[
    {
        "text": "abstract:    It is proved, using the curved line element of a\n  spherically symmetric charged object in general relativity and the\n  Schwinger discharge mechanism of quantum field theory, that the\n  orbital periods $T_{\\infty}$ of test particles around central compact\n  objects as measured by flat-space asymptotic observers are\n  fundamentally bounded from below. The lower bound on orbital periods\n  becomes universal (independent of the mass $M$ of the central compact\n  object) in the dimensionless $ME_{\\text{c}}\\gg1$ regime, in which case\n  it can be expressed in terms of the electric charge $e$ and the proper\n  mass $m_{e}$ of the lightest charged particle in nature:\n  $T_{\\infty}>{{2\\pi e\\hbar}\\over{\\sqrt{G}c^2 m^2_{e}}}$ (here\n  $E_{\\text{c}}=m^2_{e}/e\\hbar$ is the critical electric field for pair\n  production). The explicit dependence of the bound on the fundamental\n  constants of nature $\\{G,c,\\hbar\\}$ suggests that it may reflect a\n  fundamental physical property of the elusive quantum theory of\n  gravity.\naddress: The Ruppin Academic Center, Emeq Hefer 40250, Israel; ; The\nHadassah Institute, Jerusalem 91010, Israel\nauthor: Shahar Hod\ndate: 2024-10-13\ntitle: Universal lower bound on orbital periods around central compact\n  objects\n\n# Introduction\n\nThe theory of quantum gravity is notorious for its elusiveness. In\nparticular, despite the fact that the physical laws of general\nrelativity and quantum field theory are well established, it is still\nvery difficult to reveal fundamental physical principles that are\nexpected to remain valid within the framework of the yet unknown quantum\ntheory of gravity. One such principle is the holographic entropy-area\nbound, whose compact formula $S/A\\leq {{k_{\\text{B}}c^3}\\over{4G\\hbar}}$\ncontains the fundamental constants of gravity ($G$), relativity ($c$),\nand quantum theory ($\\hbar$) .\n\nThe main goal of the present compact paper is to reveal the existence of\nanother fundamental physical bound (which, admittedly, is probably far\nless important than the holographic entropy-area bound) whose formula\ncontains the three basic constants of nature. In particular, below we\nshall explicitly prove that, in curved spacetimes, the orbital periods\nof test particles around central compact objects are bounded from below\nby a fundamental limit which is expressed in terms of the basic\nconstants of nature: $G, c$, and $\\hbar$.\n\nClosed circular motions of test particles around central compact objects\nprovide valuable information on the non-trivial geometries of the\ncorresponding curved spacetimes (see and references therein). In\nparticular, an empirically measured quantity which is important for the\nanalysis of closed circular motions in curved spacetimes is the orbital\nperiod $T_{\\infty}$ around the central compact object as measured by far\naway asymptotic observers.\n\nUsing a naive flat-space argument, which ignores the intriguing\ntime-dilation/contraction effect of general relativity (below we shall\nanalyze in detail the influence of this physically important effect on\nasymptotically measured orbital periods), it is quite easy to prove that\nthe orbital period of a test particle around a (possibly charged)\ncentral compact object of radius $R$ should be bounded from below. In\nparticular, since the radius of a spherically symmetric object of mass\n$M$ and electric charge $Q$ that respects the weak (positive) energy\ncondition is expected to be bounded from below by its Schwarzschild\nradius ($R\\gtrsim M$) and also by its classical charge radius\n($R\\geq {{Q^2}/{2M}}$) , the orbital period $T\\geq 2\\pi R$ around the\ncentral compact object as measured by inertial observers is expected to\nbe bounded from below by the simple functional relation $$\\label{Eq1}\nT\\geq T^{\\text{min}}(M,Q)=2\\pi\\cdot{\\text{max}}\\{M,Q^2/2M\\}\\  .$$\n\nIntriguingly, however, it is well established that, due to quantum\ncoherence effects, local energy densities in *quantum* field theory can\nbe negative. Likewise, if the matter fields inside a compact object are\ncharacterized by a *non*-minimal coupling to gravity then negative\nenergy densities are not excluded even at the classical level . The\nappearance of regions with negative energy densities inside a compact\nobject may allow its radius $R$ to be *smaller* than its classical\nradius, thus opening the possibility for the existence of closed\ncircular trajectories around the compact object that violate the purely\nclassical lower bound ().\n\nBased on this expectation, we here raise the following physically\ninteresting question: Is there a *fundamental* quantum-gravity lower\nbound on orbital periods of test particles around central compact\nobjects?\n\nThe main goal of the present compact paper is to reveal the existence of\nsuch a bound on the orbital periods, as measured by asymptotic\nobservers, around spherically symmetric central compact objects, a bound\nwhich is valid even for compact objects that may violate the classical\nlower bound (). In particular, below we shall explicitly prove that the\nSchwinger pair-production mechanism , a purely quantum effect, sets a\nlower bound on the orbital periods of test particles around central\ncompact objects in the composed Einstein-Maxwell field theory.\n\n# Lower bound on orbital periods around central compact objects\n\n## Circular trajectories around compact objects that violate the weak energy condition\n\nIn the present section we shall determine the shortest possible orbital\nperiod, $T^{\\text{min}}_{\\infty}(M)$, around a central compact object of\ntotal gravitational mass $M$ in the composed Einstein-Maxwell field\ntheory.\n\nThe external spacetime of a spherically symmetric charged compact object\nof radius $R$, total mass $M$ , and electric charge $Q$ is characterized\nby the Reissner-Nordström curved line element $$\\begin{aligned}\n\\label{Eq2}\nds^2&=&-\\Big[1-{{2M(r)}\\over{r}}\\Big]dt^2+\\Big[1-{{2M(r)}\\over{r}}\\Big]^{-1}dr^2+\\nonumber \\\\ &&\nr^2d\\theta^2+r^2\\sin^2\\theta d\\phi^2\\ \\ \\ \\ \\ \\text{for}\\ \\ \\ \\ \\ r\\geq R\\  ,\n\\end{aligned}$$ where $$\\label{Eq3}\nM(r)=M-{{Q^2}\\over{2r}}\\$$ is the gravitational mass contained within a\nsphere of radius $r$.\n\nOur goal is to determine the *shortest* possible orbital period\n$T^{\\text{min}}_{\\infty}(M)$ as measured by asymptotic observers around\nthe central compact object. We shall therefore consider test particles\nthat move arbitrarily close to the speed of light , in which case the\nasymptotically measured orbital periods can be determined from the\ncurved line element () with the properties : $$\\label{Eq4}\nds=dr=d\\theta=0\\ \\ \\ \\ \\  \\text{and}\\ \\ \\ \\ \\ \\Delta\\phi=\\pm2\\pi\\  .$$\nSubstituting the relations () into Eq. (), one obtains the functional\nexpression $$\\label{Eq5}\nT_{\\infty}(M,Q,r)={{2\\pi r}\\over{\\sqrt{1-{{2M}\\over{r}}+{{Q^2}\\over{r^2}}}}}\\$$\nfor the orbital period around a central (possibly charged) compact\nobject as measured by asymptotic observers.\n\nAs discussed above, classical charged compact objects that respect the\nweak (positive) energy condition must be larger than their classical\ncharge radius \\[see Eq. ()\\] , $$\\label{Eq6}\nR\\geq R_{\\text{c}}={{Q^2}\\over{2M}}\\  ,$$ in which case one finds the\ndimensionless inequality $g_{tt}=1-{{2M}/{r}}+{{Q^2}/{r^2}}\\leq1$ for\nexternal circular trajectories (with radii $r\\geq R$) around the central\ncompact object. Substituting this relation into Eq. (), one obtains the\nsimple *classical* lower bound $$\\label{Eq7}\nT_{\\infty}(M,Q,r)\\geq 2\\pi r\\geq 2\\pi R\\geq2\\pi\\cdot{\\text{max}}\\{Q^2/2M,M\\}\\$$\non orbital periods around central compact objects.\n\nHowever, as emphasized above, negative energy densities are not always\nexcluded in physics. In particular, they may appear due to a non-minimal\ndirect coupling of matter fields to gravity and also due to quantum\ncoherence effects in quantum field theories . The possible existence of\nspacetime regions with negative energy densities inside a compact object\nthat violates the (classical) weak energy condition may allow its radius\n$R$ to be smaller than its classical charge radius $R_{\\text{c}}$, thus\nopening the possibility for the existence of circular trajectories,\nwhose radii lie in the regime $$\\label{Eq8}\nr\\in[R,R_{\\text{c}})\\  ,$$ that violate the classical lower bound () for\n*two* reasons: (1) The numerator of () is smaller than\n$2\\pi R_{\\text{c}}$ for circular trajectories in the regime\n$R\\leq r<R_{\\text{c}}$. (2) The local mass $M(R)=M-Q^2/2R$ contained\nwithin a charged compact object which is smaller than its classical\nradius is negative. This fact implies that the denominator of () is\nlarger than $1$ for circular trajectories in the regime\n$R\\leq r<R_{\\text{c}}$. Thus, the orbital periods that characterize\ncircular trajectories in the regime () are *blueshifted* by the factor\n$\\sqrt{1-2M/r+Q^2/r^2}>1$ as measured by asymptotic observers. This is a\ngeneral relativistic time contraction effect.\n\nBefore proceeding, it is important to emphasize that, despite the fact\nthat the local mass () contained within a compact object which is\nsmaller than its classical radius is negative, we shall assume that the\n*total* ADM mass $M$ of the spacetime as measured by asymptotic\nobservers is positive.\n\n## A fundamental lower bound on orbital periods around central compact objects\n\nIt is of physical interest to explore the physical and mathematical\nproperties of closed circular motions around central compact objects\nthat, due to quantum coherence effects or non-minimal coupling to\ngravity , may violate the classical lower bound (). In particular, one\nnaturally wonders whether orbital periods around central compact objects\nthat violate the weak energy condition are fundamentally bounded from\nbelow by the physical laws of general relativity and quantum theory?\n\nIn the present section we shall reveal the physically intriguing fact\nthat the Schwinger pair-production mechanism sets a fundamental quantum\nlower bound on the orbital period () around central compact objects, a\nbound which is valid even for compact objects that violate the weak\nenergy condition and can therefore violate the classical lower bound ().\n\nIn particular, we shall now prove that, as opposed to the unbounded\nredshift (time dilation) effect which characterizes the orbital periods\nof test particles that circle a central black hole close to its horizon\n\\[$T_{\\infty}(r\\to R_{\\text{horizon}})\\to\\infty$\\], the blueshift time\ncontraction effect, which characterizes the orbital periods of test\nparticles that circle a central compact object with negative energy\ndensities , is fundamentally (quantum mechanically) bounded from above.\n\nOur goal is to determine the shortest possible orbital period\n$T^{\\text{min}}_{\\infty}(M)$ around a central compact object of a given\nmass $M$. To this end, we first point out that, for given values of the\ngravitational mass $M>0$ of the central compact object and the radius\n$r$ of the external circular trajectory, the orbital period\n$T_{\\infty}(Q;M,r)$ as measured by asymptotic observers decreases\nmonotonically with the electric charge $Q$ of the central compact object\n\\[see Eq. ()\\]. Thus, in order to minimize the orbital period () of a\ntest particle around a central object with a given total mass $M$, one\nshould maximize its electric charge. In particular, $$\\label{Eq9}\nT^{\\text{min}}_{\\infty}(M,Q_{\\text{max}},r)=\n{{2\\pi r}\\over{\\sqrt{1-{{2M}\\over{r}}+{{Q^2_{\\text{max}}}\\over{r^2}}}}}\\  ,$$\nwhere $Q_{\\text{max}}=Q_{\\text{max}}(r)$ is the maximally allowed\nelectric charge that can be contained within a sphere of radius $r$.\n\nOne naturally wonders: What physics prevents us from making the\nexpression () for the orbital period as small as we wish? Or, in other\nwords, we ask: Is there a fundamental physical mechanism that bounds the\nelectric charge $Q_{\\text{max}}(r)$ that can be contained within a\nsphere of radius $r$? The answer is ‘yes’!\n\nIn particular, the Schwinger pair-production mechanism (a *quantum*\npolarization effect) implies the existence of a fundamental upper bound\non the electric field strength of the charged compact object :\n$$\\label{Eq10}\n{{Q}\\over{r^2}}\\leq E_{\\text{c}}\\equiv {{m^2_e}\\over{e\\hbar}}\\  ,$$\nwhere $\\{e,m_e\\}$ are respectively the electric charge and the proper\nmass of the lightest charged particle in nature. Substituting the upper\nbound () into Eq. (), one obtains the radius-dependent functional\nexpression $$\\label{Eq11}\nT^{\\text{min}}_{\\infty}(r;M,E_{\\text{c}})={{2\\pi r}\\over{\\sqrt{1-{{2M}\\over{r}}+E^2_{\\text{c}}r^2}}}\\$$\nfor the shortest possible orbital period.\n\nInterestingly, and most importantly for our analysis, one finds from Eq.\n() that, for a given total mass $M$ of the system, the orbital period is\n*minimized* for the field-independent ($E_{\\text{c}}$-*independent*)\norbital radius $$\\label{Eq12}\nr^{\\text{min}}=3M\\  .$$\n\nSince the electric charge is confined to the interior of the central\ncompact object, the electric field strength in the exterior ($r\\geq R$)\nspacetime region is a monotonically decreasing function of the orbital\nradius $r$. Thus, the assumption in Eq. () that the electric field along\nthe trajectory of the test particle saturates the quantum upper bound ()\n\\[as discussed above, for a given value $r$ of the orbital radius, the\nlarger the electric field along the trajectory of the particle, the\nshorter is the orbital period as measured by asymptotic observers, see\nEq. ()\\] corresponds to the assumption that the particle moves along a\ncircular trajectory infinitesimally close to the surface of the charged\ncompact object: $$\\label{Eq13}\nr^{\\text{min}}\\to R^+\\  .$$\n\nSubstituting () into Eq. (), one obtains the remarkably simple\nfunctional relation $$\\label{Eq14}\nT^{\\text{min}}_{\\infty}(M,E_{\\text{c}})=2\\pi\\cdot\\sqrt{{27M^2}\\over{1+27M^2E^2_{\\text{c}}}}\\$$\nfor the shortest possible orbital period around a central compact object\nof total gravitational mass $M$ as measured by asymptotic observers.\n\nIt is interesting to note that, in the dimensionless regime\n$ME_{\\text{c}}\\ll1$, the analytically derived lower bound () on orbital\nperiods around central compact objects yields the *classical*\n($\\hbar$-independent) bound $$\\label{Eq15}\nT^{\\text{min}}_{\\infty}(M,E_{\\text{c}})\\ \\to\\ 6\\sqrt{3}\\pi\\cdot M\\ \\ \\ \\ \\ \\text{for}\\ \\ \\ \\ \\ ME_{\\text{c}}\\ll1\\  .$$\nOn the other hand, in the opposite dimensionless regime\n$ME_{\\text{c}}\\gg1$ the lower bound () yields the purely *quantum*\n($\\hbar$-dependent) bound $$\\label{Eq16}\nT^{\\text{min}}_{\\infty}(M,E_{\\text{c}})\\ \\to\\ {{2\\pi}\\cdot{E^{-1}_{\\text{c}}}}\n\\ \\ \\ \\ \\ \\text{for}\\ \\ \\ \\ \\ ME_{\\text{c}}\\gg1\\  .$$ Intriguingly, the\nlower bound () is universal in the sense that it is *independent* of the\nmass $M$ of the central compact object.\n\nIt is worth stressing the fact that, in the dimensionless\n$ME_{\\text{c}}\\gg1$ regime, the value of $T^{\\text{min}}$, as given by\nthe analytically derived quantum expression (), satisfies the strong\ninequality\n$T^{\\text{min}}(M,E_{\\text{c}})={{2\\pi}/{E_{\\text{c}}}}\\ll 2\\pi\\cdot{\\text{max}}\\{Q^2/2M,M\\}$\n, and it therefore violates the classical bound () on orbital periods.\n\n# Summary and Discussion\n\nMotivated by the fact that negative energy densities may appear in\nvarious physical situations (for example, due to quantum coherence\neffects that appear in quantum field theories and also due to a possible\nnon-minimal coupling of matter fields to gravity ), in the present\ncompact paper we have raised the following physically important\nquestion: Is there a *fundamental* lower bound on the orbital periods of\ntest particles around central compact objects?\n\nIn order to address this question, we have analyzed the\nthree-dimensional functional behavior of the orbital periods\n$T_{\\infty}=T_{\\infty}(M,Q,r)$ of test particles whose velocities are\narbitrarily close to the speed of light on the physical parameters\n$\\{M,Q\\}$ that characterize the central compact object and on the radii\n$r$ of the external circular trajectories.\n\nIn particular, the main goal of the present paper was to derive a robust\nlower bound on orbital periods of test particles around spherically\nsymmetric central compact objects, a bound which is valid even for\ncompact objects that violate the weak (positive) energy condition and\nthus may violate the classical lower bound () on the radii of charged\ncompact objects and the corresponding classical lower bound () on\norbital periods as measured by asymptotic observers.\n\nIntriguingly, we have revealed the fact that the Schwinger\npair-production mechanism , a purely quantum effect, is responsible for\nthe existence of a previously unknown fundamental lower bound on the\norbital periods of test particles around central compact objects.\n\nThe main analytical results derived in this paper and their physical\nimplications are as follows:\n\n\\(1\\) We have emphasized the fact that external circular trajectories\naround (possibly charged) central compact objects that respect the\nclassical weak (positive) energy condition are characterized by the\nrelation $1-{{2M}/{r}}+{{Q^2}/{r^2}}<1$ \\[see the lower bound ()\\], in\nwhich case the orbital periods as measured by far away asymptotic\nobservers are longer than the corresponding locally measured orbital\ntimes (this is the familiar time dilation effect in general relativity).\n\nOn the other hand, we have pointed out that charged compact objects that\nviolate the classical positive energy condition may be characterized by\nthe presence of external circular trajectories in the regime\n$r\\in[R,R_{\\text{c}})$ with the property $1-{{2M}/{r}}+{{Q^2}/{r^2}}>1$,\nin which case the general relativistic time contraction effect implies\nthat the asymptotically measured orbital periods are *shorter* than the\ncorresponding locally measured orbital periods.\n\nInterestingly, we have explicitly proved that, as opposed to the\nunbounded time dilation (redshift) effect,\n$T_{\\infty}(r\\to R_{\\text{horizon}})\\to\\infty$, which characterizes the\norbital periods of test particles in the near-horizon region of a\ncentral black hole, the time contraction (blueshift) effect, which\ncharacterizes the orbital periods of test particles around central\ncompact objects with negative energy densities, is fundamentally\n(quantum mechanically) bounded from above according to the analytically\nderived functional relation ().\n\n\\(2\\) Using the curved line element () that characterizes the exterior\nspacetime region of a spherically symmetric charged compact object in\ngeneral relativity and the Schwinger discharge mechanism of quantum\nfield theory, we have explicitly proved that the orbital periods\n$T_{\\infty}(M)$ of test particles around a central compact object of\ntotal mass $M$ as measured by asymptotic observers are fundamentally\nbounded from below by the functional relation \\[see Eqs. () and ()\\]\n$$\\label{Eq17}\nT_{\\infty}(M)\\geq T^{\\text{min}}_{\\infty}(M)=\n2\\pi\\cdot\\sqrt{{27M^2}\\over{1+27M^2\\cdot({{m^2_e}/{e\\hbar}})^2}}\\  .$$\nFor a central object of total mass $M$, the minimally allowed orbital\nperiod () is obtained for the following physical parameters of the\ncompact object and the circular trajectory: $Q=9M^2E_{\\text{c}}$ and\n$r\\to R=3M$ \\[see Eqs. (), (), and ()\\].\n\nIt is worth emphasizing the fact that, as opposed to the classical lower\nbound () on orbital periods, the analytically derived lower bound () is\nvalid even in the quantum regime of matter fields that may violate the\nweak energy condition. In particular, circular trajectories around\ncentral compact objects whose radii violate the classical lower bound ()\nare still characterized by the fundamental quantum lower bound () on\ntheir orbital periods.\n\n\\(3\\) The lower bound () becomes universal (*independent* of the mass\n$M$ of the central compact object) in the dimensionless\n$ME_{\\text{c}}\\gg1$ regime, in which case it can be expressed in the\nremarkably compact form (we write here the explicit dependence of\n$T^{\\text{min}}_{\\infty}$ on the fundamental constants of nature\n$\\{G,c,\\hbar\\}$): $$\\label{Eq18}\nT^{\\text{min}}_{\\infty}\\to {{2\\pi\\hbar\\sqrt{k}e}\\over{\\sqrt{G}c^2 m^2_{e}}}\n\\ \\ \\ \\ \\ \\text{for}\\ \\ \\ \\ \\ ME_{\\text{c}}\\gg1\\  ,$$ where\n$\\epsilon_0=1/4\\pi k$ is the electric constant (vacuum permittivity).\n\nThe explicit dependence of the minimally allowed orbital time\n$T^{\\text{min}}_{\\infty}$ on the fundamental constants of gravity ($G$),\nrelativity ($c$), and quantum physics ($\\hbar$) suggests that it may\nreflect a genuine physical property of a fundamental quantum theory of\ngravity.\n\n\\(4\\) Interestingly, inspection of Eq. () reveals the fact that, in\norder for the smallest possible orbital period $T^{\\text{min}}_{\\infty}$\nto be larger than the fundamental scale set by the Planck time\n$t_{\\text{P}}=(\\hbar G/c^5)^{1/2}$, one must demand the existence of a\n*weak-gravity* bound of the form $$\\label{Eq19}\n{{Gm^2_{e}}\\over{ke^2}}\\lesssim \\alpha^{-1/2}\\  ,$$ where $\\alpha$ is\nthe dimensionless fine-structure constant.\n\n \n\n**ACKNOWLEDGMENTS**\n\nThis research is supported by the Carmel Science Foundation. I would\nlike to thank Yael Oren, Arbel M. Ongo, Ayelet B. Lata, and Alona B. Tea\nfor helpful discussions.\n"
    },
    {
        "text": "abstract: We first present an abstract principle for the interchange of\n  infimization and integration over spaces of mappings taking values in\n  topological spaces. New conditions on the underlying space and the\n  integrand are then introduced to convert this principle into concrete\n  scenarios that are shown to capture those of various existing\n  interchange rules. These results are leveraged to improve\n  state-of-the-art interchange rules for evaluating Legendre conjugates,\n  subdifferentials, recessions, Moreau envelopes, and proximity\n  operators of integral functions by bringing the corresponding\n  operations under the integral sign.\nauthor: Minh N. Bùi; Patrick L. Combettes\ndate:  \ntitle:  Interchange Rules for Integral Functions[^1]\n\n. Compliant space, convex analysis, integral function, interchange\nrules, normal integrand.\n\n# Introduction\n\nThis paper concerns the interchange of the infimization and integration\noperations in the context of the following assumption.\n\nMany problems in analysis and its applications require the evaluation of\nthe infimum over $\\ensuremath{\\mathcal X}$ of the function\n$f\\colon x\\mapsto\\int_\\Omega\\varphi({\\mkern 2mu\\cdot\\mkern 2mu},x({\\mkern 2mu\\cdot\\mkern 2mu}))d\\mu$.\nA simpler task is to evaluate the function\n$\\phi\\colon\\omega\\mapsto\\inf\\varphi(\\omega,\\ensuremath{\\mathsf X})$ and\nthen compute $\\int_\\Omega\\phi d\\mu$. In general, this provides only a\nlower bound as\n$\\inf f(\\ensuremath{\\mathcal X})\\ensuremath{\\geqslant}\\int_\\Omega\\phi d\\mu$.\nConditions under which the two quantities are equal have been\nestablished in , , and under various hypotheses on\n$\\ensuremath{\\mathsf X}$, $(\\Omega,\\ensuremath{\\EuScript F},\\mu)$,\n$\\ensuremath{\\mathcal X}$, and $\\varphi$. The resulting\ninfimization-integration interchange rule is a central tool in areas\nsuch as multivariate analysis , calculus of variations , economics ,\nstochastic processes , stochastic optimization , finance , convex\nanalysis , variational analysis , and stochastic programming . Note\nthat, in Assumption –, we do not require that\n$(\\ensuremath{\\mathsf X},\\EuScript{T}_\\ensuremath{\\mathsf X})$ be a\ntopological vector space to accommodate certain applications. For\ninstance, in , $\\ensuremath{\\mathsf X}$ is the space of càdlàg functions\non $[0,1]$ and $\\EuScript{T}_\\ensuremath{\\mathsf X}$ is the Skorokhod\ntopology. In this context,\n$(\\ensuremath{\\mathsf X},\\EuScript{T}_\\ensuremath{\\mathsf X})$ is a\nPolish space which is not a topological vector space but which satisfies\nAssumption –.\n\nOur first contribution is Theorem  below, which provides, under the\numbrella of Assumption , a broad setting for the interchange of\ninfimization and integration.\n\n**Theorem 1** (interchange principle).  * Suppose that Assumption  and\nthe following hold:*\n\n1.  * $\\inf\\varphi({\\mkern 2mu\\cdot\\mkern 2mu},\\ensuremath{\\mathsf X})$\n    is $\\ensuremath{\\EuScript F}$-measurable.*\n\n2.  * There exists a sequence $(x_n)_{n\\in\\ensuremath{\\mathbb N}}$ in\n    $\\mathcal{L}(\\Omega;\\ensuremath{\\mathsf X})$ such that the following\n    are satisfied:*\n\n    1.  *\n        $\\inf\\varphi({\\mkern 2mu\\cdot\\mkern 2mu},\\ensuremath{\\mathsf X})=\n        \\inf_{n\\in\\ensuremath{\\mathbb N}}\\varphi({\\mkern 2mu\\cdot\\mkern 2mu},x_n({\\mkern 2mu\\cdot\\mkern 2mu})+\\overline{x}({\\mkern 2mu\\cdot\\mkern 2mu}))$\n        $\\ensuremath{\\text{\\rm$\\mu$-a.e.}}$*\n\n    2.  * There exists an increasing sequence\n        $(\\Omega_k)_{k\\in\\ensuremath{\\mathbb N}}$ of finite\n        $\\mu$-measure sets in $\\ensuremath{\\EuScript F}$ such that\n        $\\bigcup_{k\\in\\ensuremath{\\mathbb N}}\\Omega_k=\\Omega$ and\n        $$\\label{e:99}\n        \\bigcup_{n\\in\\ensuremath{\\mathbb N}}\\bigcup_{k\\in\\ensuremath{\\mathbb N}}\n        \\big\\{{1_Ax_n}~|~{\\ensuremath{\\EuScript F}\\ni A\\subset\\Omega_k\\,\\,\\text{and}\\,\\,\n        \\overline{x_n(A)}\\,\\,\\text{is compact}}\\big\\}\\subset\\ensuremath{\\mathcal X}.$$*\n\n*Then $$\\label{e:1}\n\\inf_{x\\in\\ensuremath{\\mathcal X}}\\int_\\Omega\\varphi\\big(\\omega,x(\\omega)\\big)\n\\mu(d\\omega)=\n\\int_\\Omega\\inf_{\\mathsf{x}\\in\\ensuremath{\\mathsf X}}\\varphi(\\omega,\\mathsf{x})\\,\n\\mu(d\\omega).$$*\n\nTheorem  is proved in Section . The second contribution is the\nintroduction of two new tools — compliant spaces and an extended notion\nof normal integrands. This is done in Section , where these notions are\nillustrated through various examples. In Section , compliance and\nnormality are utilized to build a pathway between the abstract\ninterchange principle of Theorem  and separate conditions on\n$\\ensuremath{\\mathcal X}$ and $\\varphi$ that capture various application\nsettings. The main result of that section is Theorem , which encompasses\nin particular the interchange rules of , as well as those implicitly\npresent in . These different frameworks have so far not been brought\ntogether and we improve them in several directions, for instance by not\nrequiring the completeness of $(\\Omega,\\ensuremath{\\EuScript F},\\mu)$\nand by relaxing the assumptions on $\\ensuremath{\\mathsf X}$. This leads\nto new concrete scenarios under which holds. Our third contribution,\npresented in Section , concerns convex-analytical operations on integral\nfunctions. By combining Theorem , compliance, and normality, we broaden\nconditions for evaluating Legendre conjugates, subdifferentials,\nrecessions, Moreau envelopes, and proximity operators of integral\nfunctions by bringing the corresponding operations under the integral\nsign. These results improve state-of-the-art convex calculus rules from\n.\n\n# Notation and background\n\n## Measure theory\n\nWe set $\\overline{\\mathbb{R}}=\\left[{-}\\infty,{+}\\infty\\right]$. Let\n$(\\Omega,\\ensuremath{\\EuScript F})$ be a measurable space and let $A$ be\na subset of $\\Omega$. The characteristic function of $A$ is denoted by\n$1_A$ and the complement of $A$ is denoted by $\\complement A$. Now let\n$(\\ensuremath{\\mathsf X},\\EuScript{T}_\\ensuremath{\\mathsf X})$ be a\nHausdorff topological space with Borel $\\sigma$-algebra\n$\\ensuremath{\\EuScript B}_\\ensuremath{\\mathsf X}$. We denote by\n$\\mathcal{L}(\\Omega;\\ensuremath{\\mathsf X})$ the vector space of\nmeasurable mappings from $(\\Omega,\\ensuremath{\\EuScript F})$ to\n$(\\ensuremath{\\mathsf X},\\ensuremath{\\EuScript B}_\\ensuremath{\\mathsf X})$.\nGiven a measure $\\mu$ on $(\\Omega,\\ensuremath{\\EuScript F})$,\n$\\mathcal{L}^1(\\Omega;\\ensuremath{\\mathbb R})$ is the subset of\n$\\mathcal{L}(\\Omega;\\ensuremath{\\mathbb R})$ of integrable functions,\nand $\\mathcal{L}^1(\\Omega;\\overline{\\mathbb{R}})$ is defined likewise.\nGiven a separable Banach space\n$(\\ensuremath{\\mathsf X},\\|{\\mkern 2mu\\cdot\\mkern 2mu}\\|_\\ensuremath{\\mathsf X})$,\nwe set $\\mathcal{L}^\\infty(\\Omega;\\ensuremath{\\mathsf X})=\n\\big\\{{x\\in\\mathcal{L}(\\Omega;\\ensuremath{\\mathsf X})}~|~{\\sup\\|x(\\Omega)\\|_\\ensuremath{\\mathsf X}<\\ensuremath{{{+}\\infty}}}\\big\\}$.\n\n## Topological spaces\n\nGiven topological spaces\n$(\\ensuremath{\\mathsf Y},\\EuScript{T}_\\ensuremath{\\mathsf Y})$ and\n$(\\ensuremath{\\mathsf Z},\\EuScript{T}_\\ensuremath{\\mathsf Z})$,\n$\\EuScript{T}_\\ensuremath{\\mathsf Y}\\boxtimes\\EuScript{T}_\\ensuremath{\\mathsf Z}$\ndenotes the standard product topology.\n\nLet $(\\ensuremath{\\mathsf X},\\EuScript{T}_\\ensuremath{\\mathsf X})$ be a\nHausdorff topological space. The Borel $\\sigma$-algebra of\n$(\\ensuremath{\\mathsf X},\\EuScript{T}_\\ensuremath{\\mathsf X})$ is\ndenoted by $\\ensuremath{\\EuScript B}_\\ensuremath{\\mathsf X}$.\nFurthermore,\n$(\\ensuremath{\\mathsf X},\\EuScript{T}_\\ensuremath{\\mathsf X})$ is:\n\n-   regular if, for every closed subset $\\mathsf{C}$ of\n    $(\\ensuremath{\\mathsf X},\\EuScript{T}_\\ensuremath{\\mathsf X})$ and\n    every $\\mathsf{x}\\in\\complement\\mathsf{C}$, there exist\n    $\\mathsf{V}\\in\\EuScript{T}_\\ensuremath{\\mathsf X}$ and\n    $\\mathsf{W}\\in\\EuScript{T}_\\ensuremath{\\mathsf X}$ such that\n    $\\mathsf{C}\\subset\\mathsf{V}$, $\\mathsf{x}\\in\\mathsf{W}$, and\n    $\\mathsf{V}\\cap\\mathsf{W}=\\ensuremath{\\varnothing}$;\n\n-   a Polish space if it is separable and there exists a distance\n    $\\mathsf{d}$ on $\\ensuremath{\\mathsf X}$ that induces the same\n    topology as $\\EuScript{T}_\\ensuremath{\\mathsf X}$ and such that\n    $(\\ensuremath{\\mathsf X},\\mathsf{d})$ is a complete metric space;\n\n-   a Souslin space if there exist a Polish space\n    $(\\ensuremath{\\mathsf Y},\\EuScript{T}_\\ensuremath{\\mathsf Y})$ and a\n    continuous surjective mapping from\n    $(\\ensuremath{\\mathsf Y},\\EuScript{T}_\\ensuremath{\\mathsf Y})$ to\n    $(\\ensuremath{\\mathsf X},\\EuScript{T}_\\ensuremath{\\mathsf X})$;\n\n-   a Lusin space if there exists a topology\n    $\\widetilde{\\EuScript{T}_\\ensuremath{\\mathsf X}}$ on\n    $\\ensuremath{\\mathsf X}$ such that\n    $\\EuScript{T}_\\ensuremath{\\mathsf X}\\subset\\widetilde{\\EuScript{T}_\\ensuremath{\\mathsf X}}$\n    and\n    $(\\ensuremath{\\mathsf X},\\widetilde{\\EuScript{T}_\\ensuremath{\\mathsf X}})$\n    is a Polish space;\n\n-   a Fréchet space if it is a locally convex real topological vector\n    space and there exists a translation-invariant distance $\\mathsf{d}$\n    on $\\ensuremath{\\mathsf X}$ that induces the same topology as\n    $\\EuScript{T}_\\ensuremath{\\mathsf X}$ and such that\n    $(\\ensuremath{\\mathsf X},\\mathsf{d})$ is a complete metric space.\n\nNow let\n$\\mathsf{f}\\colon\\ensuremath{\\mathsf X}\\to\\overline{\\mathbb{R}}$. The\nepigraph of $\\mathsf{f}$ is\n$$\\mathop{\\mathrm{epi}}\\mathsf{f}=\\big\\{{(\\mathsf{x},\\xi)\\in\\ensuremath{\\mathsf X}\\times\\ensuremath{\\mathbb R}}~|~{\n\\mathsf{f}(\\mathsf{x})\\ensuremath{\\leqslant}\\xi}\\big\\},$$ $\\mathsf{f}$\nis proper if\n$\\ensuremath{{{-}\\infty}}\\notin\\mathsf{f}(\\ensuremath{\\mathsf X})\\neq\\{\\ensuremath{{{+}\\infty}}\\}$,\nand $\\mathsf{f}$ is $\\EuScript{T}_\\ensuremath{\\mathsf X}$-lower\nsemicontinuous if $\\mathop{\\mathrm{epi}}\\mathsf{f}$ is\n$\\EuScript{T}_\\ensuremath{\\mathsf X}\\boxtimes\\EuScript{T}_\\ensuremath{\\mathbb R}$-closed.\n\n## Duality\n\nThe dual of a real topological vector space\n$(\\ensuremath{\\mathsf X},\\EuScript{T}_\\ensuremath{\\mathsf X})$, that is,\nthe vector space of continuous linear functionals on\n$(\\ensuremath{\\mathsf X},\\EuScript{T}_\\ensuremath{\\mathsf X})$, is\ndenoted by\n$(\\ensuremath{\\mathsf X},\\EuScript{T}_\\ensuremath{\\mathsf X})^*$.\n\nLet $\\ensuremath{\\mathsf X}$ and $\\ensuremath{\\mathsf Y}$ be real vector\nspaces which are in separating duality via a bilinear form\n$\\langle{{{\\mkern 2mu\\cdot\\mkern 2mu}},{{\\mkern 2mu\\cdot\\mkern 2mu}}}\\rangle_{\\ensuremath{\\mathsf X},\\ensuremath{\\mathsf Y}}\\colon\\ensuremath{\\mathsf X}\\times\\ensuremath{\\mathsf Y}\\to\\ensuremath{\\mathbb R}$,\nthat is , $$\\begin{cases}\n(\\forall\\mathsf{x}\\in\\ensuremath{\\mathsf X})\\quad\n\\langle{{\\mathsf{x}},{{\\mkern 2mu\\cdot\\mkern 2mu}}}\\rangle_{\\ensuremath{\\mathsf X},\\ensuremath{\\mathsf Y}}=0\n\\quad\\Rightarrow\\quad\n\\mathsf{x}=\\mathsf{0}\n\\\\\n(\\forall\\mathsf{y}\\in\\ensuremath{\\mathsf Y})\\quad\n\\langle{{{\\mkern 2mu\\cdot\\mkern 2mu}},{\\mathsf{y}}}\\rangle_{\\ensuremath{\\mathsf X},\\ensuremath{\\mathsf Y}}=0\n\\quad\\Rightarrow\\quad\n\\mathsf{y}=\\mathsf{0}.\n\\end{cases}$$ In addition, equip $\\ensuremath{\\mathsf X}$ with a locally\nconvex topology $\\EuScript{T}_\\ensuremath{\\mathsf X}$ which is\ncompatible with the pairing\n$\\langle{{{\\mkern 2mu\\cdot\\mkern 2mu}},{{\\mkern 2mu\\cdot\\mkern 2mu}}}\\rangle_{\\ensuremath{\\mathsf X},\\ensuremath{\\mathsf Y}}$\nin the sense that\n$(\\ensuremath{\\mathsf X},\\EuScript{T}_\\ensuremath{\\mathsf X})^*\n=\\{\\langle{{{\\mkern 2mu\\cdot\\mkern 2mu}},{\\mathsf{y}}}\\rangle_{\\ensuremath{\\mathsf X},\\ensuremath{\\mathsf Y}}\\}_{\\mathsf{y}\\in\\ensuremath{\\mathsf Y}}$\nand, likewise, equip $\\ensuremath{\\mathsf Y}$ with a locally convex\ntopology $\\EuScript{T}_\\ensuremath{\\mathsf Y}$ which is compatible with\nthe pairing\n$\\langle{{{\\mkern 2mu\\cdot\\mkern 2mu}},{{\\mkern 2mu\\cdot\\mkern 2mu}}}\\rangle_{\\ensuremath{\\mathsf X},\\ensuremath{\\mathsf Y}}$\nin the sense that\n$(\\ensuremath{\\mathsf Y},\\EuScript{T}_\\ensuremath{\\mathsf Y})^*\n=\\{\\langle{{\\mathsf{x}},{{\\mkern 2mu\\cdot\\mkern 2mu}}}\\rangle_{\\ensuremath{\\mathsf X},\\ensuremath{\\mathsf Y}}\\}_{\\mathsf{x}\\in\\ensuremath{\\mathsf X}}$\n. Following , the Legendre conjugate of\n$\\mathsf{f}\\colon\\ensuremath{\\mathsf X}\\to\\overline{\\mathbb{R}}$ is\n$$\\label{e:l0d}\n\\mathsf{f}^*\\colon\\ensuremath{\\mathsf Y}\\to\\overline{\\mathbb{R}}\\colon\n\\mathsf{y}\\mapsto\\sup_{\\mathsf{x}\\in\\ensuremath{\\mathsf X}}\n\\big(\\langle{{\\mathsf{x}},{\\mathsf{y}}}\\rangle_{\\ensuremath{\\mathsf X},\\ensuremath{\\mathsf Y}}-\n\\mathsf{f}(\\mathsf{x})\\big)$$ and the Legendre conjugate of\n$\\mathsf{g}\\colon\\ensuremath{\\mathsf Y}\\to\\overline{\\mathbb{R}}$ is\n$$\\mathsf{g}^*\\colon\\ensuremath{\\mathsf X}\\to\\overline{\\mathbb{R}}\\colon\n\\mathsf{x}\\mapsto\\sup_{\\mathsf{y}\\in\\ensuremath{\\mathsf Y}}\n\\big(\\langle{{\\mathsf{x}},{\\mathsf{y}}}\\rangle_{\\ensuremath{\\mathsf X},\\ensuremath{\\mathsf Y}}-\n\\mathsf{g}(\\mathsf{y})\\big).$$ Let\n$\\mathsf{f}\\colon\\ensuremath{\\mathsf X}\\to\\overline{\\mathbb{R}}$. If\n$\\mathsf{f}$ is proper, its subdifferential is the set-valued operator\n$$\\label{e:s14}\n\\begin{aligned}\n\\partial\\mathsf{f}\\colon\\ensuremath{\\mathsf X}&\\to 2^\\ensuremath{\\mathsf Y}\\\\\n\\mathsf{x}&\\mapsto\\big\\{{\\mathsf{y}\\in\\ensuremath{\\mathsf Y}}~|~{\n(\\forall\\mathsf{z}\\in\\ensuremath{\\mathsf X})\\,\\,\n\\langle{{\\mathsf{z}-\\mathsf{x}},{\\mathsf{y}}}\\rangle_{\\ensuremath{\\mathsf X},\\ensuremath{\\mathsf Y}}\n+\\mathsf{f}(\\mathsf{x})\\ensuremath{\\leqslant}\\mathsf{f}(\\mathsf{z})}\\big\\}\n=\\big\\{{\\mathsf{y}\\in\\ensuremath{\\mathsf Y}}~|~{\n\\mathsf{f}(\\mathsf{x})+\\mathsf{f}^*(\\mathsf{y})=\n\\langle{{\\mathsf{x}},{\\mathsf{y}}}\\rangle_{\\ensuremath{\\mathsf X},\\ensuremath{\\mathsf Y}}}\\big\\}.\n\\end{aligned}$$ In addition, $\\mathsf{f}$ is convex if\n$\\mathop{\\mathrm{epi}}\\mathsf{f}$ is a convex subset of\n$\\ensuremath{\\mathsf X}\\times\\ensuremath{\\mathbb R}$, and\n$\\Gamma_0(\\ensuremath{\\mathsf X})$ denotes the class of proper lower\nsemicontinuous convex functions from $\\ensuremath{\\mathsf X}$ to\n$\\ensuremath{\\left]{-}\\infty,{+}\\infty\\right]}$. Suppose that\n$\\mathsf{f}\\in\\Gamma_0(\\ensuremath{\\mathsf X})$ and let\n$\\mathsf{z}\\in\\mathop{\\mathrm{dom}}\\mathsf{f}$. The recession function\nof $\\mathsf{f}$ is the function in $\\Gamma_0(\\ensuremath{\\mathsf X})$\ndefined by $$\\label{e:r}\n\\mathop{\\mathrm{rec}}\\mathsf{f}\\colon\\ensuremath{\\mathsf X}\\to\\ensuremath{\\left]{-}\\infty,{+}\\infty\\right]}\\colon\\mathsf{x}\\mapsto\n\\lim_{0<\\alpha\\uparrow\\ensuremath{{{+}\\infty}}}\n\\frac{\\mathsf{f}(\\mathsf{z}+\\alpha\\mathsf{x})\n-\\mathsf{f}(\\mathsf{z})}{\\alpha}.$$ Now suppose that, in addition,\n$\\ensuremath{\\mathsf X}=\\ensuremath{\\mathsf Y}$ is Hilbertian and\n$\\langle{{{\\mkern 2mu\\cdot\\mkern 2mu}},{{\\mkern 2mu\\cdot\\mkern 2mu}}}\\rangle_{\\ensuremath{\\mathsf X},\\ensuremath{\\mathsf Y}}$\nis the scalar product of $\\ensuremath{\\mathsf X}$, and let\n$\\gamma\\in\\ensuremath{\\left]0,{+}\\infty\\right[}$. The Moreau envelope of\n$\\mathsf{f}$ of index $\\gamma$ is the function in\n$\\Gamma_0(\\ensuremath{\\mathsf X})$ defined by $$\\label{e:7}\n\\prescript{\\gamma}{}{\\mathsf{f}}\\colon\\ensuremath{\\mathsf X}\\to\\ensuremath{\\mathbb R}\\colon\n\\mathsf{x}\\mapsto\n\\min_{\\mathsf{y}\\in\\ensuremath{\\mathsf X}}\\bigg(\\mathsf{f}(\\mathsf{y})\n+\\dfrac{1}{2\\gamma}\\|\\mathsf{x}-\\mathsf{y}\\|_\\ensuremath{\\mathsf X}^2\\bigg)$$\nand the proximal point of $\\mathsf{x}\\in\\ensuremath{\\mathsf X}$ relative\nto $\\gamma\\mathsf{f}$ is the unique point\n$\\mathop{\\mathrm{prox}}_{\\gamma\\mathsf{f}}\\mathsf{x}\\in\\ensuremath{\\mathsf X}$\nsuch that $$\\label{e:7b}\n\\prescript{\\gamma}{}{\\mathsf{f}}(\\mathsf{x})\n=\\mathsf{f}(\\mathop{\\mathrm{prox}}_{\\gamma\\mathsf{f}}\\mathsf{x})\n+\\dfrac{1}{2\\gamma}\n\\|\\mathsf{x}-\\mathop{\\mathrm{prox}}_{\\gamma\\mathsf{f}}\\mathsf{x}\\|_\\ensuremath{\\mathsf X}^2.$$\nThe proximity operator\n$\\mathop{\\mathrm{prox}}_{\\gamma\\mathsf{f}}\\colon\\ensuremath{\\mathsf X}\\to\\ensuremath{\\mathsf X}$\nthus defined can be expressed as $$\\label{e:8}\n\\mathop{\\mathrm{prox}}_{\\gamma\\mathsf{f}}=(\\ensuremath{\\mathrm{Id}}+\\gamma\\partial\\mathsf{f})^{-1}.$$\n\n# Proof of the interchange principle\n\nProving Theorem  necessitates a few technical facts.\n\n**Lemma 2**.  * Let $(\\Omega,\\ensuremath{\\EuScript F})$ be a measurable\nspace, let $n$ be a strictly positive integer, and let\n$(\\varrho_i)_{0\\ensuremath{\\leqslant}i\\ensuremath{\\leqslant}n}$ be a\nfamily in $\\mathcal{L}(\\Omega;\\ensuremath{\\mathbb R})$. Then there\nexists a family\n$(B_i)_{0\\ensuremath{\\leqslant}i\\ensuremath{\\leqslant}n}$ in\n$\\ensuremath{\\EuScript F}$ such that $$\\label{e:bx}\n(B_i)_{0\\ensuremath{\\leqslant}i\\ensuremath{\\leqslant}n}\\,\\,\\text{are pairwise disjoint},\n\\quad\\bigcup_{i=0}^nB_i=\\Omega,\n\\quad\\text{and}\\quad\n\\min_{0\\ensuremath{\\leqslant}i\\ensuremath{\\leqslant}n}\\varrho_i=\\sum_{i=0}^n1_{B_i}\\varrho_i.$$*\n\n. We proceed by induction on $n$. If $n=1$, we obtain by choosing\n$B_0=[\\varrho_0\\ensuremath{\\leqslant}\\varrho_1]$ and\n$B_1=\\complement B_0$. Now assume that the claim is true for $n$, let\n$\\varrho_{n+1}\\in\\mathcal{L}(\\Omega;\\ensuremath{\\mathbb R})$, and set\n$$\\varrho=\\min_{0\\ensuremath{\\leqslant}i\\ensuremath{\\leqslant}n}\\varrho_i,\n\\quad\nD=[\\varrho\\ensuremath{\\leqslant}\\varrho_{n+1}],\\quad\nC_{n+1}=\\complement D,\n\\quad\\text{and}\\quad\n\\big(\\forall i\\in\\{0,\\ldots,n\\}\\big)\\;\\;C_i=B_i\\cap D.$$ Then\n$(C_i)_{0\\ensuremath{\\leqslant}i\\ensuremath{\\leqslant}n+1}$ is a family\nof pairwise disjoint sets in $\\ensuremath{\\EuScript F}$. Additionally,\n$$\\bigcup_{i=0}^{n+1}C_i\n=C_{n+1}\\cup\\bigcup_{i=0}^nC_i\n=\\big(\\complement D\\big)\\cup\\bigcup_{i=0}^n(B_i\\cap D)\n=\\big(\\complement D\\big)\\cup D\n=\\Omega$$ and $$\\begin{aligned}\n\\min_{0\\ensuremath{\\leqslant}i\\ensuremath{\\leqslant}n+1}\\varrho_i\n=\\min\\{\\varrho,\\varrho_{n+1}\\}\n=1_D\\varrho+1_{\\complement D}\\varrho_{n+1}\n=1_D\\sum_{i=0}^n1_{B_i}\\varrho_i+1_{C_{n+1}}\\varrho_{n+1}\n=\\sum_{i=0}^{n+1}1_{C_i}\\varrho_i,\n\\end{aligned}$$ which concludes the induction argument.\n\n**Lemma 3**.  * Let $(\\Omega,\\ensuremath{\\EuScript F},\\mu)$ be a\n$\\sigma$-finite measure space such that $\\mu(\\Omega)\\neq 0$ and let\n$\\mathcal{R}$ be a nonempty subset of\n$\\mathcal{L}(\\Omega;\\overline{\\mathbb{R}})$. Then there exists an\nelement in $\\mathcal{L}(\\Omega;\\overline{\\mathbb{R}})$, denoted by\n$\\mathop{\\mathrm{ess\\,inf}}\\mathcal{R}$ and unique up to a set of\n$\\mu$-measure zero, such that $$\\label{e:od3}\n\\big(\\forall\\vartheta\\in\\mathcal{L}(\\Omega;\\overline{\\mathbb{R}})\\big)\n\\quad\\big[\\;(\\forall\\varrho\\in\\mathcal{R})\\;\\;\n\\vartheta\\ensuremath{\\leqslant}\\varrho\\,\\,\\ensuremath{\\text{\\rm$\\mu$-a.e.}}\\;\\big]\n\\quad\\Leftrightarrow\\quad\n\\vartheta\\ensuremath{\\leqslant}\\mathop{\\mathrm{ess\\,inf}}\\mathcal{R}\\,\\,\\ensuremath{\\text{\\rm$\\mu$-a.e.}}$$\nMoreover, there exists a sequence\n$(\\varrho_n)_{n\\in\\ensuremath{\\mathbb N}}$ in $\\mathcal{R}$ such that\n$\\mathop{\\mathrm{ess\\,inf}}\\mathcal{R}=\\inf_{n\\in\\ensuremath{\\mathbb N}}\\varrho_n$.*\n\n. Using Assumption , construct\n$0<\\chi\\in\\mathcal{L}^1(\\Omega;\\ensuremath{\\mathbb R})$ such that\n$\\int_\\Omega\\chi d\\mu=1$ and define\n$\\ensuremath{\\mathsf{P}}\\colon\\ensuremath{\\EuScript F}\\to[0,1]\\colon A\\mapsto\\int_A\\chi d\\mu$.\nThen $(\\forall A\\in\\ensuremath{\\EuScript F})$ $\\mu(A)=0$\n$\\Leftrightarrow$ $\\ensuremath{\\mathsf{P}}(A)=0$. Hence, the assertions\nfollow from applied in the probability space\n$(\\Omega,\\ensuremath{\\EuScript F},\\ensuremath{\\mathsf{P}})$.\n\n**Lemma 4**.  * Let $(\\Omega,\\ensuremath{\\EuScript F},\\mu)$ be a measure\nspace, let\n$(\\ensuremath{\\mathsf X},\\EuScript{T}_\\ensuremath{\\mathsf X})$ be a\nSouslin space, let\n$z\\colon(\\Omega,\\ensuremath{\\EuScript F})\\to(\\ensuremath{\\mathsf X},\\ensuremath{\\EuScript B}_\\ensuremath{\\mathsf X})$\nbe measurable, and let $E\\in\\ensuremath{\\EuScript F}$ be such that\n$\\mu(E)<\\ensuremath{{{+}\\infty}}$. Then there exists a sequence\n$(E_n)_{n\\in\\ensuremath{\\mathbb N}}$ in $\\ensuremath{\\EuScript F}$ such\nthat $$\\big[\\;(\\forall n\\in\\ensuremath{\\mathbb N})\\;\\;\nE_n\\subset E\\,\\,\\text{and}\\,\\,\n\\overline{z(E_n)}\\,\\,\\text{is compact}\\;\\big]\n\\quad\\text{and}\\quad\n\\mu(E)=\\mu\\bigg(\\bigcup_{n\\in\\ensuremath{\\mathbb N}}E_n\\bigg).$$*\n\n. We adapt the proof of , where\n$(\\ensuremath{\\mathsf X},\\EuScript{T}_\\ensuremath{\\mathsf X})$ is a\nlocally convex Souslin topological vector space. Define\n$\\nu\\colon\\ensuremath{\\EuScript B}_\\ensuremath{\\mathsf X}\\to\\ensuremath{\\left[0,{+}\\infty\\right[}\\colon\n\\mathsf{K}\\mapsto\\mu(E\\cap z^{-1}(\\mathsf{K}))$. Then $\\nu$ is a measure\non\n$(\\ensuremath{\\mathsf X},\\ensuremath{\\EuScript B}_\\ensuremath{\\mathsf X})$.\nHence, since\n$(\\ensuremath{\\mathsf X},\\EuScript{T}_\\ensuremath{\\mathsf X})$ is a\nSouslin space, it follows from that\n$$\\nu(\\ensuremath{\\mathsf X})=\\sup\\big\\{{\\nu(\\mathsf{K})}~|~{\\mathsf{K}\\,\\,\n\\text{is a compact subset of}\\,\\,(\\ensuremath{\\mathsf X},\\EuScript{T}_\\ensuremath{\\mathsf X})}\\big\\}.$$\nThus, for every $n\\in\\ensuremath{\\mathbb N}$, there exists a compact set\n$\\mathsf{K}_n$ such that\n$\\nu(\\ensuremath{\\mathsf X})\\ensuremath{\\leqslant}\\nu(\\mathsf{K}_n)+2^{-n}$.\nNow define a sequence $(E_n)_{n\\in\\ensuremath{\\mathbb N}}$ of\n$\\ensuremath{\\EuScript F}$-measurable subsets of $E$ by\n$(\\forall n\\in\\ensuremath{\\mathbb N})$ $E_n=E\\cap\nz^{-1}(\\mathsf{K}_n)$. On the one hand, for every\n$n\\in\\ensuremath{\\mathbb N}$, since $\\mathsf{K}_n$ is compact and\n$z(E_n)\\subset\\mathsf{K}_n$, $\\overline{z(E_n)}$ is compact. On the\nother hand, $$(\\forall n\\in\\ensuremath{\\mathbb N})\\quad\n\\mu\\bigg(\\bigcup_{k\\in\\ensuremath{\\mathbb N}}E_k\\bigg)\n\\ensuremath{\\leqslant}\\mu(E)\n=\\nu(\\ensuremath{\\mathsf X})\n\\ensuremath{\\leqslant}\\nu(\\mathsf{K}_n)+2^{-n}\n=\\mu(E_n)+2^{-n}\n\\ensuremath{\\leqslant}\\mu\\bigg(\\bigcup_{k\\in\\ensuremath{\\mathbb N}}E_k\\bigg)+2^{-n},$$\nwhich implies that\n$\\mu(E)=\\mu(\\bigcup_{n\\in\\ensuremath{\\mathbb N}}E_n)$.\n\n**Lemma 5**.  * Suppose that Assumption – hold. Let\n$\\psi\\colon(\\Omega\\times\\ensuremath{\\mathsf X},\\ensuremath{\\EuScript F}\\otimes\\ensuremath{\\EuScript B}_\\ensuremath{\\mathsf X})\\to\\overline{\\mathbb{R}}$\nbe measurable, let $\\mathcal{Z}$ be a nonempty at most countable subset\nof $\\mathcal{L}(\\Omega;\\ensuremath{\\mathsf X})$, and let\n$(\\Omega_k)_{k\\in\\ensuremath{\\mathbb N}}$ be an increasing sequence of\nfinite $\\mu$-measure sets in $\\ensuremath{\\EuScript F}$ such that\n$\\bigcup_{k\\in\\ensuremath{\\mathbb N}}\\Omega_k=\\Omega$. Define\n$$\\label{e:8vc}\n\\mathcal{D}=\\bigcup_{z\\in\\mathcal{Z}}\\bigcup_{k\\in\\ensuremath{\\mathbb N}}\n\\big\\{{1_Az}~|~{\\ensuremath{\\EuScript F}\\ni A\\subset\\Omega_k\\,\\,\n\\text{and}\\,\\,\\overline{z(A)}\\,\\,\\text{is compact}}\\big\\}$$ and\n$$\\label{e:8ki}\n\\mathcal{R}=\\big\\{{\\varrho\\in\\mathcal{L}^1(\\Omega;\\ensuremath{\\mathbb R})}~|~{\n(\\ensuremath{\\exists\\,}x\\in\\mathcal{D})\\,\\,\\psi\\big({\\mkern 2mu\\cdot\\mkern 2mu},x({\\mkern 2mu\\cdot\\mkern 2mu})\\big)\\ensuremath{\\leqslant}\n\\varrho({\\mkern 2mu\\cdot\\mkern 2mu})\\,\\,\\ensuremath{\\text{\\rm$\\mu$-a.e.}}}\\big\\}.$$\nSuppose that $$\\label{e:h3}\n\\psi({\\mkern 2mu\\cdot\\mkern 2mu},\\mathsf{0})\\ensuremath{\\leqslant}0.$$\nThen $\\mathcal{R}\\neq\\ensuremath{\\varnothing}$ and\n$\\mathop{\\mathrm{ess\\,inf}}\\mathcal{R}\n\\ensuremath{\\leqslant}\\inf_{z\\in\\mathcal{Z}}\\psi({\\mkern 2mu\\cdot\\mkern 2mu},z({\\mkern 2mu\\cdot\\mkern 2mu}))$\n$\\ensuremath{\\text{\\rm$\\mu$-a.e.}}$*\n\n. Take $z\\in\\mathcal{Z}$ and note that\n$(\\forall A\\in\\ensuremath{\\EuScript F})$\n$1_Az\\in\\mathcal{L}(\\Omega;\\ensuremath{\\mathsf X})$. Since\n$\\overline{z(\\ensuremath{\\varnothing})}=\\ensuremath{\\varnothing}$ is\ncompact, it results from that\n$0=1_\\ensuremath{\\varnothing}z\\in\\mathcal{D}$. Hence, by ,\n$0\\in\\mathcal{R}$. Next, thanks to Assumption , there exists\n$\\chi\\in\\mathcal{L}^1(\\Omega;\\ensuremath{\\mathbb R})$ such that\n$\\chi>0$. Let us set $$\\label{e:5h0}\n(\\forall n\\in\\ensuremath{\\mathbb N})\\quad A_n=\\Omega_n\\cap\n\\big[\\psi\\big({\\mkern 2mu\\cdot\\mkern 2mu},z({\\mkern 2mu\\cdot\\mkern 2mu})\\big)\\ensuremath{\\leqslant}n\\chi({\\mkern 2mu\\cdot\\mkern 2mu})\\big].$$\nLemma  asserts that there exists a family\n$(A_{n,k})_{(n,k)\\in\\ensuremath{\\mathbb N}^2}$ in\n$\\ensuremath{\\EuScript F}$ such that $$\\label{e:24r}\n(\\forall n\\in\\ensuremath{\\mathbb N})\\quad\n\\begin{cases}\n(\\forall k\\in\\ensuremath{\\mathbb N})\\;\\;\nA_{n,k}\\subset A_n\\,\\,\\text{and}\\,\\,\n\\overline{z(A_{n,k})}\\,\\,\\text{is compact}\n\\\\\n\\displaystyle\n\\mu(A_n)=\\mu\\bigg(\\bigcup_{k\\in\\ensuremath{\\mathbb N}}A_{n,k}\\bigg).\n\\end{cases}$$ In turn, by and , $$\\label{e:y6}\n(\\forall n\\in\\ensuremath{\\mathbb N})(\\forall k\\in\\ensuremath{\\mathbb N})\\quad\n1_{A_{n,k}}z\\in\\mathcal{D}.$$ Define $$\\label{e:c32}\n(\\forall n\\in\\ensuremath{\\mathbb N})(\\forall k\\in\\ensuremath{\\mathbb N})(\\forall m\\in\\ensuremath{\\mathbb N})\\quad\n\\varrho_{n,k,m}({\\mkern 2mu\\cdot\\mkern 2mu})=\n\\max\\big\\{\\psi\\big({\\mkern 2mu\\cdot\\mkern 2mu},1_{A_{n,k}}({\\mkern 2mu\\cdot\\mkern 2mu})z({\\mkern 2mu\\cdot\\mkern 2mu})\\big),\n-m\\chi({\\mkern 2mu\\cdot\\mkern 2mu})\\big\\}.$$ Fix temporarily\n$(n,k,m)\\in\\ensuremath{\\mathbb N}^3$. We infer from , , and that\n$$\\begin{aligned}\n(\\forall\\omega\\in\\Omega)\\quad\n\\psi\\big(\\omega,1_{A_{n,k}}(\\omega)z(\\omega)\\big)\n&=\n\\begin{cases}\n\\psi\\big(\\omega,z(\\omega)\\big),\n&\\text{if}\\,\\,\\omega\\in A_{n,k};\\\\\n\\psi(\\omega,\\mathsf{0}),\n&\\text{otherwise}\n\\end{cases}\n\\nonumber\\\\\n&\\ensuremath{\\leqslant}\n\\begin{cases}\nn\\chi(\\omega),\n&\\text{if}\\,\\,\\omega\\in A_{n,k};\\\\\n0,\n&\\text{otherwise}\n\\end{cases}\n\\nonumber\\\\\n&\\ensuremath{\\leqslant}n\\chi(\\omega).\n\\end{aligned}$$ Therefore,\n$-m\\chi\\ensuremath{\\leqslant}\\varrho_{n,k,m}\\ensuremath{\\leqslant}n\\chi$,\nwhich entails that\n$\\varrho_{n,k,m}\\in\\mathcal{L}^1(\\Omega;\\ensuremath{\\mathbb R})$. In\nturn, we derive from , , and that $\\varrho_{n,k,m}\\in\\mathcal{R}$. Thus,\nLemma  guarantees that there exists\n$B_{n,k,m}\\in\\ensuremath{\\EuScript F}$ such that $\\mu(B_{n,k,m})=0$ and\n$$\\label{e:uf}\n\\big(\\forall\\omega\\in\\complement B_{n,k,m}\\big)\\quad\n(\\mathop{\\mathrm{ess\\,inf}}\\mathcal{R})(\\omega)\\ensuremath{\\leqslant}\\varrho_{n,k,m}(\\omega).$$\nNow set $$\\label{e:a34}\nA=\\bigcap_{(n,k)\\in\\ensuremath{\\mathbb N}^2}\\complement A_{n,k},\n\\quad\nB=\\bigcup_{(n,k,m)\\in\\ensuremath{\\mathbb N}^3}B_{n,k,m},\n\\quad\\text{and}\\quad\nC=\\big[\\psi\\big({\\mkern 2mu\\cdot\\mkern 2mu},z({\\mkern 2mu\\cdot\\mkern 2mu})\\big)<\\ensuremath{{{+}\\infty}}\\big]\\cap(A\\cup B).$$\nThen $\\mu(B)=0$. Furthermore, since yields\n$[\\psi({\\mkern 2mu\\cdot\\mkern 2mu},z({\\mkern 2mu\\cdot\\mkern 2mu}))<\\ensuremath{{{+}\\infty}}]=\\bigcup_{n\\in\\ensuremath{\\mathbb N}}A_n$,\nit follows from and that\n$$\\mu\\Big(\\big[\\psi\\big({\\mkern 2mu\\cdot\\mkern 2mu},z({\\mkern 2mu\\cdot\\mkern 2mu})\\big)<\\ensuremath{{{+}\\infty}}\\big]\\cap A\\Big)\n\\ensuremath{\\leqslant}\\sum_{n\\in\\ensuremath{\\mathbb N}}\\mu(A_n\\cap A)\n\\ensuremath{\\leqslant}\\sum_{n\\in\\ensuremath{\\mathbb N}}\\mu\\bigg(A_n\\cap\n\\bigcap_{k\\in\\ensuremath{\\mathbb N}}\\complement A_{n,k}\\bigg)\n=0.$$ Hence, using , we obtain $$\\label{e:c0}\n\\mu(C)=0\\quad\\text{and}\\quad\n\\complement C\n=\\big[\\psi\\big({\\mkern 2mu\\cdot\\mkern 2mu},z({\\mkern 2mu\\cdot\\mkern 2mu})\\big)=\\ensuremath{{{+}\\infty}}\\big]\\cup\n\\big(\\complement A\\cap\\complement B\\big).$$ Now suppose that\n$\\omega\\in\\complement A\\cap\\complement B$. Then it follows from that\nthere exists $(n,k)\\in\\ensuremath{\\mathbb N}^2$ such that\n$\\omega\\in A_{n,k}\\cap\\complement B$. Therefore, we derive from , , and\nthat $$\\label{e:b9}\n(\\forall m\\in\\ensuremath{\\mathbb N})\\quad\n(\\mathop{\\mathrm{ess\\,inf}}\\mathcal{R})(\\omega)\n\\ensuremath{\\leqslant}\\varrho_{n,k,m}(\\omega)\n=\\max\\big\\{\\psi\\big(\\omega,1_{A_{n,k}}(\\omega)z(\\omega)\\big),\n-m\\chi(\\omega)\\big\\}.$$ Hence, letting\n$m\\uparrow\\ensuremath{{{+}\\infty}}$ yields\n$(\\mathop{\\mathrm{ess\\,inf}}\\mathcal{R})(\\omega)\\ensuremath{\\leqslant}\n\\psi(\\omega,1_{A_{n,k}}(\\omega)z(\\omega))\n=\\psi(\\omega,z(\\omega))$. We have thus shown that\n$\\mathop{\\mathrm{ess\\,inf}}\\mathcal{R}\\ensuremath{\\leqslant}\\psi({\\mkern 2mu\\cdot\\mkern 2mu},z({\\mkern 2mu\\cdot\\mkern 2mu}))$\n$\\ensuremath{\\text{\\rm$\\mu$-a.e.}}$ Since $\\mathcal{Z}$ is at most\ncountable, the proof is complete.\n\n. Define $$\\label{e:p9}\n\\Phi\\colon\\mathcal{L}(\\Omega;\\ensuremath{\\mathsf X})\\to\\mathcal{L}(\\Omega;\\overline{\\mathbb{R}})\\colon\nx\\mapsto\\varphi\\big({\\mkern 2mu\\cdot\\mkern 2mu},x({\\mkern 2mu\\cdot\\mkern 2mu})\\big)$$\nand note that, thanks to Assumption , $$\\label{e:o9}\n\\int_\\Omega\\inf\\varphi({\\mkern 2mu\\cdot\\mkern 2mu},\\ensuremath{\\mathsf X})\\,d\\mu\n\\ensuremath{\\leqslant}\\inf_{x\\in\\ensuremath{\\mathcal X}}\\int_\\Omega\\Phi(x)d\\mu\n\\ensuremath{\\leqslant}\\int_\\Omega\\Phi(\\overline{x})d\\mu\n<\\ensuremath{{{+}\\infty}}.$$ Hence, the interchange rule holds when\n$\\inf_{x\\in\\ensuremath{\\mathcal X}}\\int_\\Omega\\Phi(x)d\\mu=\\ensuremath{{{-}\\infty}}$\nand we assume henceforth that $$\\label{e:77}\n\\inf_{x\\in\\ensuremath{\\mathcal X}}\\int_\\Omega\\Phi(x)d\\mu\\in\\ensuremath{\\mathbb R}.$$\nNow define $$\\label{e:80}\n\\vartheta=\\max\\big\\{\\Phi(\\overline{x}),0\\big\\}$$ and $$\\label{e:72}\n\\psi\\colon\\Omega\\times\\ensuremath{\\mathsf X}\\to\\overline{\\mathbb{R}}\\colon\n(\\omega,\\mathsf{x})\\mapsto\n\\begin{cases}\n\\varphi\\big(\\omega,\\mathsf{x}+\\overline{x}(\\omega)\\big)-\n\\vartheta(\\omega),&\\text{if}\\,\\,\\vartheta(\\omega)<\\ensuremath{{{+}\\infty}};\\\\\n\\ensuremath{{{-}\\infty}},&\\text{if}\\,\\,\\vartheta(\\omega)=\\ensuremath{{{+}\\infty}}.\n\\end{cases}$$ Then we derive from Assumption  that $$\\label{e:r1}\n\\vartheta\\in\\mathcal{L}^1(\\Omega;\\overline{\\mathbb{R}})$$ and,\ntherefore, that $$\\label{e:r2}\n\\mu\\big([\\vartheta=\\ensuremath{{{+}\\infty}}]\\big)=0.$$ On the other\nhand, Assumption  ensures that the mapping\n$(\\Omega\\times\\ensuremath{\\mathsf X},\\ensuremath{\\EuScript F}\\otimes\\ensuremath{\\EuScript B}_\\ensuremath{\\mathsf X})\\to(\\ensuremath{\\mathsf X},\\ensuremath{\\EuScript B}_\\ensuremath{\\mathsf X})\\colon\n(\\omega,\\mathsf{x})\\mapsto\\mathsf{x}+\\overline{x}(\\omega)$ is\nmeasurable. Thus, it follows from Assumption , , and that\n$$\\label{e:f78}\n\\psi\\,\\,\\text{is $\\ensuremath{\\EuScript F}\\otimes\\ensuremath{\\EuScript B}_\\ensuremath{\\mathsf X}$-measurable}.$$\nAt the same time, since $$\\label{e:45}\n\\inf_{\\mathsf{x}\\in\\ensuremath{\\mathsf X}}\\psi({\\mkern 2mu\\cdot\\mkern 2mu},\\mathsf{x})\n=\\inf_{\\mathsf{x}\\in\\ensuremath{\\mathsf X}}\n\\varphi\\big({\\mkern 2mu\\cdot\\mkern 2mu},\\mathsf{x}+\\overline{x}({\\mkern 2mu\\cdot\\mkern 2mu})\\big)-\n\\vartheta({\\mkern 2mu\\cdot\\mkern 2mu})\n=\\inf_{\\mathsf{x}\\in\\ensuremath{\\mathsf X}}\\varphi({\\mkern 2mu\\cdot\\mkern 2mu},\\mathsf{x})-\\vartheta({\\mkern 2mu\\cdot\\mkern 2mu})$$\nand since Assumption  yields\n$\\inf\\varphi({\\mkern 2mu\\cdot\\mkern 2mu},\\ensuremath{\\mathsf X})<\\ensuremath{{{+}\\infty}}$,\nit results from that $$\\label{e:pd}\n\\inf\\psi({\\mkern 2mu\\cdot\\mkern 2mu},\\ensuremath{\\mathsf X})\\in\\mathcal{L}(\\Omega;\\overline{\\mathbb{R}}).$$\nLet us set\n$$\\Psi\\colon\\mathcal{L}(\\Omega;\\ensuremath{\\mathsf X})\\to\\mathcal{L}(\\Omega;\\overline{\\mathbb{R}})\\colon\nx\\mapsto\\psi\\big({\\mkern 2mu\\cdot\\mkern 2mu},x({\\mkern 2mu\\cdot\\mkern 2mu})\\big).$$\nBy and , $$\\label{e:jc}\n\\big(\\forall\\omega\\in\\complement[\\vartheta=\\ensuremath{{{+}\\infty}}]\\big)\n(\\forall x\\in\\ensuremath{\\mathcal X})\\quad\n\\big(\\Psi(x)\\big)(\\omega)\n=\\big(\\Phi(x+\\overline{x})\\big)(\\omega)-\\vartheta(\\omega).$$ Hence, upon\ninvoking , we deduce from Assumption & that $$\\begin{aligned}\n\\label{e:15}\n\\inf_{x\\in\\ensuremath{\\mathcal X}}\\int_\\Omega\\Psi(x)d\\mu\n&=\\inf_{x\\in\\ensuremath{\\mathcal X}}\\int_\\Omega\\big(\\Phi(x+\\overline{x})-\n\\vartheta\\big)d\\mu\n\\nonumber\\\\\n&=\\inf_{x\\in\\ensuremath{\\mathcal X}}\\int_\\Omega\\Phi(x+\\overline{x})d\\mu-\n\\int_\\Omega\\vartheta d\\mu\n\\nonumber\\\\\n&=\\inf_{x\\in\\ensuremath{\\mathcal X}}\\int_\\Omega\\Phi(x)d\\mu-\\int_\\Omega\\vartheta d\\mu\n\\end{aligned}$$ and, likewise, from that $$\\label{e:14}\n\\int_\\Omega\\inf\\psi({\\mkern 2mu\\cdot\\mkern 2mu},\\ensuremath{\\mathsf X})\\,d\\mu\n=\\int_\\Omega\\inf\\varphi({\\mkern 2mu\\cdot\\mkern 2mu},\\ensuremath{\\mathsf X})\\,d\\mu-\\int_\\Omega\\vartheta d\\mu.$$\nNow set $$\\label{e:d2}\n\\mathcal{D}=\\bigcup_{n\\in\\ensuremath{\\mathbb N}}\\bigcup_{k\\in\\ensuremath{\\mathbb N}}\n\\big\\{{1_Ax_n}~|~{\\ensuremath{\\EuScript F}\\ni A\\subset\\Omega_k\n\\,\\,\\text{and}\\,\\,\\overline{x_n(A)}\\,\\,\\text{is compact}}\\big\\}$$ and\n$$\\label{e:pui}\n\\mathcal{R}=\\big\\{{\\varrho\\in\\mathcal{L}^1(\\Omega;\\ensuremath{\\mathbb R})}~|~{\n(\\ensuremath{\\exists\\,}x\\in\\mathcal{D})\\,\\,\\Psi(x)\\ensuremath{\\leqslant}\\varrho\\,\\,\\ensuremath{\\text{\\rm$\\mu$-a.e.}}}\\big\\},$$\nand note that states that $$\\label{e:d0}\n\\mathcal{D}\\subset\\ensuremath{\\mathcal X}.$$ Using and , we infer from\nLemma  applied to $\\mathcal{Z}=\\{x_n\\}_{n\\in\\ensuremath{\\mathbb N}}$\nthat\n$\\mathop{\\mathrm{ess\\,inf}}\\mathcal{R}\\ensuremath{\\leqslant}\\inf_{n\\in\\ensuremath{\\mathbb N}}\\Psi(x_n)$\n$\\ensuremath{\\text{\\rm$\\mu$-a.e.}}$ In turn, we derive from , , and that\n$$\\mathop{\\mathrm{ess\\,inf}}\\mathcal{R}\n\\ensuremath{\\leqslant}\\inf_{n\\in\\ensuremath{\\mathbb N}}\\Psi(x_n)\n=\\inf_{n\\in\\ensuremath{\\mathbb N}}\\Phi(x_n+\\overline{x})-\\vartheta\n=\\inf\\varphi({\\mkern 2mu\\cdot\\mkern 2mu},\\ensuremath{\\mathsf X})-\\vartheta\n=\\inf\\psi({\\mkern 2mu\\cdot\\mkern 2mu},\\ensuremath{\\mathsf X})\\,\\,\\ensuremath{\\text{\\rm$\\mu$-a.e.}}$$\nOn the other hand, implies that $(\\forall\\varrho\\in\\mathcal{R})$\n$\\inf\\psi({\\mkern 2mu\\cdot\\mkern 2mu},\\ensuremath{\\mathsf X})\\ensuremath{\\leqslant}\\varrho({\\mkern 2mu\\cdot\\mkern 2mu})$\n$\\ensuremath{\\text{\\rm$\\mu$-a.e.}}$ Hence, and Lemma  guarantee that\n$\\inf\\psi({\\mkern 2mu\\cdot\\mkern 2mu},\\ensuremath{\\mathsf X})\\ensuremath{\\leqslant}\\mathop{\\mathrm{ess\\,inf}}\\mathcal{R}$\n$\\ensuremath{\\text{\\rm$\\mu$-a.e.}}$ Altogether,\n$\\mathop{\\mathrm{ess\\,inf}}\\mathcal{R}=\\inf\\psi({\\mkern 2mu\\cdot\\mkern 2mu},\\ensuremath{\\mathsf X})$\n$\\ensuremath{\\text{\\rm$\\mu$-a.e.}}$ Thus, we deduce from Lemma  that\nthere exists a sequence $(\\varrho_n)_{n\\in\\ensuremath{\\mathbb N}}$ in\n$\\mathcal{R}$ such that $$\\label{e:g8}\n\\inf_{n\\in\\ensuremath{\\mathbb N}}\\varrho_n({\\mkern 2mu\\cdot\\mkern 2mu})=\\inf\\psi({\\mkern 2mu\\cdot\\mkern 2mu},\\ensuremath{\\mathsf X})\\,\\,\\ensuremath{\\text{\\rm$\\mu$-a.e.}}$$\nFor every $n\\in\\ensuremath{\\mathbb N}$, it follows from and that there\nexist $\\ell_n\\in\\ensuremath{\\mathbb N}$, $k_n\\in\\ensuremath{\\mathbb N}$,\nand $\\ensuremath{\\EuScript F}\\ni A_n\\subset\\Omega_{k_n}$ such that\n$$\\label{e:o4}\n\\overline{x_{\\ell_n}(A_n)}\\,\\,\\text{is compact}\n\\quad\\text{and}\\quad\n\\Psi\\big(1_{A_n}x_{\\ell_n}\\big)\\ensuremath{\\leqslant}\\varrho_n\\,\\,\\ensuremath{\\text{\\rm$\\mu$-a.e.}}$$\nLet us set $$\\label{e:c5}\n(\\forall n\\in\\ensuremath{\\mathbb N})\\quad\\chi_n=\\min_{0\\ensuremath{\\leqslant}i\\ensuremath{\\leqslant}n}\\varrho_i.$$\nFix temporarily $n\\in\\ensuremath{\\mathbb N}$. Lemma  asserts that there\nexists a family\n$(B_{n,i})_{0\\ensuremath{\\leqslant}i\\ensuremath{\\leqslant}n}$ in\n$\\ensuremath{\\EuScript F}$ such that $$\\label{e:bi6}\n(B_{n,i})_{0\\ensuremath{\\leqslant}i\\ensuremath{\\leqslant}n}\\,\\,\\text{are pairwise disjoint},\n\\quad\\bigcup_{i=0}^nB_{n,i}=\\Omega,\\quad\\text{and}\\quad\n\\chi_n=\\sum_{i=0}^n1_{B_{n,i}}\\varrho_i.$$ Now set $$\\label{e:y8}\ny_n=\\sum_{i=0}^n1_{A_i\\cap B_{n,i}}x_{\\ell_i}.$$ For every\n$i\\in\\{0,\\ldots,n\\}$, since\n$A_i\\cap B_{n,i}\\subset A_i\\subset\\Omega_{k_i}$, implies that\n$\\overline{x_{\\ell_i}(A_i\\cap B_{n,i})}$ is compact and, therefore, and\nyield\n$1_{A_i\\cap B_{n,i}}x_{\\ell_i}\\in\\mathcal{D}\\subset\\ensuremath{\\mathcal X}$.\nConsequently, and Assumption  ensure that\n$y_n\\in\\ensuremath{\\mathcal X}$. At the same time, we derive from , ,\nand that $$\\label{e:7gf}\n\\Psi(y_n)\n=\\sum_{i=0}^n1_{B_{n,i}}\\Psi\\big(1_{A_i}x_{\\ell_i}\\big)\n\\ensuremath{\\leqslant}\\sum_{i=0}^n1_{B_{n,i}}\\varrho_i\n=\\chi_n\\,\\,\\ensuremath{\\text{\\rm$\\mu$-a.e.}}$$ Therefore, since\n$y_n\\in\\ensuremath{\\mathcal X}$, $$\\label{e:g7}\n\\inf_{x\\in\\ensuremath{\\mathcal X}}\\int_\\Omega\\Psi(x)d\\mu\n\\ensuremath{\\leqslant}\\int_\\Omega\\Psi(y_n)d\\mu\n\\ensuremath{\\leqslant}\\int_\\Omega\\chi_nd\\mu.$$ On the other hand, it\nresults from , , and that\n$\\inf_{x\\in\\ensuremath{\\mathcal X}}\\int_\\Omega\\Psi(x)d\\mu\\in\\ensuremath{\\mathbb R}$.\nThus, since\n$\\chi_n\\downarrow\\inf_{i\\in\\ensuremath{\\mathbb N}}\\varrho_i({\\mkern 2mu\\cdot\\mkern 2mu})=\n\\inf\\psi({\\mkern 2mu\\cdot\\mkern 2mu},\\ensuremath{\\mathsf X})$\n$\\ensuremath{\\text{\\rm$\\mu$-a.e.}}$ by virtue of and , and the monotone\nconvergence theorem entail that\n$$\\inf_{x\\in\\ensuremath{\\mathcal X}}\\int_\\Omega\\Psi(x)d\\mu\n\\ensuremath{\\leqslant}\\lim\\int_\\Omega\\chi_nd\\mu\n=\\int_\\Omega\\lim\\chi_n\\,d\\mu\n=\\int_\\Omega\\inf\\psi({\\mkern 2mu\\cdot\\mkern 2mu},\\ensuremath{\\mathsf X})\\,d\\mu.$$\nConsequently, since\n$\\int_\\Omega\\inf\\psi({\\mkern 2mu\\cdot\\mkern 2mu},\\ensuremath{\\mathsf X})\\,d\\mu\\ensuremath{\\leqslant}\n\\inf_{x\\in\\ensuremath{\\mathcal X}}\\int_\\Omega\\Psi(x)d\\mu$, we conclude\nthat $$\\inf_{x\\in\\ensuremath{\\mathcal X}}\\int_\\Omega\\Psi(x)d\\mu\n=\\int_\\Omega\\inf\\psi({\\mkern 2mu\\cdot\\mkern 2mu},\\ensuremath{\\mathsf X})\\,d\\mu.$$\nIn view of , , and , the proof is complete.\n\n# Compliant spaces and normal integrands\n\nThe objective of this section is to develop tools to convert the\ninterchange principle of Theorem  into interchange rules formulated in\nterms of explicit conditions on the ambient space\n$\\ensuremath{\\mathcal X}$ and the integrand $\\varphi$. Our framework\nhinges on a notion of compliant spaces and a notion of normal integrands\nin an extended sense.\n\n## Compliant spaces\n\nWe introduce the following notion of a compliant space, which\ngeneralizes and unifies the notions of decomposability employed in the\ninterchange rules of .\n\n**Proposition 6**.  * Suppose that Assumption – holds, together with one\nof the following:*\n\n1.  * $(\\ensuremath{\\mathsf X},\\EuScript{T}_\\ensuremath{\\mathsf X})$ is\n    a Souslin topological vector space and, for every\n    $A\\in\\ensuremath{\\EuScript F}$ such that\n    $\\mu(A)<\\ensuremath{{{+}\\infty}}$ and every\n    $z\\in\\mathcal{L}(\\Omega;\\ensuremath{\\mathsf X})$ such that $z(A)$ is\n    $\\EuScript{T}_\\ensuremath{\\mathsf X}$-bounded (in the sense that,\n    for every neighborhood\n    $\\mathsf{V}\\in\\EuScript{T}_\\ensuremath{\\mathsf X}$ of $\\mathsf{0}$,\n    there exists $\\alpha\\in\\ensuremath{\\left]0,{+}\\infty\\right[}$ such\n    that $z(A)\\subset\\bigcap_{\\beta>\\alpha}\\beta\\mathsf{V}$ ),\n    $1_Az\\in\\ensuremath{\\mathcal X}$.*\n\n2.  * $\\ensuremath{\\mathsf X}$ is a separable Banach space with strong\n    topology $\\EuScript{T}_\\ensuremath{\\mathsf X}$ and, for every\n    $A\\in\\ensuremath{\\EuScript F}$ such that\n    $\\mu(A)<\\ensuremath{{{+}\\infty}}$ and every\n    $z\\in\\mathcal{L}^\\infty(\\Omega;\\ensuremath{\\mathsf X})$,\n    $1_Az\\in\\ensuremath{\\mathcal X}$.*\n\n3.  * $\\ensuremath{\\mathsf X}$ is a separable Banach space with strong\n    topology $\\EuScript{T}_\\ensuremath{\\mathsf X}$,\n    $\\mu(\\Omega)<\\ensuremath{{{+}\\infty}}$, and\n    $\\mathcal{L}^\\infty(\\Omega;\\ensuremath{\\mathsf X})\\subset\\ensuremath{\\mathcal X}$.*\n\n4.  * $\\ensuremath{\\mathsf X}$ is a separable Banach space with strong\n    topology $\\EuScript{T}_\\ensuremath{\\mathsf X}$ and\n    $\\ensuremath{\\mathcal X}$ is *Rockafellar-decomposable* in the sense\n    that, for every $A\\in\\ensuremath{\\EuScript F}$ such that\n    $\\mu(A)<\\ensuremath{{{+}\\infty}}$, every\n    $z\\in\\mathcal{L}^\\infty(\\Omega;\\ensuremath{\\mathsf X})$, and every\n    $x\\in\\ensuremath{\\mathcal X}$,\n    $1_Az+1_{\\complement A}x\\in\\ensuremath{\\mathcal X}$.*\n\n5.  * $(\\ensuremath{\\mathsf X},\\EuScript{T}_\\ensuremath{\\mathsf X})$ is\n    a Souslin locally convex topological vector space and\n    $\\ensuremath{\\mathcal X}$ is *Valadier-decomposable* in the sense\n    that, for every $A\\in\\ensuremath{\\EuScript F}$ such that\n    $\\mu(A)<\\ensuremath{{{+}\\infty}}$, every\n    $z\\in\\mathcal{L}(\\Omega;\\ensuremath{\\mathsf X})$ such that\n    $\\overline{z(A)}$ is compact, and every\n    $x\\in\\ensuremath{\\mathcal X}$,\n    $1_Az+1_{\\complement A}x\\in\\ensuremath{\\mathcal X}$.*\n\n6.  * $\\ensuremath{\\mathsf X}$ is the standard Euclidean space\n    $\\ensuremath{\\mathbb R}^N$ and, for every\n    $A\\in\\ensuremath{\\EuScript F}$ such that\n    $\\mu(A)<\\ensuremath{{{+}\\infty}}$ and every\n    $z\\in\\mathcal{L}^\\infty(\\Omega;\\ensuremath{\\mathsf X})$,\n    $1_Az\\in\\ensuremath{\\mathcal X}$.*\n\n*Then $\\ensuremath{\\mathcal X}$ is compliant.*\n\n. : Let $A\\in\\ensuremath{\\EuScript F}$ be such that\n$\\mu(A)<\\ensuremath{{{+}\\infty}}$ and let\n$z\\in\\mathcal{L}(\\Omega;\\ensuremath{\\mathsf X})$ be such that\n$\\overline{z(A)}$ is compact. It results from that $z(A)$ is\n$\\EuScript{T}_\\ensuremath{\\mathsf X}$-bounded. Thus\n$1_Az\\in\\ensuremath{\\mathcal X}$.\n\n$\\Rightarrow$`<!-- -->`{=html}$\\Rightarrow$`<!-- -->`{=html}: Clear.\n\n$\\Rightarrow$`<!-- -->`{=html}: Clear.\n\n: Clear.\n\n$\\Rightarrow$`<!-- -->`{=html}: Clear.\n\n## Normal integrands\n\nWe introduce a notion of a normal integrand which unifies and extends\nthose of .\n\nThe following theorem furnishes examples of normal integrands.\n\n**Theorem 7**.  * Let\n$(\\ensuremath{\\mathsf X},\\EuScript{T}_\\ensuremath{\\mathsf X})$ be a\nSouslin space, let $(\\Omega,\\ensuremath{\\EuScript F})$ be a measurable\nspace, and let\n$\\varphi\\colon\\Omega\\times\\ensuremath{\\mathsf X}\\to\\overline{\\mathbb{R}}$\nbe such that $(\\forall\\omega\\in\\Omega)$\n$\\mathop{\\mathrm{epi}}\\varphi_\\omega\\neq\\ensuremath{\\varnothing}$.\nSuppose that one of the following holds:*\n\n1.  * $\\varphi$ is\n    $\\ensuremath{\\EuScript F}\\otimes\\ensuremath{\\EuScript B}_\\ensuremath{\\mathsf X}$-measurable\n    and one of the following is satisfied:*\n\n    1.  * There exists a measure $\\mu$ such that\n        $(\\Omega,\\ensuremath{\\EuScript F},\\mu)$ is complete and\n        $\\sigma$-finite.*\n\n    2.  * $\\Omega$ is a Borel subset of $\\ensuremath{\\mathbb R}^M$ and\n        $\\ensuremath{\\EuScript F}$ is the associated Lebesgue\n        $\\sigma$-algebra.*\n\n    3.  * For every $\\omega\\in\\Omega$, there exists\n        $\\boldsymbol{\\mathsf{V}}_\\omega\\in\n        \\EuScript{T}_\\ensuremath{\\mathsf X}\\boxtimes\\EuScript{T}_\\ensuremath{\\mathbb R}$\n        such that\n        $\\boldsymbol{\\mathsf{V}}_\\omega\\subset\\mathop{\\mathrm{epi}}\\varphi_\\omega$\n        and $\\overline{\\boldsymbol{\\mathsf{V}}_\\omega}\n        =\\overline{\\mathop{\\mathrm{epi}}\\varphi_\\omega}$.*\n\n    4.  * The functions $(\\varphi_\\omega)_{\\omega\\in\\Omega}$ are upper\n        semicontinuous.*\n\n2.  * The functions\n    $(\\varphi({\\mkern 2mu\\cdot\\mkern 2mu},\\mathsf{x}))_{\\mathsf{x}\\in\\ensuremath{\\mathsf X}}$\n    are $\\ensuremath{\\EuScript F}$-measurable and one of the following\n    is satisfied:*\n\n    1.  * $(\\ensuremath{\\mathsf X},\\EuScript{T}_\\ensuremath{\\mathsf X})$\n        is metrizable and, for every $\\omega\\in\\Omega$, there exists\n        $\\boldsymbol{\\mathsf{V}}_\\omega\\in\n        \\EuScript{T}_\\ensuremath{\\mathsf X}\\boxtimes\\EuScript{T}_\\ensuremath{\\mathbb R}$\n        such that\n        $\\boldsymbol{\\mathsf{V}}_\\omega\\subset\\mathop{\\mathrm{epi}}\\varphi_\\omega\n        =\\overline{\\boldsymbol{\\mathsf{V}}_\\omega}$.*\n\n    2.  * $(\\ensuremath{\\mathsf X},\\EuScript{T}_\\ensuremath{\\mathsf X})$\n        is a Fréchet space and, for every $\\omega\\in\\Omega$,\n        $\\varphi_\\omega\\in\\Gamma_0(\\ensuremath{\\mathsf X})$ and\n        $\\mathop{\\mathrm{int\\,dom}}\\varphi_\\omega\\neq\\ensuremath{\\varnothing}$.*\n\n    3.  * $(\\ensuremath{\\mathsf X},\\EuScript{T}_\\ensuremath{\\mathsf X})$\n        is the standard Euclidean line $\\ensuremath{\\mathbb R}$ and, for\n        every $\\omega\\in\\Omega$,\n        $\\varphi_\\omega\\in\\Gamma_0(\\ensuremath{\\mathbb R})$ and\n        $\\mathop{\\mathrm{dom}}\\varphi_\\omega$ is not a singleton.*\n\n3.  * $(\\ensuremath{\\mathsf X},\\EuScript{T}_\\ensuremath{\\mathsf X})$ is\n    a regular Souslin space, the functions\n    $(\\varphi_\\omega)_{\\omega\\in\\Omega}$ are continuous, and the\n    functions\n    $(\\varphi({\\mkern 2mu\\cdot\\mkern 2mu},\\mathsf{x}))_{\\mathsf{x}\\in\\ensuremath{\\mathsf X}}$\n    are $\\ensuremath{\\EuScript F}$-measurable.*\n\n4.  * For some separable Fréchet space\n    $(\\ensuremath{\\mathsf Y},\\EuScript{T}_\\ensuremath{\\mathsf Y})$,\n    $\\ensuremath{\\mathsf X}=(\\ensuremath{\\mathsf Y},\\EuScript{T}_\\ensuremath{\\mathsf Y})^*$,\n    $\\EuScript{T}_\\ensuremath{\\mathsf X}$ is the weak topology, the\n    functions $(\\varphi_\\omega)_{\\omega\\in\\Omega}$ are\n    $\\EuScript{T}_\\ensuremath{\\mathsf X}$-lower semicontinuous, and one\n    of the following is satisfied:*\n\n    1.  * For every closed subset $\\boldsymbol{\\mathsf{C}}$ of\n        $(\\ensuremath{\\mathsf X}\\times\\ensuremath{\\mathbb R},\\EuScript{T}_\\ensuremath{\\mathsf X}\\boxtimes\\EuScript{T}_\\ensuremath{\\mathbb R})$,\n        $\\big\\{{\\omega\\in\\Omega}~|~{\\boldsymbol{\\mathsf{C}}\n        \\cap\\mathop{\\mathrm{epi}}\\varphi_\\omega\\neq\\ensuremath{\\varnothing}}\\big\\}\\in\\ensuremath{\\EuScript F}$.*\n\n    2.  * $(\\Omega,\\EuScript{T}_\\Omega)$ is a Hausdorff topological\n        space,\n        $\\ensuremath{\\EuScript F}=\\ensuremath{\\EuScript B}_\\Omega$, and\n        $\\varphi$ is\n        $\\EuScript{T}_\\Omega\\boxtimes\\EuScript{T}_\\ensuremath{\\mathsf X}$-lower\n        semicontinuous.*\n\n    3.  * $(\\Omega,\\EuScript{T}_\\Omega)$ is a Lusin space,\n        $\\ensuremath{\\EuScript F}=\\ensuremath{\\EuScript B}_\\Omega$, and\n        $\\varphi$ is\n        $\\ensuremath{\\EuScript F}\\otimes\\ensuremath{\\EuScript B}_\\ensuremath{\\mathsf X}$-measurable.*\n\n5.  * $\\ensuremath{\\mathsf X}$ is a separable reflexive Banach space,\n    $\\EuScript{T}_\\ensuremath{\\mathsf X}$ is the weak topology,\n    $(\\Omega,\\EuScript{T}_\\Omega)$ is a Hausdorff topological space,\n    $\\ensuremath{\\EuScript F}=\\ensuremath{\\EuScript B}_\\Omega$, the\n    functions $(\\varphi_\\omega)_{\\omega\\in\\Omega}$ are\n    $\\EuScript{T}_\\ensuremath{\\mathsf X}$-lower semicontinuous, and one\n    of the following is satisfied:*\n\n    1.  * $\\varphi$ is\n        $\\EuScript{T}_\\Omega\\boxtimes\\EuScript{T}_\\ensuremath{\\mathsf X}$-lower\n        semicontinuous.*\n\n    2.  * $(\\Omega,\\EuScript{T}_\\Omega)$ is a Lusin space and $\\varphi$\n        is\n        $\\ensuremath{\\EuScript F}\\otimes\\ensuremath{\\EuScript B}_\\ensuremath{\\mathsf X}$-measurable.*\n\n6.  * $(\\ensuremath{\\mathsf X},\\EuScript{T}_\\ensuremath{\\mathsf X})$ is\n    the standard Euclidean space $\\ensuremath{\\mathbb R}^N$, $\\Omega$ is\n    a Borel subset of $\\ensuremath{\\mathbb R}^M$,\n    $\\ensuremath{\\EuScript F}=\\ensuremath{\\EuScript B}_\\Omega$,\n    $\\varphi$ is\n    $\\ensuremath{\\EuScript F}\\otimes\\ensuremath{\\EuScript B}_\\ensuremath{\\mathsf X}$-measurable,\n    and the functions $(\\varphi_\\omega)_{\\omega\\in\\Omega}$ are lower\n    semicontinuous.*\n\n7.  * $(\\ensuremath{\\mathsf X},\\EuScript{T}_\\ensuremath{\\mathsf X})$ is\n    a Polish space, the functions $(\\varphi_\\omega)_{\\omega\\in\\Omega}$\n    are lower semicontinuous, and one of the following is satisfied:*\n\n    1.  * For every $\\boldsymbol{\\mathsf{V}}\\in\n        \\EuScript{T}_\\ensuremath{\\mathsf X}\\boxtimes\\EuScript{T}_\\ensuremath{\\mathbb R}$,\n        $\\big\\{{\\omega\\in\\Omega}~|~{\\boldsymbol{\\mathsf{V}}\\cap\n        \\mathop{\\mathrm{epi}}\\varphi_\\omega\\neq\\ensuremath{\\varnothing}}\\big\\}\\in\\ensuremath{\\EuScript F}$.*\n\n    2.  * $(\\ensuremath{\\mathsf X},\\EuScript{T}_\\ensuremath{\\mathsf X})$\n        is the standard Euclidean space $\\ensuremath{\\mathbb R}^N$ and,\n        for every closed subset $\\boldsymbol{\\mathsf{C}}$ of\n        $\\ensuremath{\\mathsf X}\\times\\ensuremath{\\mathbb R}$,\n        $\\big\\{{\\omega\\in\\Omega}~|~{\n        \\boldsymbol{\\mathsf{C}}\\cap\\mathop{\\mathrm{epi}}\\varphi_\\omega\\neq\\ensuremath{\\varnothing}}\\big\\}\\in\\ensuremath{\\EuScript F}$.*\n\n8.  * There exists a measurable function\n    $\\mathsf{f}\\colon(\\ensuremath{\\mathsf X},\\ensuremath{\\EuScript B}_\\ensuremath{\\mathsf X})\\to\\overline{\\mathbb{R}}$\n    such that $(\\forall\\omega\\in\\Omega)$ $\\varphi_\\omega=\\mathsf{f}$.*\n\n*Then $\\varphi$ is normal.*\n\n. Set $\\boldsymbol{G}=\\big\\{{(\\omega,\\mathsf{x},\\xi)\\in\n\\Omega\\times\\ensuremath{\\mathsf X}\\times\\ensuremath{\\mathbb R}}~|~{\\varphi(\\omega,\\mathsf{x})\\ensuremath{\\leqslant}\\xi}\\big\\}$.\nThen $$\\label{e:s03}\n\\boldsymbol{G}=\\big\\{{(\\omega,\\mathsf{x},\\xi)\\in\n\\Omega\\times\\ensuremath{\\mathsf X}\\times\\ensuremath{\\mathbb R}}~|~{(\\mathsf{x},\\xi)\\in\\mathop{\\mathrm{epi}}\\varphi_\\omega}\\big\\}.$$\nFurther, yields $$\\label{e:s04}\n\\varphi\\,\\,\\text{is $\\ensuremath{\\EuScript F}\\otimes\\ensuremath{\\EuScript B}_\\ensuremath{\\mathsf X}$-measurable}\n\\quad\\Leftrightarrow\\quad\n\\boldsymbol{G}\\in\\ensuremath{\\EuScript F}\\otimes\\ensuremath{\\EuScript B}_\\ensuremath{\\mathsf X}\\otimes\\ensuremath{\\EuScript B}_\\ensuremath{\\mathbb R}\n=\\ensuremath{\\EuScript F}\\otimes\\ensuremath{\\EuScript B}_{\\ensuremath{\\mathsf X}\\times\\ensuremath{\\mathbb R}}.$$\nWe also note that $(\\ensuremath{\\mathsf X}\\times\\ensuremath{\\mathbb R},\n\\EuScript{T}_\\ensuremath{\\mathsf X}\\boxtimes\\EuScript{T}_\\ensuremath{\\mathbb R})$\nis a Souslin space .\n\n: Applying to the mapping $\\Upsilon\\colon\\Omega\\to\n2^{\\ensuremath{\\mathsf X}\\times\\ensuremath{\\mathbb R}}\\colon\\omega\\mapsto\\mathop{\\mathrm{epi}}\\varphi_\\omega$,\nwe deduce from and that there exist a sequence\n$(x_n)_{n\\in\\ensuremath{\\mathbb N}}$ of mappings from $\\Omega$ to\n$\\ensuremath{\\mathsf X}$ and a sequence\n$(\\varrho_n)_{n\\in\\ensuremath{\\mathbb N}}$ of functions from $\\Omega$ to\n$\\ensuremath{\\mathbb R}$ such that $$\\label{e:10ds}\n(\\forall n\\in\\ensuremath{\\mathbb N})\\quad\n(\\Omega,\\ensuremath{\\EuScript F})\\to(\\ensuremath{\\mathsf X}\\times\\ensuremath{\\mathbb R},\\ensuremath{\\EuScript B}_{\\ensuremath{\\mathsf X}\\times\\ensuremath{\\mathbb R}})\\colon\n\\omega\\mapsto\\big(x_n(\\omega),\\varrho_n(\\omega)\\big)\n\\,\\,\\text{is measurable}$$ and $$(\\forall\\omega\\in\\Omega)\\quad\n\\big\\{\\big(x_n(\\omega),\\varrho_n(\\omega)\\big)\\big\\}_{n\\in\\ensuremath{\\mathbb N}}\n\\subset\\Upsilon(\\omega)\n\\quad\\text{and}\\quad\n\\overline{\\Upsilon(\\omega)}=\n\\overline{\\big\\{\\big(x_n(\\omega),\n\\varrho_n(\\omega)\\big)\\big\\}_{n\\in\\ensuremath{\\mathbb N}}}.$$ Moreover,\nsince\n$\\ensuremath{\\EuScript B}_{\\ensuremath{\\mathsf X}\\times\\ensuremath{\\mathbb R}}=\\ensuremath{\\EuScript B}_\\ensuremath{\\mathsf X}\\otimes\\ensuremath{\\EuScript B}_\\ensuremath{\\mathbb R}$\n, it follows from that, for every $n\\in\\ensuremath{\\mathbb N}$,\n$x_n\\colon(\\Omega,\\ensuremath{\\EuScript F})\\to(\\ensuremath{\\mathsf X},\\ensuremath{\\EuScript B}_\\ensuremath{\\mathsf X})$\nand\n$\\varrho_n\\colon(\\Omega,\\ensuremath{\\EuScript F})\\to(\\ensuremath{\\mathbb R},\\ensuremath{\\EuScript B}_\\ensuremath{\\mathbb R})$\nare measurable. Altogether, $\\varphi$ is normal.\n\n$\\Rightarrow$`<!-- -->`{=html}: Take $\\mu$ to be the Lebesgue measure on\n$\\Omega$.\n\n: Let $\\{(\\mathsf{x}_n,\\xi_n)\\}_{n\\in\\ensuremath{\\mathbb N}}$ be a dense\nset in\n$(\\ensuremath{\\mathsf X}\\times\\ensuremath{\\mathbb R},\\EuScript{T}_\\ensuremath{\\mathsf X}\\boxtimes\\EuScript{T}_\\ensuremath{\\mathbb R})$\nand define $$\\label{e:rc0}\n(\\forall n\\in\\ensuremath{\\mathbb N})\\quad\n\\Omega_n=\\big[\\varphi({\\mkern 2mu\\cdot\\mkern 2mu},\\mathsf{x}_n)\\ensuremath{\\leqslant}\\xi_n\\big].$$\nOn the one hand, the\n$\\ensuremath{\\EuScript F}\\otimes\\ensuremath{\\EuScript B}_\\ensuremath{\\mathsf X}$-measurability\nof $\\varphi$ ensures that $(\\forall n\\in\\ensuremath{\\mathbb N})$\n$\\Omega_n\\in\\ensuremath{\\EuScript F}$. On the other hand, for every\n$\\omega\\in\\Omega$, since $\\boldsymbol{\\mathsf{V}}_\\omega$ is open, there\nexists $n\\in\\ensuremath{\\mathbb N}$ such that\n$(\\mathsf{x}_n,\\xi_n)\\in\\boldsymbol{\\mathsf{V}}_\\omega\n\\subset\\mathop{\\mathrm{epi}}\\varphi_\\omega$, which yields\n$\\omega\\in\\Omega_n$ and thus\n$\\Omega=\\bigcup_{k\\in\\ensuremath{\\mathbb N}}\\Omega_k$. This yields a\nsequence $(\\Theta_n)_{n\\in\\ensuremath{\\mathbb N}}$ of pairwise disjoint\nsets in $\\ensuremath{\\EuScript F}$ such that $$\\label{e:tf0}\n\\Theta_0=\\Omega_0,\\quad\n\\bigcup_{n\\in\\ensuremath{\\mathbb N}}\\Theta_n=\\Omega,\\quad\\text{and}\\quad\n(\\forall n\\in\\ensuremath{\\mathbb N})\\;\\;\\Theta_n\\subset\\Omega_n.$$ For\nevery $\\omega\\in\\Omega$, there exists a unique\n$n_\\omega\\in\\ensuremath{\\mathbb N}$ such that\n$\\omega\\in\\Theta_{n_\\omega}$. Now define\n$$z\\colon\\Omega\\to\\ensuremath{\\mathsf X}\\colon\\omega\\mapsto\\mathsf{x}_{n_\\omega}\n\\quad\\text{and}\\quad\n\\vartheta\\colon\\Omega\\to\\ensuremath{\\mathbb R}\\colon\\omega\\mapsto\\xi_{n_\\omega}.$$\nThen $$(\\forall\\mathsf{V}\\in\\EuScript{T}_\\ensuremath{\\mathsf X})\\quad\nz^{-1}(\\mathsf{V})=\\bigcup_{\\substack{n\\in\\ensuremath{\\mathbb N}\\\\\n\\mathsf{x}_n\\in\\mathsf{V}}}\\Theta_n\\in\\ensuremath{\\EuScript F},$$ which\nimplies that $z\\in\\mathcal{L}(\\Omega;\\ensuremath{\\mathsf X})$. Likewise,\n$\\vartheta\\in\\mathcal{L}(\\Omega;\\ensuremath{\\mathbb R})$. Next, define\n$$\\label{e:xxi0}\n(\\forall n\\in\\ensuremath{\\mathbb N})\\quad\nx_n\\colon\\Omega\\to\\ensuremath{\\mathsf X}\\colon\\omega\\mapsto\n\\begin{cases}\n\\mathsf{x}_n,&\\text{if}\\,\\,\\omega\\in\\Omega_n;\\\\\nz(\\omega),&\\text{if}\\,\\,\\omega\\in\\complement\\Omega_n\n\\end{cases}$$ and $$\\label{e:xxi1}\n(\\forall n\\in\\ensuremath{\\mathbb N})\\quad\n\\varrho_n\\colon\\Omega\\to\\ensuremath{\\mathbb R}\\colon\\omega\\mapsto\n\\begin{cases}\n\\xi_n,&\\text{if}\\,\\,\\omega\\in\\Omega_n;\\\\\n\\vartheta(\\omega),\n&\\text{if}\\,\\,\\omega\\in\\complement\\Omega_n.\n\\end{cases}$$ Then $(x_n)_{n\\in\\ensuremath{\\mathbb N}}$ and\n$(\\varrho_n)_{n\\in\\ensuremath{\\mathbb N}}$ are sequences in\n$\\mathcal{L}(\\Omega;\\ensuremath{\\mathsf X})$ and\n$\\mathcal{L}(\\Omega;\\ensuremath{\\mathbb R})$, respectively. Moreover, we\ndeduce from , , , and that\n$$(\\forall\\omega\\in\\Omega)(\\forall n\\in\\ensuremath{\\mathbb N})\\quad\n\\big(x_n(\\omega),\\varrho_n(\\omega)\\big)\\in\\mathop{\\mathrm{epi}}\\varphi_\\omega.$$\nOn the other hand, for every $\\omega\\in\\Omega$, since\n$\\{(\\mathsf{x}_n,\\xi_n)\\}_{n\\in\\ensuremath{\\mathbb N}}$ is dense in\n$(\\ensuremath{\\mathsf X}\\times\\ensuremath{\\mathbb R},\\EuScript{T}_\\ensuremath{\\mathsf X}\\boxtimes\\EuScript{T}_\\ensuremath{\\mathbb R})$\nand since $\\boldsymbol{\\mathsf{V}}_\\omega$ is open, we infer from , ,\nand that $$\\overline{\\big\\{\\big(x_n(\\omega),\n\\varrho_n(\\omega)\\big)\\big\\}_{n\\in\\ensuremath{\\mathbb N}}}\n=\\overline{\\big\\{(\\mathsf{x}_n,\\xi_n)\\big\\}_{n\\in\\ensuremath{\\mathbb N}}\\cap\n\\mathop{\\mathrm{epi}}\\varphi_\\omega}\n\\supset\\overline{\\big\\{(\\mathsf{x}_n,\\xi_n)\\big\\}_{n\\in\\ensuremath{\\mathbb N}}\\cap\n\\boldsymbol{\\mathsf{V}}_\\omega}\n=\\overline{\\boldsymbol{\\mathsf{V}}_\\omega}\n=\\overline{\\mathop{\\mathrm{epi}}\\varphi_\\omega}.$$ Consequently,\n$\\varphi$ is normal.\n\n$\\Rightarrow$`<!-- -->`{=html}: Set $(\\forall\\omega\\in\\Omega)$\n$\\boldsymbol{\\mathsf{V}}_\\omega=\\big\\{{(\\mathsf{x},\\xi)\\in\n\\ensuremath{\\mathsf X}\\times\\ensuremath{\\mathbb R}}~|~{\\varphi(\\omega,\\mathsf{x})<\\xi}\\big\\}$.\nNow fix $\\omega\\in\\Omega$ and\n$(\\mathsf{x},\\xi)\\in\\mathop{\\mathrm{epi}}\\varphi_\\omega$. Since the\nsequence $(\\mathsf{x},\\xi+2^{-n})_{n\\in\\ensuremath{\\mathbb N}}$ lies in\n$\\boldsymbol{\\mathsf{V}}_\\omega$ and\n$(\\mathsf{x},\\xi+2^{-n})\\to(\\mathsf{x},\\xi)$, we obtain\n$(\\mathsf{x},\\xi)\\in\\overline{\\boldsymbol{\\mathsf{V}}_\\omega}$. Hence\n$\\overline{\\boldsymbol{\\mathsf{V}}_\\omega}=\n\\overline{\\mathop{\\mathrm{epi}}\\varphi_\\omega}$. At the same time, the\nupper semicontinuity of $\\varphi_\\omega$ guarantees that\n$\\boldsymbol{\\mathsf{V}}_\\omega$ is open.\n\n$\\Rightarrow$`<!-- -->`{=html}: It suffices to show that $\\varphi$ is\n$\\ensuremath{\\EuScript F}\\otimes\\ensuremath{\\EuScript B}_\\ensuremath{\\mathsf X}$-measurable.\nLet $\\{(\\mathsf{x}_n,\\xi_n)\\}_{n\\in\\ensuremath{\\mathbb N}}$ be dense in\n$(\\ensuremath{\\mathsf X}\\times\\ensuremath{\\mathbb R},\\EuScript{T}_\\ensuremath{\\mathsf X}\\boxtimes\\EuScript{T}_\\ensuremath{\\mathbb R})$,\nlet $\\boldsymbol{\\mathsf{V}}\\in\n\\EuScript{T}_\\ensuremath{\\mathsf X}\\boxtimes\\EuScript{T}_\\ensuremath{\\mathbb R}$,\nand set $\\mathbb{K}=\\big\\{{n\\in\\ensuremath{\\mathbb N}}~|~{\n(\\mathsf{x}_n,\\xi_n)\\in\\boldsymbol{\\mathsf{V}}}\\big\\}$. Then\n$$\\label{e:vd9}\n\\overline{\\{(\\mathsf{x}_n,\\xi_n)\\}_{n\\in\\mathbb{K}}}\n=\\overline{\\{(\\mathsf{x}_n,\\xi_n)\\}_{n\\in\\ensuremath{\\mathbb N}}\\cap\n\\boldsymbol{\\mathsf{V}}}\n=\\overline{\\boldsymbol{\\mathsf{V}}}.$$ Suppose that there exists\n$\\omega\\in\\Omega$ such that $$\\label{e:tx}\n\\boldsymbol{\\mathsf{V}}\\cap\\mathop{\\mathrm{epi}}\\varphi_\\omega\\neq\\ensuremath{\\varnothing}\n\\quad\\text{and}\\quad\n(\\forall n\\in\\mathbb{K})\\;\\;\n(\\mathsf{x}_n,\\xi_n)\\notin\\mathop{\\mathrm{epi}}\\varphi_\\omega.$$ Since\n$\\boldsymbol{\\mathsf{V}}$ is open and\n$\\overline{\\boldsymbol{\\mathsf{V}}_\\omega}=\\mathop{\\mathrm{epi}}\\varphi_\\omega$,\nthere exists $(\\mathsf{y},\\eta)\n\\in\\boldsymbol{\\mathsf{V}}\\cap\\boldsymbol{\\mathsf{V}}_\\omega$.