作者机构：
[Deng, Guotai] Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Peoples R China.;[Deng, Guotai] Cent China Normal Univ, Hubei Key Lab Math Sci, 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 & Stat, Key Lab High Performance Comp & Stochast Informat, Minist Educ China, Changsha 410081, Hunan, Peoples R China.;[Ngai, Sze-Man] Georgia Southern Univ, Dept Math Sci, Statesboro, GA 30460 USA.
通讯机构：
[Ngai, SM ] H;Hunan Normal Univ, Coll Math & Stat, Key Lab High Performance Comp & Stochast Informat, Minist Educ China, Changsha 410081, Hunan, Peoples R China.;Georgia Southern Univ, Dept Math Sci, Statesboro, GA 30460 USA.
期刊：
Advances in Mathematics,2022年410:108716 ISSN：0001-8708
通讯作者：
Chuntai Liu<&wdkj&>Sze-Man Ngai
作者机构：
[Deng, Guotai] Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Peoples R China.;[Deng, Guotai] Cent China Normal Univ, Hubei Key Lab Math Sci, 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 & Stat, Key Lab High Performance Comp & Stochast Informat, HPCSIP,Minist Educ China, Changsha 410081, Hunan, Peoples R China.;[Ngai, Sze -Man] Georgia Southern Univ, Dept Math Sci, Statesboro, GA 30460 USA.
通讯机构：
[Chuntai Liu] S;[Sze-Man Ngai] K;School of Mathematics and Computer Science, Wuhan Polytechnic University, Wuhan 430023, PR China<&wdkj&>Key Laboratory of High Performance Computing and Stochastic Information Processing (HPCSIP) (Ministry of Education of China), College of Mathematics and Statistics, Hunan Normal University, Changsha, Hunan 410081, PR China<&wdkj&>Department of Mathematical Sciences, Georgia Southern University, Statesboro, GA 30460-8093, USA
关键词：
Ball-like tile;Brouwer's invariance of domain theorem;Horizontal distance;Self-affine tile;Tame ball
摘要：
We study a family of self-affine tiles in Rd (d >= 2) with noncollinear digit sets, which naturally generalizes a two-dimensional class studied originally by Deng and Lau and its extension to R3 by the authors. By using Brouwer's invariance of domain theorem, along with a tool which we call horizontal distance, we obtain necessary and sufficient conditions for the tiles to be d-dimensional tame balls. This answers positively the conjecture in an earlier paper by the authors stating that a member in a certain class of self-affine tiles is homeomorphic to a d-dimensional ball if and only if its interior is connected. (c) 2022 Elsevier Inc. All rights reserved.
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.
摘要：
<jats:title>Abstract</jats:title><jats:p>We construct a family of self-affine tiles in <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0008439519000237_inline1" /><jats:tex-math>$\mathbb{R}^{d}$</jats:tex-math></jats:alternatives></jats:inline-formula> (<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0008439519000237_inline2" /><jats:tex-math>$d\geqslant 2$</jats:tex-math></jats:alternatives></jats:inline-formula>) with noncollinear digit sets, which naturally generalizes a class studied originally by Q.-R. Deng and K.-S. Lau in <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0008439519000237_inline3" /><jats:tex-math>$\mathbb{R}^{2}$</jats:tex-math></jats:alternatives></jats:inline-formula>, and its extension to <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0008439519000237_inline4" /><jats:tex-math>$\mathbb{R}^{3}$</jats:tex-math></jats:alternatives></jats:inline-formula> by the authors. We obtain necessary and sufficient conditions for the tiles to be connected and for their interiors to be contractible.</jats:p>
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
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 {c(n)}(n=1)(infinity), be a sequence of complex numbers. In this paper we answer when the range of Sigma(infinity)(n=1)+/- c(n) 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：
[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.
