arXiv Math @arxiv_math@qoto.org

In the context of cellular networks, users located at the periphery of cells are particularly vulnerable to substantial interference from neighbouring cells, which can be represented as a two-user interference channel. This study introduces two highly effective methodologies, namely TwinNet and SiameseNet, using autoencoders, tailored for the design of encoders and decoders for block transmission and detection in interference-limited environments. The findings unambiguously illustrate that the developed models are capable of leveraging the interference structure to outperform traditional methods reliant on complete orthogonality. While it is recognized that systems employing coordinated transmissions and independent detection can offer greater capacity, the specific gains of data-driven models have not been thoroughly quantified or elucidated. This paper conducts an analysis to demonstrate the quantifiable advantages of such models in particular scenarios. Additionally, a comprehensive examination of the characteristics of codewords generated by these models is provided to offer a more intuitive comprehension of how these models achieve superior performance.

In the study of a non-convex minimization problem by Lachand-Robert and Peletier, they found that the difference between the compactly supported perturbation $u+εh$ of a strictly convex function $u$, and the $Γ$-regularization of $u+εh$, is at most $o(ε)$. Here we find that this result is optimal, albeit they expected a much stronger estimate.

In this work, we study the doubly degenerate nutrient taxis system with logistic source \begin{align} \begin{cases}\tag{$\star$}\label{eq 0.1} u_t=\nabla \cdot(u^{l-1} v \nabla u)- \nabla \cdot\left(u^{l} v \nabla v\right)+ u - u^2, \\ v_t=Δv-u v \end{cases} \end{align} in a smooth bounded domain $Ω\subset \mathbb{R}^2$, where $l \geqslant 1$. It is proved that for all reasonably regular initial data, the corresponding homogeneous Neumann initial-boundary value problem \eqref{eq 0.1} possesses a global weak solution which is continuous in its first and essentially smooth in its second component. We point out that when $l = 2$, our result is consistent with that of [G. Li and M. Winkler, Analysis and Applications, (2024)].

Results on the peridynamics equilibrium and evolution equations over the space of periodic vector-distributions in multi-spatial dimensions are presented. The associated operator considered is the linear state-based peridynamic operator for a homogeneous material. Results for weakly singular (integrable) as well as singular integral kernels are developed. The asymptotic behavior of the eigenvalues of the peridynamic operator's Fourier multipliers and eigenvalues are characterized explicitly in terms of the nonlocality (peridynamic horizon), the integral kernel singularity, and the spatial dimension. We build on the asymptotic analysis to develop regularity of solutions results for the peridynamic equilibrium as well as the peridynamic evolution equations over periodic distribution. The regularity results are presented explicitly in terms of the data, the integral kernel singularity, and the spatial dimension. Nonlocal-to-local convergence results are presented for the eigenvalues of the peridynamic operator and for the solutions of the equilibrium and evolution equations. The local limiting behavior is shown for two types of limits as the peridynamic horizon (nonlocality) vanishes or as the integral kernel becomes hyper-singular.

Sum of squares (SOS) optimization is a powerful technique for solving problems where the positivity of a polynomials must be enforced. The common approach to solve an SOS problem is by relaxation to a Semidefinite Program (SDP). The main advantage of this transormation is that SDP is a convex problem for which efficient solvers are readily available. However, while considerable progress has been made in recent years, the standard approaches for solving SDPs are still known to scale poorly. Our goal is to devise an approach that can handle larger, more complex problems than is currently possible. The challenge indeed lies in how SDPs are commonly solved. State-Of-The-Art approaches rely on the interior point method, which requires the factorization of large matrices. We instead propose an approach inspired by polynomial neural networks, which exhibit excellent performance when optimized using techniques from the deep learning toolbox. In a somewhat counter-intuitive manner, we replace the convex SDP formulation with a non-convex, unconstrained, and \emph{over parameterized} formulation, and solve it using a first order optimization method. It turns out that this approach can handle very large problems, with polynomials having over four million coefficients, well beyond the range of current SDP-based approaches. Furthermore, we highlight theoretical and practical results supporting the experimental success of our approach in avoiding spurious local minima, which makes it amenable to simple and fast solutions based on gradient descent. In all the experiments, our approach had always converged to a correct global minimum, on general (non-sparse) polynomials, with running time only slightly higher than linear in the number of polynomial coefficients, compared to higher than quadratic in the number of coefficients for SDP-based methods.

