bluuebunny/crossref_metadata_2025 huggingface dataset:
Transversally affine foliations
Authors: Furness P. M. D. , Fédida E.
Published: May 2009

Let ℱ be a smooth foliation of codimension p on a smooth manifold Mm. We can define ℱ by an atlas of coordinate charts (U, (x, y)), called leaf charts, where (x, y): U → Rm−p × Rp are coordinate functions for which the leaves of ℱ are given by y1 constant,…,yp constant, in U. Clearly, on the overlap of two such leaf charts (U, (x, y)) and (U′, (x′, y′)) we have a coordinate transformation of the form

If y′ is always affine in y, i.e.

where and Bi are constants, we shall say that ℱ is a transversally affine foliation. This notion is, in a sense, dual to that of affine foliation, see [2], in which x′ is affine in x and each leaf has an induced flat affine structure.
A note on Cauchy's root test
Authors: Murphy I. S.
Published: May 2009

Throughout this note we deal with a series Σan of positive terms. The following tests for the convergence of this series are well-known.

Test 1. (Ratio test). LetThen, if K < 1, Σanconverges, while if K > 1, Σandiverges.

Test 2. (Root test). LetThen, if K < 1, Σanconverges, while if K > 1, Σandiverges.
Support projections on Banach spaces
Authors: Spain P. G.
Published: May 2009

Each bounded linear operator a on a Hilbert space K has a hermitian left-support projection p such that and (1 – p)K = ker α* = ker αα*. I demonstrate here that certain operators on Banach spaces also have left supports.

Throughout this paper X will be a complex Banach space with norm-dual X', and L(X) will be the Banach algebra of bounded linear operators on X. Two linear subspaces Y and Z of X are orthogonal (in the sense of G. Birkhoff) if ∥ y ∥ ≦ ∥ y + z ∥ (y ∈Y, z ∈ Z); this orthogonality relation is not, in general, symmetric. It is easy to see that pX is orthogonal to (1 – p)X if and only if the norm of p is 0 or 1, when p is a projection on X.
A reflection principle for solutions to the Helmholtz equation and an application to the inverse scattering problem
Authors: Colton David
Published: May 2009

A classical result in potential theory is the Schwarz reflection principle for solutions of Laplace's equation which vanish on a portion of a spherical boundary. The question naturally arises whether or not such a property is also true for solutions of the Helmholtz equation. This has been answered in the affirmative by Diaz and Ludford ([4]; see also [10]) in the limiting case of the plane. It is the purpose of this paper to show that a reflection principle is also valid for spheres of finite radius. As an application of this result we shall study the problem of the analytic continuation of solutions to the Helmholtz equation defined in the exterior of a bounded domain in three-dimensional Euclidean space ℝ3 We shall show that through the use of the reflection principle derived in this paper, this problem can be reduced to the problem of the analytic continuation of an analytic function of two complex variables, which in turn can be performed through a variety of known methods (cf. [7]).
Representations of the ℓ1-algebra of an inverse semigroup having the separation property
Authors: Barnes Bruce A.
Published: May 2009

Let S be a semigroup, and let ℓ1(S) denote the l1-semigroup algebra of S. Beginning with the fundamental paper of E. Hewitt and H. Zuckerman [5], there has been a considerable amount of research done concerning the Banach algebra ℓ1(S) in the case when S is abelian; see the bibliography [7]. However, until recently, there was very little information known concerning ℓ1(S) when S was nonabelian and infinite. Now for certain classes of infinite nonabelian semigroups with involution, recent progress has been made in the study of the Banach *-algebra ℓ1(S) and the *-representations of l1(S). In [2], B. Barnes and J. Duncan prove that ℓ1(S) is Jacobson semisimple, study the spectrum of elements in ℓ1(S), and construct and study *-representations of ℓ1(S) when S is the free semigroup with a finite or countably infinite set of generators (and also in some cases where the generators satisfy certain relations). In [1], the present author considered the representation theory of ℓ1(S) where S is an inverse semigroup. This paper is a sequel to [1].
Topological groups with co-monoid structures
Authors: Katz Elyahu
Published: May 2009

The Eckman–Hilton duality [4] reverses arrows in diagrams, turns products to co-products, and multiplications to co-multiplications, etc. In accordance with this process, Kan [5] obtained the dual of a monoid structure in the category of groups. In this way, we obtain co-monoid structures on topological groups. The main result of this paper is that for kaω groups (see §2), we obtain a one-to-one correspondence between the co-monoid structures, and the free topological bases of the group (§3), thus obtaining topological analogues of the main results of [5].
The reckoning of certain quartic and octic Gauss sums
Authors: Berndt Bruce C. , Chowla S.
Published: May 2009

In this brief note, we evaluate certain quartic and octic Gauss sums with the use of theorems on fourth and eighth power difference sets. We recall that a subset H of a finite (additive) abelian group G is said to be a difference set of G [5, p. 64] if for some fixed natural number λ, every nonzero element of G can be written as a difference of two elements of H fin exactly λ ways.
A Lyapunov inequality and forced oscillations in general nonlinear nth order differential-difference equations
Authors: Chen Lu-San
Published: May 2009

The purpose of this paper is to consider the general nonlinear nth order differential-difference equation

and derive an inequality of Lyapunov type. Later we use this inequality to find conditions to ensure that the oscillatory solutions of equation (1) tend to zero as t → ∞. The conditions that ensure that the oscillatory solutions of equation (1) tend to zero, also cause all solutions of equation

to be non-oscillatory.
On a dual integral equation with a trigonometric kernel
Authors: Stocks D. C.
Published: May 2009

