bluuebunny/crossref_metadata_2025 huggingface dataset:
Permanence and asymptotic stability in diagonally convex reaction–diffusion systems
Authors: Vuuren Jan H. van , Norbury John
Published: November 2011

Reaction–diffusion systems are widely used to model competition in, for example, the scientific fields of biology, chemistry, medicine and industry. It is often not too difficult to establish positive uniform upper bounds on solution components of such systems, but the task of establishingstrictly positiveuniform lower bounds (when they exist) can be quite troublesome. Conditions for the existence of such lower bounds in Lotka–Volterra competitve models are already well known. A general context for the understanding of these conditions is provided in this paper, by establishing more general permanence criteria (for the nonexplosion and noncollapse to zero of solutions) in the class of diagonally convex competitive reaction–diffusion systems with zero flux Neumann boundary conditions. This class admits most famous competition models as special cases. The asymptotic (large time) behaviour of positive solutions within the bounds of permanence is also considered and it is shown that the methods of proof for asymptotic stability (and hence resilience) normally associated with order-preserving systems (such as comparison arguments) are also applicable, in a slightly generalised form, to competitive systems as long as the competitive interactions are not too strong. The general criteria for permanence obtained here provide a natural method for developing new and easily verifiable permanence conditions for a host of non Lotka–Volterra competition models, as is illustrated by considering three famous special cases. In one of these cases known results are recovered, while in the other two cases new conditions for solution permanence are established.
Radiation condition and limiting amplitude principle for acoustic propagators with two unbounded media
Authors: Zhang Bo
Published: November 2011

Consider the diffraction problem for perturbed acoustic propagators with perturbations decreasing slowly at infinity. The propagation speed is discontinuous at the interface of two unbounded media, and the interface may be an arbitrary and smooth surface locally. A Sommerfeld radiation condition is introduced for the acoustic propagator, and is then used to establish the limiting absorption principle and the resolvent estimate at low frequencies for such an operator. Furthermore, we prove the existence of a unique solution to the diffraction problem and the validity of the limiting amplitude principles for the acoustic propagator.
A Ginzburg–Landau problem with weight having minima on the boundary
Authors: Beaulieu Anne , Hadiji Rejeb
Published: November 2011

In this paper, we study the following Ginzburg–Landau functional:

where(G, C), andpis a smooth bounded and non-negative map, having minima on the boundary of Ḡ. We give the location of the singularities in the case where the degree around each singularity is equal to 1.
Liouville theorems for elliptic inequalities and applications
Authors: Birindelli Isabeau , Mitidieri Enzo
Published: November 2011

In this paper we prove nonexistence of positiveC2solutions for systems of semilinear elliptic inequalities, for polyharmonic semilinear inequalities in cones and, under better conditions on the nonlinearity, for bounded positive solutions of elliptic semilinear equations in half spaces. Using a blow-up argument, these results allow us to provea-prioribounds for a class of semilinear elliptic systems of equations in bounded domains.
Asymptotic behaviour, nonoscillation and stability in periodic first-order linear delay differential equations
Authors: Philos Ch. G.
Published: November 2011

First-order scalar linear delay differential equations with periodic coefficients and constant delays are considered, where the coefficients have a common period and the delays are multiples of this period. A basic asymptotic criterion is given. Moreover, some results on the nonoscillation and on the stability of the trivial solution are obtained. An equation, which is in a sense the characteristic equation, plays an important role in establishing the results of the paper.
Averaging lemmas without time Fourier transform and application to discretized kinetic equations
Authors: Bouchut F. , Desvillettes L.
Published: November 2011

We prove classical averaging lemmas in theL2framework with the help of the Fourier transform in variablesxandv, but nott. This method is then used to study discretized problems arising out of the numerical analysis of kinetic equations.
Trajectory singularities of general planar motions
Authors: Donelan P. S. , Gibson C. G. , Hawes W.
Published: November 2011

Local models are given for the singularities that can appear on the trajectories ofgeneral motions of the plane with more than two degrees of freedom. Versal unfoldings of these model singularities give rise to computer-generated pictures describing the family of trajectories arising from small deformations of the tracing point, and determine the local structure of the bifurcation curves.
Free quotients of infinite rank of GL2 over Dedekind domains
Authors: Mason A. W.
Published: November 2011

This paper is concerned with integral domains R, for which the factor group SL2(R)/U2(R) has a non-trivial, free quotient, where U2(R) is the subgroup of GL2(R) generated by the unipotent matrices. Recently, Krstić and McCool have proved that SL2(P[x])/U2(P[x]) has a free quotient of infinite rank, where P is a domain which is not a field. This extends earlier results of Grunewald, Mennicke and Vaserstein.

