An extension of the Rabinowitz bifurcation theorem to Lipschitz potential operators in Hilbert spaces

The main result of the paper is an extension of the bifurcation theorem of Rabinowitz to equations with continuous jointly in and of class . We also prove a bifurcation theorem for critical points of the function which is just continuous and changes at an isolated minimum (in ) to isolated maximum when passes, say, zero. The proofs of the theorems, as well as the the theorems themselves, are new, in certain important aspects, even when applied to smooth functions.

     

-perturbations of Hopf's bifurcation points and homoclinic tangencies

In this note we show that a diffeomorphism which has a Hopf's bifurcation point, can be perturbed around the bifurcation point in order to get a diffeomorphism which exhibits homoclinic tangencies. In the case this is not possible because of the typical unfolding of a Hopf's bifurcation point.

     

Local behaviour of polynomials

In this paper we study the local behaviour of a trigonometric polynomial around any of its zeros in terms of its estimated values at an adequate number of freely chosen points in . The freedom in the choice of sample points makes our results particularly convenient for numerical calculations. Analogous results for polynomials of the form are also proved.

     

The transversal homoclinic points are dense in the codimension-1 Hénon-like strange attractors

We consider the codimension-1 Hénon-like strange attractors . We prove that the transversal homoclinic points are dense in , and that hyperbolic periodic points are dense in . Moreover the hyperbolic periodic points that are heteroclinically related to the primary fixed point ( transversal intersection of stable and unstable manifolds) are dense in .

     

Local bifurcation from the second eigenvalue of the Laplacian in a square

In this work we study local bifurcation from the branch of trivial solutions for a class of semilinear elliptic equations, at the second eigenvalue of a square. We find that the bifurcation set can be locally described as the union of exactly four bifurcation branches of nontrivial solutions which cross the bifurcation point . We also compute the Morse index of the solutions in the four branches.

     

Bifurcations of the Hill's region in the three body problem

In the spatial three body problem, the topology of the integral manifolds (i.e. the level sets of energy and angular momentum , as well as center of mass and linear momentum) and the Hill's regions (the projection of the integral manifold onto position coordinates) depends only on the quantity It was established by Albouy and McCord-Meyer-Wang that, for and , there are exactly eight bifurcation values for at which the topology of the integral manifold changes. It was also shown that for each of these values, the topology of the Hill's region changes as well. In this work, it is shown that there are no other values of for which the topology of the Hill's region changes. That is, a bifurcation of the Hill's region occurs if and only if a bifurcation of the integral manifold occurs.

     

Steady-state bifurcation with Euclidean symmetry

We consider systems of partial differential equations equivariant under the Euclidean group and undergoing steady-state bifurcation (with nonzero critical wavenumber) from a fully symmetric equilibrium. A rigorous reduction procedure is presented that leads locally to an optimally small system of equations. In particular, when and and for reaction-diffusion equations with general , reduction leads to a single equation. (Our results are valid generically, with perturbations consisting of relatively bounded partial differential operators.)

In analogy with equivariant bifurcation theory for compact groups, we give a classification of the different types of reduced systems in terms of the absolutely irreducible unitary representations of . The representation theory of is driven by the irreducible representations of . For , this constitutes a mathematical statement of the `universality' of the Ginzburg-Landau equation on the line. (In recent work, we addressed the validity of this equation using related techniques.)

When , there are precisely two significantly different types of reduced equation: scalar and pseudoscalar, corresponding to the trivial and nontrivial one-dimensional representations of . There are infinitely many possibilities for each .

     

Curve-straightening and the Palais-Smale condition

This paper considers the negative gradient trajectories associated with the modified total squared curvature functional . The focus is on the limiting behavior as tends to zero from the positive side. It is shown that when spaces of curves exist in which some trajectories converge and others diverge. In one instance the collection of critical points splits into two subsets. As tends to zero the critical curves in the first subset tend to the critical points present when . Meanwhile, all the critical points in the second subset have lengths that tend to infinity. It is shown that this is the only way the Palais-Smale condition fails in the present context. The behavior of the second class of critical points supports the view that some of the trajectories are `dragged' all the way to `infinity'. When the curves are rescaled to have constant length the Euler figure eight emerges as a `critical point at infinity'. It is discovered that a reflectional symmetry need not be preserved along the trajectories. There are examples where the length of the curves along the same trajectory is not a monotone function of the flow-time. It is shown how to determine the elliptic modulus of the critical curves in all the standard cases. The modulus must satisfy when the space is limited to curves of fixed length and the endpoints are separated by the vector .

     

Computing The Differential of An Unfolded Contact Diffeomorphism

Consider a bifurcation problem, namely, its bifurcation equation. There is a diffeomorphism linking the actual solution set with an unfolded normal form of the bifurcation equation. The differential D(0) of this diffeomorphism is a valuable information for a numerical analysis of the imperfect bifurcation.The aim of this paper is to construct algorithms for a computation of D(0). Singularity classes containing bifurcation points with codim 3, corank = 1 are considered.     

