一类递归集的Hausdorff维数及Bouligand维数

本文给出递归集的Ｈａｕｓｄｏｒｆｆ维数的下界估计，并由此确定了一类递归集的维数，所获结果包含并推广了Ｂｅｄｆｏｒｄ，Ｄｅｋｋｉｎｇ及文志英、钟红柳等人的有关结果。     

Zeroth-order phase transitions and Zipf law quantization

From the standpoint of thermodynamic averaging of fission microprocesses, we investigate the origin of radioactive release in an NPP after an accident or after resource depletion. The genesis of the NPP release is interpreted as a new thermodynamic phenomenon, a zeroth-order phase transition. This problem setting results in a problem in probabilistic number theory. We prove the corresponding theorem leading to quantization of the Zipf law for the frequency of a zeroth-order phase transition with different values of the jump of the Gibbs thermodynamic potential. We introduce the notion of hole dimension. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 150, No. 1, pp. 118–142, January, 2007.     

Quantization of lie Bialgebras via the Formality of the operad of Little Disks

We give a proof of the Etingof–Kazhdan theorem on quantization of Lie bialgebras based on the formality of the chain operad of little disks and show that the Grothendieck–Teichmüller group acts non-trivially on the corresponding quantization functors. Partially supported by an NSF Grant and A. Sloan Research Fellowship Received: November 2004 Revision: April 2006 Accepted: May 2006     

Testing for additivity with B-splines

Regression splines are often used for fitting nonparametric functions, and they work especially well for additivity models. In this paper, we consider two simple tests of additivity: an adaptation of Tukey's one degree of freedom test and a nonparametric version of Rao's score test. While the Tukey-type test can detect most forms of the local non-additivity at the parametric rate of O(n-1/2), the score test is consistent for all alternative at a nonparametric rate. The asymptotic distribution of these test statistics is derived under both the null and local alternative hypotheses. A simulation study is conducted to compare their finite-sample performances with some existing kernel-based tests. The score test is found to have a good overall performance.     

A multifractal decomposition according to rate of returns

For one‐dimensional simple symmetric random walk, the Hausdorff and packing dimensions of sets of sample paths with prescribed rate of returns to the origin are determined. This gives a multifractal decomposition of the underlying sample space. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Ginzburg－Landau－Newell模型的动力学行为

木文对Ｇｉｎｚｂｕｒｇ－Ｌａｎｄａｕ－Ｎｅｗｅｄ模型的动力学行为进行了讨论，得到了该模型的整体吸引子的存在性，同时得到了此吸引子维数的下界估计和该吸引子的Ｈａｕｓｄｏｒｆｆ维数和Ｗａｃｔａｌ（分形）维数的上界估计．     

一类二维非线性Schrodinger方程的吸引子及其维数

本文研究了一类二维非线性Ｓｃｈｒｏｄｉｎｇｅｒ方程解的有限维行为，我们得到了此方程存在吸引子，并得到了此吸引子维数的上界估计     

Finite dimensional semisimple Q-algebras

A Q-algebra can be represented as an operator algebra on an infinite dimensional Hilbert space. However we don’t know whether a finite n-dimensional Q-algebra can be represented on a Hilbert space of dimension n except n = 1, 2. It is known that a two dimensional Q-algebra is just a two dimensional commutative operator algebra on a two dimensional Hilbert space. In this paper we study a finite n-dimensional semisimple Q-algebra on a finite n-dimensional Hilbert space. In particular we describe a three dimensional Q-algebra of the disc algebra on a three dimensional Hilbert space. Our studies are related to the Pick interpolation problem for a uniform algebra.     

样条空间S~μ_k（△）的维数

本文考虑了欧式空间Ｒ￣ｎ中任意单纯形剖分上的样条函数空间．证明了当ｋ≥（３μ＋１）２￣（ｎ－２）＋１时，计算任意单纯形剖分Δ上的ｋ次μ阶光滑样条空间的维数，可归结为计算每个σ－关联域（ｉ－单纯形σ∈Δ）Ｒ（σ）上的２￣（ｎ－ｉ－１）μ次μ阶光滑（ｉ≤ｎ－１）样条空间的维数。这里σ－关联域Ｒ（σ）是指Δ中所有包含σ的单纯形所成的单纯形剖分．     

The VC dimension of k‐uniform random hypergraphs

A set of vertices is shattered in a hypergraph if any of its subsets is obtained as the intersection of an edge with the set. The VC dimension is the size of the largest shattered subset. Under the binomial model of k‐uniform random hypergraphs, the threshold function for the VC dimension to be larger than a given integer is obtained. The same is done for the testing dimension, which is the largest integer d such that all sets of cardinality d are shattered. © 2006 Wiley Periodicals, Inc. Random Struct. Alg., 2007