C}( A_5\n  )$.  The dotted lines should be identified with opposite\n  orientations.  The red vertices show the direct summands of a rigid\n  object $R$, and the red and blue vertices show the direct summands\n  of a cluster tilting object $T$.} \n\\label{fig:AR_quiver}\n\\end{figure}\n\\begin{figure}\n\\[\n  \\xymatrix @+1.8pc @!0 {\n    z \\ar[dr]&& \\frac{ ux + uy + yz + z }{ uyz } \\ar[dr]&& u \\ar[dr]&& \\frac{ y + 1 }{ u } \\ar[dr]&& \\frac{ uvx + vz + xy + y }{ vxy } \\\\\n    & \\frac{ ux + yz + z }{ uy } \\ar[dr] \\ar[ur]&& \\frac{ ux + uy + z }{ yz } \\ar[dr] \\ar[ur]&& y \\ar[dr] \\ar[ur]&& \\frac{ uvx + vyz + vz + xy + xy^2 + y+y^2 }{ uvxy } \\ar[dr] \\ar[ur]\\\\\n    \\frac{ uvx + vyz + vz + y + y^2 }{ uxy } \\ar[dr] \\ar[ur]\\ar@{.}[uu] \\ar@{.}[dd] && \\frac{ ux + z }{ y } \\ar[dr] \\ar[ur]&& \\frac{ x + y }{ z } \\ar[dr] \\ar[ur]&& \\frac{ vz + xy +  y }{ vx } \\ar[dr] \\ar[ur]&& \\frac{ uvx + vyz + vz + y + y^2 }{ uxy } \\ar@{.}[uu] \\ar@{.}[dd] \\\\\n    & \\frac{ uvx + vz + y }{ xy } \\ar[dr] \\ar[ur]&& x \\ar[dr] \\ar[ur]&& \\frac{ vz + x + x^2 + xy + y }{ vxz } \\ar[dr] \\ar[ur]&& \\frac{ vz + y }{ x } \\ar[dr] \\ar[ur]\\\\\n    \\frac{ uvx + vz + xy + y }{ vxy }\\ar[ur] && v \\ar[ur]&& \\frac{ x + 1 }{ v } \\ar[ur]&& \\frac{ vz + x + y }{ xz } \\ar[ur]&& z \\\\\n                        }\n\\]\n\\caption{The Auslander-Reiten quiver of $\\mathsf{C}( A_5 )$ with values of\n  the original Caldero-Chapoton map $X$.  The map depends on the\n  cluster tilting object $T$ shown by red and blue vertices in Figure\n  \\ref{fig:AR_quiver}.}\n\\label{fig:frieze}\n\\end{figure}\n\nIt is a salient property of $X$ that it is a {\\em frieze} in\nthe sense of \\cite{AD}, that is, if $\\tau c \\rightarrow b \\rightarrow\nc$ is an AR triangle then\n\\[\n  X( \\tau c )X( c ) - X( b ) = 1,\n\\]\nsee \\cite[theorem]{DG} and \\cite[prop.\\ 3.10]{CC}.  In the case of\n$\\mathsf{C}( A_5 )$, this means that for each ``diamond'' in the AR quiver,\nof the form\n\\begin{equation}\n\\label{equ:diamond}\n\\vcenter{\n  \\xymatrix @-1.2pc {\n    & b_1 \\ar[dr] & \\\\ \\tau c \\ar[ur] \\ar[dr] & & c \\lefteqn{,} \\\\ & b_2 \\ar[ur] &\n                    }\n        }\n\\end{equation}\nwe have\n\\[\n  X( \\tau c )X( c ) - X( b_1 )X( b_2 ) = 1.\n\\]\n\nThe definition of $X$ depends on a cluster tilting object $T$.  For\ninstance, the $X$ shown in Figure \\ref{fig:frieze} depends on the $T$\nwhich has the indecomposable summands shown by red and blue vertices in\nFigure \\ref{fig:AR_quiver}.\n\nThis paper is about a modified Caldero-Chapoton map $\\rho$ which is\nmore general than $X$ in two respects: it depends on a rigid object\n$R$ and has values in a general commutative ring $A$.  An object $R$\nis rigid if $\\operatorname{Hom}( R,\\Sigma R ) = 0$.  This is much weaker than being\ncluster tilting: recall that $T$ is cluster tilting if $\\operatorname{Hom}( T ,\n\\Sigma t ) = 0 \\Leftrightarrow t \\in \\operatorname{add} T \\Leftrightarrow \\operatorname{Hom}( t ,\n\\Sigma T ) = 0$.  Our first main result gives conditions under which\n$\\rho$ is a {\\em generalised frieze}, in the sense that if $\\tau c\n\\rightarrow b \\rightarrow c$ is an AR triangle then\n\\[\n  \\rho( \\tau c )\\rho( c ) - \\rho( b ) \\in \\{\\, 0,1 \\,\\}.\n\\]\nOur second main result is that the conditions can be satisfied if\n$A$ is chosen to be a Laurent polynomial ring over the integers.\n\nGeneralised friezes with values in the integers were introduced by\ncombinatorial means in \\cite{BHJ}, and it was shown in \\cite{HJ} that\nthey can be recovered from a modified Caldero-Chapoton map.  The\ntheory of \\cite{HJ} and the original Caldero-Chapoton map are both special cases of the theory developed here.\n\nFor example, consider $\\mathsf{C}( A_5 )$ again and let $R$ be the rigid\nobject which has the indecomposable summands shown by red vertices in\nFigure \\ref{fig:AR_quiver}.  Our results imply that we can choose $A =\n\\mathbb{Z}[ u^{ \\pm 1 } , v^{ \\pm 1 } , z^{ \\pm 1 } ]$, and Figure\n\\ref{fig:generalised_frieze} shows the AR quiver of $\\mathsf{C}( A_5 )$ with\nthe values of $\\rho$ on the indecomposable objects.\n\\begin{figure}\n\\[\n  \\xymatrix @+1.8pc @!0 {\n    z \\ar[dr]\\ar@{.}[dd] && \\frac{ u+z }{ uz } \\ar[dr] && u \\ar[dr]&& \\frac{ 1 }{ u } \\ar[dr]&& \\frac{ 1+uv+vz }{ v } \\ar@{.}[dd] \\\\\n    \\begin{tikzpicture}[xscale=0.5,yscale=0.5,baseline=-0.5ex] \\fill[gray!25] (-1,0) -- (0,1) -- (1,0) -- (0,-1) -- cycle; \\end{tikzpicture}& \\frac{ u+z }{ u } \\ar[dr] \\ar[ur]&& \\frac{ u+z }{ z } \\ar[dr] \\ar[ur]&\\begin{tikzpicture}[xscale=0.5,yscale=0.5,baseline=-0.5ex] \\fill[gray!25] (-1,0) -- (0,1) -- (1,0) -- (0,-1) -- cycle; \\end{tikzpicture}& 1 \\ar[dr] \\ar[ur]&\\begin{tikzpicture}[xscale=0.5,yscale=0.5,baseline=-0.5ex] \\fill[gray!25] (-1,0) -- (0,1) -- (1,0) -- (0,-1) -- cycle; \\end{tikzpicture}& \\frac{ 1+uv+vz }{ uv } \\ar[dr] \\ar[ur]\\\\\n    \\frac{ 1+uv+vz }{ u } \\ar[dr] \\ar[ur]\\ar@{.}[dd] && u+z \\ar[dr] \\ar[ur]&& \\frac{ 1 }{ z } \\ar[dr] \\ar[ur] && \\frac{ 1+vz }{ v } \\ar[dr] \\ar[ur]&& \\frac{ 1+uv+vz }{ u } \\ar@{.}[dd] \\\\\n    & 1+uv+vz \\ar[dr] \\ar[ur]&\\begin{tikzpicture}[xscale=0.5,yscale=0.5,baseline=-0.5ex] \\fill[gray!25] (-1,0) -- (0,1) -- (1,0) -- (0,-1) -- cycle; \\end{tikzpicture}& 1 \\ar[dr] \\ar[ur]&\\begin{tikzpicture}[xscale=0.5,yscale=0.5,baseline=-0.5ex] \\fill[gray!25] (-1,0) -- (0,1) -- (1,0) -- (0,-1) -- cycle; \\end{tikzpicture}& \\frac{ 1+vz }{ vz } \\ar[dr] \\ar[ur]&& 1+vz \\ar[dr] \\ar[ur]&\\begin{tikzpicture}[xscale=0.5,yscale=0.5,baseline=-0.5ex] \\fill[gray!25] (-1,0) -- (0,1) -- (1,0) -- (0,-1) -- cycle; \\end{tikzpicture} \\\\\n    \\frac{ 1+uv+vz }{ v } \\ar[ur]&& v \\ar[ur]&& \\frac{ 1 }{ v } \\ar[ur]&& \\frac{ 1+vz }{ z } \\ar[ur] && z \\\\\n                        }\n\\]\n\\caption{The Auslander-Reiten quiver of $\\mathsf{C}( A_5 )$ with values of\n  the modified Caldero-Chapoton map $\\rho$.  The map depends on the\n  rigid object $R$ shown by red vertices in Figure\n  \\ref{fig:AR_quiver}.  The values form a generalised frieze, the grey\n  diamonds indicating where $\\rho( \\tau c )\\rho( c ) - \\rho( b_1\n  )\\rho( b_2 ) = 1$.}\n\\label{fig:generalised_frieze}\n\\end{figure}\nIn this case, the generalised frieze property means that for\neach ``diamond'' in the AR quiver, of the form \\eqref{equ:diamond}, we\nhave\n\\[\n  \\rho( \\tau c )\\rho( c ) - \\rho( b_1 )\\rho( b_2 )\n  \\in \\{\\, 0,1 \\,\\}.\n\\]\nThe solid grey diamonds in Figure \\ref{fig:generalised_frieze} indicate where the displayed expression is equal to $1$.\n\nLet us explain how $\\rho$ is defined.  Let $k$ be an algebraically\nclosed field, $\\mathsf{C}$ an essentially small $\\operatorname{Hom}$-finite $k$-linear\ntriangulated category.  Assume that $\\mathsf{C}$ has split idempotents and\nhas a Serre functor.  Note that these are the only assumptions on\n$\\mathsf{C}$ which is hence permitted to be a good deal more general than a\ncluster category.  Let $\\Sigma$ denote the suspension functor of $\\mathsf{C}$\nand write $\\mathsf{C}( -,- )$ instead of $\\operatorname{Hom}_{ \\mathsf{C} }( -,- )$.  Let $R$ be a\nrigid object of $\\mathsf{C}$, assumed to be basic for reasons of simplicity,\nwith endomorphism algebra $E = \\operatorname{End}_{ \\mathsf{C} }( R )$.  There is a functor\n\\[\n  \\begin{array}{rcl}\n    \\mathsf{C} & \\stackrel{G}{\\longrightarrow} & \\mathsf{mod}\\,E, \\\\[2mm]\n    c   & \\longmapsto                   & \\mathsf{C}( R,\\Sigma c ).\n  \\end{array}\n\\]\nLet $A$ be a commutative ring and let\n\\[\n  \\alpha : \\operatorname{obj}\\,\\mathsf{C} \\rightarrow A\n\\;\\;,\\;\\;\n  \\beta : \\operatorname{K}_0( \\mathsf{mod}\\,E ) \\rightarrow A\n\\]\nbe two maps, where $\\operatorname{obj}\\,\\mathsf{C}$ is the set of objects of $\\mathsf{C}$ and\n$\\operatorname{K}_0$ denotes the Grothendieck group of an abelian category.\n\nThe modified Caldero-Chapoton map is the map $\\rho : \\operatorname{obj}\\,\\mathsf{C}\n\\rightarrow A$ defined as follows.\n\\[\n  \\rho( c )\n  = \\alpha( c )\n    \\sum_e \\chi \\big( \\operatorname{Gr}_e ( Gc ) \\big) \\beta( e )\n\\]\nHere $c \\in \\mathsf{C}$ is an object, the sum is over $e \\in \\operatorname{K}_0( \\mathsf{mod}\\,E\n)$, and $\\chi$ denotes the Euler characteristic defined by \\'{e}tale\ncohomology with proper support.  By $\\operatorname{Gr}_e$ is denoted the\nGrassmannian of submodules $M \\subseteq Gc$ with $\\operatorname{K}_0$-class $[ M ] =\ne$. \n\nThe original Caldero-Chapoton map is the special case where $R$ is a\ncluster tilting object and $\\alpha$ and $\\beta$ are two particular\nmaps; see Remark \\ref{rmk:original_CC}.  The modified Caldero-Chapoton\nmap of \\cite{HJ} is the special case where $A = \\mathbb{Z}$ and $\\alpha$ and\n$\\beta$ are identically equal to $1$.  In general, $\\rho$ is only\nlikely to be interesting if the maps $\\alpha$ and $\\beta$ are chosen\ncarefully, and we formalise this by saying that $\\alpha$ and $\\beta$\nare {\\em frieze-like for the AR triangle} $\\Delta = \\tau c \\rightarrow\nb \\rightarrow c$ if they satisfy the technical conditions in\nDefinition \\ref{def:good}.  Observe that the conditions are trivially\nsatisfied if $\\alpha$ and $\\beta$ are identically equal to $1$.  Our\nfirst main result is the following.\n\n\n\\bigskip\n{\\bf Theorem A. }\n{\\em \nIf $\\alpha$ and $\\beta$ are frieze-like for each AR triangle in $\\mathsf{C}$,\nthen the modified Caldero-Chapoton map $\\rho : \\operatorname{obj}\\,\\mathsf{C} \\rightarrow\nA$ is a generalised frieze in the sense of \\cite[def.\\ 3.4]{HJ}.  That\nis,\n\\begin{enumerate}\n\n  \\item  $\\rho( c_1 \\oplus c_2 ) = \\rho( c_1 )\\rho( c_2 )$,\n\n\\medskip\n\n  \\item  if $\\Delta  = \\tau c \\rightarrow b \\rightarrow c$ is an AR\n    triangle then $\\rho( \\tau c )\\rho( c ) - \\rho( b ) \\in \\{\\, 0,1\n    \\,\\}$.\n\n\\end{enumerate}\n}\n\\bigskip\n\nThis follows from Proposition \\ref{pro:exponential} and Theorem\n\\ref{thm:A} which even show\n\\[\n  \\rho( \\tau c )\\rho( c ) - \\rho( b )\n  = \\left\\{\n      \\begin{array}{cl}\n        0 & \\mbox{ if $G( \\Delta )$ is a split short exact sequence, } \\\\[2mm]\n        1 & \\mbox{ if $G( \\Delta )$ is not a split short exact sequence. }\n      \\end{array}\n    \\right.\n\\]\nNote that $G( \\Delta )$ is never split exact when $R$ is a\ncluster tilting object, that is, in the case of the original\nCaldero-Chapoton map.\n\nOur second main result is that one can find frieze-like maps $\\alpha$\nand $\\beta$, and hence generalised friezes, with values in Laurent\npolynomials.\n\n\n\\bigskip\n{\\bf Theorem B. }\n{\\em\nAssume that $\\mathsf{C}$ is $2$-Calabi-Yau and that the basic rigid object\n$R$ has $r$ indecomposable summands and is a direct summand of a\ncluster tilting object.\n\nThen there are maps $\\alpha$ and $\\beta$ with values in $\\mathbb{Z}[ x_1^{\n  \\pm 1 }, \\ldots, x_r^{ \\pm 1 } ]$, using all the variables $x_1$,\n$\\ldots$, $x_r$, which are frieze-like for each AR triangle in $\\mathsf{C}$.\n\nHence there is a modified Caldero-Chapoton map $\\rho : \\operatorname{obj} \\mathsf{C}\n\\rightarrow \\mathbb{Z}[ x_1^{ \\pm 1 }, \\ldots, x_r^{ \\pm 1 } ]$, using\nall the variables $x_1$, $\\ldots$, $x_r$, which is a generalised\nfrieze. \n}\n\\bigskip\n\nThis is established in Definition \\ref{def:alpha_and_beta}, Theorem\n\\ref{thm:B}, and Remark \\ref{rmk:app}.  We leave it vague for now what\nit means to ``use all the variables $x_1, \\ldots, x_r$'', but see\nRemark \\ref{rmk:app}.  In fact, it is sometimes possible to get values\nin Laurent polynomials in more than $r$ variables.  For example, the\nbasic rigid object $R$ defined by the red vertices in Figure\n\\ref{fig:AR_quiver} has $r = 2$ indecomposable summands, but the\ncorresponding $\\rho$ has values in $\\mathbb{Z}[ u^{ \\pm 1 }, v^{ \\pm 1 },\nz^{ \\pm 1 } ]$ as shown in Figure \\ref{fig:generalised_frieze}.\n\nIt is natural to ask if the $\\mathbb{Z}$-algebra generated by the values\nof $\\rho$ is an interesting object.  In particular, one can ask how it\nis related to cluster algebras and how it is affected by mutation of\nrigid objects as defined in \\cite[sec.\\ 2]{MP}; see the questions in\nSection \\ref{sec:questions}.\n\nThe paper is organised as follows: Section \\ref{sec:conditions}\ndefines what it means for $\\alpha$ and $\\beta$ to be frieze-like and\nproves Theorem A.  Section \\ref{sec:construction} proves Theorem B.\nSection \\ref{sec:example} shows how to obtain the example in Figure\n\\ref{fig:generalised_frieze}.  Section \\ref{sec:questions} poses some\nquestions. \n\n\n\n\n\\section{The frieze-like condition on the maps $\\alpha$ and $\\beta$\nimplies that $\\rho$ is a generalised frieze}\n\\label{sec:conditions}\n\n\nThis section proves Theorem A in the introduction.  It is a consequence of Proposition\n\\ref{pro:exponential} and Theorem \\ref{thm:A}.\n\nWe start by setting up items to be used in the rest of the paper.\nInstead of a basic rigid object $R$ we will work with a rigid\nsubcategory $\\mathsf{R}$.  This is more general since we can set $\\mathsf{R} =\n\\operatorname{add}\\,R$ when $R$ is given, but not every $\\mathsf{R}$ has this form, see\n\\cite[sec.\\ 6]{JP}.  The higher generality means that the definitions\nof $G$ and $\\beta$ are different from those in the introduction.\n\n\n\\begin{Setup}\n\\label{set:blanket}\nLet $k$ be an algebraically closed field and let $\\mathsf{C}$ be an\nessentially small $k$-linear $\\operatorname{Hom}$-finite triangulated category with\nsplit idempotents.  Hence $\\mathsf{C}$ is a Krull-Schmidt category.\n\nAssume that $\\mathsf{C}$ has a Serre functor.  Hence it has AR triangles by\n\\cite[thm.\\ A]{RVdB}, and the Serre functor is $\\Sigma \\circ \\tau$\nwhere $\\Sigma$ is the suspension functor and $\\tau$ is the AR\ntranslation.\n\nLet $\\mathsf{R}$ be a full subcategory of $\\mathsf{C}$ which is closed under direct\nsums and summands, is functorially finite, and rigid, that is, $\\mathsf{C}(\n\\mathsf{R},\\Sigma \\mathsf{R} ) = 0$.\n\nThe category of $k$-vector spaces is denoted $\\mathsf{Mod}\\,k$ and the\ncategory of $k$-linear contravariant functors $\\mathsf{R} \\rightarrow\n\\mathsf{Mod}\\,k$ is denoted $\\mathsf{Mod}\\,\\mathsf{R}$.  It is a $k$-linear abelian category\nand its full subcategory of objects of finite length is denoted\n$\\mathsf{fl}\\,\\mathsf{R}$.\n\nLet $A$ be a commutative ring and let\n\\[\n  \\alpha : \\operatorname{obj}\\,\\mathsf{C} \\rightarrow A\n\\;\\;,\\;\\;\n  \\beta : \\operatorname{K}_0( \\mathsf{fl}\\,\\mathsf{R} ) \\rightarrow A\n\\]\nbe maps which are ``exponential'' in the sense that\n\\begin{align*}\n  \\alpha( 0 ) = 1 \\;\\; , \\;\\;\n  & \\alpha( c \\oplus d ) = \\alpha( c )\\alpha( d ), \\\\[2mm]\n  \\beta( 0 ) = 1 \\;\\; , \\;\\;\n  & \\beta( e + f ) = \\beta( e )\\beta( f ).\n\\end{align*}\n\\end{Setup}\n\n\n\\begin{bfhpg}\n[The modified Caldero-Chapoton map]\n\\label{bfhpg:CC}\nThere is a functor\n\\[\n  \\begin{array}{rcl}\n    \\mathsf{C} & \\stackrel{G}{\\longrightarrow} & \\mathsf{Mod}\\,\\mathsf{R}, \\\\[2mm]\n    c   & \\longmapsto                   & \\mathsf{C}( -,\\Sigma c ) |_{ \\mathsf{R} }.\n  \\end{array}\n\\]\nThe modified Caldero-Chapoton map is defined by the following\nformula.\n\\begin{equation}\n\\label{equ:CC}\n  \\rho( c )\n  = \\alpha( c )\n    \\sum_e \\chi \\big( \\operatorname{Gr}_e ( Gc ) \\big) \\beta( e )\n\\end{equation}\nThe sum is over $e \\in \\operatorname{K}_0( \\mathsf{fl}\\,\\mathsf{R} )$, and $\\operatorname{Gr}_e( Gc )$ is the\nGrassmannian of submodules $M \\subseteq Gc$ where $M$ has finite\nlength in $\\mathsf{Mod}\\,\\mathsf{R}$ and class $[ M ] = e$ in $\\operatorname{K}_0( \\mathsf{fl}\\,\\mathsf{R} )$.\nThe notation is otherwise as explained in the introduction.\n\n\nThe formula may not make sense for each $c \\in \\mathsf{C}$, but it does make\nsense if $Gc$ has finite length in $\\mathsf{Mod}\\, \\mathsf{R}$ since then $\\operatorname{Gr}_e( Gc\n)$ is finite-dimensional and non-empty for only finitely many values\nof $e$; see \\cite[1.6 and 1.8]{JP}.  When the formula makes\nsense, it defines an element $\\rho( c ) \\in A$.  Note that\n\\[\n  \\rho( 0 ) = 1.\n\\]\n\n\\end{bfhpg}\n\n\n\\begin{Proposition}\n\\label{pro:exponential}\nLet $a,c \\in \\mathsf{C}$ be objects such that $Ga,Gc$ have finite length in\n$\\mathsf{Mod}\\, \\mathsf{R}$.  Then $G( a \\oplus c )$ has finite length in $\\mathsf{Mod}\\,\n\\mathsf{R}$ and\n\\[\n  \\rho( a \\oplus c ) = \\rho( a )\\rho( c ).\n\\]\n\\end{Proposition}\n\n\\begin{proof}\nThe statement about the length of $G( a \\oplus c )$ is clear.\n\nIt is immediate that\n\\begin{equation}\n\\label{equ:product}\n  \\rho( a )\\rho( c )\n  = \\alpha( a \\oplus c )\n    \\sum_g \n    \\Big( \n      \\sum_{ e+f=g } \\chi \\big( \\operatorname{Gr}_e ( Ga ) \\times \\operatorname{Gr}_f ( Gc ) \\big)\n    \\Big) \\beta( g )\n  = ( {\\textstyle *} ).\n\\end{equation}\nIn \\cite[sec.\\ 2]{HJ} we considered a pair of morphisms $a \\rightarrow\nb \\rightarrow c$ in $\\mathsf{C}$ and introduced auxiliary spaces $X_{ e,f }$.\nIf we set $a \\rightarrow b \\rightarrow c$ equal to the canonical\nmorphisms $a \\rightarrow a \\oplus c \\rightarrow c$, then \\cite[lem.\\\n2.4(i+v)]{HJ} and \\cite[rmk.\\ 2.5]{HJ} mean that we can compute as\nfollows.\n\\[\n  ( {\\textstyle *} )\n  = \\alpha( a \\oplus c )\n    \\sum_g \\Big( \\sum_{ e+f = g } \\chi( X_{ e,f } ) \\Big) \\beta( g )\n  = \\alpha( a \\oplus c )\n    \\sum_g \\chi \\Big( \\operatorname{Gr}_g \\big( G( a \\oplus c ) \\big) \\Big)\\beta( g )\n  = \\rho( a \\oplus c )\n\\]\n\\end{proof}\n\n\n\\begin{Definition}\n[Frieze-like $\\alpha$ and $\\beta$]\n\\label{def:good}\nLet \n\\[\n  \\Delta = \\tau c \\rightarrow b \\rightarrow c\n\\]\nbe an AR triangle in $\\mathsf{C}$ and assume that $Gc$ and $G( \\tau c )$ have \nfinite length in $\\mathsf{Mod}\\, \\mathsf{R}$.  We say that {\\em $\\alpha$ and $\\beta$\nare frieze-like for $\\Delta$} if the following hold.\n\\begin{enumerate}\n\n  \\item  If $c \\not\\in \\mathsf{R} \\cup \\Sigma^{ -1 }\\mathsf{R}$ and $G( \\Delta )$\n    is a split short exact sequence, then\n\\[\n  \\alpha( b ) = \\alpha( c \\oplus \\tau c ).\n\\]\n\n\\medskip\n\n  \\item  If $c \\not\\in \\mathsf{R} \\cup \\Sigma^{ -1 }\\mathsf{R}$ and $G( \\Delta )$\n    is a non-split short exact sequence, or if $c = \\Sigma^{-1 }r\n    \\in \\Sigma^{ -1 }\\mathsf{R}$, then\n\\[\n  \\alpha( b ) = \\alpha( c \\oplus \\tau c ) \n\\;\\;,\\;\\;\n  \\alpha( c \\oplus \\tau c )\\beta \\big( [ Gc ] \\big) = 1.\n\\]\n\n\\medskip\n\n  \\item  If $c = r \\in \\mathsf{R}$, then\n\\[\n  \\alpha( c \\oplus \\tau c ) = 1\n\\;\\;,\\;\\;\n  \\alpha( b ) = \\beta \\big( [S_r] \\big) \n\\]\nwhere $S_r \\in \\mathsf{Mod}\\,\\mathsf{R}$ is the simple object supported at $r$, see\n\\cite[prop.\\ 2.2]{AusRepII}.\n\n\\end{enumerate}\n\\end{Definition}\n\n\n\\begin{Remark}\nNote that if $c \\not\\in \\mathsf{R} \\cup \\Sigma^{ -1 }\\mathsf{R}$ then $G( \\Delta )$\nis a (split or non-split) short exact sequence, see \\cite[lem.\\\n1.12(iii)]{HJ}. \n\\end{Remark}\n\n\n\\begin{Theorem}\n\\label{thm:A}\nLet \n\\[\n  \\Delta = \\tau c \\rightarrow b \\rightarrow c\n\\]\nbe an AR triangle in $\\mathsf{C}$.  Assume that $Gc$ and $G( \\tau c )$ have \nfinite length in $\\mathsf{Mod}\\, \\mathsf{R}$ and that $\\alpha$ and $\\beta$ are\nfrieze-like for $\\Delta$.  Then $Gb$ has finite length in $\\mathsf{Mod}\\, \\mathsf{R}$\nand\n\\[\n  \\rho( \\tau c )\\rho( c ) - \\rho( b )\n  =\n  \\left\\{\n    \\begin{array}{cl}\n      0 & \\mbox{ if $G( \\Delta )$ is a split short exact sequence, } \\\\[2mm]\n      1 & \\mbox{ if $G( \\Delta )$ is not a split short exact sequence. }\n    \\end{array}\n  \\right.\n\\]\n\\end{Theorem}\n\n\\begin{proof}\nIt is clear from the definition of $G$ that it is a homological\nfunctor, so $G( \\Delta )$ is an exact sequence (albeit not necessarily\nshort exact).  The statement about the length of $Gb$ follows.\n\nWe split into cases.  First some preparation:\nsetting $a = \\tau c$ in Equation \\eqref{equ:product} gives\n\\begin{equation}\n\\label{equ:product2}\n  \\rho( \\tau c )\\rho( c )\n  = \\alpha( c \\oplus \\tau c )\n    \\sum_g \n    \\Big( \n      \\sum_{ e+f=g } \\chi \\big( \\operatorname{Gr}_e ( G( \\tau c ) ) \\times \\operatorname{Gr}_f ( Gc ) \\big)\n    \\Big) \\beta( g )\n    = ( {\\textstyle *} ).\n\\end{equation}\nMoreover, we use the morphisms $a \\rightarrow b \\rightarrow c$ and\nauxiliary spaces $X_{ e,f }$ from \\cite[sec.\\ 2]{HJ} again, this time\nsetting $a \\rightarrow b \\rightarrow c$ equal to the AR triangle\n$\\tau c \\rightarrow b \\rightarrow c$.  We can then use the results of\n\\cite[sec.\\ 2]{HJ}.\n\nCase (i):  $c \\not\\in \\mathsf{R} \\cup \\Sigma^{ -1 }\\mathsf{R}$ and $G( \\Delta )$ is\na split short exact sequence.  \n\nWe start from Equation \\eqref{equ:product2} and\ncompute as follows:\n\\[\n  ( {\\textstyle *} )\n  \\stackrel{\\rm (a)}{=}\n    \\alpha( b )\n    \\sum_g \n    \\Big( \n      \\sum_{ e+f=g } \\chi ( X_{ e,f } )\n    \\Big) \\beta( g ) \\\\\n  \\stackrel{\\rm (b)}{=}\n    \\alpha( b )\n    \\sum_g \\chi \\big( \\operatorname{Gr}_g( Gb ) \\big) \\beta( g ) \\\\\n  = \\rho(b),\n\\]\nwhere (a) is by Definition \\ref{def:good}(i) and \\cite[lem.\\\n2.4(i+v)]{HJ}, and (b) is by \\cite[rmk.\\ 2.5]{HJ}.\n\nCase (ii): $c \\not\\in \\mathsf{R} \\cup \\Sigma^{ -1 }\\mathsf{R}$ and $G( \\Delta )$ is a\nnon-split short exact sequence.\n\nWe start from Equation \\eqref{equ:product2} and compute as follows:\n\\begin{align*}\n  ( {\\textstyle *} )\n  & =\n    \\alpha( c \\oplus \\tau c )\n    \\Big\\{\n      \\chi \\big( \\operatorname{Gr}_0 ( G(\\tau c) ) \\times \\operatorname{Gr}_{[Gc]} ( Gc ) \\big)\n      \\beta \\big( [Gc] \\big)  \\\\\n      & \\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\n        + \\sum_g \\Big(\n                   \\sum_{ \\substack{ e+f=g \\\\[1mm] (e,f) \\neq (0,[Gc]) }}\n                                \\chi \\big(\n                                        \\operatorname{Gr}_e ( G(\\tau c) )\n                                        \\times \\operatorname{Gr}_f ( Gc )\n                                      \\big)\n                 \\Big) \\beta( g ) \n    \\Big\\} \\\\\n  & \\stackrel{\\rm (c)}{=}\n    \\alpha( c \\oplus \\tau c ) \n    \\Big\\{ \n      \\beta \\big( [Gc] \\big) \n      + \\sum_g \\Big(\n                 \\sum_{ \\substack{ e+f=g \\\\[1mm] (e,f) \\neq (0,[Gc]) }}\n                   \\chi \\big( X_{ e,f } \\big)\n               \\Big) \\beta( g )  \n    \\Big\\} \\\\\n  & \\stackrel{\\rm (d)}{=}\n    1 + \\alpha(b) \\sum_g \\Big(\n                           \\sum_{ \\substack{ e+f=g \\\\[1mm] (e,f) \\neq (0,[Gc]) }}\n                           \\chi \\big( X_{ e,f } \\big)\n                         \\Big) \\beta( g ) \\\\\n  & \\stackrel{\\rm (e)}{=}\n    1 + \\alpha(b) \\sum_g \\Big(\n                           \\sum_{ e+f=g }\n                           \\chi \\big( X_{ e,f } \\big)\n                         \\Big) \\beta( g ) \\\\\n  & \\stackrel{\\rm (f)}{=}\n    1 + \\alpha(b) \\sum_g \\chi \\big( \\operatorname{Gr}_g( Gb ) \\big) \\beta( g ) \\\\\n  & = 1 + \\rho( b ),\n\\end{align*}\nwhere (c) follows from \\cite[lem.\\ 2.4(ii)+(iv)+(v)]{HJ}, (d) is by\nDefinition \\ref{def:good}(ii), (e) is by \\cite[lem.\\ 2.4(iii)]{HJ},\nand (f) is by \\cite[rmk.\\ 2.5]{HJ}.\n\nCase (iii): $c = \\Sigma^{ -1 }r \\in \\Sigma^{ -1 }\\mathsf{R}$.  Then\n$G( \\Delta )$ is not a split short exact sequence, but\n\\begin{equation}\n\\label{equ:Pr}\n  G( \\Delta )\n  = G( \\tau c \\rightarrow b \\rightarrow c )\n  = 0 \\rightarrow \\operatorname{rad}\\,P_r \\rightarrow P_r\n\\end{equation}\nby \\cite[lem.\\ 1.12(i)]{HJ}, where $P_r = \\mathsf{C}( -,r ) \\,|_{ \\mathsf{R} }$ is\nthe indecomposable projective object of $\\mathsf{Mod}\\,\\mathsf{R}$ associated with\n$r$, see \\cite[1.5]{HJ}.  In particular, we have $G( \\tau c ) = 0$ whence\n$\\rho( \\tau c ) = \\alpha( \\tau c )$, and this gives the first equality\nin the following computation.\n\\begin{align*}\n  \\rho( \\tau c )\\rho( c )\n  & = \\alpha( \\tau c )\\alpha( c )\n      \\sum_f \\chi \\big( \\operatorname{Gr}_f( Gc ) \\big) \\beta( f ) \\\\\n  & = \\alpha( c \\oplus \\tau c )\n      \\sum_f \\chi \\big( \\operatorname{Gr}_f( Gc ) \\big) \\beta( f ) \\\\\n  & = \\alpha( c \\oplus \\tau c )\n      \\Big\\{\n        \\chi \\big( \\operatorname{Gr}_{ [Gc] }( Gc ) \\big) \\beta \\big( [Gc] \\big)\n        + \\sum_{ f \\neq [Gc] } \\chi \\big( \\operatorname{Gr}_f( Gc ) \\big) \\beta( f )\n      \\Big\\} \\\\\n  & \\stackrel{\\rm (g)}{=}\n    \\alpha( c \\oplus \\tau c )\n      \\Big\\{\n        \\beta \\big( [Gc] \\big)\n        + \\sum_{ f \\neq [Gc] } \\chi \\big( \\operatorname{Gr}_f( Gc ) \\big) \\beta( f )\n      \\Big\\} \\\\\n  & \\stackrel{\\rm (h)}{=}\n    \\alpha( c \\oplus \\tau c )\n      \\Big\\{\n        \\beta \\big( [Gc] \\big)\n        + \\sum_{ f } \\chi \\big( \\operatorname{Gr}_f( Gb ) \\big) \\beta( f )\n      \\Big\\} \\\\\n  & \\stackrel{\\rm (j)}{=}\n    1 + \\alpha(b) \\sum_{ f } \\chi \\big( \\operatorname{Gr}_f( Gb ) \\big) \\beta( f ) \\\\\n  & = 1 + \\rho( b )\n\\end{align*}\nTo see (g), note that for $M' \\subseteq Gc$ we have\n\\begin{equation}\n\\label{equ:1.2}\n  [ M' ] = [ Gc ] \\Leftrightarrow M' = Gc\n\\end{equation}\nby \\cite[eq.\\ (1.2)]{HJ}.  Hence $\\operatorname{Gr}_{ [Gc] }( Gc )$ has only a\nsingle point whence $\\chi \\big( \\operatorname{Gr}_{ [Gc] }( Gc ) \\big) = 1$.\nTo see (h), note that Equation \\eqref{equ:Pr} says that $Gc$ is an\nindecomposable projective object with radical $Gb$.  So the proper\nsubmodules of $Gc$ are precisely all the submodules of $Gb$, whence\nEquation \\eqref{equ:1.2} implies\n\\[\n  \\operatorname{Gr}_f( Gb )\n  = \\left\\{\n      \\begin{array}{cl}\n        \\operatorname{Gr}_f( Gc ) & \\mbox{ for $f \\neq [ Gc ]$, } \\\\[2mm]\n        \\emptyset   & \\mbox{ for $f = [ Gc ]$ }\n      \\end{array}\n    \\right.\n\\]\nand (h) follows.  Finally, (j) holds by Definition\n\\ref{def:good}(ii). \n\nCase (iv): $c = r \\in \\mathsf{R}$.  Then $G( \\Delta )$ is not a split short\nexact sequence, but we have\n\\[\n  G( \\Delta )\n  = G( \\tau c \\rightarrow b \\rightarrow c )\n  = I_r \\rightarrow \\operatorname{corad}\\,I_r \\rightarrow 0\n\\]\nby \\cite[lem.\\ 1.12(ii)]{HJ}, where $I_r = \\mathsf{C}( -,\\Sigma\\tau r ) \\,|_{\n  \\mathsf{R} }$ is the indecomposable injective object of $\\mathsf{Mod}\\,\\mathsf{R}$\nassociated with $r$, see \\cite[1.10]{HJ}, and $\\operatorname{corad}$ denotes the\nquotient by the socle.  Now proceed dually to Case (iii), replacing\nDefinition \\ref{def:good}(ii) by Definition \\ref{def:good}(iii).\n\\end{proof}\n\n\n\n\n\\section{A construction of frieze-like maps $\\alpha$ and $\\beta$ with\nvalues in Laurent polynomials}\n\\label{sec:construction}\n\n\nThis section proves Theorem B in the introduction.  It is a consequence of Definition\n\\ref{def:alpha_and_beta}, Theorem \\ref{thm:B}, and Remark\n\\ref{rmk:app}. \n\n\n\\begin{Setup}\n\\label{set:2}\nWe continue to work under Setup \\ref{set:blanket} and \nhenceforth add the assumption that $\\mathsf{C}$ is a $2$-Calabi-Yau category with a\ncluster tilting subcategory $\\mathsf{T}$ which belongs to a cluster\nstructure in the sense of \\cite[sec.\\ II.1]{BIRS}, and which satisfies\n$\\mathsf{R} \\subseteq \\mathsf{T}$.\n\n\nNote that the AR translation of $\\mathsf{C}$ is\n\\[\n  \\tau = \\Sigma\n\\]  \nand that the Serre functor is $\\Sigma^2$.\n\\end{Setup}\n\n\n\\begin{Remark}\nWhen $\\mathsf{R}$ is a rigid subcategory of $\\mathsf{C}$, it is often possible to\nfind a cluster tilting subcategory $\\mathsf{T}$ with $\\mathsf{R} \\subseteq \\mathsf{T}$.\nNot always, however: if $\\mathsf{C}$ is a cluster tube, then such a $\\mathsf{T}$\ncannot be found since cluster tubes have no cluster tilting\nsubcategories, see \\cite[cor.\\ 2.7]{BMV}.\n\\end{Remark}\n\n\n\\begin{bfhpg}\n[Mutation and exchange triangles]\nLet ${\\mathsf{ind}}\\, \\mathsf{T}$ denote the set of (isomorphism classes of)\nindecomposable objects of $\\mathsf{T}$.  Each $t \\in {\\mathsf{ind}}\\, \\mathsf{T}$ has a\nmutation $t^{ {\\textstyle *} }$ which is the unique indecomposable object in\n$\\mathsf{C}$ such that $\\mathsf{T}$ remains a cluster tilting subcategory if $t$ is\nreplaced by $t^{ {\\textstyle *} }$.  There are distinguished triangles\n\\begin{equation}\n\\label{equ:exchange_triangles1}\n  t^{ {\\textstyle *} } \\rightarrow a \\rightarrow t\n\\;\\;,\\;\\;\n  t \\rightarrow a' \\rightarrow t^{ {\\textstyle *} }\n\\end{equation}\nwith $a, a' \\in \\operatorname{add}\\big( ( {\\mathsf{ind}}\\, \\mathsf{T} ) \\setminus t \\big)$, known\nas exchange triangles, see \\cite[sec.\\ II.1]{BIRS}.  \n\\end{bfhpg}\n\n\n\\begin{Definition} \n[The subgroup $N$]\n\\label{def:N}\nThe split Grothendieck group of an additive category is denoted by\n$\\operatorname{K}_0^{ \\operatorname{split} }$.  It has a relation $[ a \\oplus b ] = [ a ] + [ b ]$\nfor each pair of objects $a,b$, where $[ a ]$ is the $\\operatorname{K}_0^{ \\operatorname{split}\n}$-class of $a$.\n\nDefine a subgroup of $\\operatorname{K}_0^{ \\operatorname{split} }( \\mathsf{T} )$ as follows.\n\\begin{equation}\n\\label{equ:N}\n  N = \\bigg\\langle\\, [a] - [a'] \n      \\,\\bigg|\n        \\begin{array}{l}\n          \\mbox{ $s^{ {\\textstyle *} } \\rightarrow a \\rightarrow s$ \\;,\\;\n                 $s \\rightarrow a' \\rightarrow s^{ {\\textstyle *} }$\n                 are exchange } \\\\[2mm]\n          \\mbox{ triangles with \n                 $s \\in {\\mathsf{ind}}\\, \\mathsf{T} \\setminus {\\mathsf{ind}}\\, \\mathsf{R}$ }\n        \\end{array}\n      \\,\\bigg\\rangle\n\\end{equation}\nLet\n\\[\n  Q : \\operatorname{K}_0^{ \\operatorname{split} }( \\mathsf{T} ) \\rightarrow \\operatorname{K}_0^{ \\operatorname{split} }( \\mathsf{T} ) \/ N\n\\]\ndenote the canonical surjection.\n\\end{Definition}\n\n\n\\begin{bfhpg}\n[Simple objects, $\\operatorname{K}$-theory, and the homomorphism $\\overline{ \\theta }$]\n\\label{bfhpg:simples_and_K}\nThe inclusion functor $i : \\mathsf{R} \\rightarrow \\mathsf{T}$ induces an exact functor\n\\[\n  i^{ {\\textstyle *} } : \\mathsf{Mod}\\,\\mathsf{T} \\longrightarrow \\mathsf{Mod}\\,\\mathsf{R}\n\\;\\;,\\;\\;\n  i^{ {\\textstyle *} }( M ) = M \\circ i.\n\\]\nEach indecomposable object $t \\in {\\mathsf{ind}}\\, \\mathsf{T}$ gives rise to a simple\nobject $\\overline{S}_t \\in \\mathsf{Mod}\\,\\mathsf{T}$, and each $r \\in {\\mathsf{ind}}\\, \\mathsf{R}$\ngives rise to a simple object $S_r \\in \\mathsf{Mod}\\,\\mathsf{R}$, see \\cite[prop.\\\n2.3(b)]{AusRepII}.  It is not hard to show\n\\[\n  i^{ {\\textstyle *} }\\overline{S}_t \n  = \\left\\{\n      \\begin{array}{cl}\n        S_t & \\mbox{ if $t \\in {\\mathsf{ind}}\\, \\mathsf{R}$, } \\\\[2mm]\n        0   & \\mbox{ if $t \\in {\\mathsf{ind}}\\, \\mathsf{T} \\setminus {\\mathsf{ind}}\\, \\mathsf{R}$. }\n      \\end{array}\n    \\right.\n\\]\nSince $i^{ {\\textstyle *} }$ is exact and sends simple objects to simple objects\nor $0$, is preserves finite length so restricts to an exact functor\n\\[\n  i^{ {\\textstyle *} } : \\mathsf{fl}\\,\\mathsf{T} \\rightarrow \\mathsf{fl}\\,\\mathsf{R}.\n\\]\nLet\n\\[\n  \\kappa : \\operatorname{K}_0( \\mathsf{fl}\\,\\mathsf{T} ) \\rightarrow \\operatorname{K}_0( \\mathsf{fl}\\,\\mathsf{R} )\n\\]\nbe the induced homomorphism.  The source is a free group on the\nclasses $[ \\overline{S}_t ]$ for $t \\in {\\mathsf{ind}}\\, \\mathsf{T}$ and the target\nis a free group on the classes $[ S_r ]$ for $r \\in {\\mathsf{ind}}\\, \\mathsf{R}$.\nThe homomorphism $\\kappa$ is surjective and given by\n\\begin{equation}\n\\label{equ:kappa}\n  \\kappa\\big( [ \\overline{S}_t ] \\big)\n  = \\left\\{\n      \\begin{array}{cl}\n        [ S_t ] & \\mbox{ if $t \\in {\\mathsf{ind}}\\, \\mathsf{R}$, } \\\\[2mm]\n        0       & \\mbox{ if $t \\in {\\mathsf{ind}}\\, \\mathsf{T} \\setminus {\\mathsf{ind}}\\, \\mathsf{R}$. }\n      \\end{array}\n    \\right.\n\\end{equation}\n\nThere is a functor\n\\[\n  \\begin{array}{rcl}\n    \\mathsf{C} & \\stackrel{\\overline{G}}{\\longrightarrow} & \\mathsf{Mod}\\,\\mathsf{T}, \\\\[2mm]\n    c   & \\longmapsto                              & \\mathsf{C}( -,\\Sigma c ) |_{ \\mathsf{T} },\n  \\end{array}\n\\]\nand $i^{ {\\textstyle *} } \\overline{G} = G$ where $G$ is the functor from\nSubsection \\ref{bfhpg:CC}.\n\nWe define a homomorphism as follows,\n\\begin{equation}\n\\label{equ:overlinetheta}\n  \\overline{\\theta} : \\operatorname{K}_0( \\mathsf{fl}\\,\\mathsf{T} ) \n    \\rightarrow \\operatorname{K}_0^{\\operatorname{split}}( \\mathsf{T} )\n  \\;\\;,\\;\\; \\overline{ \\theta }\n            \\big( [ \\overline{S}_t ] \\big) = [ a ] - [ a' ],\n\\end{equation}\nwhere $a, a'$ come from the exchange triangles\n\\eqref{equ:exchange_triangles1}, see \\cite[1.5(ii)]{JP}. \n\\end{bfhpg}\n\n\n\\begin{bfhpg}\n[The homomorphism $\\theta$]\nIt is clear from Equations \\eqref{equ:N}, \\eqref{equ:kappa}, and \\eqref{equ:overlinetheta}\nthat there is a unique homomorphism $\\theta$ which makes the following\nsquare commutative.\n\\begin{equation}\n\\label{equ:square}\n\\vcenter{\n  \\xymatrix {\n    \\operatorname{K}_0( \\mathsf{fl}\\,\\mathsf{T} ) \\ar@{->>}_{\\kappa}[d] \\ar^-{ \\overline{ \\theta }}[r] & \\operatorname{K}_0^{ \\operatorname{split} }( \\mathsf{T} ) \\ar@{->>}^{Q}[d]\\\\\n    \\operatorname{K}_0( \\mathsf{fl}\\,\\mathsf{R} ) \\ar_-{ \\theta }[r] & \\operatorname{K}_0^{ \\operatorname{split} }( \\mathsf{T} ) \/ N \\\\\n                     }\n        }\n\\end{equation}\n\\end{bfhpg}\n\n\n\\begin{bfhpg}\n[Index and coindex]\n\\label{bfhpg:ind}\nFor $c \\in \\mathsf{C}$ there is a distinguished triangle $t_1 \\rightarrow t_0\n\\rightarrow c$ with $t_0, t_1 \\in \\mathsf{T}$ by \\cite[sec.\\ 1]{DK}, and the\nindex $\\operatorname{ind} c = [t_0] - [t_1]$ is a well-defined element of $\\operatorname{K}_0^{\n\\operatorname{split} }( \\mathsf{T} )$.  Similarly there is a distinguished triangle $c\n\\rightarrow \\Sigma^2 t^0 \\rightarrow \\Sigma^2 t^1$ with $t^0, t^1 \\in\n\\mathsf{T}$, and the coindex $\\operatorname{coind} c = [t^0] - [t^1]$ is a well-defined\nelement of $\\operatorname{K}_0^{ \\operatorname{split} }( \\mathsf{T} )$. \n\\end{bfhpg}\n\n\n\\begin{Definition}\n[The maps $\\alpha$ and $\\beta$]\n\\label{def:alpha_and_beta}\nRecall that $A$ is a commutative ring.  Let\n\\begin{equation}\n\\label{equ:epsilon_exponential}\n  \\varepsilon : \\operatorname{K}_0^{ \\operatorname{split} }( \\mathsf{T} ) \/ N \\rightarrow A\n\\end{equation}\nbe a map which is ``exponential'' in the sense that\n\\[\n  \\varepsilon( 0 ) = 1\n\\;\\;,\\;\\;\n  \\varepsilon( e+f ) = \\varepsilon( e )\\varepsilon( f ).\n\\]\nDefine\n\\[\n  \\alpha : \\operatorname{obj}\\,\\mathsf{C} \\rightarrow A\n\\;\\;,\\;\\;\n  \\beta : \\operatorname{K}_0( \\mathsf{fl}\\,\\mathsf{R} ) \\rightarrow A\n\\]\nby\n\\begin{equation}\n\\label{equ:alpha_beta}\n  \\alpha( c ) = \\varepsilon Q( \\operatorname{ind} c )\n\\;\\;,\\;\\;\n  \\beta( e ) = \\varepsilon \\theta( e ).\n\\end{equation}\nIt is easy to see that $\\alpha$ and $\\beta$ satisfy the conditions in\nSetup \\ref{set:blanket}, and Equation \\eqref{equ:CC} now defines a\nmodified Caldero-Chapoton map $\\rho$ with values in $A$. \n\nRemark \\ref{rmk:app} has further comments to the choice of $\\varepsilon$ which is crucial to the properties of $\\rho$. \n\\end{Definition}\n\n\n\\begin{Remark}\n\\label{rmk:original_CC}\nThe definition of $\\alpha$ and $\\beta$ is motivated by the original\nCaldero-Chapoton map which is recovered as follows when $\\mathsf{R} = \\mathsf{T}$:\nin this case, $N = 0$ and $Q$ is the identity while $\\theta =\n\\overline{ \\theta }$, so Equations \\eqref{equ:alpha_beta} read\n\\[\n  \\alpha( c ) = \\varepsilon ( \\operatorname{ind} c )\n\\;\\;,\\;\\;\n  \\beta( e ) = \\varepsilon \\overline{ \\theta }( e ).\n\\]\nThe group $\\operatorname{K}_0^{ \\operatorname{split} }( \\mathsf{T} )$ is free on the classes $[ t ]$ for\n$t \\in \\operatorname{ind}\\,\\mathsf{T}$.  Let $A$ be the Laurent polynomial ring on\ngenerators $x_t$ for $t \\in \\operatorname{ind}\\,\\mathsf{T}$ and set $\\varepsilon \\big( [ t\n] \\big) = x_t$ for $t \\in \\operatorname{ind}\\,\\mathsf{T}$.  Then the map $\\rho$ from\nEquation \\eqref{equ:CC} is the original Caldero-Chapoton map, see\n\\cite[1.8]{JP}. \n\\end{Remark}\n\n\n\\begin{Lemma}\n\\label{lem:coind}\nThe map $\\overline{\\theta}$ from Equation \\eqref{equ:overlinetheta}\nsatisfies the following. \n\\begin{enumerate}\n\n  \\item  If $\\overline{G}c$ has finite length in $\\mathsf{Mod}\\, \\mathsf{T}$ then\n\\[\n  \\overline{\\theta} \\big( [ \\overline{G}c ] \\big)\n  = - ( \\operatorname{ind} c + \\operatorname{ind} \\Sigma c).\n\\]\n\n\\smallskip\n\n  \\item Let $\\Sigma c \\stackrel{ \\varphi }{ \\longrightarrow } b\n  \\longrightarrow c$ be an AR triangle in $\\mathsf{C}$.  If $\\overline{ G }(\n  \\Sigma c )$ and $\\overline{ G }c$ have finite length in $\\mathsf{Mod}\\, \\mathsf{T}$\n  then\n\\[\n  \\operatorname{ind} b\n  = \\left\\{\n      \\begin{array}{cl}\n        - \\overline{ \\theta } \\big( [ \\overline{ G }c ] \\big)\n          & \\mbox{ if $c \\not\\in \\mathsf{T}$, } \\\\[2mm]\n        \\overline{ \\theta } \\big( [ \\overline{S}_t ] \\big)\n          & \\mbox{ if $c = t \\in \\mathsf{T}$. }\n      \\end{array}\n    \\right.\n\\]\n\\end{enumerate}\n\\end{Lemma}\n\n\\begin{proof}\nFirst note that \\cite[lem.\\ 2.1(2) and prop.\\ 2.2]{Palu} apply to the present setup by \\cite[1.3]{JP}.\n\n(i)  Combine \\cite[1.5]{JP} with \\cite[lem.\\ 2.1(2)]{Palu}.\n\n(ii)  Observe that\n\\begin{equation}\n\\label{equ:Sigmab1}\n  \\operatorname{ind} b\n  = \\overline{ \\theta } \\big( [ \\operatorname{Ker} \\overline{ G }\\varphi ] \n                              - [ \\overline{ G }c ] \\big).\n\\end{equation}\nThis can be seen by combining part (i) of the lemma with \n\\cite[prop.\\ 2.2]{Palu}.\n\n\nThe case $c \\not\\in \\mathsf{T}$: then $\\mathsf{C}( t,b ) \\rightarrow \\mathsf{C}( t,c\n)$ is surjective because $\\Sigma c \\stackrel{ \\varphi }{\n\\longrightarrow } b \\longrightarrow c$ is an AR triangle.  The long\nexact sequence \n$\\mathsf{C}( t,b )\n  \\longrightarrow \\mathsf{C}( t,c )\n  \\longrightarrow \\mathsf{C}( t,\\Sigma^2 c )\n  \\stackrel{ ( \\Sigma \\varphi )_* }{ \\longrightarrow }\n    \\mathsf{C}( t,\\Sigma b )$\nshows that $( \\Sigma \\varphi )_{ {\\textstyle *} } : \\mathsf{C}( t,\\Sigma^2 c )\n\\rightarrow \\mathsf{C}( t,\\Sigma b )$ is injective.  This implies that\n$\\overline{G}\\varphi$ is injective whence Equation \\eqref{equ:Sigmab1}\ngives $\\operatorname{ind} b = - \\overline{ \\theta } \\big( [ \\overline{ G }c ]\n\\big)$ as desired.\n\nThe case $c = t \\in \\mathsf{T}$: then \n\\[\n  \\overline{ G }( \\Sigma c\n  \\stackrel{ \\varphi }{ \\longrightarrow } b\n  \\longrightarrow c )\n  = \\overline{ I }_t\n  \\stackrel{ \\overline{ G }\\varphi }{ \\longrightarrow }\n  \\operatorname{corad} \\overline{ I }_t\n  \\rightarrow\n  0\n\\]\nby \\cite[lem.\\ 1.12(ii)]{HJ}.  Here $\\overline{ I }_t = \\mathsf{C}(\n-,\\Sigma^2 t )|_{ \\mathsf{T} }$ is the indecomposable injective object of\n$\\mathsf{Mod}\\,\\mathsf{T}$ associated with $t$, and $\\operatorname{corad}$ denotes the quotient by\nthe socle.  Hence $\\operatorname{Ker} \\overline{ G }\\varphi = \\overline{ S }_t$ and\n$\\overline{ G }c = 0$, whence Equation \\eqref{equ:Sigmab1} reads $\\operatorname{ind}\nb = \\overline{ \\theta } \\big( [ \\overline{ S }_t ] \\big)$ as desired.\n\\end{proof}\n\n\n\\begin{Theorem}\n\\label{thm:B}\nLet \n\\[\n  \\Delta \n   = \\Sigma c \\rightarrow b \\rightarrow c\n\\]\nbe an AR triangle in $\\mathsf{C}$ such that $\\overline{ G }c$ and\n$\\overline{ G }( \\Sigma c )$ have finite length in $\\mathsf{Mod}\\, \\mathsf{T}$.\n\nThen $Gc$ and $G( \\Sigma c )$ have finite length in $\\mathsf{Mod}\\, \\mathsf{R}$, and\nthe maps $\\alpha$ and $\\beta$ from Definition \\ref{def:alpha_and_beta}\nare frieze-like for $\\Delta$.\n\\end{Theorem}\n\n\\begin{proof}\nThe statement on lengths holds because $G = i^{ {\\textstyle *} }\\overline{ G }$\nand $i^{ {\\textstyle *} }$ preserves finite length, see Subsection\n\\ref{bfhpg:simples_and_K}.  We must now check the conditions of\nDefinition \\ref{def:good}.\n\nFirst, we show that\n\\begin{equation}\n\\label{equ:alpha2}\n  \\alpha( c \\oplus \\Sigma c )\\beta \\big( [ Gc ] \\big) = 1,\n\\end{equation}\nin particular establishing the second equation in Definition\n\\ref{def:good}(ii).  Equation \\eqref{equ:alpha_beta} gives\n\\begin{equation}\n\\label{equ:alpha10}\n  \\alpha( c \\oplus \\Sigma c )\n  = \\varepsilon Q \\big( \\operatorname{ind} ( c \\oplus \\Sigma c ) \\big)\n  = \\varepsilon Q( \\operatorname{ind} c + \\operatorname{ind} \\Sigma c ).\n\\end{equation}\nOn the other hand, combining Equation \\eqref{equ:alpha_beta}, the fact\nthat $[ Gc ] = [ i^{ {\\textstyle *} }\\overline{ G }c ] = \\kappa[ \\overline{ G }c\n]$, the commutative square \\eqref{equ:square}, and Lemma\n\\ref{lem:coind}(i) gives\n\\[\n  \\beta \\big( [ Gc ] \\big)\n  = \\varepsilon \\theta \\big( [ Gc ] \\big)\n  = \\varepsilon \\theta \\kappa \\big( [ \\overline{ G }c ] \\big)\n  = \\varepsilon Q \\overline{ \\theta } \\big( [ \\overline{ G }c ] \\big)\n  = \\varepsilon Q \\big( - ( \\operatorname{ind} c + \\operatorname{ind} \\Sigma c ) \\big).\n\\]\nMultiplying the last two equations proves Equation \\eqref{equ:alpha2}. \n\nSecondly, if $c = t \\in \\mathsf{T}$ then it is direct from the\ndefinition of index and coindex that\n\\[\n  \\operatorname{ind} c = [ t ]\n\\;\\;,\\;\\;\n  \\operatorname{ind} \\Sigma c = - [ t ].