bluuebunny/crossref_metadata_2025 huggingface dataset:
Π′-closure of finite Π-solvable groups
Authors: Arad Zvi
Published: April 2009

The purpose of this paper is to present a proof of the following theorem: Suppose Π is a set of dd primes, G is a finite Π-solvable group, and A is a nilpotent Π-subgroup of maximal order of G. Then G has a normal Π-complement, if and only if NG(ZJ(A)) has a normal Π-complement. (J(A) is the Thompson subgroup of A.)
Banach algebras of power series
Authors: Loy Richard J.
Published: April 2009

Let C[[t]] denote the algebra of all formal power series over the complex field C in a commutative indeterminate t with the weak topology determined by the projections pj: Σαiti ↦αj. A subalgebra A of C[[t]] is a Banach algebra of power series if it contains the polynomials and is a Banach algebra under a norm such that the inclusion map A ⊂ C[[t]] is continuous. Such algebras were first introduced in [13] when considering algebras with one generator, and studied, in a special case, in [23]. For a partial bibliography of their subsequent study and application see the references of [9] (note that the usage of the term Banach algebra of power series in [9] differs from that here), and also [2], [3], [11]. Indeed, an examination of their use in [11], under more general topological conditions than here, led the present author to the results of [14], [15], [16], [17].
The frequencies of almost periodic solutions of almost periodic differential equations
Authors: O'Brien G. C.
Published: April 2009

Almost periodic solutions of a first order almost periodic differential equation in Rp are shown to have less than p basic frequencies additional to the basic frequencies of the almost periodic right hand of the equation.
Nonsingular rings with essential socles
Authors: Ivanov G.
Published: April 2009

This paper is a study of nonsingular rings with essential socles. These rings were first investigated by Goldie [5] who studied the Artinian case and showed that an indecomposable nonsingular generalized uniserial ring is isomorphic to a full blocked triangular matrix ring over a sfield. The structure of nonsingular rings in which every ideal generated by a primitive idempotent is uniform was determined for the Artinian case by Gordon [6] and Colby and Rutter [2], and for the semiprimary case by Zaks [12]. Nonsingular rings with essential socles and finite identities were characterized by Gordon [7] and the author [10]. All these results were obtained by representing the rings in question as matrix rings. In this paper a matrix representation of arbitrary nonsingular rings with essential socles is found (section 2). The above results are special cases of this representation. A general method for representing rings as matrices is developed in section 1.
Commutative semigroup varieties with the amalgamation property
Authors: Clarke G. T.
Published: April 2009

We determine which varieties of commutative semigroups have the weak or strong amalgamation property. These are precisely the varieties of inflations of semilattices of abelian groups.
Inverse semigroups determined by their lattices of inverse subsemigroups
Authors: Jones P. R.
Published: April 2009

In a previous paper ([14]) the author showed that a free inverse semigroup is determined by its lattice of inverse subsemigroups, in the sense that for any inverse semigroup T, implies . (In fact, the lattice isomorphism is induced by an isomorphism of upon T.) In this paper the results leading up to that theorem are generalized (from completely semisimple to arbitrary inverse semigroups) and applied to various classes, including simple, fundamental and E-unitary inverse semigroups. In particular it is shown that the free product of two groups in the category of inverse semigroups is determined by its lattice of inverse subsemigroups.
Set mappings defined on pairs
Authors: Williams N. H.
Published: April 2009

A set mapping on pairs over the set S is a function f such that for each unordered pair a of elements of S,f(a) is a subset of S disjoint from a. A subset H of S is said to be free for f if x∉ f({y, z}) for all x, y, z from H. In this paper, we investigate conditions imposed on the range of f which ensure that there is a large set free for f. For example, we show that if f is defined on a set of size K+ + with always |f(a)| <k then f has a free set of size K+ if the range of f satisfies the k-chain condition, or if any two sets in the range of f have an intersection of size less than θ for some θ with θ < K.
Entire solutions of the differential equation Δ;u = f(u)
Authors: Walter Wolfgang
Published: April 2009

The existence of spherically symmetric solutions of the equation ▵u = f(u) is proved for a large class of functions f(z). Among others, functions satisfying an inequality zf(z) < 0 for |z| > A, and in particular the function f(z) = -sinh z, belong to this class.
On Fitting classes that are Schunck classes
Authors: Cossey John
Published: April 2009

An example is given to show that a class of finite soluble groups that is both a Fitting class and a Schunck class need not be a formation. The novel feature of this class is that it is defined by imposing conditions on complemented chief factors of groups in it: this technique usually does not give rise to Fitting classes that are not formations.
Dense, uniform and densely subuniform chains
Authors: Ash C. J.
Published: April 2009

A chain, or linearly ordered set, is densely subuniform if it is dense and for every order type the elements whose corresponding initial sections have this order type, if any, are dense in the chain. It is uniform if all intial sections are isomorphic. This paper gives constructions for densely subuniform chains which are not uniform. The question arises from the study of congruence-free inverse semigroups, but may also have independent interest.
The spectrum of an R-homomorphism
Authors: Wickstead A. W.
Published: April 2009

Let E be a real Banach space ordered by a closed, normal and generating cone. Suppose also that the order induced on E has the Riesz decomposition property. It is shown that if T:E → E is a positive linear operator with the property that y, z, a ∈ E with a ≧ Ty, Tz implies there is x ∈ E with x ≧ y, z and a ≧ Tx then the approximate point spectrum and spectrum of T are cyclic subsets of the complex plane. That is, if α = |α|γ lies in one of these sets then so does |α|γk for all integers k.
On a sheaf representation of a class of near-rings
Authors: Szeto George
Published: April 2009

