Study classifies Einstein-Yang-Mills spaces in 4D symmetric spaces.
problem Classifying Lorentzian symmetric spaces with Einstein-Yang-Mills properties.
method Classification based on invariant metric connections and diagonal metrics.
result Four-dimensional symmetric spaces with nontrivial isotropy groups are classified.
Proves local existence and extension principle for Einstein Yang--Mills system with spherical symmetry.
problem Local existence and stability of the spherically symmetric Einstein Yang--Mills system.
method Employed an L2-based method to prove local existence and establish an extension principle. result Established local existence and extension principle for the SSEYM with H1 data. Proves stability of Minkowski space-time in Einstein-Yang-Mills system.
problem Stability of Minkowski space-time governed by Einstein-Yang-Mills system.
method Null frame decomposition, well-posedness of Cauchy development, convergence to Minkowski space-time.
result Exterior stability of Minkowski space-time in Lorenz gauge without spherical symmetry.
Proves stability of Minkowski space-time for Einstein-Yang-Mills equations.
problem Stability of Minkowski space-time for perturbations governed by Einstein-Yang-Mills equations.
method Proves exterior energy estimates for tensorial non-linear wave equations in Minkowski space-time.
result Proves exterior stability of Minkowski space-time for Einstein-Yang-Mills equations.
Stability of Minkowski space-time in higher dimensions proven for arbitrary small perturbations.
problem Stability of Minkowski space-time solution to Einstein-Yang-Mills equations in higher dimensions.
method Global stability proof for arbitrary small perturbations using wave coordinates and gauge invariant norms.
result Global stability of Minkowski space-time in higher dimensions n≥5 for arbitrary small perturbations. Stability of Minkowski space-time in Einstein-Yang-Mills system proven.
problem Stability of Minkowski space-time in Einstein-Yang-Mills system.
method Null frame decomposition, wave coordinates, dispersive estimates.
result Solutions converge to zero Yang-Mills curvature and Minkowski space-time.
The paper explores Kaluza-Klein theories without assuming a fibration structure.
problem Exploring Kaluza-Klein theories without assuming a fibration structure.
method Variational formulations of gauge theories and Einstein--Yang-Mills equations.
result Classical solutions allow the construction of a manifold X of dimension 4 as physical space-time, leading to solutions of the Einstein--Yang-Mills systems. Study the deformation theory of Einstein-Yang-Mills system on compact manifolds.
problem Deformation theory of Einstein-Yang-Mills system on compact manifolds.
method Slice theorem, linearization analysis, essential deformation characterization.
result Realize moduli space of Einstein-Yang-Mills pairs as an analytic set in a finite-dimensional tame Fréchet manifold.
Let P be a principal U(1)-bundle over a closed manifold M. On P, one can define a modified version of the Ricci flow called the Ricci Yang-Mills flow, due to these equations being a coupling of Ricci flow and the Yang-Mills heat flow. We use maximal regularity theory and ideas of Simonett concerning the asymptoti…
Study Einstein-Yang-Mills fields on specific manifolds, proving field deformations.
problem Deforming Einstein-Yang-Mills fields over conformally compact manifolds.
method Deformation theory using 0-calculus of Mazzeo and Melrose. result Any small perturbation of boundary data can be realized as an Einstein-Yang-Mills field.
Study wormholes in Einstein-Yang-Mills theory with a phantom field.
problem Existence of wormholes in Einstein-Yang-Mills theory with a phantom scalar field.
method Rigorous mathematical proof and numerical analysis of wormhole solutions.
result Existence of an infinite sequence of symmetric wormhole solutions.
Developed a new symmetric hyperbolic formulation for Einstein-Yang-Mills system.
problem Future stability of solutions of the Einstein-Yang-Mills system with arbitrary dimension.
method Tensorial symmetric hyperbolic formulation and local well-posedness for Cauchy problem.
result Established local well-posedness for the Cauchy problem of EYM equations in the temporal gauge.
