A locally-built, LLM-digested index of recent arXiv papers in quant finance, geometry/topology, and statistical ML — keyword search served straight from SQLite on this machine.
Solves Jang equation for hyperboloidal data, proving positive mass theorem.
Study examines solutions to Jang equation on anti-de Sitter spacetimes.
Solves Jang's equation for hyperboloidal data in 4-7 dimensions, proving positive mass theorem.
We study Jang's equation on a one-parameter family of asymptotically flat, spherically symmetric Cauchy hypersurfaces in the maximally extended Schwarzschild spacetime. The hypersurfaces contain apparent horizons and are parametrized by their proximity to the singularity at . We show that on those hypersurfaces …
Paper analyzes blowup of regularized Jang solutions and constant expansion surfaces.
We prove some related results concerning blow-up solutions for the Jang equation. First: it has been shown that, given an outermost marginally outer trapped surface (MOTS) Σ, there exists a solution to Jang's equation which blows up at Σ. Here we show that in addition, large classes of spherically symmetric initial dat…
The generalized Jang equation was introduced in an attempt to prove the Penrose inequality in the setting of general initial data for the Einstein equations. In this paper we give an extensive study of this equation, proving existence, regularity, and blow-up results. In particular, precise asymptotics for the blow-up …
Proves positive mass theorem for hyperbolic manifolds with ends.
The aim of this paper is to collect some facts about the blowup of Jang's equation. First, we discuss how to construct solutions that blow up at an outermost MOTS. Second, we exclude the possibility that there are extra blowup surfaces in data sets with non-positive mean curvature. Then we investigate the rate of conve…
New proof shows spacetime energy is always positive in higher dimensions.
Let (M, g, k) be an initial data set for the Einstein equations of general relativity. We prove that there exist solutions of the Plateau problem for marginally outer trapped surfaces (MOTSs) that are stable in the sense of MOTSs. This answers a question of G. Galloway and N. O'Murchadha and is an ingredient in the pro…
We introduce a generalized version of the Jang equation, designed for the general case of the Penrose Inequality in the setting of an asymptotically flat space-like hypersurface of a spacetime satisfying the dominat energy condition. The appropriate existence and regularity results are established in the special case o…
In this paper we show how a natural coupling of the Dirac equation with the generalized Jang equation, leads to a proof of the rigidity statement in the positive mass theorem with charge, without the maximal slicing condition, provided a solution to the coupled system exists.
We extend the Jang equation proof of the positive energy theorem due to R. Schoen and S.-T. Yau from dimension to dimensions . This requires us to address several technical difficulties that are not present when . The regularity and decay assumptions for the initial data sets to which our argume…
Researchers prove a Penrose inequality for spacetime with specific conditions.
We solve the Plateau problem for marginally outer trapped surfaces in general Cauchy data sets. We employ the Perron method and tools from geometric measure theory to force and control a blow-up of Jang's equation. Substantial new geometric insights regarding the lower order properties of marginally outer trapped surfa…
In this paper, we show how to reduce the Penrose conjecture to the known Riemannian Penrose inequality case whenever certain geometrically motivated systems of equations can be solved. Whether or not these special systems of equations have general existence theories is therefore an important open problem. The key tool …
Reviewed and extended Wang-Yau quasi-local mass definitions.
We give general sufficient conditions for the existence of trapped surfaces due to concentration of matter in spherically symmetric initial data sets satisfying the dominant energy condition. These results are novel in that they apply and are meaningful for arbitrary spacelike slices, that is they do not require any au…
We show how to reduce the general formulation of the mass-angular momentum inequality, for axisymmetric initial data of the Einstein equations, to the known maximal case whenever a geometrically motivated system of equations admits a solution. This procedure is based on a certain deformation of the initial data which p…
For an admissible class of smooth compact initial data sets with boundary, we prove a comparison theorem between the Wang/Liu-Yau quasi-local mass of the boundary and the Hawking mass of strictly minimizing hulls in the Jang graphs of the domain. Using this, we prove a quasi-local Penrose inequality that involves these…
Hawking's theorem on the topology of black holes asserts that cross sections of the event horizon in 4-dimensional asymptotically flat stationary black hole spacetimes obeying the dominant energy condition are topologically 2-spheres. This conclusion extends to outer apparent horizons in spacetimes that are not necessa…
New proof shows equality in spacetime mass theorem.
Proves stability of spacetime Penrose inequality for spherical symmetric initial data.
New examples of knots with special bridge positions found.
We give a simple criterion for a Heegaard splitting to yield a Haken manifold. As a consequence, we construct many Haken manifolds, in particular homology spheres, with prescribed properties, namely Heegaard genus, Heegaard distance and Casson invariant. Along the way we give simpler and shorter proofs of the existence…
Generalizes quandle constructions and defines a multiplication that results in an abelian group.
We provide estimates on the Bartnik mass of constant mean curvature (CMC) surfaces which are diffeomorphic to spheres and have positive mean curvature. We prove that the Bartnik mass is bounded from above by the Hawking mass and a new notion we call the asphericity mass. The asphericity mass is defined by applying Hami…
Training of discrete latent variable models remains challenging because passing gradient information through discrete units is difficult. We propose a new class of smoothing transformations based on a mixture of two overlapping distributions, and show that the proposed transformation can be used for training binary lat…
Learning in models with discrete latent variables is challenging due to high variance gradient estimators. Generally, approaches have relied on control variates to reduce the variance of the REINFORCE estimator. Recent work (Jang et al. 2016, Maddison et al. 2016) has taken a different approach, introducing a continuou…
New method finds 198,846 toric-colorable seeds of Picard number 5.
Permutations and matchings are core building blocks in a variety of latent variable models, as they allow us to align, canonicalize, and sort data. Learning in such models is difficult, however, because exact marginalization over these combinatorial objects is intractable. In response, this paper introduces a collectio…
The definition of quasi-local mass for a bounded space-like region in space-time is essential in several major unsettled problems in general relativity. The quasi-local mass is expected to be a type of flux integral on the boundary two-surface and should be independent of whichever space-like region it bounds. An impor…
The paper explores new quandle systems for handlebody-links and spatial graphs.
Throughout the literature on the charged Riemannian Penrose inequality, it is generally assumed that there is no charged matter present; that is, the electric field is divergence-free. The aim of this article is to clarify when the charged Riemannian Penrose inequality holds in the presence of charged matter, and when …
Improved NiNo networks accelerate Adam training by up to 50%.
Study finds rigid properties of boundary-free hypersurfaces in specific data sets.
New bounds derived using conditional -information for machine learning models.
GL-LowPopArt improves minimax-optimal estimation for trace regression.
Study Galois groupoids of discret Painlevé equations.
Proves solvability of general inverse σ_k equations with constant coefficients.
We present an unsupervised approach for discovering semantic representations of mathematical equations. Equations are challenging to analyze because each is unique, or nearly unique. Our method, which we call equation embeddings, finds good representations of equations by using the representations of their surrounding …
Paper establishes estimates for nonlinear equations on compact manifolds.
Proves C^2,alpha estimates for elliptic equations on hyperkähler manifolds.
The paper generalizes Monge-Ampère equations and their solutions in differential geometry.
We study four distinct second-order nonlinear equations of Rabelo which describe pseudospherical surfaces. By transforming these equations to the constant-characteristic form we relate them to some well-studied integrable equations. Two of the Rabelo equations are found to be related to the sine-Gordon equation. The ot…
Introduces a new PDE involving differential forms for Kähler geometry.
Sharp sub-Gaussian bounds for subsolutions of Trudinger's equation on Riemannian manifolds.