Best Proximity Points of Generalized Semicyclic Impulsive Self-Mappings: Applications to Impulsive Differential and Difference Equations

This paper is devoted to the study of convergence properties of distances between points and the existence and uniqueness of best proximity and fixed points of the so-called semicyclic impulsive self-mappings on the union of a number of nonempty subsets in metric spaces. The convergences of distances between consecutive iterated points are studied in metric spaces, while those associated with convergence to best proximity points are set in uniformly convex Banach spaces which are simultaneously complete metric spaces. The concept of semicyclic self-mappings generalizes the well-known one of cyclic ones in the sense that the iterated sequences built through such mappings are allowed to have images located in the same subset as their pre-image. The self-mappings under study might be in the most general case impulsive in the sense that they are composite mappings consisting of two self-mappings, and one of them is eventually discontinuous. Thus, the developed formalism can be applied to the study of stability of a class of impulsive differential equations and that of their discrete counterparts. Some application examples to impulsive differential equations are also given.

Möbius-like groups of homeomorphisms of the circle

A group G → Homeo_+(S^1) is a Möbius-like group if every element of G is conjugate in Homeo(S^1) to a Mobius transformation. Our main result is: given a Mobus like like group G which has at least one global fixed point, G is conjugate in Homeo(S^1) to a Möbius group if and only if the limit set of G is all of S^1 . Moreover, we prove that if the limit set of G is not SI, then after identifying some closed subintervals of S^1 to points, the induced action of G is conjugate to an action of a Möbius group.

We also show that the above result does not hold in the case when G has no global fixed points. Namely, we construct examples of Möbius-like groups with limit set equal to S^1, but these groups cannot be conjugated to Möbius groups.

The interplay of localization and interactions in quantum many-body systems

Disorder and interactions both play crucial roles in quantum transport. Decades ago, Mott showed that electron-electron interactions can lead to insulating behavior in materials that conventional band theory predicts to be conducting. Soon thereafter, Anderson demonstrated that disorder can localize a quantum particle through the wave interference phenomenon of Anderson localization. Although interactions and disorder both separately induce insulating behavior, the interplay of these two ingredients is subtle and often leads to surprising behavior at the periphery of our current understanding. Modern experiments probe these phenomena in a variety of contexts (e.g. disordered superconductors, cold atoms, photonic waveguides, etc.); thus, theoretical and numerical advancements are urgently needed. In this thesis, we report progress on understanding two contexts in which the interplay of disorder and interactions is especially important.

The first is the so-called “dirty” or random boson problem. In the past decade, a strong-disorder renormalization group (SDRG) treatment by Altman, Kafri, Polkovnikov, and Refael has raised the possibility of a new unstable fixed point governing the superfluid-insulator transition in the one-dimensional dirty boson problem. This new critical behavior may take over from the weak-disorder criticality of Giamarchi and Schulz when disorder is sufficiently strong. We analytically determine the scaling of the superfluid susceptibility at the strong-disorder fixed point and connect our analysis to recent Monte Carlo simulations by Hrahsheh and Vojta. We then shift our attention to two dimensions and use a numerical implementation of the SDRG to locate the fixed point governing the superfluid-insulator transition there. We identify several universal properties of this transition, which are fully independent of the microscopic features of the disorder.

The second focus of this thesis is the interplay of localization and interactions in systems with high energy density (i.e., far from the usual low energy limit of condensed matter physics). Recent theoretical and numerical work indicates that localization can survive in this regime, provided that interactions are sufficiently weak. Stronger interactions can destroy localization, leading to a so-called many-body localization transition. This dynamical phase transition is relevant to questions of thermalization in isolated quantum systems: it separates a many-body localized phase, in which localization prevents transport and thermalization, from a conducting (“ergodic”) phase in which the usual assumptions of quantum statistical mechanics hold. Here, we present evidence that many-body localization also occurs in quasiperiodic systems that lack true disorder.

Branes and supersymmetric quantum field theories

Since the discovery of D-branes as non-perturbative, dynamic objects in string theory, various configurations of branes in type IIA/B string theory and M-theory have been considered to study their low-energy dynamics described by supersymmetric quantum field theories.

One example of such a construction is based on the description of Seiberg-Witten curves of four-dimensional N = 2 supersymmetric gauge theories as branes in type IIA string theory and M-theory. This enables us to study the gauge theories in strongly-coupled regimes. Spectral networks are another tool for utilizing branes to study non-perturbative regimes of two- and four-dimensional supersymmetric theories. Using spectral networks of a Seiberg-Witten theory we can find its BPS spectrum, which is protected from quantum corrections by supersymmetry, and also the BPS spectrum of a related two-dimensional N = (2,2) theory whose (twisted) superpotential is determined by the Seiberg-Witten curve. When we don’t know the perturbative description of such a theory, its spectrum obtained via spectral networks is a useful piece of information. In this thesis we illustrate these ideas with examples of the use of Seiberg-Witten curves and spectral networks to understand various two- and four-dimensional supersymmetric theories.

