Kahler-Ricci flow long-time behavior and initial data
problem Relationship between Kahler-Ricci flow and initial data
method Investigate long-time behavior
result Asymptotic profiles and non-trivial breathers
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arXiv research
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48 results for “non-trivial initial data”
The Degasperis-Procesi equation's solutions define pseudospherical metrics and can lead to surface collapse.
problem Understanding the breakdown of manifolds determined by Cauchy problems of the Degasperis-Procesi equation.
method Analyzing the pseudospherical nature of local and non-local formulations of the Degasperis-Procesi equation.
result Solutions to Cauchy problems with non-trivial initial data define an orthonormal coframe for pseudospherical metrics.
Exotic hyperbolic gluingsmath.DG
We carry out "exotic gluings" a la Carlotto-Schoen for asymptotically hyperbolic general relativistic initial data sets. In particular we obtain a direct construction of non-trivial initial data sets which are exactly hyperbolic in large regions extending to conformal infinity.
The dominant energy condition imposes a restriction on initial value pairs found on a spacelike hypersurface of a Lorentzian manifold. In this article, we study the space of initial values that satisfy this condition strictly. To this aim, we introduce an index difference for initial value pairs and compare it to its c…
We develop a gluing construction which adds scaled and truncated asymptotically Euclidean solutions of the Einstein constraint equations to compact solutions with potentially non-trivial cosmological constants. The result is a one-parameter family of initial data which has ordinary and scaled "point-particle" limits an…
We construct large families of initial data sets for the vacuum Einstein equations with positive cosmological constant which contain exactly Delaunay ends; these are non-trivial initial data sets which coincide with those for the Kottler-Schwarzschild-de Sitter metrics in regions of infinite extent. From the purely Rie…
In this note, we give a geometric characterization of the compact and totally umbilical hypersurfaces that carry a non trivial locally static Killing Initial Data (KID). More precisely, such compact hypersurfaces have constant mean curvature and are isometric to one of the following manifolds: (i) Sn the standard spher…
Study spin-0 fields on n-dimensional Minkowski spacetimes, computing asymptotic charges.
problem Analyzing spin-0 fields on Minkowski spacetimes near infinity.
method Conformal geometry and Friedrich's cylinder at spatial infinity.
result Found infinitely many well-defined asymptotic charges in even dimensions, no charges in odd dimensions.
The paper studies parallel spinor flows on 3D Cauchy hypersurfaces and provides initial data characterizations.
problem Characterizing parallel spinors on Ricci flat Lorentzian four-manifolds.
method Evolution flow defined by parallel spinors, proving preservation of constraints, solving left-invariant flows.
result Initial data characterization of parallel spinors on Ricci flat Lorentzian four-manifolds.
Study shows Brakke flow's non-triviality for smooth boundaries in codimension 1.
problem Understanding Brakke flow's non-triviality for smooth boundaries in codimension 1.
method Analyzing spacetime Brakke flow constructed by Buet et al. for initial varifolds.
result Support of mass measure of spacetime Brakke flow coincides with classical mean curvature flow's support.
We investigate the prescribed Q-curvature flow for GJMS operators with non-trivial kernel on compact manifolds of even dimension. When the total Q-curvature is negative, we identify a conformally invariant condition on the nodal domains of functions in the kernel of the GJMS operator, allowing us to prove the global ex…
This note corrects a mistake in the original book in the evolution equations of total curvature for the curve-shrinking flow in an ambient Ricci Flow. The resulting upper bound for the evolution of total curvature is an exponential bound in time. The change involves the multiplicative constant. Here we show that it dep…
A new algorithm for K-means clustering in evolving data streams.
problem Clustering of continuously arriving data in streaming scenarios with concept drift.
method Formal definition of Streaming -means, surrogate error function, algorithm for minimizing surrogate error.
result The surrogate error function effectively approximates the Streaming -means error.
Untrained neural networks can unfairly assign predictions to the same class.
problem Biasing effects in neural networks during initial training phases.
method Theoretical analysis of deep neural networks, focusing on Initial Guessing Bias (IGB).
result Model structure and preprocessing methods influence IGB.
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…
New metric tensor field on symmetric matrices simplifies eigenvector computation.
problem Complex eigenvector computation for 2x2 symmetric matrices.
method Introducing a metric tensor field on the space of symmetric matrices, resulting in a curved manifold.
result Parallel transport simplifies eigenvector computation for one-parameter families of matrices.
