Dissertations / Theses on the topic 'Normed linear spaces'

In this thesis we consider the problem of simultaneously approximating elements of a set B C X by a single element of a set K C X. This type of a problem arises when the element to be approximated is not known precisely but is known to belong to a set.Thus, best simultaneous approximation is a natural generalization of best approximation which has been studied extensively. The theory of best simultaneous approximation has been studied by many authors, see for example [4], [8], [25], [28], [26] and [12] to name but a few.

Bibliography: leaves 176-182.
Certain properties associated with these classes are stable under small perturbation, i.e. stable under additive perturbation by continuous operators whose norms are less than the minimum modulus of the relation being perturbed, and are also stable under perturbation by compact, strictly singular or strictly cosingular operators. In this work we continue the study of these classes and introduce the classes of α-Atkinson and β-Atkinson relations. These are subclasses of upper and lower semi-Fredholm relations respectively, having generalised inverses and defined in terms of the existence of continuous projections onto their ranges and nullspaces.

"A natural process in examining properties of Banach spaces is to see if a Banach space can be decomposed into simpler Banach spaces; in other words, to see if a Banach space has complemented subspaces. This thesis concentrates on three main aspects of this problem: norm of projections of a Banach space onto its finite dimensional subspaces; a class of Banach spaces, each of which has a large number of infinite dimensional complemented subspaces; and methods of finding Banach spaces which have uncomplemented subspaces, where the subspaces and the quotient spaces are chosen as well-known classical sequence spaces (finding non-trivial twisted sums)." --Abstract.
Master of Mathematical Sciences

"A natural process in examining properties of Banach spaces is to see if a Banach space can be decomposed into simpler Banach spaces; in other words, to see if a Banach space has complemented subspaces. This thesis concentrates on three main aspects of this problem: norm of projections of a Banach space onto its finite dimensional subspaces; a class of Banach spaces, each of which has a large number of infinite dimensional complemented subspaces; and methods of finding Banach spaces which have uncomplemented subspaces, where the subspaces and the quotient spaces are chosen as well-known classical sequence spaces (finding non-trivial twisted sums)." --Abstract.
Master of Mathematical Sciences

