The Experts below are selected from a list of 3471 Experts worldwide ranked by ideXlab platform
David Carfì - One of the best experts on this subject based on the ideXlab platform.
-
Summable families in Tempered Distribution spaces
arXiv: Functional Analysis, 2011Co-Authors: David CarfìAbstract:In this note we define summable families in Tempered Distribution spaces and we state some their properties and characterizations. Summable families are the analogous of summable sequences in separable Hilbert spaces, but in Tempered Distribution spaces, having elements (functional) realizable as generalized vectors indexed by real Euclidean spaces (not pointwise defined ordered families of scalars indexed by real Euclidean spaces in the sense of Distributions). Any family we introduce here is summable with respect to every Tempered system of coefficients belonging to a certain normal space of Distributions, in the sense of superpositions. The summable families we present in this note are one possible rigorous and simply manageable mathematical model for the infinite families of vector-states appearing in the formulation of the continuous version of the celebrated Principle of Superpositions in Quantum Mechanics.
-
schwartz families in Tempered Distribution spaces
arXiv: Functional Analysis, 2011Co-Authors: David CarfìAbstract:In this paper we define Schwartz families in Tempered Distribution spaces and prove many their properties. Schwartz families are the analogous of infinite dimensional matrices of separable Hilbert spaces, but for the Schwartz test function spaces, having elements (functions) realizable as vectors indexed by real Euclidean spaces (ordered families of scalars indexed by real Euclidean spaces). In the paper, indeed, one of the consequences of the principal result (the characterization of summability for Schwartz families) is that the space of linear continuous operators among Schwartz test function spaces is linearly isomorphic with the space of Schwartz families. It should be noticed that this theorem is possible because of the very good properties of Schwartz test function spaces and because of the particular structures of the Schwartz families viewed as generalized matrices; in fact, any family of Tempered Distribution, regarded as generalized matrix, has one index belonging to a Euclidean space and one belonging to a test function space, so that any Schwartz family is a matrix in the sense of Distributions. Another motivation for the introduction and study of these families is that these are the families which are summable with respect to every Tempered system of coefficients, in the sense of superpositions. The Schwartz families we present in this paper are one possible rigorous and simply manageable mathematical model for the infinite matrices used frequently in Quantum Mechanics.
Jin Lei - One of the best experts on this subject based on the ideXlab platform.
-
Mean dimension and an embedding theorem for real flows
2020Co-Authors: Gutman Yonatan, Jin LeiAbstract:We develop mean dimension theory for $\mathbb{R}$-flows. We obtain fundamental properties and examples and prove an embedding theorem: Any real flow $(X,\mathbb{R})$ of mean dimension strictly less than $r$ admits an extension $(Y,\mathbb{R})$ whose mean dimension is equal to that of $(X,\mathbb{R})$ and such that $(Y,\mathbb{R})$ can be embedded in the $\mathbb{R}$-shift on the compact function space $\{f\in C(\mathbb{R},[-1,1])|\;\mathrm{supp}(\hat{f})\subset [-r,r]\}$, where $\hat{f}$ is the Fourier transform of $f$ considered as a Tempered Distribution. These canonical embedding spaces appeared previously as a tool in embedding results for $\mathbb{Z}$-actions.Comment: 22 pages, 1 figure. To be published in Fundamenta Mathematica
-
Mean dimension and an embedding theorem for real flows
'Institute of Mathematics Polish Academy of Sciences', 2020Co-Authors: Gutman Yonatan, Jin LeiAbstract:We develop mean dimension theory for R-flows. We obtain fundamental properties and examples and prove an embedding theorem: Any real flow (X, R) of mean dimension strictly less than r admits an extension (Y, R) whose mean dimension is equal to that of (X, R) and such that (Y, R) can be embedded in the R-shift on the compact function space {f is an element of C(R, [-1,1]) : supp((f) over cap) subset of [-r , r]}, where (f) over cap is the Fourier transform of f considered as a Tempered Distribution. These canonical embedding spaces appeared previously as a tool in embedding results for Z-actions.NCN (National Science Center, Poland) 2016/22/E/ST1/00448 2013/08/A/ST1/00275 Comisión Nacional de Investigación Científica y Tecnológica (CONICYT) CONICYT FONDECYT 3190127 Basal funding AFB 17000
Gutman Yonatan - One of the best experts on this subject based on the ideXlab platform.
