We investigate monomial ideals satisfying the non-pure dual exchange property, a notion introduced in connection with componentwise polymatroidal ideals. Our contributions are twofold. First, we show that in two variables, every integrally closed monomial ideal satisfies this property; as a consequence, we characterize polymatroidal ideals in two variables. Second, for strongly stable (Borel) ideals in three variables, we establish a practical criterion: it suffices to verify the defining condition only for the Borel generators, and this verification reduces to simple inequalities involving the degrees in the second and third variables.

Zhou et al. (2025) proposed a distance Laplacian analog of Brouwer's conjecture on partial sums of Laplacian eigenvalues, asserting that for any connected graph $G$, $\sum_{i=1}^r \partial_i^L(G)\le W(G)+\binom{r+2}{3},$ where $\partial_i^L(G)$ are the eigenvalues of the distance Laplacian matrix and $W(G)$ is the Wiener index. We prove this inequality for three broad classes of graphs, thereby improving and extending existing results. First, we prove that all connected graphs of diameter at most $D$ satisfy the inequality once the order $n$ satisfies $n\ge\lceil\frac49(D+1)^3\rceil$. Second, we show that the inequality holds for every diameter-$2$ graph with the only exceptions being $K_{1,3}$ at $r=2$ and $K_{1,4}$ at $r=3$. Third, we prove that if the maximum degree is $Δ(G)=n-k$, then the inequality holds for all $n\ge N(k)$, where $N(2)=10$ and $N(k)=\lceil 5(k-1)^{3/2}\rceil$ for $k\ge 3$. Our proofs rely on decomposing the distance Laplacian matrix into Laplacian matrices of auxiliary graphs whose edges are vertex pairs at distance at least a prescribed value, together with classical eigenvalue inequalities.

The present paper is devoted to studying derivations and local derivations on the Euclidean Lie algebras $\mathfrak{e}(n)$. We give a complete desrciption of the derivation algebra of $\mathfrak{e}(n)$ for $n\geq 4$. Furthermore, we prove that every local derivations on the $\mathfrak{e}(n)$ is a derivation.

We study finite adaptability for decision-making under uncertainty, where a small set of candidate solutions is prepared in advance and the best response is selected after uncertainty is realized. While existing methods have made significant progress on exact formulations, scalability remains a persistent challenge due to (i) the combinatorial nature of assigning decisions to uncertainty realizations, and (ii) the joint optimization of uncertainty set partition and subsequent decisions. We propose a framework that makes the partition of the uncertainty set explicit and uses polyhedral partitions as the basis for policy design. Under mild regularity conditions and for general risk measures, we show that such policies converge to the optimal fully adjustable policy as the number of regions increases. Building on this result, we develop a parametric partition framework that allows flexible policy design with tractable reformulations for both robust and stochastic finite adaptability problems. To improve scalability, we introduce an approximate-learn-parallel framework that integrates partition learning with parallel optimization while preserving solution robustness. Computational experiments on classical testbeds in both robust and stochastic settings show that the proposed method scales to larger instances and yields competitive policy performance.

Let $W(n)$ be the finite-dimensional simple Lie superalgebra of fundamental type in the Cartan type series of Kac's classification result \cite{Kac77} over an algebraically closed field of characteristic $0$. Let $\mathbf{g}$ be the graded-zero part of $W(n)$ which is isomorphic to $\mathfrak{gl}(n)$. In the first part of this paper, following the basic idea of taking the ``minimal" parabolic subalgebra $\mathsf{P}$ as a working platform in \cite{DSY} we introduce the Whittaker category $\mscrw$ for representations of $W(n)$ associated with a nilpotent element $e$ in $\mathbf{g}_0$ and with $W(n)_{-1}$. This Whittaker category turns out to be close to the classical Whittaker category McDowell and Miličić-Soergel studied in \cite{Mc} and \cite{MS}, respectively (or see \cite{Back}). We finally classify the simple objects in $\mscrw$. In the second part, we introduce the finite $W$-algebra associated with $e$, we then establish a generalized Skryabin's equivalence between the representation category of the finite $W$-superalgebra and the category $\mscrw'$ of so-called weakened Whittaker modules over $W(n)$. Here $\mscrw'$ naturally contains $\mscrw$ as a full subcategory.

