Derives equilibrium law for Plateau borders in wet soap films and foams.
problem Equilibrium law for Plateau borders in wet foams and films.
method Rigorous derivation using Gauss' capillarity theory, homotopic spanning condition, and effective compactness theorems.
result Sharp regularity properties of energy minimizers for Plateau borders in wet foams and films.
Real foams can be viewed as a geometrically well-organized dispersion of more or less spherical bubbles in a liquid. When the foam is so drained that the liquid content significantly decreases, the bubbles become polyhedral-like and the foam can be viewed now as a network of thin liquid films intersecting each other at…
Study approximates Plateau's laws using the Allen-Cahn equation.
problem Approximating Plateau's laws with the Allen-Cahn equation.
method Minimizing the Allen-Cahn energy under volume and spanning constraints.
result Energy minimizing solutions approximate Plateau-type singularities.
We define a (co-)Poisson (co)algebra of curves on a bordered surface. A bordered surface is a surface whose boundary have marked points. Curves on the bordered surface are oriented loops and oriented arcs whose endpoints in the set of marked points. We define a (co-)Poisson (co)bracket on the symmetric algebra of a quo…
Gradient-free optimizers are ineffective on barren plateaus in quantum computing.
problem Effect of barren plateaus on gradient-free optimization in quantum computing.
method Numerical simulations and theoretical analysis of gradient-free optimization algorithms.
result Gradient-free optimizers are not effective in barren plateau landscapes due to exponentially suppressed cost function differences.
The study examines cross-border lending behavior from G7 countries, showing changes in driving factors after the 2008 financial crisis.
problem Understanding the factors affecting cross-border lending behavior among G7 countries.
method Employed a gravity model to analyze bilateral and global factors influencing cross-border lending.
result Driving factors for cross-border lending have changed since the 2008 financial crisis, with continent variable becoming more significant.
Combinatorial definition of bordered Floer theory for torus-boundary manifolds.
problem Defining Floer homology for manifolds with torus boundaries.
method Combinatorial definition with Z coefficients. result Recovery of combinatorial Heegaard Floer homology.
Paper proves stable minimal surfaces in 3D are flat.
problem Understanding stable minimal surfaces in 3D.
method Analyzes quadratic area growth and stability conditions.
result Stable minimal Plateau surfaces in 3D are flat.
The paper models and deforms A-infinity structures for bordered knot algebras.
problem Understanding A-infinity structures for bordered knot algebras.
method Combinatorial model and weighted deformation of A-infinity structures.
result Explicit combinatorial model for bordered knot algebras' A-infinity structure.
Paper proves uniqueness of weak solutions for Plateau flow.
problem Proving uniqueness of weak solutions for Plateau flow.
method Used natural energy condition and alternative methods from Struwe.
result Proves uniqueness of weak solutions under natural condition.
Researchers solve a Plateau problem for maximal surfaces in pseudo-hyperbolic spaces.
problem Finding maximal surfaces with given boundary curves in pseudo-hyperbolic spaces.
method Defined and proved the existence of unique solutions using asymptotic Plateau problem and analysis of pseudo-holomorphic curves.
result Existence and uniqueness of maximal surfaces with specified boundary conditions.
Invariant for 3-manifolds with torus boundary defined.
problem Defining invariants for 3-manifolds with specific boundaries.
method Module over a weighted A-infinity algebra associated to a torus.
result Invariant constructed for bordered 3-manifolds with torus boundary.
Barren plateaus are not an average-case phenomenon, but a highly non-unique problem.
problem Avoiding barren plateaus in neural network training
method First-moment framework for initialization strategies
result Many families of inequivalent initialization strategies can avoid concentration
This is a survey of bordered Heegaard Floer homology, an extension of the Heegaard Floer invariant HF-hat to 3-manifolds with boundary. Emphasis is placed on how bordered Heegaard Floer homology can be used for computations.
Solves Plateau problem for surfaces in pinched curvature manifolds.
problem Asymptotic Plateau problem for immersed surfaces in pinched curvature manifolds.
method Complete solution to asymptotic Plateau problem, providing dynamical stability of hypersurface laminations.
result Achieved complete solution to the asymptotic Plateau problem for immersed surfaces of constant extrinsic curvature in Cartan--Hadamard manifolds.
