Dissertationen zum Thema „Interval“

O objetivo deste trabalho e determinar a solução de algumas equações de coeficientes intervalares. Este estudo utiliza uma Teoria das Aproximações Intervalares, a qual foi descrita por [ACI91]. Nesta teoria a igualdade para intervalos e substituída pela relação de aproximação . Esta substituição deve-se ao fato da igualdade utilizada na Teoria Clássica dos Intervalos para resolução de equações de coeficientes intervalares não apresentar uma solução satisfatória, visto que a solução encontrada não contem todas as soluções das equações reais que compõe a equação intervalar. Pela substituição da igualdade intervalar por uma relação de aproximação é possível determinar a solução de equações de coeficientes intervalares, de maneira que esta solução contenha todas as possíveis soluções das equações reais pertencentes a equação intervalar. Apresenta-se alguns conceitos básicos, bem como analisa-se algumas propriedades no espaço solução ( /(R), +, •, C, 1). São representadas graficamente diferentes tipos de funções neste espaço intervalar, com os objetivos de obtenção da imagem, caracterização da solução e identificação gráfica da região de solução (ótima e externa), para cada tipo de função. Como a representação de intervalos de /(R) esta determinada num semiplano de eixos X - X+, onde X - representa o extremo inferior de cada intervalo e X+ representa o extremo superior dos intervalos, apresenta-se o espaço intervalar estendido /(R). Neste espaço intervalar estão definidos os intervalos não-regulares, representados no outro semi-piano de eixos X - X+ Em /(R) serão apresentados alguns conceitos fundamentais, assim como operações aritméticas e algumas considerações referentes aos intervalos não-regulares. No espaço intervalar /(R) e possível resolver equações de coeficientes intervalares de maneira análoga a resolução de equações reais no espaço real, pois este espaço intervalar possui a estrutura semelhante a de um corpo. Com isto apresenta-se a solução de equações de coeficientes intervalares lineares, obtida diretamente, assim como determina-se a Formula de Bascara Intervalar para resolução da Equação Quadrática Intervalar. Para funções que possuem grau maior que 2 apresenta-se alguns métodos iterativos intervalares, tais como o Método de Newton Intervalar, o Método da Secante Intervalar e o Método híbrido Intervalar, que permitem a obtenção do intervalo solução para funções intervalares. Por fim apresenta-se alguns conceitos básicos no espaço intervalar matricial M„,„(/(R)), bem como apresenta-se alguns métodos diretos para resolução de sistemas de equações lineares intervalares.
The aim of this work is to determine the solution set of some Equations of Interval Coefficients. The study use a Theory of Interval Approximation. The begining of this theory was described by [ACI91]. In this theory the equality for intervals is replaced by an approximation relation. When we make use of that relation to solve interval equations, it's possible to obtain an optimal solution, i.e., to get an interval solution that contain all of real solutions of the real equations envolved in the interval equation. By using the equality of Classical Interval Theory for solving interval equations we can not get an optimal solution, that is, the interval solution in the most of equations not consider some real solutions of real equations that belong to the interval equation. We present some basic concepts and analyse some properties at the interval space (1(R), E, -a x , 1). Different kind of functions are showed in this space in order to obtain the range, the solution caracterization and the graphic identification of the optimal and external solution region, for each kind of function. The representation of intervals in /(R) is determined in a half plane of axes X - , X+, where X - represent the lower endpoint and X+ represent the upper endpoint of the intervals. The nonregular intervals are defined in /(R), which are determined in an other half plane. In this interval space are presenting some specific concepts, as well as arithmetical operations and some remarks about nonregular intervals. The interval space (1(R), +, •, C, Ex , 1) have a similar structure to a field, so it's possible to solve interval coefficients equations analogously as to solve real equations in the real space. We present the solution of linear interval equations and we determine an interval formula to solve square interval equation. We present some intervals iterated methods for functions that have degree greater than 2 that allow to get an interval solution of interval functions. Finally we show some basic concepts about the interval matrix space Af,„„(IR)) and present direct methods for the resolution of linear interval sistems.

Consideramos um modelo de regressão linear simples, em que tanto a variável resposta como a independente estão sujeitas a censura intervalar. Como motivação utilizamos um estudo em que o objetivo é avaliar a possibilidade de previsão dos resultados de um exame audiológico comportamental a partir dos resultados de um exame audiológico eletrofisiológico. Calculamos intervalos de previsão para a variável resposta, analisamos o comportamento dos estimadores de máxima verossimilhança obtidos sob o modelo proposto e comparamos seu desempenho com aquele de estimadores obtidos de um modelo de regressão linear simples usual, no qual a censura dos dados é desconsiderada.
We consider a simple linear regression model in which both variables are interval censored. To motivate the problem we use data from an audiometric study designed to evaluate the possibility of prediction of behavioral thresholds from physiological thresholds. We develop prediction intervals for the response variable, obtain the maximum likelihood estimators of the proposed model and compare their performance with that of estimators obtained under ordinary linear regression models.

O estudo dos métodos intervalares é importante para a resolução de sistemas de equações lineares, pois os métodos intervalares produzem resultados dentro de limites confiáveis (do intervalo solução) e provam a existência ou não existência de soluções, portanto produzem resultados confiáveis, o que os métodos pontuais podem não proporcionar. Outro aspecto a destacar é o campo de utilizando de sistemas de equações lineares em problemas das engenharias e outras ciências, o que mostra a aplicabilidade desses métodos e por conseguinte a necessidade de elaboração de ferramentas que possibilitem a implementação desses métodos intervalares. O objetivo deste trabalho não é a elaboração de novos métodos intervalares, mas sim o de realizar uma descrição e implementação de alguns dos métodos intervalares encontrados na bibliografia pesquisada. A versão intervalar dos métodos pontuais não é simples e o calculo por métodos intervalares pode ser dispendioso, uma vez que se está tratando com vetores e matrizes de intervalos. A implementação dos métodos intervalares são foi possível graças a existência de ferramentas, como o compilador Pascal-XSC, que incorpora as suas características aspectos importantes como a aritmética intervalar, a verificação automática do resultado, o produto escalar Ótimo e a aritmética de alta exatidão. Este trabalho é dividido em duas etapas. A primeira apresenta um estudo dos métodos intervalares para a resolução de sistemas de equações lineares. São caracterizadas as metodologias de desenvolvimento desses métodos. Metodologias estas, que foram divididas em três grupos de métodos: métodos intervalares baseados em operações algébricas intervalares ou métodos diretos, métodos intervalares baseados em refinamento ou métodos híbridos e métodos intervalares baseados em interacões. São definidas as características, os métodos que as compõe e a aplicabilidade desses métodos na resolução de sistemas de equações lineares. A segunda etapa é caracterizada pela elaboração dos algoritmos referentes aos métodos intervalares estudados e sua respectiva implementação, dando origem a uma biblioteca aplicativa intervalar para a resolução de sistemas de equações lineares, implementada no PC-486 e utilizando o compilador Pascal-XSC. Para este desenvolvimento foi realizado, previamente, um estudo sobre este compilador e sobre bibliotecas disponíveis que são utilizadas na implementação da biblioteca aplicativa intervalar. A biblioteca selintp é organizada em quatro módulos: o módulo dirint (referente aos métodos diretos); o modulo refint (referente aos métodos baseados em refinamento); o módulo itrint (referente aos métodos iterativos) e o modulo equalg (para sistemas de equações de ordem 1). Por fim, através daquela biblioteca foram realizadas comparações entre os resultados obtidos (resultados pontuais, intervalares, seqüenciais e vetoriais) a rim de se realizar uma analise de desempenho quantitativa (exatidão) e uma comparação entre os resultados obtidos. Esses resultados sendo comparados com os obtidos com a biblioteca biblioteca esta que esta sendo desenvolvida para o ambiente do supercomputador Cray Y-MP do CESUP/UFRGS, como parte do projeto de Aritmética Vetorial Intervalar do Grupo de Matemática Computacional da UFRGS.
The study of interval methods is important for resolution of linear equation systems, because such methods produce results into reliable bounds and prove the existence or not existence of solutions, therefore they produce reliable results that, the punctual methods can non present,save that there is an exhaustive analysis of errors. Another aspect to emphasize is the field of utilization of linear equation systems in engineering problems and other sciences, in which is showed the applicability of that methods and, consequently, the necessity of tools elaboration that make possible the implementation of that interval methods. The goal of this work is not the elaboration of new interval methods, but to accomplish a description and implementation of some interval methods found in the searched bibliography. The interval version of punctual methods is not simple, and the calculus by interval methods can be expensive, respecting is treats of vectors and matrices of intervals. The implementation of interval methods was only possible due to the existence of tools, as the Pascal-XSC compiler, which incorporates to their features, important aspects such as the interval arithmetic, the automatic verification of the result, the optimal scalar product and arithmetic of high accuracy. This work is divided in two stages. The first presents a study of the interval methods for resolution of linear equation systems, in which are characterized the methodologies of development of that methods. These methodologies were divided in three method groups: interval methods based in interval algebraic operations or direct methods, interval methods based in refinament or hybrid methods, and interval methods based in iterations, in which are determined the features, the methods that compose them, and the applicability of those methods in the resolution of linear equation systems. The second stage is characterized for the elaboration of the algorithms relating to the interval methods studied and their respective implementation, originating a interval applied library for resolution of linear equation systems, selintp, implemented in PC-486 and making use of Pascal-XSC compiler. For this development was previously accomplished a study about compiler and avaiable libraries that are used in the inplementation of the interval applied library. The library selintp is organized in four modules: the dirint module (regarding to the direct methods); the refint module (regarding to the methods based in refinament); the itrint module (regarding to the iterative methods) and equalg module(for equation systems of order 1). At last, throu gh this library, comparisons were developed among the results obtained (punctual, interval, sequential and vectorial results) in order to be accomplished an analysis of quantitative performance (accuracy) and a comparison among the results obtained with libselint a library, that is been developed for the Cray Y-MP supercomputer environment of CESUP/UFRGS, as part of the Interval Vectorial Arithmetic project of Group of Computational Mathematics of UFRGS.

