Papers

Motivated by Postnikov's study of lattice-point enumeration in Minkowski sums of simplices, we investigate lattice points in Minkowski sums of cross polytopes and establish analogous results, together with several related consequences. In particular, we introduce the support-enumerator associated with Postnikov's notion of draconian sequences and show that it coincides with the $h^*$-polynomial of the corresponding root polytope. This provides a new interpretation of the $h^*$-polynomial and yields a simple method for computing the volume of the corresponding polytope. By exploiting the symmetry of these root polytopes, we further establish a duality property for support-enumerators, which in turn provides a proof of a conjecture by Athanasiadis and Chapoton concerning the $h$-polynomials of preorders. Consequently, we obtain a formula for the number of lattice points in Minkowski sums of cross polytopes in terms of draconian sequences and show that these polytopes are Ehrhart positive. Furthermore, this formula leads to analogous expressions for the number of lattice points on their boundaries and for their surface volumes.

Let $X$ be the Shimura curve attached to an indefinite quaternion $\mathbb{Q}$-algebra $B$ with a maximal order $O_B$. This paper investigates the reduction $X\otimes \mathbb{F}_p$ of $X$ modulo an arbitrary prime $p$, focusing particularly on its superspecial locus. We give an explicit criterion for the existence of superspecial $\mathbb{F}_q$-rational points on $X$. Furthermore, we compute both the number of geometric superspecial points and the number of $\mathbb{F}_p$-rational superspecial points, through the Eichler class number formula and the Selberg trace formula. As a key ingredient, we classify the Dieudonné modules attached to superspecial $O_B$-abelian surfaces, which generalizes Ribet's classification of admissible quaternion bimodules of rank $2$ by dropping the admissible hypothesis. These results generalize Deuring's explicit formula for supersingular elliptic curves over $\mathbb{F}_p$ and give the Shimura-curve analogue of the Ibukiyama-Katsura formulas for principally polarized superspecial abelian surfaces over $\mathbb{F}_p$.

Let $C^r_{1,k}$ be the $r$-uniform $k$-crown and put $h=r-k+2$. For a finite linear intersecting $r$-uniform hypergraph $G$, let $\tau_h(G)$ be the minimum size of a set meeting every edge of $G$ in at least $h$ vertices, and define \[ \rho_{r,k}=\sup_G\frac{|E(G)|}{\tau_h(G)}. \] We prove that every fixed pair $(G,B)$, with $B$ an $h$-fold transversal, yields \[ \operatorname{ex}^{\mathrm{lin}}_r(n,C^r_{1,k}) \ge \frac{|E(G)|}{|B|}n-O_{G,B}(\sqrt n) \] for all sufficiently large $n$. Incidence counting gives $\rho_{r,k}\le r/h$, and equality is characterized after dualization by a pairwise balanced design with a distinguished regular subfamily. For $r=q+1$, where $q$ is a prime power, truncated projective planes give \[ \frac qh\le \rho_{q+1,k}\le\frac{q+1}{h}. \] The upper endpoint is attained whenever a maximal $h$-arc exists; in particular, if $q$ is even and $h\mid q$, then $\rho_{q+1,k}=(q+1)/h$. Padding the truncated-plane construction gives \[ \rho_{r,r}=(1-o(1))\frac r2 \] and, uniformly for each fixed $\varepsilon>0$ and $\varepsilon r\le k\le r$, \[ \rho_{r,k}=(1+o(1))\frac{r}{r-k+2}. \] For nonintersecting templates, the corresponding transfer is governed by a local safe-block condition that replaces the $h$-fold transversal requirement.

