Relevant bibliographies by topics / Probabilistic number theory

Academic literature on the topic 'Probabilistic number theory'

Author: Grafiati
Published: 4 June 2021
Last updated: 1 March 2023

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Contents

  1. Journal articles
  2. Dissertations / Theses
  3. Books
  4. Book chapters
  5. Conference papers
  6. Reports

Journal articles on the topic "Probabilistic number theory":

1

Indlekofer, K. H. "Number theory—probabilistic, heuristic, and computational approaches." Computers & Mathematics with Applications 43, no. 8-9 (April 2002): 1035–61. http://dx.doi.org/10.1016/s0898-1221(02)80012-8.

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Stakėnas, V. "On some inequalities of probabilistic number theory." Lithuanian Mathematical Journal 46, no. 2 (April 2006): 208–16. http://dx.doi.org/10.1007/s10986-006-0022-2.

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Erdõs, P. "Recent problems in probabilistic number theory and combinatorics." Advances in Applied Probability 24, no. 4 (December 1992): 766–67. http://dx.doi.org/10.1017/s0001867800024654.

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Stakėnas, Vilius. "Jonas Kubilius and genesis of probabilistic number theory." Lithuanian Mathematical Journal 55, no. 1 (January 2015): 25–47. http://dx.doi.org/10.1007/s10986-015-9263-2.

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Zhang, Wen-Bin. "Probabilistic number theory in additive arithmetic semigroups II." Mathematische Zeitschrift 235, no. 4 (December 1, 2000): 747–816. http://dx.doi.org/10.1007/s002090000165.

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Elliott, P. D. T. A. "Jonas Kubilius and Probabilistic Number Theory Some Personal Reflections." Lithuanian Mathematical Journal 55, no. 1 (January 2015): 2–24. http://dx.doi.org/10.1007/s10986-015-9262-3.

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Daili, Noureddine. "DENSITIES AND NATURAL INTEGRABILITY. APPLICATIONS IN PROBABILISTIC NUMBER THEORY." JP Journal of Algebra, Number Theory and Applications 47, no. 1 (July 1, 2020): 51–65. http://dx.doi.org/10.17654/nt047010051.

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Lokutsievskiy, Lev V. "Optimal probabilistic search." Sbornik: Mathematics 202, no. 5 (May 31, 2011): 697–719. http://dx.doi.org/10.1070/sm2011v202n05abeh004162.

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Aistleitner, Christoph, and Christian Elsholtz. "The Central Limit Theorem for Subsequences in Probabilistic Number Theory." Canadian Journal of Mathematics 64, no. 6 (December 1, 2012): 1201–21. http://dx.doi.org/10.4153/cjm-2011-074-1.

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Abstract Let (nk)k≥1 be an increasing sequence of positive integers, and let f (x) be a real function satisfyingIf the distribution ofconverges to a Gaussian distribution. In the casethere is a complex interplay between the analytic properties of the function f , the number-theoretic properties of (nk)k≥1, and the limit distribution of (2).In this paper we prove that any sequence (nk)k≥1 satisfying lim contains a nontrivial subsequence (mk)k≥1 such that for any function satisfying (1) the distribution ofconverges to a Gaussian distribution. This result is best possible: for any ε > 0 there exists a sequence (nk)k≥1 satisfying lim such that for every nontrivial subsequence (mk)k≥1 of (nk)k≥1 the distribution of (2) does not converge to a Gaussian distribution for some f.Our result can be viewed as a Ramsey type result: a sufficiently dense increasing integer sequence contains a subsequence having a certain requested number-theoretic property.
10

Haik, Sumayra. "AFFINE LINES AND ADVANCED PROBABILISTIC MODEL THEORY." Mathematical Statistician and Engineering Applications 70, no. 2 (February 26, 2021): 01–14. http://dx.doi.org/10.17762/msea.v70i2.8.

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Letjbe an almost surely algebraic, meromorphic domain.Recent developments in singular logic [36, 22] have raised the question ofwhether every pseudo-Dedekind number is ultra-Gaussian and singular.We show tha
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Journal articles Dissertations / Theses Books

Dissertations / Theses on the topic "Probabilistic number theory":

1

Harper, Adam James. "Some topics in analytic and probabilistic number theory." Thesis, University of Cambridge, 2012. https://www.repository.cam.ac.uk/handle/1810/265539.

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This dissertation studies four problems in analytic and probabilistic number theory. Two of the problems are about a certain random number theoretic object, namely a random multiplicative function. The other two problems are about smooth numbers (i.e. numbers only having small prime factors), both in their own right and in their application to finding solutions to S-unit equations over the integers. Thus all four problems are concerned, in different ways, with _understanding the multiplicative structure of the integers. More precisely, we will establish that certain sums of a random multiplicative function satisfy a normal approximation (i.e. a central limit theorem) , but that the complete sum over all integers less than x does not satisfy such an approximation. This reflects certain facts about the number and size of the prime factors of a typical integer. Our proofs use martingale methods, as well as a conditioning argument special to this problem. Next, we will prove an almost sure omega result for the sum of a random multiplicative function, substantially improving the existing result of Halasz. We will do this using a connection between sums of a random multiplicative function and a certain random trigonometric sum process, so that the heart of our work is proving precise results about the suprema of a class of Gaussian random processes. Switching to the study of smooth numbers, we will establish an equidistribution result for the y-smooth numbers less than x among arithmetic progressions to modulus q, asymptotically as (logx)/(logq)-+ oo, subject to a certain condition on the relative sizes of y and q. The main point of this work is that it does not require any restrictions on the relative sizes of x and y. Our proofs use a simple majorant principle for trigonometric sums, together with general tools such as a smoothed explicit formula. Finally, we will prove lower bounds for the possible number of solutions of some S-unit equations over the integers. For example, we will show that there exist arbitrarily large sets S of prime numbers such that the equation a+ l = c has at least exp{(#S)116- �} solutions (a, c) with all their prime factors from S. We will do this by using discrete forms of the circle method, and the multiplicative large sieve, to count the solutions of certain auxiliary linear equations.
2

Hughes, Garry. "Distribution of additive functions in algebraic number fields." Title page, contents and summary only, 1987. http://web4.library.adelaide.edu.au/theses/09SM/09smh893.pdf.

