bluuebunny/crossref_metadata_2025 huggingface dataset:
The Lebesgue function for Hermite-Fejér interpolation on the extended Chebyshev nodes
Authors: Smith Simon J.
Published: April 2009

Given f ∈ C[−1, 1] and n point (nodes) in [−1, 1], the Hermite-Fejér interpolation polynomial is the polynomial of minimum degree which agrees with f and has zero derivative at each of the nodes. In 1916, L. Fejér showed that if the nodes are chosen to be zeros of Tn (x), the nth Chebyshev polynomial of the first kind, then the interpolation polynomials converge to f uniformly as n → ∞. Later, D.L. Berman demonstrated the rather surprising result that this convergence property no longer holds true if the Chebyshev nodes are extended by the inclusion of the end points −1 and 1 in the interpolation process. The aim of this paper is to discuss the Lebesgue function and Lebesgue constant for Hermite-Fejér interpolation on the extended Chebyshev nodes. In particular, it is shown that the inclusion of the two endpoints causes the Lebesgue function to change markedly, from being identically equal to 1 for the Chebyshev nodes, to having the form 2n2(1 − x2)(Tn (x))2 + O (1) for the extended Chebyshev nodes.
Stability, instability, oscillation and nonoscillation in scalar integrodifferential systems
Authors: Gopalsamy K.
Published: April 2009

Sufficient conditions are obtained for a homogeneous scalar integrodifferential system to have solutions all of which either converge to zero or oscillate or at least one of which can be nonoscillatory asymptotically.
A connected metric space without an equally spaced chain of points
Authors: Reyna J. Arias de
Published: April 2009

We construct a connected subspace M of the euclidean plane R2 containing two points A and B such that, for every pair of points {P, Q} of M\{A, B}, the three real numbers d(A, P), d(P, Q) and d(Q, B) are not the same. This solves a question posed by Väisälä.
Pseudocomplemented distributive lattices with small endomorphism monoids
Authors: Adams M.E. , Koubek V. , Sichler J.
Published: April 2009

By a result of K.B. Lee, the lattice of varieties of pseudo-complemented distributive lattices is the ω + 1 chain

where B−1, B0, B1 are the varieties formed by all trivial, Boolean, and Stone algebras, respectively. General theorems on relative universality proved in the present paper imply that there is a proper class of non-isomorphic algebras in B3 with finite endomorphism monoids, while every infinite algebra from B2 has infinitely many endomorphisms. The variety B4 contains a proper class of non-isomorphic algebras with endomorphism monoids consisting of the identity and finitely many right zeros; on the other hand, any algebra in B3 with a finite endomorphism monoid of this type must be finite.
Coefficient estimates for alpha-spiral functions
Authors: Bhatia B.L. , Rajasekaran S.
Published: April 2009

Letbelong to the class of α-spiral functions of order ρ(|α| < π/2, 0 ≤ ρ < 1). In this paper, we determine sharp coefficient estimates for functions of the formf(z)t, wheretis a positive integer. We also study the influence of the second coefficient on the other coefficients for such functions. The results obtained not only generalize the results of MacGregor, Boyd, Srivastava, Silverman and Silvia and others, but also give rise to new results.
Existence of limit cycles for a class of autonomous systems
Authors: Sofo Anthony
Published: April 2009

A proof is given for the existence of at least one stable periodic limit cycle solution for the polynomial non-linear differential equation of the form

in some cases where the Levinson-Smith criteria are not directly applicable.
A radius of convexity problem
Authors: Padmanabhan K.S. , Ganesan M.S.
Published: April 2009

The authors determine the sharp radius of convexity for functions analytic in the unit disc having power series representation of the form f(z) = z + an+1zn+1 + an+2zn+2 + … where an+1 is fixed and such that zf′(z)/f(x) = (1 + Aw(z))/(1 + Bw(z)), −1 ≤ B < 0 < A ≤ 1 where w(z) is an analytic function satisfying the conditions of Schwarz's lemma, in the case A + B ≥ 0. The estimate obtained is an improvement over the corresponding result obtained by Mogra and Juneja for functions analytic and starlike in the unit disc, with missing coefficients where the initial non-vanishing coefficient is fixed.
Cyclability of r-regular r-connected graphs
Authors: McCuaig W.D. , Rosenfeld M.
Published: April 2009

For each value of r ≥ 4, r even, we construct infinitely many r-regular, r-connected graphs whose cyclability is not greater than 6r − 4 if r = 0 (mod 4) and 8r − 5 if r ≡ 2 (mod 4).
Sharp estimates of approximation by some positive linear operators
Authors: Sahai Ashok , Prasad Govind
Published: April 2009

Recently, Varshney and Singh [Rend. Mat. (6) 2 (1982), 219–225] have given sharper quantitative estimates of convergence for Bernstein polynomials, Szasz and Meyer-Konig-Zeller operators. We have achieved improvement over these estimates by taking moments of higher order. For example, in case of the Meyer-Konig-Zeller operator, they gave the following estimate

wherein ∥·∥ stands for sup norm. We have improved this result to

We may remark here that for this modulus of continuity ) our result cannot be sharpened further by taking higher order moments.
Countable sum theorem for locally closed sets
Authors: Deo Satya , Jamwal Dalip Singh
Published: April 2009

In this paper we show that the countable sum theorem for locally-closed sets, for sheaf theoretic cohomological dimension over a given ringL, holds in all perfectly normal spaces as well as in all locally compact spaces.
Polynomials determining Dedekind domains
Authors: Hillman Jonathan A.
Published: April 2009

