bluuebunny/crossref_metadata_2025 huggingface dataset:
Supersoluble groups of automorphisms of compact Riemann surfaces
Authors: Gromadzki Grzegorz , MacLachlan Colin
Published: May 2009

Given an integer g ≥ 2 and a class of finite groups let N(g, ) denote the order of the largest group in that a compact Riemann surface of genus g admits as a group of automorphisms. For the classes of all finite groups, cyclic groups, abelian groups, nilpotent groups, p-groups (given p), soluble groups and finally for metabelian groups, an upper bound for N(g, ) as well as infinite sequences for g for which this bound is attained were found in [5, 6, 7, 8, 13], [4], [10], [15], [16], [1], [2] respectively. This paper deals with that problem for the class of finite supersoluble groups i.e. groups with an invariant series all of whose factors are cyclic. In addition, it goes further by describing exactly those values of g for which the bound is attained. More precisely we prove:
Harmonic inverse semigroups
Authors: Petrich Mario , Rankin Stuart A.
Published: May 2009

An inverse semigroup S shall be said to be harmonic if every congruence on S is determined by any one of its classes. In other words, if λ and ρ are congruences on S having a congruence class in common, then λ = ρ. The class of all harmonic semigroups contains all bisimple inverse semigroups, as proved by Žitomirskiĭ [11] and also by Schein [10], and all congruence-free inverse semigroups. Moreover, is contained in the class of all 0-simple or simple inverse semigroups, as is easy to see. We shall show that there exist non-bisimple, non-congruence-free harmonic semigroups and that there are simple inverse semigroups which are not harmonic.
A Magnus theorem for free products of locally indicable groups
Authors: Edjvet M.
Published: May 2009

A one-relator product Gof groups A and Bis defined to be the quotient of their free product A * B by the normal closure, «W»A*B, of a single element W, which is assumed to be cyclically reduced and of length at least 2. For convenience, the group Gwill be denoted by (A * B)/W.
Spectral properties of elements in different Banach algebras
Authors: Grobler J. J. , Raubenheimer H.
Published: May 2009

Let A be a Banach algebra with unit 1 and let B be a Banach algebra which is a subalgebra of A and which contains 1. In [5]Barnes gave sufficient conditions for B to be inverse closed in A. In this paper we consider single elements and study the question of how the spectrum relative to B of an element in B relates to the spectrum of the element relative to A.
Small isomorphisms between group algebras
Authors: Wood G. V.
Published: May 2009

If G1 and G2 are locally compact groups and the algebras Ll(G1) and Ll(G2) are isometrically isomorphic, then G1 and G2 are isomorphic (Wendel, 1952, [8]). There is evidence that the following generalization of Wendel's result is true.

If T is an algebra isomorphism of L1(G1) onto L1(G2) with ∥T∥ < √2, then G1, and G2 are isomorphic.
Complex doubles of bordered Klein surfaces with maximal symmetry
Authors: May Coy L.
Published: May 2009

A compact bordered Klein surface X of algebraic genus g ≥ 2 has maximal symmetry [6] if its automorphism group A(X) is of order 12(g — 1), the largest possible. The bordered surfaces with maximal symmetry are clearly of special interest and have been studied in several recent papers ([6] and [9] among others).
The distribution of cube-full numbers
Authors: Shiu P.
Published: May 2009

An elementary derivation of the asymptotic formula for the number of cube-full numbers up to x is given. This derivation is used, together with an estimation of a three dimensional exponential sum, to establish the asymptotic formula for the number of cube-full numbers in the short interval x < n < x⅔+θ where 140/1123 < θ < 1/3.
Goldie criteria for some semiprime rings
Authors: Brown K. A. , Wehrfritz B. A. F.
Published: May 2009

We principally consider rings R of the form R = S[G], generated as a ring by the subring S of R and the subgroup G of the group of units of R normalizing S. (All our rings have identities except the nilrings.) We wish to deduce that certain semiprime images of R are Goldie rings from ring theoretic information about S and group theoretic information about G. Usually the latter is given in the form that G/N has some solubility or finiteness property, where N is some specified normal subgroup of G contained in S. Note we do not assume that N = G∩S; in particular N = 〈1〉 is always an option.
On projective characters of prime degree
Authors: Higgs R. J.
Published: May 2009

All groups G considered in this paper are finite and all representations of G are defined over the field of complex numbers. The reader unfamiliar with projective representations is referred to [9] for basic definitions and elementary results.

Let Proj(G, α) denote the set of irreducible projective characters of a group G with cocycle α. In a previous paper [3] the author showed that if G is a (p, α)-group, that is the degrees of the elements of Proj(G, α) are all powers of a prime number p, then G is solvable. However Isaacs and Passman in [8] were able to give structural information about a group G for which ξ(1) divides pe for all ξ ∈ Proj(G, 1), where 1 denotes the trivial cocycle of G, and indeed classified all such groups in the case e = l. Their results rely on the fact that G has a normal abelian p-complement, which is false in general if G is a (p, α)-group; the alternating group A4 providing an easy counter-example for p = 2.
On sparsely totient numbers
Authors: Harman Glyn
Published: May 2009

Following Masser and Shiu [6] we say that a positive integer n is sparsely totient if

Here φ is the familiar Euler totient function. We write ℱ for the set of sparsely totient numbers. In [6] several results are proved about the multiplicative structure of ℱ. If we write P(n) for the largest prime factor of n then it was shown (Theorem 2 of [6]) that

and infinitely often
Algebraic ideals of semiprime Banach algebras
Authors: Giotopoulos S. , Roumeliotis M.
Published: May 2009

