bluuebunny/crossref_metadata_2025 huggingface dataset:
Enhanced lumbar spine bone mineral content in piglets fed arachidonic acid and docosahexaenoic acid is modulated by severity of growth restriction
Authors: Kohut June , Watkins Bruce , Weiler Hope
Published: May 2009

The infant born small size for gestational age (SGA) has low bone mass. Since dietary arachidonic acid (AA) and DHA enhance bone mass in normal-birth-weight piglets the objective of the present study was to test for such benefits in the SGA piglet. In the present 15 d study, two levels of dietary AA and DHA (6:1 ratio of AA:DHA diets, 0·6:0·1 or 1·2:0·2 g/100 g dietary fat) v. a control diet were tested for effects on growth, fatty acid status, whole-body and regional bone mineral content (BMC) and metabolism in SGA piglets categorised as either very low birth weight (VLBW; ≤ 1·0 kg; n 12) or low birth weight (LBW; 1·1 to 1·2 kg; n 18). Differences in outcomes for each body weight category were detected using ANOVA with post hoc Bonferroni tests. Growth was not influenced by diet, yet the LBW piglets fed 0·6:0·1 AA and DHA as g/100 g fat had elevated BMC in the spine, whereas the VLBW piglets had higher BMC of the spine if fed the higher intake of AA and DHA. In both weight categories, the higher intake of AA and DHA lowered bone resorption relative to controls, whereas bone formation was unchanged. Tissue fatty acid concentrations reflected dietary AA and DHA, especially trabecular bone of VLBW piglets. Whether the enhanced lumbar spine BMC is due to enhanced Ca absorption and thus suppression of bone resorption remains to be established.
Matrix representations of semigroups
Authors: McAlister D. B.
Published: May 2009

In a series of papers [6], [7], [8], [10], Munn has considered the problem of constructing all irreducible representations of a semigroup by matrices over a field. In [10], he showed how to construct all the irreducible representations of an arbitrary inverse semigroup from those of associated Brandt semigroups. In this paper, we generalize the method of [10] to give a construction for the irreducible representations of an arbitrary semigroup from those of certain associated semigroups
Automorphisms of the semigroup of all differentiable functions
Authors: Magill Kenneth D.
Published: May 2009

Let R denote the space of real numbers and let D(R) denote the family of all functions mapping R into R that are (finitely) differentiable at each point of R. Since the composition f o g of two differentiable functions is also differentiable and since the composition operation is associative, it follows that D(R) is a semigroup with this operation. Such semigroups have been studied previously. Nadler, in [4], has shown that the semigroup of al differentiable functions mapping the closed unit interval into itself has no idempotent elements other than the identity function and the constant functions. The proof of that result carries over easily to the semigroup D(R).
On Schur's second partition theorem
Authors: Andrews George E.
Published: May 2009

In 1926, I. J. Schur proved the following theorem on partitions [3].

The number of partitions of n into parts congruent to ±1 (mod 6) is equal to the number of partitions of n of the form 1 + …+bs = n, where bi–bi+1 ≧ 3 and, if 3 ∣ bi, then bi–bi+1 > 3.

Schur's proof was based on a lemma concerning recurrence relations for certain polynomials. In 1928, W. Gleissberg gave an arithmetic proof of a strengthened form of Schur's theorem [2]; however, the combinatorial reasoning in Gleissberg's paper becomes very intricate.
Type Radicals
Authors: Leavitt W. G.
Published: May 2009

The lower radical of a module type. For a ring R with unit, the module type t(R) was defined in [6] as follows: t(0) = 0; t(R) = d if every free R-module has invariant rank; t(R) = (c, k) for integers c, k ≧ 1 if every free R-module of rank < c has invariant rank, while a free module of rank h ≧ c has rank h + nk for any integer n ≧ 0. The module types form a lattice under the ordering 0 < (c, k) < d and (c', k') ≦ (c, k) if and only if c ' ≦ c and k' Ⅰ k. Two of the basic theorems on types are:

A. [6; Theorem 2, p. 115] If R → R' is a unit-preserving homomorphism, then t(R')≦ t(R).
Note on a pair of dual trigonometric series
Authors: Cooke J. C.
Published: May 2009

It is the purpose of this note to discuss the solution of the pair of series

where F(x) and G(x) are given and the coefficients an are to be determined.
Inductive and projective limits of normed spaces
Authors: Pym John S.
Published: May 2009

Let {Ui, Uij} be an inductive system of normed linear spaces Ui and continuous linear maps uij; Uj → Ui. (We write j ≺ i if uij: Uj → Ui.) An inductive limit of the system with respect to a class of spaces A in and maps f in is a space Uu in Uu and a system ui → Uu of maps in such that (i) whenever j ≺ i, and that (ii) if A is any space in and fi: Ui → A is any system of maps in for which then there is a unique map f: Uu → A in such that fi = fo ui for each i. If is the class of all vector spaces and is the class of linear maps, we obtain the algebraic inductive limit, which we denote simply by U. The usual choice is to take to be the class of locally convex spaces and the class of continuous linear maps; the inductive limit UL then always exists [1, § 16 C]. If is again the continuous linear mappings but contains only normed spaces, the corresponding inductive limit UN may not always exist. However, if in addition we require that contains just contractions (norm-decreasing linear mappings), then an inductive limit Uc will exist if every uij is a contraction [2]. We shall give a condition under which these limits coincide (as far as possible), and consider the corresponding condition for projective limits.
Primitive sequences whose elements have no large prime factors
Authors: Anderson Ian
Published: May 2009

