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.
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.
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.
We describe a set of conformally covariant boundary operators associated to the sixth-order GJMS operator on a conformally invariant class of manifolds which includes compactifications of Poincaré--Einstein manifolds. This yields a conformally covariant energy functional for the sixth-order GJMS operator on such manifo…
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.
We consider surfaces with boundary satisfying a sixth order nonlinear elliptic partial differential equation corresponding to extremising the L2-norm of the gradient of the mean curvature. We show that such surfaces with small L2-norm of the second fundamental form and satisfying so-called `flat boundary conditio…
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.
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.
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.
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.
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.
The paper establishes a duality between non-compact and compact symmetric pairs.
problem Understanding the relationship between non-compact and compact symmetric pairs.
method Developed a duality theorem between non-compact pseudo-Riemannian semisimple symmetric pairs and commutative compact semisimple symmetric triads.
result Explicit description of a one-to-one correspondence between non-compact and compact symmetric pairs.
New local method solves Yamabe problems on compact and non-compact manifolds.
problem Yamabe problems on compact and non-compact manifolds.
method Local method for compact and non-compact manifolds.
result Generalizes Brezis and Nirenberg's nonlinear eigenvalue problem to subsets of manifolds.
Compact curve solution emerges from non-compact curve.
problem Constructing solutions from non-compact curves.
method Slingshot solution to curve shortening flow.
result Compact embedded solution exists for a finite time.
Compact metrics on Heisenberg manifolds have a specific condition for being relatively compact.
problem Conditions for relatively compact sets of left invariant metrics on Heisenberg manifolds.
method Necessary and sufficient condition for relatively compact sets of left invariant metrics.
result A condition for a set of left invariant metrics to be relatively compact in the moduli space.
Compact Lie groups have compact isometry groups with pseudo-Riemannian metrics.
problem Characterizing isometry groups of compact Lie groups with pseudo-Riemannian metrics.
method Analyzing left-invariant pseudo-Riemannian metrics on compact Lie groups.
result Isometry groups of compact Lie groups are compact.
Study on GL(2) geometries on complex manifolds, focusing on Kähler-Einstein and Fano manifolds.
problem Characterizing compact complex manifolds with holomorphic GL(2)-geometry.
method Analyzing Kähler-Einstein and Fano manifolds, using GL(2) and SL(2) geometries.
result Only compact Kähler-Einstein manifolds with holomorphic GL(2)-geometry are covered by compact complex tori, three dimensional quadric, or three dimensional Lie ball.
Paper proves structure for compact Kähler manifolds with pseudo-effective tangent bundles.
problem Compact Kähler manifolds with pseudo-effective tangent bundles.
method Smooth or locally constant rationally connected fibration onto a quotient of a compact complex torus.
result Compact Kähler manifolds with pseudo-effective tangent bundles admit a fibration structure.
Compact embeddings for invariant functions in metric-measure spaces.
problem Embedding functions with symmetry in metric-measure spaces.
method Analyzing H-invariant functions in compact metric-measure spaces, extending to Riemannian manifolds. result Obtained compact Sobolev embeddings for critical exponents.
Compact theorem on Hamiltonian stationary submanifolds in symplectic manifolds.
problem Compactness of Hamiltonian stationary Lagrangian submanifolds in symplectic manifolds.
method Proving a compactness theorem with area and extrinsic curvature bounds.
result Uniform bounds on area and total extrinsic curvature lead to compactness of Hamiltonian stationary Lagrangian submanifolds.
Compact groups with polynomial growth have specific embeddings.
problem Understanding groups with polynomial growth.
method Embedding into semidirect products of Lie and compact groups.
result Groups can be embedded as co-compact subgroups.
Let F be a foliation of codimension 2 on a compact manifold with at least one non-compact leaf. We show that then F must contain uncountably many non-compact leaves. We prove the same statement for oriented p-dimensional foliations of arbitrary codimension if there exists a closed p form which evaluates positively on e…
Study on compact strong HKT manifolds and their properties.
problem Characterizing the structure of compact strong HKT manifolds.
method Geometric analysis, rigidity theorems, classification, and properties of Ricci foliations.
result Compact strong HKT manifolds are Hopf fibrations over compact 4-dimensional orbifolds.
