We study a categorical approach to the concept of varieties of chiral algebras. We prove that the class of chiral algebras in the variety defined by a binary quadratic operad Var, whose conformal structure is abelian, coincides with the class of differential algebras in the variety defined by the Manin black product of the operads Var and Com, where Com is the operad of associative commutative algebras.

In recent years, numerous polynomial invariants of knotoids have been constructed, some of which are defined with the signs of the crossings. In this paper, the coloring-allowed invariants of planar knotoids, which is a class of planar knotoid invariants defined with the coloring number are introduced. We demonstrate several basic properties of the coloring number and some examples of coloring-allowed invariants, among which the 4-phases functions are discussed in detail, including their invariance and properties.

The analysis of singularities in scalar curvature is a classical problem. In this survey, we investigate the behavior of scalar curvature under several geometric flows, with a focus on three specific cases: the Ricci flow, the Kähler{-}Ricci flow coupled with $(1,1)$-forms, and the Laplacian flow.

Higher-order ODE solvers have shown strong empirical promise for accelerating diffusion models through the probability flow ODE, but rigorous non-asymptotic guarantees for such acceleration remain limited. In this paper, we develop a Chebyshev--Gauss--Seidel higher-order sampler and establish a non-asymptotic convergence guarantee that allows the approximation order to grow logarithmically with the number of outer iterations. In the exact-score setting, up to logarithmic factors, the proposed sampler requires at most \[ d^{1+o_T(1)}\varepsilon^{-1/K_1} \] score functions to approximate the target distribution on \(\mathbb{R}^d\) within total variation distance \(\varepsilon\), where \(o_T(1)\to 0\) as \(T\to\infty\) and \(K_1>0\) is a sufficiently large constant. The analysis assumes only a polynomial second-moment bound on the target distribution, thereby relaxing the bounded-support condition imposed in existing higher-order theory. Moreover, the guarantee is robust to score and Jacobian estimation errors and does not require higher-order smoothness assumptions on the score estimates. Numerical experiments on anisotropic Gaussian mixture benchmarks support the predicted improvement in the accuracy--cost tradeoff under finite score-evaluation budgets.

We give formulas for the $ τ$ and $ \varepsilon $ concordance invariants of satellite knots whose patterns are braided, meaning they wind around the solid torus without reversing. Our methods lead us to define the class of squeezed patterns, analogous to squeezed knots as defined by Feller-Lewark-Lobb. We show that all braided patterns are squeezed, and we give a $ τ$ formula for squeezed patterns as well. Towards a conjecture of Hedden, we show that no squeezed pattern, and thus no braided pattern, with winding number at least 2 induces a homomorphism on the concordance group.

We study a bilevel network interdiction problem, with an attacker interdicting (attacking) certain components of a network and a defender optimizing operations over the ensuing network. We study when the defender's optimal objective is submodular or supermodular with respect to the attacker's interdiction decisions, for optimizing the bilevel integer program more efficiently. We first consider the min-cost flow (MinCF) interdiction problem and derive necessary and sufficient conditions for the supermodularity or submodularity to hold under three types of attacks, respectively on supplies/demands, flow capacities, and cost coefficients. We extend to other variants, including capacitated facility location, maximum flow (MaxF), and shortest path (SP) interdiction. The conditions hold under general network topologies and parameter settings, and depend solely on the locations of the attacks. We further incorporate additional network information (e.g., detailed parameters and special topologies) to establish less restrictive conditions. We also derive necessary and sufficient conditions for supermodularity or submodularity in SP and MaxF interdiction in series-parallel networks. Furthermore, we explore more challenging interdiction problems where the defender may make additional binary decisions (e.g., repairing or reinforcing the network) and identify conditions that preserve submodularity or recover supermodularity. Via extensive numerical studies with diverse types of attacks, we demonstrate an order-of-magnitude computational speedup achieved by exploiting these properties and generating valid inequalities, for solving network interdiction at scale.

