bluuebunny/crossref_metadata_2025 huggingface dataset:
A distributional theory of fractional calculus
Authors: Lamb W.
Published: November 2011

Synopsis

In this paper, a theory of fractional calculus is developed for certain spacesD′p,μof generalised functions. The theory is based on the construction of fractionalpowers of certain simple differential and integral operators. With the parameter μ suitably restricted, these fractional powers are shown to coincide with the Riemann-Liouville and Weyl operators of fractional integration and differentiation. Standard properties associated with fractional integrals and derivatives follow immediately from results obtained previously by the author on fractional powers of operators; see [6], [7]. Some spectral properties are also obtained.
The algebra of a commutative semigroup over a commutative ring with unity
Authors: Munn W. D.
Published: November 2011

Synopsis

A new description is provided for the nil radical of the algebra RS of a commutative semigroup S over a commutative ring R with a 1. It is shown that the Jacobson radical of RS is nil if the Jacobson radical of R is nil and that the converse holds in the case where S is periodic.
Remarks on a non-well posed problem
Authors: Medeiros L. A.
Published: November 2011

Synopsis

In this paper, it is proved that the problem (∂2u/∂t2) + ∆xu = v, u(x, 0) = u0(x), ut(x, 0) = u1(x), with homogeneous Dirichlet conditions on the boundary, is well posed provided v, u0, u1 belong to a suitable space of functions.
Fractional powers of a closed pair of operators
Authors: Dalsen Marié Grobbelaar-Van
Published: November 2011

Synopsis

In this paper we introduce the concept of fractional powers of a pair of operators between two Banach spaces. The operators need not be closed, but form a closed pair. The properties of the fractional powers are studied. An application of the theory is briefly discussed.
On the zeros of (f″ + αf)f and a result of Steinmetz
Authors: Langley J. K.
Published: November 2011

Synopsis

We characterise all functions f meromorphic of finite order in the plane such that fF has only finitely many zeros, where F = f″ + αf for some constant α. The problem is related to results of N. Steinmetz and others.
Dynamics near a symmetric Hopf bifurcation
Authors: Nagata Wayne
Published: November 2011

Synopsis

We consider the effects of a small symmetry breaking perturbation on a system of differential equations near a Hopf bifurcation point, where the unperturbed system has O(2) symmetry. We show that there exist secondary bifurcations of invariant two-tori of solutions and that the flow on the tori can be quasiperiodic or weakly resonant (phase locked), depending on the size of the perturbation.
Steady symmetric vortex pairs and rearrangements
Authors: Burton G. R.
Published: November 2011

Synopsis

We prove an existence theorem for a steady planar flow of an ideal fluid, containing a bounded symmetric pair of vortices, and approaching a uniform flow at infinity. The data prescribed are the rearrangement class of the vorticity field, and either the momentum impulse of the vortex pair, or the velocity of the vortex pair relative to the fluid at infinity. The stream function ψ for the flow satisfies the semilinear elliptic equation

in a half-plane bounded by the line of symmetry, where φ is an increasing function that is unknown a priori. The results are proved by maximising the kinetic energy over all flows whose vorticity fields are rearrangements of a specified function.
Almost convergent functions and their multipliers
Authors: Balibrea F. , Vera G.
Published: November 2011

Synopsis

In this paper we give a characterisation of the multipliers of a space of almost convergent functions with respect to invariant means related to ergodic semigroups of operators. The characterisation extends several results of the literature.
The order and level of a subgroup of GL2 over a Dedekind ring of arithmetic type
Authors: Mason A. W.
Published: November 2011

Synopsis

Let R be a commutative ring and let q be an R-ideal. Let En(R) be the subgroup of GLn(R) generated by the elementary matrices and let En(R, q) be the normal subgroup of En(R) generated by the q-elementary matrices. For each subgroup S of GLn(R) the order of S, o(S), is the R-ideal generated by xij, xii − xjj (i ≠ j), where (xij) ∈ S, and the level of S, l(S), is the largest R-ideal q0 with the property that En (R, q0) ≦ S. It is known that when n ≧ 3, the subgroup S is normalised by En(R) if and only if o(S) = l(S). It is also known that this result does not hold when n = 2. For example, there are uncountably many normal subgroups S of SL2(ℤ) such that o(S) ≠ {0} and l(S) = {0}, where ℤ is the ring of integers. In this paper we prove that, when A is a Dedekind ring of arithmetic type containing infinitely many units, the order q and level q′ of a subgroup S of GL2(A), normalised by E2(A), are closely related. It is proved that Ψ(q)≦q′, where ≦(q) = 12uq, with u the A-ideal generated by u2 − 1 (u ∈ A*), when A is contained in a number field, and Ψ(q) = q3, when A is contained in a function field.
Remarks on a paper of Olver
Authors: Fairlie D. B.
Published: November 2011

