Solves Dirac equation coupled to vector bundles.
problem Yang-Mills equations and vector bundles on Riemann surfaces.
method Analyzes coupled Dirac operators.
result Provides concrete solutions to the Dirac equation.
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48 results for “vector equations”
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.
Classifies vector equations with higher symmetries.
problem Classifying integrable vector equations with higher symmetries.
method Complete classification of isotropic vector equations of geometric type with higher symmetries.
result New examples of integrable multi-component systems and auto-Backlund transformations found.
New equations connect unit Killing vectors to initial data.
problem Characterizing initial data for Einstein vacuum with unit Killing vectors.
method Developed new equations (uKID) by eliminating scaling and using propagation identity.
result Found equations that are finite type and characterize unit normalized Killing vectors.
Investigates -equation on holomorphic vector bundles over Kähler manifolds.
problem Analyzes properties and solutions of -equation on holomorphic vector bundles.
method Introduces and studies -equation, provides algebraic and numerical criteria.
result Provides an algebraic condition (asymptotic -stability) and a numerical criterion for vortex bundles.
Monge-Ampère type equation for the Nakano positive curvature tensor of holomorphic vector bundlesmath.DG
Solves a Monge-Ampère type equation for Nakano positive curvature tensors of holomorphic vector bundles.
problem Solving Monge-Ampère type equations for Nakano positive curvature tensors of holomorphic vector bundles.
method Solves the Monge-Ampère type equation in the conformal class of a Nakano positive Hermitian metric.
result Solves the Monge-Ampère type equation for Nakano positive curvature tensors of holomorphic vector bundles.
The paper examines ellipticity of specific equations on vector bundles.
problem Investigating ellipticity of vector bundle versions of Monge-Ampère equations.
method Analyzing continuity paths and preserving ellipticity of equations.
result Not all equations preserve ellipticity along continuity paths, but does.
Study on positivity properties of vector bundle Monge-Ampère equation.
problem Analyzing positivity in vector bundle Monge-Ampère equation.
method Investigates MA-positivity and MA-semi-positive solutions for different ranks of holomorphic bundles over complex surfaces and manifolds.
result Positivity preservation in rank-two holomorphic bundles but not in higher ranks.
In this paper, position vector of a spacelike general helix with respect to standard frame in Minkowski space E are studied in terms of Frenet equations. First, a vector differential equation of third order is constructed to determine position vector of an arbitrary spacelike general helix. In terms of solution, …
In this paper, position vectors of a time-like curve with respect to standard frame of Minkowski space E are studied in terms of Frenet equations. First, we prove that position vector of every time-like space curve in Minkowski space E satisfies a vector differential equation of fourth order. The general so…
Study higher rank deformed Hermitian-Yang-Mills equations for stable vector bundles.
problem Stability conditions for higher rank vector bundles over complex manifolds.
method Establish equivalence between dHYM equations and -stability.
result Equivalence between dHYM solutions and -stability for vortex type bundles.
In this paper, we prove that the position vector of every space curve satisfies a vector differential equation of fourth order. Also, we determine the parametric representation of the position vector of general helices from the intrinsic equations and where and are th…
Solves inverse problem for Maxwell equations using vector fields.
problem Inverse problem for Maxwell equations in vacuum.
method Abstract theory of implicit differential equations over pre-symplectic manifolds.
result Provides solution for Maxwell equations using vector fields.
Proves stability of certain vector bundles on Kähler surfaces.
problem Stability of rank 2 holomorphic vector bundles on Kähler surfaces.
method Proves existence of -positive and -critical metrics leading to bundle stability.
result Proves stability results for deformed Hermitian Yang-Mills and almost Hermite-Einstein equations for rank 2 bundles.
We construct solutions of the vacuum vector constraint equations on manifolds with cylindrical ends.
Study higher rank deformed Hermitian-Yang-Mills equations for stable vector bundles.
problem Stability conditions for higher rank vector bundles over complex manifolds.
method Establish equivalence between dHYM equations and -stability.
result Equivalence between dHYM solutions and -stability for vortex type bundles.
New methods for solving hydrodynamic-type equations using quasi-rectifiable Lie algebras.
problem Solving systems of hydrodynamic-type equations.
method Introducing and analyzing quasi-rectifiable Lie algebras and vector fields.
result New methods for solving hydrodynamic-type equations.
Study Galois groupoids of vector fields, proving lower semicontinuity.
problem Computing Galois groupoids for general parameter values of Painlevé equations.
method Prove lower semicontinuity of Galois groupoids of vector fields.
result Results can compute Galois groupoids for general parameter values of Painlevé equations.
