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Author: Christophe Garban
Publisher: Cambridge University Press
ISBN: 1107076439
Category : Computers
Languages : en
Pages : 223
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Book Description
This is the first book to cover the theory of noise sensitivity of Boolean functions with particular emphasis on critical percolation.
Author: Christophe Garban
Publisher: Cambridge University Press
ISBN: 1107076439
Category : Computers
Languages : en
Pages : 223
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Book Description
This is the first book to cover the theory of noise sensitivity of Boolean functions with particular emphasis on critical percolation.
Author: Itai Benjamini
Publisher:
ISBN: 9782130515319
Category :
Languages : en
Pages : 225
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Book Description
Author: Arsen Vasilyan
Publisher:
ISBN:
Category :
Languages : en
Pages : 90
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Book Description
The noise sensitivity of a Boolean function ... is one of its fundamental properties. For noise parameter ..., the noise sensitivity is denoted as ... This quantity is defined as follows: First, pick ... uniformly at random from ... , then pick ... by flipping each ... independently with probability ... is defined to equal P... Much of the existing literature on noise sensitivity explores the following two directions: (1) Showing that functions with low noise-sensitivity are structured in certain ways. (2) Mathematically showing that certain classes of functions have low noise sensitivity. Combined, these two research directions show that certain classes of functions have low noise sensitivity and therefore have useful structure. The fundamental importance of noise sensitivity, together with this wealth of structural results, motivates the algorithmic question of approximating ... given an oracle access to the function ... We show that the standard sampling approach is essentially optimal for general Boolean functions. Therefore, we focus on estimating the noise sensitivity of monotone functions, which form an important subclass of Boolean functions, since many functions of interest are either monotone or can be simply transformed into a monotone function (for example the class of unate functions consists of all the functions that can be made monotone by reorienting some of their coordinates [22]). Specifically, we study the algorithmic problem of approximating ... for monotone ... given the promise that ... for constant ... and for in the range ... on the query complexity of any algorithm that approximates ... to within any constant factor, where ... can be any positive constant. Thus, our algorithm’s query complexity is close to optimal in terms of its dependence on ??. We introduce a novel descending-ascending view of noise sensitivity, and use it as a central tool for the analysis of our algorithm. To prove lower bounds on query complexity, we develop a technique that reduces computational questions about query complexity to combinatorial questions about the existence of "thin" functions with certain properties. The existence of such "thin" functions is proved using the probabilistic method. These techniques also yield new lower bounds on the query complexity of approximating other fundamental properties of Boolean functions: the total influence and the bias.
Author: Ryan O'Donnell
Publisher: Cambridge University Press
ISBN: 1107038324
Category : Computers
Languages : en
Pages : 445
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Book Description
This graduate-level text gives a thorough overview of the analysis of Boolean functions, beginning with the most basic definitions and proceeding to advanced topics.
Author: Itai Benjamini
Publisher: Springer Science & Business Media
ISBN: 1441996753
Category : Mathematics
Languages : en
Pages : 1199
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Book Description
This volume is dedicated to the memory of the late Oded Schramm (1961-2008), distinguished mathematician. Throughout his career, Schramm made profound and beautiful contributions to mathematics that will have a lasting influence. In these two volumes, Editors Itai Benjamini and Olle Häggström have collected some of his papers, supplemented with three survey papers by Steffen Rohde, Häggström and Cristophe Garban that further elucidate his work. The papers within are a representative collection that shows the breadth, depth, enthusiasm and clarity of his work, with sections on Geometry, Noise Sensitivity, Random Walks and Graph Limits, Percolation, and finally Schramm-Loewner Evolution. An introduction by the Editors and a comprehensive bibliography of Schramm's publications complete the volume. The book will be of especial interest to researchers in probability and geometry, and in the history of these subjects.
Author: Maria Eulália Vares
Publisher: Springer Nature
ISBN: 3030607542
Category : Mathematics
Languages : en
Pages : 819
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Book Description
This is a volume in memory of Vladas Sidoravicius who passed away in 2019. Vladas has edited two volumes appeared in this series ("In and Out of Equilibrium") and is now honored by friends and colleagues with research papers reflecting Vladas' interests and contributions to probability theory.
Author: Sergio Rajsbaum
Publisher: Springer Science & Business Media
ISBN: 3540434003
Category : Computational complexity
Languages : en
Pages : 643
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Book Description
This book constitutes the refereed proceedings of the 5th International Symposium, Latin American Theoretical Informatics, LATIN 2002, held in Cancun, Mexico, in April 2002. The 44 revised full papers presented together with a tutorial and 7 abstracts of invited contributions were carefully reviewed and selected from a total of 104 submissions. The papers presented are devoted to a broad range of topics from theoretical computer science and mathematical foundations, with a certain focus on algorithmics and computations related to discrete structures.
Author: Clay Mathematics Institute. Summer School
Publisher: American Mathematical Soc.
ISBN: 0821868632
Category : Mathematics
Languages : en
Pages : 481
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Book Description
This volume is a collection of lecture notes for six of the ten courses given in Buzios, Brazil by prominent probabilists at the 2010 Clay Mathematics Institute Summer School, ``Probability and Statistical Physics in Two and More Dimensions'' and at the XIV Brazilian School of Probability. In the past ten to fifteen years, various areas of probability theory related to statistical physics, disordered systems and combinatorics have undergone intensive development. A number of these developments deal with two-dimensional random structures at their critical points, and provide new tools and ways of coping with at least some of the limitations of Conformal Field Theory that had been so successfully developed in the theoretical physics community to understand phase transitions of two-dimensional systems. Included in this selection are detailed accounts of all three foundational courses presented at the Clay school--Schramm-Loewner Evolution and other Conformally Invariant Objects, Noise Sensitivity and Percolation, Scaling Limits of Random Trees and Planar Maps--together with contributions on Fractal and Multifractal properties of SLE and Conformal Invariance of Lattice Models. Finally, the volume concludes with extended articles based on the courses on Random Polymers and Self-Avoiding Walks given at the Brazilian School of Probability during the final week of the school. Together, these notes provide a panoramic, state-of-the-art view of probability theory areas related to statistical physics, disordered systems and combinatorics. Like the lectures themselves, they are oriented towards advanced students and postdocs, but experts should also find much of interest.
Author: Christoph Bandt
Publisher: Springer Science & Business Media
ISBN: 3034600305
Category : Mathematics
Languages : en
Pages : 292
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Book Description
Over the last fifteen years fractal geometry has established itself as a substantial mathematical theory in its own right. The interplay between fractal geometry, analysis and stochastics has highly influenced recent developments in mathematical modeling of complicated structures. This process has been forced by problems in these areas related to applications in statistical physics, biomathematics and finance. This book is a collection of survey articles covering many of the most recent developments, like Schramm-Loewner evolution, fractal scaling limits, exceptional sets for percolation, and heat kernels on fractals. The authors were the keynote speakers at the conference "Fractal Geometry and Stochastics IV" at Greifswald in September 2008.
Author: Günter Last
Publisher: Cambridge University Press
ISBN: 1107088011
Category : Mathematics
Languages : en
Pages : 315
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Book Description
A modern introduction to the Poisson process, with general point processes and random measures, and applications to stochastic geometry.
