Papers

Graph-structured optimization with linear constraints is fundamental to critical infrastructure but faces scalability limits due to massive strict hard constraints and high dimensionality. While recent projection-based methods such as Trainable Sampling Kaczmarz-Motzkin Net (T-SKM-Net) guarantee feasibility, they face high computational costs in dynamic environments by processing the entire constraint set and requiring expensive matrix factorizations. To bridge this gap, we propose the Accelerated Trainable-SKM (AT-SKM) Net framework. To concentrate computation on the active constraints and eliminate redundant calculations, we introduce a hybrid sampling strategy guided by a topology-aware heterogeneous GNN model. To efficiently handle topological shifts in graph-based constraints, we employ a Cholesky Update mechanism that theoretically reduces the equality projection complexity from O(N^3) to O(N^2) under low-rank perturbations. Experiments on random geometric graphs, N-1 Security-Constrained DC-OPF, and minimum-cost gas transport problem demonstrate that AT-SKM reduces iteration counts by up to 85% and achieves 2.95x-7.29x SKM layer speedups, while maintaining zero constraint violations.

In this paper we define a novel product on polyominoes called the tensor product of polyominoes. Using a special case of this product which we name a dilation, we construct a new class of polyominoes called the dilated closed paths. This class is a ”thick” generalization of Cisto and Navarra's class of closed paths (arXiv:2006.13935). We prove the Zig-Zag Walk Conjecture for this new class thereby fully characterizing primality for its associated polyomino ideals.

Let $S=K[x_1,\ldots,x_n]$ be a polynomial ring over a field $K$ and let $I$ be a monomial ideal of $S$. We classify linear systems of parameters of $S/I$ over any field $K$ using linear algebra and show explicit linear systems of parameters when $K$ has at least $n$ elements. If $I(G)$ is the edge ideal of a perfect graph $G$, a cycle or the complement of a cycle, we show that $S/I(G)$ has a 0-1 linear system of parameters, and for graphs with independence number equal to $2$, we characterize when $S/I(G)$ has a 0-1 linear system of parameters.

A categorification of the crystal $B(\widetilde{U}_q(\mathfrak{g}))$ of the modified quantized enveloping algebra associated to a symmetrizable Kac-Moody algebra $\mathfrak{g}$ is given. A crystal is constructed from a categorical $\mathfrak{g}$-action on abelian categories without finiteness conditions, and is used to construct the bicrystal structure on the set of self-dual indecomposable objects of the categorified quantum group. Crystals from Webster's generalized tensor product categorification are identified with tensor products of lowest and highest weight crystals. As an application, a categorification of the extremal weight module is obtained.

A meander is a configuration of two simple plane curves intersecting transversely. The orders of their intersection points define a permutation that determines the configuration. When tangencies are allowed, however, different configurations can share the same permutation. We study the combinatorial and algebraic structures arising from this non-uniqueness. We give a realization criterion and show that the realizations of each realizable permutation form an affine space over the two-element field. We describe this space using an associated graph, called the component spine. We prove that the component spine of every permutation is a cactus. We also introduce the tangency polynomial, which counts realizations by their number of tangencies, investigate its properties, and prove that it factors over the cycles and bridges of the component spine. We derive a central limit theorem for tangency counts and obtain asymptotic formulas for the number of distinct tangency polynomials.

We use hyperbolic area and covering geometry to study wandering Fatou components and singular orbits. We give proofs without quasiconformal deformation of the rational no-wandering theorem and the known no-wandering results in Bergweiler's Question~9. The framework also excludes wandering for entire functions with compact singular sets whose derived sets lie in the Fatou set. For Bergweiler's Question~8, every wandering component of a transcendental meromorphic map has a subsequence of iterates converging locally uniformly to infinity. For Bergweiler's Question~4, we establish hyperbolic separation near singular-free Baker cycles after deleting any finite set of original singular values as orbit generators. For Bergweiler's Question~11, corresponding finite-deletion boundary estimates hold along wandering tails. In higher dimension, a relative-area theorem excludes bounded wandering components of polynomial skew products over bounded bases carrying an absolutely continuous invariant probability of full support. Applied to elliptic polynomial skew products over Siegel rotation domains, it yields a local no-wandering theorem without conditions on the fiber critical orbits or continuity of fiber Julia sets. A global criterion covers regular skew products on $\mathbb P^2$ whose bounded base Fatou components eventually enter periodic Siegel disks, including $(\lambda z+z^d,q(z,w))$ for Brjuno $\lambda$ and polynomial $q$ of total degree at most $d$ with a nonzero $w^d$ coefficient.