Classifies geodesic vectors in low-dimensional Lie algebras.
problem Stability of geodesic vectors in Lie algebras.
method Complete classification of Lyapunov stable and unstable geodesic vectors.
result Classification for metric Lie algebras of dimension 3 and 4.
On the Optimal Regularity Implied by the Assumptions of Geometry II: Connections on Vector Bundlesmath.DG
Extends optimal regularity and compactness to vector bundles over non-Riemannian manifolds.
problem Optimal regularity and compactness for connections on vector bundles.
method Derive RT-equations, establish existence theory, handle curvature up to .
result Optimal regularity and compactness extended to vector bundles over non-Riemannian manifolds.
Reconstructing signature features from randomized vector fields in differential equations.
problem Reconstructing signature features from controlled differential equations with random vector fields.
method Using controlled ordinary differential equations driven by continuous bounded variation curves, the study explores the extent to which signature features can be reconstructed from the non-linear flow of these equations.
result The number of signature features that can be reconstructed from the non-linear flow of controlled ordinary differential equations with random vector fields is exponential in the hidden dimension, under certain conditions.
Study characterizes 2-Killing vector fields on complex spacetimes.
problem Characterize -Killing vector fields on multiply twisted product spacetimes.
method Determine nonlinear differential equations, find twisted functions, provide solutions, and construct examples.
result Completely describe -Killing vector fields and twisted functions on multiply twisted product spacetimes.
Geometric equation defines canonical metrics on vector bundle families.
problem Finding canonical metrics on families of holomorphic vector bundles.
method Introducing a geometric partial differential equation for families of holomorphic vector bundles.
result Construction of Hermite--Einstein metrics in adiabatic classes on product manifolds and proof of the existence of a unique solution for the Dirichlet problem.
We introduce -critical connections for holomorphic vector bundles and prove their existence under stability conditions.
problem Existence of -critical connections for holomorphic vector bundles.
method Associated geometric PDEs to Bridgeland stability conditions and used infinite dimensional moment maps.
result In the large volume limit, a sufficiently smooth holomorphic vector bundle admits a -critical connection if and only if it is asymptotically -stable.
Proves conditions for weighted Hermite-Einstein metrics on vector bundles.
problem Conditions for existence of weighted Hermite-Einstein metrics.
method Introduces weighted Hermite-Einstein equation, stability notions, and proves existence.
result Existence of weighted Hermite-Einstein metrics if and only if slope polystable.
We determine the most general group of equivalence transformations for a family of differential equations defined by an arbitrary vector field on a manifold. We also find all invariants and differential invariants for this group up to the second order. A result on the characterization of classes of these equations by t…
Steerable neural ODEs on homogeneous spaces for equivariant feature dynamics.
problem Learning continuous-time equivariant dynamics of vector-valued features on homogeneous spaces.
method Introduces steerable neural ordinary differential equations on homogeneous spaces, interpreting features as sections of associated vector bundles over .
result Steerable NODEs are -equivariant when the flow and connection are -invariant, and they incorporate existing models.
New equations on manifolds linked to torus actions, proving existence and uniqueness.
problem Existence and uniqueness of solutions for generalized Kazdan-Warner equations.
method Linear action of a torus on complex vector spaces, existence and uniqueness proof on compact manifolds.
result Existence and uniqueness of solutions for the generalized Kazdan-Warner equations.
Let (E,D,P) be a flat vector bundle with a parabolic structure over a punctured Riemann surface, (M,g). We consider a deformation of the harmonic metric equation which we call the Poisson metric equation. This equation arises naturally as the dimension reduction of the Hermitian-Yang-Mills equation for holomorphic vect…
A set of equations is developed to describe a curve in space given the curvature and the angle of rotation of the osculating plane. The set of equations has a solution (in terms of and ) that indirectly solves the Frenet-Serret equations, with a unique value of for each specified value of . Explic…
A gauge-invariant form of the nonlinear Hodge equations is studied.
Variant of Seiberg-Witten equations for multiple-spinors connects to stability of holomorphic bundles.
problem Detecting stability of holomorphic vector bundles using Seiberg-Witten equations.
method Abelian gauge-theoretic variant of Seiberg-Witten equations for multiple-spinors.
result Constructs a numerical invariant related to stability of holomorphic vector bundles.
