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Terence Tao - One of the best experts on this subject based on the ideXlab platform.
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algebraic combinatorial geometry the polynomial method in arithmetic Combinatorics incidence Combinatorics and number theory
EMS Surveys in Mathematical Sciences, 2014Co-Authors: Terence TaoAbstract:Arithmetic Combinatorics is often concerned with the problem of controlling the possible range of behaviours of arbitrary finite sets in a group or ring with respect to arithmetic operations such as addition or multiplication. Similarly, combinatorial geometry is often concerned with the problem of controlling the possible range of behaviours of arbitrary finite collections of geometric objects such as points, lines, or circles with respect to geometric operations such as incidence or distance. Given the presence of arbitrary finite sets in these problems, the methods used to attack these problems have primarily been combinatorial in nature. In recent years, however, many outstanding problems in these directions have been solved by algebraic means (and more specifically, using tools from algebraic geometry and/or algebraic topology), giving rise to an emerging set of techniques which is now known as the polynomial method. Broadly speaking, the strategy is to capture (or at least partition) the arbitrary sets of objects (viewed as points in some configuration space) in the zero set of a polynomial whose degree (or other measure of complexity) is under control; for instance, the degree may be bounded by some function of the number of objects. One then uses tools from algebraic geometry to understand the structure of this zero set, and thence to control the original sets of objects. While various instances of the polynomial method have been known for decades (e.g. Stepanov’s method, the combinatorial Nullstellensatz, or Baker’s theorem), the general theory of this method is still in the process of maturing; in particular, the limitations of the polynomial method are not well understood, and there is still considerable scope to apply deeper results from algebraic geometry or algebraic topology to strengthen the method further. In this survey we present several of the known applications of these methods, focusing on the simplest cases to illustrate the techniques. We will assume as little prior knowledge of algebraic geometry as possible.
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algebraic combinatorial geometry the polynomial method in arithmetic Combinatorics incidence Combinatorics and number theory
arXiv: Combinatorics, 2013Co-Authors: Terence TaoAbstract:Arithmetic Combinatorics is often concerned with the problem of bounding the behaviour of arbitrary finite sets in a group or ring with respect to arithmetic operations such as addition or multiplication. Similarly, combinatorial geometry is often concerned with the problem of bounding the behaviour of arbitrary finite collections of geometric objects such as points, lines, or circles with respect to geometric operations such as incidence or distance. Given the presence of arbitrary finite sets in these problems, the methods used to attack these problems have primarily been combinatorial in nature. In recent years, however, many outstanding problems in these questions have been solved by algebraic means (and more specifically, using tools from algebraic geometry and/or algebraic topology), giving rise to an emerging set of techniques which is now known as the polynomial method. While various instances of the polynomial method have been known for decades (e.g. Stepanov's method, the combinatorial nullstellensatz, or Baker's theorem), the general theory of this method is still in the process of maturing; in particular, the limitations of the polynomial method are not well understood, and there is still considerable scope to apply deeper results from algebraic geometry or algebraic topology to strengthen the method further. In this survey we present several of the known applications of these methods, focusing on the simplest cases to illustrate the techniques. We will assume as little prior knowledge of algebraic geometry as possible.
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an improved bound for the minkowski dimension of besicovitch sets in medium dimension
Geometric and Functional Analysis, 2001Co-Authors: Izabella Laba, Terence TaoAbstract:We use geometrical Combinatorics arguments, including the "hairbrush" argument of Wolff [W1], the x-ray estimates in [W2], [LT], and the sticky/plany/grainy analysis of [KLT], to show that Besicovitch sets in \( {\bold R}^n \) have Minkowski dimension at least \( {n+2 \over 2} + \varepsilon_n \) for all \( n \geq 4 \), where \( \varepsilon_n > 0 \) is an absolute constant depending only on n. This complements the results of [KLT], which established the same result for n = 3, and of [B3], [KT], which used arithmetic Combinatorics techniques to establish the result for \( n \ge 9 \). Unlike the arguments in [KLT], [B3], [KT], our arguments will be purely geometric and do not require arithmetic Combinatorics.
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an improved bound for the minkowski dimension of besicovitch sets in medium dimension
arXiv: Classical Analysis and ODEs, 2000Co-Authors: Izabella Laba, Terence TaoAbstract:We use geometrical Combinatorics arguments, including the ``hairbrush'' and x-ray arguments of Wolff and the sticky/plany/grainy analysis of Katz, Laba, and Tao, to show that Besicovitch sets in R^n have Minkowski dimension at least (n+2)/2 + \eps_n for all n > 3, where \eps_n > 0 is an absolute constant depending only on n. This complements the results of Katz, Laba, and Tao, which established the same result for n=3, and of Bourgain and Katz-Tao, arithmetic Combinatorics techniques to establish the result for n > 8. In contrast to previous work, our arguments will be purely geometric and do not require arithmetic Combinatorics.
Robert A Cordery - One of the best experts on this subject based on the ideXlab platform.
