On the distribution of the sequence of fractional parts of a slowly increasing exponential function

In this paper for a slowly increasing exponential function we study the degree of deviation of the distribution of the sequence of its fractional parts from the “ideal” uniform distribution. An estimate is given which can appropriately be called the measure of irregularity of the distribution of a given sequence. Translated fromMatematicheskie Zametki, Vol. 65, No. 1, pp. 148–152, January, 1999.     

On generalized variational inequalities

We obtain a new version of the minimax inequality of Ky Fan. As an application, an existence result for the generalized variational inequality problem with set-valued mappings defined on noncompact sets in Hausdorff topological vector spaces is given. Also, some existence results for the generalized variational inequality problem for quasimonotone and pseudomonotone mappings are obtained. Dedicated to the memory of T. Rapcsák.     

Minimal relative entropy for equivalent martingale measures by low-discrepancy sequence in Lévy process

In this paper, we focused on computing the minimal relative entropy between the original probability and all of the equivalent martin gale measure for the Lévy process. For this purpose, the quasiMonte Carlo method is used. The probability with minimal relative entropy has many suitable properties. This probability has the minimal Kullback-Leibler distance to the original probability. Also, by using the minimal relative entropy the exponential utility indifference price can be found. In this paper, the Monte Carlo and quasi-Monte Carlo methods have been applied. In the quasi-Monte Carlo method, two types of widely used lowdiscrepancy sequences, Halton sequence and Sobol sequence, are used. These methods have been used for exponential Lévy process such as variance gamma and CGMY process. In these two processes, the minimal relative entropy has been computed by Monte Carlo and quasi-Monte Carlo, and compared their results. The results show that quasi-Monte Carlo with Sobol sequence performs better in terms of fast convergence and less error. Finally, this method by fitting the variance gamma model and parameters estimation for the model has been implemented for financial data and the corresponding minimal relative entropy has been computed.     

逆极限的不变测度和一致正熵性质

In this paper, the interconnection of some ergodic properties between a continuous selfmap and its inverse limit is studied. It has been proved that (1) their invariant Borel probability measures are identical up to homeomorphism and (2) they preserve uniform positive entropy property simuitaneously. As applications, it is also proved that the upper semi-continu-ous properties of their entropy maps are restricted each other, and the entropy map of the asymptotically h-expansive continuous map is upper semi-contlnuous, at the same time a continuous map having u, p.e. is topological weakmixing.     

On tests of independence in a trivariate exponential distribution

In this paper we consider the problem of testing a null hypothesis H₀ against an alternative hypothesis H_a, related to a trivariate model. H₀ is transformed into a hypothesis concerning the equality of scale parameters of three exponential distributions, following Bhattacharyya and Johnson. Two conditional tests for the transformed hypothesis are considered, and simple expressions for the power functions of our tests are obtained.     

The conditioning of the maximum entropy covariance matrix and its inverse

The maximum entropy covariance matrix is positive definite even when the number of variables p exceeds the sample size n. However, the inverse of this matrix can have stability problems when p is close to n, although these problems tend to disappear as p increases beyond n. We analyze such problems using the variance of the latent roots in a particular metric as a condition number.     

On the Number of Birch Partitions

Birch and Tverberg partitions are closely related concepts from discrete geometry. We show two properties for the number of Birch partitions: Evenness and a lower bound. This implies the first nontrivial lower bound for the number of Tverberg partitions that holds for arbitrary q, where q is the number of partition blocks. The proofs are based on direct arguments and do not use the equivariant method from topological combinatorics.     

A note on two measures of dependence

In a paper in 1956, Dobrushin proved a central limit theorem for triangular arrays of Markov chains under certain dependence assumptions somewhat related to “Doeblin’s condition”. In a very recent paper, Peligrad proved a central limit theorem of the same type, but with dependence assumptions based on maximal correlations. This note here will give a concrete example to illustrate the extent to which Peligrad’s result goes beyond that of Dobrushin.     

On the Jensen-Steffensen inequality for generalized convex functions

Jensen-Steffensen type inequalities for P-convex functions and functions with nondecreasing increments are presented. The obtained results are used to prove a generalization of ?eby?ev’s inequality and several variants of Hölder’s inequality with weights satisfying the conditions as in the Jensen-Steffensen inequality. A few well-known inequalities for quasi-arithmetic means are generalized.     

