Dissertations / Theses on the topic 'Compactness'

In my opinion, compactness is the most important concept in mathematics. We 'll track it from the one-dimensional real line in calculus to infinite dimensional spaces of functions and surfaces and see what it can do.

The aim of this thesis is to understand the constructive scope of compactness. We show that it is possible to define, constructively, a meaningful notion of compactness in a more general setting than the uniform/metric space one. Furthermore, we show that it is not possible to define compactness constructively in a topological space. We investigate exactly what principles are necessary and sufficient to prove classically true theorems about compactness, as well as their antitheses. We develop beginnings of a constructive theory of differentiable manifolds.

Codensity monads are ubiquitous, as are various different notions of compactness and finiteness. Two such examples of "compact" spaces are compact Hausdorff Spaces and Linearly Compact Vector Spaces. Compact Hausdorff Spaces are the algebras of the codensity monad induced by the inclusion of finite sets in the category of sets. Similarly linearly compact vector spaces are the algebras of the codensity monad induced by the inclusion of finite dimensional vector spaces in the category of vector spaces. So in these two examples the notions of finiteness, compactness and codensity are intertwined. In this thesis we generalise these results. To do this we generalise the notion of ultrafilter, and follow the intuition of the compact Hausdorff case. We give definitions of general notions of "finiteness" and "compactness" and show that the algebras for the codensity monad induced by the "finite" objects are exactly the "compact" objects.

Bruning considered sets of isospectral Schrodinger operators with smooth real potentials on a compact manifold of dimension three. He showed the set of potentials associated to an isospectral set is compact in the topology of smooth functions by relating the spectrum to the trace of the heat semi-group. Similarly, we can consider the resonances of Schrodinger operators with real valued potentials on Euclidean space of whose support lies inside a ball of fixed radius that generate the same resonances as some fixed Schrodinger operator, an ``isoresonant" set of potentials. This isoresonant set of potentials is also compact in the topology of smooth functions for dimensions one and three. The basis of the result stems from the relation of a regularized wave trace to the resonances via the Poisson formula (also known as the Melrose trace formula). The second link is the small-t asymptotic expansion of the regularized wave trace whose coefficients are integrals of the potential function and its derivatives. For an isoresonant set these coefficients are equal due to the Poisson formula. The equivalence of coefficients allows us to uniformly bound the potential functions and their derivatives with respect to the isoresonant set. Finally, taking a sequence of functions in the isoresonant set we use the uniform bounds to construct a convergent subsequence using the Arzela-Ascoli theorem.

Thesis (MSc (Mathematics))--University of Stellenbosch, 2010.
ENGLISH ABSTRACT: As an introduction to point-free topology, we will explicitly show the connection between topology and frames (locales) and introduce an abstract notion, which in the point-free setting, can be thought of as a subspace of a topological space. In this setting, we refer to this notion as a sublocale and we will show that there are at least four ways to represent sublocales. By using the language of category theory, we proceed by investigating closure in the point-free setting by way of operators. We de ne what we mean by a coclosure operator in an abstract context and give two seemingly di erent examples of co-closure operators of Frm. These two examples are then proven to be the same. Compactness is one of the most important notions in classical topology and therefore one will nd a great number of results obtained on the subject. We will undertake a study into the interrelationship between three weaker compact notions, i.e. feeble compactness, pseudocompactness and countable compactness. This relationship has been established and is well understood in topology, but (to a degree) the same cannot be said for the point-free setting. We will give the frame interpretation of these weaker compact notions and establish a point-free connection. A potentially promising result will also be mentioned.
AFRIKAANSE OPSOMMING: As 'n inleiding tot punt-vrye topologie, sal ons eksplisiet die uiteensetting van hierdie benadering tot topologie weergee. Ons de nieer 'n abstrakte konsep wat, in die punt-vrye konteks, ooreenstem met 'n subruimte van 'n topologiese ruimte. Daar sal verder vier voorstellings van hierdie konsep gegee word. Afsluiting, deur middel van operatore, word in die puntvrye konteks ondersoek met behulp van kategorie teorie as taalmedium. Ons sal 'n spesi eke operator in 'n abstrakte konteks de nieer en twee o enskynlik verskillende voorbeelde van hierdie operator verskaf. Daar word dan bewys dat hierdie twee operatore dieselfde is. Kompaktheid is een van die mees belangrikste konsepte in klassieke topologie en as gevolg daarvan geniet dit groot belangstelling onder wiskundiges. 'n Studie in die verwantskap tussen drie swakker forme van kompaktheid word onderneem. Hierdie verwantskap is al in topologie bevestig en goed begryp onder wiskundiges. Dieselfde kan egter, tot 'n mate, nie van die puntvrye konteks ges^e word nie. Ons sal die puntvrye formulering van hierdie swakker konsepte van kompaktheid en hul verbintenis, weergee. 'n Resultaat wat moontlik belowend kan wees, sal ook genoem word.

Bibliography: leaves 100-107.
The main purpose of this thesis is to develop a point-free notion of E-compactness. Our approach follows that of Banascheski and Gilmour in [17]. Any regular frame E has a fine nearness and hence induces a nearness on an E-regular frame L. We show that the frame L is complete with respect this nearness iff L is a closed quotient of a copower of E. This resembles the classical definition, but it is not a conservative definition: There are spaces that may be embedded as closed subspaces of powers of a space E, but their frame of opens are not closed quotients of copowers of the frame of opens of E. A conservative definition of E-compactness is obtained by considering Cauchy completeness with respect to this nearness. Another central notion in the thesis is that of K-Lindelöf frames, a generalisation of Lindelöf frames introduced by J.J. Madden [59]. In the last chapter we investigate the interesting relationship between the completely regular K-Lindelöf frames and the K-compact frames.

Includes abstract.
Includes bibliographical references.
The aim of the thesis is to investigate aspects of the theory of such spaces, concentrating mainly, but not exclusively, on nite dimensional spaces. One of the main aims of this thesis is to investigate compactness in the setting of asymmetrically normed lattices. In order to do this, it was necessary to study convergence of sequences and left-K-sequential completeness and precompactness of subsets of such spaces.

