Dissertations / Theses on the topic 'Matrix partitions'

Neste trabalho, com base em representações por matrizes de duas linhas para alguns tipos de partição (algumas já conhecidas e outras novas), identificamos propriedades sugeridas por classificá-las de acordo com a soma dos elementos de sua segunda linha. Esta soma sempre fornece alguma propriedade da partição relacionada. Se considerarmos versões sem sinal de algumas funções Mock Theta, seu termo geral pode ser interpretado como função geradora para algum tipo de partição com restrições. Para retornar aos coeficientes originais, é possível definir um peso para cada matriz e depois somá-las para contá-los. Uma representação análoga para essas partições nos permite observar propriedades sobre elas, novamente por meio de uma classificação referente à soma dos seu elementos da segunda linha. Esta seriação é feita por meio de tabelas criadas pelo software matemático Maple, as quais nos sugerem padrões e identidades relacionadas com outros tipos de partições conhecidas e, muitas vezes, encontrando uma fórmula fechada para contá-las. Tendo as conjecturas obtidas, elas são provadas por meio de bijeções entre conjuntos ou por contagem.
In this work, based on representations by matrices of two lines for some kind of partition (some already known and other new ones), we identify properties suggested by classifying them according to the sum of its second line. This sum always provides some properties of the related partition. If we consider unsigned versions of some Mock Theta Functions, its general term can be interpreted as generating function for some kind of partition with restrictions. To come back to the original coefficients, you can set a weight for each array and so add them to evaluate the coefficients. An analogous representation for partitions allows us to observe properties, again by classificating them according to the sum of its elements on the second row. This classification is made by means of tables created by mathematical software Maple, which suggest patterns, identities related to other known types of partitions and often, finding a closed formula to count them. Having established conjectured identities, all are proved by bijections between sets or counting methods.

Cette thèse est consacrée au problème de la comparaison de deux partitions non strictes (floues/probabilistes, possibilistes) d'un même ensemble d'individus en plusieurs clusters. Sa résolution repose sur la définition formelle de mesures de concordance reprenant les principes des mesures historiques développées pour la comparaison de partitions strictes et trouve son application dans des domaines variés tels que la biologie, le traitement d'images, la classification automatique. Selon qu'elles s'attachent à observer les relations entre les individus décrites par chacune des partitions ou à quantifier les similitudes entre les clusters qui composent ces partitions, nous distinguons deux grandes familles de mesures pour lesquelles la notion même d'accord entre partitions diffère, et proposons d'en caractériser les représentants selon un même ensemble de propriétés formelles et informelles. De ce point de vue, les mesures sont aussi qualifiées selon la nature des partitions comparées. Une étude des multiples constructions sur lesquelles reposent les mesures de la littérature vient compléter notre taxonomie. Nous proposons trois nouvelles mesures de comparaison non strictes tirant profit de l'état de l'art. La première est une extension d'une approche stricte tandis que les deux autres reposent sur des approches dite natives, l'une orientée individus, l'autre orientée clusters, spécifiquement conçues pour la comparaison de partitions non strictes. Nos propositions sont comparées à celles de la littérature selon un plan d'expérience choisi pour couvrir les divers aspects de la problématique. Les résultats présentés montrent l'intérêt des propositions pour le thème de recherche qu'est la comparaison de partitions. Enfin, nous ouvrons de nouvelles perspectives en proposant les prémisses d'un cadre qui unifie les principales mesures non strictes orientées individus.

