bluuebunny/crossref_metadata_2025 huggingface dataset:
Minimal Decompositions of Complete Graphs into Subgraphs with Embeddability Properties
Authors: Beineke L. W.
Published: December 2010

Although the problem of finding the minimum number of planar graphs into which the complete graph can be decomposed remains partially unsolved, the corresponding problem can be solved for certain other surfaces. For three, the torus, the double-torus, and the projective plane, a single proof will be given to provide the solutions. The same questions will also be answered for bicomplete graphs.
On Simply Transitive Primitive Permutation Groups
Authors: Bercov R. D.
Published: December 2010

In (1) we considered finite primitive permutation groups G with regular abelian subgroups H satisfying the following hypothesis:

(*) H = A × B × C, where A is cyclic of prime power order pα ≠ 4, B has exponent pβ < pα, and C has order prime to p.

We remark that an abelian group fails to satisfy (*) (apart from the minor exception associated with the prime 2) if and only if it is the direct product of two subgroups of the same exponent.

We showed in (1) that such a group G is doubly transitive unless it is the direct product of two or more subgroups each of the same order greater than 2. This was done by showing that (in the terminology of (3)) the existence of a non-trivial primitive Schur ring over H implies such a direct decomposition of H.
Homotopy Groups of Transformation Groups
Authors: Rhodes F.
Published: December 2010

In a previous paper (2) I defined the fundamental group σ(X, x0, G) of a group Gof homeomorphisms of a space X, and showed that if the transformation group admits a family of preferred paths, then σ(X, x0, G) can be represented as a group extension of π1(X, x0) by G. In this paper the homotopy groups of a transformation group are defined. The nth absolute homotopy group of a transformation group which admits a family of preferred paths is shown to be representable as a split extension of the nth absolute torus homotopy group τn(X, x0) by G.

In § 6 it is shown that the action of G on X induces a homomorphism of Ginto a quotient group of a subgroup of the group of automorphisms of τn(X, x0). This homomorphism is used to obtain a necessary condition for the embedding of one transformation group in another, in particular, for the embedding of a discrete flow in a continuous flow with the same phase space.
On Operator Algebras and Invariant Subspaces
Authors: Davis Chandler , Radjavi Heydar , Rosenthal Peter
Published: December 2010

If is a collection of operators on the complex Hilbert space , then the lattice of all subspaces of which are invariant under every operator in is denoted by Lat . An algebra of operators on is defined (3; 4) to be reflexive if for every operator B on the inclusion Lat ⊆ Lat B implies .

Arveson (1) has proved the following theorem. (The abbreviation “m.a.s.a.” stands for “maximal abelian self-adjoint algebra”.)

ARVESON's THEOREM. Ifis a weakly closed algebra which contains an m.a.s.a.y and if Lat, then is the algebra of all operators on .

A generalization of Arveson's Theorem was given in (3). Another generalization is Theorem 2 below, an equivalent form of which is Corollary 3. This theorem was motivated by the following very elementary proof of a special case of Arveson's Theorem.
On a Question of Seidel Concerning Holomorphic Functions Bounded on a Spiral
Authors: Barth K. F. , Schneider W. J.
Published: December 2010

Let S be a spiral contained in D = {|z| < 1} such that S tends to C = {|z| = 1}. For the sake of brevity, by “f is bounded on S” we shall mean that f is holomorphic in D, unbounded, and bounded on S. The existence of such functions was first discussed by Valiron (9; 10); see also (1; 3; 8). Valiron also proved that any function that is “bounded on a spiral” must have the asymptotic value ∞ (10, p. 432). Functions that are bounded on a spiral may also have finite asymptotic values (1, p. 1254). In view of the above, Seidel has raised the question (oral communication): “Does there exist a function bounded on a spiral that has only the asymptotic value ∞?”. The following theorem answers this question affirmatively.
Three Plane Sextics and their Automorphisms
Authors: Edge W. L.
Published: December 2010

1. The sextics of the title have five cusps and are particular examples of the curve encountered by Humbert (5). The claim to notoriety of Humbert's curve of genus 5 has been that all its abelian integrals of the first kind are linear combinations of five elliptic integrals; it also has (1) the striking property that its 120 Weierstrassian points are confluent in threes at only 40 distinct points; whereas a general curve of genus 5 has, after Riemann, 12 moduli, a Humbert curve has merely 2. The three curves to be studied now have no free moduli at all, but although this tempts one to construct period matrices, such transcendental topics will not be handled here.
Unbounded Positive Definite Functions
Authors: Stewart James
Published: December 2010

