bluuebunny/crossref_metadata_2025 huggingface dataset:
On the spectra of non-self-adjoint realisations of second-order elliptic operators
Authors: Evans W. D.
Published: November 2011

Synopsis

Let τ denote the second-order elliptic expression

where the coefficients bj and q are complex-valued, and let Ω be a spherical shell Ω = {x:x ∈ ℝn, l <|x|<m} with l≧0, m≦∞. Under the conditions assumed on the coefficients of τ and with either Dirichlet or Neumann conditions on the boundary of Ω, τ generates a quasi-m-sectorial operator T in the weighted space L2(Ω;w). The main objective is to locate the spectrum and essential spectrum of T. Best possible results are obtained.
On the swirling flow between rotating coaxial disks, asymptotic behaviour, II
Authors: Kreiss Heinz Otto , Parter Seymour V.
Published: November 2011

Synopsis

Consider solutions 〈H(x, ε), G(x, ε)〉 of the von Kármán equations for the swirling flow between two rotating coaxial disks

and

We assume that |H(x, ε)| + |Hʹ(x, ε)| + |G(x, ε)|≦B. This work considers shapes and asymptotic behaviour as ε→0+. We consider the type of limit functions 〈H(x), G(x)〉 that are permissible. In particular, if 〈H(x, ε), G(x, ε)〉 also satisfy the boundary conditions H(0, ε)=H(1, ε)=0, Hʹ(0, ε)=Hʹ(1, ε)=0 then H(x) has no simple zeros. That is, there does not exist a point Z ε [0, 1] such that H(x)=0, Hʹ(z)≠0. Moreover, the case of “cells” which oscillate is studied in detail.
Almost-periodic forcing for a wave equation with a nonlinear, local damping term
Authors: Haraux Alain
Published: November 2011

Synopsis

Let Ω⊂ℝnbe a bounded open domain and T = ∂Ω. It β is a maximal monotone graph in ℝ×ℝ with 0ϵβ(0), and f: ℝ×Ω→ℝ is measurable with t→ f(t,.) S2-almost periodic as a function ℝ→L2(Ω), we consider the nonlinear hyperbolic equation

We show that:

(i) if ゲ is strictly increasing and (1) has a solution ω on ℝ with [ω, Əω/Ət] almost periodic: , for any solution of (1) there exists with u(t,.)–ω(t,.)—ξin

(ii) if β is single valued and everywhere defined, the existence of ω as above implies that, for every solution of (1), there exists Ϛ(t, x) with ә2Ϛ/әt2–0△Ϛ = in ℝ×Ω and u(t,.)–ω(t,.)—0 in as t → +∞

(iii) if β–1 is uniformly continuous and ゲ satisfies some growth assumption (depending on N), for every f as above, there exists ω solution of (1) on ℝ with [ω, Əω/Ət] almost periodic: ℝ → .
The distribution of the values of a random power series in the unit disk
Authors: Jakob Matthias , Offord A. C.
Published: November 2011

Synopsis

This is a study of the family of power series where Σ αnZn has unit radius of convergence and the εn are independent random variables taking the values ±1 with equal probability. It is shown that if

then almost all these power series take every complex value infinitely often in the unit disk.
Remarks on the singularity indices of Arnol'd and Berry
Authors: Chillingworth David , Romero-Fuster Carmen
Published: November 2011

Synopsis

We give a coordinate-free algebraic interpretation for singularity indices of quasihomogeneous functions and their unfoldings, as first exploited systematically by Berry in statistical optics. We also make some observations on geometric interpretation, on formulae for indices, and on the difficulties of rigorous treatment for non-quasihomogeneous functions.
Regular solutions of wave equations containing nonlinearities of the type L Φ L
Authors: Weyer Jürgen
Published: November 2011

Synopsis

Regular solutions of the forced nonlinear wave equation uu + L4u + LΦLu = r are studied in Hilbert spaces. L is a linear, positive, selfadjoint operator and the nonlinear nucleus Φ(u) = f(|u|2)u is generated by a C1-function f, such that LΦ(Lu) = f(|Lu|2)L2u. If the initial value data u(0) = ϕ and u1(0) = ψ belong to the domain D(Lk+4) and D(Lk+2), respectively, and if rεD(Lk), then there is a (global) solution u(t) such that u ε D(Lk+4), ut ε D(Lk+2) and uuε D(Lk) for all times t. The abstract result is applied to examples in nonlinear elasticity theory.
Subvarieties of the class of MS-algebras
Authors: Blyth T. S. , Varlet J. C.
Published: November 2011

Synopsis

In a previous publication (1983), we defined a class of algebras, denoted by MS, which generalises both de Morgan algebras and Stone algebras. Here we describe the lattice of subvarieties of MS. This is a 20-element distributive lattice. We then characterise all the subvarieties of MS by means of identities. We also show that some of these subvarieties can be described in terms of three important subsets of the algebra. Finally, we determine the greatest homomorphic image of an MS-algebra that belongs to a given subvariety.
Linearization stability and Signorini Series for the traction problem in elastostatics
Authors: Marsden J. E. , Wan Y. H.
Published: November 2011

Synopsis

This paper uses previous results of Chillingworth, Marsden and Wan on symmetry and bifurcation for the traction problem in three dimensional elastostatics to establish new results on the Signorini expansion. We show that the Signorini compatibility conditions are necessary and sufficient for linearization stability and analogies with results known for other field theories are pointed out. Under an explicit non-degeneracy condition, a new series expansion is given in which successive terms are inductively determined in pairs rather than singly. Our results include as special cases, classical results of Signorini, Tolotti and Stoppelli.
On Schauder bases in Hardy spaces
Authors: Oswald P.
Published: November 2011

Synopsis

It is proved that in the case ½<p<l the periodic Franklin system forms a Schauder basis for the real Hardy space Hp(T) defined on the one-dimensional torus.

