bluuebunny/crossref_metadata_2025 huggingface dataset:
Weak convergence on non-separable metric spaces
Authors: Pollard David
Published: April 2009

For certain problems in the weak convergence theory of probability measures on non-separable metric spaces it is convenient to consider measures defined on a σ-field B0 smaller than the Borel σ-field. A suitable theory has been developed by Dudley and Wichura. In this paper it is shown that some of the key results in that theory can be deduced directly from the better known weak convergence theory for Borel measures. This is achieved by a remetrization of the underlying space to make it separable. The σ-field B0 contains the new Borel σ-field and weak convergence in Dudley's sense becomes equivalent to convergence of restrictions of the measures to the latter σ-field.

1980 Mathematics Subject Classification (Amer. Math. Soc.): primary 60 B 10, 60 B 05.
On semigroups and groups of local polynomial functions
Authors: Nöbauer Wilfried
Published: April 2009

Let Zn be the factor ring of the integers mod n and t be a positive integer. In this paper some results are given on the structure of the semigroup of all mappings from Zn into Zn and on the structure of the group of all permutations on Zn, which, for any t elements, can be represented by a polynomial function. If n = ab and a, b are relatively prime, then this (semi)group is isomorphic to the direct product of the respective (semi)groups for a and b. Thus it is sufficient to consider only the case where n = pe, p being a prime. In this case it is proved, that the (semi)group is isomorphic to the wreath product of a certain sub(semi)group of the symmetric (semi)group on Zpe−1 by the symmetric (semi)group on Zp. Some remarks on these sub(semi)groups are given.

Subject classification (Amer. Math. Soc. (MOS) 1970): 20 B 99, 13 B 25.
Zero-set ultrafilters and Gδ-closures in uniform spaces
Authors: Curzer Howard , Hager Anthony W.
Published: April 2009

The paper examines the classes K1 and Γ1 of Hausdorff uniform spaces which are Gδ-closed in their Samuel compactifications, or completions. It is shown that the classes are epi-reflective, the reflections K1 and Γ are described, K1 and Γ1 are represented as epi-reflective hulls, membership in the classes is described by fixation of certain zero-set ultrafilters, and it is shown that k1 = Γ1 exactly on spaces without discrete sets of measurable power. The results include familiar facts about realcompact and topologically complete topological spaces and are closely connected with the theory of metric-fine uniform spaces.

Subject classification (Amer. Math. Soc. (MOS) 1970): primary 54 C 50, 54 E 15, 18 A 40; secondary 54 B 05, 54 B 10, 54 C 10, 54 C 30.
Covering a polygon with triangles: a Caratheodory-type theorem
Authors: Baker M. J. C.
Published: April 2009

Generalizations are proved for theorems of Carathéodory (1907), Kirchberger (1903) and Watson (1973), the theme of these results being how thickly the convex hull of a family of points is covered by simplexes whose vertices are chosen from the points of the family.

Subject classification (Amer. Math. Soc. (MOS) 1970): 52A35.
Unambiguous evaluations of bidecic Jacobi and Jacobsthal sums
Authors: Evans Ronald J.
Published: April 2009

For a class of primes p ≡ 1 (mod 20) for which 2 is a quintic nonresidue, unambiguous evaluations of parameters of bidecic Jacobi and Jacobsthal sums (mod p) are presented, in terms of the partition p = a2+5b2+5c2+5d2, ab = d2–c2–cd. Similar results for sums of others orders have been obtained by E. Lehmer and by K. S. Williams.

Subject classification (Amer. Math. Soc. (MOS) 1970): 10 G 05.
On two-bridged knot polynomials
Authors: Hartley R. I.
Published: April 2009

The extended diagram of a two-bridged knot is introduced, and it is shown how the coefficients of the Alexander polynomial of the knot may be read straight from this diagram. Using this result, it is shown by diagram manipulation that a conjecture of Fox about the coefficients of the Alexander polynomial of an alternating knot is true at least for two-bridged knots (which are all alternating).

1980 Mathematics subject classification (Amer. Math. Soc.): 57 M 25.
Finitely presented groups of Fibonacci type. part II
Authors: Campbell Colin M. , Robertson Edmund F.
Published: April 2009

A class of cyclically presented groups with n generators and n Fibonacci type relations is discussed. Conditions are given for these groups to be finite and metacyclic. With these conditions the presentations are reduced to the standard form for metacyclic groups with trivial Schur multiplicator. This enables certain ismorphisms between the groups to be found.

Subject classification (Amer. Math. Soc. (MOS) 1970): 20 F 05.
Cubic symmetric graphs of order twice an odd prime-power
Authors: Feng Yan-Quan , Kwak Jin Ho
Published: April 2009

An automorphism group of a graph is said to be s-regular if it acts regularly on the set of s-arcs in the graph. A graph is s-regular if its full automorphism group is s-regular. For a connected cubic symmetric graph X of order 2pn for an odd prime p, we show that if p ≠ 5, 7 then every Sylow p-subgroup of the full automorphism group Aut(X) of X is normal, and if p ≠3 then every s-regular subgroup of Aut(X) having a normal Sylow p-subgroup contains an (s − 1)-regular subgroup for each 1 ≦ s ≦ 5. As an application, we show that every connected cubic symmetric graph of order 2pn is a Cayley graph if p > 5 and we classify the s-regular cubic graphs of order 2p2 for each 1≦ s≦ 5 and each prime p. as a continuation of the authors' classification of 1-regular cubic graphs of order 2p2. The same classification of those of order 2p is also done.
Mappings on matrices: invariance of functional values of matrix products
Authors: Chan Jor-Ting , Li Chi-Kwong , Sze Nung-Sing
Published: April 2009