Let G be an abelian group, written additively. A complexvalued function ƒ, defined on G, is said to be positive definite if the inequality

1

holds for every choice of complex numbers C1, …, cn and S1, …, sn in G. It follows directly from (1) that every positive definite function is bounded. Weil (9, p. 122) and Raïkov (5) proved that every continuous positive definite function on a locally compact abelian group is the Fourier-Stieltjes transform of a bounded positive measure, thus generalizing theorems of Herglotz (4) (G = Z, the integers) and Bochner (1) (G = R, the real numbers).

If ƒ is a continuous function, then condition (1) is equivalent to the condition that

2
Groups Generated by two Parabolic Linear Fractional Transformations
Authors: Lyndon R. C. , Ullman J. L.
Published: December 2010

We are interested in the structure of a group G of linear fractional transformations of the extended complex plane that is generated by two parabolic elements A and B, and, particularly, in the question of when such a group G is free. We shall, as usual, represent elements of G by matrices with determinant 1, which are determined up to change of sign. Two such groups G will be conjugate in the full linear fractional group, and hence isomorphic, provided they have, up to a change of sign, the same value of the invariant τ = Trace(AB) – 2. We put aside the trivial case that τ = 0, where G is abelian. In the study of these groups, two normalizations have proved convenient. Sanov (17) and Brenner (3) took the generators in the form

while Chang, Jennings, and Ree (4) took them in the form
Linearly Stable Polytopes
Authors: McMullen P.
Published: December 2010

Our aim in this paper is to describe a new class of convex polytopes, which will be called linearly stable. These have properties analogous to those of projectively stable polytopes (called projectively unique by Grunbaum (1, exercise 4.8.30)), which were first investigated early in 1966 by Grunbaum and Pedes. Although many particular examples of projectively stable polytopes have been found, at present no general criteria for projective stability are known.

The main result of this paper is a theorem which enables us to classify linearly stable polytopes completely.

I wish to thank Professor G. C. Shephard for his many helpful suggestions for improvements to this paper. During the period of this research I held a research studentship, for which I would like to thank the Science Research Council.
Generalized Casimir Operators
Authors: Douglas A. J.
Published: December 2010

Throughout this paper, S will be a ring (not necessarily commutative) with an identity element ls ≠ 0s. We shall use R to denote a second ring, and ϕ: S→ R will be a fixed ring homomorphism for which ϕ1S = 1R.

In (7), Higman generalized the Casimir operator of classical theory and used his generalization to characterize relatively projective and injective modules. As a special case, he obtained a theorem which contains results of Eckmann (3) and of Higman himself (5), and which also includes Gaschütz's generalization (4) of Maschke's theorem. (For a discussion of some of the developments of Maschke's idea of averaging over a finite group, we refer the reader to (2, Chapter IX).) In the present paper, we define the Casimir operator of a family of S-homomorphisms of one R-module into another, and we again use this operator to characterize relatively projective and injective modules.
The Influence of Generalized Frattini Subgroups on the Solvability of a Finite Group
Authors: Beidleman James C.
Published: December 2010

1. The Frattini and Fitting subgroups of a finite group G have been useful subgroups in establishing necessary and sufficient conditions for G to be solvable. In [1, pp. 657-658, Theorem 1], Baer used these subgroups to establish several very interesting equivalent conditions for G to be solvable. One of Baer's conditions is that ϕ(S), the Frattini subgroup of S, is a proper subgroup of F(S), the Fitting subgroup of S, for each subgroup S ≠ 1 of G. Using the Fitting subgroup and generalized Frattini subgroups of certain subgroups of G we provide certain equivalent conditions for G to be a solvable group. One such condition is that F(S) is not a generalized Frattini subgroup of S for each subgroup S ≠ 1 of G. Our results are given in Theorem 1.
On Balanced Incomplete Block Designs with Large Number of Elements
Authors: Hanani Haim
Published: December 2010

A balanced incomplete block design (BIBD) B[k, λ; v] is an arrangement of v distinct elements into blocks each containing exactly k distinct elements such that each pair of elements occurs together in exactly λ blocks.

The following is a well-known theorem [5, p. 248].