Fix \(\eta>0\) and \(A_0>0\). Let \(q\) tend to infinity through odd primes, put \(Q=\log q\), and let \(T=T(q)\) satisfy \[ \frac{(\log Q)^{1+\eta}}{Q}\le T\le Q^{A_0}. \] Set \(I=(T,2T]\), and sum without weights over the \(q-2\) nonprincipal characters modulo \(q\). Let \(\mathcal N_q\) count nontrivial zeros in \(I\) with multiplicity, let \(\mathcal N^s_{0,q}\) and \(\mathcal N^*_{0,q}\) count simple and distinct zeros on the critical line, and let \(\mathcal N_{d,q}\) count all distinct zeros in \(I\). Uniformly in this height range, we prove unconditionally \[ \mathcal N_q=\frac{qT\log q}{2\pi}\{1+o_{\eta,A_0}(1)\}, \quad \frac{\mathcal N^s_{0,q}}{\mathcal N_q}, \frac{\mathcal N^*_{0,q}}{\mathcal N_q} \ge C_{\mathrm{MT}}-o_{\eta,A_0}(1), \quad \frac{\mathcal N_{d,q}}{\mathcal N_q}\ge C_d-o_{\eta,A_0}(1), \] where \[ C_{\mathrm{MT}}=\frac32-\frac1{\sqrt2}\cot\!\left(\frac1{\sqrt2}\right) =0.672500703679\ldots, \qquad C_d=\frac{1+C_{\mathrm{MT}}}{2}=0.836250351839\ldots. \] The proof combines Selberg's family-averaged argument estimate and zero-density deletion with a finite Gevrey Gabor compression of Weil's Hermitian form. Quantitative Fourier–Laplace estimates control exterior zeros down to the lower endpoint, while a local–remote shell decomposition gives uniform control throughout the polylogarithmic upper range. Matrix moment estimates and an inertia-based rank–trace inequality then yield the counting bounds. No form of the generalized Riemann hypothesis is assumed.

Oscillator Ising Machines (OIMs) have emerged as promising computational architectures for approximating solutions to combinatorial optimization problems. We derive and analyze the mean-field limit of an OIM model and show that it inherits the gradient flow structure of the finite-dimensional dynamics. We identify conditions under which this mean-field evolution admits an Eulerian formulation as a gradient flow on the Wasserstein space of probability measures and contrast this with a Lagrangian formulation which is always available. This gradient flow structure strongly constrains the long-time dynamics and enables a complete classification of limit solutions and their stability in the totally symmetric case. In particular, all limit solutions are fixed points whose phases cluster into at most four groups, and for almost all parameter values, only binarized fixed points – those with clusters at $0$ and/or $\pi$ – can be stable. Since binarized states are exactly those for which a feasible solution to the original problem can be read out, this shows that feasible solutions can almost always be recovered. We provide tight bounds on the parameter thresholds for which fixed points in this binarized family are stable, thereby identifying the threshold for binarization in this model, and present numerical evidence that the mean-field model correctly predicts behavioral regimes in certain large random networks, including Erdős-Rényi networks. Moreover, this problem provides a case study in the application of mean-field models, gradient flows, and optimal transport to the study of large-scale systems, and we highlight a number of features of this problem which are informative in this respect.

In the paper, we develop a structure theory for metric measure spaces with synthetic lower Ricci curvature bounds, known as RCD spaces. Under a positive injectivity-radius assumption, we recover a smooth differential structure and prove a regularity result: such spaces are $W^{1,p}_{\mathrm{loc}}\cap C^{0,\alpha}_{\mathrm{loc}}$-Riemannian manifolds whose weight functions also belong to $W^{1,p}_{\mathrm{loc}}\cap C^{0,\alpha}_{\mathrm{loc}}$, in both distance and harmonic charts for all $p<\infty$ and $\alpha \in (0,1)$. We also establish a quantitative lower bound for the harmonic radius, together with compactness theorems and quantitative geometric bounds. As a byproduct, we develop a comprehensive elliptic regularity theory in this setting. These results apply, in particular, to smooth weighted Riemannian manifolds and to RCD spaces satisfying a synthetic curvature upper bound, or CBA condition. Even in these settings, the resulting regularity statements are new. In the latter case, both the Riemannian metric and the weight function are shown to be locally Lipschitz. As further applications, we establish fibration theorems and use metric smoothing to confirm, in a synthetic framework, a conjecture of V. Kapovitch concerning almost flat manifolds with mixed curvature bounds.

