bluuebunny/crossref_metadata_2025_split huggingface dataset:
Spectral properties of two parameter eigenvalue problems II
Authors: Binding Paul , Browne Patrick J. 
Published: November 2011

SynopsisWe 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

SynopsisWe 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

SynopsisWe 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.
Equisummability for linear operators in Banach spaces
Authors: Kon M. A. , Ramm A. G. , Raphael L. A. 
Published: November 2011

SynopsisLet 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

SynopsisIn 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

SynopsisWe 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

SynopsisIn 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

SynopsisWe study the bifurcation diagram and uniqueness of solutions ofBy 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

SynopsisTheorems 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

SynopsisAny 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

SynopsisLet 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

SynopsisA 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.
Boundary regularity for linear and quasilinear variational inequalities
Authors: Lieberman Gary M. 
Published: November 2011

SynopsisA 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.
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

SynopsisLimit-point and limit-circle criteria are given for the generalised Sturm-Liouville differential expressionwhere(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.
Bifurcation from a homoclinic orbit in parabolic differential equations
Authors: Blázquez C. Miguel 
Published: November 2011

SynopsisThis 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

SynopsisA 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

SynopsisIn 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.
The set of positive solutions of semilinear equations in large balls
Authors: Gardner R. , Peletier L. A. 
Published: November 2011

SynopsisThe 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 periodic solutions of nonlinear second order vector differential equations
Authors: Habets P. , Nkashama M. N. 
Published: November 2011

SynopsisThis paper considers existence of periodic solutions for vector Liénard differential equationsIn our main result we writewhere 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 equationthe assumptions are weakened.Our approach is based on Leray-Schauder's degree theory and a priori estimates.
Growth, motion and 1-parameter families of symmetry sets
Authors: Bruce J. W. , Giblin P. J. 
Published: November 2011

SynopsisAssociated 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

SynopsisUpper 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

SynopsisVarious 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

SynopsisWe study the initial value problem for the nonlinear Schrödinger equationUnder 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.
Minimizing properties of arbitrary solutions to the Ginzburg–Landau equation
Authors: Comte M. , Mironescu P. 
Published: November 2011

We consider the Ginzburg–Landau type equationwhereGis a smooth bounded domain in ℝ2,g ∊ C∞(∂G;ℝ2/ {0}), andε> 0 is a small parameter. We prove the uniqueness of solutions to this equation under some non-vanishing assumptions onuε, or under conditions on the boundary functiong.
The wave equation, O(2, 2), and separation of variables on hyperboloids
Authors: Kalnins E. G. , Miller W. 
Published: November 2011

SynopsisWe classify group-theoretically all separable coordinate systems for the eigenvalue equation of the Laplace-Beltrami operator on the hyperboloid = 1, finding 71 orthogonal and 3 non-orthogonal systems. For a number of cases the explicit spectral resolutions are worked out. We show that our results have application to the problem of separation of variables for the wave equation and to harmonic analysis on the hyperboloid and the group manifold SL(2, R). In particular, most past studies of SL(2, R) have employed only 6 of the 74 coordinate systems in which the Casimir eigenvalue equation separates.
Transformations of second order ordinary and partial difierential operators
Authors: Ahlbrandt Calvin D. , Hinton Don B. , Lewis Roger T. 
Published: November 2011

SynopsisLiouville type transformations are given for symmetric linear ordinary and partial differential operators of second order. Explicit formulas are given for the coefficients of the transformed operators. As a corollary to the general theory we obtain an “Atkinson form” for certain first order vector partial differential operators. This leads to a generalization of the concept of “g-unitary” transformations. Applications to oscillation and spectral theories are included.
On the range and inversion of fractional integrals in weighted spaces
Authors: Andersen Kenneth F. 
Published: November 2011

SynopsisThe weight functions w(x) for which the Riemann fractional integral operator Iα is bounded from the Lebesgue space Lp(wp) into Lq(wq), l/q = l/p −, have been characterized by Muckenhoupt and Wheeden. In this paper, we prove an inversion formula for Iα in the context of these weighted spaces and we also characterize the range of Iα as a subset of Lq(wq) Similar results are proved for other fractional integrals. These results may be viewed as weighted analogues of certain results of Stein and Zygmund, Herson and Heywood, Heywood, and Kober who considered the unweighted case, w(x) = l.
On the essential spectra of linear 2nth order differential operators with complex coefficients
Authors: Race David 
Published: November 2011

SynopsisIn this paper, a formally J-symmetric, linear differential expression of 2nth order, with complex-valued coefficients, is considered. A number of results concerning the location of the essential spectrum of associated operators are obtained. These are extensions of earlier work dealing with complex Strum-Liouville operators, and include results which, in the real case, are due to Birman, Glazman and others. They lead to criteria, for the non-emptiness of the regularity field, of the corresponding minimal operator-a condition which is needed in the theory of J-selfadjoint extensions.
Existence, uniqueness and asymptotic expansion of the solution of a multipoint boundary-value problem with parameters for a singularly-perturbed system of integro-differential equations
Authors: Hristova S. G. , Bainov D. D. 
Published: November 2011

