We study the Cauchy data spaces of the strongly Callias-type operators using maximal domain on manifolds with non-compact boundary, with the aim of understanding the Atiyah-Patodi-Singer index and elliptic boundary value problems.
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48 results for “Cauchy data spaces”
Continuous family of elliptic operators' projections maintain Cauchy data spaces.
problem Maintaining Cauchy data spaces for a continuous family of elliptic operators.
method Elementary tools and classical results applied to operator graphs, Sobolev spaces, and Green's formula.
result Orthogonalized Calderón projections form a continuous family of projections.
A classical problem in general relativity is the Cauchy problem for the linearised Einstein equation (the initial value problem for gravitational waves) on a globally hyperbolic vacuum spacetime. A well-known result is that it is uniquely solvable up to gauge solutions, given initial data on a spacelike Cauchy hypersur…
Study the metric geometry of Cauchy hypersurfaces in spacetimes.
problem Properties of the space of Cauchy hypersurfaces.
method Equipped with a Hausdorff-type metric, studied completeness and local compactness.
result Generalized completeness results for spacetimes.
New method for flux quantization on phase space stacks.
problem Defining and constructing flux-quantized phase space stacks.
method Observation of flux densities and characterization of Cauchy data.
result Flux-quantized phase space stacks have classifying spaces with rational Whitehead L-infinity algebra.
The zoology of singularities for Lorentzian manifold is slightly more complicated than for Riemannian manifolds. Our present work study Cauchy-compact globally hyperbolic singular flat spacetimes with extreme BTZ-like singular lines. We use the notion of BTZ-extension of a singular spacetime introduced in a previous pa…
We consider a connection on a complex line bundle over a Riemann surface with boundary , with connection 1-form . We show that the Cauchy data space of the connection Laplacian (also called magnetic Laplacian) , with a complex valued potential, uniquely determines the…
We establish a Cauchy type inequality for the geometric intersection number between two 1-dimensional submanifolds in a surface. Some of the basic results in Thurston's theory of measured laminations on surfaces are derived from the Cauchy inequality.
Geometric approach to Dirac operator evolution on spacetimes.
problem Constructing the Cauchy evolution operator for Lorentzian Dirac operators.
method Realizing the operator as a sum of oscillatory integrals, relating to Feynman propagator.
result Relating Cauchy evolution operators to Feynman propagators and constructing Hadamard states.
The paper constructs Gromov-Hausdorff metrics for Lorentzian spaces and calculates dimensions.
problem Understanding the geometry and dimensions of Lorentzian spaces.
method Construction of Gromov-Hausdorff metrics, calculation of dimensions, and analysis of Lorentzian spaces.
result Dushnik-Miller dimension of Minkowski spaces is countably infinite.
We give a local integral formula, valid on general curved space-times, for the characteristic Cauchy problem for the Dirac equation with arbitrary spin using the method developed by Friedlander in his book "the wave equation on a curved spacetime" (1975). The results obtained by Penrose in the flat case in "Null hypers…
Introduces Cauchy-Schwarz divergence for domain adaptation.
problem Evaluating discrepancy between source and target domains in unsupervised domain adaptation.
method Introduces Cauchy-Schwarz divergence as a measure for evaluating discrepancy between marginal and conditional distributions.
result CS divergence offers a tighter generalization error bound than Kullback-Leibler divergence.
Geometric approach solves maximum likelihood for Cauchy-like distributions.
problem Estimating center and scatter robustly from heavy-tailed data.
method Geodesic convexity and symmetry spaces of noncompact type.
result Efficient numerical solution for robust estimates of location and spread.
We study the global theory of linear wave equations for sections of vector bundles over globally hyperbolic Lorentz manifolds. We introduce spaces of finite energy sections and show well-posedness of the Cauchy problem in those spaces. These spaces depend in general on the choice of a time function but it turns out tha…
We consider (flat) Cauchy-complete GH spacetimes, i.e., globally hyperbolic flat lorentzian manifolds admitting some Cauchy hypersurface on which the ambient lorentzian metric restricts as a complete riemannian metric. We define a family of such spacetimes - model spacetimes - including four subfamilies: translation sp…
Many physical systems are described by partial differential equations (PDEs). Determinism then requires the Cauchy problem to be well-posed. Even when the Cauchy problem is well-posed for generic Cauchy data, there may exist characteristic Cauchy data. Characteristics of PDEs play an important role both in Mathematics …
Defines metrics for Lorentzian spaces and explores maximal developments.
problem Defining metrics and maximal developments in Lorentzian spaces.
method Defining Lorentzian spaces and using functorial properties.
result Explicit non-spacetime example of maximal globally hyperbolic Lorentzian space.
