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Mathematics, Volume 1, Issue 3 (September 2013) – 3 articles , Pages 76-110

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Article
Effective Congruences for Mock Theta Functions
Mathematics 2013, 1(3), 100-110; https://doi.org/10.3390/math1030100 - 04 Sep 2013
Cited by 2 | Viewed by 3061
Abstract
Let M(q) = c(n)q n be one of Ramanujan’s mock theta functions. We establish the existence of infinitely many linear congruences of the form: c(An + B) 0 (mod l j ) where A is a multiple of l and an auxiliary prime, p. Moreover, we give an effectively computable upper bound on the smallest such p for which these congruences hold. The effective nature of our results is based on the prior works of Lichtenstein [1] and Treneer [2]. Full article
Article
Scattering of Electromagnetic Waves by Many Nano-Wires
Mathematics 2013, 1(3), 89-99; https://doi.org/10.3390/math1030089 - 18 Jul 2013
Cited by 7 | Viewed by 2700
Abstract
Electromagnetic wave scattering by many parallel to the zaxis, thin, impedance, parallel, infinite cylinders is studied asymptotically as a 0. Let Dm be the cross-section of the mth cylinder, a be its radius and x ^ m = (x m1 , x m2 ) be its center, 1 m M , M = M (a). It is assumed that the points, x ^ m , are distributed, so that N(Δ)= 1 2πa Δ N ( x ^ )d x ^ [1+o(1)] where N (∆) is the number of points, x ^ m , in an arbitrary open subset, ∆, of the plane, xoy. The function, N( x ^ ) 0 , is a continuous function, which an experimentalist can choose. An equation for the self-consistent (effective) field is derived as a 0. A formula is derived for the refraction coefficient in the medium in which many thin impedance cylinders are distributed. These cylinders may model nano-wires embedded in the medium. One can produce a desired refraction coefficient of the new medium by choosing a suitable boundary impedance of the thin cylinders and their distribution law. Full article
Article
On the Distribution of the spt-Crank
Mathematics 2013, 1(3), 76-88; https://doi.org/10.3390/math1030076 - 28 Jun 2013
Cited by 11 | Viewed by 3391
Abstract
Andrews, Garvan and Liang introduced the spt-crank for vector partitions. We conjecture that for any n the sequence { N S (m , n) } m is unimodal, where N S (m , n) is the number of S-partitions of size n with crank m weight by the spt-crank. We relate this conjecture to a distributional result concerning the usual rank and crank of unrestricted partitions. This leads to a heuristic that suggests the conjecture is true and allows us to asymptotically establish the conjecture. Additionally, we give an asymptotic study for the distribution of the spt-crank statistic. Finally, we give some speculations about a definition for the spt-crank in terms of “marked” partitions. A “marked” partition is an unrestricted integer partition where each part is marked with a multiplicity number. It remains an interesting and apparently challenging problem to interpret the spt-crank in terms of ordinary integer partitions. Full article
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