O presente trabalho dedica-se ao estudo de algumas consequências da representação matricial para conjuntos de partições de inteiros e funções mock theta. Na primeira parte do texto, classificamos as partições geradas por seis diferentes funções mock theta, de acordo com a soma das entradas da segunda linha das matrizes associadas, e apresentamos algumas fórmulas fechadas e identidades para essas partições. De nimos também a família ffm (q)gm 1 de funções mock theta, inspiradas pelo que chamamos de versão sem sinal da função f1(q). Fornecemos uma representação matricial análoga para as funções fm (q), o que leva a resultados interessantes a respeito das partições geradas por elas. A parte II do texto trata de uma nova abordagem que gera um conjunto diferente de partições de inteiros. A definição desse conjunto baseia-se na construção de um caminho sobre o reticulado Z2, determinado pela representao matricial para diferentes conjuntos de partições de n, e que liga a reta x + y = n a origem. As novas partições possuem apenas partes mpares distintas, com algumas restições particulares. Esse processo de construção de novas partições, chamado de Path Procedure, e aplicado a partições irrestritas, bem como para partições contadas pelas 1a e 2a Identidades de Rogers-Ramanujan e funções mock theta f5 (q) e T1(q).

On dit d'un sous-ensemble de NP qu'il présente une dichotomie s'il contient des problèmes qui sont soit résolubles en temps polynomial (dans Ptime), soit difficiles (NP-complets). La classe des problèmes de satisfaction de contraintes (CSP) finis est un sous-ensemble bien connu de NP qui présente une telle dichotomie. La classe de complexité NP n'a pas de dichotomie à moins que P = NP. Pour ces deux classes, il existe des logiques qui leur sont associées. -- NP est capturé par la logique Existentielle du second ordre (ESO) par le théorème de Fagin, c'est-à-dire qu'un problème est dans NP si et seulement s'il est exprimable par une formule ESO.-- CSP est un sous-ensemble de la logique de Feder et Vardi, le fragment monotone, monadique et sans inégalités de SNP, lui-même un fragment syntaxique de ESO (MMSNP); et, pour chaque formule de MMSNP, il existe un problème CSP équivalent via des réductions polynomiales.Ceci implique que la logique ESO, tout comme NP, n'a pas de dichotomie, à contraster avec le fait que MMSNP a une dichotomie tout comme CSP. L'objectif principal de cette thèse est d'étudier les propriétés de dichotomie de sous-ensembles de NP qui contiennent strictement CSP ou MMSNP.Feder et Vardi ont prouvé que si nous omettons une des trois propriétés qui définissent MMSNP, à savoir être monotone, monadique ou omettre les inégalités, alors la logique résultante n'a pas de dichotomie. Comme leurs preuves restent parfois sommaires, nous revisitons ces résultats et fournissons des preuves détaillées. Le fragment guardé et monotone de SNP (GMSNP) est une extension connue de MMSNP qui est obtenue en relâchant la restriction "monadique" de MMSNP. Nous définissons de manière similaire une nouvelle logique appelée MMSNP avec des inégalités gardées, en relâchant la restriction d'être "sans inégalités". Nous prouvons qu'elle est strictement plus expressive que MMSNP et qu'elle possède également une dichotomie.Il existe une logique MMSNP₂ qui étend MMSNP de la même manière que MSO₂ étend la logique monadique du second ordre (MSO). On sait que MMSNP₂ est un fragment de GMSNP et que ces deux classes ont toutes deux une dichotomie ou n'en ont pas. Nous revisitons ce résultat et le renforçons en prouvant que, en ce qui concerne le fait d'avoir une dichotomie, sans perte de généralité, on peut considérer seulement les problèmes MMSNP₂ sur des signatures à un élément, au lieu des problèmes GMSNP sur des signatures finies arbitraires.Nous cherchons à prouver l'existence d'une dichotomie pour les MMSNP₂ en construisant en temps polynomial, pour tout problème MMSNP₂, un problème MMSNP équivalent. Nous rencontrons quelques obstacles pour construire une telle équivalence. Cependant, si nous permettons aux formules MMSNP d'être composées d'un nombre dénombrable de conjonctions négatives, nous prouvons qu'une telle équivalence existe. De plus, la formule MMSNP infinie correspondante a la propriété d'être "régulière". Cette propriété de régularité signifie que, dans un certain sens, cette formule est essentiellement finie. Il est connu que les problèmes MMSNP réguliers peuvent être exprimés par CSP sur des modèles oméga-catégoriques. De plus, il existe une caractérisation de la dichotomie algébrique pour les CSP oméga-catégoriques qui décrivent des problèmes MMSNP. Si l'on parvient à étendre cette caractérisation algébrique sur les problèmes réguliers MMSNP, alors notre résultat fournirait une dichotomie algébrique pour MMSNP₂. (...)
A subset of NP is said to have a dichotomy if it contains problem that are either solvable in P-time or NP-complete. The class of finite Constraint Satisfaction Problems (CSP) is a well-known subset of NP that follows such a dichotomy. The complexity class NP does not have a dichotomy unless P = NP. For both of these classes there exist logics that are associated with them. -- NP is captured by Existential Second-Order (ESO) logic by Fagin's theorem, i.e., a problem is in NP if and only if it is expressible by an ESO sentence.-- CSP is a subset of Feder and Vardi's logic, Monotone Monadic Strict NP without inequalities (MMSNP), and for every MMSNP sentence there exists a P-time equivalent CSP problem. This implies that ESO does not have a dichotomy as well as NP, and that MMSNP has a dichotomy as well as CSP. The main objective of this thesis is to study subsets of NP that strictly contain CSP or MMSNP with respect to the dichotomy existence.Feder and Vardi proved that if we omit one of the three properties that define MMSNP, namely being monotone, monadic or omitting inequalities, then the resulting logic does not have a dichotomy. As their proofs remain sketchy at times, we revisit these results and provide detailed proofs. Guarded Monotone Strict NP (GMSNP) is a known extension of MMSNP that is obtained by relaxing the "monadic" restriction of MMSNP. We define similarly a new logic that is called MMSNP with Guarded inequalities, relaxing the restriction of being "without inequalities". We prove that it is strictly more expressive than MMSNP and that it also has a dichotomy.There is a logic MMSNP₂ that extends MMSNP in the same way as MSO₂ extends Monadic Second-Order (MSO) logic. It is known that MMSNP₂ is a fragment of GMSNP and that these two classes either both have a dichotomy or both have not. We revisit this result and strengthen it by proving that, with respect to having a dichotomy, without loss of generality, one can consider only MMSNP₂ problems over one-element signatures, instead of GMSNP problems over arbitrary finite signatures.We seek to prove the existence of a dichotomy for MMSNP₂ by finding, for every MMSNP₂ problem, a P-time equivalent MMSNP problem. We face some obstacles to build such an equivalence. However, if we allow MMSNP sentences to consist of countably many negated conjuncts, then we prove that such an equivalence exists. Moreover, the corresponding infinite MMSNP sentence has a property of being "regular". This regular property means that, in some sense, this sentence is still finite. It is known that regular MMSNP problems can be expressed by CSP on omega-categorical templates. Also, there is an algebraic dichotomy characterisation for omega-categorical CSPs that describe MMSNP problems. If one manages to extend this algebraic characterisation onto regular MMSNP, then our result would provide an algebraic dichotomy for MMSNP₂.Another potential way to prove the existence of a dichotomy for MMSNP₂ is to mimic the proof of Feder and Vardi for MMSNP. That is, by finding a P-time equivalent CSP problem. The most difficult part there is to reduce a given input structure to a structure of sufficiently large girth. For MMSNP and CSP, it is done using expanders, i.e., structures, where the distribution of tuples is close to a uniform distribution. We study this approach with respect to MMSNP₂ and point out the main obstacles. (...)