\nTherefore, we infer from that there exists a subnet\n$(\\mathsf{x}_{k(b)},\\xi_{k(b)})_{b\\in B}$ of\n$(\\mathsf{x}_n,\\xi_n)_{n\\in\\mathbb{K}}$ such that\n$(\\mathsf{x}_{k(b)},\\xi_{k(b)})\\to(\\mathsf{y},\\eta)$. This and force\n$(\\mathsf{y},\\eta)\n\\in\\overline{\\complement\\mathop{\\mathrm{epi}}\\varphi_\\omega}\n=\\overline{\\complement\n\\overline{\\boldsymbol{\\mathsf{V}}_\\omega}}\n=\\complement\\mathop{\\mathrm{int}}\n\\overline{\\boldsymbol{\\mathsf{V}}_\\omega}$, which is in contradiction\nwith the inclusion $(\\mathsf{y},\\eta)\\in\\boldsymbol{\\mathsf{V}}_\\omega$.\nHence, the $\\ensuremath{\\EuScript F}$-measurability of the functions\n$(\\varphi({\\mkern 2mu\\cdot\\mkern 2mu},\\mathsf{x}))_{\\mathsf{x}\\in\\ensuremath{\\mathsf X}}$\nyields $$\\big\\{{\\omega\\in\\Omega}~|~{\n\\boldsymbol{\\mathsf{V}}\\cap\\mathop{\\mathrm{epi}}\\varphi_\\omega\\neq\\ensuremath{\\varnothing}}\\big\\}\n=\\bigcup_{n\\in\\mathbb{K}}\\big\\{{\\omega\\in\\Omega}~|~{\n(\\mathsf{x}_n,\\xi_n)\\in\\mathop{\\mathrm{epi}}\\varphi_\\omega}\\big\\}\n=\\bigcup_{n\\in\\mathbb{K}}\n\\big[\\varphi({\\mkern 2mu\\cdot\\mkern 2mu},\\mathsf{x}_n)\\ensuremath{\\leqslant}\\xi_n\\big]\n\\in\\ensuremath{\\EuScript F}.$$ Therefore, since\n$(\\ensuremath{\\mathsf X}\\times\\ensuremath{\\mathbb R},\\EuScript{T}_\\ensuremath{\\mathsf X}\\boxtimes\\EuScript{T}_\\ensuremath{\\mathbb R})$\nis a separable metrizable space and the sets\n$(\\mathop{\\mathrm{epi}}\\varphi_\\omega)_{\\omega\\in\\Omega}$ are closed,\nand imply that\n$\\boldsymbol{G}\\in\\ensuremath{\\EuScript F}\\otimes\\ensuremath{\\EuScript B}_{\\ensuremath{\\mathsf X}\\times\\ensuremath{\\mathbb R}}$.\nConsequently, asserts that $\\varphi$ is\n$\\ensuremath{\\EuScript F}\\otimes\\ensuremath{\\EuScript B}_\\ensuremath{\\mathsf X}$-measurable.\n\n$\\Rightarrow$`<!-- -->`{=html}: Set $(\\forall\\omega\\in\\Omega)$\n$\\boldsymbol{\\mathsf{V}}_\\omega=\\mathop{\\mathrm{int}}\\mathop{\\mathrm{epi}}\\varphi_\\omega$.\nFor every $\\omega\\in\\Omega$, the assumption ensures that\n$\\mathop{\\mathrm{epi}}\\varphi_\\omega$ is closed and convex, and that\n$\\boldsymbol{\\mathsf{V}}_\\omega\\neq\\ensuremath{\\varnothing}$ . Thus\nyields $(\\forall\\omega\\in\\Omega)$\n$\\mathop{\\mathrm{epi}}\\varphi_\\omega=\\overline{\\boldsymbol{\\mathsf{V}}_\\omega}$.\n\n$\\Rightarrow$`<!-- -->`{=html}: Clear.\n\n: It results from that there exists a topology\n$\\widetilde{\\EuScript{T}_\\ensuremath{\\mathsf X}}$ on\n$\\ensuremath{\\mathsf X}$ such that $$\\label{e:8dy}\n\\EuScript{T}_\\ensuremath{\\mathsf X}\\subset\\widetilde{\\EuScript{T}_\\ensuremath{\\mathsf X}}$$\nand $$\\label{e:8dx}\n\\big(\\ensuremath{\\mathsf X},\\widetilde{\\EuScript{T}_\\ensuremath{\\mathsf X}}\\big)\\,\\,\n\\text{is a metrizable Souslin space}.$$ Set $(\\forall\\omega\\in\\Omega)$\n$\\boldsymbol{\\mathsf{V}}_\\omega=\\big\\{{(\\mathsf{x},\\xi)\\in\n\\ensuremath{\\mathsf X}\\times\\ensuremath{\\mathbb R}}~|~{\\varphi(\\omega,\\mathsf{x})<\\xi}\\big\\}$.\nThen, since implies that $$\\label{e:gf}\n(\\forall\\omega\\in\\Omega)\\quad\n\\varphi_\\omega\\,\\,\n\\text{is $\\widetilde{\\EuScript{T}_\\ensuremath{\\mathsf X}}$-continuous},$$\nit follows that $$\\label{e:r0d}\n(\\forall\\omega\\in\\Omega)\\quad\n\\boldsymbol{\\mathsf{V}}_\\omega\\in\n\\widetilde{\\EuScript{T}_\\ensuremath{\\mathsf X}}\\boxtimes\\EuScript{T}_\\ensuremath{\\mathbb R}\n\\quad\\text{and}\\quad\n\\overline{\\boldsymbol{\\mathsf{V}}_\\omega}^{\n\\widetilde{\\EuScript{T}_\\ensuremath{\\mathsf X}}\\boxtimes\\EuScript{T}_\\ensuremath{\\mathbb R}}\n=\\overline{\\mathop{\\mathrm{epi}}\\varphi_\\omega}^{\\widetilde{\\EuScript{T}_\\ensuremath{\\mathsf X}}\n\\boxtimes\\EuScript{T}_\\ensuremath{\\mathbb R}}\n=\\mathop{\\mathrm{epi}}\\varphi_\\omega.$$ On the other hand, we derive\nfrom , , and that the Borel $\\sigma$-algebra of\n$(\\ensuremath{\\mathsf X},\\widetilde{\\EuScript{T}_\\ensuremath{\\mathsf X}})$\nis $\\ensuremath{\\EuScript B}_\\ensuremath{\\mathsf X}$. Altogether,\napplying to the metrizable Souslin space\n$(\\ensuremath{\\mathsf X},\\widetilde{\\EuScript{T}_\\ensuremath{\\mathsf X}})$,\nwe deduce that $\\varphi$ is\n$\\ensuremath{\\EuScript F}\\otimes\\ensuremath{\\EuScript B}_\\ensuremath{\\mathsf X}$-measurable\nand that there exist sequences $(x_n)_{n\\in\\ensuremath{\\mathbb N}}$ in\n$\\mathcal{L}(\\Omega;\\ensuremath{\\mathsf X})$ and\n$(\\varrho_n)_{n\\in\\ensuremath{\\mathbb N}}$ in\n$\\mathcal{L}(\\Omega;\\ensuremath{\\mathbb R})$ such that\n$$(\\forall\\omega\\in\\Omega)\\quad\n\\big\\{\\big(x_n(\\omega),\\varrho_n(\\omega)\\big)\\big\\}_{n\\in\\ensuremath{\\mathbb N}}\n\\subset\\mathop{\\mathrm{epi}}\\varphi_\\omega\n\\quad\\text{and}\\quad\n\\overline{\\mathop{\\mathrm{epi}}\\varphi_\\omega}^{\n\\widetilde{\\EuScript{T}_\\ensuremath{\\mathsf X}}\\boxtimes\\EuScript{T}_\\ensuremath{\\mathbb R}}=\n\\overline{\\big\\{\\big(x_n(\\omega),\n\\varrho_n(\\omega)\\big)\\big\\}_{n\\in\\ensuremath{\\mathbb N}}}^{\n\\widetilde{\\EuScript{T}_\\ensuremath{\\mathsf X}}\\boxtimes\\EuScript{T}_\\ensuremath{\\mathbb R}}.$$\nHence, by and , $$\\begin{aligned}\n\\overline{\\big\\{\\big(x_n(\\omega),\n\\varrho_n(\\omega)\\big)\\big\\}_{n\\in\\ensuremath{\\mathbb N}}}\n\\supset\\overline{\\big\\{\\big(x_n(\\omega),\n\\varrho_n(\\omega)\\big)\\big\\}_{n\\in\\ensuremath{\\mathbb N}}}^{\n\\widetilde{\\EuScript{T}_\\ensuremath{\\mathsf X}}\\boxtimes\\EuScript{T}_\\ensuremath{\\mathbb R}}\n=\\overline{\\mathop{\\mathrm{epi}}\\varphi_\\omega}^{\n\\widetilde{\\EuScript{T}_\\ensuremath{\\mathsf X}}\\boxtimes\\EuScript{T}_\\ensuremath{\\mathbb R}}\n=\\mathop{\\mathrm{epi}}\\varphi_\\omega.\n\\end{aligned}$$ Consequently, $\\varphi$ is normal.\n\n: It follows from that\n$(\\ensuremath{\\mathsf Y}\\times\\ensuremath{\\mathbb R},\\EuScript{T}_\\ensuremath{\\mathsf Y}\\boxtimes\\EuScript{T}_\\ensuremath{\\mathbb R})$\nis a separable Fréchet space. Moreover, by ,\n$\\ensuremath{\\mathsf X}\\times\\ensuremath{\\mathbb R}=\n(\\ensuremath{\\mathsf Y}\\times\\ensuremath{\\mathbb R},\\EuScript{T}_\\ensuremath{\\mathsf Y}\\boxtimes\\EuScript{T}_\\ensuremath{\\mathbb R})^*$\nand the weak topology of\n$\\ensuremath{\\mathsf X}\\times\\ensuremath{\\mathbb R}$ is\n$\\EuScript{T}_\\ensuremath{\\mathsf X}\\boxtimes\\EuScript{T}_\\ensuremath{\\mathbb R}$.\nIn turn, arguing as in , we deduce that there exists a covering\n$(\\boldsymbol{\\mathsf{C}}_n)_{n\\in\\ensuremath{\\mathbb N}}$ of\n$\\ensuremath{\\mathsf X}\\times\\ensuremath{\\mathbb R}$, with respective\n$\\EuScript{T}_\\ensuremath{\\mathsf X}\\boxtimes\\EuScript{T}_\\ensuremath{\\mathbb R}$-induced\ntopologies\n$(\\EuScript{T}_{\\boldsymbol{\\mathsf{C}}_n})_{n\\in\\ensuremath{\\mathbb N}}$,\nsuch that, for every $n\\in\\ensuremath{\\mathbb N}$,\n$(\\boldsymbol{\\mathsf{C}}_n,\n\\EuScript{T}_{\\boldsymbol{\\mathsf{C}}_n})$ is a compact separable\nmetrizable space, hence a Polish space. We also introduce\n$$\\label{e:q0x}\n(\\forall n\\in\\ensuremath{\\mathbb N})\\quad\nQ_n\\colon\\Omega\\times\\boldsymbol{\\mathsf{C}}_n\\to\\Omega\\colon\n(\\omega,\\mathsf{x},\\xi)\\mapsto\\omega.$$ Note that, for every subset\n$\\boldsymbol{\\mathsf{C}}$ of\n$\\ensuremath{\\mathsf X}\\times\\ensuremath{\\mathbb R}$, $$\\label{e:rz2}\n\\big\\{{\\omega\\in\\Omega}~|~{\\boldsymbol{\\mathsf{C}}\n\\cap\\mathop{\\mathrm{epi}}\\varphi_\\omega\\neq\\ensuremath{\\varnothing}}\\big\\}\n=\\bigcup_{n\\in\\ensuremath{\\mathbb N}}\n\\big\\{{\\omega\\in\\Omega}~|~{\\boldsymbol{\\mathsf{C}}\\cap\n\\boldsymbol{\\mathsf{C}}_n\n\\cap\\mathop{\\mathrm{epi}}\\varphi_\\omega\\neq\\ensuremath{\\varnothing}}\\big\\}\n=\\bigcup_{n\\in\\ensuremath{\\mathbb N}}Q_n\\Big(\\boldsymbol{G}\\cap\n\\big(\\Omega\\times(\\boldsymbol{\\mathsf{C}}\\cap\n\\boldsymbol{\\mathsf{C}}_n)\\big)\\Big).$$\n\n: For every $n\\in\\ensuremath{\\mathbb N}$, set\n$$\\Omega_n=\\big\\{{\\omega\\in\\Omega}~|~{\n\\boldsymbol{\\mathsf{C}}_n\\cap\\mathop{\\mathrm{epi}}\\varphi_\\omega\\neq\\ensuremath{\\varnothing}}\\big\\},$$\ndenote by $\\ensuremath{\\EuScript F}_n$ the trace $\\sigma$-algebra of\n$\\ensuremath{\\EuScript F}$ on $\\Omega_n$, and observe that\n$$\\label{e:0f}\n\\Omega_n\\in\\ensuremath{\\EuScript F}\\quad\\text{and}\\quad\\ensuremath{\\EuScript F}_n\\subset\\ensuremath{\\EuScript F}.$$\nNow define $$\\label{e:kx}\n\\mathbb{K}=\\big\\{{n\\in\\ensuremath{\\mathbb N}}~|~{\\Omega_n\\neq\\ensuremath{\\varnothing}}\\big\\}\n\\quad\\text{and}\\quad\n(\\forall n\\in\\mathbb{K})\\;\\;\nK_n\\colon\\Omega_n\\to 2^{\\boldsymbol{\\mathsf{C}}_n}\\colon\n\\omega\\mapsto\\boldsymbol{\\mathsf{C}}_n\\cap\\mathop{\\mathrm{epi}}\\varphi_\\omega.$$\nThen $$\\label{e:ky}\n\\mathbb{K}\\neq\\ensuremath{\\varnothing}\\quad\\text{and}\\quad\n\\bigcup_{n\\in\\mathbb{K}}\\Omega_n=\\Omega.$$ Furthermore, the\n$\\EuScript{T}_\\ensuremath{\\mathsf X}\\boxtimes\\EuScript{T}_\\ensuremath{\\mathbb R}$-closedness\nof $(\\mathop{\\mathrm{epi}}\\varphi_\\omega)_{\\omega\\in\\Omega}$ guarantees\nthat $$(\\forall n\\in\\mathbb{K})(\\forall\\omega\\in\\Omega)\\quad\nK_n(\\omega)\\;\\text{is\n$\\EuScript{T}_{\\boldsymbol{\\mathsf{C}}_n}$-closed}.$$ On the other hand,\nfor every $n\\in\\mathbb{K}$ and every closed subset\n$\\boldsymbol{\\mathsf{D}}$ of $(\\boldsymbol{\\mathsf{C}}_n,\n\\EuScript{T}_{\\boldsymbol{\\mathsf{C}}_n})$, there exists a closed subset\n$\\boldsymbol{\\mathsf{E}}$ of\n$(\\ensuremath{\\mathsf X}\\times\\ensuremath{\\mathbb R},\\EuScript{T}_\\ensuremath{\\mathsf X}\\boxtimes\\EuScript{T}_\\ensuremath{\\mathbb R})$\nsuch that $\\boldsymbol{\\mathsf{D}}=\\boldsymbol{\\mathsf{C}}_n\\cap\n\\boldsymbol{\\mathsf{E}}$ and therefore, since\n$\\boldsymbol{\\mathsf{C}}_n$ is\n$\\EuScript{T}_\\ensuremath{\\mathsf X}\\boxtimes\\EuScript{T}_\\ensuremath{\\mathbb R}$-closed,\nwe deduce from that\n$$\\big\\{{\\omega\\in\\Omega_n}~|~{\\boldsymbol{\\mathsf{D}}\\cap K_n(\\omega)\n\\neq\\ensuremath{\\varnothing}}\\big\\}\n=\\Omega_n\\cap\\big\\{{\\omega\\in\\Omega}~|~{\n\\boldsymbol{\\mathsf{C}}_n\\cap\\boldsymbol{\\mathsf{E}}\n\\cap\\mathop{\\mathrm{epi}}\\varphi_\\omega\\neq\\ensuremath{\\varnothing}}\\big\\}\n\\in\\ensuremath{\\EuScript F}_n.$$ Hence, for every $n\\in\\mathbb{K}$,\nsince $(\\boldsymbol{\\mathsf{C}}_n,\n\\EuScript{T}_{\\boldsymbol{\\mathsf{C}}_n})$ is a Polish space, we deduce\nfrom that there exist measurable mappings $\\boldsymbol{y}_n$ and\n$(\\boldsymbol{z}_{n,k})_{k\\in\\ensuremath{\\mathbb N}}$ from\n$(\\Omega_n,\\ensuremath{\\EuScript F}_n)$ to\n$(\\boldsymbol{\\mathsf{C}}_n,\\ensuremath{\\EuScript B}_{\\boldsymbol{\\mathsf{C}}_n})$\nsuch that $$\\label{e:tp}\n(\\forall\\omega\\in\\Omega_n)\\quad\n\\boldsymbol{y}_n(\\omega)\\in K_n(\\omega)\\quad\\text{and}\\quad\nK_n(\\omega)\n=\\overline{\\big\\{\\boldsymbol{z}_{n,k}(\\omega)\\big\\}_{\nk\\in\\ensuremath{\\mathbb N}}}^{\\EuScript{T}_{\\boldsymbol{\\mathsf{C}}_n}}\n=\\boldsymbol{\\mathsf{C}}_n\\cap\n\\overline{\\big\\{\\boldsymbol{z}_{n,k}(\\omega)\\big\\}_{k\\in\\ensuremath{\\mathbb N}}}.$$\nIn addition, since asserts that $$\\begin{aligned}\n&\n(\\forall n\\in\\mathbb{K})\\quad\n\\big\\{{(\\omega,\\mathsf{x},\\xi)\\in\n\\Omega_n\\times\\boldsymbol{\\mathsf{C}}_n}~|~{\n(\\mathsf{x},\\xi)\\in\\boldsymbol{\\mathsf{C}}_n\\cap\\mathop{\\mathrm{epi}}\\varphi_\\omega}\\big\\}\n\\nonumber\\\\\n&\\hskip 26mm\n=\\big\\{{(\\omega,\\mathsf{x},\\xi)\\in\n\\Omega_n\\times\\boldsymbol{\\mathsf{C}}_n}~|~{\n(\\mathsf{x},\\xi)\\in K_n(\\omega)}\\big\\}\n\\nonumber\\\\\n&\\hskip 26mm\n\\in\\ensuremath{\\EuScript F}_n\\otimes\\ensuremath{\\EuScript B}_{\\boldsymbol{\\mathsf{C}}_n}\n\\nonumber\\\\\n&\\hskip 26mm\n\\subset\\ensuremath{\\EuScript F}\\otimes\\ensuremath{\\EuScript B}_{\\ensuremath{\\mathsf X}\\times\\ensuremath{\\mathbb R}},\n\\end{aligned}$$ we get from that $$\\boldsymbol{G}\n=\\bigcup_{n\\in\\mathbb{K}}\\big\\{{(\\omega,\\mathsf{x},\\xi)\\in\n\\Omega_n\\times\\boldsymbol{\\mathsf{C}}_n}~|~{\n(\\mathsf{x},\\xi)\\in\\boldsymbol{\\mathsf{C}}_n\\cap\\mathop{\\mathrm{epi}}\\varphi_\\omega}\\big\\}\n\\in\\ensuremath{\\EuScript F}\\otimes\\ensuremath{\\EuScript B}_{\\ensuremath{\\mathsf X}\\times\\ensuremath{\\mathbb R}}.$$\nThus, in the light of , $\\varphi$ is\n$\\ensuremath{\\EuScript F}\\otimes\\ensuremath{\\EuScript B}_\\ensuremath{\\mathsf X}$-measurable.\nNext, using , we construct a family $(\\Theta_n)_{n\\in\\mathbb{K}}$ of\npairwise disjoint sets in $\\ensuremath{\\EuScript F}$ such that\n$$\\label{e:yf}\n\\Theta_{\\min\\mathbb{K}}=\\Omega_{\\min\\mathbb{K}},\n\\quad\n\\bigcup_{n\\in\\mathbb{K}}\\Theta_n=\\Omega,\n\\quad\\text{and}\\quad\n(\\forall n\\in\\mathbb{K})\\;\\;\\Theta_n\\subset\\Omega_n.$$ In turn, for\nevery $\\omega\\in\\Omega$, there exists a unique\n$\\ell_\\omega\\in\\mathbb{K}$ such that $\\omega\\in\\Theta_{\\ell_\\omega}$.\nTherefore, appealing to , the mapping $$\\label{e:8uy}\n\\boldsymbol{y}\\colon\\Omega\\to\\ensuremath{\\mathsf X}\\times\\ensuremath{\\mathbb R}\\colon\n\\omega\\mapsto\\boldsymbol{y}_{\\ell_\\omega}(\\omega)$$ is well defined and,\nin view of , $$\\label{e:z0}\n(\\forall\\omega\\in\\Omega)\\quad\n\\boldsymbol{y}(\\omega)\n=\\boldsymbol{y}_{\\ell_\\omega}(\\omega)\n\\in K_{\\ell_\\omega}(\\omega)\n\\subset\n\\mathop{\\mathrm{epi}}\\varphi_\\omega.$$ Let $\\boldsymbol{\\mathsf{V}}\\in\n\\EuScript{T}_\\ensuremath{\\mathsf X}\\boxtimes\\EuScript{T}_\\ensuremath{\\mathbb R}$.\nThen, for every $n\\in\\mathbb{K}$,\n$\\boldsymbol{\\mathsf{V}}\\cap\\boldsymbol{\\mathsf{C}}_n$ is\n$\\EuScript{T}_{\\boldsymbol{\\mathsf{C}}_n}$-open and thus the\nmeasurability of\n$\\boldsymbol{y}_n\\colon(\\Omega_n,\\ensuremath{\\EuScript F}_n)\\to\n(\\boldsymbol{\\mathsf{C}}_n,\\ensuremath{\\EuScript B}_{\\boldsymbol{\\mathsf{C}}_n})$\nand ensure that $\\boldsymbol{y}_n^{-1}(\n\\boldsymbol{\\mathsf{V}}\\cap\\boldsymbol{\\mathsf{C}}_n)\n\\in\\ensuremath{\\EuScript F}_n\\subset\\ensuremath{\\EuScript F}$. Hence, we\ninfer from , , and that $$\\begin{aligned}\n\\boldsymbol{y}^{-1}(\\boldsymbol{\\mathsf{V}})\n&=\\bigcup_{n\\in\\mathbb{K}}\\big\\{{\\omega\\in\\Theta_n}~|~{\n\\boldsymbol{y}(\\omega)\\in\\boldsymbol{\\mathsf{V}}}\\big\\}\n\\nonumber\\\\\n&=\\bigcup_{n\\in\\mathbb{K}}\\big\\{{\\omega\\in\\Theta_n}~|~{\n\\boldsymbol{y}_n(\\omega)\\in\\boldsymbol{\\mathsf{C}}_n\\cap\n\\boldsymbol{\\mathsf{V}}}\\big\\}\n\\nonumber\\\\\n&=\\bigcup_{n\\in\\mathbb{K}}\\big(\\Theta_n\\cap\n\\boldsymbol{y}_n^{-1}(\n\\boldsymbol{\\mathsf{C}}_n\\cap\\boldsymbol{\\mathsf{V}})\\big)\n\\nonumber\\\\\n&\\in\\ensuremath{\\EuScript F}.\n\\end{aligned}$$ This verifies that\n$\\boldsymbol{y}\\colon(\\Omega,\\ensuremath{\\EuScript F})\\to\n(\\ensuremath{\\mathsf X}\\times\\ensuremath{\\mathbb R},\\ensuremath{\\EuScript B}_{\\ensuremath{\\mathsf X}\\times\\ensuremath{\\mathbb R}})$\nis measurable. We now define $$\\label{e:xcd}\n(\\forall n\\in\\mathbb{K})(\\forall k\\in\\ensuremath{\\mathbb N})\\quad\n\\boldsymbol{x}_{n,k}\\colon\\Omega\\to\\ensuremath{\\mathsf X}\\times\\ensuremath{\\mathbb R}\\colon\n\\omega\\mapsto\n\\begin{cases}\n\\boldsymbol{z}_{n,k}(\\omega),&\\text{if}\\,\\,\\omega\\in\\Omega_n;\\\\\n\\boldsymbol{y}(\\omega),\n&\\text{if}\\,\\,\\omega\\in\\complement\\Omega_n.\n\\end{cases}$$ It results from that\n$(\\boldsymbol{x}_{n,k})_{n\\in\\mathbb{K},k\\in\\ensuremath{\\mathbb N}}$ are\nmeasurable mappings from $(\\Omega,\\ensuremath{\\EuScript F})$ to\n$(\\ensuremath{\\mathsf X}\\times\\ensuremath{\\mathbb R},\\ensuremath{\\EuScript B}_{\\ensuremath{\\mathsf X}\\times\\ensuremath{\\mathbb R}})$.\nFurthermore, and give $$\\label{e:sd9}\n(\\forall n\\in\\mathbb{K})(\\forall k\\in\\ensuremath{\\mathbb N})\n(\\forall\\omega\\in\\Omega)\\quad\n\\boldsymbol{x}_{n,k}(\\omega)\\in\\mathop{\\mathrm{epi}}\\varphi_\\omega.$$\nFix $\\omega\\in\\Omega$ and let\n$\\boldsymbol{\\mathsf{x}}\\in\\mathop{\\mathrm{epi}}\\varphi_\\omega$. Since\n$\\bigcup_{n\\in\\mathbb{K}}(\\boldsymbol{\\mathsf{C}}_n\\cap\n\\mathop{\\mathrm{epi}}\\varphi_\\omega)=\\mathop{\\mathrm{epi}}\\varphi_\\omega$,\nthere exists $N\\in\\mathbb{K}$ such that $\\omega\\in\\Omega_N$ and\n$\\boldsymbol{\\mathsf{x}}\\in\\boldsymbol{\\mathsf{C}}_N\n\\cap\\mathop{\\mathrm{epi}}\\varphi_\\omega=K_N(\\omega)$. Thus, it results\nfrom and that $$\\boldsymbol{\\mathsf{x}}\n\\in\\overline{\\big\\{\\boldsymbol{z}_{N,k}(\\omega)\\big\\}_{k\\in\\ensuremath{\\mathbb N}}}\n=\\overline{\\big\\{\\boldsymbol{x}_{N,k}(\\omega)\\big\\}_{k\\in\\ensuremath{\\mathbb N}}}\n\\subset\\overline{\\big\\{\\boldsymbol{x}_{n,k}(\\omega)\\big\\}_{\nn\\in\\mathbb{K},k\\in\\ensuremath{\\mathbb N}}}.$$ Therefore, since\n$\\mathop{\\mathrm{epi}}\\varphi_\\omega$ is closed, it follows from and\nthat $$\\mathop{\\mathrm{epi}}\\varphi_\\omega\n=\\overline{\\big\\{\\boldsymbol{x}_{n,k}(\\omega)\\big\\}_{\nn\\in\\mathbb{K},k\\in\\ensuremath{\\mathbb N}}}.$$ At the same time, for\nevery $n\\in\\mathbb{K}$ and every $k\\in\\ensuremath{\\mathbb N}$, since\n$\\ensuremath{\\EuScript B}_{\\ensuremath{\\mathsf X}\\times\\ensuremath{\\mathbb R}}=\\ensuremath{\\EuScript B}_\\ensuremath{\\mathsf X}\\otimes\\ensuremath{\\EuScript B}_\\ensuremath{\\mathbb R}$\nand since\n$\\boldsymbol{x}_{n,k}\\colon(\\Omega,\\ensuremath{\\EuScript F})\\to\n(\\ensuremath{\\mathsf X}\\times\\ensuremath{\\mathbb R},\\ensuremath{\\EuScript B}_{\\ensuremath{\\mathsf X}\\times\\ensuremath{\\mathbb R}})$\nis measurable, there exist\n$x_{n,k}\\in\\mathcal{L}(\\Omega;\\ensuremath{\\mathsf X})$ and\n$\\varrho_{n,k}\\in\\mathcal{L}(\\Omega;\\ensuremath{\\mathbb R})$ such that\n$(\\forall\\omega\\in\\Omega)$ $\\boldsymbol{x}_{n,k}(\\omega)\n=(x_{n,k}(\\omega),\\varrho_{n,k}(\\omega))$. Altogether, $\\varphi$ is\nnormal.\n\n$\\Rightarrow$`<!-- -->`{=html}: Let $\\boldsymbol{\\mathsf{C}}$ be a\nnonempty closed subset of\n$(\\ensuremath{\\mathsf X}\\times\\ensuremath{\\mathbb R},\\EuScript{T}_\\ensuremath{\\mathsf X}\\boxtimes\\EuScript{T}_\\ensuremath{\\mathbb R})$.\nNote that the lower semicontinuity of $\\varphi$ ensures that\n$\\boldsymbol{G}$ is closed. For every $n\\in\\ensuremath{\\mathbb N}$,\nsince $\\boldsymbol{G}\\cap(\\Omega\\times(\\boldsymbol{\\mathsf{C}}\\cap\n\\boldsymbol{\\mathsf{C}}_n))$ is closed in\n$(\\Omega\\times\\boldsymbol{\\mathsf{C}}_n,\n\\EuScript{T}_\\Omega\\boxtimes\n\\EuScript{T}_{\\boldsymbol{\\mathsf{C}}_n})$, it follows from and that\n$Q_n(\\boldsymbol{G}\\cap(\\Omega\\times(\\boldsymbol{\\mathsf{C}}\\cap\n\\boldsymbol{\\mathsf{C}}_n)))$ is closed in\n$(\\Omega,\\EuScript{T}_\\Omega)$ and, therefore, that it belongs to\n$\\ensuremath{\\EuScript B}_\\Omega=\\ensuremath{\\EuScript F}$. Thus, by ,\n$\\big\\{{\\omega\\in\\Omega}~|~{\\boldsymbol{\\mathsf{C}}\n\\cap\\mathop{\\mathrm{epi}}\\varphi_\\omega\\neq\\ensuremath{\\varnothing}}\\big\\}\\in\\ensuremath{\\EuScript F}$.\n\n$\\Rightarrow$`<!-- -->`{=html}: There exists a topology\n$\\widetilde{\\EuScript{T}_\\Omega}$ on $\\Omega$ such that\n$$\\EuScript{T}_\\Omega\\subset\\widetilde{\\EuScript{T}_\\Omega}\n\\,\\,\\text{and}\\,\\,\n\\big(\\Omega,\\widetilde{\\EuScript{T}_\\Omega}\\big)\\,\\,\n\\text{is a Polish space}.$$ In addition, by , the Borel $\\sigma$-algebra\nof $(\\Omega,\\widetilde{\\EuScript{T}_\\Omega})$ is\n$\\ensuremath{\\EuScript B}_\\Omega=\\ensuremath{\\EuScript F}$. Let\n$\\boldsymbol{\\mathsf{C}}$ be a closed subset of\n$(\\ensuremath{\\mathsf X}\\times\\ensuremath{\\mathbb R},\\EuScript{T}_\\ensuremath{\\mathsf X}\\boxtimes\\EuScript{T}_\\ensuremath{\\mathbb R})$\nand fix temporarily $n\\in\\ensuremath{\\mathbb N}$. Since the\n$\\ensuremath{\\EuScript F}\\otimes\\ensuremath{\\EuScript B}_\\ensuremath{\\mathsf X}$-measurability\nof $\\varphi$ and ensure that\n$\\boldsymbol{G}\\in\\ensuremath{\\EuScript F}\\otimes\\ensuremath{\\EuScript B}_{\\ensuremath{\\mathsf X}\\times\\ensuremath{\\mathbb R}}$,\nwe have $\\boldsymbol{G}\\cap(\\Omega\\times(\\boldsymbol{\\mathsf{C}}\\cap\n\\boldsymbol{\\mathsf{C}}_n))\n=\\boldsymbol{G}\\cap(\\Omega\\times\\boldsymbol{\\mathsf{C}})\\cap\n(\\Omega\\times\\boldsymbol{\\mathsf{C}}_n)\n\\in\\ensuremath{\\EuScript B}_{\\Omega\\times\\boldsymbol{\\mathsf{C}}_n}$. At\nthe same time, for every $\\omega\\in\\Omega$, $$\\begin{aligned}\n&\\big\\{{(\\mathsf{x},\\xi)\\in\\ensuremath{\\mathsf X}\\times\\ensuremath{\\mathbb R}}~|~{\n(\\omega,\\mathsf{x},\\xi)\\in\n\\boldsymbol{G}\\cap\\big(\\Omega\\times(\\boldsymbol{\\mathsf{C}}\\cap\n\\boldsymbol{\\mathsf{C}}_n)\\big)}\\big\\}\n\\nonumber\\\\\n&\\hskip 26mm\n=\\big\\{{(\\mathsf{x},\\xi)\\in\\ensuremath{\\mathsf X}\\times\\ensuremath{\\mathbb R}}~|~{(\\mathsf{x},\\xi)\\in\n\\boldsymbol{\\mathsf{C}}\\cap\\boldsymbol{\\mathsf{C}}_n\n\\,\\,\\text{and}\\,\\,\n(\\mathsf{x},\\xi)\\in\\mathop{\\mathrm{epi}}\\varphi_\\omega}\\big\\},\n\\nonumber\\\\\n&\\hskip 26mm\n=\\boldsymbol{\\mathsf{C}}\\cap\\boldsymbol{\\mathsf{C}}_n\n\\cap\\mathop{\\mathrm{epi}}\\varphi_\\omega\n\\end{aligned}$$ is a closed subset of the compact space\n$(\\boldsymbol{\\mathsf{C}}_n,\n\\EuScript{T}_{\\boldsymbol{\\mathsf{C}}_n})$. In turn, since\n$(\\Omega,\\widetilde{\\EuScript{T}_\\Omega})$ and\n$(\\boldsymbol{\\mathsf{C}}_n,\n\\EuScript{T}_{\\boldsymbol{\\mathsf{C}}_n})$ are Polish spaces, guarantees\nthat $Q_n(\\boldsymbol{G}\\cap(\\Omega\\times(\\boldsymbol{\\mathsf{C}}\\cap\n\\boldsymbol{\\mathsf{C}}_n)))\\in\\ensuremath{\\EuScript B}_\\Omega=\\ensuremath{\\EuScript F}$.\nConsequently, we infer from that\n$\\big\\{{\\omega\\in\\Omega}~|~{\\boldsymbol{\\mathsf{C}}\n\\cap\\mathop{\\mathrm{epi}}\\varphi_\\omega\\neq\\ensuremath{\\varnothing}}\\big\\}\\in\\ensuremath{\\EuScript F}$.\n\n: Let $(\\ensuremath{\\mathsf Y},\\EuScript{T}_\\ensuremath{\\mathsf Y})$ be\nthe strong dual of $\\ensuremath{\\mathsf X}$. Then\n$(\\ensuremath{\\mathsf Y},\\EuScript{T}_\\ensuremath{\\mathsf Y})$ is a\nseparable reflexive Banach space. Consequently, follows from , and\nfollows from .\n\n$\\Rightarrow$`<!-- -->`{=html}: Let $\\EuScript{T}_\\Omega$ be the\ntopology on $\\Omega$ induced by the standard topology on\n$\\ensuremath{\\mathbb R}^M$. By , $(\\Omega,\\EuScript{T}_\\Omega)$ is a\nLusin space.\n\n: The lower semicontinuity of $(\\varphi_\\omega)_{\\omega\\in\\Omega}$\nensures that the sets\n$(\\mathop{\\mathrm{epi}}\\varphi_\\omega)_{\\omega\\in\\Omega}$ are closed.\nHence, since $(\\ensuremath{\\mathsf X}\\times\\ensuremath{\\mathbb R},\n\\EuScript{T}_\\ensuremath{\\mathsf X}\\boxtimes\\EuScript{T}_\\ensuremath{\\mathbb R})$\nis a Polish space, and yield\n$\\boldsymbol{G}\\in\\ensuremath{\\EuScript F}\\otimes\\ensuremath{\\EuScript B}_{\\ensuremath{\\mathsf X}\\times\\ensuremath{\\mathbb R}}$.\nTherefore, by , $\\varphi$ is\n$\\ensuremath{\\EuScript F}\\otimes\\ensuremath{\\EuScript B}_\\ensuremath{\\mathsf X}$-measurable.\nConsequently, we deduce the assertion from .\n\n$\\Rightarrow$`<!-- -->`{=html}: This follows from .\n\n: The $\\ensuremath{\\EuScript B}_\\ensuremath{\\mathsf X}$-measurability of\n$\\mathsf{f}$ implies that $\\varphi$ is\n$\\ensuremath{\\EuScript F}\\otimes\\ensuremath{\\EuScript B}_\\ensuremath{\\mathsf X}$-measurable.\nAt the same time, since\n$(\\ensuremath{\\mathsf X}\\times\\ensuremath{\\mathbb R},\n\\EuScript{T}_\\ensuremath{\\mathsf X}\\boxtimes\\EuScript{T}_\\ensuremath{\\mathbb R})$\nis a Souslin space, we deduce from that there exists a sequence\n$\\{(\\mathsf{x}_n,\\xi_n)\\}_{n\\in\\ensuremath{\\mathbb N}}$ in\n$\\mathop{\\mathrm{epi}}\\mathsf{f}$ such that\n$\\overline{\\{(\\mathsf{x}_n,\\xi_n)\\}_{n\\in\\ensuremath{\\mathbb N}}}\n=\\overline{\\mathop{\\mathrm{epi}}\\mathsf{f}}$. Altogether, upon setting\n$$(\\forall n\\in\\ensuremath{\\mathbb N})\\quad\nx_n\\colon\\Omega\\to\\ensuremath{\\mathsf X}\\colon\\omega\\mapsto\\mathsf{x}_n\n\\quad\\text{and}\\quad\n\\varrho_n\\colon\\Omega\\to\\ensuremath{\\mathbb R}\\colon\\omega\\mapsto\\xi_n,$$\nwe conclude that $\\varphi$ is normal.\n\n# Interchange rules with compliant spaces and normal integrands\n\nThe main result of this section is the following interchange theorem,\nwhich brings together the abstract principle of Theorem , the notion of\ncompliance of Definition , and the notion of normality of Definition .\n\n**Theorem 8**.  * Suppose that Assumption  holds, that\n$\\ensuremath{\\mathcal X}$ is compliant, and that $\\varphi$ is normal.\nThen $$\\label{e:311}\n\\inf_{x\\in\\ensuremath{\\mathcal X}}\\int_\\Omega\\varphi\\big(\\omega,x(\\omega)\\big)\n\\mu(d\\omega)=\n\\int_\\Omega\\inf_{\\mathsf{x}\\in\\ensuremath{\\mathsf X}}\\varphi(\\omega,\\mathsf{x})\\,\n\\mu(d\\omega).$$*\n\n. We apply Theorem . By virtue of the normality of $\\varphi$, per\nDefinition , we choose sequences $(z_n)_{n\\in\\ensuremath{\\mathbb N}}$ in\n$\\mathcal{L}(\\Omega;\\ensuremath{\\mathsf X})$ and\n$(\\vartheta_n)_{n\\in\\ensuremath{\\mathbb N}}$ in\n$\\mathcal{L}(\\Omega;\\ensuremath{\\mathbb R})$ such that $$\\label{e:n0d}\n(\\forall\\omega\\in\\Omega)\\quad\n\\big\\{\\big(z_n(\\omega),\\vartheta_n(\\omega)\\big)\\big\\}_{n\\in\\ensuremath{\\mathbb N}}\n\\subset\\mathop{\\mathrm{epi}}\\varphi_\\omega\n\\quad\\text{and}\\quad\n\\overline{\\mathop{\\mathrm{epi}}\\varphi_\\omega}=\\overline{\\big\\{\\big(z_n(\\omega),\n\\vartheta_n(\\omega)\\big)\\big\\}_{n\\in\\ensuremath{\\mathbb N}}}.$$ On the\nother hand, Assumption  ensures that $(\\forall\\omega\\in\\Omega)$\n$\\inf\\varphi(\\omega,\\ensuremath{\\mathsf X})<\\ensuremath{{{+}\\infty}}$.\nNow fix $\\omega\\in\\Omega$ and let\n$\\xi\\in\\left]\\inf\\varphi(\\omega,\\ensuremath{\\mathsf X}),\\ensuremath{{{+}\\infty}}\\right[$.\nThen there exits $\\mathsf{x}\\in\\ensuremath{\\mathsf X}$ such that\n$(\\mathsf{x},\\xi)\\in\\mathop{\\mathrm{epi}}\\varphi_\\omega$. Thus, in view\nof , we obtain a subnet $(\\vartheta_{k(b)}(\\omega))_{b\\in B}$ of\n$(\\vartheta_n(\\omega))_{n\\in\\ensuremath{\\mathbb N}}$ such that\n$\\vartheta_{k(b)}(\\omega)\\to\\xi$. On the other hand,\n$$(\\forall b\\in B)\\quad\n\\inf\\varphi(\\omega,\\ensuremath{\\mathsf X})\n\\ensuremath{\\leqslant}\\inf_{n\\in\\ensuremath{\\mathbb N}}\\varphi\\big(\\omega,z_n(\\omega)\\big)\n\\ensuremath{\\leqslant}\\varphi\\big(\\omega,z_{k(b)}(\\omega)\\big)\n\\ensuremath{\\leqslant}\\vartheta_{k(b)}(\\omega).$$ Hence\n$\\inf\\varphi(\\omega,\\ensuremath{\\mathsf X})\n\\ensuremath{\\leqslant}\\inf_{n\\in\\ensuremath{\\mathbb N}}\\varphi(\\omega,z_n(\\omega))\n\\ensuremath{\\leqslant}\\xi$. In turn, letting\n$\\xi\\downarrow\\inf\\varphi(\\omega,\\ensuremath{\\mathsf X})$ yields\n$\\inf\\varphi(\\omega,\\ensuremath{\\mathsf X})=\n\\inf_{n\\in\\ensuremath{\\mathbb N}}\\varphi(\\omega,z_n(\\omega))$.\nTherefore, property  in Theorem  is satisfied with\n$(\\forall n\\in\\ensuremath{\\mathbb N})$ $x_n=z_n-\\overline{x}$. At the\nsame time, property  in Theorem  follows from Assumption  and the\ncompliance of $\\ensuremath{\\mathcal X}$. Finally, since the functions\n$(\\varphi({\\mkern 2mu\\cdot\\mkern 2mu},z_n({\\mkern 2mu\\cdot\\mkern 2mu})))_{n\\in\\ensuremath{\\mathbb N}}$\nare $\\ensuremath{\\EuScript F}$-measurable by Assumption , so is\n$\\inf_{n\\in\\ensuremath{\\mathbb N}}\\varphi({\\mkern 2mu\\cdot\\mkern 2mu},z_n({\\mkern 2mu\\cdot\\mkern 2mu}))=\\inf\\varphi({\\mkern 2mu\\cdot\\mkern 2mu},\\ensuremath{\\mathsf X})$.\n\nIn the remainder of this section, we construct new scenarios for the\nvalidity of the interchange rule as instantiations of Theorem .\n\n. Note that Assumption  is satisfied. Hence, the assertion follows from\nProposition  and Theorem .\n\n. Combine Theorem  and Theorem .\n\nWhen specialized to probability in separable Banach spaces, Theorem \nyields conditions for the interchange of infimization and expectation.\nHere is an illustration.\n\n. This is a special case of Example .\n\n. We deduce from Assumption  and Theorem  that $\\varphi$ is normal.\nThus, the conclusion follows from Theorem .\n\nAn important realization of Example  is the case of Carathéodory\nintegrands.\n\n. Since $(\\ensuremath{\\mathsf X},\\EuScript{T}_\\ensuremath{\\mathsf X})$\nis a Souslin topological vector space, implies that it is a regular\nSouslin space. Thus, we deduce from Theorem  that $\\varphi$ is normal\nand, in particular, it is\n$\\ensuremath{\\EuScript F}\\otimes\\ensuremath{\\EuScript B}_\\ensuremath{\\mathsf X}$-measurable.\nHence, Assumption  is satisfied. Consequently, Example  yields the\nconclusion.\n\n# Interchanging convex-analytical operations and integration\n\nWe put the interchange principle of Theorem , compliance, and normality\nin action to evaluate convex-analytical objects associated with integral\nfunctions, namely conjugate functions, subdifferential operators,\nrecession functions, Moreau envelopes, and proximity operators. This\nanalysis results in new interchange rules for the convex calculus of\nintegral functions. Throughout this section, we adopt the following\nnotation.\n\nWe shall require the following result. Its item appears in in the\nspecial case when $(\\Omega,\\ensuremath{\\EuScript F},\\mu)$ is complete.\n\n**Lemma 9**.  * Let $(\\Omega,\\ensuremath{\\EuScript F},\\mu)$ be a\n$\\sigma$-finite measure space such that $\\mu(\\Omega)\\neq 0$, let\n$(\\ensuremath{\\mathsf X},\\EuScript{T}_\\ensuremath{\\mathsf X})$ be a\nSouslin locally convex real topological vector space, and let\n$(\\ensuremath{\\mathsf Y},\\EuScript{T}_\\ensuremath{\\mathsf Y})$ be a\nseparable locally convex real topological vector space. Suppose that\n$\\ensuremath{\\mathsf X}$ and $\\ensuremath{\\mathsf Y}$ are placed in\nseparating duality via a bilinear form\n$\\langle{{{\\mkern 2mu\\cdot\\mkern 2mu}},{{\\mkern 2mu\\cdot\\mkern 2mu}}}\\rangle_{\\ensuremath{\\mathsf X},\\ensuremath{\\mathsf Y}}\\colon\\ensuremath{\\mathsf X}\\times\\ensuremath{\\mathsf Y}\\to\\ensuremath{\\mathbb R}$\nwith which $\\EuScript{T}_\\ensuremath{\\mathsf X}$ and\n$\\EuScript{T}_\\ensuremath{\\mathsf Y}$ are compatible. Then the following\nhold:*\n\n1.  *\n    $\\langle{{{\\mkern 2mu\\cdot\\mkern 2mu}},{{\\mkern 2mu\\cdot\\mkern 2mu}}}\\rangle_{\\ensuremath{\\mathsf X},\\ensuremath{\\mathsf Y}}\\colon(\\ensuremath{\\mathsf X}\\times\\ensuremath{\\mathsf Y},\n    \\ensuremath{\\EuScript B}_\\ensuremath{\\mathsf X}\\otimes\\ensuremath{\\EuScript B}_\\ensuremath{\\mathsf Y})\\to\\ensuremath{\\mathbb R}$\n    is measurable.*\n\n2.  * Let\n    $\\ensuremath{\\mathcal X}\\subset\\mathcal{L}(\\Omega;\\ensuremath{\\mathsf X})$\n    and\n    $\\ensuremath{\\mathcal Y}\\subset\\mathcal{L}(\\Omega;\\ensuremath{\\mathsf Y})$\n    be vector subspaces such that the following are satisfied:*\n\n    1.  *\n        $(\\forall x\\in\\ensuremath{\\mathcal X})(\\forall y\\in\\ensuremath{\\mathcal Y})$\n        $\\int_\\Omega|\\langle{{x(\\omega)},{y(\\omega)}}\\rangle_{\\ensuremath{\\mathsf X},\\ensuremath{\\mathsf Y}}|\n        \\mu(d\\omega)<\\ensuremath{{{+}\\infty}}$.*\n\n    2.  *\n        $\\bigcup_{\\mathsf{x}\\in\\ensuremath{\\mathsf X}}\\big\\{{1_A\\mathsf{x}}~|~{A\\in\\ensuremath{\\EuScript F}\\,\\,\n        \\text{and}\\,\\,\\mu(A)<\\ensuremath{{{+}\\infty}}}\\big\\}\\subset\\ensuremath{\\mathcal X}$.*\n\n    3.  *\n        $\\bigcup_{\\mathsf{y}\\in\\ensuremath{\\mathsf Y}}\\big\\{{1_A\\mathsf{y}}~|~{A\\in\\ensuremath{\\EuScript F}\\,\\,\n        \\text{and}\\,\\,\\mu(A)<\\ensuremath{{{+}\\infty}}}\\big\\}\\subset\\ensuremath{\\mathcal Y}$.*\n\n    *Then $\\widetilde{\\ensuremath{\\mathcal X}}$ and\n    $\\widetilde{\\ensuremath{\\mathcal Y}}$ are in separating duality via\n    the bilinear form\n    $\\langle{{{\\mkern 2mu\\cdot\\mkern 2mu}},{{\\mkern 2mu\\cdot\\mkern 2mu}}}\\rangle$\n    defined by $$\\label{e:62}\n    (\\forall\\widetilde{x}\\in\\ensuremath{\\widetilde{\\mathcal X}})(\\forall\\widetilde{y}\\in\\ensuremath{\\widetilde{\\mathcal Y}})\\quad\n    \\langle{{\\widetilde{x}},{\\widetilde{y}}}\\rangle=\n    \\int_\\Omega\\big\\langle{{x(\\omega)},{y(\\omega)}}\\big\\rangle_{\\ensuremath{\\mathsf X},\\ensuremath{\\mathsf Y}}\\mu(d\\omega).$$*\n\n. : We deduce from that\n$(\\ensuremath{\\mathsf X},\\EuScript{T}_\\ensuremath{\\mathsf X})$ is a\nregular Souslin space. On the other hand, since\n$\\EuScript{T}_\\ensuremath{\\mathsf Y}$ and\n$\\EuScript{T}_\\ensuremath{\\mathsf X}$ are compatible with\n$\\langle{{{\\mkern 2mu\\cdot\\mkern 2mu}},{{\\mkern 2mu\\cdot\\mkern 2mu}}}\\rangle_{\\ensuremath{\\mathsf X},\\ensuremath{\\mathsf Y}}$,\nthe functions\n$(\\langle{{\\mathsf{x}},{{\\mkern 2mu\\cdot\\mkern 2mu}}}\\rangle_{\\ensuremath{\\mathsf X},\\ensuremath{\\mathsf Y}})_{\\mathsf{x}\\in\\ensuremath{\\mathsf X}}$\nare $\\ensuremath{\\EuScript B}_\\ensuremath{\\mathsf Y}$-measurable and the\nfunctions\n$(\\langle{{{\\mkern 2mu\\cdot\\mkern 2mu}},{\\mathsf{y}}}\\rangle_{\\ensuremath{\\mathsf X},\\ensuremath{\\mathsf Y}})_{\\mathsf{y}\\in\\ensuremath{\\mathsf Y}}$\nare continuous. Hence, Theorem  implies that\n$\\langle{{{\\mkern 2mu\\cdot\\mkern 2mu}},{{\\mkern 2mu\\cdot\\mkern 2mu}}}\\rangle_{\\ensuremath{\\mathsf X},\\ensuremath{\\mathsf Y}}\\colon(\\ensuremath{\\mathsf X}\\times\\ensuremath{\\mathsf Y},\n\\ensuremath{\\EuScript B}_\\ensuremath{\\mathsf X}\\otimes\\ensuremath{\\EuScript B}_\\ensuremath{\\mathsf Y})\\to\\ensuremath{\\mathbb R}$\nis measurable.\n\n: Note that guarantees that, for every $x\\in\\ensuremath{\\mathcal X}$ and\nevery $y\\in\\ensuremath{\\mathcal Y}$,\n$\\langle{{x({\\mkern 2mu\\cdot\\mkern 2mu})},{y({\\mkern 2mu\\cdot\\mkern 2mu})}}\\rangle_{\\ensuremath{\\mathsf X},\\ensuremath{\\mathsf Y}}$\nis $\\ensuremath{\\EuScript F}$-measurable. Now let\n$\\{\\mathsf{y}_n\\}_{n\\in\\ensuremath{\\mathbb N}}$ be a dense subset of\n$(\\ensuremath{\\mathsf Y},\\EuScript{T}_\\ensuremath{\\mathsf Y})$ and let\n$\\widetilde{x}\\in\\ensuremath{\\widetilde{\\mathcal X}}$ be such that\n$(\\forall\\widetilde{y}\\in\\ensuremath{\\widetilde{\\mathcal Y}})$\n$\\langle{{\\widetilde{x}},{\\widetilde{y}}}\\rangle=0$. Then, for every\n$n\\in\\ensuremath{\\mathbb N}$ and every $A\\in\\ensuremath{\\EuScript F}$\nsuch that $\\mu(A)<\\ensuremath{{{+}\\infty}}$, since ensures that\n$1_A\\mathsf{y}_n\\in\\ensuremath{\\mathcal Y}$, we deduce from that\n$\\int_A\\langle{{x(\\omega)},{\\mathsf{y}_n}}\\rangle_{\\ensuremath{\\mathsf X},\\ensuremath{\\mathsf Y}}\\mu(d\\omega)\n=\\int_\\Omega\\langle{{x(\\omega)},{1_A(\\omega)\\mathsf{y}_n}}\\rangle_{\\ensuremath{\\mathsf X},\\ensuremath{\\mathsf Y}}\n\\mu(d\\omega)=0$. Therefore, since\n$(\\Omega,\\ensuremath{\\EuScript F},\\mu)$ is $\\sigma$-finite, it follows\nthat $(\\forall n\\in\\ensuremath{\\mathbb N})$\n$\\langle{{x({\\mkern 2mu\\cdot\\mkern 2mu})},{\\mathsf{y}_n}}\\rangle_{\\ensuremath{\\mathsf X},\\ensuremath{\\mathsf Y}}=0$\n$\\ensuremath{\\text{\\rm$\\mu$-a.e.}}$ Thus $\\widetilde{x}=0$. Likewise,\n$(\\forall\\widetilde{y}\\in\\ensuremath{\\widetilde{\\mathcal Y}})$\n$\\langle{{{\\mkern 2mu\\cdot\\mkern 2mu}},{\\widetilde{y}}}\\rangle=0$\n$\\Rightarrow$ $\\widetilde{y}=0$, which completes the proof.\n\nThe main result of this section is set in the following environment,\nwhich is well defined by virtue of Lemma .\n\n**Proposition 10**.  * Suppose that Assumption  holds. Then $\\varphi^*$\nis\n$\\ensuremath{\\EuScript F}\\otimes\\ensuremath{\\EuScript B}_\\ensuremath{\\mathsf Y}$-measurable.*\n\n. According to Assumption  and Definition , there exist sequences\n$(x_n)_{n\\in\\ensuremath{\\mathbb N}}$ in\n$\\mathcal{L}(\\Omega;\\ensuremath{\\mathsf X})$ and\n$(\\varrho_n)_{n\\in\\ensuremath{\\mathbb N}}$ in\n$\\mathcal{L}(\\Omega;\\ensuremath{\\mathbb R})$ such that $$\\label{e:ef}\n(\\forall\\omega\\in\\Omega)\\quad\n\\big\\{\\big(x_n(\\omega),\\varrho_n(\\omega)\\big)\\big\\}_{n\\in\\ensuremath{\\mathbb N}}\n\\subset\\mathop{\\mathrm{epi}}\\varphi_\\omega\n\\quad\\text{and}\\quad\n\\overline{\\mathop{\\mathrm{epi}}\\varphi_\\omega}=\\overline{\\big\\{\\big(x_n(\\omega),\n\\varrho_n(\\omega)\\big)\\big\\}_{n\\in\\ensuremath{\\mathbb N}}}.$$ Set\n$$(\\forall n\\in\\ensuremath{\\mathbb N})\\quad\n\\psi_n\\colon\\Omega\\times\\ensuremath{\\mathsf Y}\\to\\ensuremath{\\mathbb R}\\colon(\\omega,\\mathsf{y})\\mapsto\n\\langle{{x_n(\\omega)},{\\mathsf{y}}}\\rangle_{\\ensuremath{\\mathsf X},\\ensuremath{\\mathsf Y}}-\\varrho_n(\\omega).$$\nThen, for every $n\\in\\ensuremath{\\mathbb N}$, Assumption – and Lemma \nensure that $\\psi_n$ is\n$\\ensuremath{\\EuScript F}\\otimes\\ensuremath{\\EuScript B}_\\ensuremath{\\mathsf Y}$-measurable.\nOn the other hand, since the functions\n$(\\langle{{{\\mkern 2mu\\cdot\\mkern 2mu}},{\\mathsf{y}}}\\rangle_{\\ensuremath{\\mathsf X},\\ensuremath{\\mathsf Y}})_{\\mathsf{y}\\in\\ensuremath{\\mathsf Y}}$\nare continuous, we derive from Assumption , , and that $$\\begin{aligned}\n\\big(\\forall(\\omega,\\mathsf{y})\\in\\Omega\\times\\ensuremath{\\mathsf Y}\\big)\\quad\n\\varphi^*(\\omega,\\mathsf{y})\n&=\\sup_{(\\mathsf{x},\\xi)\\in\\mathop{\\mathrm{epi}}\\varphi_\\omega}\\big(\n\\langle{{\\mathsf{x}},{\\mathsf{y}}}\\rangle_{\\ensuremath{\\mathsf X},\\ensuremath{\\mathsf Y}}-\\xi\\big)\n\\nonumber\\\\\n&=\\sup_{(\\mathsf{x},\\xi)\\in\\overline{\\mathop{\\mathrm{epi}}\\varphi_\\omega}}\\big(\n\\langle{{\\mathsf{x}},{\\mathsf{y}}}\\rangle_{\\ensuremath{\\mathsf X},\\ensuremath{\\mathsf Y}}-\\xi\\big)\n\\nonumber\\\\\n&=\\sup_{n\\in\\ensuremath{\\mathbb N}}\\big(\\langle{{x_n(\\omega)},{\\mathsf{y}}}\\rangle_{\\ensuremath{\\mathsf X},\\ensuremath{\\mathsf Y}}\n-\\varrho_n(\\omega)\\big)\n\\nonumber\\\\\n&=\\sup_{n\\in\\ensuremath{\\mathbb N}}\\psi_n(\\omega,\\mathsf{y}).\n\\end{aligned}$$ Thus $\\varphi^*$ is\n$\\ensuremath{\\EuScript F}\\otimes\\ensuremath{\\EuScript B}_\\ensuremath{\\mathsf Y}$-measurable.\n\nWe first investigate the conjugate and the subdifferential of integral\nfunctions.\n\n**Theorem 11**.  * Suppose that Assumption  holds. Then the following\nare satisfied:*\n\n1.  *\n    $\\mathfrak{I}_{\\varphi,\\ensuremath{\\widetilde{\\mathcal X}}}^*=\\mathfrak{I}_{\\varphi^*,\\ensuremath{\\widetilde{\\mathcal Y}}}$.*\n\n2.  * Suppose that\n    $\\mathfrak{I}_{\\varphi,\\ensuremath{\\widetilde{\\mathcal X}}}$ is\n    proper, let $\\widetilde{x}\\in\\ensuremath{\\widetilde{\\mathcal X}}$,\n    and let $\\widetilde{y}\\in\\ensuremath{\\widetilde{\\mathcal Y}}$. Then\n    $\\widetilde{y}\\in\n    \\partial\\mathfrak{I}_{\\varphi,\\ensuremath{\\widetilde{\\mathcal X}}}(\\widetilde{x})$\n    $\\Leftrightarrow$ $y(\\omega)\\in\\partial\\varphi_\\omega(x(\\omega))$\n    for $\\mu$-almost every $\\omega\\in\\Omega$.*\n\n. : In view of Assumption  and Proposition ,\n$\\mathfrak{I}_{\\varphi,\\ensuremath{\\widetilde{\\mathcal X}}}$ and\n$\\mathfrak{I}_{\\varphi^*,\\ensuremath{\\widetilde{\\mathcal Y}}}$ are well\ndefined. Further, there exist sequences\n$(z_n)_{n\\in\\ensuremath{\\mathbb N}}$ in\n$\\mathcal{L}(\\Omega;\\ensuremath{\\mathsf X})$ and\n$(\\vartheta_n)_{n\\in\\ensuremath{\\mathbb N}}$ in\n$\\mathcal{L}(\\Omega;\\ensuremath{\\mathbb R})$ such that $$\\label{e:imd}\n(\\forall\\omega\\in\\Omega)\\quad\n\\big\\{\\big(z_n(\\omega),\\vartheta_n(\\omega)\\big)\\big\\}_{n\\in\\ensuremath{\\mathbb N}}\n\\subset\\mathop{\\mathrm{epi}}\\varphi_\\omega\n\\quad\\text{and}\\quad\n\\overline{\\mathop{\\mathrm{epi}}\\varphi_\\omega}=\\overline{\\big\\{\\big(z_n(\\omega),\n\\vartheta_n(\\omega)\\big)\\big\\}_{n\\in\\ensuremath{\\mathbb N}}}.$$ Let\n$\\widetilde{y}\\in\\ensuremath{\\widetilde{\\mathcal Y}}$, define\n$\\psi\\colon\\Omega\\times\\ensuremath{\\mathsf X}\\to\\ensuremath{\\left]{-}\\infty,{+}\\infty\\right]}\\colon\n(\\omega,\\mathsf{x})\\mapsto\\varphi_\\omega(\\mathsf{x})\n-\\langle{{\\mathsf{x}},{y(\\omega)}}\\rangle_{\\ensuremath{\\mathsf X},\\ensuremath{\\mathsf Y}}$,\nand note that $(\\forall\\omega\\in\\Omega)$\n$\\mathop{\\mathrm{epi}}\\psi_\\omega\\neq\\ensuremath{\\varnothing}$.\nAssumption  and Lemma  imply that $$\\label{e:tys}\n\\psi\\,\\,\\text{is $\\ensuremath{\\EuScript F}\\otimes\\ensuremath{\\EuScript B}_\\ensuremath{\\mathsf X}$-measurable}.$$\nMoreover, using the continuity of the linear functionals\n$(\\langle{{{\\mkern 2mu\\cdot\\mkern 2mu}},{\\mathsf{y}}}\\rangle_{\\ensuremath{\\mathsf X},\\ensuremath{\\mathsf Y}})_{\\mathsf{y}\\in\\ensuremath{\\mathsf Y}}$,\nwe derive from that $$\\begin{aligned}\n(\\forall\\omega\\in\\Omega)\\quad\n\\inf\\psi(\\omega,\\ensuremath{\\mathsf X})\n&=\\inf_{(\\mathsf{x},\\xi)\\in\\mathop{\\mathrm{epi}}\\varphi_\\omega}\\big(\n\\xi-\\langle{{\\mathsf{x}},{y(\\omega)}}\\rangle_{\\ensuremath{\\mathsf X},\\ensuremath{\\mathsf Y}}\\big)\n\\nonumber\\\\\n&=\\inf_{(\\mathsf{x},\\xi)\\in\\overline{\\mathop{\\mathrm{epi}}\\varphi_\\omega}}\\big(\n\\xi-\\langle{{\\mathsf{x}},{y(\\omega)}}\\rangle_{\\ensuremath{\\mathsf X},\\ensuremath{\\mathsf Y}}\\big)\n\\nonumber\\\\\n&=\\inf_{n\\in\\ensuremath{\\mathbb N}}\\big(\\vartheta_n(\\omega)\n-\\langle{{z_n(\\omega)},{y(\\omega)}}\\rangle\\big)\n\\nonumber\\\\\n&\\ensuremath{\\geqslant}\\inf_{n\\in\\ensuremath{\\mathbb N}}\\big(\\varphi_\\omega\\big(z_n(\\omega)\\big)\n-\\langle{{z_n(\\omega)},{y(\\omega)}}\\rangle\\big)\n\\nonumber\\\\\n&=\\inf_{n\\in\\ensuremath{\\mathbb N}}\\psi\\big(\\omega,z_n(\\omega)\\big)\n\\nonumber\\\\\n&\\ensuremath{\\geqslant}\\inf\\psi(\\omega,\\ensuremath{\\mathsf X}).\n\\end{aligned}$$ Hence, $(\\forall\\omega\\in\\Omega)$\n$\\inf\\psi(\\omega,\\ensuremath{\\mathsf X})=\\inf_{n\\in\\ensuremath{\\mathbb N}}\\psi(\\omega,z_n(\\omega))$.\nCombining this with , we infer that\n$\\inf\\psi({\\mkern 2mu\\cdot\\mkern 2mu},\\ensuremath{\\mathsf X})$ is\n$\\ensuremath{\\EuScript F}$-measurable and that $\\psi$ fulfills property \nin Theorem  with $(\\forall n\\in\\ensuremath{\\mathbb N})$\n$x_n=z_n-\\overline{x}$. In turn, thanks to Assumption  and the\ncompliance of $\\ensuremath{\\mathcal X}$, property  in Theorem  is\nfulfilled. Thus, by invoking and Theorem , we obtain $$\\begin{aligned}\n\\mathfrak{I}_{\\varphi,\\ensuremath{\\widetilde{\\mathcal X}}}^*(\\widetilde{y})\n&=\\sup_{\\widetilde{x}\\in\\ensuremath{\\widetilde{\\mathcal X}}}\\big(\n\\langle{{\\widetilde{x}},{\\widetilde{y}}}\\rangle-\n\\mathfrak{I}_{\\varphi,\\ensuremath{\\widetilde{\\mathcal X}}}(\\widetilde{x})\\big)\n\\nonumber\\\\\n&=\\sup_{x\\in\\ensuremath{\\mathcal X}}\\bigg(\n\\int_\\Omega\\big\\langle{{x(\\omega)},{y(\\omega)}}\\big\\rangle_{\\ensuremath{\\mathsf X},\\ensuremath{\\mathsf Y}}\\mu(d\\omega)\n-\\int_\\Omega\\varphi\\big(\\omega,x(\\omega)\\big)\\mu(d\\omega)\n\\bigg)\n\\nonumber\\\\\n&=-\\inf_{x\\in\\ensuremath{\\mathcal X}}\\int_\\Omega\\psi\\big(\\omega,x(\\omega)\\big)\n\\mu(d\\omega)\n\\nonumber\\\\\n&=-\\int_\\Omega\\inf_{\\mathsf{x}\\in\\ensuremath{\\mathsf X}}\\psi(\\omega,\\mathsf{x})\n\\,\\mu(d\\omega)\n\\nonumber\\\\\n&=\\int_\\Omega\\varphi_\\omega^*\\big(y(\\omega)\\big)\\mu(d\\omega),\n\\end{aligned}$$ as desired.\n\n: Since the functions $(\\varphi_\\omega)_{\\omega\\in\\Omega}$ are proper by\nAssumption , we derive from , , , and the Fenchel–Young inequality that\n$$\\begin{aligned}\n\\widetilde{y}\\in\n\\partial\\mathfrak{I}_{\\varphi,\\ensuremath{\\widetilde{\\mathcal X}}}(\\widetilde{x})\n&\\Leftrightarrow\n\\mathfrak{I}_{\\varphi,\\ensuremath{\\widetilde{\\mathcal X}}}(\\widetilde{x})+\n\\mathfrak{I}_{\\varphi^*,\\ensuremath{\\widetilde{\\mathcal Y}}}(\\widetilde{y})\n=\\langle{{\\widetilde{x}},{\\widetilde{y}}}\\rangle\n\\nonumber\\\\\n&\\Leftrightarrow\n\\int_\\Omega\n\\varphi_\\omega\\big(x(\\omega)\\big)\\mu(d\\omega)\n+\\int_\\Omega\\varphi_\\omega^*\\big(y(\\omega)\\big)\\mu(d\\omega)\n=\\int_\\Omega\\big\\langle{{x(\\omega)},{y(\\omega)}}\\big\\rangle_{\\ensuremath{\\mathsf X},\\ensuremath{\\mathsf Y}}\\mu(d\\omega)\n\\nonumber\\\\\n&\\Leftrightarrow\n\\varphi_\\omega\\big(x(\\omega)\\big)\n+\\varphi_\\omega^*\\big(y(\\omega)\\big)\n=\\big\\langle{{x(\\omega)},{y(\\omega)}}\\big\\rangle_{\\ensuremath{\\mathsf X},\\ensuremath{\\mathsf Y}}\\,\\,\\ensuremath{\\text{\\rm$\\mu$-a.e.}}\n\\nonumber\\\\\n&\\Leftrightarrow\ny(\\omega)\\in\\partial\\varphi_\\omega\\big(x(\\omega)\\big)\\,\\,\\ensuremath{\\text{\\rm$\\mu$-a.e.}},\n\\end{aligned}$$ which completes the proof.\n\nA first important consequence of Theorem  is the following.\n\n**Proposition 12**.  * Suppose that Assumption  holds, that\n$(\\ensuremath{\\mathsf Y},\\EuScript{T}_\\ensuremath{\\mathsf Y})$ is a\nSouslin space, that\n$\\mathop{\\mathrm{dom}}\\mathfrak{I}_{\\varphi^*,\\ensuremath{\\widetilde{\\mathcal Y}}}\\neq\\ensuremath{\\varnothing}$,\nthat $\\ensuremath{\\mathcal Y}$ is compliant, and that\n$(\\forall\\omega\\in\\Omega)$\n$\\varphi_\\omega\\in\\Gamma_0(\\ensuremath{\\mathsf X})$. Then the following\nare satisfied:*\n\n1.  *\n    $\\mathfrak{I}_{\\varphi,\\ensuremath{\\widetilde{\\mathcal X}}}\\in\\Gamma_0(\\ensuremath{\\widetilde{\\mathcal X}})$.*\n\n2.  * Set\n    $\\mathop{\\mathrm{rec}}\\varphi\\colon\\Omega\\times\\ensuremath{\\mathsf X}\\to\\ensuremath{\\left]{-}\\infty,{+}\\infty\\right]}\\colon\n    (\\omega,\\mathsf{x})\\mapsto(\\mathop{\\mathrm{rec}}\\varphi_\\omega)(\\mathsf{x})$.\n    Then $\\mathop{\\mathrm{rec}}\\varphi$ is\n    $\\ensuremath{\\EuScript F}\\otimes\\ensuremath{\\EuScript B}_\\ensuremath{\\mathsf X}$-measurable\n    and\n    $\\mathop{\\mathrm{rec}}\\mathfrak{I}_{\\varphi,\\ensuremath{\\widetilde{\\mathcal X}}}=\\mathfrak{I}_{\\mathop{\\mathrm{rec}}\\varphi,\\ensuremath{\\widetilde{\\mathcal X}}}$.*\n\n. : Let $\\widetilde{x}\\in\\ensuremath{\\widetilde{\\mathcal X}}$ and set\n$$\\label{e:r3}\n\\psi\\colon\\Omega\\times\\ensuremath{\\mathsf Y}\\to\\ensuremath{\\left]{-}\\infty,{+}\\infty\\right]}\\colon(\\omega,\\mathsf{y})\n\\mapsto\\varphi_\\omega^*(\\mathsf{y})\n-\\langle{{x(\\omega)},{\\mathsf{y}}}\\rangle_{\\ensuremath{\\mathsf X},\\ensuremath{\\mathsf Y}}\n\\quad\\text{and}\\quad\n\\vartheta=\\inf\\psi({\\mkern 2mu\\cdot\\mkern 2mu},\\ensuremath{\\mathsf Y}).$$\nBy Assumption , $$\\label{e:fxs}\n\\varphi\\big({\\mkern 2mu\\cdot\\mkern 2mu},x({\\mkern 2mu\\cdot\\mkern 2mu})\\big)\\,\\,\\text{is $\\ensuremath{\\EuScript F}$-measurable},$$\nwhile it results from Proposition  and Lemma  that $$\\label{e:yxc}\n\\psi\\,\\,\\text{is $\\ensuremath{\\EuScript F}\\otimes\\ensuremath{\\EuScript B}_\\ensuremath{\\mathsf Y}$-measurable}.$$\nMoreover, for every $\\omega\\in\\Omega$, since\n$\\varphi_\\omega\\in\\Gamma_0(\\ensuremath{\\mathsf X})$, $\\varphi_\\omega^*$\nis proper and hence\n$\\mathop{\\mathrm{epi}}\\psi_\\omega\\neq\\ensuremath{\\varnothing}$. On the\nother hand, the Fenchel–Moreau biconjugation theorem yields\n$$\\label{e:8cp}\n(\\forall\\omega\\in\\Omega)\\quad\n\\vartheta(\\omega)\n=-\\varphi_\\omega^{**}\\big(x(\\omega)\\big)\n=-\\varphi_\\omega\\big(x(\\omega)\\big)$$ and it thus follows from that\n$\\vartheta$ is $\\ensuremath{\\EuScript F}$-measurable. Now define\n$$\\label{e:m3}\n(\\forall n\\in\\ensuremath{\\mathbb N})\\quad\nM_n\\colon\\Omega\\to 2^\\ensuremath{\\mathsf Y}\\colon\\omega\\mapsto\n\\begin{cases}\n\\big\\{{\\mathsf{y}\\in\\ensuremath{\\mathsf Y}}~|~{\\psi(\\omega,\\mathsf{y})\\ensuremath{\\leqslant}-n}\\big\\},\n&\\text{if}\\,\\,\\vartheta(\\omega)=\\ensuremath{{{-}\\infty}};\\\\\n\\big\\{{\\mathsf{y}\\in\\ensuremath{\\mathsf Y}}~|~{\\psi(\\omega,\\mathsf{y})\n\\ensuremath{\\leqslant}\\vartheta(\\omega)+2^{-n}}\\big\\},\n&\\text{if}\\,\\,\\vartheta(\\omega)\\in\\ensuremath{\\mathbb R}.\n\\end{cases}$$ Fix temporarily $n\\in\\ensuremath{\\mathbb N}$. By ,\n$\\big\\{{(\\omega,\\mathsf{y})}~|~{\\mathsf{y}\\in\nM_n(\\omega)}\\big\\}\\in\\ensuremath{\\EuScript F}\\otimes\\ensuremath{\\EuScript B}_\\ensuremath{\\mathsf Y}$.\nHence, since\n$(\\ensuremath{\\mathsf Y},\\EuScript{T}_\\ensuremath{\\mathsf Y})$ is a\nSouslin space, guarantees that there exist\n$y_n\\in\\mathcal{L}(\\Omega;\\ensuremath{\\mathsf Y})$ and\n$B_n\\in\\ensuremath{\\EuScript F}$ such that $\\mu(B_n)=0$ and\n$(\\forall\\omega\\in\\complement B_n)$ $y_n(\\omega)\\in M_n(\\omega)$. Now\nset $B=\\bigcup_{n\\in\\ensuremath{\\mathbb N}}B_n$. Then $\\mu(B)=0$ and, by\nvirtue of and ,\n$$\\big(\\forall\\omega\\in\\complement B\\big)(\\forall n\\in\\ensuremath{\\mathbb N})\\quad\n\\vartheta(\\omega)\n\\ensuremath{\\leqslant}\\inf_{k\\in\\ensuremath{\\mathbb N}}\\psi\\big(\\omega,y_k(\\omega)\\big)\n\\ensuremath{\\leqslant}\\psi\\big(\\omega,y_n(\\omega)\\big)\n\\ensuremath{\\leqslant}\n\\begin{cases}\n-n,&\\text{if}\\,\\,\\vartheta(\\omega)=\\ensuremath{{{-}\\infty}};\\\\\n\\vartheta(\\omega)+2^{-n},&\\text{if}\\,\\,\\vartheta(\\omega)\\in\\ensuremath{\\mathbb R}.\n\\end{cases}$$ Thus, letting $n\\uparrow\\ensuremath{{{+}\\infty}}$ yields\n$(\\forall\\omega\\in\\complement B)$\n$\\vartheta(\\omega)=\\inf_{n\\in\\ensuremath{\\mathbb N}}\\psi(\\omega,y_n(\\omega))$.\nConsequently, since $\\ensuremath{\\mathcal Y}$ is compliant, property  in\nTheorem  is satisfied. In turn, we deduce from , Theorem , , and\nTheorem  that $$\\begin{aligned}\n\\mathfrak{I}_{\\varphi,\\ensuremath{\\widetilde{\\mathcal X}}}(\\widetilde{x})\n&=\\int_\\Omega\\varphi\\big(\\omega,x(\\omega)\\big)\\mu(d\\omega)\n\\nonumber\\\\\n&=-\\int_\\Omega\\inf_{\\mathsf{y}\\in\\ensuremath{\\mathsf Y}}\\psi(\\omega,\\mathsf{y})\\,\n\\mu(d\\omega)\n\\nonumber\\\\\n&=-\\inf_{y\\in\\ensuremath{\\mathcal Y}}\\int_\\Omega\\psi\\big(\\omega,y(\\omega)\\big)\n\\mu(d\\omega)\n\\nonumber\\\\\n&=\\sup_{y\\in\\ensuremath{\\mathcal Y}}\\bigg(\n\\int_\\Omega\\big\\langle{{x(\\omega)},{y(\\omega)}}\\big\\rangle_{\\ensuremath{\\mathsf X},\\ensuremath{\\mathsf Y}}\\mu(d\\omega)\n-\\int_\\Omega\\varphi_\\omega^*\\big(y(\\omega)\\big)\\mu(d\\omega)\\bigg)\n\\nonumber\\\\\n&=\\sup_{\\widetilde{y}\\in\\widetilde{\\ensuremath{\\mathcal Y}}}\n\\big(\\langle{{\\widetilde{x}},{\\widetilde{y}}}\\rangle-\n\\mathfrak{I}_{\\varphi,\\ensuremath{\\widetilde{\\mathcal X}}}^*(\\widetilde{y})\\big)\n\\nonumber\\\\\n&=\\mathfrak{I}_{\\varphi,\\ensuremath{\\widetilde{\\mathcal X}}}^{**}(\\widetilde{x}).\n\\end{aligned}$$ Thus\n$\\mathfrak{I}_{\\varphi,\\ensuremath{\\widetilde{\\mathcal X}}}=\n\\mathfrak{I}_{\\varphi,\\ensuremath{\\widetilde{\\mathcal X}}}^{**}$ and,\nsince $\\mathfrak{I}_{\\varphi,\\ensuremath{\\widetilde{\\mathcal X}}}$ is\nproper, we conclude that\n$\\mathfrak{I}_{\\varphi,\\ensuremath{\\widetilde{\\mathcal X}}}\\in\\Gamma_0(\\ensuremath{\\widetilde{\\mathcal X}})$.\n\n: The normality of $\\varphi$ implies that it is\n$\\ensuremath{\\EuScript F}\\otimes\\ensuremath{\\EuScript B}_\\ensuremath{\\mathsf X}$-measurable\nand that there exists $u\\in\\mathcal{L}(\\Omega;\\ensuremath{\\mathsf X})$\nsuch that $(\\forall\\omega\\in\\Omega)$\n$u(\\omega)\\in\\mathop{\\mathrm{dom}}\\varphi_\\omega$. Hence, for every\n$n\\in\\ensuremath{\\mathbb N}$, the function\n$(\\Omega\\times\\ensuremath{\\mathsf X},\\ensuremath{\\EuScript F}\\otimes\\ensuremath{\\EuScript B}_\\ensuremath{\\mathsf X})\\to\\ensuremath{\\left]{-}\\infty,{+}\\infty\\right]}\\colon\n(\\omega,\\mathsf{x})\\mapsto\n\\varphi_\\omega(u(\\omega)+n\\mathsf{x})-\\varphi_\\omega(u(\\omega))$ is\nmeasurable. Since, by ,\n$$(\\forall\\omega\\in\\Omega)(\\forall\\mathsf{x}\\in\\ensuremath{\\mathsf X})\\quad\n(\\mathop{\\mathrm{rec}}\\varphi)(\\omega,\\mathsf{x})\n=(\\mathop{\\mathrm{rec}}\\varphi_\\omega)(\\mathsf{x})\n=\\lim_{\\ensuremath{\\mathbb N}\\ni n\\uparrow\\ensuremath{{{+}\\infty}}}\n\\dfrac{\\varphi_\\omega\\big(u(\\omega)+n\\mathsf{x}\\big)\n-\\varphi_\\omega\\big(u(\\omega)\\big)}{n},$$ it follows that\n$\\mathop{\\mathrm{rec}}\\varphi$ is\n$\\ensuremath{\\EuScript F}\\otimes\\ensuremath{\\EuScript B}_\\ensuremath{\\mathsf X}$-measurable.\nNow let $\\widetilde{x}\\in\\ensuremath{\\widetilde{\\mathcal X}}$ and\n$\\widetilde{z}\\in\\mathop{\\mathrm{dom}}\\mathfrak{I}_{\\varphi,\\ensuremath{\\widetilde{\\mathcal X}}}$.\nThen, for $\\mu$-almost every $\\omega\\in\\Omega$,\n$z(\\omega)\\in\\mathop{\\mathrm{dom}}\\varphi_\\omega$ and it thus follows\nfrom the convexity of $\\varphi_\\omega$ that the function\n$\\theta\\colon\\ensuremath{\\left]0,{+}\\infty\\right[}\\to\\ensuremath{\\left]{-}\\infty,{+}\\infty\\right]}\\colon\\alpha\\mapsto\n(\\varphi_\\omega(z(\\omega)+\\alpha x(\\omega))\n-\\varphi_\\omega(z(\\omega)))/\\alpha$ is increasing. Thus, appealing to\nand the monotone convergence theorem, we deduce from that\n$$\\begin{aligned}\n\\big(\\mathop{\\mathrm{rec}}\\mathfrak{I}_{\\varphi,\\ensuremath{\\widetilde{\\mathcal X}}}\\big)(\\widetilde{x})\n&=\\lim_{\\alpha\\uparrow\\ensuremath{{{+}\\infty}}}\n\\frac{\\mathfrak{I}_{\\varphi,\\ensuremath{\\widetilde{\\mathcal X}}}(\\widetilde{z}+\n\\alpha\\widetilde{x})-\\mathfrak{I}_{\\varphi,\\ensuremath{\\widetilde{\\mathcal X}}}(\\widetilde{z})}{\n\\alpha}\n\\nonumber\\\\\n&=\\lim_{\\alpha\\uparrow\\ensuremath{{{+}\\infty}}}\\int_\\Omega\\frac{\n\\varphi_\\omega\\big(z(\\omega)+\\alpha x(\\omega)\\big)\n-\\varphi_\\omega\\big(z(\\omega)\\big)}{\\alpha}\\mu(d\\omega)\n\\nonumber\\\\\n&=\\int_\\Omega\\lim_{\\alpha\\uparrow\\ensuremath{{{+}\\infty}}}\\frac{\n\\varphi_\\omega\\big(z(\\omega)+\\alpha x(\\omega)\\big)\n-\\varphi_\\omega\\big(z(\\omega)\\big)}{\\alpha}\\,\\mu(d\\omega)\n\\nonumber\\\\\n&=\\int_\\Omega(\\mathop{\\mathrm{rec}}\\varphi_\\omega)\\big(x(\\omega)\\big)\\mu(d\\omega),\n\\end{aligned}$$ as claimed.\n\nTwo key ingredients in Hilbertian convex analysis are the Moreau\nenvelope of and the proximity operator of . To compute them for integral\nfunctions, we first observe that, in the case of Hilbert spaces\nidentified with their duals, Assumption  can be simplified as follows.\n\n**Proposition 13**.  * Suppose that Assumption  holds and let\n$\\gamma\\in\\ensuremath{\\left]0,{+}\\infty\\right[}$. Then the following are\nsatisfied:*\n\n1.  * Let $\\widetilde{x}\\in\\ensuremath{\\widetilde{\\mathcal X}}$ and\n    $\\widetilde{p}\\in\\ensuremath{\\widetilde{\\mathcal X}}$. Then\n    $\\widetilde{p}=\\mathop{\\mathrm{prox}}_{\\gamma\n    \\mathfrak{I}_{\\varphi,\\ensuremath{\\widetilde{\\mathcal X}}}}\n    \\widetilde{x}$ $\\Leftrightarrow$\n    $p(\\omega)=\\mathop{\\mathrm{prox}}_{\\gamma\\varphi_\\omega}(x(\\omega))$\n    for $\\mu$-almost every $\\omega\\in\\Omega$.*\n\n2.  * Set\n    $\\prescript{\\gamma}{}{\\varphi}\\colon\\Omega\\times\\ensuremath{\\mathsf X}\\to\\ensuremath{\\left]{-}\\infty,{+}\\infty\\right]}\\colon\n    (\\omega,\\mathsf{x})\\mapsto\\prescript{\\gamma}{}{(\\varphi_\\omega)}\n    (\\mathsf{x})$. Then $\\prescript{\\gamma}{}{\\varphi}$ is normal and\n    $\\prescript{\\gamma}{}{\\mathfrak{I}_{\\varphi,\\ensuremath{\\widetilde{\\mathcal X}}}}\n    =\\mathfrak{I}_{\\prescript{\\gamma}{}{\\varphi},\\ensuremath{\\widetilde{\\mathcal X}}}$.*\n\n. Since Assumption  is an instance of Assumption , we first infer from\nProposition  that\n$\\mathfrak{I}_{\\varphi,\\ensuremath{\\widetilde{\\mathcal X}}}\\in\\Gamma_0(\\ensuremath{\\widetilde{\\mathcal X}})$.\n\n: We derive from and Theorem  that $$\\begin{aligned}\n\\widetilde{p}=\\mathop{\\mathrm{prox}}_{\\gamma\\mathfrak{I}_{\\varphi,\n\\ensuremath{\\widetilde{\\mathcal X}}}}\\widetilde{x}\n&\\Leftrightarrow\\widetilde{x}-\\widetilde{p}\\in\\gamma\\partial\n\\mathfrak{I}_{\\varphi,\\ensuremath{\\widetilde{\\mathcal X}}}(\\widetilde{p})\n\\nonumber\\\\\n&\\Leftrightarrow x(\\omega)-p(\\omega)\\in\\gamma\\partial\n\\varphi_\\omega\\big(p(\\omega)\\big)\\,\\,\n\\text{for $\\mu$-almost every}\\,\\,\\omega\\in\\Omega\n\\nonumber\\\\\n&\\Leftrightarrow p(\\omega)=\\mathop{\\mathrm{prox}}_{\\gamma\\varphi_\\omega}\nx(\\omega)\\,\\,\\text{for $\\mu$-almost every}\n\\,\\,\\omega\\in\\Omega.\n\\end{aligned}$$\n\n: Since\n$\\ensuremath{\\EuScript B}_{\\ensuremath{\\mathsf X}\\times\\ensuremath{\\mathbb R}}=\\ensuremath{\\EuScript B}_\\ensuremath{\\mathsf X}\\otimes\\ensuremath{\\EuScript B}_\\ensuremath{\\mathbb R}$,\nit results from Assumption  and Definition  that there exists a sequence\n$(\\boldsymbol{x}_n)_{n\\in\\ensuremath{\\mathbb N}}$ in\n$\\mathcal{L}(\\Omega;\\ensuremath{\\mathsf X}\\times\\ensuremath{\\mathbb R})$\nsuch that $$\\label{e:usn}\n(\\forall\\omega\\in\\Omega)\\quad\n\\mathop{\\mathrm{epi}}\\varphi_\\omega=\n\\overline{\\big\\{\\boldsymbol{x}_n(\\omega)\\big\\}_{n\\in\\ensuremath{\\mathbb N}}}.$$\nSet $\\boldsymbol{\\mathsf{V}}=\n\\big\\{{(\\mathsf{x},\\xi)\\in\\ensuremath{\\mathsf X}\\times\\ensuremath{\\mathbb R}}~|~{\n\\|\\mathsf{x}\\|_\\ensuremath{\\mathsf X}^2/(2\\gamma)<\\xi}\\big\\}$. Then\n$\\boldsymbol{\\mathsf{V}}$ is open and therefore, for every\n$\\boldsymbol{\\mathsf{C}}\\subset\\ensuremath{\\mathsf X}\\times\\ensuremath{\\mathbb R}$,\n$\\boldsymbol{\\mathsf{C}}+\\boldsymbol{\\mathsf{V}}\n=\\overline{\\boldsymbol{\\mathsf{C}}}+\\boldsymbol{\\mathsf{V}}$. Thus, we\nderive from and that $$\\begin{aligned}\n(\\forall\\omega\\in\\Omega)\\quad\n\\big\\{{(\\mathsf{x},\\xi)\\in\\ensuremath{\\mathsf X}\\times\\ensuremath{\\mathbb R}}~|~{\n\\prescript{\\gamma}{}{(\\varphi_\\omega)}(\\mathsf{x})<\\xi}\\big\\}\n&=\\big\\{{(\\mathsf{x},\\xi)\\in\\ensuremath{\\mathsf X}\\times\\ensuremath{\\mathbb R}}~|~{\n\\varphi_\\omega(\\mathsf{x})<\\xi}\\big\\}+\\boldsymbol{\\mathsf{V}}\n\\nonumber\\\\\n&=\\overline{\\big\\{{(\\mathsf{x},\\xi)\\in\\ensuremath{\\mathsf X}\\times\\ensuremath{\\mathbb R}}~|~{\n\\varphi_\\omega(\\mathsf{x})<\\xi}\\big\\}}+\\boldsymbol{\\mathsf{V}}\n\\nonumber\\\\\n&=\\mathop{\\mathrm{epi}}\\varphi_\\omega+\\boldsymbol{\\mathsf{V}}\n\\nonumber\\\\\n&=\\overline{\\big\\{\\boldsymbol{x}_n(\\omega)\\big\\}_{n\\in\\ensuremath{\\mathbb N}}}\n+\\boldsymbol{\\mathsf{V}}\n\\nonumber\\\\\n&=\\big\\{\\boldsymbol{x}_n(\\omega)\\big\\}_{n\\in\\ensuremath{\\mathbb N}}\n+\\boldsymbol{\\mathsf{V}}\n\\nonumber\\\\\n&=\\bigcup_{n\\in\\ensuremath{\\mathbb N}}\\big(\\boldsymbol{x}_n(\\omega)\n+\\boldsymbol{\\mathsf{V}}\\big).\n\\end{aligned}$$ Hence, for every $\\mathsf{x}\\in\\ensuremath{\\mathsf X}$\nand every $\\xi\\in\\ensuremath{\\mathbb R}$, since\n$(\\mathsf{x},\\xi)-\\boldsymbol{\\mathsf{V}}\\in\\ensuremath{\\EuScript B}_{\\ensuremath{\\mathsf X}\\times\\ensuremath{\\mathbb R}}$\nand $\\{\\boldsymbol{x}_n\\}_{n\\in\\ensuremath{\\mathbb N}}\\subset\n\\mathcal{L}(\\Omega;\\ensuremath{\\mathsf X}\\times\\ensuremath{\\mathbb R})$,\nwe obtain $$\\big\\{{\\omega\\in\\Omega}~|~{\n\\prescript{\\gamma}{}{(\\varphi_\\omega)}(\\mathsf{x})<\\xi}\\big\\}\n=\\left\\{{\\omega\\in\\Omega}~\\middle|~{\n(\\mathsf{x},\\xi)\\in\\bigcup_{n\\in\\ensuremath{\\mathbb N}}\\big(\\boldsymbol{x}_n(\\omega)\n+\\boldsymbol{\\mathsf{V}}\\big)}\\right\\}\n=\\bigcup_{n\\in\\ensuremath{\\mathbb N}}\\boldsymbol{x}_n^{-1}\\big(\n(\\mathsf{x},\\xi)-\\boldsymbol{\\mathsf{V}}\\big)\n\\in\\ensuremath{\\EuScript F},$$ which shows that\n$(\\prescript{\\gamma}{}{\\varphi})({\\mkern 2mu\\cdot\\mkern 2mu},\\mathsf{x})$\nis $\\ensuremath{\\EuScript F}$-measurable. Hence, since\n$(\\ensuremath{\\mathsf X},\\EuScript{T}_\\ensuremath{\\mathsf X})$ is a\nFréchet space, Theorem  ensures that $\\prescript{\\gamma}{}{\\varphi}$ is\nnormal. It remains to show that\n$\\prescript{\\gamma}{}{\\mathfrak{I}_{\\varphi,\\ensuremath{\\widetilde{\\mathcal X}}}}\n=\\mathfrak{I}_{\\prescript{\\gamma}{}{\\varphi},\\ensuremath{\\widetilde{\\mathcal X}}}$.\nLet $\\widetilde{x}\\in\\ensuremath{\\widetilde{\\mathcal X}}$ and set\n$\\widetilde{p}=\\mathop{\\mathrm{prox}}_{\\gamma\n\\mathfrak{I}_{\\varphi,\\ensuremath{\\widetilde{\\mathcal X}}}}\n\\widetilde{x}$. Then, by , for $\\mu$-almost every $\\omega\\in\\Omega$,\n$p(\\omega)=\\mathop{\\mathrm{prox}}_{\\gamma\\varphi_\\omega}(x(\\omega))$\nand, therefore, yields\n$\\prescript{\\gamma}{}{(\\varphi_\\omega)}(x(\\omega))\n=\\varphi_\\omega(p(\\omega))\n+\\|x(\\omega)-p(\\omega)\\|_\\ensuremath{\\mathsf X}^2/(2\\gamma)$. Hence\n$$\\begin{aligned}\n\\prescript{\\gamma}{}{\\mathfrak{I}_{\\varphi,\\ensuremath{\\widetilde{\\mathcal X}}}}(\\widetilde{x})\n&=\\mathfrak{I}_{\\varphi,\\ensuremath{\\widetilde{\\mathcal X}}}(\\widetilde{p})\n+\\frac{1}{2\\gamma}\\|\\widetilde{x}-\\widetilde{p}\\|_{\\ensuremath{\\widetilde{\\mathcal X}}}^2\n\\nonumber\\\\\n&=\\int_\\Omega\\varphi_\\omega\\big(p(\\omega)\\big)\\mu(d\\omega)\n+\\frac{1}{2\\gamma}\\int_\\Omega\n\\|x(\\omega)-p(\\omega)\\|_\\ensuremath{\\mathsf X}^2\\mu(d\\omega)\n\\nonumber\\\\\n&=\\int_\\Omega\n\\prescript{\\gamma}{}{(\\varphi_\\omega)}\\big(x(\\omega)\\big)\\mu(d\\omega)\n\\nonumber\\\\\n&=\\mathfrak{I}_{\\prescript{\\gamma}{}{\\varphi},\\ensuremath{\\widetilde{\\mathcal X}}}(\\widetilde{x}),\n\\end{aligned}$$ which concludes the proof.\n\n[^1]: Contact author: P. L. Combettes. Email:\n    [`plc@math.ncsu.edu`](mailto:plc@math.ncsu.edu). Phone: +1 919 515\n    2671. The work of M. N. Bùi was supported by NAWI Graz and the work\n    of P. L. Combettes was supported by the National Science Foundation\n    under grant DMS-1818946.\n"
    },
    {
        "text": "abstract: Via the hierarchy of correlations, we study doublon-holon pair\n  creation in the Mott state of the Fermi-Hubbard model induced by a\n  time-dependent electric field. Special emphasis is placed on the\n  analogy to electron-positron pair creation from the vacuum in quantum\n  electrodynamics (QED). We find that the accuracy of this analogy\n  depends on the spin structure of the Mott background. For Ising type\n  anti-ferromagnetic order, we derive an effective Dirac equation. A\n  Mott state without any spin order, on the other hand, does not\n  explicitly display such a quasi-relativistic behavior.\nauthor: Friedemann Queisser; Konstantin Krutitsky; Patrick Navez; Ralf\nSchützhold\ndate: 2024-10-13\ntitle: Doublon-holon pair creation in Mott-Hubbard systems in analogy to\n  QED\n\n# Introduction\n\nNon-equilibrium dynamics in strongly interacting quantum many-body\nsystems is a rich and complex field displaying many fascinating\nphenomena. As the *drosophila* of strongly interacting quantum many-body\nsystems, we consider the Mott insulator phase of the Fermi-Hubbard model\n. An optical laser can then serve as the external stimulus driving the\nsystem out of equilibrium – leading to the creation of doublon-holon\npairs, see also .\n\nThe intuitive similarity between the upper and lower Hubbard bands one\nthe one hand and the Dirac sea and the positive energy continuum in\nquantum electrodynamics (QED) on the other hand suggests analogies\nbetween doublon-holon pair creation from the Mott state and\nelectron-positron pair creation from the vacuum, see also . In the\nfollowing, we study this analogy in more detail, with special emphasis\non the space-time dependence in more than one dimension, such as the\npropagation of doublons and holons. More specifically, we strive for an\nanalytic understanding without mapping the Fermi-Hubbard Hamiltonian to\nan effective single site model, see also .\n\nOf course, analogies between electron-positron pair creation in QED and\nother systems at lower energies have already been discussed in previous\nworks. Examples include ultra-cold atoms in optical lattices as well as\nelectrons in semi-conductors graphene and $^3$He . However, as we shall\nsee below, there are important differences to the Fermi-Hubbard model\nconsidered here. First, the Mott gap arises naturally through the\ninteraction (see also ) and does not have to be introduced by hand.\nSecond, the particle-hole symmetry between the upper and lower Hubbard\nband – analogous to the $\\cal C$ symmetry in QED – is also an intrinsic\nproperty (in contrast to the valence and conduction bands in\nsemi-conductors, for example). Third, the quantitative analogy to the\nDirac equation (in 1+1 dimensions) and the resulting quasi-relativistic\nrelativistic behavior does also emerge without additional fine-tuning\n(at least in the case of Ising type spin order, see below).\n\n# Extended Fermi-Hubbard Model\n\nIn terms of the fermionic creation and annihilation operators\n$\\hat c_{\\mu s}^\\dagger$ and $\\hat c_{\\nu s}$ at the lattice sites $\\mu$\nand $\\nu$ with spin $s\\in\\{\\uparrow,\\downarrow\\}$ and the associated\nnumber operators $\\hat n_{\\mu s}$, the extended Fermi-Hubbard\nHamiltonian reads ($\\hbar=1$) $$\\begin{aligned}\n\\label{Fermi-Hubbard}\n\\hat H=-\\frac1Z\\sum_{\\mu\\nu s} T_{\\mu\\nu} \\hat c_{\\mu s}^\\dagger \\hat c_{\\nu s} \n+U\\sum_\\mu\\hat n_\\mu^\\uparrow\\hat n_\\mu^\\downarrow\n+\\sum_{\\mu s}V_\\mu\\hat n_{\\mu s}  \n\\,.\\;\n\\end{aligned}$$ Here the hopping matrix $T_{\\mu\\nu}$ equals the\ntunneling strength $T$ for nearest neighbors $\\mu$ and $\\nu$ and is zero\notherwise. The coordination number $Z$ counts the number of nearest\nneighbors $\\mu$ for a given lattice site $\\nu$ and is assumed to be\nlarge $Z\\gg1$. In order to describe the Mott insulator, the on-site\nrepulsion $U$ is also supposed to be large $U\\gg T$. Finally, the\npotential $V_\\mu(t)$ represents the external electric field, e.g., an\noptical laser.\n\n## Hierarchy of Correlations\n\nTo obtain an approximate analytical solution, we consider the reduced\ndensity matrices of one $\\hat\\rho_\\mu$ and two $\\hat\\rho_{\\mu\\nu}$\nlattice sites etc. Next, we split up the correlated parts via\n$\\hat\\rho_{\\mu\\nu}^{\\rm corr}=\\hat\\rho_{\\mu\\nu}-\\hat\\rho_{\\mu}\\hat\\rho_{\\nu}$\netc. For large $Z\\gg1$, we may employ an expansion into powers of $1/Z$\nwhere we find that higher-order correlators are successively suppressed\n. More precisely, the two-point correlator scales as\n$\\hat\\rho_{\\mu\\nu}^{\\rm corr}=\\,{\\cal O}(1/Z)$, while the three-point\ncorrelation is suppressed as\n$\\hat\\rho_{\\mu\\nu\\lambda}^{\\rm corr}=\\,{\\cal O}(1/Z^2)$ etc.\n\nVia this expansion into powers of $1/Z$, we may find approximate\nsolutions of the evolution equations $$\\begin{aligned}\n\\label{evolution}\ni\\partial_t \\hat\\rho_\\mu \n&=& \nF_1(\\hat\\rho_\\mu,\\hat\\rho_{\\mu\\nu}^{\\rm corr})\n\\,,\\nonumber\\\\\ni\\partial_t \\hat\\rho_{\\mu\\nu}^{\\rm corr} \n&=& \nF_2(\\hat\\rho_\\mu,\\hat\\rho_{\\mu\\nu}^{\\rm corr},\\hat\\rho_{\\mu\\nu\\lambda}^{\\rm corr})\n\\,.\n\\end{aligned}$$ Using $\\hat\\rho_{\\mu\\nu}^{\\rm corr}=\\,{\\cal O}(1/Z)$,\nthe first evolution equation can be approximated by\n$i\\partial_t \\hat\\rho_\\mu = F_1(\\hat\\rho_\\mu,0)+\\,{\\cal O}(1/Z)$. Its\nzeroth-order solution $\\hat\\rho_\\mu^0$ yields the mean-field background,\nwhich will be specified below.\n\nNext, the suppression\n$\\hat\\rho_{\\mu\\nu\\lambda}^{\\rm corr}=\\,{\\cal O}(1/Z^2)$ allows us to\napproximate the second equation  to leading order in $1/Z$ via\n$i\\partial_t \\hat\\rho_{\\mu\\nu}^{\\rm corr}\\approx \nF_2(\\hat\\rho_\\mu^0,\\hat\\rho_{\\mu\\nu}^{\\rm corr},0)$. In order to solve\nthis leading-order equality, it is convenient to split to fermionc\ncreation and annihilation operators in particle $I=1$ and hole $I=0$\ncontributions via $$\\begin{aligned}\n\\hat c_{\\mu s I}=\\hat c_{\\mu s}\\hat n_{\\mu\\bar s}^I=\n\\left\\{\n\\begin{array}{ccc}\n \\hat c_{\\mu s}(1-\\hat n_{\\mu\\bar s}) & {\\rm for} & I=0 \n \\\\ \n \\hat c_{\\mu s}\\hat n_{\\mu\\bar s} & {\\rm for} & I=1\n\\end{array}\n\\right.\n\\,,\n\\end{aligned}$$ where $\\bar s$ denotes the spin opposite to $s$. In\nterms of these particle and hole operators, the correlations (for\n$\\mu\\neq\\nu$) are determined by $$\\begin{aligned}\n\\label{correlations}\ni\\partial_t\n\\langle\\hat c^\\dagger_{\\mu s I}\\hat c_{\\nu s J}\\rangle^{\\rm corr}\n=\n\\frac1Z\\sum_{\\lambda L} T_{\\mu\\lambda}\n\\langle\\hat n_{\\mu\\bar s}^I\\rangle^0\n\\langle\\hat c^\\dagger_{\\lambda s L}\\hat c_{\\nu s J}\\rangle^{\\rm corr}\n\\nonumber\\\\\n-\n\\frac1Z\\sum_{\\lambda L} T_{\\nu\\lambda}\n\\langle\\hat n_{\\nu\\bar s}^J\\rangle^0\n\\langle\\hat c^\\dagger_{\\mu s I}\\hat c_{\\lambda s L}\\rangle^{\\rm corr}\n\\nonumber\\\\\n+\n\\left(U_\\nu^J-U_\\mu^I+V_\\nu-V_\\mu\\right) \n\\langle\\hat c^\\dagger_{\\mu s I}\\hat c_{\\nu s J}\\rangle^{\\rm corr}\n\\nonumber\\\\\n+\\frac{T_{\\mu\\nu}}{Z}\n\\left(\n\\langle\\hat n_{\\mu\\bar s}^I\\rangle^0\n\\langle\\hat n_{\\nu s}\\hat n_{\\nu\\bar s}^J\\rangle^0\n-\n\\langle\\hat n_{\\nu\\bar s}^J\\rangle^0\n\\langle\\hat n_{\\mu s}\\hat n_{\\mu\\bar s}^I\\rangle^0\n\\right) \n\\,,\n\\end{aligned}$$ where\n$\\langle\\hat X_\\mu\\rangle^0={\\rm Tr}\\{\\hat X_\\mu\\hat\\rho_\\mu^0\\}$ denote\nexpectation values in the mean-field background.\n\nThe evolution equations  for the correlators can be simplified by\nfactorizing them via the following effective linear equations for the\nparticle and hole operators $$\\begin{aligned}\n\\label{factorization}\n\\left(i\\partial_t-U^I_\\mu-V_\\mu\\right)\\hat c_{\\mu s I}\n=\n-\\frac1Z\\sum_{\\nu J} \nT_{\\mu\\nu} \\langle\\hat n_{\\mu\\bar s}^I\\rangle^0 \\hat c_{\\nu s J}\n\\,.\n\\end{aligned}$$ Of course, the hierarchy of correlations is not the only\nway to derive such effective evolution equations, similar results can be\nobtained by other approximation schemes, e.g., .\n\n# Ising type spin order\n\nIn order to analyze the effective equations , we have to specify the\nmean-field background $\\hat\\rho_\\mu^0$. The Mott insulator state\ncorresponds to having one particle per lattice site, which leaves to\ndetermine the remaining spin degrees of freedom. As our first example,\nwe consider anti-ferromagnetic spin order of the Ising type . To this\nend, we assume a bi-partite lattice which can be spit into two\nsub-lattices $\\cal A$ and $\\cal B$ where all neighbors $\\nu$ of a\nlattice site $\\mu\\in\\cal A$ belong to $\\cal B$ and vice versa. Then, the\nzeroth-order mean-field background reads $$\\begin{aligned}\n\\label{mean-field-Ising}\n\\hat\\rho_\\mu^0\n=\n\\left\\{ \n\\begin{array}{ccc}\n\\left|\\uparrow\\right>_\\mu\\!\\left<\\uparrow\\right| & {\\rm for} & \\mu\\in\\cal A\n\\\\\n\\left|\\downarrow\\right>_\\mu\\!\\left<\\downarrow\\right| & {\\rm for} & \\mu\\in\\cal B\n\\end{array}\n\\right. \n\\,.\n\\end{aligned}$$ This state minimizes the Ising type anti-ferromagnetic\ninteraction $\\hat S^z_\\mu\\hat S^z_\\nu$. Note that the Fermi-Hubbard\nHamiltonian  does indeed generate an effective anti-ferromagnetic\ninteraction via second-order hopping processes, but it would correspond\nto a Heisenberg type anti-ferromagnetic interaction\n$\\hat{\\mbox{\\boldmath$S$}}_\\mu\\cdot\\hat{\\mbox{\\boldmath$S$}}_\\nu$ .\nAlthough the state  does not describe the exact minimum of this\ninteraction\n$\\hat{\\mbox{\\boldmath$S$}}_\\mu\\cdot\\hat{\\mbox{\\boldmath$S$}}_\\nu$, it\ncan be regarded as an approximation or a simplified toy model for such\nan anti-ferromagnet. Alternatively, one could imagine additional spin\ninteractions between the electrons (stemming from the full microscopic\ndescription) which are not contained in the tight-binding model  and\nstabilize the state .\n\n## Effective Dirac Equation\n\nThis background  supports hole excitations $I=0$ of spin $\\uparrow$ and\nparticle excitations $I=1$ of spin $\\downarrow$ for the sub-lattice\n$\\cal A$, and vice versa for the sub-lattice $\\cal B$. For the other\nterms, such as $\\hat c_{\\mu\\in{\\cal A},s=\\uparrow,I=1}$, the right-hand\nside of Eq.  vanishes and thus they become trivial and are omitted in\nthe following.\n\nAs a result, particle excitations in the sub-lattice $\\cal A$ are tunnel\ncoupled to hole excitations of the same spin in the sub-lattice $\\cal B$\nand vice versa. Since the spin components $s=\\uparrow$ and\n$s=\\downarrow$ evolve independent of each other, we drop the spin index\nin the following. Introducing the effective spinor in analogy to the\nDirac equation $$\\begin{aligned}\n\\hat\\psi_\\mu\n=\n\\left(\n\\begin{array}{c}\n\\hat c_{\\mu  I=1} \n\\\\\n\\hat c_{\\mu  I=0} \n\\end{array}\n\\right) \n\\,,\n\\end{aligned}$$ the evolution equation can be cast into the form\n$$\\begin{aligned}\n\\label{pre-Dirac}\ni\\partial_t\\hat\\psi_\\mu\n=\n\\left(\n\\begin{array}{cc} \nV_\\mu+U & 0 \n\\\\\n0 & V_\\mu \n\\end{array}\n\\right)\n\\cdot\\hat\\psi_\\mu\n-\\frac1Z\\sum_{\\nu}T_{\\mu\\nu}\n%\\left(\n%\\begin{array}{cc} \n%0 & 1 \n%\\\\\n%1 & 0\n%\\end{array}\n%\\right)\n\\sigma_x\\cdot\\hat\\psi_\\nu\n\\,,\n%\\nn\n\\end{aligned}$$ where $\\sigma_x$ is the Pauli spin matrix. This form is\nalready reminiscent of the Dirac equation in 1+1 dimensions. To make the\nanalogy more explicit, we first apply a simple phase transformation\n$\\hat\\psi_\\mu\\to\\hat\\psi_\\mu\\exp\\{itU/2\\}$ after which the $U$ term\nreads $\\sigma_zU/2$.\n\nSince the wavelength of an optical laser is typically much longer than\nall other relevant length scales, we may approximate it (in the\nnon-relativistic regime) by a purely time-dependent electric field\n$\\mbox{\\boldmath$E$}(t)$ such that the potential reads\n$V_\\mu(t)=q\\mbox{\\boldmath$r$}_\\mu\\cdot\\mbox{\\boldmath$E$}(t)$ where\n$\\mbox{\\boldmath$r$}_\\mu$ is the position vector of the lattice site\n$\\mu$. Then we may use the Peierls transformation\n$\\hat\\psi_\\mu\\to\\hat\\psi_\\mu\\exp\\{i\\varphi_\\mu(t)\\}$ with\n$\\dot\\varphi_\\mu=V_\\mu$ to shift the potential $V_\\mu$ into\ntime-dependent phases of the hopping matrix\n$T_{\\mu\\nu}\\to T_{\\mu\\nu}e^{i\\varphi_\\mu(t)-i\\varphi_\\nu(t)}=T_{\\mu\\nu}(t)$.\nNext, a spatial Fourier transformation simplifies Eq.  to\n$$\\begin{aligned}\n\\label{Dirac-k}\ni\\partial_t\\hat\\psi_{\\mbox{\\boldmath$\\scriptstyle k$}}\n=\n\\left(\n\\frac{U}{2}\\,\\sigma_z\n-\nT_{\\mbox{\\boldmath$\\scriptstyle k$}}(t)\\sigma_x\n\\right)\\cdot\\hat\\psi_{\\mbox{\\boldmath$\\scriptstyle k$}}\n\\,,\n\\end{aligned}$$ where $T_{\\mbox{\\boldmath$\\scriptstyle k$}}(t)$ denotes\nthe Fourier transform of the hopping matrix including the time-dependent\nphases, which yields the usual minimal coupling form\n$T_{\\mbox{\\boldmath$\\scriptstyle k$}}(t)=T_{\\mbox{\\boldmath$\\scriptstyle k$}-q\\mbox{\\boldmath$\\scriptstyle A$}(t)}$.\nNote that the Peierls transformation is closely related to the gauge\ntransformation $A_\\mu\\to A_\\mu+\\partial_\\mu\\chi$ in electrodynamics.\nUsing this gauge freedom, one can represent an electric field\n$\\mbox{\\boldmath$E$}(t)$ via the scalar potential $\\phi$ as\n$\\partial_t+iq\\phi$ in analogy to the $V_\\mu$ or via the vector\npotential $\\mbox{\\boldmath$A$}$ as $\\nabla-iq\\mbox{\\boldmath$A$}$ in\nanalogy to the\n$T_{\\mbox{\\boldmath$\\scriptstyle k$}}(t)=T_{\\mbox{\\boldmath$\\scriptstyle k$}-q\\mbox{\\boldmath$\\scriptstyle A$}(t)}$.\n\nIn the absence of the electric field, the dispersion relation following\nfrom Eq.  reads $$\\begin{aligned}\n\\label{dispersion-relativistic}\n\\omega_{\\mbox{\\boldmath$\\scriptstyle k$}}=\\pm\\sqrt{\\frac{U^2}{4}+T_{\\mbox{\\boldmath$\\scriptstyle k$}}^2}\n\\,.\n\\end{aligned}$$ The positive and negative frequency solutions correspond\nto the upper and lower Hubbard bands, which are separated by the Mott\ngap. Unless the electric field is too strong or too fast, one expects\nthe main contributions to doublon-holon pair creation near the minimum\ngap, i.e., the minimum of $T_{\\mbox{\\boldmath$\\scriptstyle k$}}^2$,\ntypically at $T_{\\mbox{\\boldmath$\\scriptstyle k$}}=0$. Then, a Taylor\nexpansion\n$\\mbox{\\boldmath$k$}=\\mbox{\\boldmath$k$}_0+\\delta\\mbox{\\boldmath$k$}$\naround a zero $\\mbox{\\boldmath$k$}_0$ of\n$T_{\\mbox{\\boldmath$\\scriptstyle k$}}$ yields $$\\begin{aligned}\n\\label{Dirac-approx}\ni\\partial_t\\hat\\psi_{\\mbox{\\boldmath$\\scriptstyle k$}}\n\\approx \n\\left(\n\\frac{U}{2}\\,\\sigma_z\n-\n\\mbox{\\boldmath$c$}_{\\rm eff}\\cdot[\\delta\\mbox{\\boldmath$k$}-q\\mbox{\\boldmath$A$}(t)] \n\\sigma_x\n\\right)\\cdot\\hat\\psi_{\\mbox{\\boldmath$\\scriptstyle k$}}\n\\,,\n\\end{aligned}$$ where\n$\\mbox{\\boldmath$c$}_{\\rm eff}=\\nabla_{\\mbox{\\boldmath$\\scriptstyle k$}}T_{\\mbox{\\boldmath$\\scriptstyle k$}}|_{\\mbox{\\boldmath$\\scriptstyle k$}_0}$\ndenotes the effective propagation velocity, in analogy to the speed of\nlight. Note that the validity of this approximation does not only\nrequire $\\delta\\mbox{\\boldmath$k$}$ to be small, it also assumes that\n$q\\mbox{\\boldmath$A$}(t)$ does not become too large, e.g., that we are\nfar away from the regime of Bloch oscillations .\n\nOn the other hand, even if $q\\mbox{\\boldmath$A$}(t)$ varies over a\nlarger range, the above approximation  could still provide a reasonably\ngood description for strong-field doublon-holon pair creation. This\nprocess can be understood as Landau-Zener tunneling occurring when an\navoided level crossing is traversed with a finite speed (set by\n$\\mbox{\\boldmath$E$}=-\\dot{\\mbox{\\boldmath$A$}}$). Since this tunneling\nprocess mainly occurs in the vicinity of the minimum gap, i.e., where\n$T_{\\mbox{\\boldmath$\\scriptstyle k$}}(t)=T_{\\mbox{\\boldmath$\\scriptstyle k$}-q\\mbox{\\boldmath$\\scriptstyle A$}(t)}=0$,\nit is sufficient to consider the region around\n$\\mbox{\\boldmath$k$}-q\\mbox{\\boldmath$A$}=\\mbox{\\boldmath$k$}_0$.\n\n## Analogy to QED\n\nUp to simple phase factors\n$\\exp\\{i\\mbox{\\boldmath$k$}_0\\cdot\\mbox{\\boldmath$r$}_\\mu\\}$, Eq. \ndisplays a qualitative analogy to the Dirac equation in 1+1 dimensions,\nwhere $c_{\\rm eff}$ plays the role of the speed of light while $U/2$\ncorresponds to the mass $m_{\\rm eff}c_{\\rm eff}^2=U/2$. As a result, we\nmay now apply many of the results known from quantum electrodynamics\n(QED) . For example, the effective spinor\n$\\hat\\psi_{\\mbox{\\boldmath$\\scriptstyle k$}}$ can be expanded into\nparticle and hole contributions (first in the absence of an electric\nfield) $$\\begin{aligned}\n\\hat\\psi_{\\mbox{\\boldmath$\\scriptstyle k$}}\n=\nu_{\\mbox{\\boldmath$\\scriptstyle k$}}\\hat d_{\\mbox{\\boldmath$\\scriptstyle k$}}+v_{\\mbox{\\boldmath$\\scriptstyle k$}}\\hat h_{\\mbox{\\boldmath$\\scriptstyle k$}}^\\dagger \n\\,,\n\\end{aligned}$$ where the quasi-particles are usually referred to as\ndoublons $\\hat d_{\\mbox{\\boldmath$\\scriptstyle k$}}$ (upper Hubbard\nband) and holons $\\hat h_{\\mbox{\\boldmath$\\scriptstyle k$}}^\\dagger$\n(lower Hubbard band). The Mott state $\\left|\\rm Mott\\right>$ is then\ndetermined by\n$\\hat d_{\\mbox{\\boldmath$\\scriptstyle k$}}\\left|\\rm Mott\\right>=\\hat h_{\\mbox{\\boldmath$\\scriptstyle k$}}\\left|\\rm Mott\\right>=0$.\n\nIn the presence of an electric field, these operators can mix – as\ndescribed by the Bogoliubov transformation $$\\begin{aligned}\n\\hat d_{\\mbox{\\boldmath$\\scriptstyle k$}}^{\\rm out}\n=\n\\alpha_{\\mbox{\\boldmath$\\scriptstyle k$}}\\hat d_{\\mbox{\\boldmath$\\scriptstyle k$}}^{\\rm in}\n+\n\\beta_{\\mbox{\\boldmath$\\scriptstyle k$}}\\left(\\hat h_{\\mbox{\\boldmath$\\scriptstyle k$}}^{\\rm in}\\right)^\\dagger \n\\,,\n\\end{aligned}$$ where the structure of the Dirac equation  implies the\nnormalization\n$|\\alpha_{\\mbox{\\boldmath$\\scriptstyle k$}}|^2+|\\beta_{\\mbox{\\boldmath$\\scriptstyle k$}}|^2=1$.\nStarting in the Mott state\n$\\hat d_{\\mbox{\\boldmath$\\scriptstyle k$}}^{\\rm in}\\left|\\rm Mott\\right>=\\hat h_{\\mbox{\\boldmath$\\scriptstyle k$}}^{\\rm in}\\left|\\rm Mott\\right>=0$,\nthe $\\beta_{\\mbox{\\boldmath$\\scriptstyle k$}}$ coefficient yields the\namplitude for doublon-holon pair creation\n$\\hat d_{\\mbox{\\boldmath$\\scriptstyle k$}}^{\\rm out}\\left|\\rm Mott\\right>\\propto\\beta_{\\mbox{\\boldmath$\\scriptstyle k$}}$.\nIn analogy to QED, we may now discuss different regimes. For weak\nelectric fields $\\mbox{\\boldmath$E$}$ oscillating near resonance\n$\\omega\\approx U$, we find the usual lowest-order perturbative scaling\n$\\beta_{\\mbox{\\boldmath$\\scriptstyle k$}}\\sim|q\\mbox{\\boldmath$c$}_{\\rm eff}\\cdot\\mbox{\\boldmath$E$}|$\n. Higher orders of perturbation theory then lead us in the multi-photon\nregime, for example $n\\omega\\approx U$ with\n$\\beta_{\\mbox{\\boldmath$\\scriptstyle k$}}\\sim|q\\mbox{\\boldmath$c$}_{\\rm eff}\\cdot\\mbox{\\boldmath$E$}|^n$.\n\nNote that a completely different kind of resonances such as\n$\\omega\\approx2U$ can occur if we take higher-order correlations into\naccount , but these are beyond our effective description .\n\nIf the electric field becomes stronger and slower, we enter the\nnon-perturbative regime of the Sauter-Schwinger effect where the\npair-creation amplitude displays an exponential scaling\n$$\\begin{aligned}\n\\beta_{\\mbox{\\boldmath$\\scriptstyle k$}}\\sim\\exp\\left\\{-\\frac{\\pi U^2}{8q|\\mbox{\\boldmath$c$}_{\\rm eff}\\cdot\\mbox{\\boldmath$E$}|}\\right\\}\n\\,.\n\\end{aligned}$$ The quantitative analogy to the Dirac equation even\nallows us to directly transfer further results from QED, for example the\ndynamically assisted Sauter-Schwinger effect, where pair creation by a\nstrong and slowly varying electric field is enhanced by adding a weaker\nand faster varying field, see, e.g., .\n\n# Unordered spin state\n\nLet us compare our findings above to the case of a mean-field background\nwithout any spin order $$\\begin{aligned}\n\\label{mean-field-unordered}\n\\hat\\rho_\\mu^0\n=\n\\frac{\\left|\\uparrow\\right>_\\mu\\!\\left<\\uparrow\\right|+\\left|\\downarrow\\right>_\\mu\\!\\left<\\downarrow\\right|}{2}\n\\,, \n\\end{aligned}$$ which could arise for a finite temperature which is too\nsmall to excite doublon-holon pairs but large enough to destroy the spin\norder. Another option could be a weak magnetic disorder potential and/or\nspin frustration.\n\nIn this case, we do not have to distinguish the two sub-lattices\n$\\cal A$ and $\\cal B$ and all lattice sites can support particle and\nhole excitations. Since all expectation values\n$\\langle\\hat n_{\\mu s}^I\\rangle^0$ yield $1/2$ (instead of zero or\nunity), the analog of Eq.  now reads (after the same phase\ntransformation) $$\\begin{aligned}\n\\label{matrix-unordered}\ni\\partial_t\n\\hat\\psi_{\\mbox{\\boldmath$\\scriptstyle k$}}\n%\\left(\n%\\begin{array}{c}\n%\\hat f_{\\mathbf{k}s}\n%\\\\\n%\\hat e_{\\mathbf{k}s}^\\dagger \n%\\end{array}\n%\\right)\n=\n\\frac12\n\\left(\n\\begin{array}{cc}\nU-T_{\\bf k} & -T_{\\bf k}\n\\\\\n-T_{\\bf k} & -U-T_{\\bf k}\n\\end{array}\n\\right)\n\\cdot\n\\hat\\psi_{\\mbox{\\boldmath$\\scriptstyle k$}}\n%\n%\\left(\n%\\begin{array}{c}\n%\\hat f_{\\mathbf{k}s}\n%\\\\\n%\\hat e_{\\mathbf{k}s}^\\dagger \n%\\end{array}\n%\\right)\n\\,.\n\\end{aligned}$$ The eigenvalues of the above $2\\times2$-matrix yield the\nquasi-particle frequencies $$\\begin{aligned}\n\\label{quasi-particle-energies}\n\\omega^\\pm_\\mathbf{k}\n=\n\\frac12\\left(-T_\\mathbf{k}\\pm\\sqrt{T_\\mathbf{k}^2+U^2}\\right)\n\\,.\n\\end{aligned}$$ In view of the additional term $-T_\\mathbf{k}$ in front\nof the square root, this dispersion relation does not display the same\nquasi-relativistic form as in Eq. . Putting it another way, we find that\nEq.  is not formally equivalent to the Dirac equation (in 1+1\ndimensions).\n\nAs a consequence, the propagation of quasi-particles in the two\nmean-field backgrounds  and is quite different. For the Ising type\norder , the coherent propagation of a doublon or holon (without changing\nthe background structure) requires second-order hopping processes. Hence\n$\\omega_{\\mbox{\\boldmath$\\scriptstyle k$}}$ in Eq.  is a quadratic\nfunction of $T_{\\mbox{\\boldmath$\\scriptstyle k$}}$. For the unordered\nbackground , on the other hand, doublons and holons can propagate\ncoherently via first-order hopping processes. This is reflected in the\nlinear contribution $-T_\\mathbf{k}$ in Eq. .\n\nHowever, if we do not consider quasi-particle propagation but focus on\nthe probability for creating a doublon-holon pair in a given mode\n$\\mbox{\\boldmath$k$}$, we may again derive a close analogy to QED. To\nthis end, we apply yet another $\\mbox{\\boldmath$k$}$-dependent phase\ntransformation\n$\\hat\\psi_{\\mbox{\\boldmath$\\scriptstyle k$}}\\to e^{i\\vartheta_{\\mbox{\\boldmath$\\scriptstyle k$}}(t)}\\hat\\psi_{\\mbox{\\boldmath$\\scriptstyle k$}}$\nwith\n$\\dot\\vartheta_{\\mbox{\\boldmath$\\scriptstyle k$}}(t)=T_{\\mbox{\\boldmath$\\scriptstyle k$}}(t)/2$.\nNote that $T_{\\mbox{\\boldmath$\\scriptstyle k$}}(t)$ contains the vector\npotential $\\mbox{\\boldmath$A$}(t)$, i.e., the time integral of the\nelectric field $\\mbox{\\boldmath$E$}(t)$. Thus, the phase\n$\\vartheta_{\\mbox{\\boldmath$\\scriptstyle k$}}(t)$ involves yet another\ntime integral, which makes it a even more non-local function of time.\nAfter this phase transformation, Eq.  becomes again formally equivalent\nto the Dirac equation in 1+1 dimensions, but now with the effective\nspeed of light being reduced by a factor of two.\n\n# Conclusions\n\nWe study doublon-holon pair creation from the Mott insulator state of\nthe Fermi-Hubbard model induced by an external electric field\n$\\mbox{\\boldmath$E$}(t)$ which could represent an optical laser, for\nexample. We find that the creation and propagation dynamics of the\ndoublons and holons depends on the spin structure of the mean-field\nbackground. For Ising type anti-ferromagnetic order, we observe a\nquantitative analogy to QED. More specifically, in the vicinity of the\nminimum gap (i.e., the most relevant region for pair creation), the\ndoublons and holons are described by an effective Dirac equation in 1+1\ndimensions in the presence of an electric field\n$\\mbox{\\boldmath$E$}(t)$.\n\nAs a consequence of this quantitative analogy, we may employ the\nmachinery of QED and apply many results regarding electron-positron pair\ncreation to our set-up. For example, in the perturbative (single- or\nmulti-photon) regime with the threshold conditions $\\omega\\geq nU$ for\nthe $n$-th order, the doublon-holon pair creation amplitude\n$\\beta_{\\mbox{\\boldmath$\\scriptstyle k$}}$ yields the perturbative\nscaling\n$\\beta_{\\mbox{\\boldmath$\\scriptstyle k$}}\\sim|q\\mbox{\\boldmath$c$}_{\\rm eff}\\cdot\\mbox{\\boldmath$E$}|^n$.\n\nFor stronger and slower electric fields, we enter the non-perturbative\n(tunneling) regime in analogy to the Sauter-Schwinger effect in QED and\nthus recover the exponential dependence already discussed earlier\nregarding the dielectric breakdown of Mott insulator, see, e.g., . Note\nthat the quantitative analogy established above (see also Table )\nunambiguously determines the pair-creation exponent and pre-factor\nwithout and free fitting parameters.\n\nIf we consider the annihilation of doublon-holon pairs instead of their\ncreation, this analogy applies to the stimulated annihilation within an\nexternal field, but not to the spontaneous annihilation of an\nelectron-positron pair by emitting a pair of photons, for example. In\norder to model this process, one has to include a mechanism for\ndissipating the energy, e.g., by coupling the Fermi-Hubbard model to an\nenvironment, see also .\n\nFor a mean-field background without any spin order, on the other hand,\nthe creation and propagation of doublons and holons does not display\nsuch a quasi-relativistic behavior. The dispersion relation is different\nand the evolution equation deviates from the Dirac equation. Still, for\na purely time-dependent electric field $\\mbox{\\boldmath$E$}(t)$\nconsidered here, the doublon-holon pair creation amplitude\n$\\beta_{\\mbox{\\boldmath$\\scriptstyle k$}}$ for a given mode\n$\\mbox{\\boldmath$k$}$ can again be related to QED. After a\n$\\mbox{\\boldmath$k$}$-dependent phase transformation (which is non-local\nin time), the amplitude $\\beta_{\\mbox{\\boldmath$\\scriptstyle k$}}$ is\ngiven by the same expression, just with the effective speed of light\n$c_{\\rm eff}$ being reduced by a factor of two.\n\nIn view of the $\\mbox{\\boldmath$k$}$-dependence of the phase\ntransformation, this mapping does only work for purely time-dependent\nelectric field $\\mbox{\\boldmath$E$}(t)$. For space-time dependent\nelectric fields $\\mbox{\\boldmath$E$}(t,\\mbox{\\boldmath$r$})$, the\ndeviation of the dispersion relation and the resulting difference in\npropagation become important – which will be the subject of further\nstudies.\n\n|  |  |\n|:--:|:--:|\n| Mott insulator | QED vacuum |\n| upper Hubbard band | positive energy continuum |\n| lower Hubbard band | Dirac sea |\n| doublons & holons | electrons & positrons |\n| Mott gap $U=2m_{\\rm eff}c_{\\rm eff}^2$ | electron mass |\n| velocity $\\mbox{\\boldmath$c$}_{\\rm eff}=\\nabla_{\\mbox{\\boldmath$\\scriptstyle k$}}T_{\\mbox{\\boldmath$\\scriptstyle k$}}|_{\\mbox{\\boldmath$\\scriptstyle k$}_0}$ | speed of light $c$ |\n| Landau-Zener tunneling | Sauter-Schwinger effect |\n| Peierls transformation | gauge transformation |\n\nSketch of the analogy. The electric field $\\mbox{\\boldmath$E$}(t)$ and\nelementary charge $q$ play the same role in both cases.\n\nFunded by the Deutsche Forschungsgemeinschaft (DFG, German Research\nFoundation) – Project-ID 278162697– SFB 1242. PN thanks support from the\nEU project SUPERGALAX (Grant agreement ID: 863313).\n"
