Minimal energy local systems on curves are compact components of character varieties.
problem Characterizing local systems on surfaces with minimal energy.
method Study of minimal energy local systems on surfaces of genus g with d punctures.
result Minimal energy local systems form compact components of real relative character varieties.
In relativity, the energy of a moving particle depends on the observer, and the rest mass is the minimal energy seen among all observers. The Wang-Yau quasi-local mass for a surface in spacetime introduced in [7] and [8] is defined by minimizing quasi-local energy associated with admissible isometric embeddings of the …
In this paper, the problem of energy efficient transmission and computation resource allocation for federated learning (FL) over wireless communication networks is investigated. In the considered model, each user exploits limited local computational resources to train a local FL model with its collected data and, then,…
A2I Transformer predicts atom energies from coordinates, avoiding heavy featurization.
problem Efficiently predicting atom energies from molecular coordinates with minimal featurization.
method End-to-end model using self-attention, permutation-equivariant.
result Stable predictions with significantly smaller errors than molecular dynamics simulations.
The paper classifies energy-minimizing sets in specific domains.
problem Classifying volume-constraint local energy-minimizing sets.
method Proved a Poincaré-type inequality for stable sets.
result Relative boundary of energy-minimizing sets is smooth.
Study shows TAP free energy minimization provides better posterior inference in high-dimensional linear models.
problem Deviation from true posterior mean and underestimation of posterior uncertainty in variational inference.
method Minimization of TAP free energy in a high-dimensional asymptotic framework, showing geometric and statistical properties.
result Local minimizer of TAP free energy provides consistent estimate of posterior marginals and correctly calibrated posterior inference.
A new AI optimization method uses energy-conserving dynamics inspired by Born-Infeld theory.
problem Optimization challenges in non-convex loss functions and machine learning tasks.
method Discretization of Born-Infeld dynamics for energy-conserving Hamiltonian optimization.
result The method avoids high local minima and outperforms traditional methods in shallow valleys.
Investigates energy minimizers and critical points of scale-invariant tangent-point energies for knots.
problem Finding and characterizing minimizers and critical points of scale-invariant tangent-point energies for closed curves.
method Develops convergence and regularity theories based on fractional Sobolev spaces and new energy functionals.
result Minimizing sequences converge to locally critical embeddings in all but finitely many points, and locally critical embeddings are regular.
Paper proposes MAMRL for efficient energy dispatch in self-powered edge computing systems.
problem High energy consumption in self-powered edge computing systems.
method Developed a semi-distributed data-driven MAMRL framework to solve a two-stage linear stochastic programming problem.
result The proposed MAMRL framework reduces up to 11% non-renewable energy usage and 22.4% energy cost.
New energy measure for isolated systems in general relativity.
problem Quantifying energy in isolated systems in general relativity.
method Optimal isometric embedding and conformal Killing fields.
result Finite quasi-local energies for asymptotically flat spacetimes.
We prove several new results concerning action minimizing periodic orbits of Tonelli Lagrangian systems on an oriented closed surface M. More specifically, we show that for every energy larger than the maximal energy of a constant orbit and smaller than or equal to the Mañé critical value of the universal abelian cov…
A physics-based method improves data interpolators and regression tasks.
problem Improving accuracy and efficiency in function learning.
method Inspired by statistical mechanics, introduces corrections to minimize energy.
result Improves performance in interpolation and regression tasks, especially in high-dimensional spaces.
Let Ef be the energy of some knot τ for any f from certain class of functions. The problem is to find knots with extremal values of energy. We discuss the notion of the locally perturbed knot. The knot circle minimizes some energies Ef and maximizes some others. So, is there any energy such that the circle ne…
We prove that any limit-interface corresponding to a locally uniformly bounded, locally energy-bounded sequence of stable critical points of the van der Waals--Cahn--Hilliard energy functionals with perturbation parameter tending to 0 is supported by an embedded smooth stable minimal hypersurface in low dimensions and …
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. New algorithm reduces costs in wind energy systems by minimizing decision changes.
problem Costs associated with decision changes in wind energy systems.
method Episodic CBO with movement costs using Gaussian Process and mirror descent.
result Our algorithm consistently outperforms standard CBO in altitude optimization.
The paper studies harmonic graphs in the Heisenberg group and their properties.
problem No analogous theorem exists for H-minimal surfaces in the Heisenberg group. method Introduced intrinsic Dirichlet energy and studied its critical points (contact harmonic graphs).
result Calibration condition and construction of energy-minimizing graphs with various singularities.
