arXiv research

The paper introduces new functionals and equations for complex vector bundles.

A large class of semi-Hamiltonian systems of hydrodynamic type is interpreted as the equations governing families of critical points of functions obeying the classical linear Darboux equations for conjugate nets.The distinguished role of the Euler-Poisson-Darboux equations and associated Lauricella-type functions is em…

In this paper we study a functional equation associated to the Kummer's equation (K) of the trilogarithm. Then we apply our results to web geometry and to characterize the functions solution of (K).

Characterizes complex Hessian equations for bounded energy functions.

New variational principle found for non-variational differential equations.

Paper solves Hessian equations on Kähler manifolds.

In this paper, we study local solutions F=(F1,..,Fn) of a general functional equation of the form F1(U1(x,y))+....+Fn(Un(x,y))=0. A such equation will be called an ``abelian functional equation'' (Afe). We will restrict ourselves to the case when the inner functions Ui's are real rational functions. First we prove that…

We introduce certain energy functionals to the complex Monge-Ampere equation over a bounded domain with inhomogeneous boundary condition, and use these functionals to show the convergence of the solution to the parabolic Monge-Ampere equation.

Solves Dirichlet problem for generalized Hitchin's equation on cyclic Higgs bundles.

The sinh-Gordon equation is solved on finite, symmetric graphs.

Researchers solve metric curvature equations on manifolds with boundary.

Ancient solutions to biharmonic heat equation bounded by polynomial dimensions.

Researchers prove constant solutions for a specific Finslerian equation.

Researchers study surface area functionals in CR manifolds, deducing equations for various cases.

Study generalizes Hermitian-Einstein equation for cyclic Higgs bundles, proving existence and inequality.

The paper connects Schrödinger equations to geodesics on a 2-surface.

The Weil conjecture is a delightful theorem for algebraic varieties on finite fields and an important model for dynamical zeta functions. In this paper, we prove a functional equation of Lefschetz zeta functions for infinite cyclic coverings which is analogous to the Weil conjecture. Applying this functional equation t…

The paper proves solutions for Yamabe equations on manifolds with boundary.

The paper explores the geometry of the Spence-Kummer trilogarithm equation and its Galois analogue.

Elvet solves differential equations and variational problems with neural networks.

The paper extends portfolio theory to include contingent claim functions for option pricing.

In this paper we study the gradient estimate for positive solutions of Schrodinger equations on locally finite graph. Then we derive Harnack's inequality for positive solutions of the Schrodinger equations. We also set up some results about Green functions of the Laplacian equation on locally finite graph. Interesting …

Solves a specific Dirichlet problem on Hermitian manifolds.

Score-fPINN tackles high-dimensional FPL equations using fractional score functions.

We study algebraic solutions of the Riccati equation over the field of rational functions $\mathbb C(t)$, and over the elliptic function field $\mathbb C(\wp,\wp^\prime)$.

A quaternionic contact (qc) heat equation and the corresponding qc energy functional are introduced. It is shown that the qc energy functional is monotone non-increasing along the qc heat equation on a compact qc manifold provided certain positivity conditions are satisfied.

Defines a functional for Riemann surfaces, proving a unique solution.

GF-Net learns Green's functions for linear reaction-diffusion equations.

We prove that under certain assumptions a partial differential equation can be derived from a variational principle. It is well-known from Noether's theorem that symmetries of a variational functional lead to conservation laws of the corresponding Euler-Lagrange equation. We reverse this statement and prove that a diff…

Monogenic functions are basic to Clifford analysis. On Euclidean space they are defined as smooth functions with values in the corresponding Clifford algebra satisfying a certain system of first order differential equations, usually referred to as the Dirac equation. There are two equally natural extensions of these eq…

In this paper, we study a coupled system of equations on oriented compact 4-manifolds which we call the Bach-Merkulov equations. These equations can be thought of as the conformally invariant version of the classical Einstein-Maxwell equations in general relativity. Inspired by the work of C. LeBrun on Einstein-Maxwell…

Uniform bounds for Green's function on Kähler manifolds derived from complex Monge-Ampère equations.

Paper solves a complex equation for smooth domains.

Solves complex Monge-Ampère equations with Hölder continuous solutions in Kähler manifolds.

Paper solves bond option pricing with credit risk using Black-Scholes equations.

We prove existence results for nodal solutions of the Yamabe equation that are constant along the level sets of an isoparametric function.

In many regular cases, there exists a (properly defined) limit of iterations of a function in several real variables, and this limit satisfies the functional equation (1-z)f(x)=f(f(xz)(1-z)/z); here z is a scalar and x is a vector. This is a special case of a well-known translation equation. In this paper we present a …

We study the $2k$-Hitchin equations introduced by Ward \cite{Ward 2} from the geometric viewpoint of Higgs bundles. After an introduction on Higgs bundles and $2k$-Hitchin's equations, we review some elementary facts on complex geometry and Yang-Mills theory. Then we study some properties of holomorphic vector bundles …

We develop an integral geometry of stationary Euler equations defining some function $w$ on the Grassmannian of affine lines in the space. This function depends on a putative compactly supported solution $v$ of the system, and we deduce a linear differential equation for $w$. We prove also that the purported annulation…

Paper explores solving HJB equations using neural networks.

The paper refines optimization algorithms using Lyapunov functions and differential equations.

Study on convergence of SDEs using entropy methods.

We propose a new algorithm for solving parabolic partial differential equations (PDEs) and backward stochastic differential equations (BSDEs) in high dimension, by making an analogy between the BSDE and reinforcement learning with the gradient of the solution playing the role of the policy function, and the loss functi…

In this article it is shown that the study of harmonic diffeomorphisms, with nonvanishing Hopf differential, reduces to the study of the Beltrami equation of a certain type: the imaginary part of the logarithm of the Beltrami function coincides with the imaginary part of the logarithm of the Hopf differential, therefor…

In this paper we prove that there exists a smooth classical solution to the HJB equation for a large class of constrained problems with utility functions that are not necessarily differentiable or strictly concave. The value function is smooth if admissible controls satisfy an integrability condition or if it is contin…

Investigates webs related to cluster algebras and polylogarithms.

We show that the Green functions on flat tori can have either 3 or 5 critical points only. There does not seemto be any directmethod to attack this problem. Instead, we have to employ sophisticated non-linear partial differential equations to study it. We also study the distribution of number of critical points over th…

Proves existence and uniqueness of solutions to a quaternionic Monge-Ampère equation.