We study the dynamics of planar sets under area-preserving crystalline curvature flow. We prove under mild assumptions on $ϕ$ that the flat flow solution from regular initial data coincides with a classical ODE evolution, extending the results of \cite{Almgren1995} to the area-preserving setting. We also show that for arbitrary initial data, the flat flow converges exponentially in time to a disjoint union of Wulff shapes, and under a non-bubbling assumption, the flow eventually becomes regular. Both of these results are novel for area-preserving crystalline flow of general sets, \emph{i.e.} without assuming geometric properties such as convexity or star-shapedness. A key ingredient of independent interest is that planar almost-minimizers are Lipschitz $ϕ$-regular, which we prove by exploiting a sharp minimality estimate for distinguished line segments, as opposed to the excess decay argument given in \cite{Ambrosio2002regularity}. The novelty of our approach lies in the application of $ϕ$-minimal barriers for energy competition arguments, both for the geometric rigidity of the discretized flat flow and for the regularity of almost-minimizers.

Let $d_r(n)$ be the maximum, over all binary phylogenetic trees with $n$ leaves, of the minimum number of $r$-state characters required to define the tree. Semple and Steel proved that $d_r(n)\geq\lceil(n-3)/(r-1)\rceil$, and Bordewich and Semple proved that equality holds for each fixed $r$ and all sufficiently large $n$. We study the corresponding threshold $n_r$, the least $N$ for which equality holds for every $n\geq N$. The Bordewich--Semple construction yields an explicit polynomial upper bound of order $O(r^5)$ for this threshold. We prove the near-linear estimate \[ 3r+1\leq n_r\leq \ceil{64(r-1)\log_2(r+1)}+3\qquad(r\geq4). \] The proof constructs, for every binary phylogenetic tree with $m=n-3$ internal edges, a linked quartet certificate whose conflict graph has maximum degree at most $16\lceil\log_2(m+2)\rceil+4$. Equitable coloring then packs the certificate into exactly $\lceil m/(r-1)\rceil$ $r$-state characters once $m\geq\lceil64(r-1)\log_2(r+1)\rceil$. We also include the lower bound $n_r\geq3r+1$, obtained from the snowflake obstruction, and state the natural conjecture that this lower bound is the exact threshold for all $r\geq4$. The conjectural endpoint is consistent with the known small-state thresholds: $n_4=13$ and $n_5=16$, while the cases $r=2,3$ are also explicitly classified.

This paper presents a successive convexification framework for trajectory optimization under continuous-time Signal Temporal Logic (CT-STL) specifications. The framework employs generalized mean-based robustness (GMSR), a smooth and exact parameterization of discrete-time STL, as a logical building block for constructing differentiable CT-STL constraints in optimal control. It is integrated with time-dilation for free-final-time problems, finite-dimensional control parameterization, multiple-shooting discretization of the dynamics, and a convergence-guaranteed sequential convex programming method, prox-convex, to solve the nonconvex program. The main CT-STL realization embeds temporal aggregation into augmented continuous-time dynamics. This augmentation-based construction is largely transcription-independent, can be incorporated into existing optimal-control pipelines with minimal structural changes, and enables smooth CT-STL parameterizations with accuracy controlled by a user-selected regularization parameter. We also discuss a complementary dense-time realization that evaluates CT-STL formulas directly on the integration subnodes used for dynamics discretization, yielding a smooth and exact parameterization on the numerical trajectory representation, up to the accuracy of the integration scheme. The proposed GMSR-based formulations mitigate the locality and gradient-masking behavior of standard quantitative semantics and therefore provide a favorable landscape for gradient-based trajectory optimization. The framework is demonstrated through trajectory-optimization examples for a double-integrator system with continuous-time \always{}, \eventually{}, and \until{} specifications, and a 6-DoF quadrotor flight problem with combined \always{}, \implication{}, and \eventually{}-type specifications. The implementation is available at https://github.com/UW-ACL/TrajOpt_CT-STL.

