bluuebunny/crossref_metadata_2025 huggingface dataset:
Characterizations of Finite Projective and Affine Spaces
Authors: Kantor William M.
Published: December 2010

A well-known result of Dembowski and Wagner (4) characterizes the designs of points and hyperplanes of finite projective spaces among all symmetric designs. By passing to a dual situation and approaching this idea from a different direction, we shall obtain common characterizations of finite projective and affine spaces. Our principal result is the following.

Theorem 1.A finite incidence structure is isomorphic to the design of points and hyperplanes of a finite projective or affine space of dimension greater than or equal to4if and only if there are positive integers v, k, and y, with μ> 1and(μ– l)(v — k) ≠ (k—μ)2such that the following assumptions hold.

(I)

Every block is on k points, and every two intersecting blocks are on μ common points.

(II)

Given a point and two distinct blocks, there is a block containing both the point and the intersection of the blocks.

(III)

Given two distinct points p and q, there is a block on p but not on q.

(IV)

There are v points, and v– 2 ≧k>μ.
Generalization of Hölder's Theorem to Ordered Modules
Authors: Viswanathan T. M.
Published: December 2010

Hölder's theorem on archimedean groups states:

An ordered (abelian) group G is order isomorphic to an ordered subgroup of the ordered group R of real numbers if and only if it is archimedean.

We comprehend this theorem in the following setting: G is a Z-module and Ris the completion with respect to the open interval topology of the ordered field Q; Qitself is the ordered quotient field of the ordered domain Z.

Rephrasing the situation, we raise the following question: We start with a fully ordered domain A,let Kbe its ordered quotient field. We endow Kwith the open interval topology and consider , the topological completion of K. Is it possible to impose a compatible order structure on and if this can be done, when can we say that an ordered A-module Mis order isomorphic to an ordered A-submodule of ? In Theorem 3.1, we obtain a set of necessary and sufficient conditions for this isomorphism to hold.
Topologies on Generalized Inner Product Spaces
Authors: Prugovečki Eduard
Published: December 2010

In the present note we introduce a straightforward algebraic generalization of inner product spaces, which we appropriately name generalized inner product (GIP) spaces. In the same fashion in which different topologies :an be introduced in inner product spaces, adequate topologies can be introduced in GIP spaces in such a manner that topological vector spaces are obtained. We enumerate and derive some fundamental properties of different topologies in GIP spaces, having primarily in mind their possible later application to quantum physics.

The desirability of having in quantum physics more general structures than Hilbert spaces (in which quantum mechanics is usually formulated) is suggested by Dirac's formalism (2), which deals with “unnormalizable” vectors. Unfortunately, although this formalism is very elegant from the point of view of the facili ty of dealing with its symbolism, it completely lacks in mathematical rigour.
On Certain Discrete Inequalities involving Partial Sums
Authors: Beesack Paul R.
Published: December 2010

Our aim in this paper is to prove inequalities of the form

1

or

2

for all real values of the parameters α, β and all non-negative (in some cases all positive) xi. Obviously, an is finite in all cases, and we shall show that An is finite if α and α + β are both non-negative. In all cases, we obtain sharp values of the constants an, An (when finite), as well as bounds for these constants, and their behaviour as n → ∞. In case a < 0, we naturally consider only positive xi, otherwise the xi may be non-negative. Although we always write xi ≧ 0 in the following, this should be read as xi > 0 in case α < 0; similar remarks apply to the parameter t introduced below.
Idempotent Ideals in Perfect Rings
Authors: Michler Gerhard
Published: December 2010

All rings considered in this note have an identity element, and all R-modules are unitary.

Bass (2) defined a left perfect ring as a ring R satisfying the minimum condition on principal right ideals. A commutative ring R is perfect if and only if R is a direct sum of finitely many local rings whose radicals are T-nilpotent. Therefore, the commutative perfect rings with finite global projective dimension are just the direct sums of finitely many commutative fields, and hence they trivially satisfy the minimum condition for all ideals. However, in the non-commutative case, even hereditary perfect rings are not necessarily right or left artinian (cf. Example 3.4).

Each left perfect ring R has only finitely many idempotent (two-sided) ideals (Corollary 2.3), where the ideal X of R is called idempotent, if X = X2. Hence, it makes sense to consider minimal idempotent idealsof the left perfect ring R, i.e., ideals of R which are minimal in the set of all idemponent ideals of R.
A Characterization of Chainable Continua
Authors: Fugate J. B.
Published: December 2010

In this paper, certain results of Bing (1) and myself (2) are extended. It is well-known that a chainable compact metric continuum must be a-triodic (contain no triods), hereditarily unicoherent (the common part of each two subcontinua is connected), and each subcontinuum must be chainable. Our principal result states that a compact metric continuum M is chainable if and only if M is a-triodic, hereditarily unicoherent and each indecomposable subcontinuum of M is chainable. Some condition on the indecomposable subcontinua of M seems essential, if we consider the dyadic solenoid, 5, which is indecomposable, a-triodic and hereditarily unicoherent. Indeed, each proper subcontinuum of S is an arc. However, S is not chainable, since it cannot be embedded in the plane.
On Planar Continuous Families of Curves
Authors: Zamfirescu Tudor
Published: December 2010

In a recent paper (3), Grünbaum has found a general and unifying setting for a number of properties of some special lines associated with a planar convex body. Besides various interesting results, two conjectures are stated and two kinds of convexity and polygonal connectedness are introduced.

