Symmetric fixed points of a smoothing transformation

2003 ◽  
Vol 35 (02) ◽  
pp. 377-394 ◽  
Author(s):  
Amke Caliebe
Keyword(s):  
Fixed Point ◽  
Fixed Points ◽  
Positive Solutions ◽  
Special Class ◽  
Infinitely Divisible Distributions ◽  
Infinitely Divisible ◽  
Existence Of Positive Solutions ◽  
Fixed Point Equation ◽  
Point Equation ◽  
Smoothing Transformation

LetT= (T1,T2,…) be a sequence of real random variables with ∑j=1∞1|Tj|>0< ∞ almost surely. We consider the following equation for distributions μ:W≅ ∑j=1∞TjWj, whereW,W1,W2,… have distribution μ andT,W1,W2,… are independent. We show that the representation of general solutions is a mixture of certain infinitely divisible distributions. This result can be applied to investigate the existence of symmetric solutions forTj≥ 0: essentially under the condition that E ∑j=1∞Tj2log+Tj2< ∞, the existence of nontrivial symmetric solutions is exactly determined, revealing a connection with the existence of positive solutions of a related fixed-point equation. Furthermore, we derive results about a special class of canonical symmetric solutions including statements about Lebesgue density and moments.

Symmetric fixed points of a smoothing transformation

2003 ◽  
Vol 35 (2) ◽  
pp. 377-394 ◽  
Author(s):  
Amke Caliebe
Keyword(s):  
Fixed Point ◽  
Fixed Points ◽  
Positive Solutions ◽  
Special Class ◽  
Infinitely Divisible Distributions ◽  
Infinitely Divisible ◽  
Existence Of Positive Solutions ◽  
Fixed Point Equation ◽  
Point Equation ◽  
Smoothing Transformation

LetT= (T1,T2,…) be a sequence of real random variables with ∑j=1∞1|Tj|>0< ∞ almost surely. We consider the following equation for distributions μ:W≅ ∑j=1∞TjWj, whereW,W1,W2,… have distribution μ andT,W1,W2,… are independent. We show that the representation of general solutions is a mixture of certain infinitely divisible distributions. This result can be applied to investigate the existence of symmetric solutions forTj≥ 0: essentially under the condition that E ∑j=1∞Tj2log+Tj2< ∞, the existence of nontrivial symmetric solutions is exactly determined, revealing a connection with the existence of positive solutions of a related fixed-point equation. Furthermore, we derive results about a special class of canonical symmetric solutions including statements about Lebesgue density and moments.

scholarly journals Existence of Positive Solutions for a Kind of Fractional Boundary Value Problems

2014 ◽  
Vol 2014 ◽  
pp. 1-8
Author(s):  
Hongjie Liu ◽  
Xiao Fu ◽  
Liangping Qi
Keyword(s):  
Fixed Point ◽  
Green Function ◽  
Fixed Point Theorem ◽  
Boundary Value Problem ◽  
Positive Solutions ◽  
Boundary Value ◽  
Fractional Boundary Value Problem ◽  
Existence Of Positive Solutions ◽  
Liouville Fractional Derivative ◽  
Fractional Boundary Value Problems

We are concerned with the following nonlinear three-point fractional boundary value problem:D0+αut+λatft,ut=0,0<t<1,u0=0, andu1=βuη, where1<α≤2,0<β<1,0<η<1,D0+αis the standard Riemann-Liouville fractional derivative,at>0is continuous for0≤t≤1, andf≥0is continuous on0,1×0,∞. By using Krasnoesel'skii's fixed-point theorem and the corresponding Green function, we obtain some results for the existence of positive solutions. At the end of this paper, we give an example to illustrate our main results.

scholarly journals Existence of Positive Solutions for Fractional Differential Equation with Nonlocal Boundary Condition

2011 ◽  
Vol 2011 ◽  
pp. 1-10 ◽  
Author(s):  
Hongliang Gao ◽  
Xiaoling Han
Keyword(s):  
Differential Equation ◽  
Boundary Condition ◽  
Fixed Point ◽  
Fractional Differential Equation ◽  
Positive Solutions ◽  
Nonlocal Boundary Condition ◽  
Nonlocal Boundary ◽  
Existence Of Positive Solutions ◽  
Fractional Differential ◽  
The Fixed Point Theorem

By using the fixed point theorem, existence of positive solutions for fractional differential equation with nonlocal boundary conditionD0+αu(t)+a(t)f(t,u(t))=0,0<t<1,u(0)=0,u(1)=∑i=1∞αiu(ξi)is considered, where1<α≤2is a real number,D0+αis the standard Riemann-Liouville differentiation, andξi∈(0,1),  αi∈[0,∞)with∑i=1∞αiξiα-1<1,a(t)∈C([0,1],[0,∞)),  f(t,u)∈C([0,1]×[0,∞),[0,∞)).