Banachverbände spielen sowohl in der Theorie als auch in der Anwendung von geordneten normierten Räume eine bedeutende Rolle. Einerseits erweisen sich viele in der Praxis relevanten Räume als Banachverbände, andererseits ermöglichen die Vektorverbandsstruktur und die enge Beziehung zwischen Ordnung und Norm ein tiefes Verständnis solcher normierter Räume. An dieser Stelle setzen folgende Überlegungen an: - Die genaue Untersuchung einiger Resultate der reichhaltigen Banachverbandstheorie ließ (zu Recht) vermuten, dass in manchen Fällen die Verbandsnormeigenschaft keine notwendige Voraussetzung ist. In der Literatur gibt es bereits einige interessante Untersuchungen allgemeiner geordneter normierter Räume mit qualifizierten positiven Kegeln und in dem Zusammenhang eine Reihe wertvoller Dualitätsaussagen. An dieser Stelle sind die Eigenschaften der Normalität, der Nichtabgeflachtheit und der Regularität eines Kegels erwähnt, welche selbst im Falle eines mit einer Norm versehenen Vektorverbandes eine schwächere Relation zwischen Ordnung und Norm ergeben als die Verbandsnormeigenschaft. - In einer neueren Arbeit wurde der aus der Theorie der Vektorverbände gut bekannte Begriff der Disjunktheit bereits auf beliebige geordnete Räume verallgemeinert, wobei viele Eigenschaften disjunkter Vektoren, des disjunkten Komplements einer Menge usw., welche aus der Verbandstheorie bekannt sind, erhalten bleiben. Auf entsprechende Weise, d.h. durch das Ersetzen exakter Infima und Suprema durch Mengen unterer bzw. oberer Schranken, können der Modul eines Vektors sowie der Begriff der Solidität einer Menge für geordnete (normierte) Räume eingeführt werden. An solchen Überlegungen knüpft die vorliegende Arbeit an. Im Kapitel m-Normen ======== werden verallgemeinerte Formen der M-Norm Eigenschaft eingeführt und untersucht. AM-Räume und (approximative) Ordnungseinheit-Räume sind Beispiele für geordnete normierte Räume mit m-Norm. Die Schwerpunkte dieses Kapitels sind zum Einen Kegel- und Normeigenschaften dieser Räume und deren Charakterisierung mit Hilfe solcher Eigenschaften und zum Anderen Dualitätsaussagen, wie sie zum Teil bereits aus der Theorie der AM- und AL-Räume bekannt sind. Minimal totale Mengen ===================== Ziel dieses Kapitels ist es, den oben erwähnten verallgemeinerten Disjunktheitsbegiff für geordnete normierte Räume zu untersuchen. Eine zentrale Rolle spielen dabei totale Mengen im Dualraum und insbesondere minimal totale Mengen sowie deren Zusammenhang mit der Disjunktheit von Elementen des Ausgangsraumes. Normierte pre-Riesz Räume ========================= Wie bereits bekannt, lässt sich jeder pre-Riesz Raum ordnungsdicht in einen (bis auf Isomorphie) eindeutigen minimalen Vektorverband einbetten, die so genannte Riesz Vervollständigung. Ist der pre-Riesz Raum normiert und sein positiver Kegel abgeschlossen, dann kann eine Verbandsnorm auf der Riesz Vervollständigung eingeführt werden, welche sich in vielen Fällen als äquivalent zur Ausgangsnorm auf dem pre-Riesz Raum erweist. Es ist allgemein bekannt, dass sich dann auch stetige lineare Funktionale fortsetzen lassen. In diesem Kapitel wird nun untersucht, inwiefern sich Ordnungsrelationen auf einer Menge stetiger linearer Funktionale beim Übergang zur Menge der Fortsetzungen erhalten lassen. Die gewonnenen Erkenntnisse kommen anschließend bei Untersuchungen zur schwachen bzw. schwach*-Topologie auf geordneten normierten Räumen zur Anwendung. Hierbei werden zwei Fragestellungen behandelt. Zum Einen gilt das Augenmerk disjunkten Folgen in geordneten normierten Räumen. Als Beispiel seien ordnungsbeschränkte disjunkte Folgen in geordneten normierten Räumen mit halbmonotoner mNorm genannt, welche stets schwach gegen Null konvergieren. Zum Anderen werden monoton fallende Folgen und Netze bzw. disjunkte Folgen von stetigen linearen Funktionalen auf einem geordneten normierten Raum betrachtet
Banach lattices play an important role in the theory of ordered normed spaces. One reason is, that many ordered normed vector spaces, that are important in practice, turn out to be Banach lattices, on the other hand, the lattice structure and strong relations between order and norm allow a deep understanding of such ordered normed spaces. At this point the following is to be considered. - The analysis of some results in the rich Banach lattice theory leads to the conjecture, that sometimes the lattice norm property is no necessary supposition. General ordered normed spaces with a convenient positive cone were already examined, where some valuable duality properties could be achieved. We point out the properties of normality, non-flatness and regularity of a cone, which are a weaker relation between order and norm than the lattice norm property in normed vector lattices. - The notion of disjointness in vector lattices has already been generalized to arbitrary ordered vector spaces. Many properties of disjoint elements, the disjoint complement of a set etc., well known from the vector lattice theory, are preserved. The modulus of a vector as well as the concept of the solidness of a set can be introduced in a similar way, namely by replacing suprema and infima by sets of upper and lower bounds, respectively. We take such ideas up in the present thesis. A generalized version of the M-norm property is introduced and examined in section m-norms. ======= AM-spaces and approximate order unit spaces are examples of ordered normed spaces with m-norm. The main points of this section are the special properties of the positive cone and the norm of such spaces and the duality properties of spaces with m-norm. Minimal total sets ================== In this section we examine the mentioned generalized disjointness in ordered normed spaces. Total sets as well as minimal total sets and their relation to disjoint elements play an inportant at this. Normed pre-Riesz spaces ======================= As already known, every pre-Riesz space can be order densely embedded into an (up to isomorphism) unique vector lattice, the so called Riesz completion. If, in addition, the pre-Riesz space is normed and its positive cone is closed, then a lattice norm can be introduced on the Riesz completion, that turns out to be equivalent to the primary norm on the pre-Riesz space in many cases. Positive linear continuous functionals on the pre-Riesz space are extendable to positive linear continuous functionals in this setting. Here we investigate, how some order relations on a set of continuous functionals can be preserved to the set of the extension. In the last paragraph of this section the obtained results are applied for investigations of some questions concerning the weak and the weak* topology on ordered normed vector spaces. On the one hand, we focus on disjoint sequences in ordered normed spaces. On the other hand, we deal with decreasing sequences and nets and disjoint sequences of linear continuous functionals on ordered normed spaces

