In this paper, we develop the well-posedness theory and uncover the noise-regularization effect on scattering for the stochastic Zakharov system in dimensions $d \geq 4$ and beyond the energy space. Our focus is particularly directed at the large data regime, where the global existence and long-time dynamics of the deterministic Zakharov system remain largely open. We prove the local well-posedness of the stochastic system in the full deterministic regularity regime and establish a blow-up alternative at the endpoint regularity, which implies the persistence of regularity in the full well-posedness regime. Furthermore, we prove that for any large initial data, with high probability, non-conservative noise yields global and scattering solutions. Our proof introduces a tailored functional framework. To establish local well-posedness, we employ a refinement of adapted Fourier restriction and lateral Strichartz spaces, which allows us to control both the nonlinear interactions and the critical first-order derivative perturbations arising from rescaling transforms. To achieve the noise-regularization effect, we augment this setting with maximal function spaces. We derive new trilinear estimates for the stochastic wave nonlinearity that are crucial for the global dynamics by fully exploiting the temporal regularity of geometric Brownian motions in scaling-(sub)critical Besov spaces.

We equip the space of Cauchy hypersurfaces in a globally hyperbolic spacetime with a natural Hausdorff-type metric and study its properties, in particular completeness and local compactness, for Lorentzian manifolds and in more general synthetic Lorentzian settings. For this purpose, we also generalize results on completeness properties of spacetimes due to Beem and Takahashi.

Deterministic identification (DI) has emerged as a promising paradigm for large-scale and goal-oriented communication systems. Despite significant progress, a fundamental open problem has remained unresolved: a persistent gap between the best known lower and upper bounds on the DI capacity, as well as on the corresponding rate-reliability tradeoff bounds. In this paper, we finally close this gap for Gaussian channels $\mathcal{G}$ by constructing an optimised code that achieves the known upper bound. This allows us to establish that the linearithmic capacity for deterministic identification is $\dot{C}_{\text{DI}}(\mathcal{G})=\frac{1}{2}$. Furthermore, we analyse the rate-reliability tradeoff and show that the proposed scheme matches the known upper bounds to first order, thereby closing the existing gap in reliability performance for all admissible error decay regimes. Finally, we demonstrate the existence of an optimum universal code, which does not require knowledge of the channel parameters and yet achieves capacity.

We develop a framework for the fifth-order Kadomtsev--Petviashvili equation on $\mathbb{T}_x \times \mathbb{R}_y$ within a mean-zero KP-adapted Sobolev scale. A localized high-order feedback acting on the periodic variable yields a $5/2$--derivative gain in suitable space--time norms, leading to propagation of regularity and a unique continuation property for the linear dynamics. As a consequence, we derive an observability inequality for the adjoint system and establish exponential stabilization of the nonlinear closed-loop equation: for small initial data in $X_{s,0}$, $s>2$, solutions are global and decay exponentially in $X_s$. Combining observability with the Hilbert Uniqueness Method and a fixed-point argument, we obtain local exact controllability near the origin, with $L^2$ controls supported in the feedback region and cost linear in the data size. The analysis relies on a novel combination of unique continuation, frequency grouping, and the one-sided Fourier vanishing mechanism introduced for the Benjamin--Ono equation by Linares and Rosier in \textit{Trans. Amer. Math. Soc.} (2015)~\cite{LR}, here extended to the fifth-order Kadomtsev--Petviashvili equation.

We introduce the strong $q$-timelike Brunn-Minkowski condition $\mathsf{sTBM}_q(K,N)$ on synthetic Lorentzian spaces, for $0<q<1$. We show that, in the timelike $q$-essentially non-branching setting, the $q$-timelike curvature dimension condition $\mathsf{TCD}_q(K,N)$ is equivalent to $\mathsf{TBM}_q(K,N^+)$, and that the entropic $q$-timelike curvature dimension condition $\mathsf{TCD}_q^e(K,N)$ is equivalent to the reduced $\mathsf{sTBM}$ condition, $\mathsf{sTBM}_q^*(K,N)$. This extends, to a non-smooth setting, our earlier work in proving the equivalence between Ricci curvature and the Brunn-Minkowski inequality on $C^2$ spacetimes.