If A is a Dedekind domain and f generates a prime ideal of A[X] which is not maximal, then the domain A[X]/(f) is Dedekind if and only if f is not contained in the square of any maximal ideal of A[X]. This criterion is used find the ring of integers of a cyclotomic field, and to determine when a plane curve is normal.
The stability of LU-decompositions of block tridiagonal matrices
Authors: Mattheij R. M. M.
Published: April 2009

An investigation is made of the stability of block LU-decomposition of matricesAarising from boundary value problems of differential equations, in particular of ordinary differential equations with separated boundary conditions. It is shown that for such matrices the pivotal growth can be bounded by constants of the order of ‖A‖ and, if solution space is dichotomic, often by constants of order one. Furthermore a method to estimate the growth of the pivotal blocks is given. A number of examples support the analysis.
Systèmes dynamiques dans les espaces ultramétriques compacts
Authors: Kiventidis Thomas
Published: April 2009

We give a definition of any discrete dynamical system in compact ultrametric space we study the behaviour of its trajectories.
Generalization of Leader's fixed point principle
Authors: Som Tanmoy , Mukherjee R.N.
Published: April 2009

In J. Math. Anal. Appl. 61 (1977), 466–474, Leader has given a fixed point principle for an operator f:X → X, where X is a metric-space, based on conditional uniform equivalence of orbits. We generalize this principle for two mappings f1 and f2 to give common fixed point results in two different ways. Further we derive an f-generalized fixed point theorem for two commuting mappings.
An integral characterization of Euclidean space
Authors: Borwein J.M.
Published: April 2009

We show that recent integral versions of the classic Jordan-Von Neumann characterization of Euclidean space may he viewed as special cases of a general averaging principle over sets of isometries.
Oscillations in linear systems of differential-difference equations
Authors: Gopalsamy K.
Published: April 2009

Sufficient conditions are derived for all bounded solutions of a class of linear systems of differential difference equations to be oscillatory.
Cardinality of discrete subsets of a topological space: Corrigendum
Authors: Gauld David B. , Vamanamurthy M.K.
Published: April 2009

Theorem 1 and Corollary 1 of the original paper [Bull. Austral. Math. Soc. 25 (1982), 99–101] are false unless, for example, we assume the Generalised Continuum Hypothesis. The proof of Theorem 1 correctly shows that exp(|c|) ≤ exp(|D|) but one cannot then deduce that the cardinality of C is at most that of D. All one can deduce is that the cardinality of C is less than exp(|D|).
On closed radical orbits in homogeneous complex manifolds
Authors: Gilligan Bruce
Published: April 2009

Suppose G is a complex Lie group having a finite number of connected components and H is a closed complex subgroup of G with H° solvable. Let RG denote the radical of G. We show the existence of closed complex subgroups I and J of G containing H such that I/H is a connected solvmanifold with I° ⊃ RG, the space G/J has a Klein form SG/A, where A is an algebraic subgroup of the semisimple complex Lie group SG: = G/RG, and, unless I = J, the space J/I has Klein form , where is a Zariski dense discrete subgroup of some connected positive dimensional semisimple complex Lie group Ŝ.
Yosida functions and Picard values of integral functions and their derivatives
Authors: Huaihui Chen
Published: April 2009

In this paper we improve and generalise a result of J. Clunie by proving that if f(z) is a transcendental integral function with only zeros of order at least k + 1, then f(k)(z) assumes every finite non-zero complex value infinitely often. Also, the related criterion for normality of a family of holomorphic functions is given, and the value distribution of f2 + afk is discussed.
New inequalities for planar convex sets with lattice point constraints
Authors: Awyong Poh W. , Scott Paul R.
Published: April 2009

We obtain new inequalities relating the inradius of a planar convex set with interior containing no point of the integral lattice, with the area, perimeter and diameter of the set. By considering a special sublattice of the integral lattice, we also obtain an inequality concerning the inradius and area of a planar convex set with interior containing exactly one point of the integral lattice.
Parameterisation of developable surfaces by asymptotic lines
Authors: Ushakov Vitaly
Published: April 2009

An example of a “non-developable” surface of vanishing Gaussian curvature from W. Klingenberg's textbook is considered and its place in the theory of 2-dimensional developable surfaces is pointed out. The surface is found in explicit form. Other examples of smooth developable surfaces not allowing smooth asymptotic parametrisation are analysed. In particular, Hartman and Nirenberg's example (1959) is incorrect.
On the Darboux problem of neutral type
Authors: Bugajewski Dariusz , Zima Miroslawa
Published: April 2009

The aim of this paper is to prove uniqueness theorems for the Darboux problem of neutral type in the space L∞ and L1.
Potential operators and multipliers on locally compact Vilenkin groups
Authors: Kitada Toshiyuki
Published: April 2009

We study, under the setting of a locally compact Vilenkin group G, a weighted norm inequality for the potential operators of Riesz type and its applications to multipliers on G. We also consider the maximal operators of fractional type.
Higher-order optimality conditions for a minimax
Authors: Van Luu Do , Oettli W.
Published: April 2009

Higher-order necessary and sufficient optimality conditions for a nonsmooth minimax problem with infinitely many constraints of inequality type are established under suitable basic assumptions and regularity conditions.
On the cohomology of a class of nilpotent Lie algebras
Authors: Armstrong Grant F. , Sigg Stefan
Published: April 2009

