A systematic and integrated approach to Cantor Sets and their applications to various branches of mathematics The Elements of Cantor Sets: With applications features a thorough introduction to Cantor Sets and applies ...
A systematic and integrated approach to Cantor Sets and their applications to various branches of mathematics The Elements of Cantor Sets: With applications features a thorough introduction to Cantor Sets and applies these sets as a bridge between real analysis, probability, topology, and algebra. The author fills a gap in the current literature by providing an introductory and integrated perspective, thereby preparing readers for further study and building a deeper understanding of analysis, topology, set theory, numbertheory, and algebra. The Elements of Cantor Sets provides coverage of: Basic definitions and background theorems as well as comprehensive mathematical details A biography of Georg Ferdinand Ludwig Philipp Cantor, one of the most significant mathematicians of the last century Chapter coverage of fractals and self-similar sets, sums of Cantor Sets, the role of Cantor Sets in creating pathological functions, p-adic numbers, and several generalizations of Cantor Sets A wide spectrum of topics from measure theory to the Monty Hall Problem An ideal text for courses in real analysis, topology, algebra, and set theory for undergraduate and graduate-level courses within mathematics, computer science, engineering, and physics departments, The Elements of Cantor Sets is also appropriate as a useful reference for researchers and secondary mathematics education majors.
Linear algebra is considered an essential mathematical theory that has many engineering applications. While many theorem provers support linear spaces, they only consider finite dimensional spaces. In addition, availa...
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This article reviews latent semantic analysis (LSA), a theory of meaning as well as a method for extracting that meaning from passages of text, based on statistical computations over a collection of documents. LSA as ...
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This article reviews latent semantic analysis (LSA), a theory of meaning as well as a method for extracting that meaning from passages of text, based on statistical computations over a collection of documents. LSA as a theory of meaning defines a latent semantic space where documents and individual words are represented as vectors. LSA as a computational technique uses linear algebra to extract dimensions that represent that space. This representation enables the computation of similarity among terms and documents, categorization of terms and documents, and summarization of large collections of documents using automated procedures that mimic the way humans perform similar cognitive tasks. We present some technical details, various illustrative examples, and discuss a number of applications from linguistics, psychology, cognitive science, education, information science, and analysis of textual data in general. (C) 2013 John Wiley & Sons, Ltd. WIREs Cogn Sci 2013, 4:683-692. doi: 10.1002/wcs.1254 Conflict of interest: The author has declared no conflicts of interest for this article. For further resources related to this article, please visit the WIREs website.
Markov chains are a fundamental class of stochastic processes. They are widely used to solve problems in a large number of domains such as operational research, computer science, communication networks and manufacturi...
Markov chains are a fundamental class of stochastic processes. They are widely used to solve problems in a large number of domains such as operational research, computer science, communication networks and manufacturing systems. The success of Markov chains is mainly due to their simplicity of use, the large number of available theoretical results and the quality of algorithms developed for the numerical evaluation of many metrics of interest. The author presents the theory of both discrete-time and continuous-time homogeneous Markov chains. He carefully examines the explosion phenomenon, the Kolmogorov equations, the convergence to equilibrium and the passage time distributions to a state and to a subset of states. These results are applied to birth-and-death processes. He then proposes a detailed study of the uniformization technique by means of Banach algebra. This technique is used for the transient analysis of several queuing systems. Contents 1. Discrete-Time Markov Chains 2. Continuous-Time Markov Chains 3. Birth-and-Death Processes 4. Uniformization 5. Queues About the Authors Bruno Sericola is a Senior Research Scientist at Inria Rennes - Bretagne Atlantique in France. His main research activity is in performance evaluation of computer and communication systems, dependability analysis of fault-tolerant systems and stochastic models.
The overall aim of this thesis was to apply techniques from algebraic geometry to prob- lems in economics. algebraic geometry has found many applications in various areas of mathematics and in several other fields. We...