Banachverbände spielen sowohl in der Theorie als auch in der Anwendung von geordneten normierten Räume eine bedeutende Rolle. Einerseits erweisen sich viele in der Praxis relevanten Räume als Banachverbände, andererseits ermöglichen die Vektorverbandsstruktur und die enge Beziehung zwischen Ordnung und Norm ein tiefes Verständnis solcher normierter Räume. An dieser Stelle setzen folgende Überlegungen an: - Die genaue Untersuchung einiger Resultate der reichhaltigen Banachverbandstheorie ließ (zu Recht) vermuten, dass in manchen Fällen die Verbandsnormeigenschaft keine notwendige Voraussetzung ist. In der Literatur gibt es bereits einige interessante Untersuchungen allgemeiner geordneter normierter Räume mit qualifizierten positiven Kegeln und in dem Zusammenhang eine Reihe wertvoller Dualitätsaussagen. An dieser Stelle sind die Eigenschaften der Normalität, der Nichtabgeflachtheit und der Regularität eines Kegels erwähnt, welche selbst im Falle eines mit einer Norm versehenen Vektorverbandes eine schwächere Relation zwischen Ordnung und Norm ergeben als die Verbandsnormeigenschaft. - In einer neueren Arbeit wurde der aus der Theorie der Vektorverbände gut bekannte Begriff der Disjunktheit bereits auf beliebige geordnete Räume verallgemeinert, wobei viele Eigenschaften disjunkter Vektoren, des disjunkten Komplements einer Menge usw., welche aus der Verbandstheorie bekannt sind, erhalten bleiben. Auf entsprechende Weise, d.h. durch das Ersetzen exakter Infima und Suprema durch Mengen unterer bzw. oberer Schranken, können der Modul eines Vektors sowie der Begriff der Solidität einer Menge für geordnete (normierte) Räume eingeführt werden. An solchen Überlegungen knüpft die vorliegende Arbeit an. Im Kapitel m-Normen ======== werden verallgemeinerte Formen der M-Norm Eigenschaft eingeführt und untersucht. AM-Räume und (approximative) Ordnungseinheit-Räume sind Beispiele für geordnete normierte Räume mit m-Norm. Die Schwerpunkte dieses Kapitels sind zum Einen Kegel- und Normeigenschaften dieser Räume und deren Charakterisierung mit Hilfe solcher Eigenschaften und zum Anderen Dualitätsaussagen, wie sie zum Teil bereits aus der Theorie der AM- und AL-Räume bekannt sind. Minimal totale Mengen ===================== Ziel dieses Kapitels ist es, den oben erwähnten verallgemeinerten Disjunktheitsbegiff für geordnete normierte Räume zu untersuchen. Eine zentrale Rolle spielen dabei totale Mengen im Dualraum und insbesondere minimal totale Mengen sowie deren Zusammenhang mit der Disjunktheit von Elementen des Ausgangsraumes. Normierte pre-Riesz Räume ========================= Wie bereits bekannt, lässt sich jeder pre-Riesz Raum ordnungsdicht in einen (bis auf Isomorphie) eindeutigen minimalen Vektorverband einbetten, die so genannte Riesz Vervollständigung. Ist der pre-Riesz Raum normiert und sein positiver Kegel abgeschlossen, dann kann eine Verbandsnorm auf der Riesz Vervollständigung eingeführt werden, welche sich in vielen Fällen als äquivalent zur Ausgangsnorm auf dem pre-Riesz Raum erweist. Es ist allgemein bekannt, dass sich dann auch stetige lineare Funktionale fortsetzen lassen. In diesem Kapitel wird nun untersucht, inwiefern sich Ordnungsrelationen auf einer Menge stetiger linearer Funktionale beim Übergang zur Menge der Fortsetzungen erhalten lassen. Die gewonnenen Erkenntnisse kommen anschließend bei Untersuchungen zur schwachen bzw. schwach*-Topologie auf geordneten normierten Räumen zur Anwendung. Hierbei werden zwei Fragestellungen behandelt. Zum Einen gilt das Augenmerk disjunkten Folgen in geordneten normierten Räumen. Als Beispiel seien ordnungsbeschränkte disjunkte Folgen in geordneten normierten Räumen mit halbmonotoner mNorm genannt, welche stets schwach gegen Null konvergieren. Zum Anderen werden monoton fallende Folgen und Netze bzw. disjunkte Folgen von stetigen linearen Funktionalen auf einem geordneten normierten Raum betrachtet.
Banach lattices play an important role in the theory of ordered normed spaces. One reason is, that many ordered normed vector spaces, that are important in practice, turn out to be Banach lattices, on the other hand, the lattice structure and strong relations between order and norm allow a deep understanding of such ordered normed spaces. At this point the following is to be considered. - The analysis of some results in the rich Banach lattice theory leads to the conjecture, that sometimes the lattice norm property is no necessary supposition. General ordered normed spaces with a convenient positive cone were already examined, where some valuable duality properties could be achieved. We point out the properties of normality, non-flatness and regularity of a cone, which are a weaker relation between order and norm than the lattice norm property in normed vector lattices. - The notion of disjointness in vector lattices has already been generalized to arbitrary ordered vector spaces. Many properties of disjoint elements, the disjoint complement of a set etc., well known from the vector lattice theory, are preserved. The modulus of a vector as well as the concept of the solidness of a set can be introduced in a similar way, namely by replacing suprema and infima by sets of upper and lower bounds, respectively. We take such ideas up in the present thesis. A generalized version of the M-norm property is introduced and examined in section m-norms. ======= AM-spaces and approximate order unit spaces are examples of ordered normed spaces with m-norm. The main points of this section are the special properties of the positive cone and the norm of such spaces and the duality properties of spaces with m-norm. Minimal total sets ================== In this section we examine the mentioned generalized disjointness in ordered normed spaces. Total sets as well as minimal total sets and their relation to disjoint elements play an inportant at this. Normed pre-Riesz spaces ======================= As already known, every pre-Riesz space can be order densely embedded into an (up to isomorphism) unique vector lattice, the so called Riesz completion. If, in addition, the pre-Riesz space is normed and its positive cone is closed, then a lattice norm can be introduced on the Riesz completion, that turns out to be equivalent to the primary norm on the pre-Riesz space in many cases. Positive linear continuous functionals on the pre-Riesz space are extendable to positive linear continuous functionals in this setting. Here we investigate, how some order relations on a set of continuous functionals can be preserved to the set of the extension. In the last paragraph of this section the obtained results are applied for investigations of some questions concerning the weak and the weak* topology on ordered normed vector spaces. On the one hand, we focus on disjoint sequences in ordered normed spaces. On the other hand, we deal with decreasing sequences and nets and disjoint sequences of linear continuous functionals on ordered normed spaces.

