Papers

We characterize metric measure spaces satisfying parabolic Harnack inequalities for a doubly nonlinear equation in terms of volume doubling and Poincaré inequalities. Our approach uses purely analytical methods, based on obtaining estimates for solutions to a related Cauchy problem. This extends previous linear results to the nonlinear setting without relying on heat kernel estimates and representation formulae.

In this paper, we extend the work of \cite{Chahal} in several directions. We first determine all Heron triangles that tightly circumscribe the unit circle and the associated $\tau$-congruent numbers generated by them. We then characterize all rational right triangles that tightly circumscribe the unit ellipse and identify the corresponding congruent numbers. In addition, we study of the congruent numbers from the excircle opposite a vertex of a rational right triangle, that is, the circle tangent to one side of the triangle and to the extensions of the remaining two sides.

Periodic driving serves as an effective method for controlling the properties of physical systems. Called "Floquet engineering," it is a broad field of theoretical and experimental activity. Whereas original Floquet theory was proposed to a system of ordinary differential equations, the quantum systems with time-dependent potential require using partial differential equations. Among different methods of analysis of such systems, time series is a most common one. Though general scheme was developed in a number of works, its application to specific problems often faces significant difficulties. In particular, the class of the problems describing magnetic multilayers with time-dependent potential (e.g., rotating magnetization of some of the layers) leads to significant complication of the problem due to two-component wave function and matching conditions at the interfaces. Taking as an example a two-layer system containing magnetic layer with rotating magnetization, we construct a class of solution containing arbitrary but finite number of the terms. The structure of the solution is analyzed. In particular, we show that boundary conditions, which seem a natural generalization of that for a stationary problem, cannot be imposed in the case of rotating magnetizations.

For a graded ideal I in a graded ring, the deviation of I is defined as the difference between the minimal number of generators of I and its grade. In this article, we provide bigraded free resolutions of the symmetric algebras for specific classes of ideals of deviation two. Additionally, we study the regularity of powers of deviation two ideals generated by d-sequences. In particular, we examine the powers of Huneke-Ulrich ideals and find bounds on their regularity.

Many decentralized decision problems require multiple parties to coordinate on shared decisions while keeping objectives, constraints, and data private. Large language models (LLMs) offer a promising way to lower the barrier to participation by generating local optimization agents from natural-language specifications. In coordination settings, however, executability is not enough: a generated agent may compile, solve, and pass local checks while still being semantically wrong, for example by misrepresenting costs, mis-scoping constraints, or responding incorrectly to incentives. Such errors often surface only during coordination, as systematic behavioral failures rather than infeasibility. We propose coordination-in-the-loop verification and repair for LLM-generated optimization agents. We instantiate this idea with an Alternating Direction Method of Multipliers (ADMM)-style consensus protocol and introduce OptiLoop, a pipeline that generates local optimization agents from text, verifies them through short, bounded coordination runs against a fixed reference counterparty, extracts structured behavioral and static evidence, and applies evidence-driven repair. When failures are structural rather than implementational, OptiLoop escalates from localized code fixes to corrected-formulation repair, and it can additionally reuse episodic lessons from prior instances. On 40 held-out test scenarios, OptiLoop-Full improves objective match from 66.0% to 93.0% and social match from 68.5% to 89.0% relative to a strong local-validation baseline, while reducing mean objective gap from 15.3% to 3.5% and mean social gap from 7.6% to 2.0%. These results show that, for generated optimization agents deployed inside decentralized decision loops, correctness should be validated in the loop itself rather than through isolated execution alone.

In this paper, we consider the problem of simultaneous testing of multivariate normal means under arbitrary covariance dependence. Specifically, let $\boldsymbol{X}\sim N_n(\boldsymbol{\theta},\boldsymbol{\Sigma})$, where $\boldsymbol{\theta}\in\mathbb{R}^n$ is unknown and $\boldsymbol{\Sigma}$ is a known positive definite covariance matrix. The objective is to test $H_{0i}:\theta_i=0$ against $H_{Ai}:\theta_i\neq 0$, simultaneously for $i=1,\ldots,n$. We establish a general admissibility theorem for a broad class of monotone residual-based step-down multiple testing procedures which iteratively rank the active hypotheses using statistics obtained through locally adaptive strictly increasing transformations of suitably standardized residual statistics arising from conditional normal distributions. Our main result shows that every such procedure is admissible with respect to a vector-valued loss function whose components are the usual individual $0$–$1$ testing losses. The proof relies on a delicate geometric analysis of the induced acceptance regions together with structural invariance properties of the adaptive stagewise rejection indices. The theorem substantially extends the admissibility theory developed for the maximum residual down procedure of Cohen et al. (2009) and reveals that admissibility under dependence is fundamentally driven by the monotone ordering structure induced by the residual statistics rather than by the precise functional form of the testing rule itself.