Despite the popularity of Koopman modeling for nonlinear systems, in the presence of input variables, the evident nonexistence of a fully linear time-invariant model even in infinite dimensions makes Koopman-based control largely an open problem to date. Focusing on discrete-time systems in this paper, which eschews from using operator semigroup and infinitesimal generator notions, it is proven that nonlinear systems, if satisfying appropriate smoothness and regularity conditions, can be expressed exactly as bilinear dynamics, when the state variables and input variables are separately lifted into their reproducing kernel Hilbert spaces (RKHSs). To account for the knowledge of an equilibrium point at the origin, the RKHS is defined by a linear--radial product kernel, and hence the functions belonging to this RKHS are spanned by the multiplications of component functions and Sobolev functions. The stabilization problem, namely the determination of a feedback law that causes a Lyapunov function (expressed as a kernel sum-of-squares form) to decrease, is then posed as an infinite-dimensional optimization problem over state-dependent conditional probability measures over the input space, solved via a discretization scheme.

We give a novel proof of the Gorin-Shkolnikov identity based on Pitman's SDE representation of Brownian excursion local times. More generally, we derive a family of Gaussian identities for nonlinear functionals of excursion local times, which includes Hariya's identity. These results extend naturally to reflected Brownian bridges conditioned on their local times and to Brownian meanders.

In 2025 Cornet, Dravec, and Torres determined the domination number $γ(J(n, 3))$ of the Johnson graph for every even $n \ge 6$, expressing it as a closed form $ϕ_n$ in terms of Fort\textendash{}Hedlund covering numbers, and conjectured the same value for odd $n$. We prove this conjecture: $γ(J(n, 3)) = ϕ_n$ for every odd $n \ge 7$, completing the determination of $γ(J(n, 3))$ for all $n \ge 6$.

In this paper, we prove an invariance principle for the additive functionals of a family of voter models, which include the nearest-neighbor cases on $d$-dimensional lattices for $d$ at least $5$ or on regular trees with degree at least $3$ and some long-range cases on lattices as special examples. The proof of our main result extends a martingale decomposition method, where the Kolmogorov backward equation and the duality relationship between the voter model and the coalescing random walk play the key roles.

The concept of downward conditional monotonicity for the Markov-modulated Poisson process (MMPP) is introduced and used to derive the optimal stochastic domination of a standard Poisson point process. The maximum arrival rate for the Poisson process which allows this domination to exist is shown to be related to an eigenvalue extracted from the generator matrix of the quasi-birth--death (QBD) formulation of the MMPP. This allows derivation of survival and extinction regimes for a large family of contact processes whose infection and recovery rates vary over time according to an underlying random environment with a finite number of states. Direct comparison with standard contact processes which dominate from above and below accomplishes this.

In this paper, we consider a Gorenstein-dimensional analogue of Foxby's characterization of Gorenstein rings. We prove that a commutative Noetherian local ring is Gorenstein if it admits a complex whose depth, Gorenstein flat dimension, and Gorenstein injective dimension are all finite. This gives an affirmative answer to the original question of Christensen, Foxby, and Holm, which had remained open in this generality even for modules, and at the same time establishes its natural extension to complexes.

We consider an inverse boundary value problem for the time-independent Schrödinger equation in dimension three. We prove that the local Dirichlet-to-Neumann map defined near a boundary point uniquely determines the potential in a neighborhood of the boundary point in the interior. In particular, we show that the uniqueness question can be reduced to the injectivity of a weighted X-ray transform, which links inverse boundary value problems to integral geometry.

Conditional on the Generalized Riemann Hypothesis for $L(s,χ)$, we prove the Selberg--Tsang high-moment bound for $X_χ(t) = \operatorname{Im}\log L(\tfrac12+it,χ)$ at fixed squarefree odd conductor $q \ge 3$ and primitive non-principal character $χ$. Writing $L_T = \log\log(qT)$: for every $K > 0$ there exist constants $C_K$ and $T_0$ such that $\frac{1}{T}\int_T^{2T} |X_χ(t)|^{2k}\,dt \le (C_K\,k\,L_T)^k$ for all $T \ge T_0$ and every integer $1 \le k \le K L_T$. The proof ports Selberg's pointwise approximate formula for $S(t)$ to $L(s,χ)$ at fixed conductor under GRH, splits it into three prime-power Dirichlet polynomials, and evaluates their moments via Soundararajan's mean-value lemma. As a corollary, Markov's inequality yields a Gaussian-scale tail $\exp(-c V^2 / L_T)$ for $\sqrt{L_T} \ll V \ll L_T$ -- a GRH-conditional, fixed-conductor, imaginary-part analogue of the large-deviation upper bounds known for $\log|ζ(\tfrac12+it)|$.