    },
    {
        "text": "abstract: Let $Y$ be a (partial) minimal model of a scheme $V$ with a\n  cluster structure (of type $\\mathcal{A}$, $\\mathcal{X}$ or of a\n  quotient of $\\mathcal{A}$ or a fibre of $\\mathcal{X}$). Under natural\n  assumptions, for every choice of seed we associate a Newton–Okounkov\n  body to every divisor on $Y$ supported on $Y \\setminus V$ and show\n  that these Newton–Okounkov bodies are positive sets in the sense of\n  Gross, Hacking, Keel and Kontsevich . This construction essentially\n  reverses the procedure in loc. cit. that generalizes the polytope\n  construction of a toric variety to the framework of cluster varieties.\n  .\n  In a closely related setting, we consider cases where $Y$ is a\n  projective variety whose universal torsor $\\text{UT} _Y$ is a partial\n  minimal model of a scheme with a cluster structure of type\n  $\\mathcal{A}$. If the theta functions parametrized by the integral\n  points of the associated superpotential cone form a basis of the ring\n  of algebraic functions on $\\text{UT} _Y$ and the action of the torus\n  $T_{\\text{Pic}(Y)^*}$ on $\\text{UT} _Y$ is compatible with the cluster\n  structure, then for every choice of seed we associate a\n  Newton–Okounkov body to every line bundle on $Y$. We prove that any\n  such Newton–Okounkov body is a positive set and that $Y$ is a minimal\n  model of a quotient of a cluster $\\mathcal{A}$-variety by the action\n  of a torus.\n  .\n  Our constructions lead to the notion of the intrinsic Newton–Okounkov\n  body associated to a boundary divisor in a partial minimal model of a\n  scheme with a cluster structure. This notion is intrinsic as it relies\n  only on the geometric input, making no reference to the auxiliary data\n  of a valuation or a choice of seed. The intrinsic Newton–Okounkov body\n  lives in a real tropical space rather than a real vector space. A\n  choice of seed gives an identification of this tropical space with a\n  vector space, and in turn of the intrinsic Newton–Okounkov body with a\n  usual Newton–Okounkov body associated to the choice of seed. In\n  particular, the Newton–Okounkov bodies associated to seeds are related\n  to each other by tropicalized cluster transformations providing a wide\n  class of examples of Newton-Okoukov bodies exhibiting a wall-crossing\n  phenomenon in the sense of Escobar–Harada .\n  .\n  This approach includes the partial flag varieties that arise as\n  minimal models of cluster varieties (for example full flag varieties\n  and Grassmannians). For the case of Grassmannians, our approach\n  recovers, up to interesting unimodular equivalences, the\n  Newton–Okounkov bodies constructed by Rietsch–Williams in .\naddress: Instituto de Matemáticas Unidad Oaxaca, Universidad Nacional\nAutónoma de México, León 2, altos, Centro Histórico, 68000 Oaxaca,\nMexico;  School of Mathematics, Kavli IPMU (WPI), UTIAS, The University\nof Tokyo, Kashiwa, Japan, 277-8583; Department of Mathematics, King’s\nCollege London, Strand, London WC2R 2LS, UK;  Consejo Nacional de\nCiencia y Tecnología - Instituto de Matemáticas Unidad Oaxaca,\nUniversidad Nacional Autónoma de México, León 2, altos, Centro\nHistórico, 68000 Oaxaca, Mexico\nauthor: Lara Bossinger, Man-Wai Cheung, Timothy Magee and Alfredo Nájera\nChávez\ndate: 2024-10-13\ntitle: Newton–Okounkov bodies and minimal models for cluster varieties\n\n# Introduction\n\n## Overview\n\nCluster varieties are certain schemes constructed by gluing a (possibly\ninfinite) collection of algebraic tori using distinguished birational\nmaps called cluster transformations. These schemes were introduced in\nand can be studied from many different points of view. They are closely\nrelated to cluster algebras and $Y$-patterns defined by Fomin and\nZelevinsky in . In this paper we approach them from the perspectives of\nbirational and toric geometry, mainly following . In , the authors show\nthat certain sets called *positive polytopes* can be used to produce\ncompactifications of cluster varieties and toric degenerations of such\ncompactifications. In the trivial case where the cluster variety in\nquestion is just a torus, a positive lattice polytope is simply a usual\nconvex lattice polytope and this construction produces the toric variety\nassociated to such a polytope. One of the main goals of this paper is to\nreverse this construction in a systematic way and understand this\nprocess from the view-point of Newton–Okounkov bodies. We also study the\nwall-crossing phenomenon for Newton–Okounkov bodies arising from cluster\nstructures. We treat independently the case of the Grassmannians as, in\nthis context, we compare the Newton–Okounkov bodies we construct with\nthose constructed in and explore some consequences. Moreover, throughout\nthe text we systematically consider not only cluster varieties but also\nquotients and fibres associated to them (see § for the precise\ndefinitions of these quotients and fibres). For simplicity, in this\nintroduction our main focus is on cluster varieties. We fix once and for\nall an algebraically closed field $\\Bbbk$ of characteristic zero. Unless\notherwise stated, all the schemes we consider are over $\\Bbbk$.\n\n## The tropical spaces\n\nLet $\\mathcal{V}$ be a cluster variety. By definition, $\\mathcal{V}$ is\nendowed with an atlas of algebraic tori of the form\n$$\\mathcal{V} = \\bigcup_{\\textbf{s}} T_{L;\\textbf{s}},$$ where $L$ is a\nfixed lattice, $T_{L; \\textbf{s}}$ is a copy of the algebraic torus\n$T_L= \\mathop{\\mathrm{Spec}}(\\Bbbk[L^*])$ associated to $L$ (so\n$L^*=\\text{Hom}(L, \\mathbb{Z} )$) and the tori in the atlas are\nparametrized by *seeds $\\textbf{s}$ for* $\\mathcal{V}$. We will exploit\nthe fact that $\\mathcal{V}$ is a log-Calabi–Yau variety. This property\nimplies that $\\mathcal{V}$ is endowed with a canonical up-to-scaling\nvolume form $\\Omega$. Moreover, recall that a cluster variety is of one\nof the types: $\\mathcal{A}$ or $\\mathcal{X}$.\n\nJust like in toric geometry where one can consider the dual torus\n$T_L^{\\vee}:=T_{L^*}$, the *dual* of $\\mathcal{V}$ is a cluster variety\n$\\mathcal{V} ^{\\vee}$ whose defining atlas consists of tori of the form\n$T^\\vee_L$. It is well known that the ring\n$H^{0} (T_L,\\mathcal{O}_{T_{L}})$ of algebraic functions on $T_L$ has a\ndistinguished basis –the set of characters of $T_L$– parametrized by\n$L^*$. For nearly 10 years it was conjectured that this fact can be\ngeneralized for $\\mathcal{V}$ using this notion of duality. In order to\nstate such a generalization, we consider the integral tropicalization of\n$\\mathcal{V} ^{\\vee}$, which we denote by\n$\\mathrm{Trop} _{\\mathbb{Z} }(\\mathcal{V} ^{\\vee})$. The precise\ndefinition of $\\mathrm{Trop} _{\\mathbb{Z} }(\\mathcal{V} ^{\\vee})$ can be\nfound in §. For this introduction the key fact that we need is that a\nprime divisor $D$ on a variety birational to $\\mathcal{V} ^\\vee$\ndetermines a point of\n$\\mathrm{Trop} _{\\mathbb{Z} }(\\mathcal{V} ^{\\vee})$ if $\\Omega$ has a\npole along $D$. In Fock–Goncharov conjectured that\n$H^{0}(\\mathcal{V} , \\mathcal{O}_\\mathcal{V} )$ has a canonical vector\nspace basis parametrized by\n$\\mathrm{Trop} _{\\mathbb{Z} }(\\mathcal{V} ^{\\vee})$. Although false in\ngeneral, this conjecture does hold in many of the cases of wide\ninterest. In the authors linked this conjecture to the log Calabi–Yau\nmirror symmetry conjecture , suggesting that the canonical basis\nproposed by Fock–Goncharov is the *theta basis*. As we would like to be\nas close to toric geometry as possible we systematically assume that the\nfull Fock–Goncharov conjecture holds for the cluster variety\n$\\mathcal{V}$ under consideration. So, under under the assumption that\nthe full Fock–Goncharov conjecture holds for $\\mathcal{V}$, one may\nconsider $\\mathrm{Trop} _{\\mathbb{Z} }(\\mathcal{V} ^{\\vee})$ as\nreplacing $L^*$ and the characters of $T_L$ are replaced by the theta\nfunctions on $\\mathcal{V}$. Moreover, the real vector space\n$L^*\\otimes \\mathbb{R}$ is replaced by the real tropicalization\n$\\mathrm{Trop} _{\\mathbb{R} }(\\mathcal{V} ^{\\vee})$ and convex polyhedra\ninside $L^*\\otimes \\mathbb{R}$ are replaced by positive sets in the real\ntropical space\n$\\mathrm{Trop} _{\\mathbb{R} }(\\mathcal{V} ^{\\vee})\\supset \\mathrm{Trop} _{\\mathbb{Z} }(\\mathcal{V} ^{\\vee})$\n(see § and Definition for the definitions of\n$\\mathrm{Trop} _{\\mathbb{R} }(\\mathcal{V} ^{\\vee})$ and of positive set,\nrespectively).\n\nBesides the trivial case where $\\mathcal{V}$ is just a torus (and hence\n$\\mathcal{V} ^{\\vee}$ is just the dual torus), the tropical spaces\n$\\mathrm{Trop} _{\\mathbb{Z} }(\\mathcal{V} ^{\\vee})$ and\n$\\mathrm{Trop} _{\\mathbb{R} }(\\mathcal{V} ^{\\vee})$ do not possess a\nlinear structure (there is no natural notion of addition in these spaces\nand only multiplication by positive scalars makes sense). However, in\ncertain situations these tropical spaces do contain subsets where\naddition and scalar multiplication make sense, which we call *linear\nsubsets*. In any case, every choice of seed $\\textbf{s}^\\vee$ for\n$\\mathcal{V} ^{\\vee}$ gives rise to a bijection\n$\\mathfrak{r}_{\\textbf{s}^\\vee}:\\mathrm{Trop} _{\\mathbb{R} }(\\mathcal{V} ^{\\vee}) \\longrightarrow \\mathbb{R} ^d$\nthat restricts to a bijection\n$\\mathrm{Trop} _{\\mathbb{Z} }(\\mathcal{V} ^{\\vee}) \\overset{\\sim}{\\longrightarrow} \\mathbb{Z} ^d$,\nwhere $d$ is the dimension of both $\\mathcal{V}$ and\n$\\mathcal{V} ^\\vee$. In general, different seeds lead to different\nbijections. When we fix one such identification\n$\\mathfrak{r}_{\\textbf{s}^\\vee}$ and talk about linear subsets of\n$\\mathbb{Z} ^d$ and positive subsets of $\\mathbb{R} ^d$, what we mean is\nthat the inverse image of such a set under\n$\\mathfrak{r}_{\\textbf{s}^\\vee}$ has the given property.\n\n## Positive Newton–Okounkov bodies and minimal models\n\nNewton–Okounkov bodies are convex closed sets in real vector spaces.\nTheir systematic study was developed by Lazarsfeld–Mustaţă and\nKaveh–Khovanskii based on the work of Okounkov . This concept is a far\nreaching generalization of both the Newton polytope of a Laurent\npolynomial and the polytope of a polarized projective toric variety. In\nthe authors introduced Newton–Okounkov bodies for Cartier divisors on\nirreducible varieties. In this paper we consider Newton–Okounkov bodies\nassociated to Weil divisors in the setting of minimal models for cluster\nvarieties. More precisely, let $D$ be a Weil divisor on a\n$d$-dimensional normal variety $Y$ admitting a non-zero global section,\nthat is, the space $H^0(Y, \\mathcal{O}(D))$ is non-zero, where\n$\\mathcal{O}(D)$ is the coherent sheaf associated to $D$. The section\nring of $D$ is a graded ring\n$$R(D)=\\bigoplus_{k\\in \\mathbb{Z} _{\\geq 0}}{R}_k(D)$$ whose $k$-th\nhomogeneous component is the vector space\n$R_k(D)=H^0(Y, \\mathcal{O}(kD)) \\subset \\Bbbk (Y)$. Fix a non-zero\nelement $\\tau\\in R_1(D)$, and suppose we are given a total order on\n$\\mathbb{Z} ^d$ and a valuation $\\nu: \\Bbbk(Y)^* \\to \\mathbb{Z} ^d$.\nThen the Newton–Okounkov body associated to this data is:\n$$\\begin{split} \n\\Delta_\\nu(D,\\tau) := \\overline{\\mathop{\\mathrm{conv}}\\Bigg( \\bigcup_{k\\geq 1}  \\left\\{\\frac{\\nu\\left(f/\\tau^k\\right)}{k} \\mid f\\in R_k(D)\\setminus \\{0\\} \\right\\} \\Bigg) }\\subseteq \\mathbb{R} ^d.\n \\end{split}$$\n\nGiven a cluster variety $\\mathcal{V}$, our first goal is to use its\ncluster structure to construct Newton–Okounkov bodies associated to\ndivisors in compactifications of $\\mathcal{V}$, generalizing the\nconstruction of the polytope of a torus invariant divisors on a toric\nvariety. Hence, we need to establish the class of compactifications of\n$\\mathcal{V}$, the divisors therein and the valuations we consider.\n\nWe begin discussing valuations obtained from the cluster structure. In\ncase $\\mathcal{V}$ is a cluster $\\mathcal{A}$-variety, this is closely\nrelated to the work of Fujita and Oya . However, our approach includes\nthe cases where $\\mathcal{V}$ is a cluster $\\mathcal{X}$-variety, a\nquotient of a cluster $\\mathcal{A}$-variety, or a fibre of a cluster\n$\\mathcal{X}$-variety. In order to be able to use the cluster structure\nof $\\mathcal{V}$ to construct a valuation on $\\Bbbk(\\mathcal{V} )$\ncertain conditions (depending on whether $\\mathcal{V}$ is of type\n$\\mathcal{A}$ or of type $\\mathcal{X}$) need to be fulfilled. For\ninstances, if $\\mathcal{V}$ is of type $\\mathcal{A}$, a sufficient\ncondition is that the rectangular matrix $\\widetilde{B}$ determining the\ncluster structure of $\\mathcal{V}$ has full rank[^1]; if $\\mathcal{V}$\nis of type $\\mathcal{X}$ we need that the full Fock–Goncharov conjecture\nholds for $\\mathcal{X}$ (as we are assuming), see § for more details,\nincluding the cases of quotients of $\\mathcal{A}$ and fibres of\n$\\mathcal{X}$. In case the necessary conditions are satisfied then for\nevery $\\textbf{s}$ for $\\mathcal{V}$ we have a cluster valuation\n$$\\nu_\\textbf{s}: \\Bbbk(\\mathcal{V} ) \\setminus\\{0\\} \\to (\\mathbb{Z} ^d, <_{\\textbf{s}}).$$\nThe total order $<_{\\textbf{s}}$ on $\\mathbb{Z} ^d$ depends also on the\ntype of $\\mathcal{V}$. Moreover, in case $\\mathcal{V}$ is of type\n$\\mathcal{A}$ in the literature this valuation is generally denoted by\n$\\mathbf{g} _{\\textbf{s}}$ and called a $\\mathbf{g}$-*vector valuation*\nas it is closely related to the $\\mathbf{g}$-vectors associated to\ncluster monomials introduced in . In case $\\mathcal{V}$ is of type\n$\\mathcal{X}$ the associated cluster valuation has not been\nsystematically defined yet in the literature to the best of our\nknowledge. In this case we also denote $\\nu_{\\textbf{s}}$ by\n$\\mathbf{c} _{\\textbf{s}}$ and call it a $\\mathbf{c}$-*vector valuation*\nsince this valuation is closely related to the $\\mathbf{c}$-vectors\nassociated to $Y$-variables introduced in and more generally to\n**c**-vectors of theta functions on $\\mathcal{X}$ defined in , and\ncurrently investigated in . In any case, for every seed $\\textbf{s}$ the\ntheta basis of $H^0(\\mathcal{V} , \\mathcal{O}_{\\mathcal{V} })$ is\nadapted for the cluster valuation $\\nu_{\\textbf{s}}$. In particular, if\n$Y$ is a variety birational to $\\mathcal{V}$ and $D$ is a divisor in\n$Y$, then, upon a choice of non-zero section $\\tau \\in R_1(D)$ and a\nseed $\\textbf{s}$, we can construct a Newton–Okounkov body\n$\\Delta_{\\nu_\\textbf{s}}(D,\\tau)$. We are primarily interested in\nconditions ensuring that such a Newton–Okounkov body is a positive set.\nOn the one hand this is a condition that needs to be satisfied if one\nseeks to reverse Gross–Hacking–Keel–Kontsevich’s construction of a\ncompactification of a cluster variety from a positive set. On the other\nhand, we are further interested in describing how the change of seed\naffects the Newton–Okounkov body and positivity plays the key role in\nunderstanding this. If $\\Delta_{\\nu_\\textbf{s}}(D,\\tau)$ is positive\nthen any other $\\Delta_{\\nu_{\\textbf{s}'}}(D,\\tau)$ is obtained from\n$\\Delta_{\\nu_\\textbf{s}}(D,\\tau)$ by a composition of tropicalized\ncluster transformations. This will be discussed in more detail in the\nnext subsection of the introduction. In order to be able to show that\n$\\Delta_{\\nu_\\textbf{s}}(D,\\tau)$ is positive we restrict the class of\ncompactifications of $\\mathcal{V}$, the divisors we consider, and the\nsections we choose.\n\nOne can define a partial minimal model for $\\mathcal{V}$[^2] is an\ninclusion $\\mathcal{V} \\subset Y$ such that $Y$ is normal and $\\Omega$\nhas a simple pole along every irreducible divisorial component of the\nboundary $D=Y \\setminus \\mathcal{V}$, see . It is a minimal model if $Y$\nis projective over $\\Bbbk$. These are the kind of (partial)\ncompactifications of $\\mathcal{V}$ we consider. The main reason for this\nis that any prime divisor supported on $D$ determines a primitive point\nof $\\mathrm{Trop} _{\\mathbb{Z} }(\\mathcal{V} )$. Let $D'$ be a divisor\nsupported on $D$. We say that $R(D')$ has a *graded theta basis* if for\neach $k$ the set of theta functions on $\\mathcal{V}$ contained in\n$H^0(Y,\\mathcal{O}(kD'))$ forms a basis (see Definition ). Then we can\nprove the following result.\n\n**Theorem 1**. (Theorem ) Let $D'$ be a Weil divisor supported on the\nboundary $D$ of the minimal model $\\mathcal{V} \\subset Y$ such that\n$R(D')$ has a graded theta basis. Let $\\tau\\in R_1(D')$ be such that\n$\\nu_{\\textbf{s}}(\\tau)$ belongs to a linear subset of $\\mathbb{Z} ^d$.\nThen the Newton–Okounkov body $\\Delta_{\\nu_{\\textbf{s}}}(D',\\tau)$ is a\npositive polytope.\n\nIn Lemma  we provide sufficient conditions ensuring that $R(D)$ has a\ngraded theta basis. Moreover, the work of Mandel provides conditions\nensuring that a line bundle on a cluster $\\mathcal{X}$-variety has a\ngraded theta basis.\n\nWe further study another setting where we can use cluster structures to\nconstruct Newton–Okounkov bodies and show that they are positive\npolytopes: suppose that $Y$ is a normal projective variety such that its\nPicard group is free and finitely generated. The universal torsor of $Y$\nis a scheme $\\text{UT} _Y$ whose ring of algebraic functions is\nisomorphic to the direct sum of all the spaces of sections associated to\nall (isomorphism classes of) line bundles over $Y$. We assume that\n$\\text{UT} _Y$ is a partial minimal model of a cluster\n$\\mathcal{A}$-variety, which we denote by\n$\\mathcal{A} \\subset \\text{UT} _Y$. For example, we encounter this\nsituation frequently in the study of homogeneous spaces, where moreover\nthe ring of global functions on $\\text{UT} _Y$ has a representation\ntheoretic interpretation due to the Borel–Weil–Bott Theorem (Remark ).\nThis fact is commonly used when constructing Newton–Okounkov bodies in\nLie theory, see e.g. and the references therein.\n\nLet $D_1, \\dots, D_s$ be the irreducible divisorial components of\n$D= \\text{UT} _Y \\setminus \\mathcal{V}$ and let\n$\\vartheta ^{\\mathcal{A} ^{\\vee}}_{i}$ be the theta function on\n$\\mathcal{A} ^{\\vee}$ parametrized by the point in\n$\\mathrm{Trop} _{\\mathbb{Z} }(\\mathcal{A} )$ associated to $D_i$. The\n(theta) superpotential[^3] associated to the inclusion\n$\\mathcal{A} \\subset \\text{UT} _Y$ is\n$$W_{\\text{UT} _Y} = \\sum_{i=1}^s \\vartheta ^{\\mathcal{A} ^{\\vee}}_i.$$\nThe associated superpotential cone is the subset $\\Xi_{\\text{UT} _Y}$ of\n$\\mathrm{Trop} _{\\mathbb{R} }(\\mathcal{V} ^{\\vee})$ where the\ntropicalized superpotential takes non-negative values. Given a choice of\nseed $\\textbf{s}^\\vee$ for $\\mathcal{A} ^\\vee$, $\\Xi_{\\text{UT} _Y}$ is\nidentified with a polyhedral cone\n$\\Xi_{\\text{UT} _Y, \\textbf{s}^\\vee}\\subset \\mathbb{R} ^d$. As discussed\nin , in many cases the integral points of $\\Xi_{\\text{UT} _Y}$\nparametrize the set of theta functions on $\\mathcal{A}$ that extend to\n$\\text{UT} _Y$. This happens for example if $\\mathcal{A}$ has *theta\nreciprocity* (see Definition ), a condition that is conjectured to be\ntrue in situations more general than ours. Even stronger, in many of the\nexamples arising in nature the integral points of $\\Xi_{\\text{UT} _Y}$\nparametrize a basis of $H^0(\\text{UT} , \\mathcal{O}_{\\text{UT} _Y})$. In\n, Gross–Hacking–Keel–Kontsevich give criteria ensuring that this is\nsatisfied. These conditions hold true in many cases of interest in\nrepresentation theory, as was proven in several papers including and .\nMoreover, for special choices of seeds $\\textbf{s}^{\\vee}$, in these\ncases the cone $\\Xi_{\\text{UT} _Y, \\textbf{s}^\\vee}$ agrees with known\npolyhedral cones such as the Gelfand–Tsetlin cone, string cones or the\nKnudson–Tao hive cone. Much of the inspiration of this paper is due to\nthe representation theoretic results that precede it. In the case where\nthe integral points of $\\Xi_{\\text{UT} _Y}$ parametrize the set of theta\nfunctions on $\\mathcal{A}$ that extend to $\\text{UT} _Y$, we can\nrestrict a **g**-vector valuation $\\mathbf{g} _\\textbf{s}$ from\n$\\Bbbk(\\mathcal{A} )$ to\n$H^0(\\text{UT} _Y, \\mathcal{O}_{\\text{UT} _Y})$. Therefore, given a line\nbundle $\\mathcal{L}$ on $Y$ we can construct a Newton–Okounkov body\n$\\Delta_{\\mathbf{g} _\\textbf{s}}(\\mathcal{L} )$ in a similar way as\nbefore. In order to show that\n$\\Delta_{\\mathbf{g} _{\\textbf{s}}}(\\mathcal{L} )$ is a positve polytope\nwe need to consider torus actions on $\\mathcal{A}$ and fibrations of\n$\\mathcal{A} ^{\\vee}$ over a torus as we now explain.\n\nThe universal torsor $\\text{UT} _Y$ is endowed with the action of the\ntorus $T_{\\text{Pic}(Y)^*}$ associated to the dual of the Picard group\nof $Y$. We first need this torus action to preserve $\\mathcal{A}$ and\nthat the induced action on $\\mathcal{A}$ is cluster in the sense of\n(roughly speaking this means that the restricted action can be\nidentified with the action induced by the choice of a sublattice of the\nkernel of $\\widetilde{B}$). In such situations we have a cluster\nfibration $$w:\\mathcal{A} ^{\\vee}\\to T_{\\text{Pic}(Y)}.$$ Recall that\nthe choice of seed gives rise to the identification\n$\\mathfrak{r}_{\\textbf{s}^\\vee}:\\mathrm{Trop} _{\\mathbb{R} }(\\mathcal{A} ^{\\vee}) \\to \\mathbb{R} ^d$.\nThe tropicalization of $w$ expressed using such an identification is a\nlinear map $w^T:\\mathbb{R} ^d \\to \\text{Pic}(Y)\\otimes \\mathbb{R}$.\nUnder the conditions above the Newton–Okounkov body\n$\\Delta_{\\mathbf{g} _{\\textbf{s}}}(\\mathcal{L} )$ can be described as a\nslicing of the superpotential cone. More precisely, we have the\nfollowing result (see Definition ).\n\n**Theorem 2**. () Assume that the theta functions on $\\mathcal{V}$\nparametrized by the integral points of $\\Xi_{\\text{UT} _Y}$ form a basis\nof $H^0(\\text{UT} _Y, \\mathcal{O}_{\\text{UT} _Y})$. If the action of\n$T_{\\text{Pic}(Y)^*}$ restricts to a cluster action of\n$T_{\\text{Pic}(Y)^*}$ on $\\mathcal{A}$ then for any class\n$[\\mathcal{L} ]\\in \\text{Pic}(Y)$ the Newton–Okounkov body\n$\\Delta_{{\\bf g}_{\\textbf{s}}}(\\mathcal{L} )$ can be describe as\n$$\\Delta_{{\\bf g}_{\\textbf{s}}}(\\mathcal{L} )=\\mathrm{Trop} _{\\mathbb{R} }(w)^{-1}([ \\mathcal{L} ])\\cap \\Xi_{\\text{UT} _Y, \\textbf{s}}.$$\nIn particular, $\\Delta_{{\\bf g}_{\\textbf{s}}}(\\mathcal{L} )$ is a\npositive subset of $\\mathrm{Trop} _{\\mathbb{R} }(\\mathcal{V} ^{\\vee})$\nand $Y$ is a minimal model of the quotient of $\\mathcal{A}$ by the\naction of $T_{\\text{Pic}(Y)^*}$.\n\nThe case where $Y$ is the Grassmannian $\\text{Gr}_{n-k}(\\mathbb{C} ^n)$\nfits the framework above so it is possible to use the cluster\n$\\mathcal{A}$ structure to construct Newton–Okounkov bodies associated\nto arbitrary line bundles over $\\text{Gr}_{n-k}(\\mathbb{C} ^n)$. We show\nthat the Newton–Okounkov bodies we construct are unimodular to the\nNewton–Okounkov bodies constructed for $\\text{Gr}_{n-k}(\\mathbb{C} ^n)$\nby Rietsch and Williams in using the cluster $\\mathcal{X}$ structure on\nGrassmannians (see Theorem ). Moreover, the flow valuations of are\ninstances of $\\bf c$-vector valuations.\n\nThis comparison result already has interesting consequences related to\ntoric degenerations:\n\n1.  Given a rational polytopal Newton–Okounkov body $\\Delta$ for a (very\n    ample) line bundle $\\mathcal{L}$ over $Y$ Anderson’s main result in\n    applies and it yields a toric degeneration of $Y$ to a toric variety\n    (whose normalization is) defined by $\\Delta$. As the semigroup\n    algebras of the **g**-vector valuations are saturated, no\n    normalization is necessary.\n\n2.  The construction of Gross–Hacking–Keel–Kontsevich in associates to a\n    positive polytope $P$ a minimal model $\\mathcal{V} \\subset Y$ and\n    moreover, using Fomin–Zelevinsky’s principal coefficients, a toric\n    degneration of $Y$ to the toric variety defined by $P$. As our\n    Newton–Okounkov bodies are positive polytopes, this construction\n    applies in our setting.\n\nThe identification of the Newton–Okounkov bodies constructed by\nRietsch–Williams and our Newton–Okounkov bodies constructed from\n**g**-vectors implies the following result.\n\n**Theorem 3**. (Theorem  and Remark ) The toric degenerations of\n$\\text{Gr}_{n-k}(\\mathbb{C} ^n)$ determined by the Newton–Okounkov\npolytopes constructed by Rietsch–Williams using Anderson’s result\ncoincide with the toric degenerations of\n$\\text{Gr}_{n-k}(\\mathbb{C} ^n)$ given by Gross–Hacking–Keel–Kontsevich\nconstruction using principal coefficients.\n\n## The intrinsic Newton–Okounkov body\n\nUnderstanding how Newton–Okounkov bodies change upon changing the\nvaluation is an interesting problem that has attracted the attention of\nseveral authors, see for example . So let us return to the discussion on\nhow the Newton–Okounkov bodies constructed above transform if we change\nthe choice of seed. Given any two seeds $\\textbf{s}$ and $\\textbf{s}'$\nfor $\\mathcal{V} ^\\vee$ there is a piecewise linear bijection\n$\\mathrm{Trop} _{\\mathbb{R} }(\\mu^{\\mathcal{V} ^\\vee}_{\\textbf{s},\\textbf{s}'}):\\mathbb{R} ^d \\to \\mathbb{R} ^d$\nrelating the identifications of\n$\\mathrm{Trop} _{\\mathbb{R} }(\\mathcal{V} ^{\\vee})$ with\n$\\mathbb{R} ^d$. More precisely, we have a commutative diagram\n$$\\xymatrix{\n&\\mathrm{Trop} _{\\mathbb{R} }(\\mathcal{V} ^{\\vee})\n\\ar_{\\mathfrak{r}_{\\textbf{s}}}[dl] \\ar^{\\mathfrak{r}_{\\textbf{s}'}}[dr] & \\\\\n\\mathbb{R} ^d \\ar^{\\mathrm{Trop} _{\\mathbb{R} }(\\mu^{\\mathcal{V} ^\\vee}_{\\textbf{s},\\textbf{s}'})}[rr]&   & \\mathbb{R} ^d.\n}$$ Every map\n$\\mathrm{Trop} _{\\mathbb{R} }(\\mu^{\\mathcal{V} ^\\vee}_{\\textbf{s},\\textbf{s}'})$\nrestricts to a piecewise linear bijection of $\\mathbb{Z} ^d$ and, by\nconstruction, the maps\n$\\mathrm{Trop} _{\\mathbb{R} }(\\mu^{\\mathcal{V} ^\\vee}_{\\textbf{s},\\textbf{s}'})$\nare composition of tropicalized cluster transformations for\n$\\mathcal{V} ^\\vee$ (see § for a more concise description). For a subset\n$P\\subseteq \\mathrm{Trop} _{\\mathbb{R} }(\\mathcal{V} ^{\\vee})$ we let\n$P_{\\textbf{s}}=\\mathfrak{r}_{\\textbf{s}}(P)$. One of the main\nproperties behind our interest in showing that the Newton–Okounkov\nbodies we have constructed are positive sets is the following: if\n$P\\subseteq \\mathrm{Trop} _{\\mathbb{R} }(\\mathcal{V} ^{\\vee})$ is a\npositive set then\n$\\mathrm{Trop} _{\\mathbb{R} }(\\mu^{\\mathcal{V} ^\\vee}_{\\textbf{s},\\textbf{s}'})(P_{\\textbf{s}})=P_{\\textbf{s}'}$\nfor any two seeds, $\\textbf{s}$ and $\\textbf{s}'$. In particular, in\nthis situation the entire collection of sets\n$\\{P_\\textbf{s}\\}_{\\textbf{s}}$ parametrized by the seed for\n$\\mathcal{V} ^{\\vee}$ may be replaced by $P$, a single intrinsic object\nthat can be used to recover any $P_\\textbf{s}$ in the family.\n\nIn the case where a Newton–Okounkov body $\\Delta_{\\nu_{\\textbf{s}}}$\n(associated to a line bundle $\\mathcal{L}$ or a pair $(D',\\tau)$ as in\nthe previous subsection) is positive, any other Newton–Okounkov body\n$\\Delta_{\\nu_{\\textbf{s}'}}$ associated to the same data is also\npositive. In this situation there is a single intrinsic object\n$\\Delta_{\\mathrm{BL}} \\subset\\mathrm{Trop} _{\\mathbb{R} }(\\mathcal{V} ^{\\vee})$\nrepresenting the entire collection\n$\\{ \\Delta_{\\nu_{\\textbf{s}}}\\}_{\\textbf{s}}$. We call\n$\\Delta_{\\mathrm{BL}}$ the *intrinsic Newton–Okounkov body* (associate\nto the data we begin with). The subindex $\\mathrm{BL}$ in\n$\\Delta_{\\mathrm{BL}}$ stands for *broken line*, the choice of this\nnotation goes back to where the last three authors of this paper\nintroduce *broken line convexity*– a notion of convexity defined in a\ntropical space that ensures positivity. Broken lines are pieces of\ntropical curves in $\\mathrm{Trop} _{\\mathbb{R} }(\\mathcal{V} ^{\\vee})$\nused to define theta functions on $\\mathcal{V}$ and describe their\nmultiplication (see §). Straight line segments defining convexity in a\nlinear space are replaced by broken line segments in the tropical space\nto define broken line convexity. The main result of is that a closed set\nis broken line convex if and only if it is positive.\n\nIn the situations where we are able to show that\n$\\Delta_{\\nu_{\\textbf{s}}}\\subset \\mathbb{R} ^d$ is positive, it turns\nout that it is moreover polyhedral, a property that fails in general,\nsee e.g. . Since\n$\\Delta_{\\nu_{\\textbf{s}'}}=\\mathrm{Trop} _{\\mathbb{R} }(\\mu^{\\mathcal{V} ^\\vee}_{\\textbf{s},\\textbf{s}'})(\\Delta_{\\nu_{\\textbf{s}}})$\nany other $\\Delta_{\\nu_{\\textbf{s}'}}$ is also polyhedral. The integral\npoints of the convex bodies we consider are naturally associated to\ntheta functions, which suggests is the following question: does there\nexist a finite set of theta functions such that\n$\\Delta_{\\nu_{\\textbf{s}'}}$ is the convex hull of their images under\n$\\nu_{\\textbf{s}'}$ for any seed $\\textbf{s}'$? Such a collection of\npoints might vary as we change seeds as exhibited in the case of the\nGrassmannians in an example in and generalized to an infinite family of\nexamples in . Given the notion of broken line convexity, a slight\nreformulation of the question becomes more natural: does there exist a\nfinite set of theta functions such that the broken line convex hull of\ntheir images under $\\nu_{\\textbf{s}'}$ is $\\Delta_{\\nu_{\\textbf{s}'}}$\nfor some (and hence any) seed $\\textbf{s}'$? In fact, from the intrinsic\nNewton–Okounkov body perspective, the valuation is replaced by integral\ntropical points parametrizing theta functions and there is no reference\nto a seed at all. Using this perspective, $\\Delta_{\\mathrm{BL}}$ becomes\na broken line convex subset of\n$\\mathrm{Trop} _{\\mathbb{R} }(\\mathcal{V} ^{\\vee})$ whose integral\npoints parametrize the theta basis of the first graded piece $R_1$ of\nthe corresponding graded ring. In we give sufficient conditions ensuring\nthat $\\Delta_{\\mathrm{BL}}$ can be described as the broken line convex\nhull of a finite collection of points and describe this collection.\nApplying this result to the setting of Grassmannians we obtain that if\n$\\mathcal{L} _e$ is line bundle over\n$\\mathop{\\mathrm{Gr}}_{n-k}(\\mathbb{C} ^n)$ obtained by pullback of\n$\\mathcal{O}(1)$ under the Plücker embedding\n$\\mathop{\\mathrm{Gr}}_{n-k}(\\mathbb{C} ^n)\\hookrightarrow \\mathbb P^{\\binom{n}{k}-1}$\nthen the intrinsic Newton–Okounkov body\n$\\Delta_{\\mathrm{BL}}(\\mathcal{L} _e)$ is the broken line convex hull of\nthe ${\\bf g}$-vectors of the Plücker coordinates (Corollary ).\n\nBroken line convexity also allows to generalize the Newton polytope of a\nLaurent polynomial to the the world of cluster varieties. In particular,\nin § we introduce the *theta function analog of the Newton polytope* of\n$f$, for any $f\\in H^0(\\mathcal{V} , \\mathcal{O}_\\mathcal{V} )$. The\nintrinsic Newton–Okounkov bodies $\\Delta_{\\mathrm{BL}}$ can be described\nusing this notion. The key idea is exploiting the bijection between the\ntheta basis (a special case of an *adapted basis*) and integral tropical\npoints parametrizing them. This idea is explained for full rank\nvaluations with finitely generated value semigroup in the survey . It is\ntherefore interesting to continue studying this new class of objects.\n\n## Organization of the paper\n\nIn § we review background material on cluster varieties their quotients\nand their fibres (§), and on tropicalization (§). In § we recall the\nconstruction of cluster scattering diagrams and the theta functions on\n(quotients and fibres of) cluster varieties. In § we elaborate on the\nexistence of a theta basis on the ring of regular functions on a partial\nminimal model of (a quotient or a fibre of) a cluster variety. This\nsection largely follows . In § we recall the **g**-vector valuations for\n(quotients) $\\mathcal{A}$-varieties. We introduce **c**-vector\nvaluations for (fibres of) $\\mathcal{X}$-varieties. The main results of\nthe paper are contained in §. The study of Newton–Okoukov bodies\nassociated to Weil divisors on minimal models is treated in § while the\nNewton–Okoukov bodies for line bundles are treated in §. The intrinsic\nNewton–Okounkov body and the wall-crossing phenomenon for these are\naddressed in §. Finally, in § we apply the results of the previous\nsection to Grassmannians. One of the main technical conditions to be\nsatisfied is verified in §. In § we prove a unimodular equivalence\nbetween the Newton–Okounkov bodies we construct and those constructed by\nRietsch–Williams in . In § we describe the intrinsic Newton–Okounkov\nbodies for Grassmannians as the broken line convex hull of the\n**g**-vectors of Plücker coordinates (in arbitrary seeds).\n\n### Acknowledgements\n\nThe authors L. Bossinger and A. Nájera Chávez were partially supported\nby PAPIIT project IA100122 dgapa UNAM 2022 and by CONACyT project\nCF-2023-G-106. M. Cheung was supported by World Premier International\nResearch Center Initiative (WPI Initiative), MEXT, Japan. T. Magee was\nsupported by EPSRC grant EP/V002546/1.\n\n# Preliminaries\n\n## Cluster varieties, quotients and fibres\n\nWe briefly recall the construction of cluster varieties, their quotients\nand their fibres. The reader is invited to consult for the details we\nshall omit in this section.\n\nUnless otherwise stated, all tensor products are taken with respect to\n$\\mathbb{Z}$. Moreover, given a lattice $L$ we denote by\n$L^*:= \\mathop{\\mathrm{Hom}}(L,\\mathbb{Z} )$ its $\\mathbb{Z}$-dual and\nlet $\\langle \\cdot , \\cdot \\rangle: L\\times L^* \\to \\mathbb{Z}$ be the\ncanonical pairing given by evaluation. We further denote by\n$L_\\mathbb{R} := L \\otimes \\mathbb{R}$ the real vector space associated\nto $L$. We fix an algebraically closed field $\\Bbbk$ of characteristic\n$0$ and let $T_L:= \\text{Spec}(\\Bbbk [L^*])$ be the algebraic torus\nwhose character lattice is $L^*$.\n\n### Cluster varieties and their dualities\n\nThe **fixed data** $\\Gamma$ consist of the following:\n\n-   a finite set $I$ of **directions** and a distinguished subset\n    $I_{\\text{uf}}\\subseteq I$ of **mutable** (or **unfrozen**)\n    **directions**. Elements of $I \\setminus I_{\\text{uf}}$ are the\n    **frozen directions**;\n\n-   a lattice $N$ of rank $|I|$ together with a saturated sublattice\n    $N_{\\text{uf}}\\subseteq N$ of rank $|I_{\\text{uf}}|$;\n\n-   a skew-symmetric bilinear form\n    $\\{ \\cdot , \\cdot \\} : N \\times N \\rightarrow \\mathbb{Q}$;\n\n-   a finite index sublattice $N^\\circ \\subseteq N$ such that\n    $\\{ N, N_{\\text{uf}}\\cap N^{\\circ}\\}\\subset \\mathbb{Z}$ and\n    $\\{ N_{\\text{uf}}, N^{\\circ} \\}\\subset \\mathbb{Z}$;\n\n-   a collection of positive integers $\\{d_i\\}_{i \\in I}$ with greatest\n    common divisor $1$;\n\n-   the dual lattices $M = \\mathop{\\mathrm{Hom}}(N, \\mathbb{Z} )$ and\n    $M^{\\circ}=\\mathop{\\mathrm{Hom}}(N^{\\circ},\\mathbb{Z} )$.\n\nA ${\\bf seed}$ for $\\Gamma$ is a tuple $\\textbf{s}:= ( e_i )_{i \\in I}$\nsuch that $\\{ e_i \\}_{i\\in I}$ is a basis for $N$,\n$\\{e_i\\}_{i \\in I_{\\text{uf}}}$ is a basis for $N_{\\text{uf}}$ and\n$\\{d_i e_i \\}_{i \\in I }$ is a basis for $N^{\\circ}$. We let\n$f_i := {d_i}^{-1} e_i^*$ and observe that $\\{f_i\\}_{i\\in I}$ is a basis\nof $M^{\\circ}$. For $i,j\\in I$ we write\n$\\epsilon_{ij}:= \\lbrace e_{i},d_j e_{j} \\rbrace$ and define the matrix\n$\\epsilon=(\\epsilon_{ij})_{i,j\\in I}$. When we work with various seeds\nat the same time we introduce labels of the form $e_{i;\\textbf{s}}$,\n$f_{i;\\textbf{s}}$, $\\epsilon_{\\textbf{s}}=(\\epsilon_{ij;\\textbf{s}})$,\netc. to distinguish the data associated to $\\textbf{s}$. We can\n**mutate** a seed $\\textbf{s}=(e_i)_{i\\in I}$ in a mutable direction\n$k\\in I_{\\text{uf}}$ to obtain a new seed\n$\\mu_k(\\textbf{s})=(e'_i)_{i\\in I}$ given by $$\\label{e_mutation}\ne_i':=\\begin{cases} e_i+[\\epsilon_{ik}]_+e_k & i\\neq k,\\\\\n-e_k&i=k,\n\\end{cases}$$ where $[x]_+:= \\text{max}(0,x)$ for $x \\in \\mathbb{R}$.\n\nLet $r:=|I_{\\text{uf}}|$ and let $\\mathbb{T}_r$ denote the $r$-regular\ntree whose edges are labeled by the elements of $I_{\\text{uf}}$. We\nrefer to $r$ as the **rank** and fix it one and for all. By a common\nabuse of notation, the set of vertices of this tree is also denoted by\n$\\mathbb T_r$. We fix once and for all a distinguished vertex\n$v_0\\in \\mathbb{T}_r$ and let $%\n  \\mathrel{\\mathop{\\mathbb{T}_r}\\limits^{\n    \\vbox to0ex{\\kern-0.5\\ex@\n    \\hbox{$\\scriptstyle\\longrightarrow$}\\vss}}}$ be the unique\norientation of $\\mathbb{T}_r$ such that the $r$ edges incident to $v_0$\nare oriented in outgoing direction from $v_0$, and every vertex\ndifferent from $v_0$ has one incoming edge and $r-1$ outgoing edges. We\nwrite $v\\overset{k}{\\longrightarrow}v'\\in %\n  \\mathrel{\\mathop{\\mathbb{T}_r}\\limits^{\n    \\vbox to0ex{\\kern-0.5\\ex@\n    \\hbox{$\\scriptstyle\\longrightarrow$}\\vss}}}$ to indicate that the\nedge in between the vertices $v,v'$ of $%\n  \\mathrel{\\mathop{\\mathbb{T}_r}\\limits^{\n    \\vbox to0ex{\\kern-0.5\\ex@\n    \\hbox{$\\scriptstyle\\longrightarrow$}\\vss}}}$ is oriented from $v$ to\n$v'$ and is labeled by $k$.\n\nFix once and for all a seed $\\textbf{s}_0=(e_i\\mid i \\in I)$ and call it\nthe **initial seed**. To every vertex $v\\in \\mathbb{T}_r$ we attach a\nseed $\\textbf{s}_v$ as follows: we let $\\textbf{s}_{v_0}=\\textbf{s}_0$,\nif $v\\overset{k}{\\longrightarrow}v'\\in %\n  \\mathrel{\\mathop{\\mathbb{T}_r}\\limits^{\n    \\vbox to0ex{\\kern-0.5\\ex@\n    \\hbox{$\\scriptstyle\\longrightarrow$}\\vss}}}$ then\n$\\textbf{s}_{v'}=\\mu_k(\\textbf{s}_{v})$. For simplicity we write\n$\\textbf{s}\\in %\n  \\mathrel{\\mathop{\\mathbb{T}_r}\\limits^{\n    \\vbox to0ex{\\kern-0.5\\ex@\n    \\hbox{$\\scriptstyle\\longrightarrow$}\\vss}}}$ if\n$\\textbf{s}=\\textbf{s}_v$ for some $v\\in %\n  \\mathrel{\\mathop{\\mathbb{T}_r}\\limits^{\n    \\vbox to0ex{\\kern-0.5\\ex@\n    \\hbox{$\\scriptstyle\\longrightarrow$}\\vss}}}$.\n\nFor every seed $\\textbf{s}=(e_{i;\\textbf{s}}\\mid i\\in I)\\in %\n  \\mathrel{\\mathop{\\mathbb{T}_r}\\limits^{\n    \\vbox to0ex{\\kern-0.5\\ex@\n    \\hbox{$\\scriptstyle\\longrightarrow$}\\vss}}}$ we introduce the **seed\ntori** $\\mathcal{A} _{\\textbf{s}} = T_{N^{\\circ}}$ and\n$\\mathcal{X} _{\\textbf{s}} = T_{M}$ which are endowed with the **cluster\ncoordinates** $\\{A_{i;\\textbf{s}} := z^{f_{i;\\textbf{s}}}\\}_{i \\in I}$\nand $\\{X_{i;\\textbf{s}} := z^{e_{i;\\textbf{s}}}\\}_{i \\in I}$,\nrespectively. The **$\\mathcal{A}$-cluster transformation** associated to\n$\\textbf{s}$ and $k \\in I_{\\text{uf}}$ is the birational map\n$\\mu^{\\mathcal{A} }_{k}:\\mathcal{A} _{\\textbf{s}} \\dashrightarrow \\mathcal{A} _{\\mu_k(\\textbf{s})}$\nspecified by the pullback formula $$\\label{A_mut}\n(\\mu^{\\mathcal{A} }_{k})^*(z^m):=z^{m} (1+z^{v_{k;\\textbf{s}}})^{-\\langle d_k e_{k;\\textbf{s}},m\\rangle} \\ \\ \\text{ for }m\\in M^{\\circ},$$\nwhere $v_{k;\\textbf{s}}:=\\{e_{k;\\textbf{s}}, \\cdot \\}\\in M^{\\circ}$.\nSimilarly, the **$\\mathcal{X}$-cluster transformation** associated to\n${\\mathbf{s}}$ and $k$ is the birational map\n$\\mu^{\\mathcal{X} }_{k}:\\mathcal{X} _{\\textbf{s}} \\dashrightarrow \\mathcal{X} _{\\mu_k(\\textbf{s})}$\nspecified by the pull-back formula $$\\label{X_mut}\n(\\mu^{\\mathcal{X} }_{k})^*(z^n):=z^{n} (1+z^{e_{k;\\textbf{s}}})^{-[ n,e_{k;\\textbf{s}} ]}\\ \\ \\text{ for }n\\in N,$$\nwhere $[\\cdot, \\cdot]:N\\times N \\to \\mathbb{Q}$ is the bilinear form\ndetermined by setting $[e_i,e_j]=\\left\\{e_i, d_je_j\\right\\}$.\n\nFor seeds $\\textbf{s}, \\textbf{s}'\\in %\n  \\mathrel{\\mathop{\\mathbb{T}_r}\\limits^{\n    \\vbox to0ex{\\kern-0.5\\ex@\n    \\hbox{$\\scriptstyle\\longrightarrow$}\\vss}}}$ connected by iterated\nmutation in a sequence of directions $k_1, \\dots, k_s\\in I_{\\text{uf}}$,\nwe let $\\mu^{\\mathcal{A} }_{\\textbf{s}, \\textbf{s}'}$ (resp.\n$\\mu^{\\mathcal{X} }_{\\textbf{s}, \\textbf{s}'}$) be the composition of\ncluster transformations in the same sequence of directions and in the\nsame order. A birational transformation of the form\n$\\mu^{\\mathcal{A} }_{\\textbf{s}, \\textbf{s}'}$ (or\n$\\mu^{\\mathcal{X} }_{\\textbf{s}, \\textbf{s}'}$) can be used to glue its\ndomain and range by identifying the largest open subschemes where the\ntransformation is an isomorphism. We use this kind of gluing to define\ncluster varieties. More precisely, the cluster $\\mathcal{A}$-variety\nassociated to $\\Gamma$ and $\\textbf{s}_0$ is\n$$\\mathcal{A} _{\\Gamma,\\textbf{s}_0}:=\\bigcup\\limits_{\\textbf{s}\\in %\n  \\mathrel{\\mathop{\\mathbb{T}_r}\\limits^{\n    \\vbox to0ex{\\kern-0.5\\ex@\n    \\hbox{$\\scriptstyle\\longrightarrow$}\\vss}}}} \\mathcal{A} _{\\textbf{s}}/ \\left( \\text{gluing by } \\mu^{\\mathcal{A} }_{\\textbf{s}', \\textbf{s}''} \\right)_{\\textbf{s}',\\textbf{s}''\\in%\n  \\mathrel{\\mathop{\\mathbb{T}_r}\\limits^{\n    \\vbox to0ex{\\kern-0.5\\ex@\n    \\hbox{$\\scriptstyle\\longrightarrow$}\\vss}}}}.$$ The cluster\n$\\mathcal{X}$-variety associated to $\\Gamma$ and $\\textbf{s}_0$ is\n$$\\mathcal{X} _{\\Gamma,\\textbf{s}_0}:=\\bigcup\\limits_{\\textbf{s}\\in %\n  \\mathrel{\\mathop{\\mathbb{T}_r}\\limits^{\n    \\vbox to0ex{\\kern-0.5\\ex@\n    \\hbox{$\\scriptstyle\\longrightarrow$}\\vss}}}} \\mathcal{X} _{\\textbf{s}}/ \\left( \\text{gluing by } \\mu^{\\mathcal{X} }_{\\textbf{s}', \\textbf{s}''} \\right)_{\\textbf{s}',\\textbf{s}''\\in%\n  \\mathrel{\\mathop{\\mathbb{T}_r}\\limits^{\n    \\vbox to0ex{\\kern-0.5\\ex@\n    \\hbox{$\\scriptstyle\\longrightarrow$}\\vss}}}}.$$\n\nFrom now on an element $\\textbf{s}\\in %\n  \\mathrel{\\mathop{\\mathbb{T}_r}\\limits^{\n    \\vbox to0ex{\\kern-0.5\\ex@\n    \\hbox{$\\scriptstyle\\longrightarrow$}\\vss}}}$ will be referred to as\na seed for $\\mathcal{A}$ (or $\\mathcal{X}$). It is important to recall\nthat declaring another $\\textbf{s}\\in %\n  \\mathrel{\\mathop{\\mathbb{T}_r}\\limits^{\n    \\vbox to0ex{\\kern-0.5\\ex@\n    \\hbox{$\\scriptstyle\\longrightarrow$}\\vss}}}$ as an initial seed\ngives rise to isomorphic cluster varieties. We fix the pair\n$(\\Gamma,\\textbf{s})$ once and for all and denote\n$\\mathcal{A} _{\\Gamma, \\textbf{s}_0}$ (resp.\n$\\mathcal{X} _{\\Gamma, \\textbf{s}_0}$) simply by $\\mathcal{A}$ (resp.\n$\\mathcal{X}$).\n\n### Quotients of $\\mathcal{A}$-varieties and fibres of $\\mathcal{X}$-varieties\n\nLet\n$N^{\\perp}_{\\text{uf}}:= \\{ m\\in M \\mid \\langle n, m \\rangle=0 \\ \\forall \\  n\\in N_{\\text{uf}} \\}$.\nIn particular,\n$M/ N^{\\perp}_{\\operatorname{uf}}\\cong (N_{\\text{uf}})^*$. By a slight\nabuse of notation we also write $M^{\\circ}/ N_{\\text{uf}}^{\\perp}$. Here\n$N_{\\text{uf}}^\\perp$ is taken in $M^\\circ$ rather than $M$, so\n$M^{\\circ}/ N_{\\text{uf}}^{\\perp}$ is torsion free. Since\n$\\{ N_{\\text{uf}},N \\}\\subseteq \\mathbb{Z}$ the following homomorphisms\nare well defined $$\\begin{aligned}\n \\label{eq:p12star}\n  \\begin{matrix}\n    p_1^*: & N_{\\operatorname{uf}} & \\rightarrow & M^\\circ &\\qquad \\phantom{aaaaa} \\qquad \\qquad & p_2^* : & N & \\rightarrow& M^{\\circ}/ N^{\\perp}_{\\operatorname{uf}}. \\\\\n    & n &\\mapsto & \\{ n, \\cdot \\}\n  & \\qquad  & &  n &\\mapsto &  \\{ n, \\cdot \\} + N_{\\text{uf}}^{\\perp}\n  \\end{matrix}\n\\end{aligned}$$ The matrix representing $p_2^*$ with respect to a seed\n$\\textbf{s}\\in %\n  \\mathrel{\\mathop{\\mathbb{T}_r}\\limits^{\n    \\vbox to0ex{\\kern-0.5\\ex@\n    \\hbox{$\\scriptstyle\\longrightarrow$}\\vss}}}$ is the *extended\nexchange matrix* $\\widetilde{B}_{\\textbf{s}}$ of .\n\n**Definition 1**.\n\nA **cluster ensemble lattice map** for $\\Gamma$ is a homomorphism\n$p^*: N \\to M^\\circ$ such that $p^*|_{N_{\\text{uf}}} = p^*_1$ and the\ncomposition\n$N \\overset{p^*}{\\longrightarrow} M^\\circ \\twoheadrightarrow M^{\\circ}/ N^{\\perp}_{\\operatorname{uf}}$\nagrees with $p_2^*$, where\n$M^\\circ \\twoheadrightarrow M^{\\circ}/ N^{\\perp}_{\\operatorname{uf}}$\ndenotes the canonical projection. Note that different choices of $p^*$\ndiffer by a homomorphism\n$N/ N_{\\operatorname{uf}} \\rightarrow  N^{\\perp}_{\\operatorname{uf}}$.\n\nIn other words, given a seed $\\mathbf{s}$, the $|I|\\times|I|$ square\nmatrix $B_{p^*;\\mathbf{s}}$ associated to a cluster ensemble lattice map\n$p^*$ with respect to the bases $(e_i)_{i\\in I}$ and $(f_i)_{i\\in I}$\nsatisfies $$\\label{eq:Mp*}\nB_{p^*;\\mathbf{s}} - \\epsilon^{\\rm{tr}}_\\textbf{s}=\n\\left[\\begin{matrix}\n0 & 0 \\\\\n0 & \\ast\n\\end{matrix}\\right],$$ where the $0$ entries represent the blocks\n$I_{\\text{uf}}\\times I_{\\text{uf}}$,\n$I_{\\text{uf}}\\times (I\\setminus I_{\\text{uf}})$, and\n$(I\\setminus I_{\\text{uf}})\\times I_{\\text{uf}}$, and the $\\ast$ entry\nindicates that the\n$(I\\setminus I_{\\text{uf}})\\times(I\\setminus I_{\\text{uf}})$ block has\nno constraints. Every cluster ensemble lattice map $p^*:N\\to M^{\\circ}$\ncommutes with mutation. Therefore, $p^*$ gives rise to a **cluster\nensemble map** $$p:\\mathcal{A} \\to \\mathcal{X} .$$\n\nThe map $p:\\mathcal{A} \\to \\mathcal{X}$ yields both, torus actions on\n$\\mathcal{A}$ and fibrations of $\\mathcal{X}$ over a torus, as we\nexplain subsequently. Let $$\\label{eq:define K}\nK=\\ker(p_2^*)=\\left\\{k\\in N\\mid \\{k,n\\}=0\\,\\forall\\, n\\in N_{\\rm uf}^\\circ\\right\\} \\quad \\text{and} \\quad K^{\\circ}=K \\cap N^{\\circ}.$$\nTo obtain an action on $\\mathcal{A}$ we consider a saturated sublattice\n$$H_{\\mathcal{A} } \\subseteq K^\\circ.$$ The inclusion\n$H_{\\mathcal{A} } \\hookrightarrow N^\\circ$ gives rise to an inclusion\n$T_{H_{\\mathcal{A} }}\\hookrightarrow T_{N^{\\circ}}$ as a subgroup. Since\n$p^*$ commutes with mutation and $H_{\\mathcal{A} }\\subseteq K$ we have a\nnon-canonical inclusion\n$$T_{H_{\\mathcal{A} }}\\hookrightarrow \\mathcal{A} .$$ The action of\n$T_{H_{\\mathcal{A} }}$ on $T_{N^\\circ}$ given by multiplication extends\nto a free action of $T_{H_{\\mathcal{A} }}$ on $\\mathcal{A}$ and gives\nrise to a geometric quotient\n$\\mathcal{A} \\to \\mathcal{A} /T_{H_{\\mathcal{A} }}$. The scheme\n$\\mathcal{A} /T_{H_{\\mathcal{A} }}$ is obtained by gluing tori of the\nform\n$T_{N^{\\circ}/H_{\\mathcal{A} }}\\cong T_{N^{\\circ}}/T_{H_{\\mathcal{A} }}$;\nthe gluing is induced by the $\\mathcal{A}$-mutations used to glue the\nseed tori for $\\mathcal{A}$. More precisely, for every seed $\\textbf{s}$\nfor $\\mathcal{A}$ we let\n$(\\mathcal{A} /T_{H_{\\mathcal{A} }})_{\\textbf{s}}$ be a copy of the\ntorus $T_{N^{\\circ}/H_{\\mathcal{A} }}$. For $k\\in I_{\\text{uf}}$ the\nmutation\n$\\mu^{\\mathcal{A} /T_{H_\\mathcal{A} }}_{k}: (\\mathcal{A} /T_{H_{\\mathcal{A} }})_{\\textbf{s}}  \\dashrightarrow  (\\mathcal{A} /T_{H_{\\mathcal{A} }})_{\\mu_k(\\textbf{s})}$\nis given by $$\\label{A/T_mut}\n\\left(\\mu^{\\mathcal{A} /T_{H_{\\mathcal{A} }}}_{k}\\right)^*(z^m):=z^{m} (1+z^{v_{k;\\textbf{s}}})^{-\\langle d_k e_{k;\\textbf{s}},m\\rangle} \\ \\ \\text{ for }m\\in H_{\\mathcal{A} }^{\\perp}.$$\nLet $\\mu^{\\mathcal{A} /T_{H_{\\mathcal{A} }}}_{\\textbf{s}, \\textbf{s}'}$\ndenote the composition of mutations determined by the path in $%\n  \\mathrel{\\mathop{\\mathbb{T}_r}\\limits^{\n    \\vbox to0ex{\\kern-0.5\\ex@\n    \\hbox{$\\scriptstyle\\longrightarrow$}\\vss}}}$ connecting\n$\\textbf{s}, \\textbf{s}'\\in %\n  \\mathrel{\\mathop{\\mathbb{T}_r}\\limits^{\n    \\vbox to0ex{\\kern-0.5\\ex@\n    \\hbox{$\\scriptstyle\\longrightarrow$}\\vss}}}$. Then\n$$\\mathcal{A} /T_{H_{\\mathcal{A} }}:=\\bigcup\\limits_{\\textbf{s}\\in %\n  \\mathrel{\\mathop{\\mathbb{T}_r}\\limits^{\n    \\vbox to0ex{\\kern-0.5\\ex@\n    \\hbox{$\\scriptstyle\\longrightarrow$}\\vss}}}} (\\mathcal{A} /T_{H_{\\mathcal{A} }})_{\\textbf{s}}/ \\left( \\text{gluing by } \\mu^{\\mathcal{A} /T_{H_{\\mathcal{A} }}}_{\\textbf{s}', \\textbf{s}''} \\right)_{\\textbf{s}', \\textbf{s}'' \\in %\n  \\mathrel{\\mathop{\\mathbb{T}_r}\\limits^{\n    \\vbox to0ex{\\kern-0.5\\ex@\n    \\hbox{$\\scriptstyle\\longrightarrow$}\\vss}}}}.$$\n\nTo obtain the fibration of $\\mathcal{X}$ over a torus we consider a\nsaturated sublattice $$H_{\\mathcal{X} } \\subseteq K.$$ The inclusion\n$H_{\\mathcal{X} } \\hookrightarrow N$ induces a surjection\n$T_M:= \\mathop{\\mathrm{Spec}}(\\Bbbk[N]) \\to \\mathop{\\mathrm{Spec}}(\\Bbbk[H_{\\mathcal{X} }])=:T_{H_{\\mathcal{X} }^*}$.\nThis extends to a globally defined map $$\\label{eq:weight_map}\n    w_{H_{\\mathcal{X} }}:\\mathcal{X} \\to T_{H^*_\\mathcal{X} }.$$\n\n**Remark 2**. The subindex $\\mathcal{V}$ in the lattice\n$H_{\\mathcal{V} }$ stands for the cluster variety $\\mathcal{V}$ for\nwhich the choice of sublattice is relevant. When there is no risk of\nconfusion, we drop the subindex $\\mathcal{V}$ from $H_{\\mathcal{V} }$\n(see the end of §).\n\nWe let $\\mathcal{X} _{\\phi}$ be the fibre of the map over a closed point\n$\\phi\\in T_{H^*_{\\mathcal{X} }}$. In this work we mainly focus on the\nfibre $\\mathcal{X} _{{\\bf 1}_{T_{H^*_{\\mathcal{X} }}}}$, where\n${\\bf 1}_{T_{H^*_{\\mathcal{X} }}}\\in T_{H^*_{\\mathcal{X} }}$ is the\nidentity element. When there is no risk of confusion on the fibration we\nare considering we will denote this scheme simply by\n$\\mathcal{X} _{\\bf 1}$.\n\nThe fibre $\\mathcal{X} _{\\bf 1}$ is obtained by gluing tori isomorphic\nto $T_{H^\\perp_{\\mathcal{X} }}$ via the restrictions of the\n$\\mathcal{X}$-mutations used to glue the seed tori for $\\mathcal{X}$\n(see for a detailed treatment of this construction). As in the previous\nsituations, we have a description of the form\n$$\\mathcal{X} _{\\bf 1}:=\\bigcup\\limits_{\\textbf{s}\\in %\n  \\mathrel{\\mathop{\\mathbb{T}_r}\\limits^{\n    \\vbox to0ex{\\kern-0.5\\ex@\n    \\hbox{$\\scriptstyle\\longrightarrow$}\\vss}}}} (\\mathcal{X} _{\\bf 1})_{\\textbf{s}}/ \\left( \\text{gluing by } \\mu^{\\mathcal{X} _{\\bf 1}}_{\\textbf{s}', \\textbf{s}''} \\right)_{\\textbf{s}', \\textbf{s}'' \\in %\n  \\mathrel{\\mathop{\\mathbb{T}_r}\\limits^{\n    \\vbox to0ex{\\kern-0.5\\ex@\n    \\hbox{$\\scriptstyle\\longrightarrow$}\\vss}}}},$$ where\n$(\\mathcal{X} _{\\bf 1})_{\\textbf{s}}$ is a torus isomorphic to\n$T_{H_{\\mathcal{X} }^\\perp}$,\n$\\mu^{\\mathcal{X} _{\\bf 1}}_{k}: (\\mathcal{X} _{\\bf 1})_{\\textbf{s}}  \\dashrightarrow  (\\mathcal{X} _{\\bf 1})_{\\mu_k(\\textbf{s})}$\nis given by $$\\label{X_phi_mut}\n\\left(\\mu^{\\mathcal{X} _{\\bf 1}}_{k}\\right)^*(z^{n+H_{\\mathcal{X} }}):=z^{n+H_{\\mathcal{X} }}(1+z^{e_{k;\\textbf{s}}+H_{\\mathcal{X} }})^{-[ n,e_{k;\\textbf{s}} ]}\\ \\ \\text{ for } n+H_{\\mathcal{X} } \\in N/H_{\\mathcal{X} }$$\nand $\\mu^{\\mathcal{X} _{\\bf 1}}_{\\textbf{s},\\textbf{s}'}$ is defined as\nfor the other varieties we have introduced so far.\n\n**Definition 3**. A variety of the form\n$\\mathcal{A} /T_{H_{\\mathcal{A} }}$ is referred to as a **quotient of\n$\\mathcal{A}$**. A variety of the form $\\mathcal{X} _{\\bf 1}$ is\nreferred to as a **fibre of $\\mathcal{X}$**. A **cluster action** on\n$\\mathcal{A}$ is the action of a torus of the form\n$T_{H_{\\mathcal{A} }}$.\n\nLet $T$ be an algebraic torus endowed with a set of coordinates\n$z_1, \\dots , z_r$ and let $\\omega_T$ be its canonical bundle. A\n**volume form** on $T$ is a nowhere vanishing form in\n$H^0(T, \\omega_T)$. The **standard volume form** on $T$ is (any non-zero\nscalar multiple of)\n$$\\Omega_T= \\frac{dz_1 \\wedge \\dots \\wedge dz_r}{z_1 \\cdots z_r}.$$\n\n**Definition 4**. A **log Calabi–Yau pair** $(Y, D)$ is a smooth complex\nprojective variety $Y$ together with a reduced normal crossing divisor\n$D\\subset Y$ such that $K_X+D=0$. We say a scheme $V$ is log Calabi–Yau\nif there exists a log Calabi–Yau pair $(Y,D)$ such that $V$ is\n$Y \\setminus D$ up to codimension 2.\n\nIt follows from that any log Calabi–Yau variety $V$ is endowed with a\nunique up to scaling holomorphic volume form (*i.e.* a nowhere vanishing\nholomorphic top form) $\\Omega_V$ which has at worst a simple pole along\neach component of $D$ for any such $(Y,D)$. See for further details.\n\nAs explained in both $\\mathcal{A}$ and $\\mathcal{X}$ are log Calabi–Yau,\nthe key point being that these schemes are obtained by gluing tori via\nbirational maps that preserve the standard volume form on each seed\ntorus (endowed with cluster coordinates). For the same reason, the\nschemes of the form $\\mathcal{A} /T_{H_{\\mathcal{A} }}$ and\n$\\mathcal{X} _{\\phi}$ are also log Calabi–Yau. The canonical volume form\non $\\mathcal{A} /T_{H_{\\mathcal{A} }}$ (resp. $\\mathcal{X} _{\\phi}$) is\ninduced by (resp. the restriction of) the canonical volume form of\n$\\mathcal{A}$ (resp. $\\mathcal{X}$).\n\n### Principal coefficients, $\\mathcal{X}$ as a quotient of $\\mathcal{A}_{\\mathrm{prin}}$ and $\\mathcal{A}$ as a fibre of $\\mathcal{X}_{\\mathrm{prin}}$\n\nFor the fixed data\n$\\Gamma=\\left(I, I_{\\text{uf}}, N,N^{\\circ}, M, M^{\\circ}, \\{ \\cdot, \\cdot \\}, \\{d_i\\}_{i\\in I} \\right)$,\nwe consider its principal counterpart\n$$\\Gamma_{{\\mathrm{prin}} }=\\left(I_{{\\mathrm{prin}} }, (I_{{\\mathrm{prin}} })_{\\text{uf}}, N_{{\\mathrm{prin}} }, N_{{\\mathrm{prin}} }^{\\circ}, M_{{\\mathrm{prin}} }, M^{\\circ}_{{\\mathrm{prin}} }, \\{ \\cdot, \\cdot \\}_{{\\mathrm{prin}} }, \\{d_i\\}_{i\\in I_{{\\mathrm{prin}} }} \\right),$$\nwhere the index set $I_{\\mathrm{prin}}$ is the disjoint union of two\ncopies of $I$, its subset $(I_{\\mathrm{prin}} )_{\\text{uf}}$ is the set\n$I_{\\text{uf}}$ thought of as a subset of the first copy of $I$,\n$$N_{{\\mathrm{prin}} } = N \\oplus M^\\circ, \\quad  N_{{\\mathrm{prin}} }^{\\circ}= N^{\\circ}\\oplus M, \\quad (N_{{\\mathrm{prin}} })_{\\text{uf}}=N_{\\text{uf}}\\oplus 0, \\quad M_{{\\mathrm{prin}} } = M \\oplus N^\\circ, \\quad M_{{\\mathrm{prin}} }^{\\circ}=M^{\\circ}\\oplus N.$$\nFor $i \\in I_{{\\mathrm{prin}} }$ belonging to either the first or second\ncopy of $I$, the corresponding integer in the tuple\n$\\{d_i \\mid i\\in I_{{\\mathrm{prin}} }\\}$ is equal to integer indexed by\n$i$ for $\\Gamma$, and\n$$\\{(n_1,m_1),(n_2,m_2)\\}_{{\\mathrm{prin}} }= \\{n_1, n_2\\} + \\langle n_1,m_2 \\rangle - \\langle n_2,m_1 \\rangle.$$\nRecall that $\\textbf{s}_0=(e_i)_{i \\in I}$ is the initial seed for\n$\\Gamma$. Then the initial seed for $\\Gamma_{{\\mathrm{prin}} }$ is\n${\\textbf{s}_0}_{{\\mathrm{prin}} }=\\left((e_i,0),(0,f_i)\\right)_{i\\in I}$.\nSince $\\Gamma$ and $\\textbf{s}_0$ were already fixed, we denote the\ncluster variety\n$\\mathcal{A} _{\\Gamma_{{\\mathrm{prin}} },{{\\textbf{s}{_0}}_{{\\mathrm{prin}} }}}$\n(resp.\n$\\mathcal{X} _{\\Gamma_{{\\mathrm{prin}} },{{\\textbf{s}{_0}}_{{\\mathrm{prin}} }}}$)\nsimply by $\\mathcal{A}_{\\mathrm{prin}}$ (resp.\n$\\mathcal{X} _{{\\mathrm{prin}} }$). It is moreover worth pointing out\nthat $\\mathcal{A}_{\\mathrm{prin}}$ is in fact independent of the choice\nof initial seed $\\textbf{s}_0$ as explained in .\n\nIn the authors show that the scheme $\\mathcal{X}$ can be described as a\nquotient of $\\mathcal{A}_{\\mathrm{prin}}$ in the sense of Definition .\nTo obtain such a description we need to choose a cluster ensemble\nlattice map $p^*:N \\to M^{\\circ}$ for $\\Gamma$. This choice determines\nthe cluster ensemble map $$\\label{eq:def_p_prin}\np_{{\\mathrm{prin}} }: \\mathcal{A}_{\\mathrm{prin}}\\to \\mathcal{X}_{\\mathrm{prin}}.$$\nThe map $p_{{\\mathrm{prin}} }$ is induced by the cluster ensemble\nlattice map $$\\begin{aligned}\np_{{\\mathrm{prin}} }^*:N_{{\\mathrm{prin}} } &\\to M^\\circ_{{\\mathrm{prin}} }\\\\\n(n,m) &\\mapsto \\left(p^*(n)-m,n\\right)\n\\end{aligned}$$ for $\\Gamma_{\\mathrm{prin}}$. Set\n$K_{{\\mathrm{prin}} }:=\\ker(p_{{\\mathrm{prin}} ,2}^*)$ and\n$K_{{\\mathrm{prin}} }^\\circ:= K_{{\\mathrm{prin}} }\\cap N^\\circ_{\\mathrm{prin}}$,\nwhere $p_{{\\mathrm{prin}} ,2}^*$ corresponds to the map $p_2^*$ in for\n$\\Gamma_{{\\mathrm{prin}} }$. We let $$\\label{eq:H_Aprin} \n  H_{\\mathcal{A}_{\\mathrm{prin}}}:= \\left\\{\\left.\\left(n,-(p^*)^*(n)\\right)\\in N^\\circ_{\\mathrm{prin}} \\, \\right| \\, n \\in N^\\circ\\right\\}.$$\n\nIt is straightforward to verify that $H_{\\mathcal{A}_{\\mathrm{prin}}}$\nis a saturated sublattice of $K^\\circ_{\\mathrm{prin}}$ that is\nisomorphic to $N^\\circ$. In particular, we have a quotient\n$\\mathcal{A}_{\\mathrm{prin}}/ T_{H_{\\mathcal{A}_{\\mathrm{prin}}}}$\nendowed with an atlas of seed tori isomorphic to $T_M$ (indeed,\n$T_{N^\\circ_{{\\mathrm{prin}} }}/T_{H_{\\mathcal{A}_{\\mathrm{prin}}}}\\cong T_{N^\\circ \\oplus M}/T_{N^\\circ}\\cong T_M$).\nThere is an isomorphism $$\\label{eq:def_chi}\n    \\chi : \\mathcal{A}_{\\mathrm{prin}}/T_{H_{\\mathcal{A}_{\\mathrm{prin}}}}\\overset{\\sim}{\\longrightarrow} \\mathcal{X}$$\nrespecting the cluster tori of domain and range. The restriction of\n$\\chi$ to a seed torus is a monomial map whose pullback is given by\n$$\\begin{split} \n\\chi^*: N &\\to (H_{\\mathcal{A}_{\\mathrm{prin}}})^\\perp \n\\\\\nn &\\mapsto (p^*(n),n).\n \\end{split}$$ There is also a surjective map $$\\label{eq:def_tilde_p}\n    \\tilde{p}:\\mathcal{A}_{\\mathrm{prin}}\\to \\mathcal{X} .$$ respecting\nseed tori. The restriction of $\\tilde{p}$ to a seed torus is a monomial\nmap whose pullback is given by $$\\begin{aligned}\n    \\tilde{p}^*:  N &\\to M^\\circ_{{\\mathrm{prin}} }\\\\\n       \\ \\ n &\\mapsto  (p^*(n),n).\n\\end{aligned}$$ In particular, we have $\\tilde{p}= \\chi\\circ \\varpi$,\nwhere $$\\label{eq:def_varpi}\n\\varpi: \\mathcal{A}_{\\mathrm{prin}}\\to \\mathcal{A}_{\\mathrm{prin}}/T_{H_{\\mathcal{A}_{\\mathrm{prin}}}}$$\nis the canonical projection.\n\nIt is also possible to describe $\\mathcal{A}$ as a fibre of\n$\\mathcal{X}_{\\mathrm{prin}}$. There is an injective map\n$$\\label{eq:def_xi}\n\\xi:\\mathcal{A} \\to \\mathcal{X}_{\\mathrm{prin}}$$ respecting seed tori.\nThe restriction of $\\xi$ to a seed torus is a monomial map whose\npullback is given by $$\\begin{aligned}\n\\xi^*: N_{{\\mathrm{prin}} } &\\to M^\\circ\n\\\\\n(n,m) &\\mapsto p^*(n)-m.\n\\end{aligned}$$ Let $$\\label{eq:H_Xprin} \nH_{\\mathcal{X}_{\\mathrm{prin}}}:= \\left\\{\\left(n,p^*(n)\\right)\\in N_{\\mathrm{prin}} \\mid n \\in N\\right\\}.$$\nIt is routine to check that $H_{\\mathcal{X}_{\\mathrm{prin}}}$ is a valid\nchoice to construct a fibration of $\\mathcal{X}_{\\mathrm{prin}}$ over\nthe torus $T_{H^*_{\\mathcal{X}_{\\mathrm{prin}}}}$. Hence, we can\nconsider the fibre\n$(\\mathcal{X}_{\\mathrm{prin}})_{\\bf 1}=(\\mathcal{X}_{\\mathrm{prin}})_{{\\bf 1}_{T_{H^*_{ \\mathcal{X}_{\\mathrm{prin}}}}}}$\nassociated to this fibration. There is an isomorphism\n$$\\label{eq:def_delta}\n\\delta:\n\\mathcal{A} \\overset{\\sim}{\\longrightarrow} (\\mathcal{X}_{\\mathrm{prin}})_{\\bf 1}$$\nrespecting seed tori. The restriction of $\\delta$ to a seed torus is a\nmonomial map whose pullback is given by $$\\begin{aligned}\n\\delta^*: N_{{\\mathrm{prin}} } /H_{\\mathcal{X}_{\\mathrm{prin}}} &\\to M^\\circ\n\\\\\n(n,m) + H_{\\mathcal{X}_{\\mathrm{prin}}} &\\mapsto p^*(n)-m.\n\\end{aligned}$$ In particular, we have that $$\\xi=\\iota \\circ \\delta,$$\nwhere\n$\\iota: (\\mathcal{X}_{\\mathrm{prin}})_{\\bf 1}\\hookrightarrow \\mathcal{X}_{\\mathrm{prin}}$\nis the canonical inclusion. For later reference we also introduce the\nmap $$\\label{eq:def_rho}\n\\rho: \\mathcal{X}_{\\mathrm{prin}}\\to \\mathcal{X} .$$ respecting seed\ntori. The restriction of $\\rho$ to a seed torus is a monomial map whose\npullback is given by $$\\begin{aligned}\n    \\rho^*:  N &\\to N_{{\\mathrm{prin}} }\\\\\n       \\ \\ n & \\mapsto  (n,p^*(n)).\n\\end{aligned}$$ In particular,\n$\\rho \\circ p_{{\\mathrm{prin}} }= \\tilde{p}$. The maps we have\nconsidered so far fit into the following commutative diagram\n$$\\xymatrix{\n(\\mathcal{X}_{\\mathrm{prin}})_{\\bf 1} \\ar@{^{(}->}^{\\ \\iota}[r] & \\mathcal{X}_{\\mathrm{prin}}\\ar_{\\rho}[d] & \\mathcal{A}_{\\mathrm{prin}}\\ar_{p_{{\\mathrm{prin}} }}[l] \\ar@{->>}^{\\varpi}[d] \\ar_{\\tilde{p}}[dl] \\\\\n\\mathcal{A} \\ar^{\\delta}_{\\cong}[u] \\ar_{p}[r] \\ar^{\\xi}[ru] & \\mathcal{X} & \\mathcal{A}_{\\mathrm{prin}}/T_{H_{\\mathcal{A}_{\\mathrm{prin}}}.} \\ar_{\\cong \\ \\ }^{\\chi \\ \\ }[l]\n}$$\n\n**Remark 5**. The maps introduced in this section are associated with\n$\\Gamma$, hence, we label the maps with the subindex $\\Gamma$ to stress\nthe fixed data $\\Gamma$ they are associated with.\n\n## Tropicalization\n\nIn this section we discuss tropicalizations of cluster varieties. We\nmainly follow , and .\n\nLet $T_L$ be the torus associated to a lattice $L$. A rational function\n$f$ on $T_L$ is called positive if it can be written as a fraction\n$f=f_1/f_2$, where both $f_1$ and $f_2$ are a linear combination of\ncharacters of $T_L$ with coefficients in $\\mathbb{Z} _{>0}$. The\ncollection of positive rational functions on $T_L$ forms a semifield\ninside $\\Bbbk(T_L)$ denoted by $Q_{\\rm sf}(L)$. A rational map\n$f:T_L\\dashrightarrow T_{L'}$ between two tori is a **positive rational\nmap** if the pullback $f^*:\\Bbbk(T') \\to \\Bbbk(T)$ restricts to an\nisomorphism $f^*:Q_{\\rm sf}(L') \\to Q_{\\rm sf}(L)$. If $P$ is a\nsemifield, then the $P$ valued points of $T_L$ form the set\n$$\\label{eq:FG_tropicalization}\nT_L(P):=\\mathop{\\mathrm{Hom}}_{\\rm sf}(Q_{\\rm sf} (L), P)$$ of semifield\nhomomorphisms from $Q_{\\rm sf} (L)$ to $P$. In particular, a positive\nbirational isomorphism $\\mu:T\\dashrightarrow T'$ induces a bijection\n$$\\begin{aligned}\n\\mu_*: T(P) & \\to T'(P)\\\\\n h \\ & \\mapsto \\ h \\circ f^*.    \n\\end{aligned}$$ By a slight but common abuse of notation the sublattice\nof monomials of $Q_{\\rm sf}(L)$ is denoted by $L^*$. Considering $P$\njust as an abelian group the restriction of an element of\n$Q_{\\rm sf}(L)$ to $L^*$ determines a canonical bijection\n$T_L(P) \\overset{\\sim}{\\longrightarrow} \\mathop{\\mathrm{Hom}}_{\\rm groups} (L^*, P)$.\n\n**Remark 6**. We systematically identify $T_L(P)$ with $L\\otimes P$ by\ncomposing the canonical bijection\n$T_L(P) \\overset{\\sim}{\\longrightarrow} \\mathop{\\mathrm{Hom}}_{\\rm groups} (L^*, P)$\nwith the canonical isomorphism\n$\\mathop{\\mathrm{Hom}}_{\\rm groups}(L^*, P) \\cong L \\otimes P$.\n\nLet $\\mathcal{V}$ be a (quotient or a fibre of a) cluster variety. For\nevery $\\textbf{s}, \\textbf{s}'\\in %\n  \\mathrel{\\mathop{\\mathbb{T}_r}\\limits^{\n    \\vbox to0ex{\\kern-0.5\\ex@\n    \\hbox{$\\scriptstyle\\longrightarrow$}\\vss}}}$ the gluing map\n$\\mu^{\\mathcal{V} }_{\\textbf{s}, \\textbf{s}'}: \\mathcal{V} _\\textbf{s}\\dashrightarrow  \\mathcal{V} _\\textbf{s}'$\nis a positive rational map. So we can glue\n$\\mathcal{V} _{\\textbf{s}}(P)$ and $\\mathcal{V} _{\\textbf{s}'} (P)$\nusing $(\\mu^{\\mathcal{V} }_{\\textbf{s}, \\textbf{s}'})_*$ and define\n$$\\mathcal{V} (P):= \\coprod_{\\textbf{s}\\in %\n  \\mathrel{\\mathop{\\mathbb{T}_r}\\limits^{\n    \\vbox to0ex{\\kern-0.5\\ex@\n    \\hbox{$\\scriptstyle\\longrightarrow$}\\vss}}}} \\mathcal{V} _{\\textbf{s}}(P) / \\left(\\text{gluing by } (\\mu^{\\mathcal{V} }_{\\textbf{s}, \\textbf{s}'})_*\\right)_{\\textbf{s}, \\textbf{s}'\\in %\n  \\mathrel{\\mathop{\\mathbb{T}_r}\\limits^{\n    \\vbox to0ex{\\kern-0.5\\ex@\n    \\hbox{$\\scriptstyle\\longrightarrow$}\\vss}}}}.$$ Every point\n${\\bf a}\\in \\mathcal{V} (P)$ can be represented as a tuple\n$(a_{\\textbf{s}})_{\\textbf{s}\\in %\n  \\mathrel{\\mathop{\\mathbb{T}_r}\\limits^{\n    \\vbox to0ex{\\kern-0.5\\ex@\n    \\hbox{$\\scriptstyle\\longrightarrow$}\\vss}}}}$ such that\n$(\\mu^{\\mathcal{V} }_{\\textbf{s}, \\textbf{s}'})_*(a_\\textbf{s})=(a_{\\textbf{s}'})$\nfor all $\\textbf{s},\\textbf{s}'\\in %\n  \\mathrel{\\mathop{\\mathbb{T}_r}\\limits^{\n    \\vbox to0ex{\\kern-0.5\\ex@\n    \\hbox{$\\scriptstyle\\longrightarrow$}\\vss}}}$. Since all of the maps\n$(\\mu^\\mathcal{V} _{\\textbf{s},\\textbf{s}'})_*$ are bijections, the\nassignment $$\\label{not:tropical_space} \\begin{split} \n   \\mathfrak{r}_{\\textbf{s}}:\\mathcal{V} (P)&\\to \\mathcal{V} _{\\textbf{s}}(P)\\quad \\text{given by} \\quad {\\bf a}=(a_{\\textbf{s}})_{\\textbf{s}\\in %\n  \\mathrel{\\mathop{\\mathbb{T}_r}\\limits^{\n    \\vbox to0ex{\\kern-0.5\\ex@\n    \\hbox{$\\scriptstyle\\longrightarrow$}\\vss}}}} \\mapsto a_{\\textbf{s}}.\n  \\end{split}$$ determines an identification of $\\mathcal{V} (P)$ with\n$\\mathcal{V} _{\\textbf{s}}(P)$. If $S\\subset \\mathcal{V} (P)$ we let\n$$\\label{eq:identification}\nS_{\\textbf{s}}(P):=\\mathfrak{r}_{\\textbf{s}} (S) \\subset \\mathcal{V} _\\textbf{s}(P)$$\nand write $S_{\\textbf{s}}$ instead of $S_{\\textbf{s}}(P)$ when the\nsemifield $P$ is clear from the context.\n\nThe semifields we consider in this note are the integers, the rationals\nand the real numbers with their additive structure together with the\nsemifield operation determined by taking the maximum (respectively,\nminimum). We denote these semifields by $\\mathbb{Z} ^T$, $\\mathbb{Q} ^T$\nand $\\mathbb{R} ^T$ (respectively, $\\mathbb{Z} ^t$, $\\mathbb{Q} ^t$ and\n$\\mathbb{R} ^t$). The canonical inclusions\n$\\mathbb{Z} \\hookrightarrow \\mathbb{Q} \\hookrightarrow \\mathbb{R}$ give\nrise to canonical inclusions\n$$\\mathcal{V} (\\mathbb{Z} ^T) \\hookrightarrow \\mathcal{V} (\\mathbb{Q} ^T) \\hookrightarrow \\mathcal{V} (\\mathbb{R} ^T) \\quad \\quad \\text{ and } \\quad \\quad  \\mathcal{V} (\\mathbb{Z} ^t) \\hookrightarrow \\mathcal{V} (\\mathbb{Q} ^t) \\hookrightarrow \\mathcal{V} (\\mathbb{R} ^t).$$\nFor a set $S\\subseteq \\mathcal{V} (\\mathbb{R} ^T)$ (resp.\n$S\\subseteq \\mathcal{V} (\\mathbb{R} ^t)$) we let\n$S(\\mathbb{Z} ):= S\\cap \\mathcal{V} (\\mathbb{Z} ^T)$ (resp.\n$S(\\mathbb{Z} ):= S\\cap \\mathcal{V} (\\mathbb{Z} ^t)$). Moreover, for\n$G=\\mathbb{Z} , \\mathbb{Q}$ or $\\mathbb{R}$, there is an isomorphism of\nsemifields $G^T\\to G^t$ given by $x \\mapsto -x$ induces a canonical\nbijection $$\\begin{aligned}\n \\label{eq:imap}\n    i: \\mathcal{V} (G^T) \\rightarrow \\mathcal{V} (G^t). \n\\end{aligned}$$ Since $i$ amounts to a sign change (see Remark below),\nwe think of $i$ as an involution and denote its inverse again by $i$.\n\n**Remark 7**. The set $\\mathcal{V} (\\mathbb{Z} ^t)$ can be identified\nwith the **geometric tropicalization** of $\\mathcal{V}$, defined as\n$$\\mathcal{V} ^{\\mathrm{trop} }(\\mathbb{Z} ) \n     \\coloneqq \\{ \\text{divisorial discrete valuations } \\nu: \\Bbbk(\\mathcal{V} ) \\setminus \\{ 0\\} \\rightarrow \\mathbb Z \\mid \\nu (\\Omega_{\\mathcal{V} }) <0 \\} \\cup \\{ 0\\},$$\nwhere a discrete valuation is divisorial if it is given by the order of\nvanishing of a $\\mathbb{Z} _{>0}$-multiple of a prime divisor on some\nvariety birational to $\\mathcal{V}$.\n\n**Remark 8**. Let $G=\\mathbb{Z} , \\mathbb{Q}$ or $\\mathbb{R}$.\nIdentifying $\\mathcal{V} (G^T)$ with $\\mathcal{V} _{\\textbf{s}}(G^T)$\nvia the bijection $\\mathfrak{r}_\\textbf{s}$ the map $i$ in can be\nthought of as the multiplication by $-1$ (*cf.* Remark ).\n\nA positive rational function $g$ on $\\mathcal{V}$ is a rational function\non $\\mathcal{V}$ such that the restriction of $g$ to every seed torus\n$\\mathcal{V} _{\\textbf{s}}$ is a positive rational function.\n\n**Definition 9**. The **tropicalization** of a positive rational\nfunction $g: \\mathcal{V} \\dashrightarrow \\Bbbk$ with respect to\n$\\mathbb{R} ^T$ is the function\n$g^T:\\mathcal{V} (\\mathbb{R} ^T)\\to \\mathbb{R}$ given by\n$$\\label{eq:restriction}\n{\\bf a}\\mapsto a_{\\textbf{s}}(g),$$ where\n${\\bf a}=(a_{\\textbf{s}})_{\\textbf{s}\\in %\n  \\mathrel{\\mathop{\\mathbb{T}_r}\\limits^{\n    \\vbox to0ex{\\kern-0.5\\ex@\n    \\hbox{$\\scriptstyle\\longrightarrow$}\\vss}}}}$. The tropicalization\nof $g$ with respect to $\\mathbb{R} ^t$ is the function\n$g^t:\\mathcal{V} (\\mathbb{R} ^t)\\to \\mathbb{R}$ defined as\n$${\\bf v} \\mapsto -v_{\\textbf{s}}(g),$$\n\nwhere ${\\bf v}=(v_{\\textbf{s}})_{\\textbf{s}\\in %\n  \\mathrel{\\mathop{\\mathbb{T}_r}\\limits^{\n    \\vbox to0ex{\\kern-0.5\\ex@\n    \\hbox{$\\scriptstyle\\longrightarrow$}\\vss}}}}$. A direct computation\nshows that both $g^T$ and $g^t$ are well defined. Namely, one checks\nthat for $\\textbf{s},\\textbf{s}'\\in %\n  \\mathrel{\\mathop{\\mathbb{T}_r}\\limits^{\n    \\vbox to0ex{\\kern-0.5\\ex@\n    \\hbox{$\\scriptstyle\\longrightarrow$}\\vss}}}$\n$$a_{\\textbf{s}} (g)=a_{\\textbf{s}'}(g),$$ where in the left (resp.\nright) side of the equality we think of $g$ as a rational function on\n$\\mathcal{V} _\\textbf{s}$ (resp. $\\mathcal{V} _{\\textbf{s}'}$).\nMoreover, we have that $$\\label{eq:comparing_tropicalizations}\n    g^T({\\bf a})=g^t(i({\\bf a})),$$ for all\n${\\bf a} \\in \\mathcal{V} (\\mathbb{R} ^T)$.\n\n**Remark 10**. In order to keep notation lighter we adopt the following\nconventions:\n\n-   given a positive rational function\n    $g\\in \\Bbbk (\\mathcal{V} )=\\Bbbk(\\mathcal{V} _\\textbf{s})$ the\n    tropicalizations of $g$ with domains $\\mathcal{V} (\\mathbb{R} ^T)$\n    and $\\mathcal{V} _{\\textbf{s}}(\\mathbb{R} ^T)$ are denoted by the\n    same symbol $g^T$ for all $\\textbf{s}\\in %\n      \\mathrel{\\mathop{\\mathbb{T}_r}\\limits^{\n        \\vbox to0ex{\\kern-0.5\\ex@\n        \\hbox{$\\scriptstyle\\longrightarrow$}\\vss}}}$;\n\n-   the restriction of $g^T$ (resp. $g^t$) to\n    $\\mathcal{V} (\\mathbb{Z} ^T)$ (resp. $\\mathcal{V} (\\mathbb{Z} ^t)$)\n    is also denoted by $g^T$ (resp. $g^t$);\n\n-   when $P$ is one of $\\mathbb{Z} ^T, \\mathbb{Q} ^T$ or $\\mathbb{R} ^T$\n    (resp. $\\mathbb{Z} ^t, \\mathbb{Q} ^t$ or $\\mathbb{R} ^t$) the map\n    $(\\mu^{\\mathcal{V} }_{\\textbf{s}, \\textbf{s}'})_*$ is denoted by\n    $(\\mu^{\\mathcal{V} }_{\\textbf{s}, \\textbf{s}'})^T$ (resp.\n    $(\\mu^{\\mathcal{V} }_{\\textbf{s}, \\textbf{s}'})^t$).\n\n**Remark 11**. Later we will need to systematically consider\n$\\mathcal{V} (\\mathbb{R} ^t)$ when $\\mathcal{V}$ is a variety of the\nform $\\mathcal{A}$ or $\\mathcal{A} /T_H$ and\n$\\mathcal{V} (\\mathbb{R} ^T)$ when $\\mathcal{V}$ is a variety of the\nform $\\mathcal{X}$ or $\\mathcal{X} _{\\bf 1}$. In particular, from § on\nwe use the notation $\\mathrm{Trop} _{\\mathbb{R} }(\\mathcal{V} )$ that\ntakes into account the different kinds of tropicalizations that we use\nfor different kinds of varieties, see equation .\n\nFor latter use we record the following formulae associated to the\nmutations determined by $\\Gamma$: $$\\label{eq:tropical_A_mutation}\n    \\left(\\mu^{\\mathcal{A} }_{k}\\right)^T(n)=n+[\\langle v_k,n\\rangle]_+(-d_ke_k)$$\nand $$\\label{eq:tropical_X_mutation}\n    \\left(\\mu^{\\mathcal{X} }_{k}\\right)^T(m)=m+[\\langle d_ke_k,m \\rangle]_+v_k.$$\nIn case we tropicalize these mutations with respect to $\\mathbb{R} ^t$\nwe replace $[\\ \\cdot\\ ]_+$ by $[\\ \\cdot\\ ]_-$.\n\nFinally, if we think of $T_L (\\mathbb{R} ^T)$ (resp.\n$T_L(\\mathbb{R} ^t)$) as a vector space (see Remark ), the\ntropicalization of a positive Laurent polynomial\n$g= \\sum_{\\ell\\in L^*}c_{\\ell} z^{\\ell} \\in Q_{\\rm sf}(L)$ with respect\nto $\\mathbb{R} ^T$ (resp. $\\mathbb{R} ^t$) is the function\n$g^T:  T_L(\\mathbb{R} ^T) \\to \\mathbb{R}$ (resp.\n$g^t:  T_L(\\mathbb{R} ^t) \\to \\mathbb{R}$) given by $$\\begin{aligned}\nx &\\mapsto& - \\max \\{ \\langle \\ell , x  \\rangle \\mid \\ell\\in L^* \\text{ such that } c_{\\ell} \\neq 0 \\}\\\\\n (\\text{resp. } x &\\mapsto& \\min \\{ \\langle \\ell , x  \\rangle \\mid \\ell\\in L^* \\text{ such that } c_\\ell \\neq 0 \\}).\n\\end{aligned}$$\n\n# Theta functions and their labeling by tropical points\n\n## Fock–Goncharov duality\n\nFor\n$\\Gamma=(I, I_{\\text{uf}}, N,N^{\\circ}, M, M^{\\circ}, \\{ \\cdot, \\cdot \\}, \\{d_i\\}_{i\\in I} )$\nthe Langlands dual fixed data is\n$\\Gamma^\\vee=(I, I_{\\text{uf}}, N^\\vee, (N^\\vee)^{\\circ}, M^\\vee, (M^\\vee)^{\\circ}, \\{ \\cdot, \\cdot \\}^\\vee, \\{d^\\vee_i\\}_{i\\in I} )$,\nwhere $d:=\\text{lcm}(d_i)_{i\\in I}$,\n$$N^\\vee = N^\\circ, \\quad  (N^\\vee)^{\\circ}= d\\cdot N, \\quad  M^\\vee = M^\\circ, \\quad (M^\\vee)^{\\circ}=d^{-1}\\cdot M, \\quad \\{\\cdot, \\cdot \\}^\\vee= d^{-1}\\{\\cdot, \\cdot \\} \\quad \\text{and} \\quad d^\\vee_i:=d\\,d_i^{-1}.$$\nIf $\\textbf{s}=(e_i)_{ i\\in I}$ is a seed for $\\Gamma$ then the\nLanglands dual seed is $\\textbf{s}^\\vee:=(e_i^\\vee)_{i\\in I}$, where\n$e_i^\\vee:=d_ie_i$. We also set $v^\\vee_i:=\\{e^\\vee_i, \\cdot \\}^\\vee$\nThese constructions give rise to **Langlands dual cluster varieties**\nwhich we denote as follows $$\\begin{aligned}\n    \\begin{array}{l l l l}\n    {}^L(\\mathcal{A} _{\\Gamma;\\textbf{s}_0}) := \\mathcal{A} _{\\Gamma^\\vee;\\textbf{s}_0^\\vee} \\qquad \\qquad &     \\text{and} \\qquad \\qquad & {}^L(\\mathcal{X} _{\\Gamma; \\textbf{s}_0}) := \\mathcal{X} _{\\Gamma^\\vee; \\textbf{s}_0^\\vee}.\n\\end{array}\n\\end{aligned}$$ Since $\\Gamma$ and $\\textbf{s}_0$ were already fixed, we\ndenote ${}^L(\\mathcal{A} _{\\Gamma;\\textbf{s}_0})$ (resp.\n${}^L(\\mathcal{X} _{\\Gamma;\\textbf{s}_0})$) simply by\n${}^{L}\\mathcal{A}$ (resp. ${}^{L}\\mathcal{X}$).\n\n**Definition 12**. The **Fock–Goncharov dual** of $\\mathcal{A}$ (resp.\n$\\mathcal{X}$) is the cluster variety $\\mathcal{A} ^{\\vee}$ (resp.\n$\\mathcal{X} ^{\\vee}$) given by\n$$\\mathcal{A} ^{\\vee} := {}^L\\mathcal{X} \\qquad \\qquad      \\text{and} \\qquad \\qquad  \\mathcal{X} ^{\\vee} := {}^L\\mathcal{A} .$$\n\nIn particular, we have that\n$$\\mathcal{A}_{\\mathrm{prin}}^\\vee = {}^L(\\mathcal{X} _{\\mathrm{prin}} )=\\mathcal{X} _{(\\Gamma_{\\mathrm{prin}} )^\\vee} \\qquad \\qquad     \\mathcal{X}_{\\mathrm{prin}}^{\\vee}= {}^L(\\mathcal{A}_{\\mathrm{prin}})=\\mathcal{A} _{(\\Gamma_{\\mathrm{prin}} )^\\vee}.$$\n\n**Remark 13**. Notice that\n$\\mathcal{A} _{(\\Gamma_{{\\mathrm{prin}} })^\\vee}$ (resp.\n$\\mathcal{X} _{(\\Gamma_{{\\mathrm{prin}} })^\\vee}$) is canonically\nisomorphic to $\\mathcal{A} _{(\\Gamma^\\vee)_{{\\mathrm{prin}} }}$ (resp.\n$\\mathcal{X} _{(\\Gamma^\\vee)_{{\\mathrm{prin}} }}$). Hence, we frequently\nidentify these schemes without making reference to the canonical\nisomorphisms between them.\n\nIt is not hard to see that the map $$\\begin{aligned}\n\\label{eq:L p}\n    {(p^\\vee)^*:= -d^{-1}(p^*)^*:N^\\vee \\to (M^\\vee)^{\\circ}}\n\\end{aligned}$$ is well defined and is a cluster ensemble lattice map\nfor the Langlands dual data ${}^{L}\\Gamma$, where $(p^*)^*$ is the\nlattice map dual to $p^*$. Indeed, in the bases for $N^\\vee$ and\n$(M^\\vee)^{\\circ}$ determined by $\\textbf{s}^\\vee$, and in comparison\nwith the matrix $B_{p^*;\\mathbf{s}}$ in , the matrix of $(p^\\vee)^*$ is\nof the form\n$$B_{(p^\\vee)^*;\\textbf{s}^\\vee}= -B_{p^*;\\textbf{s}}^{\\rm{tr}}.$$ In\nparticular, we have an associated dual cluster ensemble map\n$$p^\\vee:\\mathcal{A} ^\\vee \\to \\mathcal{X} ^\\vee.