Related papers: Introduction to $\mathcal{A}$-Calculus
We present a multi-parameter non-constant-invariant class of Abel ordinary differential equations with the following remarkable features. This one class is shown to unify, that is, contain as particular cases, all the integrable classes…
We define a noncommutative differential calculus constructed from the inner derivation, then several relevant examples are showed. It is of interest to note that for certain $C^*$-algebra, this calculus is closely related to the classical…
This paper focuses on natural dualities for varieties of bilattice-based algebras.Such varieties have been widely studied as semantic models in situations where information is incomplete or inconsistent. The most popular tool for studying…
We set up a left ring of fractions over a certain ring of boundary problems for linear ordinary differential equations. The fraction ring acts naturally on a new module of generalized functions. The latter includes an isomorphic copy of the…
The purpose of this work is to complete the algebraic foundations of second-order languages from the viewpoint of categorical algebra as developed by Lawvere. To this end, this paper introduces the notion of second-order algebraic theory…
In the context of the integration over algebras introduced in a previous paper, we obtain several results for a particular class of associative algebras with identity. The algebras of this class are called self-conjugated, and they include,…
We state some elementary problems concerning the relation between difference calculus and differential calculus, and we try to convince the reader that, in spite of the simplicity of the statements, a solution of these problems would be a…
Suppose q is a complex number of modulus one and different from 1,-1. Let O(R^2_q) be the *-algebra with two hermitean generators x and y satisfying the relation xy=qyx. Using operator representations of the *-algebra O(R^2_q) on Hilbert…
Starting with a symmetric/antisymmetric matrix with integer coefficients (which we view as an analogue of a metric/form on a principal bundle over the "manifold" Spec Z) we introduce arithmetic analogues of Chern connections and their…
We extend Agler's notion of a function algebra defined in terms of test functions to include products, in analogy with the practice in real algebraic geometry, and hence the term preordering in the title. This is done over abstract sets and…
We introduce the scale calculus, which generalizes the classical differential calculus to non differentiable functions. The new derivative is called the scale difference operator. We also introduce the notions of fractal functions, minimal…
The main goals of this paper are: i) To develop an abstract differential calculus on metric measure spaces by investigating the duality relations between differentials and gradients of Sobolev functions. This will be achieved without…
Higher order derivatives of functions are structured high dimensional objects which lend themselves to many alternative representations, with the most popular being multi-index, matrix and tensor representations. The choice between them…
Recent innovations on the differential calculus for functions of non-commuting variables, begun for a quaternionic variable, are now extended to the case of a general matrix over the complex numbers. The expansion of F(X+Delta) is given to…
Finite differences have been widely used in mathematical theory as well as in scientific and engineering computations. These concepts are constantly mentioned in calculus. Most frequently-used difference formulas provide excellent…
The aim of this work is to develop a global calculus for pseudo-differential operators acting on suitable algebras of generalized functions. In particular, a condition of global hypoellipticity of the symbols gives a result of regularity…
We present a new method to explicitly define Abelian functions associated with algebraic curves, for the purpose of finding bases for the relevant vector spaces of such functions. We demonstrate the procedure with the functions associated…
In this paper, we introduce a new type of $ pq $-calculus. The $ pq $-derivative and $ pq $-integration are investigated and various properties of these concepts are given. The fundamental theorem of $ pq $-calculus and formulas of $ pq…
The aim of this paper is to inter-relate several algebraic and analytic objects, such as real-type algebraic curves, quadrature domains, functions on them and rational matrix functions with special properties, and some objects from Operator…
It is well known that unital contractive representations of the disk algebra are completely contractive. Let A denote the subalgebra of the disk algebra consisting of those functions f whose first derivative vanishes at 0. We prove that…