Papers

Very recently (2026), Bauschke and Tung extended from finite- to infinite-dimensional Hilbert spaces a result published by Meshulam in 1996 (following an earlier result of Aharoni-Duchet-Wajnryb from 1984) regarding the boundedness of infinite products of relaxed projections onto a finite family of closed affine subspaces. In the present note we extend in various ways the result of Bauschke and Tung by allowing certain perturbations and proving perturbation resilience, by considering a mixture of closed half-spaces and closed hyperplanes, and by using dynamic weighted sums of dynamic strings (of dynamic lengths) of relaxed projections in the iterative process. We also discuss the limitation to generalize the Bauschke-Tung result to arbitrary closed and convex sets by presenting a large family of counterexamples in which the associated control is not cyclic and not even almost cyclic. Along the way we establish a general theorem of independent interest regarding the uniform boundedness and uniform unboundedness of infinite products of nonexpansive mappings in a normed space setting.

de Brecht, Goubault-Larrecq, Jia and Lyu asked whether every sober convergence Choquet-complete space is domain-complete. We introduce the notion of singleton Choquet-completeness, a weakening of convergence Choquet-completeness in which the open sets chosen by player $\alpha$ are required to have a singleton intersection, but not necessarily to form a neighbourhood basis. We prove that every singleton Choquet-complete $T_1$ space is domain-representable. Consequently, every convergence Choquet-complete $T_1$ space is domain-representable and hence sober. Thus, in the $T_1$ case, the sobriety assumption in the above question is redundant, and the question reduces to whether every convergence Choquet-complete $T_1$ space is domain-complete.

This paper proposes a novel physics-guided spectral deep operator network, termed SpectONet, for solving Euler-Bernoulli beam (EBB) vibration problems. The proposed framework integrates the operator-learning capability of DeepONet with physics-informed constraints and Chebyshev-Gauss-Lobatto (CGL) sensor placement. Unlike conventional DeepONet frameworks, which commonly employ uniformly distributed sensors, SpectONet uses nonuniform spectral sensor locations with a higher concentration of points near the domain boundaries. This sampling strategy improves the finite-dimensional representation of boundary-sensitive structural responses while requiring only a limited number of branch-network inputs. The governing beam equation, together with the associated initial and boundary conditions, is incorporated into the training objective to promote physically consistent and generalizable predictions. Numerical experiments on three EBB vibration problems demonstrate the effectiveness of the proposed framework. Comparisons with strong baseline models such as Vanilla DeepONet, PI-DeepONet, PINN, and CNN-UNet show that SpectONet consistently achieves lower prediction errors across all considered evaluation metrics. In particular, SpectONet achieves an average reduction of approximately 79% in the mean relative $\mathcal{L}_2$ error compared with the best-performing baseline across the three benchmark problems. These results demonstrate the effectiveness of SpectONet as an accurate physics-guided operator-learning framework for structural vibration analysis.

We study stochastic composite nonconvex optimization over a compact convex set when gradient samples arrive along a single trajectory of a fixed ergodic Markov chain. Existing single-trajectory variance-reduction theory covers smooth unconstrained objectives; we address the projection-free composite setting using the generalized Frank-Wolfe gap. We propose MC-ALFCG, which combines a momentum conditional-gradient method with coupled capped multilevel Monte Carlo estimation and per-iteration clipping. The deepest nested average uses consecutive states from the same trajectory, yielding conditional bias $O(\tau_{\mathrm{mix}}/T)$ uniformly over the starting state, while coupling controls the gradient-difference second moment through the iterate displacement. Clipping enforces the pathwise bounds needed by the adaptive analysis. We reduce the Markovian recursion to its independent-sampling counterpart under $\sigma^2\mapsto 2\Lambda G_\sigma^2$ and $L^2\mapsto 2\Lambda L^2$, where $\Lambda=O(\tau_{\mathrm{mix}}\log T)$. For positive centered noise, the tuned method achieves expected sample complexity $\widetilde{O}((\tau_{\mathrm{mix}}^2G_\sigma+\tau_{\mathrm{mix}}^{5/2}G_\sigma^2)\varepsilon^{-3}+\tau_{\mathrm{mix}}^5\varepsilon^{-2})$. The exactly noiseless specialization achieves $\widetilde{O}(\varepsilon^{-2})$ with mixing-time-free constants, while a mixing-time-oblivious variant achieves $\widetilde{O}(\tau_{\mathrm{mix}}^6\varepsilon^{-3}+\tau_{\mathrm{mix}}^3\varepsilon^{-2})$. All guarantees are in expectation under a fixed transition kernel. Controlled numerical studies examine dependence sensitivity, a nonconvex composite instance, and clipping behavior.

