arXiv Math @arxiv_math@qoto.org

This paper introduces an innovative approach for signal reconstruction using data acquired through multi-input-multi-output (MIMO) sampling. First, we show that it is possible to perfectly reconstruct a set of periodic band-limited signals $\{x_r(t)\}_{r=1}^R$ from the samples of $\{y_m(t)\}_{m=1}^M$, which are the output signals of a MIMO system with inputs $\{x_r(t)\}_{r=1}^R$. Moreover, an FFT-based algorithm is designed to perform the reconstruction efficiently. It is demonstrated that this algorithm encompasses FFT interpolation and multi-channel interpolation as special cases. Then, we investigate the consistency property and the aliasing error of the proposed sampling and reconstruction framework to evaluate its effectiveness in reconstructing non-band-limited signals. The analytical expression for the averaged mean square error (MSE) caused by aliasing is presented. Finally, the theoretical results are validated by numerical simulations, and the performance of the proposed reconstruction method in the presence of noise is also examined.

In this paper, we construct a $p$-adic $L$-function for a $P$-ordinary Hida family of cuspidal automorphic representations on a unitary group. We first describe the geometry and representation theory involved in the study of such $P$-ordinary automorphic representations. Then, we construct a $p$-adic family of Siegel Eisenstein series and use the doubling method of Garrett and Piatetski--Shapiro-Rallis to relate certain zeta integrals to special values of standard $L$-functions. The $p$-adic interpolation of these special values is obtained by looking at the $p$-adic variation of the Fourier coefficients of the Siegel Eisenstein series.

The permutations of horse racing, where ties are possible, are counted by the $\textit{Fubini numbers}$, also called the $\textit{horse numbers}$. The $\textit{r-horse numbers}$ are a counting of such horse race finishes where some subset of $r$ horses agree to finish the race in a specific strong ordering. The $r$-horse numbers for fixed $r$ are expressed as a sum of $r$ index shifted sequences of Fubini numbers weighted with the $\textit{signed Stirling numbers of the first kind}$. Then eventual modular periodicity of $\textit{r-Fubini numbers}$ is shown and their maximum period is determined to be the $\textit{Carmichael function}$ of the modulus. The maximum period is attained in the case of an odd modulus for Fubini numbers.

In this paper, we calculate the annihilating polynomial of the colored Jones polynomial for the Hopf link and the Whitehead link and consider the link version of the AJ conjecture. We also give another annihilating polynomial of the colored Jones polynomial for those links.

In this article, we develop and present a novel regularization scheme for ill-posed inverse problems governed by nonlinear partial differential equations (PDEs). In [43], the author suggested a bi-level regularization framework. This study significantly improves upon the bi-level algorithm by sequential initialization, yielding accelerated convergence and demonstrable multi-scale effect, while retaining regularizing effect with respect to noise and allows for the usage of inexact PDE solvers. The sequential bi-level approach illustrates its universality through several reaction-diffusion applications, in which the nonlinear reaction law needs to be determined. To demonstrate the applicability of our algorithms, we moreover prove that the proposed tangential cone condition is satisfied.

This paper studies the interference broadcast channel comprising multiple multi-antenna Base Stations (BSs), each controlling a beyond diagonal Reconfigurable Intelligent Surface (RIS) and serving multiple single-antenna users. Wideband transmissions are considered with the objective to jointly design the BS linear precoding vectors and the phase configurations at the RISs in a distributed manner. We take into account the frequency selectivity behavior of each RIS's tunable meta-element, and focusing on the sum rate as the system's performance criterion, we present a distributed optimization approach that enables cooperation between the RIS control units and their respective BSs. According to the proposed scheme, each design variable can be efficiently obtained in an iterative parallel way with guaranteed convergence properties. Our simulation results demonstrate the validity of the presented distributed algorithm and showcase its superiority over a non-cooperative scheme as well as over the special case where the RISs have a conventional diagonal structure.

We investigate the structure of the minimal displacement set in weakly systolic complexes. We show that such set is systolic and that it embeds isometrically into the complex. As corollaries, we prove that any isometry of a weakly systolic complex either fixes the barycentre of some simplex (elliptic case) or it stabilizes a thick geodesic (hyperbolic case).

