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1
Louvel, Benoît. « Twisted Kloosterman sums and cubic exponential sums ». Doctoral thesis, Montpellier 2, 2008. http://hdl.handle.net/11858/00-1735-0000-0006-B3CB-A.
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Chambille, Saskia. « Exponential sums, cell decomposition and p-adic integration ». Thesis, Lille 1, 2018. http://www.theses.fr/2018LIL1I023/document.
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Dans cette thèse nous étudions des sommes exponentielles et des intégrales p-adiques, en utilisant la théorie des modèles et la géométrie. La première partie traite des sommes exponentielles dans des corps P-minimaux. La deuxième partie examine le comportement asymptotique des sommes exponentielles sur les corps p-adiques. Dans la première partie nous commençons par démontrer une théorème de décomposition cellulaire pour tous les corps P-minimaux, c.-à-d. indépendamment de l’existence des fonctions de Skolem définissables. En l’absence de ces fonctions nous introduisons les cellules en grappe régulières, inspirés par la notion classique de cellule p-adique de Denef. Notre décomposition cellulaire utilise les cellules classiques et les cellules en grappe régulières. Ensuite nous étendons la notion de fonction constructible exponentielle des structures semi-algébriques et sous-analytiques à tous les corps P-minimaux. Pour cela nous ajoutons des sommes exponentielles aux algèbres des fonctions constructibles. En utilisant notre décomposition cellulaire, nous démontrons que les fonctions constructibles exponentielles sont stables dans le contexte d’intégration. Cela signifie que l’intégration d’une fonction constructible exponentielle sur certaines de ses variables produit une fonction constructible exponentielle dans les autres variables. Dans la deuxième partie nous démontrons les conjectures d’Igusa, Denef-Sperber et Cluckers-Veys sur le comportement asymptotique des sommes exponentielles pour les polynômes dont le seuil log-canonique ne dépasse pas un demi. Nous apportons deux démonstrations ; l’une utilise l’intégration motivique et l’autre les fonctions zêtas d’IgusaIn this thesis we study p-adic exponential sums and integrals using ideas from model theory and geometry. The first part of this thesis deals with exponential sums in P-minimal fields. The second part discusses estimates for the asymptotic behaviour of exponential sums over p-adic fields. Our work on P-minimal fields starts with the proof of a cell decomposition theorem that holds in all P-minimal fields, i.e., independently of the existence of definable Skolem functions. For P-minimal fields that lack these functions, we introduce the notion of regular clustered cells. This notion is close to the classical notion of p-adic cells, that was introduced by Denef. Our cell decomposition uses both classical cells and regular clustered cells. Next, we extend the notion of exponential-constructible functions, already defined in the semi-algebraic and subanalytic setting, to all P-minimal fields. We do this by enlarging the algebras of constructible functions with exponential sums. Using our cell decomposition theorem we prove that exponential-constructible functions are stable under integration. This means that the act of integrating an exponential-constructible function over some of its variables produces an exponential-constructible function in the other variables. In our work on estimates for the asymptotic behaviour of exponential sums we prove the Igusa, Denef-Sperber and Cluckers-Veys conjectures for polynomials with log-canonical threshold at most one half. We give two different proofs, one using motivic integration, and the other one using the Igusa zeta functions
3
Watt, N. « some problems in analytic number theory ». Thesis, Bucks New University, 1988. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.384667.
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Alsulmi, Badria. « Generalized Jacobi sums modulo prime powers ». Diss., Kansas State University, 2016. http://hdl.handle.net/2097/32668.
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Allison, Gisele. « Some problems related to incomplete character sums ». Thesis, University of Nottingham, 1999. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.285601.
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Chênevert, Gabriel. « Exponential sums, hypersurfaces with many symmetries and Galois representations ». Thesis, McGill University, 2009. http://digitool.Library.McGill.CA:80/R/?func=dbin-jump-full&object_id=32386.
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The main theme of this thesis is the study of compatible systems of $\ell$-adic Galois representations provided by the étale cohomology of arithmetic varieties with a large group of symmetries. A canonical decomposition of these systems into isotypic components is proven (Section 3.1). The isotypic components are realized as the cohomology of the quotient with values in a certain sheaf, thus providing a geometrical interpretation for the rationality of the corresponding $L$-functions. A particular family of singular hypersurfaces $W_\ell^{m,n}$ of degree $\ell$ and dimension $m + n - 3$, admitting an action by a product of symmetric groups $S_m \times S_n$, arises naturally when considering the average moments of certain exponential sums (Chapter 4); asymptotics for these moments are obtained by relating them to the trace of the Frobenius morphism on the cohomology of the desingularization of the corresponding varieties, following the approach of Livné. Two other closely related classes of smooth hypersurfaces admitting an $S_n$-action are introduced in Chapter 3, and the character of the representation of the symmetric group on their primitive cohomology is computed. In particular, a certain smooth cubic hypersurface of dimension 4 is shown to carry a compatible system of 2-dimensional Galois representations. A variant of the Faltings-Serre method is developed in Chapter 5 in order to explicitly determine the corresponding modular form, whose existence is predicted by Serre's conjecture. We provide a systematic treatment of the Faltings-Serre method in a form amenable to generalization to Galois representations of other fields and to other groups besides $\GL_2$.Le thème principal de cette thèse est l'étude des systèmes compatibles de représentations galoisiennes $\ell$-adiques provenant de la cohomologie étale de variétés arithmétiques admettant beaucoup de symétries. Une décomposition canonique de ces systèmes en composantes isotypiques est obtenue (section 3.1). Les composantes isotypiques sont décrites comme la cohomologie du quotient à valeurs dans un certain faisceau, fournissant ainsi une interprétation géométrique de la rationalité des fonctions $L$ correspondantes. Une famille spécifique d'hypersurfaces $W_\ell^{m,n}$ de degré $\ell$ et dimension $m+n-3$, admettant une action du produit de groupes symétriques $S_m \times S_n$, apparaît naturellement en lien avec les moments moyens de certaines sommes exponentielles (chapitre 4); le comportement limite de ces moments est obtenu en considérant la trace du morphisme de Frobenius sur la cohomologie de la désingularisation des variétés correspondantes, suivant l'approche développée par Livné. Deux autres classes apparentées d'hypersurfaces lisses admettant une action du groupe symétrique sont introduites au chapitre 3, et le caractère de la représentation de $S_n$ sur leur cohomologie primitive est calculé. En particulier, dans le cas d'une certaine hypersurface cubique de dimension 4, un système compatible de représentations galoisiennes de dimension 2 est obtenu. Une variante de la méthode de Faltings-Serre est développée dans le chapitre 5 afin de déterminer explicitement la forme modulaire correspondante, dont l'existence est prédite par la conjecture de Serre. Nous proposons un traitement systématique de la méthode de Falting
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Pigno, Vincent. « Prime power exponential and character sums with explicit evaluations ». Diss., Kansas State University, 2014. http://hdl.handle.net/2097/18277.
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Doctor of PhilosophyDepartment of Mathematics
Christopher Pinner
Exponential and character sums occur frequently in number theory. In most applications one is only interested in estimating such sums. Explicit evaluations of such sums are rare. In this thesis we succeed in evaluating three types of sums when p is a prime and m is sufficiently large. The twisted monomial sum, the binomial character sum, and the generalized Jacobi sum. We additionally show that these are all sums which can be expressed in terms of classical Gauss sums.
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Qin, Huan. « Averages of fractional exponential sums weighted by Maass forms ». Diss., University of Iowa, 2017. https://ir.uiowa.edu/etd/5607.
