bluuebunny/crossref_metadata_2025 huggingface dataset:
Mutually conjugate solutions of formally self-adjoint differential equations
Authors: Hill J. M. , Nillsen R. V.
Published: November 2011

Synopsis

Let L be a formally self-adjoint linear differential operator of order m with strictly positive leading coefficient and let m = 2n + 1 if m is odd, m = 2n if m is even. Let y1, y2,…, yn be n given mutually conjugate solutions of Ly = 0 on I, where I is some interval, whose Wronskian is non-zero on I. Then L = (−1)nQ*Q or L = (−1)nQ*DQ where Q is a differential operator of order n, Q* is the adjoint operator and D denotes differentiation. This fact is used to construct further solutions yn+1,−, ym of Ly = 0 so that y1,…, ym is a basis for the solutions of Ly = 0 and for which yi and yn+j are mutually conjugate if i ≠ j. If y1 ≠ 0 on I the degree of L may be lowered by 2 to obtain a formally self-adjoint operator L1 for which mutually conjugate solutions are constructed. If this process is continued a factorization result is obtained which is related to a result of Pólya.
Two series inequalities
Authors: Copson E. T.
Published: November 2011

Synopsis

In 1932, Hardy and Littlewood proved the inequalities

The best possible values of the constants K1 and K2 being 1 and 4 respectively. The object of this paper is to prove analogous results for infinite series in which the derivative of the real function f is replaced by the finite difference

of a sequence {an} of real numbers.
On the possibility of defining the Chapman–Kolmogorov semi-group on L∞
Authors: Devinatz Allen , Malliavin Paul
Published: November 2011

Synopsis

If the diffusion matrix coefficient of an Itô stochastic differential equation is everywhere non-singular, then the corresponding Chapman-Kolmogorov semi-group may be defined on L∼(Rn), the space of Lebesgue equivalence classes of essentially bounded Borei measurable functions. However, if the diffusion matrix is singular at some points of Rn, it is not clear that this can always be done. We show that in certain situations it is possible to do so.
Nodal cubic surfaces
Authors: Edge W. L.
Published: November 2011

Synopsis

The cubic surfaces in, save for the elliptic cone, are, whatever their singularities, projections of del Pezzo's non-singular surface F, of order 9 in. It is explained how, merely by specifying the geometrical relation of the vertex of projection to F, each cubic surface is obtainable “at a stroke”, without using spaces of intermediate dimensions.
On a class of systems of singular integral equations on a bounded domain
Authors: Džuraev A.
Published: November 2011

Synopsis

A class of systems of two-dimensional singular integral equations with even kernels over bounded domains in the plane is studied. The applications include integral equations with the Bergman kernel function. The method of investigation is the following: the integral equation is reduced to a Riemann type boundary value problem for a first order elliptic system. This is solved by means of one-dimensional singular integral equations over the boundary curve. An adjoint problem is formulated, the Noetherian theorems are established, and a formula for the index is given.
Some ergodic theorems
Authors: Dunford Nelson
Published: November 2011

Synopsis

A general ergodic theorem is proved for semi-group operators onB-spaceX. In particularXmay be a Lebesgue spaceLp(S, Σ, μ) where (S, Σ, μ) is a positive measure space.

The discussion is based on the theory of semi-groups as developed by Hille [6] and results in the theory of product measures [3]. The reader need only be familiar with the basic concepts of these theories, as all pertinent results used in this note are proved as they are needed.
On the variation of Teichmüller's metric
Authors: Gardiner Frederick P.
Published: November 2011

Synopsis

The main result of this article is the calculation of the first derivative of Teichmüller's metric from an inequality of Reich and Strebel. Furthermore, from the same inequality one is able to calculate information about the difference quotient for the second derivative. From the techniques used here it does not seem possible to determine whether the metric isC2.
Instability results for fourth order differential equations
Authors: Skrapek W. A.
Published: November 2011

SYNOPSIS

In this paper we give sufficient conditions (Theorems 1 and 2) for the instability of the fourth order differential equation
On the point spectra of complex Sturm—Liouville operators
Authors: Knowles Ian , Race David
Published: November 2011

Synopsis

In this paper, the formally J-symmetric Sturm—Liouville operator with complex-valued coefficients is considered, and as a preliminary two criteria are established for the J-selfadjointness of certain of its extensions. The main purpose of the work is to give results on the location of the point spectra of such operators. Some of the results are extensions of results known for real-valued coefficients, whilst others are new.
Baer semigroup coordinatisations of distributive lattices
Authors: Blyth T. S.
Published: November 2011

Synopsis

In a previous publication we investigated certain idempotent residuated mappings and showed how these could be used to provide a solution to the problem of finding a Baer semigroup coordinatisation of bounded modular lattices. Here we use essentially the same idempotents to provide a coordinatisation of bounded distributive lattices. Specifically, we prove that a bounded lattice L is distributive if and only if it can be coordinatised by a Baer semigroup S such that if eS, fS, gS ∈ R(S) with eS ∩ fS = eS ∩ gS then there are idempotents ē, , ∈ S such that ēS = eS, S = fS, S = gS and ē commutes with both and .
Periodic solutions of a nonlinear wave equation
Authors: Bahri Abbas , Brezis Haïm
Published: November 2011

