arXiv Math @arxiv_math@qoto.org

Motivated by the rich properties and various applications of recurrence relations, we consider the extension of traditional recurrence relations to matrices, where we use matrix multiplication and the Kronecker product to construct matrix sequences. We provide a sharp condition, which when satisfied, guarantees that any fixed-depth matrix recurrence relation defined over a product (with respect to matrix multiplication) will converge to the zero matrix. We also show that the same statement applies to matrix recurrence relations defined over a Kronecker product. Lastly, we show that the dual of this condition, which remains sharp, guarantees the divergence of matrix recurrence relations defined over a consecutive Kronecker product. These results completely determine the stability of nontrivial fixed-depth complex-valued recurrence relations defined over a consecutive product.

In this paper, we describe the Brill--Noether theory of a general smooth plane curve and a general curve $C$ on a Hirzebruch surface of fixed class. It is natural to study the line bundles on such curves according to the splitting type of their pushforward along projection maps $C \to \mathbb{P}^1$. Inspired by Wood's parameterization of ideal classes in rings associated to binary forms, we further refine the stratification of line bundles $L$ on $C$ by fixing the splitting types of both $L$ and $L(Δ)$, where $Δ\subset C$ is the intersection of $C$ with the directrix of the Hirzebruch surface. Our main theorem determines the dimensions of these locally closed strata and, if the characteristic of the ground field is zero, proves that they are smooth.

Assuming a weighted Nash type inequality for the generator $-A$ of a Markov semigroup, we prove a weighted Nash type inequality for its fractional power and deduce non-uniform bounds on the transition kernel corresponding to the Markov semigroup generated by $-A^α$.

We study the potentially undecidable problem of whether a given 2-dimensional CW complex can be embedded into $\mathbb{R}^4$. We provide operations that preserve embeddability, including joining and cloning of 2-cells, as well as $Δ\mathrm Y$-transformations. We also construct a CW complex for which $\mathrm YΔ$-transformations do not preserve embeddability. We use these results to study 4-flat graphs, i.e., graphs that embed in $\mathbb{R}^4$ after attaching any number of 2-cells to their cycles; a graph class that naturally generalizes planarity and linklessness. We verify several conjectures of van der Holst; in particular, we prove that each of the 78 graphs of the Heawood family is an excluded minor for the class of 4-flat graphs.

In 2015 Hikami and Inoue constructed a representation of the braid group in terms of cluster algebra associated with the decomposition of the complement of the corresponding knot into ideal hyperbolic tetrahedra. This representation leads to the calculation of the hyperbolic volume of the complement of the knot that is the closure of the corresponding braid. In this paper, based on the Hikami-Inoue representation discussed above, we construct a representation for the virtual braid group. We show that the so-called "forbidden relations" do not hold in the image of the resulting representation. In addition, based on the developed method, we construct representations for the flat braid group and the flat virtual braid group.

We discuss whether classical examples of dynamical systems satisfying the shadowing property also satisfy the shadowing property for the induced map on the hyperspace of continua, obtaining both positive and negative results. We prove that transitive Anosov diffeomorphisms, or more generally continuum-wise hyperbolic homeomorphisms, do not satisfy the shadowing property for the induced map on the hyperspace of continua. We prove that dendrite monotone maps satisfy the shadowing property if, and only if, their induced map on the hyperspace of continua also satisfies it. We give some algorithms that show that there are abundant dynamical systems satisfying the shadowing property with dendrites (compact metric trees) as the underlying space. As a consequence, we show that the universal dendrite of order n admits a homeomorphism with the shadowing property.

An equivalent version of the Borodin-Kostochka Conjecture, due to Cranston and Rabern, says that any graph with $χ= Δ= 9$ contains $K_3 \lor E_6$ as a subgraph. Here we prove several results in support of this conjecture, where vertex-criticality and forbidden substructure conditions get us either close or all the way to containing $K_3 \lor E_6$.

The aim of this paper is to study the existence of eigenvalues in the gap of the essential spectrum of the one-dimensional Dirac operator in the presence of a bounded potential. We employ a generalized variational principle to prove existence of such eigenvalues, estimate how many eigenvalues there are, and give upper and lower bounds for them.

For $Δ$ a finite connected nontrivial directed multigraph, we prove: 1. $Δ$ has a directed circuit using each directed edge exactly once if and only if both each pair of distinct vertices of $Δ$ occur in a common directed circuit and in-degree$({\bf x}) =$ out-degree$({\bf x})$ for every vertex ${\bf x}$. 2. $Δ$ contains a non-circuit directed path which uses every directed edge exactly once if and only if both every pair of distinct vertices of $Δ$ occur in a common directed circuit and there are vertices ${\bf b \not= e}$ such that in-degree$({\bf e}) -$ out-degree$({\bf e}) = 1 =$ out-degree$({\bf b}) -$ in-degree$({\bf b})$ but, for every vertex ${\bf x \notin \{b,e\}}$, it happens that in-degree$({\bf x}) =$ out-degree$({\bf x})$.

The first examples of noncrossed product division algebras were given by Amitsur in 1972. His method is based on two basic steps: (1) If the universal division algebra $U(k,n)$ is a $G$-crossed product then every division algebra of degree $n$ over $k$ should be a $G$-crossed product; (2) There are two division algebras over $k$ whose maximal subfields do not have a common Galois group. In this note, we give a short proof for the second step in the case where $\chr k\nmid n$ and $p^3|n$.