$$\n\nWe proceed to introduce the Fock–Goncharov dual for a quotient of\n$\\mathcal{A}$. So consider a cluster ensemble lattice map\n$p^*:N\\to M^{\\circ}$ for $\\Gamma$ and the cluster ensemble lattice map\n$(p^\\vee)^*:N^\\vee\\to (M^\\vee)^\\circ$ for $\\Gamma^\\vee$. Recall from\nthat $K=\\ker(p_2^*)$. Similarly, we set\n\n$$K^\\vee=\\ker((p^\\vee)_2^*)=\\{k\\in N^\\circ\\mid \\{k,n\\}=0 \\text{ for all } n\\in d\\cdot N_{\\rm uf}\\},$$\nwhere $(p^\\vee)_2^*$ is the map $p^*_2$ of for $\\Gamma^\\vee$. Let\n$H_{\\mathcal{A} }\\subseteq K^\\circ$ be a saturated sublattice and\nconsider the quotient $\\mathcal{A} /T_{H_\\mathcal{A} }$. Recall from §\nthat $\\mathcal{A} /T_{H_\\mathcal{A} }$ is obtained by gluing tori of the\nform $T_{N^{\\circ}/H_{\\mathcal{A} }}$. Since\n$N^{\\circ}/H_{\\mathcal{A} }$ and\n$H_{\\mathcal{A} }^{\\perp} \\subset M^\\circ$ are dual lattices the\nFock–Goncharov dual of $\\mathcal{A} /T_{H_{\\mathcal{A} }}$ should be a\nfibre of $\\mathcal{A} ^\\vee$ obtained by gluing tori of the form\n$T_{H_{\\mathcal{A} }^{\\perp}}$. In order to construct it notice that for\n$n$ in $N_{\\text{uf}}$ we have\n$\\langle k,p^*(n)\\rangle = -{d}^{-1}\\{k,dn\\}=\\langle dk,-(p^\\vee)^*(n)\\rangle$.\nThis implies that $$K^\\circ=p^*(N_{\\rm uf})^\\perp=K^\\vee.$$ In\nparticular, $H_{\\mathcal{A} }$ is a saturated sublattice of $K^\\vee$ as\nit is saturated in $K^\\circ$. It is therefore possible to find\n$T_{H_{\\mathcal{A} }^*}$ as the base of a fibration of the form for\n$\\mathcal{A} ^{\\vee}$ as we are allowed to set\n$$H_{\\mathcal{A} ^{\\vee}}=H_{\\mathcal{A} }\\subseteq K^\\vee.$$ So\nconsider the fibration\n$$w_{H_{\\mathcal{A} }}:\\mathcal{A} ^{\\vee}\\to T_{H_\\mathcal{A} ^*}.$$\nNotice that the fibre\n$(\\mathcal{A} ^{\\vee})_{{\\bf 1}_{T_{H_\\mathcal{A} ^*}}}$ is obtained\ngluing tori of the form $T_{H_{\\mathcal{A} }^\\perp}$ as desired.\nTherefore, we define the Fock–Goncharov dual of the quotient\n$\\mathcal{A} /T_{H_\\mathcal{A} }$ as\n$$(\\mathcal{A} /T_{H_{\\mathcal{A} }})^{\\vee}:= (\\mathcal{A} ^{\\vee})_{{\\bf 1}_{T_{H_\\mathcal{A} ^*}}}=\\left({}^L\\mathcal{X} \\right)_{{\\bf 1}_{T_{H_\\mathcal{A} ^*}}}.$$\n\nSimilarly, let $H_{\\mathcal{X} }\\subseteq K$ be a saturated sublattice\nand let $w_{H_{\\mathcal{X} }}:\\mathcal{X} \\to T_{H^*_\\mathcal{X} }$ be\nthe associated fibration. Recall that\n$\\mathcal{X} _{{\\bf 1}_{T_{H^*_\\mathcal{X} }}}$ is obtained by gluing\ntori of the form $T_{H_{\\mathcal{X} }^{\\perp}}$. Its Fock–Goncharov dual\nis a quotient of $\\mathcal{X} ^{\\vee}$ glued from tori of the form\n$T_{(H^\\perp_\\mathcal{X} )^*}$ which we construct next. A direct\ncomputation shows that $d\\cdot H_{\\mathcal{X} }$ is a saturated\nsublattice of $(K^\\vee)^\\circ$. In particular, we are allowed to choose\n$$H_{\\mathcal{X} ^{\\vee}}= d\\cdot H_{\\mathcal{X} }\\subseteq (K^\\vee)^\\circ$$\nas a sublattice giving rise to a quotient\n${}^{L}\\mathcal{A} /T_{d\\cdot H_{\\mathcal{X} }}$. This quotient is\nobtained by gluing tori of the form\n$T_{d\\cdot N}/T_{d\\cdot H_\\mathcal{X} }\\cong T_{N/H_\\mathcal{X} }\\cong T_{(H^{\\perp}_\\mathcal{X} )^*}$.\nTherefore, we define the Fock–Goncharov dual of\n$\\mathcal{X} _{{\\bf 1}_{T_{H^*_{\\mathcal{X} }}}}$ as\n$$\\left(\\mathcal{X} _{{\\bf 1}_{T_{H^*_{\\mathcal{X} }}}}\\right)^{\\vee}:= \\mathcal{X} ^{\\vee}/T_{ H_{\\mathcal{X} ^{\\vee}}}={}^L\\mathcal{A} /T_{d\\cdot H_\\mathcal{X} }.$$\n\nIn what follows, when we consider a saturated sublattice $H$ of\n$K^\\circ$ and write expressions such as $\\mathcal{A} /T_{H}$ or\n$w_{H}:\\mathcal{A} ^{\\vee}\\to T_{H^*}$ we will be implicitly assuming\nthat we have set $$H_{\\mathcal{A} }= H = H_{\\mathcal{A} ^{\\vee}}.$$\nSimilarly, when $H$ is a saturated sublattice of $K$ and we write\nexpressions such as $w_{H}:\\mathcal{X} \\to T_{H^*}$,\n$\\mathcal{X} _{\\bf 1}$ or $\\left(\\mathcal{X} _{\\bf 1}\\right)^{\\vee}$ we\nwill be implicitly assuming that we have set\n$$H_{\\mathcal{X} }= H = d^{-1}\\cdot H_{\\mathcal{X} ^{\\vee}},$$\n$$\\quad  \\mathcal{X} _{\\bf 1}= \\mathcal{X} _{{\\bf 1}_{T_{H^*_{\\mathcal{X} }}}}\\quad \\quad \\text{and} \\quad \\quad \\left(\\mathcal{X} _{\\bf 1}\\right)^{\\vee}= \\mathcal{X} ^{\\vee}/T_{H_{\\mathcal{X} ^{\\vee}}}.$$\n\n**Remark 14**. Let $\\mathcal{V}$ be (a quotient of) $\\mathcal{A}$ or (a\nfibre of) $\\mathcal{X}$. In the skew-symmetric case Argüz and Bousseau\nshowed that $\\mathcal{V}$ and $\\mathcal{V} ^{\\vee}$ are mirror dual\nschemes from the point of view of . A similar result is proven for the\nskew-symmetrizable case when $\\mathcal{V}$ has dimension $2$ in with\narguments that may be generalized to arbitrary dimension.\n\n## Scattering diagrams and theta functions\n\nTheta functions are a particular class of global function on (quotients\nand fibres of) cluster varieties introduced in . In this subsection we\noutline their construction. The main case to consider is the one of\n$\\mathcal{A}_{\\mathrm{prin}}$ since scattering diagrams and theta\nfunctions for (quotients of) $\\mathcal{A}$ and (fibres of) $\\mathcal{X}$\ncan be constructed from this case.\n\n**Remark 15**. From now on, whenever we consider the variety\n$\\mathcal{A} =\\mathcal{A} _{\\Gamma,\\textbf{s}_0}$ we will assume\n$\\Gamma$ is of **full-rank**. By definition this means that the map\n$p_1^*:N_{\\text{uf}}\\to M^\\circ$ given by $n \\mapsto \\{ n , \\cdot \\}$ is\ninjective. There are various results of this article for $\\mathcal{A}$\nthat are valid even if $\\Gamma$ is not of full-rank. However, various\nkey results we shall use do need the full-rank condition (*cf.* Remark\n). Even though we are imposing full-rank assumption we will frequently\nrecall that we are assuming it to insist on the necessity of the\nassumption.\n\n### Theta functions on full-rank $\\mathcal{A}$\n\nThroughout this section we systematically identify\n$\\mathcal{A} ^{\\vee}_{\\textbf{s}^\\vee}(\\mathbb{R} ^T)$ with\n$M^\\circ_{\\mathbb{R} }$, see §. A **wall** in $M^{\\circ}_\\mathbb{R}$ is\na pair $(\\mathfrak{d}, f_{\\mathfrak{d}})$ where\n$\\mathfrak{d}\\subseteq M^{\\circ}_\\mathbb{R}$ is a convex rational\npolyhedral cone of codimension one, contained in $n^{\\perp}$ for some\n$n \\in N_{\\operatorname{uf}, \\textbf{s}}^+$, and\n$f_{\\mathfrak{d}}  = 1+ \\sum_{k \\geq 1} c_k z^{kp^*_1(n)}$ is called a\n**scattering function**, where $c_k \\in \\Bbbk$. A **scattering diagram**\n$\\mathfrak{D}$ in $M^{\\circ}_\\mathbb{R}$ is a (possibly infinite)\ncollection of walls satisfying a certain finiteness condition (see ).\nThe **support** and the **singular locus** of $\\mathfrak{D}$ are defined\nas\n$$\\mathop{\\mathrm{Supp}}(\\mathfrak{D}):= \\bigcup_{\\mathfrak{d}\\in \\mathfrak{D}} \\mathfrak{d}\\ \\ \\ \\text{and} \\ \\ \\ \\mathop{\\mathrm{Sing}}(\\mathfrak{D}):= \\bigcup_{\\mathfrak{d}\\in \\mathfrak{D}} \\partial\\mathfrak{d}\\ \\cup \\bigcup_{\\overset{\\mathfrak{d}_1,\\mathfrak{d}_2 \\in \\mathfrak{D}}{\\text{dim}(\\mathfrak{d}_1 \\cap \\mathfrak{d}_2) = |I|-2}} \\mathfrak{d}_1 \\cap \\mathfrak{d}_2.$$\n\nA wall $(\\mathfrak{d}, f_{\\mathfrak{d}})$ defines a **wall-crossing\nautomorphism** $\\mathfrak{p}_{\\mathfrak{d}}$ of $\\Bbbk (M)$ given in a\ngenerator $z^m$ by\n$\\mathfrak{p}_{\\mathfrak{d}}(z^m)=z^m f_{\\mathfrak{d}}^{\\langle n_{\\mathfrak{d}}, m \\rangle }$,\nwhere $n_{\\mathfrak{d}}$ is the primitive normal vector of the wall\n$\\mathfrak{d}$ with a choice of direction going against the flow of the\npath $\\gamma$. If we fix a scattering diagram $\\mathfrak{D}$ and a\npiecewise linear proper map\n$\\gamma:[0,1]\\to M^\\circ_{\\mathbb{R} }\\setminus \\mathop{\\mathrm{Sing}}(\\mathfrak{D})$\nintersecting $\\text{Supp}( \\mathfrak{D})$ transversely then the **path\nordered product** $\\mathfrak{p}_{\\gamma , \\mathfrak{D}}$ is defined as\nthe composition of automorphisms of the form\n$\\mathfrak{p}_{\\mathfrak{d}}$, where we consider the walls\n$\\mathfrak{d}$ that are transversely crossed by $\\gamma$. However,\nobserve that $\\gamma$ might cross an infinite number of walls,\ntherefore, we would be potentially composing an infinite number of\nautomorphisms and such infinite composition is well defined. Again, the\nreader is referred to for a detailed discussion.\n\n**Definition 16**. A scattering diagram $\\mathfrak{D}$ is **consistent**\nif for all $\\gamma$ as above $\\mathfrak{p}_{\\gamma, \\mathfrak{D}}$ only\ndepends on the endpoints of $\\gamma$. Two scattering diagrams\n$\\mathfrak{D}$ and $\\mathfrak{D}'$ are **equivalent** if\n$\\mathfrak{p}_{\\gamma, \\mathfrak{D}}= \\mathfrak{p}_{\\gamma, \\mathfrak{D}'}$\nfor all $\\gamma$.\n\nTo define cluster scattering diagrams for $\\mathcal{A}$ one first\nconsiders\n$$\\mathfrak{D}_{{\\rm in}, \\textbf{s}}^{\\mathcal{A} } := \\left\\{\\left.\\left( e_i^{\\perp} , 1+z^{ p_{1}^*\\left( e_i \\right)}\\right) \\right| \\  i \\in I_{\\text{uf}}\\right\\}.$$\nA **cluster scattering diagram** for $\\mathcal{A}$ is a consistent\nscattering diagram in $M^{\\circ}_{ \\mathbb{R} }$ containing\n$\\mathfrak{D}_{{\\rm in}, \\textbf{s}}^{\\mathcal{A} }$. By the following\ntheorem, cluster scattering diagrams for $\\mathcal{A}$ do exist\n(provided $\\Gamma$ is of full-rank).\n\n**Theorem 17**. Assume $\\Gamma$ is of full-rank. Then for every seed\n$\\textbf{s}$ there is a consistent scattering diagram\n$\\mathfrak{D}_{\\textbf{s}}^{\\mathcal{A} }$ such that\n$\\mathfrak{D}_{{\\rm in}, \\textbf{s}}^{\\mathcal{A} } \\subset \\mathfrak{D}_{\\textbf{s}}^{\\mathcal{A} }$.\nFurthermore $\\mathfrak{D}_{\\textbf{s}}^{\\mathcal{A} }$ is equivalent to\na scattering diagram all of whose scattering functions are of the form\n${ f_{\\mathfrak{d}} = (1+ z^{p_{1}^*(n)})^c}$, for some $n \\in N$, and\n$c$ a positive integer.\n\n**Definition 18**.\n\nFix a cluster scattering diagram\n$\\mathfrak{D}^{\\mathcal{A} }_{\\textbf{s}}$. Let\n$m\\in M^{\\circ} \\setminus \\{0\\}$ and\n$x_0 \\in M^{\\circ}_{\\mathbb{R} } \\setminus \\text{Supp}(\\mathfrak{D})$. A\n(generic) **broken line** for $\\mathfrak{D}^{\\mathcal{A} }_{\\textbf{s}}$\nwith initial exponent $m$ and endpoint $x_0$ is a piecewise linear\ncontinuous proper path\n$\\gamma : ( - \\infty , 0 ] \\rightarrow M^\\circ_{\\mathbb{R} } \\setminus \\mathop{\\mathrm{Sing}}(\\mathfrak{D}^{\\mathcal{A} }_{\\textbf{s}})$\nbending only at walls, with a finite number of domains of linearity $L$\nand a monomial $c_L z^{m_L} \\in \\Bbbk[M^\\circ]$ for each of these\ndomains. The path $\\gamma$ and the monomials $c_L z^{m_L}$ are required\nto satisfy the following conditions:\n\n-   $\\gamma(0) = x_0$.\n\n-   If $L$ is the unique unbounded domain of linearity of $\\gamma$, then\n    $c_L z^{m_L} = z^{m}$.\n\n-   For $t$ in a domain of linearity $L$, $\\gamma'(t) = -m_L$.\n\n-   If $\\gamma$ bends at a time $t$, passing from the domain of\n    linearity $L$ to $L'$ then $c_{L'}z^{m_{L'}}$ is a term in\n    $\\mathfrak{p}_{{\\gamma}|_{(t-\\epsilon,t+\\epsilon)},\\mathfrak{D}_t} (c_L z^{m_L})$,\n    where\n    ${\\mathfrak{D}_t = \\left\\{\\left.(\\mathfrak{d}, f_{\\mathfrak{d}}) \\in \\mathfrak{D}^{\\mathcal{A} }_{\\textbf{s}}\\right| \\gamma (t) \\in \\mathfrak{d}\\right\\}}$.\n\nWe refer to $m_L$ as the **slope** or **exponent vector** of $\\gamma$ at\n$L$ and set\n\n-   $I(\\gamma) = m$;\n\n-   $\\text{Mono} (\\gamma) = c(\\gamma)z^{F(\\gamma)}$ to be the monomial\n    attached to the unique domain of linearity of $\\gamma$ having $x_0$\n    as an endpoint.\n\n**Definition 19**.\n\nChoose a point $x_0$ in the interior of\n$\\mathcal{C}_\\textbf{s}^+:=\\{m\\in M^{\\circ}_{\\mathbb{R} }\\mid \\langle e_i, m \\rangle \\geq 0 \\text{ for all  } i \\in I_{\\text{uf}}\\}$\nand let\n$m\\in \\mathcal{A} ^\\vee_{\\textbf{s}^\\vee}(\\mathbb{Z} ^T)=M^\\circ$. The\n**theta function** on $\\mathcal{A}$ associated to $m$ is $$\\label{eq:tf}\n    \\vartheta ^{\\mathcal{A} }_{ m}:= \\sum_{\\gamma} \\text{Mono} (\\gamma),$$\nwhere the sum is over all broken lines $\\gamma$ with $I(\\gamma)=m$ and\n$\\gamma(0)=x_0$. For $m= 0$ we define $\\vartheta^{\\mathcal{A} }_{0} =1$.\nWe say $\\vartheta ^{\\mathcal{A} }_{ m}$ is **polynomial** if the sum in\nis finite.\n\n**Remark 20**. It is a nontrivial fact that\n$\\vartheta ^{\\mathcal{A} }_{m}$ is independent of the point\n$x_0\\in \\mathcal{C}_{\\textbf{s}}^+$ we have chosen, see . Moreover, in\ngeneral $\\vartheta ^\\mathcal{A} _{m}$ can be an infinite sum and in\norder to think of $\\vartheta ^{\\mathcal{A} }_{m}$ as a function on a\nspace one needs to work formally an consider a degeneration of\n$\\mathcal{A}$, see for the details. However, in case\n$\\vartheta ^{\\mathcal{A} }_{m}$ is polynomial then\n$\\vartheta ^{\\mathcal{A} }_{m}\\in H^0(\\mathcal{A} ,\\mathcal{O}_{\\mathcal{A} })$,\nthat is, $\\vartheta ^{\\mathcal{A} }_{m}$ is an algebraic function on\n$\\mathcal{A}$. The definition of $\\vartheta ^{\\mathcal{A} }_{m}$ in\ncorresponds to the expression of such function written in the\ncoordinates of the seed torus $\\mathcal{A} _{\\textbf{s}}$.\n\n### Labeling by tropical points\n\nRecall that we are identifying\n$\\mathcal{A} ^{\\vee}_{\\textbf{s}^\\vee}(\\mathbb{R} ^T)$ and\n$M^{\\circ}_{\\mathbb{R} }$. By construction, a theta function on\n$\\mathcal{A}$ is labeled by a point\n$m\\in \\mathcal{A} ^\\vee_{\\textbf{s}^\\vee}(\\mathbb{Z} ^T)=  M^{\\circ}$.\nBy , this labeling upgrades to a labeling by a point in\n$\\mathcal{A} ^\\vee(\\mathbb{Z} ^T)$. The main point being that if we let\n$m'=(\\mu^{\\mathcal{A} ^\\vee}_{k})^T(m)\\in \\mathcal{A} ^{\\vee}_{\\mu_k(\\textbf{s}^\\vee)}(\\mathbb{Z} ^T)$\nfor $k \\in I_{\\text{uf}}$ then $\\vartheta ^{\\mathcal{A} }_m$ and\n$\\vartheta ^{\\mathcal{A} }_{m'}$ correspond to the same function (see\nRemark ) expressed, however, in different cluster coordinates. This fact\nis of great importance for this paper so we would like to highlight it:\n\n*every theta function on $\\mathcal{A}$ is naturally labeled by a point\nof $\\mathcal{A} ^\\vee(\\mathbb{Z} ^T)$*.\n\nIn light of the discussion just above, from now on we label theta\nfunctions on $\\mathcal{A}$ either by elements of\n$\\mathcal{A} ^{\\vee}(\\mathbb{Z} ^T)$ or of\n$\\mathcal{A} ^{\\vee}_{\\textbf{s}^\\vee}(\\mathbb{Z} ^T)$. For sake of\nclarity, tropical points are denoted in bold font and as tuples. That\nis, ${\\bf m}=(m_{\\textbf{s}^\\vee})_{\\textbf{s}^\\vee\\in %\n  \\mathrel{\\mathop{\\mathbb{T}_r}\\limits^{\n    \\vbox to0ex{\\kern-0.5\\ex@\n    \\hbox{$\\scriptstyle\\longrightarrow$}\\vss}}}}$ denotes an element of\n$\\mathcal{A} ^\\vee(\\mathbb{Z} ^T)$ and\n$m_{\\textbf{s}^\\vee}=\\mathfrak{r}_{\\textbf{s}^\\vee}({\\bf m})$. With this\nnotation we have the following identity\n$$\\vartheta ^{\\mathcal{A} }_{\\bf m}=\\vartheta ^{\\mathcal{A} }_{m_{\\textbf{s}^\\vee}}.$$\n\nEven further, we can think of\n$\\mathop{\\mathrm{Supp}}(\\mathfrak{D}^{\\mathcal{A} }_{\\textbf{s}})$ as a\nsubset of $\\mathcal{A} ^{\\vee}_{\\textbf{s}^\\vee}(\\mathbb{R} ^T)$. By we\nhave that for every $k \\in I_{\\text{uf}}$,\n$\\mu^{\\mathcal{A} ^\\vee}_{\\textbf{s}^\\vee,\\mu_k(\\textbf{s}^\\vee)}\\left(\\mathop{\\mathrm{Supp}}(\\mathfrak{D}^{\\mathcal{A} }_{\\textbf{s}})\\right)=\\mathop{\\mathrm{Supp}}(\\mathfrak{D}^{\\mathcal{A} }_{\\mu_k(\\textbf{s})})$\nand that $\\mathfrak{D}^{\\mathcal{A} }_{\\textbf{s}}$ and\n$\\mathfrak{D}^{\\mathcal{A} }_{\\mu_k(\\textbf{s})}$ are equivalent. Hence\nthere is a well defined subset\n$\\mathop{\\mathrm{Supp}}(\\mathfrak{D}^\\mathcal{A} ) \\subset \\mathcal{A} ^{\\vee}(\\mathbb{R} ^T)$\nsuch that\n$$\\mathfrak{r}_{\\textbf{s}^\\vee}\\left(\\mathop{\\mathrm{Supp}}(\\mathfrak{D}^\\mathcal{A} ) \\right)= \\mathop{\\mathrm{Supp}}(\\mathfrak{D}^\\mathcal{A} _{\\textbf{s}})$$\nfor every $\\textbf{s}\\in %\n  \\mathrel{\\mathop{\\mathbb{T}_r}\\limits^{\n    \\vbox to0ex{\\kern-0.5\\ex@\n    \\hbox{$\\scriptstyle\\longrightarrow$}\\vss}}}$. The point here is that\n$\\mathop{\\mathrm{Supp}}(\\mathfrak{D}^\\mathcal{A} )$ is seed independent.\nSimilarly, the map\n$\\mu^{\\mathcal{A} ^\\vee}_{\\textbf{s}^\\vee,\\mu_k(\\textbf{s}^\\vee)}$\ndetermines a bijection between the set of broken lines for\n$\\mathfrak{D}^{\\mathcal{A} }_{\\textbf{s}}$ and the set of broken lines\nfor $\\mathfrak{D}^{\\mathcal{A} }_{\\mu_k(\\textbf{s})}$ (see ). In\nparticular, supports of broken lines make sense in\n$\\mathcal{A} ^\\vee(\\mathbb{R} ^T)$.\n\n**Remark 21**. It is possible to upgrade\n$\\mathop{\\mathrm{Supp}}(\\mathfrak{D}^\\mathcal{A} )$ to a scattering\ndiagram inside $\\mathcal{A} ^{\\vee}(\\mathbb{R} ^T)$. In this generality\nscattering functions are described using log Gromov–Witten invariants.\nSee for details.\n\n### The middle cluster algebra\n\nLet us recall now that broken lines also encode the multiplication of\ntheta functions. That is, given a product of arbitrary theta functions\n$\\vartheta ^{\\mathcal{A} }_p \\vartheta ^{\\mathcal{A} }_q$ with\n$p,q \\in \\mathcal{A} ^\\vee_{\\textbf{s}^\\vee}(\\mathbb{Z} ^T)$, we can use\nbroken lines to express the structure constants\n$\\alpha\\left(p,q,r\\right)$ in the expansion $$\\begin{aligned}\n \\label{eq:product}\n    \\vartheta^{\\mathcal{A} }_p \\vartheta^{\\mathcal{A} }_q = \\sum_{r\\in \\mathcal{A} ^\\vee_{\\textbf{s}^\\vee}(\\mathbb{Z} ^T)} \\alpha(p,q,r) \\vartheta^{\\mathcal{A} }_r.\n\\end{aligned}$$ We review the construction here. First, pick a general\nendpoint $z$ near $r$. Then define ()\n$$\\label{eq:multibrokenline} \\begin{split} \n    \\alpha_z (p, q, r) := \\sum_{\\substack{\\left(\\gamma^{(1)}, \\gamma^{(2)}\\right) \\\\ I(\\gamma^{(1)})= p,\\ I(\\gamma^{(2)})= q\\\\ \\gamma^{(1)}(0) = \\gamma^{(2)}(0) = z\\\\\n    F(\\gamma^{(1)}) + F(\\gamma^{(2)}) = r   }}   c(\\gamma^{(1)})\\ c(\\gamma^{(2)}),   \\end{split}$$\nwhere the sum is over all pairs of broken lines\n$\\left(\\gamma^{(1)}, \\gamma^{(2)}\\right)$ ending at $z$ with initial\nslopes $I(\\gamma^{(1)}) = p$, $I(\\gamma^{(2)}) = q$ and final slopes\nsatisfying $F(\\gamma^{(1)})+F(\\gamma^{(2)}) =r$.\nGross–Hacking–Keel–Kontsevich show that for $z$ sufficiently close to\n$r$, $\\alpha_z (p, q, r)$ is independent of $z$ and gives the structure\nconstant $\\alpha (p, q, r)$ (see ).\n\n**Definition 22**.\n\nLet\n$\\Theta(\\mathcal{A} ):= \\{ {\\bf m} \\in \\mathcal{A} ^{\\vee}(\\mathbb{Z} ^T) \\mid \\vartheta^{\\mathcal{A} }_{{\\bf m}} \\text{ is polynomial}\\}$.\nThe **middle cluster algebra** $\\text{mid}(\\mathcal{A} )$ is the\n$\\Bbbk$-algebra whose underlying vector space is\n$\\{ \\vartheta ^{\\mathcal{A} }_{{\\bf m}} \\mid {\\bf m} \\in \\Theta(\\mathcal{A} ) \\}$,\nthe multiplication of the basis elements is given by and extended\nlinearly to all $\\mathop{\\mathrm{mid}}(\\mathcal{A} )$.\n\n### Theta functions on $\\mathcal{A}_{\\mathrm{prin}}$\n\nThe data $\\Gamma_{{\\mathrm{prin}} }$ is of full-rank. Therefore, this\ncase is a particular case of §. So we can talk about scattering\ndiagrams, broken lines and theta functions for\n$\\mathcal{A}_{\\mathrm{prin}}$. The following result follows from Theorem\nand the definition of theta functions.\n\n**Lemma 23**. Fix a seed $\\widetilde{\\textbf{s}}$ for\n$\\mathcal{A}_{\\mathrm{prin}}$ and express theta functions on the cluster\ncoordinates determined by $\\widetilde{\\textbf{s}}$. For\n$(m,n)\\in \\mathfrak{r}_{\\widetilde{\\textbf{s}}}(\\Theta(\\mathcal{A}_{\\mathrm{prin}}))$\nwe have that\n$\\vartheta ^{\\mathcal{A}_{\\mathrm{prin}}}_{(m,n)}=\\vartheta ^{\\mathcal{A}_{\\mathrm{prin}}}_{(m,0)}\\vartheta ^{\\mathcal{A}_{\\mathrm{prin}}}_{(0,n)}$\nand $\\vartheta ^{\\mathcal{A}_{\\mathrm{prin}}}_{(0,n)}$ is the Laurent\nmonomial on the coefficients given by $n$.\n\nNote that, for $(m_1,n_1),(m_2,n_2) \\in M^{\\circ}_{\\rm prin}$, in\ngeneral we have that\n$\\vartheta ^{\\mathcal{A}_{\\mathrm{prin}}}_{(m_1+m_2,n_1+n_2)} \\neq \\vartheta ^{\\mathcal{A}_{\\mathrm{prin}}}_{(m_1,n_1)} \\vartheta ^{\\mathcal{A}_{\\mathrm{prin}}}_{(m_2,n_2)}$.\nThe above lemma holds because the decomposition is only separating the\nunfrozen and frozen parts (*cf.*  below).\n\n**Remark 24**. Scattering diagrams for $\\mathcal{A}_{\\mathrm{prin}}$ can\nbe used to define scattering diagrams, broken lines therein and theta\nfunctions on a variety $\\mathcal{V}$ of form $\\mathcal{A}$ (even if\n$\\Gamma$ is not of full-rank), $\\mathcal{X}$, $\\mathcal{A} /T_{H}$ and\n$\\mathcal{X} _{{\\bf 1}}$. Further, in each one of these cases we can\ndefine the associated middle cluster algebra\n$\\mathop{\\mathrm{mid}}(\\mathcal{V} )$ and the set $\\Theta(\\mathcal{V} )$\nparametrizing its theta basis. In the following subsections we explain\nthe cases of $\\mathcal{A} /T_{H}$, $\\mathcal{X}$, and\n$\\mathcal{X} _{{\\bf 1}}$ individually. We do not treat the case of\n$\\mathcal{A}$ for $\\Gamma$ when $\\Gamma$ is not of full-rank since the\nresults of § do not apply to this case.\n\n### Theta functions on $\\mathcal{A} /T_{H}$\n\nSuppose that $\\Gamma$ is of full-rank (*cf.* Remark ). Let\n$H \\subset K^\\circ$ be a saturated sublattice and consider the quotient\n$\\mathcal{A} /T_{H}$ and the fibration $w_H: \\mathcal{A} ^{\\vee}\\to H^*$\n(see the end of §). The next result shows that theta functions on\n$\\mathcal{A}$ have a well defined $T_{H}$-weight.\n\n**Proposition 25**.\n\nEvery polynomial theta function on $\\mathcal{A}$ is an eigenfunction\nwith respect to the $T_{H}$-action. For every\n${\\bf q}\\in \\Theta(\\mathcal{A} )$ the $T_H$-weight of\n$\\vartheta ^{\\mathcal{A} }_{\\bf q}$ is the image of\n${\\bf q} \\in \\mathcal{A} ^{\\vee}(\\mathbb{Z} ^T)$ under the tropicalized\nmap $w^{T}_{H}:\\mathcal{A} ^{\\vee}(\\mathbb{Z} ^T) \\to H^*$. Under the\nisomorphism $H^* \\cong M^\\circ/H^\\perp$ and in the lattice\nidentification of $\\mathcal{A} ^\\vee_{\\textbf{s}^\\vee}(\\mathbb{Z} ^T)$\nof $\\mathcal{A} ^\\vee (\\mathbb{Z} ^T)$ the map $w^T_{H}$ is given by\n$$\\begin{aligned}\n   w^{T}_{H} : \\ & \\mathcal{A} ^{\\vee}_{\\textbf{s}^\\vee}(\\mathbb{Z} ^T) \\to M^\\circ/H^{\\perp},\\\\\n     & \\ \\ \\  \\ \\ \\  q \\longmapsto q + H^{\\perp}.\n\\end{aligned}$$\n\nThe claims are essentially contained in the literature already (see for\ninstance ). The differences are that we are acting by a potentially\nsmaller torus (Gross–Hacking–Keel–Kontsevich act by $T_{K^\\circ}$ rather\nthan $T_{H}$) and, regarding the map\n$w_{H}: \\mathcal{A} ^\\vee \\to T_{H^*}$, we are including $\\Bbbk[H]$ into\n$\\Bbbk[N^\\vee]=\\Bbbk[N^\\circ]$ rather than including $\\Bbbk[K^\\circ]$\ninto $\\Bbbk[N^\\circ]$. For the convenience of the reader we give a proof\nof the statement.\n\n*Proof of Proposition .* By all scattering functions may be taken to be\nof the form $\\left(1+z^{p^*(n)}\\right)^c$ for some $n \\in N_{\\text{uf}}$\nand some positive integer $c$.[^4] For\n$q\\in \\Theta(\\mathcal{A} )_{\\textbf{s}^\\vee}$ we have that\n$\\vartheta ^{\\mathcal{A} }_q$ is as a Laurent polynomial in\n$\\Bbbk[M^\\circ]$. All monomial summands of\n$\\vartheta ^{\\mathcal{A} }_{q}$ have the form $c_m z^{q + m}$ for some\n$m \\in p^*(N_{\\text{uf}})$ and $c_m \\in \\mathbb{Z} _{>0}$. The\n$T_{H}$-weight of this monomial is obtained from the map\n$$\\begin{split} \nT_{H} \\to T_{\\mathbb{Z} }=\\Bbbk^*\\quad \\text{given by } \\quad z^{h} \\mapsto z^{\\left\\langle{q+m , h}\\right\\rangle} \\quad \\text{for $h \\in {H}$}.\n \\end{split}$$ Since $H \\subset p^*\\left(N_{\\text{uf}}\\right)^\\perp$ we\nhave that\n$z^{\\left\\langle{q+m , h}\\right\\rangle} = z^{\\left\\langle{q, h}\\right\\rangle}$.\nThat is, the $T_{H}$-weight of each monomial $z^{m'}$, $m'\\in M^\\circ$,\nis the character of $T_{H}$ given by\n$m' + H^\\perp  \\in M^\\circ/H^\\perp \\cong H^*$. Moreover, all monomial\nsummands of $\\vartheta ^{\\mathcal{A} }_q$ have the $T_{H}$-weight\n$q + H^\\perp \\in H^*$. Next, the piecewise linear map\n$(\\mu_k^{\\mathcal{A} ^\\vee})^T:M^\\circ_\\textbf{s}\\to M^\\circ_{\\mu_k(\\textbf{s})}$\nsends $m$ to $m+m'$ for some $m'\\in p^*(N_{\\text{uf}})$. So, the choice\nof torus does not affect the $T_{H}$-weight. Therefore,\n$\\vartheta ^{\\mathcal{A} }_q$ is an eigenfunction whose weight is\n$q + H^\\perp$. Furthermore, the projection\n$$\\begin{split} M^\\circ &\\to M^\\circ/H^\\perp\\quad \\text{given by} \\quad q \\mapsto q + H^\\perp  \\end{split}$$\ndualizes the inclusion $H \\hookrightarrow N^\\circ$. So, restricting to\nseed tori, this is precisely the tropicalization of the map\n${T_{M^\\circ} \\rightarrow T_{H^*}}$ whose pullback is the inclusion\n$H \\hookrightarrow N^\\circ$. Since $p^*$ commutes with mutation, we see\nthat the $T_{H}$-weight of $\\vartheta ^{\\mathcal{A} }_{\\bf q}$ is the\nimage of ${\\bf q}$ under the tropicalization of\n$w_{H}: \\mathcal{A} ^\\vee \\rightarrow T_{H^*}$. ◻\n\nEvery weight $0$ eigenfunction on $\\mathcal{A}$ induces a well defined\nfunction on $\\mathcal{A} /T_{H}$. So in order to construct a\nscattering-diagram-like structure $\\mathfrak{D}^{\\mathcal{A} /T_H}$\ndefining theta functions on $\\mathcal{A} /T_{H}$ we consider the\n**weight zero slice** inside $\\mathcal{A} ^{\\vee}(\\mathbb{R} ^T)$\ndefined as $(w^T_H)^{-1}(0)$. Observe that identifying\n$\\mathcal{A} ^\\vee$ with $M^\\vee$ via a choice of seed, then\n$(w^T_H)^{-1}(0)$ corresponds to $H^{\\perp}_{\\mathbb{R} }$. With this in\nmind, we define\n$\\mathop{\\mathrm{Supp}}(\\mathfrak{D}^{\\mathcal{A} /T_{H}})$ as\n$$\\mathop{\\mathrm{Supp}}(\\mathfrak{D}^{\\mathcal{A} /T_{H}}):=\\mathop{\\mathrm{Supp}}(\\mathfrak{D}^{\\mathcal{A} })\\cap (w^T_H)^{-1}(0).$$\nThe scattering functions attached to the walls of\n$\\mathfrak{r}_{\\textbf{s}^\\vee}(\\mathop{\\mathrm{Supp}}(\\mathfrak{D}^{\\mathcal{A} /T_{H}}))$\nare the same as the corresponding functions attached to the walls of\n$\\mathfrak{D}^{\\mathcal{A} }_\\textbf{s}$. This gives rise to a\nscattering diagram $\\mathfrak{D}^{\\mathcal{A} /T_{H}}_{\\textbf{s}}$\ninside $(\\mathcal{A} /T_{H})^\\vee_{\\textbf{s}^\\vee}(\\mathbb{R} ^T)$ for\nevery $\\textbf{s}\\in %\n  \\mathrel{\\mathop{\\mathbb{T}_r}\\limits^{\n    \\vbox to0ex{\\kern-0.5\\ex@\n    \\hbox{$\\scriptstyle\\longrightarrow$}\\vss}}}$. The broken lines for\n$\\mathfrak{D}^{\\mathcal{A} /T_{H}}_\\textbf{s}$ are the broken lines for\n$\\mathfrak{D}^{\\mathcal{A} }_\\textbf{s}$ entirely contained in\n$\\mathfrak{r}_{\\textbf{s}^\\vee}(w^{-1}_{H}(0))$.\n\nIn order to label a theta function on $\\mathcal{A} /T_{H}$ with an\nelement of $(\\mathcal{A} /T_{H})^{\\vee}(\\mathbb{Z} ^T)$ it suffices to\nconsider a bijection\n$(\\mathcal{A} /T_{H})^{\\vee}(\\mathbb{R} ^T) \\overset{\\sim}{\\longrightarrow} (w_H^T)^{-1}(0)$.\nSuch a bijection can be obtained tropicalizing the inclusion\n$\\mathfrak{i}_H:(\\mathcal{A} /T_{H})^{\\vee} \\hookrightarrow \\mathcal{A} ^\\vee$.\nIndeed, in lattice identifications of the tropical spaces given by a\nseed $\\textbf{s}$, the map\n$\\mathfrak{i}_H^T:(\\mathcal{A} /T_{H})^{\\vee}_{\\textbf{s}}(\\mathbb{Z} ^T)\\hookrightarrow \\mathcal{A} ^\\vee_{\\textbf{s}}(\\mathbb{Z} ^T)$\ncorrespond to the inclusion $H^\\perp \\hookrightarrow M^\\vee$ and\n$w^{-1}_{H}(0)(\\mathbb{Z} )$ corresponds to $H^\\perp$.\n\nIn particular, we obtain (as one should have expected) that the theta\nfunctions on $\\mathcal{A} /T_{H}$ are precisely the functions on\n$\\mathcal{A} /T_{H}$ induced by the $T_H$-weight zero theta functions on\n$\\mathcal{A}$. So we let\n$\\Theta(\\mathcal{A} /T_H)\\subset (\\mathcal{A} /T_H)^{\\vee}(\\mathbb{Z} ^T)$\nbe the preimage of $\\Theta(\\mathcal{A} )\\cap (w_H^T)^{-1}(0)$ under\n$\\mathfrak{i}^T_H$ and define the middle cluster algebra\n$\\mathop{\\mathrm{mid}}(\\mathcal{A} /T_H)$ as in the case of\n$\\mathcal{A}$ (see ). In particular, for\n${\\bf m}\\in \\Theta (\\mathcal{A} /T_H)$ the theta function\n$\\vartheta ^{\\mathcal{A} /T_H}_{\\bf m}$ is the function on\n$\\mathcal{A} /T_H$ induced by\n$\\vartheta ^{\\mathcal{A} }_{\\mathfrak{i}^T_H({\\bf m})}$. So,\n\n*every theta function on $\\mathcal{A} /T_H$ is naturally labeled by a\npoint of $(\\mathcal{A} /T_H)^\\vee(\\mathbb{Z} ^T)$*.\n\n### Theta functions on $\\mathcal{X}$\n\nRecall from § that there is an isomorphism\n$\\chi: \\mathcal{A}_{\\mathrm{prin}}/T_{H_{\\mathcal{A}_{\\mathrm{prin}}}}\\to \\mathcal{X}$,\nwhere\n$$H_{\\mathcal{A}_{\\mathrm{prin}}}=\\left\\{\\left(n,-(p^*)^*(n)\\right)\\in N^\\circ_{\\mathrm{prin}} \\mid n \\in N^\\circ\\right\\} \\subset K^{\\circ}_{{\\mathrm{prin}} }.$$\nHence, the construction of theta functions on $\\mathcal{X}$ is already\ncovered in the previous subsection. However, there is a very subtle\ndifference created by treating\n$\\mathcal{A}_{\\mathrm{prin}}/T_{H_{\\mathcal{A}_{\\mathrm{prin}}}}$ as a\ncluster $\\mathcal{X}$-variety as opposed to a quotient of\n$\\mathcal{A}_{\\mathrm{prin}}$:\n\n*every theta function on $\\mathcal{X}$ is naturally labeled by a point\nof $\\mathcal{X} ^\\vee(\\mathbb{Z} ^t)$ as opposed to\n$\\mathcal{X} ^\\vee(\\mathbb{Z} ^T)$.*\n\nIf we would proceed as in the previous subsection we would label theta\nfunctions on\n$\\mathcal{A}_{\\mathrm{prin}}/T_{H_{\\mathcal{A}_{\\mathrm{prin}}}}$ by\npoints of\n$(\\mathcal{A}_{\\mathrm{prin}}/T_{H_{\\mathcal{A}_{\\mathrm{prin}}}})^\\vee(\\mathbb{R} ^T)$.\nThe origin of the difference is made explicit by the following lemma.\n\n**Lemma 26**. There is a canonical bijection between\n$\\mathcal{X} ^{\\vee}(\\mathbb{R} ^t)$ and\n$\\left(w^T_{H_{\\mathcal{A}_{\\mathrm{prin}}}}\\right)^{-1}(0)\\subset \\mathcal{A}_{\\mathrm{prin}}^\\vee(\\mathbb{R} ^T)$.\n\n*Proof.* One can verify directly that the composition\n$\\xi^T_{\\Gamma^\\vee}\\circ i$ gives rise to the desired bijection, where\n$$i:\\mathcal{X} ^\\vee(\\mathbb{R} ^t) \\to \\mathcal{X} ^\\vee(\\mathbb{R} ^T)$$\nis the bijection discussed in § and\n$$\\xi_{\\Gamma^\\vee}^T:  \\mathcal{X} ^\\vee(\\mathbb{R} ^T) \\to \\mathcal{A}_{\\mathrm{prin}}^{\\vee}(\\mathbb{R} ^T)$$\nis the tropicalization of the map\n$\\xi_{\\Gamma^\\vee}:\\mathcal{X} ^\\vee=\\mathcal{A} _{\\Gamma^\\vee} \\to  \\mathcal{X} _{(\\Gamma^\\vee)_{{\\mathrm{prin}} }}\\cong \\mathcal{X} _{(\\Gamma_{{\\mathrm{prin}} })^\\vee}=\\mathcal{A}_{\\mathrm{prin}}^{\\vee}$\ndescribed in , see Remarks and . However, for the convenience of the\nreader we include computations that show in a rather explicit way the\nnecessity to consider $\\mathcal{X} ^\\vee(\\mathbb{Z} ^t)$ as opposed to\n$\\mathcal{X} ^\\vee(\\mathbb{Z} ^T)$. For simplicity throughout this proof\nwe denote $w_{H_{\\mathcal{A}_{\\mathrm{prin}}}}$ simply by $w$.\n\nPick a seed $\\textbf{s}=(e_i)_{i \\in I}\\in %\n  \\mathrel{\\mathop{\\mathbb{T}_r}\\limits^{\n    \\vbox to0ex{\\kern-0.5\\ex@\n    \\hbox{$\\scriptstyle\\longrightarrow$}\\vss}}}$ for $\\Gamma$ and\nconsider the seed $\\textbf{s}^\\vee$ for $\\Gamma^\\vee$. Denote by\n$\\widetilde{\\textbf{s}}^\\vee$ the seed for\n$(\\Gamma_{{\\mathrm{prin}} })^\\vee$ obtained mutating\n$\\textbf{s}_{0_{{\\mathrm{prin}} }}$ in the same sequence of directions\nneeded to obtain $\\textbf{s}$ from $\\textbf{s}_0$. Then\n$$\\left((w^T)^{-1}(0)\\right)_{\\widetilde{\\textbf{s}}^\\vee}(\\mathbb{R} ^T)=H^\\perp_{\\mathcal{A}_{\\mathrm{prin}}}=\\{(p^*(n),n)\\in M^\\circ_{{\\mathrm{prin}} ,\\mathbb{R} } \\mid n\\in N_{\\mathbb{R} }\\}\\subset M^\\circ_{{\\mathrm{prin}} , \\mathbb{R} }=M^\\circ_{\\mathbb{R} }\\oplus N_{\\mathbb{R} }$$\n(see to recall the meaning of\n$\\left((w^T)^{-1}(0)\\right)_{\\widetilde{\\textbf{s}}^\\vee}(\\mathbb{R} ^T)$).\nWe now verify that for every $k\\in I_{\\text{uf}}$ there is a commutative\ndiagram $$\\xymatrix{\n\\left((w^T)^{-1}(0)\\right)_{\\widetilde{\\textbf{s}}^{\\vee}}(\\mathbb{R} ^T) \\ar^{\\left(\\mu^{\\mathcal{A}_{\\mathrm{prin}}^\\vee}_{k}\\right)^T}[rr]  \\ar_{\\pi^{\\mathcal{X} ^\\vee}_1}[d] & & \\left((w^T)^{-1}  (0)\\right)_{\\mu_k(\\widetilde{\\textbf{s}}^{\\vee})}(\\mathbb{R} ^T) \\ar^{\\pi^{\\mathcal{X} ^\\vee}_2}[d]\n\\\\\n\\mathcal{X} ^{\\vee}_{\\textbf{s}^{\\vee}}(\\mathbb{R} ^t)\\ar^{\\left(\\mu^{\\mathcal{X} ^\\vee}_{k}\\right)^t}[rr] & & \\mathcal{X} ^{\\vee}_{\\mu_k(\\textbf{s}^{\\vee})}(\\mathbb{R} ^t),\n}$$ where the vertical maps $\\pi^{\\mathcal{X} ^\\vee}_1$ and\n$\\pi^{\\mathcal{X} ^\\vee}_2$ are both given by $(p^*(n),n)\\mapsto dn$\n(recall that\n$\\mathcal{X} ^{\\vee}_{\\textbf{s}^{\\vee}}(\\mathbb{R} ^t)=(N^\\vee)^\\circ_{\\mathbb{R} }= (d\\cdot N)_{\\mathbb{R} } =\\mathcal{X} ^{\\vee}_{\\mu_k(\\textbf{s}^{\\vee})} (\\mathbb{R} ^t)$).\nBy definition we have that $$\\begin{aligned}\n    \\left(\\mu^{\\mathcal{A}_{\\mathrm{prin}}^\\vee}_{k}\\right)^T(p^*(n),n)& \\overset{\\eqref{eq:tropical_X_mutation}}{=} & (p^*(n),n)+[\\langle (d e_k,0),(p^*(n),n)\\rangle]_+\\{d_ke_k, \\cdot \\}^{\\vee}_{{\\mathrm{prin}} } \\\\\n    &=&  (p^*(n),n)+ [p^*(n)(de_k)]_+(\\{d_ke_k, \\cdot\\}^\\vee,d_ke_k)\\\\\n    &=&(p^*(n) + [\\{n, de_k\\}]_+\\{d_ke_k, \\cdot\\}^\\vee,n+ [\\{n, de_k\\}]_+d_ke_k).\n\\end{aligned}$$ Using the facts that $d, d_k>0$ and that\n$d\\max(a,b)=\\max(da,db)$ and $\\max(a,b)=-\\min(-a,-b)$ for all\n$a,b \\in \\mathbb{R}$, we compute that $$\\begin{aligned}\n\\pi^{\\mathcal{X} ^\\vee}_2\\left(\\left(\\mu^{\\mathcal{A}_{\\mathrm{prin}}^\\vee}_{k}\\right)^T(p^*(n),n)\\right) &=&  dn+ d[\\{n, de_k\\}^\\vee]_+d_ke_k\\\\\n&=&  dn+ [\\{dn, de_k\\}^\\vee]_+d_ke_k\\\\\n &=&  dn+[-\\{de_k,dn\\}^{\\vee}]_+d_ke_k\\\\\n &=&  dn+[-\\{d_ke_k,dn\\}^{\\vee}]_+de_k\\\\\n &=&  dn-[\\{d_ke_k,dn\\}^{\\vee}]_-de_k\\\\\n  &=&  dn-[\\langle v_k^\\vee,dn\\rangle]_-de_k\\\\\n&=& dn+[\\langle v_k^\\vee,dn\\rangle]_-(-d_k^\\vee e^\\vee_k)\\\\\n&\\overset{\\eqref{eq:tropical_A_mutation}}{=}& \\left(\\mu^{\\mathcal{X} ^\\vee}_{k}\\right)^t (dn)\\\\\n&=& \\left(\\mu^{\\mathcal{X} ^\\vee}_{k}\\right)^t \\left(\\pi^{\\mathcal{X} ^\\vee}_1(p^*(n),n)\\right).\n\\end{aligned}$$ This gives the commutativity of the diagram. Notice\nmoreover that $\\pi^{\\mathcal{X} ^\\vee}_1$ and\n$\\pi^{\\mathcal{X} ^\\vee}_2$ are canonical bijections. These two facts\ntogether imply that we have a well defined bijection\n$$\\pi^{\\mathcal{X} ^\\vee}: (w^T)^{-1}(0)(\\mathbb{R} ^T) \\overset{\\sim}{\\longrightarrow} \\mathcal{X} ^{\\vee}(\\mathbb{R} ^t).$$\nThe fact that $\\xi^T_{\\Gamma^\\vee}\\circ i$ is the inverse of\n$\\pi^{\\mathcal{X} ^\\vee}$ follows from noticing that, in lattice\nidentifications of the domain and codomian of $\\xi^T_{\\Gamma^\\vee}$\ngiven by a choice of seed, we have that\n$$\\xi^T_{\\Gamma^\\vee}(dn)=(-p^*(n),-n).$$ ◻\n\nWe can now define cluster scattering diagrams for $\\mathcal{X}$ using\ncluster scattering diagrams for $\\mathcal{A}_{\\mathrm{prin}}$ and the\nquotient map $\\tilde{p}:\\mathcal{A}_{\\mathrm{prin}}\\to \\mathcal{X}$\ndescribed in and the content of Lemma . We define\n$\\mathop{\\mathrm{Supp}}(\\mathfrak{D}^{\\mathcal{X} })$ as\n$$\\mathop{\\mathrm{Supp}}(\\mathfrak{D}^{\\mathcal{X} }):=\\pi^{\\mathcal{X} ^\\vee}\\left(\\mathop{\\mathrm{Supp}}(\\mathfrak{D}^{\\mathcal{A}_{\\mathrm{prin}}})\\cap (w^T_H)^{-1}(0)\\right)\\subset \\mathcal{X} ^\\vee(\\mathbb{Z} ^t).$$\nBy definition the support of the scattering diagram\n$\\mathfrak{D}^{\\mathcal{X} }_{\\textbf{s}}$ is\n$\\mathfrak{r}_{\\textbf{s}^\\vee}\\left(\\mathop{\\mathrm{Supp}}(\\mathfrak{D}^{\\mathcal{X} })\\right)$.\nThe scattering functions attached to the walls of\n$\\mathop{\\mathrm{Supp}}(\\mathfrak{D}^{\\mathcal{X} }_\\textbf{s})$ are\nobtained by applying $\\tilde{p}^*$ to the scattering functions of the\ncorresponding walls of\n$\\mathfrak{D}^{\\mathcal{A}_{\\mathrm{prin}}}_\\textbf{s}$. We proceed in\nan analogous way to define broken lines for\n$\\mathfrak{D}^{\\mathcal{X} }_\\textbf{s}$. As in the previous cases,\nsupports of broken lines are well defined inside\n$\\mathcal{X} ^\\vee(\\mathbb{Z} ^t)$.\n\nThe labeling of a theta function on $\\mathcal{X}$ with an element of\n$\\mathcal{X} ^{\\vee}(\\mathbb{Z} ^t)$ is obtained using the bijection of\nLemma . More precisely, for\n${\\bf n} \\in \\mathcal{X} ^\\vee(\\mathbb{Z} ^t)$ with\n${\\bf n}\\in \\Theta(\\mathcal{X} )$ we have\n$$\\tilde{p}^*(\\vartheta ^\\mathcal{X} _{\\bf n})=\\vartheta ^{\\mathcal{A}_{\\mathrm{prin}}}_{\\xi^T_{\\Gamma^\\vee}\\circ i({\\bf n})}.$$\nExplicitly, in lattice identifications of the tropical spaces, we have\nthat for $dn \\in \\mathcal{X} ^\\vee_{\\textbf{s}^\\vee}(\\mathbb{Z} ^t)$\n$$\\tilde{p}^*\\left(\\vartheta ^{\\mathcal{X} }_{dn}\\right):= \n\\vartheta ^{\\mathcal{A}_{\\mathrm{prin}}}_{(p^*(n),n)}.$$\n\n**Example 27**. Let $\\epsilon\n=\n\\left(\\begin{matrix}\n0 & 2 \\\\\n-1 & 0\n\\end{matrix}\\right)$ and $d_1=1, d_2=2$. Using the above parametrization\nwe compute\n$$\\vartheta ^{\\mathcal{X} }_{2(-1,-2)}=X_1^{-1}X_2^{-2}+2X_1^{-1}X_2^{-1}+X_1^{-1}.$$\nIndeed, we have that $\\xi^T_{\\Gamma^\\vee}\\circ i(2(-1,-2))=(2,-2)$ and\n$$\\vartheta ^{\\mathcal{A}_{\\mathrm{prin}}}_{(2,-2),(-1,-2)}= \\left(\\vartheta ^{\\mathcal{A}_{\\mathrm{prin}}}_{(1,-1),(0,0)}\\right)^2 \\vartheta ^{\\mathcal{A}_{\\mathrm{prin}}}_{(0,0),(-1,-2)} = \\left(\\dfrac{A_1+t_2}{A_2}\\right)^2t_1^{-1}t_2^{-2}= \\tilde{p}^*(X_1^{-1}X_2^{-2}+2X_1^{-1}X_2^{-1}+X_1^{-1}).$$\n\n### Theta functions on $\\mathcal{X} _{\\bf 1}$\n\nAs in the previous subsections we would like to highlight that\n\n*every theta function on $\\mathcal{X} _{\\bf 1}$ is naturally labeled by\na point of $(\\mathcal{X} _{\\bf 1})^\\vee(\\mathbb{Z} ^t)$*\n\nas we now explain. The tropical space\n$(\\mathcal{X} _{\\bf 1})^{\\vee}(\\mathbb{R} ^t)$ is the quotient of\n$\\mathcal{X} ^{\\vee} (\\mathbb{R} ^t)$ by the tropicalization of the\naction of $T_H$ on $\\mathcal{X} ^{\\vee}$. In other words, since the\nvariety $(\\mathcal{X} _{\\bf 1})^{\\vee}$ is a quotient of\n$\\mathcal{X} ^\\vee$, we can consider the quotient map by\n$\\varpi_{H}: \\mathcal{X} ^\\vee \\to (\\mathcal{X} _{\\bf 1})^\\vee$ to\nobtain a surjection\n$$\\varpi_H^t:  \\mathcal{X} ^{\\vee}(\\mathbb{R} ^t) \\to (\\mathcal{X} _{\\bf 1})^{\\vee} (\\mathbb{R} ^t).$$\nThen, given\n$\\overline{\\bf n}\\in (\\mathcal{X} _{\\bf 1})^{\\vee} (\\mathbb{R} ^t)$ and\n${\\bf n}\\in (\\varpi_H^t)^{-1}(\\overline{\\bf n})$ we define\n$$\\vartheta ^{\\mathcal{X} _{\\bf 1}}_{\\overline{\\bf n}}=\\vartheta ^{\\mathcal{X} }_{\\bf n}|_{\\mathcal{X} _{\\bf 1}}.$$\nMore concretely, working in lattice identifications of the tropical\nspaces, we have that\n$\\mathcal{X} ^{\\vee}(\\mathbb{R} ^t)_{\\textbf{s}^\\vee} = N_\\mathbb{R}$\nand\n$(\\mathcal{X} _{\\bf 1})^{\\vee}_{\\textbf{s}^\\vee}(\\mathbb{R} ^t) {\\cong}N_\\mathbb{R} /H_{\\mathbb{R} }$.\nThen for every $n \\in N$\n$$\\vartheta ^{\\mathcal{X} _{\\bf 1}}_{d n + H}=\\vartheta ^{\\mathcal{X} }_{dn}|_{\\mathcal{X} _{\\bf 1}}.$$\nOne can proceed in an analogous way as in the previous cases to\nconstruct a scattering diagram like structure\n$\\mathfrak{D}^{\\mathcal{X} _{\\bf 1}}_{\\textbf{s}}$ inside\n$(\\mathcal{X} _{\\bf 1})^\\vee_{\\textbf{s}}(\\mathbb{Z} ^t)$. In turn we\nobtain a description of\n$\\vartheta ^{\\mathcal{X} _{\\bf 1}}_{\\overline{\\bf n}}$ using broken\nlines and use these to define\n$\\mathop{\\mathrm{mid}}(\\mathcal{X} _{\\bf 1})$ and\n$\\Theta(\\mathcal{X} _{\\bf 1})$.\n\n### The full Fock–Goncharov conjecture\n\nLet $\\mathcal{V}$ be a scheme of the form $\\mathcal{A}$, $\\mathcal{X}$,\n$\\mathcal{A} /T_{H}$ or $\\mathcal{X} _{{\\bf 1}}$. The **upper cluster\nalgebra** of $\\mathcal{V}$ is defined as\n$$\\text{up}(\\mathcal{V} ):=H^0(\\mathcal{V} ,\\mathcal{O}_{\\mathcal{V} }).$$\nEvery polynomial theta function on $\\mathcal{V}$ belongs to\n$\\text{up}(\\mathcal{V} )$, therefore, we have a natural $\\Bbbk$-linear\nmap $\\mathop{\\mathrm{mid}}(\\mathcal{V} )\\to \\text{up}(\\mathcal{V} )$. If\n$\\mathcal{V}$ is one of $\\mathcal{A}$ (see Remark ) or $\\mathcal{X}$ it\nwas proved in that this map is in fact an injective homomorphism of\nalgebras. These cases already imply that the same is true is\n$\\mathcal{V}$ if of the form $\\mathcal{A} /T_H$ or\n$\\mathcal{X} _{\\bf 1}$.\n\n**Remark 28**. If $\\mathcal{V} = \\mathcal{A}$, $\\mathcal{X}$,\n$\\mathcal{A} /T_{H}$ or $\\mathcal{X} _{{\\bf 1}}$ then\n$\\mathop{\\mathrm{mid}}(\\mathcal{V} )$ is an integral domain. Indeed,\n$\\mathop{\\mathrm{mid}}(\\mathcal{V} )$ is a subalgebra of\n$\\mathop{\\mathrm{up}}(\\mathcal{V} )=H^0(\\mathcal{V} ,\\mathcal{O}_{\\mathcal{V} })$\nwhich is a domain as $\\mathcal{V}$ is irreducible.\n\nAs we have seen in the previous subsections theta functions on varieties\nof the form $\\mathcal{A}$ or $\\mathcal{A} /T_H$ are naturally labeled by\nthe $\\mathbb{Z} ^T$-points of its Fock–Goncharov dual, whereas theta\nfunctions on varieties of the form $\\mathcal{X}$ or\n$\\mathcal{X} _{\\bf 1}$ are naturally labeled by the\n$\\mathbb{Z} ^t$-points of its Fock–Goncharov dual. Since we would like\nto consider all these cases simultaneously we introduce the following\nnotation. For $G= \\mathbb{Z} , \\mathbb{Q}$ or $\\mathbb{R}$ we set\n\n$$\\label{eq:unif}\n    \\mathrm{Trop} _G(\\mathcal{V} ):=\n    \\begin{cases}\n        \\mathcal{V} (G^t) &\\text{ if } \\mathcal{V} =\\mathcal{A} \\text{ or } \\mathcal{V} =\\mathcal{A} /T_H\\vspace{1mm}\\\\\n        \\mathcal{V} (G^T) \\ & \\text{ if } \\mathcal{V} =\\mathcal{X} \\text{ or } \\mathcal{V} =\\mathcal{X} _{\\bf 1}.\n    \\end{cases}$$\n\nSimilarly, for a positive rational function\n$g: \\mathcal{V} \\dashrightarrow \\Bbbk$ we let $$\\label{eq:unif_function}\n    \\mathrm{Trop} _G(g):=\n    \\begin{cases}\n        g^t &\\text{ if } \\mathcal{V} =\\mathcal{A} \\text{ or } \\mathcal{V} =\\mathcal{A} /T_H\\vspace{1mm}\\\\\n        g^T \\ &\\text{ if } \\mathcal{V} =\\mathcal{X} \\text{ or } \\mathcal{V} =\\mathcal{X} _{\\bf 1}.\n    \\end{cases}$$\n\nIn particular, if we think of the seed torus $\\mathcal{V} _\\textbf{s}$\nas a cluster variety with only frozen directions then\n$\\mathrm{Trop} _G(\\mathcal{V} _\\textbf{s})=\\mathfrak{r}_{\\textbf{s}}(\\mathrm{Trop} _G(\\mathcal{V} ))=\\mathcal{V} _{\\textbf{s}}(G^t)$,\nif $\\mathcal{V}$ is of the form $\\mathcal{A}$ or $\\mathcal{A} /T_H$ and\n$\\mathrm{Trop} _G(\\mathcal{V} _\\textbf{s})=\\mathfrak{r}_{\\textbf{s}}(\\mathrm{Trop} _G(\\mathcal{V} ))=\\mathcal{V} _{\\textbf{s}}(G^T)$,\nif $\\mathcal{V}$ is of the form $\\mathcal{X}$ or $\\mathcal{X} _{\\bf 1}$.\nFor later use we also set $$\\label{eq:Theta_seed}\n\\Theta(\\mathcal{V} )_{\\textbf{s}^\\vee}:=\\mathfrak{r}_{\\textbf{s}^\\vee}(\\Theta(\\mathcal{V} ))\\subset \\mathrm{Trop} _{\\mathbb{Z} }(\\mathcal{V} ^\\vee),$$\nsee the line just below equation . Following we introduce the following\ndefinition.\n\n**Definition 29**. Let $\\mathcal{V}$ be a scheme of the form\n$\\mathcal{A}$, $\\mathcal{X}$, $\\mathcal{A} /T_{H}$ or\n$\\mathcal{X} _{{\\bf 1}}$. We say that **the full Fock–Goncharov\nconjecture** holds for $\\mathcal{V}$ if\n\n-   $\\Theta(\\mathcal{V} )=\\mathrm{Trop} _{\\mathbb{Z} }(\\mathcal{V} ^{\\vee})$,\n    and\n\n-   the natural map\n    $\\mathop{\\mathrm{mid}}(\\mathcal{V} ) \\to \\text{up}(\\mathcal{V} )$ is\n    an isomorphism.\n\n# Bases of theta functions for partial minimal models\n\nIn , the authors obtained nearly optimal conditions ensuring that the\nfull Fock–Goncharov conjecture holds for a cluster variety. However,\nthey were able to prove that the ring of regular functions of a partial\ncompactifications of a cluster varieties has a basis of theta functions\nunder much stronger conditions. In this section we outline this\nframework, including quotients and fibres of cluster varieties, and\nrefer to for a detailed treatment. The main class of (partial)\ncompactifications we shall consider are the (partial) minimal models\ndefined below.\n\n**Definition 30**. Let $\\mathcal{V}$ be a scheme of the form\n$\\mathcal{A} , \\mathcal{X} , \\mathcal{A} /T_H$ or\n$\\mathcal{X} _{\\bf 1}$. An inclusion $\\mathcal{V} \\subset Y$ as an open\nsubscheme of a normal variety $Y$ is a **partial minimal model** of\n$\\mathcal{V}$ if the canonical volume form on $\\mathcal{V}$ has a simple\npole along every irreducible divisor of $Y$ contained in\n$Y \\setminus \\mathcal{V}$. It is a **minimal model** if $Y$ is, in\naddition, projective. We call $Y \\setminus \\mathcal{V}$ the **boundary**\nof $\\mathcal{V} \\subset Y$.\n\nFor example, if $\\mathcal{V}$ is a cluster $\\mathcal{A}$-variety with\nfrozen variables we can let these variables vanish to obtain a partial\nminimal model of $\\mathcal{V}$ as in . Similarly, if we consider a torus\nas a cluster variety (by letting $I_{\\text{uf}}= \\emptyset$) then a\npartial minimal model is simply a normal toric variety.\n\nGiven a partial minimal model $\\mathcal{V} \\subset Y$, where\n$\\mathcal{V}$ is a scheme of the form\n$\\mathcal{A} , \\mathcal{X} , \\mathcal{A} /T_H$ or\n$\\mathcal{X} _{\\bf 1}$, we would like to describe the set of theta\nfunctions on $\\mathcal{V}$ (resp. $\\mathcal{V} ^\\vee$) that extend to\n$Y$ in a similar way as the ring of algebraic functions on a normal\ntoric variety is described in toric geometry using polyhedral fans. In\norder to be able to do so we need that the pair\n$(\\mathcal{V} , \\mathcal{V} ^\\vee)$ satisfies a technical condition\n–*theta reciprocity*– that we will introduce shortly. For this, we need\nto discuss first the *tropical pairings* associated to the pair\n$(\\mathcal{V} ,\\mathcal{V} ^{\\vee})$.\n\nIn order to define the tropical pairings we temporarily assume that\n$\\mathcal{V}$ is a variety of the form $\\mathcal{A}$ or\n$\\mathcal{A} /T_{H}$ so that $\\mathcal{V} ^\\vee$ is a cluster\n$\\mathcal{X}$-variety or a fibre of a cluster $\\mathcal{X}$-variety,\nrespectively. In particular,\n$\\Theta(\\mathcal{V} )\\subset \\mathcal{V} ^\\vee(\\mathbb{Z} ^T)= \\mathrm{Trop} _{\\mathbb{Z} }(\\mathcal{V} ^\\vee)$\nand\n$\\Theta(\\mathcal{V} ^\\vee)\\subset \\mathcal{V} (\\mathbb{Z} ^t)=\\mathrm{Trop} _{\\mathbb{Z} }(\\mathcal{V} )$,\nsee . Recall from Remark  that the set $\\mathcal{V} (\\mathbb{Z} ^t)$\n(resp. $\\mathcal{V} ^\\vee(\\mathbb{Z} ^t)$) is canonically identified\nwith the geometric tropicalization\n$\\mathcal{V} ^\\mathrm{trop} (\\mathbb{Z} )$ (resp.\n$(\\mathcal{V} ^\\vee)^\\mathrm{trop} (\\mathbb{Z} )$). Therefore, we\nsystematically think of the elements of $\\mathcal{V} (\\mathbb{Z} ^t)$\n(resp. $\\mathcal{V} ^\\vee(\\mathbb{Z} ^t)$) as divisorial discrete\nvaluations on $\\Bbbk(\\mathcal{V} )$ (resp. $\\Bbbk(\\mathcal{V} ^\\vee)$).\nWe also consider the bijection\n$i : \\mathcal{V} ^\\vee(\\mathbb{Z} ^T) \\to \\mathcal{V} ^\\vee(\\mathbb{Z} ^t )$\nintroduced in § (see the comment bellow ). The **tropical pairings**\nassociated to the pair $(\\mathcal{V} ,\\mathcal{V} ^\\vee)$ are the\nfunctions\n$\\langle \\cdot , \\cdot \\rangle : \\Theta(\\mathcal{V} ^{\\vee})  \\times \\Theta (\\mathcal{V} )  \\to \\mathbb{Z}$\nand\n$\\langle \\cdot , \\cdot \\rangle^{\\vee} : \\Theta(\\mathcal{V} ^{\\vee})  \\times \\Theta (\\mathcal{V} )  \\to \\mathbb{Z}$\ngiven by\n$$\\langle {\\bf v} , {\\bf b} \\rangle = {\\bf v}(\\vartheta ^{\\mathcal{V} }_{\\bf b}) \\ \\ \\ \\ \\ \\ \\ \\text{and} \\ \\ \\ \\ \\ \\ \\ \\langle {\\bf v} , {\\bf b} \\rangle^{\\vee} = i({\\bf b}) (\\vartheta ^{\\mathcal{V} ^{\\vee}}_{\\bf v}),$$\n\n**Definition 31**. Let $\\mathcal{V}$ be a scheme of the form\n$\\mathcal{A} , \\mathcal{X} , \\mathcal{A} /T_H$ or\n$\\mathcal{X} _{\\bf 1}$. The pair $(\\mathcal{V} ,\\mathcal{V} ^\\vee)$ has\n**theta reciprocity** if\n$\\Theta(\\mathcal{V} )=\\mathrm{Trop} _{\\mathbb{Z} }(\\mathcal{V} ^\\vee)$,\n$\\Theta(\\mathcal{V} ^{\\vee})=\\mathrm{Trop} _{\\mathbb{Z} }(\\mathcal{V} )$,\nand\n$\\langle {\\bf v} , {\\bf b} \\rangle = \\langle {\\bf v} , {\\bf b} \\rangle^{\\vee}$\nfor all\n$({\\bf v},{\\bf b})\\in \\mathrm{Trop} _{\\mathbb{Z} }(\\mathcal{V} ) \\times \\mathrm{Trop} _{\\mathbb{Z} }(\\mathcal{V} ^\\vee)$.\n\n**Remark 32**. Definition shall not be considered artificial. In fact,\nan analogous conjecture for affine log Calabi–Yau varieties with maximal\nboundary is expected to hold true, see .\n\n**Lemma 33**. Let $\\mathcal{V}$ be a scheme of the form\n$\\mathcal{A} , \\mathcal{X} , \\mathcal{A} /T_H$ or $\\mathcal{X} _{\\bf 1}$\nand let $\\mathcal{V} \\subset Y$ be a (partial) minimal model. Suppose\nthat the pair $(\\mathcal{V} ,\\mathcal{V} ^\\vee)$ has theta reciprocity.\nThen for every seed $\\textbf{s}\\in %\n  \\mathrel{\\mathop{\\mathbb{T}_r}\\limits^{\n    \\vbox to0ex{\\kern-0.5\\ex@\n    \\hbox{$\\scriptstyle\\longrightarrow$}\\vss}}}$ the set of theta\nfunctions on $\\mathcal{V}$ that extend to $Y$ can be described as the\nintersection of $\\Theta(\\mathcal{V} ^\\vee)_{\\textbf{s}^\\vee}$ (see )\nwith a polyhedral cone of the vector space\n$\\mathrm{Trop} _{\\mathbb{R} }(\\mathcal{V} ^{\\vee}_{\\textbf{s}^\\vee})$\n(see the sentence bellow equation ).\n\n*Proof.* We treat the cases $\\mathcal{V} = \\mathcal{A}$ or\n$\\mathcal{A} /T_H$ as the proof is completely analogous for the cases\n$\\mathcal{V} = \\mathcal{X}$ or $\\mathcal{X} _{\\bf 1}$. Let\n$D_1, \\dots, D_s$ be the irreducible divisors of $Y$ contained in the\nboundary of $\\mathcal{V} \\subset Y$. Since $Y$ is normal, to describe\nthe theta functions on $\\mathcal{V}$ that extend to $Y$ it is enough to\ndescribe the set of theta functions that extend to $D_1, \\dots , D_s$\nsince $Y\\setminus (\\mathcal{V} \\cup D_1, \\dots , D_s)$ has co-dimension\ngreater or equal to $2$ in $Y$. Let $\\mathop{\\mathrm{ord}}_{D_j}$ be the\ndiscrete valuation on $\\Bbbk(\\mathcal{V} )\\setminus \\{ 0 \\}$ associated\nto the irreducible divisor $D_j$. Since $\\mathcal{V} \\subset Y$ is a\npartial minimal model, $\\mathop{\\mathrm{ord}}_{D_j}$ determines a point\nof $\\mathcal{V} (\\mathbb{Z} ^t)$. Since\n$\\Theta(\\mathcal{V} ^{\\vee})= \\mathcal{V} (\\mathbb{Z} ^t)$ we have\n$\\mathop{\\mathrm{ord}}_{D_j} \\in \\Theta (\\mathcal{V} ^{\\vee})$.\nTherefore,\n$\\vartheta ^{\\mathcal{V} ^{\\vee}}_{\\mathop{\\mathrm{ord}}_{D_j}}$ is a\npolynomial theta function and its tropicalization is the function\n$$(\\vartheta _{\\mathop{\\mathrm{ord}}_{D_j}}^{\\mathcal{V} ^\\vee})^t:\\mathcal{V} ^{\\vee}( \\mathbb{Z} ^t)\\to \\mathbb{Z} \\quad \\text{given by} \\quad  v \\mapsto v (\\vartheta ^{\\mathcal{V} ^{\\vee}}_{\\mathop{\\mathrm{ord}}_{D_j}}).$$\nIn other words,\n$(\\vartheta _{\\mathop{\\mathrm{ord}}_{D_j}}^{\\mathcal{V} ^{\\vee}})^t(v)=\\langle \\mathop{\\mathrm{ord}}_{D_j}, i(v) \\rangle$.\nSince $\\Theta(\\mathcal{V} )= \\mathcal{V} ^\\vee(\\mathbb{Z} ^T)$ we have\nthat $i(v)\\in \\Theta(\\mathcal{V} )$ and, therefore,\n$\\vartheta ^\\mathcal{V} _{i(v)}$ is a polynomial theta function. The\nassumption\n$\\langle{\\bf v} , {\\bf b} \\rangle = \\langle {\\bf v} , {\\bf b} \\rangle^{\\vee}$\nfor all ${\\bf v}$ and ${\\bf b}$ implies that\n$$(\\vartheta _{\\mathop{\\mathrm{ord}}_{D_j}}^{\\mathcal{V} ^{\\vee}})^t(v)= (\\vartheta ^{\\mathcal{V} }_{i(v)})^t(\\mathop{\\mathrm{ord}}_{D_j}),$$\nsince\n$$(\\vartheta _{\\mathop{\\mathrm{ord}}_{D_j}}^{\\mathcal{V} ^{\\vee}})^t(v) =\n\\langle \\mathop{\\mathrm{ord}}_{D_j}, i(v) \\rangle =\n\\langle \\mathop{\\mathrm{ord}}_{D_j}, i(v)\\rangle^{\\vee} =\n\\mathop{\\mathrm{ord}}_{D_j}(\\vartheta ^{\\mathcal{V} }_{i(v)}) =\n(\\vartheta ^{\\mathcal{V} }_{i(v)})^t(\\mathop{\\mathrm{ord}}_{D_j}).$$\nThus a theta function\n$\\vartheta ^{\\mathcal{V} }_{i(v)} \\in \\mathop{\\mathrm{mid}}(\\mathcal{V} )$\nextends to $D_j$ if and only if\n$0\\leq (\\vartheta ^{\\mathcal{V} ^{\\vee}}_{\\mathop{\\mathrm{ord}}_{D_j}})^t(v)$.\nIn particular, a theta function $\\vartheta ^\\mathcal{V} _{i(v)}$ extends\nto $Y$ if and only if\n$$i(v)\\in \\bigcap_{i=1}^s\\{b\\in\\mathcal{V} ^{\\vee}_{\\textbf{s}^\\vee}(\\mathbb{R} ^T)\\mid 0\\leq (\\vartheta ^{\\mathcal{V} ^{\\vee}}_{\\mathop{\\mathrm{ord}}_{D_j}})^T(b)\\}$$\nsince $g^T(b)=g^t(i(b))$ for every positive function $g$ on\n$\\mathcal{V}$, see . By definition of tropicalization, the set\n$\\bigcap_{i=1}^s\\{b\\in\\mathcal{V} ^{\\vee}_{\\textbf{s}^\\vee}(\\mathbb{R} ^T)\\mid 0\\leq (\\vartheta ^{\\mathcal{V} ^{\\vee}}_{\\mathop{\\mathrm{ord}}_{D_j}})^T(b)\\}$\nis a polyhedral cone of\n$\\mathcal{V} ^{\\vee}_{\\textbf{s}^\\vee}(\\mathbb{R} ^T)=\\mathrm{Trop} _{\\mathbb{R} }(\\mathcal{V} ^{\\vee}_{\\textbf{s}^\\vee})$. ◻\n\nWe now turn to the problem of understanding when the theta functions on\n$\\mathcal{V}$ that extend to a (partial) minimal model\n$\\mathcal{V} \\subset Y$ form a basis of $H^0(Y, \\mathcal{O}_Y)$. The\nfollowing notion is central.\n\n**Definition 34**. Let $\\mathcal{V}$ be a scheme of the form\n$\\mathcal{A} , \\mathcal{X} , \\mathcal{A} /T_H$ or\n$\\mathcal{X} _{\\bf 1}$. We say that the theta functions on $\\mathcal{V}$\n**respect the order of vanishing** if for all\n${\\bf v}\\in \\mathcal{V} (\\mathbb{Z} ^t)$ and\n$\\displaystyle \\sum_{{\\bf q}\\in \\Theta(\\mathcal{V} )} \\alpha_{\\bf q} \\vartheta ^{\\mathcal{V} }_{\\bf q}\\in \\mathop{\\mathrm{mid}}(\\mathcal{V} )$\nthen\n$${\\bf v}\\left(\\sum_{{\\bf q}\\in \\Theta(\\mathcal{V} )} \\alpha_{\\bf q} \\vartheta ^{\\mathcal{V} }_{\\bf q}\\right) \\geq 0 \\ \\ \\text{ if and only if }\\ \\  {\\bf v}(\\vartheta _{\\bf q})\\geq 0  \\text{ for all } {\\bf q} \\text{ such that } \\alpha_{\\bf q}\\neq 0.$$\n\nNotice that in the authors conjecture that the theta functions on\n$\\mathcal{A}_{\\mathrm{prin}}$ respect the order of vanishing. The\n**superpotential** associated to a partial minimal model\n$\\mathcal{V} \\subset Y$ is the function on $\\mathcal{V} ^\\vee$ defined\nas $$\\label{eq:def superpotential}\n    W_{Y}:=\\sum_{j=1}^n \\vartheta ^{\\mathcal{V} ^{\\vee}}_{j},$$ where\n$$\\label{eq:def superpotential_summands}\n   \\vartheta ^{\\mathcal{V} ^{\\vee}}_{j}=\\begin{cases}\n       \\vartheta ^{\\mathcal{V} ^{\\vee}}_{\\mathop{\\mathrm{ord}}_{D_j}} &\\text{ if } \\mathcal{V} =\\mathcal{A} \\text{ or } \\mathcal{V} =\\mathcal{A} /T_H\\vspace{1mm}\\\\\n       \\vartheta ^{\\mathcal{V} ^{\\vee}}_{i(\\mathop{\\mathrm{ord}}_{D_j})} \\ &\\text{ if } \\mathcal{V} =\\mathcal{X} \\text{ or } \\mathcal{V} =\\mathcal{X} _{\\bf 1}.\n   \\end{cases}$$ The **superpotential cone** associated to $W_Y$ is\n$$\\label{eq:def Xi}\n    \\Xi_Y:= \\{ {\\bf v} \\in \\mathrm{Trop} _{\\mathbb{R} }(\\mathcal{V} ^\\vee) \\mid \\mathrm{Trop} _{\\mathbb{R} }(W_Y)({\\bf v})\\geq0 \\},$$\nsee equation .\n\nWe further set\n$\\Xi_{Y;\\textbf{s}^\\vee}:= \\mathfrak{r}_{\\textbf{s}^\\vee}(\\Xi_Y)\\subset \\mathrm{Trop} _{\\mathbb{R} }(\\mathcal{V} ^{\\vee}_{\\textbf{s}^\\vee})$.\nNotice that if the theta functions on $\\mathcal{V}$ respect the order of\nvanishing then $\\Xi_{Y;\\textbf{s}}$ is precisely the polyhedral subset\nof Lemma . The next results follows at once from the definitions.\n\n**Lemma 35**. Let $\\mathcal{V}$ be a scheme of the form\n$\\mathcal{A} , \\mathcal{X} , \\mathcal{A} /T_H$ or $\\mathcal{X} _{\\bf 1}$\nand let $\\mathcal{V} \\subset Y$ be a (partial) minimal model. Suppose\nthat the full Fock–Goncharov conjecture holds for $\\mathcal{V}$, that\nthe pair $(\\mathcal{V} , \\mathcal{V} ^\\vee)$ has theta reciprocity and\nthat the theta functions on $\\mathcal{V}$ respect the order of\nvanishing. Then the set of theta functions on $\\mathcal{V}$ parametrized\nby the points of $\\Xi_Y(\\mathbb{Z} )$ is a basis of\n$H^0(Y, \\mathcal{O}(Y))$.\n\n**Lemma 36**. Suppose there is a cluster ensemble map\n$p:\\mathcal{A} \\to \\mathcal{X}$ that is an isomorphism. Then theta\nfunctions on $\\mathcal{A}$ respect the order of vanishing if and only\ntheta functions on $\\mathcal{X}$ respect the order of vanishing.\n\n*Proof.* The result follows at once from the fact that\n$p^*(\\vartheta ^{\\mathcal{X} }_{\\bf n})= \\vartheta ^{\\mathcal{A} }_{(p^{\\vee})^T\\circ i ({\\bf n})}$. ◻\n\nWe propose the following definition that allows to have the benefits of\nLemma without having to verify all its assumptions. We apply this in §.\n\n**Definition 37**. We say that $\\mathcal{V} \\subset Y$ has **enough\ntheta functions** if the full Fock–Goncharov conjecture holds for\n$\\mathcal{V}$ and the theta functions on $\\mathcal{V}$ parametrized by\n$\\Xi_{Y} (\\mathbb{Z} )$ form a basis of $H^0(Y, \\mathcal{O}_Y)$.\n\nWe now recall an important notion introduced in that can be used to\nverify in a combinatorial way that a partial minimal model\n$\\mathcal{A} \\subset Y$ has enough theta functions provided $Y$ is\nobtained by letting the frozen variables vanish.\n\n**Definition 38**. We say that a seed $\\textbf{s}=(e_i)_{ i \\in I}$ is\n**optimized** for a point ${\\bf n} \\in \\mathcal{A} (\\mathbb{Z} ^t)$ if\nunder the identification of $\\mathcal{A} (\\mathbb{Z} ^t)$ with $N^\\circ$\nafforded by $\\textbf{s}$ we have that $\\{ e_k, n_{\\textbf{s}} \\}\\geq 0$\nfor all $k \\in I_{\\text{uf}}$.\n\n**Lemma 39**.\n\nAssume that $\\mathcal{A}$ satisfies the full Fock–Goncharov conjecture.\nLet $\\mathcal{A} \\subset Y$ be a partial minimal model of $\\mathcal{A}$\nand let $D_1, \\dots , D_s$ be the irreducible divisors of $Y$ contained\nin $Y\\setminus \\mathcal{A}$. Assume that\n$p^*_2|_{N^{\\circ}}: N^{\\circ}\\to N_{\\text{uf}}^*$ is surjective and\nthat the point\n$\\mathop{\\mathrm{ord}}_{D_j}\\in \\mathcal{A} ^{\\vee}(\\mathbb{Z} ^t)$ has\nan optimized seed for every $1 \\leq j \\leq s$. Then the partial minimal\nmodel $\\mathcal{A} \\subset Y$ has enough theta functions.\n\n*Proof.* Since $p^*_2|_{N^{\\circ}}$ is surjective we have that\n$\\mathcal{A}_{\\mathrm{prin}}$ is isomorphic to $\\mathcal{A} \\times T_M$\n(see ). Consider the partial compactification\n$\\mathcal{A}_{\\mathrm{prin}}\\subset Y \\times T_M$. Its boundary is\nisomorphic to $D\\times T_M$ and the irreducible components of the\nboundary are the divisors $\\widetilde{D}_1, \\dots, \\widetilde{D}_s$,\nwhere $\\widetilde{D}_j:=D_j \\times T_M$. By hypothesis\n$\\mathop{\\mathrm{ord}}_{D_j}$ is optimized for some seed $\\textbf{s}_j$.\nLet $\\widetilde{\\textbf{s}}_j$ be the seed for\n$\\Gamma_{{\\mathrm{prin}} }$ obtained mutating\n$\\textbf{s}_{0_{{\\mathrm{prin}} }}$ in the same sequence of directions\nneeded to obtain $\\textbf{s}_j$ from $\\textbf{s}_0$. Observe that for\nevery $1\\leq j \\leq s$, under the identifications\n$$\\mathcal{A} _{{\\mathrm{prin}} ,\\widetilde{\\textbf{s}}_j}(\\mathbb{Z} ^t) = N_{\\mathrm{prin}} ^{\\circ} = \\mathcal{A} _{\\textbf{s}_j}(\\mathbb{Z} ^t) \\oplus  T_M(\\mathbb{Z} ^t),$$\nthe point $\\mathop{\\mathrm{ord}}_{\\widetilde{D}_j}$ of\n$\\mathcal{A}_{\\mathrm{prin}}(\\mathbb{Z} ^t)$ corresponds to the point\n$(\\mathop{\\mathrm{ord}}_{D_j},0)$ of\n$\\mathcal{A} (\\mathbb{Z} ^t)\\times T_M(\\mathbb{Z} ^t)$.\n\nRecall that the index set of unfrozen indices for\n$\\mathcal{A}_{\\mathrm{prin}}$ is $I_{\\text{uf}}$. In particular, for\nevery $k \\in I_{\\text{uf}}$ we have that the $k^{\\text{th}}$ element of\n$\\widetilde{\\textbf{s}}_{j}$ is of the form $( e_{k;j},0)$, where\n$e_{k;j}$ is the $k^{\\text{th}}$ element of $\\textbf{s}_j$. Then for\neach $1\\leq j\\leq s$ we compute $$\\begin{aligned}\n\\{ (e_{k;j},0),  \\mathop{\\mathrm{ord}}_{\\widetilde{D}_j}\\} & = \\{ (e_{k;j},0), (\\mathop{\\mathrm{ord}}_{D_j},0)\\} \\\\\n& = \\{e_{k;j}, \\mathop{\\mathrm{ord}}_{D_j} \\} \\geq 0.\n\\end{aligned}$$\n\nThis tells us that $\\mathop{\\mathrm{ord}}_{\\widetilde{D}_j}$ is\noptimized for $\\widetilde{\\textbf{s}}_j$. Let\n$W_{Y\\times T_M}=\\sum_{j}^{s}\\vartheta ^{\\mathcal{A}_{\\mathrm{prin}}^\\vee}_{\\mathop{\\mathrm{ord}}{\\widetilde{D}_j}}$\nbe the superpotential associated to\n$\\mathcal{A}_{\\mathrm{prin}}\\subset Y \\times T_M$. By Proposition 9.7\nand Lemma 9.10 (3) of the integral points of $\\Xi_{Y \\times T_M}$ can be\ndescribed as\n$$\\Xi_{Y \\times T_M}\\cap (\\mathbb{Z} ) = \\{ b \\in \\Theta(\\mathcal{A}_{\\mathrm{prin}}) \\mid \\mathop{\\mathrm{ord}}_{i(b)} (\\vartheta ^{\\mathcal{A}_{\\mathrm{prin}}^{\\vee}}_j)\\geq 0 \\text{ for all } j\\}.$$\nWe define $\\mathop{\\mathrm{mid}}(Y\\times T_M)$ to be the vector subspace\nof $\\mathop{\\mathrm{mid}}(\\mathcal{A}_{\\mathrm{prin}})$ spanned by the\ntheta functions parametrized by $\\Xi_{Y \\times T_M}(\\mathbb{Z} ^T)$. For\nthe convenience of the reader we point out that in the notation of the\npartial compactification $Y\\times T_M$ of $\\mathcal{A}_{\\mathrm{prin}}$\nwould be denoted by $\\overline{\\mathcal{A} }_{\\text{prin}}^{S}$ and\n$\\Xi_{Y \\times T_M}(\\mathbb{Z} )$ by\n$\\Theta(\\overline{\\mathcal{A} }_{\\text{prin}}^{S})$, where\n$S:=\\{ i(\\mathop{\\mathrm{ord}}_{\\widetilde{D}_1}),\\dots , i(\\mathop{\\mathrm{ord}}_{\\widetilde{D}_s})\\}$.\nBy we have\n$$\\mathop{\\mathrm{mid}}(Y\\times T_M)=H^0(Y\\times T_M, \\mathcal{O}_{Y\\times T_M}) \\cong H^0(Y, \\mathcal{O}_{Y})\\otimes_{\\Bbbk} H^0( T_M, \\mathcal{O}_{ T_M}).$$\nIn particular, $H^0(Y\\times T_M, \\mathcal{O}_{Y\\times T_M})$ has a theta\nbasis parametrized by $\\Theta(Y\\times T_M)$. The theta function\n$\\vartheta ^{\\mathcal{A} }_{\\mathop{\\mathrm{ord}}_{D_j}}$ is obtained\nfrom $\\vartheta ^{\\mathcal{A}_{\\mathrm{prin}}}_{\\widetilde{D}_j}$ by\nspecializing the coefficients to $1$. This implies that\n$$\\Xi_{Y}\\cap \\mathrm{Trop} _{\\mathbb{Z} }(\\mathcal{A} ^{\\vee})= \\Xi_{Y \\times T_M} \\cap \\mathrm{Trop} _{\\mathbb{Z} }(\\mathcal{A} ^{\\vee}).$$\nWe conclude that $H^0(Y, \\mathcal{O}_Y)$ has a theta basis parametrized\nby the integral point of $\\Xi_{Y}$. ◻\n\n# Valuations on middle cluster algebras and adapted bases\n\nIn the authors noticed that the so-called **g**-vectors associated to\ncluster variables can be used to construct valuations on\n$\\Bbbk(\\mathcal{A} )$ provided $\\Gamma$ is of full-rank. In this section\nwe study some properties of these valuations. We extend this approach\nfor quotients of $\\mathcal{A}$ and (fibres of) $\\mathcal{X}$.\n\nLet $\\mathcal{V}$ be a scheme of the form\n$\\mathcal{A} , \\mathcal{X} , \\mathcal{A} /T_H$ or\n$\\mathcal{X} _{\\bf 1}$. Recall from § that every theta function on\n$\\mathcal{V}$ is labeled with a point of\n$\\mathrm{Trop} _{\\mathbb{Z} }(\\mathcal{V} ^\\vee)$, see .\n\n**Definition 40**. Suppose $\\Gamma$ is of full-rank and let\n$\\textbf{s}\\in %\n  \\mathrel{\\mathop{\\mathbb{T}_r}\\limits^{\n    \\vbox to0ex{\\kern-0.5\\ex@\n    \\hbox{$\\scriptstyle\\longrightarrow$}\\vss}}}$ be a seed for $\\Gamma$.\nThe **opposite dominance order** on $M^\\circ$ defined by $\\textbf{s}$ is\nthe partial order $\\preceq_{\\textbf{s}}$ on $M^\\circ$ determined by the\nfollowing condition: $$\\label{eq:dom_order}\nm_1 \\prec_{\\textbf{s}} m_2 \\ \\Leftrightarrow \\ m_2=  m_1 + p^{\\ast}_1(n) \\text{ for some }n\\in N^+_{\\operatorname{uf}, \\textbf{s}}.$$\n\n**Remark 41**. In Definition , $m_1\\preceq_{\\textbf{s}} m_2$ means that\neither $m_1 \\prec_\\textbf{s}m_2$ or $m_1=m_2$. We will also adopt this\nnotation for other orders we consider. The dominance order was\noriginally considered in and it is the opposite order to the one given\nin Definition . This order was exploited by in his work on bases for\ncluster algebras. The full-rank condition is needed so that\n$\\preceq_{\\textbf{s}}$ is reflexive. However, observe that for every\nseed $\\textbf{s}$ such that\n$\\text{ker}(p_1^*)\\cap N^+_{\\operatorname{uf}, \\textbf{s}} = \\emptyset$,\nequation still determines a partial order on $M^\\circ$ even if $\\Gamma$\nis not of full-rank. Nonetheless, whenever we talk about an (opposite)\ndominance order in this paper we will be tacitly assuming that $\\Gamma$\nis of full-rank.\n\nIt is straightforward to verify that $\\preceq_{\\textbf{s}}$ is\n**linear**. That is, $m_1 \\preceq_\\textbf{s}m_2$ implies that\n$m_1 + m \\preceq_\\textbf{s}m_2 + m$ for all $m \\in M^\\circ$.\n\n**Definition 42**. Let $A$ be an integral domain with a $\\Bbbk$-algebra\nstructure, $L$ a lattice isomorphic to $\\mathbb{Z} ^r$ and $\\leq$ a\ntotal order on $L$. A **valuation** on $A$ with values in $L$ is a\nfunction $\\nu : A\\setminus \\{0 \\} \\to (L,<)$ such that\n\n-   $\\nu(f+g) \\geq  \\min\\{\\nu(f), \\nu(g)\\}$, unless $f+g=0$,\n\n-   $\\nu(fg)= \\nu(f) + \\nu(g)$,\n\n-   $\\nu(cf)=\\nu(f)$ for all $c \\in \\Bbbk^*$.\n\nFor $l \\in L$ we define the subspace\n$A_{\\nu \\geq l}:= \\{ x\\in A \\setminus \\{0\\} \\mid \\nu(x)\\geq l\\} \\cup \\{ 0 \\}$\nof $A$. The subspace $A_{\\nu > l}$ is defined analogously. We say that\n$\\nu$ has **1-dimensional leaves** if the dimension of the quotient\n$$\\label{eq:graded_piece}\nA_l:=A_{\\nu \\geq l} \\big{/} A_{\\nu > l}$$ is either $0$ or $1$ for all\n$l\\in L$. A basis $B$ of $A$ is **adapted** for $\\nu$ if for all\n$l\\in L$ the set $B\\cap A_{\\nu \\geq l}$ is a basis of $A_{\\nu\\geq l}$.\n\n**Lemma 43**.\n\nAssume $\\Gamma$ is of full-rank. Let\n$\\vartheta^{\\mathcal{A} }_{m_1},\\vartheta^{\\mathcal{A} }_{m_2}\\in \\text{mid}(\\mathcal{A} )$\nwith\n$m_1,m_2\\in =\\mathrm{Trop} _{\\mathbb{Z} }(\\mathcal{A} ^{\\vee}_{\\textbf{s}^\\vee})=M^\\circ$.\nThen the product\n$\\vartheta^{\\mathcal{A} }_{m_1}\\vartheta^{\\mathcal{A} }_{m_2}$ expressed\nin the theta basis of $\\text{mid}(\\mathcal{A} )$ has the following form\n$$\\vartheta^{\\mathcal{A} }_{m_1}\\vartheta^{\\mathcal{A} }_{m_2}= \\vartheta^{\\mathcal{A} }_{m_1+m_2}+ \\sum_{m_1+m_2 \\prec_{\\textbf{s}}  m}c_{m}\\vartheta^{\\mathcal{A} }_{m}.$$\n\n*Proof.* First notice that for any broken line $\\gamma$ we have that\n$$F(\\gamma)=I(\\gamma) +a_{1}p^*_1(n_{1})+ \\dots + a_{r}p^*_1(n_{r}),$$\nwhere $a_1, \\dots , a_r$ are non-negative integers and\n$n_1, \\dots , n_r \\in N^+_{\\operatorname{uf}, \\textbf{s}}$. This follows\nfrom and the bending rule of broken lines (*i.e.* (4)). In particular,\nwe have that\n$a_{1}n_{1}+ \\dots + a_{r}n_{r}\\in N^+_{\\operatorname{uf}, \\textbf{s}} \\cup \\{ 0 \\}$.\nMoreover, $a_{1}p^*_1(n_{1})+ \\dots + a_{r}p^*_1(n_{r}) = 0$ if and only\nif $a_1=\\cdots = a_r =0$. Therefore,\n$I(\\gamma) \\preceq_{\\textbf{s}} F(\\gamma)$ and $I(\\gamma)=F(\\gamma)$ if\nand only if $\\gamma$ does not bend at all.\n\nThe statement we want to prove already follows from the observations\nmade above. Indeed, by we know that $\\alpha(m_1,m_2,m)\\neq 0$ if and\nonly if there exist broken lines $\\gamma_1$ and $\\gamma_2$ such that\n$I(\\gamma_i)=m_i$ for $i \\in \\{1,2\\}$ and\n$F(\\gamma_1)+F(\\gamma_2)=m=\\gamma_1(0)=\\gamma_2(0)$. Therefore, if\n$\\alpha(m_1,m_2,m)\\neq 0$ then\n$m_1 + m_2=I(\\gamma_1) + I(\\gamma_2) \\preceq_{\\textbf{s}} m$. Moreover,\nthe equality $m_1+ m_2=m$ holds if and only if both $\\gamma_1$ and\n$\\gamma_2$ do not bend at all. This latter case can be realized in a\nunique way, therefore, $\\alpha(m_1,m_2,m_1+m_2)=1$. ◻\n\nFrom now on the symbol $\\leq_{\\textbf{s}}$ is used to denote a total\norder on $M^\\circ$ refining $\\preceq_{\\textbf{s}}$.\n\n**Definition 44**. Let\n${\\bf m}=(m_{\\textbf{s}^\\vee})\\in \\mathrm{Trop} _{\\mathbb{Z} }(\\mathcal{A} ^{\\vee})$.\nThe **g-vector of** $\\vartheta ^{\\mathcal{A} }_{\\bf m}$ **with respect\nto** $\\textbf{s}$ is $$\\label{eq:red-g-val-A}\n{\\bf g}_{\\textbf{s}}\\left(\\vartheta ^{\\mathcal{A} }_{\\bf m}\\right):= m_{\\textbf{s}^\\vee}\n\\in \\mathrm{Trop} _{\\mathbb{Z} }(\\mathcal{A} ^{\\vee}_{\\textbf{s}^\\vee}).$$\n\n**Definition 45**.\n\nAssume $\\Gamma$ is of full-rank and think of $M^{\\circ}$ as\n$\\mathrm{Trop} _{\\mathbb{Z} }(\\mathcal{A} ^{\\vee}_{\\textbf{s}^\\vee})$.\nLet\n$\\mathbf{g} _{\\textbf{s}}:\\mathop{\\mathrm{mid}}(\\mathcal{A} ) \\setminus \\{ 0\\} \\to (M^{\\circ},\\leq_{\\textbf{s}})$\nbe the map given by $$\\label{eq:g_val}\n    \\mathbf{g} _{\\textbf{s}}(f):= \\min{}_{\\leq_{\\textbf{s}}}\\{m_1, \\dots , m_t\\},$$\nwhere\n$f=c_1\\vartheta^{\\mathcal{A} }_{m_1} + \\dots + c_t\\vartheta^{\\mathcal{A} }_{m_t}$,\n$m_j\\in M^\\circ$ and $c_j\\not=0$ for all $j=1,\\dots,t$ is the expression\nof $f$ in the theta basis of $\\text{mid}(\\mathcal{A} )$.\n\n**Lemma 46**.\n\nFor every seed $\\textbf{s}$ the map $\\mathbf{g} _{\\textbf{s}}$ is a\nvaluation on $\\mathop{\\mathrm{mid}}(\\mathcal{A} )$ with 1-dimensional\nleaves and the theta basis\n$\\{ \\vartheta _{m} \\mid m\\in \\Theta (\\mathcal{A} ) \\}$ is adapted for\n$\\mathbf{g} _{\\textbf{s}}$.\n\n*Proof.* This statement follows from but for the convenience of the\nreader we give a proof here. Items (1) and (3) of Definition  follow\ndirectly from the definition of $\\mathbf{g} _{\\textbf{s}}$. For item (2)\nconsider the expressions\n$f=\\sum_{i=1}^r c_i\\vartheta^{\\mathcal{A} }_{m_i}$ and\n$g=\\sum_{j=1}^s c'_j\\vartheta^{\\mathcal{A} }_{m'_j}$ where all $c_i$ and\n$c'_j$ are non-zero. Then by $$\\begin{aligned}\n\\label{eq:fg in basis}\nfg=\\sum_{i,j} c_ic'_j\\left(\\vartheta^{\\mathcal{A} }_{m_i+m'_j} + \\sum_{m_i+m'_j\\prec_{\\textbf{s}} m}c_{m}\\vartheta^{\\mathcal{A} }_{m}\\right).\n\\end{aligned}$$ By definition of $\\mathbf{g} _{\\textbf{s}}$ we have\n$m_\\mu:=\\mathbf{g} _{\\textbf{s}}(f)\\prec_{\\textbf{s}} m_i$ for all\n$i\\in \\{1,\\dots, r\\} \\setminus \\{ \\mu \\}$ and\n$m'_\\nu:=\\mathbf{g} _{\\textbf{s}}(g)\\prec_{\\textbf{s}} m'_j$ for all\n$j\\in \\{1,\\dots,s\\}\\setminus \\{\\nu\\}$. We need to show that the term\n$\\vartheta_{m_\\mu+m'_\\nu}$ appears with non-zero coefficient in $fg$.\nAssume there exist $i\\not =\\mu$ and $j\\not=\\nu$ such that\n$m_\\mu+m'_\\nu=m_i+m'_j$. Then as $\\prec_{\\textbf{s}}$ is linear we have\n$$m_\\mu +m'_\\nu \\prec_{\\textbf{s}} m_\\mu + m'_j \\prec_{\\textbf{s}} m_i + m'_j,$$\na contradiction. Hence, the term $\\vartheta_{m_\\mu+m'_\\nu}$ appears in\nthe expression of $fg$ with coefficient $c_\\mu c'_\\nu\\not =0$ and\n$\\mathbf{g} _{\\textbf{s}}(fg)=m_\\mu+m'_\\nu=\\mathbf{g} _{\\textbf{s}}(f)+\\mathbf{g} _{\\textbf{s}}(g)$.\n\nThe fact that ${\\bf g}_{\\textbf{s}}$ has one dimension leaves follows\ndirectly from (). It is also clear from the definitions that for\n$m\\in M^{\\circ}$ the subspace $\\mathop{\\mathrm{mid}}(\\mathcal{A} )_{m}$\nas in () is isomorphic to $\\Bbbk\\cdot \\vartheta ^{\\mathcal{A} }_{m}$ if\n$m \\in \\Theta(\\mathcal{A} )$ and $0$-dimensional otherwise. In\nparticular, the fact that we have a bijection between the set of values\nof $\\mathbf{g} _\\textbf{s}$ and the elements of the theta basis is\nequivalent to the theta basis being an adapted basis, see . ◻\n\n**Corollary 47**. The image of the valuation ${\\bf g}_{\\textbf{s}}$ is\nindependent of the linear refinement $\\leq_{\\textbf{s}}$ of\n$\\preceq_{\\textbf{s}}$.\n\n*Proof.* Since the theta basis is adapted for ${\\bf g}_{\\textbf{s}}$ we\nhave\n$${\\bf g}_{\\textbf{s}}\\left(\\mathop{\\mathrm{mid}}(\\mathcal{A} )\\setminus \\{0\\}\\right)= {\\bf g}_{\\textbf{s}}\\left(\\Theta(\\mathcal{A} )\\right).$$\nThe result follows. ◻\n\n**Remark 48**.\n\nSince $\\mathop{\\mathrm{mid}}(\\mathcal{A} )$ is a domain (see Remark )\nwhose associated field of fractions is isomorphic to\n$\\Bbbk(A_i :i \\in I)$, we can extend the valuation\n${\\bf g}_{\\mathbf{s}}$ on $\\text{mid}(\\mathcal{A} )$ to a valuation on\n$\\Bbbk(A_i :i \\in I)$ by declaring\n${\\bf g}_{\\mathbf{s}} (f/g):={\\bf g}_{\\mathbf{s}} (f)- {\\bf g}_{\\mathbf{s}} (g)$.\n\nThe valuation ${\\bf g}_{\\textbf{s}}$ is called the ****g**-vector\nvaluation associated to $\\textbf{s}$**.\n\nWe now turn our attention to quotients of $\\mathcal{A}$. We keep the\nassumption that $\\Gamma$ is of full-rank and consider a saturated\nsublattice $H=H_{\\mathcal{A} }$ of $K^\\circ$. Recall from § that\n$$\\mathrm{Trop} _{\\mathbb{Z} }((\\mathcal{A} /T_H)^{\\vee}_{\\textbf{s}^\\vee})= H^{\\perp}.$$\nSince $\\Theta(\\mathcal{A} /T_{H})_{\\textbf{s}^\\vee}\\subset  H^ \\perp$,\nwe can restrict restrict the total order $\\leq_{\\textbf{s}}$ on\n$M^{\\circ}$ to $H^{\\perp}$ to obtain a **g**-vector valuation on\n$\\mathop{\\mathrm{mid}}(\\mathcal{A} /T_{H})$ associated to $\\textbf{s}$\nas in the previous cases:\n$${\\bf g}_{\\textbf{s}}: \\mathop{\\mathrm{mid}}(\\mathcal{A} /T_H)\\setminus\\{0\\} \\to \\mathrm{Trop} _{\\mathbb{Z} }((\\mathcal{A} /T_H)^{\\vee}_{\\textbf{s}^\\vee}).$$\n\n**Remark 49**. As opposed to the case of $\\mathcal{A}$, in general the\nfield of fractions of $\\mathop{\\mathrm{mid}}(\\mathcal{A} /T_H)$ might\nnot be isomorphic to $\\Bbbk(\\mathcal{A} /T_H)$. This fails for example\nif the smallest cone in\n$\\mathrm{Trop} _{\\mathbb{R} }((\\mathcal{A} /T_H)^{\\vee}_{\\textbf{s}^\\vee})$\ncontaining $\\Theta(\\mathcal{A} /T_H)_{\\textbf{s}^\\vee}$ is not\nfull-dimensional. However, the field of fractions of\n$\\mathop{\\mathrm{mid}}(\\mathcal{A} /T_H)$ is isomorphic to\n$\\Bbbk(\\mathcal{A} /T_H)$ provided $\\mathcal{A} /T_H$ satisfies the full\nFock–Goncharov conjecture. In such a case, a **g**-vector valuation on\n$\\mathop{\\mathrm{mid}}(\\mathcal{A} /T_H)$ can be extended to\n$\\Bbbk(\\mathcal{A} /T_H)$ as in .\n\nWe now treat the case of $\\mathcal{X}$. So fix a cluster ensemble\nlattice map $p^*:N \\to M^{\\circ}$ and a seed $\\textbf{s}$. Consider the\nidentifications\n$\\mathrm{Trop} _{\\mathbb{Z} }(\\mathcal{X} ^\\vee_{\\textbf{s}})= d\\cdot N$\nand\n$\\mathrm{Trop} _{\\mathbb{Z} }(\\mathcal{A} ^{\\vee}_{{\\mathrm{prin}} ,\\widetilde{\\textbf{s}}^\\vee}) = M^{\\circ}_{{\\mathrm{prin}} }=M^{\\circ}\\oplus N$\nwhere $\\widetilde{\\textbf{s}}$ is the seed for $\\Gamma_{\\mathrm{prin}}$\nobtained mutating $\\textbf{s}_{0_{\\mathrm{prin}} }$ in the same sequence\nof directions needed to obtain $\\textbf{s}$ from $\\textbf{s}_0$. Recall\nfrom § that we have an inclusion\n$\\mathrm{Trop} _{\\mathbb{Z} }(\\mathcal{X} ^\\vee_{\\textbf{s}})\\to \\mathrm{Trop} _{\\mathbb{Z} }(\\mathcal{A} ^{\\vee}_{{\\mathrm{prin}} ,\\widetilde{\\textbf{s}}^\\vee})$\ngiven by $dn \\mapsto (p^*(n),n)$.\n\n**Definition 50**. Let\n${\\bf n}=(dn_{\\textbf{s}^\\vee})\\in \\mathrm{Trop} _{\\mathbb{Z} }(\\mathcal{X} ^{\\vee})$.\nThe **c-vector of** $\\vartheta ^{\\mathcal{X} }_{\\bf n}$ with respect to\n$\\textbf{s}$ is $$\\label{eq:red-g-val-X}\n{\\bf c}_{\\textbf{s}}\\left(\\vartheta ^{\\mathcal{X} }_{\\bf n}\\right):= dn_{\\textbf{s}^\\vee}\n\\in \\mathrm{Trop} _{\\mathbb{Z} }(\\mathcal{X} ^{\\vee}_{\\textbf{s}^\\vee}).$$\n\n**Remark 51**. Observe that\n$\\mathbf{c} _\\textbf{s}(\\vartheta ^{\\mathcal{X} }_{\\bf n})$ is an\nelement of $d\\cdot N$. In practice we could work with the lattice $N$ as\nopposed to $d\\cdot N$ as they are canonically isomorphic. The lattice\n$N$ is the set where the ${\\bf c}$-vectors (in the sense of ) live.\n\n**Definition 52**. The **divisibility order** on $N$ determined by\n$\\textbf{s}$ is the partial order $\\preceq_{\\textbf{s}, \\text{div}}$\ngiven by\n$$n_1 \\preceq_{\\textbf{s}, \\text{div}} n_2 \\text{ if and only if } \nn_2- n_1 \\in N_{\\textbf{s}}^+.$$\n\n**Lemma 53**.\n\nThe restriction of $\\preceq_{\\widetilde{\\textbf{s}}^\\vee}$ to the $N$\ncomponent of $M^\\circ_{{\\mathrm{prin}} }$ coincides with the\ndivisibility order $\\prec_{\\textbf{s},\\text{div}}$ on $N$.\n\n*Proof.* Let\n$p^*_{{\\mathrm{prin}} ,1}:N_{\\operatorname{uf}, {\\mathrm{prin}} }\\to M^\\circ_{\\mathrm{prin}}$\nbe the given by $(n,m)\\mapsto \\{ (n,m), \\cdot \\}_{{\\mathrm{prin}} }$ (in\nother words, $p^*_{{\\mathrm{prin}} ,1}$ corresponds to the map $p_1^*$\nin for $\\Gamma_{{\\mathrm{prin}} }$). In particular,\n$p^*_{{\\mathrm{prin}} ,1} (n,0) = (p^*_1(n), n)$. Let $n_1,n_2 \\in N$ be\ndistinct elements such that $n_2-n_1 \\in N^+_\\textbf{s}$. Let\n$\\widetilde{m}_i=(p_1(n_i),n_i)$ for $i = 1,2$. Then\n$\\widetilde{m}_2 -\\widetilde{m}_1= (p^*_1(n_2-n_1), n_2 -n_1)$. The\nresult follows. ◻\n\nThe next result follows at once from and .\n\n**Lemma 54**. Let\n$\\vartheta ^{\\mathcal{X} }_{dn_1},\\vartheta ^{\\mathcal{X} }_{dn_2} \\in \\mathop{\\mathrm{mid}}(\\mathcal{X} )$\nwith\n$d_1n_1, d_2n_2 \\in \\mathrm{Trop} _{\\mathbb{Z} }(\\mathcal{X} ^\\vee_{\\textbf{s}^\\vee})=d\\cdot N$.\nThen the product\n$\\vartheta^{\\mathcal{X} }_{dn_1}\\vartheta^{\\mathcal{X} }_{dn_2}$\nexpressed in the theta basis of $\\mathop{\\mathrm{mid}}(\\mathcal{X} )$ is\nof the following form\n$$\\vartheta^{\\mathcal{X} }_{dn_1}\\vartheta^{\\mathcal{X} }_{dn_2}= \\vartheta^{\\mathcal{X} }_{dn_1+dn_2}+ \\sum_{n_1+n_2 \\ \\prec_{\\textbf{s}, \\text{div}} \\ n} c_{n}\\vartheta^{\\mathcal{X} }_{dn}.$$\n\nFrom now on we let $\\leq_{\\textbf{s},\\text{div}}$ be any total order\nrefining $\\preceq_{\\textbf{s}, \\text{div}}$.\n\n**Corollary 55**. Let\n${\\bf c}_{\\textbf{s}}:\\mathop{\\mathrm{mid}}(\\mathcal{X} ) \\setminus \\{ 0\\} \\to (d \\cdot N,\\leq_{\\textbf{s},\\text{div}})$\nbe the map defined by\n$${\\bf c }_{\\textbf{s}}(f):= \\min{}_{\\leq_{\\textbf{s},\\text{div}}}\\{n_1, \\dots , n_t\\},$$\nwhere\n$f=c_1\\vartheta ^{\\mathcal{X} }_{d n_1} + \\dots + c_t\\vartheta ^{\\mathcal{X} }_{d n_t}$\nis the expression of $f$ in the theta basis of\n$\\text{mid}(\\mathcal{X} )$. Then ${\\bf c }_{\\textbf{s}}$ is a valuation\nwith 1-dimensional leaves and the theta basis for\n$\\mathop{\\mathrm{mid}}(\\mathcal{X} )$ is adapted for\n${\\bf c}_\\textbf{s}$.\n\nWe now let $\\mathcal{X} _{\\bf 1}$ be the fibre of $\\mathcal{X}$\nassociated to a sublattice $H:= H_{\\mathcal{X} } \\subset K$. In order to\ndefine a **c**-vector valuation on\n$\\mathop{\\mathrm{mid}}(\\mathcal{X} _{\\bf 1})$ we need that\n$$H\\cap N^+_{\\textbf{s}}= \\emptyset.$$ Since, if this condition holds,\n$\\preceq_{\\textbf{s}, \\text{div}}$ induces a well partial order on\n$N/H =\\mathcal X_{\\bf 1,\\textbf{s}}$ defined as\n$$n_1 + H \\preceq_{\\textbf{s}, \\text{div}} n_2+H \\quad \\text{ if and only if } \\quad n_2 - n_1 \\in N^+_{\\textbf{s}}+ H.$$\nThe rest of the construction follows from the cases already treated.\n\n**Lemma 56**. Suppose $\\Gamma$ is of full-rank and let\n$p: \\mathcal{A} \\to \\mathcal{X}$ be a cluster ensemble map. Then we have\na commutative diagram $$\\xymatrix{\n\\mathop{\\mathrm{mid}}(\\mathcal{X} ) \\setminus \\{0\\} \\ar^{p^*}[r] \\ar_{{\\bf c}_{\\textbf{s}}}[d] &  \\mathop{\\mathrm{mid}}(\\mathcal{A} ) \\setminus \\{0\\} \\ar^{{\\bf g}_{\\textbf{s}}}[d] \\\\\n\\mathrm{Trop} _{\\mathbb{Z} }(\\mathcal{X} ^{\\vee}_{\\textbf{s}^\\vee}) \\ar_{(p^\\vee)^T\\circ i} [r] & \\mathrm{Trop} _{\\mathbb{Z} }(\\mathcal{A} ^{\\vee}_{\\textbf{s}^\\vee}) \n}$$\n\n*Proof.* It is enough to show that for\n${\\bf n} \\in \\Theta(\\mathcal{X} )$ we have\n$$\\mathbf{g} _{\\textbf{s}}(p^*(\\vartheta ^\\mathcal{X} _{\\bf n}))=(p^\\vee)^T\\circ i({\\bf c}_{\\textbf{s}} (\\vartheta ^\\mathcal{X} _{\\bf n}))$$\nLet $dn=\\mathfrak{r}_{\\textbf{s}^\\vee}({\\bf n})$. We have that\n$$\\vartheta ^{\\mathcal{X} }_{dn}=z^n + \\sum_{n\\prec_{\\textbf{s}}n'}a_{n'}z^{n'}.$$\nTherefore,\n$$p^*(\\vartheta ^{\\mathcal{X} }_{dn})=z^{p^*(n)} + \\sum_{n<_{\\textbf{s}, \\text{div}}n'}a_{n'}z^{p^*(n')}.$$\nWe conclude that\n$\\mathbf{g} _{\\textbf{s}}(p^*(\\vartheta ^\\mathcal{X} _{\\bf n}))=p^*(n)$.\nOn the other hand we have that\n${\\bf c}_{\\textbf{s}} (\\vartheta ^\\mathcal{X} _{\\bf n})=dn$. We compute\n$$\\begin{aligned}\n    (p^\\vee)^T\\circ i (dn)= ((p^\\vee)^*)^*(-dn)=\\left(-\\frac{1}{d}(p^*)^*)\\right)^*(-dn)=p^*(n).\n\\end{aligned}$$ The claim follows. ◻\n\nWe would like to treat **g**-vector valuations for varieties of the form\n$\\mathcal{A}$ and $\\mathcal{A} /T_H$ and **c**-vector valuations on\n$\\mathcal{X}$ and $\\mathcal{X} _{\\bf 1}$ in a uniform way. With this in\nmind we introduce the following notation.\n\n**Notation 57**.\n\nLet $\\mathcal{V}$ be a cluster variety and $\\mathcal{V} ^{\\vee}$ its\nFock–Goncharov dual. The cluster valuation on\n$\\mathop{\\mathrm{mid}}(\\mathcal{V} )$ associated to a seed\n$\\textbf{s}\\in %\n  \\mathrel{\\mathop{\\mathbb{T}_r}\\limits^{\n    \\vbox to0ex{\\kern-0.5\\ex@\n    \\hbox{$\\scriptstyle\\longrightarrow$}\\vss}}}$ is\n$$\\nu_{\\textbf{s}}:\\mathop{\\mathrm{mid}}(\\mathcal{V} )\\setminus\\{0\\} \\to (\\mathrm{Trop} _{\\mathbb{Z} }(\\mathcal{V} ^\\vee_{\\textbf{s}^\\vee}), <_{\\textbf{s}}),$$\nwhere\n$\\mathrm{Trop} _{\\mathbb{Z} }(\\mathcal{V} ^\\vee_{\\textbf{s}^\\vee})$ is\nas in and $<_{\\textbf{s}}$ is a linear order on\n$\\mathrm{Trop} _{\\mathbb{Z} }(\\mathcal{V} ^\\vee_{\\textbf{s}^\\vee})$\nrefining $\\prec_\\textbf{s}$ in case $\\mathcal{V} =\\mathcal{A}$ or\n$\\mathcal{A} /T_H$ and it refines $\\prec_{\\textbf{s},\\text{div}}$ if\n$\\mathcal{V} =\\mathcal{X}$ or $\\mathcal{X} _{\\bf 1}$.\n\n# Newton–Okounkov bodies\n\nIn this section we provide a general approach to construct\nNewton–Okounkov bodies associated to certain partial minimal models of\nvarieties with a cluster structure. In particular, we treat a situation\nthat often arises in representation theory where the universal torsor of\na projective variety has a cluster structure of type $\\mathcal{A}$. The\nNewton–Okounkov bodies we construct depend on the choice of an initial\nseed. Hence we discuss how the bodies associated to different choices of\ninitial seed are related and introduce the intrinsic Newton–Okounkov\nbody which is seed independent.\n\n## Schemes and ensembles with cluster structure\n\n**Definition 58**.\n\nWe say a smooth scheme (over $\\Bbbk$) $V$ **can be endowed with cluster\nstructure of type** $\\mathcal{V}$ if there is a birational map\n$\\Phi: \\mathcal{V} \\dashrightarrow V$ which is an isomorphism outside a\ncodimension two subscheme of the domain and range. In this setting, we\nsay that the pair $(V,\\Phi)$ is **a scheme with cluster structure of\ntype** $\\mathcal{V}$.\n\n**Remark 59**. We are straying slightly from in . Specifically, we are\nnow including $\\Phi$ as part of the data defining a scheme with cluster\nstructure. So, given two different birational maps\n$\\Phi_1:\\mathcal{V} _1 \\dashrightarrow V$ and\n$\\Phi_2: \\mathcal{V} _2 \\dashrightarrow V$ as in , we now consider\n$(V,\\Phi_1)$ and $(V,\\Phi_2)$ different as schemes with cluster\nstructure (as is the case, for example, for open positroid varieties,\nsee Remark ). Nevertheless, when the map $\\Phi$ is clear from the\ncontext or we are just dealing with a single birational map\n$\\mathcal{V} \\dashrightarrow V$, we will simply say that $V$ has a\ncluster structure of type $\\mathcal{V}$.