THEOREM 1. A necessary condition for the existence of a BIBD B[k, λ,v] is that

(1)

It is also well known [5] that condition (1) is not sufficient for the existence of B[k, λ; v].

There is an old conjecture that for any given k and λ condition (1) may be sufficient for the existence of a BIBD B[k, λ; v] if v is sufficiently large. It is attempted here to prove this conjecture in some specific cases.

There is an old conjecture that for any given k and X condition (1) may be sufficient for the existence of a BIBD B[k, λ; v] if v is sufficiently large. It is attempted here to prove this conjecture in some specific cases.
On Convex Univalent Functions
Authors: Basgoze T. , Frank J. L. , Keogh F. R.
Published: December 2010

In what follows, we suppose that ƒ(z) = Σ0∞anzn is regular for |z| < 1. Let

and

Then (see, for example, [6, pp. 235-236]), for 0 ≦ r < ρ < 1, we have:

The following results are well known.
On the Hyperplane Sections Through two Given Points of an Algebraic Variety
Authors: Kuan Wei-Eihn
Published: December 2010

1. Let k be an infinite field and let V/k be an irreducible variety of dimension ≧ 2 in a projective n-space Pn over k. Let P and Q be two k-rational points on V In this paper, we describe ideal-theoretically the generic hyperplane section of V through P and Q (Theorem 1) and prove that the section is almost always an absolutely irreducible variety over k1/pe if V/k is absolutely irreducible (Theorem 3). As an application (Theorem 4), we give a new simple proof of an important special case of the existence of a curve connecting two rational points of an absolutely irreducible variety [4], namely any two k-rational points on V/k can be connected by an irreducible curve.

I wish to thank Professor A. Seidenberg for his continued advice and encouragement on my thesis research.
The Spectra of Semi-Normal Singular Integral Operators
Authors: Putnam C. R.
Published: December 2010

Suppose that

(1.1)

and define the bounded self-adjoint operators H and J on the Hilbert space L2(0, 1) by

(1.2)

the integral being a Cauchy principal value

It is seen that

(1.3)

or, equivalently,

(1.4)

Since (Cƒ, ƒ) = π–1|(ƒ, ϕ)|2 ≧ 0, A is semi-normal. (For a discussion of such operators, see [4].)
Linearization of the Product of Jacobi Polynomials. I
Authors: Gasper George
Published: December 2010

Let Pn(α,β) be the Jacobi polynomial of degree n, order (α,β), α,β > – 1, defined by

[9, p. 67], and let Rn(α,β)(x) = Pn(α,β)(x)/Pn(αβ)(1). Then for n ≧ m,

where

Since Rn(α, β)(l) = 1, it follows that

(1)

It is known that if (the ultraspherical case) or if α = β + 1, then α = β + 1, then g(k, n, m) ≧ 0.
On R0-Closed Classes, and Finitely Generated Groups
Authors: Dark Rex , Rhemtulla Akbar H.
Published: December 2010

1.1. If a group satisfies the maximal condition for normal subgroups, then all its central factors are necessarily finitely generated. In [2], Hall asked whether there exist finitely generated soluble groups which do not satisfy the maximal condition for normal subgroups but all of whose central factors are finitely generated. We shall answer this question in the affirmative. We shall also construct a finitely generated group all of whose subnormal subgroups are perfect (and which therefore has no non-trivial central factors), but which does not satisfy the maximal condition for normal subgroups. Related to these examples is the question of which classes of finitely generated groups satisfy the maximal condition for normal subgroups. A characterization of such classes has been obtained by Hall, and we shall include his result as our first theorem.
Metacyclic Invariants of Knots and Links
Authors: Fox R. H.
Published: December 2010

To each representation ρ on a transitive permutation group P of the group G = π(S – k) of an (ordered and oriented) link k = k1 ∪ k2 ∪ … ∪ kμ in the oriented 3-sphere S there is associated an oriented open 3-manifold M = Mρ(k), the covering space of S – k that belongs to ρ. The points 01, 02, … that lie over the base point o may be indexed in such a way that the elements g of G into which the paths from oi to oj project are represented by the permutations gρ of the form , and this property characterizes M. Of course M does not depend on the actual indices assigned to the points o1, o2, … but only on the equivalence class of ρ, where two representations ρ of G onto P and ρ′ of G onto P′ are equivalent when there is an inner automorphism θ of some symmetric group in which both P and P′ are contained which is such that ρ′ = θρ.
A Characterization of the Approximately Continuous Denjoy Integral
Authors: Kubota Yöto
Published: December 2010

The approximately continuous integral which includes the Lebesgue integral has been considered by Burkill [1] and Ridder [3; 4]. I [2] also defined the AD-integral of this kind which is more general than the AP-integral of Burkill [1] and the Denjoy integral in the wide sense. But this integral is equivalent to the β-integral of Ridder.