Recently, C. Li constructed stability conditions on the derived categories of all smooth complex projective varieties. These stability conditions satisfy the support property of Kontsevich-Soibelman with respect to the lattice in cohomology generated by powers of an ample class. We extend Li's results to construct stability conditions with full support on Lefschetz type varieties whose algebraic cohomology is generated by divisor classes. This includes all smooth complex projective varieties of dimension at most 3 and smooth projective toric varieties.

We study the BFGS method with an Armijo-Wolfe line search for minimizing convex functions with Lipschitz-continuous gradients, without assuming strong convexity. We establish a global iteration complexity bound of $\mathcal{O}(k^{-1/2})$ for the smallest gradient norm among the first $k$ iterates. Moreover, when the initial sublevel set is bounded, we show that the function value gap converges at a rate of $\mathcal{O}(k^{-1})$. Our analysis leverages the classical trace-log-determinant potential function and reveals that a key inequality underlying this potential function remains valid without strong convexity.

Our purpose is to investigate the local well-posedness of the compressible Euler equations in a two-dimensional bounded corner domain with acute contact angles. This configuration describes a free surface intersecting the fixed bottom at two points, where the fluid is subject to a gravitational field and the interface between the fluid and air is influenced by capillary forces. When the contact angles are less than $\pi/2$, we establish a local existence theory for the solution, with dissipation effects occurring at the contact points. The main analytical challenge arises from contact point singularities, which renders previous methods for dealing with compressible free boundary problems inadequate. To overcome this, we first establish the geometric structure for the compressible Euler equations, an approach originally introduced by Shatah and Zeng \cite{Shatah2008} for incompressible fluids. Additionally, we provide a singularity analysis for the wave equations in the corner domain, which ensures the validity of calculations near the corner. Finally, based on the geometric structure and singularity analysis, we obtain a priori energy estimates. Using these estimates, we also prove the local well-posedness of the system in a geometric formulation. To our knowledge, this is the first result addressing compressible Euler equations with a free boundary that involves contact points.

Traditional numerical guidance methods for powered landing of reusable rockets are typically constrained by high computational complexity and inadequate real-time performance. Moreover, insufficient consideration of attitude smoothness often induces severe fluctuations in control commands; meanwhile, most existing approaches are tailored for single-thrust scenarios, failing to accommodate the guidance requirements of multi-engine thrust switching. To mitigate these limitations, this paper proposes an analytical guidance method optimized for attitude smoothness, which supports dual-thrust-mode switching. First, a corresponding optimal control problem is formulated, and it is theoretically proven that the optimal attitude command takes a concise piecewise cubic function form. This transforms complex trajectory optimization into a parametric analytical optimization problem, yielding a substantial improvement in computational efficiency. Further, a three-phase guidance framework is designed to enable adaptive determination of the guidance activation point and thrust switching point; when integrated with an aerodynamic correction strategy, this framework enhances the method's adaptability in complex flight environments particularly under high lift-to-drag ratio conditions. Simulation results demonstrate that the attitude command profile generated by the proposed method aligns closely with the theoretical optimal solution, with an ultra-short computation time, confirming its strong potential for online real-time implementation. Even under stringent conditions (e.g., limited thrust adjustment range, high lift-to-drag ratios, and parameter deviations), the method consistently achieves high-precision landing, showcasing promising prospects for engineering applications.

We establish a new local-to-global rigidity phenomenon for the Bergman metric. Namely, under natural geometric hypotheses, a local conformal identification of Bergman metrics determines the underlying complex manifold globally, up to the unavoidable ambiguity of removing Bergman-negligible subsets. More precisely, let $\Omega\subseteq\mathbb{C}^n$ be a bounded domain with a complete Bergman metric, and suppose that the Bergman metric of a complex manifold $M$ is locally conformal, via a holomorphic map $f$, to that of $\Omega$, with a constant conformal factor when $n=1$. We prove that the given local map $f$ extends to a biholomorphism $F\colon M\to D$ onto a subdomain $D\subseteq\Omega$ in two complementary settings. If $M$ is Stein, then $\Omega\setminus D$ is a closed pluripolar set. If $M$ is a bounded domain and $\Omega$ satisfies a natural symmetry condition expressed in terms of its automorphism orbits, then $\Omega\setminus D$ is Bergman-negligible. In particular, this applies when $\Omega$ is a bounded homogeneous domain and yields a characterization, up to Bergman-negligible sets, of a class of bounded domains with locally symmetric Bergman metrics, addressing a question raised by Loi–Palmieri and Zimmer. A key ingredient in the proofs is a new Calabi-type extension theorem tailored to Bergman metrics.

