As is well-known, the real quaternion division algebra H is algebraically isomorphic to a 4-by-4 real matrix algebra. But the real division octonion algebra O can not be algebraically isomorphic to any matrix algebras over the real number field R, because O is a non-associative algebra over R. However since O is an extension of H by the Cayley-Dickson process and is also finite-dimensional, som...

In this paper, we extend the notions of states and measures presented in [12] to the case of pseudo-BCK algebras and study similar properties. We prove that, under some conditions, the notion of a state in the sense of [12] coincides with the Bosbach state, and we extend to the case of pseudo-BCK algebras some results proved by J. Kühr only for pseudo-BCK semilattices. We characterize extremal ...

Multiple-valued logics are non-classical logics. They are similar to classical logic because they accept the principle of truth-functionality, namely, the truth of a compound sentence is determined by the truth values of its component sentences (and so remains unaffected when one of its component sentences is replaced by another sentence with the same truth value). But they differ from classica...

Most management science and economics students attending an advanced statistics course appreciate a quick repetition of the concept of random variables. We describe the application of the computer algebra system Derive to the familiar “throwing a die” example which turns out to be a substantial time saver, e.g. since tedious manual calculations can be avoided. Our approach also allows straightf...

Some unexpected properties of the cubic algebra generated by the covariant derivatives of a generic Yang-Mills connection over the (s+ 1)-dimensional pseudo euclidean space are pointed out. This algebra is Gorenstein and Koszul of global dimension 3 but except for s = 1 (i.e. in the 2-dimensional case) where it is the universal enveloping algebra of the Heisenberg Lie algebra and is a cubic Art...

The aim of this paper is to construct left-invariant Einstein pseudo-Riemannian Sasaki metrics on solvable Lie groups. We consider the class $\mathfrak z$-standard algebras dimension $2n+3$, which are in one-to-one correspondence with pseudo-K\"ahler nilpotent $2n$ endowed a compatible derivation, suitable sense. characterize structures and derivations giving rise Sasaki-Einstein metrics. class...

We develop new computational algorithms, based on the method of equivariant moving frames, for classifying the differential invariants of Lie symmetry pseudo-groups of differential equations and analyzing the structure of the induced differential invariant algebra. The Korteweg–deVries (KdV) and Kadomtsev–Petviashvili (KP) equations serve as illustrate examples. In particular, we deduce the fir...