In this note we prove the following

Theorem. The periodic Franklin system forms a Schauder basis in the real Hardyspace Hp(T) defined on the one-dimensional torus if ½<p< l.
On integral transformations with G-function kernels
Authors: Rooney P. G.
Published: November 2011

Synopsis

Integral transformations T of the formal form

where G is a Meijer-G function are studied on the spaces ℐμ,r of functions f such that

Results are obtained on the boundedness, representation and range of the transformations, and on when they have an inverse of the same formal form.
The breadth of the lattice of those varieties of inverse semigroups which contain the variety of groups
Authors: Reilly Norman R.
Published: November 2011

Synopsis

The relations v1 and v2 defined on the lattice ℒ of varieties of inverse semigroups by v1 if and only if and v2 if and only if , where denotes tie variety of groups, are both congruences on ℒ the class v1, is simply the lattice of varieties of grcups and is therefore known to have cardinality .

The class v2 is precisely the sublattice of ℒ consisting of those varieties containing . Each v1-class contains preciselyone element of v2. The main result of this paper establishes that the sublattice v2 of ℒ has breadth . From this it follows that the lattice ℒ/v1 also has breadth . Some consequences concerning varieties generated by fundamental inverse semigroups are also considered.
Qualitative theory for hyperbolic characteristic initial value problems
Authors: Kreith Kurt , Pagan Gordon
Published: November 2011

Synopsis

Characteristic initial value problems associated with hyperbolic equations of the form uxy + g(x, y)u = 0 are considered for (x, y)∈ℝ+× ℝ+. New criteria for the existence of a nodal line asymptotic to the axes are established, as are criteria for the existence of a zero beyond such a nodal line. Some numerical solutions are presented in graphical form and discussed relative to what is known about oscillation properties of such problems.
Entire solutions of linear differential equations and bounds for growth and index numbers
Authors: Shah S. M.
Published: November 2011

Synopsis

In this paper entire solutions of differential equations with polynomial coefficients are considered and bounds on the maximum modulus and the index are obtained, when the equation is of second order and the coefficients are of second degree.
Positive extensions of automorphisms of spin factors
Authors: Robertson A. Guyan
Published: November 2011

Synopsis

A spin factor is a JW-factor of type I2. It is shown that certain automorphisms of finite dimensional spin factors extend to extremal positive linear maps on complex matrix algebras which are not decomposable, and hence, do not preserve extreme rays of the positive cone.
A perturbation method and the limit-point case of even order symmetric differential expressions
Authors: Mergler Bernhard , Schultze Bernd
Published: November 2011

Synopsis

We give a new perturbation theorem for symmetric differential expressions (relatively bounded perturbations, with relative bound 1) and prove with this theorem a new limit-point criterion generalizing earlier results of Schultze. We also obtain some new results in the fourth-order case.
The Rothe method for the wave equation in several space dimensions
Authors: Martensen Erich
Published: November 2011

Synopsis

The interior initial-boundary value problem for the wave equation in m ≧ 1 space dimension is considered for vanishing boundary values. Certain regularity, dependent on m, is required for the solution and additionalboundary conditions, the number of which being also dependent on m, are imposed on the given right hand side. Emphazising the case m = 3, the Rothe method is applied after the problem has been rewritten as a hyperbolic first order evolution problem for m + 1 unknown functions. The sequence of discrete solutions obtained is shown to be discretely convergent to the continuous solution in the sense of uniform convergence if the solution of the continuous problem is assumed to exist. A priori estimates are derived both for the discrete solutions and the continuous solution.
On diffracted wave fronts
Authors: Wolfe Peter
Published: November 2011

Synopsis

In this paper we study the wave equation, in particular the propagation of discontinuities. Two problems are considered: diffraction of a normally incident plane pulse by a plane screen and diffraction of a spherical wave by the same screen. It is shown that when an incident wave front strikes the edge of the screen a diffracted wave front is produced. The discontinuities are precisely computed in a neighbourhood of the edge for a small time interval after the arrival of the incident wave front and a theorem of Hörmander on the propagation of singularities is used to obtain a globalresult.
Some loci in S5 associated with systems of conics in a plane
Authors: Knight A. J.
Published: November 2011

Synopsis

If the conic α, a0x2 + a1y2 + a2z2 + 2a3yz + 2a4zx + 2a5xy=0, is represented by the point (a0, a1, a2, a3, a4, a5) of S5, the transforms of α by projectivities that fix a second conic ω will generally be represented by points of a threefold in S5 containing (a0, a1,…, a5). It is shown that this threefold is in general a rational sextic belonging to an ∞2 family, that is composed of ∞1 sets of projectively equivalent threefolds. Special, exceptional members of the family are discussed.