The edge-path matrix is a square matrix where each off-diagonal entry records the maximum number of edge-disjoint paths between the corresponding pair of vertices. Akbari et al. [On edge-path eigenvalues of graphs, Linear Multilinear Algebra 70 (2022) 2998-3008] proposed two conjectures: Conjecture 1 relates the edge-path matrix to an upper bound on the number of edges in the graph, while Conjecture 2 asserts that a graph is Eulerian if and only if all entries of its edge-path matrix are even. In this paper, we prove the two conjectures.

We study holomorphic maps between complex manifolds that preserve the Bergman metric up to a positive constant. In the equal dimensional case, either the source is a Stein manifold and the target is a bounded domain in the complex Euclidean space, or the source is a bounded pseudoconvex domain and the target is a complex manifold, holomorphic Bergman isometry is a biholomorphism onto the complement of a relatively closed locally pluripolar set, and the constant is necessarily one. These include the bounded domains in the complex Euclidean space as special cases. Some applications to curvature-preserving maps are obtained. We further prove the rigidity result for local holomorphic Bergman isometries into Cartesian product of strongly pseudoconvex domains, with applications to finite analytic and modular correspondences.

Accurate parameter estimation for atmospheric turbulence channels is challenging because the probability density functions of the Gamma-Gamma (GG) and Lognormal-Rician (LR) models involve special functions and numerical integrations. This paper proposes two Gaussianization parameter estimators for GG and LR turbulence channels, i.e., the quantile-transformation (QT) estimator and the Box-Cox estimator. The QT estimator employs bidirectional cross-transformation together with higher-order statistical matching, whereas the Box-Cox estimator constructs an approximate likelihood by incorporating the transformation Jacobian. Asymptotic analysis identifies skewness as the leading-order deviation from Gaussianity under weak-to-moderate turbulence and yields asymptotic expressions for the Box-Cox power parameters. In addition, a physics-informed regularizer based on the extended Rytov-theory mapping from the Rytov variance to the GG shape parameters is introduced to improve parameter identifiability. Simulation results demonstrate that both estimators provide robust performance under noiseless and noisy conditions, while the physics-informed regularization term can improve the parameter estimation performance by at least three orders of magnitude compared with the iterative moment-based estimator and the method-of-moments/convex-optimization estimator in specific turbulence scenarios.

In this paper, we investigate the behavior of a vector field subject to a Lorentz-violating quartic potential on a fixed Schwarzschild background. We analyze the equilibrium configuration of the field and its linear perturbations. We show that the equilibrium solution and spherically symmetric perturbations are governed by a quadratic function whose properties determine their qualitative behavior. By requiring the background field to be real and finite, we derive bounds on the allowed parameters and identify critical cases in which the monopole perturbations become enhanced. We further derive asymptotic expressions for multipole perturbations near the event horizon and at spatial infinity.