Synopsis

We provide a sufficient and “almost” necessary condition for the existence of a periodic solution of the equation

whereFis nondecreasing inuand has a small linear growth as |u|→∞.
Genetic algebras for some inbreeding systems
Authors: Holgate P.
Published: November 2011

Synopsis

The paper deals with genetic algebras for populations in which given fractions of the population practise specified types of inbreeding. It is shown that the technique of linearising the quadratic transformation of genetic type frequencies, which has been studied for triangular algebras, can be extended to a much wider class, including those for inbreeding systems.
Differential Operators of infinite order
Authors: Hille Einar
Published: November 2011

Synopsis

The differential operators in question are of the form G(DZ) where G(w)is an entire function of order at most 1/n and minimal type while Dz is a linear differential operator of order n with coefficients which are entire ( = integral) functions of z, usually polynomials. This class of operators form a natural generalization of the class G(d/dz) studied during the first half of the century Muggli, Polya, Ritt and others. The class G(DZ) was introduced by the present author and his pupils in the 1940s. In fact, the present paper is partly based on a MS from that period, mostly devoted to the special case

but also containing generalizations, some of which were later worked out by Klimczak. A basic tool in this paper is the characteristic series

Examples are given showing that the domain of absolute convergence of such a series need neither be convex nor of finite connectivity, a question which has puzzled the author for forty odd years. Characteristic series arising from regular or singular boundary value problems for the operator Dz are used to study the inversion problem

for given F(z). In particular it is shown that exp (Dx)[W(z)] = 0 has the unique solution W(z) ≡ 0. Some singular boundary value problems are considered briefly.
Proof of a conjecture of Offord
Authors: Erdös Paul
Published: November 2011

Synopsis

If z1, z2 … zn are complex numbers satisfying |zi−zj|≧1 for all i, j then the number of the 2n sums where ει = ±1, which lie in any circle of radius r cannot exceed αr2n/n3/2 where αr depends only on r.
From B-completeness to countable codimensional subspaces via the closed graph theorem
Authors: Saiflu H. , Tweddle I.
Published: November 2011

Synopsis

We show that for some closed graph theorems each countable codimensional subspace of a domain space may also serve as a domain space. This provides a general principle from which we are able to extract some of the known results on the inheritance of topological vector space properties by subspaces of countable codimension. We make use of a result of Savgulidze and Smoljanov on B-completeness for which we provide a new and simpler proof.
Liouville theorems for a class of fourth order elliptic equations
Authors: Goyal Vinod B. , Schaefer Philip W.
Published: November 2011

Synopsis

Liouville type theorems are obtained for bounded entire solutions of equations of the form Δ2u − q(x)Δu + p(x)u = 0 by means of subharmonic functionals and Green type inequalities.
Reflection of singularities of solutions to the wave equation and the leading singularity of the scattering kernel
Authors: Alber H. D.
Published: November 2011

Synopsis

In this paper the reflection of singularities of solutions to the wave equation is studied using progressing wave expansions. The results are used to calculate the leading singularity of the Lax–Phillips scattering kernel, thus generalizing earlier results of Majda.
m (λ)-functions for complex Sturm-Liouville operators
Authors: Race David
Published: November 2011

Synopsis

In this paper, the formally J-symmetric Sturm-Liouville operator with complex-valued coefficients is considered and a generalisation of the Weyl limit-point, limit-circle dichotomy is sought by means of m (λ )-functions. These functions are then used to give an explicit description of all the associated J-selfadjoint operators with separated boundary conditions in the limit-circle case. A formulation of the eigenvalues of these operators, and a characterisation of which extensions are non-well-posed, are also found. Finally, the limit-point case is studied, mainly by means of an example.
B*-algebra eigenfunction expansions
Authors: Lopez Manuel
Published: November 2011

Synopsis

In this paper the Sturm-Liouville regular boundary value problem and expansion theorem is extended to functions with values in some B*-algebras.
Singular eigenvalue problems with eigenvalue parameter contained in the boundary conditions
Authors: Fulton Charles T.
Published: November 2011

Synopsis

In this paper I extend the analysis of regular problems containing the eigenvalue parameter in the boundary conditions given by Walter (1973) and myself (1977) to singular problems which involve the eigenvalue parameter linearly in a regular or a limit-circle boundary condition at the left endpoint. The formulation of the limit-circle boundary conditions follows that given in another paper by the present author in 1977, and has the advantage that a λ-dependent boundary condition at a regular endpoint becomes a special case of a λ-dependent boundary condition at a limit-circle endpoint. The simplicity of the spectrum is also built into the formulation given, and the spectral function is shown to have bounded total variation over (−∞, ∞) which is known in terms of the parameters of the λ-dependent boundary condition independently of the limit-circle/limit-point classification at the right endpoint. The theory is applied to the constant coefficient equation in [0, ∞) and the Bessel equation of order zero in (0, ∞), explicit formulae for the spectral function being obtained in each case. Finally, the question is posed as to whether the classical Weyl theory for problems not involving λ in the boundary conditions can also be formulated so as to involve spectral functions having bounded total variation.
A semilinear elliptic eigenvalue problem, II. The plasma problem
Authors: Keady Grant , Norbury John
Published: November 2011

Synopsis

This paper continues the study of the boundary value problem, for (λ, ψ)

Here δ denotes the Laplacian,kis a given positive constant, and λ1will denote the first eigenvalue for the Dirichlet problem for −δ on Ω. For λ ≦ λ1, the only solutions are those with ψ = 0. Throughout we will be interested in solutions (λ, ψ) with λ > λ1and with ψ > 0 in Ω.

