A study of a special kind of N-fixed point equation system and applications

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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Analysis of Crossword Puzzle Difficulty Using a Random Graph Process

The Mathematics of Various Entertaining Subjects ◽

10.23943/princeton/9780691164038.003.0008 ◽

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.

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A Characterization of the Compound-Exponential Type Distributions

Canadian Mathematical Bulletin ◽

10.4153/cmb-2011-086-7 ◽

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.

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A conformal fixed-point equation for the effective average action

International Journal of Modern Physics A ◽

10.1142/s0217751x19500271 ◽

2019 ◽

Vol 34 (05) ◽

pp. 1950027 ◽

Cited By ~ 5

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.

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Modified SOR-Like Method for Absolute Value Equations

Mathematical Problems in Engineering ◽

10.1155/2020/9231639 ◽

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.

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On exact sampling of stochastic perpetuities

Journal of Applied Probability ◽

10.1017/s0021900200099204 ◽

2011 ◽

Vol 48 (A) ◽

pp. 165-182 ◽

Cited By ~ 6

Author(s):

Jose H. Blanchet ◽

Karl Sigman

Keyword(s):

Fixed Point ◽

Net Present Value ◽

Positive Density ◽

Simulation Algorithm ◽

Present Value ◽

The Past ◽

Coupling From The Past ◽

Fixed Point Equation ◽

Point Equation ◽

The Right

A stochastic perpetuity takes the formD∞=∑n=0∞exp(Y1+⋯+Yn)Bn, whereYn:n≥0) and (Bn:n≥0) are two independent sequences of independent and identically distributed random variables (RVs). This is an expression for the stationary distribution of the Markov chain defined recursively byDn+1=AnDn+Bn,n≥0, whereAn=eYn;D∞then satisfies the stochastic fixed-point equationD∞D̳AD∞+B, whereAandBare independent copies of theAnandBn(and independent ofD∞on the right-hand side). In our framework, the quantityBn, which represents a random reward at timen, is assumed to be positive, unbounded with EBnp<∞ for somep>0, and have a suitably regular continuous positive density. The quantityYnis assumed to be light tailed and represents a discount rate from timenton-1. The RVD∞then represents the net present value, in a stochastic economic environment, of an infinite stream of stochastic rewards. We provide an exact simulation algorithm for generating samples ofD∞. Our method is a variation ofdominated coupling from the pastand it involves constructing a sequence of dominating processes.

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Generalized Ulam-Hyers stability and well-posedness for fixed point equation via α-admissibility

Journal of Inequalities and Applications ◽

10.1186/1029-242x-2014-418 ◽

2014 ◽

Vol 2014 (1) ◽

Cited By ~ 2

Author(s):

Supak Phiangsungnoen ◽

Poom Kumam

Keyword(s):

Fixed Point ◽

Well Posedness ◽

Fixed Point Equation ◽

Point Equation

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On exact sampling of stochastic perpetuities

Journal of Applied Probability ◽

10.1239/jap/1318940463 ◽

2011 ◽

Vol 48 (A) ◽

pp. 165-182 ◽

Cited By ~ 9

Author(s):

Jose H. Blanchet ◽

Karl Sigman

Keyword(s):

Fixed Point ◽

Net Present Value ◽

Positive Density ◽

Simulation Algorithm ◽

Present Value ◽

The Past ◽

Coupling From The Past ◽

Fixed Point Equation ◽

Point Equation ◽

The Right

A stochastic perpetuity takes the form D∞=∑n=0∞ exp(Y1+⋯+Yn)Bn, where Yn:n≥0) and (Bn:n≥0) are two independent sequences of independent and identically distributed random variables (RVs). This is an expression for the stationary distribution of the Markov chain defined recursively by Dn+1=AnDn+Bn, n≥0, where An=eYn; D∞ then satisfies the stochastic fixed-point equation D∞D̳AD∞+B, where A and B are independent copies of the An and Bn (and independent of D∞ on the right-hand side). In our framework, the quantity Bn, which represents a random reward at time n, is assumed to be positive, unbounded with EBnp <∞ for some p>0, and have a suitably regular continuous positive density. The quantity Yn is assumed to be light tailed and represents a discount rate from time n to n-1. The RV D∞ then represents the net present value, in a stochastic economic environment, of an infinite stream of stochastic rewards. We provide an exact simulation algorithm for generating samples of D∞. Our method is a variation of dominated coupling from the past and it involves constructing a sequence of dominating processes.

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A Unified Proximity Algorithm with Adaptive Penalty for Nuclear Norm Minimization

Symmetry ◽

10.3390/sym11101277 ◽

2019 ◽

Vol 11 (10) ◽

pp. 1277

Author(s):

Wenyu Hu ◽

Weidong Zheng ◽

Gaohang Yu

Keyword(s):

Singular Values ◽

Equation System ◽

Linear Constraints ◽

Nuclear Norm ◽

Nuclear Norm Minimization ◽

Norm Minimization ◽

Alternating Direction ◽

Adaptive Penalty ◽

Fixed Point Equation ◽

Point Equation

The nuclear norm minimization (NNM) problem is to recover a matrix that minimizes the sum of its singular values and satisfies some linear constraints simultaneously. The alternating direction method (ADM) has been used to solve this problem recently. However, the subproblems in ADM are usually not easily solvable when the linear mappings in the constraints are not identities. In this paper, we propose a proximity algorithm with adaptive penalty (PA-AP). First, we formulate the nuclear norm minimization problems into a unified model. To solve this model, we improve the ADM by adding a proximal term to the subproblems that are difficult to solve. An adaptive tactic on the proximity parameters is also put forward for acceleration. By employing subdifferentials and proximity operators, an equivalent fixed-point equation system is constructed, and we use this system to further prove the convergence of the proposed algorithm under certain conditions, e.g., the precondition matrix is symmetric positive definite. Finally, experimental results and comparisons with state-of-the-art methods, e.g., ADM, IADM-CG and IADM-BB, show that the proposed algorithm is effective.

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Symmetric fixed points of a smoothing transformation

Advances in Applied Probability ◽

10.1017/s0001867800012301 ◽

2003 ◽

Vol 35 (02) ◽

pp. 377-394 ◽

Cited By ~ 2

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.

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