arXiv subjects

Carlos Tomei

Publications and source records attributed to Carlos Tomei.

At least 37 records · Page 2Linked to original sources

arXiv2015

Abundance of cusps and a converse to the Ambrosetti-Prodi theorem

According to the Ambrosetti-Prodi theorem, the map $F(u)= - Δu - f(u)$ between appropriate functional spaces is a global fold. Among the hypotheses, the convexity of the function $f$ is required. We show in two different ways that, under mild conditions, convexity is indeed necessary. If $f$ is not convex, there is a point with at least four preimages under $F$. More, $F$ generically admits cusps among its critical points. We present a larger class of nonlinearities $f$ for which the critical set of $F$ has cusps. The results are true for a class of boundary conditions.

math.AP↗

arXiv2015

Geometric Aspects of Ambrosetti-Prodi operators with Lipschitz nonlinearities

For Dirichlet boundary conditions on a bounded domain, what happens to the critical set of the Ambrosetti-Prodi operator if the nonlinearity is only a Lipschitz map? It turns out that many properties which hold in the smooth case are preserved, despite of the fact that the operator is not even differentiable at some points. In particular, a global Lyapunov-Schmidt decomposition of great convenience for numerical inversion is still available.

math.AP↗

arXiv2015

Fibers and global geometry of functions

Since the seminal work of Ambrosetti and Prodi, the study of global folds was enriched by geometric concepts and extensions accomodating new examples. We present the advantages of considering fibers, a construction dating to Berger and Podolak's view of the original theorem. A description of folds in terms of properties of fibers gives new perspective to the usual hypotheses in the subject. The text is intended as a guide, outlining arguments and stating results which will be detailed elsewhere.

math.AP↗

arXiv2011

Dynamics of the symmetric eigenvalue problem with shift strategies

A common algorithm for the computation of eigenvalues of real symmetric tridiagonal matrices is the iteration of certain special maps $F_σ$ called shifted $QR$ steps. Such maps preserve spectrum and a natural common domain is ${\cal T}_Λ$, the manifold of real symmetric tridiagonal matrices conjugate to the diagonal matrix $Λ$. More precisely, a (generic) shift $s \in \RR$ defines a map $F_s: {\cal T}_Λ\to {\cal T}_Λ$. A strategy $σ: {\cal T}_Λ\to \RR$ specifies the shift to be applied at $T$ so that $F_σ(T) = F_{σ(T)}(T)$. Good shift strategies should lead to fast deflation: some off-diagonal coordinate tends to zero, allowing for reducing of the problem to submatrices. For topological reasons, continuous shift strategies do not obtain fast deflation; many standard strategies are indeed discontinuous. Practical implementation only gives rise systematically to bottom deflation, convergence to zero of the lowest off-diagonal entry $b(T)$. For most shift strategies, convergence to zero of $b(T)$ is cubic, $|b(F_σ(T))| = Θ(|b(T)|^k)$ for $k = 3$. The existence of arithmetic progressions in the spectrum of $T$ sometimes implies instead quadratic convergence, $k = 2$. The complete integrability of the Toda lattice and the dynamics at non-smooth points are central to our discussion. The text does not assume knowledge of numerical linear algebra.

math.DS↗

arXiv2011

Numerical analysis of semilinear elliptic equations with finite spectral interaction

We present an algorithm to solve $- \lap u - f(x,u) = g$ with Dirichlet boundary conditions in a bounded domain $Ω$. The nonlinearities are non-resonant and have finite spectral interaction: no eigenvalue of $-\lap_D$ is an endpoint of $\bar{\partial_2f(Ω,\RR)}$, which in turn only contains a finite number of eigenvalues. The algorithm is based in ideas used by Berger and Podolak to provide a geometric proof of the Ambrosetti-Prodi theorem and advances work by Smiley and Chun for the same problem.