Any ring of the type P[x] has Krull dimension at least 2. The purpose of this paper is to show that result of Krstić and McCool extends to some domains of Krull dimension 1, in particular to certain Dedekind domains. This result, which represents a two-dimensional anomaly is the best possible in the following sense. It is well known that SL2(R) = U2(R), when R is a domain of Krull dimension zero, i.e. when R is a field. It is already known that for some arithmetic Dedekind domains A, the factor group SL2(A)/U2(A) has a free quotient of finite (and not infinite) rank.
Flow properties for Young-measure solutions of semilinear hyperbolic problems
Authors: Mielke Alexander
Published: November 2011

For hyperbolic systems in one spatial dimension ∂tu + C∂xu = f(u), u(t, x) ∈ ℝd, we study sequences of oscillating solutions by their Young-measure limit, μ, and develop tools to study the evolution of μ directly from the Young measure, v, of the initial data. For d ≤ 2 we construct a flow mapping, St, such that μ(t) = St(v) is the unique Young-measure solution for initial value v. For d ≥ 3 we establish existence and uniqueness of Young measures that have product structure, that is the oscillations in direction of the Riemann invariants are independent. Counterexamples show that neither μ nor the marginal measures of the Riemann invariants are uniquely determined from v, except if a certain structural interaction condition for f is satisfied. We rely on ideas of transport theory and make use of the Wasserstein distance on the space of probability measures.
Stability of non-monotone waves in a three-species reaction—diffusion model
Authors: Miller Patrick D.
Published: November 2011

A stability theorem is proved for non-monotone waves in a reaction–diffusion system modelling three competing species, where few stability results currently exist for systems with more than two species. Asymptotic stability with respect to the nonlinear equations is established by showing that the spectrum of the linearized operator has no unstable eigenvalues and that the zero eigenvalue associated with translation invariance is simple. The result is obtained in a singular regime where strong pattern formation occurs and solutions to the linear equations can be separated into particular solutions related to the fast–slow structure of the underlying wave and its singular limit. In this system both the fast and slow waves can contribute an instability and the global characterization of these solutions must address certain difficulties not present in lower-dimensional systems. The topological index of Alexander, Gardner and Jones is used to count the eigenvalues of the linear operator.
Self-homotopy equivalences of product spaces
Authors: Pavešić Petar
Published: November 2011

The group of self-homotopy equivalences Aut(X ć Y) is represented as a product of two subgroups under the assumption that the self-equivalences of X ć Y can be diagonalized. Moreover, an analogous result holds for the subgroup Aut#(X ć Y) of self-equivalences, which induce identity automorphisms on homotopy groups. Other methods for the computation of Aut(X ć Y) are studied, especially when the spaces involved have an H- or coH-structure, and several examples are considered, among others, some non-simply connected H-spaces of rank 2.
On stress singularities near the boundary of a polygonal crack
Authors: Kozlov V. A. , Maz'ya V. G.
Published: November 2011

Synopsis

Two theorems on the spectrum of operator pencils generated by the three-dimensional system of anisotropic elasticity in the complement of a plane angle are proved.
Lipschitz regularity of solutions of some asymptotically convex problems
Authors: Raymond Jean-Pierre
Published: November 2011

Synopsis

We prove local Lipschitz regularity for minimisers of integral functionals of the form J(u) = ∫Ω{f(Du(x)) + g(x, u(x))} dx, where the integrand f is not convex but satisfies some asymptotic convexity assumption.
Behaviour at ±∞ for a model of laminar flames with applications to questions of flame propagation versus extinction
Authors: Avrin Joel D.
Published: November 2011

Synopsis

We consider the behaviour at x = ±∞ of solutions to reaction-diffusion equations modelling laminar flames in a premixed reactive gas. We show that if the initial data have limits at ±∞, then the solutions satisfy ODEs at ±∞ for all positive time. We then analyse the qualitative behaviour of solutions to the ODEs. Our applications include extensions of previous results on questions of flame propagation versus extinction, and a new decay result: if the initial temperature is above ignition temperature at one end of the domani and if the initial concentration vanishes at the other, then we show that the concentration decays^to zero uniformly as the time variable goes to infinity.
On theKOℤ/2-Euler class, I
Authors: Crabb M. C.
Published: November 2011

Synopsis

This paper presents a systematic account of ℤ/2-equivariantKO-theoretic methods in the study ofr-fields (that is,rlinearly independent cross-sections,r≧ 1) on a real vector bundle. Applications of the theory to the problem of immersing complex and quaternionic projective spaces in Euclidean space are given in [13].
On the inversion of the even and odd Hilbert transformations
Authors: Heywood P. , Rooney P.G.
Published: November 2011

Synopsis

The even and odd Hilbert transformations, H+ and H−, are known to be bounded from the power-weighted space ℒμp to itself for −1 < μ 1 and 0 < μ < 2 respectively. We show that they fail to be surjective on ℒ0,p and ℒ1, p respectively, and we characterise the spaces H+(ℒ0, p), H- (ℒ1, p) and find inverses for H+, H− on them.
Integrable spreads and spaces of constant curvature
Authors: Farran H. R. , Robertson S. A.
Published: November 2011

Synopsis

This paper is a continuation of [2], where we introduced the notion of global k-spreads on manifolds. Here we show that the space of all k-spreads on a manifold has the structure of an affine space, modelled on the vector space of sections of a certain vector bundle. We give some sufficient conditions for a manifold admitting an integrable k-spread to be a space of constant curvature and answer one of the questions raised in [2].
Periodic solutions for: ẋ(t) = λf(x(t),x(t – 1))
Authors: Arino O. , Benkhalti R.
Published: November 2011

Synopsis

We present a new result on the existence of periodic solutions for the equation:

for all positive parameters λ sufficiently large. Our fundamental assumption is the following monotonicity property: if ø ≧ ψ (ø and ψ are data) then x(ø) ≧ x(ψ). The proof consists in applying the global Hopf bifurcation theorem. The main steps are: (i) a classical estimation of the periods; (ii) an a priori estimate for the solutions along a connected branch; (iii) a transformation acting on periodic solutions.
Free generators in relatively free completely regular semigroups
Authors: Trotter P.G.
Published: November 2011

