When almost is not even close: Remarks on the approximability of HDTP

A growing number of researchers in Cognitive Science advocate the thesis that human cognitive capacities are constrained by computational tractability. If right, this thesis also can be expected to have far-reaching consequences for work in Artificial General Intelligence: Models and systems considered as basis for the development of general cognitive architectures with human-like performance would also have to comply with tractability constraints, making in-depth complexity theoretic analysis a necessary and important part of the standard research and development cycle already from a rather early stage. In this paper we present an application case study for such an analysis based on results from a parametrized complexity and approximation theoretic analysis of the Heuristic Driven Theory Projection (HDTP) analogy-making framework

On the Green's Function of the almost-Mathieu Operator

The square tight-binding model in a magnetic field leads to the almost-Mathieu operator which, for rational fields, reduces to a $q\times q$ matrix depending on the components $\mu$, $\nu$ of the wave vector in the magnetic Brillouinzone. We calculate the corresponding Green's function without explicit knowledge of eigenvalues and eigenfunctions and obtain analytical expressions for the diagonal and the first off-diagonal elements; the results which are consistent with the zero magnetic field case can be used to calculate several quantities of physical interest (e. g. the density of states over the entire spectrum, impurity levels in a magnetic field).Comment: 9 pages, 3 figures corrected some minor errors and typo

Computable functions, quantum measurements, and quantum dynamics

We construct quantum mechanical observables and unitary operators which, if implemented in physical systems as measurements and dynamical evolutions, would contradict the Church-Turing thesis which lies at the foundation of computer science. We conclude that either the Church-Turing thesis needs revision, or that only restricted classes of observables may be realized, in principle, as measurements, and that only restricted classes of unitary operators may be realized, in principle, as dynamics.Comment: 4 pages, REVTE

Aharonov-Bohm cages in the GaAlAs/GaAs system

Aharonov-Bohm oscillations have been observed in a lattice formed by a two dimensional rhombus tiling. This observation is in good agreement with a recent theoretical calculation of the energy spectrum of this so-called T3 lattice. We have investigated the low temperature magnetotransport of the T3 lattice realized in the GaAlAs/GaAs system. Using an additional electrostatic gate, we have studied the influence of the channel number on the oscillations amplitude. Finally, the role of the disorder on the strength of the localization is theoretically discussed.Comment: 6 pages, 11 EPS figure

Hofstadter-type energy spectra in lateral superlattices defined by periodic magnetic and electrostatic fields

We calculate the energy spectrum of an electron moving in a two-dimensional lattice which is defined by an electric potential and an applied perpendicular magnetic field modulated by a periodic surface magnetization. The spatial direction of this magnetization introduces complex phases into the Fourier coefficients of the magnetic field. We investigate the effect of the relative phases between electric and magnetic modulation on band width and internal structure of the Landau levels.Comment: 5 LaTeX pages with one gif figure to appear in Phys. Rev.

Quantized Orbits and Resonant Transport

A tight binding representation of the kicked Harper model is used to obtain an integrable semiclassical Hamiltonian consisting of degenerate "quantized" orbits. New orbits appear when renormalized Harper parameters cross integer multiples of $\pi/2$. Commensurability relations between the orbit frequencies are shown to correlate with the emergence of accelerator modes in the classical phase space of the original kicked problem. The signature of this resonant transport is seen in both classical and quantum behavior. An important feature of our analysis is the emergence of a natural scaling relating classical and quantum couplings which is necessary for establishing correspondence.Comment: REVTEX document - 8 pages + 3 postscript figures. Submitted to Phys.Rev.Let

Higher-order thoughts in action : Consciousness as an unconscious re-description process

Peer reviewedPostprin

Electronic Band Structure In A Periodic Magnetic Field

We analyze the energy band structure of a two-dimensional electron gas in a periodic magnetic field of a longitudinal antiferromagnet by considering a simple exactly solvable model. Two types of states appear: with a finite and infinitesimal longitudinal mobility. Both types of states are present at a generic Fermi surface. The system exhibits a transition to an insulating regime with respect to the longitudinal current, if the electron density is sufficiently low.Comment: 8 pages, 5 figures; to appear in Phys. Rev. B '9

Hofstadter butterfly and integer quantum Hall effect in three dimensions

For a three-dimensional lattice in magnetic fields we have shown that the hopping along the third direction, which normally tends to smear out the Landau quantization gaps, can rather give rise to a fractal energy spectram akin to Hofstadter's butterfly when a criterion, found here by mapping the problem to two dimensions, is fulfilled by anisotropic (quasi-one-dimensional) systems. In 3D the angle of the magnetic field plays the role of the field intensity in 2D, so that the butterfly can occur in much smaller fields. The mapping also enables us to calculate the Hall conductivity, in terms of the topological invariant in the Kohmoto-Halperin-Wu's formula, where each of $\sigma_{xy}, \sigma_{zx}$ is found to be quantized.Comment: 4 pages, 6 figures, RevTeX, uses epsf.sty,multicol.st

The "Artificial Mathematician" Objection: Exploring the (Im)possibility of Automating Mathematical Understanding

Reuben Hersh confided to us that, about forty years ago, the late Paul Cohen predicted to him that at some unspecified point in the future, mathematicians would be replaced by computers. Rather than focus on computers replacing mathematicians, however, our aim is to consider the (im)possibility of human mathematicians being joined by “artificial mathematicians” in the proving practice—not just as a method of inquiry but as a fellow inquirer