On an approximation property of Pisot numbers

Let 1<q<2 be a real number, m≥1 be a rational integer and l_m(q)={|P(q)|,P∈Z[X],P(q)≠0,H(P)≤m}, where Z[X] denotes the set of polynomials P with rational integer coefficients and H(P) is the height of P. The value of l_m(q) was determined for many particular Pisot numbers ([3] and [7]). In this paper we determine the infimum and the supremum of the numbers l_m(q) for any fixed m. We also determine the greatest limit point for the case m=1. This revised version was published online in June 2006 with corrections to the Cover Date.     

Nearest interval approximation of a fuzzy number

The problem of the interval approximation of fuzzy numbers is discussed. A new interval approximation operator, which is the best one with respect to a certain measure of distance between fuzzy numbers, is suggested.     

关于M-PN空间中算子方程Tx=μx+p(μ≥1)的解

在M enger PN空间中研究了一类非线性算子方程T x=μx p(μ≥1)的解,得到了几个新的定理.同时,改进和推广了若干个重要结论.     

关于算子方程Nχ=δLχ解的若干研究

利用多重拓扑度的性质以及一些不等式研究了零指标的Predholm算子和L-全连续算子方程Nχ=δχ的解,获得若干新的结果,推广了一些重要结论,并将结果应用到一类二阶微分方程的边值问题中.     

Some relations between entropy and approximation numbers

A general result is obtained which relates the entropy numbers of compact maps on Hilbert space to its approximation numbers. Compared with previous works in this area, it is particularly convenient for dealing with the cases where the approximation numbers decay rapidly. A nice estimation between entropy and approximation numbers for noncompact maps is given. Partially supported by the National 863 Project of China and the National Natural Science Foundation of China (Grant No. 19771003).     

Asymptotic Recurrence and Waiting Times for Stationary Processes

Let be a discrete-valued stationary ergodic process distributed according to P and let x=(..., x _–1, x ₀, x ₁,...) denote a realization from X. We investigate the asymptotic behavior of the recurrence time R _n defined as the first time that the initial n-block reappears in the past of x. We identify an associated random walk, on the same probability space as X, and we prove a strong approximation theorem between log R _n and . From this we deduce an almost sure invariance principle for log R _n. As a byproduct of our analysis we get unified proofs for several recent results that were previously established using methods from ergodic theory, the theory of Poisson approximation and the analysis of random trees. Similar results are proved for the waiting time W _n defined as the first time until the initial n-block from one realization first appears in an independent realization generated by the same (or by a different) process.     

Stochastic approximation with dependent noise

In this work we derive the usual limit laws (weak and strong convergence, central limit theorem, invariance principle) for stochastic approximation with stationary noise. The idea is to introduce an artificial sequence, related to the SA scheme, but which clearly obeys the desired limit law. This sequence is subtracted from the SA scheme and the remainder, which behaves more or less deterministically, is shown to vanish using simple limit arguments.     

Z-P-S空间中一类非线性算子方程解的存在性问题

拓扑度理论是研究非线性算子方程解的存在性的有力工具.利用拓扑度的方法,对Z-P-S空间中一类非线性算子方程解的存在性问题进行了研究,得到了若干新的结果.     

Estimates for the Approximation Numbers of One Class of Integral Operators. I

Asymptotic estimates are obtained for the approximation numbers of the Hardy integral operator with variable limits of integration.     

Algebraic Numbers Close to 1 in Non-Archimedean Metrics

Let 1 be an algebraic number with relatively small height. Recently, many authors, including Amoroso, Dubickas, Mignotte and Waldschmidt, stated sharp lower bounds for the quantity | – 1|. Here, we provide a p-adic analogue of their results. For instance, we give an upper bound for the absolute value of the norm of – 1, and we show that our estimate is rather sharp in terms of the degree of . Further, we discuss a generalization in several variables of our result.     

Stochastic approximation and the final value theorem

The aim here is to show how to obtain many of the well-known limit results (i.e., central limit theorem, law of the iterated logarithm, invariance principle) of stochastic approximation (SA) by a shorter argument and under weaker conditions. The idea is to introduce an artificial sequence, related to the SA scheme, and which clearly obeys the limit law. This sequence is subtracted from the SA scheme and then simple deterministic limit theory is used to show the remainder is negligible. As a consequence of this approach proofs are shorter and the meaning of conditions becomes clearer. Because the difference equations are not summed up it is simple to state results for general a_n, c_n sequences.     

A 2-groupoid Characterisation of the Cubical Homotopy Pushout

We give a 2-track-theoretical characterisation of the homotopy pushout of a 3-corner by recognising the mapping 2-simplex as an initial object in a coherent homotopy category of Hausdorff spaces under a 3-corner with morphisms expressed in terms of the 1-morphisms and 2-morphisms of a homotopy 2-groupoid.