Papers

This study proposes a metagraph-based domain-decomposed Galerkin reduced-order model (MBDD-G-ROM) for distributed-memory parallel reduced-order analysis of large-scale problems. The method represents domain-decomposed Galerkin reduced-order models over arbitrary domain decompositions using two graph levels: calculation-point graphs for interactions among discretization points and metagraphs for connectivity among local approximation-space subdomains. In the proper orthogonal decomposition (POD)-based implementation, POD computation subdomains are represented as metanodes, while metaedges encode the block-sparsity induced by overlaps between local POD basis supports. Partitioning the metagraph enables the POD computation subdomains to be decoupled from the parallel computation subdomains, allowing distributed-memory parallelization of both offline and online phases, including reduced-system assembly and iterative linear solution, without requiring the two decompositions to coincide. The metagraph also supports static load balancing through metanode weights that approximate computational costs. The method is evaluated for an unsteady diffusion equation and incompressible Navier-Stokes flow around a three-dimensional cylinder. The results show that MBDD-G-ROM preserves reduced-order solution accuracy while achieving high online parallel efficiency. A load-balancing test further demonstrates that cost-based metanode weights can improve computational efficiency.

We give a self-contained account of a version of Renault's disintegration theorem for (twisted) C$^*$-algebras of not necessarily Hausdorff étale groupoids that can be covered by countably many open bisections. As an application we discuss the $I$-norm, crossed product decompositions by inverse semigroup actions, and KMS-states and weights for general étale groupoid C$^*$-algebras.

We present a unified finite element method for the dynamics of fluidic biomembranes. The model is governed by the Navier–Stokes equations in the bulk coupled to the surface Navier–Stokes equations on the evolving biomembrane surface, with bending forces arising from the Willmore energy. By allowing the bulk mesh velocity to be independent of the fluid velocity and permitting a free tangential surface velocity, we are able to derive a unified weak formulation of the coupled bulk-surface Navier–Stokes system. To address the bending force, we consider an evolution equation for the curvature and propose a surface arbitrary Lagrangian–Eulerian (ALE) weak formulation. Discretization with either fitted or unfitted finite elements leads to well-posed fully discrete linear schemes that are unconditionally energy stable. We present a variety of numerical examples to demonstrate the favourable properties of the proposed methods.

We shall discuss the characterization of cyclicity of polynomials in Dirichlet-type spaces in the Euclidean unit ball of $\mathbb{C}^n$ which remains an open problem.

Empirical risk minimization is a standard and effective paradigm for learning predictive models by minimizing average loss. In high-stakes decision-making, however, an average-loss criterion may underrepresent rare but severe losses. Spectral risk measures (SRMs) provide a principled framework by incorporating weighted order statistics of losses, but the induced nonsmoothness and nonseparability from sorting make the resulting optimization problems challenging. We propose a relative inexact proximal augmented Lagrangian method with a semismooth Newton subproblem solver for solving SRM-based optimization problems. Exploiting a dual reformulation and properties of the Moreau envelope, we reduce the subproblems to structured dual-variable formulations, significantly simplifying computation. We provide explicit generalized Jacobian characterizations and tailor the pool adjacent violators algorithm for their efficient evaluation. Numerical results on synthetic and real-data instances show that the proposed method attains lower running times than the tested ADMM baseline while producing comparable stationarity residuals and sparse solutions.

We establish a connection between Minkowski billiards and symplectic billiards, two classes of dynamical systems that have been studied largely independently. We show that the Minkowski billiard map can be described in symplectic terms via reduction from the canonical symplectic structure on $V \times V^*$, and that symplectic billiards can be viewed as a “square root” of a symplectic version of Minkowski billiards. As an application, we recover several known results on symplectic billiards from the more general Minkowski setting, and extend some of them to higher dimensions and to periodic orbits of even period. In particular, we prove the existence of at least $(r-1)(n-1)$ $2r$-periodic symplectic billiard orbits in dimension $2n$.

