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**Mathematics**Follow

**Open Access journal**

ISSN (Online) 2227-7390

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**MDPI**[119 journals]

**Mathematics, Vol. 2, Pages 1-11: One-Dimensional Nonlinear Stefan Problems**

in Storm’s Materials**Authors:***Adriana Briozzo, María Natale***Pages:**1 - 11**Abstract:**We consider two one-phase nonlinear one-dimensional Stefan problems for a semi-infinite material x > 0; with phase change temperature Tf : We assume that the heat capacity and the thermal conductivity satisfy a Storm’s condition. In the first case, we assume a heat flux boundary condition of the type [PLEASE CHECK FORMULA IN THE PDF], and in the second case, we assume a temperature boundary condition T = Ts < Tf at the fixed face. Solutions of similarity type are obtained in both cases, and the equivalence of the two problems is demonstrated. We also give procedures in order to compute the explicit solution.**PubDate:**2013-12-27**DOI:**10.3390/math2010001**Issue No:***Vol. 2, No. 1 (2013)*

**Mathematics, Vol. 1, Pages 111-118: Sign-Periodicity of Traces of Singular**

Moduli**Authors:***Dohoon Choi, Byungchan Kim, Subong Lim***Pages:**111 - 118**Abstract:**Zagier proved that the generating functions of traces of singular values of Jm(z) are weight 3/2 weakly holomorphic modular forms. In this paper we prove that there is the sign-periodicity of traces of singular values of Jm(z).**PubDate:**2013-10-15**DOI:**10.3390/math1040111**Issue No:***Vol. 1, No. 4 (2013)*

**Mathematics, Vol. 1, Pages 76-88: On the Distribution of the spt-Crank****Authors:***George Andrews, Freeman Dyson, Robert Rhoades***Pages:**76 - 88**Abstract:**Andrews, Garvan and Liang introduced the spt-crank for vector partitions. We conjecture that for any n the sequence {NS (m, n)}m is unimodal, where NS (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.**PubDate:**2013-06-28**DOI:**10.3390/math1030076**Issue No:***Vol. 1, No. 3 (2013)*

**Mathematics, Vol. 1, Pages 89-99: Scattering of Electromagnetic Waves by**

Many Nano-Wires**Authors:***Alexander Ramm***Pages:**89 - 99**Abstract:**Electromagnetic wave scattering by many parallel to the z−axis, thin, impedance, parallel, infinite cylinders is studied asymptotically as a → 0. Let Dm be the cross-section of the m−th cylinder, a be its radius and xˆm = (xm1, xm2) be its center, 1 ≤ m ≤ M , M = M (a). It is assumed that the points, xˆm, are distributed, so that N (∆) = (1 / 2πa) * ∫∆ N (xˆ)dxˆ[1 + o(1)], where N (∆) is the number of points, xˆm, in an arbitrary open subset, ∆, of the plane, xoy. The function, N (xˆ) ≥ 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.**PubDate:**2013-07-18**DOI:**10.3390/math1030089**Issue No:***Vol. 1, No. 3 (2013)*

**Mathematics, Vol. 1, Pages 100-110: Effective Congruences for Mock Theta**

Functions**Authors:***Nickolas Andersen, Holley Friedlander, Jeremy Fuller, Heidi Goodson***Pages:**100 - 110**Abstract:**Let M(q) = ∑c(n) qn 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 lj) 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].**PubDate:**2013-09-04**DOI:**10.3390/math1030100**Issue No:***Vol. 1, No. 3 (2013)*

**Mathematics, Vol. 1, Pages 46-64: Stability of Solutions to Evolution**

Problems**Authors:***Alexander Ramm***Pages:**46 - 64**Abstract:**Large time behavior of solutions to abstract differential equations is studied. The results give sufficient condition for the global existence of a solution to an abstract dynamical system (evolution problem), for this solution to be bounded, and for this solution to have a finite limit as t → ∞, in particular, sufficient conditions for this limit to be zero. The evolution problem is: u = A(t)u + F(t; u) + b(t); t ≥ 0; u(0) = u0: (*) Here u:= du/dt , u = u(t) ∈ H, H is a Hilbert space, t ∈ R+ := [0;∞), A(t) is a linear dissipative operator: Re(A(t)u; u)**PubDate:**2013-05-13**DOI:**10.3390/math1020046**Issue No:***Vol. 1, No. 2 (2013)*