Synopsis

Some disparate ideas in the literature are drawn together. The work of P. J. Olver and his associates on Lagrangians which vanish for arbitrary variations, the so-called null Lagrangians, is viewed as a parallel development to Witten's study of topological field theories. A theorem of Olver, that all hyperjacobians are expressible as divergences, and are thus candidates for the construction of null Lagrangians, is shown to follow directly from the observation that such entities appear in a power series development of the general associative product, and this technique facilitates the construction of multi-dimensional examples.
Some remarks on tight hypersurfaces
Authors: Coghlan Leslie
Published: November 2011

Synopsis

We study the outer part of tight hypersurfaces. We explore in detail how the outer part of such hypersurfaces for n ≧ 3 is more complicated than in the case of tight surfaces in R3. We give a theorem describing tight hypersurfaces of arbitrary dimension.
Conjugacy classes in Sylow p-subgroups of GL(n,q), II
Authors: Vera-López A. , Arregi J. M.
Published: November 2011

Synopsis

In this paper, we find the canonical matrices of the conjugacy classes of the Sylow p-subgroup of GL(n, pt) consisting of all upper unitriangular matrices, whose cardinality is one of the two maximal possible values, that is, pt(n−1)(n−2)/2 and ptn(n−3)/2, as well as their number.
Existence of positive entire solutions for weakly coupled semilinear elliptic systems
Authors: Furusho Yasuhiro
Published: November 2011

Synopsis

Weakly coupled semilinear elliptic systems of the form

are considered in RN, N≧2, where k = 1, 2, …, M, u = (u1, …, uM) and λ is a real constant. The aim of this paper is to give sufficient conditions for (*) to have entire solutions whose components are positive in RN and converge to non-negative constants as |x| tends to ∞. For this purpose a new supersolution-subsolution method is developed for the system (*) without any hypotheses on the monotonicity of the non-linear terms fk with respect to u.
On successive coefficients of univalent functions
Authors: Shen Dawei
Published: November 2011

Synopsis

A coefficient difference bound for a class of univalent functions in the unit disk whose radial growth is large enough is established in this paper.
On a result of Petersson concerning the modular group
Authors: Stothers W. W.
Published: November 2011

Synopsis

Let N(u, g, h) denote the number of subgroups of index u, genus g and parabolic class number h in the classical modular group. In 1974, Petersson proved that, for g = 0 and h = 1, 2, N(u, g, h) grows exponentially with u. We obtain a formula valid for all g and h. From this we derive an asymptotic estimate, showing that Petersson's result gives the correct type of growth. The proofs use coset diagrams.
The principle of limiting absorption for Laplacians on two-point homogeneous spaces
Authors: Thompson M.
Published: November 2011

Synopsis

The present note is concerned to develop the principle of limiting absorption for the Laplacian Δ on a two-point homogeneous noncompact space M = G/H subject to a real-valued potential perturbation V. Such a property depends on the detailed structure of the Laplacian in a suitable coordinate system while V is assumed to satisfy a short-range condition.
Commuting linear differential expressions
Authors: Giertz M. , Kwong M. K. , Zettl A.
Published: November 2011

Synopsis

We consider the question: When do two ordinary linear differential expressions commute? It turns out that the set of all expressions which commute with a given one form a commutative ring. Here we study the algebraic structure of these rings. As an application a complete characterization of normal differential expressions is obtained.
Oscillatory and asymptotic behaviour of solutions of a class of linear ordinary differential equations
Authors: Kusano Takaŝi , Naito Manabu , Tanaka Kyoko
Published: November 2011

Synopsis

The equation to be considered is

where pi(t), 0≦i≦n, and q(t) are continuous and positive on some half-line [a, ∞). It is known that (*) always has “strictly monotone” nonoscillatory solutions defined on [a, ∞), so that of particular interest is the extreme situation in which such strictly monotone solutions are the only possible nonoscillatory solutions of (*). In this paper sufficient conditions are given for this situation to hold for (*). The structure of the solution space of (*) is also studied.
Travelling waves in an initial-boundary value problem
Authors: Veling E. J. M.
Published: November 2011

Synopsis

In this paper we consider the initial-boundary value problem for the semihnear diffusion equation ul=uxx+f(u) on the half-line x>0, when for 0<a<1 f(0)=f(a)=f(1)=0 and f(u)<0 on (0, a), f(u)>0 on (a, 1). For a wide class of initial and boundary values a uniformly valid asymptotic expression is given to which the solution converges exponentially. This expression is composed of a travelling wave and a solution of the stationary problem.
Entropy numbers of embedding maps between Besov spaces with an application to eigenvalue problems
Authors: Carl Bernd
Published: November 2011

Synopsis

In this paper we determine the asymptotic behaviour of entropy numbers of embedding maps between Besov sequence spaces and Besov function spaces. The results extend those of M. Š. Birman, M. Z. Solomjak and H. Triebel originally formulated in the language of ε-entropy. It turns out that the characterization of embedding maps between Besov spaces by entropy numbers can be reduced to the characterization of certain diagonal operators by their entropy numbers.