Orientador: Valeriano Antunes de Oliveira
Resumo: Neste trabalho, primeiramente, serão apresentados problemas de otimização nos quais a função objetivo é de múltiplas variáveis e de valor intervalar e as restrições de desigualdade são dadas por funcionais clássicos, isto é, de valor real. Serão dadas as condições de otimalidade usando a E−diferenciabilidade e, depois, a gH−diferenciabilidade total das funções com valor intervalar de várias variáveis. As condições necessárias de otimalidade usando a gH−diferenciabilidade total são do tipo KKT e as suficientes são do tipo de convexidade generalizada. Em seguida, serão estabelecidos problemas de controle ótimo nos quais a funçãao objetivo também é com valor intervalar de múltiplas variáveis e as restrições estão na forma de desigualdades e igualdades clássicas. Serão fornecidas as condições de otimalidade usando o conceito de Lipschitz para funções intervalares de várias variáveis e, logo, a gH−diferenciabilidade total das funções com valor intervalar de várias variáveis. As condições necessárias de otimalidade, usando a gH−diferenciabilidade total, estão na forma do célebre Princípio do Máximo de Pontryagin, mas desta vez na versão intervalar.
Abstract: In this work, firstly, it will be presented optimization problems in which the objective function is interval−valued of multiple variables and the inequality constraints are given by classical functionals, that is, real−valued ones. It will be given the optimality conditions using the E−differentiability and then the total gH−differentiability of interval−valued functions of several variables. The necessary optimality conditions using the total gH−differentiability are of KKT−type and the sufficient ones are of generalized convexity type. Next, it will be established optimal control problems in which the objective function is also interval−valued of multiple variables and the constraints are in the form of classical inequalities and equalities. It will be furnished the optimality conditions using the Lipschitz concept for interval−valued functions of several variables and then the total gH−differentiability of interval−valued functions of several variables. The necessary optimality conditions using the total gH−differentiability is in the form of the celebrated local Pontryagin Maximum Principle, but this time in the intervalar version.
Doutor

We examine the problem of counting interval graphs. We answer the question posed by Hanlon, of whether the formal power series generating function of the number of interval graphs on n vertices has a positive radius of convergence. We have found that it is zero. We have obtained a lower bound and an upper bound on the number of interval graphs on n vertices. We also study the application of interval graphs to the dynamic storage allocation problem. Dynamic storage allocation has been shown to be NP-complete by Stockmeyer. Coloring interval graphs on-line has applications to dynamic storage allocation. The most colors used by Kierstead's algorithm is 3 ω -2, where ω is the size of the largest clique in the graph. We determine a lower bound on the colors used. One such lower bound is 2 ω -1.

Neste trabalho e apresentada uma nova abordagem para a representação gráfica de intervalos. Segundo esta abordagem é possível realizar a análise visual de intervalos a partir da associação entre propriedades geométricas do piano cartesiano e de conjuntos de intervalos representados como pontos desse piano. Esta nova abordagem possibilita a representação da interpretação dual de intervalos, assim como a analise visual de relacionamentos em (IR, <=) e (IR, C). Neste contexto, a representação gráfica do conjunto de intervalos degenerados, representado pela reta y = x, constitui um caso especial desta representação,"o. Por sua vez, a relação (IR, representada pelo semiplano superior a reta y = x, denotado piano IR. A interpretação visual de operações intervalares é obtida diretamente através da aplicação da representação gráfica proposta. Além disto, operandos e operadores podem ser estudados diretamente a partir desta representação. Foram desenvolvidos experimentos de analise visual de intervalos utilizando a abordagem proposta e resultados bastante promissores foram obtidos. Estes experimentos possibilitaram a identificação de novas propriedades de intervalos assim como interpretações não usuais para operações intervalares. Esta representação pode ser utilizada também para observar o comportamento de seqüências de intervalos gerados a partir de programas baseado na aplicação da aritmética intervalar. Nesta caso, pode ser observado como os intervalos desta seqüência variam com relação ao seu ponto médio e o raio, assim como a relação entre eles. Esta representação foi utilizada com sucesso para obter a solução geométrica da equação intervalar afim e efetuando sua validação. Finalmente, analisamos a contribuição efetiva deste trabalho no contexto da aritmética intervalar.
This thesis presents a framework enabling the visual analysis of intervals, obtained by mapping geometric properties of the cartesian plane into interval sets to obtain a graphical representation. This new approach makes possible a dual interval representation and the immediate visual analysis of several relationships in (IR, <=) and (IR, C). In this sense, the set of degenerated intervals is a special case of this approach as they are represented by the straight line y=x. In turn, the order relation in (IR, C) is represented through the half-plane above the straight line y = x, denoted IR plane. Applying this framework, the visual interpretation of most interval operations is obtained directly from the graphical representation of the operands and the operations being studied. On the other hand, some experiments on interval visual analysis were developed with good final results. Thus, new properties and unusual interpretations for known operations can be developed with rather small effort. Moreover, this representation can be easily embedded into a running algorithm, to observe convergence and behavior of interval iterations, as one can easily see how intervals change with respect to midpoint and radius, as well as with respect to each other. The validation of this new approach was carried through the geometric solution of linear interval equations. This result was analyzed in order to verify the effective contribution of this geometrical representation in the context of interval arithmetic.

