Kloosterman sum identities and low-weight codewords in a cyclic code with two zeros

We apply relations of n-dimensional Kloosterman sums to exponential sums over finite fields to count the number of low-weight codewords in a cyclic code with two zeros. As a corollary we obtain a new proof for a result of Carlitz which relates one- and two-dimensional Kloosterman sums. In addition, we count some power sums of Kloosterman sums over certain subfields.     

Some transcendental functions over function fields with positive characteristicCertaines fonctions transcendantes sur des corps de fonctions de caractéristique positive

In this work we shall define two families of functions over function fields with positive characteristic and show that such a function is transcendental if and only if its generating sequence is not ultimately zero. As a result, the Carlitz exponential and the Carlitz logarithm are transcendental functions. Our proof is elementary in the sense that we only use a theorem due to H. Sharif and C. Woodcock, and to T. Harase which generalizes the theorem of Christol about automatic sequences. To cite this article: J.-Y. Yao, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 939–943.     

Multiplication formulas for Apostol-type polynomials and multiple alternating sums

We investigate multiplication formulas for Apostol-type polynomials and introduce λ-multiple alternating sums, which are evaluated by Apostol-type polynomials. We derive some explicit recursive formulas and deduce some interesting special cases that involve the classical Raabe formulas and some earlier results of Carlitz and Howard.     

An Itô formula for a family of stochastic integrals and related Wong–Zakai theorems

The aim of this paper is to generalize two important results known for the Stratonovich and Itô integrals to any stochastic integral obtained as limit of Riemann sums with arbitrary evaluating point: the ordinary chain rule for certain nonlinear functions of the Brownian motion and the Wong–Zakai approximation theorem. To this scope we begin by introducing a new family of products for smooth random variables which reduces for specific choices of a parameter to the pointwise and to the Wick products. We show that each product in that family is related in a natural way to a precise choice of the evaluating point in the above mentioned Riemann sums and hence to a certain notion of stochastic integral. Our chain rule relies on a new probabilistic representation for the solution of the heat equation while the Wong–Zakai type theorem follows from a reduction method for quasi-linear SDEs together with a formula of Gjessing’s type.     

Dirichlet series with periodic coefficients

In this paper we shall unify the results obtained so far in various scattered literature, for Dirichlet characters and the associated Dirichlet L-functions, under the paradigm of periodic arithmetic functions and the associated Dirichlet series. Notably we shall determine the Laurent coefficients of the series in question to cover Funakura’s result and proceed on to prove the Ayoub-Berndt-Carlitz-Chowla-Müller-Redmond theorem.     

Coupled fixed point results for nonlinear integral equations

In this paper, we prove some coupled coincidence point theorems for such nonlinear contraction mappings having a mixed monotone property in partially ordered metric spaces by dropping the condition of commutative. We also prove coupled common fixed point theorem for w-compatible mappings. An example of a nonlinear contraction mapping which is not applied by Lakshmikantham and ?iri?’s theorem [1] but applied by our result is given. Further, we apply our results to the existence theorem for solution of nonlinear integral equations.     

Rosenthal inequalities in noncommutative symmetric spaces

We give a direct proof of the ‘upper’ Khintchine inequality for a noncommutative symmetric (quasi-)Banach function space with nontrivial upper Boyd index. This settles an open question of C. Le Merdy and the fourth named author (Le Merdy and Sukochev, 2008 [24]). We apply this result to derive a version of Rosenthal?s theorem for sums of independent random variables in a noncommutative symmetric space. As a result we obtain a new proof of Rosenthal?s theorem for (Haagerup) Lp-spaces.     

Finding Sum of Powers on Arithmetic Progressions with Application of Cauchy’s Equation

In this paper we introduce a method to find the sum of powers on arithmetic progressions by using Cauchy’s equation and obtain a general formula. Then we apply our results to show how to determine some other sums of powers and sums of products. Our results are more general than those in [9]. Finally we discuss the sum of powers on arithmetic progressions in commmutative rings with characteristic 2 and find ‘full polynomials’.     

Representation of bi-parameter singular integrals by dyadic operators

We prove a dyadic representation theorem for bi-parameter singular integrals. That is, we represent certain bi-parameter operators as averages of rapidly decaying sums of what we call bi-parameter shifts. A new version of the product space T1 theorem is established as a consequence.     

Subordination and superordination results for analytic functions with respect to symmetrical points

The theory of symmetric functions has many applications in the investigation of fixed points, estimation of absolute values of some integrals and obtaining the results of the type of Cartan’s uniqueness theorem. In this paper, we solve some differential subordinations and superordinations involving analytic functions with respect to the symmetric points and also derive some sandwich results under certain assumptions on the parameters involved. The various results presented in this paper are shown to apply to yield the corresponding (new or known) results for many simpler function classes.     

