bluuebunny/crossref_metadata_2025 huggingface dataset:
A report on stable graphs
Authors: Holton D. A.
Published: April 2009

It is the aim of this paper to introduce a new concept relating various subgroups of the automorphism group of a graph to corresponding subgraphs. Throughout g will denote a (Michigan) graph on a vertex set V(¦V¦ =n) and Γ(g)=G will be the automorphism group of G considered as a permutation group on V.En, Cn, Dn and Sn are the identity, cyclic, dihedral, and symmetric groups acting on a set of size n, while Sp(q) is the permutation group of pq objects which is isomorphic to Sp but is q-fold in the sense that the objects are permuted q at a time [6]. H ≦ G means that H is a subgroup of G. Other group concepts can be found in Wielandt [7]. The graphs G1 ∪ G2, G1 + G2, G1 × G2, and G1[G2] along with their corresponding groups are as defined in, for example, Harary [4]. Finally we use Kn for the complete graph on n vertices.
A one-step method for the numerical solution of ordinary non-linear second-order differential equations based upon lobatto four-point quadrature formula
Authors: Sharma K. D. , Gupta R. G.
Published: April 2009

This paper describes a one-step method besed upon the Lobatto four-point quadreture formula for the numerical integration of differential: y″(x) = f(x, y(x), y'(x)); y(x0)=y0, y'(x0)=y'0. The method has a local truncation error 0(h6) in y(x) and 0(h5) in y′(x). In the case of linear second-order differential equation, a stability criterion has been developed. Theoretical and computational comparisons of the new method existing method is discussed.
On semiprime segments of rings
Authors: Mazurek R. , Törner G.
Published: April 2009

A semiprime segment of a ring R is a pair P2 ⊂ P1 of semiprime ideals of R such that ∩ In ⊆ P2 for all ideals I of R with P2 ⊂ I ⊂ P1. In this paper semiprime segments with P1 a comparizer ideal are classified as either simple, exceptional, or archimedean, extending to several classes of rings a classification known for right chain rings. These three types of semiprime segments are also characterized in terms of the pseudo-radical.
On the reduction of positive quaternary quadratic forms
Authors: Barnes E. S. , Cohn M. J.
Published: April 2009

A fundamental region for the reduction of positive quaternary quadratic forms is exhibited. It is a convex polyhedral cone with twelve edges in the 10-dimensional space of quaternary quadratic forms.
On a class of non-linear functional equations connected with modular functions
Authors: Mahler K.
Published: April 2009

Let p be a prime. This paper deals with solutions of functional equations in either formal Laurent series or in analytic functions. Examples connected to special modular functions are considered.
On transitive commutative idempotent quasigroups
Authors: Neumaier Arnold
Published: April 2009

Commutative idempotent quasigroups with a sharply transitive automorphism group G are described in terms of so-called Room maps of G. Orthogonal Room maps and skew Room maps are used to construct Room squares and skew Room squares. Very general direct and recursive constructions for skew Room maps lead to the existence of skew Room maps of groups of order prime to 30. Also some nonexistence results are proved.
On groups with all subgroups almost subnormal
Authors: Detomi Eloisa
Published: April 2009

In this paper we consider groups in which every subgroup has finite index in the nth term of its normal closure series, for a fixed integer n. We prove that such a group is the extension of a finite normal subgroup by a nilpotent group, whose class is bounded in terms of n only, provided it is either periodic or torsion-free.
On the orders of conjugacy classes in group algebras ofp-groups
Authors: Bovdi A. , Kovács L. G. , Mihovski S.
Published: April 2009

Letpbe a prime,a field ofpnelements, andGa finitep-group. It is shown here that ifGhas a quotient whose commutator subgroup is of orderpand whose centre has indexpk, then the group of normalized units in the group algebrahas a conjugacy class ofpnkelements. This was first proved by A. Bovdi and C. Polcino Milies for the casek= 2; their argument is now generalized and simplified. It remains an intriguing question whether the cardinality of the smallest noncentral conjugacy class can always be recognized from this test.
On the chromatic number of plane tilings
Authors: Coulson D.
Published: April 2009

It is known that 4 ≤ x(ℝ2) ≤ 7, where x(ℝ2) is the number of colour necessary to colour each point of Euclidean 2-space so that no two points lying distance 1 apart have the same colour. Any lattice-sublattice colouring sucheme for R2 must use at least 7 colour to have an excluded distance. This article shows that at least 6 colours are necessary for an excluded distance when convex polygonal tiles (all with area greater than some positive constant) are used as the colouring base.
Some regularity estimates for convolution semigroups on a group of polynomial growth
Authors: Dungey Nick
Published: April 2009

We study a convolution semigroup satisfying Gaussian estimates on a groupGof polynomial volume growth. IfQis a subgroup satisfying a certain geometric condition, we obtain high order regularity estimates for the semigroup in the direction ofQ. Applications to heat kernels and convolution powers are given.
Concerning non-measurable subsets of a given measurable set
Authors: Pu H. W.
Published: April 2009

Let R, μ and Mμ denote the set of real numbers, Lebesgue outer measure and the class of Lebesgue measurable subsets of R respectively. It is easy to prove that the complement Ec of E ∈ Mμ is a set of Lebesgue measure zero if the inequality holds for some δ > 0 and all intervals I of R. However, in Hewitt [1], raised a problem whether the result is still true if E is not a priori measurable set. In this paper, a negative answer to this question is given through a counter-example. Also, it is proved that for a given set E ∈ M,μ with μ(E) > 0 there is a non-measurable subset A of E satisfying μ(E)
Further results on the critical Galton-Watson process with immigration
Authors: Pakes A. G.
Published: April 2009