In the special case Ω =B(0,R) there is a branch ℱe, of explicit exact solutions which bifurcate from infinity at λ = λ1and for which the following conclusions are valid, (a) The setAψ,

is simply-connected, (b) Along ℱe, ψm→k, ‖ψ‖1→ 0 and the diameter ofAψtends to zero as λ → ∞, where

Here it is shown that the above conclusions hold for other choices of Ω, and in particular, for Ω = (−a, a)×(−b, b). (Existence is settled in Part I, and elsewhere.)

The results of numerical and asymptotic calculations when Ω = (−a, a)×(−b, b) are given to illustrate both the above, and some limitations in the conclusions of our analysis.
An index theorem and Bendixson's negative criterion for certain differential equations of higher dimension
Authors: Smith Russell A.
Published: November 2011

Synopsis

Poincaré's theorem on the sum of the indices of a plane autonomous differential equation at its critical points inside a periodic orbit is here extended to the periodic orbits of higher-dimensional equations under certain conditions. Under the same conditions, higher-dimensional extensions are obtained of some theorems of Bendixson and others which exclude periodic orbits from regions in which the differential equation haspositive divergence. The application of these results to feedback control equations is also considered.
Naturally ordered regular semigroups with a greatest idempotent
Authors: Blyth T. S. , McFadden R.
Published: November 2011

Synopsis

We consider ordered regular semigroups in which the order extends the natural order on the idempotents, and which are graced with the presence of a greatest idempotent. This implies that every element has a greatest inverse. An investigation into the properties ofthese special elements allows a description of Green's relations on the subsemigroup generated by the idempotents. This in turn leads to a complete description of the structure of idempotent-generated naturally ordered regular semigroups having a greatest idempotent. The smallest such semigroup that is not orthodox is also described. These results lead us to obtain structure theorems in the general case with the added condition that Green's relations be regular. Finally, necessary and sufficient conditions for such a semigroup to be a Dubreil-Jacotin semigroup are found.
Necessary and sufficient conditions for asymptotic decay of oscillations in delayed functional equations
Authors: Chen Lu-San , Yeh Cheh-Chih
Published: November 2011

Synopsis

This paper studies the equation

where the differential operator Ln is defined by

and a necessary and sufficient condition that all oscillatory solutions of the above equation converge to zero asymptotically is presented. The results obtained extend and improve previous ones of Kusano and Onose, and Singh, even in the usual case where

where N is an integer with l≦N≦n–1.
Rank one plus a null-Lagrangian is an inherited property of two-dimensional compliance tensors under homogenisation
Authors: Grabovsky Yury , Milton Graeme W.
Published: November 2011

Assume that the local compliance tensor of an elastic composite in two space dimensions is equal to a rank-one tensor plus a null-Lagrangian (there is only one symmetric one in two dimensions). The purpose of this paper is to prove that the effective compliance tensor has the same representation: rank-one plus the null-Lagrangian. This statement generalises the wellknown result of Hill that a composite of isotropic phases with a common shear modulus is necessarily elastically isotropic and shares the same shear modulus. It also generalises the surprising discovery of Avellaneda et al. that under a certain condition on the pure crystal moduli the shear modulus of an isotropic polycrystal is uniquely determined. The present paper sheds light on this effect by placing it in a more general framework and using some elliptic PDE theory rather than the translation method. Our results allow us to calculate the polycrystalline G-closure of the special class of crystals under consideration. Our analysis is contrasted with a two-dimensional model problem for shape-memory polycrystals. We show that the two problems can be thought of as ‘elastic percolation’ problems, one elliptic, one hyperbolic.
Internal layers in high-dimensional domains
Authors: Sakamoto Kunimochi
Published: November 2011

For a system of semilinear elliptic partial differential equations with a small parameter, denned on a bounded multi-dimensional smooth domain, we show the existence of solutions with internal layers. The high-dimensionality of the domain gives rise to quite interesting an outlook in the analysis, dramatically different from that in one-dimensional settings. Our analysis indicates, in a certain situation, an occurrence of an infinite series of bifurcation phenomena accumulating as the small parameter goes to zero. We also present a related free boundary problem with a possible approach to its resolution.
Weighted Lorentz norm inequalities for general maximal operators associated with certain families of Borel measures
Authors: Gavilán María Dolores Sarrión
Published: November 2011

Given a certain family ℱ of positive Borel measures and γ ∈ [0, 1), we define a general onesided maximal operatorand we study weighted inequalities inLp,qspaces for these operators. Our results contain, as particular cases, the characterisation of weighted Lorentz norm inequalities for some well-known one-sided maximal operators such as the one-sided Hardy–Littlewood maximal operator associated with a general measure, the one-sided fractional maximal operatorand the maximal operatorassociated with the Cesèro-α averages.
Extension of a Theorem of Zygmund
Authors: Xiao Jie
Published: November 2011

We extend Zygmund's Theorem characterising the Bloch functions via a generalised Libera transform and so we answer an open problem formulated by N. Danikas, S. Ruscheweyh and A. Siskakis. Furthermore, we show some differences between the holomorphic Zygmund class and the class of holomorphic functions whose derivatives are of logarithmic growth on the unit disk.
The lattice of congruences on direct products of cyclic semigroups and certain other semigroups
Authors: Trueman D. C.
Published: November 2011

