bluuebunny/crossref_metadata_2025 huggingface dataset:
Some factorable nil rings of characteristic two
Authors: Duncan J. , Macdonald I. D.
Published: November 2011

It is shown that there exists an uncountable number of rings R with the properties (1) if r∈R then 2r = 0 and r2 = 0, (2) if r ∈ R then there exist s, t in R with r = st.
Weyl's m-coefficient in a B*-algebra
Authors: Moodie James
Published: November 2011

Some results are given for Weyl's m-coefficient in the algebra B = C(K). Included are a discussion of a theorem of Hille, a generalisation of Weyl's L.P./L.C. classification, and an example.
A quasi-periodic boundary value problem for the Laplacian and the continuation of its resolvent
Authors: Alber H. D.
Published: November 2011

Synopsis

A quasi-periodic boundary value problem for the Helmholtz equation in an unbounded domain is considered. This problem arises from scattering of plane waves by periodic structures.

Existence and uniqueness theorems are proved, and the continuation of the resolvent of this problem to a Riemannian surface is constructed. This construction makes no use of the continuation of the resolvent kernel but runs along the following lines:

First a family of differential operators is defined, which is holomorphic in a generalized sense. Then, using a result from analytic perturbation theory about families of operators with compact resolvent, it is shown that the family of inverses of these differential operators gives the desired continuation.
Computing values of a function on [0,1] from its moments
Authors: Wimp Jet
Published: November 2011

In this work we present an algorithm for computing an integrable function almost everywhere on (0,1) when its moments are known. The method is based on the use of certain delta-shaped sequences, and can be adjusted to take advantage of the local smoothness of the function.

As an application, we give an algorithm for the pointwise inversion of the Laplace transform which utilizes the values of the image function at equidistant points.
The essential self-adjointness of differential operators with positive coefficients
Authors: Keller R. G.
Published: November 2011

We consider the formally self-adjoint 2mth-order elliptic differential operator in ℝn given by where lt is an operator of order t, and pt ≧0 for t ≧1 and establish conditions under which the operator on is essentially self-adjoint in L2. A feature is that the major conditions (including the positivity of the coefficients) have to be imposed only in an increasing sequence of annular regions surrounding the origin.
Estimates of Liouville-Green type for solutions of systems of differential equations
Authors: Hadid S. B.
Published: November 2011

Synopsis

Let P(x) and Q(x) be n × n matrices whose entries involve a single given function q(x). Under suitable conditions, estimates

of Liouville-Green type are obtained for the components yν(x) of solutions of the system

In the case n = 2, the estimate is applied to systems arising from fourth-order equations and a generalization of a result of Eastham [2] is obtained.
Green's matrix and the formula of Titchmarsh-Kodaira for singular left-definite canonical eigenvalue problems
Authors: Schultze Bernd
Published: November 2011

Synopsis

The theory of singular left-definite canonical eigenvalue problems treated by Nieβen and Schneider in is generalized to arbitrary λ∈(ℂ\ℝ∪{0}. In this enlarged theory the Green's matrix of the problem is evaluated and a natural analogue of the Titchmarsh-Kodaira formula is proved. This formula permits the explicit computation of the spectral matrix playing the main role in the expansion theorems of this theory.
On a non-regular parametric integral
Authors: Hayman W. K.
Published: November 2011

Suppose that f(t, λ) is, for fixed t in an open interval (a, b), a regular (or analytic or holomorphic) function of λ, when λ lies in a domain D of the complex plane. We assume that f is integrable with respect to t for fixed λ ∈ D, and consider the parametric integral

By hypothesis F(λ) exists for λ ∈ D and Professor W. N. Everitt has asked whether F(λ) is necessarily regular as a function of λ.
A star-shaped condition for stability of linear retarded functional differential equations†
Authors: Silkowski Richard
Published: November 2011

SYNOPSIS

Sufficient conditions are developed for asymptotic stability of the autonomous linear functional differential equation of retarded type. If the asymptotic stability of

implies the asymptotic stability of

then these conditions are also necessary. Necessary and sufficient conditions are developed for the largest cone in the region of stability. These results are illustrated with the example
Near-rings of mappings
Authors: Meldrum J. D. P. , Oswald A.
Published: November 2011

Synopsis

This paper is concerned with the structure of M = Maps(G), the near-ring of all mappings from a group G to itself which commute with a group S* of automorphisms of G. Here S is S* together with the zero endomorphism. Necessary and sufficient conditions on the pair (G, S) are obtained for M to be (i) regular, (ii) unit regular, (iii) an equivalence near-ring. These conditions take a very simple form. In the case (iii), the two-sided M-subgroups of M are determined. The next result shows that under suitable conditions, M is a simple near-ring. A definition of transitivity is given for subnear-rings of M, and some properties of transitive near-rings are proved. Finally two examples are given to show that all the classes of near-rings considered are distinct.
On eigendistributions in linear transport theory
Authors: Lekkerkerker C. G.
Published: November 2011

Synopsis

An attempt is made to provide a sound basis for the method of singular eigenfunction expansions which has been in vogue in linear transport theory for some decades. The procedure is exemplified by a treatment of the one-dimensional neutron transport equation with a degenerate scattering function. Full-range as well as half-range results are derived. At the end of the paper the implications for a certain matrix factorization problem are given.
On the asymptotic expansions of solutions of an nth order linear differential equation
Authors: Paris R. B.
Published: November 2011