If A is a semiprime Banach algebra, soc A, rad A the socle and radical of A, then Soc A ∩ rad A = (0). This elementary result enables us to prove some results concerning algebraic ideal and algebraic elements modulo the socle of A. We also deduce several conditions for A equivalent to the condition dim A <+∞.
On the number of terms in the irreducible factors of a polynomial over ℚ
Authors: Choudhry A. , Schinzel A.
Published: May 2009

All polynomials considered in this paper belong to ℚ[x] and reducibility means reducibility over ℚ. It has been established by one of us [5] that every binomial in ℚ[x] has an irreducible factor which is either a binomial or a trinomial. He has further raised the question “Does there exist an absolute constant K such that every trinomial in ℚ[x] has a factor irreducible over ℚ which has at most K terms (i.e. K non-zero coefficients)?”
Universal notions characterizing spectral decompositions
Authors: Lange Ridgley , Wang Shengwang
Published: May 2009

In this note we characterize certain types of spectral decomposition in terms of “universal” notions valid for any operator on a Banach space. To be precise, let X be a complex Banach space and let T be a bounded linear operator on X. If F is a closed set in the plane C, let X(T, F) consist of all y ∈ X satisfying thes identity

where f:C\F → X is analytic. It is then easy to see that X(T, F) is a T-invariant linear manifold in X. Moreover, if y ∈ X then

is a compact subset of the spectrum σ(T). Our aim is to give necessary and sufficient conditions for a decomposable or strongly decomposable operator in terms of X(T, F) and γ(y, T). Recall that T is decomposable if whenever G1G2 are open and cover C there exist T-invariant closed linear manifolds M1, M2 with X= M1 + M2 and σ(T | M1) ⊂ Gi(i = 1,2) (equivalently, σ(T | Mi)⊂ Ḡi, see [4, p. 57]). In this case, X(T, F) is norm closed if Fis closed and each y in X has a unique maximally defined local resolvent satisfying (1.1) on C\Fy; Fy is called the local spectrum σ(y, T) and coincides with γ(y, T). Hence T has the single valued extension property (SVEP); i.e., zero is the only analytic function f:V → X satisfying (z − T)f(z) = 0 on V. If T is decomposable and the restriction T | X(T, F) is also decomposable for each closed F, then T is called strongly decomposable. We point out that Albrecht [2] has shown by example that not every decomposable operator is strongly decomposable, while Eschmeier [6]has given a simpler construction to show that this phenomenon occurs even in Hilbert space.
Some isometric characterizations of ln∞
Authors: Zhu Jiang
Published: May 2009

The previous results on isometrically characterizing ln∞ in terms of operator ideal norms can be summarized as follows.

Let E be an n-dimensional Banach space.

(1) If λ(E) = 1, then (see [3], [5], [6]), where λ(E) is the projection constant of E (see [4]).

(2) If π(E) = n, then (see [1],[2]).

(3) If Δ2(E) = √n, then (see [8]), where Δ2(E) is the 2-dominated constant of E(see [4]).

(4) If for any linear operator T: → E, v1(T) = (T), then (see [7]), where v1 is the 1-nuclear norm (see [4]).
Centers of mass for operator-families
Authors: Takaguchi Makoto
Published: May 2009

Let Hbe a complex Hilbert space and let B(H) be the algebra of (bounded) operators on H. Let A =(A,…,An) be an n-tuple of operators on H. The joint numerical range of A is the subset W(A) of ℂn such that
Bounded endomorphisms of free P-algebras
Authors: Ševčovič Daniel
Published: May 2009

The present note deals with bounded endomorphisms of free p-algebras (pseudocomplemented lattices). The idea of bounded homomorphisms was introduced by R. McKenzie in [8]. T. Katriňák [5] subsequently studied the properties of bounded homomorphisms for the varieties of p-algebras. This concept is also an efficient tool for the characterization of, so-called, splitting as well as projective algebras in the varieties of all lattices or p-algebras. For details the reader is referred to [2], [5], [6], [7] and other references therein. Let us emphasize that the main results that are contained in the above mentioned references strongly depend on the boundedness of each endomorphism of any finitely generated free algebra in a given variety.
Regular graphs with regular neighborhoods
Authors: Šoltés Ľubomír
Published: May 2009

The existence of r-regular graphs such that each edge lies in exactly t triangles, for given integers t < r, is studied. If t is sufficiently close to r then each such connected graph has to be the complete multipartite graph. Relations to graphs with isomorphic neighborhoods are also considered.
A note on the joint operator norm of hermitian operators on Banach spaces
Authors: Chō Muneo , Huruya Tadasi
Published: May 2009

Let X be a complex Banach space and H be a hermitian operator on X. Then in [7] Sinclair proved that r(H) = ¶H¶, where r(H) and ¶H¶ are the spectral radius and the operator norm of H, respectively.