Let g denote a finite dimensional nilpotent Lie algebra over ℂ containing an Abelian ideal a of codimension 1, with z ∈ g/a. We give a combinatorial description of the Betti numbers of g in terms of the Jordan decomposition induced by ad(z)|a. As an application we prove that the filiform-nilpotent Lie algebras arising in the case t = 1 have unimodal Betti numbers.
Borel directions and iterated orbits of meromorphic functions
Authors: Qiao Jianyong
Published: April 2009

For transcendental meromorphic functions of finite order, we prove that there exist iterated orbits which tend to the Borel directions. This gives a relation between the value distribution theory and the iteration theory of meromorphic functions.
Incongruent embeddings of a bouquet into surfaces
Authors: Kim Jin Hwan , Park Young Kou
Published: April 2009

Two 2-cell embeddingsi, jof a graphGinto surfaces  and ′ are said to be congruent with respect to a subgroup Γ of Aut(G) if there are a homeomorphismh:  → ′ and an automorphism γ ∈ Γ such thath∘i=j∘ γ. In this paper, we compute the total number of congruence classes of 2-cell embeddings of any bouquet of circles into surfaces with respect to a group consisting of graph automorphisms of a bouquet.
Torsion free groups generated by a pair of rational parabolic Möbius transformations
Authors: Bamberg John , Cairns Grant
Published: April 2009

Let T be a subgroup of PSL (2, ℚ) generated by a pair of rational parabolic matrices P1, P2, and let ℐ be the Jøgensen number. We prove that T has a non-trivial element of finite order if and only if ℐ = 4/n2 or ℐ = 9/n2 for some non-zero integer n.
Addendum à quelques remarques sur des questions d'approximation diophantienne
Authors: Philippon Patrice
Published: April 2009

Dans [1, Section 3], on a montré comment les conjectures les plus définitives sur les formes linéaires de logarithmes entraînent simplement la conjecture abc (dans sa forme faible). De façcon plus pragmatique on peut également chercher les conjectures les plus accessibles sur les formes linéaires de logarithmes qui entraînent encore la conjecture abc. De ce point de vue l'énoncé suivant semble être l'amélioration la plus minime de l'inégalité de Liouville qu'il conviendrait d'établir. On y note | · |p la valeur absolue p-adique sur Q, normalisée de la façon habituelle (c'est-à-dire, |P|p = 1/P), et h (·) la hauteur logarithmique sur Q (c'est-à-dire, le logarithme du maximum des valeurs absolues des numérateur et dénominateur dans une écriture en fraction réduite).
Sharp constants in higher-order heat kernel bounds
Authors: Dungey Nick
Published: April 2009

We consider a spaceXof polynomial type and a self-adjoint operator onL2(X) which is assumed to have a heat kernel satisfying second-order Gaussian bounds. We prove that any power of the operator has a heat kernel satisfying Gaussian bounds with a precise constant in the Gaussian. This constant was previously identified by Barbatis and Davies in the case of powers of the Laplace operator onRN. In this case we prove slightly sharper bounds and show that the above-mentioned constant is optimal.
Existence of solutions for a vector saddle point problem
Authors: Kazmi K. R. , Khan S.
Published: April 2009

We establish an existence theorem for weak saddle points of a vector valued function by making use of a vector variational inequality and convex functions.
Conformally flat hypersurfaces with constant Gauss-Kronecker curvature
Authors: Defever Filip
Published: April 2009

We consider 3-dimensional conformally flat hypersurfaces of E4 with constant Gauss-Kronecker curvature. We prove that those with three different principal curvatures must necessarily have zero Gauss-Kronecker curvature.
The Tarski–Kantorovitch prinicple and the theory of iterated function systems
Authors: Jachymski Jacek , Gajek Leslaw , Pokarowski Piotr
Published: April 2009

We show how some results of the theory of iterated function systems can be derived from the Tarski–Kantorovitch fixed–point principle for maps on partialy ordered sets. In particular, this principle yields, without using the Hausdorff metric, the Hutchinson–Barnsley theorem with the only restriction that a metric space considered has the Heine–Borel property. As a by–product, we also obtain some new characterisations of continuity of maps on countably compact and sequential spaces.
On compact group extension of Bernoulli shifts
Authors: Ahn Youngho
Published: April 2009

Let ρ : G →  (H) be an irreducible unitary representation of a compact group G where  (H) is a set of unitary operators of finite dimensional Hilbert space H. For the (p1, …, PL)-Bernoulli shift, the solvability of ρ(φ(x)) g (Tx) = g (x) is investigated, where φ(x) is a step function.
A note on asymptotic uniqueness for some nonlinearities which change sign
Authors: Dancer E. N.
Published: April 2009

In this paper, we prove the uniqueness of the decaying positive solution on all ofRnfor certain second order non linear elliptic equations. This improves earlier work of a number of authors. These problems occur in the theory of peak solutions. In particular, our results apply to a number of non-smooth nonlinearities which occur as limiting equations in population problems.
On Terai's conjecture concerning Pythagorean numbers
Authors: Le Maohu A.
Published: April 2009

In this paper we prove that if a, b, c, r are fixed positive integers satisfying a2 + b2 = cr, gcd(a, b) = 1, a ≡ 3(mod 8), 2 | b, r > 1, 2 ∤ r, and c is a (x,y,z) = (2, 2,r) satisfying x > 1, y > 1 and z > 1.
Characteristic polynomials of finitely generated modules over Weyl algebras
Authors: Levin Alexander B.
Published: April 2009