Numerical peak points and numerical Šilov boundary for holomorphic functions

In this paper, we characterize the numerical and numerical strong-peak points for when is the complex space or . We also prove that for all is the numerical Šilov boundary for

     

On the exactness of an S-shaped bifurcation curve

For a class of two-point boundary value problems we prove exactness of an S-shaped bifurcation curve. Our result applies to a problem from combustion theory, which involves nonlinearities like for .

     

Boundary and Lens Rigidity of Lorentzian Surfaces

Let be a Lorentzian metric on the plane that agrees with the standard metric outside a compact set and so that there are no conjugate points along any time-like geodesic of . Then and are isometric. Further, if and are two dimensional compact time oriented Lorentzian manifolds with space--like boundaries and so that all time-like geodesics of maximize the distances between their points and and are ``boundary isometric', then there is a conformal diffeomorphism between and and they have the same areas. Similar results hold in higher dimensions under an extra assumption on the volumes of the manifolds.

     

On the number of geodesic segments connecting two points on manifolds of non-positive curvature

We prove that on a complete Riemannian manifold of dimension with sectional curvature , two points which realize a local maximum for the distance function (considered as a function of two arguments) are connected by at least geodesic segments. A simpler version of the argument shows that if one of the points is fixed and then the two points are connected by at least geodesic segments. The proof uses mainly the convexity properties of the distance function for metrics of negative curvature.

     

Statistical properties of generalized discrepancies

When testing that a sample of points in the unit hypercube comes from a uniform distribution, the Kolmogorov-Smirnov and the Cramér-von Mises statistics are simple and well-known procedures. To encompass these measures of uniformity, Hickernell introduced the so-called generalized -discrepancies. These discrepancies can be used in numerical integration through Monte Carlo and quasi-Monte Carlo methods, design of experiments, uniformity testing and goodness-of-fit tests. The aim of this paper is to derive the statistical asymptotic properties of these statistics under Monte Carlo sampling. In particular, we show that, under the hypothesis of uniformity of the sample of points, the asymptotic distribution is a complex stochastic integral with respect to a pinned Brownian sheet. On the other hand, if the points are not uniformly distributed, then the asymptotic distribution is Gaussian.

     

The geography of irreducible 4-manifolds

In this paper we investigate the existence and the uniqueness problems for simply connected irreducible -manifolds. By taking fiber sums along an embedded surface of square and by a rational blow-down procedure, we construct many new irreducible -manifolds which have infinitely many distinct smooth structures. Furthermore, we prove that all but at most finitely many lattice points lying in the non-positive signature region with are covered by these irreducible -manifolds.

     

Non-existence of a curve over of genus 5 with 14 rational points

We show that an absolutely irreducible, smooth, projective curve of genus over with rational points cannot exist.

     

Limit cycles for cubic systems with a symmetry of order 4 and without infinite critical points

In this paper we study those cubic systems which are invariant under a rotation of radians. They are written as where is complex, the time is real, and , are complex parameters. When they have some critical points at infinity, i.e. , it is well-known that they can have at most one (hyperbolic) limit cycle which surrounds the origin. On the other hand when they have no critical points at infinity, i.e. there are examples exhibiting at least two limit cycles surrounding nine critical points. In this paper we give two criteria for proving in some cases uniqueness and hyperbolicity of the limit cycle that surrounds the origin. Our results apply to systems having a limit cycle that surrounds either 1, 5 or 9 critical points, the origin being one of these points. The key point of our approach is the use of Abel equations.

     

Normal Euler classes of knotted surfaces and triple points on projections

We present a new formula relating the normal Euler numbers of embedded surfaces in -space and the number of triple points on their projections into -space. This formula generalizes Banchoff's formula between normal Euler numbers and branch points on the projections.

     

Topologically conjugate Kleinian groups

Two Kleinian groups and are said to be topologically conjugate when there is a homeomorphism such that . It is conjectured that if two Kleinian groups and are topologically conjugate, one is a quasi-conformal deformation of the other. In this paper generalizing Minsky's result, we shall prove that this conjecture is true when is finitely generated and freely indecomposable, and the injectivity radii of all points of and are bounded below by a positive constant.

     

On the mapping class group action on the cohomology of the representation space of a surface

The mapping class group of a -pointed Riemann surface has a natural action on any moduli space of parabolic bundles with the marked points as the parabolic points. We prove that under some numerical conditions on the parabolic data, the induced action of the mapping class group on the cohomology algebra of the moduli space of parabolic bundles factors through the natural epimorphism of the mapping class group onto the symplectic group.