The paper examines curvature properties of a specific magnetic spacetime metric.
problem Investigating the geometric properties of Melvin magnetic spacetime metric.
method Analyzing a Melvin type static, cylindrically symmetric spacetime metric in Weyl form, considering curvature conditions and tensor properties.
result The Melvin magnetic metric is pseudosymmetric and satisfies the Roter type condition.
We prove the existence of harmonic spinor fields in axisymmetric Riemannian 3-manifolds having nonnegative scalar curvature and asymptotic to the usual constant time hypersurface of Melvin's magnetic universe. Such a spinor can be used in the proof of the uniqueness of the magnetized Schwarzschild solution.
Sharp Minkowski inequality found for AdS-Melvin spacetime surfaces.
problem Proving a Minkowski-type inequality for surfaces in the AdS-Melvin space.
method Used weighted normal flow to prove inequality for general surfaces.
result Sharp Minkowski inequality holds for all surfaces in AdS-Melvin space.
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.
New homology spheres found with special knots.
problem Finding homology spheres with specific knots.
method Infinitely many homology spheres constructed.
result Two distinct knots with special surgeries found.
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.
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…
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.
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.
The paper identifies magnetic ground states and their role in determining the conformal class of a surface.
problem Understanding the magnetic ground states and their relation to the conformal class of a surface.
method Analyzing the magnetic Laplacian and its eigenvalues on a Riemannian surface.
result The ground state spectrum uniquely determines the volume and conformal class of the metric.
This work reconstructs knot invariants from Alexander polynomials, proving consistency with known theorems.
problem Reconstructing knot invariants from Alexander polynomials.
method Quantization, deformation, and rewriting of Alexander polynomials.
result Derives new formulae for colored superpolynomials and proves consistency with Melvin-Morton-Rozansky theorem.
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. Normal forms prove dynamical results for magnetic fields on surfaces.
problem Existence and rigidity of Zoll flows on surfaces.
method Proved normal forms for strong magnetic fields and used them to derive dynamical results.
result Flow cannot be Zoll unless specific conditions hold.
Study on magnetic systems on surfaces with closed orbits.
problem Characterizing and proving rigidity of Zoll magnetic systems.
method Characterization and proof of rigidity for magnetic systems on surfaces of positive genus.
result Only magnetic systems with constant curvature metrics and functions yield Zoll Hamiltonian flows.
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.
Study magnetic perturbations in Riemannian and Lorentzian Calderón problems.
problem Determining metrics from boundary measurements under magnetic perturbations.
method Runge approximation for Riemannian case, microlocal analysis for Lorentzian case.
result Metrics can be uniquely determined in both Riemannian and Lorentzian cases under specific perturbations.
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.
The paper defines a critical value for a magnetic system and extends solutions beyond blow-up.
problem Analyzing blow-up behavior and extending solutions for a magnetic system.
method Formulated as a magnetic geodesic equation on an infinite-dimensional Lie group, computed Mañé's critical value, established Hopf-Rinow theorem.
result Computed Mañé's critical value for the magnetic two-component Hunter-Saxton system and extended solutions beyond blow-up.
Constructs smooth integrable magnetic systems on a two-torus.
problem Creating smooth magnetic systems on a two-torus with specific properties.
method Uses Nash-Moser implicit function theorem to find zeros of an action functional.
result Characterizes Zoll magnetic systems and proves their existence.
Study magnetic geodesics on half-Lie groups, proving Hopf-Rinow theorem for energies above critical value.
problem Investigate magnetic geodesics on half-Lie groups using Riemannian and two-form structures.
method Define Mañé's critical value, prove Finsler geodesic flow equivalence, and apply Hopf-Rinow theorem.
result Hopf-Rinow theorem holds for energies above Mañé's critical value on magnetic geodesics.
Let M be a closed oriented surface endowed with a Riemannian metric g and let Ω be a 2-form. We show that the magnetic flow of the pair (g,Ω) has zero asymptotic Maslov index and zero Liouville action if and only g has constant Gaussian curvature, Ω is a constant multiple of the area form of g and the mag…
Consider a compact Riemannian manifold with boundary endowed with a magnetic field. A path taken by a particle of unit charge, mass, and energy is called a magnetic geodesic. It is shown that if everything is real-analytic, the topology, metric, and magnetic field are uniquely determined by the scattering relation of t…
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.
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.
New derivation of knot invariants from universal invariant.
problem Deriving knot invariants from universal invariant.
method Using Hopf algebra D and a Mathematica implementation to compute ZD(K). result Derivation of large-color expansion from universal invariant.
A necessary and sufficient condition for energy-momentum conservation is proved within a topological, pre-metric approach to classical electrodynamics including magnetic as well as electric charges. The extended Lorentz force, consisting of mutual actions by F=(E, B) on the electric current and G=(H, D) on the magnetic…
We consider magnetic flows on 2-step nilmanifolds M=Γ\G, where the Riemannian metric g and the magnetic field σ are left-invariant. Our first result is that when σ represents a rational cohomology class and its restriction to g=TeG vanishes on the derived algebra, then the associated…
Integrates magnetic geodesic and sub-Riemannian flows on Stefel variety, proving integrability and Lax presentations.
problem Integrability of magnetic geodesic and sub-Riemannian flows on Vn,2. method Proves integrability of magnetic geodesic and sub-Riemannian flows on Vn,2 with respect to magnetic field ηdα. result Integrable cases of a heavy rigid body with a gyrostat are derived.
