New solutions found for Ginzburg-Landau equations on complex manifolds.
problem Finding solutions to Ginzburg-Landau equations on closed manifolds.
method Bifurcation theory applied to Laplace-type operator eigenvalues.
result First nonminimal and irreducible solutions on nontrivial line bundles.
New solutions found to Ginzburg-Landau equations on surfaces.
problem Existence of novel solutions to Ginzburg-Landau equations on closed surfaces.
method 2D, critically coupled Ginzburg-Landau theory, topology of moduli space.
result Existence of nonminimal, irreducible solutions on nontrivial line bundles.
The study finds conditions for irreducible Ginzburg-Landau fields on compact 2-manifolds.
problem Conditions for existence of irreducible Ginzburg-Landau fields on compact 2-manifolds.
method Analyzes Ginzburg-Landau equations on compact 2-manifolds with specific boundary conditions.
result Existence of irreducible Ginzburg-Landau fields for certain parameter values.
We establish a glueing theorem for the Ginzburg-Landau equations in dimension n>2. To this end, we consider a nondegenerate minimal submanifold of codimension 2, and construct a one-parameter family of solutions to the Ginzburg-Landau equations such that the energy density concentrates near this submanifold. The pr…
Researchers find entire solutions to magnetic Ginzburg-Landau equations in 4D.
problem Finding entire solutions to magnetic Ginzburg-Landau equations in 4D.
method Using Lyapunov-Schmidt reduction.
result Existence of entire solutions and saddle type solutions with specific zero sets.
We find nontrivial solutions to a Ginzburg-Landau equation on compact manifolds.
problem Finding nontrivial solutions to a specific Ginzburg-Landau equation on compact manifolds.
method Using min-max techniques to construct solutions whose energy grows logarithmically with a small parameter.
result The energy of constructed solutions concentrates on a nontrivial stationary, rectifiable (n−2)-varifold. Global solution found for biharmonic wave maps into spheres.
problem Solving biharmonic wave maps into spherical targets.
method Reformulated as a conservation law, solved with Ginzburg-Landau approximation.
result Global weak solution constructed in the energy space.
Estimates Ginzburg-Landau energy to show concentration of energy on a rectifiable varifold.
problem Estimating Ginzburg-Landau energy on manifolds with nontrivial first Betti number.
method Decomposing harmonic component into integral and fractional parts, using min-max construction.
result Energy concentration on a rectifiable (n−2)-varifold as εo0. Study of Ginzburg-Landau equation on manifolds with boundary, showing energy breakdown.
problem Behavior of solutions to Ginzburg-Landau equation on manifolds with boundary.
method Asymptotic analysis, energy upper bound, convexity condition, harmonic 1-form, rectifiable (n-2)-varifold.
result Energy of solutions breaks into two parts: one captured by a harmonic 1-form and the other by a stationary rectifiable (n-2)-varifold.
Study bounds the index of minimal submanifolds using energy measures and Yang-Mills-Higgs equations.
problem Bounding the index of codimension 2 minimal submanifolds.
method Second inner variation of energy, convergence of energy measures, and stress-energy tensors.
result Bound the Morse index of the submanifold by the index of critical points.
Study finds multiple solutions for Gross-Pitaevskii equations on curved spaces.
problem Finding multiple solutions for Gross-Pitaevskii equations on Riemannian manifolds.
method Critical point theory and Γ-convergence for Ginzburg-Landau functionals, plus new isoperimetric results.
result Lower bounds on the multiplicity of solutions in terms of the topology of the velocity set.
The paper studies momentum-based minimization for Ginzburg-Landau on Euclidean spaces and graphs.
problem Minimizing the Ginzburg-Landau functional on Euclidean spaces and graphs.
method Momentum-based minimization using a convex-concave splitting-based FISTA-type time discretization.
result Momentum can lead to faster convergence if the time step size is large but not too large.
In this paper we study the Kato' inequality on locally finite graph. We also study the application of Kato inequality to Ginzburg-Landau equations on such graphs. Interesting properties of Schrodinger equation and a Liouville type theorem are also derived.
Study vortex moduli space and compute Berry curvature for Ginzburg-Landau vortices.
problem Analyzing vortex moduli space and Berry curvature in Ginzburg-Landau theory.
method Using theorems of Taubes and Bradlow, compute tangent vectors and Berry curvature in the large volume limit.
result Computed Berry curvature and holonomy for Ginzburg-Landau vortices.
The paper proves unique constant solutions for maps with p-Ginzburg-Landau energy.
problem Finding unique constant solutions for maps with p-Ginzburg-Landau energy.
method Assuming growth conditions or asymptotic conditions for the p-Ginzburg-Landau energy, the paper establishes Liouville type theorems.
result Establishes unique constant solutions for constant Dirichlet boundary value problems on starlike domains.
