The pseudo-determinant Det(A) of a square matrix A is defined as the product of the nonzero eigenvalues of A. It is a basis-independent number which is up to a sign the first nonzero entry of the characteristic polynomial of A. We prove Det(FG) = ∑ P det(FP)det(GP) for any two n×m matrices F,G. The sum to the right runs over all k × k minors of A, where k is determined by F and G. If F = G is t...

We address the reasons why the “Wick-rotated”, positive-definite, space-time metric obeys the Pythagorean theorem. An answer is proposed based on the convexity and smoothness properties of the functional spaces purporting to provide the kinematic framework of approaches to quantum gravity. We employ moduli of convexity and smoothness which are eventually extremized by Hilbert spaces. We point o...

The generalized Kullback-Leibler divergence (K-Ld) in Tsallis statistics [constrained by the additive duality of generalized statistics (dual generalized K-Ld)] is here reconciled with the theory of Bregman divergences for expectations defined by normal averages, within a measure-theoretic framework. Specifically, it is demonstrated that the dual generalized K-Ld is a scaled Bregman divergence....

We establish partition regularity of the generalised Pythagorean equation in five or more variables. Furthermore, we show how Rado’s characterisation a regular remains valid over set positive kth powers provided has at least (1+o(1))k log k thus completely describe which diagonal forms are and not, given sufficiently many In addition, prove supersaturated version theorem for linear restricted e...

With respect to multi-attribute group decision making (MAGDM) problems in which the attribute values take form of Pythagorean fuzzy numbers, a new Pythagorean fuzzy aggregation operator called Pythagorean fuzzy induced generalized ordered weighted averaging (PFIGOWA) operator is proposed. Some desirable properties and families of the proposed operator are discussed. Furthermore, based on the pr...

We formalize the notion of vector semi-inner products and introduce a class seminorms which are built from these maps. The classical Pythagorean theorem parallelogram law then generalized to that have geometric mean closed lattice for codomain. In special case this codomain is square root closed, semiprime $f$-algebra, we provide sharpening triangle inequality as well condition equality.

The data in this article was obtained from the algebraic and statistical analysis of the first 331 primitive Pythagorean triples. The ordered sample is a subset of the larger Pythagorean triples. A primitive Pythagorean triple consists of three integers a, b and c such that; [Formula: see text]. A primitive Pythagorean triple is one which the greatest common divisor (gcd), that is; [Formula: se...

In this paper we present a school project where students constructed three-dimensional pyramids using the Logo programming language, complemented with paper-and-pencil, dynamic geometry (Cabri) and spreadsheet (Excel) investigations. The aim of this project was to give, through a fun and meaningful way, and using a constructionist approach, junior secondary students (12-14 year olds), early acc...