A sequence a1 < a2 < … of positive integers is said to be primitive if no element of the sequence divides any other. The study of primitive sequences arose naturally out of investigations into the subject of abundant numbers, where sequences each of whose elements is of the form , the pi being fixed primes, are of particular importance. Such a sequence is said to be built up from the primes p1…pr. Thus Dickson [1], in an early paper on abundant numbers, proved that a primitive sequence built up from a fixed set of primes is necessarily finite.
On complemented and annihilator algebras
Authors: Alexander Freda E.
Published: May 2009

The purpose of this paper is twofold. In [6] Tomiuk gives a representation theorem for a topologically simple right complemented algebra that is also an annihilator algebra. We strengthen this and then give a converse, so as to characterise right complemented algebras among respectively primitive Banach algebras and primitive annihilator Banach algebras. Our second aim is to investigate the relationship between the different annihilator conditions—left annihilator, right annihilator, annihilator, and dual—when imposed on a complemented algebra. Tomiuk [6] has already shown that a right complemented semisimple algebra that is a left annihilator algebra is an annihilator algebra; further, a topologically simple bi-complemented algebra that is also an annihilator algebra is dual. We show that for a topologically simple right complemented algebra all four annihilator conditions are equivalent. Further, for a semi-simple Banach algebra the first three are equivalent provided it is right complemented, and if it is also left complemented, then they are equivalent to duality.
Matrices over orthomodular lattices
Authors: Bevis J. H.
Published: May 2009

In this paper elementary properties are established for matrices whose coordinates are elements of a lattice L. In particular, many of the results of Luce [4] are extended to the case where L is an orthomodular lattice, a lattice with an orthocomplementation denoted by in which a ≦ b ⇒ a ∨(a′ ∧ b) = b. Originally, these were called orthocomplemented weakly modular lattices, Foulis [2]. In Theorem 1 a characterization is given of the nucleus with respect to matrix multiplication, which is in general nonassociative. Matrices with A-1 = transpose (A) are characterized in Lemma 8. Theorems 3 and 4 respectively, give partial characterizations of zero divisors and inverses.
Remark on a criterion for common transversals
Authors: Perfect Hazel
Published: May 2009

All sets considered will be finite, and |x| will denote the cardinal number of the set X.

Let = (Ai:i∈I) be a family of subsets of a set E. A subset E′ ⊆ E is called a transversal of if there exists a bijection σ:E′→ I such that e ∈ Aσ(e) (e ∈ E′). According to a well-known theorem of P. Hall [2], the familyhas a transversal if and only iffor every subset I′ of I. Ford and Fulkerson [1] obtained (as a special case of a more general theorem) an analogous criterion for the existence of a common transversal (CT) of two families. We may state their result in the following terms.
The numerical range of an element of a normed algebra
Authors: Bonsall F. F.
Published: May 2009

Given a normed linear space X, let S(X), X′, B(X) denote respectively the unit sphere {x: ∥x∥ = 1} of X, the dual space of X, and the algebra of all bounded linear mappings of X into X. For each x ∊ S(X) and T ∊ B(X), let Dx(x) = {f e X′:∥f∥ = f(x)= 1}, and V(T; x) = {f(Tx):f∊Dx(x)}. The numerical range V(T) is then defined by
Martingale convergence theorems for sequences of Stone algebras
Authors: Wright J. D. Maitland
Published: May 2009

A vector lattice W is boundedly complete when each subset {aj:j ∊ J} of W which is bounded above by an element of W has a least upper bound in W. The least upper bound of {aj:j ∊ J} is denoted by and the greatest lower bound by whenever these exist.
Radicals of semigroup rings
Authors: Weissglass Julian
Published: May 2009

Let denote the contracted semigroup ring of the ompletely 0-simple semigroup D over the ring R. The Rees structure theory of completely 0-simple semigroups is used to obtain necessary and sufficient conditions that have zero radical (Theorem 3.8). By using Amitsur's construction of the upper π-radical [1], we are able to treat the Jacobson, Baer (prime), Levitzki (locally nilpotent) and possibly the nil radicals simultaneously. Our results generalize a theorem of Munn [6] on semigroup algebras of finite 0-simple semigroups.
Semiconvex spaces II
Authors: Iyahen S. O.
Published: May 2009

One of the concepts introduced in [2] is that of a hyperbornological space, an idea which effectively replaces that of a bornological space when semiconvex spaces are being considered. In Section 2 of the present paper, it is shown how the topology of such a space may be described in terms of bounded pseudometrices. This is used in Section 3 to tackle the problem of when a product of separated hyperbornological spaces has the same property. It is shown that, as in the classical case of bornological spaces, this problem is equivalent to one in measure theory.
Riemann's method and the characteristic value and cauchy problems for the damped wave equation
Authors: Young Eutiquio C.
Published: May 2009

Riemann's method for solving the Cauchy problem for hyperbolic differential equations in two independent variables has been extended in a number of papers [4], [5], [2] to the wave equation in space of higher dimensions. The method, which consists in the determination of a so-called Riemann function, hinges on the solution of a characteristic value problem. Accordingly, if Riemann's method is to be used in solving a characteristic value problem, one will have to consider another characteristic value problem and thus the process becomes circular. This difficulty was first overcome by Protter [7] in solving the characteristic value problem for the wave equation in three variables. There he employed a variation of Riemann's method developed by Martin [5]. Martin's result was later extended by Diaz and Martin [2] to the wave equation in an arbitrary number of variables. This made it possible to extend Protter's result to the wave equation in space of higher dimensions [8].
A covering theorem for typically real functions
Authors: Brannan D. A. , Kirwan W. E.
Published: May 2009

Let T denote the class of functions

f(z) = z+a2z2+…

that are analytic in U = {|z| <1}, and satisfy the condition

Imf(z). Imz≧ 0 (zεU).

