arXiv subjects

Nicolau C. Saldanha

Publications and source records attributed to Nicolau C. Saldanha.

52 records · Page 3Linked to original sources

arXiv2007

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

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, $\Fbd(\tilde v, \overline u) = (\tilde v, \overline v)$, where $\tilde v$ is a function of average zero and both $\overline u$ and $\overline v$ 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 $v$.

math.CA↗

arXiv2006

The asymptotics of Wilkinson's shift iteration

We study the rate of convergence of Wilkinson's shift iteration acting on Jacobi matrices with simple spectrum. We show that for AP-free spectra (i.e., simple spectra containing no arithmetic progression with 3 terms), convergence is cubic. In order 3, there exists a tridiagonal symmetric matrix P_0 which is the limit of a sequence of a Wilkinson iteration, with the additional property that all iterations converging to P_0 are strictly quadratic. Among tridiagonal matrices near P_0, the set X of initial conditions with convergence to P_0 is rather thin: it is a union of disjoint arcs X_s meeting at P_0, where s ranges over the Cantor set of sign sequences s: N -> {1,-1}. Wilkinson's step takes X_s to X_{s'}, where s' is the left shift of s. Among tridiagonal matrices conjugate to P_0, initial conditions near P_0 but not in X converge at a cubic rate.

math.NA↗

arXiv2005

Reconstruction of tridiagonal matrices from spectral data

Jacobi matrices are parametrized by their eigenvalues and norming constants (first coordinates of normalized eigenvectors): this coordinate system breaks down at reducible tridiagonal matrices. The set of real symmetric tridiagonal matrices with prescribed simple spectrum is a compact manifold, admitting an open covering by open dense sets ${\cal U}^π_Λ$ centered at diagonal matrices $Λ^π$, where $π$ spans the permutations. {\it Bidiagonal coordinates} are a variant of norming constants which parametrize each open set ${\cal U}^π_Λ$ by the Euclidean space. The reconstruction of a Jacobi matrix from inverse data is usually performed by an algorithm introduced by de Boor and Golub. In this paper we present a reconstruction procedure from bidiagonal coordinates and show how to employ it as an alternative to the de Boor-Golub algorithm. The inverse bidiagonal algorithm rates well in terms of speed and accuracy.

math.NA↗

arXiv2005

Stability of compact actions of the Heisenberg group

Let G be the Heisenberg group of real lower triangular 3x3 matrices with unit diagonal. A locally free smooth action of G on a manifold M^4 is given by linearly independent vector fields X_1, X_2, X_3 such that X_3 = [X_1,X_2] and [X_1,X_3] = [X_2, X_3] = 0. The C^1 topology for vector fields induces a topology in the space of actions of G on M^4. An action is compact if all orbits are compact. Given a compact action $θ$, we investigate under which conditions its C^1 perturbations $\tildeθ$ are guaranteed to be compact. There is more than one interesting definition of stability, and we show that in the case of the Heisenberg group, unlike for actions of R^n, the definitions do not turn out to be equivalent.

math.GT↗

arXiv2005

The topology of the monodromy map of the second order ODE

We consider the following question: given $A \in SL(2,R)$, which potentials $q$ for the second order Sturm-Liouville problem have $A$ as its Floquet multiplier? More precisely, define the monodromy map $μ$ taking a potential $q \in L^2([0,2π])$ to $μ(q) = \tildeΦ(2π)$, the lift to the universal cover $G = \widetilde{SL(2,R)}$ of $SL(2,R)$ of the fundamental matrix map $Φ: [0,2π] \to SL(2,R)$, \[ Φ(0) = I, \quad Φ'(t) = \begin{pmatrix} 0 & 1 q(t) & 0 \end{pmatrix} Φ(t). \] Let $H$ be the real infinite dimensional separable Hilbert space: we present an explicit diffeomorphism $Ψ: G_0 \times H \to H^0([0,2π])$ such that the composition $μ\circ Ψ$ is the projection on the first coordinate. The key ingredient is the correspondence between potentials $q$ and the image in the plane of the first row of $Φ$, parametrized by polar coordinates, which we call the Kepler transform. As an application among others, let $C_1 \subset L^2([0,2π])$ be the set of potentials $q$ for which the equation $-u'' + qu = 0$ admits a nonzero periodic solution: $C_1$ is diffeomorphic to the disjoint union of a hyperplane and cartesian products of the usual cone in $R^3$ with $H$.

math.CA↗

arXiv2005

The topology of critical sets of some ordinary differential operators

We survey recent work of Burghelea, Malta and both authors on the topology of critical sets of nonlinear ordinary differential operators. For a generic nonlinearity $f$, the critical set of the first order nonlinear operator $F_1(u)(t) = u'(t) + f(u(t))$ acting on the Sobolev space $H^1_p$ of periodic functions is either empty or ambient diffeomorphic to a hyperplane. For the second order operator $F_2(u)(t) = -u''(t) + f(u(t))$ on $H^2_D$ (Dirichlet boundary conditions), the critical set is ambient diffeomorphic to a union of isolated parallel hyperplanes. For second order operators on $H^2_p$, the critical set is not a Hilbert manifold but is still contractible and admits a normal form. The third order case is topologically far more complicated.