Synopsis

Let W be a semigroup with W\W2 non-empty, such that if ρ is a congruence on W with xpy for all x, y= W\W2, then zpw for all z, w= W2. We prove that the lattice of congruences on W is directly indecomposable, and conclude that a direct product of cyclic semigroups, with at least two non-group direct factors, has a directly indecomposable lattice of congruences. We find that the lattice of congruences on a direct product S1×S2×V of two non-trivial cyclic semigroups S1 and S2, one not being a group, and any other semigroup V, is not lower semimodular, and hence, not modular. We then prove that any finite ideal extension of a group by a nil semigroup has an upper semimodular lattice of congruences, and conclude that a finite direct product of finite cyclic semigroups has an upper semimodular lattice of congruences.
Proof of a conjecture of Race
Authors: Niessen Heinz-Dieter
Published: November 2011

Synopsis

The following conjecture of Race will be proved: if τ is a formally J-symmetric quasi-differential expression on a real interval I, such that for some λ = ℂ all solutions of τy = τy belong to L2(I), then λ belongs to the regularity field of the minimal operator To generated by τ in L2(l).
On a class of evolution problems
Authors: Fichera Gaetano
Published: November 2011

Synopsis

A class of evolution problems is investigated [see (1.5), (1.6), (1.7) of the present paper] which includes, as a particular case, an evolution problem previously considered by a different author. Existence and uniqueness theorems are given in several function spaces. It is shown that, when the solution is required to belong to spaces of smooth functions, the problem becomes overdetermined. The necessary and sufficient integro-differential equations, to be satisfied by the datum, for the existence of a smooth solution, are given.
Estimates for the state density for ordinary differential operators with white gaussian noise potential
Authors: Moscatelli V. B. , Thompson M.
Published: November 2011

Synopsis

The present paper is concerned with developing the existence and asymptotic properties of the state density N(λ) associated with certain higher order random ordinary differential operators A of the form

where Ao has homogeneous and ergodic coefficients with respect to the σ-algebra generated by the Wiener process q(ω, x). The analysis uses the Weyl min-max principle to determine rough upper and lower bounds for N(λ).
On a conjecture related to the number of solutions of a nonlinear Dirichlet problem
Authors: Lazer A. C. , McKenna P. J.
Published: November 2011

Synopsis

In an earlier paper (1981), the present authors made a conjecture about the number of solutions of a semilinear elliptic boundary value problem which has been investigated extensively in the past decade. The conjecture is proved in the one-dimensional case.
Lusin-type approximation of BD functions
Authors: Ebobisse François B.
Published: November 2011

The purpose of this paper is to establish a Lusin-type approximation of functions with bounded deformation by Lipschitz or C1 functions. The main ingredients inthe proof of our result are the maximal function of the measure Eu, the ‘Poincaré-type’ result by Kohn and the approximate symmetric differentiability of BD functions by Ambrosio and others.
Asymptotic solution for the perturbed Stokes problem in a bounded domain in two and three dimensions
Authors: Konate Dialla
Published: November 2011

We consider the Stokes problem with a small viscosity. When the viscosity goes to zero, the boundary-layer phenomenon can appear. In this case, the solution of the given perturbed Stokes equation cannot be properly approximated by the solution of its limiting equation ‘near’ the boundary Γ of the domain of study, say Ω To overcome this problem, we need to construct a corrector term in the neighbourhood of Γ Lions has studied this problem and has constructed a corrector for the case where Ω is a half space in ℝ2. The case where Ω is an open and bounded domain of ℝ2 or ℝ3, which remained unsolved, is the concern of this paper. The construction of the corrector to the perturbed Stokes equation depends heavily on the geometry of Ω In two dimensions, we construct the corrector in the form of a stream function, while in ℝ3 we construct it in the form of a potential vector. The corrector acts effectively in a neighbourhood of Γ that is the boundary layer. Using similar methods to those of Baranger and Tartar, we define the thickness of the boundary layer in a natural way. In addition, in this paper we study the behaviour of the corrected solution in some Hölder spaces.
Existence of multiple non-trivial solutions for nonlinear p-Laplacian problems on ℝN
Authors: Wu Aiwen
Published: November 2011

We shall prove multiplicity of non-trivial solutions for the following p-Laplacian problems

where f(x,u) is continuous, periodic in x and has superlinear growth in u.
Subdirectly irreducible double MS-algebras
Authors: Blyth T. S. , Varlet J. C.
Published: November 2011

Synopsis

We prove that there are 22 non-isomorphic subdirectly irreducible double MS-algebras and give a complete description of them.
The linear growth of localized disturbances in unstable compressible flows
Authors: Candler S. , Craik A. D. D.
Published: November 2011

Synopsis

Accurate computation of the evolution of initially localized disturbances in compressible parallel flows is a tedious task requiring superposition of a large number of Fourier modes with differing temporal growth rates. An alternative approximate method, similar to that developed by Craik (1981, 1982) for viscous incompressible flows, is presented here. This involves asymptotic evaluation, by the saddle point method, of a double Fourier integral representation of the disturbance, with the actual dispersion relation replaced by a simpler analytic expression containing several parameters which may be adjusted to approximate the flow under investigation. Limiting cases yield informative results in simple closed form: these exemplify the possible shapes into which the disturbance may evolve. In particular, ‘splitting’ of the disturbance into two dominant regions is demonstrated.
Uniqueness of positive solutions for a class of elliptic boundary value problems
Authors: Castro Alfonso , Shivaji Ratnasingham
Published: November 2011

