We provide sufficient conditions for comparing $m$-generalized order statistics with respect to the increasing concave, increasing convex, and star-shaped stochastic orders. These conditions allow us to rank classical order statistics, selected censored type-II order statistics, and records. They depend on both the parameters of the generalized order statistics and the underlying distribution. Rather than assuming a specific parametric form, we adopt a nonparametric approach and assume some stochastic transform-ordered property, that is, some suitable shape condition. This framework encompasses many relevant classes of distributions that are related, via transform order, to the generalized and the negative generalized Pareto distribution.

We study the semilinear wave equation $ V(x) \partial_t^2 u + d(t) M(x,\nabla_x) u=\tilde{V}(x) \tilde{d}(t) |u|^{p-1}u$ on $ \mathbb{R}^N \times \mathbb{R}$ and show the existence of solutions which are localized in space and in time, called rogue waves, by means of variational methods. We introduce an energy functional on a suitable Hilbert space, and provide sufficient conditions on the coefficients $V, \tilde{V}, d, \tilde{d}$, the elliptic operator $M$ and $p>1$ for the existence of a critical point. Our approach is based on a detailed analysis of the wave type operator and in particular its spectral properties. Further regularity considerations show that critical points are weak solutions to our equation. Moreover, we provide examples of the coefficients and the elliptic operator which satisfy our assumptions.

We construct probabilistic strong solutions to the cubic Schrödinger equation on the three-dimensional ball with radial initial data, which is a significant improvement of a result by Bourgain--Bulut. These solutions lie in the supercritical regime with respect to the probabilistic scaling introduced by Deng--Nahmod--Yue. We achieve this result through gauge transformations that do not modify the equation, combined with a refined modulation analysis using random averaging operators.

Estimating the logarithm of the determinant of large sparse positive definite symmetric matrices is an important task in numerical linear algebra, machine learning, Gaussian processes, and uncertainty quantification. In this work, we introduce two scalable and efficient methods for large-scale log-determinant termed the Optimal Stochastic Arnoldi with Incomplete Orthogonalization Procedure (OSA-IOP) and the Optimal Stochastic Lanczos Quadrature (OSLQ). The OSA-IOP approach extends the Incomplete Orthogonalization Procedure (IOP), originally developed for matrix exponential functions for exponential time stepping integrators, to compute the action of the matrix algorithm on a vector. We observe that combining IOP with a randomized Hutch++ algorithm, the OSA-IOP significantly reduces computational cost while maintaining high accuracy. The OSLQ method estimates log-determinants by coupling Lanczos quadrature with Hutch++ and controlled orthogonalization, leveraging Krylov subspaces as efficient quadrature mechanisms to approximate quadratic forms involving the matrix logarithm. We derive error bounds for both methods. Extensive numerical experiments on large-scale sparse matrices from real-world applications demonstrate the accuracy, robustness, and scalability of the proposed approaches.

For smooth nonconvex-strongly-concave minimax problems, existing second-order methods share a common double-loop structure where the inner maximization is solved to sufficiently high accuracy before each second-order step. First-order methods are often adopted in the inner loops for scalability, but they also undermine the condition-insensitivity of second-order methods, limiting these methods to instances with mild conditioning. To resolve this issue, we propose a novel single-loop framework based on an equivalent regularized minimization reformulation of the original problem. By deriving a new adaptive cubic-quadratic majorization to dynamically absorb the non-Lipschitz components of the reformulated Hessian, we establish a regularized Newton method with robust theoretical guarantees across multiple settings. For deterministic problems, our single-loop method matches the $\mathcal{O}(\varepsilon^{-1.5})$ global iteration complexity of the existing double-loop second-order methods, while automatically achieving a local superlinear rate that is unavailable in existing works due to the inner-loop bottleneck. For the stochastic setting, we achieve $\mathcal{O}(\varepsilon^{-3})$ gradient and $\mathcal{O}(\varepsilon^{-2})$ Hessian complexities by integrating a recursive variance reduction, strictly improving those of the double-loop methods by $\mathcal{O}(\varepsilon^{-0.5})$ factors. In both deterministic and stochastic experiments, our methods significantly outperform the benchmarks, offering substantial speedups over the double-loop methods even under mildly conditioned instances. As a byproduct of our analysis, we close a gap in stochastic second-order methods for nonconvex minimization, where the best known result contains a nontrivial technical issue.

We study the distribution of cokernels of Haar-random matrices over the $p$-adic integers with prescribed zero patterns, motivated by the Cohen--Lenstra heuristics. A central feature of our approach is that the asymptotic cokernel distribution is governed by the reductions modulo $p$ of these matrices, viewed as random matrices over the finite field $\mathbb{F}_p$. For several families of support patterns arising from stair-shaped zero regions, including general stair-shaped patterns, band matrices, and matrices with two symmetric stair-shaped zero regions, we show that convergence of the cokernel distribution to the Cohen--Lenstra distribution is equivalent to an asymptotic nonsingularity condition over $\mathbb{F}_p$. We further propose a conjecture for general support patterns and give examples showing that analogous rank-$r$ criteria fail for $r\ge 1$.

