arXiv Math @arxiv_math@qoto.org

We give a cohomological criterion for certain decomposition of Borel graphs, which is an analog of Dunwoody's work on accessibility of groups. As an application, we prove that a Borel graph $(X,G)$ with uniformly bounded degrees of cohomological dimension one is Lipschitz equivalent to a Borel acyclic graph on $X$. This gives a new proof of a result of Chen-Poulin-Tao-Tserunyan on Borel graphs with components quasi-isometric to trees.

This paper presents iterative methods for solving tensor equations involving the T-product. The proposed approaches apply tensor computations without matrix construction. For each initial tensor, these algorithms solve related problems in a finite number of iterations, with negligible errors. The theoretical analysis is validated by numerical examples that demonstrate the practicality and effectiveness of these algorithms.

We study a variant of Newton's algorithm applied to under-determined systems of non-smooth equations. The notion of regularity employed in our work is based on Newton differentiability, which generalizes semi-smoothness. The classic notion of Newton differentiability does not suffice for our purpose, due to the existence of multiple zeros and as such we extend it to uniform Newton differentiability. In this context, we can show that the distance between the iterates and the set of zeros of the system decreases super-linearly. For the special case of smooth equations, the assumptions of our algorithm are simplified. Finally, we provide some numerical examples to showcase the behavior of our proposed method. The key example is a toy model of complementarity constraint problems, showing that our method has great application potential across engineering fields.

In this paper, we investigate the problem of $\mathcal{F}$-harmonic forms and the long-time behavior of the Type IIA flow on certain nilpotent and solvable symplectic Lie algebras (and corresponding nilmanifolds and solvmanifolds). In particular, we demonstrate that the Type IIA flow in these cases are very useful to detect desired geometric structures including Lagrangian torus fibrations and harmonic almost complex structures.

We study discrete-time Markov chains on countably infinite state spaces, which are perturbed by rather general confining (i.e.\ growing at infinity) potentials. Using a discrete-time analogue of the classical Feynman--Kac formula, we obtain two-sided estimates for the $n$-step heat kernels $u_n(x,y)$ of the perturbed chain. These estimates are of the form $u_n(x,y)\asymp λ_0^nϕ_0(x)\widehatϕ_0(y)+F_n(x,y)$, where $ϕ_0$ (and $\widehatϕ_0$) are the (dual) eigenfunctions for the lowest eigenvalue $λ_0$; the perturbation $F_n(x,y)$ is explicitly given, and it vanishes if either $x$ or $y$ is in a bounded set. The key assumptions are that the chain is uniformly lazy and that the \enquote{direct step property} (DSP) is satisfied. This means that the chain is more likely to move from state $x$ to state $y$ in a single step rather than in two or more steps. Starting from the form of the heat kernel estimate, we define the intrinsic (or ground-state transformed) chains and we introduce time-dependent ultracontractivity notions -- asymptotic and progressive intrinsic ultracontractivity -- which we can link to the growth behaviour of the confining potential; this allows us to consider arbitrarily slow growing potentials. These new notions of ultracontractivity also lead to a characterization of uniform (quasi-)ergodicity of the perturbed and the ground-state transformed Markov chains. At the end of the paper, we give various examples that illustrate how our findings relate to existing models, e.g.\ nearest-neighbour walks on infinite graphs, subordinate processes or non-reversible Markov chains.

Maximal inequalities refer to bounds on expected values of the supremum of averages of random variables over a collection. They play a crucial role in the study of non-parametric and high-dimensional estimators, and especially in the study of empirical risk minimizers. Although the expected supremum over an infinite collection appears more often in these applications, the expected supremum over a finite collection is a basic building block. This follows from the generic chaining argument. For the case of finite maximum, most existing bounds stem from the Bonferroni inequality (or the union bound). The optimality of such bounds is not obvious, especially in the context of heavy-tailed random vectors. In this article, we consider the problem of finding sharp upper and lower bounds for the expected $L_{\infty}$ norm of the mean of finite-dimensional random vectors under marginal variance bounds and an integrable envelope condition.

This paper studies the numerical approximation of parametric time-dependent partial differential equations (PDEs) by proper orthogonal decomposition reduced order models (POD-ROMs). Although many papers in the literature consider reduced order models for parametric equations, a complete error analysis of the methods is still a challenge. We introduce and analyze in this paper a new POD method based on finite differences (respect to time and respect to the parameters that may be considered). We obtain a priori bounds for the new method valid for any value of time in a given time interval and any value of the parameter in a given parameter interval. Our design of the new POD method allow us to prove pointwise-in-time error estimates as opposed to average error bounds obtained typically in POD methods. Most of the papers concerning POD methods for parametric equations are just based on the snapshots computed at different times and parameter values instead of their difference quotients. We show that the error analysis of the present paper can also cover the error analysis of that case (that we call standard). Some numerical experiments compare our new approach with the standard one and support the error analysis.

