Related papers: Vinogradov's Cohomological Geometry of Partial Dif…
INTRODUCTION This papers deals with partial differential equations of second order, linear, with constant and not constant coefficients, in two variables, which admit real characteristics. I face the study of PDEs with the mentality of the…
In this note we highlight a common origin for many ubiquitous geometric structures, as well as several new ones by using only the functors of differential calculus in A.M Vinogradov's original sense, adapted to special classes of (graded)…
Let $B$ be a commutative algebra and $A$ be a $B$-algebra (determined by an algebra homomorphism $\varepsilon:B\rightarrow A$). M. D. Staic introduced a Hochschild like cohomology $H^{\bullet}((A,B,\varepsilon);A)$ called secondary…
Partial differential equations (PDEs) with spatially-varying coefficients arise throughout science and engineering, modeling rich heterogeneous material behavior. Yet conventional PDE solvers struggle with the immense complexity found in…
Understanding the role that subgradients play in various second-order variational analysis constructions can help us uncover new properties of important classes of functions in variational analysis. Focusing mainly on the behavior of the…
We discuss practical methods for computing the space of solutions to an arbitrary homogeneous linear system of partial differential equations with constant coefficients. These rest on the Fundamental Principle of Ehrenpreis-Palamodov from…
These lecture notes are a systematic and self-contained exposition of the cohomological theories naturally related to partial differential equations: the Vinogradov C-spectral sequence and the C-cohomology, including the formulation in…
Complexes and cohomology, traditionally central to topology, have emerged as fundamental tools across applied mathematics and the sciences. This survey explores their roles in diverse areas, from partial differential equations and continuum…
We describe a way of solving a partial differential equation using the differential invariants of its point symmetries. By first solving its quotient PDE, which is given by the differential syzygies in the algebra of differential…
In this present paper, we study geometric structures of rank two prolongations of implicit second-order partial differential equations (PDEs) for two independent and one dependent variables and characterize the type of these PDEs by the…
A method of constructing covariant differential calculi on a quantum homogeneous space is devised. The function algebra X of the quantum homogeneous space is assumed to be a left coideal of a coquasitriangular Hopf algebra H and to contain…
Finite element exterior calculus refers to the development of finite element methods for differential forms, generalizing several earlier finite element spaces of scalar fields and vector fields to arbitrary dimension $n$, arbitrary…
Partial differential equations (PDEs) are crucial for modeling various physical phenomena such as heat transfer, fluid flow, and electromagnetic waves. In computer-aided engineering (CAE), the ability to handle fine resolutions and large…
A new formulation of boundary value problems in gradient elasticity is presented in this work. The main outcome is the construction of partial differential systems of second order, which are typically equivalent with the well known fourth…
We analyse the singularity formation of congruences of solutions of systems of second order PDEs via the construction of \emph{shape maps}. The trace of such maps represents a congruence volume whose collapse we study through an appropriate…
This paper is the second in a series of works dedicated to studying non-linear partial differential equations via derived geometric methods. We study a natural derived enhancement of the de Rham complex of a non-linear PDE via…
A typical example of a Cartan calculus consists of the Lie derivative and the contraction with vector fields of a manifold on the derivation ring of the de Rham complex. In this manuscript, a second stage of the Cartan calculus is…
Many equations of mathematical physics are described by differential polynomials, that is by polynomials in the derivatives of a certain number of functions. However, up to the knowledge of the author, differential algebra in a modern…
Recently we have demonstrated how to use partner symmetries for obtaining noninvariant solutions of heavenly equations of Plebanski that govern heavenly gravitational metrics. In this paper, we present a class of scalar second-order PDEs…
In this paper we take some classical ideas from commutative algebra, mostly ideas involving duality, and apply them in algebraic topology. To accomplish this we interpret properties of ordinary commutative rings in such a way that they can…