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Editorial
p. 1-2
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 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. [...]

Research
p. 3-8
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 Ono about the decomposition of certain finite field hypergeometric functions into functions of lower dimension.

p. 9-30
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 $\pi $ , there are formulas for $1/\pi $ of the form Following Ramanujan's work on modular equations and approximations of $\pi $ , there are formulas for $1/\pi $ of the form $\sum _{k=0}^{\infty}\frac{{\left(\frac{1}{2}\right)}_{k}{\left(\frac{1}{d}\right)}_{k}{\left(\frac{d-1}{d}\right)}_{k}}{k{!}^{3}}(ak+1){\left({\lambda}_{d}\right)}^{k}=\frac{\delta}{\pi}$ for $d=2,3,4,6,$ where ${\u0142}_{d}$ are singular values that correspond to elliptic curves with complex multiplication, and $a,\delta $ 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.

p. 31-45
Received: 19 February 2013 / Revised: 7 March 2013 / Accepted: 7 March 2013 / Published: 14 March 2013

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Abstract: Let C_{1} and C_{2} be algebraic plane curves in ${\u2102}^{2}$ such that the curves intersect in d_{1} · d_{2} points where d_{1} , d_{2} are the degrees of the curves respectively. Oka and Sakamoto proved that π1(${\u2102}^{2}$ \ C_{1} U C_{2} )) ≅ π1 (${\u2102}^{2}$ \ C_{1} ) × π1 (${\u2102}^{2}$ \ C_{2} ) [1]. In this paper we prove the converse of Oka and Sakamoto’s result for line arrangements. Let A _{1} and A _{2} be non-empty arrangements of lines in ${\u2102}^{2}$ such that π1 (M(A _{1} U A _{2} )) ≅ π1 (M(A _{1} )) × π1 (M(A _{2} )) Then, the intersection of A _{1} and A _{2} consists of /A _{1} / · /A _{2} / points of multiplicity two.

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