Under the assumption that the X-ray transform over symmetric solenoidal 2-tensors is injective, we prove that smooth compact connected manifolds with strictly convex boundary, no conjugate points and a hyperbolic trapped set are locally marked boundary rigid.
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48 results for “local traps”
New framework tackles deep learning issues like local traps and miscalibration.
problem Local traps and miscalibration in deep neural networks.
method Sparse deep learning framework with prior annealing algorithms.
result Proposed method successfully addresses local traps and miscalibration.
The paper introduces canonical parameters for marginally trapped surfaces in Minkowski space.
problem Determining marginally trapped surfaces in Minkowski space.
method Introducing canonical parameters and proving existence and uniqueness theorems.
result Every marginally trapped surface is determined by three smooth functions.
New method constructs spacelike data leading to trapped surfaces.
problem Formation of trapped surfaces from spacelike initial data.
method Free data formalism and local existence result.
result Data can be extended to asymptotically flat Cauchy data.
The paper studies marginally trapped submanifolds in Lorentzian manifolds under null energy condition.
problem Understanding marginally trapped submanifolds in Lorentzian manifolds.
method Analyzes properties of marginally trapped submanifolds in a Lorentzian manifold satisfying the null energy condition.
result Marginally trapped submanifolds have locally volume-maximizing properties in certain null hypersurfaces.
Overview of marginally trapped surfaces in various spacetimes.
problem Understanding marginally trapped surfaces in different spacetimes.
method Differential geometric study of marginally trapped surfaces in Minkowski, de Sitter, anti-de Sitter, and Robertson-Walker spacetimes.
result General local descriptions and classifications of these surfaces.
Study examines how insurance affects households prone to proportional losses, especially those near poverty.
problem Impact of insurance on households susceptible to proportional losses, focusing on poverty traps.
method Modelled proportional capital losses with insurance, derived closed formulae and non-local differential equations.
result New formulae and methods to calculate trapping probability, constraints on parameters to prevent certainty of trapping.
We give a local characterization of codimension two submanifolds which are marginally trapped in Robertson-Walker spaces, in terms of an algebraic equation to be satisfied by the height function. We prove the existence of a large number of local solutions. We refine the description in the case of curves with null accel…
We give local, explicit representation formulas for n-dimensional spacelike submanifolds which are marginally trapped in the Minkowski space, the de Sitter and anti de Sitter spaces and the Lorentzian products of the sphere and the hyperbolic space by the real line.
Study on trapped submanifolds in spacetimes, proving rigidity and non-existence results.
problem Rigidity and non-existence of trapped submanifolds in spacetimes.
method Analysis of parabolic spacelike submanifolds with causal mean curvature in orthogonally splitted and globally hyperbolic spacetimes.
result Non-existence of local minima or maxima of certain functions on submanifolds, leading to rigidity results.
Study rigidity in Penrose's singularity theorem with weakly trapped surfaces.
problem Understand the global structure of spacetimes with weakly trapped surfaces.
method Show foliation of MOTS generating totally geodesic null hypersurfaces.
result Obtain local or global rigidity results based on assumptions.
Marginally outer trapped surfaces are widely considered as the best quasi-local replacements for event horizons of black holes in General Relativity. However, this equivalence is far from being proved, even in stationary and static situations. In this paper we study an important aspect of this equivalence, namely wheth…
Two critical questions about intergenerational outcomes are: one, whether significant barriers or traps exist between different social or economic strata; and two, the extent to which intergenerational outcomes do (or can be used to) affect individual investment and consumption decisions. We develop a model to explicit…
In a matter-filled spacetime, perhaps with positive cosmological constant, a stable marginally outer trapped 2-sphere must satisfy a certain area inequality. Namely, as discussed in the paper, its area must be bounded above by , where is a lower bound on a natural energy-momentum term. We then consider th…
Study shows no closed trapped submanifolds can be tangent to certain spacelike hypersurfaces.
problem Existence of closed trapped submanifolds in spacetime regions foliated by specific hypersurfaces.
method Introduced future convex spacelike/null hypersurfaces and proved no dimensional closed trapped submanifolds can be tangent to these hypersurfaces from their future side.
result Closed trapped submanifolds cannot be found in open spacetime regions foliated by future convex hypersurfaces.
Study analyzes household capital risk and poverty trapping, deriving a new function for capital deficit distribution.
problem Analyzing the risk of household capital falling into poverty.
method Introduced a new Gerber-Shiu function to model trapping time and capital deficit distribution.
result Derived a model for capital deficit distribution at trapping using GB distributions.
