bluuebunny/crossref_metadata_2025 huggingface dataset:
On the values of the Epstein zeta function
Authors: Smart John Roderick
Published: May 2009

Let ζ(s) = σn-s(Res >1) denote the Riemann zeta function; then, as is well known,, whereBmdenotes themth Bernoulli number, In this paper we investigate the possibility of similar evaluations of the Epstein zeta function ζq(s) at the rational integerss = k> 2. Let

be a positive definite quadratic form and

where the summation is over all pairs of integers except (0, 0). In attempting to evaluate ζq(k) we are guided by Kronecker's first limit formula [11]

where γ is Euler's constant,

is the Dedekind eta-function, and τ is the complex number in the upper half plane, ℋ, associated with Q by the formula

On the basis of (1.3) we would expect a formula involving functions of τ. This formula is stated in Theorem 1, (2.13).
A remark on the divisor function d(n)
Authors: Vaughan R. C.
Published: May 2009

Let d(n) denote the number of positive divisors of. A long time ago, Erdös and Mirsky [1] raised the question whether the equation d(n) = d(n+l) holds for infinitely many n. It does not seem easy to settle this problem, and in the present note we give a partial result.
A ratio limit theorem for contraction projections and applications
Authors: Kopp P. E.
Published: May 2009

The similarities between martingale convergence theory and pointwise ergodic theory are now well known [5, 7, 9, 10]. In [5] the similarity between the proofs of the Hopf– Dunford–Schwartz individual ergodic theorem and the martingale convergence theorem is systematically exploited to produce very general ” maximal ergodic ” inequalities for certain sequences of contractions on L1-spaces. A different approach by Rota [10] and Rao [9] leads to a unified convergence theory for martingales and Abel limits. Bishop [1] has produced ” upcrossing” inequalities which yield both theChacon-Ornstein theorem [4] and the martingale convergence theorem.
Locally convex spaces with fundamental sequences
Authors: De Wilde Marc
Published: May 2009

After Dieudonné [3], many authors have considered locally convex topological vector spaces that admit a fundamental sequence of compact, precompact,… subsets. Their work has been essentially to identify them as strong duals of Fréchet-Montel spaces, under suitable additional assumptions (barrelledness or evaluability). However, it seems that these spaces have never been characterized without additional assumptions. That is the aim of the present paper.
Pure-injective modules
Authors: Fakhruddin Syed M.
Published: May 2009

It is proved that a pure-injective module over a commutative ring with unity is a summand of a product of duals of finitely presented modules, where duals are to be understood with reference to the circle group T, with induced module structures. Using similar techniques, it is also shown that an R-module has its underlying group pure-injective precisely when it is a submodule of a product of duals of cyclic modules and also a summand as abelian group of the same product.

All rings considered are commutative with unity and all modules are unitary. Let Mod-R be the category of modules over a ring R. An exact sequence 0 → A → B → C → 0 in Mod-R is pure-exact if, for any N in Mod-R, 0 → A⊗N → B⊗N → C⊗N → 0 is exact. A module M is pure-injective if it has the injective property relative to the class of pure-exact sequences in Mod-R. A module P is FP (finitely presented) if it is the image of a finitely generated free module with a finitely generated kernel. A module M is compact if it carries a Hausdorff compact topology so that M is a topological R-module. Let T denote the circle group—the group of real numbers modulo the integers—and let X* denote the dual module Homℤ(X, T) of the module X.
On limit problems associated with some inequalities
Authors: Diananda P. H.
Published: May 2009

Let {an} be a sequence of non-negative real numbers. Suppose that

Then M1,n is the arithmetic mean, MO,n the geometric mean, and Mr,n the generalized mean of order r, of a1, a2, …, an. By a result of Everitt [1] and McLaughlin and Metcalf [5], {n(Mr,n–Ms,n)}, where r ≧ l ≧ s, is a monotonic increasing sequence. It follows that this sequence tends to a finite or an infinite limit as n → ∞. Everitt [2, 3] found a necessary and sufficient condition for the finiteness of this limit in the cases r, s = 1, 0 and r ≧ 1 > s > 0. His results are included in the following theorem.
An order-preserving representation theorem for complex Banach algebras and some examples
Authors: Thompson A. C. , Vijayakumar M. S.
Published: May 2009

Let A be a complex Banach algebra with unit e of norm one. We show that A can be represented on a compact Hausdorff space ω which arises entirely out of the algebraic and norm structures of A. This space induces an order structure on A that is preserved by the representation. In the commutative case, ω is the spectrum of A, and we have a generalization of Gelfand's representation theorem for commutative complex Banach algebras with unit. Various aspects of this representation are illustrated by considering algebras of n × n complex matrices.
Compatible tight Riesz orders and prime subgroups
Authors: Reilly N. R.
Published: May 2009