Equations for the threefold are found in terms of the mutual projection invariants of ω and α.
A multiplicity result for a class of elliptic boundary value problems
Authors: Amann Herbert , Hess Peter
Published: November 2011

Synopsis

We consider a mildly nonlinear elliptic boundary value problem depending on a parameter. Given appropriate hypotheses concerning the asymptotic behaviour of the nonlinearity, we derive lower bounds on the number of solutions. The results complement an earlier theorem due to Kazdan and Warner [6].
A simpler proof of two inequalities of Brodlie and Everitt
Authors: Beesack P. R.
Published: November 2011

Synopsis

In 1975 K. W. Brodie and W. N. Everitt dealt with integral inequalities of the form

in the two casesT= ℝ,T= ℝ+, and obtained the best possible constantsKT(μ)for all μ ε ℝ. The proof was not elementary, but an elementary proof was given in 1977 by E. T. Copson. This note shows how Copson's method can be greatly simplified so as to obtain the results in a more straightforward manner.
The second dual of a Banach algebra
Authors: Duncan J. , Hosseiniun S. A. R.
Published: November 2011

Synopsis

We give a survey of the current state of knowledge on the Arens second dual of a Banach algebra, including some simplified proofs of known results, some new results, some open problems and a full bibliography of the subject.
Bounds for the point spectra of separated Dirac operators
Authors: Evans W. D. , Harris B. J.
Published: November 2011

Synopsis

In this paper real numbers A1, A2 are obtained which are such that for λ > A1 or λ < A2 there are no non-trivial solutions of the equation

which are in L2(a,∞) for some a>0. Precise values for A1 and A2 are obtained in certain cases.
Aprioribounds for a class of nonlinear elliptic equations and applications to physical problems
Authors: Bandle Catherine
Published: November 2011

Synopsis

Upper and lower bounds for the solutions of a nonlinear Dirichlet problem are given and isoperimetric inequalities for the maximal pressure of an ideal charged gas are constructed. The method used here is based on a geometrical result for two-dimensional abstract surfaces.
Existence and multiplicity result for a class of second order elliptic equations
Authors: Hofer Helmut
Published: November 2011

Synopsis

Some second order semilinear elliptic boundary value problems of the Ambrosetti-Prodi-type are studied. Existence and multiplicity of solutions is proved in dependence on a parameter. Constructing a global strongly increasing fixed point operator in a suitable function space, observing - under appropriate conditions, which are in some sense optimal–that the fixed point operator has some properties similar to a strongly positive linear endomorphism, one unifies and improves the treatment of such problems, whether the nonlinearity is dependent on the gradient or not, and obtains some new results.
Oscillation criteria for third order hyperbolic characteristic initial value problems
Authors: Travis C. C. , Yoshida Norio
Published: November 2011

Synopsis

Sufficient conditions for oscillation of solutions to third order hyperbolic characteristic initial value problems are established. The results generalize known oscillation criteria for second order hyperbolic problems.
Groups of units of zero ternary quadratic forms
Authors: Maclachlan Colin
Published: November 2011

Synopsis

The groups of units of indefinite ternary quadratic forms with rational integer coefficients contain subgroups of index two which are isomorphic to Fuchsian groups and which, for zero forms, are commensurable with the classical modular group. This is used to obtain a family of forms whose groups are representatives of the conjugacy classes of maximal groups associated with zero forms. The signatures of the groups of the forms in this family are determined and it is shown that the group associated to any zero form is isomorphic to a subgroup of finite index in the group of one of three particular forms. This last result should be compared with the corresponding result by Mennicke on non-zero forms.
A class of bisimple, idempotent-generated congruence-free semigroups
Authors: Howie J. M.
Published: November 2011

Synopsis

Let X be a set with infinite regular cardinality m. Within the full transformation semigroup ℑ(X) a subsemigroup Sm is described which is bisimple and idempotent-generated. Its minimum non-trivial homomorphic image has both these properties and is also congruence-free. The semigroup contains an isomorphic copy of every semigroup having order less than m.
Boundary value problems for the vibrating string equation
Authors: Sleeman B. D.
Published: November 2011

Synopsis

Dirichlet, Neumann and mixed boundary value problems for the equation uxx −uyy = 0 are considered for a variety of rectangular domains. Uniqueness of solutions to these non-well-posed problems is considered by separation of variables methods. The question of uniqueness is also discussed for domains other than rectangles.
Generalised functions and their convolution
Authors: Jones D. S.
Published: November 2011

Synopsis

Some attributes of a new method of defining the convolution and multiplication of generalised functions, too singular to be encompassed by earlier theories, are developed and its extension to the case of several variables is described.
Elliptic transmutation I
Authors: Carroll Robert
Published: November 2011

Synopsis

Given and similar , modelled on radial Laplace-Beltrami operators (ρp = , in this paper we begin the study of transmutations which leads to elliptic equations Working with and transmutations Qm → −D2 for m > −½ and −D2 → for m < −½, we obtain a transmutation formulation and derivation of many results of generalized axially symmetric potential theory in the first case and in both cases generalized Hilbert transforms (different). Canonical generalizations are then automatic using general transmutation theory.
Gradient estimates for degenerate diffusion equations II
Authors: Alikakos Nicholas D. , Rostamian Rouben
Published: November 2011