We define and study the rectangular finite free heat flow, a dynamical system on polynomials that plays the role of the heat equation in the setting of rectangular finite free probability. We show several equivalent characterizations of the evolution (including PDE and gradient flow formulations), establish basic properties of the dynamics, and determine the asymptotic distributions of the polynomial roots in the long-time and high-degree limits. We also discuss connections with Calogero-Moser systems and Dunkl processes, and we show that the rectangular finite free heat flow describes the mean curvature expansion of a family of compact Lie group orbits.

Let $M$ be a simple connected binary matroid, and let $e$ and $f$ be distinct elements of $M$. It is well known that, when the only circuits containing $e$ are spanning, $M$ is a circuit with at least three elements. This paper proves that if every circuit containing $\{e,f\}$ is spanning, then the canonical tree decomposition of $M$ is a path in which each vertex is labeled by a circuit, a copy of $U_{1,3}$, or a binary spike having one non-tip element deleted.

We study parabolic second-order tangent sets of semialgebraic sets and their use in local polynomial optimization. For a basic closed semialgebraic feasible set, we compare the true parabolic tangent set with the algebraic second-order linearized set determined by gradients and Hessians of the active constraints. Under directional rank stability and semialgebraic parabolic arc-realizability, the outer, inner, and arc-generated tangent sets coincide with this algebraic model. Exact formulas are obtained for smooth hypersurfaces, regular complete intersections, smooth inequality systems, and stratified semialgebraic sets. These formulas yield algebraically checkable second-order necessary conditions and sufficient conditions for quadratic growth in polynomial optimization. Examples show how the theory detects curvature, flatness, branch dependence, and the failure of ordinary quadratic scaling.

Flexible intelligent metasurface (FIM) technology has emerged as a promising technology for enhancing wireless communication performance by dynamically reshaping the propagation environment. Compared with conventional rigid reconfigurable intelligent surfaces (RIS), an FIM is composed of multiple electromagnetic (EM) scattering units, each of which can flexibly modify its displacement in the direction normal to the surface, thereby cooperatively morphing the overall surface shape. This additional degree of freedom (DoF) enables improved beamforming and interference mitigation, particularly in complex multicell scenarios. In this paper, an optimization problem for maximizing the weighted sum-rate (WSR) in a multicell multi-user multiple-input single-output (MU-MISO) system assisted by an FIM deployed at the cell boundary is investigated. We jointly optimize the transmit beamforming at the base station (BS), the phase shift matrix, and the FIM surface shape, subject to constraints on the transmit power budget, unit-modulus reflection coefficients, and surface shape morphing range. Due to the non-convex objective function with highly coupled variables, solving the formulated optimization problem is challenging. To tackle this challenge, we propose an efficient alternating optimization framework that leverages the weighted minimum mean square error (WMMSE) method to reformulate the problem and the block coordinate descent (BCD) algorithm to iteratively update the variables. Specifically, the Riemannian conjugate gradient (RCG) algorithm is leveraged to optimize the phase shift matrix, while the projected gradient descent (PGD) method is adopted to optimize the surface shape of the FIM. Additionally, the optimal beamforming vectors are obtained in closed form.

Motivated by the B-coloring defined by Gyárfás and Sárközy, we introduce a new edge coloring called \emph{D-coloring}. For a graph $G$, a D-coloring of $G$ is a proper edge coloring such that every diamond subgraph is rainbow. The \emph{D-chromatic index} of $G$, denoted by $χ'_D(G)$, is the minimum number of colors needed for a D-coloring of $G$. Denote by $Δ$ the maximum degree of $G$. We prove that $χ'_D(G)\le \frac{9}{16}Δ^2+\frac{1}{2}Δ$, conjecture that $χ'_D(G)\le \frac{1}{2}Δ^2+\frac{1}{2}Δ$, and verify this conjecture for $Δ\le 5$.

For a graph G = (V, E) embedded in the Klein bottle, let F(G) denote the set of faces of G. A C_4-face-magic Klein bottle labeling on G is a bijection f: V(G) to {1, 2,..., |V(G)|} such that for any F in F(G) with F isomorphic C_4, the sum of all the vertex labelings along C_4 is a constant. We say that a C_4-face-magic labeling X={x_{i,j} : 0< i,j< 5} on the 4x4 Klein bottle grid graph is horizontally (or vertically) pairwise balanced if x_{2i-1,j} + x_{2i,j}=17 for 0< i <3 and 0< j \le <5 (or x_{i,2j-1} + x_{i2,j}=17 for 0< i <5 and 0< j <3). We show that the 4x4 Klein bottle grid graph has 192 C_4-face-magic labelings up to symmetries on a Klein bottle. We classify these labelings into two categories depending on whether a C_4-face-magic label preserving permutation of the labeling is either horizontally pairwise balanced or vertically pairwise balanced. These results extend known results on C_4-face-magic labelings on an mxn Klein bottle grid graph.

