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.
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)
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.
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.
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.
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. [...]
