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Mathematics by Experiment: Plausible Reasoning in the 21st Century, extended second edition, A K
 2008. EXPERIMENTATION AND COMPUTATION 19
, 2008
"... If mathematics describes an objective world just like physics, there is no reason why inductive methods should not be applied in mathematics just the same as in physics. (Kurt Gödel, 1951) Paper Revised 09–09–04 This paper is an extended version of a presentation made at ICME10, related work is elab ..."
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Cited by 41 (18 self)
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If mathematics describes an objective world just like physics, there is no reason why inductive methods should not be applied in mathematics just the same as in physics. (Kurt Gödel, 1951) Paper Revised 09–09–04 This paper is an extended version of a presentation made at ICME10, related work is elaborated in references [1–7]. 1 I shall generally explore experimental and heuristic mathematics and give (mostly) accessible, primarily visual and symbolic, examples. The emergence of powerful mathematical computing environments like Maple and Matlab, the growing
Query Merging: Improving Query Subscription Processing in a Multicast Environment
 IEEE Trans. on Knowledge and Data Engineering
"... This paper introduces techniques for reducing data dissemination costs of query subscriptions in a multicast environment. The reduction is achieved by merging queries with overlapping, but not necessarily equal, answers. The paper formalizes the querymerging problem and introduces a general fram ..."
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Cited by 20 (0 self)
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This paper introduces techniques for reducing data dissemination costs of query subscriptions in a multicast environment. The reduction is achieved by merging queries with overlapping, but not necessarily equal, answers. The paper formalizes the querymerging problem and introduces a general framework and cost model for evaluating merging. We prove that the problem is NPhard and propose exhaustive algorithms and three heuristic algorithms: the Pair Merging Algorithm, the Directed Search Algorithm and the Clustering Algorithm. We develop a simulator, which uses geographical queries as a representative example, for evaluating the different heuristics and show that the performance of our heuristics is close to optimal. KEYWORDS: Query Processing, Data Dissemination, Query Merging, Query Subscriptions, Multicast of Query Results, Geographical Queries. 1 Introduction With information dissemination (information push), data is delivered from a set of producers to a (typically) larg...
Simplicial Properties of the Set of Planar Binary Trees
 J. Alg. Combinatorics
, 1997
"... Planar binary trees appear as main ingredient of a new homology theory related to dialgebras, c.f. [L]. Here we investigate the simplicial properties of the set of these trees, which are independent of the dialgebra context though they are reflected in the dialgebra homology. The set of planar binar ..."
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Cited by 8 (3 self)
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Planar binary trees appear as main ingredient of a new homology theory related to dialgebras, c.f. [L]. Here we investigate the simplicial properties of the set of these trees, which are independent of the dialgebra context though they are reflected in the dialgebra homology. The set of planar binary trees is endowed with a natural (almost) simplicial structure which gives rise to a chain complex. Our main idea consists in decomposing the set of trees into classes, by exploiting the orientation of their leaves. This decomposition yields a chain bicomplex whose total chain complex is that of binary trees. Our main theorem concerns a further decomposition of this bicomplex. Each vertical complex is the direct sum of subcomplexes which are in bijection with the planar binary trees. This decomposition is used in the computation of dialgebra homology as a derived functor, c.f. [F2]. Introduction The planar binary trees have been widely studied for their combinatorial properties, which rela...
Generalization of Hermite ? s Divisibility Theorems and the MannShanks Primality Criterion for sFibonomial Arrays." Fibonacci Quarterly 12.2
"... In a previous paper [4] I found that two theorems of Hermite concerning factors of binomial coefficients might be extended to generalized binomial coefficients [2] , however one of my proofs imposed severe restrictions on the sequence {A} used to define the generalized coefficients Also it was fou ..."
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Cited by 4 (3 self)
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In a previous paper [4] I found that two theorems of Hermite concerning factors of binomial coefficients might be extended to generalized binomial coefficients [2] , however one of my proofs imposed severe restrictions on the sequence {A} used to define the generalized coefficients Also it was found that the MannShanks primality criterion [6] follows
Automated Counting Of Lego Towers
"... H.N.V. Temperley's method for counting vertically convex polyominoes is modified, generalized, and most importantly, programmed (in Maple). ..."
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Cited by 3 (1 self)
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H.N.V. Temperley's method for counting vertically convex polyominoes is modified, generalized, and most importantly, programmed (in Maple).
Graph Compositions I: Basic Enumeration
"... The idea of graph compositions generalizes both ordinary compositions of positive integers and partitions of nite sets. In this paper we develop formulas, generating functions, and recurrence relations for composition counting functions for several families of graphs. 1. ..."
