In this dissertation, we study the following two topics, i.e., the union closed set conjecture and the maximum edges cut in connected digraphs. The union-closed-set-conjecture-topic goes as follows. A finite family of finite sets is {\it union closed} if it contains the union of any two sets in it. Let $X_{\mathcal{F}}=\cup_{F\in\mathcal{F}}F$. A union closed family of sets is {\it separating} if for any two distinct elements in $\mathcal{F}$, there is a set in $\mathcal{F}$ containing one of them, but not the other and there does not exist an element which is contained in every set of it. Note that any union closed family $\mathcal{F}$ is a poset with set inclusion as the partial order relation. A separating union closed family $\mathcal{F}$ is {\it irreducible} ({\it normalized}) if $|X_{\mathcal{F}}|$ is the minimum (maximum, resp.) with respect to the poset structure of $\mathcal{F}$. In the part of dissertation related to this topic, we develop algorithms to transfer any given separating union closed family to a/an normalized/irreducible family without changing its poset structure. We also study properties of these two extremal union closed families in connection with the {\it Union Closed Sets Conjecture} of Frankl. Our result may lead to potential full proof of the union closed set conjecture and several other conjectures. The part of the dissertation related to the maximum edge cuts in connected digraphs goes as follows. In a given digraph $D$, a set $F$ of edges is defined to be a {\it directed cut} if there is a nontrivial partition $(X,Y)$ of $V(D)$ such that $F$ consists of all the directed edges from $X$ to $Y$. The maximum size of a directed cut in a given digraph $D$ is denoted by $\Lambda (D)$, and we let $\mathcal{D}(1,1)$ be the set of all digraphs $D$ such that $d^{+}(v)=1$ or $d^{-}(v)=1$ for every vertex $v$ in $D$. In this part of dissertation, we prove that $\Lambda (D) \geq \frac{3}{8}(|E(D)|-1)$ for any connected digraph $D\in\mathcal{D}(1,1)$, which provides a positive answer to a problem of Lehel, Maffray, and Preissmann. Additionally, we consider triangle-free digraphs in $\mathcal{D}(1,1)$ and answer their another question.

Thesis (MSc (Mathematics))--University of Stellenbosch, 2010. / ENGLISH ABSTRACT: Within lattice theory, an interesting question asked is whether a given abstract lattice may be represented concretely as subsets of a closure system on a topological space. This is true for boolean algebras, bounded distributive lattices and arbitrary bounded lattices. In particular, there are a multitude of ways to represent bounded lattices. We present some of these ideas, as well as an analysis of the differences between them. We further investigate the attempts that were made to extend the above representations to lattices endowed with operators, in particular the work done on bounded distributive lattices with operators. We then make a new contribution by extending this work to arbitrary bounded lattices with operators. We also show that the so-called sufficiency operator has a relational representation in the bounded lattice case. / AFRIKAANSE OPSOMMING: Binne die raamwerk van tralie teorie word die vraag soms gevra of ’n gegewe tralie konkreet veteenwoordig kan word as subversamelings van ’n afsluitingssisteem op ’n topologiese ruimte. Die voorgenoemde is waar vir, onder andere, boolse algebras, begrensde distributiewe tralies en algemene begrensde tralies. Daar is veral vir begrensde tralies menigte maniere om hul te verteenwoordig. Ons bied sommige van hierdie idees voor, asook ’n analiese van die verskille daarin teenwoordig. Verder ondersoek ons ook sommige van die maniere waarop tralies tesame met operatore verteenwoordig kan word. Ons sal spesiale aandag gee aan distributiewe tralies met operatore, soos gedoen in, met die idee om die voorgenoemde uit te brei na algemene begrensde tralies met operatore. Ons toon dan verder aan dat die sogenoemde voldoende operator ook ’n relasionele verteenwoordiging het in die begrensde tralie geval.