We study the leading order asymptotics of a Random Matrix theory partition function related to colored triangulations. This partition function comes from a three Hermitian matrix model that has been introduced in the physics literature. We provide a detailed and precise description of the combinatorial objects that the partition function counts that has not appeared previously in the literature. We also provide a general framework for studying the leading order asymptotics of an N dimensional integral that one encounters studying the partition function of colored triangulations. The results are obtained by generalizing well know results for integrals coming from Hermitian matrix models with only one matrix that give the leading order asymptiotics in terms of a finite dimensional variational problem. We apply these results to the partition function for colored triangulations to show that the minimizing density of the variational problem is unique, and agrees with the one proposed in the physics literature. This provides the first complete mathematically rigorous description of the leading order asymptotics of this matrix model for colored triangulations.

The major goal for physicochemical screening of pharmaceuticals is to predict human drug absorption, distribution, elimination, excretion and toxicity. These are all dependent on the lipophilicity of the drug, which is expressed as a partition coefficient i.e. a measure of a drug’s preference for the lipophilic or hydrophilic phases. The most common method of determining a partition coefficient is the shake flask method using octanol and water as partitioning media. However, this system has many limitations when modeling the interaction of ionised compounds with membranes, therefore, unreliable partitioning data for many solutes has been reported. In addition to these concerns, the procedure is tedious and time consuming and requires a high level of solute and solvent purity. Micellar liquid chromatography (MLC) has been proposed as an alternative technique for measuring partition coefficients utilising surfactant aggregates, known as micelles. This thesis investigates the application of MLC in determining micelle-water partition coefficients (logPMW) of pharmaceutical compounds of varying physicochemical properties. The effect of mobile phase pH and column temperature on the partitioning of compounds was evaluated. Results revealed that partitioning of drugs solely into the micellar core was influenced by the interaction of charged and neutral species with the surface of the micelle. Furthermore, the pH of the mobile phase significantly influenced the partitioning behaviour and a good correlation of logPMW was observed with calculated distribution coefficient (logD) values. More interestingly, a significant change in partitioning was observed near the dissociation constant of each drug indicating an influence of ionised species on the association with the micelle and retention on the stationary phase. Elevated column temperatures confirmed partitioning of drugs considered in this study was enthalpically driven with a small change in the entropy of the system because of the change in the nature of hydrogen bonding. Finally, a quantitative structure property relationship was developed to evaluate biological relevance in terms of predicting skin permeability of the newly developed partition coefficient values. This study provides a better surrogate for predicting skin permeability based on an easy, fast and cheap experimental methodology, and the method holds the predictive capability for a wider population of drugs. In summary, it can be concluded that MLC has the ability to generate partition coefficient values in a shorter time with higher accuracy, and has the potential to replace the octanol-water system for pharmaceutical compounds.

The interplay between spectrum and structure of graphs is the recurring theme of the three more or less independent chapters of this thesis. The first chapter provides a method to relate the eigensolutions of two matrices, one being the principal submatrix of the other, via an arbitrary annihilating polynomial. This is extended to lambda-matrices and to matrices the entries of which are rational functions in one variable. The extension may be interpreted as a possible generalization of other known techniques which aim at reducing the size of a matrix while preserving the spectral information. Several aspects of an application in order to reduce the computational costs of ordinary eigenvalue problems are discussed. The second chapter considers the straightforward extension of the well known concept of equitable partitions to weighted graphs, i.e. complex matrices. It provides a method to divide the eigenproblem into smaller parts corresponding to the front divisor and its complementary factor in an easy and stable way with complexity which is only quadratic in matrix size. The exploitation of several equitable partitions ordered by refinement is discussed and a suggestion is made that preserves hermiticity if present. Some generalizations of equitable partitions are considered and a basic procedure for finding an equitable partition of complex matrices is given. The third chapter deals with isospectral and unitary equivalent graphs. It introduces a construction for unitary equivalent graphs which contains the well known GM-switching as a special case. It also considers an algebra of graph matrices generated by the adjacency matrix that corresponds to the 1-dimensional Weisfeiler-Lehman stabilizer in a way that mimics the correspondence of the coherent closure and the 2-dimensional Weisfeiler-Lehman stabilizer. The algebra contains the degree matrix, the (combinatorial, signless and normalized) Laplacian and the Seidel matrix. An easy construction produces graph pairs that are simultaneously unitary equivalent w.r.t. that algebra.