Study stability of GL and YMH functionals on spheres and CP spaces.
problem Stability and critical points of Ginzburg-Landau and Yang-Mills-Higgs functionals.
method Analysis of critical points using Lawson-Simons methods.
result Lower bounds on Morse index and no stable critical points for YMH on Sn for n≥4. Study shows only rotations can be approximated by Ginzburg-Landau critical points.
problem Proving not all harmonic maps can be approximated by Ginzburg-Landau critical points.
method Rigidity theorem applied to Ginzburg-Landau energy critical points.
result Only rotations can be approximated by Ginzburg-Landau critical points.
Study local minimizers of Ginzburg-Landau functionals in high dimensions, showing energy measures converge to rectifiable measures.
problem Investigating minimizers of Ginzburg-Landau functionals in high dimensions with energy bounds.
method Analyzing minimizers with logarithmic energy bounds and considering the vacuum manifold's homotopy classes.
result Normalized energy measures converge to an (n−2)-rectifiable measure associated with a stationary varifold. Study on vortex sheet formation in Abelian gauge theories.
problem Understanding vortex sheet formation in Abelian gauge theories.
method Inspired by Allard's regularity theory, constructs approximate solutions and analyzes their perturbations.
result Establishes a geometric framework and regularity theory for the limiting defect set.
Study of vortex interactions in Ginzburg-Landau models on 2D Riemannian manifolds.
problem Characterize and quantify interactions between vortices in Ginzburg-Landau models.
method Variational Ginzburg-Landau model, Γ-limit analysis, flux quantization constraints.
result Renormalized energy between vortices determined as a Γ-limit.
We simplify modularity optimization using a Ginzburg-Landau functional and an MBO scheme.
problem Modularity optimization in network communities.
method We derive a Ginzburg-Landau functional approximation and an MBO scheme for modularity optimization.
result Our method converges to the correct energy in the limit and is faster and more stable.
Uniform small energy regularity for fractional geometric problems proved.
problem Proving regularity for fractional geometric problems.
method Analyzing parabolic boundary reaction Ginzburg-Landau problems and fractional harmonic maps to spheres.
result Uniform small energy regularity results for s∈(0,1), answering a posed question. Constructs surfaces with conical singularities using variational methods.
problem Creating Hamiltonian Stationary Surfaces with specific singularities.
method Variational methods and convergence process similar to Ginzburg-Landau analysis.
result Obtained surfaces with prescribed conical singularities related to optimal Wente constants.
Study of critical points in Ginzburg-Landau approximation with stability results.
problem Stability of critical points in Ginzburg-Landau approximation.
method Application of previous joint method with T. Rivière for upper semi-continuity of extended Morse index.
result Upper semi-continuity of extended Morse index for sequences of critical points.
Minimal submanifolds are found as energy concentration sets in variational problems.
problem Understanding the structure of minimal submanifolds in codimension two.
method Purely variational approach, extending previous work on geodesics.
result Non-degenerate minimal submanifolds can be derived from critical maps of the Ginzburg-Landau functional.
We develop a suitable generalization of Almgren's theory of varifolds in a lorentzian setting, focusing on area, first variation, rectifiability, compactness and closure issues. Motivated by the asymptotic behaviour of the scaled hyperbolic Ginzburg-Landau equations, and by the presence of singularities in lorentzian m…
For Ginzburg-Landau vortices, energy quantization holds only when density is less than 2.
problem Energy quantization in Ginzburg-Landau vortices for higher dimensions.
method Analyzing normalized energy measures and vorticity sets.
result Energy quantization only holds when density is less than 2.
Improved node classification in signed social networks using diffuse interface methods.
problem Classifying nodes in signed social networks (positive and negative interactions).
method Diffuse interface methods based on Ginzburg-Landau functional and extended graph Laplacian.
result Performance improvement in real signed social networks, outperforming state of the art.
Develops geometric framework for dissipative field equations.
problem Dissipative field equations and their geometric analysis.
method Canonical k-contact manifolds, k-contactifications, splitting results, regularity conditions, criteria for PDEs. result Explicit Hamiltonian descriptions for various nonlinear PDEs.
In this note, we study properties of the gradient map of the isoparametric polynomial. For a given isoparametric hypersurface in sphere, we calculate explicitly the gradient map of its isoparametric polynomial which turns out many interesting phenomenons and applications. We find that it should map not only the focal s…
In this paper we study spherically symmetric monopoles, which are critical points for the Yang-Mills-Higgs functional over a disk in 3 dimensions, with prescribed degree and covariant constant at the boundary. This is a 3-dimensional gauge-theory generalization of the Ginzburg-Landau model in 2 dimensions.
