bluuebunny/crossref_metadata_2025_split huggingface dataset:
Global theorems for symmetry sets of smooth curves and polygons in the plane
Authors: Banchoff T. F. , Giblin P. J. 
Published: November 2011

SynopsisThe symmetry set of a plane curve y is the locus of centres of circles which either (i) are tangent to y at at least two distinct points, or (ii) have at least 4-point (A3) contact with y somewhere. The points (ii) also lie at cusps of the evolute of γ and the symmetry set, together with the evolute, can be studied by regarding their union as a full bifurcation set. For information on symmetry sets, see [2, 4, 5, 7, 8].Here we shall use a theorem of Ozawa [9] to deduce global formulae relating three features of symmetry sets. The transitions which occur on the symmetry set of a generic one-parameter family of plane curves have been found [4], and more insight into the global formulae can be gained by tracing them through the transitions and verifying that they are invariant (see Section 2). We also briefly introduce symmetry sets of plane polygons in Section 3, and show that, contrary to first impressions, the same global formulae hold there.
Results old and new on the hyper-Bessel equation
Authors: Paris R. B. , Wood A. D. 
Published: November 2011

SynopsisWe consider a variety of integral representations, single and multiple, old and new, for solutions of the hyper-Bessel equation u(n) – zmu =0. In particular, we show how a very early multiple Laplace integral solution of Molins (1876) may be related to recent Mellin–Barnes integral representations given by the present authors by way of multiple integral solutions given by Saxton and the second author for an associated equation. Although both these multiple integral solutions may be found by elementary methods, it is not easy to find their asymptotic expansions for large z, and we show how these may conveniently be obtained from our earlier results [10].
Floquet solutions of non-linear ordinary differential equations
Authors: Hassan H. S. 
Published: November 2011

SynopsisIn this paper we study the solutions of the boundary value problemwhere t ∊ℝ, x ∊ ℝN, f is a continuous function of (t,x)and locally Lipschitz in x and ω is a fixed positive number and λ ∊ ℝ. By using degree theory we prove results on the existence of solutions of (*) and the dependence of such solutions on λ. We shall prove that (*) does not have an isolated solution, and study the topological properties of the components of solutions of (*).
Minors of the wronskian of the differential equation Lny+p(x)y=0
Authors: Elias Uri 
Published: November 2011

SynopsisThe equation studied here is Lny +p(x)y = 0, where Ln is a disconjugate differential operator and p(x) is of a fixed sign. It is shown that certain minors of the wronskian of this equation satisfy a very similar differential equation Mnz +p(x)z =0. We prove that some properties of the original equation which are essential for oscillation theory are inherited by the solutions of the second one.
Sign-definite solutions in some linear elliptic systems
Authors: Cosner Chris , Schaefer Philip W. 
Published: November 2011

SynopsisWe consider a weakly coupled set of two partial differential equations where the coupling matrix has variable elements and the principal part of each equation is the same uniformly elliptic operator. Weobtain necessary conditions that the system of equations can be decoupled. By decoupling the system and using a positivity lemma due to Hess and Kato, we determine the algebraic sign of the solution components. This work extends recent results of de Figueiredo and Mitidieri. Further, one can use these results to determine the sign of the solution to certain fourth order elliptic boundary value problems.
On subordinacy and analysis of the spectrum of Schrödinger operators with two singular endpoints
Authors: Gilbert D. J. 
Published: November 2011

SynopsisThe theory of subordinacy is extended to all one-dimensional Schrödinger operatorsfor which the corresponding differential expressionL= –d2/(dr2) +V(r) is in the limit point case at both ends of an interval (a,b), withV(r) locally integrable. This enables a detailed classification of the absolutely continuous and singular spectra to be established in terms of the relative asymptotic behaviour of solutions ofLu = xu, x εℝ, asr→aandr→b. The result provides a rigorous but straightforward method of direct spectral analysis which has very general application, and somefurther properties of the spectrum are deduced from the underlying theory.
On interpolation of strictly (co-)singular linear operators
Authors: Beucher O. J. 
Published: November 2011

SynopsisWe show that the property of linear operators to be in the surjective hull (injective hull) of the ideal of strictly singular (strictly cosingular) operators between Banach spaces is an interpolation property with respect to the real interpolation method with parameters 0 < ủ < 1 and < p < ℞.
Complex and real singularities of eigenfunction expansions
Authors: Zayed Ahmed I. 
Published: November 2011

