After introducing concepts from convex analysis, we study how to continuously transform one convex function into another. A natural choice is the arithmetic average, as it is pointwise continuous; however, this choice fails to average functions with different domains. On the contrary, the proximal average is not only continuous (in the epi-topology) but can actually average functions with disjoint domains. In fact, the proximal average not only inherits strict convexity (like the arithmetic average) but also inherits smoothness and differentiability (unlike the arithmetic average). Then we introduce a computational framework for computer-aided convex analysis. Motivated by the proximal average, we notice that the class of piecewise linear-quadratic (PLQ) functions is closed under (positive) scalar multiplication, addition, Fenchel conjugation, and Moreau envelope. As a result, the PLQ framework gives rise to linear-time and linear-space algorithms for convex PLQ functions. We extend this framework to nonconvex PLQ functions and present an explicit convex hull algorithm. Finally, we discuss a method to find primal-dual symmetric antiderivatives from cyclically monotone operators. As these antiderivatives depend on the minimal and maximal Rockafellar functions [5, Theorem 3.5, Corollary 3.10], it turns out that the minimal and maximal function in [12, p.132,p.136] are indeed the same functions. Algorithms used to compute these antiderivatives can be formulated as shortest path problems.

Disertacijoje nagrinėjamos atsitiktinių keitinių problemos yra priskirtinos tikimybinei kombinatorikai. Gauti rezultatai aprašo visiškai adityviųjų funkcijų, apibrėžtų simetrinėje grupėje, reikšmių asimptotinius skirstinius Evenso tikimybinio mato atžvilgiu, kai grupės eilė neaprėžtai didėja. Išvestos adityviųjų funkcijų laipsninių ir faktorialinių momentų formulės. Funkcijų, išreiškiančių atsitiktinio keitinio ciklų su bet kokiais apribojimais skaičius, atveju rastos būtinos ir pakankamos ribinių tikimybinių dėsnių egzistavimo sąlygos. Išsamiai išnagrinėtas konvergavimas į Puasono, quasi-Puasono, Bernulio, binominio ir kitus skirstinius, sukoncentruotus sveikųjų neneigiamų skaičių aibėje. Rezultatai apibendrinti sveikareikšmių visiškai adityviųjų funkcijų klasėje. Darbe įrodytas bendras silpnasis didžiųjų skaičių dėsnis, rastos būtinos ir pakankamos adityviųjų funkcijų sekų pasiskirstymo funkcijų konvergavimo į išsigimusį nuliniame taške dėsnį egzistavimo sąlygos. Sprendžiamos problemos yra susietos su tikimybiniais vektorių, turinčių sveikąsias neneigiamas koordinates, uždaviniais. Adicinėje tokių vektorių pusgrupėje išnagrinėti multiplikatyviųjų funkcijų vidurkiai tikimybinio mato, vadinamo Evanso atrankos formule, atžvilgiu. Gauti tikslūs viršutinieji ir apatinieji įverčiai. Iš jų išplaukia svarbios atsitiktinių keitinių tikimybių savybės. Disertacijoje plėtojami faktorialinių momentų ir kiti kombinatoriniai bei tikimybiniai metodai. / In the thesis the examining problems of random permutations are attributed to the probabilistic combinatorics. Obtained results describe asymptotical distributions of completely additive functions values defined on a symmetric group with respect to Ewens probability measure, if the group order unbounded increases. Power and factorial moments formulae of additive functions are derived. There are established necessary and sufficient conditions under which the distributions of a number of cycles with restricted lengths obey the limit probability laws. The convergence to the Poisson, quasi-Poisson, Bernoulli, binomial and other distributions, defined on the positive whole - number set are exhaustively investigated. The results are generalized on the class of whole - number completely additive functions. The general weak law of large numbers is proved in the thesis, necessary and sufficient existence conditions, under which the distributions of the sequences of additive functions converge to the degenerate at the point zero limit law are established. Examining problems are related to the probability tasks of the vectors, which have whole - numbered nonnegative coordinates. The mean values of multiplicative functions defined on those vectors’ additive semigroup with respect to the Ewens measure, called Ewens Sampling Formula, and investigated. Lower and upper sharp estimates are obtained. From the latter results follow important probabilities’ properties of random... [to full text]

Exact solution of the Schr&ouml / dinger equation with some potentials is obtained. The normal and supersymmetric cases are considered. Deformed ring-shaped potential is solved in the parabolic and spherical coordinates. By taking appropriate values for the parameter q, similar results are obtained for Hulth&eacute / n and exponential type screened potentials. Similarly, Morse, P&ouml / schl-Teller and Hulth&eacute / n potentials are solved for the supersymmetric case. Supersymmetric solution of PT-/non-PT-symmetric and non-Hermitian Morse potential is also studied. The Nikiforov-Uvarov and Hamiltonian Hierarchy methods are used in the calculations. Eigenfunctions and corresponding energy eigenvalues are calculated analytically. Results are in good agreement with ones obtained before.