Construct locally minimizing (1,2)-clusters with prescribed asymptotic geometry.
problem Minimizing clusters with prescribed asymptotic geometry.
method Develop a refined construction using the Hardt-Simon foliation.
result Produce a countably infinite family of distinct locally minimizing clusters asymptotic to a singular area-minimizing hypercone.
Minimal surfaces connect to horizons and electrostatic systems.
problem Connecting minimal surfaces to horizons and electrostatic systems.
method One-parameter min-max problem for area functional, inequality relating area and charge.
result Minimal surfaces of index one are related to unstable horizons in electrostatic systems.
Quantum vacuum energy (Casimir energy) is reviewed for a mathematical audience as a topic in spectral theory. Then some one-dimensional systems are solved exactly, in terms of closed classical paths and periodic orbits. The relations among local spectral densities, energy densities, global eigenvalue densities, and tot…
Paper proposes f-EBM for training deep EBMs using various f-divergences.
problem Training deep EBMs with intractable partition functions.
method Introduces f-EBM framework and optimization algorithm for any f-divergence.
result f-EBM outperforms contrastive divergence and other f-divergences.
Recently, supervised hashing methods have attracted much attention since they can optimize retrieval speed and storage cost while preserving semantic information. Because hashing codes learning is NP-hard, many methods resort to some form of relaxation technique. But the performance of these methods can easily deterior…
The MBO scheme for data clustering is analyzed in the large data limit, proving convergence to optimal partition problems.
problem Analyzing the MBO scheme for data clustering in the large data limit.
method Implicit gradient descent on the thresholding energy of a similarity graph.
result The MBO scheme outcomes converge to minimizers of a weighted optimal partition problem.
Researchers find optimal configurations of complex knots and links.
problem Finding the most efficient configurations of complex knots and links.
method Minimizing Möbius and Minimum Distance energies by describing them with a small number of free parameters.
result Optimal geometries for Hopf links, Borromean rings, and chain links are found.
This paper optimizes a power-to-heat system using reinforcement learning for cost minimization under uncertain conditions.
problem Optimizing a power-to-heat system with fluctuating renewable energy sources.
method Stochastic optimal control, reinforcement learning (Q-learning).
result Reinforcement learning provides an efficient solution to the optimization problem.
The geodesic equation for the right invariant L2-metric (which is a weak Riemannian metric) on each Virasoro-Bott group is equivalent to the KdV-equation. We prove that the corresponding energy functional, when restricted to paths with fixed endpoints, has no local minima. In particular solutions of KdV don't define…
This study proves energy bounds in specific AdS spacetimes.
problem Proving positive energy theorems in asymptotically locally AdS spacetimes.
method Derived positive energy theorem for spacetimes with compact, Einstein cross-sections.
result First complete proofs of BPS inequalities in AdS and locally AdS spacetimes.
AIF improves physical AI agents' performance in dynamic environments.
problem Physical AI agents are less capable than biological agents in open-ended real-world environments.
method Developed from probability theory, Bayesian machine learning, variational inference, and Active Inference (AIF), grounded in the Free Energy Principle.
result AIF minimizes variational free energy and is well-suited to physical constraints.
Geometric proof shows regularity of anisotropic minimal surfaces in 2D.
problem Regularity of anisotropic minimal surfaces in 2D.
method Geometric proof using surface energy and strict convexity.
result All anisotropic surface minimizers in 2D are locally disjoint unions of line segments.
Inspired by the work of Chen-Zhang \cite{Chen-Zhang}, we derive an evolution formula for the Wang-Yau quasi-local energy in reference to a static space, introduced by Chen-Wang-Wang-Yau \cite{CWWY}. If the reference static space represents a mass minimizing, static extension of the initial surface Σ, we observe that …
The paper proves properties of curves in Riemannian manifolds.
problem Characterizing curves in Riemannian manifolds.
method Analyzing locally minimizing and weak geodesics.
result Locally minimizing curves are weak geodesics under certain conditions.
Unified approach classifies stable and minimal elastic curves.
problem Classifying stable and minimal elastic curves under various conditions.
method Unified geometric approach using a `cut-and-paste` trick.
result Complete classification of stable closed p-elasticae and stable pinned p-elasticae. In the 1980's, Almgren developed a theory of multi-valued Dirichlet energy minimizing functions on n dimensional domains and used it, in an essential way, to bound the Hausdorff dimension of the singular sets of area minimizing rectifiable currents of dimension n and codimension ≥2. Recent work of the second …
New stabilization found in planar elasticae with degenerate diffusion.
problem Existence of local minimizers in degenerate p-elasticae. method Analysis of pinned planar p-elasticae with degenerate diffusion. result Uncountably many local minimizers with diverging energy in degenerate regime.