Synopsis

The asymptotic expansions of solutions of a class of linear ordinary differential equations of arbitrary order n are investigated for large values of the independent variable z in the complex plane. Solutions are expressed in terms of Mellin-Barnes integrals and their asymptotic expansions are subsequently determined by means of the asymptotic theory of integral functions of the hypergeometric type. Three classes of solutions are considered: (i) solutions whose behaviour is either exponentially large or algebraic for |z|→∞ in different sectors of the z-plane, (ii) solutions which are even and odd functions of z when the order n of the differential equation is even and (iii) solutions which are exponentially damped as |z|→∞ in a certain sector of the z-plane.
Stability of steady shearing flows subject to large shearing perturbations in a non-linear viscoelastic fluid
Authors: Slemrod M.
Published: November 2011

SYNOPSIS

This paper studies stability of steady state solutions in a non-linear viscoelastic fluid. The main technique is to imbed the equation of motion in singularly perturbed equations and apply an energy method and the parabolic maximum principle.
On uniform asymptotic expansions of finite Laplace and Fourier integrals
Authors: Soni Kusum
Published: November 2011

Synopsis

A uniform asymptotic expansion of the Laplace integrals ℒ(f, s) with explicit remainder terms is given. This expansion is valid in the whole complex s−plane. In particular, for s = −ix, it provides the Fourier integral expansion.
On the existence of bifurcating solutions in the presence of symmetries
Authors: Dancer E. N.
Published: November 2011

Synopsis

We study bifurcation problems in the presence of continuous groups of symmetries and obtain theorems on the existence and uniqueness of solutions. We also briefly consider some applications.
Complementary bivariational principles for linear problems involving non-self-adjoint operators
Authors: Cole R. J.
Published: November 2011

Synopsis

Bivariational principles are constructed that yield upper and lower bounds to the quantity 〈g0, f〉, where f is the solution of the equation f0−Tf = 0, g0 is a given function and T is a non-self-adjoint linear operator from a Hilbert space into itself. The theory is illustrated by an integral equation of Fredholm type.
On the stability of self-adjointness of Schrödinger operators under positive perturbations
Authors: Cycon Hans L.
Published: November 2011

Synopsis

We introduce a class of essentially self-adjoint Schrödinger operators, where essential self-adjointness is stable under positive potential perturbations. We show that this class is “stable” under certain perturbations and contains operators discussed by Simon and Kato. Finally, an extended essentially self-adjointness criterion is given.
Ramifications of Landau's inequality
Authors: Kwong Man Kam , Zettl A.
Published: November 2011

Synopsis

The problem of determining the best constant κ in the inequality ‖y′‖≦K ‖y‖ ‖y″‖ is discussed in the context of the classical Lp spaces, 1 ≦ p ≦ ∞.
Whittaker functions with both parameters large: uniform approximations in terms of parabolic cylinder functions
Authors: Olver F. W. J.
Published: November 2011

Synopsis

Asymptotic approximations are derived for the Whittaker functions Wκ,μ (z), Mκ, μ (z), Wικ, ιμ (iz) and Mικ, ιμ(iZ) for large positive values of the parameter μ that are uniform with respect to unrestricted values of the argument z in the open interval (0, ∞), and bounded real values of the ratio κ/μ. The approximations are in terms of parabolic cylinder functions, and in most instances are accompanied by strict error bounds.

The results are derived by application of a recently-developed asymptotic theory of second-order differential equations having coalescing turning points, and an extension of the general theory of equations of this kind is also included.
Products of idempotents in finite full transformation semigroups
Authors: Howie J. M.
Published: November 2011

Synopsis

If |X| = n and α is a singular mapping in J(X), define c(α) to be the number of cyclic orbits of α and f(α) to be the number of fixed points. Then α is expressible as a product of n + c(α)−f(α) idempotents of rank n − 1, and no smaller number of idempotents of rank n − 1 will suffice. The maximum possible value of n + c(α)–f(α) is [3/2(n − 1)], which is thus a best possible global lower bound for the number of idempotents required to generate a singular element of J(X).
On regular solutions of a nonlinear equation of Choquard's type
Authors: Menzala Gustavo Perla
Published: November 2011

Synopsis

We study the nonlinear equation

in ℝ3, where Δ denotes the Laplacian operator, and R and K are real-valued functions satisfying suitable conditions. We use a variational formulation to show the existence of a non-trivial weak solution of the above equation for some real number λ. Because of our assumptions on R and K we shall look for solutions which are spherically symmetric, decrease with r = |x| and vanish at infinity.
On the non-existence of semi-groups for some equations of continuum mechanics
Authors: Wilkes N. S.
Published: November 2011

Synopsis

Linear semi-group theory can be used to prove the existence of solutions to the equations of linear elasticity when the elasticity tensor is positive definite. Here, it is shown that this condition is also necessary for the existence of a semi-group. The method is also applied to linear dissipative equations.
Rearrangement inequalities
Authors: Minc Henryk
Published: November 2011

Synopsis

Rearrangement inequalities of Ruderman and Minc are generalized.
The solution of Lyapunov's matrix equation by a geometric method
Authors: Young N. J.
Published: November 2011

