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We show that there exist infinite dimensional spaces of series, every non-zero element of which, enjoys certain pathological property. Some of these properties consist on being (i) conditional convergent, (ii) divergent, or (iii) being a subspace of l(infinity) of divergent series. We also show that the space 1(1)(omega)(X) of all weakly unconditionally Cauchy series in X has an infinite dimensional vector space of non-weakly convergent series, and that the set of unconditionally convergent series on X contains a vector space E, of infinite dimension, so that if x is an element of E \ {0} then Sigma(i) parallel to x(i)parallel to = infinity., Depto. de Análisis Matemático y Matemática Aplicada, Fac. de Ciencias Matemáticas, TRUE, pub

We study a linear form in the values of Euler’s series \(F(t)=\sum\nolimits_{n=0}^\infty n!t^n\) at algebraic integer points \(α_j∈\mathbb{Z}_\mathbb{K}, j=1,…,m\), belonging to a number field \(\mathbb{K}\). In the two main results it is shown that there exists a non-Archimedean valuation \(v\vert p\) of the field \(\mathbb{K}\) such that the linear form \({\mathrm\Lambda}_v=\lambda_0+\lambda_1F_v(\alpha_1)+\dots+\lambda_mF_v(\alpha_m)\), \(\lambda_i\in{\mathbb{Z}}_\mathbb{K}\), does not vanish. The second result contains a lower bound for the v-adic absolute value of \({\mathrm\Lambda}_v\), and the first one is also extended to the case of primes in residue classes. On the way to the main results, we present explicit Padé approximations to the generalised factorial series \(\sum\nolimits_{n=0}^\infty{\left(\prod\nolimits_{k=0}^{n-1}P(k)\right)}t^n\), where \(P(x)\) is a polynomial of degree one.

Let \(ξ\) and \(m\) be integers satisfying \(ξ \ne 0\) and \(m ≥ 3\). We show that for any given integers \(a\) and \(b\), \(b \ne 0\), there are \(\frac{φ(m)}{2}\) reduced residue classes modulo \(m\) each containing infinitely many primes \(p\) such that \(a−bF_p(ξ) \ne 0\), where \(F_p(ξ) =\sum^{\infty}_{n=0} n!ξ^n\) is the p-adic evaluation of Euler’s factorial series at the point \(ξ\).

The aim of this paper is to analyze the persuasive and manipulative strategies and devices used in the political propaganda in the films of the Divergent Series. Special attention is paid to Insurgent, since the outbreak of a war between different ideological and political factions and the government is a breeding ground for manipulative, political strategies to persuade the population to side with the former or the latter, which they use to try to win. In this scenario, political discourse also serves both deterring rebellious Divergents threatening the established system and controlling the masses from the different factions. To this end, I follow the parameters of different theoretical frameworks, such as critical-political discourse analysis, Appraisal Theory –focusing on attitude or attitudinal positioning– and the phenomena of polarization and legitimation. The results will allow us to see how the political agents of the film managed to influence their audiences with their speeches by using rhetorical means, subjectivity to present situations, objects or people positively or negatively at the speaker’s convenience or interests, and different types of power., El objetivo de este artículo es analizar las estrategias y recursos de persuasión y manipulación que se emplean en la propaganda política de las películas de la trilogía Divergente. Mención especial recibe la segunda, Insurgente, pues el estallido de la guerra entre las diversas facciones políticas e ideológicas es caldo de cultivo de estrategias políticas manipulativas para persuadir a la población y atraerla hacia un lado u otro con el propósito de ganar. Con estas premisas, el discurso político sirve tanto para disuadir a los Divergentes rebeldes que amenazan el sistema establecido como para controlar a las masas de las diferentes facciones. Para el estudio sigo los parámetros de diferentes marcos teóricos, como el análisis crítico del discurso político, la Teoría de la Valoración –centrándome en la actitud– y los fenómenos de polarización y legitimación. Los resultados permiten observar cómo los agentes políticos de la película consiguen influir en la población que escucha sus discursos repletos de medios retóricos, subjetividad para presentar situaciones, objetos o personas de manera positiva o negativa según los intereses y conveniencia del emisor, y diferentes tipos de poder.

Цель статьи – получить простой и эффективный алгоритм представления корней алгебраического полинома произвольной степени в виде рациональной функции от коэффициентов, используя возможности систем компьютерной математики, а также дать обоснование его применимости и привести ряд примеров. = The purpose of the article is getting a simple and efficient algorithm for the representation of the radicals of the algebraic polynomials of the arbitrary degree through the rational function of the coefficients, using the capabilities of system of computer mathematics as well as giving justification of its application and showing some examples.

Using Padé approximations to the series \(E(z) = \sum_{k = 0}^{\infty}k!(-z)^k \), we address arithmetic and analytical questions related to its values in both p-adic and Archimedean valuations.

Цель статьи – разработать новый прямой алгоритм нахождения корня алгебраического уравнения через его коэффициенты. = The purpose of the article is to receive new direct algorithm of determination of the solution of the algebraic equation through its coefficients.

It is proved the existence of large algebraic structures –including large vector subspaces or infinitely generated free algebras– inside, among others, the family of Lebesgue measurable functions that are surjective in a strong sense, the family of nonconstant differentiable real functions vanishing on dense sets, and the family of non-continuous separately continuous real functions. Lineability in special spaces of sequences is also investigated. Some of our findings complete or extend a number of results by several authors.

It is proved the existence of large algebraic structures –including large vector subspaces or infinitely generated free algebras– inside, among others, the family of Lebesgue measurable functions that are surjective in a strong sense, the family of nonconstant differentiable real functions vanishing on dense sets, and the family of non-continuous separately continuous real functions. Lineability in special spaces of sequences is also investigated. Some of our findings complete or extend a number of results by several authors.

It is proved the existence of large algebraic structures –including large vector subspaces or infinitely generated free algebras– inside, among others, the family of Lebesgue measurable functions that are surjective in a strong sense, the family of nonconstant differentiable real functions vanishing on dense sets, and the family of non-continuous separately continuous real functions. Lineability in special spaces of sequences is also investigated. Some of our findings complete or extend a number of results by several authors.

It is proved the existence of large algebraic structures –including large vector subspaces or infinitely generated free algebras– inside, among others, the family of Lebesgue measurable functions that are surjective in a strong sense, the family of nonconstant differentiable real functions vanishing on dense sets, and the family of non-continuous separately continuous real functions. Lineability in special spaces of sequences is also investigated. Some of our findings complete or extend a number of results by several authors.