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The purpose of this study is to investigate the oscillatory behavior of the fractional exponential sum weighted by certain automorphic forms for GL(2) x GL(3) case. Automorphic forms are complex-values functions defined on some topological groups which satisfy a number of applicable properties. One nice property that all automorphic forms admit is the existence of Fourier series expansions, which allows us to study the properties of automorphic forms by investigating their corresponding Fourier coefficients. The Maass forms is one family of the classical automorphic forms, which is the major focus of this study.
Let f be a fixed Maass form for SL(3, Z) with Fourier coefficients Af(m, n). Also, let {gj} be an orthonormal basis of the space of the Maass cusp form for SL(2, Z) with corresponding Laplacian eigenvalues 1/4+kj^2, kj>0. For real α be nonzero and β>0, we considered the asymptotics for the sum in the following form Sx(f x gj, α, β) = ∑Af(m, n)λgj(n)e(αn^β)φ(n/X) where φ is a smooth function with compactly support, λgj(n) denotes the nth Fourier coefficient of gj, and X is a real parameter that tends to infinity. Also, e(x) = exp(2πix) throughout this thesis.
We proved a bound of the weighted average sum of Sx(f x gj, α, β) over all Laplacian eigenvalues, which is better than the trivial bound obtained by the classical Rankin-Selberg method. In this case, we allowed the form varies so that we can obtain information about their oscillatory behaviors in a different aspect. Similar to the proofs of the subconvexity bounds for Rankin-Selberg L-functions for GL(2) x GL(3) case, the method we used in this study includes several sophisticated techniques such as weighted first and second derivative test, Kuznetsov trace formula, and Voronoi summation formula.
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Wong, Chi-Yan, et 黃志仁. « Some results on the error terms in certain exponential sums involving the divisor function ». Thesis, The University of Hong Kong (Pokfulam, Hong Kong), 2002. http://hub.hku.hk/bib/B42577147.
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Wong, Chi-Yan. « Some results on the error terms in certain exponential sums involving the divisor function ». Click to view the E-thesis via HKUTO, 2002. http://sunzi.lib.hku.hk/hkuto/record/B42577147.
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Potts, Daniel, et Manfred Tasche. « Parameter estimation for nonincreasing exponential sums by Prony-like methods ». Universitätsbibliothek Chemnitz, 2012. http://nbn-resolving.de/urn:nbn:de:bsz:ch1-qucosa-86476.
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For noiseless sampled data, we describe the close connections between Prony--like methods, namely the classical Prony method, the matrix pencil method and the ESPRIT method.
Further we present a new efficient algorithm of matrix pencil factorization based on QR decomposition of a rectangular Hankel matrix. The algorithms of parameter estimation are also applied to sparse Fourier approximation and nonlinear approximation.
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White, Christopher J. « Finding primes in arithmetic progressions and estimating double exponential sums ». Thesis, University of Bristol, 2016. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.707745.
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Mohammadi, Nikouypasokhi Ali. « On the sum-product phenomenon in arbitrary finite fields and its applications ». Thesis, The University of Sydney, 2018. http://hdl.handle.net/2123/18980.
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This thesis establishes new quantitative results in several problems relating to the sum-product phenomenon in arbitrary finite fields. We give new estimates of exponential sums, two-variable expanders and point-line incidences. We also consider an energy variant of the sum-product problem, extending a result of Balog and Wooley to arbitrary finite fields. Our approach towards the sum-product problem relies on the so-called additive pivot technique and our results hold under certain structural assumptions, requiring that a set is not largely contained in a proper subfield. This is in contrast to other recent developments, which rely on a result on point-plane incidences due to Rudnev and hold only for sets which are bounded in size in terms of the characteristic of the field.
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Czarnecki, Kyle Jeffrey. « Resonance sums for Rankin-Selberg products ». Diss., University of Iowa, 2016. https://ir.uiowa.edu/etd/3066.
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Consider either (i) f = f1 ⊠ f2 for two Maass cusp forms for SLm(ℤ) and SLm′(ℤ), respectively, with 2 ≤ m ≤ m′, or (ii) f= f1 ⊠ f2 ⊠ f3 for three weight 2k holomorphic cusp forms for SL2(ℤ). Let λf(n) be the normalized coefficients of the associated L-function L(s, f), which is either (i) the Rankin-Selberg L-function L(s, f1 ×f2), or (ii) the Rankin triple product L-function L(s, f1 ×f2 ×f3). First, we derive a Voronoi-type summation formula for λf (n) involving the Meijer G-function. As an application we obtain the asymptotics for the smoothly weighted average of λf (n) against e(αnβ), i.e. the asymptotics for the associated resonance sums. Let ℓ be the degree of L(s, f). When β = 1/ℓ and α is close or equal to ±ℓq 1/ℓ for a positive integer q, the average has a main term of size |λf (q)|X 1/2ℓ+1/2 . Otherwise, when α is fixed and 0 < β < 1/ℓ it is shown that this average decays rapidly. Similar results have been established for individual SLm(ℤ) automorphic cusp forms and are due to the oscillatory nature of the coefficients λf (n).
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Draper, Sandra D. « Evalutaion of certain exponential sums of quadratic functions over a finite fields of odd characteristic ». [Tampa, Fla] : University of South Florida, 2006. http://purl.fcla.edu/usf/dc/et/SFE0001674.
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Bongiovanni, Alex. « Problems with power-free numbers and Piatetski-Shapiro sequences ». Kent State University / OhioLINK, 2021. http://rave.ohiolink.edu/etdc/view?acc_num=kent1618331559201676.
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Erturk, Huseyin. « Limit theorems for random exponential sums and their applications to insurance and the random energy model ». Thesis, The University of North Carolina at Charlotte, 2016. http://pqdtopen.proquest.com/#viewpdf?dispub=10111893.
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In this dissertation, we are mainly concerned with the sum of random exponentials. Here, the random variables are independent and identically distributed. Another distinctive assumption is the number of variables in this sum is a function of the constant on the exponent. Our first goal is to find the limiting distributions of the random exponential sums for new class of the random variables. For some classes, such results are known; normal distribution, Weibull distribution etc.
Secondly, we apply these limit theorems to some insurance models and the random energy model in statistical physics. Specifically for the first case, we give the estimate of the ruin probability in terms of the empirical data. For the random energy model, we present the analysis of the free energy for new class of distribution. In some particular cases, we prove the existence of several critical points for the free energy. In some other cases, we prove the absence of phase transitions.
Our results give a new approach to compute the ruin probabilities of insurance portfolios empirically when there is a sequence of insurance portfolios with a custom growth rate of the claim amounts. The second application introduces a simple method to drive the free energy in the case the random variables in the statistical sum can be represented as a function of standard exponential random variables. The technical tool of this study includes the classical limit theory for the sum of independent and identically distributed random variables and different asymptotic methods like the Euler-Maclaurin formula and Laplace method.
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Jonsson, Fredrik. « Self-Normalized Sums and Directional Conclusions ». Doctoral thesis, Uppsala universitet, Matematiska institutionen, 2012. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-162168.
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This thesis consists of a summary and five papers, dealing with self-normalized sums of independent, identically distributed random variables, and three-decision procedures for directional conclusions. In Paper I, we investigate a general set-up for Student's t-statistic. Finiteness of absolute moments is related to the corresponding degree of freedom, and relevant properties of the underlying distribution, assuming independent, identically distributed random variables. In Paper II, we investigate a certain kind of self-normalized sums. We show that the corresponding quadratic moments are greater than or equal to one, with equality if and only if the underlying distribution is symmetrically distributed around the origin. In Paper III, we study linear combinations of independent Rademacher random variables. A family of universal bounds on the corresponding tail probabilities is derived through the technique known as exponential tilting. Connections to self-normalized sums of symmetrically distributed random variables are given. In Paper IV, we consider a general formulation of three-decision procedures for directional conclusions. We introduce three kinds of optimality characterizations, and formulate corresponding sufficiency conditions. These conditions are applied to exponential families of distributions. In Paper V, we investigate the Benjamini-Hochberg procedure as a means of confirming a selection of statistical decisions on the basis of a corresponding set of generalized p-values. Assuming independence, we show that control is imposed on the expected average loss among confirmed decisions. Connections to directional conclusions are given.
