bluuebunny/crossref_metadata_2025 huggingface dataset:
On periodic solutions for singular perturbation problems
Authors: Korman Philip
Published: November 2011

Synopsis

We apply a version of the Nash–Moser method to prove existence of periodic solutions for nonlinear elliptic equations and systems, involving singular perturbations. We allow nonlinearities depending on derivatives of order two more than that of the linear part, thus extending the previous results. Our result is new even in the case of one equation in one spatial dimension.
Classification of regularly solvable first-order differential operators
Authors: Benammar M.
Published: November 2011

Synopsis

In this article the expression τφ: = pφ + qφ with complex-valued coefficients is considered. We are particularly concerned with this expression when it is not formally symmetric, i.e., τ ≠ τ+, where τ+ is the formal adjoint of τ, and especially with the operators which are regularly solvable with respect to the minimal operators generated by τ and τ+ in the sense of W. D. Evans in [3]. This article is divided into five sections: Section 1 is an introduction, Section 2 is a brief study of the regular problem, in Section 3, some preliminary results in the singular case are displayed in Section 4, the joint field of regularity in the singular case is investigated and in Section 5, we discuss the case when .
On the stability for a population growth equation with time delay
Authors: Sugie Jitsuro
Published: November 2011

Synopsis

In this paper we give a sufficient condition which guarantees that the zero solution of a population growth equation is uniformly stable.
Rank-one convexity does not imply quasiconvexity
Authors: Šverák Vladimír
Published: November 2011

We consider variational integrals

defined for (sufficiently regular) functionsu: Ω→Rm. Here Ω is a bounded open subset ofRn,Du(x) denotes the gradient matrix ofuatxandfis a continuous function on the space of all realm×nmatrices Mm×n. One of the important problems in the calculus of variations is to characterise the functionsffor which the integralIis lower semicontinuous. In this connection, the following notions were introduced (see [3], [9], [10]).
On the minimization of singular quadratic functional
Authors: Bradley John S. , Hinton Don B. , Kauffman Robert M.
Published: November 2011

Synopsis

A quadratic functional Q is considered which is defined by an integral on a subset of functions in a weighted Hilbert space. The functional Q is minimized subject to the Dirichlet index of the associated differential operator being minimal. The infimum of Q is shown to be the least point in the spectrum of a certain self-adjoint operator which arises as a Friedrichs extension.
On the Lax-Phillips scattering theory
Authors: Amrein W. O. , Wollenberg M.
Published: November 2011

Synopsis

We give a simple description of the wave operators appearing in the Lax-Phillips scattering theory. This is used to derive a relation between the scattering matrix and a kind of time delay operator and to characterize all scattering systems having the same scattering operator.
Concerning the “panneaux” of M. Sova
Authors: Donegan H. A. , McMaster T. B. M.
Published: November 2011

Synopsis

For the investigation of certain kinds of reflexivity in topological vector spaces, Sova has shown that the ideas of panneaux and of espaces pannelés are as crucial as are those of barrels and of barrelled spaces. The objectives of this note are: (i) to establish the existence of (concentrated) barrels which are not panneaux, and (ii) to observe the equivalence of the conditions “pannelé” and “barrelled” for metrizable spaces, and the failure of this equivalence for general spaces.
Smoothing effect and asymptotic behaviour for the solutions of a nonlinear time dependent system
Authors: Dias João-Paulo , Haraux Alain
Published: November 2011

Synopsis

In this paper we obtain some new results on a nonlinear parabolic system related to the equations of the nematic liquid crystals and introduced in earlier papers by J. P. Dias.

These results mainly concern the existence and uniqueness of generalized solutions for discontinuous data and also their asymptotic behaviour in various cases.
Self-adjoint subspace extensions satisfying λ-linear boundary conditions
Authors: Röh Harald
Published: November 2011

Synopsis

Let S be a symmetric subspace in a Hilbert space ℋ2 with finite equal deficiency indices and let S* be its adjoint subspace in ℋ2. We consider those self-adjoint subspace extensions ℋ of S into some larger Hilbert spaces ℋ2 = (ℋ × ℂm)2 which satisfy H⋂({0} × ℂm)2 = {{0,0}}. These extensions H are characterized in terms of inhomogeneous boundary conditions for S*; they are associated with eigenvalue problems for S* depending on λ-linear boundary conditions, which we also characterize.
A directed set of Sturm–Liouville operators
Authors: Benzinger Harold E.
Published: November 2011

Synopsis

Asymptotic estimates for the eigenvalues and eigenfunctions of a directed set of Sturm–Liouville operators are obtained. Particular attention is paid to the influence of the diffusion coefficient, as it becomes arbitrarily large.
Jacobi-type polynomials under an indefinite inner product
Authors: Mingarelli Angelo B. , Krall Allan M.
Published: November 2011

Synopsis

The polynomials which are orthogonal with respect to

when α> – 1, M>0 are considered when α<–1 and/or M<0. The Cauchy regularization of 〈·, ·〉 provides orthogonality and generates a Pontrjagin (Krein) space spanned by the polynomials. The polynomials are eigenfunctions associated with a self-adjoint, fourth order differential operator.
Some classes of linear bornological spaces
Authors: Canela Miguel A.
Published: November 2011