Our aim in this paper is to characterize the AD-integral in the following way: The AD-integral is the least general approximately continuous integral (Definition 1) which includes the Lebesgue integral and fulfils the Cauchy and Harnack conditions (Definition 2).
Compositions of Set Operations
Authors: Bressler D. W. , Cayford A. H.
Published: December 2010

The set operations under consideration are Borel operations and Souslin's operation (). With respect to a given family of sets and in a setting free of any topological structure there are defined three Borel families (Definitions 3.1) and the family of Souslin sets (Definition 4.1). Conditions on an initial family are determined under which iteration of the Borel operations with Souslin's operation () on the initial family and the families successively produced results in a non-decreasing sequence of families of analytic sets (Theorem 5.2.1 and Definition 3.5). A classification of families of analytic sets with respect to an initial family of sets is indicated in a manner analogous to the familiar classification of Borel sets (Definition 5.3).
Un Théorème De Régularité Pour Une Équation Différentielle Abstraite
Authors: Arminjon Paul
Published: December 2010

Soit Y un espace localement convexe complet, la topologie étant définie par une famille de semi-normes {pα}. Soit A un opérateur fermé de domaine D(A) dense dans Y. Nous considérons dans cet article des fonctions u de la variable réelle t à valeurs dans Y. On dit que u(t) ∈ Cl[(a, b); Y] est une solution au sens classique de l'équation différentielle abstraite (ou opérationnelle)

(1)

si pour tout t ∈ (a, b), u(t) ∈ D(A), et satisfait la relation

élément de Y, f étant une fonction continue à valeurs dans Y.

Dans leur important mémoire [1], Agmon et Nirenberg ont étudié en détail les propriétés des solutions d'une telle équation dans le cas où Y est un espace de Banach.
On the Integral Extensions of Quadratic Forms Over Local Fields
Authors: Band Melvin
Published: December 2010

Let F be a local field with ring of integers and unique prime ideal (p). Suppose that V a finite-dimensional regular quadratic space over F, W and W′ are two isometric subspaces of V (i.e. τ: W → W′ is an isometry from W to W′). By the well-known Witt's Theorem, τ can always be extended to an isometry σ ∈ O(V).

The integral analogue of this theorem has been solved over non-dyadic local fields by James and Rosenzweig [2], over the 2-adic fields by Trojan [4], and partially over the dyadics by Hsia [1], all for the special case that W is a line. In this paper we give necessary and sufficient conditions that two arbitrary dimensional subspaces W and W′ are integrally equivalent over non-dyadic local fields.
Integral Inequalities for Equimeasurable Rearrangements
Authors: Duff G. F. D.
Published: December 2010

For a real-valued functionfon the domain [0,b], the equimeasurable decreasing rearrangementf* offis defined as a functionμ–1inverse toμ, whereμ(y) is the measure of the set {x|f(x) >y}. Inequalities connected with rearrangements of sequences as well as functions play a considerable part in various branches of analysis, and, for example, the concluding chapter of Hardy, Littlewood, and Pólya [3] is devoted to rearrangement inequalities. Equimeasurable rearrangements of functions are also used by Zygmund [6, Vol. II, Chapters I and XII].
Eigenfunctions of Operator-Valued Analytic Functions
Authors: Sherman Malcolm J.
Published: December 2010

This paper is a sequel to [2], whose primary purposes are to clarify and generalize the concept introduced there of an eigenfunction of an inner function, and to answer questions raised there concerning the equivalence of several possible forms of the definition. A new definition, proposed here, leads to a complete characterization of the eigenfunctions of Potapov inner functions of normal operators, and the result is more satisfactory than [2, Theorem 3.4], although the latter is used strongly in the proof.