A tight Riesz group is a partially ordered group which satisfies a strengthened form of the Riesz interpolation property. The term “tight” was introduced by Miller in [8], and the tight interpolation property has been considered in papers by Fuchs [3], Miller [8, 9], Loy and Miller [7] and Wirth [12]. If the closure of the cone P, in the interval topology, of such a partially ordered group G contains no pseudozeros, then is itself the cone of a partial order on G. Loy and Miller found of particular interest the case in which this associated partial orderis a lattice order. This situation was then considered in reverse by A. Wirth [12] who investigated under what circumstances a lattice ordered group would permit the existence of a tight Riesz order (called a compatible tight Riesz order) for which the initial lattice order is the order defined by the closure of the cone of the tight Riesz order.Wirth gave two fundamental anduseful characterizations of those subsets of the cone of a lattice ordered group that canbe the strict cone of a compatible tight Riesz order; one is in terms of archimedean classes and the other is an elementwise characterization. Although Loy, Miller and Wirth restricted their attention to abelian groups, much of what they do carries over verbatim to nonabelian groups. In the main result of this paper (Theorem 2.6) a description of the strict cone of a compatible tight Riesz order on a lattice ordered group Gis given in terms of the prime subgroups of G.This is particularly useful when one is attempting to identify the compatible tight Riesz orders on some particular lattice ordered group or class of lattice ordered groups, since it narrows down to a convenient family of subsets the possible candidates for strict cones of compatible tight Riesz orders. These can then be tested under Wirth's criteria. This technique is illustrated in § 5, where the compatible tight Riesz orders are determined o a lattice ordered group of the type V(Γ, Gγ), where Γ is of finite width, and in § 6, where two examples are considered.
On the generalized Josephus problem
Authors: Jakóbczyk F.
Published: May 2009

The problem of Josephus and the forty Jewsis well known [1, 3]. In its most general form, this problem is equivalent to the problem of m-enumeration of a set, as described below.

Define the ordered set

We choose and remove cyclically, from left to right, each with element of Zn until the set is exhausted. The chosen elements are ordered into a new ordered set

which is therefore a permutation of Zn, obtained by what we call m-enumeration of theset Zn.
Some dual integral equations involving inverse finite Mellin transforms
Authors: Tweed John
Published: May 2009

The object of this paper is to find the solutions of some dual equations involving the inverses of certain Mellin type transforms that were first introduced by D. Naylor in his paper [1]. Because these transforms are relatively unknown we shall begin by defining them and making a note of some of their properties. The main result is contained in the following theorem.
A characterization of the definiteness of a Hermitian matrix
Authors: Au-Yeung Yik-Hoi , Yuen Tai-Kwok
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 (unitary resp.) if A = A* (AA*= identity matrix resp.). An n ×x n hermitian matrix A is said to be definite (semidefinite resp.) if uAu*vAv* ≥ 0 (uAu*vAv* ≧ 0 resp.) for all nonzero u and v in Fn. If A and B are n × n hermitian matrices, then we say that A and B can be diagonalized simultaneously into blocks of size less than or equal to m (abbreviated to d. s. ≧ m) if there exists a nonsingular matrix U with elements in F such that UAU* = diag{A1,…, Ak} and UBU* = diag{B1…, Bk}, where, for each i = 1, …, k, Ai and Bk are of the same size and the size is ≧ m. In particular, if m = 1, then we say A and B can be diagonalized simultaneously (abbreviated to d. s.).
On commutative non-self-adjoint operator algebras
Authors: Kelly R. H.
Published: May 2009

A proof is given here of a theorem of Sarason [9, Theorem 2], the proof being valid in an arbitrary (non-separable) complex Hilbert space. Sarason's proof uses a theorem and lemma of Wermer which may both fail when the separability hypothesis is omitted [3]. By using a special case of Sarason's theorem and another result of Sarason [10, Lemma 1] a simplified and shortened proof is given of a result of Scroggs [11, Corollary 1].
On a conjecture of G. Hajós
Authors: Sands A. D.
Published: May 2009

The purpose of this note is to provide by means of an example a negative answer to a conjecture of Hajós [3] concerning the factorization of finite abelian groups. This question is also raised as Problem 81 in Fuchs [2].
On the enumeration of planar trees of hexagons
Authors: Beineke L. W. , Pippert R. E.
Published: May 2009

In their paper “ The enumeration of tree-like polyhexes”, Harary and Read [6] consider structures obtained by assembling hexagons subject to certain restrictions. Their problem is introduced as a simplified hexagonal cell-growth problem.
Extended Schauder decompositions of locally convex spaces
Authors: Webb J. H.
Published: May 2009

Let E[τ] be a locally convex Hausdorff topological vector space. An extended decomposition of E[τ] is a family {Ea}α∈A of closed subspaces of E such that, for each x in E and each α in A, there exists a unique point xα in Eα, with Here convergence will have the following meaning. Let Ф denote the set of all finite subsets of A. The sum is said to be convergent to x if for each neighbourhood U of 0 in E, there is an element φ0 of Ф such that , for all φ in Ф containing φ0. It follows that is Cauchy if and only if, for each neighbourhood U of 0 in E, there is an element φ0 of Ф such that , for all φ in Ф disjoint from φ0.
Characterizations of commutativity forC*-algebras
Authors: Crabb M. J. , Duncan J. , McGregor C. M.
Published: May 2009

Letbe aC*-algebra acting on the Hilbert spaceHand letbe the self-adjoint elements of. The following characterization of commutativity is due to I. Kaplansky (see Dixmier [3, p. 58]).
An analogue of a conjecture of Sato and Tate for a Hilbert modular form
Authors: Resnikoff H. L. , Saldaña R. L.
Published: May 2009

If k denotes a number field and εm is the product of an elliptic curve ε with itself m times over k, then for each prime π where ε has non-degenerate reduction, the zeta factor ζ(επ'S) can be expressed as

Where |π| denotes the norm of π. It is a consequence of a conjecture of Tate [16] that if ε does not have complex multiplications, then the numbers are distributed according to the density function

that is, the density of the set of primes π such that – is
On the construction of ring extensions
Authors: Green Edward L. , Reiten Idun
Published: May 2009

Let ∧ denote a basic artin ring and r its radical. In most of this paper we assume that r2 = 0 and that Λ is a trivial extension Λ/r ⋉ r (see Section 1 for definition). Let P1 …, Pn be the non-isomorphic indecomposable projective (left) Λ-modules, and consider triples (Pi, Mi, ui), where the Mi, are (left) Λ-modules and ui:rPi → Mi/rMi isomorphisms. From this data we construct a new ring Г, which in “nice cases” has the property that r′3 = 0, Г/r′2 ≅ Λ, and r′ Qi ≅ Mi as (left) Λ-modules, where the Qi are the indecomposable projective (left) Г-modules and r′ is the radical of Г.
E-unitary congruences on inverse semigroups
Authors: Reilly N. R. , Munn W. D.
Published: May 2009

By an E-unitary inverse semigroup we mean an inverse semigroup in which the semilattice is a unitary subset. Such semigroups, elsewhere called ‘proper’ or ‘reduced’ inverse semigroups, have been the object of much recent study. Free inverse semigroups form a subclass of particular interest.

