Existence of analytic solutions to analytic nonlinear q-difference equations

In this paper, some analytic approaches are formulated for the existence of analytic solutions of analytic nonlinear difference equations. From the point of view of dynamical systems, analytic solutions of such kinds of equations can be generally expressed by formal power series of exponential variables, so we are interested in considering a difference equation as a q-difference equation via a suitable coordinate transformation. After stating analytic results for formal series solutions of nonlinear q-difference equations, we also derive some results for the existence of analytic solutions to autonomous rational difference equations.     

Finite analytic numerical method for solving two‐dimensional quasi‐Laplace equation

A typical power series analytic solution of quasi‐Laplace equation in the infinitesimal angle domain around the singular point of the square cells is provided in this article. Toward the singular point, the gradient of the potential variable will tend to infinity, which is described by the first term of the power series solution. Based on this analytic solution, three finite analytic numerical methods are proposed. These methods are analogous and are constructed, respectively, when considering different numbers of the terms or using different schemes to determine the relevant parameters in the power series. Numerical examples show that all of the three finite analytic numerical methods proposed can provide rather accurate solutions than the traditional numerical methods. In contrast, when using the traditional numerical schemes to solve the quasi‐Laplace equation in a strong heterogeneous medium, the refinement ratio for the grid cell needs to increase dramatically to get an accurate result. In practical applications, subdividing each origin cell into 2 × 2 or 3 × 3 subcells is enough for the finite analytical numerical methods to get relatively accurate results. The finite analytical numerical methods are also convenient to construct the flux field with high accuracy.© 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 1755–1769, 2014     

Macdonald identities and multidimensional theta-functions

A new proof of the combinatorial Macdonald identities is presented. It is shown that one may regard these identities as a decomposition of certain multidimensional theta-functions into infinite products. The proof is based on some analytical properties of theta-functions. It is briefly discussed how one can modify the proof in order to replace analytical arguments by formal ones involving only operations with formal series.Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 240, 1997, pp. 67–77.The author is grateful to A. M. Vershik and Yu. M. Bazlov for their interest and helpful comments.This research was supported by ISSEP (grant No. A96-1965).     

k解析函数的Fourier级数

讨论了复平面上k解析函数的性质,并利用k解析函数的泰勒展开定理研究了k解析函数的Fourier级数,推广了经典的解析函数的Fourier级数理论.     

Markov chain test for time dependence and homogeneity: An analytical and empirical evaluation

This paper evaluates the small and large sample properties of Markov chain time-dependence and time-homogeneity tests. First, we present the Markov chain methodology to investigate various statistical properties of time series. Considering an auto-regressive time series and its associated Markov chain representation, we derive analytical measures of the statistical power of the Markov chain time-dependence and time-homogeneity tests. We later use Monte Carlo simulations to examine the small-sample properties of these tests. It is found that although Markov chain time-dependence test has desirable size and power properties, time-homogeneity test does not perform well in statistical size and power calculations.     

Continuity of Formal Power Series Products in Nonlinear Control Theory

Formal power series products appear in nonlinear control theory when systems modeled by Chen–Fliess series are interconnected to form new systems. In fields like adaptive control and learning systems, the coefficients of these formal power series are estimated sequentially with real-time data. The main goal is to prove the continuity and analyticity of such products with respect to several natural (locally convex) topologies on spaces of locally convergent formal power series in order to establish foundational properties behind these technologies. In addition, it is shown that a transformation group central to describing the output feedback connection is in fact an analytic Lie group in this setting with certain regularity properties.

     

Embedding theorems for two-parameter groups of formal power series and related problems

The classical problem of analytic iteration is that of embedding analytic functions in one-parameter Lie groups of formal power series. The main purpose of the present paper is to consider similar problems for two-parameter groups. These problems are closely related to problems concerning conjugate power series and conjugate one parameter groups of such series, to which the first sections of the paper are devoted. As an application of the conjugacy theorems and the embedding theorems we bring an algebraic characterization of the class of the two-parameter groups.     

Analytic invariant curves for a second order difference equation modelled from macroeconomics under the Brjuno condition

In this paper, we investigate analytic invariant curves of a planar mapping which derives a second order difference equation modelled from macroeconomics under the Brjuno condition that is weaker than the Diophantine condition, and present analytic invariant curves by constructing uniformly convergent power series.     

The analytic connectivity in uniform hypergraphs: Properties and computation

The analytic connectivity (AC), defined via solving a series of constrained polynomial optimization problems, serves as a measure of connectivity in hypergraphs. How to compute such a quantity efficiently is important in practice and of theoretical challenge as well due to the non-convex and combinatorial features in its definition. In this article, we first perform a careful analysis of several widely used structured hypergraphs in terms of their properties and heuristic upper bounds of ACs. We then present an affine-scaling method to compute some upper bounds of ACs for uniform hypergraphs. To testify the tightness of the obtained upper bounds, two possible approaches via the Pólya theorem and semidefinite programming respectively are also proposed to verify the lower bounds generated by the obtained upper bounds minus a small gap. Numerical experiments on synthetic datasets are reported to demonstrate the efficiency of our proposed method. Further, we apply our method in hypergraphs constructed from social networks and text analysis to detect the network connectivity and rank the keywords, respectively.     

