The paper generalizes the Bott-Virasoro group and derives new Euler equations.
problem Understanding the generalized Bott-Virasoro group and its dynamics.
method Generalizing the Bott-Virasoro group using connection cochain and deriving Euler equations.
result New Euler equations derived from the generalized Bott-Virasoro group.
Proves solvability of general inverse σ_k equations with constant coefficients.
problem Solvability of general inverse σ_k equations with constant coefficients.
method Proves existence of unique solution if a C-subsolution exists.
result Confirms analytical conjecture for deformed Hermitian--Yang--Mills equation.
We present a contact transformation of the generalized Hunter--Saxton equation to the Euler--Poisson equation with special values of the Ovsiannikov invariants. We also find the general solution for the generalized Hunter--Saxton equation.
The paper generalizes Monge-Ampère equations and their solutions in differential geometry.
problem Understanding the structure of Monge-Ampère equations and their solutions.
method Generalizing Monge-Ampère equations to higher-order systems and proving their solutions correspond to integral manifolds of exterior differential systems.
result The Korteweg-de Vries (KdV) equation and Cauchy-Riemann equations are examples of generalized Monge-Ampère equations.
Study generalizes Hermitian-Einstein equation for cyclic Higgs bundles, proving existence and inequality.
problem Addressing Hermitian-Einstein equation for cyclic Higgs bundles.
method Introducing generalizations using subharmonic functions and proving existence, uniqueness, and convergence of heat equations.
result Existence, uniqueness, and convergence of solutions for heat equations.
Abstract: Proves compactness theorem for generalized Seiberg-Witten equations in 3D.
problem Compactness of solutions to generalized Seiberg-Witten equations in 3D.
method Abstract compactness theorem for a family of generalized Seiberg-Witten equations in dimension three.
result Recovers and extends known compactness theorems for stable flat connections and Seiberg-Witten equations.
Solves generalized Kazdan-Warner equations on foliated manifolds.
problem Existence and uniqueness of solutions to generalized Kazdan-Warner equations on foliated manifolds.
method Extends theorem to compact foliated manifolds, provides examples of PDEs.
result Solves the transverse Hitchin equation and its generalizations.
In this work we present a new approach on studying dynamical systems. Combining the two ways of expressing the uncertainty, using probabilistic theory and credibility theory, we have research the generalized fractional hybrid equations. We have introduced the concepts of generalized fractional Wiener process, generaliz…
The paper studies curvature equations and their solvability.
problem Solving curvature type equations and their Dirichlet problems.
method General class of fully nonlinear curvature equations, Christoffel-Minkowski problem, degenerate equations.
result Solvability of curvature type equations and Dirichlet problems.
We study the contact geometry of scalar second order hyperbolic equations in the plane of generic type. Following a derivation of parametrized contact-invariants to distinguish Monge-Ampere (class 6-6), Goursat (class 6-7) and generic (class 7-7) hyperbolic equations, we use Cartan's equivalence method to study the gen…
Proves convexity of level sets of general inverse σ_k equations.
problem Convexity of level sets of general inverse σ_k equations.
method Analyzes level sets of degree n general inverse σ_k equations and uses numerical conditions to verify convexity.
result Proves convexity of level sets of general inverse σ_k equations.
We give a description of recently introduced Doubrov-Ferapontov general heavenly equation in terms of closed differential Plücker two-form, rationally depending on the spectral parameter. We demonstrate that general heavenly equation is an important generating equation in the context of Takasaki hyper-Kähler hierarchy,…
Solves Dirichlet problem for generalized Hitchin's equation on cyclic Higgs bundles.
problem Existence and uniqueness of solutions to the Dirichlet problem.
method Formulated using subharmonic functions; generalizes Hitchin's equation for diagonal harmonic metrics on cyclic Higgs bundles.
result Existence and uniqueness of solutions to the Dirichlet problem.
Generalizes Hodge correlators using quantum master equation concepts.
problem Developing a mathematical framework for non-acyclic Chern-Simons theory.
method Introduces a DG Lie algebra of uni-trivalent graphs with loops satisfying a Maurer-Cartan equation.
result Arithmetic analogue of effective action and quantum master equation.
New rigidity results for a generalized Ricci-Hessian equation on manifolds.
problem Understanding rigidity in generalized Ricci-Hessian equations on manifolds.
method Proving new rigidity results related to a generalized Ricci-Hessian equation.
result New rigidity results for the generalized Ricci-Hessian equation on Riemannian manifolds.
