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Ergodic optimization and discrete weak KAM theory are two parallel theories with several results in common. For instance, the Mather set is the locus of orbits which minimize the ergodic averages of a given observable. In the favorable…
Function (linear) spaces on which an arbitrary function operates (i.e. the space is stable w.r.t. the pointwise unary operation defined by the function) were investigated, for continuous real or complex operations, by deLeeuw-Katznelson,…
A weak invariant associated with a master equation is characterized in such a way that its spectrum is not constant in time but its expectation value is conserved under time evolution generated by the master equation. Here, an intriguing…
We construct algorithms and topological invariants that allow us to distinguish the topological type of a surface, as well as functions and vector fields for their topological equivalence. In the first part we discus the main structures…
An explicit formula is given for a fundamental solution for a class of semielliptic operators. The fundamental solution is used to investigate properties of these operators as mappings between weighted function spaces. Necessary and…
The statistical mechanical approach to complex networks is the dominant paradigm in describing natural and societal complex systems. The study of network properties, and their implications on dynamical processes, mostly focus on locally…
We prove a basic property of continuous multilinear mappings between topological vector spaces, from which we derive an easy proof of the fact that a multilinear mapping (and a polynomial) between topological vector spaces is weakly…
We prove the existence of a weak solution for boundary value problems driven by a mixed local--nonlocal operator. The main novelty is that such an operator is allowed to be nonpositive definite.
We propose a novel approach for preserving topological structures of the input space in latent representations of autoencoders. Using persistent homology, a technique from topological data analysis, we calculate topological signatures of…
Recently, the study of topological structures in photonics has garnered significant interest, as these systems can realize robust, non-reciprocal chiral edge states and cavity-like confined states that have applications in both linear and…
Topological Machine Learning (TML) is an emerging field that leverages techniques from algebraic topology to analyze complex data structures in ways that traditional machine learning methods may not capture. This tutorial provides a…
In this paper, we study solution operators of physical field equations on geometric meshes from a function-space perspective. We reveal that Hodge orthogonality fundamentally resolves spectral interference by isolating unlearnable…
Stable computational algorithms for the approximate solution of the Cauchy problem for nonstationary problems are based on implicit time approximations. Computational costs for boundary value problems for systems of coupled multidimensional…
The Cauchy problem for the von Neumann hierarchy of nonlinear equations is investigated. One describes the evolution of all possible states of quantum many-particle systems by the correlation operators. A solution of such nonlinear…
This paper provides a framework to strong time periodic solutions of quasilinear evolution equations. The novelty of this approach is that zero is allowed to be a spectral value of the underlying linearized operator. This approach is then…
We investigate the continuity properties of the solution operator to the wave map system from the flat Minkowski space to a general nonflat target of arbitrary dimension, and we prove by an explicit class of counterexamples that this map is…
Let $E$ be a sublattice of a vector lattice $F$. A continuous operator $T$ from the vector lattice $E$ into a normed vector space $X$ is said to be $\tilde{o}$rder-norm continuous whenever $x_\alpha\xrightarrow{Fo}0$ implies…
In this article we present the first results on domain decomposition methods for nonlocal operators. We present a nonlocal variational formulation for these operators and establish the well-posedness of associated boundary value problems,…
This paper is a cursory study on how topological features are preserved within the internal representations of neural network layers. Using techniques from topological data analysis, namely persistent homology, the topological features of a…
We develop a homotopical framework for small categories that extends classical invarints of algebraic topology to the categorical setting. Our approach is based on the construction of genuine path category, obtained trough a localization…