In this paper we modify the classical Gröbner basis technique and prove the existence of a characteristic polynomial in two variables associated with a finitely generated module over a Weyl algebra. We determine invariants of such a polynomial and show that some of the invariants are not carried by the Bernstein dimension polynomial of the module.
Asplund spaces and a variant of weak uniform rotundity
Authors: Giles John , Vanderwerff Jon
Published: April 2009

We introduce a property formally weaker than weak uniform rotundity, which we call equatorial weak uniform rotundity. We show that an equatorially weakly uniformly rotund norm need not be weakly locally uniformly rotund. Nevertheless, we show that an equatorially weakly uniformly rotund Banach space is an Asplund space.
On the notion of compactness in supergeometry
Authors: Bruzzo Ugo , Pestov Vladimir
Published: April 2009

We discuss the problem of finding an analogue of the concept of topological space in supergeometry, motivated by the search for a procedure to compactify supermanifolds along odd coordinates. In particular, we examine the topologies arising naturally on the sets of points of locally ringed superspaces, and show that in the presence of a nontrivial odd sector such topologies are never compact. The main outcome of our discussion is the following new observation: not only the usual framework of supergeometry (the theory of locally ringed spaces), but the more general approach of the functor of points, need to be further enlarged.
Potential symmetries of inhomogeneous nonlinear diffusion equations
Authors: Sophocleous C.
Published: April 2009

In this paper potential symmetries are sought of the inhomogeneous nonlinear diffusion equations ut = x1−M[xN−1f (u) ux]x. The functional forms of f (u) that admit such symmetries are completely classified. A complete list is presented of the symmetries, which depend on the values of the parameters M and N. We give examples of similarity solutions using potential symmetries. In some cases, the potential symmetries enable us to convert non–invertible mappings of nonlinear partial differential equations to linear ones.
Some extensions of a dual of the Hahn-Banach Theorem, with applications to separation and Helly type theorems
Authors: Singer Ivan
Published: April 2009

In previous papers we have proved that if G is a ω*-closed subspace of the conjugate space B* of a normed linear space B, then every b ∈ B can be extended within B, from G to the whole B*, with an arbitrarily small increase of the norm. Here we give some extensions of this result to the case when B* is replaced by a normed linear space E and B by any linear subspace V of E*, and some applications to separation and Helly type theorems.
Computer aided determination of a Fibonacci group
Authors: Havas George
Published: April 2009

The Fibonacci group F(2, 7) has been known to be cyclic of order 29 for about five years. This was first established by computer coset enumerations which exhibit only the result, without supporting proofs. The working in a coset enumeration actually contains proofs of many relations that hold in the group. A hand proof that F(2, 7) is cyclic of order 29, based on the working in computer coset enumerations, is presented here.
On convex lattice polygons
Authors: Scott P.R.
Published: April 2009

Let π be a convex lattice polygon with b boundary points and c (≥ 1) interior points. We show that for any given c, the number b satisfies b ≤ 2c + 7, and identify the polygons for which equality holds.
Existence of solutions to quasilinear differential equations in a Banach space
Authors: Ward James R.
Published: April 2009

Initial value problems of the form x′ + A(t, x)x = f(t, x), x(0) = a, t ≥ 0, are considered in a real, separable, reflexive Banach space. Results concerning the existence of solutions on (0, ∞) are given by considering linear systems of the form x′ + A(t, u(t))x = f(t, u(t)). Here u(t) belongs to a suitable function space.
A Banach space of functions of generalized variation
Authors: Russell A.M.
Published: April 2009

In this note we show that BVk, [a, b], the linear space of functions of bounded kth variation on [a, b], is a Banach space under the norm ‖ • ‖k, where
Maximal normal subgroups of the integral linear group of countable degree
Authors: Burns R.G. , Farouqi I.H.
Published: April 2009

This paper continues the second author's investigation of the normal structure of the automorphism group г of a free abelian group of countably infinite rank. It is shown firstly that, in contrast with the case of finite degree, for each prime p every linear transformation of the vector space of countably infinite dimension over Zp, the field of p elements, is induced by an element of г Since by a result of Alex Rosenberg GL(אo, Zp ) has a (unique) maximal normal subgroup, it then follows that г has maximal normal subgroups, one for each prime.
On common fixed point theorems
Authors: Singh S.P. , Meade B.A.
Published: April 2009

The aim of this paper is to extend a recent result of S.A. Husain and V.M. Sehgal [Bull. Austral. Math. Soc. 13 (1975), 261–267]. The condition that the function should be continuous in Husain and Sehgal, op. cit., is replaced by a semicontinuity condition. Moreover, a different proof is given, the last part of which requires no continuity or semicontinuity condition whatsoever.
A formula for the exact number of primes below a given bound in any arithmetic progression
Authors: Hudson Richard H.
Published: April 2009

The formula of [E.] Meissel [Math. Ann. 2 (1870), 636–642] is generalized to arbitrary arithmetic progressions. Meissel's formula is applicable not only to computation of π(x) for large x (recently x = 1013), but also is a sieve technique (see MR36#2548), useful for studying the subtle effect of primes less then or equal to x1/2 on the behavior of primes less than or equal to x. The same is true of the generalized Meissel, with the added advantage that the behavior of primes less than or equal to x can be studied in arbitrary progressions.
Multiplicative relations in number fields
Authors: van der Poorten A.J. , Loxton J.H.
Published: April 2009

