bluuebunny/crossref_metadata_2025_split huggingface dataset:
The Hankel transform of some classes of generalized functions and connections with fractional integration
Authors: McBride Adam C. 
Published: November 2011

SynopsisIn previous papers [11,12,13], certain spaces of generalized functions were studied from the point of view of fractional calculus. In this paper, we show how a Hankel transform Hv of order v can be defined on for all complex numbers v except for those lying on a countable number of lines of the form Re v = constant in the complex v-plane. The mapping properties of Hv on are obtained. Various connections between Hv (or modifications of Hv) and operators of fractional integration are examined.
Direct summands inl-groups
Authors: Miller John Boris 
Published: November 2011

SynopsisWe discuss convexl-subgroups of anl-groupGin their role as direct summands, not so much ofGas of each other. This is done by writingA≥dBfor subgroupsA, Bto mean thatAis a direct summand ofB, and studying the properties of the resulting poset. It is shown to be a hypolattice, that is, to have local lattice properties in a certain sense. However it need not be a lattice; and there may exist meets of pairs of elements, outside the hypolattice structure. It need not be conditionally complete even whenGis conditionally complete. We look also at the map which sends a subgroup to its lattice-closure; the lattice-closed subgroups also form a hypolattice. Our main result asserts that this hypolattice is conditionally complete ifGis. The paper ends with some examples and counter examples inC(X).
A Bessel function inequality
Authors: Rawlins A. D. 
Published: November 2011

SynopsisThis paper proves some inequalities for the imaginary part of the transcendental function in a simply connected sector of the complex z-plane, where 0 < v < 1, and part of the boundary depends on v. These inequalities arose in a work of Everitt and Jones [1] which was on a general integral inequality. We give an alternative method of proving these Bessel function inequalities.
Deficiency zero groups involving Fibonacci and Lucas numbers
Authors: Campbell C. M. , Robertson E. F. 
Published: November 2011

SynopsisTwo closely related classes X(n) and Y(n) of two generator two relation groups are studied. The group presentations arise from an investigation of a Fibonacci type group of order 1512. The Reidemeister-Schreier algorithm is used to show that the groups X(n) are finite and not metabelian. The orders of these groups are determined and shown to be divisible by powers of Fibonacci numbers or by powers of Lucas numbers. In addition these groups add to the relatively few examples of non-metabelian two generator two relation groups whose orders are known precisely.
On the structure of morphism near-rings
Authors: Meldrum J. D. P. 
Published: November 2011

SynopsisThe connection between the structure of a near-ring and that of the group on which it acts is used to obtain results concerning the structure of near-rings. A generalized R series is defined for an R module, where R is a zero-symmetric left near-ring, and it is shown that all R modules have maximal R series. The idea of a near-ring which annihilates a series is introduced and some easy consequences of the definition are pointed out. Semi-primitive near-rings are introduced and a general structural result connecting the last two ideas is given. Some special cases which generalize earlier results on endomorphism near-rings are stated. Finally some of the limitations of the idea of semi-primitive near-rings are shown, and some applications are given, in particular to the endomorphism near-rings of soluble groups and of the symmetric groups.
Idempotent generators in finite full transformation semigroups
Authors: Howie J. M. 
Published: November 2011

SynopsisIt was proved by Howie in 1966 that , the semigroup of all singular mappings of a finite set X into itself, is generated by its idempotents. Implicit in the method of proof, though not formally stated, is the result that if |X| = n then the n(n – 1) idempotents whose range has cardinal n – 1 form a generating set for. Here it is shown that if n ≧ 3 then a minimal set M of idempotent generators for contains ½n(n–1) members. A formula is given for the number of distinct sets M.
The Schrödinger equation as a singular perturbation problem
Authors: de Jager E. M. , Küpper T. 
Published: November 2011

SynopsisComparisons have been made of the eigenvalues and the corresponding eigenfunctions of the eigenvalue problemsandwith φ ∈ C(-∞, +∞) and 0≦φ(x)≦C|x|i+1(1+|x|1), −∞<x<+∞ where i and l are arbitrary positive numbers with i≧2k≧2, k integer. In first approximation the eigenvalues λ and λ− and the corresponding eigenfunctions ψ and ψ are the same for ε→0; the error decreases whenever the exponent i increases.
A quasi-periodic boundary value problem for the Laplacian and the continuation of its resolvent
Authors: Alber H. D. 
Published: November 2011