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Müllner, Clemens. « Exponential sum estimates and Fourier analytic methods for digitally based dynamical systems ». Thesis, Aix-Marseille, 2017. http://www.theses.fr/2017AIXM0042/document.
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La présente thèse a été fortement influencée par deux conjectures, l'une de Gelfond et l'autre de Sarnak.En 1968, Gelfond a prouvé que la somme des chiffres modulo m est asymtotiquement équirépartie dans des progressions arithmétiques, et il a formulé trois problèmes nouveaux.Le deuxième et le troisième problèmes traitent des sommes des chiffres pour les nombres premiers et les suites polynomiales.En ce qui concerne les nombres premiers et les carrés, Mauduit et Rivat ont résolu ces problèmes en 2010 et 2009, respectivement.Drmota, Mauduit et Rivat ont réussi généraliser le résultat concernant la suite des sommes des chiffres des carrés.Ils ont démontré que chaque bloc apparaît asymptotiquement avec la même fréquence.Selon la conjecture de Sarnak, il n'y a pas de corrélation entre la fonction de Möbius et des fonctions simples.La présente thèse traite de la répartition de suites automatiques le long de sous-suites particulières ainsi que d'autres propriétés de suites automatiques.Selon l'un des résultats principaux du présent travail, toutes les suites automatiques vérifient la conjecture de Sarnak.Moyennant une approche légèrement modifiée, nous traitons également la répartition de suites automatiques le long de la suite des nombres premiers.Dans le cadre du traitement de suites automatiques générales, nous avons mis au point une nouvelle structure destinée aux automates finisdéterministes ouvrant une vision nouvelle pour les automates et/ou les suites automatiques.Nous étendons les résultat de Drmota, Mauduit et Rivat concernant les suites digitales.Cette approche peut également être considérée comme une généralisation du troisième problème de GelfondThe present dissertation was inspired by two conjectures, one by Gelfond and one of Sarnak.In 1968 Gelfond proved that the sum of digits modulo m is asymptotically equally distributed along arithmetic progressions.Furthermore, he stated three problems which are nowadays called Gelfond problems.The second and third questions are concerned with the sum of digits of prime numbers and polynomial subsequences.Mauduit and Rivat were able to solve these problems for primes and squares in 2010 and 2009 respectively.Drmota, Mauduit and Rivat generalized the result concerning the sequence of the sum of digits of squares.They showed that each block appears asymptotically equally frequently.Sarnak conjectured in 2010 that the Mobius function does not correlate with deterministic functions.This dissertation deals with the distribution of automatic sequences along special subsequences and other properties of automatic sequences.A main result of this thesis is that all automatic sequences satisfy the Sarnak conjecture.Through a slightly modified approach, we also deal with the distribution of automatic sequences along the subsequence of primes.In the course of the treatment of general automatic sequences, a new structure for deterministic finite automata is developed,which allows a new view for automata or automatic sequences.We extend the result of Drmota, Mauduit and Rivat to digital sequences.This is also a generalization of the third Gelfond problem
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Moreira, Nunes Ramon. « Problèmes d’équirépartition des entiers sans facteur carré ». Thesis, Paris 11, 2015. http://www.theses.fr/2015PA112123/document.
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Cette thèse concerne quelques problèmes liés à la répartition des entiers sans facteur carré dansles progressions arithmétiques. Ces problèmes s’expriment en termes de majorations du terme d’erreurassocié à cette répartition.Les premier, deuxième et quatrième chapitres sont concentrés sur l’étude statistique des termesd’erreur quand on fait varier la progression arithmétique modulo q. En particulier on obtient une formuleasymptotique pour la variance et des majorations non triviales pour les moments d’ordre supérieur. Onfait appel à plusieurs techniques de théorie analytique des nombres comme les méthodes de crible et lessommes d’exponentielles, notamment une majoration récente pour les sommes d’exponentielles courtesdue à Bourgain dans le deuxième chapitre.Dans le troisième chapitre on s’intéresse à estimer le terme d’erreur pour une progression fixée. Onaméliore un résultat de Hooley de 1975 dans deux directions différentes. On utilise ici des majorationsrécentes de sommes d’exponentielles courtes de Bourgain-Garaev et de sommes d’exponentielles torduespar la fonction de Möbius dues à Bourgain et Fouvry-Kowalski-MichelThis thesis concerns a few problems linked with the distribution of squarefree integers in arithmeticprogressions. Such problems are usually phrased in terms of upper bounds for the error term relatedto this distribution.The first, second and fourth chapter focus on the satistical study of the error terms as the progres-sions varies modulo q. In particular we obtain an asymptotic formula for the variance and non-trivialupper bounds for the higher moments. We make use of many technics from analytic number theorysuch as sieve methods and exponential sums. In particular, in the second chapter we make use of arecent upper bound for short exponential sums by Bourgain.In the third chapter we give estimates for the error term for a fixed arithmetic progression. Weimprove on a result of Hooley from 1975 in two different directions. Here we use recent upper boundsfor short exponential sums by Bourgain-Garaev and exponential sums twisted by the Möbius functionby Bourgain et Fouvry-Kowalski-Michel
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Swaenepoel, Cathy. « Chiffres des nombres premiers et d'autres suites remarquables ». Thesis, Aix-Marseille, 2019. http://www.theses.fr/2019AIXM0161/document.
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Dans ce travail, nous étudions la répartition des chiffres des nombres premiers. Bourgain (2015) a obtenu une formule asymptotique pour le nombre de nombres premiers avec une proportion$c > 0$ de chiffres préassignés en base 2 ($c$ est une constante absolue non précisée).Nous généralisons ce résultat à toute base $g \geq 2$ et nousdonnons des valeurs explicites pour la proportion $c$ en fonction de $g$. En adaptant, développant et précisant la stratégie introduite par Bourgain dans le cas $g=2$, nous présentons une démonstration détaillée du cas général.La preuve est fondée sur la méthode du cercle et combine des techniques d’analyse harmonique avec des résultats sur les zéros des fonctions $L$ de Dirichlet, notamment une région sans zérotrès fine due à Iwaniec.Ce travail s'inscrit aussi dans l'étude des nombres premiers dans des ensembles << rares >>.Nous étudions également la répartition des << chiffres >> (au sens de Dartyge et S\'ark\"ozy) de quelques suites remarquables dans le contexte des corps finis. Ce concept de << chiffre >> est à la base de la représentation des corps finis dans les logiciels de calcul formel.Nous étudions des suites variées comme les suites polynomiales, les générateurs ou encore les produits d'éléments de deux ensembles assez grands. Les méthodes développées permettent d'obtenir des estimations explicites très précises voire optimales dans certains cas. Les sommes d'exponentielles sur les corps finis jouent un rôle essentiel dans les démonstrations.Les résultats obtenus peuvent être reformulés d'un point de vue plus algébrique avec la fonction trace qui est très importante dans l'étude des corps finisIn this work, we study the distribution of prime numbers' digits. Bourgain (2015) obtained an asymptotic formula for the number of prime numbers with a proportion $c > 0$ of preassigned digits in base 2 ($c$ is an absolute constant not specified). We generalize this result in any base $g \geq 2$ and we provide explicit admissible values for the proportion $c$ depending on $g$.By adapting, developing and refining Bourgain's strategy in the case $g=2$, we present a detailed proof for the general case.The proof is based onthe circle method and combines techniques from harmonic analysis together with results onzeros of Dirichlet $L$-functions, notably a very sharp zero-free region due to Iwaniec.This work also falls within the study of prime numbers in sparse ``sets''.In addition, we study the distribution of the ``digits'' (in the sense of Dartyge and S\'ark\"ozy) of some sequences of interest in the context of finite fields. This concept of ``digits'' is fundamental in the representation of finite fields in computer algebra systems. We study various sequences such as polynomial sequences, generators as well as products of elements of two large enough sets.Our methods provide very sharp explicit estimates which are even optimal in some cases.Exponential sums over finite fields play an essential role in the proofs.Our results can be reformulated from a more algebraic point of view with the trace function which is of basic importance in the study of finite fields
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Shawket, Zaid Esmat. « Propriétés arithmétiques et statistiques des fonctions digitales restreintes ». Thesis, Aix-Marseille 2, 2011. http://www.theses.fr/2011AIX22059.