Finally, the entropy numbers are applied to the study of eigenvalues of operators acting on a Banach space which admit a factorization through embedding maps between Besov spaces.

The statements of this paper are obtained by results recently proved elsewhere by the author.
On the Vitali–Hahn–Saks theorem
Authors: Moltó Aníbal
Published: November 2011

Synopsis

In this paper, a class of Boolean rings containing the class discussed in papers by Seever (1968) and Faires (1976), is defined in such a way that an extension of the classical Vitali–Hahn–Saks theorem holds for exhausting additive set functions. Some new compact topological spaces K for which C(K) is a Grothendieck space are constructed and a Nikodym type theorem is deduced from it. The Boolean algebras of Seever and Faires and those we study here are defined by ‘interpolation properties’ between disjoint sequences in the algebra. We give an example at the end of the paper that illustrates the difficulties arising when we try to find a larger class of Boolean algebras, defined in terms of such properties, for which the Vitali–Hanh–Saks theorem holds.
Quantitative maximum principles and strongly coupled gradient-like reaction-diffusion systems
Authors: Alikakos Nicholas D.
Published: November 2011

Synopsis

In §§1 and 2, we consider mainly a system of reaction-diffusion equations with general diffusion matrix and we establish the stabilization of all solutions at t →∞. The interest of this problem derives from two separate facts. First, the sets that are useful for localizing the asymptotics cease to be invariant as soon as the diffusion matrix is not a multiple of the identity. Second, the set of equilibria is connected. In §3, we establish uniform L§ bounds for the solutions of a class of parabolic systems. The unifying feature in the problems considered is the lack of any conventional maximum principles.
Estimates for solutions of the Dirichlet problem for the biharmonic equation in a neighbourhood of an irregular boundary point and in a neighbourhood of infinity. Saint-Venant's principle
Authors: Kondratiev Y. A. , Oleinik O. A.
Published: November 2011

Synopsis

In this paper energy estimates for solutions of the Dirichlet problem for the biharmonicequation, expressing Saint-Venant's principle in elasticity, are proved. From these integral inequalities, estimates for the maximum modulus of solutions and the gradient of solutions with homogeneous Diriehlet's boundary conditions in a neighbourhood of an irregular boundary point or in a neighbourhood of infinity are derived. These estimates characterize the continuity of solutions and their gradients at these points.
On idempotent affine mappings
Authors: Dawlings R. J. H.
Published: November 2011

Synopsis

Let V be a vector space and End (V) the semigroup of endomorphisms of V. An affine mapping of V is a map A: V → V given by xA = xα + a, where a belongs to End (V) and a is some element of V. Let (V) be the semigroup of affine mappings of V.

Let E' denote the non-injective idempotents of End (V) and let ℰ denote the idempotents of (V). In this paper 〈ℰ〉 is determined in terms of 〈E′〉 when End (V) consists of all endomorphisms of V and when End (V) only contains the continuous endomorphisms (in which case we restrict V to being an inner product space).
The varieties of commutative semigroups for which epis are onto
Authors: Higgins P. M.
Published: November 2011

Synopsis

In a forthcoming paper, N. M. Khan gives a condition for a variety of commutative semigroups V to be saturated in the sense of Howie and Isbell (1967) (i.e. epis are onto for each S ∈ V). We show the necessity of the condition by constructing a non-saturated semigroup which is a member of every commutative variety not satisfying Khan's condition. This determination of the saturated varieties of commutative semigroups enables us then to prove that these varieties form a sublattice of the lattice of varieties of all commutative semigroups.
Lie algebras whose maximal subalgebras are modular
Authors: Varea V. R.
Published: November 2011

Synopsis

A subalgebra M of a Lie algebra L is called modular in L if M is a modular element in the lattice of the subalgebras of L. Our aim is to study the finite-dimensional Lie algebras all of whose maximal subalgebras are modular. We characterize these algebras over any field of characteristic zero.
On small cancellation theory over H.N.N. extensions
Authors: Perraud J.
Published: November 2011

Synopsis

Small cancellation theory has been extended to symmetrized subsets of free products, amalgamated free products and Higman-Neumann-Neumann (H.N.N.) extensions. We though that it was possible to obtain results on decision problems if we could define small cancellation conditions for finite subsets.

