Space Of Rational Functions Research Articles

Representation of Symmetric Functions as Gram Determinants

We introduce a representation of symmetric functions as determinants of Gram matrices on the space of rational functions in one variable. We use a bilinear map on the space of rational functions that allows us to identify rational functions with linear functionals. Using basic properties of Gram matrices and rational functions we obtain results about symmetric functions, like a general Jacobi–Trudi identity, and an expansion formula for general symmetric functions in terms of Schur functions. Our approach is suitable to study in a unified way the confluent “symmetric” (or pseudo-symmetric) functions proposed by Rota, and several kinds of generalized symmetric functions, like the factorial Schur functions and symmetric functions in several sets of indeterminates.

The space of rational maps on ℙ 1

The space of rational maps on ℙ 1

Rational maps, monopoles and skyrmions

We discuss the similarities between BPS monopoles and skyrmions, and point to an underlying connection in terms of rational maps between Riemann spheres. This involves the introduction of a new ansatz for Skyrme fields. We use this to construct good approximations to several known skyrmions, including all the minimal energy configurations up to baryon number nine, and some new solutions such as a baryon number seventeen Skyrme field with the truncated icosahedron structure of a buckyball. The new approach is also used to understand the low-lying vibrational modes of skyrmions, which are required for quantization. Along the way we discover an interesting Morse function on the space of rational maps which may be of use in understanding the Sen forms on the monopole moduli spaces.

Monopoles, braid groups, and the dirac operator

Using the relation between the space of rational functions on ℂ, the space ofSU(2)-monopoles on ℝ3, and the classifying space of the braid group, see [10], we show how the index bundle of the family of real Dirac operators coupled toSU(2)-monopoles can be described using permutation representations of Artin's braid groups. We also show how this implies the existence of a pair consisting of a gauge fieldA and a Higgs field Φ on ℝ3 whose corresponding Dirac equation has an arbitrarily large dimensional space of solutions.

Multiple solutions of the Dirichlet problem for harmonic maps from discs into 2-spheres

In this paper we study the topology of the space of maps from the disc into the 2-sphere with given boundary data by comparing to the topology of the space of rational functions. And we prove theorems about multiple solutions of the Dirichlet problem for harmonic maps in one homotopy class.

The Level Manifold of a Generalized Toda Equation Hierarchy

The finite nonperiodic Toda lattice equation induces a linear one-parameter flow on a space of rational functions. The level manifold of the Toda equation hierarchy is shown to be a product of lines. Our main results establish a generalization of this Toda hierarchy which will be called the cyclic-Toda hierarchy. It is proved that the cyclic-Toda hierarchy is completely integrable and its level manifold is diffeomorphic to a disjoint union of cylinders.

The level manifold of a generalized Toda equation hierarchy

The finite nonperiodic Toda lattice equation induces a linear one-parameter flow on a space of rational functions. The level manifold of the Toda equation hierarchy is shown to be a product of lines. Our main results establish a generalization of this Toda hierarchy which will be called the cyclic-Toda hierarchy. It is proved that the cyclic-Toda hierarchy is completely integrable and its level manifold is diffeomorphic to a disjoint union of cylinders.

A criterion for pure simplicity

Every purely simple Kronecker module M of finite rank is an extension of a finite-dimensional torsion-free module N by an infinite-dimensional torsion-free module F of rank 1, but not conversely. We prove that for each such N and F a purely simple extension M does exist. Extensions of N by F are built up using linear functionals on K( X), the space of rational functions. Our existence proof is based on a criterion in terms of such functionals for the pure simplicity of an extension M. The possibility of embedding such M back into modules of rank 1 is also explored.

Some Computational Aspects of a Method for Rational Approximation

Numerical aspects of a method for rational approximation of analytic functions on regions in the complex plane are considered. The approximation method divides the problem of computing a rational approximant into three subproblems. First, one chooses a. space of rational functions, then one selects a basis for this space, and finally one determines an element of the space by interpolation. For approximation on regions with bounded simply connected complement, we discuss the choice of space and basis from a numerical point of view. We illustrate with computed examples.