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Mathematics, Volume 1, Issue 1 (March 2013) – 4 articles , Pages 1-45

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210 KiB  
Article
A Converse to a Theorem of Oka and Sakamoto for Complex Line Arrangements
by Kristopher Williams
Mathematics 2013, 1(1), 31-45; https://doi.org/10.3390/math1010031 - 14 Mar 2013
Cited by 65 | Viewed by 4553
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 that [...] 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(A1 U A2)) ≅ 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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301 KiB  
Article
ρ — Adic Analogues of Ramanujan Type Formulas for 1/π
by Sarah Chisholm, Alyson Deines, Ling Long, Gabriele Nebe and Holly Swisher
Mathematics 2013, 1(1), 9-30; https://doi.org/10.3390/math1010009 - 13 Mar 2013
Cited by 14 | Viewed by 5706
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
141 KiB  
Article
On Matrices Arising in the Finite Field Analogue of Euler’s Integral Transform
by Michael Griffin and Larry Rolen
Mathematics 2013, 1(1), 3-8; https://doi.org/10.3390/math1010003 - 05 Feb 2013
Cited by 31 | Viewed by 4065
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 [...] 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
62 KiB  
Editorial
Mathematics—An Open Access Journal
by Sergei K. Suslov
Mathematics 2013, 1(1), 1-2; https://doi.org/10.3390/math1010001 - 28 Dec 2012
Cited by 135 | Viewed by 5870
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 [...] 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
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