We explicitly construct the additional integrals of motion, ensuring maximal superintegrability of the discrete-time rational Ruijsenaars-Schneider model. Using them, we investigate the algebraic aspects of superintegrability in both continuous- and discrete-time settings. In particular, we determine the complete structures of the polynomial symmetry algebras associated with both the rational Ruijsenaars-Schneider model and its discretization. We demonstrate that discretization leads to a nontrivial deformation of the continuous symmetry algebra with respect to the discretization parameter, thereby extending recent analogous results from the rational Calogero-Moser system to its relativistic generalization.

It is well known that the \Lovasz-Theta function of a random graph $G(n,\tfrac{1}{2})$ is $\Theta(\sqrt{n})$. More precisely, it is tightly concentrated in the interval \( [\sqrt{n},\, 2\sqrt{n}], \) where the upper bound follows from an explicit dual witness for the associated semidefinite program. Numerical evidence and heuristic arguments suggest that the true value is $(1+o(1))\sqrt{n}$. However, closing this gap has remained a longstanding challenge, resisting existing techniques even in light of recent progress on sharp algorithmic thresholds and non-asymptotic free probability. In this work, we resolve this question by proving that the \Lovasz-Theta function of $G(n,\tfrac{1}{2})$ is $(1+o_n(1))\sqrt{n}$ with high probability, determining its asymptotic value up to vanishing relative error.

In this article, we prove that the theory $T_3$ of groups of exponent $3$ has a model companion. Previous work established the existence of model companions for theories of groups of fixed finite exponent and nilpotency class at most $2$ (Saracino–Wood), and for theories of groups of prime exponent $p$ and nilpotency class at most $c<p$ (Maier). These results do not cover $T_3$, since groups of exponent $3$ may have nilpotency class $3$. Our theorem verifies the $n=3$ case of a conjecture proposed by the third author that, for each integer $n>1$, the theory $T_n$ of groups of exponent $n$ has a model companion if and only if every finitely generated group of exponent $n$ is finite, or equivalently, if and only if the Burnside problem has a positive solution for exponent $n$.

In science and engineering, many physical laws and scientific principles naturally arise as variational problems of minimizing an energy functional over an appropriate functional space. In many cases, it is more advantageous to directly discover the minimizer of the energy functional than to solve the associated Euler-Lagrange equations. Conventional numerical solvers, such as the finite-element method (FEM), often lack the flexibility to incorporate prior knowledge or noisy observational data. In recent years, machine learning approaches, particularly kernel-based methods, have attracted growing attention in scientific computing. Unlike FEM, which requires domain discretization and mesh generation, kernel-based methods construct solutions from scattered nodes, thereby avoiding the complexity of meshing, especially in high-dimensional or geometrically complex domains. This works presents a two-step procedure for variational energy minimization based on the optimal recovery formulation in a reproducing kernel Hilbert space (RKHS), providing a unified framework for seamlessly incorporating both physical constraints and noisy data. On the computational side, we employ the sparse Cholesky decomposition for Matérn kernels to alleviate the well-known cubic complexity bottleneck of kernel methods. On the theoretical side, we rigorously establish the existence and convergence theory of the proposed method using $\Gamma$-convergence theory. Numerical experiments on benchmark problems demonstrate the efficiency, robustness and accuracy of the approach.

