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 $F_\sigma$ called shifted $QR$ steps. Such maps preserve spectrum and a natural common domain is ${\cal T}_\Lambda$, the manifold of real symmetric tridiagonal matrices conjugate to the diagonal matrix $\Lambda$. More precisely, a (generic) shift s \in \RR defines a map $F_s: {\cal T}_\Lambda \to {\cal T}_\Lambda$. A strategy \sigma: {\cal T}_\Lambda \to \RR specifies the shift to be applied at $T$ so that $F_\sigma(T) = F_{\sigma(T)}(T)$. 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, $|b(F_\sigma(T))| = \Theta(|b(T)|^k)$ 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.Comment: 22 pages, 4 figures. This preprint borrows heavily from the unpublished preprint arXiv:0912.3376 but is adapted for a different audienc

São Paulo, megacidade e redesenvolvimento sustentável: uma estratégia propositiva São Paulo, megacity and sustainable redevelopment: a strategic proposal

O trabalho faz uma reflexão crítica acerca da emergência das megacidades no século XXI para, em seguida, focar a megacidade brasileira, São Paulo, seus desafios e oportunidades em busca de uma reinvenção necessária que traga a agenda de um desejável redesenvolvimento sustentável no âmbito urbano. O trabalho conclui apresentando um modelo de redesenvolvimento de um território em transformação, a Diagonal Sul, orla ferroviária paulistana. O projeto urbano estratégico apresentado busca investigar oportunidades de reorganização desse território: como implementar o projeto urbano na nossa maior megacidade? A tese é a do reaproveitamento dos vazios urbanos centrais como contraponto à expansão periférica: adensamento populacional e de atividades onde haja infraestrutura e localização central. Cidade mais compacta é cidade mais sustentável

Perspective chapter: a phenomenological approach to empathy

In this chapter, we will seek to determine the role that empathy plays in the relationship between different human beings. In the phenomenological tradition, empathy was often considered the primordial form of relationship with others. According to classical phenomenology, bodily externalizations of inner psychic phenomena play an important role here. In this chapter, we will critically evaluate this approach to empathy. We will argue that empathy presupposes a prior recognition of the other in everyday life. From this perspective, empathy would constitute a second—although fundamental—moment of intersubjective life. The experience of a shared world entails an immediately shared understanding of that world and of those who live in it. Empathy would awaken only when anonymous relationships with other human beings become problematic. In the twentieth century, this concept has been appropriated by psychology on different levels: as a device to analyze how far the lack of empathy may be at the origin of certain pathologies and as a clinical tool that allows the psychologist to gain a deeper insight into the patient’s problems. Unfortunately, dialog between the two disciplines around this issue was almost never carried out. This chapter tries to foster the possibilities of such a dialog.info:eu-repo/semantics/publishedVersio

An atlas for tridiagonal isospectral manifolds

Let ${\cal T}_\Lambda$ be the compact manifold of real symmetric tridiagonal matrices conjugate to a given diagonal matrix $\Lambda$ with simple spectrum. We introduce {\it bidiagonal coordinates}, charts defined on open dense domains forming an explicit atlas for ${\cal T}_\Lambda$. In contrast to the standard inverse variables, consisting of eigenvalues and norming constants, every matrix in ${\cal T}_\Lambda$ 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.Comment: Fixed typos; 16 pages, 3 figure