Thus T denotes the class of typically real functions introduced by W. Rogosinski [5].

One of the most striking results in the theory of functions

g(z) = z + b2z2…

that are analytic and univalent in U is the Koebe-Bieberbach covering theorem which states that {|w| <¼} ⊂ g(U). In this note we point out that the same result holds for functions in the class T, a fact which seems to have been overlooked previously. We also determine the largest subdomain of U in which every f(z) in T is univalent, extending previous results in [1] and [2].
An elementary inequality in function theory
Authors: Everitt W. N.
Published: May 2009

In the theory of self-adjoint operators in Hilbert space and of formally self-adjoint linear differential equations there are many situations involving analytic functions on the complex plane whose singularities are confined to the real axis and where the growth of the function at such singular points is strictly limited.
The a-points of Faber polynomials for a special function
Authors: Al-Amiri Hassoon S.
Published: May 2009

Let f(ζ) be a power series of the form

where lim sup |an|1/n < ∞. The Faber polynomials {fn(ζ)} (n = 0, 1, 2, …) are the polynomial parts of the formal expansion of (f(ζ))n about ζ = ∞. Series (1) defines an analytic element of an analytic function which we designate as w = f(ζ). Since at ζ = ∞ the analytic element is univalent in some neighborhood of infinity; thus the inverse of this element is uniquely determined in some neighborhood of w= ∞, and has a Laurent expansion of the form

where lim sup |bn|1/n = p < ∞. Let ζ = g(w) be this single-valued function defined by (2) in |w| > p. No analytic continuation of g(w) will be considered.
Hereditary semisimple classes
Authors: Leavitt W. G.
Published: May 2009

It is well-known (see e.g. [1, p. 5]) that a class ℳ of (not necessarily associative) rings is the semisimple class for some radical class, relative to some universal class if and only if it has the following properties:

(a)if ℳ, then every non-zero ideal I of Rhas a non-zero homomorphic image I/J∈ℳ.

(b) If R∈ but R∉ℳ, then R has a non-zero ideal I∈, where ℳ = {K ∈ | every non-zero K/H∉ℳ}. In fact ℳ is the radical class whose semisimple class is ℳ. On the other hand, if ℘ is a radical class, then ℐ℘ = {K∈/ if I is a non-zero ideal of K, then I∉℘} is its semisimple class. If a class ℳ is hereditary (that is, when R∈ℳ, then all its ideals are in ℳ), it clearly satisfies (a), but there do exist non-hereditary semisimple classes (see [2]). The condition (satisfied in all associative or alternative classes) is that ℘ is hereditary for a radical class ℘ if and only if ℘(I) ⊆ ℘(R) for all ideals I of all rings R∈ [3, Lemma 2, p. 595].
On bornological products
Authors: Robertson A. P.
Published: May 2009

It is well known that, provided that the indexing set I is not too large, the product

of a family of bornological locally convex topological vector spaces Eαis bornological. Products of bornological spaces were first studied by Mackey [3]. He reduced the problem to the study of R1, showing that this space is bornological if and only if I satisfies a certain condition, related to a problem in measure theory posed by Ulam [5]. We shall therefore call it the Mackey-Ulam condition on I. A similar study of the spaces R1 is to be found in the paper [4] by Simons; see also [1, Ch. IV, §6, exercise 3].
Homomorphic images of restrictive star semigroups
Authors: Magill Kenneth D.
Published: May 2009

LetYbe a subspace of a topological spaceX. ThenS(X, Y)denotes the semigroup, under composition, of all continuous selfmaps ofXwhich also carryYintoY. In the special caseY=X, the simpler notationS(X)is used. We have devoted several recent papers ([4], [7] and [8]) to the problem of determining whenS(Z)andS(X, Y)are isomorphic and, more generally, whenS(Z)is a homomorphic image ofS(X, Y). In this paper, we investigate the analogous problem for certain semigroups of functions on spaces which were introduced in [5]. These include semigroups of closed functions which are treated in further detail.
A necessary and sufficient condition for simultaneous diagonalization of two hermitian matrices and its application
Authors: Au-Yeung Yik-Hoi
Published: May 2009

We denote by F the field R of real numbers, the field C of complex numbers, or the skew field H of real quaternions, and by Fn an n dimensional left vector space over F. If A is a matrix with elements in F, we denote by A* its conjugate transpose. In all three cases of F, an n × n matrix A is said to be hermitian if A = A*, and we say that two n × n hermitian matrices A and B with elements in F can be diagonalized simultaneously if there exists a non singular matrix U with elements in F such that UAU* and UBU* are diagonal matrices. We shall regard a vector u ∈ Fn as a l × n matrix and identify a 1 × 1 matrix with its single element, and we shall denote by diag {A1, …, Am} a diagonal block matrix with the square matrices A1, …, Am lying on its diagonal.
Numerical range estimates for the norms of iterated operators
Authors: Crabb M. J.
Published: May 2009

Let X be a complex normed space, with dual space X′, and T a bounded linear operator on X. The numerical range V(T) of T is defined as {f(Tx): x∊X, f∊ X′, ∥x∥ = ∥f∥ = f(x) = 1}. Let ⃒V(T)⃒ denote sup {⃒λ⃒: λ∊ V(T)}. Our purpose is to prove the following theorem.
On a theorem of Iseki
Authors: Kessler Peter
Published: May 2009