Synopsis

The author has recently proposed a new algorithm for the solution of the Lyapunov matrix equation of stability theory. This algorithm is based on a formula for the solution of a special case of the equation. This formula is established in the present paper by means of a geometric interpretation. The key ideas are the uses of shift operators and non-orthogonal projections in infinite-dimensional Hilbert space.
A semilinear elliptic eigenvalue problem, I
Authors: Keady Grant , Norbury John
Published: November 2011

Synopsis

The nonlinear operator equation (N), , describes both semilinear elliptic boundary value problems (P) and their natural discretisations (Ph). (Here is a positive compact linear operator and [t]+ ≡ (t + |t|)/2 for all t ∊ ℝ.) It is proved that, for q ≧ 0 (q ≢ 0), in a Banach lattice E the equation (N) has an unbounded continuum ℱ of nontrivial solutions (λ, ψ) ∊ ℝ × E bifurcating from infinity at (λ1, ∞) (here is the first (positive) eigenvalue of ). All nontrivial solutions (λ, ψ) have λ ≧ λ1, and if maps a smaller cone Ks, into itself then there is a ℱs ⊂ℝ×Ks with similar properties to ℱ. The existence theory for (N) is applied to problems (P) and (Ph) which are defined on the simply-connected domain Ω. It is shown that the projection of ℱ on the λ-axis is either unbounded if the continuous function q > 0 except on a set of measure zero in Ω, or is bounded if q ≡ 0 on a subdomain of Ω. If is the second eigenvalue for in (Ph) then there is at most one nontrivial solution for each λ satisfying λ1 < λ < λ2; in the corresponding uniqueness result for (P) q is restricted to being strictly positive somewhere on ∂Ω Additional properties for the solutions of (P) and (Ph) are also proved.
Units of indefinite quaternion algebras
Authors: Chalk J. H. H.
Published: November 2011

Synopsis

If ℍ is a rational indefinite quaternion algebra andTis an order of ℍ, letG+(T) denote the subgroup of unitsu∊Twith normn(u) = + 1. For a certain class of orders, Eichler has determined the measure μ(G+(T)) ofG+(T), viewed as a Fuchsian group. This is extended to arbitrary orders by methods depending only upon classical number-theory and group theory. As an application, an estimate of the magnitude of a small non-trivial solution of the diophantine Pellian equation

is supplied and the restrictions on the integersDandPassociated with previous work on this question are eliminated.
The spectrum of multiparameter problems in Hilbert space
Authors: McGhee D. F. , Roach G. F.
Published: November 2011

Synopsis

In this paper, appropriate definitions are given for the spectra of multiparameter problems in Hilbert space when the operators involved are not necessarily self-adjoint. Notions of adjoint multiparameter problems are introduced, and some properties of the spectrum of a single linear operator are generalised to multiparameter settings.

The results which are achieved are compared with those in the literature which have beendeveloped for multiparameter problems involving self-adjoint operators.
Relative resolutions of homology manifolds
Authors: Martin Nigel
Published: November 2011

Synopsis

In the theory of resolutions of polyhedral homology manifolds to PL manifolds, itis natural to ask whether, when such a resolution exists, it is possible to preserve a given subspace of the original homology manifold under the resolution. The answer is provided in the affirmative so long as the codimension of the subspace is at least 3. In addition, it is shown that if the original dimension was at least 5 then a resolution may be chosen so as to induce an isomorphism of fundamental groups. As a corollary to these results we see that any homology n-manifold homology n-sphere may be PL embedded in S+3.
Cauchy and Poisson integral representations for ultradistributions of compact support and distributional boundary values
Authors: Pathak R. S.
Published: November 2011

Synopsis

Ultradistributions of compact support are represented as the boundary values of Cauchy and Poisson integrals corresponding to tubular radial domains Tc' =ℝn + iC', C'⊂⊂C, where C is an open, connected, convex cone. The Cauchy integral of is shown to be an analytic function in TC' which satisfies a certain boundedness condition. Analytic functions which satisfy a specified growth condition in TC' have a distributional boundary value which can be used to determine an distribution.
Idempotent-connected abundant semigroups
Authors: El-Qallali A. , Fountain J. B.
Published: November 2011

Synopsis

A general theory for a class of abundant semigroups is developed. For a semigroup S in this class let E be its set of idempotents and  the subsemigroup of S generated by E. When  is regular there is a homomorphism with a number of desirable properties from S onto a full subsemigroup of the Hall semigroup T. From this fact, analogues of results in the regular case are obtained for *-simple and ℐ*-simple abundant semigroups.
Quasi-adequate semigroups
Authors: El-Qallali A. , Fountain J. B.
Published: November 2011

Synopsis

The investigation of general quasi-adequate semigroups is initiated. These are semigroups which are abundant and in which the idempotents form a subsemigroup. For such a semigroup S we study the minimum good congruence γ such that S/γ is adequate. Results on γ together with results from a previous paper of the authors are used to obtain a structure theorem for a class of quasi-adequate semigroups.
Products of idempotents in the semigroup of singular endomorphisms of a finite-dimensional vector space
Authors: Dawlings R. J. H.
Published: November 2011

Synopsis

Let Mn be the semigroup, under composition, of endomorphisms of an n-dimensional vector space. Let E be the set of idempotents of Mn. It has been shown that each singular element of Mn() may be expressed as a composition of elements of E. In this paper the minimum number of idempotents needed in this composition is determined. This is given in terms of n and one parameter dependent on the element. Further, it is shown that En–1⊂, while En=.
An elliptic boundary-value problem with a discontinuous nonlinearity
Authors: Keady G.
Published: November 2011

Synopsis

We study the boundary-value problem, for(λ/k,ψ),

Here ∆ denotes the Laplacian,His the Heaviside step function and one of A or k is a given positive constant. We define

and usually omit the subscript. Throughout we are interested in solutions with ψ>0 inΩ and hence with λ/=0.

