Study proves energy critical heat equation solutions are Type I blowups for n ≥ 7.
problem Analyzing blowup behavior of energy critical nonlinear heat equations.
method Reverse inner-outer gluing mechanism and bubbling behavior analysis.
result Proves all blowups are of Type I for n ≥ 7.
Study classifies traveling solitary waves in energy-critical half-wave maps.
problem Energy-critical half-wave maps into S2. method Geometric characterization, conformal Möbius group, Jacobi operators.
result Explicit classification and detailed analysis of traveling solitary waves.
Study classifies ruled surfaces critical to Dirichlet energy.
problem Identifying ruled surfaces critical to Dirichlet energy.
method Explicit parametrization of ruled surfaces.
result Classification of ruled surfaces as critical points of Dirichlet energy.
Morse theory connects low energy submanifolds in 3-sphere.
problem Understanding low energy submanifolds in the 3-sphere.
method Morse-theoretic techniques and negative gradient flow.
result Constructs connections between low energy critical submanifolds.
We study the wave analog of the Liouville equation and the constant mean curvature equations in 2 space dimensions, which are energy critical. We exhibit a blow-up criteria for the former using tools from conformal geometry, and we exhibit finite time blow-up for the latter under suitable assumptions on the initial dat…
New result on critical points of Bethe free energy under deformation retracts.
problem Characterizing critical points of Bethe free energy for complex graphs.
method Analyzing homotopy types and deformation retracts of factor graphs.
result Critical points of Bethe free energy are invariant under deformation retracts.
Rigidity for 4D Willmore submanifolds with boundary.
problem Understanding critical points of Willmore energy with boundary conditions.
method Proving a 4-Willmore equation and establishing curvature estimates.
result Four dimensional Willmore submanifolds with totally geodesic boundary are umbilic.
Study proves uniform regularity for surface energies, critical and subcritical.
problem Establishing regularity for surface energies in critical and subcritical cases.
method Uniform ε-regularity estimates for intrinsic elliptic Lagrangians.
result Critical points of surface energies are uniformly regular for a wide class of Lagrangians.
A method to analyze maps into circles with singularities.
problem Analyzing maps with singularities into circles.
method Renormalization of Dirichlet Lagrangian for S1-harmonic maps. result Applications in Willmore energy and frame energies.
Proves smooth critical points of Möbius energy are analytic.
problem Analyticity of critical points of Möbius energy.
method Cauchy's method of majorants and gradient decomposition.
result Smooth critical points of Möbius energy are analytic.
Study of critical tori for mean curvature energies in Killing submersions.
problem Analyzing surface energies in Killing submersions.
method Symmetry reduction and binormal evolution of critical curves.
result Construction of vertical tori critical for mean curvature energies.
Analyticity of critical points for O'Hara's knot energies proved.
problem Analyzing the regularity of critical points for O'Hara's knot energies.
method Cauchy's method of majorants and a Möbius energy-inspired gradient decomposition.
result Smooth critical points of O'Hara's knot energies are analytic.
The study examines stationary surfaces with boundaries and their properties.
problem Investigating stationary surfaces with boundaries and their critical points.
method A generalized bending energy functional is considered, and the first variation is computed. Boundary-value problems are examined, and a characterization of free-boundary surfaces is given.
result Characterization of free-boundary surfaces with rotational symmetry for scaling-invariant functionals.
Study of critical points for 4D conformally invariant curvature energies.
problem Analyzing critical points of conformally invariant curvature energies in 4 dimensions.
method Using Noether's theorem and divergence-free potentials, generating an algebraic structure, and considering Palais-Smale sequences.
result Improved energy estimates for critical points under small-energy hypotheses.
Global Schrödinger map flows to Kähler manifolds proved for high dimensions with small data.
problem Global existence of Schrödinger map flows to Kähler manifolds with small data in critical Sobolev spaces.
method Decay estimates of moving frame dependent quantities in caloric gauge setting, combined with a bootstrap-iteration scheme.
result Global existence of Schrödinger map flows to Kähler manifolds with small data in critical Sobolev spaces for high dimensions.
Paper proves existence of minimal doublings on surfaces with specific properties.
problem Existence of minimal doublings on surfaces with given properties.
method Variational approach to finding nondegenerate critical points of a Coulomb-type energy.
result Proves existence of minimal doublings for surfaces of index one in a generic 3-manifold.
New foliations found for critical surfaces of Hawking energy, resolving discrepancies.
problem Finding consistent critical surfaces for the Hawking energy in non-totally geodesic spacelike hypersurfaces.
method Constructing a unique local foliation of area constrained critical surfaces of the Hawking energy in the general case of non-totally geodesic spacelike hypersurfaces.
result Discrepancy found in the small sphere limit of the Hawking energy, explained and resolved.
Model predicts web page parallelism for improved browser performance and energy.
problem Improving browser performance and energy usage through parallelism.
method Supervised learning model using web page primitives and parallelism features.
result Model predicts parallelism and optimizes performance and energy usage.
Stable type II blowup solutions found for a specific heat flow equation.
problem Stability of type II blowup solutions for a harmonic heat flow.
method Reduction to a finite-dimensional problem, modulation techniques, and contradiction argument.
result Stable type II blowup solutions constructed for the energy supercritical harmonic heat flow.
New sparsity attacks degrade DNN efficiency, raising concerns for resource-constrained systems.
problem Vulnerabilities in DNNs through energy and latency attacks.
method Proposed sparsity attacks that modify DNN inputs to reduce activation sparsity, increasing execution time and energy consumption.
result Adversarial sparsity attacks can degrade DNN efficiency by up to 1.82x in image recognition DNNs.