Continues to prove open and dense set of metrics without local limiting Carleman weights.
problem Identifying metrics without local limiting Carleman weights.
method Analyzing conformally invariant tensors and studying metrics' properties.
result Proves the set of metrics without local limiting Carleman weights is open and dense.
In this article we consider the anisotropic Calderon problem and related inverse problems. The approach is based on limiting Carleman weights, introduced in Kenig-Sjoestrand-Uhlmann (Ann. of Math. 2007) in the Euclidean case. We characterize those Riemannian manifolds which admit limiting Carleman weights, and give a c…
Study geodesic X-ray transforms on curved manifolds using Carleman estimates.
problem Invertibility of geodesic X-ray transforms on curved manifolds.
method Using Carleman estimates to show invertibility of geodesic vector field.
result Geodesic X-ray transform is invertible on negatively curved simple manifolds.
The paper proves new inequalities on the unit ball in higher dimensions.
problem Establishing new weighted inequalities on the unit ball.
method Limiting approach to prove Carleman and Huber inequalities.
result Sharp weighted Carleman and Huber inequalities on the unit ball.
Proves unique continuation at infinity for certain expanding Ricci solitons.
problem Unique continuation at the boundary of expanding Ricci solitons.
method Establishes Carleman inequalities for weighted Laplacian.
result Unique continuation at infinity for asymptotically Ricci flat Ricci expanders.
In this note we prove that a generic Riemannian manifold of dimension ≥3 does not admit any nontrivial local conformal diffeomorphisms. This is a conformal analog of a result of Sunada concerning local isometries, and makes precise the principle that generic manifolds in high dimensions do not have conformal symm…
Let C[M] be a (local) Denjoy-Carleman class of Beurling or Roumieu type, where the weight sequence M=(Mk) is log-convex and has moderate growth. We prove that the groups DiffB[M](Rn), DiffW[M],p(Rn), ${\operatorname{Diff}}{\mathcal{S}}{}_…
Paper uses Carleman estimates to study harmonic functions on surfaces at infinity.
problem Unique continuation of harmonic functions on surfaces at infinity.
method Develops Carleman estimates for surfaces in Euclidean space at infinity.
result Obtains unique continuation property for harmonic functions.
Approximates symplectic automorphisms of coadjoint orbits using Hamiltonian Carleman methods.
problem Approximating symplectic automorphisms of coadjoint orbits.
method Hamiltonian Carleman approximation for coadjoint orbits of complex Lie groups.
result Established the Hamiltonian density property for closed coadjoint orbits of all complex Lie groups.
The paper develops quantitative estimates for holomorphic sections over bounded domains.
problem Establishing precise inequalities for holomorphic sections over bounded domains.
method Develops Sobolev-type inequalities and applies them to holomorphic sections of Hermitian vector bundles.
result Quantitative Carleman-type estimates for holomorphic sections are derived, improving on previous non-quantitative results.
Paper approximates continuous functions on Jordan arcs using conformal minimal immersions.
problem Approximating continuous functions on Jordan arcs using minimal immersions.
method Conformal minimal immersions and directed holomorphic curves.
result Continuous functions on Jordan arcs can be approximated by conformal minimal immersions.
Study finds solutions to inequality decay to zero on warped cylinders.
problem Analyzing solutions to a specific inequality on warped cylindrical ends.
method New Carleman estimate for independent interest.
result Solutions decay to zero along warped cylindrical ends.
New uncertainty principle for Schrödinger equations on hyperbolic manifolds.
problem Uncertainty principle for Schrödinger equations on hyperbolic manifolds.
method General strategy of Escauriaza-Kenig-Ponce-Vega, new Carleman estimates, logarithmic convexity, new mollifier and weight function.
result Similar rigidity phenomenon as in Euclidean space persists in hyperbolic geometry.
Quantifies polynomial approximation rates for smooth functions under various distributions.
problem Approximating smooth functions with polynomials under different distributional constraints.
method Develops a quantitative analogue of Carleman's theorem using complex analysis.
result Establishes superexponential rates of approximation for certain function classes over general distributions.
We prove the exponential law A(E×F,G)≅A(E,A(F,G)) (bornological isomorphism) for the following classes A of test functions: B (globally bounded derivatives), W∞,p (globally p-integrable derivatives), S (Schwartz space), D…
Proves non-existence of periodic vacuum spacetimes.
problem The existence of time-periodic vacuum spacetimes.
method Extending a candidate Killing vector field from null infinity using Carleman estimates.
result Smooth asymptotically flat solutions are stationary in a neighborhood of infinity.
