Papers

Wegner conjectured that every finite family $\mathcal R$ of axis-parallel rectangles satisfies $\tau(\mathcal R)\le 2\nu(\mathcal R)-1$, where $\nu$ is the packing number and $\tau$ is the piercing number. Ajwani, Gajjala, Raman, and Ray recently disproved this by constructing a triangle-free counterexample on $2196\cdot 8^9$ rectangles and, using a computer-assisted package-and-port recursion, obtained a standard LP gap of $17891/8064$ for Maximum Independent Set of Rectangles. We give a simpler and hand-checkable counterexample with $64$ rectangles. It is built from an eight-rectangle gadget whose independent sets inject into four ordered slots; we then use four horizontal and four vertical copies of this gadget to form a triangle-free family with $\nu=16$ and $\tau\ge 32$. We use the same horizontal-vertical step to define recursive families of rectangles $P_r$ with $\nu(P_r)=4^{2^r}$. For the standard clique, equivalently point, relaxation we obtain a finite gap $73/32$ at $P_3$, improving the previous benchmark of $17891/8064$. We then construct recursive fractional solutions and matching piercing sets showing $\lim_r \alpha^*(P_r)/\nu(P_r)=\lim_r \tau(P_r)/\nu(P_r)=5/2$. Finally, by disjoint union with isolated rectangles, we show that every rational $t\in[1,5/2)$ occurs as a standard LP gap and also as a packing-piercing ratio for suitable rectangle families.

Low-bit quantization makes small reasoning models inexpensive to deploy but can degrade their chains of thought. This motivates decoder-side monitors that intervene when generation becomes unreliable. We show that a natural candidate, the centered token log-probability increment $\log p(w_t)+H_t$, is the wrong observable for this purpose. Under the model's own sampling law it is a mean-zero martingale by construction, so it measures sampling self-consistency rather than trajectory health and is nearly silent during confident repetition, where both $\log p(w_t)$ and entropy are close to zero. We introduce a training-free decoding controller that combines (i) a degeneration-aware alarm score fusing token uncertainty with explicit verbatim repetition and (ii) a calibrated e-process-inspired sequential detector. The raw product process is Ville-valid under a conditional-mean null, while the deployed CUSUM-floored statistic is treated as an empirical change detector because the score is history-dependent and autocorrelated. On GSM8K with DeepSeek-R1-Distill-Qwen-1.5B in FP16 and INT4, calibration turns a monitor that fires on 93–95% of generations into a selective detector of failing traces ($\phi \approx 0.3$, precision $\approx 0.6$ against a 0.38 base rate). In this pilot, the controller reduces measured verbatim-degeneration signals and yields a positive but statistically inconclusive INT4 accuracy change from 63% to 69% (paired McNemar $p=0.18$, $n=100$), at a 28% token-budget cost. We also find that non-termination, rather than looping, is the dominant failure mode on GSM8K. The main contribution is methodological: an explanation of why centered token log-probability is inadequate for decoder monitoring and a calibrated, cautiously evaluated replacement.

In the present paper, we investigate two local zero-sum problems. Let $n,k\ge 2$. We denote by $\mathsf{D}^*(n,nk)$ (resp. $\eta^{*}(n,nk)$) the smallest positive integer $\ell$ (if exists) such that, from any given $\ell$ integers not divisible by $n$, one can select some (resp. at most $n$) of them whose sum is divisible by $n$ but not by $nk$. We prove that both $\mathsf{D}^*(n,nk)$ and $\eta^{*}(n,nk)$ are equal to $2n-1$ if $\mathrm{rad}(n) \mid \mathrm{rad}(k)$ and infinite otherwise. The corresponding inverse problem is also determined. We denote by $\mathsf{D}_n^{\times}$ (resp. $\eta_n^{\times}$) the smallest positive integer $\ell$ such that, from any given $\ell$ integers coprime to $n$, one can select some (resp. at most $n$) of them whose sum $\sigma$ satisfies $\gcd(\sigma, n^2)=n$. We prove that $\mathsf{D}_n^{\times}=\eta_n^{\times}=2n-1$ if $n$ is a prime power, and determine its inverse problem.