It is proved that R is a near-ring with identity in which every element is a power of itself if and only if it is isomorphic with a near-ring of sections of a sheaf of near-fields in which every element is a power of itself. We also obtain that the Boolean spectrum is homeomorphic with the space of all completely prime ideals of R with the Zariski topology.
Abstract harmonic analysis of generalised functions on locally compact semigroups with applications to invariant means
Authors: Wong James C. S.
Published: April 2009

Let S be a locally compact semigroup and M(S) its measure algebra. It is shown that the dual M(S)* is isometrically order isomorphic to the space GL(S) of all generalised functions on S first introduced by Šreǐder (1950). Moreover, convolutions of elements in each of the spaces M(S)* and GL(S) can be defined in such a way that the above isomorphism preserves convolutions. These results on representation of functionals in M(S)* by generalised functions practically open up a new chapter in abstract harmonic analysis. As an example, some applications to invariant means on locally compact semigroups are given.
A new basis for discrete analytic polynomials
Authors: Zeilberger Doron
Published: April 2009

A new basis for discrete analytic polynomials is presented for which the series converges absolutely to a discrete analytic function in the upper right quarter lattice whenever
On certain subalgebras of a dual B*-algebra
Authors: Wong Pak-Ken
Published: April 2009

Let A be a dual B*-algebra and Ap the p-class in A. We show that the conjugate space of A1 is A**, the second conjugate space of A. We also obtain a three lines theorem for Ap (1 ≦p≦∞).
Linearization of certain uniform homeomorphisms
Authors: Weston Anthony
Published: April 2009

This article concerns the uniform classification of infinite dimensional real topological vector spaces. We examine a recently isolated linearization procedure for uniform homeomorphisms of the form φ: X →Y, where X is a Banach space with non-trivial type and Y is any topological vector space. For such a uniform homeomorphism φ, we show that Y must be normable and have the same supremal type as X. That Y is normable generalizes theorems of Bessaga and Enflo. This aspect of the theory determines new examples of uniform non-equivalence. That supremal type is a uniform invariant for Banach spaces is essentially due to Ribe. Our linearization approach gives an interesting new proof of Ribe's result.
Fourier algebra of a hypergroup. I
Authors: Muruganandam Varadharajan
Published: April 2009

In this article we study the Fourier space of a general hypergroup and its multipliers. The main result of this paper characterizes commutative hypergroups whose Fourier space forms a Banach algebra under pointwise product with an equivalent norm. Among those hypergroups whose Fourier space forms a Banach algebra, we identify a subclass for which the Gelfand spectrum of the Fourier algebra is equal to the underlying hypergroup. This subclass includes for instance, Jacobi hypergroups, Bessel-Kingman hypergroups.
Amicable orthogonal designs of order eight
Authors: Street Deborah J.
Published: April 2009

New amicable orthogonal designs of order eight are given and they are used to construct ne orthogonal designs of order 32.
A note on local polynomial functions on commutative semigroups
Authors: Grossman P. A.
Published: April 2009

Given a universal algebra A, one can define for each positive integer n the set of functions on A which can be “interpolated” at any n elements of A by a polynomial function on A. These sets form a chain with respect to inclusion. It is known for several varieties that many of these sets coincide for all algebras A in the variety. We show here that, in contrast with these results, the coincident sets in the chain can to a large extent be specified arbitrarily by suitably choosing A from the variety of commutative semigroups.
Groups fixing graphas in switching classes
Authors: Liebeck Martin W.
Published: April 2009

A permutation group G on a finite set Ω is always exposable if whenever G stabilises a switching class of graphs on Ω, G fixes a graph in the switching class. Here we consider the problem: given a finite group G, which permutation representations of G are always exposable? We present solutions to the problem for (i) 2-generator abelian groups, (ii) all abelian groups in semiregular representations. (iii) generalised quaternion groups and (iv) some representations of the symmetric group Sn.
On bisimple semigroups generated by a finite number of idempotents
Authors: Byleen Karl
Published: April 2009

Non-completely simple bisimple semigroups S which are generated by a finite number of idempotents are studied by means of Rees matrix semigroups over local submonoids eSe, e = e2 ∈ S. If under the natural partial order on the set Es of idempotents of such a semigroup S the sets ω(e) = {ƒ ∈ Es: ƒ ≤ e} for each e ∈ Es are well-ordered, then S is shown to contain a subsemigroup isomorphic to Sp4, the fundamental four-spiral semigroup. A non-completely simple hisimple semigroup is constructed which is generated by 5 idempotents but which does not contain a subsemigroup isomorphic to Sp4.
Estimates for multiple exponential sums
Authors: Loxton John H. , Smith Robert A.
Published: April 2009

We give estimates for exponential sums of the shape , where F is a polynomial with interger coefficient and each component of (x1,…, xn) in the sum runs through a complete set of residues modulo q.
Invariant subspace lattices of Lambert's weighted shifts
Authors: Yadav B. S. , Chatterjee S.
Published: April 2009

