Traditionally, Pythagorean triples (PT) consist of three positive integers, $$$(x,y,z)\in {\mathbb{Z}}_{+}^3$, such that $$$x^2+y^2=z^2$, and Pythagorean triple preserving matrices (PTPM) $$$A$ are $$$3\times 3$ matrices with entries in the real numbers $$$\text{R}$, such that the product $$$(x,y,z)A$ is also a Pythagorean triple. In this paper, we study PT and PTPM from the view of projective geometry, and extend the results concerning PT and PTPM from integers to any commutative ring with identity. In particular, we use the method of polynomial parametrization for projective conics to obtain the general form of PT over any commutative ring with identity. In addition, we view the PTPM as projective transformations and formulate the general form of a PTPM over any commutative ring with identity.

Crovisier and Rams [2] recently constructed an embedded Cantor set in $$ and showed that it could not be realized as an attractor of any iterated function system (IFS) using measure-theoretic properties. Also, an example of a locally connected continuum in $$ which is not the attractor of any IFS on $$ is constructed in a work of Kwieciński [6]. Kwieciński points out that a variation on his main construction provides an arc in $$ which is not the attractor of any IFS either. In this work, for each $$, we construct an $$-cell in $$ and show that this $$-cell cannot be the attractor of any IFS on $$. The $$ case reaffirms the result observed by Kwieciński.

In this paper, we give the taxicab version of the Heron-Tartaglia formula to calculate the volume of a tetrahedron using the taxicab distance.

The Pasting Lemma for continuous functions plays a key role in algebraic topology. Several mathematicians have established pasting lemmas for some stronger and weaker forms of continuous functions. In this paper we prove pasting lemmas for rg-continuous, gp-continuous, gc-irresolute, and gpr-continuous functions.

The aim of this paper is to introduce a new closure operator and an associated new topology in bitopological spaces. We also define some new separation axioms and a comparative study is done.

The SOR method is a well-known method obtained from a one-part splitting of the system matrix $$, using one parameter $$ for the diagonal. Using one parameter for the lower triangular matrix of $$, M. Sisler introduced a new method. Later, he combined the standard SOR method and his method to get a two-parameter method. Sisler proved that for cyclic and positive-definite matrices, if zero is an eigenvalue of the Jacobi iteration matrix, the two-parameter method is not superior to the SOR method. In this paper we generalize Sisler's method and provide a range for the second parameter on which the two-parameter method is superior to the SOR method.

The well-known SOR method is obtained from a one-part splitting of the system matrix $$, using one parameter $$ for the diagonal. A strong interest in using more than one parameter for the SOR method to improve the convergence has been developed. Sisler, Niethammer, and Hadiidimos worked on the two-parameter method in the seventies. This author has generalized Sisler's method and introduced a range for the second parameter, providing a faster two-parameter method compared to the SOR method. In this paper, we go one step further by removing the hypothesis that requires the eigenvalues of the Jacobi iteration matrix to be real. The result is an optimal value for the second parameter when the eigenvalues of the SOR method are in a certain well-defined region.

In this note, we give a simple counterexample to a version of Farkas' Lemma for multilinear forms, and provide an elementary proof of a positive result for a special case. We also mention some remaining open problems.