Thin plate splines are highly attractive smoothers. However, they have cubic computational cost, which severely limits their use in practice. As a remedy, Wood (2003) suggested thin plate regression splines (TPRS), which provide a low rank approximation. The key step of the TPRS approximation is a truncated eigendecomposition of the radial basis function (RBF) design matrix. However, as Wood (2003) writes, the optimality of the TPRS approximation is a slightly weak one. This is because the RBF coefficients are subject to orthogonality constraints and the TPRS approximation is only optimal if these constraints are ignored. To address this shortcoming, we suggest a slightly different low rank approximation. The suggested approximation is based on a truncated Demmler-Reinsch basis (TDRB), which provides a best low rank approximation of the smoother matrix in terms of Frobenius and spectral norm. We prove that the TDRB smoother achieves the optimal rate of convergence and suggest an efficient algorithm for its construction. This algorithm is based on a truncated Karhunen-Loève (KL) expansion of the equivalent Bayesian smoothness prior and it has the same computational cost as required for TPRS. We demonstrate the applicabilty of our approach through simulations and a real data example. We find that the performance is very similar to that of TPRS but the suggested approach has some advantages.

We prove that the planar balanced excited random walk $M(2,1,2)$ is recurrent. This walk takes a horizontal simple random walk step on its first departure from each vertex and a planar simple random walk step on every later departure. Moreover, the number of distinct vertices visited before time $n$, multiplied by $(\log n)/n$, converges to $\pi$ almost surely and in every $L^p$, $1\le p<\infty$, the same limit as for the planar simple random walk. More generally, we prove recurrence of balanced excited random walks in spatially inhomogeneous cookie environments whenever the total positive and negative cookie strengths at each vertex are bounded by constants $A, B$ with $A+B<1+1/(2\pi +1)$.

We study the computational complexity of peak-oriented rationalizability, a survey based revealed-preference test introduced by Seror (2026). We provide a polynomial-time algorithm for testing rationalizability and recovering a utility function, establishing that peak-oriented preference elicitation is computationally tractable. In contrast, we show that computing the peak-oriented Houtman-Maks index is NP-hard. These results delineate the precise computational boundaries of peak-oriented revealed-preference analysis.

We analyze solutions to a one-dimensional periodic linear model for peridynamic waves, focusing on the setting where the natural energy space allows for spatial discontinuities. Exploiting the high-frequency asymptotics of the peridynamic dispersion relation, we construct an explicit solution whose initial displacement has a single jump discontinuity but becomes continuous at every positive time. By time reversibility, this yields a continuous initial displacement whose evolution develops a jump at any prescribed time and is continuous at every other time. We further quantify the formation of this singularity through a sharp Hölder-norm blow-up estimate obtained via a Littlewood-Paley characterization of Hölder spaces. Finally, by superposing suitably weighted solutions, we construct an evolution exhibiting jump discontinuities on a dense set of times.

A graph $G=(V,E)$ is called smooth (respectively, strongly smooth) if for any two vertices $u,v\in V$, the distance point-shadow $v|u := \{ x\in V: d(u,x)=d(u,v)+d(v,x)\}$, respectively, the point-shadow $v/u := \{ x\in V: v\in\mathrm{conv}(u,x)\}$, is geodesically convex. Smooth graphs have been introduced by Nebeský (2005) in the context of step systems. Graphs with convex point-shadows and convex distance point-shadows also naturally occur in convexity theory. Brešar et al. (2026) recently showed that several classes of graphs are smooth and that smoothness is preserved by Cartesian products, gated amalgams, and isometric subgraphs. Weakly modular graphs comprise the most important classes of graphs from Metric Graph Theory: median, modular, Helly, bridged, and dual polar graphs. In this note, we characterize smooth and strongly smooth weakly modular graphs in terms of forbidden isometric subgraphs on 5 and 7 vertices. This settles Problem 1 of the paper by Brešar et al. We also characterize prime strongly smooth weakly modular graphs, i.e., strongly smooth weakly modular graphs that cannot be obtained from smaller graphs by Cartesian products and gated amalgams.