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Dans ce travail nous étudions les propriétés arithmétiques et statistiques d'une nouvelle classe de fonctions de comptage des chiffres appelées fonctions digitales restreintes. Nous présentons tout d'abord les principales propriétés des suites engendrées par une substitution ou un $q$-automate ainsi que la suite célèbre de Thue-Morse et ses généralisations, puis nous comparons ces notions avec celle de fonction digitale restreinte.Nous étudions ensuite les sommes d'exponentielles associées à ces fonctions digitales restreintes ainsi que leur application d'une part à l'étude de la répartition modulo 1 des fonctions digitales restreintes et d'autre part à l'étude des propriétés statistiques des suites arithmétiques définies par des fonctions digitales restreintes.Dans la dernière partie de ce travail on étudie la représentation géométrique de ces sommes d'exponentielle à la lumière des travaux antérieurs de Dekking et Mendès-France ce qui nous conduit à énoncer plusieurs problèmes ouvertsIn this work we study the arithmetic and statistic properties of a new class of digital counting functions called restricted digital functions. We first present the main properties of sequences generated by a substitution or a $q$-automate followed by presenting the famous Thue-Morse sequence and its generalizations, then we compare these notions with the one of the restricted digital function.We then study the exponential sums associated with these restricted digital function and their implementation on the one hand to the study of uniform distribution modulo 1 of these restricted digital functions and on the other, to the study of the statistical properties of the arithmetic sequences defined by restricted digital functions.In the last part of this work we study the geometric representation of these exponential sums in the light of previous works of Dekking and Mendès-France which leads us to announce several open problems
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Tahay, Pierre-Adrien. « Colonnes dans les automates cellulaires et suites généralisées de Rudin-Shapiro ». Electronic Thesis or Diss., Université de Lorraine, 2020. http://www.theses.fr/2020LORR0198.
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Cette thèse se situe à la frontière entre mathématiques et informatique théorique. Nous nous intéressons dans un premier temps aux automates finis et aux automates cellulaires. Bien qu’ils s’agissent de deux objets mathématiques assez différents, il est possible de les relier par des constructions explicites, en regardant la réalisation des suites automatiques dans les diagrammes espace-temps des automates cellulaires. Dans un second temps, nous étudions les corrélations discrètes de certaines suites automatiques, appelées suites généralisées de Rudin–Shapiro, qui se comportent comme des suites aléatoires pour la corrélation discrète d’ordre 2, bien qu’elles soient déterministes. Après une introduction des objets d’étude, que nous illustrons par plusieurs exemples, nous rappelons le résultat de Rowland et Yassawi, qui ont montré en 2015 qu’il était possible de construire de manière explicite toute suite p-automatique, dans le cas où p est un nombre premier, en colonne d’un automate cellulaire linéaire, à partir d’une configuration initiale finie. En utilisant leur méthode, nous obtenons différentes constructions de suites automatiques de référence, puis nous établissons un moyen explicite de construire toute une famille de suites p-automatiques, appelées suites généralisées de Rudin–Shapiro, que nous étudions dans la deuxième partie de la thèse, dans un cadre plus général. Nous nous intéressons également au cas de certaines suites non-automatiques, telles que l’indicatrice des polynômes et le mot de Fibonacci, que nous réussissons à construire en colonne d’automates cellulaires non-linéaires. Puis nous obtenons des résultats sur des recodages binaires, permettant de réduire le nombre de symboles dans les automates cellulaires. Grâce à un recodage binaire, nous avons également construit explicitement une suite 3-automatique sur un alphabet binaire, en colonne d’un automate cellulaire à 2 états, non-périodique à partir d’un certain rang, ce qui répond à une question posée par Rowland et Yassawi. Dans la deuxième partie de cette thèse, nous reprenons les travaux de Grant, Shallit et Stoll, qui ont établi en 2009 des résultats sur les corrélations discrètes de suites infinies sur des alphabets finis. En exploitant les propriétés de récursivité de la suite classique de Rudin–Shapiro, ils construisent une famille de suites déterministes sur des alphabets plus grands, pour lesquelles ils montrent que dans le cas où la taille de l’alphabet est sans facteur carré, la moyenne empirique des coefficients de corrélation d’ordre 2 a la même limite que dans le cas de suites où les lettres sont tirées aléatoirement, de manière uniforme et indépendamment. De plus, ils arrivent à quantifier explicitement le terme d’erreur. En généralisant leur construction à l’aide de la théorie des matrices de différence, nous arrivons à établir un résultat similaire pour des alphabets de taille quelconque ainsi qu’une amélioration du terme d’erreur dans certains cas. Tout comme Grant et al., nous nous servons de la théorie des sommes d’exponentielles pour démontrer notre résultat sur les corrélations discrètes d’ordre 2 de nos suites généralisées de Rudin–Shapiro. Dans la troisième partie, nous terminons par une approche combinatoire de ces questions, qui nous a permis d’obtenir une amélioration du terme d’erreur dans le cas où la taille de l’alphabet est un produit d’au moins deux nombres premiers distincts, et de généraliser certains de nos résultatsThis thesis is at the interface between mathematics and theoretical computer science. In the first part, our main objects are finite automata and cellular automata. While relatively different in nature, it is possible to link both by explicit constructions. More specifically, it is possible to realise automatic sequences in the space-time diagrams of cellular automata. In the second part, we study discrete correlation properties of so-called generalised Rudin–Shapiro sequences. These are automatic sequences, hence deterministic, but show similar properties as random sequences with respect to their discrete correlation of order 2. After introducing the objects of study, illustrated by several examples, we first recall the result of Rowland and Yassawi. They showed in 2015 via an algebraic approach that it is possible to construct explicitly any p-automatic sequence (p is a prime number) as a column of a linear cellular automaton with a finite initial configuration. By using their method, we obtain several constructions of classical automatic sequences, and an explicit way to build a family of p-automatic sequences that we study in a more general context in the second part of the thesis. We also investigate several non-automatic sequences, such as the characteristic sequence of integer-valued polynomials and the Fibonacci word, which both can be realised as columns of non-linear cellular automata. We end this part by some results about binary recodings in order to reduce the number of symbols in the cellular automata. Under a binary recoding, we give explicitly a 3-automatic sequence on a binary alphabet, as a column of a cellular automaton with 2 states, that is not eventually periodic. This answers a question asked by Rowland et Yassawi. In the second part of the thesis, we take up research from 2009 of Grant, Shallit, and Stoll about discrete correlations of infinite sequences over finite alphabets. By using the recursivity properties of the classical Rudin–Shapiro sequence, they built a family of deterministic sequences over larger alpha- bets, called generalised Rudin–Shapiro sequences, for which they showed that when the size of the alphabet is squarefree, the empirical means of the discrete correlation coefficients of order 2 have the same limit as in the case of random sequences where each letter is independently and uniformly chosen. Moreover, they gave explicit error terms. We extend their construction by means of difference matrices and establish a similar result on alphabets of arbitrary size. On our way, we obtain an improvement of the error term in some cases. The methods stem, as those used by Grant et al., from the theory of exponential sums. In the third part, we use a more direct combinatorial approach to study correlations. This allows for an improvement of the error term when the size of the alphabet is a product of at least two distinct primes, and allows to generalise some of our results of the second part
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Esquincalha, Agnaldo da Conceição. « Estimação de parâmetros de sinais gerados por sistemas lineares invariantes no tempo ». Universidade do Estado do Rio de Janeiro, 2009. http://www.bdtd.uerj.br/tde_busca/arquivo.php?codArquivo=1238.
