bluuebunny/crossref_metadata_2025_split huggingface dataset:
Approximation to inverses of normal operators
Authors: Mika J. , Pack D. C. 
Published: November 2011

SynopsisFor many purposes, and in particular for the calculation of upper and lower bounds to bilinear forms 〈g0,f〉, where f is the solution to an operator equation and g0 is known, it isuseful to obtain an approximation to the inverse of the operator.For a normal operator A acting in a complex Hilbert space with bounded inverse A−1, we use a direct approach through fundamental results in functional analysis and derive a recipe for the ‘best formula’ for A−1 of form B = βI with β constant and I the identity operator. Examples illustrate that this leads to improved results for certain classes of operator.The investigation is extended to the representation of A−1 by a polynomial in A withcomplex coefficients. For polynomials of first and higher orders the application is restricted to bounded self-adjoint operators; at first order, explicit formulae are given.Application of the method to Fredholm operators is considered in detail, and its use for the point-wise solution to Fredholm integral equations is illustrated.
Finite speed of propagation and asymptotic rates for some nonlinear higher order parabolic equations with absorption
Authors: Bernis F. 
Published: November 2011

SynopsisThe “energy solutions” to the equationhave finite speed of propagation if l < q < 2 or l < r < 2. If 1 <r <2 (Vq <1) support u(· t) is uniformly bounded for t >0 (localisation property) and if q<2 ≦ r, sharp upper bounds of the interface (or free boundary) are obtained. We use a weighted energy method, the weights being powers of the distance to a variable half-space. We also study decay rates as t→∞ and extinction in finite time for bounded and unbounded domains (with null Dirichlet boundary conditions). Our equation includes the porous media equation with absorption. Analogous results hold if (−Δ)m is replaced by an appropriate quasilinear elliptic operator.
On prespectrality of generalised derivations
Authors: Hladnik Milan 
Published: November 2011

SynopsisIn this paper it is proved that, for scalar-type operators a and b on an infinite dimensional separable complex Hilbert space H, the generalised derivation Δa,b, defined for bounded linear operators x onℋ by the equation Δa,bx = ax − xb, is a (scalar-type) prespectral operator of the class (the trace class operators on ℋ) if and only if at least one of the spectra σ(a)or σ(b)is finite. It is shown also that the same condition is necessary and sufficient for Δa,b restricted to any one of the von Neumann-Schatten classes(p≠2) to be a spectral operator (of scalar type). Our results may be compared with those of J. Anderson and C. Foiaş, who established in [1] that, for scalar-type a, b, Δa,b is a (scalar-type) spectral operator if and only if both spectra, σ(a) and σ(b), are finite. However, we use different and more direct methods to show the existence or nonexistence of the spectral resolution of identity for Δa,b.
On the asymptotic properties of solutions to a differential equation in a case of bifurcation without eigenvalues
Authors: Magnus R. J. 
Published: November 2011

SynopsisThe semi-linear equation Δu − λu + h(x)uσ = 0 is studied on all of d-dimensional Euclidean space. In the bifurcation problem a non-trivial solution is sought for small λ which tends to zero with λ. The asymptotic dependence of the solution on λ is examined. For fixed λ = 1 the existence of non-degenerate non-trivial solutions is proved for generic measurable h(x) sufficiently near to a constant, provided d = 1 or 3. The two problems are seen to be interdependent. The bifurcation problem at λ = 0 is particularly interesting as the linearised equation is of non-Fredholm type.
On two-dimensional slow viscous flows past obstacles in a half-plane
Authors: Fischer T. M. , Hsiao G. C. , Wendland W. L. 
Published: November 2011

SynopsisWe consider a cylinder with arbitrary cross section moving in a viscous incompressible fluid parallel to a plane wall. Formal asymptotic expansions of the solution for small Reynolds numbers are constructed by using boundary integral equations of the first kind. In contrast to the problem without a wall, we show that there exists a unique solution to the zeroth order problem. However, the problem considered here is still singular in the sense that we find the Stokes paradox in the next higher order problem. A justification of the formal asymptotic expansion for the first two terms is established rigorously.
Properties of some ordinary differential equations related to free boundary problems
Authors: Caginalp Gunduz , Hastings Stuart 
Published: November 2011