Potential-driven steady-state flow in networks is an abstract problem which manifests in various engineering applications, such as transport of natural gas, water, electric power through infrastructure networks or flow through fractured rocks modelled as discrete fracture networks. In general, while the problem is simple when restricted to a single edge of a network, it ceases to be so for a large network. The resultant system of nonlinear equations depends on the network topology and in general there is no numerical algorithm that offers guaranteed convergence to the solution (assuming a solution exists). Some methods offer guarantees in cases where the network topology satisfies certain assumptions but these methods fail for larger networks. On the other hand, the Newton-Raphson algorithm offers a convergence guarantee if the starting point lies close to the (unknown) solution. It would be advantageous to compute the solution of the large nonlinear system through the solution of smaller nonlinear sub-systems wherein the solution algorithms (Newton-Raphson or otherwise) are more likely to succeed. This article proposes and describes such a procedure, an hierarchical network partitioning algorithm that enables the solution of large nonlinear systems corresponding to potential-driven steady-state network flow equations.

We give a uniform estimate and an inequality for solutions of an equation with Dirichlet boundary condition.

In this article, we compute $δ$-invariants of Du Val del Pezzo surfaces of degree 1.

This paper presents a mathematical model for determining the movement of the bearing in an internal gear pump. The paper also performs simulation calculations to find the movement trajectory of the shaft with the given input data.

This paper is devoted to study of the limiting behaviour of an elastic material with periodically distributed rigid inclusions of size ε, as the small parameter ε goes to zero. We address here the case with inclusions of the same size as the period of the structure. The body in consideration here is suppose to be clamped on one part of its exterior boundary and submitted to given tractions on the other. By means of the well known two-scale convergence techniques, one convergence result is proved.

We study the representations of a class of non-commutative polynomial algebras truncated at degree 3, with one additional relation. We determine the irreducible components of their varieties of representations. We do this by showing that the subvarieties with fixed radical or socle layering are irreducible. We then compare the coverings we get using radical and socle layerings to determine the components.

The linear canonical transform (LCT) serves as a powerful generalization of the Fourier transform (FT), encapsulating various integral transforms within a unified framework. This versatility has made it a cornerstone in fields such as signal processing, optics, and quantum mechanics. Extending this concept to quaternion algebra, the Quaternion Fourier Transform (QFT) emerged, enriching the analysis of multidimensional and complex-valued signals. The Quaternion Linear Canonical Transform (QLCT), a further generalization, has now positioned itself as a central tool across various disciplines, including applied mathematics, engineering, computer science, and statistics. In this paper, we introduce the Two Dimensional Quaternion Linear Canonical Transform (2DQLCT) as a novel framework for probability modeling. By leveraging the 2DQLCT, we aim to provide a more comprehensive understanding of probability distributions, particularly in the context of multi-dimensional and complex-valued signals. This framework not only broadens the theoretical underpinnings of probability theory but also opens new avenues for researchers

Thomassen's chord conjecture from 1976 states that every longest cycle in a $3$-connected graph has a chord. The circumference $c(G)$ and induced circumference $c'(G)$ of a graph $G$ are the length of its longest cycles and the length of its longest chordless cycles, respectively. In $2017$, Harvey proposed a stronger conjecture: Every $2$-connected graph $G$ with minimum degree at least $3$ has $c(G)\geq c'(G)+2$. This conjecture implies Thomassen's chord conjecture. We observe that wheels are the unique hamiltonian graphs for which the circumference and the induced circumference differ by exactly one. Thus we need only consider non-hamiltonian graphs for Harvey's conjecture. In this paper, we propose a conjecture involving wheels that is equivalent to Harvey's conjecture on non-hamiltonian graphs. A graph is $\ell$-holed if its all holes have length exactly $\ell$. Furthermore, we prove that Harvey's conjecture holds for $\ell$-holed graphs and graphs with a small induced circumference. Consequently, Thomassen's conjecture also holds for this two classes of graphs.