Sacerdote and Schupp (1974) defined the small cancellation condition C'(l/6) for symmetrized subsets of an H.N.N. extension. We define this condition for finite subsets, with the following properties:

For each finite subset X, there is a symmetrized subset X1 with the same normal closure and, if X1 satisfies C'(l/6), then X satisfies C'(l/6).

For some H.N.N. extensions, we can decide whether any finite subset satisfies C'(l/6), and, in this case, we can solve the word problem for the corresponding quotient.
Absence of singular continuous spectrum for two-body Schrödinger operators with long-range potentials (a new proof)
Authors: Cycon Hans L.
Published: November 2011

Synopsis

We prove the absence of singular continuous spectra for Schrödinger operators − Δ + V with long-range potential V such that V and (1 + r)1+ε(∂/∂r) V is (− Δ)-compact by using a modified “Mourre type” estimate and by Kato-Lavine's H-smoothness theory.
On the compatibility of the natural order on a regular semigroup
Authors: Blyth T. S. , Gomes Gracinda M. S.
Published: November 2011

Synopsis

A notable achievement in the algebraic theory of semigroups has been the discovery by Nambooripad of the natural order on a regular semigroup. He has shown that this order is compatible with multiplication if and only if the semigroup is locally inverse, in the sense that every local submonoid is an inverse semigroup. In this paper we determine precisely when the natural order is compatible on the right (respectively left) with multiplication; this is so if and only if every local submonoid is ℒ-unipotent (respectively ℛ-unipotent).
Growth of localized disturbances on a vortex sheet
Authors: Craik A. D. D.
Published: November 2011

Synopsis

Linearized local disturbances on a vortex sheet are known to develop singularities after a finite term in some cases but not in others. A simple test for the appearance of such singularities is given in terms of the Fourier transform of the initial disturbance. Such singularities are a consequence of the artificiality of the vortex sheet model and should not be regarded as physically meaningful.
A note on expansions for the two-dimensional, compressible, laminar boundary layer equations near a point of zero skin friction
Authors: Wilks G.
Published: November 2011

Synopsis

The first non-arbitrary coefficient, α12, of the Buckmaster expansions is evaluated in the context of the extended Goldstein-Stewartson theory. Leading terms of the next order contributions to the skin friction and heat transfer coefficients are also obtained.
On holomorphic extensions from spheres in ℂ2
Authors: Globevnik Josip
Published: November 2011

Synopsis

A theorem of Rudin states that if B is the open unit ball in ℂN, N > 1, if 0<ρ < 1, if is the family of all complex lines in ℂN at a distance ρ from the origin and if f ∈ C(∂B) is such that for every Λ∈ the function f|Λ∂B has a continuous extension to Λ ∩ B which is holomorphic in Λ ∩ B, then f has a continuous extension to B which is holomorphic in B. In this paper we show that when N = 2, the theorem still holds if is replaced by a considerably smaller family.
Stability regions for multi-parameter systems of periodic second order ordinary differential equations
Authors: Browne Patrick J. , Sleeman B. D.
Published: November 2011

Synopsis

This paper studies the stability regions associated with the multi-parameter system

where the functions qr(xr), ars(xr) are periodic and the system is subjected to periodic or semi-periodic boundary conditions.
On optimal mild solutions of non-homogeneous differential equations in Banach spaces
Authors: Zaidman S.
Published: November 2011

Synopsis

Consider mild solutions on the real line of non-homogeneous differential equations in a Banach space: u′(t) = Au(t) + f(t), where A is the infinitesimal generator of a C0-semigroup.

We prove an existence result for optimal solutions (as defined in the text) in reflexive spaces and an uniqueness fact in uniformly convex B-spaces.
Equations de Schrödinger non linéaires en dimension deux
Authors: Cazenave Thierry
Published: November 2011

Synopsis

This paper is devoted to the study of some non linear Schrödinger equations in two dimensions, arising in non linear optics; in particular, it is concerned with solutions to the Cauchy problem. The problem of global existence and regularity of the solutions, the asymptotic behaviour of global solutions, and the blow-up of non global solutions are studied.
On the divergence in rth mean of a class of eigenfunction expansions
Authors: Pitts C. G. C.
Published: November 2011

Synopsis

We consider the expansion of a function in Lr (the class of measurable functions whose rth powers are Lebesgue integrable over some interval) in terms of the eigenfunctions arising from a singular Sturm-Liouville problem defined over an infinite or semi-infinite interval. We show that if l ≦ r ≦ inline1 or if r ≧ 4 there exists f in Lr whose eigenfunction expansion is divergent in the rth mean sense, and that the terms of the series form an unbounded sequence in Lr The result extends some work of Askey and Wainger concerning the Hermite series expansions of functions in Lr(–∞, ∞).
Subgroups of SL2ℤ generated by elementary matrices
Authors: Tuler Randy
Published: November 2011

