Papers

Recent well-posedness results for evolutionary partial differential equations with state-dependent inhomogeneity are extended to a larger problem class incorporating nonautonomous material behaviour. This generalization facilitates the formulation of a framework able to accommodate delayed boundary value problems related to evolutionary partial differential equations. The results permit the ad hoc treatment of Dirichlet- and Neumann-like boundary conditions with state-dependent delay. Complex boundary conditions involving nonautonomous delay in the material law and state-dependent delay in the forcing term can be accommodated by use of extended state spaces. This approach provides the first systematic treatment addressing well-posedness of several classical boundary conditions involving state-dependent delay. The viability and versatility of the theory is showcased by applications to parabolic and hyperbolic partial differential equations with Dirichlet, Neumann, Robin, Wentzell-Robin and Leontovich boundary conditions.

We study a family of highly symmetric instances of the stable marriage problem built from regular actions of finite groups. Given a finite group G of order n and an ordering A of its elements, we define the regular anti-phase template P(G,A). These templates have n canonical stable matchings. We show that the anti-phase condition is canonical: among automorphism-maximal profiles, the anti-phase templates are exactly those satisfying a constant rank-sum identity. This gives a structural characterization rather than an ad hoc definition. We prove a generator criterion and its exact converse: the stable set has size n if and only if each adjacent quotient generates the group. This result holds for all finite groups and does not require commutativity. We further establish a counting lower bound for the number of stable matchings in terms of subgroup indices. The bound is sharp for groups of order at most 5 and for all groups of order 4; in particular, it yields at least 10 stable matchings for the Klein group, with equality confirmed by enumeration. Finally, we show that cyclic profiles do not always produce chains; the structure depends on the ordering. All computational claims are verified by an accompanying script.

The Moore-Penrose pseudoinverse of a matrix can be defined and calculated using its singular value decomposition. There are also direct methods for computing matrix pseudo-inverses (those that avoid eigenvalue computations), but these are often rank-revealing, poorly conditioned, or otherwise limited in practice. In this paper, we demonstrate that direct methods can overcome these limitations. In particular, we reinterpret several existing direct methods and introduce new variations that are appropriate for large scale or sparse multilinear regression applications.

We study the graphs $MECS_{b,n}^a \cong \overline{K}_a \join \big(n(K_b+K_2)\big)$, obtained by attaching an independent set of size $a$ to $n$ disjoint copies of the block $K_b+K_2$. For $n=1$, we get $MECS_{b,1}^a$, and recover the results of one-block case studied in [Anand, Gupta, Rather and Singh, Homological invariants of some complete split-like graphs, Beitr. Algebra Geom. (2025)]. Using Hochster's formula, tensor products of minimal free resolutions over disjoint variable sets, and the Betti-number formula for graph joins, we derive explicit descriptions of the independence complex, independence polynomial and its analytic properties, Hilbert series, linear and quadratic Betti strands, regularity, projective dimension, and several structural invariants of $MECS_{b,n}^a$. We further classify the well-covered and unmixed members, compute induced matching numbers, show that the family is never Cohen–Macaulay, and record algorithmic procedures for evaluating the Betti data.

The robustness of approximate Riemann solutions has been a crucial topic in computational magnetohydrodynamics, from both theoretical and practical perspectives. Recently, the widely used HLLD approximate Riemann solution was revised, becoming significantly more robust under strong magnetic fields. Yet, simplifications were needed as the compressible slow magnetoacoustic mode is not included in the HLLD scheme, and as a result non-physical density distribution has been found when having strong slow shocks. In this note, a simple fix is introduced for the estimated intermediate internal energy, providing robust results in several representative test cases, preserving pressure-positivity when the magnetic field is enhanced by a factor of 1000.

In this paper, we introduce and study the notion of generalized $\xi$-parallel maps between Riemannian manifolds, where $\xi$ is a vector field on the target manifold. We define the associated energy functional and derive its first variation formula, leading to the Euler–Lagrange equation characterizing generalized $\xi$-parallel maps. We establish fundamental properties of this new class of maps and investigate its relationship with harmonic and biharmonic maps. Furthermore, we study generalized $\xi$-parallel curves in space forms and examine generalized $\xi$-parallel submanifolds of space forms admitting concircular vector fields.