Nikiforov conjectured that, for every fixed $k\ge2$ and all sufficiently large $n$, the unique $n$-vertex $C_{2k+2}$-free graph with maximum adjacency spectral radius is $S^+_{n,k}$, where $S_{n,k}=K_k\vee\overline K_{n-k}$ and $S^+_{n,k}$ is obtained from $S_{n,k}$ by adding one edge inside the independent part. Cioabă, Desai and Tait proved this conjecture for $n\ge k^{O(k)}$. Later, Li and Ning raised the problem of determining the optimal exponent $γ=γ(k)$ such that the same conclusion holds for $n\ge Ω(k^{γ(k)})$. We prove a stronger uniform theorem for Nikiforov's matrices $A_α(G)=αD(G)+(1-α)A(G)$. More precisely, for every $ε>0$ there are constants $C_ε$ and $k_ε$ such that for all $0\leα\le1-ε$, $k\ge k_ε$ and $n\ge C_εk$, every $n$-vertex $C_{2k+2}$-free graph $G$ satisfies $ρ_α(G)\leρ_α(S^+_{n,k})$, with equality if and only if $G\cong S^+_{n,k}$. In particular, the case when $α=0$ answers the problem of Li and Ning in the linear range, and the $A_α$-spectral even-cycle threshold is linear in $k$, uniformly for all $α$ bounded away from $1$. Our proof introduces a weighted rooted Erdős--Gallai type path lemma, which may be of independent interest in Perron-vector methods for spectral extremal graph problems. The same method also yields asymptotically tight $A_α$-spectral bounds for two local forbidden-subgraph families, namely $(K_1\vee P_\ell)$-free graphs and $F_s$-free graphs, where $F_s$ denotes the friendship graph.

We present a differentiable framework for end-to-end mutual information (MI) optimization over linear Gaussian directed acyclic graphs (DAGs). The framework targets network-wide design under global constraints, such as a total transmit power budget, and covers MIMO precoding, amplify-and-forward relays, RIS-aided channels, and branching/merging topologies within a common linear Gaussian model. Its core ingredient is a \emph{K-recursion} that analytically propagates all node-pair covariances along the DAG in topological order, including non-adjacent cross-covariances that are necessary for correctly handling branching and merging paths. The resulting covariances yield a closed-form log-determinant expression for the end-to-end MI as a smooth function of the controllable factors. Complex-valued reverse-mode automatic differentiation on this K-recursion then returns the exact Wirtinger gradient at every controllable factor in a single backward sweep, and projected gradient ascent (PGA) is used to maximize the MI under the global constraints. Because no closed-form gradient expression per topology is required, the same topology-agnostic implementation applies to any linear Gaussian DAG. A single topology-agnostic implementation is applied to four representative DAG classes: single-link MIMO, a diamond DAG, a two-hop AF relay, and input-covariance shaping. The same implementation reaches the classical water-filling optimum in the settings where it is available and yields MI improvements in non-single-link topologies without using topology-specific gradient formulas. A further experiment on a multi-layer Gaussian network (11 nodes, 5 layers) illustrates applicability to nontrivial multi-layer topologies for which no closed-form gradient is available.

Let $K \subset \mathbb{R}^{2}$ be a compact set. For $θ\in S^{1}$, let $\mathrm{Vis}_θ(K) \subset K$ be the visible part of $K$ in direction $θ$. We prove that $\operatorname{dim}_{\mathrm{H}} \mathrm{Vis}_θ(K) \leq \tfrac{3}{2}$ for $\mathcal{H}^{1}$ almost every $θ\in S^{1}$. The previous record was $\operatorname{dim}_{\mathrm{H}}\mathrm{Vis}_θ(K) \leq 11/6 \approx 1.833$, due to D. Dąbrowski. Our main tool is a variant of a recent incidence lower bound theorem due to Cohen, Pohoata, and Zakharov where, roughly speaking, lines have been replaced by rays, and $δ^{\varepsilon}$-separated incidences are replaced by $1$-separated incidences.

Let $(T(n),Ω)$ be the covering of the generalized Kronecker quiver $K(n)$, where $Ω$ is a bipartite orientation. Given a regular Auslander--Reiten component $\cD$ of $\modd(T(n),Ω)$, we introduce two invariants: the width $\cW(\cD)$ and the number of flow modules $b(\cD)$. We show that $\cW(\cD)\geq \frac{b(\cD)+1}{2}$. In particular, we get $\{\cW(\cD)| \cD \text{ is a regular component} \}=\mathbb{N}$.

This paper explores stability dichotomies of thermoelastic Timoshenko-Boltzmann systems with hereditary memory, comparing Fourier's parabolic (no thermal relaxation) and Cattaneo's hyperbolic (with thermal relaxation) heat conduction via semigroup theory and spectral analysis. The Fourier-semigroup achieves exponential stability through the $δ$-condition plus a specific parameter condition, while sole $δ$-condition ensures $1/2$-order polynomial stability. The Cattaneo-semigroup retains this polynomial stability but requires a thermal relaxation-included modified parameter condition for exponential stability. As thermal relaxation time approaches zero, Cattaneo's modified parameter converges to Fourier's, formalizing their asymptotic connection. Results show thermal relaxation alters exponential criteria but preserves polynomial decay, supporting stability analysis under time-dependent thermal loading in structural mechanics.

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.