In this paper, we obtain an explicit form of the currently best known inequality for linear forms in the logarithms of algebraic numbers. The results complete our previous investigations (Bull. Austral. Math. Soc. 15 (1976), 33–57) which were conditional on a certain independence condition on the algebraic numbers. The extra work needed to obtain unconditional results centres on the properties of multiplicative relations in number fields. In particular, we show that a set of multiplicatively dependent algebraic numbers always satisfies a relation with relatively small exponents.
Stability of impulsively perturbed systems
Authors: Mohana Rao M. Rama , Hari Rao V. Sree
Published: April 2009

Until recently most authors have devoted their research to the theory of perturbed systems under continuous perturbations. In this paper, Liapunov's second method is employed to investigate sufficient conditions for integral and integral asymptotic stability of ordinary differential systems with respect to impulsive perturbations.
Varieties of a closed category
Authors: Day B.J.
Published: April 2009

In this article closed variety structures generated by a monoidal class of identities are examined. This leads to closed enrichments of various results of Y. Diers [Ann. Soc. Sci. Bruxelles Sér. I 90 (1976), 159–172] on varieties of a category.
Orthodox bands of modules
Authors: Pastijn F.
Published: April 2009

In this paper we shall consider orthodox bands of commutative groups, together with a ring of endomorphisms. We shall generalize the concept of a left module by introducing orthodox bands of left modules; we shall also deal with linear mappings, the transpose of a linear mapping and with the dual of an orthodox band of left modules.
On a sufficient optimality condition over convex feasible regions
Authors: Alders C.D. , Sposito V.A.
Published: April 2009

In this note a sufficient optimality condition is established for nonlinear programming problems over arbitrary cone domains. A Kuhn-Tucker type sufficient condition is established if the programming problem has a pseudoconvex objective function and a convex feasible region.
Dirichlet's diophantine approximation theorem
Authors: Cusick T.W.
Published: April 2009

One form of Dirichlet's theorem on simultaneous diophantine approximation asserts that if α1, α2, …, αn are any real numbers and m ≥ 2 is an integer, then there exist integers q, p1, p2, …, pn such that 1 ≤ q < m and |qαi.-pi| ≤ m–1/n holds for 1 < i < n. The paper considers the problem of the extent to which this theorem can be improved by replacing m–1/n by a smaller number. A general solution to this problem is given. It is also shown that a recent result of Kurt Mahler [Bull. Austral. Math. Soc. 14 (1976), 463–465] amounts to a solution of the case n = 1 of the above problem. A related conjecture of Mahler is proved.
On quotients of certain countable groups
Authors: Pride Stephen J.
Published: April 2009

A group A is said to be SQ-universal if every countable group is embeddable in some quotient of A. It is well-known that the number of non-isomorphic factor groups of a countable sp-universal group B is the power of the continuum. Since B has such an abundance of non-isomorphic quotients it is natural to ask what types of quotients B must have and what types B need not have. This note is concerned with these questions.
The boundedness principle characterizes second category subsets
Authors: Broughan Kevin A.
Published: April 2009

Converses are proved for the Osgood (the Principle of Uniform Boundedness), Dini, and other well known. theorems. The notion of a continuous step function on a topological space is defined and a class of spaces identified for which each lower semicontinuous function is the pointwise limit of a monotonically increasing sequence of step functions.
Using product designs to construct orthogonal designs
Authors: Robinson Peter J.
Published: April 2009

This paper produces new types of designs, called product designs, which prove extremely useful for constructing orthogonal designs. An orthogonal design of order 2t and type

is constructed. This design often meets the Radon bound for the number of variables.

We also show that all orthogonal designs of order 2t and type (a, b, c, d, 2t-a-b-c-d), with 0 < a + b + c + d < 2t, exist for t = 5, 6, and 7.
Groups of automorphisms of linearly ordered sets: Corrigenda
Authors: Hickman J.L.
Published: April 2009

Dr Isidore Fleischer has shown by a very simple counterexample [1] that Theorem 9 (p. 30) of my paper [3] is incorrect. The error occurs in the proof of Theorem 8, and is due to my having interpreted certain group constructions as cartesian products, whereas they are in fact wreath products. The correct version of Theorem 8 is obtained by replacing cartesian products with wreath products in the appropriate places in Definition 5. The result, however, is easier to see than to state.
Transformations on tensor products and the torsionless property in abelian groups: Corrigendum
Authors: Webb Cary
Published: April 2009

Professor Paul L. Sperry, when reviewing the paper [1] for Mathematical Reviews, has drawn the author's attention to an error in the formulation of Theorem 4.

The hypotheses on the abelian groups Ci, i ∈ I, in Theorem 4 were incorrectly stated. (They implied, for one thing, that the slender group A is divisible, an absurdity.) The correct formulation is as follows.
Rearranging transfinite series of ordinals
Authors: Komjath Péter
Published: April 2009

A new, simplified proof is given of a theorem of J.L. Hickman (to be published; see also J. London Math. Soc. (2) 9 (1974), 239–244), giving an upper bound for the sums of a well-ordered series of ordinals under permutation.
Optimal reduction in replacement systems
Authors: Staples John
Published: April 2009

High level programming languages can generally be compiled in many different ways. Thus it is natural to ask if there is a best way, for example in the sense of achieving complete execution of the program in a minimum number of steps; such a complete, minimal execution procedure will be called optimal.

