Papers

In this Brief Communication, we establish the notion of critical damping for generic two-degree-of-freedom system. For given set of masses and stiffnesses, the critical damping corresponds to the real eigenvalue with maximal multiplicity, equal to minus geometrical mean of the eigenfrequencies. This case corresponds to the fastest possible asymptotic decay rate for generic initial conditions. The damping matrix for the critical case is unique up to reflection of one modal coordinate and, generically, non-diagonal. Quite surprisingly, for large difference of the eigenfrequencies, it is also not positive definite. Therefore, physical realization of the critical case will require active elements that provide negative effective damping.

We study the expansion of the real field by the graphs of power functions on the unit circle. Under a natural number-theoretic conjecture, we prove that adding such dense subsets does not increase the topological complexity of definable sets: every open definable set remains semialgebraic. The proof uses a two-sorted structure that separates the linear and algebraic data, inspired by Zilber's raising to powers. Using Hrushovski's amalgamation method, we construct and axiomatize a class of rich structures, and then show that the intended structure is a model. This provides a new example of a tame expansion of the real field by dense trajectories.

In this paper, we construct a class of non-weight modules over the BMS-Kac-Moody algebra by taking tensor products of finitely many irreducible modules $\Phi(\lambda,\a,\b,\r,h(t))$ with irreducible restricted modules. We obtain the necessary and sufficient conditions for these tensor product modules to be irreducible, and determine the corresponding conditions for two such modules to be isomorphic. Moreover, we compare these modules with other known non-weight modules, showing that these irreducible modules are new.

We study exact Kullback–Leibler (KL) projection for low-rank factorizations whose two nonnegative factors have prescribed row marginals and a shared, learned column marginal. For arbitrary positive row marginals of equal total mass, the joint KL projection reduces exactly to a strictly convex gauge-fixed dual with only $r-1$ effective variables; its Hessian is a sum of categorical covariance terms and admits $O((n+m)r)$ matrix-free Hessian–vector products. The projection theorem is objective-independent. We then specialize this geometry to doubly stochastic (DS) graph learning through $W=U\operatorname{Diag}(g)^{-1}V^\top$, where row-simplex factors with a common column mass induce an exactly DS graph without materializing an $n\times n$ optimization variable. Combined with observed-edge sparse fitting, a stochastic anchor-reduced manifold regularizer, and Bregman backtracking, the resulting mirror-descent method preserves exact feasibility at every accepted step. Under a nonvanishing latent-mass condition, it satisfies sufficient decrease and an $O(1/N)$ mirror-stationarity bound, while strictly positive accumulation points are KKT stationary. Matched clustering experiments show competitive accuracy, feasibility residuals near numerical precision, and favorable anytime behavior without a dense learned graph.

A low-Earth-orbit satellite has only minutes to downlink imagery on a limited power budget, and some moments in a pass carry far better channels than others. Spending more power on the good moments and skipping hopeless ones should deliver more images, but the channel cannot be measured in time, so the schedule is planned in advance from weather statistics using a safety margin. That margin is sized so the forecast is statistically reliable, and the scheduler is judged by whether it beats uniform power on average. Both targets are wrong for a satellite that gets one attempt per pass: on real Sentinel-2 imagery, a scheduler gaining one to two images per pass on average still delivers fewer than uniform power on a quarter to a third of individual passes, and a genie knowing the weather exactly does no better. We instead size the margin by measuring, on recorded passes, how often it would have backfired, keeping only settings that provably backfire rarely. The procedure declines to schedule our learned codec and certifies scheduling for codecs with a sharper cliff.

