arXiv Math @arxiv_math@qoto.org

In this paper, we first give a new result characterizing the strongly connected digraphs with a diameter equal to that of their line digraphs. Then, we introduce the concepts of the inner diameter and inner radius of a digraph and study their behaviors in its iterated line digraphs. Furthermore, we provide a method to characterize sequences of integers (corresponding to the inner diameter or the number of vertices of a digraph and its iterated line digraphs) that satisfy some conditions. Among other examples, we apply the method to the cyclic Kautz digraphs, square-free digraphs, and the subdigraphs of De Bruijn digraphs. Finally, we present some tables with new sequences that do not belong to The On-Line Encyclopedia of Integer Sequences.

In this paper, we provide a criterion on uniform large deviation principles (ULDP) for stochastic differential equations under locally weak monotone conditions and Lyapunov conditions, which can be applied to stochastic systems with coefficients of polynomial growth and possible degenerate driving noises, including the stochastic Hamiltonian systems. The weak convergence method plays an important role in obtaining the ULDP. This result extends the scope of applications of the main theorem in \cite{WYZZ}.

We prove a formula for the involutive concordance invariants of the cabled knots in terms of that of the companion knot and the pattern knot. As a consequence, we show that any iterated cable of a knot with parameters of the form (odd,1) is not smoothly slice as long as either of the involutive concordance invariants of the knot is nonzero. Our formula also gives new bounds for the unknotting number of a cabled knot, which are sometimes stronger than other known bounds coming from knot Floer homology.

Let $E$ be an elliptic curve defined over $\mathbb{Q}$, and let $K$ be an imaginary quadratic field. Consider an odd prime $p$ at which $E$ has good supersingular reduction with $a_p(E)=0$ and which is inert in $K$. Under the assumption that the signed Selmer groups are cotorsion modules over the corresponding Iwasawa algebra, we prove that the Mordell-Weil ranks of $E$ are bounded over any subextensions of the anticyclotomic $\mathbb{Z}_p$-extension of $K$. Additionally, we provide an asymptotic formula for the growth of the $p$-parts of the Tate-Shafarevich groups of $E$ over these extensions.

It is established ground states and multiplicity of solutions for a nonlocal Schrödinger equation $(-Δ)^s u + V(x) u = λa(x) |u|^{q-2}u + b(x)f(u)$ in $\mathbb{R}^N,$ $u \in H^s(\mathbb{R}^N),$ where $0<s<\min\{1,N/2\},$ $1<q<2$ and $λ>0,$ under general conditions over the measurable functions $a,$ $b$, $V$ and $f.$ The nonlinearity $f$ is superlinear at infinity and at the origin, and does not satisfy any Ambrosetti-Rabinowitz type condition. It is considered that the weights $a$ and $b$ are not necessarily bounded and the potential $V$ can change sign. We obtained a sharp $λ^*> 0$ which guarantees the existence of at least two nontrivial solutions for each $λ\in (0, λ^*)$. Our approach is variational in its nature and is based on the nonlinear Rayleigh quotient method together with some fine estimates. Compactness of the problem is also considered.

We define and study a multiplicity free covering of a graded manifold. As an application of our research we give a new conceptual proof of the theorem about equivalence of categories of graded manifolds and symmetric $n$-fold vector bundles.

We prove that $\operatorname{vol}(S^{d})/8$ is the highest volume of a pair of $d$-dimensional isospectral and non-isometric spherical orbifolds for any $d\geq5$. Furthermore, we show that $\operatorname{vol}(S^{2n-1})/11$ is the highest volume of a pair of $(2n-1)$-dimensional isospectral and non-isometric spherical space forms if either $n\geq11$ and $n\equiv 1\pmod 5$, or $n\geq7$ and $n\equiv 2\pmod 5$, or $n\geq3$ and $n\equiv 3\pmod 5$.

Let \begin{equation*} S_{0}=0,\quad S_{n}=X_{1}+...+X_{n},\ n\geq 1, \end{equation*} be a random walk whose increments belong without centering to the domain of attraction of a stable law with scaling constants $a_{n}$, that provide convergence as $n\rightarrow \infty $ of the distributions of the elements of the sequence $\left\{ S_{n}/a_{n},n=1,2,...\right\} $ to this stable law. Let $L_{r,n}=\min_{r\leq m\leq n}S_{m}$ be the minimum of the random walk on the interval $[r,n]$. It is shown that \begin{equation*} \lim_{r,k,n\rightarrow \infty }\mathbf{P}\left( L_{r,n}\leq ya_{k}|S_{n}\leq ta_{k},L_{0,n}\geq 0\right) ,\, t\in \left( 0,\infty \right), \end{equation*} can have five different expressions, the forms of which depend on the relationships between the parameters $r,k$ and $n$.