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Zhao, Wenzhong. "Probabilistic databases and their application." Lexington, Ky. : [University of Kentucky Libraries], 2004. http://lib.uky.edu/ETD/ukycosc2004d00183/wzhao0.pdf.

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Thesis (Ph. D.)--University of Kentucky, 2004.
Title from document title page (viewed Jan. 7, 2005). Document formatted into pages; contains x, 180p. : ill. Includes abstract and vita. Includes bibliographical references (p. 173-178).
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Lloyd, James Robert. "Representation, learning, description and criticism of probabilistic models with applications to networks, functions and relational data." Thesis, University of Cambridge, 2015. https://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.709264.

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Li, Xiang, and 李想. "Managing query quality in probabilistic databases." Thesis, The University of Hong Kong (Pokfulam, Hong Kong), 2011. http://hub.hku.hk/bib/B47753134.

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In many emerging applications, such as sensor networks, location-based services, and data integration, the database is inherently uncertain. To handle a large amount of uncertain data, probabilistic databases have been recently proposed, where probabilistic queries are enabled to provide answers with statistical guarantees. In this thesis, we study the important issues of managing the quality of a probabilistic database. We first address the problem of measuring the ambiguity, or quality, of a probabilistic query. This is accomplished by computing the PWS-quality score, a recently proposed measure for quantifying the ambiguity of query answers under the possible world semantics. We study the computation of the PWS-quality for the top-k query. This problem is not trivial, since directly computing the top-k query score is computationally expensive. To tackle this challenge, we propose efficient approximate algorithms for deriving the quality score of a top-k query. We have performed experiments on both synthetic and real data to validate their performance and accuracy. Our second contribution is to study how to use the PWS-quality score to coordinate the process of cleaning uncertain data. Removing ambiguous data from a probabilistic database can often give us a higher-quality query result. However, this operation requires some external knowledge (e.g., an updated value from a sensor source), and is thus not without cost. It is important to choose the correct object to clean, in order to (1) achieve a high quality gain, and (2) incur a low cleaning cost. In this thesis, we examine different cleaning methods for a probabilistic top-k query. We also study an interesting problem where different query users have their own budgets available for cleaning. We demonstrate how an optimal solution, in terms of the lowest cleaning costs, can be achieved, for probabilistic range and maximum queries. An extensive evaluation reveals that these solutions are highly efficient and accurate.
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Computer Science
Master
Master of Philosophy
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Rotondo, Pablo. "Probabilistic studies in number theory and word combinatorics : instances of dynamical analysis." Thesis, Sorbonne Paris Cité, 2018. http://www.theses.fr/2018USPCC213/document.

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L'analyse dynamique intègre des outils propres aux systèmes dynamiques (comme l'opérateur de transfert) au cadre de la combinatoire analytique, et permet ainsi l'analyse d'un grand nombre d'algorithmes et objets qu'on peut associer naturellement à un système dynamique. Dans ce manuscrit de thèse, nous présentons, dans la perspective de l'analyse dynamique, l'étude probabiliste de plusieurs problèmes qui semblent à priori bien différents : l'analyse probabiliste de la fonction de récurrence des mots de Sturm, et l'étude probabiliste de l'algorithme du “logarithme continu”. Les mots de Sturm constituent une famille omniprésente en combinatoire des mots. Ce sont, dans un sens précis, les mots les plus simples qui ne sont pas ultimement périodiques. Les mots de Sturm ont déjà été beaucoup étudiés, notamment par Morse et Hedlund (1940) qui en ont exhibé une caractérisation fondamentale comme des codages discrets de droites à pente irrationnelle. Ce résultat relie ainsi les mots de Sturm au système dynamique d'Euclide. Les mots de Sturm n'avaient jamais été étudiés d'un point de vue probabiliste. Ici nous introduisons deux modèles probabilistes naturels (et bien complémentaires) et y analysons le comportement probabiliste (et asymptotique) de la “fonction de récurrence” ; nous quantifions sa valeur moyenne et décrivons sa distribution sous chacun de ces deux modèles : l'un est naturel du point de vue algorithmique (mais original du point de vue de l'analyse dynamique), et l'autre permet naturellement de quantifier des classes de plus mauvais cas. Nous discutons la relation entre ces deux modèles et leurs méthodes respectives, en exhibant un lien potentiel qui utilise la transformée de Mellin. Nous avons aussi considéré (et c'est un travail en cours qui vise à unifier les approches) les mots associés à deux familles particulières de pentes : les pentes irrationnelles quadratiques, et les pentes rationnelles (qui donnent lieu aux mots de Christoffel). L'algorithme du logarithme continu est introduit par Gosper dans Hakmem (1978) comme une mutation de l'algorithme classique des fractions continues. Il calcule le plus grand commun diviseur de deux nombres naturels en utilisant uniquement des shifts binaires et des soustractions. Le pire des cas a été étudié récemment par Shallit (2016), qui a donné des bornes précises pour le nombre d'étapes et a exhibé une famille d'entrées sur laquelle l'algorithme atteint cette borne. Dans cette thèse, nous étudions le nombre moyen d'étapes, tout comme d'autres paramètres importants de l'algorithme. Grâce à des méthodes d'analyse dynamique, nous exhibons des constantes mathématiques précises. Le système dynamique ressemble à première vue à celui d'Euclide, et a été étudié d'abord par Chan (2005) avec des méthodes ergodiques. Cependant, la présence des puissances de 2 dans les quotients change la nature de l'algorithme et donne une nature dyadique aux principaux paramètres de l'algorithme, qui ne peuvent donc pas être simplement caractérisés dans le monde réel.C'est pourquoi nous introduisons un nouveau système dynamique, avec une nouvelle composante dyadique, et travaillons dans ce système à deux composantes, l'une réelle, et l'autre dyadique. Grâce à ce nouveau système mixte, nous obtenons l'analyse en moyenne de l'algorithme
Dynamical Analysis incorporates tools from dynamical systems, namely theTransfer Operator, into the framework of Analytic Combinatorics, permitting the analysis of numerous algorithms and objects naturally associated with an underlying dynamical system.This dissertation presents, in the integrated framework of Dynamical Analysis, the probabilistic analysis of seemingly distinct problems in a unified way: the probabilistic study of the recurrence function of Sturmian words, and the probabilistic study of the Continued Logarithm algorithm.Sturmian words are a fundamental family of words in Word Combinatorics. They are in a precise sense the simplest infinite words that are not eventually periodic. Sturmian words have been well studied over the years, notably by Morse and Hedlund (1940) who demonstrated that they present a notable number theoretical characterization as discrete codings of lines with irrationalslope, relating them naturally to dynamical systems, in particular the Euclidean dynamical system. These words have never been studied from a probabilistic perspective. Here, we quantify the recurrence properties of a ``random'' Sturmian word, which are dictated by the so-called ``recurrence function''; we perform a complete asymptotic probabilistic study of this function, quantifying its mean and describing its distribution under two different probabilistic models, which present different virtues: one is a naturaly choice from an algorithmic point of view (but is innovative from the point of view of dynamical analysis), while the other allows a natural quantification of the worst-case growth of the recurrence function. We discuss the relation between these two distinct models and their respective techniques, explaining also how the two seemingly different techniques employed could be linked through the use of the Mellin transform. In this dissertation we also discuss our ongoing work regarding two special families of Sturmian words: those associated with a quadratic irrational slope, and those with a rational slope (not properly Sturmian). Our work seems to show the possibility of a unified study.The Continued Logarithm Algorithm, introduced by Gosper in Hakmem (1978) as a mutation of classical continued fractions, computes the greatest common divisor of two natural numbers by performing division-like steps involving only binary shifts and substractions. Its worst-case performance was studied recently by Shallit (2016), who showed a precise upper-bound for the number of steps and gave a family of inputs attaining this bound. In this dissertation we employ dynamical analysis to study the average running time of the algorithm, giving precise mathematical constants for the asymptotics, as well as other parameters of interest. The underlying dynamical system is akin to the Euclidean one, and was first studied by Chan (around 2005) from an ergodic, but the presence of powers of 2 in the quotients ingrains into the central parameters a dyadic flavour that cannot be grasped solely by studying this system. We thus introduce a dyadic component and deal with a two-component system. With this new mixed system at hand, we then provide a complete average-case analysis of the algorithm by Dynamical Analysis
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Pariente, Cesar Alberto Bravo. "Um método probabilístico em combinatória." Universidade de São Paulo, 1996. http://www.teses.usp.br/teses/disponiveis/45/45132/tde-07052010-163719/.