SynopsisA quasi-periodic boundary value problem for the Helmholtz equation in an unbounded domain is considered. This problem arises from scattering of plane waves by periodic structures.Existence and uniqueness theorems are proved, and the continuation of the resolvent of this problem to a Riemannian surface is constructed. This construction makes no use of the continuation of the resolvent kernel but runs along the following lines:First a family of differential operators is defined, which is holomorphic in a generalized sense. Then, using a result from analytic perturbation theory about families of operators with compact resolvent, it is shown that the family of inverses of these differential operators gives the desired continuation.
Computing values of a function on [0,1] from its moments
Authors: Wimp Jet 
Published: November 2011

In this work we present an algorithm for computing an integrable function almost everywhere on (0,1) when its moments are known. The method is based on the use of certain delta-shaped sequences, and can be adjusted to take advantage of the local smoothness of the function.As an application, we give an algorithm for the pointwise inversion of the Laplace transform which utilizes the values of the image function at equidistant points.
Estimates of Liouville-Green type for solutions of systems of differential equations
Authors: Hadid S. B. 
Published: November 2011

SynopsisLet P(x) and Q(x) be n × n matrices whose entries involve a single given function q(x). Under suitable conditions, estimatesof Liouville-Green type are obtained for the components yν(x) of solutions of the systemIn the case n = 2, the estimate is applied to systems arising from fourth-order equations and a generalization of a result of Eastham [2] is obtained.
Green's matrix and the formula of Titchmarsh-Kodaira for singular left-definite canonical eigenvalue problems
Authors: Schultze Bernd 
Published: November 2011

SynopsisThe theory of singular left-definite canonical eigenvalue problems treated by Nieβen and Schneider in is generalized to arbitrary λ∈(ℂ\ℝ∪{0}. In this enlarged theory the Green's matrix of the problem is evaluated and a natural analogue of the Titchmarsh-Kodaira formula is proved. This formula permits the explicit computation of the spectral matrix playing the main role in the expansion theorems of this theory.
On a non-regular parametric integral
Authors: Hayman W. K. 
Published: November 2011

Suppose that f(t, λ) is, for fixed t in an open interval (a, b), a regular (or analytic or holomorphic) function of λ, when λ lies in a domain D of the complex plane. We assume that f is integrable with respect to t for fixed λ ∈ D, and consider the parametric integralBy hypothesis F(λ) exists for λ ∈ D and Professor W. N. Everitt has asked whether F(λ) is necessarily regular as a function of λ.
A star-shaped condition for stability of linear retarded functional differential equations†
Authors: Silkowski Richard 
Published: November 2011

SYNOPSISSufficient conditions are developed for asymptotic stability of the autonomous linear functional differential equation of retarded type. If the asymptotic stability ofimplies the asymptotic stability ofthen these conditions are also necessary. Necessary and sufficient conditions are developed for the largest cone in the region of stability. These results are illustrated with the example
Near-rings of mappings
Authors: Meldrum J. D. P. , Oswald A. 
Published: November 2011

SynopsisThis paper is concerned with the structure of M = Maps(G), the near-ring of all mappings from a group G to itself which commute with a group S* of automorphisms of G. Here S is S* together with the zero endomorphism. Necessary and sufficient conditions on the pair (G, S) are obtained for M to be (i) regular, (ii) unit regular, (iii) an equivalence near-ring. These conditions take a very simple form. In the case (iii), the two-sided M-subgroups of M are determined. The next result shows that under suitable conditions, M is a simple near-ring. A definition of transitivity is given for subnear-rings of M, and some properties of transitive near-rings are proved. Finally two examples are given to show that all the classes of near-rings considered are distinct.
On eigendistributions in linear transport theory
Authors: Lekkerkerker C. G. 
Published: November 2011

SynopsisAn attempt is made to provide a sound basis for the method of singular eigenfunction expansions which has been in vogue in linear transport theory for some decades. The procedure is exemplified by a treatment of the one-dimensional neutron transport equation with a degenerate scattering function. Full-range as well as half-range results are derived. At the end of the paper the implications for a certain matrix factorization problem are given.
On the asymptotic expansions of solutions of an nth order linear differential equation
Authors: Paris R. B. 
Published: November 2011

