Displaying similar documents to “Rectilinearization of functions definable by a Weierstrass system and its applications”

A decomposition of a set definable in an o-minimal structure into perfectly situated sets

Wiesław Pawłucki (2002)

Annales Polonici Mathematici

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A definable subset of a Euclidean space X is called perfectly situated if it can be represented in some linear system of coordinates as a finite union of (graphs of) definable 𝓒¹-maps with bounded derivatives. Two subsets of X are called simply separated if they satisfy the Łojasiewicz inequality with exponent 1. We show that every closed definable subset of X of dimension k can be decomposed into a finite family of closed definable subsets each of which is perfectly situated and such...

Definable Davies' theorem

Asger Törnquist, William Weiss (2009)

Fundamenta Mathematicae

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We prove the following descriptive set-theoretic analogue of a theorem of R. O. Davies: Every Σ¹₂ function f:ℝ × ℝ → ℝ can be represented as a sum of rectangular Σ¹₂ functions if and only if all reals are constructible.

A Simple Proof of the Polar Decomposition Theorem

Paweł Wójcik (2017)

Annales Mathematicae Silesianae

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In this expository paper, we present a new and easier proof of the Polar Decomposition Theorem. Unlike in classical proofs, we do not use the square root of a positive matrix. The presented proof is accessible to a broad audience.

The generalized equations of Riccati and their applications to the theory of linear differential equations

T. Iwiński

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CONTENTSIntroduction.............................................................................................................................. 31. Definition of the Riccati equation of the n-th order....................................................... 62. Theorems on the existence of solutions of R equations. Relations between the solutions of linear differential equations and the solutions of the corresponding R equations..................................................................................................

Decomposition into special cubes and its applications to quasi-subanalytic geometry

Krzysztof Jan Nowak (2009)

Annales Polonici Mathematici

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The main purpose of this paper is to present a natural method of decomposition into special cubes and to demonstrate how it makes it possible to efficiently achieve many well-known fundamental results from quasianalytic geometry as, for instance, Gabrielov's complement theorem, o-minimality or quasianalytic cell decomposition.

Componentwise and Cartesian decompositions of linear relations

S. Hassi, H. S. V. de Snoo, F. H. Szafraniec

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Let A be a, not necessarily closed, linear relation in a Hilbert space ℌ with a multivalued part mul A. An operator B in ℌ with ran B ⊥ mul A** is said to be an operator part of A when A = B +̂ ({0} × mul A), where the sum is componentwise (i.e. span of the graphs). This decomposition provides a counterpart and an extension for the notion of closability of (unbounded) operators to the setting of linear relations. Existence and uniqueness criteria for an operator part are established...

Conservation Rules of Direct Sum Decomposition of Groups

Kazuhisa Nakasho, Hiroshi Yamazaki, Hiroyuki Okazaki, Yasunari Shidama (2016)

Formalized Mathematics

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In this article, conservation rules of the direct sum decomposition of groups are mainly discussed. In the first section, we prepare miscellaneous definitions and theorems for further formalization in Mizar [5]. In the next three sections, we formalized the fact that the property of direct sum decomposition is preserved against the substitutions of the subscript set, flattening of direct sum, and layering of direct sum, respectively. We referred to [14], [13] [6] and [11] in the formalization. ...

On a decomposition for pairs of commuting contractions

Zbigniew Burdak (2007)

Studia Mathematica

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A new decomposition of a pair of commuting, but not necessarily doubly commuting contractions is proposed. In the case of power partial isometries a more detailed decomposition is given.

On selection of interface weights in domain decomposition methods

Čertíková, Marta, Šístek, Jakub, Burda, Pavel

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Different choices of the averaging operator within the BDDC method are compared on a series of 2D experiments. Subdomains with irregular interface and with jumps in material coefficients are included into the study. Two new approaches are studied along three standard choices. No approach is shown to be universally superior to others, and the resulting recommendation is that an actual method should be chosen based on properties of the problem.

Quantifier elimination in quasianalytic structures via non-standard analysis

Krzysztof Jan Nowak (2015)

Annales Polonici Mathematici

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The paper is a continuation of an earlier one where we developed a theory of active and non-active infinitesimals and intended to establish quantifier elimination in quasianalytic structures. That article, however, did not attain full generality, which refers to one of its results, namely the theorem on an active infinitesimal, playing an essential role in our non-standard analysis. The general case was covered in our subsequent preprint, which constitutes a basis for the approach presented...

A variant of the reciprocal super Catalan matrix

Emrah Kılıç, Ilker Akkus, Gonca Kızılaslan (2015)

Special Matrices

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Recently Prodinger [8] considered the reciprocal super Catalan matrix and gave explicit formulæ for its LU-decomposition, the LU-decomposition of its inverse, and obtained some related matrices. For all results, q-analogues were also presented. In this paper, we define and study a variant of the reciprocal super Catalan matrix with two additional parameters. Explicit formulæ for its LU-decomposition, LUdecomposition of its inverse and the Cholesky decomposition are obtained. For all...