Synopsis

Uniqueness of non-negative solutions conjectured in an earlier paper by Shivaji is proved. Our methods are independent of those of that paper, where the problem was considered only in a ball. Further, our results apply to a wider class of nonlinearities.
On the topological structure of the set of generalized solutions of the catenary problem
Authors: Alexander J. C. , Reeken Michael
Published: November 2011

Synopsis

The global topological structure of the space of configurations of a non-rotating elastic string under compression and tension is studied. The part of the string under tension is specified by a measurable subset of the interval. The set of such intervals, with the Hausdorff topology, is considered a parameter space for the equation satisfied by the string, and the solutions are shown to form an infinitedimensional continuum over this parameter space. A new global topological theorem is needed, since the parameter space is not Euclidean. The topological theorem is based on the fixed-point transfer.
On normal families and a result of Drasin
Authors: Langley J. K.
Published: November 2011

Synopsis

We prove the following: suppose that a and b are complex numbers, with a non-zero, and that n is an integer not less than 5. Then, if F is a family of functions meromorphic in a plane domain Dsuch that, for each f in F, the equation

has no solutions in D, then F is normal in D.
Large time estimates for solutions to the porous medium equation with nonintegrable data via comparison
Authors: Alikakos Nicholas D. , Rostamian Rouben
Published: November 2011

Synopsis

We consider the Cauchy problem for the porous medium equation in one space dimension, with initial data which are locally integrable. We measure the asymptotic behaviour of the initial data near infinity in an integral sense and relate this to the pointwise rate of growth or decay of solution for large time. The emphasis is on a novel comparison method wherein the initial data are rearranged on the ×-axis to form a sequence of Dirac δ-masses. By using the explicit solution in the latter case, we derive upper and lower bounds for the solution to the original problem by comparisons.
On the periodic solution of the Van der Pol oscillator with large damping
Authors: El-Abbasy E. M.
Published: November 2011

Synopsis

Littlewood showed that the forced Van der Pol oscillator with 0<b<⅔ and k large normally has subharmonic solutions of order 2n + l where n ≅ O([⅔−b]k). Numerical experiments suggest that n ≅ (⅔ –b)k/3 as k →∞. A refinement of Littlewood's calculation is given which leads to this result.
Asymptotic solutions to linear differential equations with coefficients of the power order of growth, 1
Authors: Lantsman M. H.
Published: November 2011

Synopsis

We consider a method for determining the asymptotic solution to a sufficiently wide class of ordinary linear homogeneous differential equations in a sector of a complex plane or of a Riemann surface for large values of the independent variable z. The main restriction of the method is the condition that the coefficients in the equation should be analytic and single-valued functions in the sector for | z | ≫ 1 possessing the power order of growth for |z| → ∞. In particular, the coefficients can be any powerlogarithmic functions. The equation

can be taken as a model equation. Here ai are complex numbers, aij are real numbers (i = 1,2,…, n; j = 0, 1, …, m) and ln1 Z≡ln z, lnsz= lnlnS−1z = S = 2, … It has been shown that the calculation of asymptotic representations for solution to any equation in the class considered may be reduced to the solution of some algebraic equations with constant coefficients by means of a simple and regular procedure. This method of asymptotic integration may be considered as an extension (to equations with variable coefficients) of the well known integration method for linear differential equations with constant coefficients. In this paper, we consider the main case when the set of all roots of the characteristic polynomial possesses the property of asymptotic separability.
Deficiency zero presentations for certain perfect groups
Authors: Campbell C. M. , Robertson E. F. , Kawamata T. , Miyamoto I. , Williams P. D.
Published: November 2011

Synopsis

In this paper we give two generator, two relation presentations for the following perfect groups which were not previously known to have deficiency zero: SL(2, 32), SL(2, 64), SL(2, 27), SL(2, 49), Â7, Ŝz(8), SL(2, 5) × SL(2, 5), SL(2, 5) × SL(2, 25). We also give two generator, two relation presentations for three other finite perfect groups, two having SL(2, 7) as an image and one having SL(2, 5) as an image. We also discuss presentations for certain other perfect groups which were known to have deficiency zero and find some neat new presentations for them.
Boundary conditions and the Cayley transform
Authors: Zahreddine Z.
Published: November 2011

Synopsis

Self-adjoint operators in L2(0, 1) associated with a formally symmetric differential operator regular in [0, 1] can be determined by boundary conditions or as extensions of the minimal operator. These extensions are determined by extensions of the Cayley transform of the minimal operator. This paper establishes an explicit expression for the extension of the Cayley transform in terms of the boundary conditions and vice versa.
The Steiner system S(5,8, 24) constructed from dual affine planes
Authors: Kadir G. A. , Key J. D.
Published: November 2011

Synopsis

We construct firstly a single tactical configuration which has the structure of the dual of the affine plane of order 4, and show how to obtain a further set of 3 such dual planes which, together with , satisfy a certain set of intersection properties. This set of 4 dual planes is used to extend the 20 points of to the Steiner system = S(5, 8, 24). The construction leads to the production of involutions of the type which fix the points of an octad. It is shown that 3 involutions each of this type suffice to generate M24, each of the simple Mathieu groups inside M24, the Todd group, and all the intransitive maximal subgroups of M24.
Dynamical modelling of phase transitions by means of viscoelasticity in many dimensions
Authors: Rybka Piotr
Published: November 2011