SynopsisThe asymptotic expansions of solutions of a class of linear ordinary differential equations of arbitrary order n are investigated for large values of the independent variable z in the complex plane. Solutions are expressed in terms of Mellin-Barnes integrals and their asymptotic expansions are subsequently determined by means of the asymptotic theory of integral functions of the hypergeometric type. Three classes of solutions are considered: (i) solutions whose behaviour is either exponentially large or algebraic for |z|→∞ in different sectors of the z-plane, (ii) solutions which are even and odd functions of z when the order n of the differential equation is even and (iii) solutions which are exponentially damped as |z|→∞ in a certain sector of the z-plane.
On uniform asymptotic expansions of finite Laplace and Fourier integrals
Authors: Soni Kusum 
Published: November 2011

SynopsisA uniform asymptotic expansion of the Laplace integrals ℒ(f, s) with explicit remainder terms is given. This expansion is valid in the whole complex s−plane. In particular, for s = −ix, it provides the Fourier integral expansion.
Complementary bivariational principles for linear problems involving non-self-adjoint operators
Authors: Cole R. J. 
Published: November 2011

SynopsisBivariational principles are constructed that yield upper and lower bounds to the quantity 〈g0, f〉, where f is the solution of the equation f0−Tf = 0, g0 is a given function and T is a non-self-adjoint linear operator from a Hilbert space into itself. The theory is illustrated by an integral equation of Fredholm type.
On the stability of self-adjointness of Schrödinger operators under positive perturbations
Authors: Cycon Hans L. 
Published: November 2011

SynopsisWe introduce a class of essentially self-adjoint Schrödinger operators, where essential self-adjointness is stable under positive potential perturbations. We show that this class is “stable” under certain perturbations and contains operators discussed by Simon and Kato. Finally, an extended essentially self-adjointness criterion is given.
Ramifications of Landau's inequality
Authors: Kwong Man Kam , Zettl A. 
Published: November 2011

SynopsisThe problem of determining the best constant κ in the inequality ‖y′‖≦K ‖y‖ ‖y″‖ is discussed in the context of the classical Lp spaces, 1 ≦ p ≦ ∞.
Whittaker functions with both parameters large: uniform approximations in terms of parabolic cylinder functions
Authors: Olver F. W. J. 
Published: November 2011

SynopsisAsymptotic approximations are derived for the Whittaker functions Wκ,μ (z), Mκ, μ (z), Wικ, ιμ (iz) and Mικ, ιμ(iZ) for large positive values of the parameter μ that are uniform with respect to unrestricted values of the argument z in the open interval (0, ∞), and bounded real values of the ratio κ/μ. The approximations are in terms of parabolic cylinder functions, and in most instances are accompanied by strict error bounds.The results are derived by application of a recently-developed asymptotic theory of second-order differential equations having coalescing turning points, and an extension of the general theory of equations of this kind is also included.
Products of idempotents in finite full transformation semigroups
Authors: Howie J. M. 
Published: November 2011

SynopsisIf |X| = n and α is a singular mapping in J(X), define c(α) to be the number of cyclic orbits of α and f(α) to be the number of fixed points. Then α is expressible as a product of n + c(α)−f(α) idempotents of rank n − 1, and no smaller number of idempotents of rank n − 1 will suffice. The maximum possible value of n + c(α)–f(α) is [3/2(n − 1)], which is thus a best possible global lower bound for the number of idempotents required to generate a singular element of J(X).
On regular solutions of a nonlinear equation of Choquard's type
Authors: Menzala Gustavo Perla 
Published: November 2011

SynopsisWe study the nonlinear equationin ℝ3, where Δ denotes the Laplacian operator, and R and K are real-valued functions satisfying suitable conditions. We use a variational formulation to show the existence of a non-trivial weak solution of the above equation for some real number λ. Because of our assumptions on R and K we shall look for solutions which are spherically symmetric, decrease with r = |x| and vanish at infinity.
On the non-existence of semi-groups for some equations of continuum mechanics
Authors: Wilkes N. S. 
Published: November 2011

SynopsisLinear semi-group theory can be used to prove the existence of solutions to the equations of linear elasticity when the elasticity tensor is positive definite. Here, it is shown that this condition is also necessary for the existence of a semi-group. The method is also applied to linear dissipative equations.
The solution of Lyapunov's matrix equation by a geometric method
Authors: Young N. J. 
Published: November 2011