The research work presented in this thesis concern to the stability analysis of linear time-delay systems with low-order controllers. This thesis is divided into three parts.The first part of the thesis focus on the study of linear SISO (single-input/single-output) systems with input/output delays, where the feedback loop is closed with a controller of PID-type. Inspired by the geometrical approach developed by Gu et al. we propose an analytical method to find the stability regions of all stabilizing controllers of PID-type for the time-delay system. Based on this same approach, we propose an algorithm to calculate the degree of fragility of a given controller of PID- type (PI, PD and PID).The second part of the thesis focuses on the stability analysis of linear systems under an NCS (Networked System Control) based approach. More precisely, we first focus in the stabilization problem by taking into account the induced network delays and the effects induced by the sampling period. To carry out such an analysis we have adopted an eigenvalue perturbation-based approach.Finally, in the third part of the thesis we tackle certain problems concerning to the behavior of the zeros of a certain class of sampled-data SISO systems. More precisely, given a continuous-time system, we obtain the sampling intervals guaranteeing the invariance of the number of unstable zeros in each interval. To perform such an analysis, we adopt an eigenvalue perturbation-based approach.

A semigroup is simply a set with an associative binary operation; computational semigroup theory is the branch of mathematics concerned with developing techniques for computing with semigroups, as well as investigating semigroups with the help of computers. This thesis explores both sides of computational semigroup theory, across several topics, especially in the finite case. The central focus of this thesis is computing and describing maximal subsemigroups of finite semigroups. A maximal subsemigroup of a semigroup is a proper subsemigroup that is contained in no other proper subsemigroup. We present novel and useful algorithms for computing the maximal subsemigroups of an arbitrary finite semigroup, building on the paper of Graham, Graham, and Rhodes from 1968. In certain cases, the algorithms reduce to computing maximal subgroups of finite groups, and analysing graphs that capture information about the regular I-classes of a semigroup. We use the framework underpinning these algorithms to describe the maximal subsemigroups of many families of finite transformation and diagram monoids. This reproduces and greatly extends a large amount of existing work in the literature, and allows us to easily see the common features between these maximal subsemigroups. This thesis is also concerned with direct products of semigroups, and with a special class of semigroups known as Rees 0-matrix semigroups. We extend known results concerning the generating sets of direct products of semigroups; in doing so, we propose techniques for computing relatively small generating sets for certain kinds of direct products. Additionally, we characterise several features of Rees 0-matrix semigroups in terms of their underlying semigroups and matrices, such as their Green's relations and generating sets, and whether they are inverse. In doing so, we suggest new methods for computing Rees 0-matrix semigroups.

Etude theorique des divers aspects des phenomenes critiques bidimensionnels par des methodes essentiellement analytiques: transformation en modeles de champ libre, exploitation de la symetrie conforme. Determination des valeurs exactes d'exposants critiques, et des expressions des fonctions de partition et des fonctions de correlation. Test d'un certain nombre de resultats par des calculs de matrice de transfert ou des simulations de monte carlo

This thesis investigates the representations of the symmetric group on the homology of the dual matroid of a complete graph. These representations arise as follows: with each graph we can associate a matroid, by taking the set of edges of the graph as ground set and the edge sets of simple cycles as the circuits of the matroid. We focus on the dual of the matroid of the complete graph. We calculate the homology of the simplicial complex L associated with this matroid. Permuting the vertices of the complete graph induces a permutation on the edge set which is a vertex map of the simplicial complex. This vertex map sends independents to independents, thus inducing a simplicial map from the polytope of L to itself, hence on the homology spaces of L. This defines a representation of the symmetric group on the homology Hi(L,C). We show that the above representation is induced from a primitive representation of the cyclic subgroup of order n.