In this paper, we introduce the Surplus Parking Gathering Problem ($\mathcal{SPG}$), a new coordination problem for robots deployed on an infinite grid. The input consists of a set of designated parking nodes, each associated with a prescribed capacity, while the total number of robots exceeds the total parking capacity. The objective is to saturate every parking node exactly according to its capacity while gathering all remaining surplus robots at a common grid node that is not specified a priori. The robots are assumed to be autonomous, anonymous, oblivious, identical, disoriented, and homogeneous. We consider the asynchronous (async) model with global visibility and global strong multiplicity detection. We first establish necessary conditions for the solvability of $\mathcal{SPG}$ by characterizing the initial configurations that admit no deterministic distributed algorithm. For all the remaining solvable configurations, we present a deterministic distributed algorithm that correctly solves the problem. The proposed algorithm proceeds in several phases and avoids collisions throughout its execution. We prove that the algorithm terminates in finite time and, upon termination, every parking node is saturated according to its prescribed capacity while all surplus robots are gathered at a uniquely determined gathering node. We further analyze the move complexity of the proposed algorithm, obtaining an $O(n(a+b)+n^2)$ upper bound together with an $\Omega(n(a+b))$ worst-case lower bound for the $\mathcal{SPG}$ problem.

We present the full-detailed construction of new geometric quotient structures for affine real algebraic varieties equipped with a linearizable action of a compact Lie group G. These quotients are functors from the category of arc-symmetric/AS-sets equipped with a linearizable action of G and equivariant continuous maps with AS-graph to the category of AS-sets. We furthermore provide a complete review of the results employed in the constructions, including properties of equivariant real algebraic geometry with respect to polynomial group actions, as well as an introduction to semialgebraic arc-symmetric sets and AS-sets of compact affine Nash manifolds.

We study the mixing properties of discrete-time and continuous-time dissipative dynamical systems driven by bounded mixing random forces. The continuous-time systems are reduced to discrete-time random dynamical systems generated by time-one maps, so that the main analysis is carried out in the discrete setting. We introduce a class of mixing random forcings whose regular conditional distributions with respect to the past satisfy natural regularity, recurrence, and non-degeneracy assumptions, extending the framework previously developed for more restrictive classes of processes in a paper by Kuksin-Shirikyan in GAFA (2025). Under a linearised controllability assumptions on the system, we prove exponential mixing in the total variation metric for finite-dimensional phase spaces. We then establish an infinite-dimensional counterpart yielding exponential mixing in the dual-Lipschitz metric under suitable amendments of restrictions on the system and the random forcing. Our approach is based on lifting the dynamics to an appropriate Markov process on an infinite-dimensional history space and applying a Doeblin coupling argument through the method of Kantorovich functional. As applications, we derive exponential mixing for a broad class of ordinary differential equations driven by mixing random processes with bounded continuous trajectories. As an application of our result to PDEs we discuss the randomly perturbed primitive equations of atmospheric dynamics.

We investigate the equality case of the spin positive mass theorem with charge in the Riemannian setting. This leads naturally to the notion of charged parallel spinor, which plays a central role in the analysis of extremal charged manifolds. As an application, we characterize the equality case of the mass–charge inequality in terms of the extremal Reissner–Nordström geometry for asymptotically flat manifolds with connected boundary and for manifolds with a single asymptotically cylindrical end.

A critical transition is a sudden, abrupt and potentially unforeseen change in the state of a complex system often associated with catastrophic collapses and irreversability. Here we introduce a novel, interpretable early-warning signal of bifurcation-induced critical transitions. The proposed indicator can be readily computed from timeseries observations by maximum likelihood estimation and it has the advantage of providing immediate insight on the state of the system. We showcase its applicability by assessing its performance using models of ecosystems and ocean circulation collapse.

We study polynomials in several complex variables that are stable (i.e. they have no zeros in the unit ball $\mathbb{B}_n$), but vanish on the unit sphere along submanifolds of dimension at most one. We characterize the zeros of such polynomials on the unit sphere in terms of peak sets for the space $A^\infty(\mathbb{B}_n).$ Furthermore, we explicitly construct a polynomial in $\mathbb{C}^3$ that provides a negative answer to the question of whether the equivalent conditions of this characterization universally hold for all stable polynomials. As an application of the developed theory, we obtain a characterization of a certain class of cyclic polynomials in the Dirichlet-type space $D_{n-\frac{1}{2}}(\mathbb{B}_n).$ Next, we examine polynomials whose local zero set, in a neighborhood of points on the unit sphere, coincides with the graph of a holomorphic function. In this particular case, we achieve a characterization of the set of zeros lying on the unit sphere whenever the zero set has local maximum dimension $n-1$. Lastly, we discuss potential generalizations and open problems.