For a commuting n-tuple T = (T1,…, Tn) of operators on X, we denote the (Taylor) joint spectrum of T by σ(T) (see [9]) and define the joint operator norm ¶T¶ and the joint spectral radius r(T) by

and

respectively.
One sided invertibility and localisation
Authors: Hajarnavis C. R.
Published: May 2009

In general, a prime ideal P of a prime Noetherian ring need not be classically localisable. Since such a localisation, when it does exist, is a striking property; sufficiency criteria which guarantee it are worthy of careful study. One such condition which ensures localisation is when P is an invertible ideal [5, Theorem 1.3]. The known proofs of this result utilise both the left as well as the right invertiblity of P. Such a requirement is, in practice, somewhat restrictive. There are many occasions such as when a product of prime ideals is invertible [6] or when a non-idempotent maximal ideal is known to be projective only on one side [2], when the assumptions lead to invertibilty also on just one side. Our main purpose here is to show that in the context of Noetherian prime polynomial identity rings, this one-sided assumption is enough to ensure classical localisation [Theorem 3.5]. Consequently, if a maximal ideal in such a ring is invertible on one side then it is invertible on both sides [Proposition 4.1]. This result plays a crucial role in [2]. As a further application we show that for polynomial identity rings the definition of a unique factorisation ring is left-right symmetric [Theorem 4.4].
On a result of Johnson about Schur multipliers
Authors: Wiegold James
Published: May 2009

The purpose of this short note is to give a new and shorter proof of the following theorem of Johnson [1], and to extend it somewhat.

Theorem 1. Let G be a finite non-cyclic p-group possessing a non-empty subset X such that, for each x in X, G′ is a complement for  in G. Then the Schur multiplier of G is non-trivial.
New characterizations of approximately Gorenstein rings
Authors: Johnson Johnny A. , Taylor Monty B.
Published: May 2009

In [1], a condition was sought on a commutative Noetherian ringRsuch that wheneverRsatisfied the condition andRwas cyclically pure in anR-algebraS, thenRwas necessarily pure inS. Such a condition was found and turned out not only to be a sufficient condition but also a necessary condition. Rings satisfying this condition were called approximately Gorenstein. In this paper, we give new equivalent conditions for a ring to be approximately Gorenstein. Our conditions involve a certain metric topology defined on the lattice of ideals of a commutative Noetherian ring as well as the concept of a principal system. The notion of principal systems was defined in [9] and was suggested by Macaulay's theory of inverse systems [7]. These ideas have proved to be useful in many research probems in commutative algebra.
On the fundamental group of a Lie semigroup
Authors: Neeb Karl-Hermann
Published: May 2009

The simplest type of Lie semigroups are closed convex cones in finite dimensional vector spaces. In general one defines a Lie semigroup to be a closed subsemigroup of a Lie group which is generated by one-parameter semigroups. If W is a closed convex cone in a vector space V, then W is convex and therefore simply connected. A similar statement for Lie semigroups is false in general. There exist generating Lie semigroups in simply connected Lie groups which are not simply connected (Example 1.15). To find criteria for cases when this is true, one has to consider the homomorphism

induced by the inclusion mapping i:S→G, where S is a generating Lie semigroup in the Lie group G. Our main results concern the description of the image and the kernel of this mapping. We show that the image is the fundamental group of the largest covering group of G, into which S lifts, and that the kernel is the fundamental group of the inverse image of 5 in the universal covering group G. To get these results we construct a universal covering semigroup S of S. If j: H(S): = S ∩ S-1 →S is the inclusion mapping of the unit group of S into S, then it turns out that the kernel of the induced mapping

may be identfied with the fundamental group of the unit group H(S)of S and that its image corresponds to the intersection H(S)0 ⋂π1(S), where π1(s) is identified with a central subgroup of S.
The Least Commutative Congruence on a simple regular ω-semigroup
Authors: Bonzini C. , Cherubini A. , Piochi B.
Published: May 2009

Piochi in [10] gives a description of the least commutative congruence γ of an inverse semigroup in terms of congruence pairs and generalizes to inverse semigroups the notion of solvability. The object of this paper is to give an explicit construction of λ for simple regular ω-semigroups exploiting the work of Baird on congruences on such semigroups. Moreover the connection between the solvability classes of simple regular ω-semigroups and those of their subgroups is studied.
Some examples of noncommutative local rings
Authors: Jordan D. A.
Published: May 2009

In this paper we construct examples which answer three questions in the general area of noncommutative Noetherian local rings and rings of finite global dimension. The examples are formed in the same basic way, beginning with a commutative polynomial ring A over a field k and a k-derivation δ of A, taking the skew polynomial ring R = A[x;δ] and localizing at a prime ideal of the form IR, where I is a prime ideal of A invariant under δ. The localization is possible by a result of Sigurdsson [13].
Some subgroups of SL(3, z) generated by involutions
Authors: Humphries Stephen P.
Published: May 2009

For R a commutative ring with identity 1 we let SL(n, R) denote the group of n × n integral matrices with determinant 1. A transvection T is an element of SL(n, R) which we represent (see [1]) as a pair (φ d) where φ ∈ (Rn)*, the dual space of Rn, d ∈ Rn, φ(d) = 0, and for all x ∈ Rn we have

T(x) = + φ(x) d.

Throughout this paper an involution is an element Y of SL(n, R) which has order two. Let n = 3 and R = Z and let C = diag(–1, –1, –1) be the central element of GL(3, Z).
Uncountable existentially closed groups in locally finite group classes
Authors: Leinen Felix
Published: May 2009

In this paper, will always denote a local class of locally finite groups, which is closed with respect to subgroups, homomorphic images, extensions, and with respect to cartesian powers of finite -groups. Examples for x are the classes L ℐπ of all locally finite π-groups and L(ℐπ ∩ ) of all locally soluble π-groups (where π is a fixed set of primes). In [4], a wreath product construction was used in the study of existentially closed -groups (=e.c. -groups); the restrictive type of construction available in [4] permitted results for only countable groups. This drawback was then removed partially in [5] with the help of permutational products. Nevertheless, the techniques essentially only permitted amalgamation of -groups with locally nilpotent π-groups. Thus, satisfactory results could be obtained for Lp-groups (resp. locally nilpotent π-groups) [6], while the theory remained incomplete in all other cases.
On permutation polynomials whose difference is linear
Authors: Stothers W. W.
Published: May 2009