In the special case Ω = B(0, R), denoting the explicit exact solutions by ℑe, the following statements are true, (a) The set Aψ, issimply-connected, (b) Along ℑe, the diameter of Aψtendsto zero when the area of Aψ, tends to zero.

For doubly-symmetrised solutions in domains Ω such as rectangles, it is shown that the statements (a) and (b) above remain true.
Locally minimising solutions of − Δu = u(1 − |u|2) in R2
Authors: Sandier Etienne
Published: November 2011

We prove that locally minimising solutions of − Δu = u(1 − |u|2) in R2, i.e. solutions that minimise the action in any bounded domain of R2, are such that ∫R2(1 − |u|2)2(x) dx < + ∞. We prove a similar property for locally minimising solutions in a half-plane.
On a semilinear equation in ℝ2involving bounded measures
Authors: Vazquez Juan L.
Published: November 2011

Synopsis

We study the semilinear equation –Δu+ β(u) =fin ℝ2, where β is a continuous increasing real function with β(0) = 0 andfis a bounded Radon measure. We show the existence of a solution, which is unique in the appropriate class, provided that each of the point masses contained infdoes not exceed some critical value denned in terms of the growth of (β at ∞ This condition is shown to be necessary for the existence of solutions, even locally. The one-dimensional situation is also discussed.
Singularities of dispersion relations
Authors: Dangelmayr G.
Published: November 2011

Synopsis

Generic singularities occurring in dispersion relations are discussed within the framework of imperfect bifurcation theory and classified up to codimension four. Wave numbers are considered as bifurcation variables x =(x1,…, xn) and the frequency is regarded as a distinguished bifurcation parameter λ. The list of normal forms contains, as special cases, germs of the form ±λ +f(x), where f is a standard singularity in the sense of catastrophe theory. Since many dispersion relations are ℤ(2)-equivariant with respect to the frequency, bifurcation equations which are ℤ(2)-equivariant with respect to the bifurcation parameter are introduced and classified up to codimension four in order to describe generic singularities which occur at zero frequency. Physical implications of the theory are outlined.
Sturm—Liouville problems and discontinuous eigenvalues
Authors: Everitt W. N. , Möller M. , Zettl A.
Published: November 2011

If a Sturm—Liouville problem is given in an open interval of the real line, then regular boundary value problems can be considered on compact sub-intervals. For these regular problems, all with necessarily discrete spectra, the eigenvalues depend on both the end-points of the compact intervals, and upon the choice of the real separated boundary conditions at these end-points. These eigenvalues are not, in general, continuous functionsof the end-points and boundary conditions. The paper shows the surprising form of these discontinuities. The results have applications to the approximations of singular Sturm—Liouville problems by regular problems, and to the theoretical aspects of the Sleign2 Computer program.
Undercompressive shocks and Riemann problems for scalar conservation laws with non-convex fluxes
Authors: Hayes Brian , Shearer Michael
Published: November 2011

The Riemann initial value problem is studied for scalar conservation laws whose fluxes have a single inflection point. For a regularization consisting of balanced diffusive and dispersive terms, the travelling wave criterion is used to select admissible shocks. In some cases, the Riemann problem solution contains an undercompressive shock. The analysis is illustrated by exploring parameter space for the Buckley–Leverett flux. The boundary of the set of parameters for which there is a physical solution of the Riemann problem for all data is computed. Within the region of acceptable parameters, the solution hasseveral different forms, depending on the initial data; the different forms are illustrated by numerical computations. Qualitatively similar behaviour is observed in Lax–Wendroff approximations of solutions of the Buckley–Leverett equation with no dissipation or dispersion.
Degenerate Lyapunov functionals of a well-known prey–predator model with discrete delays
Authors: He Xue-Zhong
Published: November 2011

It is commonly believed that, as far as stabilities are concerned, ‘small delays are negligible in some modelling processes’. However, to have an affirmative answerfor this common belief is still an open problem for many nonlinear equations. In this paper, the classical Lotka–Volterra prey–predator equation with discrete delays

is considered, and, by using degenerate Lyapunov functionals method, an affirmative answer to this open problem on both local and global stabilities of the prey–predator delay equations is given. It is shown that degenerate Lyapunov functional method is a powerful tool for studying the stability of such nonlinear delay systems. A detailed and explicit procedure of constructing such functionals is provided. Furthermore, some explicit estimates on the allowable sizes of the delays are obtained.
Adjoint action of a finite loop space. II
Authors: Iwase Norio , Kono Akira
Published: November 2011