Synopsis

We establish upper and lower bounds for various norms of solutions and their gradients for the equation ut = div (|∇u|m−1 ∇u) in ℝN in terms of the norms of the initial data. Based on the L∞ estimate of ∇u, we conclude that u(x, t) is Lipschitz continuous in space-time, for all t>0, whenever u(x,0) is in L1(ℝN).
Analogues of the bicyclic semigroup in simple semigroups without idempotents
Authors: Jones Peter R.
Published: November 2011

Synopsis

By an important theorem of Andersen, any semigroup, containing idempotents, which is simple but not completely simple contains a copy of the bicyclic semigroup B = 〈a, b | ab = 1〉. In this paper the semigroups A = 〈a, b | a2b = a〉 and C = 〈a, b | a2b = a, ab2 = b〉 are shown to play a similar role in various classes of simple semigroups without idempotents, particularly in those for which Green's relation is nontrivial. For example it is shown that every right simple semigroup without idempotents is a union of copies of A; every finitely generated simple semigroup without idempotents contains either A or C. In a generalisation of a different sort it is shown that the bicyclic semigroup divides every simple semigroup without idempotents.

Similar results are obtained for 0-simple semigroups without nonzero idempotents.
Some second-order systems of partial differential equations of composite type
Authors: Wei Lin
Published: November 2011

Synopsis

The Cauchy problem and the Dirichlet-Cauchy type problem of some second-order systems of partial differential equations of composite type of two unknown functions are investigated. Such systems possess some of the characteristics not only of elliptic but also of hyperbolic systems in the same domain. Representations of the solutions are found for the upper half plane. To this end, the composite systems are reduced to the canonical form by means of successive applications of three kinds of linear transformations. Function theoretic methods are used to obtain representation formulae. Furthermore, some composite systems of 2m-unknown function are also considered.
E-unitary regular semigroups
Authors: Szendrei Mária B.
Published: November 2011

Synopsis

Let S be an E-unitary regular semigroup and V a variety of bands. We prove that S can be embedded into a semidirect product of a band from V by a group if and only if S can be embedded in a canonical way into the semidirect product of the free band in V over a well-determined partial semigroup by the greatest group homomorphic image of S. Moreover, we show that every E-unitary regular semigroup with regular band of idempotents E can be embedded into a semidirect product of a band B by a group, where B belongs to the variety of bands generated by E.
Quasilinear Hamiltonian systems
Authors: Rogulski Jan S.
Published: November 2011

Synopsis

We consider quasilinear systems of 2N partial differential equations with 2N unknown functions depending on n + 1 variables as evolution systems on the space L2(Rn, RN) × L2(Rns, RN) endowed with a symplectic form induced by the standard scalar product on L2(Rn, RN). The necessary and sufficient conditions for such a system to be a Hamiltonian system are derived. The main purpose of this paper is to propose a straightforward link between the symplectic approach formulated by Chernoff, Hughes and Marsden and the multisymplectic formulations of evolution systems created by Kijowski and developed by Gawedzki and Kondracki. A general method of constructing the multisymplectic form and the Hamiltonian form for these systems is given.
Global theorems for symmetry sets of smooth curves and polygons in the plane
Authors: Banchoff T. F. , Giblin P. J.
Published: November 2011

Synopsis

The symmetry set of a plane curve y is the locus of centres of circles which either (i) are tangent to y at at least two distinct points, or (ii) have at least 4-point (A3) contact with y somewhere. The points (ii) also lie at cusps of the evolute of γ and the symmetry set, together with the evolute, can be studied by regarding their union as a full bifurcation set. For information on symmetry sets, see [2, 4, 5, 7, 8].

Here we shall use a theorem of Ozawa [9] to deduce global formulae relating three features of symmetry sets. The transitions which occur on the symmetry set of a generic one-parameter family of plane curves have been found [4], and more insight into the global formulae can be gained by tracing them through the transitions and verifying that they are invariant (see Section 2). We also briefly introduce symmetry sets of plane polygons in Section 3, and show that, contrary to first impressions, the same global formulae hold there.
A fourth order limit-3 expression with nonempty essential spectrum
Authors: Schultze Bernd
Published: November 2011

Synopsis

It is shown that even in the fourth order case there exist real symmetric ordinary differential expressions with nonempty essential spectrum which are not in the limit-point case.
Boundary regularity and normal derivatives of logarithmic potentials
Authors: Král J.
Published: November 2011

Synopsis

We consider the weak Neumann problem for logarithmic potentials in plane domains. We prove that this problem can be treated by the Fredholm–Radon method if and only if the boundary of the corresponding domain is formed by finitely many curves fulfilling specified regularity conditions.
Results old and new on the hyper-Bessel equation
Authors: Paris R. B. , Wood A. D.
Published: November 2011

Synopsis

We consider a variety of integral representations, single and multiple, old and new, for solutions of the hyper-Bessel equation u(n) – zmu =0. In particular, we show how a very early multiple Laplace integral solution of Molins (1876) may be related to recent Mellin–Barnes integral representations given by the present authors by way of multiple integral solutions given by Saxton and the second author for an associated equation. Although both these multiple integral solutions may be found by elementary methods, it is not easy to find their asymptotic expansions for large z, and we show how these may conveniently be obtained from our earlier results [10].
Floquet solutions of non-linear ordinary differential equations
Authors: Hassan H. S.
Published: November 2011

