Mathematics Commons^™

Zeros Of The Jost Function For A Class Of Exponentially Decaying Potentials, Daphne Gilbert, Alain Kerouanton

Articles

We investigate the properties of a series representing the Jost solution for the differential equation $-y''+q(x)y=lambda y$, $x geq 0$, $q in mathrm{L}({mathbb{R}}^{+})$. Sufficient conditions are determined on the real or complex-valued potential $q$ for the series to converge and bounds are obtained for the sets of eigenvalues, resonances and spectral singularities associated with a corresponding class of Sturm-Liouville operators. In this paper, we restrict our investigations to the class of potentials $q$ satisfying $|q(x)| leq ce^{-ax}$, $x geq 0$, for some $a>0$ and $c$ greator than 0.

The Simultaneous Onset And Interaction Of Taylor And Dean Instabilities In A Couette Geometry, C. P. Hills, A. P. Bassom

Articles

The fluid flow between a pair of coaxial circular cylinders generated by the uniform rotation of the inner cylinder and an azimuthal pressure gradient is susceptible to both Taylor and Dean type instabilities. The flow can be characterised by two parameters: a measure of the relative magnitude of the rotation and pressure effects and a non-dimensional Taylor number. Neutral curves associated with each instability can be constructed but it has been suggested that these curves do not cross but rather posses `kinks'. Our work is based in the small gap, large wavenumber limit and considers the simultaneous onset of Taylor …

On The Integrability Of A Class Of Nonlinear Dispersive Wave Equations, Rossen Ivanov

Articles

We investigate the integrability of a class of 1+1 dimensional models describing nonlinear dispersive waves in continuous media, e.g. cylindrical compressible hyperelastic rods, shallow water waves, etc. The only completely integrable cases coincide with the Camassa-Holm and Degasperis-Procesi equations.

On The Dressing Method For The Generalised Zakharov-Shabat System, Rossen Ivanov

Articles

The dressing procedure for the Generalised Zakharov-Shabat system is well known for systems, related to sl(N) algebras. We extend the method, constructing explicitly the dressing factors for some systems, related to orthogonal and symplectic Lie algebras. We consider ’dressed’ fundamental analytical solutions with simple poles at the prescribed eigenvalue points and obtain the corresponding Lax potentials, representing the soliton solutions for some important nonlinear evolution equations.

Torsion-Free Weakly Transitive Abelian Groups, Brendan Goldsmith, Lutz Strungmann

Articles

We introduce the notion of weak transitivity for torsion-free abelian groups. A torsion-free abelian group G is called weakly transitive if for any pair of elements x, y ∈ G and endomorphisms ϕ, ψ ∈ End(G) such that xϕ = y, yψ = x, there exists an automorphism of G mapping x onto y. It is shown that every suitable ring can be realized as the endomorphism ring of a weakly transitive torsion-free abelian group, and we characterize up to a number-theoretical property the separable weakly transitive torsion-free abelian groups.

On The Cotorsion Images Of The Baer-Specker Group, Brendan Goldsmith, T. Kelly, S, Wallutis

Articles

No abstract available

Quasi-Minimal Abelian Groups, Brendan Goldsmith, S. O. Hogain, S. Wallutis

Articles

An abelian group $G$ is said to be quasi-minimal (purely quasi-minimal, directly quasi-minimal) if it is isomorphic to all its subgroups (pure subgroups, direct summands, respectively) of the same cardinality as $G$. Obviously quasi-minimality implies pure quasi-minimality which in turn implies direct quasi-minimality, but we show that neither converse implication holds. We obtain a complete characterisation of quasi-minimal groups. In the purely quasi-minimal case, assuming GCH, a complete characterisation is also established. An independence result is proved for directly quasi-minimal groups.

On The Empirical Balanced Truncation For Nonlinear Systems, Marissa Condon, Rossen Ivanov

Articles

Novel constructions of empirical controllability and observability gramians for nonlinear systems for subsequent use in a balanced truncation style of model reduction are proposed. The new gramians are based on a generalisation of the fundamental solution for a Linear Time-Varying system. Relationships between the given gramians for nonlinear systems and the standard gramians for both Linear Time-Invariant and Linear Time-Varying systems are established as well as relationships to prior constructions proposed for empirical gramians. Application of the new gramians is illustrated through a sample test-system.

The Spectral Function For Sturm-Liouville Problems Where The Potential Is Of Wigner-Von Neumann Type Or Slowly Decaying, Daphne Gilbert, B.J. Harris, S.M. Riehl

Articles

We consider the linear, second-order, differential equation (∗) with the boundary condition (∗∗)

We suppose that q(x) is real-valued, continuously differentiable and that q(x)→0 as x→∞ with q∉L1[0,∞). Our main object of study is the spectral function ρα(λ) associated with () and (). We derive a series expansion for this function, valid for λ⩾Λ0 where Λ0 is computable and establish a Λ1, also computable, such that () and () with α=0, have no points of spectral concentration for λ⩾Λ1. We illustrate our results with examples. In particular we consider the case of the Wigner–von Neumann potential.

