bluuebunny/crossref_metadata_2025 huggingface dataset:
On the Matrix Equation X′X = A
Authors: Maxfield John E. , Minc Henryk
Published: January 2009

If X is a matrix with non-negative entries then X′X is positive semi-definite with non-negative entries. Conversely, if A is positive semi-definite then there exist matrices Y, not necessarily with non-negative entries, such that Y′Y = A. In the present paper we investigate whether, given a positive semidefinite matrix A with non-negative entries, the equation X′X = A has a solution X with non-negative entries. An equivalent statement of the problem is: Can a positive semi-definite matrix with non-negative entries be expressed as a sum of rank 1 positive semi-definite matrices with non-negative entries? We answer the question in the affirmative for n≦4 and quote the following example due to M.
A Proof of Kirkman's Hypothesis
Authors: Watson G. N.
Published: January 2009

The theorem which I propose to establish first attracted my attention while I was turning over the pages of a volume of Cayley's Collected Mathematical Papers (Cayley, 1). The enunciation of the theorem (with no attempt towards a proof) had been published earlier by Kirkman (3) in a lengthy paper on combinatorial analysis (one of the three-score papers of which Kirkman was the author); among the topics discussed in this paper was the enumeration of the total number of different ways D(r, k) in which a (convex) polygon of r sides can be dissected into k+l parts by drawing k non-intersecting diagonals (i.e., no two diagonals may cross each other except at a vertex or outside the polygon).
An Analogue of a Result of Jacobsthal
Authors: Cohen Eckford
Published: January 2009

Jacobsthal (4)has proved that the n×n matrix

is invertible with the inverse,

Here μ(x) denotes the Möbius function for positive integral x and is assumed to be 0 for other values; [x] has its usual meaning as the number of positive integers ≦x.
A Cyclic Inequality and an Extension of it. II
Authors: Diananda P. H.
Published: January 2009

Throughout this paper, unless otherwise stated,nandLstand for positive integers and α,t,x,x1,x2, … for positive real numbers. Let

where

and
Embedding Theorems for Groups
Authors: Chehata C. G.
Published: January 2009

By a partial endomorphism of a group G we mean a homomorphic mapping μ of a subgroup A of G onto a subgroup B of G. If μ is denned on the whole of G then it is called a total endomorphism. We call a partial endomorphism totally extendable (or extendable) if there exists a supergroup G*⊇G with a total endomorphism μ* which extends μ in the sense that gμ* = gμ, whenever the right-hand side is defined (3).
Thermal Stress in Anisotropic Cylinders
Authors: Eason G.
Published: January 2009

In recent years considerable attention has been given to problems of thermal stress in isotropic materials. Much of this work has been devoted to statical problems although there has been some work on problems with time dependence, for example the quasi-static solutions obtained by Sternberg (1) and Eason and Sneddon (2). A good deal of interest has also been shown in statical thermal stress problems when the material is anisotropic. For the type of material considered here statical problems have been investigated by Grechushnikov and Brodovskii (3) and Sirotin (4) among others. Little attention has been given, however, to time dependent thermal stress problems when the material is anisotropic.
An Application of the Addition Theorem for Determinants
Authors: Jack Henry
Published: January 2009

The integral evaluated in this note was suggested by the famous one connected with the Poincaré polynomials of the classical groups (see (1)).

Let X be an n × n matrix whose elements depend on k parameters. Denote by a manifold in Euclidean space of dimension n2, with the property that if X ∈ , then so does XI−i for 1≦i≦n, where I−i is the unit matrix I altered by a minus sign in the (i, i)th place. Suppose further that there exists on a measure which is invariant under the transformation X→XI−i. Such manifolds and measures exist. For example (see (2), § 5), the set of all proper and improper n×n orthogonal matrices H is such a manifold, the H depending on ½n(n−1) parameters because of the orthogonality and normality of the columns of H. Since the set of all H is a compact topological group, an invariant measure exists.
Note on a Difference-Product Inequality
Authors: Aitken A. C.
Published: January 2009

L. J. Mordell has recently considered (1) the squared modulus of a complex difference-product, namely

under the conditions

and also under the quite different condition

He proves that under (2) the maximum of δ is nn, and is attained when and only when the zr are vertices of a regular n-gon on the unit circle.
Summation and uncountable bi-orthogonal systems in locally convex spaces
Authors: Joiner Harry F.
Published: January 2009

The purpose of this paper is to extend to locally convex spaces and to uncountable systems several well-known results concerning infinite series, biorthogonal sequences, and Schauder bases. Section 2 gives three extensions of the theorem of Orlicz (10) and Pettis (11) and some lemmas that will be needed later. The third section introduces the notions of a summability basis and a summability basis of subspaces, and two main theorems are proved, including a simplification of Retherford and McArthur's proof (12) of a theorem of Nikol'skiĭ (9). Section 4 investigates the positive cone of an uncountable biorthogonal system, particularly conditions equivalent to the regularity of this cone.
A note on free inverse semigroups
Authors: O'Carroll L.
Published: January 2009

Recently Scheiblich (7) and Munn (3), amongst others, have given explicit constructions for FIA, the free inverse semigroup on a non-empty set A. Further, Reilly (5) has investigated the free inverse subsemigroups of FIA. In this note we generalise two of Reilly's lesser results, and also characterise the surjective endomorphisms of FIA. The latter enables us to determine the group of automorphisms of FIA, and to show that if A is finite then FIA is Hopfian (a result proved independently by Munn (3)). Finally, we give an alternative proof of Reilly's main theorem, which uses Munn's theory of birooted trees.
On supports of semigroups of measures
Authors: Chow H. L.
Published: January 2009

