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We analyze the universality and generalization of graph neural networks (GNNs) on attributed graphs, i.e., with node attributes. To this end, we propose pseudometrics over the space of all attributed graphs that describe the fine-grained…

Deep learning's performance has been extensively recognized recently. Graph neural networks (GNNs) are designed to deal with graph-structural data that classical deep learning does not easily manage. Since most GNNs were created using…

The imposition of boundary conditions upon a quantized field can lead to singular energy densities on the boundary. We treat the boundaries as quantum mechanical objects with a nonzero position uncertainty, and show that the singular energy…

In this paper, we study the unitarizations in the spaces of holomorphic sections of equivariant holomorphic line bundles over a bounded homogeneous domain under the action of a connected algebraic group acting transitively on the domain. We…

We study small oscillations of highly connected systems, which represent a limit opposite to the more familiar case of disordered crystals. As a concrete example we analyze the vibrational spectra of composite pendula. Remarkably, these…

Graph neural networks (GNNs) integrate deep architectures and topological structure modeling in an effective way. However, the performance of existing GNNs would decrease significantly when they stack many layers, because of the…

Random matrix ensembles are introduced that respect the local tensor structure of Hamiltonians describing a chain of $n$ distinguishable spin-half particles with nearest-neighbour interactions. We prove a central limit theorem for the…

We consider the problem of quantization of the bosonic membrane via the large $N$ limit of its matrix regularizations $H_N$ in Fock space. We prove that there exists a choice of the Fock space frequency such that $ H_N$ can be written as a…

The Nielsen number $N(f)$ is a lower bound for the minimal number of fixed points among maps homotopic to $f$. When these numbers are equal, the map is called Wecken. A recent paper by Brimley, Griisser, Miller, and the second author…

We show that a system of a domain wall coupled to a scalar field has static negative energy density at certain distances from the domain wall. This system provides a simple, explicit example of violation of the averaged weak energy…

Let $u_n$ be a sequence of mappings from a closed Riemannian surface $M$ to a general Riemannian manifold $N$. If $u_n$ satisfies \beno \sup_{n}\big(\|\nabla u_n\|_{L^2(M)}+\|\tau(u_n)\|_{L^{p}(M)}\big)\leq \Lambda\quad \text{for…

Adapting \cite{strz3}, we define generalized $p$-harmonic maps into Riemannian homogeneous targets, a notion of solutions not belonging to the energy space. Restricting our attention to the subcritical range $p$ greater than the domain…

This paper investigates positive harmonic functions on a domain which contains an infinite cylinder, and whose boundary is contained in the union of parallel hyperplanes. (In the plane its boundary consists of two sets of vertical…

Let $G$ be a nonempty bounded domain in a finite-dimensional Euclidean space. The main results are general estimates from below at points from $G$ for an arbitrary subharmonic function $u\not\equiv -\infty$ on the closure of the domain $G$…

In the present note we describe geometrically the homology classes in the total space of a surface bundle over a surface in terms of the holonomy map. We treat the cases where the base surface is closed or has one boundary component. We…

The paper presents a generalization bound for quantum neural networks based on a dynamical Lie algebra. Using covering numbers derived from a dynamical Lie algebra, the Rademacher complexity is derived to calculate the generalization bound.…

This is a report on the derivation and application of a generalized version of the Wulff construction in two dimensions. The construction is used to find the shape of a domain containing an XY-like order parameter. In such a domain the…

Generalization and approximation capabilities of message passing graph neural networks (MPNNs) are often studied by defining a compact metric on a space of input graphs under which MPNNs are H\"older continuous. Such analyses are of two…

Generalized superstatistics, i.e., a "statistics of superstatistics," is proposed. A generalized superstatistical system comprises a set of superstatistical subsystems and represents a generalized hyperensemble. There exists a random…

Graph Neural Networks (GNNs) have been shown to be effective models for different predictive tasks on graph-structured data. Recent work on their expressive power has focused on isomorphism tasks and countable feature spaces. We extend this…