arXiv Math @arxiv_math@qoto.org

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

Channel pruning is a promising method for accelerating and compressing convolutional neural networks. However, current pruning algorithms still remain unsolved problems that how to assign layer-wise pruning ratios properly and discard the least important channels with a convincing criterion. In this paper, we present a novel channel pruning approach via information theory and interpretability of neural networks. Specifically, we regard information entropy as the expected amount of information for convolutional layers. In addition, if we suppose a matrix as a system of linear equations, a higher-rank matrix represents there exist more solutions to it, which indicates more uncertainty. From the point of view of information theory, the rank can also describe the amount of information. In a neural network, considering the rank and entropy as two information indicators of convolutional layers, we propose a fusion function to reach a compromise of them, where the fusion results are defined as ``information concentration''. When pre-defining layer-wise pruning ratios, we employ the information concentration as a reference instead of heuristic and engineering tuning to provide a more interpretable solution. Moreover, we leverage Shapley values, which are a potent tool in the interpretability of neural networks, to evaluate the channel contributions and discard the least important channels for model compression while maintaining its performance. Extensive experiments demonstrate the effectiveness and promising performance of our method. For example, our method improves the accuracy by 0.21% when reducing 45.5% FLOPs and removing 40.3% parameters for ResNet-56 on CIFAR-10. Moreover, our method obtains loss in Top-1/Top-5 accuracies of 0.43%/0.11% by reducing 41.6% FLOPs and removing 35.0% parameters for ResNet-50 on ImageNet.

Orbital angular momentum (OAM), which is considered as a novel way to achieve high capacity, has been attracted much attention recently. OAM signals emitted by uniform circular array (UCA) are widely regarded to go through the Bessel-form channels. However, the channel gains corresponding to the Bessel-form channels are with low signal-to-noise-ratio (SNR) on OAM-modes and it is difficult to achieve high capacity using all OAM modes. To achieve maximum capacity offered by OAM multiplexing for wireless backhaul communications, in this paper we propose the optimal UCA design, which selects the optimal OAM-modes and radius of receive UCA. We formulate the capacity maximization problem and divide it into two subproblems for obtaining the corresponding optimal UCA design for OAM multiplexing based wireless backhaul communications. In particular, the optimal radius of the receive UCA is firstly derived. Then, we propose an mode selection scheme to choose appropriate OAM-modes for data transmission to maximize the capacity. Extensive simulations obtained validate that the capacity of OAM multiplexing can be significantly increased with our developed scheme.

We study various acyclicity conditions on higher-categorical pasting diagrams in the combinatorial framework of regular directed complexes. We present an apparently weakest acyclicity condition under which the $ω$-category presented by a diagram shape is freely generated in the sense of polygraphs. We then consider stronger conditions under which this $ω$-category is equivalent to one obtained from an augmented directed chain complex in the sense of Steiner, or consists only of subsets of cells in the diagram. Finally, we study the stability of these conditions under the operations of pasting, suspensions, Gray products, joins and duals.

In this short note, we describe finite groups all of whose non-trivial cyclic subgroups have the same Chermak-Delgado measure.

Let K be a global field, that is, a number field or a global function field. It is known that the answer to the question in the title over K is "Yes" when K has no real embeddings. We show that otherwise the answer is "No". Namely, we show that when K is a number field admitting a real embedding, it is impossible to define a group structure on the first Galois cohomology sets H^1(K,G) for all reductive K-groups G in a functorial way.

The concept of $S$-characters of finite groups was introduced by Zhmud as a generalisation of transitive permutation characters. Any non-trivial $S$-character takes a zero value on some group element. By a deep result depending on the classification of finite simple groups a non-trivial transitive permutation character even vanishes on some element of prime power order. We present an example that this does not generalise to $S$-characters, thereby answering a question posed by J.-P. Serre.

Non-orthogonal multiple access (NOMA) is a promising technology for next-generation wireless communication systems due to its enhanced spectral efficiency. In this paper, we consider an uplink NOMA system operating together with a high-dimensional absorptive reconfigurable intelligent surface (A-RIS). We aim to minimize the total power transmitted by the users in order to meet signal-to-interference-plus-noise constraints at the base station in the presence of a jammer. We propose an iterative algorithm to solve the high-dimensional non-convex optimization problem using linear programming to find the transmit powers and a fractional programming algorithm based on the Dinkelbach algorithm with a sequential convex relaxation procedure to optimize the reflection coefficients. We show that our algorithm converges on large optimization problems, with a jammer comprising as many as $64$ antennas, and an A-RIS with $128$ elements. Our numerical results show that, compared with a standard RIS that reflects all impinging energy, the A-RIS can dramatically reduce the users' required transmit power and successfully mitigate interference from the jammer. The absorption capability of the A-RIS is in particular useful in cases when the number of jammer antennas is of the same order as the number of A-RIS elements.

We define a notion of a homotopy chiral algebra (HCA), which means a chiral algebra up to higher homotopies, and prove that the Cech complex of a sheaf of chiral algebras admits a structure of a HCA.

In this article, we use the Bruhat and Schubert cells to calculate the cellular homology of the maximal compact subgroup $K$ of a connected semisimple Lie group $G$ whose Lie algebra is a split real form. We lift to the maximal compact subgroup the previously known attaching maps for the maximal flag manifold and use it to characterize algebraically the incidence order between Schubert cells. We also present algebraic formulas to compute the boundary maps which extend to the maximal compact subgroups similar formulas obtained in the case of the maximal flag manifolds. Finally, we apply our results to calculate the cellular homology of $\mbox{SO}(3)$ as the maximal compact subgroup of $\mbox{SL}(3, \mathbb{R})$ and the cellular homology of $\mbox{SO}(4)$ as the maximal compact subgroup of the split real form $G_2$.

Variable selection is a problem of statistics that aims to find the subset of the $N$-dimensional possible explanatory variables that are truly related to the generation process of the response variable. In high-dimensional setups, where the input dimension $N$ is comparable to the data size $M$, it is difficult to use classic methods based on $p$-values. Therefore, methods based on the ensemble learning are often used. In this review article, we introduce how the performance of these ensemble-based methods can be systematically analyzed using the replica method from statistical mechanics when $N$ and $M$ diverge at the same rate as $N,M\to\infty, M/N\toα\in(0,\infty)$. As a concrete application, we analyze the power of stability selection (SS) and the derandomized knockoff (dKO) with the $\ell_1$-regularized statistics in the high-dimensional linear model. The result indicates that dKO provably outperforms the vanilla knockoff and the standard SS, while increasing the bootstrap resampling rate in SS might further improve the detection power.