bluuebunny/crossref_metadata_2025 huggingface dataset:
Spectral analysis of perturbed multiplication operators occurring in polymerisation chemistry
Authors: Kokholm Niels Jørgen
Published: November 2011

Synopsis

We consider a mathematical model for the motion of a marked monomer in a system of reacting polymers at equilibrium. A well-posed integro-differential initial value problem for the probability of finding the marked monomer in a molecule of a given length is formulated. We prove exponential convergence of the probability to a unique equilibrium distribution. A quite complete spectral analysis is carried out for a self adjoint operator, which is a perturbation of a multiplication operator by an integral operator and is related to the generator of the time evolution.
The asymptotic behaviour of the solutions of the Kassoy problem with a modified source term
Authors: Budd C. , Qi Y.
Published: November 2011

Synopsis

We study the asymptotic behaviour as x →∞ of the solutions of the ordinary differential equation problem

This equation generalises the ordinary differential equation obtained by studying the blow-up of the similarity solutions of the semilinear parabolic partial differential equation vt=vxx = ev. We show that if λ≦1, all solutions of (*) tend to —∞ as rapidly as the function —exp (x2/4) (E- solutions). However, if λ>1, then there also exists a solution which tends to –∞, like 2λlog(x) (L-solutions). Thus, the case λ = 1, for which (*) reduces tothe Kassoy equation, is the borderline between two quite different forms of asymptotic behaviour of the function u(x).
Normal and osculating maps for submanifolds of RN
Authors: Gasparini I. Cattaneo , Romani G.
Published: November 2011

Synopsis

Let Mn be a manifold supposed “nicely curved” isometrically immersed in ℝn+p. Starting from a generalised Gauss map associated to the splitting of the normal bundle defined from the values of the fundamental forms of M of order k (k ≧ 0), we give necessary and sufficient conditions for the map to be totally geodesic and harmonic . For k = 0 is the classical Gauss map and our formula reduces to Ruh–Vilm's formula with a more precise formulation due to the consideration of the splitting of the normal bundle.

We also give necessary conditions for M, supposed complete, to admit an isometric immersion with . This theorem generalises a theorem of Vilms on the manifolds with second fundamental forms parallel (case k = 0). The result is interesting as the class of manifolds satisfying the condition is larger than the class of manifolds satisfying .
On boundary feedback stabilisability of a viscoelastic beam
Authors: Leugering Guenter
Published: November 2011

Synopsis

It is shown that a cantilevered beam with weak viscoelastic damping of Boltzmann-type can be uniformly stabilised by velocity feedback applied as a shearing force at the free end of the beam. Estimates for the viscoelastic energy are derived using the energy multiplier method. The energy decay is related to the decay of the relaxation modulus associated with the viscoelastic material.
A nonlinear pseudoparabolic equation
Authors: Ang Dang Dinh , Thanh Tran
Published: November 2011

Synopsis

The authors prove results on uniqueness and global existence of initial and boundary value problems for the nonlinear pseudoparabolic equation

with nonhomogeneous boundary conditions. A salient feature of the paper is that F and its partial derivatives are allowed to be unbounded. In the special case b(x, t)= α2 (a positive constant), it is proved that the corresponding solution uα, under appropriate conditions on the data (which are satisfied, for example, by the Benjamin–Bona–Mahony equation), uα→ u0 the solution corresponding to β = 0, on sufficiently small time interval. A result on the asymptotic behaviour of the solution is given for t → ∞.
Real roots of polynomials and right inverses for partial differential operators in the space of tempered distributions
Authors: Langenbruch Michael
Published: November 2011

Synopsis

LetP(D)be a partial differential operator with constant coefficients. IfP(D)has a continuous linear right inverse in the space of tempered distributions, thenPis the product of a polynomial without real roots and a real polynomial admitting a right inverse. If the polynomialPis real and irreducible, thenP(D)admits a right inverse in the tempered distributions if and only ifP(×)has the property of zeros of R. Thorn.
The Friedrichs extension of regular ordinary differential operators
Authors: Niessen Heinz-Dieter , Zettl Anton
Published: November 2011

Synopsis

For regular symmetric ordinary differential expressions we show that (i) the minimal operator is bounded below and (ii) the Friedrichs extension is determined by Dirichlet boundary conditions. Both proofs are based on elementary inequalities.
On a linear, partly hyperbolic model of viscoelastic flow past a plate
Authors: Fraenkel L. E.
Published: November 2011

Synopsis

The paper presents solutions, for a class of shear relaxation functions, ofa linear problem formulated by Joseph [8] to elucidate the steady, supercritical flow of a viscoelastic fluid past a semi-infinite flat plate. The velocity (U, 0) at infinity is parallel to the plate, and‘supercritical flow ‘means that U is greater than the propagation speed C of shear waves. As a result, the vorticity satisfies a hyperbolic equation and is confined to the region downstream of a shock wave from the leading edge of the plate. The disturbance velocity fieldextends upstream of the shock and is continuous across it. In contrast to the case of a Newtonian fluid, the solutions are unique under the condition that the functions representing the vorticity on the two sides of the platebelong to a certain Banach space.
A variational approach to a nonlinear Steklov problem
Authors: Ermens B. , Mawhin J.
Published: November 2011

Synopsis

We study the Steklov problem which consists in finding a complex function w(z) = U(z) + iV(z) holomorphic in the open unit disc G of the complex plane, continuous on its closure, such that V(0) = 0 and verifying on its boundary the condition

(d/ds)V(eis)+g(s, U(eis))=h(s),

where g and h are given functions. Using the Hilbert transform, the problem is reduced to the search for periodic solutions of an equivalent singular integro-differential equation which is treated by the direct method of the calculus of variations. When g(s,.) is non-decreasing, we obtain a necessary and sufficient condition for the solvability. The case of a non-monotone nonlinearity is also considered.
Complicated dynamics of scalar reaction diffusion equations with a nonlocal term
Authors: Fiedler Bernold , Poláčik Peter
Published: November 2011

Synopsis

We consider the dynamics of scalar equations ut, = uxx + f(x, u) + c(x)α(u), 0 < x < l, where α denotes some weighted spatial average and Dinchlet boundary conditions are assumed. Prescribing f, c, α appropriately, it is shown that complicated dynamics can occur. Specifically, linearisations at equilibria can have any number of purely imaginary eigenvalues. Moreover, the higher order terms of the reduced vector field in an associated centre manifold can be prescribed arbitrarily, up to any finite order. These results are in marked contrast with the case α = 0, where bounded solutions are known to converge to equilibrium.
Serrin integrals and second order problems of plasticity
Authors: Cesari L. , Yang Wei H.
Published: November 2011

Synopsis

We use the modern tools of the duality principles and the calculus of variations to formulate, analyse and solve a class of plasticity problems involving second order partial derivatives. The Serrin-type integrals can most appropriately facilitate the existence statements for the extrema from either side of the duality relation in a larger class of BV functions, and interpret the solutions with possible discontinuities on sets of measure zero. The exact solutions of a beam and numerical solutions of a circular plate are presented to demonstrate the theoretical conclusions.
Asymptotic stability criteria for delay-differential equations
Authors: Becker L.C. , Burton T.A.
Published: November 2011

Synopsis

This paper is concerned with the problem of showing uniform stability and equiasymptotic stability of thezero solution of functional differential equations with either finite or infinite delay. The investigations are based on Liapunov's direct method and attention is focused on those equations whose right-hand sides are unbounded for bounded state variables.
On the vibrations of a square membrane
Authors: Komornik V.
Published: November 2011

Synopsis

We establish that if Ω is an open square and if P is an arbitrary point of Ω, then the solutions of the wave equation with Dirichlet boundary condition utt − Δu = 0 in R × Ω, u = 0 on R × Γ can remain strictly positive during an arbitrarily large time. In fact stronger results are proved, considering several points of Ω simultaneously and prescribing different, more general, behaviour of the solution at these points.
On the asymptotic behaviour at infinity of solutions of the traction boundary value problem
Authors: Orazov B.B.
Published: November 2011

Synopsis

Korn's inequalities are proved for star-shaped domains and it is shown how the constants in these inequalities depend on the dimensions of the domain. These inequalities are then used to prove a generalisation of Saint-Venant's Principle for nonlinear elasticity and additionally to establish the asymptotic behaviour of solutions to the traction boundary value problem for a non-prismatic cylinder.
The existence of bitangent spheres
Authors: Hai-bao Duan , Rees Elmer
Published: November 2011

Synopsis

Given any point x on a smooth, closed submanifold M of Euclidean space, there is a bitangent hyperplane or sphere that is tangent to M at x.
Remarks concerning a free boundary problem arising in the theory of liquid drops and in plasma physics
Authors: Barrett John W. , Elliott Charles M.
Published: November 2011

Synopsis

We consider a generalisation of the liquid drop problem, introduced in [1, Part II], by allowing the upper and lower surfaces to have different surface tension coefficients γv and γu. We study the existence, uniqueness and regularity of this problem. In addition, we show that as γv/γu →0, the solution of this problem converges to the solution of the “plasma problem”.
On a uniqueness theorem of L. Amerio and G. Prouse
Authors: Haraux Alain
Published: November 2011

Synopsis

We give two generalizations of a theorem of L. Amerio and G. Prouse concerning uniqueness of the almost-periodic solution of the wave equation with a local multivalued damping term and almost-periodic forcing.

Counterexamples are given which show that these results may fail if some hypotheses of the theorems are dropped. They also show that the two generalizations are relatively independent.
Resolution of a semilinear equation in L1
Authors: Gallouët Thierry , Morel Jean-Michel
Published: November 2011

Synopsis

Let Ω = ℝN or Ω be a bounded regular open set of ℝN and let γ(x, S): Ω × ℝ → ℝ be a continuous nondecreasing function in s, measurable in x, such that γ(x, 0) = 0 almost everywhere. We solve, for f ∈ L1(Ω), the problem (P): −Δu + γ(., u) = f in Ω, u = 0 on ∂Ω. (In fact, for this result, instead of assuming that γ is nondecreasing in s we need only that γ(x, s)s≧0.) We deduce an ‘almost’ necessary and sufficient condition on , in order that (P) has a solution. Roughly speaking, this condition is f = −ΔV + g, with g ∈ L1(Ω) and γ(., V) ∈ L1(Ω)
Exotic equilibrium states of the elastic string
Authors: Reeken Michael
Published: November 2011

Synopsis

A detailed description of the totality of distributional solutions of the catenary problem for nonlinear elastic strings is given.
Left orders in strongly regular rings
Authors: Ánh Pham Ngoc , Márki László
Published: November 2011

Synopsis

In this paper we characterise left orders in strongly regular rings, both for classical left orders and left orders in the sense of Fountain and Gould [2] where the ring of quotients need not have an identity.
On the semigroups of partial one-to-one order-decreasing finite transformations
Authors: Umar Abdullahi
Published: November 2011

Synopsis

Let In be the symmetric inverse semigroup on Xn = {1,…, n}, let Sln be the subsemigroup of strictly partial one-to-one self-maps of Xn and let = { α ∊ SIn: x} ≦ x = U = ∅= be the semigroup of all partial one-to-one decreasing maps including the empty or zero map of Xn. In this paper it is shown that is an (irregular, for n ≧ 2) type A semigroup with n D*-classes and D* = I*. Further, it is shown that is generated by the n(n + l)/2 quasi-idempotents in
Stable determination of a crack from boundary measurements
Authors: Alessandrini Giovanni
Published: November 2011

Synopsis

We treat the problem of determining a crack inside a conductor when two pairs of current and voltage boundary measurements are given. We prove a theorem of continuous dependence from the data.
The equation utt − Δu = |u|p for the critical value of p
Authors: Schaeffer Jack
Published: November 2011

Synopsis

The equation utt − Δu = |u|p is considered in two and three space dimensions. Smooth Cauchy data of compact support are given at t = 0. For the case of three space dimensions, John has shown that solutions with sufficiently small data exist globally in time if but that small data solutions blow up in finite time if Glassey has shown the two dimensional case is similar. This paper shows that small data solutions blow up in finite time when p is the critical value, in three dimensions and in two.
Spectral functions of a symmetric linear relation with a directing mapping, II
Authors: Langer H. , Textorius B.
Published: November 2011

Synopsis

The results of part I (see [5]) are applied to pairs of formally symmetric differential expressions, to Hermitian differential systems and to a reduced operator moment problem.
Nondassical eigenvalue distribution of one-dimensional Schrödinger operators
Authors: Klaus M.
Published: November 2011

Synopsis

We consider differential operators of the form H = −d2/dx2 + q(x) acting on u ∈ L2(0,∞) with boundary condition u(0) = 0. The potential q(x) is such that H has essential spectrum [0,∞) and an infinite sequence of negative eigenvalues converging to zero. Let n(E) denote the number of eigenvalues of H which are less than E. Under certain conditions on q(x), the well-known formula n(E)∼(2φ)−1 vol {x, p | p2 + q(x)<E}, E↑0, holds. We shall study the validity of this formula for potentials which show oscillatory behaviour as x →∞, like e.g. q(x) = −(1 + x)−α(a + b sin x) with 0<α <2, a≧0, b≠0. We shall obtain the leading-order behaviour of both n(E) and vol n(E)∼(2φ)−1 vol {x, p | p2 + q(x)<E} as E↑0 for a certain class of q's, and we shall see that the classical formula fails in most cases, but there are some noteworthy exceptions.
Symmetry sets
Authors: Bruce J. W. , Giblin P. J. , Gibson C. G.
Published: November 2011

Synopsis

For a smooth manifold M ⊆ ℝn, the symmetry set S(M) is defined to be the closure of the set of points u∈ℝn which are centres of spheres tangent to M at two or more distinct points. (The idea has its origin in the theory of shape recognition.) The connexion with singularities is that S(M) can be described alternatively as the levels bifurcation set of the family of distance-squared functions on M. In this paper a multi-germ version of the standard uniqueness result for versal unfoldings of potential functions is used to obtain a complete list of local normal forms (up to diffeomorphism) for the symmetry sets of generic plane curves, generic space curves, and generic surfaces in 3-space. For these cases the authors verify that M can be recovered as the envelope of a family of spheres centred at smooth points of S(M).
A general monotone scheme for elliptic systems with applications to ecological models
Authors: Korman Philip , Leung Anthony W.
Published: November 2011

Synopsis

We consider weakly-coupled elliptic systems of the type

with each fi being either an increasing or a decreasing function of each uj. Assuming the existence of coupled super- and subsolutions, we prove the existence of solutions, and provide a constructive iteration scheme to approximate the solutions. We then apply our results to study the steady-states of two-species interaction in the Volterra–Lotka model with diffusion.
Minimum action solutions of non-linear elliptic equations in unbounded domains containing a plane
Authors: Kavian Otared
Published: November 2011

Synopsis

Let d ≧ 1 be an integer and ω ⊂ℝd a smooth bounded domain and consider the elliptic equation − Δu = g(u) on Ω = ℝ2 × ω. We prove that under (almost) necessary and sufficient conditions on the continuous function g: ℝm→ ℝm the above equation has a minimum-action solution.
A nilpotent-generated semigroup associated with a semigroup of full transformations
Authors: Howie John M. , Marques-Smith M. Paula O.
Published: November 2011

Synopsis

Let X be a set with infinite regular cardinality m and let ℱ(X) be the semigroup of all self-maps of X. The semigroup Qm of ‘balanced’ elements of ℱ(X) plays an important role in the study by Howie [3,5,6] of idempotent-generated subsemigroups of ℱ(X), as does the subset Sm of ‘stable’ elements, which is a subsemigroup of Qm if and only if m is a regular cardinal. The principal factor Pm of Qm, corresponding to the maximum ℱ-class Jm, contains Sm and has been shown in [7] to have a number of interesting properties.

Let N2 be the set of all nilpotent elements of index 2 in Pm. Then the subsemigroup (N2) of Pm generated by N2 consists exactly of the elements in Pm/Sm. Moreover Pm/Sm has 2-nilpotent-depth 3, in the sense that
Lattice isomorphisms of Malcev algebras
Authors: Elduque Alberto
Published: November 2011

Synopsis

The lattice of subalgebras of a Malcev algebra determines to a great extent the structureof the algebra. It is shown that conditions such as nilpotency, solvability or semisimplicity are almost characterised by means of conditions on this lattice. This enables us to study the relationship between Malcev algebras with isomorphic lattices of subalgebras.
Almost-everywhere injectivity in nonlinear elasticity
Authors: Qi Tang
Published: November 2011

Synopsis

This paper gives a sufficient condition for almost-everywhere injectivity for nonlinear three dimensional elasticity similar to that of Claret-Necas [8], namely.

We prove that this relation is maintained under the weak convergence of minimising sequences for nonlinear elasticity problems. The existence and partial regularity of an “inverse” function are proved.
Painlevé analysis of the damped, driven nonlinear Schrödinger equation
Authors: Clarkson Peter A.
Published: November 2011

Synopsis

In this paper we apply the Painlevé tests to the damped, driven nonlinear Schrödinger equation

where a(x, t) and b(x, t) are analytic functions of x and t, todetermine under what conditions the equation might be completely integrable. It is shown that (0.1) can pass the Painlevé tests only if

where α0(t),α1(t) and β(t) are arbitrary, real analytic functions of time. Furthermore, it is shown that in this special case, (0.1) may be transformed into the original nonlinear Schrödinger equation, which is known to be completely integrable.
Local existence and blow-up criterion for the Boussinesq equations
Authors: Chae Dongho , Nam Hee-Seok
Published: November 2011

Synopsis

In this paper, we prove local existence and uniqueness of smooth solutions of the Boussinesq equations. We also obtain a blow-up criterion for these smooth solutions. This shows that the maximum norm of the gradient of the passive scalar controls the breakdown of smooth solutions of the Boussinesq equations. As an application of this criterion, we prove global existence of smooth solutions in the case of zero external force.
Positive definite temperature functions and a correspondence to positive temperature functions
Authors: Chung Soon-Yeong
Published: November 2011

Synopsis

Positive definite temperature functionsu(x, t) in ℝn+1= {(x, t)|x∈ ℝn,t> 0} are characterised by

where μ is a positive measure satisfying that for every ℰ > 0,

is finite. A transformis introduced to give an isomorphism between the class ofall positive definite temperature functions and the class of all possible temperature functions inThen correspondence given bygeneralises the Bochner–Schwartz Theorem for the Schwartz distributions and extends Widder's correspondence characterising some subclass of the positive temperature functions by the Fourier-Stieltjes transform.
The critical exponents of parabolic equations and blow-up inRn
Authors: Qi Yuan-wei
Published: November 2011

In this paper we study the Cauchy problem inRnof general parabolic equations which take the formut= Δum+ts|x|σupwith non-negative initial value. Heres≧ 0,m> (n− 2)+/n,p> max (1,m) and σ > − 1 ifn= 1 or σ > − 2 ifn≧ 2. We prove, among other things, that forp≦pc, wherepc≡m+s(m− 1) + (2 + 2s+ σ)/n> 1, every nontrivial solution blows up in finite time. But forp>pca positive global solution exists.
Free-discontinuity problems generated by singular perturbation
Authors: Alicandro Roberto , Braides Andrea , Gelli Maria Stella
Published: November 2011

We show that some free discontinuity problems can be obtained as a limit of nonconvex local functionals with a singular perturbation of higher order.
A perturbed Hopf bifurcation with reflection symmetry
Authors: Nagata Wayne
Published: November 2011

Synopsis

We study the effects of a small symmetry breaking perturbation on a system of differential equations at a coupled Hopf bifurcation with O(2) symmetry, where the perturbation breaks the continuous rotation symmetry, but retains a reflection (Z2) symmetry. It is shown that for a large range of parameter values, the invariant manifolds of the unperturbed bifurcation persist and that for some values of normal form coefficients there are secondary bifurcations of nonsymmetric periodic standing wave solutions.
A global existence theorem for the Cauchy problem of nonlinear wave equations
Authors: Gao Jianmin , Xu Lichen
Published: November 2011

Synopsis

In this paper we consider the global existence (in time) of the Cauchy problem of the semilinear wave equation utt – Δu = F(u, Du), x ∊ Rn, t > 0. When the smooth function F(u, Du) = O((|u| + |Du|)k+1) in a small neighbourhood of the origin and the space dimension n > ½ + 2/k + (1 + (4/k)2)½/2, a unique global solution is obtained under suitable assumptions on initial data. The method used here is associated with the Lorentz invariance of the wave equation and an improved Lp–Lq decay estimate for solutions of the homogeneous wave equation. Similar results can be extended to the case of “fully nonlinear wave equations”.
On smooth solutions to the initial value problem for the mixed nonlinear Schrödinger equations
Authors: Boling Guo , Shaobin Tan
Published: November 2011

Synopsis

Solutions to the initial value problem for the mixed nonlinear Schrödinger equation

are considered. Conditions on the constants α,β, γ, function g(·) and initial data u(x, 0) are given so that, for this problem, the unique existence of smooth solutions is proved. In addition, the decay behaviours of the smooth solutions as |x|→+∞ are discussed.
On the absolutely continuous spectrum of a vector-matrix Dirac system
Authors: Clark Stephen L.
Published: November 2011

A Dirac system is considered which has a matrix-valued long-range, short-range and oscillatory potentials. The system has one singular endpoint at infinity. Additional conditions on the potential are given which guarantee particular asymptotic behaviour of an energy functional associated with a certain set of solutions. This asymptotic behaviour guarantees the existence of a purely absolutely continuous spectrum outside a gap containing the origin.
Existence and multiplicity of homoclinic orbits for potentials on unbounded domains
Authors: Caldiroli Paolo
Published: November 2011

We study the system in RN, where V is a potential with a strict local maximum at 0 and possibly with a singularity. First, using a minimising argument, we can prove the existence of a homoclinic orbit when the component Ω of {x ∈ RN: V(x) < V(0)} containing 0 is an arbitrary open set; in the case Ω unbounded we allow V(x) to go to 0 at infinity, although at a slow enough rate. Then we show that the presence of a singularity in Ω implies that a homoclinic solution can also be found via a minimax procedure and, comparing the critical levels of the functional associated to the system, we see that the two solutions are distinct whenever the singularity is ‘not too far’ from 0.
Switched diffusion processes and systems of elliptic equations: a Dirichlet space approach
Authors: Chen Z. Q. , Zhao Z.
Published: November 2011

The switched diffusion process associated with a weakly coupled system of elliptic equations is studied via a Dirichlet space approach and is applied to prove the existence theorem of the Cauchy initial problem for the system. A representation theorem for the solution of the Dirichlet boundary value problem and a generalised Skorohod decomposition for the reflecting switched diffusion process are obtained.
On the asphericity of one-relator relative presentations
Authors: Edjvet Martin
Published: November 2011

We study the 1-relator relative presentation 〈H, x|xaxbx−1c〉 where H is a group, a, b, c ∈ H, x ∉ H and b, c ≠ 1. We give necessary and sufficient conditions for this presentation to be aspherical apart from two outstanding special cases which remain open.
Nilpotent local automorphisms of an independence algebra
Authors: Lima Lucinda M.
Published: November 2011

Let A be a strong independence algebra of infinite rank m. Let ℒ(A) be the inverse monoid of all local automorphisms of A. The Baer–Levi semigroup B over the algebra A is defined to be the subsemigroup of ℒ(A) consisting of all the elements α with dom α = A and corank im α = m. Let Km be the subsemigroup of ℒ(A) generated by B−1B. Then Km is inverse and it is generated (as a semigroup) by N2, the subset of ℒ(A) consisting of all nilpotent elements of index 2. The 2-nilpotent depth, Δ2(Km) of Km is defined to be the smallest positive integer t such that Km = N2∪…∪(N2)t. In fact, Δ2(Km) is either 2 or 3 and a criterion is found which distinguishes between the two cases.

If N denotes the set of all nilpotents in ℒ(A), then the subsemigroup generated by N is also Km. In fact, Km is proved to be exactly N2.
A necessary and sufficient condition for nonattainment and formation of microstructure almost everywhere in scalar variational problems
Authors: Friesecke Gero
Published: November 2011

For scalar variational problems

subject to linear boundary values, we determine completely those integrandsW: ℝn→ ℝ for which the minimum is not attained, thereby completing previous efforts such as a recent nonexistence theorem of Chipot [9] and unifying a large number of examples and counterexamples in the literature.

As a corollary, we show that in case of nonattainment (and providedWgrows superlinearly at infinity), every minimising sequence converges weakly but not strongly inW1,1(Ω) to a unique limit, namely the linear deformation prescribed at the boundary, and develops fine structure everywhere in Ω, that is to say every Young measure associated with the sequence of its gradients is almost-nowhere a Dirac mass.

Connections with solid–solid phase transformations are indicated.
The direct and inverse problem in Newtonian scattering
Authors: Astaburuaga M. A. , Fernández Claudio , Cortés Víctor H.
Published: November 2011

Synopsis

In this paper we study the direct and inverse scattering problem on the phase space for a classical particle moving under the influence of a conservative force. We provide a formula for the scattering operator in the one-dimensional case and we settle the properties of the potential that can be deduced from it. We also study the question of recovering the shape of the barriers which can be seen from −∞ and ∞. An example is given showing that these barriers are not uniquely determined by the scattering operator.
Maps between p-completed classifying spaces II
Authors: Wojtkowiak Zdzisław
Published: November 2011

We investigate maps between p-completed classifying spaces of compact connected Lie groups. Let G and G′ be two connected compact Lie groups. For a space X, let Xp be a p-completion of X. If p does not divide the order of the Weyl group of G, we give descriptions of the set of homotopy classes [(BG)p, (BG′)p] in terms of K-theory and in terms of “admissible” maps of Adams and Mahmud.
On the analysis and control of hyperbolic systems associated with vibrating networks
Authors: Lagnese J. E. , Leugering G. , Schmidt E. J. P. G.
Published: November 2011

In this paper a general linear model for vibrating networks of one-dimensional elements is derived. This is applied to various situations including nonplanar networks of beams modelled by a three-dimensional variant on the Timoshenko beam, described for the first time in this paper. The existence and regularity of solutions is established for all the networks under consideration. The methods of first-order hyperbolic systems are used to obtain estimates from which exact controllability follows for networks containing no closed loops.
Geometry related to the key del Pezzo surface and the associated mapping of plane cubics
Authors: Edge W. L.
Published: November 2011

Synopsis

When the points of a projective space [9] map ternary cubics the maps of perfect cubes are the points of a del Pezzo surface F. Several manifolds linked with F are listed and the cubics mapped on these are noted. A new approach leads to the first of Aronhold's two invariants.
Rees matrix covers for a class of abundant semigroups
Authors: Lawson Mark V.
Published: November 2011

Synopsis

Recently considerable attention has been paid to the study of locally inverse regular semigroups. McAlister [14] obtained a description of such semigroups as locally isomorphic images of regular Rees matrix semigroups over an inverse semigroup. The class of abundant semigroups originally arose from ‘homological’ considerations in the theory of S-systems: they are the semigroup theoretic analogue of PP-rings. Cancellative monoids, full subsemigroups of regular semigroups as well as the multiplicative semigroups of PP rings are abundant. The aim of this paper is to show how the structure theory described above for regular semigroups may be generalised to a class of abundant semigroups.
Operators preserving orthogonality are isometries
Authors: Koldobsky Alexander
Published: November 2011

Synopsis

We prove that every operator preserving orthogonality in a real Banach space is an isometry multiplied by a constant.
Linear equations in B(ℤ)*
Authors: Filali M.
Published: November 2011

Synopsis

Let B(ℤ)* be the Banach dual of the space of all bounded complex-valued functions on ℤ. For each n ε ℤ, let Ln be the translation operator on B(ℤ) and Tn be its adjoint operator on B(ℤ)*. This paper concerns itself with equations of the form

where (an)nεℤ is a sequence of complex numbers.
Inequalities for non-decreasing sequences
Authors: Alzer Horst
Published: November 2011

Synopsis

In this paper we prove an extension of inequality (1.1) due to A. Meir.
Sharp uniform bounds for steady potential fluid-Poisson systems
Authors: Gamba Irene M.
Published: November 2011

We consider steady potential hydrodynamic-Poisson systems with a dissipation term (viscosity) proportional to a small parametervin a two- or three-dimensional bounded domain. We show here that for any smooth solution of a boundary value problem which satisfies that the speed, denoted by |∇φv|, has an upper coarse bound, uniform in the parameterv, then a sharper, correct uniform bound is obtained: the viscous speed |∇φv| is bounded pointwise, at pointsx0in the interior of the flow domain, by cavitation speed (given by Bernoulli's Law at vacuum states) plus a term ofthat depends on. The exponent is β = 1 for the standard isentropic gas flow model and β = 1/2 for the potential hydrodynamic Poisson system. Both cases are considered to have a γ-pressure law with 1<γ<2 in two space dimensions and 1 < γ< 3/2 in three space dimensions. These systems have cavitation speeds which take not necessarily constant values. In fact, for the potential hydrodynamic-Poisson systems, cavitation speed is a function that depends on the potential flow function and also on the electric potential.

In addition, we consider a two-dimensional boundary value problem which has been proved to have a smooth solution whose speed is uniformly bounded. In this case, we show that the pointwise sharper bound can be extended to the section of the boundary ∂Ω\∂3Ω, where ∂3Ω is called the outflow boundary. The exponent β varies between 1 and 1/8 depending on the location ofx0at the boundary and on the curvature of the boundary atx0. In particular, our estimates apply to classical viscous approximation to transonic flow models.
The second Hamiltonian structure for the periodic Toda lattice
Authors: Beffa Gloria Marí
Published: November 2011

In this paper we study the Poisson geometry of the second Hamiltonian structure for the periodic N Toda lattice, around a certain family of singularities. We show that their singular leaves are not isolated and that the regular codimension of the leaves at points of this kind is always equal to three. This result is based on a rather unexpected result about a certain Toeplitz matrix.
Local minimisers of a three-phase partition problem with triple junctions
Authors: Sternberg Peter , Zeimer William P.
Published: November 2011

We establish the existence of isolated local minimisers to the problem of partitioning certain two-dimensional domains into three subdomains having least interfacial area. The solution we exhibit has the special property that the three boundaries of the minimising partition meet at a common point or “triple junction”. The configuration represents a likely candidate for a stable equilibrium in the dynamical problem of two-dimensional motion by curvature and also leads to the existence of local minimisers possessing triple junction structure to the energy associated with the vector Ginzburg–Landau and Cahn–Hilliard evolutions.
Existence of changing sign solutions for some semilinear problems with jumping nonlinearities at zero
Authors: Dancer E. N. , Du Yihong
Published: November 2011

We study the existence of changing sign solutions of an elliptic semilinear boundary value problem, which arises as a limiting equation of the two species Lotka–Volterra competing equations system. Using variational methods and a result of D'Aujourd'hui, we find conditions which are both sufficient and necessary for this existence problem.
Finite semilattices whose non-invertible endomorphisms are products of idempotents
Authors: Adams M. E. , Bulman-Fleming Sydney , Gould Matthew , Wildsmith Amy
Published: November 2011

For a finite semilattice S, is is proved that if every noninvertible endomorphism is a product of idempotents, then S is a chain; the converse was proved, independently, by A. Ya. Aĭzenštat and J. M. Howie. For a finite pseudocomplemented semilattice S, with pseudocomplementation regarded as a unary operation, it is proved that all noninvertible endomorphisms are products of idempotents if and only if S is Boolean or a chain.
Invariant algebraic curves and conditions for a centre
Authors: Christopher C. J.
Published: November 2011

Conditions for the existence of a centre in two-dimensional systems are considered along the lines of Darboux. We show how these methods can be used in the search for maximal numbers of bifurcating limit cycles. We also extend the method to include more degenerate cases such as are encountered in less generic systems. These lead to new classes of integrals. In particular, the Kukles system is considered, and new centre conditions for this system are obtained.
Harmonic maps from noncompact Riemannian manifolds with non-negative Ricci curvature outside a compact set
Authors: Wang Youde
Published: November 2011

In this paper we prove the uniqueness and existence of harmonic maps of finite energy from a complete, noncompact Riemannian manifold (M, g) with Sobolev constant S2(M) > 0 and Ricci curvature Ric (M) ≧ 0 outside some compact subset, into a complete manifold of nonpositive curvature or a regular ball. In particular, we prove the uniqueness and existence of bounded harmonic functions on (M, g).
Variation of constants for hybrid systems of functional differential equations
Authors: Hale Jack K. , Huang Wenzhang
Published: November 2011

The objective is to derive a variation of constants formula for systems of functional differential equations (or delay differential equations) coupled with functional equations (or difference equations). The difficulties arise because of the constraints imposed by the functional equations.
Approximate controllability of the semilinear heat equation
Authors: Fabre Caroline , Puel Jean-Pierre , Zuazua Enrike
Published: November 2011

This article is concerned with the study of approximate controllability for the semilinear heat equation in a bounded domain Ω when the control acts on any open and nonempty subset of Ω or on a part of the boundary. In the case of both an internal and a boundary control, the approximate controllability in LP(Ω) for 1 ≦ p < + ∞ is proved when the nonlinearity is globally Lipschitz with a control in L∞. In the case of the interior control, we also prove approximate controllability in C0(Ω). The proof combines a variational approach to the controllability problem for linear equations and a fixed point method. We also prove that the control can be taken to be of “quasi bang-bang” form.
Asymptotic behaviour of weak solutions to a boundary value problem for dynamic viscoelastic equations with memory
Authors: Liang Jin , Tiehu Qin
Published: November 2011

The paper discusses the asymptotic behaviour of weak solutions u(t, x), as t → ∞, to the boundary value problem for one-dimensional viscoelastic equations with singular memory. The changes of phase are admitted for the problem. One of our results is that ut(t, ·)⇀0 weakly in L2(0,1) as t → ∞.
Limit analysis for a class of nonconvex problems
Authors: Buttazzo Giuseppe , Faina Loris
Published: November 2011

Synopsis

The problem

is considered, where X is a normed space, F: X →] –∞, + ∞] is a (possibly non-convex) functional and L ∈ X'. We look for the values of γy for which the infimum above is attained. Applications to nonconvex functionals denned on measures and on the BV space are given.
On the continuity of the polyconvex, quasiconvex and rank-oneconvex envelopes with respect to growth condition
Authors: Gangbo Wilfrid
Published: November 2011

Synopsis

LetCf, Pf, QfandRfbe respectively the convex, polyconvex, quasi-convex and rank-one-convex envelopes of a given functionf. If fp: RNxM→R andfq(ξ) behaves as |ξ|qat infinityq∈ (1, ∞), we show that. This is the case for (Pfp)pprovided thatq≠1,…, min (N, M), otherwise. In the last part of this work, we show thatf(ξ) = g(|ξ|) does not imply in generalPf = Qf.
Bifurcation of the periodic differential equations with regularisable infinity
Authors: Ke-ying Guan
Published: November 2011

A class of scalar periodic differential equations x ]f(x, t, A) with regularisable infinity in which the Riccati equation is included, can be treated as autonomous systems on the torus T2. Through this geometric interpretation, the properties of the set of bifurcation points of λ (∊R) are studied. When the results obtained are applied to the Riccati equation, they can match the well-known properties of the spectral set of the corresponding Hill's equation.
The stationary thermistor problem with a current limiting device
Authors: Cimatti Giovanni
Published: November 2011

Synopsis

The electrical heating of a conductor connected with a current limiting device of total resistance R is studied under mixed boundary conditions. A theorem of existence is given using a transformation which permits the reduction of the nonlinear system governing the problem to the classical mixed boundary value problem for the Laplace equation.
Generalised characteristics in hyperbolic systems of conservation laws with special coupling
Authors: Dafermos C. M. , Geng X.
Published: November 2011

Synopsis

Using the theory of generalised characteristics, we study the structure of BV solutions of genuinely nonlinear systems of two conservation laws whose shock and rarefaction wave curves of the first family are straight lines. We also establish a priori estimates on the variation of the solution similar to those obtained earlier by Glimm and Lax.
Symmetry breaking and multiple solutions for a Neumann problem in an exterior domain
Authors: Zelati Vittorio Coti , Esteban Maria J.
Published: November 2011

Synopsis

In this paper we prove existence of multiple positive solutions for a Neumann problem in ℝN/(0, R), R large, with a superquadratic and odd nonlinearity. The proof is based on the fact that in such a situation the minimum of the corresponding energy functional (which is achieved) is not an even function and that there is quite a large gap (for large R) between such a minimum and the minimum of the same functional on even functions. In the set of functions whose energy lies in such a gap, we can apply index theory to prove the desired multiplicity result.
Bifurcation of steady-state solutions in predator-prey and competition systems
Authors: Blat J. , Brown K. J.
Published: November 2011

Synopsis

We discuss steady-state solutions of systems of semilinear reaction-diffusion equations which model situations in which two interacting species u and v inhabit the same bounded region. It is easy to find solutions to the systems such that either u or v is identically zero; such solutions correspond to the case where one of the species is extinct. By using decoupling and global bifurcation theory techniques, we prove the existence of solutions which are positive in both u and v corresponding to the case where the populations can co-exist.
Sur les sous-groupes planaires des groupes des dispersions des équations difiérentielles linéaires du deuxième ordre
Authors: Borůvka O.
Published: November 2011

Synopsis

A group consisting of real continuous functions of one real variable on the interval j = (−∞, ∞) is called planar if through each point of the plane j × j there passes just one element s ∈ .

Every differential oscillatory equation (Q): y″ = Q(t)y (t ∈ j = (−∞, ∞), Q ∈ C(0)) admits functions, called the dispersions of (Q), that transform (Q) into itself. These dispersions are integrals of Kummer's equation (QQ): −{X, t} + Q(X)X′2(t) = Q(t) and form a three-parameter group , known as the dispersion group of (Q). The increasing dispersions of (Q) form a three-parameter group invariant in . The centre of the group is an infinite cyclic group , whose elements, the central dispersions of (Q), describe the position of conjugate points of (Q).

The present paper contains new results concerning the algebraic structure of the group . It provides information on the following: (1) the existence and properties of planar subgroups of a given group and (2) the existence and properties of the groups containing a given planar group . The results obtained are: the planar subgroups of a given group form a system depending on two constants, SQ, such that for all ∈SQ. The equations (Q) whose groups contain the given planar group form a system dependent on one constant, QS, such that for all (Q)∈QS.
Lower bounds for the spectrum of a second order linear differential equation with a coefficient whose negative part is p-integrable
Authors: Harris B. J.
Published: November 2011

Synopsis

We consider ihe differential expression M[y]: = −y″ + qy on [0, ∞) where q_∈ Lp [0, ∞) for some p ≧ 1. It is known that M, together with the boundary conditions y(0) = 0 or y′(0) = 0, defines linear operators on L2 [0, ∞). We obtain lower bounds for the spectra of these operators. Our bounds depend on the Lp norm of q_ and extend results of Everitt and Veling.
On the limit-n classification of powers of positive coefficient ordinary differential expressions
Authors: Kauffman Robert M.
Published: November 2011

Synopsis

Let , with each pi a finite sum of real multiples of real powers of x, and with D = d/dx. Suppose that pi≧0, pN > 0, and p0≧ε > 0 on the interval [1, ∞). Suppose that there exists a j such that degree pi − 2j > degree pi−2i for all i ≠ j. We compute the deficiency index d(Ln) in L2[l, ∞) of any power of L. It is known that d(L) = N. We give an exact formula for d(Ln) which, if degree pi − 2j > 0, shows that d(Ln) = nN for all n if and only if j = 0, and that the limit as n approaches infinity of d(Ln)/n is N + j. We show in addition that if degree pi − 2i ≦ 0 for all i, d(Ln) = nN for all n.
Uniqueness of positive solutions of some semilinear Sturm–Liouville problems on the half line
Authors: Toland J. F.
Published: November 2011

Synopsis

This note gives a simple proof of uniqueness for positive solutions of certain non-linear boundary value problems on ℝ+ which are typified by the equation

with boundary conditions u′(0) = u(+∞) = 0. In the autonomous case (r ≡ 1), this is easy to see, by quadrature. The proof here supposes r to be non-increasing on ℝ+.
p-Summing operators on injective tensor products of spaces
Authors: Montgomery-Smith Stephen , Saab Paulette
Published: November 2011

Synopsis

Let X, Y and Z be Banach spaces, and let Πp (Y, Z) (1 ≦ p < ∞) denote the space of p-summing operators from Y to Z. We show that, if X is a ℒ∞-space, then a bounded linear operator is 1-summing if and only if a naturally associated operator T#: X → Πl (Y, Z) is 1-summing. This result need not be true if X is not a ℒ∞-space. For p > 1, several examples are given with X = C[0, 1] to show that T# can be p-summing without T being p-summing. Indeed, there is an operator T on whose associated operator T# is 2-summing, but for all N ∈ N, there exists an N-dimensional subspace U of such that T restricted to U is equivalent to the identity operator on . Finally, we show that there is a compact Hausdorff space K and a bounded linear operator for which T#: C(K) → Π1 (l1, l2) is not 2-summing.
Cauchy problem for an extended model of combustion
Authors: Lu Yun-guang
Published: November 2011

Synopsis

This paper considers the Cauchy problem for an extended model of combustion (u + qz)t + f(u)x = 0, zt + kg(u)z = 0 with Lp bounded initial data, where g(u) is a piecewise Lipschitz continuous function and its discontinuous points have no finite limit point. The integral representation gives a definition of a weak solution in Lp space. Some existence results are obtained based on a simplified method of compensated compactness in which the weak continuity theorem of 2 * 2 determinants plays a more important role, but the idea of Young measures has been avoided.
On singular differential operators with positive coefficients
Authors: Schultze Bernd
Published: November 2011

Synopsis

A class of singular real formally self-adjoint differential expressions M on I = [a, = ∞) (a ∈ ℝ), i.e. expressions of the form My = with pj ≧ 0 (j = 0, …, n – 1), pn > 0 is constructed with the following property: For every integer k with 0 ≦ k < n/2 there exists an expression M in this class such that the deficiency index of T0(M) – the minimal operator associated with M – is n + 2k. This generalises a result in [3] and proves part of the McLeod's conjecture.
On the spectrum of the periodic Dirac operator
Authors: Harris B. J.
Published: November 2011

Synopsis

In an earlier paper we considered periodic Dirac operators and obtained criteria for them to be self-adjoint and for their spectra to be devoid of eigenvalues of finite multiplicity. The question of the existence of eigenvalues of infinite multiplicity was left open. In this article we obtain further criteria for self-adjointness and show that under these conditions periodic Dirac operators do not possess eigenvalues of infinite multiplicity. We also obtain a spectral gap result.