Gauss’ lemma

Gauss Lemma and Law of Quadratic Reciprocity

The papers [20], [10], [9], [11], [4], [1], [2], [17], [8], [19], [7], [16], [13], [21], [22], [5], [18], [3], [15], [6], and [23] provide the terminology and notation for this paper. For simplicity, we adopt the following convention: i, i1, i2, i3, j, a, b, x denote integers, d, e, n denote natural numbers, f , f ′ denote finite sequences of elements of Z, g, g1, g2 denote finite sequences of ...

Gauss elimination algorithm for interval matrix

A simple proof of Zariski's Lemma

‎Our aim in this very short note is to show that the proof of the‎ ‎following well-known fundamental lemma of Zariski follows from an‎ ‎argument similar to the proof of the fact that the rational field‎ ‎$mathbb{Q}$ is not a finitely generated $mathbb{Z}$-algebra.

Generalizations of Borel-Cantelli Lemma

The Borel-Cantelli Lemma is very important in the probability theory‎. ‎In this paper‎, ‎we first describe the general case of the Borel-Cantelli Lemma‎. ‎The first part of this lemma‎, ‎assuming convergence and the second part includes divergence and independence assumptions‎. ‎In the following‎, ‎we have brought generalizations of the first and second part of this lemma‎. ‎In most generalizat...

Symmetric Gauss - Lobattoand Modified Anti - Gauss Rules