Markov Chains with Positive Transitions Are Not Determined by Any p-Marginals

 We prove that if is a finite valued stationary Markov Chain with strictly positive probability transitions, then for any natural number p, there exists a continuum of finite valued non Markovian processes which have the p-marginal distributions of X and with positive entropy, whereas for an irrational rotation and essentially bounded real measurable function f with no zero Fourier coefficient on the unit circle with normalized Lebesgue measure, the process is uniquely determined by its three-dimensional distributions in the class of ergodic processes. We give also a family of Gaussian non-Markovian dynamical systems for which the symbolic dynamic associated to the time zero partition has the two-dimensional distributions of a reversible mixing Markov Chain. (Received 22 July 1999; in revised form 24 February 2000)     

Involution Semigroups Are Not Globally Determined

The purpose of this paper is to present an example of nonisomorphic involution semigroups which are globally isomorphic, i.e. which have isomorphic power algebras (elements of which are the nonempty subsets of the original algebras). As an immediate consequence, we obtain that (involution) semirings are not globally determined. September 7, 1999     

Integer Polynomials with Roots mod p for all Primes p

Let f(X) be an integer polynomial which is a product of two irreducible factors. Assume that f(X) has a root mod p for all primes p. If the splitting field of f(X) over the rationals is a cyclic extension of the stem fields, then the Galois group of f(X) over the rationals is soluble and of bounded Fitting length. Moreover, the fixed groups of the stem extensions are in, some sense, unique.     

p-Adic ideals of p-rank d and the p-adic Nullstellensatz

In this paper we introduce and we study the notion of p-adic ideal of p-rank d and the one of p-adic radical of p-rank d of an ideal, by analogy with the notion of real ideal and the one of real radical of an ideal in the real case. Using those two notions and following more closely the real approach of Bochnak et al. (Géométrie algébrique réelle, Springer, Berlin 1987) we give a new proof of the p-adic Nullstellensatz.     

Relative ( pn , p , pn , n )-difference sets with GCD ( p , n )=1

Let p be an odd prime. We first get some non-existence and structural results on (pn,p,pn,n) relative difference sets with gcd(p,n)=1 through a group ring approach. We then give a construction of (p(p+1),p,p(p+1),p+1) relative difference sets with p a Mersenne prime. Supported by National Natural Science Foundation of China (10331030).     

An ergodic Markov chain is not determined by any p-marginals

We prove that if X = (X_n)_nεZ is a finite state space ergodic Markov chain, then for any natural number p, there exist ergodic non-Markovian processes Y = (Y_n)_nεZ with positive entropy, such that for all integers n₁, …, n_p, the joint distribution of Y_n1, …, Y_np is identical to the joint distribution of X_n1, …, X_np.     

The Group Ring of SL2(p) over p-adic Integers for p Odd

Let p > 2 be a prime, R =  _p[ζ_p^f − 1], K =  _p[ζ_p^f − 1], and G = SL₂(p^f). The group ring RG is calculated nearly up to Morita equivalence: The projections of RG into the simple components of KG are given explicitly and the endomorphism rings and homomorphism bimodules between the projective indecomposable RG-lattices are described.     

Perturbed Markov Chains with Damping Component

The paper is devoted to studies of regularly and singularly perturbed Markov chains with damping component. In such models, a matrix of transition probabilities is regularised by adding a special damping matrix multiplied by a small damping (perturbation) parameter ε. We perform a detailed perturbation analysis for such Markov chains, particularly, give effective upper bounds for the rate of approximation for stationary distributions of unperturbed Markov chains by stationary distributions of perturbed Markov chains with regularised matrices of transition probabilities, asymptotic expansions for approximating stationary distributions with respect to damping parameter, explicit coupling type upper bounds for the rate of convergence in ergodic theorems for n-step transition probabilities, as well as ergodic theorems in triangular array mode.

     

Markov Chains with finite convergence time

We study the properties of finite ergodic Markov Chains whose transition probability matrix P is singular. The results establish bounds on the convergence time of P^m to a matrix where all the rows are equal to the stationary distribution of P. The results suggest a simple rule for identifying the singular matrices which do not have a finite convergence time. We next study finite convergence to the stationary distribution independent of the initial distribution. The results establish the connection between the convergence time and the order of the minimal polynomial of the transition probability matrix. A queuing problem and a maintenance Markovian decision problem which possess the property of rapid convergence are presented.     

