Ho1der Estimate of Harmonic Functions on a Class of p.c.f. Self-Similar Sets

In this paper we establish sharp H/51der estimates of harmonic functions on a class of connected post critically finite （p.c.f.） self-similar sets, and show that functions in the domain of Laplacian enjoy the same property. Some weU-known examples, such as the Sierpinski gasket, the unit interval, the level 3 Sierpinski gasket, the hexagasket, the 3-dimensional Sierpinski gasket, and the Vicsek set are also considered.     

A new lower bound of the Hausdorff measure of the Sierpinski gasket

For the Sierpinski gasket, by using a sort of cover consisting of special regular hexagons, we define a new measure that is equivalent to the Hausdorff measure and obtain a lower bound of this measure. Moreover, the following lower bound of the Hausdroff measure of the Sierpinski gasket has been achieved H^s（S）≥0.670432,where S denotes the Sierpinski gasket, s = dimn（S） = log23, and H^s（S） denotes the s-dimensional Hausdorff measure of S. The above result improves that developed in .     

A Grain of Dust Falling Through a Sierpinski Gasket

In this paper we analyze the downward random motion of a particle in a vertical, bounded, Sierpinski gasket G, where at each layer either absorption or delays are considered. In the case of motion with absorption the explicit distribution of the position of the descending particle in the pre-gasket Gn is obtained and the limiting case of the Sierpinski gasket discussed. For the delayed downward motion we derive a representation of the random time needed to arrive at the base of Gn in terms of independent binomial random variables （containing the contribution of delays at different layers with different geometrical structures）.     

The uv-Decomposition on a Class of D.C. Functions and Optimality Conditions

In this paper,the UV-theory and P-differential calculus are employed to study second-order ex-pansion of a class of D.C.functions and minimization problems.Under certain conditions,some properties ofthe U-Lagrangian,the second-order expansion of this class of functions along some trajectories are formulated.Some first and second order optimality conditions for the class of D.C.optimization problems are given.     

Energy and Laplacian of fractal interpolation functions

In this paper, we first characterize the finiteness of fractal interpolation functions(FIFs) on post critical finite self-similar sets. Then we study the Laplacian of FIFs with uniform vertical scaling factors on the Sierpinski gasket(SG). As an application, we prove that the solution of the following Dirichlet problem on SG is a FIF with uniform vertical scaling factor 1/5: Δu = 0 on SG\{q_1, q_2, q_3}, and u(q_i) = a_i, i = 1, 2, 3, where q_i, i = 1, 2, 3, are boundary points of SG.     

SOME GEOMETRIC PROPERTIES OF BROWNIAN MOTION ON SIERPINSKI GASKET

Let {X(t),t≥0} be Brownian motion on Sierpinski gasket,The Hausdorff and packing dimensions of the image of a ompact set are studied,The uniform Hausdorff and packing dimensions of the inverse image are also discussed.     

本刊英文版2012年55卷第3期摘要

Martin boundary and exit space on the Sierpinski gasket;On higher-dimensional contrast structure of singularly perturbed Dirichlet problem;On the limit behavior of the magnetic Zakharov system;On representations of real Jacobi groups;The mean value involving Dedekind sums and two-term exponential sums     

An approximation method to estimate the Hausdorff measure of the Sierpinski gasket

In this paper, we firstly define a decreasing sequence {P^n(S)} by the generation of the Sierpinski gasket where each P^n(S) can be obtained in finite steps. Then we prove that the Hausdorff measure H^8(S) of the Sierpinski gasket S can be approximated by {P^n(S)} with P^n(S)/(1 1/2^n-3)s ≤ H^8(S)≤ Pb(S).An algorithm is presented to get P^n(S) for n≤ 5. As an application, we obtain the best lower bound of H^8(S) till now: H^8(S) ≥ 0.5631.     

The boundedness of bilinear singular integral operators on Sierpinski gaskets

In the paper we give the boundedness estimate of bilinear singular integral operators on Sierpinski gasket inspired from [1].     