Synopsis

In this paper, two classes of linear bornological spaces are considered, the Kolmogorov spaces and the spaces of type b. These spaces satisfy conditions which are weakenings of the definition of infratopological linear bornological spaces. Various properties of these spaces are proved, and two examples are given, showing the independence of the two conditions introduced.
Borel sets and the spectrum of an operator on an F-space
Authors: Słodkowski Zbigniew
Published: November 2011

Synopsis

It is shown that the spectrum of a linear operator T on a complete linear metric space X is a Borel set. If, in addition, X is locally convex or separable, then the spectrum of T is a Gδσ set.
On the swirling flow between rotating coaxial disks, asymptotic behaviour, I
Authors: Kreiss Heinz Otto , Parter Seymour V.
Published: November 2011

Synopsis

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

and

We also assume that |H(x, ε)|≦B√(ε) while |G(x, ε)|≦B. This work considers the shapes and asymptotic behaviour as ε→0+. We consider the kind of limit functions that are permissible. The only possible limits (interior) for G(x, ε) are constants. If that limit constant is not zero, then ε−½H(x, ε) will also tend to a constant.
Periodic and constant solutions of matrix Riccati differential equations: n = 2
Authors: Sánchez David A.
Published: November 2011

Synopsis

Several formulas are developed which can be used to determine constant solutions and the possible periods of periodic solutions (if any) of autonomous homogeneous matrix Riccati differential equations. These formulas are used to analyse some 2 × 2 cases, as well as to discuss the existence of periodic solutions under weak periodic forcing.
Some properties of periodic B-spline collocation matrices
Authors: Lee S. L. , Micchelli C. A. , Sharma A. , Smith P. W.
Published: November 2011

Synopsis

In three recent papers by Cavaretta et al., progress has been made in understanding the structure of bi-infinite totally positive matrices which have a block Toeplitz structure. The motivation for these papers came from certain problems of infinite spline interpolation where total positivity played an important role.

In this paper, we re-examine a class of infinite spline interpolation problems. We derive new results concerning the associated infinite matrices (periodic B-spline collocation matrices) which go beyond consequences of the general theory. Among other things, we identify the dimension of the null space of these matrices as the width of the largest band of strictly positive elements.
On the number of simple eigenvalues of a graph
Authors: Rowlinson Peter
Published: November 2011

Synopsis

Let Γ be a graph with n points, and let G be the group of automorphisms of Γ. An orbit of G on which G acts as an elementary abelian 2-group is said to be exceptional. It is shown that the number of simple eigenvalues of Γ is at most (5n+4t)/9, where t is the number of points of Γ lying in exceptional orbits of G.
On the existence of nodal domains for elliptic differential operators
Authors: Müller-Pfeiffer E.
Published: November 2011

Synopsis

We prove that the selfadjoint elliptic differential equation (1) has rectangular nodal domains if the quadratic form of the equation takes on negative values. The existence of nodal domains is closely connected with the position of the smallest point of the spectrum of the corresponding selfadjoint operator (Friedrichs extension). If the smallest point of a second order selfadjoint differential operator with Dirichlet boundary conditions is an eigenvalue, then this eigenvalue is strictly increasing when the (possibly unbounded) domain, where the coefficients of the differential operator are denned, is shrinking (Theorem 4).
A fixed point theorem for α-condensing maps on a sphere
Authors: Massatt Paul
Published: November 2011

Synopsis

This paper shows that ifSis a sphere in a Banach space andf:S→Sis an α-contraction, thenfhas a fixed point. The paper generalizes a result of R. D. Nussbaum which holds for α-contractions only. The proof uses the Browder nonrepulsive fixed point theorem and is motivated by recent work of M. Martelli and G. D. Cooperman.
A two point boundary value problem for neutral functional differential equations
Authors: Sficas Y. G. , Ntouyas S. K.
Published: November 2011

Synopsis

This paper is concerned with the existence of solutions of a two point boundary value problem for neutral functional differential equations. We consider the problem

where M and N are n × n matrices. This is examined by using the “shooting method”. Also, an example is given to illustrate how our result can be applied to yield the existence of solutions of a periodic boundary value problem.
L2-lower bounds to solutions of one-body Schrödinger equations
Authors: Froese R. , Herbst I. , Hoffmann-Ostenhof M. , Hoffmann-Ostenhof T.
Published: November 2011

The asymptotic behaviour of L2-solutions of one-body Schrödinger equations (–δ+V–E)ψ = 0 in ΩR = {x ∊ Rn||x|>R} is investigated. We show, for example, that if V tends to zero in a certain sense for |x|→∞, then either |x|γ exp for some γ>0 or ψ has compact support. Related results are given for potentials tending to infinity for |x|→∞.
Solvability of cardinal spline interpolation problems
Authors: Goodman T. N. T.
Published: November 2011

Synopsis

We consider interpolation by piecewise polynomials, where the interpolation conditions are on certain derivatives of the function at certain points of a periodic vector x, specified by a periodic incidence matrix G. Similarly, we allow discontinuity of certain derivatives of the piecewise polynomial at certain points of x, specified by a periodic incidence matrix H. This generalises the well-known cardinal spline interpolation of Schoenberg. We investigate conditions on G, H and x under which there is a unique bounded solution for any given bounded data.
On left quasinormal orthodox semigroups
Authors: Gomes Gracinda M. S.
Published: November 2011

Synopsis

The existence of a smallest inverse congruence on an orthodox semigroup is known. It is also known that a regular semigroupSis locally inverse and orthodox if and only if there exists a local isomorphism fromSonto an inverse semigroupT.

