New energy model reveals knotted rod configurations.
problem Modeling elastic rods with stretch or inflation.
method Introduced a generalized functional on framed curves, used correspondence to infinite-dimensional Grassmann manifold.
result Explicit parameterizations of all periodic critical framed curves.
Study p-Willmore disks with boundary energies, finding equilibrium configurations.
problem Finding equilibrium configurations for p-Willmore disks with boundary energies.
method Model boundary as Kirchhoff elastic rod, interior term dependent on mean and Gaussian curvatures. Study among topological disks and p-Willmore examples.
result Equilibrium configurations for p-Willmore disks with boundary energies.
Develops control and observer methods for complex systems.
problem Controlling and observing infinite-dimensional systems with boundary actuation.
method Energy-Casimir method and port-Hamiltonian system representation.
result Control law and observer designed for Kirchhoff-Love plate example.
Bicycle paths form geodesics in 3D subspaces, related to Kirchhoff rods.
problem Optimizing bicycle paths between two points.
method Variational equations and geometric analysis of bicycle paths.
result Bicycle geodesics are contained in 3D subspaces and relate to Kirchhoff rods.
Introduces Lax-Kirchhoff moduli spaces for quivers and Lie groups.
problem Constructing moduli spaces for quivers and Lie groups.
method Introduces Lax equations and Kirchhoff conditions, constructs slices, and uses Marsden-Weinstein reduction.
result Proves M(Γ) is a finite-dimensional smooth symplectic manifold with a Hamiltonian action of G∂Γ. SHAKE-GNN scales GNNs for large graphs with multi-scale representations.
problem Scaling Graph Neural Networks (GNNs) to large graphs.
method SHAKE-GNN uses a hierarchy of Kirchhoff Forests for stochastic multi-resolution graph decompositions.
result SHAKE-GNN achieves competitive performance on large-scale graph classification benchmarks.
Determinants of theta curves and symmetric graphs are studied.
problem Understanding the determinants of theta curves and symmetric graphs.
method Combinatorial approach using Kirchhoff's Matrix Tree Theorem and spanning tree enumeration.
result The determinant of a simple theta curve is the product of the determinants of its constituent knots.
New approach to electric group for knots and links.
problem No previous publication of electric invariant for knots and links.
method Simple and general approach to electric group for oriented knots and links, using proper colouring of knot diagrams.
result Each homomorphism from the electric group to an arbitrary finite group can be described by a proper colouring of the diagram.
Theory of point vortices extended to closed surfaces.
problem Extending point vortex dynamics to closed surfaces.
method Unified theory of point vortex dynamics on the plane, sphere, and closed surfaces.
result Comprehensive guide to point vortex dynamics on closed surfaces with genus zero and vanishing total vorticity.
Neural model parses non-projective dependency trees efficiently.
problem Parsing non-projective dependency trees.
method Probabilistic parsing model using neural representations and Kirchhoff's Matrix-Tree Theorem.
result State-of-the-art parsing performance on nine datasets.
The paper analyzes defects on structured surfaces and calculates stress and shape.
problem Analyzing defects on structured surfaces and their effects on stress and shape.
method Classified and quantified defects, derived strain incompatibility relations, and applied to shells.
result Determined internal stress field and deformed shape for shells with defects.
The abstract discusses the existence of complex structures on spheres and their implications.
problem The existence of complex structures on the six sphere and their implications.
method Analyzing the parallelism and H-space multiplication on the seven sphere associated with almost complex structures on the six sphere.
result The integrability condition of the almost complex structure on the six sphere does not imply the homotopy associativity of the multiplication on the seven sphere.
The paper explores variational problems on Riemannian manifolds with special foliations, proving existence results.
problem Variational problems on Riemannian manifolds with singular Riemannian foliations.
method Application of Palais' Principle of Symmetric Criticality and Rellich-Kondrachov-Hebey-Vaugon Embedding Theorem.
result Existence of countably infinite weak solutions to variational problems.
Physics-guided neural network improves power flow analysis.
problem Infeasibility of traditional numerical approaches due to outdated or unavailable PF equations.
method Proposes a physics-guided neural network to learn PF mappings from historical data while constraining by physical laws.
result Physics-guided neural network achieves better performance and generalizability than unconstrained data-driven approaches.