A combinatorial game is a two-player game without hidden information or chance elements. The disjunctive sum $G + H$ of games $G$ and $H$ is the game in which $G$ and $H$ are played in parallel, and a player makes a move on exactly one of $G$ and $H$ in a turn. The ordinal sum $G \colon H$ is similar to the disjunctive sum, but once the left game $G$ is played, the right game $H$ is discarded and can no longer be played. It is known that the outcome of a mixture of disjunctive sums and ordinal sums, such as $(G_1 \colon G_2) + ((G_3 + G_4) \colon G_5)$, is determined by the variation sets, the set of Grundy numbers of all options, of the components in the normal-play. In this paper, we propose a generalization of an ordinal sum, called an ordinal sum with substitution $G \colon_{\widehat{H}} H$, which is the game made by combining $G$, $H$, and $\widehat{H}$ in the following way: the games $G$ and $H$ are played in parallel; a player makes a move on exactly one of $G$ and $H$ in a turn; each time the left game $G$ is played, the right game $H$ is replaced with $\widehat{H}$. We investigate their fundamental properties and prove a simple formula for the variation sets of ordinal sums with substitution. Apply the formula, we give an explicit expression of the Grundy number of a chain of ordinal sums with substitution consisting of nimbers. We also provide an example illustrating the generalization of ordinal sums with substitution to poset structures.

Let $M_n$ be an $n\times n$ signed random combinatorial matrix whose rows are independent and uniformly distributed over the set of $\{-1,0,1\}$-vectors with exactly $n/2$ zero coordinates. Despite the dependence induced by the row constraints, we prove that there exist constants $C,c > 0$ such that for any $\varepsilon\ge0$, \begin{align*} \textbf{P}\left(s_{n}(M_n)\le {\varepsilon}{n^{-1/2}}\right)\le C\varepsilon+e^{-cn}. \end{align*} In particular, the probability that $M_n$ is singular is exponentially small. Our approach builds on the Combinatorial Least Common Denominator (CLCD) introduced by Tran and develops the method in the present constrained setting.

We study the problem of identifying change points in high-dimensional generalized linear models, and propose an approach based on sample-weighted empirical risk minimization. Our method, Weighted ERM, encodes priors on the change points via weights assigned to each sample, to obtain weighted versions of standard estimators such as M-estimators and maximum-likelihood estimators. Under mild assumptions on the data, we obtain a precise asymptotic characterization of the performance of our method for general Gaussian designs, in the high-dimensional limit where the number of samples and covariate dimension grow proportionally. We show how this characterization can be used to efficiently construct a posterior distribution over change points. Numerical experiments on both simulated and real data illustrate the efficacy of Weighted ERM compared to existing approaches, demonstrating that sample weights constructed with weakly informative priors can yield accurate change point estimators. Our method is implemented as an open-source package, weightederm, available in Python and R.

Let $F$ be a local non-archimedian field of odd residue characteristic and let $G=PGL(2)$. In this paper we study an analog of irreducible cuspidal representations of the group $G(F)$ when $F$ is replaced by the field $K=F((t))$. The story turns out to be similar to the classical case, but also with some differences. We present a construction of such representations essentially (up to a small subtlety) starting from a quadratic extension $L$ of $K$ and a character $θ:L^*/K^*\to \mathbb C^*$ which is not Galois invariant. We also show that the restriction of the representations we construct to the group $P(K)$ (here $P$ is a Borel subgroup of $PGL(2)$) is irreducible. However, contrary to the classical case it turns out that these restrictions are not isomorphic to the "standard" irreducible cuspidal representation of $P(K)$. In the Appendix we propose a notion of cuspidality for smooth representations of the group $H(K)$ for an arbitrary split reductive group $H$.

