Inexact Inverse Subspace Iteration with Preconditioning Applied to Quadratic Matrix Polynomials

An inexact variant of inverse subspace iteration is used to find a small invariant pair of a large quadratic matrix polynomial. It is shown that linear convergence is preserved provided the inner iteration is performed with increasing accuracy. A preconditioned block GMRES solver is employed as inner iteration. The preconditioner uses the strategy of “tuning” which prevents the inner iteration from increasing and therefore results in a substantial saving in costs. The accuracy of the computed invariant pair can be improved by the addition of a post-processing step involving very few iterations of Newton’s method. The effectiveness of the proposed approach is demonstrated by numerical experiments.

CUBIC SURFACES WITH A GALOIS INVARIANT PAIR OF STEINER TRIHEDRA

We present a method to construct non-singular cubic surfaces over ℚ with a Galois invariant pair of Steiner trihedra. We start with cubic surfaces in a form generalizing that of Cayley and Salmon. For these, we develop an explicit version of Galois descent.

DIELECTRON PRODUCTION IN C+C AT 1 AGeV WITH HADES

The production of e+e- - pairs has been measured for C + C collisions at a beam energy of 1 AGeV. Preliminary invariant pair mass spectra are constructed utilizing a probabilistic analysis method for the identification of single e+/e- tracks. The results are compared to predictions based on sources with known production cross sections, and to earlier measurements.

Computer Simulation Study of Classical Spin Models in Two Dimensions with Long-Range Ferromagnetic Interactions Anisotropic in Spin Space

We have considered a classical spin system, consisting of 3-component unit vectors, associated with a two-dimensional lattice {uk, k ∈ Z2}, and interacting via a translationally invariant pair potential, of the long-range ferromagnetic form, anisotropic in spin space [Formula: see text] here a ≥ 0, b ≥ 0, σ > 2, ∊ is a positive constant setting energy and temperature scales (i.e. T*=kBT/∊), xj denotes dimensionless coordinates of lattice sites, and uj,α cartesian spin components; our discussion has been specialized to the extreme, O(2)-symmetric, case 0=a < b. When 2 < σ < 4, the potential model can be proven to support an ordering transition taking place at finite temperature; on the other hand, when σ ≥ 4 a Berezinskiǐ–Kosterlitz–Thouless-like transition takes place. Two potential models defined by σ=3 and σ=4, respectively, have been characterized quantitatively by Monte Carlo simulation. For σ=3, comparison is also reported with other theoretical treatments, such as Molecular Field and Two Site Cluster approximations.

Computer Simulation Evidence for Berezhinskiǐ–Kosterlitz–Thouless-Like Transitions in One Dimension

We have considered classical spin systems, consisting of n-component unit vectors (n=2,3), associated with a one-dimensional lattice {uk, k ∈ Z}, and interacting via translationally and rotationally invariant pair potentials, of the long-range and ferromagnetic form [Formula: see text] here ε is a positive constant setting energy and temperature scales (i.e. T*=k B T/ε). Available theorems entail the existence of an ordering transition at finite temperature when 0<σ<1, and its absence for σ≥1. When σ=1, spin-wave arguments and previous simulation results suggest the existence of a transition to a low-temperature phase with slow decay of correlations and infinite susceptibility, i.e. a Berezhinskiǐ–Kosterlitz–Thouless-like transition. We have reexamined this case by a more detailed simulation study, and especially addressed the sample-size dependence of calculated observables: simulation results suggest the existence of such a transition, and the proposed estimates for the transition temperatures are Θ2=0.775±0.025 and Θ3=0.500±0.025, when n=2 and n=3, respectively.

Computer Simulation Study of a Disordered Two-Dimensional Nematogenic Lattice Model with Long-Range Interactions Isotropic in Spin Space

We have considered a classical spin system, consisting of 3-component unit vectors, associated with a two-dimensional lattice {uk, k∈ Z2}, and interacting via translationally invariant pair potentials, isotropic in spin space, and of the long-range form [Formula: see text] Here ∊ is a positive constant setting energy and temperature scales (i.e. T*=k B T /∊), P2 denotes the second Legendre polynomial, and xj are dimensionless coordinates of the lattice sites. Available theorems entail the existence of an ordering transition at finite temperature when 0 < σ < 2, and its absence when σ ≥ 2. We have studied the border case σ=2, by means of computer simulation. Similarly to the nearest-neighbour counterpart of the present model, and to other long-range models, we found evidence suggesting a transition to a low-temperature phase with slow decay of correlations and infinite susceptibility, i.e. a Berezhinski[Formula: see text]–Kosterlitz–Thouless-like transition; the transition temperature was estimated to be Θ=1.112 ± 0.005.

COMPUTER SIMULATION STUDY OF A CLASSICAL HEISENBERG MODEL IN TWO DIMENSIONS WITH LONG-RANGE FERROMAGNETIC INTERACTIONS ISOTROPIC IN SPIN SPACE

We have considered a classical spin system, consisting of 3-component unit vectors, associated with a two-dimensional lattice {uk, k ∈ Z2}, and interacting via a translationally and rotationally invariant pair potential, of the long-range ferromagnetic form [Formula: see text] Here ∊ is a positive constant setting energy and temperature scales (i.e. T* = k B T/∊), and xj denotes dimensionless coordinates of the lattice sites. This potential model is known rigorously to possess an ordering transition at finite temperature, and has been characterized quantitatively by Monte Carlo simulation, whose results suggest a continuous transition taking place at [Formula: see text]; comparison is also reported with other theoretical treatments, such as Spherical Model, Molecular Field and Two Site Cluster approximations.

COMPUTER SIMULATION STUDY OF A DISORDERED ONE-DIMENSIONAL NEMATOGENIC LATTICE MODEL WITH LONG-RANGE INTERACTIONS ISOTROPIC IN SPIN SPACE

We have considered a classical spin system, consisting of 3-component unit vectors, associated with a one-dimensional lattice {uk, k ∈ Z}, and interacting via translationally invariant pair potentials, isotropic in spin space, and of the long-range form [Formula: see text] where ∊ is a positive constant setting energy and temperature scales (i.e. T* = kBT/∊). Extending previous rigorous results, one can prove the existence of an ordering transition at finite temperature when 0 < σ < 1, and its absence when σ ≥ 1. We have studied the border case σ = 1, by means of computer simulation. Similarly to the magnetic counterparts of the present model, we found evidence suggesting a transition to a low-temperature phase with slow decay of correlations and infinite susceptibility, i.e. a Berezhinskiǐ–Kosterlitz–Thouless-like transition; the transition temperature was estimated to be Θ = 0.475 ± 0.005.