Let S denote a compact semitopological semigroup (i.e. the multiplication is separately continuous) and P(S) the set of probability measures on S. Then P(S) is a compact semitopological semigroup under convolution and the weak * topology (4). Let Γ be a subsemigroup of P(S) and where supp μ is the support of μ ∈P(S). In the case in which S is commutative it was shown by Glicksberg in (4) that S(Γ) is an algebraic group in S if Γ is an algebraic group. For a general semigroup S, Pym (7) considered Γ = {η}, η being an idempotent, and established that S(Γ) is a topologically simple subsemigroup of S, i.e. every ideal of S(Γ) is dense in S(Γ). In this note we prove that if Γ is a simple subsemigroup of P(S) (a semigroup is simple if it contains no proper ideal) which contains an idempotent then S(Γ) is a topologically simple subsemigroup of S. We also give an example to show that our conclusion (hence also Pym's) is best possible in the sense that S(Γ) is not simple in general
The osculating spaces of a certain curve in [n]
Authors: Edge W. L.
Published: January 2009

The curve in question is the non-singular intersection Γ of the n − 1 quadric primals

where it is presumed that no two of the n + 1 numbers aj are equal. Define

then it will be seen that the osculating prime of Γ at x = ξ is
Finite dimensionality, nilpotents and quasinilpotents in Banach algebras
Authors: Duncan J. , Tullo A. W.
Published: January 2009

In this note we present some rather loosely connected results on Banach algebras together with some illustrative examples. We consider various conditions on a Banach algebra which imply that it is finite dimensional. We also consider conditions which imply the existence of non-zero nilpotents, and hence the existence of finite dimensional subalgebras. In the setting of Banach algebras quasinilpotents figure more prominently than nilpotents. We give an example of a non-commutative Banach algebra in which 0 is the only quasinilpotent; this resolves a problem of Hirschfeld and Zelazko (4).
Spectral approximants of normal operators
Authors: Halmos P. R.
Published: January 2009

For each non-empty subset Λ of the complex plane, let (Λ) be the set of all those operators (on a fixed Hilbert space H) whose spectrum is included in Λ. The problem of spectral approximation is to determine how closely each operator on H can be approximated (in the norm) by operators in (Λ). The problem appears to be connected with the stability theory of certain differential equations. (Consider the case in which Λ is the right half plane.) In its general form the problem is extraordinarily difficult. Thus, for instance, even when Λ is the singleton {0}, so that (Λ) is the set of quasinilpotent operators, the determination of the closure of (Λ) has been an open problem for several years (3, Problem 7).
A nonselfadjoint dynamical system
Authors: Kreith Kurt
Published: January 2009

This paper concerns criteria for assuring that every solution of a real fourth order nonselfadjoint differential equation

is oscillatory at x = ∞. Our technique is a generalisation of that used by Whyburn (1) for the study of the selfadjoint equation,

combined with the theory of H-oscillation of vector equations as introduced by Domšlak (2) and studied by Noussair and Swanson (3). Whyburn's technique consists of representing (1.2) as a dynamical system of the form

and then studying (1.3) in terms of polar coordinates in the y, z-plane. In Section 2 below we show how to represent (1.1) as a dynamical system of the form
On (von Neumann) regular rings
Authors: Ming R. Yue Chi
Published: January 2009

Throughout, A denotes an associative ring with identity and “module” means “left, unitary A-module”. In (3), it is proved that A is semi-simple, Artinian if A is a semi-prime ring such that every left ideal is a left annihilator. A natural question is whether a similar result holds for a (von Neumann) regular ring. The first proposition of this short note is that if A contains no non-zero nilpotent element, then A is regular iff every principal left ideal is the left annihilator of an element of A. It is well-known that a commutative ring is regular iff every simple module is injective (I. Kaplansky, see (2, p. 130)). The second proposition here is a partial generalisation of that result.
Some results involving weak compactness in C(X), C(υX) and C(X)′
Authors: Tweddle I.
Published: January 2009

The main aim of the present note is to compare C(X) and C(υX), the spaces of real-valued continuous functions on a completely regular space X and its real 1–1 compactification υX, with regard to weak compactness and weak countable compactness. In a sense to be made precise below, it is shown that C(X) and C(υX) have the same absolutely convex weakly countably compact sets. In certain circumstances countable compactness may be replaced by compactness, in which case one obtains a nice representation of the Mackey completion of the dual space of C(X) (Theorems 5, 6, 7).
On metacyclic Fibonacci groups
Authors: Campbell C. M. , Robertson E. F.
Published: January 2009

Let Fn be the free group on {ai: i ∈ Zn}, where the set of congruence classes mod n is used as an index set for the generators. Let φ be the permutation (1, 2, 3, …, n) of Zn and denote by θ the automorphism of Fn induced by φ, namely
The hyperosculating spaces to certain curves in [n]
Authors: Dye R. H.
Published: January 2009

Let Γ be an irreducible and non-singular curve in [n] (n ≧ 3) which is the complete intersection of n − 1 primals of order m (m ≧ 2) with a common “self-polar” simplex S: by this I mean that the rth polar of each vertex of S with respect to any one of the defining primals is the opposite face of S counted m−r times, for r = 1, 2, …, m − 1. The various such Γ constitute the curves of the title; they were encountered in (2). When m = 2, Γ is the intersection of n − 1 quadrics with a common self-polar simplex in the familiar classical sense.
The number of primes representable as the sum of two square-free squares
Authors: Keates M.
Published: January 2009