We nearly settle a natural extremal question about set systems over $[n]$: the tradeoff between the {size} (number of sets) and the number of {full chains}. This question was initially raised by Johnson, Leader, and Russell [Combin.~Probab.~Comp., 2015] as a counterpart to Sperner-type results in combinatorics. Recently, a framework introduced by Ameli, Nederlof, and Wang, and independently by Dallant and Kozma [FOCS 2026] linked this question to the space- and time-complexity of Bellman-Held-Karp-style dynamic programming algorithms for permutation problems such as the traveling salesman (TSP). Precisely, they showed that a space-time product $\gamma^{n+o(n)}$ is feasible for the TSP, whenever a set system of (normalized) size $S$ and chain density $D$ exists, with $ \gamma = S^2/D$. In this paper we show an essentially {optimal} bound of $\gamma \approx 3.1819$ for this quantity, closing the gap between the previous best lower and upper bounds of $\gamma \geq 3.015$ and $ \gamma \leq 3.572$ respectively. This implies a TSP algorithm with space-time product $O(3.1819^n)$ for input size $n$, as well as a limit to further improvements in this broad framework. More generally, we can obtain close to optimal values $D$ for any feasible value $S$, effectively settling the question of the number of full chains at every size. The crucial step towards our results is casting the extremal combinatorics question as an {information~vs.~entropy} tradeoff involving two random variables. This reformulation {exactly} captures the optimal tradeoff for the combinatorial problem, leading to a framework in which primal-dual certificates can be derived, proving rigorous upper and lower bounds on $\gamma$. We also give a further application of our techniques, improving a bound of Duffus, Sands, and Winkler on the minimum size of fibres in the Boolean lattice.

This paper addresses the optimal control of epidemic dynamics under conflicting socio-economic objectives. We propose an economic Model Predictive Control (MPC) framework, applied to an extended SEIR-V (Susceptible-Exposed-Infected-Recovered-Vaccinated) compartmental model to govern the spread of an infectious disease while minimizing economic disruption. The control problem is formulated as a constrained nonlinear optimization problem, in which the controller dynamically adjusts social interaction levels (transmission rate beta) and vaccination efforts to minimize a composite cost function that penalizes fatalities, healthcare capacity violations, and economic losses. We conduct a rigorous sensitivity analysis of the prediction horizon N, demonstrating that the closed loop is robust to the horizon choice and that N = 35 days minimizes the realized cost. Furthermore, both the closed-loop solution and an open-loop turnpike analysis across diverse initial conditions reveal that the celebrated "Hammer and Dance" mitigation strategy emerges naturally as the mathematical optimum: the optimal trajectories anchor to a unique suppression turnpike (maximum lockdown) to drive hospitalizations toward the disease-free equilibrium before progressively reopening the economy. Through a turnpike-based argument we establish practical asymptotic stability of the optimal operating point, providing a mathematically grounded decision-support tool for pandemic policy.

We propose an a posteriori error estimator that relies on an equilibrated flux reconstruction to enable analysis-aware interface coarsening decisions in elliptic interface problems. Interface coarsening consists of simplifying an internal interface that separates regions characterized by different physical properties in order to facilitate the meshing process and reduce the total number of degrees of freedom. This process extends the concept of defeaturing, where small features on the boundary of a computational domain are removed. Here, the features lie on an interior interface instead. The focus is on a Laplace problem with a discontinuous diffusion coefficient. The estimator accounts for both the modeling error arising from the interface coarsening and the numerical error from the discrete approximation of the solution to the coarsened-interface problem. It is localized on the mesh elements, and its constants explicitly track the contrast between the diffusion coefficients. The impact of different features can be assessed individually, yielding local contributions that indicate which features to coarsen and which to retain for a given mesh size. The use of an equilibrated flux reconstruction allows us to sharply bound the bulk numerical source of error. We prove the reliability of the estimator and verify it across several numerical examples, including the case where an internal interface is fully removed.

We show that a Nakai–Moishezon-type criterion associated with the complex $2$-Hessian equation produces a Gauduchon class. In complex dimension three, this numerical criterion is equivalent to the existence of a smooth $2$-admissible representative and hence to the solvability of the $2$-Hessian equation. As consequences of these results, we prove the corresponding conjectures of Murakami for the complex Hessian equation and of Székelyhidi for the Hessian quotient equation in dimension three. We also establish a boundary version of the above results.