SynopsisThe author has recently proposed a new algorithm for the solution of the Lyapunov matrix equation of stability theory. This algorithm is based on a formula for the solution of a special case of the equation. This formula is established in the present paper by means of a geometric interpretation. The key ideas are the uses of shift operators and non-orthogonal projections in infinite-dimensional Hilbert space.
A semilinear elliptic eigenvalue problem, I
Authors: Keady Grant , Norbury John 
Published: November 2011

SynopsisThe nonlinear operator equation (N), , describes both semilinear elliptic boundary value problems (P) and their natural discretisations (Ph). (Here is a positive compact linear operator and [t]+ ≡ (t + |t|)/2 for all t ∊ ℝ.) It is proved that, for q ≧ 0 (q ≢ 0), in a Banach lattice E the equation (N) has an unbounded continuum ℱ of nontrivial solutions (λ, ψ) ∊ ℝ × E bifurcating from infinity at (λ1, ∞) (here is the first (positive) eigenvalue of ). All nontrivial solutions (λ, ψ) have λ ≧ λ1, and if maps a smaller cone Ks, into itself then there is a ℱs ⊂ℝ×Ks with similar properties to ℱ. The existence theory for (N) is applied to problems (P) and (Ph) which are defined on the simply-connected domain Ω. It is shown that the projection of ℱ on the λ-axis is either unbounded if the continuous function q > 0 except on a set of measure zero in Ω, or is bounded if q ≡ 0 on a subdomain of Ω. If is the second eigenvalue for in (Ph) then there is at most one nontrivial solution for each λ satisfying λ1 < λ < λ2; in the corresponding uniqueness result for (P) q is restricted to being strictly positive somewhere on ∂Ω Additional properties for the solutions of (P) and (Ph) are also proved.
Units of indefinite quaternion algebras
Authors: Chalk J. H. H. 
Published: November 2011

SynopsisIf ℍ is a rational indefinite quaternion algebra andTis an order of ℍ, letG+(T) denote the subgroup of unitsu∊Twith normn(u) = + 1. For a certain class of orders, Eichler has determined the measure μ(G+(T)) ofG+(T), viewed as a Fuchsian group. This is extended to arbitrary orders by methods depending only upon classical number-theory and group theory. As an application, an estimate of the magnitude of a small non-trivial solution of the diophantine Pellian equationis supplied and the restrictions on the integersDandPassociated with previous work on this question are eliminated.
The spectrum of multiparameter problems in Hilbert space
Authors: McGhee D. F. , Roach G. F. 
Published: November 2011

SynopsisIn this paper, appropriate definitions are given for the spectra of multiparameter problems in Hilbert space when the operators involved are not necessarily self-adjoint. Notions of adjoint multiparameter problems are introduced, and some properties of the spectrum of a single linear operator are generalised to multiparameter settings.The results which are achieved are compared with those in the literature which have beendeveloped for multiparameter problems involving self-adjoint operators.
Relative resolutions of homology manifolds
Authors: Martin Nigel 
Published: November 2011

SynopsisIn the theory of resolutions of polyhedral homology manifolds to PL manifolds, itis natural to ask whether, when such a resolution exists, it is possible to preserve a given subspace of the original homology manifold under the resolution. The answer is provided in the affirmative so long as the codimension of the subspace is at least 3. In addition, it is shown that if the original dimension was at least 5 then a resolution may be chosen so as to induce an isomorphism of fundamental groups. As a corollary to these results we see that any homology n-manifold homology n-sphere may be PL embedded in S+3.
Cauchy and Poisson integral representations for ultradistributions of compact support and distributional boundary values
Authors: Pathak R. S. 
Published: November 2011

SynopsisUltradistributions of compact support are represented as the boundary values of Cauchy and Poisson integrals corresponding to tubular radial domains Tc' =ℝn + iC', C'⊂⊂C, where C is an open, connected, convex cone. The Cauchy integral of is shown to be an analytic function in TC' which satisfies a certain boundedness condition. Analytic functions which satisfy a specified growth condition in TC' have a distributional boundary value which can be used to determine an distribution.
Idempotent-connected abundant semigroups
Authors: El-Qallali A. , Fountain J. B. 
Published: November 2011