Coarse finite-volume averages do not generally form a predictive state: realized interface fluxes close a conservative update, but distinct fine-grid states with identical parent averages can generate different future coarse histories. We analyze this failure for periodic scalar advection discretized by a first-order upwind finite-volume method with forward Euler time integration. We derive a finite-horizon observation-rank law: each additional observation exposes one new child-cell layer and contributes one fewer independent direction than the number of parent cells, until the unresolved layers are exhausted. Centralized prediction therefore requires one fewer additional coordinate than the number of parent cells per exposed layer, whereas product-local prediction requires one coordinate per layer in each parent. An anchored flux-divergence queue attains the centralized bound, identifies the periodic flux gauge, and, at saturation, forms a minimal autonomous predictive state with the parent averages. We then distinguish exact observability from stable recoverability. The collar-to-queue map becomes rapidly ill-conditioned as the Courant number decreases, so algebraically visible delayed information may fall below a prescribed numerical tolerance. For Courant numbers strictly between zero and one, we prove contraction of the nonconstant component under bounded arithmetic perturbations, with separate control of mean drift, an explicit perturbation neighborhood, and a grid-dependent decay time. Numerical experiments illustrate the rank ladder, effective-rank loss, queue conditioning, step-function flattening, and delayed coarse separation followed by dissipative decay. The results provide a solvable benchmark for assessing state sufficiency in coarse, reduced, multiscale, and learned scientific models.

We present an asymptotic analysis of two diffusive capture problems in general smooth closed three-dimensional geometries with multiple small reactive boundary patches of arbitrary shapes. (i) The narrow escape problem seeks to determine the escape rate of Brownian particles from an enclosed region through small boundary windows. (ii) The related Berg-Purcell (or narrow entrance) problem seeks to resolve the capture rate for signaling molecules diffusing outside the cell and entering through localized reactions at membrane-bound receptors. We obtain matched asymptotic solutions of these two problems and thus address the long-standing challenge of describing the role that curvature and local reactivities play in modulating diffusive capture rates. Our explicit expansions quantify local effects on diffusive capture through the sizes, shapes, and reactivities of the patches together with the principal curvatures of the manifold at each patch. In turn, we examine global effects on diffusive capture such as the spatial configuration of patches on the manifold, as encloded by the associated surface Neumann Green's function and its regular part. The accuracy of our asymptotic formulas is validated against a full numerical solution for an ellipsoidal domain. Overall, our results yield new insights on how geometry and stochasticity combine to shape the dynamics of various biological processes.

Let $(E,h)$ be a singular Hermitian vector bundle on a complex manifold $X$, and let \[ \mathcal E(h)_x=\{F\in\mathcal O(E)_x:|F|_h^2\in L^1_{\mathrm{loc},x}\} \] be its multiplier submodule sheaf. We introduce a finite plurisubharmonic Gram scalarization condition under which the higher-rank integrability problem reduces to finitely many scalar multiplier ideals. This reduction yields coherence and strong openness without imposing a general positivity hypothesis. When the scalar weights and the varying weight have analytic singularities, it also gives a theory of module jumping numbers, including a simultaneous-residue criterion for actual jumps. Finally, we study the induced Skoda filtration: Artin–Rees yields eventual periodicity, Tor controls whether periodicity starts at the scalar threshold, and a direct-image quotient measures the obstruction to descent.

We establish counting and sampling analogues of the Lovász Local Lemma: we give efficient algorithms for approximately counting and exactly sampling satisfying assignments of general constraint satisfaction problems (CSPs) in the local lemma regime $$4 \mathrm{e} p (D+1)^2\leq 1, $$ where $p$ is the maximum constraint violation probability and $D$ is the maximum dependency degree. This condition is tight up to constant factors under $\mathbf{NP}\neq\mathbf{RP}$, matching known hardness bounds for counting and sampling in natural subclasses of CSPs. Our key ingredient is a novel $2$-tree expansion for constraint marginal probabilities that exhibits exponential decay of correlations throughout this regime. This expansion yields deterministic polynomial-time approximate counting for fixed local parameters, randomized approximate counting with quadratic cost, and exact sampling in expected near-linear time when the local parameters are fixed.