An important structure theorem for E-unitary inverse semigroups has been obtained by McAlister [4, 5]. From a triple (G, ) consisting of a group G, a partially ordered set and a subset , satisfying certain conditions, he constructs an E-unitary inverse semigroup P(G, ). A semigroup of this type is called a P-semigroup. The structure theorem states that every E-unitary inverse semigroup is, to within isomorphism, of this form. A second theorem asserts that every inverse semigroup is isomorphic to a quotient of a Psemigroup by an idempotent-separating congruence. We refer below to these results as McAlister's Theorems A and B respectively. A triple (C, ) of the type used to construct a P-semigroup is here termed a “McAlister triple”. It is shown further, in [5], that there is essentially only one such triple corresponding to a given E-unitary inverse semigroup.
Embedding inverse semigroups in wreath products
Authors: Houghton C. H.
Published: May 2009

Any extension of a group A by a group B can be embedded in their wreath product A Wr B. Here we consider generalizations of this result for inverse semigroups.

Suppose S is an inverse semigroup and ρ0 is a congruence on S. We put T = S/ρ0 and denote the natural map from S to T by ρ. The kernel of ρ is the inverse image ETρ−1 of the semilattice ET of idempotents of T. First we show that if each ρ0-class of idempotents of S is inversely well-ordered, then S can be embedded in K Wr T, the standard wreath product of K and T. In general, not all elements of K Wr T have inverses. However, we can define a wreath product W(K, T) which is an inverse semigroup and which contains S when the previous condition holds. If ρ0 is idempotent-separating and S is 0-bisimple, K is the union of zero and a family of isomorphic groups. In this case, we can replace K by a single component group G of K, augmented by zero, and show that S can be embedded in W(G0, T). These results are analogous to the extension theories of D'Alarcao [1] and Munn [3] and they give conditions under which all inverse semigroup extensions of an inverse semigroup A by an inverse semigroup T are contained in a semigroup with structure depending only on A and T.
On various types of barrelledness and the hereditary property of (DF)-spaces
Authors: Husain T. , Wong Yau-Chuen
Published: May 2009

Recently, Levin and Saxon [5], De Wilde and Houet [2] defined the σ-barrelledness while Husain [3] defined the countable barrelledness and countable quasibarrelledness. It is well-known that barrelled spaces are countably barrelled, and countably barrelled spaces are σ-barrelled. It is natural to ask whether there is some condition for σ-barrelled (resp. countably barrelled) spaces to be countably barrelled (resp. barrelled). Using the concept of S-absorbent sequences of sets, we are able to give such conditions in Theorem 2.5 and Corollaries 2.6 and 2.7.
Abelian Ergodic Theorems for Vector-Valued Functions
Authors: Kopp P. E.
Published: May 2009

The definition of Ωf,a on p. 57 should read a
The number of Hamiltonian circuits in large, heavily edged graphs
Authors: Sheehan J. , Wright E. M.
Published: May 2009

G is a graph on n nodes with q edges, without loops or multiple edges. We write α = q/n and β for the maximum degree of any node of G. We write

and H for the number of Hamiltonian circuits (H.c.) in the complement of G, or, what is the same thing, the number of those H.c. in the complete graph Kn which have no edge in common with G. Our object here is to prove the following theorem.
On polytopes with small faces
Authors: Lillington J. N.
Published: May 2009

In this paper all sets considered are assumed to be compact subsets of Euclidean Space En. A number of results concerning the total edge-lengths of polyhedra have been given by various authors, many of which are mentioned in references in [1]. In [1], it was conjectured that all polytopes inscribed in the unit sphere and containing its centre have total edge-length greater than 2n. This was proved true for simplicial polytopes and shown to be best possible in the sense that there exist simplices with the stated property and with total edge-length arbitrarily close to 2n. In this paper we shall show that the bound is not always best possible if the magnitudes of the faces of such polytopes are restricted and we shall also give some related results on surface areas. This work was carried out while the author was a research student at Royal Holloway College, London and is a revised version of part of the author's thesis approved for the Ph.D. degree.
An algorithm for determining defining relations of a subgroup
Authors: McLain D. H.
Published: May 2009

Suppose that G is a finitely presented group, and that we are given a set of generators for a subgroup H of finite index in G. In this paper we describe an algorithm by which a set of defining relations may be found for H in these generators.

The algorithm is suitable for programming on a digital computer. It appears to have significant computational advantages over the method of Mendelsohn [8] (which is based on the Schreier-Reidemeister results, see for example [4, pp. 91–95]) in those cases where the generators of H are given as other than the familiar Schreier-Reidemeister generators.
Derivations and pure states
Authors: Sproston J. P.
Published: May 2009

In this note we point out and discuss a relationship between the following two properties which a C*-algebra A may have:

(D) all derivations on A are inner;

(P) the set of pure states of A is w*-compact.