We investigate the geometric properties of marginally trapped surfaces (surfaces which have null mean curvature vector) in the spaces of oriented geodesics of Euclidean 3-space and hyperbolic 3-space, endowed with their canonical neutral Kaehler structures. We prove that every rank one surface in these four manifolds i…
Paper studies stochastic optimization methods with momentum, proving convergence and avoiding traps.
problem Optimizing non-convex functions with momentum.
method Unified analysis of stochastic gradient descent variants, including S-NAG and Adam.
result Convergence to critical points and avoidance of undesired critical points like local maxima or saddle points.
New positive mass theorems for ALH manifolds with toroidal ends.
problem Proving positive mass theorems for asymptotically locally hyperbolic manifolds.
method Utilizes properties of marginally outer trapped surfaces and a new technique involving μ-bubbles.
result Obtained new positive mass theorems for asymptotically locally hyperbolic manifolds without boundary.
Eight different refinements of trapped surfaces are proposed, of three basic types, each intended as potential stability conditions. Minimal trapped surfaces are strictly minimal with respect to the dual expansion vector. Outer trapped surfaces have positivity of a certain curvature, related to surface gravity. Increas…
Study of marginally trapped surfaces in a perturbed Schwarzschild spacetime.
problem Understanding marginally trapped surfaces in perturbed Schwarzschild spacetime.
method Developed a method to study spacelike surfaces in a double null coordinate system.
result For every incoming null hypersurface nearly spherically symmetric, there exists a unique embedded marginally trapped surface.
The study proves Strichartz and spectral projection theorems on specific types of curved surfaces.
problem Proving Strichartz and spectral projection theorems on curved surfaces.
method Using large negative curvature neighborhoods, the study proves theorems on asymptotically conic and Euclidean ends surfaces.
result The study proves theorems without loss of interval on specific types of curved surfaces.
In a recent paper, Eichmair, Galloway and Pollack have proved a Gannon-Lee-type singularity theorem based on the existence of marginally outer trapped surfaces (MOTS) on noncompact initial data sets for globally hyperbolic spacetimes. However, one might wonder whether the corresponding incomplete geodesics could still …
Study geodesics in conformally compact manifolds, showing smoothness and asymptotic behavior.
problem Analyzing geodesics in conformally compact manifolds with varying curvature.
method Examining asymptotic behavior and regularity of geodesics near boundary.
result Non-trapped geodesics extend to conformal infinity with regularity, endpoints smooth on initial conditions.
Generic singularities found in spacetimes with weakly trapped submanifolds.
problem Existence of singularities in spacetimes with weakly trapped submanifolds.
method Using Whitney topologies and singularity theorems, the study shows genericity of singularities in spacetimes with weakly trapped submanifolds.
result Generic singularities are found in spacetimes with weakly trapped submanifolds.
Constructs initial data for multiple black holes with specified ADM parameters.
problem Forming multiple black holes with specific ADM parameters.
method Smooth, asymptotically flat vacuum initial data with prescribed ADM energy, momentum, and angular momentum.
result Maximal development of data results in spacetimes containing multiple black holes.
Study improves curvature estimate for stable marginally outer trapped hypersurfaces with a free boundary.
problem Curvature estimate for stable marginally outer trapped hypersurfaces with a free boundary.
method Iteration argument based on uniform area bound.
result Improved curvature estimate for stable marginally outer trapped hypersurfaces.
We show that any vacuum initial data set containing a marginally outer trapped surface S and satisfying a "no KIDs" condition can be perturbed near S so that S becomes strictly outer trapped in the new vacuum initial data set. This, together with the results in [9], gives a precise sense in which generic initial data c…
Wariness and Poverty Trapsq-fin.CP
Wariness affects poverty traps and equilibrium diversity in economic models.
problem The impact of wariness on poverty traps and equilibrium diversity in economic models.
method An overlapping generations model to explore the effects of wariness on poverty traps and equilibrium diversity.
result Wariness can amplify or mitigate the likelihood of poverty traps and can lead to multiple intertemporal equilibria.
A marginally trapped surface in the four-dimensional Minkowski space is a spacelike surface whose mean curvature vector is lightlike at each point. In the present paper we find all marginally trapped surfaces with pointwise 1-type Gauss map. We prove that a marginally trapped surface is of pointwise 1-type Gauss map if…
This paper considers some fundamental questions concerning marginally trapped surfaces, or apparent horizons, in Cauchy data sets for the Einstein equation. An area estimate for outermost marginally trapped surfaces is proved. The proof makes use of an existence result for marginal surfaces, in the presence of barriers…
Paper proves trapped surface formation for EMCSF system without symmetry assumptions.
problem Formation of trapped surfaces for the Einstein--Maxwell--charged scalar field system.
method Scale-critical trapped surface formation result established from past null infinity.
result Focusing of gravitational waves, concentration of electromagnetic fields, or condensation of scalar fields can lead to trapped surface formation.