The purpose of this paper is to generalize a result of K. Iseki [1]. In his note, Iseki proves that, in a normal space S, for every countable discrete collection ℋ = {H1, H2, …} of sets from S, there exists a countable collection = {U1, U2, …} of mutually disjoint open sets from S such that Hi ⊂ Ui for every i.
Finitely generated commutative semigroups
Authors: McAlister D. B. , O'Carroll L.
Published: May 2009

Since all the semigroups considered in this paper are commutative, we shall use the terms “semigroup” and “group” where we actually mean “commutative semigroup” and “commutative group”. Some basic results from the theory of semigroups are required and will be used without explicit mention; these results may be found in [1, § 4.3]. We shall denote the additive semigroups of integers, positive integers, negative integers, positive rationals by Z, Z+, Z-, Q+respectively.
Matrix transformations of some sequence spaces—II
Authors: Rao K. Chandrasekhara
Published: May 2009

This paper is a continuation of [1]. We begin with the notations for the sequence spaces considered in this paper. Let Γ be the space of sequences x = {xp} of complex numbers such that |xp|1/p⃗0 as p⃗∞. Γ can also be regarded as the space of integral functions f(z) = . The sequence space Γ is a vector space over the complex numbers with seminorms
On successive approximations for nonexpansive mappings in Banach spaces
Authors: Kirk W. A.
Published: May 2009

Let X be a Banach space and K a convex subset of X. A mapping Tof K into K is called a nonexpansive mapping if | T(x) – T(y) | ≦ | x – y | for all x, yεK.
On a functional equation for the exponential function of a complex variable
Authors: Haruki Hiroshi
Published: May 2009

The following result is well known in the theory of analytic functions; see [1].

Theorem A. Suppose that f(z) is an entire function of a complex variable z. Then f(z) satisfies the functional equation

where z = x + iy (x, y real), if and only if f(z) = aexp(sz), where a is an arbitrary complex constant and s is an arbitrary real constant.
On the flat overrings of an integral domain
Authors: Wajnryb Bronislaw , Zaks Abraham
Published: May 2009

The present paper deals with relations between flat overrings and quotient rings. Weare mainly concerned with Richman's results [10] on flat overrings and withthose of Davis [2], Gilmer [3], Gilmer and Heinzer [4], Gilmer and Ohm [5], and Mott [8], on rings with the QR propertyand with the property (#) defined in Section 1. Some of their results are generalized, and it is shown that certain theorems, which at first glance seem to have nothing in common, are in fact particular cases of a single more general theorem.
Pure-complete subgroups of direct sums of Prüfer groups
Authors: Hill Paul
Published: May 2009

Suppose that G is a p-primary abelian group. The subgroup G[p] = {x∈G:px=0} is called the socle of G and any subgroup S of G[p] is called a subsocle of G. If each subsocle of G supports a pure subgroup, then G is said to be pure-complete [1]. It is well known that, if G a direct sum of cyclic groups, then G is necessarily pure-complete. Further results about pure-complete groups are contained in [1] and [3].
On the inclusion of a bounded convergence field in the space of almost convergent sequences
Authors: Atalla Robert E.
Published: May 2009

Let T = (tmn) be a regular matrix, and CTbe its bounded convergence field. Necessary and sufficient conditions for CT to contain the space of almost convergent sequences are well known. (See, e.g., [7, p.62]). G. M. Petersen has suggested as a problem for research the discovery of necessary and sufficient conditions for the reverse inclusion: When is CT contained in the space of almost convergent sequences? [7, p. 137, research problem 9]. In this paper we deal with this question in a more general context. First we need some notation.
On embeddable finite amalgams of groups
Authors: Majeed Abdul
Published: May 2009

In this paper, a problem of B. H. Neumann and Hanna Neumann [7] about the finite embeddability of an embeddable finite amalgam is discussed. After proving a “reduction theorem” for a finite amalgam to have a finite embedding, we examine some known embeddable amalgams (cf. [3]) as regards their embeddability in a finite group. Since the existence of the generalised free product and the embeddability of an amalgam are synonymous terms, Theorem 3.1 generalises a result in [4]. A sufficient condition for an amalgam of type S to have a finite embedding is also given.
On the range of finite embeddings of a finite amalgam
Authors: Majeed Abdul
Published: May 2009

Certain questions about the range of finite embeddings of a finite amalgam were discussed in [3]. Another pertinent question is the following.

If a finite reduced amalgam has both an infinite and a finite embedding, does it have a maximal finite embedding such that all other finite embeddings are its homomorphic images? We give a counter example to answer this question in the negative. The finite amalgam considered will involve a group from the family of groups of the type (l, m; n, k) discussed by Coxeter [2] having the following presentation

These groups may be regarded as factor groups of

which is known to be finite if

and infinite otherwise.
Almost commutative bands
Authors: Hall T. E.
Published: May 2009

To find a “ description of the structure of bands which is complete modulo semilattices ” (from page 25 of [1]) seems to be a very difficult problem. As far as the author is aware, the only class of bands (except for rectangular bands) for which this problem has been solved (see [4] and [3]) is the class of all bands satisfying a generalization of commutativity, namely the condition that efgh = egfh for all elements e, f, g and h.
A direct proof of Leutbecher's Lemma
Authors: Wohlfahrt Klaus
Published: May 2009

Using the theory of group extensions, A. Leutbecher [1] proved this

Lemma. Let G bea group, and w some 2-cocycle of a trivial G-module M. The cohomology class ofw will contain symmetric cocycles if and only if w is semisymmetric.