LetMn, be the algebra of alln × nmatrices over a field F, wheren≧ 2. LetSbe a subset ofMncontaining all rank one matrices. We study mappings φ:S → Mn, such thatF(φ (A)φ (B)) =F(A B)for various families of functionsFincluding all the unitary similarity invariant functions on real or complex matrices. Very often, these mappings have the form A ↦ μ(A)S(σ (aij))S-1for allA= (aij) ∈Sfor some invertibleS ∈ Mn, field monomorphism σ of F, and an F*-valued mapping μ defined onS. For real matrices, σ is often the identity map; for complex matrices, σ is often the identity map or the conjugation map: z ↦ z. A key idea in our study is reducing the problem to the special case whenF:Mn→ {0, 1} is defined byF(X)= 0, ifX= 0, andF(X)= 1 otherwise. In such a case, one needs to characterize φ:S → Mnsuch that φ(A) φ (B) = 0 if and only ifAB= 0. We show that such a map has the standard form described above on rank one matrices inS.
A Dirichlet series expansion for the p-adic zeta-function
Authors: Delbourgo Daniel
Published: April 2009

We prove that the p-adic zeta-function constructed by Kubota and Leopoldt has the Dirichlet series expansion Where the convergence of the first summation is for the p-adic topology. The proof of this formula relates the values of p(–s, ω1+σ) for s ∈ Zp, with a branch of the ‘sth-fractional derivative’, of a suitable generating function.
Cotorsion modules and relative pure-injectivity
Authors: Mao Lixin , Ding Nanqing
Published: April 2009

Let R be a ring. A right R-module C is called a cotorsion module if Ext1R (F, C) = 0 for any flat right R-module F. In this paper, we first characterize those rings satisfying the condition that every cotorsion right (left) module is injective with respect to a certain class of right (left) ideals. Then we study relative pure-injective modules and their relations with cotorsion modules.
Homomorphisms of the algebra of locally integrable functions on the half line
Authors: Grabiner Sandy
Published: April 2009

Let φ be a continuous nonzero homomorphism of the convolution algebra L1loc(R+) and also the unique extension of this homomorphism to Mloc(R+). We show that the map φis continuous in the weak* and strong opertor topologies on Mloc, considered as the dual space of Cc(R+) and as the multiplier algebra of L1loc. Analogous results are proved for homomorphism from L1 [0, a) to L1 [0, b). For each convolution algebra L1 (ω1), φ restricts to a continuous homomorphism from some L1 (ω1) to some L1 (ω2), and, for each sufficiently large L1 (ω2), φ restricts to a continuous homomorphism from some L1 (ω1) to L1 (ω2). We also determine which continuous homomorphisms between weighted convolution algebras extend to homomorphisms of L1loc. We also prove results on convergent nets, continuous semigroups, and bounded sets in Mloc that we need in our study of homomorphisms.
Automatic continuity for Banach algebras with finite-dimensional radical
Authors: Pham Hung Le
Published: April 2009

The paper [3] proved a necessary algebraic condition for a Banach algebra A with finite-dimensional radical R to have a unique complete (algebra) norm, and conjectured that this condition is also sufficient. We extend the above theorem. The conjecture is confirmed in the case where A is separable and A/R is commutative, but is shown to fail in general. Similar questions for derivations are discussed.
Hanna Neumann
Authors: Newman M. F. , Wall G. E.
Published: April 2009

Fellowship of the Australian Academy of Science and Fellowship of the Australian College of Education are formal recognition of Hanna Neumann's impact on a country she had first set foot in only in August 1963. But then`Hanna Neumann was a remarkable person. Throughout her life she had won the love and respect of many people. The extent of this can not really be measured, however some indications can be given. A memorial meeting held in Canberra overflowed a large lecture theatre even though it was virtually vacation time. A collection of papers dedicated to her memory and a fund to provide some form of memorial to her have both drawn quite overwhelming support from many parts of the world. One finds a tremendous list of words describing her memorable qualities: warm, enthusiastic, inspiring, energetic, firm, tactful, sympathetic, efficient, patient, shrewd, humble, peace-loving, courageous, gracious, … No words can hope to evoke more than a pale shadow of such a person; this story must be read in such a light.
Kelvin-wave diffraction by a gap
Authors: Buchwald V. T. , Miles J. W.
Published: April 2009

The problem of diffraction of a Kelvin wave by a narrow opening in a barrier separating two semi-infinite sheets of water is formulated as an integral equation. An approximate solution is obtained on the hypothesis that the gap is small compared with the length of the incident wave. The scattered disturbance comprises both Kelvin and Poincaré waves, and asymptotic formulate for their amplitudes are obtained. The results, which are relevant to tidal diffraction by a strait, reveal that the phase shift in the diffracted Kelvin wave is much larger than the corresponding shift induced by either a narrow channel or an inland sea opening into a semi-infinite ocean.
Inequalities relating different definitions of discrepancy
Authors: Smyth C. J.
Published: April 2009

