bluuebunny/crossref_metadata_2025 huggingface dataset:
A nonlinear complementarity problem for monotone functions
Authors: Nanda Sribatsa , Patel Ujagar
Published: April 2009

In this note we prove that for a monotone function that fixes the origin, the complementarity problem for Cn always admits a solution. If, moreover, the function is strictly monotone, then zero is the unique solution. These results are stronger than known results in this direction for two reasons: firstly, there is no condition on the nature of the cone and secondly, no feasibility assumptions are made.
A nonlinear complementarity problem in mathematical programming in Hilbert space
Authors: Nanda Sribatsa , Nanda Sudarsan
Published: April 2009

In this paper we prove the following existence and uniqueness theorem for the nonlinear complementarity problem by using the Banach contraction principle. If T: K → H is strongly monotone and lipschitzian with k2 < 2c < k2+1, then there is a unique y ∈ K, such that Ty ∈ K* and (Ty, y) = 0 where H is a Hilbert space, K is a closed convex cone in H, and K* the polar cone.
A family of inequalities for convex sets
Authors: Scott P.R.
Published: April 2009

Let K be a bounded, closed convex set in the euclidean plane. We denote the diameter, width, perimeter, area, inradius, and circumradius of K by d, w, p, A, r, and R respectively. We establish a number of best possible upper bounds for (w−2r)d, (w−2r)R,(w−2r)p, (w−2r)A in terms of w and r. Examples are:
On geometric duality for state-constrained control problems
Authors: Jefferson T.R. , Scott C.H.
Published: April 2009

For convex optimal control problems without explicit pure state constraints, the structure of dual problems is now well known. However, when these constraints are present and active, the theory of duality is not highly developed. The major difficulty is that the dual variables are not absolutely continuous functions as a result of singularities when the state trajectory hits a state constraint. In this paper we recognize this difficulty by formulating the dual probram in the space of measurable functions. A strong duality theorem is derived. This pairs a primal, state constrained convex optimal control problem with a dual convex control problem that is unconstrained with respect to state constraints. In this sense, the dual problem is computationally more attractive than the primal.
Stein quasigroups II: algebraic aspects
Authors: Pelling M.J. , Rogers D.G.
Published: April 2009

This paper furthers the foundation of the theory of quasigroups obeying the law x(xy) = yx by studying their algebraic properties. Much information is obtained by analysing the cycle decomposition of left translations regarded as permutations, and other results are obtained by representation in terms of abelian groups with an operation.
Composition operators on a functional Hilbert space
Authors: Singh R.K. , Sharma S.D.
Published: April 2009

Let T be a mapping from a set X into itself and let H(X) be a functional Hilbert space on the set X. Then the composition operator CT on H(X) induced by T is a bounded linear transformation from H(X) into itself defined by CTf = f ∘ T. In this paper composition operators are characterized in the case when H(X) = H2(π+) in terms of the behaviour of the inducing functions in the vicinity of the point at infinity. An estimate for the lower bound of ∥CT∥ is given. Also the invertibility of CT is characterized in terms of the invertibility of T.
On the stability of barrelled topologies, I
Authors: Robertson W.J. , Yeomans F.E.
Published: April 2009

This note investigates, for locally convex topological vector spaces, the question of how far the property of being barrelled is stable under small increase in the size of the dual space. If the dual F of a barrelled space E is enlarged by a finite dimensional vector space M, then E remains barrelled under the new Mackey topology τ(E, F+M). We discuss what happens when M has countable dimension.
Characterization and enumeration of hypotraversable graphs
Authors: Mohanty S.P. , Rao Daljit
Published: April 2009

A graph G on p ≥ 2 vertices is said to be hypotraversable if G is not traversable but G − v is traversable, that is, G − v has an open trail which contains all the vertices and edges of G − v, for each vertex v of G. In this paper we first characterize hypotraversable graphs and then, using this characterization, obtain a complete enumeration of these graphs on p vertices for each positive integer p.
On the boundary values of the solutions of linear elliptic equations
Authors: Chabrowski J. , Thompson H.B.
Published: April 2009

The purpose of this article is to investigate the traces of weak solutions of a linear elliptic equation. In particular, we obtain a sufficient condition for a solution belonging to the Sobolev space to have an L2-trace on the boundar.
Nonuniform dichotomy of evolutionary processes in Banach spaces
Authors: Preda Petre , Megan Mihail
Published: April 2009

In this paper we study nonuniform dichotomy concepts of linear evolutionary processes which are defined in a general Banach space and whose norms can increase no faster than an exponential. Connections between the dichotomy concepts and (B, D) admissibility properties are established. These connections have been partially accomplished in an earlier paper by the authors for the case when the process was a semigroup of class C0 and (B, D) = [(Lp, Lq).
Toward a classification of dynamical symmetries in classical mechanics
Authors: Prince Geoff
Published: April 2009

A one-parameter group on evolution space which permutes the classical trajectories of a Lagrangian system is called a dynamical symmetry. Following a review of the modern approach to the “symmetry-conservation law” duality an attempt is made to classify such invariance groups according to the induced transformation of the Cartan form. This attempt is fairly successful inasmuch as the important cases of Lie, Noether and Cartan symmetries can be distinguished. The theory is illustrated with a presentation of results for the classical Kepler problem.
Real hardy spaces of an annulus
Authors: Wang Hwai-chiuan
Published: April 2009