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Coordena??o de Aperfei?oamento de Pessoal de N?vel Superior
In this work we use Interval Mathematics to establish interval counterparts for the main tools used in digital signal processing. More specifically, the approach developed here is oriented to signals, systems, sampling, quantization, coding and Fourier transforms. A detailed study for some interval arithmetics which handle with complex numbers is provided; they are: complex interval arithmetic (or rectangular), circular complex arithmetic, and interval arithmetic for polar sectors. This lead us to investigate some properties that are relevant for the development of a theory of interval digital signal processing. It is shown that the sets IR and R(C) endowed with any correct arithmetic is not an algebraic field, meaning that those sets do not behave like real and complex numbers. An alternative to the notion of interval complex width is also provided and the Kulisch- Miranker order is used in order to write complex numbers in the interval form enabling operations on endpoints. The use of interval signals and systems is possible thanks to the representation of complex values into floating point systems. That is, if a number x 2 R is not representable in a floating point system F then it is mapped to an interval [x;x], such that x is the largest number in F which is smaller than x and x is the smallest one in F which is greater than x. This interval representation is the starting point for definitions like interval signals and systems which take real or complex values. It provides the extension for notions like: causality, stability, time invariance, homogeneity, additivity and linearity to interval systems. The process of quantization is extended to its interval counterpart. Thereafter the interval versions for: quantization levels, quantization error and encoded signal are provided. It is shown that the interval levels of quantization represent complex quantization levels and the classical quantization error ranges over the interval quantization error. An estimation for the interval quantization error and an interval version for Z-transform (and hence Fourier transform) is provided. Finally, the results of an Matlab implementation is given
Neste trabalho utiliza-se a matem?tica intervalar para estabelecer os conceitos intervalares das principais ferramentas utilizadas em processamento digital de sinais. Mais especificamente, foram desenvolvidos aqui as abordagens intervalares para sinais, sistemas, amostragem, quantiza??o, codifica??o, transformada Z e transformada de Fourier. ? feito um estudo de algumas aritm?ticas que lidam com n?meros complexos sujeitos ? imprecis?es, tais como: aritm?tica complexa intervalar (ou retangular), aritm?tica complexa circular, aritm?tica setorial e aritm?tica intervalar polar. A partir da?, investiga-se algumas propriedades que ser?o relevantes para o desenvolvimento e aplica??o no processamento de sinais discretos intervalares. Mostra-se que nos conjuntos IR e R(C), seja qual for a aritm?tica correta adotada, n?o se tem um corpo, isto ?, os elementos desses conjuntos n?o se comportam como os n?meros reais ou complexos com suas aritm?ticas cl?ssicas e que isso ir? requerer uma avalia??o matem?tica dos conceitos necess?rios ? teoria de sinais e a rela??o desses com as aritm?ticas intervalares. Tamb?m tanto ? introduzido o conceito de amplitude intervalar complexa, como alternativa ? defini??o cl?ssica quanto utiliza-se a ordem de Kulisch-Miranker para n?meros complexos afim de que se escreva n?meros complexos intervalares na forma de intervalos, o que torna poss?vel as opera??es atrav?s dos extremos. Essa rela??o ? utilizada em propriedades de somas de intervalos de n?meros complexos. O uso de sinais e sistemas intervalares foi motivado pela representa??o intervalar num sistema de ponto flutuante abstrato. Isto ?, se um n?mero x 2 R n?o ? represent?vel em um sistema de ponto flutuante F, ele ? mapeado para um intervalo [x;x], tal que x ? o maior dos n?meros menores que x represent?vel em F e x ? o menor dos n?meros maiores que x represent?vel em F. A representa??o intervalar ? importante em processamento digital de sinais, pois a imprecis?o em dados ocorre tanto no momento da medi??o de determinado sinal, quanto no momento de process?-los computacionalmente. A partir da?, define-se sinais e sistemas intervalares que assumem tanto valores reais quanto complexos. Para isso, utiliza-se o estudo feito a respeito das aritm?ticas complexas intervalares e mostram-se algumas propriedades dos sistemas intervalares, tais como: causalidade, estabilidade, invari?ncia no tempo, homogeneidade, aditividade e linearidade. Al?m disso, foi definida a representa??o intervalar de fun??es complexas. Tal fun??o estende sistemas cl?ssicos a sistemas intervalares preservando as principais propriedades. Um conceito muito importante no processamento digital de sinais ? a quantiza??o, uma vez que a maioria dos sinais ? de natureza cont?nua e para process?-los ? necess?rio convert?-los em sinais discretos. Aqui, este processo ? descrito detalhadamente com o uso da matem?tica intervalar, onde se prop?em, inicialmente, uma amostragem intervalar utilizando as id?ias de representa??o no sistema de ponto flutuante. Posteriormente, s?o definidos n?veis de quantiza??o intervalares e, a partir da?, ? descrito o processo para se obter o sinal quantizado intervalar e s?o definidos o erro de quantiza??o intervalar e o sinal codificado intervalar. ? mostrado que os n?veis de quantiza??o intervalares representam os n?veis de quantiza??o cl?ssicos e o erro de quantiza??o intervalar representa o e erro de quantiza??o cl?ssico. Uma estimativa para o erro de quantiza??o intervalar ? apresentada. Utilizando a aritm?tica retangular e as defini??es e propriedades de sinais e sistemas intervalares, ? introduzida a transformada Z intervalar e s?o analisadas as condi??es de converg?ncia e as principais propriedades. Em particular, quando a vari?vel complexa z ? unit?ria, define-se a transformada de Fourier intervalar para sinais discretos no tempo, al?m de suas propriedades. Por fim, foram apresentadas as implementa??es dos resultados que foram feitas no software Matlab

Interval uncertainty affects our decision making activities on a daily basis making the data structure of intervals of real numbers more and more popular in theoretical models and related software applications. Natural questions for people or businesses that face interval uncertainty in their data when dealing with cooperation are how to form the coalitions and how to distribute the collective gains or costs. The theory of cooperative interval games is a suitable tool for answering these questions. In this thesis, the classical theory of cooperative games is extended to cooperative interval games. First, basic notions and facts from classical cooperative game theory and interval calculus are given. Then, the model of cooperative interval games is introduced and basic definitions are given. Solution concepts of selection-type and interval-type for cooperative interval games are intensively studied. Further, special classes of cooperative interval games like convex interval games and big boss interval games are introduced and various characterizations are given. Some economic and Operations Research situations such as airport, bankruptcy and sequencing with interval data and related interval games have been also studied. Finally, some algorithmic aspects related with the interval Shapley value and the interval core are considered.

A statistical experiment is a mathematical object that provides a framework for statistical inference, including hypothesis testing and parameter estimation, from observations of an empirical phenomenon. When observations in the continuum of real numbers are not empirically measurable to infinite precision and when conventional floating-point computations used in the inference procedure are not exact, the statistical experiment can become epistemologically invalid. The family of measures of the conventional statistical experiment indexed by a compact finite dimensional continuum is extended to the complete metric space of all compact subsets (of a certain form) of the index set. This is accomplished by the natural interval extension of the likelihood function. The extended experiment allows a statistical decision made with the aid of a computer to be equivalent to a numerical proof of its global optimality. Three open problems in computational statistics were solved using the extended experiment: (1) parametric bootstraps of likelihood ratio test statistics for finite mixture models, (2) rigorous maximum likelihood estimates of the branch lengths of a phylogenetic tree with a fixed topology or shape and (3) Monte Carlo sampling from a multi-modal target density with sharp peaks or witches’ hats.

Thesis (S.M.)--Massachusetts Institute of Technology, Dept. of Earth, Atmospheric, and Planetary Sciences, 1998.
Includes bibliographical references (leaves 55-56).
by Hafiz Jaman Alshammery.
S.M.

This thesis continues the study of interval orders and related structures, containing results on both the labeled and unlabeled variants. Following a result of Eriksen and Sjöstrand (2014) we identify a link between structures following the Fishburn distribution and Mahonian structures. This is used to detail a technique for the construction of Fishburn structures (structures in bijection with unlabeled interval orders) from appropriate Mahonian structures. This technique is introduced on a bivincular pattern of Bousquet-Mélou et al. (2010) and then used to introduce a previously unconsidered class of matchings; explicitly, zero alignment matchings according to the number of arcs which are both right-crossed and left-nesting. The technique is then used to identify a statistic on the factorial posets of Claesson and Linusson (2011) following the Fishburn distribution. Factorial posets mapped to zero by this statistic are canonically labeled factorial posets which may alternatively be viewed as unlabeled interval orders. As a consequence of our approach we find an identity for the Fishburn numbers in terms of the Mahonian numbers and discuss linear combinations of Fishburn patterns in a manner similar to that of the Mahonian combinations of Babson and Steingrímsson (2001). To study labeled interval orders we introduce ballot matrices, a signed combinatorial structure whose definition naturally follows from the generating function for labeled interval orders. A sign reversing involution on ballot matrices is defined. Adapting a bijection of Dukes, Jelínek and Kibitzke (2011), we show that matrices fixed under this involution are in bijection with labeled interval orders and that they decompose to a pair consisting of a permutation and an inversion table. To fully classify such pairs results pertaining to the enumeration of permutations having a given set of ascent bottoms are given. This allows for a new formula for the number of labeled interval orders.

Thesis (M.S.)--West Virginia University, 2009.
Title from document title page. Document formatted into pages; contains vii, 50 p. : ill. Includes abstract. Includes bibliographical references (p. 37-41).

