Study extends continuity equation for Gauduchon metrics.
problem Continuity equation for Gauduchon metrics.
method Solution to Gauduchon conjecture by Székelyhidi, Tosatti, and Weinkove.
result Extended interval of maximal existence for continuity equation.
Study continuity equation on Hopf and Inoue surfaces, proving estimates and convergence.
problem Analyzing the continuity equation on specific complex surfaces.
method Extended La Nave-Tian's continuity equation to Hermitian setting, proving estimates and Gromov-Hausdorff convergence.
result Proved a priori estimates for solutions on Hopf and Inoue surfaces, and convergence of Inoue surfaces to a circle.
Study on Hölder continuity of complex Monge-Ampère solutions on Stein spaces.
problem Understanding continuity of solutions to complex Monge-Ampère equations on Stein spaces.
method Analyzing solutions with Lp densities and Hölder boundary data on Stein spaces with isolated singularities. result Solutions are Hölder continuous outside singular points if boundary data is Hölder continuous.
Paper proves uniform continuity bounds for complex Monge-Ampère solutions.
problem Estimating the continuity of solutions to complex Monge-Ampère equations.
method PDE-based approach from fully non-linear equations in Kähler geometry.
result Uniform and sharp estimate for the modulus of continuity.
Study on stability and continuity of solutions to complex Monge-Ampère equations on compact Hermitian manifolds.
problem Stability and continuity of solutions to degenerate complex Monge-Ampère equations.
method Analysis of Hölder continuity and global continuity of solutions.
result Established uniform diameter bound for the twisted Chern-Ricci flow.
Study on continuity of solutions for complex Monge-Ampère equations with movable singularities.
problem Continuity of solutions with prescribed singularities for complex Monge-Ampère equations.
method Strong continuity methods with movable singularities, including Kähler-Einstein metrics.
result Sufficient conditions for strong continuity of solutions and openness results for Fano type equations.
Solves complex Monge-Ampère equations with Hölder continuous solutions in Kähler manifolds.
problem Finding Hölder continuous solutions to complex Monge-Ampère equations.
method Analyzes the complex Monge-Ampère equation in Kähler manifolds using Sobolev spaces and Hölder continuity.
result Hölder continuity of solutions is equivalent to the measure's Hölder continuity in a complex Sobolev space.
Study Galois groupoids of discret Painlevé equations.
problem Computing Galois groupoids for discret Painlevé equations.
method Using semi-continuity theorem for Galois groupoid in confluence of difference to differential equations.
result Computed Galois groupoids for discret Painlevé equations.
Continuous solutions found for complex geometry equations.
problem Finding solutions to complex geometry equations on Hermitian manifolds.
method Proving existence of continuous quasi-plurisubharmonic solutions for specific measures.
result Existence of continuous quasi-plurisubharmonic solutions for measures dominated by capacity.
Sharp Lipschitz bounds and gradient estimates for fully nonlinear parabolic equations.
problem Understanding moduli of continuity for fully nonlinear parabolic equations.
method Proving moduli of continuity of viscosity solutions are subsolutions of one-dimensional parabolic equations.
result Sharp Lipschitz bounds and gradient estimates for fully nonlinear parabolic equations with bounded initial data.
Continuity of solutions to complex Monge-Ampère equations on compact Kähler spaces proved.
problem Continuity of solutions to complex Monge-Ampère equations on compact Kähler spaces.
method Analyzing bounded solutions on reduced, locally irreducible compact Kähler spaces.
result Proves continuity of solutions, affirming conjectures and solving open problems.
Proves Hölder continuity of complex Monge-Ampère solutions.
problem Global Hölder continuity of solutions to complex Monge-Ampère equation.
method Analyzes Dirichlet problem on strictly pseudoconvex domains or Hermitian manifolds.
result Proves global Hölder continuity of solutions under given conditions.
Study on existence and properties of continuous solutions to complex Hessian equations.
problem Existence and properties of continuous solutions to complex Hessian equations.
method Established new capacity estimates and weak stability estimates for the m-Hessian measure. result Existence of continuous solutions to the complex Hessian equation under certain conditions.
Note on gradient estimates for complex Monge-Ampere equation.
problem Gradient estimates for solutions of complex Monge-Ampere equation.
method Estimates Lp and L∞ for gradient in terms of continuity of the right-hand side. result Gradient estimates for solutions of complex Monge-Ampere equation.
In this paper, we investigate the moduli of continuity for viscosity solutions of a wide class of nonsingular quasilinear evolution equations and also for the level set mean curvature flow, which is an example of singular degenerate equations. We prove that the modulus of continuity is a viscosity subsolution of some o…
Solves complex Monge-Ampère equation for Kähler-Ricci solitons.
problem Complex Monge-Ampère equation for shrinking gradient Kähler-Ricci solitons.
method Aubin continuity path, implementing another continuity method.
result Shows existence of solution for the equation.