In this article we use real methods to study the Dirichlet and the boundary value problems of an annulus. Then we establish various properties of real Hardy spaces on the annulus, and give some applications.
Function spaces on the unit circle
Authors: Loy Richard J.
Published: April 2009

In this note we give some negative results concerning the question of whether certain integrable functions on the unit circle with mean value zero are expressible as finite sums of differences g – gα of integrable functions g, where gα denotes the translate of g by α.
Krull dimension and torsion radicals
Authors: Boyle A.K. , Serven R.J.
Published: April 2009

Let R be a ring with Krull dimension α and let τα be defined on a module MR by . Equivalent conditions for τα to be a torsion radical are given. The relationship between τα-criticals and α-criticals is also explored.
A note on coverings and Kervaire complexes
Authors: Brick Stephen G.
Published: April 2009

In the combinatorial category, a two-complexXis said to beKervaireif any set of equations modelled onX, over any group, has a solution in a larger group. The Kervaire-Laudenbach conjecture speculates that ifH2(X) = 0 thenXis Kervaire. We show that the validity of this conjecture would imply that all aspherical two-complexes are Kervaire. In particular, any two-complex homotopically equivalent to a two-manifold (≠S2,RP2) would be Kervaire. We show that this is indeed the case for certain such two-complexes. We generalise this to staggered two-complexes, and, more generally, one-relator extensions of Kervaire complexes. We obtain similar results for diagrammatically reducible two-complexes. Our proofs make use of covering spaces.
The number of factors in a paperfolding sequence
Authors: Allouche Jean-Paul
Published: April 2009

We prove that the number of factors of length k in any paperfolding sequence is equal to 4k once k ≥ 7.
Ishikawa's iterations of real Lipschitz functions
Authors: Deng Lei , Ding Xie Ping
Published: April 2009

In this paper, we consider Ishikawa's iteration scheme to compute fixed points of real Lipschitz functions. Two general convergence theorems are obtained. Our results generalise the result of Hillam.
A variational inequality in non-compact sets and its applications
Authors: Kim Won Kyu , Tan Kok-Keong
Published: April 2009

In this note, we shall prove a new variational inequality in non-compact sets and as an application, we prove a generalisation of the Schauder-Tychonoff fixed point theorem.
A selection theorem and its applications
Authors: Ding Xie Ping , Kim Won Kyu , Tan Kok-Keong
Published: April 2009

In this paper, we first prove an improved version of the selection theorem of Yannelis-Prabhakar and next prove a fixed point theorem in a non-compact product space. As applications, an intersection theorem and two equilibrium existence theorems for a non-compact abstract economy are given.
Small embeddings of partial directed cycle systems
Authors: Lindner C.C. , Rodger C.A.
Published: April 2009

In this paper, a generalisation of the Andersen, Hilton, Rodger Theorem for embedding partial idempotent latin squares is proved. This result is then used to prove that a partial directed m-cycle system of order n can be embedded in a directed m-cycle system of order (2n + 1)m if m is odd, of order 2nm if m ≥ 8 is even, 12n + 1 if m = 6 and approximately 2n + √2n if m = 4.
The classification of groups with the small squaring property on 3-sets
Authors: Longobardi P. , Maj M.
Published: April 2009

Let G be a group and k an integer greater than 1. We say that G has the square property of k-sets and we write G ∈ DS(k) if |X2| < k2 for any subset X of G of order k. The groups in DS(2) are exactly the Dedekind groups, as Freiman showed. The class .DS(3) has been recently studied by Berkovich, Freiman and Praeger. In this paper we complete the classification of DS(3)-groups by characterising finite 2-groups of exponent 4 in DS(3).
Large chaos in smooth functions of zero topological entropy
Authors: López Víctor Jiménez
Published: April 2009

For any α ∈ (0, 1), examples of C∞ functions fα: [0, 1] → [0, 1] with zero topological entropy and possessing a δ−scrambled set of Lebesgue measure α are given. This answers a question posed by Smítal.
On two problems of H. Neumann and J. Wiegold about linked products of groups
Authors: Ivanov S.V. , Ol'shanskii A.Yu.
Published: April 2009

The authors give affirmative answers to two questions of H. Neumann and Wiegold on linked embeddings of groups.
On orders of directly indecomposable finite rings
Authors: Hirano Yasuyuki , Sumiyama Takao
Published: April 2009

Let R be a directly indecomposable finite ring. Let p be a prime, let m be a positive integer and suppose the radical of R has pm elements. Then we show that . As a consequence, we have that, for a given finite nilpotent ring N, there are up to isomorphism only finitely many finite rings not having simple ring direct summands, with radical isomorphic to N. Let R* denote the group of units of R. Then we prove that (1 − 1/p)m+1 ≤ |R*| / |R| ≤ 1 − 1/pm. As a corollary, we obtain that if R is a directly indecomposable non-simple finite 2′-ring then |R| < |R*| |Rad(R)|.
On strongly right bounded finite rings II
Authors: Xue Weimin
Published: April 2009

An associative ring R is called a BT-ring if R is strongly right bounded, but not right duo, and not strongly left bounded. We show that the order of the smallest BT-rings (without unity) is 16, while we prove earlier that the order of the smallest unitary BT-rings is 32.
Ishikawa and Mann iteration methods for nonlinear strongly accretive mappings
Authors: Osilike M.O.
Published: April 2009