Adjoint actions of compact simply connected Lie groups are studied by Kozima and the second author based on the series of studies on the classification of simple Lie groups and their cohomologies. At odd primes, the first author showed that there is a homotopy theoretic approach that will prove the results of Kozima and the second author for any 1-connected finite loop spaces. In this paper, we use the rationalization of the classifying space to compute the adjoint actions and the cohomology of classifying spaces assuming torsion free hypothesis, at the prime 2. And, by using Browder's work on the Kudo–Araki operations Q1 for homotopy commutative Hopf spaces, we show the converse for general 1-connected finite loop spaces, at the prime 2. This can be done because the inclusion j: G > BAG satisfies the homotopy commutativity for any non-homotopy commutative loop space G.
The Hardy–Rellich inequality for polyharmonic operators
Authors: Owen Mark P.
Published: November 2011

The Hardy-Rellich inequality given here generalizes a Hardy inequality of Davies, from the case of the Dirichlet Laplacian of a region Ω ⊆ ℝN to that of the higher-order polyharmonic operators with Dirichlet boundary conditions. The inequality yields some immediate spectral information for the polyharmonic operators and also bounds on the trace of the associated semigroups and resolvents.
Asymptotics for the spectral heat function and bounds for integrals of Dirichlet eigenfunctions
Authors: van den Berg M. , Watson S. P.
Published: November 2011

We introduce the spectral heat function H associated with the Dirichlet-Laplace–Beltrami operator ΔM on a compact smooth Riemannian manifold M with a non-empty smooth boundary. We obtain two-term asymptotics for H without assuming any billiard conditions on M. As a corollary, we obtain estimates for the integral of the k′th Dirichlet eigenfunction of ΔM over M for k→∞.
The accelerated convergence of heat transfer series in high frequency oscillating boundary layer flows of compressible fluids
Authors: Gribben R. J.
Published: November 2011

Synopsis

Series, previously derived, for the heat transfer at the surface of a cylinder fixed in a compressible gas which flows with a high frequency oscillation about a steady mean velocity, are considered in the range of frequencies for which the convergence is slow. Different methods of accelerated convergence, suitable for Fourier and power series, are applied over different parts of the cycle and graphs of the heat transfer over a period of the oscillation are obtained.
Remark on Hilbert's boundary value problem for Beltrami systems
Authors: Begehr Heinrich
Published: November 2011

Synopsis

The Schauder continuation method for nonlinear problems is based on appropriate a priori estimates for related linear equations. Recently, in a paper by the present author and G. C. Hsiao, the Hilbert boundary value problem with positive index for nonlinear elliptic systems in the plane was solved by this method but the constructive derivation of the a priori estimate necessarily required a restriction on the ellipticity condition. This is because the norm of the generalized Hilbert transform in the case of positive index is too big. Here, as in a forthcoming paper by G.C. Wen, an indirect and therefore non-constructive proof of the a priori estimate is given which does not require any further restrictions and allows the Hilbert boundary value problem to be solved for nonlinear elliptic systems in general.
Quasispectral subspaces of quasinilpotent operators
Authors: Omladič Matjaž
Published: November 2011

Synopsis

We give a definition of “quasispectral maximalč subspaces for a quasinilpotent, but not nilpotent, bounded Banach space operator. The definition applies to a class of operators, close to the Volterra operator.
Relaxation for some dynamical problems
Authors: Dacorogna B.
Published: November 2011

Synopsis

In this article, we study the functionalWhere Ω ⊂ ĝnis a bounded open set andu: Ω ×(0,T)→ ĝmand whenF: Rnm→R fails to be quasiconvex. We show that with respect to strong convergence of ∂u/∂tand weak convergence of ∇×u, the above functional behaves aswhereQFis the lower quasiconvex envelope ofF.
On a nonlinear eigenvalue problem occurring in population genetics
Authors: Clément Ph. , Peletier L. A.
Published: November 2011

Synopsis

We discuss the nonlinear eigenvalue problem

withr(–×) = –r(×) and r'≧0.

For ε =h=0, the solution to Problem P is wellknown, and every solution, exceptu= 0 andu=1, is unstable with respect to the corresponding parabolic problem.
Convergence theorems for intermediate problems
Authors: Beattie Christopher A. , Greenlee W. M.
Published: November 2011

Synopsis

General convergence theorems are established for the lower part of the intermediate operator spectra arising in lower bound methods of Aronszajn type.Convergence of the T*T method is considered in detail, as are constructions that allow perturbations that are nonclosable in the underlying Hilbert space. Problems with essential spectra are admitted, as is the use of infinite rank projections capable of displacing essential spectra. These last features are necessary for a theory applicable to Schrodinger operators corresponding to atomic systems.
Lattices of congruences on free finitely generated commutative semigroups and direct products of cyclic monoids
Authors: Trueman D. C.
Published: November 2011

Synopsis

Let S11× V1 be a direct product of a cyclic monoid S11with S1 not a group, and a semigroup V1 with an adjoined identity. We prove thatboth the lattice of congruences on (S11 × V1)/{(1,1)} and the lattice of congruences on (S11 × V1) are neither lower semimodular nor uppersemimodular. We then prove that the lattice of congruences on a free finitely generated commutative semigroup with more than one generator is neither lower semimodular nor upper semimodular.
Effects of commensurable delays in the stability of difference equations
Authors: Ferreira José M.
Published: November 2011

Synopsis

This paper studies how, in a difference equation, uniform asymptotic stability for commensurable delays implies uniform asymptotic stability globally in the delays. A result is also given on renorming the space C= C([ – ∥r∥, 0]; ℂn so that a difference operator has norm less than one if it is uniformly asymptotically stable globally in the delays.
The arithmetic of certain quartic curves
Authors: Cassels J. W. S.
Published: November 2011