\n\\]\nInserting into Equation \\eqref{equ:alpha10} gives\n\\begin{equation}\n\\label{equ:alpha3}\n  c \\in \\mathsf{T}\n  \\; \\Rightarrow \\;\n  \\alpha( c \\oplus \\Sigma c ) = 1,\n\\end{equation}\nin particular establishing the first equation in Definition\n\\ref{def:good}(iii).\n\nThirdly, suppose $c \\not\\in \\mathsf{R}$.  We will show\n\\begin{equation}\n\\label{equ:alpha}\n  \\alpha( b ) = \\alpha( c \\oplus \\Sigma c ),\n\\end{equation}\nestablishing Definition \\ref{def:good}(i) as well as the first\nequation in Definition \\ref{def:good}(ii).\n\nThe case $c = t \\in \\mathsf{T}$: Note that $c \\not\\in \\mathsf{R}$ implies\n$\\overline{ \\theta } \\big( [ \\overline{ S }_t ] \\big) \\in N$ by\nEquations \\eqref{equ:overlinetheta} and \\eqref{equ:N}.  Using\nEquation \\eqref{equ:alpha_beta} and Lemma \\ref{lem:coind}(ii)\ntherefore gives \n\\[\n  \\alpha( b )\n  = \\varepsilon Q ( \\operatorname{ind} b )\n  = \\varepsilon Q \\overline{ \\theta }\\big( [ \\overline{ S }_t ] \\big)\n  = \\varepsilon( 0 )\n  = 1.\n\\]\nCombining with Equation \\eqref{equ:alpha3} shows Equation\n\\eqref{equ:alpha}. \n\nThe case $c \\not\\in \\mathsf{T}$: combining the two parts of Lemma\n\\ref{lem:coind} shows $\\operatorname{ind} b = \\operatorname{ind} c + \\operatorname{ind} \\Sigma c$.  Applying\n$\\varepsilon Q$ shows $\\alpha( b ) = \\alpha( c )\\alpha( \\Sigma c )$\nwhich is equivalent to Equation \\eqref{equ:alpha}.\n\nFinally, suppose $c = r \\in \\mathsf{R}$.  We show that\n\\[\n  \\alpha( b ) = \\beta \\big( [ S_r ] \\big),\n\\]\nestablishing the second equation in Definition \\ref{def:good}(iii).\nLemma \\ref{lem:coind}(ii) says \n\\[\n  \\operatorname{ind} b = \\overline{ \\theta } \\big( [ \\overline{ S }_r ] \\big).\n\\]\nApplying $\\varepsilon Q$ gives the first of the following equalities. \n\\[\n  \\alpha( b )\n  = \\varepsilon Q \\overline{ \\theta } \\big( [ \\overline{ S }_r ] \\big)\n  = \\varepsilon \\theta \\kappa \\big( [ \\overline{ S }_r ] \\big)\n  = \\varepsilon \\theta \\big( [ S_r ] \\big)\n  = \\beta \\big( [ S_r ] \\big)\n\\]\nThe other equalities are by the commutative diagram \\eqref{equ:square}\nand Equations \\eqref{equ:kappa} and \\eqref{equ:alpha_beta}.\n\\end{proof}\n\n\n\\begin{Remark}\n\\label{rmk:app}\nThe maps $\\alpha$ and $\\beta$ from Definition\n\\ref{def:alpha_and_beta}, and hence the modified Caldero-Chapoton map\n$\\rho$ from Equation \\eqref{equ:CC}, depend on the map $\\varepsilon :\n\\operatorname{K}_0^{ \\operatorname{split} }( \\mathsf{T} ) \/ N \\rightarrow A$.  The possible choices of\n$\\varepsilon$ are determined by the structure of $\\operatorname{K}_0^{ \\operatorname{split} }( \\mathsf{T}\n) \/ N$ which we do not know in general.\n\nHowever, let us suppose that ${\\mathsf{ind}}\\, \\mathsf{R}$ and ${\\mathsf{ind}}\\, \\mathsf{T}$ are\nfinite, with $r$, respectively $r+s$, objects.  This is the situation from\nTheorem B in the introduction if we set $R$, respectively $T$, equal\nto the direct sum of the indecomposable objects in $\\operatorname{ind}\\,\\mathsf{R}$,\nrespectively $\\operatorname{ind}\\,\\mathsf{T}$.\nThen we can set $A = \\mathbb{Z}[ x_1^{ \\pm 1 }, \\ldots, x_r^{ \\pm 1\n} ]$ and use all the variables $x_1, \\ldots, x_r$, thereby proving\nTheorem B.\n\nNamely, $\\operatorname{K}_0^{ \\operatorname{split} }( \\mathsf{T} )$ is a free abelian group\non $r+s$ generators, one per object in ${\\mathsf{ind}}\\, \\mathsf{T}$, and the\nsubgroup $N$ has $s$ generators, one per object in ${\\mathsf{ind}}\\, \\mathsf{T}\n\\setminus {\\mathsf{ind}}\\, \\mathsf{R}$.  So $\\operatorname{K}_0^{ \\operatorname{split} }( \\mathsf{T} ) \/ N$ has a\nquotient group $F$ which is free abelian of rank $( r+s ) - s = r$.\nThe desired map\n$\\varepsilon : \\operatorname{K}_0^{ \\operatorname{split} }( \\mathsf{T} ) \/ N \\rightarrow \\mathbb{Z}[ x_1^{ \\pm 1\n}, \\ldots, x_r^{ \\pm 1 } ]$ can be obtained by sending each generator\nof $F$ to a generator of $\\mathbb{Z}[ x_1^{ \\pm 1 }, \\ldots, x_r^{ \\pm 1 }\n]$.\n\nNote that $\\operatorname{K}_0^{ \\operatorname{split} }( \\mathsf{T} )$ may have a quotient group which is\nfree abelian of rank $n > r$, see the example in Section\n\\ref{sec:example}.  In this case, the above method means that\nwe can even set $A = \\mathbb{Z}[ x_1^{ \\pm 1 }, \\ldots, x_n^{ \\pm 1 } ]$ and\nuse all the variables $x_1, \\ldots, x_n$.\n\\end{Remark}\n\n\n\n\n\\section{Example: a modified Caldero-Chapoton map on the cluster\ncategory of Dynkin type $A_5$}\n\\label{sec:example}\n\n\nThis section shows how to obtain the example in Figure\n\\ref{fig:generalised_frieze} in the introduction.\n\n\\begin{Setup}\nLet the category $\\mathsf{C}$ of Setups \\ref{set:blanket} and \\ref{set:2} be\n$\\mathsf{C}( A_5 )$, the cluster category of Dynkin type $A_5$.  There is a\nbijection between ${\\mathsf{ind}}\\, \\mathsf{C}$ and the diagonals of a $8$-gon, see\n\\cite[secs.\\ 2 and 5]{CCS}.  We let ${\\mathsf{ind}}\\, \\mathsf{R}$, respectively\n${\\mathsf{ind}}\\, \\mathsf{T}$, be given by the red diagonals, respectively all the\nred and blue diagonals, in Figure \\ref{fig:8gon}.  These data satisfy\nour assumptions, see \\cite[sec.\\ 1]{BMRRT}.\n\\begin{figure}\n\\[\n  \\begin{tikzpicture}[auto]\n    \\node[name=s, shape=regular polygon, regular polygon sides=8,\n    minimum size=5cm, draw] at (0,0) {}; \n    \\node[name=t, shape=regular polygon, regular polygon sides=8, minimum size=5.8cm] at (0,0) {}; \n    \\draw[shift=(t.corner 1)] node {$1$};\n    \\draw[shift=(t.corner 2)] node {$2$};\n    \\draw[shift=(t.corner 3)] node {$3$};\n    \\draw[shift=(t.corner 4)] node {$4$};\n    \\draw[shift=(t.corner 5)] node {$5$};\n    \\draw[shift=(t.corner 6)] node {$6$};\n    \\draw[shift=(t.corner 7)] node {$7$};\n    \\draw[shift=(t.corner 8)] node {$8$};\n    \\draw[very thick, blue] (s.corner 1) to (s.corner 7);\n    \\draw[very thick, blue] (s.corner 2) to (s.corner 4);\n    \\draw[very thick, red] (s.corner 2) to (s.corner 5);\n    \\draw[very thick, red] (s.corner 2) to (s.corner 7);\n    \\draw[very thick, blue] (s.corner 5) to (s.corner 7);\n  \\end{tikzpicture} \n\\]\n\\caption{The diagonals of the $8$-gon correspond to the indecomposable\nobjects of $\\mathsf{C}( A_5 )$.  The red diagonals define ${\\mathsf{ind}}\\, \\mathsf{R}$ and\nall the red and blue diagonals define ${\\mathsf{ind}}\\, \\mathsf{T}$.}   \n\\label{fig:8gon}\n\\end{figure}\n\\end{Setup}\n\n\n\\begin{bfhpg}\n[Some properties of $\\mathsf{C}$]\nWe denote diagonals and their corresponding indecomposable objects by\npairs of vertices, so $\\{\\, 2,7 \\,\\}$ is both a red diagonal in Figure\n\\ref{fig:8gon} and an object of ${\\mathsf{ind}}\\, \\mathsf{R}$.  The AR quiver of\n$\\mathsf{C}$ and the objects of $\\operatorname{ind}\\,\\mathsf{R}$, respectively $\\operatorname{ind}\\,\\mathsf{T}$, are\nshown in Figure \\ref{fig:AR_quiver} in the introduction. \n\nAt the level of objects, the suspension functor $\\Sigma$ is given by\n$\\Sigma \\{\\, i,j \\,\\} = \\{\\, i-1,j-1 \\,\\}$.  Note that vertex numbers\nare taken modulo $8$.\n\nIf $x, y \\in {\\mathsf{ind}}\\, \\mathsf{C}$ then\n\\begin{equation}\n\\label{equ:C7_Homs}\n  \\mathsf{C}( x,\\Sigma y )\n  =\n  \\left\\{\n    \\begin{array}{cl}\n      k & \\mbox{ if the diagonals corresponding to $x$ and $y$ cross, } \\\\[2mm]\n      0 & \\mbox{ if not. }\n    \\end{array}\n  \\right.\n\\end{equation}\nIf $i,k,j,\\ell$ are four vertices in anticlockwise\norder on the polygon, then $\\{\\, i,j \\,\\}$ and $\\{\\, k,\\ell \\,\\}$ are\ncrossing diagonals, and there are the following non-split distinguished\ntriangles,\n\\begin{equation}\n\\label{equ:exchange_triangles}\n  \\{\\, i,j \\,\\}\n  \\rightarrow \\{\\, i,\\ell \\,\\} \\oplus \\{\\, j,k \\,\\}\n  \\rightarrow \\{\\, k,\\ell \\,\\}\n  \\;\\; , \\;\\;\n  \\{\\, k,\\ell \\,\\}\n  \\rightarrow \\{\\, i,k \\,\\} \\oplus \\{\\, j,\\ell \\,\\}\n  \\rightarrow \\{\\, i,j \\,\\},\n\\end{equation}\nwhere a pair of neighbouring vertices must be interpreted as $0$.\n\\end{bfhpg}\n\n\n\\begin{bfhpg}\n[$\\operatorname{K}$-theory]\nThe category $\\mathsf{T}$ has the following indecomposable objects.\n\\[\n  \\{\\, 1,7 \\,\\}\n  \\;\\;,\\;\\; \\{\\, 2,4 \\,\\}\n  \\;\\;,\\;\\; \\{\\, 2,5 \\,\\}\n  \\;\\;,\\;\\; \\{\\, 2,7 \\,\\}\n  \\;\\;,\\;\\; \\{\\, 5,7 \\,\\}\n\\]\nTheir $\\operatorname{K}_0^{ \\operatorname{split} }$-classes are free generators of $\\operatorname{K}_0^{ \\operatorname{split}\n}( \\mathsf{T} )$.  To save parentheses, the classes are denoted $[ 1,7 ]$\netc.  The objects in ${\\mathsf{ind}}\\, \\mathsf{T} \\setminus {\\mathsf{ind}}\\, \\mathsf{R}$ are\n\\[  \n    \\{\\, 1,7 \\,\\} \\;\\;,\\;\\; \\{\\, 2,4 \\,\\} \\;\\;,\\;\\; \\{\\, 5,7 \\,\\},\n\\]\nand Equation \\eqref{equ:exchange_triangles} means that they sit in the\nfollowing exchange triangles.\n\\[\n  \\xymatrix @R=1ex {\n    \\{\\, 2,8 \\,\\} \\ar[r] & 0 \\ar[r] & \\{\\, 1,7 \\,\\}\n    & \\{\\, 1,7 \\,\\} \\ar[r] & \\{\\, 2,7 \\,\\} \\ar[r] & \\{\\, 2,8 \\,\\} \\\\\n    \\{\\, 3,5 \\,\\} \\ar[r] & 0 \\ar[r] & \\{\\, 2,4 \\,\\}\n    & \\{\\, 2,4 \\,\\} \\ar[r] & \\{\\, 2,5 \\,\\} \\ar[r] & \\{\\, 3,5 \\,\\} \\\\\n    \\{\\, 2,6 \\,\\} \\ar[r] & \\{\\, 2,7 \\,\\} \\ar[r] & \\{\\, 5,7 \\,\\}\n    & \\{\\, 5,7 \\,\\} \\ar[r] & \\{\\, 2,5 \\,\\} \\ar[r] & \\{\\, 2,6 \\,\\}\n            }\n\\]\nAccordingly, the subgroup $N$ of Definition \\ref{def:N} is\n\\[\n  N = \\big\\langle\\,\n        - [ 2,7 ] \\;,\\; - [ 2,5 ] \\;,\\; [ 2,7 ] - [ 2,5 ]\n      \\,\\big\\rangle \\\\[2mm]\n    = \\big\\langle\\,\n        [ 2,5 ] \\;,\\; [ 2,7 ]\n      \\,\\big\\rangle,\n\\]\nand $\\operatorname{K}_0^{ \\operatorname{split} }( \\mathsf{T} ) \/ N$ is the free abelian group generated\nby\n\\[\n  [ 1,7 ] + N\n  \\;\\;,\\;\\; [ 2,4 ] + N\n  \\;\\;,\\;\\; [ 5,7 ] + N.\n\\]\n\nThe category $\\mathsf{fl}\\, \\mathsf{R}$ has the simple objects\n\\[\n  S_{\\{\\, 2,5 \\,\\}} \\;\\;,\\;\\; S_{\\{\\, 2,7 \\,\\}}\n\\]\nwhose $\\operatorname{K}_0$-classes are free generators of $\\operatorname{K}_0( \\mathsf{fl}\\, \\mathsf{R} )$, and\n$\\mathsf{fl}\\, \\mathsf{T}$ has the simple objects\n\\[\n  \\overline{ S }_{\\{\\, 1,7 \\,\\}}\n  \\;\\;,\\;\\; \\overline{ S }_{\\{\\, 2,4 \\,\\}}\n  \\;\\;,\\;\\; \\overline{ S }_{\\{\\, 2,5 \\,\\}}\n  \\;\\;,\\;\\; \\overline{ S }_{\\{\\, 2,7 \\,\\}}\n  \\;\\;,\\;\\; \\overline{ S }_{\\{\\, 5,7 \\,\\}}\n\\]\nwhose $\\operatorname{K}_0$-classes are free generators of $\\operatorname{K}_0( \\mathsf{fl}\\, \\mathsf{T} )$.  To\nsave parentheses, the simple objects will be denoted $S_{ 2,5 }$,\nrespectively $\\overline{ S }_{ 1,7 }$, etc.\n\\end{bfhpg}\n\n\n\\begin{Definition}\nLet the map\n\\[\n  \\varepsilon : \\operatorname{K}_0^{ \\operatorname{split} }( \\mathsf{T} ) \/ N\n                \\rightarrow\n                \\mathbb{Z}[ u^{ \\pm 1 } , v^{ \\pm 1 } , z^{ \\pm 1 } ]\n\\]\nbe given by\n\\begin{equation}\n\\label{equ:epsilon}\n  \\varepsilon \\big( [ 1,7 ] + N \\big) = u   \n\\;\\;,\\;\\;\n  \\varepsilon \\big( [ 2,4 ] + N \\big) = v   \n\\;\\;,\\;\\;\n  \\varepsilon \\big( [ 5,7 ] + N \\big) = z.  \n\\end{equation}\nEquation \\eqref{equ:alpha_beta} defines the maps $\\alpha$ and $\\beta$, \nand the modified Caldero-Chapton map $\\rho$ is defined by Equation\n\\eqref{equ:CC}.\n\\end{Definition}\n\n\n\\begin{Example}\nLet us compute $\\rho \\big( \\{\\, 4,6 \\,\\} \\big)$.\n\nEquation \\eqref{equ:alpha_beta} gives\n\\[\n  \\alpha \\big( \\{\\, 4,6 \\,\\} \\big)\n  = \\varepsilon Q( \\operatorname{ind} \\{\\, 4,6 \\,\\} )\n  = ( {\\textstyle *} ).\n\\]\nNow $\\{\\, 4,6 \\,\\} = \\Sigma \\{\\, 5,7 \\,\\}$ so $\\operatorname{ind} \\{\\, 4,6 \\,\\} =\n\\operatorname{ind} \\Sigma \\{\\, 5,7 \\,\\} = - [ 5,7 ]$, where the last equality is\ndirect from the definition of index because $\\{\\, 5,7 \\,\\} \\in \\mathsf{T}$,\nsee Subsection \\ref{bfhpg:ind}.  Hence Equation \\eqref{equ:epsilon}\ngives\n\\[\n  ( {\\textstyle *} )\n  = \\varepsilon Q \\big( -[ 5,7 ] \\big)\n  = \\varepsilon \\big( -[ 5,7 ] + N \\big)\n  = z^{ -1 }.\n\\]\n\n\nWe have $G \\big( \\{\\, 4,6 \\,\\} \\big) = \\mathsf{C}( -,\\Sigma \\{\\, 4,6 \\,\\}\n)|_{ \\mathsf{R} }$.  Moreover, $\\mathsf{R} = \\operatorname{add} \\big\\{\\, \\{\\, 2,5 \\,\\} , \\{\\, 2,7\n\\,\\} \\,\\big\\}$, and it is direct from Equation\n\\eqref{equ:C7_Homs} that $G \\big( \\{\\, 4,6 \\,\\} \\big)$ is supported\nonly at $\\{\\, 2,5 \\,\\}$ where it has the value $k$.  That is,\n\\[\n  G \\big( \\{\\, 4,6 \\,\\} \\big) = S_{ 2,5 }.\n\\]\nIt follows that the only non-empty Grassmannians appearing in Equation\n\\eqref{equ:CC} when computing $\\rho \\big( \\{\\, 4,6 \\,\\} \\big)$ are\n$\\operatorname{Gr}_0 \\big( G\\{\\, 4,6 \\,\\} \\big)$ and $\\operatorname{Gr}_{ [ S_{ 2,5 } ] } \\big(\nG\\{\\, 4,6 \\,\\} \\big)$, and it is clear that each is a point so has\nEuler characteristic $1$.\n\nFinally, Equations \\eqref{equ:kappa} and \\eqref{equ:alpha_beta} and\ndiagram \\eqref{equ:square} give \n\\[\n  \\beta \\big( [ S_{ 2,5 } ] \\big)\n  = \\varepsilon \\theta \\big( [ S_{ 2,5 } ] \\big)\n  = \\varepsilon \\theta \\kappa \n    \\big( [ \\overline{ S }_{ 2,5 } ] \\big)\n  = \\varepsilon Q \\overline{ \\theta }\n    \\big( [ \\overline{ S }_{ 2,5 } ] \\big)\n  = ( {\\textstyle *} {\\textstyle *} ).\n\\]\nEquation \\eqref{equ:exchange_triangles} gives exchange triangles\n\\[\n  \\xymatrix @R=1ex {\n    \\{\\, 4,7 \\,\\} \\ar[r] & \\{\\, 2,4 \\,\\} \\oplus \\{\\, 5,7 \\,\\} \\ar[r] & \\{\\, 2,5 \\,\\}\n    & \\{\\, 2,5 \\,\\} \\ar[r] & \\{\\, 2,7 \\,\\} \\ar[r] & \\{\\, 4,7 \\,\\}\n                   }\n\\]\nand Equation \\eqref{equ:overlinetheta} gives $\\overline{ \\theta }\n\\big( [ \\overline{ S }_{ 2,5 } ] \\big) = [ 2,4 ] + [ 5,7 ] -\n[ 2,7 ]$ whence Equation \\eqref{equ:epsilon} gives\n\\[\n  ( {\\textstyle *} {\\textstyle *} )\n  = \\varepsilon Q\n    \\big( [ 2,4 ] + [ 5,7 ] - [ 2,7 ] \\big)\n  = vz.\n\\]\n\nHence Equation \\eqref{equ:CC} says\n\\begin{align*}\n  \\rho \\big( \\{\\, 4,6 \\,\\} \\big)\n  & = \\alpha \\big( \\{\\, 4,6 \\,\\} \\big)\n      \\sum_e \\chi \\big( \\operatorname{Gr}_e( G\\{\\, 4,6 \\,\\} ) \\big) \\beta( e ) \\\\\n  & = z^{ -1 } \\cdot \\Big(\n      \\chi \\big( \\operatorname{Gr}_0( S_{ 2,5 } ) \\big) \n         \\beta( 0 )\n      + \\chi \\big( \\operatorname{Gr}_{[ S_{ 2,5 } ]}( S_{ 2,5 } ) \\big) \n         \\beta \\big( [S_{ 2,5 }] \\big)\n                    \\Big) \\\\\n  & = z^{ -1 } \\cdot ( 1 + vz ) \\\\\n  & = \\frac{ 1+vz }{ z }.\n\\end{align*}\n\nSimilar computations for the other indecomposable objects finally\nproduce the generalised frieze in Figure \\ref{fig:generalised_frieze}\nin the introduction.\n\\end{Example}\n\n\n\n\n\\section{Questions}\n\\label{sec:questions}\n\n\nWe end the paper with some questions.\n\n\\begin{enumerate}\n\n\\item  The group $\\operatorname{K}_0^{ \\operatorname{split} }( \\mathsf{T} )$ is free abelian on ${\\mathsf{ind}}\\,\n  \\mathsf{T}$, and the subgroup $N$ of Definition \\ref{def:N} is generated by\n  all expressions $[a] - [a']$ where $s^{ {\\textstyle *} } \\rightarrow a\n  \\rightarrow s$ and $s \\rightarrow a' \\rightarrow s^{ {\\textstyle *} }$ are\n  exchange triangles with $s \\in {\\mathsf{ind}}\\, \\mathsf{T} \\setminus {\\mathsf{ind}}\\, \\mathsf{R}$.\n\n\\medskip\n\\noindent\nWhat is the rank $n$ of the quotient $\\operatorname{K}_0^{ \\operatorname{split} }( \\mathsf{T} ) \/ N$?\nNote that when $n$ is finite, it is the largest integer such that the\nmethod of Remark \\ref{rmk:app} results in a modified Caldero-Chapoton\nmap $\\rho : \\operatorname{obj} \\mathsf{C} \\rightarrow \\mathbb{Z}[ x_1^{ \\pm 1 }, \\ldots, x_n^{ \\pm\n  1 } ]$ using all the variables $x_1, \\ldots, x_n$.\n\n\\medskip\n\n\\item  Consider the $\\mathbb{Z}$-subalgebra of $\\mathbb{Z}[ x_1^{ \\pm 1 }, \\ldots, x_n^{\n    \\pm 1 } ]$ generated by the values of the modified Caldero-Chapoton map $\\rho$. \n\n\\medskip\n\\noindent\n  What is its relation to the cluster algebra?\n\n\\medskip\n\n\\item  Let $T$ be a cluster tilting object and use it to define a\n  Caldero-Chapoton map $X$.  If $T$ is subjected to cluster mutation,\n  then the values of $X$ change in a well-understood way, see\n  \\cite[proof of cor.\\ 5.4]{Palu}.\n\n\\medskip\n\\noindent\n  There is a notion of mutation of rigid objects due to\n  \\cite[sec.\\ 2]{MP}.  What happens to the values of the modified\n  Caldero-Chapoton map under such mutation?\n\n\\end{enumerate}\n\n\n\n\n\\medskip\n\\noindent\n{\\bf Acknowledgement.}\nThis paper is a direct continuation of \\cite{HJ}.   Both papers grew\nout of \\cite{BHJ} with Christine Bessenrodt, and we are grateful to\nher for the fruitful collaboration.\n\nWe thank the referee for several interesting comments and Robert Marsh\nfor answering a question about rigid subcategories.\n\nPart of this work was done while Peter J\\o rgensen was visiting the\nLeibniz Universit\\\"{a}t Hannover.  He thanks Christine Bessenrodt,\nThorsten Holm, and the Institut f\\\"{u}r Algebra, Zahlentheorie und\nDiskrete Mathematik for their hospitality.  He gratefully acknowledges\nsupport from Thorsten Holm's grant HO 1880\/5-1, which falls under the\nresearch priority programme SPP 1388 {\\em Darstellungstheorie} of the\nDeutsche Forschungsgemeinschaft (DFG).\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:intro}\n\nOntologies are witnessing an increasing popularity outside specialized AI\ncommunities. While this is mostly due to Semantic Web\napplications~\\cite{ber01}, we must also credit their ability to\ncope with taxonomies and part-whole relationships, to\nhandle  heterogeneous attributes, and their provision for various\nautomated reasoning services --- see, e.g.,~\\cite{staab2013handbook}.  \nThese features have been recognized since long time in system engineering,\nthe community encompassing all areas of research devoted to design,\nimplementation, monitoring and diagnosis of technical processes.\nFor instance, in the operations and maintenance sub-community, the use\nof ontologies is explicitly advocated\\footnote{See, e.g., the MIMOSA\nopen standard architecture at \\url{www.mimosa.org}.}.  \nAlso standards like ISO 13374\n(\\textit{Condition monitoring and diagnostics of \nmachines -- Data processing, communication and presentation}) suggest\nthe use of ontologies for several tasks, mostly related to data\nconceptualization. However, the adoption of ontologies\nfaces some challenges, mostly due to speed and reliability constraints\nimposed by industrial settings.  \n\nHere we investigate this issue by considering four contributions of\nours to application domains wherein ontologies provide key\ncapabilities in system engineering. The first case study is\nabout an on-board rolling-stock condition analyzer, i.e., a\nsystem to perform fault detection and\nclassification~\\cite{AmbrosiGT09}. The second one is about monitoring \nan intermodal logistic system~\\cite{CasuCT13}. The third one is about an\nontology-based framework to generate diagnostic-decision support\nsystems~\\cite{CicalaLOT16}. Finally, a fourth case study is an \napplication to computer-automated design of elevator\nsystems.\nIn the following, we briefly introduce each case study, giving details\nabout its context, underlying motivation and intended objectives. The \nultimate goal of the paper is to discuss and compare the results\nobtained to assess the effectiveness of ontologies in such\napplication domains.  \n\n\\noindent\n\\textit{Ontologies for condition analysis.}\nWe introduced an ontology-based condition analyzer\n(CA)~\\cite{AmbrosiGT09} in the context of the EU project  \nIntegrail\\footnote{More details about Integrail\nat \\url{http:\/\/www.integrail.eu\/}.}.      \nOur CA collects signals from control logic installed on locomotives,\nand it leverages an ontology to correlate\nobserved data, symptoms and faults. \nThe CA must mate two competing\nneeds: $(i)$ railway regulations\nrequire hardware which is highly reliable, and whose performances are\nthus far even from desktop workstations; $(ii)$\nontology-related tools, e.g., description logic reasoners,\nhave relatively large memory, processor and storage footprints.\nIn this experience, the main \ngoal was thus to check whether reasoning with ontologies can provide\nuseful diagnostic feedback in a resource-restricted scenario. \n\n\\noindent\n\\textit{Ontologies for system monitoring.}\nIn~\\cite{CasuCT13} we provided strong evidence of practical uses\nfor ontologies in complex systems engineering by implementing\na monitor for \\emph{Intermodal Logistics Systems} (ILSs), i.e.,\nsystems supporting the movement of containerized goods. In\nparticular, we considered combination of rail and road transport,\nwhere rail transport is provided by short-distance shuttle trains,\nand network coverage is\nachieved through connections at specialized terminals. In this\nexperience, the main goal was to gather data about terminal operations\nand compute global performances indicators, where access to data is\nmediated by an ontology --- ontology-based data access\n(OBDA)~\\cite{cal05}. Here, unlike the CA case study, \nthe ability to handle large amount of data is crucial, but reasoning\nis limited to SPARQL query answering.