Let B(H) be the Banach algebra of all (bounded linear) operators on an infinite-dimensional separable complex Hilbert space H and let be a bounded sequence of positive real numbers. For a given injective operator A in B(H) and a non-zero vector f in H, we put We define a weighted shift Tw with the weight sequence on the Hilbert space 12 of all square-summable complex sequences by . The main object of this paper is to characterize the invariant subspace lattice of Tw under various nice conditions on the operator A and the sequence .
Factorization structures and subcategories of the category of topological spaces
Authors: Nakagawa Ryosuke
Published: April 2009

Various investigations on (E, M)-categories are given by many authors. Recently, Herrlich (to appear) and Strecker (1972) gave interesting results on relations between factorizations (E, M) on nice categories and epireflective subcategories of . Factorization (E, M) treated by them satisfy that E ⊂ Epi. In this paper we shall consider factorization structures (E, M) such that E⊃Epi in the category of all topological spaces and study relations between these structures and some subcategories of . Our results are quite different from those in Herrlich (to appear) and Strecker (1972) and are rather related with the results in Herrlich (1969). Herrlich (1969) defined limit operators on and gave a one-to-one correspondence between suitable limit operators and bicoreflective subcategories of . We shall consider two kinds of factorization structures on and show that the first ones correspond to bicoreflective subcategories which are closed under closed embeddings, the second ones correspond to bireflective subcategories which satisfy that the classes of all reflections are closed under closed embeddings (see Proposition 11) and these correspondences are one-to-one.
The Universality of the variety of quasigroups
Authors: Pigozzi Don
Published: April 2009

The variety of quasigroups is universal for varieties of algebras of the most general kind in the sense that each such variety can be interpreted in a natural way in a suitably chosen subvariety of quasigroups. More precisely, for any algebra〈A, f0, f1, f2, …〉wheref0, f1, f2, …is an arbitrary finite or infinite sequence of operations of finite rank, there exists a quasigroup 〈B,.〉 and polynormial operationsF0, F1, F2, … over 〈B,·〉 such that〈A, f0, f1, …〉is a subalgebra of〈B, F0, F1, …〉satisfying exactly the same identities. Moreover, if there are only finitely many f0, f1, …, then 〈B1〉 can be taken so that its identities are recursive in those of 〈A, f0, f1, …〉, If 〈A, f0, f1, …〉is a free algebra with an infinite number of free generators, thenBcan also be taken to coincide withA. This universal property of quasigroups has a number of consequences for their equational metatheory.
Even every Join-extension solves a Universal problem
Authors: Fleischer Isidore
Published: April 2009

Generalization to arbitrary join-extensions of a poset of a recent characterization of those which are complete lattices.
Chains in generalized Boolean lattices
Authors: Byrd Richard D. , Mena Roberto A.
Published: April 2009

A chain C in a distributive lattice L is called strongly maximal in L if and only if for any homomorphism φ of L onto a distributive lattice K, the chain (Cφ)0 is maximal in K, where (Cφ)0 = Cφ if 0 ∉ K, and (Cφ)0 = Cφ ∪ {0}, otherwise. Gratzer (1971, Theorem 28) states that if B is a generalized Boolean lattice R-generated by L and C is a chain in L, then C R-generates B if and only if C is strongly maximal in L. In this note (Theorem 4.6), we prove the following assertion, which is not far removed from Gratzer's statement: let B be a generalized Boolean lattice R-generated by L and C be a chain in L. If 0 ∈ L, then C generates B if and only if C is strongly maximal in L. If 0 ∉ L, then C generates B if and only if C is strongly maximal in L and [C)L = L. In Section 5 (Example 5.1) a counterexample to Gratzer's statement is provided.
Upper radicals and essential ideals
Authors: Olson Dwight M. , Jenkins Terry L.
Published: April 2009

For any class of rings it is shown that the class (M) of all rings each nonzero homomorphic image of which contains either a nonzero -ideal or an essential ideal is a radical class. If is a class of simple rings the upper radical generated by , (M), is shown to be equal to (M) where ' is the class of simple rings complementary to .
Applications of p-adic interpolation to exponential polynomials and sums of powers
Authors: Laohakosol Vichian , Pitman Jane
Published: April 2009

An approach to p-adic interpolation via divided differences is used to give alternative proofs of results of van der Poorten on p-adic exponential polynomials and to derive a p-adic analogue of Turan's first main theorem on sums of powers.
Isomorphisms between semigroups of isotone maps
Authors: Bandelt Hans-J.
Published: April 2009

Every poset with 0 is determined by various semigroups of isotone selfmaps which preserve 0. Two theorems along these lines are given and applied to some recent results concerning relation semigroups on topological spaces.
Multiplicative functions and Ramanujan's τ-function
Authors: Elliott P. D. T. A.
Published: April 2009

It is proved that (|τ(n)|n−11/2)δ has a mean-value for 0 <δ > < 2, where τ(n) is Ramanujan's function from modular arithmetic. Some further results are conjectured.
On a relation between injectors and certain complemented chief factors of finite soluble groups
Authors: Makan A. R.
Published: April 2009

The Fitting class of finite soluble π-groups, where π is an arbitrary set of primes, has the property that each complement of an -avoided, complemented chief factor of any finite soluble group G contains an -injector of G. In other words, each -avoided, complemented chief factor of G is -complemented in the sense of Hartley (see [2]).
On the semigroup of Ck selfmaps of Rn
Authors: Wood G. R.
Published: April 2009