We show that if $G$ is a $d$-regular Vizing-class-1 graph, then the proper additive chromatic index of $G$, denoted $\eta'_p(G)$, is equal to its chromatic index. This verifies that a strengthening of the Additive Coloring Conjecture of Czerwiński et al. holds for line graphs of $d$-regular Vizing-class-1 graphs. We show that if $G$ is a $d$-regular Vizing-class-2 graph, $\eta'_{p}(G)\leq \frac{(2^{\lceil \log_2 (d+1)\rceil})^2+2}{3}$, and if $G$ is a $d$-regular Vizing-class-2 graph that admits a proper edge-coloring with a smallest color class of size $r$ and $\text{girth}(G)\geq 6r-5$, then $\eta_p'(G)\leq 2d$, among other results.

Let $\calP$ be a general pencil of curves of degree $d$ in the projective plane. In this paper we review the computation of the number of curves in $\calP$ that have a hyperflex line, a flex bitangent line or a tritangent line. Then we focus on the curves in the dual plane described by the flex tangents and the bitangents of the curves of $\calP$ and the curves in the original plane described by the flexes and the points of bitangencies of the curves in $\calP$. Some of these curves have been studied already: we mainly focus here on the ones that still have not been treated systematically, and we compute their degree, genus, and singularities.

The picture-hanging puzzle, popularized by Demaine et al. (2014), asks for a way to wrap a wire around $n$ nails such that the picture hangs as long as fewer than $k$ nails are removed, but falls as soon as any $k$ are removed. Solutions correspond to words in the free group $F_n$. We give explicit, deterministic, polynomial-length constructions for two regimes: $2$-out-of-$n$ with word length at most $\tfrac{8}{3}n^{\log_2 6} - 4n^2$, and $(n-2)$-out-of-$n$ with word length $6n\log_2(n/2)$, both for $n$ a power of two. These improve on Wästlund's quasi-polynomial deterministic construction in their respective regimes. We also report, via exhaustive computer search, the exact minimum length of $16$ for the $2$-out-of-$4$ puzzle, attained by two structurally distinct solutions. As an additional contribution, we observe that the natural workshop realization with carabiners on a flat board introduces an over/under ambiguity at every wire crossing; a wrong choice can produce a Whitehead link, which is topologically distinct from the intended commutator.

We prove that the derived category of a branched double cover is equivalent to a category of matrix factorizations for a fiberwise quadratic potential on the associated line bundle. This requires the linear fiber coordinate to have odd cohomological degree, so we work with matrix factorizations which no longer have the traditional splitting into even and odd parts.

We show that the completed volumes introduced by Duriev-Goujard-Yakovlev as an approximation to compute Masur-Veech volumes via Witten-Kontsevich's combinatorial classes agrees with the top intersection of the tautological class on the double ramification cycle, computable as a coefficient of a Chiodo class. For the proof we describe the components of the double ramification cycle and their excess intersection classes to the extent seen by the top tautological intersection. This gives a recursion computing completed volumes in terms of volumes appearing in a certain set of level graphs, not only for quadratic differentials. It also completes the work of Duriev-Goujard-Yakovlev solving the technically most involved case of strata with two singularities.

We study the semiclassical spectral theory of a one-dimensional Dirac operator describing waves at the interface between topologically distinct media. We derive a modified Bohr-Sommerfeld quantization condition for the squared operator via a systematic formal WKB construction producing approximate eigenpairs. Our result differs from the standard result by the half-integer shift depending on the pseudo-spin index which allows for recovering the topologically protected zero mode. We verify our result for the solvable Pöschl–Teller potential and provide numerical computations confirming the convergence of eigenvalues and eigenfunctions to their WKB approximations.