LetP1, P2…, pN be N points in the unit s-dimensional closed square Q = [0, 1]s. For any measurable set S ⊆ Q, we define δ(S), the discrepancy of S, by , where V(S) is the s-dimensional volume of S, and n(S), is the number of indices i for which pi∈S. Let , where the supermum is taken over all s-balls B ∈ Q, and , the supermum in this case being taken over all convex sets C ∈ Q. Clearly Dc ≧ Dk. In this paper we establish Theorem , where φ1is a constant depending only on s.
A computer application to finite p-groups
Authors: Macdonald I. D.
Published: April 2009

The first and still the best known computer application to groups exploits coset enumeration and this has been very thoroughly studied; see for instance [2] and [8]. No doubt this is because the algorithm is simple in the sense of programming. The Underlying mathematics is far from simple, touching as it does no logical difficulties akin to the word problem for groups, and this is reflected in the facts that random access to large tables is required and that there is no indication at any stage (for example when storage space is exhausted) whether the algorithm would be completed at any later stage. Efficient computation depends on choosing a subgroup of small index m in the group under examination, for group elements will be represented as permutations of degree m, and the larger m is the more tedious it will be to check properties like orders of group elements. Yet in many cases m may have to be fairly large so that the subgroup is “corefree” i.e. the representation is faithful.
Generalized Gaussian estimates and riesz means of Schrödinger groups
Authors: Blunck Sönke
Published: April 2009

We show that generalized Gaussian estimates for selfadjoint semigroups (e-tA)t∈ R+ onL2implyLpboundedness of Riesz means and other regularizations of the Schrödinger group (eitA)t∈ R. This generalizes results restricted to semigroups with a heat kernel, which are due to Sjöstrand, Alexopoulos and more recently Carron, Coulhon and Ouhabaz. This generalization is crucial for elliptic operators A that are of higher order or have singular lower order terms since, in general, their semigroups fail to have a heat kernel.
The weighted g-Drazin inverse for operators
Authors: Dajić A. , Koliha J. J.
Published: April 2009

The paper introduces and studies the weighted g-Drazin inverse for bounded linear operators between Banach spaces, extending the concept of the weighted Drazin inverse of Rakočević and Wei (Linear Algebra Appl. 350 (2002), 25–39) and of Cline and Greville (Linear Algebra Appl. 29 (1980), 53–62). We use the Mbekhta decomposition to study the structure of an operator possessing the weighted g-Drazin inverse, give an operator matrix representation for the inverse, and study its continuity. An open problem of Rakočević and Wei is solved.
Perfect effect algebras are categorically equivalent with Abelian interpolation po-groups
Authors: Dvurečenskij Anatolij
Published: April 2009

We introduce perfect effect algebras and we show that every perfect algebra is an interval in the lexicographical product of the group of all integers with an Abelian directed interpolation po-group. To show this we introduce prime ideals of effect algebras with the Riesz decomposition property (RDP). We show that the category of perfect effect algebras is categorically equivalent to the category of Abelian directed interpolation po-groups. Moreover, we prove that any perfect effect algebra is a subdirect product of antilattice effect algebras with the RDP.
On systems of diagonal forms
Authors: Knapp Michael P.
Published: April 2009

In this paper we consider systems of diagonal forms with integer coefficients in which each form has a different degree. Every such system has a nontrivial zero in every p-adic field Qp provided that the number of variables is sufficiently large in terms of the degrees. While the number of variables required grows at least exponentially as the degrees and number of forms increase, it is known that if p is sufficiently large then only a small polynomial bound is required to ensure zeros in Qp. In this paper we explore the question of how small we can make the prime p and still have a polynomial bound. In particular, we show that we may allow p to be smaller than the largest of the degrees.
Quasi-Frobenius quotient rings of group rings
Authors: Horn Andreas
Published: April 2009

The purpose of this paper is an extension of a theorem of Hughes (1973). He showed: Let R be a ring which has a right artinian right quotient ring and let G be a group which has a (transfinite) ascending normal series with each factor either finite or cyclic, but only a finite number of finite factors. Then the group ring RG has a right artinian right quotient ring.
Intersections of α-spaces
Authors: Fowler Northrup
Published: April 2009

Let be an infinite r.e. repère, an infinite dimensional r.e. space such that ≦ L(). A condition is derived that is both necessary and sufficient for the existence of an infinite subset β ⊂ such that L(β)∪ is not an α-space. Examples which satisfy this condition are exhibited, proving that the class of α-spaces is not closed under intersections.
Stieltjes-type integrals
Authors: Russell A. M.
Published: April 2009

Burkill (1957) introduced extended Riemann-Stieltjes integrals of the form ∫baƒ(x)dkg(x)/dxk-1 to provide an alternative approach to the theory of distributions. We will denote Burkill's integral by (B)∫baƒ(x)dkg(x)/dxk-1 Burkill's paper (1957) partially motivated the study of the Riemann-Stieltjes integral in Russell (1970). He showed that the two integrals are not equivalent.
On a theorem of J. A. Green
Authors: O'Reilly Michael F.
Published: April 2009

Let G be a finite group, K a field of characteristic p and Γ the group algebra of G over k. Let e = ∑g∈Gαgg, αg ∈k, be a primitive central idempotent of Γ; let supp e = {g ∈ G: αg≠0}. We provide a short proof of a slightly stronger version of Theorem 5 of Green (1968).
Delay differential equations of odd order satisfying property Pk
Authors: Terry Raymond D.
Published: April 2009