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This dissertation explores two themes central to the field of non-equilibrium statistical physics. The first is centered around the use of random walks, first-passage processes, and Brownian motion to model basic stochastic search processes found in biology and ecological systems. The second is centered around clustered networks: how clustering modifies the nature of transition in the appearance of various graph motifs and their use in modeling social networks. In the first part of this dissertation, we start by investigating properties of intermediate crossings of Brownian paths. We develop simple analytical tools to obtain probability distributions of intermediate crossing positions and intermediate crossing times of Brownian paths. We find that the distribution of intermediate crossing times can be unimodal or bimodal. Next, we develop analytical and numerical methods to solve a system of 𝑁 diffusive searchers which are reset to the origin at stochastic or periodic intervals. We obtain the optimal criteria to search for a fixed target in one, two and three dimensions. For these two systems, we also develop efficient ways to simulate Brownian paths, where the simulation kernel makes maximal use of first-passage ideas. Finally we develop a model to understand foraging in a resource-rich environment. Specifically, we investigate the role of greed on the lifetime of a diffusive forager. This lifetime shows non-monotonic dependence on greed in one and two dimensions, and surprisingly, a peak for negative greed in 1d. In the second part of this dissertation, we develop simple models to capture the non-tree-like (clustering) aspects of random networks that arise in the real world. By 'clustered networks', we specifically mean networks where the probability of links between neighbors of a node (i.e., 'friends of friends') is positive. We discuss three simple and related models. We find a series of transitions in the density of graph motifs such as triangles (3-cliques), 4-cliques etc as a function of the clustering probability. We also find that giant 3-cores emerge through first- or second-order, or even mixed transitions in clustered networks.

We present a controlled microscopic study of hole dynamics in both a gapped and a gapless quantum spin liquid. Our approach is complementary to previous phenomenological works on lightly doped quantum spin liquids as we introduce mobile holes into the ground state of the exactly solvable Kitaev honeycomb model. In the spatially anisotropic (Abelian) gapped phase of the model, we address the properties of a single hole [its internal degrees of freedom as well as its hopping properties], a pair of holes [their absolute and relative particle statistics as well as their interactions], and the collective state for a finite density of holes. Our main result is that the holes in the doped model possess internal degrees of freedom as they can bind the fractional excitations of the undoped model and that the resulting composite holes with different excitations bound are distinct fractional particles with fundamentally different single-particle properties and different experimental signatures in the multi-particle ground state at finite doping. For example, some hole types are free to hop in two dimensions, while others are confined to hop in one dimension only. Also, distinct hole types have different particle statistics and, in particular, some of them exhibit non-trivial (anyonic) relative statistics. At finite doping, the respective hopping dimensionalities manifest themselves in an electrical conductivity that is either approximately isotropic or extremely anisotropic. In the gapless phase of the model, we consider a single hole and address the possibility of a coherent quasiparticle description. Our main result is that a mobile hole has a finite quasiparticle weight which vanishes in the stationary limit. Although this result is obtained in terms of an approximate variational state, we argue that it is also applicable for the exact ground state of the doped model.

This thesis focuses on Ibram Lassaw's open-lattice works by first discussing works from 1945 to 1950 to outline the conceptual and formal themes that contributed to the later style. The open-lattice form presents a complicated interplay between geometric and biomorphic forms, heaviness and lightness, tangibility and remoteness, and openness--which creates a partial boundary whereby the viewer is able to visually penetrate the form, while still being removed bodily. This thesis attempts to root Lassaw's open-lattice works and his metallic accretion process within the Abstract Expressionism movement by comparing the similar bodily experience of viewing Lassaw's works to those of Jackson Pollock's, for example, with a focused attention on material characteristics. This embodied approach offers a new and highly-appropriate language by which to discuss Lassaw's textural open-lattice works. A video of Lassaw's sculpting process is included with this thesis as a supplemental file.