We extend the univariate Newton interpolation algorithm to arbitrary spatial dimensions and for any choice of downward-closed polynomial space, while preserving its quadratic runtime and linear storage cost. The generalisation supports any choice of the provided notion of non-tensorial unisolvent interpolation nodes, whose number coincides with the dimension of the chosen-downward closed space. Specifically, we prove that by selecting Leja-ordered Chebyshev-Lobatto or Leja nodes, the optimal geometric approximation rates for a class of analytic functions -- termed Bos--Levenberg--Trefethen functions -- are achieved and extend to the derivatives of the interpolants. In particular, choosing Euclidean degree results in downward-closed spaces whose dimension only grows sub-exponentially with spatial dimension, while delivering approximation rates close to, or even matching those of the tensorial maximum-degree case, mitigating the curse of dimensionality. Several numerical experiments demonstrate the performance of the resulting multivariate Newton interpolation compared to state-of-the-art alternatives and validate our theoretical results.

We study the $2k$-th moment of central values of the family of Dirichlet $L$-functions to a fixed prime modulus and establish sharp upper bounds for all real $k \in [0,2]$.

We study $\mathbb{E}_n$-Koszul duality for pairs of algebras of the form $\mathrm{C}_{\bullet}(Ω^{n}_*X;\Bbbk) \leftrightarrow \mathrm{C}^{\bullet}(X;\Bbbk)$, and the closely related question of $n$-affineness for Betti stacks. It was expected, but not known, that $\mathbb{E}_n$-Koszul duality should induce a kind of Morita equivalence between categories of iterated modules. We establish this rigorously by proving that the $(\infty,n)$-category of iterated modules over $\mathrm{C}_{\bullet}(Ω_*^{n+1}X;\Bbbk)$ is equivalent to the $(\infty,n)$-category of quasi-coherent sheaves of $(\infty,n-1)$-categories on $\mathrm{cSpec}(\mathrm{C}^{\bullet}(X;\Bbbk))$, where $\mathrm{cSpec}(\mathrm{C}^{\bullet}(X;\Bbbk))$ is the cospectrum of $\mathrm{C}^{\bullet}(X;\Bbbk)$. By the monodromy equivalence, these categories are also equivalent to the category of higher local systems on $X$, $n\mathbf{LocSysCat}^{n-1}(X;\Bbbk)$. Our result is new already in the classical case $n=1$, although it can be seen to recover well known formulations of $\mathbb{E}_1$-Koszul duality as a Morita equivalence of module categories (up to appropriate completions of the $t$-structures). We also investigate (higher) affineness properties of Betti stacks. We give a complete characterization of $n$-affine Betti stacks, in terms of the $0$-affineness of their iterated loop space. As a consequence, we prove that $n$-truncated Betti stacks are $n$-affine; and that $π_{n+1}(X)$ is an obstruction to $n$-affineness.

As semantic communication (SemCom) gains increasing attention as a novel communication paradigm, ensuring the security of transmitted semantic information over open wireless channels becomes crucial. Existing secure SemCom solutions often lack explicit control over security. To address this, we propose a coding-enhanced jamming approach for secure SemCom over wiretap channels. This approach integrates deep joint source and channel coding (DeepJSCC) with neural network-based digital modulation, enabling controlled jamming through two-layer superposition coding. The outer constellation sequence encodes the source image, while the inner constellation sequence, derived from a secret image, acts as the jamming signal. By minimizing the mutual information between the outer and inner constellation sequences, the jamming effect is enhanced. The jamming signal is superposed on the outer constellation sequence, preventing the eavesdropper from recovering the source image. The power allocation coefficient (PAC) in the superposition coding can be adjusted to control system security. Experiments show that our approach matches existing methods in security while significantly improving reconstruction performance across varying channel signal-to-noise ratios (SNRs) and compression ratios.

Banyaga and Hurtubise defined the Morse-Bott-Smale chain complex as a quotient of a large chain complex by introducing five degeneracy relations. In this paper, we unify the five conditions into only one degeneracy condition. This allows for a simpler definition of Morse-Bott homology and more computable examples. Moreover, we show that our chain complex for a Morse-Smale function is quasi-isomorphic to the Morse-Smale-Witten chain complex. As a result, we obtain another proof of the Morse Homology Theorem.

