Related papers: A dynamical approach to nonhomogeneous spectra

Given a connected semisimple Lie group $G$ and an arithmetic subgroup $\Gamma$, it is well-known that each irreducible representation $\pi$ of $G$ occurs in the discrete spectrum $L^2_{\text{disc}}(\Gamma\backslash G)$ of…

The Cayley sum graph $\Gamma_A$ of a set $A \subseteq \mathbb{Z}_n$ is defined to have vertex set $\mathbb{Z}_n$ and an edge between two distinct vertices $x, y \in \mathbb{Z}_n$ if $x + y \in A$. Green and Morris proved that if the set $A$…

We present in this paper the motivation and theory of nonlinear spectral representations, based on convex regularizing functionals. Some comparisons and analogies are drawn to the fields of signal processing, harmonic analysis and sparse…

We explore in this paper sufficient conditions for the $H$-property to hold, with a particular focus on the so-called line graphons. A graphon is a symmetric, measurable function from the unit square $[0,1]^2$ to the closed interval…

We study one class of continuous functions $f$ defined on segment $[0,1]$ by equality $$ f(x)=\delta_{\alpha_1(x)1}+\sum^{\infty}_{k=2}\left[\delta_{\alpha_k(x)k}\prod^{k-1}_{j=1}g_{\alpha_j…

We study stochastic homogenisation of free-discontinuity surface functionals defined on piecewise rigid functions which arise in the study of fracture in brittle materials. In particular, under standard assumptions on the density, we show…

A fractional matching of a graph $G$ is a function $f:E(G) \to [0,1]$ such that for any $v\in V(G)$, $\sum_{e\in E_G(v)}f(e)\leq 1$ where $E_G(v) = \{e \in E(G): e$ is incident with $v$ in $G\}$. The fractional matching number of $G$ is…

Let $\Gamma$ be a discrete subgroup of $\textrm{PGL}(d,\R)$ and fix some euclidean norm $\|\ \|$ on $\R^d.$ Let $N_\Gamma(t)$ be the number of elements in $\Gamma$ whose operator norm is $\leq t.$ In this article we prove an asymptotic for…

In this paper we present a general procedure that allows for the reduction or expansion of any network (considered as a weighted graph). This procedure maintains the spectrum of the network's adjacency matrix up to a set of eigenvalues…

In the present paper we consider Riemannian coverings $(X,g) \to (M,g)$ with residually finite covering group $\Gamma$ and compact base space $(M,g)$. In particular, we give two general procedures resulting in a family of deformed coverings…

Spatial non-homogeneities can synchronize clusters of spatially-extended oscillators in different frequency plateaus. Motivated by physiological rhythms, we fully characterize the phase diagram of a Ginzburg-Landau (GL) model with a…

Precise understanding of nonlinear evolution of cosmological perturbations during inflation is necessary for the correct interpretation of measurements of non-Gaussian correlations in the cosmic microwave background and the large-scale…

The asymptotic spectrum of graphs, introduced by Zuiddam (arXiv:1807.00169, 2018), is the space of graph parameters that are additive under disjoint union, multiplicative under the strong product, normalized and monotone under homomorphisms…

Let $g\in L^2(\mathbb{R})$ be a strictly decreasing continuous function supported on $\mathbb{R}_+$ such that for all $t > 0$ we have $g(x+t)\le q(t)g(x)$ for some $q(t)<1$. We prove that the Gabor system…

Let mu(G) and mu_min(G) be the largest and smallest eigenvalues of the adjacency matricx of a graph G. We refine quantitatively the following two results on graph spectra. (i) if H is a proper subgraph of a connected graph G, then…

A novel general framework for the study of $\Gamma$-convergence of functionals defined over pairs of measures and energy-measures is introduced. This theory allows us to identify the $\Gamma$-limit of these kind of functionals by knowing…

Given parameters $\alpha\geq 1,\beta\geq 0$, a subgraph $G'=(V,H)$ of an $n$-vertex unweighted undirected graph $G=(V,E)$ is called an $(\alpha,\beta)$-spanner if for every pair $u,v\in V$ of vertices, $d_{G'}(u,v)\leq \alpha…

Let $g(n)$ be the largest positive integer $k$ such that there are distinct primes $p_i$ for $1\leq i\leq k$ so that $p_i |n+i$. This function is related to a celebrated conjecture of C.A. Grimm. We establish upper and lower bounds for…

A tuple of commuting Hilbert space operators $(S_1, \dots, S_{n-1}, P)$ having the closed symmetrized polydisc \[ \Gamma_n = \left\{ \left(\sum_{i=1}^{n}z_i, \sum\limits_{1\leq i<j\leq n} z_iz_j, \cdots, \prod_{i=1}^{n}z_i\right) :…

Results for $\beta$-functions and anomalous dimensions in general scalar fermion theories are presented to three loops. Various constraints on the individual coefficients for each diagram following from supersymmetry are analysed. The…