A weak concept of compactness for nonmonotonic logics is proposed, which is suitable for several nonmonotonic logics, for which Makinsons smallest cumulative extension as well as Freund/Lehmanns canonical extension fail.

The work describes the modern concepts of sustainable development emphasizing on compact city concept. The methodology for obtaining numerical expressions of compactness is developed and examples are shown.
Darbe analizuojama kompaktiško miesto teorija ir jos raida, miestų vystymosi tendencijos ir urbanistines kryptis. Nagrinėjami taikomi miesto formos įvertinimo metodai, parinkti metodai urbanistinei miesto erdvinės struktūros analizei. Įvertintas 7 didžiausių Lietuvos miestų kompaktiškumas ir darnumo aspektu įvertinta Kauno miesto erdvinė struktūra.

We present three criteria for compactness in the context of apartness spaces and Bishop-style constructive mathematics. Each of our three criteria can be summarised as requiring that there is a positive distance between any two disjoint closed sets. Neat locatedness and the product apartness give us three variations on this theme. We investigate how our three criteria relate to one another and to several existing compactness criteria, namely classical compactness, completeness, total boundedness, the anti-Specker property, and Diener's neat compactness.

Thesis (MSc (Mathemathical Sciences))--Stellenbosch University, 2009.
We have the familiar Kuratowski-Mr owka theorem in topology, where compactness is characterised by a closure and a projection-map (X is compact i p : X Y ! Y is a closed mapping, for any space Y , i.e. p(A) = p(A) A X Y ). Using this as our starting point, we generalise compactness to a categorical setting. We then generalise even further to "asymmetric" compactness. Then we discuss a functional approach to compactness, where we do not explicitly mention closure operators. All this provides economical proofs as well as applications in di erent areas of mathematics.

In this thesis we present a new result concerning the theory of group invariant persistent homology. This theory adapts persistent homology in the presence of the action on a space of functions Phi of a subgroup G of the group H of all self-homeomorphisms of a topological space X. Its model is based on a space of suitable operators defined on Phi. After describing the mathematical setting and recalling some basic results, we prove that the space of these operators is compact with respect to a suitable topology. In order to prove this result, we require that Phi, G, X are compact.

Sustainable urban development is mentioned together with the concept of urban form in contemporary planning literature. The main reason behind this is a need for determining an ideal physical development scheme and its main principles of urban future in a broad term. Besides, the operational side of urban planning requires a concrete set of design codes in order to transform urban space in both macro and mezzo scale. At this point, the concept of urban compactness and the idea of Compact City have come into the agenda of planning. In the last decade, the model of compact city has become a prototype of sustainable urban form in developed countries. It is also argued whether compact urbanity is a nostalgic metaphor or an engineering solution. It has emerged as a reaction to the negative consequences of urban sprawl and suburbanization as the anti-urbanist urban phenomena in Western geographies. Hence, the relevance of urban compactness should be examined for developing and underdeveloped countries and their settlement structures. The basic motivation of the thesis is to examine the relevance and validity of urban compactness in the case of Turkey as a developing Eurasian country. For this end, the evolution of urban compactness as a fact and an idea in the historical context of developed countries and it&rsquo
s meaning for the developing world
Ankara is examined as a case study by re-reading its planning history and the transformation of its urban form from the point of view of compactness.