"A natural process in examining properties of Banach spaces is to see if a Banach space can be decomposed into simpler Banach spaces; in other words, to see if a Banach space has complemented subspaces. This thesis concentrates on three main aspects of this problem: norm of projections of a Banach space onto its finite dimensional subspaces; a class of Banach spaces, each of which has a large number of infinite dimensional complemented subspaces; and methods of finding Banach spaces which have uncomplemented subspaces, where the subspaces and the quotient spaces are chosen as well-known classical sequence spaces (finding non-trivial twisted sums)." --Abstract.
Master of Mathematical Sciences

It is suggested to discriminate between different state space models for a given time series by means of a Bayesian approach which chooses the model that minimizes the expected loss. Practical implementation of this procedures requires a fully Bayesian analysis for both the state vector and the unknown hyperparameters which is carried out by Markov chain Monte Carlo methods. Application to some non-standard situations such as testing hypotheses on the boundary of the parameter space, discriminating non-nested models and discrimination of more than two models is discussed in detail. (author's abstract)
Series: Forschungsberichte / Institut für Statistik

Model diagnostics for normal and non-normal state space models is based on recursive residuals which are defined from the one-step ahead predictive distribution. Routine calculation of these residuals is discussed in detail. Various tools of diagnostics are suggested to check e.g. for wrong observation distributions and for autocorrelation. The paper also covers such topics as model diagnostics for discrete time series, model diagnostics for generalized linear models, and model discrimination via Bayes factors. (author's abstract)
Series: Forschungsberichte / Institut für Statistik