Lattice-based post-quantum cryptography relies on the hardness of the Learning With Errors (LWE) and Module Learning With Errors (MLWE) problems. This work introduces a constructive framework for encoding small MLWE instances as Quadratic Unconstrained Binary Optimization (QUBO) models suitable for quantum annealing. The formulation jointly represents secret coefficients and explicit error variables within a unified binary optimization structure, enabling their simultaneous recovery from the ground-state solution. Beyond the encoding, we develop a stability analysis of the resulting optimization landscape under additive perturbations. We show that the admissible noise region forms a convex polytope defined by competing candidate secrets, and establish an equivalent characterization in terms of the QUBO energy gap between the optimal and second-best solutions. Numerical experiments on low-dimensional benchmark instances using exact simulation demonstrate correct recovery of both secret and discretized error vectors, and confirm consistency between geometric stability regions and energy-gap behavior. We further quantify the scaling of logical variables and embedding overhead with increasing MLWE dimensions to assess feasibility on quantum annealing architectures. The results establish a systematic connection between MLWE problems and quantum optimization while providing a framework for analyzing robustness properties of QUBO formulations. Although current quantum annealing hardware remains insufficient for cryptographically relevant parameters, the proposed methodology offers a structured basis for studying lattice-based problems in quantum optimization settings without implying a practical threat to standardized post-quantum schemes.

This paper introduces, analyzes, and applies variants of the enumeration framework symmetric lexicographic symmetric-subset reverse search for the enumeration of symmetric feasible subsets of a finite set up to symmetry. The framework is implemented in detail for three applications: cocircuits, circuits, and triangulations of point configurations. There are two new methods presented and analyzed to check the lexicographic minimality of a subset in its orbit: the critical-element method and the modified switch-table method. Moreover, new application-dependent methods to reduce the number of necessary enumeration nodes are introduced: rank-pruning for cocircuits and lex-pruning for triangulations. With a C++-implementation of the ideas in the software package TOPCOM, in all three applications known benchmarks can be computed faster by a large margin. The following new numbers could be computed for the first time (among others): the number of cocircuits of the 9-cube, the number of circuits of the 8-cube, and the number of all triangulations of the product of a 5- and a 3-simplex, as well as the number of all triangulations of a point configuration in dimension six with 17~points with disconnected flip-graph (constructed by Santos). Moreover, for Santos's triangulation it has computationally been checked that its flip-graph component is indeed purely non-regular. Furthermore, in another instance in dimension five with 26 points (also constructed by Santos), a flaw has been detected: Santos's triangulation can be heuristically flipped to a regular triangulation in the original point configuration. In a mildly modified version of the point configuration, the heuristics cannot flip Santos's triangulation to a regular triangulation anymore.

Recently, Chudnovsky, Cook, Davies, Oum, and Tan obtained the first finite bound on the chromatic number of t-perfect graphs, showing that they are 199053-colorable. We improve this bound to 130 by refining their proof. The original proof establishes that every graph with large odd girth and large chromatic number contains a certain structure called an r-arithmetic rope, and that its existence in a certain leveling of a graph with large odd girth would imply an odd wheel as a t-minor, a known obstruction to t-perfectness. While their technique requires a lower bound on the chromatic number that is exponential in r, we show that the existence of an r-arithmetic rope can already be guaranteed under a linear bound. Using a slightly weakened notion of arithmetic ropes allows us to reduce the bound even further.

We use methods of Persistent Homology Theory to study invariant manifolds of dynamical systems. We first establish connections between the persistence diagrams of two sets which are close to each other, with respect to the Hausdorff distance. We then apply these results to study properties of limit sets of specific dynamical systems, by using the persistence diagram of a numerically obtained sample set. Under mild assumptions, we show how to use numerical data to state analytical results concerning the geometry of the limit sets.