\n\nLet $V=(V,\\Phi)$ be a scheme with a cluster structure of type\n$\\mathcal{V}$. Since $V$ is normal and isomorphic to $\\mathcal{V}$ up to\nco-dimension $2$ then $V$ and $\\mathcal{V}$ have isomorphic rings of\nregular functions. In turn, we can talk about polynomial theta functions\non $V$ which we denote by $\\vartheta ^V_{\\bf v}$ for\n${\\bf v}\\in \\Theta (\\mathcal{V} )$. Moreover, recall that $\\mathcal{V}$\nis log Calabi–Yau. By $V$ is also log Calabi–Yau. Hence, $V$ has a\ncanonical volume form whose pullback by $\\Phi$ coincides with the\ncanonical volume form on $\\mathcal{V}$. Moreover, a (partial) minimal\nmodel $V\\subset Y$ and its boundary can be defined as in Definition .\n\n**Definition 60**. An inclusion $V \\subset Y$ as an open subscheme of a\nnormal variety $Y$ is a **partial minimal model** of $V$ if the\ncanonical volume form on $V$ has a simple pole along every irreducible\ndivisor of $Y$ contained in $Y \\setminus V$. It is a **minimal model**\nif $Y$ is, in addition, projective. We call $Y \\setminus V$ the\n**boundary** of $V  \\subset Y$.\n\n**Definition 61**. Suppose $\\Phi:\\mathcal{V} \\dashrightarrow V$ endows\n$V$ with a cluster structure of type $\\mathcal{V}$ and that the cluster\nvaluation $\\nu_{\\textbf{s}}$ extends to $\\Bbbk(\\mathcal{V} )$. Then the\n**cluster valuation**\n$\\nu^{\\Phi}_{\\textbf{s}}:\\Bbbk(V)^*\\to \\mathrm{Trop} _{\\mathbb{Z} }(\\mathcal{V} ^\\vee)$\nis given by\n$$\\nu^{\\Phi}_{\\textbf{s}}(f)=  \\nu_{\\textbf{s}}(\\Phi^*(f)).$$\n\n**Definition 62**. Suppose\n$\\Phi_{\\mathcal{A} }:\\mathcal{A} \\dashrightarrow V_1$ and\n$\\Phi_{\\mathcal{X} }:\\mathcal{X} \\dashrightarrow V_2$ endow $V_1$ (resp.\n$V_2$) with cluster structures of type $\\mathcal{A}$ (resp.\n$\\mathcal{X}$). We say that $V_1 \\overset{\\tau}{\\to} V_2$ is a cluster\nensemble structure if there exists a cluster ensemble map\n$p:\\mathcal{A} \\to \\mathcal{X}$ such that the following diagram commutes\n$$\\xymatrix{\n    V_1 \\ar^{\\tau}[r] & V_2 \\\\\n    \\mathcal{A} \\ar@{-->}^{\\Phi_{\\mathcal{A} }}[u] \\ar_p[r] & \\mathcal{X} \\ar@{-->}_{\\Phi_{\\mathcal{X} }}[u].\n    }$$\n\n## Newton–Okounkov bodies for Weil divisors supported on the boundary\n\nThroughout this section we let $\\mathcal{V}$ be a scheme of the form\n$\\mathcal{A}$, $\\mathcal{X}$, $\\mathcal{A} /T_{H}$ or\n$\\mathcal{X} _{{\\bf 1}}$. Whenever we talk about a cluster valuation on\n$\\mathop{\\mathrm{mid}}(\\mathcal{V} )$ we are implicitly assuming we are\nin a setting where such valuation exist, see §.\n\n**Definition 63**. A closed subset\n$S\\subseteq \\mathrm{Trop} _{\\mathbb{R} }(\\mathcal{V} ^{\\vee})$ is\n**positive** if for any positive integers $d_1, d_2$, any\n$p_1\\in d_1\\cdot S(\\mathbb{Z} )$, $p_2\\in d_2\\cdot S(\\mathbb{Z} )$ and\nany $r \\in \\mathrm{Trop} _{\\mathbb{Z} }(\\mathcal{V} ^{\\vee})$ such that\n$\\alpha (p_1,p_2,r)\\neq 0$, we have that\n$r \\in (d_1 +d_2)\\cdot S(\\mathbb{Z} )$.\n\n**Remark 64**. We can also define positive sets inside\n$\\mathrm{Trop} _{\\mathbb{R} }(\\mathcal{V} ^{\\vee})_{\\textbf{s}^\\vee}$ in\nexactly the same way they are defined in Definition . In particular we\nhave that $S\\subset \\mathrm{Trop} _{\\mathbb{R} }(\\mathcal{V} ^{\\vee})$\nis positive if and only if\n$\\mathfrak{r}_{\\textbf{s}^\\vee}(S)\\subset \\mathrm{Trop} _{\\mathbb{R} }(\\mathcal{V} ^\\vee_{\\textbf{s}})$\nis positive.\n\nIn the authors discuss how positive sets give rise to both, partial\nminimal models of cluster varieties and toric degenerations of such. In\nthis section we study the inverse problem. Namely, we let $(V,\\Phi)$ be\na scheme with a cluster structure of type $\\mathcal{V}$ and construct\nNewton–Okounkov bodies associated to a partial minimal model\n$V \\subset Y$ (see §). Then we show that under suitable hypotheses these\nNewton–Okounkov bodies are positive sets. We let $D_1, \\dots , D_s$ be\nthe irreducible divisors of $Y$ contained in the boundary of\n$V\\subset Y$ and let $D:=\\bigcup_{j=1}^s D_j$.\n\nGiven a Weil divisor $D'$ on $Y$ we denote by $R(D')$ the associated\n**section ring**. Recall that $R(D')$ can be described as the\n$\\mathbb{Z} _{\\geq 0}$-graded ring whose $k^{\\mathrm{th}}$ homogeneous\ncomponent is\n$$R_k(D') := H^0(Y, \\mathcal{O}(kD'))= \\left\\{  f\\in \\Bbbk(Y)^* \\mid \\text{div}(f)+kD'\\geq 0 \\right\\}\\cup \\{ 0\\},$$\nwhere $\\text{div}(f)$ is the principal divisor associated to $f$. Even\nmore concretely, if $D'=c_1  D'_1 + \\cdots + c_{s'}D'_{s'}$, where\n$D'_1, \\dots , D'_{s'}$ are distinct prime divisors of $Y$ and\n$c_1, \\dots , c_{s'}$ are non-negative integers, then $R_k(D')$ is the\nvector space consisting of the rational functions on $Y$ that are\nregular on the complement of $\\bigcup_{j=1}^{s'} D'_j$ and whose order\nof vanishing along every prime divisor $D'_j$ is bounded below by\n$-kc_j$. The multiplication of $R(D')$ is induced by the multiplication\non $\\Bbbk(Y)$.\n\n**Definition 65**.\n\nLet $\\nu:\\Bbbk(Y)\\setminus \\{ 0 \\} \\to L$ be a valuation, where\n$(L, < )$ is a linearly ordered lattice. Let $D'$ be a Weil divisor on\n$Y$ having a non-zero global section. For a choice of non-zero section\n$\\tau \\in R_1 (D')$ the associated **Newton–Okounkov body** is\n$$\\begin{split} \n\\Delta_\\nu(D',\\tau) := \\overline{\\mathop{\\mathrm{conv}}\\Bigg( \\bigcup_{k\\geq 1}  \\left\\{\\frac{\\nu\\left(f/\\tau^k\\right)}{k} \\mid f\\in R_k(D')\\setminus \\{0\\} \\right\\} \\Bigg) }\\subseteq L\\otimes \\mathbb{R} ,\n \\end{split}$$ where $\\mathop{\\mathrm{conv}}$ denotes the convex hull\nand the closure is taken with respect to the standard topology of\n$L\\otimes \\mathbb{R}$.\n\nFrom now on we assume that $D'$ has a non-zero global section. We would\nlike to use a cluster valuation\n$\\nu^{\\Phi}_{\\textbf{s}}: \\Bbbk(V)\\setminus \\{ 0\\} \\to (\\mathrm{Trop} _{\\mathbb{Z} }(\\mathcal{V} ^{\\vee}_{\\textbf{s}^\\vee}),<_{\\textbf{s}})$\nto construct Newton–Okounkov bodies. Notice that if $\\mathcal{V}$\nsatisfies the full Fock–Goncharov conjecture, then it is possible to do\nso as we can extend $\\nu_{\\textbf{s}}$ from\n$\\mathop{\\mathrm{mid}}(\\mathcal{V} )=\\mathop{\\mathrm{up}}(\\mathcal{V} )$\nto $\\Bbbk(\\mathcal{V} ) = \\Bbbk(Y)$. Observe, moreover, that if $D'$ is\nsupported on $D$ (that is $D'=\\sum_{j=1}^s c_jD_j$ for some integers\n$c_1,\\dots , c_s$) then every graded piece $R_k(D')$ is contained in\n$H^{0}(V,\\mathcal{O}_V)\\cong H^{0}(\\mathcal{V} ,\\mathcal{O}_{\\mathcal{V} })$,\nso elements of $R_k(D')$ can be described using the theta basis for\n$H^0(\\mathcal{V} ,\\mathcal{O}_{\\mathcal{V} })$. Moreover,\n$\\mathop{\\mathrm{ord}}_{D_j}\\in \\mathcal{V} (\\mathbb{Z} ^t)$, so we can\ndefine $\\vartheta ^{\\mathcal{V} }_j$ as in .\n\n**Definition 66**. Assume $\\mathcal{V}$ satisfies the full\nFock–Goncharov conjecture and that $D'$ is of the form\n$D'=\\sum_{j=1}^s c_jD_j$. We say that $R(D')$ **has a graded theta\nbasis** if for every integer $k\\geq 0$ the set of theta functions on\n$\\mathcal{V}$ parametrized by the integral points of\n$$P_k(D'):= \\bigcap_{j=1}^s \\left\\{b\\in \\mathrm{Trop} _{\\mathbb{R} }(\\mathcal{V} ^{\\vee}) \\mid  \\mathrm{Trop} _{\\mathbb{R} }(\\vartheta ^{\\mathcal{V} ^\\vee}_j)(b) \\geq -kc_j\\right\\}$$\nis a basis for $R_k(D')$.\n\nThe reader should notice that in case $\\mathcal{V}$ has theta\nreciprocity (see Definition ), then the definition of $P_k(D')$ becomes\nvery natural from the perspective of toric geometry, see §. We now\nintroduce a notion that allows us to make a good choice for the section\n$\\tau$.\n\n**Definition 67**. A subset $L\\subset \\Theta(\\mathcal{V} )$ is\n**linear** if\n\n-   for any $a,b\\in L$ there exists a unique $r\\in\\Theta(\\mathcal{V} )$\n    such that $\\alpha(a,b,r)\\neq 0$ and moreover, $r\\in L$,\n\n-   for each $a\\in L$ there exists a unique $b\\in L$ such that\n    $\\vartheta ^{\\mathcal{V} }_a \\vartheta ^{\\mathcal{V} }_b=1$.\n\nWe further say that a linear subset $L$ **acts linearly** on\n$\\Theta(\\mathcal{V} )$ if for any $a\\in L$ and\n$b \\in \\Theta(\\mathcal{V} )$ there exists a unique\n$r\\in \\mathrm{Trop} _{\\mathbb{Z} }(\\mathcal{V} ^{\\vee})$ such that\n$\\alpha(a,b,r)\\neq 0$.\n\nFor example, if $\\mathcal{V} =\\mathcal{A}$ then\n$\\mathfrak{r}_{\\textbf{s}}^{-1}(N_{\\text{uf}}^\\perp)$ is linear and acts\nlinearly on $\\Theta(\\mathcal{V} )$. If $\\mathcal{V} =\\mathcal{X}$ then\n$\\mathfrak{r}_{\\textbf{s}}^{-1}(\\ker(p_2^*))$ is linear and acts\nlinearly on $\\Theta(\\mathcal{V} )$.\n\n**Theorem 68**. Let $V\\subset Y$ be a partial minimal model. Assume the\nfull Fock–Goncharov conjecture holds for $\\mathcal{V}$. Let\n$D'=\\sum_{j=1}^s c_j D_j$ be a Weil divisor on $Y$ supported on $D$ such\nthat $R(D')$ has a graded theta basis. Let $\\tau\\in R_1(D')$ be such\nthat $\\nu^{\\Phi}_{\\textbf{s}}(\\tau)$ belongs to a linear subset of\n$\\mathrm{Trop} _{\\mathbb{Z} }(\\mathcal{V} ^{\\vee})$ acting linearly on\n$\\mathrm{Trop} _{\\mathbb{Z} }(\\mathcal{V} ^{\\vee})$. Then the\nNewton–Okounkov body\n$\\Delta_{\\nu^{\\Phi}_{\\textbf{s}}}(D',\\tau)\\subset \\mathrm{Trop} _{\\mathbb{R} }(\\mathcal{V} ^{\\vee}_{\\textbf{s}^\\vee})$\nis a positive set.\n\n*Proof.* To make notation lighter, throughout this proof we denote\n$\\Delta_{\\nu_{\\textbf{s}}}(D',\\tau)$ simply by $\\Delta$,\n$P_k(D')_\\textbf{s}$ by $P_k$ and $\\nu^{\\Phi}_{\\textbf{s}}$ by\n$\\nu_{\\textbf{s}}$. We work in the lattice identification\n$\\mathrm{Trop} _{\\mathbb{Z} }(\\mathcal{V} ^{\\vee}_{\\textbf{s}^\\vee})$ of\n$\\mathrm{Trop} _{\\mathbb{Z} }(\\mathcal{V} ^{\\vee})$. The linear subset\nof the statement corresponds to a sublattice\n$L \\subseteq \\mathrm{Trop} _{\\mathbb{Z} }(\\mathcal{V} ^{\\vee}_{\\textbf{s}^\\vee})$.\n\nConsider $d_1, d_2 \\in \\mathbb{Z} _{>0}$ and\n$p_1\\in d_1\\Delta(\\mathbb{Z} )$, $p_2\\in d_2\\Delta(\\mathbb{Z} )$. We\nhave to show that for any\n$r \\in \\mathrm{Trop} _{\\mathbb{Z} }(\\mathcal{V} ^{\\vee}_{\\textbf{s}^\\vee})$\nwith $\\alpha (p_1,p_2,r)\\neq 0$ then\n$r \\in (d_1 +d_2)\\Delta(\\mathbb{Z} )$. For this it is enough to show\nthat $k\\Delta = P_k - k\\nu_{\\textbf{s}}(\\tau)$ for all\n$k \\in \\mathbb{Z} _{>0}$ as we now explain.[^5] If this is the case then\nfor $i=1,2$, the point $p_i+d_i\\nu_\\textbf{s}(\\tau)$ belongs to\n$P_{d_i}(\\mathbb{Z} )$. By hypothesis\n$\\vartheta ^V_{p_i+d_i \\nu_{\\textbf{s}}(\\tau)}\\in R_{d_i}(D')$. In\nparticular, the product\n$\\vartheta ^V_{p_1+d_1 \\nu_{\\textbf{s}}(\\tau)}\\vartheta ^V_{p_2+d_2 \\nu_{\\textbf{s}}(\\tau)}$\nmust belong to $R_{d_1+d_2}(D')$ and this product must be expressed as a\nlinear combination of theta functions that belong to $R_{d_1+d_2}(D')$.\nTo finish we just need to convince ourselves that\n$$\\alpha(p_1+d_1\\nu_\\textbf{s}(\\tau),p_2+d_2\\nu_\\textbf{s}(\\tau), r+(d_1+d_2)\\nu_\\textbf{s}(\\tau))\\neq 0$$\nas this would imply\n$$r+(d_1+d_2)\\nu_\\textbf{s}(\\tau)\\in P_{d_1+d_2}(\\mathbb{Z} )=(d_1+d_2)\\Delta(\\mathbb{Z} )+ (d_1+d_2)\\nu_\\textbf{s}(\\tau) .$$\nHowever, this follows at once from the fact that $\\nu_\\textbf{s}(\\tau)$\nbelongs to the linear subset $L$. Indeed, the condition\n$\\alpha(p_1,p_2,r)\\neq 0$ implies the existence of a pair of broken\nlines $\\gamma_1, \\gamma_2$ such that $I(\\gamma_i)=p_i$ and\n$F(\\gamma_1)+F(\\gamma_2)=r$. Since $\\nu_\\textbf{s}(\\tau)\\in L$ we can\nconstruct new broken lines $\\gamma'_1$ and $\\gamma'_2$ such that\n$I(\\gamma'_i)=p_i+d_i\\nu_\\textbf{s}(\\tau)$ and\n$F(\\gamma'_1)+F(\\gamma'_2)=r+(d_1+d_2)\\nu_\\textbf{s}(\\tau)$ by changing\nthe direction of all the domains of linearity of $\\gamma_i$ by\n$d_i\\nu_\\textbf{s}(\\tau)$.\n\nWe now proceed to show that $k\\Delta= P_k-k\\nu_\\textbf{s}(\\tau)$ for all\n$k \\in \\mathbb{Z} _{>0}$. First notice that $aP_1= P_a$ for all\n$a\\in \\mathbb{R} _{\\geq 0}$ (if $g$ is a positive Laurent polynomial\nthen $g^T(ax)=ag^T(x)$ provided $a$ is non-negative). Since $P_k$ is\nclosed and convex in order to show that\n$k \\Delta  \\subset P_k- k\\nu_\\textbf{s}(\\tau)$ it is enough to show that\n$\\frac{k}{k'}\\ \\nu_\\textbf{s}(f/\\tau^{k'})=\\frac{k}{k'}\\ \\nu_\\textbf{s}(f)-k \\nu_\\textbf{s}(\\tau)$\nbelongs to $P_k-k\\nu_\\textbf{s}(\\tau)$ for all $k'\\geq 1$ and all\n$f\\in R_{k'}(D')\\setminus \\{0\\}$. This follows at once from the fact\nthat $\\frac{k}{k'}\\nu_\\textbf{s}(f)\\in P_k$ as $\\frac{k}{k'}P_{k'}=P_k$.\nTo obtain the reverse inclusion it is enough to show that the inclusion\nholds at the level of rational points, namely,\n$P_k(\\mathbb{Q} )-k\\nu_\\textbf{s}(\\tau)\\subset k\\Delta(\\mathbb{Q} )$.\nIndeed, since $P_k$ is a finite intersection of rational hyperplanes in\n$\\mathrm{Trop} _{\\mathbb{R} }(\\mathcal{V} ^{\\vee}_{\\textbf{s}^\\vee})$ it\ncan be described as the convex hull of its rational points. If\n$x\\in P_k(\\mathbb{Q} )$ then\n$\\frac{x}{k}\\in \\frac{1}{k}P_k(\\mathbb{Q} )=P_1(\\mathbb{Q} )$. Let\n$d\\in \\mathbb{Z} _{>0}$ be such that\n$x':=\\frac{dx}{k} \\in \\mathrm{Trop} _{\\mathbb{Z} }(\\mathcal{V} ^{\\vee}_{\\textbf{s}^\\vee})$.\nIn particular, $x'\\in P_{d}(\\mathbb{Z} )_{\\textbf{s}}$ which gives that\n$d^{-1}\\nu_\\textbf{s}(\\frac{\\vartheta _{x'}}{\\tau^{d}})\\in \\Delta$.\nFinally, notice that\n$d^{-1}\\nu_\\textbf{s}(\\frac{\\vartheta _{x'}}{\\tau^{d}})=d^{-1}(\\nu_\\textbf{s}(\\vartheta _{x'})-d\\nu_\\textbf{s}(\\tau))=d^{-1}x'-\\nu_\\textbf{s}(\\tau)$\nwhich implies $x-k\\nu_\\textbf{s}(\\tau) \\in k\\Delta$. ◻\n\nIn Theorem the assumption that $R(D')$ has a graded theta basis might\nseem rather strong. We now provide a situation in which this hypothesis\nholds and in the next subsection we treat a more robust framework in\nwhich this condition follows directly from the equivariant nature of\ntheta functions.\n\n**Lemma 69**. Let $V\\subset Y$ be a minimal model. Assume\n$D=\\sum_{j=1}^n D_j$ is ample with $D'=cD$ very ample for some\n$c\\in \\mathbb{Z} _{>0}$. Assume further that the image of the embedding\nof $Y$ into a projective space given by $D'$ is projectively normal. If\n$\\mathcal{V}$ has theta reciprocity and the theta functions on\n$\\mathcal{V}$ respect the order of vanishing (see Definition ), then\n$R(D')$ has a graded theta basis.\n\n*Proof.* It is enough to treat the case $\\mathcal{V} =V$. Consider the\naffine cone $\\widetilde{Y}$ of the embedding of $Y$ into a projective\nspace given by $D'$. We consider the canonical projection\n$\\widetilde{Y}\\setminus \\{ 0\\} \\overset{\\pi}{\\to } Y$ and let\n$\\mathcal{V} ':= \\pi^{-1}(\\mathcal{V} )$. Observe that\n$\\mathcal{V} '\\cong \\mathcal{V} \\times \\mathbb{C} ^*$. We may think of\n$\\mathcal{V} '$ as the cluster variety obtained from $\\mathcal{V}$ by\nadding a frozen index and extending trivially the bilinear form in the\nfixed data defining $\\mathcal{V}$. In particular,\n$\\text{up}(\\mathcal{V} ')= \\text{up}(\\mathcal{V} )[x^{\\pm 1}]$, where\n$x$ is the coordinate for the $\\mathbb{C} ^*$ component. Notice that the\ntheta functions on $\\mathcal{V} '$ are of the form\n$\\vartheta ^{\\mathcal{V} '}_{(p,h)}=\\vartheta ^{\\mathcal{V} '}_{(0,h)}\\vartheta ^{\\mathcal{V} '}_{(p,0)} =x^h\\vartheta ^{\\mathcal{V} }_p$,\nwhere $\\vartheta ^{\\mathcal{V} }_p$ is a theta function on $\\mathcal{V}$\nand $h \\in \\mathbb{Z} =\\mathrm{Trop} _{\\mathbb{Z} }(\\mathbb{C} ^*)$. An\nanalogous description holds for the theta functions on\n$(\\mathcal{V} ')^\\vee \\cong \\mathcal{V} ^\\vee \\times \\mathbb{C} ^*$.\nNamely, these theta functions are of the form\n$x^h\\vartheta _q^{\\mathcal{V} ^\\vee}$ for some $h\\in \\mathbb{Z}$. We\nconsider the inclusion $R(D')\\hookrightarrow \\text{up}(\\mathcal{V} ')$\ngiven by sending a homogeneous element $f\\in R_k(D')$ to $x^kf$. The map\nis well defined since $f$ is regular on $\\mathcal{V}$. Moreover, if we\nlet $\\widetilde{D}_j:= \\pi^{-1}(D_j)$ then for all $j$ we have\n$\\mathop{\\mathrm{ord}}_{\\widetilde{D}_j}\\left(x^{k}\\right)=k$ and\n$\\mathop{\\mathrm{ord}}_{\\widetilde{D}_j}\\left(\\vartheta ^{\\mathcal{V} '}_{(p,0)}\\right)=\\mathop{\\mathrm{ord}}_{D_j}\\left(\\vartheta ^V_p\\right)$.\nIn particular, thinking of $\\mathrm{Trop} _{\\mathbb{Z} }(\\mathcal{V} ')$\nas $\\mathrm{Trop} _{\\mathbb{Z} }(\\mathcal{V} )\\times \\mathbb{Z}$ we have\n$\\mathop{\\mathrm{ord}}_{\\widetilde{D}_k}=(\\mathop{\\mathrm{ord}}_{D_k},1)$.\nSince theta functions on $\\mathcal{V}$ respect the order of vanishing,\nthe same holds for the theta functions on $\\mathcal{V} '$. This implies\nthat for every $a \\in \\mathbb{Z}$ and every $j$,\n$\\mathop{\\mathrm{ord}}_{D_j}\\left(\\sum_q \\alpha_q \\vartheta _q^{\\mathcal{V} }\\right)\\geq a$\nif and only if\n$\\mathop{\\mathrm{ord}}_{D_j}(\\vartheta _q^{\\mathcal{V} })\\geq a$ for all\n$q$ such that $\\alpha_q \\neq 0$. To see this there is only one\nimplication to be checked (the other follows from the axioms of\nvaluations). So assume\n$\\mathop{\\mathrm{ord}}_{D_j}\\left(\\sum_q \\alpha_q \\vartheta _q^{\\mathcal{V} }\\right)\\geq a$.\nSince\n$\\mathop{\\mathrm{ord}}_{D_j}\\left(\\sum_q \\alpha_q \\vartheta _q^{\\mathcal{V} }\\right)=\\mathop{\\mathrm{ord}}_{\\widetilde{D}_j}\\left(\\sum_q \\alpha_q \\vartheta _q^{\\mathcal{V} }\\right)$\nand $x^{-a}\\vartheta _q^{\\mathcal{V} }$ is a theta function on\n$\\mathcal{V} '$ for all $q$ we have the following $$\\begin{aligned}\n    \\mathop{\\mathrm{ord}}_{D_j}\\left(\\sum_q \\alpha_q \\vartheta _q^{\\mathcal{V} }\\right)\\geq a & \\Longleftrightarrow   \\mathop{\\mathrm{ord}}_{\\widetilde{D}_j}\\left(\\sum_q \\alpha_q \\vartheta _q^{\\mathcal{V} }\\right)\\geq a \\\\\n    & \\Longleftrightarrow   \\mathop{\\mathrm{ord}}_{\\widetilde{D}_j}\\left(x^{-a}\\sum_q \\alpha_q \\vartheta _q^{\\mathcal{V} }\\right) \\geq 0 \\\\\n    & \\Longleftrightarrow   \\mathop{\\mathrm{ord}}_{\\widetilde{D}_j}(x^{-a} \\vartheta _q^{\\mathcal{V} })\\geq 0  \\text{ for all } q \\text{ such that } \\alpha_q\\neq 0 \\\\\n     & \\Longleftrightarrow   \\mathop{\\mathrm{ord}}_{\\widetilde{D}_j}( \\vartheta _q^{\\mathcal{V} })\\geq a  \\text{ for all } q \\text{ such that } \\alpha_q\\neq 0 \\\\\n     & \\Longleftrightarrow   \\mathop{\\mathrm{ord}}_{D_j}( \\vartheta _q^{\\mathcal{V} })\\geq a  \\text{ for all } q \\text{ such that } \\alpha_q\\neq 0.\n\\end{aligned}$$ Since $D'$ is very ample and $Y$ is projectively normal\nin its embedding given by $D'$ we have that\n$H^0(\\widetilde{Y}, \\mathcal{O}_{\\widetilde{Y}}) \\cong R(D') \\hookrightarrow \\text{up}(\\mathcal{V} ')$.\nIn particular, if we express $f \\in R_k(D')$ as\n$f= \\sum_q \\alpha_q \\vartheta ^{\\mathcal{V} }_q$, we have that\n$\\mathop{\\mathrm{ord}}_{D_j}\\left(\\vartheta ^\\mathcal{V} \\right)\\geq -kc$\nfor all $j$ and all $q$ such that $\\alpha_q \\neq 0$. This means that\n$\\vartheta _q^{\\mathcal{V} } \\in R_k(D')$ for all such $q$. In\nparticular, the theta functions of $\\mathcal{V}$ that lie in $R_k(D')$\nhave to be a basis a of $R_k(D')$. By theta reciprocity, such theta\nfunctions are precisely those parametrized by $P_k(D')$. ◻\n\n**Remark 70**. If $R(D')$ is finitely generated and the semigroup\ngenerated by the image of $\\nu^\\mathcal{V} _{\\textbf{s}}$ is of\nfull-rank and finitely generated then there is a one parameter toric\ndegeneration of $Y$ to the toric variety associated to\n$\\Delta_{\\nu^\\mathcal{V} _{\\textbf{s}}}(D',\\tau)$ [^6]. As explained in\nfor cluster varieties of type $\\mathcal{A}$ (regardless of the full-rank\nassumption) a polyhedral positive set defines a partial compactification\n$\\mathcal{A} _{\\text{prin}} \\subset \\overline{\\mathcal{A} }_{\\text{prin}}$.\nThis compactification comes with a flat morphism\n$\\overline{\\mathcal{A} }_{\\text{prin}}\\to \\mathbb A^r$ having\n$\\overline{\\mathcal{A} }=Y$ as fibre over ${\\bf 1}=(1, \\dots , 1)$ and\nwhose fibre over $0$ is the toric variety associated to the positive\nset. Therefore, both constructions can be used to degenerate varieties\nwith a cluster structure to the same toric variety. However, the variety\ngiven by the latter construction contains various intermediate fibres\nthat lie in between $\\mathcal A=\\mathcal{V}$ and a toric variety.\nMoreover, while Anderson’s degenerations produces a\n$(\\Bbbk^*)$-equivariant family, for the latter degeneration this is the\ncase if and only if $\\Gamma$ is of full-rank.\n\n## Newton–Okounkov bodies for line bundles via universal torsors\n\nIn this section we consider a particularly nice geometric situation that\narises often in representation theory. We let $Y$ be an irreducible\nnormal projective scheme whose Picard group $\\text{Pic}(Y)$ is free of\nfinite rank $\\rho \\in \\mathbb{Z} _{>0}$ (recall that $\\text{Pic}(Y)$ is\nalways abelian). Following (see also , , or ), we consider the universal\ntorsor of $Y$ and the associated Cox ring (*cf.* Remark ). For the\nconvenience of the reader we recall these concepts. We begin by\nconsidering the quasi-coherent sheaf of $\\mathcal{O}_Y$-modules\n$$\\bigoplus_{[\\mathcal{L} ] \\in \\text{Pic}(Y)} \\mathcal{L} .$$ In\nessence, the universal torsor of $Y$ is obtained by applying a relative\nspectrum construction (also denoted by **Spec**) to this sheaf. However,\nthe choice of the representative $\\mathcal{L}$ in the class\n$[\\mathcal{L} ]$ prevents this sheaf from having a natural\n$\\mathcal{O}_Y$-algebra structure. To address this situation one can\nproceed as in and consider line bundles\n$\\mathcal{L} _1, \\dots, \\mathcal{L} _{\\rho}$ whose isomorphism classes\nform a basis of $\\text{Pic}(Y)$. For\n$v=(v_{1},\\dots, v_{\\rho})\\in \\mathbb{Z} ^{\\rho}$ we let\n$\\mathcal{L} ^{v}= \\mathcal{L} _1^{\\otimes v_1}\\otimes \\cdots \\otimes \\mathcal{L} _{\\rho}^{\\otimes v_{\\rho}}$\nand consider the quasi-coherent sheaf\n$$\\bigoplus_{v \\in \\mathbb{Z} ^{\\rho}}\\mathcal{L} ^{v}.$$ This sheaf has\na natural structure of a reduced $\\mathcal{O}_Y$-algebra that is locally\nof finite type over $\\mathcal{O}_Y$ (the component associated to the\nzero element of $\\text{Pic}(Y)$). This means that for sufficiently small\naffine open subsets $U$ of $Y$, the space\n$\\bigoplus_{v \\in \\mathbb{Z} ^{\\rho}}\\mathcal{L} ^{v}(U)$ is a finitely\ngenerated $\\mathcal{O}_Y(U)$-algebra. The universal torsor of $Y$ is\nobtained by gluing the affine schemes\n$\\text{Spec}\\left(\\bigoplus_{v \\in \\mathbb{Z} ^{\\rho}}\\mathcal{L} ^{v}(U)\\right)$.\n\n**Definition 71**. The **universal torsor** of $Y$ is\n$$\\text{UT} _Y= \\textbf{Spec}\\left(\\bigoplus_{v \\in \\mathbb{Z} ^{\\rho}}\\mathcal{L} ^{v} \\right).$$\nThe **Cox ring** of $Y$ is\n$$\\text{Cox}(Y)= H^0 (\\text{UT} _Y,\\mathcal{O}_{\\text{UT} _Y}).$$\n\nUniversal torsors can be used to generalize the construction of a\nprojective variety from its affine cone as follows. Observe that the\ninclusion of $\\mathcal{O}_Y$ as the degree $0$ part of\n$\\bigoplus_{v \\in \\mathbb{Z} ^{\\rho}}\\mathcal{L} ^{v}$ gives rise to an\naffine regular map $\\text{UT} _Y\\to Y$. Since $\\text{Cox}(Y)$ is\n$\\text{Pic}(Y)$-graded there is an action of\n$T_{\\text{Pic}(Y)^*}= \\text{Spec}(\\mathbb{C} [\\text{Pic}(Y)])$ on\n$\\text{UT} _Y$. This action is free and the map $\\text{UT} _Y\\to Y$ is\nthe associated quotient map (see ).\n\n**Remark 72**. The notion of a Cox ring associated to a projective\nvariety (satisfying some technical assumptions) was first introduced in\n. This notion was generalized in for any divisorial variety with only\nconstant globally invertible functions, in particular, for any\nquasi-projective variety (over very general ground fields). However, in\nthe term *Cox ring* was not used. The importance of considering\nuniversal torsors and Cox rings in the context of cluster varieties was\npointed out in (see also ) and satisfactorily pursued in representation\ntheoretic contexts where Cox rings arise naturally, see for example .\n\n**Remark 73**. For simplicity we are assuming that $\\text{Pic}(Y)$ is\nfree. In case it has torsion we can still construct a universal torsor\nwhich might not be unique as it depends on the choice of a *shifting\nfamily* as in (see for a related discussion). Generalizations of the\nresults of this section to the torsion case shall be treated elsewhere.\n\n**Remark 74**. If $Y$ is smooth we can construct the Cox ring of $Y$ and\nthe universal torsor (still assuming that $\\text{Pic(Y)}$ is torsion\nfree) in an equivalent way. The Cox ring can be defined as\n$\\text{Cox}(Y)=\\bigoplus_{v\\in \\mathbb{Z} ^{\\rho}} H^0 (Y, \\mathcal{L} ^v)$.\nIf $\\text{Cox}(Y)$ is finitely generated over $\\mathcal{O}_Y$-algebra\nthen the universal torsor $\\text{UT} _Y$ is obtained from\n$\\text{Spec}(\\text{Cox}(Y))$ by removing the unstable locus of the\nnatural $T_{\\text{Pic(Y)}^*}$-action on $\\text{Spec}(\\text{Cox}(Y))$.\n\nFrom now on we assume $V\\subset \\text{UT} _Y$ is a partial minimal model\nwhere $(V,\\Phi)$ is a scheme with a cluster structure of type\n$\\mathcal{A}$. In most of the result of this section we assume that\n$V\\subset \\text{UT} _Y$ has enough theta functions. Under certain\nconditions that we discuss next, it is possible to show that $Y$ is a\nminimal model for a scheme with a cluster structure given by a quotient\nof $\\mathcal{A}$ and construct Newton–Okounkov bodies for elements of\n$\\text{Pic}(Y)$. The key point is to relate the action of\n$T_{\\text{Pic}(Y)^*}$ on $\\text{UT} _Y$ with the torus actions on\n$\\mathcal{A}$ arising from cluster ensemble maps.\n\n**Lemma 75**. Let $p:\\mathcal{A} \\to \\mathcal{X}$ be a cluster ensemble\nmap and $H\\subset K^{\\circ}$ be a saturated sublattice. Consider the\nquotient $\\mathcal{A} /T_H$ and the fibration\n$w_H:\\mathcal{A} ^\\vee \\to T_{H^*}$ (see §). Then the set\n$$\\left\\{ \\vartheta ^{\\mathcal{A} }_{\\bf m} \\in \\mathop{\\mathrm{mid}}(\\mathcal{A} )  \\mid {\\bf m} \\in \\left(\\mathrm{Trop} _{\\mathbb{Z} }(w_H)\\right)^{-1}(q) \\cap \\Theta(\\mathcal{A} )\\right\\}$$\nconsists precisely of the polynomial theta functions on $\\mathcal{A}$\nwhose $T_H$-weight is $q$. Moreover, for every $q \\in H^*$ the set\n$\\left(\\mathrm{Trop} _{\\mathbb{R} }(w_H)\\right)^{-1}(q)\\subset \\mathrm{Trop} _{\\mathbb{R} }(\\mathcal{A} ^{\\vee})$\nis positive.\n\n*Proof.* The first claim follows from . So we only need to show that\n$\\left(\\mathrm{Trop} _{\\mathbb{R} }(w_H)\\right)^{-1}(q)$ is positive. In\norder to show this it is convenient to work with a condition equivalent\nto positivity called broken line convexity, see §. We work in the\nlattice identification\n$\\mathrm{Trop} _{\\mathbb{R} }(\\mathcal{A} ^{\\vee}_{\\textbf{s}^\\vee})$ of\n$\\mathrm{Trop} _{\\mathbb{R} }(\\mathcal{A} ^{\\vee})$. We first argue that\nthe set\n$\\left(\\mathrm{Trop} _{\\mathbb{R} }(w_H)\\right)^{-1}(0)_{\\textbf{s}^\\vee}$\nis positive. First notice that any linear segment $L$ of a broken line\nsegment contained in\n$\\left(\\mathrm{Trop} _{\\mathbb{R} }(w_H)\\right)^{-1}(0)_{\\textbf{s}^\\vee}$\nhas itself tangent direction in\n$\\left(\\mathrm{Trop} _{\\mathbb{Z} }(w_H)\\right)^{-1}(0)_{\\textbf{s}^\\vee}$.\nLet\n$m\\in \\left(\\mathrm{Trop} _{\\mathbb{Z} }(w_H)\\right)^{-1}(0)_{\\textbf{s}^\\vee}$\nbe the tangent direction of $L$. The tangent direction of the following\nlinear segment is of form $m+cp^*(n)$ for some $n\\in N^+_{\\textbf{s}}$\nand $c\\in  \\mathbb{Z} _{\\geq 0}$. For any $h\\in H^\\circ$ we have\n$$\\langle m+cp^*(n),h\\rangle =\\langle m,h\\rangle + c\\{n,h\\}=0,$$ as\n$H^\\circ\\subset K^\\circ$. So the next tangent direction also belongs to\n$\\left(\\mathrm{Trop} _{\\mathbb{Z} }(w_H)\\right)^{-1}(0)_{\\textbf{s}^\\vee}$.\nWe conclude that the set\n$\\left(\\mathrm{Trop} _{\\mathbb{R} }(w_H)\\right)^{-1}(0)_{\\textbf{s}^\\vee}$\nis broken line convex and by the main result of (see Theorem below) the\nset\n$\\left(\\mathrm{Trop} _{\\mathbb{Z} }(w_H)\\right)^{-1}(0)_{\\textbf{s}^\\vee}$\nis positive. This already implies that for any\n$x\\in \\left(\\mathrm{Trop} _{\\mathbb{R} }(w_H)\\right)^{-1}(q)_{\\textbf{s}^\\vee}$\nthe set\n$x+ \\left(\\mathrm{Trop} _{\\mathbb{R} }(w_H)\\right)^{-1}(0)_{\\textbf{s}^\\vee}$\nremains positive. Indeed, let\n$y, z \\in x+ \\left(\\mathrm{Trop} _{\\mathbb{R} }(w_H)\\right)^{-1}(0)_{\\textbf{s}^\\vee}$.\nThen\n$y- z \\in \\left(\\mathrm{Trop} _{\\mathbb{R} }(w_H)\\right)^{-1}(0)_{\\textbf{s}^\\vee}$.\nIn other words, any line segment within the set\n$x+\\left(\\mathrm{Trop} _{\\mathbb{R} }(w_H)\\right)^{-1}(0)_{\\textbf{s}^\\vee}$\nhas tangent direction in\n$\\left(\\mathrm{Trop} _{\\mathbb{R} }(w_H)\\right)^{-1}(0)_{\\textbf{s}^\\vee}$.\nTherefore, after bending it will remain in the set\n$x+\\left(\\mathrm{Trop} _{\\mathbb{R} }(w_H)\\right)^{-1}(0)_{\\textbf{s}^\\vee}$.\nFinally, observe that\n$x+\\left(\\mathrm{Trop} _{\\mathbb{R} }(w_H)\\right)^{-1}(0)_{\\textbf{s}^\\vee}=\\left(\\mathrm{Trop} _{\\mathbb{R} }(w_H)\\right)^{-1}(q)_{\\textbf{s}^\\vee}$. ◻\n\nHaving in mind Proposition  and the action of the $T_{\\text{Pic}(Y)^*}$\non $\\text{UT} _Y$ we introduce the following notion.\n\n**Definition 76**.\n\nThe pair $(p,H)$ has the **Picard property** with respect to\n$V\\subset \\text{UT} _Y$ if\n\n-   $H$ and $\\text{Pic}(Y)^*$ have the same rank, and\n\n-   the action of $T_{H}$ on $\\mathcal{A}$ coincides with the action of\n    $T_{\\text{Pic}(Y)^*}$ on $\\text{UT} _Y$ restricted to the image of\n    $\\Phi:\\mathcal{A} \\dashrightarrow V$.\n\nRecall the definitions of the superpotential and its associated cone of\ntropical points from and in §. The following result adapts the content\nof Proposition to this framework.\n\n**Lemma 77**. Suppose that $V \\subset \\text{UT} _Y$ is a partial minimal\nmodel with enough theta functions and that $(p,H)$ has the Picard\nproperty with respect to this model. Then for every class\n$[\\mathcal{L} ]\\in \\text{Pic}(Y)\\cong H^*$ we have that the theta\nfunctions parametrized by the integral points of the set\n$\\left(\\mathrm{Trop} _{\\mathbb{R} }(w_H)\\right)^{-1}\\left([\\mathcal{L} ]\\right)\\cap \\Xi_{\\text{UT} _Y}$\nis a basis for $H^0(Y, \\mathcal{L} )$. In particular,\n$\\mathop{\\mathrm{Cox}}(Y)$ has a basis of theta functions which are\n$T_{\\text{Pic}(Y)^*}$-eigenfunctions.\n\nWe consider the section ring\n$R(\\mathcal{L} )=\\bigoplus_{k\\geq 0} R_k(\\mathcal{L} )$. The\n$k^{\\mathrm{th}}$ homogeneous component is defined as\n$R_k(\\mathcal{L} )=H^0(Y, \\mathcal{L} ^{\\otimes k})$. The product of\n$R(\\mathcal{L} )$ is given by the tensor product of sections. Fix a seed\n$\\textbf{s}\\in %\n  \\mathrel{\\mathop{\\mathbb{T}_r}\\limits^{\n    \\vbox to0ex{\\kern-0.5\\ex@\n    \\hbox{$\\scriptstyle\\longrightarrow$}\\vss}}}$, a linear dominance\norder $<_{\\textbf{s}}$ on\n$\\mathrm{Trop} _{\\mathbb{Z} }(\\mathcal{A} ^{\\vee}_{\\textbf{s}^\\vee})$\nand consider the valuation\n$\\mathbf{g} ^{\\Phi}_{\\textbf{s}}:\\Bbbk(V)\\setminus \\{ 0 \\} \\to (\\mathrm{Trop} _{\\mathbb{Z} }(\\mathcal{A} ^{\\vee}_{\\textbf{s}^\\vee}), <_{\\textbf{s}}).$\nObserve that $R_k(\\mathcal{L} )\\subset \\mathop{\\mathrm{Cox}}(Y)$ for all\n$k$. Hence we can define the Newton–Okounkov body $$\\begin{split} \n\\Delta_{\\mathbf{g} ^{\\Phi}_{\\textbf{s}}}(\\mathcal{L} ) := \\overline{\\mathop{\\mathrm{conv}}\\Bigg( \\bigcup_{k\\geq 1}\\left\\{ \\frac{1}{k}\\mathbf{g} ^{\\Phi}_{\\textbf{s}} (f) \\mid f\\in R_k(\\mathcal L)\\setminus \\{0\\} \\right\\} \\Bigg) }\\subseteq \\mathrm{Trop} _{\\mathbb{Z} }(\\mathcal{A} ^\\vee_{\\textbf{s}^\\vee})=M^{\\circ}_{\\mathbb{R} }.\n \\end{split}$$\n\n**Theorem 78**.\n\nSuppose that $V \\subset \\text{UT} _Y$ is a partial minimal model with\nenough theta functions and that $(p,H)$ has the Picard property with\nrespect to this model. Then for any line bundle $\\mathcal{L}$ on $Y$\n$$\\Delta_{{\\bf g}^{\\Phi}_{\\textbf{s}}}(\\mathcal{L} )=\\left(\\mathrm{Trop} _{\\mathbb{R} }(w_H)\\right)^{-1}\\left([\\mathcal{L} ]\\right)_{\\textbf{s}^\\vee}\\cap \\Xi_{\\text{UT} _Y, \\textbf{s}^\\vee}.$$\nIn particular, $\\Delta_{{\\bf g}^{\\Phi}_{\\textbf{s}}}(\\mathcal{L} )$ is a\npositive subset of\n$\\mathrm{Trop} _{\\mathbb{R} }(\\mathcal{A} ^{\\vee}_{\\textbf{s}^\\vee})$.\n\n*Proof.* To make notation lighter, throughout this proof we let\n$S=\\left(\\mathrm{Trop} _{\\mathbb{R} }(w_H)\\right)^{-1}\\left([\\mathcal{L} ]\\right)_{\\textbf{s}^\\vee}\\cap \\Xi_{\\text{UT} _Y,\\textbf{s}^\\vee}$\nand denote $\\mathbf{g} ^{\\Phi}_{\\textbf{s}}$ simply by\n$\\mathbf{g} _{\\textbf{s}}$. Observe that\n$[\\mathcal{L} ^{\\otimes k}]=k[\\mathcal{L} ]$ in $\\text{Pic}(Y)$.\nTherefore, by Lemma we have that\n${\\bf g}_{\\textbf{s}}(R_k(\\mathcal{L} ))\\subseteq \\left(\\mathrm{Trop} _{\\mathbb{R} }(w_H)\\right)^{-1}\\left(k[\\mathcal{L} ]\\right)_{\\textbf{s}^\\vee}$\nfor all $k\\geq 1$. In particular,\n$\\dfrac{1}{k}{\\bf g}_{\\textbf{s}}(R_k(\\mathcal{L} ))\\subseteq \\left(\\mathrm{Trop} _{\\mathbb{R} }(w_H)\\right)^{-1}\\left([\\mathcal{L} ]\\right)_{\\textbf{s}^\\vee}$\nfor all $k \\geq 1$. Since\n$\\left(\\mathrm{Trop} _{\\mathbb{R} }(w_H)\\right)^{-1}\\left([\\mathcal{L} ]\\right)_{\\textbf{s}^\\vee}$\nis closed in\n$\\mathrm{Trop} _{\\mathbb{R} }(\\mathcal{A} ^{\\vee}_{\\textbf{s}^\\vee})$\nand convex we have that\n$\\Delta_{{\\bf g}_{\\textbf{s}}}(\\mathcal{L} )\\subseteq \\left(\\mathrm{Trop} _{\\mathbb{R} }(w_H)\\right)^{-1}\\left([\\mathcal{L} ]\\right)_{\\textbf{s}^\\vee}$.\nLet $\\mathbb B_k$ be the theta basis of $R_k(\\mathcal{L} )$, see . Since\nthe theta basis is adapted for ${\\bf g}_{\\textbf{s}}$ we have that\n${\\bf g}_{\\textbf{s}}(R_k(\\mathcal{L} ))={\\bf g}_{\\textbf{s}}(\\mathbb B_k)$.\nSince $\\mathcal{A} \\subseteq \\text{UT} _Y$ has enough theta functions,\nevery theta function $\\vartheta \\in \\mathbb B_k$ is a global function on\n$\\text{UT} _Y$, therefore, we have that\n${\\bf g}_{\\textbf{s}}(\\vartheta ) \\in \\Xi_{\\text{UT} _Y}$. Since\n$\\Xi_{\\text{UT} _Y}$ is closed in\n$\\mathrm{Trop} _{\\mathbb{R} }(\\mathcal{A} ^{\\vee}_{\\textbf{s}^\\vee})$,\nconvex and closed under positive scaling then\n$\\Delta_{{\\bf g}_{\\textbf{s}}}(\\mathcal{L} )\\subseteq \\Xi_{\\text{UT} _Y,\\textbf{s}^\\vee}$.\nHence, $\\Delta_{{\\bf g}_{\\textbf{s}}}(\\mathcal{L} )\\subseteq S$. To see\nthe reverse inclusion we notice that the set of rational points of $S$\ncoincide with the set\n$\\bigcup_{k\\geq 1} \\frac{1}{k}  \\mathbf{g} _{\\textbf{s}}(\\mathbb B_k)=  \\bigcup_{k\\geq 1} \\frac{1}{k}  \\mathbf{g} _{\\textbf{s}}\\left(R_k(\\mathcal{L} )\\right)$.\nSince $S$ can be expressed as the closure of its set of rational points\nwe have that\n$S\\subseteq \\Delta_{\\mathbf{g} _{\\textbf{s}}}(\\mathcal{L} )$. Finally,\nsince\n$\\left(\\mathrm{Trop} _{\\mathbb{R} }(w_H)\\right)^{-1}\\left([\\mathcal{L} ]\\right)_{\\textbf{s}^\\vee}$\nand $\\Xi_{\\text{UT} _Y,\\textbf{s}^\\vee}$ are positive sets then\n$S=\\Delta_{{\\bf g}_{\\textbf{s}}}(\\mathcal{L} )$ is an intersection of\npositive sets. Hence, it is positive. ◻\n\n**Remark 79**. Under the assumptions of we have that $Y$ is a minimal\nmodel with enough theta functions for an open subscheme $V'\\subset Y$\nwith a cluster structure given by a birational map\n$\\Phi':\\mathcal{A} /T_{H}\\dashrightarrow V'$ induced by $\\Phi$. To\nrelate the Newton–Okounkov bodies constructed in this section with those\nconstructed in the former we let $\\mathcal{L}$ be isomorphic to\n$\\mathcal{O}(D')$ for some Weil divisor $D'$ on $Y$ satisfying the\nframework of §. Under the identification\n$\\mathrm{Trop} _{\\mathbb{R} }(\\mathcal{A} ^{\\vee}_{\\textbf{s}^\\vee}) = M^\\circ_\\mathbb{R}$\nwe realize\n$\\mathrm{Trop} _{\\mathbb{R} }((\\mathcal{A} /T_H)^{\\vee}_{\\textbf{s}^\\vee})$\nas the subset of $M^\\circ_\\mathbb{R}$ orthogonal to $H$ (see §). For any\n$\\tau \\in R_1(D')$ we have\n$\\Delta_{\\mathbf{g} ^{\\Phi'}_\\textbf{s}}(D',\\tau)\\subset M_{\\mathbb{R} }^\\circ\\cap \\left(\\mathrm{Trop} _{\\mathbb{R} }(w_H)\\right)^{-1}(0)_{\\textbf{s}^\\vee}$\nand by construction\n$$\\Delta_{\\mathbf{g} ^{\\Phi'}_\\textbf{s}}(D',\\tau) =\\Delta_{\\mathbf{g} ^{\\Phi}_\\textbf{s}}(\\mathcal{L} )- \\mathbf{g} ^{\\Phi}_{\\textbf{s}}(\\tau).$$\n\n**Example 80**.\n\nAn important class of examples is provided by the base affine spaces.\nConsider $G=SL_{n+1}(\\Bbbk)$ and $B\\subset G$ a Borel subgroup with\nunipotent subgroup $U\\subset B$. Then $G/U$ is a universal torsor for\n$G/B$. Moreover, $G/U$ carries a cluster structure induced by the double\nBruhat cell $G^{e,w_0}:=B_-\\cap Bw_0B$, where $B_-\\subset G$ is the\nBorel subgroup opposite to $B$ (i.e. $B\\cap B^-=:T$ is a maximal torus)\nand $w_0$ the longest element in this Weyl group $S_n$ is identified\nwith a matrix representative in $N_G(T)/C_G(T)$ (the normalizer of $T$\nmodulo the centralizer of $T$). The cluster structure on $G^{e,w_0}$ was\nintroduced by Berenstein–Fomin–Zelevinsky in and it follows that (up to\nco-dimension 2) $G^{e,w_0}$ agrees with the corresponding\n$\\mathcal A$-cluster variety. By there is an embedding\n$G^{e,w_0}\\hookrightarrow G/U$ compatible with the cluster structure. In\nparticular, $G/U$ is a partial compactification of the\n$\\mathcal A$-cluster variety $G^{e,w_0}$ obtained by adding the locus\nwhere frozen variables are allowed to vanish. Magee further proved in\nthat the full Fock–Goncharov conjecture holds and a cluster ensemble map\nsatisfying is provided in . Hence, we obtain a ${\\bf g}$-vector\nvaluation ${\\bf g}_{\\textbf{s}}$ on $H^0(G/U,\\mathcal O_{G/U})$ for\nevery choice of seed $\\textbf{s}$.\n\nIn particular, applies: recall that the Picard group of $G/B$ is\nisomorphic to the lattice spanned by the fundamental weights\n$\\omega_1,\\dots,\\omega_{n}$. Let $\\Lambda$ denote the dominant weights,\n*i.e.* its elements are $\\lambda=a_1\\omega_1+\\dots+a_n\\omega_n$ with\n$a_i\\in \\mathbb Z_{\\ge 0}$ and let $\\mathcal L_\\lambda\\to G/B$ be the\nassociated line bundle. The ring of regular functions on the\nquasi-affine variety $G/U$ coincides with the Cox ring of the flag\nvariety:\n$$H^0(G/U,\\mathcal O_{G/U})\\cong \\bigoplus_{\\lambda \\in \\Lambda} H^0 (G/B,\\mathcal L_\\lambda).$$\nHence, we may restrict the ${\\bf g}$-vector valuations\n${\\bf g}_{\\textbf{s}}$ for all seeds $\\textbf{s}$ to the section ring of\nany line bundle on $G/B$. The resulting Newton–Okounkov polytopes\ncoincide with slices of the tropicalization of the superpotential\ncorresponding to the compactification. It has been shown in that for\ncertain choices of seeds these polytopes are unimodularly equivalent to\nLittelmann’s string polytopes (see ).\n\n**Example 81**. Grassmannians also form a distinguished class of\nexamples fitting this framework. We treat this class separately in §.\n\n## The intrinsic Newton–Okounkov body\n\nIn the situation of § or §, we can choose two seeds\n$\\textbf{s}, \\textbf{s}'%\n  \\mathrel{\\mathop{\\mathbb{T}_r}\\limits^{\n    \\vbox to0ex{\\kern-0.5\\ex@\n    \\hbox{$\\scriptstyle\\longrightarrow$}\\vss}}}$ to obtain two\nNewton–Okounkov bodies, say $\\Delta_{\\nu_\\textbf{s}}$ and\n$\\Delta_{\\nu_\\textbf{s}'}$ (these are associated to a line bundle\n$\\mathcal{L}$ in case we are in a framework as in § or to a divisor $D'$\nand a section $\\tau$ in case our framework is as in §). In the same\nspirit as in (see also and ), in this section we show that if one of\n$\\Delta_{\\nu_{\\textbf{s}}}$ or $\\Delta_{\\nu_{\\textbf{s}'}}$\n(equivalently both) is a positive set then these Newton–Okounkov bodies\nare related to each other by a distinguished piecewise linear\ntransformation and, moreover, any such Newton–Okounkov body can be\nintrinsically described as a *broken line convex hull* (see Theorems and\nbelow). In order to obtain the last assertion we rely on . Along the way\nwe introduce a theta function analog of the Newton polytope associated\nto a regular function on a torus.\n\nWe start by considering Newton–Okounkov bodies associated to Weil\ndivisors as in §. Let $\\mathcal{V}$ be a scheme of the form\n$\\mathcal{A}$, $\\mathcal{X}$, $\\mathcal{A} /T_{H}$ or\n$\\mathcal{X} _{\\bf 1}$ and $(V, \\Phi)$ a scheme with a cluster structure\nof type $\\mathcal{V}$. Denote by\n$\\mathbb{B}_{\\vartheta }(\\mathcal{V} )=\\{\\vartheta ^{\\mathcal{V} }_{\\bf v}\\mid {\\bf v}\\in \\Theta(\\mathcal{V} )\\}$\nthe theta basis of $\\mathop{\\mathrm{mid}}(\\mathcal{V} )$. We begin by\nobserving that a cluster valuation $\\nu_{\\textbf{s}}$ on\n$\\mathop{\\mathrm{mid}}(\\mathcal{V} )$ can be thought of as an extension\nof the composition of the seed-independent map $$\\begin{aligned}\n    \\label{eq:nu_seed_free}\n    \\nu: \\mathbb{B}_{\\vartheta }(\\mathcal{V} )  &\\to&   \\mathrm{Trop} _{\\mathbb{Z} }(\\mathcal{V} ^\\vee)\\\\\n    \\nonumber\n\\vartheta ^{\\mathcal{V} }_{\\bf v} &\\mapsto &{\\bf v},\n\\end{aligned}$$ with the identification\n$\\mathfrak{r}_{\\textbf{s}^\\vee}:\\mathrm{Trop} _{\\mathbb{Z} }(\\mathcal{V} ^\\vee) \\to  \\mathrm{Trop} _{\\mathbb{Z} }(\\mathcal{V} ^\\vee_{\\textbf{s}^\\vee})$.\nIf $\\mathbb B_{\\vartheta }(V)$ denotes the set of polynomial theta\nfunctions on $V$ then we can define\n$\\nu^{\\Phi} : \\mathbb B_{\\vartheta }(V) \\to \\mathrm{Trop} _{\\mathbb{Z} }(\\mathcal{V} ^\\vee)$\nanalogously. Moreover, even though\n$\\mathrm{Trop} _{\\mathbb{Z} }(\\mathcal{V} ^{\\vee})$ may not have a\nlinear structure, if\n$\\Theta (\\mathcal{V} )= \\mathrm{Trop} _{\\mathbb{Z} }(\\mathcal{V} ^{\\vee})$\nand $L\\subseteq \\mathrm{Trop} _{\\mathbb{Z} }(\\mathcal{V} ^{\\vee})$ is a\nlinear subset acting linearly on\n$\\mathrm{Trop} _{\\mathbb{Z} }(\\mathcal{V} ^{\\vee})$ (see Definition )\nthen for every $y\\in L$ we have a well defined “subtraction\" function\n$$\\begin{split}  (\\ \\cdot \\ )-y: \\mathrm{Trop} _{\\mathbb{Z} }(\\mathcal{V} ^{\\vee})  &\\to   \\mathrm{Trop} _{\\mathbb{Z} }(\\mathcal{V} ^{\\vee})\\\\\nx &\\mapsto x-y,  \\end{split}$$ where $-y$ is the unique point of\n$\\mathrm{Trop} _{\\mathbb{Z} }(\\mathcal{V} ^{\\vee})$ such that\n$\\vartheta _y\\vartheta _{-y}=1$ and $x-y$ is the unique point of\n$\\mathrm{Trop} _{\\mathbb{Z} }(\\mathcal{V} ^{\\vee})$ such that\n$\\vartheta _{x}\\vartheta _{-y}=\\vartheta _{x-y}$.\n\nWe now define our notion of convexity. Recall from § that we might think\nof supports of broken lines as seed independent objects. In light of\nthis we consider the following.\n\n**Definition 82**. A closed subset $S$ of\n$\\mathrm{Trop} _{\\mathbb{R} }(\\mathcal{V} )$ is **broken line convex**\nif for every pair of rational points $s_1, s_2$ in $S(\\mathbb{Q} )$,\nevery segment of a broken line with endpoints $s_1$ and $s_2$ is\nentirely contained in $S$.\n\n**Remark 83**. The broken lines considered in Definition include those\nthat are *non-generic*. Namely, broken lines that are obtained as limits\nof the generic broken lines introduced in . See for details.\n\nThe main result of asserts that positivity of a set is equivalent to its\nbroken line convexity:\n\n**Theorem 84**.\n\nLet $\\mathcal{V}$ be a variety of the form $\\mathcal{A}$, $\\mathcal{X}$,\n$\\mathcal{A} /T_{H}$ or $\\mathcal{X} _{\\bf 1}$. Then a closed subset $S$\nof $\\mathrm{Trop} _{\\mathbb{R} }(\\mathcal{V} )$ is is broken line convex\nif and only if it is positive.\n\nMorally, this means that broken line convexity in\n$\\mathrm{Trop} _{\\mathbb{R} }(\\mathcal{V} ^\\vee)$ play the same role in\ndescribing partial minimal models of $\\mathcal{V}$ that usual convexity\nin $M_\\mathbb{R}$ plays in describing normal toric varieties\n$T_N \\subset X$. One appealing feature of the broken line convexity\nnotion is that it makes no reference to any auxiliary data– given\n$\\mathcal{V}$, we can talk about broken line convexity in\n$\\mathrm{Trop} _{\\mathbb{R} }(\\mathcal{V} ^{\\vee})$. In contrast, the\nNewton–Okounkov bodies we discussed in § and § are convex bodies whose\nconstruction depends upon a choice of seed $\\textbf{s}$. More generally,\na usual Newton–Okounkov body depends not only on the geometric data of a\nprojective variety together with a divisor but also on the auxiliary\ndata of a choice of valuation. Broken line convexity makes no reference\nto any such auxiliary data and will lead us to an intrinsic version of a\nNewton–Okounkov body.\n\n**Definition 85**. Let\n$S \\subset\\mathrm{Trop} _{\\mathbb{R} }(\\mathcal{V} ^{\\vee})$ be a set.\nThe **broken line convex hull of $S$**, denoted by\n$\\mathop{\\mathrm{conv_{BL}}}(S)$, is the intersection of all broken line\nconvex sets containing $S$.\n\n**Remark 86**. We can also define broken line convexity and broken line\nconvex hulls inside\n$\\mathrm{Trop} _{\\mathbb{R} }(\\mathcal{V} ^{\\vee}_{\\textbf{s}^\\vee})$ in\nexactly the same way they are defined in Definitions and . In\nparticular, we have that\n$S\\subset \\mathrm{Trop} _{\\mathbb{R} }(\\mathcal{V} ^{\\vee})$ is broken\nline convex if and only if\n$\\mathfrak{r}_{\\textbf{s}^\\vee}(S)\\subset \\mathrm{Trop} _{\\mathbb{R} }(\\mathcal{V} ^\\vee_{\\textbf{s}})$\nis broken line convex.\n\nUsing this convexity notion, we describe a set analogous to the Newton\npolytope of a function on a torus.\n\n**Definition 87**. Given a regular function\n${f= \\sum_{{\\bf v} \\in \\mathrm{Trop} _{\\mathbb{Z} }(\\mathcal{V} ^{\\vee})}} a_{\\bf v} \\vartheta ^{V}_{\\bf v}$\non $V$, we define the **$\\vartheta$-function analogue of the Newton\npolytope of $f$** to be\n$$\\begin{split}  \\mathop{\\mathrm{Newt_{\\vartheta }}}(f) := \\mathop{\\mathrm{conv_{BL}}}\\left\\{ {\\bf v} \\in \\mathrm{Trop} _{\\mathbb{Z} }(\\mathcal{V} ^{\\vee}) \\mid a_{\\bf v} \\neq 0 \\right\\}.  \\end{split}$$\n\nThis leads to an intrinsic version of the Newton–Okounkov bodies we have\nconstructed. So consider a partial minimal model $V \\subset Y$ and let\n$D'$ be a divisor on $Y$ supported on the boundary of $V\\subset Y$.\n\n**Definition 88**. Assume that $R(D')$ has a graded theta basis (see\nDefinition ). Then the associated **intrinsic Newton–Okounkov body** is\n$$\\begin{split} \n\\Delta_{\\mathrm{BL}}(D'):= \\mathop{\\mathrm{conv_{BL}}}\\Bigg( \\bigcup_{k\\geq 1} \\Bigg(\\bigcup_{f \\in R_k(D')} \\frac{1}{k} \\mathop{\\mathrm{Newt_{\\vartheta }}}(f) \\Bigg)  \\Bigg)\\subseteq \\mathrm{Trop} _{\\mathbb{R} }(\\mathcal{V} ^\\vee).\n \\end{split}$$\n\nIn order to describe how the different realizations of intrinsic\nNewton–Okounkov bodies are related we record the tropicalization of the\ngluing map\n$\\mu^{\\mathcal{V} ^\\vee}_k:\\mathcal{V} ^{\\vee}_\\textbf{s}\\dashrightarrow \\mathcal{V} ^{\\vee}_{\\textbf{s}'}$\nin terms of the fixed data $\\Gamma$ and inital seed\n$\\textbf{s}_0=(e_i)_{i\\in I}$ defining $\\mathcal{V}$.\n$$\\mathrm{Trop} _{\\mathbb{R} }\\left(\\mu^{\\mathcal{A} ^\\vee}_{k}\\right)(m)=\\begin{cases} m + \\langle d_ke_k, m \\rangle  v_k & \\text{if } \\langle e_k, m \\rangle \\geq 0,\\\\\nm & \\text{if } \\langle e_k, m \\rangle \\leq 0,\n\\end{cases}$$ for $m \\in M^{\\circ}$.\n$$\\mathrm{Trop} _{\\mathbb{R} }\\left(\\mu^{\\mathcal{X} ^\\vee}_{k}\\right)(n)=\\begin{cases} n + \\{n,d_ke_k \\} e_k & \\text{if } \\{  n,e_K \\}\\geq 0,\\\\\nn & \\text{if } \\{ n,e_K\\} \\leq 0,\n\\end{cases}$$ for $n \\in N$.\n$$\\mathrm{Trop} _{\\mathbb{R} }\\left(\\mu^{(\\mathcal{X} _{\\bf 1})^\\vee}_{k}\\right)(n+H)=\\begin{cases} n + \\{n,d_ke_k \\}e_k + H & \\text{if } \\{ n, e_k \\}\\geq 0,\\\\\nn + H& \\text{if } \\{ n, e_k \\} \\leq 0,\n\\end{cases}$$ for $n + H \\in N/H$.\n$$\\mathrm{Trop} _{\\mathbb{R} }\\left(\\mu^{(\\mathcal{A} /T_H)^\\vee}_{k}\\right) =  \\mathrm{Trop} _{\\mathbb{R} }\\left(\\mu^{\\mathcal{A} ^\\vee}_{k}\\right) \\mid_{H^\\perp}.$$\n\n**Theorem 89**.\n\nLet $(V,\\Phi)$ be a scheme with a cluster structure of type\n$\\mathcal{V}$ and let $V \\subset Y$ be a partial minimal model. Assume\nthat the full Fock–Goncharov conjecture holds for $\\mathcal{V}$ and that\nthere exists a theta function $\\tau \\in R_1(D')$ such that\n$\\nu^{\\Phi}_{\\textbf{s}}(\\tau)$ lies in a linear subset of\n$\\mathrm{Trop} _{\\mathbb{Z} }(\\mathcal{V} ^\\vee)$. If\n$\\Delta_{\\nu^{\\Phi}_{\\textbf{s}}}(D',\\tau)$ is positive then for every\nseed $\\textbf{s}\\in %\n  \\mathrel{\\mathop{\\mathbb{T}_r}\\limits^{\n    \\vbox to0ex{\\kern-0.5\\ex@\n    \\hbox{$\\scriptstyle\\longrightarrow$}\\vss}}}$ we have that\n$\\mathfrak{r}_{\\textbf{s}^\\vee}(\\Delta_{\\mathrm{BL}}(D')-\\nu^{\\Phi}(\\tau))= \\Delta_{\\nu^{\\Phi}_{\\textbf{s}}}(D',\\tau)$.\nIn particular, for any other seed $\\textbf{s}'\\in %\n  \\mathrel{\\mathop{\\mathbb{T}_r}\\limits^{\n    \\vbox to0ex{\\kern-0.5\\ex@\n    \\hbox{$\\scriptstyle\\longrightarrow$}\\vss}}}$ we have that\n$$\\Delta_{\\nu_{\\textbf{s}'}}(D', \\tau )= \\mathrm{Trop} _{\\mathbb{R} }\\left(\\mu^{\\mathcal{V} ^\\vee}_{\\textbf{s},\\textbf{s}'}\\right)\\left(\\Delta_{\\nu_{\\textbf{s}}}(D', \\tau)\\right).$$\n\n*Proof.* It is enough to treat the case $V= \\mathcal{V}$. We consider\nthe broken line convex hull of\n$$S=\\bigcup_{k\\geq 1}\\left\\{\\dfrac{\\nu_{\\textbf{s}}(f)}{k}-\\nu_{\\textbf{s}}(\\tau)\\mid f\\in R_k(D') \\setminus\\{0\\}\\right\\}$$\nin\n$\\mathrm{Trop} _{\\mathbb{R} }(\\mathcal{V} ^{\\vee}_{\\textbf{s}^\\vee})$.\nSince all line segments of\n$\\mathrm{Trop} _{\\mathbb{R} }(\\mathcal{V} ^{\\vee}_{\\textbf{s}^\\vee})$\ncan be thought of as a segment of a broken line and\n$\\Delta_{\\nu_{\\textbf{s}}}(D', \\tau)$ is closed we have that\n$\\Delta_{\\nu_{\\textbf{s}}}(D', \\tau)\\subseteq \\mathop{\\mathrm{conv_{BL}}}(S)$.\nBy $\\Delta_{\\nu_{\\textbf{s}}}(D', \\tau)$ is broken line convex. Since\n$S\\subset \\Delta_{\\nu_{\\textbf{s}}}(D', \\tau)$ we have the reverse\ninclusion. The last statement follows from the fact that broken line\nconvex sets are preserved by\n$\\mathrm{Trop} _{\\mathbb{R} }(\\mu^{\\mathcal{V} ^{\\vee}}_k)$. ◻\n\nThere is an analogous result for line bundles fitting the framework of\n§.\n\n**Definition 90**. Let $Y$ be a projective variety such that\n$\\text{Pic}(Y)$ is free of finite rank. Assume $(V, \\Phi)$ is a scheme\nwith a cluster structure of type $\\mathcal{A}$ and that\n$V \\subset \\text{UT} _Y$ is a partial minimal model with enough theta\nfunctions. Let $(p, H)$ have the Picard property (see ). The **intrinsic\nNewton–Okounkov body associated to a class\n$[ \\mathcal{L} ]\\in \\text{Pic}(Y)\\cong H^*$** is $$\\begin{split} \n\\Delta_{\\mathrm{BL}}(\\mathcal{L} ):= \\mathop{\\mathrm{conv_{BL}}}\\Bigg( \\bigcup_{k\\geq 1} \\Bigg(\\bigcup_{f \\in R_k(\\mathcal{L} )} \\frac{1}{k} \\mathop{\\mathrm{Newt_{\\vartheta }}}(f) \\Bigg)  \\Bigg)\\subseteq \\mathrm{Trop} _{\\mathbb{R} }(\\mathcal{A} ^{\\vee}).\n \\end{split}$$\n\nIn this case we have the following theorem whose proof is completely\nanalogous to the proof of . Moreover, it uses the fact that\n$\\nu_{\\textbf{s}}(\\mathcal{L} )$ is a positive set, as shown in .\n\n**Theorem 91**.\n\nKeep the assumptions of Definition . For every seed $\\textbf{s}\\in %\n  \\mathrel{\\mathop{\\mathbb{T}_r}\\limits^{\n    \\vbox to0ex{\\kern-0.5\\ex@\n    \\hbox{$\\scriptstyle\\longrightarrow$}\\vss}}}$ we have that\n$\\Delta_{\\nu^{\\Phi}_{\\textbf{s}}}(\\mathcal{L} )=\\mathfrak{r}_{\\textbf{s}^\\vee}(\\Delta_{\\mathrm{BL}}(\\mathcal{L} ))$.\nIn particular, for every $\\textbf{s}' \\in %\n  \\mathrel{\\mathop{\\mathbb{T}_r}\\limits^{\n    \\vbox to0ex{\\kern-0.5\\ex@\n    \\hbox{$\\scriptstyle\\longrightarrow$}\\vss}}}$ we have that\n$$\\Delta_{\\nu^\\Phi_{\\textbf{s}'}}(\\mathcal{L} )= \\left(\\mu^{\\mathcal{V} ^\\vee}_{\\textbf{s}^\\vee, \\textbf{s}'^\\vee}\\right)^T(\\Delta_{\\nu^\\Phi_{\\textbf{s}}}(\\mathcal{L} )).$$\n\n*Proof.* We showed in that $\\Delta_{\\nu_{\\textbf{s}}}(\\mathcal{L} )$ is\na positive set. The proof of this result is completely analogous to the\nproof of . ◻\n\nIn either situation (divisors or line bundles) we are of course free to\ncompute the intrinsic Newton–Okounkov body as a usual Newton–Okounkov\nbody in any vector space realization of\n$\\mathrm{Trop} _{\\mathbb{R} }(\\mathcal{V} ^{\\vee})$. However, the\nintrinsic definition has certain advantages as we now explain. For\nsimplicity, from now on we concentrate on line bundles as in ; the\nreader can make the appropriate changes for the case of divisors as in .\nIt is often the case that\n$\\Delta_{\\mathrm{BL}}(\\mathcal{L} ) = \\mathop{\\mathrm{conv_{BL}}}\\Big( \\bigcup_{k=1}^\\ell \\frac{1}{k} \\nu^{\\Phi}\\left(R_k(\\mathcal{L} )\\right) \\Big)$\nfor some finite $\\ell$, meaning in these cases the infinite union\nreduces to finite union. Consider such an instance and let\n$\\ell_{\\textbf{s}}$ be the smallest integer such that\n$\\Delta_{\\nu^{\\Phi}_{\\textbf{s}}}(\\mathcal{L} )=\\mathop{\\mathrm{conv}}\\Big( \\bigcup_{k=1}^{\\ell_{\\textbf{s}}} \\frac{1}{k} \\nu^{\\Phi}_{\\textbf{s}}\\left(R_k(\\mathcal{L} )\\right) \\Big)$.\nThen the corresponding $\\ell$ for the intrinsic Newton–Okounkov body is\nat most $\\min_{\\textbf{s}}\\left\\{\\ell_{\\textbf{s}}\\right\\}$. Moreover,\nwe can give conditions indicating when $\\ell$ has been attained. We will\nstart with a condition that, after adopting a slightly different\nperspective on theta functions, becomes tautological.[^7] We will then\nadapt this condition to give a sufficient criterion that is more likely\nto be known for a given minimal model (and a known line bundle or Weil\ndivisor).\n\n**Proposition 92**.\n\nLet $\\mathcal{L}$ be as in . Suppose there exists a positive integer\n$\\ell$ such that for all $h>\\ell$, each theta function $\\vartheta ^V_r$\nin $R_h(\\mathcal{L} )$ appears as a summand (with non-zero coefficient)\nof some product $\\vartheta ^V_p \\vartheta ^V_q$, where\n$\\vartheta ^V_p \\in R_i(\\mathcal{L} )$ and\n$\\vartheta ^V_q \\in R_j(\\mathcal{L} )$ for some positive integers $i$\nand $j$ with $i+j =h$. Then $$\\begin{split} \n\\Delta_{\\mathrm{BL}}(\\mathcal{L} ) = \\mathop{\\mathrm{conv_{BL}}}\\Bigg( \\bigcup_{k=1}^{\\ell} \\Bigg(\\bigcup_{f \\in R_k(\\mathcal{L} )} \\frac{1}{k} \\mathop{\\mathrm{Newt_{\\vartheta }}}(f) \\Bigg)  \\Bigg) .\n \\end{split}$$\n\n*Proof.* This is an immediate consequence of results in . We adopt the\nterminology and conventions of *loc. cit.* for this proof. In\nparticular, we allow non-generic broken lines (see Remark ).\n\nSince the structure constant $\\alpha(p,q,r)$ is non-zero, there exists a\npair of broken lines $\\left(\\gamma_1,\\gamma_2\\right)$ with\n$I(\\gamma_1) = p$, $I(\\gamma_2) = q$, $\\gamma_1(0)=\\gamma_2(0) = r$, and\n$F(\\gamma_1)+ F(\\gamma_2) = r$. Then the construction of yields a broken\nline segment from $\\frac{p}{i}$ to $\\frac{q}{j}$ passing through\n$\\frac{r}{h}$. As a consequence, we have\n$$\\begin{split} \\frac{r}{h} \\in \\mathop{\\mathrm{conv_{BL}}}\\Bigg( \\bigcup_{k=1}^{\\max(i,j)} \\Bigg(\\bigcup_{f \\in R_k(\\mathcal{L} )} \\frac{1}{k} \\mathop{\\mathrm{Newt_{\\vartheta }}}(f) \\Bigg)  \\Bigg) .  \\end{split}$$\nBy hypothesis, $R_k(\\mathcal{L} )$ has a basis of theta functions for\nall $k$, so $$\\begin{split} \n\\mathop{\\mathrm{conv_{BL}}}\\Bigg(\\bigcup_{f \\in R_h(\\mathcal{L} )} \\frac{1}{h} \\mathop{\\mathrm{Newt_{\\vartheta }}}(f) \\Bigg) =  \\mathop{\\mathrm{conv_{BL}}}\\left( \\frac{r}{h} \\mid \\vartheta ^V_r \\in R_h(\\mathcal{L} )\\right) .\n \\end{split}$$ We have just seen that each such $\\frac{r}{h}$ is\ncontained in\n$$\\begin{split} \\mathop{\\mathrm{conv_{BL}}}\\Bigg( \\bigcup_{k=1}^{h-1} \\Bigg(\\bigcup_{f \\in R_h(\\mathcal{L} )} \\frac{1}{h} \\mathop{\\mathrm{Newt_{\\vartheta }}}(f) \\Bigg)  \\Bigg),  \\end{split}$$\nso $$\\begin{split} \n\\mathop{\\mathrm{conv_{BL}}}\\Bigg(\\bigcup_{f \\in R_h(\\mathcal{L} )} \\frac{1}{h} \\mathop{\\mathrm{Newt_{\\vartheta }}}(f) \\Bigg) \\subset  \\mathop{\\mathrm{conv_{BL}}}\\Bigg( \\bigcup_{k=1}^{h-1} \\Bigg(\\bigcup_{f \\in R_k(\\mathcal{L} )} \\frac{1}{k} \\mathop{\\mathrm{Newt_{\\vartheta }}}(f) \\Bigg)  \\Bigg).\n \\end{split}$$ As this holds for all $h>\\ell$, we conclude that\n$$\\begin{split} \n\\Delta_{\\mathrm{BL}}(\\mathcal{L} ) = \\mathop{\\mathrm{conv_{BL}}}\\Bigg( \\bigcup_{k=1}^{\\ell} \\Bigg(\\bigcup_{f \\in R_k(\\mathcal{L} )} \\frac{1}{k} \\mathop{\\mathrm{Newt_{\\vartheta }}}(f) \\Bigg)  \\Bigg) .\n \\end{split}$$ ◻\n\n**Remark 93**. In dimension 2, Mandel showed that the assumption in\nimplies that $r=p+q$ in some seed. It is a very interesting problem to\ndetermine if this holds for higher dimensions.\n\nNote that as we have (by assumption) a theta basis for\n$R(\\mathcal{L} )$, the condition of is implied by the following\ncondition:\n\n**Condition 1**.\n\nThere exists a positve integer $\\ell$ such that for all $h>\\ell$, the\nnatural map\n$R_i (\\mathcal{L} ) \\otimes R_j(\\mathcal{L} ) \\to R_h (\\mathcal{L} )$ is\nsurjective for some positive integers $i$ and $j$ with $i+j =h$.\n\n**Remark 94**. The is satisfied in our main class of examples coming\nfrom representation theory: recall the setting of where line bundles\n$\\mathcal L_\\lambda$ of the full flag variety $G/B$ are indexed by\ndominant weights $\\lambda$. By the Borel–Weil–Bott Theorem the graded\npieces $R_i(\\mathcal L_\\lambda)$ of the section rings of these line\nbundles satisfy $$R_i(\\mathcal L_\\lambda)\\cong  V(i\\lambda)^*,$$ where\n$V(i\\lambda)$ is the irreducible $G$-representation of highest weight\n$i\\lambda$ and $i\\ge 0$. By work of Baur the tensor product\n$V(i\\lambda)\\otimes V(j\\lambda)$ contains among its irreducible\ncomponents the unique component of maximal weight, called Cartan\ncomponent, which is $V((i+j)\\lambda)$. Hence,\n$$R_i(\\mathcal L_\\lambda)\\otimes R_j(\\mathcal{L}_\\lambda)\\cong V(i\\lambda)^*\\otimes V(j\\lambda)^*\\twoheadrightarrow V((i+j)\\lambda)^*\\cong R_{i+j}(\\mathcal L_\\lambda).$$\nAlthough in we only treat the case of $SL_{n+1}(\\Bbbk)$ it is worth\nnoticing that the Borel–Weil(–Bott) Theorem holds for semisimple Lie\ngroups and algebraic groups over $\\Bbbk$ and Baur’s result holds for\nirreducible representations of connected, simply-connected complex\nreductive groups. Notice further that these observations also hold for\npartial flag varieties, *i.e.* quotient $G/P$ by parabolic subgroups\n$P\\subset G$ as the cohomology of an equivariant line bundles on $G/P$\nis equal to the cohomology of its pullback along the natural projection\n$G/B\\twoheadrightarrow G/P$. So the cohomology of the line bundle on\n$G/P$ can be calculated using the usual Borel–Weil(–Bott) Theorem for\n$G/B$, by the Leray spectral sequence.\n\n# The case of the Grassmannian\n\nWe now consider in detail the case of the Grassmannians. Throughout this\nsection we work over the complex numbers, fix two positive integers\n$k<n$ and let $$Y= \\mathop{\\mathrm{Gr}}_{n-k}(\\mathbb{C} ^n)$$ be the\ncorresponding Grassmannian. Let $\\widetilde{Y}$ be the affine cone of\n$Y$ in its Plücker embedding\n$Y\\hookrightarrow \\mathbb{P}^{\\binom{n}{n-k}-1}$ and $\\mathcal{L} _e$ be\nthe bundle over $Y$ obtained by pullback of $\\mathcal{O}(1)$ under this\nembedding. By definition, the Plücker coordinates are a basis for\n$H^0(Y,\\mathcal{L} _e)$. It is well known that\n$\\mathop{\\mathrm{Pic}}({Y})$ is free of rank one and $[\\mathcal{L} _e]$\nis a generator. Moreover, the universal torsor of $Y$ is\n$$\\text{UT} _{Y}\\cong \\widetilde{Y}\\setminus\\{ 0\\}$$ and the action of\n$T_{\\mathop{\\mathrm{Pic}}({Y})^*}$ on $\\text{UT} _{Y}$ coincides with\nthe diagonal action of $\\mathbb{C} ^*$. Plücker coordinates are denoted\nby $p_{J}$ where $J\\in \\binom{[n]}{n-k}$ is an $n-k$-element subset of\n$\\{1, \\dots , n\\}$. Working with cyclic intervals, we let\n$D_i=\\{ p_{[i+1, i+k]} =0 \\}$ and consider the divisor\n$$D=\\bigcup_{i=1}^nD_{i} \\subset Y.$$ For any $i$ the line bundle\n$\\mathcal{O}_Y(D_i)$ is isomorphic to $\\mathcal{L} _e$ and the Weil\ndivisor $\\sum_{i=1}^nD_i$ is anticanonical. We let\n$\\widetilde{D}_i \\subset \\text{UT} _Y$ be the preimage of $D_i$ under\nthe quotient map $\\text{UT} _Y\\twoheadrightarrow Y$ and set\n$\\widetilde{D}= \\bigcup_{i=1}^n \\widetilde{D}_i$. The divisor\n$\\sum_{i=1}^n\\widetilde{D}_i$ is anticanonical and\n$(\\text{UT} _Y, \\widetilde{D})$ is a log Calabi–Yau pair. It follows\nfrom the work of Scott that the log Calabi–Yau variety\n$\\text{UT} _Y \\setminus \\widetilde{D}$ has a cluster structure of type\n$\\mathcal{A}$ which is skew-symmetric (that is, for its fixed data all\n$d_i=1$ and $N=N^\\circ$) and such that the frozen variables are\nprecisely the Plücker variables $\\{ p_{[i+1,i+k]}\\}_{i=1}^{n}$. This\ncluster structure is given by an inclusion\n$$\\mathcal{A} \\hookrightarrow \\text{UT} _Y  \\setminus \\widetilde{D}.$$\nSince $\\widetilde{D}$ is the locus in $\\text{UT} _{Y}$ where the frozen\nvariables vanish, we have that $\\mathcal{A} \\subset \\text{UT} _Y$ is a\npartial minimal model (see the example below Definition ). In (see also\n), the authors show that $\\mathcal{A}$ has a seed with a maximal green\nsequence so we can use to conclude that the full Fock–Goncharov\nconjecture holds for $\\mathcal{A}$. Proposition 9.4 in together with\nimply that $\\mathcal{A} \\subset \\text{UT} _Y$ is a partial minimal model\nwith enough theta functions in the sense of Definition . In the\nfollowing subsection we exhibit a cluster ensemble lattice map $p^*$ for\n$\\mathcal{A}$ such that for $K := \\ker(p^*)$, the pair $(p,K)$ has the\nPicard property in the sense of with respect to\n$\\mathcal{A} \\subset \\text{UT} _Y$. These considerations allow us to\napply to all the results of § and §. In particular, we can think of the\nGrassmannian as a minimal model for the quotient $\\mathcal{A} /T_K$.\n\n**Remark 95**. The variety $Y \\setminus D$ is usually called the open\npositroid variety. This variety can be endowed with a cluster structure\nof any of the kinds we consider in this paper: $\\mathcal{A}$,\n$\\mathcal{X}$, a quotient of $\\mathcal{A}$ or a fibre of $\\mathcal{X}$.\n\n## The Picard property\n\nIn this section we verify that the Picard property () holds for a\ncertain choice of cluster ensemble map and sublattice. This condition is\nnecessary in order to apply to the Grassmannian.\n\nWe rely on background from but recall important notions below. For\nbackground on plabic graphs we refer the reader to *loc. cit.*. Recall,\nthat plabic graphs[^8] are combinatorial objects encoding those seeds\nwhose associated $\\mathcal{A}$-cluster variables are Plücker\ncoordinates. To simplify the exposition we do not distinguish between a\nplabic graph and its associated seed. Given an index set\n$J\\in\\binom{[n]}{n-k}$ we construct a Young diagram $\\mu_J$ inside an\n$(n-k)\\times k$ grid inside a rectangle. Let $w_J$ be the path along\nedges of the grid from north east to south west corner whose south steps\nare in $J$. Then $\\mu_J$ is the Young diagram (inside the rectangle\nattached to the north west corner) whose south east border is $w_J$.\nAmong all plabic graphs there is a particularly symmetric one know as\nthe **rectangles plabic graph**\n$G_{\\Yboxdim 4pt \\yng(1)}:=G^{\\rm rec}_{k,n}$. The associated cluster\nvariables are naturally indexed by *rectangular* Young diagrams\n(together with the *empty* rectangle, denoted by $\\varnothing$). In what\nfollows we focus on this plabic graph as the initial seed and denote by\n$$\\label{eq_seed}\n\\textbf{s}_{\\Yboxdim 4pt \\yng(1)}=(e_\\varnothing)\\cup(e_{i\\times j}\\mid  1\\le i\\le n-k, 1\\le j\\le k),$$\nthe induced basis of $N=N^\\circ\\cong \\mathbb Z^{k(n-k)+1}$. Let\n$\\{f_\\varnothing\\}\\cup\\{f_{i\\times j}\\mid  1\\le i\\le n-k, 1\\le j\\le k\\}$\ndenote the corresponding basis of $M^\\circ=M$. We write $N_{\\textbf{s}}$\nrespectively $M_\\textbf{s}$ whenever we think of the lattices together\nwith a choice of basis induced by a seed $\\textbf{s}$.\n\nWe start by defining a lattice map\n$$\\psi:N_{\\textbf{s}_{\\Yboxdim 4pt \\yng(1)}} \\to  M_{\\textbf{s}_{\\Yboxdim 4pt \\yng(1)}}$$\nwhich is given with respect to the bases induced by\n$s_{\\Yboxdim 4pt \\yng(1)}$ as follows: for $i\\times j$ a mutable vertex\nand $a\\times b$ with either $a=n-k$ or $b=k$ a frozen vertex we define\n$$\\begin{aligned}\n    e_{i\\times j} &\\mapsto& f_{(i-1)\\times(j-1)} - f_{(i-1)\\times j} + f_{i\\times(j+1)} - f_{(i+1)\\times (j+1)} + f_{(i+1)\\times j} - f_{i\\times (j-1)} \\\\\n   % e_{1\\times 1} &\\mapsto& f_{1\\times 2} - f_{2\\times 2} + f_{2\\times 1} - f_{\\varnothing}\\\\\n    e_{a\\times b} &\\mapsto& f_{a\\times b} - f_{(a-1)\\times b} + f_{(a-1)\\times(b-1)} - f_{a\\times (b-1)} \\\\\n    e_{\\varnothing} &\\mapsto& f_{\\varnothing} - f_{1\\times k} + f_{1\\times 1} - f_{(n-k)\\times 1}\n\\end{aligned}$$ with the convention that $f_{0\\times j}=f_{i\\times 0}=0$\nwhenever $i,j\\not =0$ and $f_{0\\times 0}=f_{\\varnothing}$. We may\npresent the map pictorially by recording the coefficient of the basis\nelement $e_{i\\times j}$ in the $i\\times j$’th position of the grid (with\nan extra position $0\\times 0$ representing the vertex $\\varnothing$).\n$$\\begin{aligned}\n\\label{eq:pictorial p*}\n\\begin{tikzpicture}[scale=.4]\n\\node at (-3,0){$e_{i\\times j}$};\n\\draw[dashed,opacity=.5] (-1.5,.5) -- (1.5,.5);\n\\draw[dashed,opacity=.5] (-1.5,-0.5) -- (1.5,-0.5);\n\\draw[dashed,opacity=.5] (-0.5,-1.5) -- (-0.5,1.5);\n\\draw[dashed,opacity=.5] (0.5,-1.5) -- (0.5,1.5);\n\\node[opacity=.5] at (-1,-1) {\\small $0$};\n\\node[opacity=.5] at (-1,0) {\\small$0$};\n\\node[opacity=.5] at (-1,1) {\\small$0$};\n\\node[opacity=.5] at (0,-1) {\\small$0$};\n\\node at (0,0) {\\small$1$};\n\\node[opacity=.5] at (0,1) {\\small$0$};\n\\node[opacity=.5] at (1,-1) {\\small$0$};\n\\node[opacity=.5] at (1,0) {\\small$0$};\n\\node[opacity=.5] at (1,1) {\\small$0$};\n\n\\node at (2.5,0) {$\\mapsto$};\n\n\\begin{scope}[xshift=5cm]\n\\draw[dashed,opacity=.5] (-1.5,.5) -- (1.5,.5);\n\\draw[dashed,opacity=.5] (-1.5,-0.5) -- (1.5,-0.5);\n\\draw[dashed,opacity=.5] (-0.5,-1.5) -- (-0.5,1.5);\n\\draw[dashed,opacity=.5] (0.5,-1.5) -- (0.5,1.5);\n\\node[opacity=.5]  at (-1,-1) {\\small$0$};\n\\node at (-1,0) {\\small$-1$};\n\\node at (-1,1) {\\small$1$};\n\\node at (0,-1) {\\small$1$};\n\\node[opacity=.5]  at (0,0) {\\small$0$};\n\\node at (0,1) {\\small$-1$};\n\\node at (1,-1) {\\small$-1$};\n\\node at (1,0) {\\small$1$};\n\\node[opacity=.5]  at (1,1) {\\small$0$};\n\\end{scope}\n\n\n\\begin{scope}[xshift=15cm]\n\\node at (-3,0){$e_{a\\times b}$};\n\\draw[dashed,opacity=.5] (-1.5,.5) -- (1.5,.5);\n\\draw[dashed,opacity=.5] (-1.5,-0.5) -- (1.5,-0.5);\n\\draw[dashed,opacity=.5] (-0.5,-1.5) -- (-0.5,1.5);\n\\draw[dashed,opacity=.5] (0.5,-1.5) -- (0.5,1.5);\n\\node[opacity=.5] at (-1,-1) {\\small $0$};\n\\node[opacity=.5] at (-1,0) {\\small$0$};\n\\node[opacity=.5] at (-1,1) {\\small$0$};\n\\node[opacity=.5] at (0,-1) {\\small$0$};\n\\node at (0,0) {\\small$1$};\n\\node[opacity=.5] at (0,1) {\\small$0$};\n\\node[opacity=.5] at (1,-1) {\\small$0$};\n\\node[opacity=.5] at (1,0) {\\small$0$};\n\\node[opacity=.5] at (1,1) {\\small$0$};\n\n\\node at (2.5,0) {$\\mapsto$};\n\n\\begin{scope}[xshift=5cm]\n\\draw[dashed,opacity=.5] (-1.5,.5) -- (1.5,.5);\n\\draw[dashed,opacity=.5] (-1.5,-0.5) -- (1.5,-0.5);\n\\draw[dashed,opacity=.5] (-0.5,-1.5) -- (-0.5,1.5);\n\\draw[dashed,opacity=.5] (0.5,-1.5) -- (0.5,1.5);\n\\node[opacity=.5]  at (-1,-1) {\\small$0$};\n\\node at (-1,0) {\\small$-1$};\n\\node at (-1,1) {\\small$1$};\n\\node[opacity=.5]  at (0,-1) {\\small$0$};\n\\node at (0,0) {\\small$1$};\n\\node at (0,1) {\\small$-1$};\n\\node[opacity=.5]  at (1,-1) {\\small$0$};\n\\node[opacity=.5]  at (1,0) {\\small$0$};\n\\node[opacity=.5]  at (1,1) {\\small$0$};\n\\end{scope}\n\\end{scope}\n\\end{tikzpicture}\n\\end{aligned}$$ All entries in the grid above *not* corresponding to\nvertices in the particular case considered should simply be neglected. A\nstraightforward computation reveals the following\n\n**Proposition 96**.\n\nWe have\n$\\ker(\\psi)=\\langle{(1,1,\\dots,1)}\\rangle=K_{\\textbf{s}_{\\Yboxdim 4pt \\yng(1)}}$\nand $\\psi(N_{\\textbf{s}_{\\Yboxdim 4pt \\yng(1)}})={(1,1,\\dots,1)}^\\perp$.\nSo, the induced map $\\psi:N/K\\to K^\\perp$ is a lattice isomorphism.\n\nIn fact, $\\psi$ defines a cluster ensemble lattice map (Definition ), so\nwe obtain $$\\begin{aligned}\n\\label{eq:p-map Gr}\n    p:\\mathcal{A} \\to\\mathcal{X} , \\quad \\text{determined by }\\quad  (p\\vert_{\\mathcal{A} _{\\textbf{s}_{\\Yboxdim 4pt \\yng(1)}}})^*=\\psi.\n\\end{aligned}$$ There is a combinatorial way to obtain the map $\\psi$ by\nintroducing *frozen arrows* to the quiver of the initial seed to *close\ncycles* involving frozen vertices (see Figure ). These arrows are used\nto determine the submatrix denoted by $*$ in .\n\nAs a direct consequence of and we observe that the action of $T_K$ on\n$\\mathcal{A}$ coincides with the $\\mathbb{C} ^*$-action (of\nsimultaneously scaling Plücker coordinates) on $\\text{UT} _Y$ restricted\nto $\\mathcal{A}$. In particular:\n\n**Corollary 97**. The Picard property holds for $(p,K)$ with respect to\n$\\mathcal A\\hookrightarrow\\text{UT} _Y$.\n\n## Valuations and Newton–Okounkov bodies\n\nThis subsection is the core of our application to the Grassmannian. We\nshow in Theorem  that certain Newton–Okounkov bodies as they appear in\n(see also Remark ) are unimodularly equivalent to Newton–Okounkov bodies\nof Rietsch–Williams. We first introduce the combinatorics that govern\nRietsch–Williams’ flow valuation and the ${\\bf g}$-vector valuation in\nthis case.\n\n### The flow valuation\n\nBased on Postnokiv’s *boundary measurement map* for plabic networks\nRietsch–Williams associate a **flow valuation** to every plabic graph\n$G$ or more generally every seed $\\textbf{s}$ making use of the\n$\\mathcal{X}$-type cluster structure on the Grassmannian We denote it by\n$$\\mathop{\\mathrm{val}}_\\textbf{s}:\\mathbb C(Y)\\setminus \\{0\\}\\to \\mathbb Z^{(n-k)\\times k}.$$\nThe valuation is defined as the multidegree of the lowest degree summand\n(with respect a fixed graded lexicographic order) on Laurent polynomials\nin $\\mathcal{X}$ variables and then extended to rational functions in\nthe natural way. The lattice is of dimension $(n-k)k$ (as apposed to\n$(n-k)k+1$ which is the number of vertices), as the the variable\ncorresponding to $\\varnothing$ never appears (more details below in §).\nNotice that it therefore coincides with our definition of a **c**-vector\nvaluation for cluster $\\mathcal{X}$ varieties (Corollary ). For\n$G=G_{\\Yboxdim 4pt \\yng(1)}$ we simply write\n$\\mathop{\\mathrm{val}}_G=\\mathop{\\mathrm{val}}_{\\Yboxdim 4pt \\yng(1)}$.\nThe flow valuation with respect to the rectangles plabic graph can be\ncomputed in a particularly explicit way as Rietsch–Williams show in . We\nbriefly summarize some of their findings.\n\n**Proposition 98**. For $J\\in\\binom{[n]}{n-k}$, the valuation\n$\\mathop{\\mathrm{val}}_{\\Yboxdim 4pt \\yng(1)}(p_J)$ can be represented\nby a *GT tableau* (defined as follows, see ) of size $(n-k)\\times k$\nwhose $(i\\times j)^{\\text{th}}$ entry represents the coefficient of the\ncorresponding basis element. The entries of the GT tableau are obtained\nas in four steps:\n\n-   draw the Young diagram $\\mu_J$ whose south border is the path $w_J$\n    associated to $J$ in the $(n-k)\\times k$-rectangle;\n\n-   draw another copy of $w_{J}$ shifted by *one step south* and *one\n    step east* (this implies that some steps of the new path $w_J^1$ lie\n    outside of the $(n-k)\\times k$-rectangle);\n\n-   continue repeating Step 2 until the new copy of $w_J$ lies\n    *entirely* outside of the $(n-k)\\times k$-rectangle;\n\n-   lastly, place an $i$ inside every box (that is part of the\n    $(n-k)\\times k$-rectangle) in between the paths $w_{J}^{i-1}$ and\n    $w_{J}^i$.\n\nAll other boxes are filled with zeros.\n\nRietsch–Williams compute the Newton–Okounkov bodies associated to this\nvaluation. In our notation they are of form\n$\\Delta_{\\mathop{\\mathrm{val}}_{\\Yboxdim 4pt \\yng(1)}}(D_{n-k},p_{(n-k)\\times k})$,\nwhere $p_{(n-k)\\times k}=p_{[1,n-k]}$ is the Plücker coordinate (and\nhence section of $\\mathcal L_e$) associated to the frozen vertex\n$(n-k)\\times k$.\n\n**Example 99**. The procedure of is depicted in Figure  for\n$J=\\{3,4,7,9,11,12\\}\\subset [13]$.\n\n### A combinatorial description of ${\\bf g}$-vectors\n\nIn this subsection we consider the cluster variety\n$\\mathcal{A} ^{\\rm op}$ whose initial quiver is obtained by opposing the\ninitial quiver for $\\mathcal{A}$. It is well known that $\\mathcal{A}$\nand $\\mathcal{A} ^{\\rm op}$ are isomorphic (in general opposing the\nquiver gives rise to isomorphic cluster $\\mathcal{A}$-varieties). We\nalso have a partial minimal model\n$\\mathcal{A} ^{\\mathop{\\mathrm{op}}} \\hookrightarrow \\text{UT} _Y$. We\nwrite $\\textbf{s}_{\\Yboxdim 4pt \\yng(1)}^{\\rm op}$ to denote the seed\n$\\textbf{s}_{\\Yboxdim 4pt \\yng(1)}$ of equation thought of as the\ninitial seed for $\\mathcal{A} ^{\\rm op}$. Notice that $-\\psi$ determines\na cluster ensemble map\n$p^{\\rm op}:\\mathcal{A} ^{\\rm op} \\to \\mathcal{X} ^{\\rm op}$ so that the\nPicard property holds for $(p^{\\rm op},K)$ with respect to\n$\\mathcal{A} ^{\\rm op}\\hookrightarrow \\text{UT}_Y$.\n\nIn this setting an explicit combinatorial formula to compute\n**g**-vectors of Plücker coordinates can be deduced from the\ncategorification of the Grassmannian cluster algebra developed in . We\nlearned about it from Bernhard Keller in private email communication.\nThe below formula describes ${\\bf g}$-vectors with respect to the seed\n$\\textbf{s}_{\\Yboxdim 4pt \\yng(1)}^{\\rm op}$ for the cluster variety\n$\\mathcal{A} ^{\\rm op}$ which we think of as another cluster structure\non $\\text{UT} _{Y}$.\n\n**Corollary 100**.\n\n(Hook formula for **g**-vectors) Consider the seed\n$\\textbf{s}_{\\Yboxdim 4pt \\yng(1)}^{\\rm op}$ and\n$J \\in \\binom{[n]}{n-k}$. We let $i_1\\times j_1,\\dots ,i_s \\times j_s$\nbe the rectangles corresponding to the turning points in the path $w_J$\nthat cuts out $\\mu_J$ inside the $(n-k)\\times k$-rectangle. Then\n$$\\begin{aligned}\n\\label{eq:g-vectors for Grec}\n{\\bf g}_{\\Yboxdim 4pt \\yng(1)^{\\rm op}}(p_J):={\\bf g}^{\\mathcal{A} ^{\\rm op}}_{\\textbf{s}_{\\Yboxdim 4pt \\yng(1)}^{\\rm op}}(p_J)= \\sum_{p=1}^{s}f_{i_{p}\\times j_{p}}-f_{i_{p}\\times j_{p+1}},\n\\end{aligned}$$ where we set $f_{i_s\\times j_{s+1}}:=0$.\n\n**Example 101**. The Consider $n-k=4$, $n=9$, and $J=\\{2,4,6,7\\}$. We\nhave that $\\mu_{J}=\\Yboxdim 4pt \\yng(4,3,2,2)$ and by\n$${\\bf g}_{\\Yboxdim 4pt \\yng(1)^{\\rm op}}\\left(p_{\\Yboxdim 2pt \\yng(4,3,2,2)}\\right)= f_{\\Yboxdim 2pt \\yng(4)}- f_{\\Yboxdim 2pt \\yng(3)}+f_{\\Yboxdim 2pt \\yng(3,3)}-f_{\\Yboxdim 2pt \\yng(2,2)}+f_{\\Yboxdim 2pt \\yng(2,2,2,2)}.$$\n\n### Equality of the Newton–Okounkov bodies\n\nThe aim of this section is to identify the Newton–Okounkov bodies of\nflow valuations with Newton–Okounkov bodies of **g**-vector valuations\nfor $\\mathcal{A} ^{\\rm op}$. We use a particular cluster ensemble\nlattice map for the identification and work in the initial seed\n$\\textbf{s}^{\\rm op}_{\\Yboxdim 4pt \\yng(1)}$ (whose quiver is opposite\nto the quiver depicted in Figure  for $n=9,k=5$).\n\nWe think of the open positroid variety inside\n$Y=\\text{Gr}_{n-k}(\\mathbb{C} ^n)$ as the quotient of the cluster\nvariety $\\mathcal{A} ^{\\rm op}$ by the torus $T_{K}$. We choose a\nsection\n$$\\sigma: N/K \\to N \\quad \\text{with image} \\quad N  \\cap f_{\\varnothing}^\\perp;$$\nthat is, a coset $n\\mod K$ is sent to its unique representative\nsatisfying $\\langle n,f_{\\varnothing}\\rangle=0$. It is not hard to see\nthat $\\sigma$ induces an isomorphism between the rings of rational\nfunctions $\\mathbb C(T_{M/\\langle f_{\\varnothing}\\rangle})$ and\n$\\mathbb C(T_{K^\\perp})$ that commutes with cluster $\\mathcal X$\nmutation. We use $\\sigma$ to realize\n$\\mathrm{Trop} _{\\mathbb{Z} }(({\\mathcal{X} _{\\bf 1}^\\vee})_{\\textbf{s}^\\vee})=\\mathrm{Trop} _{\\mathbb{Z} }((\\mathcal{A} ^{\\rm op}/T_K)_{\\textbf{s}^{\\rm op}})= N/K$\ninside\n$\\mathrm{Trop} _\\mathbb{Z} (\\mathcal{A} ^{\\rm op}_{\\textbf{s}^{\\rm op}})=N =\\mathrm{Trop} _\\mathbb{Z} (\\mathcal{X} ^\\vee_{\\textbf{s}^\\vee})$\nfor every seed. Moreover, the dual of $\\sigma$ induces an isomorphism of\nlattices $$\\sigma^*:M/\\langle f_{\\varnothing}\\rangle \\to K^\\perp.$$\nNotice that $T_{K^\\perp}=\\pi^{-1}(\\bf 1)$ where $\\pi:T_M\\to T_{K^*}$ is\nthe restriction of $\\mathcal{X} \\to T_{K^*}$ to a cluster chart. As\nalluded to above, we obtain an isomorphism of cluster\n$\\mathcal{X}$-varieties\n$$\\sigma^*:\\mathcal{X} _{\\setminus \\varnothing}\\to \\mathcal{X} _{\\bf 1},$$\nwhere $\\mathcal{X} _{\\setminus \\varnothing}$ is the\n$\\mathcal{X}$-variety associated with the initial data obtained by\ndeleting the index $\\varnothing$ upon realizing\n$M/\\langle f_{\\varnothing}\\rangle$ as\n$\\langle f_{i\\times j}\\mid 1\\le i\\le n-k,1\\le j\\le k\\rangle \\subset M$.\nGiven a seed $\\textbf{s}$ we denote the corresponding seed of\n$\\mathcal{X} _{\\setminus\\varnothing}$ by\n$\\textbf{s}_{\\setminus\\varnothing}$. In particular, we have\n$\\mathrm{Trop} _{\\mathbb{Z} }((\\mathcal X^\\vee_{\\setminus\\varnothing})_{\\textbf{s}^\\vee_{\\setminus \\varnothing}})=N_{\\textbf{s}}\\cap f_{\\varnothing}^\\perp$.\nThe flow valuation is defined on ring of rational functions on the\npositroid variety which coincides with\n$$\\mathbb C(\\mathcal{X} _{\\setminus \\varnothing}) \\cong \\mathbb C(x_{i\\times j}:1\\le i\\le n-k,1\\le j\\le k).$$\n\nThe next result follows from the preceding discussion and Corollary .\n\n**Proposition 102**. For every choice of seed $\\textbf{s}$ the diagram\ncommutes: $$\\xymatrix{\n\\mathbb C(\\mathcal{X} _{\\bf 1})\\setminus \\{0\\} \\ar[d]_{(\\sigma^*)^*}\\ar[r]^{\\mathbf{c} _{\\textbf{s}}} & N/K\\ar[d]^{\\sigma}\\\\\n\\mathbb C(\\mathcal{X} _{\\setminus \\varnothing})\\setminus \\{0\\} \\ar[r]_{\\mathop{\\mathrm{val}}_{\\textbf{s}}} & N_{\\textbf{s}}\\cap f_{\\varnothing}^\\perp.\n}$$\n\nThe flow valuation is a ${\\bf c}$-vector valuation for the variety\n$\\mathcal{X} _{\\setminus \\varnothing}$ as both are defined by picking\nthe lowest degree exponent of a Laurent polynomial with respect to the\nsame order. That is:\n$$\\mathop{\\mathrm{val}}_{\\textbf{s}}= \\mathbf{c} ^{\\mathcal{X} _{\\setminus \\varnothing}}_{\\textbf{s}\\setminus\\varnothing}.$$\nAlternatively, in light of Proposition we may think of the flow\nvaluation as a ${\\bf c}$-vector valuation for $\\mathcal{X} _{\\bf 1}$.\nOur aim now is to identify the images of\n$\\mathop{\\mathrm{val}}_\\textbf{s}$ with those of a $\\bf g$-vector\nvaluation for $\\mathcal{A} ^{\\rm op}$, or more precisely a\n${\\bf g}$-vector valuation for $\\mathcal{A} ^{\\rm op}/T_{K}$. To avoid\nconfusion we introduce the following notation $$\\begin{aligned}\n\\label{eq:homogenized g vector op}\n\\bar {\\bf g}_{\\Yboxdim 4pt \\yng(1)^{\\rm op}}:R\\setminus \\{0\\}\\longrightarrow K^\\perp\\cong M_{\\textbf{s}}/\\langle f_{(n-k)\\times k}\\rangle \n\\end{aligned}$$ defined for a homogeneous element\n$h\\in R_q\\setminus \\{0\\}$ by $$h\\longmapsto \n{\\bf g}_{\\Yboxdim 4pt \\yng(1)^{\\rm op}}\\left(\\frac{h}{\np_{(n-k)\\times k}^q}\\right),$$ where\n${\\bf g}_{\\Yboxdim 4pt \\yng(1)^{\\rm op}}(p_{(n-k)\\times k})=f_{(n-k)\\times k}$.\nNotice that $\\bar {\\bf g}_{\\Yboxdim 4pt \\yng(1)^{\\rm op}}$ is the\nrestriction of\n${\\bf g}_{\\Yboxdim 4pt \\yng(1)^{\\rm op}}: \\mathbb C(Y)\\setminus \\{0\\}\\to M$\nto the section ring $R\\hookrightarrow \\mathbb C(Y)$ where the embedding\nis defined by $R_q\\ni h\\mapsto h/p_{(n-k)\\times k}^q$.\n\n**Theorem 103**.\n\n*Proof.* We prove the claim in several steps. First, we need to describe\n$\\mathop{\\mathrm{val}}_{\\Yboxdim 4pt \\yng(1)^{\\rm op}}(p_J)$.\nFortunately, this is straightforward using . Let us analyze the image of\nthe *$i$-strip*, *i.e.* the image of the elements of form\n$-ie_{a\\times b}$ corresponding to a box in position $a\\times b$ of the\ngrid lying between the path $w_J^i$ and $w_J^{i-1}$. We deduce\n\nNotice that unless $i=1$ all non zero entries in the picture cancel with\nthe images of the $(i-1)$- and the $(i+1)$-strips. When $i=1$ however,\nthe entry $i=1$ above the path $w_J^{0}=w_J$ stays. Hence, for every\ncorner in $w_{J}$ corresponding to a south step followed by a west step\n$-\\psi\\left(\\mathop{\\mathrm{val}}_{\\Yboxdim 4pt \\yng(1)}(p_J)\\right)$\nhas coefficient $1$ for $f_{a\\times b}$ where $a\\times b$ corresponds to\nthe box whose south east corner coincides with this corner of $w_J$. The\ncase of a corner in $w_{J}$ corresponding to a west step followed by a\nsouth step is very similar, with the only difference that the signs\nchange. In particular,\n$-\\psi\\left(\\mathop{\\mathrm{val}}_{\\Yboxdim 4pt \\yng(1)}(p_J)\\right)$\nhas coefficient $-1$ for $f_{a\\times b}$ where $a\\times b$ corresponds\nto the box whose south east corner is adjacent to this corner of $w_J$.\n\nIt is left to analyze the parts of\n$-\\psi\\left(\\mathop{\\mathrm{val}}_{\\Yboxdim 4pt \\yng(1)}(p_J)\\right)$\ncorresponding to *frozen* vertices. The arguments here are very similar,\nthe only special case being the south east corner of the\n$(n-k)\\times k$-rectangle. Hence, we restrict our attention to this case\nand omit the others.\n\nConsider the vertex in position $(n-k)\\times k$ and assume in\n$\\mathop{\\mathrm{val}}_{\\Yboxdim 4pt \\yng(1)}(p_J)$ the corresponding\nentry is $i$. Notice, that coefficient for the vertex\n$(n-k-1)\\times (k-1)$ necessarily is $i-1$. So, applying $-\\psi$ we see\nthat\n\nObserve that the entries $1$ and $-i$ cancel by similar arguments as\nabove. The only non-zero coefficient in this picture is $-1$ for\n$f_{(n-k)\\times k}$. Summarizing, we have $$\\begin{aligned}\n-\\psi\\left(\\mathop{\\mathrm{val}}_{\\Yboxdim 4pt \\yng(1)}(p_J)\\right) &=& \\sum_{\\begin{smallmatrix}\n  \\text{south to west}\\\\\n  \\text{corners of }w_J\n\\end{smallmatrix} } f_{a'\\times b'} -f_{(n-k)\\times k}\n-\\sum_{\\begin{smallmatrix}\n  \\text{west to south}\\\\\n  \\text{corners of }w_J\n\\end{smallmatrix} } f_{a\\times b}\\\\\n&\\overset{\\text{Equation~\\eqref{eq:g-vectors for Grec}}}{=}& {\\bf g}_{\\Yboxdim 4pt \\yng(1)^{\\rm op}}(p_{J}) - f_{(n-k)\\times k}=\\bar{\\bf g}_{\\Yboxdim 4pt \\yng(1)^{\\rm op}}(p_J).