Synopsis

An elementary matrix has ones down the main diagonal and at most one element off the diagonal that differs from zero. We study the subgroups of SL2ℤ generated by sets of elementary matrices. Specifically, we give a stringent condition that the entries of a matrix belonging to such a group must satisfy.
Semilinear elliptic equations in unbounded domains of Rn
Authors: Donato Patrizia , Migliaccio Lucia , Schianchi Rosanna
Published: November 2011

Synopsis

We study, in unbounded domains Ω⊂Rn, an elliptic semilinear problem with homogeneous boundary conditions. We assume that the nonlinear term f(x, u, Du) satisfies some condition of quadratic growth with respect to Du. We prove, in the framework of weighted Sobolev spaces, that, if and are respectively a subsolution and a supersolution of our problem, then there exists a least solution ū and a greatest solution û in the ordered interval and we obtain some multiplicity results.
Weighted norm inequalities of sum form involving derivatives
Authors: Kwong Man Kam , Zettl A.
Published: November 2011

Synopsis

Here we obtain the inequality

under very general conditions on the non-negative weight functions u, v, w, for general p, l≦p<∞ and for both bounded and unbounded intervals I.
Some subsemigroups of infinite full transformation semigroups
Authors: Howie J. M.
Published: November 2011

Synopsis

As in an earlier paper by the author, three cardinal numbers, the shift, the defect and the collapse, are associated with each element of the full transformation semigroup ℑ(X), where X is an infinite set. It is shown that the elements of finite shift and non-zero defect form a subsemigroup F of ℑ(X). Moreover, if E(F) denotes the set of idempotents in F then 〈E(F)〉 = F, but (E(F))n ⊂F for every finite n. For each infinite cardinal m not exceeding ∣X∣ the set Qm of balanced elements of weight m, i.e. those with shift, defect and collapse all equal to m, forms a subsemigroup of ℑ(X). Moreover, (E(Qm))4=Qm,(E(Qm))3⊂Qm.
Covariants of pencils of binary cubics
Authors: Newstead P. E.
Published: November 2011

Synopsis

The invariants of pencils of binary cubics were described in an earlier paper by the author. In this note, we give a complete set of covariants for such pencils, and show how each type of pencil may be denned by the identical vanishing of covariants.
On the number of solutions of right-definite problems with a convergent Dirichlet integral
Authors: Eastham M. S. P.
Published: November 2011

Synopsis

A recently developed asymptotic theory of higher-order differential equations is applied to problems of right-definite type to determine the numbers M+, M− of linearly independent solutions with a convergent Dirichlet integral, M+ and M− referring to the usual upper and lower λ.-half-planes. Particular attention is given to the phenomenon noted by Karlsson in which one of M+ and M− is maximal but not the other. Conditions are given under which M+ (say) is maximal and M− is the same, one less, and two less.
Geršgorin theory and the definiteness of determinantal operators
Authors: Farenick D. R. , Browne Patrick J.
Published: November 2011

Synopsis

Let Aij, l≦j≦k, be bounded Hermitean operators on Hilbert spaces Hi, 1≦i≦k, and let be the induced operators on . An important operator for multiparameter theory is δ: H →H denned by δ = det the determinant being expanded formally. Various definiteness properties of δ are critical for multiparameter spectral theory.

We use the operators Aij to construct a numerical matrix δ(δ) upon which we use Geršgorin theory to investigate the non-singularity and definiteness of δ. Diagonal dominance properties of the array [Aij] are also discussed.
Finitely generated GEn-rings
Authors: Mason A. W.
Published: November 2011

Synopsis

Suslin and Cohn have proved that the polynomial ring Z[t1, …,td], where Z is the ring of rational integers and d >0, is a GEn-ring if and only if n ≧ 3. (A commutative ring R with identity is called a GEn-ring if and only if SLn(R) is generated by elementary matrices, where n ≧ 2.) In this paper we consider the following question:

Given algebraic numbers α1, …,αd, for which n (if any) is the ring A =Z [α1, …,αd a GEn-ring?

By standard results from algebraic K-theory it follows that (a) A is GEn for all n ≧ 2, (b) A is not GEn for any n ≧ 2, or (c) A is GEn if and only if n ≧ 3. Examples of each type are provided. In particular, it is shown that if each αi is real or at least one αi is not an algebraic integer, then A is of type (a).
Global higher bifurcations in coupled systems of nonlinear eigenvalue problems
Authors: Cantrell Robert Stephen
Published: November 2011

Synopsis

Coexistent steady-state solutions to a Lotka–Volterra model for two freely-dispersing competing species have been shown by several authors to arise as global secondary bifurcation phenomena. In this paper we establish conditions for the existence of global higher dimensional n-ary bifurcation in general systems of multiparameter nonlinear eigenvalue problems which preserve the coupling structure of diffusive steady-state Lotka–Volterra models. In establishing our result, we mainly employ the recently-developed multidimensional global multiparameter theory of Alexander–Antman. Conditions for ternary steady-state bifurcation in the three species diffusive competition model are given as an application of the result.
Existence for the modified Kassoy problem
Authors: Bebernes J. , Troy W.
Published: November 2011