The Ramsey community number $r_k$ is the smallest size at which a network is better described by communities than by none, under a Bayesian detection rule. On the diamond hierarchical lattice we show that $r_k$ is an exact renormalization-group crossing: the block-model sufficient statistics obey a linear map with eigenvalues $\{bs,b\}$, the degree-corrected evidence density flows to $\ln K$ at a community fixed point, and $r_k$ is the generation at which the running evidence clears the detection threshold. Degree correction advances detection by two generations. We derive $r_k(b,s;q)$ in closed form for the whole family. Finally, placing on the lattice the Reichardt–Bornholdt community Hamiltonian – whose ground state is the partition itself – we find an exact community-ordered phase: below the ferromagnetic critical temperature the two hubs lock into opposite communities for any resolution $\gamma>0$, a staggered order that persists as $n\to\infty$. Allowing each nested sub-community its own label, the optimal partition is a hierarchy of $q_{\rm opt}\sim\sqrt{n}$ communities, so the number of Potts states that best describes the network grows with the network. This hierarchy orders thermally level by level, through a cascade of first-order transitions whose temperatures fall as $1/\ln q$, so every stable level persists as $n\to\infty$: the emergent partition is detectable, optimal, and thermodynamically ordered.

The Picard iteration converges to the unique fixed point of a $q$-contractive operator at a linear rate $q^N$, and a lower bound with an affine construction shows that no deterministic method querying only operator values can do better. But what about higher-order methods that query derivatives? A single Jacobian evaluation reveals an affine map entirely, so the affine construction says nothing about higher-order methods. In this work, we show that finite-order derivative information still does not accelerate the worst-case complexity for smooth contractive fixed-point computation. This contrasts with higher-order smooth minimization, where higher-order derivatives do improve worst-case rates for convex and non-convex minimization.

In this paper we study the mathematical table problem from a geometric-topological point of view. We prove a zero-existence theorem on a cylinder, which gives a new proof of Fenn's square-table theorem under Fenn's boundary conditions, and establish a variant under different boundary conditions. We also prove that every square table admits a horizontal placement on saddle surfaces. Finally, we show that almost every level set of a smooth Fenn graph contains a rectangle similar to any prescribed rectangle and an orientation-preserving similar copy of every prescribed cyclic quadrilateral.

Brešar showed that a finite dimensional unital associative algebra is zero product determined if and only if it is generated by idempotents. For the analogue of zero Jordan product determined algebras, only one direction was known: over a field of characteristic not 2, every algebra generated by idempotents is zero Jordan product determined. Whether the converse holds has remained an open problem. In this paper, we answer this question affirmatively in the finite dimensional case. Some related open problems are stated at the end.

We use symmetric gauge theory to develop a general class of cutting-plane algorithms for semidefinite programming. We formulate a separation problem based on spectral normalizations induced by gauges and derive a closed-form separation oracle. This oracle yields an implementable cut-generation procedure that, by varying the gauge, recovers standard cut families and generates new ones with tunable spectral structure. We embed the oracle within Kelley's method and characterize convergence as a function of the chosen gauge and initial conic relaxation. Numerical experiments on small and large instances of box-constrained quadratic programming and sparse principal component analysis illustrate the versatility and performance of the proposed framework.

In 2008, Beckner (Proc. Amer. Math. Soc. 136(5), 1871-1885) proved two inequalities of the Herbst type, which are the critical forms of the Stein-Weiss inequality. In 2018, Chen et al. (Tran. Amer. Math. Soc. 370(12), 8429-8450) established the reversed Stein-Weiss inequality. In this paper, we are concerned about its critical case and give a reversed Herbst inequality. Namely, $$ \left|\int_{\mathbb{R}^n}\int_{\mathbb{R}^n}|x-y|^{\alpha/q'-n}|y|^{\alpha/q'}g(x)h(y)dxdy\right| \geq C_{n,\alpha,p,q'}\|g\|_{L^{q'}(\mathbb{R}^n)}\|h\|_{L^p(\mathbb{R}^n)} $$ holds for any nonnegative functions $g \in L^{q'}(\mathbb{R}^n)$ and $h \in L^p(\mathbb{R}^n)$, where $n\geq 1$, $p, q' \in (0,1)$, $\alpha>n$ satisfying ${1}/{p}+{1}/{q'}-{2\alpha}/(q'n)=1$. Such an inequality is not covered by the reversed Stein-Weiss inequality. Meanwhile, we prove the existence of extremal functions of this inequality. Finally, we study the Euler-Lagrange system satisfied by those extremal functions $$ \left\{\begin{matrix} u(x)=\int_{\mathbb{R}^n}|x-y|^{\beta-n}v^{-p_2}(y)|y|^{\beta}dy, v(x)=\int_{\mathbb{R}^n}|x-y|^{\beta-n}u^{-p_1}(y)|x|^{\beta}dy. \end{matrix}\right. $$ We obtain necessary conditions for the existence of positive solutions, and investigate their integrability and asymptotic behavior when $|x| \to 0$ and $|x| \to \infty$.