The property Pk(K = 0,1,…,n) is formulated. For k = 0, n this property reduces to conditions A and B defined by Kiguradze (1962) for a class of ordinary differential equations. Sufficient conditions are then given which guarentee that a class of delay differential equations of odd order possesses property Pk. The property PK is also seen to be useful in reducing the number of types of positve solutions of a related nonhomogeneous delay differential equation.
On numerical ranges of operators on locally convex spaces
Authors: Giles J. R. , Joseph G. , Koehler D. O. , Sims B.
Published: April 2009

Numerical range theory for linear operators on normed linear spaces and for elements of normed algebras is now firmly established and the main results of this study are conveniently presented by Bonsall and Duncan in (1971) and (1973). An extension of the spatial numerical range for a class of operators on locally convex spaces was outlined by Moore in (1969) and (1969a), and an extension of the algebra numerical range for elements of locally m-convex algebras was presented by Giles and Koehler (1973). It is our aim in this paper to contribute further to Moore's work by extending the concept of spatial numerical range to a wider class of operators on locally convex spaces.
Twenty-step algorithm for determining the asymptotic number of trees of various speces
Authors: Harary Frank , Robinson Robert W. , Schwenk Allen J.
Published: April 2009

The technique for finding for asymptotic number of unlabled trees of various sorts was developed by pólya (1973) and perfected by Otter (1848). Modern presentations are available in the book of Harary and Palmer (1973; Chapter 9), and in the paper of Bender (to appear).
Completion of normed algebras of polunomials
Authors: Dales H. G. , McClure J. P.
Published: April 2009

Let P be the algebra of polynomials in one inderminate x over the complex field C. Suppose ∥ · ∥ is a norm on P such that the coefficient functionals cj: ∑αix1 → αj (j = 0,1,2,…) are all continuous with respect to ∥·∥, and Let K ⊂ C be the set of characters on P which are ∥·∥-continuous. then K is compact, C\K is connected, and 0∈K. K. Let A be the completion of P with respect to ∥·∥. Then A is a singly generated Banach algebra, with space of characters (homeomorphic with) K. The functionals cj have unique extensions to bounded linear functionals on A, and the map a →∑Ci(a)xi (a ∈ A) is a homomorphism from A onto an algebra of formal power series with coefficients in C. We say that A is an algebra of power series if this homomorphism is one-to-one, that is if a ∈ A and a≠O imply cj(a)≠ 0 for some j.
Normal transformation semigroups
Authors: Symons J. S. V.
Published: April 2009

Many transformation semigroups S over a set X have the inner-automorphism property: their automorphisms are precisely those mappings of the form g-1 · g: α → g-1αg(α∈S) where g is a fixed permutation of S. For example, the full transformation semigroup, any two sided ideal of this semigroup, and, with some exceptions for small cardinals, the alternating group and the symmetric group itself. See Malcev (1952) and Scott (1965, Chapter 11). In this paper we determine all transformation semigroups over finite X with this property. In fact, we do more: we characterize all transformation semigroups invariant under the mappings g-1 · g. We refer to these as -normal transformation semigroups.
Product semigroups
Authors: DeVun E. E.
Published: April 2009

In earlier work Borrego, Cohen and DeVun (1971a, b) topological semigroups which were uniquely representable in terms of two subsets were investigated. In this note we extend the definition and prove similar results for a larger class of semigroups.
Rudin-Shapiro sequences for arbitrary compact groups
Authors: McMullen J. R. , Price J. F.
Published: April 2009

Let G be a compact group. A sequence of functions in L∞ (G) is said to be a Rudin-Shapiro sequence (briefly, an RS-sequence) if the following conditions hold: (1) (2) (3) The main purpose here is to prove the following theorem: Theorem: Theorem. Let G be an infinite compact group. Then G has an RS-sequence consisting of trigonometric polynomials.
On the zeros of Hilbert spaces of analytic functions
Authors: Kon S. H.
Published: April 2009

An attempt is made to characterise the zeros of some Hilbert spaces of analytic functions means of their kernel functions. Results on the zeros of functions in DΦ and their uniqueness s are included, in particular we give an affirmative answer to a question of Shapiro and Shield
Some sequence spaces and almost convergence
Authors: Nanda Sudarsan
Published: April 2009

In this paper we investigate some new sequence spaces which naturally emerge from the concept of almost convergence. Just as ordinary, absolute and strong summability, it is expected that almost convergence must give rise to almost, absolutely almost and strongly almost summability. Almost and absolutely almost summable sequences have been discussed by several authors. The object of this paper is to introduce the spaces of strongly almost summable sequences which happen to be complete paranormed spaces under certain conditions. Some topological results, characterisation of strongly almost regular matrices, uniqueness of generalized limits and inclusion relations of such sequences have been discussed.
Geometric and probabilistic aspects of some combinatorial identities
Authors: Speed T. P.
Published: April 2009

For positive integers a, b and n define the combinational expression We give geometric and probabilistic interpretations of these expressions (and their multidimensional extensions) and find new, simple proofs of the convolution identities known to hold for such expressions.
Hadamard matrices and submatrices
Authors: Vijayan K.
Published: April 2009

Shrinkande and Bhagwan Das (1970) showed how to extend a (4t – 1, 4t) row-orthogonal matrix with entries ± 1 to a Hadamard matrix of order 4t. Using a slightly different approach we consider extensions of (4t – k, 4t) row-orthogonal matrix to a Hadamard matrix of order 4t.
The Schur-Zassenhaus theorem in locally finite groups
Authors: Hartley B.
Published: April 2009