Synopsis

In this paper we study the solutions of the boundary value problem

where t ∊ℝ, x ∊ ℝN, f is a continuous function of (t,x)and locally Lipschitz in x and ω is a fixed positive number and λ ∊ ℝ. By using degree theory we prove results on the existence of solutions of (*) and the dependence of such solutions on λ. We shall prove that (*) does not have an isolated solution, and study the topological properties of the components of solutions of (*).
Minors of the wronskian of the differential equation Lny+p(x)y=0
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. It is shown that certain minors of the wronskian of this equation satisfy a very similar differential equation Mnz +p(x)z =0. We prove that some properties of the original equation which are essential for oscillation theory are inherited by the solutions of the second one.
A rank-one perturbation result on the spectra of certain operators
Authors: Datko R.
Published: November 2011

Synopsis

In this note we show that certain perturbations, involving rank-one stabilising operators, which correspond to small delays in some feedback control problems, shift a part of the point spectrum of the unperturbed operator into the right-half plane.
Sign-definite solutions in some linear elliptic systems
Authors: Cosner Chris , Schaefer Philip W.
Published: November 2011

Synopsis

We consider a weakly coupled set of two partial differential equations where the coupling matrix has variable elements and the principal part of each equation is the same uniformly elliptic operator. Weobtain necessary conditions that the system of equations can be decoupled. By decoupling the system and using a positivity lemma due to Hess and Kato, we determine the algebraic sign of the solution components. This work extends recent results of de Figueiredo and Mitidieri. Further, one can use these results to determine the sign of the solution to certain fourth order elliptic boundary value problems.
On subordinacy and analysis of the spectrum of Schrödinger operators with two singular endpoints
Authors: Gilbert D. J.
Published: November 2011

Synopsis

The theory of subordinacy is extended to all one-dimensional Schrödinger operatorsfor which the corresponding differential expressionL= –d2/(dr2) +V(r) is in the limit point case at both ends of an interval (a,b), withV(r) locally integrable. This enables a detailed classification of the absolutely continuous and singular spectra to be established in terms of the relative asymptotic behaviour of solutions ofLu = xu, x εℝ, asr→aandr→b. The result provides a rigorous but straightforward method of direct spectral analysis which has very general application, and somefurther properties of the spectrum are deduced from the underlying theory.
On interpolation of strictly (co-)singular linear operators
Authors: Beucher O. J.
Published: November 2011

Synopsis

We show that the property of linear operators to be in the surjective hull (injective hull) of the ideal of strictly singular (strictly cosingular) operators between Banach spaces is an interpolation property with respect to the real interpolation method with parameters 0 < ủ < 1 and < p < ℞.
Necked states of non-linearly elastic plates
Authors: Negrón-Marrero Pablo V.
Published: November 2011

Synopsis

In this paper we study the equilibrium equations for axisymmetric deformations of isotropic circular plates in tension. We give results on the global multiplicity of solutions and study the stability of the trivial homogeneous solution for large displacements.
Complex and real singularities of eigenfunction expansions
Authors: Zayed Ahmed I.
Published: November 2011

Synopsis

Let θ(t, λ) and (t,λ) be the solutions to the differential equation y“+(λ − q(t))y =0, − ∞ <t < ∞ such that θ(0, λ)) = l, θ'(0, λ)) = 0, ø(0, λ) = 0 and ø'(0, λ) = −1. It is known that for a function f(t)∈ L2(−∞,∞), we have

where for some measures dξ, dη; and dζ independent of f.

In the first part of this paper we devise a technique to locate the singular points of f(t) (t is complex) under the assumptions that E(λ), F(λ) are of order O(e-cvλ), as λ →∞and q(t) is a sufficiently nice function. In this case all the singularities of fare off the real axis. In the second part of the paper we relax the restriction on E(λ) and f(λ) so that they are of order O(λk) as λ →∞ for some constant k, and we show that in this case f(t) is a generalised function whose analytic representation is holomorphic in the upper and lower half planes. We then devise a technique to continue analytically across the real axis and locate its singularities thereon.
On the minimal eigenvalue of a positive definite operator determinant
Authors: Volkmer H.
Published: November 2011

Synopsis

In this paper we study a problem in multilinear algebra which consists of finding small values of a certain quotientμ/α. Here μ is the minimal eigenvalue of a positive definite operator determinant Δ of the type introduced by F. V. Atkinson, and α is the minimum of the quadratic form corresponding to Δ with respect to all decomposable tensors of unit norm. Our results are connected with earlier results of P. Binding.
Generalised Liouville-Green asymptotic approximations for second-order differential equations
Authors: Cassell J. S.
Published: November 2011

Synopsis

The equation {r(x)y'}' = q(x)y is considered with the coefficients q and rtaking complex values. Conditions are obtained for there to be solutions with the asymptotic form , . These involve the coefficients and their derivatives up to some order N.
Construction of solutions for compressible, isentropic Navier-Stokes equations in one space dimension with nonsmooth initial data
Authors: Hoff David
Published: November 2011

We prove the global existence of weak solutions for the Cauchy problem for the Navier-Stokes equations for one-dimensional, isentropic flow when the initial velocity is in L2 and the initial density is in L2 ∩ BV. Solutions are obtained as limits of approximations obtained by building heuristic jump conditions into a semi-discrete difference scheme. This allows for a rather simple analysis in which pointwise control is achieved through piecewise H1 and total variation estimates.
Radicals of semigroup algebras of commutative and cancellative semigroups
Authors: Puczyłowski Edmund R.
Published: November 2011