Let q be a power of a prime p, and let Sqbe the set of permutations of {0, 1,…, q – 1). As Sq is isomorphic to the group of permutations of Fq, the field of q elements, each element of 5, can be regarded as a polynomial over Fq. Various authors (e.g. [1], [2], [3]) have considered functions f(x) such that

f(x) ∈ Sq, and (f(x) + λ×) ∈ Sq

for some λ ∈ Fq when λ = 1, f(x) is a complete mapping polynomial ([3]).
On Schachermayer's example about the Banach-Saks property
Authors: Nunez Carmelo
Published: May 2009

A Banach space (X, ∥.∥) is said to have the Banach-Saks property (B.S.P.) if, for every bounded sequence (xn) in X, we can choose a subsequence () of (xn) such that the sequence

converges in the X-norm. This property, that a Banach space may enjoy or not, has been extensively studied.
Proximity and similarity of operators II
Authors: Burnap Charles , Lambert Alan
Published: May 2009

In this paper we continue the examination of the question of similarity of operators A and B begun in reference [3]. In that article, a similarity result was obtained based on a measure of closeness, or proximity, of the uniformly continuous semigroups etA and etB, t>0. The operators considered were elements of ℬ(ℋ), the algebra of bounded operators on a Hilbert space ℬWe now wish to relax this requirement and replace ℬ(ℋ) by a complex Banach algebra ℬ with unit I. In Section 2 we give a necessary condition for the similarity of A, B ∈ ℋ. We then give a condition sufficient to guarantee A and B are approximately similar (as defined in reference [5]). In Section 3 we restrict our attention to the case where ℋ = ℋ(ℋ). There we give a condition which guarantees A, B ∈ ℋ(ℋ) are intertwined by a Fredholm operator. This leads naturally into a discussion of proximity-similarity in the Calkin algebra si. This is the subject of Section 4. Following reference [7] we define a metric p on N(ℋ), the normal elements of ℋ We show (N(ℋ), p) is a complete metric space and that the unitary orbit of ℋ (N(ℋ) p)is the p-connected component of a in N (ℋ).
Conjugacy of free finite group actions on infranilmanifolds
Authors: Sadowski Michał
Published: May 2009

In this note we give the proof of the following result (previously known for homotopically trivial and free actions on infranilmanifolds [3, Theorem 5.6]).

Theorem 1. Let G be a finite group acting freely and smoothly on a closed infranilmanifold M. Assume that dim M≠3, 4. Then the action of G is topologically conjugate to an affine action.
On the location of the roots of polynomial congruences
Authors: Hooley C.
Published: May 2009

We have indicated in our tract [9] that several interesting problems in the theory of numbers are related to results about the evenness of the distribution of the roots v of a polynomial congruence

where f(x) = a0xn + … + an is an irreducible polynomial having integral coefficients and degree n≧2. We alluded, for example, to our work on the Chebyshev problem of the greatest prime factor of n2 – D [8], in which an essential component was our earlier demonstration [6] of the uniform distribution, modulo 1, of v/k when f(x) = x2 – D. But, having pointed out that the quantitative descriptions of such uniformity had to be very sharp for substantial applications, we then noted with regret that little more than mere uniform distribution was obtained in our generalization [7] of [6] to congruences of higher degree. Indeed, it has only been for certain cubic polynomials that results have been produced that are comparable in power with those for quadratic polynomials, and even these depend on the assumption of the unproved hypothesis R* regarding the size of incomplete Kloosterman sums [10].
Categorical universality of regular double p-algebras
Authors: Koubek V. , Sichler J.
Published: May 2009

An algebra A = (L; ∨, ∧, *, +, 0, 1) of type (2, 2, 1, 1, 0, 0) is a doublep-algebra if (L; ∨, ∧, 0, 1) is a (0, l)-lattice in which * and + are unary operations of pseudocomplementation and dual pseudocomplementation determined by the respective requirements that x ≤ a* be equivalent to x ∧ a = 0, and that x ≥ a+ if and only if x ∨ a = 1.
On the kernels of representations of finite groups II
Authors: Koshitani Shigeo
Published: May 2009

About fifteen years ago I. M. Isaacs and S. D. Smith [9] gave several character-theoretic characterizations of finite p-solvable groups G with p-length one, where p is a prime number. They proved that for a finite group G with a Sylow p-subgroup P, the following four conditions (a)–(d) are equivalent.
Prime and semiprime semigroup rings of cancellative semigroups
Authors: Okniński Jan
Published: May 2009

Let S be a cancellative semigroup. This paper is motivated by the problem of finding a description of semigroup rings K[S] over a field K that are semiprime or prime. Results of this type are well-known in the case of a group ring K[G], cf. [8]. The description, as well as the proofs, involve the FC-centre of G defined as the subset of all elements with finitely many conjugates in G. In [4], [5] Krempa extended the FC-centre techniques to the case of an arbitrary cancellative semigroup S. He defined a subsemigroup Δ(S) of S which coincides with the FC-centre in the case of groups, and can be used to describe the centre and to study special elements of K[S]. His results were strengthened by the author in [7], where Δ(S) was also applied in the context of prime and semiprime algebras K[S]. However, Δ(S) itself is not sufficient to characterize semigroup rings of this type. We note that in [2], [3] Dauns developed a similar idea for a study of the centre of semigroup rings and certain of their generalizations.
Weakly stable Banach spaces and the Banach-Saks properties
Authors: Zachariades Theodosis
Published: May 2009