We reduce the embedding problem for hypo SU(2) and SU(3)-structures to the embedding problem for hypo G2-structures into parallel Spin(7)-manifolds. The latter will be described in terms of gauge deformations. This description involves the intrinsic torsion of the initial G2-structure and allows us to prove that the ev…
FibeRed reduces complex data dimensions while preserving topology.
problem Hard embedding of topologically complex datasets in low-dimensional Euclidean space.
method Modeling datasets with vector bundles, reducing fibers while preserving topology.
result FibeRed learns topologically faithful embeddings in lower dimensions than existing methods.
This paper classifies Kähler manifolds with specific Einstein-type properties.
problem Classifying gradient Einstein-type Kähler manifolds with .
method Unified framework of Einstein-type manifolds, focusing on classification with .
result Complete classification of non-trivial, complete gradient Einstein-type Kähler manifolds with .
Study confirms asymptotic behavior of logarithmic balanced metric near infinity.
problem Asymptotic behavior of logarithmic balanced metric near infinity.
method Non-trivial refinement of tools from previous work.
result Partial confirmation of conjecture on asymptotic behavior.
Study uses random matrix theory to improve tensor approximation accuracy.
problem Improving tensor approximation accuracy in the presence of noise.
method Random matrix theory applied to tensor unfoldings.
result Characterizes spectral behavior of tensor unfoldings and predicts reconstruction performance.
Near-Optimal differentially private low-rank trace regression with guaranteed private initializationstat.ML
Paper develops DP methods for low-rank matrix estimation with near-optimal performance.
problem Estimating a low-rank matrix under differential privacy constraints.
method Introduced computationally efficient DP-initialization and Riemannian optimization-based DP-RGrad algorithm.
result DP-RGrad achieves near-optimal convergence rate under weak differential privacy constraints.
Third in a series, this paper constructs non-trivial Cayley fibrations with conical singularities.
problem Constructing non-trivial Cayley fibrations with conical singularities.
method Using gluing methods and stability results for weak and usual fibrations.
result Construction of examples of Cayley fibrations on twisted connected sum manifolds.
How Many Links Fit in a Box?math.GT
New upper bound for non-trivial links in a cube, geometric approach.
problem Upper limit for disjoint non-trivial links in a unit cube.
method Geometric argument, no multi-linear properties used.
result Proves an upper bound for all non-trivial links.
Sharp analysis of power iteration for tensor PCA, improving convergence and stopping criteria.
problem Analyzing the power iteration algorithm for tensor PCA to improve convergence and stopping criteria.
method Sharp bounds on the number of iterations, revealing a smaller algorithmic threshold, proposing a stopping criterion.
result Sharp bounds on the number of iterations required for power method to converge, revealing a smaller algorithmic threshold than previously conjectured.
The appearance of non trivial torsion for some Ricci dependent theories in the Palatini formalismgr-qc
The study reveals non-trivial torsion in certain gravity theories with Ricci-dependent Lagrangians.
problem Exploring the appearance of non-trivial torsion in gravity theories with Ricci-dependent Lagrangians.
method Investigates theories with Lagrangians of the form L(L(R_{μν})), analyzing the equations of motion and the role of torsion.
result Theories with Ricci-dependent Lagrangians admit non-zero torsion even in simple space-time dimensions, suggesting its importance.
Study on learning rates in neural networks of varying depth.
problem Dependence of maximal update learning rate on network depth.
method Analysis of random fully connected ReLU networks with mean-field weight initialization.
result Maximal update learning rate scales like with network depth.
Study how untrained policies explore in RL environments.
problem Challenges in reinforcement learning, especially sparse or adversarial reward structures.
method Theoretical and empirical analysis of untrained deep neural policies in a toy model.
result Untrained policies generate correlated actions and non-trivial state-visitation distributions.
Study of Dirac-Witten operator on Lorentzian manifolds under dominant energy condition.
problem Detecting non-trivial homotopy groups in spaces of initial data under strict dominant energy condition.
method Use index theory and Lorentzian Hitchin's α-invariant to analyze Dirac-Witten operator.
result Kernel of Dirac-Witten operator is non-trivial only if fundamental group is virtually solvable of derived length at most 2.