Investigates how FDI and R&D affect host countries' growth.
problem Host countries may fall into a middle-income trap if they focus solely on FDI.
method Optimal growth model with FDI and R&D.
result R&D investment is crucial for sustained growth in host countries.
No trapped surfaces can form under low-regularity bounds in certain spacetimes.
problem Existence of trapped surfaces in low regularity solutions to Einstein's equations.
method Analyzing the initial data in Besov norm and extending to smallness.
result No trapped surfaces can exist initially when the Cauchy data are close to Minkowski spacetime data.
The concept of closed trapped surface is of paramount importance in General Relativity and other gravitational theories. However, it is a purely geometrical object. With the aim of bringing this concept to closer attention by the mathematical community, I introduce the generalized idea of trapped submanifold by using t…
Paper proves trapped surface formation for Einstein-Maxwell-charged scalar field system.
problem Formation of trapped surfaces in Einstein-Maxwell-charged scalar field system.
method Generalized Christodoulou's approach for spherical symmetry and improved for Minkowskian data.
result Improved bound on trapped surface formation for Minkowskian data.
In the present paper we provide new examples of marginally trapped surfaces and tubes in FLRW spacetimes by using a basic relation between these objects and CMC surfaces in 3-manifolds. We also provide a new method to construct marginally trapped surfaces in closed FLRW spacetimes, which is based on the classical Hopf …
The expectation-maximization (EM) algorithm has been widely used in minimizing the negative log likelihood (also known as cross entropy) of mixture models. However, little is understood about the goodness of the fixed points it converges to. In this paper, we study the regions where one component is missing in two-comp…
Paper proves existence of anisotropic dynamical horizons in gravitational collapse.
problem Existence of apparent horizons in gravitational collapse.
method Scale-critical hyperbolic method and non-perturbative elliptic techniques.
result Smooth and spacelike apparent horizons emerge from general initial data in gravitational collapse.
We derive integral and sup-estimates for the curvature of stably marginally outer trapped surfaces in a sliced space-time. The estimates bound the shear of a marginally outer trapped surface in terms of the intrinsic and extrinsic curvature of a slice containing the surface. These estimates are well adapted to situatio…
New system studies trapped light paths in Euclidean space.
problem Trapping of light paths in Euclidean space with negative refractive index.
method Introduces wind-tree tiling billiards system to study trajectories of rays in Euclidean space with rectangular obstacles.
result Almost every configuration of the system traps trajectories with initial vertical direction in an infinite strip.
The paper explores rigid geometric structures near surfaces with equality in area-charge inequalities.
problem Geometric constraints near surfaces with equality in area-charge inequalities.
method Investigation of equality in area-charge inequalities for spherical minimal surfaces and MOTS within the Einstein-Maxwell equations framework.
result Equality in area-charge inequalities imposes rigid geometric structures, including normal electric and magnetic fields and isometric Riemannian products.
HiSS sampling overcomes local mode traps in rugged discrete spaces.
problem Sampling multimodal discrete distributions with gradient-based methods.
method Integrates Metropolis-within-Gibbs framework with logistic convolution.
result HiSS outperforms alternatives on various tasks, including Ising models and binary neural networks.
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…
The purpose of the present work is to study (marginally) trapped submanifolds lying in a null hypersurface. Let $(M,g,N)\to\Bm(c)$ be a null hypersurface of a space-time with constant sectional curvature , endowed with a Screen Integrable and Conformal rigging . The (Marginally) Trapped Submanifolds we are intere…
Reviewed and extended Wang-Yau quasi-local mass definitions.
problem Clarifying and extending definitions of quasi-local mass.
method Reviewed existing definitions and extended positivity proof.
result Extended positivity proof for Wang-Yau quasi-local energy.
We show that degenerate horizons exhibit a new trapping effect. Specifically, we obtain a non-degenerate Morawetz estimate for the wave equation in the domain of outer communications of extremal Reissner-Nordstrom up to and including the future event horizon. We show that such an estimate requires 1) a higher degree of…
Small deformations of marginally (outer) trapped surfaces are considered by using their stability operator. In the case of spherical symmetry, one can use these deformations on any marginally trapped round sphere to prove several interesting results. The concept of 'core' of a black hole is introduced: it is a minimal …