SynopsisAn asymptotic expansion in powers of the small parameter of the solution of a singularly-perturbed system of integro-differential equations with a non-linear boundary condition set at several points of the interval considered is constructed and verified.
Extensions of states of C* -algebras, II
Authors: Archbold R. J. , Bunce J. W. , Gregson K. D. 
Published: November 2011

SynopsisLet A be a unital C*-algebra and let B be an abelian C*-subalgebra containing the identity of A. For any pure state h of B let Fh be the set of states of A which restrict to h on B. Necessary and sufficient conditions are given for an element x in A to have the property that, for each h, x is unable to distinguish between distinct elements of Fh. By specializing, this leads to a new proof of a theorem giving necessary and sufficient conditions for Fh to be a singleton for each h.It is also shown that if A is postliminal and π(B) is a maximal abelian C*-subalgebra of π(B) for each irreducible representation π of A then Fh is a Choquet simplex for each h.
A non-linear Goursat problem for a 4th order polyvibrating equation
Authors: Borzymowski Andrzej 
Published: November 2011

SynopsisIn the paper the existence is proved of a solution of a non-linear Goursat problem for a 4th order partial differential equation with the boundary conditions given on four curves emanating from a common point. The problem is reduced to a system of integro-functional equations and then Schauder's fixed point theorem is applied.
Existence and uniqueness for two-point boundary value problems
Authors: Baxley John V. , Brown Sarah E. 
Published: November 2011

SynopsisBoundary value problems associated with y″ = f(x, y, y′) for 0 ≦ x ≦ 1 are considered. Using techniques based on the shooting method, conditions are given on f(x, y,y′) which guarantee the existence on [0, 1] of solutions of some initial value problems. Working within the class of such solutions, conditions are then given on nonlinear boundary conditions of the form g(y(0), y′(0)) = 0, h(y(0), y′(0), y(1), y′(1)) = 0 which guarantee the existence of a unique solution of the resulting boundary value problem.
Superassociative systems with given semigroup of inner right translations
Authors: Länger H. 
Published: November 2011

SynopsisLetnbe some fixed positive integer and let (A,f) be some fixed algebra of typen+ 1. (A,f) is called ann-dimensional superassociative system iff(f(x0,…,xn), ȳ) =f(x0,f(x1, ȳ), …,f(xn, ȳ)) for anyx0, …,xn∈Aand for any ȳ ∈An. The semigroup ({f(., ā) | ā ∈An},∘) is called the semigroup of inner right translations of (A,f). In the present note a theorem is derived in order to determine alln-dimensional superassociative systems with a given semigroup of inner right translations. As an example, using this method all two-element superassociative systems are determined.
A lower bound for electrostatic capacity in the plane
Authors: Fraenkel L. E. 
Published: November 2011

SynopsisThis note presents a lower bound, in terms of the diameter ratio of the inner and outer conductors, for the electrostatic capacity of certain two-dimensional condensers. We use double Steiner symmetrization to prove that the minimizing condenser consists of a line segment placed symmetrically within a circle; the capacity of this condenser is known explicitly.
Completely simple semigroups: free products, free semigroups and varieties
Authors: Jones P. R. 
Published: November 2011

SynopsisThe class CS of completely simple semigroups forms a variety under the operations of multiplication and inversion (x−1 being the inverse of x in its ℋ-class). We determine a Rees matrix representation of the CS-free product of an arbitrary family of completely simple semigroups and deduce a description of the free completely simple semigroups, whose existence was proved by McAlister in 1968 and whose structure was first given by Clifford in 1979. From this a description of the lattice of varieties of completely simple semigroups is given in terms of certain subgroups of a free group of countable rank. Whilst not providing a “list” of identities on completely simple semigroups it does enable us to deduce, for instance, the description of all varieties of completely simple semigroups with abelian subgroups given by Rasin in 1979. It also enables us to describe the maximal subgroups of the “free” idempotent-generated completely simple semigroups T(α, β) denned by Eberhart et al. in 1973 and to show in general the maximal subgroups of the “V-free” semigroups of this type (which we define) need not be free in any variety of groups.
Estimates on the support of the solutions of some nonlinear elliptic and parabolic problems
Authors: Díaz J. I. , Herrero M. A. 
Published: November 2011

SynopsisWe obtain existence and uniqueness of solutions with compact support for some nonlinear elliptic and parabolic problems including the equations of one-dimensional motion of a non-newtonian fluid. Precise estimates for the support of these solutions are obtained, and the optimality of our hypotheses is discussed.
Integrable-square solutions of a singular ordinary differential equation
Authors: Gilbert Richard C. 
Published: November 2011

SynopsisAbsolutely square integrable solutions are determined for the equation= λywhere the ζn−r(x) are holomorphic in a sector of the complex plane and have asymptotic expansions asxapproaches infinity. It is shown that the number of such solutions depends upon the roots of the characteristic equation and their multiplicity, and upon the sign of the derivative of the characteristic polynomial. Application is made to formally symmetric ordinary differential operators.
An analogue of Langton's theorem on valuative criteria for vector bundles
Authors: Mehta V. B. , Ramanathan A. 
Published: November 2011

SynopsisWe prove in this paper that on a non-singular projective variety, the χ-semistable functor is proper and the χ-stable functor is separated. This result was proved for μ-stability and μ-semistability by Langton. An essential part of our proof consists in defining a notion of stability between the μ and χ definitions and then proceeding by induction.