math.AP↗

arXiv2010

Convergence rates to deflation of simple shift strategies

The computation of eigenvalues of real symmetric tridiagonal matrices frequently proceeds by a sequence of QR steps with shifts. We introduce simple shift strategies, functions sigma satisfying natural conditions, taking each n x n matrix T to a real number sigma(T). The strategy specifies the shift to be applied by the QR step at T. Rayleigh and Wilkinson's are examples of simple shift strategies. We show that if sigma is continuous then there exist initial conditions for which deflation does not occur, i.e., subdiagonal entries do not tend to zero. In case of deflation, we consider the rate of convergence to zero of the (n, n-1) entry: for simple shift strategies this is always at least quadratic. If the function sigma is smooth in a suitable region and the spectrum of T does not include three consecutive eigenvalues in arithmetic progression then convergence is cubic. This implies cubic convergence to deflation of Wilkinson's shift for generic spectra. The study of the algorithm near deflation uses tubular coordinates, under which QR steps with shifts are given by a simple formula.

math.NA↗

arXiv2009

Cut-and-paste of quadriculated disks and arithmetic properties of the adjacency matrix

We define cut-and-paste, a construction which, given a quadriculated disk obtains a disjoint union of quadriculated disks of smaller total area. We provide two examples of the use of this procedure as a recursive step. Tilings of a disk $Δ$ receive a parity: we construct a perfect or near-perfect matching of tilings of opposite parities. Let $B_Δ$ be the black-to-white adjacency matrix: we 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↗

arXiv2008

The Asymptotics of Wilkinson's Iteration: Loss of Cubic Convergence

One 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 $2\times 2$ principal minor closest to the corner entry. It has been a long-standing conjecture that the rate of convergence of the algorithm is cubic. In contrast, we show that there exist matrices for which the rate of convergence is strictly quadratic. More precisely, let $T_X$ be the $3 \times 3$ matrix having only two nonzero entries $(T_X)_{12} = (T_X)_{21} = 1$ and let $T_L$ be the set of real, symmetric tridiagonal matrices with the same spectrum as $T_X$. There exists a neighborhood $U \subset T_L$ of $T_X$ which is invariant under Wilkinson's shift strategy with the following properties. For $T_0 \in U$, the sequence of iterates $(T_k)$ exhibits either strictly quadratic or strictly cubic convergence to zero of the entry $(T_k)_{23}$. In fact, quadratic convergence occurs exactly when $\lim T_k = T_X$. Let $X$ be the union of such quadratically convergent sequences $(T_k)$: the set $X$ has Hausdorff dimension 1 and is a union of disjoint arcs $X^σ$ meeting at $T_X$, where $σ$ ranges over a Cantor set.

math.SP↗

arXiv2008

The geometry of the critical set of nonlinear periodic Sturm-Liouville operators

We study the critical set C of the nonlinear differential operator F(u) = -u" + f(u) defined on a Sobolev space of periodic functions H^p(S^1), p >= 1. Let R^2_{xy} \subset R^3 be the plane z = 0 and, for n > 0, let cone_n be the cone x^2 + y^2 = tan^2 z, |z - 2 pi n| < pi/2; also set Sigma = R^2_{xy} U U_{n > 0} cone_n. For a generic smooth nonlinearity f: R -> R with surjective derivative, we show that there is a diffeomorphism between the pairs (H^p(S^1), C) and (R^3, Sigma) x H where H is a real separable infinite dimensional Hilbert space.

math.FA↗

arXiv2008

An atlas for tridiagonal isospectral manifolds

Let ${\cal T}_Λ$ be the compact manifold of real symmetric tridiagonal matrices conjugate to a given diagonal matrix $Λ$ with simple spectrum. We introduce {\it bidiagonal coordinates}, charts defined on open dense domains forming an explicit atlas for ${\cal T}_Λ$. In contrast to the standard inverse variables, consisting of eigenvalues and norming constants, every matrix in ${\cal T}_Λ$ now lies in the interior of some chart domain. We provide examples of the convenience of these new coordinates for the study of asymptotics of isospectral dynamics, both for continuous and discrete time.

math.SP↗

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

The complete integrability of a Lie-Poisson system proposed by Bloch and Iserles

We establish the Liouville integrability of the differential equation $\dot S(t)= [N,S^2(t)],$ recently considered by Bloch and Iserles. Here, $N$ is a real, fixed, skew-symmetric matrix and $S$ is real symmetric. The equation is realized as a Hamiltonian vector field on a coadjoint orbit of a loop group, and sufficiently many commuting integrals are presented, together with a solution formula for their related flows in terms of a Riemann-Hilbert factorization problem. We also answer a question raised by Bloch and Iserles, by realizing the same system on a coadjoint orbit of a finite dimensional Lie group.

math.CA↗

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

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↗

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↗