Here we have called w symmetric or semisymmetric according as w(h, g) = w(g, h) for all g, h ∈G or only for those with hg = gh. In one direction, the proof reduces to observing that 2-coboundaries of trivial G-modules are semisymmetric. The nontrivial part of the lemma also admits of a straightforward proof, as follows.
Simultaneous dual integral equations
Authors: Lowndes J. S.
Published: May 2009

Lowengrub [l] has considered simultaneous dual integral equations of the form

where i = 1,2 …n, I1= {x:0 ≦ x x <1}, I2 {= x:0 ≦ x >1}, the cIJ are constants, the f1(x) are known functions and the functions φ(x) are to be determined.

denotes the modified operator of the Hankel transform with the inversion formula
Arithmetic progressions in finite sets of real numbers
Authors: Klotz W.
Published: May 2009

In this paper we investigate the structure of a set of n reals that contains a maximal number of l-term arithmetic progressions. This problem has been indicated by J. Riddell. Let l and n be positive integers with 2 ≦ l ≦ n. By F1(n) we denote the maximal number of l-term arithmetic progressions that a set of n reals can contain. A set of n reals containing F1(n)l-progressions will be called an Fl,(n)-set.
The stability of pure weights under conditioning
Authors: Foulis D. J. , Randall C. H.
Published: May 2009

In [1], we showed how a collection of physical operations or experiments could be represented by a nonempty set of nonempty sets satisfying certain conditions (irredundancy and coherence) and we called such sets . We also introduced “complete stochastic models” for the empirical universe of discourse represented by such a manual , namely, the so-called weight functions for . These weight functions form a convex set the extreme points of which are called pure weights. We also showed that there is a so-called logic ∏() affiliated with a manual and that each weight function for induces a state on this logic.
Quasi-homology and universal coefficients
Authors: Hsu N. C.
Published: May 2009

In order to study an arbitrary sequence of modules and homomorphisms, we propose a definition of “homology” modules, or what we call quasi-homology modules, for such a sequence. Then we seek partial analogues of the universal coefficient theorems to make some propaganda for the notion.
Delange's characterization of the sine function
Authors: Ching Chin-Hung , Chui Charles K.
Published: May 2009

In [2], H. Delange gives the following characterization of the sine function.

Theorem A. f(x)=sin x is the only infinitely differentiable real-valued function on the real line such that f'(O)= 1 and

for all real x and n = 0,1,2,….

It is clear that, if f satisfies (1), then the analytic continuation of f is an entire function satisfying

for all z in the complex plane. Hence f is of at most order one and type one. In this note, we prove the following theorem.
On the numerical radius of an element of a normed algebra
Authors: Mocanu GH.
Published: May 2009

Let A be a unital normed algebra over the complex field ℂ, A' the dual space of A, i.e., the Banach space of all continuous linear functionals on A, and let S be the set of all states on A, i.e.,
On an integral equation
Authors: Homentcovschi D.
Published: May 2009

We shall solve the equation

where 0 <a<b, and f(x) is a continuous function on the interval (a, b).
Multiplicative forms and nonassociative algebras
Authors: Odoni R. W. K.
Published: May 2009

In [1], we introduced the notion of multiplicative forms on associative algebras of finite rank over integral domains D, and obtained a complete classification when D ⊆, the complex field. We propose here to remove the hypothesis of associativity, using a refinement of the technique of Schafer [2]. In [l], it was noted that multiplicative forms extend uniquely under the adjunction of an identity when is associative but not unitary; this appears difficult to verify in the general case, so that some mild restriction on is required. We shall assume that is biregular, that is that contains elements eL, eR such that the linear maps x eL x and x xeR, are bijective on We can then (§1) reduce the biregular case to the unitary case, which is handled in §2.
Counting word-trees
Authors: Mitchell J. M. O.
Published: May 2009

In his recent study of free inverse semigroups, Munn [2] introduced and used extensively the concept of a word-tree. In this note the number of such trees is found.
Transform solutions of Dirichlet problems in quarter-spaces
Authors: Sneddon I. N.
Published: May 2009

In a recent paper, Boudjelkha and Diaz [1] have considered the solutions of the Dirichlet problems for Laplace's equation

and for Helmholtz's equation

in¼–ℝn. The purpose of this brief note is to show that their formulae may be derived easily by the use of the theory of multiple Fourier transforms.
On (v, k, λ)-configurations with v=4pe
Authors: McFarland R. L.
Published: May 2009

A (v, k, λ)-configuration, also called a symmetric balanced incomplete block design, is an arrangement of v distinct objects called points or varieties into v subsets called lines or blocks such that each line contains exactly k points and each pair of distinct lines contains exactly λ points in common. To avoid certain trivial configurations, one assumes that 0<λ<k<v–1.
Twisted group rings which are semi-prime Goldie rings
Authors: Reid A.
Published: May 2009

In this paper we examine when a twisted group ring,Rγ(G), has a semi-simple, artinian quotient ring. In §1 we assemble results and definitions concerning quotient rings, Ore sets and Goldie rings and then, in §2, we defineRγ(G). We prove a useful theorem for constructing a twisted group ring of a factor group and establish an analogue of a theorem of Passman. Twisted polynomial rings are discussed in §3 and I am indebted to the referee for informing me of the existence of [4]. These are used as a tool in proving results in §4.
Inverse semigroups as extensions of semilattices
Authors: O'Carroll Liam
Published: May 2009

