Although twenty five centuries old, the Pythagorean theorem appears vigorous and ubiquitous. A key to the distance formula in Descartes’s method of coordinates, the theorem is implicitly present in all scientific models and engineering computations involving spatial relationships or trigonometry. An invisible companion to the dot-product operation, it is inherent in equations of mathematical ph...

ON THE DIAGONALS OF PROJECTIONS IN MATRIX ALGEBRAS OVER VON NEUMANN ALGEBRAS Soumyashant Nayak Richard V. Kadison The main focus of this dissertation is on exploring methods to characterize the diagonals of projections in matrix algebras over von Neumann algebras. This may be viewed as a non-commutative version of the generalized Pythagorean theorem and its converse (Carpenter’s Theorem) studie...

A topological approach of determining the best one out of four solutions of the Hermitian interpolation problem for quintic Pythagorean hodograph curves is presented. An existence theorem is proved together with complete analysis of uniqueness/nonuniqueness properties. A simple formula of obtaining the best solution without using any curve fairing or curvature integral is also presented.

We present a new proof of the Pythagorean theorem which suggests a particular decomposition of the elements of a topological algebra in terms of an “inverse norm” addressing unital algebraic structure rather than simply vector space structure . One consequence is the unification of Euclidean norm, Minkowski norm, geometric mean, and determinant, as expressions of this entity in the context of d...

A topological approach to identifying the “good” interpolant among the four distinct solutions to the first–order Hermite interpolation problem for planar quintic Pythagorean–hodograph curves is presented. An existence theorem is proved, together with a complete analysis of uniqueness/non– uniqueness properties. A simple formula for finding the “good” solution, without appealing to curve fairne...