In the analysis of parametrized nonautonomous evolutionary equations, bounded entire solutions are natural candidates for bifurcating objects. Appropriate explicit and sufficient conditions for such branchings, however, require to combine contemporary functional analytical methods from the abstract bifurcation theory for Fredholm operators with tools originating in dynamical systems. This paper establishes alternatives classifying the shape of global bifurcating branches of bounded entire solutions to Carathéodory differential equations. Our approach is based on the parity associated to a path of index 0 Fredholm operators, the global Evans function as a recent tool in nonautonomous bifurcation theory and suitable topologies on spaces of Carathéodory functions.

We compute the equivariant K-theory of torus fixed points of Cherkis bow varieties of affine type A. We deduce formulas for the generating series of the Euler numbers of these varieties and observe their modularity in certain cases. We also obtain refined formulas on the motivic level for a class of bow varieties strictly containing Nakajima quiver varieties. These series hence generalise results of Nakajima-Yoshioka. As a special case, we obtain formulas for certain Zastava spaces. We define a parabolic analogue of Nekrasov's partition function and find an equation relating it to the classical partition function.

In this paper, we investigate the subelliptic nonlocal Brezis-Nirenberg problem on stratified Lie groups involving critical nonlinearities, namely, \begin{align*} (-Δ_{\mathbb{G}, p})^s u&= μ|u|^{p_s^*-2}u+λh(x, u) \quad \text{in}\quad Ω, \\ u&=0\quad \text{in}\quad \mathbb{G}\backslash Ω, \end{align*} where $(-Δ_{\mathbb{G}, p})^s$ is the fractional $p$-sub-Laplacian on a stratified Lie group $\mathbb{G}$ with homogeneous dimension $Q,$ $Ω$ is an open bounded subset of $\mathbb{G},$ $s \in (0,1)$, $\frac{Q}{s}>p\geq2,$ $p_s^*:=\frac{pQ}{Q-ps}$ is subelliptic fractional Sobolev critical exponent, $μ, λ>0$ are real parameters and $h$ is a lower order perturbation of the critical power $|u|^{p_s^*-2}u$. Utilising direct methods of the calculus of variation, we establish the existence of at least one weak solution for the above problem under the condition that the real parameter $λ$ is sufficiently small. Additionally, we examine the problem for $μ= 0$, representing subelliptic nonlocal equations on stratified Lie groups depending on one real positive parameter and involving a subcritical nonlinearity. We demonstrate the existence of at least one solution in this scenario as well. We emphasize that the results obtained here are also novel for $p=2$ even for the Heisenberg group.

A generalization of Ripà's square spiral solution for the $n \times n \times \cdots \times n$ Points Upper Bound Problem. Additionally, we provide a non-trivial lower bound for the $k$-dimensional $n_1 \times n_2 \times \cdots \times n_k$ Points Problem. In this way, we can build a range in which, with certainty, all the best possible solutions to the problem we are considering will fall. Finally, we give a few characteristic numerical examples in order to appreciate the fineness of the result arising from the particular approach we have chosen.

We prove that the Lalescu sequence is monotonically decreasing.

This study uses the Lotka Volterra Predator-Prey model to offer a notion of piecewise patterns for the various piecewise derivatives. Using the piecewise derivatives, we produced numerical solutions that are referred to as the Adams-Bashforth method. The computer results show piecewise patterns in the Lotka Volterra Predator-Prey model's real-world behaviours. The Lotka-Volterra model looks into the relationships between competition and abundance between two competing species. Changes in the abundance of one species are modelled as a function of the abundance of its competitors, but the competitive mechanism is given and evaluated. This notion led some scholars to label certain mathematical expressions as "phenomenological" and to propose a different theoretical framework that gives resources special consideration. The Lotka-Volterra model, often called the predator-prey model or the Lotka-Volterra model, is a nonlinear mathematical expression that is frequently used to analyse the dynamical behaviours of biological systems in which two species interact, one as a predator and the other as prey. The variation in the populations over time is illustrated by the mathematical statement.

