Noise statistics oblivious algorithm improves robust regression with sparse outliers.
problem Robust regression with sparse outliers in the presence of Gaussian noise.
method Developed a noise statistics oblivious algorithm called RRT-GARD by modifying GARD.
result RRT-GARD performs nearly as well as GARD with known noise statistics.
Orthogonal matching pursuit (OMP) is a widely used compressive sensing (CS) algorithm for recovering sparse signals in noisy linear regression models. The performance of OMP depends on its stopping criteria (SC). SC for OMP discussed in literature typically assumes knowledge of either the sparsity of the signal to be e…
Global calculus for manifolds with boundary, solving evolution problems.
problem Global solvability of evolution problems on manifolds with boundary.
method Established global functional calculus and Gårding inequality for pseudo-differential operators without local coordinates.
result Global solvability for a class of evolution problems.
The study connects curvature operators' positivity to manifold topology.
problem Positivity of curvature operators and their geometric implications.
method Analysis of Garding cones and positivity properties of curvature operators.
result Shifted cone conditions on curvature operators constrain manifold topology.
Formula establishes determinant majorization for symmetric matrices.
problem Determining determinant majorization for symmetric matrices.
method Establishes determinant majorization formula for symmetric matrices using invariant Garding-Dirichlet polynomials.
result Formula F(A)N1≥det(A)n1 for symmetric matrices. The paper establishes a majorization result for symmetric matrices.
problem Majorization of symmetric matrices under certain conditions.
method Using Garding-Dirichlet operator and properties of I-central operators. result A definitive inequality for symmetric matrices.
Study estimates eigenvalues for concave Hessian operators on convex domains.
problem Estimating eigenvalues for concave elliptic Hessian operators.
method Investigates Dirichlet eigenvalue problem for a broad class of concave elliptic Hessian operators.
result Existence and properties of the first nonzero eigenvalue and eigenfunction.
Novel boundary integral equations for Dirac operators in 3D Lipschitz domains.
problem Developing equations for Dirac operators in complex 3D domains.
method First-kind boundary integral equations, generalized Garding inequalities, Fredholm operators, finite dimensional kernels, Betti numbers.
result Finite dimensional kernels equal to the sum of Betti numbers, explaining the bilinear forms.
The paper proves curvature estimates for specific hypersurfaces in hyperbolic space.
problem Proving curvature estimates for hypersurfaces with prescribed asymptotic boundary in hyperbolic space.
method Refined curvature estimates and solving for specific curvature functions.
result Existence of smooth complete hypersurfaces with controlled curvature.
This paper presents a simple, self-contained account of Garding's theory of hyperbolic polynomials, including a recent convexity result of Bauschke-Guler-Lewis-Sendov and an inequality of Gurvits. This account also contains new results, such as the existence of a real analytic arrangement of the eigenvalue functions. I…
Solves curvature problems on manifolds with negative curvature.
problem Prescribed curvature problems on closed manifolds with negative curvature.
method Investigates fully nonlinear prescribed curvature problems for modified Schouten tensor on closed Riemannian manifolds with negative curvature.
result Proves solvability of curvature problems under certain conditions.
Groupoids provide a more appropriate framework for differential geometry than principal bundles. Synthetic differential geometry is the avant-garde branch of differential geometry, in which nilpotent infinitesimals are available in abundance. The principal objective in this paper is to show within our favorite framewor…
With a view towards applications in the theory of infinite-dimensional representations of finite-dimensional Lie supergroups, we introduce a new category of supermanifolds. In this category, supermanifolds of `maps' and `fields' (fibre bundle sections) exist. In particular, loop supergroups can be realised globally in …
Let Qcn+1 be the complete simply-connected (n+1)-dimensional space form of curvature c. In this paper we obtain a new characterization of geodesic spheres in Qcn+1 in terms of the higher order mean curvatures. In particular, we prove that the geodesic sphere is the only complete bounded …
Develops global pseudo-differential calculus on homogeneous vector bundles.
problem Global theory of subelliptic pseudo-differential operators on homogeneous vector bundles.
method Global symbolic calculus, complex functional calculus, Hörmander system of vector-fields.
result Global pseudo-differential calculus on homogeneous vector bundles.
The paper solves pseudo-convexity for special Lagrangian equations, with applications in mirror symmetry.
problem Existence of solutions to the Special Lagrangian Potential Equation.
method Explicitly computing boundary conditions for pseudo-convexity in potential theory.
result Interesting pseudo-convexity results for various related equations, including deformed Hermitian-Yang-Mills and curvature equations.
Estimates for polynomial operators using determinant majorization and subharmonics.
problem Bounding solutions of polynomial operators on Euclidean domains.
method Combines Alexandrov estimate and determinant majorization, using subharmonics and semiconvex approximation.
result Includes classical Alexandrov-Bakelman-Pucci estimate for linear operators.
The paper proves Liouville rigidity for Hessian equations, characterizing geometric conditions for constant solutions.
problem Characterizing geometric conditions for constant solutions in Hessian equations.
method Recursive geometric condition (Liouville admissibility) and anisotropic constructions.
result The Liouville-type property is characterized as a geometric property of the admissible set.
New asymptotics found for multivariate sequences with a pole.
problem Discontinuous drop in exponential growth rate for multivariate generating functions.
method Homology computations on algebraic varieties, topological explanations, symbolic algebraic computation.
result Explicit determination of dominant asymptotics for multivariate generating functions.
Solves Gursky-Streets equations for σk Yamabe problem in dimensions n≥2k.
problem Solving the σk Yamabe problem in dimensions n≥2k. method Introduced and solved the Gursky-Streets equations with uniform C1,1 estimates using concavity of the operator and Garding's theory of hyperbolic polynomials. result Established the uniqueness of the solution to the degenerate equations for the first time.
The paper proves existence of solutions to a Loewner-Nirenberg problem on Riemannian manifolds.
problem Existence of solutions to a specific nonlinear problem on Riemannian manifolds.
method Proves existence of viscosity solutions using approximating cones and limit of smooth solutions.
result Existence of a Lipschitz viscosity solution to the Loewner-Nirenberg problem.
Proves well-posedness of mixed Robin boundary value problem on manifolds with bounded geometry.
problem Well-posedness of mixed Robin boundary value problem on manifolds with boundary and bounded geometry.
method Analyzes operators on sections of a vector bundle with bounded geometry, proving regularity and well-posedness in Sobolev spaces.
result Main result is well-posedness in Sobolev spaces Hs(M;E) for s≥0 on non-compact manifolds.