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O presente trabalho é um esforço de apresentar, organizado em forma de survey, um conjunto de resultados que ilustram a aplicação de um certo método probabilístico. Embora não apresentemos resultados novos na área, acreditamos que a apresentação sistemática destes resultados pode servir para a compreensão de uma ferramenta útil para quem usa dos métodos probabilísticos na sua pesquisa em combinatória. Os resultados de que falaremos tem aparecido na última década na literatura especializada e foram usados na investigação de problemas que resitiram a outras aproximações mais clássicas. Em vez de teorizar sobre o método a apresentar, nós adotaremos a estratégia de apresentar três problemas, usando-os como exemplos práticos da aplicação do método em questão. Surpeendentemente, apesar da dificuldade que apresentaram para ser resolvidos, estes problemas compartilham a caraterística de poder ser formulados muito intuitivamente, como veremos no Capítulo 1. Devemos advertir que embora os problemas que conduzem nossa exposição pertençam a áreas tão diferentes quanto teoria de números, geometria e combinatória, nosso intuito é fazer énfase no que de comum tem as suas soluções e não das posteriores implicações que estes problemas tenham nas suas respectivas áreas. Ocasionalmente comentaremos sim, outras possíveis aplicações das ferramentas usadas para solucionar estes problemas de motivação. Os problemas de que trataremos tem-se caracterizado por aguardar várias décadas em espera de solução: O primeiro, da teoria de números, surgiu na pesquisa de séries de Fourier que Sidon realizava a princípios de século e foi proposto por ele a Erdös em 1932. Embora tenham havido, desde 1950, diversos avanços na pesquisa deste problema, o resultado de que falaremos data de 1981. Já o segundo problema, da geometria, é uma conjectura formulada em 1951 por Heilbronn e refutada finalmente em 1982. O último problema, de combinatória, é uma conjectura de Erdös e Hanani de 1963, que foi tratada em diversos casos particulares até ser finalmente resolvida em toda sua generalidade em 1985.
The following work is an effort to present, in survey form, a collection of results that illustrate the application of a certain probabilistic method in combinatorics. We do not present new results in the area; however, we do believe that the systematic presentation of these results can help those who use probabilistic methods comprenhend this useful technique. The results we refer to have appeared over the last decade in the research literature and were used in the investigation of problems which have resisted other, more classical, approaches. Instead of theorizing about the method, we adopted the strategy of presenting three problems, using them as practical examples of the application of the method in question. Surpisingly, despite the difficulty of solutions to these problems, they share the characteristic of being able to be formulated very intuitively, as we will see in Chapter One. We should warn the reader that despite the fact that the problems which drive our discussion belong to such different fields as number theory, geometry and combinatorics, our goal is to place emphasis on what their solutions have in common and not on the subsequent implications that these problems have in their respective fields. Occasionally, we will comment on other potential applications of the tools utilized to solve these problems. The problems which we are discussing can be characterized by the decades-long wait for their solution: the first, from number theory, arose from the research in Fourier series conducted by Sidon at the beginning of the century and was proposed by him to Erdös in 1932. Since 1950, there have been diverse advances in the understanding of this problem, but the result we talk of comes from 1981. The second problem, from geometry, is a conjecture formulated in 1951 by Heilbronn and finally refuted in 1982. The last problem, from combinatorics, is a conjecture formulated by Erdös and Hanani in 1963 that was treated in several particular cases but was only solved in its entirety in 1985.
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Schimit, Pedro Henrique Triguis. "Modelagem e controle de propagação de epidemias usando autômatos celulares e teoria de jogos." Universidade de São Paulo, 2010. http://www.teses.usp.br/teses/disponiveis/3/3139/tde-05122011-153541/.