We discuss several issues regarding material homogeneity and strain compatibility for materially uniform thin elastic shells from the viewpoint of a 3-dimensional theory, with small thickness, as well as a 2-dimensional Cosserat theory. A relationship between inhomogeneity and incompatibility measures under the two des…
We consider a family of compact manifolds which shrinks with respect to an appropriate parameter to a graph. The main result is that the spectrum of the Laplace-Beltrami operator converges to the spectrum of the (differential) Laplacian on the graph with Kirchhoff boundary conditions at the vertices. On the other hand,…
We study the stability of symmetric trajectories of a particle on the Lie group SO(3) whose motion is governed by an SO(3)×SO(2) invariant metric and an SO(2)×SO(2) invariant potential. Our method is to reduce the number of degrees of freedom at {\em singular} values of the SO(2)×SO(2) momentu…
Knot Theory is currently a very broad field. Even a long survey can only cover a narrow area. Here we concentrate on the path from Goeritz matrices to quasi-alternating links. On the way, we often stray from the main road and tell related stories, especially if they allow as to place the main topic in a historical cont…
The paper defines surface area for graphs and derives spectral estimates.
problem Understanding connectivity measures and spectral properties of graphs.
method Introducing surface area concepts related to inverse degree and deriving spectral bounds.
result An upper bound on the second eigenvalue for planar graphs.
Paper derives a formula for the determinant of Dirichlet-to-Neumann operator on Riemann surfaces.
problem Bounding asymptotics of a conformal invariant under degeneration of Riemann surfaces.
method Meyer-Vietoris formula, gluing, height function on moduli space, properness of height function, Steklov isospectral metrics, Laplacian with Dirichlet/Neumann boundary conditions.
result Properness of height function on moduli space of genus zero hyperbolic surfaces implies compactness theorem for Steklov isospectral metrics.
The paper compares PINN methods for solving drift-diffusion equations on metric graphs.
problem Solving drift-diffusion equations on metric graphs using machine learning.
method Comparison of physics-informed neural networks (PINNs) for solving drift-diffusion equations on metric graphs.
result PINNs offer a flexible and versatile tool for solving parameter identification or optimization problems on metric graphs.
The aim of this paper is to give a formulation of the dynamics of nonlinear RLC circuits as a geometric Birkhoffian system and to discuss in this context the concepts of regularity, conservativeness, dissipativeness. An RLC circuit, with no assumptions placed on its topology, will be described by a family of Birkhoffia…
Extended Möbius energy formula for generalized O'Hara's energies.
problem Maintaining Möbius invariance in O'Hara's energies.
method Extended cosine formula for generalized O'Hara's energies.
result Condition for right circle minimization under length-constraint.
This paper decomposes generalized O'Hara's energies into components.
problem Decomposing generalized O'Hara's energies to understand their components.
method Using an analogue of Doyle-Schramm's cosine formula, the paper derives a decomposition for generalized O'Hara energies.
result Derives a decomposition for generalized O'Hara energies into three components.
Versatile model for High Energy Physics events.
problem Modeling complex interactions in high-energy physics data.
method Energy-based probabilistic model with multi-purpose architecture.
result Achieves success in diverse applications like simulation, anomaly detection, and particle identification.
Stability of a new map derived from the equator map is analyzed.
problem Stability of a new map derived from the equator map.
method Detailed stability analysis of the generalized equator map as a critical point of the extrinsic k-energy and p-energy.
result Established generalizations of classical (in)stability results.
The paper examines geometric curvatures in generalized Riemannian spaces.
problem Understanding the physical meaning of scalar curvatures in generalized Riemannian spaces.
method Developed Madsen's formulae for pressures and energy-densities, analyzed with different concepts of generalized Riemannian spaces.
result Linearities of energy-momentum tensor, pressure, energy-density, and state-parameter are examined.
We introduce the "Energy-based Generative Adversarial Network" model (EBGAN) which views the discriminator as an energy function that attributes low energies to the regions near the data manifold and higher energies to other regions. Similar to the probabilistic GANs, a generator is seen as being trained to produce con…
Paper proposes new loss functions for training energy networks.
problem Challenges in computing gradients for training energy networks.
method Proposes generalized Fenchel-Young losses for efficient gradient computation.
result Demonstrates the calibration of excess risk for linear-concave energies.
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.
Improves sample quality of generative models using energy-based methods.
problem Low sample quality in generative models.
method Constructs an energy function on latent space, trains an energy-based model, and generates improved samples.
result Significant improvement in sample quality with minimal computational overhead.
Derives energy-momentum tensor from Standard Model, examines energy conditions.
problem Validating energy conditions in the context of the Standard Model.
method Geometric variational problem on globally hyperbolic manifold, deriving energy-momentum tensor.
result Validates various energy conditions in general relativity.