We prove that the set of integrable functions on the unit circle for which the analogue of Paley's theorem for $H^1$ fails is residual in $L^1(\mathbb T)$. Moreover, we establish algebraic genericity and spaceability results in several Hardy-type function spaces under prescribed conditions on Taylor coefficients, extending phenomena considered in \cite{pandis2024some,NestoridisGenericH1}.

We develop new tools to compute the index of symmetry in the context of homogeneous fibrations. As a consequence of our results, we determine the index of symmetry of every homogeneous space diffeomorphic to a compact rank-one symmetric space, and of every homogeneous $S^1$-bundle over a compact irreducible symmetric space. Moreover, we construct irreducible homogeneous metrics whose leaves of symmetry are symmetric spaces of arbitrarily large rank.

This paper is devoted to the analysis of a semilinear suspension bridge model with pointwise localized dissipation. The main contribution of the work is the development of a robust semigroup framework that substantially simplifies the stability analysis of coupled systems with localized damping. The core of the approach relies on a comparison principle involving the essential growth bound of the associated semigroups together with the invariance of the essential spectrum under compact perturbations. This methodology allows us to reduce the stability analysis of the full coupled system to that of suitably chosen auxiliary systems, thereby avoiding the delicate multiplier techniques and observability arguments commonly used in the literature. As a first consequence, we establish the exponential stability of the corresponding linear homogeneous system. The semilinear problem is then treated as a locally Lipschitz perturbation of the exponentially stable linear semigroup, yielding global well-posedness and exponential decay of solutions. Finally, we investigate the long-time dynamics of the semilinear system. By combining the dissipative structure of the model with suitable regularization properties of the nonlinear semigroup, we prove the existence of a global compact attractor and an exponential attractor with finite fractal dimension. These results provide a complete description of the asymptotic behavior of solutions and show that the proposed framework is sufficiently flexible to be adapted to other coupled systems with localized dissipation.

This article investigates the consensus tracking problem of multi-agent systems under jointly connected topology through automated synthesis of Lyapunov functions. Based on the proposed distributed nonlinear control protocol, several consensus criteria for first-order multi-agent systems are established and certified by the construction and synthesis of more general polynomial Lyapunov functions. By employing sum-of-squares decomposition for multivariate polynomials, we can efficiently synthesize polynomial Lyapunov functions to achieve consensus verification in polynomial time, although the widely used quadratic Lyapunov functions do not exist. Moreover, polynomial coupling functions for our proposed protocol are concomitantly generated. Furthermore, the distributed nonlinear control protocol is extended to deal with second-order multi-agent systems, while ensuring second-order consensus verification. Finally, an example is presented to demonstrate the efficacy of our method.

We explore Seshadri constants associated to weighted blow-ups of complex projective varieties and demonstrate how to use this notion to construct symplectic embeddings of ellipsoids. We illustrate the utility of this point of view by providing constructions of full fillings of $\mathbb{CP}^2$ by ellipsoids corresponding to all of the exceptional (post-Fibonacci) steps of the McDuff–Schlenk staircase and some non-obvious embeddings of ellipsoids in ellipsoids.

We study scattering resonances of finite and infinite periodic two- and three-dimensional systems of high-contrast resonators beyond the subwavelength regime. At each fixed admissible nonzero reference frequency, we introduce frequency-dependent capacitance matrices and derive quantitative asymptotic expansions of hybridized Fabry–Pérot resonant frequencies and their corresponding eigenmodes in terms of the material contrast parameter. We provide a partial differential equation (PDE) formulation of the frequency-dependent capacitance matrix analogous to the one for the capacitance matrix in the subwavelength regime. Based on this PDE formulation, we establish key properties of the frequency-dependent capacitance matrix and estimate its norm at high frequencies for a single smooth resonator with nontrapping exterior in three dimensions, identifying a regime in which the reduced residual remains perturbative. For infinite periodic resonator arrays, we prove bandgap opening under a uniform exterior non-resonance condition and generic Dirac cones for a honeycomb scaling family. Our results extend the use of discrete approximations as a powerful tool for characterizing the resonant properties of finite and infinite periodic systems of high-contrast resonators at arbitrarily high frequencies and understanding their anomalous localization and transport properties arising from strong coupling to a discrete set of eigenfrequencies.