Consider a Galton-Watson process in which each individual reproduces independently of all others and has probability aj (j = 0, 1, …) of giving rise to j progeny in the following generation, and in which there is an independent immigration component where bj, (j = 0, 1, …) is the probability that j individuals enter the population at each generation. Defining Xn (n = 0, 1, …) to be the population size at the n- th generation, it is known that {Xn} defines a Markov chain on the non-negative integers.
Connectedness in the scale of a uniform space
Authors: Leslie G. C. , Kent D. C.
Published: April 2009

Associated with each uniform space (S, U) there is a uniform lattice (P, V, ≦) called the scale of (S, U). The scale was first introduced by D. Bushaw (see [2] and [3]) for the purpose of studying stability in topological dynamics. Further properties of the scale were investigated in [4], and this paper pursues one of the research problems suggested in the latter paper.
On the semigroups of Fredholm mappings
Authors: Yamamuro Sadayuki
Published: April 2009

LetE1andE2be real Banach spaces and letL(E1) andL(E2) be the Banach algebras of all continuous linear mappings onE1andE2respectively. It is a well- known result of M. Eidelheit [1] thatL(E1) thatL(E2) are isomorphic as rings if and only ifE1andE2are topologically and algebraically isomorphic. It is easy to see that the essential part of his proof is the following fact.
Relative amenability and the non-amenability of B(l1)
Authors: Read C. J.
Published: April 2009

In this paper we begin with a short, direct proof that the Banach algebraB(l1) is not amenable. We continue by showing that various direct sums of matrix algebras are not amenable either, for example the direct sum of the finite dimensional algebrasis no amenable for 1 ≤p≤ ∞,p≠ 2. Our method of proof naturally involves free group algebras, (by which we mean certain subalgebras ofB(X)for some spaceXwith symmetric basis—not necessarilyX = l2) and we introduce the notion of ‘relative amenability’ of these algebras.
Valuated Butler groups of finite rank
Authors: Fuchs L. , Rangaswamy K. M.
Published: April 2009

Valuated Butler groups of finite rank are investigated. The valuated B2-groups are both epic images and pure subgroups of completely decomposable valuated groups of finite rank (Theorem 3.1). However, there are fundamental changes in the theory of Butler groups when valuations are involved. We introduce valuated B1-groups and show that they are valuated B2-groups. Surprisingly, valuated B2-groups of rank greater than 1 need not be valuated B1 -groups, unless they carry a special kind valuation, see Theorem 7.1. Theorem 6.5 gives a full characterization of valuated B1 -groups of finite rank, generalizing Bican's characterization of Butler groups.
Characterizing rings by a direct decomposition property of their modules
Authors: Huynh Dinh Van , Rizvi S. Tariq
Published: April 2009

A module M is said to satisfy the condition (℘*) if M is a direct sum of a projective module and a quasi-continuous module. In an earlier paper, we described the structure of rings over which every (countably generated) right module satisfies (℘*), and it was shown that such a ring is right artinian. In this note some additional properties of these rings are obtained. Among other results, we show that a ring over which all right modules satisfy (℘*) is also left artinian, but the property (℘*) is not left-right symmetric.
Omitted rays and wedges of fractional Cauchy transforms
Authors: Hibschweiler R. A. , Macgregor T. H.
Published: April 2009

For α > 0 let α denote the set of functions which can be expressed where μ is a complex-valued Borel measure on the unit circle. We show that if f is an analytic function in Δ = {z ∈ : |z| < 1} and there are two nonparallel rays in /f(Δ) which do not meet, then f ∈ α where απ denotes the largest of the two angles determined by the rays. Also if the range of a function analytic in Δ is contained in an angular wedge of opening απ and 1 < α < 2, then f ∈ α.
Free algebras in varieties of BL-algebras generated by a BLn-chain
Authors: Busaniche Manuela , Cignoli Roberto
Published: April 2009

Free algebras with an arbitrary number of free generators in varieties of BL-algebras generated by one BL-chain that is an ordinal sum of a finite MV-chain Ln, and a generalized BL-chain B are described in terms of weak Boolean products of BL-algebras that are ordinal sums of subalgebras of Ln, and free algebras in the variety of basic hoops generated by B. The Boolean products are taken over the Stone spaces of the Boolean subalgebras of idempotents of free algebras in the variety of MV-algebras generated by Ln.

2000 Mathematics subject classification: primary 03G25, 03B50, 03B52, 03D35, 03G25, 08B20.
Varieties of topological groups and left adjoint functors
Authors: Morris Sidney A.
Published: April 2009

In [6] and [2] Markov and Graev introduced their respective concepts of a free topological group. Graev's concept is more general in the sense that every Markov free topological group is a Graev free topological group. In fact, if FG(X) is the Graev free topological group on a topological space X, then it is the Markov free topological group FM(Y) on some space Y if and only if X is disconnected. This, however, does not say how FG(X) and FM(X) are related.
On one relator groups and HNN extensions
Authors: McCool James , Schupp Paul E.
Published: April 2009