We present an interpretation of Deepcode, a learned feedback code that showcases higher-order error correction relative to an earlier interpretable model. By interpretation, we mean succinct analytical encoder and decoder expressions (albeit with learned parameters) in which the role of feedback in achieving error correction is easy to understand. By higher-order, we mean that longer sequences of large noise values are acted upon by the encoder (which has access to these through the feedback) and used in error correction at the decoder in a two-stage decoding process.

We define the Cartesian, Categorical, and Lexicographic, and Strong products of quantum graphs. We provide bounds on the quantum chromatic number of these products in terms of the quantum chromatic number of the factors. To adequately describe bounds on the lexicographic product of quantum graphs, we provide a notion of a quantum $b$-fold chromatic number for quantum graphs.

Given a non-invertible dynamical system with a transfer operator, we show there is a minimal cover with a transfer operator that preserves continuous functions. We also introduce an essential cover with even stronger continuity properties. For one-sided sofic subshifts, this generalizes the Krieger and Fischer covers, respectively. Our construction is functorial in the sense that certain equivariant maps between dynamical systems lift to equivariant maps between their covers, and these maps also satisfy better regularity properties. As applications, we identify finiteness conditions which ensure that the thermodynamic formalism is valid for the covers. This establishes the thermodynamic formalism for a large class of non-invertible dynamical systems, e.g. certain piecewise invertible maps. When applied to semi-étale groupoids, our minimal covers produce étale groupoids which are models for $C^*$-algebras constructed by Thomsen. The dynamical covers and groupoid covers are unified under the common framework of topological graphs.

Deep neural networks (DNNs) were shown to facilitate the operation of uplink multiple-input multiple-output (MIMO) receivers, with emerging architectures augmenting modules of classic receiver processing. Current designs consider static DNNs, whose architecture is fixed and weights are pre-trained. This induces a notable challenge, as the resulting MIMO receiver is suitable for a given configuration, i.e., channel distribution and number of users, while in practice these parameters change frequently with network variations and users leaving and joining the network. In this work, we tackle this core challenge of DNN-aided MIMO receivers. We build upon the concept of hypernetworks, augmenting the receiver with a pre-trained deep model whose purpose is to update the weights of the DNN-aided receiver upon instantaneous channel variations. We design our hypernetwork to augment modular deep receivers, leveraging their modularity to have the hypernetwork adapt not only the weights, but also the architecture. Our modular hypernetwork leads to a DNN-aided receiver whose architecture and resulting complexity adapts to the number of users, in addition to channel variations, without retraining. Our numerical studies demonstrate superior error-rate performance of modular hypernetworks in time-varying channels compared to static pre-trained receivers, while providing rapid adaptivity and scalability to network variations.

We consider a nonlocal aggregation diffusion equation incorporating repulsion modelled by nonlinear diffusion and attraction modelled by nonlocal interaction. When the attractive interaction kernel is radially symmetric and strictly increasing on its domain it is previously known that all stationary solutions are radially symmetric and decreasing up to a translation; however, this result has not been extended to accommodate attractive kernels that are non-decreasing, for instance, attractive kernels with bounded support. For the diffusion coefficient $m>1$, we show that, for attractive kernels that are radially symmetric and non-decreasing, all stationary states are radially symmetric and decreasing up to a translation on each connected subset of their support. Furthermore, for $m>2$, we prove analytically that stationary states have an upper-bound independent of the initial data, confirming previous numerical results given in the literature.

This note studies projective planes having a collineation group fixing a flag $(\infty,L_\infty )$ and transitive on the flags $(w,W )$ with $w\notin L_\infty $ and $\infty\notin W$.

We provide inequalities enabling to bound the error between the exact solution and an approximated solution of an eigenvalue problem, obtained by subspace projection, as in the reduced basis method. We treat self-adjoint operators and degenerate cases. We apply the bounds to the eigenvector continuation method, which consists in creating the reduced space by using basis vectors extracted from perturbation theory.

This paper introduces an auto-stabilized weak Galerkin (WG) finite element method with a built-in stabilizer for Poisson equations. By utilizing bubble functions as a key analytical tool, our method extends to both convex and non-convex elements in finite element partitions, marking a significant advancement over existing stabilizer-free WG methods. It overcomes the restrictive conditions of previous approaches and is applicable in any dimension $d$, offering substantial advantages. The proposed method maintains a simple, symmetric, and positive definite structure. These benefits are evidenced by optimal order error estimates in both discrete $H^1$ and $L^2$ norms, highlighting the effectiveness and accuracy of our WG method for practical applications.

We consider the use of multipreconditioning, which allows for multiple preconditioners to be applied in parallel, on high-frequency Helmholtz problems. Typical applications present challenging sparse linear systems which are complex non-Hermitian and, due to the pollution effect, either very large or else still large but under-resolved in terms of the physics. These factors make finding general purpose, efficient and scalable solvers difficult and no one approach has become the clear method of choice. In this work we take inspiration from domain decomposition strategies known as sweeping methods, which have gained notable interest for their ability to yield nearly-linear asymptotic complexity and which can also be favourable for high-frequency problems. While successful approaches exist, such as those based on higher-order interface conditions, perfectly matched layers (PMLs), or complex tracking of wave fronts, they can often be quite involved or tedious to implement. We investigate here the use of simple sweeping techniques applied in different directions which can then be incorporated in parallel into a multipreconditioned GMRES strategy. Preliminary numerical results on a two-dimensional benchmark problem will demonstrate the potential of this approach.