In this note we formally solve the following dual integral equations:

where h is a constant and the Fourier cosine transform of u–1 φ(u) is assumed to exist. These dual equations arise in a crack problem in elasticity theory.
Congruences on an orthodox semigroup via the minimum inverse semigroup congruence
Authors: Eberhart Carl , Williams Wiley
Published: May 2009

It is well known that the lattice Λ(S) of congruences on a regular semigroup S contains certain fundamental congruences. For example there is always a minimum band congruence β, which Spitznagel has used in his study of the lattice of congruences on a band of groups [16]. Of key importance to his investigation is the fact that β separates congruences on a band of groups in the sense that two congruences are the same if they have the same meet and join with β. This result enabled him to characterize θ-modular bands of groups as precisely those bands of groups for which ρ⃗(ρ∨β, ρ∧β)is an embedding of Λ(S) into a product of sublattices.
Embedding inverse semigroups of homeomorphisms on closed subsets
Authors: Baird Bridget Bos
Published: May 2009

All topological spaces here are assumed to be T2. The collection F(Y)of all homeomorphisms whose domains and ranges are closed subsets of a topological space Y is an inverse semigroup under the operation of composition. We are interested in the general problem of getting some information about the subsemigroups of F(Y) whenever Y is a compact metric space. Here, we specifically look at the problem of determining those spaces X with the property that F(X) is isomorphic to a subsemigroup of F(Y). The main result states that if X is any first countable space with an uncountable number of points, then the semigroup F(X) can be embedded into the semigroup F(Y) if and only if either X is compact and Y contains a copy of X, or X is noncompact and locally compact and Y contains a copy of the one-point compactification of X.
E-unitary inverse semigroups over semilattices
Authors: McAlister D. B.
Published: May 2009

An inverse semigroup is called E-unitary if the equations ea = e = e2 together imply a2 = a. In a previous paper [4], the author showed that any E-unitary inverse semigroup is isomorphic to a semigroup constructed from a triple (G, ℋ, ) consisting of a down-directed partially ordered set ℋ, an ideal and subsemilattice of ℋ and a group G acting on ℋ, on the left, by order automorphisms in such a way that ℋ = G. This semigroup is denoted by P(G, ℋ, ); it consists of all pairs (a, g)∈ × G such that g−1a ∈ , under the multiplication
Subgroups of infinite index in the modular group
Authors: Stothers W. W.
Published: May 2009

The modular group Г is the group of integral bilinear transformations of the extended complex plane which preserve the upper half-plane. It has the presentation 〈x, y:x2 = y3 = 1〉, and the generators can be chosen so that u = xy maps z to z + 1.
Fourier-Stieltjes Transforms which vanish at infinity off certain sets
Authors: Pigno Louis
Published: May 2009

In this paper G is a nondiscrete compact abelian group with character group Г and M(G) the usual convolution algebra of Borel measures on G. We designate the following subspaces of M(G) employing the customary notations: Ma(G) those measures which are absolutely continuous with respect to Haar measure; MS(G) the space of measures concentrated on sets of Haar measure zero and Md(G) the discrete measures.
On the spectra of prespectral operators
Authors: Nagy B.
Published: May 2009

The spectrum of a prespectral operator was investigated by Dowson in [4]. The question was left open there whether, if a prespectral operator has closed range, the same is true for its scalar part. In this paper we answer this in the affirmative and point out some consequences concerning the essential spectra of prespectral operators. Also, following Taylor and Halberg [11], we present the state diagram of a prespectral operator, which will show, in a sense, the sharpness of the results of the spectral theory of such operators.
On right unipotent semigroups II
Authors: Venkatesan P. S.
Published: May 2009

We describe two congruences α and γ contained in ℒ on an arbitrary orthodox semigroup. Let S be a right unipotent semigroup. We show that (i) α is an inverse semigroup congruence and γ is the finest fundamental inverse semigroup congruence on S, (ii) S is a union of groups if and only if ℒ on S and (iii) S is a band of groups if and only if ℒ on S.
Ranges of Lyapunov transformations for Hilbert space
Authors: Kyle J.
Published: May 2009

Interest in the ranges of Lyapunov transformations began with Taussky in [5]. Recently in a series of papers, Loewy has studied the ranges of Lyapunov transformations on matrices. In particular in [2] and [3], the following result was obtained.
Idempotents in completely 0-simple semigroups
Authors: Howie J. M.
Published: May 2009

The structure theorem for completely 0-simple semigroups established by Rees [5] in 1940 has proved a very powerful tool in the investigation of such semigroups. In this paper the theorem is applied to an investigation of the subsemigroup of a completely 0-simple semigroup generated by its idempotents. Previous work on this problem has been carried out by Kim [4], but the present note offers a more direct approach.
Whitehead groups of semidirect products of free groups
Authors: Choo Koo-Guan
Published: May 2009

Let G be a group. We denote the Whitehead group of G by Wh G and the projective class group of the integral group ring ℤ(G) of G by . Let α be an automorphism of G and T an infinite cyclic group. Then we denote by G ×αT the semidirect product of G and T with respect to α. For undefined terminologies used in the paper, we refer to [3] and [7].
An acyclic analogue to Heawood's theorem
Authors: Albertson Michael O. , Berman David M.
Published: May 2009

The concept of acyclic coloring was introduced by Grünbaum [5] and is a generalization of point arboricity.