2009 - 2010
Abstract Nello studio di vari problemi ellittici con soluzioni in spazi di Sobolev S( ) (con o senza peso) definiti su un aperto di Rn, non necessariamente limitato o regolare, spesso risulta necessario stabilire risultati di regolarit`a e stime a priori per le soluzioni di tali problemi. Questi risultati si basano molte volte sulla limitatezza e l’eventuale compattezza dell’operatore di moltiplicazione u −! g u (i) definito su uno spazio di Sobolev S( ) e a valori in uno spazio di Lebesgue Lp( ) con un opportuno p 2 [1,+1[ e dove g `e un’assegnata funzione definita in uno spazio normato V . `E necessario, quindi, ottenere una stima del tipo kg ukLp( ) c · kgkV · kukS( ) , (ii) dove la costante c 2 R+ dipende dalle propriet`a di regolarit`a di , dagli esponenti di sommabilit`a e la funzione g soddisfa opportune condizioni. Se L `e l’operatore differenziale associato al problema ellittico, stime del tipo (ii) permettono, ad esempio, di provare immediatamente la limitatezza dell’operatore, dove 1 2 Abstract g rappresenta uno dei coefficienti dell’operatore stesso. Tuttavia, altri tipi di risultati non si riescono ad ottenere direttamente per l’operatore L, a causa della natura non necessariamente regolare dei suoi coefficienti. Risulta dunque necessario introdurre una classe di operatori Lh, i cui coefficienti, pi`u regolari, approssimano i coefficienti dell’operatore L. Questa “ deviazione ” dei coefficienti di Lh da quelli di L, deve essere fatta controllando le norme dei coefficienti approssimanti con quelle dei coefficienti dati. Dunque, `e necessario ottenere stime dove la dipendenza dai coefficienti `e espressa solo in termini delle loro norme (in tal caso, per esempio, non ci sono problemi nel passaggio a limite). In altre parole, se g rappresenta un coefficiente di L e gh un coefficiente pi`u regolare della classe approssimante, `e necessario avere un “ buon controllo ” sulla differenza g − gh. L’introduzione delle decomposizioni per funzioni appartenenti ad opportuni spazi funzionali, che rappresentano l’ambiente dei coefficienti dell’operatore differenziale L, gioca un ruolo molto rilevante in questo processo di approssimazione. Nel presente lavoro, si costruiscono decomposizioni per funzioni appartenenti ad opportuni spazi funzionali, la cui introduzione `e legata alla risolubilit`a di alcuni problemi ellittici del tipo sopra menzionato. Come applicazione, si ottengono risultati di limitatezza e compattezza per un operatore di moltiplicazione definito in uno spazio di Sobolev (con o senza peso) . L’idea della decomposizione consiste nello scrivere una funzione g, appartenente ad un opportuno spazio funzionale, come somma di una funzione gh, pi`u regolare, e di una rimanente funzione g − gh, la cui norma `e controllata dal modulo di continuit`a della funzione g. Nella prima parte del lavoro si approfondisce lo studio di alcuni spazi Abstract 3 funzionali pesati, la cui introduzione `e legata alla risolubilit`a di problemi di Dirichlet per equazioni differenziali lineari del secondo ordine di tipo ellittico, in domini non regolari, con soluzioni in spazi di Sobolev con peso.Come applicazione, usando le decomposizioni per funzioni appartenenti a tali spazi funzionali pesati, si provano risultati di immersione e compattezza sull’operatore (i), definito in uno spazio di Sobolev con peso. La struttura espositiva del Capitolo 1 e del Capitolo 2 rispecchia la progressione delle considerazioni svolte. Nel Capitolo 1 si studiano alcune propriet`a e applicazioni degli spazi di Sobolev con peso. Siano un dominio di Rn, k 2 N, 1 p < +1 e il vettore peso le cui componenti sono funzioni misurabili su . Lo spazio di Sobolev con peso Wk,p( ; ) `e l’insieme delle funzioni u = u(x) definite a.e. su , le cui derivate (nel senso delle distribuzioni) @ u, di ordine | | k, sono tali che: Z |@ u(x)|p (x) dx < +1. Nel Capitolo 2, si considera una classe di funzioni peso, denotata con A( ), e si definiscono i corrispondenti spazi di Sobolev con pesoWk,p s ( ) su aperti di Rn. Precisamente, una funzione peso : ! R+ appartiene alla classe A( ) se e solo se esiste una costante 2 R+, indipendente da x and y, tale che : −1 (y) (x) (y) , 8 y 2 , 8 x 2 \ B(y, (y)), 4 Abstract Per k 2 N0, s 2 R e 1 p +1, si denota con Wk,p s ( ) lo spazio delle distribuzioni u su tali che s+| |−k @ u 2 Lp( ) per | | k, munito della seguente norma : kukWk,p s ( ) = X | | k k s+| |−k @ ukLp( ) , dove la funzione peso appartiene alla classe A( ). Nel Capitolo 2 si approfondisce, inoltre, lo studio degli spazi funzionali pesati Kr t (r 2 [1,+1[, t 2 R) e di alcuni suoi sottospazi. Sia r 2 [1,+1[ e t 2 R, si denota con Kr t ( ) la classe delle funzioni g, appartenenti a Lrl oc( ), tali che : sup t−n r (x) kgkLr( \B(x, (x))) < +1, dove la funzione peso appartiene alla classe A( ). Si prova, facilmente, che gli spazi L1 t ( ) e C1 o ( ) sono sottoinsiemi di Kr t ( ) (lo spazio L1 t ( ) `e costituito dalle funzioni g tali che t g 2 L1( )). Si possono, pertanto, definire le chiusure di L1 t ( ) e di C1 o ( ) in Kr t ( ) (denotate rispettivamente con Krt( ) e Krt ( )). Si costruiscono, inoltre, opportune decomposizioni per funzioni g 2 Krt ( ) e per funzioni g 2 Krt ( ), da cui si ottengono risultati di immersione sull’operatore di moltiplicazione (i), definito su uno spazio di Sobolev con peso Wk,p s ( ) e a valori in Lq( ) e dove il fattore moltiplicativo g appartiene ad un opportuno sottospazio di Kr t ( ). L’utilizzo delle decomposizioni in tali risultati consente di evidenziare come la parte meno regolare (g−gh) della funzione g in Krt ( ) o in Krt ( ), influenzi la stima. Infine, uno studio approfondito degli spazi Krt ( ) ha condotto all’introduzione di Abstract 5 un nuovo sottospazio di Kr t ( ), denotato con Krt ( ) . Si sono esaminate le relazione che intercorrono tra Krt ( ), Krt ( ) e Krt ( ) ed in particolare si sono individuate opportune condizioni sulla funzione peso 2 A( ) affinch`e si abbia Krt ( ) = Krt ( ). Nel Capitolo 3 si approfondisce lo studio degli spazi di tipo Morrey. Anche in questo caso, utilizzando le decomposizioni per funzioni appartenenti ad opportuni sottospazi di tipo Morrey, si ottiene un risultato di compattezza per l’operatore (i), definito su un classico spazio di Sobolev. Sia un aperto non limitato di Rn, n 2.Per p 2 [1,+1[ e 2 [0, n[, si considera lo spazio Mp, ( ) costituito dalle funzioni g in Lp loc( ) tali che: kgkp Mp, ( ) = sup 2]0,1] x2 − Z \B(x, ) |g(y)|p dy < +1, dove B(x, ) `e la sfera aperta di Rn di centro x e raggio . Lo spazio di tipo Morrey Mp, ( ) rappresenta una generalizzazione del classico spazio di Morrey Lp, e contiene strettamente lo spazio Lp, (Rn) se = Rn. La sua introduzione `e legata alla risolubilit`a di problemi di tipo ellittico con coefficienti discontinui su domini non limitati. Nella prima parte del Capitolo 3, si rivolge l’attenzione alle propriet`a di densit`a degli spazi di tipo Morrey. Si forniscono, infatti, utili lemmi di caratterizzazione per funzioni appartenenti alle chiusure di L1( ) e C1 o ( ) in Mp, ( ) ( denotate rispettivamente con fMp, ( ) e Mp, 0 ( )). Utilizzando tali lemmi di caratterizzazione, si costruiscono decomposizioni per funzioni in fMp, ( ) e in Mp, 0 ( ) che consentono di provare un 6 Abstract risultato di compattezza sul seguente operatore di moltiplicazione u 2 Wk,p( ) ! g u 2 Lq( ) con q 2 [p,+1[ e g appartenente ad un opportuno sottospazio di Mp, ( ). Infine, un’attenta disamina degli spazi Mp, ( ) e dei suoi sottospazi conduce all’introduzione di un nuovo spazio funzionale pesato di tipo Morrey Mp, ( ), dove il peso appartiene ad una classe di funzioni peso, denotata con G( ). Precisamente, fissato d 2 R+, una funzione peso : ! R+ appartiene alla classe G( , d) se e solo se esiste una costante 2 R+, indipendente da x and y, tale che −1 (y) (x) (y) , 8 y 2 , 8 x 2 (y, d). Si pone G( ) = [ d>0 G( , d). Siano 2 G( ) \ L1( ) e d un numero reale positivo tale che 2 G( , d). Fissato un sottoinsieme misurabile secondo Lebesgue E di , per p 2 [1,+1[ e 2 [0, n[ si denota con Mp, ( ) lo spazio delle funzioni g 2 Mp, ( ) tali che lim h!+1 sup E2 ( ) sup x2 2]0,d] − (x)|E(x, )| 1h kg EkMp, ( ) = 0, Abstract 7 Un’attenta analisi delle relazioni che intercorrono tra Mp, ( ), fMp, ( ) e Mp, 0 ( ), ha consentito di provare le seguenti inclusioni Mp, 0 ( ) Mp, ( ) fMp, ( ) . In particolare si sono individuate opportune condizioni sulla funzione peso affich`e si abbia Mp, 0 ( ) = Mp, ( ). Si precisa che i risultati ottenuti nel Capitolo 2 possono trovare applicazione nello studio di problemi al contorno per equazioni ellittiche su domini non regolari (ad esempio, domini con frontiera singolare), con soluzioni in spazi di Sobolev pesati Wk,p s , per provare che gli operatori differenziali associati al corrispondente problema ellittico (i cui coefficienti di ordine inferiore appartengono ad opportuni spazi Kr t ) hanno rango chiuso o sono semi-fredholmiani. I risultati contenuti nel Capitolo 3, invece, possono essere utili, per esempio, nello studio di problemi di Dirichlet per equazioni ellittiche su domini non limitati (la cui frontiera `e sufficientemente regolare), con soluzioni in classici spazi di Sobolev, per stabilire stime a priori sul corrispondente operatore differenziale associato al problema ellittico, i cui coefficienti di ordine inferiore appartengono a spazi di tipo Morrey Mp, . Si precisa, inoltre, che gli spazi Krt ( ) e Mp, ( ) possono essere utilizzati nello studio di alcuni problemi al contorno per equazioni di tipo ellittico con coefficienti discontinui appartenenti a tali spazi. [a cura dell'autore]
IX n.s.