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Estuda-se o espalhamento de doenças contagiosas utilizando modelos suscetível-infectado-recuperado (SIR) representados por equações diferenciais ordinárias (EDOs) e por autômatos celulares probabilistas (ACPs) conectados por redes aleatórias. Cada indivíduo (célula) do reticulado do ACP sofre a influência de outros, sendo que a probabilidade de ocorrer interação com os mais próximos é maior. Efetuam-se simulações para investigar como a propagação da doença é afetada pela topologia de acoplamento da população. Comparam-se os resultados numéricos obtidos com o modelo baseado em ACPs aleatoriamente conectados com os resultados obtidos com o modelo descrito por EDOs. Conclui-se que considerar a estrutura topológica da população pode dificultar a caracterização da doença, a partir da observação da evolução temporal do número de infectados. Conclui-se também que isolar alguns infectados causa o mesmo efeito do que isolar muitos suscetíveis. Além disso, analisa-se uma estratégia de vacinação com base em teoria dos jogos. Nesse jogo, o governo tenta minimizar os gastos para controlar a epidemia. Como resultado, o governo realiza campanhas quase-periódicas de vacinação.
The spreading of contagious diseases is studied by using susceptible-infected-recovered (SIR) models represented by ordinary differential equations (ODE) and by probabilistic cellular automata (PCA) connected by random networks. Each individual (cell) of the PCA lattice experiences the influence of others, where the probability of occurring interaction with the nearest ones is higher. Simulations for investigating how the disease propagation is affected by the coupling topology of the population are performed. The numerical results obtained with the model based on randomly connected PCA are compared to the results obtained with the model described by ODE. It is concluded that considering the topological structure of the population can pose difficulties for characterizing the disease, from the observation of the time evolution of the number of infected individuals. It is also concluded that isolating a few infected subjects can cause the same effect than isolating many susceptible individuals. Furthermore, a vaccination strategy based on game theory is analyzed. In this game, the government tries to minimize the expenses for controlling the epidemic. As consequence, the government implements quasi-periodic vaccination campaigns.
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Silva, Everton Juliano da. "Uma demonstração analítica do teorema de Erdös-Kac." Universidade de São Paulo, 2014. http://www.teses.usp.br/teses/disponiveis/45/45131/tde-24032015-132813/.

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Em teoria dos números, o teorema de Erdös-Kac, também conhecido como o teorema fundamental de teoria probabilística dos números, diz que se w(n) denota a quantidade de fatores primos distintos de n, então a sequência de funções de distribuições N definidas por FN(x) = (1/N) #{n <= N : (w(n) log log N)/(log log N)^(1/2)} <= x}, converge uniformemente sobre R para a distribuição normal padrão. Neste trabalho desenvolvemos todos os teoremas necessários para uma demonstração analítica, que nos permitirá encontrar a ordem de erro da convergência acima.
In number theory, the Erdös-Kac theorem, also known as the fundamental theorem of probabilistic number theory, states that if w(n) is the number of distinct prime factors of n, then the sequence of distribution functions N, defined by FN(x) = (1/N) #{n <= N : (w(n) log log N)/(log log N)^(1/2)} <= x}, converges uniformly on R to the standard normal distribution. In this work we developed all theorems needed to an analytic demonstration, which will allow us to find an order of error of the above convergence.
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Shi, Lingsheng. "Numbers and topologies." Doctoral thesis, Humboldt-Universität zu Berlin, Mathematisch-Naturwissenschaftliche Fakultät II, 2003. http://dx.doi.org/10.18452/14871.

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In der Ramsey Theorie fuer Graphen haben Burr und Erdos vor nunmehr fast dreissig Jahren zwei Vermutungen formuliert, die sich als richtungsweisend erwiesen haben. Es geht darum diejenigen Graphen zu charakterisieren, deren Ramsey Zahlen linear in der Anzahl der Knoten wachsen. Diese Vermutungen besagen, dass Ramsey Zahlen linear fuer alle degenerierten Graphen wachsen und dass die Ramsey Zahlen von Wuerfeln linear wachsen. Ein Ziel dieser Dissertation ist es, abgeschwaechte Varianten dieser Vermutungen zu beweisen. In der topologischen Ramseytheorie bewies Kojman vor kurzem eine topologische Umkehrung des Satzes von Hindman und fuehrte gleichzeitig sogenannte Hindman-Raeume und van der Waerden-Raeume ein (beide sind eine Teilmenge der folgenkompakten Raeume), die jeweils zum Satz von Hindman beziehungsweise zum Satz von van der Waerden korrespondieren. In der Dissertation wird zum einen eine Verstaerkung der Umkehrung des Satzes von van der Waerden bewiesen. Weiterhin wird der Begriff der Differentialkompaktheit eingefuehrt, der sich in diesem Zusammenhang ergibt und der eng mit Hindman-Raeumen verknuepft ist. Dabei wird auch die Beziehung zwischen Differentialkompaktheit und anderen topologischen Raeumen untersucht. Im letzten Abschnitt des zweiten Teils werden kompakte dynamische Systeme verwendet, um ein klassisches Ramsey-Ergebnis von Brown und Hindman et al. ueber stueckweise syndetische Mengen ueber natuerlichen Zahlen und diskreten Halbgruppen auf lokal zusammenhaengende Halbgruppen zu verallgemeinern.
In graph Ramsey theory, Burr and Erdos in 1970s posed two conjectures which may be considered as initial steps toward the problem of characterizing the set of graphs for which Ramsey numbers grow linearly in their orders. One conjecture is that Ramsey numbers grow linearly for all degenerate graphs and the other is that Ramsey numbers grow linearly for cubes. Though unable to settle these two conjectures, we have contributed many weaker versions that support the likely truth of the first conjecture and obtained a polynomial upper bound for the Ramsey numbers of cubes that considerably improves all previous bounds and comes close to the linear bound in the second conjecture. In topological Ramsey theory, Kojman recently observed a topological converse of Hindman's theorem and then introduced the so-called Hindman space and van der Waerden space (both of which are stronger than sequentially compact spaces) corresponding respectively to Hindman's theorem and van der Waerden's theorem. In this thesis, we will strengthen the topological converse of Hindman's theorem by using canonical Ramsey theorem, and introduce differential compactness that arises naturally in this context and study its relations to other spaces as well. Also by using compact dynamical systems, we will extend a classical Ramsey type theorem of Brown and Hindman et al on piecewise syndetic sets from natural numbers and discrete semigroups to locally connected semigroups.
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Books on the topic "Probabilistic number theory":

1

Tenenbaum, Gerald. Introduction to analytic and probabilistic number theory. Cambridge: Cambridge University Press, 1995.