Starting with the most general four-dimensional spacetime possessing two commuting Killing vectors and a nontrivial Killing tensor, we analytically integrate Einstein-Yang-Mills equations for a completely arbitrary gauge group. It is assumed that the gauge field inherits the symmetries of the background and is aligned …
In this paper, we prove a convergence theorem for sequences of Einstein Yang-Mills systems on U(1)-bundles over closed n-manifolds with some bounds for volumes, diameters, L2-norms of bundle curvatures and L2n-norms of curvature tensors. This result is a generalization of earlier compactness the…
Proves energy estimates for tensorial wave equations, decoupling components for stability proof.
problem Proving stability of (1+3)-Minkowski space-time with various non-linearities. method Decouples energy estimates for tensorial wave equations, exploiting tensorial structure and Lie derivatives.
result Decoupled energy estimates for tensorial solutions, allowing new stability proofs.
For any positive integer n and any Lie group G, given a definite symmetric bilinear form on Rn and an Ad-invariant scalar product on the Lie algebra of G, we construct a variational problem on fields defined on an arbitrary oriented (n+dimG)-dimension…
Existence of an infinite sequence of harmonic maps between spheres of certain dimensions was proven by Bizon and Chmaj. This sequence shares many features of the Bartnik-McKinnon sequence of solutions to the Einstein-Yang-Mills equations as well as sequences of solutions that have arisen in other physical models. We ap…
Study proves global existence and decay for complex wave equations.
problem Global existence and decay for quasilinear wave equations with weak-null condition.
method Novel decoupling of higher order energy estimates, focusing on tangential components.
result Established global existence and decay for solutions with small data.
The paper introduces a trilinear functional to recover torsion in spectral triples.
problem Recovering torsion in noncommutative spectral triples.
method Introduces a trilinear functional for spectral triples and demonstrates its application to recover torsion.
result The trilinear functional recovers the torsion of the linear connection in canonical spectral triples.
Study Galois groupoids of discret Painlevé equations.
problem Computing Galois groupoids for discret Painlevé equations.
method Using semi-continuity theorem for Galois groupoid in confluence of difference to differential equations.
result Computed Galois groupoids for discret Painlevé equations.
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 an unsupervised approach for discovering semantic representations of mathematical equations. Equations are challenging to analyze because each is unique, or nearly unique. Our method, which we call equation embeddings, finds good representations of equations by using the representations of their surrounding …
Paper establishes estimates for nonlinear equations on compact manifolds.
problem Estimating solutions to fully nonlinear equations with gradient terms on compact almost Hermitian manifolds.
method Establishes second order estimates and proves existence of solutions for specific equations.
result Proves existence of solutions for various equations, including Monge-Ampère and Hessian 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 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.
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…
Introduces a new PDE involving differential forms for Kähler geometry.
problem Solving a unified PDE for various important equations in Kähler geometry.
method Introduces a fully nonlinear PDE with differential form Λ and proves solvability conditions.
result Generalizes previous works and proves a conjecture for the dHYM equation.
Sharp sub-Gaussian bounds for subsolutions of Trudinger's equation on Riemannian manifolds.
problem Bounding weak subsolutions of Trudinger's equation on Riemannian manifolds.
method Proving sub-Gaussian upper bounds for weak subsolutions.
result The upper bounds are sharp for specific classes of manifolds, including \(\mathbb{R}^{n}\).
In this paper we perform a blow-up and quantization analysis of the following nonlocal Liouville-type equation \begin{equation}(-Δ)^\frac12 u= κe^u-1~\mbox{in S1,} \end{equation} where (−Δ)21 stands for the fractional Laplacian and κ is a bounded function. We interpret the above equation as the prescri…
The paper derives gradient estimates for porous medium and fast diffusion equations on metric measure spaces.
problem Gradient estimates for porous medium and fast diffusion equations on metric measure spaces.
method Derives Li-Yau and Souplet-Zhang type gradient estimates for the given equations.
result Gradient estimates for the equations on complete noncompact metric measure spaces with compact boundary.
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.
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.
