Study boundary value problems for elliptic operators on manifolds.
problem Characterize and analyze boundary conditions for first-order elliptic differential operators.
method Develops a new framework for elliptic boundary conditions, proving equivalence and regularity of solutions.
result Elliptic boundary conditions yield a Fredholm operator on compact manifolds.
Formula for conditional covariance matrices simplified for elliptical distributions.
problem Analytical formula for conditional covariance matrices in elliptical distributions.
method Analytical formula for conditional covariance matrices of elliptically distributed random vectors based on linear combinations of marginal variables.
result Simplified formula for conditional covariance matrices, introducing univariate invariant.
A guide for solving first-order elliptic boundary value problems.
problem Solving first-order elliptic boundary value problems on manifolds.
method Operator methods and general elliptic boundary conditions.
result Characterization of a new subclass of elliptic boundary conditions.
Unified derivation of stochastic order conditions for elliptical distributions.
problem Classifying multivariate elliptical distributions based on stochastic orders.
method Established an identity for comparing expectations of functions of elliptical vectors and used it to derive conditions for stochastic orders.
result Unified derivation of conditions for various stochastic orders in multivariate elliptical distributions.
Study calculates tail risk for various mixture distributions.
problem Estimating tail risk for complex distribution mixtures.
method Analyzes tail conditional expectation for location-scale mixtures of elliptical distributions.
result Developed methods for calculating tail risk in various distributions.
Proves solutions to elliptic equations on Hermitian manifolds with optimal conditions.
problem Solving elliptic equations on Hermitian manifolds with boundary conditions.
method Derives quantitative boundary estimates and proves existence of solutions.
result Proves existence of solutions under almost optimal structural conditions.
Constructs Lie groupoid integrating elliptic tangent bundles and Poisson structures.
problem Existence and integration of elliptic tangent bundles and Poisson structures.
method Explicit construction of Lie groupoids and local models for symplectic integration.
result Necessary and sufficient topological condition for integration of elliptic tangent bundles.
The paper finds sign-changing solutions for a specific type of elliptic equation.
problem Existence of sign-changing solutions for a Yamabe type equation.
method Investigates a critical elliptic equation with a Yamabe type operator on a compact manifold with boundary.
result Existence of sign-changing solutions assured under certain geometric conditions.
Rigidity of elliptic genera proven for non-spin manifolds with S1-action.
problem Rigidity of elliptic genera for non-spin manifolds with S1-action. method Analysis of universal covering spin condition and π2(M) for rigidity. result Rigidity of elliptic genera is proven for spin universal coverings but not for non-spin universal coverings.
In this article we introduce local gauge conditions under which many curvature tensors appearing in conformal geometry, such as the Weyl, Cotton, Bach, and Fefferman-Graham obstruction tensors, become elliptic operators. The gauge conditions amount to fixing an n-harmonic coordinate system and normalizing the determi…
Study shows conditions for rational ellipticity of manifolds with symmetries.
problem Conditions for rational ellipticity of manifolds with symmetries.
method Analyzes conditions on compact simply connected manifolds with G-actions. result Proves rational ellipticity of M/G if M satisfies certain conditions. Study finite-type solutions of elliptic sinh-Gordon equation with Durham boundary conditions.
problem Finite-type solutions of elliptic sinh-Gordon equation with Durham boundary conditions.
method Determine rationality criteria for Durham conditions and analyze spectral curve properties.
result Rationality criteria are sufficient for finite-type solutions with complementary boundary conditions.
First time projective elliptic genera constructed for oriented manifolds.
problem Defining and proving properties of projective elliptic genera.
method Topological construction and analytic interpretation via fractional index theorem.
result Modularity properties of projective elliptic genera proven.
The paper calculates moments and conditional risks for skewed elliptical distributions.
problem Estimating moments and tail conditional risks for skewed elliptical distributions.
method Derives explicit expressions for multivariate doubly truncated moments and conditional risks for generalized skew-elliptical distributions.
result Explicit formulas for multivariate doubly truncated moments and conditional risks are derived for various skewed elliptical distributions.
Paper finds conditions for finite-dimensional vector spaces of cross-sections.
problem Finite-dimensional vector spaces of cross-sections for holomorphic bundles.
method Analyzes complex vector spaces of holomorphic cross-sections over elliptic orbits.
result Provides a sufficient condition for vector spaces to be finite dimensional.