\n\\end{aligned}$$ This implies\n$-\\psi(\\Delta_{\\mathop{\\mathrm{val}}_{\\Yboxdim 4pt \\yng(1)}}(D_{n-k},p_{(n-k)\\times k})=\\Delta_{\\bar{\\bf g}_{\\Yboxdim 4pt \\yng(1)}^{\\rm op}}(D_{n-k},p_{(n-k)\\times k})$. ◻\n\n**Remark 104**. The attentive reader might notice that the Theorem  and\nthe discussion preceding it closely resemble Lemma . However, the\ndifference in convention choices in and the present paper yield the\nnecessity of a non-trivial change of coordinates. To avoid lengthening\nthe exposition even more we decided to give the result in a single seed\nbut allude to the fact that Theorem  indeed is an instance of Lemma \n(after non-trivial changes of cluster coordinates). After making the\nappropriate change of coordinate one can show that the map $-\\psi$ may\nbe described as the tropicalization of a cluster ensemble map. In\nparticular, Theorem  can be extended to all seeds.\n\n## The intrinsic Newton–Okounkov body for Grassmannians\n\nAs before, let $\\mathcal{L} _e$ be the bundle over\n$\\text{Gr}_{n-k}(\\mathbb{C} ^n)$ obtained by pullback of\n$\\mathcal{O}(1)$ under the Plücker embedding\n$\\text{Gr}_{n-k}(\\mathbb{C} ^n)\\hookrightarrow \\mathbb{P}^{\\binom{n}{k}-1}$.\nRecall the definition of the intrinsic Newton–Okounkov body from\nDefinition .\n\n**Corollary 105**. Consider the partial minimal model\n$\\mathcal{A} ^{\\rm op}\\hookrightarrow \\text{UT} _{Y}$, the minimal model\n$\\mathcal{A} ^{\\mathop{\\mathrm{op}}}/T_K \\hookrightarrow Y$ and the map\n${\\bf g}:\\mathbb{B}_{\\vartheta }(\\mathcal{A} ^{\\mathop{\\mathrm{op}}})\\to \\mathrm{Trop} _{\\mathbb{Z} }((\\mathcal{A} ^{\\mathop{\\mathrm{op}}})^\\vee)$\nof . Then $$\\begin{split} \n\\Delta_{\\mathrm{BL}}(\\mathcal{L} _e) = \\mathop{\\mathrm{conv_{BL}}}\\Bigg( \\left\\{ \\mathbf{g} \\left(p_J\\right) \\mid J \\in \\binom{[n]}{n-k}\\right\\} \\Bigg).\n \\end{split}$$\n\n*Proof.* The Newton–Okounkov polytope for the flow valuation with\nrespect to $\\textbf{s}_{\\Yboxdim 4pt \\yng(1)}$ is the convex hull of the\nimages of Plücker coordinates (see ). So by Theorem  the same is true\nfor the Newton–Okounkov body\n$\\Delta_{\\bar{\\mathbf{g} }_{\\Yboxdim 4pt \\yng(1)^{\\mathop{\\mathrm{op}}}}}(D_{n-k},p_{(n-k)\\times k})$.\nBy Theorem ,\n$\\Delta_{\\bar{\\mathbf{g} }_{\\Yboxdim 4pt \\yng(1)^{\\mathop{\\mathrm{op}}}}}(D_{n-k},p_{(n-k)\\times k})$\nis positive, hence it is broken line convex. Therefore, the broken line\nconvex hull of the set\n$\\left\\{ \\bar{\\mathbf{g} }_{\\Yboxdim 4pt \\yng(1)^{\\mathop{\\mathrm{op}}}} \\left(p_J\\right) \\mid J \\in \\binom{[n]}{n-k}\\right\\}$\nin the lattice\n$\\mathrm{Trop} _\\mathbb{R} ({\\mathcal{A} ^{\\mathop{\\mathrm{op}}}/T_{K}}^{\\vee}_{\\textbf{s}_{\\Yboxdim 4pt \\yng(1)^{\\mathop{\\mathrm{op}}}}})$\ncoincides with its convex hull. We take into account Remark  to get that\n$\\Delta_{\\mathbf{g} _{\\Yboxdim 4pt \\yng(1)^{\\mathop{\\mathrm{op}}}}}(\\mathcal{L} _e) = \\Delta_{\\bar{\\mathbf{g} }_{\\Yboxdim 4pt \\yng(1)^{\\mathop{\\mathrm{op}}}}}(D_{n-k},p_{(n-k)\\times k}) + \\mathbf{g} _{\\Yboxdim 4pt \\yng(1)^{\\mathop{\\mathrm{op}}}}(p_{(n-k)\\times k})$.\nThis implies that\n$\\Delta_{\\mathbf{g} _{\\Yboxdim 4pt \\yng(1)^{\\mathop{\\mathrm{op}}}}}(\\mathcal{L} _e)$\nis the broken line convex hull of the set\n$\\left\\{ \\mathbf{g} _{\\Yboxdim 4pt \\yng(1)^{\\mathop{\\mathrm{op}}}} \\left(p_J\\right) \\mid J \\in \\binom{[n]}{n-k}\\right\\}$.\nBeing a broken line convex set is independent of the choice of seed, the\nresult follows. ◻\n\n**Remark 106**. In the proof of Corollary , we implicitly use the\n$\\mathcal{A}$ cluster structure to view the intrinsic Newton–Okounkov\nbody $\\Delta_{\\mathrm{BL}}(\\mathcal{L} _e)$ as the broken line convex\nhull of tropical points indexing Plücker coordinates. We could\nalternatively use the $\\mathcal{X}$ cluster structure as\nRietsch–Williams do, and define theta functions with the corresponding\n$\\mathcal{X}$ scattering diagram. By identifying the Rietsch–Williams\nvaluation with the $\\mathbf{c}$-vector valuation (Corollary ) and noting\nthat there is a cluster ensemble automorphism of the open positroid\nvariety (see , ), we can apply Lemma  to give a completely analogous\nstatement to Corollary  which uses the Rietsch–Williams valuation rather\nthan the $\\mathbf{g}$-vector valuation. In fact, in § we present an\nexample of an explicit computation related to the intrinsic\nNewton–Okounkov body $\\Delta_{\\mathrm{BL}}(\\mathcal{L} _e)$ defined via\nthe $\\mathcal{X}$ scattering diagram.\n\n### Example\n\nIn this subsection, we will give an example of the intrinsic\nNewton–Okounkov body for the case of\n$\\mathop{\\mathrm{Gr}}_3\\left(\\mathbb{C} ^6\\right)$ and compare this to a\nNewton–Okounkov body of . In particular, in , Rietsch–Williams discuss a\nnon-integral vertex appearing in the Newton–Okounkov body\n$\\Delta_{\\mathop{\\mathrm{val}}_{G}}(D_{3},p_{123})$ associated to the\nplabic graph $G$ of Figure . We illustrate how this non-integral vertex\nin the usual Newton–Okounkov body framework corresponds to a point in\nthe interior of a broken line segment in\n$\\Delta_{\\mathrm{BL}}(\\mathcal{L} _e)$ and thus is not a genuine vertex\nfrom the intrinsic Newton–Okounkov body perspective. Here, to facilitate\ncomparison with , we will view the open positroid variety as\n$\\mathcal{X} _{\\mathbf{1}}$ (up to codimension 2). So, the scattering\ndiagram we use to define $\\Delta_{\\mathrm{BL}}(\\mathcal{L} _e)$ in the\nsubsection will be\n$\\mathfrak{D}^{\\mathcal{X} _{\\mathbf{1}}}_{\\text{in},\\textbf{s}_G}$ for\na particular choice of initial seed $\\textbf{s}_G$. The choice of seed\nis encoded by the plabic graph illustrated in Figure .\n\nRecall that the Young diagrams in Figure  label the network parameters\nused in flow polynomial expressions (see ). The $\\mathcal{A}$-cluster\ndetermined by trips in the plabic graph $G$ consists of the Plücker\ncoordinates whose indices are given in Figure  (see ).\n\nAccording to , a non-integral vertex in the Newton–Okounkov polytope\ncomes from half the valuation of the flow polynomial for the element\n$f = (p_{124} p_{356} - p_{123} p_{456}) / {p_{123}^2}$. They compute\n$\\frac{1}{2}\\mathop{\\mathrm{val}}_G(f)$ and express its entries in\ntabular form (see ) as we have reproduced in Table . The function\n$p_{123}^2\\, f$ is one of the two $\\mathcal{A}$ cluster variables that\nare not Plücker coordinates, see *e.g.* . It is obtained by mutation at\n$\\Yboxdim 4pt \\yng(2,1)$.\n\n```latex\n\\begin{tabular}{|C|C|C|C|C|C|C|C|C|} \n\\hline\n  \\vphantom{\\Yboxdim 4pt \\yng(1,1,1,1)} \\Yboxdim 4pt \\yng(3,3,3)  & \\Yboxdim 4pt \\yng(3,3,2)  & \\Yboxdim 4pt \\yng(2,2,2)  & \\Yboxdim 4pt \\yng(1,1,1)  & \\Yboxdim 4pt \\yng(3,3)  & \\Yboxdim 4pt \\yng(2,1)  & \\Yboxdim 4pt \\yng(1,1)  & \\Yboxdim 4pt \\yng(3)  & \\Yboxdim 4pt \\yng(2)\\\\\n\\hline\n     \\vphantom{\\Yboxdim 4pt \\yng(1,1,1)_{\\Yboxdim 4pt \\yng(1,1,1)}}\\frac{3}{2} &     \\frac{3}{2} & 1 &   \\frac{1}{2} & 1 & \\frac{1}{2} &\\frac{1}{2} &\\frac{1}{2} &\\frac{1}{2} \\\\\n\\hline\n\\end{tabular}\n```\n\nNote we can re-interpret the expression for $f$ as the expansion of a\nproduct of theta functions. All Plücker coordinates are $\\mathcal{A}$\ncluster variables, and all $\\mathcal{A}$ cluster monomials are theta\nfunctions. Then $$\\begin{split} p_{124}\\, p_{356} = \np_{123}^2\\, f + p_{123}\\, p_{456}, \\end{split}$$ and the right hand side\nis a sum of two theta functions. This means there are only two balanced\npairs of broken lines contributing to the product. We will see that the\npair with no bending corresponds to the summand $p_{123}\\, p_{456}$,\nwhile the other involves a maximal bend at an initial wall. (Since the\nbend is at an initial wall, we are able to see the relevant broken line\nsegment without constructing the consistent scattering diagram.)\n\nWe interpret the Rietsch–Williams valuation as being valued in\n$\\mathrm{Trop} _{\\mathbb{Z} }((\\mathcal{X} _{\\mathbf{1}})^{\\vee})$ (see\nProposition ) and consider broken lines in the associated $\\mathcal{X}$\ncluster scattering diagram. The choice of seed identifies\n$\\mathrm{Trop} _{\\mathbb{Z} }((\\mathcal{X} _{\\mathbf{1}})^\\vee)$ with\n$N^\\vee / {\\left\\langle{(1,1,\\dots, 1)}\\right\\rangle}$ and we draw the\nscattering in\n$\\left( N^\\vee / {\\left\\langle{(1,1,\\dots, 1)}\\right\\rangle}\\right) \\otimes \\mathbb{R}$.\n\nWe will use Figure to define the fixed data and the seed data for the\ncluster structure. The initial scattering diagram for the $\\mathcal{X}$\nvariety is\n$$\\mathfrak{D}^{\\mathcal{X} _{}}_{\\text{in},\\textbf{s}_G}= \\left\\{   \\left( (v_{\\mu})^{\\perp}, \\  1+ z^{e_{\\mu}}\\right) \\mid \\mu \\in \\{ \\Yboxdim 4pt \\yng(2), \\Yboxdim 4pt \\yng(2,1), \\Yboxdim 4pt \\yng(3,3,2), \\Yboxdim 4pt \\yng(1,1) \\}\\right\\}.$$\nTo get initial scattering diagram for the fibre over $\\mathbf{1}$ we\ntake the quotient of the support\n$\\mathfrak{D}^{\\mathcal{X} }_{\\text{in},\\textbf{s}_G}$ by\n$\\left(\\mathbb{R} \\cdot (1,1,\\dots, 1) \\right)$. (Observe that\n$(v_{\\mu})^{\\perp}$ is invariant under translations by\n$\\mathbb{R} \\cdot (1,1,\\dots, 1)$.)\n\n$$\\mathfrak{D}^{\\mathcal{X} _{\\mathbf{1}}}_{\\text{in},\\textbf{s}_G}= \\left\\{   \\left( (v_{\\mu})^{\\perp}/ \\left(\\mathbb{R} \\cdot (1,1,\\dots, 1) \\right) , \\  1+ z^{e_{\\mu}}\\right) \\mid \\mu \\in \\{ \\Yboxdim 4pt \\yng(2), \\Yboxdim 4pt \\yng(2,1), \\Yboxdim 4pt \\yng(3,3,2), \\Yboxdim 4pt \\yng(1,1) \\}\\right\\}$$\n\nAll pertinent valuations may be found in . We record them here using the\nordering of Table . We choose the representative whose coefficient of\n$e_\\varnothing$ is $0$ and do not record this entry.\n$$\\begin{split} \\mathop{\\mathrm{val}}_{G}({p}_{124}) = (1,0,0,0,0,0,0,0,0) \\end{split}$$\n$$\\begin{split} \\mathop{\\mathrm{val}}_{G}({p}_{356}) = (2,2,2,1,2,1,1,1,1) \\end{split}$$\n$$\\begin{split} \\mathop{\\mathrm{val}}_{G}({p}_{123}) = (0,0,0,0,0,0,0,0,0)  \\end{split}$$\n$$\\begin{split} \\mathop{\\mathrm{val}}_{G}({p}_{456}) = (3,2,2,1,2,1,1,1,1)  \\end{split}$$\nSo, we have\n$\\mathop{\\mathrm{val}}_{G}({p}_{124}) + \\mathop{\\mathrm{val}}_G( p_{356} ) = \\mathop{\\mathrm{val}}_{G}( p_{123} \\, p_{456}  )$.\nThe summand $p_{123} \\, p_{456}$ in the product $p_{124} \\, p_{356}$\ncorresponds to the straight broken line segment from\n$\\mathop{\\mathrm{val}}_{G}({p}_{356})$ to\n$\\mathop{\\mathrm{val}}_G( p_{124} )$, whose midpoint is\n$\\frac{1}{2}  \\mathop{\\mathrm{val}}_{G}( p_{123} \\, p_{456}  )$.\n\nThe bending wall for the other broken line segment is\n$\\left((v_{\\Yboxdim 2pt \\yng(3,3,2)})^{\\perp}, 1+z^{e_{\\Yboxdim 2pt \\yng(3,3,2)}}\\right)$.\nNote that\n$v_{\\Yboxdim 2pt \\yng(3,3,2)}= f_{\\Yboxdim 2pt \\yng(3,3,3)} + f_{\\Yboxdim 2pt \\yng(2,1)}-f_{\\Yboxdim 2pt \\yng(3,3)} - f_{\\Yboxdim 2pt \\yng(2,2,2)}$,\nand\n$\\frac{1}{2}\\mathop{\\mathrm{val}}(f) = \\frac{1}{2}\\mathop{\\mathrm{val}}(p_{123}^2\\, f)$\nis perpendicular to this vector. So,\n$\\frac{1}{2}\\mathop{\\mathrm{val}}(p_{123}^2\\, f)$ lies in the support of\nthis wall. We will see that there is a broken line segment from\n$\\mathop{\\mathrm{val}}_{G}(p_{356})$ to\n$\\mathop{\\mathrm{val}}_{G}(p_{124})$ passing through\n$\\frac{1}{2}\\mathop{\\mathrm{val}}(f)$ and bending maximally here, as\ndepicted in Figure .\n\nRecall that the exponent vector of the decoration monomial along\n$\\ell_i$ is the negative of the velocity vector there. Traveling along\n$\\ell_1$, this velocity vector is positively proportional to\n$$\\frac{1}{2}\\mathop{\\mathrm{val}}_G(f) - \\mathop{\\mathrm{val}}_G(p_{356}) = \n-\\left(\\frac{1}{2}, \\frac{1}{2}, 1, \\frac{1}{2},  1, \\frac{1}{2} , \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{2}\\right).$$\nWe can take a broken line with exponent vector $v_1=(1,1,2,1,2,1,1,1,1)$\nalong $\\ell_1$. The possible bendings after crossing the wall correspond\nto summands of\n$$\\begin{split} z^{v_1}\\left(1+z^{e_{\\Yboxdim 2pt \\yng(3,3,2)}}\\right)^{-\\left\\langle{v_1,v_{\\Yboxdim 2pt \\yng(3,3,2)}}\\right\\rangle}=z^{v_1}\\left(1+z^{e_{\\Yboxdim 2pt \\yng(3,3,2)}}\\right)^{2}. \\end{split}$$\nThe maximal bending corresponds to the summand\n$z^{v_1+ 2 e_{\\Yboxdim 2pt \\yng(3,3,2)}} = z^{(1,3,2,1,2,1,1,1,1)}$. Let\nus call $v_2 := (1,3,2,1,2,1,1,1,1)$. Then observe that\n$\\frac{1}{2}\\mathop{\\mathrm{val}}_G(f) - \\frac{1}{2}v_2 = \\mathop{\\mathrm{val}}_G(p_{124})$.\nSo, we have a broken line segment $\\gamma$ traveling from\n$\\mathop{\\mathrm{val}}_G(p_{356})$ to\n$\\frac{1}{2}\\mathop{\\mathrm{val}}_G(f)$ with decoration monomial\n$z^{v_1}$, bending maximally and continuing to\n$\\mathop{\\mathrm{val}}_G(p_{124})$ with decoration monomial $z^{v_2}$.\nPrecisely $\\frac{1}{2}$ a unit of time is spent in each straight\nsegment. From this perspective, $\\frac{1}{2}\\mathop{\\mathrm{val}}_G(f)$\nis not a genuine vertex; $\\frac{1}{2}\\mathop{\\mathrm{val}}_G(f)$ is in\nthe relative interior of the support of $\\gamma$, and the endpoints of\n$\\gamma$ are in the Newton–Okounkov body.\n\n[^1]: A weaker condition is enough to construct a valuation for a given\n    seed, see Remark .\n\n[^2]: Throughout the text we consider more generally (partial) minimal\n    models for schemes $V$ with a cluster structure given by a\n    birational map $\\mathcal{V} \\dashrightarrow V$.\n\n[^3]: If $Y$ is Fano, then this should be considered as the\n    superpotential used for mirror symmetry purposes.\n\n[^4]: That is, the equivalence class of consistent scattering diagrams\n    for $\\mathcal{A}$ contains a representative whose scattering\n    functions are of this form.\n\n[^5]: In fact, it is enough that the equality holds at the level of\n    integral points, namely,\n    $k\\Delta(\\mathbb{Z} )= P_k(\\mathbb{Z} ) - k\\nu_{\\textbf{s}}(\\tau)$.\n    However, we are able to show the stronger condition\n    $k\\Delta= P_k - k\\nu_{\\textbf{s}}(\\tau)$.\n\n[^6]: That is, there is a scheme $\\mathcal Y$ and a flat morphism\n    $\\mathcal{Y}\\to \\mathbb A^1$ whose generic fibre is isomorphic to\n    $Y$ and special fibre isomorphic to the toric variety associated to\n    $\\Delta_{\\nu_{\\textbf{s}}}(D',\\tau)$.\n\n[^7]: This perspective is essentially the *jagged path* description of\n    theta functions rather than the broken line description. See for\n    example .\n\n[^8]: To be precise, we are only interested in reduced plabic graphs\n    with trip permutation $\\pi_{k,n}$.\n"
    },
    {
        "text": "abstract:  *We consider initial boundary value problems with the\n  homogeneous Neumann boundary condition. Given an initial value, we\n  establish the uniqueness in determining a spatially varying\n  coefficient of zeroth-order term by a single measurement of Dirichlet\n  data on an arbitrarily chosen subboundary. The uniqueness holds in a\n  subdomain where the initial value is positive, provided that it is\n  sufficiently smooth which is specified by decay rates of the Fourier\n  coefficients. The key idea is the reduction to an inverse elliptic\n  problem and relies on elliptic Carleman estimates.*  \n  **Key words.** inverse coefficient problem, parabolic equation,\n  uniqueness, initial boundary value problem, inverse elliptic problem,\n  Carleman estimate  \n  **AMS subject classifications.**35R30, 35K15\nauthor: $^1$ O.Y.  Imanuvilov and $^{2,3,4}$ M. Yamamoto\ntitle:  Inverse parabolic problem with initial data by a single\n  measurement\n\n# Introduction\n\nLet $\\Omega$ be a bounded domain in $\\Bbb R^n$ with $C^2-$ boundary\n$\\partial\\Omega$. We set $\\nu=(\\nu_1(x),\\dots,\\nu_n(x))$ be an outward\nunit normal vector to $\\partial\\Omega$, and\n$$\\partial_k = \\frac{\\partial}{\\partial x_k}, \\quad 1\\le k\\le n, \\quad\n\\partial_t = \\frac{\\partial}{\\partial t}.$$ We assume\n$$a_{ij}=a_{ji} \\in C^2(\\overline{\\Omega}) \\quad \\mbox{for all $1\\le i,j\\le n$},\n\\quad c_1, c_2 \\in C(\\overline{\\Omega}),$$ and there exists a constant\n$\\kappa_1>0$ such that $$\\sum_{i,j=1}^n a_{ij}(x)\\xi_i\\xi_j\n\\ge \\kappa_1\\vert \\xi\\vert^2 \\quad \\mbox{for all $x \\in \\overline{\\Omega}$ and\n$\\xi=(\\xi_1, ... \\xi_n) \\in \\Bbb R^n$}.$$ We set\n$$\\partial_{\\nu_A}v: = \\sum_{i,j=1}^na_{ij}(x)\\nu_i(x)\\partial_jv(x), \\quad \nx\\in \\partial\\Omega.$$ Moreover let\n$$A(x,D)v(x) = - \\sum_{i,j=1}^n\\partial_i(a_{ij}(x)\\partial_jv(x)) \\quad \n\\mbox{with} \\quad\n\\mathcal{D}(A):= \\{ v\\in H^2(\\Omega);\\, \\partial_{\\nu_{A_0}} v= 0\\quad\n\\mbox{on $\\partial\\Omega$}\\}.                \\eqno{(1.1)}$$\n\nIn this article, we consider the folowing  \n**Inverse problem.**  \n*Let $$\\left\\{ \\begin{array}{rl}\n& \\partial_tu = -A(x,D)u + c_1(x)u \\quad \\mbox{in $\\Omega\\times (0,T)$}, \\\\\n& \\partial_{\\nu_A}u = 0 \\quad\\mbox{on $\\partial\\Omega$}, \\\\\n& u(x,0) = a(x), \\quad x\\in \\Omega,\n\\end{array}\\right.\n                             \\eqno{(1.2)}$$ and\n$$\\left\\{ \\begin{array}{rl}\n& \\partial_tv = -A(x,D)v + c_2(x)v \\quad \\mbox{in $\\Omega\\times (0,T)$}, \\\\\n& \\partial_{\\nu_A}v = 0 \\quad\\mbox{on $\\partial\\Omega$}, \\\\\n& v(x,0) = a(x), \\quad x\\in \\Omega.\n\\end{array}\\right.\n                                    \\eqno{(1.3)}$$ Let an initial value\n$a$ be suitably given and $\\gamma \\subset \\partial\\Omega$ be an\narbitrarily chosen non-empty connected relatively open subset of\n$\\partial\\Omega$. Then\n$$\\mbox{$u=v$ on $\\gamma \\times (0,T)$ implies $c_1=c_2$ in $\\Omega$?}$$*\n\nThis inverse problem has been intensively studied in the literature.\nMost general results are obtained for the case when a time of\nobservation $t_0$ belongs to the open interval $(0,T)$. In this case,\nbased on the method introduce in Bukhgeim and Klibanov , Imanuvilov and\nYamamoto proved the uniqueness and the Lipschitz stability in\ndetermination of coefficients corresponding to the zeroth order term.\nRecently in Imanuvilov and Yamamoto , the authors proved a conditional\nLipschitz stability estimate as well as the uniqueness for the case\n$t_0=T$. See also Huang, Imanuvilov and Yamamoto .\n\nIn case an observation is taken at the initial moment $t_0=0,$ to the\nauthors’ best knowledge, the question of uniqueness of a solution of\ninverse problem is open in general. Limited to the one-dimensional case,\na result in this direction was obtained by Suzuki and Suzuki and\nMurayama . Klibanov proved the uniqueness of determination of zeroth\norder term coefficient in the case when $a_{ij}=\\delta_{ij}$ (the case\nof the Laplace operator). Also it is assumed that the domain of\nobservation $\\Gamma=\\partial\\Omega$. The method proposed in is based on\nan integral transform and subsequent reduction of the original problem\nto the problem of determination of a coefficient of zeroth order term\nfor a hyperbolic equation. After that, the method in is applied. It\nshould be mentioned that the method introduced by Bukhgeim and Klibanov\nis based on Carleman estimates and the Carleman type estimates for\nhyperbolic equations are subjected to so-called non-trapping conditions.\nTherefore both assumptions made in are critically important for the\napplication of this method except of the one dimensional case. In , the\nauthors extended results of to the case of general second order\nhyperbolic equation. The main purpose of the current work is to remove\nthe non-trapping assumptions and prove the uniqueness without any\ngeometric constraints on the observation subbondary $\\gamma$.  \nHenceforth we set\n$$-A_1(x,D) = -A(x,D) + c_1(x), \\quad -A_2(x,D) = -A(x,D) + c_2(x)$$\nwith the domains $\\mathcal{D}(A_1) = \\mathcal{D}(A_2) = \\mathcal{D}(A)$.\nIt is known that the spectrum $\\sigma(A_k)$ of $A_k$, $k=1,2$, consists\nentirely of eigenvalues with finite multiplicities.\n\nBy changing $\\widetilde{u}:= e^{Mt}u$ with some constant $M$, it\nsuffices to assume that there exists a constant $\\kappa_2>1$ such that\n$(A_1u,u)_{L^2(\\Omega)} \\ge \\kappa_2\\Vert u\\Vert^2_{L^2(\\Omega)}$ for\n$u \\in \\mathcal{D}(A_1)$ and\n$(A_2v,v)_{L^2(\\Omega)} \\ge \\kappa_2\\Vert v\\Vert^2_{L^2(\\Omega)}$ for\n$v\\in \\mathcal{D}(A_2)$.\n\nThen, setting\n$$\\sigma(A_1)= \\{\\lambda_k\\}_{k\\in \\mathbb{N}}, \\quad \\sigma(A_2) = \\{ \\mu_k\\}_{k\\in \\mathbb{N}},$$\nwe can number as\n$$1 < \\lambda_1 < \\lambda_2 < \\cdots, \\qquad 1<\\mu_1 < \\mu_2 < \\cdots.$$\n\nLet $P_k$ be the projection for $\\lambda_k$, $k \\in \\mathbb{N}$ which is\ndefined by $$P_k = \\frac{1}{2\\pi\\sqrt{-1}} \\int_{\\gamma(\\lambda_k)}\n(z-A_1)^{-1} dz, \\quad\nQ_k = \\frac{1}{2\\pi\\sqrt{-1}} \\int_{\\gamma(\\mu_k)}\n(z-A_2)^{-1} dz,$$ where $\\gamma(\\lambda_k)$ is a circle centered at\n$\\lambda_k$ with sufficiently small radius such that the disc bounded by\n$\\gamma(\\lambda_k)$ does not contain any points in\n$\\sigma(A_1)\\setminus \\{\\lambda_k\\}$, and $\\gamma(\\mu_k)$ is a similar\nsufficiently small circle centered at $\\mu_k$. Then\n$P_k:L^2(\\Omega) \\longrightarrow L^2(\\Omega)$ is a bounded linear\noperator to a finite dimensional space and $P_k^2 = P_k$ and\n$P_kP_{\\ell} = 0$ for $k, \\ell\\in \\mathbb{N}$ with $k \\ne \\ell$. Then\n$P_kL^2(\\Omega) = \\{ b\\in \\mathcal{D}(A_1);\\, A_1b=\\lambda_kb\\}$, and we\nhave $a = \\sum_{k=1}^{\\infty} P_ka$ in $L^2(\\Omega)$ for each\n$a \\in L^2(\\Omega)$ (e.g., Agmon , Kato ). Setting\n$m_k:= \\mbox{dim}\\, P_kL^2(\\Omega)$, we have $m_k<\\infty$, and we call\n$m_k$ the multiplicity of $\\lambda_k$. Similarly let $n_k$ and $Q_k$ be\nthe multiplicity and the eigenprojection for $\\mu_k$.  \nMoreover we set $Q:= \\Omega\\times (0,T)$, and\n$$H^{2,1}(Q):= \\{ w\\in L^2(Q);\\, \nw, \\, \\partial_iw,\\, \\partial_i\\partial_jw,\\, \\partial_tw \\in L^2(Q)\n\\,\\, \\mbox{for $1\\le i,j\\le n$}\\}.$$\n\nLet $$\\Gamma=\\{x\\in \\gamma;\\, \\vert a(x)\\vert >0\\}.$$\n\nWe assume that $$\\Gamma\\ne \\emptyset. \\eqno{(1.4)}$$ For\n$a\\in C(\\overline{\\Omega})$, we set\n$$\\Omega_0 := \\{ x\\in \\Omega;\\, \\vert a(x)\\vert > 0\\}.$$\n\nFor $\\Gamma$, we define\n$$\\mbox{$\\omega: = \\{ x\\in \\Omega_0;\\,$ there exist a point $x_*\\in \\Gamma$ \nand}$$ $$\\mbox{a smooth curve $\\ell \\in C^\\infty[0,1]$ such that  \n$\\ell(\\xi) \\in \\Omega_0$ for $0<\\xi\\le 1$ and $\\ell(0)=x_*$, $\\ell(1)=x\\}$}.\n                                    \\eqno{(1.5)}$$ We remark that the\ndefinition (1.5) implies $\\ell \\setminus \\{x_*\\}\n\\subset \\omega$.\n\nIn (1.5), replacing smooth curves by piecewise smooth curves, we still\nhave the same definition for $\\omega$.\n\nWe note that $\\omega$ is not necessarily a connected set. However, if in\naddition we suppose that\n$$\\mbox{$\\Gamma$ is a connected subset of $\\partial\\Omega$},  \\eqno{(1.6)}$$\nthen one can verify that $\\omega \\subset \\Omega$ is a domain, that is, a\nconnected open set. Indeed, choosing $x, \\widetilde{x}\\in \\omega$\narbitrarily, we will show that we can find a piecewise smooth curve\n$L \\subset \\omega$ connecting $x$ and $\\widetilde{x}$ as follows. First\nwe can choose smooth curves $\\ell, \\widetilde{\\ell}\n\\subset \\Omega_0 \\cup \\Gamma$ and points\n$x_*, \\widetilde{x}_* \\in \\Gamma$ such that $\\ell$ connects $x$ and\n$x_*$, $\\widetilde{\\ell}$ connects $\\widetilde{x}$ and\n$\\widetilde{x}_*$. The definition implies that $\\ell \\setminus \\{x_*\\},\n\\widetilde{\\ell} \\setminus \\{ \\widetilde{x_*}\\} \\subset \\omega$. Since\n$\\vert a\\vert >0$ in $\\Gamma$, we can find a smooth curve\n$\\widetilde{\\gamma} \\subset \\Omega_0$ connecting $x_*$ and\n$\\widetilde{x}_*$. Therefore, since $\\widetilde{\\gamma} \\subset \\omega$,\nit follows that $x$ and $\\widetilde{x}$ can be connected by a piecewise\nsmooth curve $L \\subset \\omega$ composed by\n$\\ell, \\widetilde{\\ell}, \\widetilde{\\gamma}$, which means that $\\omega$\nis a connected set. Moreover, if $x\\in \\omega$, then we see that any\npoint $\\widetilde{x} \\in \\Omega_0$ which is sufficiently close to $x$,\ncan be connected to some point $\\widetilde{x}_*\n\\in \\Gamma$ by some smooth curve in $\\Omega_0$. Therefore, $\\omega$ is a\nconnected and open set, that is, $\\omega$ is a domain. $\\blacksquare$\n\nWe can understand that $\\omega$ is the maximal set such that all the\npoints of $\\omega$ is connected by a curve in $\\Omega_0$ to $\\Gamma$. By\n(1.4), we note that $\\omega \\ne \\emptyset$.  \n**Examples.**  \n(i) Under condition (1.4), we have $\\omega = \\Omega$ if\n$\\Omega_0 = \\Omega$. In general, if $\\{ x\\in \\Omega;\\, a(x) = 0\\}$ has\nno interior points, then $\\omega = \\Omega_0$.  \n(ii) Assume that (1.4) and (1.6) hold true. Let subdomains\n$D_1, ..., D_m \\subset \\Omega$ satisfy\n$\\overline{D_1}, ..., \\overline{D_2} \\subset \\Omega$ and $a=0$ on\n$\\overline{D_k}$ for $1\\le k \\le m$ and $\\vert a\\vert > 0$ in\n$\\Omega\\setminus \\overline{\\bigcup_{k=1}^m D_k}$. Then\n$\\omega = \\Omega\\setminus \\overline{\\bigcup_{k=1}^m D_k}$.  \n(iii) Assume that (1.4) and (1.6) hold true. Let sudomains $D_1, D_2$\nsatisfy $\\overline{D_1} \\subset D_2$, $\\overline{D_2} \\subset \\Omega$,\n$a=0$ in $\\overline{D_2 \\setminus D_1}$ and $\\vert a\\vert > 0$ in\n$D_1 \\cup (\\Omega\\setminus \\overline{D_2})$. Then\n$\\omega = \\Omega\\setminus \\overline{D_2}$. We note that $D_1$ is not\nincluded in $\\omega$ although $\\vert a\\vert > 0$ in $D_1$.  \nNow we state the main uniqueness result.  \n**Theorem 1.**  \n*Let $a\\in C(\\overline\\Omega),$ $u,v \\in H^{2,1}(Q)$ satisfy (1.2) and\n(1.3) respectively and $\\partial_tu, \\partial_tv \\in H^{2,1}(Q)$ and let\n(1.4) hold true. Assume  \n**Condition 1:** there exists a function $\\theta \\in C[1,\\infty)$\nsatisfying\n$$\\lim_{\\eta\\to\\infty} \\frac{\\theta(\\eta)}{\\eta^{\\frac{2}{3}}} = +\\infty$$\nand\n$$\\sum_{k=1}^{\\infty}e^{\\theta(\\lambda_k)} \\Vert P_ka\\Vert^2_{L^2(\\Omega)} < \\infty \n\\quad \\mbox{or}\n\\quad \\sum_{k=1}^{\\infty}e^{\\theta(\\mu_k)} \\Vert Q_ka\\Vert^2_{L^2(\\Omega)} < \\infty.\n                                         \\eqno{(1.7)}$$ Then,\n$$u=v \\quad \\mbox{on $\\gamma \\times (0,T)$}$$ implies $c_1=c_2$ on\n$\\overline{\\omega}$.*\n\nAs is seen by the proof, without the assumption (1.7), we can prove at\nleast the coincidence of the eigenvalues of $A_1$ and $A_2$ of\nnon-vanishing modes:  \n**Corollary.**  \n*Let $u=v$ on $\\gamma \\times (0,T)$. Then\n$$\\{ \\lambda_k;\\, k\\in \\mathbb{N}, \\, P_ka \\ne 0 \\quad \\mbox{in $\\Omega$}\\}\n= \\{ \\mu_k;\\, k\\in \\mathbb{N}, \\, Q_ka \\ne 0 \\quad \\mbox{in $\\Omega$}\\}$$\nand if $P_ka \\ne 0$ in $\\Omega$ for $k\\in \\mathbb{N}$, then\n$$P_ka = Q_ka \\quad \\mbox{on $\\gamma$},$$ after suitable re-numbering of\n$k$.*\n\nThe corollary means that $u=v$ on $\\gamma \\times (0,T)$ implies that\nthere exists $N_1 \\in \\mathbb{N}\\cup \\{\\infty\\}$ such that we can find\nsequences $\\{i_k\\}_{1\\le k\\le N_1}, \\, \\{j_k\\}_{1\\le k\\le N_1}\n\\subset \\mathbb{N}$ satisfying $$\\left\\{ \\begin{array}{rl}\n& \\lambda_{i_k} = \\mu_{j_k}, \\quad P_{i_k}a \\ne 0, \\,\\, \nQ_{j_k}a \\ne 0 \\quad \\mbox{in $\\Omega$},\\quad\n   P_{i_k}a = Q_{j_k}a = 0 \\quad \\mbox{on $\\gamma$}\n\\quad \\mbox{for $1\\le k \\le N_1$}, \\\\\n& P_ia = 0 \\quad\\mbox{in $\\Omega$ if $i\\not\\in \\{ i_k\\}_{1\\le k\\le N_1}$},\n\\quad\n Q_ja = 0 \\quad\\mbox{in $\\Omega$ if $j\\not\\in \\{ j_k\\}_{1\\le k\\le N_1}$}.\n\\end{array}\\right.$$ We remark that even in the case $N_1=\\infty$, we\nmay have $\\{ i_k\\}_{1\\le k \\le N_1} \\subsetneqq \\mathbb{N}$.  \n**Remark.** In (1.7), consider a function $\\theta(\\eta) = \\eta^p$.\nTheorem 1 asserts the uniqueness if the initial value $a$ is smooth in\nthe sense (1.7). We emphasize that in (1.7), the critical exponent of\n$\\lambda_k$ should be greater than $\\frac{2}{3}$. If we assume the\nstronger condition $p=1$, that is,\n$$\\sum_{k=1}^{\\infty}e^{\\sigma \\lambda_k} \\Vert P_ka\\Vert^2_{L^2(\\Omega)} < \\infty \\quad \\mbox{and}\n\\quad \\sum_{k=1}^{\\infty}e^{\\sigma \\mu_k} \\Vert Q_ka\\Vert^2_{L^2(\\Omega)} < \\infty\n                                         \\eqno{(1.8)}$$ with some\nconstant $\\sigma>0$, then the uniqueness is trivial because we can\nextend the solutions $u(\\cdot,t)$ and $v(\\cdot,t)$ to the time interval\n$(-\\delta, 0)$ with small $\\delta > 0$. Indeed, since\n$\\sum_{k=1}^{\\infty}\\vert e^{\\frac{1}{2}\\sigma \\lambda_k}\\vert^2 \\Vert P_ka\\Vert^2_{L^2(\\Omega)} \n< \\infty$, we can verify that\n$u(\\cdot,t) = \\sum_{k=1}^{\\infty}e^{-\\lambda_kt}P_ka$ in $L^2(\\Omega)$\nfor $t > -\\frac{1}{2}\\sigma$. Therefore we can extend $u(\\cdot,t)$ to\n$\\left(-\\frac{\\sigma}{2},\\, 0\\right)$ in $L^2(\\Omega)$ and also to\n$(-\\delta, 0)$ with sufficiently small $\\delta>0$. The extension of\n$v(\\cdot,t)$ is similarly done. Therefore, under (1.8), our inverse\nproblem is reduced to the case where the spatial data of $u,v$ are given\nat an intermediate time of the whole time interval under consideration,\nwhich has been already solved in Bukhgeim and Klibanov , Imanuvilov and\nYamamoto , Isakov .\n\nCondition corresponding to the case $p=\\frac{1}{2}$ in (1.7) appears in\nthe controllability of a parabolic equation. We know that a function\n$a(\\cdot)$ in $\\Omega$ satisfying the condition (1.7) with\n$\\theta(\\eta) = \\eta^{\\frac{1}{2}}$ and $c_1\\equiv 0$ belongs to the\nreachable set\n$$\\{ u(\\cdot,0);\\, b\\in L^2(\\Omega),\\, h \\in L^2(\\partial\\Omega\\times (-\\tau,0)\\},$$\nwhere $u$ is the solution to $$\\left\\{ \\begin{array}{rl}\n& \\partial_tu = \\Delta u \\quad \\mbox{in $\\Omega\\times (-\\tau,0)$}, \\\\\n& \\partial_{\\nu}u = h \\quad \\mbox{on $\\partial\\Omega\\times (-\\tau,0)$},\\\\\n& u(\\cdot,-\\tau) = b \\quad \\mbox{in $\\Omega$}\n\\end{array}\\right.$$ (Theorem 2.3 in Russell ). See also (1.9) stated\nbelow.  \nThe article is composed of four sections. In Section 2, we show Carleman\nestimates for elliptic operator. In Section 3, we prove the uniqueness\nfor our inverse problem first under a condition: there exists a constant\n$\\sigma_1>0$ such that\n$$\\sum_{k=1}^{\\infty}e^{\\sigma_1 \\lambda_k^{\\frac{1}{2}}}\\Vert P_ka\\Vert^2_{L^2(\\Omega)} \n+ \\sum_{k=1}^{\\infty}e^{\\sigma_1 \\mu_k^{\\frac{1}{2}}} \\Vert Q_ka\\Vert^2_{L^2(\\Omega)} < \\infty,\n                                         \\eqno{(1.9)}$$ and next by\nproving that (1.7) yields (1.9), we complete the proof of Theorem 1. In\nSection 4, we prove a Carleman estimate used for deriving (1.9) from\n(1.7).\n\n# Key Carleman estimate\n\nThe proof of Theorem 1 relies essentially on the reduction of our\ninverse parabolic problem to an inverse elliptic problem. After the\nreduction, we prove the uniqueness by the method developed in or, , ,\nand so we need a relevant Carleman estimate for an elliptic equation.\nFor the statement of Carleman estimate, we introduce a weight function.\n\nWe arbitrarily fix $y \\in \\omega$. For $y$, we construct a non-empty\ndomain $\\omega_y \\subset \\Omega$ satisfying $$\\left\\{ \\begin{array}{rl}\n&\\mbox{(i)} \\,\\,y \\in \\omega_y, \\quad \\omega_y \\subset \\omega. \\\\\n&\\mbox{(ii)} \\,\\,\\mbox{$\\partial\\omega_y$ is of $C^{\\infty}$-class.}\\\\\n&\\mbox{(iii)} \\,\\, \\mbox{$\\partial\\omega_y \\cap \\Gamma$ has interior points \nin the topology of $\\partial\\Omega$.} \\\\\n&\\mbox{(iv)} \\,\\, \\vert a(x)\\vert > 0 \\quad \\mbox{for all \n  $x\\in \\overline{\\omega_y}$}.\n\\end{array}\\right.\n                                        \\eqno{(2.1)}$$ Indeed, since\n$y\\in \\omega$, by the definition of $\\omega$, we can find\n$y_*\\in \\Gamma$ and a smooth curve $\\ell \\in C^{\\infty}[0,1]$ such that\n$\\ell(1) = y$ and $\\ell(0) = y_*$, $\\ell(\\xi) \\in \\omega$ for\n$0<\\xi \\le 1$. Then as $\\omega_y$, we can choose a sufficiently thin\nneighborhood of the curve $\\{\\ell(\\xi);\\, 0<\\xi\\le 1\\}$ which is\nincluded in $\\omega$.\n\nFor the proof of Theorem 1, we will show that if $y\\in\\omega$ then\n$y\\notin \\mbox{supp}\\, f.$ This of course implies that $f=0$ on\n$\\omega$. First we establish a Carleman estimate in\n$\\omega_y \\times (-\\tau,\\tau)$ with a constant $\\tau > 0$.\n\nWe know that there exists a function $d\\in C^2(\\overline{\\omega_y})$\nsuch that\n$$\\vert \\nabla d(x)\\vert > 0 \\quad \\mbox{for $x\\in \\overline{\\omega_y}$}, \\quad\nd(x) > 0  \\quad \\mbox{for $x\\in \\omega_y$}, \\quad\nd(x) = 0 \\quad \\mbox{for $x\\in \\partial\\omega_y\\setminus \\Gamma$}.\n                                                           \\eqno{(2.2)}$$\nThe existence of such $d$ is proved for example in Imanuvilov . See also\nFursikov and Imanuvilov .\n\nFor a constant $\\tau>0$, we set\n$${\\mathcal Q}_\\tau:= \\omega_y \\times (-\\tau, \\, \\tau),$$\n$\\partial_0 := \\frac{\\partial}{\\partial t}$, and\n$$\\alpha(x,t) := e^{\\lambda(d(x) - \\beta t^2)}, \n\\quad (x,t)\\in {\\mathcal Q}_{\\tau}                           \\eqno{(2.3)}$$\nwith an arbitrarily chosen constant $\\beta > 0$ and sufficiently large\nfixed $\\lambda> 0$. Then  \n**Lemma 2.1 (elliptic Carleman estimate).**  \n*There exists a constant $s_0 > 0$ such that we can find a constant\n$C>0$ such that $$\\begin{aligned}\n& \\int_{{\\mathcal Q}_\\tau} \\left\\{ \\frac{1}{s}\\sum_{i,j=0}^n \\vert \\partial_i\\partial_jw\\vert^2\n+ s\\vert \\partial_tw\\vert^2 + s\\vert \\nabla w\\vert^2\n+ s^3\\vert w\\vert^2\\right) e^{2s\\alpha} dxdt\\\\\n\\le& C\\int_{{\\mathcal Q}_\\tau} \\vert \\partial_t^2w - A_1w\\vert^2 e^{2s\\alpha} dxdt\n\\end{aligned}$$ for all $s \\ge s_0$ and\n$w\\in H^2_0({\\mathcal Q}_\\tau)$.*   \nHere we recall that\n$-A_1w = \\sum_{i,j=1}^n\\partial_i(a_{ij}(x)\\partial_jw)\n+ c_1(x)w$. The constants $s_0>0$ and $C>0$ can be chosen uniformly\nprovided that $\\Vert c_1\\Vert_{L^{\\infty}(\\omega)} \\le M$: arbitrarily\nfixed constant $M>0$.\n\nWe note that Lemma 2.1 is a Carleman estimate for the elliptic operator\n$\\partial_t^2 - A_1w$. Since $(\\nabla \\alpha, \\, \\partial_t\\alpha)\n= (\\nabla d, \\, -2\\beta t) \\ne (0,0)$ on\n$\\overline{{\\mathcal Q}_{\\tau}}$ by (2.2), the proof of the lemma relies\ndirectly on integration by parts and standard, similar for example to\nthe proof of Lemma 7.1 (p.186) in Bellassoued and Yamamoto . See also\nHörmander , Isakov , where the estimation of the second-order\nderivatives is not included but can be be derived by the a priori\nestimate for the elliptic boundary value problem.\n\nFor the proof of Theorem 1, we further need another Carleman estimate in\n$\\Omega$ for an elliptic equation. We can find\n$\\rho\\in C^2(\\overline{\\Omega})$ such that\n$$\\rho(x) > 0 \\quad \\mbox{for $x \\in\\Omega$}, \\quad\n\\vert \\nabla \\rho(x) \\vert > 0 \\quad \\mbox{for $x \\in\\overline{\\Omega}$}, \\quad\n\\partial_{\\nu_A}\\rho(x) \\le 0 \\quad \\mbox{for $x\\in \\partial\\Omega\\setminus \\gamma$}.\n                                                      \\eqno{(2.4)}$$ The\nconstruction of $\\rho$ can be found in Lemma 2.3 in for example.\nMoreover, fixing a constant $\\lambda>0$ large, we set\n$$\\psi(x):= e^{\\lambda(\\rho(x) - 2\\Vert \\rho\\Vert_{C(\\overline{\\Omega})})}, \\quad \nx\\in \\Omega.$$ Then  \n**Lemma 2.2.**  \n*There exist constants $s_0>0$ and $C>0$ such that\n$$\\int_{\\Omega} (s^3\\vert g\\vert^2 + s\\vert \\nabla g\\vert^2)e^{2s\\psi(x)} dx\n\\le C\\int_{\\Omega} \\vert A_2g\\vert^2 e^{2s\\psi(x)} dx \n+ cs^3\\int_{\\gamma} (\\vert g\\vert^2 + \\vert \\nabla g\\vert^2) e^{2s\\psi} dS$$\nfor all $s>s_0$ and $g \\in H^2(\\Omega)$ satisfying $\\partial_{\\nu_A}g=0$\non $\\partial\\Omega$.*\n\nWe postpone the proof of Lemma 2.2 to Section 4.\n\n# Proof of Theorem 1.1\n\nWe divide the proof into four steps. In Steps 1-3, we assume the\ncondition (1.9) to prove the conclusion on uniqueness in Theorem 1.1.  \n**First Step.**  \nWe write $u(t):= u(\\cdot,t)$ and $v(t):= v(\\cdot,t)$ for $t>0$. We\nrecall that $$u(t) = \\sum_{k=1}^{\\infty}e^{-\\lambda_kt}P_ka, \\quad\nv(t) = \\sum_{k=1}^{\\infty}e^{-\\mu_kt}Q_ka  \\quad \\mbox{in $H^2(\\Omega)$ for $t>0$.}$$\n\nWe can choose subsets $\\mathbb{N}_1, \\mathbb{M}_1 \\subset \\mathbb{N}$\nsuch that\n$$\\mathbb{N}_1:= \\{k \\in \\mathbb{N};\\, P_ka \\not\\equiv 0 \\quad \\mbox{in $\\Omega$}\\}, \\quad\n\\mathbb{M}_1:= \\{k \\in \\mathbb{N};\\, Q_ka \\not\\equiv 0 \\quad \\mbox{in $\\Omega$}\\}.\n                                           \\eqno{(3.1)}$$ We note that\n$\\mathbb{N}_1 = \\mathbb{N}$ or $\\mathbb{M}_1 = \\mathbb{N}$ may happen.\n\nWe can renumber the sets $\\mathbb{N}_1$ and $\\mathbb{M}_1$ as\n$$\\mathbb{N}_1 = \\{1, ...., N_1\\}, \\quad \\mathbb{M}_1=\\{ 1, ...., M_1\\},$$\nwhere $N_1 = \\infty$ or $M_1 = \\infty$ may occur. By\n$^{\\sharp}\\mathbb{N}_1$ we mean the cardinal number of the set\n$\\mathbb{N}_1$. Wote that\n$$\\lambda_1< \\lambda_2 < \\cdots < \\lambda_{N_1} \\quad \\mbox{if $^{\\sharp}\\mathbb{N}_1 < \\infty$} \\quad\n\\lambda_1< \\lambda_2 < \\cdots \\quad \\mbox{if $^{\\sharp}\\mathbb{N}_1 = \\infty$}$$\nand\n$$\\mu_1< \\mu_2 < \\cdots < \\mu_{M_1} \\quad \\mbox{if $^{\\sharp}\\mathbb{M}_1 < \\infty$} \\quad\n\\mu_1< \\mu_2 < \\cdots \\quad \\mbox{if $^{\\sharp}\\mathbb{M}_1 = \\infty$}.$$\n\nAssuming that $u=v$ on $\\gamma \\times (0,T)$, by the time analyticity of\n$u(t)$ and $v(t)$ for $t>0$ (e.g., Pazy ), we obtain\n$$\\sum_{k=1}^{N_1} e^{-\\lambda_kt}P_ka = \\sum_{k=1}^{M_1} e^{-\\mu_kt}Q_ka \n\\quad \\mbox{on $\\gamma \\times (0,\\infty)$}.    \n                                                \\eqno{(3.2)}$$ We will\nprove that $\\lambda_1 = \\mu_1$. Assume that $\\lambda_1 < \\mu_1$. Then\n$$P_1a + \\sum_{k=2}^{N_1} e^{-(\\lambda_k-\\lambda_1)t}P_ka \n= \\sum_{k=1}^{M_1} e^{-(\\mu_k-\\lambda_1)t}Q_ka \n\\quad \\mbox{on $\\gamma \\times (0,\\infty)$}.$$ Since\n$\\lambda_k - \\lambda_1 > 0$ for $2\\le k \\le N_1$ and\n$\\mu_k - \\lambda_1 > 0$ for $1\\le k \\le M_1$, letting $t\\to \\infty$, we\nsee that $P_1a=0$ on $\\Gamma$. Therefore $$\\left\\{ \\begin{array}{rl}\n& (A_1-\\lambda_1)P_1a = 0 \\quad \\mbox{in $\\Omega$}, \\\\\n& P_1a\\vert_{\\Gamma} = 0, \\quad \\partial_{\\nu_A}P_1a\\vert_{\\partial\\Omega} = 0.\n\\end{array}\\right.$$ The unique continuation for the elliptic equation\n$A_1P_1a = \\lambda_1P_1a$, (see e.g. ) yields that\n$$P_1a = 0 \\quad \\mbox{in $\\Omega$}.$$ This is a contradiction by\n$1 \\in \\mathbb{N}_1$. Thus the inequality $\\lambda_1 < \\mu_1$ is\nimpossible. Similarly we can see that the inequality $\\lambda_1 > \\mu_1$\nis impossible. Therefore $\\lambda_1 = \\mu_1$ follows.\n\nBy (3.2) and $\\lambda_1 = \\mu_1$, we have $$P_1a - Q_1a \n= -\\sum_{k=2}^{N_1} e^{-(\\lambda_k-\\lambda_1)t}P_ka \n+ \\sum_{k=2}^{M_1} e^{-(\\mu_k-\\lambda_1)t}Q_ka \n\\quad \\mbox{on $\\gamma \\times (0,\\infty)$}.$$ Hence, by\n$\\lambda_k-\\lambda_1 > 0$ and $\\mu_k - \\lambda_1 = \\mu_k - \\mu_1 > 0$\nfor all $k \\ge 2$, letting $s \\to \\infty$ we obtain $P_1a = Q_1a$ on\n$\\gamma$.\n\nIn view of (3.2), we obtain $$\\sum_{k=2}^{N_1} e^{-\\lambda_kt}P_ka \n= \\sum_{k=2}^{M_1} e^{-\\mu_kt}Q_ka \n\\quad \\mbox{on $\\gamma \\times (0,\\infty)$}.$$ Repeating the same\nargument as much as possible, we reach\n$$N_1 = M_1, \\quad \\lambda_k = \\mu_k, \\quad\nP_ka = Q_ka \\quad \\mbox{on $\\gamma \\times (0,\\infty)$ for\n$1\\le k \\le N_1$}.                              \\eqno{(3.3)}$$  \n**Second Step.**\n\nWe consider two initial boundary value problems for elliptic equations:\n$$\\left\\{ \\begin{array}{rl}\n& \\partial_t^2w_1 - A_1w_1 = 0 \\quad \\mbox{in $\\Omega\\times (0,\\tau)$}, \\\\\n& \\partial_{\\nu_A}w_1 = 0 \\quad \\mbox{on $\\partial\\Omega\\times (0,\\tau)$}, \\\\\n& w_1(x,0) = a(x), \\quad \\partial_tw_1(x,0) = 0, \\quad x\\in \\Omega\n\\end{array}\\right.\n                                 \\eqno{(3.4)}$$ and\n$$\\left\\{ \\begin{array}{rl}\n& \\partial_t^2w_2 - A_2w_2 = 0 \\quad \\mbox{in $\\Omega\\times (0,\\tau)$}, \\\\\n& \\partial_{\\nu_A}w_2 = 0 \\quad \\mbox{on $\\partial\\Omega\\times (0,\\tau)$}, \\\\\n& w_2(x,0) = a(x), \\quad \\partial_tw_2(x,0) = 0, \\quad x\\in \\Omega.\n\\end{array}\\right.\n                                \\eqno{(3.5)}$$ Since we have the\nspectral representations by (1.9), we can obtain\n$$e^{-tA_1^{\\frac{1}{2}}}a = \\sum_{k=1}^{\\infty}e^{-\\lambda_k^{\\frac{1}{2}}t}P_ka, \\quad\ne^{tA_1^{\\frac{1}{2}}}a = \\sum_{k=1}^{\\infty}e^{\\lambda_k^{\\frac{1}{2}}t}P_ka \\quad \n\\mbox{in $L^2(\\Omega)$ for $t>0$}$$ and similar representations hold for\n$e^{\\pm tA_2^{\\frac{1}{2}}}a$.\n\nThen by the assumption (1.9) on $a$, we see that\n$$w_1(t) = \\frac{1}{2}(e^{-tA_1^{\\frac{1}{2}}}a + e^{tA_1^{\\frac{1}{2}}}a), \\quad\nw_2(t) = \\frac{1}{2}(e^{-tA_2^{\\frac{1}{2}}}a + e^{tA_2^{\\frac{1}{2}}}a)$$\nin $H^2(\\Omega\\times (0,\\tau))$ for $t \\in (0,\\tau)$, satisfy (3.4) and\n(3.5) respectively if $\\tau>0$ is chosen sufficiently small.\n\nIn view of (3.3) and the definition (3.1) of $\\mathbb{N}_1$, the\nspectral representations imply $$\\begin{aligned}\n& w_1(x,t) = \\frac{1}{2}\\sum_{k=1}^{N_1} (e^{-\\lambda_k^{\\frac{1}{2}}t}P_ka\n+ e^{\\lambda_k^{\\frac{1}{2}}t}P_ka)\n+ \\frac{1}{2}\\sum_{k\\in \\mathbb{N}\\setminus \\{1, ..., N_1\\}}(e^{-\\lambda_k^{\\frac{1}{2}}t}P_ka\n+ e^{\\lambda_k^{\\frac{1}{2}}t}P_ka)\\\\\n=& \\frac{1}{2}\\sum_{k=1}^{N_1} (e^{-\\lambda_k^{\\frac{1}{2}}t}P_ka\n+ e^{\\lambda_k^{\\frac{1}{2}}t}P_ka) \\quad \\mbox{in $\\Omega\\times (0,\\tau)$,}\n\\end{aligned}$$ and\n$$w_2(x,t) = \\frac{1}{2}\\sum_{k=1}^{N_1} (e^{-\\lambda_k^{\\frac{1}{2}}t}Q_ka\n+ e^{\\lambda_k^{\\frac{1}{2}}t}Q_ka) \\quad \\mbox{in $\\Omega\\times (0,\\tau)$}.$$\nTherefore (3.3) yields\n$$w_1 = w_2 \\quad \\mbox{on $\\gamma \\times (0,\\tau)$.}       \\eqno{(3.6)}$$\n\nNow we reduce our inverse problem for the parabolic equations to the one\nfor elliptic equations for (3.4) and (3.5). This is the essence of the\nproof.  \n**Third Step.**  \nBy (1.9), we can readily verify further regularity\n$\\partial_tw_1, \\partial_tw_2\n\\in H^2(0,\\tau;H^2(\\Omega))$. Setting $y:= w_1 - w_2$ and $R:=w_2$ in\n$\\Omega\\times (0,\\tau)$ and $f:= c_2-c_1$ in $\\Omega$, by (3.4) -(3.6)\nwe have $$\\left\\{ \\begin{array}{rl}\n& \\partial_t^2y - A_1y = f(x)R(x,t) \\quad \\mbox{in $\\Omega\\times (0,\\tau)$}, \\\\\n& \\partial_{\\nu_A}y = 0 \\quad \\mbox{on $\\partial\\Omega\\times (0,\\tau)$}, \\\\\n& y=0 \\quad \\mbox{on $\\gamma \\times (0,\\tau)$}, \\\\\n& y(x,0) = \\partial_ty(x,0) = 0, \\quad x\\in \\Omega.\n\\end{array}\\right.\n                                                   \\eqno{(3.7)}$$\n\nNow we will prove that for any pair $(y,f)$ solving problem (3.7) we\nhave $y\\notin \\mbox{supp}, f.$ Since $y$ was chosen as an arbitrary\npoint from $\\omega$ this implies $$f=0\\quad \\mbox{in}\\quad \\omega.$$ The\nargument relies on .\n\nWe set $$\\widetilde{y}(x,t) = \n\\left\\{ \\begin{array}{rl}\n& y(x,t), \\quad 0<t<\\tau, \\\\\n& y(x,-t), \\quad -\\tau<t<0\n\\end{array}\\right.$$ and $$\\widetilde{R}(x,t) = \n\\left\\{ \\begin{array}{rl}\n& R(x,t), \\quad 0<t<\\tau, \\\\\n& R(x,-t), \\quad -\\tau<t<0.\n\\end{array}\\right.$$ Then we see that\n$\\widetilde{y} \\in H^3(-\\tau,\\tau;H^2(\\omega_y))$ by\n$y(\\cdot,0) = \\partial_ty(\\cdot,0) = 0$ in $\\omega_y$, and\n$$\\left\\{ \\begin{array}{rl}\n& \\partial_t^2\\widetilde{y} - A_1\\widetilde{y} = f(x)\\widetilde{R}(x,t) \n\\quad \\mbox{in ${\\mathcal Q}_{\\tau}:= \\omega_y\\times (-\\tau,\\tau)$}, \\\\\n& \\widetilde{y}= \\partial_{\\nu_A}\\widetilde{y} = 0 \\quad \\mbox{on $(\\partial\\omega_y \\cap \\Gamma) \n\\times (-\\tau,\\tau)$}, \\\\\n& \\widetilde{y}(x,0) = \\partial_t\\widetilde{y}(x,0) = 0, \\quad x\\in \\omega_y.\n\\end{array}\\right.$$ Setting $\\widetilde{z}:= \\partial_t\\widetilde{y}$,\nwe have $\\widetilde{z} \\in H^2(Q_{\\tau})$ and\n$$\\left\\{ \\begin{array}{rl}\n& \\partial_t^2\\widetilde{z} - A_1\\widetilde{z} = f(x)\\partial_t\\widetilde{R}(x,t) \n\\quad \\mbox{in ${\\mathcal Q}_{\\tau}$}, \\\\\n& \\widetilde{z} = \\partial_{\\nu_A}\\widetilde{z} = 0 \\quad \\mbox{on $(\\partial\\omega_y \\cap \\Gamma)\n\\times (-\\tau,\\tau)$}, \\\\\n& \\widetilde{z}(x,0) = 0, \\quad x\\in \\omega_y\n\\end{array}\\right.\n                                     \\eqno{(3.8)}$$ and\n$$\\partial_t\\widetilde{z}(x,0) = f(x)a(x), \\quad x\\in \\omega_y.     \\eqno{(3.9)}$$\nIn the weight function $d(x) - \\beta t^2$ in Lemma 2.1, for $\\tau>0$ we\nchoose $\\beta > 0$ sufficiently large, so that\n$$\\Vert d\\Vert_{C(\\overline{\\omega_y})} - \\beta \\tau^2 < 0,   \\eqno{(3.10)}$$\nWe choose constants $\\delta_1, \\delta_2 > 0$ such that\n$$\\Vert d\\Vert_{C(\\overline{\\omega_y})} - \\beta \\tau^2 < 0 < \\delta_1 < \\delta_2 \n   \\quad \\mbox{and}\\quad d(y)>\\delta_2                           \\eqno{(3.11)}$$\nand we define $\\chi \\in C^{\\infty}(\\overline{{\\mathcal Q}_{\\tau}})$\nsatisfying $$\\chi(x,t) = \n\\left\\{ \\begin{array}{rl}\n& 1, \\quad d(x) - \\beta t^2 > \\delta_2, \\\\\n& 0, \\quad d(x) - \\beta t^2 < \\delta_1. \n\\end{array}\\right.\n                                              \\eqno{(3.12)}$$ In\nparticularl, $d(y) > \\delta_2$ implies $$\\chi(y,0)=1.$$\n\nSetting $z:= \\chi \\widetilde{z}$ on $\\overline{{\\mathcal Q}_{\\tau}}$, we\nsee that\n$$z = \\partial_{\\nu_A}z = 0 \\quad \\mbox{on $\\partial\\omega_y \\times (-\\tau, \\tau)$}\n                                                  \\eqno{(3.13)}$$ and\n$$z = \\partial_t z = 0 \\quad \\mbox{on $\\omega_y \\times \\{ \\pm \\tau\\}$}.\n                                                \\eqno{(3.14)}$$ Indeed,\n$(x,t) \\in (\\partial\\omega_y \\setminus \\Gamma) \\times (-\\tau, \\tau)$\nimplies $d(x) - \\beta t^2 = -\\beta t^2 \\le 0 < \\delta_1$ by (2.2), and\nso the definition (3.12) of $\\chi$ yields that $\\chi(x,t) = 0$ in a\nneighborhood of such $(x,t)$. For\n$(x,t) \\in (\\partial\\omega_y \\cap \\Gamma) \\times (-\\tau,\\tau)$, by (3.8)\nwe see that $z(x,t) = \\partial_{\\nu_A}z(x,t) =  0$, which verifies\n(3.13). Moreover, on $\\omega_y\\times \\{ \\pm \\tau\\}$, by (3.11) we have\n$$d(x) - \\beta t^2 \\le \\Vert d\\Vert_{C(\\overline{\\omega_y})} - \\beta\\tau^2\n< 0 < \\delta_1,$$ so that $\\chi(x,t) = 0$ in a neighborhood of such\n$(x,t)$. Thus (3.14) has been verified. $\\blacksquare$\n\nConsequently we prove that $z\\in H^2_0(Q_{\\tau})$. Moreover, we can\nreadily obtain $$\\left\\{ \\begin{array}{rl}\n& \\partial_t^2z - A_1z = \\chi(\\partial_t\\widetilde{R})f + R_0(x,t), \\quad \n(x,t) \\in {\\mathcal Q}_{\\tau}, \\\\\n& z = \\vert \\nabla z\\vert = 0 \\quad \\mbox{on $\\partial{\\mathcal Q}_{\\tau}$},\n\\end{array}\\right.\n                                                   \\eqno{(3.15)}$$ where\n$R_0$ is a linear combination of $\\nabla \\widetilde{z}$,\n$\\partial_t\\widetilde{z}$, whose coefficients are linear combinations of\n$\\nabla\\chi$ and $\\partial_t\\chi$. Therefore (3.12) implies\n$$R_0(x,t) \\ne 0 \\quad \\mbox{only if $\\delta_1 \\le d(x) - \\beta t^2\n\\le \\delta_2$}.                              \\eqno{(3.16)}$$\n\nTherefore we can apply Lemma 2.1 to (3.15) with (3.16):\n$$\\int_{{\\mathcal Q}_{\\tau}} \\left( \\frac{1}{s}\\vert \\partial_t^2z\\vert^2\n+ s\\vert \\partial_tz\\vert^2 \\right) e^{2s\\alpha} dxdt \n                                                          \\eqno{(3.17)}$$\n$$\\begin{aligned}\n\\le & C\\int_{{\\mathcal Q}_{\\tau}} \\vert \\chi(\\partial_t\\widetilde{R})f \\vert^2e^{2s\\alpha} dxdt\n+ C\\int_{{\\mathcal Q}_{\\tau}} \\vert R_0(x,t)\\vert^2 e^{2s\\alpha} dxdt \\\\\n\\le & C\\int_{{\\mathcal Q}_{\\tau}}\\chi^2 \\vert f\\vert^2 e^{2s\\alpha} dxdt \n+ Ce^{2se^{\\lambda\\delta_2}}\n\\end{aligned}$$ for all large $s>0$.\n\nOn the other hand, since $\\partial_tz(\\cdot,-\\tau) = 0$ in $\\omega_y$ by\n(3.14), we have $$\\begin{aligned}\n& \\int_{\\omega_y} \\vert \\partial_tz(x,0)\\vert^2 e^{2s\\alpha(x,0)} dx \n= \\int^0_{-\\tau} \\partial_t\\left( \\int_{\\omega_y} \\vert \\partial_tz(x,t)\\vert^2\ne^{2s\\alpha(x,t)} dx \\right) dt\\\\\n=& \\int^0_{-\\tau} \\int_{\\omega_y} \\{ 2(\\partial_tz)(x,t)\\partial_t^2z(z,t)\n+ \\vert \\partial_tz \\vert^2 2s(\\partial_t\\alpha))\\} e^{2s\\alpha(x,t)} dxdt.\n\\end{aligned}$$ Since\n$$\\vert (\\partial_tz)(\\partial_t^2z)\\vert \\le \\frac{1}{2}\n\\left( s\\vert \\partial_tz\\vert^2 + \\frac{1}{s}\\vert \\partial_t^2z\\vert^2\n\\right) \\quad \\mbox{in $\\mathcal{Q}_\\tau$},$$ in terms of (3.17) we\nobtain\n$$\\int_{\\omega_y} \\vert \\partial_tz(x,0)\\vert^2 e^{2s\\alpha(x,0)} dx \n\\le C\\int_{{\\mathcal Q}_{\\tau}} \\left( \\frac{1}{s} \\vert \\partial_t^2z(x,t)\\vert^2 \n+ s\\vert \\partial_tz\\vert^2 \\right) e^{2s\\alpha} dxdt \n                                                  \\eqno{(3.18)}$$\n$$\\le C\\int_{{\\mathcal Q}_{\\tau}}\\vert \\chi\\vert^2\\vert f\\vert^2 e^{2s\\alpha} \ndxdt + Ce^{2se^{\\lambda\\delta_2}}$$ for all large $s>0$.\n\nMoreover, we have $$\\begin{aligned}\n& \\partial_tz(x,0) = \\partial_t(\\chi\\widetilde{z})(x,0)\n= (\\partial_t\\chi)(x,0)\\widetilde{z}(x,0) + \\chi(x,0)\\partial_t\\widetilde{z}(x,0)\\\\\n=& \\chi(x,0)f(x)a(x)\n\\end{aligned}$$ by (3.9) and $\\widetilde{z}(x,0) = 0$ for\n$x \\in \\omega_y$ in (3.8). Therefore, in terms of (2.1)-(iv), we obtain\n$$\\vert \\partial_tz(x,0)\\vert \\ge C\\vert \\chi(x,0)f(x)\\vert, \\quad \nx\\in \\overline{\\omega_y}.$$ Consequently (3.18) implies\n$$\\int_{\\omega_y} \\vert\\chi(x,0)\\vert^2 \\vert f(x) \\vert^2 e^{2s\\alpha(x,0)} dx \n\\le C\\int_{\\mathcal{Q}_{\\tau}} \\vert \\chi f\\vert^2 e^{2s\\alpha} dxdt \n+ Ce^{2se^{\\lambda\\delta_2}}             \\eqno{(3.19)}$$ for all large\n$s>0$.\n\nMoreover, we see $$\\begin{aligned}\n&\\int_{{\\mathcal Q}_{\\tau}} \\vert \\chi f\\vert^2 e^{2s\\alpha} dxdt \n= \\int^{\\tau}_{-\\tau} \\int_{\\omega_y} \\vert\\chi(x,0)f(x)\\vert^2 e^{2s\\alpha} \ndxdt\\\\\n= & \\int_{\\omega_y} \\vert \\chi(x,0) f(x)\\vert^2 e^{2s\\alpha(x,0)} \n\\left( \\int^{\\tau}_{-\\tau} e^{2s(\\alpha(x,t) - \\alpha(x,0))} dt\\right)dx.\n\\end{aligned}$$ Since\n$$\\int^{\\tau}_{-\\tau} e^{2s(\\alpha(x,t) - \\alpha(x,0))} dt\n= \\int^{\\tau}_{-\\tau} e^{2se^{\\lambda d(x)}(e^{-\\lambda\\beta t^2} - 1)} dt\n\\le \\int^{\\tau}_{-\\tau} e^{Cs(e^{-\\lambda\\beta t^2} - 1)} dt\n= o(1)$$ as $s \\to \\infty$ by the Lebesgue convergence theorem. Hence,\n(3.19) yields\n$$\\int_{\\omega_y} \\vert \\chi(x,0)f(x)\\vert^2 e^{2s\\alpha(x,0)} dx\n\\le o(1)\\int_{\\omega_y} \\vert \\chi(x,0)f(x)\\vert^2 e^{2s\\alpha(x,0)} dx\n+ Ce^{2se^{\\lambda\\delta_2}},$$ and we can absorb the first term on the\nright-hand side into the left-hand side to reach\n$$\\int_{\\omega_y} \\vert \\chi(x,0)f(x)\\vert^2 e^{2s\\alpha(x,0)} dx\n\\le Ce^{2se^{\\lambda\\delta_2}}             \\eqno{(3.20)}$$ for all large\n$s>0$.\n\nHenceforth we set\n$B(y,\\varepsilon):= \\{ x;\\, \\vert x-y\\vert < \\varepsilon\\}$. Then, we\ncan choose $\\delta_3 > \\delta_2$ and a sufficiently small\n$\\varepsilon> 0$ such that $B(y,\\varepsilon) \\subset \\omega_y$ and\n$d(x) \\ge \\delta_3$ for all $x\\in B(y,\\varepsilon)$. This is possible,\nbecause $d(y) > \\delta_2$ in (3.11) and $\\omega_y$ is an open set\nincluding $y$.\n\nWe shrink the integration region of the left-hand side of (3.20) to\n$B(y,\\varepsilon)$ and obtain\n$$\\int_{B(y,\\varepsilon)} \\vert \\chi(x,0)f(x)\\vert^2 e^{2s\\alpha(x,0)} dx\n\\le Ce^{2se^{\\lambda\\delta_2}}$$ for all large $s>0$. Since\n$d(x) \\ge \\delta_3 > \\delta_2$ for $x\\in B(y,\\varepsilon)$, the\ncondition (3.12) yields $\\chi(x,0) = 1$ and\n$\\alpha(x,0) = e^{\\lambda d(x)}\n\\ge e^{\\lambda\\delta_3}$ for all $x\\in B(y,\\varepsilon)$. Therefore,\n$$\\left( \\int_{B(y,\\varepsilon)} \\vert f(x)\\vert^2 dx\\right) e^{2se^{\\lambda\\delta_3}}\n\\le Ce^{2se^{\\lambda\\delta_2}},$$ that is,\n$\\Vert f\\Vert^2_{L^2(B(y,\\varepsilon))} \\le e^{-2s(e^{\\lambda\\delta_3}\n- e^{\\lambda\\delta_2})}$ for all large $s>0$.\n\nIn terms of $\\delta_3 > \\delta_2$, letting $s\\to \\infty$, we see that\nthe right-hand side tends to $0$, and so $f=0$ in $B(y,\\varepsilon)$.\nSince $y$ is arbitrarily chosen, we reach $f=c_2-c_1 = 0$ in $\\omega$.\nThus the conclusion of Theorem 1.1 is proved under condition (1.9).\n$\\blacksquare$  \n**Fourth Step.**  \nWe will complete the proof of Theorem 1.1 by demonstrating that (1.7)\nimplies (1.9). Without loss of generality, we can assume\n$$\\sum_{k=1}^{\\infty}e^{\\theta(\\lambda_k)}\\Vert P_ka\\Vert^2_{L^2(\\Omega)} < \\infty.\n                                     \\eqno{(3.21)}$$ It suffices to\nprove that there exists a constant $\\sigma_1>0$ such that\n$$\\sum_{k=1}^{\\infty} e^{\\sigma_1\\lambda_k^{\\frac{1}{2}}}\\Vert Q_ka\\Vert^2_{L^2(\\Omega)} < \\infty,\n                                                     \\eqno{(3.22)}$$\nwith the assumption that the set of $k\\in \\mathbb{N}$ such that\n$Q_ka\\ne 0$ in $\\Omega$ is infinite. For simplicity, we can consider the\ncase where $P_ka \\ne 0$ in $\\Omega$ for all $k\\in \\mathbb{N}$. We can\nargue similarly in the rest cases. Then, by Corollary which was already\nproved in First Step, we choose a subset\n$\\mathbb{M}_1 \\subset \\mathbb{N}$ such that\n$$\\{ \\lambda_i\\}_{i\\in\\mathbb{N}} = \\{ \\mu_j\\}_{j\\in \\mathbb{M}_1}, \\quad\nQ_ja = 0 \\quad \\mbox{in $\\Omega$ for $j\\in \\mathbb{N}\\setminus \\mathbb{M}_1$}.$$\nNow it suffices to prove (3.22) in the case where $Q_ka\\ne 0$ in\n$\\Omega$ for all $k\\in \\mathbb{N}$.\n\nAfter re-numbering, we can obtain\n$$\\lambda_k = \\mu_k, \\quad P_ka = Q_ka \\quad \\mbox{on $\\gamma$ for all\n$k\\in \\mathbb{N}$}.                              \\eqno{(3.23)}$$ The\ntrace theorem and the a priori estimate for an elliptic operator yields\n$$\\Vert P_ka\\Vert_{H^1(\\Gamma)} \\le C\\Vert P_ka\\Vert_{H^2(\\Omega)}\n\\le C(\\Vert A_1P_ka\\Vert_{L^2(\\Omega)} + \\Vert P_ka\\Vert_{L^2(\\Omega)})\n= C(\\lambda_k+1)\\Vert P_ka\\Vert_{L^2(\\Omega)}.           \\eqno{(3.24)}$$\nHere and henceforth $C>0$ denotes generic constants which are\nindependent of $s>0$ and $k\\in \\mathbb{N}$.\n\nSince $A_2Q_k = \\lambda_kQ_k$, by (3.23) we apply Lemma 2.2 to have\n$$s^3\\int_{\\Omega} \\vert Q_ka\\vert^2 e^{2s\\psi} dx \n\\le C\\int_{\\Omega} \\lambda_k^2\\vert Q_ka\\vert^2 e^{2s\\psi} dx\n+ Cs^3\\int_{\\gamma} (\\vert Q_ka\\vert^2 + \\vert \\nabla (Q_ka)\\vert^2)\ne^{2s\\psi} dx                               \\eqno{(3.25)}$$\n$$\\le C\\int_{\\Omega} \\lambda_k^2\\vert Q_ka\\vert^2 e^{2s\\psi} dx\n+ Cs^3e^{2sM}\\Vert P_ka\\Vert^2_{H^1(\\gamma)}$$ for all large $s>0$. Here\nwe set $M:= \\max_{x\\in \\overline{\\Gamma}} \\psi(x)$.\n\nWe choose $s>0$ sufficiently large and set\n$s_k:= s^*\\lambda_k^{\\frac{2}{3}}$ for $k\\in \\mathbb{N}$. Then, using\n(3.24), we obtain $$\\begin{aligned}\n& ({s^*}^3\\lambda_k^2 - C\\lambda_k^2)\\int_{\\Omega} \\vert Q_ka\\vert^2\ne^{2s_k\\psi} dx\n\\le C{s^*}^3\\lambda_k^2e^{2s_kM}\\Vert P_ka\\Vert^2_{H^1(\\Omega)}\\\\\n\\le& C{s^*}^3\\lambda_k^2e^{2s_kM}(\\lambda_k+1)^2\\Vert P_ka\\Vert^2_{L^2(\\Omega)}.\n\\end{aligned}$$ Since $\\psi \\ge 0$ in $\\Omega$ and we can take $s^*>0$\nsufficiently large, we see\n$${s^*}^3\\lambda_k^2 \\Vert Q_ka\\Vert^2_{L^2(\\Omega)}\n\\le C{s^*}^3\\lambda_k^2\\lambda_k^2 e^{2s_kM}\\Vert P_ka\\Vert^2_{L^2(\\Omega)},$$\nthat is,\n$$\\Vert Q_ka\\Vert^2_{L^2(\\Omega)} \\le C\\lambda_k^2 e^{C_1\\lambda_k^{\\frac{2}{3}}}\n\\Vert P_ka\\Vert^2_{L^2(\\Omega)},$$ where we set $C_1:= 2s^*M$. Here we\nnote that $s^*$ and $M$, and so the constant $C_1$ are independent of\n$k\\in \\mathbb{N}$.\n\nTherefore, since we can find a constant $C_2>0$ such that\n$\\eta^2 e^{C_1\\eta^{\\frac{2}{3}} + \\sigma_1\\eta^{\\frac{1}{2}}}\n\\le C_2e^{C_2\\eta^{\\frac{2}{3}}}$ for all $\\eta \\ge 0$, we see\n$$\\sum_{k=1}^{\\infty}e^{\\sigma_1\\lambda_k^{\\frac{1}{2}}}\\Vert Q_ka\\Vert^2_{L^2(\\Omega)}\n\\le C\\sum_{k=1}^{\\infty}\\lambda_k^2e^{C_1\\lambda_k^{\\frac{2}{3}}+\\sigma_1\\lambda_k^{\\frac{1}{2}}}\n\\Vert P_ka\\Vert^2_{L^2(\\Omega)}\n\\le C_2\\sum_{k=1}^{\\infty}e^{C_2\\lambda_k^{\\frac{2}{3}}}\\Vert P_ka\\Vert^2_{L^2(\\Omega)}.$$\nMoreover,\n$\\lim_{k\\to\\infty} \\frac{\\theta(\\lambda_k)}{\\lambda_k^{\\frac{2}{3}}}\n= \\infty$ yields that for the constant $C_2>0$ we can choose\n$N\\in \\mathbb{N}$ such that\n$C_2\\lambda_k^{\\frac{2}{3}} \\le \\theta(\\lambda_k)$ for $k \\ge N$.\nConsequently,\n$$\\sum_{k=N}^{\\infty} e^{\\sigma_1\\lambda_k^{\\frac{1}{2}}}\\Vert Q_ka\\Vert^2_{L^2(\\Omega)}\n\\le C\\sum_{k=N}^{\\infty} e^{\\theta(\\lambda_k)}\\Vert P_ka\\Vert^2_{L^2(\\Omega)}\n< \\infty,$$ and so\n$$\\sum_{k=1}^{\\infty} e^{\\sigma_1\\lambda_k^{\\frac{1}{2}}}\\Vert Q_ka\\Vert^2_{L^2(\\Omega)}\n< \\infty.$$ Thus (3.21) completes the proof of Theorem 1.1.\n$\\blacksquare$\n\n# Appendix: Proof of Lemma 2.2\n\nWe can prove the lemma by integration by parts similarly to Lemma 7.1\n(p.186) in Bellassoued and Yamamoto ) for example, but here we derive\nfrom a Carleman estimate for the parabolic equation by Imanuvilov .\n\nWe set $Q:= \\Omega\\times (0,T)$. We choose $\\ell \\in C^{\\infty}[0,T]$\nsuch that $$\\left\\{ \\begin{array}{rl}\n& \\ell(t) = 1 \\quad \\mbox{for $\\frac{T}{4}\\le t\\le \\frac{3}{4}T$},\\\\\n& \\ell(0) = \\ell(T) = 0,\\\\\n& \\mbox{$\\ell$ is strictly increasing on $\\left[ 0, \\, \\frac{T}{4}\\right]$\nand strictly decreasing on $\\left[ \\frac{T}{4}, \\,T\\right]$}.\n\\end{array}\\right.\n                                \\eqno{(4.1)}$$ In particular,\n$\\ell(t) \\le 1$ for $0\\le t\\le T$. Choosing $\\lambda>0$ sufficiently\nlarge, we set\n$$\\alpha(x,t) := \\frac{e^{\\lambda\\rho(x)} - e^{2\\lambda\\Vert \\rho\\Vert_{C(\\overline{\\Omega})}}}\n{\\ell(t)}, \\quad\n\\varphi(x,t) := \\frac{e^{\\lambda\\rho(x)}}{\\ell(t)}, \\quad (x,t)\\in \\Omega\\times (0,T).$$\nThen we know  \n**Lemma 4.1**  \n*There exist constants $s_0>0$ and $C>0$ such that $$\\begin{aligned}\n& \\int_Q (s\\varphi\\vert \\nabla U\\vert^2 + s^3\\varphi^3\\vert U\\vert^2)\ne^{2s\\alpha} dxdt \\\\\n\\le& C\\int_Q \\vert \\partial_tU - A_2U\\vert^2 e^{2s\\alpha} dxdt\n+ C\\int_{\\gamma} (\\vert \\partial_tU\\vert^2 + s\\varphi\\vert \\nabla U\\vert^2\n+ s^3\\varphi^3\\vert U\\vert^2) e^{2\\alpha} dSdt\n\\end{aligned}$$ for all $s \\ge s_0$ and $U\\in H^{2,1}(Q)$ satisfying\n$\\partial_{\\nu_A}U = 0$ on $\\partial\\Omega\\times (0,T)$.*\n\nThe proof is found in Chae, Imanuvilov and Kim .\n\nWe apply Lemma 4.1 to $g(x)$ satisfying $\\partial_{\\nu_A}g = 0$ on\n$\\partial\\Omega$ to obtain\n$$\\int_Q (s\\varphi(x,t)\\vert \\nabla g(x)\\vert^2 + s^3\\varphi^3(x,t)\\vert g(x)\\vert^2)\ne^{2s\\alpha(x,t)} dxdt \n                                       \\eqno{(4.2)}$$\n$$\\le C\\int_Q \\vert A_2g\\vert^2 e^{2s\\alpha(x,t)} dxdt\n+ C\\int^T_0 \\int_{\\gamma} (s\\varphi(x,t)\\vert \\nabla g(x)\\vert^2\n+ s^3\\varphi^3\\vert g(x)\\vert^2) e^{2\\alpha} dSdt$$ for all $s \\ge s_0$.\nMoreover, in terms of (4.1) and $e^{\\lambda\\rho(x)} \n= e^{2\\lambda\\Vert \\rho\\Vert_{C(\\overline{\\Omega})}}\\psi(x)$ for\n$x\\in \\Omega$, we have\n$$\\int_Q (s\\varphi(x,t)\\vert \\nabla g(x)\\vert^2 + s^3\\varphi^3(x,t)\\vert g(x)\\vert^2)\ne^{2s\\alpha(x,t)} dxdt\n                                        \\eqno{(4.3)}$$\n$$\\ge \\int^{\\frac{3}{4}T}_{\\frac{T}{4}} \\int_{\\Omega}\n (se^{\\lambda\\rho(x)}\\vert \\nabla g(x)\\vert^2 \n+ s^3e^{3\\lambda\\rho(x)}\\vert g(x)\\vert^2) \n\\exp( 2s(e^{\\lambda\\rho(x)} - e^{2\\lambda\\Vert \\rho\\Vert_{C(\\overline\\Omega)}}) ) dxdt$$\n$$\\ge C\\frac{T}{2}\n\\int_{\\Omega}  (s\\vert \\nabla g(x)\\vert^2 + s^3\\vert g(x)\\vert^2) \ne^{2s(e^{2\\lambda\\Vert\\rho\\Vert_{C(\\overline\\Omega)}}\\psi(x))} dx \ne^{-2se^{2\\lambda\\Vert \\rho\\Vert_{C(\\overline\\Omega)}}}$$\n$$\\ge C\\frac{T}{2}\n\\int_{\\Omega}  (s\\vert \\nabla g(x)\\vert^2 + s^3\\vert g(x)\\vert^2) \ne^{2s\\psi(x)} dx \ne^{-2se^{2\\lambda\\Vert \\rho\\Vert_{C(\\overline\\Omega)} }}.$$ Here $C>0$\ndepends on $\\lambda$ but not on $s>0$. By $e^{2s\\alpha(x,t)} \\le 1$ in\n$Q$, (4.2) and (4.3), we obtain\n$$C\\frac{T}{2}\\int_{\\Omega}  (s\\vert \\nabla g(x)\\vert^2 + s^3\\vert g(x)\\vert^2) \ne^{2s\\psi(x)} dx e^{-2se^{2\\lambda\\Vert \\rho\\Vert_{C(\\overline\\Omega)}}}\n                                        \\eqno{(4.4)}$$\n$$\\le C\\int_Q \\vert A_2g\\vert^2 e^{2s\\alpha(x,t)} dxdt\n+ C\\int^T_0 \\int_{\\gamma} (s\\varphi(x,t)\\vert \\nabla g(x)\\vert^2\n+ s^3\\varphi^3(x,t)\\vert g(x)\\vert^2) \ne^{2s\\alpha(x,t)} dSdt.$$ Since $\\sup_{(x,t)\\in \\gamma \\times (0,T)} \n\\vert (s\\varphi)^ke^{2s\\alpha(x,t)}\\vert \n< \\infty$ for $k=1,3$ and\n$$e^{2s\\psi(x)} = e^{2se^{\\lambda(\\rho(x) - 2\\Vert \\rho\\Vert_{C(\\overline{\\Omega})})}}\n\\ge e^{2se^{-\\lambda\\Vert \\rho\\Vert_{C(\\overline\\Omega)}}},$$ we can\nfind a constant $C_1 = C_1(\\lambda) > 0$ such that\n$$(s\\varphi(x,t))^k e^{2s\\alpha(x,t)} \\le C_1e^{2s\\psi(x)}, \\quad\n(x,t) \\in \\gamma \\times (0,T).$$ Therefore $$\\begin{aligned}\n& \\int_{\\Omega} (s\\vert \\nabla g\\vert^2 + s^3\\vert g\\vert^2)\ne^{2s\\psi(x)} dx \\\\\n\\le &Ce^{2se^{2\\lambda\\Vert \\rho\\Vert_{C(\\overline{\\Omega})}}}\n\\left( \\int_{\\Omega} \\vert A_2g\\vert^2 e^{2s\\psi} dx \n+ \\int_{\\gamma} (s\\vert \\nabla g\\vert^2 + s^3\\vert g(x)\\vert^2) \ne^{2s\\psi} dS\\right).\n\\end{aligned}$$ Substituting (4.3) and (4.4) into (4.2), we complete the\nproof of Lemma 2.2. $\\blacksquare$\n\n**Acknowledgements.** The work was supported by Grant-in-Aid for\nScientific Research (A) 20H00117 of Japan Society for the Promotion of\nScience.\n"
    },
    {
        "text": "abstract: We present a new algorithm by which the Adomian polynomials\n  can be determined for scalar-valued nonlinear polynomial functional in\n  a Hilbert space. This algorithm calculates the Adomian polynomials\n  without the complicated operations such as parametrization, expansion,\n  regrouping, differentiation, etc. The algorithm involves only some\n  matrix operations. Because of the simplicity in the mathematical\n  operations, the new algorithm is faster and more efficient than the\n  other algorithms previously reported in the literature. We also\n  implement the algorithm in the MATHEMATICA code. The computing speed\n  and efficiency of the new algorithm are compared with some other\n  algorithms in the one-dimensional case.\nauthor: Mithun Bairagi\ntitle: A New Algorithm to determine Adomian Polynomials for nonlinear\n  polynomial functions\n\n# introduction\n\nThe Adomian Decomposition Method (ADM) has gained huge attention in\ndifferent fields of science and engineering for solving nonlinear\nfunctional equations. In practice, many nonlinear problems do not admit\nexact solutions, and in most cases, we have to find approximate\nsolutions by employing numerical or analytical approximation techniques.\nThe ADM is a reliable technique for solving wide classes of nonlinear\nsystems, including ordinary differential, partial differential,\nintegro-differential, algebraic, differential-algebraic,\nnon-integer-order differential, integral equations, and so on ). This\ntechnique can provide an analytical approximation to the exact solutions\nin the series form that converge very rapidly . The Adomian\ndecomposition method coupled with the Laplace transform, develops a\npowerful method called the Laplace Adomian decomposition method (LADM).\nLADM has also been used in numerous articles to find the numerical\nsolution of fractional-order nonlinear differential equations, as can be\nseen in .\n\nFollowing , let us recall the basic ideas of the Adomian Decomposition\nMethod. We consider a nonlinear ODE in order $p$ with independent\nvariable $x$ (real and scalar) and dependent variable $u$ in the general\nform $$\\mathcal{F}u=g(x),$$ where $\\mathcal{F}$ is the nonlinear\noperator from a Hilbert space $H$ into $H$. In ADM, $\\mathcal{F}$ is\nassumed to be decomposed into $$\\label{nlode}\n    Lu+Ru+Nu=g(x),$$ where $L$ is the highest-order linear differential\noperator $L[.]=\\frac{d^p}{dx^p}[.]$ which is assumed to be invertible,\n$R$ is a linear differential operator containing the linear derivatives\nof less order than $L$, $N$ is a nonlinear operator containing all other\nnonlinear terms, $g(x)\\in H$ is a given analytic function. Here we\nshould note that the choice of the operator $L$ is not generally unique\n. For example, in , A. Wazwaz has chosen the linear differential\noperator $L[.]$ as $L[.]=x^{-2}\\frac{d}{dx}\\left(x^2\\frac{d}{dx}\\right)$\nfor the Lane-Emden equation. It is also notable that $u$ is a scalar\nfunction of real variable $x$ in Eq. . For a system of differential\nequations, $u$ will be a vector-valued function. However, in this paper,\nour studies are restricted to single ODE where $u$ is a scalar-valued\nfunction. The principle step of the decomposition method is to suppose a\nseries solution defined by $$\\label{seriesSolu}\n        u=\\sum_{i=0}^{\\infty}u_i,$$ and then the ADM scheme\ncorresponding to the functional equation converges rapidly to $u\\in H$\nwhich is the unique solution to the functional equation . Equation\ndecomposes the nonlinear term $Nu$ into an infinite series $$\\label{Nu}\n        Nu=\\sum_{i=0}^{\\infty}A_i,$$ where $A_i$ are the so-called\nAdomian polynomials which depend on the solution components\n$u_0,u_1,\\ldots,u_i$. For a given nonlinear functional $Nu=F(u)$ ($F(u)$\nis assumed to be an analytic function of variable $u$ in Hilbert space\n$H$), the Adomian polynomials are determined by the following\ndefinitional formula introduced by G. Adomian : $$\\label{admdef}\n        A_M=\\left.\\frac{1}{M!}\\frac{d^M }{d \\lambda^M}{F\\left(\\sum_{k=0}^{\\infty}u_k\\lambda^k\\right)}\\right|_{\\lambda=0},\\;\\; \\;\\;M=0,1,2,\\ldots,$$\nwhere the analytic parameter $\\lambda$ is simply a grouping parameter.\nAn important property of Adomian polynomial $A_M$ is that it depends by\nconstruction only on the solution components $(u_0,u_1,\\ldots,u_M)$ and\ndoes not depend on higher-order solution components $u_k$ with $k>M$ .\nTherefore, the higher-order terms for $k>M$ do not contribute in\nsummation in Eq. .\n\nMain step of ADM is to determine the Adomian polynomials of the\nnonlinear term $Nu$. Using the definitional formula it is difficult to\ncalculate higher-order Adomian terms due to the complexity in\ncalculations of higher-order derivatives. Later, many authors have\ndeveloped several convenient algorithms for fast generation of the\none-variable and the multi-variable Adomian polynomials. Adomian and\nRach produced a recurrence rule that provides a systematic computational\nprocedure to determine Adomian polynomials. Later, Rach in his paper\nestablished simple symmetry rules (which is called Rach’s rule) in\nAdomian and Rach’s algorithms, by which Adomian’s polynomials can be\ndetermined quickly to higher orders. Using the algorithm presented by\nWazwaz in , we need to collect the terms from the expansion, which takes\na large computational time for higher orders. Applying the algorithm in\n, we require to compute the derivative after substitution in a\nrecurrence relation between the Adomian polynomials. Recently in , the\nauthors modified the formula to determine the Adomian polynomials for\nnonlinear polynomial functionals. In , Duan has developed more efficient\nand fast recurrence algorithms for the rapid generation of the Adomian\npolynomials for one-variable (which is the one-dimensional case in our\nstudies) and multi-variable cases. Duan’s Corollary 1 algorithm (called\nindex recurrence algorithm) and Duan’s Corollary 3 algorithm do not\ninvolve the differentiation operator in determining the reduced\npolynomials in one dimension. We only require the operations of addition\nand multiplication, which make these algorithms faster and more\nefficient techniques.\n\nIn this work, we have presented a new algorithm for fastest computations\nof Adomian polynomials for scalar-valued nonlinear polynomial functional\n(with index as positive integers) in a Hilbert space $H$ with the help\nof matrix formulations rather than recurrence processes. Our proposed\nalgorithm does not require complex mathematical operations such as\nparametrization, expansion, regrouping, and differentiation. In this\nalgorithm, the higher-order Adomian polynomials can be determined\nthrough few matrix operations, making it faster and more efficient than\nthe other existing algorithms in the literature. We have generalized the\nnew algorithm in two dimensions where the solution $u$ depends on\ntwo-state variables such as $t,x$.\n\nThe paper is organized as follows: In Sec. we present our algorithm to\ndetermine Adomian polynomials for nonlinear polynomial functional. In\nSec. , we apply our algorithms to the polynomial functions, and the\ncomputation times are compared with some other popular algorithms\npreviously reported in the literature. In Sec. , we discuss our results\nand make some conclusions on our works. We list the MATHEMATICA code for\nthe new algorithms in Listing for one-dimensional case and in Listing\nfor two-dimensional case in Appendix: , respectively. We have also\nlisted the MATHEMATICA code for some other algorithms which are Duan’s\nCorollary 1 algorithm and Duan’s Corollary 3 algorithm with the\none-dimensional case in Listings , in Appendix: .\n\n# description of our proposed algorithm\n\nIn this section, we have described a new algorithm for calculating the\nAdomian polynomials. This algorithm is only applicable for scalar-valued\nnonlinear polynomial functional (with index as positive integers) in a\nHilbert space $H$ for the two-dimensional case. In order to increase the\ncalculating efficiency in this algorithm, all the mathematical\noperations are performed in the matrix forms.\n\nLet us now consider a nonlinear polynomial functional $F$ depends on two\ndifferent functions $u$ and $v$ in $H$. The functions $u$ and $v$ can be\nexpanded into the following two-dimensional series $$\\label{uvSeries}\n        u=\\sum_{i=0}^{\\infty}\\sum_{j=0}^{\\infty}u_{ij}\\;\\;\\;\\text{and}\\;\\;\\;v=\\sum_{i=0}^{\\infty}\\sum_{j=0}^{\\infty}v_{ij}\\;.$$\nTo illustrate our algorithm, we take the nonlinearity $F$ in the simple\nform $$\\label{nlF}\n        F=uv.$$ And this nonlinear function can be decomposed by a\nseries $$F=\\sum_{i=0}^{\\infty}\\sum_{j=0}^{\\infty}A_{ij},$$ where\n$A_{ij}$ are called Adomian polynomials of the components\n$u_{ij},v_{ij}\\;(i=0,1,\\ldots,j=0,1,\\ldots)$. Now, we divide the\nalgorithm into six main steps (labeled from Step-1 to Step-6), and to\nillustrate each step, we have used the nonlinear polynomial function .\n\n-   **Step-1** (Express the functions $u$ and $v$ in the matrix forms):\n    In this step, the functions $u$ and $v$ are expressed in the matrix\n    forms. For computations in computer, we truncate the infinite series\n    up to the finite terms $i=m,j=n$. We can increase the accuracy in\n    our results by increasing the values of $m,n$ as far as possible.\n    The functions $u,v$ in the Eq. can be expressed by\n    $(m+1)\\times (n+1)$ matrices $$\\label{soluMatrix}\n                U=\n                \\begin{pmatrix}\n                    u_{00}&  u_{01}& \\ldots& u_{0l}& \\ldots& u_{0n}\\\\ \n                    \\vdots&  \\vdots& \\ldots& \\vdots& \\ldots& \\vdots\\\\\n                    u_{k0}&  u_{k1}& \\ldots& u_{kl}& \\ldots& u_{kn}\\\\ \n                    \\vdots&  \\vdots& \\ldots& \\vdots& \\ldots& \\vdots\\\\\n                    u_{m0}&  u_{m1}& \\ldots& u_{ml}& \\ldots& u_{mn}\n                \\end{pmatrix} \\;\\;\\text{and}\\;\\; %\\overset{r_1+r_2}{\\longrightarrow} \n                V=\n                \\begin{pmatrix}\n                    v_{00}&  v_{01}& \\ldots& v_{0l}& \\ldots& v_{0n}\\\\ \n                    \\vdots&  \\vdots& \\ldots& \\vdots& \\ldots& \\vdots\\\\\n                    v_{k0}&  v_{k1}& \\ldots& v_{kl}& \\ldots& v_{kn}\\\\ \n                    \\vdots&  \\vdots& \\ldots& \\vdots& \\ldots& \\vdots\\\\\n                    v_{m0}&  v_{m1}& \\ldots& v_{ml}& \\ldots& v_{mn}\n                \\end{pmatrix}. %\\underset{\\overset{r_1-4r_2}{\\longrightarrow}}{\\overset{r_1+r_2}{\\longrightarrow}}$$\n\n-   **Step-2** (Extracting the submatrices from the matrices $U$ and\n    $V$): The Adomian polynomials corresponding to any matrix elements\n    (let the matrix elements $u_{kl},v_{kl}$ located at row $k+1$,\n    column $l+1$) in Eq. , depend on the other matrix elements whose row\n    number ($r$) and column number ($c$) are less than or equal to $k+1$\n    and $l+1$ respectively, but do not depend on the matrix elements\n    located at $r>k+1$ and $c>l+1$. In order to calculate the Adomian\n    polynomials for the elements $u_{kl}$ and $v_{kl}$ in $U$ and $V$,\n    we extract the submatrices formed by the elements with rows\n    $r\\leq k+1$ and columns $c\\leq l+1$ of the matrices $U$ and $V$ in\n    Eq. . These submatrices are given by $$\\label{subMatrix}\n                U[{0,1,\\ldots,k;0,1,\\ldots,l}]=\n                \\begin{pmatrix}\n                    u_{00}&  u_{01}& \\ldots& u_{0l}\\\\ \n                    \\vdots&  \\vdots& \\ldots& \\vdots\\\\\n                    u_{k0}&  u_{k1}& \\ldots& u_{kl}\\\\               \n                \\end{pmatrix} \\;\\;\\text{and}\\;\\; %\\overset{r_1+r_2}{\\longrightarrow} \n                V[{0,1,\\ldots,k;0,1,\\ldots,l}]=\n                \\begin{pmatrix}\n                    v_{00}&  v_{01}& \\ldots& v_{0l}\\\\ \n                    \\vdots&  \\vdots& \\ldots& \\vdots\\\\\n                    v_{k0}&  v_{k1}& \\ldots& v_{kl}         \n                \\end{pmatrix}. %\\underset{\\overset{r_1-4r_2}{\\longrightarrow}}{\\overset{r_1+r_2}{\\longrightarrow}}$$\n\n-   **Step-3** (Flipping the submatrix): In this step, all the matrix\n    elements of any one of the submatrices in Eq. are flipped\n    horizontally and then vertically or vice versa. Here we perform the\n    flipping operation on the submatrix\n    $V[{0,1,\\ldots,k;0,1,\\ldots,l}]$. The flipping operation along\n    horizontal axis can be shown in the following way\n    $$\\ensuremath{\\overset{\\xrightarrow[\\hphantom{ \\begin{pmatrix}\n                v_{00}&  v_{01}& \\ldots& v_{0l}\\\\ \n                \\vdots&  \\vdots& \\ldots& \\vdots\\\\\n                v_{k0}&  v_{k1}& \\ldots& v_{kl}         \n            \\end{pmatrix}}]{\\text{flipping horizontally}}}{ \\begin{pmatrix}\n                v_{00}&  v_{01}& \\ldots& v_{0l}\\\\ \n                \\vdots&  \\vdots& \\ldots& \\vdots\\\\\n                v_{k0}&  v_{k1}& \\ldots& v_{kl}         \n            \\end{pmatrix}}}\\longrightarrow\n            \\begin{pmatrix}\n            v_{0l}&  v_{0l-1}& \\ldots&  v_{00}\\\\ \n                \\vdots&  \\vdots& \\ldots& \\vdots\\\\\n                v_{kl}&  v_{kl-1}& \\ldots& v_{k0}           \n            \\end{pmatrix}=V[{0,1,\\ldots,k;l,l-1,\\ldots,0}].$$ Then, the\n    flipping operation along vertical axis is performed on the above\n    flipped submatrix, which can be shown as\n    $$\\text{\\tiny flipping vertically}\\ensuremath{\\left\\downarrow\\vphantom{    \\begin{pmatrix}\n                    v_{0l}&  v_{0l-1}& \\ldots&  v_{00}\\\\ \n                    \\vdots&  \\vdots& \\ldots& \\vdots\\\\\n                    v_{kl}&  v_{kl-1}& \\ldots& v_{k0}           \n            \\end{pmatrix}}\\right.{    \\begin{pmatrix}\n                    v_{0l}&  v_{0l-1}& \\ldots&  v_{00}\\\\ \n                    \\vdots&  \\vdots& \\ldots& \\vdots\\\\\n                    v_{kl}&  v_{kl-1}& \\ldots& v_{k0}           \n            \\end{pmatrix}}}\\longrightarrow\n            \\begin{pmatrix}\n                v_{kl}&  v_{kl-1}& \\ldots&  v_{k0}\\\\ \n                \\vdots&  \\vdots& \\ldots& \\vdots\\\\\n                v_{0l}&  v_{0l-1}& \\ldots& v_{00}           \n            \\end{pmatrix}=V[{k,k-1,\\ldots,0;l,l-1,\\ldots,0}].$$\n\n-   **Step-4** (Element-wise matrices multiplication): In the\n    element-wise multiplication (also known as the Hadamard product),\n    each element $i,j$ in the two matrices are multiplied together. We\n    perform the element-wise multiplication between two matrices\n    $U[{0,1,\\ldots,k;0,1,\\ldots,l}]$ and\n    $V[{k,k-1,\\ldots,0;l,l-1,\\ldots,0}]$, given by\n    $$U[{0,1,\\ldots,k;0,1,\\ldots,l}]\\circ V[{k,k-1,\\ldots,0;l,l-1,\\ldots,0}]=W[{0,1,\\ldots,k;0,1,\\ldots,l}]$$\n    and in the matrix notation the above equation can be expressed by\n    $$\\begin{pmatrix}\n                u_{00}&  u_{01}& \\ldots& u_{0l}\\\\ \n                \\vdots&  \\vdots& \\ldots& \\vdots\\\\\n                u_{k0}&  u_{k1}& \\ldots& u_{kl}\\\\               \n            \\end{pmatrix}\\circ \\begin{pmatrix}\n                                    v_{kl}&  v_{kl-1}& \\ldots&  v_{k0}\\\\ \n                                    \\vdots&  \\vdots& \\ldots& \\vdots\\\\\n                                    v_{0l}&  v_{0l-1}& \\ldots& v_{00}           \n                                \\end{pmatrix}\n             =  \\begin{pmatrix}\n                    u_{00} v_{kl}&  u_{01} v_{kl-1}& \\ldots& u_{0l}v_{k0}\\\\ \n                    \\vdots&  \\vdots& \\ldots& \\vdots\\\\\n                    u_{k0}v_{0l}&  u_{k1}v_{0l-1}& \\ldots& u_{kl}v_{00}         \n                \\end{pmatrix}.$$ Here the symbol $\\circ$ denotes the\n    element-wise multiplication between two matrices.\n\n-   **Step-5** (Summation over matrix elements): In this step, we take\n    summation over all the elements of the matrix\n    $W[{0,1,\\ldots,k;0,1,\\ldots,l}]$ and this summation is $$\\label{Akl}\n            A_{kl}=\\sum_{i=0}^{k}\\sum_{j=0}^{l}W_{ij} = u_{00} v_{kl}+  u_{01} v_{kl-1}+\\ldots+u_{kl}v_{00}.$$\n    Here $A_{kl}$ is the Adomian polynomial for the two matrix elements\n    $u_{kl},v_{kl}$. In the Adomian polynomial $A_{kl}$, notably, the\n    sum of the first index at subscripts of the components of $u,v$ in\n    each term in $A_{kl}$ are same. Similarly, the sum of the second\n    index of the components of $u,v$ in each term in $A_{kl}$ are also\n    same (here for the first index, the sum is $k$ and for the second\n    index, the sum is $l$), which obey the important property of the\n    Adomian polynomial given in .\n\n-   **Step-6** (Constructing Adomian matrix): Repeating the previous\n    steps from Step-1 to Step-5, the Adomian polynomials corresponding\n    to each matrices elements in Eq. are determined. All the calculated\n    Adomian polynomials are stored in a matrix and can be expressed by\n    $$\\label{adomianM}\n            A=\n            \\begin{pmatrix}\n                A_{00}&  A_{01}& \\ldots& A_{0l}& \\ldots& A_{0n}\\\\ \n                \\vdots&  \\vdots& \\ldots& \\vdots& \\ldots& \\vdots\\\\\n                A_{k0}&  A_{k1}& \\ldots& A_{kl}& \\ldots& A_{kn}\\\\ \n                \\vdots&  \\vdots& \\ldots& \\vdots& \\ldots& \\vdots\\\\\n                A_{m0}&  A_{m1}& \\ldots& A_{ml}& \\ldots& A_{mn}\n            \\end{pmatrix}.$$ We call the matrix $A$ in as Adomian matrix\n    for the given polynomial nonlinearity .\n\nWe present the pseudo-code for the algorithms described in Step-1 to\nStep-6 in Listing which compute the Adomian matrix of Eq. . Here, it is\nworthwhile to note how a few simple matrix operations in Step-1 to\nStep-6 generate the Adomian polynomials of Eq. . It is clear from Step-1\nto Step-6 that only $4(m+1)(n+1)-(m+n+2)$ number of matrix operations\n($2(m+1)(n+1)-(m+n+2)$ number of flippings, $(m+1)(n+1)$ number of\nelement-wise matrices multiplications and $(m+1)(n+1)$ number of matrix\nsummations) are required to compute the Adomian matrix of Eq. with\n$i=m,j=n$ in Eq. . This simplicity in mathematical operations enhances\nthe computing efficiency of this algorithm.\n\n``` numberLines\ninput: Functions $u$ and $v$ of Eq. $\\eqref{nlF}$\noutput: Adomian matrix $A$\nfunction AdomianMatrix($u,v$)\n    Express $u$ in matrix form $U$: $U$ $\\gets$ Matrix($\\sum_{i=0}^{m}\\sum_{j=0}^{n}u_{ij}$)\n    Express $v$ in matrix form $V$: $V$ $\\gets$ Matrix($\\sum_{i=0}^{m}\\sum_{j=0}^{n}v_{ij}$)\n    for $k\\gets m$ to $k\\geq 0$ do\n        for $l\\gets n$ to $l\\geq 0$ do\n            $U[{0,1,\\ldots,k;0,1,\\ldots,l}]$ $\\gets$ the submatrix of $U$ $\\text{for}$ the elements $U_{kl}$\n            $V[{0,1,\\ldots,k;0,1,\\ldots,l}]$ $\\gets$ the submatrix of $V$ $\\text{for}$ the elements $V_{kl}$\n            $V[{k,k-1,\\ldots,0;l,l-1,\\ldots,0}]$ $\\gets$ $V[{0,1,\\ldots,k;0,1,\\ldots,l}]$ are flipped horizontally and then vertically\n            Element-wise multiplication: $W[{0,1,\\ldots,k;0,1,\\ldots,l}]$ $\\gets$ $U[{0,1,\\ldots,k;0,1,\\ldots,l}]\\circ V[{k,k-1,\\ldots,0;l,l-1,\\ldots,0}]$\n            $A_{kl}$ $\\gets$ $\\sum_{i=0}^{k}\\sum_{j=0}^{l}W_{ij}$\n        end for\n    end for\n    return A\nend function    \n```\n\n``` numberLines\ninput: Functions $u^{(1)},u^{(2)},u^{(3)},\\ldots, u^{(P-2)},u^{(P-1)},u^{(P)}$ of Eq. $\\eqref{genF}$\noutput: Adomian matrix $A$\nfunction AdomianMatrix2($u^{(1)},u^{(2)},u^{(3)},\\ldots, u^{(P-2)},u^{(P-1)},u^{(P)}$)\n    Express $u^{(1)},u^{(2)},u^{(3)},\\ldots, u^{(P-2)},u^{(P-1)},u^{(P)}$ in matrix forms: $U^{(P)}$ $\\gets$ Matrix($\\sum_{i=0}^{m}\\sum_{j=0}^{n}u_{ij}^{(P)}$)\n    $A$ $\\gets$ $U^{(P)}$\n    for $k\\gets P$ to $k\\geq 2$ do\n        A $\\gets$ AdomianMatrix($U^{(k-1)}$,$A$)    \n    end for\n    return A\nend function        \n```\n\n## $F$ in general form\n\nLet us now consider the nonlinear polynomial functional $F$ in the\nfollowing general form $$\\label{genF}\n    F=u^{(1)}u^{(2)}u^{(3)}\\ldots u^{(P-2)}u^{(P-1)}u^{(P)},$$ where $F$\ndepends on $P$ number of two-dimensional functions\n$u^{(1)},u^{(2)},u^{(3)},\\ldots ,u^{(P)}$. For $P=2$ and\n$u^{(1)}=u,u^{(2)}=v$, Eq. is reduced to Eq. . The algorithms presented\nin the Step-1 to Step-6 also work for Eq. in the following way. Let\n$U^{(1)},U^{(2)},U^{(3)},\\ldots ,U^{(P)}$ are the matrix forms of the\ntwo-dimensional functions $u^{(1)},u^{(2)},u^{(3)},\\ldots ,u^{(P)}$\nrespectively. In order to determine the Adomian matrix of Eq. , at\nfirst, we will start to determine the Adomian matrix for the first two\nmatrices $U^{(1)},U^{(2)}$ or for the last two matrices\n$U^{(P-1)},U^{(P)}$ using the algorithms presented in the Step-1 to\nStep-6. Let $A^{(P-1)(P)}$ is the Adomian matrix of the last two\nmatrices $U^{(P-1)}$ and $U^{(P)}$. Next, we determine the Adomian\nmatrix of the two matrices $A^{(P-1)(P)}$ and the previous one\n$U^{(P-2)}$. This process is continued up to first matrices $U^{(1)}$.\nAfter completing this process, finally, we will get the Adomian matrix\nof $F$ given in Eq. . We present this process in pseudo-code in Listing\nwhich determines the Adomian matrix of Eq. .\n\nNow, we consider the nonlinear polynomial functional $F$ in the more\ngeneral and complicated form (a sum raised to a power) $$\\label{FpN}\n    F=\\left(u^{(1)}+u^{(2)}+u^{(3)}+\\ldots+ u^{(P-2)}+u^{(P-1)}+u^{(P)}\\right)^\\mathcal{N}$$\nwhere the power index $\\mathcal{N}$ is a positive integer number. In\nthis case, at first, we expand Eq. in sum of product terms. Then we can\neasily determine the Adomian matrix of each term of the expansion using\nthe above algorithms for Eq. . Finally, simply adding all the Adomian\nmatrices of each term, we get the Adomian matrix of Eq. .\n\nIn a one-dimensional case, the series have only one index (say $i$).\nTherefore, all the matrices are one dimension, and in this case, in\nStep-3, we have to perform only a horizontal flipping operation. Besides\nthis, all the algorithms described from Step-1 to Step-6 are identical\nin a one-dimensional case. In the following, we call the new algorithm\npresented by us the Adomian matrix algorithm.\n\n# Software implementation and comparisons with other algorithms\n\nWe have implemented the algorithm described in Sec. (called Adomian\nmatrix algorithm) into MATHEMATICA code in Listings (one-dimensional\ncase), (two-dimensional case) of Appendix: , respectively. These\nMATHEMATICA programs can determine one-dimensional (using Listing ) and\ntwo-dimensional (using Listing ) Adomian polynomials of the following\npolynomial functional $$\\label{FN}\n    F=u^\\mathcal{N},$$ where the power index $\\mathcal{N}$ is an\npositive integer number that represents the order of nonlinearity. To\ndetermine the Adomian polynomials of Eq. , we have to input the power\nindex $\\mathcal{N}$ and the order of the Adomian matrix in the function\narguments (detailed descriptions of these function arguments are given\nin the Appendix) of the MATHEMATICA functions, and these functions print\nthe Adomian polynomials in the output cell of the MATHEMATICA notebook.\n\nMATHEMATICA codes for some other algorithms such as Duan’s Corollary 1\nalgorithm , Duan’s Corollary 3 algorithm for one-dimensional case are\nalso presented in Listings , of Appendix: . The MATHEMATICA programs in\nListings and Listings are taken from Appendix: A.1 in and from Appendix:\nA in respectively. Here to make the programs more faster we have\nmodified the programs (given in , ) which work only with the polynomial\nfunctional and evaluate the differentiation of using the factorial\nformula\n$\\frac{d^iF}{du^i}=\\frac{\\mathcal{N}!}{(\\mathcal{N}-i)!}u^{\\mathcal{N}-i}$.\n\nWe have compared the Adomian matrix algorithm with other algorithms by\nemploying the MATHEMATICA programs given in Listings , , , and using the\npolynomial functional . In Table , we have shown the comparisons between\nthe computing speeds (measured in seconds) of the Adomian matrix\nalgorithm (3rd column) and two different other algorithms (4th and 5th\ncolumns) for the one-dimensional case using the MATHEMATICA programs\ngiven in Listings , , in Appendix: . We measure the computing times by\nMATHEMATICA 9.0 on the laptop with Intel(R) Core(TM) i5-7200U CPU $@$\n2.50 GHz and 8 GB RAM, using the MATHEMATICA command `Timing[]` with\nsuppressing output (i.e., the results are retained in memory). Table\ndisplays that the Adomian matrix algorithm is faster and more efficient\nthan the other two algorithms: Duan’s Corollary 1 algorithm and Duan’s\nCorollary 3 algorithms . For example, we observe that in calculating the\nfirst $50$ Adomian polynomials, the Adomian matrix algorithm is almost\n$10^4$ times faster for $\\mathcal{N}=3$ and almost $10^3$ times faster\nfor $\\mathcal{N}=10$ in comparison to the other two algorithms.\nMoreover, in calculating the first $100$ Adomian polynomials, the\nAdomian matrix algorithm spends the time $\\sim 10^{-2}$ s, but, notably,\nthe other two algorithms are unable to give results within an elapsed\ntime of $600$ s.\n\nWe have also checked the computation efficiency of the Adomian matrix\nalgorithm in the two-dimensional cases using the MATHEMATICA code in\nListing . For example, the Adomian polynomials of Eq. in the order of\n$40\\times40$ are generated within $2.6$ s for $\\mathcal{N}=3$ and within\n$19.5$ s for $\\mathcal{N}=10$.\n\n```latex\n\\begin{tabularx}{14.2cm}{|>{\\centering\\arraybackslash}p{2.1cm}|>{\\centering\\arraybackslash}p{2.9cm}|>{\\centering\\arraybackslash}p{2.4cm}|>{\\centering\\arraybackslash}p{3cm}|Y|}\n            \\hline\n             Nonlinearity \\newline\n             $(\\mathcal{N})$& Number of Adomian\\newline polynomials $(n)$& Adomian matrix\\newline algorithm&  Duan’s Corollary 1\\newline algorithm \\cite{Duan}& Duan’s Corollary 3\\newline algorithm \\cite{Duan1} \\\\\n            \\hline\n            \\multirow{4}{*}{3}&  10&  0.00047&  0.0020&  0.0025\\\\\n            \\cline{2-5}\n            &  30&  0.002&  0.83&  0.76\\\\\n            \\cline{2-5}\n            &  50&  0.0047&  62&  46\\\\\n            \\cline{2-5}\n            &  100&  0.017&  $\\boldsymbol{\\times}$&  $\\boldsymbol{\\times}$\\\\\n            \\hline\n            \\multirow{4}{*}{5}&  10&  0.00078&  0.0026&  0.0025\\\\\n            \\cline{2-5}\n            &  30&  0.0039&  0.87&  0.68\\\\\n            \\cline{2-5}\n            &  50&  0.0092&  62.5&  46.4\\\\\n            \\cline{2-5}\n            &  100&  0.037&  $\\boldsymbol{\\times}$&  $\\boldsymbol{\\times}$\\\\\n            \\hline\n            \\multirow{4}{*}{10}&  10&  0.0033&  0.0037&  0.0029\\\\\n            \\cline{2-5}\n            &  30&  0.012&  0.96&  0.65\\\\\n            \\cline{2-5}\n            &  50&  0.026&  62.7&  46.7\\\\\n            \\cline{2-5}\n            &  100&  0.095&  $\\boldsymbol{\\times}$&  $\\boldsymbol{\\times}$\\\\\n            \\hline\n        \\end{tabularx}\n```\n\n# Conclusion\n\nWe have presented a new algorithm (called the Adomian matrix algorithm)\nto determine the Adomian polynomials for scalar-valued nonlinear\npolynomial functional (with index as positive integers) in a Hilbert\nspace $H$. The computations in the Adomian matrix algorithm do not need\ncomplicated mathematical operations such as parametrization, expansion,\nregrouping, differentiation, and so on. It is clear from Step-1 to\nStep-6 in Sec. that the Adomian polynomials are determined entirely by\nsome simple matrix operations. Because of the simplicity in mathematical\noperations, the algorithm is more efficient for the fast generation of\nthe Adomian polynomials. We have designed two MATHEMATICA programs\n(one-dimensional case in Listing and two-dimensional case in Listing )\nbased on the Adomian matrix algorithm, and compared its efficiency in\ncomputations for the one-dimensional cases with other two popular and\npowerful algorithms, which are Duan’s Corollary 1 algorithm and Duan’s\nCorollary 3 algorithms . We have observed that the computation\nefficiency of the Adomian matrix algorithm is better than the other two\nalgorithms. For example, in calculating the first $50$ Adomian\npolynomials in one dimension with the nonlinearity index $\\mathcal{N}=3$\nin Eq. , the Adomian matrix algorithm is almost $10^4$ times faster than\nthe other two algorithms. For $\\mathcal{N}=10$, we are able to find the\nfirst $100$ Adomian polynomials using this new algorithm in just\n$10^{-2}$ s, whereas for $\\mathcal{N}=3$ and $n=100$, the other two\nalgorithms fail to produce any results until $600$ s have passed.\nTherefore, we can conclude that the Adomian matrix algorithm can be used\nto determine a large number of Adomian polynomials of nonlinear\npolynomial functionals that make the solutions more accurate.\n\n# Mathematica programs for one-dimensional case \n\nThe following three MATHEMATICA programs can determine one-dimensional\nAdomian polynomials of the nonlinear function . The function arguments\nN\\_ and n\\_ represent the nonlinear power index $\\mathcal{N}$ in Eq. and\nthe number of first Adomian polynomials, respectively.\n\n``` mathematica\nAdomMatAlgo1D[N_, n_] := Module[{h, j, k},\n    u =.;\n    mat = Table[Subscript[u, h], {h, 0, n - 1}];\n    temmat = Table[Subscript[u, h], {h, 0, n - 1}];\n    For[j = 1, j <= N - 1, j++,\n      For[k = n, k >= 1, k--,\n        mat[[k]] = Total[temmat[[;; k]]*Reverse[mat[[;; k]]]];\n       ];\n     ];\n    mat\n   ]\n```\n\n``` mathematica\nDuanIndexAlgoAdom[N_, n_] := Module[{Apoly, Zpoly, dirClt}, \n    Subscript[Apoly, 0] = Subscript[u, 0]^N;\n    Zpoly = Table[0, {i, 1, n - 1}, {j, 1, i}];\n    Do[Zpoly[[suInd, 1]] = Subscript[u, suInd], {suInd, 1, n - 1}];\n    For[i = 2, i <= n - 1, i++, \n      For[j = 2, j <= i, j++, \n        Zpoly[[i, j]] = Expand[Subscript[u, 1]*Zpoly[[i - 1, j - 1]]];\n        If[Head[Zpoly[[i, j]]] === Plus, \n          Zpoly[[i, j]] = Map[#/Exponent[#, Subscript[u, 1]] &, Zpoly[[i, j]]], \n          Zpoly[[i, j]] = Map[#/Exponent[#, Subscript[u, 1]] &, Zpoly[[i, j]], {0}]]];\n      For[j = 2, j <= Floor[i/2], j++, \n        Zpoly[[i, j]] = Zpoly[[i, j]] + (Zpoly[[i - j, j]] /. \n        Subscript[u, sub_] -> Subscript[u, sub + 1])]];\n    dirClt = Table[Factorial[N]/Factorial[N - j]*(Subscript[u, 0]^(N - j)), {j, 1, n - 1}];\n    Do[Subscript[Apoly, suInd] = Take[dirClt, suInd].Zpoly[[suInd]], {suInd, 1, n - 1}];\n    Table[Subscript[Apoly, suInd], {suInd, 0, n - 1}]]\n```\n\n``` mathematica\nDuanCoro3AlgoAdm[N_, n_] := Module[{cPoly, i, k, j, derClt}, \n   Table[cPoly[i, k], {i, 1, n - 1}, {k, 1, i}];\n   derClt = Table[Factorial[N]/Factorial[N - k]*(Subscript[u, 0]^(N - k)), \n                 {k,1, n - 1}];\n   Apoly[0] = Subscript[u, 0]^N;\n   For[i = 1, i <= n - 1, i++, \n     cPoly[i, 1] = Subscript[u, i];\n     For[k = 2, k <= i, k++, \n       cPoly[i, k] = Expand[1/i*Sum[(j + 1)*Subscript[u, j + 1]*cPoly[i - 1 - j, k - 1], \n                           {j, 0,i - k}]]];\n     Apoly[i] = Take[derClt, i].Table[cPoly[i, k], {k, 1, i}]];\n   Table[Apoly[i], {i, 0, n - 1}]]\n```\n\n# Mathematica programs for two-dimensional case \n\nThe following MATHEMATICA program can determine two-dimensional Adomian\npolynomials of the nonlinear function . The function arguments N\\_, m\\_\nand n\\_ represent the nonlinear power index $\\mathcal{N}$ in Eq. , the\nnumber of rows and number of columns in the Adomian matrix,\nrespectively.\n\n``` mathematica\nAdomMatAlgo2D[N_, m_, n_] := Module[{g, h, j, k, l},\n        u =.;\n        mat = Table[Subscript[u, g, h], {g, 0, m - 1}, {h, 0, n - 1}];\n        temmat = Table[Subscript[u, g, h], {g, 0, m - 1}, {h, 0, n - 1}];\n        For[j = 1, j <= N - 1, j++,\n          For[k = m, k >= 1, k--,\n            For[l = n, l >= 1, l--,\n              mat[[k, l]] = Total[temmat[[;; k, ;; l]]*Reverse[Reverse[mat[[;; k, ;; l]], 1], 2], 2];\n             ];\n           ];\n         ];\n        mat\n       ]\n```\n"
