Paper studies non-associativity in quantum systems with magnetic fields.
problem Non-associativity of magnetic translations in quantum systems.
method Quantum field theory approach with n-component fermions. result Non-associativity described by a 3-cocycle of Rn with values in S1. Let M be a real 2m-torus equipped with a translation-invariant metric h and a translation-invariant symplectic form w; the latter we interpret as a magnetic field on M. The Hamiltonian flow of half the norm-squared function induced by h on T^*M (the "kinetic energy") with respect to the twisted symplectic form w_{T^*M}…
We study in detail Hodge-Helmholtz decompositions in non-smooth exterior domains filled with inhomogeneous and anisotropic media. We show decompositions of alternating differential forms belonging to weighted Sobolev spaces into irrotational and solenoidal forms. These decompositions are essential tools, for example, i…
Graph neural networks predict solid-state NMR parameters from atomic structures.
problem Efficiently predicting NMR parameters from atomic structures for complex materials.
method Graph neural networks applied to tensor quantities for anisotropic magnetic shielding and electric field gradient.
result Improved accuracy in predicting NMR properties from diverse and complex materials.
Challenge to separate Earth's magnetic field from vehicle's magnetic field for accurate navigation.
problem Separate Earth's magnetic field from vehicle's magnetic field for accurate magnetic navigation.
method Use machine learning (ML) and integrate physics of magnetic navigation (SciML) to remove aircraft magnetic field from total magnetic field.
result A model can be constructed to effectively remove aircraft magnetic field from the dataset.
Extends E. Hopf's theorem to magnetic systems without conjugate points.
problem Proving magnetic curvature non-positive for magnetic systems without conjugate points.
method Using magnetic curvature introduced by the first author, proving magnetic flatness conditions.
result Magnetic flatness is a rigid condition with specific metric and curvature properties.
A magnetic field is defined by the property that its divergence is zero in a three dimensional oriented Riemannian manifold. Each magnetic field generates a magnetic flow whose trajectories are curves called as magnetic curves. In this paper, we give a new variational approach to studies the magnetic flow asociated wit…
Compatibility equations adapted to magnetic geometry.
problem No specific problem stated; magnetic geometry is the setting.
method Established compatibility equations in magnetic geometry.
result Analogues of Gauss, Ricci, and Codazzi-Mainardi equations in magnetic geometry.
This paper studies Hamilton-Jacobi equations for magnetic systems with constraints.
problem Understanding dynamics of magnetic systems with geometric constraints.
method Developed Hamilton-Jacobi equations for magnetic systems with nonholonomic constraints.
result Revealed relationships between magnetic structures, constraints, and dynamics.
Magnetic Brunn-Minkowski inequalities on Riemannian manifolds
problem Establishing equivalence between Brunn-Minkowski inequalities and magnetic Ricci curvature
method Using magnetic geodesics
result Proving a sharp, undistorted Brunn-Minkowski inequality
Study shows finiteness of magnetic hypersurfaces on closed manifolds.
problem Understanding the finiteness of magnetic hypersurfaces on closed manifolds.
method Introduced a dynamical version of the second fundamental form to generalize a previous result.
result Real-analytic negatively s-curved magnetic systems on closed real-analytic manifolds have only finitely many closed totally s-magnetic hypersurfaces. The paper studies magnetic curves in C-manifolds and their properties.
problem Understanding magnetic trajectories in C-manifolds. method Proving magnetic trajectories are θα-slant curves and providing parametrizations. result Normal magnetic curves in C-manifolds are θα-slant curves with specific curvature functions. New tensors help solve magnetic flow integrability.
problem Integrability of magnetic flows on specific manifolds.
method Introduced magnetic Killing symmetric tensors to construct first integrals.
result Proved integrability of invariant magnetic flows on 2-step nilmanifolds.
Study magnetic geodesics on Heisenberg groups and manifolds.
problem Dynamics of magnetic flows on Heisenberg groups.
method Explicit description of magnetic geodesics, determination of lengths.
result Density of periodic magnetic geodesics and marked magnetic length spectrum rigidity.
Paper introduces magnetic Hodge Laplacian for differential forms.
problem No specific problem stated; general spectral analysis of differential forms.
method Introduced magnetic Hodge Laplacian, discussed spectral results.
result Similarities and differences with magnetic Laplacian on functions.
The study characterizes Sasakian manifolds from magnetic Hopf surfaces.
problem Characterizing Sasakian manifolds from magnetic Hopf surfaces.
method Using a unit Killing vector field and Lie dragging a magnetic curve, the study characterizes Sasakian structures.
result If a magnetic Hopf surface is a constant mean curvature surface, then the manifold M is a Sasakian manifold.