Following our previous works on $C^*$-graph algebras and the associated Cuntz-Krieger graph families, in this paper we will try to have a look at the colored version of these structures and to see what a $C^*$-colored graph algebra might mean by employing some constructive examples very close to the toy example used in our previous works, and we also will try to study their graph theoretical properties as possible.

In this paper, by virtue of a determinantal formula for derivatives of the ratio between two differentiable functions, in view of the Faà di Bruno formula, and with the help of several identities and closed-form formulas for the partial Bell polynomials $\operatorname{B}_{n,k}$, the author establishes thirteen Maclaurin series expansions of the functions \begin{align*} &\ln\frac{\operatorname{e}^x+1}{2}, && \ln\frac{\operatorname{e}^x-1}{x}, && \ln\cosh x, \\ &\ln\frac{\sinh x}{x}, && \biggl[\frac{\ln(1+x)}{x}\biggr]^r, && \biggl(\frac{\operatorname{e}^x-1}{x}\biggr)^r \end{align*} for $r=\pm\frac{1}{2}$ and $r\in\mathbb{R}$ in terms of the Dirichlet eta function $η(1-2k)$, the Riemann zeta function $ζ(1-2k)$, and the Stirling numbers of the first and second kinds $s(n,k)$ and $S(n,k)$. presents four determinantal expressions and three recursive relations for the Bernoulli numbers $B_{2n}$. finds out three closed-form formulas for the Bernoulli numbers $B_{2n}$ and the generalized Bernoulli numbers $B_n^{(r)}$ in terms of the Stirling numbers of the second kind $S(n,k)$, and deduce two combinatorial identities for the Stirling numbers of the second kind $S(n,k)$. acquires two combinatorial identities, which can be regarded as diagonal recursive relations, involving the Stirling numbers of the first and second kinds $s(n,k)$ and $S(n,k)$. recovers an integral representation and a closed-form formula, and establish an alternative explicit and closed-form formula, for the Bernoulli numbers of the second kind $b_n$ in terms of the Stirling numbers of the first kind $s(n,k)$. obtains three identities connecting the Stirling numbers of the first and second kinds $s(n,k)$ and $S(n,k)$.

We show a construction for dense 3-uniform linear hypergraphs without $3\times 3$ grids, improving the lower bound on its Turán number.

We investigate random partitions of complete graphs defined by Poissonian emsembles of Markov loops

In this study, we discuss the relationship between two families of density-power-based divergences with functional degrees of freedom -- the Hölder divergence and the functional density power divergence (FDPD) -- based on their intersection and generalization. These divergence families include the density power divergence and the $γ$-divergence as special cases. First, we prove that the intersection of the Hölder divergence and the FDPD is limited to a general divergence family introduced by Jones et al. (Biometrika, 2001). Subsequently, motivated by the fact that Hölder's inequality is used in the proofs of nonnegativity for both the Hölder divergence and the FDPD, we define a generalized divergence family, referred to as the $ξ$-Hölder divergence. The nonnegativity of the $ξ$-Hölder divergence is established through a combination of the inequalities used to prove the nonnegativity of the Hölder divergence and the FDPD. Furthermore, we derive an inequality between the composite scoring rules corresponding to different FDPDs based on the $ξ$-Hölder divergence. Finally, we prove that imposing the mathematical structure of the Hölder score on a composite scoring rule results in the $ξ$-Hölder divergence.

Nonlinear spectral problems arise across a range of fields, including mechanical vibrations, fluid-solid interactions, and photonic crystals. Discretizing infinite-dimensional nonlinear spectral problems often introduces significant computational challenges, particularly spectral pollution and invisibility, which can distort or obscure the true underlying spectrum. We present the first general, convergent computational method for computing the spectra and pseudospectra of nonlinear spectral problems. Our approach uses new results on nonlinear injection moduli and requires only minimal continuity assumptions: specifically, continuity with respect to the gap metric on operator graphs, making it applicable to a broad class of problems. We use the Solvability Complexity Index (SCI) hierarchy, which has recently been used to resolve the classical linear problem, to systematically classify the computational complexity of nonlinear spectral problems. Our results establish the optimality of the method and reveal that Hermiticity does not necessarily simplify the computational complexity of these nonlinear problems. Comprehensive examples -- including nonlinear shifts, Klein--Gordon equations, wave equations with acoustic boundary conditions, time-fractional beam equations, and biologically inspired delay differential equations -- demonstrate the robustness, accuracy, and broad applicability of our methodology.

A convergent power series solution is obtained for the SIR model, using an asymptotically motivated gauge function. For certain choices of model parameter values, the series converges over the full physical domain (i.e., for all positive time). Furthermore, the radius of convergence as a function of nondimensionalized initial susceptible and infected populations is obtained via a numerical root test.