Проблематика. Статтю присвячено дослідженню методу типу Пуассона–Абеля підсумовування рядів Фур’є, а саме методу \[\sigma _{k}(r,\alpha )\] множників підсумовування таких рядів.Мета дослідження. Дослідження методу \[\sigma _{k}(r,\alpha )\] множників підсумовування рядів Фур’є.Методика реалізації. Для дослідження методу \[\sigma _{k}(r,\alpha )\] множників підсумовування рядів Фур’є застосовано методику дослідження методу Пуассона–Абеля підсумовування таких рядів.Результати дослідження. У роботі показано, що застосування методу \[\sigma _{k}(r,\alpha )\] множників підсумовування ряду Фур’є періодичної функції f(t) приводить до згортки цієї функції з ядрами \[De(r,\alpha ,t)\] при цілих значеннях параметра r ці ядра являють собою поліноміальні нормалізовані базисні B-сплайни порядку r – 1 (r = 1, 2, ...). Також показано, що при \[\alpha \rightarrow 0\] метод підсумовування \[\sigma _{k}(r,\alpha )\] множниками є F-ефективним.Висновки. Встановлено, що ядра \[De(r,\alpha ,t)\] можна розглядати як тригонометричне подання нормалізованих B-сплайнів порядку ( а множники \[\sigma _{k}(r,\alpha )\] є коефіцієнтами Фур’є цих сплайнів. Відкривається можливість використання як ядер \[De(r,\alpha ,t)\] інших типів фінітних функцій із наперед заданими властивостями. Множники ж підсумовування для таких ядер отримуються як коефіцієнти Фур’є цих ядер. Метод підсумовування тригонометричних рядів \[\sigma _{k}(r,\alpha )\] множниками може бути застосовано для підсумовування тригонометричних розбіжних рядів, коефіцієнти яких \[a_{k},b_{k}\] мають порядок зростання \[O(k^{\beta }), -1< \beta < \infty .\] “Узагальненим” сумам таких рядів можна надавати певних властивостей гладкості., Проблематика. В статье рассматривается метод типа Пуассона–Абеля суммирования рядов Фурье, а именно метод \[\sigma _{k}(r,\alpha )\] множителей суммирования таких рядов.Цель исследования. Исследование метода \[\sigma _{k}(r,\alpha )\] множителей суммирования рядов Фурье.Методика реализации. Для исследования метода \[\sigma _{k}(r,\alpha )\] множителей суммирования рядов Фурье использовалась методика исследования метода Пуассона-Абеля суммирования таких рядов.Результаты исследования. В работе показано, что применение метода \[\sigma _{k}(r,\alpha )\] множителей суммирования ряда Фурье периодической функции f(t) приводит к свертке этой функции с ядрами \[De(r,\alpha ,t)\] при целых значениях параметра r эти ядра являются полиномиальными нормализованными базисными B-сплайнами порядка r – 1 (r = 1, 2, ...). Также показано, что при \[\alpha \rightarrow 0\] метод суммирования \[\sigma _{k}(r,\alpha )\] множителями является F-эффективным.Выводы. Показано, что ядра \[De(r,\alpha ,t)\] можно рассматривать как тригонометрическое представление нормализованных базисных B-сплайнов порядка ( а множители \[\sigma _{k}(r,\alpha )\] являются коэффициентами Фурье этих сплайнов. В качестве ядер \[De(r,\alpha ,t)\] можно использовать и другие типы финитных функций с наперед заданными свойствами; множители же суммирования для таких ядер являются коэффициентами Фурье этих ядер. Метод суммирования тригонометрических рядов \[\sigma _{k}(r,\alpha )\] множителями может быть использован для суммирования тригонометрических расходящихся рядов, коэффициенты которых \[a_{k},b_{k}\] имеют порядок возрастания \[O(k^{\beta }), -1< \beta < \infty .\] “Обобщенным” суммам таких рядов можно сообщать необходимые свойства гладкости., Background. The method of Poisson-Abel type of summation of Fourier series, namely, the method of summation with \[\sigma _{k}(r,\alpha )\] factors is considered in this paper.Objective. Investigation of method of summation of Fourier series with \[\sigma _{k}(r,\alpha )\] factors.Methods. We apply the analysis of Poisson-Abel method of summation of such series for investigation of method of summation of Fourier series with \[\sigma _{k}(r,\alpha )\] factors.Results. It is proved in this paper that application of method of summation with \[\sigma _{k}(r,\alpha )\] factors of Fourier series of periodical function f(t) derives to the convolution of this function with kernels \[De(r,\alpha ,t)\] if the parameter r is integer, these kernels become polynomial normalized basic B-splines of order ( Also it is proved that for \[\alpha \rightarrow 0\] the method of summation with \[\sigma _{k}(r,\alpha )\] multipliers is F-effective.Conclusions. We prove that kernels \[De(r,\alpha ,t)\] may be considered as trigonometric representation of normalized basic B-splines of order r – 1 (r = 1, 2, ...) (and the factors \[\sigma _{k}(r,\alpha )\] are Fourier coefficients of these splines. Other types of finite functions with given properties also may be used as kernels \[De(r,\alpha ,t);\] factors of summation for these kernels are the Fourier coefficients of these kernels. Method of summation of trigonometric series with \[\sigma _{k}(r,\alpha )\] factors may be applied for summation of trigonometric divergent series with coefficients \[a_{k},b_{k}\] which have the order of increasing \[O(k^{\beta }), -1< \beta < \infty .\] One can impose the needed properties of smoothness to “generalized” sums of these series.

The steady problem of free surface flow due to a submerged line source is revisited for the case in which the fluid depth is finite and there is a stagnation point on the free surface directly above the source. Both the strength of the source and the fluid speed in the far field are measured by a dimensionless parameter, the Froude number. By applying techniques in exponential asymptotics, it is shown that there is a train of periodic waves on the surface of the fluid with an amplitude which is exponentially small in the limit that the Froude number vanishes. This study clarifies that periodic waves do form for flows due to a source, contrary to a suggestion by Chapman & Vanden-Broeck (2006, J. Fluid Mech., 567, 299--326). The exponentially small nature of the waves means they appear beyond all orders of the original power series expansion; this result explains why attempts at describing these flows using a finite number of terms in an algebraic power series incorrectly predict a flat free surface in the far field.