Synopsis

The shape of radicals of semigroups algebras of commutative and cancellative semigroups is studied. The questions asto when a radical of those algebras is homogeneous and if homogeneous radicals have more regular form are examined.
Growth properties of solutions to a linear differential equation
Authors: Jones Gary D.
Published: November 2011

Synopsis

Dominance properties of solutions to Lny + p(x)y = 0, where Ln is a disconjugate operator, are compared to dominance properties of solutions to its adjoint.
Shooting methods and topological transversality
Authors: Buffoni B.
Published: November 2011

We show that shooting methods for homoclinic or heteroclinic orbits in dynamical systems may automatically guarantee the topological transversality of the stable and unstable manifolds. The interest of such results is twofold. First, these orbits persist under perturbations which destroy the structure allowing the shooting method and, second, topological transversality is often sufficient when some kind of transversality is required to obtain chaotic dynamics. We shall focus on heteroclinic solutions in the extended Fisher–Kolmogorov equation.
Essential instability of pulses and bifurcations to modulated travelling waves
Authors: Sandstede B. , Scheel A.
Published: November 2011

Reaction-diffusion systems on the real line are considered. Localized travelling waves become unstable when the essential spectrum of the linearization about them crosses the imaginary axis. In this article, it is shown that this transition to instability is accompanied by the bifurcation of a family of large patterns that are a superposition of the primary travelling wave with steady spatially periodic patterns of small amplitude. The bifurcating patterns can be parametrized by the wavelength of the steady patterns; they are time-periodic in a moving frame. A major difficulty in analysing this bifurcation is its genuinely infinite-dimensional nature. In particular, finite-dimensional Lyapunov–Schmidt reductions or centre-manifold theory do not seem to be applicable to pulses having their essential spectrum touching the imaginary axis.
Asymptotically self-similar global solutions of a semilinear parabolic equation with a nonlinear gradient term
Authors: Snoussi S. , Tayachi S. , Weissler F. B.
Published: November 2011

We study the existence and the asymptotic behaviour of global solutions of the semilinear parabolic equation u(0) = ϧwhere a, b ∈ℝ, q > 1, p > 1. Forq=2p/(p+1) and ½ 1(p-1)>1 (equivalently, q > (n + 2)/(n + 1)), we prove the existence of mild global solutions for small initial data with respect to some norm. Some of those solutions are asymptotically self-similar.
Positivity for elliptic and parabolic systems
Authors: Auchmuty J. F. G.
Published: November 2011

Synopsis

The positivity of solutions of initial-boundary value problems for weakly-coupled semilinear parabolic or elliptic systems of equations is studied. Conditions on the coupling terms are described which ensure that the solutions of the parabolic systems remain positive whenever the initial conditions are positive. For elliptic systems involving a parameter, conditions on the coupling terms are described which imply that solution branches which contain a positive solution, in fact, contain only positive solutions. Applications of these theorems to certain reaction-diffusion equations arising in the modelling of biological phenomena are given.
Inequalities and separation for Schrödinger type operators in L2(Rn)
Authors: Everitt W. N. , Giertz M.
Published: November 2011

Synopsis

The symmetric differential expression M determined by Mf = − Δf;+qf on G, where Δ is the Laplacian operator and G a region of n-dimensional real euclidean space Rn, is said to be separated if qfϵL2(G) for all f ϵ Dt,; here D1 ⊂ L2(G) is the maximal domain of definition of M determined in the sense of generalized derivatives. Conditions are given on the coefficient q to obtain separation and certain associated integral inequalities.
Isometric folding of Riemannian manifolds
Authors: Robertson S. A.
Published: November 2011

Synopsis

When a sheet of paper is crumpled in the hands and then crushed flat against a desk-top, the pattern of creases so formed is governed by certain simple rules. These rules generalize to theorems on folding Riemannian manifolds isometrically into one another. The most interesting results apply to the case in which domain and codomain have the same dimension. The main technique of proof combines the notion of volume with Hopf's concept of the degree of a map.
Finite metacyclic groups all extensions of which split
Authors: Rose John S.
Published: November 2011

Synopsis

The groups of the title are characterized arithmetically.
Some elliptic balls which avoid a Nyquist set in ℂn+1
Authors: Smith Russell A.
Published: November 2011

Synopsis

Conditions are obtained for certain elliptic balls in ℂn+1 to have empty intersection with the Nyquist set of a vector polynomial G(z). Such conditions are shown to yield explicit criteria for the existence of periodic solutions of non-autonomous scalar differential equations of the form A* G(D)y = p.
Some results on the Gamma function and other hypertranscendental functions
Authors: Bank Steven B.
Published: November 2011

Synopsis

This paper deals with conditions which guarantee that a meromorphic function on the plane cannot satisfy any algebraic differential equation having coefficients in a given field of meromorphic functions. Some of the conditions are of growth type, while others depend on a representation for the function.
On mean p-valent functions in an ellipse
Authors: Elhosh M. M.
Published: November 2011

Synopsis

Modulus and coefficient bounds for functions mean p-valent in the interior of an ellipse, analogous to known bounds for the unit disc, are established in this paper.
Equivalence, adjoints and symmetry of quasi-differential expressions with matrix-valued coefficients and polynomials in them
Authors: Frentzen Hilbert
Published: November 2011