Synopsis

Let F(X, Y, Z) be a non-singular quadratic form with rational coefficients. The curve EF(x2, y2, z2) = 0 is of genus 3. A procedure is described for deciding whether there is an effective divisor on E of degree 3 defined over the rationals. There is such a divisor if and only if there is a point on E defined over some algebraic number field of odd degree. An example is constructed for which there is no such divisor although (i) there are points on E defined over all p-adic fields and over the reals and (ii) there are infinitely many rational points on each of the three curves F(X, y2, z2) = 0, F(x2, Y, z2) = 0 and F(x2, y2, Z) = 0.
On Opial type inequalities in two independent variables
Authors: Pachpatte B. G.
Published: November 2011

Synopsis

The aim of the present paper is to establish some new integral inequalities of Opial type in two independent variables. Our results are the two independent variable generalizations of some of the inequalities recently established by the present author and in special cases yield the two independent variable analogue of the Opial inequality and its generalization given by G. S. Yang.
Existence and continuity of a weak solution to the problem of a free boundary in a concentrated capacity
Authors: Shillor M.
Published: November 2011

Synopsis

A weak (enthalpy) formulation of the problem of a free boundary moving in the thermal concentrated capacity is given. The problem is to solve the heat equation in a given domain, while on a part of the boundary of this domain the solution (or rather its trace) solves a Stefan problem with forced convection. The existence of a global weak solution is proved by the method of finite differences. Some regularity is obtained from this proof, and the continuity of the temperature is proved. The uniqueness, which is related to the existence of mushy regions, is discussed. A classical enthalpy formulation is conjectured.
On some nonlinear elliptic equations involving derivatives of the nonlinearity
Authors: Carrillo J. , Chipot M.
Published: November 2011

Synopsis

We give some results on existence and uniqueness for the solution of elliptic boundary value problems of type

when the βi are not necessarily smooth.
An Ω-estimate for the lattice rest of a convex planar domain
Authors: Nowak Werner Georg
Published: November 2011

Synopsis

Let D be a compact convex planar domain containing the origin, the boundary of which is of class C∞ and has finite non-vanishing curvature throughout. For the number A(i) of lattice points in the “blown up” domain √tD, the estimate

is established. This is a generalization of Hardy's classical result for the circle problem. The proof is based on asymptotic formulae for certain exponential integrals due to E. Hlawka.
Existence conditions for higher order eigensets of multiparameter operators
Authors: Binding Paul , Browne Patrick J. , Turyn Lawrence
Published: November 2011

Synopsis

We consider classes of self-adjoint operators for which the nonpositive part of the spectrum consists of eigenvalues ρ0(λ)≦ ρ1(») ≦ … repeated according to multiplicity. The sets of λ where ρi(λ) is negative and zero are labelled Ni and Zi respectively, and Pi = ℝk\(Ni ∪ Zi). We study conditions on the Vj sufficient to ensure nonemptiness of at least one of Ni, Zi and Pi for all T or for all positive definite T, as well as conditions which are necessary in the sense that failure permits emptiness for at least one T.

As an example of our results, we show in the Sturm–Liouville case

with L∞ coefficients and separated end conditions, that nonemptiness of Zi for all T (i.e. for all p > 0, all q and all boundary data) is equivalent to the i-independent condition that the ftj do not vanish simultaneously on a set of positive measure.
Tangencies between stable and unstable manifolds
Authors: Battelli Flaviano , Palmer Kenneth J.
Published: November 2011

Synopsis

In this paper perturbation theory is used to construct systems in four dimensions having two dimensional stable and unstable manifolds which touch along a homoclinic orbit but only with a second order contact.
The extension problem and the mod 2 cohomology of the space of loops on Spin (N)
Authors: Kuribayashi Katsuhiko
Published: November 2011

Synopsis

Let {Er, dr} be a spectral sequence converging to a Hopf algebra H*. We give a method of reconstructing H* from E∞**. By using our method, we determine the mod 2 cohomology of the space of loops on a simply-connected space whose mod 2 cohomology is isomorphic to that of Spin(N) as an algebra over the Steenrod algebra.
A class of deficiency zero soluble groups of derived length 4
Authors: Doostie H. , Jamali A. R.
Published: November 2011

Synopsis

In this paper we study a class of 2-generator 2-relator groups G(m) and show that they are all finite. Moreover, two infinite subclasses are soluble of derived length 4.
Function spaces related to gauge groups
Authors: Sutherland W. A.
Published: November 2011

Synopsis

Components in the function space of maps from a space X to the classifying space BG of a topological group G can sometimes be distinguished up to homotopy type by a Samelson product method. When X is a closed Riemann surface and G is a unitary group, this method is nearly sufficient to classify the components up to homotopy type.
Regularity and large time behaviour of solutions of a conservation law without convexity
Authors: Dafermos C. M.
Published: November 2011

Synopsis

Using the method of generalized characteristics, we discuss the regularity and large time behaviour of admissible weak solutions of a single conservation law, in one space variable, with one inflection point.