Let X be a real Banach space with a uniformly convex dual, X*, and let C be a nonempty closed convex and bounded subset of X. Let T: C → C be a strongly accretive and a continuous mapping. For any f ∈ C, let S: C → C be defined by Sx = f + x – Tx for each x ∈ C. Then, the iteration process xo ∈ C,

under suitable conditions on the real sequence converges strongly to a solution of the equation Tx = f in C. Furthermore, if T is strongly accretive and Lipschitz with Lipschitz constant L ≥ 1 then the iteration process x0 ∈ C,

under suitable conditions on the real sequences and converges strongly to a solution of the equation Tx = f in C. Explicit error estimates are obtained.
Some notes on the method of moving planes
Authors: Dancer E.N.
Published: April 2009

In this paper, we obtain a version of the sliding plane method of Gidas, Ni and Nirenberg which applies to domains with no smoothness condition on the boundary. The method obtains results on the symmetry of positive solutions of boundary value problems for nonlinear elliptic equations. We also show how our techniques apply to some problems on half spaces.
On Gaussian elimination and determinant formulas for matrices with chordal inverses
Authors: Bakonyi Mihály
Published: April 2009

In this paper a formula is obtained for the entries of the diagonal factor in the U D L factorisation of an invertible operator matrix in the case when its inverse has a chordal graph. As a consequence, in the finite dimensional case a determinant formula is obtained in terms of some key principal minors. After a cancellation process this formula leads to a determinant formula from an earlier paper by W.W. Barrett and C.R. Johnson, deriving in this way a different and shorter proof of their result. Finally, an algorithmic method of constructing minimal vertex separators of chordal graphs is presented.
Hecke groups and continued fractions
Authors: Rosen David , Schmidt Thomas A.
Published: April 2009

The Hecke groups

are Fuchsian groups of the first kind. In an interesting analogy to the use of ordinary continued fractions to study the geodesics of the modular surface, the λ-continued fractions (λF) introduced by the first author can be used to study those on the surfaces determined by the Gq. In this paper we focus on periodic continued fractions, corresponding to closed geodesics, and prove that the period of the λF for periodic has nearly the form of the classical case. From this, we give: (1) a necessary and sufficient condition for to be periodic; (2) examples of elements of ℚ(λq) which also have such periodic expansions; (3) a discussion of solutions to Pell's equation in quadratic extensions of the ℚ(λq); and (4) Legendre's constant of diophantine approximation for the Gq, that is, γq such that < γq/Q2 implies that P/Q of “reduced finite λF form” is a convergent of real α ∉ Gq(∞).
Finite Hilbert transforms and compactness
Authors: Okada Susumu
Published: April 2009

It is shown that for the finite Hilbert transform Tp on the Banach space Lp(]–1, 1[), 1 < p < ∞, the linear operator is not strictly singular whenever n is a positive integer.
Characteristic subgroups of relatively free groups
Authors: Bryant Roger M.
Published: April 2009

A simple new proof is given of a result of Vaughan-Lee which implies that if G is a relatively free nilpotent group of finite rank k and nilpotency class c with c < k then the characteristic subgroups of G are all fully invariant. It is proved that the condition c < k can be weakened to c < k + p − 2 when G has p–power exponent for some prime p. On the other hand it is shown that for each prime p there is a 2-generator relatively free p-group G which is nilpotent of class 2p such that the centre of G is not fully invariant.
Topological and order-topological orthomodular lattices
Authors: Riecanová Zdenka
Published: April 2009

The necessary and sufficient conditions for atomic orthomodular lattices to have the MacNeille completion modular, or (o)-continuous or order topological, orthomodular lattices are proved. Moreover we show that if in an orthomodular lattice the (o)-convergence of filters is topological then the (o)-convergence of nets need not be topological. Finally we show that even in the case when the MacNeille completion of an orthomodular lattice L is order-topological, then in general the (o)-convergence of nets in does not imply their (o)-convergence in L. (This disproves, also for the orthomodular and order-topological case, one statement in G.Birkhoff's book.)
Periods for triangular maps
Authors: Alsedà Lluís , Llibre Jaume
Published: April 2009

We study the sets of periods of triangular maps on a cartesian product of arbitrary spaces. As a consequence we extend Kloeden's Theorem (in a 1979 paper) to a class of triangular maps on cartesian products of intervals and circles. We also show that, in some sense, this is the more general situation in which the Sharkovskiĭ ordering gives the periodic structure of triangular maps.
On the signature of generalised Seifert fibrations
Authors: Pang Peter Y.
Published: April 2009

In this note, we prove a signature product formula for generalised Seifert fibrations. We also discuss how this result can be viewed using the theory of minimal models.
Posets of ordered compactifications
Authors: Richmond Thomas A.
Published: April 2009

If (X′, τ′, ≤′) is an ordered compactification of the partially ordered topological space (X, τ, ≤) such that ≤′ is the smallest order that renders (X′, τ′, ≤′) a T2-ordered compactification of X, then X′ is called a Nachbin (or order-strict) compactification of (X, τ, ≤). If (X′, τ′, ≤*) is a finite-point ordered compactification of (X, τ, ≤), the Nachbin order ≤′ for (X′, τ′) is described in terms of (X, τ, ≤) and X′. When given the usual order relation between compactifications (ordered compactifications), posets of finite-point Nachbin compactifications are shown to have the same order structure as the poset of underlying topological compactifications. Though posets of arbitrary finite-point ordered compactifications are shown to be less well behaved, conditions for their good behavior are studied. These results are used to examine the lattice structure of the set of all ordered compactifications of the ordered topological space (X, τ, ≤).
Projectivity and duality in a Morita context
Authors: Zhengping Zhou
Published: April 2009

The concepts of closed submodule, left and right annihilators are generalised and a necessary and sufficient condition on any Morita context is given so that the duality and projectivity between these sets exist. As a special case, we get Hutchinson's necessary and sufficient condition under a weaker hypothesis on the context.
Eigenvalue estimates and isoperimetric inequalities for cone-manifolds
Authors: Hodgson Craig , Tysk Johan
Published: April 2009

This paper studies eigenvalue bounds and isoperimetric inequalities for Rieman-nian spaces with cone type singularities along a codimension-2 subcomplex. These “cone-manifolds” include orientable orbifolds, and singular geometric structures on 3-manifolds studied by W. Thurston and others.