In this paper, we study the Seshadri constants and Gromov widths of polarized toric varieties whose moment polytopes are generalized permutohedra. We show that both invariants coincide with the lattice width of the moment polytope, and provide an explicit formula for them in terms of the defining submodular function.

We introduce a measure-based nudging framework for assimilating macroscopic observations into microscopic mean-field particle dynamics. The central difficulty is a representation mismatch: the forecast is a labeled particle system, while the observations specify only a smoothed, permutation-invariant density. To address this mismatch, we define the forecast-observation discrepancy as a quadratic functional on probability measures after applying the same smoothing operator used by the observation process. The Wasserstein gradient of this functional induces a transport velocity on state space, which yields a particle-level correction without constructing particle-to-particle matching, linearizing the dynamics, or estimating ensemble covariances. For a fixed observation scale, we prove well-posedness of the assimilated McKean-Vlasov dynamics and propagation of chaos for the interacting particle approximation. Under exact smoothed observations and an observability condition at the kernel scale, we establish an $L^2$-stability estimate showing exponential decay up to a bias floor controlled by model misspecification. Numerical experiments on linear, bimodal, chaotic, kinetic, and collective-motion systems demonstrate that the method can recover macroscopic structure from incomplete density-level observations.

This paper studies information rates of noisy duplication channels with memory, motivated by nanopore DNA sequencing. In nanopore sequencing, the measured signal is affected by both inter-symbol interference (ISI), caused by multiple DNA bases residing in the pore, and random sample duplications, where variable translocation speed causes each base to generate a random number of samples. These two effects make direct theoretical analysis difficult. To address this, we derive a new decomposition of the information rate into two interpretable terms: one associated with the intrinsic memory of an auxiliary ISI channel, and another that captures the uncertainty in the segment boundaries caused by random duplications. This decomposition separates the dominant channel distortions and replaces the direct analysis of the full channel with two more readily interpretable components. We then study the second term through a soft alignment functional closely related to Soft-DTW, which enables strong AEP results and an alternative proof of the Markov-constrained coding theorem based on strong information stability. Finally, we develop a lower bound on the information rate that depends on the distribution of jump distances between adjacent nanopore levels. This bound gives a simple geometric explanation of channel synchronisability and provides a tractable framework for computing achievable rates of Oxford nanopore sequencers.

Let $K_\mathrm{KR}$ denote the optimal Kalton--Roberts constant for approximately additive real-valued set functions on algebras of sets. Kalton and Roberts proved $K_\mathrm{KR}\le89/2$, and Bondarenko, Prymak, and Radchenko improved the upper bound to $38.8$. We prove that $$K_\mathrm{KR}\le\frac{694,198,146,664,396,294,486,127,753}{34,994,834,677,886,019,996,000,000}\,\approx 19.837.$$ Thus the original Kalton--Roberts upper bound is more than halved. The proof changes the source collections fed into the expander-recombination step however still uses expander graphs as the other proofs do. The four expander families used in the final recombination are certified by exact rational interval arithmetic, and the proof has been formalised in Lean.

This paper focuses on parallel-plate electrostatic actuators, which are fundamental structures found in Micro-Electro-Mechanical Systems (MEMS) for numerous modern devices. It considers the equations used to model these actuators. In particular, we investigate the behavior of the solutions to a MEMS model that is represented by a second-order ordinary differential equation, where the spring is characterized as soft, linear, or hard. We present the mathematical structure underlying the pull-in and touchdown phenomena that characterize the behavior of systems incorporating soft and hard spring, based on the behavior of linear springs. The pull-in phenomenon corresponds to the degeneration of the solutions to the model equation. The touchdown phenomenon corresponds to the finite-time singularity of the solutions. These global dynamics and structures are systematically revealed through geometric approaches based on Poincaré-type compactification, the center manifold theorem, and blow-up technique. These methods successfully induce and resolve the dynamics at infinity.