Synopsis

We prove that there exists an ᾱ > 0 such that for each 0 ≦ α ≦ ᾱ, there is at least one β = β(α) < 0 such that y“ − (x/2)y' + ey − 1 =0, y(0) = α, y'(0) = β has a solution y(x, α,β) satisfying the asymptotic property y(x, α,β)∼ −2 In x + Kα as x → ∞.
An exact method for the calculation of certain Titchmarsh-Weyl m-functions
Authors: Harris B. J.
Published: November 2011

Synopsis

We consider the second order, linear, differential equation

where q is real-valued. In the case we calculate the Titchmarsh-Weyl m-function associated with (*).
On the fixed point index in locally convex spaces
Authors: Furi M. , Pera M. P.
Published: November 2011

Synopsis

Let E be a Hausdorff locally convex space, Q a convex closed subset of E and U an open subset of Q. We develop an index theory for a class of locally compact maps f: U → E for which the usual assumption f(U) ⊂ Q is replaced by an appropriate “pushing condition”. Moreover, from this index theory, we deduce a general continuation principle and some global results for nonlinear eigenvalue problems.
Large-time behaviour of solutions to hyperbolic–parabolic systems of conservation laws and applications
Authors: Kawashima Shuichi
Published: November 2011

Synopsis

We study the large-time behaviour of solutions to the initial value problem for hyperbolic-parabolic systems of conservation equations in one space dimension. It is proved that under suitable assumptions a unique solution exists for all time t ≧ 0, and converges to a given constant state at the rate t − ¼ as t → ∞. Moreover, it is proved that the solution approaches the superposition of the non-linear and linear diffusion waves constructed in terms of the self-similar solutions to the Burgers equation and the linear heat equation at the rate t − ½ +α, α > 0, as t →∞. The proof is essentially based on the fact that for t → ∞ the solution to the hyperbolic-parabolic system is well approximated by the solution to a semilinear uniformly parabolic system whose viscosity matrix is uniquely determined from the original system. The results obtained are applicable straightforwardly to the equations of viscous (or inviscid) heat-conductive fluids.
On a counterexample of a rank 1 convex function which is not polyconvex in the case N = 2
Authors: Aubert G.
Published: November 2011

Synopsis

J. M. Ball has introduced the notion of polyconvexity to study nonlinear problems in elasticity and he has shown that polyconvexity implies rank 1 convexity. In this paper we prove by a counterexample that the converse of this implication is false in two dimensions.
Studies in multiplicative solutions to linear differential equations
Authors: Arscott F. M.
Published: November 2011

Synopsis

Given an ordinary linear differential equation whose singularities are isolated, a solution is called multiplicative for a closed path C if, when continued analytically along C, it returns to its starting-point merely multiplied by a constant. This paper first classifies such paths into three types, then investigates combinations of two such paths, in which a number of qualitatively different situations can arise. A key result is also given relating to a three-path combination. There are applications to special functions and Floquet theory for periodic equations.
A remark on the equation of a vibrating plate
Authors: Stahel Andreas
Published: November 2011

Synopsis

We consider the von Karman equations, which describe a vibrating plate either with a clamped boundary or with completely free boundary. In both cases we obtain a unique, classical solution. As the main tool we use a set of integral equations, which we deduce from the well known “variations of constants” formula.
Volterra integral equations and a new Gronwall inequality (Part I: The linear case)
Authors: Norbury J. , Stuart A. M.
Published: November 2011

Synopsis

We present a generalisation of the continuous Gronwall inequality and show its use in bounding solutions of discrete inequalities of a form that arise when analysing the convergence of product integration methods for Volterra integral equations. We then use these ideas to prove convergence of a numerical method which is effective in approximating Volterra integral equations of the second kind with weakly singular kernels.
H-spaces with generating subspaces
Authors: Iwase Norio
Published: November 2011

Synopsis

For an H-space with a generating subspace, we construct a space whose K-cohomology is a direct sum of a truncated polynomial algebra and an ideal, which enables technical restrictions to be removed from several known results in the homotopy theory of H-spaces.
Some general constructions of tight mappings
Authors: Coghlan Leslie
Published: November 2011