We obtain orbital counting results for relatively Anosov groups with respect to linear functionals with finite critical exponent. Our counting results follow from an equidistribution result and rely crucially on previous equidistribution results obtained in our proof of counting results for periods. Our results generalize earlier work of Sambarino in the setting of Anosov groups.

We study contact surgery numbers for contact structures on the 3-torus. We show that all contact structures obtained via contact surgery along a Legendrian Borromean ring are overtwisted, and that this construction yields infinitely many pairwise non-contactomorphic contact structures on $\mathbb{T}^3$. We prove an obstruction on the maximal Thurston-Bennequin invariant of 3-component links to produce $\mathbb{T}^3$ by Dehn surgery. We also compare Legendrian surgery presentations of the Stein-fillable contact structure on $\mathbb{T}^3$. In this setting, we exhibit a non-destabilizable Legendrian knot with no graded normal ruling whose associated Stein filling has nonzero full symplectic homology.

We introduce the radial part of the Laplace-Beltrami operator on partial flag manifolds and study its eigenfunctions, the elementary zonal spherical functions. These eigenfunctions can be related to orthogonal polynomials on the simplex of Hermitian matrices invariant under simultaneous conjugation by unitary matrices with respect to the complex matrix-variate Dirichlet distribution. Using a matrix-variate version of Koornwinder's method for constructing orthogonal polynomials in multiple variables from orthogonal polynomials in one variable, we construct a family of pairwise orthogonal polynomials in terms of Hermitian Jacobi polynomials and conjecture that some of these are elementary zonal spherical functions. Furthermore, we recall the definition of multivariate Schur polynomials and show how they can be used to obtain information about elementary zonal spherical functions. In particular, we give some partial results in the direction that the radial part of the Laplace-Beltrami operator on partial flag manifolds is strongly triangular with respect to the multivariate Schur polynomials. The elementary zonal spherical functions can then be constructed by applying the Gram-Schmidt orthogonalisation process to the multivariate Schur functions.

In this paper, we study the following critical Hartree problem \begin{equation}\label{equationabstract} \begin{cases} \displaystyle-\Delta u+(Q+\varepsilon V)u= A_{3,\mu} \left(\int_{\Omega}\frac{u^{6-\mu}(y)}{|x-y|^{\mu}}dy\right) u^{5-\mu} &\mathrm{~in~}\Omega,\\ \displaystyle u>0&\mathrm{~in~}\Omega,\\u=0&\mathrm{~on~}\partial\Omega,\end{cases} \end{equation} where $\Omega\subset\mathbb{R}^3$ is a bounded open set, $0<\mu<2$, the exponent $6-\mu$ is the upper critical exponent in the sense of the Hardy–Littlewood–Sobolev inequality and $A_{3,\mu}>0$ is a normalization constant. The function $Q$ is assumed to be critical in the sense of Hebey and Vaugon, and the solutions $u_{\varepsilon}$ of \eqref{equationabstract} are assumed to be an optimizing sequence for the Hardy–Littlewood–Sobolev inequality. Under a natural nondegeneracy assumption, we derive a precise asymptotic expansion of \(u_{\varepsilon}\), determine the exact blow-up rate, and identify the concentration point. We also obtain the pointwise blow-up behavior both near and away from the concentration point.