LetSbe an inverse semigroup with semilattice of idempotentsE, and letρbe a congruence onS. Thenρis said to beidempotent-determined[2], or I.D. for short, if (a, b) ∈ р anda∈Eimply thatb∈E. If, further,ρis a group congruence, then clearlyρis the minimum group congruence onS, and in this caseSis said to beproper[8]. LetT=S/ρ.
On Tricomi's relation for the Hilbert transformation
Authors: Rooney P. G.
Published: May 2009

Tricomi [2] has shown that if φi ∈ Lpi (− ∞, ∞), i = 1, 2, where 1 < pi < ∞, (pi)–l + (P2)–1 < 1, and if H denotes the Hilbert transformation, that is

where the symbol (P) denotes that the integral is taken in the Cauchy principal value sense, then
Abelian ergodic theorems for vector-valued functions
Authors: Kopp P. E.
Published: May 2009

This note contains extensions of the Abelian ergodic theorems in [3] and [6] to functions which take their values in a Banach space. The results are based on an adaptation of Rota's maximal ergodic theorem for Abel limits [8]. Convergence theorems for continuous parameter semigroups are deduced by the approximation technique developed in [3], [6]. A direct application of the resolvent equation also enables us to deduce a convergence theorem for pseudo-resolvents.
Kernel functors for which the associated idempotent kernel functor is stable
Authors: Manocha J. N.
Published: May 2009

Let R be a ring with unity and let denote the category of unital right R-modules. A preradical γ of is a functor γ: → such that

(i) γ(M) ⊆ M for each R-module M,

(ii) for f:M → N, γ(f) is the restriction of f to γ(M).
Properties and applications of a certain operator associated with the Kontorovich-Lebedev transform
Authors: Ben-Menahem Ari
Published: May 2009

The integral

arises in problems of scalar wave propagation in welded elastic wedges. In (1.1), Kim(β1r) is the modified Bessel function of the second kind and m, τ are real. It is shown that Q(τ, m) is a generalized function that includes a complex shift operator. We shall investigate the properties of this operator and establish a new integral transform based on the kernel Q(τ, m).
On the cohomology of Fuchsian groups
Authors: Patterson S. J.
Published: May 2009

The object of this paper is to redevelop the classical theory of multipliers of Fuchsian groups [16] and to attempt a classification. The language which appears most appropriate is that of group extensions and the cohomology of groups. This viewpoint is not entirely novel [12] but the entire theory has never been based on it before.
Divisibility in categories of a class which includes the category of binary relations
Authors: Fitz-Gerald D. G.
Published: May 2009

Green's relations are essential for “co-ordinatizing” semigroups. Jacqueline Klasa, in applying cognate ideas to categories [4, 5], has shown that divisibility in suitably-behaved categories may be described in terms of subobjects and quotients.

Here it is shown that adjoint functors which are onto objects preserve divisibility (in a certain sense). The inclusion functor of the category of sets into the category R of binary relations is such a functor. A slight modification of its right adjoint allows the representation of R as a full subcategory in a category CSL of complete semilattice morphisms.
On a class of generalized functions
Authors: Rowlands K.
Published: May 2009

Let be the complex linear space of all infinitely differentiable functions φ on the interval J = (a,b)(− ∞ ≦ a < 0 < b ≦ + ∞) such that φ(k)(0) = 0 for all non-negative integers k. Krabbe ([2], [3]) has defined a class of generalized functions on Jas an algebra ℳ of linear operators on and has developed an operational calculus for these operators. Shultz ([6], Theorem 2.18) has recently shown that ℳ is isomorphic to , where (resp. ) is the set of all distributions on J whose supports are contained in J+ = [0, b) (J- = (a,0]). In this paper we combine some of the ideas developed in [4] with results established in an earlier paper by Shultz to give an easier proof of the above result. Our methods also give a more direct proof of the main result (Theorem 1.22) of [2].
On the closed graph theorem
Authors: Popoola J. O. , Tweddle I.
Published: May 2009

Our main purpose is to describe those separated locally convex spaces which can serve as domain spaces for a closed graph theorem in which the range space is an arbitrary Banach space of (linear) dimension at most c, the cardinal number of the real line R. These are the δ-barrelled spaces which are considered in §4. Many of the standard elementary Banach spaces, including in particular all separable ones, have dimension at most c. Also it is known that an infinite dimensional Banach space has dimension at least c (see e.g. [8]). Thus if we classify Banach spaces by dimension we are dealing, in a natural sense, with the first class which contains infinite dimensional spaces.
The purity of a completion
Authors: Cox S. H.
Published: May 2009

This note establishes two statements from R. M. Fossum's review [1] of a paper by E. A. Magarian [2]. Firstly, if A → B is a pure homomorphism (of commutative rings) then A[[x1,…,xs]] → B[[x1,…, xs]] is pure. Secondly, if Rn → R is a directed family of pure homomorphisms then ∪ Rn → R is pure. A consequence is that if Rn → R is a directed family of pure homomorphisms and if R is Noetherian, then ∪ Rn[[x1,…,xs]]is Noetherian.
Extreme operators on H∞
Authors: McDonald John N.
Published: May 2009