Fourier neural operators (FNOs) provide efficient nonlocal spectral learning, but varying geometries and independently chosen discretizations remain difficult to accommodate. We introduce the ambient-domain extension Fourier neural operator (ADEx-FNO), a deterministic framework that incorporates geometry without modifying the defining Fourier-operator layers. Each physical domain is embedded in a fixed ambient hypercube and represented by a signed distance function. Inputs and solution fields are deterministically extended to the ambient domain, transferred to a common, potentially nonuniform rectilinear latent grid, processed by the FNO, then interpolated to an independently chosen target discretization and restricted to the physical domain. All geometry-transfer operations lie outside the optimization procedure and require no trainable graph, point-cloud, deformation, or geometry-decoding modules. ADEx-FNO achieves relative l2 errors of 0.32%-0.77% on held-out smooth-domain nonlinear Poisson and advection-reaction-diffusion problems in 2D and 3D, and is also evaluated on unseen nonsmooth geometries. A single ADEx-FNO inference is then used to initialize conventional CFD solvers. For all 29 converged 2D and 3D RANS cases, pseudo-time iterations decrease, with mean reductions of 44.17% and 43.03%, respectively, with comparable gains across three mesh resolutions. URANS cases reduce post-window physical-time advances by 18.52%-27.51%. In transfer from 2D URANS training data to DNS at different Mach and Reynolds numbers, the bootstrap interval decreases by 23.47%-48.21%, depending on the target statistic. In all CFD tests, ADEx-FNO provides only the initial field; the governing-equation solver controls the subsequent solution, while physical or statistical consistency is assessed separately from computational savings.

We study identifiability of mixing measures in infinite mixture models. We show that, in many common cases, lack of identifiability can be characterized in terms of certain differential structures of the kernel family with respect to its parameters. In our main results, we prove that when the kernel is annihilated by a non-trivial differential or difference-differential operator over the parameter space, there exist infinitely many distinct mixing measures yielding the same mixture density. We give verifiable conditions for such operators to exist, covering many common cases, including the location-scale Gaussian, location-scale Student-t, Gamma, Beta, Dirichlet, negative binomial and non-central Chi-squared families. Furthermore, our conditions apply to any exponential family whose parameter dimension exceeds the dimension of its sufficient statistic and, more generally, to kernels with polynomially-growing score functions. We complement our results with a minimax lower bound on the estimation error for the mixing measure in the Wasserstein distance under non-identifiability. On the flips side, we describe three classes of kernels for which identifiability is preserved and nonparametric statistical inference remains possible.

Let $G$ be a torsion-free one-ended hyperbolic group and let $\phi\in\operatorname{Aut} (G)$. We prove that the mapping torus, or suspension, $$ M:=G_{\phi}=G\rtimes_\phi \mathbb{Z} $$ has solvable conjugacy problem. This builds on the pioneering work of Préaux who solved the conjugacy problem for all (geometrisable) three-manifolds. We view $M$ as a generalisation of a fibred three-manifold. Our proof begins with the canonical JSJ tree of $G$, whose suspension gives a graph-of-groups decomposition of $ M $. We refine each suspended QH vertex using a Nielsen–Thurston reduction system for its induced monodromy, taking account of non-orientable surfaces and orientation-reversing monodromy. Equivalently, this is a further geometric JSJ decomposition for each vertex which is a fibred three-manifold, but allowing Klein bottles as well as tori in the splitting. We then `block' pieces of this refined JSJ together according to whether they share a central element, glued along an elementary vertex, by a sequence of folding operations. This new splitting is cocompact and acylindrical; in particular certain local `solution sets' turn out to be rational subsets of virtually abelian subgroups - the virtually abelian groups in question are products of the edge groups. Our version of Préaux's graph-of-groups argument then reduces global conjugacy to effective intersection and non-emptiness for these rational sets.

Let $\overrightarrow{C}$ be any orientation of the cycle $C_{\ell}$ which is not directed. We prove that, for every integer $\ell\ge3$ and every $\mu>0$, there is a real $\gamma$ such that every sufficiently large oriented graph $D$ with $\ell\mid |D|$, minimum semidegree at least $(1/4+\mu)|D|$ and independence number at most $\gamma |D|$ has a $\overrightarrow{C}$-factor. The constant $1/4$ is asymptotically tight. This proof establishes Ramsey-Turán type lattice absorption lemmas and an almost covering theorem via the oriented tree embedding lemma under chromatic number constraints.

