Dean S. Clark

Global asymptotic behavior of a two-dimensional difference equation modelling competition

We investigate the global asymptotic behavior of solutions of the system of difference equations xn+1 = xn/ a + cyn, yn+1 = yn/ b + dxn, n =0,1,..., where the parameters a and b are in (0, 1), c and d are arbitrary positive numbers and the initial conditions x0 and y0 are arbitrary nonnegative numbers. We show that the stable manifold of this system separates the positive quadrant into basins of attraction of two types of asymptotic behavior. In the case where a = b we find an explicit equation for the stable manifold.

Necessary and sufficient conditions for the Robbins-Monro method

This paper examines the relation between convergence of the Robbins-Monro iterates Xn+1= Xn-an[latin small letter f with hook](Xn)+an[xi]n, [latin small letter f with hook]([theta])=0, and the laws of large numbers Sn=an[Sigma]n-1j=0 [xi]j-->0 as n-->+[infinity]. If an is decreasing at least as rapidly as c/n, then Xn-->[theta] w.p. 1 (resp. in Lp, p[greater-or-equal, slanted]1) implies Sn-->0 w.p. 1 (resp. in Lp, p[greater-or-equal, slanted]1) as n-->+[infinity]. If an is decreasing at least as slowly as c[+45 degree rule]n and limn-->+[infinity]a n=0, then Sn-->0 w.p. 1 (resp. in Lp, p[greater-or-equal, slanted]2) implies Xn-->[theta] w.p. 1 (resp. in Lp, p[greater-or-equal, slanted]2) as n -->+[infinity]. Thus, there is equivalence in the frequently examined case an[reverse similar, equals]c[+45 degree rule]n. Counter examples show that the LLN must have the form of Sn, that the rate of decrease conditions are sharp, that the weak LLN is neither necessary nor sufficient for the convergence in probability of Xn to [theta] when an[reverse similar, equals]c[+45 degree rule]n.

Estimates of the orthogonal polynomials with weight exp(-x m ), m an even positive integer

Etablissement de quelques estimations des polynomes orthogonaux de poids exp(−X m ), ou m est un entier positif pair

Short proof of a discrete Gronwall inequality

Abstract We give an elementary proof of a generalization of the classical discrete Gronwall inequality x n ⩽ a n + ∑ j = n 0 n − 1 b j x j , n = n 0 ,…, N , implies x n ⩽a ∗ ∏ j = n 0 n −1 (1+b j ) a ∗ = max {a j : j = n 0 ,…,N} , n = n 0 ,…, N ) which improves the description of the multiplier a ∗ to a minimum, rather than a maximum, over a certain subset of indices in { n 0 ,…, N }.

Periodic solutions of arbitrary length in a simple integer iteration

We prove that all solutions to the nonlinear second-order difference equation in integers yn+1 = ⌈ayn⌉-yn-1, {a ∈ ℝ:|a|<2, a≠0,±1}, y0, y1 ∈ ℤ, are periodic. The first-order system representation of this equation is shown to have self-similar and chaotic solutions in the integer plane.

Avoiding-sequences with minimum sum

Abstract A circular sequence of positive integers of length n is a sequence of n terms in which the first and last are considered consecutive. For x≤n a positive integer, such a sequence is x-avoiding if no set of consecutive terms sums to x. We show that an x-avoiding circular sequence of length n satisfies a1+a2+⋯+an≥2n, and give a simple necessary and sufficient condition for equality. Minimizing sequences are exhibited when the minimum sum is known.

Herbert and the Hungarian Mathematician: Avoiding Certain Subsequence Sums

Dean Clark has worked as a musician, economist, mathemati? cian, and scientific writer. His graduate degrees are in eco? nomics and applied mathematics, both from Brown University. He has taught at Roger Williams College, the University of Zurich (Switzerland), Hofstra University, and the University of Rhode Island, where he is currently Associate Professor of Mathematics. He is a jazz bassist, arranger and composer, and his compositions are occasionally performed in concert by URl jazz ensembles.

Second order difference equations related to the collatz 3n+1 conjecture

Consider the following two-term analog of the Collatz 3n+1 iteration:Start with two positive integers. If the sum is even, take the average. If the sum is odd, multiply the difference by 3/2 and add 1/2. Iterate on the last two terms of the sequence. More generally, for b≥1 an odd integer and :1,x 2eIN, we study the second order difference equations for their asymptotic properties, sensitivity to initial conditions, and predictability of the limit of {Xn }, which is shown necessarily to exist only when b = 1,3 or 5.

Inductive Tiling of the Plane by Penrose Aperiodic Rhombi

(1995). Inductive Tiling of the Plane by Penrose Aperiodic Rhombi. The College Mathematics Journal: Vol. 26, No. 4, pp. 266-267.

Circular avoiding sequences with prescribed sum

Abstract For given positive integers x, n , and s an x -avoiding circular sequence (of positive integers) of length n and sum s has no set of consecutive terms summing to x , even if wraparound is allowed. A necessary and sufficient condition for the existence of such a sequence is obtained. An effective method to construct avoiding sequences is given. For the cases of most interest the number of avoiding sequences is found.