We consider the intersection $\mathcal{Q}$ of $m$ equal-radius balls in $\mathbb{R}^n$ whose centers lie on a common sphere centered at the origin, with the origin contained in the convex hull of the centers. Assuming $\mathcal{Q}$ possesses a unique vertex $x^{\star}$ lying on the hyperplane $1_{n \times 1}^T \cdot x=0$, under a combinatorial-stability assumption, we give a polynomial-time algorithm that recovers $x^\star$. The algorithm constructs a one-parameter family of auxiliary polytopes (max-indicator polytopes) whose vertices move radially under a controlled translation of a quadratic center. The distinguished vertex remains fixed, while all others travel along fixed rays. By comparing a pair of symmetrically perturbed polytopes with the unperturbed one and solving a linear number of linear programs, the algorithm isolates $x^\star$.

For all $\delta>0$ and infinitely many $n \in \mathbb N$, we show that there exists a set $L$ of $n$ lines in $\mathbb R^2$ such that there are no intersecting quadruples, but for every subset $L' \subset L$ such that $|L'| \geq n^{\frac{4}{5}+\delta}$, there exist three lines from $L'$ with a common point of intersection. This gives an improved bound for a dual form of a theorem of Balogh and Solymosi. As a consequence, we derive an improved lower bound for the Hadwiger-Debrunner number $HD_2(p,3)$. We also give, for all $0 \leq s \leq 1$ and arbitrarily large $n \in \mathbb N$, a construction of a point set $S \subset [n]^3$ with cardinality $|S|\geq n^{3-s}$, such that $S$ contains $O(n^{6-4s})$ collinear triples. This shows that a supersaturation lemma of Balogh and Solymosi is optimal, up to logarithmic factors.

We show that abelian categories can be characterized as structures consisting of a colax algebra and a lax algebra connected by a lax mixed rewriting rule. To this end we develop a theory of lax rewriting rules for pairs of lax-lax and colax-lax algebras over 2-monads.

The concentration–compactness channels below the ground-state threshold are investigated for the three-dimensional nonradial focusing energy-critical nonlinear Schrödinger equation. After one-sided normalization, a minimal critical element falls into four classes: the finite-time, rapid-cascade, bounded-scale finite-mass, and residual quasi-soliton channels. The first three classes are rigorously excluded. The main result shows, without radial symmetry, zero momentum, or a slow spatial center, that every finite-mass bounded-scale almost-periodic solution is identically zero. Consequently, any minimal counterexample to below-threshold scattering must lie in the residual quasi-soliton channel; if its scale is bounded, then it has infinite mass at every time.

Numerical Ricci-flat metrics on Calabi-Yau manifolds are becoming increasingly accurate. However, they often lack the interpretability required to extract theoretical insights. In this paper, we introduce a novel variant of Donaldson's algorithm based on the Moore-Penrose pseudo-inverse that operates on the global sections of the ambient space rather than the manifold itself. This approach allows us to use the canonical monomial basis to compute interpretable balanced metrics even at large degrees $k$. Applying our ambient algorithm to multiple families, including the Dwork family and complete intersection Calabi-Yau manifolds, we discover that the metric parameters obey novel power laws near the Large Complex Structure Limit (LCSL). We connect these to the Gromov-Hausdorff metric collapse predicted by the SYZ conjecture.

Let $\ell$ be a line not passing through any vertex of a triangle $ABC$ and not parallel to any side. Line $\ell$ meets the sidelines $BC$, $CA$, $AB$ of $\triangle ABC$ at points $D$, $E$, $F$, respectively. We consider three of the triangles that are formed: $\triangle AEF$, $\triangle BFD$, and $\triangle CDE$. Placing a fixed triangle center (such as the incenter, centroid, or orthocenter) in each of these three triangles determines a central triangle. We investigate when the reference triangle and its central triangle are perspective, i.e., when the lines $AD$, $BE$, and $CF$ are concurrent. A computer search over the first 1000 centers in the Encyclopedia of Triangle Centers suggested numerous examples of concurrence. We give elementary geometric proofs for the circumcenter, orthocenter, and Clawson point, develop a general criterion for concurrence, and identify several operations, including isogonal and isotomic conjugation, that preserve this property. Our main result is a complete characterization of the center functions whose associated cevians are concurrent for every transversal $\ell$. This yields an explicit normal form for such centers. We also show that concurrence depends only on the direction of the transversal, and we investigate the special case in which the transversal is parallel to the Euler line.