We study systems of coupled Ornstein-Uhlenbeck diffusions with an absorbing failure state on finite graphs that converge locally weakly to a unimodular Galton-Watson tree. A single failure can trigger a cascade of threshold crossings, and the central question is when the cascade is subcritical or supercritical and what the cluster looks like near criticality. Under an explicitly stated front-decoupling hypothesis (the saturated-coupling regime), verified exactly on the Bethe lattice, the failed cluster of the root converges in law to the genealogy of a finite-type Galton-Watson process whose mean matrix M is computed from a single matrix-valued many-to-one formula. Conditional on this reduction we prove that criticality occurs at rho(M) = 1, that the quenched and annealed growth rates are the tangent at 0 and the value at 1 of a single convex real-analytic pressure phi(beta) = log rho(M(beta)), making the quenched/annealed gap a chord-tangent inequality, that on survival the end boundary of the failed cluster has Hausdorff dimension log rho(M), that at criticality P(|S| = n) ~ C n^{-3/2} and the conditioned cluster rescaled by n^{-1/2} converges to Aldous' Continuum Random Tree under a moment hypothesis, and that the subcritical domain is open and contractible but not convex, with the spectral gap 1 - rho(M) giving quantitative stability. Several natural claims in this setting are shown to be false, including the Hilbert-Schmidt/Krein-Rutman operator programme, the scalar many-to-one identity, and the spectral-radius/dimension identity. The reduction itself is proved unconditionally in the dissipative regime kappa < 1, where a pathwise front-tracking argument shows that back-reaction echoes decay geometrically in graph distance and beat the volume growth of the failed cluster; the strong-coupling regime remains open and is stated as a conjecture.

Let $\lambda>-1/2$, $\lambda\neq0$, and let $\Delta_\lambda=-\frac{d^2}{dx^2}-\frac{2\lambda}{x}\frac{d}{dx}$ be the Bessel operator on $\mathbb R_+=(0,\infty)$ studied by Muckenhoupt and Stein (TAMS 1965). Andersen and Kerman (Studia Math. 1981) proved that for $1<p<\infty$, the Bessel Riesz transform $R_\lambda=\frac{d}{dx}\Delta_\lambda^{-1/2}$ is bounded on $L^p(\mathbb R_+,w(x)\,dx)$ if and only if $w$ is in the intrinsic class $A_{p,\lambda}$. However, the sharp quantitative weighted bound via $[w]_{A_{p,\lambda}}$ was not addressed before. In this paper, we give a positive answer to this question by proving the sharp quantitative estimate $$ \|R_\lambda f\|_{L^p(\mathbb R_+,w\,dx)} \le C_{p,\lambda} [w]_{A_{p,\lambda}}^{\max\{1,1/(p-1)\}} \|f\|_{L^p(\mathbb R_+,w\,dx)}. $$ Moreover, for a real-valued function $b$ in the Bessel BMO space ${\rm BMO}_\lambda$, the sharp weighted bound for the Riesz commutator is $$ \|[b,R_\lambda]f\|_{L^p(\mathbb R_+,w\,dx)} \le C_{p,\lambda}\|b\|_{{\rm BMO}_\lambda} [w]_{A_{p,\lambda}}^{2\cdot\max\{1,1/(p-1)\}} \|f\|_{L^p(\mathbb R_+,w\,dx)}. $$ The argument uses the exact conjugation $U(x)=x^{p-2\lambda-1}w(x), \ d\nu_\lambda=x^{2\lambda+1}\,dx, \ [U]_{A_p(d\nu_\lambda)}=[w]_{A_{p,\lambda}},$ which reduces the Andersen–Kerman estimate to an $A_p$ weighted estimate for the auxiliary operator $$ \mathcal R_\lambda F(x)=\frac{1}{x}R_\lambda(yF(y))(x) $$ on the space of homogeneous type $(\mathbb R_+,|x-y|,d\nu_\lambda)$, whose kernel is a standard Calderón–Zygmund kernel with respect to $\nu_\lambda$.

In \cite {ChenFengLiXiXu2026}, a reverse projection inequality for general convex bodies was established in three dimensions. In this paper, we completely characterize the equality cases of this inequality. We further show that the reverse projection inequality fails in every dimension $n\geq9$. The counterexamples are constructed as full-dimensional polytopes with at most $n+2$ facets.

Sections 2–3 of this note were prepared primarily for the author's private use, as a means of organizing the materials in (generally non-holomorphic) conformal field theory (CFT) on 2-dimensions in the language of the vertex operator algebra, and also of collecting subtle details. The choice of topics is intended to supplement introductory physics textbooks on the subject. Section 4 deals with (generally non-holomorphic) Type 0 CFTs in the language of vertex operator superalgebra (SVOA). Formulations of crossing symmetry and intertwining superoperators [resp. CFT vertex superoperators] of states in the Ramond (SVOA-twisted) sector [resp. the Ramond–Ramond sector] are presented by using two auxiliary parameters. The formulation allows systematic generalization to the class of simple current extensions characterized by four conditions: (1) the vanishing third cohomology, (2) skew-symmetric bimultiplicative form, (3) fixed-point-free action of simple currents, and (4) one additional condition described in the main text.

Let $1<p<\infty$ and let $T$ be a positive strongly Kreiss bounded operator on an $L^p$-space. We prove that there exist constants $C,\kappa>0$ such that \[ \|T^N\| \leq C \left( \log(N+1) \right)^\kappa, \qquad N \geq 1. \] This answers a question raised by Arnold and Cuny concerning positive strongly Kreiss bounded operators on $L^p$-spaces. It also shows that the polynomial exponent appearing in the corresponding problem of Deng, Lorist and Veraar has infimum zero. The proof is elementary: positivity yields a local $\ell^p$-estimate for the orbit on intervals of length of order $\sqrt N$; a positive vector-valued lifting of $T$ then produces a self-improving estimate which can be iterated down to the logarithmic scale.

We construct, for the first time, a family of higher-order finite element differential forms on tensor-product sparse grids. The construction starts from compatible one-dimensional spaces of continuous piecewise polynomials and discontinuous piecewise polynomials of one degree lower, linked by differentiation in each coordinate direction. Their hierarchical decompositions combine Alpert multiwavelets with their integrated counterparts. We establish commuting canonical interpolation operators and corresponding approximation error bounds under mixed Sobolev regularity. For the sparse-grid de Rham complex of arbitrary polynomial degree on the unit cube in arbitrary dimension, we also prove exactness, polynomial-degree-robust stable discrete potentials, and an H(d)-bounded commuting projection by developing a novel stable homotopy operator. Numerical experiments for curl-curl source problems on a cube and a Maxwell eigenproblem on a square-annulus illustrate the effectiveness of the proposed higher-order sparse-grid method.

A compact two-stage fourth-order two-derivative IMEX method may retain its classical non-commuting fourth-order expansion and L-stable stiff damping while losing high-order accuracy when the relaxation time is comparable with the time step. We ask whether uniform fourth-order accuracy can be recovered without redesigning the original two-stage, two-implicit-solve core. For linear relaxation systems with a separated slow spectral subspace, we introduce a slow–fast response correction (SFRC): the exact slow projection is advanced by the unchanged compact IMEX step and projected back to the slow subspace, while the fast complement is advanced by its exact semigroup. For fixed nonzero relaxation time, the correction is $O(\dt^5)$, so the base mixed expansion through degree four is unchanged, and the pure-fast response remains L-stable. An abstract slow–fast argument reduces the full-state error to a slow-block defect and converts a uniform $O(\dt^5)$ local estimate into a uniform $O(\dt^4)$ global estimate. For each Jin–Xin Fourier mode, the required local estimate is proved by an exact projected-amplification factorization and a uniform bound in $x=\eps/\dt$. In particular, the unchanged compact core already has a uniform fifth-order one-step defect after restriction to the exact finite-$\eps$ slow eigenspace. A two-dimensional three-variable model yields the same mechanism on fixed Fourier grids. Symbolic verification and one- and two-dimensional experiments corroborate the analytical predictions. SFRC is therefore a rigorous linear benchmark identifying the finite-$\eps$ slow–fast information sufficient to reconcile compactness, mixed fourth-order accuracy, stiff damping, and uniform fourth-order time accuracy.