In this paper, we show the existence of a smallestR-unipotent congruence ρ on an orthodox semigroupSand give its expression in the case whereSis also left quasinormal. Finally, we prove that a regular semigroupSis left quasinormal and orthodox if and only if there exists a local isomorphism fromSonto anR-unipotent semigroupT.
Bloch waves and multiparameter spectral theory
Authors: Rynne B. P. , Sleeman B. D.
Published: November 2011

Synopsis

We study the spectral theory of a multiparameter system of periodic Schrödinger operators. Bloch waves are generalized eigenfunctions of these operators and are used to give eigenfunction expansion theorems and to derive some properties of the spectrum of the system.
On vectors of weight 16 in the code of a projective plane of order 10
Authors: Prince Alan R.
Published: November 2011

Synopsis

The problem of constructing certain 39-line starts for a projective plane of order 10, assuming that there is a vector of weight 16 in the associated binary code, is considered.
Towers of submanifolds of Grassmannians
Authors: Rees Elmer
Published: November 2011

Synopsis

There is a conjecture that a tower of smooth subvarieties V(n) with fixed codimension l in Gk(ℂn) must be a standard example. It is shown that even under topological hypotheses, all cohomological invariants of such a tower must coincide with those of standard examples.
A non-probabilistic approach to Poisson spaces
Authors: Paterson Alan L. T.
Published: November 2011

Synopsis

Using techniques from probability theory, it has been established that if μ is a probability measure on a separable, locally compact group, then the space of μ-harmonic functions on the group can be identified with C(X) for some compact, Hausdorff space X. The space X is known as the Poisson space of μ. We generalise this result in the context of a measure μ on a locally compact semigroup S, in particular establishing the existence of a Poisson space for non-separable groups. The proof is non-probabilistic, and depends on properties of projections on C(K)(K compact Hausdorff). We then show that if S is compact and the support of μ generates S, then the Poisson space associated with μ, is X, where X×G×Y is the Rees product representing the kernel of S.
Admissibility criteria for shock wave solutions of a system of conservation laws of mixed type
Authors: Shearer Michael
Published: November 2011

Synopsis

The following system of conservation laws is considered:

where σ: ℝ→ℝ is a smooth function monotonically increasing except in an interval. Two criteria for the admissibility of shocks are shown to be independent in the sense that there are shocks satisfying each and violating the other. This contrasts with the corresponding situation for strictly hyperbolic systems (σ'(u)>0 for all u), for which the two criteria are equivalent.
A congruence-free semigroup associated with an infinite cardinal number
Authors: Marques M. Paula O.
Published: November 2011

Synopsis

Let X be a set with infinite cardinality m and let Qm be the semigroupof balanced elements in ℐ(X), as described by Howie. If I is the ideal{αεQm:|Xα|<m} then the Rees factor Pm = Qm/I is O-bisimple and idempotent-generated. Its minimum non-trivial homomorphic image has both these properties and is congruence-free. Moreover, has depth 4, in the sense that [E()]4 = , [E()]3≠
Continuous data dependence in linear elastodynamics on unbounded domains without definiteness conditions on the elasticities
Authors: Galdi Giovanni P. , Rionero Salvatore
Published: November 2011

Synopsis

By suitably coupling convexity and weight function methods, we prove uniqueness and continuous dependence theorems in linear elastodynamics in unbounded domains without definiteness conditions on the elasticities. The class of solutions considered allows the “growth” at large spatial distances.
Parameterization of the M(λ) function for a Hamiltonian system of limit circle type
Authors: Hinton D. B. , Shaw J. K.
Published: November 2011

Synopsis

The authors continue their study of Titchmarch-Weyl matrix M(λ) functions for linear Hamiltonian systems. A representation for the M(λ) function is obtained in the case where the system is limit circle, or maximum deficiency index, type. The representation reduces, in a special case, to a parameterization for scalar m-coefficients due to C. T. Fulton. A proof that matrix M(λ) functions are meromorphic in the limit circle case is given.
The Hilbert boundary value problem for nonlinear elliptic systems
Authors: Begehr Heinrich , Hsiao George C.
Published: November 2011

Synopsis

The Hilbert boundary value problem for a first order nonlinear elliptic system in the plane with linear boundary conditions of nonnegative index is (under suitable side conditions uniquely) solved by use of the Newton imbedding method. This constructive method is based on an a priori estimate which arises from an integral representation formula for C1-functions first developed by Haack and Wendland. The approximation procedure yields an error estimate too.
Cardinal interpolation by Dm-splines
Authors: Goodman T. N. T. , Lee S. L.
Published: November 2011

Synopsis

Duchon (1978) considered interpolation in ℝn by “Dm-splines”, which are interpolating functions having, in a sense, minimum energy. The purpose of this paper is to consider the analogous interpolation at the lattice of points in ℝn with integer co-ordinates, generalising aspects of Schoenberg's (1973) theory of cardinal spline interpolation. Following Schoenberg, we prove that higher order “basic” splines can be written as convolutions of lower order ones, using a new notion of convolution due to Jones (1982).
Operators of minimal norm via modified Green's functions
Authors: Kleinman R. E. , Roach G. F.
Published: November 2011

