with |α| + |β| = 1. The physical state of n particles is described by a vector in the Hilbert space H⊗n, which consists of all complex linear combinations of products |x1〉⊗|x2〉⊗· · ·⊗|xn〉, where |xi〉 ranges over the two basis vectors of the i-th copy of the Hilbert space H. Typically one uses the convention xi ∈ {0, 1} and the basis vectors of the 2-dimensional complex vector spaceH⊗n are writt...

According to celebrated Hurwitz theorem, there exists four division algebras consisting of R (real numbers), C (complex numbers), H (quaternions) and O (octonions). Keeping in view the utility of octonion variable we have tried to extend the three dimensional vector analysis to seven dimensional one. Starting with the scalar and vector product in seven dimensions, we have redefined the gradient...

In this paper, we study the Grassmannian of n-dimensional subspaces of a 2n-dimensional vector space and its infinite-dimensional analogues. Such a Grassmannian can be endowed with two binary relations (adjacent and distant), with pencils (lines of the Grassmann space) and with so-called Zreguli. We analyse the interdependencies among these different structures. Mathematics Subject Classificati...

We construct an example of an A∞ algebra structure defined over a finite dimensional graded vector space. Introduction A∞ algebras (or sha algebras) and L∞ (or sh Lie algebras) have been topics of current research. Construction of small examples of these algebras can play a role in gaining insight into deeper properties of these structures. These examples may prove useful in developing a deform...

This study considers four means of defining differential operators for extracting local aspects of shape in ill-specified environments: fuzzy differentiation as kernel smoothing; differentiation in the sense of weak or generalized derivatives; differentiation for fuzzy functions between normed spaces; and fuzzy differentiation for mappings between fuzzy manifolds. More consideration is given to...

A reflection in a real vector space equipped with a positive definite symmetric bilinear form is any automorphism that sends some nonzero vector v to its negative and pointwise fixes its orthogonal complement, and a finite reflection group is a discrete group generated by such transformations. We note two important classes of groups which occur as finite reflection groups: for a 2-dimensional v...