Rational Generalized Moonshine From Abelian Orbifoldings Of The Moonshine Module, Rossen Ivanov, Michael Tuite

Articles

We consider orbifoldings of the Moonshine Module with respect to the abelian group generated by a pair of commuting Monster group elements with one of prime order p = 2, 3, 5, 7 and the other of order pk for k = 1 or k prime. We show that constraints arising from meromorphic orbifold conformal field theory allow us to demonstrate that each orbifold partition function with rational coefficients is either constant or is a hauptmodul for an explicitly found modular fixing group of genus zero. We thus confirm in the cases considered the Generalised Moonshine conjectures for all rational …

K(Π, 1) For Artin Groups Of Finite Type, Colum Watt, Thomas Brady

Articles

This paper is a continuation of a programme to construct new K(π, 1)’s for Artin groups of finite type which began in [4] with Artin groups on 2 and 3 generators and was extended to braid groups in [3]. These K(π, 1)’s differ from those in [6] in that their universal covers are simplicial complexes. In [4] a complex is constructed whose top-dimensional cells correspond to minimal factorizations of a Coxeter element as a product of reflections in a finite Coxeter group. Asphericity is established in low dimensions using a metric of non-positive curvature. Since the nonpositive curvature condition is …

A Partial Order On The Orthogonal Group, Colum Watt, Thomas Brady

Articles

We define a natural partial order on the orthogonal group and completely describe the intervals in this partial order. The main technical ingredient is that an orthogonal transformation induces a unique orthogonal transformation on each subspace of the orthogonal complement of its fixed subspace.

On Unit Sum Numbers Of Rational Groups, Brendan Goldsmith, Christopher Meehan, S. Wallutis

Articles

The unit sum numbers of rational groups are investigated: the importance of the prime 2 being an automorphism of the rational group is discussed and other results are achieved by considering the number and distribution of rational primes which are, or are not, automorphisms of the group. Proof is given of the existence of rational groups with unit sum numbers greater than 2 but of finite value .

On Torsion And Mixed Minimal Abelian Groups, Brendan Goldsmith, S. O. Hogain

Articles

An abelian group is said to be minimal if it is isomorphic to all its subgroups of finite index. We obtain a complete characterisation of such groups in the torsion case; in the case of mixed groups of rank 1 we obtain a characterisation for some large classes of such groups.

Flow Patterns In A Two-Roll Mill, Christopher Hills

Articles

The two-dimensional flow of a Newtonian fluid in a rectangular box that contains two disjoint, independently-rotating, circular boundaries is studied. The flow field for this two-roll mill is determined numerically using a finite-difference scheme over a Cartesian grid with variable horizontal and vertical spacing to accommodate satisfactorily the circular boundaries. To make the streamfunction numerically determinate we insist that the pressure field is everywhere single-valued. The physical character, streamline topology and transitions of the flow are discussed for a range of geometries, rotation rates and Reynolds numbers in the underlying seven-parameter space. An account of a preliminary experimental study of …

Some Irrational Generalised Moonshine From Orbifolds, Rossen Ivanov, Michael Tuite

Articles

We verify the Generalised Moonshine conjectures for some irrational modular functions for theMonster centralisers related to the Harada-Norton, Held, M12 and L3(3) simple groups based on certain orbifolding constraints. We find explicitly the fixing groups of the hauptmoduls arising in each case.

The Problem Of A Viscoelastic Cylinder Rolling On A Rigid Half-Space, John Murrough Golden, G.A.C. Graham

Articles

The problem of a viscoelastic cylinder rolling on a rigid base, propelled by a line force acting at its centre, is solved in the noninertial approximation. The method used is based on a decomposition of hereditary integrals developed by the authors in previous work, and on the viscoelastic Kolosov-Muskhelishvili equations which are used to generate a Hilbert problem. In this formulation, the problem reduces to a nonsingular integral equation in space and time, which simplifies under steady-state conditions and for exponential decay materials, to algebraic form. There are also two subsidiary conditions.

