Topological rigidity of linear cellular automaton shifts

We prove that topologically isomorphic linear cellular automaton shifts are algebraically isomorphic. Using this, we show that two distinct such shifts cannot be isomorphic. We conclude that the automorphism group of a linear cellular automaton shift is a finitely generated abelian group.     

On Miki's identity for Bernoulli numbers

We give a short proof of Miki's identity for Bernoulli numbers,

     

An identity of symmetry for the Bernoulli polynomials

The main purpose of this paper is to prove an identity of symmetry for the higher order Bernoulli polynomials. It turns out that the recurrence relation and multiplication theorem for the Bernoulli polynomials which discussed in [F.T. Howard, Application of a recurrence for the Bernoulli numbers, J. Number Theory 52 (1995) 157-172], as well as a relation of symmetry between the power sum polynomials and the Bernoulli numbers developed in [H.J.H. Tuenter, A symmetry of power sum polynomials and Bernoulli numbers, Amer. Math. Monthly 108 (2001) 258-261], are all special cases of our results.     

Some representations of the multivariate Bernoulli and binomial distributions

Multivariate but vectorized versions for Bernoulli and binomial distributions are established using the concept of Kronecker product from matrix calculus. The multivariate Bernoulli distribution entails a parameterized model, that provides an alternative to the traditional log-linear model for binary variables.     

Some rigorous results for the Greenberg-Hastings Model

In this paper, we obtain some rigorous results for a cellular automaton known as the Greenberg-Hastings Model. The state space is {0, 1, 2}^{Z d} . The dynamics are deterministic and discrete time. A site which is 1 changes to 2, a site which is 2 changes to 0, and a site which is 0 changes to a 1 if one of its 2d neighbors is a 1. In one dimension, we compute the exact asymptotic rate at which the system dies out when started at random and compute the topological entropy. In two or more dimensions we show that starting from a nontrivial product measure, the limit exists as 3m and is Bernoulli shift. Finally, we investigate the behavior of the system on a large finite box.     

任何序贯样本情形下伯努利参数的置信限

设（xn,n≥1）是独立同分布的随变量列,xn=0或1且P（xn=1)=p(n≥1),这里p是未知的,设τ是（xn,n≥1）的任何有限停止时间,对任何序贯样本（x1,x2,…,x)和r∈(0,1),本文给出了p的r水平最优置信下限和最优置信上限,还讨论了一些有关的问题。     

Higher-order convolutions for Bernoulli and Euler polynomials

We prove convolution identities of arbitrary orders for Bernoulli and Euler polynomials, i.e., sums of products of a fixed but arbitrary number of these polynomials. They differ from the more usual convolutions found in the literature by not having multinomial coefficients as factors. This generalizes a special type of convolution identity for Bernoulli numbers which was first discovered by Yu. Matiyasevich.     

Tangential flatness and global rigidity of higher rank lattice actions

We establish the continuous tangential flatness for orientable
weakly Cartan actions of higher rank lattices. As a corollary, we obtain the global rigidity of Anosov Cartan actions.

     

Shortened recurrence relations for Bernoulli numbers

Starting with two little-known results of Saalschütz, we derive a number of general recurrence relations for Bernoulli numbers. These relations involve an arbitrarily small number of terms and have Stirling numbers of both kinds as coefficients. As special cases we obtain explicit formulas for Bernoulli numbers, as well as several known identities.     

Extended Zeilberger's algorithm for identities on Bernoulli and Euler polynomials

We present a computer algebra approach to proving identities on Bernoulli polynomials and Euler polynomials by using the extended Zeilberger's algorithm given by Chen, Hou and Mu. The key idea is to use the contour integral definitions of the Bernoulli and Euler numbers to establish recurrence relations on the integrands. Such recurrence relations have certain parameter free properties which lead to the required identities without computing the integrals. Furthermore two new identities on Bernoulli numbers are derived.     

Strong rigidity of generalized Bernoulli actions and computations of their symmetry groups

We study equivalence relations and II₁ factors associated with (quotients of) generalized Bernoulli actions of Kazhdan groups. Specific families of these actions are entirely classified up to isomorphism of II₁ factors. This yields explicit computations of outer automorphism and fundamental groups. In particular, every finitely presented group is concretely realized as the outer automorphism group of a continuous family of non stably isomorphic II₁ factors.     

Expression for restricted partition function through Bernoulli polynomials

Explicit expressions for restricted partition function W(s,d ^m ) and its quasiperiodic components W _j (s,d ^m ) (called Sylvester waves) for a set of positive integers d ^m ={d ₁,d ₂,…,d _m } are derived. The formulas are represented in a form of a finite sum over Bernoulli polynomials of higher order with periodic coefficients.      

A generating function for a class of generalized Bernoulli polynomials

First we derive a generating function and a Fourier expansion for a class of generalized Bernoulli polynomials. Then we derive formulas that allow certain Dirichlet series to be evaluated in terms of these generalized Bernoulli polynomials.      

Variational approach to shape derivatives for a class of Bernoulli problems

The shape derivative of a functional related to a Bernoulli problem is derived without using the shape derivative of the state. The gradient information is combined with level set ideas in a steepest descent algorithm. Numerical examples show the feasibility of the approach.     

Riccati方程与Bernoulli方程的一种解关系(英文)

给出Ｒｉｃｃａｔｉ方程和Ｂｅｒｎｏｕｌｌｉ方程的统一求积方法,揭示两类方程的一种解关系．     

A Bernoulli polynomial approach with residual correction for solving mixed linear Fredholm integro-differential-difference equations

In this study, an approximate method based on Bernoulli polynomials and collocation points has been presented to obtain the solution of higher order linear Fredholm integro-differential-difference equations with the mixed conditions. The method we have used consists of reducing the problem to a matrix equation which corresponds to a system of linear algebraic equations. The obtained matrix equation is based on the matrix forms of Bernoulli polynomials and their derivatives by means of collocations. The solutions are obtained as the truncated Bernoulli series which are defined in the interval [a,b]. To illustrate the method, it is applied to the initial and boundary values. Also error analysis and numerical examples are included to demonstrate the validity and applicability of the technique.     

退化高阶伯努利多项式与广义等幂和多项式的对称等式

研究了退化伯努利多项式与广义等幂和多项式的对称关系,获得了关于多个退化高阶伯努利多项式与广义等幂和多项式的若干对称关系.     

Some convergence results for asynchronous algorithms

Resumé Nous présentons dans cet article des résultats de convergence des algorithmes asynchrones basés essentiellement sur la notion classique de contraction.Nous généralisons, en particulier, tous les résultats de convergence de ces algorithmes qui font l'hypothèse de contraction en norme vectorielle qui récemment a été très souvant utilisée.Par ailleurs, l'hypothèse de contraction en norme vectorielle peut se trouver difficile, voire impossible à vérifier pour certains problèmes qui peuvent être cependant abordés dans le cadre de la contraction classique que nous adoptons.

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Some convergence results for asynchronous algorithms

Summary In this paper we present convergence results for the asynchronous algorithms based essentially on the notion of classical contraction.We generalize, in particular, all convergence results for those algorithms which are based on the vectorial norm hypothesis, in wide spread use recently.Certain problems, for which the vectorial norm hypothesis can be difficult or even impossible to verify, can nontheless be tackled within the scope of the classical contraction that we adopte.