We present a field-theoretic construction of a class of artificial neural networks from solitonic degrees of freedom in nonlinear scalar field theory. The purpose is not to rename a standard neural layer in the language of solitons, but to start from a continuum action, restrict the theory to a nonperturbative sector containing localized stable solutions, perform a collective-coordinate reduction, and derive the neural layer as the finite-dimensional input-output map of the reduced solitonic dynamics. In this construction, the computational unit is a projected collective coordinate of a localized field configuration rather than an elementary point variable; the activation function is the solitonic response profile or scattering map; the weight matrix is the Hessian or overlap matrix of an effective interaction energy among solitons; the bias is induced by external sources, vacuum asymmetry, or boundary forcing; and depth is a discrete evolution parameter on the solitonic moduli space. We develop the construction explicitly for the \(\phi^4\) kink, where the \(\tanh\) activation and the logistic sigmoid arise from the kink profile, and then derive the multilayer feedforward form from an operator-splitting approximation to collective-coordinate gradient flow. We emphasize the novelty criterion: the neural architecture is obtained only after specifying the field action, the solitonic ansatz, the moduli-space metric, the interaction functional, and the projection map. The result is a controlled route from nonlinear field theory to neural-network structure, with robustness tied to the energetic and topological stability of the solitonic sector.

We prove an elementary bounded-differences inequality for functions of non-isotropic Gaussian vectors. Specifically, if $f$ has bounded coordinate differences and $X\sim\mathcal N(\mu,\Sigma)$, then the resulting concentration bound depends on the condition number $\kappa(\Sigma)$. As an application, we answer a question of Simone Bombari concerning the subgaussianity of sign-quantized linear maps $Y=\mathrm{sgn}(Wx)$. In the special case where $f$ is the coordinatewise sign function, an argument was initially suggested to us by Gemini 3.5 Flash without attribution. We subsequently discovered that it closely resembles an earlier argument of Barber and Kolar [Ann. Statist. 46 (2018), Lemma 4.5]. This revision corrects the attribution and documents the episode as an instance of AI-assisted mathematical discovery.

Recently there has been a lot of progress in the development of economic nonlinear model predictive control (NMPC) schemes for multistage optimal power flow (OPF) problems. However, the additional inclusion of discrete decision variables to model generator runtimes and generator startup costs can amount to large scale mixed-integer nonlinear programs (MINLPs) that are computationally very challenging. This work investigates the practical approach that replaces the nonlinear AC power flow equations by convex quadratic approximations. In combination with the discrete generator dynamics this leads to a mixed-integer quadratically constrained program (MIQCP) which is of significantly lower complexity and can be solved in reasonable time by off-the-shelf solvers such as CPLEX. We further show that simple terminal constraints are not sufficient to guarantee recursive feasibility of the NMPC scheme if constraints on generator runtime and on the number of generator startup events are present. To address this challenge we propose the use of additional time-coupled constraints and prove the resulting recursive feasibility property. Based on the assumption of periodic dissipativity of the underlying system we can prove stability of the proposed controller. To illustrate our results, we present simulations of a realistic 6-bus microgrid under different demand scenarios.

In one of his posthumous papers, conserved in Göttingen, Riemann considers the derivatives of $\log\zeta(s)$ at the point $1/2$, giving explicit values for them. Around 2010 we shared Riemann's value of the second derivative with some mathematicians. From that time I have been asked several times for references. So I decided to write this. Specially explaining the wonderful formulas \[\frac{\zeta'(\frac12)}{\zeta(\frac12)}=\frac{\pi}{4}+\frac{\gamma}{2}+\frac{\log(8\pi)}{2},\quad \frac{\zeta''(\frac12)}{\zeta(\frac12)}-\Bigl(\frac{\zeta'(\frac12)}{\zeta(\frac12)}\Bigr)^2=8-\frac{\pi^2}{4}-2G+2\sum_{n=1}^\infty\frac{1}{\alpha_n^2}\]