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Um intervalo é um tipo de dado complexo usado na agregação de informações ou na representação de dados imprecisos. Este trabalho apresenta duas novas representações para intervalos com o objetivo de se construir novos métodos de agrupamento e regressão linear para este tipo de dado. O agrupamento por nuvens dinâmicas define partições nos dados e associa protótipos a cada uma destas partições. Os protótipos resumem a informação das partições e são usados na minimização de um critério que depende de uma distância, responsável por quantificar a proximidade entre instâncias e protótipos. Neste sentido, propõe-se a formulação de uma nova distância híbrida entre intervalos baseando-se em distâncias para pontos. Os pontos utilizados são obtidos dos intervalos através de um mapeamento. Também são propostas duas versões com pesos para a distância criada: uma com pesos no hibridismo e outra com pesos adaptativos. Na regressão linear, propõe-se a representação dos intervalos através da equação paramétrica da reta. Esta parametrização permite o ajuste dos pontos nas variáveis regressoras que dão as melhores estimativas para os limites da variável resposta. Antes da realização da regressão, um critério é calculado para a verificação da coerência matemática da predição, na qual o limite superior deve ser maior ou igual ao inferior. Se o critério mostra que a coerência não é garantida, propõe-se a aplicação de uma transformação sobre a variável resposta. Assim, este trabalho também propõe algumas transformações que podem ser aplicadas a dados intervalares, no contexto de regressão. Dados sintéticos e reais são utilizados para comparar os métodos provenientes das representações propostas e aqueles presentes na literatura.
An interval is a complex data type used in the information aggregation or in the representation of imprecise data. This work presents two new representations of intervals in order to construct a new cluster method and a new linear regression method for this kind of data. Dynamic clustering defines partitions into the data and it defines prototypes associated with each one of these partitions. The prototypes summarize the information about the partitions and they are used in a minimization criterion which depends on a distance, which is responsible for quantifying the proximity between instances and prototypes. In this way, it is proposed a new hybrid distance between intervals based on a family of distances between points. Points are obtained from the interval through a mapping. Also, it is proposed two versions of the hybrid distance, both with weights: one with weights in hybridism and other with adaptive weights. In linear regression, it is proposed to represent the intervals through the parametric equation of the line. This parametrization allows to find the set of points in the regression variables corresponding to the best estimates for the response variable limits. Before the regression construction, a criterion is computed to verify the mathematical consistency of prediction, where the upper limit must be greater than or equal to the lower. If the test shows that consistency is not guaranteed, then the application proposes a transformation of the response variable. Therefore, this work also proposes some transformations that can be applied to interval data in the regression context. Synthetic and real data are used to compare the proposed methods and those one proposed on literature.

In this study, for the well-known Analytic Hierarchy Process (AHP) method a new approach to interval priority weight generation from interval comparison matrix is proposed. This method can be used for both inconsistent and consistent matrices. Also for the problems having more than two hierarchical levels a synthesizing heuristic is presented. The performances of the methods, interval generation and synthesizing, are compared with the methods that are already available in the literature on randomly generated matrices.

An Interval Routing Scheme (IRS) represents the routing tables in a network in a space-efficient way by labeling each vertex with an unique integer address and the outgoing edges at each vertex with disjoint subintervals of these addresses. An IRS that has at most k intervals per edge label is called a k-IRS. In this thesis, we propose a new type of interval routing scheme, called an Ordered Interval Routing Scheme (OIRS), that uses an ordering of the outgoing edges at each vertex and allows nondisjoint intervals in the labels of those edges. Our results on a number of graphs show that using an OIRS instead of an IRS reduces the size of the routing tables in the case of optimal routing, i. e. , routing along shortest paths. We show that optimal routing in any k-tree is possible using an OIRS with at most 2^k-1 intervals per edge label, although the best known result for an IRS is 2^k+1 intervals per edge label. Any torus has an optimal 1-OIRS, although it may not have an optimal 1-IRS. We present similar results for the Petersen graph, k-garland graphs and a few other graphs.

Acumen is a language and tool for modeling and simulating cyber-physical systems. It allows the user to conduct simulations using a technique called rigorous simulation that produces results with explicit error bounds, expressed as intervals. This feature can be useful when designing and testing systems where the reliability of results or taking uncertainty into account is important. Unfortunately, analyzing these simulation results can be difficult, as Acumen supports only two ways of presenting them: raw data tables and 2D-plots. These views of the data make certain kinds of analysis cumbersome, such as understanding correlations between variables. This is especially true when the model in question is large. This project proposes a new way of visualising rigorous simulation results in Acumen. The goal of this project is to create a method for visualising intervallic values in 3D, and implement it in Acumen. To achieve that, every span of values is represented as a series of overlapping objects. This family of objects, which constitutes an under-approximation of the true simulation result, is then wrapped inside a semi-translucent box that is a conservative over-approximation of the simulation result. The resulting implementation makes for a combination of mathematical correctness (rigour), and mediation of intervals in question. It enables the user to explore the results of his rigorous simulations as conveniently as with the existing, non-rigorous simulation methods, using the 3D visualisation to simplify the study of real-life problems. To our knowledge, no existing software features visualisation of interval-based simulation results, nor is there any convention for doing this. Some ways in which the proposed solution could be improved are suggested at the end of this report

Floating point arithmetic has become prevalent in virtually every scientific computation, yet suffers from rounding, truncation and catastrophic cancellation errors to the extent that the result may be completely inaccurate. Whilst modern processors do not have the capabilities to natively support interval arithmetic, it can be implemented in software though such a method of implementation results in larger program size and often more than doubles execution times. Embedded systems do not have the luxury of utilising software implementations of interval arithmetic and so a hardware based alternative must be sought. This thesis examines the algorithms traditionally used for interval multiplication and proposes a new method, free of the caveats which plague traditional multiplication computation. The inherent parallelism of these algorithms is examined to determine their throughput and latency characteristics, given a selection of platforms which exhibit variations in the number of input buses, FPUs and result buses. A software implementation of each method of interval multiplication is described, alongside their relative performances. Given the data-dependant nature of interval computation and the current trend towards vector processing architectures, these software implementations include reformulations of both tradi- tional methods of interval multiplication to provide vectorisation of the software and thus completely eliminate branching. Realised as co-processors to the Altera Nios Il, this thesis describes the design decisions, specification, area requirements, power utilisation and per-cycle performance of hardware implementations of the brute-force, nine case and herein presented integer-based interval multipliers.

Orientador: Geraldo Nunes Silva
Banca: Weldon A. Lodwick
Banca: Yurilev Chalco Cano
Banca: Ulcilea Alves Severino Leal
Banca: Valeriano Antunes de Oliveira
Resumo: Esta Tese trabalha com alguns conceitos fundamentais da analise intervalar e suas aplicações. Em primeiro lugar, a Tese aborda a álgebra de funções de valor intervalar gH diferenciáveis. Especificamente, damos condições para a gH- diferenciabilidade da soma e gH-diferença de duas funções de valor intervalar gH-diferenciáveis; também para o pro duto e composição de uma função real diferenciável e uma função de valor intervalar gH diferenciável. Em segundo lugar, a Tese e dedicada a obtenção de condições necessárias e suficientes para problemas de otimização com funções objetivas de valor intervalar. Essas funções objetivas são obtidas a partir de funções contínuas usando aritmética intervalar restrita. Damos um conceito de derivada para esta classe de funções de valor intervalar e, em seguida, introduzimos o conceito de ponto estacionário. Encontramos as condições necessárias com base na definição dos pontos estacionários e provamos que essas condições também são suficientes nas noções de convexidade generalizada. Obtemos também condições necessárias e suficientes para o problema de otimização intervalar com restrições. E, finalmente, lidamos com o espaço quociente de intervalos I em relação a família de intervalos simétricos e dado um conceito de diferenciabilidade para funções de classes de equivalência, fazemos uma comparação com outros conceitos de diferenciabilidade. Alguns exemplos e contraexemplos ilustram os resultados obtidos
Abstract: This Thesis works with some fundamentals concepts of interval analysis and it applica tions. First of all, the thesis deals with the algebra of gH-diferentiable interval-valued functions. Specifically, we give conditions for the gH-diferentiability of the sum and gH-diference of two gH-diferentiable interval-valued functions; also for the product and composition of a diferentiable real function and a gH-diferentiable interval-valued func tion. Second, the thesis is devoted to obtaining necessary and sucient conditions for optimization problems with interval-valued objective functions. These objective func tions are obtained from continuous functions by using constrained interval arithmetic. We give a concept of derivative for this class of interval-valued functions and then we introduce the concept of stationary point. We find necessary conditions based on the stationary points definition and we prove that these conditions are also sucient under generalized convexity notions. We obtain the necessary and sucient conditions for con strained interval-valued optimization problem. And finally, we deal with the quotient space of intervals I with respect to the family of symmetric intervals and given a concept of di↵erentiability for equivalence classes-valued functions, we make a comparison with other concepts of diferentiability. Some examples and counterexamples illustrates the obtained results.
Doutor