In this paper an asymptotic formula is obtained for the number of primes representable as the sum of two square-free squares. The precise result is:

Theorem 1.Let N(x) be the number of primes not exceeding x represented by the quadratic formy2 + z2, where y and z are square-free. Let w be a fixed arbitrarily large number. Then

where

and
A Harnack inequality for degenerate elliptic equations on minimal surfaces
Authors: Cooper Frances
Published: January 2009

Harnack inequalities are known to be of great importance in the theory of quasilinear elliptic partial differential equations. In the case of such equations defined over a domain Ω in Rn, inequalities of this type have been proved for solutions of second-order equations in divergence form which are of either elliptic or degenerate elliptic structure. More recently Bombieri and Giusti (2) have proved a Harnack inequality for solutions of linear elliptic equations on a minimal surface in Rn+1. The equations are of the form

where summation over i, j = 1, …, n+l is understood, and δ = (δ1, …, δn+1) is the tangential derivative on S. In (2), the inequality is used to give much simplified proofs of some classical results on minimal surfaces, and to generalise some more recent ones.
A summability method due to linear differential equations and a uniqueness property of solutions of singular differential equations
Authors: Gingold H.
Published: January 2009

The purpose of this paper is to expose a method which will match a function f(z) existing in a domain D to a formal series whose radius of convergence may be zero. This matching process has to be done in a “natural way”, and has to be “regular”, which means that if a power series converges absolutely in the circle E = {z | |z|<r} then the summability function f(z) produced by our method in the domain D and matched to will coincide with in the domain E∩D. Euler, in his time, matched the function to the power series .
A note on train algebras
Authors: Abraham Victor M.
Published: January 2009

Train algebras were first introduced by Etherington in (1) and proved very useful in dealing with problems in mathematical genetics. The types of algebras which arose were commutative, non-associative and finite-dimensional. It proved convenient in the general theory to regard them as defined over the complex numbers. We remind the reader of some basic definitions. A baric algebra is one which admits a non-trivial homomorphism into its coefficient field K. A (principal) train algebra is baric and has a rank equation in which the coefficients of a general element x depend only on its baric value, generally called the weight of x. A special train algebra (STA) is a baric algebra in which the nilideal is nilpotent and all its right powers are ideals; the nilideal being the set of elements of A of weight zero. In (2) Etherington showed that in a baric algebra one can always take a very simple basis consisting of a distinguished element of unit weight and all other basis elements of weight zero.
A note on “simplification and scaling”
Authors: Riley N.
Published: January 2009

In a recent paper Segel (1) points out that the diverse techniques (which “comprise the core of the applied mathematicians art (or craft)”) of the applied mathematician, although in general reliably proven, are “rarely explicitly delineated but rather are transmitted indirectly and informally”. In his article Segel aims to clarify two such techniques, namely:

(i) Scaling—or how to choose dimensionless variables in such a way that the relative size of the various terms in an equation is explicitly indicated by the magnitudes of the dimensionless parameters which precede them,

(ii) Simplification—a procedure in which a term is neglected under the assumption that it is small, and the consistency of the assumption checked later.
A note on the Todd-Coxeter coset enumeration algorithm
Authors: Beetham M. J. , Campbell C. M.
Published: January 2009

In (8) Todd and Coxeter described an algorithm for enumerating the cosets of a finitely generated subgroup of finite index in a finitely presented group. Several authors ((1), (2), (5), (6), (7)) have discussed a modification of the algorithm to give also a presentation of the subgroup in terms of the given generators.
Some inequalities for norm unitaries in Banach algebras
Authors: Crabb M. J. , Duncan J.
Published: January 2009

Let A be a complex unital Banach algebra. An element u∈A is a norm unitary if

(For the algebra of all bounded operators on a Banach space, the norm unitaries arethe invertible isometries.) Given a norm unitary u∈A, we have Sp(u)⊃Γ, where Sp(u) denotes the spectrum of u and Γ denotes the unit circle in C. If Sp(u)≠Γ we may suppose, by replacing eiθu, that . Then there exists h ∈ A such that
Generalised Euler constants
Authors: Knopfmacher J.
Published: January 2009

Let the Laurent expansion of the Riemann zeta function ξ(s) about s=1 be written in the form

It has been discovered independently by many authors that, in terms of this notation, the coefficient
Numerical ranges of derivations
Authors: Kyle J.
Published: January 2009

In this paper we shall examine the relationship between the numerical range of aninner derivation, and that of its implementing element.