Differential equations arise in numerous applications, particularly within scientific and technical contexts. Systems of stiff, time-dependent ordinary differential equations constitute the focus of this work. Exponential integrators are designed to solve such equations by integrating the linear part exactly, while simultaneously approximating the nonlinear part through a linear combination of $\varphi$-functions. By utilizing an augmented stiffness matrix, state-of-the-art methods like KIOPS and RK2EXPINT solve the linear part and evaluate linear combinations of $\varphi$-functions for the nonlinear part in a single step, effectively reducing the computational effort to a single matrix exponential evaluation. However, these classical approaches assume that the system is represented by matrices and vectors, potentially not utilizing the underlying high-dimensional structure. Tensors address this limitation and offer significant storage efficiency through well-established decompositions like the Tensor Train (TT) format. This work provides a general framework for solving stiff, time-dependent systems directly within the TT format. Specifically, KIOPS-TT and RK2EXPINT-TT are developed as extensions of the original KIOPS and RK2EXPINT algorithms. This involves reformulating the scheme of explicit exponential Runge-Kutta integrators for tensors and augmenting the stiffness tensor to compute linear combinations of $\varphi$-functions acting on tensors through a single evaluation of the exponential function using Krylov subspace methods. Furthermore, it is shown that the underlying theory of the matrix methods remains valid, thereby enabling the transfer of key theorems to the tensor case. Numerical experiments confirm significant speed-ups for KIOPS-TT and RK2EXPINT-TT in low-rank scenarios compared to their classical counterparts.

We are interested in the long-time behaviour of the ecological dynamics of two competing species in a spatially heterogeneous environment consisting of two habitat types. Our goal is to provide conditions for the persistence of the two populations. First, we consider a spatially continuous model, formalized as an infinite-dimensional system of integro-differential equations. We show that if each species would persist if it were alone, then mutual invasibility of each other's monospecific equilibrium is a sufficient condition for long time survival of both species. Second, we introduce a finite-dimensional system of ordinary differential equations which approximate the spatial dynamics by averaging over a finite number of habitat types. We derive an analogous sufficient condition for stable coexistence, and show that in this case, there exists a positive coexistence equilibrium. Finally, we complete our theoretical result using a simulation study. Our results indicate that mutual invasibility also is a necessary condition for stable coexistence in both models. In addition, we show that the finite-dimensional model underestimates species' persistance, which indicates that spatial heterogeneity promotes survival.

The theory of $k$-record values (Type 2 $k$-records) plays an important role in the study of partial extremes and in statistical inference based on record data. A common approach reduces the analysis of $k$-records associated with a distribution function $F$ to that of ordinary records from the transformed distribution $F_{1:k}(x)=1-(1-F(x))^k$. This representation is widely used to derive distributional and inferential results, often without an explicit construction of the underlying stochastic mechanism, and relies on a structural property of order statistics that, although classical, is typically invoked without proof. We give a direct derivation of the probabilistic structure of $k$-record processes based on the sequence of running order statistics $U_n=X_{n-k+1:n}$, the $k$-th largest among the first $n$ observations. We show that $(U_n)$ forms a Markov chain with respect to its natural filtration, with an explicit transition kernel. The key step is a conditional-independence property of upper order statistics, which we isolate and prove: conditionally on $U_n$, the $k-1$ observations exceeding $U_n$ are distributed as order statistics from the distribution truncated at $U_n$, independently of the whole past trajectory. Under continuity of $F$, the usual Type 2 $k$-record times coincide almost surely with the record times of $(U_n)$. This yields a transparent construction of the $k$-record process as the record process of a Markov chain, and classical distributional results – including the representation through $F_{1:k}$ and the joint density of the first $m$ $k$-record values – are recovered in a unified framework. We also treat the exponential case, in which the $k$-record values form a random walk with independent exponential increments and Gamma-distributed marginals, and record a corresponding characterisation of the exponential distribution.