We first give a precise definition of “cone-manifold” and prove some basic results on the geometry of these spaces. We then generalise results of S.-Y. Cheng on upper bounds of eigenvalues of the Laplacian for disks in manifolds with Ricci curvature bounded from below to cone-manifolds, and characterise the case of equality in these estimates.

We also establish a version of the Lévy-Gromov isoperimetric inequality for cone-manifolds. This is used to find lower bounds for eigenvalues of domains in cone-manifolds and to establish the Lichnerowicz inequality for cone-manifolds. These results enable us to characterise cone-manifolds with Ricci curvature bounded from below of maximal diameter.
Embedding up to homotopy type in Euclidean space
Authors: Dranišnikov A.N. , Repovš D.
Published: April 2009

We give 8 short proof of the classical Stallings theorem that every finite n-dimensional cellular complex embeds up to homotopy in the 2n-dimensional Euclidean space. As an application we solve a problem of M. Kreck.
Ramanujan's remarkable summation formula and an interesting convolution identity
Authors: Adiga S. Bhargava Chandrashekar , Somashekara D.D.
Published: April 2009

In this note we obtain a convolution identity for the coefficients Bn(α, θ, q) defined by

using Ramanujan's 1Ψ1 summation. The identity contains as special cases convolution identities of Kung-Wei Yang and a few more interesting analogue.
Locally Lipschitz functions are generically pseudo-regular on separable Banach spaces
Authors: Giles J.R. , Sciffer Scott
Published: April 2009

For a locally Lipschitz function on a separable Banach space the set of points of Gâteaux differentiability is dense but not necessarily residual. However, the set of points where the upper Dini derivative and the Clarke derivative agree is residual. It follows immediately that the set of points of intermediate differentiability is also residual and the set of points where the function is Gâteaux but not strictly differentiable is of the first category.
Equationally defined radical classes
Authors: McConnell N.R. , Stokes Timothy
Published: April 2009

We consider universal classes of multioperator groups, and give a sufficient condition for a subclass defined by algebraic elementwise rules to be a radical class.
On the generalised Ruscheweyh class of analytic functions of complex order
Authors: Ahuja O.P.
Published: April 2009

The main object of this paper is to identify various classes of analytic functions which are starlike, convex, pre-starlike, Ruscheweyh class of orderα, β-spiral-like,β-convex-spiral-like, starlike of complex order, complex of complex order, and others as special cases of a family of analytic functions of complex order in the unit disk. This makes a uniform treatment possible. Finally, we derive sharp estimates for all coefficients of the functions from the family.
On locally Lipschitz vector-valued invex functions
Authors: Yen N.D. , Sach P.H.
Published: April 2009

The four types of invexity for locally Lipschitz vector-valued functions recently introduced by T. W. Reiland are studied in more detail. It is shown that the class of restricted K-invex in the limit functions is too large to obtain desired optimisation theorems and the other three classes are contained in the class of functions which are invex 0 in the sense of our previous joint paper with B. D. Craven and T. D. Phuong. We also prove that the extended image of a locally Lipschitz vector-valued invex function is pseudoconvex in the sense of J. Borwein at each of its points.
Asymptotically stable stationary solutions to the reaction-diffusion equations
Authors: Dikansky Arnold
Published: April 2009

We assume that there exists an asymptotically stable stationary solution of a Galerkin approximation for the reaction-diffusion system. It is shown that there exists a nearby stationary solution of the full reaction-diffusion system provided the order of the Galerkin approximation is high enough. The Lyapunov second method is used to prove the asymptotic stability of the stationary solution.
Biconcave-function characterisations of UMD and Hilbert spaces
Authors: Lee Jinsik Mok
Published: April 2009

Suppose that X is a real or complex Banach space with norm |·|. Then X is a Hilbert space if and only if

for all x in X and all X-valued Bochner integrable functions Y on the Lebesgue unit interval satisfying EY = 0 and |x − Y| ≤ 2 almost everywhere. This leads to the following biconcave-function characterisation: A Banach space X is a Hilbert space if and only if there is a biconcave function η: {(x, y) ∈ X × X: |x − y| ≤ 2} → R such that η(0, 0) = 2 and

If the condition η(0, 0) = 2 is eliminated, then the existence of such a function η characterises the class UMD (Banach spaces with the unconditionally property for martingale differences).
Small zeros of quadratic L-functions
Authors: Özlük Ali E. , Snyder C.
Published: April 2009