Solves area-minimizing surface problem for finite curves in H^2xR.
problem Asymptotic Plateau problem for area-minimizing surfaces.
method Complete solution for finite curves in $\BHH$.
result Fairly complete solution for finite curves in $\BHH$.
New model for visual cortex border completion using bicycle wheel motions.
problem Understanding border completion in the visual cortex V1.
method Sub-Riemannian Hamiltonian formalism and bicycle wheel analogy.
result Analogies between visual cortex border completion and bicycle wheel motions.
Unique solutions found for Plateau problems in smooth and continuous calibrations.
problem Finding unique solutions to the Plateau problem for specific types of currents.
method Boundary regularity theory for area-minimizing currents and unique continuation argument.
result Every compactly supported smoothly or continuously calibrated integral current is the unique solution to the Plateau problem for its boundary data.
Study on minimal submanifolds in curved spaces with unique solution to asymptotic Plateau problem.
problem Minimal submanifolds in negatively curved spaces with small curvature.
method Analysis of spheres at infinity and asymptotic Plateau problem.
result Complete minimal submanifolds bound a class of spheres with uniquely solvable asymptotic Plateau problem.
The paper proves rigidity of bordered polyhedral surfaces using variational principles.
problem Determining the rigidity of bordered polyhedral surfaces.
method Using the variational principle, the paper shows that bordered polyhedral surfaces are determined by boundary values and discrete curvatures on interior edges.
result The paper re-proves the classical result that two Euclidean or hyperbolic cyclic polygons are congruent if their side lengths are equal.
The paper solves fractional combinatorial flows for prescribed hyperbolic bordered surfaces.
problem Finding hyperbolic bordered surfaces with prescribed boundary lengths.
method Fractional combinatorial Calabi flow and generalized combinatorial Yamabe flow.
result The flows converge to a hyperbolic surface with prescribed boundary lengths.
Spherical Plateau problem studies minimal surfaces in quotients of spheres.
problem Minimal surfaces in quotients of spheres.
method Metric currents, barycenter map method.
result Intrinsic uniqueness of solutions for negatively curved manifolds.
Survey of bordered Floer homology for 3-manifolds with boundary.
problem Computing invariants for 3-manifolds with boundary.
method Extends Heegaard Floer homology to include boundary.
result Powerful computational tools for 3-manifolds with boundary.
This paper forecasts renewable energy prospects in South America through cross-border interconnection.
problem Lack of renewable energy integration across South American countries.
method Long-term scenario forecasting methodology applied to raw data from typical countries.
result Promoting cross-border interconnection towards renewables can optimize energy supply, reduce costs, and balance the energy matrix.
Solves Plateau's Problem in Heisenberg group for graphs.
problem Plateau's Problem in the Heisenberg group for intrinsic graphs.
method Geometric construction and calibration argument.
result Solves Plateau's Problem under smallness conditions.
Solves the asymptotic Plateau problem in hyperbolic space for specific curvature.
problem Existence of complete hypersurfaces with prescribed asymptotic boundary.
method Curvature estimates.
result Solves the problem for a wider range of curvature values.
The paper defines contact invariants in bordered Floer homology.
problem Tackles contact invariants in bordered Floer homology.
method Defines contact invariants in bordered sutured Floer homology.
result Contact invariants derived in bordered sutured modules and bimodules.
Quantum models avoiding barren plateaus can also be efficiently simulated classically.
problem Understanding the limitations of barren plateaus in quantum computing.
method Analyzing commonly used models and their ability to be simulated classically.
result Many quantum models with barren plateau-free landscapes can also be efficiently simulated classically.
Noise causes learning plateaus in neural networks.
problem Plateau phenomena in online learning due to vanishing gradients.
method Analysis of stochastic gradient descent in multi-layer perceptrons.
result Noise induces synchronisation leading to strong plateaus.
Detects handlebodies and mapping class extensions using bordered Floer homology.
problem Detecting handlebodies and mapping class extensions over compression bodies.
method Combining bordered Floer homology with ideas from Casson-Long and train tracks.
result Algorithm to detect mapping class extensions over any compression body.
Plateau's problem is to show the existence of an area minimizing surface with a given boundary, a problem posed by Lagrange in 1760. Experiments conducted by Plateau showed that an area minimizing surface can be obtained in the form of a film of oil stretched on a wire frame, and the problem came to be called Plateau's…
Introduction to contact invariant in bordered Floer homology.
problem Understanding the contact invariant in bordered Floer homology.
method Construction relies on special foliated open books, with a procedure to obtain such books and a local proof of vanishing for overtwisted structures.
result Explicit bordered computation of the vanishing of the contact invariant for overtwisted structures.