We explicitly determine all magnetic curves corresponding to the Killing magnetic fields on the 3-dimensional Euclidean space.
Study of magnetic geodesics on Heisenberg nilmanifolds.
problem Existence and properties of closed magnetic geodesics on Heisenberg nilmanifolds.
method Analyzing conditions for the existence of closed magnetic geodesics on compact quotients of Heisenberg nilmanifolds.
result Existence of contractible closed magnetic geodesics for any energy level below the Mañé critical value.
Study magnetic Steklov eigenvalues on manifolds with boundary.
problem Eigenvalue problem for magnetic Steklov operators on compact manifolds.
method Equivalent characterizations, bounds, comparison results.
result Established bounds for the smallest eigenvalue of magnetic Steklov operators.
Study magnetic potentials on Anosov manifolds using spectral data.
problem Recover magnetic potentials from spectral data on Anosov manifolds.
method Utilize principal wave trace invariants and magnetic Schrödinger operator.
result Spectral data uniquely determines magnetic and electric potentials on Anosov manifolds.
Examples are presented of how the geometric notion of the mean curvature is used for general magnetic field configurations and magnetic surfaces. It is shown that the mean magnetic curvature is related to the variation of the absolute value of the magnetic field along its lines. Magnetic surfaces of constant mean curva…
Study magnetic curvature on Lie groups, extending Milnor's work.
problem Exploring magnetic curvatures on Lie groups.
method Computing magnetic curvatures and analyzing algebraic properties.
result Extending results from Milnor's classic paper on left-invariant metrics.
In this paper we study rigidity aspects of Zoll magnetic systems on closed surfaces. We characterize magnetic systems on surfaces of positive genus given by constant curvature metrics and constant magnetic functions as the only magnetic systems such that the associated Hamiltonian flow is Zoll, i.e. every orbit is clos…
Study shows magnetic trajectories in Berger spheres are homogeneous.
problem Homogeneity of contact magnetic trajectories in Berger spheres.
method Proved every contact magnetic trajectory is a product of a homogeneous geodesic and a charged Reeb flow.
result Contact magnetic trajectories in Berger spheres are homogeneous.
Extends magnetic flow theory results to higher dimensions.
problem Magnetic flows on manifolds of arbitrary dimension.
method New Pestov identities and adapted Riemannian geometry concepts.
result Established tensor tomography and ergodicity results for magnetic flows.
The paper explores how magnetic systems' spectra can identify metrics and 1-forms.
problem Can the marked magnetic action spectrum of magnetic systems with Anosov flow determine the metric and 1-form?
method The paper addresses this question in two settings: locally for systems with close metrics and 1-forms, and for metrics in the same conformal class.
result The paper answers the question affirmatively in both settings.
Study magnetic flows on 3D contact sub-Riemannian manifolds using Rumin complex.
problem Understanding magnetic flows on 3D contact sub-Riemannian manifolds.
method Introducing horizontal magnetic flows via closed Rumin differential two-forms and analyzing the lifted sub-Riemannian structure.
result Horizontal magnetic flows can be interpreted as geodesic flows on a suitably lifted structure, which is of Engel type when the magnetic field is non-vanishing.
Metrics are isometric for certain Anosov magnetic systems.
problem Isometry of metrics for Anosov magnetic systems.
method Conjugacy isotopic to the identity, volume-preserving conjugacy, cohomology class.
result Isometric metrics for conjugate Anosov magnetic systems.
Study on magnetic Dirac operators and their spectrum.
problem Understanding the spectrum of magnetic Dirac operators.
method Analysis of magnetic Dirac operators over complete Riemannian manifolds.
result Find sufficient conditions for maximal or discrete spectrum.
A quasiclassical method approximates magnetic monopole eigenvalues.
problem Describing eigenvalues of magnetic Laplacian in non-exact magnetic fields.
method Applying multidimensional WKB method to nontrivial line bundles.
result Demonstrated for Dirac magnetic monopole on a sphere.
Researchers compute trace formula for magnetic Laplacian on hyperbolic surfaces.
problem Analyzing the magnetic Laplacian on compact hyperbolic surfaces.
method Computed the trace formula for magnetic Laplacian energies above the Mane critical level.
result Asymptotic behavior of trace formula coefficients near the Mane critical level.
Homogeneous magnetic trajectories in a special linear group proven.
problem Proving homogeneity of magnetic trajectories in a specific group.
method Using contact magnetic curves and geodesics.
result Every contact magnetic trajectory is a product of a homogeneous geodesic and a charged Reeb flow.
