The study proves estimates for transverse nonlinear equations on Sasakian manifolds with applications in geometry.
problem Estimating transverse fully nonlinear equations on Sasakian manifolds.
method Proving a priori estimates for transverse fully nonlinear equations.
result The study proves estimates for transverse fully nonlinear equations on Sasakian manifolds and gives geometric applications.
Method solves inverse problems for semilinear equations with power nonlinearities.
problem Solving inverse problems for semilinear equations with power nonlinearities.
method Higher order linearizations based on a nonlinear Dirichlet-to-Neumann map.
result Solves inverse problems for certain semilinear equations in dimensions 2 and n≥3.
Geometrically characterizes virtual nonlinear nonholonomic constraints using symplectic methods.
problem Characterizing virtual nonlinear nonholonomic constraints geometrically.
method Geometric characterization using symplectic structures and Chetaev equations.
result A unique control law exists to satisfy virtual constraints, and closed-loop dynamics are projections of uncontrolled dynamics.
Finsler geometry connects to higher-spin fields and curved-space Fronsdal equations.
problem Eliminating non-transverse modes in Finsler dynamics for higher spins.
method Parameterizing Finsler geometry in terms of symmetric tensors, analyzing linear and nonlinear terms, and examining gauge transformations.
result Finsler dynamics leads to the curved-space Fronsdal equation for all spins, plus a Stueckelberg-like coupling.
Simple perturbation of Vafa-Witten equations leads to transversality.
problem Transversality of Vafa-Witten moduli space.
method Simple perturbation of Vafa-Witten equations, proving transversality for SU(2) or SO(3) structure groups. result For generic perturbation parameter, the full rank part of the moduli space satisfies transversality.
We study the transversal wave equation on a compact Riemannian foliated manifold. As applications, we get an Egorov's type theorem for transversally elliptic operators, state a relationship between the singularities of the Fourier transform of the spectrum distribution function of a transversally elliptic operator and …
The abstract discusses transversality for infinite dimensional manifolds.
problem Transversality for infinite dimensional Fréchet manifolds.
method Constructing degree of nonlinear Fredholm mappings using transversality results.
result Proof of rank theorem, invariance of domain theorem, and Bursuk-Ulam type theorem.
Study proves interaction of three impulsive gravitational waves, showing local solution and Lipschitz continuity.
problem Interaction of three impulsive gravitational waves in Einstein vacuum equations.
method Geometric estimates and wave estimates to prove local solution and continuity.
result Local solution to Einstein vacuum equations with three impulsive gravitational waves, Lipschitz continuity.
Adapts PDE method to prove L∞ estimates for complex Hessian equations.
problem Proving L∞ estimates for complex Hessian equations on transverse Kähler manifolds. method Adapts PDE approach of Guo-Phong-Tong and Guo-Phong-Tong-Wang [17, 18].
result Obtains L∞ estimate for transverse complex Monge-Ampère equations. This paper establishes transversality for perturbed Vafa-Witten moduli spaces on 4-manifolds.
problem Transversality of Vafa-Witten moduli spaces on 4-manifolds with C≡0. method Constructing perturbation terms to ensure the moduli space is a smooth manifold of dimension zero.
result For generic perturbation terms, the moduli space of solutions is a smooth manifold of dimension zero.
Geometric flow on curves in S^3 generates YO equations solutions.
problem Modeling short wave-long wave interaction.
method Simple geometric flow on curves in S3. result Constructs transverse curves for YO equations periodic solutions.
Researchers solve an inverse problem for a semilinear elliptic equation on complex manifolds.
problem Determining an unknown function in a semilinear elliptic equation on complex manifolds.
method Analyzing higher order linearizations and interactions of Gaussian quasimode solutions.
result An unknown smooth function can be uniquely determined from the Dirichlet-to-Neumann map.
We show that formal isomorphism of intransitive linear Lie equations along transversal to the orbits can be extended to neighborhoods of these transversal. In analytic cases, the word formal is dropped from theorems. Also, we associate an intransitive Lie algebra with each intransitive linear Lie equation, and from the…
The wave equation utt=c2uxx is generally regarded as a linear approximation to the equation describing the amplitude of a transversely vibrating elastic string in the plane. But, as is shown in \cite{BC96}, the assumption of transverse vibration in fact implies that the wave equation describes the vibration…
Higher-order optimization problems naturally appear when investigating the effects of a patent with finite length, as in the pioneering work of Futagami and Iwaisako (2007). In this paper, we establish the Euler equations and transversality conditions necessary for analyzing such higher-order optimization problems. We …
The Stone-Weierstrass theorem aids in solving inverse problems on specific manifolds.
problem Inverse problems on holomorphically separable Kähler manifolds and conformally transversally anisotropic manifolds.
method Application of the Stone-Weierstrass theorem to show uniqueness in inverse problems.
result Generalization and simplification of earlier results in inverse problems.