Energy Matching unifies flow matching and energy-based models for generative modeling.
problem Inability of flow-based models to integrate partial observations and priors.
method Energy Matching framework that integrates flow matching and energy-based models.
result Substantially outperforms existing EBMs on CIFAR-10 and ImageNet generation.
Paper finds relations between Willmore-type energies, weighted areas, and vertical potential energies for cylindrical critical points.
problem Tackles relations between three types of energy functions for cylindrical critical points.
method Uses differential equations and critical point analysis for Willmore-type energies and weighted areas.
result Generating curves coincide for Willmore-type energies and weighted areas, and similar results hold for Willmore-type energies and vertical potential energies.
Reduces energy for 4D submanifolds in R^n.
problem Energy reduction for 4D submanifolds in R^n.
method Connected sum energy reduction for fourth-order Willmore energy.
result Established a connected sum energy reduction for the fourth-order Willmore energy.
New relation found between ADM mass and generalized Komar energy for dynamical spacetimes.
problem Finding equality between ADM mass and Komar energy in dynamical spacetimes.
method Constructing a generalized Komar energy from the normal evolution vector and proving equality under specific conditions.
result Equality between ADM mass and generalized Komar energy for dynamical asymptotically-flat spacetimes.
GEBM combines energy function and base distribution for better generative modeling.
problem Improving generative models with better quality samples and performance.
method Alternating training between energy function and base distribution, using MCMC for sampling.
result GEBMs produce higher quality samples and better performance than GANs.
Paper tackles energy sharing in ZECs using DRL.
problem Improving energy status of ZECs through agent-based energy sharing.
method Modelled as a multi-agent environment, solved with DRL.
result Agents learn to collaborate and improve ZEC's energy status over time.
The Hawking energy is nonnegative and rigid on area-constrained surfaces in general relativity.
problem The rigidity and positivity of the Hawking energy on specific surfaces in general relativity.
method Evaluation of the Hawking energy on area-constrained critical surfaces under the dominant energy condition.
result The Hawking energy is nonnegative and rigid on area-constrained surfaces, including charged and cosmological constant variants.
The hyperbolic positive energy theorem links causal properties to energy-momentum vectors in asymptotically hyperbolic spaces.
problem Establishing the causal-future-directed character of energy-momentum vectors in hyperbolic spaces.
method Analyzing n-dimensional asymptotically hyperbolic Riemannian manifolds with spherical conformal infinity, focusing on the dominant energy condition. result The causal-future-directed character of the energy-momentum vector can be traced back to that of asymptotically Euclidean initial data sets.
Graph Energy Matching improves generation quality for molecular graphs.
problem Discrete energy-based models struggle with efficient and high-quality sampling for graph generation.
method Inspired by transport-map optimization, Graph Energy Matching learns a permutation-invariant potential energy to guide sampling.
result GEM matches or surpasses discrete diffusion baselines on molecular graph benchmarks.
The spectral geometry of mesh matrices of graphs is explored, leading to new formulas and eigenvalue estimates.
problem Understanding the spectral properties of mesh matrices of graphs.
method Definition and study of mesh matrices, introduction of mesh Laplacian, derivation of characteristic polynomial formulas.
result Mesh Laplacian eigenvalues are all real and greater than or equal to 1, with a smallest positive eigenvalue estimated.
Successful implementation of California's Renewable Portfolio Standard (RPS) mandating 33 percent renewable energy generation by 2020 requires inclusion of a robust strategy to mitigate increased risk of energy deficits (blackouts) due to short time-scale (sub 1 hour) intermittencies in renewable energy sources. Of the…
Study on weather forecasting errors for solar energy forecasting.
problem Uncertainty in weather forecasting for solar PV generation.
method Comparison of forecasted and observed weather data, statistical metrics, and sensitivity test.
result Identified influential weather variables improving solar PV generation forecasting.
Unified framework for training generator, energy model, and inference model.
problem Training of generator, energy model, and inference model in a unified probabilistic formulation.
method Divergence Triangle framework integrating variational learning, adversarial learning, wake-sleep algorithm, and contrastive divergence.
result Unified training of generator, energy model, and inference model without costly Markov chain Monte Carlo methods.
Inference problems in graphical models can be represented as a constrained optimization of a free energy function. It is known that when the Bethe free energy is used, the fixedpoints of the belief propagation (BP) algorithm correspond to the local minima of the free energy. However BP fails to converge in many cases o…
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.
New estimates for Hitchin's equations at high energy.
problem Solutions to Hitchin's self-duality equations at high energy.
method New estimates and asymptotic decoupling phenomenon.
result Generalization to arbitrary Higgs bundles.