A proper k-coloring of the vertices of a graph Gis said to be acyclic if G contains no two-colored cycle. The acyclic chromatic number of a graph G, denoted by a(G), is the minimum value of k for which G has an acyclic k-coloring. Let a(n) denote the maximum value of the acyclic chromatic number among all graphs of genus n. In [5], Grünbaum conjectured that a(0) = 5 and proved that a(0)≤9. The conjecture was proved by Borodin [3] after the upper bound was improved three times in [7], [1] and [6]. In [2], we proved that a(1)≤a(0) + 3. The purpose of this paper is to prove the following

Theorem. Any graph of genus n>0 can be acyclically colored with 4n + 4 colors.

It is not known for any n>0 whether a(n)>H(n), the Heawood number [8].
On Pontryagin duality
Authors: Day B. J.
Published: May 2009

The main aim of this article is to discuss the relationship between Pontryagin duality and pro-objects. The basic idea arises from K. H. Hofmann's articles [7] and [8] where it is shown that the elementary abelian (Lie) groups are “dense” in the category of locally compact hausdorff abelian groups.
Triple orthogonal series
Authors: Feinerman Robert
Published: May 2009

In a number of recent papers, we have developed an abstract approach to dual orthogonal series (see [1], [2], [3], and [4]). Such series arise in crack theory, heat transfer, etc. In this paper, we generalize these results to triple orthogonal series. We also show, via a counterexample, that, surprisingly, the results in the dual case are not generalizable as completely as expected.
Remarks on the commutativity of the radicals of group algebras
Authors: Koshitani Shigeo
Published: May 2009

Let K be an arbitrary field with characteristic p > 0, G a finite group of order pag′ with (p, g′) = 1, P a p-Sylow subgroup of G and G′ the commutator subgroup of G. For a ring R denote by J(R) the Jacobson radical of R and by Z(R) the centre of R. We write KG for the group algebra of G over K.
Endomorphism monoids of distributive double p-algebras
Authors: Adams M. E. , Sichler J.
Published: May 2009

A distributive p-algebra is an algebra 〈L; ∨, ∧, *, 0, 1〉 for which 〈L, ∨, ∧, 0, 1〉 is a bounded distributive lattice and * is a unary operation on L such that a ∧ x = 0 if and only if x ≤ a* (i.e. a pseudocomplementation). A distributive double p-algebra is an algebra 〈L; ∨, ∧, *, +, 0, 1〉 in which the deletion of + gives a distributive p-algebra and the deletion of * gives a dual distributive p-algebra, that is a ∨ (x = 1 if and only if x ≥ a+.
Diagrams associated with subgroups of Fuchsian groups
Authors: Stothers W. W.
Published: May 2009

The use of graphs in the study of groups is well-established. In this paper, we wish to indicate how certain graph-like objects may be used in a similar way. A diagram is a pseudograph which may have some free edges, i.e. edges with just one end.
On the derivatives at the origin of entire harmonic functions
Authors: Armitage D. H.
Published: May 2009

If f is an entire function in the complex plane such that

where 0 ≤ α < 1, and all the derivatives of f at 0 are integers, then it is easy to show that f is a polynomial (see e.g. Straus [10]). The best possible result of this type was proved by Pólya [9]. The main aim of this paper is to prove two analogous results for harmonic functions defined in the whole of the Euclidean space Rn, where n ≥ 2 (i.e. entire harmonic functions).
On the approximation of π by special algebraic numbers
Authors: Braune Erhard
Published: May 2009

Suppose that m0 is an integer, m0≥3, ρ = exp(2πi/m0), K = ℚ(ρ, i), v denotes the degree of K, ξ∈K has degree N over ℚ. The length, where is the (irreducible) minimal polynomial for with ξ relatively prime integer coefficients. Feldman [2, p. 49] proved that there is an absolute constant c0>0 such that

From [2, p. 49, Notes 1 and 2] we know that v = φ(m0) or v = 2φ(m0), and φ(m0)≥ c1m0(log log m0)−1 (c1 > an absolute constant), where φ(m0) denotes Euler's function.
On the Stability of Barrelled Topologies II
Authors: Tweddle I. , Yeomans F. E.
Published: May 2009

If E is a Hausdorff barrelled space, which does not already have its finest locally convex topology, then the continuous dual E′ may be enlarged within the algebraic dual E*. Robertson and Yeomans [10] have recently investigated whether E can retain the barrelled property under such enlargements. Whereas finite-dimensional enlargements of the dual preserve barrelledness, they have shown that this is not always so for countable-dimensional enlargements E′+M. In fact, if E contains an infinitedimensional bounded set, there always exists a countable-dimensional M for which the Mackey topology τ(E, E′+M) is not barrelled [10, Theorem 2].
The conjugate hull of an inverse semigroup
Authors: Petrich Mario
Published: May 2009

There is a large body of literature on inverse semigroups. This literature contains a considerable amount of information concerning congruences on these semigroups, which is not surprising in view of the demonstrated fact that congruences on inverse semigroups play a decisive role in most of the existing structure theorems. In addition, for an inverse semigroup of known structure, finding its congruence lattice, or even certain properties of this lattice, often gives information about these semigroups not apparent in their structure theorems.
A problem in lie rings
Authors: Wiegold James
Published: May 2009

An important step in the proof of Kostrikin's fundamental theorem [2] on finite groups of prime exponent is the following result.

Theorem 1. Let L be a Lie algebra of characteristic p satisfying the t-th Engel condition for some t<p, and suppose that L is generated by elements that are right-Engel of length 2. Then L is locally nilpotent.
Results on common fixed points on complete metric spaces
Authors: Fisher Brian
Published: May 2009

The following theorem was proved in [1].

