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208 KiB

Open AccessArticle

On the Distribution of the spt-Crank

by George E. Andrews, Freeman J. Dyson and Robert C. Rhoades

Mathematics 2013, 1(3), 76-88; https://doi.org/10.3390/math1030076 - 28 Jun 2013

Cited by 13 | Viewed by 5339

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 [...] Read more.

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

201 KiB

Open AccessArticle

Scattering of Electromagnetic Waves by Many Nano-Wires

by Alexander G. Ramm

Mathematics 2013, 1(3), 89-99; https://doi.org/10.3390/math1030089 - 18 Jul 2013

Cited by 8 | Viewed by 4200

Abstract

Electromagnetic wave scattering by many parallel to the z−axis, thin, impedance, parallel, infinite cylinders is studied asymptotically as a → 0. Let D_m be the cross-section of the m−th cylinder, a be its radius and [...] Read more.

{\hat{x}}_{m} {= (x}_{m1} {, x}_{m2})

be its center, 1 ≤ m ≤ M , M = M (a). It is assumed that the points,

{\hat{x}}_{m}

, are distributed, so that

N (Δ) = \frac{1}{2 π a} \int_{Δ} N (\hat{x}) d \hat{x} [1 + o (1)]

where N (∆) is the number of points,

{\hat{x}}_{m}

, in an arbitrary open subset, ∆, of the plane, xoy. The function,

N (\hat{x}) \geq 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

201 KiB

Open AccessArticle

Effective Congruences for Mock Theta Functions

by Nickolas Andersen, Holley Friedlander, Jeremy Fuller and Heidi Goodson

Mathematics 2013, 1(3), 100-110; https://doi.org/10.3390/math1030100 - 4 Sep 2013

Cited by 2 | Viewed by 4821

Abstract

Let

M (q) = \sum {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 (A n + B) \equiv 0 (mod l^{j})

where A [...] Read more.

Let

M (q) = \sum {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 (A n + B) \equiv 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

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

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