In the present paper, we shall prove one of Grünbaum's conjectures (§ 3, Theorem 1); we consider the other in § 4 and establish some related results in §§ 5 and 6. Six-partite problems are studied in this general setting (§ 7) and a question raised by Ceder (2) is answered. We give a generalization of the notion of a continuous family of curves in § 8, and discuss some new kinds of connectedness in § 9.
A Lower Bound for the Scholz-Brauer Problem
Authors: B. Stolarsky Kenneth
Published: December 2010

In (6)Scholz asked if the inequality

1.1

held for all positive integers q, where l(n)is the number of multiplications required to raise xto the nth power (a precise definition of l(n)in terms of addition chains is given in § 2). Soon afterwards, Brauer (2) showed, among other things, that l(n) ∼(log n)/(log2). This suggests the problem of Calculating

1.2

It can be deduced from (2) that θ≦ 1. If θ <1, (1.1) follows immediately for infinitely many q.My main result,Theorem 5 of § 4, merely shows that θ is slightly larger than ⅓.Actually, I know of no case where (1.1) is not in fact an equality; a tedious calculation verifies this for 1 ≦ q≦ 8.
A First Approximation to {X, Y}
Authors: Samuel Brown Benson
Published: December 2010

If ℭ and ℭ′ are classes of finite abelian groups, we write ℭ + ℭ′ for the smallest class containing the groups of ℭ and of ℭ′. For any positive number r, ℭ < r is the smallest class of abelian groups which contains the groups Zp for all primes p less than r.

Our aim in this paper is to prove the following theorem.

THEOREM. Iƒ ℭ is a class of finite abelian groups and

(i) πi(Y) ∈ℭ for i < n,

(ii) H*(X; Z) is finitely generated,

(iii) Hi(X;Z)∈ ℭ for i > n + k,

Then

This statement contains many of the classical results of homotopy theory: the Hurewicz and Hopf theorems, Serre's (mod ℭ) version of these theorems, and Eilenberg's classification theorem. In fact, these are all contained in the case k = 0.
Remarque Sur La Moyenne Arithmetique de Fonctions Univalentes Convexes
Authors: Labelle G. , Rahman Q. I.
Published: December 2010

Soient f(z) et g (z) deux fonctions univalentes normalisées convexes dans |z|. Nous avons démontré récemment (3) que leur moyenne géométrique F = (fg)1/2 applique le disque sur un domaine convexe et l'estimation est précise.

Nous voulons ici considérer le même problème pour la moyenne arithmétique des deux fonctions f(z) et g (z).

THÉORÈME. Soient f(z) et g(z) deux fonctions univalentes normalisées convexes dans |z|. Si K représente la plus petite racine positive de l'équation

1

alors la moyenne arithmétique applique |z| < κ sur un domaine convexe.

Il nous faut, pour la preuve du théorème, le lemme suivant.

LEMME. Si la function
The Asymptotic behaviour Of μ(z, β, α)
Authors: Wyman M. , R.Wong
Published: December 2010

The function μ(z, β, α) defined by

where

plays an important role in Volterra's theory of convolution-logarithms, and also in the Paley-Wiener inversion formula for the Laplace transformation. These and other properties of pi are briefly described in (3).

Our aim in this paper is to find the asymptotic behaviour of μ as |z| < ∞, a result which, as far as we are aware, has not been obtained. It is of interest to note that the procedure to be used also gives, with some minor modification, the asymptotic behaviour of μ as z → 0, a result that is well known (3, p. 219). When this behaviour becomes known, it becomes possible to write a significant generalization of Watson's Lemma.

In 1906, Barnes (1) published a significant paper containing many asymptotic results of major importance. It is of passing interest to note that this paper contains a general theorem from which the result of Watson's Lemma, published in 1918, can easily be obtained. It would seem that this latter result is misnamed in mathematical literature, and might well be called Barnes’ Lemma.
The Generalized Kulikov Criterion
Authors: Megibben Charles
Published: December 2010

In 1941, Kulikov (5) showed that a p-primary abelian group G is a direct sum of cyclic groups if and only if G is the union of an ascending sequence of subgroups each of which has a finite bound on the heights of its elements. An easy reformulation of the Kulikov criterion is: A p-primary abelian group G is a direct sum of cyclic groups if and only if G[p] = ⴲn<ωSn where, for each n, the non-zero elements of Sn have precisely height n. This statement suggests the consideration of reduced p-groups G such that G[p] = ⴲa<λSα where, for each α, Sα – {0} ⊆ pαG – pα+lG. We shall call such p-groups summable (the term principal p-group has been used by Honda (4)). Recall that the length of a reduced p-group G is the first ordinal λ such that pλG = 0.
A Classification Of n-Abelian Groups
Authors: Alperin J. L.
Published: December 2010

The concept of an abelian group is central to group theory. For that reason many generalizations have been considered and exploited. One, in particular, is the idea of an n-abelian group (6). If n is an integer and n > 1, then a group G is n-abelian if, and only if,

(xy)n = xnyn

for all elements x and y of G. Thus, a group is 2-abelian if, and only if, it is abelian, while non-abelian n-abelian groups do exist for every n > 2.

Many results pertaining to the way in which groups can be constructed from abelian groups can be generalized to theorems on n-abelian groups (1; 2). Moreover, the case of n = p, a prime, is useful in the study of finite p-groups (3; 4; 5). Moreover, a recent result of Weichsel (9) gives a description of all p-abelian finite p-groups.
Indices of Function Spaces and their
Relationship to Interpolation
Authors: Boyd David W.
Published: December 2010

A special case of the theorem of Marcinkiewicz states that if T is a linear operator which satisfies the weak-type conditions (p, p) and (q,q), then T maps Lr continuously into itself for any r with p < r < q. In a recent paper (5), as part of a more general theorem, Calderόn has characterized the spaces X which can replace Lr in the conclusion of this theorem, independent of the operator T. The conditions which X must satisfy are phrased in terms of an operator S(σ) which acts on the rearrangements of the functions in X.