In his work [5] on subgroups of one relator groups, Moldavanski observed that if G is a one relator group whose defining relator R is cyclically reduced and has exponent sum zero on some generator occurring in it, then G is an HNN extension of a one relator group H whose defining relator is shorter than R. This observation, together with Britton's Lemma, can be used to give rather easy proofs of the basic results on one relator groups. To exposit this point of view, we give here a proof of the Freiheitssatz, the solvability of the word problem for one relator groups, and the theorem classifying elements of finite order in one relator groups. In particular, the solution obtained for the word problem is often easy to apply. We also give a proof of the “Spelling Theorem” of Newman [6].
A property of the complex semigroup algebra of a free monoid
Authors: Crabb M. J. , McGregor C. M. , Munn W. D.
Published: April 2009

It is shown that the complex semigroup algebra of a free monoid of rank at least two is *-primitive, where * denotes the involution on the algebra induced by word-reversal on the monoid.
A note on upper radicals in rings
Authors: Leavitt W. G.
Published: April 2009

A class (calledc-radicals) is defined such that, given ac-radicalP, there is in any classM′ a certain internal criterion that its upper radicalUM′ =P, ForPa non-c-radical (called aq-radical) there exists no smallest classMsuch thatUM=P, andPis aq-radical if and only if for someMwithP=UMthere exists 0 ≠R∈Msuch that when an image¯ ofRhas a non-zero image inMthere exists an infinite chain of epimorphisms¯ →R1→R2→ … with allR1∈Mand noRi, the image of anyRj, withj>i. Several examples of such rings are constructed including a ring all of whose images are primitive. Thus all radicals contained in the Jacobson radical areq-radicals.
Oscillatory and asymptotic properties of homogeneous and nonhomogeneous delay differential equations of even order
Authors: Terry Raymond D.
Published: April 2009

In this paper we consider the (non)oscillation properties of two general nonhomogeneous nonlinear delay differential equations of order 2n using as background and motivation the techniques previously applied to the associated homogeneous delay differential equations H+ and H−. The equations N+ and N− are each reduced to homogeneous form by the introduction of transformations u(t) = y(t) – R(t) and v(t) = R(t) — y(t), where R(t) is a solution of the associated nonhomogeneous differential equation (N). We first extend certain results for the equation H+ and then develop a classification of the positive solutions of equation H−. Using this classification and the one developed by Terry (1974) for H+ we develop a natural classification of the positive solutions of N+ and N− according to the sign properties of the derivatives of u(t) and v(t). For each choice of R(t), it is seen that there are 2n + 1 types of positive solutions of N+or N–. An intermediate Riccati transformation is employed to obtain integral criteriafor the nonexistence of some of these solutions. Analysis of the Taylor remainder results in sufficient conditions for the nonexistence of other such solutions.
Multiplicators and metabelian groups
Authors: Brumslag Gilbert
Published: April 2009

We prove here that the multiplicator of certain simply described metabelian groups is not finitely generated
Symmetric square roots of the infinite identity matrix
Authors: Pearce C. E. M. , Potts R. B.
Published: April 2009

Some non-trivial real, symmetric square roots of the infinite identity matrix are exhibited. These may be found either from the use of involutory integral transforms and a set of real orthonormal functions or by an algebraic factorisation procedure. The two approaches are shown to be equivalent.
A 4-chromatic graph with a special plane drawing
Authors: Wormald Nicholas
Published: April 2009

A graph is found which is 4-chromatic, has girth 5, and can be obtained by taking a setSof points in the plane and joining two of the points whenever their distance is 1. The points inSare not found explicitly, but their existence is demonstrated by use of a computer.

Subject classification (Amer. Math. Soc. (MOS) 1970): 05 C 15, 05 C 10.
Extremal graphs of diameter 3
Authors: Caccetta L.
Published: April 2009

This paper is concerned with graphs of order n and diameter at most 3 having the property that by deleting any s or fewer vertices (edges) the resulting subgraphs (partial graphs) have duameter at most. λ. A graph satisfying the above constraints and having minimum number of edges is said to be extramal. A characterization of extremal graphs is presented for the case s = 1.

Subject classification (Amer. Math. Soc. (MOS) 1970): 05 C 35.
On product varieties of inverse semigroups
Authors: Bales J. L.
Published: April 2009

This paper extends results on product varieties of groups to inverse semigroups. We show that if is a variety of groups and any inverse semigroup variety, then ∘ is a variety. We give a characterization of the identities of∘in terms of the identities of and ofWe show that if does not contain the variety of all groups then it has uncountably many supervarieties. Finally we show that ifis another variety of groups then

Subject classifiaction (Amer. Math. Soc. (MOS) 1970): 20 M 05.
The free centre-by-metabelian groups
Authors: Gupta Chander Kanta
Published: April 2009

Let be the free centre-by-metabelian group of rank n. In this paper, our main result is the following

Theorem. For n ≧ 4, Gn has a finite elementary abelian subgroup Hn of rank . More precisely, Hn is a minimal fully invariant subgroup contained in the centre of Gn and Gn/Hn is isomorphic to a group of 3 x 3 matrices over a finitely generated integral domain of characteristic zero.
One-relater groups with center
Authors: Meskin Stephen , Pietrowski A. , Steinberg Arthur
Published: April 2009

abstract

Many one-relator groups with center have been shown to be of the form A necessary and a sufficient condition for the sequence (P1, Q1, P2, Q2,…,Pt, Qt) are given in order for groups of the above form to be one-relator groups.
Oriented Tait graphs
Authors: Szekeres G.
Published: April 2009