Let A1 and A2 be sup-norm algebras, each containing the constant functions. Let P(A1, A2) denote the set of bounded linear operators from A1 to A2 which carry 1 into 1 and have norm 1. Several authors have considered the problem of describing the extreme points of P(A1, A2). In the case where A1 is the algebra of continuous complex functions on some compact Hausdorff space, and A2 is the algebra of complex scalars, Arens and Kelley proved that the extreme operators in P(A1, A2) are exactly the multiplicative ones (see [1]). It was shown by Phelps in [6] that if A1 is self-adjoint, then every extreme point of P(A1, A1) is multiplicative. In [4], Lindenstrauss, Phelps, and Ryff exhibited non-multiplicative extreme points of P(A, A) and P(H∞, H∞), where A and H∞ are, respectively, the disk algebra, and the algebra of bounded analytic functions on the open unit disk D. The extreme multiplicative operators in P(A, A) were described in [6]. Rochberg proved in [8] that, if T is a member of P(A, A) which carries the identity on D into an extreme point of the unit ball of A, then T is multiplicative and is an extreme point of P(A, A). Rochberg's paper [9] is a study of certain extremal subsets of P(A, A), namely, those of the form K(F, G) = {T∊P(A, A):TF=G}, where F and G are inner functions in A. We proved in [5] that, if F is non-constant, then K(F, G) contains an extreme point of P(A, A).
States which have a trace-like property relative to a C*-subalgebra of B(H)
Authors: Robertson Guyan
Published: May 2009

In what follows, B(H) will denote the C*-algebra of all bounded linear operators on a Hilbert space H. Suppose we are given a C*-subalgebra A of B(H), which we shall suppose contains the identity operator 1. We are concerned with the existence of states f of B(H) which satisfy the following trace-like relation relative to A:

Our first result shows the existence of states f satisfying (*), when A is the C*-algebra C*(x) generated by a normaloid operator × and the identity. This allows us to give simple proofs of some well-known results in operator theory. Recall that an operator × is normaloid if its operator norm equals its spectral radius.
Order and Schwartz distributions
Authors: Feldman W. A. , Porter J. F.
Published: May 2009

The space of Schwartz distributions on the unit circle Г in the plane is topologically a considerable generalization of the space of regular, finite Borel measures on Г. However, the order structure of is usually taken to be the same as that of : there are no “positive” distributions which are not measures. This perhaps warrants consideration, since the order structure of generates its topology. In this paper we construct a system of order structures for which is a more natural complement in the intermediate stages to the topology of and which provides an interpretation of with its Schwartz topology as a quotient of a generalized base norm space V′. Where denotes the space of continuous functions on Г with its supremum norm topology, V′ is the dual of . The space contains the infinitely differentiable functions on Г with their usual topology, and (via the pointwise ordering on ) in its product ordering is realized as a generalized order unit space. Some consequences for harmonic functions are discussed.
Optimisation of quadratic forms associated with graphs
Authors: Waller Derek A.
Published: May 2009

Quadratic forms associated with graphs were introduced over a century ago by Jordan [4]. We are concerned with the optimisation of such quadratic forms, following Motzkin and Straus [5], and we use the setting of categories and functors to express the nice interplay between the algebra and the graph theory. Applications to interchange graphs are also obtained.
Primitive Ore extensions
Authors: Jordan D. A.
Published: May 2009

Apart from simple Ore extensions such as the Weyl algebras, the best known example of a primitive Ore extension is the universal enveloping algebra U(g) of the 2-dimensional solvable Lie algebra g over a field k of characteristic zero, see [4, p. 22]. U(g) is a polynomial algebra over k in two indeterminates x and y with multiplication subject to the relation xy – yx = y, and may be regarded either as an Ore extension of k [x] by the k-automorphism which maps x to x – 1 or as an Ore extension of k[y] by the derivation yd/dy. The argument suggested in [4, p. 22] to prove the primitivity of U(g) can easily be generalised [6] to show that, if α is an automorphism of the ring R then the following conditions are sufficient for R[x, α] to be primitive: (i) no power αs, s ≧ 1, of α is inner; (ii) the only ideals of R invariant under α are 0 and R. These conditions are necessary and sufficient for the simplicity of the skew Laurent polynomial ring R[x, x–1, α] but are not necessary for the primitivity of R[x, α] (the ordinary polynomial ring D[x] over a division ring D not algebraic over its centre is easily seen to be primitive).
On the radical of a group algebra over a commutative ring
Authors: Potter Gloria
Published: May 2009

Several people, including Wallace [4] and Passman [3], have studied the Jacobson radical of the group algebra F[G] where F is a field and G is a multiplicative group. In [4], for instance, Wallace proves that if G is an abelian group with Sylow p-subgroup P and if F is a field of characteristic p, then the Jacobson radical of F[G] equals the right ideal generated by the radical of F[P]. In this paper we shall study group algebras over arbitrary commutative rings. By a reduction to the case of a semi-simple commutative ring, we obtain Theorem 1 whose corollary contains a generalization of Wallace's theorem. Theorem 2, on the other hand, uses the first theorem to obtain results related to the main theorem of [3].
On a new class of polynomials
Authors: Panda Rekha
Published: May 2009

The present paper incorporates a preliminary study of a new generalization of several known polynomial systems belonging to (or providing extensions of) the families of the classical Jacobi, Hermite and Laguerre polynomials. It is shown how suitable specializations will yield a number of known or new results in the theory of the special functions considered.
A note on regular local Noether lattices II
Authors: Johnson Johnny A.
Published: May 2009

Let (R, M) be a local ring and let R* be the M-adic ring completion of R. It is well known that R is a regular local ring if and only if R* is a regular local ring. The purpose of the note is to show that this result is essentially a consequence of a more general theory concerning local Noether lattices which was developed in [6].
Solvable-by-finite subgroups of GL(2, F)
Authors: Majeed Abdul , Mason A. W.
Published: May 2009

