No abstract

The homotopy type of spaces of locally convex curves with fixed endpoints in Spin n+1 , the universal covering the orthogonal group SO n+1 for n ≥ 2, has been determined for n = 2 but is in general not known. The results in this paper have as one important aim trying to make progress in this problem. In the process, we prove a related conjecture of B. Shapiro and M. Shapiro regarding the behavior of fundamental systems of solutions to linear ordinary differential equations. We define the itinerary of a locally convex curve Γ : [0, 1] → Spin n+1 as a (finite) word w in the alphabet S n+1 {e} of non-trivial permutations. This word encodes the succession of non-open Bruhat cells of Spin n+1 pierced by Γ(t) as t ranges from 0 to 1. We prove that, for each word w, the subspace of curves of itinerary w is an embedded contractible (globally collared topological) submanifold of finite codimension, thus defining a stratification of the space of curves. We show how to obtain explicit (topologically) transversal sections for each of these submanifolds. We also study the neighboring relation between strata. This is an important step in the construction of abstract cell complexes mapped into the original space of curves by weak homotopy equivalences, which we cover in a follow-up paper.

A smooth curve γ : [0, 1] → S 2 is locally convex if its geodesic curvature is positive at every point. J. A. Little showed that the space of all locally convex curves γ with γ(0) = γ(1) = e 1 and γThe space L −1,c is known to be contractible. We prove that L +1 and L −1,n are homotopy equivalent to (ΩS 3 ) ∨ S 2 ∨ S 6 ∨ S 10 ∨ · · · and (ΩS 3 ) ∨ S 4 ∨ S 8 ∨ S 12 ∨ · · · , respectively. As a corollary, we deduce the homotopy type of the components of the space Free(S 1 , S 2 ) of free curves γ : S 1 → S 2 (i.e., curves with nonzero geodesic curvature). We also determine the homotopy type of the spaces Free([0, 1], S 2 ) with fixed initial and final frames.

Abstract. This paper contains a description of a program designed to find all the solutions of systems of two real equations in two real unknowns which uses detailed information about the critical set of the associated function from the plane to the plane. It turns out that the critical set and its image are highly structured, and this is employed in their numerical computation. The conceptual background and details of implementation are presented. The most important features of the program are the ability to provide global information about the function and the robustness derived from such topological information.

We consider the operator F (u) = u ′ + f (t, u(t)) acting on periodic real valued functions. Generically, critical points of F are infinite dimensional Morin-like singularities and we provide operational characterizations of the singularities of different orders. A global Lyapunov-Schmidt decomposition of F converts F into adapted coordinates, F(ṽ,ū) = (ṽ,v), whereṽ is a function of average zero and bothū andv are numbers. Thus, global geometric aspects of F reduce to the study of a family of one-dimensional maps: we use this approach to obtain normal forms for several nonlinearities f . For example, we characterize autonomous nonlinearities giving rise to global folds and, in general, we show that F is a global fold if all critical points are folds. Also, f (t, x) = x 3 − x, or, more generally, the Cafagna-Donati nonlinearity, yield global cusps; for F interpreted as a map between appropriate Hilbert spaces, the requested changes of variable to bring F to normal form can be taken to be diffeomorphisms. A key ingredient in the argument is the contractibility of both the critical set and the set of non-folds for a generic autonomous nonlinearity. We also obtain a numerical example of a polynomial f of degree 4 for which F contains butterflies (Morin singularities of order 4)-it then follows that F (u) = v has six solutions for some v.1991 Mathematics Subject Classification. Primary 58C27, 34B15, 34L30; Secondary 47H15.

We consider the nonlinear Sturm-Liouville differential operator F (u) = −u ′′ + f (u) for u ∈ H 2 D ([0, π]), a Sobolev space of functions satisfying Dirichlet boundary conditions. For a generic nonlinearity f : R → R we show that there is a diffeomorphism in the domain of F converting the critical set C of F into a union of isolated parallel hyperplanes. For the proof, we show that the homotopy groups of connected components of C are trivial and prove results which permit to replace homotopy equivalences of systems of infinite dimensional Hilbert manifolds by diffeomorphisms. IntroductionConsider the nonlinear Sturm-Liouville problem −u ′′ (t) + f (u(t)) = g(t), u(0) = u(π) = 0 and for any smooth nonlinearity f : R → R denote by F the differential operatoris the Sobolev space of functions u(t) with square integrable second derivatives which satisfy Dirichlet boundary conditions u(0) = u(π) = 0 2000 Mathematics Subject Classification. Primary 34L30, 58B05, Secondary 34B15, 46T05. Keywords and phrases Sturm-Liouville, nonlinear differential operators, infinite dimensional manifolds.D. Burghelea was supported by an NSF grant. N. Saldanha and C. Tomei acknowledge the support of CNPq, Faperj and Finep.

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We study self-affine tilings of R n with special emphasis on the two-digit case. We prove that in this case the tile is connected and, if n 3, is a lattice-tile.