Interference alignment is an efficient spectrum reuse scheme for multiuser wireless networks. In this letter, assuming no channel state information at the transmitters, we investigate a fluid antenna system (FAS)-enabled interference alignment scheme over a multiuser interference channel. In the proposed system, the additional spatial degrees of freedom provided by the FAS are exploited to nullify the real components of the effective combined interference such that those are reserved for interference-free desired signal detection. The interference alignment conditions and corresponding outage probabilities are theoretically derived. Simulations verify that the theoretically derived outage probabilities agree well with the numerical results. Furthermore, the proposed system is able to achieve a higher average sum rate than conventional single-user transmission schemes over non-interfering channels.

We study variable screening for high-frequency continuous-time regression, targeting the support of the spot regression coefficient. Under coordinatewise truncation and finite-variation jumps, uniform entrywise bounds for local correlation estimates yield the rate $\{(\log p)/n\}^{\alpha/(2\alpha+1)}$ under a local averaged $\alpha$-Hölder condition, up to confidence and grid factors. These bounds imply sure screening for the marginal support. To retain the regression support under local dependence, we propose partial-residual spot iterative screening with pathwise Lepski adaptation. Under sparse eigenvalues and standardized beta-min, the procedure has the sure screening property and requires only correlation columns associated with the current active set. A list-size minimax lower bound applies to both support targets and matches the localization and multiplicity order of the marginal-screening bound when $\log(p/d)\asymp\log p$. Simulations and a high-frequency factor application illustrate the procedures.

Loeb measure theory stands as one of the most influential concepts in nonstandard analysis, underpinning nearly all applications in probability, stochastic processes, and mathematical economics. The paper resolves a fundamental open problem in Loeb measure theory originally posed by Keisler and Sun: let $(\Omega,\mathcal{F},\mu)$ and $(\Omega,\mathcal{G},\nu)$ be two Loeb equivalent internal probability spaces, and $\mathcal H$ be the internal algebra generated from $\mathcal{F}\cup\mathcal{G}$. Does there exist an internal probability measure $P$ on $\mathcal H$ such that $(\Omega,\mathcal{H},P)$ is Loeb equivalent to $(\Omega,\mathcal{F},\mu)$? While arXiv:2112.13955 recently provided a positive answer for hyperfinite probability spaces, the problem remained open for general internal probability spaces. We establish the existence of such an internal probability measure for all internal probability spaces.

In this paper, we study isolated singularities of the following semilinear elliptic equation $-\Delta u+\frac12 x\cdot \nabla u+\frac{1}{q-1}u-u^q=0$ in $\Omega \setminus \{0\}$, where $n\ge 3$, $\Omega \subset \mathbb{R}^n$ is a domain, $0 \in \Omega$ and $q>1$. This equation arises in the study of blow-up profiles of semilinear heat equations. For $\frac{n}{n-2}< q < \frac{n+2}{n-2}$, we establish a complete classification of isolated singularities for nonnegative solutions and characterize the precise asymptotic behavior of singular solutions. Our results improve those of Guedda and Kirane (Trans. Amer. Math. Soc., 1995: 3595-3603), where analogous results were obtained only for radially symmetric positive solutions. In addition, we also derive the asymptotic behavior of solutions in the Serrin critical case $q=\frac{n}{n-2}$ and the supercritical case $q>\frac{n+2}{n-2}$.

The random-cluster model with cluster weight $0<q<1$ is expected to exhibit negative dependence, but without FKG even pairwise negative correlation remains open on general graphs. We study the model on the infinite $\Delta$-regular tree with wired boundary conditions. Write $\widehat p:=p/(p+q(1-p))$ and $p_{\mathsf c}:=q/(\Delta+q-2)$. Classical results identify the product wired state for $p\le p_{\mathsf c}$ and construct a percolative all-wired limit for $p>p_{\mathsf c}$. We prove that the supercritical wired DLR specification has a unique Gibbs measure, namely this all-wired limit. Consequently, the wired DLR phase diagram is complete: the unique measure is Bernoulli bond percolation with parameter $\widehat p$ for $p\le p_{\mathsf c}$, while it percolates for $p>p_{\mathsf c}$. We also establish negative dependence across wired branches. On a finite wired tree, the vector of branch-connectivity indicators satisfies conditional negative association under positive external fields (CNA+). Hence bounded increasing observables supported on disjoint collections of branches incident to a common vertex have nonpositive covariance. The same inequality holds in the unique infinite-volume wired measure for every $p$, with equality for $p\le p_{\mathsf c}$.