Synopsis

The exterior Dirichlet and Neumann problems can be treated very satisfactorily by using a fundamental solution which is modified by adding radiating spherical wave functions. It has been shown [3], that the coefficients of these added terms can be chosen to ensure that the associated boundary integral equation formulation of the problem was uniquely solvable and, in addition, that the modified Green's function was a least squares best approximation to the exact Green's function for the problem. Here we show that the coefficients can be chosen to ensure not only unique solvability but also minimization of the norm of the modified integral operator. This leads to a constructive method of solution. The theory is illustrated when the boundary is a sphere and when it is a perturbation of a sphere.
Cohen elements in Banach algebras
Authors: Sinclair Allan M.
Published: November 2011

Synopsis

The definition of Cohen elements in a commutative Banach algebra with a countable bounded approximate identity given by Esterle is modified slightly to be more analogous to the invertible elements in a unital Banach algebra. With the modified definition the n1-Cohen factorization results that were proved by Esterle are shown tohold in the semigroup of Cohen elements. If is the algebra of continuous complex valued functions vanishing at infinity on a σ-compact locally compact Hausdorff space X, then the Cohen elements in are identified and a natural quotient of a subsemigroup of Cohen elements is shown to be a group, isomorphic to the abstract index group of C(X∪{∞}).
Non-linear functional differential equations and abstract integral equations
Authors: Kappel F. , Schappacher W.
Published: November 2011

Synopsis

The equivalence between solutions of functional differential equations and an abstract integral equation is investigated. Using this result we derive a general approximation result in the state space C and consider as an example approximation by first order spline functions. During the last twenty years C1-semigroups of linear transformations have played an important role in the theory of linear autonomous functional differential equations (cf. for instance the discussion in [9, Section 7.7]). Applications of non-linear semigroup theory to functional differential equations are rather recent beginning with a paper by Webb [17]. Since then a considerable number of papers deal with problems in this direction. A common feature of the majority of these papers is that as a first step with the functional differential equation there is associated a non-linear operator A in a suitable Banach-space. Then appropriate conditions are imposed on the problem such that the conditions of the Crandall-Liggett-Theorem [5] hold for the operator A. This gives a non-linear semigroup. Finally the connection of this semigroup tothe solutions of the original differential equation has to be investigated [c.f. 8, 15, 18]. To solve thislast problem in general is the most difficult part of this approach.

In the present paper we consider the given functional differential equation as a perturbation of the simple equationx = 0. The solutions of this equation generate a very simple C1-semigroup. The solutions of the original functional differential equation generate solutions of an integral equation which is the variation of constants formula for the abstract Cauchy problem associated with the equation x = 0. Under very mild conditions we can prove a one-to-one correspondence between solutions of the given functional differential equation and solutions of the integral equation in the Lp-space setting. In the C-space setting the integral equation inthe state space has to be replaced by a ‘pointwise’ integral equation. Using the pointwise integral equation together with a theorem which guarantees continuous dependence of fixed points on parameters we show under rather weak hypotheses that the original functional differential equation can be approximated by a sequence of ordinary differential equations. Using 1st order spline functions we finally get results which are very similar to those obtained in [1 and 11] in the L2-space setting.
Effects of a sharp pressure rise on a compressible laminar boundary layer, when the Prandtl number is σ = 0.72
Authors: Curle N.
Published: November 2011

Synopsis

Following an earlier paper (Curle 1978) we consider a compressible laminar boundary layer with uniform pressure when the distance x along the wall satisfies x < x0 and a prescribed large adverse pressure gradient when x > x0. The viscosity and absolute temperature are again taken to be proportional, but the Prandtl number is no longer assumed to be unity. After applying the Illingworth-Stewartson transformation, the transformed external velocity u1(x) is chosen so that

is large and constant, where Ts is the stagnation temperature, Tw is the (constant) Wall temperature and u0 is the upstream value of u1(x).

The flow reacts to this sharp pressure rise mainly in a thin inner sublayer, so inner and outer asymptotic expansions are derived and matched for functions F and S which determine the stream function and the temperature.

The skin friction, heat transfer, displacement thickness and momentum thickness are determined as functions of , andinvolve two parameters B1, B2, which depend upon the Mach number and the walltemperature. Detailed numerical calculations are presented here for σ = 0.72. In particular, it is seen that the heat transfer rate varies roughly like σ⅓ except near to separation, where it varies like σ¼.
On the deficiency indices of a fourth order singular differential operator
Authors: Wood Alastair D.
Published: November 2011

Synopsis

We consider the operatorL[y] =y(4)+ ((ax2+bx+c)y′)′ +dyon the half-line [0, ∞). This paper shows that the deficiency indices are independent of the real numbersb, canddwhena≠ 0. They depend only on the sign ofaand are (2,2) ifa< 0 and (3, 3) ifa> 0. In the casea=0 the sign ofbmust be considered.
The point-continuous spectrum of second-order, ordinary differential operators
Authors: Thurlow Christine
Published: November 2011

Synopsis

Given any countably infinite set of isloated points on the ℷ -axis, it is shown that there is a continuous q(x) such that these points constitute exactly the point-continuous spectrum for the equation yn″(x) + (ℷ —q(x))y(x) = 0(0≦x<∞) with some homogenous boundary condition at x = 0. This extends a result given by Eastham and McLeod for countably infinite sets of isolated points on the positive ℷ-axis.
Comportement à l'infini pour certains systèmes dissipatifs non linéaires
Authors: Haraux A.
Published: November 2011

Synopsis

In the Hilbert space framework, we give some results concerning the behaviour when t goes to infinity for solutions of equations of the form:

A is assumed to be a maximal monotone operator and F(t) is a periodic function.

