Magistro darbe nagrinėjamas Oilerio sandaugų reikšmių pasiskirstymas analizinių funkcijų erdvėje. Taip pat gaunamas išreikštinis ribinio mato pavidalas. / In the Master work, we investigate the value-distribution of Euler products in the space of analytic functions. Also, we give an explicit form of the limit measure.

Tegul s=σ+it yra kompleksinis kintamasis. Oilerio sandaugos yra apibrėžiamos pagal pirminius skaičius, taip pat yra reikalaujama, kad funkcija L(s) tenkintų papidomas sąlygas. Mes įrodome diskrečią ribinę teoremą tikimybinių matų silpno konvergavimo prasme kompleksinėje plokštumoje C Oilerio sandaugoms. / Let s=σ+it be a complex variable. The Euler products L(s) is defined by the prime number. If the function L(s) satisfies some additional hypotheses. In the Master work we prove the discrete limit theorem in the sense of weakly convergent probability measures for the Euler products on the complex plane.

Tegul s yra kompleksinis kintamasis. Oilerio sandaugos apibrėžiamos pagal pirminius p skaičius. Funkcija L(s) turi tenkinti hipotezes. Magistro darbe, įrodome diskrečią ribinę teoremą silpno tikimybinių matų konvergavimo prasme Oilerio sandaugoms kompleksinėje plokštumoje. Gauta mato išreikštinė forma. / Let s be a complex variable. The Euler products is defined by the prime number p. The Function L(s) satisfies some additional hypoteses. In Master work, we prove the discrete limit theorem in the sense of weakly convergent probability measures for the Euler products on the complex plane. Then the probability measure weakly converges to the distribution of one explicitly given complex-valued random element as N-> infinity.

Dans une première partie, nous proposons un nouveau modèle de chemostat dans lequel la population bactérienne est représentée de manière individu-centrée, structurée en masse, et la dynamique du substrat est modélisée par une équation différentielle ordinaire. Nous obtenons un processus markovien que nous décrivons à l'aide de mesures aléatoires. Nous déterminons, sous une certaine renormalisation du processus, un résultat de convergence en loi de ce modèle individu-centré hybride vers la solution d'un système d'équations intégro-différentielles. Dans une seconde partie, nous nous intéressons à des modèles de dynamiques adaptatives du chemostat. Nous reprenons le modèle individu-centré étudié dans la première partie, auquel nous ajoutons un mécanisme de mutation. Sous des hypothèses de mutations rares et de grande population, les résultats asymptotiques obtenus dans la première partie nous permettent de réduire l'étude d'une population mutante à un modèle de croissance-fragmentation-soutirage en milieu constant. Nous étudions la probabilité d'extinction de cette population mutante. Nous décrivons également le modèle déterministe associé au modèle individu-centré hybride avec mutation et nous comparons les deux approches, stochastique et déterministe; notamment nous démontrons qu'elles mènent au même critère de possibilité d'invasion d'une population mutante dans une population résidente.Nous présentons des simulations numériques illustrant les résultats mathématiques obtenus. / In the first part, we propose a new chemostat model in which the bacterial population is mass structured and individual-based and the substrate dynamics are modelized by an ordinary differential equation. We obtain a Markovian process which we describe as random measures. We determine, under a certain normalization of the process, a result of convergence in distribution towards the solution of a system of integro-differential equations. In the second part, we are interested in adaptive dynamic models of the chemostat. We add a mutation mechanism to the individual-based model which was studied in the first part. Under rare mutations and large population size hypotheses, the asymptotical result of the first part allows us to reduce the study of the mutant population to a growth-fragmentation-washout model in a constant environment. We study the extinction probability of this mutant population. We also describe the deterministic model related to the hybrid individual-based model with mutations and we compare these two approaches (stochastic and deterministic). In particular we prove that the two approaches lead to the same invasion criteria of a mutant population in a resident population.We present numeric simulations in order to illustrate the mathematical results.

