Gowers norms and pseudorandom measures of subsets

Let $A \subset {{\Bbb Z}_N}$, and ${f_A}(s) = \left\{ {\begin{array}{*{20}{l}}{1 - \frac{{|A|}}{N},}&{{\rm{for}}\;s \in A,}\\{ - \frac{{|A|}}{N},}&{{\rm{for}}\;s \notin A.}\end{array}} \right.$ We define the pseudorandom measure of order k of the subset A as follows, P_k(A, N) = $\begin{array}{*{20}{c}}{\max }\\D\end{array}$|$\mathop \Sigma \limits_{n \in {\mathbb{Z}_N}}$f_A(n + c₁)f_A(n + c₂) … f_A(n + c_k)|, where the maximum is taken over all D = (c₁, c₂, . . . , c_k) ∈ ${\mathbb{Z}^k}$ with 0 ≤ c₁ < c₂ < … < c_k ≤ N - 1. The subset A ⊂ ${{\mathbb{Z}_N}}$ is considered as a pseudorandom subset of degree k if P_k(A, N) is “small” in terms of N. We establish a link between the Gowers norm and our pseudorandom measure, and show that “good” pseudorandom subsets must have “small” Gowers norm. We give an example to suggest that subsets with “small” Gowers norm may have large pseudorandom measure. Finally, we prove that the pseudorandom subset of degree L(k) contains an arithmetic progression of length k, where L(k) = 2·lcm(2, 4, . . . , 2|$\frac{k}{2}$|), for k ≥ 4, and lcm(a₁, a₂, . . . , a_l) denotes the least common multiple of a₁, a₂, . . . , a_l.     

Boundary behavior of harmonic functions on metric measure spaces with non-negative Ricci curvature

Let (X, d, μ) be a metric measure space with non-negative Ricci curvature. This paper is concerned with the boundary behavior of harmonic function on the (open) upper half-space X\times {\text{ℝ}}_{+}. We derive that a function f of bounded mean oscillation (BMO) is the trace of harmonic function u(x,t) on X\times {\text{ℝ}}_{+},u(x,0)=f\left(x\right), whenever u satisfies the following Carleson measure condition where \nabla =({\nabla }_x,{\partial }_t) denotes the total gradient and B(x_B,r_B) denotes the (open) ball centered at x_B with radius r_B. Conversely, the above condition characterizes all the harmonic functions whose traces are in BMO space.     

Exponential distributions on semigroups

Three fundamental characterizations of the standard exponential distribution on [0, ) are the remaining life, memoryless and constant failure properties. Analogs of these properties are studied for distributions on a class of semigroups in which the semigroup operation replaces addition, a compatible partial order replaces the ordinary order, and a left-invariant measure replaces Lebesgue measure. Partial characterizations of exponential distributions on such semigroups are obtained and the semigroup formulation provides new characterizations of certain aging properties studied in reliability-increasing failure rate, new better than used, and increasing failure rate average.     

Rate conservation laws: A survey

We survey the rate conservation law, RCL for short, arising in queues and related stochastic models. RCL was recognized as one of the fundamental principles to get relationships between time and embedded averages such as the extended Little's formulaH=G, but we show that it has other applications. For example, RCL is one of the important techniques for deriving equilibrium equations for stochastic processes. It is shown that the various techniques, including Mecke's formula for a stationary random measure, can be formulated as RCL. For this purpose, we start with a new definition of the rate with respect to a random measure, and generalize RCL by using it. We further introduce the notion of quasi-expectation, which is a certain extension of the ordinary expectation, and derive RCL applicable to the sample average results. It means that the sample average formulas such asH=G can be obtained as the stationary RCL in the quasi-expectation framework. We also survey several extensions of RCL and discuss examples. Throughout the paper, we would like to emphasize how results can be easily obtained by using a simple principle, RCL.     

有阻尼Sine－Gordon方程的全局吸引子的维数

本文通过引入新范数,得到有阻尼Ｓｉｎｅ－Ｇｏｒｄｏｎ方程的Ｄｉｒｉｃｈｌｅｔ问题的全局吸引子的维数的一个估计．结果表明：当“阻尼”与“扩散”同时增大或正弦项系数减小时,吸引子的维数减小．特别地,得到了零维吸引子存在的参数条件．     

Spectrum of positive entropy multidimensional dynamical systems with a mixed time

It is shown that if an abelian countable group is such that is a finite group and every aperiodic positive entropy action of on a Lebesgue probability space has a countable Haar spectrum in the subspace , where denotes the Pinsker -
algebra of , then every aperiodic positive entropy action of on has the same property. A positive answer to the question of J.P. Thouvenot is obtained as a corollary.

     

HYPERFINITE TRANSVERSAL THEORY. II

We continue the investigation of validity of Hall's theorem in the case of the Loeb space of an internal, uniformly distributed, hyperfinite measure space initiated in1992 by the author. Some new classes of graphs are introduced for which the measure theoretic version of Hall's theorem still holds.

     

Carleson measure characterization of Bloch functions

We give several equivalences of Bloch functions and little Bloch functions. Using these results we obtain the generalized Carleson measure characterization of Bloch functions and the generalized vanishing Carleson measure characterization of little Bloch functions, that is,f B if and only if |D f(z)| ^p (1-|z|²)^p-1 dm(z) is a generalized Carleson measure;f B ₀ if and only if |D f(z)| ^p (1-|z|²)^p-1 dm(z) is a generalized vanishing Carleson measure, whereD f( > 0) is the fractional derivative of analytic functionf of order, m denotes the normalised Lebesgue measure.Supported partly by the Young Teacher Natural Science Foundation of Shandong Province.     

MICELLIZATION STUDIES OF DODECYL BENZENESULFONIC ACID AND ITS INTERACTION WITH POLYVINYLPYRROLIDONE

Dodecyl benzenesulfonic acid (DBSA) surfactant was used in the present study to find the effect of concentration on its electrical conductance in solution from 293-323K above and below the critical micelle concentration (CMC). The micellization parameters i.e. degree of counter ion binding (β), aggregation number (n) and number of counter ion micelle(m) were measured. The interaction of DBSA with polyvinylpyrrolidone (PVP) was also studied at 293K throughconductance and surface tension measure ments. A number of important parameters i.e. critical aggregation concentration (CAC), Gibb‘s free energy (△G) and binding ratio (R) were determined and the effect of NaCl on the CAC and polymer saturation point (PSP) was also investigated.     

Repleteness and regular lattice measures

Let X be an abstract set and L a lattice of subsets of X. To each lattice regular measure µ, we associate two induced measures and on suitable lattices of the Wallman space IR(L) and another measure µ on the spaceIR(L). We will investigate the reflection of smoothness properties of p onto , and µ; and try to set some new criterion for repleteness and measure repleteness.