Series Of Theorems Research Articles

Linear nested Artin approximation theorem for algebraic power series

We give an elementary proof of the nested Artin approximation theorem for linear equations with algebraic power series coefficients. Moreover, for any Noetherian local subring of the ring of formal power series, we clarify the relationship between this theorem and the problem of the commutation of two operations for ideals: the operation of replacing an ideal by its completion and the operation of replacing an ideal by one of its elimination ideals. In particular we prove that a Grothendieck conjecture about morphisms of analytic/formal algebras and Artin’s question about linear nested approximation problem are equivalent.

Uniqueness Theorems for Series by Vilenkin System

In this paper, we prove uniqueness theorems and restoration formulas for coefficients of series by Vilenkin system. The series is assumed to be convergent in measure and the distribution function of the majorant of partial sums satisfies some necessary condition.

The Riemann-Cantor uniqueness theorem for unilateral trigonometric series via a special version of the Lusin-Privalov theorem

Using Baire's theorem, we give a very simple proof of a special version of the Lusin-Privalov theorem and deduce via Abel's theorem the Riemann-Cantor theorem on the uniqueness of the coefficients of pointwise convergent unilateral trigonometric series.

Cauchy, infinitesimals and ghosts of departed quantifiers

Procedures relying on infinitesimals in Leibniz, Euler and Cauchy have been interpreted in both a Weierstrassian and Robinson's frameworks. The latter provides closer proxies for the procedures of the classical masters. Thus, Leibniz's distinction between assignable and inassignable numbers finds a proxy in the distinction between standard and nonstandard numbers in Robinson's framework, while Leibniz's law of homogeneity with the implied notion of equality up to negligible terms finds a mathematical formalisation in terms of standard part. It is hard to provide parallel formalisations in a Weierstrassian framework but scholars since Ishiguro have engaged in a quest for ghosts of departed quantifiers to provide a Weierstrassian account for Leibniz's infinitesimals. Euler similarly had notions of equality up to negligible terms, of which he distinguished two types: geometric and arithmetic. Euler routinely used product decompositions into a specific infinite number of factors, and used the binomial formula with an infinite exponent. Such procedures have immediate hyperfinite analogues in Robinson's framework, while in a Weierstrassian framework they can only be reinterpreted by means of paraphrases departing significantly from Euler's own presentation. Cauchy gives lucid definitions of continuity in terms of infinitesimals that find ready formalisations in Robinson's framework but scholars working in a Weierstrassian framework bend over backwards either to claim that Cauchy was vague or to engage in a quest for ghosts of departed quantifiers in his work. Cauchy's procedures in the context of his 1853 sum theorem (for series of continuous functions) are more readily understood from the viewpoint of Robinson's framework, where one can exploit tools such as the pointwise definition of the concept of uniform convergence. Keywords: historiography; infinitesimal; Latin model; butterfly model

Uniqueness theorem for multiple Franklin series

The paper presents the proof of the uniqueness theorem formultiple series in the Franklin systemthat converge inmeasure and whosemajorant of cubic partial sums with numbers 2 ν satisfies a certain necessary condition. This result is new in the one-dimensional case as well.

On an intermediate value theorem for polynomials and power series over a valued field

ABSTRACTThe paper proves an intermediate value theorem for polynomials and power series over a valued field with an additive divisible valuation group and infinite residue field. A deeper description of the behavior of the valuation for power series is obtained using Hensel’s lemma.

SOME LIMITING BEHAVIOR FOR ASYMPTOTICALLY NEGATIVE ASSOCIATED RANDOM VARIABLES

In this paper, we study the almost sure convergence for sequences of asymptotically negative associated (ANA) random variables. As a result, we extend the classical Khintchine–Kolmogorov convergence theorem, Marcinkiewicz strong law of large numbers, and the three series theorem for sequences of independent random variables to sequences of ANA random variables without necessarily adding any extra conditions.

On maximal subalgebras and a generalized Jordan–Hölder theorem for Lie algebras

ABSTRACTThe purpose of this paper is to continue the study of chief factors of a Lie algebra and to prove a further strengthening of the Jordan–Hölder theorem for chief series.

Central limit theorems for sequential and random intermittent dynamical systems

We establish self-norming central limit theorems for non-stationary time series arising as observations on sequential maps possessing an indifferent fixed point. These transformations are obtained by perturbing the slope in the Pomeau–Manneville map. We also obtain quenched central limit theorems for random compositions of these maps.

A converse theorem for metaplectic Eisenstein series on the double and triple covers of SL2(ℚ(−D))

In this work, we prove a converse theorem for metaplectic Eisenstein series on the [Formula: see text]th metaplectic cover of the group [Formula: see text], where [Formula: see text] is an imaginary quadratic number field containing the [Formula: see text]th roots of unity. This is an analog to previous converse theorems relating certain double Dirichlet series to the Mellin transforms of Eisenstein series of half-integer weight. We also propose a way to generalize this result to any number field.

