摘要：
In this paper, we study Rademacher series with d-dimensional vector-valued coefficients. We first employ a new combinatorial technique to present a sufficient condition for the Rademacher range of a sequence with a unique direction equal to
$${\mathbb R}^2$$
. This result also gives a positive answer to the question that whether the Rademacher range of
$$\{(n^{-1},n^{-1}\ln ^{-1}(n+1))\}$$
is
$${\mathbb R}^2$$
. Next, by constructing homogeneous Cantor sets, we prove that, for each
$$s\in [1,d]$$
, there exists a sequence with a unique direction such that its Rademacher range of Hausdorff dimension s is dense in
$${\mathbb R}^d$$
but not equal to
$${\mathbb R}^d$$
.
期刊：
CANADIAN MATHEMATICAL BULLETIN-BULLETIN CANADIEN DE MATHEMATIQUES,2019年62(4):727-740 ISSN：0008-4395
通讯作者：
Deng, Guotai
作者机构：
[Deng, Guotai] Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Hubei, Peoples R China.;[Deng, Guotai] Cent China Normal Univ, Hubei Key Lab Math Sci, Wuhan 430079, Hubei, Peoples R China.;[Liu, Chuntai] Wuhan Polytech Univ, Sch Math & Comp Sci, Wuhan 430023, Hubei, Peoples R China.;[Ngai, Sze-Man] Hunan Normal Univ, Key Lab High Performance Comp & Stochast Informat, Minist Educ China, Coll Math & Stat, Changsha 410081, Hunan, Peoples R China.;[Ngai, Sze-Man] Georgia Southern Univ, Dept Math Sci, Statesboro, GA 30460 USA.
通讯机构：
[Deng, Guotai] C;Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Hubei, Peoples R China.;Cent China Normal Univ, Hubei Key Lab Math Sci, Wuhan 430079, Hubei, Peoples R China.
期刊：
FRACTALS-COMPLEX GEOMETRY PATTERNS AND SCALING IN NATURE AND SOCIETY,2018年26(5):1850061 ISSN：0218-348X
通讯作者：
Liu, Chuntai
作者机构：
[Liu, Chuntai] Wuhan Polytech Univ, Sch Math & Comp Sci, Wuhan 430023, Hubei, Peoples R China.
通讯机构：
[Liu, Chuntai] W;Wuhan Polytech Univ, Sch Math & Comp Sci, Wuhan 430023, Hubei, Peoples R China.
关键词：
Augment Tree;Cantor Set;Iterated Function System;Lipschitz Equivalence;Self-Similar Set
摘要：
Self-similarity and Lipschitz equivalence are two basic and important properties of fractal sets. In this paper, we consider those properties of the union of Cantor set and its translate. We give a necessary and sufficient condition that the union is a self-similar set. Moreover, we show that the union satisfies the strong separation condition if it is of the self-similarity. By using the augment tree, we characterize the Lipschitz equivalence between Cantor set and the union of Cantor set and its translate.
期刊：
TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY,2018年370(2):1321-1350 ISSN：0002-9947
通讯作者：
Deng, Guotai
作者机构：
[Deng, Guotai] Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Hubei, Peoples R China.;[Deng, Guotai] Cent China Normal Univ, Hubei Key Lab Math Sci, Wuhan 430079, Hubei, Peoples R China.;[Liu, Chuntai] Wuhan Polytech Univ, Sch Math & Comp Sci, Wuhan 430023, Hubei, Peoples R China.;[Ngai, Sze-Man] Hunan Normal Univ, Coll Math & Comp Sci, Changsha 410081, Hunan, Peoples R China.;[Ngai, Sze-Man] Georgia Southern Univ, Dept Math Sci, Statesboro, GA 30460 USA.
通讯机构：
[Deng, Guotai] C;Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Hubei, Peoples R China.;Cent China Normal Univ, Hubei Key Lab Math Sci, Wuhan 430079, Hubei, Peoples R China.
期刊：
HOUSTON JOURNAL OF MATHEMATICS,2016年42(1):179-210 ISSN：0362-1588
通讯作者：
Deng, Guotai
作者机构：
[Deng, Guotai] Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Peoples R China.;[Liu, Chuntai] Wuhan Polytech Univ, Sch Math & Comp Sci, Wuhan 430023, Peoples R China.;[Ngai, Sze-Man] Hunan Normal Univ, Coll Math & Comp Sci, Changsha 410081, Hunan, Peoples R China.;[Ngai, Sze-Man] Georgia So Univ, Dept Math Sci, Statesboro, GA 30460 USA.
通讯机构：
[Deng, Guotai] C;Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Peoples R China.
关键词：
Graph-directed iterated function systems;finite type condition;finite boundary type condition;Hausdorff dimension;box dimension
摘要：
We Set up a framework for dimensions of the boundary of each component F-i of a graph self-similar family defined by a graph-directed iterated function system satisfying the socalled graph finite boundary type condition. We show that partial derivative F-i has the same Hausdortf and box dimension, with the corresponding Hauselorff measure being pustve and sigma-finite. These results are natural extensions of existing ones concerning the dimensions of the boundary of a self similar tile.
期刊：
ANNALES ACADEMIAE SCIENTIARUM FENNICAE-MATHEMATICA,2015年40(1):135-148 ISSN：1239-629X
通讯作者：
He, Xing-Gang
作者机构：
[He, Xing-Gang] Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Peoples R China.;[Liu, Chun-Tai] Wuhan Polytech Univ, Sch Math & Comp Sci, Wuhan 430023, Peoples R China.
通讯机构：
[He, Xing-Gang] C;Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Peoples R China.
关键词：
Hausdorff dimension;Infinite bernoulli convolution;Moran function system;Rademacher series
摘要：
Let {cn}∞ n=1 be a sequence of complex numbers. In this paper we answer when the range of Σ ∞ n=1 ±cn is dense or equal to the complex plane. Some examples are given to explain our results. As its application, we calculate the Hausdorff dimension of the level sets of a Rademacher series with complex coefficients.
作者机构：
[Liu, Chuntai] Wuhan Polytech Univ, Sch Math & Comp Sci, Wuhan 430023, Peoples R China.;[Li, Haixiong] Hubei Univ Educ, Sch Math & Stat, Wuhan 430205, Peoples R China.
通讯机构：
[Liu, Chuntai] W;Wuhan Polytech Univ, Sch Math & Comp Sci, Wuhan 430023, Peoples R China.
关键词：
Hausdorff dimension;Catalan numbers;local level sets;Moran set;symbolic space
摘要：
AbstractTakagi’s function is a continuous nowhere-differentiable function constructed by Takagi in 1903. In this paper, we study the local level sets of Takagi’s function. By using the tools of Moran sets and symbolic spaces, we establish the Hausdorff dimension of local level sets for x∈[0,1]\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$x\in [0,1]$$\end{document} and obtain the range of the Hausdorff dimension. In addition, we derive a lower bound for the Hausdorff dimension of the set of points x\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$x$$\end{document} whose local level set has Hausdorff dimension s\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$s$$\end{document}.
作者机构：
[刘春苔] School of Mathematics and Computer Science, Wuhan Polytechnic University
关键词：
数学教学;教学方法;课堂教学;微分方程
摘要：
The discrete frame and the continuous frame in a Hilbert space are discussed. By the tool excess of a sequence, a sufficient and necessary condition is presented under which a discrete frame is equivalent to a continuous frame with respect to the whole natural numbers. with a positive Borel measure. And some examples are given to illuminate the condition.