**Mathematics, Vol. 1, Pages 65-75: On the Class of Dominant and Subordinate**

Products**Authors:***Alexander Berkovich, Keith Grizzell***Pages:**65 - 75**Abstract:**In this paper we provide proofs of two new theorems that provide a broad class of partition inequalities and that illustrate a na¨ıve version of Andrews’ anti-telescoping technique quite well. These new theorems also put to rest any notion that including parts of size 1 is somehow necessary in order to have a valid irreducible partition inequality. In addition, we prove (as a lemma to one of the theorems) a rather nontrivial class of rational functions of three variables has entirely nonnegative power series coefficients.**PubDate:**2013-05-15**DOI:**10.3390/math1020065**Issue No:***Vol. 1, No. 2 (2013)*

**Mathematics, Vol. 1, Pages 3-8: On Matrices Arising in the Finite Field**

Analogue of Euler’s Integral Transform**Authors:***Michael Griffin, Larry Rolen***Pages:**3 - 8**Abstract:**In his 1984 Ph.D. thesis, J. Greene defined an analogue of the Euler integral transform for finite field hypergeometric series. Here we consider a special family of matrices which arise naturally in the study of this transform and prove a conjecture of Ono about the decomposition of certain finite field hypergeometric functions into functions of lower dimension.**PubDate:**2013-02-05**DOI:**10.3390/math1010003**Issue No:***Vol. 1, No. 1 (2013)*

**Mathematics, Vol. 1, Pages 9-30: ρ — Adic Analogues of**

Ramanujan Type Formulas for 1/π**Authors:***Sarah Chisholm, Alyson Deines, Ling Long, Gabriele Nebe, Holly Swisher***Pages:**9 - 30**Abstract:**Following Ramanujan’s work on modular equations and approximations of π, there are formulas for 1/π of the form [PLEASE CHECK FORMULA IN THE PDF] for d = 2, 3, 4, 6, where λd are singular values that correspond to elliptic curves with complex multiplication, and α, δ are explicit algebraic numbers. In this paper we prove a ρ-adic version of this formula in terms of the so-called Ramanujan type congruence. In addition, we obtain a new supercongruence result for elliptic curves with complex multiplication.**PubDate:**2013-03-13**DOI:**10.3390/math1010009**Issue No:***Vol. 1, No. 1 (2013)*

**Mathematics, Vol. 1, Pages 31-45: A Converse to a Theorem of Oka and**

Sakamoto for Complex Line Arrangements**Authors:***Kristopher Williams***Pages:**31 - 45**Abstract:**Let C1 and C2 be algebraic plane curves in C2 such that the curves intersect in d1 · d2 points where d1, d2 are the degrees of the curves respectively. Oka and Sakamoto proved that π1(C2 \ C1 U C2)) ≅ π1 (C2 \ C1) × π1 (C2 \ C2) [1]. In this paper we prove the converse of Oka and Sakamoto’s result for line arrangements. Let A1 and A2 be non-empty arrangements of lines in C2 such that π1 (M(A1 U A2)) ≅ π1 (M(A1)) × π1 (M(A2)) Then, the intersection of A1 and A2 consists of /A1/ · /A2/ points of multiplicity two.**PubDate:**2013-03-14**DOI:**10.3390/math1010031**Issue No:***Vol. 1, No. 1 (2013)*

**Mathematics, Vol. 1, Pages 1-2: Mathematics—An Open Access Journal****Authors:***Sergei Suslov***Pages:**1 - 2**Abstract:**As is widely known, mathematics plays a unique role in all natural sciences as a refined scientific language and powerful research tool. Indeed, most of the fundamental laws of nature are written in mathematical terms and we study their consequences by numerous mathematical methods (and vice versa, any essential progress in a natural science has been accompanied by fruitful developments in mathematics). In addition, the mathematical modeling in various interdisciplinary problems and logical development of mathematics on its own should be taken into account. [...]**PubDate:**2012-12-28**DOI:**10.3390/math1010001**Issue No:***Vol. 1, No. 1 (2012)*