The relationship is deduced from a comparison of two existing theorems ((I) and (II) below).
Remarks on Weak Compactness in L1(μ,X)
Authors: Diestel J.
Published: May 2009

Let (Ω,Σ,μ) be a finite measure space and X a Banach space. Denote by L1 (μ,X) the Banach space of (equivalence classes of) μ-strongly measurable X-valued Bochner integrable functions f:Ω→X normed by

The problem of characterizing the relatively weakly compact subsets of L1(Ω, X) remains open. It is known that for a bounded subset of L1(μ, X) to be relatively weakly compact it is necessary that the set be uniformly integrable; recall that K ⊆ L1, (μ, X) is uniformly integrable whenever given ε >0 there exists δ > 0 such that if μ (E) ≦ δ then ∫E∥f∥ dμ ≦ δ, for all f ∈ K. S. Chatterji has noted that in case X is reflexive this condition is also sufficient [4]. At present unless one assumes that both X and X* have the Radon-Nikodym Property (see [1]), a rather severe restriction which, for purposes of potential applicability, is tantamount to assuming reflexivity, no good sufficient conditions for weak compactness in L1(μ, X) exist. This note puts forth such sufficient conditions; the basic tool is the recent factorization method of W. J. Davis, T. Figiel, W. B. Johnson and A. Pelczynski [3].
Solution of an integral equation
Authors: Lowndes J. S.
Published: May 2009

In this paper we obtain the general solution of the equation

whose kernel is

where 0<α, β<1, μ, η, σ > 0, δ>0, are real parameters, z = max(x, y), φ(t) and g(y) are prescribed functions and ƒ(x) is to be determined.
Quotients of F-spaces
Authors: Kalton N. J.
Published: May 2009

Let X be a non-locally convex F-space (complete metric linear space) whose dual X′ separates the points of X. Then it is known that X possesses a closed subspace N which fails to be weakly closed (see [3]), or, equivalently, such that the quotient space X/N does not have a point separating dual. However the question has also been raised by Duren, Romberg and Shields [2] of whether X possesses a proper closed weakly dense (PCWD) subspace N, or, equivalently a closed subspace N such that X/N has trivial dual. In [2], the space Hp (0<p<1) was shown to have a PCWD subspace; later in [9], Shapiro showed that ℓp (0<p<1) and certain spaces of analytic function have PCWD subspaces. Our first result in this note is that every separable non-locally convex F-space with separating dual has a PCWD subspace.
On groups with decomposable commutator subgroups
Authors: Guralnick Robert M.
Published: May 2009

Let G be a group. We define λ(G) to be the smallest integer n such that every element of the commutator subgroup G′ is a product of n commutators. Ito [4] has shown that λ (An)= 1 for all n. Thompson [7] has shown that λ (SLn(q))= 1 for all n and q. In fact, there is no known simple group G such that λ(G)>1. However, there do exist such perfect groups (cf. [7]).
An application of some fractional integrals
Authors: Lowndes J. S.
Published: May 2009

1. A discussion of the Erdélyi-Kober operator

and its adjoint operator Kη,α together with an account of some of their applications can be found in the book by Sneddon [5].
Semigroups satisfying minimal conditions II
Authors: Hall T. E. , Munn W. D.
Published: May 2009

In this paper we continue the investigation of minimal conditions on semigroups begun by J. A. Green [3] and taken up by Munn [5]. A unified account of the results in [3] and [5], together with some additional material, is presented in the text-book by Clifford and Preston [1, §6.6]. All terminology and notation not introduced explicitly willbe as in [1].
Uniform distribution of sequences in rings of integral matrices
Authors: Niederreiter Harald , Shiue Jau-Shyong
Published: May 2009

For various discrete commutative rings a concept of uniform distribution has already been introduced and studied, for example, for the ring of rational integers by Niven [9] (see also Kuipers and Niederreiter [2, Ch. 5]), for the rings of Gaussian and Eisenstein integers by Kuipers, Niederreiter, and Shiue [3], for rings of algebraic integers by Lo and Niederreiter [4], [7], and for finite fields by Gotusso [1] and Niederreiter and Shiue [8]. In the present paper, we shall show that a satisfactory theory of uniform distribution can also be developed in a noncommutative setting, namely for matrix rings over the rational integers.
Symmetries and pseudo-symmetries of hyperelliptic surfaces
Authors: Singerman David
Published: May 2009

1. Let X be a closed Riemann surface of genus g ≥ 2 and let Aut X denote the group of automorphisms of X where, in this paper, an automorphism means a conformal or anticonformal self-homeomorphism. X is called hyperelliptic if it admits a conformal automorphism J of order 2 such that X/H has genus 0, where H = 〈J〉 is the group of order 2 generated by J. Thus X is a two-sheeted covering of the sphere which is branched over 2g + 2 points and J is the sheet-interchange map. J is the unique conformal automorphism of order 2 such that X/〈J〉 has genus 0 and it follows that if U ∈ Aut X, then UJU−1 = J. Thus J is central in Aut X and H ≤ Aut X. (Cf. [8])
Decomposition algebras of Riesz operators
Authors: Murphy G. J. , West T. T.
Published: May 2009

Let H be a Hilbert space and let B denote the Banach algebra of all bounded linear operators on H with K denoting the closed ideal of compact operators in B. If T ∈ B, σ(T) and r(T) will denote the spectrum and spectral radius of T, respectively, and π the canonical mapping of B onto the Calkin algebra B/K.
A Class of infinite soluble groups with an A-group condition
Authors: Tomkinson M. J.
Published: May 2009

