We uncover the structure of the space of symmetric functions in non-commutative variables by showing that the underlined Hopf algebra is both free and co-free. We also introduce the Hopf algebra of quasi-symmetric functions in non-commutative variables and define the product and coproduct on the monomial basis of this space and show that this Hopf algebra is free and cofree. In the process of l...

We uncover the structure of the space of symmetric functions in non-commutative variables by showing that the underlined Hopf algebra is both free and co-free. We also introduce the Hopf algebra of quasi-symmetric functions in non-commutative variables and define the product and coproduct on the monomial basis of this space and show that this Hopf algebra is free and cofree. In the process of l...

We uncover the structure of the space of symmetric functions in non-commutative variables by showing that the underlined Hopf algebra is both free and co-free. We also introduce the Hopf algebra of quasi-symmetric functions in non-commutative variables and define the product and coproduct on the monomial basis of this space and show that this Hopf algebra is free and cofree. In the process of l...

By classical results of Herglotz and F. Riesz, any bounded analytic function in the complex unit disk has a unique inner–outer factorization. Here, is called inner or outer if multiplication by this defines an isometry dense range, respectively, as linear operator on Hardy Space, H2, functions with square-summable Taylor series. This factorization can be further refined; θ decomposes uniquely p...

Let G be a group with identity e. Let R be a G-graded commutative ring and let M be a graded R-module. The graded classical prime spectrum Cl.Specg(M) is defined to be the set of all graded classical prime submodule of M. The Zariski topology on Cl.Specg(M); denoted by ϱg. In this paper we establish necessary and sufficient conditions for Cl.Specg(M) with the Zariski topology to be a Noetherian...

The Newtonian Potential is computed exactly in a theory that is fundamentally Non Commutative in the space-time coordinates. When the dispersion for the distribution of the source is minimal (i.e. it is equal to the non commutative parameter θ), the behavior for large and small distances is analyzed. PACS number: 11.10.Nx

In this paper some classes of ternary algebraic structures (semi-heaps, heaps) are considered. The connection between heaps (laterally commutative heaps) and corresponding algebraic and geometric structures is presented. The equivalence of heap existence and the Desargues system on the same set is directly proved. It is the starting point for an analogous result about a laterally commutative he...

Commutative H∗-algebras are characterized without postulating the existence of Hilbert space structure.

The noncommutative soliton is characterized by the use of the projection operators in non-commutative space. By using the close relation with the K-theory of C∗-algebra, we consider the variations of projection operators along the commutative directions and identify their topological charges. When applied to the string theory, it gives the modification of the brane charges due to tachyon backgr...

In this paper we prove that the Grosse-Wulkenhaar type non-commutative orientable complex scalar φ3 theory, with two non-commutative coordinates and the third one commuting with the other two, is renormalizable to all orders in perturbation theory. Our proof relies on a multiscale analysis in x space.