Differentiable simulation is a key component in learning, control, and inverse problems, where gradients through nonlinear implicit solvers are required. Existing approaches either rely on unrolled automatic differentiation, whose memory grows with solver depth, or on equation-level implicit differentiation, which assembles global Jacobians and solves large sparse adjoint systems, discarding the locality of the forward solver – and differentiating the converged equation rather than the finite computation that actually ran. We propose solver-level differentiation, which differentiates the executed solver itself. When a solver is composed of block implicit updates, its discrete adjoint is obtained by applying the corresponding adjoint updates in reverse order, yielding a reverse-sweep formulation whose backward pass mirrors the forward solver. From an operator perspective, the forward pass realizes an approximate inverse through ordered local solves, and the backward applies its transpose through reverse local adjoint solves, constructing no global system. We instantiate this idea on Vertex Block Descent, yielding a differentiable solver whose reverse colored Gauss-Seidel sweeps are composed entirely of local $3\times 3$ adjoint solves. The backward matches automatic differentiation through the identical executed forward to machine precision at every solver depth, where the equation-level adjoint is off by 37% after one sweep; in a controlled same-codebase, same-GPU comparison it is 33x faster and uses 71x less memory than unrolled automatic differentiation; and the same construction is exact on projective dynamics and extended position-based dynamics. We scale differentiable elastodynamics to $10^6$ contact-coupled soft bodies (8M vertices) on one GPU. Overall, this work highlights solver structure as a practical organizing principle for efficient differentiable simulation.

We present a unified proof and certificate package establishing seven exact finite Zarankiewicz values for the forbidden graph $K_{3,3}$: $z(12,18;3)=108$, $z(13,17;3)=110$, $z(13,18;3)=116$, $z(14,17;3)=118$, $z(14,18;3)=124$, $z(15,17;3)=126$, and $z(15,18;3)=132$. The witnesses form a connected family generated by an explicit $13 \times 18$ matrix with 116 ones. Deleting one row or one column and adding either of exactly two admissible weight-eight rows for this fixed labeled core produces the remaining witnesses. The exceptional upper-bound closure, $z(12,18;3)\leq108$, combines the published uniqueness of the extremal $12 \times 17$ graph with 103 edges and an exhaustive rejection of all $\binom{12}{6}=924$ possible degree-six column extensions. The supplement contains all witnesses, a portable verifier, a machine-readable report, and integrity hashes. Prior numerical ingredients and the role of the present work are separated cell by cell.

In this paper, we establish a Heintze–Karcher inequality for closed embedded hypersurfaces in a class of warped product manifolds endowed with radial density. As a consequence, we prove Alexandrov-type theorem for constant weighted mean curvature hypersurfaces in such spaces. In particular, we prove that a closed embedded $\lambda$-self-expander in the Euclidean space must be a round sphere centered at the origin.

Let $(\mathscr U,w)$ be a smooth separated Deligne–Mumford stack over $\mathbb{C}$. Assume that its coarse space is quasi-projective. We prove $E_1$-degeneration at every rational index for its irregular Hodge filtration. The result holds on every good stack compactification and every NC rational stack compactification. In particular, the filtration and its graded dimensions can be computed on any such compactification. The proof uses finite flat descent for Kontsevich complexes. It reduces the projective stack case to the theorem of Esnault–Sabbah–Yu for smooth projective varieties.