Magill, Jr. and Yamamuro have been responsible in recent years for a number of papers showing that the property that every automorphism is inner is held by many semigroups of functions and relations on topological spaces. Following [9], we say a semigroup has the Magill property if every automorphism is inner. we say a semigroup has the Magill property if every automorphism is inner. That the semigroup of Fréchet differentiable selfmaps, D of a finite dimensional Banach space E, had the Magill property was shown in [10], while a lengthy result in [6] extended this to the semigroup of k times Fréchet differentiable selfmaps, Dk, of a Fréchet Montel space (FM-space). In the latter paper it was noted that with a little additional effort the semigroup Ck, of k times continuously Fréchet differentiable selfmaps of FM-space, could be shown to possess the Magill property. It is the purpose of this paper to present a simpler proof of this result in the case where the underlying space is finite dimensional.
Subgroups of HNN groups
Authors: Cohen D. E.
Published: April 2009

The purpose of this paper is to give a more precise form of Theorem 1 of [2], which gives a structure theorem for subgroups of HNN groups; we prove the following.

Let H be a subgroup of the HNN group . Then H is an HNN group whose base is a tree product of groups H ∪ wAw-1 where w runs over a set of double coset representatives of (H,A); the amalgamated and associated subgroups are all of the form H ∊ vUiv-l for some v. We can be more precise about which subgroups occur and about the tree product. We will also obtain stronger forms of other results in [1] and [2].
Each join-completion of a partially ordered set in the solution of a universal problem
Authors: Schmidt Jürgen
Published: April 2009

The main result of this paper is the theorem in the title. Only special cases of it seem to be known so far. As an application, we obtain a result on the unique extension of Galois connexions. As a matter of fact, it is only by the use of Galois connexions that we obtain the main result, in its present generality.
On the embedding of a group in a join of given groups
Authors: Hall P.
Published: April 2009

1. In their fundamental paper of 1949, Higman, Neumann and Neumann proved for the first time that a countable group can always be embedded in some 2-generator group: [1], Theorem IV. Two kinds of improvement of this result have recently appeared. In [4], Theorem 2, Dark has shown that the embedding can always be made subnormally. On the other hand, in [2], Theorem 2.1, Levin has shown that the two generators can be given preassigned orders m > 1 and n > 2; and in [3], Miller and Schupp prove that the 2-generator group can also be made to satisfy several additional requirements, such as being complete and Hopfian.
A basis for the laws of a class of simple groups
Authors: Southcott Bruce
Published: April 2009

This paper presents a basis for the laws which hold in each of the finite simple groups, PSL(2, 2n), n> ≥ 2, thus partially solving a problem raised by Cossey, Macdonald and Street [3]. They considered the more general problem of finding bases for the laws which hold in PSL(2, pn), and succeded in finding a number of general laws, and in completing bases for pn ≤ 11. The solution of the general problem appears to be very difficult.
On groups generated by three-dimensional special unitary groups II
Authors: Phan Kok-Wee
Published: April 2009

We shall determine in this paper groups of types Dn, E6, E7 and E8 generated by SU(3, q)'s, q odd, q > 3. These groups are defined in Phan (1975). [We shall refer to this paper as I]. Acquaintance with the results of I is assumed. The identification of groups of type D4 is similar to that of SU(n, q). We actually construct an isomorphism from the universal group of type D4 onto Spin+(8, q). This direct approach does not appear to be feasible for groups of type Dn with n ≧ 5. Fortunately Wong's recent result (1974) is applicable here. But his theorem requires that the characteristic of the field be odd; hence unlike the unitary case, we assume that q is odd and q 3. Using Wong's theorem, we proceed to show by induction that groups of type Dn are homomorphic images of Spin+(2n, q) or Spin −(2n, q) according as n is even or n is odd.
A note on 2-distributors
Authors: Bozapalides Symeon
Published: April 2009

We construct an associative tensor product for 2-distributors by means of cartesian coends and we prove the usual adjointness property for a suitable Hom.
Splitting properties in Archimedeanl-groups
Authors: Anderson Marlow , Conrad Paul , Kenny Otis
Published: April 2009

Throughout this paper anl-group will always mean an archimedean lattice-ordered group and we shall confine our attention to such groups. Anl-groupsplitsif it is a cardinal summand of eachl-group that contains it as anl-ideal. Suppose thatGis anl-subgroup of anl-groupH. ThenGislargeinHorHis anessential extensionofGif for eachl-ideal L≠0 ofH, L∩G≠0.Gisessentiallyclosed if it does not admit any proper essential extension. Conrad (1971) proved that each essentially closedl-group splits, but not conversely.
Directed packings of pairs into quadruples
Authors: Skillicorn David B.
Published: April 2009

A directed packing of pairs into quadruples is a collection of 4-subsets of a set of cardinality ν with the property that each ordered pair of elements appears at most once in a 4-subset (or block). The maximal number of blocks with this property is denoted by DD(2, 4, ν). Such a directed packing may also be thought of as a packing of transtivie tournaments into the complete directed graph on ν points. It is shown that, for all but a finite number of values of ν, DD(2, 4, ν) is maximal.
Complete mappings of finite fields
Authors: Niederreiter Harald , Robinson Karl H.
Published: April 2009

We discuss complete mapping polynomials of finite fields, which are a special class of permutation polynomials. Complete mapping polynomials of small degree are classified. Results are obtained on a class of complete mapping binomials and on permutation binomials.
Components of an open set
Authors: Mandelkern Mark
Published: April 2009