Let L = HK be a semidirect product of a normal locally finite π′-group H by a locally finite π′-group K, where π, is a set of primes. Suppose CK(H) = 1 and L is Sylow π-sparse (which in the countable case just says that the Sylow π-subgroups of L are conjugate). This paper completes the characterization of those groups which can occur as K—this had previously been obtained under the assumption that L is locally soluble. The answer is the same—essentially that the groups occurring are those having a subgroup of finite index which is a subdirect product of so-called “pinched” groups.
A generalization of a theorem of Totten
Authors: Mullin R. C. , Vanstone S. A.
Published: April 2009

An (r, 1) system is pair (V, F) where V is a v-set and F is a family of non-null subsets of V (b in number) which satisfy the following.

(i) Every pair of distinct members of V occur in precisely one member of F;

(ii) Every member of V occurs in precisely r members of F.

A pseudo parallel complement PPC(n, α) is an (n+1, 1) system with ν = n2 – αn and b ≦ n2 + n – α in which there are at least n – α blocks of size n. A pseudo intersection complement PIC (n, α) is an (n+1, 1) system with ν=n2 – αn + α – 1 b ≦ n2+n – α in which there are at least n – α+1 blocks of size n-1. It is shown that for α ≧ 4, every PIC(n, α) can be embedded in a PPC(n, α – 1) and that for n > (α4 + 2α3 + 2α2 + α)/2 every PPC(n, α) can be embedded in a projective plant of order n. The latter generalizes a result of Totten (who proves the result for α = 1).
Characterization of a family of simple groups by their character table, II
Authors: Herzog Marcel , Wright David
Published: April 2009

It is shown that the simple groups G2(q), q = 3f, are characterized by their character table. This result completes characterization of the simple groups G2(q), q odd, by their character table.
The subdirect decomposition theorem for classes of structures closed under direct limits
Authors: Caicedo Xavier
Published: April 2009

By a theorem of G. Birkhoff, every algebra in an equationally defined class of algebras K is a subdirect product of subdirectly irreducible algebras of K. In this paper we show that this result is true for any class of structures. not necessarily algebraic, closed under isomorphisms and direct limits. Quasivarieties in the sense of Malcev are examples of such classes of structures. This includes Birkhoffs result as a particular case.
Linear spaces with line range {n−1, n, n + 1} and at most n2 points
Authors: Batten Lynn Margaret
Published: April 2009

We characterize all finite linear spaces with p ≤ n2 points where n ≥ 8 for p ≠ n2 − 1 and n ≥ 23 for p = n2−1, and the line range is {n−1, n, n+1}. All such linear spaces are shown to be embeddable in finite projective planes of order a function of n. We also describe the exceptional linear spaces arising from p < n2−1 and n ≥ 4.
Random labelled trees and their branching networks
Authors: Grimmett G. R.
Published: April 2009

A random rooted labelled tree on n vertices has asymptotically the same shape as a branching-type process, in which each generation of a branching process with Poisson family sizes, parameter one, is supplemented by a single additional member added at random to one of the families in that generation. In this note we use this probabilistic representation to deduce the asymptotic distribution of the distance from the root to the nearest endertex other than itself.
Lattice-ordered modules of quotients
Authors: Steinberg Stuart A.
Published: April 2009

Let Q be the ring of quotients of the f-ring R with respect to a positive hereditary torsion theory and suppose Q is a right f-ring. It is shown that if the finitely-generated right ideals of R are principal, then Q is an f-ring. Also, if QR is injective, Q is an f-ring if and only if its Jacobson radical is convex. Moreover, a class of po-rings is introduced (which includes the classes of commutative po-rings and right convex f-rings) over which Q(M) is an f-module for each f-module M.
C-Commutativity
Authors: Cheatham T. , Enochs E.
Published: April 2009

An associative ring R with identity is said to be c-commutative for c ∈ R if a, b ∈ R and ab = c implies ba = c. Taft has shown that if R is c-commutative where c is a central nonzero divisor]can be omitted. We show that in R[x] is h(x)-commutative for any h(x) ∈ R [x] then so is R with any finite number of (commuting) indeterminates adjoined. Examples adjoined. Examples are given to show that R [[x]] need not be c-commutative even if R[x] is, Finally, examples are given to answer Taft's question for the special case of a zero-commutative ring.
Weighted shifts and commuting normal extension
Authors: Lubin Arthur
Published: April 2009

The main result of this paper shows that the existence of commuting normal extension (c.n.e.) for an arbitrary family of commuting subnormal operators can be determined by considering appropriate families of multivariable weighted shifts. In proving this some known criteria for c.n.e. are generalized. It is also shown that a family of jointly quasi-normal operators has c.n.e.
Algebras stably equivalent to Nakayama algebras of Loewy length at most 4
Authors: Reiten Idun
Published: April 2009

Two artin algebras Λ and Λ′ are said to be stably equivalent if their categories of finitely generated modules modulo projectives are equivalent. In this paper a characterization is given of the artin algebras stably equivalent to Nakayama algebras of Loewy length (at most) four. The proof is an illustration of the technique of using irreducible maps to study problems about stable equivlence.
Large abelian subgroups of Chevalley groups
Authors: Barry Michael J. J.
Published: April 2009