SynopsisSome second order ordinary differential equations of the form ξ2ϕ″ + ξ2(N − 1)″′/r + ½(ϕ − ϕ3) + ½k = 0 are studied. Properties such as existence and monotonicity of solutions are considered for N ≧ 1, ξ > 0 and two sets of boundary conditions. For N = 1, some explicit results are obtained for small ξ. These ODE's arise from a phase field approach to free boundary problems involving a phase transition.
Hopf bifurcation and symmetry: travelling and standing waves on the circle
Authors: Gils Stephan A. van , Mallet-Paret John 
Published: November 2011

SynopsisIn this paper we consider Hopf bifurcation in the presence of O(2) symmetry. The system of reaction diffusion equations ut, = D(µ)uxx + f(µ, u) provided with periodic boundary conditions may serve as a model problem. We prove the bifurcation of a torus of standing waves and two circles of travelling waves and we compute the stability (with asymptotic phase) of these periodic solutions, giving explicit formulae. Finally we demonstrate how a small perturbation which breaks part of the symmetry leads to secondary bifurcation.
Solutions to some problems in amenable semigroups
Authors: Ayyaswamy S. K. , Ramakrishnan P. V. 
Published: November 2011

SynopsisThis paper discusses a few problems on the size of the set of invariant means of an amenable semigroup posed by Maria M. Klawe, Alan L. T. Paterson and M. Rajagopalan and P. V. Ramakrishnan ([4], [5], [8] and [9]).
Inverse-monotone nonlinear differential operators of the second order
Authors: Schröder Johann 
Published: November 2011

SynopsisThis paper provides a survey on a class of methods to obtain sufficient conditions for the inversemonotonicity of second-order differential operators. Pointwise differential inequalities as well as weak differential inequalities are treated. In particular, the theory yields results on the relation between inverse-mo no tone operators and monotone definite operators, i.e. monotone operators in the Browder–Minty sense. This presentation is restricted to ordinary differential operators. Most methods explained here can also be applied to elliptic-parabolic partial differential operators in essentially the same way.
Bifurcation in the Duffing equation with independent parameters, II
Authors: Hale Jack K. , Rodrigues Hildebrando M. 
Published: November 2011

SynopsisIn a previous paper, the authors gave a complete description of the number of even harmonic solutions of Duffing's equation without damping for the parameters varying in a full neighbourhood of the origin in the parameter space. In this paper, the analysis is extended to the case of an independent small damping term. It is also shown that all solutions of the undamped equation are even functions of time.
The G-invariant implicit function theorem in infinite dimensions
Authors: Dancer E. N. 
Published: November 2011

SynopsisOur basic theorem is a version of the implicit function theorem in the case of continuous groups of symmetries. The result is sufficiently general to cover a great many applications. It generalizes some earlier work of the author and corrects and improves some work of Vanderbauwhede. We also consider the breaking of symmetries problem and the variational case. Finally, we apply our results to study the periodic solutions of an ordinary differential equation.
Optimal lower bounds for the spectrum of a second order linear differential equation with ap-integrable coefficient
Authors: Veling E. J. M. 
Published: November 2011

SynopsisIn this note the differential expressionM[y] ≡ − y” + qy, q∈Lp(ℝ+) for some p ≧ l, is considered on [0,∞) together with the boundary condition either y(0) = 0 or y'(0) = 0. Lower bounds are given for the spectrum of the self-adjoint operatorsTgenerated by M[·] and these boundary conditions. The bounds depend on theLp-norm of the coefficientqand they improve results of Everitt and Eastham. The bounds are optimal.
Limit-point (LP) criteria for real symmetric differential expressions of order 2n
Authors: Kurss H. , Meyer G. 
Published: November 2011

SynopsisAn interval-type LP criterion foris derived in which “positive” coefficients play a prominent role. When pn = 1 and all the other pi are zero this reduces to a result of Ismagilov (1962). Successive specializations are obtained with the growth of the pi constrained by monomials in x. Previous LP criteria of Everitt (1968) and Hinton (1972, 1974) are shown to be special cases.
Generalisation of Dedekind's problem of the enumeration of coherent structures
Authors: Ansell J. , Bendell A. , Humble S. 
Published: November 2011