Synopsis

We study the equations of viscoelasticity in a multidimensional setting for the ‘no-traction’ boundary data. For the sake of modelling phase transitions we do not assume elliptieity of the stored energy function W. We construct dynamics in W1,2(Ωℝn) globally in time. Next, we study the question of stability for a class of equilibria. Moreover, we show a certain kind of decay in time of solutions for arbitrary initial conditions.
Positive perturbations of self-adjoint Schrödinger operators
Authors: Kappeler Th.
Published: November 2011

Synopsis

In this paper, we prove that a positive perturbation T = T0 + q (q ≧ 0 and in ) of an essentially self-adjoint Schrödinger operator T0 = −Δ + q0 on is again essentially self-adjoint if T is relatively bounded with respect to T0. An application of the method of the proof to positive approximations of elements u ≧ 0 in D(T) by a positive sequence in is given.
Bernstein's theorem for polynomial maps of complex topological vector spaces
Authors: Włodarczyk K.
Published: November 2011

Synopsis

The well-known theorem of Bernstein is extended to generalized polynomial maps of closed balanced convex sets in complex Hausdorff topological vector spaces.
Nonlinear problems with strong resonance at infinity: an abstract theorem and applications
Authors: Capozzi A. , Salvatore A.
Published: November 2011

Synopsis

In this paper, we consider the equation

where A is a linear operator, N = ψ′ with ψ ∈ C1(E, R), and E is an Hilbert space. We suppose that N has a derivative at infinity N′(∞) and that 0 belongs to the spectrum of A–N′(∞). We prove an abstract theorem for multiplicity of solutions for the above equation. We then apply this theorem to the study of periodic solutions of Hamiltonian systems and of semilinear wave equations when the period is prescribed.
On the sign of the solutions to some semilinear Dirichlet problems
Authors: Cafagna Vittorio , Donati Flavio
Published: November 2011

Synopsis

We study the sign of solutions to a class of semilinear Dirichlet problems when the nonlinearity is, for instance, a concave-convex function which interacts with the spectrum of the linear part. We are able to prove, in some cases, exact multiplicity results for positive and negative solutions. For the proofs we employ a device which splits the given problem into two others satisfying a suitable version of the Ambrosetti–Prodi result.
On the linearised theory of buckling for a compressed viscoelastic rod
Authors: Reynolds David W.
Published: November 2011

Synopsis

Results are obtained on the existence, multiplicity, blow-up and asymptotic behaviour of the solutions to the integro-differential equation

which arises in the linearised, quasi-static theory of buckling for a viscoelastic rod.
Extension and amalgamation in rings
Authors: Renshaw J.
Published: November 2011

Synopsis

It is proved that if a ringRhas theextension propertyin containing ringSi, then the amalgam [R; Si,] is strongly embeddable. Using a result of P. M. Cohn, a necessary and sufficient condition for a ring to be an amalgamation base is then given. It is also shown thatRis alevelsubring of a ringSif and only ifS/Ris flat. From this, a classical result of P. M. Cohn on flat amalgams is proved.
Singularities of the inverses of Fredholm operators
Authors: Ramm A. G.
Published: November 2011

Synopsis

Let B(k) be a linear bounded mapping of a Banach space X into a Banach space Y meromorphic in the parameter k on a connected domain of the complex plane. Under certain assumptions on B(k), more general than previously considered, the singularities of the inverse operator are described.
Bubble growth in porous media and Hele–Shaw cells
Authors: Howison S. D.
Published: November 2011

Synopsis

We consider the characterisation of a class of free boundary problems arising in the flow of a viscous liquid in a porous medium (or, in two dimensions, a Hele–Shaw cell). Injected air forms a bubble which grows as time increases; it is shown that three kinds of behaviour can occur. Firstly, the solution may cease to exist in finite time; secondly, the solution may exist for all time and the free boundary may have one or more limit points as t tends to infinity; and thirdly, the bubble may exist for all time and fill the whole space as t tends to infinity. Two-dimensional explicit examples arc given of all three types of behaviour, and it is proved that the only solutions of the third kind are those in which the bubble is always elliptical; the proof uses the theory of null quadrature domains. It is shown that solutions for ellipsoidal bubbles exist in three dimensions and it is conjectured that the only three-dimensional null quadrature domains with finite complement are those whose complement is an ellipsoid.
Ck centre unstable manifolds
Authors: Chow Shui-Nee , Lu Ke ning
Published: November 2011

Synopsis

We consider the existence and smoothness of global centre unstable manifolds for finite and infinite dimensional flows or maps. We show that every global centre unstable manifold can be expressed as a graph of a Ck map, provided that the nonlinearities are Ck smooth. The proofs are based on a lemma by D. Henry on a necessary and sufficient condition for a Lipschitz map to be continuously differentiable.
On the maximum of solutions for a semilinear elliptic problem
Authors: Sweers Guido
Published: November 2011

