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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 38 (15 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
Convolution polynomials
 The Mathematica Journal 2,4 (Fall
, 1992
"... Abstract. The polynomials that arise as coefficients when a power series is raised to the power x include many important special cases, which have surprising properties that are not widely known. This paper explains how to recognize and use such properties, and it closes with a general result about ..."
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Cited by 21 (1 self)
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Abstract. The polynomials that arise as coefficients when a power series is raised to the power x include many important special cases, which have surprising properties that are not widely known. This paper explains how to recognize and use such properties, and it closes with a general result about approximating such polynomials asymptotically. A family of polynomials F0(x),F1(x),F2(x),... forms a convolution family if Fn(x) has degree ≤ n and if the convolution condition Fn(x + y) = Fn(x)F0(y) + Fn−1(x)F1(y) + · · · + F1(x)Fn−1(y) + F0(x)Fn(y) holds for all x and y and for all n ≥ 0. Many such families are known, and they appear frequently in applications. For example, we can let Fn(x) = x n /n!; the condition (x + y) n n! n∑ k=0 x k k! y n−k (n − k)! is equivalent to the binomial theorem for integer exponents. Or we can let Fn(x) be the binomial coefficient () x n; the corresponding identity ( ) n∑ x + y x y n k n − k is commonly called Vandermonde’s convolution. k=0 How special is the convolution condition? Mathematica will readily find all sequences of polynomials that work for, say, 0 ≤ n ≤ 4: F[n_,x_]:=Sum[f[n,j]x^j,{j,0,n}]/n! conv[n_]:=LogicalExpand[Series[F[n,x+y],{x,0,n},{y,0,n}]
Evolving Hierarchical and Recursive Teleoreactive Programs through Genetic Programming
 Programming, EuroGP 2003, LNCS 2610
, 2003
"... Teleoreactive programs and the triple tower architecture have been proposed as a framework for linking perception and action in agents. The triple tower architecture continually updates the agents knowledge of the world and evokes actions according to teleoreactive control structures. This paper ..."
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Cited by 8 (0 self)
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Teleoreactive programs and the triple tower architecture have been proposed as a framework for linking perception and action in agents. The triple tower architecture continually updates the agents knowledge of the world and evokes actions according to teleoreactive control structures. This paper uses block stacking problems to demonstrate how genetic programming may be used to evolve hierarchical and recursive teleoreactive programs.
On the integrality of nth roots of generating functions
 J. COMBINATORIAL
, 2006
"... Motivated by the discovery that the eighth root of the theta series of the E8 lattice and the 24th root of the theta series of the Leech lattice both have integer coefficients, we investigate the question of when an arbitrary element f ∈ R (where R = 1 + xZ[[x]]) can be written as f = g n for g ∈ R, ..."
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Cited by 5 (1 self)
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Motivated by the discovery that the eighth root of the theta series of the E8 lattice and the 24th root of the theta series of the Leech lattice both have integer coefficients, we investigate the question of when an arbitrary element f ∈ R (where R = 1 + xZ[[x]]) can be written as f = g n for g ∈ R, n ≥ 2. Let Pn: = {g n  g ∈ R} and let µn: = n ∏ pn p. We show among other things that (i) for f ∈ R, f ∈ Pn ⇔ f (mod µn) ∈ Pn, and (ii) if f ∈ Pn, there is a unique g ∈ Pn with coefficients mod µn/n such that f ≡ gn (mod µn). In particular, if f ≡ 1 (mod µn) then f ∈ Pn. The latter assertion implies that the theta series of any extremal even unimodular lattice in Rn (e.g. E8 in R8) is in Pn if n is of the form 2i3j5k (i ≥ 3). There do not seem to be any exact analogues for codes, although we show that the weight enumerator of the rth order ReedMuller code of length 2m is in P2r (and similarly that the theta series of the BarnesWall lattice BW2m is in P2m). We give a number of other results and conjectures, and establish a conjecture of Paul D. Hanna that there is a unique element f ∈ Pn (n ≥ 2) with coefficients restricted to the set {1, 2,..., n}.
The Number of Hierarchical Orderings
 Order
, 2003
"... An ordered setpartition (or preferential arrangement) of n labeled elements represents a single "hierarchy" ..."
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Cited by 2 (2 self)
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An ordered setpartition (or preferential arrangement) of n labeled elements represents a single "hierarchy"
Evolving TeleoReactive Programs for Block Stacking Using Indexicals through Genetic Programming
, 2002
"... This paper demonstrates how stronglytyped genetic programming may be used to evolve valid teleoreactive programs that solve the general blockstacking problem using indexicals. ..."
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Cited by 2 (1 self)
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This paper demonstrates how stronglytyped genetic programming may be used to evolve valid teleoreactive programs that solve the general blockstacking problem using indexicals.
Parity theorems for statistics on lattice paths and Laguerre configurations
, 2004
"... We examine the parity of some statistics on lattice paths and Laguerre configurations, giving both algebraic and combinatorial treatments. For the former, we evaluate qgenerating functions at q = −1; for the latter, we define appropriate paritychanging involutions on the associated structures. In ..."