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Tenenbaum, Gerald. Jie xi yu gai lü shu lun dao yin =: Jiexi yu gailü shulun daoyin. 8th ed. Beijing: Gao deng jiao yu chu ban she, 2011.

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Suciu, Dan. Probabilistic databases. San Rafael, Calif. (1537 Fourth Street, San Rafael, CA 94901 USA): Morgan & Claypool, 2011.

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Kubilius, Jonas. Analiziniai ir tikimybiniai metodai skaičių teorijoje: Trečiosios tarptautines konferencijos J. Kubiliaus garbei darbų rinkinys / redaktoriai, A. Dubickas, A. Laurinčikas, E. Manstavičius = Analytic and probabilistic methods in number theory : proceedings of the third international conference in honour of J. Kubilius, Palanga, Lithuania, 24-28 September 2001. Vilnius: TEV, 2002.

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Laurincikas, E., E. Manstavicius, and V. Stakenas, eds. Analytic and Probabilistic Methods in Number Theory. Berlin, Boston: DE GRUYTER, 1997. http://dx.doi.org/10.1515/9783110944648.

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Tenenbaum, Gerald. Introduction à la théorie analytique et probabiliste des nombres. 2nd ed. Paris: Société Mathématique de France, 1995.

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Jonas, Kubilius, Laurinčikas Antanas, Manstavičius E, and Stakėnas V, eds. Analytic and probabilistic methods in number theory: Proceedings of the second international conference in honour of J. Kubilius, Palanga, Lithuania, 23-27 September 1996. Vilnius, Lithuania: TEV, 1997.

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Knopfmacher, John. Number theory arising from finite fields: Analytic and probabilistic theory. New York: Marcel Dekker, 2001.

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Gut, Allan. Probability: A graduate course. New York: Springer, 2013.

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Boultbee, R. Rounded numbers. [Toronto, Ont.?: s.n.], 1990.

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Book chapters on the topic "Probabilistic number theory":

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Murty, M. Ram, and V. Kumar Murty. "Probabilistic Number Theory." In The Mathematical Legacy of Srinivasa Ramanujan, 149–53. India: Springer India, 2012. http://dx.doi.org/10.1007/978-81-322-0770-2_11.

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Boston, Nigel. "A probabilistic generalization of the Riemann zeta function." In Analytic Number Theory, 155–62. Boston, MA: Birkhäuser Boston, 1996. http://dx.doi.org/10.1007/978-1-4612-4086-0_8.

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Elliott, P. D. T. A. "Progress in Probabilistic Number Theory." In Grundlehren der mathematischen Wissenschaften, 423–48. New York, NY: Springer New York, 1985. http://dx.doi.org/10.1007/978-1-4613-8548-6_25.

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Kubilius, Jonas. "Recent Progress in Probabilistic Number Theory." In Asymptotic Methods in Probability and Statistics with Applications, 507–19. Boston, MA: Birkhäuser Boston, 2001. http://dx.doi.org/10.1007/978-1-4612-0209-7_36.

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Indlekofer, Karl-Heinz. "A New Method in Probabilistic Number Theory." In Probability Theory and Applications, 299–308. Dordrecht: Springer Netherlands, 1992. http://dx.doi.org/10.1007/978-94-011-2817-9_19.

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Kubilius, J. "On some inequalities in the probabilistic number theory." In Lecture Notes in Mathematics, 214–20. Berlin, Heidelberg: Springer Berlin Heidelberg, 1988. http://dx.doi.org/10.1007/bfb0078476.

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Guan, Ji, and Nengkun Yu. "A Probabilistic Logic for Verifying Continuous-time Markov Chains." In Tools and Algorithms for the Construction and Analysis of Systems, 3–21. Cham: Springer International Publishing, 2022. http://dx.doi.org/10.1007/978-3-030-99527-0_1.

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AbstractA continuous-time Markov chain (CTMC) execution is a continuous class of probability distributions over states. This paper proposes a probabilistic linear-time temporal logic, namely continuous-time linear logic (CLL), to reason about the probability distribution execution of CTMCs. We define the syntax of CLL on the space of probability distributions. The syntax of CLL includes multiphase timed until formulas, and the semantics of CLL allows time reset to study relatively temporal properties. We derive a corresponding model-checking algorithm for CLL formulas. The correctness of the model-checking algorithm depends on Schanuel’s conjecture, a central open problem in transcendental number theory. Furthermore, we provide a running example of CTMCs to illustrate our method.
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Devroye, Luc, László Györfi, and Gábor Lugosi. "Uniform Laws of Large Numbers." In A Probabilistic Theory of Pattern Recognition, 489–506. New York, NY: Springer New York, 1996. http://dx.doi.org/10.1007/978-1-4612-0711-5_29.

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Fullér, Robert, István Á. Harmati, and Péter Várlaki. "On Probabilistic Correlation Coefficients for Fuzzy Numbers." In Aspects of Computational Intelligence: Theory and Applications, 249–63. Berlin, Heidelberg: Springer Berlin Heidelberg, 2013. http://dx.doi.org/10.1007/978-3-642-30668-6_17.

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Noschinski, Lars. "Proof Pearl: A Probabilistic Proof for the Girth-Chromatic Number Theorem." In Interactive Theorem Proving, 393–404. Berlin, Heidelberg: Springer Berlin Heidelberg, 2012. http://dx.doi.org/10.1007/978-3-642-32347-8_27.