Finite soluble groups in which all the Sylow subgroups are abelian were first investigated by Taunt [8] who referred to them as A-groups. Locally finite groups with the same property have been considered by Graddon [2]. By the use of Sylow theorems it is clear that every section (homomorphic image of a subgroup) of an A-group is also an A-group and hence every nilpotent section of an A-group is abelian. This is the characterization that we use here in considering groups which are not, in general, periodic.
On the group ring of a free product with amalgamation
Authors: Jordan Camilla R.
Published: May 2009

Let G = A*HB be the free product of the groups A and B amalgamating the proper subgroup H and let R be a ring with 1. If H is finite and G is not finitely generated we show that any non-zero ideal I of R(G) intersects non-trivially with the group ring R(M), where M = M(I) is a subgroup of G which is a free product amalgamating a finite normal subgroup. This result compares with A. I. Lichtman's results in [6] but is not a direct generalisation of these.
Perfect codes in the graphs Ok and L(Ok)
Authors: Smith D. H.
Published: May 2009

In [6] the question of the existence of perfect e-codes in the infinite family of distance-transitive graphs Ok was considered. It was pointed out that it is difficult to rule out completely any particular value of e because of the difficulty of working with the sphere packing condition. For e= 1, 2, 3 it can be seen from the results of [6] that the condition given by the generalisation of Lloyd's theorem is satisfied for infinitely many values of k. We shall show that this is not the case for e=4 and we shall prove that there are no perfect 4-codes in Ok.
On the group ring of a free product with amalgamation
Authors: Jordan Camilla R.
Published: May 2009

Let G = A*HB be the free product of the groups A and B amalgamating the proper subgroup H and let Rbe a ring with 1. If His finite and G is not finitely generated we show that any non–zero ideal I of R(G) intersects non-trivially with the group ring R(M), where M = M(I) is a subgroup of G which is a free product amalgamating a finite normal subgroup. This result compares with A. I. Lichtman's results in [6] but is not a direct generalisation of these.
Perfect codes in the graphs Ok and L(Ok)
Authors: Smith D. H.
Published: May 2009

In [6] the question of the existence of perfect e-codes in the infinite family of distance-transitive graphs Ok was considered. It was pointed out that it is difficult to rule out completely any particular value of [6] because of the difficulty of working with the sphere packing condition. For e = 1, 2, 3 it can be seen from the results of [6] that the condition given by the generalisation of Lloyd's theorem is satisfied for infinitely many values of k. We shall show that this is not the case for e = 4 and we shall prove that there are no perfect 4-codes in Ok.
Solution of Irving's Ramsey problem
Authors: Harborth Heiko , Nitzschke Heinz-Michael
Published: May 2009

In [1] the following question was posed by R. W. Irving (see also Conjecture 4.10 in [4]): Is there an edge 2-colouring of the complete bipartite graph K13.17 with no monochromatic K3.3? We give a negative answer in this note (Theorem 2). Furthermore we prove Conjecture 4.11 (i) of [4] (Theorem 1), that is, any edge 2-coloured K2n+i,4n-3 contains a monochromatic K2,n with the 2 and n vertices a subset of the 2n +1 and 4n–3 vertices, respectively. Theorem 1 is a consequence of Satz 4 in [3], however, we give a direct proof here.
The group of isometries on Hardy spaces of the n-ball and the polydisc
Authors: Berkson Earl , Porta Horacio
Published: May 2009

Let C be the complex plane, and U the disc |Z| < 1 in C. Cn denotes complex n-dimensional Euclidean space, <, > the inner product, and | · | the Euclidean norm in Cn;. Bn will be the open unit ball {z ∈ Cn:|z| < 1}, and Un will be the unit polydisc in Cn. For l ≤ p < ∞, p ≠ 2, Gp(Bn) (resp., Gp(Un)) will denote the group of all isometries of Hp(Bn) (resp., Hp(Un)) onto itself, where Hp(Bn) and HP(Un) are the usual Hardy spaces.
Curves with zero derivative in F-spaces
Authors: Kalton N. J.
Published: May 2009

Let X be an F-space (complete metric linear space) and suppose g:[0, 1] → X is a continuous map. Suppose that g has zero derivative on [0, 1], i.e.

for 0≤t≤1 (we take the left and right derivatives at the end points). Then, if X is locally convex or even if it merely possesses a separating family of continuous linear functionals, we can conclude that g is constant by using the Mean Value Theorem. If however X* = {0} then it may happen that g is not constant; for example, let X = Lp(0, 1) (0≤p≤1) and g(t) = l[0,t] (0≤t≤1) (the characteristic function of [0, t]). This example is due to Rolewicz [6], [7; p. 116].
Sur une question de localisation
Authors: Malliavin-Brameret M. P.
Published: May 2009

II est connu [4] que si A = U() est l'algèbre enveloppante d'une algèbre de Lie nilpotente de dimension finie sur un corps F de caractéristique 0, tout idéal (complètement) premier P a pourlocalisé R = Ap un anneau régulier au sens de [5]; c'est-à-dire que le radical de Jacobson de R est engendrè par une suite centralisante régulière de longueur n = K-dim R, soit (z1…, zn). Dans le cas très particulier où P est l'idéal d'augmentation de U() il suffit de prendre pour (z1…, zn) l'image dans U()p d'une base de sur F adaptée à la suite centrale ascendante de .
Rational period functions of the modular group II
Authors: Knopp Marvin I.
Published: May 2009

In the earlier article [7], I began the study of rational period functions for the modular group Γ(l) = SL(2, Z) (regarded as a group of linear fractional transformations) acting on the Riemann sphere. These are rational functions q(z) which occur in functional equations of the form

where k∈Z and F is a function meromorphic in the upper half-plane ℋ, restricted in growth at the parabolic cusp ∞. The growth restriction may be phrased in terms of the Fourier expansion of F(z) at ∞:

with some μ∈Z. If (1.1) and (1.2) hold, then we call F a modular integral of weight 2k and q(z) the period of F.
An embedding theorem for free inverse semigroups
Authors: Munn W. D.
Published: May 2009

In this note it is shown that if S is a free inverse semigroup of rank at least two and if e, f are idempotents of S such that e > f then S can be embedded in the partial semigroup eSe/fSf. The proof makes use of Scheiblich's construction for free inverse semigroups [7, 8] and of Reilly's characterisation of a set of free generators in an inverse semigroup [4, 5].
Hereditariness of strong and stable radicals
Authors: Puczyłowski E. R.
Published: May 2009

The aim of this paper is to discuss some relations among hereditary, strong and stable radicals. In particular we investigate hereditariness of lower strong and stable radicals. Some facts obtained are related to some results and questions of [2, 6, 7].