We provide a symbolic classification of generalized Seifert fiber spaces, which were introduced by Mitsuishi and Yamaguchi in the classification of collapsing Alexandrov $3$-spaces. Additionally, we show that the canonical double branched cover of a non-manifold generalized Seifert fiber space is a Seifert manifold and compute its symbolic invariants in terms of those of the original space.

In this companion article to [HKK24], we apply the theory of equivariant Poincaré duality developed there in the special case of cyclic groups $C_p$ of prime order to remove, in a special case, a technical condition given by Davis--Lück [DL24] in their work on the Nielsen realisation problem for aspherical manifolds. Along the way, we will also give a complete characterisation of $C_p$--Poincaré spaces as well as introduce a genuine equivariant refinement of the classical notion of virtual Poincaré duality groups which might be of independent interest.

Downlink channel temporal prediction is a critical technology in massive multiple-input multiple-output (MIMO) systems. However, existing methods that rely on fixed-step historical sequences significantly limit the accuracy, practicality, and scalability of channel prediction. Recent advances have shown that large language models (LLMs) exhibit strong pattern recognition and reasoning abilities over complex sequences. The challenge lies in effectively aligning wireless communication data with the modalities used in natural language processing to fully harness these capabilities. In this work, we introduce Csi-LLM, a novel LLM-powered downlink channel prediction technique that models variable-step historical sequences. To ensure effective cross-modality application, we align the design and training of Csi-LLM with the processing of natural language tasks, leveraging the LLM's next-token generation capability for predicting the next step in channel state information (CSI). Simulation results demonstrate the effectiveness of this alignment strategy, with Csi-LLM consistently delivering stable performance improvements across various scenarios and showing significant potential in continuous multi-step prediction.

Channel knowledge map (CKM) is a promising technology to enable environment-aware wireless communications and sensing. Link state map (LSM) is one particular type of CKM that aims to learn the location-specific line-of-sight (LoS) link probability between the transmitter and the receiver at all possible locations, which provides the prior information to enhance the communication quality of dynamic networks. This paper investigates the LSM construction for cellularconnected unmanned aerial vehicles (UAVs) by utilizing both the expert empirical mathematical model and the measurement data. Specifically, we first model the LSM as a binary spatial random field and its initial distribution is obtained by the empirical model. Then we propose an effective binary Bayesian filter to sequentially update the LSM by using the channel measurement. To efficiently update the LSM, we establish the spatial correlation models of LoS probability on the location pairs in both the distance and angular domains, which are adopted in the Bayesian filter for updating the probabilities at locations without measurements. Simulation results demonstrate the effectiveness of the proposed algorithm for LSM construction, which significantly outperforms the benchmark scheme, especially when the measurements are sparse.

We consider the eigenvalue problem for the Schrödinger operator on bounded, convex domains with mixed boundary conditions, where a Dirichlet boundary condition is imposed on a part of the boundary and a Neumann boundary condition on its complement. We prove inequalities between the lowest eigenvalues corresponding to two different choices of such boundary conditions on both planar and higher-dimensional domains. We also prove an inequality between higher order mixed eigenvalues and pure Dirichlet eigenvalues on multidimensional polyhedral domains.

Unmanned Aerial Vehicles (UAVs), due to their low cost and high flexibility, have been widely used in various scenarios to enhance network performance. However, the optimization of UAV trajectories in unknown areas or areas without sufficient prior information, still faces challenges related to poor planning performance and low distributed execution. These challenges arise when UAVs rely solely on their own observation information and the information from other UAVs within their communicable range, without access to global information. To address these challenges, this paper proposes the Qedgix framework, which combines graph neural networks (GNNs) and the QMIX algorithm to achieve distributed optimization of the Age of Information (AoI) for users in unknown scenarios. The framework utilizes GNNs to extract information from UAVs, users within the observable range, and other UAVs within the communicable range, thereby enabling effective UAV trajectory planning. Due to the discretization and temporal features of AoI indicators, the Qedgix framework employs QMIX to optimize distributed partially observable Markov decision processes (Dec-POMDP) based on centralized training and distributed execution (CTDE) with respect to mean AoI values of users. By modeling the UAV network optimization problem in terms of AoI and applying the Kolmogorov-Arnold representation theorem, the Qedgix framework achieves efficient neural network training through parameter sharing based on permutation invariance. Simulation results demonstrate that the proposed algorithm significantly improves convergence speed while reducing the mean AoI values of users. The code is available at https://github.com/UNIC-Lab/Qedgix.

