arXiv research

A guide for solving first-order elliptic boundary value problems.

Study calculates tail risk for various mixture distributions.

Proves solutions to elliptic equations on Hermitian manifolds with optimal conditions.

We study boundary value problems for first-order elliptic differential operators on manifolds with compact boundary. The adapted boundary operator need not be selfadjoint and the boundary condition need not be pseudo-local. We show the equivalence of various characterisations of elliptic boundary conditions and demonst…

Constructs Lie groupoid integrating elliptic tangent bundles and Poisson structures.

The paper finds sign-changing solutions for a specific type of elliptic equation.

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…

Rigidity of elliptic genera proven for non-spin manifolds with $S^1$-action.

Study shows conditions for rational ellipticity of manifolds with symmetries.

Study finite-type solutions of elliptic sinh-Gordon equation with Durham boundary conditions.

In this paper, we construct for the first time the projective elliptic genera for a compact oriented manifold equipped with a projective complex vector bundle. Such projective elliptic genera are rational q-series that have topological definition and also have analytic interpretation via the fractional index theorem in…

The paper calculates moments and conditional risks for skewed elliptical distributions.

In this short note we provide an analytical formula for the conditional covariance matrices of the elliptically distributed random vectors, when the conditioning is based on the values of any linear combination of the marginal random variables. We show that one could introduce the univariate invariant depending solely …

Let ${\bf X}$ and ${\bf X}$ be two $n$-dimensional elliptical random vectors, we establish an identity for $E[f({\bf Y})]-E[f({\bf X})]$, where $f: \Bbb{R}^n \rightarrow \Bbb{R}$ fulfilling some regularity conditions. Using this identity we provide a unified derivation of sufficient and necessary conditions for classif…

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…

Rational ellipticity proven for $G$-manifolds with specific quotient properties.

The paper studies elliptic operators on manifolds with boundary.

This study explores the index theory of Heisenberg elliptic and transversally Heisenberg elliptic operators using $KK$-theory.

Solves open problems for fully nonlinear elliptic equations on manifolds.

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.

New calculus solves boundary value problems for elliptic operators.

Special Liouville metrics with Ricci-like conditions are determined by elliptic functions.

Uniform K-homology theory applied to elliptic operators on manifolds with boundary.

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.

Study constructs transverse metrics using transformations commuting with elliptic operators.

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 $\partial [D]$ = 0 in $K_1(\partial M)$, where $[D]$ is the relative $K$-cycle in $K_…

Study of elliptic boundary value problems on non-compact manifolds.

We define abstract Sobolev type spaces on $\mathsf{L}^p$-scales, $p\in [1,\infty)$, on Hermitian vector bundles over possibly noncompact manifolds, which are induced by smooth measures and families $\mathfrak{P}$ 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.

In this paper we consider the complex vector spaces of holomorphic cross-sections of homogeneous holomorphic vector bundles over elliptic adjoint orbits, and provide a sufficient condition for the vector spaces to be finite dimensional in view of root systems.

We develop a general method of proving the ellipticity of boundary value problems for the stationary vacuum space time, by showing that the stationary vacuum field equations are elliptic subjected to a geometrically natural collection of boundary conditions in the projection formalism. Using this we prove the manifold …

Elliptical processes generalize Gaussian and Student-t models with fat tails and computational efficiency.

Paper shows string cobordism at 24 dims can be determined by elliptic genus.

Study fully nonlinear elliptic equations on complex manifolds.

Proves approximation and interpolation for regular immersions directed by algebraically elliptic cones.

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. …

We study a class of fully nonlinear elliptic equations on closed Hermitian manifolds. Under the assumption of cone condition, we derive the $L^\infty$ estimate directly.

Bayesian PINNs learn elliptic PDEs with near-minimax posterior contraction rate.

Uniqueness found for elliptic equations with drift on manifolds.

Extends a theorem for first-order elliptic operators on manifolds.

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 $M\times \mathbb R_+$, where $M$ is a complete noncompact Riemannian manifold. Under specific assumptions, we establish existence of solutions satisfying prescribed conditions at infinity, depending on the…

Researchers find a method to construct projective structures on a specific surface.

Estimates gaps between eigenvalues for elliptic operators on manifolds.

Constructs a new type of metric for elliptic surfaces.

We show that one can define a spectral curve for the Cauchy-Riemann operator on a punctured elliptic curve if one imposes appropriate boundary conditions. Algebraic curves of the type thus obtained appear as irreducible components of spectral curves of minimal tori with planar ends in R^3. It appears that these curves …