In [9] J. L. Krivine and B. Maurey introduced the class of stable Banach spaces: a separable Banach space is called stable if for every pair of bounded sequences (xn)n, (yn)n and for every pair of ultrafilters on the natural numbers we have
One-relator products of torsion-free groups
Authors: Brodskiĭ S. D. , Howie James
Published: May 2009

If A and B are torsion-free groups, and W is a cyclically reduced word of even length in A*B, it is generally conjectured that a Freiheitssatz holds, namely that each of A and B are embedded via the natural map into the one-relator product group G = (A*B)/N(W), where N denotes normal closure. If W has length 2, then G is a free product of A and B with infinite cyclic amalgamation, and the result is obvious. The purpose of this note is to prove the Freiheitssatz in some special cases.
Approximating Fredholm operators on a nonseparable Hilbert space
Authors: Bouldin Richard
Published: May 2009

This paper obtains a simple formula for the distance from a given operator to the set of invertible operators without requiring the underlying space to be separable. That formula is used to compute the distance to the Fredholm operators with a given index. These results require the further study of the concepts of essential nullity and essential deficiency, which permitted us to characterize the closure of the invertible operators. We also introduce a parameter called the modulus of Fredholmness.
Proper left type-A monoids revisited
Authors: Fountain John , Gomes Gracinda M. S.
Published: May 2009

The relation ℛ* is defined on a semigroup S by the rule that ℛ*b if and only if the elements a, b of S are related by the Green's relation ℛ in some oversemigroup of S. A semigroup S is an E-semigroup if its set E(S)of idempotents is a subsemilattice of S. A left adequate semigroup is an E-semigroup in which every ℛ*-class contains an idempotent. It is easy to see that, in fact, each ℛ*-class of a left adequate semigroup contains a unique idempotent [2]. We denote the idempotent in the ℛ*-class of a by a+.
Eta-products which are simultaneous eigenforms of Hecke operators
Authors: Biagioli Anthony J. F.
Published: May 2009

The Dedekind eta-function is defined for any τ in the upper half-plane by

where x = exp(2πiτ) and x1/24 = exp(2πiτ/24). By an eta-product we shall mean a function

where N ≥ 1 and eachrδ ∈ ℤ. In addition, we shall always assume that is an integer. Using the Legendre-Jacobi symbol (—), we define a Dirichlet character ∈ by

when a is odd. If p is a prime for which ∈(p) ≠ 0and if F is a function with a Fourier series

then we define a Hecke operator Tp by

where

and
Isomorphisms between endomorphism rings of projective modules
Authors: García José Luis , Simón Juan Jacobo
Published: May 2009

Let R and S be arbitrary rings, RM and SN countably generated free modules, and let φ:End(RM)→End(sN) be an isomorphism between the endomorphism rings of M and N. Camillo [3] showed in 1984 that these assumptions imply that R and S are Morita equivalent rings. Indeed, as Bolla pointed out in [2], in this case the isomorphism φ must be induced by some Morita equivalence between R and S. The same holds true if one assumes that RM and SN are, more generally, non-finitely generated free modules.
One sided invertibility and localisation II
Authors: Hajarnavis C. R.
Published: May 2009

The aim of this paper is to generalise the results of [7] from the prime to the semiprime case. It was shown, for instance, that if M is the annihilator of a simple right module S of projective dimension 1 over a Noetherian prime polynomial identity (PI) ring R then M is either an invertible ideal or an idempotent ideal [7, Proposition 4.2]. One of the main applications of this result was that a prime Noetherian affine PI ring of global dimension less than or equal to 2 is a finite module over its centre. It turns out that this theorem is valid more generally when the ring is semiprime [1, Theorem A]. Clearly this requires [7, Proposition 4.2] also to be strengthened to the semiprime case. We do this by showing that a right invertible maximal ideal in a semiprime Noetherian PI ring is also left invertible (Theorem 3.5).
Faithful linear representations of certain free nilpotent groups
Authors: Wehrfritz B. A. F.
Published: May 2009

Brian Hartley asked me whether a free (nilpotent of class 2 and exponent p2)-group of countable rank has a faithful linear representation of finite degree, p here being a prime of course. The answer is yes. The point is that this then yields via work of F. Leinen and M. J. Tomkinson, see [3,3.6] an image of a linear p-group, which is not even finitary linear. The question of which relatively free groups have faithful linear representations dates back at least to work of W. Magnus in the 1930's, see [4, pp. 33, 34 and the final comment on p. 40] for a discussion of this. Our construction, which works more generally, is a further contribution. We write ℜc for the variety of nilpotent groups of class at most c and (ℭ9 for the variety of groups of exponent dividing q.
Product submanifolds with pointwise 3-planar normal sections
Authors: Arslan Kadri , West Alan
Published: May 2009

Let M be a smooth m-dimensional submanifold in (m + d)-dimensional Euclidean space ℝm+d For x ∊ M and a non-zero vector X in TXM, we define the (d + l)-dimensional affine subspace E(x, X) ofℝm+d by
Groups of small symmetric genus
Authors: May Coy L. , Zimmerman Jay
Published: May 2009

Group actions on compact surfaces have received considerable attention during the past century. The surface has often carried an analytic structure and been considered a Riemann surface or, equivalently, a complex algebraic curve.
A note on groups with non-central norm
Authors: Bryce R. A. , Rylands L. J.
Published: May 2009

