Related papers: On analysis in differential algebras and modules
An introduction to applied mathematics written for students in engineering and science. Focus is on a rigorous presentation that also builds understanding by discussion, analogy, and examples. Discussion of concepts involved in modeling…
In this text, we wish to provide the reader with a short guide to recent works on the theory of dilatations in Commutative Algebra and Algebraic Geometry. These works fall naturally into two categories: one emphasises foundational and…
Differentially-algebraic (D-algebraic) functions are solutions of polynomial equations in the function, its derivatives, and the independent variables. We revisit closure properties of these functions by providing constructive proofs. We…
I give a short review of the theory of twisted symmetries of differential equations, emphasizing geometrical aspects. Some open problems are also mentioned.
Differentiable physics provides a new approach for modeling and understanding the physical systems by pairing the new technology of differentiable programming with classical numerical methods for physical simulation. We survey the rapidly…
This paper surveys some applications of moduli theory to issues concerning the distribution of rational points on algebraic varieties. It will appear on the proceedings of the Fano Conference.
An extended summary of the lecture course given at the V School on Geometry and Physics, Bia\l owe\.za 2016, in which an algebraic approach to differentiation and integration that is characteristic for non-commutative geometry is described.
These notes contain a survey of some aspects of the theory of differential modules and complexes as well as of their generalization, that is, the theory of $N$-differential modules and $N$-complexes. Several applications and examples coming…
This is a survey on the usage of the module theoretic notion of a "retractable module" in the study of algebras with actions. We explain how classical results can be interpreted using module theory and end the paper with some open…
In this paper, we present methods to simplify reducible linear differential systems before solving. Classical integrals appear naturally as solutions of such systems. We will illustrate the methods developed in a previous paper on several…
Intended for mathematical physicists interested in applications of the division algebras to physics, this article highlights some of their more elegant properties with connections to the theories of Galois fields and quadratic residues.
Course material for mathematical methods of theoretical physics intended for an undergraduate audience.
The paper describes different approaches to generalize the trapezoidal method to fractional differential equations. We analyze the main theoretical properties and we discuss computational aspects to implement efficient algorithms. Numerical…
We view difference algebra as the study of algebraic objects in the topos of difference sets. The methods of topos theory and categorical logic enable us to develop difference homological algebra, identify a solid foundation for difference…
This book intends to give the main definitions and theorems in mathematics which could be useful for workers in theoretical physics. It gives an extensive and precise coverage of the subjects which are addressed, in a consistent and…
The role of differential equations in the process of calculating Feynman integrals is reviewed. An example of a diagram is given for which the method of differential equations was introduced, the properties of the inverse-mass-expansion…
This paper is a short introduction to orthogonal polynomials, both the general theory and some special classes. It ends with some remarks about the usage of computer algebra for this theory.
These notes include introductory material on the notion of splitting fields for modules over a k-algebra where k is a field.
General concept of ternary algebras is introduced in this article, along with several examples of its realization. Universal envelope of such algebras is defined, as well as the concept of tri-modules over ternary algebras. The universal…
This paper surveys some selected topics in the theory of conformal metrics and their connections to complex analysis, partial differential equations and conformal differential geometry.