Synopsis

Quasi-differential expressions with matrix-valued coefficients, which generalize those of Shin and Zettl, are considered with regard to equivalence, adjoints and symmetry. The characterization results imply that in the scalar case the class of quasi-differential expressions considered here coincides with that of Shin and is equivalent to that of Zettl. Furthermore polynomials in quasi-differential expressions are defined as expressions of the same kind and shown to coincide with the usual ones. Finally it is indicated that the known general results for the deficiency indices carry over to quasi-differential expressions.
Global invertibility of Sobolev functions and the interpenetration of matter
Authors: Ball J. M.
Published: November 2011

Synopsis

A global inverse function theorem is established for mappingsu: Ω → ℝn, Ω ⊂ ℝnbounded and open, belonging to the Sobolev spaceW1.p(Ω),p>n. The theorem is applied to the pure displacement boundary value problem of nonlinear elastostatics, the conclusion being that there is no interpenetration of matter for the energy-minimizing displacement field.
On the location of the Weyl circles
Authors: Atkinson F. V.
Published: November 2011

Synopsis

The paper deals with explicit estimates concerning certain circles in the complex plane which were associated with Sturm–Liouville problems by H. Weyl. By the use of Riccati equations instead of linear integral equations, improvements are obtained for results of Everitt and Halvorsen concerning the behaviour of the Titchmarsh–Weyl m-coefficient.
Stable methods for solving the inverse scattering problem for a cylinder
Authors: Colton David , Kirsch Andreas
Published: November 2011

Synopsis

It is shown that the inverse scattering problem for an infinite cylinder can be stabilized by assuming a priori that the unknown boundary of the cylindrical cross section lies in a compact family of continuously differentiable simple closed curves. A constructive method for determining the shape of this boundary is given under the assumption that an initial approximation is known and that the scattering cross section is known forn distinct incoming plane waves in the resonant region.
The uniform exponential stability of a class of linear differential–difference equations in a Hilbert space
Authors: Datko Richard
Published: November 2011

Synopsis

A necessary and sufficient condition is given for the uniform exponential stability of certain autonomous differential–difference equations whose phase space is a Hilbert space. It is shown that this property is preserved when the delays depend homogeneously on a nonnegative parameter.
On the existence of weak solutions for quasilinear parabolic initial-boundary value problems
Authors: Landes Rüdiger
Published: November 2011

Synopsis

Structure conditions on a strongly nonlinear operator A(u) are given, under whichthe initial-Dirichlet-boundary value problem for has weak solutions.
On the convergence of the method of finite differences applied to the solution of the initial value problem for functional-differential neutral type equations
Authors: Kazakova N. G. , Bainov D. D.
Published: November 2011

Synopsis

When solving practically the neutral type equations the derivatives are replaced by finite differences while the members of integral type are replaced by quadrature formulae. The paper deals with the convergence of a natural class of methods applied to the Cauchy problem for functional-differential neutral type equations. It is not obligatory for the approximated operators to be compact.
Norm inequalities of product form in weighted Lp spaces
Authors: Kwong M. K. , Zettl A.
Published: November 2011

Synopsis

The classical inequality ‖y′‖2≦K‖y‖ ‖y″‖ and its higher order analogues are extended from the Lp spaces to the weighted spaces, for appropriate w.
Multiparameter definiteness conditions
Authors: Binding Paul
Published: November 2011

Synopsis

Several definiteness conditions in the multiparameter spectral literature are discussed. It is shown that some of these conditions permit simplifying transformations of the eigenvalues, leading to further definiteness properties. Geometrical equivalents for the algebraic conditions are established in terms of separation of convex cones. As a result, the relationship between the standard left and right definiteness conditions is clarified.
Some integrodifferential inequalities in n-independent variables
Authors: Chen Lu-San , Yeh Cheh-Chih
Published: November 2011

Synopsis

This note presents some integrodifferential inequalities in n-independent variables which are generalizations of the integrodifferential inequalities recently established by Pachpatte in two independent variables.
Uniform basis properties of exponential solutions of functional differential equations of retarded type
Authors: Ho L. F.
Published: November 2011

Synopsis

We define spaces of functions on an interval naturally related to functional differential equations of retarded type in the sense that exponential solutions of such equations form a uniform basis (or Riesz basis) in these spaces. Embedding theorems relating these spaces to the Sobolev spaces are established using a result on Carleson's measure defined on a half plane.
Local algebras and finite projective planes
Authors: Prince Alan R.
Published: November 2011

Synopsis

An analogue of the group algebra of a finite group is defined for any finite projective plane, in the case where the characteristic of the scalar field divides the order of the plane.
A new proof that regular biordered sets come from regular semigroups
Authors: Easdown David
Published: November 2011

Synopsis

A new proof is given that all regular biordered sets come from regular semigroups.
Complementary bounds for inner products associated with non-linear equations
Authors: Cole R. J. , Mika J. , Pack D. C.
Published: November 2011