One of Calderόn's results implies that if X is a function space in the sense of Luxemburg (9), then X must be a rearrangement-invariant space.
Transversal Theory and Matroids
Authors: Welsh D. J. A.
Published: December 2010

In this paper I use techniques developed by Mirsky and Perfect (5) to generalize the extremely close relationship between transversal theory and the theory of matroids or independence structures. I extend in two directions a fundamental theorem of Rado (8) and use the techniques of Mirsky and Perfect to obtain easy proofs of known and unknown results about systems of representatives with repetition.

2. Basic concepts. In this section I review the results used subsequently. Throughout the paper, S will denote a finite set and A will denote the collection of subsets of S, {Ai i ∈ I}, where I is a finite index set. |K| will denote the cardinality of a set K and I use the notation
On Division Near-Rings
Authors: Steve Ligh
Published: December 2010

The following results (9, Exercise 26, p. 10; 1, Theorem 9.2; 8, Theorem III. 1.11) are known.

(A) Let R be a ring with more than one element. Then R is a division ring ifand only if for every a ≠0 in R, there exists a unique b in R such that aba = a.

(B) Let R be a near-ring which contains a right identity e ≠ 0. Then R is adivision near-ring if and only if it contains no proper R-subgroups.

(C) Let R be a finite near-ring with identity. Then R is a division near-ringif and only if the R-module R+ is simple.

In this paper we will show that (A) can be generalized to distributively generated near-rings. We also will extend (B) and (C) to a larger class of near-rings.
On Self-Adjoint Factorization of Operators
Authors: Radjavi Heydar
Published: December 2010

The main result of this paper is that every normal operator on an infinitedimensional (complex) Hilbert space ℋ is the product of four self-adjoint operators; our Theorem 4 is an actually stronger result. A large class of normal operators will be given which cannot be expressed as the product of three self-adjoint operators.

This work was motivated by a well-known resul t of Halmos and Kakutani (3) that every unitary operator on ℋ is the product of four symmetries, i.e., operators that are self-adjoint and unitary.

1. By “operator” we shall mean bounded linear operator. The space ℋ will be infinite-dimensional (separable or non-separable) unless otherwise specified. We shall denote the class of self-adjoint operators on ℋ by and that of symmetries by .
A Class of Three-Generator, Three-Relation, Finite Groups
Authors: Wamsley J. W.
Published: December 2010

Mennicke (2) has given a class of three-generator, three-relation finite groups. In this paper we present a further class of three-generator, threerelation groups which we show are finite.

The groups presented are defined as:

with α|γ| ≠ 1, β|γ| ≠ 1, γ ≠ 0.

We prove the following result.

THEOREM 1. Each of the groups presented is a finite soluble group.

We state the following theorem proved by Macdonald (1).

THEOREM 2. G1(α, β, 1) is a finite nilpotent group.

1. In this section we make some elementary remarks.
On the Addition of Relations in an Abelian Category
Authors: Hilton P. J. , Wu Y.-C.
Published: December 2010

In [1] the category of relations in an abelian category was studied. A relation from A to B appeared as an equivalence class of pairs of -morphisms (ϕ, ψ),

Moreover, this equivalence class may be written as the composite , where is canonically embedded in and is the image of Γ in the canonical involution on . Every equivalence class has an essentially unique minimal representative (ϕ0,ψ0), characterized by the property that

where

and if and only if the square

is exact.
Montel Algebras on the Plane
Authors: Meyers W. E.
Published: December 2010

The results of Rudin in [7] show that under certain conditions, the maximum modulus principle characterizes the algebra A (G) of functions analytic on an open subset G of the plane C (see below). In [2], Birtel obtained a characterization of A(C) in terms of the Liouville theorem; he proved that every singly generated F-algebra of continuous functions on C which contains no non-constant bounded functions is isomorphic to A(C) in the compact-open topology. In this paper we show that the Montel property of the topological algebra A (G) also characterizes it. In particular, any Montel algebra A of continuous complex-valued functions on G which contains the polynomials and has continuous homomorphism space M (A) homeomorphic to G is precisely A(G).
Axioms for Absolute Geometry. III
Authors: Rigby J. F.
Published: December 2010

This paper is a continuation of [1; 2]. In [2], I stated that I had been unable to construct examples of planes satisfying various conditions. Some of the examples that I have since constructed are given below. A discussion of one-dimensional absolute geometries, with examples, will be given in a separate paper. The relevant parts of [1] and [2] are [1, § 1, § 2 up to 2.4; 2, § 2]. We shall use the notation and terminology of [1; 2]; the axioms Cl*-C4* and C4** (referred to below) can all be found in [1].

We shall show here that spaces of dimension greater than 1 exist, both Archimedean and non-Archimedean, satisfying Cl*-C4*, in which not all points are isometric, and that C4** does not follow from Cl*-C4* in non- Archimedean geometries of dimension greater than 1.
On Group Rings
Authors: Coleman D. B.
Published: December 2010

Let R be a commutative ring with unity and let G be a group. The group ring RG is a free R-module having the elements of G as a basis, with multiplication induced by

The first theorem in this paper deals with idempotents in RG and improves a result of Connell. In the second section we consider the Jacobson radical of RG, and we prove a theorem about a class of algebras that includes RG when G is locally finite and R is an algebraically closed field of characteristic zero. The last theorem shows that if R is a field and G is a finite nilpotent group, then RG determines RP for every Sylow subgroup P of G, regardless of the characteristic of R.
Point-Transitive Actions by the Unit Interval
Authors: Borrego J. T. , DeVun E. E.
Published: December 2010

An action is a continuous function α: T × X → X, where T is a semigroup, X is a Hausdorff space, and α(t1, α(t2, x)) = α(t1,t2x) for all t1, t2 ∈ T and x ∈ X . If, for an action α, Q(α) = {x ∈ X| α(T × {x}) = X} is non-empty, then α is called a point-transitive action. Our aim in this note is to classify the point-transitive actions of the unit interval with the usual, nil, or min multiplications.