Tese (doutorado) - Universidade Federal de Santa Catarina, Centro Tecnológico, Programa de Pós-Graduação em Engenharia Mecânica, 2014 / Made available in DSpace on 2015-06-02T04:04:41Z (GMT). No. of bitstreams: 1 333800.pdf: 7783075 bytes, checksum: c6e9b3bfd105f63cfef6d87840c8f43c (MD5) Previous issue date: 2014 / Na ausência de excitação, neurônios permanecem em repouso. O estado de repouso é caracterizado pela existência de diferenças de potencial elétrico e concentrações iônicas através da membrana do axônio. No entanto, os mecanismos responsáveis e a magnitude dos fluxos durante o estabelecimento do estado de repouso, após uma perturbação externa, não estão ainda completamente elucidados. O estado de repouso não é um estado de equilíbrio, mas um estado dinâmico estacionário. Sabe-se da existência de proteínas, classificadas genericamente como bombas e canais, que transportam íons através da membrana modificando o potencial através da mesma. No entanto, há várias hipóteses contraditórias sobre a magnitude e relação entre os fluxos provocados por tais proteínas e sobre a importância de outros fatores, como, por exemplo, a existência de cargas fixas. Um dos motivos para a dificuldade em escolher as hipóteses que tornam a modelagem mais consistente reside nas escalas de espaço e tempo diminutas dos fenômenos envolvidos o que dificulta a obtenção de medições locais e controladas. Nesse sentido, o desenvolvimento de modelos consistentes permite a avaliação das características e dos efeitos das várias hipóteses e contribui para o estabelecimento de modelos mais confiáveis. Nesse trabalho, utiliza-se, o método lattice Boltzmann para tratar o transporte iônio e o campo elétrico nos meios intra e extra celulares nas vizinhanças da membrana celular de um axônio. Uma caracterização físico-química da região vizinha à membrana de um axônio é feita baseada nos dados experimentais existentes. De posse dessa caracterização, utiliza-se o método numérico para a simulação da eletrodifusão iônica de uma porção de axônio quando uma perturbação a partir do estado de repouso é introduzida. Várias hipóteses são usadas, as soluções numéricas são comparadas com soluções existentes na literatura e com medições e, com base nas simulações, conclui-se quais hipóteses são mais adequadas para se explicar a manutenção do repouso. Além disso, também se observa que são necessários mais experimentos para obter maior confiabilidade sobre os modelos constitutivos de fluxos através dos canais e, principalmente, das bombas.<br> / Abstract: In the absence of excitation, neurons remain in a state called the resting state. The resting state is characterized by differences in electrical potential and ionic concentrations across the axon membrane. However, there is still a lacking of a complete understanding of the local mechanisms responsible and the magnitudes of the fluxes involved during the changes towards the resting state after the neuron suffers an external disturbance. The resting state is most likely a dynamic state, notan equilibrium one. Known mechanisms rely on the action of proteins called generically pumps and channels. These biological devices selectively transport ionic charges across the membrane. Contradictory hypothesis on the magnitude and relations among the charge transfers and on the effect of other parameters, such as the existence of fixed charges, have been advanced. The main difficult in sorting out the validity of these hypothesis rests on the inherent difficult in obtaining local and controlled measurements. In this sense, a reliable and consistent theoretical formulation may come as an aid to evaluate the effects and features of each mechanism and underlying hypothesis. In this work, the Lattice Boltzmann method is used to simulate the transport of ions and the electrical field in the intra and extra cellular spaces in the surroundings of an axon membrane. A physical-chemical characterization of the system is made based on the experimental evidence available in the literature. This characterization is then used in the numerical simulation of the electro - diffusion problem in a section of an axon when a perturbation from the resting state is introduced. Many hypotheses are used and the numerical solutions are compared with solutions and measurements found in the literature. From the results, the validity of the different hypothesis is assessed. The need for additional measurements to increase the reliability of the models for the constitutive fluxes through channels and pumps is evidenced.

Orientador: Elso Drigo Filho / Banca: Carla Goldman / Banca: José Roberto Ruggiero / Resumo: Neste trabalho foi analisada uma rede unidimensional composta por sistemas massamola com um potencial de Rosen-Morse on site. Esse tipo de rede é usado para estudar propriedades termodinâmicas do DNA, particularmente sua desnaturação térmica. No contexto do presente trabalho, o potencial de Rosen-Morse simula as ligações de hidrogênio entre a dupla fita da molécula. A partir do gráfico do estiramento médio dos pares de base versus temperatura se observou a desnaturação térmica do sistema. Esse resultado mostra que é possível obter transição de fase com um potencial sem uma barreira infinita, porém assimétrico. Outro resultado obtido é a forma da curva, que mostrou uma transição ligeiramente mais abrupta em comparação com a curva de transição feita para o potencial de Morse on site usado no modelo original de Peyrard-Bishop. Esse comportamento é encontrado em modelos que buscam uma melhor aproximação entre o modelo e os resultados experimentais de desnaturação térmica para o DNA / Abstract: In this work, it was analyzed a one-dimensional lattice formed by mass-spring systems with an additional Rosen-Morse potential on site. This kind of lattice is used to study thermodynamic properties of DNA, in particular the thermal denaturation. In the context of this work, the Rosen-Morse potential simulates the hydrogen bounds between the double helix of the DNA. The graphic of the average base pairs stretching in function of temperature gives information about the thermal denaturation of the macromolecule. This result shows that it is possible to obtain phase transition using an asymmetric potential on site without an infinite barrier. The graphic also showed a sharp denaturation in comparison with the transition curve obtained when the Morse potential on site is used in the original Peyrard- Bishop model. This behavior improves the model / Mestre

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