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Conference papers on the topic "Probabilistic number theory":

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Flodin, Larkin, and Arya Mazumdar. "Probabilistic Group Testing with a Linear Number of Tests." In 2021 IEEE International Symposium on Information Theory (ISIT). IEEE, 2021. http://dx.doi.org/10.1109/isit45174.2021.9517841.

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Pigott, Ronald. "Advanced Probabilistic Design of Multi-Degree of Freedom Systems Subjected to a Number of Discreet Excitation Frequencies." In ASME 1997 International Mechanical Engineering Congress and Exposition. American Society of Mechanical Engineers, 1997. http://dx.doi.org/10.1115/imece1997-0031.

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Abstract From the beginning, engineers have focused on the special case of determinism in the design process, and an enormous methodology has been developed to support this approach. Today, however, customers are demanding greater reliability and are imposing greater penalties for failure. In order to achieve higher reliability, and in order to assess the risk of failure, probabilistic approaches will almost certainly have to be employed. While designers have always used probability in their work, it has usually been done with risk represented in a single factor of safety. This paper focuses on the application of probability theory to the design of multi-degree of freedom systems that are subjected to a number of discreet excitation frequencies. Problems of this nature are often encountered in rotating machines where the excitation frequencies are multiples of the speed of rotation. In a previous paper (Pigott, 1996), only variability in the system natural frequencies was considered, and the probability of exceeding a certain vibratory stress was determined. It was shown that, at the reliability levels of interest, it is necessary to consider multiple mode contributions to the total vibratory stress. In this paper, the previous approach is extended to include variability in the magnitude of the excitation force, system damping, and system fatigue strength leading to the determination of a total probability of failure due to high cycle fatigue. The results of applying the deterministic analysis, the first level of probabilistic analysis (Pigott, 1996) and the full probabilistic analysis are presented for a sample system. The benefits from using the probabilistic approach for component design are also discussed.
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Kersten, Daniel, and David C. Knill. "Reflectance estimation and lightness constancy: a probabilistic approach." In OSA Annual Meeting. Washington, D.C.: Optica Publishing Group, 1987. http://dx.doi.org/10.1364/oam.1987.thc2.

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The image of a natural scene can be considered a product of reflectance and effective illumination functions. One goal of image understanding is the estimation of the reflectance and illumination from image luminance data. Human observers are remarkably good at inferring reflectance changes from an image. One well-known aspect of this ability is lightness constancy. However, because there are two unknowns for every data point in the image, this problem is ill-posed, and it does not have a unique solution without prior constraints on the class of reflectance and illumination functions. We use Markov random fields to model and thereby constrain the class of reflectance and illumination functions. Our computational goal is taken to be the maximization of the posterior distribution of the reflectance and illumination conditional on the image. This goal is achieved using stochastic relaxation. This approach provides a general framework for quantifying the computational theory apart from specific algorithmic implementations for a number of problems of early vision.
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He, Jian Wen, and Ying Min Low. "Probabilistic Assessment of the Clashing Between Flexible Marine Risers." In ASME 2010 29th International Conference on Ocean, Offshore and Arctic Engineering. ASMEDC, 2010. http://dx.doi.org/10.1115/omae2010-20046.

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Flexible marine risers are compliant to external forces from waves, current and platform motions, and clashing between risers is an important concern. In deepwater developments where the number of connected risers is large, it is not economical to space them too far apart. In this regard, it is necessary to establish the probability of riser clashing throughout the service life; however, at present there appears to be no systematic procedure for assessing this risk. This paper presents a novel procedure for estimating the probability of riser clashing based on post-processing results obtained from time domain simulations of flexible risers subjected to random wave loads. First, an efficient technique is employed to sieve out critical pairs among riser elements. From these element pairs, the time history of a normalized clearance parameter is derived from the nodal displacements of the elements. Subsequently, the mean up-crossing rate of this parameter is extracted and extrapolated to the threshold of clashing using extreme value theory. As this method is still in its early developmental stage, critical effects such as vortex-induced vibrations and wake interference will not be considered in the present work.
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Karge, Jonas, and Sebastian Rudolph. "The More the Worst-Case-Merrier: A Generalized Condorcet Jury Theorem for Belief Fusion." In 19th International Conference on Principles of Knowledge Representation and Reasoning {KR-2022}. California: International Joint Conferences on Artificial Intelligence Organization, 2022. http://dx.doi.org/10.24963/kr.2022/21.

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In multi-agent belief fusion, there is increasing interest in results and methods from social choice theory. As a theoretical cornerstone, the Condorcet Jury Theorem (CJT) states that given a number of equally competent, independent agents where each is more likely to guess the true out of two alternatives, the chances of determining this objective truth by majority voting increase with the number of participating agents, approaching certainty. Past generalizations of the CJT have shown that some of its underlying assumptions can be weakened. Motivated by requirements from practical belief fusion scenarios, we provide a significant further generalization that subsumes several of the previous ones. Our considered setting simultaneously allows for heterogeneous competence levels across the agents (even tolerating entirely incompetent or even malicious voters), and voting for any number of alternatives from a finite set. We derive practical lower bounds for the numbers of agents needed to give probabilistic guarantees for determining the true state through approval voting. We also demonstrate that the non-asymptotic part of the CJT fails in our setting for arbitrarily high numbers of voters.
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Skelonis, Carl D., M. Brett Shelton, and Glenn T. Burney. "A Probabilistic Gas Touched Length Analysis." In ASME 2011 Power Conference collocated with JSME ICOPE 2011. ASMEDC, 2011. http://dx.doi.org/10.1115/power2011-55298.

