On a Question of Haskell P. Rosenthal Concerning a Characterization of c0 and lp

The following property of a normalized basis in a Banach spaceis considered: any normalized block sequence of the basis hasa subsequence equivalent to the basis. Under uniformity or othernatural assumptions, a basis with this property is equivalentto the unit vector basis of c₀ or l_p. An analogous problem concerningspreading models is also addressed. 2000 Mathematics SubjectClassification 46B20 (primary), 46B15 (secondary).     

An SR1/BFGS SQP algorithm for nonconvex nonlinear programs with block-diagonal Hessian matrix

We present a quasi-Newton sequential quadratic programming (SQP) algorithm for nonlinear programs in which the Hessian of the Lagrangian function is block-diagonal. Problems with this characteristic frequently arise in the context of optimal control; for example, when a direct multiple shooting parametrization is used. In this article, we describe an implementation of a filter line-search SQP method that computes search directions using an active-set quadratic programming (QP) solver. To take advantage of the block-diagonal structure of the Hessian matrix, each block is approximated separately by quasi-Newton updates. For nonconvex instances, that arise, for example, in optimum experimental design control problems, these blocks are often found to be indefinite. In that case, the block-BFGS quasi-Newton update can lead to poor convergence. The novel aspect in this work is the use of SR1 updates in place of BFGS approximations whenever possible. The resulting indefinite QPs necessitate an inertia control mechanism within the sparse Schur-complement factorization that is carried out by the active-set QP solver. This permits an adaptive selection of the Hessian approximation that guarantees sufficient progress towards a stationary point of the problem. Numerical results demonstrate that the proposed approach reduces the number of SQP iterations and CPU time required for the solution of a set of optimal control problems.     

Absolutely continuous spectrum for random Schrödinger operators on the Bethe strip

The Bethe strip of width m is the cartesian product $\mathbb {B}\times \lbrace 1,\ldots ,m\rbrace$, where $\mathbb {B}$ is the Bethe lattice (Cayley tree). We prove that Anderson models on the Bethe strip have “extended states” for small disorder. More precisely, we consider Anderson‐like Hamiltonians $H_\lambda =\frac{1}{2} \Delta \otimes 1 + 1 \otimes A\,+\,\lambda \mathcal {V}$ on a Bethe strip with connectivity K ≥ 2, where A is an m × m symmetric matrix, $\mathcal {V}$ is a random matrix potential, and λ is the disorder parameter. Given any closed interval $I\subset \big (\!-\!\sqrt{K}+a_{{\rm max}},\sqrt{K}+a_{\rm {min}}\big )$, where a_min and a_max are the smallest and largest eigenvalues of the matrix A, we prove that for λ small the random Schrödinger operator H_λ has purely absolutely continuous spectrum in I with probability one and its integrated density of states is continuously differentiable on the interval I.     

“It’s better to forget physics”: The Idea of the Tactical Nuclear Weapon in the Early Cold War

Physics in Perspective - The American physicist John Wheeler once told his colleague Richard Feynman that, in case of war, “it’s better to forget physics and tell the admirals and...     

Quaternionic monopoles

We present the simplest non-abelian version of Seiberg-Witten theory: Quaternionic monopoles. These monopoles are associated withSpin ^h (4)-structures on 4-manifolds and form finite-dimensional moduli spaces. On a Kähler surface the quaternionic monopole equations decouple and lead to the projective vortex equation for holomorphic pairs. This vortex equation comes from a moment map and gives rise to a new complex-geometric stability concept. The moduli spaces of quaternionic monopoles on Kähler surfaces have two closed subspaces, both naturally isomorphic with moduli spaces of canonically stable holomorphic pairs. These components intersect along a Donaldson instanton space and can be compactified with Seiberg-Witten moduli spaces. This should provide a link between the two corresponding theories.Partially supported by: AGE-Algebraic Geometry in Europe, contract No ERBCHRXCT940557 (BBW 93.0187), and by SNF, nr. 21-36111.92     

Information Shift Dynamics Described by Tsallis q = 3 Entropy on a Compact Phase Space

Recent mathematical investigations have shown that under very general conditions, exponential mixing implies the Bernoulli property. As a concrete example of statistical mechanics that are exponentially mixing we consider the Bernoulli shift dynamics by Chebyshev maps of arbitrary order N≥2, which maximizes Tsallis q=3 entropy rather than the ordinary q=1 Boltzmann-Gibbs entropy. Such an information shift dynamics may be relevant in a pre-universe before ordinary space-time is created. We discuss symmetry properties of the coupled Chebyshev systems, which are different for even and odd N. We show that the value of the fine structure constant αel=1/137 is distinguished as a coupling constant in this context, leading to uncorrelated behaviour in the spatial direction of the corresponding coupled map lattice for N=3.     

Optically active Babinet planar metamaterial film for terahertz polarization manipulation

A planar Babinet‐inverted dimer metamaterial possessing strong optical activity is proposed and characterized. An original fabrication method to produce large area (up to several cm²) freely suspended flexible metallic membranes is implemented to fabricate the metamaterial. Its optical properties are characterized by terahertz time‐domain spectroscopy, revealing anisotropic transmission with high optical activity. A simple coupled resonator model is applied to explain the principal optical features of the dimers, with predictive power of positions and number of resonances through a parametrical model. The model is validated for correct polarization‐dependent quantitative results on the optical activity in transmission spectra. The fabrication method presented in this work as well as the slit dimer design has great potential for exploitation in terahertz optics.     

Prewavelet analysis of the heat equation

Summary. We consider the heat equation in a smooth domain of R with Dirichlet and Neumann boundary conditions. It is solved by using its integral formulation with double-layer potentials, where the unknown , the jump of the solution through the boundary, belongs to an anisotropic Sobolev space. We approximate by the Galerkin method and use a prewavelet basis on , which characterizes the anisotropic space. The use of prewavelets allows to compress the stiffness matrix from to when N is the size of the matrix, and the condition number of the compressed matrix is uniformly bounded as the initial one in the prewavelet basis. Finally we show that the compressed scheme converges as fast as the Galerkin one, even for the Dirichlet problem which does not admit a coercive variational formulation. Received April 16, 1999 / Published online August 2, 2000