碩士
國立臺灣科技大學
管理學院MBA
98
This paper employs three block-modeling and partitioning methodologies: matrix joining process, CONCOR and permutation with partition through matrix joining process, and CONCOR and permutation with partition based on sign change for partitioning a virtual community into sub-communities. We also design a robust check to measure and compare the performance of those three methods in terms of error rates. Our empirical results reveal that (1) CONCOR plus permutation would favor the situation where there are only two subgroups. (2) The variable percentage of missing is not a determinant for error rates. (3) The error rates are not much affected by matrix size. (4) Maximizing X_C^2 value as the criterion function for matrix joining process might be too stringent since it considers both ties sent to and received from every pair of actors in the evaluated subgroup.

碩士
國立臺灣科技大學
管理學院MBA
98
This paper employs three block-modeling and partitioning methodologies: matrix joining process, CONCOR and permutation with partition through matrix joining process, and CONCOR and permutation with partition based on sign change for partitioning a virtual community into sub-communities. We also design a robust check to measure and compare the performance of those three methods in terms of error rates. Our empirical results reveal that (1) CONCOR plus permutation would favor the situation where there are only two subgroups. (2) The variable percentage of missing is not a determinant for error rates. (3) The error rates are not much affected by matrix size. (4) Maximizing X_C^2 value as the criterion function for matrix joining process might be too stringent since it considers both ties sent to and received from every pair of actors in the evaluated subgroup.