All rings in the paper are associative. Fundamental definitions and properties of radicals may be found in [9]. Definitions of hereditary and strong radicals are used as in Sands [7]. We say that a radical S is left (right) stable if

(ρ): for every ring R and every left (right) ideal I of R it follows S(I)⊆S(R).
A characterization of spectral operators on Hilbert spaces
Authors: Albrecht Ernst
Published: May 2009

Let H be a complex Hilbert space and denote by B(H) the Banach algebra of all bounded linear operators on H. In [5; 6] J. Ph. Labrousse proved that every operator S∈B(H) which is spectral in the sense of N. Dunford (see [3]) is similar to a T∈B(H) with the following property

Conversely, he showed that given an operator S∈B(H) such that its essential spectrum (in the sense of [5; 6]) consists of at most one point and such that S is similar to a T∈B(H) with the property (1), then S is a spectral operator. This led him to the conjecture that an operator S∈B(H) is spectral if and only if it is similar to a T∈B(H) with property (1). The purpose of this note is to prove this conjecture in the case of operators which are decomposable in the sense of C. Foias (see [2]).
Essential normal and conjugate extensions of inverse semigroups
Authors: Pastijn Francis
Published: May 2009

In the following we use the notation and terminology of [6] and [7]. If S is an inverse semigroup, then Es denotes the semilattice of idempotents of S. If a is any element of the inverse semigroup, then a−1 denotes the inverse of a in S. An inverse subsemigroup S of an inverse semigroup S′ is self-conjugate in S′ if for all x ∈ S′,x−1Sx ⊆ S; if this is the case, S′ is called a conjugate extension of S. An inverse subsemigroup S of S′ is said to be a full inverse subsemigroup of S′ if Es = Es′. If S is a full self-conjugate inverse subsemigroup of the inverse semigroup S′, then S is called a normal inverse subsemigroup of S′, or, S′ is called a normal extension of S.
Rings which resemble rings of entire functions
Authors: Rush David E.
Published: May 2009

Since Helmer's 1940 paper [9] laid the foundations for the study of the ideal theory of the ring A(ℂ) of entire functions, many interesting results have been obtained for the rings A(X) of analytic functions on non-compact connected Riemann surfaces. For example, the partially ordered set Spec (A(ℂ) of prime ideals of A(ℂ) has been described by Henrikson and others [2], [10], [11]. Also, it has been shown by Ailing [4] that Spec(A(ℂ))sSpec(A(X)) as topological spaces for any non-compact connected Riemann surface X. Many results on the valuation theory of A(X) have also been obtained [1], [2]. In this note we show that a large portion of the results on the rings A(X) extend to the W-rings with complete principal divisor space which were defined by J. Klingen in [15], [16]. Therefore, many properties of A(ℂ) are shared by its non-archimedian counterparts studied by M. Lazard, M. Krasner, and others [8], [17], [18].
Trigonometric sums over primes II
Authors: Harman Glyn
Published: May 2009

We write e(x) for e2πix, ∥x∥ for the distance of x from the nearest integer and use A ≫ B to mean |A|<c |B|, where c is a positive constant depending at most on k and e. The letter p always denotes a prime number; P2 represents a number with precisely two prime factors. We continue the investigations started in [6] and will make many references to the analysis there. Here we prove the following theorems.
A description of the projective Stone algebras
Authors: Düntsch Ivo
Published: May 2009

In his book [6] Grätzer sets a guideline for the research on Stone algebras; in view of his and Chen's triple characterization of Stone algebras [3], he considers a problem for these algebras solved if it can be reduced to a problem for Boolean algebras and one for distributive lattices with 1. In the same book he lists as Problem 53: describe the projective Stone algebras. For a Stone algebra L, let BL be its centre and DL its dense set. In order that L be projective, BL has to be a Boolean retract of BF for some free Stone algebra F, and DL has to be a retract of DF in the category of distributive lattices with 1. These conditions are, however, not sufficient, so one hopes to characterize the projective Stone algebras by adding further conditions, and in this spirit we in fact arrive at a description of these algebras. We also show, however, that every such algebra is a retract of some DF, F a free Stone algebra, so we conclude that there is no nice structural characterization of the projective Stone algebras along the line of Grätzer's programme.
Ramanujan congruences for p-k (mod 11')
Authors: Gordon Basil
Published: May 2009

Denote by

the Euler product, and by

the partition generating function. More generally, if k is any integer, put

so that p(n) = p−1(n). In [3], Atkin proved the following theorem.
Indefinite quadratic polynomials
Authors: Cook R. J.
Published: May 2009

Let

be an indefinite quadratic form with real coefficients. A well-known result, due to Birch, Davenport and Ridout [1], [5] and [6], states that if n ≥21 then for any ε > 0 there is an integer vector x ≠O such that