We study routing games in which travelers optimize over routes that are remembered or surfaced, rather than over a fixed exogenous action set. The paper develops a tractable design theory for endogenous recall and then connects it back to an explicit finite-memory micro model. At the micro level, each traveler carries a finite memory state, receives surfaced alternatives, chooses via a logit rule, and updates memory under a policy such as LRU. This yields a stationary Forgetful Wardrop Equilibrium (FWE); existence is proved under mild regularity, and uniqueness follows in a contraction regime for the reduced fixed-point map. The paper's main design layer is a stationary salience model that summarizes persistent memory and interface effects as route-specific weights. Salience-weighted stochastic user equilibrium is the unique minimizer of a strictly convex potential, which yields a clean optimization and implementability theory. In this layer we characterize governed implementability under ratio budgets and affine tying constraints, and derive constructive algorithms on parallel and series-parallel networks. The bridge between layers is exact for last-choice memory (B=1): the micro model is then equivalent to the salience model, so any interior salience vector can be realized by an appropriate surfacing policy. For larger memories, we develop an explicit LRU-to-TTL-to-salience approximation pipeline and add contraction-based bounds that translate surrogate-map error into fixed-point and welfare error. Finally, we define a Recall Braess Paradox, in which improving recall increases equilibrium delay without changing physical capacity, and show that it can arise on every two-terminal network with at least two distinct s-t paths. Targeted experiments support the approximation regime, governed-design predictions, and the computational advantages of the reduced layer.

Electric car-sharing systems are pivotal for sustainable urban mobility, but their strategic design is complicated by operational constraints, particularly those arising from the charging needs of electric vehicles. The success of these systems hinges on integrating long-term investment decisions (such as station locations, charger capacities, and fleet size) with daily operational realities, including vehicle routing to serve user trip requests and battery management. While existing integrated models address this strategic-operational link, they have prioritized economic efficiency, overlooking the critical dimension of service equity. This paper addresses this gap by making fairness a central design principle, operationalized through two distinct paradigms, namely, service-rate disparity and max-min fairness, measured explicitly via realized group service rates rather than static spatial accessibility. To capture demand heterogeneity, we adopt a multi-day representative-demand setting, and develop a bi-objective trajectory-based formulation that jointly optimizes revenue and service equity. We develop a solution framework in which a branch-and-price algorithm solves the single-objective variants of the models, embedded within an exact bi-objective procedure to generate the Pareto frontier and complemented by a diving-heuristic-based approach for obtaining high-quality frontier approximations for larger instances. Through extensive computational experiments, including a Vienna-based real-data case study, we provide key managerial insights into the fundamental trade-offs between revenue, equity, and system design, demonstrating that the proposed framework can serve as a useful decision-support tool for designing station-based electric car-sharing systems that are both economically viable and socially inclusive.

General first-order methods (GFOM) are a flexible class of iterative algorithms which update a state vector by matrix-vector multiplications and entrywise nonlinearities. A long line of work has sought to understand the large-n dynamics of GFOM, mostly focusing on "very random" input matrices and the approximate message passing (AMP) special case of GFOM whose state is asymptotically Gaussian. Yet, it has long remained unknown how to construct iterative algorithms that retain this Gaussianity for more structured inputs, or why existing AMP algorithms can be as effective for some deterministic matrices as they are for random matrices. We analyze diagrammatic expansions of GFOM via the limiting traffic distribution of the input matrix, the collection of all limiting values of permutation-invariant polynomials in the matrix entries, to obtain the following results: 1. We calculate the traffic distribution for the first non-trivial deterministic matrices, including (minor variants of) the Walsh-Hadamard and discrete sine and cosine transform matrices. This determines the limiting dynamics of GFOM on these inputs, resolving parts of longstanding conjectures of Marinari, Parisi, and Ritort (1994). 2. We design a new AMP iteration which unifies several previous AMP variants and generalizes to new input types, whose limiting dynamics are Gaussian conditional on some latent random variables. The asymptotic dynamics hold for a large and natural class of traffic distributions (encompassing both random and deterministic input matrices) and the algorithm's analysis gives a simple combinatorial interpretation of the Onsager correction, answering questions posed recently by Wang, Zhong, and Fan (2022).

In this work we examine the problem of data-driven prediction. That is, given a LTI system with unknown dynamics, we wish to use data collected from the system to predict the system's output response to a given sequence of known inputs. Current methods for predicting require strong conditions on the data such as persistency of excitation. We examine this problem with the goal of finding weaker conditions that still enable prediction. We approach the problem from the data informativity perspective and formally define the notion of informativity for unique prediction. We provide sufficient conditions for informativity for unique prediction and design algorithms to compute the unique output trajectory of the unknown system given known inputs. We demonstrate the results with a numerical example showing that unique output prediction is possible without being able to uniquely identify the unknown data-generating system.