Chromatography is one of the most studied methods, due to its simplicity, versatility and high reproducibility, to separate and purify molecules that can have therapeutic, industrial and biotechnological interest. In recent years, the development of new chromatographic matrices has been continuously increased in order to afford rapid and efficient separations and decrease the use of resources. Gellan gum is a natural anionic exopolysaccharide and, in the presence of divalent cations, has the ability to form thermos-reversible strong gels resistant to temperature and extreme acidic conditions. In this work, it was proposed the preparation of gellan gum microbeads to be used as a stable chromatographic stationary phase. In order to produce the matrix, a low-cost water-in-oil emulsion technique was adopted. To obtain optimal conditions to the bead formulation, experimental design was applied, which allowed the optimization of the experimental conditions and to produce microbeads with the smallest diameter possible. Due to the negative charge of gellan gum, it was possible to study the interactions established with three model proteins (BSA, a-chymotrypsin and lysozyme) and with a therapeutic complex protein, SCOMT. In the model protein assays, MES buffer with pH 6.2 was used, which conferred negative charge to BSA and positive charge to a-chymotrypsin and lysozyme, due to its isoelectric points. Thus, BSA did not bind to the matrix while the other two proteins were retained to the gellan gum, being eluted with an increase of ionic strength. Regarding to prepurified SCOMT sample, a buffer with pH 4.0 was used under equilibrium conditions, conferring positive charge to the protein, thus, it also interacted with the column and was majorly eluted by pH manipulation (by changing the buffer pH to 6.4), allowing the elimination of some protein contaminants. Dynamic binding capacity assay of the gellan gum microbeads was made, in order to characterize this support as a novel chromatographic matrix. The values of DBC of the gellan gum stationary phase to 10 % and 50 % of breakthrough were 2.43 mg/mL and 4.73 mg/mL, respectively. These DBC values are satisfactory when compared to commercial resins used in affinity chromatography, taking into account that protein interaction only occurred at the gellan bead surface. These results indicated that gellan gum microbeads obtained by waterin- oil emulsion technique can be used as an innovative and promising chromatographic support due to its gelling ability and versatility to interact with different biomolecules.
A cromatografia é um dos métodos mais usados para a separação e purificação de diferentes biomoléculas terapêuticas. É uma técnica que tem sido muito explorada, ao longo dos últimos anos, nas áreas da indústria farmacêutica e biotecnológica para a obtenção de proteínas e ácidos nucleicos com elevado grau de pureza. Os suportes cromatográficos têm sido alvo de grandes desenvolvimentos a fim de encontrar uma matriz que reúna as características ideais como porosidade, estabilidade química e física, elevada eficiência de transferência de massa, baixo custo de produção associado, biocompatibilidade, hidrofilicidade, boa capacidade de manutenção do fluxo e capacidade de reutilização após vários ensaios cromatográficos. A goma de gelana é um polímero polissacárido natural linear que tem uma vasta gama de aplicações em diversas áreas, desde a indústria alimentar (agente espessante e gelificante), cosmética (loções, cremes, maquilhagem, etc), indústria biotecnológica (como um substituinte do agar), indústria farmacêutica (sistema de entrega direcionada de fármacos e microencapsulações) e medicina (construção de “scaffolds” para regeneração de tecidos). Devido às suas propriedades como versatilidade, biocompatibilidade, biodegradabilidade e elevada estabilidade, a gelana tem atraído grande interesse por parte do mercado e tem ganho crescente importância entre as diferentes indústrias. A gelana possui a capacidade de, em determinadas condições e na presença de catiões divalentes, sofrer uma alteração conformacional formando uma forte rede tridimensional devido às interações entre a gelana, os iões e as moléculas de água dando origem a um gel termoreversível, resistente a altas temperaturas e a extremas condições ácidas. Tendo em conta as características apresentadas, o presente trabalho teve como objetivo otimizar a formulação de uma matriz cromatográfica pela técnica de emulsão água-em-óleo a fim de ser usada em diversos processos cromatográficos. A inovação deste projeto tem como base a produção de microesferas através de um