There has been a recent push towards adoption of the in-place soil stiffness as a means of assessing compactness of pavement geomaterials. The Humboldt GeoGauge is a relatively new and promising instrument that is portable, that provides instantaneous results and that does not require handling of radioactive materials. Unlike the nuclear gage, which yields the soil unit weight and water content, the GeoGauge yields soil stiffness corresponding to very low strains. Based on a series of low-strain soil stiffness measurements made under controlled laboratory conditions on compacted silts from Oahu, the variation of modulus with water content, dry unit weight, degree of saturation, volume change upon wetting, shear strength and soil plasticity is discussed. These results help advance the understanding of the role of stiffness in assessing compactness of cohesive soils. For compacted partly saturated soils, the dry unit weight can be related to stiffness and water content. This relationship is derived herein. Using this relationship, measured values of stiffness and water content can then be used to predict the dry unit weight in the field. This work involved testing of tropical soils, which can undergo irreversible changes upon drying, resulting in permanent alterations in soil properties. Upon drying, the tropical cohesive soils tested became less plastic, coarser (downward shift in grain size and higher sand equivalent), and exhibits a higher maximum dry unit weight and lower optimum water content.
x, 87 leaves

This work focuses on two dimensional simply-connected complexes. In particular, I examine sufficient conditions on such a complex for convex hulls of compact sets to be compact. I show that there exist complexes where convex hulls of compact sets are not compact if no restriction is placed on the curvature. I prove that convex hulls of compact sets are compact for most nonpositively curved planar complexes, and for negatively curved simply-connected complexes composed of triangles.

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Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES
The main goal of this work is to analyze concentration-compactness principles for fractional Sobolev spaces based on the concentration compactness principle of P.-L. Lions and in the pro le decomposition for weak convergence in Hilbert spaces due to K. Tintarev and K.-H Fieseler. As application, we address questions on compactness of the associated energy functional to the following nonlocal elliptic problems, $' ''''''&' ''''''% p qsu fpx; uq in RN; p qsu 􀀀 apxqu fpx; uq in RN; $&% p qsu 􀀀 V pxqu 􀀀 Kpxq u fpx; uq 􀀀 gpx; uq in R3; p q Kpxqu2 in R3; where 0   s   1; 0     1; 2 􀀀 4s ¥ 3; ¡ 0 and Kpxq ¥ 0 belongs to a suitable Lebesgue space. We obtain existence results for a wide class of possible singular potentials apxq; not necessarily bounded away from zero and for oscillatory nonlinearities in both subcritical and critical growth range that may not satisfy the Ambrosetti-Rabinowitz condition.
O objetivo principal deste trabalho é analisar princípios de concentração de compacidade para espaços de Sobolev fracionários baseados na concentração de compacidade de P.-L. Lions e no per l de decomposição para convergência fraca em espaços de Hilbert devido a K. Tintarev e K.-H Fieseler. Como aplicação, abordamos questões sobre a compacidade do funcional energia associado aos seguintes problems elípticos não locais, $' ''''''&' ''''''% p qsu fpx; uq em RN; p qsu 􀀀 apxqu fpx; uq em RN; $&% p qsu 􀀀 V pxqu 􀀀 Kpxq u fpx; uq 􀀀 gpx; uq em R3; p q Kpxqu2 em R3; onde 0   s   1; 0     1; 2 􀀀 4s ¥ 3; ¡ 0 e Kpxq ¥ 0 pertence a um espaço de Lebesgue adequado. Obtemos resultados de existência para uma vasta classe de potenciais apxq possivelmente singulares, não necessariamente limitados por baixo por uma constante positiva e para não linearidades oscilatórias em ambos os crescimentos subcríticos e críticos que podem não satisfazer a condição de Ambrosetti-Rabinowitz.