Two new triangles of q-integers via q-Eulerian polynomials of type A and B

The classical Eulerian polynomials can be expanded in the basis t ^k?1(1+t) ^n+1?2k (1≤k≤?(n+1)/2?) with positive integral coefficients. This formula implies both the symmetry and the unimodality of the Eulerian polynomials. In this paper, we prove a q-analogue of this expansion for Carlitz’s q-Eulerian polynomials as well as a similar formula for Chow–Gessel’s q-Eulerian polynomials of type B. We shall give some applications of these two formulas, which involve two new sequences of polynomials in the variable q with positive integral coefficients. It is an open problem to give a combinatorial interpretation for these polynomials.     

A Bernstein—Chernoff deviation inequality, and geometric properties of random families of operators

In this paper we first describe a new deviation inequality for sums of independent random variables which uses the precise constants appearing in the tails of their distributions, and can reflect in full their concentration properties. In the proof we make use of Chernoff's bounds. We then apply this inequality to prove a global diameter reduction theorem for abstract families of linear operators endowed with a probability measure satisfying some condition. Next we give a local diameter reduction theorem for abstract families of linear operators. We discuss some examples and give one more global result in the reverse direction, and extensions. This research was partially supported by BSF grant 2002-006.     

A note on Herglotz’s theorem for time series on function spaces

In this article, we prove Herglotz’s theorem for Hilbert-valued time series. This requires the notion of an operator-valued measure, which we shall make precise for our setting. Herglotz’s theorem for functional time series allows to generalize existing results that are central to frequency domain analysis on the function space. In particular, we use this result to prove the existence of a functional Cramér representation of a large class of processes, including those with jumps in the spectral distribution and long-memory processes. We furthermore obtain an optimal finite dimensional reduction of the time series under weaker assumptions than available in the literature. The results of this paper therefore enable Fourier analysis for processes of which the spectral density operator does not necessarily exist.     

Integrals over polytopes, multiple zeta values and polylogarithms, and Euler’s constant

Let T be the triangle with vertices (1, 0), (0, 1), (1, 1). We study certain integrals over T, one of which was computed by Euler. We give expressions for them both as linear combinations of multiple zeta values, and as polynomials in single zeta values. We obtain asymptotic expansions of the integrals, and of sums of certain multiple zeta values with constant weight. We also give related expressions for Euler’s constant, and study integrals, one of which is the iterated Chen (Drinfeld-Kontsevich) integral, over some polytopes that are higher-dimensional analogs of T. The latter leads to a relation between certain multiple polylogarithm values and multiple zeta values.     

Projections, entropy and sumsets

In this paper we shall generalize Shearer??s entropy inequality and its recent extensions by Madiman and Tetali, and shall apply projection inequalities to deduce extensions of some of the inequalities concerning sums of sets of integers proved recently by Gyarmati, Matolcsi and Ruzsa. We shall also discuss projection and entropy inequalities and their connections.     

Reflections on the U(1) problem in general relativity

After reviewing the known stability results for vacuum, U(1) symmetric solutions to Einstein’s field equations, we shall describe some mathematical ideas and techniques that were unavailable during the earlier research in this area and which might conceivably be exploited to shed new light on the large data global Cauchy problem for this interesting class of spacetimes.     

Asymptotic structural characteristics of fuzzy measure and their applications

In fuzzy measure theory, as Sugeno's fuzzy measures lose additivity in general, the concept ‘almost’, which is well known in classical measure theory, splits into two different concepts, ‘almost’ and ‘pseudo-almost’. In order to replace the additivity, it is quite necessary to investigate some asymptotic behaviors of a fuzzy measure at sequences of sets which are called ‘waxing’ and ‘waning’, and to introduce some new concepts, such as ‘autocontinuity’, ‘converse-autocontinuity’ and ‘pseudo-autocontinuity’. These concepts describe some asymptotic structural characteristics of a fuzzy measure.In this paper, by means of the asymptotic structural characteristics of fuzzy measure, we also give four forms of generalization for both Egoroff's theorem, Riesz's theorem and Lebesgue's theorem respectively, and prove the almost everywhere (pseudo-almost everywhere) convergence theorem, the convergence in measure (pseudo-in measure) theorem of the sequence of fuzzy integrals. In the last two theorems, the employed conditions are not only sufficient, but also necessary.     

Some extensions of Carlitz on cyclic sums and generating functions

Recently, the author (Proc. Amer. Math. Soc., 57 1976, 271–275) derived two theorems involving double series, which gave as a consequence new and known generating functions for the Jacobi polynomial. The method of proof differed from that of previous workers. Using an extension of this procedure, we present in this paper two theorems for double and m-dimensional series which generalize our previous work. These formulas also yield new generating functions for the Jacobi polynomial and extend some formulas of Carlitz (Boll. U.M.I. (3), 16 1961, 150–155) and others. A feature of this work is the inclusion of the Jacobi polynomial within the framework of m-dimensional cyclic sums, thus generalizing a main result of Carlitz (SIAM Rev., 6 1964, 20–30).     

On the converse of Abel's theorem in characteristic p

Darboux's and Griffiths' converse of Abel's theorem says (in effect) that any addition law like those obtained via Abel's theorem from sums of Abelian integrals on algebraic curves must in fact arise from this sort of algebraic situation. In this paper, we prove a characteristic p version of the result for plane cubic curves.