For any group S let Ab(S) = {A∣A is an abelian subgroup of S of maximal order}. Let G be a Chevalley group of type An, Bn, Cn, or Dn over a finite field of characteristic p and let. In this paper Ab(U) is determined for all such groups.
The Gaussian law and lacunary sets of characters
Authors: Dudley E. , Hall P.
Published: April 2009

Salem and Zygmund (1947, 1948), Baker (1972) and Dudley (1975) have shown that certain lacunary sets P of characters of a compact abelian group have sequences of the form where фk∈P converge to the normal distribution if suitably normalized. In this paper, a theorem of probability due to McLeish (1974) is applied to clarify and extend the previous results.
The funnel boundary of multivalued dynamical systems
Authors: Kloeden P. E.
Published: April 2009

Properties of the funnel boundary are investigated for multivalued dynamical systems defined axiomatically in terms of attainability set mappings on complete, locally compact metric state spaces. The set of regular boundary events is shown to be dense in the funnel boundary and theorems of Fukuhara and Zaremba on peripheral attainability are generalized to the systems considered here.
The laws of some nilpotent groups of small rank
Authors: Chau T. C.
Published: April 2009

We shall take for granted the basic terminology currently in use in the theory of varieties of groups. Kovács, Newman, Pentony [2] and Levin [3] prove that if m is an integer greater than 2, then the variety Νm of all nilpotent groups of class at most m is generated by its free group Fm-1(Νm) of rank m – 1 but not by its free group Fm–2(Νm) of rank m — 2. That is, the free groups Fk(Nm), 2≦k ≦ m – 2, do not generate Nm. In general little is known of the varieties generated by them. The purpose of the present paper is to record the varieties of the free groups Fk(Nm) of the nilpotent varieties Nm of all nilpotent groups of class at most m for 2 ≦ k ≦ m – 2 and 5 ≦ m ≦ 6. This is done by describing a basis for the laws in these groups, that is a set of laws the fully invariant closure of which is the set of all laws for Fk(Nm). The set of laws, which, together with the appropriate nilpotency law, form a basis for the relevant groups Fk(Nm) are listed below: .
Odd order nilpotent groups of class two with cyclic centre
Authors: Leong Y. K.
Published: April 2009

The isomorphism problem for finite groups of odd order and nilpotency class 2 with cyclic centre will be solved using some results of Brady [1], [2]. Since a finite nilpotent group is the direct product of its Sylow subgroups, we only need to consider finite q-groups where q is a prime. It has been shown in [1] and [2] that a finite q-group of nilpotency class 2 with cyclic centre is a central product either of two-generator subgroups with cyclic centre or of two-generator subgroups with cyclic centre and a cyclic subgroup, and that the q-groups of class 2 on two generators with cyclic centre comprise the following list: , and if q = 2 we have as well .
Direct products and the Hopf property
Authors: Jones J. M. Tyrer
Published: April 2009

In [1] Neumann and Dey prove that the free product of two finitely generated Hopf groups is Hopf and ask whether a similar result holds for direct products. It is the purpose of this paper to show that this is not the case. We prove

THEOREM A. There exists a finitely generated group G satisfying the following conditions:

(i) G is isomorphic to a proper direct factor of itself;

(ii) G is the direct product of two Hopf groups.
Walking around the Brauer tree
Authors: Green J. A.
Published: April 2009

LetG be a finite group, and k a field of finite characteristic p, such that the polynomial x¦G¦ –1 splits completely in k[x]. Let Β be a kG-block which has defect group D which is cylclic of order pd (d ≧ 1). Brauer showed in a famous paper [2] that, in case d = 1, the decomposition matrix of Β is determined by a certain positive integer e which divides p − 1, and a tree Г, a connected acyclic linear graph of e + 1 vertices and e edges. Twenty-five years later Dade ([3]) extended Brauer's theorem to the general case.
Poincare's conjecture and the homeotopy group of a closed, orientable 2-manifold
Authors: Birman Joan S.
Published: April 2009

In 1904 Poincaré [11] conjectured that every compact, simply-connected closed 3-dimensional manifold is homeomorphic to a 3-sphere. The corresponding result for dimension 2 is classical; for dimension ≧ 5 it was proved by Smale [12] and Stallings [13], but for dimensions 3 and 4 the question remains open. It has been discovered in recent years that the 3-dimensional Poincaré conjecture could be reformulated in purely algebraic terms [6, 10, 14, 15] however the algebraic problems which are posed in the references cited above have not, to date, proved tractable.
Generating groups of certain soluble varieties
Authors: Gupta Narain , Levin Frank
Published: April 2009

Any variety of groups is generated by its free group of countably infinite rank. A problem that appears in various forms in Hanna Neumann's book [7] (see, for intance, sections 2.4, 2.5, 3.5, 3.6) is that of determining if a given variety B can be generated by Fk(B), one of its free groups of finite rank; and if so, if Fn(B) is residually a k-generator group for all n ≧ k. (Here, as in the sequel, all unexplained notation follows [7].)
Free products of hopfian lattices
Authors: Grätzer G. , Sichler J.
Published: April 2009

In this paper we are going to prove the following results:

Theorem 1. There exist two bounded hopfian lattices such that their {0,1}- free product is not hopfian.

Theorem 2. There exist two hopfian lattices such that their free product is not hopfian.