Synopsis

In this paper we study some properties of a semilinear elliptic eigenvalue problem with nondefinite right-hand side. In the first part we show that every solution will have its maximum in some specified interval J. If the domain is inside a cone in ℝN with N > 1, then J is strictly smaller than in the one-dimensional case. In the second part we show, for bounded domains, that if the maximum is inside some subinterval of J, then for any eigenvalue there will be at most one solution.
Parallelisability in Banach spaces: a parallelisable dynamical system with uniformly bounded trajectories
Authors: Garay B. M.
Published: November 2011

Synopsis

In the Banach space of real sequences which converge to zero with the supremum norm, we construct a parallelisable dynamical system with uniformly-bounded trajectories.
On Schur complements andn-metaharmonic functions
Authors: Nichols Edward C. , Smith Ronald L.
Published: November 2011

Synopsis

Several properties of symmetric matrices and positive definite matrices are derived. These are used to improve an estimate given for regular solutions of then-metaharmonic differential equation by Chow and Dunninger [1].
Maximal smoothness of solutions to certain Euler–Lagrange equations from nonlinear elasticity
Authors: Bauman Patricia , Phillips Daniel , Owen Nicholas C.
Published: November 2011

Synopsis

We investigate the maximal smoothness of stationary states for the multiple integral =

Such variational problems are motivated by the study of nonlinear elasticity. Assuming certain structure conditions for γ and given a stationary state , we derive an a priori LP estimate for for any p < ∞ in terms of and where . As a consequence, we show that a C1,β stationary state necessarily satisfies det and is of class C2, β in Ω. Nevertheless, singular stationary states do exist: we construct a nonsmooth C1 solution for a particular γ in two dimensions such that det in Ω and det vanishes at precisely one point in Ω.
Lower semicontinuity and continuity of functions of measures with respect to the strict convergence
Authors: Delladio S.
Published: November 2011

Synopsis

Let Ω be an open subset of Rn. It is well known that, given a suitable real-valued function f on Ω × Rk and a Rk -valued Borel measure µ on Ω, then one can define a real-valued measurefµ on Ω. The object of this note is to define the Ψ-strict convergence of the Rk-valued Borel measures µj to the Rk-valued Borel measure µ, where Ψ: Ω × Rk → [0, + ∞] is a continuous function which is positively homogeneous and convex in the Rk-variable, and to investigate the lower semicontinuity and continuity of the map µ → fμ with respect to the Ψ-strict convergence; here f is positively homogeneous in the Rk-variable and satisfies one suitable convexity condition (related to Ψ).
Ext spaces for general linear and symmetric groups
Authors: Martin Stuart
Published: November 2011

Synopsis

We consider the links between Ext1 groups of simple modules for the symmetric group, and Ext1 of simple modules for the general linear group.
Conical functions with one or both parameters large
Authors: Dunster T. M.
Published: November 2011

Synopsis

Uniform asymptotic expansions are derived for conical functions, Legendre functions of order µ and degree −½ + iτ, where µ and τ are non-negative real parameters. As τ → ∞, expansions are furnished for the conical functions which involve Bessel functions of order µ. These expansions are uniformly valid for 0 ≦ µ ≦ Aτ (A an arbitrary positive constant), and are also uniformly valid for Re (z) ≧ 0 in the complex argument case, and 0 ≦ z < ∞ in the real argument case. The case µ → ∞ is also considered, and expansions are furnished which are uniformly valid in the same z regions for 0 ≦ τ ≧ Bµ (B an arbitrary positive constant); in the cases where Re(z) ≧ 0 and 1 ≦ z < ∞, the expansions involve Bessel functions of purely imaginary order iτ, and in the case where 0 ≦ z < 1 the expansions involve elementary functions.
Topologically nilpotent Banach algebras and factorisation
Authors: Dixon P. G.
Published: November 2011

Synopsis

A Banach algebra A is said to be topologically nilpotent if sup {‖x1x2…xn‖1/n: xi ∈ A, ‖xi‖ ≦ 1 (1 ≦ i ≦ n)} tends to zero as n → ∞. A Banach algebra A is uniformly topologically nil if sup {‖xn‖ 1/n: x ∈ A, ‖x‖ ≦ 1} tends to zero as n → ∞. These notions are equivalent for commutative algebras and a topological version of the Nagata-Higman Theorem gives a partial result for the non-commutative case. Topologically nilpotent algebras have a strong non-factorisation property and this yields theorems of the type “factorisation implies the existence of arbitrarily slowly decreasing powers”. Extensions of topologically nilpotent algebras by topologically nilpotent algebras are topologically nilpotent.
The Signorini perturbation scheme in an abstract setting
Authors: Chillingworth D. R. J.
Published: November 2011

Synopsis

The Signorini perturbation scheme is a series expansion algorithm that locates solution branches for a class of symmetry-breaking bifurcation problems in nonlinear elastostatics. The relationship of the formal steps in the algorithm to geometric aspects of the problem is brought out in work of J. E. Marsden and Y.-H. Wan, where an abstract formulation is also considered. In this paper, the abstract algorithm and its geometry are explored further: the logical structure is clarified, and it is shown how the scheme adapts to the presence of additional symmetry constraints.
The group of homotopy self-equivalences of non-simply-connected spaces using Postnikov decompositions
Authors: Rutter J. W.
Published: November 2011

