Minimal submanifolds confined in space are highly restricted.
problem Understanding minimal submanifolds in confined spaces.
method Analyzing structural restrictions and volume growth properties.
result Proper minimal immersions with sublinear height growth must have Euclidean volume growth.
Minimal submanifolds either fill space or are confined with geometric restrictions.
problem Understanding the behavior of minimal submanifolds in space.
method A dichotomy principle and volume doubling theorem.
result Quantitative restrictions on confined minimal submanifolds, including volume growth and optimal density rates.
We prove mean curvature estimates and a Jorge-Koutroufiotis type theorem for submanifolds confined into either a horocylinder of N X L or a horoball of N, where N is a Cartan-Hadamard manifold with pinched curvature. Thus, these submanifolds behave in many respects like submanifolds immersed into compact balls and into…
We study the mean curvature flow of complete space-like submanifolds in pseudo-Euclidean space with bounded Gauss image, as well as that of complete submanifolds in Euclidean space with convex Gauss image. By using the confinable property of the Gauss image under the mean curvature flow we prove the long time existence…
Let L be a special Lagrangian submanifold of a compact, Calabi-Yau manifold M with boundary lying on the symplectic, codimension 2 submanifold W. It is shown how deformations of L which keep the boundary of L confined to W can be described by an elliptic boundary value problem, and two results about minimal…
The paper studies submanifolds in weighted manifolds, establishing parabolicity criteria and describing bounded curvature submanifolds.
problem Characterizing submanifolds of bounded mean curvature in weighted manifolds.
method Comparisons of h-capacity, criteria for h-parabolicity, and use of geometric functions. result Established criteria for h-parabolicity and h-hyperbolicity of submanifolds. New minimal surfaces grow area very quickly.
problem Understanding minimal surfaces with rapid area growth.
method Examples of minimal immersions in Euclidean space.
result Proper minimal surfaces with rapid area growth found.
Study reveals weak knotting in confined polymers, not dominated by any single knot type.
problem Characterizing knotting in open, confined polymers.
method Modeling open curves as virtual knots, comparing lattice walks and ideal chains in confined and unconfined conditions.
result Weak knotting is a common feature in confined polymers, not dominated by any single knot type.
Study confined subgroups in groups with contracting elements, showing their growth rate is strictly greater than half of the ambient growth rate.
problem Understanding the growth rate of confined subgroups in groups with contracting elements.
method Through boundary actions, analyzing the Hopf decomposition and quotient growth.
result Confined subgroups have a growth rate strictly greater than half of the ambient growth rate.
Fast algorithm samples confined polygons efficiently.
problem Sampling confined random equilateral closed polygons efficiently.
method Uses symplectic geometry to sample moment polytope, leading to a linear-time algorithm.
result Explicit formulas for expected distances and total curvature of vertices to the origin.
CONFINE enhances neural networks' interpretability without sacrificing accuracy.
problem Lack of interpretability in deep neural networks, especially in healthcare.
method CONFINE uses conformal prediction to generate prediction sets with robust uncertainty estimates.
result CONFINE achieves correct efficiency up to 3.3% higher than original accuracy.
Study geometric quantum confinement on special incomplete Riemannian manifolds.
problem Characterize quantum confinement on Grushin-type manifolds.
method Constant-fibre direct integral scheme combined with Weyl's analysis.
result Fully characterizes essential self-adjointness of Laplace-Beltrami operator.
5D SCFTs can have confining vacua with strings and unbroken symmetries.
problem Investigating phases of 5D SCFTs by varying couplings.
method Using geometric realisation of M-theory on metrically conical Calabi-Yau threefolds.
result Many 5D SCFTs have couplings leading to massive, confining vacua with strings and unbroken symmetries.
New framework shows C∗-simplicity for groups without certain subalgebras.
problem Characterizing C∗-simplicity of groups. method Introducing confined subalgebras and Uniformly Recurrent States.
result A countable discrete group is C∗-simple if it has no non-trivial amenable confined subalgebras. Extends confining subset theory to describe hyperbolic actions of solvable groups with higher rank abelianizations.
problem Describing hyperbolic actions of solvable groups with higher rank abelianizations.
method Extends confining subset theory to apply to solvable groups with higher rank abelianizations.
result Complete description of hyperbolic actions of generalized solvable Baumslag-Solitar groups.