Much of this paper is taken from the author's doctoral thesis (5) written at theUniversity of Newcastle upon Tyne with the helpful guidance of Professor B. E.Johnson. The author also acknowledges the financial support of the Science ResearchCouncil during the period of this research.
Radicals in Categories
Authors: Holcombe Michael , Walker Roland
Published: January 2009

The study of radicals in general categories has followed several lines of development. The problem of defining radical properties in general categories has been considered by Kurosh and Shul'geifer, see (7). Under mild conditions on their categories they obtain sufficient conditions for the existence of radical functors which are closely related to radical properties. Another approach is by Maranda (5) and Dickson (3) who studied idempotent radical functors and torsion theories in abelian categories. Our aim has been to study radical functors in as general a category as possible. To this end we introduce the concept of an R-category. The categories of rings, modules, near-rings, groups and Jordan algebras are all examples of R-categories.
Lattice isomorphisms of inverse semigroups
Authors: Jones P. R.
Published: January 2009

A largely untouched problem in the theory of inverse semigroups has been that of finding to what extent an inverse semigroup is determined by its lattice of inverse subsemigroups. In this paper we discover various properties preserved by lattice isomorphisms, and use these results to show that a free inverse semigroup ℱℐx is determined by its lattice of inverse subsemigroups, in the strong sense that every lattice isomorphism of ℱℐx upon an inverse semigroup T is induced by a unique isomorphism of ℱℐx upon T. (A similar result for free groups was proved by Sadovski (12) in 1941. An account of this may be found in Suzuki's monograph on the subject of subgroup lattices (14)).
On central traces and groups of symmetries of order unit Banach spaces
Authors: Chu Cho-Ho
Published: January 2009

A central trace on an order-unit Banach space A(K) is a centre-valued module homomorphism invariant under the group of symmetries of A(K).

The concept of central traces has been crucial in the theory of types for convex sets established in (4), (5). In von Neumann algebras, they are precisely the canonical centre-valued traces and their existence hinges on a fundamental theorem (Dixmier's approximation process) in von Neumann algebras. On the other hand, the existence of central traces in finite dimensional spaces is an easy consequence of Ryll-Nardzewski's fixed point theorem (5).
On Near-Rings of Quotients
Authors: Oswald A.
Published: January 2009

In (2), Holcombe investigated near-rings of zero-preserving mappings of a group Γ which commute with the elements of a semigroup S of endomorphisms of Γ and examined the question: under what conditions do near-rings of this type have near-rings of right quotients which are 2-primitive with minimum condition on right ideals? In the first part of this paper (§2) we investigate further properties of near-rings of this type. The second part of the paper (§3) deals with those near-rings which have semisimple near-rings of right quotients. Our results here are analogous to those of Goldie (1); in particular, with a suitable definition of finite rank we prove that a near-ring which has a semisimple near-ring of right quotients has finite rank
On Left Bipotent Near-Rings
Authors: Jat J. L. , Choudhary S. C.
Published: January 2009

A near-ring N is defined to be left bipotent if Na = Na2 for each a in N. Many properties of such near-rings are proved in Section 1, and results of Chandran (4) are generalised. Most of the results are different from, and contrary to, the ring case. Necessary and sufficient conditions have also been obtained under which such near-rings become regular. Section 2 deals with left bipotent near-rings without zero divisors. Some structure theorems for direct sum decompositions and J(N) = (0) are proved and it is shown that for a left bipotent S-near-ring, the singular ‘set’ S(N) = 0. Necessary examples and counter examples are supplied.
Asymptotic Behaviour of Eigen-Values of Certain Integral Equations
Authors: Reade J. B.
Published: January 2009

The integral equations we consider are

and

where 0 < α < 1. The asymptotic behaviour of the eigen-values of the latter equation is already known (see (1) and (4)). The former equation has been studied by many authors but as yet no explicit statement seems to have been made about the behaviour of its eigen-values.
Centralisers in Wreath Products
Authors: Meldrum J. D. P.
Published: January 2009

In this paper, the centraliser of an arbitrary element of a wreath product is determined. One application of this is to find the breadth of a wreath product (Theorems 21 and 22), a problem which was raised in discussion with Dr. I. D. Macdonald. Another application is to groups generated by elements generating their own centralisers (Theorem 20).

Let A and B be two groups. Define

AB = {f : B → A; f(b) = e for all but a finite number of elements of B} to be a group by defining the product pointwise
Logarithms in multiplier algebras
Authors: Wood G. V.
Published: January 2009

In (3) it is shown that, for a locally compact abelian group G and x∈G, δx has a logarithm in M(G) if and only if x has finite order. Since M(G) can be identified with the multipliers of L1(G), one might expect a similar result for the algebras of multipliers on Lp(G) for 1 < p < ∞. However, in contrast, it is shown in (2) that for a locally compact abelian group G and 1 < p < ∞, every translation operator on Lp(G) has a logarithm in the multiplier algebra. Here we consider whether the same results are true for non-abelian groups.
Generators for subgroups of wreath products
Authors: Hulse J. A.
Published: January 2009

IfGis a group, let

m(G)

be the least cardinal such thatGmay be generated bym(G) elements, and let

m,(G)
Commutators in Banach algebras
Authors: Pták Vlastimil
Published: January 2009

In a recent paper (6) the present author has shown that, for an element a of a Banach algebra A, the condition

for all x∈A and some constant α is equivalent to [x, a]∈Rad a for all x∈A; it turns out that α may be replaced by |α|σ It is the purpose of the present note to investigate a related condition
A quasi-equivalence between Borel summability and convergence for Fourier-Laguerre series at the end-point
Authors: Lorch Lee , Tzimbalario Jean
Published: January 2009

A suitable function f(x) 0 ≤ x < ∞, can be expanded into a Fourier series of Laguerre polynomials , whose interval of orthogonality is 0 ≤ x < ∞. The usual problems as to convergence and, lacking convergence, summability, and also the asymptotic behaviour of Lebesgue constants, arise for such developments. A summary of work on these convergence and summability problems, together with extensive references to the literature, can be found in the standard treatise by G. Szegö (5, especially Chapter IX) to whom many of these results are due.
Semi-embeddings of Banach space
Authors: Lotz Heinrich P. , Peck N. T. , Porta Horacio
Published: January 2009

It is a most implausible fact that a one-to-one operator from c0 into a Banach space which maps the unit ball of c0 onto a closed set is necessarily an isomorphism.