Let $H=-\Delta+V(|x|)$ be a nonnegative radial Schrödinger operator on $\mathbb{R}^d$, $d\ge 2$, whose positive harmonic function satisfies $U(r)\simeq r^{-\sigma_0}$ for $0<r\le 1$ and $U(r)\simeq r^{-\sigma_\infty}$ for $r\ge 1$, with $-d/2<\sigma_0,\sigma_\infty<d/2$. Assuming the two-sided ground-state heat-kernel estimate of Ishige, Kabeya, and Ouhabaz, we determine the maximal open range in which the kernel of $H^{-s/2}$ admits the clean two-sided estimate $K_s^H(x,y)\simeq |x-y|^{s-d}U(|x|)U(|y|)/[U(|x|+|x-y|)U(|y|+|x-y|)]$, namely $0<s<\min\{d,d-2\sigma_0,d-2\sigma_\infty\}$. In this range we give a complete necessary-and-sufficient classification of the broken-power estimate $\|w_{-\beta_0,-\beta_\infty}H^{-s/2}f\|_{L^{q,v}}\lesssim \|w_{\alpha_0,\alpha_\infty}f\|_{L^{p,u}}$ for $1<p,q<\infty$ and $1\le u,v\le\infty$. The result covers signed ground-state exponents, the full range $q<p$, all one-sided weight equalities, both scale equalities, and simultaneous endpoint corners. Scale equality is governed by $u\le v$, including when $q<p$; an input or output power endpoint requires respectively $u=1$ or $v=\infty$; and at a same-side power/scale corner the only admissible pair is $(u,v)=(1,\infty)$. The proof combines clean-kernel analysis, local Lorentz-Hardy-Littlewood-Sobolev estimates, rank-one endpoint arguments, geometric annular sequence spaces, a triangular matrix theorem, and a nine-block decomposition.

In this paper, we investigate the existence, multiplicity, and semiclassical concentration of normalized solutions to a critical Schrödinger–Poisson system with a singular Hardy potential in \(\mathbb{R}^3\). More precisely, we consider \[ \begin{cases} -\varepsilon^2\Delta u+ \left(V(x)-\dfrac{\kappa\varepsilon^2}{|x|^2}\right)u -\phi |u|^3u =\lambda u+\mu |u|^{q-2}u+|u|^4u, & \text{in } \mathbb{R}^3, \\[1mm] -\varepsilon^2\Delta\phi=|u|^5, & \text{in } \mathbb{R}^3, \end{cases} \] under the prescribed mass constraint \[ \int_{\mathbb{R}^3}|u|^2\,dx=a^2\varepsilon^3, \] where \(a,\mu>0\), \(q\in(2,10/3)\), \(\varepsilon>0\) is a small semiclassical parameter, and \(0<\kappa<1/4\). The parameter \(\lambda\in\mathbb{R}\) appears as a Lagrange multiplier associated with the mass constraint, while \(V:\mathbb{R}^3\to(0,+\infty)\) is a continuous electric potential whose minimum set is assumed to be nonempty and compact. The main difficulty stems from the simultaneous presence of the inverse-square Hardy singularity, the mass constraint, the critical local nonlinearity, and the nonlocal Poisson interaction. By combining the Hardy inequality, constrained variational methods, suitable truncation arguments, and concentration-compactness techniques, we first establish the existence of a normalized ground state for sufficiently small mass and sufficiently small \(\varepsilon\). We then employ Ljusternik–Schnirelmann category theory to obtain multiple normalized solutions whose number is related to the topology of the minimum set of \(V\). Finally, we show that the corresponding semiclassical states concentrate near the global minimum set of the electric potential as \(\varepsilon\to0\).

Let $X_1,\,X_2,\,\ldots,\,X_N$, $N\in\mathbb N$ be independent, discrete, integer-valued random variables. Assume that $X_j\geqslant m_j$ almost surely for each $j=1,\,2,\,\ldots,\,N$, where $m_1,\,m_2,\,\ldots,\,m_N\in\mathbb{Z}$ satisfy $m_1+\cdots+m_N<0$. Furthermore, suppose that the sequence $X_1,\,X_2,\,\ldots$ is periodic in distribution, i.e. $X_k{\buildrel d \over =} X_{k+N}$ for all $k\in\mathbb N$. We derive computable representations for the distribution functions of $\max\{X_1,\,X_1+X_2,\,\ldots\}$, $\max\{X_2,\,X_2+X_3,\,\ldots\}$, $\ldots$, $\max\{X_N,\,X_N+X_{N+1},\,\ldots\}$. The obtained formulas are based on a linear recurrence whose initial values are determined from a linear system that involves the roots of an associated characteristic equation and the distributions of $X_1,\,X_2,\,\ldots,\,X_N$. Several examples are presented, including a biseasonal-biased Rademacher random walk for which the distribution, generating functions, and all moments admit explicit closed-form expressions. In addition, we identify and correct several inaccuracies in the results reported in \cite{Grigutis2024}.

