A locally-built, LLM-digested index of recent arXiv papers in quant finance, geometry/topology, and statistical ML — keyword search served straight from SQLite on this machine.
Paper solves a class of differential equations with specific solutions.
Developed a theory of local convexity for second order differential equations on Lie algebroids.
New variational principle found for non-variational differential equations.
Three types of equations of mathematical physics, namely, the equations, which describe any physical processes, the equations of mechanics and physics of continuous media, and field-theory equations are studied in this paper. In the first and second case the investigation is reduced to the analysis of the nonidentical …
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 terms of the horizontal (characteristic) cohomology. Applications to computing invar…
Study geometry and PDEs from group-determinants and representation theory.
Extends Feller theory to non-locally compact spaces for stochastic equations.
Introduces geometric control theory for students.
The theory of Lie remarkable equations, i.e. differential equations characterized by their Lie point symmetries, is reviewed and applied to ordinary differential equations. In particular, we consider some relevant Lie algebras of vector fields on and characterize Lie remarkable equations admitted by the …
These are lecture notes of a course on symmetry group analysis of differential equations, based mainly on P. J. Olver's book 'Applications of Lie Groups to Differential Equations'. The course starts out with an introduction to the theory of local transformation groups, based on the Stefan-Sussman theory on the integrab…
Linearized Einstein equations simplified via Calabi operator.
The theory of quasi-Lie systems, i.e. systems of first order ordinary differential equations which can be related via a generalised flow to Lie systems, is extended to systems of partial differential equations and its applications to obtaining -dependent superposition rules and integrability conditions are analysed.…
We give an abstract formulation of the formal theory partial differential equations (PDEs) in synthetic differential geometry, one that would seamlessly generalize the traditional theory to a range of enhanced contexts, such as super-geometry, higher (stacky) differential geometry, or even a combination of both. A moti…
New method solves differential equations on manifolds, with applications in physics.
Introduces internal Lagrangians for differential equations and connects them to presymplectic structures.
The properties of the Riemann extensions of nonriemannian spaces defined by the first order systems of differential equations are considered.
Developed a theory of ultradifferentiable sheafs with applications.
Logic approach finds real singularities in differential equations.
Notes on relative algebroids for geometric problems.
A method to derive Lagrangians from field equations in metric-affine theories of gravity.
One applies the symmetry group theory for study the partial differential equations of Tzitzeica surfaces theory. One finds infinitesimal symmetries, Lagrangians and a new solution of Titzeica equation.
Study on symmetries of differential equations using gauge transformations and coverings.
The article concerns the geometrical theory of general systems of partial differential equations in the \emph{absolute sense}, i.e., without any additional structure and subject to arbitrary change of variables in the widest possible meaning. The main result describes the composition series $Ω^0\subsetΩ^1\subset\cd…
Solutions to a differential equation link to contact structures.
A new method uses PDEs to predict spatiotemporal phenomena.
We show that classical Wilczynski--Se-ashi invariants of linear systems of ordinary differential equations are generalized in a natural way to contact invariants of non-linear ODEs. We explore geometric structures associated with equations that have vanishing generalized Wilczynski invariants and establish relationship…
Generalized differential geometry uses infinitesimals to solve singularities in differential equations.
Traditional models of macroeconomic dynamics are fundamentally incorrect. The reason lies in a misunderstanding of peculiarities of the analysis of infinitesimal quantities. However, even those types of solutions that are envisaged by the above-mentioned models are nonrepresentative in the sense of the reflection of re…
We apply Gauge Theory of Arbitrage (GTA) {hep-th/9710148} to derivative pricing. We show how the standard results of Black-Scholes analysis appear from GTA and derive correction to the Black-Scholes equation due to a virtual arbitrage and speculators reaction on it. The model accounts for both violation of the no-arbit…
Backlund transformations are used to search for solutions, particularly soliton solutions, of non-linear differential equations. In this paper we present an invariant geometrical theory of Backlund transformations for second order evolution equations with one space variable. The main concept is that of connection defin…
The classical Galois theory deals with certain finite algebraic extensions and establishes a bijective order reversing correspondence between the intermediate fields and the subgroups of a group of permutations called the Galois group of the extension. It has been the dream of many mathematicians at the end of the nine…
Characterizes algebraic integrability and minimality of Lie equations for non-commutative pseudogroups.
Systems of partial differential equations lie at the heart of physics. Despite this, the general theory of these systems has remained rather obscure in comparison to numerical approaches such as finite element models and various other discretisation schemes. There are, however, several theoretical approaches to systems…
Survey explores cohomology's roles in applied math and sciences.
Lectures on PDEs for creating surfaces with specific curvature.
The quantum differential equations can be regarded as examples of equations with certain universal properties which are of wider interest beyond quantum cohomology itself. We present this point of view as part of a framework which accommodates the KdV equation and other well known integrable systems. In the case of qua…
Neural differential equations combine deep learning and differential equations for modeling complex systems.
Solves inverse problem for Maxwell equations using vector fields.
The article provides a modest survey of the absolute theory of general systems of (partial) differential equations. The equations are relieved of all additional structures and subject to quite arbitrary change of the variables. An abstract mathematical theory in the Bourbaki sense with its own concepts and technical to…
Exterior differential systems are given, and their Cartan characters calculated, for Maxwell and SU(2)-Yang-Mills equations in dimensions from three to six.
The main purpose of this article is to introduce a comprehensive, unified theory of the geometry of all connections. We show that one can study a connection via a certain, closely associated second-order differential equation. One of the most important results is our extended Ambrose-Palais-Singer correspondence. We ex…
An optimal control problem associated with the dynamics of the orientation of a bipolar molecule in the plane can be understood by means of tools in differential geometry. For first time in the literature -symplectic formalism is used to provide the optimal control problems associated to some families of partial dif…
Conservation laws, heirarchies, scattering theory and Bäcklund transformations are known to be the building blocks of integrable partial differential equations. We identify these as facets of a theory of Poisson group actions, and apply the theory to the ZS-AKNS nxn heirarchy (which includes the non-linear Schrödinger …
This is an elementary and self--contained review of twistor theory as a geometric tool for solving non-linear differential equations. Solutions to soliton equations like KdV, Tzitzeica, integrable chiral model, BPS monopole or Sine-Gordon arise from holomorphic vector bundles over $T\CP^1$. A different framework is pro…
Study uses neural networks to solve complex equations efficiently.
The paper discusses how to improve machine learning models using partial differential equations.
Survey uses Milnor fibrations to classify first integrals of differential systems.
This paper extends transfer operator theory to McKean-Vlasov equations.