SynopsisIn the context of reliability theory, two definitions are given for coherent functions of n variables, where both function and variables can take any of l possible levels. The enumeration problem for such functions is discussed and several recursive bounds are derived. In the case of l = 2 (the Dedekind problem) a recursive upper bound is derived which is better than the previous best explicit upper bound forn < 15, and also provides a systematic improvement on this bound for larger values of n.
Nonlinear initial value problems with a singular perturbation of hyperbolic type
Authors: Geel R. 
Published: November 2011

SynopsisThis paper deals with initial value problems in ℝ2 which are governed by a hyperbolic differential equation consisting of a nonlinear first order part and a linear second order part. The second order part of the differential operator contains a small factor ε and can therefore be considered as a perturbation of the nonlinear first order part of the operator.The existence of a solution u together with pointwise a priori estimates for this solution are established by applying a fixed point theorem for nonlinear operators in a Banach space.It is shown that the difference between the solution u and the solution w of the unperturbed nonlinear initial value problem (which follows from the original problem by putting ε = 0) is of order ε, uniformly in compact subsets of ℝ2 where w is sufficiently smooth.
On the completeness of eigenvectors of right definite multiparameter problems
Authors: Volkmer Hans 
Published: November 2011

SynopsisWe prove that there exists a complete system of eigenvectors of the eigenvalue problemfor self-adjoint operators Tr and Vrs on separable Hilbert spaces Hr. It is assumed that(i) the operators Tr have discrete spectrum;(ii) the operators Vrs are bounded and commute for each r;(iii) the operators Vrs have the definite sign factor property.This theorem generalizes and improves a result of Cordes for two-parameter problems. The proof of the theorem depends on an approximation of the given eigenvalue problem by simpler problems, a technique which is related to Atkinson's proof of his expansion theorem.
On conditional diffusion processes
Authors: Lyons T. J. , Zheng W. A. 
Published: November 2011

SynopsisIf Xt is the diffusion process associated with a second-order uniformly elliptic operator L in divergence form, then without assuming smoothness in L we prove that for each x and y in ℝd,where p is the fundamental solution to the heat equation associated with L. This allows one to control p when bounded drift terms are added to L; and also allows one to do Stratonovich integration with respect to the process conditioned to start at any point; previous work only dealt with quasi-every starting point.
A Hilbert space approach to distributions
Authors: Picard Rainer H. 
Published: November 2011

SynopsisA compact chain of Sobolev type Hilbert spaces , n integer, is introduced that is invariant withrespect to the Fourier transform ℱ. The spaces are related to powers of the adjoint of the so-called tempered derivative introduced in the sequential approach to distributions. It turns out that the intersection of all these Hilbert spaces coincides with the space of rapidly decaying C∞-functions and their union leads to the space of tempered distributions. Moreover, the naturally induced convergence concepts coincide with the usual ones. The approach provides not only a new and arguably more elementary approach to distributions it also provides a deeper insight into the action of the Fourier transform which is a unitary mapping in each space of the chain. Finally the Schwartz distributions are incorporated in the approach as locally tempered distributions.
On the growth of solutions to second order differential equations in Banach spaces
Authors: Fattorini H. O. 
Published: November 2011

SynopsisWe obtain estimates for the exponential growth of the solutions tou″(t) = (A+ ζ2I)u(t) in terms of the exponential growth of the solutions tou″(t) =Au(t), where ζ is an arbitrary complex number. Estimates in exponentially weightedL2norms are also considered in Hilbert space.
Hopf bifurcation at multiple eigenvalues with zero eigenvalue
Authors: López-Gómez J. 
Published: November 2011

SynopsisWe give local results related to Hopf bifurcation for parabolic equations. The linear part about the equilibrium point can have zero eigenvalues. In our results the information about the perturbation is essential and it is possible to obtain bifurcation even if some ‘i’ or zero remains on the imaginary axis for all values of the parameter.
On some infinite integrals involving logarithmic exponential and powers
Authors: Chaudhry M. Aslam , Ahmad Munir 
Published: November 2011

SynopsisIn this paper we define an integral function Iμ(α; a, b) for non-negative integral values of μ byIt is proved that the function Iμ(α; a, b) satisfies a functional recurrence-relation which is then exploited to evaluate the infinite integralSome special cases of the result are also discussed.
Blow-up time involved with perturbed Hamilton–Jacobi equations
Authors: Díaz G. 
Published: November 2011