We study the periodicity and $p$-adic convergence of sequences of values of the translated division polynomials $(\Psi_n(P))_{n\geq 0}$, $(\Phi_n(P))_{n\geq 0}$, and $(\Omega_n(P))_{n\geq 0}$ evaluated at a pair of points on an elliptic curve. We prove that these sequences are purely periodic over finite fields. We then use their periodicity properties to prove that certain subsequences are $\mathbb{Z}_p$-Cauchy. Applying this result to Somos 4 and Somos 5 sequences yields analogous convergence results. We also establish pure periodicity modulo prime powers for the even- and odd-indexed subsequences of Somos 5 sequences under suitable hypotheses.

We study collinearly complete finite sets of points arising from configurations of pairwise skew $n$-planes in $\mathbb {P}^{2n+1}_K$. To such a configuration we associate a groupoid generated by the natural collinearity correspondences between the $n$-planes, and we investigate the geometry of its finite orbits. In characteristic zero, we prove a rigidity result for the case in which the associated group is finite cyclic. After a suitable normalization, the matrices defining the configuration are simultaneously diagonalizable and admit a common two-block decomposition. Consequently, the orbit of a general point meets each $n$-plane in a collinear set, and the full orbit is contained in a distinguished projective $3$-space. This reduces the geometry of such orbits to the classical case of skew lines in $\mathbb {P}^3_{\mathbb C}$: inside the distinguished $\mathbb{P}^3_{\mathbb C}$, the orbit is geproci. We also prove that finite unions of general orbits are cut out set-theoretically from the union of the $n$-planes by a reducible surface. Finally, we show that this characteristic-zero rigidity fails in positive characteristic. In characteristic $2$, we construct cyclic examples whose orbit slices are Fano plane configurations, and we exhibit a genuinely higher-dimensional finite non-cyclic example in projective $5$-space with associated group ${\rm PGL}_3(\mathbb F_2)\cong {\rm PSL}_2(\mathbb F_7)$.

We find all zero-dimensional spaces of newforms of weight $2$ and squarefree level $N$ with a fixed Atkin–Lehner sign pattern. As an application, we classify unexpected primes of good reduction of Atkin–Lehner quotients of modular and Shimura curves of squarefree levels.

In this paper, we introduce and analyse a model in which a shared, age-structured resource is exploited by two competing populations through age-dependent interaction kernels. It provides a general framework for studying resource-mediated competition in epidemiology, immunology and ecology. We use integrated semigroups theory to establish the well-posedness of the model and its fundamental properties, including positivity and the existence of a global attractor. Then, by means of Lyapunov functionals, persistence results and generalised LaSalle principle, we fully characterize the asymptotic dynamics in terms of the competitors's basic reproduction numbers and invasive fitnesses. Our results demonstrate that in addition to the classical competitive exclusion, the system can exhibit globally asymptotically stable coexistence, an outcome allowed by a mechanism of age-mediated niche partitioning.

The class of parametric statistical models that can be treated as Riemannian manifolds is considerably larger than the classical Fisher-Rao setting allows, once one works in the space of tempered distributions. A law is represented by a tempered distribution T in S'(R^k), while an instrument - a positive Schwartz kernel, a weak regular inference function, or a weak Stein representation - extracts information from the law without being part of it. Any instrument with full-rank sensitivity and positive-definite variability induces the Godambe information G = S^T V^{-1} S, a Riemannian metric on the parameter space; the Fisher-Rao manifold is recovered exactly when the score is an admissible instrument, and every Godambe metric is dominated by the Fisher metric in the Loewner order whenever the latter exists. Four examples lie outside the Fisher-Rao class for four different reasons: a location model built on the Cantor distribution (an undominated family - no likelihood, no score, and no Fisher information exist at all), the uniform scale model (parameter-dependent support), the shifted exponential model (transform-based inference), and a stratified finite mixture (a provably biased score in a dominated model); a lattice stochastic heat equation driven by alpha-stable noise provides a fifth, dynamical example, whose closed-form weak Godambe information stabilises at a rate governed by the spectral gap of the discrete Laplacian. Quadratic Stein discrepancies induce the same local geometry, and reproducing-kernel constructions generate a hierarchy of geometries. Because there is no canonical instrument, the model carries a family of Godambe metrics; we discuss the inferential, diagnostic, geometric, and computational roles of its members, and show that weak inferential separation (nonformation) appears geometrically as block-diagonality of the Godambe metric.