The study sharpens local Bernstein estimates for Laplace eigenfunctions on compact manifolds.
problem Understanding local growth properties of Laplace eigenfunctions on compact Riemannian manifolds.
method Refined Donnelly-Fefferman method based on L2--Carleman estimates, combined with elliptic regularity and patching of local Carleman estimates. result Almost sharp local Lp--Bernstein inequalities for p∈[1,∞]. Extends Killing vector fields to beyond compact Cauchy horizons.
problem Proves existence of Killing vector fields beyond compact Cauchy horizons.
method New unique continuation theorem for wave equations through smooth compact lightlike hypersurfaces; novel Carleman type estimate.
result Killing vector field exists on both sides of the horizon.
Researchers transform equations and define integral operators on a ball.
problem Transforming equations from half space to ball.
method Identify Poisson kernel, define extension operator, prove inequalities.
result Uniqueness of extremal functions in limit case.
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 give a simple proof of weak Unique Continuation Property for perturbed Dirac operators, using the Carleman inequality. We apply the result to a class of perturbations of the Seiberg-Witten monopole equations that arise in Floer theory.
The paper develops theory for holomorphic null curves in SL2(C).
problem Developing theory for holomorphic null curves in SL2(C).
method Establish Runge, Mergelyan, Mittag-Leffler, and Carleman type theorems for holomorphic null immersions.
result Proves every open Riemann surface admits a proper holomorphic null embedding into SL2(C).
Study solves Calderón problem on complex manifolds using harmonic functions.
problem Solving the linearized Calderón problem on complex manifolds.
method Constructing Morse holomorphic functions with prescribed critical points.
result Positive answer to the Calderón problem on specific complex manifolds.
New method forecasts stock option prices accurately.
problem Accurate forecasting of stock option prices.
method Solving the ill-posed Black-Scholes equation using the Quasi-Reversibility Method.
result Good forecasting results demonstrated on market data.
We give a simple, direct proof of the backward uniqueness of solutions to a class of second-order geometric evolution equations including the Ricci and cross-curvature flows. The proof, based on a classical argument of Agmon-Nirenberg, uses the logarithmic convexity of a certain energy quantity in the place of Carleman…
Paper reveals how minimal surfaces' volumes can deduce their Riemannian structure.
problem Determining the Riemannian structure of minimal surfaces from their volumes.
method Analysis of Dirichlet-Neumann map and Carleman estimates.
result Volumes of minimal surfaces determine their Riemannian structure.
We prove uniqueness results for a Calderon type inverse problem for the Hodge Laplacian acting on graded forms on certain manifolds in three dimensions. In particular, we show that partial measurements of the relative-to-absolute or absolute-to-relative boundary value maps uniquely determine a zeroth order potential. T…
Geometric theory developed for ultradifferentiable functions and their wavefront sets.
problem Developing a geometric theory for ultradifferentiable functions.
method Using Dyn'kin's Theorem and Bony's Theorem, the ultradifferentiable wavefront set is defined and its properties are proven.
result Microlocal elliptic regularity theorem for ultradifferentiable vector bundles is proven.
The strong unique continuation property for Einstein metrics can be concluded from the well-known fact that Einstein metrics are analytic in geodesic normal coordinates. Here we give a proof of the same result that given two Einstein metrics with the same Ricci curvature on a fixed manifold, if they agree to infinite o…
We consider Calderon's inverse problem with partial data in dimensions n≥3. If the inaccessible part of the boundary satisfies a (conformal) flatness condition in one direction, we show that this problem reduces to the invertibility of a broken geodesic ray transform. In Euclidean space, sets satisfying the flat…
Empower efficient representation of distributions through moment-preserving methods.
problem Representing high-dimensional probability measures efficiently and accurately.
method Empower efficient representation of distributions through moment-preserving methods.
result Empowers efficient and accurate representation of high-dimensional probability measures.
The paper extends spectral estimates to hyperbolic surfaces with hyperbolic ends.
problem Proving a necessary condition for observability of the heat semigroup on manifolds.
method Propagation of smallness estimates of Carleman and Logunov-Malinnikova type.
result Established spectral estimates for surfaces with hyperbolic ends, proving the thickness condition is necessary.