We study asymptotic properties of divergent formal solutions appearing in the linearization problem of a sigular vector field without a Diophantine condition or an existence of additional first integrals. We will give an asymptotic meaning to divergent formal solutions constructed by a singular perturbative solution (cf. [6])., This research was partly supported by Grant-in-Aid for Scientific Research (No. 20540172), JSPS, Japan.

This thesis is a combination of three pieces of work: 1, We explore some axioms of divergent series and their relations with conformal field theory. As a consequence we obtain another way of calculating $L(0,\chi)$ and $L(-1,\chi)$ for $\chi$ being a Dirichlet character. We hope this discussion is also of interest to physicists doing renormalization theory for a reason indicated in the Introduction section. We consider a twist of the oscillator representation of the Virasoro algebra by a group of Dirichlet characters first introduced by Bloch in \cite{Bloch}, and use this to give a 'physical interpretation' of why the values of certain divergent series should be given by special L values. Furthermore, we use this to show that some fractional powers which are crucial for some infinite products to have peculiar modular transformation properties are expressed explicitly by certain linear combinations of $L(-1, \chi)$'s for appropriately chosen $\chi$'s, and can be understood physically as a kind of 'vacuum Casimir energy' in our settings. 2, We give an attempt to reinterpret Tate's thesis by a sort of conformal field theory on a number field. Based on this and the existence of a hypothetical 3-dimensional gauge theory, we give a physical interpretation of the Gauss quadratic reciprocity law by a sort of S-duality. 3, We find a complete list of positive definite symmetric matrices with integer entries $\begin{bmatrix}a&b\\b&d\end{bmatrix}$ such that all complex solutions to the system of equations\begin{align}1-x_1=x_1^ax_2^b\nonumber \\ 1-x_2=x_1^bx_2^d\nonumber\end{align}are real. This result is related to Nahm's conjecture in rank 2 case. (In each chapter, except for equations, chapter number is omitted from quotation. For example, theorem 1.1.1 will be quoted as theorem 1.1.)

This thesis is a combination of three pieces of work: 1, We explore some axioms of divergent series and their relations with conformal field theory. As a consequence we obtain another way of calculating $L(0,\chi)$ and $L(-1,\chi)$ for $\chi$ being a Dirichlet character. We hope this discussion is also of interest to physicists doing renormalization theory for a reason indicated in the Introduction section. We consider a twist of the oscillator representation of the Virasoro algebra by a group of Dirichlet characters first introduced by Bloch in \cite{Bloch}, and use this to give a 'physical interpretation' of why the values of certain divergent series should be given by special L values. Furthermore, we use this to show that some fractional powers which are crucial for some infinite products to have peculiar modular transformation properties are expressed explicitly by certain linear combinations of $L(-1, \chi)$'s for appropriately chosen $\chi$'s, and can be understood physically as a kind of 'vacuum Casimir energy' in our settings. 2, We give an attempt to reinterpret Tate's thesis by a sort of conformal field theory on a number field. Based on this and the existence of a hypothetical 3-dimensional gauge theory, we give a physical interpretation of the Gauss quadratic reciprocity law by a sort of S-duality. 3, We find a complete list of positive definite symmetric matrices with integer entries $\begin{bmatrix}a&b\\b&d\end{bmatrix}$ such that all complex solutions to the system of equations\begin{align}1-x_1=x_1^ax_2^b\nonumber \\ 1-x_2=x_1^bx_2^d\nonumber\end{align}are real. This result is related to Nahm's conjecture in rank 2 case. (In each chapter, except for equations, chapter number is omitted from quotation. For example, theorem 1.1.1 will be quoted as theorem 1.1.)

This paper gives an account of two articles which illustrate the use of Resurgence theory for estimating the diference between two complex invariant manifolds associated with the area-preserving H?enon map for the first one, and with a Hamiltonian system which stems from the rapidly forced pendulum for the second one., Peer Reviewed