SPECTRAL PROPERTY OF CERTAIN MORAN MEASURES WITH THREE-ELEMENT DIGIT SETS

Fractals ◽  
2019 ◽  
Vol 27 (04) ◽  
pp. 1950068 ◽  
Author(s):  
XIU-QUN FU ◽  
XIN-HAN DONG ◽  
ZONG-SHENG LIU ◽  
ZHI-YONG WANG
Keyword(s):  
Spectral Property ◽  
Orthonormal Basis ◽  
Probability Measures ◽  
Finite Support ◽  
Equal Distribution ◽  
Infinite Convolution

Let [Formula: see text],[Formula: see text][Formula: see text],[Formula: see text][Formula: see text], satisfy [Formula: see text]. Let [Formula: see text] be the infinite convolution of probability measures with finite support and equal distribution. In this paper, we show that if [Formula: see text], then there exists a discrete set [Formula: see text] such that [Formula: see text] is an orthonormal basis for [Formula: see text].

On the spectrality of self-affine measures with four digits on ℝ2

2021 ◽  
pp. 2150004
Author(s):  
Ming-Liang Chen ◽  
Zhi-Hui Yan
Keyword(s):  
Spectral Property ◽  
Orthonormal Basis ◽  
Zero Set ◽  
Real Matrix ◽  
Exponential Functions ◽  
Bernoulli Convolution ◽  
Digit Set ◽  
Similar Characterization ◽  
Equivalent Characterization ◽  
Matrix Formula

In this paper, we study the spectral property of the self-affine measure [Formula: see text] generated by an expanding real matrix [Formula: see text] and the four-element digit set [Formula: see text]. We show that [Formula: see text] is a spectral measure, i.e. there exists a discrete set [Formula: see text] such that the collection of exponential functions [Formula: see text] forms an orthonormal basis for [Formula: see text], if and only if [Formula: see text] for some [Formula: see text]. A similar characterization for Bernoulli convolution is provided by Dai [X.-R. Dai, When does a Bernoulli convolution admit a spectrum? Adv. Math. 231(3) (2012) 1681–1693], over which [Formula: see text]. Furthermore, we provide an equivalent characterization for the maximal bi-zero set of [Formula: see text] by extending the concept of tree-mapping in [X.-R. Dai, X.-G. He and C. K. Lai, Spectral property of Cantor measures with consecutive digits, Adv. Math. 242 (2013) 187–208]. We also extend these results to the more general self-affine measures.

Spectrality of a Class of Moran Measures

2020 ◽  
Vol 63 (2) ◽  
pp. 366-381
Author(s):  
Ming-Liang Chen ◽  
Jing-Cheng Liu ◽  
Juan Su ◽  
Xiang-Yang Wang
Keyword(s):  
Probability Measure ◽  
Orthonormal Basis ◽  
Borel Probability Measure ◽  
Borel Probability ◽  
Positive Integers ◽  
Moran Measure ◽  
Infinite Convolution

AbstractLet $\{M_{n}\}_{n=1}^{\infty }$ be a sequence of expanding matrices with $M_{n}=\operatorname{diag}(p_{n},q_{n})$, and let $\{{\mathcal{D}}_{n}\}_{n=1}^{\infty }$ be a sequence of digit sets with ${\mathcal{D}}_{n}=\{(0,0)^{t},(a_{n},0)^{t},(0,b_{n})^{t},\pm (a_{n},b_{n})^{t}\}$, where $p_{n}$, $q_{n}$, $a_{n}$ and $b_{n}$ are positive integers for all $n\geqslant 1$. If $\sup _{n\geqslant 1}\{\frac{a_{n}}{p_{n}},\frac{b_{n}}{q_{n}}\}<\infty$, then the infinite convolution $\unicode[STIX]{x1D707}_{\{M_{n}\},\{{\mathcal{D}}_{n}\}}=\unicode[STIX]{x1D6FF}_{M_{1}^{-1}{\mathcal{D}}_{1}}\ast \unicode[STIX]{x1D6FF}_{(M_{1}M_{2})^{-1}{\mathcal{D}}_{2}}\ast \cdots \,$ is a Borel probability measure (Cantor–Dust–Moran measure). In this paper, we investigate whenever there exists a discrete set $\unicode[STIX]{x1D6EC}$ such that $\{e^{2\unicode[STIX]{x1D70B}i\langle \unicode[STIX]{x1D706},x\rangle }:\unicode[STIX]{x1D706}\in \unicode[STIX]{x1D6EC}\}$ is an orthonormal basis for $L^{2}(\unicode[STIX]{x1D707}_{\{M_{n}\},\{{\mathcal{D}}_{n}\}})$.

scholarly journals The Equivalence of Two Extremum Problems

1960 ◽  
Vol 12 ◽  
pp. 363-366 ◽  
Author(s):  
J. Kiefer ◽  
J. Wolfowitz
Keyword(s):  
Finite Number ◽  
Dimensional Euclidean Space ◽  
Probability Measures ◽  
Compact Set ◽  
Finite Support ◽  
Compact Topological Space ◽  
Extremum Problems ◽  
Linearly Independent ◽  
Borel Field ◽  
Image Position

Let f1 , …, fk be linearly independent real functions on a space X, such that the range R of (f1, …, fk) is a compact set in k dimensional Euclidean space. (This will happen, for example, if the fi are continuous and X is a compact topological space.) Let S be any Borel field of subsets of X which includes X and all sets which consist of a finite number of points, and let C = {ε} be any class of probability measures on S which includes all probability measures with finite support (that is, which assign probability one to a set consisting of a finite number of points), and which are such thatis defined. In all that follows we consider only probability measures ε which are in C.

Distribution of Aedes aegypti and Aedes albopictus in different eco zones of Thiruvananthapuram city with special reference to dengue viremia in humans

ENTOMON ◽  
2018 ◽  
Vol 43 (4) ◽  
pp. 223-230
Author(s):  
S. Sunil Kumar ◽  
D.A. Evans ◽  
K. Muthulakshmi ◽  
T. DilipKumar ◽  
R. Heera Pillai ◽  
...  
Keyword(s):  
Aedes Aegypti ◽  
Aedes Albopictus ◽  
Dengue Shock Syndrome ◽  
Single Step ◽  
Equal Distribution ◽  
Distribution Study ◽  
Urban Zone ◽  
High Prevalence ◽  
Vector Management ◽  
Efficient Vector

Mosquito index study of three ecologically different ecozones of the Thiruvananthapuram district, Kerala showed sharp difference on the proportionate distribution of Aedes aegypti and Aedes albopictus. Human dengue viremia (HDV) was very high in those ecozones where A.aegypti density was high and HDV was low where A.albopictus was high. In a coastal zone of Thiruvananthapuram city, A. aegypti was the most abundant vector and in a hilly, arid suburban zone, A.albopictus was the abundant vector. In the urban zone both species of mosquitoes showed equal distribution. Study on the circulating serotypes in the serum of HDV by Single step single tube Multiplex PCR showed all the four serotypes viz DENV1, DENV2, DENV3 and DENV4 in patients of Thiruvananthapuram city, which indicated the possibility of Dengue Shock Syndrome, unless there is efficient vector management. Among the four dengue serotypes, Type 1 was the most abundant virus. Abundance of microhabitats in Thiruvananthapuram city, which support A. aegypti may be the reason for high prevalence of dengue fever in the urban zone.

Global Justice in Lutheran Political Theology

2016 ◽  
Vol 3 (1) ◽  
pp. 45-58
Author(s):  
Carl-Henric Grenholm
Keyword(s):  
Political Philosophy ◽  
Global Justice ◽  
Political Theology ◽  
Critical Examination ◽  
Equal Distribution ◽  
Theory Of Justice ◽  
Sharp Distinction ◽  
Main Thesis ◽  
Theories Of Justice ◽  
Influential Theory

The purpose of this article is to examine the contributions that might be given by Lutheran political theology to the discourse on global justice. The article offers a critical examination of three different theories of global justice within political philosophy. Contractarian theories are criticized, and a thesis is that it is plausible to argue that justice can be understood as liberation from oppression. From this perspective the article gives an analysis of an influential theory of justice within Lutheran ethics. According to this theory justice is not an equal distribution but an arrangement where the subordinate respect the authority of those in power. This theory is related to a sharp distinction between law and gospel. The main thesis of the article is that Lutheran political theology should take a different approach if it aims to give a constructive contribution to theories of justice. This means that Lutheran ethics should not be based on Creation and reason alone – it should also be based on Christology and Eschatology.

LIFTING THE FUNCTOR ITO THE CATEGORY OF UNIFORM SPACES

2020 ◽  
Vol 4 (1) ◽  
pp. 29-39
Author(s):  
Dilrabo Eshkobilova ◽  
Keyword(s):  
Compact Support ◽  
Probability Measures ◽  
Uniform Spaces ◽  
Continuous Maps ◽  
Uniformly Continuous

Uniform properties of the functor Iof idempotent probability measures with compact support are studied. It is proved that this functor can be lifted to the category Unif of uniform spaces and uniformly continuous maps

Sign in / Sign up

Export Citation Format

Share Document