Synopsis

We present several general methods of constructing tight hypersurfaces f: Mn → En+1, n ≧ 2. We prove a smoothing lemma, which allows us to approximate tight continuous hypersurfaces by C∞ tight ones. We show that given a tight immersion or C∞-stable map f: M2 → E3 of a compact surface M2 other than S2, there is a tight C∞-stable map g: M2 # RP2:→ E3. We prove that given C∞ tight immersions f: Mn → En+1 and g: Nn → En+1 of compact n-manifolds Mn and Nn into En+1, there is a tight C∞ immersion of Mn # Nn into En+1. Two other methods involve hypersurfaces of rotation and sets in En+1 at a fixed distance from a tightly embedded n-manifold with boundary in En. One consequence of these methods is that the outer part of C∞ tight hypersurfacesf: Mn → En+1, n ≧ 3, is far more complicated than in the case of tight surfaces in En. For example, given any C∞n-manifold M with boundary tightly embedded in En, there is a tight immersion f:Nn → En+1 of a closed n-manifold having M as a topset. Kuiper's theorem describing tight immersions of surfaces into E3 does not generalise to the case of hypersurfaces f: Mn → En+1, n ≧ 3, without substantial restrictions on Mn and/or f.
A note on the one-dimensional maximal function
Authors: Bernal Antonio
Published: November 2011

Synopsis

In this note, we consider the Hardy-Littlewood maximal function on R for arbitrary measures, as was done by Peter Sjögren in a previous paper. We determine the best constant for the weak type inequality.
A weighted gradient inequality
Authors: Sinnamon Gordon
Published: November 2011

Synopsis

Necessary and sufficient conditions on weights u and v are given for the inequality

to hold for all g ∊ .
The perturbed test function method for viscosity solutions of nonlinear PDE
Authors: Evans Lawrence C.
Published: November 2011

Synopsis

The method of viscosity solutions for nonlinear partial differential equations (PDEs) justifies passages to limits by in effect using the maximum principle to convert to the corresponding limit problem for smooth test functions. We describe in this paper a “perturbed test function” device, which entails various modifications of the test functions by lower order correctors. Applications include homogenisation for quasilinear elliptic PDEs and approximation of quasilinear parabolic PDEs by systems of Hamilton-Jacobi equations.
Weighted norm inequalities involving gradients
Authors: Carton-Lebrun C. , Heinig H. P.
Published: November 2011

Synopsis

Letthen, for certain weight functionsuandvand indicesp,q, it is shown that ∥Tαf∥q, u≦C∥grad f ∥p, v'q>n/ α holds. For α=l,p=qandu=v≡ l this reduces to a result of M. Weiss. In addition we establishn-dimensional weighted Hardy—Littlewood type inequalities ofthe formfor large classes of weights.
Liouville-type results for fourth order elliptic equations
Authors: Goyal Vinod B.
Published: November 2011

Synopsis

Liouville type theorems are obtained for the solutions to elliptic equations of the form Δ2u −q(x)Δu + p(x)f(u)=0 by means of two subharmonic functionals and Green type inequalities.
On a characterisation of Dirac measures
Authors: Escobedo Miguel
Published: November 2011

Synopsis

Assume f ∈ C(RM, SN−1) (M ≧ 1, N ≧ 1) is periodic with period one in each variable and satisfies:

Then, every probability measure on [0, 1)M such that

is a Dirac measure at some point of [0, 1)M.
Multiplicity results for nonlinear elliptic equations involving critical Sobolev exponent
Authors: Capozzi A. , Palmieri G.
Published: November 2011

Synopsis

In this paper we study the following boundary value problem

where Ω is a bounded domain in Rn, n≧3, x ∈Rn, p* = 2n/(n – 2) is the critical exponent for the Sobolev embedding is a real parameter and f(x, t) increases, at infinity, more slowly than .

By using variational techniques, we prove the existence of multiple solutions to the equations (0.1), in the case when λ belongs to a suitable left neighbourhood of an arbitrary eigenvalue of −Δ, and the existence of at least one solution for any λ sufficiently large.
Weighted norm inequalities for the Hankel- and -transformations
Authors: Emara S. A. , Heinig H. P.
Published: November 2011

Synopsis

We give conditions on pairs of non-negative weight functions u and v which are sufficient that, for 1<p, q <∞,

where T is the Hankel-or the K-transformation.

The proofs are based on a weighted Marcinkiewicz interpolation theorem for linear operators. In the case that T is the Hankel transformation and 1<p≦q <∞, the result is similar to a weighted estimate of Heywood and Rooney [9], but with different weight conditions.
Approximation to inverses of normal operators
Authors: Mika J. , Pack D. C.
Published: November 2011

Synopsis

For many purposes, and in particular for the calculation of upper and lower bounds to bilinear forms 〈g0,f〉, where f is the solution to an operator equation and g0 is known, it isuseful to obtain an approximation to the inverse of the operator.

For a normal operator A acting in a complex Hilbert space with bounded inverse A−1, we use a direct approach through fundamental results in functional analysis and derive a recipe for the ‘best formula’ for A−1 of form B = βI with β constant and I the identity operator. Examples illustrate that this leads to improved results for certain classes of operator.