In view of A. Andreotti and H. Grauert's vanishing theorem for q-complete domains in C^n, (Théorème de finitude pour la cohomologie des espaces complexes, Bull. Soc. Math. France 90 (1962), 193--259,) we re-prove a vanishing result by J.-P. Sha, (p-convex Riemannian manifolds, Invent. Math. 83 (1986), no. 3, 437--447,)…
Smooth functions on fat closed sets extend to smooth functions on all of space.
problem Extending smooth functions from fat closed sets to the entire space.
method Investigating arc-smooth functions on fat closed sets with Hölder boundary and subanalytic properties.
result Arc-smooth functions on fat closed sets extend to smooth functions on all of space.
A new weighted MCC measure improves classifier performance evaluation.
problem Lack of measures sensitive to observation weights in multiclass classification.
method Proposes weighted versions of Pearson-Matthews Correlation Coefficient (MCC) for binary and multiclass classification.
result Weighted MCC values are higher for classifiers that perform better on highly weighted observations.
Develops theory of weightings for Lie groupoids and algebroids.
problem Understanding differential geometry of weightings for Lie groupoids and algebroids.
method Extending work on weighted manifolds, defining weighted submanifolds, and developing theories of linear weightings and multiplicative weightings.
result Characterizes infinitesimally multiplicative weightings for Lie algebroids and classifies multiplicative weightings of Lie groupoids.
Stability of weighted extremal manifolds proven through blowups.
problem Stability of weighted extremal manifolds.
method Blowup technique to analyze weighted extremal Kähler manifolds.
result Proves weighted extremal manifolds are relatively weighted K-polystable.
The paper extends spin geometry to weighted manifolds and defines a new mass for Ricci flow.
problem Generalizing spin geometry to weighted manifolds and defining a new mass.
method Investigates spectral properties of the weighted Dirac operator and defines a new mass.
result Defines a new mass for weighted asymptotically Euclidean manifolds and shows its monotonicity under Ricci flow.
Paper generalizes CR Obata theorem to weighted Sasakian manifolds.
problem Deriving eigenvalue estimates for weighted Kohn Laplacian.
method Derived weighted CR Reilly's formula and applied to Sasakian manifolds.
result CR Obata theorem proven for weighted Sasakian manifolds.
New mass and staticity concepts derived from weighted curvature maps.
problem Deriving mass and staticity concepts for weighted manifolds.
method Developed a weighted curvature map and its adjoint, leading to weighted mass and static metrics.
result Equivalence and uniqueness theorems for weighted static manifolds and Penrose inequality.
The study explores weightings on submanifolds and their geometric properties.
problem Understanding weightings on submanifolds and their geometric implications.
method Detailed exploration of weighted normal bundles, weighted deformation spaces, and weighted blow-ups.
result A description of weightings in terms of subbundles of higher tangent bundles, leading to new concepts for Lie algebroids and groupoids.
Proves existence and uniqueness of weighted metrics for smooth spaces.
problem Existence and uniqueness of weighted metrics for smooth metric measure spaces.
method Proves existence and uniqueness using weighted ambient metrics and Poincaré metrics.
result Existence and uniqueness of weighted metrics for smooth metric measure spaces.
Defines and proves properties of weighted renormalized volume coefficients.
problem None explicitly stated; focuses on mathematical definitions and proofs.
method Defines weighted renormalized volume coefficients and proves their variational nature and polynomial representation.
result Weighted renormalized volume coefficients are variational and can be expressed as polynomials of specific tensors.
A new weighted FDA method improves face recognition accuracy.
problem Equal treatment of all class pairs in FDA leads to suboptimal performance.
method Cosine-weighted and automatically weighted FDA methods are proposed.
result Improved face recognition accuracy through weighted FDA.
New invariants help solve existence of weighted cscK metrics.
problem Existence of weighted cscK metrics in K-stability.
method Introduced weighted analytic delta invariant and beta invariant.
result Sufficient condition for existence of weighted cscK metrics.
Proves positive mass theorem for non-spin weighted manifolds.
problem Proving the positive mass theorem for non-spin weighted manifolds.
method Establishing density theorem and generalizing Geroch conjecture.
result Proves positive weighted mass theorem for non-spin weighted manifolds.
Derives integral formulae on weighted manifolds.
problem No specific problem stated; focuses on mathematical derivations.
method Introduces weighted mean sigma-r curvature and uses weighted Newton transformations.
result Derives integral formulae generalizing previous work.
Method measures weight similarity in neural networks using normalization and statistical inference.
problem Quantifying weight similarity in non-convex neural networks.
method Chain normalization rule and hypothesis-training-testing statistical inference.
result Weights of identical neural networks converge to similar local solutions.