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…
Sharp growth tightness proven for group quotients.
problem Growth behavior of group quotients by confined subgroups.
method Statistically convex-cocompact action with contracting elements.
result Sharp growth tightness proven, with applications to uniformly recurrent subgroups.
Study on membranes under confinement, proving existence and regularity of minimizers.
problem Existence and regularity of minimizers for constrained Helfrich energy.
method Elliptic system analysis, careful study of measure-valued Lagrange multiplier.
result Optimal regularity for solutions throughout branch points, rigid behavior for unit ball minimizers.
The study examines knot probabilities in confined lattice polygons.
problem Determining the relative knotting probabilities in confined lattice knots.
method Used Monte Carlo algorithms to enumerate conformations of lattice knots in a confined volume.
result Relative knotting probabilities are small, with the model dominated by unknots.
This work uses scientific constraints to validate neural network predictions in fusion physics.
problem Verifying the scientific plausibility of neural network predictions in fusion physics.
method Using known scientific constraints as a validation tool.
result Validated neural network predictions in fusion physics using scientific constraints.
Neural networks' optimization dynamics are confined to a single basin despite connected basins in the loss landscape.
problem Neural networks' optimization dynamics are confined to a single basin despite connected basins in the loss landscape.
method Identifying entropic barriers arising from the interplay between curvature variations along low-loss paths and noise in optimization dynamics.
result Curvature-induced entropic forces bias noisy dynamics back toward the endpoints, explaining the confinement and connectivity of solutions.
We consider the problem of minimizing the bending or elastic energy among Jordan curves confined in a given open set Ω. We prove existence, regularity and some structural properties of minimizers. In particular, when Ω is convex we show that a minimizer is necessarily a convex curve. We also provide an example of a…
Study of bound states in quantum layers with confining potentials.
problem Investigating bound states in quantum layers with confining potentials.
method Developed a general approach using parallel coordinates based on the surface but outside its cut locus.
result Discrete eigenvalues exist for certain quantum layers with positive total Gauss curvature.
Classifies quantum particle behavior on a special cylinder.
problem Quantum confinement and transmission on a Grushin cylinder.
method Characterizes self-adjoint realizations of the Laplace-Beltrami operator.
result Identifies physically meaningful extensions of the Hamiltonian.
Hydrogen atom confined in an inverted-Gaussian potential, with detailed numerical methods and results.
problem Studying hydrogen atom in a specific potential.
method Three numerical methods: Lagrange-mesh, fourth order finite differences, and finite element method.
result Accurate numerical results for hydrogen atom energies and eigenfunctions, improving previous literature.
Geometric QCD framework establishes stable vacuum for quark confinement.
problem Quark confinement in QCD.
method Geometric construction of stable vacuum using Hodge-dual surfaces.
result Existence and stability of the Hodge-dual surface in 4D ensures quark confinement.
The study characterizes hypersurfaces in weighted cylinders and generalizes confinement properties.
problem Characterizing hypersurfaces in weighted Riemannian products.
method Analyzing parabolic hypersurfaces with boundary in weighted cylinders.
result Generalized confinement properties of hypersurfaces in weighted cylinders.
We consider the quantum completeness problem, i.e. the problem of confining quantum particles, on a non-complete Riemannian manifold M equipped with a smooth measure ω, possibly degenerate or singular near the metric boundary of M, and in presence of a real-valued potential V∈Lloc2(M). The main …
Study properties of self-similar solutions in Mean and Inverse Mean Curvature Flows.
problem Analyze geometric and potential theoretic properties of self-similar solutions.
method Analysis of extrinsic distance function and Brownian motion behavior.
result Parabolic MCF-solitons are self-shrinkers and parabolic IMCF-solitons are self-expanders.
Improved neural network surrogates for ICF using manifold and cycle consistency.
problem Modeling and predicting complex physical processes in inertial confinement fusion.
method Training neural network surrogates that are consistent with the physical manifold and cyclically consistent.
result Surrogates are superior in predictive performance, more resilient to sampling artifacts, and more data efficient.
We consider an open domain with a compact boundary in an Euclidean space and a Schroedinger operator with magnetic field on this domain. We give sufficient conditions on the rate of growth of the magnetic field near the boundary which guarantees essential self-adjointness of this operator. From the physical point of vi…
We study geometric properties of complete non-compact bounded self-shrinkers and obtain natural restrictions that force these hypersurfaces to be compact. Furthermore, we observe that, to a certain extent, complete self-shrinkers intersect transversally a hyperplane through the origin. When such an intersection is comp…
Seq2seq models predict complex multi-physics systems' time evolution.
problem Predicting the time-evolution of complex multi-physics systems.
method Sequence-to-sequence models applied to multi-physics simulations.
result Seq2seq models accurately emulate complex systems and predict their evolution.
