Functions and Analysis

Seeking the Analytic Quaternion

Authors: Colin Walker
Comments: 12 Pages.

By combining the complex analytic Cauchy-Riemann derivative with the Cayley-Dickson construction of a quaternion, possible formulations of a quaternion derivative are explored with the goal of finding an analytic quaternion derivative having conjugate symmetry. Two such analytic derivatives can be found. Although no example is presented, it is suggested that this finding may have significance in areas of quantum mechanics where quaternions are fundamental, especially regarding the enigmatic phenomenon of complementarity, where a quantum process seems to present two essential aspects.
Category: Functions and Analysis

Set Theory NC_{∞^{}}^{} Based on Bivalent Infinitary Logic with Restricted Modus Ponens Rule. Basic Real Analysis on External Non-Archimedean Field ℝ_{c}^{}. Basic Complecs Analysis on External Field ℂ_{c}^{}=ℝ_{c}^{}[i].

Authors: Jaykov Foukzon
Comments: 132 Pages.

In this paper we deal with set theory NC_{∞}^{} based on gyper infinitary logic with Restricted Modus Ponens Rule [1]-[3].The main goal of this paper is to present basic analysis on external non Archimedean field ℝ_{c}^{}.The non Archimedean external field ℝ_{c}^{} consist of Cauchy hyperreals.The non-Archimedean external field ℝ_{c}^{#}≠┊^{∗}ℝ┊ is obtained as generalized Cauchy completion of non-Archimedean field ℚ^{#} or ^{∗}ℚ.In order to obtain such completion we deal with external hyper infinite Cauchy sequences {x_{n}}_{n∈ℕ^{#}},{x_{n}}_{n∈┊^{∗}ℕ┊}.We have emphasised that such external Cauchy sequences defined external hyperreal numbers in natural way. Basic Analysis on External Non-Archimedean Field ℝ_{c}^{#} is considered.
Category: Functions and Analysis

Maximality of Linear Operators

Authors: Mohammed Meziane
Comments: 80 Pages.

In this thesis, we show some maximality results about non-necessarily bounded linear operators.
Category: Functions and Analysis

Square Roots of Boudned Operators

Authors: Mohammed Hichem Mortad
Comments: 10 Pages.

This is part of some lectures about square roots of bounded operators.
Category: Functions and Analysis

An Operator Theory Problem Book

Authors: Mohammed Hichem Mortad
Comments: 18 Pages.

This constitutes the preface and references of the book "an operator theory problem book".
Category: Functions and Analysis

Positive Operators. Square Root

Authors: Mohammed Hichem Mortad
Comments: 41 Pages.

This is Chapter 5 of the manuscript "an operator theory problem book".
Category: Functions and Analysis

PhD's Thesis of Youcef Naas

Authors: Youcef Nass
Comments: 73 Pages.

Le troisième chapitre concerne le cas L p (0, 1; X). Plus précisément, on s’intéresse à l’équation différentielle abstraite du second ordre de type elliptique (1) avec les conditions aux limites de type mêlé (4) où A est un opérateur linéaire fermé sur un espace de Banach complexe X et u0, u 0 1 sont des éléments donnés dans X. Ici f ∈ L p (0, 1; X), 1 < p < ∞, 5 INTRODUCTION INTRODUCTION et X a la proporiété géométrique dite UMD. On suppose que A est un opérateur Bip et on montre que (1)-(4) admet une unique solution stricte, sous certaines hypothèses naturelles d’ellipticité de l’opérateur et de régularité sur les données, on donne alors, une représentation explicite de la solution stricte. La formule de représentation de la solution est donnée par deux méthodes, la première se base sur le calcul fonctionnel de Dunford et la deuxième sur la méthode de Krein[27], l’unicité de la représentation est démontrée. Dans ce chapitre, on fait une nouvelle approche du problème (1)-(4) en utilisant le théorème de Mikhlin. Dans cette partie on utilise les techniques des multiplicateurs de Fourier et la théorie de Mikhlin pour majorer les puissances imaginaires pures d’opérateurs. Le quatrième chapitre illustre notre théorie abstraite par quelques exemples concrets d’applications en EDP dans le cas des espaces L p et C α .
Category: Functions and Analysis