The reader is referred to [5; 7; 9] for information concerning the general theory of semigroups. All semigroups which are considered here are compact and Abelian and all spaces are compact Hausdorff. Actions by semigroups have been studied in [1; 3; 8].
On the Zeros of Functions with Derivatives in H1 and H∞
Authors: Wells James
Published: December 2010

Let {zk},0 < |zk| < 1, be a given sequence of points in the open unit disc D = {z: |z| < 1} and let E be its set of limit points on the unit circle T. In this note we consider the problem of finding conditions on the sequence {zk} which will ensure the existence of a function f analytic in D satisfying

(A)

and whose derivative f′ belongs to the Hardy class H1 or, alternatively, whose derivatives of all orders are bounded in D. We shall prove the following two theorems.

THEOREM 1. If

(1)

(2)

and

(3)

then there is a function f analytic in D which satisfies (A) and its derivative f′ belongs to H1.
On the Vanishing of a (G, σ) Product in a (G, σ) Space
Authors: Singh K.
Published: December 2010

In this paper, we shall construct a vector space, called the (G, σ) space, which generalizes the tensor space, the Grassman space, and the symmetric space. Then we shall determine a necessary and sufficient condition that the (G, σ) product of the vectors x1, x2, …, xn is zero.

1. Let G be a permutation group on I = {1, 2, …, n} and F, an arbitrary field. Let σ be a linear character of G, i.e., σ is a homomorphism of G into the multiplicative group F* of F.

For each i ∈ I, let Vi be a finite-dimensional vector space over F. Consider the Cartesian product W = V1 × V2 × … × Vn.

1.1. Definition. W is called a G-set if and only if Vi = Vg(i) for all i ∊ I, and for all g ∊ G.
Integral Formulas for Submanifolds and their Applications
Authors: Yano Kentaro
Published: December 2010

Liebmann [12] proved that the only ovaloids with constant mean curvature in a 3-dimensional Euclidean space are spheres. This result has been generalized to the case of convex closed hypersurfaces in an m-dimensional Euclidean space by Alexandrov [1], Bonnesen and Fenchel [3], Hopf [4], Hsiung [5], and Süss [14].

The result has been further generalized to the case of closed hypersurfaces in an m-dimensional Riemannian manifold by Alexandrov [2], Hsiung [6], Katsurada [7; 8; 9], Ōtsuki [13], and by myself [15; 16].

The attempt to generalize the result to the case of closed submanifolds in an m-dimensional Riemannian manifold has been recently done by Katsurada [10; 11], Kôjyô [10], and Nagai [11].
A New Bound for Nil U-Rings
Authors: Biggs R. G.
Published: December 2010

A U-ring is a ring in which every subring is a meta ideal. A meta ideal of a ring R is a subring I of R which lies in a chain of subrings,

with the properties:

(1) Iλ is an ideal of Iλ+1 for all λ < β;

(2) If α is a limit ordinal number, then Iα = ∪λ<αIλ.

Freidman [3] proved that every nil U-ring is a locally nilpotent ring. Since there are many locally nilpotent rings which are not U-rings, the class of locally nilpotent rings is not a very good bound for the class of nil U-rings. This paper establishes a new bound for nil U-rings based on a property of the multiplicative semigroup of the ring.
Coefficient Estimates for a Class of Star-Like Functions
Authors: Brannan D. A. , Clunie J. , Kirwan W. E.
Published: December 2010

In this note we continue the study, initiated in [1], of the class S*(α) of functions

(1.1)

that are analytic and univalent in the unit disc U and satisfy the condition

(1.2)

S*(1) is the frequently studied class of univalent star-like functions. For each α, S*(α) is a subclass of the class K(α) of close-to-convex functions of order α introduced by Pommerenke [4]. Properties of the class S*(α) proved useful in studying the coefficient behaviour of bounded univalent functions that are analytic and map U onto a convex domain [1]. In this note we investigate the problem of determining

but we are able to give only a partial solution.
A Local Ratio Theorem
Authors: Akcoglu M. A. , Chacon R. V.
Published: December 2010

Let Tt, t > 0, be a strongly continuous semigroup of positive linear contractions on the L1-space of a σ-finite measure space . We denote the integral ∫0tTsƒ ds, ƒ ∈ L1, by S0tƒ, which is denned as the limit of Riemann sums, in the norm topology of L1. It is easy to see that, given ƒ ∈ L1+, there exists a function F on the product space X× (0, ∞), measurable with respect to the usual product σ-field, such that for every t ≧ 0, ∫0tF(·, s) ds gives a representation of S0tƒ. We write S0tƒ(x) for ∫0tF(x, S) ds, with a fixed choice of F.
Hewitt Realcompactifications of Products
Authors: McArthur William G.
Published: December 2010

The Hewitt realcompactification vX of a completely regular Hausdorff space X has been widely investigated since its introduction by Hewitt [17]. An important open question in the theory concerns when the equality v(X × Y) = vX × vY is valid. Glicksberg [10] settled the analogous question in the parallel theory of Stone-Čech compactifications: for infinite spaces X and Y, β(X × Y) = βX × β Y if and only if the product X × Y is pseudocompact. Work of others, notably Comfort [3; 4] and Hager [13], makes it seem likely that Glicksberg's theorem has no equally specific analogue for v(X × Y) = vX × vY. In the absence of such a general result, particular instances may tend to be attacked by ad hoc techniques resulting in much duplication of effort.
A Structural Approach to Noether Lattices
Authors: Johnson E. W. , Johnson J. A. , Lediaev J. P.
Published: December 2010

In this paper we explore the extent to which embedding and isomorphism questions about a Noether lattice ℒ can be reduced to questions about simpler structures associated with ℒ.