Let $E \subset \mathbb R^d$, $d \ge 2$, be compact, and let $\phi(x,y)$ be a smooth function satisfying the Phong–Stein rotational curvature condition on $\{\phi(x,y)=1\}$. We prove that if $\dim_{\mathcal H}(E)>1$, then $$ \left|\bigcup_{x \in E} \{y : \phi(x,y)=1\}\right|>0. $$ This extends the positivity theorem of Mitsis ($d\geq3$) and Wolff ($d=2$) for spheres to a general variable coefficient setting via $L^2$ estimates for Fourier integral operators. The argument also shows that positivity is stable under finite-order degeneracies of the Monge–Ampère determinant through the weighted averaging theory of Sogge and Stein. We next consider variable level sets $$ \Sigma_x=\{y:\phi(x,y)=t(x)\}, $$ where $t(x)$ is measurable. A maximal operator argument yields positivity under the condition $\dim_{\mathcal H}(E)>2$. We show that this loss reflects a genuine geometric obstruction related to Kakeya-type compression phenomena. In contrast, under a direct geometric intersection hypothesis controlling overlaps of the hypersurfaces $\Sigma_x$, we recover the full threshold $\dim_{\mathcal H}(E)>1$ for arbitrary measurable selections $t=t(x)$. At the endpoint $\dim_{\mathcal H}(E)=1$, we obtain positivity under the additional assumption that $E$ is $1$-rectifiable with $\mathcal H^1(E)>0$. We also show that positivity of Lebesgue measure does not in general imply interior regularity: even for large or rectifiable parameter sets, the resulting unions may have empty interior. Finally, we discuss extensions to higher co-dimension families and the role of geometric structure in preventing compression phenomena.

This paper revisits the notion of classical orthogonal polynomials from a broader functional-analytic point of view. It is intended neither as a survey of known results nor as a review of the literature, but rather as a conceptual reappraisal of the subject from a perspective in which certain persistent distortions become plainly visible. The theory is developed on subsets of the complex plane that are stable under half-step translations, and both classicality and orthogonality are understood in the continuous dual of a suitable locally convex space of polynomials. The repeated reappearance of ostensibly new families of classical orthogonal polynomials arising from exotic maps, algebraically equivalent families artificially separated, unnecessary parameter restrictions inherited from positive-definite models, apparently distinct phenomena associated with root-of-unity values of $q$ in the $q$-exponential case, naive $q\to -1$ limits in that same setting, finite truncations of otherwise infinite orthogonal polynomial sequences, and geometric recastings of the local half-step relation in terms of plane conic curves all make the need for a broader structural framework increasingly clear. The present paper seeks to articulate such a framework in a way that allows the reader to distinguish the genuinely new from the merely artificial, without resorting to an exhaustive case-by-case examination of prior work. In the ordinary quadratic and non-torsion $q$-exponential regimes, the infinite regularity criteria retain their full force as if-and-only-if statements; finite resonances, by contrast, are governed by the exact moment system and the non-vanishing of its Hankel determinants.

Given a smooth, oriented, simply-connected $4$-manifold $M$, the homological Nielsen realization problem asks: when does a finite group of isometries $G\leq O(H_2(M;\mathbb{Z}))$ preserving the intersection form lift isomorphically to a finite group of orientation-preserving diffeomorphisms? We study this question for the smooth, positive-definite 4-manifolds $M_n:=\#_n\mathbb{CP}^2$. Even though every isometry of $H_2(M_n;\mathbb{Z})$ is induced by some orientation-preserving diffeomorphism, not necessarily of finite order, we show that Nielsen realization is sparse: as $n\to\infty$, a random subgroup of $O(H_2(M_n;\mathbb{Z}))$ is asymptotically almost never realizable in $\mathrm{Diff}^+(M_n)$; the same is true for random odd order elements of $O(H_2(M_n;\mathbb{Z}))$. We present both positive realization results in certain cases and a range of obstructions to realization in other cases. The proofs combine equivariant connected-sum constructions, fixed-point theory for group actions on 4-manifolds, finite group actions on surfaces, analytic combinatorics, and previous work of Hambleton–Tanase.

Using direct methods of the calculus of variations we establish the existence of an infinite class of spherically-symmetric solutions to the multi-field Schrödinger-Poisson system. This is achieved by proving that the energy functional admits a global minimum when restricted to the set of vector-valued wave functions in the Sobolev space $H^1$ which are invariant with respect to a suitable representation of the rotation group and whose components have fixed $L^2$-norms. Additionally, we show that these minima correspond to solutions which are orbital stable with respect to perturbations of the wave function within this set. The generalization to include an external potential and some important properties of the minima are also discussed.