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Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES
The increased consumption of hyperlipidic diet has been an increase in obesity rates and levels of serum cholesterol and triglycerides in a large part of the population, as well as, has been linked with the development of neurodegenerative diseases, such as Alzheimer's disease. On the other hand, several studies demonstrated the importance of lipids in brain structure and activity. Epilepsy is a pathology related to the brain activity disorder, with high rate of refractoriness to conventional therapeutics, in these cases hyperlipidic diet has been used such an alternative treatment. Therefore, the investigation of possible interference from hyperlipidemic diets in TLE can add new perspectives in understanding the behavior and treatment of this pathology. In the present study we used mathematical computational methods to analyze electrographic patterns of rats in status epilepticus induced by pilocarpine fed with hyperlipidic diet. These rats were analyzed through electrographic parameters using ECoG records and determining: energies of power spectrum in the frequency of delta, theta, alpha and beta waves; Lempel-Ziv complexity; and fractal dimension of phase space. Status epilepticus induced changes in the encephalographic pattern measured by distribution of main brain waves using power spectrum, Lempel-Ziv complexity and fractal dimension of phase space. Hyperlipidic diet in normal rats also changed the values of brain waves energy in power spectrum and Lempel-Ziv complexity; however, fractal dimension of phase space showed no significant differences due to hyperlipidic diet treatment. Despite the hyperlipidic diet reduced brain activity before pilocarpine administration, the nutritional status did not change the encephalographic pattern during status epilepticus. In conclusion, hyperlipidic diet induced slower brain waves and decreased the complexity of brain activity, opposite effects of status epilepticus. Therefore, the mathematical methods were effective to detect brain hyperactivity caused by status epilepticus and reduced brain activity induced by hyperlipidic diet.
O aumento do consumo de dietas hiperlipídicas vem elevando os índices de obesidade e os níveis de colesterol e triglicerídeos de grande parte da população, além de estar relacionado ao desenvolvimento de doenças neurodegenerativas, como a doença de Alzheimer. Por outro lado muitas pesquisas têm comprovado a importância dos lipídeos na estrutura e atividade do cérebro. A epilepsia é uma patologia relacionada à desordem da atividade cerebral, com alto índice de refratariedade a medicamentos convencionais, nesses casos, o consumo de dietas hiperlipídica vem sendo utilizado como uma terapia alternativa. A investigação de possíveis interferências de dietas hiperlipídicas na ELT pode acrescentar novas perspectivas na compreensão do comportamento e tratamento desta condição patológica. Nesse trabalho foram analisados ratos em status epilepticus induzido pela pilocarpina submetidos à dieta hiperlipídica. Esses ratos foram analisados através de parâmetros eletrográficos utilizando os registros de ECoG e determinando as energias do seu espectro de potência nas freqüências das ondas delta, teta, alfa e beta; a complexidade de Lempel-Ziv e a dimensão fractal do espaço de fase. O status epilepticus induziu alterações no padrão encefalográfico mensuradas pela distribuição de energia das principais ondas cerebrais utilizando o espectro de potência, a complexidade de Lempel-Ziv e a dimensão fractal do espaço de fase. A dieta hiperlipídica, em ratos normais, também alterou os valores da energia das ondas cerebrais no espectro de potência e na complexidade de Lempel-Ziv; entretanto, a dimensão fractal do espaço de fase não revelou diferenças significativas devido ao tratamento com a dieta hiperlipídica. Apesar da dieta hiperlipídica ter reduzido a atividade cerebral antes da administração da pilocarpina, a condição nutricional não influenciou o padrão encefalográfico durante o status epilepticus. Em conclusão, a dieta hiperlipídica causou uma desaceleração das ondas cerebrais e diminuição da complexidade da atividade cerebral, efeitos contrários aos do status epilepticus. Portanto, os métodos matemáticos utilizados foram eficientes na detecção da hiperatividade cerebral causada pelo status epilepticus e redução da atividade cerebral induzida pela dieta hiperlipídica.