    },
    {
        "text": "abstract: A blocking semioval is a set of points in a projective plane\n  that is both a blocking set (i.e., every line meets the set, but the\n  set contains no line) and a semioval (i.e., there is a unique tangent\n  line at each point). The smallest size of a blocking semioval is known\n  for all finite projective planes of order less than 11; we investigate\n  the situation in $PG(2,11)$.\nauthor: Jeremy M. Dover\ntitle: **On the minimum blocking semioval in $PG(2,11)$**\n\n# Introduction\n\nIn a projective plane $\\pi$, a **semioval** is a set of points $S$ such\nthat there is a unique tangent line (i.e., line with one point of\ncontact) at each point. The size of a semioval ${\\cal S}$ in $PG(2,q)$\nis known to satisfy $q+1 \\le |{\\cal S}| \\le q\\sqrt{q}+1$. As defined by\nKiss , a semioval ${\\cal S}$ is *small* if it satisfies\n$|{\\cal S}| \\le 3q+3$. Several articles, including Kiss, et al.  and\nBartoli , investigate the spectrum of sizes for small semiovals in\nDesarguesian planes.\n\nA set of points $S$ in $\\pi$ is called a **blocking set** if every line\nof $\\pi$ meets $S$ in at least one point, but $S$ contains no line. A\nset of points in $\\pi$ is called a **blocking semioval** if it is both a\nblocking set and a semioval. The smallest possible size of a blocking\nsemioval is known for all planes of order less than 11. In the smallest\ncases, the author  showed there are no blocking semiovals in $PG(2,2)$,\nand the smallest possible sizes in $PG(2,q)$ for $q=3,4,5,7$ are\n$6, 9, 11, {\\rm and}\\; 16$, respectively. In $PG(2,8)$ the author \nexhibited a blocking semioval of size 19, and showed that no smaller\nblocking semiovals can exist.\n\nRegarding the planes of order 9, the author  showed that no plane of\norder 9 can have a blocking semioval of size 20 or smaller. Nakagawa and\nSuetake  had previously shown the existence of blocking semiovals of\nsize 21 in the three non-Desarguesian planes of order 9, and Dover,\nMellinger and Wantz  exhibited a blocking semioval of size 21 in\n$PG(2,9)$.\n\nIn this article, we continue this program by studying the smallest\nblocking semioval in $PG(2,11)$.\n\n# Some introductory analysis\n\nTo begin scoping the problem, we first appeal to the two best known\nlower bounds for the size of blocking semiovals. For any projective\nplane of order $q$, the author  shows that a blocking semioval must have\nat least $2q + \\sqrt{2q-\\frac{47}{4}}-\\frac12$ points. Specializing to\nthe $q=11$ case, we calculate that a blocking semioval in $PG(2,11)$\nmust have at least 25 points. Since $PG(2,11)$ is Desarguesian, the\nlower bound of Héger and Takáts  also applies, which states that a\nblocking semioval in $PG(2,q)$ must have at least $\\frac94q-3$ points.\nHowever, this result provides a bound less than 25, from which we\nconclude that the smallest possible size for a blocking semioval in\n$PG(2,11)$ is 25 points.\n\nLet ${\\cal S}$ be a putative blocking semioval of size 25 in $PG(2,11)$.\nThe author  (Proposition 2.1 and Theorem 2.2) shows that ${\\cal S}$\ncannot fully contain a line, nor can it have an $11$-secant. We then\napply Theorem 3.1 from Dover , which states that if ${\\cal S}$ has an\n(11-$k$)-secant for $1\\leq k < 10$, then ${\\cal S}$ has at least\n$11 \\frac{3k+4}{k+2} - k$ points. Evaluating all of these bounds as $k$\nvaries from 1 to 10, we find that ${\\cal S}$ cannot have any 7-secants,\n8-secants or 9-secants.\n\nOne possibility is that ${\\cal S}$ could have a 10-secant, in which case\nTheorem 4.2 in  shows that there is only one 10-secant. On the other\nhand, ${\\cal S}$ could have no lines meeting it in more than 6 points.\nWe deal with these cases separately.\n\n# The 10-secant case\n\nSuppose that a blocking semioval ${\\cal S}$ of size 25 in $PG(2,11)$ has\na 10-secant $\\ell_{10}$. Let $Q$ and $R$ be the two points on\n$\\ell_{10}$ that are not in ${\\cal S}$. If $P$ is a point of ${\\cal S}$\nnot on $\\ell_{10}$, then the tangent line to ${\\cal S}$ through $P$ must\npass through either $Q$ or $R$; let ${\\cal S}_Q$ (resp. ${\\cal S}_R$) be\nthe set of points in ${\\cal S}$ not on $\\ell_{10}$ whose tangent passes\nthrough $Q$ (resp. $R$). If we denote the set of points in ${\\cal S}$\nnot on $\\ell_{10}$ as ${\\cal S}'$, we have\n${\\cal S}' = {\\cal S}_Q \\dot\\cup {\\cal S}_R$, so we must have\n$|{\\cal S}_Q| + |{\\cal S}_R| = 15$.\n\nSince ${\\cal S}$ is a blocking set, every line through $Q$ must meet\n${\\cal S}$ in at least one point. One of these is $\\ell_{10}$, but the\nremaining 11 lines through $Q$ must be covered by the 15 points of\n${\\cal S}'$. The lines through $Q$ and points of ${\\cal S}_Q$ are\ntangents to ${\\cal S}$ and thus contain only a single point of\n${\\cal S}_Q$, and no points of ${\\cal S}_R$; hence there are\n$|{\\cal S}_Q|$ of these. A line $m$ through $Q$ containing a point $X$\nof ${\\cal S}_R$ cannot be a tangent to ${\\cal S}$, since the tangent to\n${\\cal S}$ at $X$ meets $\\ell_{10}$ in $R$. Hence $m$ contains at least\n2 points of ${\\cal S}$, but these points cannot be on $\\ell_{10}$, and\nby the above cannot be in ${\\cal S}_Q$, so $m$ must contain at least 2\npoints of ${\\cal S}_R$.\n\nSince every line through $Q$ is covered by ${\\cal S}$, we have:\n$$12 \\leq 1 + |{\\cal S}_Q| + \\frac12|{\\cal S}_R|$$\n\nUsing the fact that $|{\\cal S}_Q| = 15 - |{\\cal S}_R|$, we can\nsubstitute in this inequality to obtain $|{\\cal S}_Q| \\geq 7$. However,\nthe exact same argument holds for $R$, showing $|{\\cal S}_R| \\geq 7$,\nfrom which we conclude that one of $|{\\cal S}_Q|$ and $|{\\cal S}_R|$ is\n7 and the other is 8. Since our choice of labelling $Q$ and $R$ is\narbitrary, we assume without loss of generality that $|{\\cal S}_Q| = 8$\nand $|{\\cal S}_R| = 7$.\n\nSince $|{\\cal S}_Q|$ contains 8 points, eight lines through $Q$ are\ntangents to ${\\cal S}$, and one is $\\ell_{10}$, so the remaining three\nlines through $Q$ must meet ${\\cal S}_R$ in at least 2 points each. As\n$|{\\cal S}_R| = 7$, this implies that these remaining 3 lines consist of\ntwo 2-secants and a 3-secant to ${\\cal S}$. Similar analysis shows that\n$R$ lies on $\\ell_{10}$, seven tangents and four 2-secants to\n${\\cal S}$.\n\nWith a structure this well-defined, we have a reasonable hope that a\ncomputer search can identify whether or not a blocking semioval of this\nform exists in $PG(2,11)$. We develop a search strategy by picking items\nin a potential blocking semioval in order, using Magma  at each stage to\ndetermine the number of projectively inequivalent options we have that\nrespect previous choices.\n\nWe begin by coordinatizing $PG(2,11)$ with homogeneous coordinates such\nthat $Q = (1,0,0)$ and $R = (0,1,0)$. The automorphism group leaving $Q$\nand $R$ invariant is transitive on lines through $Q$ other than\n$\\ell_{10}$, thus we may assume that the 3-secant to ${\\cal S}$ through\n$Q$ is $[0,1,0]$.\n\n**Pick two other lines through $Q$ to be 2-secants to ${\\cal S}$.** We\nuse Magma to compute the group of automorphisms in $PG(2,11)$ that leave\n$Q$ and $R$ invariant, as well as the line $[0,1,0]$, and calculate the\norbits of pairs of lines through $Q$. As a result of this calculation,\nwe find that we may always take one of our 2-secants through $Q$ to be\n$[0,1,1]$, and then the other must be one of\n$[0,1,2],[0,1,3],[0,1,5],[0,1,7],[0,1,10]$, giving 5 possible options\nfor this pair of lines.\n\n**Pick three points on the 3-secant to be in ${\\cal S}_R$.** For each of\nthe five selections of non-tangent lines through $Q$ above, we again use\nMagma to calculate the group of automorphisms that leave each of $Q$,\n$R$, and the three non-tangent lines through $Q$ fixed. This group is\ndoubly-transitive on the points of the 3-secant through $Q$ distinct\nfrom $Q$, allowing us to assume that $(0,0,1)$ and $(1,0,1)$ are in\n${\\cal S}$. However, this is the extent to which the automorphism group\ncan help us, and we must consider each of the other 9 points on this\nline as candidates to be in ${\\cal S}$, leaving 9 options for this point\nset.\n\n**Pick two points on $[0,1,1]$ to be in ${\\cal S}_R$.** We have already\npicked three points on the 3-secant to be in ${\\cal S}_R$, and the lines\njoining each of these points to $R$ cannot contain any other point of\n${\\cal S}$. This eliminates three points of $[0,1,1]$ from\nconsideration, as well as $Q$, so there are ${8 \\choose 2} = 28$\ncandidate point pairs.\n\n**Pick two points on the other 2-secant through $Q$ to be in\n${\\cal S}_R$.** As before, we may not pick points of this line that\nalready lie on lines connecting points of ${\\cal S}_R$ and $R$, which\neliminates 6 possible points; there can be no overlap since all of these\nlines intersect at $R$. Hence there are ${6 \\choose 2} = 15$ candidate\npoint pairs.\n\n**Pick the points of ${\\cal S}_Q$.** There will be four remaining lines\nthrough $R$ which do not pass through a point of ${\\cal S}_R$. Order\nthem arbitrarily. On the first, pick two points to be in ${\\cal S}_Q$,\nnoting that the line between either of these points and $Q$ must be a\ntangent. As above, this choice affects the number of candidate points\nfor each of the remaining 2-secants through $R$. These point selections\ncan be made in ${8 \\choose 2} {6 \\choose 2} {4 \\choose 2} = 2520$ ways.\n\nThere are $47,638,000 \\approx 2^{25.5}$ possible configurations, which\nis a managable number for computer search. For each of these\nconfigurations, we must check if the resulting set, when added to the\nten points on $\\ell_{10}$ distinct from $Q$ and $R$, forms a blocking\nsemioval. However, the way we’ve constructed the set ensures that it is\na blocking set, and that every point of ${\\cal S}'$ has a unique tangent\nline; all that remains to check at this stage is whether there is a\nunique tangent line at each of the points of $\\ell_{10}$ distinct from\n$Q$ and $R$, or equivalently if each of the points of $\\ell_{10}$\ndistinct from $Q$ and $R$ lies on exactly two lines not blocked by\n${\\cal S}'$, as one of those two will be $\\ell_{10}$.\n\nWe executed this search in Magma, and we found no blocking semioval of\nthis form. Hence we conclude that there is no blocking semioval of size\n25 in $PG(2,11)$ with a 10-secant.\n\n# Constraint Programming\n\nIn order to confirm the results of the previous section, and to set up\nfor further searching, we have developed a constraint programming model\nto search for blocking semiovals, implemented using Google’s OR-Tools \nBoolean satisfiability (SAT) solver, using Python. For each point\n$(x,y,z)$, we create a Boolean variable $p\\_x\\_y\\_z$ which is true if\n$(x,y,z)$ is in our blocking semioval and false otherwise. Mechanically,\nwe also need to create a 0/1-valued integer variable tied to the value\nof the Boolean (0 for false, 1 for true) to calculate line intersection\nsizes with the blocking semioval, but for purposes of the model\ndescription we will conflate these two variables. Similarly, for each\nline $[a,b,c]$ we create a Boolean variable $\\ell\\_a\\_b\\_c$ which is\ntrue if and only if the line is tangent to our blocking semioval.\n\nGiven these variable definitions, every blocking semioval corresponds to\na solution of the constraint program defined with only three types of\nconstraints: $$\\begin{aligned}\n\\sum_{(x,y,z) \\in [a,b,c]} p\\_x\\_y\\_z &=& 1\\; {\\rm if} \\; \\ell\\_a\\_b\\_c\\\\\n\\sum_{(x,y,z) \\in [a,b,c]} p\\_x\\_y\\_z &>& 1\\; {\\rm if} \\; \\neg\\ell\\_a\\_b\\_c\\\\\n\\sum_{[a,b,c] \\ni (x,y,z)} \\ell\\_a\\_b\\_c &=& 1\\; {\\rm if} \\; p\\_x\\_y\\_z\\\\\n\\end{aligned}$$ The first equation type asserts that if $\\ell$ is a\ntangent, then only one of its points lies on the blocking semioval. The\nsecond equation type asserts that all non-tangent lines meet the\nblocking semioval in more than one point. The final equation type\nasserts that for each point of the blocking semioval, only one of the\nlines through it is a tangent. Applying these conditions across all\npoints and lines in the plane, we see that any solution of this\nconstraint program is a blocking semioval, and vice versa.\n\nThe SAT problem is known to be NP-complete in general, but there are\nnumerous algorithms and implementations which can solve many specific\ninstances. In particular, we have modeled the specific search of the\nlast section using constraint programming by adding the additional\nconditions derived to our generic model: that the total number of points\nmust be 25, that certain points must be or not be on the blocking\nsemioval, and that certain lines must meet the blocking semioval in 1, 2\nor 3 points. By way of comparison, the Magma search conducted in the\nprevious section took approximately 53 minutes to determine that there\nwas no blocking semioval of the claimed form. Google’s OR-Tools SAT\nsolver reached the same conclusion in 47 seconds; excerpts of the code\nare given in Appendix A. Using the constraint programming approach will\nbe particularly valuable for us in the next section.\n\n# The 6-secant case\n\nWe have concluded that if ${\\cal S}$ is a blocking semioval $PG(2,11)$\nwith 25 points, ${\\cal S}$ cannot have a 10-secant. Hence if ${\\cal S}$\nexists, no line can meet ${\\cal S}$ in more than 6 points. Let $x_i$\ndenote the number of lines of $PG(2,11)$ meeting ${\\cal S}$ in exactly\n$i$ points. From , we have the following relations, specialized to\n$PG(2,11)$: $$\\begin{aligned}\nx_2 + x_5 + 3x_6 & = & 123 \\label{eq1}\\\\\nx_3 - 3x_5 - 8x_6 & = & -89 \\label{eq2}\\\\\nx_4 + 3x_5 + 6x_6 & = & 74 \\label{eq3}\n\\end{aligned}$$\n\nNote that Equation , when reduced modulo 3, shows that\n$x_4 \\equiv 2 \\pmod{3}$, and in particular must be at least 2. Hence\n$3x_5 + 6x_6 \\le 72$ by Equation . On the other hand Equation  shows\nthat $3x_5 + 8 x_6 \\ge 89$. Hence $2 x_6 \\ge 17$, from which we conclude\n$x_6 \\geq 9$. We proceed through with a series of small propositions\nthat will gradually restrict the possible structure of ${\\cal S}$.\n\n**Proposition 1**.  * Let ${\\cal S}$ be a blocking semioval in\n$PG(2,11)$ with 25 points and no 10-secant. Then no point of ${\\cal S}$\nlies on more than three 6-secants to ${\\cal S}$, and at least four\npoints of ${\\cal S}$ lie on exactly three 6-secants to ${\\cal S}$.*\n\n*Proof.* Let ${\\cal L}_6$ be the set of 6-secants to ${\\cal S}$, and\ndefine $y_i$ to be the number of points of ${\\cal S}$ lying on exactly\n$i$ 6-secants in ${\\cal L}_6$. First note that $y_i = 0$ for all\n$i \\ge 5$. Indeed if $P \\in {\\cal S}$ lies on five (or more) 6-secants,\nthen each of these 6-secants contains 5 points of ${\\cal S}$ distinct\nfrom $P$, giving at least $25+1 = 26$ points on ${\\cal S}$.\n\nWe also note that $y_4 = 0$. If $P \\in {\\cal S}$ lies on four 6-secants,\nthen these four 6-secants contain 20 points of ${\\cal S}$ distinct from\n$P$. One of the other lines through $P$ is a tangent to ${\\cal S}$ at\n$P$, leaving seven additional lines through $P$, each of which must\ncontain at least one other point of ${\\cal S}$ distinct from $P$. This\nforces ${\\cal S}$ to have at least 28 points, again a contradiction.\nHence no point of ${\\cal S}$ can lie on more than three 6-secants in\n${\\cal L}_6$.\n\nCounting points of ${\\cal S}$ and flags of points of ${\\cal S}$ lying on\nlines of ${\\cal L}_6$, we obtain the following relations on the $y_i$:\n$$\\begin{aligned}\ny_0 + y_1 + y_2 + y_3 & = & 25\\\\\ny_1 + 2y_2 + 3y_3 & = & 6x_6 \\ge 54\n\\end{aligned}$$\n\nAs all of the $y_i$ are non-negative, we certainly have\n$50 + y_3 = 2y_0 + 2y_1 + 2y_2 + 2y_3 + y_3 \\ge  y_1 + 2y_2 + 3y_3 \\ge 54$,\nfrom which we derive that $y_3 \\ge 4$. ◻\n\n**Proposition 2**.  * Let ${\\cal S}$ be a blocking semioval in\n$PG(2,11)$ with 25 points and no 10-secant. Then there exists a pair of\npoints $Q,R \\in {\\cal S}$ such that $Q$ and $R$ each lie on three\n6-secants to ${\\cal S}$, and the line $\\overline{QR}$ is also a 6-secant\nto ${\\cal S}$.*\n\n*Proof.* Let ${\\cal Y}_3$ be the set of points in ${\\cal S}$ that lie on\nexactly three 6-secants. Let $P_1$, $P_2$, $P_3$, and $P_4$ be four\ndistinct points of ${\\cal Y}_3$, which must exist by Proposition . If\nany pair of the $P_i$ lie together on a 6-secant to ${\\cal S}$, we are\ndone, so assume otherwise. The three 6-secants through $P_1$ each\ncontains 5 points of ${\\cal S}$ in addition to $P_1$, and we define\n${\\cal S}_1$ to be this set of 15 points lying on a 6-secant through\n$P_1$, distinct from $P_1$. Note that by assumption none of $P_2$, $P_3$\nor $P_4$ can be in ${\\cal S}_1$, as otherwise two points of ${\\cal Y}_3$\nwould lie together on a 6-secant.\n\nNow consider the three 6-secants through $P_2$. Again, none of these\nlines can contain any of $P_1$, $P_3$ or $P_4$. Moreover, each of these\nlines meets ${\\cal S}_1$ in at most three points, one from each of the\n6-secants through $P_1$. Hence each 6-secant through $P_2$ contains at\nleast two points distinct from the $P_i$ and ${\\cal S}_1$. But there are\nfour $P_i$, and 15 points in ${\\cal S}_1$. Since ${\\cal S}$ only has 25\npoints, each 6-secant through $P_2$ must contain exactly two points\ndistinct from the $P_i$ and ${\\cal S}_1$, and thus exactly three points\nof ${\\cal S}_1$. In particular, exactly nine points of ${\\cal S}_1$ lie\non a 6-secant through $P_1$ and a 6-secant through $P_2$. Call these\npoints ${\\cal S}_{12}$.\n\nThe same analysis holds for the three 6-secants through $P_3$. Hence\nthere are exactly nine points of ${\\cal S}_1$ that lie on a 6-secant\nthrough $P_1$ and a 6-secant through $P_3$, which we denote\n${\\cal S}_{13}$. Since ${\\cal S}_1$ only has 15 points, its 9-point\nsubsets ${\\cal S}_{12}$ and ${\\cal S}_{13}$ must intersect in at least\nthree points, and any point of that intersection is a point of\n${\\cal S}$ that lies on exactly three 6-secants (it cannot lie on more,\nby Proposition ) and lies on a 6-secant together with another point of\n${\\cal Y}_3$. This completes the proof. ◻\n\nFrom Proposition , our blocking semioval ${\\cal S}$ must contain two\npoints $Q$ and $R$ which each lie on three 6-secants, one of which is\ntheir common line $n$. Let $\\ell_1$ and $\\ell_2$ be the other two\n6-secants through $Q$ and $m_1$ and $m_2$ be the other two 6-secants\nthrough $R$. Define ${\\cal I}$ to be the set of four points which are\nthe intersections of $\\ell_i$ and $m_j$ for $i,j \\in \\{1,2\\}$. Figure \nprovides a graphical depiction of these definitions.\n\n**Proposition 3**.  * Let ${\\cal S}$ be a blocking semioval in\n$PG(2,11)$ with 25 points and no 10-secant. Define the points $Q$ and\n$R$, the lines $n$, $\\ell_1$, $\\ell_2$, $m_1$ and $m_2$, and the set\n${\\cal I}$ as above. Then at least three points of ${\\cal I}$ must lie\nin ${\\cal S}$.*\n\n*Proof.* We begin by counting the number of points in the subset\n${\\cal S}_6$ of ${\\cal S}$ consisting of the points of ${\\cal S}$ that\nlie on our five noted 6-secants. There are six points on $n$, and then\neach of $\\ell_1$, $\\ell_2$, $m_1$ and $m_2$ contributes five additional\npoints (not counting $Q$ and $R$), with the points of ${\\cal I}$ counted\ntwice. Hence ${\\cal S}_6$ contains $26 - |{\\cal I}|$ points, showing\nimmediately that ${\\cal I}$ contains at least one point of ${\\cal S}$.\n\nBut recall that ${\\cal S}$ must have at least nine 6-secants. Except for\nthe five 6-secants defining ${\\cal S}_6$, no 6-secant can contain more\nthan five points of ${\\cal S}_6$. So all of the additional 6-secants, of\nwhich there are at least four, must contain a point of ${\\cal S}$ that\nis not in ${\\cal S}_6$. However, they cannot all contain the same point\noff ${\\cal S}_6$, since by Proposition  a point of ${\\cal S}$ can lie on\nat most three 6-secants. So there must be at least two points of\n${\\cal S}$ not in ${\\cal S}_6$, forcing ${\\cal S}_6$ to contain at most\n23 points, from which we conclude that at least three points of\n${\\cal I}$ lie in ${\\cal S}$. ◻\n\nDefine ${\\cal S}^*$ to be the set of points in ${\\cal S}$ that lie on\none or more of the lines $n$, $\\ell_1$, $\\ell_2$, $m_1$ and $m_2$.\nNotice that the configuration given in Figure  shows five 6-secants to\nour putative blocking semioval ${\\cal S}$ (as well as ${\\cal S}^*$); but\nrecall that ${\\cal S}$ must have at least nine 6-secants. As the points\nof ${\\cal S}^*$ are on the union of five lines, any 6-secant to\n${\\cal S}$ not shown must contain at least one point not in\n${\\cal S}^*$. On the other hand, ${\\cal S}^*$ contains either 22 or 23\nof the points of the blocking semioval, depending on whether ${\\cal I}$\nhas 4 or 3 points in ${\\cal S}$, respectively. Let’s address the latter\ncase first.\n\n**Proposition 4**.  * Let ${\\cal S}$ be a blocking semioval in\n$PG(2,11)$ with 25 points and no 10-secant. Define the points $Q$ and\n$R$, the lines $n$, $\\ell_1$, $\\ell_2$, $m_1$ and $m_2$, and the set\n${\\cal I}$ as above. If ${\\cal I}$ has three points in ${\\cal S}$, then\nthere exists a point $P \\in {\\cal S}\\setminus {\\cal S}^*$ which lies on\nat least two 6-secants to ${\\cal S}$ such that neither of these\n6-secants contains any point of ${\\cal S}\\setminus {\\cal S}^*$ except\n$P$.*\n\n*Proof.* Since ${\\cal I}$ has three points in ${\\cal S}$, ${\\cal S}^*$\ncontains 23 points of ${\\cal S}$, leaving two points of ${\\cal S}$ not\nin ${\\cal S}^*$, which we call $A$ and $B$. Since ${\\cal S}$ has at\nleast nine 6-secants and only five of these contain points strictly\nwithin ${\\cal S}^*$, there must be at least four 6-secants to ${\\cal S}$\nwhich contain at least one of $A$ or $B$. It is possible that the line\ncontaining $A$ and $B$ is a 6-secant to ${\\cal S}$, but this leaves\nthree 6-secants to ${\\cal S}$ that contain exactly one of $A$ or $B$.\nThe pigeonhole principle completes the proof, as one of these two points\nmust lie on two such 6-secants. ◻\n\nThough a seemingly simple result, Proposition  is incredibly powerful\nfrom a computational perspective. In order to set up for a search, let\nus coordinatize $PG(2,11)$ such that the points of ${\\cal I}$ are\n$\\ell_2 \\cap m_2 = (1,0,0)$, $\\ell_2 \\cap m_1 = (0,1,0)$,\n$\\ell_1 \\cap m_2 = (0,0,1)$ and $\\ell_1 \\cap m_1 = (1,1,1)$, with\n$(1,1,1)$ not in ${\\cal S}$. This forces $Q=(1,1,0)$ and $R=(1,0,1)$ to\nalso be in ${\\cal S}$. Any automorphism which fixes this configuration\nmust leave $(1,1,1)$ and $(1,0,0)$ fixed, but it could permute $(0,1,0)$\nand $(0,0,1)$, thus we do have a non-trivial automorphism group of order\n2 which we can use to narrow the search space for our point\n$P \\in {\\cal S}$ whose existence is guaranteed by Proposition . Using\nMagma, we find that there are 45 orbits of candidate points for $P$.\n\nOnce $P$ is chosen, we need to pick two lines through it to be\n6-secants. But in order to be 6-secants, these lines must meet each of\n$n$, $\\ell_1$, $\\ell_2$, $m_1$ and $m_2$ in ${\\bf distinct}$ points,\nwhich must all be in ${\\cal S}$. Thus we need to pick two lines through\n$P$ out of 6 or 7 candidates (depending on whether or not $P$ lies on a\nline containing two points of ${\\cal I}$), which can be done in either\n15 or 21 ways. Running out all the possibilities with Magma, we find\nthat there are 760 possible configurations for $P$ and its two\n6-secants. Combined with the 5 points we coordinatized above, we have\n760 starting configurations, each of which provides 16 out of 25 points\non a putative blocking semioval. This puts us in the realm of easy\ncomputation.\n\nFor each of these 760 starting configurations, we create a constraint\nprogramming model as in Section . Starting with the basic relations used\nto define a blocking semioval, we add the additional constraints:\n\n1.  the total number of points on the blocking semioval is 25;\n\n2.  the points $\\{(1,0,0),(0,1,0),(0,0,1),(1,1,0),(1,0,1)\\}$ are in the\n    blocking semioval;\n\n3.  the point $(1,1,1)$ is not in the blocking semioval;\n\n4.  the point $P$ is in the blocking semioval; and\n\n5.  the points of intersection of the two lines chosen to be 6-secants\n    to the blocking semioval with each of $\\ell_1$, $\\ell_2$, $m_1$,\n    $m_2$ and $n$ are in the blocking semioval.\n\nThe SAT solver runs through these cases at a rate of roughly two per\nsecond, and in each case determined that the model was infeasible, from\nwhich we can conclude that there is no blocking semioval of the form\ndiscussed here with just three points of ${\\cal I}$ in the blocking\nsemioval. Now we follow a similar program to Proposition  for the case\nwhere all four points of ${\\cal I}$ lie in ${\\cal S}$.\n\n**Proposition 5**.  * Let ${\\cal S}$ be a blocking semioval in\n$PG(2,11)$ with 25 points and no 10-secant. Define the points $Q$ and\n$R$, the lines $n$, $\\ell_1$, $\\ell_2$, $m_1$ and $m_2$, and the set\n${\\cal I}$ as above. If all four points of ${\\cal I}$ lie in ${\\cal S}$,\nthen there exists a point $P \\in {\\cal S}\\setminus {\\cal S}^*$ which\nlies on at least two 6-secants to ${\\cal S}$ such that at least one of\nthese 6-secants contains no point of ${\\cal S}\\setminus {\\cal S}^*$\nexcept $P$.*\n\n*Proof.* Since all four points of ${\\cal I}$ lie in ${\\cal S}$,\n${\\cal S}^*$ contains 22 points of ${\\cal S}$, leaving three points of\n${\\cal S}$ not in ${\\cal S}^*$. As before ${\\cal S}$ has at least nine\n6-secants and only five of these contain points strictly within\n${\\cal S}^*$, so there must be at least four 6-secants to ${\\cal S}$\nwhich contain at least one point not in ${\\cal S}^*$. Since there are\nonly three points in ${\\cal S}\\setminus {\\cal S}^*$, at most three of\nour four 6-secants can contain more than one point of\n${\\cal S}\\setminus {\\cal S}^*$, so there must exist a point\n$P \\in {\\cal S}\\setminus {\\cal S}^*$ which lies on a 6-secant that\ncontains no other point of ${\\cal S}\\setminus {\\cal S}^*$.\n\nIf there is another 6-secant through $P$ we are done. If no other\n6-secant passes through $P$, then the remaining three 6-secants must\npass through either of $A$ or $B$, the two points of\n${\\cal S}\\setminus {\\cal S}^*$ distinct from $P$. Again using the\npigeonhole principle, at least one of these points (say $A$) must lie on\nat least two 6-secants, and at most one of those 6-secants can contain\nanother point (necessarily $B$) of ${\\cal S}\\setminus {\\cal S}^*$. In\nthis case $A$ lies on at least two 6-secants, at least one of which\ncontains no other point of ${\\cal S}\\setminus {\\cal S}^*$, proving the\nclaim. ◻\n\nWhile not as powerful as Proposition , it gets the job done. We again\ncoordinatize $PG(2,11)$ as above, but in this case all of $(1,0,0)$,\n$(0,1,0)$, $(0,0,1)$ and $(1,1,1)$ are in ${\\cal S}$, as well as\n$(1,1,0)$ and $(1,0,1)$. There is an automorphism group of order 8 that\nfixes $\\{(1,1,0),(1,0,1)\\}$ and $\\{(1,0,0),(0,1,0),(0,0,1),(1,1,1)\\}$ as\nsets, which allows us to narrow down the choice of our point $P$, whose\nexistence is guaranteed by Proposition , to one of 15 possible orbits.\nWe then choose a line to be the 6-secant which contains no other point\nof ${\\cal S}\\setminus {\\cal S}^*$, which we can pick in 6 or 7 ways, as\ndescribed above. However, our remaining 6-secant could be any other line\nthrough $P$ except $\\ell_{10}$, yielding 1,056 cases.\n\nFor each of these 1,056 starting configurations, we again create a\nconstraint programming model. In addition to the basic relations used to\ndefine a blocking semioval, we add:\n\n1.  the total number of points on the blocking semioval is 25;\n\n2.  the points $\\{(1,0,0),(0,1,0),(0,0,1),(1,1,1),(1,1,0),(1,0,1)\\}$ are\n    in the blocking semioval;\n\n3.  the point $P$ is in the blocking semioval;\n\n4.  the points of intersection of the first line chosen to be a 6-secant\n    to the blocking semioval with each of $\\ell_1$, $\\ell_2$, $m_1$,\n    $m_2$ and $n$ are in the blocking semioval; and\n\n5.  the second line chosen to be a 6-secant to the blocking semioval\n    contains six points of the blocking semioval.\n\nThe SAT solver resolves each of these case in approximately 4 seconds,\nand again determined that each case was infeasible. This allows us to\nstate our main result:\n\n**Theorem 6**. *There is no blocking semioval with exactly 25 points in\n$PG(2,11)$.*\n\n# Conclusion\n\nAs part of testing our constraint programming code to double check the\n10-secant case, we commented out the constraint forcing the number of\npoints in the blocking semioval to be 25, assuming the code would find a\nvertexless triangle or some other known blocking semioval. Instead, we\nfound a blocking semioval in $PG(2,11)$ with 26 points, well smaller\nthan the 29 points contained in the smallest previously known blocking\nsemioval in that plane. The coordinates for the points of this blocking\nsemioval are given in Table .\n\n|  |  |  |  |  |  |  |  |  |\n|:--:|:--:|:--:|:--:|:--:|:--:|:--:|:--:|:--:|\n| $(1,1,0)$ | $(1,2,0)$ | $(1,3,0)$ | $(1,4,0)$ | $(1,5,0)$ | $(1,6,0)$ | $(1,7,0)$ | $(1,8,0)$ | $(1,9,0)$ |\n| $(1,10,0)$ | $(0,0,1)$ | $(1,0,2)$ | $(1,0,6)$ | $(1,0,7)$ | $(1,0,8)$ | $(1,0,10)$ | $(1,1,3)$ | $(1,2,3)$ |\n| $(1,3,5)$ | $(1,4,4)$ | $(1,5,9)$ | $(1,6,5)$ | $(1,7,1)$ | $(1,8,4)$ | $(1,9,1)$ | $(1,10,9)$ |  |\n\nPoints of a 26-point semioval in $PG(2,11)$\n\nThe stabilizer $G$ of this blocking semioval has order 5, and has three\nfixed points $(1,0,0)$, $(0,1,0)$ and $(0,0,1)$. It is generated by a\ncollineation induced by the matrix: $$\\left(\\begin{matrix}\n1 & 0 & 0\\\\\n0 & 9 & 0\\\\\n0 & 0 & 4\n\\end{matrix}\\right)$$\n\nThe three lines of the triangle generated by the fixed points are a\n10-secant ($[0,0,1]$), a tangent ($[1,0,0]$, at $(0,0,1)$), and a\n6-secant which contains $(0,0,1)$ and a 5-point orbit under $G$. Using\nthis combinatorial structure and an analogous group structure, we\nattempted to generalize this blocking semioval to larger planes,\nparticularly $PG(2,19)$, using a fairly straightforward exhaustion in\nMagma using all possible pairs of squares in $GF(19)$ to define our\nautomorphism group. This search was unsuccessful; however, analyzing the\nlist of blocking semiovals in $PG(2,7)$  shows that the unique smallest\nblocking semioval in that plane, of size 16, has an analogous\nautomorphism group and combinatorial structure.\n\nTable  provides a summary of the size of the smallest blocking semiovals\nin small Desarguesian planes, where these values are known.\n\n|   Plane   | Minimum Size |   Plane    | Minimum Size |\n|:---------:|:------------:|:----------:|:------------:|\n| $PG(2,2)$ |     none     | $PG(2,3)$  |      6       |\n| $PG(2,4)$ |      9       | $PG(2,5)$  |      11      |\n| $PG(2,7)$ |      16      | $PG(2,8)$  |      19      |\n| $PG(2,9)$ |      21      | $PG(2,11)$ |      26      |\n\nSizes of smallest known blocking semiovals\n\n# Appendix A: Constraint programming primitives\n\nThe constraint program needed to define the incidences in $PG(2,11)$ is\nrather lengthy; we actually developed a Magma program to write the\nconstraint program. Here we provide some excerpts from the constraint\nprograms used in this paper. The code here is excerpted from a Python\nprogram using Google’s OR Tools library. Note that sums with ellipses\ncontinue over all points on a line or lines through a point, and are\nredacted for readability considerations.\n\n    ###\n    #Variables describing a point\n    ###\n    # Boolean is true/false as (1,0,0) is in/not in the blocking semioval\n    p_1_0_0 = model.NewBoolVar(\"point_1_0_0\")\n    # Integer value of the Boolean variable\n    p_1_0_0_int = model.NewIntVar(0,1,\"P![1,0,0]\")\n    #Constraints to tie the Boolean and integer variables together\n    model.Add(p_1_0_0_int == 1).OnlyEnforceIf(p_1_0_0)\n    model.Add(p_1_0_0_int == 0).OnlyEnforceIf(p_1_0_0.Not())\n\n    ###\n    #Variables describing a line\n    ###\n    # Boolean is true/false as [1,0,0] is/is not tangent to blocking semioval\n    l_1_0_0 = model.NewBoolVar(\"line_1_0_0\")\n    # Integer value of the Boolean variable\n    l_1_0_0_int = model.NewIntVar(0,1,\"line_1_0_0_int\")\n    #Constraints to tie the Boolean and integer variables together\n    model.Add(l_1_0_0_int == 1).OnlyEnforceIf(l_1_0_0)\n    model.Add(l_1_0_0_int == 0).OnlyEnforceIf(l_1_0_0.Not())\n\n    ###\n    #Blocking semioval constraints\n    ###\n    # Only one point on [1,0,0] is on the blocking semioval if it is a tangent\n    model.Add(p_0_1_0_int+p_0_1_1_int+ ... == 1).OnlyEnforceIf(l_1_0_0)\n    # More than one point on [1,0,0] is on the blocking semioval if not\n    model.Add(p_0_1_0_int+ ... > 1).OnlyEnforceIf(l_1_0_0.Not())\n    # Only one line through (1,0,0) is tangent to the blocking semioval\n    model.Add(l_0_1_0_int+l_0_1_1_int+ ... == 1).OnlyEnforceIf(p_1_0_0)\n\n    ###\n    #Additional constraints (not necessarily used simultaneously)\n    ###\n    # Force [0,0,1] to be a 6-secant\n    model.Add(p_1_0_0_int+p_1_1_0_int+p_0_1_0_int+ ... == 6)\n    # Assert (1,0,0) is in the blocking semioval, while (1,1,1) is not\n    model.AddBoolAnd([p_1_0_0,p_1_1_1.Not()])\n    # Assert the blocking semioval has 25 points\n    model.Add(p_1_0_0_int+p_0_1_0_int+p_0_0_1_int+ ... == 25)\n"
    },
    {
        "text": "abstract: Let $F$ be a nilpotent group acted on by a group $H$ via\n  automorphisms and let the group $G$ admit the semidirect product $FH$\n  as a group of automorphisms so that $C_G(F) = 1$. We prove that the\n  order of $\\gamma_\\infty(G)$, the rank of $\\gamma_\\infty(G)$ are\n  bounded in terms of the orders of $\\gamma_{\\infty}(C_G(H))$ and $H$,\n  the rank of $\\gamma_{\\infty}(C_G(H))$ and the order of $H$,\n  respectively in cases where either $FH$ is a Frobenius group; $FH$ is\n  a Frobenius-like group satisfying some certain conditions; or\n  $FH=\\langle \\alpha,\\beta\\rangle$ is a dihedral group generated by the\n  involutions $\\alpha$ and $\\beta$ with $F =\\langle \\alpha\\beta\\rangle$\n  and $H =\\langle\\alpha \\rangle$.\naddress: Department of Academic Areas, Instituto Federal de Goiás,\nFormosa-GO 73813-816, Brazil; Department of Mathematics, University of\nBrasília, Brasília-DF 70910-900, Brazil; Department of Mathematics,\nMiddle East Technical University, 06800, Ankara/Turkey\nauthor: Eliana Rodrigues; Emerson de Melo; Gülin Ercan\ntitle: NILPOTENT RESIDUAL  \n  OF A FINITE GROUP\n\n# Introduction\n\nThroughout all groups are finite. Let a group $A$ act by automorphisms\non a group $G$. For any $a \\in A$, we denote by $C_G(a)$ the set\n$\\{x\\in G : x^a=x\\},$ and write $C_G(A)=\\bigcap_{a\\in A}C_G(a).$ In this\npaper we focus on a certain question related to the strong influence of\nthe structure of such fixed point subgroups on the structure of $G$, and\npresent some new results when the group $A$ is a Frobenius group or a\nFrobenius-like group or a dihedral group of automorphisms.\n\nIn what follows we denote by $A^\\#$ the set of all nontrivial elements\nof $A$, and we say that $A$ acts coprimely on $G$ if $(|A|,|G|)=1$.\nRecall that a Frobenius group $A=FH$ with kernel $F$ and complement $H$\ncan be characterized as a semidirect product of a normal subgroup $F$ by\n$H$ such that $C_F(h)=1$ for every $h \\in H^\\#$. Prompted by Mazurov’s\nproblem $17.72$ in the Kourokva Notebook , some attention was given to\nthe situation where a Frobenius group $A=FH$ acts by automorphisms on\nthe group $G$. In the case where the kernel $F$ acts fixed-point-freely\non $G$, some results on the structure of $G$ were obtained by Khukhro,\nMakarenko and Shumyatsky in a series of papers , , , , , , . They\nobserved that various properties of $G$ are in a certain sense close to\nthe corresponding properties of the fixed-point subgroup $C_G(H)$,\npossibly also depending on $H$. In particular, when $FH$ is metacyclic\nthey proved that if $C_G(H)$ is nilpotent of class $c$, then the\nnilpotency class of $G$ is bounded in terms of $c$ and $|H|$. In\naddition, they constructed examples showing that the result on the\nnilpotency class of $G$ is no longer true in the case of non-metacyclic\nFrobenius groups. However, recently in it was proved that if $FH$ is\nsupersolvable and $C_G(H)$ is nilpotent of class $c$, then the\nnilpotency class of $G$ is bounded in terms of $c$ and $|FH|$.\n\nLater on, as a generalization of Frobenius group the concept of a\nFrobenius-like group was introduced by Ercan and Güloğlu in , and their\naction studied in a series of papers , ,,,,. A finite group $FH$ is said\nto be Frobenius-like if it has a nontrivial nilpotent normal subgroup\n$F$ with a nontrivial complement $H$ such that $FH/F'$ is a Frobenius\ngroup with Frobenius kernel $F/F'$ and complement $H$ where $F'=[F,F]$.\nSeveral results about the properties of a finite group $G$ admitting a\nFrobenius-like group of automorphisms $FH$ aiming at restrictions on $G$\nin terms of $C_G(H)$ and focusing mainly on bounds for the Fitting\nheight and related parameters as a generalization of earlier results\nobtained for Frobenius groups of automorphisms; and also new theorems\nfor Frobenius-like groups based on new representation-theoretic results.\nIn these papers two special types of Frobenius-like groups have been\nhandled. Namely, Frobenius-like groups $FH$ for which $F'$ is of prime\norder and is contained in $C_F(H)$; and the Frobenius-like groups $FH$\nfor which $C_F(H)$ and $H$ are of prime orders, which we call Type I and\nType II, respectively throughout the remainder of this paper.\n\nIn Shumyatsky showed that the techniques developed in can be used in the\nstudy of actions by groups that are not necessarily Frobenius. He\nconsidered a dihedral group $D=\\langle \\alpha, \\beta \\rangle$ generated\nby two involutions $\\alpha$ and $\\beta$ acting on a finite group $G$ in\nsuch a manner that $C_G(\\alpha \\beta)=1$. In particular, he proved that\nif $C_G(\\alpha)$ and $C_G(\\beta)$ are both nilpotent of class $c$, then\n$G$ is nilpotent and the nilpotency class of $G$ is bounded solely in\nterms of $c$. In , a similar result was obtained for other groups. It\nshould also be noted that in an extension of about the nilpotent length\nobtained by proving that the nilpotent length of a group $G$ admitting a\ndihedral group of automorphisms in the same manner is equal to the\nmaximum of the nilpotent lengths of the subgroups $C_G(\\alpha)$ and\n$C_G(\\beta)$.\n\nThroughout we shall use the expression “$(a,b,\\dots )$-bounded” to\nabbreviate “bounded from above in terms of $a, b,\\dots$ only”. Recall\nthat the rank $\\mathbf r(G)$ of a finite group $G$ is the minimal number\n$r$ such that every subgroup of $G$ can be generated by at most $r$\nelements. Let $\\gamma_\\infty(G)$ denote the *nilpotent residual* of the\ngroup $G$, that is the intersection of all normal subgroups of $G$ whose\nquotients are nilpotent. Recently, in , de Melo, Lima and Shumyatsky\nconsidered the case where $A$ is a finite group of prime exponent $q$\nand of order at least $q^3$ acting on a finite $q'$-group $G$. Assuming\nthat $|\\gamma_\\infty(C_G(a))| \\leq m$ for any $a \\in A^\\#$, they showed\nthat $\\gamma_\\infty(G)$ has $(m,q)$-bounded order. In addition, assuming\nthat the rank of $\\gamma_\\infty(C_G(a))$ is at most $r$ for any\n$a \\in A^\\#$, they proved that the rank of $\\gamma_\\infty(G)$ is\n$(m,q)$-bounded. Later, in , it was proved that the order of\n$\\gamma_\\infty(G)$ can be bounded by a number independent of the order\nof $A$.\n\nThe purpose of the present article is to study the residual nilpotent of\nfinite groups admitting a Frobenius group, or a Frobenius-like group of\nType I and Type II, or a dihedral group as a group of automorphisms.\nNamely we obtain the following results.\n\n**Theorem A** Let $FH$ be a Frobenius, or a Frobenius-like group of Type\nI or Type II, with kernel $F$ and complement $H$. Suppose that $FH$ acts\non a finite group $G$ in such a way that $C_G(F)=1$. Then\n\n-   $|\\gamma_\\infty(G)|$ is bounded solely in terms of $|H|$ and\n    $|\\gamma_{\\infty}(C_G(H))|$;\n\n-   the rank of $\\gamma_\\infty(G)$ is bounded in terms of $|H|$ and the\n    rank of $\\gamma_{\\infty}(C_G(H))$.\n\n**Theorem B** Let $D= \\langle\\alpha, \\beta \\rangle$ be a dihedral group\ngenerated by two involutions $\\alpha$ and $\\beta$. Suppose that $D$ acts\non a finite group $G$ in such a manner that $C_G(\\alpha\\beta)=1$. Then\n\n-   $|\\gamma_\\infty(G)|$ is bounded solely in terms of\n    $|\\gamma_{\\infty}(C_G(\\alpha))|$ and $|\\gamma_\\infty(C_G(\\beta))|$;\n\n-   the rank of $\\gamma_\\infty(G)$ is bounded in terms of the rank of\n    $\\gamma_{\\infty}(C_G(\\alpha))$ and $\\gamma_\\infty(C_G(\\beta))$.\n\nThe paper is organized as follows. In Section 2 we list some results to\nwhich we appeal frequently. Section 3 is devoted to the proofs of two\nkey propositions which play crucial role in proving Theorem A and\nTheorem B whose proofs are given in Section 4.\n\n# Preliminaries\n\nIf $A$ is a group of automorphisms of $G$, we use $[G,A]$ to denote the\nsubgroup generated by elements of the form $g^{-1}g^a$, with $g \\in G$\nand $a \\in A$. Firstly, we recall some well-known facts about coprime\naction, see for example , which will be used without any further\nreferences.\n\n**Lemma 1**.  * Let $Q$ be a group of automorphisms of a finite group\n$G$ such that $(|G|,|Q|) = 1$. Then*\n\n-   *$G= C_G(Q)[G,Q]$.*\n\n-   *$Q$ leaves some Sylow $p$-subgroup of $G$ invariant for each prime\n    $p \\in \\pi(G)$.*\n\n-   *$C_{G /N}(Q) =  C_G(Q) N /N$ for any $Q$-invariant normal subgroup\n    $N$ of $G$.*\n\nWe list below some facts about the action of Frobenius and\nFrobenius-like groups. Throughout, a non-Frobenius Frobenius-like group\nis always considered under the hypothesis below.\n\n**Hypothesis\\*** Let $FH$ be a non-Frobenius Frobenius-like group with\nkernel $F$ and complement $H$. Assume that a Sylow $2$-subgroup of $H$\nis cyclic and normal, and $F$ has no extraspecial sections of order\n$p^{2m+1}$ such that $p^m + 1 = |H_1|$ for some subgroup $H_1 \\leq H$.\n\nIt should be noted that Hypothesis\\* is automatically satisfied if\neither $|FH|$ is odd or $|H| = 2$.\n\n**Theorem 2**.  * Suppose that a finite group $G$ admits a Frobenius\ngroup or a Frobenius-like group of automorphisms $FH$ with kernel F and\ncomplement H such that $C_G(F)=1$. Then $C_G(H)\\ne 1$ and $\\mathbf r(G)$\nis bounded in terms of $\\mathbf r(C_G(H))$ and $|H|.$*\n\n**Proposition 3**.  * Let $FH$ be a Frobenius, or a Frobenius-like group\nof Type I or Type II. Suppose that $FH$ acts on a $q$-group $Q$ for some\nprime $q$ coprime to the order of $H$ in case $FH$ is not Frobenius. Let\n$V$ be a $kQFH$-module where $k$ is a field with characteristic not\ndividing $|QH|.$ Suppose further that $F$ acts fixed-point freely on the\nsemidirect product $VQ$. Then we have $C_V(H)\\ne 0$ and\n$$Ker(C_Q(H) \\ \\textrm{on} \\ C_V(H)) = Ker(C_Q(H) \\ \\textrm{on} \\ V).$$*\n\n*Proof.* See Proposition 2.2 when $FH$ is Frobenius; Proposition C when\n$FH$ is Frobenius-like of Type I; and Proposition 2.1 when $FH$ is\nFrobenius-like of Type II. It can be easily checked that Proposition 2.2\nis valid when $C_Q(F)=1$ without the coprimeness condition\n$(|Q|,|F|)=1.$ ◻\n\nThe proof of the following theorem can be found in and in .\n\n**Theorem 4**.  * Let $D= \\langle\\alpha, \\beta \\rangle$ be a dihedral\ngroup generated by two involutions $\\alpha$ and $\\beta$. Suppose that\n$D$ acts on a finite group $G$ in such a manner that\n$C_G(\\alpha\\beta)=1$. Then*\n\n-   *$G = C_G(\\alpha)C_G(\\beta)$;*\n\n-   *the rank of $G$ is bounded in terms of the rank of $C_G(\\alpha)$\n    and $C_G(\\beta)$;*\n\n**Proposition 5**. *Let $D =\\langle \\alpha,\\beta\\rangle$ be a dihedral\ngroup generated by the involutions $\\alpha$ and $\\beta.$ Suppose that\n$D$ acts on a $q$-group $Q$ for some prime $q$ and let V be a\n$kQD$-module for a field $k$ of characteristic different from $q$ such\nthat the group $F =\\langle \\alpha\\beta\\rangle$ acts fixed point freely\non the semidirect product $VQ$. If $C_Q(\\alpha)$ acts nontrivially on\n$V$ then we have $C_V (\\alpha)\\ne 0$ and\n$Ker(C_Q(\\alpha) \\ \\textrm{on} \\ C_V(\\alpha)) = Ker(C_Q(\\alpha) \\ \\textrm{on} \\ V)$.*\n\n*Proof.* This is Proposition C in . ◻\n\nThe next two results were established in .\n\n**Lemma 6**.  * Suppose that a group $Q$ acts by automorphisms on a\ngroup $G$. If $Q=\\langle q_1,\\ldots , q_n \\rangle$, then\n$[G,Q]=[G,q_1]\\cdots [G,q_n].$*\n\n**Lemma 7**.  * Let $p$ be a prime, $P$ a finite $p$-group and $Q$ a\n$p'$-group of automorphisms of $P$.*\n\n-   *If $|[P,q]|\\leq m$ for every $q\\in Q$, then $|Q|$ and $|[P,Q]|$ are\n    $m$-bounded.*\n\n-   *If $r([P,q])\\leq m$ for every $q\\in Q$, then $r(Q)$ and $r([P,Q])$\n    are $m$-bounded.*\n\nWe also need the following fact whose proof can be found in .\n\n**Lemma 8**.  * Let $G$ be a finite group such that\n$\\gamma_\\infty(G) \\leq F(G)$. Let $P$ be a Sylow $p$-subgroup of\n$\\gamma_\\infty(G)$ and $H$ be a Hall $p'$-subgroup of $G$. Then\n$P= [P,H]$.*\n\n# Key Propositions\n\nWe prove below a new proposition which studies the actions of Frobenius\nand Frobenius-like groups and forms the basis in proving Theorem A.  \n\n**Proposition 9**.  * Assume that $FH$ be a Frobenius group, or a\nFrobenius-like group of Type I or Type II with kernel $F$ and complement\n$H$. Suppose that $FH$ acts on a $q$-group $Q$ for some prime $q$. Let\n$V$ be an irreducible $\\mathbb{F}_pQFH$-module where $\\mathbb{F}_p$ is a\nfield with characteristic $p$ not dividing $|Q|$ such that $F$ acts\nfixed-point-freely on the semidirect product $VQ$. Additionaly, we\nassume that $q$ is coprime to $|H|$ in case where $FH$ is not Frobenius.\nThen $\\mathbf  r([V,Q])$ is bounded in terms of\n$\\mathbf  r([C_V(H), C_Q(H)])$ and $|H|$.*\n\n*Proof.* Let $\\mathbf  r([C_V(H), C_Q(H)])=s.$ We may assume that\n$V=[V,Q]$ and hence $C_V(Q)=0$. By Clifford’s Theorem,\n$V=V_1\\oplus \\cdots \\oplus V_t$, direct sum of of $Q$-homogeneous\ncomponents $V_i$ , which are transitively permuted by $FH$. Set\n$\\Omega =\\{V_1,\\dots, V_t\\}$ and fix an $F$-orbit $\\Omega_1$ in\n$\\Omega$. Throughout, $W=\\Sigma_{U\\in \\Omega_1}U.$\n\nNow, we split the proof into a sequence of steps.  \n*(1) We may assume that $Q$ acts faithfully on $V$. Furthermore\n$Ker(C_Q(H) \\ \\textrm{on} \\ C_V(H)) = Ker(C_Q(H) \\ \\textrm{on} \\ V)=1$.*\n\n*Proof.* Suppose that $Ker(Q \\ \\rm{on} \\ V)\\neq 1$ and set\n$\\overline{Q} =Q/Ker(Q \\ \\rm{on} \\ V)$. Note that since $C_Q(F)=1$, $F$\nis a Carter subgroup of $QF$ and hence also a Carter subgroup of\n$\\overline{Q}F$ which implies that $C_{\\overline{Q}}(F)=1$. Notice that\nthe equality $\\overline{C_Q(H)}=C_{\\overline{Q}}(H)$ holds in case $FH$\nis Frobenius (see Theorem 2.3). The same equality holds in case where\n$FH$ is non-Frobenius due to the coprimeness condition $(q,|H|)=1.$ Then\n$[C_V(H),C_Q(H)]=[C_V(H),C_{\\overline{Q}}(H)]$ and so we may assume that\n$Q$ acts faithfully on $V$. Notice that by Proposition we have\n$$Ker(C_Q(H) \\ \\textrm{on} \\ C_V(H)) = Ker(C_Q(H) \\ \\textrm{on} \\ V)=1$$\nestablishing the claim. ◻\n\n*(2) We may assume that $Q=\\langle c^F \\rangle$ for any nonidentity\nelement $c\\in C_{Z(Q)}(H)$ of order $q$. In particular $Q$ is abelian.*\n\n*Proof.* We obtain that $C_{Z(Q)}(H)\\ne 1$ as $C_Q(F)=1$ by Proposition\n. Let now $1\\ne c \\in C_{Z(Q)}(H)$ of order $q$ and consider\n$\\langle c^{FH} \\rangle=\\langle c^F \\rangle$, the minimal $FH$-invariant\nsubgroup containing $c$. Since $V$ is an irreducible $QFH$-module on\nwhich $Q$ acts faithfully we have that $V=[V,\\langle c^F \\rangle]$. Thus\nwe may assume that $Q=\\langle c^F \\rangle$ as claimed. ◻\n\n*(3) $V=[V,c]\\cdot [V,c^{f_1}] \\cdots[V,c^{f_n}]$ where $n$ is a\n$(s,|H|)$-bounded number. Hence it suffices to bound\n$\\mathbf r([W,c])$.*\n\n*Proof.* Notice that the group $C_Q(H)$ embeds in the automorphism group\nof $[C_V(H),C_Q(H)]$ by step $\\it(1)$. Then $C_Q(H)$ has $s$-bounded\nrank by Lemma . This yields by Theorem that $Q$ has $(s,|H|)$-bounded\nrank. Thus, there exist $f_1=1,\\ldots ,f_n$ in $F$ for an\n$(s,|H|)$-bounded number $n$ such that\n$Q=\\langle  c^{f_1},\\ldots,c^{f_n} \\rangle$. Now\n$V=[V,c]\\cdot [V,c^{f_2}] \\cdots[V,c^{f_n}]=\\prod_{i=1}^n [V,c]^{f_i}$\nby Lemma . This shows that we need only to bound $\\mathbf r([V,c])$\nsuitably. In fact it suffices to show that $\\mathbf r([W,c])$ is\nsuitably bounded as $V=\\Sigma_{h\\in H}W^h.$ ◻\n\n*(4) $H_1=Stab_H(\\Omega_1)\\ne1$. Furthermore the rank of the sum of\nmembers of $\\Omega_1$ which are not centralized by $c$ and contained in\na regular $H_1$-orbit, is suitably bounded.*\n\n*Proof.* Fix $U\\in \\Omega_1$ and set $Stab_F(U)=F_1$. Choose a\ntransversal $T$ for $F_1$ in $F.$ Let $W=\\sum_{t\\in T}U^{t}$ where $T$\nis a transversal for $F_1$ in $F$ with $1\\in T.$ Then we have\n$V=\\sum_{h\\in H}W^h$. Notice that $[V,c]\\ne 0$ by $\\it(1)$ which implies\nthat $[W,c]\\neq 0$ and hence $[U^t,c]=U^t$ for some $t\\in T$. Without\nloss of generality we may assume that $[U,c]=U.$\n\nSuppose that $Stab_H(\\Omega_1)=1$. Then we also have $Stab_H(U^t)=1$ for\nall $t\\in T$ and hence the sum $X_t=\\sum_{h\\in H}U^{th}$ is direct for\nall $t\\in T.$ Now, $U\\leq X_1$. It holds that\n$$C_{X_t}(H)=\\{ \\sum_{h\\in H}v^{h} \\ : \\ v\\in U^t\\}.$$ Then\n$|U|=|C_{X_1}(H)|=|[C_{X_1}(H),c]|\\leq |[C_V(H),C_Q(H)]|$ implies\n$\\mathbf  r(U)\\leq s.$ On the other hand $V=\\bigoplus_{t\\in T}X_t$ and\n$$[C_V(H),c]=\\bigoplus \\{ [C_{X_t}(H),c] : t\\in T \\,\\, \\text{with }\\,\\,[U^t,c]\\ne 0\\}\\leq [C_V(H),C_Q(H)].$$\nIn particular, $\\{t\\in T : [U^t,c]\\ne 0\\}$ is suitably bounded whence\n$\\mathbf  r([W,c])$ is $(s,|H|)$-bounded. Hence we may assume that\n$Stab_H(\\Omega_1)\\ne 1.$\n\nNotice that every element of a regular $H_1$-orbit in $\\Omega_1$ lies in\na regular $H$-orbit in $\\Omega$. Let $U\\in \\Omega_1$ be contained in a\nregular $H_1$-orbit of $\\Omega_1.$ Let $X$ denote the sum of the members\nof the $H$-orbit of $U$ in $\\Omega$, that is $X=\\bigoplus_{h\\in H}U^h$.\nThen $C_X(H)=\\{ \\sum_{h\\in H}v^{h} \\ : \\ v\\in U\\}$. If $[U,c]\\ne 0$ then\nby repeating the same argument in the above paragraph we show that\n$\\mathbf  r(U)\\leq s$ is suitably bounded. On the other hand the number,\nsay $m$, of all $H$-orbits in $\\Omega$ containing a member $U$ such that\n$[U,c]\\ne 0$ is suitably bounded because\n$m\\leq \\mathbf r([C_V(H),c])\\leq s.$ It follows then that the rank of\nthe sum of members of $\\Omega_1$ which are not centralized by $c$ and\ncontained in a regular $H_1$-orbit, is suitably bounded. ◻\n\n*(5) We may assume that $FH$ is not Frobenius.*\n\n*Proof.* Assume the contrary that $FH$ is Frobenius. Let\n$H_1=Stab_H(\\Omega_1)$ and pick $U\\in \\Omega_1$. Set $S=Stab_{FH_1}(U)$\nand $F_1=F\\cap S$. Then $|F:F_1|=|\\Omega_1|=|FH_1:S|$ and so\n$|S:F_1|=|H_1|$. Since $(|F_1|,|H_1|)=1$, by the Schur-Zassenhaus\ntheorem there exists a complement, say $S_1$ of $F_1$ in $S$ with\n$|H_1|=|S_1|$. Therefore there exists a conjugate of $U$ which is\n$H_1$-invariant. There is no loss in assuming that $U$ is\n$H_1$-invariant. On the other hand if $1\\neq h\\in H_1$ and $x\\in F$ such\nthat $U^{xh}=U^x$, then $[h,x]\\in Stab_{F}(U)=F_1$ and so\n$F_1x=F_1x^h=(F_1x)^h$. This implies that $F_1x\\cap C_F(h)$ is nonempty.\nNow the Frobenius action of $H$ on $F$ forces that $x\\in F_1$. This\nmeans that for each $x\\in F\\setminus F_1$ we have $Stab_{H_1}(U^x)=1$.\nTherefore $U$ is the unique member of $\\Omega_1$ which is\n$H_1$-invariant and all the $H_1$-orbits other than $\\{U\\}$ are regular.\nBy $\\it(4)$, the rank of the sum of all members of $\\Omega_1$ other than\n$U$ is is suitably bounded. In particular $\\mathbf r(U)$ and hence\n$\\mathbf r([W,c])$ is suitably bounded in case where $[U^x,c]\\ne 0$ for\nsome $x\\in F\\setminus F_1$. Thus we may assume that $c$ is trivial on\n$U^x$ for all $x\\in F\\setminus F_1$. Now we have $[W,c]=[U,c]=U.$\n\nDue to the action by scalars of the abelian group $Q$ on $U$, it holds\nthat $[Q,F_1]\\leq C_Q(U)$. We also know that $c^x$ is trivial on $U$ for\neach $x\\in F\\setminus F_1$. Since $C_Q(F)=1$, there are prime divisors\nof $|F|$ different from $q.$ Let $F_{q'}$ denote the $q'$-Hall subgroup\nof $F.$ Clearly we have $C_Q(F_{q'})=1$. Let now\n$y=\\prod_{f\\in F_{q'}}c^f$. Then we have\n$$1=y=(\\prod_{f\\in F_1\\cap F_{q'} }c^f)(\\prod_{f\\in F_{q'}\\setminus F_1}c^f)\\in c^{|F_1\\cap F_{q'}|}C_Q(U).$$\nAs a consequence $c\\in C_Q(U)$, because $q$ is coprime to $|F_{q'}|$.\nThis contradiction establishes the claim. ◻\n\n*(6) We may assume that the group $FH$ is Frobenius-like of Type II.*\n\n*Proof.* On the contrary we assume that $FH$ is Frobenius-like of Type\nI. By $\\it(4),$ we have $H_1=Stab_H(\\Omega_1)\\ne 1$. Choose a\ntransversal $T_1$ for $H_1$ in $H.$ Now $V=\\bigoplus_{h\\in T_1}W^h.$\nAlso we can guarantee the existence of a conjugate of $U$ which is\n$H_1$-invariant by means of the Schur-Zassenhaus Theorem as in $\\it(5)$.\nThere is no loss in assuming that $U$ is $H_1$-invariant.\n\nSet now $Y=\\Sigma_{x\\in F'}U^x$ and $F_2=Stab_F(Y)$ and $F_1=Stab_F(U).$\nClearly, $F_2=F'F_1$ and $Y$ is $H_1$-invariant. Notice that for all\nnonidentity $h\\in H$, we have $C_F(h)\\leq F'\\leq F_2$ . Assume first\nthat $F=F_2$. This forces that we have $V=Y$. Clearly, $Y\\ne U$, that is\n$F'\\not \\leq F_1$, because otherwise $Q=[Q,F]=1$ due to the scalar\naction of the abelian group $Q$ on $U$. So $F'\\cap F_1=1$ which implies\nthat $|F:F_1|$ is a prime. Then $F_1\\unlhd F$ and $F'\\leq F_1$ which is\nimpossible. Therefore $F\\ne F_2$.\n\nIf $1\\neq h\\in H$ and $t\\in F$ such that ${Y}^{th}={Y}^{t}$ then\n$[h,t]\\in F_2$. Now, $F_2t=F_2t^h=(F_2t)^h$ and this implies the\nexistence of an element in $F_2t\\cap C_F(h)$. Since\n$C_F(h)\\leq F'\\leq F_2$ we get $t\\in F_2$. In particular, for each\n$t\\in F\\setminus F_2$ we have $Stab_{H}(Y^t)=1$.\n\nLet $S$ be a transversal for $F_2$ in $F$. For any $t\\in S\\setminus F_2$\nset $Y_t=Y^t$ and consider $Z_t=\\Sigma_{h\\in H}{Y_t}^h$. Notice that\n$V=Y\\oplus \\bigoplus_{t\\in S\\setminus F_2}Z_t$. As the sum $Z_t$ is\ndirect we have $$C_{Z_t}(H)=\\{ \\sum_{h\\in H}v^{h} \\ : \\ v\\in Y_t\\}$$\nwith $|C_{Z_t}(H)|=|Y_t|.$ Then\n$\\mathbf r([Y_t,c])=\\mathbf r([C_{Z_t}(H),c])\\leq s$ for each\n$t\\in S\\setminus F_2$ with $[Y_t,c]\\ne 0$. On the other hand,\n$$\\Sigma\\{\\mathbf r([C_{Z_t}(H),c]) : t\\in S \\,\\,\\text{with}\\,\\,[Y_t,c]\\ne 0\\}\\leq \\mathbf r([C_V (H), c])\\leq s$$\nwhence $|\\{t\\in S\\setminus F_2 : [Y_t,c]\\ne 0\\}|$ is suitably bounded.\nSo the claim is established if there exists $t\\in S\\setminus F_2$ such\nthat $[Y_t,c]\\ne 0$, since we have\n$V=Y\\oplus \\bigoplus_{t\\in S\\setminus F_2}Z_t$. Thus we may assume that\n$c$ is trivial on $\\bigoplus_{t\\in S\\setminus F_2}Z_t$ and hence\n$[V,c]=[Y,c].$\n\nThere are two cases now: We have either $F'\\cap F_1=1$ or $F'\\leq F_1.$\nFirst assume that $F'\\leq F_1.$ Then we get $F_1=F_2$ because\n$F_2=F'F_1.$ Now $U=Y.$ Due to the action by scalars of the abelian\ngroup $Q$ on $U$, it holds that $[Q,F_1]\\leq C_Q(U)$. From this point on\nwe can proceed as in the proof of step $\\it(5)$ and observe that\n$C_Q(F_{q'})=1$. Letting now $y=\\prod_{f\\in F_{q'}}c^f$, we have\n$$1=y=(\\prod_{f\\in F_1\\cap F_{q'} }c^f)(\\prod_{f\\in F_{q'}\\setminus F_1}c^f)\\in c^{|F_1\\cap F_{q'}|}C_Q(U).$$implying\nthat $c\\in C_Q(U)$, because $q$ is coprime to $|F_{q'}|$.\n\nThus we have $F_1\\cap F'=1$. First assume that $H_1=H$. Then $Y$ is\n$H$-invariant and $F_1H$ is a Frobenius group. Note that $C_U(F_1)=1$ as\n$C_V(F)=1$, and hence $C_{Y}(F_1)=1$ since $F'\\leq Z(F).$ We consider\nnow the action of $QF_1H$ on $Y$ and the fact that\n$\\mathbf r([C_{Y}(H),C_Q(H)])\\leq s.$ Then step $\\it(5)$, we obtain that\n$\\mathbf r(Y)=\\mathbf r([Y,Q])$ is $(s,|H|)$-bounded. Next assume that\n$H_1\\ne H.$ Choose a transversal for $H_1$ in $H$ and set\n$Y_1=\\Sigma_{h\\in T_1}Y^h$. Clearly this sum is direct and hence\n$$C_{Y_1}(H)=\\{ \\sum_{h\\in T_1}v^{h} \\ : \\ v\\in Y\\}$$ with\n$|[C_{Y_1}(H),c]|=|[Y,c]|.$ Then\n$\\mathbf r([Y,c])=\\mathbf r([C_{Y_1}(H),c])\\leq s$ establishing claim\n$\\it(6)$. ◻\n\n*(7) The proposition follows.*\n\n*Proof.* From now on $FH$ is a Frobenius-like group of Type II, that is,\n$H$ and $C_F(H)$ are of prime orders. By step $\\it(4)$ we have\n$H=H_1 =Stab_H( \\Omega_1)$ since $|H|$ is a prime. Now $V=W$. We may\nalso assume by the Schur-Zassenhaus theorem as in the previous steps\nthat there is an $H$-invariant element, say $U$ in $\\Omega$. Let $T$ be\na transversal for $F_1=Stab_F(U)$ in $F$. Then\n$F= \\bigcup_{t\\in T}{F_1}t$ implies $V=\\bigoplus_{t\\in T}U^t$. It should\nalso be noted that we have $|\\{t\\in T : [U^t,c]\\ne 0\\}|$ is suitably\nbounded as\n$$[C_V(H),c]=\\bigoplus \\{ [C_{X_t}(H),c] : t\\in T \\,\\, \\text{with }\\,\\,[U^t,c]\\ne 0\\}\\leq [C_V(H),C_Q(H)]$$\nwhere $X_t=\\bigoplus_{h\\in H}U^{th}$.\n\nLet $X$ be the sum of the components of all regular $H$-orbits on\n$\\Omega$, and let $Y$ denote the sum of all $H$-invariant elements of\n$\\Omega$. Then $V=X\\oplus Y.$ Suppose that ${U}^{th}={U}^t$ for $t\\in T$\nand $1\\ne h\\in  H$. Now $[t,h]\\in F_1$ and so the coset $F_1t$ is fixed\nby $H$. Since the orders of $F$ and $H$ are relatively prime we may\nassume that $t\\in C_F(H).$ Conversely for each $t\\in C_F(H)$, ${U}^t$ is\n$H$-invariant. Hence the number of components in $Y$ is\n$|T\\cap C_F(H)|=|C_F(H):C_{F_1}(H)|$ and so we have either\n$C_F(H)\\leq F_1$ or not.\n\nIf $C_F(H)\\not\\leq F_1$ then $C_{F_1}(H)=1$ whence $F_1H$ is Frobenius\ngroup acting on $U$ in such a way that $C_{U}(F_1)=1$. Then\n$\\mathbf  r(U)$ is $(s,|H|)$-bounded by step $\\it(5)$ since\n$\\mathbf  r([C_{U}(H),C_Q(H)])\\leq s$ holds. This forces that\n$\\mathbf r([V,c])$ is bounded suitably and hence the claim is\nestablished.\n\nThus we may assume that $C_F(H)\\leq F_1.$ Then $Y=U$ is the unique\n$H$-invariant $Q$-homogeneous component. If $[U^t,c]\\ne 0$ for some\n$t\\in F\\setminus F_1$ we can bound $\\mathbf r(U)$ and hence\n$\\mathbf r([V,c])$ suitably. Thus we may assume that $c$ is trivial on\n$U^t$ for each $t\\in F\\setminus F_1.$ Due to the action of the abelian\ngroup $Q$ on $U$, it holds that $[Q,F_1]\\leq C_Q(U)$. From this point on\nwe can proceed as in the proof of step $\\it(5)$ and observe that\n$C_Q(F_{q'})=1$. Letting now $y=\\prod_{f\\in F_{q'}}c^f$, we have\n$$1=y=(\\prod_{f\\in F_1\\cap F_{q'} }c^f)(\\prod_{f\\in F_{q'}\\setminus F_1}c^f)\\in c^{|F_1\\cap F_{q'}|}C_Q(U).$$implying\nthat $c\\in C_Q(U)$, because $q$ is coprime to $|F_{q'}|$. This final\ncontradiction completes the proof of Proposition 3.1. ◻\n\n ◻\n\nThe next proposition studies the action of a dihedral group of\nautomorphisms and is essential in proving Theorem B.\n\n**Proposition 10**.  * Let $D= \\langle\\alpha, \\beta \\rangle$ be a\ndihedral group generated by two involutions $\\alpha$ and $\\beta$.\nSuppose that $D$ acts on a $q$-group $Q$ for some prime $q$. Let $V$ be\nan irreducible $\\mathbb{F}_pQD$-module where $\\mathbb{F}_p$ is a field\nwith characteristic $p$ not dividing $|Q|$. Suppose that $C_{VQ}(F)=1$\nwhere $F=\\langle \\alpha\\beta\\rangle$. If\n$max\\{\\mathbf r([C_V(\\alpha), C_Q(\\alpha)]),\\mathbf r([C_V(\\beta), C_Q(\\beta)])\\}\\leq s$,\nthen $\\mathbf r([V,Q])$ is $s$-bounded.*\n\n*Proof.* We set $H=\\langle \\alpha\\rangle$. So $D=FH$. By Lemma and\nTheorem , we have $[V,Q]=[V, C_Q(\\alpha)][V,C_Q(\\beta)]$. Then it is\nsufficient to bound the rank of $[V,C_Q(H)]$. Following the same steps\nas in the proof of Proposition by replacing Proposition 2.3 by\nProposition 2.4, we observe that $Q$ acts faithfully on $V$ and\n$Q=\\langle c^F\\rangle$ is abelian with $c\\in C_{Z(Q)}(H)$ of order $q$.\nFurthermore\n$Ker(C_Q(H) \\ \\textrm{on} \\ C_V(H)) = Ker(C_Q(H) \\ \\textrm{on} \\ V)=1$.\nNote that it suffices to bound $\\mathbf r([V,c])$ suitably.\n\nLet $\\Omega$ denote the set of $Q$-homogeneous components of the\nirreducible $QD$-module $V.$ Let $\\Omega_1$ be an $F$-orbit of $\\Omega$\nand set $W=\\Sigma_{U\\in \\Omega_1}U.$ Then we have $V=W+W^{\\alpha}$.\nSuppose that $W^{\\alpha}\\ne W$. Then for any $U\\in \\Omega_1$ we have\n$Stab_H(U)=1$. Let $T$ be a tranversal for $Stab_F(U)=F_1$ in $F$ . It\nholds that $V=\\Sigma_{t\\in T}X_t$ where $X_t=U^t+U^{t\\alpha}.$ Now\n$[V,c]=\\Sigma_{t\\in T}[X_t,c]$ and $C_V(H)=\\Sigma_{t\\in T} C_{X_t}(H)$\nwhere $C_{X_t}(H)=\\{w+w^{\\alpha} : w\\in U^t\\}$. Since $[V,c]\\ne 0$ there\nexists $t\\in T$ such that $[U^t,c]\\ne 0$, that is $[U^t,c]=U^t.$ Then\n$[C_{X_t}(H),c]=C_{X_t}(H).$ Since $\\mathbf  r([C_V(H),C_Q(H)])\\leq s$\nwe get $\\mathbf  r(U)=\\mathbf  r(C_{X_t}(H))\\leq s$. Furthermore it\nfollows that $|\\{t\\in T : [U^t,c]\\ne 0\\}|$ is $s$-bounded and as a\nconsequence $\\mathbf  r([V,c])$ is suitably bounded. Thus we may assume\nthat $W^{\\alpha}=W$ which implies that $\\Omega_1=\\Omega$ and $H$ fixes\nan element, say $U$, of $\\Omega$ as desired.\n\nLet $U^t\\in \\Omega$ be $H$-invariant. Then $[t,\\alpha]\\in F_1.$ On the\nother hand $t^{-1}t^{\\alpha}=t^{-2}$ since $\\alpha$ inverts $F$. So\n$F_1t$ is an element of $F/F_1$ of order at most $2$ which implies that\nthe number of $H$-invariant elements of $\\Omega$ is at most $2$. Let now\n$Y$ be the sum of all $H$-invariant elements of $\\Omega$. Then\n$V=Y\\oplus \\bigoplus_{i=1}^m X_i$ where $X_1,\\ldots X_m$ are the sums of\nelements in $H$-orbits of length $2.$ Let $X_i=U_i\\oplus U_i^{\\alpha}$.\nNotice that if $[U_i,c]\\ne 0$ for some $i$, then we obtain\n$\\mathbf  r(U)=\\mathbf  r(U_i)\\leq s$ by a similar argument as above. On\nthe other hand we observe that the number of $i$ for which\n$[U_i,c]\\ne 0$ is $s$-bounded by the the hypothesis that\n$\\mathbf r([C_V(H),c])\\leq s$. It follows now that $\\mathbf r([V,c])$ is\nsuitably bounded in case where $[U_i,c]\\ne 0$ for some $i$.\n\nThus we may assume that $c$ centralizes $\\bigoplus_{i=1}^m X_i$ and that\n$[U,c]=U$. Due to the scalar action by scalars of the abelian group $Q$\non $U$, it holds that $[Q,F_1]\\leq C_Q(U)$. As $F_1\\unlhd FH$, we have\n$[Q,F_1]\\leq  C_Q(V)=1$. Clearly we have $C_Q(F_{q'})=1$ where $F_{q'}$\ndenotes the Hall $q'$-part of $F$ whose existence is guaranteed by the\nfact that $C_Q(F)=1.$ Let now $y=\\prod_{f\\in F_{q'}}c^f$. Then we have\n$$1=y=(\\prod_{f\\in F_1\\cap F_{q'} }c^f)(\\prod_{f\\in F_{q'}\\setminus F_1}c^f)\\in c^{|F_1\\cap F_{q'}|}C_Q(U).$$\nAs a consequence $c\\in C_Q(U)$, because $q$ is coprime to $|F_{q'}|$.\nThis contradiction completes the proof of Proposition . ◻\n\n# Proofs of theorems\n\nFirstly, we shall give a detailed proof for Theorem A part (b). The\nproof of Theorem A (a) can be easily obtained by just obvious\nmodifications of the proof of part (b).\n\nFirst, we assume that $G = PQ$ where $P$ and $Q$ are $FH$-invariant\nsubgroups such that $P$ is a normal $p$-subgroup for a prime $p$ and $Q$\nis a nilpotent $p'$-group with $|[C_P(H),C_Q(H)]|=p^s$. We shall prove\nthat $\\mathbf r(\\gamma_{\\infty}(G))$ is $((s,|H|)$-bounded. Clearly\n$\\gamma_{\\infty}(G)=[P,Q]$. Consider an unrefinable $FH$-invariant\nnormal series $$P=P_{1}>P_{2}>\\cdots>P_{k}>P_{k+1}=1.$$ Note that its\nfactors $P_i/P_{i+1}$ are elementary abelian. Let $V=P_{k}$. Since\n$C_V(Q)=1$, we have that $V=[V,Q]$. We can also assume that $Q$ acts\nfaithfully on $V$. Proposition yields that $\\mathbf r(V)$ is\n$(s, |H|)$-bounded. Set $S_i=P_i/P_{i+1}$. If $[C_{S_i}(H),C_Q(H)]=1$,\nthen $[S_i,Q]=1$ by Proposition . Since $C_P(Q)=1$ we conclude that each\nfactor $S_i$ contains a nontrivial image of an element of\n$[C_P(H),C_Q(H)]$. This forces that $k \\leq s$. Then we proceed by\ninduction on $k$ to obtain that $\\mathbf r([P,Q])$ is an\n$(s,|H|)$-bounded number, as desired.\n\nLet $F(G)$ denote the Fitting subgroup of a group $G$. Write\n$F_{0}(G)=1$ and let $F_{i+1}(G)$ be the inverse image of\n$F(G/F_{i}(G))$. As is well known, when $G$ is soluble, the least number\n$h$ such that $F_{h}(G)=G$ is called the Fitting height $h(G)$ of $G$.\nLet now $r$ be the rank of $\\gamma_{\\infty}(C_G(H))$. Then $C_G(H)$ has\n$r$-bounded Fitting height (see for example Lemma 1.4 of ) and hence $G$\nhas $(r,|H|)$-bounded Fitting height.\n\nWe shall proceed by induction on $h(G)$. Firstly, we consider the case\nwhere $h(G)=2$. Indeed, let $P$ be a Sylow $p$-subgroup of\n$\\gamma_{\\infty}(G)$ and $Q$ an $FH$-invariant Hall $p'$-subgroup of\n$G$. Then, by the preceeding paragraphs and Lemma , the rank of\n$P=[P,Q]$ is $(r,|H|)$-bounded and so the rank of $\\gamma_{\\infty} (G)$\nis $(r,|H|)$-bounded. Assume next that $h(G)>2$ and let $N=F_2(G)$ be\nthe second term of the Fitting series of $G$. It is clear that the\nFitting height of $G/\\gamma_{\\infty} (N)$ is $h-1$ and\n$\\gamma_{\\infty} (N)\\leq \\gamma_{\\infty}(G)$. Hence, by induction we\nhave that $\\gamma_{\\infty}(G)/\\gamma_{\\infty} (N)$ has $(r,|H|)$-bounded\nrank. As a consequence, it holds that\n$${\\bf r}(\\gamma_{\\infty}(G))\\leq {\\bf r}( \\gamma_{\\infty}(G)/\\gamma_{\\infty} (N))+{\\bf r}(\\gamma_{\\infty}(N))$$\ncompleting the proof of Theorem A(b).\n\nThe proof of Theorem B can be directly obtained as in the above argument\nby replacing Proposition by Proposition ; and Proposition by Proposition\n.\n"
    },
    {