We classify, up to almost diffeomorphism, the smooth closed oriented manifolds homotopy equivalent to each of the sphere products $S^{4k-1}\times S^{4k}$, $S^{4k}\times S^{4k}$, and $S^{4k}\times S^{4k+1}$. In each case we realize the image of the normal-invariant map in the smooth surgery exact sequence by explicit families of manifolds: sphere bundles over $S^{4k}$; pinch maps and Milnor plumbings of disk bundles; and Novikov sphere bundles together with connected sums of homotopy spheres.

We consider a two-dimensional piecewise $C^2$ domain that cuts through a quasi-uniform fixed polygonal background mesh, for instance made of quadrilaterals. A simple procedure based on convex hulls gives rise to a rather small number of polygonal boundary elements of various shapes, including elements with small edges and large aspect ratios; this is the computational mesh for a virtual element method (VEM), a trimmed background mesh. We classify all possible geometric configurations and study their stability and approximability properties. This entails deriving robust stabilization mechanisms and interpolation estimates for anisotropic elements and elements with small cuts, as well as a weak maximum principle for enhanced virtual elements; these contributions have intrinsic interest for VEM theory on geometric flexibility. We prove that the resulting VEM is uniformly stable in $H^1$, and also show optimal order-regularity error estimates in $H^1$ and $L^2$. Insightful numerical experiments corroborate and complement our theory. The proposed method is suitable for treating ALE formulations of problems in moving domains.

Fix a prime $p$ and a constant $\frac{1}{p}<α\leq 1$. Consider the random directed Erdős--Rényi bipartite graph $\vec G(n,\lceilαn\rceil ,v)$ with bipartition $(V_1,V_2)$ of sizes $|V_1|=n$ and $|V_2|=\lceilαn\rceil$, and edge probability $0<v<1$. Bhargava, DePascale and Koenig conjectured a limiting distribution for the $p$-Sylow subgroup of the sandpile group of $\vec G(n,\lceilαn\rceil,v)$ as $n\to\infty$. We prove this conjecture. Similar results have previously been proved by computing the expected number of surjections from the random abelian $p$-group onto $H$, for each finite abelian $p$-group $H$. However, in the case of $p$-Sylow subgroups of sandpile groups of random directed bipartite graphs, these surjective moments often diverge to infinity, despite the conjectured limiting distribution having finite moments. We resolve this by restricting to a high-probability subset of graphs on which the surjective moments are well-behaved, and discarding a rare exceptional set of graphs whose contribution to the distribution vanishes but whose contribution to the surjective moments often diverges. Computing the conditional surjective moments on the good set and applying Wood's universality theorem yields the desired convergence in distribution.

We classify Galois self-covers $f: \mathbf{P}^n \to \mathbf{P}^n$ and show that the essential dimension of every such nontrivial cover attains its maximum possible value $n$.

We investigate a singular limit of a system of Patlak-Keller-Segel (PKS) equations modeling the evolution of multiple interacting species. Our primary motivation is the Differential Adhesion Hypothesis (DAH), introduced by Malcolm Steinberg in 1962, which posits that cell populations self-organize by minimizing adhesion energy, in a manner analogous to fluids minimizing surface tension. Our starting point is a continuum model describing the evolution of the density distributions of $N$ distinct species, representing different cell types. These species interact through attractive nonlocal forces governed by interaction kernels of similar form but different strengths (the $N\times N$ matrix of interaction coefficients encodes the key properties of the system). These attractive forces are balanced by a nonlinear pressure, depending on the total density and strong enough to prevent concentration. In the limit of short-range interactions, we establish sufficient conditions on the interaction matrix for cell sorting to take place (i.e., the spontaneous separation of the different species). We then prove a general $Γ$-convergence result for the associated energy functional, showing convergence to an interfacial energy where the surface tension coefficients are determined by a geodesic problem. A detailed analysis of this problem allows us to identify regimes in which engulfment occurs (more adhesive species cluster together and are surrounded by less adhesive ones) which is a key feature of the DAH. In the second part of the paper, we analyze the asymptotic behavior of solutions to the PKS system in the combined long-time and short-range interaction limit. Under a standard energy convergence assumption, we prove the convergence to a multiphase Hele--Shaw problem with surface tension (and contact angle conditions at triple junctions).

