A simple axiomatic characterization of the noncommutative Itô algebra is given and a pseudo-Euclidean fundamental representation for such algebra is described. It is proved that every quotient Itô algebra has a faithful representation in a Minkowski space and is canonically decomposed into the orthogonal sum of quantum Brownian (Wiener) algebra and quantum Lévy (Poisson) algebra. In particular,...

This article is a further contribution to our research [1] into a class of infinite-dimensional Lie algebras L∞(N+, N−) generalizing the standard W∞ algebra, viewed as a tensor operator algebra of SU(1, 1) in a group-theoretic framework. Here we interpret L∞(N+, N−) either as a infinite continuation of pseudo-unitary symmetries U(N+, N−), or as a “higher-U(N+, N−)-spin extension” of the diffeom...

We survey a recent extension of the moving frames method for infinite-dimensional Lie pseudo-groups. Applications include a new, direct approach to the construction of Maurer–Cartan forms and their structure equations for pseudogroups, and new algorithms, based on constructive commutative algebra, for uncovering the structure of the algebra of differential invariants for pseudogroup actions.

In the paper [8], we introduced the notions of pseudo-hyperplane and pseudoembedding of a point-line geometry and proved that every generalized quadrangle of order (s, t), 2 ≤ s <∞, has faithful pseudo-embeddings. The present paper focuses on generalized quadrangles of order (3, t). Using the computer algebra system GAP [12] and invoking some theoretical relationships between pseudo-hyperplanes...

BCI is one of the most intriguing technologies among other HCI systems, mostly because of its capability of recording brain activities. Spelling BCIs, which help paralyzed people to maintain communication, are one of the striking topics in the field of BCI. In this scientific a spelling BCI system with high transfer rate and accuracy that uses SSVEP signals is proposed.In addition, we suggested...

We construct cocycles on the Lie algebra of pseudoand q-pseudodifferential symbols of one variable and on their close relatives: the sine-algebra and the Poisson algebra on two-torus. A “quantum” Godbillon-Vey cocycle on (pseudo)differential operators appears in this construction as a natural generalization of the Gelfand-Fuchs 3-cocycle on periodic vector fields. We describe a nontrivial embed...

We construct cocycles on the Lie algebra of pseudoand q-pseudodifferential symbols of one variable and on their close relatives: the sine-algebra and the Poisson algebra on two-torus. A ``quantum'' Godbillon Vey cocycle on (pseudo)differential operators appears in this construction as a natural generalization of the Gelfand Fuchs 3-cocycle on periodic vector fields. A nontrivial embedding of th...

BCK and BCI-algebras, two classes of algebras of logic, were introduced by Imai and Iseki [1], Iseki [2] and Iseki and Tanaka [3] and have been extensively studied by various other researchers [4, 5]. It is known that the class of BCK-algebras is a proper subclass of the class of BCI-algebras. In [6, 7] a wider class of abstract algebras was introduced by Q.P.Hu and X.Li.They have shown that th...

Gelfand, Retakh, Serconek and Wilson, in [3], defined a graded algebra AΓ attached to any finite ranked poset Γ a generalization of the universal algebra of pseudo-roots of noncommutative polynomials. This algebra has since come to be known as the splitting algebra of Γ. The splitting algebra has a secondary filtration related to the rank function on the poset and the associated graded algebra ...