In Theorem 2 free product (coproduct, sum) has its usual meaning (see, for instance, [4]); in Theorem 1 we use the usual definition but all lattices are assumed to be bounded (that is, having a least element 0 and largest element 1) and all homomorphisms are assumed to be {0, l}-homomorphisms (that is, homomorphisms preserving 0 and 1).
Upper and lower fields for profinite groups of given cardinality
Authors: Krakowski Don
Published: April 2009

Profinite groups (that is, compact totally disconnected topological groups) have been characterized as Galois groups in Krakowski (1971) and Leptin (1955). They can, in fact, be realized as groups of permutations on sets of transcendentals where every transcendental has a finite orbit under the group action and the fixed field is generated by the invariant rational functions on these orbits. If G = {G←x ¦ α∈A} we define the cardinality of G as ¦ G¦ = ¦A¦. In this paper we construct, for every cardinal number c and profinite group G with ¦G¦ ≦ c two universal fields kc and Kc so that G ≃ Gal (KcKG) and G ≃ (KGKc) for fields KG and kG which depend on G. See Cassels and Frohlich (1967) for a description of the basic properties of profinite groups.
The Number of Unitarily k-Free Divisors of An Integer
Authors: Suryanarayana D. , Rao R. Sita Rama Chandra
Published: April 2009

Let k be a fixed integer ≧ 2. A positive integer n is called unitarily k-free, if the multiplicity of each prime factor of n is not a multiple of k; or equivalently, if n is not divisible unitarily by the k-th power of any integer > 1. By a unitary divisor, we mean as usual, a divisor d> 0 of n such that (d, n/d) = 1. The interger 1 is also considered to be unitarily k-free. The concept of a unitarily k-free integer was first introduced by Cohen (1961; §1). Let denote the set of unitarily k-free integers. When k = 2, the set coincides with the set Q* of exponentially odd integers (that is, integers in whose canonical representation each exponent is odd) discussed by Cohen himself in an earlier paper (1960; §1 and §6). A divisor d > 0 of the positive integer n is called a unitarily k-free divisor of n if d ∈ . Let (n) denote the number of unitarily k-free divisors of n.
On the boundary behavior of holomorphic functions in the unit disc
Authors: Yoshida Hidenobu
Published: April 2009

Seidel (1959) established various boundary properties of holomorphic functions with spiral asymptotic paths. Especially, Theorem 4 which is the fundamental result of the paper; was generalized by Faust (1962), using essentially the same method that was employed by Seidel.
On the generalized riemann integral and stochastic integral
Authors: Lee Tack-Wang
Published: April 2009

In Lee (submitted), the GW-integral (the generalized Riemann integral using Wiener measure) is defined. The object of this article is to define stochastic integral in the set up given in Lee (submitted). We also investigate the connection between the stochastic integral defined with the Legesgue counter part, the Paley-Wiener-Zygmund integral in Paley, Weiner and Zygmund (1933). Applications of the stochastic integral will be explained elsewhere.
The hardy class of functions of bounded argument rotation
Authors: Miller Sanford S. , Mocanu Petru T.
Published: April 2009

The Hardy classes for functions of bounded argument rotation and their derivatives are determined. In addition, if , then growth conditions for an are obtained.
Bounded index and summability methods
Authors: Fricke G. H. , Powell R. E.
Published: April 2009

An entire function f(z) is of bounded index if there exists a non-negative integer N such that . The least such integer N is called the index of f (se Lepson (1966)).
Notes on differential calculus in topological linear spaces, III
Authors: Yamamuro S.
Published: April 2009

Throughout this note, let E, F and G be locally convex Hausdorff spaces over the real number field R. We denote real numbers by Greek letters. The sets of all continuous semi-norms on E and F will be denoted by P(E) and P(F) respectively, and A will always stand for an open subset of E.
A functional calculus for sesquihermitian operators on quaternionic Hilbert space
Authors: Powers N. C.
Published: April 2009

A continuous real-linear operator A = A0 + i1A1 + i2A2 + i3A3 on a quaternionic Hilbert space is called sesquihermitian if the linear operators Av are Hermitian; this condition is independent of the choice of quaternion basis (i1,i2,i3). The joint spectral distribution of the Av provides a functional calculus for sesquihermitian operators and real-valued C∞-functions on . This calculus is independent of the quaternion basis and extends naturally to quaternion-valued functions to give a continuous quaternion-linear mapping from the algebra of these functions to that of sesquihermitian operators. The mapping is not, in general, multiplicative unless the Av commute, in which case it agrees with that for several commuting operators on complex Hilbert space.
On a characteristic property of point processes
Authors: Rao B. L. S. Prakasa
Published: April 2009

This note is concerned with a certain property of point processes. We prove that ifN1,N2andN3are three independent point processes, then the bivariate point process (N1+N3+N2+N3) uniquely determines the point processesN1,N2andN3.
Some remarks on fixing subgraphs and smoothly embeddable subgraphs
Authors: Grant Douglas D.
Published: April 2009

In this note we specify the conditions under which the line graph of a fixing subgraph of a graph G is a smoothly embeddable subgraph of the line graph of G, and vice-versa.
Ideals in a ring of exponential polynomials
Authors: Laird P. G.
Published: April 2009

An exponential polynomial is a finite linear combination of terms unea: t→tneat where n is any non-negative integer and a is any complex number. The set X of exponetial polynomials is clearly a vector space over the field of complex numbers C and this set is identical with the set of solutions to all homogeneous linear ordinary differential equations with constant coefficients.
Ends of locally compact groups and their coset spaces
Authors: Houghton C. H.
Published: April 2009