Synopsis

A subset Y of a free completely regular semigroup FCRx freely generates a free completely regular subsemigroup if and only if (i) each -class of FCRx contains at most one element of Y, (ii) {Dy;y ∊ Y} freely generates a free subsemilattice of the free semilattice FCRx/), and (iii) Y consists of non-idempotents. A similar description applies in free objects of some subvarieties of the variety of all completely regular semigroups.
Some modified Michaelis–Menten equations having stable closed trajectories
Authors: Smith Russell A.
Published: November 2011

A 3-dimensional autonomous ordinary differential equation is studied which models certain cellular biochemical reactions. Extended Poincaré-Bendixson theory is used to obtain algebraic conditions on the parameters which are sufficient for the existence of at least one stable closed trajectory. Similar conditions are also obtained for the absence of chaos and for the global convergence of solutions to a critical point.
Lower bounds for normal structure coefficients
Authors: Benavides T. Domínguez , Acedo G. López
Published: November 2011

Synopsis

Using some new expressions for the weakly convergent sequences coefficient WCS(X) the lower boundedness

is proved, where δ(-) is the (Clarkson) modulus of convexity. We also define a modulus of noncompact convexity concerning nearly uniformly convex spaces which is used to obtain another lower bound for WCS(X). The computation of this modulus in Ip-spaces shows that our second lower bound is the best possible in these spaces.
A note on lower limit of series and potential theory
Authors: Chevallier N.
Published: November 2011

Synopsis

We are looking for positive sequences (bn) which have the following property: If (an) is a positive sequence such that , then
Invariant subspaces and new explicit solutions to evolution equations with quadratic nonlinearities
Authors: Galaktionov Victor A.
Published: November 2011

We present new explicit solutions to some classes of quasilinear evolution equations arising in different applications, including equations of the Boussinesq type:

and quasilinear heat equations:

The method is based on construction of finite-dimensional linear functional subspaces which are invariant with respect to spatial operators having quadratic nonlinearities. The corresponding nonlinear evolution equations on invariant subspaces are shown to be equivalent to finite-dimensional dynamical systems. Examples of two-, three- and five- dimensional invariant subspaces are given. Some generalisations toN-dimensional quadratic operators are also considered.
Decay rates for the solutions of model equations for bore propagation
Authors: Rajopadhye S. V.
Published: November 2011

We study the large-time behaviour of solutions to the Korteweg-de Vries-Burgers equation with bore-like initial data. This work relies on the methods of Amick, Bona and Schonbeck to obtain sharp rates of temporal decay of certain norms of the solution, thus obtaining an improvement over results of Naumkin and Shishmarev.
Analysis of bifurcations in reaction–diffusion systems with no-flux boundary conditions: the Sel'kov model
Authors: Furter J. E. , Eilbeck J. C.
Published: November 2011

A plot of the bifurcation diagram for a two-component reaction-diffusion equation with no-flux boundary conditions reveals an intricate web of competing stable and unstable states. By studying the one-dimensional Sel'kov model, we show how a mixture of local, global and numerical analysis can make sense of several aspects of this complex picture. The local bifurcation analysis, via the power of singularity theory, gives us a framework to work in. We can then fill in the details with numerical calculations, with the global analytic results fixing the outline of the solution set. Throughout, we discuss to what extent our results can be applied to other models.
Bounds for vertical points of solutions of prescribed mean curvature type equations, I
Authors: Atkinson F. V. , Peletier L. A.
Published: November 2011

Synopsis

Conditions are obtained under which radial solutions of generalised capillary-type equations exhibit a first vertical point, together with bounds and asymptotic estimates for these points.
The Stefan problem with mushy regions: continuity of the interface
Authors: Bertsch M. , Klaver M. H. A.
Published: November 2011

Synopsis

In this paper we consider a one-dimensional Stefan problem with a source term. Under the assumption that the initial profile is monotone, we obtain continuity of the free boundaries between the solid, the mushy and the liquid region.
Multiparameter bifurcation for some particular reaction–diffusion systems
Authors: Esquinas J. , López-Gómez J.
Published: November 2011

Synopsis

In some cases, a reaction–diffusion system can be transformed into an abstract equation where the linear part is given by a polynomial of a linear operator, say Multiparameter bifurcation for this equation is considered as the coefficients of the operator polynomial in are varied.
Quadratic forms, weighted eigenfunctions and boundary value problems for non-linear second order ordinary differential equations
Authors: Fonda Alessandro , Mawhin Jean
Published: November 2011

Synopsis

Some known results for different kinds of boundary value problems for second order ordinary differential equations are generalised. Different approaches are compared with one another, using topological and variational methods and the theory of weighted eigenvalue problems.
Some existence, uniqueness and nonuniqueness theorems for solutions of parabolic equations with discontinuous nonlinearities
Authors: Feireisl Eduard , Norbury John
Published: November 2011

Synopsis

We consider the problem

where H stands for the maximal monotone graph associated with the Heaviside step function. It is shown that the problem possesses at least one (strong) solution belonging to an appropriate function space. Moreover, we prove:

(i) There is a smooth initial function u0, u0≧1, where the equality holds at exactly one point such that there are at least two different solutions corresponding to the initial data u0.

(ii) The comparison principle: The relation for any x ≠ 0, implies u1(t)>u2(t), t≧0 for any u1, u2 solving the problem with the initial data , respectively.