We study the distribution of the imaginary parts of zeros near the real axis of quadratic L-functions. More precisely, let K(s) be chosen so that |K(1/2 ± it)| is rapidly decreasing as t increases. We investigate the asymptotic behaviour of

as D → ∞. Here denotes the sum over the non-trivial zeros p = 1/2 + iγ of the Dirichlet L-function L(s, χd), and χd = () is the Kronecker symbol. The outer sum is over all fundamental discriminants d that are in absolute value ≤ D. Assuming the Generalized Riemann Hypothesis, we show that for
A formula on the subdifferential of the deconvolution of convex functions
Authors: Volle M.
Published: April 2009

It is known that, under suitable assumptions, the subdifferential ∂(f □ g) of the infimal convolution of two convex functions f and g coincides with the parallel sum of ∂ f and ∂ g. We prove that a similar formula holds for the subdifferential of the deconvolution of two convex functions: under appropriate hypothesis it coincides with the parallel star-difference of the sub-differentials of the functions.
Isometric results on a measure of non-compactness for operators on Banach spaces
Authors: Dilworth S. J.
Published: April 2009

For each λ ≥ 1 a class of Banach spaces φλ is defined. Isometric results are obtained on the equivalence between a measure of non-compactness and the essential norm of a linear operator defined on a φλ space. Best values of λ for the classical Banach spaces and for spaces with unconditional basis are investigated. For the space c of convergent sequences the non-existence of a λ-unconditional basis with λ < 2 is deduced.
Weierstrass elliptic difference equations
Authors: Potts Renfrey B.
Published: April 2009

The Weierstrass elliptic function satisfies a nonlinear first order and a nonlinear second order differential equation. It is shown that these differential equations can be discretized in such a way that the solutions of the resulting difference equations exactly coincide with the corresponding values of the elliptic function.
Holomorphic curves in the complex quadric
Authors: Jensen Gary R. , Rigoli Marco , Yang Kichoon
Published: April 2009

A local theory of holomorphic curves in the complex hyperquadric is worked out using the method of moving frames. As a consequence a complete global characterization of totally isotropic curves is obtained.
Submanifolds with finite type Gauss map
Authors: Chen Bang-yen , Piccinni Paolo
Published: April 2009

In this paper we study the following problem: To what extent does the type of the Gauss map of a submanifold of Em determine the submanifold? Several results in this respect are obtained. In particular, submanifolds with 1-type Gauss map are characterized. Surfaces with 1-type Gauss map and minimal surfaces of Sm−1 with 2-type Gauss map are completely classified. Some applications are also given.
Exponents of the class groups of imaginary Abelian number fields
Authors: Earnest A. G.
Published: April 2009

It is a classical result, deriving from the Gaussian theory of genera of integral binary quadratic forms, that there exist only finitely many imaginary quadratic fields for which the ideal class group is a group of exponent two. This finiteness has been shown to extend to all those totally imaginary quadratic extensions of any fixed totally real algebraic number field. In this paper we put forward the conjecture that there exist only finitely many imaginary abelian algebraic number fields which have ideal class groups of exponent two, and we examine the extent to which existing methods can be brought to bear on this conjecture. One consequence of the validity of the conjecture would be a proof of the existence of finite abelian groups which do not occur as the ideal class group of any imaginary abelian field.
Asymptotic linearity of solutions of nonlinear differential equations
Authors: Chen Shaozhu
Published: April 2009

In this paper we establish sufficient or necessary conditions for the nonlinear differential equation u″ + f (t, u) = 0 to have solutions which are asymptotic to lines with non-zero slopes and correct some formulations in theorems obtained by D.S. Cohen and J. Tong.
Finite soluble groups have large centralisers
Authors: Cossey John
Published: April 2009

We say that a finite groupGhas a large centraliser ifGcontains a non-central elementxwith |CG(x)| > |G|½. We prove that every finite soluble group has a large centraliser, confirming a conjecture of Bertram and Herzog.
On degenerate fully nonlinear elliptic equations in balls
Authors: Trudinger Neil S.
Published: April 2009

We establish derivative estimates and existence theorems for the Dirichlet and Neumann problems for nonlinear, degenerate elliptic equations of the form F (D2u) = g in balls. The degeneracy arises through the possible vanishing of the function g and the degenerate Monge-Ampère equation is covered as a special case.
The minimal faithful degree of a fundamental inverse semigroup
Authors: Easdown D.
Published: April 2009

This paper shows that the smallest size of a set for which a finite fundamental inverse semigroup can be faithfully represented by partial transformations of that set is the number of join irreducible elements of its semilattice of idempotents.
Asymptotic equivalence of ordinary and operator-differential equations
Authors: Simeonov P. S. , Bainov D. D.
Published: April 2009

The paper considers problems connected with the asymptotic equivalence of the system of ordinary differential equations

and the system of operator differential equations

The generality of the operator At guarantees a number of its important implementations. By a specific choice of the operator At the system (2) can be one of the concentrated delay, a system of distributed delay, a system with maxima, etcetera.
Extensions of continuous functions: reflective functors
Authors: Hunsaker W. N. , Naimpally S. A.
Published: April 2009

The purpose of this paper is to develop a general technique for attacking problems involving extensions of continuous functions from dense subspaces and to use it to obtain new results as well as to improve some of the known ones. The theory of structures developed by Harris is used to get some general results relating filters and covers. A necessary condition is derived for a continuous functionf:X→Yto have a continuous extensionf̅: λx→ λywhere λZdenotes a given extension of the spaceZ. In the case of simple extensions,f̅is continuous and in the case of strict extensionsf̅is θ-continuous. In the case of strict extensions, sufficient conditions for uniqueness off̅are derived. These results are then applied to several extensions considered by Banaschewski, Fomin, Katĕtov, Liu-Strecker, Blaszczyk-Mioduszewski, Rudölf, etc.
On Hau's lemma
Authors: Loh W.K.A.
Published: April 2009

Let f ∈ ℤ[X] and let q be a prime power pl(l ≥ 2). Hua stated and proved that

for some unspecified constant C > 0 depending on the derivative f′ of f; M denoting the maximum multiplicity of the roots of the congruence
p−tf′(x) ≡ 0 (mod p),
where t is an integer chosen so that the polynomial p−tf′(x) is primitive. An explicit value for C was given by Chalk for p ≥ 3. Subsequently, Ping Ding (in two successive articles) obtained better estimates for p ≥ 2.