Schrödinger bridges for time series (SBTS) generate synthetic paths by projecting, in relative entropy, a Brownian reference onto the path laws that match the joint distribution of the data on the observation grid. The Brownian reference, however, fixes the quadratic variation of the generated paths, which is restrictive when stochastic volatility, correlated noise, or rank-deficient covariance structures must be reproduced. We introduce "Triangular-Reference Schrödinger Bridges for Time Series" (TR-SBTS), which keeps the entropy-projection backbone of SBTS but replaces the Brownian reference by a triangular, volatility-informed, intervalwise frozen reference on a state augmented with latent covariance descriptors. The construction remains a single entropy projection on the augmented state: the minimiser is the \(h\)-transform of the reference, and on each frozen interval the optimal drift has the logarithmic-gradient form \(b^\star(t,x)=A\,\nabla\log H(t,x)\), intrinsic to the active covariance directions when the frozen covariance \(A\) is degenerate. We prove stability of the frozen approximation and consistency of the associated regularised kernel estimators, describe a reference-aware Nadaraya–Watson implementation of the conditional next-increment law, and evaluate the construction on numerical experiments.

We study the minimum number of inflection points among generic immersed closed plane curves with a fixed embedded shadow. The word immersed is essential: a genuinely embedded Jordan curve has inflection minimum zero. For tree-like shadows, inflection criterion converts inflection-free realizability into a finite coorientation problem on the building polygons of the shadow. We sharpen this viewpoint into an exact finite formula for the minimum number of normalized inflections and record a dynamic-programming computation on the block tree. We then push the method beyond the tree-like case. For every embedded shadow the same coorientation model gives a universal lower bound. For a natural larger class, called tree–necklace shadows, in which the non-tree-like blocks are separated annular cycles, the lower bound is exact after imposing an explicit $\mathbb Z_2$ holonomy condition around each necklace. We also record the algorithmic status of the exact minimization problem and formulate a likely NP-hardness problem for unrestricted shadows. Finally, we introduce a related invariant: the minimum possible least multiplicity of the Gauss map, equivalently the smallest guaranteed number of oriented parallel tangencies. This “parallel-tangent load” is controlled by the same inflection folds but is not determined by their number alone.

We discuss a spectral property for the virial operator of the 2D Zakharov-Kuznetsov (ZK) equation. This is a crucial ingredient to establish blow-up or asymptotic stability of solitary waves in higher-dimensional problems. This model in 3D setting was originally introduced by Zakharov and Kuznetsov in plasma physics, and is also a higher-dimensional generalization of the well-known Korteweg-de Vries (KdV) equation. The problem of stability of solitary waves in ZK equation or stable blow-up in modified ZK (or KdV-type) equation is an important physical question, for which virial operators and their spectral properties are the essential elements of the analysis. In this paper we investigate this problem analytically and reduce it to verifying numerically only some signs of inner products and certain eigenvalues.

In many operational contexts, solutions to NP-hard combinatorial optimization problems, modeled by means of Mixed-Integer Linear Programming (MILP), may become infeasible due to unpredictable disruptions. Typically, reoptimizing by solving the MILP formulation on the perturbed instance is not possible as new solutions must be obtained in a very short computing time, while simple repairing heuristics may result in low-quality solutions. To bridge this gap, we propose a learning-to-reoptimize framework, and apply it to the Lot Sizing Problem (LSP) under machine breakdown disruptions. We design a fix-and-optimize strategy aided by a Graph Neural Network (GNN) that efficiently computes a new solution within the neighborhood of a repaired solution. By representing the instance, the original solution and the disruption as a feature graph, we train a GNN to predict the likelihood that specific binary variables require to be modified. These predictions guide the selection of a small subset of variables to be reoptimized by an MILP solver, while the other variables are hard-fixed. Numerical experiments on a large dataset demonstrate that our approach handles effectively different problem sizes, and that it significantly outperforms a baseline alternative approach, yielding larger cost reductions within the same limited time budget.