L p -dual Quermassintegral sums

In this paper, we first introduce a concept of L _p -dual Quermassintegral sum function of convex bodies and establish the polar projection Minkowski inequality and the polar projection Aleksandrov-Fenchel inequality for L _p -dual Quermassintegral sums. Moreover, by using Lutwak’s width-integral of index i, we establish the L _p -Brunn-Minkowski inequality for the polar mixed projection bodies. As applications, we prove some interrelated results. This work was partially supported by the National Natural Science Foundation of China (Grant No. 10271071), Zhejiang Provincial Natural Science Foundation of China (Grant No. Y605065) and Foundation of the Education Department of Zhejiang Province of China (Grant No. 20050392)     

Large Deviations for Empirical Measures of Not Necessarily Irreducible Countable Markov Chains with Arbitrary Initial Measures

All known results on large deviations of occupation measures of Markov processes are based on the assumption of （essential） irreducibility. In this paper we establish the weak＊ large deviation principle of occupation measures for any countable Markov chain with arbitrary initial measures. The new rate function that we obtain is not convex and depends on the initial measure, contrary to the （essentially） irreducible case.     

Oscillators in resonance p:q:r

A canonical transformation is proposed to handle Hamiltonian systems made of the addition or subtraction of three harmonic oscillators in p:q:r resonance. This transformation is an extension of the classical Lissajous transformation for the 1:1 resonance. Our extended Lissajous variables consist of three pairs of action-angle variables, which makes possible the application of perturbation theories without encountering small divisors. A set of functions, related with the Lissajous variables, are found, and are used to describe the phase flow of reduced space after normalization.     

Infimal convolution and Muckenhoupt A p (.) condition in variable L p spaces

We study the Hardy-Littlewood maximal operator M on . Under the assumptions that the exponent p satisfies and is constant outside some large ball, we prove that if and only if . Received: 2 June 2006 Revised: 28 November 2006     

Continuous-Time Controlled Markov Chains with Discounted Rewards

This paper studies denumerable state continuous-time controlled Markov chains with the discounted reward criterion and a Borel action space. The reward and transition rates are unbounded, and the reward rates are allowed to take positive or negative values. First, we present new conditions for a nonhomogeneous Q(t)-process to be regular. Then, using these conditions, we give a new set of mild hypotheses that ensure the existence of -optimal (0) stationary policies. We also present a martingale characterization of an optimal stationary policy. Our results are illustrated with controlled birth and death processes.     

Positive solutions of singular p-Laplacian dynamic equations with sign changing nonlinearity

Let be a time scale such that . By the Schauder fixed-point theorem and the upper and lower solution method, we present some existence criteria of the positive solution of m-point singular p-Laplacian dynamic equation with boundary conditions , where φ_p(s)=|s|^p-2s with p>1, is continuous for i=1,2,…,m-1 and nonincreasing if . The nonlinear term may be singular in its dependent variable and is allowed to change sign. Our results are new even for the corresponding differential and difference equations . As an application, an example is given to illustrate our result.     

Lefschetz Formulae for p -Adic Groups

Abstract In this paper, Lefschetz formulae for torus actions on p-adic groups are proven. These are similar to comparable formulae for real Lie groups. Applications lie in the realm of dynamical zeta functions.     

Meromorphic p -valent function with fixed second negative coefficients

In this paper we consider the class consisting of analytic functions with negative coefficients and fixed second coefficient. The object of the present paper is to give coefficient estimates, convex linear combinations, some distortion theorems and radii of starlikeness and convexity for f(z) in the class . The work is supported by UKM grant ST-028-2003 and IRPA grant 09-02-02-80 EA208, Malysia. This paper is that of a talk on the “International Conference at Analysis in Theory and Applications” held in Nanjing, P. R. China, July, 2004.     

Optimum computation of p bilinear forms

We wish to answer the following question: p matrices B_i, of the same dimension, being given, what is the minimum number of multiplications we have to perform to obtain the p values X^tB_iY, for arbitrary vectors X and Y? We will give first a precise definition of the class of algorithms we consider to evaluate these p bilinear forms, and of our optimization criteria. Then we will characterize an optimal algorithm of this class, and relate the minimum number of multiplications used to the tensorial rank of the p matrices B_i. Properties of this number are given. Finally, the paper will conclude with a proof of the optimality of Strassen's algorithm to perform the product of two 2×2 matrices.     

Dense Relations Are Determined by Their Endomorphism Monoids

We introduce the class of dense relations on a set X and prove that for any finitary or infinitary dense relation on X, the relational system (X,) is determined up to semi-isomorphism by the monoid End(X,) of endomorphisms of (X,). In the case of binary relations, a semi-isomorphism is an isomorphism or an anti-isomorphism.