This article provides a better result, based upon a more precise form of Hua's main lemma, previously overlooked.
Groups covered by finitely many nilpotent subgroups
Authors: Endimioni Gérard
Published: April 2009

LetGbe a finitely generated soluble group. Lennox and Wiegold have proved thatGhas a finite covering by nilpotent subgroups if and only if any infinite set of elements ofGcontains a pair {x, y} such that (x, y) is nilpotent. The main theorem of this paper is an improvement of the previous result: we show thatGhas a finite covering by nilpotent subgroups if and only if any infinite set of elements ofGcontains a pair {x, y} such that [x,ny] = 1 for some integern= n(x, y) ≥ 0.
Finitely generated ideals in the disk algebra
Authors: Mortini Raymond
Published: April 2009

Let f1, …, fN ∈ A(D). It is shown that the ideal I(f1,…, fN) generated by the functions f1 (j = 1,…, N) equals the ideal

if and only if the functions fj have no common zero on the boundary of the unit disk D.
Torsion-free Abelian groups torsion over their endomorphism rings
Authors: Faticoni Theodore G.
Published: April 2009

We use a variation on a construction due to Corner 1965 to construct (Abelian) groups A that are torsion as modules over the ring End (A) of group endomorphisms of A. Some applications include the failure of the Baer-Kaplansky Theorem for Z[X]. There is a countable reduced torsion-free group A such that IA = A for each maximal ideal I in the countable commutative Noetherian integral domain, End (A). Also, there is a countable integral domain R and a countable. R-module A such that (1) R = End(A), (2) T0 ⊗RA ≠ 0 for each nonzero finitely generated (respectively finitely presented) R-module T0, but (3) T ⊗RA = 0 for some nonzero (respectively nonzero finitely generated). R-module T.
Schrödinger operators with magnetic and electric potentials
Authors: Kaiqi Yu
Published: April 2009

In the present paper, we consider Schrödinger operators which are formally given by . In Section 2 and 3 we prove that P has a regularly accretive extension which is a self-adjoint extension of P and it is the only self-adjoint realisation of P in L2 (RN) when satisfies = (a1, a2, …, aN) ∈ , aj, real-valued, , real-valued and the negative part V-:= max(0, -V) satisfys , with constants 0 ≤ C1 < 1, C2 ≥ 0 independent of V. In Section 4, we prove that P is essential self-adjoint on when , V sat0isfy ; V = V1 + V2, V real-valued, , i = 1, 2, V1(x) ≥ –C |x|2, for x ∈ RN with C ≥ 0 and 0 ≥ V2 ∈ KN.
A note on intermediate normalising extensions
Authors: Dăscălescu S. , van Wyk L.
Published: April 2009

We prove that the following ring-theoretic properties are shared by the two rings involved in a normalising extension R ⊂ S, and that these properties are inherited by any intermediate extension: semilocal, left perfect, semiprimary. This transfer fails for the nilpotency of the Jacobson radical. However, if the normalising set is a basis for the left R-module S, then the nilpotency of the Jacobson radical behaves in the same way as the three properties mentioned above.
On some inequalities for elementary symmetric functions
Authors: Lin Mi , Trudinger Neil S.
Published: April 2009

In this note, we prove certain inequalities for elementary symmetric funtions that are relevant to the study of partial differential equations associated with curvature problems.
Projective and affine transformations of a complex symmetric connection
Authors: Blažić Novica , Bokan Neda
Published: April 2009

Let M be a compact complex manifold and ∇ an arbitrary complex (not necessarily Riemannian) connection. In this paper we study the relation between the geometry of (M, ∇) and the topology of M, that is, we are interested in the following problem: To what extent does the topology of M determine the relations between the group of holomorphically projective transformations, the group of projective transformations and the group of affine transformations on M? Under assumptions on the Ricci-type tensors of ∇ and Chern numbers of M we show that a holomorphically projective transformation and a projective transformation are in fact affine transformations on M. A family of interesting examples of connections of this kind are constructed. Also, the case when M is a Kähler manifold is studied.
Polynomials and functions with finite spectra on locally compact Abelian groups
Authors: Basit B. , Pryde A.J.
Published: April 2009

In this paper we define polynomials on a locally compact Abelian group G and prove the equivalence of our definition with that of Domar. We explore the properties of polynomials and determine their spectra. We also characterise the primary ideals of certain Beurling algebras on the group of integers Z. This allows us to classify those elements of that have finite spectrum. If ϕ is a uniformly continuous function with bounded differences then there is a Beurling algebra naturally associated with ϕ. We give a condition on the spectrum of ϕ relative to this algebra which ensures that ϕ is bounded. Finally we give spectral conditions on a bounded function on ℝ that ensure that its indefinite integral is bounded.
Minimax theorems in probabilistic metric spaces
Authors: Cho Y.J. , Chang S.S. , Jung J.S. , Kang S.M. , Wu X.
Published: April 2009