In § 1, we use a variation of Dilworth's congruence approach [2] to associate a collection of semi-local Noether lattices with a given Noether lattice ℒ. We show that these semi-localizations determine ℒ to within isomorphism (Corollary 1.5); thus embedding and isomorphism questions about ℒ are largely reduced to the semi-local case.

In § 2, we consider the influence on a semi-local Noether lattice ℒ of the substructure ∂ ℒ consisting of all elements, all of whose associated primes are maximal. Here we find that if ∂ ℒ can be embedded in a semi-local Noether lattice ℒ*, then ℒ can be embedded in an extension of ℒ*.
On Open Extensions of Maps
Authors: Franklin S. P. , Kohli J. K.
Published: December 2010

In recent years there has been some interest in trying to improve the behaviour of maps by extending their domains. For example, in 1953 Whyburn showed that every map is the restriction of a compact map [17]. Similarly, Krolevec proved in 1967 that each locally perfect map can be extended to a perfect map [12] and in an as yet unpublished paper, Dickman obtained the same result for arbitrary maps [4]. In this paper we show that every map can be extended to an open map so that certain properties of the domain and range are preserved in the new domain. These results are then used to obtain analogues and improvements of recent theorems of Arhangel'skiï, Ĉoban, Hodel, and Proizvolov.
Induced Homotopy Equivalences on Mapping Spaces and Duality
Authors: Heath Philip R.
Published: December 2010

In this paper we present three results involving function constructions, together with an example which shows the usefulness of our considerations. The first result, Theorem 1, states roughly that homotopy is self related through function space constructions. Section 1 is devoted to the precise statement and proof of this theorem.

In order to clarify the results of § 2 we draw an analogy. Recall that under certain mild restrictions on the spaces involved, a map p: E → B is a fibration, and a map i: A → X is a cofibration, if and only if the induced maps p*: EZ → BZ and i*: ZX → ZA are fibrations for all spaces Z. The two theorems of § 2 give, in a more general category, analogous results for the notions of limit and colimit. Finally, in § 3 we suggest the importance of this type of result by using it, together with Theorem 1 and the above analogy, to deduce a result from its dual.
Conjecture of D. R. Hughes Extended to Generalized André Planes
Authors: Narayana Ra M. L.
Published: December 2010

In 1967 Foulser [1] defined a class of translation planes, called generalized André planes or λ-planes and discussed the associated autotopism collineation groups. While discussing these collineation groups he raised the following question:

“Are there collineations of a λ plane which move the axes but do not interchange them?”.

In this context, Foulser mentioned a conjecture of D. R. Hughes that among the André planes, only the Hall planes have collineations moving the axes without interchanging them. Wilke [4] answered Foulser's question partially by showing that the conjecture of Hughes is indeed correct. Recently, Foulser [2] has shown that possibly with a certain exception the Hall planes are the only generalized André planes which have collineations moving the axes without interchanging them. Our aim in this paper is to give an alternate proof, which is completely general, and is in the style of the original problem.
A Maximum Principle for Dirichlet-Finite Harmonic Functions on Riemannian Spaces
Authors: Kwon Y. K. , Sario L.
Published: December 2010

Representations of harmonic functions by means of integrals taken over the harmonic boundary ΔR of a Riemann surface R enable one to study the classification theory of Riemann surfaces in terms of topological properties of ΔR (cf. [6; 4; 1; 7]). In deducing such integral representations, essential use is made of the fact that the functions in question attain their maxima and minima on ΔR.

The corresponding maximum principle in higher dimensions was discussed for bounded harmonic functions in [3]. In the present paper we consider Dirichlet-finite harmonic functions. We shall show that every such function on a subregion G of a Riemannian N-space R attains its maximum and minimum on the set , where ∂G is the relative boundary of G in R and the closures are taken in Royden's compactification R*. As an application we obtain the harmonic decomposition theorem relative to a compact subset K of R* with a smooth ∂(K ∩ R).
Maps with Locally Flat Singular Sets
Authors: Timourian J. G.
Published: December 2010

A map f : M → N is topologically equivalent tog: X → Y if there exist homeomorphisms α: M → X and β: N → Y such that βfα–1 = g. At x ∊ M, f is locally topologically equivalent to g if, for every neighbourhood W ⊂ M of x, there exist neighbourhoods U ⊂ W of x and V of f(x) such that f|U: U → V is topologically equivalent to g.

1.1. Definition. Given a map f: M → N and x ∊ M, let F be the component of f–1(f(x)) containing x. The singular set Af is defined as follows: x ∊ M – Af if and only if there are neighbourhoods U of F and V of f(x) such that f| U: U → F is topologically equivalent to the product projection map of V × F onto V.
The Collineation Group of the Veblen-Wedderburn Plane of Order Nine
Authors: Stevenson Frederick W.
Published: December 2010

In this paper we prove that the order of the collineation group of the Veblen-Wedderburn plane of order nine is 311,040. This result was stated by Hall [3] in 1943 and proved by Pierce [9] in 1964. Hall assumed that there were 10 · 8 · 6 · 4 · 2 = 3840 collineations which permute points on the ideal line L and 81 collineations which leave L pointwise fixed. In 1955 André [1] verified this assumption. When it was realized that a harmonic homology with axis L had been overlooked, the number of central collineations with axis L doubled and hence the order of the collineation group became 3840 · 162 = 622,080. This latter figure has been assumed to be correct as recently as 1965 ([6]).