Koopman operator theory is a key tool in data assimilation of complex dynamical systems, with the potential to be applied to multimodal data. We formulate the problem of learning Koopman eigenfunctions from observations at arbitrary, possibly non-vanishing, time intervals as an optimization problem. Analysis of the formulation reveals aliasing induced by oscillatory dynamics and the sampling pattern, making an inherent identifiability limit explicit. The analysis also uncovers phase alignment near the true Koopman frequency, which creates a steep loss valley and demands careful optimization. We further show that irregular sampling can break aliasing and lead to phase cancellation. Numerical results demonstrate the efficacy of the proposed method under large regular time intervals compared to generator extended dynamic mode decomposition, and support the idea that irregular sampling can help recover the true Koopman spectrum.

In this paper, we propose a novel distributed algorithm to optimize the emergent macroscopic behavior of large-scale multi-agent systems via microscopic actions. We cast this task as a bilevel optimization problem, where the upper level formalizes the desired macroscopic target behavior through a suitable performance criterion, which is shaped in the lower level by leveraging a compressed aggregate representation estimating the macroscopic state. More precisely, the macroscopic state is parametrized by an exponential-family of distributions and constructed from the multi-agent microscopic configuration. The proposed algorithm integrates a distributed estimation mechanism, through which each agent reconstructs the macroscopic state locally, with a hypergradient-based update of the microscopic states aimed at improving the collective macroscopic behavior. We prove convergence to the set of stationary points of the bilevel problem via timescale separation arguments. Numerical simulations validate the effectiveness of the proposed method.

In this paper we introduce a new dynamical condition, the comb geometric control condition, which is sufficient for observability of the Schrödinger equation in Euclidean space. We provide examples which show this condition is strictly weaker than the observation set being open and periodic. We also prove for the fractional Schrödinger equation that for observation functions which are uniformly continuous, the geometric control condition is equivalent to observability and implies arbitrary time observability. The proofs rely on uncertainty principles for frequency localized functions which are proved using a semiclassical propagation of singularities approach.

We compare two classes of polynomial automorphisms, strongly nilpotent and Pascal finite. We conclude that every strongly nilpotent automorphism is a Pascal finite one, but not vice versa. We observe that Nagata's automorphism is Pascal finite, but not strongly nilpotent. Considering Vasyunin example leads us to conclusion that not every quadratic polynomial automorphism is Pascal finite.

In this paper, we analyze the asymptotic behavior of the point process of exceedances in a spatio-temporal setting whose points are given by the rescaled occurrence times, the sites and the rescaled values of exceedances. Here, the exceedances over a high threshold are flexibly defined via site-dependent risk functionals. Exploiting the framework of stationary regularly varying multivariate time series, we merge and extend the results from the literature in order to show weak convergence of the considered point processes of extremes and to explicitly determine its limit distribution.

We analyse the eigenvectors of the adjacency matrix of a random inhomogeneous graph constructed from a specified degree sequence. We assume that the empirical degree sequence has bounded mean and variance. We show that near the edges of the spectrum, the eigenvectors are semilocalized in the sense that their mass concentrates around a small set of resonant vertices. For the extremal eigenvalues, we establish localization around a single vertex. In order to obtain effective estimates in the presence of highly inhomogeneous degrees, we introduce a new economical pruning procedure that carefully extracts a forest from the original graph, whose adjacency matrix is compared to that of the original graph using a suitably constructed local coupling to random trees with independent edges.

Heterogeneous network data with rich nodal information become increasingly prevalent across multidisciplinary research, yet accurately modeling complex nodal heterogeneity and simultaneously selecting influential nodal attributes remains an open challenge. This problem is central to many applications in economics and sociology, when both nodal heterogeneity and high-dimensional individual characteristics highly affect network formation. We propose a statistically grounded, unified deep neural network approach for modeling nodal heterogeneity in random networks with high-dimensional nodal attributes, namely ``NetworkNet''. A key innovation of NetworkNet lies in a tailored neural architecture that explicitly parameterizes attribute-driven heterogeneity, and at the same time, embeds a scalable attribute selection mechanism. NetworkNet consistently estimates two types of latent heterogeneity functions, i.e., nodal expansiveness and popularity, while simultaneously performing data-driven attribute selection to extract influential nodal attributes. By unifying classical statistical network modeling with deep learning, NetworkNet delivers the expressive power of DNNs with methodological interpretability, algorithmic scalability, and statistical rigor with a non-asymptotic approximation error bound. Empirically, simulations demonstrate strong performance in both heterogeneity estimation and high-dimensional attribute selection. We further apply NetworkNet to a large-scale author-citation network among statisticians, revealing new insights into the dynamic evolution of research fields and scholarly impact.

