Related papers: Analyzability in the context of PDEs and applicati…
The concept of moment differentiation is extended to the class of moment summable functions, giving rise to moment differential properties. The main result leans on accurate upper estimates for the integral representation of the moment…
We generalise classical reconstruction results in algebra, using the language of monads, monoidal categories, module categories, as well as various notions of duality, such as closedness, Grothendieck--Verdier duality (also known as…
In this note we discuss some formal properties of universal linearization operator, relate this to brackets of non-linear differential operators and discuss application to the calculus of auxiliary integrals, used in compatibility…
This paper is divided into two parts. The first is a review, through categorical lenses, of the classical theory of regular-singular differential systems over $C((x))$ and $\mathbb P^1_C\smallsetminus\{0,\infty\}$, where $C$ is…
We revisit the standard axioms of domain theory with emphasis on their relation to the concept of partiality, explain how this idea arises naturally in probability theory and quantum mechanics, and then search for a mathematical setting…
The predicate complementary to the well-known Godel's provability predicate is defined. From its recursiveness new consequences concerning the incompleteness argumentation are drawn and extended to new results of consistency, completeness…
We use the Calogero equation to illustrate the following two aspects of the Painleve analysis of nonlinear PDEs. First, if a nonlinear equation passes the Painleve test for integrability, the singular expansions of its solutions around…
We introduce a new version of arithmetic in all finite types which extends the usual versions with primitive notions of extensionality and extensional equality. This new hybrid version allows us to formulate a strong form of extensionality,…
The general theory of (nonlinear) partial differential equations originated by S. Lie had a significant development in the past 30-40 years. Now this theory has solid foundations, a proper language, proper techniques and problems, and a…
Analogies have had and continue to have an important role in the development of theoretical physics. They may start from similarities of physical concepts followed by similarities in the mathematical formalization or it may be a purely…
We analyze the degree-structure induced by large reducibilities under the Axiom of Determinacy. This generalizes the analysis of Borel reducibilities given in references [1], [6] and [5] e.g. to the projective levels.
We introduce some analytic relations on the set of partial differential equations of two variables. It relies on a new comparison method to give rough asymptotic estimates for solutions which obey different partial differential equations.…
The present paper reviews some intriguing connections which link together a new renormalization technique, the theory of *-representations of infinite dimensional *-Lie algebras, quantum probability, white noise and stochastic calculus and…
We study the systems of ordinary differential equations which are implicit with respect to the higher derivatives, appearing in the linear form, and their solutions near the singular points. The invertibility of the higher derivatives…
Partial differential equations (PDEs) are at the heart of many mathematical and scientific advances. While great progress has been made on the theory of PDEs of standard types during the last eight decades, the analysis of nonlinear PDEs of…
This is a survey on rectifiability. I discuss basic properties of rectifiable sets, measures, currents and varifolds and their role in complex and harmonic analysis, potential theory, calculus of variations, PDEs and some other topics.
Partial Differential Equations (PDEs) describe phenomena ranging from turbulence and epidemics to quantum mechanics and financial markets. Despite recent advances in computational science, solving such PDEs for real-world applications…
This paper presents new results from a detailed study of the structure of autocatalytic sets. We show how autocatalytic sets can be decomposed into smaller autocatalytic subsets, and how these subsets can be identified and classified. We…
An identity that relates multipolar solutions of the Einstein equations to Newtonian potentials of bars with linear densities proportional to Legendre polynomials is used to construct analytical potential-density pairs of infinitesimally…
In this paper we introduce and investigate a new kind of functional (including ordinary and evolutionary partial) differential equations. The main goal of this paper is to explore our new philosophy by some examples on functional ODEs and…