and Applied Analysis 3 A finite subset of j consecutive positive integers starting with 1 is denoted by j : {1, 2, . . . , j}. The set R−h : −h, 0 ∪ R0 will be used to define the solution of functional differential equations on R0 including its initial condition in −h, 0 . C i R0 ,Rm0 is the set of m-vector real functions of class C i and definition domain R0 and PC i R0 ,Rm0 is the set of m-ve...

In this paper, we introduce the notion of an action $Y_X$as a generalization of the notion of a module,and the notion of a norm $vt: Y_Xto F$, where $F$ is a field and $vartriangle(xy)vartriangle(y') =$ $ vartriangle(y)vartriangle(xy')$ as well as the notion of fuzzy norm, where $vt: Y_Xto [0, 1]subseteq {bf R}$, with $bf R$  the set of all real numbers. A great many standard mappings on algebr...

In this paper we demonstrate how the geometrically motivated algorithm to determine whether a twogenerator real Möbius group acting on the Poincaré plane is or is not discrete can be interpreted as a non-Euclidean Euclidean algorithm. That is, the algorithm can be viewed as an application of the Euclidean division algorithm to real numbers that represent hyperbolic distances. In the case that t...

the aim of this article is to extend the study of the lorentz transformation of order (m,n) from m=1 and n>=1 to all m,n>=1, obtaining algebraic structures called a bi-gyrogroup and a bi-gyrovector space. a bi-gyrogroup is a gyrogroup each gyration of which is a pair of a left gyration and a right gyration. a bi-gyrovector space is constructed from a bi-gyrocommutative bi-gyrogroup that admits ...

We derive multi-level concentration inequalities for polynomials in independent random variables with an α-sub-exponential tail decay. A particularly interesting case is given by quadratic forms f(X1,…,Xn)=⟨X,AX⟩, which we prove Hanson–Wright-type explicit dependence on various norms of the matrix A. consequence these a two-level inequality variables, such as Poisson chaos. provide applications...