Synopsis

We give here a group extension sequence for calculating, for a non-simply-connected space X, the group of self-homotopy-equivalence classes which induce the identity automorphism of the fundamental group, that is the kernel of the representation → aut (π1(X)). This group extension sequence gives in terms of , where Xn is the n-th stage of a Postnikov decomposition. As special cases, we calculate for non-simply-connected spaces having at most two non-trivial homotopy groups, in dimensions 1 and n, as the unit group of a semigroup structure on ; and we calculate up to extension for non-simply-connected spaces having at most three non-trivial homotopy groups. The group is, for nice spaces, isomorphic to the groups and of self-homotopy-equivalence classes of X in the categories top*M and top M, respectively, where X→M = K1(π1(X)) is a top fibration which determines an isomorphism of the fundamental group; and our results are obtained initially in topM.
Smooth conjugacy of centre manifolds
Authors: Burchard Almut , Deng Bo , Lu Kening
Published: November 2011

Synopsis

In this paper, we prove that for a system of ordinary differential equations of classCr+1,1,r≧0 and two arbitrary Cr+1, 1local centre manifolds of a given equilibrium point, the equations when restricted to the centre manifolds areCrconjugate. The same result is proved for similinear parabolic equations. The method is based on the geometric theory of invariant foliations for centre-stable and centre-unstable manifolds.
A class of comparison theorems for non-linear hyperbolic initial value problems
Authors: Kreith Kurt
Published: November 2011

Synopsis

An exponential form of the Riccati transformation is used to establish zeros of solutions of a class of characteristic initial value problems.
A monotone method for a system of nonlinear parabolic differential equations
Authors: Chandra Jagdish , Dressel Francis G. , Norman Paul Dennis
Published: November 2011

Synopsis

A monotone iteration scheme for the solution of the initial boundary problems associated with a system of semilinear parabolic differential equations has been developed that does not require the nonlinearities to be quasimonotone. The class of equations to which this scheme applies includes physical models that describe combustion processes involving Arrhenius reaction terms.
On the structure of the set of solutions of certain holomorphic two-point boundary value problems
Authors: Hassan H. S. , Lloyd N. G.
Published: November 2011

Synopsis

Suppose that f: ℝ×ℂN→ℂN is holomorphic in z and continuous in t, and that Φ: ℂN×ℂN→ℂN is holomorphic. Boundary value problems of the form

are considered. The particular interest is in the structure and topological properties of the set of solutions. The paper is motivated by the corresponding properties of the set of periodic solutions of ż = f(t, z) when f is periodic in t. Consideration of this complex equation gives information about the periodic solutions of the real equation ẋ = f(t, x).
Finite groups with a certain centralizer of an element of order 3
Authors: Prince A. R.
Published: November 2011

Synopsis

A characterization of the symmetric group of degree 9 is given in terms of the centralizer of an element of order 3.
Global regularity and formation of singularities of solutions to first order quasilinear hyperbolic systems
Authors: Da-tsin(Li Ta-tsien) Lee
Published: November 2011

Synopsis

For the Cauchy problem for strictly hyperbolic systems with general eigenvalues, we obtain existence of global smooth solutions under certain conditions on the composition of the eigenvalues and the initial data; on the other hand, we give a sufficient condition which guarantees that singularities of the solution must occur in a finite time and describe certain applications. The present paper includes the corresponding results in earlier papers by several authors as special cases.
Orthogonal polynomials satisfying fourth order differential equations
Authors: Krall Allan M.
Published: November 2011

Synopsis

These polynomials, which are intimately connected with the Legendre, Laguerre and Jacobi polynomials, are orthogonal with respect to Stieltjes weight functions which are absolutely continuous on (− 1, 1), (0, ∞) and (0, 1), respectively, but which have jumps at some of the intervals' ends. Each set satisfies a fourth order differential equation of the form Ly = λny, where the coefficients of the operator L depends only upon the independent variable. The polynomials also have other properties, which are usually associated with the classical orthogonal polynomials.
A note on linear hyperbolic evolution equations
Authors: Grabmüller Hans
Published: November 2011

Synopsis

Given the linear hyperbolic evolution equation (P0) on a reflexive Banach space, we present a new method for an existence proof of unbounded solutions admitting an exponential growth rate as time tends to infinity. Utilizing abstract Wiener—Hopf techniques, an operational calculus is developed for the construction of the resolving operator associated with the problem under consideration. The results are based upon the fundamental hypothesis that the spectral set of the time-independent mapping A is contained in the interior of a parabola. The distance of the focus from the vertex of this parabola turns out to be a measure for the growth rate. Applicability of the results is shown in the case where A is a non-symmetric perturbation of a self-adjoint partial differential operator.
The assimilation of a strong, two-dimensional laminar jet into an aligned uniform stream
Authors: Wilks Graham , Hunt Roland
Published: November 2011

Synopsis

A comprehensive appraisal of the title problem is presented in terms of a characterizing nondimensional co-ordinate ξ which is based upon the half excess of the momentum of the jet, J. Perturbation features of the problem appear as regular and singular boundary conditions in ξ upstream and downstream respectively. The conservation of momentum excess provides a monitor on the consistency of regular and singular perturbation series solutions. In particular the conservation constraint on the downstream singular perturbation solution confirms the inadequacy of expansions in inverse half powers of ξ and justifies formally the introduction of logarithmic terms.

The formulation provides the basis for a complete numerical integration over the semi-infinite region. Accordingly detailed knowledge of velocity excess along the axis of the jet is obtained and an undetermined coefficient in the asymptotic downstream perturbation solution may be estimated.