Theorem 1. Let S and T be continuous, commuting mappings of a complete, bounded metric space (X, d) into itself satisfying the inequality

for all x, y in X, where 0≤c<1 and p, p′, q, q′≥0 are fixed integers with p+p′, q+q′≥1. Then S and T have a unique common fixed point z. Further, if p′ or q′ = 0, then z is the unique fixed point of S and if p or q = 0, then z is the unique fixed point of T.
A common generalization of local cohomology theories
Authors: Bijan-Zadeh M. H.
Published: May 2009

Throughout this note all rings considered will be commutative and noetherian and will have non-zero identity elements. A will always denote such a ring and the category of all A-modules and all A-homomorphisms will be denoted by .
Killing rational characteristic classes by surgery
Authors: Papastavridis Stavros
Published: May 2009

Let fr:Xr → BO(r) be a sequence of fibrations with maps gr:Xr → Xr+1 such that the usual diagram commutes. For such a situation R. Lashof defines the concept of an X-structure on manifolds (see [3]), and proves a Thom-isomorphism for the cobordism groups of such manifolds. Let n, m be positive integers which are fixed throughout this paper. If r is very big in comparison with n + m then Xr is a simply connected CW-complex and the map (gr)*:H*(Xr; Q)→ H*(Xr+l; Q) is an isomorphism up to dimension n. Let γ be the pull-back over Xr of the universal r-linear bundle (which is, of course, a bundle over BO(r)). If r is very big in comparison with n + m, then we put X = Xr, and we assume that γ is orientable and oriented. The elements of H*(X; Q) of dimension not greater than n, will be called rational universal X-characteristic classes. It is well-known that many of the usual classes of manifolds may be described in terms of X-structures, (e.g. SO, SU, Spin-manifolds etc.).
On non-Hurwitz groups and non-congruence subgroups of the modular group
Authors: Cohen Jeffrey
Published: May 2009

In this note homomorphisms of (2, 3, n) = 〈x, y: x2 = y3 = (xy)n = 1) into PSL3(q) are considered. Of particular interest is (2, 3, 7) whose finite factors are referred to as Hurwitz groups. It is known (see [3]) that for certain q, PSL2(q) is a Hurwitz group, so that one might suppose that PSL3(q) is a natural place to search for new Hurwitz groups. This intuition turns out to be ill-founded, for as we shall see all Hurwitz subgroups of PSL3(q) have already been discovered in [3].
Generalized Ramsey theory for graphs XII: Bipartite Ramsey sets
Authors: Harary Frank , Harborth Heiko , Mengersen Ingrid
Published: May 2009

Following the notation in Faudree and Schelp [3], we write G → (F, H) to mean that every 2-coloring of E(G), the edge set of G, contains a green (the first color) F or a red (the second color) H. Then the Ramsey number r(F, H) of two graphs F and H with no isolated vertices has been defined as the minimum p such that Kp → (F, H).
The algebraic structure of the semigroup of binary relations on a finite set
Authors: Kim Ki Hang , Roush Fred William
Published: May 2009

In this paper we study some questions proposed by B. Schein [8] regarding the semigroup of binary relationsBxfor a finite setX: what is the ideal structure ofBx, what are the congruences onBx, what are the endomorphisms ofBx? For |X| =nit is convenient to regardBxas the semigroupBnofn×n(0, l)-matrices under Boolean matrix multiplication.
The boundary of the numerical range
Authors: de Barra G.
Published: May 2009

In [1] it was shown that for a compact normal operator on a Hilbert space the numerical range was the convex hull of the point spectrum. Here it is shown that the same holds for a semi-normal operator whose point spectrum satisfies a density condition (Theorem 1). In Theorem 2 a similar condition is shown to imply that the numerical range of a semi-normal operator is closed. Some examples are given to indicate that the condition in Theorem 1 cannot be relaxed too much.
Strongly closed bounded Boolean algebras of projections
Authors: Gillespie T. A.
Published: May 2009

It is known that every complete Boolean algebra of projections on a Banach space X is strongly closed and bounded and that, although the converse of this result fails in general, it is valid if X is weakly sequentially complete [1, XVII. 3, pp. 2194–2201]. In the present note it is shown that this converse is in fact valid precisely when X contains no subspace isomorphics to the sequence space c0. More explicitly, the following two results are proved. In both, X may be a real or complex space, but c0 will consist of the null sequences in the underlying scalar field.
Subgroups of infinite index in the modular group II
Authors: Stothers W. W.
Published: May 2009

Let H be a subgroup of Γ, the modular group. Let h be the number of orbits of under the action of H. In each orbit, the stabilizers are H-conjugate. Let U be the mapping z↦z + 1. Each stabilizer is Γ-conjugate to 〈Uc〉 for some non-negative integer c. The integer c is the cusp-width of the orbit. Let h0 be the number of orbits with non-trivial stabilizer, i.e. with c>0. The sequence (c(1), …, c(h0)) of non-zero cuspwidths is the cusp-split of H. Clearly, h0<h, and h∞ = h−h0 is the number of orbits with trivial stabilizer.
Subgroups of infinite index in the modular group III
Authors: Stothers W. W.
Published: May 2009

As in [4], a specification is a list (r, s, t1 h0, hmc (l), …, c(ho)) such that

(i) each of r, s, t1 h0, h∞ is a non-negative integer,

(ii) for each i, c(i) is a positive integer,

(iii) if h∞ = 0 then ho = ∞,

(iv) if h∞ = 1 and t1 + h0 is finite then t1 is even,

(v) r + s + t1 + h0 + h∞ = ∞.
Modules over polycyclic groups have many irreducible images
Authors: Brown Kenneth A.
Published: May 2009

