The articles [8], [11], [10], [5], [1], [7], [12], [4], [3], [2], [9], and [6] provide the notation and terminology for this paper. In this paper i, j, k, k1, k2, n, m, i1, i2, j1, j2 are natural numbers. Let us consider i1, i2. We say that i1 and i2 are adjacent if and only if: (Def. 1) i2 = i1 + 1 or i1 = i2 + 1. One can prove the following propositions: (1) For all i1, i2 such that i1 and i2...

Let p be a large prime number, K,L, M, λ be integers with 1 ≤ M ≤ p and gcd(λ, p) = 1. The aim of our paper is to obtain sharp upper bound estimates for the number I2(M ; K, L) of solutions of the congruence xy ≡ λ (mod p), K + 1 ≤ x ≤ K + M, L + 1 ≤ y ≤ L + M and for the number I3(M ; L) of solutions of the congruence xyz ≡ λ (mod p), L + 1 ≤ x, y, z ≤ L + M. Using the idea of Heath-Brown from...

Bordetella pertussis toxin (PTx), also known as islet-activating protein, induces insulin secretion by ADP-ribosylation of inhibitory G proteins. PTx-induced insulin secretion may result either from inactivation of Gα(o) proteins or from combined inactivation of Gα(o), Gα(i1), Gα(i2), and Gα(i3) isoforms. However, the specific role of Gα(i2) in pancreatic β-cells still remains unknown. In globa...

1. For any instance I1 of π1, I2 = f(I1) is an instance of π2 such that OPTπ2(I2) ≤ OPTπ1(I1). 2. For any feasible solution S2 of I2, S1 = g(I1, S2) (g maps S2 into an instance of I1) we have Costπ1(I1, S1) ≤ Costπ2(I2, S2). Note that OPTπ2(I2) ≤ OPTπ1(I1) ≤ Costπ1(I1, S1) ≤ Costπ2(I2, S2). Therefore, if there is an approximation factor ∆ for π2 then there is an approximation factor ∆ for π1 as...

In this paper we introduce and investigate I2-convergence, I∗ 2 -convergence, I2-limit points, and I2-cluster points of a double sequence in a fuzzy 2-normed linear space. We prove a decomposition theorem for I2-convergence of double sequences. The notions of I2-double Cauchy and I∗ 2 -double Cauchy sequence are defined, and some of their properties are studied.

(6) Let G be a column Y-constant line Y-increasing matrix over E2 T and i1, i2, j1, j2 be natural numbers. Suppose 〈i1, j1〉 ∈ the indices of G and 〈i2, j2〉 ∈ the indices of G. If (G◦ (i1, j1))2 = (G◦ (i2, j2))2, then j1 = j2. (7) Let G be a line X-constant column X-increasing matrix over E2 T and i1, i2, j1, j2 be natural numbers. Suppose 〈i1, j1〉 ∈ the indices of G and 〈i2, j2〉 ∈ the indices o...

Alternative splicing (AS) is one of the most significant components of the functional complexity of human UDP-glucuronosyltransferase enzymes (UGTs), particularly for the UGT1A gene, which represents one of the best examples of a drug-metabolizing gene regulated by AS. Shorter UGT1A isoforms [isoform 2 (i2)] are deficient in glucuronic acid transferase activity but function as negative regulato...

The purpose of this article is to describe and clarify a tool that is at the core of the definition phase of most quality improvement projects. This tool is called the critical to quality (CTQ) flowdown. It relates high-level strategic focal points to project objectives. In their turn project objectives are linked to and decomposed into CTQs, which are made operational in the form of measuremen...