[Formulae and special characters can not be reproduced here. Please see the pdf version of the Abstract for an accurate reproduction.] The thesis provides a comprehensive study on the behaviour of sand with a small amount of fines [i.e.1/3 plastic fines and 2/3 non-plastic fines]. The work is carried out experimentally under axi-symmetric conditions using the triaxial apparatus. Conventional drained triaxial tests are conducted on isotropic consolidated specimens and all the tests indicate that sand with fines does not manifest any unusual behaviour under drained conditions. However in undrained shearing the so-called ???reverse behaviour??? is noted. The study demonstrates that the reverse behaviour conforms to the critical state framework because significant changes in the position and geometry of the critical state [CS] and isotropic consolidation [IL] lines are caused by the presence of fines. These changes cannot be adequately modelled by the intergranular void ratio as proposed by Thevanayagam and Mohan (2000). This study also demonstrates that the original state parameter [special character] as proposed by Been and Jefferies (1985) is not an adequate parameter to predict the undrained behaviour trend. A new parameter termed ???modified state parameter??? [special character] is proposed to account for the combined effects of density and confining pressure. The suitability of the modified state parameter to characterise the response of parent sand and sand with fines is assessed for a range of void ratios and confining pressures. The effect of drained stress history is an important factor affecting the subsequent undrained response. Drained pre-shearing to failure is found to improve considerably the subsequent undrained response to the extent that liquefaction may not occur. Different drained pre-shearing histories have different effects on the undrained response. However in these tests [special character] has limitations in quantifying the subsequent undrained stress-strain response. Hence, a new framework of ???yielding parameter??? [special character] extends the capability of [special character] and additional data is presented to demonstrate the suitability of this concept. The implementation of [special character] depends on whether the previous stress state reached during the stress history is below or at failure. The effects of drained pre-shearing on the position and movement of failure surface are investigated. It is found that drained pre-shearing to failure at larger confining pressures has the effect of shifting upwards part of the drained failure surface. The shift at larger stress ratios [special character] may be described in terms of dilatancy and modified state parameter at failure. The so-called strain path tests are conducted to study the influence of strain increment ratio on the deformation behaviour of sand with fines. It is found that the value of [special character] has significant effects on the stress-strain behaviour. Along negative [special character] paths the soil strain softens in the form of decreasing the shear resistance before reaching the failure state. In contrast, along positive [special character] paths the soil strain hardens to an asymptotic stress ratio. The asymptotic stress ratio decreases with increasing [special character] along a linear relationship. The framework of [special character] cannot quantify the stress-strain response along positive and negative strain paths. Consequently an ???instability parameter??? [special character] is proposed to extend the capabilities of [special character] and the reliability of this parameter is further assessed. The behaviour along a range of positive and negative [special character] paths is investigated on pre-sheared specimens. In negative [special character] the effective stress paths reach a surface located higher than the monotonic failure surface on which they trace downward towards the origin of stress space. The results indicate this surface may be the same as the drained failure envelope which has been shifted as a result of drained pre-shearing. In positive [special character] paths a large improvement is noted in both the strength and stiffness of the soil. Note The parameters [special character] and [special character] are all generalisations of [special character] so that the behaviour under complex conditions can be characterised.

abstract: Three-dimensional (3D) inductors with square, hexagonal and octagonal geometries have been designed and simulated in ANSYS HFSS. The inductors have been designed on Silicon substrate with through-hole via with different width, spacing and thickness. Spice modeling has been done in Agilent ADS and comparison has been made with results of custom excel based calculator and HFSS simulation results. Single ended quality factor was measured as 12.97 and differential ended quality factor was measured as 15.96 at a maximum operational frequency of 3.65GHz. The single ended and differential inductance was measured as 2.98nH and 2.88nH respectively at this frequency. Based on results a symmetric octagonal inductor design has been recommended to be used for application in RF biosensing. A system design has been proposed based on use of this inductor and principle of inductive sensing using magnetic labeling. / Dissertation/Thesis / M.S. Electrical Engineering 2012

Neste trabalho estudamos o problema de Dirichlet assintótico para a equação das superfícies mínimas em uma superfície de Cartan-Hadamard rotacionalmente simétrica e mostramos que o problema e unicamente solúvel para qualquer dado contínuo em seu bordo assintótico. / In this work we study the asymptotic Dirichlet problem for the minimal surface equation on rotationally symmetric Cartan-Hadamard surfaces. We prove that the problem is uniquely solvave for any continuous asymptotic boundary data.