SynopsisIn this paper we consider the evolution of positive bounded uniformly continuous data u0 by perturbed equations likeUnder general assumptions on u0, existence, uniqueness and regularity of the evolution u in the setare studied, where the blow-up function is given by .The exact blow-up rate of u is obtained. Uniqueness, regularity, decay at infinity of the function , as well a s a representation formula for the case m = 1, are also proved.
The validity of modulation equations for extended systems with cubic nonlinearities
Authors: Kirrmann Pius , Schneider Guido , Mielke Alexander 
Published: November 2011

SynopsisModulation equations play an essential role in the understanding of complicated systems near the threshold of instability. Here we show that the modulation equation dominates the dynamics of the full problem locally, at least over a long time-scale. For systems with no quadratic interaction term, we develop a method which is much simpler than previous ones. It involves a careful bookkeeping of errors and an estimate of Gronwall type.As an example for the dissipative case, we find that the Ginzburg–Landau equation is the modulation equation for the Swift–Hohenberg problem. Moreover, the method also enables us to handle hyperbolic problems: the nonlinear Schrodinger equation is shown to describe the modulation of wave packets in the Sine–Gordon equation.
On the commutativity of certain quasi-differential expressions II
Authors: Frentzen H. , Race D. , Zettl A. 
Published: November 2011

SynopsisWe consider the question: when do two ordinary, linear, quasi-differential expressions commute? For classical differential expressions, answers to this question are well known. The set of all expressions which commute with a given such expression form a commutative ring. For quasi-differential expressions less is known and such an algebraiastructure can no longer be exploited. Using the theory of very general quasi-differential expressions with matrix-valued coefficients, we prove some general results concerning commutativity of such expressions. We show how, when specialised to scalar expressions, these results include a proof of the conjecture that if a 2nth-order scalar,J-symmetric (or real symmetric) quasi-differential expression commutes with a second order expression having the same properties, then the former must be annth-order polynomial in the latter. This result was conjectured in a paper by Race and Zettl, to which this paper is a sequel.
On the periodic Fučik spectrum and a superlinear Sturm–Liouville equation
Authors: Figueiredo Djairo G. , Ruf Bernhard 
Published: November 2011

SynopsisIn the first part of the paper a variational characterisation of the periodic eigenvalues (the so-called Fučik spectrum) of a semilinear, positive homogeneous Sturm–Liouville equation is given. The proof relies on the S1-invariance of the equation.In the second part a nonlinear Sturm–Liouville equation with, typically, an exponential nonlinearity is considered. It is proved that under certain conditions this equation is solvable for arbitrary forcing terms. The proof uses a comparison of the minimax levels of the functional associated to this equation with suitable values in the Fučik spectrum.
Boundary integral equations for the scattering of electromagnetic waves by a homogeneous dielectric obstacle
Authors: Martin P. A. , Ola Petri 
Published: November 2011

SynopsisTime-harmonic electromagnetic waves are scattered by a homogeneous dielectric obstacle. The corresponding electromagnetic transmission problem is reduced to a single integral equation over S for a single unknown tangential vector field, where S is the interface between the obstacle and the surrounding medium. In fact, several different integral equations are derived and analysed, including two previously-known equations due to E. Marx and J. R. Mautz, and two new singular integral equations. Mautz's equation is shown to be uniquely solvable at all frequencies. A new uniquely solvable singular integral equation is also found. The paper also includes a review of methods using pairs of coupled integral equations over S. It is these methods that are usually used in practice, although single integral equations seem to offer some computational advantages.
Some inequalities for arithmetic and geometric means
Authors: Alzer Horst 
Published: November 2011

Let An and Gn (respectively, A′n and G′n) be the weighted arithmetic and geometric means of x1, …, xn (respectively, 1 – x1, …, 1 – xn). We present sharp upper and lower bounds for the differences and . And we determine the best possible constants r and s such thatholds for all xi ∈ [a, b] (i = 1, …, n; 0 < a < b < 1). Our theorems extend and sharpen results of Fan, Cartwright and Field, McGregor and the author.
The long-time behaviour of solutions to parabolic problems on unbounded intervals: the influence of boundary conditions
Authors: Fašangová Eva , Feireisl Eduard 
Published: November 2011