The norm K(G) of a group G is the subgroup of elements of G which normalize every subgroup of G. Under the name kern this subgroup was introduced by Baer [1]. The norm is Dedekindian in the sense that all its subgroups are normal. A theorem of Dedekind [5] describes the structure of such groups completely: if not abelian they are the direct product of a quaternion group of order eight and an abelian group with no element of order four. Baer [2] proves that a 2-group with non-abelian norm is equal to its norm.
Pythagorean ratios in arithmetic progression, part II. Four Pythagorean ratios
Authors: Lagrange Jean , Leech John
Published: May 2009

As in [3] let {a, b}designate the Pythagorean ratio (a2 − b2)/2ab between the sides of a rational right angled triangle. The principal result of [3] is that {a, b}is the arithmetic mean of two Pythagorean ratios, and hence is the middle term of a three term arithmetic progression, if and only if a /b is the geometric mean of two Pythagorean ratios. Here in Part II we study sets of four Pythagorean ratios in arithmetic progression. We show that sets of four in consecutive places in an arithmetic progression are closely related to sets of four in the first, second, third and fifth places in a progression; any one of the former sets determines two of the latter sets, and either one of the latter sets determines the other and the former. We construct an infinite sequence of sets of four ratios in consecutive places of arithmetic progressions, the last term of each set being the first term of the next set. These sets are related to solutions of the Diophantine equations r2 = 5p2q2 ± 4(p4 − 2q4). Computer searches, in addition to exhibiting enough members of this sequence to enable us to identify it, also exhibited two sets which do not belong to this sequence.
Barrelled subspaces of spaces with subseries decompositions or Boolean rings of projections
Authors: Drewnowski Lech , Florencio Miguel , Paúl Pedro J.
Published: May 2009

The research presented in this paper started by extending a theorem of Swetits [18]about barrelledness of subspaces of metrizable AK-spaces to general AK-spaces of scalar sequences. The extension reads as follows.

(1) A subspace λ0 of a barrelled AK-space λ such that λ0 ⊃ φ is barrelled if and only if its dualis weak* sequentially complete. If in addition λ0 is monotone, then it is barrelled if and only ifequals the Köthe dualof λ0.

As an easy consequence of this extension, we obtained the following result of Elstrodt and Roelcke [8, Corollary 3.4].

(2) If λ is a barrelled monotone AK-space, then also its subspace ℒ(λ), consisting of all sequences in λ with zero-density support, is barrelled.
On an arithemtical inequality
Authors: Srinivasan S.
Published: May 2009

Here we extend an arithmetical inequality about multiplicative functions obtained by K. Alladi, P. Erdős and J. D. Vaaler, to include also the case of submultiplicative functions. Also an alternative proof of an extension of a result used for this purpose is given.

Let Uk, for integral k, denote the set {1,2,…, k}, and Vk denote the collection of all subsets of Uk. In the following, all unspecified sets like A,…, are assumed to be subsets of Uk. Let σ = {Si} and τ = {Tj} be two given collections of subsets of Uk. Set

and

Let ′ denote complementation in Uk (but for in the proof of (3) where it denotes complementation in C). For any collection p of subsets of Uk, let p′ denote the collection of the complements of members of p.
Symmetries of surfaces: an extension of Kulkarni's theorem
Authors: Jones Gareth A.
Published: May 2009

In [6] Kulkarni considered the set of values of g for which a given finite group G acts faithfully as a group of orientation-preserving self-homeomorphisms of a compact, connected, orientable surface σg of genus g. Let us denote this set by (G). Then Kulkarni showed that there exists a positive integer Kdepending only on the order d = |G| of G, the exponent e= exp G of G and the structure of a Sylow 2-subgroup G2 of G, satisfying:

Theorem 1. (Kulkarni [6]) (G) consists of all but finitely many non-negative integers g ≡ 1 mod K.
Fitting classes of certain metanilpotent groups
Authors: Heinenken Hermann
Published: May 2009

There are two families of group classes that are of particular interest for clearing up the structure of finite soluble groups: Saturated formations and Fitting classes. In both cases there is a unique conjugacy class of subgroups which are maximal as members of the respective class combined with the property of being suitably mapped by homomorphisms (in the case of saturated formations) or intersecting suitably with normal subgroups (when considering Fitting classes). While it does not seem too difficult, however, to determine the smallest saturated formation containing a given group, the same problem regarding Fitting classes does not seem answered for the dihedral group of order 6. The object of this paper is to determine the smallest Fitting class containing one of the groups described explicitly later on; all of them are qp-groups with cyclic commutator quotient group and only one minimal normal subgroup which in addition coincides with the centre. Unlike the results of McCann [7], which give a determination “up to metanilpotent groups”, the description is complete in this case. Another family of Fitting classes generated by a metanilpotent group was considered and described completely by Hawkes (see [5, Theorem 5.5 p. 476]); it was shown later by Brison [1, Proposition 8.7, Corollary 8.8], that these classes are in fact generated by one finite group. The Fitting classes considered here are not contained in the Fitting class of all nilpotent groups but every proper Fitting subclass is. They have the following additional properties: all minimal normal subgroups are contained in the centre (this follows in fact from Gaschiitz [4, Theorem 10, p. 64]) and the nilpotent residual is nilpotent of class two (answering the open question on p. 482 of Hawkes [5]), while the quotient group modulo the Fitting subgroup may be nilpotent of any class. In particular no one of these classes consists of supersoluble groups only.
On 𝓕-subnormal subgroups and Frattini-like subgroups of a finite group
Authors: Ballester-Bolinches A. , Pérez-Ramos M. D.
Published: May 2009

Throughout the paper we consider only finite groups.