First, we examine how the geometry of a Seiberg-Witten curve serves as a useful tool for identifying various limits of the parameters of the Seiberg-Witten theory, including Argyres-Seiberg duality and Argyres-Douglas fixed points. Next, we consider the low-energy limit of a two-dimensional N = (2, 2) supersymmetric theory from an M-theory brane configuration whose (twisted) superpotential is determined by the geometry of the branes. We show that, when the two-dimensional theory flows to its infra-red fixed point, particular cases realize Kazama-Suzuki coset models. We also study the BPS spectrum of an Argyres-Douglas type superconformal field theory on the Coulomb branch by using its spectral networks. We provide strong evidence of the equivalence of superconformal field theories from different string-theoretic constructions by comparing their BPS spectra.

Random propagation in complex systems : nonlinear matrix recursions and epidemic spread

This dissertation studies long-term behavior of random Riccati recursions and mathematical epidemic model. Riccati recursions are derived from Kalman filtering. The error covariance matrix of Kalman filtering satisfies Riccati recursions. Convergence condition of time-invariant Riccati recursions are well-studied by researchers. We focus on time-varying case, and assume that regressor matrix is random and identical and independently distributed according to given distribution whose probability distribution function is continuous, supported on whole space, and decaying faster than any polynomial. We study the geometric convergence of the probability distribution. We also study the global dynamics of the epidemic spread over complex networks for various models. For instance, in the discrete-time Markov chain model, each node is either healthy or infected at any given time. In this setting, the number of the state increases exponentially as the size of the network increases. The Markov chain has a unique stationary distribution where all the nodes are healthy with probability 1. Since the probability distribution of Markov chain defined on finite state converges to the stationary distribution, this Markov chain model concludes that epidemic disease dies out after long enough time. To analyze the Markov chain model, we study nonlinear epidemic model whose state at any given time is the vector obtained from the marginal probability of infection of each node in the network at that time. Convergence to the origin in the epidemic map implies the extinction of epidemics. The nonlinear model is upper-bounded by linearizing the model at the origin. As a result, the origin is the globally stable unique fixed point of the nonlinear model if the linear upper bound is stable. The nonlinear model has a second fixed point when the linear upper bound is unstable. We work on stability analysis of the second fixed point for both discrete-time and continuous-time models. Returning back to the Markov chain model, we claim that the stability of linear upper bound for nonlinear model is strongly related with the extinction time of the Markov chain. We show that stable linear upper bound is sufficient condition of fast extinction and the probability of survival is bounded by nonlinear epidemic map.

Properties of convergence of a class of iterative processes generated by sequences of self-mappings with applications to switched dynamic systems

This article investigates the convergence properties of iterative processes involving sequences of self-mappings of metric or Banach spaces. Such sequences are built from a set of primary self-mappings which are either expansive or non-expansive self-mappings and some of the non-expansive ones can be contractive including the case of strict contractions. The sequences are built subject to switching laws which select each active self-mapping on a certain activation interval in such a way that essential properties of boundedness and convergence of distances and iterated sequences are guaranteed. Applications to the important problem of stability of dynamic switched systems are also given.

Adaptive image sequence coding based on global and local compensability analysis

Our study of a novel technique for adaptive image sequence coding is reported. The number of reference frames and the intervals between them are adjusted to improve the temporal compensability of the input video. The bits are distributed more efficiently on different frame types according to temporal and spatial complexity of the image scene. Experimental results show that this dynamic group-of-picture (GOP) structure coding scheme is not only feasible but also better than the conventional fixed GOP method in terms of perceptual quality and SNR. (C) 1996 Society of Photo-Optical Instrumentation Engineers.

Results on proximal and generalized weak proximal contractions including the case of iteration-dependent range sets

This paper presents some further results on proximal and asymptotic proximal contractions and on a class of generalized weak proximal contractions in metric spaces. The generalizations are stated for non-self-mappings of the forms for and , or , subject to and , such that converges uniformly to T, and the distances are iteration-dependent, where , , and are non-empty subsets of X, for , where is a metric space, provided that the set-theoretic limit of the sequences of closed sets and exist as and that the countable infinite unions of the closed sets are closed. The convergence of the sequences in the domain and the image sets of the non-self-mapping, as well as the existence and uniqueness of the best proximity points, are also investigated if the metric space is complete. Two application examples are also given, being concerned, respectively, with the solutions through pseudo-inverses of both compatible and incompatible linear algebraic systems and with the parametrical

Self-formation of quasiperiodic void structure in CaF2 induced by femtosecond laser irradiation

We report the self-formation of quasiperiodic void structure with the length of several hundred micrometers inside the CaF2 crystal. The quasiperiodical voids along the propagation direction of the laser beam were formed spontaneously after the irradiation of a single femtosecond laser beam which was focused at a fixed point inside the crystal sample. The length of the void array varied with the focal depth beneath the sample surface. The possible mechanism of the self-formed void structure was discussed. (c) 2007 American Institute of Physics.