Study expanding Ricci solitons on vector bundles, reducing to Higgs bundle equations.
problem Understanding the long-time behavior of Ricci flows on manifolds.
method Prove equations dimension-reduce to twisted harmonic-Einstein equations, establish correspondence with G-Higgs bundles.
result Produce infinite families of new non-locally homogeneous examples, complete description in dimension 4.
Randers pp-wavesgr-qc
Study of Randers spacetimes and their Finsler gravity solutions.
problem Analyzing Finsler gravity field equations for Randers spacetimes.
method Examined Berwald-type Randers spacetimes and Finsler gravity field equations, showing equivalence to Einstein gravity.
result Found exact solutions for vacuum Finsler gravity are composed of pp-waves and 1-forms.
In this paper we find solutions to a certain class of vector-valued parabolic Allen-Cahn equation that as develops as interface a given triod evolving under curve shortening flow.
Paper presents an action principle for Einstein-Weyl equations in 3D.
problem Finding an action principle for Einstein-Weyl equations.
method Metric affine f(R) gravity action plus additional terms involving Lagrange multipliers and gravitational Chern-Simons contributions.
result The Weyl vector dynamics is governed by a special case of the generalized monopole equation.
New method learns vector fields from noisy time series data.
problem Learning vector fields from noisy time series data.
method Neural network architecture with tensor products of one-dimensional neural shape functions for vector field approximation, alternating minimization for noise handling.
result Neural shape function architecture robust to noise, learning accurate vector fields from data with up to 10% Gaussian noise.
Euler's equations for a two-dimensional system can be written in Hamiltonian form, where the Poisson bracket is the Lie-Poisson bracket associated to the Lie algebra of divergence free vector fields. We show how to derive the Poisson brackets of 2d hydrodynamics of ideal fluids as a reduction from the one associated to…
Using the characterization of last multipliers as solutions of the Liouville's transport equation, new results are given in this approach of ODE by providing several new characterizations, e.g. in terms of Witten and Marsden differentials or adjoint vector field. Applications to Hamiltonian vector fields on Poisson man…
The Kaczmarz algorithm is popular for iteratively solving an overdetermined system of linear equations. The traditional Kaczmarz algorithm can approximate the solution in few sweeps through the equations but a randomized version of the Kaczmarz algorithm was shown to converge exponentially and independent of number of …
The paper proves quasi-Einstein structures on manifolds admit Killing vector fields and provides new examples.
problem Classifying quasi-Einstein structures and understanding their properties.
method Analyzing quasi-Einstein equations and exploring their connections to Hitchin's equations.
result A class of quasi-Einstein structures on closed manifolds must admit a Killing vector field.
New equations simplify gauge-theoretic Khovanov homology solutions.
problem Solving the Haydys-Witten equations for Khovanov homology.
method Introduced decoupled version of Haydys-Witten equations; investigated asymptotic behavior.
result Decoupled equations simplify analysis of full equations on manifolds with ends and boundaries.
Symplectic method solves infinite-dimensional Schrödinger equations.
problem Solving Schrödinger equations on infinite-dimensional Hilbert spaces with unbounded Hamiltonians.
method Analytic vectors, manifolds modelled on normed spaces, symplectic differential geometry, Marsden--Weinstein reduction.
result Mapped -dependent Schrödinger equations onto projective spaces.
We employ Riemannian jets which are adapted to the Riemannian geometry to obtain the existence-uniqueness of viscosity solutions to the -Laplace equation in Riemannian vector fields. Due to the differences between Euclidean jets and Riemannian jets, the Euclidean method of proof is not valid in this environm…
Study local perturbations of vector bundles with polynomial curvature solutions.
problem Existence and stability of solutions to geometric PDEs under deformations.
method Geometric invariant theory, moment map framework, polystability conditions.
result Existence and uniqueness of solutions under local polystability conditions.
Formula found for energy slope in complex geometry.
problem Calculating the asymptotic slope of a K-energy.
method Established a formula for the asymptotic slope.
result Found a formula for the asymptotic slope of α-K-energy.
Minimal volume vector fields on surfaces via calibrations.
problem Finding minimal volume vector fields on Riemann surfaces.
method Theory of calibrations to write the equation of minimal volume vector fields.
result Equation of minimal volume vector fields on Riemann surfaces.
Paper proposes a new approach to optimal transport for vector and matrix densities.
problem Optimal transport for vector and matrix densities with positivity and action transitivity constraints.
method Gauge-theoretic approach using semi-direct product groups of diffeomorphisms and gauge transformations.
result Bures-type metrics on semi-direct product groups relate to Wasserstein-type metrics on vector and matrix densities via Riemannian submersions.