Unitary Geometric Algebra

The geometric significance of the imaginary unit in a complex geometric algebra has troubled the author for 40?years. In the unitary geometric algebra presented here, the imaginary i is a unit (pseudo) vector with square minus one which anti commutes with all of the real vectors. The resulting natural hermitian inner product and hermitian outer product induce a grading of the algebra into complex k-vectors. Basic orthogonality relationships are studied.     

On subdivisions of simplicial complexes: Characterizing localh-vectors

A well-known combinatorial invariant of simplicial complexes is theh-vector, which has been the subject of much combinatorial research. This paper deals withlocal h-vectors, recently defined by Stanley as a tool for studyingh-vectors of simplicial subdivisions. The face-vector of any simplicial complex can only increase when the complex is subdivided; how does theh-vector change? Motivated by this question, Stanley derived certain useful properties of localh-vectors. In this paper we use mainly geometric arguments to show that these properties characterize localh-vectors, andregular localh-vectors.     

The Mother Minkowski Algebra of Order m

It is found that all polynomials of up to degree m have an encoding as m-vectors in a geometric algebra referred to as the Mother Minkowski algebra of order m. It is then shown that all conformal transformations may be applied to these m-vectors, the results of which, when converted back into polynomial form, give us the transformed surfaces in terms of the zero sets of the original and final polynomials.     

Balanced subdivision and enumeration in balanced spheres

We study here the affine space generated by the extendedf-vectors of simplicial homology (d – 1)-spheres which are balanced of a given type. This space is determined, and its dimension is computed, by deriving a balanced version of the Dehn-Sommerville equations and exhibiting a set of balanced polytopes whose extendedf-vectors span it. To this end, a unique minimal complex of a given type is defined, along with a balanced version of stellar subdivision, and such a subdivision of a face in a minimal complex is realized as the boundary complex of a balanced polytope. For these complexes, we obtain an explicit computation of their extendedh-vectors, and, implicitly,f-vectors.Supported in part by NSF Grant DMS-8403225.     

Fredholm vector fields and a transversality theorem

We define the notion of a Fredholm vector field and prove a transversality result giving conditions under which a vertical family of such vector fields generically have nondegenerate zeros. Many geometric objects like minimal surfaces, geodesics, and harmonic maps arise as the zeros of a Fredholm vector field.     

Resonance in Beverton–Holt population models with periodic and random coefficients

An increase in the mean population density in a fluctuating environment is known as resonance. Resonance has been observed in laboratory experiments and has been studied in discrete-time population models. We investigate this phenomenon in the Beverton–Holt model with either periodic or random variables for two biologically relevant coefficients: the intrinsic growth rate and the carrying capacity. Three types of resonance are defined: arithmetic, geometric and harmonic. Conditions are derived for each type of resonance in the case of period-2 coefficients and some results for period p>2. For period 2, regions in parameter space where each type of resonance occurs are shown to be subsets of each other. For the case of random coefficients with constant intrinsic growth rate, it is shown that the three types of resonance do not occur. Numerical examples illustrate resonance and attenuance (decrease in the mean population density) in the Beverton–Holt model when the coefficients are discrete random variables.     

On several new Hilbert-type inequalities involving means operators

In this paper, we establish several new Hilbert-type inequalities with a homogeneous kernel, involving arithmetic, geometric, and harmonic mean operators in both integral and discrete case. Such inequalities are derived by virtue of some recent results regarding general Hilbert-type inequalities and some well-known classical inequalities. We also prove that the constant factors appearing in established inequalities are the best possible. As an application, we consider some particular settings and compare our results with previously known from the literature.     

Variational formulations for scattering in a three‐dimensional acoustic waveguide

Variational formulations for direct time‐harmonic scattering problems in a three‐dimensional waveguide are formulated and analyzed. We prove that the operators defined by the corresponding forms satisfy a Gårding inequality in adequately chosen spaces of test and trial functions and depend analytically on the wavenumber except at the modal numbers of the waveguide. It is also shown that these operators are strictly coercive if the wavenumber is small enough. It follows that these scattering problems are uniquely solvable except possibly for an infinite series of exceptional values of the wavenumber with no finite accumulation point. Furthermore, two geometric conditions for an obstacle are given, under which uniqueness of solution always holds in the case of a Dirichlet problem. Copyright © 2007 John Wiley & Sons, Ltd.     

Parameters of AG codes from vector bundles

We investigate the parameters of the algebraic–geometric codes constructed from vector bundles on a projective variety defined over a finite field. In the case of curves we give a method of constructing weakly stable bundles using restriction of vector bundles on algebraic surfaces and illustrate the result by some examples.