We present new small models for the dg properads that govern pre-CY algebras: one for general pre-CY algebras and one for pre-CY algebras without curvature terms. These small models have only four and, respectively, three generators with valencies $\leq 4$ (in contrast to the original properads which have infinitely many generators with arbitrarily large valencies). We prove that the valency upper bound $4$ is sharp. We study the cohomology group of the deformation complex of the dg properad governing general pre-CY algebras and prove that it contains the totality $\prod_{g\geq 1}H^\bu(\M_{g,1})$ of cohomology groups of moduli spaces $M_{g,1}$ of genus $g$ algebraic curves with one marked point. As an application of this result we prove that the Tradler-Zeinalian properad of $V^{(d)}$-algebras is not Koszul.

In this article, we study several Bump–Friedberg type periods beyond the cuspidal spectrum. We first consider the twisted Bump–Friedberg period on $\textnormal{GL}_{2n}$, as well as a variant on $\textnormal{GL}_1\times \textnormal{GL}_{2n}$. Under suitable regularity conditions on the cuspidal datum, these periods extend continuously to automorphic functions of uniform moderate growth. Such extensions are characterized by entire Whittaker-type zeta integrals. We then introduce a Bump–Friedberg type period on $\textnormal{GL}_{2n+1}$, integrating over the subgroup $\textnormal{SL}_{n+1}\times \textnormal{GL}_n$. For certain Eisenstein series, we evaluate this period as a finite sum of products of special values of $L$-functions and normalized local zeta integrals. Assuming the expected global Langlands correspondence, the sum is indexed by the fixed points of the extended $L$-parameter on the conjectural dual variety, and the resulting $L$-factors agree with the tangent space prediction of the global numerical conjecture of Ben-Zvi-Sakellaridis-Venkatesh.

The limit states of canonical ensembles of a large number of interacting bosons at a given temperature, which are confined by harmonic forces, are studied in the framework of the resolvent algebra. It is shown that the limits satisfy the KMS condition or are ground states, regardless of the type of interaction. In case of attractive forces, where the ensembles collapse, observables that become meaningless in the limit disappear from the limit representations. For repulsive forces, this can also happen if condensates with an infinite number of particles in the same state (proper condensates) appear in the limit. The resulting structures and their interpretation are illustrated by a simple model. The study of vanishing harmonic forces (thermodynamic limit) involves changes of the dynamics. It is conveniently based on derivations acting on the algebra. They are given by the commutator of the Hamiltonians with the elements of the algebra. To ensure that the images remain in the algebra, the interaction must be regularized. This is accomplished in a manner that has only a minor impact on the dynamics and may be of broader interest. With this input a relation between the strength of the confining harmonic forces and the number of particles in the ensembles is derived from the condition that the limit states are to be stationary (invariant) under the adjoint action of the unconfined, spatially homogeneous limit dynamics. This relation encompasses the conditions that are frequently used in studies of Bose-Einstein condensates.

Reciprocal matrices are a fundamental tool in the Analytic Hierarchy Process (AHP), where priority vectors are typically derived from pairwise comparisons. The efficiency of a positive vector, in the sense of Pareto optimality, can be characterized through the strong connectivity of a directed graph $G_{A,w}$ associated with a reciprocal matrix $A$ and a vector $w$. In this paper, we investigate the structure of the condensation digraph of $G_{A,w}$ in the case where $w$ is inefficient, with particular emphasis on the Perron vector. We provide a characterization of this structure and derive several constructive results. In particular, we show how efficiency can be achieved by modifying a single pair of reciprocal entries, and we construct an augmented reciprocal matrix whose efficient vector extends $w$. Finally, we propose a procedure to transform $w$ into an efficient vector for $A$.