A voz humana apresenta flutuações na frequência, amplitude e formato de onda. Esse comportamento característico pode ser estudado usando técnicas de análise não linear, além das técnicas convencionais. O objetivo desse trabalho é analisar sinais de vozes normais e patológicas (com nódulos e edemas) usando seção de Poincaré de vários trechos do espaço de fase reconstruído e calcular a dispersão em relação ao ponto médio da seção e em relação à distribuição dos pontos sobre os eixos coordenados. Essa dispersão será calculada utilizando o conceito estatístico de desvio padrão. Foram analisados 48 sinais de voz humanas, divididos em 3 grupos (16 normais, 16 com nódulo e 16 com edema de Reinke). Foram selecionados trechos de 500m do sinal temporal nas regiões de maior estacionariedade, descartando os trechos iniciais e finais do sinal para evitar possíveis transitórios. A partir do espaço de fase bidimensional, a seção de Poincaré foi traçada em 10 trechos distintos da trajetória. Em seguida, foi gerado o espaço de fase em 3 dimensões contendo os pontos da seção. Foi feita uma rotação tridimensional dos pontos utilizando a reta tangente à trajetória de modo que a seção ficasse paralela ao plano x = O. Da seção resultante foram extraídas as componentes principais e em seguida calculado o desvio padrão da dispersão e o desvio padrão dos pontos projetados no plano em relação aos eixos coordenados (y;z). A validação da ferramenta desenvolvida para esse estudo foi realizada utilizando um sinal senoidal inserindo gradativamente Jitter e Shimmer, onde se verificou uma variação proporcional da média da dispersão. Os resultados obtidos para esse conjunto de vozes mostraram que o desvio padrão da dispersão e o desvio padrão em relação aos eixos coordenados dos pontos de vozes normais é menor do que os encontrados para vozes com edema e com nódulo. Concluiu-se que a ferramenta proposta conseguiu diferenciar vozes normais das vozes patológicas. Portanto, a ferramenta é promissora para análise e avaliação desse grupo vozes.
The human voice, normal or pathological, has fluctuations in the frequency, amplitude and waveform. This characteristic behavior can be studied using techniques of nonlinear analysis, in addition to conventional techniques. The aim of this study is to analyze signals of normal and pathological voices (with nodules and edema) using the Poincaré section of several parts of the reconstructed phase space and calculate the dispersion in relation to the midpoint of the section and in relation to the distribution of points on coordinate axes. This dispersion is calculated using the statistical concept of standard deviation. We analyzed 48 human voice signals divided into 3 groups (16 normal, 16 with nodules and 16 with Reinke\'s edema). It was selected 500m signal frames presenting good stationarity, discarding the initial and final portions of the signal to avoid possible transient. From the two-dimensional phase space, the Poincaré section was drawn on 10 different stretches of the path. It was then generated the three-dimensional phase space containing the points of the section. We conducted a three dimensional rotation of the points using the tangent to the trajectory so that the section stayed parallel to the plane. From the resulting section, principal components were extracted and then calculated the standard deviation of the dispersion and the standard deviation of the coordinate axes of the projected points of the section in the plan. The validation tool developed for this study was performed using a sinusoidal signal gradually inserting jitter and shimmer, where there was a proportional variation of the dispersion media. The results for this set of voices showed that the standard deviation of the dispersion and the standard deviation related to the coordinate axes of the points of normal voices is smaller than those found for voices with edema and nodule. It was concluded that the proposal was promising tool for analyzing and evaluating this group voices.

The main topic of the paper is on-line filtering for non-Gaussian dynamic (state space) models by approximate computation of the first two posterior moments using efficient numerical integration. Based on approximating the prior of the state vector by a normal density, we prove that the posterior moments of the state vector are related to the posterior moments of the linear predictor in a simple way. For the linear predictor Gauss-Hermite integration is carried out with automatic reparametrization based on an approximate posterior mode filter. We illustrate how further topics in applied state space modelling such as estimating hyperparameters, computing model likelihoods and predictive residuals, are managed by integration-based Kalman-filtering. The methodology derived in the paper is applied to on-line monitoring of ecological time series and filtering for small count data. (author's abstract)
Series: Forschungsberichte / Institut für Statistik

Chen, Donoho a Saunders (1998) studují problematiku hledání řídké reprezentace vektorů (signálů) s použitím speciálních přeurčených systémů vektorů vyplňujících prostor signálu. Takovéto systémy (někdy jsou také nazývány frejmy) jsou typicky vytvořeny buď rozšířením existující báze, nebo sloučením různých bazí. Narozdíl od vektorů, které tvoří konečně rozměrné prostory, může být problém formulován i obecněji v rámci nekonečně rozměrných separabilních Hilbertových prostorů (Veselý, 2002b; Christensen, 2003). Tento funkcionální přístup nám umožňuje nacházet v těchto prostorech přesnější reprezentace objektů, které, na rozdíl od vektorů, nejsou diskrétní. V této disertační práci se zabývám hledáním řídkých representací v přeurčených modelech časových řad náhodných veličin s konečnými druhými momenty. Numerická studie zachycuje výhody a omezení tohoto přístupu aplikovaného na zobecněné lineární modely a na vícerozměrné ARMA modely. Analýzou mnoha numerických simulací i modelů reálných procesů můžeme říci, že tyto metody spolehlivě identifikují parametry blízké nule, a tak nám umožňují redukovat původně špatně podmíněný přeparametrizovaný model. Tímto významně redukují počet odhadovaných parametrů. V konečném důsledku se tak nemusíme starat o řády modelů, jejichž zjišťování je většinou předběžným krokem standardních technik. Pro kratší časové řady (100 a méně vzorků) řídké odhady dávají lepší predikce v porovnání s těmi, které jsou založené na standardních metodách (např. maximální věrohodnosti v MATLABu - MATLAB System Identification Toolbox (IDENT)). Pro delší časové řady (500 a více) obě techniky dávají v podstatě stejně přesné predikce. Na druhou stranu řešení těchto problémů je náročnější, a to i časově, nicméně výpočetní doba je stále přijatelná.