In this paper the term semi-embedding denotes a one-to-one operator from one Banach space into another, which maps the closed unit ball of the domain onto a closed set. In the first section we study semi-embeddings in conjunction with weak compactness; in the second section we apply our results to the case of semi-embeddings defined on C(X), X compact.
Spectral radius formulae
Authors: Murphy G. J. , West T. T.
Published: January 2009

If A is a complex Banach algebra (not necessarily unital) and x∈A, σ(x) will denote the spectrum and spectral radius of x in A. If I is a closed two-sided ideal in A let x + I denote the coset in the quotient algebra A/I containing x. Then
Factorisable right adequate semigroups
Authors: El-Qallali Abdulsalam
Published: January 2009

On a semigroup S the relation ℒ* is defined by the rule that (a, b) ∈ ℒ* if and only if the elements a, b of S are related by Green's relation ℒ in some oversemigroup of S. It is well known that for a monoid S, every principal right ideal is projective if and only if each ℒ*-class of S contains an idempotent. Following (6) we say that a semigroup with or without an identity in which each ℒ*-class contains an idempotent and the idempotents commute is right adequate. A right adequate semigroup S in which eS ∩ aS = eaS for any e2 = e, a ∈ S is called right type A. This class of semigroups is studied in (5).
The normal subgroup structure of the infinite general linear group
Authors: Arrell David G.
Published: January 2009

The classification of the normal subgroups of the infinite general linear group GL(Ω, R) has received much attention and has been studied in, for example, (6), (4) and (2). The main theorem of (6) gives a complete classification of the normal subgroups of GL(Ω, R) when R is a division ring, while the results of (2) require that R satisfies certain finiteness conditions. The object of this paper is to produce a classification, along the lines of that given by Wilson in (7) or by Bass in (3) in the finite dimensional case, that does not require any finiteness assumptions. However, when R is Noetherian, the classification given here reduces to that given in (2).
Radicals of rings and their subrings
Authors: PuczyŁowski E. R.
Published: January 2009

It is a fundamental fact in the theory of radicals of associative rings that if S is a radical and I is a two-sided ideal of R then S(I)⊆S(R). In view of this result it seems to be interesting to investigate radicals satisfying such or similar connections for other type of subrings. There are many works devoted to similar problems (2, 8, 8, 10). In this paper we try to get a uniform description of some facts in this area.
Lie algebras all of whose maximal subalgebras have codimension one
Authors: Towers David
Published: January 2009

Let denote the class of finite-dimensional Lie algebras L (over a fixed, but arbitrary, field F) all of whose maximal subalgebras have codimension 1 in L. In (2) Barnes proved that the solvable algebras in are precisely the supersolvable ones. The purpose of this paper is to extend this result and to give a characterisation of all of the algebras in . Throughout we shall place no restrictions on the underlying field of the Lie algebra.
Whitehead squares in Thom complexes
Authors: Sutherland W. A.
Published: January 2009

One of the aims of this paper is to examine the following conjecture, attributed to Mahowald on p. 255 of (2), Part 2. Let M be a closed connected smooth manifold of odd dimension q (q≠l,3,7) and with tangent bundle τ. Let the inclusion of a compactified fibre in the Thorn complex of τ be written μ: Sq → Tτ.
Multiplicative transfers in ordinary cohomology
Authors: Steiner Richard
Published: January 2009

In [10] Segal shows that the groups of units in certain ordinary cohomology rings are the zeroth terms of generalised cohomology theories. Geometric methods then give a multiplicative transfer on these groups of units for fibrations with finite fibres; see Kahn and Priddy [6] and Adams ([1], 4). On the other hand Evens [5] by manipulations with cochains has constructed a multiplicative transfer in the cohomology of a groupGand a subgroupHof finite index. Now it is well known that the algebraic cohomology ofGandHcan be identified with the topological cohomology of their classifying spacesBGandBH, and that there is a fibrationBH→BGwith finite fibres. This suggests that Evens' algebraic transfer and the geometric transfer derived from Segal's work may be related. In the present paper I confirm this by constructing a common generalisation; I also describe some of its properties.
A counterexample in the theory of Hermitian liftings
Authors: Legg D. A.
Published: January 2009

In [3], [8], and [2], it was shown that if is an essentially Hermitian operator on l P, 1≦ p<∞, or on Lp[0,1], 1< p<∞, then T is a compact perturbation of a Hermitian operator. In [1], this result was established for operators on Orlicz sequence space l M, where 2∉[α M,β M] (the associated interval for M). In that same paper, it was conjectured that this result does not in general hold if 2∈[α M,β M]. In this paper, we show that this conjecture is correct by exhibiting an Orlicz sequence space l M and an essentially Hermitian operator on l M which is not a compact perturbation of a Hermitian operator.
Centraliser near-ring representations
Authors: Maxson C. J. , Smith K. C.
Published: January 2009