When F = 0, under a compactness assumption for trajectories of (1), we give the complete description of the asymptotic behaviour, e.g. every trajectory is asymptotic to an almost-periodic solution of (1). When F ≠ cst, the compactness hypothesis being too restrictive, we concentrate our efforts on the case of the equation:

with Dirichlet boundary condition) and get weak convergence to particular solutions of the equation when β is either univalued or strictly monotone. The methods used in these cases seem of general interest for hyperbolic equations of dissipative type with periodic forcing term.
Stability theory for quasi-linear wave equations with linear damping
Authors: Dickey R. W.
Published: November 2011

Synopsis

Large time behaviour of solutions to a damped quasi-linear wave equation are studied. Conditions are obtained which guarantee the global existence of a classical solution. The asymptotic behaviour of this solution is studied in the case of a unique equilibrium solution and in the case of multiple equilibria. The results are applied to various special examples.
A differentiability property and its application to the spectral theory of differential operators
Authors: Kauffman Robert M.
Published: November 2011

Synopsis

For an ordinary differential operation Lλ of order 2N which depends differentiably on a parameter λ, we study the differentiability with respect to λ of all solutions to Lλf = 0 which are in L2[a,∞). Applications to spectral theory are given, including a formula for the rate of change with respect to the end-point a of the spectrum of the weighted eigenvalue problem Lf = λwf, f∈L2[a,∞), f[i](a) = 0 for i ≦N − 1. The weight w may be a function or an operator. The formula seems new even when w = 1.
A variational inequality approach to Hele-Shaw flow with a moving boundary
Authors: Elliott C. M. , Janovský V.
Published: November 2011

Synopsis

This paper is concerned with the study of a mathematical model of the injection of fluid into a finite Hele–Shaw cell. The mathematical problem is one of solving Laplace's equation in an unknown region whose boundary changes with time. By a transformation of the dependent variable, an elliptic variational inequality formulation of the moving boundary problem is obtained. The variational inequality is shown to have a unique solution up to the time at which the cell is filled. Regularity results for the solution of the inequality are obtained by studying a penalty approximation of the inequality.
An analogue of Maschke's theorem for certain representations of A6 over GF(2)
Authors: Prince Alan R.
Published: November 2011

Synopsis

Certain representations of the alternating group of degree 6 over the field of 2 elements are shown to be completely reducible.
The independence of certain associative laws
Authors: Länger Helmut
Published: November 2011

Synopsis

Forn≧ 3 the independence of the associative laws

forn-ary operations (…) is proved.
Enumeration of smooth labelled graphs
Authors: Wright E. M.
Published: November 2011

Synopsis

An (n,q) graph is a graph on n labelled points andqlines without loops or multiple lines. We write ν(n,q) for the number of smooth (n,q) graphs, i.e. connected graphs without end points, and ν =V(Z,Y) = ∑n,qν(n,q)ZnYq/n! for the exponential generating function of ν(n,q). We use the Riddell “core and mantle” method to find an explicit form forV(not, as usual with this method, only a functional equation). From this we deduce a partial differential equation satisfied byV.We interpret this equation in purely combinatorial terms. We writeVk= ∑nν(n,n+k)Xn/n! and find a recurrence formula forVkfor successivek.We use these and other results to find an asymptotic expansion for ν(n,q) asn→∞ when (q/n) −logn−log logn→ + ∞ and an asymptotic approximation to ν(n,n+k) when 0 <k=oand to log ν(n,n+k) whenk<(1−ε).
Linear evolution equations in two Banach spaces
Authors: Sauer Niko
Published: November 2011

Synopsis

We consider the ordinary differential equation [Bu(t)]′ = Au(t) with A and B linear operators with domains in a Banach space X and ranges in a Banach space Y. The initial condition is that the limit as (t → 0 of Bu(t) is prescribed in Y. We study the properties of the “solution operator” S(t) which maps the initial state in Y to the solution u(t) at time t. The notion of infinitesimal generator A of S(t) is introduced and the relationships between S(t), an associated semi-group E(t), the operators A and B and some other operators are studied. In particular a pair of operators Ao and Bo, derived from A and B, determine the family S(t) of operators. These so-called “generating pairs” are characterized. The operators A and B and Ao and Bo need not be closed, but form so-called closed pairs which is a weaker condition. We also discuss two applications of the theory.
Oscillation of second-order hyperbolic equations with non-integrable coefficients
Authors: Hsiang Wu-Teh , Kwong Man Kam
Published: November 2011

Synopsis

Some sufficient conditions are obtained on the coefficient g and the initial values Φ and ψfor the solution ot the non-linear hyperbolic equation

to change sign in the first quadrant. An example is given to show that is not sufficient in the linear case.
A perturbation theorem for exponential dichotomies
Authors: Palmer Kenneth J.
Published: November 2011

Synopsis

Suppose the linear equation x' = A(t)x has an exponential dichotomy and suppose B(t) is close to A(t) in the following sense: on any interval of length 2T, B(t) is close to some translate A(t + τ) of A(t) (actually the conditions in the paper are slightly weaker than this). Then if T is sufficiently large, the equation y' = B(t)y also has an exponential dichotomy. This generalises the usual roughness theorem for exponential dichotomies.
Spectral properties of two parameter eigenvalue problems II
Authors: Binding Paul , Browne Patrick J.
Published: November 2011