Recall that a Noetherian ring R is a Hilbert ring if the Jacobson radical of every factor ring of R is nilpotent. As one of the main results of [13], J. E. Roseblade proved that if J is a commutative Hilbert ring and G is a polycyclic-by-finite group then JG is a Hilbert ring. The main theorem of this paper is a generalisation of this result in the case where all the field images of J are absolute fields—we shall say that J is absolutely Hilbert. The result is stated in terms of the (Gabriel–Rentschler–) Krull dimension; the definition and basic properties of this may be found in [5]. Let M be a finitely generated right module over the ring R. We write AnnR(M) (or just Ann(M)) for the ideal {r ∈ R: Mr = 0}, the annihilator of M in R. If M is also a left module, its left annihilator will be denoted l-AnnR(M). If R is a group ring JG, put
A note on the modulus of convexity
Authors: Plant Andrew T.
Published: May 2009

In [1, Corollary 5], Figiel gives an elegant demonstration that the modulus ofconvexity δ in real Banach space X is nondecreasing, where

It is deduced from this that in fact δ(ɛ)/ɛ is nondecreasing [Proposition 3]. During the course of the proof [Lemma 4] it is stated that if v ∊ Sx is a local maximum on Sx of φ ∈Sx*, then v is a global maximum (φ(v) = 1). This is false; it could be that v is a global minimum. It is easy to construct such an example in R2 endowed with the maximum norm. What is true is that v is a global maximum of |φ|.
Inverse semigroups whose full inverse subsemigroups form a chain
Authors: Jones P. R.
Published: May 2009

The structure of semigroups whose subsemigroups form a chain under inclusion was determined by Tamura [9]. If we consider the analogous problem for inverse semigroups it is immediate that (since idempotents are singleton inverse subsemigroups) any inverse semigroup whose inverse subsemigroups form a chain is a group. We will therefore, continuing the approach of [5, 6], consider inverse semigroups whose full inverse subsemigroups form a chain: we call these inverse ▽-semigroups.
On some generalisations of the Riemann-Liouville and Weyl fractional integrals and their applications
Authors: Lowndes J. S.
Published: May 2009

1. For functions f ∈ LLoc[0, ∞) the Riemann-Liouville operator of fractional integration I∞ is defined by

and its adjoint operator, the Weyl operator Kα, is defined by

for functions f ∈ LLoc[0, ∞) having a suitable behaviour at infinity.
On the fractional parts of αn2 and βn
Authors: Baker R. C.
Published: May 2009

We denote by ‖…‖ the distance to the nearest integer. Let α and β be real. W. M. Schmidt [5] proved that for ε > 0 and N>c1(ε) there is a natural number n such that

This extends a theorem of H. Heilbronn [4] and also sharpens a theorem of H. Davenport [3].
A note on Jacobson's conjecture for right Noetherian rings
Authors: Brown K. A. , Lenagan T. H.
Published: May 2009

In 1956, Jacobson asked whether the intersection of the powers of the Jacobson radical, J(R), of a right Noetherian ring R, must always be zero [4, p. 200]. His question was answered in the negative by I. N. Herstein [3], who noted that , where Z(2) denotes the ring of rational numbers with denominator prime to 2, affords a counterexample. In contrast, the ring , though similar in appearance to R1, satisfies . (Here, k denotes a field.)
Generation of the lower central series
Authors: Guralnick Robert M.
Published: May 2009

Let G be a group. The rth term LrG of the lower central series of G is the subgroup generated by the r-fold commutators

where , and for r > 1,
A confluent hypergeometric integral equation
Authors: Love E. R. , Prabhakar T. R. , Kashyap N. K.
Published: May 2009

Recently there have appeared papers ([7], [8]; also see [9]) in which integral equations with kernels involving the confluent hypergeometric function

have been studied. These equations are mainly Volterra equations of the first kind except that they have infinite domain (0, ∞). The rest are of the related type with integrals over (x, ∞) instead of (0, x); and all are convolution equations.
Quotient rings of semiprime rings with bounded index
Authors: Hannah John
Published: May 2009

We say that a ringRhasbounded indexif there is a positive integernsuch thatan= 0 for each nilpotent elementaofR. Ifnis the least such integer we sayRhas indexn. For example, any semiprime right Goldie ring has bounded index, and so does any semiprime ring satisfying a polynomial identity [10, Theorem 10.8.2]. This paper is mainly concerned with the maximal (right) quotient ringQof a semiprime ringRwith bounded index. Several special cases of this situation have already received attention in the literature. IfRsatisfies a polynomial identity [1], or if every nonzero right ideal ofRcontains a nonzero idempotent [18] then it is known thatQis a finite direct product of matrix rings over strongly regular self-injective rings, the size of the matrices being bounded by the index ofR. On the other hand ifRis reduced (that is, has index 1) thenQis a direct product of a strongly regular self-injective ring and a biregular right self-injective ring of type III ([2] and [15]; the terminology is explained in [6]). We prove the following generalization of these results (see Theorems 9 and 11).
Universal fields of fractions for polycyclic group algebras
Authors: Passman D. S.
Published: May 2009

Let G be a polycyclic-by-finite group and let K[G] denote its group algebra over the field K. In this paper we discuss localization in K[G] and in particular we prove that every faithful completely prime ideal is localizable. Furthermore, using a sequence of localizations, we show that, for G polyinfinite cyclic, the classical right quotient ring (K[G]) is in fact a universal field of fractions for K[G]. Finally we offer an example of a domain K[G] which does not have a universal field of fractions.
Elementary equivalence and the commutator subgroup
Authors: van den Dries L. P. D. , Glass A. M. W. , Macintyre Angus , Mekler Alan H. , Poland John
Published: May 2009