Chapter 1 We study different classes of compact sets. In particular, the class of convex-compact sets is analyzed in depth. Using these classes of sets, we provide compactness criteria by checking on a quite relaxed set of conditions. In order to ensure that we are really dealing with more general notions, we pay attention to separate the classes introduced. We also provide some stability results of the classes of compact sets used. Some Valdivia and Orihuela theorems are pushed further and an extension of a theorem due to Howard is provided. Chapter 2 We formulate some results on Banach disks and prove that every convex, relatively convex-compact subset of a locally convex space is contained in a Banach disk. We study in which cases some properties, such as separability and reflexivity, are preserved by passing to the generated Banach space. Chapter 3 The drop property, the property (alpha) and the condition (beta) are analyzed. A single technique provides short proofs of some results about drop properties on locally convex spaces. It is shown that the quasi-drop property is equivalent to a drop property for countably closed sets. We prove that the drop and quasi-drop properties, the property (alpha) and the condition (beta) are separably determined. We also study the relation between drop property, property (alpha), condition (beta), compactness and reflexivity.
Monterde Pérez, I. (2009). Some compactness criteria in locally convex and banach spaces [Tesis doctoral no publicada]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/14569
Palancia

This dissertation consists of two parts. In the first part we show that for 1 k 1, a complex manifold M of dimension at least k in the boundary of a smooth bounded pseudoconvex domain in Cn is an obstruction to compactness of the @- Neumann operator on (p, q)-forms for 0 p k n, provided that at some point of M, the Levi form of b has the maximal possible rank n − 1 − dim(M) (i.e. the boundary is strictly pseudoconvex in the directions transverse to M). In particular, an analytic disc is an obstruction to compactness of the @-Neumann operator on (p, 1)-forms, provided that at some point of the disc, the Levi form has only one vanishing eigenvalue (i.e. the eigenvalue zero has multiplicity one). We also show that a boundary point where the Levi form has only one vanishing eigenvalue can be picked up by the plurisubharmonic hull of a set only via an analytic disc in the boundary. In the second part we obtain a weaker and quantified version of McNeal’s Property ( eP) which still implies the existence of a Stein neighborhood basis. Then we give some applications on domains in C2 with a defining function that is plurisubharmonic on the boundary.

For H^1 bounded sequences, we introduce a technique, related to the Kadec-Pelczynski -decomposition for L^1 sequences, that allows us to prove compactness theorems. Roughly speaking, a bounded sequence in H^1 can be split into two sequences, one of which is weakly compact, the other forms the singular part. If the martingales are continuous then the singular part tends to zero in the semi-martingale topology. In the general case the singular parts give rise to a process of bounded variation. The technique allows to give a new proof of the optional decomposition theorem in Mathematical Finance. (author's abstract)
Series: Report Series SFB "Adaptive Information Systems and Modelling in Economics and Management Science"

In 1998, it was proved by Ben-David and Litman that a concept space has a sample compression scheme of size $d$ if and only if every finite subspace has a sample compression scheme of size $d$. In the compactness theorem, measurability of the hypotheses of the created sample compression scheme is not guaranteed; at the same time measurability of the hypotheses is a necessary condition for learnability. In this thesis we discuss when a sample compression scheme, created from compression schemes on finite subspaces via the compactness theorem, have measurable hypotheses. We show that if $X$ is a standard Borel space with a $d$-maximum and universally separable concept class $\m{C}$, then $(X,\CC)$ has a sample compression scheme of size $d$ with universally Borel measurable hypotheses. Additionally we introduce a new variant of compression scheme called a copy sample compression scheme.

The goal of this thesis is to give an exposition of the following recent result of Freeman, Lennard, Odell, Turett and Randrianantoanina. A Banach space has the Schur property if and only if every weakly compact set is contained in the closed convex hull of a weakly null sequence. This result complements an old result of Grothendieck (now called the Grothendieck Compactness Principle) stating that every norm compact subset of a Banach space is contained in the closed convex hull of a norm null sequence. We include many of the relevant definitions and preliminary results which are required in the proofs of both of these theorems.

Solar radiation offers phenomenal potential for energy conversion with energy densities on the order of 1000W/m2 in locations with regularly clear skies. As always, the difficulty lies in finding a solar-electric conversion technology capable of producing electricity at a competitive cost. The SolarCAT (Solar Compressed Air Turbine) system produces electricity by releasing stored compressed air through a series of turbines with solar dish concentrators providing the required heat for efficient conversion to electricity. To minimize impact on capital cost, high recuperator effectiveness targets are sought but unlike typical fuel-fired micro-turbines, raising the recuperator effectiveness of the solar power system yields a benefit in overall system capital cost. Improving efficiency lowers the size and cost of the largest element of the system, namely the dish. In this study potential techniques for achieving a highly compact heat-transfer media were reviewed. Folded fin, packed beds, micro-tubes, lattice frame structures, metal foams, woven textile, and micro-machining techniques were assessed. Textile structures were selected as an appropriate medium to replace the internal folded fin of the SolarCAT recuperator. The relatively long flow (>150mm) path through the proposed screen wafers requires a model for fully-developed forced convective flow between parallel plates. A mathematical model was developed by integrating the results from the work of several authors in the field of textiles and porous media. #100 mesh sintered screen wafers were brazed between two 0.25mm stainless steel sheets and destructively tested to assess their tensile strength. Although iii optimization of the braze parameters was not completed, it was found that many samples survived exposure to internal pressures in excess of 50MPa. This study found that the use of sintered screen wafers to replace the internal folded fin of the SolarCAT recuperator would have advantages over the current design with respect to both overall recuperator effectiveness, size, and cost. Textile structures can be tailored to have wide range of fluid and heat-transfer properties depending on the application. The manufacturing process is relatively simple and could be cost-effective for high-volume production.

Electoral districts drawn by independent commissions are seen by political reformers to be preferable to those drawn by state legislatures. The overtly partisan interests of elected officials, say the reformers, lead to oddly-shaped, and gerrymandered districts. To test this, shapes of districts in states with commissions are compared to those within the same state prior to the commission’s establishment. Additionally, shapes of districts in states with commissions are compared to those in a selected group of states without commissions. This study tests hypotheses on two methods of measuring compactness, Reock and Polsby-Popper, and coterminosity, the congruence of district lines and pre-existing political boundaries. The study finds that each state with a commission shows no significant difference in mean compactness compared to its pre-commission form. However, in aggregate, all post-commission districts show a significant increase in mean Reock compactness compared to all pre-commission districts, and all districts in states with commissions show significantly less Polsby-Popper compactness than districts in non-commission states. The study also finds no significant difference in coterminosity between commission states and non-commission states. Though the true effect of commissions may not be discernible from averages, other redistricting criteria also need to be controlled for and evaluated over time.