SynopsisLet θ(t, λ) and (t,λ) be the solutions to the differential equation y“+(λ − q(t))y =0, − ∞ <t < ∞ such that θ(0, λ)) = l, θ'(0, λ)) = 0, ø(0, λ) = 0 and ø'(0, λ) = −1. It is known that for a function f(t)∈ L2(−∞,∞), we havewhere for some measures dξ, dη; and dζ independent of f.In the first part of this paper we devise a technique to locate the singular points of f(t) (t is complex) under the assumptions that E(λ), F(λ) are of order O(e-cvλ), as λ →∞and q(t) is a sufficiently nice function. In this case all the singularities of fare off the real axis. In the second part of the paper we relax the restriction on E(λ) and f(λ) so that they are of order O(λk) as λ →∞ for some constant k, and we show that in this case f(t) is a generalised function whose analytic representation is holomorphic in the upper and lower half planes. We then devise a technique to continue analytically across the real axis and locate its singularities thereon.
On the minimal eigenvalue of a positive definite operator determinant
Authors: Volkmer H. 
Published: November 2011

SynopsisIn this paper we study a problem in multilinear algebra which consists of finding small values of a certain quotientμ/α. Here μ is the minimal eigenvalue of a positive definite operator determinant Δ of the type introduced by F. V. Atkinson, and α is the minimum of the quadratic form corresponding to Δ with respect to all decomposable tensors of unit norm. Our results are connected with earlier results of P. Binding.
Generalised Liouville-Green asymptotic approximations for second-order differential equations
Authors: Cassell J. S. 
Published: November 2011

SynopsisThe equation {r(x)y'}' = q(x)y is considered with the coefficients q and rtaking complex values. Conditions are obtained for there to be solutions with the asymptotic form , . These involve the coefficients and their derivatives up to some order N.
Radicals of semigroup algebras of commutative and cancellative semigroups
Authors: Puczyłowski Edmund R. 
Published: November 2011

SynopsisThe shape of radicals of semigroups algebras of commutative and cancellative semigroups is studied. The questions asto when a radical of those algebras is homogeneous and if homogeneous radicals have more regular form are examined.
Positivity for elliptic and parabolic systems
Authors: Auchmuty J. F. G. 
Published: November 2011

SynopsisThe positivity of solutions of initial-boundary value problems for weakly-coupled semilinear parabolic or elliptic systems of equations is studied. Conditions on the coupling terms are described which ensure that the solutions of the parabolic systems remain positive whenever the initial conditions are positive. For elliptic systems involving a parameter, conditions on the coupling terms are described which imply that solution branches which contain a positive solution, in fact, contain only positive solutions. Applications of these theorems to certain reaction-diffusion equations arising in the modelling of biological phenomena are given.
Inequalities and separation for Schrödinger type operators in L2(Rn)
Authors: Everitt W. N. , Giertz M. 
Published: November 2011

SynopsisThe symmetric differential expression M determined by Mf = − Δf;+qf on G, where Δ is the Laplacian operator and G a region of n-dimensional real euclidean space Rn, is said to be separated if qfϵL2(G) for all f ϵ Dt,; here D1 ⊂ L2(G) is the maximal domain of definition of M determined in the sense of generalized derivatives. Conditions are given on the coefficient q to obtain separation and certain associated integral inequalities.
Isometric folding of Riemannian manifolds
Authors: Robertson S. A. 
Published: November 2011

SynopsisWhen a sheet of paper is crumpled in the hands and then crushed flat against a desk-top, the pattern of creases so formed is governed by certain simple rules. These rules generalize to theorems on folding Riemannian manifolds isometrically into one another. The most interesting results apply to the case in which domain and codomain have the same dimension. The main technique of proof combines the notion of volume with Hopf's concept of the degree of a map.
Some elliptic balls which avoid a Nyquist set in ℂn+1
Authors: Smith Russell A. 
Published: November 2011

SynopsisConditions are obtained for certain elliptic balls in ℂn+1 to have empty intersection with the Nyquist set of a vector polynomial G(z). Such conditions are shown to yield explicit criteria for the existence of periodic solutions of non-autonomous scalar differential equations of the form A* G(D)y = p.
Equivalence, adjoints and symmetry of quasi-differential expressions with matrix-valued coefficients and polynomials in them
Authors: Frentzen Hilbert 
Published: November 2011