If G and H are elementarily equivalent groups (that is, no elementary statement of group theory distinguishes between G and H) then the definable subgroups of G are elementarily equivalent to the corresponding ones of H. But G′ of G, consisting of all products of commutators [a, b] = a−1b−1ab of elements a and b of G, may not be definable. Must G′ and H′ be elementarily equivalent?
Isometric representation of M(G) on B(H)
Authors: Ghahramani F.
Published: May 2009

In a recent paper, E. Størmer, among other things, proves the existence of an isometric isomorphism from the measure algebra M(G) of a locally compact abelian group G into BB(L2(G)), ([6], Proposition 4.6). Here we give another proof for this result which works for non-commutative G as well as commutative G. We also prove that the algebra L1(G, λ), with λ the left (or right) Haar measure, is not isometrically isomorphic with an algebra of operators on a Hilbert space. The proofs of these two results are taken from the author's Ph.D. thesis [4], submitted to the University of Edinburgh before Størmer's paper. The author wishes to thank Dr. A. M. Sinclair for his help and encouragement.
M-spaces with quasi-interior points
Authors: Feldman W. A. , Porter J. F.
Published: May 2009

In this paper we characterize those locally convex lattices which can be represented as dense sublatices containing 1 in a space C(X) and whose topologies can be recognized as topologies of uniform convergence on selections of compact subsets of X. Here C(X) is the lattice of continuous real-valued functions on a completely regular space X. The class of such locally convex lattices includes the classical order unit spaces investigated by Kakutani [3], arbitrary products of order unit spaces, for example ∏ L∞ and the order partition spaces studied in [1].
Square roots in finite full transformation semigroups
Authors: Snowden Mary , Howie J. M.
Published: May 2009

Let X be a finite set and let (X) be the full transformation semigroup on X, i.e. the set of all mappings from X into X, the semigroup operation being composition of mappings. This paper aims to characterize those elements of (X) which have square roots. An easily verifiable necessary condition, that of being quasi-square, is found in Theorem 2, and in Theorems 4 and 5 we find necessary and sufficient conditions for certain special elements of (X). The property of being compatibly amenable is shown in Theorem 7 to be equivalent for all elements of (X) to the possession of a square root.
Analytic functions of a prespectral operator
Authors: Al-Khezi S.
Published: May 2009

The purpose of this note is to present a unified treatment of the material contained in Chapter 10 of [2] on roots and logarithms of prespectral operators. Our main result gives a sufficient condition for an analytic function of a prespectral operator of class Γ to be prespectral of class Γ. A result in the opposite direction for spectral operators has been obtained by Apostol [1]. Terminology and notation in this paper are as in [2].
Arens regularity and retractions
Authors: Arikan Nilgün
Published: May 2009

In this paper a characterisation of the regularity of a normed algebra A is given in terms of retractions onto A** from A4*. The second dual A** of a normed algebra A possesses two natural Banach algebra multiplications, say ° and *. Each of ° and * extends the original algebra multiplication on A; see (2). An algebra A is called regular if and only if F * G = F ° G for all F, G ∈ A**. See (1). The existing results in the Arens regularity theory can be found in a recent survey (2). Denoting the nth dual of A by An*, and en the natural embedding of An* in its second dual A(n+2)*, we can naturally represent the second dual A** of A as a Banach space retract of A4* in two different ways:

Our main results say that A** is in fact a Banach algebra retract of A4* (i.e. the maps involved are homomorphisms) in either of these cases if and only if A is regular.
Representations of infinite soluble groups
Authors: Musson Ian M.
Published: May 2009

The purpose of this paper is to study the following two questions.

(1) When does the group algebra of a soluble group have infinite dimensional irreducible modules?

(2) When is the group algebra of a torsion free soluble group primitive?

In relation to the first question, Roseblade [13] has proved that if G is a polycyclic group and k an absolute field then all irreducible kG-modules are finite dimensional. Here we prove a converse.
Averaging operators in non commutative Lp spaces I
Authors: Barnett Christopher
Published: May 2009

The origin of the theory of averaging operators is explained in [1]. The theory has been developed on spaces of continuous functions that vanish at infinity by Kelley in [3] and on the Lp spaces of measure theory by Rota [5]. The motivation for this paper arose out of the latter paper. The aim of this paper is to prove a generalisation of Rota's main representation theorem (every average is a conditional expectation) in the context of a ‘non commutative integration’. This context is as follows. Let be a finite von Neumann algebra and ϕ a faithful normal finite trace on such that ϕ(I) = 1, where I is the identity of . We can construct the Banach spaces Lp (, ϕ), where 1 ≤ p < °, with norm ∥x∥p = ϕ(÷x÷p)1/p, of possibly unbounded operators affiliated with , as in [9]. We note that is dense in Lp(, ϕ). These spaces share many of the features of the Lp spaces of measure theory; indeed if is abelian then Lp(,ϕ) is isometrically isomorphic to Lp of some measure space.
Strong limits of normal operators
Authors: Conway John B. , Hadwin Donald W.
Published: May 2009

In [1, Theorem 3.3], E. Bishop proved that an operator S on a Hilbert space ℋ is subnormal if and only if there is a net of normal operators {Nα} that converges to S strongly (that is, ‖(Nα–S) f‖→ 0 for every f in ℋ). The proof that such a net exists if S is subnormal is not so difficult; in fact, a sequence of normal operators converging strongly to S can be found. Bishop's proof of the converse, however, is rather complicated and involves, among other things, some complicated arguments using operator-valued measures. The purpose of this note is to provide an easier proof of this part of the theorem. Our interest in finding such a proof was aroused by Paul Halmos.
A Zeuthen Segre formula for even dimensional submanifolds of real projective space
Authors: Bruce J. W.
Published: May 2009

In this paper we generalise results of Craveiro de Carvalho ([3]) in two ways. First we prove the following fact.