A elaboração desta dissertação se constituiu na tentativa de pensar uma terceira margem no hospital psiquiátrico através de uma etnografia que enfrenta problemas que estão na base da própria consolidação da disciplina antropológica e dos pressupostos filosóficos que a constituíram. Analisa-se se aqueles que estão no hospital psiquiátrico na situação de pacientes seriam diferentes demais para que a disciplina que se propõe pensar a diferença possa levá-los a sério como objeto e, simultaneamente, sujeitos de pesquisa. A descrição etnográfica visa abranger a multiplicidade subjetiva produzida por internados a partir de uma Oficina de Criatividade e os efeitos ontológicos produzidos por maiorias morais que reivindicam um acesso privilegiado à realidade e ao bom juízo. Atento a processos subjetivos inusitados potencialmente capazes de problematizar os modelos dominantes de codificação, os quais poderiam ser desconsiderados em nome da autoridade de saberes especializados no campo da saúde mental, o efeito etnográfico multiplica o tecido político ao evidenciar processos de singularização por parte de usuários e moradores de um hospital-que-foi-hospício. / The formulation of this research consisted to attempt to think the third bank in the psychiatric hospital through ethnography facing problems that underlie the consolidation itself of anthropological discipline and its philosophical presuppositions. Those at the psychiatric hospital in the situation of patients would be too different for the discipline that think about the difference may take them seriously as an object and simultaneously research subjects? The ethnographic description addresses the subjective multiplicity produced from a Creativity Workshop inside the institution and the ontological effects produced by moral majorities who claim a privileged access to reality and good judgment. Attentive to unusual subjective processes potentially able to question the dominant models of coding which could be disregarded in the name of the authority of specialized knowledge in the mental health field, the ethnographic effect multiplies the political texture to emphasize subjective processes by users and hospitalized of a hospital that was hospice.

A formação e propagação de fissuras é um fenômeno observado em diversos materiais utilizados na engenharia, como concreto, metais, cerâmicas e rochas. Tendo em vista a grande influência que fissuras têm no comportamento global da estrutura o objetivo deste trabalho consiste na implementação de um modelo de fissura com descontinuidades fortes incorporadas a fim de analisar o processo de fratura em materiais quase-frágeis. A descontinuidade no campo de deslocamentos (descontinuidade forte) é representada através da introdução de graus de liberdade adicionais no interior do elemento finito, sendo esta abordagem denominada enriquecimento elementar (E-FEM). Nestes modelos a fissura pode se propagar em qualquer direção dentro do elemento finito, evitando a necessidade de redefinição da malha em cada etapa, além de fornecer resultados relativamente independentes da malha de elementos finitos utilizada. Por serem internos a cada elemento finito, os graus de liberdade adicionais podem ser eliminados da solução global por condensação estática. Desta forma as descontinuidades são definidas em nível de elemento e o modelo pode ser facilmente implementado em códigos computacionais existentes. O modelo implementado foi proposto por Dvorkin, Cuitiño e Gioia (1990), o qual pertence à classe de modelos com formulação assimétrica estaticamente e cinematicamente consistente (SKON). Esta formulação é caracterizada por garantir o movimento de corpo rígido entre as partes do elemento além de assegurar a continuidade de tensões na linha de fissura, resultando numa matriz de rigidez assimétrica. Diferentes relações constitutivas podem ser utilizadas para descrever o comportamento das regiões com e sem fissura. Portanto, para a região não fissurada, utilizouse um modelo constitutivo elástico linear e para a região fissurada foi analisada a performance de dois modelos constitutivos distintos: linear e exponencial. A capacidade de representar o comportamento de elementos estruturais fissurados foi ilustrada através de exemplos de tração e flexão comparados com outros modelos de fissura existentes e com resultados experimentais. Em relação aos modelos constitutivos para a linha da fissura, o modelo linear não se mostrou adequado por superestimar as tensões de pico além de apresentar um ramo de amolecimento mais frágil. Já o modelo exponencial mostrou-se bastante eficiente representando de forma correta o comportamento de materiais quase-frágeis. / The formation and propagation of cracks is a phenomenon observed in many materials used in engineering, such as concrete, metals, ceramics and rocks. In view of the influence of cracks in the global behavior of the structure, the aim of this work is the implementation of an embedded strong discontinuity model in order to analyze the fracture process in quasi-brittle materials. The discontinuity in the displacement field (strong discontinuity) is represented by the introduction of additional degrees of freedom within the finite element. This approach is called elemental enrichment (E-FEM). The embedded models allow the propagation of crack in any direction within the finite element, avoiding the need of remeshing and providing objective results (mesh independent). The additional degrees of freedom are introduced into the finite element, then these degrees can be eliminated from the global solution by static condensation and the model can be easily implemented in existent computational codes. The model used here was proposed by Dvorkin, Cuitiño and Gioia (1990), which belongs to the statically and kinematically optimal non-symmetric (SKON) formulation. In this formulation, the kinematics that allows for relative rigid body motion and the enforcement of the traction continuity are introduced at element level, resulting a non-symmetric formulation. Different constitutive relations can be used to describe the behavior of the zones with and without cracks. For the zone without cracks it was used a linear elastic model and for the cracked zone it was analyzed the behavior of two different constitutive models: linear and exponential. The ability of the model to represent the behavior of cracked structural elements was illustrated by bending and tensile tests and the results were compared with numerical and experimental data. Regarding the constitutive models for the fracture zone, it was concluded that the linear model was not suitable because it overestimated the maximum stress and promoted a britller softening. In contrast, the exponential model proved to be very efficient to represent the behavior of quasi-brittle materials.