A triple $(S, +, \cdot)$ is called a left semibrace if $(S, +)$ is a semigroup, $(S, \cdot)$ is a group with identity $e$ and $x(y+z)=(xy)+(x(x^{-1}+z))$ for all $x, y, z\in S$, where $x^{-1}$ is the inverse of $x$ in the group $(S, \cdot)$. A left semibrace $(S,+,\cdot)$ is called strong if $x+y(e+z)=x+yz$ for all $x, y, z\in S$. Left semibraces and strong left semibraces are investigated extensively in literature. However, the structure of additive semigroups of general left semibraces still remains mysterious. In this note, as generalizations of near left braces, we introduce near left semibraces as follows. A quadruple $(S,+, \cdot, \mu)$ is called a near left semibrace if $(S, +)$ is a semigroup, $(S, \cdot)$ is a group, $\mu: S\to S$ is a map and $x(y+z)=(xy)+\mu(x)+xz$ for all $x, y, z\in S$. We first show that the additive semigroups of both left semibrace and near left semibraces are rectangular groups and obtain some new characterizations of strong left semibraces. In particular, we prove that a strong left semibrace can induce a near left semibrace, and vice versa. As a consequence, near left semibraces can provide set-theoretical solutions for the Yang-Baxter equation. Next, we obtain a structure theorem for all left semibraces by the generalized matched products of right zero left semibraces and right cancellative left semibraces. Finally, we consider a new map associated to a near left semibrace and give a sufficient and necessary condition under which such a map forms a set-theoretic solution of the Yang-Baxter equation. Our result improve and enrich some results obtained by Jespers and Van Antwerpen in [Forum Math. 31 (2019) 241–263], by Catino, Colazzo and Stefanelli in [Mediterr. J. Math. 17 (2020) 58] and by Catino, Mazzotta and Stefanelli in [J. Algebra 573 (2021) 576–619].

In this paper, we prove that for $p\geq 13$ and $5\leq n\leq p-3$, every reduction simple fusion system on the finite $p$-group $S(n,p)$ is block-exotic. Furthermore, we establish the block-exoticity of three infinite families of reduction simple exotic fusion systems.

We prove the existence of full dimensional KAM tori for infinite dimensional mechanical systems exhibiting long range interactions, under a Diophantine condition of Bourgain type [Bourgain2005JFA], in which the radius of the invariant tori satisfies a slower decay.

We study finite-horizon stochastic optimal control problems and approximate the resulting time-discrete formulation by a direct policy-learning problem over neural-network feedback maps. We prove a quantitative convergence estimate, in an averaged sense, for the error between the time-discrete value and the value induced by an approximately optimized neural policy. The bound separates the approximation of near-optimal feedback policies, the localization of stochastic trajectories on compact sets, and the optimization tolerance in training. The analysis does not require transition-density assumptions and covers possibly degenerate diffusions and deterministic controlled dynamics in a unified framework. Numerical experiments are provided for a degenerate stochastic radial target problem, a Hamilton–Jacobi–Bellman benchmark, and a gas storage problem, illustrating the approach and separating the main error sources: time discretization, restriction to piecewise-constant policies, neural-network approximation, and Monte Carlo evaluation.

We establish a homological rigidity phenomenon for the category of representations of quivers over the virtual field $\mathbb{F}_1$, which is inherently non-additive and does not admit classical homological algebra tools. We prove that all higher Yoneda extension groups vanish beyond degree two for arbitrary quivers, including infinite ones. Consequently, the global dimension of the category is universally bounded by 2. Moreover, we obtain a complete classification of quivers according to their homological dimension, which is determined solely by the underlying orientation structure.

We introduce locally subcartesian closed categories: categories with pullbacks equipped with a coherent choice of subobjects of pullbacks, such that the resulting affine base-change functors have right adjoints. We develop the basic theory, emphasizing the analogy with locally cartesian closed categories, and justify the terminology by showing that every slice category is monoidal closed with jointly monic projections. We also extend the theory of polynomials, showing that such a category gives rise to a bicategory of polynomials in the sense of Street, biequivalent to a 2-category of subcartesian polynomial functors. We illustrate this theory with the Lawvere quantale as a basic example and the category of nominal sets as a richer one. This work suggests a natural categorical semantics for an extensional dependent affine type theory with bunched contexts and affine implication.

In this paper, we study the nonparametric estimation of the squared diffusion coefficient of an ergodic diffusion process on non-compact supports, when the two coefficients of the stochastic differential equation belong to a space larger than the space of Hölder continuous functions. The estimators are constructed by projection onto finite-dimensional spaces, thereby avoiding truncation of the dimension. We establish risk bounds of non-adaptive estimators and explicit rates of convergence for bounded and unbounded diffusion coefficients. A model selection procedure is performed, followed by the derivation of risk bounds using Talagrand's inequality. The theoretical results are completed with a numerical study over simulated data.

