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Volume 50 Issue 03

Sun, 31 Aug 2008 23:00:00 GMT

Glasgow Mathematical Journal, Volume 50 Issue 03

Glasgow Mathematical Journal publishes original research papers in any branch of pure and applied mathematics. An international journal, its policy is to feature a wide variety of research areas, which in recent issues have included ring theory, group theory, functional analysis, combinatorics, differential equations, differential geometry, number theory, algebraic topology, and the application of such methods in applied mathematics. The journal has a web-based submission system for articles. For details of how to to upload your paper see GMJ - Online Submission Guidelines or go directly to the submission site .

COMPLETE CONVERGENCE FOR WEIGHTED SUMS OF SEQUENCES OF AANA RANDOM VARIABLES

Research Articles
GUANG-HUI CAI, BAO-CAI GUO,
Glasgow Mathematical Journal, Volume 50 Issue 03 , pp 351-357

Abstract
Let Xn, n Zygmund type strong laws of large numbers for an AANA sequence of random variables are obtained. The results obtained generalize the results of Kim, Ko and Lee (Kim, T. S., Ko, M. H. and Lee, I. H. 2004. On the strong laws for asymptotically almost negatively associated random variables. Rocky Mountain J. of Math. 34, 979 989.).

A TOPOLOGISED MEASURE HOMOLOGY

Research Articles
RICARDO BERLANGA,
Glasgow Mathematical Journal, Volume 50 Issue 03 , pp 359-369

Abstract
A homology theory based on measures, first mentioned by Thurston, is naturally defined here as a functor into the category of locally convex topological vector spaces. It is proved that the first homology space is Hausdorff.

ON FIXED POINTS OF DOUBLY SYMMETRIC RIEMANN SURFACES

Research Articles
GRZEGORZ GROMADZKI, EWA KOZŁOWSKA-WALANIA,
Glasgow Mathematical Journal, Volume 50 Issue 03 , pp 371-378

Abstract
In this paper, we study ovals of symmetries and the fixed points of their products on Riemann surfaces of genus g 2. We show how the number of these points affects the total number of ovals of symmetries. We give a generalisation of Bujalance, Costa and Singerman's theorems in which we show upper bounds for the total number of ovals of two symmetries in terms of g, the order n and the number m of the fixed points of their product, and we show their attainments for n holding some divisibility conditions. Finally, we give an upper bound for m in terms of n and g, and we study conditions under which it has given parity.

GENUS 2 SEMI-REGULAR COVERINGS WITH LIFTING SYMMETRIES

Research Articles
YOLANDA FUERTES, ALEXANDER MEDNYKH,
Glasgow Mathematical Journal, Volume 50 Issue 03 , pp 379-394

Abstract
In this paper, we obtain algebraic equations for all genus 2 compact Riemann surfaces that admit a semi-regular (or uniform) covering of the Riemann sphere with more than two lifting symmetries. By a lifting symmetry, we mean an automorphism of the target surface which can be lifted to the covering. We restrict ourselves to the genus 2 surfaces in order to make computations easier and to make possible to find their algebraic equations as well. At the same time, the main ingredient (Main Proposition) depends neither on the genus, nor on the order of the group of lifting symmetries. Because of this, the paper can be thought as a generalisation for the non-normal case to the question of lifting automorphisms of a compact Riemann surface to a normal covering, treated, for instance, by E. Bujalance and M. Conder in a joint paper, or by P. Turbek solely.

DOUBLE HILBERT TRANSFORMS ALONG POLYNOMIAL SURFACES IN R 3

Research Articles
SANJAY PATEL,
Glasgow Mathematical Journal, Volume 50 Issue 03 , pp 395-428

Abstract
We obtain a necessary and sufficient condition on a polynomial P(s, t) so that the (global) double Hilbert transforms along polynomial surfaces (s, t, P(s, t)) in R3 are bounded on Lp for 1 p .

A NOTE ON L 2 -SUMMAND VECTORS IN DUAL SPACES

Research Articles
ANTONIO AIZPURU, FRANCISCO J GARCÍA-PACHECO,
Glasgow Mathematical Journal, Volume 50 Issue 03 , pp 429-432

Abstract
It is shown that every L2-summand vector of a dual real Banach space is a norm-attaining functional. As consequences, the L2-summand vectors of a dual real Banach space can be determined by the L2-summand vectors of its predual; for every n , every real Banach space can be equivalently renormed so that the set of norm-attaining functionals is n-lineable; and it is easy to find equivalent norms on non-reflexive dual real Banach spaces that are not dual norms.