Martin boundary and exit space on the Sierpinski gasket

We define a new Markov chain on the symbolic space representing the Sierpinski gasket (SG),and show that the corresponding Martin boundary is homeomorphic to the SG while the minimal Martin boundary is the three vertices of the SG.In addition,the harmonic structure induced by the Markov chain coincides with the canonical one on the SG.This suggests another approach to consider the existence of Laplacians on those self-similar sets for which the problem is still not settled.     

The Convergence of the Steepest Descent Algorithm for D.C.Optimization

Some properties of a class of quasi-differentiable functions(the difference of two finite convex functions) are considered in this paper.And the convergence of the steepest descent algorithm for unconstrained and constrained quasi-differentiable programming is proved.     

H ¨older Estimate of Harmonic Functions on a Class of p.c.f. Self-Similar Sets

In this paper we establish sharp H ¨older estimates of harmonic functions on a class of connected post critically finite (p.c.f.) self-similar sets, and show that functions in the domain of Laplacian ...     

ON CONE D.C. OPTIMIZATION AND CONJUGATE DUALITY

This paper derives first order necessary and sufficient conditions for unconstrained cone d.c. programming problems where the underlined space is partially ordered with respect to a cone. These conditions are given in terms of directional derivatives and subdifferentials of the component functions. Moreover, conjugate duality for cone d.c. optimization is discussed and weak duality theorem is proved in a more general partially ordered linear topological vector space (generalizing the results in [11]).     

Bounds of the Hausdorff Measure of Sierpinski carpet

By means of the idea of [2]（Jia Baoguo,J.Math.Anal.Appl.In press） and the self.similarity of Sierpinski carpet, we obtain the lower and upper bounds of the Hausdorff Measure of Sierpinski carpet, which can approach the Hausdorff Measure of Sierpinski carpet infinitely.     

二元正交对称不可分尺度函数的参数化

Let I be the 2 × 2 identity matrix, and M a 2 × 2 dilation matrix with M2 = 2I. First, we present the correlation of the scaling functions with dilation matrix M and 2I. Then by relating the properties of scaling functions with dilation matrix 2I to the properties of scaling functions with dilation matrix M, we give a parameterization of a class of bivariate nonseparable orthogonal symmetric compactly supported scaling functions with dilation matrix M. Finally, a construction example of nonseparable orthogonal symmetric and compactly supported scaling functions is given.     

Spectral Self-Affine Measures on the Generalized Three Sierpinski Gasket

The self-affine measure μM,Dassociated with an iterated function system{φd(x)=M~(-1)(x + d)}_(d∈D) is uniquely determined. It only depends upon an expanding matrix M and a finite digit set D. In the present paper we give some sufficient conditions for finite and infinite families of orthogonal exponentials. Such research is necessary to further understanding the non-spectral and spectral of μM,D. As an application,we show that the L~2(μM,D) space has infinite families of orthogonal exponentials on the generalized three Sierpinski gasket. We then consider the spectra of a class of self-affine measures which extends several known conclusions in a simple manner.     

The ranks of Maiorana-McFarland bent functions

In this paper, the ranks of a special family of Maiorana-McFarland bent functions are discussed. The upper and lower bounds of the ranks are given and those bent functions whose ranks achieve these bounds are determined. As a consequence, the inequivalence of some bent functions are derived. Furthermore, the ranks of the functions of this family are calculated when t 6.     

A Numerical Calculation Algorithm of the Inclination Function Based on the Representation of SO(3) Group

In the determination of the Earth gravity field in satellite geodesy, the inclination functions represent the projection of data observed along the orbital plane of a satellite orbit into the sphere in the terrestial reference frame. The inclination functions in this work is studied from a group theoretical perspective. The inclination functions are proved to generate a representation of the SO(3) group. An orthogonal relation of the inclination functions is derived and some recurrence relations...     

Construction and Dimension Analysis for a Class of Fractal Functions

In this paper, we construct a class of nowhere differentiable continuous functions by means of the Cantor series expression of real numbers. The constructed functions include some known nondifferentiable functions, such as Bush type functions. These functions are fractal functions since their graphs are in general fractal sets. Under certain conditions, we investigate the fractal dimensions of the graphs of these functions, compute the precise values of Box and Packing dimensions, and evaluate the Hausdorff dimension. Meanwhile, the Holder continuity of such functions is also discussed.