The four colour conjecture is well known to be equivalent to the proposition that every trivalent planar graph without an isthmus (i.e. an edge whose removal disconnects the graph) has an edge colouring in three colours ([1], p. 121). By an edge colouring we mean an assignment of colours to the edges of the graph so that no two edges of the same colour meet at a common vertex, and the graph is n-valent if n edges meet at each vertex. An edge colouring by three colours is called a Tait colouring; a trivalent graph which has a Tait colouring can be split in three edge-disjoint 1-factors, i.e. spanning monovalent subgraphs.
On the latent roots of a doubly stochastic matrix
Authors: Ledermann Walter
Published: April 2009

Under certain general conditions an n x n stochastic matrix P = (pij), where

is known to possess the “ergodic” property. This means that there exists a finite matrix E such that
The nilpotent length of finite soluble groups
Authors: Frick M.
Published: April 2009

Throughout this paper a “group” will mean a “finite soluble group”.
Various varieties
Authors: MacDonald Sheila Oates
Published: April 2009

The study of varieties of universal algebras2 which was initiated by Birkhoff in 1935, [2], has received considerable attention during the past decade; the question of particular interest being: “Which varieties have a finite basis for their laws?” In that paper Birkhoff showed that the laws of a finite algebra which involve a bounded number of variables are finitely based, so it is not altogether surprising that finite algebras have received their share of this attention.
The perfect septenary forms with Δ4=2
Authors: Stacey K. C.
Published: April 2009

The aim of this paper is to enumerate all equivalence classes of perfect septenary forms with Δ4 = 2. This is an important section of the complete enumeration of perfect septenary forms by the method which was outlined in Stacey (1975). There are nine equivalence classes of these forms, four of which were announced for the first time in Stacey (1975). This work forms part of the author's D. Phil, thesis and was done at Oxford under the supervision of Dr. B. J. Birch.
Free products with amalgamation of commutative inverse semigroups
Authors: Imaoka Teruo
Published: April 2009

A class of algebras A is said to have the strong amalgamation property if for any indexed set of algebras {Ai: i ∈ J} from A, each having an algebra U ∈ A as a subalgebra, there exist an algebra B ∈ A and monomorphisms φi: Ai → B (for each i ∈ J) such that (i) φi|U = φi| U for all i,j∈J, (ii) φi(Ai) ∩ φi(Aj) =φi(U) for atl i, j ∈ J with i ≠ j, where φi|U denotes the restriction of φi to U. Omitting condition (ii) gives us the definition of the weak amalgamation property.
On the solubility of a product of permutable subgroups
Authors: Lennox John C.
Published: April 2009

Conditions are investigated under which a product G = HK of soluble subgroups H, K, each permutable in G, is soluble. It is shown in particular that this is true if one of the subgroups is nilpotent or if H ∩ K is subnormal in both H and K.
A class of modules over a locally finite group I
Authors: Hartley B.
Published: April 2009

Let G be a locally finite group, let k be a field of characteristic p ≧ 0, and let V be a (right) kG-module, not necessarily of finite dimension over k. We say that V is an Mc-modle over kG if, for each p′-subgroup H of G, the set of centrarlizers in V of subgroups of H satisfies the minimal condition under the relation of setheoretic inclusion. Here, p′ denotes the set of all peimes different from p, and in particular 0' denotes the set of all primes. It is straightforward to verify that V is an Mc-module over kG if and only if each p′-subgroup H of G contains a finite subgroup F such that CV(F) = CV(H).
Finite and infinite cyclic extensions of free groups
Authors: Karrass A. , Pietrowski A. , Solitar D.
Published: April 2009

Using Stalling's characterization [11] of finitely generated (f. g.) groups with infinitely many ends, and subgroup theorems for generalized free products and HNN groups (see [9], [5], and [7]), we give (in Theorem 1) a n.a.s.c. for a f.g. group to be a finite extension of a free group. Specifically (using the terminology extension of and notation of [5]), a f.g. group G is a finite extension of a free group if and only if G is an HNN group where K is a tree product of a finite number of finite groups (the vertices of K), and each (associated) subgroup Li, Mi is a subgroup of a vertex of K.
A theorem on generation of finite orthogonal groups
Authors: Wong W. J.
Published: April 2009

Presentation in terms of generators and relations for the classical finite simple groups of Lie type have been given by Steinberg and Curtis [2,4]. These presentations are useful in proving characterzation theorems for these groups, as in the author's work on the projective symplectic groups [5]. However, in some cases, the application is not quite instantaneous, and an intermediate result is needed to provide a presentation more suitable for the situation in hand. In this paper we prove such a result, for the orthogonal simple groups over finite fields of odd characteristic. In a subsequent article we shall use this to give a characterization of these groups in terms of the structure of the centralizer of an involution.
Abelian groups whose subgroup lattice is the union of two intervals
Authors: Breaz Simion , Călugăreanu Grigore
Published: April 2009

In this note we characterize the abelian groups G which have two different proper subgroups N and M such that the subgroup lattice L(G)=[0, M]∪ [N, G] is the union of these intervals.
Heat Kernels on homogeneous spaces
Authors: Smulders C. M. P. A.
Published: April 2009

Let a1… adbe a basis of the Lie algebragof a connected Lie groupGand letMbe a Lie subgroup of,G. Ifdxis a non-zero positive quasi-invariant regular Borel measure on the homogeneous spaceX = G/MandS: X × G→ C is a continuous cocycle, then under a rather weak condition ondxandSthere exists in a natural way a (weakly*) continuous representationUofGinLp(X;dx) for allp ε[1,].