In this article, we describe the geography of the Teichmüller stack of \cite{LMStacks} and of one of its variants we introduce here, giving some answers to questions as: which points are orbifold points? What are the different local models of special points?... We give a rough description in the general case, and we use the compacity of the cycle spaces to get a much more detailed picture in the Kähler setting.

In this article, we study the long-time dynamics of threshold solutions for the focusing energy-critical inhomogeneous Schrödinger equation and classify the corresponding threshold solutions in dimensions $d=3,4,5$. We first show the existence of special threshold solutions $W^\pm$ by constructing a sequence of approximate solutions in suitable Lorentz space, which exponentially approach the ground state $W$ in one of the time directions. We then prove that solutions with threshold energy either behave as in the subthreshold case or agree with $W, W^+$ or $W^-$ up to the symmetries of the equation. The proof relies on detailed spectral analysis of the linearized Schrödinger operator, the relevant modulation analysis, the global Virial analysis, and the concentration compactness argument in the Lorentz space.

This survey revisits classical combinatorial optimization algorithms and extends them to two-stage stochastic models, particularly focusing on client-element problems. We reformulate these problems to optimize element selection under uncertainty and present two key sampling algorithms: SSA and Boost-and-Sample, highlighting their performance guarantees. Additionally, we explore correlation-robust optimization, introducing the concept of the correlation gap, which enables approximations using independent distributions with minimal accuracy loss. This survey analyzes and presents foundational combinatorial optimization methods for researchers at the intersection of this field and reinforcement learning.

We study the behavior of shallow water waves propagating over bathymetry that varies periodically in one direction and is constant in the other. Plane waves traveling along the constant direction are known to evolve into solitary waves, due to an effective dispersion. We apply multiple-scale perturbation theory to derive an effective constant-coefficient system of Boussinesq-type equations that not only accurately describe these waves but also predict their full two-dimensional shape in some detail. Numerical experiments confirm the good agreement between the effective equations and the variable-bathymetry shallow water equations.

In this paper, for any odd $n$ and any integer $m\geq1$ with $n>4m$, we study the fundamental solution of the higher order Schrödinger equation \begin{equation*} \mathrm{i}\partial_tu(x,t)=((-Δ)^m+V(x))u(x,t),\quad t\in \mathbb{R},\,\,x\in \mathbb{R}^n, \end{equation*} where $V$ is a real-valued $C^{\frac{n+1}{2}-2m}$ potential with certain decay. Let $P_{ac}(H)$ denote the projection onto the absolutely continuous spectrum space of $H=(-Δ)^m+V$, and assume that $H$ has no positive embedded eigenvalue. Our main result says that $e^{-\mathrm{i}tH}P_{ac}(H)$ has integral kernel $K(t,x,y)$ satisfying \begin{equation*} |K(t, x,y)|\le C(1+|t|)^{-(\frac{n}{2m}-σ)}(1+|t|^{-\frac{n}{2 m}})\left(1+|t|^{-\frac{1}{2 m}}|x-y|\right)^{-\frac{n(m-1)}{2 m-1}},\quad t\neq0,\,x,y\in\mathbb{R}^n, \end{equation*} where $σ=2$ if $0$ is an eigenvalue of $H$, and $σ=0$ otherwise. A similar result for smoothing operators $H^\fracα{2m}e^{-\mathrm{i}tH}P_{ac}(H)$ is also given. The regularity condition $V\in C^{\frac{n+1}{2}-2m}$ is optimal in the second order case, and it also seems optimal when $m>1$.

In this paper, we study the optimal filtering problem for a interacting particle system generated by stochastic differential equations with interaction. By using Malliavin calculus, we construct the differential equation of the covariance process and transform the filter problem to an optimal control problem. Finally we give the necessary condition that the coefficient of the optimal filter should satisfy