New boundary operators for sixth-order GJMS operator on manifolds.
problem Developing boundary operators for sixth-order GJMS operator.
method Conformally covariant boundary operators and fractional GJMS operators.
result New realization of fractional GJMS operators and Sobolev trace inequalities.
A fast, accurate method for pricing American options with free boundaries.
problem Pricing American options with free boundaries efficiently and accurately.
method A sixth-order compact finite difference scheme with a dynamic staggered boundary scheme and 3(2) R-K Bogacki-Shampine time stepping.
result An efficient sixth-order compact scheme for pricing American options with free boundaries.
Classifies solutions to critical sixth order equations with a singularity.
problem Classifying entire positive singular solutions to critical sixth order equations.
method Integral sliding methods, qualitative analysis of ODEs, topological two-parameter shooting technique.
result Solutions are given by a singular radial factor times a periodic solution to a sixth order IVP with constant coefficients.
The paper finds infinitely many metrics with constant sixth order Q-curvature on spheres and related manifolds.
problem Finding constant sixth order Q-curvature metrics on spheres and related manifolds.
method Classical bifurcation technique and Riemannian covering.
result Infinitely many constant sixth order Q-curvature metrics on spheres and related manifolds.
Compactness of metrics with positive sixth order Q-curvature on a sphere with punctures.
problem Compactness of conformally flat singular metrics with constant, positive sixth order Q-curvature.
method Introduced necksize concept, used moving planes and blow-up arguments, proved upper and lower bounds, introduced homological invariant.
result A subsequence of metrics converges with respect to Gromov--Hausdorff metric if punctures remain separated and necksize is bounded away from zero.
The paper resolves compactness and non-compactness for fourth- and sixth-order Q-curvature problems.
problem Compactness and non-compactness of fourth- and sixth-order Q-curvature problems.
method Transformed linearized equations into overdetermined systems revealing algebraic structures.
result Proves compactness for fourth-order Q-curvature problems in dimensions 5 to 24, sixth-order in 7 to 26.
We study the anti-self-dual equation for non-diagonal SU(2)-invariant metrics and give an equivalent ninth-order system. This system reduce to a sixth-order system if the metric is in the conformal class of scalar-flat-Kaehler metric.
It is known that the spectrum of the Laplace operator on functions of a closed Riemannian manifold does not determine the integrals of the individual fourth order curvature invariants scal2, ∣ric∣2, ∣R∣2, which appear as summands in the second heat invariant a2. We study the an…
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.
Rigidity theorem for ideal surfaces with flat boundary conditions.
problem Characterizing surfaces with flat boundary conditions.
method Analyzing a sixth order nonlinear elliptic PDE and flat boundary conditions.
result Surfaces with small second fundamental form and flat boundary conditions are planar.
For certain metrics, the paper finds that the sixth-order Q-curvature is positive in some dimensions but negative in others.
problem Analyzing the positivity and non-positivity of the sixth-order Q-curvature for conformal metrics.
method Examining specific conditions on scalar curvature and Q-curvature, constructing examples to demonstrate the behavior of the sixth-order Q-curvature.
result The sixth-order Q-curvature can be positive in some dimensions but negative in others, depending on the metric.
A Levi nondegenerate real analytic hypersurface M of C^2 represented in local coordinates (z, w) in C^2 by a complex defining equation of the form w = Theta (z, \bar z, \bar w) which satisfies an appropriate reality condition, is spherical if and only if its complex graphing function Theta satisfies an explicitly writt…
We study analysis aspects of the sixth order GJMS operator Pg6. Under conformal normal coordinates around a point, the expansions of Green's function of Pg6 with pole at this point are presented. As a starting point of the study of Pg6, we manage to give some existence results of prescribed Q-curvature pr…
New method simplifies ideal curve flow with length constraint.
problem Analyzing ideal curve flow with length constraint.
method Introduced length constraint to simplify sixth order curvature flow.
result Flow exists for all time and converges to a round circle.
New operators and curvatures derived from embedded manifolds.
problem Finding obstructions and coupling extrinsic operators.
method Explicit computation of extrinsic Paneitz operator and its applications.
result New extrinsically-coupled fourth and sixth order operators.
Feature normalization prevents collapse in non-contrastive learning dynamics.
problem Non-contrastive learning can collapse into a single point due to lack of repulsive force.
method Extended previous theory based on L2 loss to cosine loss, considering feature normalization.
result Cosine loss induces stable equilibrium, preventing collapse even with insufficient repulsive force.
In a recent expository article (Notices of the AMS, 58 (2011), no. 1, 20-27), Ezhov, McLaughlin and Schmalz showed how to perform in an effective way Tanaka's prolongation procedure valid generally for filtered structures of constant type when the distribution is equipped with an integrable complex structure, so as to …
DS-GDA solves nonconvex-nonconcave problems without regularity conditions.
problem Nonconvex-nonconcave minimax optimization challenges.
method Doubly smoothed gradient descent ascent method (DS-GDA).
result Achieves convergence on various nonconvex-nonconcave problems.