        "text": "abstract: In the setting of real square matrices, it is known that, if\n  $A$ is a singular irreducible $M$-matrix, then the only nonnegative\n  vector that belongs to the range space of $A$ is the zero vector. In\n  this paper, we prove an analogue of this result for the Lyapunov and\n  the Stein operators.\n\n*A.M. Encinas $^1$, Samir Mondal $^2$ and K.C. Sivakumar $^2$   \n*\n\n*Keywords:* $M$-matrix, Singular irreducible $M$-matrix, Almost\nmonotonicity, Lyapunov operator, Stein operator.\n\n*AMS Subject Classifications:* 15A48, 15A23, 15A09, 15A18\n\n# Introduction\n\nThe set of all real matrices of order $n \\times n$ will be denoted by\n$\\mathbf{M}_n\\,(\\mathbb{R})$. For a matrix\n$X \\in \\mathbf{M}_n\\,(\\mathbb{R})$, we use $X\\geq 0$ to denote the fact\nthat all the entries of $X$ are nonnegative. If all the entries of $X$\nare positive, we denote that by $X >0$. We use the same notation for\nvectors.\n\nAs usual, for any $A\\in \\mathbf{M}_n\\,(\\mathbb{R})$, $N(A)$ and $R(A)$\ndenote the null and the range spaces of\n$A\\in \\mathbf{M}_n\\,(\\mathbb{R})$, respectively. The *rank* of $A$ is\n${\\rm rk}(A)$, the dimension of $R(A)$. In addition, $\\rho(A)$ denotes\nthe *spectral radius* of $A$; that is, the maximum of the absolute\nvalues of its eigenvalues.\n\nA matrix $A\\in\\mathbf{M}_n\\,(\\mathbb{R})$ is said to be *reducible* if\nthere is a permutation matrix $P\\in\\mathbf{M}_n\\,(\\mathbb{R})$ such that\n$P^{T}AP$ has the form $$%\n\\begin{bmatrix}\nS_{11} & S_{12}\\\\\n0 & S_{22}%\n\\end{bmatrix}$$ for some square matrices $S_{11}$ and $S_{22}$ of order\nat least one. A matrix is *irreducible* if it is not reducible.\n\nThe *group inverse* of a matrix $A\\in \\mathbf{M}_n\\,(\\mathbb{R})$ is the\nunique matrix $X \\in \\mathbf{M}_n\\,(\\mathbb{R})$, if it exists, that\nsatisfies the equations $AXA=A, XAX=X$ and $AX=XA$. If it exists, then\nthe group inverse is denoted by $A^{\\#}$. Of course, when $A$ is\nnonsingular, then $A^\\#$ exists and moreover $A^\\#=A^{-1}$.\n\nA necessary and sufficient condition for the group inverse of a matrix\n$A$ to exist is that $A$ has index $1$; that is, $R(A^2)=R(A)$, which is\nequivalent to the condition $N(A^2)=N(A)$, see . Another\ncharacterization is that $A^{\\#}$ exists iff $R(A)$ and $N(A)$ are\ncomplementary subspaces of $\\mathbb{R}^n$. It is easy to show that any\nsymmetric matrix has group inverse, that a nilpotent matrix $A$ (viz.,\n$A^n=0$), does not have group inverse, whereas the group inverse of an\nidempotent matrix (viz., $A^2=A$), (exists and) is itself. We refer the\nreader to for more details.\n\nIf ${\\cal S}^n(\\mathbb{R})$ is the subspace of symmetric matrices in\n$\\mathbf{M}_n\\,(\\mathbb{R})$, for $X \\in {\\cal S}^n(\\mathbb{R}),$ let us\nsignify $X \\succeq 0$ to denote the fact that $X$ is a positive\nsemidefinite matrix. We use $X \\succ 0,$ when $X$ is positive definite.\n\n**Definition 1**. *A matrix $A\\in \\mathbf{M}_n\\,(\\mathbb{R})$ is called\na $Z$-matrix, if all the off-diagonal entries of $A$ are nonpositive.\nAny $Z$-matrix $A$ has the representation $A=sI-B,$ where $s \\geq 0$ and\n$B\\ge 0$. If $s\\ge \\rho(B)$, then $A$ is called an *$M$-matrix*.*\n\nThe notion of $M$-matrix introduced by A. Ostrowski in 1935 in honor of\nH. Minkowski who worked with this class of matrices around 1900.\n$M$-matrices possess many interesting nonnegativity properties. For\ninstance, in the representation as above, let $s > \\rho(B).$ Then,\n$A=s\\big(I-\\frac{B}{s}\\big)$ and considering the *Neumann series* for\n$\\frac{B}{s}$; that is,\n$$\\Big(I-\\frac{B}{s}\\Big)^{-1}=\\sum_{m=0}^{\\infty} \\Big(\\frac{B}{s}\\Big)^m \\geq 0.$$\nwe conclude that $A$ is invertible and moreover\n$A^{-1}=\\frac{1}{s}\\Big(I-\\frac{B}{s}\\Big)^{-1}\\ge 0$. In such a case,\nwe shall refer to $A$ as an *invertible $M$-matrix*. Otherwise, we call\n$A$, a *singular $M$-matrix*.\n\nHere is a characterization for invertible $M$-matrices. For a proof, we\nrefer the reader to the book , where fifty different characterizations\nare given.\n\n**Theorem 2**.  *   \nLet $A$ be a $Z$-matrix. Then the following statements are equivalent:  \n(a) $A$ is an invertible $M$-matrix.  \n(b) There exists $x>0$ such that $Ax>0$.  \n(c) For every $q >0,$ there exiss $x>0$ such that $Ax=q.$  \n(d) $A$ is monotone, i.e. $Ax \\geq 0 \\Longrightarrow x\\geq 0$.  \n$A$ is inverse-nonnegative, i.e. $A$ is invertible and $A^{-1}\\ge 0$.  \n(f) $A$ is a $P$-matrix, i.e. all principal minors of $A$ are\npositive.  \n(g) $A$ is positive stable i.e. if the real part of each of its\neigenvalues is positive.  \nSuppose that $A$ has the representation $A=sI-B$, with $B \\geq 0$ and\n$s\\ge \\rho(B)$. Then each of the above statements is equivalent to:  \n(h) $s<\\rho(B)$.  \nWhen in addition, $A$ is irreducible the above statements are equivalent\nto:  \n(e’) $A$ is inverse-positive, i.e. $A$ is invertible and $A^{-1}> 0$.*\n\nWe will have the occasion to consider versions of Theorem for two\nspecial classes of operators on $S^n(\\mathbb{R})$ (Theorem and Theorem\n), in the next section.\n\nNext, let us turn our attention to the case singular $M$-matrices. A\ndistinguished subclass of such matrices, due to their relevance in many\napplications, is the set of *singular irreducible $M$-matrices*. For\nsuch matrices, we us recall a well known result, which will serve to\nmotivate the contents of this article.\n\n**Theorem 3**. *,  \nLet $A \\in \\mathbf{M}_n\\,(\\mathbb{R})$ be a singular irreducible\n$M$-matrix. Then the following hold:  \n(a) $\\rm rk(A)=n-1.$  \n(b) There exists a vector $x >0$ such that $Ax=0.$  \n(c) $A^{\\#}$ exists and is nonnegative on $R(A)$, i.e.\n$x \\geq 0, x \\in R(A) \\Longrightarrow A^{\\#}x \\geq 0$.  \n(d) All the principal submatrices of $A$, except $A$ itself is an\ninvertible $M$-matrix.  \n(e) $A$ is almost monotone, i.e. $Ax \\geq 0 \\Longrightarrow Ax=0.$*\n\nWe remark that in statement (c) was established as $A$ has \"property c\",\nand that the equivalence with (c) in Theorem was proved in . Recall that\na singular $M$-matrix $A$ is said to have \"property c\", if $A=sI-B$,\nwhere $B \\geq 0, s>0$ and the matrix $\\frac{1}{s}B$ is semi-convergent.\nSemi-convergence of a matrix $X$ means that the matrix sequence\n$\\{X^k\\}$ converges. Note that for an invertible $M$-matrix, such a\nsequence converges to zero. One of the most prominent situations where\nsuch matrices arise concerns irreducible Markov processes, where one\ndeals with matrices of the form $I-T$, where $T$ is an irreducible\ncolumn stochastic matrix. We refer the reader to for additional results\nthat characterize singular irreducible $M$-matrices.\n\n**Definition 4**. *A matrix $A$ is referred to as *range monotone (see,\nfor instance *), if $$Ax \\geq 0, x \\in R(A) \\Longrightarrow x \\geq 0.$$\nIf $A$ is range monotone, then we say that $A$ has the *range\nmonotonicity* property.*\n\nObserve that range monotonicity of $A$ is equivalent to:\n$A^2x \\geq 0 \\Longrightarrow Ax\\ge 0.$ Moreover, shows that statement\n$(c)$ of Theorem is equivalent to the range monotonicity of $A$.\n\nLet us recall a notion that is stronger than range monotonicity. This\nwill play a central role in this article.\n\n**Definition 5**. *Let $A \\in \\mathbf{M}_n\\,(\\mathbb{R})$. Then $A$ is\ncalled trivially range monotone if\n$$Ax \\geq 0, x \\in R(A) \\Longrightarrow x = 0.$$*\n\nTrivial range monotonicity of $A$ is the same as:\n$A^2x \\geq 0 \\Longrightarrow Ax=0.$ It is easy to verify that the matrix\n$A=\\begin{pmatrix}\n~~1 & -1 \\\\\n-1 & ~~1\n\\end{pmatrix}$ is trivially range monotone.\n\n**Theorem 6**.  * Let $A \\in \\mathbf{M}_n\\,(\\mathbb{R})$ be a singular\nirreducible $M$-matrix. Then $A$ is trivially range monotone.*\n\n**Proof.** Suppose that $Ax \\geq 0, x \\in R(A).$ Then by the almost\nmonotonicity of $A$ (item (e), Theorem ), it follows that $Ax=0$. So,\n$x \\in N(A)\\cap R(A)$ and since the group inverse $A^{\\#}$ exists, we\nhave $x=0.$\n\nIn this work, we shall be interested in the problem of determining\ntrivial range monotonicity for the Lyapunov and the Stein operators on\nthe space of real symmetric matrices. We consider four classes of\nmatrices that give rise to specific instances of these operators, for\nwhich we give affirmative/negative answers. The main results are\npresented in Theorem , Theorem and Theorem . Pertinent counterexamples\nare presented in Section , followed by a summary table consolidating the\nfindings.\n\n# Preliminaries\n\nA nonempty subset $C$ of a finite dimensional Hilbert space $H$ is said\nto be a *proper cone* if $C+C=C, \\alpha C \\subseteq C$, for all\n$\\alpha \\geq 0$, $C \\cap -C =\\{0\\}$ and $C$ has a nonempty interior.\nGiven a cone $C$, we define its *dual cone* $C^*$ by\n$$C^*:=\\{ y \\in \\mathbb{R}^n: \\langle x,y\\rangle \\geq 0, ~\\forall x \\in C\\}.$$\nIt follows that $C^*$ is indeed a proper cone. A well known example of a\nproper cone is the nonnegative orthant $\\mathbb{R}^n_+$, viz., the cone\nof nonnegative vectors. Let the space of all real symmetric matrices of\norder $n,$ be denoted by ${\\cal S}^n$. Then ${\\cal S}^n$ is a real\nHilbert space with the trace inner product, i.e.,\n$\\langle A,B \\rangle = tr(AB),~A,B \\in {\\cal S}^n.$ In ${\\cal S}^n,$ the\nset of all positive semidefinite matrices denoted by ${\\cal S}^n_+$, is\na cone. Both these cones satisfy the condition of self-duality, viz.,\n$C^*=C,$ when $C=\\mathbb{R}^n_+$ or ${\\cal S}^n_+$.\n\nNext, let us observe that if $A$ is a $Z$-matrix, then\n$\\langle Ae_j,e_i \\rangle = a_{ij} \\leq 0 ,\\;i\\neq j$, where $e_i$\ndenotes the $i$th standard basis vector of $\\mathbb{R}^n$. These basis\nvectors are mutually orthogonal and positive, as well. It then follows\nthat an equivalent formulation for a matrix $A$ to be a $Z$-matrix is:\n\n$x \\geq 0, \\,y \\geq 0$`<!-- -->`{=html} and\n$\\langle x,y \\rangle =0 \\;\\Longrightarrow \\; \\langle Ax,y \\rangle \\leq 0$.\n\nThis in turn is the same as saying (with $C=\\mathbb{R}^n_+$):\n\n$x \\in C, \\,y \\in C^*$ and\n$\\langle x,y \\rangle =0 \\; \\Longrightarrow \\; \\langle Ax,y \\rangle \\leq 0$.\n\nMotivated by the reformulation above, a linear operator\n$T:{\\cal S}^n \\longrightarrow {\\cal S}^n$ is called a *$Z$-operator* if\nit satisfies:\n\n$X \\succeq 0, Y \\succeq 0$ and\n$\\langle X,Y \\rangle =0 \\; \\Longrightarrow \\; \\langle T(X),Y \\rangle \\leq 0$,\n\nwhere $U \\succeq 0$ stands for the fact that the symmetric matrix $U$ is\npositive semidefinite. When $-U \\succeq 0,$ we will use the notation\n$U \\preceq 0.$ In this article, we shall be concerned with two important\nclasses of operators on ${\\cal S}^n$. These are the Lyapunov operator\nand the Stein operator. Let us first look at their definitions. Given\n$A \\in \\mathbf{M}_n\\,(\\mathbb{R})$, the *Lyapunov operator* $L_A$ on\n${\\cal S}^n$ is the operator $$L_A(X):=AX+XA^T, \\;X \\in {\\cal S}^n$$ and\nthe *Stein operator* $S_A$ is defined by\n$$S_A(X):=X-AXA^T, \\;X \\in {\\cal S}^n.$$\n\nLet $L$ denote either the Lyapunov or the Stein operator for a given\nmatrix $A$. Then $L$ satisfies the following (independent of $A$):\n\n$X \\succeq 0,\\, Y \\succeq 0$ and\n$\\langle X,Y \\rangle =0 \\; \\Longrightarrow \\; \\langle L(X),Y \\rangle \\leq  0.$\n\nThus, both these operators could be thought of as analogues of\n$Z$-matrices, for linear maps on ${\\cal S}^n$.\n\nAn operator $T$ on ${\\cal S}^n$ will be called a *positive stable\n$Z$-operator* if $T$ is an invertible $Z$-operator and\n$T^{-1}({\\cal S}^n_+) \\subseteq {\\cal S}^n_+$. Note that this\ngeneralizes what we know for an invertible $M$-matrix, namely that it is\ninvertible and that its inverse is nonnegative, i.e. leaves the cone\n$\\mathbb{R}^n_+$ invariant.\n\nThe following result is a version of (items (b) and (c) of) Theorem for\nthe Lyapunov operator . The notation $U \\succ 0$ denotes the fact that\nthe symmetric matrix $U$ is positive definite.\n\n**Theorem 7**.  * ()   \nFor $A \\in \\mathbf{M}_n\\,(\\mathbb{R})$, the following statements are\nequivalent:  \n(a) There exists $E \\succ 0$ such that $L_A(E) \\succ 0$.  \n(b) For every $Q \\succ 0$ there exists $X \\succ 0$ such that\n$L_A(X)=Q$.  \n(c) $A$ is positive stable.*\n\nA version for the Stein operator is stated next. The original result\nholds for complex matrices . We present only the real case. Recall that\na matrix $A$ is called Schur stable if all its eigenvalues lie in the\nopen unit disc of the complex plane.\n\n**Theorem 8**.  * ()  \nFor $A \\in \\mathbf{M}_n\\,(\\mathbb{R})$, the following statements are\nequivalent:  \n(a) There exists $E \\succ 0$ such that $S_A(E) \\succ0$.  \n(b) For every $Q \\succ 0$ there exists $X \\succ 0$ such that\n$S_A(X)=Q$.  \n(c) $A$ is Schur stable.*\n\nThese results may be considered as analogues of Theorem . Let $L$ stand\nfor either the Lyapunov operator or the Stein operator. Then, $(b)$ of\nTheorem or Theorem states that the range of the operator $L$ contains\nthe open set of all (symmetric) positive definite matrices. Hence $L$ is\nsurjective and so, it is injective, too. Thus, $L$ is invertible. By the\nsame statement, it also follows that\n$L^{-1}({\\cal S}^n_+) \\subseteq {\\cal S}^n_+$. Thus, these two operators\nare examples of positive stable $Z$-operators.\n\n# Main Results\n\nWe are interested in identifying some matrix classes for which the\nLyapunov operator and/or the Stein operator are/is trivially range\nmonotone. Let us define this notion, first. Motivated by the definitions\nfor matrices, we call a linear operator\n$T:{\\cal S}^n \\rightarrow {\\cal S}^n$ *range monotone* if\n$$T(X) \\succeq 0, X \\in R(T) \\Longrightarrow X \\succeq 0$$ and refer to\n$T$ as *trivially range monotone* if\n$$T(X) \\succeq 0, X \\in R(T) \\Longrightarrow X =0.$$ As in the case of\nmatrices, one may observe that range monotonicity of a linear operator\n$T$ is equivalent to the condition:\n$T^2(X) \\succeq 0 \\Longrightarrow T(X) \\succeq 0$ and that trivial range\nmonotonicity is the same as: $T^2(X) \\succeq 0 \\Longrightarrow T(X) =0.$\n\nNext, we give an example of a range monotone operator and also one which\nis not range monotone.\n\n**Example 9**. *Let $A=\n\\begin{pmatrix}\n1 & 1 \\\\\n0 & 0\n\\end{pmatrix}.$ The corresponding Lyapunov operator is given by\n$$L_A(X)=\n\\begin{pmatrix}\n2(x_{11}+x_{12}) & x_{12}+x_{22} \\\\\nx_{12}+x_{22} & 0\n\\end{pmatrix},$$ where $X= \\begin{pmatrix}\nx_{11} & x_{12} \\\\\nx_{12} & x_{22}\n\\end{pmatrix}.$ The choice $x_{11}=x_{22}=1, x_{12}=-1$ shows that $L_A$\nis not invertible. It is easily seen that if $X \\in R(L_A)$, then\n$x_{22}=0.$ Next, for such an $X$, if $L_A(X)$ is a positive\nsemidefinite matrix, then $X= \\begin{pmatrix}\nx_{11} & 0 \\\\\n0 & 0\n\\end{pmatrix}$ is also positive semidefinite. Thus, $L_A$ is a range\nmonotone operator. It is not trivially range monotone, as\n$X= \\begin{pmatrix}\n1 & 0 \\\\\n0 & 0\n\\end{pmatrix} \\in R(L_A) \\cap {\\cal S}^n_+.$*\n\n**Example 10**.  * Let $A=\n\\begin{pmatrix}\n1 & 1 \\\\\n0 & 1\n\\end{pmatrix}.$ Then the associated Stein operator is given by\n$$S_A(X)=-\n\\begin{pmatrix}\n2b +c & c \\\\\nc & 0\n\\end{pmatrix},$$ given $X=\n\\begin{pmatrix}\na & b \\\\\nb & c\n\\end{pmatrix}.$ Set $X=\\begin{pmatrix}\n-1 & 0 \\\\\n~~0 & 0\n\\end{pmatrix}$ and $Y=\\begin{pmatrix}\n0 & \\frac{1}{2} \\\\\n\\frac{1}{2} & 0\n\\end{pmatrix}.$ Then $S_A(Y)=X,$ so that $X \\in R(S_A)$. Further,\n$S_A(X)=0$. This shows that $S_A$ is not range monotone.*\n\nIt is easy to prove that if an operator is idempotent, then it is range\nmonotone. The only fact that is used, is that an idempotent operator\nacts like the identity operator on its range space. Let us record this\nstatement.\n\n**Lemma 11**.  * Let $T:V \\rightarrow V$ be idempotent. Then $T$ is\nrange monotone.*\n\nAs a first step, we address the question of when the Lyapunov and the\nStein operators are idempotent. The relevant results are Theorem and\nTheorem . It appears that this question has not been addressed, to the\nbest of our knowledge.\n\nLet us fix some more notation. The all ones vector in $\\mathbb{R}^n$ is\ndenoted by $e$. For any $j=1,\\ldots,n$, we define the symmetric matrix\n$E_{jj}=[0,\\ldots,\\stackrel{j\\atop \\downarrow}{e_j},\\ldots,0]$ and for\nany $1\\le i<j\\le n$ we also define the symmetric matrix\n$E_{ij}=[0,\\ldots,\\stackrel{i\\atop \\downarrow}{e_j},\\ldots,\\stackrel{j\\atop \\downarrow}{e_i},\\ldots,0]$.\nIt is clear that $\\{E_{ij}\\}_{1\\le i\\le j\\le n}$ is a basis of\n$\\mathcal{S}^n$. Thus, if\n$A=[c_1,\\ldots,c_n]\\in \\mathbf{M}_n\\,(\\mathbb{R})$, we have,\n$AE_{jj}=[0,\\ldots,\\stackrel{j\\atop \\downarrow}{c_j},\\ldots,0]$, for any\n$j$ whereas,\n$AE_{ij}=[0,\\ldots,\\stackrel{i\\atop \\downarrow}{c_j},\\ldots,\\stackrel{j\\atop \\downarrow}{c_i},\\ldots,0]$,\nfor any $1\\le i<j\\le n$.  \nIn what follows, we collect some properties of the Lyapunov operator. We\nonly prove the second item, as the others are straightforward.\n\n**Lemma 12**. *  \n(a) For any $A,B\\in \\mathbf{M}_n\\,(\\mathbb{R})$ and any\n$s,t\\in \\mathbb{R}$, $L_{tA+sB}=tL_A+sL_B$.  \n(b) $L_A$ is the zero operator iff $A=0$ and $L_A$ is the identity\noperator iff $A=\\frac{1}{2}I$.  \n(c) For any $A\\in \\mathbf{M}_n\\,(\\mathbb{R})$, $L_A(I)=A+A^T$ and hence\n$L_A(A+A^T)=L_A^2(I).$  \n(d) For any orthogonal matrix $P$, we have:\n$L_{PAP^T}(PXP^T)=PL_A(X)P^T.$*\n\n**Proof.** (b) Let $c_j=(c_{1j},\\ldots,c_{nj})^T$ be the $j$-th column\nof $A$, for $1 \\leq j \\leq n.$ Then\n$$L_A(E_{jj})=[0,\\ldots,c_j,\\ldots,0]+[0,\\ldots,c_j,\\ldots,0]^T.$$\nTherefore, when $L_A=0,$ we infer that $c_{ij}=0$, $j\\not=i$ and\n$2c_{jj}=0$. When $L_A=I$, then $c_{ij}=0$, $j\\not=i$ and $2c_{jj}=1$.\n\n**Definition 13**. *A square matrix $A\\in \\mathbf{M}_n\\,(\\mathbb{R})$ is\ncalled *Lyapunov idempotent* or *$L$-idempotent*, for short, iff $L_{A}$\nis idempotent; that is, $L_A^2=L_A$.*\n\nIt is clear that $A$ is $L$-idempotent when $A=0$ and also when\n$A=\\frac{1}{2}I$. Moreover, when $L_A$ is idempotent, then $L_{tA}$ is\nnot, except when $A=0$ or $t\\neq 0,1$: Since $L_{tA}=tL_A$ for any\n$t\\in \\mathbb{R}$, we have that $L_{tA}^2=t^2L_A^2=t^2L_A$. Therefore,\nif $tA$ is $L$-idempotent, then $tL_A=L_{tA}=L_{tA}^2=t^2L_A$, which\nimplies that either $t=0$ or $L_A=tL_A$. When $t\\not=0$, the above\nequality implies that either $t=1$ or $L_A=0$, which is equivalent to:\n$A=0$.\n\n**Theorem 14**.  * A matrix $A\\in \\mathbf{M}_n\\,(\\mathbb{R})$ is\n$L$-idempotent iff it is diagonal and moreover\n$\\sigma(A)\\subseteq \\big\\{0,\\frac{1}{2}\\big\\}$. In particular, the only\nnonsingular $L$-idempotent matrix is $\\frac{1}{2}I$.*\n\n**Proof.**First, $A$ is $L$-idempotent iff for any $X\\in \\mathcal{S}^n$\nwe have that $$A^2X+2AXA^T+X(A^T)^2=AX+XA^T.$$ In particular, if\n$A^2=[\\hat c_1,\\ldots,\\hat c_n]$, choosing $X=E_{jj}$, $j=1,\\ldots,n$,\nwe obtain, for the left hand side, the expression\n$$[0,\\ldots,\\hat c_j,\\ldots,0]+[0,\\ldots,\\hat c_j,\\ldots,0]^T+2[c_{1j} c_j,\\ldots,c_{jj} c_j,\\ldots,c_{nj} c_j]^T,$$\nwhereas, the right hand side reduces to\n$$[0,\\ldots,c_j,\\ldots,0]+[0,\\ldots,c_j,\\ldots,0]^T.$$ This implies that\n$2c_{ij}c_{kj}=0$ for $i,k\\not=j$. In particular, $c_{ij}^2=0$ for\n$i\\not=j$ and so $c_j=c_{jj} e_j$, which in turn, implies that $A$ must\nbe diagonal and hence symmetric. Now, let $d_1,\\ldots,d_n$ be the\ndiagonal entries of $A$. Finally, by setting $X=I$ in the identity, we\nconclude that $4A^2=2A$, that is $d_j(2d_j-1)=0$ for any $j=1,\\ldots,n$,\ncompleting the proof.\n\n**Lemma 15**.  * The Stein operator satisfies the following basic\nproperties:  \n(a) For any $A\\in \\mathbb{R}^{n \\times n}$, $S_A$ is linear.  \n(b) If $P$ is an orthogonal matrix, then,\n$S_{PAP^T}(PXP^T)=PS_A(X)P^T.$  \n(c) $S_A$ is the identity operator iff $A=0$ and $S_A$ is the zero\noperator iff $A=\\pm I$.*\n\n**Proof.**We only prove the property (c), since the others are\nstraightforward. For the first part, clearly, $S_A$ is the identity iff\n$AXA^T=0$ for any symmetric matrix $X$. In particular, taking $X=I$ we\nget that $AA^T=0$ and so $A=0$.  \nLet us prove the second part. Observe that, if for any $j=1,\\ldots,n$,\n$c_j=(c_{1j},\\ldots,c_{nj})^T$ is the $j$-th column of $A$, then\n$$S_A(E_{jj})=[0,\\ldots,e_j,\\ldots,0]-[c_{1j} c_j,\\ldots,c_{jj} c_j,\\ldots,c_{nj} c_j]^T.$$\nTherefore, when $S_A=0$, this implies that $c_{jj} c_j=e_j$. Thus,\n$c_{jj}^2=1$ and $c_{ij}=0$, $i\\not=j$.\n\nFinally, when $i<j$, taking into account that\n$A=[c_{11} e_1,\\ldots,c_{nn} e_n]$, we obtain that\n$$0=S_A(E_{ij}) =[0,\\ldots,e_j,\\ldots,e_i,\\ldots,0]-[0,\\ldots,c_{ii} c_{jj} e_j,\\ldots,c_{jj}c_{ii} e_i\\ldots, 0],$$\nwhich implies that $c_{ii}c_{ij}=1$, thereby showing that $A=\\pm I$.\n\n**Definition 16**. *A square matrix $A\\in \\mathbf{M}_n\\,(\\mathbb{R})$ is\ncalled $S$-idempotent iff $S_{A}$ is idempotent; that is, $S_A^2=S_A$.*\n\nIt is clear that $A$ is $S$-idempotent when $A=0$ and also when\n$A=\\pm I$.\n\n**Theorem 17**.  * A matrix $A\\in \\mathbf{M}_n\\,(\\mathbb{R})$ is\n$S$-idempotent iff $A^2=\\pm A$. In particular, the only nonsingular\n$S$-idempotent matrices are $\\pm I$.*\n\n**Proof.**First, for any $X\\in \\mathcal{S}^n(\\mathbb{R})$ we have that\n$$\\begin{aligned}\n    S^2_A(X)=S_A(S_A(X))&=&S_A(X)-AS_A(X)A^T\\\\\n    & =& X-AXA^T-A(X-AXA^T)A^T\\\\\n    &=& X-2AXA^T+A^2X(A^T)^2\n\\end{aligned}$$ and hence $A$ is $S$-idempotent iff\n$$AXA^T=A^2X(A^T)^2$$ for any $X\\in \\mathcal{S}^n(\\mathbb{R})$, which in\nparticular implies that $$AE_{jj}A^T=A^2E_{jj}(A^T)^2$$ for any\n$j=1,\\ldots,n$. Therefore, if for any $j=1,\\ldots,n$ we consider\n$c_j=(c_{1j},\\ldots,c_{nj})^T$, the $j$-th column of $A$ and\n$\\hat c_j=(\\hat c_{1j},\\ldots,\\hat c_{nj})^T$, the $j$-th column of\n$A^2$, then\n$$[c_{1j} c_j,\\ldots,c_{nj} c_j]=[\\hat c_{1j}\\hat c_j,\\ldots,\\hat c_{nj}\\hat c_j] ,\\hspace{.25cm} j=1,\\ldots,n.$$\n\nFor a fixed $j=1,\\ldots,n$, the above equality implies that $\\hat c_j=0$\niff $c_j=0$ and moreover $\\hat c_j=d_j c_j$, where $d_j\\in \\mathbb{R}$\nand $d_j\\not=0$. Moreover, for any $k=1,\\ldots,n$ we also have that\n$c_{kj} c_j=\\hat c_{kj}d_j c_j$, so that when $c_j\\not=0$ this implies\nthat $c_{kj}=d_j\\hat c_{kj}$. Hence $c_j=d_j\\hat c_j$. In conclusion, if\n$c_j\\not=0$ we have $c_j=d_j\\hat c_j=d_j^2 c_j$ and hence $d_j^2=1$.\nSince this statement is also true when $c_j=0$ because then\n$\\hat c_j=0$, we conclude that $A^2=AD$, where $D^2=I$.\n\nFinally, suppose that $d_id_j=-1$ when $1\\le i<j\\le n$. Since\n$$AE_{ij}A^T=A^2E_{ij}(A^T)^2,\\hspace{.15cm}1\\le i\\le j\\le n.$$ we also\nhave\n$$[c_{1j} c_i+c_{1i} c_j,\\ldots,c_{nj} c_i+c_{ni} c_j]=[\\hat c_{1j}\\hat c_i+\\hat c_{1i}\\hat c_j,\\ldots,\\hat c_{nj}\\hat c_i+\\hat c_{ni}\\hat c_j],$$\nand hence, for any $k=1,\\ldots,n$\n$$c_{kj} c_i+c_{ki} c_j=d_id_j (c_{kj} c_i+c_{ki}\\\nc_j)=-c_{kj} c_i-c_{ki} c_j$$ which implies that\n$c_{kj} c_i+c_{ki} c_j=0$, which in turn, leads us to conclude that\n$c_{kj}c_{ki}=0$. If $c_i\\not=0$, then $c_{ki}\\not=0$ for some\n$k=1,\\ldots,n$ and hence the above identities imply that $c_{kj}=0$ and\nhence $c_j=0$. So, when $d_id_j=-1$ then either $c_i$ or $c_j$ must be\nthe zero vector.\n\nLet $i=\\min\\{k=1,\\ldots,n: c_k\\not=0\\}$. Then,\n$$AD=[d_1 c_1,\\ldots,d_i c_i,\\ldots,d_nc_n]=d_i[c_1,\\ldots, c_i,\\ldots,c_n]=d_iA,$$\nsince when $1\\le j<i$, $0=c_j=d_j c_j=d_i c_j$, whereas when $j>i$ and\n$d_j\\not=d_i$, then $c_j=0$. Hence $0= c_j=d_j c_j=d_i c_j$.\n\nIn view of Lemma , Theorem and Theorem , we obtain the following result.\nWe skip the proof.\n\n**Theorem 18**.  * Let $A \\in \\mathbf{M}_n\\,(\\mathbb{R}).$ We then have\nthe following:  \n(a) If $A$ is a diagonal matrix, whose diagonal entries are either $0$\nor $\\frac{1}{2}$, then the Lyapunov operator $L_A$ is range monotone.  \n(c) If $A$ satisfies $A^2=\\pm A$, then the Stein operator $S_A$ is range\nmonotone.*\n\nLet us briefly discuss the notion of the group inverse of a linear\noperator on a finite dimensional vector space $V$ . Let\n$T:V \\rightarrow V$ be linear. $T$ is said to have a group inverse\n$L:V \\rightarrow V$ if $TLT=T, LTL=L$ and $TL=LT$. The group inverse of\n$T$ need not exist, but when it does exist, it is unique and will be\ndenoted by $T^{\\#}.$ We make use of the following in our discussion.\n\n**Theorem 19**.  * Let $T:V \\rightarrow V$ be linear. Then the following\nare equivalent:  \n(a) $T^{\\#}$ exists.  \n(b) The subspaces $R(T)$ and $N(T)$, of $V$, are complementary.  \n(c) $R(T^2)=R(T).$  \n(d) $N(T^2)=N(T).$*\n\nLet us recall that an operator $T$, on an inner product space, is called\nnormal if $TT^{\\ast}=T^{\\ast}T$, where $T^{\\ast}$ denotes the adjoint of\n$T$. It is easy to see that if $T$ is normal, then $R(T^{\\ast})=R(T)$.\nSo, if $T$ is normal, then the subspaces $R(T)$ and $N(T)$ are\ncomplementary. Thus, we have the following consequence of Theorem .\n\n**Corollary 20**. *Let $T:V \\rightarrow V$ be linear, where $V$ is a\nfinite dimensional inner product space. If $T$ is a normal operator,\nthen $T^{\\#}$ exists.*\n\n**Definition 21**. *An operator\n$T:{\\cal S}^n \\longrightarrow {\\cal S}^n$, is called generalized\n$k$-potent ($k \\geq 2$) if there exists a constant $\\alpha$ such that\n$T^k=\\alpha T.$*\n\n**Proposition 22**.  * Let $T:{\\cal S}^n \\rightarrow {\\cal S}^n$ be a\ngeneralized $k$-potent. Then $T^{\\#}$ exists. In particular, any\nidempotent operator is group invertible.*\n\n**Proof.** We have $T^k=\\alpha T$, for some $k \\geq 2.$ Let $T^2(x)=0.$\nThen $0=T^k(x)=0=\\alpha T(x)=0,$ proving that $T(x)=0.$ Thus\n$N(T^2)=N(T).$ So, the group inverse of $T$ exists.\n\nIn the next result, we identify four classes of matrices for which the\nLyapunov as well as the Stein operators are group invertible. We shall\nmake use of the following observation. Let\n$A \\in \\mathbb{R}^{n \\times n}$. Then, $L_A^{\\ast}=L_{A^T}$ and\n$S_A^{\\ast}=S_{A^T}.$\n\n**Theorem 23**. *  \nThe group inverses $L_A^{\\#}$ and $S_A^{\\#}$ exist, if $A$ satisfies any\nof the following conditions:  \n(a) $A^2=-I$.  \n(b) $A^2=I.$  \n(c) $A^T=-A$.  \n(d) $A^T=A.$*\n\n**Proof.** (a) Let $A^2=-I$. First, we consider the Lyapunov operator\n$L_A$. We have $$L_A(X)=AX+XA^T, X \\in {\\cal S}^n.$$ Thus,\n$$\\begin{aligned}\nL_{A}^2(X) & = & A^2X + AXA^T + AXA^T+X(A^T)^2 \\\\\n & = & -2(X-AXA^T). \n \n\\end{aligned}$$ So, $$\\begin{aligned}\nL_A^3(X) & = & -2(L_A(X) - L_A(AXA^T)\\\\\n& = & -2(AX + XA^T - A^2XA^T - AX(A^T)^2) \\\\\n& = & -4(AX+XA^T) \\\\\n& = & -4L_A(X).\n\\end{aligned}$$ Thus, $L_A^3=-4L_A$ and so by Proposition , $L_A^\\#$\nexists.  \nNext, we take the case of the Stein operator. We have\n$$S_A(X)=X-AXA^T, X \\in {\\cal S}^n.$$ So, $$\\begin{aligned}\nS_A^2(X) & = & S_A(X-AXA^T) \\\\\n& = & X-AXA^T-AXA^T+A^2X(A^T)^2\\\\\n&=& 2(X-AXA^T)\\\\\n& =& 2S_A(X).\n\\end{aligned}$$ So, $S_A^2=S_A$ and again, by Proposition it follows\nthat $S_A^{\\#}$ exists. In fact, in this case, $S_A^{\\#}=2S_A$. This\nproves (a).  \n(b) Let $A^2=I.$ Then, a calculation done as earlier, leads to the\nformula $$L_{A}^2(X)=2(X+AXA^T).$$ This, in turn, implies that\n$$L_{A}^3=4L_A.$$ The conclusion on the existence of the group inverse\nof $L_A$, now follows.  \nFor the Stein operator, just as in the case $A^2=-I,$ it follows that\n$$S_A^2=2S_A$$ and so, $S_A^{\\#}$ exists.  \n(c) Since $A^T=-A,$ we have $L_A(X)=AX-XA.$ Now, for\n$X,Y \\in {\\cal S}^n,$ $$\\begin{aligned}\n \\langle X, L_A(Y)\\rangle & = & \\rm tr (X(AY-YA))\\\\\n & = & \\rm tr(XAY) - \\rm tr (XYA)\\\\\n & = & -(\\rm tr (XYA) - \\rm tr (XAY))\\\\\n & = & -(\\rm tr (AXY) - \\rm tr (XAY))\\\\\n & = & -(\\rm tr((AX-XA)Y))\\\\\n & = & -\\langle L_A(X), Y\\rangle.\n\\end{aligned}$$ Hence $L_A^*=-L_{A},$ which means that\n$R(L_A)=R(L_A^{\\ast})$. Thus $R(L_A)$ and $N(L_A)$ are complementary\nsubspaces. Thus, $L_A^\\#$ exists.\n\nNext, let us consider the Stein operator. Since $A^T=-A,$ we have\n$S_A(X)=X+AXA.$ Now, $$\\begin{aligned}\n\\langle X, S_A(Y)\\rangle & = & \\rm tr(X(Y+AYA))\\\\\n& = & \\rm tr(XY+XAYA) \\\\\n&=& \\rm tr(XY) + \\rm tr(AXAY) \\\\\n&=& \\rm tr((X+AXA)Y)\\\\\n&=& \\langle S_A(X), Y\\rangle.\n\\end{aligned}$$ Hence $S_A^*=S_{A}.$ Once again, the subspaces $R(S_A)$\nand $N(S_A)$ are complementary and so $S_A^\\#$ exists.  \n(d) When $A=A^T,$ as was remarked earlier, we have $L_A^*=L_{A^T}=L_A$\nand $S_A^*=S_{A^T}=S_A.$ Thus, $L_A^\\#$ and $S_A^\\#$ exist.\n\nLet $A \\in \\mathbf{M}_n\\,(\\mathbb{R})$. We say that $A$ is *nonnegative\nstable*, if all the eigenvalues of $A$ have a nonnegative real part. $A$\nwill be referred to as *Shur semi-stable*, if all its eigenvalues lie in\nthe closed unit disc of the complex plane.\n\n**Remark 24**.  * Some remarks are in order. Let matrix $A$ be nilpotent\n(so that $A$ is nonnegative stable). Then it is easy to show that the\noperator $L_A$ is also nilpotent. As noted earlier, it follows that the\ngroup inverse of $L_A$ does not exist. So, $L_A$ is not trivially range\nmonotone. Nevertheless, we identify a class of matrices $A$, which are\nnonnegative stable, for which the associated Lyapunov operator is\ntrivially range monotone. Example shows that not all Schur semi-stable\nmatrices are trivially range monotone. However, we identify a class of\nSchur semi-stable matrices $A$, for which the Stein operator is\ntrivially range monotone. Both these are presented in the next result.*\n\nIn what follows, we consider the question of trivial range monotonicity\nof the Lyapunov and the Stein operators corresponding to matrices that\nsatisfy one of the first three conditions of Theorem .\n\n**Theorem 25**.  * For the Lyapunov operator $L_A$ and Stein operator\n$S_A$ the following are true:  \n$(a)$ Let $A^2=-I.$ Then $L_A$ and $S_A$ are trivially range monotone.  \n$(b)$ Let $A^2=I.$ Then $S_A$ is trivially range monotone.  \n$(c)$ Let $A^T=-A.$ then $L_A$ is trivially range monotone.*\n\n**Proof.** First, we observe that both the Lyapunov and the Stein\noperators are group invertible, under any of the three conditions on the\nmatrix $A$, as above (in view of Theorem ). Thus in all these cases, the\nrange space and the null space (of either the Lyapunov operator or the\nStein operator) are complementary subspaces of ${\\cal S}^n.$ We shall\nmake repeated use of this fact, in our proofs.\n\n$(a)$ Let $L_A(X)\\succeq 0$ with $X\\in R(L_A).$ Then\n$0 \\preceq L_A^3(X) =-4L_A(X).$ This means that $L_A(X)=0.$ Thus,\n$X\\in R(L_A) \\cap N(L_A)=\\{0\\}$, proving that $L_A$ is trivially range\nmonotone.\n\nNext, we show the trivial range monotonicity of the Stein operator. Let\n$X\\in R(S_A)$ so that there exists $Y \\in \\mathcal{S}^n$ with\n$S_{A}(Y)=X$. Also, let $S_A(X)\\succeq 0.$ Then, from the calculation as\nearlier, we have $$\\begin{aligned}\n0 \\preceq S_A(X)=S_A^2(Y)=2S_A(Y)=2(Y-AYA^T).\n\\end{aligned}$$ Set $Z:=Y-AYA^T.$ Then, $Z \\succeq 0$. This means that\n$$0 \\preceq AZA^T= AYA^T-A^2Y(A^T)^2=AYA^T-Y=-Z.$$ Thus, $Z=0$, i.e.\n$S_A(X)=0,$ which in turn implies that that $X=0,$ as\n$X\\in R(S_A) \\cap N(S_A).$\n\n$(b)$ The proof of the trivial range monotonicity of the Stein operator,\nis entirely similar to the second part of the above result and is\nskipped.\n\n$(c)$ Let $Z:=L_A(X)=AX-XA \\succeq 0,$ with $X\\in R(L_A).$ Then all the\neigenvalues of $Z$ are non-negative. Further,\n$\\rm tr (Z)=\\rm tr(AX-XA)=0$ and so, all the eigenvalues of $Z$ are\nzero. Since $Z$ is diagonalizable, $0=Z=L_A(X).$ Therefore $X=0,$ as\n$X\\in R(L_A) \\cap N(L_A).$ This shows the trivial range monotonicity\nproperty of $L_A$.  \n\nLet matrix $A$ be such that either $A^2=I$ or $A^2=-I.$ Then $A^{-1}$\nexists and shares the same property as that of $A$. If $A$ is\nskew-symmetric, then $A^{\\#}$ exists (since $R(A)$ and $N(A)$ are\ncomplementary) and $A^{\\#}$ is also skew-symmetric. Thus we have the\nfollowing immediate consequence of Theorem .\n\n**Corollary 26**. *Let $A^2=-I.$ Then $L_{A^{-1}}$ and $S_{A^{-1}}$ are\ntrivially range monotone. If $A^2=I,$ then $S_{A^{-1}}$ is trivially\nrange monotone. Let $A^T=-A.$ Then $L_{A^{\\#}}$ is trivially range\nmonotone.*\n\nThe first example below illustrates (a) of Theorem .\n\n**Example 27**.  * Let $A=\n\\begin{pmatrix}\n~~0 & 1 \\\\\n-1 & 0\n\\end{pmatrix},$ so that $A^2=-I.$ For any $X=\n\\begin{pmatrix}\na & b \\\\\nb & c\n\\end{pmatrix} \\in {\\cal S}^2,$ we have $$S_A(X)=\n\\begin{pmatrix}\na-c & 2b \\\\\n2b & c-a\n\\end{pmatrix}.$$ If $S_A(X) \\succeq 0,$ then $a=c$ and so $b=0$. Thus,\n$X=\\begin{pmatrix}\na & 0 \\\\\n0 & a\n\\end{pmatrix}.$ If we impose the condition that $X \\in R(S_A),$ then\n$a= -a$, proving that $X=0.$ Thus, $S_A$ is trivially range monotone.*\n\nThe next example illustrates item (b) of Theorem .\n\n**Example 28**.  * Let $A=\n\\begin{pmatrix}\n0 & 1 \\\\\n1 & 0\n\\end{pmatrix},$ so that $A^2=I.$ For any $X=\n\\begin{pmatrix}\na & b \\\\\nb & c\n\\end{pmatrix} \\in {\\cal S}^2,$ we have $$S_A(X)=\n\\begin{pmatrix}\na-c & 0 \\\\\n0 & c-a\n\\end{pmatrix}.$$ If $S_A(X) \\succeq 0,$ then $a=c$ and so\n$X=\\begin{pmatrix}\na & b \\\\\nb & a\n\\end{pmatrix}.$ Further, if $X \\in R(S_A),$ then $a= -a$ and $b=0$,\nproving that $X=0.$ Thus, $S_A$ is trivially range monotone.*\n\nItems (a) (for the Lyapunov operator) and (c) of Theorem are\nillustrated, next.\n\n**Example 29**.  * Let $A=\n\\begin{pmatrix}\n~~0 & 1 \\\\\n-1 & 0\n\\end{pmatrix}.$ Then $A$ is skew-symmetric and $A^2=-I.$ For any $X=\n\\begin{pmatrix}\na & b \\\\\nb & c\n\\end{pmatrix} \\in {\\cal S}^2,$ we have $$L_A(X)=\n\\begin{pmatrix}\n2b & c-a \\\\\nc-a & -2b\n\\end{pmatrix}.$$ The requirement that $L_A(X) \\succeq 0$ yields $b=0$\nand so $a=c$. Thus $X=a I$. The condition that $X \\in R(L_A)$ then\nimplies that $a=0$, so that $X=0,$ proving the trivial range\nmonotonicity of $L_A.$*\n\n# Counterexamples\n\nThe Lyapunov operator, is not trivially range monotone, in general, when\n$A$ is an involutory matrix.\n\n**Example 30**.  * Let $A=\n\\begin{pmatrix}\n0 & 1 \\\\\n1 & 0\n\\end{pmatrix},$ so that $A^2=I$ and for any $X=\n\\begin{pmatrix}\na & b \\\\\nb & c\n\\end{pmatrix} \\in {\\cal S}^2,$ we have $$L_A(X)=\n\\begin{pmatrix}\n2b & a+c \\\\\na+c & 2b\n\\end{pmatrix}.$$ Note that, if $X=\\begin{pmatrix}\n1 & 0 \\\\\n0 & -1\n\\end{pmatrix},$ then $L_A(X)=0,$ proving that $L_A$ is singular. Next,\ntake $Y=A$ and $U=\n\\begin{pmatrix}\n1 & 0 \\\\\n0 & 0\n\\end{pmatrix}.$ Then $L_A(U)=Y$ so that $Y \\in R(L_A).$ We have\n$L_A(Y)=2I \\succeq 0$, but $Y \\nsucceq 0.$ Thus, the Lyapunov operator\nis not even range monotone, if $A$ is involutory.*\n\nThe conclusion above holds for matrices of higher order, too. For\ninstance, if $A$ is involutory and if one sets $\\tilde{A}\n=\\begin{pmatrix}\nA & 0\\\\\n0 & \\pm 1\n\\end{pmatrix},$ then $L_{\\tilde{A}}$ not range monotone. We omit the\ndetails.\n\n**Example 31**.  * Consider the case, when $A^T=-A.$ For\n$n=1, S_A(X)=X.$ Here, $S_A$ is even monotone. For $n=2,$ (after\nnormalizing) $A$ must be of the following form $$\\begin{pmatrix}\n~0 & \\pm 1 \\\\\n\\mp 1 & ~0\n\\end{pmatrix}.$$ Then, for any symmetric matrix $X=\\begin{pmatrix}\na & b \\\\\nb & d\n\\end{pmatrix},$ we have $$S_A(X)=\n\\begin{pmatrix}\na-d & 2b \\\\\n2b & d-a\n\\end{pmatrix}.$$ $S_A$ is singular since $S_A(I)=0.$ Next, if\n$S_A(X) \\succeq 0,$, then $a=d$ and so $b=0$. Further, if\n$X \\in R(S_A),$ then we have $a=-d$, and so $a=d=0,$ proving that $X=0$.\nThis proves the trivial range monotonicity of $S_A,$ for $n=2.$*\n\nIn the next two examples, we show that the Stein operator is not\ntrivially range monotone for $n=3,4$, for a skew-symmetric matrix $A$.\n\n**Example 32**.  * Let $A=\\frac{1}{\\sqrt 2}\n\\begin{pmatrix}\n~0 & ~1 & ~0 \\\\\n-1 & ~0 & ~1 \\\\\n~0 & -1 & ~0 \n\\end{pmatrix},$ so that for any symmetric matrix $X=\n\\begin{pmatrix}\na & b & c\\\\\nb & d & e \\\\\nc & e & f \n\\end{pmatrix},$ we have $$S_A(X)= \\frac{1}{2}\\begin{pmatrix}\n2a-d & \\ 3b-e & 2c+d \\\\\n3b-e & 2d+2c-a-f & 3e-2b \\\\\n2c+d & 3e-2b & 2f-d\n\\end{pmatrix}.$$ $S_A$ is singular, since $S_A(X)=0,$ for $X= \n\\begin{pmatrix}\n-1 & ~0 & ~1\\\\\n~0 & -2 & ~0\\\\\n~1 & ~0 & -1\n\\end{pmatrix}.$ Next, if $U=\n\\begin{pmatrix}\n2 & 0 & 1 \\\\\n0 & 1 & 0 \\\\\n1 & 0 & 2\n\\end{pmatrix}$ and $Y=\\frac{1}{2}\n\\begin{pmatrix}\n3 & 0 & 3 \\\\\n0 & 0 & 0 \\\\\n3 & 0 & 3\n\\end{pmatrix}$ then $S_A(U)=Y$ so that $Y \\in \\ R(S_A).$ We have\n$S_A(Y)=\\frac{1}{2}\n\\begin{pmatrix}\n3 & 0 & 3 \\\\\n0 & 0 & 0 \\\\\n3 & 0 & 3\n\\end{pmatrix}\\succeq 0$ and $Y\\succeq 0.$ Since $Y \\neq 0$, we conclude\nthat the Stein operator is not trivially range monotone.*\n\n**Example 33**.  * Let $A=\n\\begin{pmatrix}\n~0 & 1 & ~0 & 0 \\\\\n-1 & 0 & ~0 & 0 \\\\\n~0 & 0 & ~0 & 2 \\\\\n~0 & 0 & -2 & 0\n\\end{pmatrix},$ so that for any symmetric matrix $X=\n\\begin{pmatrix}\na & b & c & d \\\\\nb & e & f & g \\\\\nc & f & h & i \\\\\nd & g & i & j\n\\end{pmatrix},$ we have $$S_A(X)=\n\\begin{pmatrix}\na-e & 2b & c-2g & d+2f \\\\\n2b & e-a & f+2d & g-2c \\\\\nc-2g & f+2d & h-4j & 5i \\\\\nd+2f & g-2c & 5i & j-4h\n\\end{pmatrix}.$$ It is clear that, if $X=\n\\begin{pmatrix}\n1 & 0 & 0 & 0 \\\\\n0 & 1 & 0 & 0 \\\\\n0 & 0 & 0 & 0 \\\\\n0 & 0 & 0 & 0\n\\end{pmatrix},$ then $S_A(X)=0,$ proving that $S_A$ is singular. Next,\nif $Y=\n\\begin{pmatrix}\n0 & 0 & ~0 & ~0 \\\\\n0 & 0 & ~0 & ~0 \\\\\n0 & 0 & -3 & ~0 \\\\\n0 & 0 & ~0 & -3\n\\end{pmatrix}$ and $U=\n\\begin{pmatrix}\n0 & 0 & 0 & 0 \\\\\n0 & 0 & 0 & 0 \\\\\n0 & 0 & 1 & 0 \\\\\n0 & 0 & 0 & 1\n\\end{pmatrix},$ then $S_A(U)=Y$ so that $Y \\in R(S_A).$ We have $S_A(Y)=\n\\begin{pmatrix}\n0 & 0 & ~0 & ~0 \\\\\n0 & 0 & ~0 & ~0 \\\\\n0 & 0 & ~9 & ~0 \\\\\n0 & 0 & ~0 & ~9\n\\end{pmatrix} \\succeq 0.$ However, $Y \\nsucceq 0,$ proving that the\nStein operator is not even range monotone.*\n\n**Example 34**.  * For $n = 5,$ consider the skew symmetric block\ndiagonal matrix $B,$ whose leading principal sub-block is $A,$ as in\nexample and whose trailing principal sub-block matrix is\n$\\begin{pmatrix}\n 0 & -1 \\\\\n 1 & ~0\n\\end{pmatrix}.$ Then, by the argument given as earlier, it may be shown\nthat $S_B$ is not trivially range monotone. This inductive argument\nallows us to conclude that, *in general*, $S_A$ is not trivially range\nmonotone when $A$ is a skew symmetric matrix, of any order $n \\geq 3$.*\n\nThis finishes the discussion for skew-symmetric case.\n\nNext, let $A$ be symmetric. Example , shows that $L_A$ is not trivially\nrange monotone. The following example shows that $S_A$ is not trivially\nrange monotone.\n\n**Example 35**.  * Consider the symmetric matrix $A=\n\\begin{pmatrix}\n1 & 0 \\\\\n0 & 2\n\\end{pmatrix},$ so that for any $X=\n\\begin{pmatrix}\na & b \\\\\nb & c\n\\end{pmatrix}\\in \\mathcal{S}^2,$ $S_A$ is given by $$S_A(X)=\n\\begin{pmatrix}\n~0 & -b \\\\\n-b & -3c\n\\end{pmatrix}.$$ $S_A$ is not invertible, as $S_A(X)=0,$ for $X=\n\\begin{pmatrix}\n1 & 0\\\\\n0 & 0\n\\end{pmatrix}.$ Note that, by item $(d)$ of Proposition , $S_A^{\\#}$\nexists. Next, if $Y=\n \\begin{pmatrix}\n 0 & ~0 \\\\\n 0 & -1\n \\end{pmatrix}$ then $S_A(U)=Y$, where $U=\n\\begin{pmatrix}\n0 & 0 \\\\\n0 & \\frac{1}{3}\n\\end{pmatrix},$ showing that $Y \\in R(S_A).$ Also, $Y \\nsucceq 0,$\nwhereas, $S_A(Y)=\n\\begin{pmatrix}\n0 & 0 \\\\\n0 & 3\n\\end{pmatrix}\\succeq 0$.*\n\nWe summarize the findings in the following table. Note that, while \"Yes\"\nstands for an affirmative answer, \"No\" means that counterexamples exist\nto show that the answers are negative, in general.\n\n| **Matrix classes** | **Trivial Range Monotonicity** |  |\n|:--:|:--:|:--:|\n| 2-3 | $L_A$ | $S_A$ |\n| $A^2=-I$ | Yes \\[Theorem (a)\\] | Yes \\[Theorem (a)\\] |\n| $A^2=I$ | No \\[Example \\] | Yes \\[Theorem (b)\\] |\n| $A^T=-A$ | Yes \\[Theorem (c)\\] | Yes ($n=2$) \\[Example \\] |\n|  |  | No ($n\\geq 3$) \\[Examples , , \\] |\n| $A^T=A$ | No \\[Example \\] | No \\[Example \\] |\n\n# Concluding Remarks\n\nWe have studied four matrix classes in the context of the notion of\ntrivial range monotonicity of the associated Lyapunov and the Stein\noperators. The motivation, as was mentioned in the introduction, is to\nobtain a generalization of the trivial range monotonicity property of\nsingular irredudcible $M$-matrices. It would be interesting to bring the\nmatrix classes for which we could obtain affirmative results, under a\nmore general framework. Similarly, to unify the matrix classes for which\nnegative results for operator analogues, have been proved. It would be\nanother interesting question to study the notion of irreducibility of\nthe Lyapunov operator and the Stein operator. It is pertinent to point\nto the fact that there are different (possibly nonequivalent) ways of\ndefining irreducibility of a \"positive\" operator. So one could propose\nsuch a notion for the Stein operator (which is the difference: $I$ minus\na positive operator $X \\rightarrow AXA^T$), simply by introducing the\nassumption of irreducibility for the second term, which is positive\n(meaning, that the matrix $AXA^T$ is symmetric and positive\nsemidefinite, whenever so is $X$). However, it is not clear how one\ncould propose irreducibility for the Lyapunov operator and the likes of\nit. While we will pursue these questions in future, we note that,\nmotivated by the considerations of this article, the notion of\nirreducibility has been investigated for $Z$-operators over Euclidean\nJordan Algebras .\n\n# Acknowledgements\n\nSamir Mondal acknowledges funding received from the Prime Minister’s\nResearch Fellowship (PMRF), Ministry of Education, Government of India,\nfor carrying out this work. The first and the third authors thank\nASEM-DUO for financial support enabling the former’s visit to India and\nthe latter, to Spain. The first author was also partially supported by\nthe Spanish I+D+i program under project PID2021-122501NB-I00. The\nauthors thank Prof. M.S. Gowda for his suggestions and comments that\nhave helped in a clearer presentation of the material. He suggested the\nnomenclature \"positive stable $Z$-operators\".\n"
    },
    {
        "text": "abstract: In this paper, we have obtained a generalization of\n  Grothendieck’s theorem for the space of continuous mappings\n  $C_{\\lambda,\\mu}(X,Y)$ where $Y$ is a complete uniform space with the\n  uniformity $\\mu$ endowed with the topology of uniform convergence on\n  the family $\\lambda$ of subsets of $X$. A new topological game is\n  defined - the Asanov-Velichko game, which makes it possible to single\n  out a class of topological spaces of the Grothendieck type.\n  .\n  The developed technique is used to generalize the Grothendieck theorem\n  for the space of continuous mappings endowed with the set-open\n  topology.\naddress: Krasovskii Institute of Mathematics and Mechanics,\nYekaterinburg, Russia; Ural Federal University, Yekaterinburg, Russia\nauthor: Mikhail Al’perin; Alexander V. Osipov\nbibliography: \\<your-bib-database\\>.bib\ntitle: Generalization of the Grothendieck Theorem\n\n# Introduction\n\nIn 1952 Grothendieck proved the following result.\n\n**Theorem 1**. *(Grothendieck) Let $X$ be a compact space, $Y$ be a\nmetrizable space. Then each relatively countably compact subspace of\n$C_p(X,Y)$ is relatively compact.*\n\nThis theorem has played an important role in topology and functional\nanalysis. One of the important applications of Grothendieck’s theorem is\nin finding topological properties of semitopological groups which imply\n(paratopological) topological groups or more generally when separate\ncontinuity implies joint continuity .\n\nGrothendieck’s theorem has been generalized many times . The following\nsuccessful generalizaion of this theorem is due to M.O. Asanov and N.V.\nVelichko .\n\n**Theorem 2**. *If a space $X$ is countably compact, then each bounded\nsubset of $C_p(X)$ is relatively compact.*\n\nLet us recall two theorems proved by A.V. Arhangel’skii .\n\n**Theorem 3**.  * If a Tychonoff space $X$ contains an everywhere dense\n$\\sigma$-countably pracompact subspace $Y$, then every pseudocompact\nsubspace $P$ of $C_p(X)$ is an Eberlein compactum.*\n\n**Theorem 4**.  * If a Tychonoff space $X$ contains an everywhere dense\n$\\sigma$-pseudocompact subspace $Y$, then every countably compact\nsubspace $P$ of $C_p(X)$ is an Eberlein compactum.*\n\nIn this paper, we investigated a generalization of Grothendieck’s\ntheorem for the space of continuous mappings $C_{\\lambda,\\mu}(X,Y)$\nwhere $Y$ is a complete uniform space with uniformity $\\mu$ endowed with\nthe topology of uniform convergence on the family $\\lambda$ of subsets\nof $X$. Also the developed technique is used to generalize the\nGrothendieck theorem for the space of continuous mappings\n$C_{\\lambda}(X, \\mathbb{R})$ endowed with the set-open ($\\lambda$-open)\ntopology.\n\n# Notation and terminology\n\nThe set of positive integers is denoted by $\\mathbb{N}$ and\n$\\omega=\\mathbb{N}\\cup \\{0\\}$. Let $\\mathbb{R}$ be the real line, we put\n$\\mathbb{I}=[0,1]\\subset \\mathbb{R}$, and let $\\mathbb{Q}$ be the\nrational numbers. We denote by $\\overline{A}$ (or $Cl_X A$) the closure\nof $A$ (in $X$).\n\nLet $X$ be a topological space, $\\lambda\\subseteq 2^X$, $(Y,\\mu)$ be a\nuniform space. A topology on $C(X,Y)$ generated by the uniformity\n$$\\nu=\\{\\langle A,M\\rangle \\subseteq C(X,Y)\\times\nC(X,Y):A\\in\\lambda,M\\in\\mu\\}$$ where\n$$\\langle A,M\\rangle = \\{\\langle f,g\\rangle \\in C(X,Y)\\times\nC(X,Y):\\forall x\\in A~\\langle f(x),g(x)\\rangle\\in M\\}$$ is called\n*topology of uniform convergence on elements of $\\lambda$* and denote by\n$C_{\\lambda,\\mu}(X,Y)$.\n\nThe well-known fact that if $\\lambda$ is a family of all compact subsets\nof $X$ or all finite subsets of $X$ then the topology on $C(X,Y)$\ninduced by the uniformity $\\nu$ of uniform convergence on elements of\n$\\lambda$ depends only on the topology induced on $Y$ by the uniformity\n$\\mu$ (see ). In these cases, we will use the notation $C_c(X,Y)$ and\n$C_p(X,Y)$, respectively. If $Y=\\mathbb{R}$ then $C_c(X)$ and $C_p(X)$,\nrespectively.\n\nIn case, if $(Y,\\rho)$ is a metric space and the uniformity $\\mu$ is\ninduced by the metric $\\rho$, then for $C_{\\lambda,\\mu}(X,Y)$, we will\nuse the notation $C_{\\lambda,\\rho}(X,Y)$ and $C_{\\lambda,\\rho}(X)$ for\nthe case $Y=\\mathbb{R}$.\n\nIf $X\\in\\lambda$, we write $C_\\mu(X,Y)$ in place of\n$C_{\\lambda,\\mu}(X,Y)$ and $C_{\\mu}(X)$ in place of\n$C_{\\mu}(X,\\mathbb{R})$.\n\n**Remark 5**. For the topology of uniform convergence on elements of\n$\\lambda$, we assume that the following natural conditions holds:\n\n\\(1\\) if $A\\in\\lambda$ and $A'\\subseteq A$ then $A'\\in\\lambda$.\n\n\\(2\\) if $A_1, A_2\\in\\lambda$ then $A_1\\bigcup A_2\\in\\lambda$.\n\n\\(3\\) if $A\\in\\lambda$ then $\\overline{A}\\in\\lambda$.\n\nNote that $C_{\\lambda,\\mu}(X,Y)$ is Hausdorff if and only if the set\n$\\bigcup \\lambda$ is dense in $X$, i.e., (see ).\n\nThen we have an additional condition on $\\lambda$.\n\n\\(4\\) $\\lambda$ is a cover of $X$.\n\nThe *set-open topology* on a family $\\lambda$ of non-empty subsets of\nthe set $X$ is a generalization of the compact-open topology and of the\ntopology of pointwise convergence. This topology, first introduced by\nArens and Dugunji in , is one of the important topologies on $C(X,Y)$.\n\nIf $A\\subseteq X$ and $V\\subseteq Y$, then $[A,V]$ is defined by as\n$[A,V]=\\{f\\in C(X,Y): f(A)\\subseteq V\\}$.\n\nLet $X$ and $Y$ be topological spaces, $\\lambda\\subseteq 2^X$ . A\ntopology on $C(X,Y)$ is called a *$\\lambda$-open topology* (set-open\ntopology) provided the family $\\{[A,V]: A\\in \\lambda$ and $V$ is open in\n$Y \\}$ form a subbase for the topology. The function space $C(X,Y)$,\nprovided with this topology, is denoted by $C_{\\lambda}(X,Y)$.\n\nRecall that a subset $A$ of a topological space $X$ is called\n\n$\\bullet$ *relatively compact* if $A$ has compact closure in $X$.\n\n$\\bullet$ *bounded* if every continuous function on $X$ is bounded on\n$A$.\n\n$\\bullet$ *relatively countably compact* if each sequence of $A$ has\nlimit point in $X$.\n\n$\\bullet$ *pseudocompact* if any continuous real-valued the function on\n$A$ is bounded.\n\nA topological space $X$ is called *countably pracompact* if there is a\nsubspace $Y\\subseteq X$ which is everywhere dense in $X$ and countably\ncompact in $X$ in the following sense: every infinite set $A\\subseteq Y$\nhas a limit point in $X$.\n\nA space is called $\\sigma$-compact ($\\sigma$-countably compact, etc.) if\nit is the union of a countable set of compact (respectively, countably\ncompact, etc.) subspaces.\n\n*A $G_\\delta$-neighborhood of a point $x$* in a space $X$ is called the\nset $V$ containing $x$ which is intersection of a countable number of\nopen sets.\n\nA notation and terminology we follow almost without exception\nEngelking’s book .\n\n# Asanov-Velichko game and compactness in $C_p(X,Y)$\n\nLet $X$ be a topological space, $\\lambda$ be a family of subsets of $X$,\n$A$ be a non-closed subset of $X$ and $x\\in\n\\overline{A}\\setminus A$.\n\nThe following game on the space $X$ will be called the *Asanov–Velichko\ngame generated by the family $\\lambda$*.\n\nPlayer ONE chooses at the $n$-th step of the game a neighborhood $V_n$\nof the point $x$. The TWO player chooses at the $n$-th step of the game\na subset $S_n$ of the set $A$ which lies in the closure of some $T_n\\in\n\\lambda$. The game is played with a countable number of steps. We will\nsay that the TWO player won if the set $\\overline{\\bigcup\\{S_n:n\\in\n\\mathbb{N}\\}}\\cap (\\bigcap\\{V_n: n\\in \\mathbb{N}\\})\\ne \\emptyset$ and\nwon by ONE otherwise. This game will be denote by $AV_{\\lambda}(X)$.\n\nThe following definition generalizes the notions of weakly $p$- and\nweakly $q$-spaces introduced in .\n\n**Definition 6**. The space $X$ is called *Asanov-Velichko generated by\nthe family $\\lambda$ ($AV_{\\lambda}$-space)* if for any non-closed\nsubset $A$ of $X$ there is a point $x\\in \\overline{A}\\setminus A$ for\nwhich the TWO player has a winning strategy in the game\n$AV_{\\lambda}(X)$.\n\nIn what follows, we will need two definitions.\n\n**Definition 7**. Let $X$ be a topological space. The family $\\lambda$\nof subsets $X$ will be called *countably invariant* if $\\lambda$\ncontains all singletons of $X$ and the fact that $A_n\\in \\lambda$ for\nevery $n\\in\n\\mathbb{N}$ imply $\\bigcup \\{A_n:n\\in \\mathbb{N}\\}\\in\n\\lambda$.\n\n**Definition 8**. () Property ${\\cal P}$ of subsets topological space\n$X$ is called a *continuously invariant property* if the facts that $A$\nis a subset of $X$ with the property ${\\cal P}$ and $f:X\\rightarrow Y$\nis a continuous map imply that $f(A)$ has the property ${\\cal P}$ in\n$f(X)$.\n\nThe following proposition makes it easy to prove generalizations of the\nGrothendieck theorem.\n\n**Proposition 9**.  * Let $X$ be a $AV_{\\lambda}$-space where $\\lambda$\nis a countably invariant family of its subspaces. Let ${\\cal P}$ be some\ncontinuously invariant property subsets of $X$, and the following\ncondition is satisfied:*\n\n*$(\\gamma )$ for every $Z\\in \\lambda$ and every $B\\subseteq \\pi\n_Z(C_p(X))$, having the property ${\\cal P}$, $B$ is compact, then for\nevery $F\\subseteq C_p(X)$, having the property ${\\cal\nP}$, $F$ is relatively compact.*\n\n*Proof.* Note that to prove the proposition it is enough to prove that\n$\\overline{F}^{C_p(X)}=\\overline{F}^{R^{X}}$. Indeed, for of any\n$x\\in X$ the set $\\pi_x(F)$ is compact and, therefore,\n$\\overline{F}^{R^{X}}\\subseteq \\prod\\{\\pi_x(F):x\\in X\\}$ is compact.\n\nSuppose $\\overline{F}^{C_p(X)}\\neq\n\\overline{F}^{\\mathbb{R}^{X}}$, that is, there is a discontinuous\nfunction $f\\in \\overline{F}^{\\mathbb{R}^{X}}$. Then there is a closed\nsubset $M$ of $\\mathbb{R}$ such that $C=f^{-1}(M)$ is not closed in $X$.\nSince $X$ is a $AV_{\\lambda}$-space, there is\n$x_0\\in \\overline{C}\\setminus C$, for which the TWO player has a winning\nstrategy in the game $AV_{\\lambda}(X)$. Let $\\epsilon =\\rho\n(f(x_0),M)$, then $\\epsilon >0.$\n\nThe facts that $\\lambda$ is countably invariant and the condition\n$(\\gamma)$ imply that $\\pi_{x_0}(F)$ is compact. Therefore, there is\n$f_1\\in F$ such that $\\pi_{x_0}(f)=\\pi_{x_0}(f_1)$, i.e.,\n$f_1(x_0)=f(x_0)$. Let $V_1=f^{-1}_1(\\{r\\in \\mathbb{R}:\\rho\n(f(x_0),r)<1\\})$. Let $S_1\\subseteq C$ and $T_1\\in \\lambda$ sets\nselected by the TWO player according to the winning strategy in the game\n$AV_{\\lambda}(X)$.\n\nLet now we have already chosen the functions $f_1,\\ldots ,f_n$, the\nneighborhoods $V_1,\\ldots ,V_n$ of the point $x_0$, the subsets\n$S_1,\\ldots\n,S_n$ of $C$, and $T_1,\\ldots ,T_n$, belonging to $\\lambda$, such that\n$S_i\\subseteq \\overline{T_i}$ for every $i=1,\\ldots,n$. By condition\n$(\\gamma)$, the projection of the set $F$ onto\n$L=\\bigcup\\{T_i:i=1,\\ldots ,n\\}\\cup \\{x_0\\}$ is compact, hence, there is\n$f_{n+1}\\in F$ such that $\\pi_L(f)=\\pi_L(f_{n+1})$. Let\n$V_{n+1}=f^{-1}_{n+1}(\\{r\\in\n\\mathbb{R}:\\rho(f(x_0),r)<\\frac{1}{n+1}\\})\\cap V_n$. Let\n$S_{n+1}\\subseteq C$ and $T_{n+1}\\in \\lambda$ sets chosen TWO player.\n\nLet $\\{f_i:i\\in \\mathbb{N}\\}$, $\\{V_i:i\\in\n\\mathbb{N}\\}$, $\\{S_i:i\\in \\mathbb{N}\\}$ and $\\{T_i:i\\in\n\\mathbb{N}\\}$ be constructed. Since $X$ is a $AV_{\\lambda}$-space, there\nis $x_{\\infty}\\in \\overline{\\bigcup\n\\{S_n:n\\in \\mathbb{N}\\}}\\cap (\\bigcap \\{V_n:n\\in \\mathbb{N}\\})$.\n\nConsider $Y=\\bigcup \\{T_i:i\\in \\mathbb{N}\\}\\cup\n\\{x_{\\infty}\\}\\cup\\{x_0\\}$. By condition $(\\gamma)$, $\\pi_Y(F)$ is\ncompact, that is, there exists $f_{\\infty}\\in F$ such that\n$\\pi_Y(f)=\\pi _Y(f_{\\infty})$. Then the following are performed\nconditions:\n\n\\(1\\) $f_{\\infty}(x_{\\infty})=f(x_0)$;\n\n\\(2\\) $f_{\\infty}(x)=f(x)$ for every $x\\in \\bigcup \\{T_i:i\\in\n\\mathbb{N}\\}$ because $\\pi_Y(f)=\\pi_Y(f_{\\infty})$ and\n$\\pi_Y(f_{\\infty})$ is a limit function for the set\n$\\pi_Y(\\{f_i:i\\in \\mathbb{N}\\})$.\n\nLet $T=(\\bigcup\\{T_i:i\\in \\mathbb{N}\\})\\setminus \\{x\\in X:\n\\rho(f(x),f(x_{\\infty}))<\\frac{3}{4}\\epsilon\\}$.\n\nBy conditions (1) and (2), $\\rho (f_{\\infty}(x),f_{\\infty}(x))\\geq\n\\frac{3}{4}\\epsilon$ for every $x\\in T$. If we prove that\n$x_{\\infty}\\in \\overline{T}$, then we obtain a contradiction with the\nfact that $f_{\\infty}$ is continuity.\n\nIndeed, let $O(x_{\\infty})$ be an arbitrary neighborhood of point\n$x_{\\infty}$, then there are $x'$ and $n\\in \\mathbb{N}$ such that\n$x'\\in S_n\\cap O(x_{\\infty})$. Let $H=T_n\\cup\n\\{x'\\}$. By condition $\\pi_H(F)$ is compact, then there exists $f'\\in F$\nsuch that $\\pi_H(f)=\\pi_H(f')$. Since $f'$ is continuous, there is a\npoint $x^{''}\\in T_n$ for which\n$\\rho(f'(x^{''}),f'(x'))<\\frac{1}{4}\\epsilon$, but then\n\n$\\rho (f(x^{''}),f(x_0))\\geq\\rho(f(x'),f(x_0))-\\rho\n(f'(x^{''}),f'(x'))\\geq\\epsilon -\\frac{1}{4}\\epsilon\n=\\frac{3}{4}\\epsilon$, i.e., $x^{''}\\in T\\cap O(x_{\\infty})$. ◻\n\nRecall that Eberlein compacta are compact subsets of Banach spaces in\nthe weak topology.\n\nThe following theorem is an obvious consequence of Theorem and\nProposition .\n\n**Theorem 10**.  * Let $X$ be a Tychonoff $AV_{\\lambda_c}$-space where\n$\\lambda_c$ is family of all its subspaces that have dense\n$\\sigma$-countably pracompact subspace. Then every pseudocompact\nsubspace in $C_p(X)$ is relatively compact.*\n\nIt also follows from Theorem and Proposition the following result.\n\n**Theorem 11**.  * Let $X$ be a Tychonoff $AV_{\\lambda_p}$-space where\n$\\lambda_p$ is family of all its subspaces with dense\n$\\sigma$-pseudocompact subspace. Then each countably pracompact subspace\nin $C_p(X)$ is relatively compact.*\n\nThere is an interesting parallel between these two theorems and the\nresults (Theorems and ) A.V. Archangel’skii in ( and ).\n\n**Definition 12**. Let $\\lambda$ be a family of subspaces of a\ntopological space $X$. The space $X$ is *functionally generated by the\nfamily $\\lambda$* if the following condition holds: for every\ndiscontinuous function $f:X\\rightarrow \\mathbb{R}$ there is an\n$A\\in \\lambda$ such that the function $\\pi_A(f)$ cannot be extended to a\nrealvalued continuous function on all of $X$.\n\n**Proposition 13**. *(Proposition 4.10 in ) Let a Tychonoff space $X$ be\nfunctionally generated by a family $\\lambda$ of subspaces of it, let\n${\\cal P}$ be some continuously invariant property, and let the\nfollowing condition holds:*\n\n*$(\\alpha)$ if $Y\\in \\lambda$, then for every $B\\subseteq\n\\pi_Y(C_p(X))$ with the property ${\\cal P}$, the closure $B$ in\n$\\pi_Y(C_p(X))$ is compact.*\n\n*Then any subset of $C_p(X)$ with the property ${\\cal\nP}$, is relatively compact.*\n\nAlthough the condition $(\\alpha)$ is somewhat different from the\ncondition $(\\gamma)$, similarity between Theorems and and two the\nfollowing two theorems is striking.\n\n**Theorem 14**. *(Theorem 8.1 in ). If a Tychonoff space $X$ is\nfunctionally generated by the family $\\lambda_c$ of all its closed\nsubspaces that have an everywhere dense $\\sigma$-countably pracompact\nsubspace, then the closure in $C_p(X)$ of every pseudocompact subspace\nis compact.*\n\n**Theorem 15**. *(Theorem 8.3 in ). If a Tychonoff space $X$ is\nfunctionally generated by the family $\\lambda_p$ of its own closed\nsubspaces that contain an everywhere dense $\\sigma$-pseudocompact\nsubspace, then the closure in $C_p(X)$ of every countably pracompact\nsubspace is compact.*\n\nAs we will see later, a certain parallelism between the results using\nfunctional generation and generation by Asanov-Velichko, is observed\nfurther.\n\n# The Grothendieck theorem for spaces $C_{\\lambda,\\mu}(X,Y)$\n\nIn this section, we prove two theorems that can be considered basic in\nthis paper.\n\nRecall that a topological space $X$ is called a *$\\mu$-space* if every\nbounded subset of $X$ is relatively compact . The class of $\\mu$–spaces\narose in connection with the study of the question of barrels of spaces\nof continuous real-valued functions $C_c(X)$ with compact-open topology.\nIn 1973 H. Buchwolter proved that $C_c(X)$ is barreled if and only if\n$X$ is a $\\mu$-space. Later the notion of $\\mu$-space acquired a value\nof its own. The class of $\\mu$-spaces is very wide, it includes all\nDieudonné complete spaces, in particular, all metrizable spaces.\n\n**Lemma 16**.  * Let $X$ be a Tychonoff space, $Y$ be a Tychonoff\n$\\mu$–space with a countable pseudocharacter and $f:X \\rightarrow Y$ be\na condensation (= a continuous bijection). Then $X$ is a $\\mu$-space.*\n\n*Proof.* Let $f_1:\\nu X\\rightarrow \\nu Y$ be a continuous extension of\nthe function $f$ from the Hewitt realcompactification $\\nu X$ of $X$\nonto the Hewitt realcompactification $\\nu Y$ of $Y$ (Theorem 3.11.16 in\n)). Now let $A$ be a bounded subset of $X$, then $\\overline{A}^{\\nu X}$\nis compact (Proposition 6.9.7 in ). Let us prove that\n$\\overline{A}^{X}=\\overline{A}^{\\nu X}$, i.e. $\\overline{A}^{\\nu\nX}\\setminus X=\\emptyset$. Let this not be the case and there is a point\n$x\\in\n\\overline{A}^{\\nu X}\\setminus X$. The set $f(A)$ is bounded in $Y$ and\n$\\overline{f(A)}^Y$ is compact, hence,\n$f_1(\\overline{A}^{\\nu X})=\\overline{f(A)}^Y\\subset Y$. Then\n$V=f^{-1}_1(f_1(x))$ is a $G_{\\delta}$-neighbourhood of $x$ because a\npseudocharacter $\\psi(f_1(x),Y)$ of the point $f_1(x)$ in $Y$ is\ncountable, and, hence, $\\psi(f_1(x),\\nu\nY)$ is countable, too. Since the mapping $f$ is one-to-one, $V\\cap X$\ncontains only one point $x_1$. Let $O(x)$ be a neighborhood of the point\n$x$ in $\\nu X$ which does not contain $x_1$. But then $V_1=V\\cap O(x)$\nis a $G_\\delta$-neighbourhood of the point $x$ and is disjoint with $X$,\nwhich contradicts the following assertion (Theorem 3.11.11 in ): for any\npoint $x\\in \\nu X$ and any of its neighborhood $V\\subset \\nu X$ the\nintersection of $V\\cap X$ is not empty. ◻\n\n**Definition 17**. Let $X$ be a topological space, $\\lambda$ be a family\nof subsets of $X$. A subset $A$ of $X$ is called *$\\lambda$-separable*\nif there is a countable subfamily $\\lambda_1$ of the family $\\lambda$\nsuch that $A\\subseteq \\overline{\\bigcup\\{B:B\\in\n\\lambda_1\\}}$. If $A=X$ then $X$ is called a *$\\lambda$-separable\nspace*.\n\n**Lemma 18**.  * Let $X$ be a Tychonoff $\\lambda$-separable space for\nsome family $\\lambda$ of subsets $X$. Then $C_{\\lambda,\\rho}(X)$ is\nsubmetrizable.*\n\n*Proof.* Let $\\lambda_1$ be a countable subfamily of $\\lambda$ such that\n$X=\\overline{\\bigcup\\{B:B\\in\n\\lambda_1\\}}$. Let $X_1=\\bigcup \\{B:B\\in \\lambda_1\\}$ and\n$\\lambda_2=\\{A\\cap X_1:A\\in \\lambda\\}$. It is obvious that the mapping\n$\\pi_{X_1}:C_{\\lambda,\\rho}(X)\\rightarrow\nC_{\\lambda_2,\\rho}(X_1)$ where $\\pi_{X_1}(f)=f\\upharpoonright X_1$ for\nevery $f\\in C_{\\lambda,\\rho}(X)$ is a condensation because $X_1$ is\ndense in $X$. On the other hand, $\\lambda_1\\subseteq\\lambda_2$ and,\nhence, the identity mapping\n$e:C_{\\lambda_1,\\rho}(X)\\rightarrow C_{\\lambda_2,\\rho}(X_1)$ is a\ncondensation, too. The mapping $f=e\\circ\\pi_{X_1}$ is a condensation\nfrom $C_{\\lambda,\\rho}(X)$ onto the metrizable space\n$C_{\\lambda_1,\\rho}(X_1)$ ($C_{\\lambda_1,\\rho}(X_1)$ is metrizable by\nProposition 4.9 in ). ◻\n\n**Corollary 19**. Let $X$ be a Tychonoff $\\lambda$–separable space for\nsome family $\\lambda$ of subsets of $X$. Then for every function $h\\in\nC_{\\lambda,\\rho}(X)$ there is the function\n$f:C_{\\lambda,\\rho}(X)\\rightarrow \\mathbb{R}$ such that $f(h)=0$ and\n$f(h_1)>0$ for any $h_1\\in C_{\\lambda,\\rho}(X)$.\n\n*Proof.* Let $g:C_{\\lambda,\\rho}(X)\\rightarrow Z$ be a condensation of\nthe space $C_{\\lambda,\\rho}(X)$ onto a metric space $Z$. Let\n$h\\in C_{\\lambda,\\rho}(X)$ and $x=g(h)$. Then $f=f_1\\circ g$ where the\nmapping $f_1:Z \\rightarrow \\mathbb{R}$ such that $f_1(y)=d(x,y)$ for\nevery $y\\in Z$. ◻\n\n**Proposition 20**.  * Let $X$ be a Tyhonoff space, $\\lambda$ be a cover\nof $X$. If $F$ is a bounded subset of $C_{\\lambda,\\rho}(X)$ then for\nevery $\\lambda$-separable $Y\\subseteq X$ there exists\n$F_1\\subseteq C_{\\lambda,\\rho}(X)$ such that $F\\subseteq F_1$ and\n$\\pi_Y(F_1)=\\overline{\\pi_Y(F)}^{C_{\\lambda_Y,\\rho}(Y)}$ is a metrizable\ncompact space.*\n\n*Proof.* Since $Y$ is $\\lambda$-separable, there exists a condensation\n$g:C_{\\lambda_Y,\\rho}(Y)\\rightarrow Z$ where $Z$ is a metrizable space\n(Lemma ). Since $Z$ is a metrizable space, it is a $\\mu$-space with a\ncountable pseudocharacter. By Lemma , $C_{\\lambda_Y,\\rho}(Y)$ is a\n$\\mu$-space. The set $\\pi_Y(F)$ is bounded in $C_{\\lambda_Y,\\rho}(X)$\nand, hence, it is relatively compact. Since the compact set\n$\\overline{\\pi_Y(F)}^{C_{\\lambda_Y,\\rho}(Y)}$ condenses into $Z$, then\nit is metrizable. For every point $h\\in\n\\overline{\\pi_Y(F)}^{C_{\\lambda_Y,\\rho}(Y)}$ the following condition\nholds: $\\pi^{-1}_Y(h)\\cap C_{\\lambda,\\rho}(X)\\neq \\emptyset$.\n\nIndeed, if this were not the case for some $h_1$, then there would be a\ncontinuous function $\\phi\n:C_{\\lambda,\\rho}(Y)\\rightarrow \\mathbb{R}$ such that $\\phi\n(h_1)=0$ and $\\phi(h)>0$ for $h\\neq h_1$ (Corollary ). Putting\n$\\phi_1=\\frac{1}{x}$ for every $x\\in (0,+\\infty)$,\n$\\phi_1:(0,+\\infty)\\rightarrow \\mathbb{R}$ we would have a continuous\nfunction $\\phi_1\\circ \\phi\\circ\n\\pi_Y:C_{\\lambda,\\rho}(X)\\rightarrow \\mathbb{R}$ which is unbounded on\n$F$. Thus, we have $F_1=\\bigcup \\{\\pi^{-1}_Y(h): h\\in\n\\overline{\\pi_Y(F)}^{C_{\\lambda_Y,\\rho}(Y)}\\}$. ◻\n\nIf $\\lambda$ is a family of subsets of a topological space $X$, then the\nfamily of all countable unions of elements of $\\lambda$ denote by\n$\\sigma\\lambda$.\n\nLet $X$ be a topological space, $\\lambda\\subseteq 2^X$. Let\n$Q=\\{B\\subseteq X:\\, \\overline{A}\\cap B$ is closed in $\\overline{A}$ for\nall $A\\in\\lambda\\}$. Let $X_{\\lambda}$ be a set $X$ with the topology\n$\\tau=\\{X\\setminus B:\\, B\\in Q\\}$. Further, we will call such space as\n*$\\lambda$-leader* of $X$.\n\nLet $X$ be a Tychonoff space, $\\lambda\\subseteq 2^X$. Let\n$X_{\\tau\\lambda}$ be a Tychonoff modification of $\\lambda$-leader\n$X_{\\lambda}$ of the space $X$. Further, we will call such space\n*$\\lambda_f$-leader* of $X$ and denote by $X_{\\tau\\lambda}$ (see more\nabout $\\lambda$- and $\\lambda_f$-leaders of $X$ in ).\n\n**Theorem 21**.  * Let $X$ be a Tychonoff $AV_{\\sigma \\lambda}$-space\nfor some a cover $\\lambda$ of $X$. Then $C_{\\lambda,\\rho}(X)$ is a\n$\\mu$-space.*\n\n*Proof.* Let $e:X_{\\tau\\lambda}\\rightarrow X$ be a natural condensation\nof the $\\lambda_f$-leader $X_{\\tau\\lambda}$ on $X$. Denote by\n$Z=C_{e^{-1}(\\lambda),\\rho}(X_{\\tau\\lambda})$. Let $F$ is a bounded\nsubset of $C_{\\lambda,\\rho }(X)$, then $F_0=\\overline{F}^{Z}$ is compact\nbecause $Z$ is a complete uniform space (a $\\mu$-space).\n\nIt suffices to show that $F_0\\subset C_{\\lambda,\\rho}(X)$. Let\n$f\\in F_0$. Let us prove that $f\\in C_{\\lambda,\\rho}(X)$.\n\nAssume the contrary, i.e. that $f$ is a discontinuous function from $X$\nto $\\mathbb{R}$. Then there is a closed set $B\\subset\n\\mathbb{R}$ such that $A=f^{-1}(B)$ is not closed in $X$. Since $X$ is a\n$AV_{\\sigma \\lambda}$-space, there is a point\n$x_0\\in \\overline{A}\\setminus A$ for which the TWO player has a winning\nstrategy in the game $AV_{\\sigma \\lambda}(X)$. Put $\\epsilon\n=\\rho (f(x_0),B)$. Since $B$ is closed, $\\epsilon\n>0.$\n\nThe fact that $f\\in F_0$ implies that there is a function $f_1\\in F$\nsuch that $|f(x_0)-f_1(x_0)|<\\frac{1}{2}$. Since $\\lambda$ is a cover of\n$X$, $\\{x_0\\}\\in \\lambda$ (Remark ).\n\nLet $V_1=\\{x\\in X:\n|f(x_0)-f_1(x)|<\\frac{1}{2}$. Then $V_1$ is a neighborhood of $x_0$. Let\n$S_1\\subseteq A$ and $T_1=\\bigcup\\{A^{i}_1\\in \\lambda :i\\in\n\\mathbb{N}\\}\\in \\sigma \\lambda$ is a set chosen by the TWO player in\naccording to the winning strategy in the game $AV_{\\sigma \\lambda}(X)$.\nLet $P_1=A_1\\in \\lambda$.\n\nSuppose that the functions $f_1,..., f_n$, the neighborhoods\n$V_1, ...,V_n$ of $x_0$, the subsets $S_1, ...,S_n$ of $A$, the elements\n$T_1, ...,T_n$ of the family $\\sigma \\lambda$ such that\n$S_{j}\\subseteq \\overline{T_{j}}$ for every $j=1,...,n$ and\n$T_{j}=\\bigcup\\{A^{i}_{j}\\in \\lambda: i\\in \\mathbb{N}\\}\\in\n\\sigma\\lambda$ are constructed. Put $P_n=\\bigcup\\limits_{i=1}^n\n\\bigcup\\limits_{j=1}^{n}A^{i}_{j}\\cup\\{x_0\\}\\in \\lambda$ (see Remark ).\nThen there is a function $f_{n+1}$ such that\n$|f(x)-f_{n+1}(x)|<(\\frac{1}{2})^{n+1}$ for every $x\\in P_n$. Let\n$V_{n+1}=\\{x\\in X:\n|f(x_0)-f_{n+1}(x)|<(\\frac{1}{2})^{n+1}\\}$. Then $V_{n+1}$ is a\nneighborhood of $x_0$. Let sets $S_{n+1}\\subseteq A$ and\n$T_{n+1}=\\bigcup\\{A^{i}_{n+1}\\in \\lambda : i\\in \\mathbb{N}\\}\\in\n\\sigma \\lambda$ are chosen by the TWO player according to the winning\nstrategy in the game $AV_{\\sigma \\lambda}(X)$. Put\n$P_{n+1}=\\bigcup\\limits_{i=1}^{n+1}\\bigcup\\limits_{j=1}^{n+1}A^{i}_{j}\\cup\n\\{x_0\\}\\in \\lambda$.\n\nContinuing this process, by induction we construct countable sets\n$\\{f_i: i\\in \\mathbb{N}\\}$, $\\{V_i:i\\in \\mathbb{N}\\}$, $\\{S_i: i\\in\n\\mathbb{N}\\}$ and $\\{T_i: i\\in \\mathbb{N}\\}$. Since $X$ is an\n$AV_{\\sigma \\lambda}$-space, there is a point\n$x_{\\omega }\\in \\overline{\\bigcup\\{S_i:i\\in\n\\mathbb{N}\\}}\\cap(\\bigcap\\{V_i:i\\in \\mathbb{N}\\})$.\n\nPut $Y=\\bigcup\\{T_i:i\\in\n\\mathbb{N}\\}\\cup\\{x_0\\}\\cup\\{x_{\\omega}\\}$. There is a function\n$f_{\\omega}\\in C_{\\lambda ,\\rho}(X)$ such that $\\pi_Y(f_{\\omega\n})\\in \\overline{\\pi _Y(\\{f_i:i\\in\n\\mathbb{N}\\})}^{C_{\\lambda_Y,\\rho}(Y)}$ (see Proposition ). Then two\nconditions holds:\n\n\\(1\\) $f_{\\omega }(x_{\\omega })=f(x_0)$ because $x_{\\omega }\\in\n\\bigcap\\{V_i:i\\in \\mathbb{N}\\}$;\n\n\\(2\\) $f_{\\omega }(x)=f(x)$ for every $x\\in \\bigcup\\{T_i:i\\in\n\\mathbb{N}\\}$.\n\nLet\n$T=(\\bigcup\\{T_i:i\\in \\mathbb{N}\\})\\setminus\\{x\\in X: |f(x_0)-f(x)|<\\frac{3}{4 }\\epsilon\\}$.\n\nBy conditions (1) and (2), $|f_{\\omega }(x_0)-f_{\\omega }(x)|\\geqslant\n\\frac{3}{4}\\epsilon$ for every $x\\in T$. If we get that\n$x_{\\omega }\\in \\overline{T}$, then we obtain a contradiction with\ncontinuity of $f_{\\omega}$, and the theorem will be proved.\n\nIndeed, let $O(x_{\\omega })$ be an arbitrary neighborhood of point\n$x_{\\omega }$. Then there are a point $x_1$ and $n\\in\n\\mathbb{N}$ such that $x_1\\in S_n\\cap O(x_{\\omega})$.\n\nThe set $H=T_n \\cup \\{x_1\\}$ is $\\lambda$-separable and, by Proposition\n, there is a function $f'\\in C_{e^{-1}(\\lambda),\\rho}(X_{\\tau\\lambda})$\nsuch that $\\pi_H(f')=\\pi_H(f)$. Since $f'$ is continuous, there exists a\npoint $x_2\\in T_n$ such that $|f'(x_1)-f'(x_2)|<\\frac{1}{4}\\epsilon$.\nBut then\n\n$|f(x_2)-f(x_0)|\\geqslant|f(x_1)-f(x_0)|-|f'(x_1)-f'(x_2)|\\geqslant\\epsilon\n-\\frac{1}{4}\\epsilon=\\frac{3}{4}\\epsilon$ , i.e. $x_2\\in T\\cap\nO(x_{\\omega })$. ◻\n\nTheorem is a generalization of the Asanov-Velichko theorem in . It is\ninteresting to note that in Theorem we can $AV_{\\lambda}$-space replace\nto a space which functional generated by the family $\\lambda$. This\nparallelism forces us to find out relationship between classes of\n$AV_{\\lambda}$-spaces and functionally generated by the same family\n$\\lambda$ of subsets.\n\n**Proposition 22**. *There is a Tychonoff space $X$ functionally\ngenerated by the family $\\lambda_c$ of all non-empty of its countable\nsubsets, but is not $AV_{\\lambda_c}$-space.*\n\n*Proof.* Let $X=\\prod\\{R_{\\alpha }:\\alpha \\in \\Lambda\\}$ where\n$R_{\\alpha }=\\mathbb{R}$ for all $\\alpha\\in \\Lambda$ where $|\\Lambda|$\nis a uncountable Ulam non-measurable cardinal . Then $X$ is functionally\ngenerated of the family $\\lambda_c$ of all its countable subsets (see ).\nLet us prove that $X$ is not $AV_{\\lambda_c}$-space.\n\nLet us introduce some notation: let $y=(y_{\\alpha })\\in\n\\prod\\{R_{\\alpha }: \\alpha \\in \\Lambda\\}$, $supp(y)=\\{\\alpha : y_{\\alpha\n}\\neq 0\\}$; the point $y$ for which $supp(y)=\\emptyset$ will be denoted\nby ${\\bf 0}$; if $M$ is a subset of $X$ then\n$supp(M)=\\bigcup\\{supp(y): y\\in\nM\\}$.\n\nLet $n\\in \\mathbb{N}$ and $A=\\{y\\in X: |supp(y)|\\leqslant n$ and\n$(\\forall \\alpha :\ny_{\\alpha }\\neq 0\\Leftrightarrow y_{\\alpha }=n)\\}$.\n\nLet us prove the following fact: there is no such point $x\\in\n\\overline{A}\\setminus A$ that the TWO player has a winning strategy in\nthe game $AV_{\\lambda _c}(X)$.\n\n\\(1\\) Let us check that $\\overline{A}\\setminus A=\\{{\\bf 0}\\}$. Let $x\\in\nX\\setminus A$ and $x\\neq {\\bf 0}$, then three cases are possible:\n\n\\(a\\) $\\exists \\alpha^{0}\\in supp(x)$ such that $x_{\\alpha^0}$ is not an\ninteger number;\n\n\\(b\\) $\\exists \\alpha^{0}_1,\\alpha ^{0}_2\\in supp(x)$ such that\n$x_{\\alpha^0_1}=i\\neq j=x_{\\alpha^0_2}$ where $i,j\\in \\mathbb{N}$;\n\n\\(c\\) $\\exists \\alpha^{0}_1,\\ldots ,\\alpha ^{0}_{n+1}\\in\nsupp(x)$ such that $x_{\\alpha^0_1}=\\ldots =x_{\\alpha^0_{n+1}}=n.$\n\nIn all three cases, the point $x$ has a neighborhood $V$ such that\n$V\\cap A=\\emptyset$.\n\n\\(a\\) $V=\\{y\\in X: |x_{\\alpha^0 }-y_{\\alpha^0}|<\\epsilon \\}$ where\n$\\epsilon=\\min \\{|x_{\\alpha^0}-n|: n\\in \\mathbb{N}\\}$;\n\n\\(b\\) $V=\\{y\\in X: |x_{\\alpha^0_1}-y_{\\alpha^0_1}|<\\frac{1}{2}$ and\n$|x_{\\alpha^0_2}-y_{\\alpha^0_2}|<\\frac{1}{2}\\}$;\n\n\\(c\\) $V=\\{y\\in X: |y_{\\alpha^0\n_i}-x_{\\alpha^0_i}|<\\frac{1}{2}$ for every $i=1,\\ldots,n+1\\}$.\n\nThus, we have proved that $\\overline{A}\\setminus A=\\{{\\bf 0}\\}$.\n\n\\(2\\) Prove that the TWO player does not have a winning strategy in the\ngame $AV_{\\lambda _c}(X)$ on the set $A$ and at the point ${\\bf 0}$.\nMoreover, we prove that the ONE player has a winning strategy.\n\nLet us check that the TWO player at any step of the game can choose at\nmost than a countable set $S_i$. Note that $S_i$ contains at most than a\ncountable number of points having the same $supp$. On the other hand, if\n$y\\in \\overline{T_i}$, then there is $y'\\in\n\\overline{T_i}$ such that $supp(y)=supp(y')$. Since $T_i\\in\n\\lambda_c$ and, hence, it is countable, $S_i$ is countable, too.\n\nNow let the set $S_1$ be chosen by the TWO player at the first step of\nthe game. Let $H_1=supp(S_1)$. We renumber the set\n$H_1=\\{\\alpha^{1}_i: i\\in \\mathbb{N}\\}$. Take as neighborhood of the\npoint ${\\bf 0}$ chosen by the ONE player, the set\n$V_1=\\{x\\in X: |x_{\\alpha^{1}_1}|<\\frac{1}{2}\\}$. Let the game already\npassed $m-1$ steps and the set $S_m\\subseteq A$ is chosen by the TWO\nplayer at the $m$-th step of the Asanov-Velichko game, let\n$H_m=supp(S_m)=\\{\\alpha ^{m}_i:i\\in \\mathbb{N}\\}$.\n\nLet $V_m=\\{x\\in X:\\forall i,\\forall j:i\\leqslant j\\leqslant m,\n|x_{\\alpha^{i}_{j}}|<\\frac{1}{2}\\}$. And so on. At the end of the game\nwe get: for every $x\\in \\bigcap \\{V_m: m\\in \\mathbb{N}\\}$ $|x_{\\alpha\n}|<\\frac{1}{2}$ for any $\\alpha \\in H=\\bigcup \\{H_m:m\\in \\mathbb{N}\\}$.\nOn the other hand, for every $x\\in\n\\overline{\\bigcup \\{S_m: m\\in \\mathbb{N}\\}}$ there is $\\alpha \\in\nH$ such that $x_{\\alpha }\\geqslant 1$. So $\\overline{\\bigcup\n\\{S_m: m\\in \\mathbb{N}\\}}\\cap (\\bigcap \\{V_m: m\\in\n\\mathbb{N}\\})=\\emptyset$. ◻\n\n**Proposition 23**. *There is a Tychonoff $AV_{\\lambda_c}$-space $X$\nwhere $\\lambda_c$ is the family of all non-empty countable subsets, but\nis not functionally generated same family.*\n\n*Proof.* Let $X=T(\\omega_1+1)$ be the set of all ordinals less than\n$\\omega_1+1$ in the order topology (see Example 3.1.27 in ). The space\n$X$ is compact and, hence, is a $AV_{\\lambda_c}$-space.\n\nBut $X$ is not functionally generated by the family $\\lambda_c$, because\nthe function\n\n$$f(\\alpha)= \\left\\{\n\\begin{array}{lcr}\n0, \\ \\ \\ \\ if \\, \\alpha <\\omega_1, \\\\\n1, \\, \\, \\, \\, if \\, \\, \\, \\, \\alpha=\\omega_1\\\\\n\\end{array}\n\\right.$$\n\nis discontinuous, but $f\\upharpoonright S$ is continuous for any\ncountable set $S$. ◻\n\nTo conclude this section, we present a theorem that reduces the problem\nof compactness of subsets in $C_{\\lambda,\\mu}(X,Y)$ to the case in\n$C_p(X,Y)$.\n\nA continuously invariant property ${\\cal P}$ is of *boundedness type* if\nand only if it implies boundedness, that is, if every subspace $Y$ of\n$X$ with the property ${\\cal P}$ is bounded in $X$ (Proposition 2.15 in\n).\n\n**Theorem 24**.  * Let $X$ be a Hausdorff topological space, let\n$\\lambda$ be a cover of $X$, and let $Y$ be a complete Hausdorff uniform\nspace with uniformity $\\mu$. Then, if ${\\cal P}$ is a boundedness type\nproperty such that any subset of $C_p(X,Y)$ that has this property, is\nrelatively compact in $C_p(X,Y)$, then every subset of\n$C_{\\lambda ,\\mu}(X,Y)$ with the property ${\\cal P}$, is relatively\ncompact in $C_{\\lambda,\\mu}(X,Y)$.*\n\n*Proof.* Let $F\\subseteq C_{\\lambda,\\mu}(X,Y)$ has the property\n${\\cal P}$ and $e:X_{\\tau\\lambda}\\rightarrow X$ be a natural\ncondensation of the $\\lambda_f$-leader $X_{\\tau\\lambda}$ of $X$ onto\n$X$. By Corollary 4.3 in , the map\n$e^{\\#}:C_{\\lambda,\\mu}(X,Y)\\rightarrow\nC_{e^{-1}(\\lambda),\\mu}(X_{\\tau\\lambda},Y)$ is an embedding, and the set\n$e^{\\#}(F)$ has the property ${\\cal P}$ (moreover, $e^{\\#}(F)$ is\nbounded). Since $C_{e^{-1}(\\lambda),\\mu}(X_{\\tau\\lambda},Y)$ is a\ncomplete uniform space (see Proposition 4.7 in ) (hence, a $\\mu$-space),\na closure of the set $e^{\\#}(F)$ in the space\n$C_{e^{-1}(\\lambda),\\mu}(X_{\\tau\\lambda},Y)$ is compact.\n\nIf we now prove that $e^{\\#}(\\overline{F}^{C_{\\lambda,\\mu}(X,Y)})=\n\\overline{e^{\\#}(F)}^{C_{e^{-1}(\\lambda),\\mu}(X_{\\tau\\lambda},Y)}$ then,\ndue to the fact that the mapping $e^{\\#}$ is an embedding, we get that\n$\\overline{F}^{C_{\\lambda,\\mu}(X,Y)}$ is compact.\n\nNote that the set $F$ has the property ${\\cal P}$ in space $C_p(X,Y)$.\nSince the identity mapping\n$id_{C(X,Y)}:C_{\\lambda,\\mu}(X,Y)\\rightarrow C_p(X,Y)$ is continuous,\nthe space $C_p(X,Y)$ has a weaker topology than the topology of\n$C_{\\lambda,\\mu}(X,Y)$ and, by the hypotheses of the theorem, $F$ is\nrelatively compact in $C_p(X,Y)$. Since $\\overline{F}^{C_p(X,Y)}$ is\ncompact, the set $e^{\\#}(\\overline{F}^{C_p(X,Y)})$ is compact in\n$C_p(X_{\\tau\\lambda},Y)$. On the other hand, since the mapping\n$e^{\\#}:C_p(X,Y)\\rightarrow C_p(X_{\\tau\\lambda},Y)$ is continuous (see ,\nCorollary 2.8), $e^{\\#}(\\overline{F}^{C_p(X,Y)})=\n\\overline{e^{\\#}(F)}^{C_p(X_{\\tau\\lambda},Y)}$.\n\nSince the identity mapping\n$id_{C(X_{\\tau\\lambda},Y)}:C_{e^{-1}(\\lambda),\\mu}(X_{\\tau\\lambda\n},Y)\\rightarrow C_p(X_{\\tau\\lambda},Y)$ is continuous (see ),\n$\\overline{e^{\\#}(F)}^{C_{e^{-1}(\\lambda),\\mu}(X_{\\tau\\lambda},Y)}\\subseteq\n\\overline{e^{\\#}(F)}^{C_p(X_{\\tau\\lambda},Y)}$, i.e.\n$\\overline{e^{\\#}(F)}^{C_{e^{-1}(\\lambda),\\mu}(X_{\\tau\\lambda},Y)}\\subseteq\ne^{\\#}(C_{\\lambda,\\mu}(X,Y))$. ◻\n\nTheorem generates the following corollaries.\n\n**Corollary 25**. Let $X$ be a Tychonoff space and let $\\lambda$ be a\ncover of $X$. If $X$ is an $AV_{\\gamma_c}$-space or functionally\ngenerated by the family $\\gamma_c$ of all its subspaces that have an\neverywhere dense $\\sigma$-countably pracompact subset, then each\npseudocompact subset in $C_{\\lambda ,\\rho}(X)$ is relatively compact.\n\n*Proof.* It suffices to refer to Theorems and of this paper and Theorem\n8.1 in . ◻\n\n**Corollary 26**. Let $X$ be a Tychonoff space and let $\\lambda$ be a\ncover of $X$. If $X$ is an $AV_{\\gamma_p}$-space or functionally\ngenerated by the family $\\gamma_p$ of all its subspaces that have an\neverywhere dense $\\sigma$-pseudocompact subset, then each countably\npracompact subset in $C_{\\lambda ,\\rho}(X)$ is relatively compact.\n\n*Proof.* It suffices to refer to Theorems and of this paper and Theorem\n8.3 in . ◻\n\n**Corollary 27**. Let $X$ be a Tychonoff space and let $\\lambda$ be a\ncover of $X$. If $X$ is an $AV_{\\gamma}$-space or functionally generated\nby the family $\\gamma$ of all its countable subsets, then the space\n$C_{\\lambda,\\rho}(X)$ is $\\mu$-space.\n\n*Proof.* It suffices to refer to Theorem of this paper, Theorem 1 in and\nTheorem 2.16 in . ◻\n\nTo indicate the breadth of the classes of spaces that cover these\ncorollaries, we indicate that Corollary applies to\n\n\\(1\\) $q$-spaces in the sense E. Michael , in particular, countably\ncompact spaces;\n\n\\(2\\) quasi-$k$–spaces in the sense of Ju. Nagata , in particular,\n$k$-spaces;\n\n\\(3\\) spaces of countable tightness, in particular, sequential and\nFrechet–Urysohn spaces;\n\n\\(4\\) locally separable spaces;\n\n\\(5\\) spaces of countable functional tightness (see and ).\n\nCorollary also applies to $\\sigma$-pseudocompact spaces.\n\n# Grothendieck’s theorem for $C_\\lambda(X)$\n\nThe purpose of this section is to prove the theorem analogous to Theorem\n. For this we need two lemmas.\n\n**Lemma 28**. *Let $X=\\lim\\limits_\\leftarrow \\{X_{\\alpha\n},\\pi^{\\alpha1}_{\\alpha 2},A\\}$ be a inverse limit of the system\n$\\{X_{\\alpha\n},\\pi^{\\alpha1}_{\\alpha 2},A\\}$ of Tychonoff spaces $X_{\\alpha }$, and\nlet for every $\\alpha$ any subset of $X_{\\alpha }$ with the property of\nboundedness type ${\\cal P}$ be relatively compact. Then any\n$B\\subseteq X$ with the property ${\\cal P}$, is relatively compact.*\n\n*Proof.* Let $B\\subseteq X$ have the property ${\\cal P}$, then for every\n$\\alpha \\in A$, $\\pi_{\\alpha }(B)$ has the property ${\\cal P}$ and,\ntherefore, $\\overline{\\pi_{\\alpha}(B)}^{X_{\\alpha}}$ is compact.\nObviously\n$\\overline{B}^{X}=\\lim \\limits_{\\leftarrow}\\{\\overline{\\pi_{\\alpha}(B)}^{X_{\\alpha}},\\pi^{\\alpha 1}_{\\alpha 2},A\\}$\nis compact because it is a closed subset of a compact set. ◻\n\nThe following result was proved by D. Preiss and P. Simon.\n\n**Proposition 29**. *(Theorem 5 in ) Let $X$ be an Eberlein compact, $x$\nnon-isolated point of $X$. Then there exists a sequence\n$\\{U_n: n\\in \\mathbb{N}\\}$ of open sets in $X$, which converges to $x$.*\n\n**Lemma 30**.  * Let $X$ be a regular space, $f:X\\rightarrow Y$ be a\ncondensation and a closure of any pseudocompact $A\\subseteq Y$ is an\nEberlein compactum, then a closure of any pseudocompact $B\\subseteq X$\nis an Eberlein compactum.*\n\n*Proof.* Let $B$ be a pseudocompact subset of $X$. Since $f(B)$ is\npseudocompact, $A=\\overline{f(B)}$ is an Eberlein compactum. Let us\nprove that $f(B)$ is closed.\n\nSuppose that $y_0\\in A\\setminus f(B)$, then, by Proposition , there is a\nsequence $\\{U_n: n\\in \\mathbb{N}\\}$ of open sets in $A$ such that any\nneighborhood of the point $y_0$ contains all members of this sequence,\nstarting from some. Let $y_n\\in\nU_n\\cap f(B)$ ($U_n\\cap f(B)\\neq \\emptyset$ because $f(B)$ is dense in\n$A$). Obviously, the sequence $\\{y_n: n\\in\n\\mathbb{N}\\}$ is discrete in $f(B)$, which contradicts pseudocompactness\nof $f(B)$.\n\nLet us now prove that $f|_{B}$ is a closed mapping (and, hence, a\nhomeomorphism). Let $C$ be an arbitrary closed subset of $B$. Since $X$\nis regular, there is a family $\\gamma$ of open sets in $B$ such that\n$C=\\bigcap\\{\\overline{U}^{B}: U\\in \\gamma\\}$. Since $\\overline{U}^B$ is\npseudocompact, the set $f(\\overline{U}^B)$ is pseudocompact , and,\nhence, is closed in $Y$. Therefore $f(C)=f(\\bigcap\\{\\overline{U}^B:U\\in\n\\gamma\\})=\\bigcap\\{f(\\overline{U}^B):U\\in \\gamma\\}$ is a closed set in\n$Y$. Thus, we have proved that $f(B)$ is an Eberlein compact and\n$f|_{B}$ is a homeomorphism. Hence, $B$ is closed and is an Eberlein\ncompactum. ◻\n\n**Corollary 31**. If a Tychonoff space $X$ contains a dense\n$\\sigma$-countably pracompact subspace, and $\\lambda$ is a family of\nsubsets of $X$ containing all finite subsets of $X$, then every\npseudocompact subset of $C_{\\lambda}(X)$ is an Eberlein compactum.\n\n**Theorem 32**.  * Let $X$ be a Tychonoff space, $\\lambda$ be a family\ncontaining all non-empty finite subsets of $X$ and closed under finite\nintersections and unions. Let ${\\cal P}$ be a boundedness type property\nimplying pseudocompactness such that for each $A\\in \\lambda$ any subset\nof the space $C_p(\\overline{A})$ with the property ${\\cal P}$, is an\nEberlein compactum. Then any subset of $C_{\\lambda}(X)$ with the\nproperty ${\\cal P}$ is relatively compact provided that any subset of\n$C_p(X)$ with the property ${\\cal P}$ is relatively compact.*\n\n*Proof.* Let $e:X_{\\tau\\lambda}\\rightarrow X$ be a natural condensation\nof a $\\lambda_f$-leader $X_{\\tau\\lambda}$ of $X$ onto $X$. By Theorem\n5.1 in , the map $e^{\\#}:C_{\\lambda}(X)\\rightarrow\nC_{e^{-1}(\\lambda)}(X_{\\tau\\lambda})$ is embedding.\n\nLet $F\\subseteq C_{\\lambda}(X)$ has the property ${\\cal P}$. Denote by\n$F_1$ the closure of the set $e^{\\#}(F)$ in the space\n$C_{e^{-1}(\\lambda)}(X_{\\tau\\lambda})$. Let us prove that $F_1$ is\ncompact.\n\nBy Theorem 5.2 in , $C_{e^{-1}(\\lambda)}(X_{\\tau\\lambda})$ is\nhomeomorphic to an inverse limit of the system\n$S_{*}(X,\\lambda, \\mathbb{R})$, and the set $\\pi_{\\overline{A}}(F_1)$\nhas the property ${\\cal P}$ in\n$C_{\\lambda_{\\overline{A}}}(\\overline{A})$ for every $A\\in \\lambda$. On\nthe other hands, $C_{\\lambda_{\\overline{A}}}(\\overline{A})$ condense\nonto $C_p(\\overline{A})$. Therefore, by Lemma , the closure of the set\n$\\pi_{\\overline{A}}(F_1)$ into\n$C_{\\lambda_{\\overline{A}}}(\\overline{A})$ is compact. Hence, $F_1$ is\ncompact as a closed subset of an inverse limit of the system consisting\nof compact sets.\n\nNote that the set $F$ has the property ${\\cal P}$ in space $C_p(X)$\n(which has a weaker topology than the topology in $C_{\\lambda}(X)$),\nand, by assumption, is relatively compact. Therefore,\n$e^{\\#}(\\overline{F}^{C_p(X)})$ is compact and is closed in\n$C_p(X_{\\tau\\lambda})$. On the other hands,\n$e^{\\#}(\\overline{F}^{C_p(X)})\\subseteq\n\\overline{e^{\\#}(F)}^{C_p(X_{\\tau\\lambda})}$ due to continuity $e^{\\#}$\n(, Corollary 2.8.). But then\n$e^{\\#}(\\overline{F}^{C_p(X)})=\\overline{e^{\\#}(F)}^{C_p(X_{\\tau\\lambda\n})}$.\n\nSince $F_1\\subseteq \\overline{e^{\\#}(F)}^{C_p(X_{\\tau\\lambda\n})}=e^{\\#}(\\overline{F}^{C_p(X)})$, $F_1\\subseteq\ne^{\\#}(C_{\\lambda }(X))$ and, hence, $F_1$ is homeomorphic to\n$\\overline{F}^{C_{\\lambda}(X)}$ which implies that $F$ is relatively\ncompact in $C_{\\lambda}(X)$. ◻\n\n**Corollary 33**. Let $X$ be a Tychonoff space and let $\\lambda$ be a\ncover of $X$ containing all finite subsets of $X$. If $X$ is an\n$AV_{\\gamma}$-space or functionally generated by the family $\\gamma$ of\nall its subspaces that have an everywhere dense $\\sigma$-countably\npracompact subset and $\\lambda\\subseteq\\gamma$, then each pseudocompact\nsubset in $C_{\\lambda}(X)$ is relatively compact.\n\n*Proof.* It suffices to refer to Theorems and of this paper and Theorem\n1.1 in . ◻\n\n**Corollary 34**. Let $X$ be a Tychonoff space and let $\\lambda$ be a\ncover of $X$ containing all finite subsets of $X$. If $X$ is an\n$AV_{\\gamma}$-space or functionally generated by the family $\\gamma$ of\nall its subspaces that have an everywhere dense $\\sigma$-pseudocompact\nsubset, then each countably pracompact subset in $C_{\\lambda}(X)$ is\nrelatively compact.\n\n*Proof.* It suffices to refer to Theorems and of this paper and Theorem\n8.3 in . ◻\n"
    }
]