The Cauchy problem for harmonic maps from Minkowski space with its standard flat metric to a certain non-constant curvature Lorentzian 2-metric is studied. The target manifold is distinguished by the fact that the Euler-Lagrange equation for the energy functional is Darboux integrable. The time evolution of the Cauchy …
On the Generalizations of the Cauchy-Schwarz-Bunyakovsky Inequality with Applications to Elasticitymath.FA
The paper generalizes the Cauchy-Schwarz-Bunyakovsky inequality and applies it to elasticity problems.
problem Applying the Cauchy-Schwarz-Bunyakovsky inequality to elasticity problems.
method Presentation of discrete and integral forms, n-dimensional generalizations, and strengthened CBS inequality.
result The strengthened CBS inequality is crucial for elasticity problems.
A multisymplectic setting for classical field theories subjected to non-holonomic constraints is presented. The infinite dimensional setting in the space of Cauchy data is also given.
First-order jet bundles can be put at the foundations of the modern geometric approach to nonlinear PDEs, since higher-order jet bundles can be seen as constrained iterated jet bundles. The definition of first-order jet bundles can be given in many equivalent ways - for instance, by means of Grassmann bundles. In this …
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.
Principal Component Analysis (PCA) has wide applications in machine learning, text mining and computer vision. Classical PCA based on a Gaussian noise model is fragile to noise of large magnitude. Laplace noise assumption based PCA methods cannot deal with dense noise effectively. In this paper, we propose Cauchy Princ…
Lorentz harmonic maps into the hyperbolic plane and spacelike surfaces in anti-de Sitter 3-spacemath.DG
Study Lorentz harmonic maps and spacelike surfaces in anti-de Sitter space.
problem Analyzing the relationship between Lorentz harmonic maps and spacelike surfaces.
method Using loop group techniques, develop DPW-type representations and solve Cauchy problems.
result Establish a correspondence between Lorentz harmonic maps and spacelike immersions, leading to families of surfaces of constant Gauss curvature.
The paper studies how certain surfaces evolve in space-time.
problem Preserving the space-like condition of non-compact hypersurfaces.
method Prescribed mean curvature flow in generalized Robertson-Walker spaces.
result The flow preserves space-like condition and exists for infinite time.
Solutions to the wave equation on de Sitter-Schwarzschild space with smooth initial data on a Cauchy surface are shown to decay exponentially to a constant at temporal infinity, with corresponding uniform decay on the appropriately compactified space.
Paper introduces robust kernel ridge regression using Cauchy loss for handling various noise types.
problem Developing robust regression methods for noisy data.
method Introduces kernel Cauchy ridge regressor (KCRR) using Cauchy loss function.
result Establishes almost minimax-optimal convergence rate for KCRR in terms of -risk.
We develop the first steps of a parabolic pluripotential theory in bounded strongly pseudo-convex domains of Cn. We study certain degenerate parabolic complex Monge-Amp{è}re equations, modelled on the K{ä}hler-Ricci flow evolving on complex algebraic varieties with Kawamata log-terminal singularities. Under natural ass…
On the Cauchy problem for the heat equation on Riemannian manifolds with conical singularitiesmath.AP
We study the existence and regularity of solutions to the Cauchy problem for the inhomogeneous heat equation on compact Riemannian manifolds with conical singularities. We introduce weighted Hölder and Sobolev spaces with discrete asymptotics and we prove existence and maximal regularity of solutions to the Cauchy prob…
Summary of a talk given at the International Seminar "Analysis of spectral invariants and related operator theory", Tokyo University of Science, Unga Campus, 5-6 Oct. 2009
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.
Properties of the Cauchy-Riemann-Fueter equation for maps between quaternionic manifolds are studied. Spaces of solutions in case of maps from a K3-surface to the cotangent bundle of a complex projective space are computed. A relationship between harmonic spinors of a generalized nonlinear Dirac operator and solutions …
On the Cauchy problem for the Fadaray tensor on globally hyperbolic manifolds with timelike boundarymath.AP
Study well-posedness of Faraday tensor problem on specific spacetime manifolds.
problem Well-posedness of the Cauchy problem for the Faraday tensor on globally hyperbolic manifolds with timelike boundary.
method Existence of Green operators for the operator and a suitable pre-symplectic structure on the space of solutions.
result Existence of Green operators and pre-symplectic structure for the operator .