Plug-and-Play (PnP) methods have emerged as a highly effective paradigm for solving imaging inverse problems by replacing traditional proximity operators of regularization terms with highly expressive deep denoisers. While empirically successful, establishing rigorous convergence guarantees for PnP algorithms remains a major challenge. Existing provable approaches based on the Gradient-Step (GS) denoiser suffer from theoretical and practical limitations, such as restrictive bounds on the regularization parameter, rigid step-size rules, and the inability to handle nonconvex data-fidelity terms. In this paper, we introduce PnP-IPA (Plug-and-Play Inexact Proximal Algorithm), a novel optimization scheme that overcomes these bottlenecks. We propose a new splitting strategy that evaluates the proximal operator of the scaled implicit regularizer inexactly. To enable adaptive step-size selection without exact objective evaluations, we design a novel surrogate merit function that successfully drives an Armijo-like backtracking line-search. Relying on the Kurdyka-Lojasiewicz property, we establish global convergence to a stationary point of the nonconvex objective without imposing any assumption on the regularization parameter. Extensive numerical experiments on image deblurring under both Gaussian and Cauchy noise demonstrate the practical advantages of PnP-IPA. By effectively lifting previous theoretical constraints, our method allows for optimal parameter tuning, yielding state-of-the-art restoration quality and robust convergence even in nonconvex regimes.

Toric quasifolds are nonrational generalizations of toric manifolds and orbifolds. We introduce the notion of nonrationality degree, an invariant that measures the extent to which a toric quasifold fails to be Hausdorff. We do so by first defining analogous notions for the underlying quasilattice and the corresponding quasitorus. We conclude by applying the nonrationality degree to reframe Gordan's lemma in the nonrational setting.

In this paper, we study a viscous conservation law with discontinuous flux. We define a weak solution concept, show its existence via an explicit formula, and prove that the weak solution is unique. In addition, we obtain the large-time behavior of the solution. The asymptotic limit of the solution is a steady-state solution. Here, the large-time asymptotic governed by the convection terms either converges to a constant steady state or to a non-constant steady state, depending on the behavior of the flux functions at a threshold value, which is the meeting point of two fluxes. This phenomenon does not occur in the single-flux case.

Nonautonomous saddle node bifurcations commonly appear in mathematical models, however, an optimal control theory tailored to such systems is not yet well established. We present conceptual work on the nonautonomous saddle node normal form. We develop an approach that reformulates the optimal control problem into an unconstrained minimization problem by exploiting a connection between bifurcation and rate-induced overshoots. The key component is a rate function that assigns the critical rate to every nonautonomous forcing. We show that in the saddle node normal form, the critical rate is unique, and hence the rate function is well defined. While this reformulation is valid in any case, the rate function is often unknown. To resolve this, we additionally present an approximation of the rate function that leads to a closed form expression of the approximated control. This makes the approach appealing in situations, in which we need repeated solves of the control problem.

The theory of simplicial complexes is a cornerstone of topology, offering a sophisticated tool for computing invariants. We present a formalization of abstract simplicial complexes and stellar subdivisions in the Lean proof assistant. We adopt a purely combinatorial framework in order to provide a cohesive foundation for studying the theory of stellar subdivisions as seen in many contexts of combinatorial topology. In particular, we provide formalizations of morphisms between abstract simplicial complexes; several crucial constructions and operations on complexes, such as links and joins; and perform a comprehensive study of how stellar subdivisions interact with these operations. We state and prove a number of identities commonly used in the study of triangulated manifolds, such as deriving equivalences between links in an abstract simplicial complex $K$ and in a stellar subdivision $\sigma_s K$, including results with no references in the standard literature. To our knowledge, this is the first formalization of stellar subdivisions in any proof assistant.

This paper presents a new analysis of the Chebyshev projection for multivariate analytic functions, drawing on pluripotential theory. It is proved that in any downward closed convex polynomial space, the Chebyshev projection achieves the same exponential convergence rate as the best polynomial approximation. This result enables a precise quantification of the exponential convergence rate of the Chebyshev projection. The analysis is then extended to several related topics, including tensorized Chebyshev interpolation, tensor product Gauss–Legendre quadrature, Padua interpolation and cubature, and Chebyshev-Galerkin method, with the corresponding exponential convergence rate established in each case. Supporting numerical experiments are provided to validate the theoretical results.

