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33
Spectral distributions of adjacency and Laplacian matrices of weighted random graphs
 Ann. Appl. Probab
, 2010
"... Abstract In this paper, we investigate the spectral properties of the adjacency and the Laplacian matrices of random graphs. We prove that (i) the law of large numbers for the spectral norms and the largest eigenvalues of the adjacency and the Laplacian matrices; (ii) under some further independent ..."
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Abstract In this paper, we investigate the spectral properties of the adjacency and the Laplacian matrices of random graphs. We prove that (i) the law of large numbers for the spectral norms and the largest eigenvalues of the adjacency and the Laplacian matrices; (ii) under some further independent conditions, the normalized largest eigenvalues of the Laplacian matrices are dense in a compact interval almost surely; (iii) the empirical distributions of the eigenvalues of the Laplacian matrices converge weakly to the free convolution of the standard Gaussian distribution and the Wigner’s semicircular law; (iv) the empirical distributions of the eigenvalues of the adjacency matrices converge weakly to the Wigner’s semicircular law. 1
From granular matter to generalized continuum
 Lecture Notes in Mathematics
, 2006
"... Summary. Following a cursory review and synthesis of multipolar continua, the rudiments of graph theory, and granular mechanics, a graphtheoretic, energybased homogenization is proposed for the systematic derivation of multipolar stress and kinematics in granular media. This provides a weakly non ..."
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Summary. Following a cursory review and synthesis of multipolar continua, the rudiments of graph theory, and granular mechanics, a graphtheoretic, energybased homogenization is proposed for the systematic derivation of multipolar stress and kinematics in granular media. This provides a weakly nonlocal hierarchy of multipolar field equations for quasistatic mechanics based on polynomial representations of the kinematics of the type employed in past works. As an improvement on those works, a method is proposed for avoiding ”overfitting ” of fluctuations based on the socalled ”Generalized Additive Method ” of statistics. Among other results, it is shown that the standard formula for Cauchy stress in granular media may break down owing to multipolar effects, and that granular rotations in the typical granular medium should not lead to Cosserat effects, as the lowestorder departure from the simplecontinuum model. Key words: granular media, continuum models,homogenization, multipolar, micromorphic, graph theory,polynomial overfitting
Discrete Mathematics for Combinatorial Chemistry
, 1998
"... The aim is a description of discrete mathematics used in a project devoted to the implementation of a software package for the simulation of combinatorial chemistry. ..."
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The aim is a description of discrete mathematics used in a project devoted to the implementation of a software package for the simulation of combinatorial chemistry.
Ibn Sīnā on analysis: 1. Proof search. Or: Abstract State Machines as a tool for history of logic
"... and I have removed some personal references. The 11th century ArabicPersian logician Ibn Sīnā (Avicenna) in section 9.6 of his book Qiyās gives what appears to be a proof search algorithm for syllogisms. We confirm that it is indeed a proof search State Machine from Ibn Sīnā’s text. The paper also ..."
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and I have removed some personal references. The 11th century ArabicPersian logician Ibn Sīnā (Avicenna) in section 9.6 of his book Qiyās gives what appears to be a proof search algorithm for syllogisms. We confirm that it is indeed a proof search State Machine from Ibn Sīnā’s text. The paper also contains a translation of the passage from Ibn Sina’s Arabic, and some notes on the text and translation. 1
Historical Projects in Discrete Mathematics and Computer Science
"... A course in discrete mathematics is a relatively recent addition, within the last 30 or 40 years, to the modern American undergraduate curriculum, born out of a need to instruct computer science majors in algorithmic thought. The roots of discrete mathematics, however, are as old as mathematics itse ..."
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A course in discrete mathematics is a relatively recent addition, within the last 30 or 40 years, to the modern American undergraduate curriculum, born out of a need to instruct computer science majors in algorithmic thought. The roots of discrete mathematics, however, are as old as mathematics itself, with the notion of counting a discrete operation, usually cited as the first mathematical development
Designing student projects for teaching and learning discrete mathematics and computer science via primary historical sources
, 2009
"... ..."
Trees and Term Rewriting in 1910: On a Paper by Axel Thue
"... Many of Axel Thue's ideas have been influential in theoretical computer science. In particular, Thue systems, semiThue systems and his work on the combinatorics of words are wellknown. Here we consider his 1910 paper which contains many notions and ideas about trees, term rewriting and word proble ..."
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Many of Axel Thue's ideas have been influential in theoretical computer science. In particular, Thue systems, semiThue systems and his work on the combinatorics of words are wellknown. Here we consider his 1910 paper which contains many notions and ideas about trees, term rewriting and word problems which are surprisingly modern and have later come to play important roles in mathematics, logic, and computer science.
A short survey of automated reasoning
"... Abstract. This paper surveys the field of automated reasoning, giving some historical background and outlining a few of the main current research themes. We particularly emphasize the points of contact and the contrasts with computer algebra. We finish with a discussion of the main applications so f ..."
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Abstract. This paper surveys the field of automated reasoning, giving some historical background and outlining a few of the main current research themes. We particularly emphasize the points of contact and the contrasts with computer algebra. We finish with a discussion of the main applications so far. 1 Historical introduction The idea of reducing reasoning to mechanical calculation is an old dream [75]. Hobbes [55] made explicit the analogy in the slogan ‘Reason [...] is nothing but Reckoning’. This parallel was developed by Leibniz, who envisaged a ‘characteristica universalis’ (universal language) and a ‘calculus ratiocinator ’ (calculus of reasoning). His idea was that disputes of all kinds, not merely mathematical ones, could be settled if the parties translated their dispute into the characteristica and then simply calculated. Leibniz even made some steps towards realizing this lofty goal, but his work was largely forgotten. The characteristica universalis The dream of a truly universal language in Leibniz’s sense remains unrealized and probably unrealizable. But over the last few centuries a language that is at least adequate for
EUCLIDEAN DISTANCE GEOMETRY AND APPLICATIONS
"... Abstract. Euclidean distance geometry is the study of Euclidean geometry based on the concept of distance. This is useful in several applications where the inputdataconsistsofanincompleteset of distances, and the output is a set of points in Euclidean space that realizes the given distances. We surv ..."
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Abstract. Euclidean distance geometry is the study of Euclidean geometry based on the concept of distance. This is useful in several applications where the inputdataconsistsofanincompleteset of distances, and the output is a set of points in Euclidean space that realizes the given distances. We survey some of the theory of Euclidean distance geometry and some of its most important applications, including molecular conformation, localization of sensor networks and statics. Key words. Matrix completion, barandjoint framework, graph rigidity, inverse problem, protein conformation, sensor network.