Alternating-Time Temporal Logic (ATL), introduced by Alur, Henzinger and Kupferman, is a logic involving coalitions of agents performing actions which cause a state change in a turn-based time system. There have been game theoretic ex- tensions on ATL, and they are very good at specifying systems of multiple agents cooperating or competing in a game-like situation. Unfortunately neither ATL nor its extensions are able to capture the idea of gradual change, or duration of actions or events. The concurrent game model of ATL operates like a turn based game, with sets of agents taking their turn, and then the environment changing based on their actions, before they take their next turn. The fact that some actions take longer than others, or that sometimes a state changes gradually, rather than immediately, is not representable in ATL. As an example, take a train entering a tunnel. Before the train enters the tunnel, it is outside the tunnel, after it has entered the tunnel, it is inside the tunnel, but for the few seconds it takes the train to enter the tunnel, it is neither inside nor outside the tunnel. ATL cannot represent this basic intuitive truth. A family of logics called Interval Logic (IL) use finite state sequences called “intervals”, which allow it to describe a more continuous model of time, rather than a discrete state based one such as ATL. This allows it to capture the idea of gradual change, of a train entering a tunnel, and the fact that actions and events have various durations. Most of the IL formulations do however not have any way of distinguishing multiple agents acting at the same time. Both of these logics - ATL and IL - are useful for specific things, but combining them might produce new applications which are not possible when only using the one or the other. In this dissertation we present one such possible combination, called Agent Interval Temporal Logic (AITL). AITL combines the notion of agents, coalitions and strategies from ATL with the interval based model of time from IL, thus creating a new logic which might have some powerful applications in a wide range of areas in which gradual change and multiple agents acting at the same time can both be accommodated.
Dissertation (MA)--University of Pretoria, 2020.
Centre for Artificial Intelligence Research at CSIR
Philosophy
MA
Unrestricted

Fundamentada a importância da utilização da Teoria dos Intervalos em computação científica, é realizada uma revisão da Teoria Clássica dos Intervalos, com críticas sobre as incompatibilidades encontradas como motivos de diversas dificuldades para desenvolvimento da própria teoria e, consequentemente, das Técnicas Intervalares. É desenvolvida uma nova abordagem para a Teoria dos Intervalos de acordo com a Teoria dos Domínios e a proposta de [ACI 89], obtendo-se os Domínios Intervalares da Matemática Computacional. Introduz-se uma topologia (Topologia de Scott) compatível com a idéia de aproximação, gerando uma ordem de informação, isto é, para quaisquer intervalos x e y, diz-se que se x -c y , então y fornece mais (no mínimo tanto quanto) informação, sobre um real r, do que x. Prova-se que esta ordem de informação induz uma topologia To (topologia de Scott) , que é mais adequada para uma teoria computacional que a topologia da Hausdorff introduzida por Moore [MOO 66]. Cada número real r é aproximado por intervalos de extremos racionais, os intervalos de informação, que constituem o espaço de informação II(Q), superando assim a regressão infinita da abordagem clássica. Pode-se dizer que todo real r é o supremo de uma cadeia de intervalos com extremos racionais “encaixados”. Assim, os reais são os elementos totais de um domínio contínuo, chamado de Domínio dos Intervalos Reais Parciais, cuja base é o espaço de informação II (Q). Cada função contínua da Análise Real é o limite de sequências de funções contínuas entre elementos da base do domínio. Toda função contínua nestes domínios constitui uma função monotônica na base e é completamente representada em termos finitos. É introduzida uma quasi-métrica que induz uma topologia compatível com esta abordagem e provê as propriedades quantitativas, além de possibilitar a utilização da noção de sequências, limites etc, sem que se precise recorrer a conceitos mais complexos. Desenvolvem-se uma aritmética, critérios de aproximação e os conceito de intervalo ponto médio, intervalo valor absoluto e intervalo diâmetro, conceitos compatíveis com esta abordagem. São acrescentadas as operações de união, interseção e as unárias. Apresenta-se um amplo estudo sobre a função intervalar e a inclusão de imagens de funções, com ênfase na obtenção de uma extensão intervalar natural contínua. Esta é uma abordagem de lógica construtiva e computacional.
The importance of Interval Theory for scientific computation is emphasized. A review of the Classical Theory is macle, including a discussion about some incompatibities that cause problems in developing interval algorithms. A new approach to the Interval Theory is developed in the light of the Theory of Domains and according to the ideas by Acióly [ACI 89], getting the Interval Domains of Computational Mathematics. It is introduced a topology (Scott Topology), which is associated with the idea of approximation, generating an information order, that is, for any intervals x and y one says that if x -c y, then "the information given by y is better or at least equal than the one given by x". One proves that this information order induces a To topology (Scott's topology) which is more suitable for a computation theory than that of Hausdorff introduced by Moore [MOO 66]. This approach has the advantage of being both of constructive logic and computational. Each real number is approximated by intervals with rational bounds, named information intervals of the Information Space II(Q), eliminating the infinite regression found in the classical approach. One can say that every real a is the supreme of a chain of rational intervals. Then, the real numbers are the total elements of a continuous domain, named the Domain of the Partial Real Intervals, whose basis is the information space II (Q). Each continuous function in the Real Analysis is the limit of sequences of continuous functions among any elements which belong to the base of the domain. In these same domains, each continuous function is monotonic on the base and it is completely represented by finite terms. It is introduced a quasi-metric that leads to a compatible topology and supplies the quantitative properties. An arithmetic, some approximation criteria, the concepts of mean point interval, absolute value interval and width interval are developed and set operations are added. The ideas of interval functions and the inclusion of ranges of functions are also presented, and a continuous natural interval extension is obtained.

Routing messages between pairs of nodes is one of the most fundamental tasks in any distributed computing system. An Interval Routing Scheme (IRS) is a well-known, space-efficient routing strategy for routing messages in a network. In this scheme, each node of the network is assigned an integer label and each link at each node is labeled with an interval. The interval assigned to a link l at a node v indicates the set of destination addresses of the messages which should be forwarded through l at v. When studying interval routing schemes, there are two main problems to be considered: a) Which classes of networks do support a specific routing scheme? b) Assuming that a given network supports IRS, how good are the paths traversed by messages? The first problem is known as the characterization problem and has been studied for several types of IRS. In this thesis, we study the characterization problem for various schemes in which the labels assigned to the vertices are d-ary integer tuples (d-dimensional IRS) and the label assigned to each link of the network is a list of d 1-dimensional intervals. This is known as Multi-dimensional IRS (MIRS) and is an extension of the the original IRS. We completely characterize the class of network which support MIRS for linear (which has no cyclic intervals) and strict (which has no intervals assigned to a link at a node v containing the label of v) MIRS. In real networks usually the costs of links may vary over time (dynamic cost links). We also give a complete characterization for the class of networks which support a certain type of MIRS which routes all messages on shortest paths in a network with dynamic cost links. The main criterion used to measure the quality of routing (the second problem) is the length of routing paths. In this thesis we also investigate this problem for MIRS and prove two lower bounds on the length of the longest routing path. These are the only known general results for MIRS. Finally, we study the relationship between various types of MIRS and the problem of drawing a hypergraph. Using some of our results we prove a tight bound on the number of dimensions of the space needed to draw a hypergraph.