A classical theorem states that any open set on the real line is a countable union of disjoint open invervals. Here the numerical content of this theorem is investigated with the methods of constructive topology.
Equidistant permutation arrays: a bound
Authors: Van Rees G. H. J. , Vanstone S. A.
Published: April 2009

An equidistant permutation array is a ν × r array A(r, λ;ν) defined on a r-set X such that every row of A is a permutation of X and any two distinct rows agree in precisely λ common columns. Define In this paper, we show that where n = r − λ. Certain results pertaining to irreducible equidistant permutation arrays are also established.
On simplices and lattice points
Authors: Scott P. R.
Published: April 2009

Let S be a simplex in En which is homothetic to a given simplex S*, which contains no point of the integral lattice in its interior, and which has maximal volume V(S). We conjecture that V(S) > nn/n!, and establish the conjecture for n < 3.
Infinitely many identities of kolberg type
Authors: Hirschhorn Michael D.
Published: April 2009

O. Kolberg has shown that if the partition generating function is split into five interlocking series, then certain algebraic relations hold among these series. We show that the same phenomenon occurs whenever the number of such series is not a power of two.
Mesures coniques et intégrale de Daniell
Authors: Becker Richard
Published: April 2009

Let X be a weakly complete proper cone contained in a weak space E and h(E) the Riesz space generated by the continuous linear forms on E. A positive conical measure μ on X is a positive linear form on h(E)|x. G. Choquet has proved μ is a Daniell integral on E when E is weakly complete, but μ is not generally a Daniell integral on X. However we give an integration theory for functions on X and compare this theory with the classical Daniell theory. The case where μ is maximal in the sense of G. Choquet is remarkable.
Higher order elliptic boundary value problems in spaces with homogeneous norms
Authors: Pryde A. J.
Published: April 2009

We consider general boundary value problems for homogeneous elliptic partial differential operators with constant coefficients. Under natural conditions on the operators, these problems give rise to isomorphisms between the appropriate spaces with homogeneous norms. We also consider operators which are not properly elliptic and boundary systems which do not satisfy the complementing condition and determine when they give rise to left or right invertible operators. A priori inequalities and regularity results for the corresponding boundary value problems in Sobolev spaces are then readily obtained.
Logical stability in group theory
Authors: Baldwin J. T. , Saxl Jan
Published: April 2009

This paper investigates the logical stability of various groups. Theorem 1: If a group G is stable and locally nilpotent then it is solvable. Theorem 2: Every non-Abelian variety of groups is unstable.
A short proof of a recent theorem of G. Szekeres
Authors: Little Charles H. C.
Published: April 2009

The purpose of this note is to apply the work of Kasteleyn (1967) on the enumeration of I-factors of a graph to derive a quick proof of a theorem of Szekeres (1973). In the following, G is understood to be a finite, directed graph. If u, v are adjacent vertices of G, we denote by (u, v) an edge of G directed from u to v. Let{f1, f2,…, fm} be a set of 1-factors of G, and for all i write where n is half the number of vertices of G. (Here we regard a 1-factor as a set of edges). Then Kasteleyn associates with fi a plus sign if ui1vi1ui2vi2…uinvin is an even permutation of u11v11u12v12…u1nv1n, and a minus sign otherwise. The symmetric difference of two 1-factors is a collection of circuits, called alternating circuits. An alternating circuit of G is said to be clockwise even if the number of its edges that are directed in agreement with the clockwise sense is even; otherwise it is clockwise odd. Since the length of any alternating circuit is even, these definitions are not dependent on the sense designated as clockwise. It follows from the work of Kasteleyn that two given 1-factors in a directed graph agree in sign if and only if the number of clockwise even alternating circuits in their symmetric difference is even.
Tripleableness of pro-c-groups
Authors: Chong-Keang Lim
Published: April 2009

Let C be a nontrivial full subcategory of the category F of finite discrete groups and continuous homomorphisms, closed under subobjects, quotient and finite products. We consider the category PC of pro-C-groups and continuous homomorphisms (i.e. inverse limits of C-groups) which forms a variety in category PF of profinite groups and continuous homomorphisms. The study of pro-Cgroups is motivated by their occurrence as Galois groups of filed extensions in algebraic number thory (see Serre (1965)). The purpose of this paper is to study the tripleableness of the forgetful functors from PC to various underlying categories. It is also shown that PC is equivalent to the category of algebras of the theory of the forgetful functor from C to S (the category of sets and mappings).
A generalization of convex functions via support properties
Authors: Ben-Tal Aharon , Ben-Israel Adi
Published: April 2009

With respect to a given family of functionsF, a function is said to beF-convex, if it is supported, at each point, by some member ofF. For particular choices ofFone obtains the convex functions and the generalized convex functions in the sense of Beckenbach.F-convex functions are characterized and studied, retaining some essential results of classical convexity.
Lattices, complements and tight Riesz orders
Authors: Cameron Neil
Published: April 2009

It is proved that there exists no compatible tight Riesz order on a complemented modular lattice. An example is provided of a complemented lattice with a compatible tight Riesz order.
Homogeneous graphs and stability
Authors: Gardiner A.
Published: April 2009