SynopsisQuasi-differential expressions with matrix-valued coefficients, which generalize those of Shin and Zettl, are considered with regard to equivalence, adjoints and symmetry. The characterization results imply that in the scalar case the class of quasi-differential expressions considered here coincides with that of Shin and is equivalent to that of Zettl. Furthermore polynomials in quasi-differential expressions are defined as expressions of the same kind and shown to coincide with the usual ones. Finally it is indicated that the known general results for the deficiency indices carry over to quasi-differential expressions.
Global invertibility of Sobolev functions and the interpenetration of matter
Authors: Ball J. M. 
Published: November 2011

SynopsisA global inverse function theorem is established for mappingsu: Ω → ℝn, Ω ⊂ ℝnbounded and open, belonging to the Sobolev spaceW1.p(Ω),p>n. The theorem is applied to the pure displacement boundary value problem of nonlinear elastostatics, the conclusion being that there is no interpenetration of matter for the energy-minimizing displacement field.
On the location of the Weyl circles
Authors: Atkinson F. V. 
Published: November 2011

SynopsisThe paper deals with explicit estimates concerning certain circles in the complex plane which were associated with Sturm–Liouville problems by H. Weyl. By the use of Riccati equations instead of linear integral equations, improvements are obtained for results of Everitt and Halvorsen concerning the behaviour of the Titchmarsh–Weyl m-coefficient.
Stable methods for solving the inverse scattering problem for a cylinder
Authors: Colton David , Kirsch Andreas 
Published: November 2011

SynopsisIt is shown that the inverse scattering problem for an infinite cylinder can be stabilized by assuming a priori that the unknown boundary of the cylindrical cross section lies in a compact family of continuously differentiable simple closed curves. A constructive method for determining the shape of this boundary is given under the assumption that an initial approximation is known and that the scattering cross section is known forn distinct incoming plane waves in the resonant region.
On the convergence of the method of finite differences applied to the solution of the initial value problem for functional-differential neutral type equations
Authors: Kazakova N. G. , Bainov D. D. 
Published: November 2011

SynopsisWhen solving practically the neutral type equations the derivatives are replaced by finite differences while the members of integral type are replaced by quadrature formulae. The paper deals with the convergence of a natural class of methods applied to the Cauchy problem for functional-differential neutral type equations. It is not obligatory for the approximated operators to be compact.
Multiparameter definiteness conditions
Authors: Binding Paul 
Published: November 2011

SynopsisSeveral definiteness conditions in the multiparameter spectral literature are discussed. It is shown that some of these conditions permit simplifying transformations of the eigenvalues, leading to further definiteness properties. Geometrical equivalents for the algebraic conditions are established in terms of separation of convex cones. As a result, the relationship between the standard left and right definiteness conditions is clarified.
Uniform basis properties of exponential solutions of functional differential equations of retarded type
Authors: Ho L. F. 
Published: November 2011

SynopsisWe define spaces of functions on an interval naturally related to functional differential equations of retarded type in the sense that exponential solutions of such equations form a uniform basis (or Riesz basis) in these spaces. Embedding theorems relating these spaces to the Sobolev spaces are established using a result on Carleson's measure defined on a half plane.
Complementary bounds for inner products associated with non-linear equations
Authors: Cole R. J. , Mika J. , Pack D. C. 
Published: November 2011

SynopsisFunctionals are found that give upper and lower bounds to the inner product 〈g0, f〉 involving the unknown solution f of a non-linear equation T[f] = f0, with f∈H, a real Hilbert space, g0 a given function in H and f0 a given function in the range of the non-linear operator T. The method depends upon a re-ordering of terms in the expansion of T[f] about a trial function so as to transfer the non-linearity to a secondary problem that requires its own particular treatment and to enable earlier results obtained for linear operators to be used for the main part. First, bivariational bounds due to Barnsley and Robinson are re-derived. The new and more accurate bounds are given under relaxed assumptions on the operator T by introducing a third approximating function. The results are obtained from identities, thus avoiding some of the conditions imposed by the use of variational methods. The accuracy of the new method is illustrated by applying it to the problem of the heat contained in a bar.
Existence of multiple positive solutions of inhomogeneous semilinear elliptic problems in unbounded domains
Authors: Zhu Xi-Ping , Zhou Huan-Song 
Published: November 2011