For a non-negative function ū(x), we study the long-time behaviour of solutions of the heat equationwith the Dirichlet or Neumann boundary conditions at x = 0. We find a critical parameter λD > 0 such that the solution subjected to the Dirichlet boundary condition tends to a spatially localized wave travelling to infinity in the space variable. On the other hand, there exists a λN > 0 such that the corresponding solution of the Neumann problem converges to a non-trivial strictly positive stationary solution. Consequently, the dynamics is considerably influenced by the choice of boundary conditions.
Heat flow with tangent penalisation converging to mean curvature motion
Authors: Zhai Jian 
Published: November 2011

In this paper, we prove that mean curvature motion can be regarded as the singular limit of the following model:where ε > 0 is a small parameter and . This model is related to the Landau–Lifshitz equation of ferromagnetism. Local existence of classical solutions of the Dirichlet problem and global existence of the travelling wave solutions are also obtained.
An elliptic boundary-value problem with a discontinuous nonlinearity, II
Authors: Keady G. , Kloeden P. E. 
Published: November 2011

SynopsisLet Ω be a bounded domain in ℝ2. The study, begun in Keady [13], of the boundary-value problem, for (λ/k, ψ),is continued. Here Δ denotes the Laplacian, H is the Heaviside step function and one of λ or k is a given positive constant. The solutions considered always have ψ > 0 in Ω and λ/k > 0, and have coresIn the special case Ω = B(0, R), a disc, the explicit exact solutions of the branch τe have connected cores A and the diameter of A tends to zero when the area of A tends to zero. This result is established here for other convex domains Ω and solutions with connected cores A.An adaptation of the maximum principles and of the domain folding arguments of Gidas, Ni and Nirenberg [9] is an important step in establishing the above result.
Semilinear evolution equations and fractional powers of a closed pair of operators
Authors: Dalsen Marié Grobbelaar-Van 
Published: November 2011

SynopsisThe nonlinear evolution problem [Bu(t)]′ =A(t,Bu)u+f(t,Bu) withBa constant linear operator andA=A(t,Bu) a time-dependent nonlinear operator from one Banach space to another, is studied. Existence and uniqueness results are obtained by making use of the theory of B-evolutions and the fractional powers ofAandB. Two examples are presented in which the theory is applied to nonlinear equations with dynamic boundary conditions.
An elementary proof of Gorny's inequality
Authors: Fabry C. 
Published: November 2011

SynopsisGorny's inequality provides upper bounds for the sup-norms ∥f(k) of a function f over an interval [a, b] for k = 1, …, n − 1, assuming the sup-norms of f and f(n) to be given. We present a simple proof of that inequality and obtain sharper estimates of the constants contained in that inequality, compared with the original verison of Gorny.
On a quasilinear wave equation with nonlinear damping
Authors: Hai Dang Dinh 
Published: November 2011

SynopsisWe prove the global existence and uniqueness of the solution of the initial and boundary value problem for the equationby using the classical Galerkin method when the forcing term and the initial data are in some sense small. The asymptotic behaviour of the solution as t → ∞ is also considered.
Unique factorisation of normal elements in polynomial rings
Authors: Jordan D.A. 
Published: November 2011

SynopsisWe consider the class of Noetherian UFN-rings, that is, Noetherian prime rings such that every non-zero ideal contains a non-zero normal element and such that the monoid of non-zero normal elements is a unique factorisation monoid. We ask whether this class of rings, a generalisation of the commutative Noetherian unique factorisation domains (UFD), is closed under polynomial extensions. The general question is apparently difficult and remains open. However, we give positive answers in special cases, in particular, for algebras over an infinite field and for domains.
On the uniqueness and ordering of steady states of predator-prey systems
Authors: Li Lige 
Published: November 2011

SynopsisThis paper discusses the relationship between the uniqueness and the ordering of strictly positive solutions of elliptic predator-prey interacting systems. If (ū, v) and (u#, v#) are two such solutions with ū ≧ u# or v ≧ v#, then ū ≡= u#, v ≡v#. When the positive solutions are numerically close to the extreme case, the solution is unique.