We study finitely generated groups of bounded type arising from tile inflation processes over Bratteli diagrams and containing the alternating group of the associated tail groupoid. Under a localization condition on finitely many singular germs, we show that the topological full group is obtained by adjoining finitely many finitary transformations. The number of adjoined transformations can be taken to be the dimension of a quotient of the mod-$2$ dimension group, and equality with the topological full group is characterized by surjectivity of the parity map restricted to the finitary elements of the original group. Under an additional parity condition on the AF truncations arising in the localization, the original group is near full, and its index is the order of the same parity quotient. We give a criterion for localization in terms of finitely many representatives of singular germs preserving arbitrarily deep cylinders, and an odd-order criterion which implies the parity condition. As an application, we study a fragmentation group of the modified LMS-group associated with the Penrose tiling and prove that it coincides with its topological full group.

We propose Dynamic Constraint Learning (DCL), a data-driven framework for constrained optimization when constraint functions are unknown and cannot be queried during optimization. At each iteration, the method learns a local surrogate from nearby data and solves a subproblem within a data-supported trust region. Compared with offline global constraint learning, the approach uses local surrogates that adapt to the data distribution during optimization and can achieve solution quality comparable to that of global models while using simpler local models and smaller optimization subproblems. We demonstrate the performance of DCL on a synthetic test problem and two case studies from the literature.

This work addresses the problem of locating Road Side Units (RSUs) in vehicular networks with the aim of improving overall traffic efficiency. We propose a heuristic approach for determining the placement of RSUs on a road network, while vehicle dynamics are modeled through a microscopic follow-the-leader interaction in which drivers continuously adapt their speed according to the distance from the vehicle ahead. The RSUs share the collected information among themselves to cooperatively estimate the network state, which is then communicated to vehicles, allowing them to compute their fastest paths. We also assume that not all drivers comply with the provided recommendations, meaning that only a fraction of users follow the suggested routes. Since the aim is to improve network performance while minimizing the number of deployed units, the methodology adopts a nested optimization scheme: an outer module explores the optimal number k of RSUs, while an inner module identifies the best placement configuration for each k. To evaluate the effectiveness and the scalability of the proposed method, we tested two networks with different topologies and sizes. The results show a decrease in Total Travel Time (TTT) compared to the baseline scenario (i.e., without RSUs) of up to 36% on the smaller network and up to 20% on the larger one. Stochastic vehicle origin-destination pairs have also been considered.

A poset model of a topological space $X$ is a poset $P$ such that $X$ is homeomorphic to the maximal point space of $P$ (the set Max($P$) of all maximal points of $P$ equipped with the relative Scott topology of $P$). Xi and Zhao proved that if a space has a dcpo model satisfying Lawson condition, it must be coherent and well-filtered. It is still open wether every coherent and well filtered space $T_1$ has a dcpo model satisfying Lawson condition. In this paper, we answer this problem. In another paper, Xi and Zhao proved that every Hausdorff k-space has a bounded complete dcpo model. It is, however, still unknown whether it is true that if a Hausdorff space is a k-space if it has a bounded complete dcpo model. We will construct a Hausdorff space which is not a k-space but has a bounded complete dcpo model.

We study sequential prediction of individual binary sequences under zero-one loss. No computable master can compete on every sequence with all total computable predictors. We therefore consider rational-valued primitive-recursive forecasters, a broad syntactically enumerable class containing finite-state, context-based, and Prediction by Partial Matching (PPM) rules. We construct a computable probabilistic predictor with an explicit sublinear regret bound relative to every primitive-recursive forecaster on every individual sequence. We further prove that the PPM predictor is primitive recursive. Consequently, our predictor attains the infinite-past Bayes error on every Martin-Löf random realization of every computable stationary ergodic binary source. This optimality extends to finitely many independent such sources interleaved according to an arbitrary primitive-recursive schedule. Finally, we establish strict separations from finite-state prediction and from every fixed primitive-recursive predictor. keywords: Universal prediction, individual sequences, prediction with expert advice, primitive recursive functions, Kolmogorov complexity, PPM.