In an interval society, voters are represented by intervals on the real line, corresponding to their approval sets on a linear political spectrum. I imagine the society to be a representative democracy, and ask how to choose members of the society as representatives. Following work in mathematical psychology by Coombs and others, I develop a measure of the compatibility (political similarity) of two voters. I use this measure to determine the popularity of each voter as a candidate. I then establish local “agreeability” conditions and attempt to find a lower bound for the popularity of the best candidate. Other results about certain special societies are also obtained

The purpose of this study was to examine the effect of swimming distance and rest interval on the intensity of swimming (relative to VO2 max) and the contributions of the three energy systems (aerobic, anaerobic, and alactic) during these interval sets. Nine male college swimmers performed fourteen different interval training sets. Distances were 25, 50, 100, or 200 yards with rest intervals of 10 seconds, 1, and 3 minutes. During these sets only the distance to be swum and the rest interval for the set was given. No qualitative information from the coach was provided. These interval sets were performed by the swimmers with the influence from timers being kept minimal. Oxygen cost during the swim was determined from the velocity of the swim based on a linear regression for swimming velocity and oxygen uptake for each swimmer. The same interval sets were completed with pace controled where venous blood samples were obtained 1, 3, 5, and 7 minutes after the completion of each training set. 81ood samples were analyzed for lactate accumulation, blood pH, p0.,, pCO2, and hemoglobin. From these values bicarbonate, base excess, and blood oxygen saturation were calculated using equations developed by Siggard-Anderson. The results of this study do indicate that there is a predictable relationship between swimming distance and rest interval on swimming intensity (relative to VO., max). There was a curvilinear L relationship between swimming intensity (relative to VO max) and rest interval for 50, 100, and 200 yard interval sets. The correlation at these distances were r-0.96, 0.93, and 0.94 respectively. There was a linear relationship between intensity and the distance swum for the 10 second, 1, and 3 minute rest intervals. The correlation for these rest intervals were r= 0.99, 0.99, and 1.00 respectively. There was an increase in the relative contribution of aerobic energy as the distance of the swim increased for all three rest intervals. At a given swiming distance there was a greater contribution of non-aerobic energy as the rest interval increased. Contrary to continuous swimming, greater swimming velocity does not directly correspond to greater contributions of anaerobic energy during intermittent swimming. The distance and rest interval during intermittent training greatly effect the relative contributions of the three energy systems. The intensity of the swim and the relative contributions of each energy system should be considered when planning specific training regimens.

In this thesis we give a proof of complex bounds for real analytic interval maps, for all possible combinatorics. We begin by constructing a sequence of intervals, known as the enhanced nest, that covers both non-renormalizable and infinitely renormalizable maps. This is a generalisation of the nest introduced in [KSvS1]. In Chapter 3, we prove the combinatorial and geometric properties of the enhanced nest, known as real bounds. The results from this chapter extend the ones in [KSvS1] to maps with odd critical points. In Chapter 4, we make use of Poincar´e disks based on intervals from the enhanced nest to construct quasi box-mappings associated to (real) first return maps. Key in this part of the proof are the pullbacks along monotone branches, that are controlled with the aid of fundamental domains, and the pullbacks between two consecutive levels from the enhanced nest of different combinatorial type, one non-terminating and one terminating. We use similar techniques to the ones developed in [LevS2] to obtain complex box-mappings from the quasi box-mappings that we constructed. Finally, we follow the arguments in [KSvS1] to prove complex bounds for the complex box-mappings we constructed.

An intensive examination was conducted to test the credibility of current traffic signal change interval policies founded on a kinematic equation developed nearly 30 years ago. The investigation involved the review of relevant literature as well as an extensive collection of data. The literature review and data analysis revealed that current change interval policies rely on the disproven assumption that traffic decelerates at a constant rate. The data analysis also demonstrated that traffic approach speed and deceleration distance affect the manner in which deceleration occurs. Based on the data analysis, an alternative treatment of the kinematic equation is proposed using surrogate deceleration rates. The surrogate rates offer a pragmatic set of input for the kinematic equation. Therefore, rather than yielding a change interval based on an inaccurate assumption, agencies can implement change intervals which are responsive to local traffic.

The mapping problem is a major player in mobile robotics, and it is essential for many real applications such as disaster response or nuclear decommissioning. Generally, the robotic mapping is addressed under the umbrella of simultaneous localization and mapping (SLAM). Several probabilistic techniques were developed in the literature to approach the SLAM problem, and despite the good performance, their convergence proof is only limited to linear Gaussian models. This thesis proposes an interval SLAM (i-SLAM) algorithm as a new approach that addresses the robotic mapping problem in the context of interval methods. The noise of the robot sensor is assumed bounded, and without any prior knowledge of its distribution, we specify soft conditions that guarantee the convergence of robotic mapping for the case of nonlinear models with non-Gaussian noise. A new theory about compact sets is developed in the context of real analysis to conclude such conditions. Then, a case study is presented where the performance of i-SLAM is compared to the probabilistic counterparts in terms of accuracy and efficiency. Moreover, this work presents an application for i-SLAM using an RGB-D sensor that operates in unknown environments. Interval methods and computer vision techniques are employed to extract planar landmarks in the environment. Then, a new hybrid data association approach is developed using a modified version of bag-of-features method to uniquely identify different landmarks across timesteps. Finally, the results obtained using the proposed data association approach are compared to the typical least-squares approaches, thus demonstrating the consistency and accuracy of the proposed approach.

Process control charts are widely used to display sample data from a process for purposes of determining whether a process is in control, for bringing an out-of-control process into control, and for monitoring a process to make sure that it stays in control. The usual practice in maintaining a control chart is to take samples from the process at fixed length sampling intervals. This research investigates the modification of the standard practice where the sampling interval or time between samples is not fixed but can vary depending on what is observed from the data. Variable sampling interval process control procedures are considered for monitoring the outcome of a production process. The time until the next sample depends on what is being observed in the current sample. Sampling is less frequent when the process is at a high level of quality and vise versa. Properties such as the average number of samples until signal, average time to signal and the variance of the time to signal are developed for the variable sampling interval Shewhart and cusum charts. A Markov chain is utilized to approximate the average time to signal and the corresponding variance for the cusum charts. Properties of the variable sampling interval Shewhart chart are investigated through Renewal Theory and Markov chain approaches for the cases of a sudden and gradual shift in the process mean respectively. Also considered is the case of a shift occurring in the time between two samples without the simplifying assumption that the process mean remains the same from time zero onward. For such a case, the adjusted time to signal is developed for both the Shewhart and cusum charts in addition to the variance of the adjusted time to signal. Results show that the variable sampling interval control charts are considerably more efficient than the corresponding fixed sampling interval control charts. It is preferable to use only two sampling intervals which keeps the complexity of the chart to a reasonable level and has practical implications. This feature should make such charts very appealing for use in industry and other fields of application where control charts are used.
Ph. D.

A Thesis submitted to The University of Arizona College of Medicine - Phoenix in partial fulfillment of the requirements for the Degree of Doctor of Medicine.
Objectives: Interpregnancy interval (IPI), the time period between the end of one pregnancy and the conception of the next, can have a significant impact on maternal and infant outcomes. This study examines the relationship between interpregnancy interval and neonatal outcomes of low birth weight, preterm birth, and specific neonatal morbidities. Study Design: Retrospective cohort study comparing neonatal outcomes across 6 categories of IPI using data on 202,600 cases identified from Arizona birth certificates and the Newborn Intensive Care Program data. Comparisons between groups were made using odds ratios and 95% confidence intervals, and multivariable logisitic regression analysis. Results: Interpregnancy intervals of < 12 months and ≥ 60 months were associated with low birth weight, preterm birth, and small for gestational age births. The shortest and longest IPI categories were also associated with specific neonatal morbidities, including periventricular leukomalacia, bronchopulmonary dysplasia, intraventricular hemorrhage, apnea bradycardia, respiratory distress syndrome, transient tachypnea of the newborn, and suspected sepsis. Relationships between interpregnancy interval and specific neonatal morbidities did not remain significant when adjusted for birth weight and gestational age. Conclusions: Significant differences in neonatal outcomes (preterm birth, low birth weight, and small for gestational age) were observed between IPI categories. Consistent with previous research, interpregnancy intervals < 12 months and ≥ 60 months appear to be associated with increased risk of poor neonatal outcomes. Any difference in specific neonatal morbidities between IPI groups appears to be mediated through increased risk of low birth weight and preterm birth by IPI.