In this paper, we consider a three-dimensional non-homogeneous incompressible fluid in a bounded periodically perforated domain at the critical Stokes-capacity scale. The density obeys a transport equation. The viscosity depends on a quasi-static temperature perturbation, which solves a uniformly elliptic transmission problem with insulated outer boundary conditions and zero spatial mean. The limiting momentum equation contains a Brinkman resistance term generated by the critical perforation regime. For each fixed perforated domain, we construct global finite-energy weak solutions and derive a quantitative relative-energy stability estimate with respect to regular solutions of the homogenized system. As long as a regular effective solution exists, the squared relative error between any microscopic weak solution and the effective solution is bounded by the initial mismatch plus $O(\eps^{2})$. For well-prepared data, the density and uncorrected velocity converge at order $O(\eps)$ in $L^\infty(0,T;L^2)$, the temperature at the same order in both $L^\infty((0,T)\times\Omega)$ and $L^2(0,T;H^1)$, and the corrected velocity in $L^2(0,T;H^1)$. The key estimates are an $O(\eps)$ cell-capacity residual in the dual energy norm, an $O(\eps)$ corrector-gradient bound in $L^{6/5}$, and fixed-domain thermal stability. The critical boundary layers retain order-one viscous energy, represented in the limit by the Brinkman dissipation, so the uncorrected velocity gradient does not converge strongly to the effective gradient whenever the effective velocity is nonzero. Finally, we prove local existence of regular effective solutions for smooth data satisfying a finite set of boundary compatibility conditions, which closes the stability argument.

Let $F$ be a totally real field of degree $n$ and discriminant $\Delta_F$, and let $X_1(\eta)$ be the cover of the Hilbert modular variety parametrizing abelian varieties with real multiplication by $\mathcal O_F$, together with a torsion point having annihilator $\eta$. Let $L=K_{\overline X_1(\eta)}+D$ be the log-canonical bundle on a smooth toroidal compactification, and let $H_L$ be an associated multiplicative height. We prove that rational points on $X_1(\eta)$ become sparser as $|\mathrm{Nm}(\eta)|\to\infty$, with $(\eta,\Delta_F)=1$. More precisely, for every number field $K$, set $$ N_{\eta,K}(B)=\#\{x\in X_1(\eta)(K):H_L(x)\leq B\}. $$ If $|\mathrm{Nm}(\eta)|\ge 5^n$ and $(\eta,\Delta_F)=1$, we prove $$ \limsup_{B\to\infty}\frac{\log\max\{1,N_{\eta,K}(B)\}}{\log B} \leq\delta_{\eta,K,n},\qquad \delta_{\eta,K,n}\ll_{[K:\mathbb Q],n}|\mathrm{Nm}(\eta)|^{-1/(2n)}. $$ In particular, $\delta_{\eta,K,n}\to0$ uniformly when $n$ and $[K:\mathbb Q]$ are bounded and $|\mathrm{Nm}(\eta)|\to\infty$. The main geometric result is a uniform lower bound, growing with the level, for the log-canonical degree of subvarieties of $X_1(\eta)$. Combining this estimate with recent progress derived from determinant-method, due to Ellenberg–Lawrence–Venkatesh and Brunebarbe–Maculan, we obtain the sparsity result above. We also prove that, for sufficiently large $|\mathrm{Nm}(\eta)|$, every subvariety of $X_1(\eta)$ is of general type, and establish a higher-dimensional generalization of the geometric torsion theorem of Bakker–Tsimerman. Namely, for a family of abelian varieties with real multiplication over a quasi-projective base of arbitrary dimension, we bound the torsion subgroup of its Mordell–Weil group in terms of the canonical volume of the base, uniformly in the totally real multiplication field of fixed degree.