(iii) For a “reasonable” set of initial data the solution is uniquely determined. Moreover, the free boundary {(x, t)| u(x, t) = 1} is regular and on its complement the equation holds in a classical sense.
Phenomena of critical exponent in ℝ2
Authors: Adimurthi  , Srikanth P. N. , Yadava S. L.
Published: November 2011

Synopsis

In this paper we make an attempt to explain the critical phenomena in ℝ2. We do this by exhibiting a class of functions having growth and for which

do not admit a solution when R is sufficiently small, where B(R) denotes the ball of radius R in ℝ2.
On the vibrations of rectangular plates
Authors: Haraux Alain , Komornik Vilmos
Published: November 2011

Synopsis

We prove that, for every non-empty open subset of a rectangular plate, there exists a positive number T such that no vibration of this plate with fixed edges can remain strictly above the rest position at all points of the subset during a period T. Moreover, the result remains valid for every non-empty segment parallel to one of the sides of the plate. Our proof is based on some results of non-harmonic analysis.
Boundary controllability of nonlinear beams to bounded states
Authors: Lagnese J. E.
Published: November 2011

Synopsis

Considered is a nonlinear model which describes the dynamic behaviour of the not necessarily small longitudinal and transverse displacements of a thin, cantilevered beam. The motion of the beam is driven by a bending moment, an axial force and a vertical shear force that act at the free end of the beam. The main goal is to describe the reachable set of the system, that is, the set of all states that can be reached by varying the end forces within a certain set of admissible controls.
Analytic semigroups generated by ultraweak operators
Authors: Vespri Vincenzo
Published: November 2011

Synopsis

Let Ω be a regular open subset ofRN. We present an improved generation result for nonvariational operators inL1(Ω). This result is obtained by studying ultraweak operators and by proving generation of analytic semigroups inLp(Ω)(l<p≦∞) and in. We also characterise interpolation and extrapolation spaces.
Normal/superconducting transitions in Landau–Ginzburg theory
Authors: Chapman S. J. , Howison S. D. , McLeod J. B. , Ockendon J. R.
Published: November 2011

Synopsis

The Landau–Ginzburg equations governing a normal/superconducting transition layer are considered. Existence, uniqueness and monotonicity of a solution are proved.
Existence and stability of steady flows of weakly viscoelastic fluids
Authors: Guillopé Colette , Saut Jean-Claude
Published: November 2011

Synopsis

We consider steady flows of viscoelastic fluids for which the extrastress tensor is given by a differential constitutive equation and is such that the retardation time is large (weaklyviscoelastic fluids).

We show the existence of a unique viscoelastic steady flow close to a given Newtonian flow and investigate its linear stability.

As an example, we consider the Bénard problem for viscoelastic fluids and we prove that there exists a nontrivial linearly stable flow of a weakly viscoelastic fluid in a container heated from below.
Fibrewise category
Authors: James I. M. , Morris J. R.
Published: November 2011

Synopsis

The purpose of this paper is to introduce a fibrewise generalisation of category, in the sense of Lusternik–Schnirelmann. This reduces to the classical concept when the space is a point. Fibrewise category may be compared with equivariant category, which has been the subject of some recent research [1,7,8]. Many variations on the basic idea of category have been discussed in the literature, for example the concept of category of a map, but since the generalisations to the fibrewise case are fairly routine they are not considered here.
Decay of solutions of a higher order multidimensional nonlinear Korteweg–de Vries–Burgers system
Authors: Linghai Zhang
Published: November 2011

We study decay estimates for the solutions to the initial value problem for a higher order multidimensional nonlinear Korteweg–de Vries–Burgers system. The method is integral estimation.
Localising and seminormal composition operators on L2
Authors: Campbell James T. , Hornor William E.
Published: November 2011

Let (X, ∑, μ) denote a σ-finite measure space. We show that the kernel condition on a weighted composition operator acting on L2(X, ∑, μ), which is necessary for hyponormality of the adjoint, implies that a certain subset of X has the localising property defined by Lambert. For operators satisfying this condition, we find a reducing subspace whose orthocomplement in L2 is annihilated by both the operator and its adjoint, allowing us to obtain characterisations of seminormality for the operator by looking only at the restriction to the reducing subspace. This simplifies the analysis significantly, giving transparent characterisations for the hyponormality and quasinormality of the adjoint, as well as a characterisation of normality for the operator which does not require the computation of any conditional expectations. Several examples are given. We then characterise the semi-hyponormal class for both the operator and its adjoint.
Cockcroft properties of graphs of 2-complexes
Authors: Gilbert N.D. , Howie James
Published: November 2011

Necessary and sufficient conditions are obtained for the 2-skeleton of the total space of a graph of 2-complexes to be Cockcroft, or L-Cockcroft for some subgroup L of the fundamental group. These conditions are used to construct new examples of Cockcroft and absolutely Cockcroft 2-complexes.
Global branches of positive weak solutions of semilinear elliptic problems over nonsmooth domains
Authors: Healey Timothy J. , Kielhöfer Hansjörg , Stuart Charles A.
Published: November 2011

We consider the nonlinear eigenvalue problem posed by a parameter-dependent semilinear second-order elliptic equation on a bounded domain with the Dirichlet boundary condition. The coefficients of the elliptic operator are bounded measurable functions and the boundary of the domain is only required to be regular in the sense of Wiener. The main results establish the existence of an unbounded branch of positive weak solutions.
Rearrangements of functions on unbounded domains
Authors: Douglas R. J.
Published: November 2011