In the case of a standard linear model, …

Eddy Structures Induced Within A Wedge By A Honing Circular Arc, C. P. Hills

Articles

In this paper we outline an expeditious numerical procedure to calculate the Stokes flow in a corner due to the rotation of a scraping circular boundary. The method is also applicable to other wedge geometries. We employ a collocation technique utilising a basis of eddy (similarity) functions introduced by Moffatt (1964) that allows us to satisfy automatically the governing equations for the streamfunction and all the boundary conditions on the surface of the wedge. The circular honing problem thereby becomes one-dimensional requiring only the satisfaction of conditions on the circular boundary. The advantage of using the Moffatt eddy functions as …

Eddies Induced In Cylindrical Containers By A Rotating End Wall, Christopher Hills

Articles

The flow generated in a viscous liquid contained in a cylindrical geometry by a rotating end wall is considered. Recent numerical and experimental work has established several distinct phases of the motion when fluid inertia plays a significant role. The current paper, however, establishes the nature of the flow in the thus far neglected low Reynolds number regime. Explicitly, by employing biorthogonality relations appropriate to the current geometry, it is shown that a sequence of exponentially decaying eddies extends outward from the rotating end wall. The cellular structure is a manifestation of the dominance of complex eigensolutions to the homogeneous …

Rotary Honing: A Variant Of The Taylor Paint-Scraper Problem, Christopher Hills, H. Moffatt

Articles

The three-dimensional Row in a corner of fixed angle α induced by the rotation in its plane of one of the boundaries is considered. A local similarity solution valid in a neighbourhood of the centre of rotation is obtained and the streamlines are shown to be closed curves. The effects of inertia are considered and are shown to be significant in a small neighbourhood of the plane of symmetry of the flow. A simple experiment confirms that the streamlines are indeed nearly closed; their projections on planes normal to the line of intersection of the boundaries are precisely the 'Taylor' …

Bounds For The Points Of Spectral Concentration Of Sturm-Liouville Problems, Daphne Gilbert, B.J. Harris

Articles

§1. Introduction. We consider the spectral function ρα(λ) associated with the Sturm–Liouville equation

with the boundary condition

Transitive And Fully Transitive Groups, Steve Files, Brendan Goldsmith

Articles

The notions of transitivity and full transitivity for abelian p-groups were introduced by Kaplansky in the 1950s. Important classes of transitive and fully transitive p-groups were discovered by Hill, among others. Since a 1976 paper by Corner, it has been known that the two properties are independent of one another. We examine how the formation of direct sums of p-groups affects transitivity and full transitivity. In so doing, we uncover a far-reaching class of p-groups for which transitivity and full transitivity are equivalent. This result sheds light on the relationship between the two properties for all p-groups.

On Unit Sum Number Of Rings And Modules, Brendan Goldsmith, S. Pabst, A. Scott

Articles

No abstract available

Describing Ideals Of Endomorphism Rings, Brendan Goldsmith, Simone Pabst

Articles

No abstract available.

On Transitive And Fully Transitive Abelian P-Groups, D. Carroll, Brendan Goldsmith

Articles

It is a long standing problem of A.L.S. Corner as to whether there exists a non-transitive fully transitive Abelian p-group with finite Ulm subgroups. The principal result of this paper shows that no such group exists for a special class of p-groups if the Ulm subjgroup is the direct sum of two cyclic groups. Some results on summands of trnsitive and fully transitive groups are obtained.

The Trace Of An Abelian Group:An Application To Digraphs, J. M. Burns, Brendan Goldsmith

Articles

The trace of a finite Abelian group is used to give a short proof of a result of Arlinghaus and Harary on the digraph number of such a group

On Endomorphisms And Automorphisms Of Some Pure Subgroups Of The Baer-Specker Group, A. L.S. Corner, Brendan Goldsmith

Articles

The endomoprhism algebras of certain pure subgroups of the Baer-Specker group are investigated to answer a question of Irwin. Automorphism groups of these Abelian groups are investigated and a realization theme is established.

On Quasi-Permutation Representations Of Finite Groups, J. M. Burns, Brendan Goldsmith, B. Hartley, R. Sandling

Articles

In [6], Wong defined a quasi-permutation group of degree n to be a finite group G of automorphisms of an n-dimensional complex vector space such that every element of G has non-negative integral trace. The terminology derives from the fact that if G is a finite group of permutations of a set ω of size n, and we think of G as acting on the complex vector space with basis ω, then the trace of an element g ∈ G is equal to the number of points of ω fixed by g. In [6] and [7], Wong studied the extent …

Cotorsion-Free Algebras As Endomorphism Algebras In L - The Discrete An Topological Cases., R. Gobel, Brendan Goldsmith

Articles

The discrete algebras A over a commutative ring R which can be realised as the full endomorphism algebra of a torsion-free R-module have been investigated by Dugas and Gobel under the additional set-theoretic axiom of constructibility, V = L. Many interesting results have been obtained for cotorsion-free algebras but the proofs involve rather elaborate calculations in linear algebra. Here these results are rederived in a more natural topological setting and substantial generalizations to topological algebras (which could not be handled in the previous linear algebra approach) are obtained. The results obtained are independent of the usual Zermelo-Fraenkel set theory ZFC.

The Walker Endomorphism Algebra Of A Mixed Module, Brendan Goldsmith, P. Zanardo

Articles

Archimedian valuation domains R are characterised in terms of the endomorphism algebras of non-splitting mixed modules of rank 1, in the Walker category.