\n\n\\noindent\n\\textit{Ontologies for diagnostic support system generation.}\nDiagnostic Decision Support Systems (DDSSs) help humans to \ninfer the health status of physical systems. In~\\cite{CicalaLOT16} we\nintroduced DiSeGnO --- for ``Diagnostic Server Generation\nthrough Ontology'' ---  to generate customized DDSSs.\nAs in the ILS monitoring case study, since it is\nexpected that large quantities of data should be handled, \nthe ontology language is\nrestricted to those designed for tractable reasoning --- see,\ne.g.,~\\cite{cal05}. In this case, ontology-based  \nreasoning is not leveraged, as DiSeGnO generates relational databases\nfrom the domain ontology and then computes diagnostic rules\nwith \\textsc{Ptolemy II}{}~\\cite{Ptolemy}, an open-source software simulating\nactor-based models.  \n\n\\noindent\n\\textit{Ontologies for computer-automated design.}\nAs mentioned in~\\cite{ByeOPHS16}, the first scientific report of\nintelligent computer-automated design (CautoD) is the paper by\nKamentsky and Liu~\\cite{KamentskyL63}, who created a computer program\nfor designing character-recognition logic circuits satisfying given\nhardware  constraints. In mechanical design --- see,\ne.g.,~\\cite{rao2012mechanical} --- the term usually refers to\ntechniques that mitigate the effort in exploring alternative\nsolutions for structural implements. In our \\textsc{LiftCreate}{}\nCautoD program for elevator systems\\footnote{Part of the \\textsc{AiLift}{}\nsoftware suite \\url{www.ailift.it}.}, ontologies  \nsupport intelligent design creation and optimization by managing\ndetailed part-whole taxonomies, wherein different relations among\ncomponents can be expressed. This case study provides thus yet another\napplication of ontologies, mostly oriented to intelligent computation\nand data persistency.\n\nOverall, the case studies considered witness the great flexibility\nthat ontologies provide in handling diverse application scenarios,\nfrom condition analysis of locomotives, to automated design of\nelevators, considering both cases wherein they provide the basis for\nlogic reasoning services, or just advanced data-modeling capabilities.  \nThe rest of the paper is structured as follows. In\nSections~\\ref{sec:condition}, \\ref{sec:ilog}, \\ref{sec:ondaBrief}\nand \\ref{sec:elevator} we sketch the design, the implementation and\nthe results obtained in the case studies described above.\nSection~\\ref{sec:concl} concludes the paper by summarizing the results\nand providing some discussion thereof. \n\n\n\n\\section{Rolling stock condition analysis}\n\\label{sec:condition}\n\\begin{figure}[t!]\n\\begin{center}\n  \\includegraphics[scale=0.27]{OntologyTraction.png}\n  \\caption{\\label{fig:e414ont}A portion of the E414 ontology regarding\n    traction faults. Concepts are nodes and\n    object properties are edges: white nodes are SP3A\n    concepts, grey ones are E414-specific concepts.}\n\\end{center}\n\\end{figure}\n\nThe CA prototype described in~\\cite{AmbrosiGT09} focuses on \nfault detection on Trenitalia E414 locomotive.\nThe main task \nof the CA is to perform fault classification according to \npriority for maintenance, and impact on mission-related and\nsafety-related aspects.\nHere, we focus on traction groups as an example of\nsubsystem that can generate a faulty condition.\nOur ontology for the E414 locomotive is written in OWL 2 language\nand it builds on the SP3A core ontology \n--- see~\\cite{AmbrosiGT09} for details. \nIn particular, the E414 ontology leverages the SP3A concepts of {\\sc\n  ObservationData}, i.e., process variables, and {\\sc Observation},\ni.e., sequences of observation data from which individuals of class\n{\\sc Symptom} and {\\sc Fault} arise. {\\sc Symptom} individuals\nare related to {\\sc Observation} individuals via the {\\sc\n  refersToObservation} property and to {\\sc Fault} individuals via the\n{\\sc refersToFault} property. {\\sc Fault} is a concept whose\nindividuals are defined in terms of the necessary {\\sc hasSymptom}\nrelationship with {\\sc Symptom} individuals.\nTwo subclasses of {\\sc Fault} are defined: {\\sc\n  PriorityFault} and {\\sc   NonPriorityFault}, with obvious\nmeaning.\nIn Figure~\\ref{fig:e414ont} we show a portion of the E414 ontology\nrelated to  traction faults, where concepts have been specialized \nin subclasses whose individuals correspond to actual signals and\nsubsystems. Fault and symptom classification is obtained by\na Description Logic (DL) reasoner considering the patterns \nobserved. For instance, in the case of {\\sc\n  TractionHighTemperatureObservation}, three ranges of temperatures\nare defined that correspond to ``interesting'' patterns: from 70 to 80\ndegrees, from 80 to 130 degrees, and over 130 degrees. It is \npostulated that observations falling in the second and in the\nthird ranges are to be considered mission critical, while the ones in\nthe first category are only maintenance critical.\n\n\n\n\nA detailed description of the CA architecture can be found\nin~\\cite{AmbrosiGT09}. Here we provide some intuition on how the\nanalyzer works considering high temperatures in the traction groups. \nWhen the temperature of a group is higher than 70 \ndegrees for at least 3 consecutive samples read from the field bus,\nthe CA starts tracking a potential anomalous pattern.\nOnce such a pattern is detected, the corresponding\nindividuals in the classes \\textsc{TractionObservationData} \nand \\textsc{TractionHighTemperatureObservation}\nare recorded. {\\sc Symptom} individuals\nare built along with all the properties required by the ontology\nspecification. For example, if an observation of the class\n\\textsc{TractionHighTemperatureObservation} has been created,   \na specific individual \n\\textsc{TractionHighTemperatureObservation} is related to  a new\n\\textsc{Symptom} individual by the\n\\textsc{refersToObservation} property.\n{\\sc Fault} individuals for each {\\sc Symptom} individual are created \ntogether with the {\\sc causedBySymptom} property.\n{\\sc Fault} as well as {\\sc Symptom} individuals are built of generic\ntype, leaving their classification to the DL reasoner. Once\nthe classification is done, the CA publishes the results, transmitting\nthem to external agents. As an example, let us assume that $i$ is an\nindividual of the class \\textsc{TractionHighTemperatureObservation}\nwhose property \\textsc{isAt} is set to the constant\n\\textsc{\\_130degrees}, $s$ is the \\textsc{Symptom} individual related\nto $i$, and $f$ is the \\textsc{Fault} individual related to $s$. \nThe E414 ontology postulates that all symptoms such that the\ncorresponding observation is an instance of\n\\textsc{TractionHighTemperatureObservation} related by \\textsc{isAt}\nto the constant \\textsc{\\_130degrees}\nare also an instance of \\textsc{TractionTotalMissionImpactSymptom},\nwhich is a subclass of \\textsc{Symptom}. Therefore, a reasoner \ncan infer that $s$ belongs to \n\\textsc{MissionRelatedSymptom}\n\n\n\\begin{table}[t!]\n\\caption{\\label{tab:results} Results with \n  (a) lazy and (b) eager implementations of the CA.}\n\\begin{center}\n\\tiny\n    \\begin{tabular}{ | c | c | c | c | }\n    \\hline\t\t\t\n      Scenario & Memory Consumption [MB] & CPU Time [ms]  & Amortized CPU Time [ms] \\\\\n    \\hline\n      1a & 38 & 90 & ND \\\\\n      2a & 74 & 25373 & 25373 \\\\\n      3a & 106 & 1053656 & 210731 \\\\\n      4a & OUT OF MEMORY & 3253637 & 191390 \\\\\n    \\hline\n      1b & 37 & 90 & ND \\\\\n      2b & 72 & 21506 & 21506 \\\\\n      3b & 104 & 86938 & 17387 \\\\\n      4b & 105 & 279523 & 16442 \\\\\n    \\hline\n    \\end{tabular}\n\\end{center}\n\\end{table}\n\nOut of the three sets of experiments performed in~\\cite{AmbrosiGT09},\nwe report just those to ensure that the CA\nimplementation fits the constraints. To this end, we ran several tests\nusing different fault scenarios\\footnote{Sets  \n  of multidimensional time series (3600 samples at 1Hz) corresponding\n  to 52  process variables are generated.\n \n \n \n  Simulations run on EN50155-compliant embedded devices \n  with 1GHz Socket 370 FC-PGA Celeron Processor with 256MB \n  of main memory and a 1GB SSD running Linux Blue Cat (kernel 2.6) and\n  Sun Java Virtual Machine implementation (JRE 1.6). The DL reasoner is \\textsc{Pellet}{}~\\cite{sirin07}.}.\nTable~\\ref{tab:results} shows the results obtained by running the CA\non four different scenarios --- the first includes no\nfault, the second includes only one fault, the third includes five\ncontemporary faults, and the last 17 contemporary faults --- \nusing two different configurations.\nConfiguration (a) is ``lazy'', i.e., it keeps all the individuals,\nwhile configuration (b) is ``eager'', i.e., it  \ndeletes individuals as soon as possible.\nAs we can see in Table~\\ref{tab:results}, the eager version results in\na great improvement over the lazy one, both in terms of memory\nconsumption and in terms of computation time.\nIn particular, in the second column of Table~\\ref{tab:results} we can\nnotice that the eager version performs reasonably well, even in the\nfourth test case (worst-case scenario). In the same scenario,\nthe lazy version exceeds the amount of available memory.\nAs we can see in the rightmost column of Table~\\ref{tab:results}, the\namortized computation time over a single scenario\ndecreases with the number of concurrent\nobservations detected in the round.\nManaging a round of samples without detected observations\ntakes only 90 ms, which leaves enough time for other activities, and\nallows the CA to process all the incoming signals in due course.\n\n\n\n\n\n\n\n\n\n\n\\section{Monitoring of intermodal systems}\n\\label{sec:ilog}\n\\begin{figure}[t!]\n\\centerline{\\scalebox{.26}{\\includegraphics{iLogDot}}}\n\\caption{\\label{fig:iLog_dot} ILS ontology describing the design of the\n  OBDA solution. Ellipses denote concepts with datatype properties;\n  directed edges are object properties; dotted edges are concept\n  inclusions.} \n\\end{figure}\n\nIn~\\cite{CasuCT13} we provided evidence that ontology-based data\naccess (OBDA)~\\cite{cal05}  \nis of practical use in the context of \\emph{Intermodal Logistics\n  Systems} (ILSs). The investigation focuses on the opportunity to build a monitoring information\nsystem (MIS) using OBDA instead of relational databases\n(RDBs). The application scenario is an ILS relying on a\nlogic akin to computer networks, i.e., frequent short-distance trains\nwith a fixed composition and a predefined daily schedule to cover some\ngeographical area. \\emph{Intermodal Transport Units} (ITUs) enter\nthe network at some terminal and travel to their destination according\nto a predefined route, usually boarding more than one train along the\nway. Terminals collect ITUs from areas of approximately 150Km in\nradius in order to minimize road transport. The MIS is a key\nenabler to minimize delivery time, maximize rolling-stock and network\nutilization and, ultimately, reduce the economic overhead of\ntransportation for the final customer. The main goal of the MIS is to\ncompute \\emph{Key Performance Indicators} (KPIs) to perform tactical and \nstrategical decision making about the network.\n\nIn Figure~\\ref{fig:iLog_dot} we present a graphical outline of the\nontology at the heart of our OBDA solution to monitor the ILS.\nThe ontology --- ILS ontology in the following --- is compliant with\nthe OWL 2 QL profile described in the official W3C's recommendation\nas \\emph{``[the sub-language of OWL 2] aimed at applications that use \n  very large volumes of instance data, and where query answering is\n  the most important reasoning task.''}. Given the ILS application\ndomain, OWL 2 QL guarantees that conjunctive query answering and\nconsistency checking can be implemented efficiently\nwith respect to the size of data and ontology, respectively. The\nrestrictions that OWL 2 QL implies did not hamper the modeling\naccuracy of our ILS ontology. In Figure~\\ref{fig:iLog_dot} we can\npinpoint classes related to freight forwarding such as \n\\textbf{Customer}, i.e., companies forwarding their goods through the\nnetwork, \\textbf{RequestForWork}, i.e., the main document witnessing\nthat a given customer has issued a request for transporting a number\nof ITUs, \\textbf{TransportOrder}, i.e., the ``bill of transit''\nassociated to each ITU, as well as entities related to physical\nelements such as \\textbf{ITU}, \\textbf{Terminal} and \\textbf{Train}. Also\n``logical'' entities are modeled such as \\textbf{Route}, i.e., a\nsequence of terminals and railway connections serviced regularly by\none or more scheduled trains and \\textbf{ScheduledStop}, i.e.,\nterminals associated to a given route with a given\nschedule. \\textbf{Event} is the main monitoring entity, as the\ncalculation of most KPIs relies on the exact recording of events at\nspecific locations. \n\n\\begin{figure}[t!]\n\\begin{center}\n  \\scalebox{.27}{\\includegraphics{q3heavy.pdf}}\n\\end{center}\n\\caption{\\label{fig:results} Computation time of a KPI with different\n    query processors: SQL (square), \\textsc{ARQ}{} (circle), \\textsc{Pellet}{}\n    (hourglass), \\textsc{Quest}{} (triangle). In each plot, the $x$ axis\n    displays the number of simulation days from 1 to 15, the $y$ axis\n    displays the CPU time (in milliseconds on a logarithmic\n    scale).} \n\\end{figure}\n\nTo assess OBDA performances, in~\\cite{CasuCT13} we obtained  different\nartificial utilization scenarios by changing \nthe number of ITUs shipped daily from each terminal. Considering  \ntypical usage patterns, we postulated that a\nprovision of 10 to 50 ITUs is to be shipped daily from each terminal,\nwith 40 to 50 ITUS corresponding to a heavy utilization.\nScenarios are simulated for an increasing number of days to\nevaluate scalability, and all of them share common settings as far as\nnumber of train travels, number of cars per train, and timetabling are\nconcerned. Unexpected delays as well as the number of customers per\nterminal follow a probabilistic model --- see~\\cite{CasuCT13} for more\ndetails. In Figure~\\ref{fig:results} we display the results\\footnote{\n  All results are obtained on a family of identical\n  Intel-based PCs, featuring a Core2Duo 2.13 GHz CPU, 4GB of RAM and\n  running Ubuntu Linux 10.04 (64 bit edition).} obtained\nin the case of an heavy utilization scenario to compute a specific\nKPI, namely the average number of ITUs unloaded per hour.\nThe performance of four different query-answering systems\nare reported: a SQL query on a native RDB implementation, and a SPARQL\nquery on the ontology store. The SPARQL query can be answered by three\ndifferent systems, namely \\textsc{ARQ}{} (the default query processor in the\n\\textsc{Jena}{} library), \\textsc{Pellet}{} (the same DL reasoner that we consider in\nSection~\\ref{sec:condition}) and \\textsc{Quest}{}~\\cite{mur12}. The latter is the only\nreasoner exploiting the fact that SPARQL queries can be compiled\non-the-fly into SQL queries for an equivalent RDB representation of\nthe ontology stored in the main memory. As we can observe in\nFigure~\\ref{fig:results}, OBDA-based solutions show higher overall\ncomputation times than the RDB-based solution --- from 1 to 2 orders\nof magnitude --- together with an apparently growing trend associated\nto the time span of the simulation. However, as we have shown\nin~\\cite{CasuCT13}, a trend test performed on the results obtained\nwith the best OBDA solutions for various KPIs, displays no statistically\nsignificant increase in the CPU time required to answer various\nqueries with respect to the number of days. Considering that for most\nKPIs we can adopt an ``eager'' solution similar to that considered in\nSection~\\ref{fig:results}, we can conclude that OBDA is practically\nfeasible for monitoring medium-to-large scale systems.\n\n\n\n\\section{Diagnostic support systems generation}\n\\label{sec:ondaBrief}\n\\begin{figure}[!t]\n\\centering\n\\scalebox{0.34}{\\includegraphics{ondaFramework.png}}\n\\caption{\\label{fig:ondaModel}Functional architecture and work-flow of DiSeGnO framework.}\n\\end{figure}\n\nIn~\\cite{CicalaLOT16} we introduced an approach to compile ontology-based\ndescriptions of equipment into diagnostic decision support systems\n(DDSSs). The tool DiSeGnO, whose functional architecture and work-flow\nis sketched in Figure~\\ref{fig:ondaModel}, fulfills  this task in\nthree phases: in the \\textsf{USER} phase, a domain ontology and\ndiagnostic rules model are designed by the user; in the\n\\textsf{DiSeGnO} phase, the system reads and analyzes the  ontology\nand the rules to output the actual DDSS; in the \\textsf{DDSS} phase,\ninput web services receive data from the observed physical system and\nrecord them in the generated data  store. According to the ISO 13374-1\nstandard a DDSS consists of six modules of which DiSeGnO implements three:\n\\textit{Data Manipulation} to  perform signal analysis and \ncompute meaningful descriptors, \\textit{State Detection}\nto check conformity to reference patterns, and \\textit{Health\n  Assessment} to diagnose faults and rate the current  \nhealth of the equipment or process.\nAs shown in Figure~\\ref{fig:ondaModel}, the ontology description is\ncreated by a system architect in the \\textsf{USER} phase. \nThe ontology must be written using OWL 2 QL language\\footnote{While\n  this can be accomplished in several ways,  the tool\n  \\textsc{prot\\'eg\\'e}{}~\\cite{gennari2003evolution} is suggested  because it \nis robust, easy to use, and it provides, either directly or through\nplug-ins, several add-ons that facilitate ontology design and\ntesting.} as in the case study shown in Section~\\ref{sec:ilog}. \nThe diagnostic computation model must be a sound\nactor diagram generated by \\textsc{Ptolemy II}{}~\\cite{Ptolemy} which describes\nthe processing to be applied to incoming data in order to generated\ndiagnostic events --- here we focus on the ontology part, but more\ndetails on the rule handling part can be found in~\\cite{CicalaLOT16}.\nThe \\textsf{DiSeGnO} phase contains the actual DDSS generation\nsystem which consists of the \\textsf{Data Store Generator}, i.e.,\na piece of software that creates a relational  database \nby mapping the domain ontology to suitable tables, \nand the \\textsf{Web Services Generator}, i.e., a module  \nthat creates interface services for incoming and outgoing events.\nFinally, in the \\textsf{DDSS} phase, \ndata is acquired and stored in the internal database,\nthe rules engine processes data and generates diagnostic events which\nare then served to some application.  \n\n\\begin{figure}[t!]\n\\centering\n\\scalebox{0.25}{\\includegraphics{ontology.pdf}}\n\\caption{\\label{fig:hvac_onto} Domain ontology for HVAC monitoring.\n  Formalism is the same as in Figure~\\ref{fig:iLog_dot}.}\n\\end{figure}\n\nAn example of a DiSeGno-compliant equipment description \nis shown in Figure~\\ref{fig:hvac_onto}. The ontology is related to a \nHeating Ventilation and Air Conditioning (HVAC) appliance and it is\ndivided into  a \\emph{static} and a \\emph{dynamic} part. In the static\npart, which is not updated while monitoring, the ontology contains a\ndescription of the observed physical system. In the\nHVAC ontology we have \\textbf{System} and\n\\textbf{DataSource},  \nrelated by the \\textbf{isInSystem} property. \\textbf{hasSubsystem}\nrelationship indicates that one \\textbf{System} could be composed  by\none or more \\textbf{SystemComponent} which are themselves subclasses\nof  \\textbf{System}. Finally, \\textbf{DataSource} is the \nclass of elements that can generate diagnostic-relevant information.   \nThe dynamic part describes \\emph{events}, including both\nthe ones generated by the  observed system and its components, and\nthose output by the DDSS. An event is always associated to a\ntime-stamp and it can be either \\emph{incoming} to the DDSS from the\nobserved system, or \\emph{outgoing} from the DDSS\\footnote{This\n  distinction is fundamental, because DiSeGnO must know which \nevents have to be associated with input and output web services,\nrespectively.}.\nThe main concepts in the dynamic part of the HVAC\nontology are \\textbf{DDSS} which \\textbf{receives} instances of\n\\textbf{IncomingEvent} and \\textbf{sends} instances of\n\\textbf{OutgoingEvent}. Notice that \\textbf{IncomingEvent} instances\nare connected to \\textbf{DataSource} instances by the role\n\\textbf{generates}, denoting that all incoming events\nare generated by some data source.