método de emulsão água-em-óleo, usando dois líquidos imiscíveis, a água e um óleo alimentar vegetal. Trata-se de uma técnica que não requer o uso de instrumentos complexos, é de fácil manuseamento, tem baixo custo associado e permite o fácil controlo dos parâmetros associados a este processo. A preparação das esferas foi feita de acordo com os seguintes parâmetros: concentração da goma de gelana (1 % - 2,5 %), velocidade de agitação (250 rpm – 750 rpm) e temperatura (20 ºC – 100 ºC). Considerando que todos estes fatores afetam a estabilidade e estrutura das microesferas, foram feitas várias formulações utilizando um desenho experimental, estratégia que permitiu obter as condições ótimas no sentido de se produzirem micropartículas com o mínimo diâmetro possível. As condições ótimas fornecidas pelo desenho foram 1,41 % de concentração de gelana, 749,47 rpm de velocidade de agitação e 99,20 ºC para obter microesferas com um tamanho de 277,08 µm. Para a formulação deste suporte cromatográfico, foi necessário a presença de catiões divalentes numa solução de reforço, para onde as esferas foram transferidas a fim de aumentar a estabilidade. O catião escolhido foi o bário, que devido ao seu grande tamanho iónico é considerado bastante eficaz na gelificação da gelana, pois contribui largamente para a formação da estável e rígida rede tridimensional. Além disso, o bário também permitiu a formação de esferas uniformes de tamanho reduzido com um pequeno ratio de inchaço. Para a validação do modelo fornecido pelo software do desenho experimental, foi necessário efetuar cinco réplicas de dois pontos previstos dados pelo programa. A visualização das esferas produzidas foi feita através de microscopia ótica que permitiu a medição dos diâmetros das micropartículas e através de microscopia eletrónica de varrimento, que tornou possível a análise estrutural das esferas num estado liofilizado, bem como a análise topográfica e morfológica da sua superfície. Além disso, esta última metodologia também permitiu avaliar a porosidade das esferas, de onde foi constatado a ausência de canais interiores, sugerindo que qualquer interação estabelecida com a matriz de goma de gelana se dê apenas na sua superfície. Aproveitando a natureza aniónica do polímero de gelana e o facto de não ter que ser funcionalizada com ligandos como os suportes cromatográficos comerciais, foi possível explorar diferentes interações com as proteínas modelo albumina sérica bovina (BSA), a-quimotripsina e lisozima, bem como com um extracto parcialmente purificado da proteína catecol-O-metiltransferase humana na isoforma solúvel (hSCOMT). Nos ensaios com as proteínas modelo, o tampão usado foi o MES a pH 6.2, o que conferiu carga negativa à BSA e cargas positivas à a-quimotripsina e lisozima devido aos seus pontos isoelétricos. Assim, a BSA não ligou à matriz tendo sido eluída após a injeção da amostra na coluna, ainda com a passagem do tampão sem sal, enquanto as outras duas proteínas interagiram devido à oposição de cargas, tendo sido eluídas com o aumento da força iónica. Quanto à amostra pré-purificada de SCOMT, foi usado um tampão de pH 4.0 nas condições de equilíbrio, o que conferiu carga positiva a esta proteína. Esta condição permitiu a interação desta proteina com a coluna, tendo sido maioritariamente eluída através da manipulação do pH (alterando o pH do tampão para 6.4), permitindo assim a eliminação de algumas proteínas contaminantes presentes na amostra. A fim de se caracterizar melhor esta nova matriz cromatográfica, foi determinada a capacidade dinâmica de ligação utilizando uma estratégia de saturação da coluna com uma solução de lisozima 0,5 mg/mL a 1 mL/min. Os valores obtidos na capacidade de ligação das microesferas de goma de gelana a 10 % e 50 % da curva de saturação foram 2,43 mg/mL e 4,73 mg/mL, respetivamente. Comparando com outras matrizes cromatográficas comerciais e tendo em conta que apenas a área de superfície das microesferas de gelana é funcional, estes valores estão dentro do esperado. Estes estudos permitiram concluir que a estabilidade da matriz cromatográfica de gelana foi incrementada com o desenvolvimento deste projecto de mestrado e que permitiu a interação com proteínas de várias naturezas, através de estratégias de eluição com manipulação de força iónica e pH. Em suma, os dados apresentados manifestam uma versatilidade da gelana em interagir com diferentes biomoléculas e, devido à sua capacidade de gelificação, foi possível a elaboração deste inovador e promissor suporte cromatográfico para a cromatografia de troca catiónica, a partir de microesferas da goma de gelana.