Let Aibe the infinitesimal generator with respect toUand the directionai, for alli∈ { 1… d}. We considern–th order strongly elliptic operatorsH= ΣcαAαwith complex coefficientscα. We show that the semigroupSgenerated by the closure ofHhas a reduced heat kernelKand we derive upper bounds forkand all its derivatives.
Decomposability of finite rings
Authors: Mainwaring Diane , Pearson K. R.
Published: April 2009

Let p be a prime, let R be a finite p-ring with identity and suppose that the radical of R has pm elements. If R is indecomposable as a ring then there are at most m+1 minimal ideals in R/Rad R.

1980 Mathematics subject classification (Amer. Math. Soc): 16 A 44.
On a theorem of Ambrose
Authors: Wraith David J.
Published: April 2009

A Riccati inequality involving the Ricci curvature can be used to deduce many interesting results about the geometry and topology of manifolds. In this note we use it to present a short alternative proof to a theorem of Ambrose.
Elements of minimal breadth in finite p-groups and lie algebras
Authors: Mann Avinoam
Published: April 2009

Let G be a finite p-group, and let M(G) be the subgroup generated by the non-central conjugacy classes of G of minimal size. We prove that this subgroup has class at most 3. A similar result is noted for nilpotent Lie algebras.
Subnomality in soluble minimax groups
Authors: McCaughan D. J.
Published: April 2009

A subgroup H of a group G is said to be subnormal in G if there is a finite chain of subgroups, each normal in its successor, connecting H to G. If such chains exist there is one of minimal length; the number of strict inclusions in this chain is called the subnormal index, or defect, of H in G. The rather large class of groups which have an upper bound for the subnormal indices of their subnormal subgroups has been inverstigated to same extent, mainly with a restriction to solublegroups — for instance, in [10] McDougall considered soluble p-groups in this class. Robinson, in [14], restricted his attention to wreath products of nilpotent groups but extended his investigations to the strictly larger class of groups in which the intersection of any family of subnormal subgroups is a subnormal subgroup. These groups are said to have the subnormal intersection property.
A system of PDEs for Riemannian spaces
Authors: Senarath Padma , Thornley Gillian , van Brunt Bruce
Published: April 2009

Matsumoto [10] remarked that some locally projectively flat Finsler spaces of non-zero constant curvature may be Riemannian spaces of non-zero constant curvature. The Riemannian connection, however, must be metric compatible, and this requirement places restrictions on the geodesic coefficients for the Finsler space in the form of a system of partial differential equations. In this paper, we derive this system of equations for the case where the geodesic coefficients are quadratic in the tangent space variablesyi, and determine the solutions. We recover two standard Remannian metrics of non-zero constant curvature from this class of solutions.
Semi-stable and stable cacti
Authors: McAvaney K. L.
Published: April 2009

Holton (1973) introduced the following concept. A graph G is semi-stable if there exists a point v in G for which Γ(Gv) = Γ(G)v: where Γ(G) is the automorphism group of G, Gv is the graph G with v deleted and Γ(G)v is the subgroup of Γ(G) that fixes v. We say G is semi-stable at v. A partial stabilising sequence in Gs1, v2,…vk of its points such that Γ(G)v1v2…vi ═ Γ(Gv1v2…v1) for i = 1,2,…,k. If there exists a partial stabilising sequence in G for which k equals the number of points of G then G is said to be stable (Holton (1973a)). Most natation and terminology in what follows is explained in Harary (1969).
Principal solutions of positive linear Hamiltonian systems
Authors: Hinton Don
Published: April 2009

The Hamiltonian system Y′ = BY + CZ, Z′ = – AY – B*Z is considered where the coefficients are continuous on I = [a, ∞, C = C* ≧ 0, and A = A* ≦ 0. A solution (Y, Z) satisfying Y*Z = Z*Y is defined to be principal (coprincipal) provided that (i) Y−1 exists on I (Z−1 exists on I) and (ii) as t→∞ ( as t → ∞). Three conditions are given which are separtely equivalent to the condition that a solution is principal iff it is coprincipal. For a self-adjoint scalar operator L of order 2n, this problem is related to the deficiency index problem and to a problem of Anderson and Lazer (1970) which concerns the number of lnearly independent solutions of L (y) =0 satisfying y(k) ∈ (a, ∞) (k = 0, …, n).
Subgroups of finitely presented metabelian groups of finite rank
Authors: Boler James
Published: April 2009

Let G be a finitely generated metabelian group whose derived group G′ has finite rank. It is shown that G can be embedded in a finitely presented metabelian group H with H′ of finite rank.
On the measure of totally real algebraic integers
Authors: Smyth C. J.
Published: April 2009

For totally real algebraic integers x of degree D(x), we examine the stucture of the set of values M(x)1D(x). where M(x) is the measure of x. We find a small limit point λ of this set, and show that the set is everywhere dense in (λ x).
Ideals of free inverse semigroups
Authors: Munn W. D.
Published: April 2009

It is shown that no proper ideal of a free inverse semigroup is free and that every isomorphism between ideals is induced by a unique automorphism of the whole semigroup. In addition, necessary and sufficient conditions are given for two principal ideals to be isomorhic.
Semi-Markov processes on a general state space: α-theory and quasi-stationarity
Authors: Arjas E. , Nummelin E. , Tweedie R. L.
Published: April 2009