Various notions of compactness in a fuzzy topological space have been introduced by different authors. The aim of this thesis is to compare them. We find that in a T₂ space (in the sense that no fuzzy net converges to two fuzzy points with different supports) all these notions are equivalent for the whole space. Furthermore, for N-compactness and f-compactness (being the only notions that are defined for an arbitrary fuzzy subset) we have equivalence under a stronger condition, namely, a T₂ space in the sense that every prime prefilter has an adherence that is non-zero in at most one point

Decision tree is one of the most popular algorithms in the domain of explainable AI. From its structure, it is simple to induce a set of decision rules which are totally understandable for a human. That is why there is currently research on improving decision or mapping other models into a tree. Decision trees generated by C4.5 or ID3 tree suffer from two main issues. The first one is that they often have lower performances in term of accuracy for classification tasks or mean square error for regression tasks compared to state-of-the-art models like XGBoost or deep neural networks. On almost every task, there is an important gap between top models like XGboost and decision trees. This thesis addresses this problem by providing a new method based on data augmentation using state-of-the-art models which outperforms the old ones regarding evaluation metrics. The second problem is the compactness of the decision tree, as the depth increases the set of rules becomes exponentially big, especially when the splitted attribute is a categorical one. Standards solution to handle categorical values are to turn them into dummy variables or to split on each value producing complex models. A comparative study of current methods of splitting categorical values in classification problems is done in this thesis. A new method is also studied in the case of regression.
Beslutsträd är en av de mest populära algoritmerna i den förklarbara AI-domänen. I själva verket är det från dess struktur verkligen enkelt att framställa en uppsättning beslutsregler som är helt förståeliga för en vanlig användare. Därför forskas det för närvarande på att förbättra beslut eller kartlägga andra modeller i ett träd. Beslutsträd genererat av C4.5 eller ID3-träd lider av två huvudproblem. Den första är att de ofta har lägre prestanda när det gäller noggrannhet för klassificeringsuppgifter eller medelkvadratfel för regressionsuppgiftens noggrannhet jämfört med modernaste modeller som XGBoost eller djupa neurala nätverk. I nästan varje uppgift finns det faktiskt ett viktigt gap mellan toppmodeller som XGboost och beslutsträd. Detta examensarbete tar upp detta problem genom att tillhandahålla en ny metod baserad på dataförstärkning med hjälp av modernaste modeller som överträffar de gamla när det gäller utvärderingsmätningar. Det andra problemet är beslutsträdets kompakthet, allteftersom djupet ökar, blir uppsättningen av regler exponentiellt stor, särskilt när det delade attributet är kategoriskt. Standardlösning för att hantera kategoriska värden är att förvandla dem till dummiesvariabler eller dela på varje värde som producerar komplexa modeller. En jämförande studie av nuvarande metoder för att dela kategoriska värden i klassificeringsproblem görs i detta examensarbete, en ny metod studeras också i fallet med regression.

Master’s thesis is devoted to the study of cardinal invariants in the F-compact spaces class. Here and throughout the paper, the concept ”compact” would mean a compact Hausdorff space. In my thesis I have tried to present and explain all necessary concepts and statements necessary for the reader to get acquainted with F-compact spaces class. In order to understand the idea of F-compact spaces, it is necessary to understand what the inverse spectrum is from itself, it is necessary to know about the cardinality of sets and to understand that two infinite sets can have different cardinalities, know about closed and open sets, and much else that you will find in this paper. In the thesis the analysis of the scientific literature sources is presented; the theorems about the relationship between the characteristics of cardinality invariants in the F-compact spaces class are investigated; the relationships between the properties of perfect normality and hereditary normality in the F - compact spaces class of countable spectral height are studied. In the process of the investigation some propositions were found, proved and filled in the missing fragments of evidence. Conclusion: At present, the method of fully closed mappings (which is used in constructing of F - compact spaces ) is the most productive method of constructing counterexamples in general topology. I believe, that this paper will be interesting to all who wants to go beyond the ordinary, habitual way of thinking, because only by studying topology we can speak clearly and precisely about things related to the idea of continuity and infinity!

This Doctoral Thesis consists of five chapters, which deal with a new Sobolev type function space called the space with multiweighted derivatives. This space is a generalization of the usual one dimensional Sobolev space. As basis for this space serves some differential operators containing weight functions.Chapter 1 is an introduction, where, in particular, the importance to study function spaces with weights is discussed and motivated. In Chapter 2 we prove some new estimates for each function in a Tchebychev system. In order to be able to study compactness of the embeddings from Chapter 3 such estimates are crucial.In Chapter 3 we rewrite and present some results of L. D. Kudryavtsev, where he investigated one dimensional Sobolev spaces. Moreover, in this chapter we rewrite and discuss some analogous results by B. L. Baidel'dinov for generalized Sobolev spaces. These results are not available in the Western literatures in this way and they are crucial for the proofs of the main results in Chapter 4. In Chapter 4 we prove some embedding theorems for these new generalized Sobolev spaces. The main results of Kudryavtsev and Baidel'dinov about characterization of the behavior of functions at a singularity take place in weak degeneration of the spaces. However, with the help of our new embedding theorems we can extend theseresults to the case of strong degeneration.The main aim of Chapter 5 is to establish boundedness and compactness of the embedding considered in Chapter 4.In Chapter 4 basically only sufficient conditions for boundedness of this embedding were obtained. In Chapter 5 we obtain necessary and sufficient conditions for boundedness and compactness of this embedding and the main results are proved in a different way.
Godkänd; 2009; 20090429 (zamira); DISPUTATION Ämnesområde: Matematik/Mathematics Opponent: Professor Victor Burenkov, Cardiff University, United Kingdom Ordförande: Professor Lars-Erik Persson, Luleå tekniska universitet Tid: Fredag den 12 juni 2009, kl 10.00 Plats: D 2214, Luleå tekniska universitet