In recent years this question has been studied and answered for several simple models of programming languages, and a technique for proving procedures to be optimal has gradually emerged. Even for model languages the proofs of optimality become intricate; thus it is natural to emphasize the simplicity of the underlying technique by generalising it to an abstract system. That is the purpose of this paper. The general method to be given applies to prove all theorems (on optimal executions for model languages) which are known to the author. None of the previously known proofs has explicitly used the method, but in no case is it particularly difficult to modify the known proof so as to conform with the method.
Subdirectly irreducible Rees matrix semigroups
Authors: Zitarelli David E.
Published: April 2009

Minimal congruences on a Rees matrix semigroup S having at least one proper congruence are described. Necessary and sufficient conditions for S to te subdirectly irreducible are given in two cases according to whether the structure group of S is trivial.
Theory of existence and uniqueness for the nonlinear Maxwell-Boltzmann equation I
Authors: Glikson Aleksander
Published: April 2009

A review of the development of the theory of existence and uniqueness of solutions to initial-value problems for mostly reduced versions of the nonlinear Maxwell-Boltzmann equation with a cut-off of intermolecular interaction, precedes the formulation and discussion of a somewhat generalized initial-value problem for the full nonlinear Maxwell-Boltzmann equation, with or without a cut-off. This is followed by a derivation of a new existence-uniqueness result for a particular Cauchy problem for the full nonlinear Maxwell-Boltzmann equation with a cut-off, under the assumption that the monatomic Boltzmann gas in the unbounded physical space X is acted upon by a member of a broad class of external conservative forces with sufficiently well-behaved potentials, defined on X and bounded from below. The result represents a significant improvement of an earlier theorem by this author which was until now the strongest obtained for Cauchy problems for the full Maxwell-Boltzmann equation. The improvement is basically due to the introduction of equivalent norms in a Banach space, the definition of which is connected with an exponential function of the total energy of a free-streaming molecule.
On the central limit theorem and iterated logarithm law for stationary processes
Authors: Heyde C.C.
Published: April 2009

It has recently emerged that a convenient way to establish central limit and iterated logarithm results for processes with stationary increments is to use approximating martingales with stationary increments. Functional forms of the limit results can be obtained via a representation for the increments of the stationary process in terms of stationary martingale differences plus other terms whose sum telescopes and disappears under suitable norming. Results based on the most general form of such a representation are here obtained.
Convergence of elements in random normed spaces
Authors: Taylor Robert Lee
Published: April 2009

For a random normed space of mappings into a separable normed linear space, convergence of identically distributed elements in i the random norm (norm distribution) is shown to be equivalent to convergence in measure in the weak linear topology. Convergence in measure in each coordinate of a Schauder basis is also shown to be a necessary and sufficient condition for convergence in the random norm topology. These results have laws of large numbers for random elements in separable normed linear spaces as almost immediate corollaries and illustrate some of the recently obtained laws of large numbers for random elements. Similar results are also given for elements which need not have the same norm distributions, and the results are extended to linear metric spaces. Finally, applications of the results to stochastic processes are considered.
Polynomials associated with groups of exponent four
Authors: Newman M.F. , Weston K.W. , Yuan Tah-Zen
Published: April 2009

Complicated groups of exponent four have been constructed from the ring of polynomials in associating non-commuting indeterminates with coefficients from the field of two elements. The justification of these constructions depends on a computational reduction result. In this note a further reduction is obtained. The expressions involved seem to have an interesting combinatorial structure.
Sub-prime radical classes determined by zerorings
Authors: Gardner B.J.
Published: April 2009

It is shown that the correspondence which associates with each radical class τ of abelian groups the (radical) class of prime radical rings with additive groups in τ gives a complete classification of those radical classes of rings which are determined (as lower radicals) by zerorings.
On irreducible spaces
Authors: Boone James R.
Published: April 2009

A constructive proof of a theorem of Worrell and Wicke, which states that every open covering of a θ-refinable space has a σ-distributively point-finite open refinement that covers the space minimally, is presented. Spaces for which every open covering has an open refinement which covers the space minimally are characterized by the use of discrete closed collections and some open questions relating to spaces of this type are included.
Nilpotent measures on compact semigroups
Authors: Chow H.L.
Published: April 2009

LetSbe a compact semigroup andP(S) the set of probability measures onS. SupposeP(S) has zero θ and define a measure μ εP(S) nilpotent if μn→ θ. It is shown that any measure with support containing that of θ is nilpotent, and the set of nilpotent measures is convex and dense inP(S). A measure μ is called mean-nilpotent if (μ + μ2+ … + μn)/n→ θ, and can be characterized in terms of its support.
Non-local rings whose ideals are all quasi-injective: Addendum
Authors: Ivanov G.
Published: April 2009

In [1] I made a Remark which, when subsequently challenged by Anne Koehier, I was unable to justify. While writing up my PhD thesis, I tried to both prove and disprove the statement but failed on both counts. However, I did reduce it to a highly plausible statement which is presented here.
Constructions for amicable orthogonal designs
Authors: Wallis Jennifer Seberry
Published: April 2009

Infinite families of amicable orthogonal designs are constructed with

(i) both of type (1, q) in order q + 1 when q ≡ 3 (mod 4) is a prime power,

(ii) both of type (1, q) in order 2(q + 1) where q ≡ 1 (mod 4) is a prime power or q + 1 is the order of a conference matrix,

(iii) both of type (2, 2q) in order 2(q + 1) when q ≡ 1 (mod 4) is a prime power or q + 1 is the order of a conference matrix.
Maximal ideals of semigroups of endomorphisms
Authors: Cezus F.A. , Magill K.D. , Subbiah S.
Published: April 2009

The maximal left ideal, the maximal right ideal and the maximal two-sided ideal are all characterized for quite general semigroups of endomorphisms. These results are then applied to such semigroups as the semigroup of all continuous selfmaps of a topological space, the semigroup of all closed selfmaps of a topological space, and the semigroup of all linear endomorphisms of a vector space.
A note on zero-sets in the Stone-Čech compactification
Authors: Rudd D.
Published: April 2009

The ring C(X) is the ring of all continuous real-valued functions on a completely regular Hausdorff space X, and βX is the Stone-Čech compactification of X.