Let \(q\) be an odd prime power and let \(V=\F_q^{2m}\), equipped with \(Q(x)=x_1^2+\cdots+x_{2m}^2\). We develop a semidefinite Delsarte framework for the two-set Erdős–Falconer distance problem over \(V\). The framework reduces the natural \(q^m\)-scale positive-proportion theorem to a uniform \(L^1\) anti-concentration statement for positive convex combinations of classical Kloosterman sums. Assuming this Kloosterman anti-concentration conjecture, we prove that, for every \(0<\alpha<\frac12\), there exists a constant \(C_{m,\alpha}\) such that \[ \min\{|E|,|F|\}\ge C_{m,\alpha}q^m \quad\Longrightarrow\quad |\Delta_Q^\times(E,F)|>\alpha(q-1) \] for all \(E,F\subset\F_q^{2m}\). More generally, a \(q^{-\theta}\)-level version of the Kloosterman input yields the geometric threshold \(q^{m+\theta}\). Unconditionally, we establish a uniform \(q^{-\frac13}\)-level \(L^1\) lower bound whenever the support has size at most \(\alpha(q-1)\), with \(0<\alpha<\frac12\). Consequently, \[ \min\{|E|,|F|\}\ge C_{m,\alpha}q^{m+\frac13} \quad\Longrightarrow\quad |\Delta_Q^\times(E,F)|>\alpha(q-1). \] The proof uses positive semidefinite \(2\times2\) Gram matrices on quadratic frequency shells, the shell Fourier transform in even dimension, and a minimax separation argument that produces a uniform signed combination of Kloosterman columns. We also provide evidence for the constant-scale Kloosterman conjecture and discuss its limitations near full support.

The article examines Nikolskii and Besov spaces with norms defined using "$L_p$-averaged" mixed moduli of continuity for functions of appropriate orders, instead of mixed moduli of continuity of known orders for certain mixed derivative functions. The work provides upper and lower estimates for a quantity of the best accuracy of approximation in the S.B.Stechkin problem for partial differential operators on a classes of such functions defined in domains of a certain kind. These estimates are not weaker, but in some cases even stronger than those derived by the author in the problem under consideration for the aforementioned classes of functions on cube $ I^d. $ It also significantly broadens the class of Nikolskii and Besov spaces of mixed smoothness for which mentioned estimates in the problem under consideration have been derived.

A $2$-knot is an embedding of a $2$-sphere into the $4$-sphere. Similar to the case of embedding circles into the $3$-sphere, this allows the $2$-sphere to be knotted. In this short paper, we present an algorithm to generate triangulations of the exteriors of $2$-knots obtained by spinning $1$-knots. We give an implementation of the algorithm in Regina and present triangulations of exteriors obtained from all $1$-knots with up to eight crossings.

Whether the solutions of conformal equations in the whole space are radially symmetric is an interesting topic. Chen-Li-Ou proved the radial symmetry for integral systems of the Hardy-Littlewood-Sobolev type and the Stein-Weiss type by the method of moving planes in integral form. In 2015, Dou-Zhu obtained the radial symmetry of extremal functions of the reversed Hardy-Littlewood-Sobolev inequality by the method of moving spheres, and Liu proved the radial symmetry of solutions of the Euler-Lagrange system by the method of moving planes developed by Dou-Guo-Zhu. In this paper, we also use the method of moving planes to prove the radial symmetry of positive solutions of the Euler-Lagrange system satisfied by the extremal functions of the reversed Stein-Weiss inequality established by Chen-Liu-Lu-Tao in 2018.

