Mathematics publications (MU)
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Items in this collection are the scholarly output of the Department of Mathematics faculty, staff, and students, either alone or as co-authors, and which may or may not have been published in an alternate format. Items may contain more than one file type.
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Item Turing-type instabilities in a mathematical model of notch and retinoic acid pathways(2006) Bani-Yaghoub, Majid; Amundsen, David E.; University of Missouri-Columbia. College of Arts and Sciences. Department of Mathematics.In this paper we employ Turing Theory to study the effects of Notch and Retinoic Acid (RA) pathways on neuronal differentiation. A mathematical model consisting of two reaction-diffusion subsystems is presented such that each subsystem is compelled by the level of RA and activated Notch utilized in the experiment. We hypothesize an interaction between RA and Notch pathways. This interaction is reflected in the model by considering a perturbation to both subsystems. The conditions for the existence of Turing instabilities are established and compared for both cases where the two subsystems are either perturbed or unperturbed. For these two cases we present numerical simulations for Turing instabilities and Turing bifurcations. The study of Turing mechanism in interacting signaling pathways might bring some insight into the recent biological findings of neuronal differentiation.Item On integers with a special divisibility property(Masarykova Universita, 2006) Banks, William David, 1964-; Luca, FlorianIn this note, we study those positive integers n which are divisible by the Carmichael function.Item Non-residues and primitive roots in Beatty sequences(Australian Mathematical Society, 2006) Banks, William David, 1964-; Shparlinski, Igor E.We study multiplicative character sums taken on the values of a non-homogeneous Beatty sequence Bα,β = {⌊αn + β⌋ : n = 1,2,3,…}, where α,β ∈ R, and α is irrational. In particular, our bounds imply that for every fixed ε > 0, if p is sufficiently large and p1/2+ε ≤ N ≤ p, then among the first N elements of Bα,β, there are N/2+o(N) quadratic non-residues modulo p. When more information is available about the Diophantine properties of α, then the error term o(N) emits a shaper estimate.Item Coincidences in the values of the Euler and Carmichael functions(Polish Academy of Sciences, Institute of Mathematics, 2006) Banks, William David, 1964-; Friedlander, J. B. (John B.); Luca, Florian; Pappalardi, Francesco; Shparlinski, Igor E.The Euler function has long been regarded as one of the most basic of the arithmetic functions. More recently, partly driven by the rise in importance of computational number theory, the Carmichael function has drawn an ever-increasing amount of attention. A large number of results have been obtained, both about the growth rate and about various arithmetical properties of the values of these two functions; see for example [2, 3, 5-7, 10-18, 20, 22, 23] and the references therein.Item Incomplete exponential sums and Diffie-Hellman triples(Cambridge University Press, 2006) Banks, William David, 1964-; Friedlander, J. B. (John B.); Koniagin, S. V. (Sergeĭ Vladimirovich); Shparlinski, Igor E.Let p be a prime and 79 an integer of order t in the multiplicative group modulo p. In this paper, we continue the study of the distribution of Diffie-Hellman triples (V-x, V-y, V-xy) by considering the closely related problem of estimating exponential sums formed from linear combinations of the entries in such triples. We show that the techniques developed earlier for complete sums can be combined, modified and developed further to treat incomplete sums as well. Our bounds imply uniformity of distribution results for Diffie-Hellman triples as the pair (x, y) varies over small boxes.