Researchers studying the inverse Galois problem realized 25 of the 26 sporadic finite simple groups as Galois groups over $\mathbb{Q}$ during 1984-1989. We complete this program by proving that the last remaining sporadic group, the Mathieu group $M_{23}$, occurs as a Galois group over $\mathbb{Q}$. In fact, we produce an explicit degree $23$ polynomial with rational coefficients whose splitting field has Galois group $M_{23}$ over $\mathbb{Q}$. To accomplish this, we discover an unanticipated splitting of the Nielsen class associated to a non-rigid triple of conjugacy classes of $M_{23}$. We compute the Belyi maps associated with this Nielsen class to construct an explicit regular Galois extension of $\mathbb{Q}(t)$ with Galois group $M_{23}$. Essential for our computation is the numerical Belyi map algorithm developed and implemented by Costa, Klug, Musty, Schiavone, Sijsling, and Voight, inspired by ideas of Hejhal and Stark. As a bonus, one specialization of the regular extension yields an $M_{22}$-extension of $\mathbb{Q}$.

Two identical scalar dynamical systems coupled through a scalar diffusion equation are studied herein, with respect to bifurcations from a symmetric steady-state to symmetric and asymmetric steady-states and to in-phase and anti-phase oscillations. Numerical continuations based on the developed theory show the shape of the bifurcation branches and the attracting nonlinear states far from bifurcation onset and confirm their, for the most part, derived stability. This study extends the work on the dynamical properties of a single scalar dynamical system coupled to its own delay through an adjacent diffusion field and quantifies further the delay of diffusive information transmission between two such Wentzell boundaries. The mathematical system is motivated by modeling biological membranes with yet unknown effective local fluxes that are coupled to bulk diffusion.

We prove that the Feldman–Ilmanen–Knopf (FIK) blowdown singularity has a nonempty open nonlinear formation basin in the full space of smooth Riemannian metrics and, on a smaller neighborhood, local marked first-order asymptotic moduli. Given a closed connected oriented Riemannian four-manifold, an implantation point, and a positive time bound, its oriented blow-up admits a relatively $C^{2,\alpha}$-open set of smooth metrics whose Ricci flows form, before that time, a localized FIK singularity with global Type-I curvature control. No symmetry, Kähler condition, or finite-dimensional tuning is imposed. Curvature stays uniformly bounded outside the implantation region; fixed-convention marked rescalings converge along the full singular-time sequence to the ancient FIK flow; and the exceptional sphere has sharp FIK asymptotics for area, intrinsic diameter, and curvature. To our knowledge, this is the first open full-metric formation theorem for a finite-time singularity modeled on a noncompact, noncylindrical shrinker. A self-contained spectral certification identifies the weighted FIK operator's nine-dimensional nonnegative space with the scaling and diffeomorphism directions, and exact modulation removes this block. On a smaller little-Hölder $h^{2,\alpha}$ neighborhood, a fixed positive-time restart yields the marked first-profile coordinate $\mathfrak{A}_1=\lambda_\infty^{-\gamma_1}V_\infty\in E_1$. This amplitude is a split $C^1$ submersion and locally the projection onto $E_1$. Equality of amplitudes characterizes marked first-order agreement; a transverse disk uniquely realizes each sufficiently small amplitude; normalized two-flow differences converge to the corresponding Jacobi field; and the amplitude determines the first quadratic scale and phase response. This gives a complete local marked first-order profile coordinate for FIK blowdown.

In this paper, we consider a random geometric graph (RGG)~\(G\) formed by~\(n\) vertices distributed uniformly in the unit square~\(S\) on the plane and equip each edge of~\(G\) with an independent weight. We assume that the adjacency distance between vertices is larger than the connectivity threshold and estimate the maximum weight of a path connecting two fixed points~\(O_1\) and~\(O_2\) in~\(S,\) using edges of~\(G\) that satisfy length and weight constraints. Our strategy is to first divide the space between~\(O_1\) and~\(O_2\) into small squares and identify nice squares containing heavy edges. We then use an iterative stitching procedure to connect these heavy edges and estimate the weight of the resulting path~\(P.\) Finally, we invoke a multi-level scaling procedure along with weight segmentation to establish an upper bound for the maximum weight of any path between~\(O_1\) and~\(O_2\) and thereby demonstrate the near-optimality of~\(P.\) We also illustrate our results using examples involving edge weights satisfying power law and exponential decay.