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Field measurements of the steamside oxide thickness for high temperature (> 850F) boiler tubing subject to the accumulation of creep damage often are made to support deterministic assessments of the remaining life. Most often, these inspections are undertaken to understand the condition of the tubing at some particular location along a circuit, often as a result of a tube failure. The life assessment is based on relationships that have been developed between oxide growth kinetics and temperature. Unfortunately, because of variability in the oxide-temperature relationships reflecting different original data sets, and because of the inherent uncertainty in materials properties where heat-specific test data is not available, there typically exists a broad range of uncertainty in the deterministic assessment results. Large utility-type boilers typically contain a number of high temperature sections, including various stages of superheat and reheat, each of which will contain miles of tubing. Since the temperature derived from an oxide thickness measurement is relevant only to the specific location where the measurement was made, the deterministically derived life calculation is also specific to that location. As a result, the attempt to draw conclusions regarding the condition of an entire superheater or reheater section from measurements made at only one or two locations in those sections is fraught with difficulties. It is for this reason that the Probabilistic Gas Touched Length Analysis model has been developed. This model makes it possible to calculate creep damage accumulation/remaining life at any point along the steam path. Oxide thickness data and operating data are the primary operating inputs into the model, which performs heat transfer calculations at user-defined locations along the length of the tube circuit. The model applies statistical methods to evaluate variations in operating conditions as well as in physical and mechanical properties using a Monte Carlo simulation to generate values for the probability of failure at selected locations. This paper will discuss the limitations of the existing approach to estimating the remaining life of high temperature boiler tubing and present the underpinning theory of the gas touched length analysis model. A case study showing the analysis results is included.
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O’Neil, D. A., J. H. Selverian, and K. S. Kim. "Plasticity Considerations in Probabilistic Ceramic-to-Metal Joint Design." In ASME 1994 International Gas Turbine and Aeroengine Congress and Exposition. American Society of Mechanical Engineers, 1994. http://dx.doi.org/10.1115/94-gt-229.

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A new probabilistic failure criterion was developed for the design of high-temperature ceramic-to-metal joints. The essential feature of the theory is the inclusion of the energy dissipated during plastic deformation of the adjacent braze layer in the joint. A large number of bi-material interface fracture simulations were performed for different crack positions and orientations near the bimaterial interface to determine the effect on stresses in the ceramic near the interface. The effective stress values were then ported to a probabilistic failure analysis code, which permitted simple inclusion of the new failure criterion. Brazed joints were made and failure tested in torsion to verify the failure criterion. Results show that the new failure criterion more closely approximates the failure of the ceramic-to-metal joints over the entire range of ultimate loads, an is a significant improvement in the failures criteria previously used for this type of joint design. Aspects of the failure criterion, material systems, residual stresses, mechanical behavior, and strength predictions will be presented.
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Cui, Wei, and Jianjun Wang. "Probabilistic Analysis of Gas Turbine Disk Multi-Crack Propagation." In ASME 2011 Turbo Expo: Turbine Technical Conference and Exposition. ASMEDC, 2011. http://dx.doi.org/10.1115/gt2011-45439.

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An approach of predicting probabilistic life of multi-source surface crack growth problem is described in the paper. Multi-source circumferential surface cracks occur occasionally on aircraft engine turbine disk. A serious of semi-elliptical cracks which located in a certain radius but different circumferential position on turbine disk was caused by low-cycle fatigue. The residual life should be predicted according to the damage tolerance design theory. A typical turbine disk was studied as a numerical example by using finite element method. The existing experimental result in literature shows that the surface crack propagates faster on circumferential direction than on axial direction, and the surface cracks are going to merge on circumferential direction, which made the multi-crack a closed ring crack. The experiment shows that the multi-cracks would merge into a 2/3 circle crack after certain number cycles which agrees the numerical simulation. Then the crack model was reasonable simplified to axial direction 2-D crack problem in meridian plane of turbine disk. This 2-D axisymmetric simplification significantly reduces the probabilistic analysis computational time. The randomness of materials, load and geometric imperfection is considered in sensitivity analysis by using response surface method. Then the probabilistic life is predicted by considering the major random variables of initial crack length and engine speed. The probabilistic life analysis result is also compared with the existing experimental results.
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Zhang, Zhe, Chao Jiang, G. Gary Wang, and Xu Han. "An Efficient Reliability Analysis Method for Structures With Epistemic Uncertainty Using Evidence Theory." In ASME 2014 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. American Society of Mechanical Engineers, 2014. http://dx.doi.org/10.1115/detc2014-35623.

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Evidence theory has a strong ability to deal with the epistemic uncertainty, based on which the uncertain parameters existing in many complex engineering problems with limited information can be conveniently treated. However, the heavy computational cost caused by its discrete property severely influences the practicability of evidence theory, which has become a main difficulty in structural reliability analysis using evidence theory. This paper aims to develop an efficient method to evaluate the reliability for structures with evidence variables, and hence improves the applicability of evidence theory for engineering problems. A non-probabilistic reliability index approach is introduced to obtain a design point on the limit-state surface. An assistant area is then constructed through the obtained design point, based on which a small number of focal elements can be picked out for extreme analysis instead of using all the elements. The vertex method is used for extreme analysis to obtain the minimum and maximum values of the limit-state function over a focal element. A reliability interval composed of the belief measure and the plausibility measure is finally obtained for the structure. Two numerical examples are investigated to demonstrate the effectiveness of the proposed method.
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Bos, Mark, Olger Koop, and Ernst Bolt. "Safety Level of a Probabilistic Admittance Policy." In ASME 2011 30th International Conference on Ocean, Offshore and Arctic Engineering. ASMEDC, 2011. http://dx.doi.org/10.1115/omae2011-49357.

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The main factor restricting admission of deep-draught ships to ports is the risk of bottom contact. In the approach channels to Rotterdam and IJmuiden such ships are subject to tidal windows because of the limited depths in the approach channels. Some decades ago probabilistic methods were introduced for the allocation of tidal windows for the Euro-Maas channel to Rotterdam and later also for the IJ channel to IJmuiden. These methods are being further developed. The increased accuracy of computational methods and of forecasted wave conditions and water levels may lead to an increased accessibility of the port. Recent developments have resulted in the new tidal window advice program Protide (PRObabilistic TIdal window DEtermination). The objective of the work presented in this paper is the verification and validation of Protide for the Euro-Maas channel to Rotterdam and for the IJ channel to IJmuiden. The probability of bottom contact during channel transit is simulated for time series of ten years of measured wave data and water levels. The fleet of ships is represented by a limited number of ships and for each ship all possible tidal windows for the ten year period are determined with Protide. A database is developed with motion response characteristics for each of the representative ships. For all possible arrival times in the ten-year period, indicated as safe by Protide, a channel transit is simulated. The probability of bottom contact during the simulated transit is computed from the motion response characteristics of the ship and with the measured wave spectra and the measured water level. Two safety criteria are applied. Firstly the probability of bottom contact during a single channel transit should not be excessive. Secondly the total probability of bottom contact during a long period of application of the admittance policy should be limited. To determine the probability that a certain ship is in a certain section of the channel some elements from queuing theory are applied. The probability that the ship is present in each channel section combined with the probability of bottom contact results in the probability of bottom contact for the ship in the channel. This leads to a long term probability of bottom contact for the channel. This paper presents the analysis method and selected simulation results for Euro-Maas channel in detail.