We show that an Osserman-type inequality holds for spacelike surfaces of constant mean curvature (CMC) 1 with singularities and with elliptic ends in de Sitter 3-space. An immersed end of a CMC 1 surface is an ``elliptic end'' if the monodromy representation at the end is diagonalizable with eigenvalues in the unit cir…
The paper finds inequalities for eigenvalues of fourth order elliptic operators on Riemannian manifolds.
problem Eigenvalue inequalities for fourth order elliptic operators on Riemannian manifolds.
method Analyzes eigenvalues of fourth order elliptic operators in divergence form with Dirichlet boundary conditions on bounded domains in compact Riemannian manifolds.
result General inequalities for eigenvalues are derived.
The paper studies elliptic operators on manifolds with boundary.
problem Characterizing boundary conditions for elliptic operators on manifolds.
method Using Calderón projectors and mixed order Sobolev spaces, the paper describes the space of boundary values and characterizes Fredholm and regular realisations.
result Characterization of boundary conditions for elliptic operators leading to Fredholm and regular realisations.
Rational ellipticity proven for G-manifolds with specific quotient properties.
problem Conditions for rational ellipticity in G-manifolds. method Proving rational ellipticity based on quotient properties and manifold submetries.
result Compact, simply connected G-manifolds with certain quotient properties are rationally elliptic. Method proves ellipticity of vacuum spacetime boundary problems.
problem Proving ellipticity of boundary value problems for stationary vacuum spacetimes.
method Developed a general method using the projection formalism.
result Proved manifold theorem for moduli space of stationary vacuum spacetimes.
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.
This study explores the index theory of Heisenberg elliptic and transversally Heisenberg elliptic operators using KK-theory.
problem Analyzing the index theory of Heisenberg elliptic and transversally Heisenberg elliptic operators.
method Applying Kasparov's methodology and examining specific conditions using Fourier transform of the nilpotent group C∗-algebra. result Demonstrated enhanced methods for analyzing hypoellipticity and defined transversal Heisenberg ellipticity in a KK-theoretic context. A version of the nonlinear Hodge equations is introduced in which the irrotationality condition is weakened. An elliptic estimate for solutions is derived.
Abstracts a construction of boundary triplets for self-adjoint elliptic problems.
problem Computing the index of families of self-adjoint elliptic boundary problems.
method Abstract axiomatic version of boundary triplets and their applications.
result Analytic proof of index theorem and computation of index differences.
New calculus solves boundary value problems for elliptic operators.
problem Boundary value problems for 0-elliptic operators.
method Developed a new calculus called symbolic 0-calculus to handle boundary value problems.
result Construct left and right parametrices for 0-elliptic operators with boundary conditions.
Uniform K-homology theory applied to elliptic operators on manifolds with boundary.
problem Developing a theory to study boundary conditions for elliptic operators on non-compact manifolds.
method Theory of relative uniform K-homology, developing a relative index map.
result Uniform K-homology classes of boundary conditions and their connection to the higher ρ-invariant.
Special Liouville metrics with Ricci-like conditions are determined by elliptic functions.
problem Characterizing Liouville metrics with Ricci-like conditions in complex space forms.
method Analyzing necessary conditions for induced metrics of parallel mean curvature surfaces and proving the existence of specific Liouville metrics.
result Explicit determination of special Liouville metrics with Ricci-like conditions by elliptic functions.
Study constructs transverse metrics using transformations commuting with elliptic operators.
problem Existence of transverse metrics in foliation theory.
method Applying the Average Method to construct a transverse metric.
result Pseudogroup of local transformations equicontinuous and quasi-analytic.
We prove a version of differential Harnack inequality for a family of sub-elliptic diffusions on Sasakian manifolds under certain curvature conditions.
We discuss the ζ−regularized determinant of elliptic boundary value problems on a line segment. Our framework is applicable for separated and non-separated boundary conditions.
In this paper, we discuss the following conjecture raised by Baum-Douglas: For any first-order elliptic differential operator D on smooth manifold M with boundary $\p M$, D possesses an elliptic boundary condition if and only if ∂[D] = 0 in K1(∂M), where [D] is the relative K-cycle in $K_…
Study of elliptic boundary value problems on non-compact manifolds.
problem Analyzing elliptic differential operators on manifolds with non-compact boundaries.
method Regularity theory and trace theorems for sections in the maximal domain under various assumptions.
result Systematic study of local and nonlocal boundary conditions, including the Atiyah-Patodi-Singer condition.