Machine discovers PDEs from spatiotemporal data without prior knowledge.
problem Discovering PDEs from complex spatiotemporal data without prior knowledge.
method Sparse Spatiotemporal System Discovery (extS3extd) using Sparse Bayesian Learning. result Automatically discovers ten types of PDEs from simulation data.
This paper studies the convergence of penalized energy to harmonic maps in Riemannian manifolds.
problem Analyzing the convergence of penalized energy to harmonic maps in Riemannian manifolds.
method Using the penalized energy functional and weak convergence techniques, the paper proves the energy identity for Ginzburg-Landau approximation of harmonic maps.
result The defect measure ν can be expressed as the sum of energies of harmonic spheres for arbitrary manifolds.
We present a graph-based variational algorithm for classification of high-dimensional data, generalizing the binary diffuse interface model to the case of multiple classes. Motivated by total variation techniques, the method involves minimizing an energy functional made up of three terms. The first two terms promote a …
At critical coupling, the interactions of Ginzburg-Landau vortices are determined by the metric on the moduli space of static solutions. The asymptotic form of the metric for two well separated vortices is shown here to be expressible in terms of a Bessel function. A straightforward extension gives the metric for N vor…
We analyze an elastic surface energy which was recently introduced by G. Napoli and L.Vergori to model thin films of nematic liquid crystals. We show how a novel approach that takes into account also the extrinsic properties of the surfaces coated by the liquid crystal leads to considerable differences with respect to …
Study heat flow for half-harmonic maps and harmonic maps with free boundary.
problem Integrability and regularity of half-harmonic maps and harmonic maps with free boundary.
method Introduced a heat flow associated to half-harmonic maps and constructed weak solutions via Ginzburg-Landau approximation.
result Proved partial regularity of weak solutions in space and time.
In my previaou paper of K. Horihata, we have proposed a Ginzburg-Landau system with a time-dependent parameter and then passing to the limit we have constructed a harmonic heat flow into spheres. Thanks to this scheme, we establish a few energy inequalities of our flow: (i) monotonical inequalities and (ii) a reverse P…
Extends diffuse interface methods to graphs and hypergraphs with non-smooth potentials.
problem Semi-supervised learning on graphs and hypergraphs.
method Generalizes diffuse interface methods using non-smooth potential functions and hypergraph Laplacians.
result The diffuse interface method can be applied to both graph and hypergraph data.
Defines renormalised energies for singular harmonic maps into compact manifolds.
problem Analyzing harmonic maps with singularities in planar domains.
method Introduces renormalised energies and synharmony to study singularities and minimising configurations.
result Renormalised energies are coercive and Lipschitz-continuous, and associated with minimising singular harmonic maps.
We present two graph-based algorithms for multiclass segmentation of high-dimensional data. The algorithms use a diffuse interface model based on the Ginzburg-Landau functional, related to total variation compressed sensing and image processing. A multiclass extension is introduced using the Gibbs simplex, with the fun…
New model suggests universe emerges from single particle quantum mechanics.
problem Exploring how the universe might arise from a single particle in quantum mechanics.
method Novel spontaneous symmetry breaking acting on probability distributions of Hamiltonians.
result Evidence supports the hypothesis that nature seeks tensor decompositions.
A new method clusters signed networks using a modified MBO scheme.
problem Clustering signed networks with mixed positive and negative edge weights.
method Adapted MBO scheme for graph-based diffuse interface model.
result Method outperforms state-of-the-art approaches on various datasets.
We study global monotone solutions of the free boundary problem that arises from minimizing the energy functional I(u)=∫∣∇u∣2+V(u), where V(u) is the characteristic function of the interval (−1,1). This functional is a close relative of the scalar Ginzburg-Landau functional $J(u) = \int |\nabla u|^…
Bayesian models for uncertainty in graph-based data classification.
problem Uncertainty quantification in high-dimensional data classification.
method Bayesian models based on graph semi-supervised learning.
result Unified framework for various classification methods.
Motivated by the sigma model limit of multicomponent Ginzburg-Landau theory, a version of the Faddeev-Skyrme model is considered in which the scalar field is coupled dynamically to a one-form field called the supercurrent. This coupled model is investigated in the general setting where physical space is an oriented Rie…
Equation embeddings learn from surrounding words to represent math equations.
problem Math equations are hard to analyze due to their uniqueness.
method Equation embeddings use surrounding words to find good representations of equations.
result Equation embeddings provide better models than existing word embedding approaches.
Paper finds new equations for pseudospherical surfaces with isometric immersions.
problem Identifying equations with isometric immersions for pseudospherical surfaces.
method Provided families of second order non-linear PDEs with local isometric immersions in E^3.
result Found equations with principal curvatures depending on finite-order jets of solutions.