Reports on the topic "Probabilistic number theory":

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Mazzoni, Silvia, Nicholas Gregor, Linda Al Atik, Yousef Bozorgnia, David Welch, and Gregory Deierlein. Probabilistic Seismic Hazard Analysis and Selecting and Scaling of Ground-Motion Records (PEER-CEA Project). Pacific Earthquake Engineering Research Center, University of California, Berkeley, CA, November 2020. http://dx.doi.org/10.55461/zjdn7385.

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This report is one of a series of reports documenting the methods and findings of a multi-year, multi-disciplinary project coordinated by the Pacific Earthquake Engineering Research Center (PEER) and funded by the California Earthquake Authority (CEA). The overall project is titled “Quantifying the Performance of Retrofit of Cripple Walls and Sill Anchorage in Single-Family Wood-Frame Buildings,” henceforth referred to as the “PEER–CEA Project.” The overall objective of the PEER–CEA Project is to provide scientifically based information (e.g., testing, analysis, and resulting loss models) that measure and assess the effectiveness of seismic retrofit to reduce the risk of damage and associated losses (repair costs) of wood-frame houses with cripple wall and sill anchorage deficiencies as well as retrofitted conditions that address those deficiencies. Tasks that support and inform the loss-modeling effort are: (1) collecting and summarizing existing information and results of previous research on the performance of wood-frame houses; (2) identifying construction features to characterize alternative variants of wood-frame houses; (3) characterizing earthquake hazard and ground motions at representative sites in California; (4) developing cyclic loading protocols and conducting laboratory tests of cripple wall panels, wood-frame wall subassemblies, and sill anchorages to measure and document their response (strength and stiffness) under cyclic loading; and (5) the computer modeling, simulations, and the development of loss models as informed by a workshop with claims adjustors. This report is a product of Working Group 3 (WG3), Task 3.1: Selecting and Scaling Ground-motion records. The objective of Task 3.1 is to provide suites of ground motions to be used by other working groups (WGs), especially Working Group 5: Analytical Modeling (WG5) for Simulation Studies. The ground motions used in the numerical simulations are intended to represent seismic hazard at the building site. The seismic hazard is dependent on the location of the site relative to seismic sources, the characteristics of the seismic sources in the region and the local soil conditions at the site. To achieve a proper representation of hazard across the State of California, ten sites were selected, and a site-specific probabilistic seismic hazard analysis (PSHA) was performed at each of these sites for both a soft soil (Vs30 = 270 m/sec) and a stiff soil (Vs30=760 m/sec). The PSHA used the UCERF3 seismic source model, which represents the latest seismic source model adopted by the USGS [2013] and NGA-West2 ground-motion models. The PSHA was carried out for structural periods ranging from 0.01 to 10 sec. At each site and soil class, the results from the PSHA—hazard curves, hazard deaggregation, and uniform-hazard spectra (UHS)—were extracted for a series of ten return periods, prescribed by WG5 and WG6, ranging from 15.5–2500 years. For each case (site, soil class, and return period), the UHS was used as the target spectrum for selection and modification of a suite of ground motions. Additionally, another set of target spectra based on “Conditional Spectra” (CS), which are more realistic than UHS, was developed [Baker and Lee 2018]. The Conditional Spectra are defined by the median (Conditional Mean Spectrum) and a period-dependent variance. A suite of at least 40 record pairs (horizontal) were selected and modified for each return period and target-spectrum type. Thus, for each ground-motion suite, 40 or more record pairs were selected using the deaggregation of the hazard, resulting in more than 200 record pairs per target-spectrum type at each site. The suites contained more than 40 records in case some were rejected by the modelers due to secondary characteristics; however, none were rejected, and the complete set was used. For the case of UHS as the target spectrum, the selected motions were modified (scaled) such that the average of the median spectrum (RotD50) [Boore 2010] of the ground-motion pairs follow the target spectrum closely within the period range of interest to the analysts. In communications with WG5 researchers, for ground-motion (time histories, or time series) selection and modification, a period range between 0.01–2.0 sec was selected for this specific application for the project. The duration metrics and pulse characteristics of the records were also used in the final selection of ground motions. The damping ratio for the PSHA and ground-motion target spectra was set to 5%, which is standard practice in engineering applications. For the cases where the CS was used as the target spectrum, the ground-motion suites were selected and scaled using a modified version of the conditional spectrum ground-motion selection tool (CS-GMS tool) developed by Baker and Lee [2018]. This tool selects and scales a suite of ground motions to meet both the median and the user-defined variability. This variability is defined by the relationship developed by Baker and Jayaram [2008]. The computation of CS requires a structural period for the conditional model. In collaboration with WG5 researchers, a conditioning period of 0.25 sec was selected as a representative of the fundamental mode of vibration of the buildings of interest in this study. Working Group 5 carried out a sensitivity analysis of using other conditioning periods, and the results and discussion of selection of conditioning period are reported in Section 4 of the WG5 PEER report entitled Technical Background Report for Structural Analysis and Performance Assessment. The WG3.1 report presents a summary of the selected sites, the seismic-source characterization model, and the ground-motion characterization model used in the PSHA, followed by selection and modification of suites of ground motions. The Record Sequence Number (RSN) and the associated scale factors are tabulated in the Appendices of this report, and the actual time-series files can be downloaded from the PEER Ground-motion database Portal (https://ngawest2.berkeley.edu/)(link is external).

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