单束飞秒激光在氧化铝晶体中诱导自组装微米点阵

报道了单束飞秒激光在氧化铝晶体中诱导自组装微米点阵的有关研究进展。当单束飞秒激光被透镜聚焦到氧化铝晶体的内部某固定点， 微米点阵就在聚焦点的下方自动生成。我们发现能否产生自组装点阵和光束的聚焦点距离样品表面的深度有关。通过比较在氧化铝晶体和氟化钙晶体中能够产生点阵的深度，我们发现在氧化铝中较浅的位置即能诱导出点阵，而在氟化钙中则要求深度较深。具体的机理在文中进行了讨论。

Self-organized microvoid array perpendicular to the femtosecond laser beam in CaF2 crystals

Microvoid arrays were self-organized when femtosecond laser beam was tightly focused at a fixed point inside CaF2 crystal sample. Except void array grown below the focal point which had been reported before, we found another void array grown vertical to the laser propagation direction. This result has potential application in the fabrication of integrated micro-optic elements and photonic crystals. The possible mechanism of the phenomenon was proposed and verified experimentally.

Review of NRA tracking projects on salmon, Salmo salar L., and sea trout, Salmo trutta L

The report briefly outlines the programme of the National Rivers Authority (NRA), placing the Fisheries programme in the context of the work of the NRA as a whole, and viewing the tracking work against the broader requirements of the NRA Fisheries research programme. All regions of England and Wales are considered. Two techniques currently exist for studying the detailed timing and extent of movements of adult salmon: tracking of individually identifiable fish, and counting the numbers of fish moving past a fixed point in the river. The development of tracking techniques and the integrated use of tracking and fish counters is briefly reviewed in Section 3. Further details of these techniques are given in Appendices. Section 4 summarises and assesses completed and current NRA tracking studies. Section 5 discusses the scientific content of these studies in relation to similar work carried out elsewhere in the UK. The NRA programme of tracking studies is evaluated in Section 6. Section 7 discusses future fisheries projects and Section 8 details the future development of tracking techniques. Finally, recommendations arising out of this review are summarised in Section 9.

Description of NRA tracking studies: tracking techniques and fish counter technology

The study began on the 7th January 1991 and was completed in June 1991. Two reports have been produced. This report published as R&D Note 33 describes NRA tracking studies, tracking techniques and fish counter technology. The second report published as R&D Note 34 evaluates NRA tracking studies and recommends future research. The latter will be used solely for NRA management purposes. This report briefly outlines the programme of the NRA, placing the Fisheries programme in the context of the work of the NRA as a whole, and viewing the tracking work against the broader requirements of the NRA Fisheries research programme. Two techniques currently exist for studying the detailed timing and extent of movements of adult salmon: tracking of individually identifiable fish, and counting the numbers of fish moving past a fixed point in the river. Further details of these techniques and their development are given in Sections 2 and 3. Section 4 summarises and assesses completed and current NRA tracking Studies. Complete project descriptions for the studies are contained in Appendix A. Section 5 discusses the scientific content of these studies in relation to similar work carried out elsewhere in the UK. Section 6 details the future development of tracking techniques. Tracking work on migratory salmonids has tended to concentrate largely upon the movements of adult salmon. Much of this report will therefore be concerned with salmon tracking studies. NRA studies involving sea trout are referred to where appropriate. The methodological problems of sea trout tracking studies are summarised in Section 2.1.3.

Ingestão de nutrientes segundo variáveis demográficas e socioeconômicas em adultos brasileiros: Inquérito Nacional de Alimentação - Brasil, 2008-2009