We study the spectral properties of a class of unbounded linear operators on a non-Archimedean Hilbert space $E_ω$. More precisely, we consider operators of the form \[ T=D+F,\qquad F=\sum_{j=1}^{m} u_j\otimes v_j, \] where $D$ is an unbounded diagonal operator and $F$ is a finite-rank perturbation. This work extends the spectral analysis of Diagana and McNeal for rank-one perturbations of diagonal operators to the case of arbitrary finite-rank perturbations. The main objective is to describe the spectrum, point spectrum, and essential spectrum of such operators in terms of the diagonal sequence associated with $D$ and the Fredholm properties of $λI-T$. The theory of Fredholm operators plays a central role, particularly in the computation of the essential spectrum and in the study of stability under finite-rank perturbations.

We establish optimal inequalities relating the systole and the inradius to the volume of finite-volume hyperbolic 3-manifolds. In the cusped orientable case, we refine a theorem of Gendulphe by proving a sharp systole-volume inequality whose unique extremal manifold is the figure-eight knot complement. Excluding the figure-eight knot complement, we obtain a stronger inequality whose extremal manifold is the sister of the figure-eight knot complement. We also establish analogous optimal systole-volume inequalities for closed orientable hyperbolic 3-manifolds, where the extremal manifolds are the Weeks-Matveev-Fomenko manifold, the manifold Vol3, and the Meyerhoff manifold. In the second part of the article, we study the inradius. We prove optimal inradius-volume inequalities for orientable and nonorientable cusped hyperbolic 3-manifolds, identifying respectively the sister of the figure-eight knot complement and the Gieseking manifold as the extremal cases. We also prove that the Gieseking manifold is the unique cusped hyperbolic 3-manifold of minimal inradius, thereby completing a result of Gendulphe, who had previously established the corresponding lower bound.

Given $R$ a finite commutative ring with identity and $k \in \mathbb{N}$, we consider the $k$-th unitary Cayley graph $G_R(k)=Cay(R,U_{R,k})$ with $U_{R,k} = \{ x^k: x \in R^*\}$, and its symmetrized version $\mathcal{G}_R(k) = Cay(R,T_{R,k})$, with $T_{R,k}=U_{R,k} \cup (-U_{R,k})$. If $R$ is a local ring with maximal ideal $\frak m$, we give the blow-up decompositions for the graphs: namely, we have $G_R(k)= (G_{R/\frak m}(k))^{(|\frak m|)}$ and $\mathcal{G}_R(k)= (\mathcal{G}_{R/\frak m}(k))^{(|\frak m|)}$ for any $k$ such that $(k,|R|)=1$. If the ring $R$ has Artin decomposition $R=R_1 \times \cdots \times R_s$ in local rings $R_i$, we give the Kronecker product decompositions $G_R(k) = G_{R_1}(k) \otimes \cdots \otimes G_{R_s}(k)$ and $\mathcal{G}_R(k) = \mathcal{G}_{R_1}(k) \otimes \cdots \otimes \mathcal{G}_{R_s}(k)$. In further $(k,|R|)=1$, these decompositions can be given in terms of generalized Paley (GP) graphs over finite fields, that is $G_{R_i}(k) = Γ(k_i,q_i)$ and similarly for $\mathcal{G}_{R_i}(k)$, for $i=1,\ldots,s$. Also, the reduced graphs correspond to the graphs of the reduced rings, i.e.\@ $\big(G_{R}(k)\big)_{red}\simeq G_{R_{red}}(k)$ and $\big(\mathcal{G}_{R}(k)\big)_{red} \simeq \mathcal{G}_{R_{red}}(k)$. By using these decompositions in terms of GP-graphs, we study some basic structural properties of the graphs such as directedness, bipartiteness and connectedness.