Recently [3] we have quantified this result, obtaining a function g(n) such that g(n)→ ½ as n n→ ∞ and such that for any η > 0 and all large enough X there is an integer vector x satisfying

where |x| = max |xi|and the implicit constant in Vinogradov's ≪-notation is independent of X.
A remark on a conjecture of Marcus on the generalized numerical range
Authors: Au-Yeung Yik-Hoi , Ng Kam-Chuen
Published: May 2009

Let A be an n × n complex matrix and c = (c1… cn) єℂn. Define the c-numerical range of A to be the set is an orthonormal set in , where * denotes the conjugate transpose. Westwick [8[ proved that if c … cn are collinear, then Wc(A) is convex. (Poon [6] gave another proof.) But in general for n ≧3, Wc(A) may fail to be convex even for normal A (for example, see Marcus [4] or Lemma 3 in this note) though it is star-shaped (Tsing [7]). In the following, we shall assume that A is normal. Let W(A) = {diag UAU*: U is unitary}. Horn [3] proved that if the eigenvalues of A are collinear, then W(A) is convex. Au-Yeung and Sing [2] showed that the converse is also true. Marcus [4] further conjectured (and proved for n = 3) that if Wc(A) is convex for all cєℂn then the eigenvalues of A are collinear. Let λ = (λ1, …, λn єℂn. We denote by the vector λ1, …, λn and by [λ] the diagonal matrix with λ1, …, λn lying on its diagonal. Since, for any unitary matrix U,. Wc(A) = Wc (UAU*), the Marcus conjecture reduces to: if Wc([λ]) is convex for all c єℂn then λ1, … λn are collinear. For the case n = 3, Au-Yeung and Poon [1] gave a complete characterization on the convexity of the set Wc([λ]) in terms of the relative position of the points , where σ є S3 the permutation group of order 3. As an example they showed that if λ1, λ2, λ3 are not collinear, then is not convex (Lemma 3 in this note gives another proof). We shall show that for the case n = 4, is not convex if λ1, λ2. λ3. λ4 are not collinear. Thus for n = 3, 4 the Marcus conjecture is answered and improved.
Sums of two regular elements
Authors: Robson J. C.
Published: May 2009

A well-known fact concerning a prime right Goldie ring R, proved in [4, Section 5], is that an essential right ideal is generated by the regular elements which it contains. There is a modification of that proof which shows that each element of R is the sum of at most two regular elements. This suggested that the recent results of Chatters and Ginn [1] concerning rings generated by their regular elements might possibly be refined a little, since their arguments actually show that elements of R are sums of at most three regular elements.
On the strongly countable-dimensionality of μ-spaces
Authors: Mizokami T.
Published: May 2009

Nagata in [3] defined strongly countable-dimensional spaces which are the countable union of closed finite-dimensional subspaces. Walker and Wenner in [7] characterized such metric spaces as follows: a space X is a strongly countable-dimensional metric space if and only if there exists a finite-to-one closed mapping of a zero-dimensional metric space onto X with weak local order.
On square-full integers in a short interval
Authors: Shiu P.
Published: May 2009

A positive integer nis called a square-full integer if p2 divides n whenever p is a prime divisor of n. For x > 1 we denote by Q(x) the number of square-full integers not exceeding x. Bateman and Grosswald [1] proved that
Some constructions of rings of finite global dimension
Authors: Robson J. C.
Published: May 2009

In the study of idealizer rings in [14], the global dimensions of certain subrings were calculated. This work was extended by Goodearl [9]. Here it is shown, in Section 1, that the techniques involved can be used rather more generally than was apparent there. The results concern a ring S with a right ideal A such that SA = S, and a subring R of S containing A. They describe gl dim R in terms of gl dim S and the projective dimensions of simple subfactors of S/R or R/A, provided that these R-modules are sufficiently well conditioned.
Hörmander's Carleson theorem for the ball
Authors: Power S. C.
Published: May 2009

Let denote the unit ball in ℂ2 and let Sdenote its boundary, the unit sphere. For z ∈ B and δ>0, the following non isotropic balls are defined, where

A finite positive Borel measure μ, on B is called a Carleson measure if there exists a constant C for which

Here σ denotes normalized surface area measure on S. The following theorem was obtained by Hörmander [6] as a special case of more general variants for strictly pseudoconvex domains in ℂn. Recently Cima and Wogen [3] derived it from a Carleson measure theorem for Bergman spaces of the ball. A different direct approach to the Bergman context, and related settings, is given in Leucking [7].
A note on congruences on orthodox semigroups
Authors: McAlister D. B.
Published: May 2009

C. Eberhart and W. Williams [3] showed that the least inverse semigroup congruence , on an orthodox semigroup S, plays a very important role in determining the structure of the lattice of congruences on S. In this note we show that their results can be applied to give an explicit construction for the idempotent separating congruences on S in terms of idempotent separating congruences on S/.
Orientable and non-orientable Klein surfaces with maximal symmetry
Authors: Singerman David
Published: May 2009

Let X be a bordered Klein surface, by which we mean a Klein surface with non-empty boundary. X is characterized topologically by its orientability, the number k of its boundary components and the genus p of the closed surface obtained by filling in all the holes. The algebraic genus g of X is defined by

.