A tese descreve a ingestão de nutrientes segundo variáveis demográficas e socioeconômicas em adultos brasileiros, com base nos dados da primeira avaliação nacional do consumo alimentar individual, o Inquérito Nacional de Alimentação (INA), realizado entre 2008 e 2009. Um total de 34.003 indivíduos com pelo menos 10 anos de idade participaram do estudo. O presente estudo incluiu 21.003 indivíduos adultos, de 20 a 59 anos de idade, com exceção das mulheres gestantes e lactantes (n=1.065). O consumo alimentar individual foi estimado utilizando dois dias de registros alimentares não consecutivos. O consumo usual de nutrientes foi estimado pelo método do National Cancer Institute que permitiu a correção da variabilidade intraindividual. As prevalências de ingestão inadequada de nutrientes foram estimadas segundo o sexo e faixas etárias utilizando o método da necessidade média estimada como ponte de corte. A inadequação de sódio foi avaliada pelo consumo acima do nível de ingestão máximo tolerável. Os resultados são apresentados na forma de dois artigos. No primeiro artigo, estimaram-se as prevalências de inadequação segundo as cinco grandes regiões (Norte, Nordeste, Sudeste, Sul e Centro-Oeste) e a situação do domicílio (urbano e rural). Observaram-se prevalências de inadequação maiores ou iguais a 70% para cálcio entre os homens e magnésio, vitamina A, sódio em ambos os sexos. Prevalências maiores ou iguais a 90% foram encontradas para cálcio entre as mulheres e vitaminas D e E em ambos os sexos. No geral, os grupos com maior risco de inadequação de micronutrientes foram as mulheres e os que residem na área rural e na região Nordeste. No segundo artigo, estimaram-se as prevalências de inadequação do consumo segundo renda e escolaridade. A renda foi caracterizada pela renda mensal familiar per capita e a escolaridade definida pelo número de anos completos de estudo. Ambas variáveis foram categorizadas em quartis. Modelos de regressão linear simples e mutuamente ajustados foram estimados para verificar a associação independente entre o consumo de nutrientes e as variáveis socioeconômicas. Foram testadas as interações entre renda e escolaridade. Verificou-se que a inadequação da maioria dos nutrientes diminuiu com o aumento da renda e escolaridade; porém, o consumo excessivo de gordura saturada e o baixo consumo de fibra aumentaram com ambas variáveis. Grande parte dos nutrientes foi independentemente associada à renda e escolaridade, contudo, o consumo de ferro, vitamina B12 e sódio entre mulheres foi associado somente com a educação. Observou-se interação entre renda e escolaridade na associação com o consumo de sódio em homens, fósforo em mulheres e cálcio em ambos os sexos. Os achados indicam que melhorar a educação é um passo importante na melhoria do consumo de nutrientes no Brasil, além da necessidade de formulação de estratégias econômicas que permitam que indivíduos de baixa renda adotem uma dieta saudável. Nossos resultados mostram também um grande desafio das ações de saúde pública na área de nutrição, com importantes inadequações de consumo em toda população adulta brasileira e particularmente em grupos populacionais e regiões mais vulneráveis do país.

Formulação teórica dos fundamentos da otimização global topográfica com análise de desempenho e aplicações à estabilidade de fases de misturas termodinâmicas

Métodos de otimização que utilizam condições de otimalidade de primeira e/ou segunda ordem são conhecidos por serem eficientes. Comumente, esses métodos iterativos são desenvolvidos e analisados à luz da análise matemática do espaço euclidiano n-dimensional, cuja natureza é de caráter local. Consequentemente, esses métodos levam a algoritmos iterativos que executam apenas as buscas locais. Assim, a aplicação de tais algoritmos para o cálculo de minimizadores globais de uma função não linear,especialmente não-convexas e multimodais, depende fortemente da localização dos pontos de partida. O método de Otimização Global Topográfico é um algoritmo de agrupamento, que utiliza uma abordagem baseada em conceitos elementares da teoria dos grafos, a fim de gerar bons pontos de partida para os métodos de busca local, a partir de pontos distribuídos de modo uniforme no interior da região viável. Este trabalho tem dois objetivos. O primeiro é realizar uma nova abordagem sobre método de Otimização Global Topográfica, onde, pela primeira vez, seus fundamentos são formalmente descritos e suas propriedades básicas são matematicamente comprovadas. Neste contexto, propõe-se uma fórmula semi-empírica para calcular o parâmetro chave deste algoritmo de agrupamento, e, usando um método robusto e eficiente de direções viáveis por pontos-interiores, estendemos o uso do método de Otimização Global Topográfica a problemas com restrições de desigualdade. O segundo objetivo é a aplicação deste método para a análise de estabilidade de fase em misturas termodinâmicas,o qual consiste em determinar se uma dada mistura se apresenta em uma ou mais fases. A solução deste problema de otimização global é necessária para o cálculo do equilíbrio de fases, que é um problema de grande importância em processos da engenharia, como, por exemplo, na separação por destilação, em processos de extração e simulação da recuperação terciária de petróleo, entre outros. Além disso, afim de ter uma avaliação inicial do potencial dessa técnica, primeiro vamos resolver 70 problemas testes, e então comparar o desempenho do método proposto aqui com o solver MIDACO, um poderoso software recentemente introduzido no campo da otimização global.