By amalgamating the approaches of Tweedie (1974) and Nummelin (1977), an α-theory is developed for general semi-Markov processes. It is shown that α-transient, α-recurrent and α-positive recurrent processes can be defined, with properties analogous to those for transient, recurrent and positive recurrent processes. Limit theorems for α-positive recurrent processes follow by transforming to the probabilistic case, as in the above references: these then give results on the existence and form of quasistationary distributions, extending those of Tweedie (1975) and Nummelin (1976).
Identities and left cancellation in distributively generated near-rings
Authors: John David
Published: April 2009

Given a semigroup S, we define {N(S), +, ·} to be the ‘free’ distributively generated near-ring. Since all words in N(S) can be expressed as the sum and difference of elements of S, we are able to define a length function on the words of N(S). The following theorems then follow: Theorem 1. N(S) contains a multiplicatice identity e if and only if e ∈ S. Theorem 2. If S is the free semigroup in the cariety of all semi—groups then N(S) is left cancellatice.
Minkowski's fundamental inequality for reduced positive quadratic forms(II)
Authors: Barnes E. S.
Published: April 2009

A convex polytope D (α) was defined in Barnes (1978) as the set of Minkowski-reduced forms with prescribed diagonal coefficients α1, α2,…αn. A local minimum of the determinant D(f) over D(α) must occur at a vertex of D(α). Here a criterion is obtained for a given vertex to provide a local minimum, completely analogous to Voronoï's criterion for a perfect form to be extreme.
A series of BIB designs
Authors: Sinha Kishore
Published: April 2009

A series of balanced incomplete block (BIB) designs with parameters , is constructed, where , w any integer.
Some finitely based varieties of rings
Authors: Evans Trevor
Published: April 2009

The results in this paper are consequences of an attempt many years ago to extend to loops some form of the theorem of Lyndon [12] that any nilpotent group has finitely based identities. Having failed in this, we looked for other algebras for which a similar approach might work. The algebra has to belong to a variety in which finitely generated algebras are finitely related and we must be able to bound the number of variables needed in a basis. Commutative Moufang loops, because of the extensive commutator calculus available (Bruck, [4]), provide one example (Evans, [6]). Here we give two examples from rings, namely associative rings satisfying xn = x (more generally, satisfying an identity x2 · p(x) = x) and nilpotent (non-associative) rings. We are also able to extend some results of Higman [9] on product varieties and we show that for associative rings the product of a nilpotent variety and a finitely based bariety is finitely based.
Charles Ernest Weatherburn
Authors: Gamblen F.
Published: April 2009

“I have received from Einstein himself a copy of his recent papers, and I shall always consider it an honour to have done so. But still I am at liberty to say that the theory does not impress me as holding the secret of the laws of nature. I feel much more disposed to trust the classical theory, modified perhaps so as to recognise the principle of equivalence. Einstein's theory is wonderful—wonderful in its complexity and in the mathematical difficulties overcome. Newton's is even more wonderful—wonderful in its simplicity and in its agreement with nearly all the experimental evidence of two and a half centuries. I cannot agree with those who would make nature more akin to the complex than to the simple.”
Free modules over some modular group rings
Authors: Carlson Jon F.
Published: April 2009

Let K be a field and G a finite group with subgroup H. We say that (G H) is a K-free pair if whenever M is a finitely generated KG-module whose restriction MH is a free KH-module, then M is a free KG-module. In this paper pairs of groups with this property will be investigated.
Locally soluble groups with min-n.
Authors: Heineken H. , Wilson J. S.
Published: April 2009

It was shown by Baer in [1] that every soluble group satisfying Min-n, the minimal condition for normal subgroups, is a torsion group. Examples of non-soluble locally soluble groups satisfying Min-n have been known for some time (see McLain [2]), and these examples too are periodic. This raises the question whether all locally soluble groups with Min-n are torsion groups. We prove here that this is not the case, by establishing the existence of non-trivial locally soluble torsion-free groups satisfying Min-n. Rather than exhibiting one such group G, we give a general method for constructing examples; the reader will then be able to see that a variety of additional conditions may be imposed on G. It will follow, for instance, that G may be a Hopf group whose normal subgroups are linearly ordered by inclusion and are all complemented in G; further, that the countable groups G with these properties fall into exactly isomorphism classes. Again, there are exactly isomorphism classes of countable groups G which have hypercentral nonnilpotent Hirsch-Plotkin radical, and which at the same time are isomorphic to all their non-trivial homomorphic images.
Some Properties of stochastic compactness
Authors: Maller R. A.
Published: April 2009

The aim of this paper is to show that some of the known properties of distributions in the domain of attraction of a stable law have counterparts for distributions which are stochastically compact in the sense of Feller. This enables us to unify the ideas of Feller and Doeblin, who first studied the concept of stochastic compactness, and give new characterizations of stochastic compactness and the domain of attraction of the normal distribution.
Ramsey numbers for certain k-graphs
Authors: Burr Stefan A. , Duke Richard A.
Published: April 2009

We are interested here in the Ramsey number r(T, C), where C is a complete k-uniform hypergraph and T is a “tree-like” k-graph. Upper and lower bounds are found for these numbers which lead, in some cases, to the exact value for r(T, C) and to a generalization of a theorem of Chváta1 on Ramsey numbers for graphs. In other cases we show that a determination of the exact values of r(T, C) would be equivalent to obtaining a complete solution to existence question for a certain class of Steiner systems.
Relations, homogeneity and group quotients
Authors: Warner M. W.
Published: April 2009