A characterisation is provided for the weak closure of the set of rearrangements of a function on an unbounded domain. The extreme points of this convex, weakly compact set are classified. This result is used to study the maximising sequences of a variational problem for steady vortices.
Modelling the effective conductivity function of an arbitrary two–dimensional polycrystal using sequential laminates
Authors: Clark Karen E. , Milton Graeme W.
Published: November 2011

The effective conductivity tensor σ*of a two-dimensional polycrystalline material depends on the conductivity tensor σ0of the pure crystal from which the polycrystal is constructed and on the geometrical configuration of grains in the polycrystal, represented by a rotation fieldR(x) giving the orientation of the crystal at each pointx.Here it is established that the dependence of σ*on σ0in any polycrystal, with R (x) held fixed, can be mimicked exactly by a polycrystal constructed by sequential lamination. It is first shown that the effective conductivity function is perturbed only slightly if we truncate the Hilbert space of fields in the polycrystal to a finitedimensional space. Then the structure of this finite-dimensional space of fields is shown to be isomorphic to the structure of the finite-dimensional space of fields associated with the sequential laminate. In particular, there is an operation which corresponds to peeling away the layers in the sequential laminate and successively reducing the dimension of the space of fields.
Multiple homoclinic orbits for autonomous, singular potentials
Authors: Bessi Ugo
Published: November 2011

We consider the problem

where u ∊ Rn, n ≧ 2, and V ∊ C2(Rn∖e, R) is a potential having an absolute maximum at 0 and such that V(x) → − ∞ as x → e. We prove that, under some conditions on V, this problem has at least n − 1 geometrically distinct solutions.
Singular non-selfadjoint differential operators
Authors: Ibrahim Sobhy El-sayed
Published: November 2011

A characterisation is obtained of all the regularly solvable operators and their adjoints generated by general ordinary quasidifferential expressions inThe domains of these operators are described in terms of boundary conditions involving thesolutions ofM[u] =λwuand the adjoint equationat both singular end-pointsaandb.These results are an extension of those proved in [3], by Evans and Ibrahim, to the case of two singular end-points of the interval (a, b), and a generalisation of those in [10] and [13] concerning selfadjoint andJ-selfadjoint differential operators, whereJdenotes complex conjugation.
Approximation of Young measures by functions and application to a problem of optimal design for plates with variable thickness
Authors: Bonnetier E. , Conca C.
Published: November 2011

Given a parametrised measure and a family of continuous functions (φn), we construct a sequence of functions (uk) such that, as k→∞, the functions φn(uk) converge to the corresponding moments of the measure,in the weak * topology. Using the sequence (uk) corresponding to a dense family of continuous functions, a proof of the fundamental theorem for Young measures is given.

We apply these techniques to an optimal design problem for plates with variable thickness. The relaxation of the compliance functional involves three continuous functions of the thickness. We characterise a set of admissible generalised thicknesses, on which the relaxed functional attains its minimum.
Any large solution of a non-linear heat conduction equation becomes monotonic in time
Authors: Galaktionov Victor A. , Posashkov Sergey A.
Published: November 2011

Synopsis

In this paper we prove a certain monotonicity in time of non-negative classical solutions of the Cauchy problem for the quasilinear uniformly parabolic equation u1 = (ϕ(u))xx + Q(u) in wT = (0, T] × R1 with bounded sufficiently smooth initial function u(0, x) = uo(x)≧0 in Rl. We assume that ϕ(u) and Q(u) are smooth functions in [0, +∞) and ϕ′(u) >0, Q(u) > 0 for u > 0. Under some additional hypothesis on the growth of Q(u)ϕ′(u) at infinity, it is proved that if u(to, xo) becomes sufficiently large at some point (to, xo) ∈ wT, then ut(t, x0) ≧0 for all t ∈ [t0, T]. The proof is based on the method of intersection comparison of the solution with the set of the stationary solutions of the same equation. Some generalisations of this property for a quasilinear degenerate parabolic equation are discussed.
The nonwandering set of a special system of differential equations
Authors: Pliss V. A.
Published: November 2011

Synopsis

In the theory of non-linear oscillations there occur systems with a small parameter in the derivatives and discontinuous forcing terms. Here we study such a system.
On Harper's mod pH-space of rank 2
Authors: Kono Akira
Published: November 2011

Synopsis

Let X be Harper's mod pH-space of rank 2. In this paper we determine the Charn character of K*(X) and the stable homotopy type of X.
Localising sets for sigma-algebras and related point transformations
Authors: Lambert Alan
Published: November 2011

Synopsis

Each sigma-finite subalgebra from the sigma-algebra of a measure space induces a conditional expectation operator which acts on L2 as well as the set of almost everywhere nonnegative measurable functions. The concept of localising set is introduced and shown to be closely related to certain functional equations involving . Localising sets are shown to arise naturally in the study of weighted point transformations f→ϕ. f°T, where ϕ is a measurable function and T is a measurable self-map of the state space. A complete characterisation of localising sets related to such transformations is given when the underlying measure space is completely atomic.
On the spectral analysis of self adjoint operators generated by second order difference equations
Authors: Smith Dale T.
Published: November 2011