The study finds continuous solutions to complex Hessian equations on compact Hermitian manifolds.
problem Finding continuous solutions to complex Hessian equations on compact Hermitian manifolds.
method Deriving an L∞-estimate for bounded solutions to the complex m-th Hessian equations on compact Hermitian manifolds, assuming a positive right-hand side in the Orlicz space Lmn(logL)n(h∘log∘logL)n. result Establishing the existence of continuous solutions to the complex Hessian equation under the prescribed assumptions.
This paper proves Hölder continuity for complex Monge-Ampère equations on Kähler varieties.
problem Establishing Hölder estimates on singular Kähler varieties.
method Geometric regularization based on partial C0 estimate. result Uniform Hölder continuity for complex Monge-Ampère equations on Kähler varieties.
Pathwise uniqueness shown for specific stochastic equations.
problem Stochastic Volterra equations with singular kernels and Hölder coefficients.
method Established pathwise uniqueness through Hölder continuity of coefficients.
result Pathwise uniqueness and existence of unique strong solutions.
Researchers find explicit solutions to complex Monge-Ampère equation.
problem Solving complex Monge-Ampère equation with constant right-hand side.
method Explicit pluripotential and viscosity solutions.
result Presented solutions lie in Wloc1,2∩Wloc2,1 and are not Dini continuous. The paper examines ellipticity of specific equations on vector bundles.
problem Investigating ellipticity of vector bundle versions of Monge-Ampère equations.
method Analyzing continuity paths and preserving ellipticity of equations.
result Not all equations preserve ellipticity along continuity paths, but σ2 does. Research proves unique continuation for Einstein-vacuum equations on aAdS spacetimes.
problem Establishing rigorous mathematical statements for AdS/CFT correspondence.
method Novel Carleman estimates and unique continuation results for wave equations on aAdS spacetimes.
result Proved a unique continuation result for the Einstein-vacuum equations from aAdS conformal boundaries.
We solve the Dirichlet problem for the complex Monge-Ampère equation on a strictly pseudoconvex with the right hand side being a positive Borel measure which is dominated by the Monge-Ampère measure of a Hölder continuous plurisubharmonic function. If the boundary data is continuous, then the solution is continuous. If…
Study continuity and Hölder estimates for solutions on Stein spaces.
problem Continuity and Hölder estimates for solutions to degenerate complex Monge-Ampère equations.
method Prove continuity up to the boundary and local Hölder estimates on the regular locus.
result Local Hölder estimates on the regular locus for solutions to degenerate complex Monge-Ampère equations.
Convex solutions to a specific equation are smooth when the phase is smooth enough.
problem Regularity of solutions to the Lagrangian mean curvature equation.
method Showed regularity for convex solutions under Hölder continuity conditions on the phase.
result Convex viscosity solutions are regular if the Lagrangian phase is Hölder continuous.
We extend the continuity equation of La Nave-Tian to Hermitian metrics and establish its interval of maximal existence. The equation is closely related to the Chern-Ricci flow, and we illustrate this in the case of elliptic bundles over a curve of genus at least two.
We show that a positive Borel measure of positive finite total mass, on compact Hermitian manifolds, admits a Holder continuous quasi-plurisubharmonic solution to the Monge-Ampere equation if and only if it is dominated locally by Monge-Ampere measures of Holder continuous plurisubharmonic functions.
Derives EoM for DNNs to describe GD dynamics precisely.
problem Gaps between differential equations and actual DNN learning dynamics due to discretization error.
method Starts from GF, derives counter term to cancel discretization error, obtains EoM.
result EoM precisely describes GD dynamics of DNNs, highlights differences between continuous and discrete GD.
Study on heat equation and eigenfunctions on RCD spaces, proving unique continuation.
problem Unique continuation for caloric functions and eigenfunctions on RCD spaces.
method Establish weak unique continuation theorem for caloric functions and eigenfunctions on compact RCD(K,2) spaces.
result Existence of non-trivial eigenfunctions and caloric solutions vanishing up to infinite order at one point.
We study Hölder continuity of solutions to the Monge-Ampère equations on compact Kähler manifolds. In [DNS] the authors have shown that the measure ωun is moderate if u is Hölder continuous. We prove a theorem which is a partial converse to this result.
Study of continuous symmetries in Nahm data and BPS monopoles.
problem Solutions to Nahm's equations with continuous symmetries.
method Classification of Ansätze and construction of Nahm data.
result Construction of new BPS monopoles with spherical symmetry.