We give an affirmative answer to Tingley's problem for Schatten–von Neumann classes $C_p$ by a unified approach for all $0<p\ne 2<\infty$, extending results due to Fernández-Polo et al.

This study investigates the Lie algebra structure and geometric properties of hybrid numbers, under a specific non-associative multiplication table. Within the scope of the study, the structure constants of the system were derived through commutator relations, and 4x4 matrix representation was constructed. By defining the Gram matrix, complex and hybrid conjugates, the connection of the eigenvalues of Gram matrices to the golden ratio has been obtained. Theoretical analysis reveal that this hybrid Lie algebra possesses a semi-simple structure and that the characteristic spectrum of its Killing form is directly related to the Golden Ratio ({\phi}). Furthermore, it is proven that the Complex and Hyperbolic conjugation operations are symmetry transformations that leave the Killing form invariant.

We consider the following additive Latin transversal problem. Given a multiset $A=(a_1,\dots,a_k)$ of elements of $\mathbb Z_m$ and a set $B\subseteq\mathbb Z_m$ of cardinality $k$, the task is to order $B$ as $b_1,\dots,b_k$ so that the sums $a_i+b_i$ are pairwise distinct. When $k=m$, Hall proved that a solution exists if and only if $\sum_{i=1}^m a_i\equiv 0 \pmod m$; moreover, his theorem yields a polynomial-time construction. Alon proved that a solution always exists when $m$ is prime and $k<m$, but no polynomial-time construction is known in general. Our main algorithmic contribution is a direct randomized algorithm for Color-Counted Matching: given an edge-colored graph and prescribed target counts for the colors, find a matching using exactly the prescribed number of edges of each color. If $q$ is the sum of the target counts and $h$ is the number of colors, our base-$(q+1)$ reduction to Exact Red Matching, combined with the algorithm of Mulmuley-Vazirani-Vazirani, gives a randomized algorithm with running time $\left(|V|^2+|E|(q+1)^{h-1}\right)^{O(1)} $ for an input graph $(V,E)$. Thus the dependence on the target matching size is $q^{O(h)}$, up to polynomial factors in the graph size. In contrast, applying the general matching-ILP theorem of Lassota and Ligthart as a black box yields a $q^{O(h^2)}$ dependence for the corresponding fixed-size color-counted instances. Applying this primitive to additive Latin transversals with $s=|\operatorname{supp}(A)|$, we obtain an algorithm in randomized time $(k+\log m)^{O(s)}$. In particular, additive Latin transversals are randomized polynomial-time constructible for every fixed support size.

We present an algorithm for solving nonlinear least-squares problems subject to a mix of nonlinear and linear constraints. The nonlinear constraints are handled by reformulating the objective as the augmented Lagrangian function while linear constraints are handled directly. Each iteration consists of approximately solving a linearly constrained problem by means of a gradient projection technique. Our approach also involves a structured approximation of the augmented Lagrangian Hessian. We show global convergence of the method and assess the performance through numerical experiments.

We make use of the free data formalism developed in \cite{CK25,CK26} to provide a direct construction of short-pulse type Cauchy data. The construction of the spacelike short-pulse follows from the local existence result established in \cite{CK26}. The forward integration construction in \cite{CK26} allows us to show that such data can be extended to a set of asymptotically flat Cauchy data. This greatly extends the result of Li–Yu \cite{LiYu}.

Audience: Tech companies such as NExT leadership; public transport operators; autonomous transit developers; charging partners; public authorities, and industrial partners considering pilots. Content: The strongest aspect of Modular Autonomous Vehicles (MAVs) is not only that vehicles can physically couple. A MAV is composed of Modular Autonomous Units (MAUs), and modularity creates operating advantages across line-level operations, intermodal services, network-level operations, stochastic real-time control, and electrified vehicle operations and charging infrastructure. Research basis and disclaimer: The brief summarizes scientific findings of four studies conducted by Dr. Xia and Dr. Sharif Azadeh on modular transit operations, including a charging infrastructure study using technical data associated with NExT MAUs.