This research announcement gives a brief report of the main results in our paper "PU(2) monopoles, I: Regularity, Uhlenbeck compactness, and transversality" (Journal of Differential Geometry, to appear). We describe the existence of perturbations for the PU(2) monopole equations, yielding both useful transversality pro…
We prove an analogue of the Hitchin-Kobayashi correspondence for compact, oriented, taut Riemannian foliated manifolds with transverse Hermitian structure. In particular, our Hitchin-Kobayashi theorem holds on any compact Sasakian manifold. We define the notion of stability for foliated Hermitian vector bundles with tr…
The paper smooths out equations on 4-manifolds to show smooth moduli spaces.
problem Understanding the structure of solutions to Vafa-Witten equations on 4-manifolds.
method Constructing perturbation terms to ensure transversality and showing moduli spaces are smooth.
result For generic perturbation, the general part of the moduli space is a smooth 0-dimensional manifold.
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.
We prove a general result about the short time existence and uniqueness of second order geometric flows transverse to a Riemannian foliation on a compact manifold. Our result includes some flows already existing in literature, as the transverse Ricci flow, the Sasaki-Ricci flow and the Sasaki J-flow and motivates the s…
Note on advancements in nonlinear elliptic equations' regularity theory.
problem Nonlinear elliptic equations and their regularity.
method De Giorgi-Nash-Moser theory, Krylov-Safonov theory, Evans-Safonov theory.
result Contributions to Hilbert's 19th problem and fully nonlinear equations.
Study solves inverse problems for equations with fractional nonlinearities.
problem Solving inverse problems for semilinear elliptic equations with fractional power nonlinearities.
method Higher order linearization method adapted for fractional order.
result Results of previous studies remain valid for general power nonlinearities.
In this paper, we introduce two notions on a surface in a contact manifold. The first one is called degree of transversality (DOT) which measures the transversality between the tangent spaces of a surface and the contact planes. The second quantity, called curvature of transversality (COT), is designed to give a compar…
We prove the existence of perturbations for the PU(2) monopole equations, yielding transversality on the complement of the anti-self-dual or reducible solutions, and the existence of an Uhlenbeck compactification for the moduli space of solutions to these perturbed PU(2) monopole equations. In December 1994, V. Pidstri…
The paper provides estimates for positive solutions to a nonlinear equation under geometric flow.
problem Analyzing positive solutions to a nonlinear equation under geometric flow.
method Gradient estimates for positive solutions under geometric flow on manifolds.
result Gradient estimates for positive solutions to a nonlinear equation under geometric flow.
The paper examines Sasaki-Ricci solitons and their transverse rigidity properties.
problem Understanding the rigidity properties of Sasaki-Ricci solitons as singularity models.
method Established fundamental equations and criteria for transverse rigidity, proving key results about scalar curvature and Weyl tensor.
result Low-dimensional Sasaki-Ricci solitons with constant scalar curvature are Sasaki-Einstein, and those with harmonic Weyl tensor are finite quotients of the sphere.
Paper proves existence of ambient manifolds for null hypersurfaces solving Einstein equations.
problem Analyzing transverse expansion of metrics at null hypersurfaces.
method Covariant approach proving existence of ambient manifolds given asymptotic expansion and constraint equations.
result Existence of ambient manifolds solving Einstein equations to infinite order at null hypersurfaces.
Study of instantons on Sasakian 7-manifolds using gauge fields.
problem Understanding instantons on Sasakian manifolds.
method Fredholm theory, cohomological conditions, index of a transverse elliptic operator.
result Moduli space of selfdual contact instantons is Kähler.
We prove the existence of foliations transverse to pseudo-Anosov flows using veering triangulations.
problem Existence of foliations transverse to pseudo-Anosov flows in 3-manifolds.
method Combinatorial approach via veering triangulations and branched surfaces.
result Existence of transverse foliations for certain pseudo-Anosov flows.
The purpose of this paper is to construct the early exercise boundary for a class of nonlinear Black--Scholes equations with a nonlinear volatility depending on the option price. We review a method how to transform the problem into a solution of a time depending nonlinear parabolic equation defined on a fixed domain. R…
Paper quantifies neural operators' efficiency for solving nonlinear parabolic PDEs.
problem Quantifying the efficiency of neural operators for solving nonlinear parabolic PDEs.
method Deriving approximation rates by transferring PDEs to integral equations and leveraging Picard's iteration.
result Neural operators can efficiently approximate solution operators of nonlinear PDEs without exponential complexity growth.