Freudenthal [5, 7] defined a compactification of a rim-compact space, that is, a space having a base of open sets with compact boundary. The additional points are called ends and Freudenthal showed that a connected locally compact non-compact group having a countable base has one or two ends. Later, Freudenthal [8], Zippin [16], and Iwasawa [11] showed that a connected locally compact group has two ends if and only if it is the direct product of a compact group and the reals.
Metanilpotent fitting classes
Authors: Bryce R. A. , Cossey John
Published: April 2009

Hawkes showed in [10] that classes of metanilpotent groups which are both formations and Fitting classes are saturated and subgroup closed; more, he characterized all such classes as those local formations with a local definition consisting of saturated formations (of nilpotent groups). In [3] we showed that those “Fitting formations” which are subgroup closed are also saturated, without restriction on nilpotent length; indeed such classes are, roughly speaking, recursively definable as local formations using a local definition consisting of such classes. It is natural to ask how these hypotheses may be weakened yet still produce the same classes of groups. Already in [10] Hawkes showed that Fitting formations need be neither subgroup closed nor saturated; and in [3] we showed that a saturated Fitting formation need not be subgroup closed (though a Fitting formation of groups of nilpotent length three is saturated if and only if it is subgroup closed).
The Fourier multiplier problem for spaces of continuous functions with P-summable transforms
Authors: Bloom Lynette M.
Published: April 2009

In this paper we consider spaces Ap, p ∈ [1, 2], and multipliers (Ap, Aq), p ∈ [1, 2], q ∈ [1, 2]. In 4.4 and 6.1 we identify (Ap, Aq) for p ∈ [1, 2], q ∈ [p, 2], and in 7.3 we identify (A2, A1). In 7.1 we give a sufficient condition, and in 7.5 a necessary condition, for membership of (Ap, Aq), p ∈ [1, 2], q ∈ [1, 2]. We give, in 7.2, a necessary condition for membership of (A2, Aq), q ∈[l,2). We include constructive proofs of some strict inclusion results for Ap, p ∈ [1,2], (3.1 and 3.2), and also, in 5.3, for (AP,AP), p∈[l,2].
Two remarks on the homology of group extensions
Authors: Hilton Peter , Stammbach Urs
Published: April 2009

In this note we apply a particular technique to obtain information on the homology homomorphism ε*: H*(G; A) → H* (Q; A) associated with a group extension and a Q-module A. The technique consists of using ε itself to pull-back (0.1); that is, we construct the pull-back extension induced from (0.1) by ε. This, however, is nothing but the semidirect product, NG, of N and G, with G operating on the left on N by conjugation. Thus we obtain from (0.1) the commutative diagram where ε1 is the projection and et is the multiplication ε1(n, x) = nx,n∈N,x∈G. We now apply the Lyndon-Hochschild-Serre spectral sequence functor to (0.2) and carry out computations in dimensions 2 and 3.
On Malcev conditions
Authors: Neumann Walter D.
Published: April 2009

This note gives a way of looking at Malcev conditions for varieties as ideals in a certain lattice. Though this viewpoint (so far) yields no new results, we feel it puts Walter Taylor's results [3] characterizing “Malcev definable classes of varieties” into a clearer perspective and is therefore worth mentioning. The author is grateful to R. Wille for pointing out W. Taylor's paper to him.
A characterization of finite p-soluble groups of p-length one by commutator identities
Authors: Brandl Rolf
Published: April 2009

A classical result of M. Zorn states that a finite group is nilpotent if and only if it satisfies an Engel condition. If this is the case, it satisfies almost all Engel conditions. We shall give a similar description of the class of p-soluble groups of p-length one by a sequence of commutator identities.
Integral quadratic forms and orthogonal designs
Authors: Eades Peter
Published: April 2009

Warren W. Wolfe obtained necessary conditions for the existence of orthogonal designs in terms of rational matrices. In this paper it is shown that these necessary conditions can be obtained in terms of integral matrices. In the integral form, Wolfe's theory is more useful in the construction of orthogonal designs.
On the reconstruction conjecture for separable graphs
Authors: Krishnamoorthy V. , Parthasarathy K. R.
Published: April 2009

Some sufficient conditions for the reconstructability of separable graphs are given proceeding along the lines suggested by Bondy, Greenwell and Hemminger. It is shown that the structure and automorphism group of a central block plays an important role in the reconstruction.
Facial reduction for a cone-convex programming problem
Authors: Borwein Jon M. , Wolkowicz Henry
Published: April 2009

In this paper we study the abstract convex program

where S is an arbitrary convex cone in a finite dimensional space, Ω is a convex set and p and g are respectively convex and S (on Ω). We use the concept of a minimal cone for (P) to correct and strengthen a previous characterization of optimality for (P), see Theorem 3.2. The results presented here are used in a sequel to provide a Lagrange multiplier theorem for (P) which holds without any constraint qualification.
On arithmetic functions and divisors of higher order
Authors: Alladi Krishnaswami
Published: April 2009

We discuss properties of arithmetic functions of higher order defined through the introduction of a new concept of divisor of higher order. We shall construct an infinite sequence of Euler-like functions and the well known Euler function will be the first member of this sequence. Asymptotic estimates of such functions are given and a study of error functions associated with the Euler-like sequence is made. We would like to mention that the familiar number theoretic functions become only the first members of an infinite sequence of functions of similar behaviour.