In this paper, new minimax theorems for mixed lower-upper semicontinuous functions in probabilistic metric spaces are given. As applications, we utilise these results to show the existence of solutions of abstract variational inequalities, implicit variational inequalities and saddle point problems, and the existence of coincidence points in probabilistic metric spaces.
A note on Ohm's rationality criterion for conics
Authors: Hajja Mowaffaq
Published: April 2009

An irreducible quadratic polynomial P(X, Y) in two variables over a field k is called a conic over k. It is called rational if its function field is simple transcendental over k (equivalently if P is parameterisable by rational functions). Ohm's rationality criterion states that P is rational if and only if (i) the locus of P is non-empty and (ii) k is algebraically closed in the function field of P. To show the irredundancy of (ii), Ohm gives an example of a non-rational conic with a non-empty locus. That locus, however, consists of a single point.

In this note, we show that a better example cannot exist by showing that if the locus of a conic contains more than one point then it is rational. We also show that the only rational conic whose locus consists of one point is the conic XY + 1 over the field of two elements.
Area-diameter and area-width relations for covering plane sets
Authors: Vassallo Salvatore
Published: April 2009

An area-diameter relation and an area-width relation for plane lattice-point-free-convex bodies is proved. This implies relations on covering sets with respect to general lattices.
Quantised affine algebras and parameter-dependent R-matrices
Authors: Bracken Anthony J. , Gould Mark D. , Zhang Yao-Zhong
Published: April 2009

Let Uq(G(1)) be a quantised non-twisted affine Lie algebra with Uq(G) the corresponding quantised simple Lie algebra. Using the previously obtained universal R-matrices for and , explicitly spectral-dependent universal R-matrices for Uq(A1) and Uq(A2) are determined. These spectral-dependent universal R-matrices are evaluated in some concrete representations; well-known results for the fundamental representations are reproduced, and an explicit formula for the spectral-dependent R-matrix associated with the V(3) ⊗ V(6) module is derived, where V(3) and V(6) carry the 3- and 6-dimensional representations of Uq(A2), respectively.
A note on symmetry and ambiguity
Authors: Mollin R.A. , van der Poorten A.J.
Published: April 2009

We present the relationship between quadratic irrationals whose continued fraction expansion has symmetric period, and ambiguous ideal cycles in real quadratic number fields.
Best approximation theorems for composites of upper semicontinuous maps
Authors: Park Sehie
Published: April 2009

Let (E, τ) be a Hausdorff topological vector space and (X, ω) a weakly compact convex subset of E with the relative weak topology ω. Recently, there have appeared best approximation and fixed point theorems for convex-valued upper semicontinuous maps F: (X, ω) → 2(E, τ) whenever (E, τ) is locally convex. In this paper, these results are extended to a very broad class of multifunctions containing composites of acyclic maps in a topological vector space having sufficiently many linear functionals. Moreover, we also obtain best approximation theorems for classes of multifunctions defined on approximatively compact convex subsets of locally convex Hausdorff topological vector spaces or closed convex subsets of Banach spaces with the Oshman property.
Higher order Gateaux smooth bump functions on Banach spaces
Authors: McLaughlin David P. , Vanderwerff Jon D.
Published: April 2009

For Г uncountable and p ≥ 1 odd, it is shown ℓp(г) admits no continuous p-times Gateaux differentiable bump function. A space is shown to admit a norm with Hölder derivative on its sphere if it admits a bounded bump function with uniformly directionally Hölder derivative. Some results on smooth approximation are obtained for spaces that admit bounded uniformly Gateaux differentiable bump functions.
On group uniformities on the square of a space and extending pseudometrics
Authors: Tkačnko Michael G.
Published: April 2009

We give some conditions under which, for a given pair (d1,d2) of continuous pseudometrics respectively onXandX3, there exists a continuous semi-normNon the free topological groupF(X) such thatN(x · y−1) =d1(x, y) andN(x · y · t−1·z−1) ≥d2((x, y), (z, t)) for allx, y, z, t∈X. The “extension” results are applied to characterise thin subsets of free topological groups and obtain some relationships between natural uniformities onX2and those induced by the group uniformities *V, V* and *V* ofF(X).
Extension of Hölder's inequality (I)
Authors: Kwon E.G.
Published: April 2009

A continuous form of Hölder's inequality is established and used to extend the inequality of Chuan on the arithmetic-geometric mean inequality.
On additive maps of prime rings
Authors: Brešar Matej , Hvala Bojan
Published: April 2009

Let R be a prime ring of characteristic not 2, C be the extended centroid of R, and f: R → R be an additive map. Suppose that [f(x), x2] = 0 for all x ∈ R. Then there exist λ ∈ C and an additive map ζ: R → C such that f(x) = λx + ζ(x) for all x ∈ R. In particular, if f(x)2 = x2 for all x ∈ R, then ζ = 0 and either λ = 1 or λ= -1.
Strange triangular maps of the square
Authors: Forti G.L. , Paganoni L. , Smítal J.
Published: April 2009

We show that continuous triangular maps of the square I1, F: (x, y) → (f(x), g(x, y)), exhibit phenomena impossible in the one-dimensional case. In particular: (1) A triangular map F with zero topological entropy can have a minimal set containing an interval {a} × I, and can have recurrent points that are not uniformly recurrent; this solves two problems by S.F. Kolyada.