Finiteness of the right little finitistic dimension of a finite dimensional algebra $\La$ is known to be equivalent to existence of a (possibly infinite dimensional) tilting right $\La$-module $T_f$ whose tilting class is $\{ T_f \}^{\perp_\infty} = (\mathcal P ^{< \infty})^{\perp}$, \cite{AT}. We use this equivalence to interpret the recent surprising results of Cummings \cite{C} concerning the asymmetry of left and right finitistic dimensions of the triangular matrix algebras $\tilde{A}$ built from arbitrary basic finite dimensional algebras $A$. In particular, we determine the structure of the tilting right $\tilde{A}$-module $T_f$ in the case when the algebra $A$ has finite right global dimension.

We compare two existence principles for maximal ideals, a classical one for vector lattices with a strong unit and a second, newly introduced one for vector lattices with a strong truncation. Although the latter strictly generalizes the former, we show that the two statements are equivalent over ZF set theory.

Nivat's conjecture asserts that every two-dimensional configuration $η: \mathbb{Z}^2 \to \mathcal{A}$ whose rectangular pattern complexity satisfies $P_η(k,n) \leq kn$ for some $k,n \in \mathbb{N}$ is periodic. A theorem of Cyr and Kra \cite{CyrKra16} establishes the conjecture in the short-rectangle case $P_η(k,n) \leq kn$, with $k \leq 3$. Using the algebraic framework of Kari-Szabados \cite{KariSzabados20} and recent advances on periodic decompositions and one-sided nonexpansive directions \cite{Colle23,Colle22}, we extend the Cyr-Kra result to the case $P_η(4,n) \leq 4n$: every configuration satisfying this complexity bound is periodic. The key new ingredient is an intermediate structural theorem of independent interest: for any non-periodic configuration with low convex pattern complexity and integer-valued alphabet $\mathcal{A}$ contained in $\mathbb{Z}_+$, there exist a configuration $\vartheta$ in the orbit closure of $η$, a $\mathbb{Z}$-minimal periodic decomposition $\vartheta = \vartheta_1+\cdots+\vartheta_m$, a prime $p \in \mathbb{N}$ with $\mathcal{A} \subset [[p]]$, and pairs of disjoint half-planes $U_i,V_i \subset \mathbb{Z}^2$ such that the reductions modulo $p$ of the components $\vartheta_i$ are fully periodic on $U_i$ and on $V_i$ simultaneously, for each $1 \leq i \leq m$.

Submodular set functions, characterized by the diminishing-returns property, provide a unifying combinatorial framework for many subset-selection problems in decision and control. Although exact maximization is NP-hard in general, the structural properties of submodular functions enable simple greedy algorithms that achieve constant-factor approximation guarantees for monotone objectives, with randomized greedy-based variants extending such guarantees to the non-monotone case. This survey reviews the theory, algorithms, and applications of submodular optimization with a focus on systems and control. We cover the structural properties of submodular functions, including curvature and the submodularity ratio, the constraint families that arise in practice (matroids, knapsack, and $p$-systems), and the main approximation algorithms for monotone and non-monotone submodular maximization, with up-to-date approximation ratios and hardness results. We then survey applications across sensor scheduling, multi-agent coordination, robust submodular optimization, leader-follower systems, distributed submodular optimization, game theory, system theory, resource allocation, social networks, and informative path planning. The survey emphasizes practically implementable greedy-based algorithms and instance-dependent refinements via curvature and the submodularity ratio. We close with observations on canonical control-theoretic objectives: certain functionals are submodular (the log-determinant and rank of the controllability Gramian, and the log-determinant of the Kalman filter information matrix), whereas closely related objectives fail to be sub- or supermodular (the steady-state Kalman filter error covariance, and the average control energy obtained from the inverse Gramian). We also highlight the cross-cutting open directions that follow.