Synopsis

We consider eigenvalues λ =(λ1, λ2) ∈R2 for the problem W(λ)x = 0, x ≠ 0, x ∈ H, where W(λ) = R + λ1V1 + λ2V2), and R, V1, V2 are self-adjoint operators on a separable Hilbert space H, R being bounded below with compact resolvent and V1, V2 being bounded. The i-th eigencurve Z1 is the set of eigenvalues λ, for which the i-th eigenvalue (counted according to multiplicity and in increasing order) of W(λ) vanishes. We study monotonic and asymptotic properties of Zi, and we give formulae for any asymptotes that exist. Additional results are given in the finite dimensional case.
Some remarks concerning quasiconvexity and strong convergence
Authors: Evans L. C. , Gariepy R. F.
Published: November 2011

Synopsis

We show that the weak convergence of a sequence of functions in a Sobolev space plus the convergence of appropriately quasiconvex “energies” imply, in fact, strong convergence. This assertion makes rigorous, for example, the heuristic principle that “quasiconvexity damps out oscillations in the gradients” of minimising sequences in the calculus of variations.
Order of singularity applied to linear operators
Authors: Scroggs James E.
Published: November 2011

Synopsis

We extend the notion of order of singularity for complex-valued functions, as originally set forth by Hadamard, to Banach algebra-valued functions. Restricting our attention to linear operators whose spectral radius is one, we obtain a connection between the rate of growth of the norm of iterates of a linear operator and the rate of growth of the norm of the resolvent of the operator near the spectrum of the operator. In the finite dimensional case, we obtain an upper bound on the size of the Jordan block corresponding to an eigenvalue of maximum modulus.
Homogenisation of a viscoelastic equation with non-periodic coefficients
Authors: Mascarenhas Maria-Luisa
Published: November 2011

Synopsis

The homogenisation of a linearly viscoelastic composite material is performed without any periodicity assumptions and with no restrictions on the initial data. A term of fading memory is evidenced in the expression of the homogenised stress tensor.
Equisummability for linear operators in Banach spaces
Authors: Kon M. A. , Ramm A. G. , Raphael L. A.
Published: November 2011

Synopsis

Let A and B be closed linear operators on a Banach space X. Assume that ε(εI – A)−1f→f as |ε|→ ∞ for all f in X, ζ∊∑ ⊂ℂ. Under what conditions on B − A does the same relationship hold for B? When does [ε(εI − A)−1 − ε(εI − B)−1 ] f→ 0 in some stronger norm than that of X? The questions are discussed in an abstract setting and the results are generalised to other analytic functions of A. Applications are given to second order elliptic operators.
An elliptic-parabolic free boundary problem: continuity of the interface
Authors: Hulshof J.
Published: November 2011

Synopsis

In this paper we establish continuity of the interface of the weak solution to an elliptic-parabolic problem. The physical background is the theory of partially saturated fluid flows in porous media. Our method is based on the maximum principle for parabolic equations. An essential assumption is that the flow is one-dimensional.
Volterra integral equations and a new Gronwall inequality (Part II: The nonlinear case)
Authors: Norbury J. , Stuart A. M.
Published: November 2011

Synopsis

We consider nonlinear singular Volterra integral equations of the second kind. We generalise the transformation method introduced in Part I of this paper [6] to cope with both the nonlinearity and slightly more general singular kernels. We also consider a particular class of nonlinear equation for which the solution behaviour is known. Using this a priori knowledge, we propose a modification of the transformation technique which results in a numerical method with good asymptotic stability properties. Applying the general theory of Part I of this paper, we prove convergence of this scheme.
Existence and uniqueness of stationary solutions of some reaction–diffusion equations arising in chemical reactor kinetics
Authors: Swetina Jörg
Published: November 2011

Synopsis

In this paper we give a proof on existence of non-negative solutions of weakly coupled systems of non-linear elliptic PDE's which model isothermal chemical reactions in a bounded volume Ω. The boundaries of this “diffusion reactor” will be allowed to be differently permeable to different species, giving rise to homogeneous Neumann, or inhomogeneous mixed, boundary conditions. We assume a “conservation of mass” condition and a second condition, the choice of which is motivated in an example.

In the second section we prove that the PDE system that describes the reaction:

in the diffusion reactor admits a unique stationary solution.

For this proof we must show invertibility for a class of non inverse-positive linear elliptic operators.
Global bifurcation analysis and uniqueness for a semilinear problem
Authors: Tertikas Achilles
Published: November 2011

Synopsis

We study the bifurcation diagram and uniqueness of solutions of

By using a rescaling technique and the Implicit Function Theorem, we establish the global bifurcation diagram. Uniqueness is proved by a separation argument to complete the bifurcation picture of the problem. Our study suggests that the bifurcation diagrams have different behaviour at λ = 0, depending on whether g(∞) > 0 or g(∞) < 0 in L∞ norm, but quite similar behaviour in Lp or W2,1.
Spectral properties of diagonally dominant infinite matrices, part I
Authors: Farid F. O. , Lancaster P.
Published: November 2011