The author proves a theorem which leads to a characterization of those zero-sets in X whose closures (in βX) are zero-sets in βX, and relates this characterization to the ideals in the ring C(X).
Jackson's Theorem for finite products and homomorphic images of locally compact abelian groups
Authors: Bloom Walter R.
Published: April 2009

Let G be a Hausdorff locally compact abelian group. The author has shown (Bull. Austral. Math. Soc. 10 (1974), 59–66) that, given ε > 0 and a certain base {Vi}i∈I of symmetric open neighbourhoods of zero, the algebra L1(G) admits a bounded positive approximate unit {ki}i∈I such that for every p–th integrable function f on G,

where ω(p; f; Vi) denotes the mean modulus of continuity with exponent p of f. The purpose of this paper is to obtain {ki}i∈I (as above) with a simple dependence of supp on {ki}i∈I on Vi; this is achieved for finite products and homomorphic images of groups for which such a simple dependence is known. The results obtained are used to give a simplified proof of the classical Jackson's Theorem for the circle group, and an analogue of this theorem for the a-adic solenoid.
An embedding theorem for ordered groups
Authors: Fox Colin D.
Published: April 2009

We show that if the normal, closure of an element a, of an orderable group, G, is abelian, then G can be embedded in an orderable group, G#, which contains an n-th root of a for every positive integer, n. Furthermore, every order of G extends to an order of G#.
On some integrodifferential equations in Banach spaces
Authors: Pachpatte B.G.
Published: April 2009

This paper is concerned with the stability, boundedness, and asymptotic behavior of solutions of integrodifferential systems of the form

We shall also investigate the behavioral relationships between the solutions of two integrodifferential systems related to this system.
Secretive prime-power groups of large rank
Authors: Wall G.E.
Published: April 2009

A question of L.G. Kovács, Joachim Neubüser, B.H. Neumann (J. Austral. Math. Soc. 12 (1971), 287–300) on the existence of ‘secretive’ prime-power groups of large rank is settled affirmatively by proving the following result: given a prime p and integer d ≥ 2, there exists a finite p–group P with cyclic centre and minimal number of generators d and having the property that every element not in its Frattini subgroup has a non-trivial power in its centre.
A note on a paper of Williamson
Authors: Lavoie J.L. , Tremblay R.
Published: April 2009

An elementary proof is given of a result obtained by concerning the sum of an infinite series involving a 4F3.
Compatible tight Riesz orders on groups of integer-valued functions
Authors: Davis Gary
Published: April 2009

A construction due to Reilly is extended to show that there is a correspondence between compatible tight Riesz orders on ZX and non-principal filters on X. The maximal compatible tight Riesz orders are in one-to-one correspondence with non-principal ultra-filters, and are dual prime subsets of the positive set of ZX. Conversely every dual prime algebraic Riesz order is maximal.
Pseudo-distributive near-rings
Authors: Heatherly Henry E. , Ligh Steve
Published: April 2009

In the study of the theory of rings, matrix rings, group rings, algebras, and so on, play a very important role. However, the analogous systems may not exist in the theory of near-rings. Recently Ligh obtained a necessary and sufficient condition for the set of n × n matrices with entries from a near-ring to be a near-ring. This opens the door for the study of other structures such as group near-rings, algebras, and so on. In this paper we initiate a study of the basic properties of pseudo-distributive near-rings, which is exactly the class of near-rings needed to carry out the construction of matrix near-rings, group near-rings, polynomials with near-ring coefficients, and so on.
Special morphisms for zero-set spaces
Authors: Gilmour Christopher R.A.
Published: April 2009

The author obtains characterizations of the quotients, epimorphisms and extreme monomorphisms in the category of separated zero-set spaces and zero-set maps (defined by Hugh Gordon [Pacific J. Math. 36 (1971), 133–157]). The method employed, that of initiality constructions, is also used to elucidate the relationship between zero-set spaces and certain other topological structures by means of forgetful functors and their right inverses. Characterizations of pseudocompactness for zero-set spaces then follow.
Equationally complete varieties of generalized groups
Authors: Page W.F.
Published: April 2009

In previous work, Page and Butson [Algebra Universalis 3 (1973), 112–126] characterized all equationally complete classes (atoms) of m–semigroups (universal algebras with one m–ary associative operation), and hence m–groups, in the commutative case. The further characterization of the non-commutative m-group atoms was thought to hinge upon a conjecture by Page [PhD thesis, University of Miami, 1973] that a weaker form of commutativity held true. In this paper that conjecture is proved and then used to study the special case m = 4. Two additional infinite sets of atoms are thereby determined, although it is not proved that these examples exhaust the remaining atoms for m = 4.
A Schreier theorem for free topological groups
Authors: Nickolas Peter
Published: April 2009