In this paper, we investigate several coefficient problems for the subclass $\mathcal{S}_{ex}^{\ast}$ of normalized analytic starlike functions defined by the subordination condition $\frac{zf'(z)}{f(z)} \prec e^{αz}$ ($0 < α\le 1$). We first obtain sharp upper bounds for the initial inverse logarithmic coefficients $Γ_1, Γ_2$, and $Γ_3$. We then establish sharp upper and lower bounds for the consecutive difference $|Γ_2| - |Γ_1|$ and determine the sharp bounds for the second-order inverse logarithmic Hankel determinant $H_{2,1}(F_{f^{-1}}/2)$. Furthermore, sharp upper and lower bounds for the third-order Hermitian--Toeplitz determinant $T_{3,1}(f)$ are derived. Finally, we provide a complete solution for the generalized Fekete--Szegő functional $|a_3 - λa_2^2| - μ|a_2|$ within this class. In each case, the bounds are shown to be sharp and the corresponding extremal functions are explicitly given.

We establish higher order boundary Harnack estimates for solutions to kinetic Fokker-Planck equations with absorbing incoming boundaries. Unlike classical elliptic and parabolic equations with Dirichlet data, we show that the quotient of two solutions for kinetic equations is not $C^{\infty}$ up to the boundary. Instead, we develop a general theory showing that, near the grazing set, the quotient of two solutions is $C^{3/2}$ if the domain and data are sufficiently smooth, and $C^{1,1}$ in the absence of source terms. These exponents are optimal.

Let $X$ be a smooth projective curve over a field $k$ with an action of a finite group $G$. A well-known result of Chevalley and Weil describes the $k[G]$-module structure of cohomologies of $X$ in the case when the characteristic of $k$ does not divide $\# G$. In case when $G$ has a cyclic $p$-Sylow subgroup, it is known that the $G$-structures of the module of holomorphic differentials and of de Rham cohomology of $X$ are completely determined by the ramification data of the cover $X \to X/G$. In this article we extend this result to the crystalline cohomology of $X$. Also, we provide an explicit description of the structure of the crystalline cohomology when $G = \mathbb Z/p^n$. The main used tool is the theory of Yakovlev diagrams - algebraic objects that classify the $G$-modules over Witt vectors.

We show that the partitions that always appear with nonnegative $e$-coefficients in chromatic symmetric functions of finite graphs are precisely the hook partitions.

An element of a group is called generalized torsion if a finite product of its conjugates equals the identity. We characterize the generalized torsion subset of upper triangular matrix groups over commutative rings and determine when these groups satisfy generalized identities in terms of torsion and exponent properties of the unit group of the underlying ring. As applications, we construct new families of infinite finitely generated solvable groups satisfying generalized identities of prescribed lengths and provide a counterexample to a question of Sherman.

We study the frog model with death and drift on $\mathbb{D}_{m,d}$, the free product of $d+1$ copies of the complete graph of order $m$. Active and inactive particles are located at the vertices of $\mathbb{D}_{m,d}$. Each active particle performs a $α$-biased random walk towards the root of $\mathbb{D}_{m,d}$, dying after a random lifetime with a geometric distribution of parameter $1-p$. Each inactive particle remains dormant until an active particle visits its location. We present conditions on the parameters $α$ and $p$ for the process to die out almost surely and to survive with positive probability. Our proofs are based on comparisons of the model with simple and multi-type branching processes.

We analyze the complexity of Bidirectional Stochastic Dual Dynamic Programming (BSDDP) algorithm applied to multistage stochastic optimization problems with strongly convex cost functions. Under standard regularity assumptions and without discount factor, we establish an explicit complexity bound on the expected number of iterations needed to obtain an epsilon-optimal policy. Our bound depends on the strong convexity constants and Lipschitz constants of the stagewise cost functions but is independent of the dimensionality of the state space. This improves upon the complexity of SDDP for convex problems from (Lan, 2020), particularly when the number of stages is moderate and the state dimension is large. The analysis further reveals that the complexity decreases as the strong convexity of the cost functions increases, highlighting the algorithm efficiency in structured problem instances.