by Chi-keung Ng.
Thesis (M.Phil.)--Chinese University of Hong Kong, 1991.
Includes bibliographical references.
Introduction --- p.1
Chapter Chapter 0. --- Preliminary --- p.4
Chapter 0.1 --- Topological vector spaces
Chapter 0.2 --- Ordered vector spaces
Chapter 0.3 --- Ordered normed spaces
Chapter 0.4 --- Ordered topological vector spaces
Chapter 0.5 --- Ordered bornological vector spaces
Chapter Chapter 1. --- Results on Ordered Normed Spaces --- p.23
Chapter 1.1 --- Results on e∞-spaces and e1-spaces
Chapter 1.2 --- Complemented subspaces of ordered normed spaces
Chapter 1.3 --- Half injections and Half surjections
Chapter 1.4 --- Strict quotients and strict subspaces
Chapter Chapter 2. --- Helley's Selection Theorem and Local Reflexivity Theorem of order type --- p.55
Chapter 2.1 --- Helley's selection theorem of order type
Chapter 2.2 --- Local reflexivity theorem of order type
Chapter Chapter 3. --- Operator Modules and Ideal Cones --- p.68
Chapter 3.1 --- Operator modules and ideal cones
Chapter 3.2 --- Space cones and space modules
Chapter 3.3 --- Injectivity and surjectivity
Chapter 3. 4 --- Dual and pre-dual
Chapter Chapter 4. --- Topologies and Bornologies Defined by Operator Modules and Ideal Cones --- p.95
Chapter 4.1 --- Generalized polars
Chapter 4.2 --- Topologies and bornologies defined by β and ε
Chapter 4. 3 --- Injectivity and generating topologies
Chapter 4.4 --- Surjectivity and generating bornologies
Chapter 4.5 --- The solid property and the generating topologies
Chapter 4.6 --- The solid property and the generating bornologies
Chapter Chapter 5. --- Semi-norms and Bounded disks defined by Operator Modules and Ideal Cones --- p.129
Chapter 5.1 --- Results on semi-norms
Chapter 5.2 --- Results on bounded disks
References --- p.146
Notations --- p.149

Tesis (Lic. en Matemática)--Universidad Nacional de Córdoba. Facultad de Matemática, Astronomía y Física, 2010.
En el capítulo I presentamos el 'Teorema de convexidad de Riesz-Thorin' y diferentes aplicaciones. Y concluimos con el 'Teorema de interpolación de Riesz-Stein'. En el capítulo II, hacemos un breve estudio de 'operadores de tipo débil y función distribución', definimos la clase de Marcinkievicz y demostramos los 'Teoremas de interpolación de Marcinkievicz. Caso diagonal y caso general'. Damos algunas aplicaciones. Concluimos este capítulo con las 'Condiciones de Kolmogoroff y Zygmund'. Este trabajo contiene un Apéndice donde destacamos variados resultados matemáticos, fundamentales para el entendimiento de los dos capítulos.

碩士
義守大學
電機工程學系碩士班
94
The classical binary classification problem is considered in this thesis. Necessary and sufficient condition is proposed to guarantee the linear binary separability of the training data in the Euclidean normed space. A suitable hyperplane that correctly classifies the training data is also constructed provided that the necessary and sufficient condition is satisfied. Based on the main result, we present an easy-to-check criterion for the linear binary separability of the training set. Finally, two numerical examples are given to illustrate the use of the main result.