math.FA↗

arXiv2004

Nilpotent pseudogroups of functions on an interval

A near-identity nilpotent pseudogroup of order m >= 1 is a family f_1, ..., f_n: (-1,1) -> R of C^2 functions for which: |f_i - id|_{C^1} < epsilon for some small positive real number epsilon < 1/10^{m+1} and commutators of the functions f_i of order at least m equal the identity. We present a classification of near-identity nilpotent pseudogroups: our results are similar to those of Plante, Thurston, Farb and Franks for nilpotent groups. As an application, we classify certain foliations of nilpotent manifolds.

math.DS↗

arXiv2004

Homotopy and cohomology of spaces of locally convex curves in the sphere

We discuss the homotopy type and the cohomology of spaces of locally convex parametrized curves gamma: [0,1] -> S^2, i.e., curves with positive geodesic curvature. The space of all such curves with gamma(0) = gamma(1) = e_1 and gamma'(0) = gamma'(1) = e_2 is known to have three connected components X_{-1,c}, X_1, X_{-1}. We show several results concerning the homotopy type and cohomology of these spaces. In particular, X_{-1,c} is contractible, X_1 and X_{-1} are simply connected, pi_2(X_{-1}) contains a copy of Z and pi_2(X_1) contains a copy of Z^2. Also, H^n(X_{1}, R) and H^n(X_{-1}, R) are nontrivial for all even n. More, dim H^{4n-2}(X_1, R) >= 2 and dim H^{4n}(X_{-1}, R) >= 2 for all positive n.

math.GT↗

arXiv2003

Functions from $R^2$ to $R^2$: a study in nonlinearity

We study the geometry of functions from the plane to the plane. For a large special class we are able to count preimages and compute them. Both numerical and theoretical aspects are discussed. Some of the tools used are Whitney's classification of critical points, rotation numbers, covering maps and continuation methods.

math.NA↗

arXiv2003

Arithmetic properties of the adjacency matrix of quadriculated disks

Let $Δ$ be a bicolored quadriculated disk with black-to-white matrix $B_Δ$. We show how to factor $B_Δ= L\tilde DU$, where $L$ and $U$ are lower and upper triangular matrices, $\tilde D$ is obtained from a larger identity matrix by removing rows and columns and all entries of $L$, $\tilde D$ and $U$ are equal to 0, 1 or -1.

math.CO↗

arXiv2002

Tilings of quadriculated annuli

Tilings of a quadriculated annulus A are counted according to volume (in the formal variable q) and flux (in p). We consider algebraic properties of the resulting generating function Phi_A(p,q). For q = -1, the non-zero roots in p must be roots of unity and for q > 0, real negative.

math.CO↗

arXiv2001

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 differential operator $F(u) = -u'' + f(u)$ for $u \in H^2_D([0, π])$, a Sobolev space of functions satisfying Dirichlet boundary conditions. For a generic nonlinearity $f: \RR \to \RR$ 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.

math.FA↗

arXiv2001

Two results on tilings of quadriculated annuli

We provide a more informal explanation of two results in our manuscript "Tilings of quadriculated annuli". Tilings of a quadriculated annulus $A$ are counted according to volume (in the formal variable q) and flux (in p). The generating function Phi_A(p,q) is such that, for q = -1, the non-zero roots in p are roots of unity and for q > 0, real negative.

math.CO↗

arXiv2000

Singular polynomials of generalized Kasteleyn matrices

Kasteleyn counted the number of domino tilings of a rectangle by considering a mutation of the adjacency matrix: a Kasteleyn matrix K. In this paper we present a generalization of Kasteleyn matrices and a combinatorial interpretation for the coefficients of the characteristic polynomial of KK^\ast (which we call the singular polynomial), where K is a generalized Kasteleyn matrix for a planar bipartite graph. We also present a q-version of these ideas and a few results concerning tilings of special regions such as rectangles.

math.CO↗

arXiv1998

An overview of domino and lozenge tilings

We consider tilings of quadriculated regions by dominoes and of triangulated regions by lozenges. We present an overview of results concerning tileability, enumeration and the structure of the space of tilings.

math.CO↗

arXiv1997

Spectra of semi-regular polytopes

We compute the spectra of the adjacency matrices of the semi-regular polytopes. A few different techniques are employed: the most sophisticated, which relates the 1-skeleton of the polytope to a Cayley graph, is based on methods akin to those of Lovász and Babai ([L], [B]). It turns out that the algebraic degree of the eigenvalues is at most 5, achieved at two 3-dimensional solids.

math.CO↗