Synopsis

Theorems of Gersgorin-type are established for a diagonally dominant, unbounded, infinite matrix operator A acting on lp for some l ≦p≦∞. The results are established using an approximating sequence of infinite matrices An that converges to A in the generalised sense as n → ∞. This constructive approach admits approximation of the spectral properties of A by those of An.
The quartic projections of Castelnuovo's normal surface with hyperelliptic prime sections
Authors: Edge W. L.
Published: November 2011

Synopsis

Any non-ruled quartic surface with a double line in space of three dimensions has plane sections of genus 2 and so is a projection of a non-singular duodecimic surface in space of eleven dimensions. This was accepted as an established fact by 1890, but there seems not to be any account of such a projection in action. The following pages are submitted to aid the removal of this century-old anomaly.
Maps between p-completed classifying spaces
Authors: Adams J. Frank , Wojtkowiak Zdzisław
Published: November 2011

Synopsis

Let G and G' be two connected compact Lie groups with maximal tori T and T'. For a space X, let Xp be the p-completion of X. We will associate to each topological map f:(BG)p→(BG')p an “admissible map” ϕ:π1(T)⊗zZp→π1(T′)⊗zZp. We then show that the study of “admissible maps” in the p-complete case may be reduced to their study in the p-local case.
A Caccioppoli inequality and partial regularity in the calculus of variations
Authors: Gariepy Ronald F.
Published: November 2011

Synopsis

A new proof is given of the partial regularity of minimisers under the principal assumptions of uniform strict quasiconvexity and polynomial growth. The proof is based on a Caccioppoli inequality and an “indirect” blow-up argument; this avoids the technical complications of a “direct” argument.
Heat equation and the principle of not feeling the boundary
Authors: van den Berg M.
Published: November 2011

Synopsis

We prove a lower bound for the Dirichlet heat kernel pD(x,y;t), where x and y are a visible pair of points in an open set D in ℝm.
Boundary regularity for linear and quasilinear variational inequalities
Authors: Lieberman Gary M.
Published: November 2011

Synopsis

A method of Jensen is extended to show that the second derivatives of the solutions of various linear obstacle problems are bounded under weaker regularity hypotheses on the dataof the problem than were allowed by Jensen. They are, in fact, weak enough that the linear results imply the boundedness of the second derivatives for quasilinear problems as well. Comparisons are made with previously known results, some of which are proved by similar methods. Both Dirichlet and oblique derivative boundary conditions are considered. Corresponding results for parabolic obstacle problems are proved.
An Lp inequality and its applications to fixed point theory and approximation theory
Authors: Xu Hong-Kun , Xu Zong-Ben
Published: November 2011

Synopsis

An Lp inequality for l < p < 2 is established and applications to fixed points of uniformly Lipschitz mappings and strongly unique best approximations are given.
Limit-point and limit-circle criteria for Sturm-Liouville equations with intermittently negative principal coefficients
Authors: Everitt W. N. , Knowles I. W. , Read T. T.
Published: November 2011

Synopsis

Limit-point and limit-circle criteria are given for the generalised Sturm-Liouville differential expression

where

(i)p, q, andware real-valued on[a, b),

(ii)p−1, q, ware locally Lebesgue integrable on[a, b),

(iii)w> 0 almost everywhere on[a, b)and the principal coefficientpis allowed toassume both positive and negative values.
Radicals and one-sided ideals
Authors: Sands A.D.
Published: November 2011

The correspondence between radicals of associative rings and A-radicals is studied. It is shown that corresponding to each A-radical there is an interval of radicals and that each radical belongs to exactly one such interval. The question of the nature of the radical of a one-sided ideal is considered. It is shown that the radicals such that the radical of a one-sided ideal is always a one-sided ideal are those which contain their associated A-radicals. Radicals such that the radical of a one-sided ideal always equals the intersection of a left ideal and a right ideal are described, as are those A-radicals such that every associated radical has this property.
Bifurcation from a homoclinic orbit in parabolic differential equations
Authors: Blázquez C. Miguel
Published: November 2011

Synopsis

This paper considers autonomous parabolic equations which have a homoclinic orbit to an isolated equilibrium point. We study these systems under autonomous perturbations. Firstly we prove that the perturbation under which the homoclinicorbit persists forms a submanifold of codimension one. Then, if a perturbation of this manifold is considered, we prove that a unique stable periodic orbit arises from the homoclinic orbit under certain conditions for the eigenvalues of thesaddle point.
Some homomorphisms of general and special linear groups
Authors: Mason A. W.
Published: November 2011

Synopsis

A ring epimorphism θ:A →B extends in a natural way to a homomorphism γn: GLn(A)→GLn(B) and, when A is commutative, to a homomorphism σn:SLn(A)→SLn(B), where n ≧ 1. In this paper we consider the question: when are γn and σn surjective (or non-surjective)?
Metrizability of precompact subsets in (LF)-spaces
Authors: Cascales B. , Orihuela J.
Published: November 2011

Synopsis

In this paper we prove that every precompact subset in any (LF)-space has a metrizable completion. As a consequence every (LF)-space is angelic and in this way the answer to a question posed by K. Floret [3] is given. Some contributions to the general problem of regularity in inductive limits posed by K. Floret [3] are also given. Particularly, extensions of well-known results of H. Neuss and M. Valdivia are provided in the general setting of (LF)-spaces. It should also be noted that our results hold for inductive limits of an increasing sequence of metrizable spaces.
Asymptotics of the Titchmarsh-Weyl m-coefficient for integrable potentials
Authors: Kaper Hans G. , Kwong Man Kam
Published: November 2011