Synopsis

In this paper, I shall consider operators generated by difference equations of the form

where Δ is the forward difference operator, and a, p, and r are sequences of real numbers. The connection between the oscillation constant of this equation and the bottom of the essential spectrum of self-adjoint extensions of the operator generated by the equation is given, as well as various other information about the spectrum of such extensions. In particular, I derive conditions for the spectrum to have only countably many eigenvalues below zero, and a simple criterion for the invariance of the essential spectrum.
On eigencurves of elliptic boundary value problems
Authors: Binding P. A. , Browne P. J. , Huang Y. X. , Picard R. H.
Published: November 2011

Synopsis

Let T be a selfadjoint uniformly elliptic partial differential operator on a bounded domain in Rn, and let S be a (possibly indefinite) L∞ multiplication operator. Estimates of the form σλ + o(λ) and σλ + β + o(1) are sought for the eigenvalues μ(λ) of λS – T as λ→ ±∞. A necessary and sufficient condition is also obtained for existence of linear eigencurves, i.e. μ(λ) = σλ + β.
Integral inequalities for concave functions with applications to special functions
Authors: Brenner J. L. , Alzer Horst
Published: November 2011

Synopsis

In this paper we prove refinements, extensions and counterparts of known integral inequalities for concave functions. Furthermore, we apply our results to find new inequalities for some special functions and mean values.
Hypergeometric groups
Authors: Alperin Roger C.
Published: November 2011

Synopsis

We introduce and begin an investigation of the family of hypergeometric groups. We determine which hypergeometric groups of Euclidean type are infinite.
The existence of infinitely many solutions all bifurcating from λ = 0
Authors: Rother Wolfgang
Published: November 2011

Synopsis

We consider the non-linear differential equation

and state conditions for the function q such that (*) has infinitely many distinct pairs of (weak) solutions such that holds for all k ∈ ℕ. The main tools are results from critical point theory developed by A. Ambrosetti and P. H. Rabinowitz [1].
Radial solutions of a semilinear elliptic problem
Authors: Herrero M. A. , Velázquez J. J. L.
Published: November 2011

Synopsis

We analyse the set of nonnegative, global, and radial solutions (radial solutions, for short) of the equation

where 0 < p < 1, and is a radial and almost everywhere nonnegative function. We show that radial solutions of (E) exist if f(r) = o(r2p/1−1−p) or if f(r) ≈ cr2p/1−p as r → ∞, where

When f(r) = c*r2p/1−p + h(r) with h(r) = o(r2p/1−p) as r → ∞, radial solutions continue to exist if h(r) is sufficiently small at infinity. Existence, however, breaks down if h(r) > 0,

Whenever they exist, radial solutions are characterised in terms of their asymptotic behaviour as r → ∞.
Symmetry breaking for semilinear elliptic equations on sectorial domains in ℝ2
Authors: Lin Song-Sun
Published: November 2011

Synopsis

We first study the Poisson equation Δu =fin Ώω,and, where Ωω= {(rcos θ,rsin θ): 0<r<1, θ ∈(0,ω)} is a sector in ℝ2, ω ∈ (0, 2π), Г0= {(cos θ, sin θ): θ ∈ (0, ω)} and Г1= ∂Ωω− Г0,band λ are in ℝ1. We obtain Schauder-type estimates and Fredholm alternative theory for the problem. We then study the symmetry breaking problem for the Gel'fand equation Δu+ λeu= 0 in Ωωand obtain a complete picture about the relationships among three parameters λ,b, and ω in the problem.
An ergodic theorem for asymptotically nonexpansive mappings
Authors: Krüppel Manfred , Górnicki Jaroslaw
Published: November 2011

The purpose of this paper is to prove the following (nonlinear) mean ergodic theorem: Let E be a uniformly convex Banach space, let C be a nonempty bounded closed convex subset of E and let T: C → C be an asymptotically nonexpansive mapping. If

exists uniformly in r = 0, 1, 2,…, then the sequence {Tnx} is strongly almost-convergent to a fixed point y of T, that is,

uniformly in i = 0, 1, 2, ….
Each univariate complex polynomial has a ‘big’ factor
Authors: Glesser Ph. , Mignotte M. , Petkovic M.
Published: November 2011

We consider the following problem: Let P be a monic polynomial of degree n with complex coefficients. What can be the maximum ‘size’ of a monic divisor Q of P? Here the size of a polynomial R is the maximum ||R|| of the moduli of its values on the unit circle. In 1991, B. Beauzamy proved that there exists a divisor Q with ||Q|| ≧ e∈n−1, ∈ = 0.0019, when all the roots of P belong to the unit circle. Using a very recent result of D. Boyd, we obtain a general result which, in the same case, gives ||Q||≧βn; here β = 1.38135 … is optimal.
Cesàro averaging operators
Authors: Stempak Krzysztof
Published: November 2011

We define a family of Cesàro operators , Reα≧0, and consider the question of their boundedness on Hp spaces. We also consider discrete versions of these operators acting on sequence spaces.
Time-dependent solutions of the Navier–Stokes equations for spatially-uniform velocity gradients
Authors: Craik A. D. D.
Published: November 2011

Classes of exact solutions of the Navier–Stokes equations for incompressible fluid flow are explored. These have spatially-uniform velocity gradients at each instant, but often display complex temporal behaviour. Particular illustrative cases are described and related to previously-known solutions.
M-function behaviour for a periodic Dirac system
Authors: Clemence Dominic P.
Published: November 2011

