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.
The Giroux correspondence and the notion of a near force-free magnetic field are used to topologically characterize near force-free magnetic fields which describe a variety of physical processes, including plasma equilibrium. As a byproduct, the topological characterization of force-free magnetic fields associated with…
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.
Mechanically interprets Gaussian curvature using spinning disks and curved surfaces.
problem Understanding Gaussian curvature through mechanical means.
method Considering the motion of a spinning disk constrained to a curved surface, relating gyroscopic force to magnetic force and Gaussian curvature.
result The gyroscopic force on a spinning disk is equal to the magnetic force on a point charge moving in a magnetic field normal to the surface, with magnitude equal to the Gaussian curvature.
Magnetic manifold HMC improves sampling on constrained manifolds.
problem Sampling from distributions restricted to embedded manifolds.
method Introduces magnetic manifold HMC, a generalization of HMC for constrained manifolds.
result Magnetic manifold HMC outperforms canonical manifold-constrained HMC.
Study of gyroscopic Chaplygin systems and magnetic flows on spheres.
problem Integrability and Hamiltonization of magnetic geodesic flows on spheres.
method Analysis of gyroscopic Chaplygin systems with magnetic forces, Hamiltonization, invariant measure existence.
result Integrable magnetic geodesic flows on spheres Sn−1 for n>3. 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.
Model predicts EMF of Ni-Mn-Ga MSMA, improved with GRNN.
problem Predicting the electromotive force (EMF) of Ni-Mn-Ga MSMA under various conditions.
method Developed a new constitutive model for Ni-Mn-Ga single crystals, incorporating magnetic easy axis offset. Used GRNN to enhance model predictions.
result GRNN improves model predictions of EMF, capturing more experimental features.
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.
New electromagnetic curvature defined via Jacobi-Maupertuis, showing positive curvature for non-zero magnetic force.
problem Defining and analyzing electromagnetic curvature.
method Using Jacobi-Maupertuis reparametrization and energy analysis.
result Positive electromagnetic Ricci curvature for non-zero magnetic force and small potential.
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…
Study preserves symplectic structure in forced discrete mechanical systems.
problem Preserving symplectic structure in forced discrete mechanical systems.
method Analyzes a specific type of forced discrete mechanical system (Q,Ld,fd), preserving a symplectic structure on QimesQ. result The preserved symplectic structure can be seen as Marsden-Weinstein reduction of the canonical symplectic structure.
The second order differential equation dtDγ˙(t)=Fγ(t)(γ˙(t))−∇V(γ(t)) on a Lorentzian manifold describes, in particular, the dynamics of particles under the action of a electromagnetic field F and a conservative force −∇V. We provide a first study on the extendability of its solu…
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.
Given a submersion π:Q→M with an Ehresmann connection H, we describe how to solve Hamiltonian systems on M by lifting our problem to Q. Furthermore, we show that all solutions of these lifted Hamiltonian systems can be described using the original Hamiltonian vector field on M along with a gener…
PS-VAE extracts multi-parameter MRI biomarkers with uncertainty quantification.
problem Uncertainty in inverse problems limits clinical acceptance of quantitative MRI methods.
method Physics-Structured Variational Autoencoder (PS-VAE) integrating physics simulator and self-supervised learning.
result PS-VAE provides full covariance of inter-parameter correlations and accelerates multi-parametric MRI quantification.
Study of thermo-metric properties and magnetic field effects on manifold curvature.
problem Understanding the geometry and curvature of thermo-metric spaces under magnetic field constraints.
method Differential geometry and combinatorial analysis of magnetic field effects.
result Discovered a small yet significant difference in even and odd dimensional CV manifolds.
Study proves uniqueness for ray transform on surfaces with obstacles.
problem Uniqueness of functions and 1-forms on surfaces with reflecting obstacles.
method Broken ray transform on twisted geodesics with nonpositive curvature and reflecting boundary.
result Proves uniqueness result for sums of functions and 1-forms.
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.
The paper estimates magnetic Laplacian eigenvalues using Ricci curvature and magnetic fields.
problem Estimating eigenvalues of the magnetic Laplacian on manifolds.
method Lichnerowicz and Buser type estimates related to Ricci curvature and magnetic fields.
result Relates eigenvalues, magnetic fields, Ricci curvature, and Cheeger type constants.
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. We propose a solution to the image deconvolution problem where the convolution kernel or point spread function (PSF) is assumed to be only partially known. Small perturbations generated from the model are exploited to produce a few principal components explaining the PSF uncertainty in a high dimensional space. Unlike …
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.
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.
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.
Study shows magnetic billiards are algebraically non-integrable in most cases.
problem Non-integrability of magnetic billiards in convex domains.
method Analyzing polynomial integrals of motion in magnetic billiard flows.
result Magnetic billiards are algebraically non-integrable for most smooth boundary curves.
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.
Study magnetic geodesics on surfaces avoiding lightlike points.
problem Understanding geodesics on surfaces with singularities.
method Analytical proof and numerical solutions.
result Magnetic geodesics avoid points with lightlike tangent planes.
We explicitly determine all magnetic curves corresponding to the Killing magnetic fields on the 3-dimensional Euclidean space.
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.
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.
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.
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.