This PhD thesis is devoted to investigate weighted differential Hardy inequalities and Hardy-type inequalities with the kernel when the kernel has an integrable singularity, and also the additivity of the estimate of a Hardy type operator with a kernel.The thesis consists of seven papers (Papers 1, 2, 3, 4, 5, 6, 7) and an introduction where a review on the subject of the thesis is given. In Paper 1 weighted differential Hardy type inequalities are investigated on the set of compactly supported smooth functions, where necessary and sufficient conditions on the weight functions are established for which this inequality and two-sided estimates for the best constant hold. In Papers 2, 3, 4 a more general class of -order fractional integrationoperators are considered including the well-known classical Weyl, Riemann-Liouville, Erdelyi-Kober and Hadamard operators. Here 0 <  < 1. In Papers 2 and 3 the boundedness and compactness of two classes of such operators are investigated namely of Weyl and Riemann-Liouville type, respectively, in weighted Lebesgue spaces for 1 < p ≤ q < 1 and 0 < q < p < ∞. As applications some new results for the fractional integration operators of Weyl, Riemann-Liouville, Erdelyi-Kober and Hadamard are given and discussed.In Paper 4 the Riemann-Liouville type operator with variable upper limit is considered. The main results are proved by using a localization method equipped with the upper limit function and the kernel of the operator. In Papers 5 and 6 the Hardy operator with kernel is considered, where the kernel has a logarithmic singularity. The criteria of the boundedness and compactness of the operator in weighted Lebesgue spaces are given for 1 < p ≤ q < ∞ and 0 < q < p < ∞, respectively. In Paper 7 we investigated the weighted additive estimates for integral operators K+ and K¯ defined by K+ ƒ(x) := ∫ k(x,s) ƒ(s)ds,  K¯ ƒ(x) := ∫ k(x,s)ƒ(s)ds. It is assumed that the kernel k of the operators K+and K- belongs to the general Oinarov class. We derived the criteria for the validity of these addittive estimates when 1 ≤ p≤ q < ∞

Understanding the mechanisms used by HIV-1 to evade antibody neutralisation may contribute to the design of a high-coverage vaccine. This thesis explores the mechanism used by a Tier 3 virus leading to its high antibody neutralisation resistance phenotype. This thesis also describes how the glycans at the base of the V3 loop contribute to (i) breadth and potency in a cohort of unselected HIV-1-infected individuals and (ii) the selective pressures resulting from the V3/glycans shielding the virus from neutralisation and the glycans themselves being targets of broad antibody responses. HIV-1 isolates that are highly resistant to broadly neutralising antibodies could limit the efficacy of an antibody-based vaccine. For this reason, it is important to understand the mechanisms behind high HIV-1 resistance to neutralising antibodies. Chapter 2 and Chapter 3 of this thesis describe virus 253-11, a highly neutralisation resistant virus, which is particularly resistant to commonly-elicited, anti-membrane proximal external region (MPER) antibodies in sera. To further understand its resistance, mutations in the MPER were introduced that are known to delay fusion following CD4-binding and thus increase the time the virus spends in the open conformation. Interestingly, we found that these mutations affect the 253-11 Envelope (Env) spike before CD4-binding by destabilising the closed trimer structure. From these data, we hypothesized that the neutralisation resistance of 253-11 was due to an unusually tight, compact pre-fusion Env trimer that resists transient changes to the open conformation. The open conformation frequently exposes narrowly-neutralising antibody epitopes. Because the unliganded 253-11 Env presumably transitions infrequently into the open conformation, it would be able to evade these responses. 253-11 was sensitive to most but not all of the most potent broadly neutralising antibodies (bnAbs) tested, most likely because those broadly neutralising antibodies can access their epitopes in the pre-fusion Env conformation. To gain further information about the structure of the 253-11 Env, we designed a recombinant 253-11 SOSIP trimer and found it to be stable and predominantly adopt a closed conformation. The crystal structure of the SOSIP trimer revealed structural elements likely responsible for 253-11 Env compactness including the inward disposition of the heptad repeat helices and gp120 protomers towards the trimer axis. Taken together, the data from Chapter 2 and Chapter 3 highlight an underappreciated Env compactness mechanism of HIV-1 resistance to neutralising antibodies and these data may be useful in HIV-1 immunogen design research. Previous candidate HIV vaccines have failed to induce wide-coverage neutralising antibodies capable of substantially protecting vaccinees. A key approach in HIV immunogen development has been to define and model epitopes recognised by anti-HIV bnAbs. Candidate immunogen models identified by bnAbs include the V3/glycans, the V2/apex and the MPER epitopes. Autoreactivity and polyreactivity of anti-V3/glycan and anti-MPER antibodies are thought to pose both direct and indirect barriers to achieving neutralisation breadth. Chapter 4 of this thesis explored which of these bnAb epitopes were associated with breadth and potency in a South African cohort of chronically HIV-infected individuals. The study found that antibodies targeting the V3/glycans were associated with breadth and potency. In contrast, antibodies to the V2/apex were not associated with neutralisation breadth/potency. This suggests that auto/polyreactivity are not critical factors in the development of breadth and potency and that the V3/glycans should remain a high-priority vaccine candidate. Since targeting the V3/glycans was associated with breadth and potency in this cohort, the study continued to look at this epitope to investigate the role of these glycans in neutralisation resistance of Tier 2 viruses. The HIV-1 Env is surrounded by glycans that often prevent antibody neutralisation, leading to the term the "glycan shield", however some bnAbs have evolved to recognise these carbohydrates. Chapter 4 of this thesis describes how the N-linked glycan at position N301 is critical for maintaining neutralisation resistance of one subtype C virus (Du156.12), but not for another subtype-matched virus (CAP45.2.00.G3). Thus, the loss of the N301 glycan may have a substantial antibody-related fitness cost for some viruses but not others. The N301 glycan, as well as glycans at positions 332 and 334, are the primary targets of the anti-V3/glycan class of neutralising antibodies, which may select for loss of the targeted glycan. The evidence presented in Chapter 4 suggests that in some viruses, loss of the N301 glycan may result in evasion of anti-V3/glycan antibody responses while maintaining overall neutralisation resistance. This phenomenon may impair efficacy of passively-infused anti-V3/glycan bnAbs or a therapeutic vaccine.