On analytic solutions to the generalized Cauchy problem with data on three surfaces for a quasilinear system

The generalized Cauchy problem with data on three surfaces is under consideration for a quasilinear analytic system of the third order. Under some simplifying assumption, we find necessary and sufficient conditions for existence of a solution in the form of triple series in the powers of the independent variables. We obtain convenient sufficient conditions under which the data of the generalized Cauchy problem has a unique locally analytic solution. We give counterexamples demonstrating that in the case we study it is impossible to state necessary and sufficient conditions for analytic solvability of the generalized Cauchy problem. We also show that the analytic solution can fail to exist even if the generalized Cauchy problem with data on three surfaces has a formal solution since the series converge only at a sole point, the origin.     

The Möbius inversion formula for Fourier series applied to Bernoulli and Euler polynomials

Hurwitz found the Fourier expansion of the Bernoulli polynomials over a century ago. In general, Fourier analysis can be fruitfully employed to obtain properties of the Bernoulli polynomials and related functions in a simple manner. In addition, applying the technique of Möbius inversion from analytic number theory to Fourier expansions, we derive identities involving Bernoulli polynomials, Bernoulli numbers, and the Möbius function; this includes formulas for the Bernoulli polynomials at rational arguments. Finally, we show some asymptotic properties concerning the Bernoulli and Euler polynomials.     

Zeta functions of discrete self-similar sets

In this paper we study a class of countable and discrete subsets of a Euclidean space that are “self-similar” with respect to a finite set of (affine) similarities. Any such set can be interpreted as having a fractal structure. We introduce a zeta function for these sets, and derive basic analytic properties of this “fractal” zeta function. Motivating examples that come from combinatorial geometry and arithmetic are given particular attention.     

Some polynomials associated with the $${\vavec{}}$$-Whitney numbes

In the present article, we study three families of polynomials associated with the r-Whitney numbers of the second kind. They are the r-Dowling polynomials, r-Whitney–Fubini polynomials and the r-Eulerian–Fubini polynomials. Then we derive several combinatorial results by using algebraic arguments (Rota’s method), combinatorial arguments (set partitions) and asymptotic methods.     

Inclusion and neighborhood properties for certain classes of multivalently analytic functions of complex order associated with the convolution structure

Making use of the familiar convolution structure of analytic functions, in this paper we introduce and investigate two new subclasses of multivalently analytic functions of complex order. Among the various results obtained here for each of these function classes, we derive the coefficient inequalities and other interesting properties and characteristics for functions belonging to the class introduced here.     

Transseries for a class of nonlinear difference equations *

Given a nonlinear analytic difference equation of level 1 with a formal power series solution ? ₀ we associate with it a stable manifold of solutions with asymptotic expansion ? ₀. This manifold can be represented by means of Borel summable series. All solutions with asymptotic expansion ? ₀ in some sector can be written as certain exponential series which are called transseries. Some of their properties are investigated: are resurgence properties and Stokes transition. Analogous problems for differential equations have been studied by Costin in [7]     

On Stirling numbers and Euler sums

In this paper, we propose another yet generalization of Stirling numbers of the first kind for noninteger values of their arguments. We discuss the analytic representations of Stirling numbers through harmonic numbers, the generalized hypergeometric function and the logarithmic beta integral. We present then infinite series involving Stirling numbers and demonstrate how they are related to Euler sums. Finally, we derive the closed form for the multiple zeta function ζ(p, 1,…, 1) for Re(p)>1.     

Euler's relation,möbius functions and matroid identities

We give a short combinatorial proof of the Euler relation for convex polytopes in the context of oriented matroids. Using counting arguments we derive from the Euler relation several identities holding in the lattice of flats of an oriented matroid. These identities are proven for any matroid by Möbius inversion.     

Analytic invariant curves for a planar mapping

This paper is concerned with the existence of analytic invariant curves for a planar mapping. By constructing a convergent power series solution of an auxiliary equation, analytic solutions of the original equation are obtained. In this paper, we discuss not only the general case, but also the critical cases as well, in particular, the case where β is a unit root is discussed.     

On universal formal power series

The point source of this work is Seleznev's theorem which asserts the existence of a power series which satisfies universal approximation properties in C^∗. The paper deals with a strengthened version of this result. We establish a double approximation theorem on formal power series using a weighted backward shift operator. Moreover we give strong conditions that guarantee the existence of common universal series of an uncountable family of weighted backward shift with respect to the simultaneous approximation. Finally we obtain results on admissible growth of universal formal power series. We especially prove that you cannot control the defect of analyticity of such a series even if there exist universal series in the well-known intersection of formal Gevrey classes.