\nAlso every \\textbf{OutgoingEvent} instance, i.e., every\ndiagnostic event, \\textbf{relatesTo} some instance of\n\\textbf{DataSource}, because the end user must be able to\nreconstruct which data source(s) provided information that caused\ndiagnostic rules to fire a given diagnostic event.\n\\textbf{OutgoingEvent} specializes to \\textbf{AlarmEvent}, \\textbf{FaultEvent}\nand \\textbf{DescriptorEvent}. Every \\textbf{OutgoingEvent} instance is\nconnected to one of \\textbf{DiagnosticIndicator} instances, \ni.e. \\textbf{Alarm}, \\textbf{Fault} and \\textbf{Descriptor} sub-concepts,\n by \\textbf{reports} relation, in order to have a reference message\n about the diagnostic rules.\n\n\n\\section{Computer-automated design of elevators}\n\\label{sec:elevator}\n\\begin{figure}[t!]\n\\begin{center}\n\\scalebox{.22}{\\includegraphics{Elevator.png}}\\\\\n\\scalebox{.26}{\\includegraphics{HydraulicElevator.png}}\n\\end{center}\n\\caption{\\label{fig:elev_onto} Ontologies describing the\n  implements handled by \\textsc{LiftCreate}{} (top) and \n  the components of \\textsf{OnePistonHydraulicElevator}\n  (bottom). Concepts are rectangles, concept inclusion is denoted by\n  solid arrows, and HAS-A object properties are denoted by diamond-based arrows.}  \n\\end{figure}\n\nOur latest ontology-based application is in the field of\ncomputer-automated design (CautoD) which differs\nfrom ``classical'' computer-aided design (CAD) in that it\nis oriented to replace some of the designer's capabilities and not just\nto support a traditional work-flow with computer graphics\nand storage capabilities. Nevertheless, CautoD programs most\noften include CAD facilities to visualize technical drawings related\nto the implements of interest. \nIn particular, our \\textsc{LiftCreate}{} program is oriented to automating\ndesign of elevators, taking the designer from the very first\nmeasurements to a complete project which guarantees feasibility within\na specific normative framework. \\textsc{LiftCreate}{} works in three steps. In\nthe first step,  the user is asked to enter relevant parameters\ncharacterizing the project, and an overall ``design philosophy'' to be\nimplemented. For instance, if the size of the elevator's shaft is\nknown and fixed in advance, \\textsc{LiftCreate}{} can generate solutions which\nmaximize payload,  door size, or car size. A design philosophy is just\na set of heuristics which, e.g., prioritize door size over other\nelements, still keeping into account hard constraints, e.g., payload\nand car size should not fall below some threshold. In the second\nphase, \\textsc{LiftCreate}{} retrieves components from a database of parts and\nexplores the (combinatorial) space of potential solutions, either\nusing heuristic search techniques, or resorting to optimizations\ntechniques --- like those suggested, e.g., in~\\cite{ByeOPHS16}. In the\nthird phase, a set of feasible designs is proposed to the user, sorted\naccording to decreasing relevance considering the initial design\nphilosophy. For instance, if door size is to be maximized, the first\nalternatives shown to the user are those with the widest doors,\nperhaps at the expense of payload or car size.\n\nThe main issue with \\textsc{LiftCreate}{} work-flow is that even simple\nversions of elevators consists of a large number of components,\nincluding car frame, car, doors (car and landing doors), emergency\nbrakes, pistons or cables, motors and control logic. In order to\nexplore the space of potential designs, components cannot be\nsolely available as drawing elements, like in classical CAD solutions,\nbut they must be handled as first class data inside \\textsc{LiftCreate}{}\nlogic. This aspect required us to organize the taxonomy related to\ndifferent kinds of elevators and, for each elevator kind, to structure\nthe components in a part-whole hierarchy. In\nFigure~\\ref{fig:elev_onto} we show a fragment of the taxonomy for\nelevators and an example of part-whole structure for a specific\nelevator kind. In particular, in\nFigure~\\ref{fig:elev_onto} (top), we see that \\textsc{LiftCreate}{} classifies\n\\textbf{Elevator} individuals in two main subclasses corresponding to\nhydraulic-based (\\textbf{HydraulicElevator}) and rope-based\n(\\textbf{RopeElevator}) designs. Both subclasses feature additional\npartitions to handle specific design requirements, e.g., rope\nelevators can be provided with a reduction gearbox or not, and the\ndrive can be direct of reeved. For one leaf class of the taxonomy,\nnamely \\textbf{OnePistonDirectHydraulicElevator}, in\nFigure~\\ref{fig:elev_onto} (bottom) we show the detailed part-whole\ndiagram, from which we learn that, e.g., the only peculiar aspects of\nsuch subclass is to have only one \\textbf{Piston}, whereas the\nremaining components are common to \\textbf{HydraulicElevator} or\n\\textbf{Elevator}. Also we can see that the car frame is specific of hydraulic elevators (\\textbf{CarFrameHydra}) and it is comprised of several parts, including\n\\textbf{CarRails}, \\textbf{Buffer} and \\textbf{Ropes}. The\nrelationships encoded in such part-whole hierarchy are\ninstrumental to \\textsc{LiftCreate}{} when it comes to handle drawing, storage\nand retrieval of designs, but also to reason about the various\ntrade-offs of a design when searching in the space of potential solutions.\n\n\n\\section{Conclusions}\n\\label{sec:concl}\nConsidering the experiences herein outlined, we summarize some\nlessons learned in applying ontologies for systems engineering. First\nand foremost, while ontologies provide an effective tool for\nconceptualizing scenarios as diverse as those considered, \nsome ontology-based tools, e.g., DL reasoners, are untenable\nunless small-to-medium scale systems are considered. In the case of\nE414 ontology reasoning with an expressive ontology required us to\nimplement strategies to ``forget'' data to avoid cluttering\nthe reasoner. In the ILS ontology, where SPARQL queries for KPIs are\nthe only reasoning requested and the usage of OWL 2 QL profile banned\nexpressive but hard-to-compute constructs, scaling \nstill requires discarding data using a recency approach. On the other hand, \nin DiSeGnO and \\textsc{LiftCreate}{}, ontologies merely provide means for\nconceptualizing data and, as such, flexibility is gained without\nsacrificing performances. The second take-home message is that\nsublanguages of OWL 2 are adequate for most modeling purposes. \nWith the only exception of E414 ontology, the ones herein\nconsidered fit OWL 2 QL constraints which allowed us\nto combine in a  natural way subclassing (``IS-A'' relationships) with \nother kind of object properties (including ``HAS-A''). However, the fact that OWL 2 QL  \nontologies can be compiled to relational databases --- as in the case\nof DiSeGnO --- or handled trough an object-persistency module --- as\nin the case of \\textsc{LiftCreate}{} --- makes their use transparent to other\nsystem components. Third, and final point, with the exception of ILS\nmonitoring, none of our applications required the integration of\ndifferent data sources which is indeed one of the main tasks which\nontologies are advocated for. Nevertheless, our experience witnesses\nthat even in single-source data modeling, ontologies provide an\nexcellent mean to bridge the gap between domain experts and\ncomputer software designers. \n\n\n\n\n\n\n\\bibliographystyle{unsrt}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\\section{Introduction}\n\nProprioception, or the ability to perceive the relative positioning of\nneighboring body parts as well as the muscle effort deployed to\nproduce it, is a fundamental human sense. Together with vision and\ntactile sensing, it plays a unique and important role in human hand\nperception \\cite{blanchard2013differential}. For robotic manipulation,\nproprioception is translated as the combination of joint position and\ntorque sensing (assuming a hand comprised exclusively of revolute joints).\n\nCompared to vision and tactile sensing, both of which have been\nstudied extensively in the context of manipulation, we believe that\ngrasping with proprioception is still an\nimportant area to advance. On one hand, vision and tactile sensing\nhave intrinsic limitations, such as occlusion for vision and hardware\ncomplexity for tactile sensing. On the other hand, when all these\nsenses are available, they can still complement each other.  Demonstrating\nmanipulation capabilities based exclusively on proprioception becomes\na useful exercise: we believe the more a hand can do with only one\nsensing modality, the more versatile it will be when multi-modal\nsensory information gets integrated.\n\nAs we show here, proprioception is promising in providing the hand\nwith the ability to adapt to the previously unknown shape of the\nobject, and to execute stable grasps. We note that there are multiple\nways for hands to adapt to an object: while fully-actuated\nhands use sensor information (such as proprioception here) to perform\nactive adaptation, underactuated hands are good examples of passive\nadaptation without the use of sensing. However, the former have the\nadvantage of versatility: proprioceptive grippers can provide\ncompliance similar to passive mechanisms, but can also change behavior\nat runtime and selectively execute different types of grasps. Of\ncourse, the price paid for the additional versatility is the increased\ncomplexity of the sensory setup.\n\nIn this study, we explore the problem of \\textit{grasping using only\nproprioceptive feedback}, without any contact information or\nknowledge of object pose and properties. To the best of our knowledge,\nwe are the first to show that a robot hand can perform all the\nfollowing tasks using proprioception exclusively:\n\\begin{itemize}\n\\item execution of fingertip grasps for unknown objects;\n\\item execution of enveloping grasps for unknown objects;\n\\item on-demand transitions between fingertip and enveloping grasps.\n\\end{itemize}\nOur main contributions are to provide methods for the tasks above,\nand, in the process, demonstrate their effectiveness by\nexperiment. Our results indicate that the proprioceptive gripper is\nmore versatile in the range of fingertip grasps it can perform,\ncompared to our two baselines: an emulated underactuated gripper\ncommanding fixed torques to the joints, as well as a physically\nconstructed underactuated gripper. In addition, our gripper also\ndisplays the ability to execute enveloping grasps and to transition to\nthem from fingertip grasps. Both examples of increased versatility\nwere achieved using proprioception as the only available sensing\nmodality.\n\n\\section{Related Work}\n\nResearchers have been exploring real-time sensing and control as\nan alternative to vision-based planning in manipulation. Assuming\nobject information is available, model-based controllers can perform\ngrasping or in-hand manipulation. For example, Yoshikawa et\nal. presented studies (e.g. \\cite{yoshikawa2000control}) on hybrid\nforce-position control for manipulation.  Arimoto et\nal. \\cite{arimoto2000dynamics} derived the dynamics of a dual-finger\ngripper and proposed a controller which can regulate the object\nposition and orientation. Caccavale et al. \\cite{caccavale2013grasp}\nproposed an impedance controller to keep track of desired object\ntrajectory and ensure the grasp quality simultaneously. Unlike our\napproach, these methods require complete information of the\nhand-object system.\n\nWhen the models of the objects are not available, researchers either\nrelied on assumption about the contacts, or used sensor-based\ntechniques for grasping. For example, Schneider and\nCannon \\cite{schneider1992object} studied object impedance control\nusing multiple manipulators. Arimoto et al. \\cite{arimoto2005two} and Yoshida et al. \\cite{yoshida2007blind}\nstudied ``blind grasping'' using two fingertips. However, these studies\nassume the contacts only happen at the end points or the end hemispheres\nof the fingertips. Wang et al. \\cite{wang2007switching} proposed a\ncontroller that can search appropriate finger contact locations using\nhaptic feedback and can switch between control modes for different\nsurfaces. Platt et al. \\cite{platt2010null} presented a study on\nchanging the contact configuration by following the gradient of\ngrasping objective functions using six-axis loadcell data. Hsiao et\nal. \\cite{hsiao2010contact} proposed a contact-reactive method using\ntactile sensing to deal with uncertainty. However, these methods\nrequire contact sensing methods, such as tactile sensors or in-finger\nload cells. In contrast, our approach does\n  not make any assumptions about contact location or state, and does not\nrequire tactile sensing data.\n\nTorque measurement is often used for grasp control. Researchers have\ndeveloped several robotic hands with force or torque sensing. For\nexample, the Robonaut Hand \\cite{lovchik1999robonaut} and the DLR Hand\nII \\cite{butterfass2001dlr} have strain gauges or force-torque sensors\nembedded in their fingers. The hand of the DOMO robot\n\\cite{edsinger2004domo} and the hand of the Obrero robotic platform\n\\cite{torres2005obrero}\nmake use of the Series Elastic Actuators (SEA), which are a type of\nactuators with elastic components in series with the motor to sense\nthe torque \\cite{pratt1995series}. Furthermore, the DLR Hand-Arm\nSystem \\cite{grebenstein2011dlr} incorporates the Variable Impedance\nActuators, which are SEAs whose spring stiffnesses are actively\ncontrolled. These hardware designs offer high performance, but at the\ncost of high complexity and large overall packages.\n\nAs an alternative, researchers have developed underactuated hands that do\nnot require sophisticated sensing and control, and this types of hands are good baselines to compare against. Underactuated hands can\nadapt to the object and make a grasp by the virtue of\ncarefully-designed torque ratios between joints. The Harvard Hand\n\\cite{dollar2010highly}, iHY Hand \\cite{odhner2014compliant}, Robotiq\nHand \\cite{birglen2004kinetostatic}, and Velo Gripper\n\\cite{ciocarlie2014velo} are good examples in this\ncategory. However, even though underactuation simplifies control, it\ngenerally does not provide as much dexterity as full actuation. Many\nof the hands above can only perform certain types of grasps, or lack the flexibility to choose the\nconfiguration after making the grasp.\n\n\n\n\\section{Hardware Platform}\n\nWhile joint position sensing is ubiquitous for fully-actuated robot\nhands, torque sensing and control is not common in commercially available\nmanipulators. This compelled us to design our own hardware testbed. We\nimplemented torque sensing and control with Series Elastic Actuators\n(SEA), a method known for high-fidelity torque control, shock\nprotection, and human-safety~\\cite{pratt1995series}. \n\n\\subsubsection{SEA Module}\n\nSimilar to the design from Ates et al. \\cite{ates2014servosea}, we developed a simple and compact SEA module (Fig.~\\ref{fig:sea}). A position-driven servo (gray) is used as the driving motor, which receives position commands and returns the current position (measured by the built-in potentiometer), i.e., $\\theta_{motor}$  can be measured. A torsion spring (orange) is used as the elastic component, connecting the motor shaft (purple) and the pulley shaft (blue). A Spectra cable is tied on the pulley to transmit the force to the finger joint. An absolute magnetic encoder (in green) is mounted on the end of the pulley shaft to measure $\\theta_{pulley}$. In steady-state, the force in the tendon can be calculated as the product of spring stiffness and deflection divided by pulley radius: \n\\begin{equation} \\label{eq:sea}\nF_{tendon} = K_{spring} \\cdot (\\theta_{pulley} - \\theta_{motor}) \/ R_{pulley}\n\\end{equation}\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width = 80 mm]{images\/hand.pdf}\\\\\n\\caption{Schematics of the gripper.}\n\\label{fig:hand}\n\\vspace{-4mm}\n\\end{figure}\n\n\n\n\\subsubsection{Gripper Design}\n\nThe gripper consists of two fingers and each finger has two links, shown in Fig.~\\ref{fig:hand}. \nIn each joint, flexion is\npowered by the tendon (shown as colored dash lines in\nFig.~\\ref{fig:hand}) connected to the SEA pulley, and the extension is\ndriven by a restoring spring. The tendon connected to the\ndistal joint (the red dash line) goes right through the axis of the\nproximal joint so that the torques of proximal and distal joints are\nfully decoupled.\n\n\n\\subsubsection{SEA-level Control}\n\nThere are three SEA-level control modes: motor position control, pulley position control and torque control, and they can be switched online. \nWe note that we built joint-level or hand-level controllers on top of these SEA-level controllers. For example, joint position and torque control can be achieved by SEA pulley position and torque control with a simple linear conversion.\n\n\n\n\\section{Fingertip Grasping}\n\nFingertip grasps commonly refer to grasps where only the most\ndistal links of each finger make contact with the object. We note that it does not necessarily mean the contacts are located in the very end of the fingers, so contact locations are still unknown. This type of\ngrasp is important not only for precision tasks, but also for cases\nwhere a more stable enveloping grasp is not immediately available\nbecause of the environment (an object laying on a table, against a\nbackdrop, etc.). In this section, we introduce a control algorithm\nwhich can perform stable fingertip grasps for unknown objects.\n\n\\subsection{Problem Statement}\n\nThe objective of our algorithm is to\nincrease the torques applied at the joints (and implicitly the contact forces) in a ``stable'' fashion after making initial contacts. In other words, we need to find an\nincrease in joint torques that produces no net wrench on the\nobject. For a hand with multi-link fingers, this is not straightforward given that the joint torques need to be coordinated in the absence of information on object\nshape and contact locations. It is necessary to note that we are not\nsolving contact planning problem, and we wish to develop reactive control strategies that do not require pre-planning.\n\nOur insight is that proprioception alone can characterize grasp stability. For an SEA-powered proprioceptive gripper, an\nunbalanced net wrench will produce object movement against the\ncompliant elements, which we can measure. Therefore, \\textit{we\n  formulate the goal of grasp stability as the one of minimizing\n  object movement while applying forces}. Since we do not have a direct measure of object\nmovement in Cartesian coordinates by using only proprioception, we use the change of joint position in joint space (measured by SEA)\nduring grasping as a proxy for object movement.\n\nFor intuition, consider the analogies shown in Fig.~\\ref{fig:analogy}:\nFigure (a) shows a (simplified) scenario in which the motors (red dots) are\ndriving the torsion springs, and the springs are pushing the fingers\n(blue dots) to make a grasp (gray dots). (b) shows a simpler\none-dimensional abstraction using linear springs, with similar\ncolor-coding as in (a). Here, we actively control the positions\n$x_{m1}$ and $x_{m2}$ (which translate to motor positions\n$\\theta_{motor}$ on the real gripper) to apply forces, and measure\n$x_{j1}$ and $x_{j2}$ (which translate to pulley positions\n$\\theta_{pulley}$ linearly mapped to joint positions). We aim to keep\n$x_{j1}$ and $x_{j2}$ constant as we squeeze.\n\n\\subsection{MIMO Grasping Controller}\n\nOur key insight is that the problem of grasping unknown objects can be solved even in the absence of\ncontact information, by using proprioception as inputs for a\nmulti-input-multi-output (MIMO) proportional-integral (PI)\ncontrol. Without knowledge of object geometry, contact locations and contact states, it\nis impossible to fully model the dynamic system analytically. However,\na proprioceptive platform still provides sensory access to the\nvariables that characterize the grasp stability, and the PI control framework provides ways to regulate these variables even though the analytical relationship is not constructed . We\nthus aim to use a feedback scheme operating exclusively in\nthe sensory space of the robot, without explicitly modeling the physics of the gripper and the object.\n\nFig.