In the paper we study sequences of random functions which are defined by some interpolation procedures for a given random function. We investigate the problem in what sense and under which conditions the sequences converge to the prescribed random function. Sufficient conditions for convergence of moment characteristics, of finite dimensional distributions and for weak convergence of distributions in spaces of continuous functions are given. The treatment of such questions is stimulated by an investigation of Monte Carlo simulation procedures for certain classes of random functions. In an appendix basic facts concerning weak convergence of probability measures in metric spaces are summarized.

<p>H-mere i H-distribucije su mikrolokalni objekti koji se koriste za ispitivanje jake konvergencije slabo konvergentnog niza u prostorima Lebega i prostorima Soboljeva. H-mere su uveli Tartar i&nbsp; Zerar (koji ih zove mikrolokalne mere defekta), u radovima [34] i [19]. H-mere su Radonove mere koje daju informacije o mogu &acute; cim oblastima jake konvergencije slabo konvergentnog<em> L</em>² niza. Da bismo mogli da posmatramo i slabo konvergentne<em> L</em>^p nizove za 1 &lt; p &lt; &infin;, Antonić&nbsp; i Mitrović u radu [11] uvode H-distribucije.</p><p>U disertaciji dajemo konstrukciju H-distribucija za slabo konvergentne nizove u <em>W</em>^-k,p prostorima, kad je 1 &lt; p &lt; &infin;, k &isin; ℕ&nbsp;i pokazujemo da kada je H-distribucija pridružena slabo konvergetnim nizovima jednaka nuli za sve test funkcije, onda imamo lokalno jaku konverenciju datog niza.</p><p>Takođe je pokazan i lokalizacijski princip, koji nam daje oblast u kojoj imamo lokalno jaku&nbsp; konvergenciju slabo konvergentnog niza. H-mere i H-distribucije deluju na test funkcije &phi;&nbsp;i &psi;&nbsp;(odgovarajuće regularnosti) koje su definisane na ℝ^d i S^d-1 (jedinična sfera u ℝ^d), pri&nbsp; čemu je funkcija &psi;, koju zovemo množilac, ograničena. U disertaciji uvodimo i H-distribucije sa neograničenim simbolom, pri čemu posmatramo slabo&nbsp; konvergentne nizove u Beselovim H^p_-s prostorima, gde je 1 &lt; p &lt; &infin;; s &isin; ℝ. U ovom delu koristimo teoriju pseudo-diferencijalnih operatora i dokazujemo kompaktnost komutatora [<i>A</i>_&psi;, T_&phi;] za razne klase množioca &psi;,&nbsp; &scaron;to je potrebno za dokaz postojanja H-distribucija. Takođe pokazujemo odgovarajuću verziju lokalizacijskog principa.</p> / <p>H-measures and H-distributions are microlocal tools that can be used to investigate strong conver-gence of weakly convergent sequences in the Lebesgue and Sobolev spaces.</p><p>H-measures are introduced by Tartar and G&eacute;rard (as microlocal defect measures) in papers [34] and [19]. H-measures are Radon measures and they provide information about the set of points where given weakly convergent sequence in <em>L</em>² converges strongly. In paper [11], Antonić and Mitrović introduced&nbsp; H-distributions in order to work with weakly convergent <em>L</em>^p sequences.</p><p>In this thesis we give construction of H-distributions for weakly convergent <em>W^-</em>^k,p sequences, where 1 &lt; p &lt; &infin;; k &isin;&nbsp;N. We show that if the H-distribution corresponding to given weakly convergent sequence is equal to zero, then we have locally strong convergence of the sequence. We also prove localization principle.</p><p>H-measures and H-distributions act on test functions &phi; and &psi;&nbsp;(regular enough) which are defined on ℝ^d and ^d-1 (unit sphere in ℝ^d ) and the function &psi;, which is called multiplier, is bounded. We also introduce H-distributions with unboundedmultipliers and in this&nbsp; case we assume that weakly convergent sequences are in Bessel potential spaces H^p_-s , where 1 &lt; p &lt; &infin;, s &isin; ℝ. Theory of pseudo-differential operators is used in construction of H-distributions with unbounded multipliers. We prove compactness of the commutator [<em>A</em>_{<em>&psi;</em>},T_&phi; ] for different classes of multipliers y and appropriate version of localization principle.</p>