Study of harmonic maps into principal bundles with applications to magnetic interactions.
problem Understanding harmonic mappings from Riemannian manifolds into principal bundles.
method Characterization and analysis of Kaluza-Klein harmonic maps and generalized magnetic maps.
result Existence and properties of generalized magnetic maps, including non-trivial examples.
The paper studies magnetic curvature and proves the existence of closed orbits on low energy levels.
problem Existence of closed magnetic geodesics on low energy levels.
method Derived magnetic curvature operator and used Bonnet-Myers argument.
result Established the existence of a contractible periodic orbit on closed manifolds.
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.
Global singularities propagate in magnetic mechanical systems on Riemannian manifolds.
problem Propagation of singularities in magnetic mechanical systems.
method Combines reduction from magnetic to Riemannian systems, analysis of reparameterized flows, and regularization techniques.
result Invariant singular set under generalized gradient flow dynamics.
For a compact Riemannian manifold with boundary, endowed with a magnetic potential α, we consider the problem of restoring the metric g and the magnetic potential α from the values of the Mañé action potential between boundary points and the associated linearized problem. We study simple magnetic systems. In this…
We study the motion of a charge on a conformally flat Riemannian torus in the presence of magnetic field. We prove that for any non-zero magnetic field there always exist orbits of this motion which have conjugate points. We conjecture that the restriction of conformal flatness of the metric is not essential for this r…
A closer look at an example introduced by Livingston & Melvin and later studied by Miyazaki shows that a plumbing of two fibered ribbon knots (along their fiber surfaces) may be algebraically slice yet not ribbon.
We consider open manifolds which are interiors of a compact manifold with boundary, and Riemannian metrics asymptotic to a conformally cylindrical metric near the boundary. We show that the essential spectrum of the Laplace operator on functions vanishes under the presence of a magnetic field which does not define an i…
The main result presented here is that the flow associated with a riemannian metric and a non zero magnetic field on a compact oriented surface without boundary, under assumptions of hyperbolic type, cannot have the same length spectrum of topologically corresponding periodic orbits as the geodesic flow associated with…
New magnetic flow rigidity theorem for negative curvatures.
problem Rigidity of magnetic flows under constant periodic orbit lengths.
method Smooth families of diffeomorphisms and magnetic flows.
result Generalization of Guillemin-Kazhdan result to magnetic flows.
The article finds non-trivial Zoll magnetic systems for surfaces of any genus.
problem Finding non-trivial Zoll magnetic systems for surfaces of any genus.
method Twistor theoretic approach, constructing holomorphic blow-down maps into ruled surfaces.
result Construction of non-trivial Zoll magnetic systems for surfaces of any genus.
In this paper we consider the lens rigidity problem with partial data for conformal metrics in the presence of a magnetic field on a compact manifold of dimension ≥3 with boundary. We show that one can uniquely determine the conformal factor and the magnetic field near a strictly convex (with respect to the magne…
Study magnetic Laplacian eigenvalues on contact manifolds.
problem Characterize spectral properties of magnetic fields on contact manifolds.
method Analyze first eigenvalue of magnetic horizontal Laplacian, provide upper bounds, and use topological conditions.
result Equality in upper bounds implies Heisenberg left-invariant nilmanifold structure and unique determination of manifold Chern class.
In recent years, defending adversarial perturbations to natural examples in order to build robust machine learning models trained by deep neural networks (DNNs) has become an emerging research field in the conjunction of deep learning and security. In particular, MagNet consisting of an adversary detector and a data re…
To a knot in 3-space, one can associate a sequence of Laurent polynomials, whose nth term is the nth colored Jones polynomial. The Volume Conjecture for small angles states that the value of the n-th colored Jones polynomial at $e^{\a/n}$ is a sequence of complex numbers that grows subexponentially, for a fixed s…
We consider magnetic geodesic flows of the normal metrics on a class of homogeneous spaces, in particular (co)adjoint orbits of compact Lie groups. We give the proof of the non-commutative integrability of flows and show, in addition, for the case of (co)adjoint orbits, the usual Liouville integrability by means of ana…
We consider a Riemannian cylinder endowed with a closed potential 1-form A and study the magnetic Laplacian with magnetic Neumann boundary conditions associated with those data. We establish a sharp lower bound for the first eigenvalue and show that the equality characterizes the situation where the metric is a product…
Enhances NOAA's Geospace model with machine learning for predicting ground magnetic perturbations.
problem Predicting the probability of ground magnetic field changes.
method Combining physics-based model with machine learning (boosted ensemble of classification trees).
result The ML-enhanced model consistently improves probabilistic forecast metrics.
We compute the average Tristram---Levine signature of any graph link with positive weights in a three sphere, generalizing the results of Kirby and Melvin. The main tools are the Neumann's algorithm for computing the equivariant signatures of graph links and the Reciprocity Law for Dedekind sums.