This thesis introduces a new mathematical object: collection of arithmetic progressions with elements satisfying the inverse property, \j-th terms of i-th and (i+1)-th progressions are multiplicative inverses of each other modulo (j+1)-th term of i-th progression". Such a collection is uniquely de ned for any pair (a; d) of co-prime integers. The progressions of the collection are ordered. Thus we call it a sequence rather than a collection. The results of the thesis are on the following number theoretic, computational and cryptographic aspects of the defined sequence and its generalizations. The sequence is closely connected to the classical Euclidean algorithm. Precisely, certain consecutive progressions of the sequence form \groupings". The difference between the common differences of any two consecutive progressions of a grouping is same. The number of progressions in a grouping is connected to the quotient sequence of the Euclidean algorithm on co-prime input pairs. The research community has studied extensively the behavior of the Euclidean algorithm. For the rst time in the literature, the connection (proven in the thesis) shows what the quotients of the algorithm signify. Further, the leading terms of progressions within groupings satisfy a mirror image symmetry property, called \symmetricity". The property is subject to the quotient sequence of the Euclidean algorithm and divisors of integers of the form x2 y2 falling in specific intervals. The integers a, d are the primary quantities of the defined sequence in a computational sense. Given the two, leading term and common difference of any progression of the sequence can be computed in time quadratic in the binary length of d. On the other hand, the inverse computational question of finding (a; d), given information on some terms of the sequence, is interesting. This problem turns out to be hard as it requires finding solutions to an nearly-determined system of multivariate polynomial equations. Two sub-problems arising in this context are shown to be equivalent to the problem of factoring integers. The reduction to the factoring problem, in both cases, is probabilistic. Utilizing the computational difficulty of solving the inverse problem, and the sub-problems (mentioned above), we propose a symmetric-key cryptographic scheme (SKCS), and a public key cryptographic scheme (PKCS). The PKCS is also based on the hardness of the problem of finding square-roots modulo composite integers. Our proposal uses the same algorithmic and computational primitives for effecting both the PKCS and SKCS. In addition, we use the notion of the sequence of arithmetic progressions to design an entity authentication scheme. The proof of equivalence between one of the inverse computational problems (mentioned above) and integer factoring led us to formulate and investigate an independent problem concerning the largest divisor of integer N bounded by the square-root of N. We present some algorithmic and combinatorial results. In the course of the above investigations, we are led to certain open questions of number theoretic, combinatorial and algorithmic nature. These pertain to the quotient sequence of the Euclidean algorithm, divisors of integers of the form x2 y2 p in specific intervals, and the largest divisor of integer N bounded by N.

We consider linear and semilinear stochastic partial differential equations that in some sense can be viewed as being at the "endpoints" of the classical variational theory by Krylov and Rozovskii [25]. In terms of regularity of the coeffcients, the minimal assumption is boundedness and measurability, and a unique L2- valued solution is then readily available. We investigate its further properties, such as higher order integrability, boundedness, and continuity. The other class of equations considered here are the ones whose leading operators do not satisfy the strong coercivity condition, but only a degenerate version of it, and therefore are not covered by the classical theory. We derive solvability in Wmp spaces and also discuss their numerical approximation through finite different schemes.