Let Γ be a graph with finite vertex set V. Γ is homogeneous if whenever U1, U2 ⊆ V are such that the vertex subgraphs (U1), (U2) are isomorphic, then every isomorphism from (U1) to (U2) extends to an automorphism of Γ; homogeneous graphs were studied by Sheehan (1974) and were classified by the author. Γ is locally homogeneous if whenever U ⊆ V, then every automorphism of (U) extends to an automorphism of Γ. We prove that every locally homogeneous graph is homogeneous.
The free exact category on a left exact one
Authors: Carboni A. , Magno R. Celia
Published: April 2009

We give an explicit one step description of the free (Barr) exact category on a left exact one. As an application we give a new two step construction of the free abelian category on an additive one.
The dilogarithm in algebraic fields
Authors: Lewin L.
Published: April 2009

Special numerical formulae for the dilogarithm of powers of a base quantity have been shown to exist by Watson, Coxeter and others. Abel's equation for the dilogarithm was put in this “exponent” form, as a result of which a four-variable, symmetrical equation, also in exponent form, was deduced. From these equations a large number of special numerical results were produced, from which certain properties of a general structural nature emerged, and enabled two new results to be predicted.

The algebraic bases for these results can be grouped in trigonometric or non-trigonometric form, and for the former it seems to be necessary to examine the properties of the dilogarithm in the complex plane. Even so, there are some identities that seem to be outside the scope of the present methods.

It is speculated that certain factorization relations, which can be identified from the equations, may play a substantial role in the results; but so far no analytic derivation of this property has appeared possible.
Essential ideals, homomorphically closed classes and their radicals
Authors: Jenkins T. L. , le Roux H. J.
Published: April 2009

Olson and Jenkins defined E(M) to be the class of all rings each nonzero homomorphic image of which contains either a nonzero M-ideal or an essential ideal where M, is any class of rings. E(M) was proven to be a radical class and various classes M were considered. Here the class E(M) is partitioned into two classes: H the class of all rings each nonzero homomorphic image of which has a proper essential ideal and the class H(M) of all rings each nonzero homomorphic image of which contains an M-ideal. It is shown that H is a radical class and under certain conditions H(M) is also a radical class. Various properties placed on M yield several well-known radical classes and an infinite number of supernilpotent nonspecial radical classes is constructed.
When is the algebra of regular sets for a finitely additive borel measure a α-algebra?
Authors: Armstrong Thomas E.
Published: April 2009

It is shown that hte algebra of regular sets for a finitely additive Borel measure μ on a compact Hausdroff space is a σ-algebra only if it includes the Baire algebra and μ is countably additive onthe σ-algebra of regular sets. Any infinite compact Hausdroff space admits a finitely additive Borel measure whose algebra of regular sets is not a σ-algebra. Although a finitely additive measure with a σ-algebra of regular sets is countably additive on the Baire σ-algebra there are examples of finitely additive extensions of countably additive Baire measures whose regular algebra is not a σ-algebra. We examine the particular case of extensions of Dirac measures. In this context it is shown that all extensions of a {0, 1}-valued countably additive measure from a σ-algebra to a larger σ-algebra are countably additive if and only if the convex set of these extensions is a finite dimensional simplex.
Embedding categories with exactness into their abelianization
Authors: Adelman Murray
Published: April 2009

We give a partial answer to the following questions: Given an additive category and a class of sequences, under what conditions is the universal functor to the abelian category faithful and what other sequences are taken to exact sequences?

The “answers” to these questions appear as theorems 2.2 and 3.2 respectively and amount to a weakening of the condition that there be enough relative projectives.

We then characterize those additive categories with exactness which have the property that all relative exact sequences are determined by a small set of functors into the category of abelian groups.
A new proof of the absence of limit cycles in a quadratic autonomous system
Authors: Matlak R. F.
Published: April 2009

The absence of limit cycles in the system (S): x′ = − y + dx2 + exy + fy2, y′ = x, (′ ≡ d/dt), for 0 ≦ f2 < 2, was shown by Yeh et al. (1963). The general case was established by Kukles and Šakhova using various special results and relying on an extraneous auxiliary system. In this paper we give a new proof of the theorem; in it, we draw mainly on the basic properties of the characteristic exponent of a cycle and use an intrinsic system, viz. the case d = − f(≠ 0) of (S) as an auxiliary system.
Promonoidal functor categories
Authors: Day B. J.
Published: April 2009

In this article a completion process for promonoidal categories is used to determine a sufficient condition for the existence of a promonoidal convolution structure on the category of functors between two promonoidal categories. This, in turn, leads to a partial closed structure on the 2-category of promonoidal categories, promonoidal functors, and promonoidal natural transformations.
Sylow subgroups of transitive permutation groups II
Authors: Praeger Cheryl E.
Published: April 2009

Let G be a transitive permutation group on a finite set of n points, and let P be a Sylow p-subgroup of G for some prime p dividing |G|. We are concerned with finding a bound for the number f of points of the set fixed by P. Of all the orbits of P of length greater than one, suppose that the ones of minimal length have length q, and suppose that there are k orbits of P of length q. We show that f ≦ kp − ip(n), where ip(n) is the integer satisfying 1 ≦ ip(n) ≦ p and n + ip(n) ≡ 0(mod p). This is a generalisation of a bound found by Marcel Herzog and the author, and this new bound is better whenever P has an orbit of length greater than the minimal length q.
Extensions of semi-hereditary rings
Authors: Evans M. W.
Published: April 2009

