the decomposition $gamma=bh$ of a group $gamma$ into a subset $b$ and a subgroup $h$ of $gamma$ induces, under general conditions, a group-like structure for $b$, known as a gyrogroup. the famous concrete realization of a gyrogroup, which motivated the emergence of gyrogroups into the mainstream, is the space of all relativistically admissible velocities along with a binary operation given by t...

We reconsider some families of orthogonal polynomials, within the framework of the so called monomiality principle. We show that the associated operational formalism allows the framing of the polynomial orthogonality using an algebraic point of view. Within such a framework, we introduce families of pseudo-orthogonal polynomials, namely polynomials, not orthogonal under the ordinary definition,...

Following our approach to metric Lie algebras developed in a previous paper we propose a way of understanding pseudo-Riemannian symmetric spaces which are not semi-simple. We introduce cohomology sets (called quadratic cohomology) associated with orthogonal modules of Lie algebras with involution. Then we construct a functorial assignment which sends a pseudo-Riemannian symmetric space M to a t...

We prove that the set of orthogonal separable coordinates on an arbitrary (pseudo-)Riemannian manifold carries a natural structure of a projective variety, equipped with an action of the isometry group. This leads us to propose a new, algebraic geometric approach to the classification of orthogonal separable coordinates by studying the structure of this variety. We give an example where this ap...

The pseudo-angle characterizes the pseudo-conformal group G." Let S2m be a fixed 2m dimensional manifold contained in 2;2 so that m . n. LetP be an arbitrary point of S2m. Suppose that S2,1n is an arbitrary (2n 1) dimensional manifold in Z2., which contains P, and let the intersection of S2m and S2,-1 be an S2m-j. Thus S2m is not contained in S2,1. The pseudo-angle between any curve C, in S2m, ...

Pseudo orthogonal bases are a certain type of frames proposed in the engineering field, whose concept is equivalent to a normalized tight frame in the frame terminology. This paper shows that pseudo orthogonal bases play an essential role in neural network learning. One of the most important issues in neural network learning is “what training data provides the optimal generalization capability?...

Pseudo orthogonal bases are a certain type of frames proposed in the engineering field, whose concept is equivalent to a tight frame with frame bound 1 in the frame terminology. This paper shows that pseudo orthogonal bases play an essential role in neural network learning. One of the most important issues in neural network learning is “what training data provides the optimal generalization cap...