new form of the Hyperbolic Pythagorean Theorem, which has a striking t e intuitive appeal and offers a strong contrast to its standard form, is presented. I xpresses the square of the hyperbolic length of the hypotenuse of a hyperbolic s o right angled triangle as the "Einstein sum" of the squares of the hyperbolic length f the other two sides, Fig. 1, thus completing the long path from Pythago...

The generalized Kullback-Leibler divergence (K-Ld) in Tsallis statistics subjected to the additive duality of generalized statistics (dual generalized K-Ld) is 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. The Pyth...

by repeated partial integration. The topic is usually reserved for more advanced calculus courses. The purpose of this note is to show that (1) can be derived using only the mathematics taught in elementary school, that is, basic algebra, the Pythagorean theorem, and the formula π · r 2 for the area of a circle of radius r . Viggo Brun gives an account of Wallis’s method in [1] (in Norwegian). ...

This paper introduces optimally-blended quadrature rules for isogeometric analysis and analyzes the numerical dispersion of the resulting discretizations. To quantify the approximation errors when we modify the inner products, we generalize the Pythagorean eigenvalue theorem of Strang and Fix. The proposed blended quadrature rules have advantages over alternative integration rules for isogeomet...

1. Introduction. The unit disc furnished with the Poincaré metric provides a model (the Poincaré model) for the hyperbolic geometry. The linear fractional transformations of the unit disc onto itself constitute the group of motions. An analogous phenomenon arises in the unit ball of any complex Hubert space when it is furnished with the Carathéodory-Reiffen metric. The purpose of this note is t...

One of the great pleasures in mathematics occurs when one considers several different proofs of a single result. In fact, when one considers the myriad proofs of the Pythagorean theorem and the irrationality of √ 2 constructed over the centuries, it seems we humans can never be satisfied with just one proof. Why do we continue to devise new approaches to known results? There is something in the...

1. I N T R O D U C T I O N Rational parametrization of curves and surfaces is a very interesting problem in mathematics. For curves, it is shown that a curve of genus zero is a rational by Noether. As for surfaces, there is Noether 's theorem that an algebraic surface S is rational if and only if it contains an irreducible rational curve C with dim ICI >1. Although there exist some techniques t...

Introduction. This paper is concerned with several disconnected developments in distance geometry. §1 deals with the congruent imbedding of metric spaces in euclidean or Hubert spaces. By showing that the validity of the Pythagorean theorem insures the essentially euclidean character of the metric, the basic role this theorem plays in euclidean geometry is seen to be fully justified. In §2 the ...