Hattori (1960) defined a right R-module A to be torsion-free if for all a ∈ A and x ∈ R, ax = 0 implies that there exist elements {x1, x2, …, xn} ⊆ R with xix = 0 for all 1 ≦ i ≦ n and {a1, a2, …, an} ⊆ A such that a = aixi. Left torsion-free is defined similarly. It is shown that for a ring R, these torsion-free modules are the torsion-free class of a hereditary torsion theory, corresponding to a perfect topology, if and only if the left flat epimorphic hull of R is a regular ring which is both left and right torsion-free. A class of right semi-hereditary rings for which the torsion-free modules of Hattori satisfy the above property are found and this class of rings is discussed.
Positive derivations on f-rings
Authors: Colville P. , Davis G. , Keimel K.
Published: April 2009

Throughout this paper A will denote an f-ring i.e. a lattice-ordered ring in the sense of Birkhoff and Pierce (1956) in which for all x, y, z ∈ A, x ∧ y = 0 implies x ∧ zy = 0 = x ∧ yz.
Embeddings into groups with only a few defining relations
Authors: Boone W. W. , Collins D. J.
Published: April 2009

It is a trivial consequence of Magnus' solution to the word problem for one-relator groups [9] and the existence of finitely presented groups with unsolvable word problem [4] that not every finitely presented group can be embedded in a one-relator group. We modify a construction of Aanderaa [1] to show that any finitely presented group can be embedded in a group with twenty-six defining relations. It then follows from the well-known theorem of Higman [7] that there is a fixed group with twenty-six defining relations in which every recursively presented group is embedded.
Some infinite factor groups of Burnside groups
Authors: Britton J. L.
Published: April 2009

Let Bd(e) denote the Burnside group with d ≥ 2 generators a1,.a2, …, a, and exponent e > 0, i.e., the free group of rank d of the Burnside variety of exponent e. It is known that Bd(e) is infinite for all sufficiently large odd values of e; cf. Novikov and Adyan [3] or Britton [1]. In particular Bd(pk), where p is an odd prime, is infinite for all sufficiently large k. It is not known whether or not Bd(2k) is infinite for all sufficiently large k infiniteness would imply that Bd(n) is infinite for all sufficiently large n, as has been conjectured by Novikov [2].
Spaces of ideals of distributive lattices II. Minimal prime ideals
Authors: Speed T. P.
Published: April 2009

This paper, the second of a sequence beginning with [14], deals with the relationship between a distributive lattice L = (L; ∨, ∧, 0) with zero, and certain spaces of minimal prime ideals of L. Similar studies of minimal prime ideals in commutative semigroups [8] and in commutative rings [6] inspired this work, and many of our results are similar to ones in these two articles. However the nature of our situation enables many of these results to be pushed deeper and thus to arrive at a more satisfactory state; indeed with the insight obtained from the simpler lattice situation, one can return to some topics considered in [6], [8] and give complete accounts. We do not do this in the present paper, but leave the details to the reader, see e.g. [15]. Also a study of minimal prime ideals illuminates some topics in the theory of distributive lattices, particularly Stone lattices.
A Banach lattice not weakly projectable
Authors: Strzelecki E.
Published: April 2009

In [4] a concept of a weakly projectable vector lattice has been introduced. Stone vector lattices [3] and thus all special types of them, like Riesz [5], σ-complete and complete vector lattices are weakly projectable. Moreover C[0, 1] is weakly projectable but not Stone [4]. As we see the collection W of weakly projectable vector lattices is quite large. This explains to some extent the difficulty in producing examples of vector lattices which do not belong to W. In this note an example of a Banach lattice [1] which is not weakly projectable is described.
Irreducible representations of the group of movements of the Euclidean plane
Authors: Symons James
Published: April 2009

It is well known that a wide range of Special Function Theory can be realized by considering unitary representations of certain topological groups.

In this approach it is very important to determine all irreducible continuous unitary representations of the group in question.

For the group of movements this problem was initiated by Vilenkin [6]. Rather restrictive conditions were imposed in this paper and while he returned to the problem in [7], it was still not solved in full generality (among other things the representation space was assumed separable). The first complete solution appears to have been given by Thoma [5]. Here, the method was to show each irreducible continuous unitary representation equivalent to a particular representation in a space of square integrable functions.
Discrete structure spaces of ƒ-rings
Authors: Colville Peter D.
Published: April 2009

Birkhoff and Pierce [2] introduced the concept of an ƒ-ring and showed that an l-ring is an f-ring if and only if it is a subdirect product of totallyordered rings. An l-ideal of an f-ring R is an algebraic ideal which is at the same time a lattice ideal of R. Structure spaces (i.e. sets of prime ideals endowed with the so-called hull-kernel or Stone topology) for ordinary rings have been studied by many authors. In this paper we consider certain analogues for ƒ-rings, and give characterisations of ƒ-rings for which these structure spaces are discrete.
On multiplicative representations of integers
Authors: Erdös P. , Szemerédi A.
Published: April 2009

Let 1 ≦ a1 < … < ak ≦ x; b1 < … b1 ≦ x. Assume that the number of solutions of a1b1 = m is less than c. The authors prove that then . They also give a simple proof of Szemerédi's theorem: If the products aibj are all distinct then . They conjecture that (2) holds for c2 = 1 + ε if x > x0(ε). Several other unsolved problems are stated.
Existence criteria for supplementary difference sets
Authors: Lehmer Emma
Published: April 2009