In this paper, we prove the $L^p(p>1)$-uncertainty principles for any $1<p<n$, including the classical Heisenberg-Pauli-Weyl inequality, Caffarelli-Kohn-Nirenberg interpolation inequality and Hardy inequality in $\mathbb R^n$ as special cases, on $n(\geq 2)$-dimensional forward complete and noncompact Finsler measure spaces $(M, F, \mathfrak{m})$ with curvatures bounded from above or below by constants. Further, we characterize the sharpness of $L^p$-uncertainty principles in terms of the reversibility of $F$ and the bounds of flag (or Ricci) curvature and S-curvature induced by the measure $\mathfrak m$ and obtained some rigidity results, which generalize the related ones in [HKZ] in Finslerian case and [KKPZ] in Riemannian case.

Given a fence poset P , we define a new statistic on permutations, denoted by blP, that provides a combinatorial interpretation of the coefficients of the order polynomial of P , answering a question of Ferroni, Morales, and Panova (2025). Using the fact that the base polytope of a lattice path matroid can be decomposed into order polytopes of fence posets, we also obtain a combinatorial interpretation of the coefficients of the Ehrhart polynomial of the base polytope of Schubert matroids, answering a question of Stanley (1999). As an application of this statistic, we establish the first nontrivial lower bound for the linear coefficient of the Ehrhart polynomial of an order polytope. Finally, we conjecture generalizations of this statistic to skew-shape posets and circular fence posets.

Let $(R,m,k)$ be an excellent local noetherian domain with field of fractions $K$. Let $$ \nu:K^*\twoheadrightarrow\Gamma $$ be a valuation centered at $R$ and let $R_\nu$ be the corresponding valuation ring of $K$, dominating $R$. Denote by $\widehat R$ the $m$-adic completion of $R$. In the applications of valuation theory to commutative algebra and the study of singularities, one is often induced to replace $R$ by its $m$-adic completion $\widehat R$ and $\nu$ by a suitable extension $\widehat\nu_-$ to $\frac{\widehat R}P$ for a suitably chosen prime ideal $P$, such that $$ P\cap R=(0). $$ In a previous article we gave a systematic description of all such extensions $\widehat\nu_-$ and defined the notion of tight extensions that are of particular interest for applications (see Herrera, Olalla, Spivakovsky and Teissier, Extending a valuation centered in a local domain to its formal completion, Proc. London Math. Soc. (3) 105 (2012) 571–621). If $\widehat\nu_-$ is a tight extension then its graded algebra is birational to that of $\nu$ (the converse is not known and might not be true). In particular, the value group of $\widehat\nu_-$ is $\Gamma$. The existence of tight extensions was conjectured by the last author (see Teissier, Valuations, deformations, and toric geometry, Fields Institute Communications, 33, 2003, 361-459). In the present paper we give a proof of Teissier's conjecture. An intended application of this result is an important step in two recent approaches to local uniformization in positive characteristic.

In commutative ring theory there are multiple equivalent definitions of integrality. These notions diverge when working with idempotent semirings. In this paper we present these different definitions of integrality for semirings and explore the relations between them. As a tool, we prove a Cayley-Hamilton theorem over additively idempotent semirings, which may be of broader interest. In examples, we compute integral closures of coordinate semirings in their total semiring of fractions and integral closures of sub-semirings of coordinate semirings. Such computation gives avenues to defining and understanding the normalization of tropical varieties as well as computing normalization of varieties tropically.

In this paper, we investigate the weak observability of second-order infinite-dimensional evolution systems generated by skew-adjoint operators of the form $iA_0$, where $A_0$ is a self-adjoint elliptic operator. We first establish a spectral characterization of weak observability by introducing the notion of spectral coercivity for the observation operator and proving its equivalence to a suitable resolvent estimate. Our main result reveals a direct link between resolvent estimates for the elliptic operator $A_0$ and the weak observability of the associated evolution generator $A$. More precisely, we prove that a resolvent inequality for $A_0$ implies a Hautus-type spectral observability estimate for $A$, which guarantees the weak observability of the system. This provides a unified spectral framework for weak observability based on the coercivity properties of the observation operator. As an application, we establish explicit weak observability estimates for the wave equation on a rectangular domain under several geometric configurations of the observation region. The analysis combines frequency-domain methods, resolvent estimates, and Fourier analysis, yielding new insights into the interplay between resolvent inequalities, spectral coercivity, and weak observability in infinite-dimensional systems.

We prove existence and uniqueness in BV for a Vlasov-Poisson boundary value problem which takes into account a nonlinear boundary condition at the boundary on the electric potential. It models the dynamical interaction between a plasma and an electrically isolated metallic wall. The nonlinear boundary condition is the so-called floating potential equation of plasma physics.