The logical system MTL (for Monoidal t-norm Logic) is a formalism of the logic of left-continuous t-norms, which are operations that arise in the study of fuzzy sets and fuzzy logic. The objective is to investigate the important results on MTL and collect them together in a coherent form. The main results considered will be the completeness results for the logic with respect to MTL-algebras, MTL-chains (linearly ordered MTL-algebras) and standard MTL-algebras (left-continuous t-norm algebras). Completeness of MTL with respect to standard MTL-algebras means that MTL is indeed the logic of left-continuous t-norms. The logical system BL (for Basic Logic) is an axiomatic extension of MTL; we will consider the same completeness results for BL; that is we will show that BL is complete with respect to BL-algebras, BL-chains and standard BL-algebras (continuous t-norm algebras). Completeness of BL with respect to standard BL-algebras means that BL is the logic of continuous t-norms.

Let H be a separable Hilbert space over the complex field. The class S := {N|M : N is normal on H and M is an invariant subspace for Ng of subnormal operators. This notion was introduced by Halmos. The minimal normal extension Ň of a subnormal operator S was introduced by σ (S) and then Bram proved that Halmos. Halmos proved that σ(Ň) (S) is obtained by filling certain number of holes in the spectrum (Ň) of the minimal normal extension Ň of a subnormal operator S. Let σ (S) := σ (Ň) be the spectrum of the minimal normal extension Ň of S; which is called the normal spectrum of a subnormal operator S: This notion is due to Abrahamse and Douglas. We give several well-known characterization of subnormality. Let C* (S1) and C* (S2) be the C*- algebras generated by S1 and S2 respectively, where S1 and S2 are bounded operators on H: Next we give a characterization for subnormality which is purely C - algebraic. We also establish an intrinsic characterization of the normal spectrum for a subnormal operator, which enables us to answer the fol-lowing two questions. Let II be a *- representation from C* (S1) onto C* (S2) such that II(S1) = S2. If S1 is subnormal, then does it follow that S2 is subnormal? What is the relation between σ (S1) and σ (S2)? The first question was asked by Bram and second was asked by Abrahamse and Douglas. Answers to these questions were given by Bunce and Deddens.

Let H be a separable Hilbert space over the complex field. The class S := {N|M : N is normal on H and M is an invariant subspace for Ng of subnormal operators. This notion was introduced by Halmos. The minimal normal extension Ň of a subnormal operator S was introduced by σ (S) and then Bram proved that Halmos. Halmos proved that σ(Ň) (S) is obtained by filling certain number of holes in the spectrum (Ň) of the minimal normal extension Ň of a subnormal operator S. Let σ (S) := σ (Ň) be the spectrum of the minimal normal extension Ň of S; which is called the normal spectrum of a subnormal operator S: This notion is due to Abrahamse and Douglas. We give several well-known characterization of subnormality. Let C* (S1) and C* (S2) be the C*- algebras generated by S1 and S2 respectively, where S1 and S2 are bounded operators on H: Next we give a characterization for subnormality which is purely C - algebraic. We also establish an intrinsic characterization of the normal spectrum for a subnormal operator, which enables us to answer the fol-lowing two questions. Let II be a *- representation from C* (S1) onto C* (S2) such that II(S1) = S2. If S1 is subnormal, then does it follow that S2 is subnormal? What is the relation between σ (S1) and σ (S2)? The first question was asked by Bram and second was asked by Abrahamse and Douglas. Answers to these questions were given by Bunce and Deddens.

Скрипник К. В. Дослідження властивостей компактних операторів у банахових просторах : кваліфікаційна робота магістра спеціальності 111 "Математика" / наук. керівник І. В. Красікова. Запоріжжя : ЗНУ, 2020. 50 с.
UA : Робота викладена на 50 сторінках друкованого тексту, містить 12 джерел. Об’єкт дослідження: компактні оператори у банахових просторах. Мета роботи: дослідити властивості компактних операторів у банахових просторах. Метод дослідження: аналітичний. У кваліфікаційній роботі досліджуються властивості компактних операторів, заданих у банахових просторах. Весь теоретичний матеріал проілюстровано прикладами та задачами. Розглянуто застосування компактних операторів до розв’язання інтегральних рівнянь.
EN : The work is presented on 50 pages of printed text, 12 references. The object of the study is compact operators in Banach space. The aim of the study is to study the properties of the compact operators in the Banach space. The method of research is analytical. In the qualification paper the properties of compact operators are investigated, given in Banach spaces. All theoretical material is illustrated with examples and tasks. The application of compact operators to the solution of integral equations is considered.