scholarly journals The size of random fragmentation intervals

2008 ◽  
Vol DMTCS Proceedings vol. AI,... (Proceedings) ◽  
Author(s):  
Rafik Aguech
Keyword(s):  
Fixed Point ◽  
Explicit Solution ◽  
Fragment Size ◽  
Fragmentation Process ◽  
Contraction Method ◽  
Point Solution ◽  
Fixed Point Equation ◽  
International Audience ◽  
Point Equation ◽  
Wasserstein Space

International audience Two processes of random fragmentation of an interval are investigated. For each of them, there is a splitting probability at each step of the fragmentation process whose overall effect is to stabilize the global number of splitting events. More precisely, we consider two models. In the first model, the fragmentation stops which a probability $p$ witch can not depend on the fragment size. The number of stable fragments with sizes less than a given $t \geq 0$, denoted by $K(t)$, is introduced and studied. In the second one the probability to split a fragment of size $x$ is $p(x)=1-e^{-x}$. For this model we utilize the contraction method to show that the distribution of a suitably normalized version of the number of stable fragments converges in law. It's shown that the limit is the fixed-point solution (in the Wasserstein space) to a distributional equation. An explicit solution to the fixed-point equation is easily verified to be Gaussian.

scholarly journals A Note on the Approximation of Perpetuities

2007 ◽  
Vol DMTCS Proceedings vol. AH,... (Proceedings) ◽  
Author(s):  
Margarete Knape ◽  
Ralph Neininger
Keyword(s):  
Fixed Point ◽  
Error Bounds ◽  
Computer Experiments ◽  
Distribution Functions ◽  
Selection Algorithm ◽  
Fixed Point Equation ◽  
International Audience ◽  
Point Equation ◽  
Approximate Distribution

International audience We propose and analyze an algorithm to approximate distribution functions and densities of perpetuities. Our algorithm refines an earlier approach based on iterating discretized versions of the fixed point equation that defines the perpetuity. We significantly reduce the complexity of the earlier algorithm. Also one particular perpetuity arising in the analysis of the selection algorithm Quickselect is studied in more detail. Our approach works well for distribution functions. For densities we have weaker error bounds although computer experiments indicate that densities can also be well approximated.

scholarly journals Distribution of inter-node distances in digital trees

2005 ◽  
Vol DMTCS Proceedings vol. AD,... (Proceedings) ◽  
Author(s):  
Rafik Aguech ◽  
Nabil Lasmar ◽  
Hosam Mahmoud
Keyword(s):  
Fixed Point ◽  
Mellin Transform ◽  
Search Trees ◽  
Digital Trees ◽  
Digital Search Trees ◽  
Point Solution ◽  
Fixed Point Equation ◽  
International Audience ◽  
Bounded Variance ◽  
Digital Search

International audience We investigate distances between pairs of nodes in digital trees (digital search trees (DST), and tries). By analytic techniques, such as the Mellin Transform and poissonization, we describe a program to determine the moments of these distances. The program is illustrated on the mean and variance. One encounters delayed Mellin transform equations, which we solve by inspection. Interestingly, the unbiased case gives a bounded variance, whereas the biased case gives a variance growing with the number of keys. It is therefore possible in the biased case to show that an appropriately normalized version of the distance converges to a limit. The complexity of moment calculation increases substantially with each higher moment; A shortcut to the limit is needed via a method that avoids the computation of all moments. Toward this end, we utilize the contraction method to show that in biased digital search trees the distribution of a suitably normalized version of the distances approaches a limit that is the fixed-point solution (in the Wasserstein space) of a distributional equation. An explicit solution to the fixed-point equation is readily demonstrated to be Gaussian.

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 Limit distribution of distances in biased random tries

2006 ◽  
Vol 43 (02) ◽  
pp. 377-390 ◽  
Author(s):  
Rafik Aguech ◽  
Nabil Lasmar ◽  
Hosam Mahmoud
Keyword(s):  
Fixed Point ◽  
Metric Space ◽  
Limit Distribution ◽  
Analytical Techniques ◽  
Random Variable ◽  
Wasserstein Metric ◽  
Contraction Method ◽  
Point Solution ◽  
Mean And Variance

Thetrieis a sort of digital tree. Ideally, to achieve balance, the trie should grow from an unbiased source generating keys of bits with equal likelihoods. In practice, the lack of bias is not always guaranteed. We investigate the distance between randomly selected pairs of nodes among the keys in a biased trie. This research complements that of Christophi and Mahmoud (2005); however, the results and some of the methodology are strikingly different. Analytical techniques are still useful for moments calculation. Both mean and variance are of polynomial order. It is demonstrated that the standardized distance approaches a normal limiting random variable. This is proved by the contraction method, whereby the limit distribution is shown to approach the fixed-point solution of a distributional equation in the Wasserstein metric space.

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 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.

scholarly journals On exact sampling of stochastic perpetuities

2011 ◽  
Vol 48 (A) ◽  
pp. 165-182 ◽  
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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