Stability and Hölder continuity of solutions to complex Monge-Ampère equations on compact Hermitian manifolds.
problem Establishing Hölder continuity of solutions to complex Monge-Ampère equations.
method Stability result for solutions in Lp space, Hölder continuity proof. result Solutions are Hölder continuous with the same exponent as in the Kähler case.
The paper explores continuous limits of pentagram maps and their relation to KdV equations.
problem Understanding the continuous limits of pentagram maps and their associated KdV equations.
method Quantum calculus and geometric constructions to derive continuous limits and Lax representations.
result Continuous limits of pentagram maps yield specific KdV equations, providing a geometric interpretation.
This paper concerns the continuous time mean-variance portfolio selection problem with a special nonlinear wealth equation. This nonlinear wealth equation has a nonsmooth coefficient and the dual method developed in [6] does not work. We invoke the HJB equation of this problem and give an explicit viscosity solution of…
New geometric transformations link discrete and continuous curve motions.
problem Establishing a connection between discrete and continuous curve motions.
method Infinitesimal Darboux transformations of smooth curves.
result Alternate geometric interpretation for semi-discrete mKdV equation.
Sufficient condition for log-continuity of complex Monge-Ampère solutions.
problem Ensuring log-continuity of solutions to complex Monge-Ampère equations.
method Analyzing compact Kähler manifolds and line bundles, providing sufficient conditions for log-continuity.
result Log-continuity of solutions to complex Monge-Ampère equations with Lp right-hand sides. Let (X,ω) be an n-dimensional compact Kähler manifold. We study degenerate complex Hessian equations of the form (ω+ddcφ)m∧ωn−m=F(x,φ)ωn. Under some natural conditions on F, this equation has a unique continuous solution. When (X,ω) is rational homogeneous we further show that the solu…
Continuity of Kähler-Einstein potentials at singularities proven.
problem Regularity of solutions to degenerate complex Monge-Ampère equations on singular spaces.
method Investigation of Dirichlet problem and global continuity of solutions.
result Kähler-Einstein potentials are continuous at isolated singularities.
Study Neumann problem for special Lagrangian type equations.
problem Neumann problem for special Lagrangian type equations.
method Uniform a priori estimates, continuity method, direct proof of boundary double normal derivative estimates.
result Existence result for Neumann problem of special Lagrangian type equations.
Solves complex equation for specific geometric solitons.
problem Solving complex Monge-Ampère equation for specific geometric solitons.
method Aubin continuity path and continuity method.
result Initial value of the path parameter has a solution and is open to all.
Derives effective continuous dynamics for adaptive SGD methods.
problem Analyzing noise in adaptive SGD methods.
method Stochastic modified equations framework and Malladi's scaling rules.
result Sampling-induced noise in SGD limits to independent Brownian motions.
Continuity of complex Monge-Ampère potentials on Kähler manifolds.
problem Continuity of solutions to complex Monge-Ampère equations on compact Kähler manifolds.
method Extending DiNezza-Lu's approach to big cohomology classes, proving continuity on Zariski open sets.
result Singular Kähler-Einstein metrics have continuous potentials on the ample locus outside of the non-klt part.
We construct explicit solutions to continuous motion of discrete plane curves described by a semi-discrete potential modified KdV equation. Explicit formulas in terms the τ function are presented. Bäcklund transformations of the discrete curves are also discussed. We finally consider the continuous limit of discrete …
Continuous-time mean-variance portfolio selection model with nonlinear wealth equations and bankruptcy prohibition is investigated by the dual method. A necessary and sufficient condition which the optimal terminal wealth satisfies is obtained through a terminal perturbation technique. It is also shown that the optimal…
New method infers hidden states in continuous-time phenomena better than traditional models.
problem Traditional HSMM's are limited to discrete time grids and cannot handle irregularly spaced data.
method Formulated integro-differential forward and backward equations for CTSMC's, introduced scalable Viterbi-type algorithm.
result Efficiently solved equations for posterior marginals and path estimates.
The study proves the existence of k-convex hypersurfaces for specific curvature equations.
problem Proving the existence of k-convex hypersurfaces for Hessian curvature equations. method Combining a priori estimates with the continuity method, and establishing a constant rank theorem.
result Existence and uniqueness of k-convex hypersurfaces for both nonhomogeneous and homogeneous Hessian curvature equations. Paper proves existence of knot solutions for specific equations.
problem Existence of knot solutions for twisted Kapustin-Witten equations.
method Explicit solutions for t=1, continuity argument in t.
result Proves existence of model knot solutions for any twisting parameter.
We prove that on compact Kähler manifolds solutions to the complex Monge-Ampère equation, with the the right hand side in Lp,p>1, are Hölder continuous.