We determine exactly which proper multiplicative subgroups of a prime field can be represented as a restricted sumset of the form $A\mathbin{\widehat{+}} A=\{a+a':a,a'\in A,\ a\ne a'\}$. We prove that a proper multiplicative subgroup $H\le\mathbb F_p^*$ cannot satisfy $H=A\mathbin{\widehat{+}} A$ whenever $|H|\ge7$, and that this threshold is sharp. In fact, such a decomposition exists precisely when $|H|\in\{1,3,6\}$, and we classify all decompositions in these exceptional cases. This gives a sharp, complete resolution of the restricted-sumset analogue of the generalized Sárközy conjecture over prime fields. This significantly extends and refines previous results of Shkredov and Yip.

We study the reverse Cheeger inequality, which bounds from above the ratio of the first Dirichlet Laplacian eigenvalue to the square of the Cheeger constant. We first extend this inequality from convex sets to a broader class. We then analyze maximizing sequences among convex bodies. To quantify domain collapse, we introduce the notion of principal widths. For three-dimensional convex bodies, we prove that if the ratio of the first principal width to the second principal width vanishes along a sequence of convex bodies, then such sequence is maximizing. Finally, in arbitrary dimensions, we prove that the same property holds for the class of rhomboid-like sets.

To better understand the landscape of well-behavedness laws for lenses on strict Cartesian symmetric monoidal categories, we define tagged lenses. We prove that tagged lenses form a strict symmetric monoidal category equipped with a strict symmetric monoidal functor into the category of putget lenses. We then observe compositional entailment for the other two lens laws, identifying change-dependence and first-last dependence of tags as sufficient conditions for getput and putput, respectively.

Orthogonal ring patterns are natural generalizations of circle patterns. Bobenko-Hoffmann-Rörig and Bobenko established the variational principles of the classical combinatorial curvature for the Euclidean, hyperbolic and spherical orthogonal ring patterns. Bobenko-Hoffmann-Rörig's work and Bobenko's work imply the rigidity of Euclidean and hyperbolic orthogonal ring patterns on closed surfaces, while the rigidity of spherical orthogonal ring patterns on closed surfaces is not known. In this paper, we study the spherical orthogonal ring patterns on closed surfaces with cellular decompositions satisfying certain necessary conditions. Using a modification of the combinatorial total geodesic curvature introduced by Nie in \cite{Nie}, we prove the rigidity of spherical orthogonal ring patterns on closed surfaces by variational principles.

The optimal design of equilibrium particle distributions generated by external electric fields is investigated in the framework of stationary Maxwellian equilibria of the Vlasov-Poisson system. The resulting normalized Poisson-Boltzmann equation is analyzed, and existence and uniqueness of the self-consistent equilibrium potential are established by two complementary approaches. A Schauder fixed-point argument based on uniform estimates for the normalized nonlinear source and a variational approach based on a strictly convex free-energy functional are developed. Building upon these analytical results, a unified optimal design framework is formulated for equilibrium densities. Valley-potential, quadratic tracking, and Kullback-Leibler design criteria are treated within the same optimization framework. First-order optimality conditions are derived, and a projected nonlinear conjugate-gradient algorithm is proposed for the numerical solution of the resulting optimization problems.

This paper addresses statistical inference for stochastic partial differential equations. We study the stochastic Allen-Cahn equation driven by space-time white noise and analyze its mild solution. Our main focus is the asymptotic behavior of the spatial quadratic variation of the solution, for which we establish the exact limiting value. As an application, we develop parameter estimation procedures based on these asymptotic results. We prove that the unique solution can be decomposed as u = X + Y where X denotes the solution of the linear stochastic heat equation and Y accounts for the nonlinear effects. Exploiting a detailed analysis of the heat kernel and its scaling behavior, we derive Hölder continuity properties of Y in both spatial and temporal variables, showing that Y exhibits substantially higher regularity than X. This decomposition and the resulting regularity estimates are key ingredients in the development of parameter estimation procedures based on the asymptotic behavior of quadratic variations of the solution.

We study the combinatorial Yamabe flow on infinitely triangulated surfaces with piecewise hyperbolic metrics. Under the assumptions of uniformly bounded vertex degree and an $\epsilon$-uniformly nondegenerate initial metric, we establish the short-time existence and uniqueness of smooth solutions to the combinatorial Yamabe flow. To address the potential degeneration of triangles along the evolution, we introduce an extended flow with generalized curvature, and establish the global existence of solutions to the extended flow.Furthermore, under an integrability condition, we establish the uniqueness of solutions to this extended flow, which follows from the stability property of the solutions. These results provide a local well-posedness theory for the hyperbolic combinatorial Yamabe flow on infinitely triangulated surfaces.