Addressing a question posed by Erdős and Hajnal, Chen and Ma proved that, for all $n \ge 600$, the complete bipartite graph $K_{n,n+1}$ is the unique graph on $2n+1$ vertices with at least $n^2+n$ edges that contains no two vertices of equal degree joined by a path of length three. In this paper, we extend this result and show that, for all $n \ge 11$, $K_{n,n+1}$ is the unique $(2n+1)$-vertex graph with at least $n^2+n$ edges that avoids two equal-degree vertices joined by a path of length five. This confirms the very next case of a general conjecture of Chen and Ma on paths of odd length with equal-degree endpoints.

The Borsuk problem asks for the smallest number of subsets with strictly smaller diameters into which any bounded set in the $d$-dimensional space can be decomposed. It is a classical problem in combinatorial geometry that has been subject of much attention over the years, and research on variants of the problem continues nowadays in a plethora of directions. In this work, we propose a formulation of the problem in the context of graphs. Depending on how the graph is partitioned, we consider two different settings dealing either with the usual notion of diameter in abstract graphs, or with the diameter in the context of continuous graphs, where all points along the edges, instead of only the vertices, must be taken into account when computing distances. We present complexity results, exact computations and upper bounds on the parameters associated to the problem.

An edge subset \( S \subseteq E(G) \) is called a 3-restricted edge-cut if $G-S$ is disconnected and each component of \( G - S \) contains at least three vertices. The 3-restricted edge-connectivity of a graph \( G \), denoted by \( λ_3(G) \), is defined as the minimum cardinality among all 3-restricted edge-cuts if there are at least one; otherwise, \( λ_3(G) = +\infty \). It is proved that $λ_3(G)\leqξ_3(G)$ if $G$ has a 3-restricted edge-cut, where $ξ_3(G) = \min \{ |[X, V(G) \setminus X]_G||X \subseteq V(G),|X| = 3 \text{ and } G[X] \text{ is connected}\}.$ If \( λ_3(G) = ξ_3(G) \), then \( G \) is said to be maximally 3-restricted edge-connected. The strong product of graphs \( G \) and \( H \), denoted by \( G \boxtimes H \), is the graph with the vertex set $ V(G)\times V(H) $ and the edge set $ \{(x_{1},y_{1})(x_{2},y_{2})|x_{1}=x_{2}\text{ and }y_{1}y_{2}\in E(H);\text{ or }y_{1}=y_{2} $ and $ x_{1}x_{2}\in E(G) $; or $ x_{1}x_{2}\in E(G) $ and $ y_{1}y_{2}\in E(H)\}$. In this paper, we prove that \( G \boxtimes C_{n} \) is maximally 3-restricted edge-connected, and determine the 3-restricted edge-connectivity of \( G \boxtimes K_{n} \), where \( G \) is a maximally edge-connected graph, \( C_{n} \) and \( K_{n} \) are the cycle and the complete graph of order \( n \), respectively.

Let \((M^n,g)\) be a smooth closed Riemannian manifold of dimension \(n \ge 5\) with positive Yamabe invariant and semi-positive \(Q\)-curvature. We establish a precompactness result in the \(C^α\)-Hölder topologie on the space of Riemannian metrics, for some \(α>0\), for the set of metrics \(\tilde{g}\) conformal to \(g\), with volume equal to that of the standard sphere \(\mathbb{S}^n\), whose \(Q\)-curvature is nonnegative and uniformly bounded in \(L^p(M,\tilde{g})\) for some \(p > \frac{n}{4}\), and whose first positive eigenvalue of the Laplace-Beltrami operator satisfies \( λ_1(M,\tilde{g}) \ge n + \frac{1}Λ \) for some positive constant \(Λ\).