AN INEQUALITY RELATED TO THE GEHRING–HALLENBECK THEOREM ON RADIAL LIMITS OF FUNCTIONS IN THE HARMONIC BERGMAN SPACES

Research Articles
MIROSLAV PAVLOVIĆ,
Glasgow Mathematical Journal, Volume 50 Issue 03 , pp 433-435

Abstract

A NOTE ON THE UNIQUENESS OF POSITIVE SOLUTIONS OF ROBIN PROBLEM

Research Articles
QIUYI DAI, YUXIA FU,
Glasgow Mathematical Journal, Volume 50 Issue 03 , pp 437-445

Abstract
This note is devoted to prove some uniqueness results of positive solutions of a Robin problem for semi-linear elliptic equations and systems.

2 -SYMMETRIC CRITICAL POINT THEOREMS FOR NON-DIFFERENTIABLE FUNCTIONS

Research Articles
PASQUALE CANDITO, ROBERTO LIVREA, DUMITRU MOTREANU,
Glasgow Mathematical Journal, Volume 50 Issue 03 , pp 447-466

Abstract
In this paper, some min max theorems for even and C1 functionals established by Ghoussoub are extended to the case of functionals that are the sum of a locally Lipschitz continuous, even term and a convex, proper, lower semi-continuous, even function. A class of non-smooth functionals admitting an unbounded sequence of critical values is also pointed out.

CROSS-CONSTRAINED VARIATIONAL PROBLEM AND THE NON-LINEAR KLEIN–GORDON EQUATIONS

Research Articles
ZAIHUI GAN, JIAN ZHANG,
Glasgow Mathematical Journal, Volume 50 Issue 03 , pp 467-481

Abstract
In this paper, we put forward a cross-constrained variational method to study the non-linear Klein Gordon equations with an inverse square potential in three space dimensions. By constructing a type of cross-constrained variational problem and establishing so-called cross-invariant manifolds of the evolution flow, we establish some new types of invariant sets for the equation and derive a sharp threshold of blowup and global existence for its solution. Finally, we give an answer to the question how small the initial data are for the global solution to exist.

DIMENSION ESTIMATE OF THE EXPONENTIAL ATTRACTOR FOR THE CHEMOTAXIS–GROWTH SYSTEM

Research Articles
MESSOUD EFENDIEV, ETSUSHI NAKAGUCHI, KOICHI OSAKI,
Glasgow Mathematical Journal, Volume 50 Issue 03 , pp 483-497

Abstract
In this paper, we study an upper bound of the fractal dimension of the exponential attractor for the chemotaxis growth system in a two-dimensional domain. We apply the technique given by Eden, Foias, Nicolaenko and Temam. Our results show that the bound is estimated by polynomial order with respect to the chemotactic coefficient in the equation similar to our preceding papers.

INVARIANT SUBMANIFOLDS OF CONTACT (κ, μ)-MANIFOLDS

Research Articles
BENIAMINO CAPPELLETTI MONTANO, LUIGIA DI TERLIZZI, MUKUT MANI TRIPATHI,
Glasgow Mathematical Journal, Volume 50 Issue 03 , pp 499-507

Abstract
Invariant submanifolds of contact ( )-manifolds are studied. Our main result is that any invariant submanifold of a non-Sasakian contact ( )-manifold is always totally geodesic and, conversely, every totally geodesic submanifold of a non-Sasakian contact ( )-manifold, 0, such that the characteristic vector field is tangent to the submanifold is invariant. Some consequences of these results are then discussed.

A CLASS OF EXCHANGE RINGS

Research Articles
TSIU-KWEN LEE, YIQIANG ZHOU,
Glasgow Mathematical Journal, Volume 50 Issue 03 , pp 509-522

Abstract
It is well known that a ring R is an exchange ring iff, for any a R, a a)R for some e2 = e R iff, for any a R, a a) for some e2 = e R. The paper is devoted to a study of the rings R satisfying the condition that for each a R, a a)R for a unique e2 = e R. This condition is not left 236) satisfy this condition. These rings are characterized as the semi-boolean rings with a restricted commutativity for idempotents, where a ring R is semi-boolean iff R/J(R) is boolean and idempotents lift modulo J(R) (or equivalently, R is an exchange ring for which any non-zero idempotent is not the sum of two units). Various basic properties of these rings are developed, and a number of illustrative examples are given.