Semidefinite relaxations of binary optimization problems often fail Slater's condition. Full facial reduction restores strict feasibility but may require auxiliary problems as costly as the original relaxation. We analyze partial facial reduction based on tractable inner approximations of the positive semidefinite cone: the nonnegative diagonal, diagonally dominant (DD), and scaled diagonally dominant (SDD) cones. For this class of SDP relaxations, we prove that every terminal face identified by SDD-partial FR is described by variables fixed to zero or one and groups of variables forced to be equal. The face admits a full-column-rank \(\{0,1\}\) basis matrix with disjoint column supports, yielding a natural reduced formulation that can preserve sparsity. For polyhedral inner approximations, we characterize the face exposed at each step through the inequalities active on the associated outer relaxation. We also establish sharp limitations. We construct a family with singularity degree two for which DD- and SDD-partial FR require the maximum possible number of steps. We further show that factor-width restrictions can prevent detection of dense valid affine equalities. These results characterize both the structure recovered by tractable partial FR and the cost of restricting the exposing-matrix search.

We prove that the resonances for a generic smooth expanding map are simple and that zero is a regular value for the associated resonant states. Moreover, the real-valued resonant states are Morse functions. Using Nash-Moser theory, we also prove that the resonant states for a generic smooth expanding map of large enough degree (depending on the dimension of the manifold the map is acting on) have the same properties as generic smooth functions.

The probability-flow ordinary differential equation (PF-ODE) associated with a diffusion process is widely used in score-based generative modeling as a deterministic sampler that reproduces the marginal distributions of the diffusion. The validity of this marginal-matching property depends on the well-posedness of an ordinary differential equation whose velocity field is constructed from the score function of the diffusion. We examine the precise mathematical relation between a diffusion process, the Fokker-Planck equation and the associated PF-ODE under weak regularity assumptions on the drift and score. We establish existence and uniqueness of the marginal density flow as a solution of the Fokker–Planck equation under minimal regularity assumptions. We then study the corresponding Lagrangian problem using the DiPerna-Lions-Ambrosio theory of regular Lagrangian flows. We prove existence, uniqueness and stability of the flow, and show that it transports the initial distribution onto the diffusion marginals, under Sobolev or bounded-variation regularity of the score together with one-sided bounds on the divergence of the probability-flow velocity. We identify sufficient conditions for the required regularity in diffusion models relevant for applications. Our analysis underlines a fundamental distinction between Eulerian and Lagrangian descriptions. We construct a counterexample in which the Fokker–Planck equation has a unique density flow while the associated PF-ODE fails to admit a regular Lagrangian flow from the initial time, demonstrating that uniqueness of the density evolution does not in general imply the existence of a deterministic probability-flow representation. Finally, we derive stability estimates for probability-flow trajectories under learned score approximations. Our findings have implications for the training and deployment of score-based diffusion models.

Characterizing when the symbolic and ordinary powers of an ideal coincide is a central problem in commutative algebra, and ideals satisfying this property are called Simis ideals. In this article, we investigate the Simis property of monomial ideals by studying the recent conjecture of Mendez, Pinto, and Villarreal on monomial ideals with minimal irreducible decomposition. Let $I$ be a monomial ideal, and let $\mathcal{F}(I)$ denote the collection of supports of the minimal generators of $I$. Assuming that $\mathcal{F}(I)=\mathcal{F}(\sqrt{I})$, we prove that if $I$ admits more than one minimal generator with a fixed support, then it is not Simis. Using this reduction, we establish the Mendez-Pinto-Villarreal conjecture for two broad classes of monomial ideals, namely support-$3$ monomial ideals and monomial ideals whose associated simplicial complexes are simplicial forests. Finally, we study the Cohen-Macaulay property of monomial ideals whose associated simplicial complexes are grafted and satisfy $\mathcal{F}(I)=\mathcal{F}(\sqrt{I})$.