We study the existence of directed Hamilton cycles in random digraphs with $m$ edges where we condition on minimum in- and out-degree $\d \ge k+1$, where $k \ge 1$. Denote such a random graph by $D_{n,m}^{(δ\geq k+1)}$. Let $m=cn$ and $c\ge c_k$, where $c_k$ is a sufficiently large constant. We prove that w.h.p. $D_{n,m}^{(δ\geq k+1)}$ contains $k$ edge disjoint Hamilton cycles.

We investigate quantum state transfer on a class of bipartite graphs, namely the butterfly graphs, within the framework of discrete-time quantum walks. These graphs facilitate the construction of scalable quantum networks that enable communication between a sender and a receiver via perfect state transfer. Our analysis demonstrates that state transfer occurs across different butterfly graphs, thereby extending the known families of networks that support high-fidelity quantum state transfer. In addition to the ideal noiseless dynamics, we further investigate the robustness of quantum state transfer in the presence of non-Markovian environmental noise, specifically, random telegraph noise, modified Ornstein-Uhlenbeck noise, which are examples of unital noise and non-Markovian amplitude damping noise, non-unital noise. These noise models capture different types of system-environment interactions and memory effects that influence the coherence of the quantum walk. These findings contribute to the theoretical understanding of how butterfly graph constructions influence quantum transport phenomena.

First, we consider generalized wave and scattering operators and derive modifications of commutation relations (between scattering operators and unperturbed operators) when the corresponding deviation factors behave as $\exp\{i t {\mathcal C}_{\pm}\}$ for $t\to \pm \infty$. Then, we construct so called secondary generalized scattering operators for the related case of linear divergence in QED, which gives a positive answer (in that case) to the well-known problem of J. R. Oppenheimer regarding scattering operators in QED: "Can the procedure be freed of the expansion in $\varepsilon$ and carried out rigorously?"

We perform a rigorous examination of the sharp interface limit of a coupled Navier-Stokes and mass-conserving Allen-Cahn system in a two-dimensional, bounded, and smooth domain as the parameter $\varepsilon > 0$, representing the thickness of the diffuse interface, tends to zero. We prove the convergence of solutions from the mass-conserving Navier-Stokes/Allen-Cahn system to those of its sharp interface limit. In this limit, the interface evolves according to mass-conserving mean curvature flow with a convection term and is coupled to a two-phase Navier-Stokes system with surface tension. Our approach entails the construction of an approximate solution for the limiting system through the use of matched asymptotic expansions, complemented by a special ansatz for the leading-order term. In order to estimate the error between this approximate solution and the exact solution, we employ a refined spectral estimate for the linearized Allen-Cahn operator near the approximate solution.

Non-Terrestrial Networks (NTN) have emerged as a key enabler to fully realize the vision of integrated, intelligent, and ubiquitous connectivity in 6G systems. However, several operational challenges, including severe Doppler effects, interference, and latency, hinder the seamless integration of NTN and Terrestrial Networks (TN). In this context, Integrated Sensing and Communication (ISAC), which unifies sensing and communication functionalities within a common framework, offers great potential to address these challenges while enabling new network capabilities. Due to its complementary functionalities, ISAC can play a pivotal role in enhancing NTN performance, although its practical adoption requires a fundamental rethinking of existing architectural and standardization frameworks. Motivated by this need, this article examines key aspects of ISAC-enabled NTN, including architectural design principles, application scenarios, standardization challenges, and key performance tradeoffs. Finally, a representative case study is presented to illustrate major technical challenges and highlight promising future research directions for ISAC-enabled NTN.

We introduce a game-theoretical framework for the doubly nonlinear parabolic equation \[ |\partial_t u|^{p-2} \partial_t u - Δ_p u = 0. \] where $Δ_p u = \nabla \cdot ( |\nabla u |^{p-2} \nabla u)$ with $p>2$ is the standard $p-$Laplacian. A key feature to our approach is a new asymptotic mean value formula (AMVF) for the $p-$Laplacian that is robust even when the gradient vanishes and is independent of the sign of the $p-$Laplacian. This new AMVF leads naturally to a dynamic programming principle (DPP) whose solutions converge to the viscosity solution of the boundary value problem for the differential equation. In addition, solutions to the DPP coincide with value functions for a stochastic, two-players, zero-sum game that we introduce and analyze here.