Texte intégralRésumé :
Fundação Carlos Chagas Filho de Amparo a Pesquisa do Estado do Rio de JaneiroNesta dissertação é apresentado um estudo sobre a recuperação de sinais modelados por somas ponderadas de exponenciais complexas. Para tal, são introduzidos conceitos elementares em teoria de sinais e sistemas, em particular, os sistemas lineares invariantes no tempo, SLITs, que podem ser representados matematicamente por equações diferenciais, ou equações de diferenças, para sinais analógicos ou digitais, respectivamente. Equações deste tipo apresentam como solução somas ponderadas de exponenciais complexas, e assim fica estabelecida a relação entre os sistemas de tipo SLIT e o modelo em estudo. Além disso, são apresentadas duas combinações de métodos utilizadas na recuperação dos parâmetros dos sinais: métodos de Prony e mínimos quadrados, e métodos de Kung e mínimos quadrados, onde os métodos de Prony e Kung recuperam os expoentes das exponenciais e o método dos mínimos quadrados recupera os coeficientes lineares do modelo. Finalmente, são realizadas cinco simulações de recuperação de sinais, sendo a última, uma aplicação na área de modelos de qualidade de água.
A study on the recovery of signals modeled by weighted sums of complex exponentials complex is presented. For this, basic concepts of signals and systems theory are introduced. In particular, the linear time invariant systems (LTI Systems) are considered, which can be mathematically represented by differential equations or difference equations, respectively, for analog or digital signals. The solution of these types of equations is given by a weighted sum of complex exponentials, so the relationship between the LTI Systems and the model of study is established. Furthermore, two combinations of methods are used to recover the parameters of the signals: Prony and least squares methods, and Kung and least squares methods, where Prony and Kung methods are used to recover the exponents of the exponentials and the least square method is used to recover the linear coefficients of the model. Finally, five simulations are performed for the recovery of signals, the last one being an application in the area of water quality models.
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Borenstein, Evan. « Additive stucture, rich lines, and exponential set-expansion ». Diss., Atlanta, Ga. : Georgia Institute of Technology, 2009. http://hdl.handle.net/1853/29664.
Texte intégralRésumé :
Thesis (Ph.D)--Mathematics, Georgia Institute of Technology, 2009.Committee Chair: Croot, Ernie; Committee Member: Costello, Kevin; Committee Member: Lyall, Neil; Committee Member: Tetali, Prasad; Committee Member: Yu, XingXing. Part of the SMARTech Electronic Thesis and Dissertation Collection.
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Wullers, Dominik [Verfasser]. « Moving Sum versus Exponentially Weighted Moving Average Tests / Dominik Wullers ». Hamburg : Helmut-Schmidt-Universität, Bibliothek, 2015. http://d-nb.info/1073154947/34.
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Ryan, Anne Garrett. « Surveillance of Poisson and Multinomial Processes ». Diss., Virginia Tech, 2011. http://hdl.handle.net/10919/26549.
Texte intégralRésumé :
As time passes, change occurs. With this change comes the need for surveillance. One may be a technician on an assembly line and in need of a surveillance technique to monitor the number of defective components produced. On the other hand, one may be an administrator of a hospital in need of surveillance measures to monitor the number of patient falls in the hospital or to monitor surgical outcomes to detect changes in surgical failure rates. A natural choice for on-going surveillance is the control chart; however, the chart must be constructed in a way that accommodates the situation at hand. Two scenarios involving attribute control charting are investigated here. The first scenario involves Poisson count data where the area of opportunity changes. A modified exponentially weighted moving average (EWMA) chart is proposed to accommodate the varying sample sizes. The performance of this method is compared with the performance for several competing control chart techniques and recommendations are made regarding the best preforming control chart method. This research is a result of joint work with Dr. William H. Woodall (Department of Statistics, Virginia Tech). The second scenario involves monitoring a process where items are classified into more than two categories and the results for these classifications are readily available. A multinomial cumulative sum (CUSUM) chart is proposed to monitor these types of situations. The multinomial CUSUM chart is evaluated through comparisons of performance with competing control chart methods. This research is a result of joint work with Mr. Lee J. Wells (Grado Department of Industrial and Systems Engineering, Virginia Tech) and Dr. William H. Woodall (Department of Statistics, Virginia Tech).Ph. D.
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Powell, Kevin James. « Topics in Analytic Number Theory ». BYU ScholarsArchive, 2009. https://scholarsarchive.byu.edu/etd/2084.
Texte intégralRésumé :
The thesis is in two parts. The first part is the paper “The Distribution of k-free integers” that my advisor, Dr. Roger Baker, and I submitted in February 2009. The reader will note that I have inserted additional commentary and explanations which appear in smaller text. Dr. Baker and I improved the asymptotic formula for the number of k-free integers less than x by taking advantage of exponential sum techniques developed since the 1980's. Both of us made substantial contributions to the paper. I discovered the exponent in the error term for the cases k=3,4, and worked the case k=3 completely. Dr. Baker corrected my work for k=4 and proved the result for k=5. He then generalized our work into the paper as it now stands. We also discussed and both contributed to parts of section 3 on bounds for exponential sums. The second part represents my own work guided by my advisor. I study the zeros of derivatives of Dirichlet L-functions. The first theorem gives an analog for a result of Speiser on the zeros of ζ'(s). He proved that RH is equivalent to the hypothesis that ζ'(s) has no zeros with real part strictly between 0 and ½. The last two theorems discuss zero-free regions to the left and right for L^{(k)}(s,χ).
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Ruan, Ning. « Global optimization for nonconvex optimization problems ». Thesis, Curtin University, 2012. http://hdl.handle.net/20.500.11937/1936.