For a 2 × 2 periodic system with a perturbation P whose first moment is finite, Jy′ = [λI + R(x) + P(x)]y, we study the behaviour of the Titchmarsh–Weyl m(λ)-coefficient at the spectral gap endpoints. Assuming gap nondegeneracy, our main result is that as λ → λ0, (λ − λ0)½(m(λ) → c ≠ 0 if and only if λ0 is a φ-half-bound state, which follows from an analysis of Jost-type functions.
Two-parameter strong laws and maximal inequalities forU-statistics
Authors: McConnell Terry R.
Published: November 2011

Synopsis

We provide necessary and sufficient conditions for two-parameter convergence in the strong law of large numbers forU-statistics. We also obtain weak-type (1,1) inequalities for one and two-sampleU-statistics of order 2 which are, in a sense, best possible.
Regularisation of Abel's integral equation
Authors: Hai Dang Dinh , Ang Dang Dinh
Published: November 2011

Synopsis

Consider the Abel integral operator

where 0 < α < 1. Supposeuis inH1(0, 1) ofH1-norm ≦E, andfis an element ofL2(0, 1) such that ∥Au–f∥L−2< ε. We give a regularised approximate solutionuβ(f) of the equation

which satisfies

and can be computed simply by performing some integrations. The preceding error estimate can be sharpened by strengthening regularity conditions onu
A class of contracted inverse semigroup rings
Authors: Munn W. D.
Published: November 2011

Synopsis

Contracted inverse semigroup rings are studied subject to the restriction that the semilattices of the given inverse semigroups satisfy a certain finiteness condition, introduced in 1980 by Teply, Turman and Quesada. Results are obtained on semiprimitivity, primitivity, primeness, decomposition into a direct sum of ideals, and chain conditions on one-sided ideals.
Periodic solutions of polynomial non-autonomous differential equations
Authors: Devlin J.
Published: November 2011

Let ℐN denote the set of (N + 1)-tuples of C1 ω-periodic functions; the point P = (pN,…, p1p0) ϵ ℐN is identified with the differential equation

We examine the way in which the total number of ω-periodic solutions can vary as P traverses a path in ℐN.
A note on semiprime Malcev superalgebras
Authors: Elduque Alberto
Published: November 2011

Synopsis

Prime Malcev superalgebras over fields of characteristic not two and three have been studied by Shestakov [8]. He obtains the remarkable result that if these superalgebras have a nonzero odd part then they are Lie superalgebras. The main purpose of this note is to extend this result to fields of characteristic three. To this aim, it is enough to use adequately a result of Filippov [3]. Commutative and anticommutative superalgebras will be considered too, showing that they are prime, semiprime or simple as superalgebras if and only if they are as algebras. Finally, some conclusions for finite-dimensional semisimple Malcev superalgebras will be deduced. Any such superalgebra is the direct sum of a semisimple Lie superalgebra and a direct sum of simple non-Lie algebras.
Existence and finite dimensionality of the global attractor for a class of nonlinear dissipative equations
Authors: Alarcón Eduardo Arbieto
Published: November 2011

Synopsis

In this work we discuss the Cauchy problem for a class of nonlinear dissipative equations as well as the existence of a global attractor and we estimate its dimension in the sense of the Hausdorff (or fractal) dimension.
Hofer–Zehnder symplectic capacity for two-dimensional manifolds
Authors: Jiang Mei-Yue
Published: November 2011

Synopsis

In this paper, we show that the Hofer-Zehnder symplectic capacity of a two-dimensional manifold is same as the area of the manifold. Using this fact, we also get a result on the symplectic capacity of the cotangent bundle of the torus with its canonical symplectic form.
Isoperimetric bounds for higher eigenvalue ratios for the n-dimensional fixed membrane problem
Authors: Ashbaugh Mark S. , Benguria Rafael D.
Published: November 2011

Synopsis

We give several results which extend our recent proof of the Payne-Pólya–Weinberger conjecture to ratios of higher eigenvalues. In particular, we show that for a bounded domain Ω⊂ℝn the eigenvalues of its Dirichlet Laplacian obey where λm denotes the mth eigenvalue and jp,k denotes the kth positive zero of the Bessel function Jp(x). Certain extensions of this result are given, the most general being the bound where k≧2 and l(m) denotes the number of nodal domains of an mth eigenfunction. Our results imply certain further conjectures of Payne, Pólya, and Weinberger concerning λ3/λ2 and λ4/λ3. In addition, we find a resonably good bound on λ4/λ1. We also briefly discuss extensions to Schrödinger operators and other elliptic eigenvalue problems.
First-order elliptic systems degenerated at the boundary
Authors: Dzhuraev Abduhamid
Published: November 2011

Synopsis

In this paper we state that the bounded solutions of general first order linear elliptic systems of two equations in bounded multiply-connected plane domains, degenerated at the boundary, are determined in domains without any boundary conditions, provided the boundary is not characteristic for this system. The explicit formula for calculating the index of the system is derived.
Regularity of the solutions for elliptic problems on nonsmooth domains in ℝ3. Part II: Regularity in neighbourhoods of edges
Authors: Guo Benqi , Babuška Ivo
Published: November 2011

This paper is the second in a series of three devoted to the analysis of the regularity of solutions of elliptic problems on nonsmooth domains in ℝ3. The present paper concentrates on the regularity of solutions of the Poisson equation in neighbourhoods of edges of a polyhedral domain in the framework of the weighted Sobolev spaces and countably normed spaces. These results can be generalised to elliptic problems arising from mechanics and engineering, for instance, the elasticity problem on polyhedral domains. Hence, the results are not only important to understand comprehensively the qualitative and quantitative aspects of the behaviours of the solution and its derivatives of all orders in neighbourhoods of edges, but also essential to design an effective computation and analyse the optimal convergence of the finite elements solutions for these problems.
Justification of mean-field coupled modulation equations
Authors: Schneider Guido
Published: November 2011