-
Mean dimension and an embedding theorem for real flows
2020Co-Authors: Gutman Yonatan, Jin LeiAbstract:We develop mean dimension theory for $\mathbb{R}$-flows. We obtain fundamental properties and examples and prove an embedding theorem: Any real flow $(X,\mathbb{R})$ of mean dimension strictly less than $r$ admits an extension $(Y,\mathbb{R})$ whose mean dimension is equal to that of $(X,\mathbb{R})$ and such that $(Y,\mathbb{R})$ can be embedded in the $\mathbb{R}$-shift on the compact function space $\{f\in C(\mathbb{R},[-1,1])|\;\mathrm{supp}(\hat{f})\subset [-r,r]\}$, where $\hat{f}$ is the Fourier transform of $f$ considered as a Tempered Distribution. These canonical embedding spaces appeared previously as a tool in embedding results for $\mathbb{Z}$-actions.Comment: 22 pages, 1 figure. To be published in Fundamenta Mathematica
-
Mean dimension and an embedding theorem for real flows
'Institute of Mathematics Polish Academy of Sciences', 2020Co-Authors: Gutman Yonatan, Jin LeiAbstract:We develop mean dimension theory for R-flows. We obtain fundamental properties and examples and prove an embedding theorem: Any real flow (X, R) of mean dimension strictly less than r admits an extension (Y, R) whose mean dimension is equal to that of (X, R) and such that (Y, R) can be embedded in the R-shift on the compact function space {f is an element of C(R, [-1,1]) : supp((f) over cap) subset of [-r , r]}, where (f) over cap is the Fourier transform of f considered as a Tempered Distribution. These canonical embedding spaces appeared previously as a tool in embedding results for Z-actions.NCN (National Science Center, Poland) 2016/22/E/ST1/00448 2013/08/A/ST1/00275 Comisión Nacional de Investigación Científica y Tecnológica (CONICYT) CONICYT FONDECYT 3190127 Basal funding AFB 17000
Takhanov Rustem - One of the best experts on this subject based on the ideXlab platform.
-
Dimension reduction as an optimization problem over a set of generalized functions
2019Co-Authors: Takhanov RustemAbstract:Classical dimension reduction problem can be loosely formulated as a problem of finding a $k$-dimensional affine subspace of ${\mathbb R}^n$ onto which data points ${\mathbf x}_1,\cdots, {\mathbf x}_N$ can be projected without loss of valuable information. We reformulate this problem in the language of Tempered Distributions, i.e. as a problem of approximating an empirical probability density function $p_{\rm{emp}}({\mathbf x}) = \frac{1}{N} \sum_{i=1}^N \delta^n (\bold{x} - \bold{x}_i)$, where $\delta^n$ is an $n$-dimensional Dirac delta function, by another Tempered Distribution $q({\mathbf x})$ whose density is supported in some $k$-dimensional subspace. Thus, our problem is reduced to the minimization of a certain loss function $I(q)$ measuring the distance from $q$ to $p_{\rm{emp}}$ over a pertinent set of generalized functions, denoted $\mathcal{G}_k$. Another classical problem of data analysis is the sufficient dimension reduction problem. We show that it can be reduced to the following problem: given a function $f: {\mathbb R}^n\rightarrow {\mathbb R}$ and a probability density function $p({\mathbf x})$, find a function of the form $g({\mathbf w}^T_1{\mathbf x}, \cdots, {\mathbf w}^T_k{\mathbf x})$ that minimizes the loss ${\mathbb E}_{{\mathbf x}\sim p} |f({\mathbf x})-g({\mathbf w}^T_1{\mathbf x}, \cdots, {\mathbf w}^T_k{\mathbf x})|^2$. We first show that search spaces of the latter two problems are in one-to-one correspondence which is defined by the Fourier transform. We introduce a nonnegative penalty function $R(f)$ and a set of ordinary functions $\Omega_\epsilon = \{f| R(f)\leq \epsilon\}$ in such a way that $\Omega_\epsilon$ `approximates' the space $\mathcal{G}_k$ when $\epsilon \rightarrow 0$. Then we present an algorithm for minimization of $I(f)+\lambda R(f)$, based on the idea of two-step iterative computation
V Krylov - One of the best experts on this subject based on the ideXlab platform.
-
image ordered texture segmentation in the space of coefficients of transform with Tempered Distribution scaling functions
International Conference on Experience of Designing and Applications of CAD Systems in Microelectronics, 2007Co-Authors: Marina Polyakova, V KrylovAbstract:In this paper Tempered Distribution scaling functions with compact support is defined. The model of image processing based on signal-semantic transform with the sequence of local integrable functions which approximate the Tempered Distribution is proposed.