Analysis of Crossword Puzzle Difficulty Using a Random Graph Process

2015 ◽  
Author(s):  
John K. McSweeney
Keyword(s):  
New York ◽  
Stochastic Process ◽  
Fixed Point ◽  
Network Structure ◽  
New York Times ◽  
Final Solution ◽  
Crossword Puzzle ◽  
Fixed Point Equation ◽  
Point Equation ◽  
Network Properties

This chapter quantifies the dynamics of a crossword puzzle by using a network structure to model it. Specifically, the chapter determines how the interaction between the structure of cells in the puzzle and the difficulty of the clues affects the puzzle's solvability. It first builds an iterative stochastic process that exactly describes the solution and obtains its deterministic approximation, which gives a very simple fixed-point equation to solve for the final solution proportion. The chapter then shows via simulation on actual crosswords from the Sunday edition of The New York Times that certain network properties inherent to actual crossword networks are important predictors of the final solution size of the puzzle.

A Characterization of the Compound-Exponential Type Distributions

2011 ◽  
Vol 54 (3) ◽  
pp. 464-471
Author(s):  
Tea-Yuan Hwang ◽  
Chin-Yuan Hu
Keyword(s):  
Fixed Point ◽  
Exponential Type ◽  
Existence And Uniqueness ◽  
Fixed Point Equation ◽  
Point Equation

AbstractIn this paper, a fixed point equation of the compound-exponential type distributions is derived, and under some regular conditions, both the existence and uniqueness of this fixed point equation are investigated. A question posed by Pitman and Yor can be partially answered by using our approach.

scholarly journals A conformal fixed-point equation for the effective average action

2019 ◽  
Vol 34 (05) ◽  
pp. 1950027 ◽  
Author(s):  
Oliver J. Rosten
Keyword(s):  
Fixed Point ◽  
Conformal Invariance ◽  
Legendre Transform ◽  
Fixed Point Equation ◽  
Average Action ◽  
Point Equation

A Legendre transform of the recently discovered conformal fixed-point equation is constructed, providing an unintegrated equation encoding full conformal invariance within the framework of the effective average action.

scholarly journals Modified SOR-Like Method for Absolute Value Equations

2020 ◽  
Vol 2020 ◽  
pp. 1-6
Author(s):  
Cui-Xia Li ◽  
Shi-Liang Wu
Keyword(s):  
Fixed Point ◽  
Nonlinear Equation ◽  
Computational Efficiency ◽  
Numerical Experiments ◽  
Absolute Value Equations ◽  
Convergence Conditions ◽  
Absolute Value ◽  
Fixed Point Equation ◽  
Point Equation ◽  
Better Than

In this paper, based on the work of Ke and Ma, a modified SOR-like method is presented to solve the absolute value equations (AVE), which is gained by equivalently expressing the implicit fixed-point equation form of the AVE as a two-by-two block nonlinear equation. Under certain conditions, the convergence conditions for the modified SOR-like method are presented. The computational efficiency of the modified SOR-like method is better than that of the SOR-like method by some numerical experiments.

scholarly journals Positive Solutions for Discontinuous Systems via a Multivalued Vector Version of Krasnosel’skiĭ’s Fixed Point Theorem in Cones

Mathematics ◽  
2019 ◽  
Vol 7 (5) ◽  
pp. 451
Author(s):  
Rodrigo López Pouso ◽  
Radu Precup ◽  
Jorge Rodríguez-López
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Integral Operator ◽  
Positive Solutions ◽  
Point Theorem ◽  
Discontinuous Systems ◽  
Second Order Differential Equations ◽  
Existence Of Positive Solutions ◽  
Vector Version ◽  
Krasnosel’Skii’S Fixed Point Theorem

We establish the existence of positive solutions for systems of second–order differential equations with discontinuous nonlinear terms. To this aim, we give a multivalued vector version of Krasnosel’skiĭ’s fixed point theorem in cones which we apply to a regularization of the discontinuous integral operator associated to the differential system. We include several examples to illustrate our theory.

scholarly journals A Relaxed Self-Adaptive Projection Algorithm for Solving the Multiple-Sets Split Equality Problem

2020 ◽  
Vol 2020 ◽  
pp. 1-12
Author(s):  
Haitao Che ◽  
Haibin Chen
Keyword(s):  
Fixed Point ◽  
Equation System ◽  
Projection Algorithm ◽  
Step Size ◽  
Split Equality Problem ◽  
Fixed Point Equation ◽  
Point Equation ◽  
Equality Problem ◽  
Adaptive Step ◽  
Self Adaptive

In this article, we introduce a relaxed self-adaptive projection algorithm for solving the multiple-sets split equality problem. Firstly, we transfer the original problem to the constrained multiple-sets split equality problem and a fixed point equation system is established. Then, we show the equivalence of the constrained multiple-sets split equality problem and the fixed point equation system. Secondly, we present a relaxed self-adaptive projection algorithm for the fixed point equation system. The advantage of the self-adaptive step size is that it could be obtained directly from the iterative procedure. Furthermore, we prove the convergence of the proposed algorithm. Finally, several numerical results are shown to confirm the feasibility and efficiency of the proposed algorithm.

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