We are interested in reflection symmetric (x↦–x) evolution problems on the infinite line. In the systems which we have in mind, a trivial ground state loses stability and bifurcates into a temporally oscillating, spatial periodic pattern. A famous example of such a system is the Taylor-Couette problem in the case of strongly counter-rotating cylinders. In this paper, we consider a system of coupled Kuramoto–Shivashinsky equations as a model problem for such a system. We are interested in solutions which are slow modulations in time and in space of the bifurcating pattern. Multiple scaling analysis is used in the existing literature to derive mean-field coupled Ginzburg–Landau equations as approximation equations for the problem. The aim of this paper is to give exact estimates between the solutions of the coupled Kuramoto–Shivashinsky equations and the associated approximations.
On the evaluation of the norm of an integral operator associated with the stability of one-electron atoms
Authors: Burenkov V. I. , Evans W. D.
Published: November 2011

The norm of an integral operator occurring in the partial wave decomposition of an operator B introduced by Brown and Ravenhall in a model for relativistic one-electron atoms is determined. The result implies that B is non-negative and has no eigenvalue at 0 when the nuclear charge does not exceed a specified critical value.
Products of nilpotent linear transformations
Authors: Sullivan R. P.
Published: November 2011

In this paper we characterise the linear transformations of an infinite-dimensional vector space that can be written as the product of nilpotent transformations. This and a linear version of Malcev's congruence on transformation semigroups are then used to construct a new class of congruence-free semigroups.
Elliptic solutions and blow-up in an integrable Hénon–Heiles system
Authors: Eilbeck J. C. , Enol'skii V. Z.
Published: November 2011

We consider an integrable case of the Henon-Heiles system and use an isomorphism with the two-gap KdV-flow to construct families of real elliptic trajectories which are associated with two-gap elliptic solitons of the KdV equation. Some of these solutions exhibit blow-up in finite time.
On a family of one-relator pro-p-groups
Authors: Gildenhuys D. , Ivanov S. , Kharlampovich O.
Published: November 2011

The problem of describing one-relator pro-p-groups of cohomological dimension two (along the lines of Lyndon's description of discrete one-relator groups of cohomological dimension two) is still open. The known method of passing by means of a suitable p-filtration to a graded Lie algebra is not applicable to the family of one-relator pro-p-groups presented in this article, since the relators cannot be separated from the p-th powers in the free pro-p-group. In terms of the p-filtrations, the relators come arbitrarily close to a p-th power, yet the groups they define have cohomological dimension two.
Closed geodesics in Riemannian manifolds with convex boundary
Authors: Candela Anna Maria , Salvatore Addolorata
Published: November 2011

In this paper we look for closed geodesies on a noncomplete Riemannian manifold ℳ. We prove that if ℳ has convex boundary, then there exists at least one closed nonconstant geodesic on it.
Weierstrass condition for the general basic variational problem
Authors: Hüsseinov Farhad
Published: November 2011

The Weierstrass necessary condition for a multidimensional basic variational problem due to Hadamard is strengthened.
On the existence of a timelike trajectory for a Lorentzian metric
Authors: Masiello Antonio
Published: November 2011

We show the existence of a timelike periodic trajectory for a time-dependent Lorentzian metric on ℝn× ℝ.
Change of stability for Schrödinger semigroups
Authors: Brown K. J. , Daners D. , López-Gómez J.
Published: November 2011

In this paper we analyse the change of stability of Schrödinger semigroups with indefinite potentials when a coupling parameter varies. Generically, the change of stability takes place at a principal eigenvalue associated with the problem. The uniqueness of the principal eigenvalue is shown for several classes of potentials.
Lipschitz conditions satisfied by Hankel transforms
Authors: Heywood P. , Rooney P. G.
Published: November 2011

It is shown that Hankel transforms of functions on certain weighted LP spaces satisfy Lipschitz and integral Lipschitz conditions. In particular, Fourier-cosine and Fourier-sine transforms satisfy such Lipschitz conditions on such spaces.
Unbounded solution components for nonlinear Hill's equations
Authors: Mrziglod Thomas
Published: November 2011

A class of nonlinear Hill's equations on ℝ is considered, where the nonlinearity is concentrated on a compact interval [−N, N]. For values of the parameter λ not in the spectrum of the linearised equation (which is purely continuous) an equivalent nonlinear Sturm–Liouville problem on [−N, N] with parameter-dependent boundary conditions at x = ± N is given. Extending this problem to all real values of the parameter in a suitable way makes it possible to prove the existence of unbounded solution components for both the extended Sturm–Liouville problem and the original problem. The complicated structure of the extended problem results in new phenomena. For example, the number of zeros of different functions in the same solution component may be different.
Poincaré–Birkhoff results for Lagrangian systems with subquadratic potential
Authors: Cingolani Silvia , Pisani Lorenzo
Published: November 2011

In this paper, using a recent generalisation of Morse Theory, we study the existence of periodic solutions of the Lagrangian equation (1.1) with subquadratic potential and asymptotically flat, nonconstant, time-dependent metric on ℝN. In Section 3, we get also an ‘alternative result’ about the minimal period or the existence of infinitely many solutions.