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Cited by 3 (0 self)
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The idea of graph compositions generalizes both ordinary compositions of positive integers and partitions of nite sets. In this paper we develop formulas, generating functions, and recurrence relations for composition counting functions for several families of graphs. 1.
Constraint Based Event Recognition for Information Extraction
, 1995
"... A common feature of news reports is the reference to events other than the one which is central to the discourse. Previous research has suggested Gricean explanations for this; more generally, the phenomenon has been referred to simply as "journalistic style". Whatever the underlying reasons, recent ..."
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Cited by 2 (0 self)
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A common feature of news reports is the reference to events other than the one which is central to the discourse. Previous research has suggested Gricean explanations for this; more generally, the phenomenon has been referred to simply as "journalistic style". Whatever the underlying reasons, recent investigations into information extraction have emphasised the need for a better understanding of the mechanisms that can be used to recognise and distinguish between multiple events in discourse. Existing information extraction systems approach the problem of event recognition in a number of ways. However, although frameworks and techniques for black box evaluations of information extraction systems have been developed in recent years, almost no attention has been given to the evaluation of techniques for event recognition, despite general acknowledgment of the inadequacies of current implementations. Not only is it unclear which mechanisms are useful, but there is also little consensus as...
COMBINATORIC RESULTS FOR GRAPHICAL ENUMERATION AND THE HIGHER CATALAN NUMBERS
, 2007
"... Abstract. We summarize some combinatoric problems solved by the higher Catalan numbers. These problems are generalizations of the combinatoric problems solved by the Catalan numbers. The generating function of the higher Catalan numbers appeared recently as an auxiliary function in enumerating maps ..."
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Abstract. We summarize some combinatoric problems solved by the higher Catalan numbers. These problems are generalizations of the combinatoric problems solved by the Catalan numbers. The generating function of the higher Catalan numbers appeared recently as an auxiliary function in enumerating maps and explicit computations of the asymptotic expansion of the partition function of random matrices in the unitary ensemble case. We give combinatoric proofs of the formulas for the number of genus 0 and genus 1 maps. 1. Higher Catalan Numbers The Catalan numbers solve a number of classical combinatoric problems such as the “Euler Polygon Division Problem”: how many ways are there to divide a marked polygon with j + 2 sides into triangles using edges and diagonals [3, 7, 8, 12, 16] (see figure 1). Figure 1. A polygon with 4 + 2 = 6 sides divided into 4 triangles using edges and diagonals They count the number of rightleft paths along a 1Dimensional integer lattice which stay to the right of 0 and go from 0 to 0 in 2j steps; equivalently they count Dyck paths from (0, 0) to (2j, 0) [1, 4, 15, 18]. They count the number of ways for 2j customers to line up, with j customers having only a 2dollar bill and j customers having only a 1dollar bill, to purchase 1dollar widgets, so that each customer receives exact change. They count the number of noncrossing handshakes possible across a round table between n people [5]. In this paper we will explore a generalization of the Catalan numbers, the higher Catalan numbers. We will show that this generalization solves enumerative problems that are natural generalizations of the problems solved by the Catalan numbers. We will then highlight their appearance in recent results on map enumeration problems.
The Catalan numbers
, 1982
"... Cn = ( 2 n n)/(n + l) belong to the class of advanced counting numbers that appear as naturally and almost as frequently as the binomial coefficients $ due to the extensive variety of combinatorial objects counted by them (see [1]9 [2])« The purpose of this note is to give a combinatorial proof of ..."
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Cn = ( 2 n n)/(n + l) belong to the class of advanced counting numbers that appear as naturally and almost as frequently as the binomial coefficients $ due to the extensive variety of combinatorial objects counted by them (see [1]9 [2])« The purpose of this note is to give a combinatorial proof of the following property of the Catalan sequence using a lattice path interpretation. Theorem Cn is odd if and only if n = 2 r 1 for some positive integer v. Proof: The proof is based mainly on the following observation: If X is a finite set and a is an involution on X with fixed point set X a s then z  =  j a  (mod 2); i.e.s \x \ and  j a  have the same parity,, Now let Dn denote the set of lattice paths in the first quadrant from the origin to the point (2ns 0) with the elementary steps x: (a, b)» • (a + 1, b + 1) xi (a, b)* (a + 1, b 1). It is well known that \Dn \ = Cn (see [2], [3]). Define a: Dn * Dn by reflecting these paths about the line x = n * The fixed point set D % of a consists of all paths in Dn symmetric with respect to the line x n. \ w i\ Now define an involution 3 on D % as follows: for wW^uuWi = = n a n d u e 1^21 ~ ~ * { x * ^ 3 s e t e & % with 1 w1uuw2 i f ^i 4 Dni I w otherwise. Of course the set Dni is empty unless n is odd. Hence9 we can put 2 Cni ~ 0 f ° r n even.