PROPOSITION 1. Given any smooth submanifold M of real projective space ℙn, for L in an open dense subset of the space of codimension 2 subspaces of ℙnwe have

(a) L meets M transversally and

(b) the pencil of hyperplanes through L have at worst Morse (A1) contact with M.
A simple proof of Hermite's theorem on the zeros of a polynomial
Authors: Young N. J.
Published: May 2009

In 1856 Hermite showed how to determine by purely rational operations the number of zeros of a given polynomial lying in a specified half plane [1]: one inspects the signature of a certain Hermitian form. This type of result is still of interest for practical applications, and several authors have provided alternatives to Hermite's original, highly computational proof (for example [2, 3]). Recently V. Pták and the author gave a simple matrix-theoretic proof and generalization of a class of Hermite-type theorems [4]. This class included the Schur-Cohn test for zeros in a circle, but not, to our regret, the original theorem of Hermite. The purpose of this note is to show that a slight modification of our method does indeed provide a simple proof of Hermite's theorem.
A problem on the Riesz-Dunford operator calculus and convex univalent functions
Authors: Hwang J. S.
Published: May 2009

In his paper [3], Ky Fan asked whether if f is a convex univalent function in the unit disk, with f(0) = 0 and f'(0) = 1, then is it true that the set of f(A) is a convex set of operators, when A runs through all proper contractions on a Hilbert space? We answer this question in the negative.
Group rings with finite central endomorphism dimension
Authors: Wehrfritz B. A. F.
Published: May 2009

Let F be any field. Denote by the class of all groups G such that every irreducible FG-module has finite dimension over F and by the class of all groups G such that every irreducible FG-module has finite dimension over its endomorphism ring. Clearly
Isotopies of generic plane curves
Authors: Bruce J. W.
Published: May 2009

The aim of this paper is to explore some facets of the geometry of generic isotopies of plane curves. Our major tool will be the paper of Arnol'd [1] on the evolution of wavefronts. The sort of questions one can ask are: in a generic isotopy of a plane curve how are vertices created and destroyed? How does the dual evolve? How can the Gauss map change? In attempting to answer these questions we are taking advantage of the fact that these phenomena are all naturally associated with singularities of type Ak. Now the bifurcation set of an Ak+1 singularity and the discriminant set of an Ak singularity coincide. So we can apply Arnol'd's results on one parameter families of Legendre (discriminant) singularities (e.g. the duals) to get information on one parameter families of Lagrange (bifurcation) singularities (e.g. the evolutes). For bifurcation sets of functions with singularities other than those of type Ak one runs up against problems with smooth moduli—see [4].
Semilocal semigroup rings
Authors: Okniński Jan
Published: May 2009

Semilocal and related classes of group rings have been investigated by many authors (cf. [10]). In particular, the following results have been obtained.

Theorem A[4,10]. Let K be a field and G a group.

(i) If ch K = 0, then K[G] is semilocal if and only if G is finite.

(ii) If ch K = p>0 and G is locally finite, then K[G] is semilocal if and only if G contains a p-subgroup of finite index.

In the case of semigroup rings some stronger conditions have been studied. Munn examined the semisimple artinian situation [6]. Zelmanov showed that if K[G]is artinian then G must be finite [11].
An alternative proof of Diestel's theorem
Authors: Khan M. Ali
Published: May 2009

We show that Diestel's theorem on weak compactness of subsets of L1,(μ, X) can be derived as a simple corollary of James's theorem. It is a pleasure to acknowledge several stimulating conserversations with Dave Emmons and the remarks of an anonymous referee. Errors are, of course, solely mine.
On the monodromy groups of lifted Euler equations
Authors: Kuiken Kathryn , Masterson John T.
Published: May 2009

In [13], Poincaré asked the following question: Which abstract groups can appear as monodromy groups [14] of second order, linear, homogeneous differential equations with meromorphic coefficients (which might depend on one or more parameters) on ℂ? In the present paper, we initiate a classification of monodromy groups of differential equations on compact Riemann surfaces of genus 1. We proceed as follows: Let

be the general Euler equation [1] which has two regular singular points at 0 and λ in the extended complex plane C. Further, let yu(z)(v – 1,2) be an arbitrary but fixed pair of linearly independent solutions to (1) valid in a neighborhood of some ordinary point. Analytic continuation of each solution along a closed loop A in C–{0, λ}, starting and ending at some fixed base point, produces a new solution yv,A (v = 1, 2) which can be expressed as

where the constants avlx (v1 ν = l,2) in C depend on the homotopy class [A] of A. Clearly, y1,A(z)/y2,A(z) = To(y1(z)/y2(z)) where the Möbius transformation T:w→ (a11W + a12)/(a21W + a22) depends on [A]. The set of all Mobius transformations T belonging to every possible closed loop A in C–{0,λ} forms a group G, called the monodromy group of the Euler equation (1). G is generated by the Mobius transformation belonging to a simple, closed loop Ao encircling 0. Hence, G is cyclic.
The Galois group of f(xr)
Authors: Cohen S. D. , Stothers W. W.
Published: May 2009