In this paper, we study qualitative properties of distribution solutions of a fourth order equation $$ -\Delta u(x)+a^2\Delta^2u(x)=u^q(x), \quad u(x)>0 \ \ in \ \ \mathbb{R}^3, $$ where $a>0$ and $q>0$. It is the static equation of a mixed dispersion Schrodinger equation, and also the Euler-Lagrange equation satisfied by extremal functions of an embedding inequality. We obtain some Liouville theorems and the corresponding related critical exponents, which imply the best constant of the embedding inequality cannot be attainable. We also obtain some regularity results (involving differentiability, integrability, radial symmetry) and asymptotics at infinity of distribution solutions. Here an equivalent integral equation with the Coulomb potential $|x|^{-1}(1-e^{-|x|/a})$ plays a key role. In addition, we also use the Pohozaev identity in integral form to obtain the Liouville theorem of this integral equation. Such the Pohozaev identity still works to handle the Allen-Cahn-type integral equation.

Families of classical field theories often depend on geometric parameters. A basic question is whether the corresponding classical observables can be compared by flat parallel transport, and whether this comparison survives quantization. We study this problem for families of shifted Poisson structures. To such a family we attach a Poisson transverse controller $\mathbb U_\pi$, a homotopy stabilizer encoding transverse symmetries which preserve the Poisson Maurer–Cartan element up to coherent homotopy. Its flat splittings are precisely transversal shifted Poisson unfoldings, and they act on vertical symmetries, Poisson cohomology and local deformation theory. We then formulate the $\hslash$-adic lifting problem for quantized objects; its degree-two obstruction classes are the transport anomalies. The construction is realized for star-products, BV observables, factorization algebras and AKSZ theories. For the Poisson sigma model, anomaly-free quantization of the flat transport makes the Cattaneo–Felder/Kontsevich boundary product horizontal over the parameter space.

We introduce a compensated Piola graph method for Lagrangian stability estimates in critical homogeneous Besov spaces below the usual product threshold. For $d\ge2$, $2d\le p<\infty$, and $s=d/p$, we establish local Hadamard well-posedness and a continuation criterion in the scaling-critical Besov phase space $\dot B^{s-1}_{p,1}(\mathbb R^d)^d\times \dot B^s_{p,1}(\mathbb R^d)$, for a class of incompressible parabolic systems coupled to nondiffusive internal variables. The main obstruction is that the formal Piola product between the inverse deformation gradient and the Lagrangian velocity is only borderline at $p=2d$ and is not continuous in general for $p>2d$. We replace this product by a closed solenoidal Piola graph, using a compensated divergence structure that survives in the high-integrability range. The principle applies to viscous non-resistive MHD, Hookean incompressible viscoelasticity, and nondiffusive Oldroyd–B systems with affine objective terms. In particular, it closes the previously untreated high-$p$ Hadamard well-posedness range for critical non-resistive MHD and for the nondiffusive Oldroyd–B systems considered here; combined with the known low-$p$ MHD theory, it gives the finite-$p$, $q=1$ critical Besov picture for non-resistive MHD.

Given a graph $L$, the maximal anti-Ramsey function $\chiS(n,e,L)$ denotes the minimum integer $\chiS$ for which there exists an $n$-vertex graph $G$ with at least $e$ edges admitting an edge-coloring with $\chiS$ colors in which each copy of $L$ in $G$ is rainbow. In 1989, Burr, Erdős, Graham, and Sós posed the following problem: Is it true that for all $\epsilon>0$, there exists $c(\epsilon)>0$ such that for all sufficiently large $n$, $ \chiS\left(n,\binom{n}{2}-\lfloor n^{2-\epsilon}\rfloor,P_4\right)>c(\epsilon)n^2. $ Very recently, Li, Ning, and Xie gave a negative answer to the problem for all $0< \epsilon< 1/2$. In this note, we establish that a quadratic lower bound holds in the complementary regime $ \epsilon\geq 1/2$. More specifically, we prove that for all $\epsilon\ge 1/2$ and sufficiently large $n$, there is an absolute constant $c>0$ such that $ \chiS\left(n,\binom{n}{2}-\lfloor n^{2-\epsilon}\rfloor,P_4\right)>c n^2. $