The investigation is extended to the representation of A−1 by a polynomial in A withcomplex coefficients. For polynomials of first and higher orders the application is restricted to bounded self-adjoint operators; at first order, explicit formulae are given.

Application of the method to Fredholm operators is considered in detail, and its use for the point-wise solution to Fredholm integral equations is illustrated.
Finite speed of propagation and asymptotic rates for some nonlinear higher order parabolic equations with absorption
Authors: Bernis F.
Published: November 2011

Synopsis

The “energy solutions” to the equation

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

Synopsis

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

Synopsis

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

Synopsis

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

Synopsis

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

Synopsis

In this paper we consider Hopf bifurcation in the presence of O(2) symmetry. The system of reaction diffusion equations ut, = D(µ)uxx + f(µ, u) provided with periodic boundary conditions may serve as a model problem. We prove the bifurcation of a torus of standing waves and two circles of travelling waves and we compute the stability (with asymptotic phase) of these periodic solutions, giving explicit formulae. Finally we demonstrate how a small perturbation which breaks part of the symmetry leads to secondary bifurcation.
Time-periodic solutions to semilinear parabolic equations
Authors: Grindrod Peter , Rynne Bryan P.
Published: November 2011

Synopsis

We consider a class of non-linear evolution equations subject to a periodic forcing term. Using bifurcation theory we obtain results on the existence and number of periodic solutions. The theory applies to semi-linear diffusion equations defined on bounded or unbounded domains.
Solutions to some problems in amenable semigroups
Authors: Ayyaswamy S. K. , Ramakrishnan P. V.
Published: November 2011

Synopsis

This paper discusses a few problems on the size of the set of invariant means of an amenable semigroup posed by Maria M. Klawe, Alan L. T. Paterson and M. Rajagopalan and P. V. Ramakrishnan ([4], [5], [8] and [9]).
Smooth solution for one-dimensional inhomogeneous Heisenberg chain equations
Authors: Ding Shijin , Guo Boling , Su Fengqiu
Published: November 2011

Using the viscosity vanishing method, we obtain the existence of a smooth solution to the inhomogeneous Heisenberg chain equations in one dimension. The uniqueness of the solution to the viscosity equation is also given.
Cα estimation on the boundary for the gradient of the shape derivative of a Neumann problem in ℝ3
Authors: Novruzi A.
Published: November 2011

In this paper we establish a non-global Cα estimation on the boundary for the gradient of the solution of a Neumann exterior problem in ℝ3. The boundary data of this problem are with small compact support and with C0 norm tending to infinity when the diameter of support data goes to zero. This problem appears, for instance, in the numerical shape optimization problems when the Newton method is used. It turns out that the gradient decreases when the distance to the boundary data supports increases (Saint Venant's principle), which, when used in applications, leads to a strong reduction of the gradient computational cost.
Inverse-monotone nonlinear differential operators of the second order
Authors: Schröder Johann
Published: November 2011

Synopsis

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

Synopsis

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

Synopsis

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

Synopsis

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

Synopsis

An interval-type LP criterion for

is derived in which “positive” coefficients play a prominent role. When pn = 1 and all the other pi are zero this reduces to a result of Ismagilov (1962). Successive specializations are obtained with the growth of the pi constrained by monomials in x. Previous LP criteria of Everitt (1968) and Hinton (1972, 1974) are shown to be special cases.
Asymptotic behaviour of an elastoviscoplastic material submitted to a cyclic load
Authors: Wesfreid Eva
Published: November 2011

Synopsis

We study the quasistatic behaviour of an elastoviscoplastic material submitted to a cyclic load. We prove the weak convergence of the solution of the stress evolution problem towards a periodical one when t → + ∞.
Generalisation of Dedekind's problem of the enumeration of coherent structures
Authors: Ansell J. , Bendell A. , Humble S.
Published: November 2011

Synopsis

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

Synopsis

This paper deals with initial value problems in ℝ2 which are governed by a hyperbolic differential equation consisting of a nonlinear first order part and a linear second order part. The second order part of the differential operator contains a small factor ε and can therefore be considered as a perturbation of the nonlinear first order part of the operator.

The existence of a solution u together with pointwise a priori estimates for this solution are established by applying a fixed point theorem for nonlinear operators in a Banach space.

It is shown that the difference between the solution u and the solution w of the unperturbed nonlinear initial value problem (which follows from the original problem by putting ε = 0) is of order ε, uniformly in compact subsets of ℝ2 where w is sufficiently smooth.
A characterization of endomorphism algebras
Authors: Länger Helmut
Published: November 2011

Synopsis

The result that any monoid is isomorphic to some endomorphism monoid is generalized to super-associative systems.