This paper develops a decision-theoretic interpretation of local expected utility on Abstract Wiener space. The framework is intended for economic acts that are naturally stochastic paths, such as income paths, consumption paths, investment payoffs, insurance losses, or laboratory stimuli unfolding over a continuum of contingencies. In this setting the object of choice is an act or payoff path, while Hermite functions serve as state-feature coordinates rather than objects of preference. Abstract Wiener space supplies a Gaussian reference measure on the Banach space of paths and a Hilbert subspace on which local utility is represented by orthogonal coefficients. The associated Wiener-integral weights are generally signed stochastic functionals; they become admissible decision weights only after a specified nonnegative normalization. This distinction clarifies the relation between the model, state-dependent utility, subjective expected utility, rank-dependent utility, and cumulative prospect theory. The main representation result gives finite primitive conditions under which projected path-valued acts admit a normalized local-utility representation. The corresponding elicitation result identifies composite utility-weight coefficients from projected acts and recovers normalized decision weights when the local utility scale is independently elicited. The paper also states testable restrictions: normalized weights must be nonnegative, add to one, preserve monotonicity, and satisfy additional rank or gain-loss restrictions when the subjective expected utility, rank-dependent utility, or cumulative prospect theory submodels are imposed. Numerical illustrations show how the representation operates in finite approximations.

PEJWAK is introduced as a profile-echoing self-anchored aggregation architecture for multi-criteria decision analysis. Each criterion is coupled geometrically with an endogenous reference derived from the same alternative before final aggregation, so criterion identity is preserved while its expressed contribution is conditioned by the alternative's profile. A general formulation separates the anchor function, anchoring kernel, retention exponents, contribution coefficients, and outer aggregation rule. The canonical member uses a weighted-arithmetic self-anchor, sets each retention exponent equal to the corresponding criterion-importance value, and derives contribution coefficients through a normalized square-root map. Analytical results establish continuity, monotonicity, diagonal calibration, internality, full-domain joint continuity, exact boundary behavior, nonseparability, failure of bisymmetry in the equal-weight binary restriction, and an exact logarithmic retention law. A deterministic MCDA audit provides complete criterion-level decomposition, exact weight-stability intervals and rank transitions, contribution-rule rank phases, set-dependence diagnostics, and exact permutation-based rank-affinity analysis. Canonical PEJWAK handles exact zeros without epsilon regularization: an active zero removes only its own criterion contribution whenever the remaining active scores sustain a positive anchor. The architecture therefore supplies an auditable profile-conditioned aggregation mechanism between direct weighted aggregation and parameter-intensive nonadditive interaction models.

We present a computational framework for robust topology optimization that integrates polygonal finite-element discretizations, spatially correlated non-Gaussian material modeling, and non-intrusive polynomial-chaos surrogates. Spatial uncertainty in Young's modulus is represented as a homogeneous non-Gaussian random field obtained via a memoryless transformation of a truncated Karhunen-Loève expansion, ensuring physical admissibility through positivity of stiffness while preserving the prescribed autocovariance. Polygonal finite elements provide a stable discretization for density-based optimization on unstructured meshes and mitigate checkerboard artefacts and mesh bias, while the sparse polynomial-chaos expansion enables efficient estimation of low-order statistical moments required by the robust objective at a fraction of the cost of intrusive or Monte Carlo approaches. Numerical studies on a cantilever and a curved beam show that introducing non-Gaussian material variability leads to systematic load-path redistribution and a reallocation of 6-12% of the structural volume, together with a reduction in compliance scatter. The non-intrusive surrogate reproduces intrusive reference results within 3% using an order of magnitude fewer full finite-element analyses. These results demonstrate that the proposed framework offers a physically consistent and computationally efficient route to topology-optimized designs that remain reliable under realistic material uncertainty.

We obtain density Ramsey theorems for configurations consisting of the vertices of a simplex $\Delta_o$ together with their barycenter. We prove that any subset $A\subseteq\mathbb{R}^n$ of positive upper density contains an isometric copy of all sufficiently large dilates of $\Delta_o$ together with its barycenter. As this configuration is non-spherical such results are not possible with respect to the quadratic Euclidean metric, we consider general metrics $\rho$ defined by a positive-definite, homogeneous forms of even degree at least four. We prove the analogous result in the discrete setting, for subsets $A$ of the integer lattice $\mathbb{Z}^n$, under some natural and necessary congruence restrictions on the scales $\lambda$ at which the set $A$ can contain an isometric copy of the simplex.