J. C. Beidleman and H. Smith [3] have proposed the following question: “If G is a group and Ha subnormal subgroup of G containing Φ(G), the Frattini subgroup of G, such that H/Φ(G)is supersoluble, is H necessarily supersoluble? “In this paper, we give not only an affirmative answer to this question but also we see that the above result still holds if supersoluble is replaced by any saturated formation containing the class of all nilpotent groups.
A note on the conditional chromatic polynomial
Authors: Voloshin V.
Published: May 2009

In this note we consider a finite graph without loops and multiple edges. The colouring of a graph G in λ colours is the colouring of its vertices in such a way that no two of adjacent vertices have the same colours and the number of used colours does not exceed λ [1, 4]. Two colourings of graph G are called different if there exists at least one vertex which changes colour when passing from one colouring to another.
Limits of covering spaces and residual properties of groups
Authors: Corson Jon Michael
Published: May 2009

The purpose of this paper is to point out a flaw in H. B. Griffiths' covering space approach to residual properties of groups [3]. One is led to this paper from Lyndon and Schupp's book [4, pp. 114, 141] where it is cited for covering space methods and a proof that F-groups are residually finite. However the main result of [3] is false.
On quasi-permutation representations of finite groups
Authors: Burns J. M. , Goldsmith B. , Hartley B. , Sandling R.
Published: May 2009

In [6], Wong defined a quasi-permutation group of degree n to be a finite group G of automorphisms of an n-dimensional complex vector space such that every element of G has non-negative integral trace. The terminology derives from the fact that if G is a finite group of permutations of a set ω of size n, and we think of G as acting on the complex vector space with basis ω, then the trace of an element g ∈ G is equal to the number of points of ω fixed by g. In [6] and [7], Wong studied the extent to which some facts about permutation groups generalize to the quasi-permutation group situation. Here we investigate further the analogy between permutation groups and quasipermutation groups by studying the relation between the minimal degree of a faithful permutation representation of a given finite group G and the minimal degree of a faithful quasi-permutation representation. We shall often prefer to work over the rational field rather than the complex field.
On dilation equations and the Hölder continuity of the de Rham functions
Authors: Pan Yibiao
Published: May 2009

We use a simple approximation method to prove the Holder continuity of the generalized de Rham functions.

1. Consider the following dilatation equation

where |α|<l/2. Suppose that f is an integrable solution of (1); then f must satisfy

where is the Fourier transform of f, and

which immediately leads to
Pure-semisimplicity is preserved under elementary equivalence
Authors: Zayed Maher
Published: May 2009

In the present note, Σr denotes the class of all right pure semisimple rings (= right pure global dimension zero). It is known that if R ∈ Σr, then R is right artinian and every indecomposable right R-module is finitely generated. The class Σr is not closed under ultraproducts [4]. While Σr is closed under elementary descent (i.e. if S ∈ Σr and R is an elementary subring of S then R ∈ σr) [4], it is an open question whether right pure-semisimplicity is preserved under the passage to ultrapowers [4, Prob. 11.16]. In this note, this question is answered in the affirmative.
Complete slices and homological properties of tilted algebras
Authors: Assem Ibrahim , Coelho Flávio Ulhoa
Published: May 2009

It is reasonable to expect that the representation theory of an algebra (finite dimensional over a field, basic and connected) can be used to study its homological properties. In particular, much is known about the structure of the Auslander-Reiten quiver of an algebra, which records most of the information we have on its module category. We ask whether one can predict the homological dimensions of a module from its position in the Auslander-Reiten quiver. We are particularly interested in the case where the algebra is a tilted algebra. This class of algebras of global dimension two, introduced by Happel and Ringel in [7], has since then been the subject of many investigations, and its representation theory is well understood by now (see, for instance, [1], [7], [8], [9], [11], [13]).In this case, the most striking feature of the Auslander-Reiten quiver is the existence of complete slices, which reproduce the quiver of the hereditary algebra from which the tilted algebra arises. It follows from well-known results that any indecomposable successor (or predecessor) of a complete slice has injective (or projective, respectively) dimension at most one, from which one deduces that a tilted algebra is representation-finite if and only if both the projective and the injective dimensions of almost all (that is, all but at most finitely many non-isomorphic) indecomposable modules equal two (see (3.1) and (3.2)). On the other hand, the authors have shown in [2, (3.4)] that a representation-infinite algebra is concealed if and only if both the projective and the injective dimensions of almost all indecomposable modules equal one (see also [14]). This leads us to consider, for tilted algebras which are not concealed, the case when the projective (or injective) dimension of almost all indecomposable successors (or predecessors, respectively) of a complete slice equal two. In order to answer this question, we define the notions of left and right type of a tilted algebra, then those of reduced left and right types (see (2.2) and (3.4) for the definitions).
Injective homogeneity and the Auslander–Gorenstein property
Authors: Yi Zhong
Published: May 2009

In this paper we refer to [13] and [16] for the basic terminology and properties of Noetherian rings. For example, an FBNring means a fully bounded Noetherian ring [13, p. 132], and a cliqueof a Noetherian ring Rmeans a connected component of the graph of links of R[13, p. 178]. For a ring Rand a right or left R–module Mwe use pr.dim.(M) and inj.dim.(M) to denote its projective dimension and injective dimension respectively. The right global dimension of Ris denoted by r.gl.dim.(R).
Representations and positive definite functions on topological semigroups
Authors: Baker J. W. , Lashkarizadeh-Bami M.
Published: May 2009