Data Envelopment Analysis (DEA) is a widely used non-parametric method for evaluating the relative efficiency of decision-making units (DMUs). However, the classic Charnes-Cooper-Rhodes (CCR) model often suffers from a lack of discriminatory power, particularly when the number of input and output variables is large relative to the number of DMUs. Under such conditions, many DMUs are evaluated as "CCR-efficient" with a score of unity, making it difficult to rank or distinguish between top performers. To address this limitation, we propose a novel Maximum-Margin DEA (MM-DEA) model that integrates the concept of "margin maximization" – inspired by structural risk minimization in machine learning – directly into the efficiency measurement framework. Like the traditional super-efficiency model, the proposed MM-DEA model excludes the target DMU from the reference set used to evaluate it; unlike the super-efficiency model, however, MM-DEA imposes an explicit upper bound on the target DMU's own score, ensuring that every score remains within the standard $[0,1]$ range rather than potentially exceeding one. We theoretically demonstrate that the MM-DEA model is a generalized extension of the CCR model, reducing to the latter when the trade-off parameter $\alpha$ is zero. Furthermore, we show that the fractional programming formulation of MM-DEA can be rigorously transformed into a linear programming (LP) problem, ensuring computational efficiency. In our numerical experiments, MM-DEA produced a unique ranking for a high-dimensional synthetic dataset on which the CCR model failed to discriminate among any of the DMUs. Our results suggest that the "margin" serves as a critical metric for managerial resilience and competitive advantage.

In this paper, we study maximum weight spanning trees of the random geometric graph (RGG)~\(G\) formed by~\(n\) vertices where each edge is independently either open or closed with a certain probability and is also equipped with an independent random positive weight. We use segmentation and iterative path construction to obtain deviation bounds for the order of growth of the maximum weight of a spanning tree in terms of an inverse of the edge weight complementary cumulative distribution function (ccdf) and also illustrate our results for the special cases of power law and exponential decay. We then use martingale difference methods to individually estimate the variance contribution due to randomness in vertex locations and edge states/weights and determine sufficient conditions for~\(L^2-\)convergence of the maximum weight, appropriately scaled and centred.

In this article, we establish elliptic gradient estimates for positive solutions of the Lane-Emden system on metric measure spaces satisfying the synthetic Ricci curvature-dimension condition $\mathrm{RCD}^*(-K,N)$. Our approach combines the weak differential calculus and the Bochner inequality available in the $\mathrm{RCD}^*(-K,N)$ setting with suitable auxiliary function arguments, extending classical gradient estimate techniques to nonsmooth spaces. As an application, we prove a Liouville-type theorem for positive solutions under appropriate geometric assumptions. This result helps us to identify constraints for which constant solutions exist. We also mention some cases where nonconstant solutions may exist in sequel. These results generalize corresponding results from the smooth Riemannian setting and contribute to the study of nonlinear elliptic systems on spaces with synthetic Ricci curvature lower bounds.

Andrews and Dastidar (2026) introduced $SOME(n)$ and $DSOME(n)$ functions related to partitions of a positive integer $n$, where $SOME(n)$ is the sum of all the odd parts in the partitions of $n$ minus the sum of all the even parts and $DSOME(n)$ is the sum of all the odd parts in the partitions of $n$ into distinct parts minusthe sum of all the even parts in the same partitions. The purpose of this paper is to establish the conjecture$SOME(\lambda)\equiv0\pmod{5^\alpha}$, $\alpha\ge 1$ and $\lambda\ge0$ are integers such that $24\lambda\equiv1\pmod{5^\alpha}$ due to Andrews and Dastidar, and the conjecture $DSOME(50n+21)\equiv0\pmod{8}$ due to Baruah and Gogoi (2026). In the process, we establish some new infinite families of congruences modulo 2, 4, and 8 for $DSOME(n)$.