We show that when the initial data are C∞ then, generically, the solution is C∞ except: (a) on a finite set of C∞ arcs across which it experiences jump discontinuities (genuine shocks or left contact discontinuities); (b) on a finite set of straight line characteristic segments across which its derivatives of order m, m = 1, 2,…, experience jump discontinuities (weak waves of order m); and (c) on the finite set of points of interaction of shocks and weak waves. Weak waves of order 1 are triggered by rays grazing upon contact discontinuities. Weak waves of order m, m ≥ 2, are generated by the collision of a weak wave of order m − 1 with a left contact discontinuity.

We establish sharp decay rates for solutions with initial data of the following types: (a) with bounded primitive; (b) with primitive having sublinear growth; (c) in L1; (d) of compact support; and (e) periodic.
Sheaf representations of strongly harmonic rings
Authors: Simmons Harold
Published: November 2011

Synopsis

We describe five different sheaf representations of a ring, all of which are full and four of which are faithful. We give a characterization of strongly harmonic rings, and show that for such rings, the four faithful representations agree.
Travelling waves in radially symmetric parabolic systems
Authors: Bates Peter W.
Published: November 2011

Synopsis

It is shown that all travelling or standing plane-wave solutions to certain radially symmetric parabolic systems may be found by solving a related scalar ordinary differential equation (ODE). The radially symmetric systems considered here are those whose reaction term is radially directed and points inward near infinity. The stability of these waves is also discussed. Many systems arising in the physical sciences are included in the class studied and so the classification and stability of the travelling waves has physical significance.
Certain 3-decompositions of complete graphs, with an application to finite fields
Authors: Rowlinson Peter
Published: November 2011

Synopsis

A necessary condition is obtained for a complete graph to have a decomposition as the line-disjoint union of three isomorphic strongly regular subgraphs. The condition is used to determine the number of non-trivial solutions of the equation x3+y3 = z3 in a finite field of characteristic p ≡ 2 mod 3.
A global invertibility theorem for manifolds with boundary
Authors: Weinstein Alan
Published: November 2011

Synopsis

A local homeomorphism from a compact, connected manifold with boundary to a simply connected manifold without boundary is shown to be one-to-one if it is one-to-one on each component of the boundary.
A transmission problem for the plate equation
Authors: Leis R. , Roach G. F.
Published: November 2011

Synopsis

A scattering theory is developed for transmission problems associated with the plate equation. Asymptotic methods of solution for large time are examined as are questions concerning regularity of solution, nature of the associated spectrum and existence of appropriate wave operators. It is shown that in contrast to solutions of the wave equation, signals can propagate with an infinite dispersion velocity.
Homoclinic solutions for coupled systems of differential equations
Authors: Grindrod P. , Sleeman B. D.
Published: November 2011

Synopsis

Topological ideas based on the notion of flows and Wazewski sets are used to establish the existence of homoclinic orbits to a class of Hamiltonian systems. The results, as indicated, are applicable to a variety of reaction diffusion equations including models of bundles of unmyelinated nerve axons.
Corrigenda: Resolution of a semilinear equation in L1
Authors: Gallouët Thierry , Morel Jean-Michel
Published: November 2011

Page 277, line 14: Read “measure” instead of “distribution”.

Page 287, line 1: Read “(Ω)” instead of “′(Ω)”,

line 13: Read “explanation” instead of “generalization”,

line 23: Read “In fact, Corollary 1” instead of “In fact, Corollary 2”.

We thank Professor L. Véron for directing our attention to these mistakes.
A note on a resonance problem
Authors: Lupo Daniela , Solimini Sergio
Published: November 2011

Synopsis

In this paper, we prove the existence of at least one solution to the problem

where ∆k is an eigenvalue of the linear part, h is orthogonal to the eigenspace corresponding to ∆R and g is a nonlinear perturbation which can be, for instance, a continuous periodic real function with mean value zero. We employ the techniques used by the second author in a previous paper in which the same result was obtained in the case in which ∆R is assumed to be simple. The final result is obtained by using variational methods and in particular a suitable version of the saddle point theorem of P. Rabinowitz.
Critical behaviour of nonlinear elliptic boundary value problems suggested by exothermic reactions
Authors: Wiebers Henning
Published: November 2011

Synopsis

We consider a class of semilinear elliptic boundary value problems depending on a parameter, which arise in the theory of combustion. Based on the results in another paper by the same author, a rigorous quantitative connection is shown between the solution set of the boundary value problem and that of a simple scalar equation (the Semenov approximation).
Hölder continuous dependence in nonlinear elastodynamics
Authors: Chiriţă Stan
Published: November 2011

Synopsis

In this paper we establish conditions to prove that if classical solutions to the initial boundary value problems for nonlinear elastodynamics exist, then they depend Hölder continuously on their initialdata and body forces.
Rees matrix semigroups over inverse semigroups
Authors: Pastijn F. J. , Petrich Mario
Published: November 2011

Synopsis

A Rees matrix semigroup over an inverse semigroup contains a greatest regular subsemigroup. The regular semigroups obtained in this manner are abstractly characterized here. The greatest completely simple homomorphic image and the idempotent generated part of such semigroups are investigated. Rectangular bands of semilattices of groups and some special cases are characterized in several ways.
Generalizations of the Landau–Hadamard inequality and inequalities for quadratic polynomials of operators
Authors: Boyadzhiev Khr. N.
Published: November 2011