Let V be a group, written additively but not necessarily abelian, and let S be a semigroup of endomorphisms of V. The set C( S; V)={ f: V→ V| fσ=σ f for all σ∈ S and f(0)=0} forms a zero-symmetric near-ring with identity under the operations of function addition and composition, called the centraliser near-ring determined by S and V. Centraliser near-rings are very general, for if N is any zero-symmetric near-ring with 1 then there exists a group V and a semigroup S of endomorphisms of V such that N≃ C( S; V).
The approximation of bivariate functions by sums of univariate ones using the L1-metric
Authors: Light W. A. , McCabe J. H. , Phillips G. M. , Cheney E. W.
Published: January 2009

We shall study a special case of the following abstract approximation problem: givena normed linear space E and two subspaces, M1 and M2, of E, we seek to approximate f ∈ E by elements in the sum of M1 and M2. In particular, we might ask whether closest points to f from M = M1 + M2 exist, and if so, how they are characterised. If we can define proximity maps p1 and p2 for M1 and M2, respectively, then an algorithm analogous to the one given by Diliberto and Straus [4] can be defined by the formulae
Groups whose irreducible representations have finite degree II
Authors: Wehrfritz B. A. F.
Published: January 2009

If F is a (commutative) field let denote the class of all groups G such that every irreducible FG-module has finite dimension over F. The introduction to [7] contains motivation for considering these classes and surveys some of the results to date concerning them. In [7] for every field F we determined the finitely generated soluble groups in . Here, for fields F of characteristic zero, we determine, at least in principle, the soluble groups in . Our main result is the following.
Closed ideals and bi-orthogonal systems in radical Banach algebras of power series
Authors: Thomas Marc P.
Published: January 2009

In this paper we investigate the weighted convolution algebras lp(ωn), where 1≦p<∞ and {ωn} is a sequence of positive weights satisfying the following conditions. If p = 1 we require ω0 = 1, and ωt+s≦ωtωs.Then
Small fractional parts of quadratic forms
Authors: Baker R. C. , Harman G.
Published: January 2009

Let ‖x‖ denote the distance of x from the nearest integer. In 1948 H. Heilbronn proved [5] that for ε>0 and N>c1(ε) the inequality

holds for any real α. This result has since been generalised in many different directions, and we consider here extensions of the type: For ε>0, N>c2{ε, s) and a quadratic formQ(x1,…, xs) there exist integersn1,…,nsnot all zero with |n1|,…,|ns≦N and with
Angular momentum, convex Polyhedra and Algebraic Geometry
Authors: Atiyah M. F.
Published: January 2009

The three families of classical groups of linear transformations (complex, orthogonal, symplectic) give rise to the three great branches of differential geometry (complex analytic, Riemannian and symplectic). Complex analytic geometry derives most of its interest from complex algebraic geometry, while symplectic geometry provides the general framework for Hamiltonian mechanics.

These three classical groups “intersect” in the unitary group and the three branches of differential geometry correspondingly “intersect” in Kähler geometry, which includes the study of algebraic varieties in projective space. This is the basic reason why Hodge was successful in applying Riemannian methods to algebraic geometry in his theory of harmonic forms.
Pairs of additive equations III: quintic equations
Authors: Cook R. J.
Published: January 2009

We consider R simultaneous equations of additive type

where the coefficients aij are integers. Artin's conjecture, for additive forms, is that the equations (1) have a non-trivial solution in integers x1,…,xN provided that they have a non-trivial real solution, which is clearly satisfied when k is odd, and
Maximum idempotents in naturally ordered regular semigroups
Authors: McAlister D. B. , McFadden R.
Published: January 2009

We shall denote by ω the natural partial order on the idempotents E = E(S) of a regular semigroup S, so that in E,

A partially ordered semigroup S(≦) is called naturally partially ordered [9] if the imposed partial order ≦ extends ω in the sense that

No assumption is made about the reverse implication.
In which metric spaces are parallel bodies of closed sets closed?
Authors: Beer Gerald
Published: January 2009

Let X be a metric space with metric d and for each x in X let Bλ[x] denote the closed ball of radius λ about x. Following Valentine [15] if K⊂X and λ is positive, then we call the set Bλ[K]=∪x∈KBλ[x] the λ-parallel body of K. The following fact is obvious.
Polynomial approximation of an entire function and rate of growth of Taylor coefficients
Authors: Freund M. , Görlich E.
Published: January 2009

The best uniform approximation of a functionfon [-1,1] by real algebraic polynomials satisfies

if and only if ƒ is the restriction to [- 1,1] of an entire function (Bernstein [2], p. 113, see also [12], pp. 83–85). For such functions ƒ the rate of best approximation has been characterized by Varga [24], Reddy [24], Shah [21], and Kapoor and Nautiyal [10] in terms of order and type of ƒ , lower order and type, and in terms of more general concepts of order. On the other hand, order and type of ƒ are connected with the Taylor coefficients, i.e. with the rate of growth of the sequence(see [23], p. 41 or [3], pp. 11/12; cf. also [19], [20], [6], [7], [8]) and this has been extended to iterated orders by Schonhage [17], Sato [16], Reddy [14], Juneja, Kapoor, and Bajpai [9] (also [22], [13]), and to generalized orders by Seremeta [18], Bajpai, Gautam, and Bajpai [1] as well as Kapoor and Nautiyal [10]. Combining the two kinds of characterizations (as done, e.g., by Reddy [15], p. 105) approximation theorems in terms of the sequenceare obtained. But in such results the rate ofbest approximation is always described by a limit relation, e.g. of the form, and this causes a considerable loss of precision, as will be discussed in more detail in Section 3 (in this respect cf. also the remark by Bernstein [2], pp. 114/115).
Boundary problems for Riccati and Lyapunov equations
Authors: Jódar Lucas
Published: January 2009