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Cited by 2 (2 self)
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We examine the parity of some statistics on lattice paths and Laguerre configurations, giving both algebraic and combinatorial treatments. For the former, we evaluate qgenerating functions at q = −1; for the latter, we define appropriate paritychanging involutions on the associated structures. In addition, we furnish combinatorial proofs for a couple of related recurrences. 1
On Squares, Cubes,... in Z[[x]] ∗
, 2005
"... Motivated by the discovery that the eighth root of the theta series of the E8 lattice and the 24th root of the theta series of the Leech lattice both have integer coefficients, we investigate the question of when an arbitrary element f ∈ Z[[x]] ∗ can be written as f = g n for g ∈ Z[[x]] ∗ , n ≥ 2. ..."
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Motivated by the discovery that the eighth root of the theta series of the E8 lattice and the 24th root of the theta series of the Leech lattice both have integer coefficients, we investigate the question of when an arbitrary element f ∈ Z[[x]] ∗ can be written as f = g n for g ∈ Z[[x]] ∗ , n ≥ 2. Let Pn: = {g n  g ∈ Z[[x]] ∗ } and let µn: = n ∏ pn p. We show among other things that (i) for f ∈ Z[[x]] ∗ , f ∈ Pn ⇔ f (mod µn) ∈ Pn, and (ii) if f ∈ Pn, up to sign there is a unique g ∈ Pn with coefficients mod µn/n such that f ≡ g n (mod µn). In particular, if f ≡ 1 (mod µn) then f ∈ Pn. The latter assertion implies that the theta series of any extremal even unimodular lattice in R n (e.g. E8 ∈ R 8) is in Pn if n is of the form 2 i 3 j 5 k (i ≥ 3). There do not seem to be any exact analogues for codes, although we show that the weight enumerator of the rth order ReedMuller code of length 2 m is in P2 r. We give a number of other results and conjectures, and establish a conjecture of Paul D. Hanna that there is a unique element f ∈ Pn (n ≥ 2) with coefficients restricted to the set {1, 2,..., n}. (1)
Bracket notation for the ‘coefficient of ’ operator
, 1993
"... When G(z) is a power series in z, many authors now write ‘[z n] G(z) ’ for the coefficient of z n in G(z), using a notation introduced by Goulden and Jackson in [5, p. 1]. More controversial, however, is the proposal of the same authors [5, p. 160] to let ‘[z n /n!] G(z)’ denote the coefficient of z ..."
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When G(z) is a power series in z, many authors now write ‘[z n] G(z) ’ for the coefficient of z n in G(z), using a notation introduced by Goulden and Jackson in [5, p. 1]. More controversial, however, is the proposal of the same authors [5, p. 160] to let ‘[z n /n!] G(z)’ denote the coefficient of z n /n!, i.e., n! times the coefficient of z n. An alternative generalization of [z n] G(z), in which we define [F(z)] G(z) to be a linear function of both F and G, seems to be more useful because it facilitates algebraic manipulations. The purpose of this paper is to explore some of the properties of such a definition. The remarks are dedicated to Tony Hoare because of his lifelong interest in the improvement of notations that facilitate manipulation. Informal introduction. In this paper ‘[z 2 + 2z 3] G(z) ’ will stand for the coefficient of z 2 plus twice the coefficient of z 3 in G(z), when G(z) is a function of z for which such coefficients are well defined. More generally, if F(z) = f0 + f1z + f2z 2 + · · · and G(z) = g0 + g1z + g2z 2 + · · ·, we will let [F(z)] G(z) = f0g0 + f1g1 + f2g2 + · · · be the “dot product ” of the vectors (f0, f1, f2,...) and (g0, g1, g2,...), assuming that the infinite sum exists. Still more generally, if F(z) = · · ·+f−2z −2 +f−1z − +f0+f1z+f2z 2 + · · · and G(z) = · · ·+g−2z −2 + g−1z − + g0 + g1z + g2z 2 + · · · are doubly infinite series, we will write
DISCRETE MATHEMATICS Generalized
, 1994
"... To Leonard Carlitz in his eightyninth year The theory of modular binomial lattices enables the simultaneous combinatorial analysis of finite sets, vector spaces, and chains. Within this theory three generalizations of Stifling numbers of the second kind, and of Lah numbers, are developed. 1. Stirli ..."
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To Leonard Carlitz in his eightyninth year The theory of modular binomial lattices enables the simultaneous combinatorial analysis of finite sets, vector spaces, and chains. Within this theory three generalizations of Stifling numbers of the second kind, and of Lah numbers, are developed. 1. Stirling numbers and their formal generalizations The notational conventions of this paper are as follows: N = {0,1,2....}, P = {1,2,...}, [0] =0, and In] = {1..... n} for n ~ P. Empty sums take the value 0 and empty products the value 1. Also, x ° = x ° = x ° = 1 for all x (including x = 0), and forn~P,x ~=x(x 1)...(xn+ 1) andx n=x(x+ 1).(x+n1). As enumerator of partitions of In] with k blocks, the Stirling number of the second kind S(n, k) plays a central role in elementary combinatorics. Not surprisingly, apart from the boundary values S(n,O) = J,.o and S(n,k) = 0 for 0 ~< n < k, there are many representations of these numbers. From the standpoint of generalizations pursued in this paper these representations fall naturally into three classes: