bluuebunny/crossref_metadata_2025 huggingface dataset:
Sharpness of embeddings in logarithmic Bessel-potential spaces
Authors: Edmunds David E. , Gurka Petr , Opic Bohumír
Published: November 2011

This paper is a continuation of [4], where embeddings of certain logarithmic Bessel-potential spaces (modelled upon generalised Lorentz-Zygmund spaces) in appropriate Orlicz spaces (with Young functions of single and double exponential type) were derived. The aim of this paper is to show that these embedding results are sharp in the sense of [8].
Partial and complete linearisations at stationary points of infinite-dimensional dynamical systems with foliations and applications
Authors: Rivera W. M.
Published: November 2011

In this paper we discuss C1-linearisations of diffeomorphisms and flows on Banach spaces. Strong foliations of the neighbourhood of the fixed point composed of leaves based on successively larger subspaces (similar to those in [14]) are constructed. Generalised gap conditions which involve the width and separation of vertical bands containing the spectrum of a linear operator are imposed to achieve maximal smoothness. The method of proof generalises that of Hartman and of Mora and Solá-Morales. Our theorems apply to weakly coupled systems of damped wave and beam equations.
Sturm–Liouville problems with indefinite weights and Everitt's inequality
Authors: Volkmer Hans
Published: November 2011

It is shown that spectral properties of Sturm–Liouville eigenvalue problems with indefinite weights are related to integral inequalities studied by Everitt. A result of Beals on indefinite problems leads to a sufficient condition for the validity of such an inequality. A Baire category argument is used to show that, in general, the inequality under consideration does not hold.
Commutative 2-local ring spectra
Authors: Astey L.
Published: November 2011

A theorem is proved characterising representable, multiplicative commutative cohomology theories that split as sums of singular cohomologies after localisation at 2. This theorem is shown to be equivalent to one proved by Würgler and Pazhitnov and Rudyak, for which we provide a simplified proof. We also provide a simple proof of a related theorem of Boardman.
Equations with an infinite number of explicit conservation laws
Authors: Fairlie D. B.
Published: November 2011

A large class of first-order partial nonlinear differential equations in two independent variables which possess an infinite set of polynomial conservation laws derived from an explicit generating function is constructed. The conserved charge densities are all homogeneous polynomials in the unknown functions which satisfy the differential equations in question. The simplest member of the class of equations is related to the Born–Infeld Equation in two dimensions. It is observed that some members of this class possess identical charge densities. This enables the construction of a set of multivariable equations with an infinite number of conservation laws.
Normal forms for periodic retarded functional differential equations
Authors: Faria Teresa
Published: November 2011

This paper addresses the computation of normal forms for periodic retarded functional differential equations (FDEs) with autonomous linear part. The analysis is based on the theory previously developed for autonomous retarded FDEs. Adequate nonresonance conditions are derived. As an illustration, the Bogdanov–Takens and the Hopf singularities are considered.
Asymptotic behaviour of the solutions of the supercooled Stefan problem
Authors: Yi Fahuai
Published: November 2011

In this paper, we prove that the Hele–Shaw problem with kinetic condition and surface tension is the limit case of the supercooled Stefan problem in the classical sense when specific heat ε goes to zero. The method is the use of a fixed-point theorem; the key is to construct a workable function space. The main feature is to obtain the existence and the uniform estimates with respect to ε > 0 at the same time for the solutions of the supercooled Stefan problem. For the sake of simplicity, we only consider the case of one phase, although the method used here is also applicable in the case of two phases.
Remarks on spherically symmetric solutions of the compressible Euler equations
Authors: Chen Gui-Qiang
Published: November 2011

Some evidence indicates that spherically symmetric solutions of the compressible Euler equations blow up near the origin at some time under certain circumstances (cf. [4,19]). In this paper, we observe a criterion forL∞Cauchy data of arbitrarily large amplitude to ensure the existence ofL∞spherically symmetric solutions in the large, which model outgoing blast waves and large-time asymptotic solutions. The equilibrium states of the solutions and their asymptotic decay to such states are analysed. Some remarks on global spherically symmetric solutions are discussed.
Some new properties of Sobolev mappings: intersection theoretical approach
Authors: Isobe Takeshi
Published: November 2011

In this paper, we give some new examples of the energy gap phenomenon for functionals defined in Sobolev spaces. Our result is independent of that of Giaquinta, Modica and Soucek. We also give some new characterisations of Sobolev maps which can be approximated by smooth maps.
Global well-posedness and conservation laws for the water wave interaction equation
Authors: Ogawa Takayoshi
Published: November 2011

Interaction equations of long and short water wave are considered. It is shown that the Cauchy problem for

is locally well posed in the largest space where the three conservations

can be justified. HereE(u,v) is the energy functional associated to the system. By these conservation laws, we establish the global well-posedness of the system in the largest class of initial data.
A variational approach to multiplicity in elliptic problems near resonance
Authors: Ramos M. , Sanchez L.
Published: November 2011

We consider the nonlinear elliptic problem ± (Δu+ λu) +f(x,u) =h(x) in Ω,u= 0 on ∂Ω, where Ω is a bounded smooth domain in ℝN, λ is near the first eigenvalue andh(x) is orthogonal to the first eigenfunction. We give some conditions of existence of positive solutions and of multiple solutions in terms of the primitive offwith respect tou.
Boundary value problems for elliptic pseudodifferential operators II
Authors: Taira Kazuaki
Published: November 2011

The purpose of this paper is to study boundary value problems for elliptic pseudodifferential operators which originate from the problem of existence of Markov processes in probability theory, generalising some results of our previous work. Our approach has a great advantage of intuitive interpretation of sufficient conditions for the unique solvability of boundary value problems in terms of Markovian motion. In fact, we prove that if a Markovian particle moves incessantly both by jumps and continuously in the state space, not being trapped in the set where no reflection phenomenon occurs, then our boundary value problem is uniquely solvable in the framework of Sobolev spaces of LP style.
On the cohomology of the classifying space of a certain gauge group
Authors: Tsukuda Shuichi
Published: November 2011

The cohomology of the classifying space of the gauge group of the nontrivial SO(3) bundle over S2 has no torsion.
On gradients of approximate travelling waves for generalised KPP equations
Authors: Zhao H. Z.
Published: November 2011

In this paper we use stochastic semiclassical analysis and the logarithmic transformation to study the gradients of the approximate travelling wave solutions for the generalised KPP equations with Gaussian and Dirac delta initial distributions. We apply the logarithmic transformation to the nonlinear reaction diffusion equations and obtain a Maruyama–Girsanov–Cameron–Martin formula for the driftμ2∇loguμ,uμbeing a solution of a generalised KPP equation. We obtain thatμ2|∇ loguμ(t,x)| is bounded and the trough is flat. The difficult problem in this paper is to prove that the corresponding crest is flat. A probabilistic approach is used in this paper to treat this problem successfully.
Periodic homogenisation of Hamilton–Jacobi equations: 2. Eikonal equations
Authors: Concordel Marie C.
Published: November 2011

Homogenisation of the first-order Hamilton-Jacobi equations when H is periodic in the second variable, leads to an effective Hamiltonian H satisfying: uε converges, as ε → 0, to the solution u of ut + H(Du) = 0. In our first paper, we assumed that H is convex and we derived a variational formula giving H. In this second paper, we consider eikonal equations, i.e. H(p, x) = ½|p|2 – V(x). Using our variational formula, we compute explicitly the effective Hamiltonian in several cases, and we study precisely the lack of strict convexity for H (‘flat part’ around the origin).
Lower semicontinuity in Sobolev spaces below the growth exponent of the integrand
Authors: Kristensen Jan
Published: November 2011

Let there be given a non-negative, quasiconvex functionFsatisfying the growth condition

for somep∈]1, ∞[. For an open and bounded set Ω⊂ℝm, we show that if

then the variational integral

is lower semicontinuous on sequences ofW1, pfunctions converging weakly inW1, q. In the proof, we make use of an extension operator to fix the boundary values. This idea is due to Meyers [26] and Maly [22], and the main contribution here is contained in Lemma 4.1, where a more efficient extension operator than the one in [22] (and in [14]) is used. The properties of this extension operator are in a certain sense best possible.
Dynamic cavitation with shocks in nonlinear elasticity
Authors: Pericak-Spector K. A. , Spector Scott J.
Published: November 2011

The hyperbolic system of conservation laws that govern the motion of a homogeneous isotropic, nonlinearly elastic body is shown to have a discontinuous solution for a class of stored-energy functions of slow growth. This solution is admissible by the usual entropy criterion and is in fact preferred by the entropy-rate criterion over the smooth equilibrium solution to the same problem. The existence of such a dissipative solution shows that the equilibrium solution is dynamically unstable. This instability cannot be ascertained by linearisation.
Periodic trajectories in static space-times
Authors: Greco Carlo
Published: November 2011

Synopsis

Let R ×Nequipped with the warped Lorentzian metricf2dt2⊕(−h), where (N,h) is a Riemannian manifold, andf:N→ ]0, ∞[ is a smooth function. Then R ×Nis called a standard static space-time, and in this paper we look for non-trivial periodic trajectories on R ×NforNcompact.
Blow up for a diffusion-advection equation
Authors: Alikakos Nicholas D. , Bates Peter W. , Grant Christopher P.
Published: November 2011

Synopsis

These results describe the asymptotic behaviour of solutions to a certain non-linear diffusionadvection equation on the unit interval. The “no flux” boundary conditions prescribed result in mass being conserved by solutions and the existence of a mass-parametrised family of equilibria. A natural question is whether or not solutions stabilise to equilibria and if not, whether they blow up in finite time. Here it is shown that for non-linearities which characterise “fast association” there is a criticalmass such that initial data which have supercritical mass must lead to blow up in finite time. It is also shown that there exist initial data with arbitrarily small mass which also lead to blow up in finite time.
On some equivalences of positive definite binary quadratic forms
Authors: Arenas A.
Published: November 2011

Synopsis

We prove that for primitive positive definite binary quadratic forms having fundamental discriminant the concepts of genus and of ℚ-equivalence coincide.
Bifurcation for quasilinear elliptic equations on Rn with natural growth conditions
Authors: Daomin Cao , Gongbao Li , Shusen Yan
Published: November 2011

Synopsis

We consider the following eigenvalue problem:

We prove the existence of H1(Rn)∩L∞(Rn) bifurcation at λ=0 but only require aij(x, t) (i,j= 1, 2, …,n) and f(x, t) to satisfy certain conditions in theneighbourhood of Rn × {0}.
Weak continuity and weak lower semicontinuity for some compensation operators
Authors: Pedregal Pablo
Published: November 2011

Synopsis

We study a special class of linear differential operators well-behaved with respect to weakconvergence. Questions related to weak lower semicontinuity, associated Young measures, weak continuity and quasi-convexity are addressed. Specifically, it is shown that the well-known necessary conditions for weak lower semicontinuity are also sufficient in this case. Some examples are given, including a discussion on how well the operator curl fits inthis context.
A class of C∞-integrable Hamiltonian systems
Authors: Oliva W. M. , Castilla M. S. A. C.
Published: November 2011

Synopsis

We discuss the C∞ complete integrability of Hamiltonian systems of type q = —grad V(q) = F(q), in which the closure of the cone generated (with nonnegative coefficients) by the vectors F(q), q ϵ ℝn, does not contain a line. The components of the asymptotic velocities are first integrals and the main aim is to prove their smoothness as functions of the initial conditions. The Toda-like system with potential V(q)=ΣNi=1 exp(fi∣ q) is a special case of the considered systems ifthe cone C(f1,…,fN)={ΣNi=1cifi,ci≧0} does notcontain a line. In any number of degrees of freedom, if C(f1,…,fN) has amplitude not too large (ang (fi, fj ≦π/2i,j=1,2,…, N), the first integrals are C∞ functions. In two degrees of freedom, without restriction on the amplitude of the cone, C∞-integrability is proved even in a case in which it is known that there is no other meromorphic integral of motion independent of energy. In three degrees of freedom the C∞-integrability of a deformation of the classic nonperiodic Toda system is proved. Some other examples are also discussed.
Heteroclinic cycles involving periodic solutions in mode interactions with O(2) symmetry
Authors: Melbourne I. , Chossat P. , Golubitsky M.
Published: November 2011

Synopsis

In this paper we show that in O(2) symmetric systems, structurally stable, asymptoticallystable, heteroclinic cycles can be found which connect periodic solutions with steady states and periodic solutions with periodic solutions. These cycles are found in the third-order truncated normal forms of specific codimension two steady-state/Hopf and Hopf/Hopf mode interactions.

We find these cycles using group-theoretic techniques; in particular, we look for certainpatterns in the lattice of isotropy subgroups. Once the pattern has been identified, the heteroclinic cycle can be constructed by decomposing the vector field on fixed-point subspaces into phase/amplitude equations (it is here that we use the assumption of normal form). The final proof of existence (and stability) relies on explicit calculations showing that certain eigenvalue restrictions can be satisfied.
Homoclinic orbits for a class of Hamiltonian systems
Authors: Rabinowitz Paul H.
Published: November 2011

Synopsis

Consider the second order Hamiltonian system:

where q ∊ ℝn and V ∊ C1 (ℝ ×ℝn ℝ) is T periodic in t. Suppose Vq (t, 0) = 0, 0 is a local maximum for V(t,.) and V(t, x) | x| → ∞ Under these and some additional technical assumptions we prove that (HS) has a homoclinic orbit q emanating from 0. The orbit q is obtained as the limit as k → ∞ of 2kT periodic solutions (i.e. subharmonics) qk of (HS). The subharmonics qk are obtained in turn via the Mountain Pass Theorem.
Boundary conditions for general ordinary differential operators and their adjoints
Authors: Evans W. D. , Ibrahim Sobhy E.
Published: November 2011

Synopsis

A characterisation is obtained of all the regularly solvable operators and their adjoints generated by a general differential expression in . The domains of these operators are described in terms of boundary conditions involving the solutions of Mu = λwu and the adjoint equation . The results include those of Sun Jiong [15] concerning self-adjoint realisations of a symmetric M when the minimal operator has equal deficiency indices: if the deficiency indices are unequal the maximal symmetric operators are determined by the results herein. Another special case concerns the J -self-adjoint operators, where J denotes complex conjugation, and for this we recover the results of Zai-jiu Shang in [16].
Solvability of boundary value problems for vector differential systems
Authors: Erbe L. H. , Liu Xinzhi , Wu Jianhong
Published: November 2011

Synopsis

Brouwer topological degree theory, the shooting type method, the disconjugacy theory of Hamiltonian systems and the Liapunov-Razumikhin technique of Volterra integrodifferential equations are employed to establish some solvability results for the 2n-dimensional differential system

subject to one of the following boundary conditions:

(i) x(0) = Qx(l), Qg(l x(1), qy(1)= g(0, x(0), y(0)),

(ii) Blx(O) = B2g(O, x(O), y(O)), C1x(l) = −C2g(l, x(l), y(l)),

where Q, Bi, Ci, i = 1, 2, are n x n real matrices. An application is given to the second order equation xn = h(t, x, x') subject to certain nonlinear boundary conditions.
Examples of rank-one convex functions
Authors: Šverák Vladimír
Published: November 2011

Synopsis

We construct rank-one convex functions which are not convex and have linear growth at infinity. We show that these functions could be useful in some problems concerning weak convergence of gradients if we were able to prove that they are quasiconvex. This question, however, seems to be open.
Partial regularity of multiple variational integrals of any order
Authors: Xiangao Liu
Published: November 2011

Synopsis

We prove Cm, μ almost everywhere regularity for minimisers of functional of the form ∫f(x, su, Dmu)dx, where f is uniformly strictly quasiconvex.
Compact boundedness in periodic functional differential equations with infinite delay
Authors: Kato Junji
Published: November 2011

Synopsis

It is the aim of this article to consider some problems arising from the non local-compactness of the phase space for functional differential equations. The compact boundedness, that is, the boundedness depending on each compact set involving the initial values, is proved to be implied from the ultimate boundedness for periodic systems of functional differential equations on Cγ: = {φ ∊ C((–∞,0]) Note that it is known that the compactness cannot be dropped in the above. An example is also given to show that the asymptotic stability is not necessarily uniform even for periodic functional differential equations on Co.
Lower semicontinuity of multiple integrals and the Biting Lemma
Authors: Ball J. M. , Zhang K.-W.
Published: November 2011

Synopsis

Weak lower semicontinuity theorems in the sense of Chacon's Biting Lemma are proved for multiple integrals of the calculus of variations. A general weak lower semicontinuity result is deduced for integrands which are acomposition of convex and quasiconvex functions. The “biting”weak limit of the corresponding integrands is characterised via the Young measure, and related to the weak* limit in the sense of measures. Finally, an example is given which shows that the Young measure corresponding to a general sequence of gradients may not have an integral representation of the type valid in the periodic case.
On the asymptotic decay of L2-solutions of one-body Schrödinger equations in unbounded domains
Authors: Hoffmann-Ostenhof M.
Published: November 2011

Synopsis

The asymptotic decay of L2-solutions of Schrödinger equations (-Δ+V)ψ=0 in ΔR= {x εRn∣∣x∣=r>R} is investigated, where V(x) = V1(r) + V2(x) with V1→ ∞ for r↑∞ and with some ε > 0 for large r. Under additional assumptions on the decay of V1, pointwise upper bounds to |ψ |and lower bounds to the spherical average of ψ are given showing the same asymptotics for r→ ∞. For the case V→ const. > 0 for r→ ℝ (investigated in [8] a simplified treatment is given.
Corrigendum: Spectral properties of two parameter eigenvalue problems II
Authors: Binding Paul , Browne Patrick J. , Picard R. H.
Published: November 2011

Synopsis

There are some mistakes in [1, Section 4], and since the main result, Theorem 4.4, is central to the theory and has already been applied in various contexts, we felt it advisable to give a complete statement and proof. The applications of vTheorem 4.4 made to date have fortunately been in situations where the results are correct. For convenience, we restate our notation.
A factorisation procedure for two by two matrix functions on the circle with two rationally independent entries
Authors: Prößdorf S. , Speck F.-O.
Published: November 2011

Synopsis

The aim of this paper is the explicit canonical or standard factorisation of matrix functions with Wiener algebra elements. The present approach covers all regular 2 × 2 matrices where two entries are arbitrary and the remaining two are linear combinations of the former with rational coefficient functions. It is based on the knowledge of how to factorise scalar functions and rational matrix functions. In general, one also needs the approximation of any scalar Wiener algebra function with a rational function. However, this can be easily circumvented in many applications by intuitive manipulations with rational matrix functions.
On the Gegenbauer transformation
Authors: Heywood P. , Rooney P. G.
Published: November 2011

Synopsis

The Gegenbauer transformation Gλk is defined for λ > −1/2, k = 0, 1, 2, …, by

where, if being the Gegenbauer polynomial of index λ and degree k, and L0k is the Tchebichef polynomial of degree k. The transformation is studied on the spaces Lµ, p denned by the norm

and its boundedness and range on these spaces is determined and inversion formulae are found.
Asymptotic behaviour of solutions of elliptic equations with critical exponents and Neumann boundary conditions
Authors: Budd C. , Knaap M. C. , Peletier L. A.
Published: November 2011

Synopsis

Asymptotic estimates are established for nontrivial positive radial eigenfunctions of the nonlinear eigenvalue problem −Δu= λ(up−uq) in the unit ballBin ℝN(N> 2) with Neumann boundary conditions, as the supremum norm tends to infinity. Herepis the critical Sobolev exponent (N+ 2)/(N− 2) and 0 <q<p− 1 = 4/(N− 2).
Normal structure coefficients of Lp(Ω)
Authors: Benavides T. Domínguez
Published: November 2011

Synopsis

Let X be a uniformly convex Banach space, and N(X) the normal structure coefficient of X. In this paper it is proved that N(X) can be calculated by considering only sets whose points are equidistant from their Chebyshev centre. This result is applied to prove that N(LP(Ω)) = min {21−1/p, 21/p}, Ω being a σ-finite measure space. The computation of N(Lp) lets us also calculate some other coefficients related to the normal structure.
Nontrivial full bounded solutions of time-periodic semilinear parabolic PDEs
Authors: Mrozek Marian , Rybakowski Krzysztof P.
Published: November 2011

Synopsis

Consider the semilinear evolution equation

(P) u + Au = f(t,u)

where A is a sectorial operator on a Banach space and f is ω-periodic in t. Using a time-discrete Conley index developed in a previous paper [6], we prove a few existence results on bounded solutions of (P) defined for all t ∊ R. More specific results are given for time-periodic scalar parabolic equations.
On the effective elasticity of a two-dimensional homogenised incompressible elastic composite
Authors: Lipton Robert
Published: November 2011

Synopsis

The set of effective elasticity tensors for all two-dimensional mixtures of two isotropic incompressible elastic materials taken in prescribed proportion is described. In two dimensions the effective tensors are completely characterised by bounds on their eigenvalues.
On an extension of the theorem of V. A. Ambarzumyan
Authors: Chakravarty N.K. , Acharyya Sudip Kumar
Published: November 2011

Synopsis

The Ambarzumyan theorem connecting the Sturm–Liouville problem and the corresponding problem associated with the Fourier differential equation is extended to a class of second order matrix differential systems.
Schauder estimates and existence theory for entire solutions of linear elliptic equations
Authors: Begehr Heinrich , Hile G.N.
Published: November 2011

Synopsis

Existence as well as uniqueness theorems under certain growth conditions are given for entire solutions to linear elliptic equations in the n-dimensional space (2≦n). Introducing a proper norm, the proofs are based on a priori estimates. These estimates could be used to solve nonlinear equationsin the space, but the conditions on the nonlinearity have to be strong as the a priori estimates applyonly to classical, not to weak, solutions.
Some Borsuk-Ulam-type theorems for maps from Riemannian manifolds into manifolds
Authors: Hai-bao Duan
Published: November 2011

Synopsis

Suppose f: M →N is a continuous map from a Riemannian manifold (M, d) into a manifold N. The main result of this paper is to give some conditions under which f identifies a pair of cut points. This result leads to generalisations of the classical Borsuk-Ulam theorem. As a consequence some topological properties of locally symmetric spaces are discovered.
The gradient theory of phase transitions for systems with two potential wells
Authors: Fonseca Irene , Tartar Luc
Published: November 2011

Synopsis

In this paper we generalise the gradient theory of phase transitions to the vector valued case. We consider the family of perturbations

of the nonconvex functional

where W:RN→R supports two phases and N ≧1. We obtain the Γ(L1(Ω))-limit of the sequence

Moreover, we improve a compactness result ensuring the existence of a subsequence of minimisers of Eε(·) converging in L1(Ω) to a minimiser of E0(·) with minimal interfacial area.
An antiradical for near-rings
Authors: Hartney J. F. T.
Published: November 2011

Synopsis

Throughout this article, the near-ring N is assumed to be zero symmetric and to satisfy the right distributive law. We construct an ideal called the socle-ideal of N which is antiradical in the sense that it is a direct sum of minimal left ideals and annihilates one or more of the radicals of N. We then use this socle-ideal to obtain a decomposition theorem for the s-radical JS(N) of N.
Abelian theorems for the distributional Mellin transform
Authors: Lavoine Jean , Misra O. P.
Published: November 2011

Synopsis

This work provides an account of the asymptotic behaviour in terms of Abelian theorems for the Mellin and inverse Mellin transforms in a distributional setting.
On some properties of matrices associated with linear ordinary quasi-differential expressions
Authors: Everitt W. N. , Key Jennifer D.
Published: November 2011

Synopsis

In the general theory of ordinary linear quasi-differential equations, the set of Shin–Zettl matrices plays an important role. This paper displays certain properties of these matrices and their behaviour under a special form of transformation. Essentially, the problems can be considered within the framework of linear algebra.
Multiplicity results for a nonlinear Dirichlet problem
Authors: Solimini S.
Published: November 2011

Synopsis

This paper deals with some multiplicity results for elliptic problems with jumping nonlinearities. Our results are concerned with the case in which only one eigenvalue of the linear problem is jumped and it is simple. The main tool used is the Leray–Schauder topological degree. We consider a parametrized problem and prove the existence of two or three distinct solutions for suitable values of the parameter.
Monogenic inverse semigroups and their C* -algebras
Authors: Conway J. B. , Duncan J. , Paterson A. L. T.
Published: November 2011

Synopsis

By using the Halmos-Wallen description of power partial isometries on Hilbert space, we give a complete description of all monogenic inverse semigroups,ℐ. We also describe the fullC*-algebraC*ℐand the reducedC*-algebra C*(ℐ)with particular emphasis on the case of the free monogenic inverse semigroupℑℐt.
Memory effect phenomena and Г-convergence
Authors: Mascarenhas Maria Luisa
Published: November 2011

Synopsis

It is known that the parametric equation u'∊+ a∊u∊ = f, u∊ (0)= 0,with α ≦ a∊ ≦ β, for all ∊ > 0 and almost everywhere in a bounded domain Ω of ℝN, and f in L∞((0, T) × Ω), shows, at the limit, a memory effect. In this work the associated minimisation problem is considered and we describe how the memory effect appears in the Γ-limit, for the weak topology H1:(0, T; L2(Ω)) of the corresponding functional. The sequence a∊ has no dependence in time.
Exact controllability for a model of a multidimensional flexible structure
Authors: Puel Jean Pierre , Zuazua Enrique
Published: November 2011

Synopsis

A simple model of a vibrating multidimensional structure made of a n-dimensional body and a one-dimensional straight string is introduced. In both regions (n-dimensional body and a onedimensional string) the state is assumed to satisfy the wave equation. Simple boundary conditions are introduced at the junction. These conditions, in the absence of control, ensure conservation of the total energy of the system and imply some rigidity of the boundary of the n-d body on a neighbourhood of the junction. The exact boundary controllability of the system is proved by means of a Dirichlet control supported on a subset of the boundary of the n-d domain which excludes the junction region. Some extensions are discussed at the end of the paper.
On the equation ut = ∆uα + uβ
Authors: Qi Yuan-Wei
Published: November 2011

Synopsis

The Cauchy problem of ut, = ∆uα + uβ, where 0 < α < l and α>1, is studied. It is proved that if 1< β<α + 2/n then every nontrivial non-negative solution is not global in time. But if β>α+ 2/n there exist both blow-up solutions and global positive solutions which decay to zero as t–1/(β–1) when t →∞. Thus the famous Fujita result on ut = ∆u + up is generalised to the present fast diffusion equation. Furthermore, regarding the equation as an infinite dimensional dynamical system on Sobolev space W1,s (W2.s) with S > 1, a non-uniqueness result is established which shows that there exists a positive solution u(x, t) with u(., t) → 0 in W1.s (W2.s) as t → 0.
The distribution of genera among quadratic spaces over global fields
Authors: Earnest A. G.
Published: November 2011

Synopsis

For a Hasse domain R in a global field F, the distribution of genera of R-lattices with specified invariants among the nonisometric quadratic spaces over F which contain them is studied. It is shown that such genera are equally distributed for spaces of dimension 2, but that this is not generally the case for spaces of dimension exceeding 2. Theresult in the binary case yields an extension of a dimension exceeding 2. The result in the binary case yields an extension of a theorem of Gauss.
The G-closure of two well-ordered, anisotropic conductors
Authors: Grabovsky Yury
Published: November 2011

Synopsis

We give a complete solution to the G-closure problem for mixtures of two well-ordered possibly anisotropic conductors. Both the G-closure with fixed volume fractions and the full G-closure are computed. The conductivity tensors are considered in a fixed frame and no rotations are allowed.
Existence and uniqueness of travelling waves for a neural network
Authors: Ermentrout G. Bard , McLeod J. Bryce
Published: November 2011

Synopsis

A one-dimensional scalar neural network with two stable steady states is analysed. It is shown that there exists a unique monotone travelling wave front which joins the two stable states. Some additional properties of the wave such as the direction of its velocity are discussed.
Nonlinear dynamical theory of the elastica
Authors: Caflisch Russel E. , Maddocks John H.
Published: November 2011

Synopsis

The dynamical behaviour of a slender rod is analyzed here in terms of a generalization of Euler's elastica theory. The model includes a linear stress-strain relation but nonlinear geometric terms. Properties of the rod may vary along its length and various boundary conditions are considered. A rotational inertia term that is neglected in many theories is retained, and is essential to the analysis. By use of the equivalence of an energy and a Sobolev norm, and by reformulation of the equations as a semilinear system, global existence of solutions is proved for any smooth initial data. Equilibrium solutions that are stable in the static sense of minimizing the potential energy are then proved to be stable in the dynamic sense due to Liapounov.
Boundary value conditions for wave fronts in reaction-diffusion systems
Authors: Fraile J. M. , Sabina J.
Published: November 2011

Synopsis

In this paper, we introduce a new class of solutions of reaction-diffusion systems, termed directional wave front solutions. They have a propagating character and the propagation direction selects some distinguished boundary points on which we can impose boundary conditions. The Neumann and Dirichlet problems on these points are treated here in order to prove some theorems on the existence of directional wave front solutions of small amplitude, and to partially establish their asymptotic behaviour.
K-analytic uniform structures
Authors: Canela Miguel A.
Published: November 2011

Synopsis

This article deals with the uniform spaces (X, μ) such that μ is a K-analytic subset of 2X×X. G. Godefroy considered this situation for X countable, in his study of certain compact sets of measurable functions, and some of his results are extended here. We prove that the uniformity of an Eberlein compact is K-analytic, and give some applications.
Balanced big Cohen-Macaulay modules and ring extensions
Authors: O'Carroll Liam
Published: November 2011

Synopsis

Let A and B be commutative Noetherian local rings such that B contains A and B is flat and integral over A. It is shown that if M is a balanced big Cohen-Macaulay A-module (that is, every system of parameters for A is an M-sequence), then M⊗AB is a balanced big Cohen-Macaulay B-module. An example of a ring A is given such that, if B is the completion of A, then the analogous result is false in this case. This answers a question posed by Riley in the negative.
The commutative ring of a finite projective plane
Authors: Prince Alan R.
Published: November 2011

Synopsis

The ring of a finite projective plane, introduced by the author in a previous paper, is shown to be a Gorenstein ring. Its integral closure and conductor are also determined.
Asymptotic solutions to linear differential equations with coefficients of the power order of growth, 2
Authors: Lantsman M. H.
Published: November 2011

Synopsis

We consider linear differential equations of the form F(x, z)≡ xn + a1(z)x(n-1)+…+an(z)x = 0 with power-logarithmic coefficients or coefficients which are asymptotically similar to power-logarithmic functions in a central sector S of a complex plane for z →∞, z∈S. The main result of this paper is that in a sufficiently small central sector SE⊂S there is a fundamental system of solutions {xi(z) = exp [∫γi(z)dz)} where each function γi(z) is equivalent to a power-logarithmic function or has an estimate of the form O(z−∞). Furthermore, a precise estimate is obtained for a partial solution of a nonhomogeneous equation F(x, z) = α(z), where the function α(z) grows like a power.
The extreme points of some classes of polynomials
Authors: Brannan D. A. , Clunie J. G.
Published: November 2011

Synopsis

We study the extreme points of two classes of polynomials of degree at most n:

It turns out that f ∈ Ext if and only if Re f(eiθ) has exactly 2n zeros in [0, 2π). On the other hand, if f∈Hn and 1−|f(eiθ)|2 has 2n zeros in [0, 2π), then either f ∈ Ext Hn or else f(z) = α + βzn where |α|+|β| = l and αβ≠0; if 1−|f(eiθ)|2 has 2m zeros, 2n, then f may or may not belong to Ext Hn.
On an inequality of Hardy-Littlewood type: I
Authors: Evans W. D. , Everitt W. N.
Published: November 2011

Synopsis

This paper gives a complete analysis of the integral inequality

where the integral on the left may be only conditionally convergent. It is shown that the best possible number K is 4 and all cases of equality are identified in terms of Hankel-Bessel functions of order ⅓.
The asymptotic form of the Titchmarsh–Weyl m-function associated with a second order differential equation with locally integrable coefficient
Authors: Harris B. J.
Published: November 2011

Synopsis

We derive an asymptotic expansion for the Titchmarsh–Weyl m-function associated with the second order linear differential equation

in the case where the only restriction on the real-valued function q is
On the existence of periodic solutions for autonomous retarded functional differential equations on R2
Authors: Dos Reis J. G. , Baroni R. L. S.
Published: November 2011

Synopsis

Let Ca be the set of all the continuous functions from the interval [−r, 0] on the sphere of radius a, on the plane. We prove, under certains conditions, that a retarded autonomous differential equation that leaves Ca invariant has a non-constant periodic solution.
Remarks on the quasi-steady one phase Stefan problem
Authors: Louro Bento , Rodrigues José-Francisco
Published: November 2011

Synopsis

This paper presents some regularity results on the solution and on the free boundary for the one phase Stefan problem with zero specific heat in the framework of the variational inequalities formulation. In particular we show the Hölder continuity of the free boundary. Estimates on the rate of convergence when the specific heat vanishes are given for the variational solutions and for the free boundaries.
Linearisation at infinity and Lipschitz estimates for certain problems in the calculus of variations
Authors: Chipot Michel , Evans Lawrence C.
Published: November 2011

Synopsis

We demonstrate local Lipschitz regularity for minimisers of certain functionals which are appropriately convex and quadratic near infinity. The proof employs a blow-up argument entailing a linearisation of the Euler—Lagrange equations “at infinity”. As a application, we prove that minimisers for the relaxed optimal design problem derived by Kohn and Strang [3] are locally Lipschitz.
An expansion theory for non-self-adjoint boundary-value problems
Authors: Walker Philip W.
Published: November 2011

Synopsis

This paper deals with two-point boundary-value problems for ordinary differential equations and the operators which they induce in the appropriate Hilbert space. The problems arenot required to be self-adjoint. No auxiliary condition such as Birkhoff-regularity is imposed. If T is such an operator, it may well have no meaningful spectral structure. It is shown, however, that when T is composed with its adjoint, the result is a non-negative self-adjoint differential operator. The eigenvalues and eigenfunctions of this composite operator are used to delineate the domain, action, range, and generalised inverse of T.
Uniqueness and non-uniqueness of normal forms for vector fields
Authors: Baider Alberto , Churchill Richard
Published: November 2011

Synopsis

The use of normal forms in the study of equilibria of vector fields and Hamiltonian systems is a well-established practice and is described in standard references (e.g. [1], [7] or [10]). Also well known is the fact that such normal forms are not unique, and the relationship between distinct normal forms of the same vector field has also been investigated, in particular by M. Kummer [8] and A. Brjuno [2,3] (also see [12]). In this paper we use this relationship to extract invariants of the vector field directly from an arbitrary normal form. The treatment is sufficiently general to handle the vector field and Hamiltonian cases simultaneously, and applications in these contexts are presented.

The formulation of our main result (Theorem 1.1) is reminiscent of, and was heavily influenced by, work of Shi Songling on planar vector fields [11]. Additional inspiration was provided by M. Kummer's contributions to the 1:1 resonance problem in [9]. The authors are grateful to Richard Cushman for comments on an earlier version of this paper.
The change in index of a graph resulting from the attachment of a pendant edge
Authors: Bell F. K. , Rowlinson P.
Published: November 2011

Synopsis

Conditions are obtained under which a graph G', obtained from a graph G by the attachment of a pendant edge, may be regarded as a perturbation of G. In this situation the index of G' is expressed as the sum of a convergent series whose terms are determined by the spectrum and certain angles of G.
Polynomial interpolation at points of a geometric mesh on a triangle
Authors: Lee S. L. , Phillips G. M.
Published: November 2011

Synopsis

In an earlier paper [8], I. J. Schoenberg discussed polynomial interpolation in one dimension at the points of a geometric progression, which was originally proposed by James Stirling. In the present paper, these ideas are generalised to two-dimensional polynomial interpolation at the points of a geometric mesh on a triangle. A Lagrange form is obtained for this interpolating polynomial and an algorithm is derived for evaluating it efficiently.
Asymptotic behaviour of solutions to the Becker-Döring equations for arbitrary initial data
Authors: Ball J. M. , Carr J.
Published: November 2011

Synopsis

The asymptotic behaviour of solutions to the Becker-Doring cluster equations is determined without the decay assumptions on the initial data made in [3].
The critical Sobolev exponent in two dimensions
Authors: McLeod Bryce , McLeod Kevin
Published: November 2011

Synopsis

The object of the paper is to investigate solutions of equations of the form

with

and in particular to look at the asymptotic behaviour of these solutions as γ ↑∞. It is found that, if tγ is the first zero of ϑ, then

while tγ is bounded below if p < 2. This answers questions raised by Brezis concerning solutions of −Δu = eup in R2
A perturbation result of semilinear elliptic equations in exterior strip domains
Authors: Hsu Tsing-san , Wang Hwai-chiuan
Published: November 2011

Synopsis

In this paper we show that if the decay of nonzero ƒ is fast enough, then the perturbation Dirichlet problem−Δu + u = up+ ƒ(z)in Ω has at least two positive solutions, where

a boundedC1,1domainS= × ωRn, Dis a boundedC1,1domain inRm+nsuch thatD ⊂⊂ Sand Ω = S\D. In case ƒ ≡ 0, we assert that there is a positive higher-energy solution providing thatDis small.
The spectrum and eigenspaces of a meromorphic operator-valued function
Authors: Magnus Robert
Published: November 2011

Synopsis

It is shown how to associate eigenvectors with a meromorphic mapping defined on a Riemann surface with values in the algebra of bounded operators on a Banach space. This generalises the case of classical spectral theory of a single operator. The consequences of the definition of the eigenvectors are examined in detail. A theorem is obtained which asserts the completeness of the eigenvectors whenever the Riemann surface is compact. Two technical tools are discussed in detail: Cauchy-kernels and Runge's Approximation Theorem for vector-valued functions.
A variational problem on subsets of ℝn
Authors: Vornicescu Mihai
Published: November 2011

Synopsis

We give an existence theorem for a nonconvex minimum problem of the Calculus of Variations.
Spectral analysis of dissipative Schrödinger operators
Authors: Allahverdiev B. P. , Canoǧlu Ahmet
Published: November 2011

Dissipative Schrodinger operators are studied in L2(0, ∞) which are extensions of symmetric operators with defect index (2, 2). We construct a selfadjoint dilation and its incoming and outgoing spectral representations, which makes it possible to determine the scattering matrix according to the scheme of Lax and Phillips. With the help of the incoming spectral representation, we construct a functional model of the dissipative operator and construct its characteristic function in terms of solutions of the corresponding differential equation. On the basis of the results obtained regarding the theory of the characteristic function, we prove a theorem on completeness of the system of eigenfunctions and associated functions of the dissipative operator.
Renormalised solutions of nonlinear parabolic problems with L1 data: existence and uniqueness
Authors: Blanchard D. , Murat F.
Published: November 2011

In this paper we prove the existence and uniqueness of a renormalised solution of the nonlinear problem

where the data f and u0 belong to L1(Ω × (0, T)) and L1 (Ω), and where the function a:(0, T) × Ω × ℝN → ℝN is monotone (but not necessarily strictly monotone) and defines a bounded coercive continuous operator from the space into its dual space. The renormalised solution is an element of C0 ([ 0, T] L1 (Ω)) such that its truncates TK(u) belong to with

this solution satisfies the equation formally obtained by using in the equation the test function S(u)φ, where φ belongs to and where S belongs to C∞(ℝ) with
Existence and uniqueness of discontinuous solutions for a hyperbolic system
Authors: Huang Feimin
Published: November 2011

In this paper, we prove the global existence and uniqueness of solutions to the Cauchy problem of a hyperbolic system, which probably contains so-called δ-waves.
Global smooth solutions for a class of quasilinear hyperbolic systems with dissipative terms
Authors: Yang Tong , Zhu Changjiang , Zhao Huijiang
Published: November 2011

In this paper we prove an existence theorem of global smooth solutions for the Cauchy problem of a class of quasilinear hyperbolic systems with nonlinear dissipative terms under the assumption that only the C0-norm of the initial data is sufficiently small, while the C1-norm of the initial data can be large. The analysis is based on a priori estimates, which are obtained by a generalised Lax transformation.
Remarks on the Hardy–Littlewood maximal function
Authors: Aldaz J. M.
Published: November 2011

We answer questions of A. Carbery, M. Trinidad Menárguez and F. Soria by proving, firstly, that for the centred Hardy–Littlewood maximal function on the real line, the best constant C for the weak type (1, 1) inequality is strictly larger than 3/2, and secondly, that C is strictly less than 2 (known to be the best constant in the noncentred case).
Spectral asymptotics with a remainder estimate of the Neumann Laplacian on horns: the case of the rapidly growing counting function
Authors: Boyarchenko S. I. , Levendorskiĭ S. Z.
Published: November 2011

We study the Neumann Laplacian in unbounded regions of the form Ω = {(t, x) | t >O,f(t)−1x ∊ Ω′}, where Ω′ ⊂ ℝn−1 is a bounded open set with the Lipschitz boundary and f decays in such a way that the spectrum of is discrete but the counting function N(λ, ) of the spectrum grows faster than a power of λ, a typical example being f(t) = exp (– t In … In t), for t ≧ t0. We compute the principal term of the asymptotics of N(λ, ), with a remainder estimate.
Existence and regularity results for oblique derivative problems for heat equations in an angle
Authors: Garroni M. G. , Solonnikov V. A. , Vivaldi M. A.
Published: November 2011

An initial-boundary-value problem is considered for the heat equation in an infinite angle dθr ⊆ R2 × [0, ∞) with the oblique derivative boundary conditions on the faces λi of the angle:

with either h0 + h1 > 0, or h0 + h1 ≦ 0. The unique solvability of such a problem is proved in appropriate weighted Sobolev spaces according to the sign of h0 + h1. Estimates of the solution are obtained under ‘natural’ restrictions on the opening of the angle.
P-th moment growth bounds of infinite-dimensional stochastic evolution equations
Authors: Liu Kai
Published: November 2011

The aim of this paper is to investigate the p-th moment growth bounds wilh a general rate function λ(t) of the strong solution for a class of stochastic differential equations in infinite dimensional space under various sufficient hypotheses. The results derived here extend the usual situations to some extent, containing for example the polynomial or iterated logarithmic growth cases studied by many authors. In particular, more generalised sufficient conditions, ensuring the p-th moment upper-bound of sample paths given by solutions of a class of nonlinear stochastic evolution equations, are captured. Applications to parabolic itô equations are also considered.