The resolution problem of the system

where U(t), A, B, D and Uo are bounded linear operators on H and B* denotes the adjoint operator of B, arises in control theory, [9], transport theory, [12], and filtering problems, [3]. The finite-dimensional case has been introduced in [6,7], and several authors have studied the infinite-dimensional case, [4], [13], [18]. A recent paper, [17],studies the finite dimensional boundary problem

where t ∈[0,b].In this paper we consider the more general boundary problem

where all operators which appear in (1.2) are bounded linear operators on a separable Hilbert space H. Note that we do not suppose C = −B* and the boundary condition in (1.2) is more general than the boundary condition in (1.1).
Existence and uniqueness in the theory of bending of elastic plates
Authors: Constanda Christian
Published: January 2009

Kirchhoff's kinematic hypothesis that leads to an approximate two-dimensional theory of bending of elastic plates consists in assuming that the displacements have the form [1]

In general, the Dirichlet and Neumann problems for the equilibrium equations obtained on the basis of (1.1) cannot be solved by the boundary integral equation method both inside and outside a bounded domain because the corresponding matrix of fundamental solutions does not vanish at infinity [2]. However, as we show in this paper, the method is still applicable if the asymptotic behaviour of the solution is suitably restricted.
On the strong maximum principle for parabolic differential equations
Authors: Walter Wolfgang
Published: January 2009

In a recent paper [2], D. Colton has given a new proof for the strong maximum principle with regard to the heat equation ut = Δu. His proof depends on the analyticity (in x) of solutions. For this reason it does not carry over to the equation

or to more general equations. But in order to tread mildly nonlinear equations such asut = Δu + f(u) which are important in many applications, it is essential to have the strong maximum principle at least for equation (*). It should also be said that this proof uses nontrivial facts about the heat equation.
On transversality
Authors: Bruce J. W.
Published: January 2009

The notion of transversality has proved of immense value in differential topology. The Thom transversality lemma and its many variants show that transversality is a dense,and often open, property. In one parameter families the occurrence of non-transversality is inevitable; for example one cannot pull two linked curves in ℝ3 apart without a non transverse intersection. The aim of this note is to prove the following. In any generic family of mappings each map in the family fails to satisfy some fixed transversality conditions at worst at isolated points, and even at these points in rather special sorts of way. So, returning to the above example, given two space curves C1 and C2 without a (necessarily non-transverse) intersection we expect, in any genericisotopy of C2, that it will meet C1 if at all, at isolated points In particular generically we do not expect C1 and C2, any time, to have an arc in common
Multipliers of Bergman spaces into Lebesgue spaces
Authors: Luecking Daniel H.
Published: January 2009

Let U be the open unit disk in the complex plane endowed with normalized Lebesgue measure m. will denote the usual Lebesgue space with respect to m, with 0<p<+∞. The Bergman space consisting of the analytic functions in will be denoted . Let μ be some positivefinite Borel measure on U. It has been known for some time (see [6] and [9]) what conditions on μ are equivalent to the estimate: There is a constant C such that

provided 0<p≦q.
Homotopy of the exceptional Lie group G2
Authors: Oka Shichiro
Published: January 2009

Let G be one of the following compact simply connected Lie groups: SU(3), Sp(2), G2. In the first two cases there is a well known stable decomposition of G as Q ∨ Sd where d = dim G and Q is a certain subspace of G. For SU(3), Q is the stunted complex quasiprojective space Σ(ℂP2/ℂP1) which fits into a cofibration sequence S3→Q→S5 with stable attaching map η:S5 → S4 For Sp(2), Q is the quaternionic quasi-projective space ℍℚ1 and fits into a cofibration sequence S3→Q→S7 with stable attaching map 2ν:S7→S4 (Here η and ν are generators of respectively.)
On the “largeness” of one-relator groups
Authors: Edjvet M.
Published: January 2009

If G is a one-relator group on at least 3 generators, or is a one-relator group with torsion on at least 2 generators, then it follows from results in [1] and [6] that G has a subgroup of finite index which can be mapped homomorphically onto F2, the free group of rank 2. In the language of [2], G is equally as large as F2, written G⋍F2.
Strongly exposed points in bases for the positive cone of ordered Banach spaces and characterizations of l1(Г)
Authors: Polyrakis Ioannis A.
Published: January 2009

The study of extreme, strongly exposed points of closed, convex and bounded sets in Banach spaces has been developed especially by the interconnection of the Radon–Nikodým property with the geometry of closed, convex and bounded subsets of Banach spaces [5],[2] . In the theory of ordered Banach spaces as well as in the Choquet theory, [4], we are interested in the study of a special type of convex sets, not necessarily bounded, namely the bases for the positive cone. In [7] the geometry (extreme points, dentability) of closed and convex subsets K of a Banach space X with the Radon-Nikodým property is studied and special emphasis has been given to the case where K is a base for acone P of X. In [6, Theorem 1], it is proved that an infinite-dimensional, separable, locally solid lattice Banach space is order-isomorphic to l1 if, and only if, X has the Krein–Milman property and its positive cone has a bounded base.
Approximation methods and the generalised Fuller index for semi-flows in Banach spaces
Authors: Potter A. J. B.
Published: January 2009