A number of theorems are established about positive definite functions and representations of certain topological semigroups. In particular we establish theorems which show that measurable positive definite functions and measurable representations can each be decomposed into the sum of two parts one of which is continuous and the other of which is “small”.
On residually transcendental valued function fields of conics
Authors: Khanduja Sudesh K.
Published: May 2009

Let K/Kobe a finitely generated field extension of transcendence degree 1. Let u0 be a valuation of Koand v a valuation of Kextending v0such that the residue field of vis a transcendental extension ofthe residue field k0of vo/such a prolongation vwill be called a residually transcendental prolongation of v0. Byan element with the uniqueness propertyfor (K, v)/(K0, v0) (or more briefly for v/v0)we mean an element / of Khaving u-valuation 0 which satisfies (i) the image of tunder the canonicalhomomorphism from the valuation ring of vonto the residue field of v(henceforth referred to as the v-residue ot t) is transcendental over ko; that is vcoincides with the Gaussian valuation on the subfield K0(t) defined by (ii) vis the only valuation of K (up to equivalence) extending the valuation .
Models for joint isometries
Authors: Attele K. R. M. , Lubin A. R.
Published: May 2009

An N-tuple ℐ= (T1…, TN) of commuting contractions on a Hilbert space H is said to be a joint isometry if for all x in H, or, equivalently, if Athavale in [1] characterized the joint isometries as subnormal N-tuples whose minimal normal extensions have joint spectra in the unit sphere S2N−X a geometric perspective of this is given in [4]. Subsequently, V. Müller and F.-H. Vasilescu proved that commuting N-tuples which are joint contractions, i.e. , can be represented as restrictions of certain weighted shifts direct sum a joint isometry. In this paper we adapt the canonical models of [3], and also construct a new canonical model, which completes the previous descriptions by showing joint isometries are indeed restrictions of specific multivariable weighted shifts [2].
A note on comonotonic additivity
Authors: Parker June M.
Published: May 2009

The axiom of comonotonic independence for a preference ordering was introduced by Schmeidler [9]. It leads to the comonotonic additivity for the functional representing the preference ordering, which is necessarily a Choquet integral.

The aim of this paper is to illuminate the concepts of comonotonicity, comonotonic independence and comonotonic additivity. For example the seemingly weaker condition of weak comonotonic independence used by Chateauneuf in [2] is seen to be equivalent to comonotonic independence. Comonotonic additivity is characterized as additivity on chains of sets. From this the characterization of Choquet integrals in [4], [1], [8] follows easily.
A nontrivial T1-space admitting a unique quasi-proximity
Authors: Künzi Hans-Peter A. , Watson Stephen
Published: May 2009

We construct a T1-space that is not hereditarily compact, although each of its open sets is the intersection of two compact open sets. The search for such a space was motivated by a problem in the theory of quasi-proximities.
Substitutional lemma for G-spaces of 1-dimensional groups
Authors: Pérez Juan Antonio
Published: May 2009

Let G be a compact Lie group and X a G-CW complex. We are interested in the calculation of the Borel cohomology of X

where EG is a universal free G-space and we use on the right hand side cellular cohomology. For an introduction to G-CW complexes see Matumoto [4] and for a good exposition on Borel cohomology see for instance torn Dieck [2], We want to replace X with an ordinary CW complex Y in order to find an ordinary CW structure on the Borel construction EG ΧGY so we can use cellular chains to compute the Borel cohomology of X. For every compact Lie group one has an extension

where G0 is the identity component, so for our case G0 is isomorphic to the circle group . We are dealing with the case in which π0(G) is isomorphic to C2, the cyclic group of order
A note concerning the ideal of nuclear operators
Authors: Barnes Bruce A.
Published: May 2009

Let be a Banach algebra of bounded linear operators such that contains every operator with finite dimensional range. Then contains every nuclear operator.
Some finiteness conditions concerning intersections of conjugates of subgroups
Authors: Lennox John C. , Longobardi Patrizia , Maj Mercede , Smith Howard , Wiegold James
Published: May 2009

In [3], a group G was said to be a CF-group if, for every subgroup H of G, H/CoreGH is finite. It was shown there that a locally finite CF-group G is abelian-by-finite and that there is a bound for the indices |H: CoreGH| as H runs through all subgroups of G. (Groups for which such a bound exists were referred to in [3] as BCF-groups.) The CF-property was further investigated in [10], one of the main results there being that nilpotent CF-groups are (again) abelian-by-finite and BCF. In the present paper, we shall discuss the CF-property in conjunction with some related properties, which are defined as follows.
Compact Hermitian surfaces of pointwise constant holomorphic sectional curvature
Authors: Sekigawa Kouei , Koda Takashi
Published: May 2009

LetM= (M, J, g) be an almost Hermitian manifold andU(M)the unit tangent bundle ofM. Then the holomorphic sectional curvatureH=H(x) can be regarded as a differentiable function onU(M). If the functionHis constant along each fibre, thenMis called a space of pointwise constant holomorphic sectional curvature. Especially, ifHis constant on the wholeU(M), thenMis called a space of constant holomorphic sectional curvature. An almost Hermitian manifold with an integrable almost complex structure is called a Hermitian manifold. A real 4-dimensional Hermitian manifold is called a Hermitian surface. Hermitian surfaces of pointwise constant holomorphic sectional curvature have been studied by several authors (cf. [2], [3], [5], [6] and so on).