~\\ref{fig:mimo} shows the block diagram of the MIMO control loop. Here, the controller is constructed on top\nof the low-level sensing, so we consider the joint angles and torques\nare already obtained from SEA measurements. The reference vector\n$\\bm{u}$ consists of desired joint angle values ($\\theta^{p1}_{des}$,\n$\\theta^{d1}_{des}$, $\\theta^{p2}_{des}$, $\\theta^{d2}_{des}$, where\nthe superscripts $p$ represent proximal and $d$ represent distal\njoints) and reference joint torques ($\\tau^{p1}_{des}$,\n$\\tau^{d1}_{des}$, $\\tau^{p2}_{des}$, $\\tau^{d2}_{des}$ ). The feedback vector $\\bm{y}$ has the same structure, but contains actual measured\nmeasured values. The desired joint angles (first half of the reference\nvector $\\bm{u}$) are extracted by a feedforward matrix $\\bm{F}$ and\nused as a feedforward term. The error between the reference $\\bm{u}$\nand feedback $\\bm{y}$ is fed into a MIMO PI block (a combination of\nmany P and PI controllers) which is a $4\\times8$ matrix. The output of\nthe PI block and the feedforward term are summed up as motor position\ncommand $\\bm{c}$ (a $ 4 \\times 1$ vector) and sent to the motors:\n\\begin{equation} \\label{eq:mimo}\n\\bm{c}(t)=\\bm{F u}(t)+ {\\bm{K}_{p}}  \\bm{e}(t)+ {\\bm{K}_{i}} \\int_{0}^{t}{ \\bm{e}(\\tau) d\\tau} \n\\end{equation}\nHere,\n$\n\\bm{F}=\\left[ \\begin{matrix} {\\bm{I}_{4\\times 4}} & {\\bm{O}_{4\\times 4}}\\end{matrix} \\right]\n$\nis the feedforward matrix,\n$\\bm{K}_p$ ($4\\times 8$ matrix with all entries being non-zero) and \n$\\bm{K}_i=\\left[ \\begin{matrix} {\\bm{K}_{4\\times 4}} & {\\bm{O}_{4\\times 4}}\\end{matrix} \\right]$ (where $\\bm{K}$ is a matrix with all entries being non-zero)\nare proportional and integral gain matrices, and $\\bm{e}$ is the $ 8 \\times 1$ error vector between $\\bm{y}$ and $\\bm{u}$. After that, the actual motor position vector $\\bm{d}$ goes to the black-box system of the gripper and the unknown object.\n\nIn the reference vector $\\bm{u}$, the desired joint angle values  ($\\theta^{p1}_{des}$, $\\theta^{d1}_{des}$, $\\theta^{p2}_{des}$, $\\theta^{d2}_{des}$) \nare equal to those in the initial touch configuration, while the reference torques ($\\tau^{p1}_{des}$, $\\tau^{d1}_{des}$, $\\tau^{p2}_{des}$, $\\tau^{d2}_{des}$ ) \nare chosen using the maximum motor torques. A special design of this controller is that we require the joint angles to be regulated exactly to the set points, but do not require the torques to be so. We allow and make use of the steady-state error of pure proportional control (we note that entries in the right half of the integral gain $\\bm{K}_i$ are set to be zeros). In this way, the reference torques do not need to be a legal set of torques that result in equilibrium --- actually, we are not able to design such a legal set of torques due to the absence of contact or object information. We let the law of dynamics decide the steady-state values for torques, and let the system balance itself automatically. The effectiveness of increasing joint torques is shown in section VI.\n\nFrom a practical standpoint, the tuning process of the MIMO PI controller is not\nas complicated as it would seem based on the number of parameters. First, due to gripper symmetry, the number of parameters is cut by half. Second, we formulate\nevery gain as a product of a baseline value ($b_i$ in (\\ref{eq:kp})(\\ref{eq:ki}) )  and a weight\ncoefficient ($w_i$ in (\\ref{eq:kp})(\\ref{eq:ki}) ). The baseline values are set to be the same if the input\nentries corresponding to those gains have same physical\ndimensionality, and the weight coefficients are tuned based on its\nrelative importance. Third, conventional tuning heuristics for the gains of single-input-single-output systems also apply here. The structures of the gain matrices are as follows:\n\\begin{equation} \\label{eq:kp}\n\\bm{K}_p=\\left[ \\begin{smallmatrix}\n{w_1 b_1} & {w_2 b_2}  & {w_2 b_1}  &  {w_2 b_2}  & {w_3 b_3}  & {w_4 b_4} & {w_4 b_3} & {w_4 b_4} \\\\\n{w_2 b_1} & {w_1 b_2}  & {w_2 b_1}  &  {w_2 b_2}  & {w_4 b_3}  & {w_3 b_4} & {w_4 b_3} & {w_4 b_4} \\\\\n{w_2 b_1} & {w_2 b_2}  & {w_1 b_1}  &  {w_2 b_2}  & {w_4 b_3}  & {w_4 b_4} & {w_3 b_3} & {w_4 b_4} \\\\\n{w_2 b_1} & {w_2 b_2}  & {w_2 b_1}  &  {w_1 b_2}  & {w_4 b_3}  & {w_4 b_4} & {w_4 b_3} & {w_3 b_4} \\\\\n\\end{smallmatrix} \\right]\n\\end{equation}\n\\begin{equation} \\label{eq:ki}\n\\bm{K}_i=\\left[ \\begin{smallmatrix}\n{w_1 b_5} & {w_2 b_6}  & {w_2 b_5}  &  {w_2 b_6}  & ~~0~~ & ~~0~~ & ~~0~~ & ~~0~~ \\\\\n{w_2 b_5} & {w_1 b_6}  & {w_2 b_5}  &  {w_2 b_6}  &   0 & 0  & 0  &     0\\\\\n{w_2 b_5} & {w_2 b_6}  & {w_1 b_5}  &  {w_2 b_6}  &   0  & 0  & 0  &    0 \\\\\n{w_2 b_5} & {w_2 b_6}  & {w_2 b_5}  &  {w_1 b_6}  & 0 & 0 & 0 & 0 \\\\\n\\end{smallmatrix} \\right]\n\\end{equation}\nWe pick $b_1 = 0.2$, $b_2 = 0.5$, $b_3 = 4.0$, $b_4 = 8.0$, $b_5 = 1.0$, $b_6 = 1.0$, $w_1 = 1.0$, $w_2 = 0.3$, $w_3 = 1.0$, $w_4 = 0.5$ for our hardware.\n\n\n\\section{Enveloping Grasping and Transitions}\n\nAn enveloping grasp is the one where both distal and proximal links\nmake contact with the object around its circumference. This type of\ngrasp is generally considered more stable than a fingertip grasp\nbecause it can resist disturbances in a wider range of directions. The\ninability to envelop is also one of the main shortcomings of simple\nparallel grippers. In contrast, some of the more recent underactuated\nhands are optimized explicitly for effective enveloping grasps of a\nwide range of objects (e.g. \\cite{ciocarlie2014velo}).\n\nWhen using our proprioceptive gripper, we found that stable\nenveloping grasps for unknown objects are easier to obtain than fingertip grasps. The\nmechanism is generally fully constrained and all the links are\ncounterbalancing each other. A simple joint torque control scheme, or the MIMO Grasping Controller, can fulfill this task.\n\nA very important ability of this gripper, further underlining its\nversatility, is to \\textit{transition} between grasp types when holding unknown objects. After\nexecuting a stable fingertip grasp (using the MIMO Grasping\nController), the gripper can switch to joint torque control with the\ntorque ratio (between distal and proximal joints) being 0.5 to 1.0,\nthus bringing the object into the hand and creating an enveloping\ngrasp. We illustrate this behavior with several experiments in the\nfollowing section.\n\n\n\n\n\\section{Experiments and Results}\nIn this section, we demonstrate the merits of the\nproprioception-enabled gripper by several experiments. We validated\nits capability of performing fingertip grasps, enveloping grasps, and\nthe transition from the former to the latter. We note that in this\nstudy we only consider a two-dimensional scenario, in which the\nobjects are confined to move only in the plane of the fingers.\n\n\\subsection{Fingertip Grasp}\nThe goal of this experiment is to test the hypothesis that the MIMO Grasping Controller is effective in fingertip grasping for unknown objects. We compare against two baselines: a (fully-actuated) gripper running a Fixed Torque Ratio Controller, and a physical underactuated gripper. \n\nThe Fixed Torque Ratio Controller, where the torques applied to proximal and distal joints always follow a certain ratio, can be thought of as an emulation of a common type of underactuated grippers (tendon-pulley-driven, without special designs such as stoppers or clutches). When this kind of grippers make grasps, the configuration-dependent torques from the extension springs can be ignored, as they are usually much smaller than the flexing torques from the tendons. Therefore, the net joint torques in proximal and distal joint have a configuration-independent and design-time-fixed ratio, which is the ratio between the joint pulley radii.\n\nFurthermore, we understand that this emulation is subject to limited control bandwidth and may have unrealistic behavior compared to its physical counterpart. Thus, we also built a physically underactuated gripper testbed for comparison. This testbed has same specs as the fully-actuated proprioceptive gripper, except that the proximal and distal joints are driven by a single tendon wrapping around the joint pulleys. In this design, we can alter the torque ratio by physically changing the pulleys between experiments. We perform torque control for proximal joints in our experiments, thus the torques on the distal joints are defined by the physically determined ratios. However, in this setup, we lose the ability to measure joint positions by SEA readings because, in underactuated mechanisms, joint positions are determined not only by actuator positions but also by contact forces which here are unknown. \n\n\n\n\\subsubsection{Experiment Protocol}\n\nOur experiment proceeds as follows. We execute the grasping in two phases. In the first phase (approaching and touching), the fingers are set in torque control mode with very low reference torques so that they stop when they touch the object. In the second phase(squeezing), the gripper executes the MIMO Grasping Controller or the Fixed Torque Ratio Controller in the fully-actuated testbed, or the  joint torque control on proximal joints in the underactuated testbed for comparison.\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=55 mm]{images\/experiment_setup.pdf}\\\\\n\\caption{Object sizes, object locations and initial touch poses.}\n\\label{fig:expsetup}\n\\vspace{-3mm}\n\\end{figure}\n\n\\begin{figure}[t]\n\\hspace{-6mm}\\includegraphics[width=100 mm]{images\/pose_photo.pdf}\n\\caption{Pose changes in different controllers.}\n\\label{fig:pose}\n\\vspace{-3mm}\n\\end{figure}\n\nThere are a lot of factors that may influence the performance. To have a well-rounded comparison, we swept the following dimensions:\n\\begin{itemize}\n\\item \\textit{Controllers.} The torque ratio is a key parameter for both the Fixed Torque Ratio Controller, and the physically underactuated gripper. We tested the MIMO Grasping Controller against the other two baselines with three different ratios between the distal and proximal joint: 0.3, 0.4 and 0.5.\n\\item \\textit{Objects.} We selected four objects for the test: a big cylinder (diameter: 67mm), a big box (side length: 57mm), a small cylinder (diameter: 47mm) and a small box (side length: 39 mm). All objects have negligible friction with the table.\n\\item \\textit{Object locations.} We swept three locations along the center line of the gripper within the range of fingertip grasp: 100 mm, 120 mm and 140 mm from the palm.\n\\item \\textit{Initial touch poses.} We tested three different distal joint angles for the initial touch: 0, 30 and 60 degrees. We note that we only include this dimension for MIMO Grasping Controller and Fixed Torque Ratio Controller test, and not for the physically underactuated hand because the distal joint angles of initial touch cannot be explicitly controlled in runtime.\n\\item \\textit{Friction coefficient.} We tested the controllers with\n  two fingertip materials: rubber (high friction, $\\mu = 1.2$) and\n  vinyl plastic (low friction, $\\mu = 0.4$).\n\\end{itemize}\n\nTo sum up, we swept all five dimensions and conducted 360 grasping experiments. Fig.~\\ref{fig:expsetup} shows the three object locations (shown as the crosshairs), three initial touch poses (colored fingers), and the sizes of the objects relative to the gripper (orange shapes).\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics{images\/torque_increase.pdf}\n\\caption{Increases of torque magnitude during squeezing}\n\\label{fig:trq}\n\\vspace{-3mm}\n\\end{figure}\n\n\\subsubsection{Performance Metric}\n\n\\begin{itemize}\n\\item \\textit{Success rate.} The success rate is our primary performance metric. We define a ``success\" if the gripper finally settles down in equilibrium with the object in hand after squeezing. This definition includes three scenarios: (1) the gripper keeps the object in fingertips near initial touch pose, without converting to enveloping grasp or reaching joint limits, (2) the gripper holds the object but reconfigures to an enveloping grasp, and (3) the gripper keeps the object in fingertips but reaches a mechanical joint limit (thus the joint torque ratio changes). These cases are all considered successful but still need to be distinguished. Case (1) is the most desirable, while (2) and (3) mean the grasp is not stable at initial pose and relies on reconfiguration to be balanced.\n\n\\item \\textit{Gripper pose change.} We believe it is also useful to keep the object in the same pose as when first contact is made. We thus use a secondary performance metric that evaluates how much the object moves in the hand during the squeezing process, with less movement considered better. Without access to object pose in Cartesian space, we measure this as the change in gripper pose between initial touch and final grasp (Euclidian distance in four-dimensional joint space). This metric is only calculated and averaged for the successful cases. Besides, it is not computed for physically underactuated gripper because the joint angles during grasping are not accessible for the reasons mentioned above.\n\\end{itemize}\n\n\\begin{figure*}[t]\n\\centering\n\\includegraphics[width=1.0\\textwidth]{images\/result.pdf}\n\\caption{Experiment results of fingertip grasping compared to Fixed Torque Ratio Controller.}\n\\label{fig:result}\n\\vspace{-3mm}\n\\end{figure*}\n\n\n\\subsubsection{Results}\nFig.~\\ref{fig:pose} shows the photos of some typical scenarios in the experiments. (a) and (b) shows the resting pose and the initial touch. (c) shows the successful grasp near initial touch pose. The other images show cases in which the object was kept in fingertip grasp but a joint limit was reached (d), the grasp was transformed into an enveloping one (e), and the object was squeezed out of the hand (f). \n\nFig.~\\ref{fig:trq} shows that all controllers are effective in increasing joint torques. The horizontal axis shows different controllers (or grippers), the vertical axis is the magnitude of the four-dimensional torque vector ($\\tau^{p1}_{meas}$, $\\tau^{d1}_{meas}$, $\\tau^{p2}_{meas}$, $\\tau^{d2}_{meas}$) which indicates how ``strong'' the grasp is, and bar colors distinguish between initial touch and final grasp. We can see that there is a significant increase in the torque magnitude, and the torque levels in different cases are similar.\n\n\nThe results of the experiments comparing against Fixed Torque Ratio Controller are visualized as multiple bar charts in Fig.~\\ref{fig:result}. In each plot, the bar colors show four different controllers,\nthe vertical axis is one of the performance metrics \nand the horizontal axis represents another dimension which is different in each plot (from (a) (f) to (d) (i): different objects, object locations, initial poses, and friction coefficients). In each bar in the first row showing the success rate, the pure-color area, the dotted area, and the line-shaded area represent, respectively, successful fingertip grasp without reaching joint limit, successful grasps but converted to enveloping, as well as successful fingertip grasp but joint angles reached limits.\n\nSimilarly, the results comparing against physically underactuated grippers are shown in Fig.~\\ref{fig:result_ua}. Here, the initial touch pose dimension is not available, so there are three dimensions (from (a) to (c): different objects, object locations, and friction coefficients). Also, the pose change metric is not available due to the absence of joint angle information. All other plotting rules are the same as Fig.~\\ref{fig:result}.\n\n\n\n\nAs shown in Fig.~\\ref{fig:result} (e)(j) and Fig.~\\ref{fig:result_ua} (d), the overall success rates are 91.67\\% for MIMO Grasping Controller, 72.22\\%, 76.39\\%, 66.67\\% for Fixed Torque Ratio Controller with torque ratio of 0.3, 0.4, 0.5, respectively, and 87.50\\%, 75.00\\%, 87.50\\% for physically underactuated gripper with torque ratio of 0.3, 0.4, 0.5, respectively. Even when the overall success rates are close (for example, Fig.~\\ref{fig:result_ua} (d)), the types of the resulting grasps are significantly different. Besides, the gripper pose change metric for the MIMO controller is 9.96, compared to 41.55, 31.93, and 81.14 (degrees) respectively for the Fixed Torque Ratio controllers. \n\n\n\\subsection{Enveloping Grasps and Transitions}\n\nWe performed a second experiment to show that this gripper can perform enveloping grasp with either MIMO Grasping Controller or Fixed Torque Ratio Controller. We tested on two objects (big cylinder and big box), two object locations (60mm and 80mm from the palm), and two controllers mentioned above. The success rate is 100\\%.\n\nThe last experiment is to show the performance of the transition from fingertip grasp to enveloping grasp. We first created fingertip grasps using the MIMO Grasping Controller, and then switched to Fixed Torque Ratio Controller with a ratio of 0.5. We tested on two objects (big cylinder and big box), three object locations (100, 120 and 140mm), three initial poses (0, 30 and 60 degrees) with the low friction fingertips. We found the success rate was 83.33\\%.\n\n\n\\begin{figure*}[t]\n\\centering\n\\includegraphics[width=0.8 \\textwidth]{images\/result_ua.pdf}\\\\\n\\caption{Experiment results of fingertip grasping compared to physically underactuated gripper.}\n\\label{fig:result_ua}\n\\vspace{-2mm}\n\\end{figure*}\n\n\\section{Discussion and Conclusion}\n\nOverall, the results of the previous section support our hypotheses: the proprioceptive gripper running MIMO Grasping Controller is effective at executing stable fingertip grasps in a variety of situations and outperforms the baselines. Furthermore, the proprioceptive gripper exhibits versatility in being able to perform multiple types of grasps and also to transition between them on-demand.\n\nBased on Fig.~\\ref{fig:result} and \\ref{fig:result_ua}, the MIMO Grasping Controller outperforms the baselines in fingertip grasping, and usually succeeds without transforming to an enveloping grasp or reaching joint limits. In contrast, the emulated and physical underactuated gripper often transform to an enveloping grasp or reach joint limits, thus relying on gripper reconfiguration. The second row of Fig.~\\ref{fig:result} ((f) to (j)) gives similar intuition: for the Fixed Torque Ratio Controller, most cases have a large pose change. While the end-result is stable, it is different from the originally intended grasp. This might be unimportant or detrimental depending on the application.\n\nIt is also interesting to notice that, in different conditions, the optimal torque ratio for the emulated or physical underactuated gripper is different. We take this to mean that there is no one clearly preferable pre-set torque ratio, which could be physically implemented in a mechanical design, in order to obtain ideal performance in all these cases. In contrast, the proprioception-enabled gripper has the flexibility to alter torques at run-time.\n\nLooking at how specific variables affect performance we can gain additional insights. From Fig.~\\ref{fig:result} and~\\ref{fig:result_ua} (a) and (b) we can see that the success rates for Fixed Torque Ratio Controller are low if the objects are small and close to the palm. This is because the emulated or physical underactuated gripper tends to transform the initial unstable fingertip grasps to enveloping when contacts are close to distal joints, but cannot cage the object if it is small because the distal links are fighting against each other --- a common issue for underactuated grippers. In contrast, the MIMO Grasping Controller does not suffer from this because it does not perform the conversion.\n\n\nIn the transitioning experiment, the high success rate shows the\nproprioceptive gripper can indeed perform the conversion between grasp types\n\\textit{on-demand}. Though underactuated grippers also\noccasionally perform such transitions, they occur unintentionally\nand without giving the user an option to select the desired type of\ngrasp.\n\nIt is important to also highlight the limitations of this\nstudy. Due to high dimensionality of the brute-force sweep in\nour experiment, we cannot afford to cover a larger range with a finer\nresolution for each dimension. In particular, we are unable to explore\nmore possibilities for physically implemented torque ratios. The\nevaluation of the controller is primarily experimental and would\nbenefit from additional stability analysis, carried out for example\nfor representative cases and grasps.\n\nOverall, we claim that proprioceptive manipulators, using active\nsensing and control such as the MIMO Grasping Controller, represent a\npromising way towards more versatile grasping and manipulation for\nunknown objects. Future work will include the extension of the operation to three-dimensional cases, optimization \/ learning of the control gains, and the inclusion of hand\nposition to our set of actively controlled variables. We are aiming to\nfurther explore these possibilities.