We describe the Ruijsenaars' action-angle duality in classical many-body integrable systems through the spectral duality transformation relating the classical spin chains and Gaudin models. For this purpose, the Lax matrices of many-body systems are represented in the multi-pole (Gaudin-like) form by introducing a fictitious spectral parameter. This form of Lax matrices is also interpreted as classical-classical version of quantum-classical duality.

Diagrammatic sets admit a notion of internal equivalence in the sense of coinductive weak invertibility, with similar properties to its analogue in strict $ω$-categories. We construct a model structure whose fibrant objects are diagrammatic sets in which every round pasting diagram is equivalent to a single cell -- its weak composite -- and propose them as a model of $(\infty, \infty)$-categories. For each $n < \infty$, we then construct a model structure whose fibrant objects are those $(\infty, \infty)$-categories whose cells in dimension $> n$ are all weakly invertible. We show that weak equivalences between fibrant objects are precisely morphisms that are essentially surjective on cells of all dimensions. On the way to this result, we also construct model structures for $(\infty, n)$-categories on marked diagrammatic sets, which split into a coinductive and an inductive case when $n = \infty$, and prove that they are Quillen equivalent to the unmarked model structures when $n < \infty$ and in the coinductive case of $n = \infty$. Finally, we prove that the $(\infty, 0)$-model structure is Quillen equivalent to the classical model structure on simplicial sets. This establishes the first proof of the homotopy hypothesis for a model of $\infty$-groupoids defined as $(\infty, \infty)$-categories whose cells in dimension $> 0$ are all weakly invertible.

We prove that the solution map for a family of non-linear transport equations in $\mathbb{R}^n$, with a velocity field given by the convolution of the density with a kernel that is smooth away from the origin and homogeneous of degree $-(n-1)$, is continuous in both the little Hölder class and the little Zygmund class. For particular choices of the kernel, one recovers well-known equations such as the 2D Euler or the 3D quasi-geostrophic equations.

A basic question about the directed landscape is how much of it can be reconstructed simply by knowing the shapes of its geodesics. We prove that the directed landscape can be reconstructed from the shapes of its semi-infinite geodesics. In order to show this result, we make use of the Busemann process of the directed landscape together with a novel ``coordinate system" and ``differential distance" to capture the behaviour of semi-infinite geodesics.

Given any $r$-edge coloring of $K_{n,n}$, how large is the maximum (over all $r$ colors) sized monochromatic subgraph guaranteed to be? We give answers to this problem for $r \leq 8$, when $r$ is a perfect square, and when $r$ is one less than a perfect square all up to a constant additive term that depends on $r$. We give a lower bound on this quantity that holds for all $r$ and is sharp when $r$ is a perfect square up to a constant additive term that depends on $r$. Finally, we give a construction for all $r$ which provides an upper bound on this quantity up to a constant additive term that depends on $r$, and which we conjecture is also a lower bound.

Let $G$ be a plane graph and let $C$ be a cycle in $G$. For each finite face of $G$, count the number of edges of $C$ the face contains. We call this the Slitherlink signature of $C$. The symmetric difference $A$ of two cycles with the same signature is totally even, meaning every vertex is incident to an even number of edges in $A$ and every face contains an even number of edges in $A$. In this paper, we completely characterize totally even subsets in the triangular grid, and count the number of edges in any totally even subset of the triangular grid. We also show that the size of the symmetric difference of two cycles with the same signature in the triangular grid is divisible by $12$; this is best possible since 12 is the greatest common divisor of all the sizes of the symmetric difference between two cycles with the same signature in a triangular grid.