This paper establishes the global proof of the Second Edge Theorem: as sample size tends to infinity, sample-dependent information-geometric manifolds—formed by the parameter space, sample-scaled Fisher metric, and dual alpha-connections—undergo metric-topological collapse onto the flat tangent space of Conventional Statistics at the true parameter. We first prove a universal tensor valence scaling law, under which tensor fields of valence one through four degenerate at rates determined by sample size. Score fluctuations stabilize, the Fisher metric freezes to its true-parameter value, affine connections dissolve, and Riemann curvature is annihilated. We then bridge geometric collapse with statistical decision theory, showing that Cheeger–Gromov flattening and Le Cam risk condensation are dual projections of the same asymptotic phase transition. A Fisher-compatible Ehresmann connection extends these results to over-parameterized and singular models, yielding horizontal leaf-space collapse and uniform local asymptotic normality. Unifying the First and Second Edge Theorems yields a nested dual-edge hierarchy: Conventional Statistics is the boundary of Information Geometry, which is itself the boundary of Statistical Mechanics and Geometry. Thus, Conventional Statistics is not a heuristic approximation but the unique zero-curvature thermodynamic attractor of regular parametric information manifolds. This redefines modern statistics as a dynamic non-equilibrium field theory of finite-sample fluctuations, phase transitions, and gauge-invariant interactions.

We propose a fully discrete method for linear fourth-order parabolic equations with Dirichlet boundary conditions, combining an implicit LBRFD multistep scheme in time with a mixed interior penalty discontinuous Galerkin (IPDG) method in space. The temporal discretization employs equispaced linear barycentric rational interpolants and incorporates a startup procedure. To facilitate the spatial discretization, the original problem is reformulated through an auxiliary variable. For certain parameter pairs $(n,d)$, the LBRFD method is shown to be $A(\alpha)$-stable and to possess a wider stability angle than the corresponding BDF$p$ method of the same order. Stability and a priori error estimates are established via a $G$-energy technique and the discrete Grönwall lemma. The theoretical analysis yields a total $L^2$ error estimate of order $h^{k-1}+\tau^p$, where $p=d$ if $n-d$ is even and $p=d+1$ if $n-d$ is odd. The reduced spatial convergence rate is attributed to boundary contributions on $\partial\Omega$. Despite this theoretical prediction, numerical experiments confirm the stability and demonstrate optimal convergence of order $h^{k+1}+\tau^p$.

Let $G$ be a finite group. For every subgroup $H\leq G$, let $\pi_H$ be the permutation character of the action of $G$ on the left cosets of $H$. We prove that the characters $\pi_H$, with $H$ running through representatives of the conjugacy classes of subgroups of $G$, are linearly independent if and only if $G$ is cyclic. In particular, no finite insoluble group has the property asked for in Kourovka Notebook Problem~11.9. The proof uses only the fixed-point formula for a coset action and an elementary triangular-matrix argument.

In this paper, we establish optimal weighted norm inequalities for commutators of singular integral operators with rough kernels. While classical Calderón-Zygmund theory relies heavily on pointwise gradient smoothness, we operate under the strictly weaker log-Dini regularity condition assumed merely on the $L^{1}\left( \mathcal{S}^{n-1}\right) $ spherical restriction of the kernel. First, we prove that these rough commutators are bounded on the weighted Lebesgue spaces $L^{p}\left( w\right) $ for the full range of Muckenhoupt weights $w\in A_{p}$ $\left( 1<p<\infty \right) $. Our primary contribution establishes a sharp weighted endpoint estimate at the critical value $p=1$. For any weight $w\in A_{1}$, we demonstrate that the commutator satisfies a weak-type inequality with a precise $L\log L$ logarithmic loss, successfully recovering the classical smooth behavior in the absence of traditional kernel regularity. The proofs rely on a meticulous refinement of microlocal decompositions combined with a direct, localized sparse domination framework involving Orlicz averages. Finally, we settle the question of optimality by constructing a rigorous counterexample based on the oscillatory properties of lacunary Fourier series. This construction proves that the log-Dini condition is sharp, confirming that the logarithmic regularity cannot be relaxed without losing the operator's fundamental boundedness.