Synopsis

We give generalizations of the Landau–Hadamard inequality ‖u′‖2 ≦ K ‖u‖ ‖u″‖ replacing u” by the second-order differential expression u″ − (α + β)u′ + αβu (α, β ∈ ℂ). The new functional inequalities are then used to obtain similar inequalities for dissipative and skew-Hermitian operators.
A non-linear Goursat problem for a high order polyvibrating equation
Authors: Borzymowski Andrzej
Published: November 2011

Synopsis

This paper proves the existence of a solution of a non-linear Goursat problem for a partial differential equation of order 2p (p ≧ 2) with the boundary conditions given on 2p curves emanating from a common point. The problem is reduced to a system of integro-differential-functional equations and then Schauder's fixed point theorem is applied.
A repeated transformation in the asymptotic solution of linear differential systems
Authors: Eastham M. S. P.
Published: November 2011

Synopsis

A sequence of transformations of the type Y = (I + o(1))Z is developed for the system Y′(x) = {Λ(x) + R(x)}Y(x), where Λ is diagonal. The transformations bring in the derivatives of R in succession until the Levinson form is obtained when a given derivative is reached. This theory covers rapidly varying coefficients and it extends results which are known for constant Λ.
Minors of the Wronskian of the differential equation Lny + p(x)y = 0. II. Dominance of solutions
Authors: Elias Uri
Published: November 2011

Synopsis

The equation studied here is Lny + p(x)y = 0, where Ln is a disconjugate differential operator and p(x) is of a fixed sign. We define a basis of the solution space and order its elements according to their relative magnitudes near infinity. Our method is independent of the possible oscillation or nonoscillation of the solutions and it is achieved by utilising the fact that some minors of the Wronskian never vanish.
Non-negative solutions for a class of non-positone problems
Authors: Castro Alfonso , Shivaji R.
Published: November 2011

Synopsis

In the recent past many results have been established on non-negative solutions to boundary value problems of the form

where λ>0, f(0)>0 (positone problems). In this paper we consider the impact on the non-negative solutions when f(0)<0. We find that we need f(u) to be convex to guarantee uniqueness of positive solutions, and f(u) to be appropriately concave for multiple positive solutions. This is in contrast to the case of positone problems, where the roles of convexity and concavity were interchanged to obtain similar results. We further establish the existence of non-negative solutions with interior zeros, which did not exist in positone problems.
Positive solutions of elliptic systems with bounded nonlinearities
Authors: Noussair Ezzat S. , Swanson Charles A.
Published: November 2011

Synopsis

Semilinear elliptic partial differential systems of second order with weak coupling are considered in exterior domains Ω ⊆ ℝN, N≧3. Conditions on the nonlinearities are given which guarantee the existence of solutions u with positive components in Ω such that u|∂Ω = 0 and u(x)→0 uniformly as |x|→∞. Asymptotic decay estimates for the solutions are established, including an exponential decay law under extra hypotheses.
Oscillation criteria and the discreteness of the spectrum of self-adjoint, even order, differential operators
Authors: Došlý Ondřej
Published: November 2011

Synopsis

This paper deals with the oscillation properties of self-adjoint differential equations

The oscillation criteria are derived, which allows a unified approach to the investigation of (*) near a finite or infinite singularity. These criteria are used to study spectral properties of singular differential operators associated with (*).
Maximisation and minimisation on classes of rearrangements
Authors: Burton G. R. , McLeod J. B.
Published: November 2011

Synopsis

Maximisation and minimisation of the Dirichlet integral of a function vanishing on the boundary of a bounded domain are studied, subject to the constraint that the Laplacean be a rearrangement of a given function. When the Laplacean is two-signed, non-existence of minimisers is proved, and some information on the limits of minimising sequences obtained; this contrasts with the known existence of minimisers in the one-signed case. When the domain is a ball and the Laplacean is one-signed, maximisers and minimisers are shown to be radial and monotone. Existence of maximisers is proved subject additionally to a finite number of linear constraints, with particular reference to ideal fluid flows of prescribed angular momentum in a disc.
Initial boundary value problems for isentropic gas dynamics
Authors: Takeno Shigeharu
Published: November 2011

Synopsis

For piston problems for a system of isentropic gas dynamics, convergence theorems of a difference scheme are obtained by compensated compactness theory and by analysis of the difference scheme.
Homogenisation of a class of fourth order equations with application to incompressible elasticity
Authors: Francfort G. A.
Published: November 2011

Synopsis

Upon formalising an analogy between two-dimensional Stokes flow and two-dimensional isotropic conductivity, we exhibit a class of fourth order equations which behave “isomorphically” like isotropic conductivity from the standpoint of homogenisation and from that of corresponding bounding methods on possible effective behaviours. In particular, Lipton's result on the G-closure problem for mixtures of two incompressible elastic materials is recovered in the two-dimensional case.
The gap phenomenon for variational integrals in Sobolev spaces
Authors: Giaquinta M. , Modica G. , Souček J.
Published: November 2011

Synopsis

We show that a gap phenomenon occurs for general variational integrals for mappings from a domain Rn into a Riemannian manifold if has a non-trivial topology.
Initial conditions for integrable-square solutions to singular differential equations
Authors: Walker Philip W.
Published: November 2011

Synopsis

This paper deals with determining in a constructive manner those members of a linear space of functions which are of integrable-square. The space considered is the set of solutions to an ordinary differential equation, and the solutions of integrable-square are delineated by way of initial conditions. Numerical procedures for implementing the construction are discussed, and application is made to the deficiency index problem. Results from some specific computations are given.