The fundamental limits of over-the-air computation (AirComp) are explored in a two-transmitter, two-receiver MIMO Gaussian network, where both receivers demand the same aggregation of source symbols originating at the two transmitters. An AirComp degrees of freedom (ACDoF) metric is defined, constrained by an asymptotic mean-squared error threshold. For a generic MIMO setting where the two transmitters are equipped with $M_1, M_2$ antennas, and the two receivers with $N_1, N_2$ antennas, the AirComp DoF value is shown to be almost surely equal to $\min\{M_1,M_2,N_1,N_2,(1/3)\max\{M_1+M_2,N_1+N_2\}\}$. For SISO settings results are extended beyond generic channels to arbitrary channel realizations. For finite signal-to-noise ratio(SNR) settings, an iterative alternating optimization algorithm is explored.

We explicitly characterise the full class of e-processes for testing conditional non-parametric hypotheses, defined by finitely many conditional constraints. Our main result is a complete-class theorem: every e-process for such a hypothesis is point-wise dominated by a predictable product of affine one-step e-variables. Therefore, for a broad class of conditional testing problems, arbitrary e-processes can be replaced without loss by test supermartingales. This extends previous complete-class results from single-step constrained testing and bounded one-dimensional conditional mean testing to a broader conditional sequential setting.

In this paper, the first of a two-part series, we explore 2-cell embeddings of cubic graphs, particularly those with small genus. Using local rotations, we introduce a new way of describing the space of 2-cell embeddings and their mutual relationship for any fixed (cubic) graph. We introduce the unstable dual of an embedding of a cubic graph, a subgraph of the dual graph, and describe how the genus of the corresponding embedding can be recovered from properties of the unstable dual. Finally, we characterize the unstable duals of embeddings with genus at most 2 of cubic cyclically 5-edge connected planar graphs and use these to generate those of genus 3 that have connectivity at most 2.

We investigate a new class of infinite-horizon backward stochastic differential equations for ergodic optimal control where the cost and state dynamics are time and path-dependent. The state process is defined on an unbounded underlying domain and satisfies an extended dissipativity condition. In contrast with the time-homogeneous Markovian setting, the optimal ergodic cost in our framework is characterized by the asymptotic behavior of a deterministic function, rather than by a single real constant. We obtain well-posedness, verification and stability properties, which extend the previous results in the literature on the Markov case.

In this work we propose, by linearizing the equations of fully nonlinear poroelasticity, a consistent model in which only the solid phase is incompressible. This reformulation circumvents some inconsistency issues encountered in standard primal formulations of nonlinear poroelasticity while still retaining its key physical coupling mechanisms. We show a well-posed and consistent discretization strategy and also formulate scalable solvers based on a Schur complement formalism. A distinctive feature of the model is that it allows for a lowest order, inf-sup stable family of Finite Elements (FE) spaces. Numerical tests in two and three dimensions are provided to validate the proposed method and solver framework.

The transition function of the unlabeled infinitely-many-neutral-alleles diffusion model, as expressed by Zhou (2015), is derived from the transition function of the labeled infinitely-many-neutral-alleles diffusion model, slightly simplifying the derivation by Feng (2010).

Quantum locally recoverable codes (QLRCs) have recently gained attention as a framework for achieving efficient quantum storage with local recovery capabilities. Analogous to their classical counterparts, QLRCs allow a lost qudit to be reconstructed using only a small subset of other qudits, thereby reducing the resource and operational overhead in recovery. In this work, we extend the study of QLRCs by considering $(r,δ)$ QLRCs characterized by locality parameter $r$ and local distance $δ\geq 2$. We present constructions of both random and explicit $(r,δ)$ QLRCs, including explicit families based on the quantum Tamo--Barg construction. We also present an efficient decoding algorithm for these quantum Tamo--Barg codes. Furthermore, we introduce quantum \emph{hierarchical} locally recoverable codes (QHLRCs), which extend local recovery to multiple hierarchical levels. For any integer $h\geq 2$, we construct both random and explicit $h$-level QHLRCs, the latter being $h$-level quantum Tamo--Barg codes, and establish a Singleton-like bound for these codes using a CSS framework built from dual-containing classical codes. These results advance the theoretical foundations of quantum erasure recovery and contribute to the design of efficient quantum storage architectures.

We settle the projectedness problem for the compactified unpunctured supermoduli stack in every genus at least two. In genus two, the odd component is split, whereas the even component is non-projected. In every genus $g\geq 3$, both compactified parity components are non-projected.