Related papers: The Gateaux Derivative and Integral over Banach Al…

We prove implicit function theorems for mappings on topological vector spaces over valued fields. In the real and complex cases, we obtain implicit function theorems for mappings from arbitrary (not necessarily locally convex) topological…

Starting with a finite-dimensional complex Lie algebra, we extend scalars using suitable commutative topological algebras. We study Birkhoff decompositions for the corresponding loop groups. Some results remain valid for loop groups with…

Here we consider a perturbation of continuous mappings on Banach spaces and investigate their image under various conditions. Consequently, we study the solvability of some classes of equations and inclusions. For these, we start by the…

Motivated by the fundamental results of the geometric algebra we study quadrilateral lattices in projective spaces over division rings. After giving the noncommutative discrete Darboux equations we discuss differences and similarities with…

It is often noted that many of the basic concepts of differential geometry, such as the definition of connection, are purely algebraic in nature. Here, we review and extend existing work on fully algebraic formulations of differential…

For a Banach algebra $A$, we say that an element $M$ in $A\otimes^\gamma A$ is a hyper-commutator if $(a\otimes 1)M=M(1\otimes a)$ for every $a\in A$. A diagonal for a Banach algebra is a hyper-commutator which its image under diagonal…

To any directed graph we associate an algebra with edges of the graph as generators and with relations defined by all pairs of directed paths with the same origin and terminus. Such algebras are related to factorizations of polynomials over…

Let $\cal A$ and $\cal B$ be Banach algebras. A linear map $T:{\cal A} \rightarrow {\cal B}$ is called separating or disjointness preserving if $ab=0$ implies $Ta\;Tb = 0$ for all $a,b\in {\cal A}$. In this paper, we study a new class of…

The central structure in various versions of noncommutative geometry is a differential calculus on an associative algebra. This is an analogue of the calculus of differential forms on a manifold. In this short review we collect examples of…

The aim of this work is to describe subsets of Banach limits in terms of a certain functional characteristic. We compute radii and cardinalities for some of these subsets.

Different index concepts for linear differential-algebraic equations are defined in the general Banach space setting, and compared. For regular finite-dimensional linear differential-algebraic equations, all these indices exist and are…

We study the concept of extended derivations of algebras which expands diverse definitions of generalized derivations given in the literature. We concentrate on the family of the anti-commutative algebras and classify such spaces of…

A classification of commutative integral domains consisting of ordinary differential operators with matrix coefficients is established in terms of morphisms between algebraic curves.

The main goal of the paper is the discussion of a deeper interaction between matrix theory over polynomial rings over a field and typical methods of commutative algebra and related algebraic geometry. This is intended in the sense of…

We study the existence of global implicit functions for equations defined on open subsets of Banach spaces. The partial derivative with respect to the second variable is only required to have a left inverse instead of being invertible.…

Classes of Banach spaces that are finitely, strongly finitely or elementary equivalent are introduced. On sets of these classes topologies are defined in such a way that sets of defined classes become compact totally disconnected…

In this paper, we will define generalized critical point, ordered extreme and order monotone property of single-valued mappings in partially ordered Banach spaces. In particular, we will find the explicit formulas of Gateaux and Frechet…

In the book, I considered differential equations of order $1$ over Banach $D$\Hyph algebra: differential equation solved with respect to the derivative; exact differential equation; linear homogeneous equation. I considered examples of…

We consider regulated curves in a Banach bundle whose projection on the basis is continuous with regulated derivative. We build a Banach manifold structure on the set of such curves. This result was previously obtained for the case of…

This paper points out the usefulness of the concept of derivation along a map in many problems in Geometry and Physics. In particular it will be shown that this approach allows us to translate the usual concepts arising in Geometrical…