If g≥2 it is known that if G is a group of automorphisms of X then |G|≤12(g-l) and that the upper bound is attained for infinitely many values of g ([4], [5]). A bordered Klein surface for which this upper bound is attained is said to have maximal symmetry. A group of 12(g-l) automorphisms of a bordered Klein surface of algebraic genus g is called an M*-group and it is known that a finite group G is an M*-group if and only if it is generated by 3 non-trivial elements T1, T2, T3 which obey the relations

([4]).
Bishop's Condition (β)
Authors: Snader Jon C.
Published: May 2009

In 1959, Bishop [4] published a seminal paper in which he studied various types of spectral decompositions or “duality theories” that an arbitrary bounded linear operator on a reflexive Banach space might have. In the course of his investigations, he isolated the following analytic property which he called condition (β).
A Ramsey-type property in additive number theory
Authors: Burr S. A. , Erdös P.
Published: May 2009

Let A be a sequence of positive integers. Define P(A) to be the set of all integers representable as a sum of distinct terms of A. Note that if A contains a repeated value, we are free to use it as many times as it occurs in A. We call A complete if every sufficiently large positive integer is in P(A), and entirely complete if every positive integer is in P(A). Completeness properties have received considerable, if somewhat sporadic, attention over the years. See Chapter 6 of [3] for a survey.
The quadratic reciprocity law and the elementary theta function
Authors: Eichler Martin
Published: May 2009

This note points out a new aspect of the well-known relationship between the subjects mentioned in the title. The following result and its generalization in totally real algebraic number fields is central to the discussion. Let denote the Legendre symbol for relatively prime numbers a and b ℇ ℤ and a substitution of the modular subgroup Γ0(4). Then, if γ>0 and b≡1 mod 2,

with

and

According to (1), the Legendre symbol behaves somewhat like a modular function ﹙apart from the known behaviour under and ﹚. (1) follows (see below) from the functional equation

with

provided that

Here we used and always will use the abbreviation

and ℇδ means the absolutely least residue of δ mod 4. In the proof, Hecke [4] assumed γ>0 (see also Shimura [5]).
On the representations of numbers by binary cubic forms
Authors: Hooley C.
Published: May 2009

There are not a few situations in the theory of numbers where it is desirable to have as sharp an estimate as possible for the number r(n) of representations of a positive integer n by an irreducible binary cubic form

A variety of approaches are available for this problem but, as they stand, they are all defective in that they introduce unwanted factors into the estimate. For instance, an estimate involving the discriminant of f(x, y) is obtained if we adopt the Lagrange procedure [5] of using congruences of the type f(σ, 1)≡0, mod n, to reduce the problem to one where n=1. Alternatively, following Oppenheim (vid. [2]), Greaves [3], and others, we may appeal to the theory of factorization of ideals, which leads to unwanted logarithmic factors owing to the involvement of algebraic units. Having had need, however, in some recent work on quartic forms [4] for an estimate without such extraneous imperfections, we intend in the present note to prove that

uniformly with respect to the coefficients of f(x, y), where ds(n) denotes the number of ways of expressing n as a product of s factors.
Character sums and small eigenvalues for Г0(p)
Authors: Iwaniec Henryk
Published: May 2009

Let Δ denote the Laplace operator acting on the space L2(Г/H) of automorphic functions with respect to a congruence group Г, square integrable over the fundamental domain F=Г/H. It is known that Δ has a point spectrum

with (Weyl's law)

and it has a purely continuous spectrum on [¼,∞) of finite multiplicity equal to the number of inequivalent cusps. The eigenpacket of the continuous spectrum is formed by the Eisenstein series Ea(z, s) on s = ½+it where a ranges over inequivalent cusps. The eigenfunctions ui(z) with positive eigenvalues are Maass cusp forms.
Diophantine approximation on Hecke groups
Authors: Lehner J.
Published: May 2009

If α is a real irrational number, there exist infinitely many reduced rational fractions p/q for which

and √5 is the best constant possible. This result is due to A. Hurwitz. The following generalization was proposed in [2]. Let Г be a finitely generated fuchsian group acting on H+, the upper half of the complex plane. Let ℒ be the limit set of Г P and the set of cusps (parabolic vertices). Assume ∞∊P. Then if α∊ℒ–P, we have

for infinitely many p/q∊Г(∞), where k depends only on Г. Attention centers on

k running over the set for which (1.2) holds. We call hthe Hurwitz constant for Г. When Г=Г(1), the modular group, (1.2) reduces to (1.1) and h(Г(l))=√5. A proof of (1.2) when Г is horocyclic (i.e., ℒ=ℝ, the real axis) was furnished by Rankin [4]; he also found upper and lower bounds for h. See also [3, pp. 334–5], where the theorem is proved for arbitrary finitely generated Г.
Macbeath's curve and the modular group
Authors: Wohlfahrt K.
Published: May 2009

The theory of algebraic curves associated with subgroups of finite index in the modular group Γ is highly developed for such subgroups of Γ as may be defined by means of congruences in the ring ℤ of rational integers. The situation in he case of non-congruence subgroups of Γ, on the other hand, is drastically different. It reduces to a few isolated examples, two of which may be found in [12]. Related research by A. O. L. Atkin and H. P. F. Swinnerton-Dyer began in [1].
Numerical ranges of powers of hermitian elements
Authors: Crabb M. J. , McGregor C. M.
Published: May 2009

An element k of a unital Banach algebra A is said to be Hermitian if its numerical range

is contained in ℝ; equivalently, ∥eitk∥ = 1(t ∈ ℝ)—see Bonsall and Duncan [3] and [4]. Here we find the largest possible extent of V(kn), n ∈ ℕ, given V(k) ⊆ [−1, 1], and so ∥k∥ ≤ 1: previous knowledge is in Bollobás [2] and Crabb, Duncan and McGregor [7]. The largest possible sets all occur in a single example. Surprisingly, they all have straight line segments in their boundaries. The example is in [2] and [7], but here we give A. Browder's construction from [5], partly published in [6]. We are grateful to him for a copy of [5], and for discussions which led to the present work. We are also grateful to J. Duncan for useful discussions.