Extending BTZ models to complete hyperbolic surfaces.
problem Extending BTZ models to complete hyperbolic surfaces.
method Proving a parametrization result for globally hyperbolic Cauchy-maximal and Cauchy-compact locally Minkowski manifolds with extreme BTZ.
result The tangent bundle of the Teichmüller space parametrizes globally hyperbolic Cauchy-maximal and Cauchy-compact locally Minkowski manifolds with extreme BTZ.
The Cauchy problem for the homogeneous (real and complex) Monge-Ampere equation (HRMA/HCMA) arises from the initial value problem for geodesics in the space of Kahler metrics. It is an ill-posed problem. We conjecture that, in its lifespan, the solution can be obtained by Toeplitz quantizing the Hamiltonian flow define…
New cosmological spacetimes without CMC Cauchy surfaces found.
problem Finding CMC Cauchy surfaces in cosmological spacetimes.
method Generalized Bartnik's construction to connected sums of three-manifolds.
result Cosmological spacetimes without CMC Cauchy surfaces for any compact three-manifolds.
The paper extends completeness notions to low-regularity spacetimes.
problem Defining completeness conditions for spacetimes with low-regularity metrics.
method Extending Beem's completeness notions to Lorentzian length spaces and proving relationships between them.
result Equivalence of completeness conditions for globally hyperbolic -spacetimes under certain conditions.
Geometric formulation of the Cauchy invariants for incompressible Euler flow in flat and curved spacesphysics.flu-dyn
Cauchy invariants are now viewed as a powerful tool for investigating the Lagrangian structure of three-dimensional (3D) ideal flow (Frisch & Zheligovsky, Commun. Math. Phys., vol. 326, 2014, pp. 499-505, Podvigina et al., J. Comput. Phys., vol. 306, 2016, pp. 320-342). Looking at such invariants with the modern tools …
This note summarizes results that were obtained by the author in his habilitation thesis (arXiv:1607.08792) concerning the development of a spectral theory for simply periodic, 2-dimensional, complex-valued solutions of the sinh-Gordon equation. Spectral data for such solutions are defined for periodic Cauchy data on a…
Symmetries of vacuum spacetimes with a compact Cauchy horizon of constant non-zero surface gravitymath.DG
We prove that any smooth vacuum spacetime containing a compact Cauchy horizon with surface gravity that can be normalised to a non-zero constant admits a Killing vector field. This proves a conjecture by Moncrief and Isenberg from 1983 under the assumption on the surface gravity and generalises previous results due to …
Proposes a new latent variable model for hyperspherical latent spaces.
problem Efficiently modeling heavy-tailed distributions in hyperspherical latent spaces.
method Introduces spherical Cauchy (spCauchy) latent variables and applies Möbius transformations.
result Shows spCauchy recovers vMF geometry in high-concentration limits and avoids complex evaluations.
We study the following problem: Given initial data on a compact Cauchy horizon, does there exist a unique solution to wave equations on the globally hyperbolic region? Our main results apply to any spacetime satisfying the null energy condition and containing a compact Cauchy horizon with surface gravity that can be no…
The study classifies compact Cauchy horizons in vacuum spacetimes.
problem Classifying compact Cauchy horizons in vacuum spacetimes.
method Complete classification theorem based on topology and null generators.
result Different cases of compact Cauchy horizons with specific manifolds and spacetime properties.
A large class of vacuum space-times is constructed in dimension 4+1 from hyperboloidal initial data sets which are not small perturbations of empty space data. These space-times are future geodesically complete, smooth up to their future null infinity, and extend as vacuum space-times through their Cauchy horizon. Dime…
New vacuum spacetimes without CMC Cauchy surfaces found.
problem Finding vacuum cosmological spacetimes without CMC Cauchy surfaces.
method Extended construction of [6] using spatial topologies .
result Obtained a large class of vacuum cosmological spacetimes.
Paper proves existence of Hadamard states for Maxwell equations.
problem Proving existence of Hadamard states for Maxwell equations on spacetime.
method Introducing Cauchy radiation gauge and new Hodge decomposition.
result Existence of Hadamard states for Maxwell equations proven.
Proves well-posedness of the Cauchy problem for the Dirac operator on non-compact spacetimes.
problem Proving well-posedness of the Cauchy problem for the Dirac operator on non-compact spacetimes.
method Analyzes globally hyperbolic manifolds with complete spacelike Cauchy hypersurfaces.
result Proves well-posedness of the Cauchy problem for the Dirac operator.
We initiate a series of works where we study the interior of dynamical rotating vacuum black holes without symmetry. In the present paper, we take up the problem starting from appropriate Cauchy data for the Einstein vacuum equations defined on a hypersurface already within the black hole interior, representing the exp…