We analyze a generalized version of the Black-Scholes equation depending on a parameter a∈(−∞,0). It satisfies the martingale condition and coincides with the Black-Scholes equation in the limit case a↗0. We show that the generalized equation is exactly solvable in terms of Hermite polynomials a…
The paper introduces new functionals and equations for complex vector bundles.
problem Complex vector bundle equations and their solutions.
method Introducing generalized Donaldson's functionals and discussing their Euler-Lagrange equations.
result Uniqueness of solutions to complex vector bundle equations.
New proof shows solutions to Lichnerowicz equation exist.
problem Existence of solutions to the Lichnerowicz equation in general relativity.
method A new proof approach.
result Existence and uniqueness of solutions proven.
Generalized diffusion type equations are considered and point symmetry analysis is applied to them. The equations with extremal order point symmetry algebras are described. Some old geometrical results are rederived in connection with theory of these equation.
Geometrically interprets two equations, showing their equivalence and providing solutions.
problem Equivalence and solutions of generalized Proudman-Johnson and r-Hunter-Saxton equations.
method Geometric interpretation through Finsler metrics and isometries.
result Equivalence of periodic and non-periodic cases as geodesic equations.
Solves a generalized Monge-Ampère equation on Kähler surfaces, proving a conjecture.
problem Existence and uniqueness of solutions to a generalized Monge-Ampère equation.
method Analyzes the equation's dependence on almost Kähler structure and proves Donaldson's conjecture.
result Proves Donaldson's conjecture for tamed almost complex 4-manifolds.
Growth of spinors in 4D and 3D generalized Seiberg-Witten equations.
problem Proving growth of spinors in GSW equations on R4 and R3. method Unified framework of GSW equations, averaged L2-norm, curvature decay assumption, Yang-Mills-Higgs energy. result Growth of spinors in GSW equations on R4 and R3 faster than a power of the radius under suitable curvature decay. Paper generalizes sub-slope definition and solves complex equations on compact manifolds.
problem Solving complex equations on compact almost Hermitian manifolds.
method Generalized sub-slope definition and proved existence of solutions for a class of equations.
result Solved complex Hessian quotient and deformed Hermitian-Yang-Mills equations.
Study geometric singular solutions of generalized Monge-Ampère equations.
problem Solving generalized Monge-Ampère equations on a plane.
method Using exterior differential systems and Cauchy characteristics.
result Criteria for geometric singular solutions to be equivalent to specific types.
Derives equations of motion for systems with angular momentum on Finsler geometries.
problem Equations of motion for dynamical systems with angular momentum on Finsler geometries.
method Apply Souriau's Principle of General Covariance to derive diffeomorphism invariant equations of motion.
result Generalizes Mathisson-Papapetrou-Dixon equations to Finsler geometries and finds conserved quantities.
Fundamental solutions found for p-Laplace equations in Heisenberg and Grushin spaces.
problem Finding solutions to p-Laplace equations with drift terms in specific geometric spaces.
method Analyzing fundamental solutions in the Heisenberg group and Grushin-type planes.
result Natural generalizations of Beals, Gaveau, and Greiner's solutions for the Laplace equation with drift term.
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…
Motion of curves and surfaces in R3 lead to nonlinear evolution equations which are often integrable. They are also intimately connected to the dynamics of spin chains in the continuum limit and integrable soliton systems through geometric and gauge symmetric connections/equivalence. Here we point out the fact that…
Proves existence and uniqueness of weak solutions for specific equations.
problem Existence and uniqueness of solutions for generalized Monge-Ampère and deformed Hermitian-Yang-Mills equations.
method Combines viscosity-theoretic and pluripotential-theoretic techniques.
result Existence and uniqueness of weak solutions in boundary cases.
We study four distinct second-order nonlinear equations of Rabelo which describe pseudospherical surfaces. By transforming these equations to the constant-characteristic form we relate them to some well-studied integrable equations. Two of the Rabelo equations are found to be related to the sine-Gordon equation. The ot…
This paper classifies symmetries of biharmonic heat equations on surfaces of revolution.
problem Investigating symmetries of biharmonic heat equations on surfaces of revolution.
method Lie symmetry analysis to classify symmetries and derive invariant solutions.
result The biharmonic heat equation on a surface of revolution has the same Lie symmetries as the harmonic heat equation.
This paper characterizes Lie symmetries for a general Lienard-type equation.
problem Characterizing Lie symmetries for a general Lienard-type equation.
method Analyzing the Lie symmetry group of the general Lienard-type equation u¨=∑k=0nfku˙k for n≥4. result The paper provides a condition for the existence of another Lie symmetry and characterizes when the equation admits such symmetries.