Study on Killing magnetic curves in Heisenberg group geometry.
problem Understanding Killing magnetic curves in Heisenberg group.
method Presentation of Heisenberg group geometry and geodesics, study of Killing magnetic curves with explicit formulas.
result Explicit formulas for Killing magnetic curves in Heisenberg group.
We interpret magnetic billiards as Finsler ones and describe an analog of the string construction for magnetic billiards. Finsler billiards for which the law "angle of incidence equals angle of reflection" are described. We characterize the Finsler metrics in the plane whose geodesics are circles of a fixed radius. Thi…
This study explores magnetic trajectories on the Heisenberg group, finding symmetries and solutions.
problem Understanding magnetic geodesics on the Heisenberg group with invariant Lorentz force.
method Analyzing the Heisenberg Lie group with a non-commutative product, deriving magnetic equations, identifying symmetries, and solving variational problems.
result Magnetic trajectories are solutions to a variational problem, providing explicit examples of Lagrangians.
Homogeneous magnetic paths found in Heisenberg space.
problem Understanding magnetic geodesics in the Heisenberg group.
method Proving homogeneity of magnetic geodesics derived from the canonical contact structure.
result Magnetic geodesics in the Heisenberg group are homogeneous.
Study magnetic curves in Sasakian manifolds, classifying and parametrizing them.
problem Classify and parameterize pseudo-Hermitian magnetic curves in Sasakian manifolds.
method Define and classify pseudo-Hermitian magnetic curves, construct parametrizations.
result Complete classification theorem for pseudo-Hermitian magnetic curves in Sasakian manifolds.
Study magnetic geodesics on Kähler potentials using variational methods.
problem Understanding magnetic geodesics on Kähler potentials.
method Variational method for a generalized Landau-Hall functional.
result Magnetic geodesic equation and its relation to a perturbed complex Monge-Ampère equation.
The article studies eigenvalues and spectrum of magnetic Dirac operators.
problem Fundamental mathematical properties of magnetic Dirac operators remain unexplored.
method Eigenvalue estimates and explicit spectrum computation for specific cases.
result Explicit computation of spectrum for magnetic fields on specific manifolds.
We consider billiard ball motion in a convex domain of the Euclidean plane bounded by a piece-wise smooth curve influenced by the constant magnetic field. We show that if there exists a polynomial in velocities integral of the magnetic billiard flow then every smooth piece γ of the boundary must be algebraic and eith…
Two non-local asymptotic invariants of magnetic fields for the ideal magnetohydrodynamics are introduced. The velocity of variation of the invariants for a non-ideal magnetohydrodynamics with a small magnetic dissipation is estimated. By means of the invariants the spectra of electromagnetic fields are investigated. A …
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.
Anosov magnetic flows on surfaces are characterized.
problem Characterizing Anosov magnetic flows on surfaces.
method Using Wojtkowski's quotient bundle, necessary and sufficient conditions are derived.
result Necessary and sufficient conditions for Anosov magnetic flows on surfaces are established.
Study magnetic geodesics on odd spheres, computing critical energy values.
problem Understanding magnetic geodesics on odd-dimensional spheres.
method Explicit computation and analysis of submanifolds and symmetries.
result Energy values determine magnetic geodesic connectivity on spheres.
Physics-informed ML models improve turbulence understanding in fusion plasmas.
problem Improving turbulence modeling in fusion plasma devices.
method Physics-informed deep learning framework constrained by PDEs.
result Direct quantitative comparisons of turbulent fields between theory and gyrokinetic models.
Researchers prove integrability of magnetic systems on spheres up to dimension 6.
problem Integrability of magnetic systems on spheres restricted to their surface.
method Proved complete integrability for n ≤ 6, noncommutative integrability for n ≥ 7, conjectured integrability for all n.
result Complete integrability of magnetic flows on spheres for n ≤ 6, noncommutative integrability for n ≥ 7.
New findings on magnetic geodesic flows and periodic motions.
problem Characterizing superintegrable systems in magnetic geodesic flows.
method Analyzing rotationally symmetric magnetic geodesic flows.
result All sufficiently slow motions in a central magnetic field are periodic under specific curvature and homogeneity conditions.
Study magnetic trajectories on 2-step nilpotent Lie groups.
problem Understanding magnetic trajectories on specific Lie groups.
method Formulated magnetic equation, found solutions for invariant Lorentz forces, computed examples in Heisenberg groups.
result Interesting magnetic trajectories involving elliptic integrals found in Heisenberg groups.