We introduce a new secondary operation on Lawson homology of smooth complex projective varieties. It refines the Lawson intersection product by retaining null-homotopy data that are invisible at the level of the graded Lawson intersection algebra. Intrinsically, the operation is the Toda bracket in multiplicative morphic cohomology transported through Friedlander–Lawson duality; in an associative cochain model it is represented by the classical Massey formula. For every defined triple, the affine bracket modulo its full indeterminacy gives a well-defined secondary invariant. The new structure is genuinely nontrivial and strictly finer than its singular-homology realization. On a smooth projective eightfold we construct integral Lawson–Massey brackets whose quotient classes are nonzero, while specified values have exact order two and map to zero under the Lawson cycle map to singular homology. A square-zero perturbation makes all three associated singular-homology classes nonzero without changing the secondary values. Infinitely many such values remain independent even after quotienting by the sum of all their indeterminacies, over one fixed singular-homology triple. Projection-formula transfers preserve the phenomenon under projective bundles and odd-degree generically finite morphisms, and cyclic triple covers give examples with ample canonical bundle in every dimension at least eight. The same defining systems yield nonzero motivic Massey values, scalar-indecomposable higher Chow torsion, and obstructions to integral and two-local formality.

In this article, we prove a Hochschild-Kostant-Rosenberg type theorem for the category of coherent matrix factorizations associated to a Landau-Ginzburg model (X,f), where X is a quasi-smooth derived Deligne-Mumford stack over an algebraically closed field of characteristic 0.

A $C^*$-category is to a $C^*$-algebra what a groupoid is to a group. An example is the category of Hilbert spaces and bounded linear maps, and indeed, every $C^*$-category can be represented as a sub-$*$-category of this. Analogously, a $W^*$-category is a $C^*$-category where every object has a predual, and we might expect a representation theory on Hilbert spaces where the resulting morphism spaces are $\sigma$-weakly closed. We identify a gap in the literature here, stemming from the (perhaps surprising) fact that a $C^*$-algebra can have a non-isometric predual and yet not be a $W^*$-algebra. Using the theory of dual TROs (ternary rings of operators) we repair this gap, and along the way, also give further justification to arguments in the literature about dual TROs. Motivated by the theory of Dual Banach Algebras, we offer an alternative axiomatisation of $W^*$-categories where we allow non-isometric preduals of the hom spaces, but require that they are additionally bimodules in a certain sense. This involves the theory of self-dual Hilbert $C^*$-modules.

We construct a function $\Phi $ such that the orbit under the representation of PSL(2,Z) is an orthonormal basis for the Bergman space with weight $\alpha =12$. Moreover, we show that $\Phi$ is effectively computable as a holomorphic function on the upper half-plane (in the precise sense of computable analysis), by providing an effective procedure. This constructs a wandering and tracelike vector for PSL(2,Z), whose abstract existence was proved by Sir Vaughan Jones in his last paper, where the corresponding construction was left as a problem. The function is built using an orthonormalization and modularization method, and it displays modular reminiscencies, despite not being modular itself. provides a computable implementing vector for the abstract anti-isomorphism between the von Neumann algebra $M_{12}(\Gamma )$ and its commutant, which is generated, in Rădulescu's sense, by cusp-form Toeplitz operators, while Voiculescu's results provide a random matrix model for $M_{12}(\Gamma )$.

This paper develops a divergence-free, inf-sup stable, optimally convergent and pressure-robust finite element method for the three-dimensional Stokes problem on curved domains. The geometry is approximated by an isoparametric tetrahedral mesh, while the velocity space is obtained from Scott–Vogelius spaces on an Alfeld split by the Piola transform. The discrete inf-sup condition is proved using suitable face bubble functions. The insufficient accuracy of quadrature rules on curved triangular interfaces leads to a consistency error that may cause suboptimal convergence. The remedy is to introduce a suitable consistency correction without stabilization terms. Moreover, commuting operators on curved domains are constructed from local commuting interpolants and shown to be globally conforming. The operators are used to approximate the load and obtain a pressure-robust discretization. Numerical examples are provided to validate the theoretical results.