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light scattering as a poisson process and first passage probability
Journal of Statistical Mechanics: Theory and Experiment, 2020Co-Authors: Claude Zeller, Robert A CorderyAbstract:A particle entering a scattering and absorbing medium executes a random walk through a sequence of scattering events. The particle ultimately achieves first-passage, leaving the medium or it is absorbed. The Kubelka-Munk model describes a flux of particles moving perpendicular to the surface of a plane-parallel medium. The particle path alternates between the positive direction into the medium and the negative direction back towards the surface. Backscattering events from the positive to the negative direction occur at local maxima or peaks, while backscattering from the negative to the positive direction occur at local minima or valleys. The probability of a particle avoiding absorption as it follows its path decreases exponentially with the path-length \(\lambda\). The reflectance of a semi-infinite slab is therefore the Laplace transform of the distribution of path-length that ends with a first-passage out of the medium. In the case of a constant scattering rate the random walk is a Poisson process. We verify our results with two iterative calculations, one using the properties of iterated convolution with a symmetric kernel and the other via direct calculation with an exponential step-length distribution. We present a novel demonstration, based on fluctuation theory of sums of random variables, that the first-passage probability as a function of the number of peaks in the alternating path is a step-length distribution-free combinatoric expression. Counting paths with backscattering on the real half-line results in the same Catalan number coefficients as Dyck paths on the whole numbers. Including a separate forward-scattering Poisson process results in an expression related to counting Motzkin paths. We therefore connect walks on the real line to discrete path Combinatorics.
Claude Zeller - One of the best experts on this subject based on the ideXlab platform.
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light scattering as a poisson process and first passage probability
Journal of Statistical Mechanics: Theory and Experiment, 2020Co-Authors: Claude Zeller, Robert A CorderyAbstract:A particle entering a scattering and absorbing medium executes a random walk through a sequence of scattering events. The particle ultimately achieves first-passage, leaving the medium or it is absorbed. The Kubelka-Munk model describes a flux of particles moving perpendicular to the surface of a plane-parallel medium. The particle path alternates between the positive direction into the medium and the negative direction back towards the surface. Backscattering events from the positive to the negative direction occur at local maxima or peaks, while backscattering from the negative to the positive direction occur at local minima or valleys. The probability of a particle avoiding absorption as it follows its path decreases exponentially with the path-length \(\lambda\). The reflectance of a semi-infinite slab is therefore the Laplace transform of the distribution of path-length that ends with a first-passage out of the medium. In the case of a constant scattering rate the random walk is a Poisson process. We verify our results with two iterative calculations, one using the properties of iterated convolution with a symmetric kernel and the other via direct calculation with an exponential step-length distribution. We present a novel demonstration, based on fluctuation theory of sums of random variables, that the first-passage probability as a function of the number of peaks in the alternating path is a step-length distribution-free combinatoric expression. Counting paths with backscattering on the real half-line results in the same Catalan number coefficients as Dyck paths on the whole numbers. Including a separate forward-scattering Poisson process results in an expression related to counting Motzkin paths. We therefore connect walks on the real line to discrete path Combinatorics.
Steven Skiena - One of the best experts on this subject based on the ideXlab platform.
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computational discrete mathematics Combinatorics and graph theory with mathematica by sriram pemmaraju professor steven skiena
2014Co-Authors: Sriram Pemmaraju, Steven SkienaAbstract:If you are searching for the book by Sriram Pemmaraju;Professor Steven Skiena Computational Discrete Mathematics: Combinatorics and Graph Theory with Mathematica ® in pdf form, then you've come to faithful website. We furnish complete release of this ebook in doc, ePub, DjVu, PDF, txt formats. You may reading Computational Discrete Mathematics: Combinatorics and Graph Theory with Mathematica ® online by Sriram Pemmaraju;Professor Steven Skiena or downloading. Too, on our site you may reading the manuals and another art eBooks online, either downloading theirs. We want invite attention what our website not store the book itself, but we provide reference to website wherever you may load either read online. So if you want to downloading pdf Computational Discrete Mathematics: Combinatorics and Graph Theory with Mathematica ® by Sriram Pemmaraju;Professor Steven Skiena, in that case you come on to the correct site. We have Computational Discrete Mathematics: Combinatorics and Graph Theory with Mathematica ® DjVu, txt, PDF, ePub, doc forms. We will be pleased if you get back to us again.
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computational discrete mathematics Combinatorics and graph theory with mathematica
2003Co-Authors: Sriram Pemmaraju, Steven SkienaAbstract:With examples of all 450 functions in action plus tutorial text on the mathematics, this book is the definitive guide to Experimenting with Combinatorica, a widely used software package for teaching and research in discrete mathematics. Three interesting classes of exercises are provided--theorem/proof, programming exercises, and experimental explorations--ensuring great flexibility in teaching and learning the material. The Combinatorica user community ranges from students to engineers, researchers in mathematics, computer science, physics, economics, and the humanities. Recipient of the EDUCOM Higher Education Software Award, Combinatorica is included with every copy of the popular computer algebra system Mathematica.
Seth Sullivant - One of the best experts on this subject based on the ideXlab platform.
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polyhedral Combinatorics of upgma cones
Advances in Applied Mathematics, 2013Co-Authors: Ruth Davidson, Seth SullivantAbstract:Distance-based methods such as UPGMA (Unweighted Pair Group Method with Arithmetic Mean) continue to play a significant role in phylogenetic research. We use polyhedral Combinatorics to analyze the natural subdivision of the positive orthant induced by classifying the input vectors according to tree topologies returned by the algorithm. The partition lattice informs the study of UPGMA trees. We give a closed form for the extreme rays of UPGMA cones on n taxa, and compute the spherical volumes of the UPGMA cones for small n.
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polyhedral Combinatorics of upgma cones
arXiv: Populations and Evolution, 2012Co-Authors: Ruth Davidson, Seth SullivantAbstract:Distance-based methods such as UPGMA (Unweighted Pair Group Method with Arithmetic Mean) continue to play a significant role in phylogenetic research. We use polyhedral Combinatorics to analyze the natural subdivision of the positive orthant induced by classifying the input vectors according to tree topologies returned by the algorithm. The partition lattice informs the study of UPGMA trees. We give a closed form for the extreme rays of UPGMA cones on n taxa, and compute the normalized volumes of the UPGMA cones for small n. Keywords: phylogenetic trees, polyhedral Combinatorics, partition lattice