Let f(x) be an irreducible polynomial of degree n with coefficients in a field L and r be an integer prime to the characteristic of L. The object of this paper is to describe the galois group g of f(xr) over L when the galois group G of f(x) itself over L is either the full symmetric group Snor the alternating group An. We shall call f standard if G = Sn or An with |G|>2.
The motions of algebraic differential equations
Authors: Rubel Lee A.
Published: May 2009

We confine ourselves, for simplicity, to first-order algebraic differential equations (ADE's), although analogous considerations may be made for higher-order ADE's:

P(x, y(x), y'(x)) = 0. (*)

A motion of (*) is a change of independent variable that takes solutions to solutions, that is, a suitable map <p of the underlying interval I into itself so that if y is a solution of (*) then y ° φ is a solution of (*), i.e.
A characterization of subnormal operators
Authors: Lambert Alan
Published: May 2009

In this note a characterization of subnormality of operators on Hilbert space is given. The characterization is in terms of a sequence of polynomials in the operator and its adjoint reminiscent of the binomial expansion in commutative algebras. As such no external Hilbert spaces are needed, nor is it necessary to introduce forms dependent on arbitrary sequences of vectors from the Hilbert space.
On certain subgroups of a join of subnormal subgroups
Authors: Smith Howard
Published: May 2009

1. Introduction: Suppose the group G is generated by subnormal subgroups H and K, and that A, B are normal subgroups of finite index in H, Krespectively. The following question has been asked by J. C. Lennox: Under what circumstances is the subgroupJ = (A, B) subnormal in G? In particular, it is of interest to know when J has finite index in G, for, if this is the case, we may factor out by the normal core of J in G and apply Wielandt's theorem on joins of subnormal subgroups of finite groups [11] to deduce that J is subnormal in G. Here we prove the following result.
On the transcendence degree of group algebras of nilpotent groups
Authors: Lorenz Martin
Published: May 2009

Let G be a finitely generated (f.g.) torsion-free nilpotent group. Then the group algebra k[G] of G over a field k is a Noetherian domain and hence has a classical division ring of fractions, denoted by k(G). Recently, the division algebras k(G) and, somewhat more generally, division algebras generated by f.g. nilpotent groups have been studied in [3] and [5]. These papers are concerned with the question to what extent the division algebra determines the group under consideration. Here we continue the study of the division algebras k(G) and investigate their Gelfand–Kirillov (GK–) transcendence degree.
On some small varieties of distributive Ockham algebras
Authors: Beazer R.
Published: May 2009

J. Berman [2] initiated the study of a variety k of bounded distributive lattices endowed with a dual homomorphic operation paying particular attention to certain subvarieties km, n. Subsequently, A. Urquhart [8] named the algebras in k distributive Ockham algebras, and developed a duality theory, based on H. A. Priestley's order-topological duality for bounded distributive lattices [6], [7]. Amongst other things, Urquhart described the ordered spaces dual to the subdirectly irreducible algebras in Sif. This work was developed further still by M. S. Goldberg in his thesis and the paper [5]. Recently, T. S. Blyth and J. C. Varlet [3], in abstracting de Morgan and Stone algebras, studied a subvariety MS of the variety k1.1. The main result in [3]is that there are, up to isomorphism, nine subdirectly irreducible algebras in MS and their Hasse diagrams are exhibited. The methods employed in [3] are purely algebraic and can be generalized to show that, up to isomorphism, there are twenty subdirectly irreducible algebras in k1.1. In section 3 of this paper, we take a short cut to this result by utilizing the results of Urquhart and Goldberg. Our basic method is simple: the results of Goldberg [5] are applied to k1,1 to produce a certain eight-element algebra B1 in k1,1, whose lattice reduct is Boolean and whose subalgebras are, up to isomorphism, precisely the subdirectly irreducibles in k1.1. We then pick out of the list of twenty such algebras those belonging to the variety MS. In section 4, we sketch a purely algebraic proof along the lines followed by Blyth and Varlet in [3].
On Ramanujan and Dirichlet series with Euler products
Authors: Raghavan S.
Published: May 2009

In his unpublished manuscripts (referred to by Birch [1] as Fragment V, pp. 247–249), Ramanujan [3] gave a whole list of assertions about various (transforms of) modular forms possessing naturally associated Euler products, in more or less the spirit of his extremely beautiful paper entitled “On certain arithmetical functions” (in Trans. Camb. Phil. Soc. 22 (1916)). It is simply amazing how Ramanujan could write down (with an ostensibly profound insight) a basis of eigenfunctions (of Hecke operators) whose associated Dirichlet series have Euler products, anticipating by two decades the famous work of Hecke and Petersson. That he had further realized, in the event of a modular form f not corresponding to an Euler product, the possibility of restoring the Euler product property to a suitable linear combination of modular forms of the same type as f, is evidently fantastic.
Finite abelian surface coverings
Authors: Jassim S. A.
Published: May 2009

LetGbe a finite abelian group, andYbe a closed surface. The problems of classifying and enumerating the free and effectiveG-actions onYmodulo selfhomeomorphisms ofYandX=Y/Gcan be transferred into ones of classifying regularG-coverings onX. P. A. Smith [7], proved that for any prime numberpthere arepr(r–1)/2equivalence classes of free (ℤp)ractions onYprovided thatrℤgenus ofX. This paper is devoted to the classification and the enumeration of regularG-covering surfaces, whenGis any finite abelian group. Recently, A. Edmonds [2] classified theG-actions on closed surfaces by theirG-bordism classes in the set(G) of free orientedG-cobordism classes of free orientedG-surfaces.