(2) In the class of mappings satisfying Per(F) = Fix(F), there are non-chaotic maps with positive sequence topological entropy and chaotic maps with zero sequence topological entropy.
A simple proof of the Beckenbach-Lorentz inequality
Authors: Mond B. , Pečarić J.E.
Published: April 2009

One of the well-known generalisations of the Hölder inequality was given by Beckenbach. An inverse to this inequality for the discrete case has appeared in the literature. Here we give a simple proof of the inverse to the Beckenbach inequality that is applicable to both the integral and discrete cases.
Chaos in sociobiology
Authors: Christie J.R. , Gopalsamy K. , Li Jibin
Published: April 2009

It is shown that the dynamical game theoretic mating behaviour of males and females can be modelled by a planar system of autonomous ordinary differential equations. This system occurs in modelling “the battle of the sexes” in evolutionary biology. The existence of a heteroclinic cycle and a continuous family of periodic orbits of the system is established; then the dynamical characteristics of a time-periodic perturbation of the system are investigated. By using the well-known Melnikov's method, a sufficient condition is obtained for the perturbed system to have a transverse heteroclinic cycle and hence to possess chaotic behaviour in the sense of Smale. Finally, subharmonic Melnikov theory is used to obtain a criterion for the existence of subharmonic periodic orbits of the perturbed system.
On an extension of Hölder's inequality
Authors: Pearce C.E.M. , Peĉarić J.
Published: April 2009

An new extension of Hölder's inequality is derived. This is shown to follow from a generalisation of Steffensen's inequality.
A note on periodic points and recurrent points of maps of dendrites
Authors: Kato Hisao
Published: April 2009

Let f: X → X be a map of a continuum X. Let P(f) denote the set of all periodic points of f and R(f) denote the set of all recurrent points of f. In [2], Coven and Hedlund proved that if f: I → I is a map of the unit interval I = [0, 1], then CI(P(f)) = CI(R(f)). In [7], Ye generalised this result to maps of a tree. It is natural to ask whether the result generalises to maps of a dendrite. (A dendrite is a locally connected continuum which contains no simple closed curve.) The aim of this paper is to show that the answer is negative.
An imbedding theorem for anisotropic Orlicz-Sobolev spaces
Authors: Hardy G. , Thompson H.B.
Published: April 2009

Let G be a convex function of m variables, let ω be a domain in ℝn, and let LG(ω) denote the vector-valued Orlicz space determined by G. We give an imbedding theorem for the space of weakly differentiable functions u provided with the norm ∥(u, Du)∥G, where m = n + 1 and Du denotes the gradient of u. This theorem is a variant of an imbedding theorem by N.S. Trudinger for the completion of in the norm ∥Du∥G, where m=n.
Natural duality via a finite set of relations
Authors: Zádori László
Published: April 2009

We present a duality theorem. We give a necessary and sufficient condition for any set of algebraic relations to entail the set of all algebraic relations in Davey and Werner's sense. The main result of the paper states that for a finite algebra a finite set of algebraic relations yields a duality if and only if the set of all algebraic relations can be obtained from it by using four types of relational constructs. Finally, we prove that a finite algebra admits a natural duality if and only if the algebra has a near unanimity term operation, provided that the algebra possesses certain 2k-ary term operations for some k. This is a generalisation of a theorem of Davey, Heindorf and McKenzie.
Totally disconnected groups and proofs of conjectures of Hofmann and Mukherjea
Authors: Willis George
Published: April 2009

Some structure theorems for totally disconnected groups are described. These theorems produce a certain positive integer valued function, called the scale function, on each totally disconnected group. The scale function has properties connected to the structure of the group and can be used to prove some conjectures of Hofmann and Mukherjea.
A note on growth sequences of finite simple groups
Authors: Erfanian Ahmad , Wiegold James
Published: April 2009

The aim of this paper is to give a new precise formula for h(n, A), where A is a finite non-abelian simple group, h(n, A) is the maximum number such that Ah(n, A) can ke generated by n elements, and n ≥ 2. P. Hall gave a formula for h(n, A) in terms of the Möbius function of the subgroup lattice of A; the new formula involves a concept called cospread associated with that of spread as explained in Brenner and Wiegold (1975).
A link between two characterisations of complete matrix rings
Authors: Groenewald N.J. , van Wyk L.
Published: April 2009

We establish a link between the two characterizations, independently obtained in 1991 by Puchs and Robson, of complete matrix rings in terms of the existence of nilpotent elements.
Positive definite functions and relative property (T) for subgroups of discrete groups
Authors: Bates Teresa , Robertson Guyan
Published: April 2009

Relative Property (T) for a subgroup H of a group G and its connection with positive definite functions are studied. A relation with the Haagerup approximation property is established. We show that if H is a non-normal subgroup of a group G which has Property (T) and G/H is amenable as a graph then H has finite index in G.
On group uniformities on square of a space and extending pseudometrics II
Authors: Tkačenko Michael G.
Published: April 2009

We find topological conditions on a space X under which the left (right, or two-sided) uniformity of the free topological group F(X) induces the universal uniformity or the product uniformity ux × ux on the square of X. Special attention is given to kω metrisable spaces. The main technical tool in the paper is an extension of certain continuous pseudometrics from X2 to F(X) considered by the author in the previous volume of this journal.