Let $1<m<n$, $q>0$, and $p\in\mathbb R$. We study positive $C^1_{\mathrm{loc}}$ weak solutions of \[-\Delta_m u=u^p|Du|^q\qquad\text{in }\mathbb R^n.\] If $0<q<m-1$, we prove that every such solution is constant provided \[p+q-m+1<\frac{(m-1)(m-q)^2}{(n-m)(m-1-q)}.\] If $q\geqslant m-1$, the same conclusion holds for every $p\in\mathbb R$. For $0<q<m$, the proof uses an auxiliary function and reduces the main differential estimate to a lower bound for a function of one variable. An explicit construction and a local maximum-principle argument then prove constancy without differentiating across $\{|Du|=0\}$. When $m=2$, our result answers the natural Liouville problem left open by Bidaut-Véron, García-Huidobro and Véron for the whole range $0<q<1$. The case $q=m$ is handled by an increasing change of the dependent variable, in the spirit of the related work of Bidaut-Véron, while the case $q>m$ follows from the theorem of Lu and Zhu.

In this paper, we reconstruct a wedge and its post-shock flow from a prescribed leading shock and a uniform supersonic incoming state. Full Euler Rankine–Hugoniot relations supply the shock data, while a reduced interior model combines homogeneous acoustic characteristic equations, entropy transport, and the Bernoulli relation. A hodograph transformation yields a linear hyperbolic equation for the physical coordinate. Comparing a shock-to-wedge mass bound with a hodograph non-degeneracy threshold gives a sufficient, data-dependent reconstruction criterion. Under the stated assumptions, this criterion ensures a unique \(C^1\) post-shock flow and a \(C^2\) wedge throughout the characteristic domain determined by the prescribed shock segment. Numerical reconstructions examine empirical grid convergence, Mach-number dependence, and discrepancies between the reduced-model and full-Euler inverse reconstructions.

We minimize $\mathbb E[h(Y-X)]$ over martingale couplings of $\text{Unif}[-1,1]$ and $\text{Unif}[-2,2]$ satisfying $Y\geq X-k$, where $h\in C^1([-3,3])$ has convex derivative. Feasibility holds exactly for $k\geq1$. For each such $k$, we construct a coupling that minimizes all costs in this class. For $1<k<3$, its support consists of two graphs, with $D(x)=x-k$ on $[k-2,1]$. The maps admit an explicit parametrization for $2\leq k<3$ and are determined by scalar equations with unique admissible roots for $1<k<2$. We prove optimality by a dual inequality, using an analytic estimate in the latter range. At $k=1$ the optimizer is $Y=X\pm1$ with equal probabilities; for $k\geq3$ it is the ordinary left-curtain coupling. In the unconstrained problem, left-monotonicity identifies the left-curtain coupling, which is optimal for this cost class. Under the constraint, a discrete example shows that the corresponding support condition, even together with every two-source comparison, does not imply optimality.

We study the behavior of isolated singularities of stationary harmonic maps with generic boundary data. For round sphere targets, we prove that, for $4\leqslant n\leqslant7$, every stable stationary harmonic map from a bounded smooth $n$-dimensional domain to round $(n-1)$-sphere with generic smooth boundary data has only radial projections composed with orthogonal transformations as tangent maps at its singularities; for 7-dimensional domains and round $k$-sphere targets with $k\geqslant7$, every stable stationary harmonic map with generic smooth boundary data is smooth. These results follow from a general minimum-index principle for closed real-analytic target manifolds. Under suitable target hypotheses ensuring strong compactness and excluding lower-dimensional singularity models, we show that, in the first domain dimension in which singularities can occur, generic boundary data force the link of every singular tangent map to attain the smallest possible Morse index. This principle applies both to stable stationary harmonic maps and, under the corresponding stronger target hypotheses, to all stationary harmonic maps. If no non-constant tangent map in the relevant class has a link attaining this minimum, generic smoothness follows. As an application, we establish a rigidity at infinity theorem. For $3\leqslant n\leqslant7$, we show that any energy-minimizing map from $n$-Euclidean space to round $(n-1)$-sphere admitting the radial projection as a blow-down limit must itself be a translate of the radial projection.