A matching value for cooperative games

A cooperative game describes a situation in which a finite set of n players can generate certain profits by their cooperation. How to allocate the profits effectively among each player is very important and it will affect the stability and the sustainable development of their cooperation. In this paper, by analyzing the characteristics of the existing solutions to solve the profit allocation problem, some concepts such as cooperative unit, perfect matching pair are firstly defined, a construction method of matching order is proposed. Then a new solution called matching value is defined, its basic properties are discussed by a series of theorems. Finally, a matching value under total coalition structure is defined and its axiomatization is proved by three axioms.

Theory of fractional-ordered thermoelastic diffusion

In this note, the traditional theory of thermoelastic diffusion is replaced by fractional ordered thermoelasticity based on fractional conservation of mass, fractional Taylor series and fractional divergence theorem. We replace the integer-order Taylor series approximation for flux with the fractional-order Taylor series approximation which can remove the restriction that the flux has to be linear, or piece-wise linear and the restriction that the control volume must be infinitesimal. There are two important distinctions between the traditional thermoelastic diffusion, and its fractional equivalent. The first is that the divergence term in the heat conduction and mass diffusion equations are the fractional divergence, and the second is the appearance of strain tensor term in the fractional equation is in the form of “incomplete fractional-strain measures”.

A computational method for solving a class of two dimensional Volterra integral equations

The use of the Geometric Series theorem, together with Schauder bases in certain Banach spaces, allows us to design a numerical algorithm in order to solve an important type of two-dimensional linear integral equations. A study of the convergence and error is presented here. All the calculations can be easily implemented and the efficiency of this method will be shown with numerical results.

Dimension counts for limit linear series on curves not of compact type

We first prove a generalized Brill–Noether theorem for linear series with prescribed multivanishing sequences on smooth curves. We then apply this theorem to prove that spaces of limit linear series have the expected dimension for curves of pseudocompact type, whenever the gluing conditions in the definition of limit linear series impose the maximal codimension. Finally, we investigate these gluing conditions in specific families of curves, showing expected dimension in several cases, each with different behavior. One of these families sheds new light on the work of Cools, Draisma, Payne and Robeva in tropical Brill–Noether theory, and suggests directions of further work in that setting.

The Billard Theorem for Multiple Random Fourier Series

We propose a generalization of a classical result on random Fourier series, namely the Billard Theorem, for random Fourier series over the d-dimensional torus. We provide an investigation of the independence with respect to a choice of a sequence of partial sums (or method of summation). We also study some probabilistic properties of the resulting sum field such as stationarity and characteristics of the marginal distribution.

The Hardy–Littlewood theorem for multiple fourier series with monotone coefficients

It was proved earlier that, for multiple Fourier series whose coefficients are monotone in each index, the classicalHardy–Littlewood theorem is not valid for p ≤ 2m/(m+1), where m is the dimension of the space. We establish how the theorem must be modified in this case.

A Tauberian theorem for multiple power series

Multiple sequences are considered. The notion of weak one-sided oscillation of such a sequence along a sequence 0 \\ \\forall\\,j=1,\\dots,n,\\ k\\in \\mathbb N\\bigr\\} \\end{equation*} ?> such that as for is introduced. The asymptotic behaviour of the sequence (for fixed positive numbers ) is deduced from the asymptotic behaviour as of the generating function , , of the multiple sequence under consideration for (where are positive and fixed). The Tauberian theorem thus established generalizes several Tauberian theorems due to the author, which were established while investigating certain classes of random substitutions and random maps of a finite set to itself. Karamata's well-known Tauberian theorem for the generating functions of sequences was the starting point for research in this direction. Bibliography: 36 titles.

A Tauberian theorem for multiple power series

Рассматриваются кратные последовательности $\{a(i)\geqslant 0, i\in Z_+^n\}$. Вводится понятие односторонней слабой осцилляции таких последовательностей вдоль последовательности $$ \{m=m(k)=(m_1(k),…,m_n(k)), m_j(k)>0 \forall j=1,…,n, k\in \mathbb N\} $$ такой, что $m_j(k)\to\infty$ для всех $j=1,…,n$ при $k\to\infty$. Из асимптотики производящей функции $A(s)$, $s\in[0,1)^n$, изучаемой кратной последовательности при $s=(e^{-\lambda_1/m_1},…,e^{-\lambda_n/m_n})$ и $k\to\infty$ ($\lambda_1,…,\lambda_n$ - положительны и фиксированы) выводится асимптотика для последовательности $a(x_1m_1,…, x_nm_n)$ (числа $x_1,…,x_n$ - положительны и фиксированы). Доказанная тауберова теорема обобщает несколько тауберовых теорем, полученных автором ранее при исследовании некоторых классов случайных подстановок и случайных отображений конечного множества в себя. При этом исходным результатом в этом направлении является известная тауберова теорема Караматы для производящих функций последовательностей. Библиография: 36 названий.

Moment inequalities for m-negatively associated random variables and their applications

The moment inequalities for m-NA random variables, especially the Marcinkiewicz–Zygmund type inequality and Rosenthal type inequality are established and the Khintchine–Kolmogorov convergence theorem and the three series theorem for m-NA random variables are also obtained. As one application of the moment inequalities, we study the large deviation for least squares estimator in nonlinear regression models under some general conditions. As another application, we investigate the strong consistency for least squares estimator in multiple linear regression models based on m-NA random variables.

Uniqueness theorems for series in the Franklin system

Uniqueness theorems for series in the Franklin system