The Absolute Value of Unbounded Linear Operators

Authors: Imene Boucif
Comments: 61 Pages. [Corrections made by viXra Admin to conform with the requirements on the Submission Form]

Dans cette thèse en théorie des opérateurs, on s’intéresse aux opérateurs normaux, aux opérateurs autoadjoints, aux opérateurs positifs, à la valeur absolue d’un opérateur, à l’inégalité triangulaire. Cette thèse s’articule autour des relations du type |AB| = |A||B|, |A||B| = |B||A|, |A ± B| ≤ |A| + |B|. Dans la première partie on fournit des définitions et des notions élémentaires en théorie des opérateurs. Après cette introduction, dans ce chapitre on s’intéresse aux opérateurs positifs bornés, à la racine carrée d’un opérateur positif, à la valeur absolue d’un opérateur borné, et on donne des résultats sur l’inégalité triangulaire et d’autres relations concernant la somme et le produit de la valeur absolue dans le cas borné. Dans le dernier chapitre, on s’intéresse au cas des opérateurs non-bornés. On commence d’abord par des définitions et des propriétés primordiales des opérateurs non-bornés, ensuite on donne des résultats sur la somme et le produit de la valeur absolue dans le cas des opérateurs non-bornés. On fournit cette partie par quelques exemples. On obtient, comme conséquence intéressante, une caractérisation de l’inversibilité pour la classe des opérateurs normaux non-bornés. On obtient également une preuve très simple de l’inclusion dans R du spectre des opérateurs non-bornés autoadjoints.

In this thesis in operator theory, we are interested in operators normal, to self-assistant operators, to positive operators, to the absolute value of an operator, with triangular inequality. This thesis revolves around the relationships of the type | AB | = | A || B |, | A || B | = | B || A |, | A ± B | ≤ | A | + | B |. In the first part we provide definitions and elementary notions in operator theory. After this introduction, in this chapter we are interested in bounded positive operators, to the square root of a positive operator, to the value absolute of a bounded operator, and we give results on the triangular inequality and other relations concerning the sum and the product of the absolute value in the case thick headed. In the last chapter, we are interested in the case of unbounded operators. We start first with definitions and primordial properties of unbounded operators, then we give results on the sum and the product of the value absolute in the case of unbounded operators. We provide this part with a few examples. As an interesting consequence, we obtain a characterization of invertibility for the class of unbounded normal operators. We obtain also a very simple proof of the inclusion in R of the spectrum of unbounded self-adjoining operators.
Category: Functions and Analysis

Fractional Distance: The Topology of the Real Number Line with Applications to the Riemann Hypothesis

Authors: Jonathan W. Tooker
Comments: 147 Pages. Remedied a few errata. Grammar and punctuation updates.

Recent analysis has uncovered a broad swath of rarely considered real numbers called real numbers in the neighborhood of infinity. Here, we extend the catalog of the rudimentary analytical properties of all real numbers by defining a set of fractional distance functions on the real number line and studying their behavior. The main results are (1) to prove with modest axioms that some real numbers are greater than any natural number, (2) to develop a technique for taking a limit at infinity via the ordinary Cauchy definition reliant on the classical epsilon-delta formalism, and (3) to demonstrate an infinite number of non-trivial zeros of the Riemann zeta function in the neighborhood of infinity. We define numbers in the neighborhood of infinity with a Cartesian product of Cauchy equivalence classes of rationals. We axiomatize the arithmetic of such numbers, prove the operations are well-defined, and then make comparisons to the similar axioms of a complete ordered field. After developing the many underlying foundations, we present a basis for a topology.
Category: Functions and Analysis

Kepler's Laws as Properties of the Kinematic Equations of Motion of a Point Along Curves of the Second Order

Authors: Viktor Strohm
Comments: 13 Pages.

Differential equations of motion on curves of the second order are inferred. Solutions to equations are made by computer programs. The results of the calculation are compared with Kepler's laws.
Category: Functions and Analysis