Paper solves Hessian equations on Kähler manifolds.
problem Solving Hessian equations on Kähler manifolds.
method Combines elementary symmetric functions; provides sufficient and necessary condition.
result Generalizes results for Hessian and Hessian quotient equations.
Proves C^2,alpha estimates for elliptic equations on hyperkähler manifolds.
problem Elliptic equations on hypercomplex manifolds.
method Proves C^2,alpha estimates under suitable assumptions.
result Solutions to specific elliptic equations on hyperkähler manifolds satisfy C^2,alpha estimates.
The paper introduces new equations in Kähler geometry and proves their solutions and convexity.
problem Solving equations in Kähler geometry and understanding their geometric implications.
method Using moment map pictures to motivate and prove solutions for the equations.
result The Mabuchi functional for certain equations is shown to be convex.
Proves existence and uniqueness of loop equation solutions for semisimple Frobenius manifolds.
problem Existence and uniqueness of solutions to loop equations in generalized Frobenius manifolds.
method Proves existence and uniqueness of solutions to loop equations for semisimple generalized Frobenius manifolds.
result Existence and uniqueness of solutions to loop equations for semisimple generalized Frobenius manifolds.
We associate an integrable generalized complex structure to each 2-dimensional symplectic Monge-Ampère equation of divergent type and, using the Gualtieri ∂ˉ operator, we characterize the conservation laws and the generating function of such equation as generalized holomorphic objects.
Scientific documents rely on both mathematics and text to communicate ideas. Inspired by the topical correspondence between mathematical equations and word contexts observed in scientific texts, we propose a novel topic model that jointly generates mathematical equations and their surrounding text (TopicEq). Using an e…
In this paper, a type of integrable evolution equation--the generalized Landau-Lifshitz equation into Sn is considered. We deal with this equation from a geometric point of view by rewriting it in a geometric form. Through the geometric energy method, we show the global well-posedness of the corresponding Cauchy pro…
StarNet trains deep models without gradients using linear equations.
problem Training deep generative models with gradients.
method Solving determined systems of linear equations.
result Least-square bounds for latent codes and model parameters.
The paper explores solutions to the distributional Bellman equation in reinforcement learning.
problem Distributional reinforcement learning considers complete return distributions, not just expected returns.
method Study existence and uniqueness of solutions to general distributional Bellman equations, linking them to multivariate affine equations.
result Any solution to a distributional Bellman equation can be derived from a multivariate affine distributional equation.
Paper transforms a complex equation into simpler forms for analysis.
problem Analyzing a fourth-order dispersive flow equation on Kähler manifolds.
method Developed the generalized Hasimoto transformation to simplify the equation.
result Explicit expressions derived for three examples of compact Kähler manifolds.
Introduces a new 2C extension of the heavenly equation.
problem Solving the general heavenly equation and its extensions.
method Infinite hierarchy of nonlocal symmetries and recursion operator.
result Solutions correspond to 4D hyper-para-Hermitian metrics.
The binormal (or vortex filament) equation provides the localized induction approximation of the 3D incompressible Euler equation. We present explicit solutions of the binormal equation in higher-dimensions that collapse in finite time. The local nature of this phenomenon suggests the appearance of singularity in nearb…
The generalized Jang equation was introduced in an attempt to prove the Penrose inequality in the setting of general initial data for the Einstein equations. In this paper we give an extensive study of this equation, proving existence, regularity, and blow-up results. In particular, precise asymptotics for the blow-up …
Paper proves a Liouville theorem for a generalized elliptic equation on H-type groups.
problem Proving a Liouville theorem for a generalized elliptic equation on H-type groups.
method Proof based on an a priori integral estimate and a generalized differential identity.
result Obtained a Liouville type theorem for the semilinear subcritical elliptic equation on H-type groups.
Normality equations describe Newtonian dynamical systems admitting normal shift of hypersurfaces. They were first derived in Euclidean geometry, then in Riemannian geometry. Recently they were rederived in more general case, when geometry of manifold is given by generalized Legendre transformation. As appears, in this …
Paper finds new equations for pseudospherical surfaces with isometric immersions.
problem Identifying equations with isometric immersions for pseudospherical surfaces.
method Provided families of second order non-linear PDEs with local isometric immersions in E^3.
result Found equations with principal curvatures depending on finite-order jets of solutions.