After recalling the notion of higher roots (or hyper-roots) associated with "quantum modules" of type $(G, k)$, for $G$ a semi-simple Lie group and $k$ a positive integer, following the definition given by A. Ocneanu in 2000, we study the theta series of their lattices. Here we only consider the higher roots associated with quantum modules (aka module-categories over the fusion category defined by the pair $(G,k)$) that are also "quantum subgroups". For $G=SU{2}$ the notion of higher roots coincides with the usual notion of roots for ADE Dynkin diagrams and the self-fusion restriction (the property of being a quantum subgroup) selects the diagrams of type $A_{r}$, $D_{r}$ with $r$ even, $E_6$ and $E_8$; their theta series are well known. In this paper we take $G=SU{3}$, where the same restriction selects the modules ${\mathcal A}_k$, ${\mathcal D}_k$ with $mod(k,3)=0$, and the three exceptional cases ${\mathcal E}_5$, ${\mathcal E}_9$ and ${\mathcal E}_{21}$. The theta series for their associated lattices are expressed in terms of modular forms twisted by appropriate Dirichlet characters.

For love dynamical models, a new idea combining piecewise concept for integer-order, stochastic, and fractional derivatives is presented in order to capture the chaos and several crossover emotional scenerios. Under the assumptions of linear growth and Lipschitz condition, the fixed-point theorem explain the uniqueness and existence to the models under the investigation. The piecewise derivatives were approximated utilising the Lagrange interpolation method, and the computer results were demonstrated numerically for several values of order $α$. It was observed that the recently presented new idea in love dynamical models can represent disordered emotional patterns in passionate loving partnerships.

In this note, we consider some Burgers-like equations involving Laguerre derivatives and demonstrate that it is possible to construct specific exact solutions using separation of variables. We prove that a general scheme exists for constructing exact solutions for these Burgers-like equations, extending to more general cases, including nonlinear time-fractional equations. Exact solutions can also be obtained for KdV-like equations involving Laguerre derivatives. We finally consider a particular class of Burgers equations with variable coefficients whose solution can be obtained similarly.

Recently, Drema and N. Saikia (2023) and M. P. Saikia, Sarma, and Sellers (2023) proved several congruences modulo powers of $2$ for overpartition triples with odd parts. In this paper, we study further divisibility properties of overpartition $k$-tuples with odd parts using elementary means as well as properties of modular forms. In particular, we prove several congruences modulo multiples of $3$, and an infinite family of congruences modulo powers of $3$; we also prove some cases of a conjecture of Saikia, Sarma, and Sellers.

Waterbomb style tessellations have been explored in the past by artists such as Ronald D. Resch, Benjamin Parker and Mitya Miller. Generalised waterbomb tessellations are still underexplored in origami design. We have explored various sets of criteria for generalising waterbomb tessellations in order to enumerate valid patterns. We only consider triangular waterbomb tessellations, other polygons will be explored in future papers. In our search we have uncovered some new waterbomb tessellations, which could offer new uses in representational and geometric origami design. We conclude by discussing foldability properties and possible generalisations.

In this article, the authors give the correct answer to the following problem, which is presented in the well-known problem book "CHALLENGING MATHEMATICAL PROBLEMS WITH ELEMENTARY SOLUTIONS"? by A. M. Yaglom and L. M. Yaglom. There are six long blades of grass with the ends protruding above and below, and you will tie together the six upper ends in pairs and then tie together the six lower ends in pairs. What is the probability that a ring will be formed when the blades of grass are tied at random in this fashion? The solution in the above book needs to be corrected, and we will present a correct answer in this article. Therefore, we are the first persons to present a correct?answer to a problem in a book published in the USSR? in 1954. By following the original idea of this problem book, we present the correct answer without using knowledge of higher knowledge, although we used a very basic knowledge of the Knot theory.

Let $a, b\in \mathbb{N}$ be relatively prime. Previous work showed that exactly one of the two equations $ax + by = (a-1)(b-1)/2$ and $ax + by + 1 = (a-1)(b-1)/2$ has a nonnegative, integral solution; furthermore, the solution is unique. Let $F_n$ be the $n$th Fibonacci number. When $(a,b) = (F_n, F_{n+1})$, it is known that there is an explicit formula for the unique solution $(x,y)$. We establish formulas to compute the solution when $(a,b) = (F_n^2, F_{n+1}^2)$ and $(F_n^3, F_{n+1}^3)$, giving rise to some intriguing identities involving Fibonacci numbers. Additionally, we construct a different pair of equations that admits a unique positive (instead of nonnegative), integral solution.