SynopsisBy using the concentration-compactness method of Lions [14, 16] and the mountain pass theorem of Ambrosetti and Rabinowitz [3], through a careful inspection of the energy balance for some sequence of approximated solutions, we show that under suitable conditions on f and h, the inhomogeneous problem. −Δu + c2u = λ(f(u) + h(x)) for x ∈ Ω (Ω is an exterior domain in ℝN, N≧ 3) and has at least two positive solutions.
Saint-Venant's principle on unbounded regions
Authors: Knops R. J. , Rionero S. , Payne L. E. 
Published: November 2011

SynopsisWe consider an anisotropic non-homogeneous linear elastic material in equilibrium and occupying an open region with non-compact boundary. In both the linearised and classical linear theories the asymptotic behaviour of the solution is determined and a clear relationship established with Saint-Venant's principle on such regions. Although the treatment is discussed with special reference to elasticity, it is equally applicable to general systems of elliptic differential equations, and thus reveals a relationship with the classical theorems of Phragmèn-Lindelöf and Liouville.
Whitney-sums (fibre-joins) in over space theory and obstruction theory for cohomology with local coefficients
Authors: Rutter John W. 
Published: November 2011

SynopsisThe generalised Whitney sum (fibre-join) and the h-fibre-join can be defined in topM, the category of spaces over M. We note here some general properties of these constructions, and, as a specific example, we consider the relation between them and the extensions to the topM category of the top h-fibre-sequences F∗ΩB→E ∪ CF→B determined by top fibrations F→E→B. As an application we obtain the truncated local coefficient cohomology sequence for a top fibration which is topM principal fibration: this situation applies, for example, to the various stages of the Postnikov decomposition of a non-simply connected space X, and in this case we have M = K1(π1(X)).
Flows on centre manifolds for scalar functional differential equations
Authors: Hale Jack K. 
Published: November 2011

SynopsisBy assuming that a linear scalar functional differential equation (FDE) has only the zero eigenvalue on the imaginary axis, it is shown that the flows on the centre manifolds of all Cr-perturbations of this equation coincide with the flows obtained from scalar ordinary differential equations (ODEs) of order m, where m is the multiplicity of the zero eigenvalue. Furthermore, it is shown that the above situation can be realized through differential difference equations with m – 1 fixed distinct delays.
A global branch of solutions to a semilinear equation on an unbounded interval
Authors: Stuart C. A. 
Published: November 2011

SynopsisFor a semilinear second order differential equation on (0, ∞), conditions are given for the bifurcation and asymptotic bifurcation in Lp of solutions to the Neumann problem. Bifurcation occurs at the lowest point of the spectrum of the linearised problem. Under stronger hypotheses, there is a global branch of solutions. These results imply similar conclusions for the same equation on R with appropriate symmetry.
On the strong limit-point and Dirichlet properties of second order differential expressions
Authors: Race D. 
Published: November 2011

SynopsisSecond order differential expressions of the form w−1(−(pf′)′ + qf) are considered at a singular end-point. Some of the known relationships between properties such as Dirichlet, weak Dirichlet and strong limit-point, are extended to incorporate an arbitrary, positive weight function and complexvalued coefficients.
On the existence of eigenvalues for some nonlinear elliptic and hyperbolic problems
Authors: Basile Nicola , Mininni Michele 
Published: November 2011

SynopsisIn this paper some eigenvalue problems for elliptic as well as hyperbolic equations are solved. The main tool used is an abstract critical point theorem on an unbounded manifold of the form {u | (Lu, u) = constant} (where L is a nonpositive selfadjoint operator), which makes use of a linking type argument on a manifold.
The existence of infinitely many bifurcating branches
Authors: Ruppen Hans-Jörg 
Published: November 2011

SynopsisWe consider the non-linear problem −Δu(x)−f(x, u(x)) = λu(x) for x ∈ℝN and u ∈ W1,2(ℝN). We show that, under suitable conditions on f, there exist infinitely many branches all bifurcating from the lowest point of the continuous spectrum λ = 0. The method used in the proof is based on a theorem of Ljusternik-Schnirelman type for the free case.
The resolvent problem for the Stokes system in exterior domains: uniqueness and non-regularity in Hölder spaces
Authors: Deuring Paul 
Published: November 2011

