Nicolau C. Saldanha

Spaces of domino tilings

We consider the set of all tilings by dominoes (2×1 rectangles) of a surface, possibly with boundary, consisting of unit squares. Convert this set into a graph by joining two tilings by an edge if they differ by aflip, i.e., a 90° rotation of a pair of side-by-side dominoes. We give a criterion to decide if two tilings are in the same connected component, a simple formula for distances, and a method to construct geodesics in this graph. For simply connected surfaces, the graph is connected. By naturally adjoining to this graph higher-dimensional cells, we obtain a CW-complex whose connected components are homotopically equivalent to points or circles. As a consequence, for any region different from a torus or Klein bottle, all geodesics with common endpoints are equivalent in the following sense. Build a graph whose vertices are these geodesics, adjacent if they differ only by the order of two flips on disjoint squares: this graph is connected.

Remarks on Self-Affine Tilings

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.

Morin singularities and global geometry in a class of ordinary differential operators

(∗) u′(t) + f(t, u(t)) = g(t), where the unknown u is a real function on S1 and the nonlinearity f : S1×R → R can assume a number of forms. Our approach is to study the global geometry of the operator F : B1 → B0, u → u′ + f(t, u) where the domain is either C1(S1) (the Banach space of periodic functions with continuous derivatives) or the Hilbert space H1(S1) of periodic functions with square integrable derivative. Ideally, we search for global changes of variables in both domain and image taking the operator F to a simple normal form. This goal has been achieved in previous occasions, starting with the seminal work of A. A. Ambrosetti and G. Prodi ([AP]) and its geometric interpretation by M. S. Berger and P. T. Church ([BC]), who showed that the operator associated to a certain nonlinear Dirichlet problem gives rise to a global fold between infinite

The numerical inversion of functions from the plane to the plane

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.

Results on infinite-dimensional topology and applications to the structure of the critical set of nonlinear Sturm–Liouville operators

We consider the nonlinear Sturm-Liouville dieren tial operator F (u) = u 00 + f(u) for u 2 H 2 D ([0; ]), a Sobolev space of functions satisfying Dirichlet boundary conditions. For a generic nonlinearity f : R! R we show that there is a dieomorphism 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 innite dimensional Hilbert manifolds by dieomorphisms.

The Asymptotics of Wilkinson’s Shift: Loss of Cubic Convergence

AbstractOne of the most widely used methods for eigenvalue computation is the QR iteration with Wilkinson’s shift: Here, the shift s is the eigenvalue of the bottom 2×2 principal minor closest to the corner entry. It has been a long-standing question whether the rate of convergence of the algorithm is always cubic. In contrast, we show that there exist matrices for which the rate of convergence is strictly quadratic. More precisely, let

An atlas for tridiagonal isospectral manifolds

T_{ {\mathcal {X}}}

The Topology of Critical Sets of Some Ordinary Differential Operators

be the 3×3 matrix having only two nonzero entries

The accumulated distribution of quadratic forms on the sphere

(T_{ {\mathcal {X}}})_{12}=(T_{ {\mathcal {X}}})_{21}=1

Spectra of regular polytopes

and let