Random walks and polygons are used to model polymers. In this paper we consider the extension of writhe, self-linking number and linking number to open chains. We then study the average writhe, self-linking and linking number of random walks and polygons over the space of configurations as a function of their length. W…
Transfer learning improves fusion simulation accuracy.
problem Calibrate fusion simulation models to experimental data.
method Hierarchical transfer learning using deep neural networks.
result Calibrated models predict Omega experiments more accurately.
Modeling GDP growth rates using Lévy flights with confining potential.
problem Understanding the impact of firm size fluctuations on GDP.
method Combining microscopic firm growth rates with macroscopic GDP, using Lévy-stable fluctuations and a confining potential.
result The model accurately predicts 200 years of US GDP growth rates.
New theorem links symmetries to first integrals in plasma physics.
problem Understanding the relationship between symmetries and first integrals in divergence-free fields.
method Developed a Noether-type Theorem reformulation for three-dimensional divergence-free vector fields.
result Converse of the Noether-type Theorem holds on the toroidal region, proving the existence of flux coordinates.
We present atomistic molecular dynamics simulations of two Polyethylene systems where all entanglements are trapped: a perfect network, and a melt with grafted chain ends. We examine microscopically at what level topological constraints can be considered as a collective entanglement effect, as in tube model theories, o…
Deep learning compares turbulence models in plasma physics.
problem Predicting edge plasma turbulence in magnetic fusion reactors.
method Physics-informed deep learning framework for comparing two-fluid and gyrokinetic models.
result Good overall agreement between two-fluid theory and gyrokinetic models in turbulent field fluctuations.
New bounds on stick number of knots found using random polygon generation.
problem Understanding the minimum number of segments needed to build a polygonal knot.
method Monte Carlo approach to generating and analyzing large ensembles of random polygons.
result Improved bounds on stick number for over 40% of knots with 10 or fewer crossings.
Wealth redistribution through Fokker-Planck equation controls preserves Gini coefficient.
problem Preserving Gini coefficient through proportional wealth tax.
method Formulating optimal redistribution as a control problem for Fokker-Planck equation.
result Progressive taxes redistribute within policy-relevant timescales.
A closed equilateral random walk in 3-space is a selection of unit length vectors giving the steps of the walk conditioned on the assumption that the sum of the vectors is zero. The sample space of such walks with n edges is the (2n−3)-dimensional Riemannian manifold of equilateral closed polygons in R3…
Frustration causes buckling-like behavior in tubular foldable mechanisms.
problem Kinematic coupling and geometric confinement in tubular foldable mechanisms.
method Analytical and numerical solutions of exact kinematics.
result Frustration propagates axially as if by buckling in tubular states.
This paper computes fixed point Floer cohomology for Dehn twists on surfaces.
problem Computing fixed point Floer cohomology for Dehn twists.
method Developed tools for computing fixed point Floer cohomology and product for Dehn twists in all dimensions.
result Splitting of the product and differential into local and Morse-theoretic contributions.
This paper surveys parallel submanifolds in Riemannian and pseudo-Riemannian manifolds.
problem Understanding parallel submanifolds in Riemannian and pseudo-Riemannian manifolds.
method Comprehensive survey of parallel submanifolds.
result Extrinsic invariants of parallel submanifolds do not vary from point to point.
Model explains surprising properties of neural network training landscapes.
problem Understanding the geometry of neural network loss landscapes.
method Developed a simple theoretical model of gradients and Hessians.
result Unified model accounts for 4 surprising properties of neural loss landscapes.
The study characterizes submanifolds in product spaces.
problem Limited studies on pseudo-umbilical submanifolds.
method Using projections from product structure, conditions for submanifolds to be invariant, anti-invariant, or semi-invariant are derived.
result Necessary and sufficient conditions for pseudo-umbilical submanifolds in locally product Riemannian manifolds.
Researchers describe a specific type of submanifolds in Euclidean space.
problem Understanding inhomogeneous almost symmetric submanifolds.
method Completely describing submanifolds as unions of parallel symmetric submanifolds.
result Described inhomogeneous properly embedded almost symmetric submanifolds as unions of symmetric submanifolds.