We describe a method to reduce partial differential equations of Monge-Ampère type in 4 variables to complex partial differential equations in 2 variables. To illustrate this method, we construct explicit holomorphic solutions of the special lagrangian equation, the real Monge-Ampère equations and the Plebanski equatio…
Probabilistic grammars improve equation discovery from data.
problem Discovering scientific laws from data using equations.
method Proposed probabilistic context-free grammars to encode soft constraints and a Monte-Carlo algorithm.
result Probabilistic grammars lead to more efficient equation discovery.
In this paper, we provide families of second order non-linear partial differential equations, describing pseudospherical surfaces (pss equations), with the property of having local isometric immersions in E^3, with principal curvatures depending on finite-order jets of solutions of the differential equation. These equa…
The paper proves constant rank theorems for special Lagrangian equations.
problem Understanding saddle solutions and Liouville type results for special Lagrangian equations.
method Argument based on saddle solutions and Liouville type results for the special Lagrangian equation.
result Obtained constant rank theorems for saddle solutions to the special Lagrangian equation and the quadratic Hessian equation.
Study solves HJB equations for time-inconsistent control problems.
problem Time-inconsistent deterministic linear quadratic control problems.
method Characterized solutions using Riccati equations with integral terms, proving uniqueness.
result Uniqueness of solutions to equilibrium HJB equations proved.
We introduce a class of overdetermined systems of partial differential equations of finite type on (pseudo)-Riemannian manifolds that we call the generalised Ricci soliton equations. These equations depend on three real parameters. For special values of the parameters they specialise to various important classes of equ…
In this thesis, we consider the suitability of using the charged cold fluid model in the description of ultra-relativistic beams. The method that we have used is the following. Firstly, the necessary notions of kinetic theory and differential geometry of second order differential equations are explained. Then an averag…
Combines geometric hydrodynamics with magnetic systems to derive new equations and prove well-posedness.
problem Deriving new equations for magnetic systems and proving their well-posedness.
method Introducing the magnetic Euler-Arnold equation and proving well-posedness for specific equations.
result Local and global well-posedness results for the magnetic Euler-Arnold equation associated with the global quasi-geostrophic equations.
We determine the Lie point symmetries of the Fokker-Planck equation and provide examples of solutions of this equation. The Fokker-Planck equation admits a conserved form, hence there is an auxiliary system associated to this equation and whose point symmetries give rise to potential symmetries of the Fokker-Planck equ…
Studies projective geometry and partial differential equations prolongation.
problem Understanding the prolongation of overdetermined geometric partial differential equations.
method Introduction to differential geometry and tractor calculus, study of prolongation of equations.
result Recovery of projective tractor and cotractor connections via partial differential equations prolongation.
Develops equivariant connections for Yang-Mills equations, simplifying interactions modeling.
problem Simplifying interactions modeling in Yang-Mills equations for different bundles.
method Introduces SO+(p,q)-equivariance to reduce Yang-Mills equations. result Models electroweak interaction and interactions with differential and wave equations.
The paper derives gradient estimates for solutions of certain equations on metric measure spaces.
problem Gradient estimates for solutions of specific nonlinear and elliptic equations on metric measure spaces.
method Derives Li-Yau and Hamilton's type gradient estimates for positive solutions.
result Gradient estimates for positive solutions of the equations on complete noncompact metric measure spaces.
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.
We come up with infinite-dimensional prequantum line bundles and moment map interpretations of three different sets of equations - the generalised Monge-Amp`ere equation, the almost Hitchin system, and the Calabi-Yang-Mills equations. These are all perturbations of already existing equations. Our construction for the g…
Study gradient estimates for nonlinear parabolic equations on Riemannian manifolds.
problem Estimating gradients for nonlinear parabolic equations on Riemannian manifolds.
method Analyzes Fisher-KPP, parabolic Allen-Cahn, and Newell-Whitehead equations on complete noncompact Riemannian manifolds.
result Gradient estimates for positive solutions and Liouville theorem for ancient solutions.
Study a modified Laplacian equation in spacetime.
problem Analyzing a perturbed Laplacian equation in spacetime.
method Examining the equation \( \Delta u + P |
abla u| = h |
abla u| \) in an initial data set.
result Identified new properties of the modified equation.