\n\n\\section*{APPENDIX}\n\n\n\n\n\n\n\\bibliographystyle{bib\/IEEEtran}  \n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\nOver the past decade, advances in data collection and increasing access to computational resources have led to a revolution in the use of data-driven techniques for the solution of complex inverse problems. One such problem is that of turbulence, the multiscale nature of which causes extreme computational demands for most practical systems. As a result, turbulence requires the use multiple modeling approximations for the higher wavenumbers which remain unsupported by computational degrees of freedom. One such modeling approach is that of large eddy simulation (LES) \\cite{sagaut2006large}, which attempts to simulate the evolution of the smaller wavenumbers while the unresolved frequencies are modeled by an algebraic or differential equation. As such, the basic premise of LES is extendable to many partial differential equation systems with quadratic non-linearities. The procedure of modeling these smaller scales is often denoted \\emph{closure} due to insufficient knowledge about higher-order wavenumber interactions with the coarse-grained system \\cite{berselli2006mathematics} and remains vital for the accurate computation of many applications \\cite{hickel2014subgrid,yu2016dynamic,zhou2018structural}. From an LES point of view, the closure problem may be considered to be dominated by commutative errors in the calculation of the non-linear term as well as the defects associated with commutative errors stemming from the dynamic term. In this study, we focus on the former.\n\nThere are two main schools of thought when it comes to the LES of the Navier-Stokes equations. The first of these promotes the use of explicit closures. Explicit LES argues for the utilization of closures in the form of sub-grid models specified as algebraic or differential equations for the unresolved scales. These are built on intuitive reasoning of how the losses of coarse graining the Navier-Stokes equations may be incorporated into an LES deployment. Some of the most notable sub-grid closure strategies are those given by the eddy-viscosity hypothesis. Within the context of the Navier-Stokes equations, it is generally accepted that the finer scales are dissipative at the Kolmogorov length scales \\cite{kolmogorov1941local} and therefore, most turbulence models seek to specify a sub-grid dissipation \\cite{frisch1995turbulence}. Most sub-grid models can be traced back to the seminal work of Smagorinsky \\cite{smagorinsky1963general}, where a model was proposed based on the concepts of an effective eddy-viscosity determined by an \\emph{a priori} specified mixing length and a $k^{-5\/3}$ scaling recovery for the kinetic energy content in the wavenumber domain. Similar hypotheses have also been used for two-dimensional turbulence \\cite{leith1968diffusion} (often utilized as a test-bed for geophysical scenarios, for instance see works by Pearson \\textit{et al.}\\cite{pearson2018log,pearson2017evaluation}), for approximating the $k^{-3}$ cascade in two-dimensional turbulence and generally have their roots in dimensional analysis related to the cascade of enstrophy. These models may also be classified as \\emph{functional} due to the phenomenological nature of their deployment and represent the bulk of explicit LES turbulence models used in practical deployments. Explicit LES closures may also be specified through the specification of a low-pass spatial filter to account for the unresolved scales \\cite{bardina1980improved,stolz1999approximate,layton2003simple,mathew2003explicit} where phenomenology is bypassed but ansatz are provided for the bulk dissipative nature of the smaller scales through the control of a characteristic filter-width. In either scenario, (i.e., whether structural or functional), the choice of the phenomenology (or dissipation control parameter) plays a key role in the successful calculation of accurate \\emph{a posteriori} statistics. In contrast, the implicit LES (or ILES) approach utilizes numerical dissipation to model the unresolved scales in a turbulent flow \\cite{grinstein2007implicit,el2017investigation,Margolin2018}. In essence, the predominantly dissipative effects of the smallest scales are replicated through an artificial numerical dissipation via a biased discretization used in the calculation of the non-linear advective term \\cite{thornber2007implicit,debonis2013solutions}. The ILES approach is popular due to reduced algorithmic complexity and represents a union of turbulence modeling and shock capturing mechanisms but is often criticized due to the difficulties involved in quantifying the correct amount of dissipation in a turbulent flow evolution. This results in ILES methods often proving robust and stable but overly dissipative. In this work, we propose a machine learning algorithm to enable selective dissipation within an ILES deployment through the use of explicit LES concepts during the training of the learning framework. \n\n\n\nThe past few years have seen a rapid increase in the use of data-driven techniques for the spatio-temporal modeling of dynamical systems \\cite{schmidt2009distilling,bright2013compressive,xiao2015non,ma2015using,gautier2015closed,brunton2016discovering,schaeffer2017learning,raissi2017machine,mohan2018deep,raissi2018hidden,rudy2018deep,san2018neural,wan2018data,kim2018deep,muravleva2018application,jin2018prediction}. When it comes to turbulence, some widely used strategies for inference include symbolic regression \\cite{weatheritt2016novel,weatheritt2017development,weatheritt2017hybrid}, where functional model-forms for Reynolds-averaged Navier-Stokes (RANS) deployments were generated through evolutionary optimization against high-fidelity data. Other techniques incorporating Bayesian ideologies have also been used, for instance by Xiao \\textit{et al.}\\cite{xiao2016quantifying} where an iterative ensemble Kalman method was used to assimilate prior data for quantifying model form uncertainty in RANS models. In Wang \\textit{et al.}\\cite{wang2017physics,wang2017comprehensive} and Wu \\textit{et al.}\\cite{wu2018data}, random-forest regressors were utilized for RANS turbulence-modeling given direct numerical simulation (DNS) data. In Singh and Duraisamy \\cite{singh2016using} and Singh \\textit{et al.}\\cite{singh2017machine}, an ANN was utilized to predict a non-dimensional correction factor in the Spalart-Allmaras turbulence model through a field-inversion process using experimental data.  Bypassing functional formulations of a turbulence model (a focus of this study) was also studied from the RANS point of view by Tracey \\textit{et al.} \\cite{tracey2015machine}. Ling and Templeton \\cite{ling2015evaluation} utilized support vector machines, decision trees and random forest regressors for identifying regions of high RANS uncertainty. A deep-learning framework where Reynolds-stresses would be predicted in an invariant subspace was developed by Ling \\textit{et al.} \\cite{ling2016reynolds}. Machine learning of invariance properties has also been discussed in the context of turbulence modeling \\cite{ling2016machine}. The reader is directed to a recent review by Duraisamy \\textit{et al.}\\cite{duraisamy2018turbulence}, for an excellent review of turbulence modeling using data-driven ideas.\n\nAs shown above, the use of data-driven ideologies and in particular artificial neural networks (ANNs) has generated significant interest in the turbulence modeling community for addressing long-standing challenges (also see \\cite{sotgiu2018turbulent,zhang2018machine,zhu2019machine,zhang2019application,raissi2019deep} for recent progress). One motivation for the popularity of ANNs is that a multilayered ANN may be optimally trained to universally approximate any non-linear function \\cite{hornik1989multilayer}. In addition, the deployment of ANNs is amenable to integration within existing computational frameworks. Greater accessibility to data and ever-improving computing capabilities has also motivated the development of advanced ANN architectures for large-scale learning of complicated physical phenomena such as turbulence. Within the context of LES (and associated with the scope of this paper) there are several investigations into sub-grid modeling using data-driven techniques. In one of the first studies of the feasibility of using learning from DNS based high-fidelity data, Sarghini \\textit{et al.}\\cite{sarghini2003neural} utilized ANNs for estimating Smagorinsky model-form coefficients within a mixed sub-grid model for a turbulent channel flow. ANNs were also used for wall-modeling by Milano and Koumotsakos \\cite{milano2002neural} where it was used to reconstruct the near wall field and compared to standard proper-orthogonal-decomposition techniques. An alternative to ANNs for sub-grid predictions was proposed by King \\textit{et al.}\\cite{king2016autonomic} where \\emph{a priori} optimization was utilized to minimize the $L^2$-error between true and modeled sub-grid quantities in a least-squares sense using a parameter-free Volterra series. Maulik and San \\cite{maulik2017neural} utilized an extreme-learning-machine (a variant of a single-layered ANN) to obtain maps between low-pass spatially filtered and deconvolved variables in an \\emph{a priori} sense. This had implications for the use of ANNs for turbulence modeling without model-form specification. A more in-depth investigation was recently undertaken by Fukami \\textit{et al.}\\cite{fukami2018super} where convolutional ANNs were utilized for reconstructing from downsampled snapshots of turbulence. Maulik \\textit{et al.} \\cite{maulik2018deconvolution} also deployed a data-driven convolutional and deconvolutional operation to obtain closure terms for two-dimensional turbulence. Gamahara and Hattori \\cite{gamahara2017searching}, utilized ANNs for identifying correlations with grid-resolved quantities for an indirect method of model-form identification in turbulent channel flow. The study by Vollant \\textit{et al.} \\cite{vollant2017subgrid} utilized ANNs in conjuction with optimal estimator theory to obtain functional forms for sub-grid stresses. In Beck \\textit{et al.}\\cite{beck2018neural}, a variety of neural network architectures such as convolutional and recurrent neural networks are studied for predicting closure terms for decaying homogeneous isotropic turbulence. A least-squares based truncation is specified for stable deployments of their model-free closures. Model-free turbulence closures are also specified by Maulik \\textit{et al.}\\cite{maulik2018deconvolution,maulik2019subgrid} and Wang \\textit{et al.}\\cite{wang2018investigations}, where sub-grid scale stresses are learned directly from DNS data and deployed in \\emph{a posteriori} assessments. King \\textit{et al.}\\cite{king2018deep} studied generative-adversarial networks and the LAT-NET \\cite{hennigh2017lat} for \\emph{a priori} recovery of statistics such as the intermittency of turbulent fluctuations and spectral scaling. A detailed discussion of the potential benefits and challenges of deep learning for turbulence (and fluid dynamics in general) may be found in the article by Kutz \\cite{kutz2017deep}.\n\nWhile a large majority of the LES-based frameworks presented above utilize a least-squares error minimization technique for constructing maps to sub-grid stresses \\emph{directly} for theoretically optimal LES \\cite{langford1999optimal,moser2009theoretically,labryer2015framework}, this work is novel in that it utilizes sub-grid statistics (pre-computed from DNS data) to train a classifier. This classifier determines whether a location requires dissipation or not through \\emph{a priori} experience in the learning phase. Once classified, the non-linear term at this particular point is evaluated using one of two schemes. If it is determined that the point requires no sub-grid closure, a symmetric and second-order accurate, energy and enstrophy conserving Arakawa-scheme \\cite{arakawa1981potential} is utilized for the non-linear term computation. If dissipation is necessary, an upwinding scheme is utilized instead. Therefore this study may be interpreted as a machine learning framework for devising hybrid schemes for non-linear term computation with a view to reconstructing turbulence statistics in a superior fashion. Therefore, this study is similar to that employed by Ling and Kurzawski \\cite{ling2017data} for adaptively determining RANS corrections. We note that the classification framework devised in this study is also deployed in an aligned work to switch between functional and structural explicit LES hypotheses spatio-temporally \\cite{maulik2018online} thus proving that high-fidelity DNS statistics may be qualitatively utilized to inform modeling strategies through conditional probability predictions. The article shall describe how the proposed framework is effective in moderating the larger dissipation of an upwinded-scheme through assessments on the Kraichnan turbulence test-case. \n\n\\section{Turbulence modeling equations}\n\nThe governing equations for two-dimensional turbulence are given by the Navier-Stokes equations in the vorticity-stream function formulation. In this formulation, our non-dimensional governing equation for incompressible flow may be represented as\n\\begin{align}\n\\label{eq1}\n\\frac{\\partial \\omega}{\\partial t} + J(\\omega,\\psi) = \\frac{1}{Re} \\nabla^2 \\omega,\n\\end{align}\nwhere $Re$ is the Reynolds number, $\\omega$ and $\\psi$ are the vorticity and stream function respectively connected to each other through the Poisson equation given by\n\\begin{align}\n\\label{eq2}\n\\nabla^2 \\psi = - \\omega.\n\\end{align}\nIt may be noted that the Poisson equation implicitly ensures a divergence-free flow evolution. The non-linear term (denoted the Jacobian) is given by\n\\begin{align}\n\\label{eq3}\nJ(\\omega,\\psi) = \\frac{\\partial \\psi}{\\partial y} \\frac{\\partial \\omega}{\\partial x} - \\frac{\\partial \\psi}{\\partial x} \\frac{\\partial \\omega}{\\partial y}.\n\\end{align}\nThe stream function and the two-dimensional velocity components are related as \n\\begin{align}\n\\label{eq3a}\nu &= \\frac{\\partial \\psi}{\\partial y}, \\quad v = -\\frac{\\partial \\psi}{\\partial x}.\n\\end{align}\n\nA reduced-order implementation of the aforementioned governing laws (i.e., an LES) is obtained through\n\\begin{align}\n\\label{eq4}\n\\frac{\\partial \\bar{\\omega}}{\\partial t} + J(\\bar{\\omega},\\bar{\\psi}) = \\frac{1}{Re} \\nabla^2 \\bar{\\omega},\n\\end{align}\nwhere the overbarred variables are now evolved on a grid with far fewer degrees of freedom. Due to the reduction in supported frequencies, the non-linear Jacobian fails to capture inter-eddy interactions at different wavenumbers. If it is assumed that the finer scales of vorticity are generally dissipative in nature for two-dimensional turbulence (based on Kraichnan's cascade of enstrophy \\cite{kraichnan1967inertial}), dissipative models may be embedded into the coarse-grained evolution of the vorticity evolution equation to recover some portion of the effect of the finer scales. Explicit LES closures embed dissipation into the vorticity evolution in the form of eddy-viscosity phenomenology or through structural arguments of scale-similarity. However ILES manipulates the computation of the non-linear Jacobian term to add numerical dissipation to mimic that of the unresolved frequencies. The latter framework, while numerically robust, suffers from difficulties associated with \\emph{directed} dissipation where it is often very easy to be over-dissipative in regions where sub-grid dissipation may not be as pronounced. In this article, we introduce a hybrid ILES framework that focuses upwinding at areas where high probability of sub-grid dissipation necessity is detected.\n\n\n\\section{Non-linear Jacobian computation}\n\nThe study utilizes two types of non-linear term computation schemes. Our first choice is symmetric, second-order accurate and conserves energy and enstrophy to minimize numerical dissipation. This is given by the well-known second-order Arakawa scheme \\cite{arakawa1981potential} as detailed below. The non-linear term in Equation \\ref{eq4} may be numerically calculated on a coarse grid using \n\\begin{align}\nJ^A (\\bar{\\omega},\\bar{\\psi}) = \\frac{1}{3} \\left( J_1 (\\bar{\\omega}, \\bar{\\psi}) + J_2 (\\bar{\\omega}, \\bar{\\psi}) + J_3 (\\bar{\\omega}, \\bar{\\psi}) \\right)\n\\end{align}\nwhere $J^A(\\bar{\\omega},\\bar{\\psi})$ will henceforth refer to the Arakawa discretization. The individual terms on the right hand side of the above equation are given as \n\\begin{align}\n\\begin{split}\n J_1 (\\bar{\\omega},\\bar{\\psi}) & = \\frac{1}{4 \\Delta x \\Delta y} \\left[ (\\bar{\\omega}_{i+1,j}-\\bar{\\omega}_{i-1,j}) (\\bar{\\psi}_{i,j+1} - \\bar{\\psi}_{i,j-1}) \\right. \\\\ \n& \\left.  - (\\bar{\\omega}_{i,j+1}-\\bar{\\omega}_{i,j-1}) (\\bar{\\psi}_{i+1,j} - \\bar{\\psi}_{i-1,j}) \\right],\n\\end{split}\n\\end{align}\n\n\\begin{align}\n\\begin{split}\n & J_2 (\\bar{\\omega},\\bar{\\psi}) = \\frac{1}{4 \\Delta x \\Delta y} \\left[ \\bar{\\omega}_{i+1,j} (\\bar{\\psi}_{i+1,j+1}-\\bar{\\psi}_{i+1,j-1}) \\right. \\\\ \n & \\left. - \\bar{\\omega}_{i-1,j} (\\bar{\\psi}_{i-1,j+1}-\\bar{\\psi}_{i-1,j-1})  - \\bar{\\omega}_{i,j+1} (\\bar{\\psi}_{i+1,j+1}-\\bar{\\psi}_{i-1,j+1})\\right. \\\\\n & \\left. + \\bar{\\omega}_{i,j-1} (\\bar{\\psi}_{i+1,j-1}-\\bar{\\psi}_{i-1,j-1}) \\right],\n\\end{split}\n\\end{align}\n\n\\begin{align}\n\\begin{split}\n& J_3 (\\bar{\\omega},\\bar{\\psi}) = \\frac{1}{4 \\Delta x \\Delta y} \\left[ \\bar{\\omega}_{i+1,j+1} (\\bar{\\psi}_{i,j+1} - \\bar{\\psi}_{i+1,j}) \\right. \\\\\n& \\left. - \\bar{\\omega}_{i-1,j-1} (\\bar{\\psi}_{i-1,j}-\\bar{\\psi}_{i,j-1}) - \\bar{\\omega}_{i-1,j+1} (\\bar{\\psi}_{i,j+1}-\\bar{\\psi}_{i-1,j}) \\right. \\\\\n& \\left. + \\bar{\\omega}_{i+1,j-1} (\\bar{\\psi}_{i+1,j}-\\bar{\\psi}_{i,j-1}) \\right].\n\\end{split}\n\\end{align}\nThe aforementioned scheme is utilized when our proposed classifier recognizes that no dissipation is necessary. \n\nA numerically dissipative computation of the non-linear term allows for that stabilization of noise accumulation at the grid cut-off wavenumbers. Although there are many different methodologies for upwind based dissipation with varying degrees of complexity, in this article, we utilize a conventional upwind-biased scheme as detailed in the following \\cite{hoffmann2000computational}. Our ILES Jacobian is computed as \n\\begin{align}\n\\begin{split}\nJ^I(\\bar{\\omega},\\bar{\\psi}) =& \\bar{u}_{i,j} \\frac{\\bar{\\omega}_{i+1,j} - \\bar{\\omega}_{i-1,j}}{2 \\Delta x} + \\frac{1}{2} (\\bar{u}^{+} \\bar{\\omega}_x^{-} + u^{-} \\bar{\\omega}_x^{+}) \\\\\n& + \\bar{v}_{i,j} \\frac{\\bar{\\omega}_{i,j+1} - \\bar{\\omega}_{i,j-1}}{2 \\Delta y} + \\frac{1}{2} (\\bar{v}^{+} \\bar{\\omega}_y^{-} + \\bar{v}^{-} \\bar{\\omega}_y^{+}),\n\\end{split}\n\\end{align}\nwhere \n\\begin{alignat}{2}\n\\bar{u}^{-} &= \\min(\\bar{u}_{i,j},0), \\quad \\bar{u}^{+} &= \\max(\\bar{u}_{i,j},0), \\\\\n\\bar{v}^{-} &= \\min(\\bar{v}_{i,j},0), \\quad \\bar{v}^{+} &= \\max(\\bar{v}_{i,j},0).\n\\end{alignat}\nIn addition,\n\\begin{align}\n\\begin{split}\n\\bar{\\omega}_x^{-} &= \\frac{\\bar{\\omega}_{i-2,j} - 3 \\bar{\\omega}_{i-1,j} + 3 \\bar{\\omega}_{i,j} - \\bar{\\omega}_{i+1,j} }{3 \\Delta x}, \\\\\n\\bar{\\omega}_x^{+} &= \\frac{\\bar{\\omega}_{i-1,j} - 3 \\bar{\\omega}_{i,j} + 3 \\bar{\\omega}_{i+1,j} - \\bar{\\omega}_{i+2,j} }{3 \\Delta x}, \\\\\n\\bar{\\omega}_y^{-} &= \\frac{\\bar{\\omega}_{i,j-2} - 3 \\bar{\\omega}_{i,j-1} + 3 \\bar{\\omega}_{i,j} - \\bar{\\omega}_{i,j+1} }{3 \\Delta y}, \\\\\n\\bar{\\omega}_y^{+} &= \\frac{\\bar{\\omega}_{i,j-1} - 3 \\bar{\\omega}_{i,j} + 3 \\bar{\\omega}_{i,j+1} - \\bar{\\omega}_{i,j+2} }{3 \\Delta y}.