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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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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
Applications Of The Classical Umbral Calculus
 Algebra Universalis
"... We describe applications of the classical umbral calculus to bilinear generating functions for polynomial sequences, identities for Bernoulli and related numbers, and Kummer congruences. 1. ..."
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We describe applications of the classical umbral calculus to bilinear generating functions for polynomial sequences, identities for Bernoulli and related numbers, and Kummer congruences. 1.
Overview of some general results in combinatorial enumeration
, 2008
"... This survey article is devoted to general results in combinatorial enumeration. The first part surveys results on growth of hereditary properties of combinatorial structures. These include permutations, ordered and unordered graphs and hypergraphs, relational structures, and others. The second part ..."
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This survey article is devoted to general results in combinatorial enumeration. The first part surveys results on growth of hereditary properties of combinatorial structures. These include permutations, ordered and unordered graphs and hypergraphs, relational structures, and others. The second part advertises five topics in general enumeration: 1. counting lattice points in lattice polytopes, 2. growth of contextfree languages, 3. holonomicity (i.e., Precursiveness) of numbers of labeled regular graphs, 4. frequent occurrence of the asymptotics cn −3/2 r n and 5. ultimate modular periodicity of numbers of MSOLdefinable structures. 1
ON A CONJECTURE OF WILF
"... Abstract. Let n and k be natural numbers and let S(n, k) denote the Stirling numbers of the second kind. It is a conjecture of Wilf that the alternating sum nX (−1) j S(n, j) j=0 is nonzero for all n> 2. We prove this conjecture for all n � ≡ 2 and � ≡ 2944838 mod 3145728 and discuss applications of ..."
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Abstract. Let n and k be natural numbers and let S(n, k) denote the Stirling numbers of the second kind. It is a conjecture of Wilf that the alternating sum nX (−1) j S(n, j) j=0 is nonzero for all n> 2. We prove this conjecture for all n � ≡ 2 and � ≡ 2944838 mod 3145728 and discuss applications of this result to graph theory, multiplicative partition functions, and the irrationality of padic series. 1.
On psiumbral extension of Stirling numbers and Dobinskilike formulas
 Advan. Stud. Contemp. Math
"... ψumbral extensions of the Stirling numbers of the second kind are considered and the resulting new type of Dobinskilike formulas are discovered. These extensions naturally encompass the two well known qextensions.The further consecutive ψumbral extensions of CarlitzGould qStirling numbers are ..."
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ψumbral extensions of the Stirling numbers of the second kind are considered and the resulting new type of Dobinskilike formulas are discovered. These extensions naturally encompass the two well known qextensions.The further consecutive ψumbral extensions of CarlitzGould qStirling numbers are therefore realized here in a twofold way. The fact that the umbral qextended Dobinski formula may also be interpreted as the average of powers of random variable Xq with the qPoisson distribution singles out the qextensions which appear to be a kind of ”bifurcation point ” in the domain of ψumbral extensions as expressed by Observations 2.1 and 2.2 as tackled with the paper closing down question.
Approximate Constructions In Finite Fields
"... this paper are new, we do not give complete detailed proofs but indicate the underlying ideas. Here we present a list of possible applications (which is certainly incomplete). We start from pointing out some general purpose applications: ffl Coding Theory : AP1, AP3, AP6 ..."
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this paper are new, we do not give complete detailed proofs but indicate the underlying ideas. Here we present a list of possible applications (which is certainly incomplete). We start from pointing out some general purpose applications: ffl Coding Theory : AP1, AP3, AP6
THE PERIOD OF THE BELL NUMBERS MODULO A PRIME
"... Abstract. We discuss the numbers in the title, and in particular whether the minimum period of the Bell numbers modulo a prime p can be a proper divisor of Np = (p p − 1)/(p − 1). It is known that the period always divides Np. The period is shown to equal Np for most primes p below 180. The investig ..."
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Abstract. We discuss the numbers in the title, and in particular whether the minimum period of the Bell numbers modulo a prime p can be a proper divisor of Np = (p p − 1)/(p − 1). It is known that the period always divides Np. The period is shown to equal Np for most primes p below 180. The investigation leads to interesting new results about the possible prime factors of Np. For example, we show that if p is an odd positive integer and m is a positive integer and q = 4m 2 p + 1 is prime, then q divides p m2 p − 1. Then we explain how this theorem influences the probability that q divides Np. 1.
Article electronically published on March 1, 2010 THE PERIOD OF THE BELL NUMBERS MODULO A PRIME
"... Abstract. We discuss the numbers in the title, and in particular whether the minimum period of the Bell numbers modulo a prime p can be a proper divisor of Np =(p p − 1)/(p − 1). It is known that the period always divides Np. The period is shown to equal Np for most primes p below 180. The investiga ..."
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Abstract. We discuss the numbers in the title, and in particular whether the minimum period of the Bell numbers modulo a prime p can be a proper divisor of Np =(p p − 1)/(p − 1). It is known that the period always divides Np. The period is shown to equal Np for most primes p below 180. The investigation leads to interesting new results about the possible prime factors of Np. For example, we show that if p is an odd positive integer and m is a positive integer and q =4m 2 p + 1 is prime, then q divides p m2 p − 1. Then we explain how this theorem influences the probability that q divides Np. 1.
CONGRUENCES FOR A WIDE CLASS OF INTEGERS BY USING GESSEL f S METHOD
, 1992
"... Let Pn,n = 0,1,2,..., be a sequence of integers that is defined by its exponential generating function/^) as That i s ^ ) is a Hurwitz series in x. As regards Bell numbers [fix) = exp{exp{x}l}], Lunnon, Pleasants, & Stephens [4] and ..."
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Let Pn,n = 0,1,2,..., be a sequence of integers that is defined by its exponential generating function/^) as That i s ^ ) is a Hurwitz series in x. As regards Bell numbers [fix) = exp{exp{x}l}], Lunnon, Pleasants, & Stephens [4] and
A CONGRUENCE FOR A CLASS OF EXPONENTIAL NUMBERS
, 1983
"... A sequence of exponential generating function as numbers, say Pn, is defined by i t s exponential ^2>Pnx n /.n \ = exp{^(x)} ..."
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A sequence of exponential generating function as numbers, say Pn, is defined by i t s exponential ^2>Pnx n /.n \ = exp{^(x)}