scholarly journals On Absolute Continuity of Conjugations between Circle Maps with Break Points

2012 ◽  
Vol 2012 ◽  
pp. 1-12 ◽  
Author(s):  
Habibulla Akhadkulov ◽  
Mohd Salmi Md Noorani
Keyword(s):  
Absolute Continuity ◽  
Irrational Rotation ◽  
Piecewise Smooth ◽  
Necessary Condition ◽  
Circle Maps ◽  
Rotation Numbers ◽  
Break Points ◽  
The Absolute ◽  
Sufficient And Necessary Condition

LetT1andT2be piecewise smooth circle homeomorphisms with break points and identical irrational rotation numbers. We provide one sufficient and necessary condition for the absolute continuity of conjugation map betweenT1andT2.

Conjugation between circle maps with several break points

2015 ◽  
Vol 36 (8) ◽  
pp. 2351-2383 ◽  
Author(s):  
ABDELHAMID ADOUANI
Keyword(s):  
Invariant Measure ◽  
Lebesgue Measure ◽  
Break Point ◽  
Irrational Rotation ◽  
Singular Function ◽  
Absolutely Continuous ◽  
Circle Maps ◽  
Almost Everywhere ◽  
Rotation Numbers ◽  
Break Points

Let$f$and$g$be two class$P$-homeomorphisms of the circle$S^{1}$with break point singularities, which are differentiable maps except at some singular points where the derivative has a jump. Assume that$f$and$g$have irrational rotation numbers and the derivatives$\text{Df}$and$\text{Dg}$are absolutely continuous on every continuity interval of$\text{Df}$and$\text{Dg}$, respectively. We prove that if the product of the$f$-jumps along all break points of$f$is distinct from that of$g$then the homeomorphism$h$conjugating$f$and$g$is a singular function, i.e. it is continuous on$S^{1}$, but$\text{Dh}(x)=0$ almost everywhere with respect to the Lebesgue measure. This result generalizes previous results for one and two break points obtained by Dzhalilov, Akin and Temir, and Akhadkulov, Dzhalilov and Mayer. As a consequence, we get in particular Dzhalilov–Mayer–Safarov’s theorem: if the product of the$f$-jumps along all break points of$f$is distinct from$1$, then the invariant measure$\unicode[STIX]{x1D707}_{f}$is singular with respect to the Lebesgue measure.

ONE SUFFICIENT AND NECESSARY CONDITION ON BALANCED BOOLEAN FUNCTIONS WITH σf = 22n + 2n+3(n ≥ 3)

2014 ◽  
Vol 25 (03) ◽  
pp. 343-353 ◽  
Author(s):  
YU ZHOU ◽  
LIN WANG ◽  
WEIQIONG WANG ◽  
XINFENG DONG ◽  
XIAONI DU
Keyword(s):  
Lower Bound ◽  
Boolean Function ◽  
Lower Bounds ◽  
Boolean Functions ◽  
Sum Of Squares ◽  
Necessary Condition ◽  
Resilient Function ◽  
The Absolute ◽  
Global Avalanche Characteristics ◽  
Sufficient And Necessary Condition

The Global Avalanche Characteristics (including the sum-of-squares indicator and the absolute indicator) measure the overall avalanche characteristics of a cryptographic Boolean function. Son et al. (1998) gave the lower bound on the sum-of-squares indicator for a balanced Boolean function. In this paper, we give a sufficient and necessary condition on a balanced Boolean function reaching the lower bound on the sum-of-squares indicator. We also analyze whether these balanced Boolean functions exist, and if they reach the lower bounds on the sum-of-squares indicator or not. Our result implies that there does not exist a balanced Boolean function with n-variable for odd n(n ≥ 5). We conclude that there does not exist a m(m ≥ 1)-resilient function reaching the lower bound on the sum-of-squares indicator with n-variable for n ≥ 7.

scholarly journals A Variant of the Necessary Condition for the Absolute Continuity of Symmetric Multivariate Mixture

Mathematics ◽  
2021 ◽  
Vol 9 (13) ◽  
pp. 1505
Author(s):  
Evgeniy Anatolievich Savinov
Keyword(s):  
Absolute Continuity ◽  
Joint Distribution ◽  
Sufficient Conditions ◽  
Random Variables ◽  
Independent Random Variables ◽  
Necessary Condition ◽  
Conditional Distributions ◽  
One Dimensional ◽  
The Absolute

Sufficient conditions are given under which the absolute continuity of the joint distribution of conditionally independent random variables can be violated. It is shown that in the case of a dimension n>1 this occurs for a sufficiently large number of discontinuity points of one-dimensional conditional distributions.

scholarly journals Cauchy matrix and Liouville formula of quaternion impulsive dynamic equations on time scales

2020 ◽  
Vol 18 (1) ◽  
pp. 353-377 ◽  
Author(s):  
Zhien Li ◽  
Chao Wang
Keyword(s):  
Time Scales ◽  
Quaternion Algebra ◽  
Matrix Exponential ◽  
Dynamic Equations ◽  
Necessary Condition ◽  
Quaternion Matrix ◽  
Exponential Functions ◽  
Cauchy Matrix ◽  
Adjoint Matrix ◽  
Sufficient And Necessary Condition

Abstract In this study, we obtain the scalar and matrix exponential functions through a series of quaternion-valued functions on time scales. A sufficient and necessary condition is established to guarantee that the induced matrix is real-valued for the complex adjoint matrix of a quaternion matrix. Moreover, the Cauchy matrices and Liouville formulas for the quaternion homogeneous and nonhomogeneous impulsive dynamic equations are given and proved. Based on it, the existence, uniqueness, and expressions of their solutions are also obtained, including their scalar and matrix forms. Since the quaternion algebra is noncommutative, many concepts and properties of the non-quaternion impulsive dynamic equations are ineffective, we provide several examples and counterexamples on various time scales to illustrate the effectiveness of our results.

scholarly journals Sufficient and necessary conditions for oscillation of linear fractional-order delay differential equations

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Qiong Meng ◽  
Zhen Jin ◽  
Guirong Liu
Keyword(s):  
Differential Equation ◽  
Fractional Order ◽  
Sufficient Conditions ◽  
Necessary Condition ◽  
Multiple Delays ◽  
All Solutions ◽  
Sufficient And Necessary Conditions ◽  
Delay Differential ◽  
T Tau ◽  
Sufficient And Necessary Condition

AbstractThis paper studies the linear fractional-order delay differential equation $$ {}^{C}D^{\alpha }_{-}x(t)-px(t-\tau )= 0, $$ D − α C x ( t ) − p x ( t − τ ) = 0 , where $0<\alpha =\frac{\text{odd integer}}{\text{odd integer}}<1$ 0 < α = odd integer odd integer < 1 , $p, \tau >0$ p , τ > 0 , ${}^{C}D_{-}^{\alpha }x(t)=-\Gamma ^{-1}(1-\alpha )\int _{t}^{\infty }(s-t)^{- \alpha }x'(s)\,ds$ D − α C x ( t ) = − Γ − 1 ( 1 − α ) ∫ t ∞ ( s − t ) − α x ′ ( s ) d s . We obtain the conclusion that $$ p^{1/\alpha } \tau >\alpha /e $$ p 1 / α τ > α / e is a sufficient and necessary condition of the oscillations for all solutions of Eq. (*). At the same time, some sufficient conditions are obtained for the oscillations of multiple delays linear fractional differential equation. Several examples are given to illustrate our theorems.

scholarly journals Research on the Relationship between Shared Leadership and Individual Creativity-Qualitative Comparative Analysis on the Basis of Clear Set

2021 ◽  
Vol 13 (10) ◽  
pp. 5445
Author(s):  
Muyun Sun ◽  
Jigan Wang ◽  
Ting Wen
Keyword(s):  
Comparative Analysis ◽  
Environmental Factors ◽  
Shared Leadership ◽  
Management Practices ◽  
Qualitative Comparative Analysis ◽  
Necessary Condition ◽  
The Individual ◽  
The Impact ◽  
The Relationship ◽  
Sufficient And Necessary Condition

Creativity is the key to obtaining and maintaining competitiveness of modern organizations, and it has attracted much attention from academic circles and management practices. Shared leadership is believed to effectively influence team output. However, research on the impact of individual creativity is still in its infancy. This study adopts the qualitative comparative analysis method, taking 1584 individuals as the research objects, underpinned by a questionnaire-based survey. It investigates the influence of the team’s shared leadership network elements and organizational environmental factors on the individual creativity. We have found that there are six combination of conditions of shared leadership and organizational environmental factors constituting sufficient combination of conditions to increase or decrease individual creativity. Moreover, we have noticed that the low network density of shared leadership is a sufficient and necessary condition of reducing individual creativity. Our results also provide management suggestions for practical activities during the team management.

scholarly journals A nonlocal game for witnessing quantum networks

2019 ◽  
Vol 5 (1) ◽  
Author(s):  
Ming-Xing Luo
Keyword(s):  
Computational Complexity ◽  
Main Idea ◽  
Bell Inequalities ◽  
Theoretical Computer Science ◽  
Graph Representation ◽  
Necessary Condition ◽  
Quantum Networks ◽  
Theoretical Computer ◽  
Sufficient And Necessary Condition ◽  
Unified Method

Abstract Nonlocal game as a witness of the nonlocality of entanglement is of fundamental importance in various fields. The well-known nonlocal games or equivalent linear Bell inequalities are only useful for Bell networks consisting of single entanglement. Our goal in this paper is to propose a unified method for constructing cooperating games in network scenarios. We propose an efficient method to construct multipartite nonlocal games from any graphs. The main idea is the graph representation of entanglement-based quantum networks. We further specify these graphic games with quantum advantages by providing a simple sufficient and necessary condition. The graphic games imply a linear Bell testing of the nonlocality of general quantum networks consisting of EPR states. It also allows generating new instances going beyond CHSH game. These results have interesting applications in quantum networks, Bell theory, computational complexity, and theoretical computer science.

New characterizations of g-frames and g-Riesz bases

2018 ◽  
Vol 16 (06) ◽  
pp. 1850053 ◽  
Author(s):  
Dongwei Li ◽  
Jinsong Leng ◽  
Tingzhu Huang
Keyword(s):  
Riesz Basis ◽  
Riesz Bases ◽  
Gram Matrix ◽  
Necessary Condition ◽  
Topological Properties ◽  
Bessel Sequence ◽  
Bessel Sequences ◽  
Sufficient And Necessary Condition

In this paper, we give some new characterizations of g-frames, g-Bessel sequences and g-Riesz bases from their topological properties. By using the Gram matrix associated with the g-Bessel sequence, we present a sufficient and necessary condition under which the sequence is a g-Bessel sequence (or g-Riesz basis). Finally, we consider the excess of a g-frame and obtain some new results.

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