Synopsis

Functionals are found that give upper and lower bounds to the inner product 〈g0, f〉 involving the unknown solution f of a non-linear equation T[f] = f0, with f∈H, a real Hilbert space, g0 a given function in H and f0 a given function in the range of the non-linear operator T. The method depends upon a re-ordering of terms in the expansion of T[f] about a trial function so as to transfer the non-linearity to a secondary problem that requires its own particular treatment and to enable earlier results obtained for linear operators to be used for the main part. First, bivariational bounds due to Barnsley and Robinson are re-derived. The new and more accurate bounds are given under relaxed assumptions on the operator T by introducing a third approximating function. The results are obtained from identities, thus avoiding some of the conditions imposed by the use of variational methods. The accuracy of the new method is illustrated by applying it to the problem of the heat contained in a bar.
Corrigenda: Cauchy and Poisson integral representations for ultradistributions of compact support and distributional boundary values
Authors: Pathak R. S.
Published: November 2011

Page 56: In the estimate , C1 = CSn/(2πσ)n, h = (2πσ)−1 where σ depends on y. Therefore, {fy(t)} may not be bounded in as y → 0, y ∈ C′.
Existence of multiple positive solutions of inhomogeneous semilinear elliptic problems in unbounded domains
Authors: Zhu Xi-Ping , Zhou Huan-Song
Published: November 2011

Synopsis

By using the concentration-compactness method of Lions [14, 16] and the mountain pass theorem of Ambrosetti and Rabinowitz [3], through a careful inspection of the energy balance for some sequence of approximated solutions, we show that under suitable conditions on f and h, the inhomogeneous problem. −Δu + c2u = λ(f(u) + h(x)) for x ∈ Ω (Ω is an exterior domain in ℝN, N≧ 3) and has at least two positive solutions.
Saint-Venant's principle on unbounded regions
Authors: Knops R. J. , Rionero S. , Payne L. E.
Published: November 2011

Synopsis

We consider an anisotropic non-homogeneous linear elastic material in equilibrium and occupying an open region with non-compact boundary. In both the linearised and classical linear theories the asymptotic behaviour of the solution is determined and a clear relationship established with Saint-Venant's principle on such regions. Although the treatment is discussed with special reference to elasticity, it is equally applicable to general systems of elliptic differential equations, and thus reveals a relationship with the classical theorems of Phragmèn-Lindelöf and Liouville.
Whitney-sums (fibre-joins) in over space theory and obstruction theory for cohomology with local coefficients
Authors: Rutter John W.
Published: November 2011

Synopsis

The generalised Whitney sum (fibre-join) and the h-fibre-join can be defined in topM, the category of spaces over M. We note here some general properties of these constructions, and, as a specific example, we consider the relation between them and the extensions to the topM category of the top h-fibre-sequences F∗ΩB→E ∪ CF→B determined by top fibrations F→E→B. As an application we obtain the truncated local coefficient cohomology sequence for a top fibration which is topM principal fibration: this situation applies, for example, to the various stages of the Postnikov decomposition of a non-simply connected space X, and in this case we have M = K1(π1(X)).
Flows on centre manifolds for scalar functional differential equations
Authors: Hale Jack K.
Published: November 2011

Synopsis

By assuming that a linear scalar functional differential equation (FDE) has only the zero eigenvalue on the imaginary axis, it is shown that the flows on the centre manifolds of all Cr-perturbations of this equation coincide with the flows obtained from scalar ordinary differential equations (ODEs) of order m, where m is the multiplicity of the zero eigenvalue. Furthermore, it is shown that the above situation can be realized through differential difference equations with m – 1 fixed distinct delays.
A global branch of solutions to a semilinear equation on an unbounded interval
Authors: Stuart C. A.
Published: November 2011

Synopsis

For a semilinear second order differential equation on (0, ∞), conditions are given for the bifurcation and asymptotic bifurcation in Lp of solutions to the Neumann problem. Bifurcation occurs at the lowest point of the spectrum of the linearised problem. Under stronger hypotheses, there is a global branch of solutions. These results imply similar conclusions for the same equation on R with appropriate symmetry.
On the strong limit-point and Dirichlet properties of second order differential expressions
Authors: Race D.
Published: November 2011

Synopsis

Second order differential expressions of the form w−1(−(pf′)′ + qf) are considered at a singular end-point. Some of the known relationships between properties such as Dirichlet, weak Dirichlet and strong limit-point, are extended to incorporate an arbitrary, positive weight function and complexvalued coefficients.
On the existence of eigenvalues for some nonlinear elliptic and hyperbolic problems
Authors: Basile Nicola , Mininni Michele
Published: November 2011

Synopsis

In this paper some eigenvalue problems for elliptic as well as hyperbolic equations are solved. The main tool used is an abstract critical point theorem on an unbounded manifold of the form {u | (Lu, u) = constant} (where L is a nonpositive selfadjoint operator), which makes use of a linking type argument on a manifold.
The existence of infinitely many bifurcating branches
Authors: Ruppen Hans-Jörg
Published: November 2011

Synopsis

We consider the non-linear problem −Δu(x)−f(x, u(x)) = λu(x) for x ∈ℝN and u ∈ W1,2(ℝN). We show that, under suitable conditions on f, there exist infinitely many branches all bifurcating from the lowest point of the continuous spectrum λ = 0. The method used in the proof is based on a theorem of Ljusternik-Schnirelman type for the free case.
Global a priori estimates for a viscous reactive gas
Authors: Bebernes J. , Bressan A.
Published: November 2011

Synopsis

A priori estimates on the solution to the complete system of equations governing a heat-conductive, viscous reactive perfect gas confined between two infinite parallel plates are derived. From these estimates, global existence of both weak and classical solutions is obtained.