M.I.Graev has shown that subgroups of free topological groups need not be free. Brown and Hardy, however, have proved that any open subgroup of the free topological group on a kw-space is again a free topological group: indeed, this is true for any closed subgroup for which a Schreier transversal can be chosen continuously. This note provides a proof of this result more direct than that of Brown and Hardy. An example is also given to show that the condition stated in the theorem is not a necessary condition for freeness of a subgroup. Finally, a sharpened version of a particular case of the theorem is obtained, and is applied to the preceding example.
Multifunctions of Souslin type: Corrigendum
Authors: Leese S.J.
Published: April 2009

In [1], Theorem 6, a sufficient condition is given for a multifunction to be “of Souslin type”. However, the proof contains an error; we are required to prove that the multifunction Ω: S→ P×NN defined by

has values which are closed subsets of P×NN (the notation is explained in [1]). The “proof” of this fact given in [1] is manifestly incorrect as it appears to assume that NN has the discrete topology.
Fuzzy machines in a category
Authors: Arbib Michael A. , Manes Ernest G.
Published: April 2009

“Fuzzy theories” and “distributive laws” are used to define “fuzzy systems” in an arbitrary category. The resulting minimal realization theory provides new insights even in classical cases (so that, for non-deterministic sequential machines, the minimal realization problem is reformulated in terms of the structure of join-irreducibles in finite lattices). The definition of “fuzzy theory” is of independent interest and meshes well with philosophical aspects of fuzzy set theory.
Some fixed-point theorems on locally convex linear topological spaces
Authors: Tarafdar E.
Published: April 2009

Let (E, τ) be a locally convex linear Hausdorff topological space. We have proved mainly the following results.

(i) Let f be nonexpansive on a nonempty τ-sequentially complete, τ-bounded, and starshaped subset M of E and let (I-f) map τ-bounded and τ-sequentially closed subsets of M into τ-sequentially closed subsets of M. Then f has a fixed-point in M.

(ii) Let f be nonexpansive on a nonempty, τ-sequentially compact, and starshaped subset M of E. Then f has a fixed-point in M.

(iii) Let (E, τ) be τ-quasi-complete. Let X be a nonempty, τ-bounded, τ-closed, and convex subset of E and M be a τ-compact subset of X. Let F be a commutative family of nonexpansive mappings on X having the property that for some f1 ∈ F and for each x ∈ X, τ-closure of the set

contains a point of M. Then the family F has a common fixed-point in M.
On common fixed points for a family of mappings
Authors: Husain S.A. , Sehgal V.M.
Published: April 2009

The purpose of this paper is to obtain some common fixed point theorems for a family of mappings in a complete metric space. The results herein improve some of the recent theorems of Kiyoshi Iséki (Bull. Austral. Math. Soc. 10 (1974), 365–370).
Nonstandard topological extensions
Authors: Herrmann Robert A.
Published: April 2009

This paper investigates the nonstandard theory of filters on a non-empty meet-semi-lattice of sets and applies this theory to the general study of topological extensionsYfor a spaceX.In particular, we apply this theory to Baire and quasi-H-closed extensions as well as Wallman type compactifications. Whereas these extensions have previously teen obtained and studied as types of ultrafilter extensions, we study them as subsets of an enlargement ofX.SinceX⊂Y⊂ ◯ and the elements ofXandY-Xare of the same set-theoretic type, these extensions appear more natural from the nonstandard viewpoint.
Asymptotic behavior of nonoscillatory solutions of second order functional differential equations
Authors: Kusano Takaŝi , Onose Hiroshi
Published: April 2009

The asymptotic behavior of nonoscillatory solutions of the second order functional differential equation

is studied. First, in the case when a(t)is oscillatory, sufficient conditions are given in order that all bounded non-oscillatory solutions of (*) approach zero as t → ∞. Secondly, in the case when a(t) is nonnegative, conditions are provided under which all nonoscillatory solutions of (*) tend to zero as t → ∞.
Centralizers involving Mathieu groups
Authors: Curran M.J.
Published: April 2009

A simple group G cannot contain a central involution t with CG(t) = 〈t〉 × M, where M is isomorphic to a simple Mathieu group.
A question of Babai on groups
Authors: Hickman J.L. , Neumann B.H.
Published: April 2009

László Babai raised the question whether every infinite group G contains a set B of elements, of cardinal equal to the order of G, such that for elements a, b, c of B the equation a = bc−1b implies a = c. We show that the answer is affirmative for certain classes of groups, including the soluble groups and the countable groups, if the Axiom of Choice is assumed, but negative, even for abelian groups, if a different axiom, compatible with Zermelo-Fraenkel set theory but incompatible with the Axiom of Choice, is assumed.
On the transcendency of the solutions of a special class of functional equations
Authors: Mahler Kurt
Published: April 2009

Let a(z) and b(w) be two rational functions in z or W with algebraic coefficients, where a(0) = 0 and let

Assume that 0 < |z| < 1, that z is not a pole of for n ≥ 0, that w is neither a pole nor a zero of b(w, n) for n ≥ 1, and that the series

for fixed w is a transcendental function of z. Then, if z and W are algebraic numbers, f(z, w) is a transcendental number.
On convergence in capacity
Authors: Meyer Burnett
Published: April 2009

The (logarithmic) capacity or transfinite diameter is originally defined for compact sets in the complex plane. An extension may be made by defining the capacity of a given arbitrary set in the plane as the supremum of the capacities of all compact sets contained in the given set. Convergence in capacity is defined analogously to convergence in measure. It is shown in this paper that properties of convergence in capacity are also analogous to those of convergence in measure.