Texte intégralRésumé :
Duality is one of the most successful ideas in modern science [46] [91]. It is essential in natural phenomena, particularly, in physics and mathematics [39] [94] [96]. In this thesis, we consider the canonical duality theory for several classes of optimization problems.The first problem that we consider is a general sum of fourth-order polynomial minimization problem. This problem arises extensively in engineering and science, including database analysis, computational biology, sensor network communications, nonconvex mechanics, and ecology. We first show that this global optimization problem is actually equivalent to a discretized minimal potential variational problem in large deformation mechanics. Therefore, a general analytical solution is proposed by using the canonical duality theory.The second problem that we consider is a nonconvex quadratic-exponential optimization problem. By using the canonical duality theory, the nonconvex primal problem in n-dimensional space can be converted into a one-dimensional canonical dual problem, which is either a concave maximization or a convex minimization problem with zero duality gap. Several examples are solved so as to illustrate the applicability of the theory developed.The third problem that we consider is quadratic minimization problems subjected to either box or integer constraints. Results show that these nonconvex problems can be converted into concave maximization dual problems over convex feasible spaces without duality gap and the Boolean integer programming problem is actually equivalent to a critical point problem in continuous space. These dual problems can be solved under certain conditions. Both existence and uniqueness of the canonical dual solutions are presented. A canonical duality algorithm is presented and applications are illustrated.The fourth problem that we consider is a quadratic discrete value selection problem subjected to inequality constraints. The problem is first transformed into a quadratic 0-1 integer programming problem. The dual problem is thus constructed by using the canonical duality theory. Under appropriate conditions, this dual problem is a maximization problem of a concave function over a convex continuous space. Theoretical results show that the canonical duality theory can either provide a global optimization solution, or an optimal lower bound approximation to this NP-hard problem. Numerical simulation studies, including some relatively large scale problems, are carried out so as to demonstrate the effectiveness and efficiency of the canonical duality method. An open problem for understanding NP-hard problems is proposed.The fifth problem that we consider is a mixed-integer quadratic minimization problem with fixed cost terms. We show that this well-known NP-hard problem in R2n can be transformed into a continuous concave maximization dual problem over a convex feasible subset of Rn with zero duality gap. We also discuss connections between the proposed canonical duality theory approach and the classical Lagrangian duality approach. The resulting canonical dual problem can be solved under certain conditions, by traditional convex programming methods. Conditions for the existence and uniqueness of global optimal solutions are presented. An application to a decoupled mixed-integer problem is used to illustrate the derivation of analytic solutions for globally minimizing the objective function. Numerical examples for both decoupled and general mixed-integral problems are presented, and an open problem is proposed for future study.The sixth problem that we consider is a general nonconvex quadratic minimization problem with nonconvex constraints. By using the canonical dual transformation, the nonconvex primal problem can be converted into a canonical dual problem (i.e., either a concave maximization problem with zero duality gap). Illustrative applications to quadratic minimization with multiple quadratic constraints, box/integer constraints, and general nonconvex polynomial constraints are discussed, along with insightful connections to classical Lagrangian duality. Conditions for ensuring the existence and uniqueness of global optimal solutions are presented. Several numerical examples are solved.The seventh problem that we consider is a general nonlinear algebraic system. By using the least square method, the nonlinear system of m quadratic equations in n-dimensional space is first formulated as a nonconvex optimization problem. We then prove that, by using the canonical duality theory, this nonconvex problem is equivalent to a concave maximization problem in Rm, which can be solved by well-developed convex optimization techniques. Both existence and uniqueness of global optimal solutions are discussed, and several illustrative examples are presented.The eighth problem that we consider is a general sensor network localization problem. It is shown that by the canonical duality theory, this nonconvex minimization problem is equivalent to a concave maximization problem over a convex set in a symmetrical matrix space, and hence can be solved by combining a perturbation technique with existing optimization techniques. Applications are illustrated and results show that the proposed method is potentially a powerful one for large-scale sensor network localization problems.
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Wei, Mu-Hsin. « Estimation of the discrete spectrum of relaxations for electromagnetic induction responses ». Thesis, Georgia Institute of Technology, 2011. http://hdl.handle.net/1853/39534.
Texte intégralRésumé :
This thesis presents a robust method for estimating the relaxations of a metallic object from its electromagnetic induction (EMI) response. The EMI response of a metallic object can be accurately modeled by a sum of real decaying exponentials. However, it is diffcult to obtain the model parameters from measurements when the number of exponentials in the sum is unknown or the terms are strongly correlated. Traditionally, the time constants and residues are estimated by nonlinear iterative search that often leads to unsatisfactory results. In this thesis, a constrained linear method of estimating the parameters is formulated by enumerating the relaxation parameter space and imposing a nonnegative constraint on the parameters. The resulting algorithm does not depend on a good initial guess to converge to a solution. Using tests on synthetic data and laboratory measurement of known targets the proposed method is shown to provide accurate and stable estimates of the model parameters.
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Giscard, Pierre-Louis. « A graph theoretic approach to matrix functions and quantum dynamics ». Thesis, University of Oxford, 2014. http://ora.ox.ac.uk/objects/uuid:ceef15b0-eed2-4615-a9f2-f9efbef470c9.
Texte intégralRésumé :
Many problems in applied mathematics and physics are formulated most naturally in terms of matrices, and can be solved by computing functions of these matrices. For example, in quantum mechanics, the coherent dynamics of physical systems is described by the matrix exponential of their Hamiltonian. In state of the art experiments, one can now observe such unitary evolution of many-body systems, which is of fundamental interest in the study of many-body quantum phenomena. On the other hand the theoretical simulation of such non-equilibrium many-body dynamics is very challenging. In this thesis, we develop a symbolic approach to matrix functions and quantum dynamics based on a novel algebraic structure we identify for sets of walks on graphs. We begin by establishing the graph theoretic equivalent to the fundamental theorem of arithmetic: all the walks on any finite digraph uniquely factorise into products of prime elements. These are the simple paths and simple cycles, walks forbidden from visiting any vertex more than once. We give an algorithm that efficiently factorises individual walks and obtain a recursive formula to factorise sets of walks. This yields a universal continued fraction representation for the formal series of all walks on digraphs. It only involves simple paths and simple cycles and is thus called a path-sum. In the second part, we recast matrix functions into path-sums. We present explicit results for a matrix raised to a complex power, the matrix exponential, matrix inverse, and matrix logarithm. We introduce generalised matrix powers which extend desirable properties of the Drazin inverse to all powers of a matrix. In the third part, we derive an intermediary form of path-sum, called walk-sum, relying solely on physical considerations. Walk-sum describes the dynamics of a quantum system as resulting from the coherent superposition of its histories, a discrete analogue to the Feynman path-integrals. Using walk-sum we simulate the dynamics of quantum random walks and of Rydberg-excited Mott insulators. Using path-sum, we demonstrate many-body Anderson localisation in an interacting disordered spin system. We give two observable signatures of this phenomenon: localisation of the system magnetisation and of the linear magnetic response function. Lastly we return to the study of sets of walks. We show that one can construct as many representations of series of walks as there are ways to define a walk product such that the factorisation of a walk always exist and is unique. Illustrating this result we briefly present three further methods to evaluate functions of matrices. Regardless of the method used, we show that graphs are uniquely characterised, up to an isomorphism, by the prime walks they sustain.
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Pasca, Bogdan Mihai. « Calcul flottant haute performance sur circuits reconfigurables ». Phd thesis, Ecole normale supérieure de lyon - ENS LYON, 2011. http://tel.archives-ouvertes.fr/tel-00654121.
Texte intégralRésumé :
De plus en plus de constructeurs proposent des accélérateurs de calculs à base de circuits reconfigurables FPGA, cette technologie présentant bien plus de souplesse que le microprocesseur. Valoriser cette flexibilité dans le domaine de l'accélération de calcul flottant en utilisant les langages de description de circuits classiques (VHDL ou Verilog) reste toutefois très difficile, voire impossible parfois. Cette thèse a contribué au développement du logiciel FloPoCo, qui offre aux utilisateurs familiers avec VHDL un cadre C++ de description d'opérateurs arithmétiques génériques adapté au calcul reconfigurable. Ce cadre distingue explicitement la fonctionnalité combinatoire d'un opérateur, et la problématique de son pipeline pour une précision, une fréquence et un FPGA cible donnés. Afin de pouvoir utiliser FloPoCo pour concevoir des opérateurs haute performance en virgule flottante, il a fallu d'abord concevoir des blocs de bases optimisés. Nous avons d'abord développé des additionneurs pipelinés autour des lignes de propagation de retenue rapides, puis, à l'aide de techniques de pavages, nous avons conçu de gros multiplieurs, possiblement tronqués, utilisant des petits multiplieurs. L'évaluation de fonctions élémentaires en flottant implique souvent l'évaluation en virgule fixe d'une fonction. Nous présentons un opérateur générique de FloPoCo qui prend en entrée l'expression de la fonction à évaluer, avec ses précisions d'entrée et de sortie, et construit un évaluateur polynomial optimisé de cette fonction. Ce bloc de base a permis de développer des opérateurs en virgule flottante pour la racine carrée et l'exponentielle qui améliorent considérablement l'état de l'art. Nous avons aussi travaillé sur des techniques de compilation avancée pour adapter l'exécution d'un code C aux pipelines flexibles de nos opérateurs. FloPoCo a pu ainsi être utilisé pour implanter sur FPGA des applications complètes.
