相关论文: Harmonic Analysis and Localization Technique
In the case of systems composed of identical particles, a typical instance in quantum statistical mechanics, the standard approach to separability and entanglement ought to be reformulated and rephrased in terms of correlations between…
In this paper, we explore connections between interpretable machine learning and learning theory through the lens of local approximation explanations. First, we tackle the traditional problem of performance generalization and bound the…
The bispectral problem is motivated by an effort to understand and extend a remarkable phenomenon in Fourier analysis on the real line: the operator of time-and-band limiting is an integral operator admitting a second-order differential…
The purpose of this paper is to study local cohomology in the noncommutative algebraic geometry framework of Artin and Zhang. The noncommutative spaces are obtained by base change of a Grothendieck category that is locally noetherian or…
In this work, we review the connection between the subjects of homogenization and nonlocal modeling and discuss the relevant computational issues. By further exploring this connection, we hope to promote the cross fertilization of ideas…
Recently, the spectral localizer framework has emerged as an efficient approach to classifying topology in photonic systems featuring local nonlinearities and radiative environments. In nonlinear systems, this framework provides rigorous…
In this paper, local monomialization theorems are proven for analytic morphisms of complex and real analytic spaces. This gives the generalization of the local monomialization theorem for morphisms of algebraic varieties over a field of…
This book is mainly an exposition of the author's works and his joint works with his former students on explicit representations of finite-dimensional simple Lie algebras, related partial differential equations, linear orthogonal algebraic…
In the present article the reduced integral representation of partitions in terms of harmonic products has been derived first by using hypergeometry and the new concept of fractional sum and secondly by studying the Fourier series of the…
Lecture notes from 2008 CMI/ETH Summer School on Evolution Equations. These notes are an informal introduction to the applications of microlocal methods in the study of linear evolution equations and spectral theory. Calculi of…
We establish Connes's local trace formula (related to the explicit formulae of number theory) for the quaternions. This is done as an application of a study of the central operator H = log(|x|) + log(|y|) in the context of invariant…
In this paper we introduce a new way of deforming convolution algebras and Fourier algebras on locally compact groups. We demonstrate that this new deformation allows us to reveal some informations of the underlying groups by examinining…
We calculate the local Fourier transforms for formal connections. In particular, we verify an analogous conjecture suggested in Laumon's paper: "Transformation de Fourier, constantes d'equations fonctionnelles et conjecture de Weil, 2.6.3".
We propose a new positional encoding method for a neural network architecture called the Transformer. Unlike the standard sinusoidal positional encoding, our approach is based on solid mathematical grounds and has a guarantee of not losing…
It is shown that the modulation spaces $M_{p}^{w}$ can be characterized by the approximation behavior of their elements using Local Fourier bases. In analogy to the Local Fourier bases, we show that the modulation spaces can also be…
We define a Fourier transform and a convolution product for functions and distributions on Heisenberg--Clifford Lie supergroups. The Fourier transform exchanges the convolution and a pointwise product, and is an intertwining operator for…
Laumon introduced the local Fourier transform for $\ell$-adic Galois representations of local fields, of equal characteristic $p$ different from $\ell$, as a powerful tool to study the Fourier-Deligne transform of $\ell$-adic sheaves over…
Motivated by Fredholm theory, we develop a framework to establish the convergence of spectral methods for operator equations $\mathcal L u = f$. The framework posits the existence of a left-Fredholm regulator for $\mathcal L$ and the…
We study Fourier theory on quantum Euclidean space. A modified version of the general definition of the Fourier transform on a quantum space is used and its inverse is constructed. The Fourier transforms can be defined by their Bochner's…
A periodic linear graph operator acts on states (functions) defined on the vertices of a graph equipped with a free translation action. Fourier transform with respect to the translation group reveals the central spectral objects, Bloch and…