A set with a relation is isomorphic to a group quotient under the condition described as weak homogeneity, and to the quotient of a group with relation preserved by right and left translations if the homogeneity is strengthened. A method of constructing these group quotients and, furthermore, all such very homogeneous spaces, is described and an illustrative example given.
On groups generated by three-dimensional special unitary groups I
Authors: Phan Kok-Wee
Published: April 2009

This paper is the result of our attempt to carry over the method of our characterization of the linear groups PSL(n, q) (Phan (1972)) to that of the unitary groups PSU (n, q). It appears desirable to have available a result on the generation of the unitary groups more closely related to the splitting of the underlying hermitian spaces of these groups into orthogonal direct sum of anisotropic lines. Another motivation comes also from our work on the unitary groups in which we obtained a fusion pattern different from that of the unitary group when the dimension of the underlying space is eight. This fusion pattern produces a group whose structure is not immediately apparent. In order to identify this group, the result of this paper appears to be a necessary step.
Certain semigroups embeddabable in topological groups
Authors: Dzinotyiweyi Heneri A. M.
Published: April 2009

In this paper we study commutative topological semigroups S admitting an absolutely continuous measure. When S is cancellative we show that S admits a weaker topology J with respect to which (S, J) is embeddable as a subsemigroup with non-empty interior in some locally compact topological group. As a consequence, we deduce certain results related to the existence of invariant measures on S and for a large class of locally compact topological semigroups S, we associate S with some useful topological subsemigroup of a locally compact group.
Topological properties of the scale of a uniform space
Authors: Richardson G. D. , Wolf E. M.
Published: April 2009

Let (S. U) be a uniform space. This space can be embedded in a complete, uniform lattice called the scale of (S. U). We prove that the scale is compact if and only if S is finite or U = {S × S}. We prove that this statement remains true if compact is replaced by countably compact, totally bounded. Lindelof, second countable, or separable. In the last section of this paper, we investigate the cardinality of the scale and the retracted scale.
Quotient bounded elements in locally convex algebras
Authors: Bhatt Subhash J.
Published: April 2009

The quotient bounded and the universally bounded elements in a calibrated locally convex algebra are defined and studied. In the case of a generalized B*-algebra A, they are shown to form respectively b* and B*-algebras, both dense in A. An internal spatial characterization of generalized B*-algebras is obtained. The concepts are illustrated with the help of examples of algebras of measurable functions and of continuous linear operators on a locally convex space.
Inverse semigroups with certain universal properties
Authors: Lausch H.
Published: April 2009

Free groups and free inverse semigroups are characterized by certain properties which are usually called universal. Category-theoretically they arise from left adjoints of forgetful functors which assign to each structure its underlying set. The purpose of this note is to give a construction of a wealth of inverse semigroups with certain universal properties which subsumes free groups and free inverse semigroups and incidentally elucidates some well-known constructions of free inverse semigroups (Scheiblich (1972)). Categorytheoretical terminology will be freely taken from (Mac Lane (1971)).
A Criterion for automorphisms of certain groups to be inner
Authors: Dyer Joan L.
Published: April 2009

Let R be a normal subgroup of the free group F, and set G = F/[R, R]. We assume that F/R is a torsion-free group which is either solvable and not cyclic, or has a non-trivial center and is not cyclic-by-periodic. Then any automorphism of G whose restriction to R/[R, R] is trivial is an inner automorphism, determined by some element of R/[R, R]. This result extends a theorem of Šmel'kin (1967).
A Fourier inversion method for the estimation of a density and its derivatives
Authors: Blum J. , Susarla V.
Published: April 2009

During the last two decades a variety of methods have been developed for the problem of estimation of unknown density f wrt Lebesgue measure and its vth dervative g(= f(v)) using i.i.d random variables X1, …, Xn when X1 ∼ f. For example, see Wegman (1972). In almost all the papers on the estimation of f(x) or g(x), various authors assumed the existence of derivatives of f of order r(>v) at x to obtain rates for the mean-square convergences and other desirable properties for their estimators. Here it is shown that if

then estimators gˇ(x) can be constructed for which E[(gˇ(x)-g(x))2] = O(n−(2a-δ)/(2a+2v+1)) for anygiveng δ > 0. Simiar statements hold for almost sure convergence of gˇ(x). It can also be shown that (gˇ(x1), gˇ(x2)) is asymptotically bivariate normal under certain conditions for x1, ≠ x2. If (Al) is satisfied with a ≦ l, then our estimators have all the desirable properties while other methods are not applicable in this situation since they require differentiability conditions on g. (For example, see Susarla and Kumar (1975) and its references.) Our estimators are defined by using the inversion theorem for some absolutely integrable characteristic functions. The motivation for our estimators is given in O'Bryan and Susarla (1975, 76) and Susarla and O'Bryan (1975).
On doubly transitive permutation groups of degree prime squared plus one
Authors: Chillag David
Published: April 2009

A doubly transitive permutation group of degreep2+ 1, pa prime, is proved to be doubly primitive forp≠ 2. We also show that if such a group is not triply transitive then either it is a normal extension ofP S L(2,p2) or the stabilizer of a point is a rank 3 group.
Generalized products of weakly m — n compact spaces, I
Authors: Dissanayake U. N. B.
Published: April 2009