This article is concerned with the asymptotic behaviour of m(λ), the Titchmarsh-Weyl m-coefficient, for the singular eigenvalue equation y“ + (λ − q(x))y = 0 on [0, ∞), as λ →∞ in a sector in the upper half of the complex plane. It is assumed that the potential function q is integrable near 0. A simplified proof is given of a result of Atkinson [7], who derived the first two terms in the asymptotic expansion of m(λ), and a sharper error bound is obtained. Theproof is then generalised to derive subsequent terms in the asymptotic expansion. It is shown that the Titchmarsh-Weyl m-coefficient admits an asymptotic power series expansion if the potential function satisfies some smoothness condition. A simple method to compute the expansion coefficients is presented. The results for the first few coefficients agree with those given by Harris [9].
The set of positive solutions of semilinear equations in large balls
Authors: Gardner R. , Peletier L. A.
Published: November 2011

Synopsis

The exact number of positive solutions of Δu + f(u) = 0 on finite balls in ℝN is determined. The assumptions about f(u) are similar to those imposed by Serrin and the second author in a previous study of uniqueness of the positive solution when the spatial domain is all of ℝN (see [7, 8]). For finite balls of sufficiently large radius it is shown here that there are exactly two positive and, hence, radial solutions. To this end, we first prove the linear nondegeneracy of the positive solution of ℝN. This is obtained by applying the technique of monotone separation of graphs [7] to the linearised equations. Somewhat sharper estimates are required here (see Part I, Section 2).
On the behaviour of eigenfunction expansions in the complex domain
Authors: Freiling Gerhard
Published: November 2011

Synopsis

By using asymptotic estimates for the eigenvalues and eigenfunctions or irregular boundary value problems, we state necessary conditions for the pointwise convergence and for the divergence of the corresponding eigenfunction expansions.
On periodic solutions of nonlinear second order vector differential equations
Authors: Habets P. , Nkashama M. N.
Published: November 2011

Synopsis

This paper considers existence of periodic solutions for vector Liénard differential equations

In our main result we write

where Q(t, x) is a symmetric matrix and h(t, x) is sublinear. The key assumption relates the asymptotic behaviour as x →+ ∞ of the eigenvalues of Q(t, x) to the spectrum of the linear operator −d2/dt2 Several choices for Q(t, x) are considered which lead to known theorems and extend others. In the case of the Duffing equation

the assumptions are weakened.

Our approach is based on Leray-Schauder's degree theory and a priori estimates.
Corrigendum: Bifurcations near a multiple eigenvalue of the rectangular plate problem
Authors: Sadovský Zoltán
Published: November 2011

Page 48: The scaling (3.28) has been omitted from the last line of the page. The text should read:

On considering equation (3.25), we choose (ζ1; 0) as a new parameter plane. Then we may introduce a scaling
Growth, motion and 1-parameter families of symmetry sets
Authors: Bruce J. W. , Giblin P. J.
Published: November 2011

Synopsis

Associated to every plane curve there is the locus of centres of circles bitangent to that curve, the so-called symmetry set of the curve. We can view this set as the spine of our curve, which can be recovered by taking the envelope of circles of varying radii along this spine. Varying the symmetry set in some isotopy while keeping the radius function fixed may be viewed as crudely modelling motion of the original curve viewed as a biological object. Fixing the symmetry set and varying the radius function can be considered to model growth crudely. In this paper we shall describe the generic changes in the curves which take place in the process of growth and motion, and outline the corresponding results for spheres centred on a space curve. We also use the idea of a stratified Morse function to describe the generic changes which occur in one parameter families of (full) bifurcation sets in the plane. Applying this to the bifurcation set of distance squared functions we find all the transitions of a symmetry set (and evolute) which occur in a generic isotopy of a plane curve.
Some applications of Hausdorff dimension inequalities for ordinary differential equations
Authors: Smith Russell A.
Published: November 2011

Synopsis

Upper bounds are obtained for the Hausdorff dimension of compact invariant sets of ordinary differential equations which are periodic in the independent variable. From these are derived sufficient conditions for dissipative analytic n-dimensional ω-periodic differential equations to have only a finite number of ω-periodic solutions. For autonomous equations the same conditions ensure that each bounded semi-orbit converges to a critical point. These results yield some information about the Lorenz equation and the forced Duffing equation.
Mixed boundary value problems of the type of Sommerfeld's half-plane problem
Authors: Speck F.-O.
Published: November 2011

Synopsis

Various physical problems in diffraction theory lead us to study modifications of the Sommerfeld half-plane problem governed by two proper elliptic partial differential equations in complementary ℝ3 half-spaces Ω± and we allow different boundary or transmission conditions on two half-planes, which together form the common boundary of Ω±.
L∞C(ℝn)-decay of classical solutions for nonlinear Schrödinger equations
Authors: Hayashi Nakao , Tsutsumi Masayoshi
Published: November 2011

Synopsis

We study the initial value problem for the nonlinear Schrödinger equation

Under suitable regularity assumptions on f and ø and growth and sign conditions on f, it is shown that the maximum norms of solutions to (*) decay as t→² ∞ at the same rate as that of solutions to the free Schrödinger equation.