We study mean field equations with multiple positive singularities on flat tori. We establish a uniform uniqueness criterion in terms of the total singular mass and the scale-invariant spectral quantity $\lambda_1(\mathbb T)|\mathbb T|$, where $\lambda_1(\mathbb T)$ denotes the first positive eigenvalue of the Laplacian. Combining nodal-set analysis for Jacobi fields with a cylindrical Alexandrov–Bol type inequality and sharp small-capacity asymptotics, we obtain the optimal leading coefficient $8/\pi^2$ in the total-mass uniqueness bound as $\lambda_1(\mathbb T)|\mathbb T|\to0$. Sharpness is demonstrated by examples with two symmetric singularities on degenerating rectangular tori that admit at least three distinct solutions. For the case of two singularities with total mass $2\rho$, we further derive uniqueness and multiplicity results near $\rho=4\pi$ from the critical-point structure of the associated symmetrized Green function. In particular, when its only critical points are the four nondegenerate two-torsion points, the equation admits a unique solution for $\rho$ just below $4\pi$ and exactly three solutions for $\rho$ just above $4\pi$.

We give a constructive numerical treatment of Monge's optimal transport problem in 1D, 2D, 3D, following Evans-Gangbo: the map is built from a Kantorovich potential $u$ and density $a$ solving $-\mathrm{div}(a\nabla u) = f^+-f^-$, via $\dot T = -a\nabla u(T)/[(1-t)f^+(T)+tf^-(T)]$, $T(0,x)=x$. In 1D this gives a closed form. In 2D–left unfinished in an earlier report–we obtain $(u,a)$ via a convex Beckmann flow problem (second-order-cone programming), cross-checked against a dual LP, then build the map by direct integration or a ray-based 1D "clock" ODE with explicit density per ray. Validation uses strong duality, mass conservation, and vanishing-regularization limits, on a translation-invariant case and two 2D examples (a radial cone, a checkerboard) checked by symmetry; both show boundary-concentrated mass and non-unique flux where supports touch. We extend to 3D (theory unchanged), matching the 2D answer to six digits and reproducing the same phenomena on a radial ball, exposing a limit of the 2D visualization style. Movies illustrating the transport maps are available as ancillary files on the arXiv page of this submission.

With the help of a summation formula for $q$-series and the creative microscoping method, we shall establish two parametric $q$-supercongruences. They are both modulo the third power of a cyclotomic polynomial. When $q\to1$, one of them is able to engender the following conclusion: for any prime $p\equiv2\pmod{3}$ and any nonnegative integer $s$ subject to $ s\leq (p-2)/3$, \[\sum_{k=s}^{(p+1)/3+s}(6k-1)\frac{(-\frac{1}{3})_{k-s}(-\frac{1}{3})_{k+s}(-\frac{1}{3})_{k}}{(k-s)!(k+s)!k!} \equiv 0\pmod{p^3}.\]

By the paper \cite{Pet2009upper}, \cite{LiNan2026}, we know that there exists $C(n)>0$, for any complete compact or non-compact Riemannian n manifold $M$ with non-negative sectional curvature, without boundary and any $R>0$, \begin{equation} R^{2-n}\int_{B(p,R)} Scal \le C(n). \end{equation} Based on this result, we will prove that (1) There exists constant $C(n)$. If $M$ is a complete, n dimensional, non-compact Riemannian manifold with non-negative sectional curvature, then for any $p\in M$, $R>0$, \begin{equation} R^{2-n}\int_{B(p,R)} Scal \ dvol \le C(n)(1-v(M)), \end{equation} where $Scal$ is the scalar curvature and $v(M)=\lim\limits_{R\to \infty} \frac{volB(p,R)}{volB(0,R)}$ is the asymptotic volume ratio. (2) There exists constant $C(n)$. If $M$ is a complete, n dimensional Riemannian manifold with sectional curvature $\ge 1$, then \begin{equation} \int_M (Scal-n(n-1)) \ dvol \le C(n)(1-\frac{vol(M)}{vol(S^n(1))}). \end{equation}