arXiv Math @arxiv_math@qoto.org

Recent advances in bipartite consensus on matrix-weighted networks, where agents are divided into two disjoint sets with those in the same set agreeing on a certain value and those in different sets converging to opposite values, have highlighted its potential applications across various fields. Traditional approaches often depend on the existence of a positive-negative spanning tree in matrix-weighted networks to achieve bipartite consensus, which greatly restricts the use of these approaches in engineering applications. This study relaxes that assumption by allowing weak connectivity within the network, where paths can be weighted by semidefinite matrices. By analyzing the algebraic constraints imposed by positive-negative trees and semidefinite paths, we derive new sufficient conditions for achieving bipartite consensus. Our findings are validated by numerical simulations.

This paper develops and analyzes a class of semi-discrete and fully discrete weak Galerkin finite element methods for unsteady incompressible convective Brinkman-Forchheimer equations. For the spatial discretization, the methods adopt the piecewise polynomials of degrees $m\ (m\geq1)$ and $m-1$ respectively to approximate the velocity and pressure inside the elements, and piecewise polynomials of degree $m$ to approximate their numerical traces on the interfaces of elements. In the fully discrete method, the backward Euler difference scheme is used to approximate the time derivative. The methods are shown to yield globally divergence-free velocity approximation. Optimal a priori error estimates in the energy norm and $L^2$ norm are established. A convergent linearized iterative algorithm is designed for solving the fully discrete system. Numerical experiments are provided to verify the theoretical results.

Friends-and-strangers graphs, coined by Defant and Kravitz, are denoted by $\mathsf{FS}(X,Y)$ where $X$ and $Y$ are both graphs on $n$ vertices. The graph $X$ represents positions and edges mark adjacent positions while the graph $Y$ represents people and edges mark friendships. The vertex set of $\mathsf{FS}(X,Y)$ consists of all one-to-one placements of people on positions, and there is an edge between any two placements if it is possible to swap two people who are friends and on adjacent positions to get from one placement to the other. Previous papers have studied when $\mathsf{FS}(X,Y)$ is connected. In this paper, we consider when $\mathsf{FS}(X,Y)$ is $k$-connected where a graph is $k$-connected if it remains connected after removing any $k-1$ or less vertices. We first consider $\mathsf{FS}(X,Y)$ when $Y$ is a complete graph or star graph. We find tight bounds on their connectivity, proving their connectivity equals their minimum degree. We further consider the size of the connected components of $\mathsf{FS}(X,\mathsf{Star}_n)$ where $X$ is connected. We show that asymptotically similar conditions as the conditions mentioned by Bangachev are sufficient for $\mathsf{FS}(X,Y)$ to be $k$-connected. Finally, we consider when $X$ and $Y$ are independent Erdős--Rényi random graphs on $n$ vertices and edge probability $p_1$ and $p_2,$ respectively. We show that for $p_0 = n^{-1/2+o(1)},$ if $p_1p_2\geq p_0^2$ and $p_1,$ $p_2 \geq w(n) p_0$ where $w(n) \rightarrow 0$ as $n \rightarrow \infty,$ then $\mathsf{FS}(X,Y)$ is $k$-connected with high probability. This is asymptotically tight as we show that below an asymptotically similar threshold $p_0'=n^{-1/2+o(1)}$, the graph $\mathsf{FS}(X,Y)$ is disconnected with high probability if $p_1p_2 \leq (p_0')^2$.

We extend to the $n$-dimensional ellipsoid contained in $\R^{n+1},$ the Darboux theory of integrability for polynomial vector fields in the $n$-dimensional sphere (Llibre et al., 2018). New results on the maximum number of invariant parallels and meridians of polynomial vector fields $\X$ on the invariant $n-$dimensional ellipsoid, as a function of its degree, are provided. Our results extend the known result on the upper bound for the number of invariant hyperplanes that a polynomial vector field $\Y$ in $\R^n$ can have in function of the degree of $\Y$.

In this paper, we provide different splitting methods for solving distributionally robust optimization problems in cases where the uncertainties are described by discrete distributions. The first method involves computing the proximity operator of the supremum function that appears in the optimization problem. The second method solves an equivalent monotone inclusion formulation derived from the first-order optimality conditions, where the resolvents of the monotone operators involved in the inclusion are computable. The proposed methods are applied to solve the Couette inverse problem with uncertainty and the denoising problem with uncertainty. We present numerical results to compare the efficiency of the algorithms.

We prove vanishing results for Witten genera of string generalized complete intersections in homogeneous $\text{Spin}^c$-manifolds and in other $\text{Spin}^c$-manifolds with Lie group actions. By applying these results to Fano manifolds with second Betti number equal to one we get new evidence for a conjecture of Stolz.

For a closed negatively monotone symplectic manifold, we construct quasi-isometric embeddings from the Euclidean spaces to its Hamiltonian diffeomorphism group, assuming it contains an incompressible heavy Lagrangian. We also show the super-heaviness of its skeleton with respect to a Donaldson hypersurface.

A cryptarithm (or alphametic) is a mathematical puzzle in which numbers are represented with words in such a way that identical letters stand for equal digits and distinct letters for unequal digits. An alphametic puzzle is usually given in the form of an equation that needs to be solved, such as SEND + MORE = MONEY. Alternatively, here we will consider cryptarithms constrained not by an equation but by a particular subsequence of natural numbers, for example perfect squares or primes. Such a cryptarithm has a unique solution if there is exactly one term in the sequence that has the corresponding pattern of digits. We will call such terms cryptarithmically unique. Here we estimate the density of such terms in an arbitrary sequence for which the overall density of terms among integers is known. In particular, among all perfect squares below 10^12, slightly less than one half are cryptarithmically unique, their density increasing toward larger numbers. Cryptarithmically unique prime numbers, however, are initially very scarce. Combinatorial estimates suggest that their density should drop below 10^-300 for decimal lengths of approximately 1829 digits, but then it recovers and is asymptotic to unity for very large primes. Finally, we introduce and discuss primonumerophobic digit patterns that no prime number happens to have.

We prove that the rational cohomology ring of moduli space of multiscale differentials in genus 0 is generated by the boundary divisors. The main idea is the technique of the Chow-Künneth generation Property and the observation that the intersection of a collection of boundary divisors in the moduli space is irreducible. We observe that the relations between the boundary strata in cohomology are generated by the pullback of the WDVV relations and the relations between the torus-invariant subvarieties in the fiber over $\overline{M}_{0,n}$. We also characterize the cases in which the moduli space is a smooth variety, and in these cases, we prove that the integral cohomology ring is generated by the boundary divisors.

A set of n non-collinear points in the Euclidean plane defines at least n different lines. Chen and Chvtal in 2008 conjectured that the same results is true in metric spaces for an adequate definition of line. More recently, it was conjectured in 2018 by Aboulker et al. that any large enough bridgeless graph on n vertices defines a metric space that has at least n lines. We study the natural extension of Aboulker et al.'s conjecture into the context of quasi-metric spaces defined by digraphs of low diameter. We prove that it is valid for quasi-metric spaces defined by bipartite digraphs of diameter at most three, oriented graphs of diameter two and, digraphs of diameter three and directed girth four.

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.