A topological space X is said to be weakly-Lindelöf if and only if every open cover of X has a countable sub-family with dense union. We know that products of two Lindelöf spaces need not be weakly-Lindelöf. In this paper we obtain non-trivial sufficient conditions on small sub-products to ensure the producitivity of the property weakly-Lindelöf with respect to arbitrary products.
Reducing two person, zero sum games with underlying symmetry
Authors: Pearson K. R.
Published: April 2009

We consider two person, zero sum games with several symmetries. Where such symmetries are present there is a group acting on the strategies of the game. We show how to use this action to produce a reduced game with a smaller matrix, but having the same value as the original game, and how to obtain optimal strategies for the original game from optimal strategies of the reduced game. An analysis of a simplified version of the popular game Mastermind is given to illustrate the theory developed.
On a class of near-rings
Authors: Santhakumari C.
Published: April 2009

It is well known that in a commutative Noetherian ring with identity every ideal has a representation as a finite intersection of primary ideals. The object of the present paper is to generalize this result to a class of near-rings called Q-near-rings which includes rings with dense quasi-centre and consequently all commutative rings.
Conformal variation of a hypersurface of constant mean curvature in an einstein space
Authors: Smith Geoffrey Howard
Published: April 2009

Let a compact orientable manifold be immersed as a hypersurface of constant mean curvature in an Einstein space. It is shown that the immersion is totally umbilical if and only if there exists a conformal variation of the immersion whose variation vector is nowhere tangential to the hypersurface.
Almost distributive lattices
Authors: Swamy U. Maddana , Rao G. C.
Published: April 2009

The concept of ‘Almost Distributive Lattices’ (ADL) is introduced. This class of ADLs includes almost all the existing ring theoretic generalisations of a Boolean ring (algebra) like regular rings, P-rings, biregular rings, associate rings, P1-rings, triple systems, etc. This class also includes the class of Baer-Stone semigroups. A one-to-one correspondence is exhibited between the class of relatively complemented ADLs and the class of Almost Boolean Rings analogous to the well-known Stone's correspondence. Many concepts in distributive lattices can be extended to the class of ADLs through its principal ideals which from a distributive lattice with 0. Subdirect and Sheaf representations of an ADL are obtained.
Some determinants that should be better known
Authors: van der Poorten A. J.
Published: April 2009

Since it is no longer fashionable to publish per se results which depend only on algebraic manipulation, many useful and complicated results of quite general interest languish unseen hidden as lemmata in specialist papers. In particular, the theory of transcendental numbers is rich in ingenious techniques for evaluating determinants. These techniques are apparently not well known even to workers in the field of transcendental numbers, let alone to researchers on other areas where the results might find application. This note accordingly discusses a variety of interesting results on determinants and is to be viewed as an appendix to the encyclopaedic volumes of Muir (1911, 1933) which the researcher might approach in order to obtain information in this area. For reasons of motivation a brief mention is made of the context in which the determinants arise.
Reducibility for real analytic quasi-periodic linear systems
Authors: O'Brien G. C.
Published: April 2009

In this paper we examine the linear differential equations x′ = Ax + P(φ)x, φ′ = ω where x ∈ Rn, φ ∈ Rm, A and ω are constant and P(φ) is real analytic and periodic in φ. We use the method of accelerated convergence to overcome the small divisors problem and reduce this system to the system y′ = By, φ′ = ω with constant coefficients.

This problem has already been examined by Mitropolśki and Samolenko but the calculations and details in their work are formidable and difficult to follow. Besides being simpler our method provides more precise estimates at all stages and can be extended to the differentiable case to provide a significant improvement over previous results.
On strongly exposing functionals
Authors: Lau Ka-Sing
Published: April 2009

Let X be a real Banach space and let K be a bounded closed convex subset of X. We prove that the set of strongly exposing functions K^ of K is a (norm) dense G8 in X* if and only if for any bounded closed convex subset C such that K⊄C, there exists a point x in K which is a strongly exposed point of conv (C ∪ K). As an application, we show that if X* is weakly compact generated, then for any weakly compact subset K in X, the set K^ is a dense G8 in X*.
Abelian Unipotent Subgroups of Finite Unitary and Symplectic Groups
Authors: Wong W. J.
Published: April 2009

If G is the unitary group U(V) or the symplectic group Sp(V) of a vector space V over a finite field of characteristic p, and r is a positive integer, we determine the abelian p-subgroups of largest order in G whose fixed subspaces in V have dimension at least r, with the restriction that we assume p ≠ 2 in the symplectic case. In particular, we determine the abelian subgroups of largest order in a Sylow p-subgroup of G. Our results complement earlier work on general linear and orthogonal groups.
Sur les mesures coniques localisables
Authors: Becker Richard
Published: April 2009

Let X be a weakly complete proper cone, contained in an Hausdorff locally convex space E, with continuous dual E′. A positive linear form on the Risz space of functions on X generated by E′ is called a conical measure on X. Let M+ (X) be the set of all conical measures on X. G. Choquet asked the question: when is every conical measure on X given by a Radon measure on (X\0)? Let L be the class of such X. In this paper we show that the fact that X ∈ L only depends, in some sense, on the cofinal subsets of the space E′|x ordered by the order of functions on X. We derive that X ∈ L is equivalent to M+ (M) ∈ L. We show that is closed under denumerable products.
The lattice of stable marriages and permutations
Authors: Hwang J. S.
Published: April 2009

Recently, we have introduced the notion of stable permutations in a Latin rectangle L(r×c) of r rows and c columns. In this note, we prove that the set of all stable permutations in L (r×c) forms a distributive lattice which is Boolean if and only if c ≤ 2.