Data-driven optimization often requires collecting data to estimate uncertain model parameters before solving the underlying decision problem. In practice, however, data acquisition may incur non-negligible costs, making it critical to determine when to stop additional data collection. In this paper, we study an optimal stopping problem for sequential data collection in stochastic optimization under parameter uncertainty. We propose a benefit-driven stopping framework that balances information gain and sampling cost. We model the unknown distribution parameter within a Bayesian learning framework and update beliefs sequentially as new observations are collected. At each iteration, the decision maker evaluates the expected marginal benefit of additional data relative to the unit sampling cost and determines whether to continue sampling or stop and implement the optimization decision. Based on this framework, we develop several stopping policies. The proposed policies are evaluated through a newsvendor problem with exponentially distributed demand. Numerical experiments compare the policies with fixed-budget and hindsight benchmark strategies. The results show that benefit-driven stopping rules can substantially reduce unnecessary data collection while achieving near-optimal decision performance, demonstrating the effectiveness of adaptive stopping in data-driven optimization.

Anisotropic consensus-based optimization (CBO), a multi-agent metaheuristic derivative-free optimization method, which reliably finds global minima of nonsmooth and nonconvex objective functions while being amenable to rigorous theoretical analysis, automatically detects and exploits additively separable structures of high-dimensional objective functions. This enables the algorithm to mitigate the curse of dimensionality where the objective function decomposes additively into lower-dimensional components. In this paper, we show this property proving that the computational complexity of anisotropic CBO depends exponentially only on the intrinsic dimension $\mathbf{d}$ of the objective function, rather than the ambient dimension $D\gg\mathbf{d}$. Additionally, we demonstrate that the computational complexity depends only on tractability conditions of the lower-dimensional components rather than on the full energy landscape, allowing for a more refined description of the objective function and algorithmic complexity, as the objective function landscape is captured directly at the level of the individual components. Our results highlight the effectiveness of anisotropic CBO for additively separable objective functions provided sufficient alignment between the structure of the anisotropic noise and the separability structure of the objective. This motivates the design of an enhanced algorithm that learns during optimization how to effectively explore the loss landscape by aligning the noise with the structure of the objective function, which we leave for future research. Numerical experiments validate our theoretical results, accentuating the influence of the intrinsic dimensionality $\mathbf{d}$, the level of separability, and the complexity and non-convexity of the objective within the separable components on performance and computational complexity of the anisotropic CBO algorithm.

Equitable renewable-energy planning is a sequential decision problem, but the decision variables available to a public planner differ sharply between mature and emerging economies. In the former the government largely builds generation, while in the latter it steers private investment through incentives and quotas. We formulate socially-equitable renewable-energy budget allocation as a Markov Decision Process (MDP) and, using a single problem-agnostic solver interface, compare the same policies across the two settings: eight U.S. cities (a mature economy) and West Java, Indonesia (an emerging economy). The results show that across both settings, a receding-horizon value-iteration policy dominates. In the U.S., it reaches 66% renewable penetration while cutting the underserved low-income population by 96% versus a random baseline. In West Java it closes the low-access gap while crowding in about seven times the private capital of a naive market-chasing heuristic. More interestingly, this heuristic, which is mildly sub-optimal in the U.S., could yield catastrophic outcomes in Indonesia, by underserving every low-access region, because chasing attractive markets and serving the underserved diverge once the planner acts through private developers.

The Neural Tangent Kernel (NTK) is one powerful tool for analyzing the training dynamics of neural networks in the over-parameterized regime. Recently, the theoretical framework has been extended to physics-informed neural networks (PINNs) for solving linear PDEs, one highly popular class of neural PDE solvers. In the analysis, the positivity of the associated NTK plays a fundamental role. However, establishing the positivity of the NTK for PINNs is highly challenging, due to the presence of multiple differential operators. In this work, we propose a new theoretical framework, called Differential Neural Tangent Kernel (DNTK), for analyzing PINNs through the lens of the NTK, and establish the positivity of the infinite width DNTK for both shallow and deep neural networks for a wide class of activation functions, including RePU and smooth but non-polynomial activations, for all linear differential operators. These theoretical results lay the foundation for the analysis of gradient type algorithms for training PINNs.

