Prime Ideal Theory for General Algebras

We introduce ideals, radicals and prime ideals in arbitrary algebras with at least one binary operation, and we show that various separation lemmas and prime ideal theorems are special instances of one general theorem which, in turn, is equivalent to the Boolean Prime Ideal Theorem (or Ultrafilter Principle).     

Tables of curves with many points

These tables record results on curves with many points over finite fields. For relatively small genus () and a small power of or we give in two tables the best presently known bounds for , the maximum number of rational points on a smooth absolutely irreducible projective curve of genus over a field of cardinality . In additional tables we list for a given pair the type of construction of the best curve so far, and we give a reference to the literature where such a curve can be found.

     

The Division Relation: Congruence Conditions and Axiomatisability

We examine a universal algebraic abstraction of the semigroup theoretic concept of “divides:” a divides b in an algebra A if for some n ∈ ω, there is a term t(x, y ₁,…, y _n ) involving all of the listed variables, and elements c ₁,…, c _n such that t ^A (a, c ₁,…, c _n ) = b. The first order definability of this relation is shown to be a very broad generalisation of some familiar congruence properties, such as definability of principal congruences. The algorithmic problem of deciding when a finitely generated variety has this relation definable is shown to be equivalent to an open problem concerning flat algebras. We also use the relation as a framework for establishing some results concerning the finite axiomatisability of finitely generated varieties.     

Experimental and numerical study on the propagation of impulsive sound around buildings

Propagation of impulsive sound around buildings and induced structural loading are investigated experimentally and numerically. Experiments were conducted on a rectangular building at Virginia Tech using sonic booms generated by shaped charges with an explosive weight of 0.78 kg, constructed from detonation cord. These experiments were simulated with a three-dimensional numerical model, in the context of geometrical acoustics (GA), by combining the image source method for the reflected field (specular reflections) with an extension of the Biot–Tolstoy–Medwin (BTM) method for the diffracted field. In this model, it is assumed that the acoustic propagation is linear and that all surfaces are acoustically rigid. This numerical model is validated against a boundary element (BE) solution and experimental data, showing a good overall agreement. The key advantages of this GA modeling approach for this application include the ability to model large three-dimensional domains over a wide frequency range and also to decompose the sound field into direct, reflected, and diffracted components, thus providing a better understanding of the sound-propagation mechanisms. Finally, this validated numerical model is used to investigate sound propagation around a cluster of six rectangular buildings, for a range of elevated source positions simulating sonic booms from aircraft.     

Evolution of Orbits and Quantum Correlation Functions by Quadratic Hamiltonians

Quantum systems with quadratic Hamiltonians are considered. Some results about the time evolution of homogeneous polynomials and of quantum correlation functions are given. The image of arbitrary orbit of Weyl–Heisenberg group under this time evolution is shown to be again an orbit of this group. For quantum free particle it is shown that its time evolution intersects arbitrary such orbit at most once. A result about existence of more orbits having the same dispersion of some quantum position is presented.PACS: 02.20.Qs, 02.30.Sa     

Asymmetric bound states of spiral pairs in excitable media

We present a stable regime of asymmetric bound states for spiral pairs in a generic numerical model of a homogeneous excitable medium. In this regime, one spiral tip (slave) rotates around the other (master). Master-slave dynamics occur for both same-chirality and opposite-chirality spiral pairs in a range of parameters and initial conditions. We study the dependency of master-slave characteristics on the medium's excitation threshold and present a phenomenological model that accounts for the qualitative properties of master-slave dynamics.     

Rigidity percolation in particle-laden foams

We study the viscoelastic behavior of aqueous foam mixed with solid noncolloidal particles. We show that adding a tiny amount of grains can enhance the elastic and loss shear moduli by more than 1 order of magnitude. The scaling of these moduli with solid volume fraction is in qualitative agreement with that predicted by an effective-medium rigidity percolation model. We present a simple model, based on capillary attraction, to explain the particle-size dependence of the threshold.     

Uniqueness of reconstruction of multiphase morphologies from two-point correlation functions

The restoration of the spatial structure of heterogeneous media, such as composites, porous materials, microemulsions, ceramics, or polymer blends from two-point correlation functions, is a problem of relevance to several areas of science. In this contribution we revisit the question of the uniqueness of the restoration problem. We present numerical evidence that periodic, piecewise uniform structures with smooth boundaries are completely specified by their two-point correlation functions, up to a translation and, in some cases, inversion. We discuss the physical relevance of the results.     

A Minimax Principle for the Eigenvalues in Spectral Gaps

A minimax principle is derived for the eigenvalues in the spectralgap of a possibly non-semibounded self-adjoint operator. Itallows the nth eigenvalue of the Dirac operator with Coulombpotential from below to be bound by the nth eigenvalue of asemibounded Hamiltonian which is of interest in the contextof stability of matter. As a second application it is shownthat the Dirac operator with suitable non-positive potentialhas at least as many discrete eigenvalues as the Schrödingeroperator with the same potential.