Conditions are found for the existence of supplementary difference sets consisting of cosets of e-th power residues modulo a prime p. For e = 4, 6 and 8 all known pairs of sets are listed in the summary.
Hadamard matrices of Williamson type
Authors: Whiteman Albert Leon
Published: April 2009

Let p be a prime ≡ 1 (mod 4) and put v = p(p + 1)/2. It is proved in this paper that there exist four symmetric circulant matrices A, B, C, D of order υ such that where Iv is the identity matrix of order υ. This result is used to construct Hadamard matrices of order 4υ that are of the type originally prescribed by Williamson.
On the permanent of Schur's matrix
Authors: Graham R. L. , Lehmer D. H.
Published: April 2009

Schur's matrix Mnis ordinarily defined to be thenbynmatrix (εjk), 0 ≦j, k < n, where ε = exp (2 πi/n). This matrix occurs in a variety of areas including number theory, statistics, coding theory and combinatorics. In this paper, we investigatePn, thepermanentofMn, which is define bywhere π ranges over alln! permutations on {0,1, …,n– 1}.Pnoccurs, for example, in the study of circulants. Specifically, letXndenote thenbyncirculant matrix(xi, j)withxi, j= xi, j, where the subscript is reduced modulon. The determinant ofXnis a homogeneous polynomial of degreenin thexiand can be written asThenPn= A(1,1, … 1). Typical of the results established in this note are: (i)P2n= 0 for alln, (ii)Pp≡p! (modp3)forpa prime >3. (iii) IfpadividesnthendividesPn. Also, a table of values ofPnis given for 1 ≦n≦ 23.
Strong Carmichael numbers
Authors: Lehmer D. H.
Published: April 2009

A composite number N is called a pseudoprime for the base a in case . An odd pseudoprime N is called strong for the base a.
Real order-automorphism groups
Authors: Weinberg E. C.
Published: April 2009

By using the concept of tame embeddings of chains, a characterization is given of the subobjects of the lattice-ordered groups of order-automorphisms of the chains of rational and real numbers.
Nilpotent and semi-n-abelian groups
Authors: Kowol G.
Published: April 2009

A group G is called semi-n-abelian, if for every g ∈ G there exists at least one a(g) ∈ G-which depends only on g-such that (gh)n = a-1(g)gnhna(g) for all h ∈ G; a group G is called n-abelian, if a(g) = e for all g ∈ G. According to Durbin the following holds for n-abelian groups: If G is n-abelian for at lesast 3 consecutive integers, then G in n-abelian for all integers and these groups are exactly the abelian groups. In this paper this problem is generalized to the semi-n-abelian case: If a finite group G is semi-n-abelian for at least 4 consecutive integers then G is semi-n-abelian for all integers and these groups are exactly the nilpotent groups, where the Sylow-2-subgroup is abelian, the Sylow-3-subgroup is any element of the Levi-variety ([[g, h], h] = e ∀ g, h ∈ G) and the Sylow-p-subgroup (p < 3) is of class <2. As a consequence we get a description of all finite (3-)groups, which are elements of the Levi-variety.
Remarks on multipliers on certain function spaces on groups
Authors: Yamaguchi Hiroshi
Published: April 2009

Let G be LCA group with an algebraically ordered dual Ĝ. Suppose also that the semigroup P of positive elements in Ĝ is not dense in Ĝ. Subspaces (G) (1 < s < ∞) are defined analogous to the Hardy spaces on the circle group, and the question whether every multiplier from into can be extended to a multiplier from L3(G) into Lq(G) is investigated. If we suppose that s ≠ ∞, the it is shown that such an extension is possible if and only if (s, q) ∈ (1, ∞) × [1, ∞] ∪ {(1, ∞)}. (the negative result for (1, 1) was obtained in a previous paper.)
Regular Rees matrix semigroups and regular Dubreil-Jacotin semigroups
Authors: Mcalister D. B.
Published: April 2009

A partially ordered semigroup S is said to be a Dubreil-Jacotin semigroup if there is an isotone homomorphism θ of S onto a partially ordered group such that {} has a greatest member. In this paper we investigate the structure of regular Dubreil-Jacotin semigroups in which the imposed partial order extends the natural partial order on the idempotents. The main tool used is a local structure theorem which is introduced in Section 2. This local structure theorem applies to many other contexts as well.
On the Schur-Baer property
Authors: Moghaddam Mohammad Reza R.
Published: April 2009

In 1957 P. Hall conjectured that every (finitely based) variety has the property that, for every group G, if the marginal factor-group is finite, then the verbal subgroup is also finite. The content of this paper is to present a precise bound for the order of the verbal subgroup of a G when the marginal factor-group is of order Pn (p a prime and n > 1) with respect to the variety of polynilpotent groups of a given class row. We also construct an example to show that the bound is attained and furthermore, we obtain a bound for the order of the Baer-invariant of a finite p-group with respect to the variety of polynilpotent groups.
Growth sequences of finite groups V
Authors: Meier D. , Wiegold James
Published: April 2009

A short and easy proof that the minimum number of generators of the nth direct power of a non-trival finite group of order s having automorphism group of order a is more than logsn + logsa, n > 1. On the other hand, for non-abelian simple G and large n, d(Gn) is within 1 + e of logsn + logsa.
Positive definite functionals in Banach spaces
Authors: Hamedani G. G.
Published: April 2009

We establish a version of Bochner Theorem due to S. Boylan for Banch spaces with a basis.