Study compact symplectic solvmanifolds' hard Lefschetz property.
problem Compact symplectic solvmanifolds' hard Lefschetz property.
method Analysis of compact symplectic solvmanifolds as quotients of solvable Lie groups by lattices.
result Characterization of conditions for the hard Lefschetz property.
Paper classifies compact symmetric triads using double Satake diagrams and canonical forms.
problem Classifying compact symmetric triads.
method Introducing double Satake diagrams and canonical forms, proving their existence and properties.
result Existence and properties of canonical forms for compact simple symmetric triads.
In this paper, we establish some compactness results of conformally compact Einstein metrics on 4-dimensional manifolds. Our results were proved under assumptions on the behavior of some local and non-local conformal invariants, on the compactness of the boundary metrics at the conformal infinity, and on the topology…
Compact shrinkers with curvature pinching conditions proven.
problem Ensuring shrinkers are compact under curvature pinching conditions.
method Various curvature pinching conditions applied to shrinkers with positive Ricci curvature and asymptotically nonnegative sectional curvature.
result Shrinkers with curvature pinching conditions are proven to be compact.
No conformal product structures on compact manifolds with constant curvature.
problem Existence of conformal product structures on compact manifolds with constant curvature.
method Analyzing non-flat manifolds and irreducible, compact locally symmetric spaces of non-positive curvature.
result Compact non-flat manifolds with constant sectional curvature admit no conformal product structure.
We study compact Riemannian manifolds for which the light between any pair of points is blocked by finitely many point shades. Compact flat Riemannian manifolds are known to have this finite blocking property. We conjecture that amongst compact Riemannian manifolds this finite blocking property characterizes the flat m…
We study Bott-Chern cohomology on compact complex non-Kähler surfaces. In particular, we compute such a cohomology for compact complex surfaces in class VII and for compact complex surfaces diffeomorphic to solvmanifolds.
In this paper, we establish compactness results of some class of conformally compact Einstein 4-manifolds. In the first part of the paper, we improve the earlier results obtained by Chang-Ge. In the second part of the paper, as applications, we derive some compactness results under perturbation conditions when the L^2-…
Extended Vaisman theorem to compact spaces with singularities.
problem Generalizing Vaisman's theorem to spaces with singularities.
method Extended Vaisman's theorem to compact complex spaces with singularities.
result Vaisman's theorem extended to compact spaces with singularities.
We classify compact homogeneous geometries of irreducible spherical type and rank at least 2 which admit a transitive action of a compact connected group, up to equivariant 2-coverings. We apply our classification to polar actions on compact symmetric spaces.
A transversely holomorphic foliation on a compact complex manifold, exhibits a compact stable leaf if and only if the set of compact leaves is not a zero measure subset of the manifold.
When studying the causal propagation of a field in a globally hyperbolic spacetime M, one often wants to express the physical intuition that it has compact support in spacelike directions, or that its support is a spacelike compact set. We compare a number of logically distinct formulations of this idea, and of the com…
The study classifies all compact 5D polytopes with 9 facets.
problem Classifying compact hyperbolic Coxeter polytopes.
method Complete classification through mathematical analysis.
result A complete list of compact hyperbolic Coxeter 5D polytopes with 9 facets.
The study classifies all compact hyperbolic polytopes with eight facets.
problem Classifying compact hyperbolic Coxeter four-polytopes with specific numbers of facets.
method Complete classification through mathematical analysis.
result The complete classification of compact hyperbolic Coxeter four-polytopes with eight facets.
Classifies π1-injective maps between non-compact surfaces.
problem Characterizing maps with injective fundamental groups.
method Proper homotopy classification of maps.
result All π1-injective proper maps are classified. Circle packings on compact surfaces simplified.
problem Simplifying circle packings on complex surfaces.
method Uniformisation of weighted maps.
result Unified approach to circle packings.
Thirty years after the birth of foliations in the 1950's, André Haefliger has introduced a special property satisfied by holonomy pseudogroups of foliations on compact manifolds, called compact generation. Up to now, this is the only general property known about holonomy on compact manifolds. In this article, we give a…
Develops Lie algebraic approach for compact complex homogeneous manifolds.
problem Proves important results on compact complex homogeneous manifolds.
method Uses standard results in Lie theory to associate a canonical abelian Lie algebra with a given integrable complex structure.
result Provides a new method of associating a canonical abelian Lie algebra with a given integrable complex structure.