We will develop various methods, some are of geometric nature and some are of algebraic nature, to detect the various achiralities of knots and links in . For example, we show that the twisted Whitehead double of a knot is achiral if and only if the double is the unknot or the figure eight knot, and we show that a…
A knot in the 3-sphere is called doubly slice if it is a slice of an unknotted 2-sphere in the 4-sphere. We give a bi-sequence of new obstructions for a knot being doubly slice. We construct it following the idea of Cochran-Orr-Teichner's filtration of the classical knot concordance group. This yields a bi-filtration o…
New neural method for inferring Markov jump processes.
problem Inference in Markov jump processes is challenging.
method Variational inference using neural ODEs and backpropagation.
result Trains neural representations of data to approximate process rates.
We prove the precise scaling, at finite depth and width, for the mean and variance of the neural tangent kernel (NTK) in a randomly initialized ReLU network. The standard deviation is exponential in the ratio of network depth to width. Thus, even in the limit of infinite overparameterization, the NTK is not determinist…
CDS combines PT and diffusion for efficient sampling from multimodal distributions.
problem Sampling from unnormalized multimodal distributions efficiently.
method Conditional Diffusion Sampling (CDS) using Conditional Interpolants and Parallel Tempering.
result CDS achieves a superior trade-off between sample quality and density evaluation cost.
Online SGD achieves consistent estimation in high-dimensional non-convex inference tasks.
problem Consistent estimation in high-dimensional non-convex optimization problems.
method Online stochastic gradient descent (SGD) on non-convex losses.
result Nearly sharp thresholds for sample complexity in high-dimensional settings.
Study on neural network initialization with shaped infinite depth-and-width networks.
problem Understanding the distribution of random covariance matrices in shaped infinite-depth-and-width networks.
method Introduced the Neural Covariance SDE to model the distribution of the random covariance matrix.
result Identified the precise scaling of the activation function necessary for a non-trivial limit.
Paper refutes conjecture on tensor power iteration convergence in overcomplete models.
problem Understanding convergence of tensor power iteration in overcomplete random tensors.
method Analysis of tensor power iteration dynamics from random initialization.
result Polynomially many steps are necessary for convergence, refutes logarithmic conjecture.
In this paper, we consider the heat flow for Yang-Mills connections on . In the equivariant setting, the Yang-Mills heat equation reduces to a single semilinear reaction-diffusion equation for which an explicit self-similar blowup solution was found by Weinkove \cite{Wei04}. We prove …
Study active learning of PTFs with derivative access.
problem Active learning of polynomial threshold functions (PTFs).
method Algorithm for active learning degree- univariate PTFs with derivative access.
result Computational efficient algorithm for active learning degree- univariate PTFs.
-wave initial datagr-qc
Initial data for -wave spacetimes constructed in 4D.
problem Characterizing initial data for -wave spacetimes.
method Constructs a vacuum initial data set with extra conditions related to CKID.
result Data development is a subset of a vacuum -wave.
We show that wave maps from two-dimensional Minkowski space to hyperbolic spaces $\H^m$ are globally smooth in time if the initial data is smooth, conditionally on some reasonable claims concerning the local theory of such wave maps, as well as the self-similar and travelling (or stationary solutions); w…
In this paper we introduce a new invariant of virtual knots and links that is non-trivial for infinitely many virtuals, but is trivial on classical knots and links. The invariant is initially be expressed in terms of a relative of the bracket polynomial and then extracted from this polynomial in terms of its exponents,…
We solve the equivalence problem for the orthogonally separable webs on the three-sphere under the action of the isometry group. This continues a classical project initiated by Olevsky in which he solved the corresponding canonical forms problem. The solution to the equivalence problem together with the results by Olev…
We study the Ricci flow for initial metrics with positive isotropic curvature (strictly PIC for short). In the first part of this paper, we prove new curvature pinching estimates which ensure that blow-up limits are uniformly PIC in all dimensions. Moreover, in dimension , we show that blow-up limits are wea…
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…
Surveying methods to create spaces with non-trivial self covers.
problem Creating spaces with non-trivial self covers.
method Surveying techniques for constructing spaces, including low and high dimension continua.
result Discussion of open problems related to self covering phenomena.
Study shows non-trivial knot Floer homology for specific knots.
problem Proving non-triviality of knot Floer homology for certain knots.
method Analyzing Alexander grading and proving non-triviality in specific knots.
result Non-trivial knot Floer homology for certain knots, answering a question posed by Baldwin and Vela-Vick.
Paper proves rigidity of initial data sets with boundary and capillary MOTS.
problem Rigidity of initial data sets with boundary and capillary MOTS.
method Estimates area of MOTS, proves rigidity for 3D, extends to high dimensions using Yamabe constant.
result Rigidity results for initial data sets with boundary and capillary MOTS.