We provide a partially functorial and constructive reconstruction procedure for formal schemes from symmetric monoidal categories of nuclear modules. More precisely, for a formal scheme $\mathfrak{X}$, we show that the torsion subcategory $D_{\mathrm{tors}}(\mathfrak{X})$ can be recovered as the maximal strongly compactly generated localizing tensor ideal of Efimov's category $\mathrm{Nuc}^{\mathrm{Ef}}(\mathfrak{X})$, and similarly for the Clausen–Scholze category $\mathrm{Nuc}^{\mathrm{CS}}(\mathfrak{X})$. Combining this with the Balmer spectrum, we reconstruct $\mathfrak{X}$ from the corresponding symmetric monoidal category of nuclear modules. Moreover, for formal schemes topologically of finite type over a field or over $\mathbb{Z}$, the contravariant functor $\mathfrak{X} \mapsto \mathrm{Nuc}^{\mathrm{Ef}}(\mathfrak{X})$ is fully faithful; in the affine case, the analogous statement holds for $\mathfrak{X} \mapsto \mathrm{Nuc}^{\mathrm{CS}}(\mathfrak{X})$.

Anyons in $2+1$ dimensions are not only characterized by exotic braiding statistics but also by intricate fusion properties. Two anyons may fuse into multiple topological charge sectors, and associativity of fusing three anyons to produce a fixed charge sector is governed by $F$-symbols. While braiding invariants, such as the modular data, are typically complex valued, a complete description of general anyon models requires understanding the arithmetic properties of its fusion associativity data as well. The $F$-symbols for many of the most common $2+1$d topological orders, including all Abelian anyon models as well as Fibonacci and Ising anyons, can be made real valued. We show this phenomenon is not universal by exhibiting braided fusion categories whose $F$-symbols cannot be made real. We call such $F$-symbols "inherently complex." The examples we study lack a charge-conjugation symmetry and our results are therefore consistent with the converse of a statement proved in a companion work linking real $F$-symbols in braided fusion categories with the existence of a suitable charge-conjugation symmetry. We analyse the smallest-rank braided fusion categories we know of with inherently complex $F$-symbols: ${\rm Rep}(\mathbb{Z}_7\rtimes\mathbb{Z}_3)$ and ${\rm Rep}(\mathbb{Z}_5\rtimes\mathbb{Z}_4)$. Consequently, the corresponding $\mathcal Z({\rm Rep}(\mathbb{Z}_7\rtimes\mathbb{Z}_3))$ and $\mathcal Z({\rm Rep}(\mathbb{Z}_5\rtimes\mathbb{Z}_4))$ anyon models also have inherently complex $F$-symbols. Our presentation connects these examples with recent results on classifying anyons beyond modular data.

For $\alpha \in \mathbb{C}$, the monogenic semidomain generated by $\alpha$ is the smallest subsemiring $S_\alpha$ of the complex field $\mathbb{C}$ containing $\alpha$. We initiate a systematic study of the arithmetic and factorizations of the monogenic semidomains $S_q$ generated by rational parameters $q$. After some preliminaries, we introduce and investigate the monoid of technical fractions $T_q$, which is a divisor-closed submonoid of the multiplicative monoid of $S_q$ that encodes a significant amount of arithmetic information about $S_q$. We then study several fundamental factorization properties of $S_q$: the bounded factorization (BF) and finite factorization (FF) properties, the unique factorization (UF) property, and the half-factorial (HF) property. First, we prove that $S_q$ satisfies the UF property if and only if it satisfies the HF property, which happens when $q \in \mathbb{N} \cup \mathbb{N}^{-1}$. We determine all the positive rational values of the parameter $q$ for which $S_q$ satisfies the FF property. Then we show that, over the class of rational monogenic semidomains, the BF property is equivalent to the ascending chain condition on principal ideals. Finally, we prove that $S_q$ is a Krull semidomain if and only if it is root-closed, which happens precisely when $S_q$ satisfies the UF property.