京都大学 / 0048 / 新制・課程博士 / 博士(理学) / 甲第19468号 / 理博第4128号 / 新制||理||1594(附属図書館) / 32504 / 京都大学大学院理学研究科数学・数理解析専攻 / (主査)准教授 矢野 孝次, 教授 上田 哲生, 教授 重川 一郎 / 学位規則第4条第1項該当 / Doctor of Science / Kyoto University / DFAM

Consider the variational inequality (VI) x* ∈C, ‹Fx*, x - x* ›≥0, x∈C (*) where C is a nonempty closed convex subset of a real Hilbert space H and F : C¡÷ H is a monotone operator form C into H. It is known that if F is strongly monotone and Lipschitzian, then VI (*) is equivalently turned into a fixed point problem of a contraction; hence Banach's contraction principle applies. However, in the case where F is inverse strongly monotone, VI (*) is equivalently transformed into a fixed point problem of a nonexpansive mapping. The purpose of this paper is to present some results which apply iterative methods for nonexpansive mappings to solve VI (*). We introduce Mann's algorithm and Halpern's algorithm and prove that the sequences generated by these algorithms converge weakly and respectively, strongly to a solution of VI (*), under appropriate conditions imposed on the parameter sequences in the algorithms.

Iterative methods are extensively used to solve linear and nonlinear problems arising from both pure and applied sciences, and in particular, in fixed point theory and optimization. An iterative method which is used to find a fixed point of an operator or an optimal solution to an optimization problem generates a sequence in an iterative manner. We are in a hope that this sequence can converge to a solution of the problem under investigation. It is therefore quite naturally to require that the distance of this sequence to the solution set of the problem under investigation be decreasing from iteration to iteration. This is the idea of Fejer-monotonicity. In this paper, We consider quasi-Fejer monotone sequences; that is, we consider Fejer monotone sequences together with errors. Properties of quasi-Fejer monotone sequences are investigated, weak and strong convergence of quasi-Fejer monotone sequences are obtained, and an application to the convex feasibility problem is included.

The aim of our work is to obtain joint universality theorems for periodic Hurwitz zeta-functions. We prove two joint universality theorems for periodic Hurwitz zeta-function. In the first theorems, the set L is linearly independent over the field of national numbers, then the periodic Hurwitz zeta-functions are universality. In the second joint universality theorem, we consider the use then parameter alpha corresponds general periodic sequence. Then the set L is linearly independent over the field of national numbers and the rank hypothesis in this theorem is weaker then that in A. Laurinčikas (2008) work. Then the second periodic Hurwitz zeta-functions are universal too. / Magistro darbe yra nagrinėjamas Hurvico dzeta funkcijų rinkinio jungtinis universalumas. Yra įrodytos dvi jungtinės universalumo teoremos. Pirmoji teorema tvirtina, kad jei aibė L yra tiesiškai nepriklausoma virš racionaliųjų skaičių kūno, tai periodinės Hurvico dzeta funkcijos yra universalios. Ši teorema žymiai susilpnina sąlygas, kurioms esant, buvo gautas analogiškas rezultatas A. Javtoko ir A. Laurinčiko 2008 m. darbe. Antroje teoremoje yra nagrinėjamas atvejis, kai kiekvieną skaičių alpha atitinka periodinių sekų rinkinys. Kai sistema L yra tiesiškai nepriklausoma virš racionaliųjų skaičių kūno ir galioja vieno rango tipo sąlyga, silpnesnė negu A. Laurinčiko darbe (2008), tai periodinių Hurvico dzeta funkcijų rinkinys yra taip pat universalus.