We define abstract Sobolev type spaces on Lp-scales, p∈[1,∞), on Hermitian vector bundles over possibly noncompact manifolds, which are induced by smooth measures and families P of linear partial differential operators, and we prove the density of the corresponding smooth Sobolev sect…
Paper proves linearity of solutions to degenerate elliptic equations in 3D.
problem Determining linearity of degree-one homogeneous solutions to degenerate elliptic equations in 3D.
method Analyzes degenerate ellipticity condition and uses geometric properties of geodesic arcs.
result Proves linearity of solutions under specific degenerate ellipticity condition.
Elliptic theory explains indicial weights for non-linear geometry problems.
problem Understanding indicial weights for elliptic operators on non-compact manifolds.
method Developed an elliptic theory for indicial weights, proving Fredholm conditions.
result An elliptic theory exists even when the weight is indicial.
Elliptical processes generalize Gaussian and Student-t models with fat tails and computational efficiency.
problem Need for models with fat tails and computational tractability.
method Represent elliptical distributions as continuous mixtures of Gaussian distributions, derive closed-form expressions for marginal and conditional distributions.
result Elliptical processes offer advantages in robust regression compared to Gaussian processes.
Paper shows string cobordism at 24 dims can be determined by elliptic genus.
problem Determining string cobordism at dimension 24.
method Using elliptic genus as an invariant.
result Shows string cobordism at 24 dims can be determined by elliptic genus.
Proves approximation and interpolation for regular immersions directed by algebraically elliptic cones.
problem Approximation and interpolation for regular immersions directed by algebraically elliptic cones.
method Uses homotopy-theoretic necessary and sufficient conditions for approximation and interpolation.
result Homotopy-theoretic conditions for approximation and interpolation are satisfied in many cases of interest.
The boundary-value problem for Laplace-type operators acting on smooth sections of a vector bundle over a compact Riemannian manifold with generalized local boundary conditions including both normal and tangential derivatives is studied. The condition of strong ellipticity of this boundary-value problem is formulated. …
Study fully nonlinear elliptic equations on complex manifolds.
problem Solving fully nonlinear elliptic equations on complex manifolds.
method Derive C2,α-estimate and prove existence theorems for solutions and Dirichlet problems. result Existence theorems for solutions on closed Hermitian manifolds with unbounded conditions.
Bayesian PINNs learn elliptic PDEs with near-minimax posterior contraction rate.
problem Learning elliptic PDEs with noisy data and non-homogeneous boundary conditions.
method Bayesian approach with a Hölder space prior on neural network weights.
result Posterior contracts at near-minimax rate without prior knowledge of solution smoothness.
We study a class of fully nonlinear elliptic equations on closed Hermitian manifolds. Under the assumption of cone condition, we derive the L∞ estimate directly.
Uniqueness found for elliptic equations with drift on manifolds.
problem Finding unique solutions to elliptic equations with drift on manifolds.
method Investigation in weighted Lebesgue spaces, focusing on conditions for uniqueness.
result Sharp conditions on drift term for uniqueness in polynomial volume growth manifolds.
Extends a theorem for first-order elliptic operators on manifolds.
problem Proving the relative index theorem for general first-order elliptic operators.
method Using boundary value problems and graphical decomposition of elliptically regular boundary conditions.
result Proves the relative index theorem for general first-order elliptic operators.
The Nahm pole boundary condition for certain gauge theory equations in four and five dimensions is defined by requiring that a solution should have a specified singularity along the boundary. In the present paper, we show that this boundary condition is elliptic and has regularity properties analogous to more standard …
We investigate existence and uniqueness of bounded solutions of parabolic equations with unbounded coefficients in M×R+, where M is a complete noncompact Riemannian manifold. Under specific assumptions, we establish existence of solutions satisfying prescribed conditions at infinity, depending on the…
Estimates gaps between eigenvalues for elliptic operators on manifolds.
problem Estimating the gaps between consecutive eigenvalues for elliptic differential operators.
method Analyzes a class of second-order elliptic differential operators in divergence form with Dirichlet boundary conditions.
result Estimates for the upper bound of gaps between eigenvalues, with results matching known best estimates for specific cases.
Researchers find a method to construct projective structures on a specific surface.
problem Constructing projective structures with given holonomy and tameness conditions.
method Grafting circular triangles determined by a natural framing of the representation.
result All structures satisfying the conditions can be obtained through this method.