Here it is proved that there are 1920 collineations which move points on L and 162 collineations which leave L pointwise fixed, thus giving the figure 311,040. Pierce's proof of this fact is established from a different viewpoint.
A Spectral Theory for Duality Systems of Operators on a Banach Space
Authors: Stampfli J. G.
Published: December 2010

This note is an addendum to my earlier paper [8]. The class of adjoint abelian operators discussed there was small because the compatibility relation between the operator and the duality map was too restrictive. (In effect, the relation is appropriate for Hilbert space, but ill-suited for other Banach spaces where the unit ball is not round.) However, the techniques introduced in [8] permit us to readily obtain a spectral theory (of the Dunford type) for a wider class of operators on Banach spaces, as we shall show.

A duality system for the operator T is an ordered sextuple

(i) T is a bounded linear operator mapping the Banach space B into B,

(ii) ϕ is a duality map from B to B*. Thus, for x ∊ B, ϕ(x) = x* ∊ B*, where ‖x‖ = ‖x*‖ and x*(x) = ‖x‖2. The existence of ϕ follows easily from the Hahn-Banach Theorem.
On Existence of Distinct Representative Sets for Subsets of a Finite Set
Authors: Clements G. F.
Published: December 2010

LetSbe a finite set and letS1,S2, …,Stbe subsets ofS,not necessarily distinct. Does there exist a set of distinct representatives (SDR) forS1,S2, …,St? That is, does there exist a subset {a1,a2, …,at} ofSsuch thatai∊Si, 1 ≦i≦t, andai≠ajifi≠j? The following theorem of Hall [2;6, p. 48] gives the answer.

THEOREM.The subsets S1,S2, …,Sthave an SDR if and only if for each s,1 ≦s≦t, |Si1∪Si1∪ … ∪Sis| ≧sfor eachs-comhination {i1,i2, …,is}of the integers1, 2, …,t.

(Here and below, |A| denotes the number of elements inA.)
The Dimensions of Irreducible Tensor Representations of the
Orthogonal and Symplectic Groups
Authors: King R. C.
Published: December 2010

It is well known [13] that the irreducible tensor
representations (IRs) of the unitary, orthogonal, and symplectic groups in
an n-dimensional space may be specified by means of Young
tableaux associated with partitions (σ)s = (σ1, σ2, …, σp) with σ1 + σ2 + … + σp = s. Formulae for the dimensions of the
corresponding representations have been established [1; 8; 9;
13] in terms of the row lengths of these tableaux. It has been
shown [12] for the unitary group, U(n), that
the formula may be written as a quotient whose numerator is a polynomial in
n containing s factors, and whose
denominator is a number independent of n, which likewise
may be expressed as a product of s factors. This formula is
valid for all n.
Radical Regularity in Differential Rings
Authors: Gorman Howard E.
Published: December 2010

In [1], we discussed completions of differentially finitely generated modules over a differential ring R. It was necessary that the topology of the module be induced by a differential ideal of R and it was natural that this ideal be contained in J(R), the Jacobson radical of R. The ideal to be chosen, called Jd(R), was the intersection of those ideals which are maximal among the differential ideals of R. The question as to when Jd(R) ⊆ J(R) led to the definition of a class of rings called radically regular rings. These rings do satisfy the inclusion, and we showed in [1, Theorem 2] that R could always be “extended”, via localization, to a radically regular ring in such a way as to preserve all its differential prime ideals.

In the present paper, we discuss the stability of radical regularity under quotient maps, localization, adjunction of a differential indeterminate, and integral extensions.
Congruence-Preserving Isomorphisms of the Translation Group associated with a Translation Plane
Authors: Radó F.
Published: December 2010

Let II, II′ be projective translation planes, their sets of points, l∞, l∞′ the improper lines, and T, T′ the corresponding translation groups. T is an Abelian group, simply transitive on . The set of the subgroups Ts = {τ|τ ∈ T, cen τ = S} for all S ∈ l∞ is called the congruence of II (cen τ = centre of τ). An injective map , where , is said to be a collineation of when and three points in are collinear if and only if their images are collinear; the set of these φ is denoted by and for we write
Asymptotic Behaviour of Disconjugate nth Order Differential Equations
Authors: Willett D.
Published: December 2010

An ordered set (u1, …, un) of positive Cn(a, b)-solutions of the linear differential equation

1.1

will be called fundamental principal system on [a, b) provided that

1.2

and

1.3

A system (u1, …, un) satisfying just (1.2) will be called a principal system on [a, b). In any principal system (u1, …, un) the solution u1 will be called a minimal solution.
Uniform Finite Generation of the Isometry Groups of Euclidean and Non-Euclidean Geometry
Authors: Lowenthal Franklin
Published: December 2010

A connected Lie group H is generated by a pair of oneparameter subgroups if every element of H can be written as a finite product of elements chosen alternately from the two one-parameter subgroups. If, moreover, there exists a positive integer n such that every element of H possesses such a representation of length at most n, then H is said to be uniformly finitely generated by the pair of one-parameter subgroups. In this case, define the order of generation of H as the least such n; otherwise define it as infinity.