In [3] Fuller introduced an index (now called the Fuller index) in order to study periodic solutions of ordinary differential equations. The objective of this paper is to give a simple generalisation of the Fuller index which can be used to study periodic points of flows in Banach spaces. We do not claim any significant breakthrough but merely suggest that the simplistic approach, presented here, might prove useful for the study of non-linear differential equations. We show our results can be used to study functional differential equations.
Two remarks on the Franklin system
Authors: Wojtaszczyk P.
Published: January 2009

The aim of this note is to present two observations about the classical Franklin system. First we show that the Franklin system, when considered in the space generated by special atoms (as defined and studied by Soares de Souza in [11] and ]12]) is an unconditional basis equivalent to the unit vector basis in l1. In our second result we give conceptually simpler proofs and some extensions of the results of Bočkariov's [1] about the conjugate Franklin system.
Inverse multiparameter eigenvalue problems for matrices II
Authors: Browne Patrick J. , Sleeman B. D.
Published: January 2009

This is a sequel to our previous paper [4] where we initiated a study of inverse eigenvalue problems for matrices in the multiparameter setting. The one parameter version of the problem under consideration asks for conditions on a given n × n symmetric matrix A and on n given real numbers s1≦s2≦…≦sn under which a diagonal matrix V can be found so that A + V has sl,…,sn as its eigenvalues. Our motivation for this problem and our method of attack on it in [4]p comes chiefly from the work of Hadeler [5] in which sufficient conditions were given for existence of the desired diagonal V. Hadeler's approach in [5] relied heavily on the Brouwer fixed point theorem and this was also our main tool in [4]. Subsequently, using properties of topological degree, Hadeler [6] gave somewhat different conditions for the existence of the diagonal V. It is our desire here to follow this lead and to use degree theory to give some results extending those in [6] to the multiparameter case.
Note on strongly regular near-rings
Authors: Hongan Motoshi
Published: January 2009

Let S be a semigroup. An element a of S is called right (resp. left) regular if a=a2x (resp. a=xa2) for some x∈S. If a is regular and right (resp. left) regular, a is called strongly right (resp. left) regular. As is well known, if a is strongly regular (i.e., right and left regular) then it is regular, more precisely, there exists uniquely an element x such that a= a2x,x= x2a and ax=xa, and a is contained in a subgroup of S (and conversely).
Groups–St Andrews 1985
Authors: Robertson Edmund F. , Campbell Colin M.
Published: January 2009

The conference Groups–St Andrews 1985 was held at the University of St Andrews from 27 July to 10 August 1985. The conference received financial support from the Edinburgh Mathematical Society, the London Mathematical Society and the British Council. There were 366 participants from 43 countries registered for the conference. Although the conference did not specialize in a particular area of group theory, a glance at Mathematical Reviews shows that the work of the participants is mainly under classifications 20D, 20E and 20F. In part this is because the conference followed an earlier conference [6] which was primarily based on topics falling under 20F.
Superimprimitive 2-generator finite groups
Authors: Macbeath A. M.
Published: January 2009

In his thesis, A. A. Hussein Omar, motivated by the study of possible shapes of generic Dirichlet regions for a surface group, made a detailed study for g = 2,3 of the groups generated by pairs (μ, τ) of regular (i.e. fixed-point-free) permutations of order 2,3 respectively and of degree n = 6(2g − 1), such that μ ْ τ is an n-cycle. He observed that, for g = 2,3, precisely one pair generates what he calls a superimprimitive group, and raised the question whether such pairs exist for all g, and, if so, whether they areunique. Our main result is that they do always exist, but that, for large values of g, theyare far from unique. (For details and some motivation for the notation, see [4, 5].)
When endomorphisms of G inducing automorphisms of G/V are automorphisms
Authors: Macedońska O.
Published: January 2009

Let G denote a relatively free group of a finite or countably infinite rank with a fixed set of free generators x1,x2,…,G′ the commutator subgroup, and V a verbal subgroup belonging to G′. Following H. Neumann [6] we shall use the vector representation for endomorphisms of G. Vector v = (ν1, ν2,…) represents an endomorphism v such that xiv = νi for all i. The identity map is represented by l=(x1,x2…). We need also thetrivial endomorphism 0 = (e, e,…). The length of vectors is equal to the rank of G. We shall consider the near-ring of vectors, with addition and multiplication given below u + v=(ulν1, u2ν2,…) where uiνi; is a product in G, and uv = (u1v, u2v,…) where uiv isthe image of ui, under the endomorphism v. There is only one distributivity law (u + v)w =uw + vw.
PSL(2, q) as an image of the extended modular group with applications to group actions on surfaces
Authors: Singerman David
Published: January 2009

The modular group PSL(2, ℤ), which is isomorphic to a free product of a cyclicgroupof order 2 and a cyclic group of order 3, has many important homomorphic images. Inparticular, Macbeath [7] showed that PSL(2, q) is an image of the modular group if q ≠ 9. (Here, as usual, q is a prime power.) The extended modular group PGL(2, ℤ) contains PSL{2, ℤ) with index 2. It has a presentation

the subgroup PSL(2, ℤ) being generated by UV and VW.
On a Certain Type of Space-Tableau
Authors: Walls Nancy
Published: January 2009

An integer n may be partitioned into the set of integers (α1, α2, …, αl), where Whenever αi−1>αi the substitution of αi+1 for αi, where αi may be αi+1≡0, would give a partition of n+1 whilst the substitution of αi−1—1 for αi−1 would give a partition of n—1; partitions of n+1 and of n—1 so arising will be said to be associated with the given partition of n.