ShotgunCSP predicts crystal structures using machine learning, achieving high accuracy with minimal computation.
problem Predicting stable or metastable crystal structures of large systems.
method Noniterative screening using transfer learning and generative models.
result ShotgunCSP achieves 93.3% accuracy in benchmark tests with 90 different crystal structures.
Controller seeks informative system observations to predict nonlinear dynamics.
problem Predicting nonlinear dynamics with uncertain parameters.
method Expected free energy minimization for balancing goal state and informative observations.
result Controller improves performance in uncertain parameter scenarios.
The goal of the present paper is to establish some kind of regularity of an energy minimizer map between Riemannian polyhedra. More precisely, we will show the hölder continuity of local energy minimizers between Riemannian polyhedra with the target spaces without focal points. With this new result, we also complete ou…
We study on which compact Sasakian 3-manifolds the Reeb field, which is a Beltrami field with eigenvalue 2, is an energy minimizer in its adjoint orbit under the action of volume preserving diffeomorphisms. This minimization property for Beltrami fields is relevant because of its connections with the phenomenon of magn…
New inequality shows energy growth and decay in geometric problems.
problem Understanding energy behavior in geometric problems.
method Introduced a symmetric (log-)epiperimetric inequality.
result Energy growth and decay observed in geometric problems.
Study minimizes CR surfaces in Heisenberg group with rotational symmetry.
problem Minimizing CR surfaces with vanishing CR invariant energy E1 in Heisenberg group. method Proved local uniqueness, classified global surfaces with rotational symmetry, computed second variation.
result Clifford torus is not a local minimizer of E1. Lipschitz mappings found between Riemann surfaces with specific properties.
problem Finding globally Lipschitz mappings between doubly connected Riemann surfaces.
method Using a result from Iwaniec, Kovalev, and Onninen, the minimizer of the energy functional is shown to be locally Lipschitz and globally Lipschitz.
result The minimizer of the energy functional is a globally Lipschitz mapping.
It is well-known that normal extremals in sub-Riemannian geometry are curves which locally minimize the energy functional. Most proofs of this fact do not make, however, an explicit use of relations between local optimality and the geometry of the problem. In this paper, we provide a new proof of that classical result,…
The article analyzes the stability of a curve shortening flow for planar networks.
problem Stability analysis of anisotropic curve shortening flow for planar networks.
method Used Lojasiewicz-Simon gradient inequality to derive stability results.
result For initial data close to an energy minimizer, the flow exists globally and converges to a different energy minimum.
Recent innovations in Information and Communication Technologies (ICT) provide new opportunities and challenges for integration of distributed energy resources (DERs) into the energy supply system as active market players. By increasing integration of DERs, novel market platform should be designed for these new market …
Study on dynamic curves with elastic energy and spontaneous curvature.
problem Modeling and analyzing dynamic planar curves with elastic energy.
method Gradient flow of inclination angle, nonlocal quasilinear system, local well-posedness, global existence, convergence.
result Local well-posedness, global existence, convergence of the flow for weak regularity initial data.
New metric reduces Weyl's energy in manifold connected sums.
problem Minimizing Weyl's energy in connected sums of manifolds.
method Interplay of Bach-flat and locally conformally flat metrics, topology of Z. result Existence of a metric with lower Weyl energy than gM. Derives energy and momentum conservation laws for Vlasov-Maxwell systems using Euler-Poincaré formulation.
problem Challenges in deriving energy and momentum conservation laws for Vlasov-Maxwell systems due to mixed Eulerian and Lagrangian variables.
method Uses Euler-Poincaré formulation to derive conservation laws for Vlasov-Maxwell-type systems, focusing on symmetries generated by isometries and time translation.
result Derives energy and momentum conservation laws for a generic class of Vlasov-Maxwell-type systems, providing a new derivation in the spirit of the Euler-Poincaré machinery.
Study of phase separation and geometry on a closed elastic curve, including dynamics and free energy minimization.
problem Free energy and dynamics of a closed elastic filament coupled to a scalar concentration field.
method Analytical and numerical simulations of coupled Willmore flow and Cahn--Hilliard gradient flow on differential geometry.
result Qualitative changes in free energy landscape due to closure constraint, leading to metastable and stable multi-domain morphologies.