A Pythagorean Triple Preserving Matrix (PTPM) is a $3 \times 3$ matrix such that if it is multiplied by a Pythagorean Triple, the result is also a Pythagorean Triple. Necessary and sufficient conditions for a Pythagorean Triple Preserving Matrix to be symmetric are given. Monoids of Symmetric Pythagorean Triple Preserving Matrices (SPTPM) with positive integer entries will be developed, and to ensure that the set is closed under matrix multiplication, the focus will be on finding commutative SPTPM's.

A natural bijection from $\mathbb{Z} ^2$ onto the set of all Pythagorean triples ${\mathcal P} = \{ (a,b,c) \in \mathbb{Z} ^3 : a^2 + b^2 = c^2 \}$ is given (Theorem 6). Consequently, all algebraic structures of $\mathbb{Z} ^2$ are carried in a natural way onto ${\mathcal P}$ (Theorems 7, 8, and 9). This solves the open problem of defining ring operations under which ${\mathcal P}$ is essentially a different ring than the one constructed by B. Dawson (Example, Section 5). This article and the enumeration of its sections and theorems is a continuation of the author's paper [3].

For $C^r$ differentiable manifolds $E$ and $B$ of the same dimension, and $\pi \colon E \to B$ a local $C^r$ diffeomorphism and onto, we examine, as with the topological case, the extent to which some basic properties of $\pi$ are inherited by the induced map $p$ on spaces of fiber transferring maps with respect to $\pi$.

Let $R$ be a left Artinian ring. Dlab and Ringel have shown that $R$ is hereditary if and only if every chain of idempotent ideals can be refined to a heredity chain [1]. In particular, if $R$ is a basic hereditary ring, then every primitive ideal is a heredity ideal. The converse to this is clearly false. (See Example 1). We will introduce a class of rings that includes serial rings and monomial algebras, for which the converse does hold.

We examine rings having the property that, for each minimally generated module, the number of elements in a minimal generator set is invariant.

Lower and upper bounds are given to the ratio of the error of the perturbed solution and the distance of the unperturbed solution to the null space of the matrix for general consistent systems of linear equations and least squares problems.

In this paper a singular integral equation of Abel's type is solved in a closed form by integration over contours in the complex plane.