For the isometry group of the spherical geometry, or equivalently for the rotation group SO(3), the order of generation is always finite.
On the Lattice of Primitive Convergence Structures
Authors: Atherton C. R.
Published: December 2010

Let S be any set and denote by F(S) the collection of all fiters on S. The collection A(S) of all mappings from F(S) to 2s, 2s being ordered by the dual of its usual ordering, may be regarded as a product of complete Boolean algebras and is, therefore, a complete atomic Boolean algebra [4]. A(S) is called the lattice of primitive convergence structures on S. If q ∈ A(S) and , then is said to q-converge to a point x ∈ S if . The collection of all topologies on S may be identified with a subset of A(S); this subset of A(S) will be denoted by T(S). A more specialized class of primitive convergence structures, and one which properly contains T(S), is C(S), the subcomplete lattice of all convergence structures on S.
An Internal Characterization of Realcompactness
Authors: Harris Douglas
Published: December 2010

A space is realcompact if it is a homeomorph of a closed subspace of a product of real lines. Many external characterizations of realcompactness have appeared, but there seems to be no simple internal characterization. We provide such a characterization in terms of the existence of a collection of covers of a certain type and use it to examine realcompact extensions of a space and to characterize the Q-closure of a space in a compac tification.

A structure on X is a collection of covers of X that forms a filter under refinement ordering; the members of a structure are called gauges. A balanced refinement of a gauge a is a gauge β with cardinal not greater than that of a such that for each B ∈ β there is A ∈ α such that {A, X – B} is also a gauge; thus a balanced refinement is certainly a refinement.
Generalized Hughes Planes
Authors: Dembowski Peter
Published: December 2010

The projective planes discovered in 1957 by Hughes [3] were originally described by means of a nearfield F satisfying the following conditions:

(a) F is finite,

(b) the centre and kernel of F coincide,

(c) F is of rank 2 over its kernel.

(The definitions of these terms will be given in § 2; the terminology used throughout the paper is that of [1].)

Rosati [5] showed in 1960 that condition (a) is not necessary, thus constructing the first “infinite Hughes planes”. Condition (b), however, plays an essential part also in Rosati's work.

The aim in this paper is to show that condition (b) is superfluous as well. For the finite case, this has been remarked by Ostrom [4] without proof; here we shall show that a “generalized Hughes plane” can be constructed over any nearfield satisfying condition (c) only.
On the Primitivity of the Group Algebra
Authors: Rosenberg Alan
Published: December 2010

LetGbe a group andFa field of arbitrary characteristic. In [4] Kaplansky asks under what conditions isF[G] primitive, whereF[G] is the group algebra ofGoverF. We give some necessary conditions onGthatF[G] be primitive and propose a conjecture.

Definition.A ringRis primitive if it has a faithful irreducible right module.

The above should really be considered as a definition of right primitive. One can analogously define left primitive and the two properties are not equivalent. For our purposes, the two concepts are equivalent, for the group algebra possesses a nice involution.

If we assume thatF[G] is primitive, there are some immediate restrictions onG. First of allGcannot be Abelian since the only primitive commutative rings are fields. (I exclude of course the case whenGconsists of one element.)
An Extension of a Theorem of Janko on Finite Groups with Nilpotent Maximal Subgroups
Authors: Randolph John W.
Published: December 2010

Throughout this paper G will denote a finite group containing a nilpotent maximal subgroup S and P will denote the Sylow 2-subgroup of S. The largest subgroup of S normal in G will be designated by core (S) and the largest solvable normal subgroup of G by rad(G). All other notation is standard.

Thompson [6] has shown that if P = 1 then G is solvable. Janko [3] then observed that G is solvable if P is abelian, a condition subsequently weakened by him [4] to the assumption that the class of P is ≦ 2 . Our purpose is to demonstrate the sufficiency of a still weaker assumption about P.
A Note on Supersoluble Groups
Authors: Johnson D. L.
Published: December 2010

Let H be a subgroup of a group G (all groups considered throughout this article are finite); then H will be called primitive if the subgroup

is distinct from H. Such subgroups, which are also called meet-irreducible, arise naturally in connection with minimal permutation representations of groups and in other contexts; for example, every subgroup of a group G can be written as an intersection of primitive subgroups of G, and the set of all primitive subgroups of G is characterized by its minimality with respect to this property. While maximal subgroups are always primitive, most groups contain non-maximal subgroups which are primitive (see remark at end of article). Note that a subgroup H of an abelian group G is primitive if, and only if, G/H is cyclic of prime-power order.
The Asymptotic Ratio Set and Direct Integral Decompositions of a Von Neumann Algebra
Authors: Nielsen Ole A.
Published: December 2010

The fact that any von Neumann algebra on a separable Hilbert space has an essentially unique direct integral decomposition into factors means that there is a global as well as a local aspect to any partial classification of von Neumann algebras. More precisely, suppose that J is a statement about von Neumann algebras which is either true or false for any given von Neumann algebra. Then a von Neumann algebra is said to satisfy J globally if it satisfies J, and to satsify J locally if almost all the factors appearing in some (and hence in any) central decomposition of it satisfy J . In a recent paper [3], H. Araki and E. J. Woods introduced the notion of the asymptotic ratio set of a factor, and by means of this they made remarkable progress in the classification of factors.
Existence Theorems for Some Weak Abstract Variable Domain Hyperbolic Problems
Authors: Carroll Robert , State Emile
Published: December 2010

In this paper we prove some existence theorems for some weak problems with variable domains arising from hyperbolic equations of the type

1.1

where A = {A(t)} is, for example, a family of elliptic differential operators in space variables x = (x1, …, xn). Thus let H be a separable Hilbert space and let V(t) ⊂ H be a family of Hilbert spaces dense in H with continuous injections i(t): V(t) → H (0 ≦ t ≦ T < ∞). Let V’ (t) be the antidual of V(t) (i.e. the space of continuous conjugate linear maps V(t) → C) and using standard identifications one writes V(t) ⊂ H ⊂ V‘(t).
Subgroups of HNN Groups and Groups with one Defining Relation
Authors: Karrass A. , Solitar D.
Published: December 2010

HNN groups have appeared in several papers, e.g., [3; 4; 5; 6; 8]. In this paper we use the results in [6] to obtain a structure theorem for the subgroups of an HNN group and give several applications.