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Shubin, Andrei. « Topics in Equidistribution and Exponential Sums ». Thesis, 2022. https://thesis.library.caltech.edu/14306/1/shubin_andrei_2021.pdf.
Texte intégralRésumé :
In this thesis, we consider a few problems connected to the exponential sums which is one of the most important topics in analytic number theory.
In the first part, we study the distribution of prime numbers in special subsets of integers and, in particular, the distribution of these primes in arithmetic progressions, small gaps between them, the behavior of the corresponding exponential sums over primes, and related questions. Big progress was made on these questions in recent years. The famous works of Zhang and Maynard gave the proof of existence of bounded gaps between consecutive primes. Applying the sieve of Selberg-Maynard-Tao and an analogue of the Bombieri-Vinogradov theorem, we obtain similar results for a large class of subsets of primes and improve some of the previous results. The proof of the analogue of the Bombieri-Vinogradov theorem is also connected to a breakthrough work of Bourgain, Demeter, and Guth on the proof of Vinogradov Mean Value Conjecture via l2-decoupling. Their result, in particular, has led to a significant improvement of the classical van der Corput estimates for a large class of exponential sums.
In the second part, we study the behavior of higher moments of Gauss sum twisted by a Mobius function. The moments of exponential sums are very important in number theory and harmonic analysis as they appear in many other problems. The sum with the Mobius function is of independent interest because of the famous Sarnak Conjecture which is on the edge of number theory, analysis, and dynamical systems. The bound we obtain for Lp-norm of the sum confirms that the Mobius function is uncorrelated with the quadratic phase αn2 for most α ϵ [0; 1].
In the third part, we study the distribution of lattice points on the surface of 3-dimensional sphere, which is known as Linnik problem. It turns out that the variance for such points is closely related to the behavior of certain GL(2) L-functions estimated at the central point 1/2. To evaluate the moments of these L-functions, we apply similar techniques used to evaluate the moments of Riemann zeta function on the critical line in the breakthrough works of Soundararajan and Harper. Their results have led to the sharp upper bounds for all positive moments of zeta function conditionally on Riemann Hypothesis and similar bounds for a broad class of L-functions in families conditionally on the corresponding Grand Riemann Hypothesis. We apply similar methods to get sharp upper bound for the variance of lattice points on the sphere. The connection of Weyl sums on the sphere to the sums of special values of GL(2) L-functions is a big output of the Langlands program, which has also gotten a lot of attention in recent years.
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Louvel, Benoît [Verfasser]. « Twisted Kloosterman sums and cubic exponential sums / vorgelegt von Benoît Louvel ». 2008. http://d-nb.info/101056661X/34.
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von, der Ohe Ulrich. « On the reconstruction of multivariate exponential sums ». Doctoral thesis, 2017. https://repositorium.ub.uni-osnabrueck.de/handle/urn:nbn:de:gbv:700-2017120716391.
Texte intégralRésumé :
We develop a theory concerning the reconstruction of multivariate exponential sums first over arbitrary fields and then consider the special cases of multivariate exponential sums over the fields of real and complex numbers.
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Zhao, Xiaomei. « Asymptotic Estimates for Rational Spaces on Hypersurfaces in Function Fields ». Thesis, 2010. http://hdl.handle.net/10012/5284.
Texte intégralRésumé :
The ring of polynomials over a finite field has many arithmetic properties similar to those of the ring of rational integers. In this thesis, we apply the Hardy-Littlewood circle method to investigate the density of rational points on certain algebraic varieties in function fields. The aim is to establish asymptotic relations that are relatively robust to changes in the characteristic of the base finite field. More notably, in the case when the characteristic is "small", the results are sharper than their integer analogues.
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Anghel, Catalina Voichita. « The Self Power Map and its Image Modulo a Prime ». Thesis, 2013. http://hdl.handle.net/1807/35765.
Texte intégralRésumé :
The self-power map is the function from the set of natural numbers to itself which sends the number $n$ to $n^n$. Motivated by applications to cryptography, we consider the image of this map modulo a prime $p$. We study the question of how large $x$ must be so that $n^n \equiv a \bmod p$ has a solution with $1 \le n \le x$, for every residue class $a$ modulo $p$. While $n^n \bmod p$ is not uniformly distributed, it does appear to behave in certain ways as a random function. We give a heuristic argument to show that the expected $x$ is approximately ${p^2\log \phi(p-1)/\phi(p-1)}$, using the coupon collector problem as a model. Rigorously, we prove the bound $x
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« A study of correlation of sequences ». Chinese University of Hong Kong, 1993. http://library.cuhk.edu.hk/record=b5887751.
Texte intégralRésumé :
by Wai Ho Mow.Thesis (Ph.D.)--Chinese University of Hong Kong, 1993.
Includes bibliographical references (leaves 116-124).
Chapter 1 --- Introduction --- p.1
Chapter 1.1 --- Spread Spectrum Technique --- p.2
Chapter 1.1.1 --- Pulse Compression Radars --- p.3
Chapter 1.1.2 --- Spread Spectrum Multiple Access Systems --- p.6
Chapter 1.2 --- Definitions and Notations --- p.8
Chapter 1.3 --- Organization of this Thesis --- p.12
Chapter 2 --- Lower Bounds on Correlation of Sequences --- p.15
Chapter 2.1 --- Welch's Lower Bounds and Sarwate's Generalization --- p.16
Chapter 2.2 --- A New Construction and Bounds on Odd Correlation --- p.23
Chapter 2.3 --- Known Sequence Sets Touching the Correlation Bounds --- p.26
Chapter 2.4 --- Remarks on Other Bounds --- p.27
Chapter 3 --- Perfect Polyphase Sequences: A Unified Approach --- p.29
Chapter 3.1 --- Generalized Bent Functions and Perfect Polyphase Sequences --- p.30
Chapter 3.2 --- The General Construction of Chung and Kumar --- p.32
Chapter 3.3 --- Classification of Known Constructions ...........； --- p.34
Chapter 3.4 --- A Unified Construction --- p.39
Chapter 3.5 --- Desired Properties of Sequences --- p.41
Chapter 3.6 --- Proof of the Main Theorem --- p.45
Chapter 3.7 --- Counting the Number of Perfect Polyphase Sequences --- p.49
Chapter 3.8 --- Results of Exhaustive Searches --- p.53
Chapter 3.9 --- A New Conjecture and Its Implications --- p.55
Chapter 3.10 --- Sets of Perfect Polyphase Sequences --- p.58
Chapter 4 --- Aperiodic Autocorrelation of Generalized P3/P4 Codes --- p.61
Chapter 4.1 --- Some Famous Polyphase Pulse Compression Codes --- p.62
Chapter 4.2 --- Generalized P3/P4 Codes --- p.65
Chapter 4.3 --- Asymptotic Peak-to-Side-Peak Ratio --- p.66
Chapter 4.4 --- Lower Bounds on Peak-to-Side-Peak Ratio --- p.67
Chapter 4.5 --- Even-Odd Transformation and Phase Alphabet --- p.70
Chapter 5 --- Upper Bounds on Partial Exponential Sums --- p.77
Chapter 5.1 --- Gauss-like Exponential Sums --- p.77
Chapter 5.1.1 --- Background --- p.79
Chapter 5.1.2 --- Symmetry of gL(m) and hL(m) --- p.80
Chapter 5.1.3 --- Characterization on the First Quarter of gL(m) --- p.83
Chapter 5.1.4 --- Characterization on the First Quarter of hL(m) --- p.90
Chapter 5.1.5 --- Bounds on the Diameters of GL(m) and HL(m) --- p.94
Chapter 5.2 --- More General Exponential Sums --- p.98
Chapter 5.2.1 --- A Result of van der Corput --- p.99
Chapter 6 --- McEliece's Open Problem on Minimax Aperiodic Correlation --- p.102
Chapter 6.1 --- Statement of the Problem --- p.102
Chapter 6.2 --- A Set of Two Sequences --- p.105
Chapter 6.3 --- A Set of K Sequences --- p.110
Chapter 7 --- Conclusion --- p.113
Bibliography --- p.124
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Pototskaia, Vlada. « Application of AAK theory for sparse approximation ». Doctoral thesis, 2017. http://hdl.handle.net/11858/00-1735-0000-0023-3F4B-1.