The distribution of a generalized least squares estimator with covariance adjustment

The distribution theory is developed for a generalized least squares estimator of the growth curve model. A special case of the estimator is the maximum likelihood estimator which is weighted by the sample covariance matrix. The distribution of two conditional forms of the estimator are derived and from these its density is obtained. Two general pivots and their distributions are derived from the conditional forms and special cases of these are investigated. The results obtained are linked to carlier work.     

Approximating conditional density functions using dimension reduction

We propose to approximate the conditional density function of a random variable Y given a dependent random d-vector X by that of Y given θ^τX, where the unit vector θ is selected such that the average Kullback-Leibler discrepancy distance between the two conditional density functions obtains the minimum. Our approach is nonparametric as far as the estimation of the conditional density functions is concerned. We have shown that this nonparametric estimator is asymptotically adaptive to the unknown index θ in the sense that the first order asymptotic mean squared error of the estimator is the same as that when θ was known. The proposed method is illustrated using both simulated and real-data examples.     

加法与乘法下的模糊熵

由于模糊集关于普通加法与乘法运算构成的代数系统是分配格、布尔代数,这就使得模糊集的应用受到一定的限制,为了扩展模糊集的应用,本文引进了模糊集的加法与乘法运算,接着研究了模糊集关于加法与乘法的运算性质以及模糊熵,距离测度,相似性测度在集合关于加法运算,乘法运算下的一些性质,并且对加法与乘法下的模糊熵与普通模糊熵作了对比.     

Fine-grained and coarse-grained entropy in problems of statistical mechanics

We consider dynamical systems with a phase space Γ that preserve a measure μ. A partition of Γ into parts of finite μ-measure generates the coarse-grained entropy, a functional that is defined on the space of probability measures on Γ and generalizes the usual (ordinary or fine-grained) Gibbs entropy. We study the approximation properties of the coarse-grained entropy under refinement of the partition and also the properties of the coarse-grained entropy as a function of time. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 151, No. 1, pp. 120–137, April, 2007.     

Equivalent measures of dependence

The maximal correlation between a pair of σ-fields $\text{A}$ and $\text{B}$ becomes arbitrarily small as sup{|P(A ? B) ? P(A) P(B)|/[P(A) P(B)]^1/2, A ∈ $\text{A}$, B ∈ $\text{B}$, P(A) > 0, P(B) > 0} becomes sufficiently small.     

Wavelet characterization of Hörmander symbol classS ρ,δ m and applications

In this paper, we characterize the symbol in Hormander symbol classS _ρ ^m ,δ (m ∈ R, ρ, δ ≥ 0) by its wavelet coefficients. Consequently, we analyse the kerneldistribution property for the symbol in the symbol classS _ρ ^m ,δ (m ∈R, ρ > 0, δ≥ 0) which is more general than known results ; for non-regular symbol operators, we establish sharp L²-continuity which is better than Calderón and Vaillancourt’s result, and establishL ^p (1 ≤p ≤∞) continuity which is new and sharp. Our new idea is to analyse the symbol operators in phase space with relative wavelets, and to establish the kernel distribution property and the operator’s continuity on the basis of the wavelets coefficients in phase space.     

On the dependence of the growth rate on the length of the defining relator

Let { } be a sequence of finitely presented groups with generating setA={a1, …, am}, and letRk be the symmetrized set of words over the alphabetA∪A−1 obtained from the defining words and their inverses by all cyclic shifts. We shall assume that the words inRk are cyclically irreducible, and their lengths tend to ∞ ask increases. In the paper, it is proved that ifRk satisfies the small cancellation conditionC'(1/6) and the number of relators increases not very rapidly with increasingk, then the growth rate ψ(Gk) tends to 2m−1 ask→∞. Translated fromMatematicheskie Zametki, Vol. 65, No. 4, pp. 611–617, April, 1999.     

The equality cases for the inequalities of Oppenheim and Schur for positive semi-definite matrices

In this paper, necessary and sufficient conditions for equality in the inequalities of Oppenheim and Schur for positive semidefinite matrices are investigated. Supported by National Natural Science Foundation of China (No. 10531070), National Basic Research Program of China 973 Program (No. 2006CB805900), National Research Program of China 863 Program (No. 2006AA11Z209) and the Natural Science Foundation of Shanghai (Grant No. 06ZR14049).