This thesis contains a description of algorithm, MW, for bounding the global minimizers and globally minimum value of a twice continuously differentiable function f :Rn → R1 R1 in a compact sub-interval of Rn. The algorithm MW is similar to the algorithm of Hansen (Han-80a] in that interval arithmetic is used together with certain of Hansen's ideas, but is different from Hansen's algorithm in that MW bounds the Kuhn Tucker points corresponding to the global minimizers of f in the given sab-interval. The Kuhn Tucker points are bounded with prescribed precision by using either of the algorithms KMSW [SheW-85c] or MAP [SheW-85b]. Numerical results which are obtained from Triplex [BaCM-82a] [MorC-83a] implementations of H and MW axe presented.

Microbiome is the catalog of microbes and their genes. And a biomarker is a substance that can be objectively measured and which can act as an indicator of pathogenic processes, biological processes or pharmacological responses to a therapeutic intervention. Also, the time between physiological death and the examination of the dead body is known as the Post mortem Interval (PMI). Determination of PMI can be a complex problem due to it being influenced by a number of intrinsic and extrinsic factors. Some of the intrinsic factors include age, sex, and pathological and physiological states of the corpse. While the extrinsic factors include temperature, humidity and insect activity. Recently, molecular changes such as protein, RNA and DNA degradation have been studied quiet widely and are seem to be producing promising results in the field of PMI estimation. More specifically, studying RNA degradation after death is considered quiet useful for precise PMI estimation. Some of the different types of RNA that aid in PMI estimation include miRNAs, circRNAs, 18S-rRNA and so on. This study focuses on the potential for estimating PMI using microbiome biomarkers as a tool rather than the tradition PMI estimation by studying various stages of decomposition.

The role of spatial reasoning in the development of systems in the domain of Artificial Intelligence is increasing. One particular approach, qualitative spatial reasoning, investigates the usage of abstract representation to facilitate the representation of and the reasoning with spatial information.This thesis investigates the usage of intervals along global axes as the under-lying representational and reasoning mechanism for a spatial reasoning system. Aspects that are unique to representing spatial information (flow and multi-dimensionality) are used to provide a method for classifying relations between objects at multiple levels of granularity. The combination of these two mechanisms (intervals and classification) provide the basis for the development of a querying system that allows qualitative queries about object relations in multi-dimensional space to be performed upon the representation.The second issue examined by this thesis is the problem of representing intervals when all the interval relations may not be known precisely. A three part solution is proposed. The first shows how the simplest situation, where all relations are explicit and primitive, can be represented and integrated with the above mentioned querying system. The second situation demonstrates how, for interval relations that are primitive but are not all explicitly known, an effective point based representation may be constructed. Finally, when relations between intervals are disjunctions of possible primitive interval relations, a representation is presented which allows solutions to queries to be constructed from consistent data.Our contribution is two-fold:1. a method of classifying the spatial relations and the means of querying these relations;2. a process of efficiently representing incomplete interval information and the means of efficiently querying this information.The work presented in this thesis demonstrates the utility of a multi-dimensional qualitative spatial reasoning system based upon intervals. It also demonstrates how an interval representation may be constructed for datasets that have variable levels of information about relationships between intervals represented in the dataset.

The role of spatial reasoning in the development of systems in the domain of Artificial Intelligence is increasing. One particular approach, qualitative spatial reasoning, investigates the usage of abstract representation to facilitate the representation of and the reasoning with spatial information.This thesis investigates the usage of intervals along global axes as the under-lying representational and reasoning mechanism for a spatial reasoning system. Aspects that are unique to representing spatial information (flow and multi-dimensionality) are used to provide a method for classifying relations between objects at multiple levels of granularity. The combination of these two mechanisms (intervals and classification) provide the basis for the development of a querying system that allows qualitative queries about object relations in multi-dimensional space to be performed upon the representation.The second issue examined by this thesis is the problem of representing intervals when all the interval relations may not be known precisely. A three part solution is proposed. The first shows how the simplest situation, where all relations are explicit and primitive, can be represented and integrated with the above mentioned querying system. The second situation demonstrates how, for interval relations that are primitive but are not all explicitly known, an effective point based representation may be constructed. Finally, when relations between intervals are disjunctions of possible primitive interval relations, a representation is presented which allows solutions to queries to be constructed from consistent data.Our contribution is two-fold:1. a method of classifying the spatial relations and the means of querying these relations;2. a process of efficiently representing incomplete interval information and the means of efficiently querying this information.The work presented ++
in this thesis demonstrates the utility of a multi-dimensional qualitative spatial reasoning system based upon intervals. It also demonstrates how an interval representation may be constructed for datasets that have variable levels of information about relationships between intervals represented in the dataset.

A microcomputer based hardware, the Time Interval Board, was designed and the software interface control program was developed. The board measures time intervals between consecutive pulses from a discriminator output. The data is stored in on-board 16K x 16 memory. The microcomputer empties and processes the data when the on-board memory is filled. Data collection continues until the preset collection period is finished or a forced end is initiated. During this period, control is passed between the hardware and the microcomputer via the interface circuit. The designed hardware is IBM PC compatible.

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O objeto de estudo deste trabalho é o processo intervalar na construção composicional de Almeida Prado em suas três fases pós-ruptura, chamadas de Pós-Tonal, Síntese e Pós-Moderna. A concepção intervalar do compositor é demonstrada por análises realizadas em três obras, uma de cada fase pós-ruptura: Sonata n. 3, Cartas Celestes I e Noturno n. 7, respectivamente. A análise tem sua metodologia baseada em um recorte arbitrário que buscou compreender três intervalos básicos que podem ser somados a outros três intervalos enarmônicos como ampliação das possibilidades intervalares na escrita. O desenvolvimento teórico da pesquisa aprofundou-se a partir de um foco e uma análise intervalar que, conceitualmente, buscaram uma superação terminológica de uma formulação anterior nomeada como expressividade intervalar (SANT’ANA, 2009) para a atualização terminológica definida como intervalo característico. O embasamento teórico para essa concepção intervalar deu-se a partir das proposições de Boulanger (1926), Costère (1954, 1962), Forte (1973), Straus ([1990] 2013), Pousseur ([2005] 2009) e Menezes (2002). Como contextualização crítico-histórica, foi realizada uma revisão sucinta de tendências teórico-analíticas e seus problemas, reconsiderando-se as observações de Schoenberg ([1922] 2001, [1954] 2004), Kerman ([1985] 1987), Cook (1987, 2007) e Kramer (2015). Para se validar a construção e o caráter da ferramenta teórico-analítica aqui empreendida, utilizou-se a teoria de Lacey (2008) associada à visão teórico-musicológica desses autores. Com o processo analítico, apresentado no quarto capítulo, procurou-se demonstrar a grande incidência do(s) intervalo(s) característico(s) na construção estrutural das obras, desvelando-se um número variado de estratégias composicionais adotadas por Almeida Prado associadas a esses intervalos. Ainda no quarto capítulo, foi aplicada a ferramenta denominada régua intervalar, que pode ajudar na melhor compreensão dos intervalos com mais potencial de dissonância, os quais agem por meio de interação e contraste frente ao background aparentemente tonal dos intervalos mais consonantes ligados aos primeiros sete parciais (harmônicos) da série harmônica. Assim, em uma pesquisa futura, a ferramenta e a proposição analítica aqui desenvolvidas podem contribuir na compreensão e na sistematização dos tipos de objetos harmônicos pensados como entidades tímbricas que são moldados a partir dessa concepção intervalar.
The object of study this work is the interval process in the compositional construction of Brazilian pianist and composer Almeida Prado in his three post-rupture phases, called PostTonal, Synthesis and Post-Modern. The intervalar conception of the composer is demonstrated by analyses carried out in three works - one from each post-rupture phase (Sonata n. 3, Cartas Celestes I and Noturno n. 7). The analysis has its methodology based on an arbitrary crop that sought to understand three basic intervals that can be added to three other enharmonic intervals as an extension of the interval possibilities in writing. The theoretical development of the research was deepened from a focus and an interval analysis that, conceptually, sought a terminological overcoming of an earlier formulation named as intervalar expressivity (SANT'ANA, 2009) for the terminological update defined as a characteristic interval. The theoretical basis for this interval conception came from the propositions of Boulanger (1926), Costère (1954, 1962), Forte (1973), Straus ([1990] 2013), Pousseur ([2005] 2009) and Menezes (2002). As a critical-historical context, a brief review of theoretical-analytical trends and their problems was carried out, reconsidering the observations of Schoenberg ([1922] 2001, [1954] 2004), Kerman ([1985], 1987), Cook (1987, 2007) and Kramer (2015). To validate the construction and character of the theoreticalanalytical tool undertaken here, Lacey (2008)’s theory was used associated with the theoretical-musicological view, methodologically adopted in this work. Like the analytical process contained in the fourth chapter, one tried to demonstrate the great incidence of the characteristic interval (s) in the structural construction of the works, revealing a varied number of compositional strategies adopted by Almeida Prado associated to these intervals. Still in the fourth chapter, the tool called interval rule was applied, which can help in better understanding the intervals with more potential of dissonance that act through interaction and contrast against the seemingly tonal background of the more consonant intervals linked to the first seven (harmonic) partials of the harmonic series. Thus, in a future research, the tool and the analytical proposition developed here, can contribute to the comprehension and systematization of the types of harmonic objects thought as timbral entities that are shaped from this intervalar conception.