ON THE SOLVABILITY OF BILINEAR EQUATIONS IN FINITE FIELDS

Research Articles
IGOR E. SHPARLINSKI,
Glasgow Mathematical Journal, Volume 50 Issue 03 , pp 523-529

Abstract

BOUNDARY VALUE PROBLEMS VIA AN INTERMEDIATE VALUE THEOREM

Research Articles
GERD HERZOG, ROLAND LEMMERT,
Glasgow Mathematical Journal, Volume 50 Issue 03 , pp 531-537

Abstract
We use an intermediate value theorem for quasi-monotone increasing functions to prove the existence of the smallest and the greatest solution of the Dirichlet problem u , u(1) = E E is quasi-monotone increasing in its second variable with respect to a regular cone.

ULTRAPOWERS OF BANACH ALGEBRAS AND MODULES

Research Articles
MATTHEW DAWS,
Glasgow Mathematical Journal, Volume 50 Issue 03 , pp 539-555

Abstract
The Arens products are the standard way of extending the product from a Banach algebra to its bidual . Ultrapowers provide another method which is more symmetric, but one that in general will only give a bilinear map, which may not be associative. We show that if is Arens regular, then there is at least one way to use an ultrapower to recover the Arens product, a result previously known for C*-algebras. Our main tool is a principle of local reflexivity result for modules and algebras.

RATIONAL POINTS ON CERTAIN DEL PEZZO SURFACES OF DEGREE ONE

Research Articles
MACIEJ ULAS,
Glasgow Mathematical Journal, Volume 50 Issue 03 , pp 557-564

Abstract
Let and let us consider a del Pezzo surface of degree one given by the equation . In this paper we prove that if the set of rational points on the curve Ea,b : Y2 = X3 + 135(2a 1350(5a + 2b 26) is infinite then the set of rational points on the surface f is dense in the Zariski topology.

EXISTENCE AND MULTIPLICITY OF SOLUTIONS FOR A NEUMANN PROBLEM INVOLVING VARIABLE EXPONENT GROWTH CONDITIONS

Research Articles
MARIA-MAGDALENA BOUREANU, MIHAI MIHĂILESCU,
Glasgow Mathematical Journal, Volume 50 Issue 03 , pp 565-574

Abstract
In this paper we study a non-linear elliptic equation involving p(x)-growth conditions and satisfying a Neumann boundary condition on a bounded domain. For that equation we establish the existence of two solutions using as a main tool an abstract linking argument due to Br zis and Nirenberg.

POLYNOMIAL DECAY FOR SOLUTIONS OF HYPERBOLIC INTEGRODIFFERENTIAL EQUATIONS

Research Articles
T. BÁRTA,
Glasgow Mathematical Journal, Volume 50 Issue 03 , pp 575-581

Abstract
We consider a linear integrodifferential equation of second order in a Hilbert space and show that the solution tends to zero polynomially if the decay of the convolution kernel is polynomial. Both polynomials are of the same order.

APPROXIMATION OF BANACH SPACE VALUED NON-ABSOLUTELY INTEGRABLE FUNCTIONS BY STEP FUNCTIONS

Research Articles
B. BONGIORNO, L. DI PIAZZA, K. MUSIAŁ,
Glasgow Mathematical Journal, Volume 50 Issue 03 , pp 583-593

Abstract
The approximation of Banach space valued non-absolutely integrable functions by step functions is studied. It is proved that a Henstock integrable function can be approximated by a sequence of step functions in the Alexiewicz norm, while a Henstock Pettis and a Denjoy Pettis integrable function can be only scalarly approximated in the Alexiewicz norm by a sequence of step functions. In case of Henstock Pettis and Denjoy Pettis integrals the full approximation can be done if and only if the range of the integral is norm relatively compact.

BOUNDS ON THE DIMENSION OF MANIFOLDS WITH INVOLUTION FIXING F n F 2

Research Articles
PEDRO L. Q. PERGHER, FÁBIO G. FIGUEIRA,
Glasgow Mathematical Journal, Volume 50 Issue 03 , pp 595-604

Abstract
Let Mm be a closed smooth manifold with an involution having fixed point set of the form Fn F2, where Fn and F2 are submanifolds with dimensions n and 2, respectively, where n the stable cobordism class of F2. In this paper, we determine the upper bound for m in terms of the pair (n, 3) was completely solved in a previous paper of the authors. The existence of these upper bounds is guaranteed by the famous 5/2-theorem of Boardman, which establishes that, under the above hypotheses, m 5/2n.