\n\\end{split}\n\\end{align}\nNote that velocity components are recovered using\n\\begin{align}\n\\begin{split}\n\\bar{u}_{i,j} &= \\frac{\\bar{\\psi}_{i,j+1}-\\bar{\\psi}_{i,j-1}}{2 \\Delta y} \\\\\n\\bar{v}_{i,j} &= -\\frac{\\bar{\\psi}_{i+1,j}-\\bar{\\psi}_{i-1,j}}{2 \\Delta x},\n\\end{split}\n\\end{align}\nwhere the second-order accurate reconstruction of the velocity leads to overall second-order accuracy for non-linear Jacobian reconstruction using the upwinded procedure outlined above. We also note that our Poisson equation given by Equation \\ref{eq2} is solved using a spectrally-accurate scheme. \n\nWith the choice of one of the two aforementioned schemes, a point in space-time may or may not have an artificial dissipation imparted to it numerically. However, we mention the caveat that switching between these two schemes would mean that the kinetic energy and enstrophy preserving property of the Arakawa scheme is lost. \n\n\\section{Machine learning for scheme selection}\n\nWe now discuss the procedure of utilizing DNS data for learning to classify one of the two dissipation scenarios. Of these two options, one is given by the choice of the Arakawa scheme and the other by our upwinded computation of the Jacobian (i.e., when the classification framework has determined that the point does not require sub-grid dissipation or vice-versa respectively). This switching between scenarios is spatio-temporally dynamic. We proceed by outlining our training strategy through the utilization of DNS data. Five equidistant snapshots of DNS data at $Re=32000$ (i.e., at $t=0,1,2,3,4$) and at $N^2 = 2048^2$ degrees of freedom (from 40000 available snapshots) are utilized to compute the grid-filtered variables (denoted FDNS) (at $N^2 = 256^2$ degrees of freedom) through the application of a spectral cut-off filter. Perfect closure values \n\\begin{align}\n\\Pi = J(\\bar{\\omega},\\bar{\\psi})-\\overline{J(\\omega,\\psi)}\n\\end{align}\nare then obtained (the reader is directed to \\citep{maulik2019subgrid} for details related to the calculation of these quantities). Note here, that the Kraichnan turbulence problem is transient with the evolution of vorticity represented in Figure \\ref{Fig1} representing different closure needs over time evolution. \n\nWe proceed by introducing the \\emph{a priori} eddy-viscosity given by\n\\begin{align}\n\\nu_e^a = \\frac{\\Pi}{\\nabla^2 \\bar{\\omega}}\n\\end{align}\nwhere the right-hand side of the above equation may be calculated from DNS snapshots. The \\emph{a priori} eddy-viscosity is centered at zero (corresponding to where closure modeling is unnecessary) and spreads out in the negative and positive directions (a hallmark of isotropic turbulence). We segregate this \\emph{a priori} estimate of sub-grid effects into three categories as follows. The \\emph{a priori} eddy-viscosities calculated from the DNS data are compared with a Gaussian distribution where values lying less than a distance of 1\\% of the standard-deviation from the mean (which is zero) are labeled as those requiring no dissipation (due to the low strength of the \\emph{a priori} eddy-viscosity). For posterity, we label these points as $k=1$. Positive values lying beyond this range are labeled as those requiring sub-grid dissipation and are labeled $k=2$. Negative values less than 1\\% of the standard-deviation are also considered to require no dissipation and are labeled $k=3$. This three-category segregation stems from a learning hypothesis that seeks to identify regions in a flow evolution that require structural, functional or no-closure modeling hypothesis. We link labels of negative or nearly-zero eddy-viscosities to the Arakawa classification and positive eddy-viscosities to the upwinded classification. The positive eddy-viscosity prediction would indicate that the sub-grid term at a point is predominantly dissipative in nature at which point the numerical dissipation of the upwinded scheme would be utilized. We note here that the concept of an \\emph{a priori} eddy-viscosity lies firmly within the explicit LES hypothesis. The classifier is therefore instrumental in moderating ILES deployments through a decision making process that recognizes the dissipative (or forcing) nature of the sub-grid quantities. \n\nWe note that the choice of 1\\% as the decision parameter for switching between hypothesis is motivated by a sensitivity study that showed the highest classification accuracy for the ANN framework. Larger choices of this hyper-parameter would result in a classifier that would be prone to classify most points in the `no-model' zone. However, we clarify that the choice of this value is also correlated with the architecture of the ANN. A potential extension of the proposed hypothesis is to combine architecture search algorithms with varying value of the decision hyper-parameters for larger classification accuracies. In addition, the three-category framework is derived from an aligned study \\cite{maulik2018online} where sub-grid models are determined according to negative, positive and nearly-zero \\emph{a priori} eddy-viscosities and utilizes the same learning. This enables use to determine a unified framework for switching between turbulence model hypotheses as well as numerical dissipation scenarios. However, we would like to emphasize that, for the purpose of switching between the Arakawa and upwinded Jacobian computation, a simple two-class framework would also suffice. \n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=\\textwidth]{Figure_1-eps-converted-to.pdf}\n\\caption{Time evolution of the Kraichnan turbulence case with DNS ($N^2 = 2048^2$) contours for vorticity of $t=1$ (top-left), $t=2$ (top-right), $t=3$ (bottom-left), $t=4$ (bottom-right). One can discern the dissipation of vorticity as the system evolves.}\n\\label{Fig1}\n\\end{figure*}\n\nA one-hot labeling of our eddy-viscosity classes is utilized for a classification deployment and a schematic for this hypothesis segregation and labeling is shown in Figure \\ref{Segregation}. The labels indicate the conditional probability of a point belonging to each possible class. As such, the training labels are given by a value of 1 for the particular class that a point belongs to and zeros for other choices. This is because there is no ambiguity in the class a training sample belongs to. Each label for the \\emph{a priori} eddy-viscosity is also associated with a corresponding input kernel of grid-resolved quantities. This kernel is given by a local stencil of vorticity and stream function. There are 9 inputs each for vorticity and stream function given by a query of the field quantity at a point on the coarse grid, 4 adjacent points in each dimension ($x,y$) and the 4 diagonally adjacent points. Each sample of our training data thus consists of 18 inputs of vorticity and stream function and outputs given by one-hot labels for the choice of closure modeling strategy. We then utilize an ANN to establish a relationship between these inputs and outputs. Mathematically, if our input vector $\\mathcal{P}$ resides in a $P$-dimensional space and our desired output $\\mathcal{Q}$ resides in a $Q$-dimensional space, this framework establishes a map $\\mathbb{M}$ as follows:\n\\begin{align}\n\\label{eq6}\n\\mathbb{M} : \\{ \\mathcal{P}_1, \\mathcal{P}_2, \\hdots, \\mathcal{P}_P\\} \\in \\mathbb{R}^P \\rightarrow \\{ \\mathcal{Q}_1, \\mathcal{Q}_2, \\hdots, \\mathcal{Q}_Q\\} \\in \\mathbb{R}^Q.\n\\end{align}\nAccordingly, the framework utilized in this article leads to the following relation:\n\\begin{align}\n\\label{eq7}\n\\mathbb{M} : \\{ \\textbf{p} \\} \\in \\mathbb{R}^{18}  \\rightarrow \\{ P(\\textbf{q}|\\textbf{p})\\} \\in \\mathbb{R}^3,\n\\end{align}\nwhere \n\\begin{align}\n\\begin{gathered}\n\\textbf{p}_{i,j} = \\{ \\bar{\\omega}_{i,j}, \\bar{\\omega}_{i,j+1}, \\bar{\\omega}_{i,j-1}, \\hdots, \\bar{\\omega}_{i-1,j-1}, \\\\ \\bar{\\psi}_{i,j}, \\bar{\\psi}_{i,j+1}, \\bar{\\psi}_{i,j-1}, \\hdots, \\bar{\\psi}_{i-1,j-1} \\}\n\\end{gathered}\n\\end{align}\nis our input vector for each query of the machine learning framework and where \n\\begin{align}\nP(\\textbf{q}|\\textbf{p})_{i,j} = \\{ P(J^k(\\bar{\\omega},\\bar{\\psi})_{i,j}| \\textbf{p}_{i,j})\\},\n\\end{align}\nis the conditional probability of a Jacobian computation (given by a connection to the explicit closure hypothesis). Note that $i,j$ refer to the spatial indices on the coarse-grid (i.e., the point of deployment). The indices $k=1$ and $k=3$ refer to the Arakawa non-linear Jacobian computation and $k=2$ refers to the upwinded computation instead (see Figure \\ref{Segregation}). Our optimal map $\\mathbb{M}$ is then trained by minimizing the categorical cross-entropy loss-function\n\\begin{align}\nE(\\textbf{w}) = -\\sum_{n=1}^{N} \\sum_{k=1}^{K} \\{ t_{nk} \\log(y_{nk}) + (1-t_{nk})\\log(1-y_{nk})\\},\n\\end{align}\nwhere $\\textbf{w}$ are the variable weight and bias parameters of the network, $N$ refers to the total number of samples and $K=3$ is the total number of classification scenarios (i.e., negative, positive or nearly-zero \\emph{a priori} eddy-viscosities). Here, $t_{nk}$ refers to the true label of class $k$ and sample $n$ and $y_{nk}$ refers to a corresponding prediction of the learning framework. One-hot encoding ensures that $t_{nk}$ values are always binary \\cite{Bishop:2006:PRM:1162264} and the outputs of the ANN may be interpreted as conditional-probabilities. Our optimal architecture is given by five 40-neuron hidden layers (obtained via grid-search hyper-parameter tuning). All hidden layers utilize ReLU units to impart non-linearity to the layer-wise transformations. For reference, our architecture is trained using the open-source deep learning software Tensorflow and is optimized with the use of ADAM, a popular gradient-descent based optimizer \\cite{kingma2014adam}. Figure \\ref{Loss_history} shows the progress to convergence for our framework with our optimally trained network displaying approximately 79\\% accuracy in classifying points to their correct labels. To summarize this section, we train a deep ANN to estimate probabilities of negative, positive or nearly-zero eddy-viscosities which are utilized to decide the choice of the Jacobian computation. We clarify that the decision to deploy a particular hypothesis is obtained by utilization of the classification scenario which has the highest conditional probability.\n\n\\begin{figure}\n\\centering\n\\includegraphics[trim={3cm 16cm 6cm 1cm},clip,width=\\columnwidth]{Schematic-eps-converted-to.pdf}\n\\caption{Hypothesis segregation and one-hot labeling for our proposed framework. The learning predicts conditional probabilities for the three segregated \\emph{a priori} eddy-viscosity classes which are utilized for Jacobian calculation decisions spatio-temporally.}\n\\label{Segregation}\n\\end{figure}\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=\\columnwidth]{Loss_history-eps-converted-to.pdf}\n\\caption{Learning rate and convergence of our classification framework training. 2000 epochs were sufficient for converged validation loss.}\n\\label{Loss_history}\n\\end{figure}\n\n\\section{Results}\n\n\\subsection{\\emph{A posteriori} deployment}\n\nIn this section, we detail the results from an \\emph{a posteriori} deployment of the classification framework (denoted ML henceforth) for the Kraichnan test-case. In the LES evolution of the problem, a considerably coarser grid is used (at $N^2=256^2$). We remark that the forward deployment of our framework needs to overcome the challenge of numerical errors and is a robust test of the generalizability and robustness of our learning. Our LES results are assessed using angle-averaged kinetic energy spectra and through structure functions of vorticity. In addition, qualitative comparisons are also provided through visual examinations of the vorticity contours. We remark that the LES deployment is performed from $t=0$ to $t=4$ which spans the training regime data obtained from DNS. In what follows we note that DNS refers to a high-fidelity evolution of the governing equations (i.e., at $N^2=2048^2$ degrees of freedom), UNS refers to results obtained using the Arakawa scheme alone and ILES refers to a simulation where the non-linear Jacobian at all points in space and time are upwinded. Figure \\ref{Fig2} shows the \\emph{a posteriori} performance of the proposed framework at $Re=32000$ in terms of energy spectra predictions. The reader may find an exact definition of the kinetic-energy spectra in Maulik and San \\cite{maulik2017stable}. We note that the training data was obtained for the same Reynolds number as well. The prediction of the proposed framework is seen to agree remarkably well with DNS. It is apparent that the switching of schemes using the classifier has obtained an optimal balance between both techniques.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=\\columnwidth]{Figure_2-eps-converted-to.pdf}\n\\caption{The \\emph{a posteriori} performance of proposed framework (ML) for $Re=32000$ and at $t=4$ in terms of angle-averaged kinetic energy spectra. Comparisons with DNS, the Arakawa scheme (UNS) and the upwinded scheme (ILES) show that ML provides directed dissipation adequately.}\n\\label{Fig2}\n\\end{figure}\n\nVorticity contours for LES resolution assessments are shown in Figure \\ref{Fig3}, where it is apparent that the proposed framework optimally balances the energy-conserving and dissipative natures of the Arakawa and upwinded schemes respectively. This is verified by qualitative examination with FDNS contours obtained by spectrally filtering the DNS snapshot for $Re=32000$ at $t=4$.\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=\\textwidth]{Figure_3-eps-converted-to.pdf}\n\\caption{Contours for the vorticity at LES resolution and at $t=4$. In the top-left, we have predictions from the ML approach. The top-right field has been obtained using ILES, the bottom-left field is obtained from UNS and the bottom right shows FDNS contours obtained by spectral cut-off filtering of DNS. }\n\\label{Fig3}\n\\end{figure*}\n\nA second statistically significant quantity of interest studied in this investigation is the vorticity structure function \\cite{grossmann1992structure} given by\n\\begin{align}\nS_\\omega^x = \\langle |\\omega(x+r,y) - \\omega(x,y)|^2 \\rangle \\\\\nS_\\omega^y = \\langle |\\omega(x,y+r) - \\omega(x,y)|^2 \\rangle,\n\\end{align}\nwhere the angle-brackets indicate ensemble averaging and $x,y$ indicate a position on the grid with $r$ being a certain distance from this location. Figures \\ref{Fig4} and \\ref{Fig5} show the structure functions obtained from \\emph{a posteriori} deployments of the UNS, ILES and ML frameworks compared against those obtained from the final time FDNS snapshot. It is clear that the proposed framework balances between UNS and ILES deployments well to recover appropriate trends. We can thus claim that our learning is appropriate for hybrid deployments of dissipative and conservative frameworks for two-dimensional turbulence. Before moving on, we would like to point out to the reader here that the proposed methodology for closure does not require any post-processing prior to deployment in the forward simulation as utilized in several data-driven turbulence modeling studies\\cite{beck2018neural,maulik2019subgrid}.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=\\columnwidth]{Figure_4-eps-converted-to.pdf}\n\\caption{\\emph{A posteriori} vorticity structure functions in $x$ direction of our proposed framework (ML), the Arakawa scheme (UNS) and the upwind scheme (ILES) with statistics obtained from an FDNS snapshot at $t=4$. It is apparent that the ML method stabilizes the UNS result optimally.}\n\\label{Fig4}\n\\end{figure}\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=\\columnwidth]{Figure_5-eps-converted-to.pdf}\n\\caption{\\emph{A posteriori} vorticity structure functions in $y$ direction of our proposed framework (ML), the Arakawa scheme (UNS) and the upwind scheme (ILES) with statistics obtained from an FDNS snapshot at $t=4$. It is apparent that the ML method stabilizes the UNS result optimally.}\n\\label{Fig5}\n\\end{figure}\n\n\n\\subsection{Validation of learning}\n\nIn this section, we proceed with a rigorous validation of our learning for deployment in regimes that are not a part of the training data. This is to ensure that the framework has truly learnt a classification based on the underlying physical hypothesis used for data segregation and is not memorizing data. This ensures that our classifier can be used in a more generalizable fashion. Figure \\ref{Fig6} shows kinetic energy spectra obtained from the forward deployment of the ML framework for a $Re=64000$ which represents a classification task that the framework has not previously seen (although the physics of the test-case remains similar). As observed, the proposed method performs quite well in this out-of-training data range as well. We note that a similar resolution ($N^2=256^2$) is utilized for this deployment. In contrast, Figure \\ref{Fig7} shows the performance of the ML technique for a reduced resolution of $N^2=128^2$ but utilizing the same Reynolds number of 32000. The kinetic energy spectra show a successful stabilization of the flow evolution at this reduced resolution although some forcing to the large scales is observed. This suggests that the classification framework may be improved by sampling from different resolutions. \n\n\\begin{figure}\n\\centering\n\\includegraphics[width=\\columnwidth]{Figure_6-eps-converted-to.pdf}\n\\caption{The \\emph{a posteriori} performance of proposed framework (ML) for $Re=64000$ and at $t=4$ in terms of energy spectra. This represents deployment of our learning at a different Reynolds number than that used for generating training data.}\n\\label{Fig6}\n\\end{figure}\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=\\columnwidth]{Figure_7-eps-converted-to.pdf}\n\\caption{The \\emph{a posteriori} performance of proposed framework (ML) for $Re=32000$, $t=4$ and at $N^2 = 128^2$ in terms of energy spectra. This represents deployment of our learning at a different resolution than that used for generating training data.}\n\\label{Fig7}\n\\end{figure}\n\n\n\\section{Concluding remarks}\n\nIn this article, we have proposed a neural network based classifier that enables us to take decisions on the choice of non-linear term computation in the LES evolution of the Kraichnan turbulence test-case. The classifier outputs conditional probabilities for the presence (or absence) of eddy-viscosities within three different ranges during deployment and is used to switch between the Arakawa and upwind computation of the non-linear Jacobian for a hybrid upwinded deployment that optimally directs dissipation on the coarse-grained flow field. Our machine learning framework is trained by calculating \\emph{a priori} eddy-viscosities which are projected onto a Gaussian distribution and segregated into three categories. Each category is devised to capture a unique behavior of the underlying sub-grid terms with negative and nearly-zero eddy-viscosity classes signifying absence of sub-grid dissipation. An optimally trained classifier is then utilized to identify if a point requires sub-grid dissipation based on if it is placed in the positive eddy-viscosity category. If so, the upwind Jacobian is calculated for imparting numerical dissipation. \n\nWe perform \\emph{a posteriori} assessments on the Kraichnan turbulence test-case through statistical quantities such as the angle-averaged kinetic energy spectra and the vorticity structure functions. It is observed that the proposed framework is successful in balancing the dissipative nature of the upwind scheme and the energy-conserving Arakawa scheme to give excellent agreement with DNS statistics. Validation for out-of-training regimes also indicate that the framework is able to learn the link between grid-resolved quantities at a coarse resolution and the nature of the sub-grid forcing. \n\nOur conclusions therefore point toward the possibility of using classifiers for the unified deployment of numerical schemes with varying dissipation through the decision making process described above. A key strength of our hypothesis stems from the fact that an ILES deployment is moderated by concepts drawn from the explicit LES ideology (i.e., that of an \\emph{a priori} eddy-viscosity). The successful deployment of our method thus points towards the possibility of deploying directed numerical dissipation that preserves the statistics of turbulence without sacrificing the shock-capturing ability of many non-oscillatory schemes. Our future work lies in that particular direction. \n\n\n\\begin{acknowledgements}\nThis material is based upon work supported by the U.S. Department of Energy, Office of Science, Office of Advanced Scientific Computing Research under Award Number DE-SC0019290. OS gratefully acknowledges their support. Disclaimer: This report was prepared as an account of work sponsored by an agency of the United States Government. Neither the United States Government nor any agency thereof, nor any of their employees, makes any warranty, express or implied, or assumes any legal liability or responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Reference herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, or otherwise does not necessarily constitute or imply its endorsement, recommendation, or favoring by the United States Government or any agency thereof. The views and opinions of authors expressed herein do not necessarily state or reflect those of the United States Government or any agency thereof.\n\\end{acknowledgements}\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}