The overall aim of this thesis was to apply techniques from algebraic geometry to prob- lems in economics. algebraic geometry has found many applications in various areas of mathematics and in several other fields. We have encountered three major approaches to employing these tools to economics. First, there are the symbolic methods from computeralgebra (Greuel and Pfister, 2002). One possible avenue of approach here is Grobner bases, which have already been used to great effect in integer programming (Lo- era et al., 2006) and also in economics (Kubler and Schmedders, 2010). Second, there is the numerical algebraic geometry route. There one uses Berstein's or Bezout's theorem, which give information on the isolated solutions of a square system of polynomial equa- tions (Sommese and Wampler, 2005). The basic idea is to construct a homotopy and trace the paths leading to those isolated solutions. It is a very active field of research and applications range from optimal control (Rostalski et al., 2011) to biology (Hao et al., 2011). Lastly there is the real algebraic geometry route. It was recently discovered (Parrilo, 2000; Lasserre, 2001b) that representation results for positive polynomials can be used to relax polynomial optimization problems into convex optimization problems. Since then it has been shown that this is a promising approach to solving various prob- lems, for instance in combinatorial optimization (Lasserre, 2001a) and also game theory (Laraki and Lasserre, 2012). Over recent years I have looked at the last two of these approaches. The results have been presented in the form of several papers, two of which have already been published and the last of which is being revised at the time of writing. The first paper is entitled "Finding all pure-strategy equilibria in games with contin- uous strategies" (Judd et al., 2012). Static and dynamic games are widely used tools for policy experiments and estimation studies. The multiple Nash equilibria in such m
We consider the problem of computing the rank of an m x n matrix A over a field. We present a randomized algorithm to find a set of r = rank(A) linearly independent columns in (O) over tilde(vertical bar A vertical ba...
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We consider the problem of computing the rank of an m x n matrix A over a field. We present a randomized algorithm to find a set of r = rank(A) linearly independent columns in (O) over tilde(vertical bar A vertical bar + r(omega)) field operations, where vertical bar A vertical bar denotes the number of nonzero entries in A and omega < 2.38 is the matrix multiplication exponent. Previously the best known algorithm to find a set of r linearly independent columns is by Gaussian elimination, with deterministic running time O(mnr(omega-2)). Our algorithm is faster when r < max{m, n}, for instance when the matrix is rectangular. We also consider the problem of computing the rank of a matrix dynamically, supporting the operations of rank one updates and additions and deletions of rows and columns. We present an algorithm that updates the rank in (O) over tilde (mn) field operations. We show that these algorithms can be used to obtain faster algorithms for various problems in exact linear algebra, combinatorial optimization and dynamic data structure.
We address the issue of simplifying symbolic polynomials on non-commutative variables. The problem is motivated by applications in optimization and various problems in systems and control. We develop theory for polyno...
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We address the issue of simplifying symbolic polynomials on non-commutative variables. The problem is motivated by applications in optimization and various problems in systems and control. We develop theory for polynomials which are linear in a subset of the variables and develop algorithms to produce representations which have the minimal possible number of terms. The results can handle polynomial matrices as well as block-matrix variables. (C) 2012 Elsevier Inc. All rights reserved.
In this paper, we study closed form evaluation for some special Hankel determinants arising in combinatorial analysis, especially for the bidirectional number sequences. We show that such problems are directly connect...
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In this paper, we study closed form evaluation for some special Hankel determinants arising in combinatorial analysis, especially for the bidirectional number sequences. We show that such problems are directly connected with the theory of quasi-definite discrete Sobolev orthogonal polynomials. It opens a lot of procedural dilemmas which we will try to exceed. A few examples deal with Fibonacci numbers and power sequences will illustrate our considerations. We believe that our usage of Sobolev orthogonal polynomials in Hankel determinant computation is quite new. (C) 2012 Elsevier Inc. All rights reserved.
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