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Mathematics, Volume 1, Issue 1 (March 2013), Pages 1-45

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Editorial

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Open AccessEditorial Mathematics—An Open Access Journal
Mathematics 2013, 1(1), 1-2; doi:10.3390/math1010001
Received: 18 December 2012 / Revised: 21 December 2012 / Accepted: 21 December 2012 / Published: 28 December 2012
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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 [...] Read more.
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. [...] Full article

Research

Jump to: Editorial

Open AccessArticle On Matrices Arising in the Finite Field Analogue of Euler’s Integral Transform
Mathematics 2013, 1(1), 3-8; doi:10.3390/math1010003
Received: 6 January 2013 / Revised: 15 January 2013 / Accepted: 22 January 2013 / Published: 5 February 2013
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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 [...] Read more.
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. Full article
Open AccessArticle ρ — Adic Analogues of Ramanujan Type Formulas for 1/π
Mathematics 2013, 1(1), 9-30; doi:10.3390/math1010009
Received: 18 February 2013 / Revised: 26 February 2013 / Accepted: 1 March 2013 / Published: 13 March 2013
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Abstract
Following Ramanujan's work on modular equations and approximations of π, there are formulas for 1/π of the form Following Ramanujan's work on modular equations and approximations of π, there are formulas for 1/π of the form [...] Read more.
Following Ramanujan's work on modular equations and approximations of π, there are formulas for 1/π of the form Following Ramanujan's work on modular equations and approximations of π, there are formulas for 1/π of the form k = 0 ( 1 2 ) k ( 1 d ) k ( d - 1 d ) k k ! 3 ( a k + 1 ) ( λ d ) k = δ π for d=2,3,4,6, where łd are singular values that correspond to elliptic curves with complex multiplication, and a,δ are explicit algebraic numbers. In this paper we prove a p-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. Full article
Open AccessArticle A Converse to a Theorem of Oka and Sakamoto for Complex Line Arrangements
Mathematics 2013, 1(1), 31-45; doi:10.3390/math1010031
Received: 19 February 2013 / Revised: 7 March 2013 / Accepted: 7 March 2013 / Published: 14 March 2013
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Abstract
Let C1 and C2 be algebraic plane curves in 2 such that the curves intersect in d1 · d2 points where d1, d2 are the degrees of the curves respectively. Oka and Sakamoto proved [...] Read more.
Let C1 and C2 be algebraic plane curves in 2 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( 2 \ C1 U C2)) ≅ π1 ( 2 \  C1) × π1 ( 2 \  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 2 such that π1 (M(A1A2)) ≅  π1 (M(A1)) ×  π1 (M(A2)) Then, the intersection of A1 and A2 consists of /A1/ ·  /A2/ points of multiplicity two. Full article

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