Search results for "singular"

Geometric Singular Perturbation Theory Beyond Normal Hyperbolicity

2001

Geometric Singular Perturbation theory has traditionally dealt only with perturbation problems near normally hyperbolic manifolds of singularities. In this paper we want to show how blow up techniques can permit enlarging the applicability to non-normally hyperbolic points. We will present the method on well chosen examples in the plane and in 3-space.

Multiple Canard Cycles in Generalized Liénard Equations

2001

AbstractThe paper treats multiple limit cycle bifurcations in singular perturbation problems of planar vector fields. The results deal with any number of parameters. Proofs are based on the techniques introduced in “Canard Cycles and Center Manifolds” (F. Dumortier and R. Roussarie, 1996, Mem. Amer. Math. Soc., 121). The presentation is limited to generalized Liénard equations εx+α(x, c)x+β(x, c)=0.

New solvability conditions for the Neumann problem for ordinary singular differential equations

2000

Proof I: Upper Bounds

2019

In this chapter we study upper bounds on singular values and determinants of certain operators related to Pδ. The bounds are not probabilistic; they only depend on a certain smallness of the perturbation.

Collision Orbits in the Isosceles Rectilinear Restricted Problem

1995

In the study of the Collinear Three-Body Problem, McGehee (1974) introduced a new set of coordinates which had the effect of blowing up the triple collision singularity. Subsequently, his method has been used to analyze some other collision or singularities. Recently, Wang (1986) introduced another transformation which differs from the McGehee’s coordinates in the fact that the blowing-up factor is now the potential function, U, instead of the moment of inertia, I. Meyer and Wang (1993) have applied this method to the Restricted Isosceles Three-body Problem with positive energy and Cors and Llibre (1994) to the hyperbolic restricted three-body problem. In this paper we study the singulariti…

Homeomorphic graph manifolds: A contribution to the μ constant problem

1999

Abstract We give a characterization, in terms of homological data in covering spaces, of those maps between (3-dimensional) graph manifolds which are homotopic to homeomorphisms. As an application we give a condition on a cobordism between graph manifolds that guarantees that they are homeomorphic. This in turn is applied to give a partial result on the μ -constant problem in (complex) dimension three.

Cyclicity of common slow–fast cycles

2011

Abstract We study the limit cycles of planar slow–fast vector fields, appearing near a given slow–fast cycle, formed by an arbitrary sequence of slow parts and fast parts, and where the slow parts can meet the fast parts in a nilpotent contact point of arbitrary order. Using the notion slow divergence integral, we delimit a large subclass of these slow–fast cycles out of which at most one limit cycle can perturb, and a smaller subclass out of which exactly one limit cycle will perturb. Though the focus lies on common slow–fast cycles, i.e. cycles with only attracting or only repelling slow parts, we present results that are valid for more general slow–fast cycles. We also provide examples o…

Locally Supported Wavelets on Manifolds with Applications to the 2D Sphere

1999

Abstract In this paper we present a construction principle for locally supported wavelets on manifolds once a multiresolution analysis is given. The wavelets provide a stable (or unconditional) basis for a scale of Sobolev spaces H s , 0 ≤ s ≤ s . We examine a fast wavelet transform with almost optimal complexity. For the two-dimensional sphere we construct a multiresolution analysis generated by continuous splines that are bilinear with respect to some special spherical grid. In our approach the poles are not exceptional points concerning the approximation power or the stability of the wavelet basis. Finally we present some numerical applications to singularity detection and the analysis o…

Maximal potentials, maximal singular integrals, and the spherical maximal function

2014

We introduce a notion of maximal potentials and we prove that they form bounded operators from L to the homogeneous Sobolev space Ẇ 1,p for all n/(n − 1) < p < n. We apply this result to the problem of boundedness of the spherical maximal operator in Sobolev spaces.

cuBool: Bit-Parallel Boolean Matrix Factorization on CUDA-Enabled Accelerators

2018

Boolean Matrix Factorization (BMF) is a commonly used technique in the field of unsupervised data analytics. The goal is to decompose a ground truth matrix C into a product of two matrices A and $B$ being either an exact or approximate rank k factorization of C. Both exact and approximate factorization are time-consuming tasks due to their combinatorial complexity. In this paper, we introduce a massively parallel implementation of BMF - namely cuBool - in order to significantly speed up factorization of huge Boolean matrices. Our approach is based on alternately adjusting rows and columns of A and B using thousands of lightweight CUDA threads. The massively parallel manipulation of entries …