Un problème courant en logistique, gestion d’entrepôt, industrie manufacturière ou gestion d’énergie dans les centres de données est de placer des objets dans un espace limité, ou conteneur. Ce problème est appelé problème de placement. De nombreux travaux dans la littérature gèrent le problème de placement en considérant des objets de formes particulières ou en effectuant des approximations polygonales. L’objectif de cette thèse est d’autoriser toute forme qui admet une définition mathématique (que ce soit avec des inégalités algébriques ou des fonctions paramétrées). Les objets peuvent notamment être courbes et non-convexes. C’est ce que nous appelons le problème de placement générique. Nous proposons un cadre de résolution pour résoudre ce problème de placement générique, basé sur les techniques d’intervalles. Ce cadre possède trois ingrédients essentiels : un algorithme évolutionnaire plaçant les objets, une fonction de chevauchement minimisée par cet algorithme évolutionnaire (coût de violation), et une région de chevauchement qui représente un ensemble pré-calculé des configurations relatives d’un objet (par rapport à un autre) qui créent un chevauchement. Cette région de chevauchement est calculée de façon numérique et distinctement pour chaque paire d’objets. L’algorithme sous-jacent dépend également du fait qu’un objet soit représenté par des inégalités ou des fonctions paramétrées. Des expérimentations préliminaires permettent de valider l’approche et d’en montrer le potentiel
A common problem in logistic, warehousing, industrial manufacture, newspaper paging or energy management in data centers is to allocate items in a given enclosing space or container. This is called a packing problem. Many works in the literature handle the packing problem by considering specific shapes or using polygonal approximations. The goal of this thesis is to allow arbitrary shapes, as long as they can be described mathematically (by an algebraic equation or a parametric function). In particular, the shapes can be curved and non-convex. This is what we call the generic packing problem. We propose a framework for solving this generic packing problem, based on interval techniques. The main ingredients of this framework are: An evolutionary algorithm to place the objects, an over lapping function to be minimized by the evolutionary algorithm (violation cost), and an overlapping region that represents a pre-calculated set of all the relative configurations of one object (with respect to the other one) that creates an overlapping. This overlapping region is calculated numerically and distinctly for each pair of objects. The underlying algorithm also depends whether objects are described by inequalities or parametric curves. Preliminary experiments validate the approach and show the potential of this framework

The correlation coefficient (CC) is a standard measure of the linear association between two random variables. The CC plays a significant role in many quantitative researches. In a bivariate normal distribution, there are many types of interval estimation for CC, such as z-transformation and maximum likelihood estimation based methods. However, when the underlying bivariate distribution is unknown, the construction of confidence intervals for the CC is still not well-developed. In this thesis, we discuss various interval estimation methods for the CC. We propose a generalized confidence interval and three empirical likelihood-based non-parametric intervals for the CC. We also conduct extensive simulation studies to compare the new intervals with existing intervals in terms of coverage probability and interval length. Finally, two real examples are used to demonstrate the application of the proposed methods.

We propose the use of cyclic interval graphs as an alternative representation for register allocation. The "thickness" of the cyclic interval graph captures the notion of overlap between live ranges of variables relative to each particular point of time in the program execution. It is effective in capturing the regular periodic nature of loop-carried dependent live ranges found in loops.
As most scientific code spend a lot of time executing loop structures, it is most crucial to perform well when register allocating for it. A good spilling algorithm and a close to optimal coloring algorithm is invaluable in minimizing the cost that may be incurred while performing register allocation for loop structures. Minimization of spill code often greatly increases the performance of the code. Our proposed spilling and coloring scheme is very well suited to these loop structures, and could be used to reap maximum benefit when used in tandem with loop scheduling algorithms (NG93, Nin93).
A collection of real program loops are used to test the effectiveness of our approach.

The following set of experiments investigates the fundamental mechanism proposed to underlie interval timing and addresses a key question in timing research pertaining to the underlying functional architecture. Numerous models have been proposed in an attempt to illustrate and explain timing performance, some based on dedicated features employing a specialised mechanism, whilst others suggest that time perception is inherent in neural dynamics. An influential set of models posit that the brain contains a mechanism akin to a mental stopwatch which can be started, stopped and paused at will. This premise was tested in the first experimental chapter, the expected decline in performance was calculated in line with model predictions. The observed deterioration significantly exceeded the calculated predictions indicating that human timing is not akin to that of a ‘stopwatch’ when timing short durations and is incongruent with the predictions of a pacemaker accumulator type mode that can be paused at will. Psychophysical methods have revealed that interval timing conforms to a fundamental property in sensory processing known as Weber’s Law. Lawful relationships such as these are important because they inform and constrain models of human interval timing. The adherence to this property was investigated across a range of durations using two comparable tasks in the following experiment. Although violations across certain durations were observed, these were not mirrored in both the utilised tasks to a statistically significant degree. The results could tentatively be argued to suggest certain constraints on the scalar model albeit a firm conclusion cannot be asserted. The second question pertaining to multimodal processing across a range of tasks and durations, indicative of the underlying architecture of interval timing, (i.e. ‘one clock or many’) was addressed via transfer of learning and correlation in the two final experiments. Perceptual learning and the generalisation to untrained durations and temporal tasks was assessed in the first of these two chapters. Training was observed to improve performance at a few of the practised durations with a more global improvement for one participant across untrained durations. Generalisation to the motor tasks from the perceptual task was observed consistently in two of the longer trained durations for all three participants pointing to a partially shared or overlapping interval timing structure. The next study further addressed some of the dichotomies reported in timing literature, with a key focus on explicit and implicit timing. Particular attention was also bestowed on timing in language: the language task based on phoneme closure duration and not overly reliant on contextual cues was observed to have a significant association with both motor and perceptual timing tasks. The results of the conducted experiments when taken together point to independent mechanisms which nonetheless possess a significant overlap.

In numerical mathematics, there is a need for methods which provide a user with the solution to his problem without requiring him to understand the mathematics underlying the method of solution. Such a method involves computable tests to determine whether or not a solution exists in a given region, and whether, if it exists, such a solution may be found by using the given method. Two valuable tools for the implementation of such methods are interval mathematics and symbolic computation. In. practice all computers have memories of finite size and cannot perform exact arithmetic. Therefore, in addition to the error which is inherent in a given numerical method, namely truncation error, there is also the error due to rounding. Using interval arithmetic, computable tests which guarantee the existence of a solution to a given problem in a given region, and the convergence of a particular iterative method to this solution, become practically realizable. This is not possible using real arithmetic due to the accumulation of rounding error on a computer. The advent of packages which allow symbolic computations to be carried out on a given computer is an important advance for computational numerical mathematics. In particular, the ability to compute derivatives automatically removes the need for a user to supply them, thus eliminating a major source of error in the use of methods requiring first or higher derivatives. In this thesis some methods which use interval arithmetic and symbolic computation for the solution of systems of nonlinear algebraic equations are presented. Some algorithms based on the symmetric single-step algorithm are described. These methods however do not possess computable existence, uniqueness, and convergence tests. Algorithms which do possess such tests, based on the Krawczyk-Moore algorithm are also presented. A simple package which allows symbolic computations to be carried out is described. Several applications for such a package are given. In particular, an interval form of Brown's method is presented.

Thesis (M.S.)--University of Wisconsin--Madison, 1989.
Typescript. eContent provider-neutral record in process. Description based on print version record. Includes bibliographical references (leaf 45).