Sequences are sufficient to describe topological properties in metric spaces or, more generally, topological spaces having a countable base for the topology. However, filters or nets are needed in more abstract spaces. Nets are more natural extension of sequences but are generally less friendly to work with since quite often two nets have distinct directed sets for domains. Operations involving filters are set theoretic and generally certain to filters on the same set. The concept of a filter was introduced by H. Cartan in 1937 and an excellent treatment of the subject can be found in N. Bourbaki (1940).
M.S.
Department of Mathematics
Arts and Sciences
Mathematics

Automobile use is associated with significant problems such as air pollution and obesity. Decisions to use the automobile or its alternatives, including walk, bicycle, and public transit, are believed to be associated with urban form. However, in contrast to the hypothesis that compact urban form significantly reduces automobile travel, previous studies reported only a modest effect on travel behavior. These studies, largely built on microeconomic utility theory, are not sufficient for assessing the effect of compactness, for several reasons: (1) The studies postulate that travel invokes only disutility, but travel may also provide intrinsic utility or benefits insomuch as people travel for its own sake; (2) the studies have traditionally focused on how urban compactness reduces the distance between trip origin and destination and accordingly reduces trip time, but urban compactness also increases congestion and reduces trip speed, and thus increases trip time; and (3) the studies have mostly examined automobile commuting, but people travel for various purposes, using different travel modes, and the impact of urban compactness on the utility of non-automobile non-commuting travel has not been duly examined. On this ground, to better explain the effects that urban compactness has on travel behavior, this dissertation refines the concept of travel utility using two additions to the microeconomic utility theory: activity-based utility theory of derived travel demand and approaches to positive utility of travel. Accordingly, it designs a conceptual model that specifies travel utility as an intermediary between urban compactness and travel behavior and examines the behavior associated with and utility derived from travel mode choices for alternative purposes of travel. Twenty individual models are derived from the conceptual model and tested within the context of Seoul, Korea, using a confirmatory approach of structural equation modeling and data from geographic information systems and a structured sample survey, which is initially designed and validated by semi-structured interviews and subsequent statistical tests. By comparing the individual models, this research concludes that the urban compactness effect on travel behavior, represented by trip frequencies and supplemented by mode shares, is better explained when travel utility is considered and if travel purposes are separately examined. Major empirical findings are that urban compactness affects travel behavior mainly by increasing the benefits of travel in comparison to its modest effect on the cost reduction and people’s behavioral response to urban compactness is to shift modes of commuting travel, decrease travel for shopping, and increase travel for leisure. These purpose-specific findings have implications for transportation planners and public health planners by assisting them in linking plans and policies concerning urban compactness to travel purposes.

Die vorliegende Dissertation befasst sich mit der mathematischen Modellierung semi-klassischer Licht-Materie-Interaktion. Im semiklassischen Bild wird Materie durch eine Dichtematrix "rho" beschrieben. Das Konzept der Dichtematrizen ist quantenmechanischer Natur. Auf der anderen Seite wird Licht durch ein klassisches elektromagnetisches Feld "(E,H)" beschrieben. Wir stellen einen mathematischen Rahmen vor, in dem wir systematisch dissipative Effekte in die Liouville-von-Neumann-Gleichung inkludieren. Bei unserem Ansatz sticht ins Auge, dass Lösungen der resultierenden Gleichung eine intrinsische Liapunov-Funktion besitzen. Anschließend koppeln wir die resultierende Gleichung mit den Maxwell-Gleichungen und erhalten ein neues selbstkonsistentes, dissipatives Modell vom Maxwell-Bloch-Typ. Der Fokus dieser Arbeit liegt auf der intensiven mathematischen Studie des dissipativen Modells vom Maxwell-Bloch-Typ. Da das Modell Lipschitz-Stetigkeit vermissen lassen, kreieren wir eine regularisierte Version des Modells, das Lipschitz-stetig ist. Wir beschränken unsere Analyse im Wesentlichen auf die Lipschitz-stetige Regularisierung. Für regularisierte Versionen des dissipativen Modells zeigen wir die Existenz von Lösungen des zugehörigen Anfangswertproblems. Der Kern des Existenzbeweises besteht aus einem Resultat von ``compensated compactness'''', das von P. Gérard bewiesen wurde, sowie aus einem Lemma vom Rellich-Typ. In Teilen folgt dieser Beweis dem Vorgehen einer älteren Arbeit von J.-L. Joly, G. Métivier und J. Rauch.
This thesis deals with the mathematical modeling of semi-classical matter-light interaction. In the semi-classical picture, matter is described by a density matrix "rho", a quantum mechanical concept. Light on the other hand, is described by a classical electromagnetic field "(E,H)". We give a short overview of the physical background, introduce the usual coupling mechanism and derive the classical Maxwell-Bloch equations which have intensively been studied in the literature. Moreover, We introduce a mathematical framework in which we state a systematic approach to include dissipative effects in the Liouville-von-Neumann equation. The striking advantage of our approach is the intrinsic existence of a Liapunov function for solutions to the resulting evolution equation. Next, we couple the resulting equation to the Maxwell equations and arrive at a new self-consistent dissipative Maxwell-Bloch type model for semi-classical matter-light interaction. The main focus of this work lies on the intensive mathematical study of the dissipative Maxwell-Bloch type model. Since our model lacks Lipschitz continuity, we create a regularized version of the model that is Lipschitz continuous. We mostly restrict our analysis to the Lipschitz continuous regularization. For regularized versions of the dissipative Maxwell-Bloch type model, we prove existence of solutions to the corresponding Cauchy problem. The core of the proof is based on results from compensated compactness due to P. Gérard and a Rellich type lemma. In parts, this proof closely follows the lines of an earlier work due to J.-L. Joly, G. Métivier and J. Rauch.

O principal objetivo desta dissertacão é estudar a produtividade da compacidade enumerável em grupos topológicos. Vários contra-exemplos foram descritos.
The main aim of this thesis is to study the productivity of coutable compactness in topological groups. Several counterexamples were described.