The interplay between spectrum and structure of graphs is the recurring theme of the three more or less independent chapters of this thesis. The first chapter provides a method to relate the eigensolutions of two matrices, one being the principal submatrix of the other, via an arbitrary annihilating polynomial. This is extended to lambda-matrices and to matrices the entries of which are rational functions in one variable. The extension may be interpreted as a possible generalization of other known techniques which aim at reducing the size of a matrix while preserving the spectral information. Several aspects of an application in order to reduce the computational costs of ordinary eigenvalue problems are discussed. The second chapter considers the straightforward extension of the well known concept of equitable partitions to weighted graphs, i.e. complex matrices. It provides a method to divide the eigenproblem into smaller parts corresponding to the front divisor and its complementary factor in an easy and stable way with complexity which is only quadratic in matrix size. The exploitation of several equitable partitions ordered by refinement is discussed and a suggestion is made that preserves hermiticity if present. Some generalizations of equitable partitions are considered and a basic procedure for finding an equitable partition of complex matrices is given. The third chapter deals with isospectral and unitary equivalent graphs. It introduces a construction for unitary equivalent graphs which contains the well known GM-switching as a special case. It also considers an algebra of graph matrices generated by the adjacency matrix that corresponds to the 1-dimensional Weisfeiler-Lehman stabilizer in a way that mimics the correspondence of the coherent closure and the 2-dimensional Weisfeiler-Lehman stabilizer. The algebra contains the degree matrix, the (combinatorial, signless and normalized) Laplacian and the Seidel matrix. An easy construction produces graph pairs that are simultaneously unitary equivalent w.r.t. that algebra.