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Chiapparoli, Paula Mercedes. « Distribución de pesos de códigos cíclicos a partir de sumas exponenciales y curvas algebraicas ». Bachelor's thesis, 2020. http://hdl.handle.net/11086/17503.
Texte intégralRésumé :
Tesis (Lic. en Matemática)--Universidad Nacional de Córdoba, Facultad de Matemática, Astronomía, Física y Computación, 2020.Este trabajo trata sobre el espectro o distribución de pesos de códigos lineales y cíclicos. Esto es en general una tarea ardua y sólo se conoce el espectro de algunas familias de códigos. Estudiaremos distintas formas de encontrar dichas distribuciones de pesos a través de diferentes caminos. Primero veremos resultados generales para códigos lineales, que en particular dan una respuesta general al caso de los códigos MDS. Luego, nos enfocaremos en códigos cíclicos generales viéndolos como códigos traza (combinando los teoremas de Delsarte y las identidades de MacWilliams). A partir de aquí haremos uso de dos estrategias generales, una que involucra ciertas sumas exponenciales (Gauss, Weil y/o Kloosterman) y otra basada en el conteo de puntos racionales de curvas algebraicas asociadas a los códigos (típicamente de Artin-Schreier). Usaremos estas técnicas para obtener los espectros de familias de códigos muy conocidas como Hamming, BCH y Reed-Muller. Finalmente, aplicaremos estos métodos a dos familias de códigos menos conocidos como los códigos de Melas y de Zetterberg. En los casos binario y ternario, el cálculo de dichos espectros se puede realizar usando curvas elípticas y la traza de operadores de Hecke de ciertas formas modulares asociadas a ellas. El trabajo contiene numerosos ejemplos, muchos de ellos nuevos.
This work deals with the spectrum or weight distribution of linear and cyclic codes. This is in general a difficult task and the spectrum is only known for some families of codes. We will study different ways to find these distributions through different ways. We will first see general results for linear codes, which in particular give a general answer to the case of MDS codes. Then, we will focus on general cyclic codes by viewing them as trace codes (combining Delsarte's theorems and MacWilliams identities). From this point on we will use two general strategies, one that involves certain exponential sums (Gauss, Weil or Kloosterman) and another one based on counting the number of rational points of algebraic curves (typically Artin-Schreier) associated with the codes. We will use these techniques to obtain the spectra of well-known families of codes such as Hamming, BCH, and Reed-Muller codes. Finally, we will apply these methods to two lesser known code families, the Melas codes and the Zetterberg codes. In the binary and ternary cases, the computation of the mentioned spectra can be performed by using elliptic curves and the trace of Hecke operators of certain modular forms associated to them. The work contains several examples, many of them new.
Fil: Chiapparoli, Paula Mercedes. Universidad Nacional de Córdoba. Facultad de Matemática, Astronomía, Física y Computación; Argentina.
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Melzer, Ines. « Fast and approximate computation of Laplace and Fourier transforms ». Doctoral thesis, 2016. https://repositorium.ub.uni-osnabrueck.de/handle/urn:nbn:de:gbv:700-2016040414362.
Texte intégralRésumé :
In this thesis, we treat the computation of transforms with asymptotically smooth and oscillatory kernels. We introduce the discrete Laplace transform in a modern form including a generalization to more general kernel functions. These more general kernels lead to specific function transforms. Moreover, we treat the butterfly fast Fourier transform. Based on a local error analysis, we develop a rigorous error analysis for the whole butterfly scheme. In the final part of the thesis, the Laplace and Fourier transform are combined to a fast Fourier transform for nonequispaced complex evaluation nodes. All theoretical results on accuracy and computational complexity are illustrated by numerical experiments.
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Klimovsky, Anton [Verfasser]. « Sums of correlated exponentials : two types of Gaussian correlation structures / Anton Klymovskiy (Klimovsky) ». 2008. http://d-nb.info/990507866/34.
Texte intégral
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Ko, Ching-Hui, et 柯瀞惠. « On the exponential-, log-normal- and inverse Gaussian- based combination methods and a rank truncated sum method for combining p values ». Thesis, 2012. http://ndltd.ncl.edu.tw/handle/81403156299167568127.
Texte intégralRésumé :
碩士輔仁大學
統計資訊學系應用統計碩士班
100
Combination test procedure is a commonly used method in meta-analysis. It is developed for increasing the testing power through combining various tests on the basis of the observed p-values. Fisher's combination procedure is the typical one of those methods. It is known that the inverse normal method and the rank truncated product method have better power performance than Fisher's method in some situations. Accordingly, in this study, our first aim is to develop inverse-type combination methods other than Fisher's method. The minimum inverse exponential method, the inverse log-normal method and the inverse inverse-Gaussian method were thus investigated. Our second aim is to develop an alternative truncated-type method other than product method. Based on the Edgington's method, the rank truncated sum method was thus proposed. Simulation studies indicated that the three new quantile combination methods have higher power performance than Fisher's combination procedure, Tippett's method and inverse normal method in many situations. On the other hand, the rank truncated sum method has higher power than the modified truncated product method and the rank truncated product method in several situations.
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El-Khatib, Mayar. « Highway Development Decision-Making Under Uncertainty : Analysis, Critique and Advancement ». Thesis, 2010. http://hdl.handle.net/10012/5741.
Texte intégralRésumé :
While decision-making under uncertainty is a major universal problem, its implications in the field of transportation systems are especially enormous; where the benefits of right decisions are tremendous, the consequences of wrong ones are potentially disastrous.
In the realm of highway systems, decisions related to the highway configuration (number of lanes, right of way, etc.) need to incorporate both the traffic demand and land price uncertainties. In the literature, these uncertainties have generally been modeled using the Geometric Brownian Motion (GBM) process, which has been used extensively in modeling many other real life phenomena. But few scholars, including those who used the GBM in highway configuration decisions, have offered any rigorous justification for the use of this model.
This thesis attempts to offer a detailed analysis of various aspects of transportation systems in relation to decision-making. It reveals some general insights as well as a new concept that extends the notion of opportunity cost to situations where wrong decisions could be made. Claiming deficiency of the GBM model, it also introduces a new formulation that utilizes a large and flexible parametric family of jump models (i.e., Lévy processes). To validate this claim, data related to traffic demand and land prices were collected and analyzed to reveal that their distributions, heavy-tailed and asymmetric, do not match well with the GBM model. As a remedy, this research used the Merton, Kou, and negative inverse Gaussian Lévy processes as possible alternatives.
Though the results show indifference in relation to final decisions among the models, mathematically, they improve the precision of uncertainty models and the decision-making process. This furthers the quest for optimality in highway projects and beyond.