We will present a brief study of the homology of cubical sets, with two main purposes. First, this combinatorial structure is viewed as representing directed spaces, breaking the intrinsic symmetries of topological spaces. Cubical sets have a directed homology, consisting of preordered abelian groups where the positive cone comes from the structural cubes. But cubical sets can also express topo...

In this paper, we first present the new concept of $2$-normed algebra. We investigate the structure of this algebra and give some examples. Then we apply a fixed point theorem to prove the stability and hyperstability of $(alpha, beta, gamma)$-derivations in $2$-Banach algebras.

Disclaimer: This dissertation does not contain plagiarised material; except where otherwise stated all theorems are the author's. Acknowledgement: Many thanks to Joel Feinstein for guidance with the literature, useful suggestions and comments on this work.

It is shown that the matrix normed structure of a non-unital operator algebra determines that of its unitization. This makes the study of certain unital operator algebras much easier and provides several interesting

We describe the alternative elements in An = R n the CayleyDickson algebras for n ≥ 4. Also we “measure” the failure of An with n ≥ 4 of being a normed algebra in terms of the alternative elements.

As mathematicians found out in the last century, there are only four normed division algebras1 over ℝ: real numbers themselves, complex numbers, quaternions and octonions. Whereas very well-known most of their properties carry to (apart from fact that these not commutative), octonions different harder handle since they even associative. However, can be used for several interesting topological c...

Octonions are introduced through some spin representations. The groups G2, F4 and the E series appear in a natural manner; one way to understand octonions is as the “second coming” of the reals, but with the spinors instead of vectors. Some physical applications in M and F -theory as putative “theories of everything” are suggested. 1 The Seven Sphere . Consider the real, complex and quaternion ...

One proves that the Mazur-Ulam theorem can be extended in the framework of metric spaces as long as a well behaved concept of midpoint is available. This leads to the new concept of Mazur-Ulam space. Besides the classical case of real normed spaces, other examples such as Sym++(n,R), the space of all n × n dimensional positive definite matrices, appear as cones attached to suitable Euclidean Jo...

In this paper, the gradual real numbers are considered and the notion of the gradual normed linear space is given. Also some topological properties of such spaces are studied, and it is shown that the gradual normed linear space is a locally convex space, in classical sense. So the results in locally convex spaces can be translated in gradual normed linear spaces. Finally, we give an examp...