SynopsisWe consider the resolvent problem for the Stokes system in exterior domains, under Dirichlet boundary conditions:where Ω is a bounded domain in ℝ3. It will be shown that in general there is no constant C > 0 such that for with , div u = 0, and for with . However, if a solution (u, π) of problem (*) exists, it is uniquely determined, provided that u(x) and ∇π(x) decay for large values of |x|. These assertions imply a non-existence result in Hölder spaces.
The -radical in structural matrix near rings, II
Authors: van Wyk Leon 
Published: November 2011

SynopsisWhether the -radical of a structural matrix near-ring (B, R) is the sum of two non-trivial ideals, one of which is nilpotent, is an open problem. However, it is known that ((B, R)) contains two ideals and , which are respectively precisely the two ideals, the sum of which is the Jacobson radical, in the case where the underlying near-ring is a ring. We strengthen our conjecture that and are the sought-after ideals by showing that (B, R)/≅(C, R) in the near-ring case, where C is the largest symmetric Boolean matrix such that C≦B, and by showing that is nilpotent.
The exact coupled bounds for effective tensors of electrical and magnetic properties of two-component two-dimensional composites
Authors: Cherkaev A. V. , Gibiansky L. V. 
Published: November 2011

SynopsisThis paper describes the effective properties of a two-dimensional two-component dielectric composite. Each component is supposed to be isotropic and is characterised by a pair of constants of electrical and magnetic permeabilities. Effective properties of an arbitrary anisotropic composite depend on its microstructure and are characterised by a pair of symmetric tensors of electrical and magnetic permeabilities. Here we construct the so-called Gm-closure which is the set of all possible values of these pairs, corresponding to all possible microstructures with prescribed volume fractions of the given components. The exact bounds of the Gm-closure are obtained and the microstructure corresponding to each point of the Gm-closure set is determined.
Symmetry properties of positive solutions of elliptic equations in an infinite sectorial cone
Authors: Chu Chie-Ping , Wang Hwai-Chiuan 
Published: November 2011

SynopsisWe prove symmetry properties of positive solutions of semilinear elliptic equations Δu + f(u) = 0 with Neumann boundary conditions in an infinite sectorial cone. We establish that any positive solution u of such equations in an infinite sectorial cone ∑α in ℝ3 is spherically symmetric if the amplitude α of ∑α is not greater than π.
Existence of multiple positive solutions for a semilinear equation with critical exponent
Authors: Yinbing Deng 
Published: November 2011

SynopsisIn this paper we discuss the problemWe show that for c > 0 there exists a positive constant μ* such that (*)μ possesses at least one solution if μ ∈ (0, μ*) and no solutions if μ > μ*. Furthermore, there exists a positive constant μ**≦μ* such that (*)μ possesses at least two solutions if μ∈(0, μ**), 2<N<6. For N≧6, μ∈(0,μ**), we show that problem (*)μ possesses a unique solution if f(x) is radial with f′(r) < 0(r = |x|).
Water waves for small surface tension: an approach via normal form
Authors: Iooss Gérard , Kirchgässner Klaus 
Published: November 2011

SynopsisIn this paper we determine the possible crest-forms of permanent waves of small amplitude which exist on the free surface of a two-dimensional fluid layer under the influence of gravity and surface tension when the Froude number is close to 1. The Bond number b, measuring surface tension, is assumed to satisfy b < ⅓. We find one-parameter families of periodic waves of two different types, quasiperiodic waves and solitary waves with oscillations at infinity. The existence of true solitary waves is established for a sequence of systems approximating the full Euler equations in every algebraic order of − 1.
The existence of a class of cosmic string solutions with infinite energy
Authors: Yang Yisong 
Published: November 2011

SynopsisRecent studies have indicated that in a certain critical coupling phase the Einstein–matter–gauge equations may be reduced to a Bogomol'nyi–Einstein system and the solutions are cosmic strings with finite separation. In this paper, we view the spacetime as known and treat the dynamics as being determined by the matter–gauge sector in an asymptotically Euclidean geometry. It is shown that the model admits a continuous family of infinite energy solutions which decay to the symmetric vacuum faster than any exponential functions.