In a recent paper [5] Tits proves that a linear group over a field of characteristic zero is either solvable-by-finite or else contains a non-cyclic free subgroup. In this note we determine all the infinite irreducible solvable-by-finite subgroups of GL(2, F), where F is an algebraically closed field of characteristic zero. (Every reducible subgroup of GL(2, F) is metabelian.) In addition, we prove that an irreducible subgroup of GL(2, F) has an irreducible solvable-by-finite subgroup if and only if it contains an element of zero trace.
Circles of numbers
Authors: Burmester M. , Forcade R. , Jacobs E.
Published: May 2009

Arrange any n integers around a circle. The following procedure can be used to obtain another circle of n integers. For each adjacent pair of the first integers, form the absolute value of their difference and place it between them; then remove the original numbers. This procedure can be repeated over and over. When n = 4 this always leads eventually to a circle of zeros. On the other hand when n = 3, unless the original numbers are equal, this never happens. We treat below the general case and related problems, using for convenience a slightly different formulation. Surprisingly there is enough structure to lead to some interesting mathematics.
Category of sequences of zeros and ones in some FK spaces
Authors: Devos Robert
Published: May 2009

Let s denote the space of all complex valued sequences and let E∞ be all eventually zero sequences. An FK space is a locally convex vector subspace of s which is also a Fréchet space (complete linear metric) with continuous coordinates. A BK space is a normed FK space. Some discussion of FK spaces is given in [11]. Well-known examples of BK spaces are the spaces m, c, c0 of bounded, convergent, null sequences respectively, all with and
Ranges of Lyapunov Transformations for operator algebras
Authors: Kyle J.
Published: May 2009

In this paper we shall extend results obtained in [5] to the W*-algebra setting.

Let be a C*-algebra and let + denote the set of positive elements in . Given a fixed element A in , the Lyapunov transformation LA corresponding to A is the mapping of into itself which sends X to AX+XA*. We are interested in characterizing those Bin for which
On the representation of a number as a sum of squares
Authors: Gundlach Karl-Bernhard
Published: May 2009

It is well known that the number Ak(m) of representations of a positive integer m as the sum of k squares of integers can be expressed in the form

where Pk(m) is a divisor function, and Rk(m) is a remainder term of smaller order. (1) is a consequence of the fact that

is a modular form for a certain congruence subgroup of the modular group, and

with

where Ek(z) is an Eisenstein series and is a cusp form (as was first pointed out by Mordell [9]). The result (1) remains true if m is taken to be a totally positive integer from a totally real number field K and Ak(m) is the number of representations of m as the sum of k squares of integers from K (at least for 2|k, k>2, and for those cases with 2+k which have been investigated). then are replaced by modular forms for a subgroup of the Hilbert modular group with Fourier expansions of the form (10) (see section 2).
Semigroups of continuous selfmaps for which Green's and ℐ relations coincide
Authors: Magill K. D.
Published: May 2009

For algebraic terms which are not defined, one may consult [2]. The symbol S(X) denotes the semigroup, under composition, of all continuous selfmaps of the topological space X. When X is discrete, S(X) is simply the full transformation semigroup on the set X. It has long been known that Green's relations and ℐ coincide for [2, p. 52] and F. A. Cezus has shown in his doctoral dissertation [1, p. 34] that and ℐ also coincide for S(X) when X is the one-point compactification of the countably infinite discrete space. Our main purpose here is to point out the fact that among the 0-dimensional metric spaces, Cezus discovered the only nondiscrete space X with the property that and ℐ coincide on the semigroup S(X). Because of a result in a previous paper [6] by S. Subbiah and the author, this property (for 0-dimensional metric spaces) is in turn equivalent to the semigroup being regular. We gather all this together in the following
Finite quasi-injective groups
Authors: Bertholf Dennis , Walls Gary
Published: May 2009

It is well known that the category of finite groups has no non-trivial injective objects. In general, a group is said to be quasi-injective if for every subgroup H of G and homomorphism f:H → G there exists an endomorphism F:G → G such that F|H = G. In other words, a group is quasi-injective whenever each homomorphism from a subgroup into the group can be extended to the whole group.
The twisted group algebra of a finite nilpotent group over a number field
Authors: Opolka Hans
Published: May 2009

Let G be a finite group with neutral element e which operates trivially on the multiplicative group R* of a commutative ring with identity 1. Let H2(G, R*) = Z2(G, R*)/B2(G, R*) denote the second cohomology group of G with respect to the trivial G-module R*. With every factor system (2-cocycle) f ∈ Z2(G, R*) we associate the so called (central) twisted group algebra (R, G, f) of G over R (see [4, Chapter V, 23.7] or [13, §4] for a definition). If f is cohomologous to f', then the R-algebras (R, G, f) and (R, G, f′) are isomorphic. Hence, up to R-algebra isomorphism, (R, G, f) is determined by the cohomology class f∈H2(G, R*) determined by f. If R = k is a field of characteristic not dividing the order |G| of G, then a computation of the discriminant of (k, G, f) shows that (k, G, f) is semisimple (see [13, 4.2]).
Congruences on simple ω-semigroups
Authors: Petrich Mario
Published: May 2009

An inverse semigroup whose idempotents form an ω-chain e0 > e1 > e2 > … is called briefly an ω-semigroup. A structure theorem for simple ω-semigroups was established by Kočin [7]; a related structure theorem for simple, and also general, ω-semigroups was proved by Munn [10]. These results represent an extension of the structure theorem for bisimple a ω-semigroups due to Reilly [14].