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…
Study solves equations on tori for Calabi-Yau problems.
problem Solving equations on tori for Calabi-Yau problems.
method Study of fully nonlinear equations on flat tori.
result Solvability of equations on tori established.
First-order jet bundles can be put at the foundations of the modern geometric approach to nonlinear PDEs, since higher-order jet bundles can be seen as constrained iterated jet bundles. The definition of first-order jet bundles can be given in many equivalent ways - for instance, by means of Grassmann bundles. In this …
Study gradient estimates for nonlinear parabolic equations on Riemannian manifolds.
problem Estimating gradients for nonlinear parabolic equations on Riemannian manifolds.
method Analyzes Fisher-KPP, parabolic Allen-Cahn, and Newell-Whitehead equations on complete noncompact Riemannian manifolds.
result Gradient estimates for positive solutions and Liouville theorem for ancient solutions.
The paper studies frequency monotonicity for solutions of nonlinear equations under Ricci flow.
problem Frequency monotonicity for positive solutions of nonlinear equations under Ricci flow.
method Obtained parabolic frequency monotonicity for solutions of two nonlinear parabolic equations with bounded Ricci curvature.
result Established integral type Harnack inequalities using parabolic frequency monotonicity.
The paper proves gradient estimates for nonlinear parabolic equations on smooth metric measure spaces.
problem Proving gradient estimates for nonlinear parabolic equations on smooth metric measure spaces.
method Using Souplet-Zhang type estimates and properties of Bakry-Emery Ricci tensor and weighted mean curvature.
result Gradient estimates for nonlinear parabolic equations on smooth metric measure spaces with Dirichlet boundary condition.
The nonlinear equations describing all the nonsingular pencils of metrics of constant Riemannian curvature are derived and the integrability of these nonlinear equations by the method of inverse scattering problem is proved. It is proved that all the nonsingular pairs of compatible metrics of constant Riemannian curvat…
The paper studies equations in conformal geometry with gradient and existence results.
problem Equations in conformal geometry on closed smooth Riemannian manifolds.
method Local gradient and second derivative estimates, existence result.
result Proved local gradient and second derivative estimates for solutions.
The paper extends Li-Yau-Hamilton estimates to evolving Kähler metrics and nonlinear heat equations.
problem Deriving estimates for nonlinear heat equations on evolving Kähler metrics.
method Generalized matrix Li-Yau-Hamilton estimates to Kähler manifolds with evolving metrics and nonlinear heat equations.
result Extended Li-Yau-Hamilton estimates to evolving Kähler metrics and nonlinear heat equations.
Solves open problems for fully nonlinear elliptic equations on manifolds.
problem Solving fully nonlinear elliptic equations on manifolds.
method Analytic slope invariant and Nakai-Moishezon criterion.
result Solves open problems including hessian and hessian quotient equations.
The abstract discusses transforming Vaisman metrics into Kähler-Einstein structures.
problem Transforming Vaisman metrics into Kähler-Einstein structures.
method Using the transverse Kähler-Ricci flow on the canonical foliation of a closed Vaisman manifold.
result Direct proof of short time existence of transverse Kähler-Ricci flow on Vaisman manifolds.
The paper studies fully nonlinear equations on Hermitian manifolds, proving existence and interior estimates.
problem Proving existence and interior estimates for fully nonlinear equations on Hermitian manifolds.
method Derives interior estimates and establishes the existence of smooth solutions for the Dirichlet problem and equations on closed manifolds.
result Derives interior estimates and establishes the existence of smooth solutions for the Dirichlet problem and equations on closed manifolds.
MC-MCL improves MCL for nonlinear clustering.
problem Nonlinear clustering in data science.
method MC-MCL combines MCL with Minimum Curvilinearity for nonlinear distances.
result MC-MCL outperforms classical MCL and baseline clustering algorithms in nonlinear datasets.
Paper establishes estimates for nonlinear equations on compact manifolds.
problem Estimating solutions to fully nonlinear equations with gradient terms on compact almost Hermitian manifolds.
method Establishes second order estimates and proves existence of solutions for specific equations.
result Proves existence of solutions for various equations, including Monge-Ampère and Hessian equations.
Study transverse knots and symplectic surfaces using Seiberg-Witten monopole equations.
problem Investigate transverse knots and symplectic surfaces via Seiberg-Witten monopole equations.
method Equivariant Seiberg-Witten theory on branched covers, introducing a novel slice-torus invariant.
result Determine the value of slice-torus invariant for certain Montesinos knots within a deviation of 2.
Study improves understanding of solutions to complex equations in geometry.
problem Understanding solutions to fully nonlinear elliptic equations.
method Obtained local second derivative estimates for strong solutions.
result Improved estimates for W2,p-strong solutions.