arXiv research

Ancient solutions to heat equations on graphs are shown to be analytic in time.

By a classical result, solutions of analytic elliptic PDEs, like the Laplace equation, are analytic. In many instances, the properties that come from being analytic are more important than analyticity itself. Many important equations are degenerate elliptic and solutions have much lower regularity. Still, one may hope …

Real analytic solutions found for special Lagrangian equation.

Sharp time analyticity proved for heat equation on Ricci solitons.

Extends machine learning models for analytic boundary conditions in differential equations.

We prove the analyticity in time for solutions of two parabolic equations in the whole space, without any decaying or vanishing conditions. One of them involves solutions to the heat equation of exponential growth of order $2$ on $\M$. Here $\M$ is $\R^d$ or a complete noncompact manifold with Ricci curvature bounded f…

Using Maple, we compute some analytical solutions of a modified Black-Scholes equation, recently proposed, in the case of the European put option. We show that the modified Black-Scholes equation with the European put option is exactly solvable in terms of associated Laguerre polynomials. We make some numerical experim…

Analyticity of heat equation extended to Bakry-Émery Ricci curvature manifolds.

We develop analytical methods for nonlinear Dirac equations. Examples of such equations include Dirac-harmonic maps with curvature term and the equations describing the generalized Weierstrass representation of surfaces in three-manifolds. We provide the key analytical steps, i.e., small energy regularity and removable…

Characterizes Bonnet surfaces using analytic conditions.

We investigate the duality between local (complex analytic) projective structures on surfaces and two dimensional (complex analytic) neighborhoods of rational curves having self-intersection +1. We study the analytic classification, existence of normal forms, pencil/fibration decomposition, infinitesimal symmetries. We…

It is well known that generic solutions of the heat equation are not analytic in time in general. Here it is proven that ancient solutions with exponential growth are analytic in time in ${\M} \times (-\infty, 0]$. Here $\M=\R^n$ or is a manifold with Ricci curvature bounded from below. Consequently a necessary and suf…

Analytic Euler fields on non-torus bundles have periodic orbits.

Analytic saddle spheres in S^3 are equators.

The systems of complex analytic second order ordinary differential equations whose solutions close up to become rational curves (after analytic continuation) are characterized by the vanishing of an explicit differential invariant, and turn out to provide an infinite dimensional family of integrable systems.

This paper derives the non-analytic solution to the Fokker-Planck equation of fractional Brownian motion using the method of Laplace transform. Sequentially, by considering the fundamental solution of the non-analytic solution, this paper obtains the transition probability density function of the random variable that i…

Data-driven approach learns effective equations for phase field interfaces.

Paper extends Poincaré's work to stochastic differential equations.

Analyzing static solutions in Finsler gravity, extending known results.

Paper explores non-uniqueness and uniqueness class for wave equations on graphs.

Analytic proof solves differential geometry problem.

Proves solvability of general inverse σ_k equations with constant coefficients.

In arXiv:0805.2192, we set up a gauge-theoretic equation on symplectic 6-manifolds, which is a version of the Hermitian-Einstein equation perturbed by Higgs fields, and call Donaldson-Thomas equation, to analytically approach the Donaldson-Thomas invariants. In this article, we consider the equation on compact Kähler t…

Paper connects algebraic and analytic methods for braid group representations.

Symplectic method solves infinite-dimensional Schrödinger equations.

Modified perturbation method removes non-smoothness in solving Black-Scholes equations.

Let $\mathcal{M}_1$ denote the space of solutions $z(x,y)$ to an elliptic, real analytic Monge-Ampère equation ${\rm det} (D^2 z)=\varphi(x,y,z,Dz)>0$ whose graphs have a non-removable isolated singularity at the origin. We prove that $\mathcal{M}_1$ is in one-to-one correspondence with $\mathcal{M}_2\times Z_2$, where…

Paper develops semi-analytic method for American options in time-dependent jump-diffusion models.

Proves analyticity of quasinormal modes in Kerr and Kerr-de Sitter spacetimes.

Paper equates torsions on wedge singularities.

Survey on geometric, analytic, and topological aspects of 4D equations.

Generalizes crystallographic properties to all dimensions.

This is a survey on the analytic theory of linear wave equations on globally hyperbolic Lorentzian manifolds. There is no claim of originality.

In this survey paper we discuss recent advances on short interest rate models which can be formulated in terms of a stochastic differential equation for the instantaneous interest rate (also called short rate) or a system of such equations in case the short rate is assumed to depend also on other stochastic factors. Ou…

Analytical solution found for a three-layer network with a specific activation function.

We give an analytic approach to the translating soliton equation with a special emphasis in the study of the Dirichlet problem in convex domains of the plane.

We give a classification of non-removable isolated singularities for real analytic solutions of the prescribed mean curvature equation in Minkowski $3$-space.

We discuss a semi-analytical method for solving SABR-type equations based on path integrals. In this approach, one set of variables is integrated analytically while the second set is integrated numerically via Monte-Carlo. This method, known in the literature as Conditional Monte-Carlo, leads to compact expressions fun…

In this paper, we study Higgs bundles on non-compact Hermitian manifolds. Under some assumptions for the underlying Hermitian manifolds which are not necessarily Kähler, we solve the Hermitian-Einstein equation on analytically stable Higgs bundles.

This paper proposes a new method to learn integration schemes for complex ODEs.

We show that for any $δ\in [0,1)$ there exists a homogeneous order $2-δ$ analytic outside zero solution to a uniformly elliptic Hessian equation in R^5.

New proof shows all conformal vector fields on complex hyperbolic space are Killing.

Study noncompact manifolds' Chern scalar curvatures, proving existence and multiplicity.

Paper studies compactifications of Higgs bundles and self-duality equations.

This paper investigates analytic properties of American option prices under the finite moment log-stable (FMLS) model. Under this model the price of American options is characterised by the free boundary problem of a fractional partial differential equation (FPDE) system. Using the technique of approximation we prove t…

New framework for calculating multivariate risk measures using Wishart process.

A step by step procedure to derive analytically the exact dynamical evolution equations of the probability density functions (PDF) of well known kinetic wealth exchange economic models is shown. This technique gives a dynamical insight into the evolution of the PDF, e.g., allowing the calculation of its relaxation time…

The paper equates the index of a vector bundle to the manifold's index and introduces an analytic torsion.