SynopsisA general theory for a class of abundant semigroups is developed. For a semigroup S in this class let E be its set of idempotents and <E> the subsemigroup of S generated by E. When <E> is regular there is a homomorphism with a number of desirable properties from S onto a full subsemigroup of the Hall semigroup T<E>. From this fact, analogues of results in the regular case are obtained for *-simple and ℐ*-simple abundant semigroups.
Quasi-adequate semigroups
Authors: El-Qallali A. , Fountain J. B. 
Published: November 2011

SynopsisThe investigation of general quasi-adequate semigroups is initiated. These are semigroups which are abundant and in which the idempotents form a subsemigroup. For such a semigroup S we study the minimum good congruence γ such that S/γ is adequate. Results on γ together with results from a previous paper of the authors are used to obtain a structure theorem for a class of quasi-adequate semigroups.
Products of idempotents in the semigroup of singular endomorphisms of a finite-dimensional vector space
Authors: Dawlings R. J. H. 
Published: November 2011

SynopsisLet Mn be the semigroup, under composition, of endomorphisms of an n-dimensional vector space. Let E be the set of idempotents of Mn. It has been shown that each singular element of Mn() may be expressed as a composition of elements of E. In this paper the minimum number of idempotents needed in this composition is determined. This is given in terms of n and one parameter dependent on the element. Further, it is shown that En–1⊂<E>, while En=<E>.
An elliptic boundary-value problem with a discontinuous nonlinearity
Authors: Keady G. 
Published: November 2011

SynopsisWe study the boundary-value problem, for(λ/k,ψ),Here ∆ denotes the Laplacian,His the Heaviside step function and one of A or k is a given positive constant. We defineand usually omit the subscript. Throughout we are interested in solutions with ψ>0 inΩ and hence with λ/=0.In the special case Ω = B(0, R), denoting the explicit exact solutions by ℑe, the following statements are true, (a) The set Aψ, issimply-connected, (b) Along ℑe, the diameter of Aψtendsto zero when the area of Aψ, tends to zero.For doubly-symmetrised solutions in domains Ω such as rectangles, it is shown that the statements (a) and (b) above remain true.
On a semilinear equation in ℝ2involving bounded measures
Authors: Vazquez Juan L. 
Published: November 2011

SynopsisWe study the semilinear equation –Δu+ β(u) =fin ℝ2, where β is a continuous increasing real function with β(0) = 0 andfis a bounded Radon measure. We show the existence of a solution, which is unique in the appropriate class, provided that each of the point masses contained infdoes not exceed some critical value denned in terms of the growth of (β at ∞ This condition is shown to be necessary for the existence of solutions, even locally. The one-dimensional situation is also discussed.
Singularities of dispersion relations
Authors: Dangelmayr G. 
Published: November 2011

SynopsisGeneric singularities occurring in dispersion relations are discussed within the framework of imperfect bifurcation theory and classified up to codimension four. Wave numbers are considered as bifurcation variables x =(x1,…, xn) and the frequency is regarded as a distinguished bifurcation parameter λ. The list of normal forms contains, as special cases, germs of the form ±λ +f(x), where f is a standard singularity in the sense of catastrophe theory. Since many dispersion relations are ℤ(2)-equivariant with respect to the frequency, bifurcation equations which are ℤ(2)-equivariant with respect to the bifurcation parameter are introduced and classified up to codimension four in order to describe generic singularities which occur at zero frequency. Physical implications of the theory are outlined.
Degenerate Lyapunov functionals of a well-known prey–predator model with discrete delays
Authors: He Xue-Zhong 
Published: November 2011

It is commonly believed that, as far as stabilities are concerned, ‘small delays are negligible in some modelling processes’. However, to have an affirmative answerfor this common belief is still an open problem for many nonlinear equations. In this paper, the classical Lotka–Volterra prey–predator equation with discrete delaysis considered, and, by using degenerate Lyapunov functionals method, an affirmative answer to this open problem on both local and global stabilities of the prey–predator delay equations is given. It is shown that degenerate Lyapunov functional method is a powerful tool for studying the stability of such nonlinear delay systems. A detailed and explicit procedure of constructing such functionals is provided. Furthermore, some explicit estimates on the allowable sizes of the delays are obtained.