We shall use the terminology and notation of [6]. In particular, if K is a group and {φi} is a collection of isomorphisms of subgroups {Li} into K, then we call the group

1

the HNN group with base K, associated subgroups { Li,φi(Li)} and free part the group generated by t1, t2, …. (We usually denote φi(Li) by Mi or L–i.) The notion of a tree product as defined in [6] will also be needed.
Disconjugacy Criteria for Nonselfadjoint Differential Equations of Even Order
Authors: Kreith Kurt
Published: December 2010

Disconjugacy criteria have been established for linear selfadjoint differential equations of order 2n by Sternberg [4] and Ahlbrandt [1]. Such differential equations can be written in the form

1.1

where it is assumed that the coefficients are real and that Pn(x) ≠ 0. We shall be interested in nontrivial solutions v(x) of (1.1), which satisfy

1.2

for distinct points α and β. The smallest β> α such that (1.2) is satisfied nontrivially by a solution of (1.1), is denoted by μ1(α) and called the first conjugate point of x = α with respect to (1.1). If no such conjugate point exists we write μ1(α) = ∞, and say that (1.1) is disconjugate on [α, ∞).

The principal purpose of this paper is to generalize these disconjugacy criteria to the general linear nonselfadjoint differential equation of the form

1.3
Klein's Oscillation Theorem for Periodic Boundary Conditions
Authors: Howe A.
Published: December 2010

Multiparameter eigenvalue problems for systems of linear differential equations with homogeneous boundary conditions have been considered by Ince [4] and Richardson [5, 6], and more recently Faierman [3] has considered their completeness and expansion theorems. A survey of eigenvalue problems with several parameters, in mathematics, is given by Atkinson [1].

We consider the two differential equations:

1a

1b

where p1’(x), q1(x), A1(x), B1(x) and p2’(y), q2(y), A2(y), B2(y) are continuous for x ∈ [a1, b1] and y ∈ [a2, b2] respectively, and p1 (x) > 0(x ∈ [a1, b1]), p2(y) > 0 (y ∈ [a2, b2]), p1(a1) = p1(b1), p2(a2) = p2(b2). The differential equations (1) will be subjected to the periodic boundary conditions.

2a

2b

Let us consider a single differential equation
Linear Combinations of Univalent Functions with Complex Coefficients
Authors: Stump Robert K.
Published: December 2010

Let U be the class of all normalized analytic functions

where z ∈ E = {z : |z| < 1} and ƒ is univalent in E. Let K denote the sub-class of U consisting of those members that map E onto a convex domain. MacGregor [2] showed that if ƒ1 ∈ K and ƒ2 ∈ K and if

1

then F ∉ K when λ is real and 0 < λ < 1, and the radius of univalency and starlikeness for F is .

In this paper, we examine the expression (1) when ƒ1 ∈ K, ƒ2 ∈ K and λ is a complex constant and find the radius of starlikeness for such a linear combination of complex functions with complex coefficients.
Generators of Monothetic Groups
Authors: Armacost D. L.
Published: December 2010

A topological group G is called monothetic if it contains a dense cyclic subgroup. An element x of G is called a generator of G if x generates a dense cyclic subgroup of G. We denote by E(G) the set of generators of G; the complement of E(G) in G, consisting of the “non-generators” of G, we write as N(G) Throughout this paper we consider only locally compact abelian (LCA) groups satisfying the T2 separation axiom (note that a monothetic group is automatically abelian). In [1] certain problems of measurability concerning the set E(G) are discussed. In this paper we shall consider some algebraic and topological properties of the sets E(G) and N(G)
A Geometric Approach to the Heine-Borel Theorem
Authors: Killgrove R. B. , Frand Jason , Giles William , Bray Henry
Published: December 2010

In a topological plane with strong enough topological properties one can use [6] open triangular regions to define a base for the topology. Similarly, one can use these regions to define boundedness of a set. In this setting we show that in the absolute plane geometry, the holding of the Heine-Borel theorem is equivalent to every four points being contained in some such region and that this second condition is equivalent to the parallel postulate. Thus we give two new conditions equivalent to the parallel postulate.
On the Unitary Equivalence of Certain Classes of Non-Normal Operators. I
Authors: Tam P. K.
Published: December 2010

The following (so-called unitary equivalence) problem is of paramount importance in the theory of operators: given two (bounded linear) operators A1, A2 on a (complex) Hilbert space , determine whether or not they are unitarily equivalent, i.e., whether or not there is a unitary operator U on such that U*A1U = A2. For normal operators this question is completely answered by the classical multiplicity theory [7; 11]. Many authors, in particular, Brown [3], Pearcy [9], Deckard [5], Radjavi [10], and Arveson [1; 2], considered the problem for non-normal operators and have obtained various significant results. However, most of their results (cf. [13]) deal only with operators which are of type I in the following sense [12]: an operator, A, is of type I (respectively, II1, II∞, III) if the von Neumann algebra generated by A is of type I (respectively, II1, II∞, III).