We construct symplectic submanifolds of symplectic manifolds with contact border. The boundary of such submanifolds is shown to be a contact submanifold of the contact border. We also give a topological characterization of the constructed submanifolds by means of a ``relative Lefschetz hyperplane Theorem''. We sketch s…
Paper finds invariant solutions for Plateau problem in hyperbolic space.
problem Finding minimal surfaces with specific symmetries in hyperbolic space.
method Proved existence of foliations by invariant minimal surfaces, used to solve the Plateau problem.
result Existence of invariant minimal surfaces solving the asymptotic Plateau problem.
This study investigates abrupt learning dynamics in Transformers, revealing plateau formation and internal representation collapse.
problem Abrupt learning in Transformers, particularly during the loss plateau.
method Investigates mechanisms of abrupt learning in shallow Transformers, focusing on attention maps and hidden states.
result Reveals plateau formation, internal representation collapse, and strong repetition bias in outputs.
We define a sutured cobordism category of surfaces with boundary and 3-manifolds with corners. In this category a sutured 3-manifold is regarded as a morphism from the empty surface to itself. In the process we define a new class of geometric objects, called bordered sutured manifolds, that generalize both sutured 3-ma…
We give a solution of Plateau's problem for singular curves possibly having self-intersections. The proof is based on the solution of Plateau's problem for Jordan curves in very general metric spaces by Alexander Lytchak and Stefan Wenger and hence works also in a quite general setting. However the main result of this …
Framework extends sutured manifold invariants to pairs with torus boundaries.
problem Extending sutured manifold invariants to pairs with torus boundaries.
method Establishing a framework for extending invariants of sutured manifolds to pairs of sutured manifolds differing by attaching a basic slice along a torus boundary.
result Framework recovers bordered-sutured Floer homology for a corresponding bordered-sutured three-manifold.
Study shows how knot Floer homology and bordered Floer theory are linked.
problem Understanding the relationship between knot Floer homology and bordered Floer theory.
method Proves local equivalence and establishes algebraic formulas.
result Involutive knot Floer homology and bordered Floer theory determine each other under certain conditions.
This is a survey of old and recent results about the asymptotic Plateau problem. Our aim is to give a fairly complete picture of the field, and present the current situation.
A formula for bordered Floer homology of concordances and satellites
problem Computing knot Floer homology for concordances and satellites
method Combinatorial method for bordered Floer homology
result Computation of knot Floer homology cobordism map
This paper extends the decorated Teichmüller theory developed before for punctured surfaces to the setting of ``bordered'' surfaces, i.e., surfaces with boundary, and there is non-trivial new structure discovered. The main new result identifies the arc complex of a bordered surface up to proper homotopy equivalence wit…
We compute bordered Floer homology CFDD of (2,2n)-torus link complement, and discuss assorted examples and type-DD structure homotopy equivalence.
In this paper we show how to recover the relative Q-grading in Heegaard Floer homology from the noncommutative grading on bordered Floer homology.
We show that bordered Heegaard Floer homology detects incompressible surfaces and bordered-sutured Floer homology detects partly boundary parallel tangles and bridges, in natural ways. For example, there is a bimodule Lambda so that the tensor product of CFD(Y) and Lambda is Hom-orthogonal to CFD(Y) if and only if the …
Knot Floer homology is a knot invariant defined using holomorphic curves. In more recent work, taking cues from bordered Floer homology,the authors described another knot invariant, called "bordered knot Floer homology", which has an explicit algebraic and combinatorial construction. In the present paper, we extend the…
This research solves Plateau's problem for CRPC surfaces.
problem Constructing surfaces with constant ratio of principal curvatures.
method Proposed a family of surfaces containing a given minimal surface without flat points.
result Obtained a partial solution to Plateau's problem for CRPC surfaces.
New energy model avoids self-intersections in curve optimization.
problem Avoiding self-intersections in curve optimization under elastic boundary energies.
method Introduced Möbius-Plateau energy to minimize curve variations.
result Screw-like solutions are plentiful, ribbon-like solutions have constraints.