Let be an inner function in the sense of Lax; i.e., is almost everywhere (a.e.) a unitary operator on a separable Hilbert space and belongs weakly to the Hardy class H2. An analytic function q (which will have to be a scalar inner function) was defined to be an eigenfunction of if the set of z in the disk {z: |z| ≦ 1} for which is invertible is a set of linear measure 0 on the circle {z: |z| = 1}.
Existence Theorems for Some Non-Linear Equations of Evolution
Authors: Clements John C.
Published: December 2010

In recent years considerable attention has been focused on non-linear hyperbolic differential equations with the object of establishing the existence of global solutions. It is our aim here to establish the existence of weak solutions of boundary value problems for non-linear equations of the form

(1-1)

where d is a real constant called the damping coefficient, u(t) is a vector-valued function defined on a subinterval of the real line into a space of complex-valued functions u(x) defined on a bounded domain Ω in the real Euclidean space EN of N dimensions, ut(t) ≡ du(t)/dt, and A(t) is the family of partial differential operators of order 2m (m = 1, 2, …) on Ω given in generalized divergence form by

(1-2)

with
Linear Transformations on Matrices: the Invariance of the Third Elementary Symmetric Function
Authors: Beasley Leroy B.
Published: December 2010

Let T be a linear transformation on Mn the set of all n × n matrices over the field of complex numbers, . Let A ∈ Mn have eigenvalues λ1, …, λn and let Er(A) denote the rth elementary symmetric function of the eigenvalues of A :

Equivalently, Er(A) is the sum of all the principal r × r subdeterminants of A. T is said to preserve Er if Er[T(A)] = Er(A) for all A ∈ Mn. Marcus and Purves [3, Theorem 3.1] showed that for r ≧ 4, if T preserves Er then T is essentially a similarity transformation; that is, either T: A → UAV for all A ∈ Mn or T: A → UAtV for all A ∈ Mn, where UV = eiθIn, rθ ≡ 0 (mod 2π).
Ordered Groups Satisfying the Maximal Condition Locally
Authors: Hursey R. J. , Rhemtulla A. H.
Published: December 2010

Let denote the class of all (fully) ordered groups satisfying the maximal condition on subgroups, and let denote the class of all locally groups. In this paper we investigate the family of convex subgroups of groups.

It is well known (see [1, pp. 51, 54]) that every convex subgroup of an is normal in G, and for any jump D –< C in the family of convex subgroups, [G′, C] ⊆ D. We observe that these properties are also true for any group and record, without proof, the following.

THEOREM 1. Any convex subgroup of angroup G is normal in G, and for any jump D –< C in the family of convex subgroups, [G′, C] ⊆ D.
Group Representations and Cardinal Algebras
Authors: Deliyannis Platon C.
Published: December 2010

Our aim in this work is to show that the global theory of group representations (as presented in [4] for example) is actually and naturally a part of the theory of cardinal algebras. In this sense it is analogous to the work of Kaplansky [2] and Loomis [3] on operator rings. In his paper Loomis proposed an abstract scheme for representation theory; it appears, however, that the idea of abstracting the equivalence class of a representation is more suitable. It is the set of all these classes that forms the cardinal algebra which we study.

The connection between cardinal algebras and operator theory has been made explicit by Fillmore [1], where he worked out a dimension theory for a class of cardinal algebras satisfying certain conditions. In the next section we shall discuss the relation between the two systems; we could say that the main difference lies in our introducing such axioms as to make possible the study of type III cases.
On the Rank Numbers of an Arc
Authors: Turgeon J.
Published: December 2010

The kth rank number, rankkB, of a differentiable arc B in real projective n-space is the least upper bound of the number of osculating k-spaces of B which meet an (n – k – l)-flat, k = 0, 1, …, n – 1. The number rank0B is called the order of B; cf. 1.1-1.3. It has been conjectured by Peter Scherk that

(0.1)

equality holding if and only if B has the order n; cf. [2, p. 396]. In this paper we prove the following results.

THEOREM 1. If B is a differentiable elementary arc, then (0.1) holds for k = 0, 1, …, n – 1.

THEOREM 2. If B is a differentiable elementary arc and order B > n, then rankkB > (k + 1) (n – k) for k = 1, …, n – 2.

By a theorem of Park [3, p. 38], every differentiable arc contains a subarc of order n. This eliminates the assumption that B is elementary from Theorem 1. We do not know whether it can be dropped from Theorem 2.
The Maximum Modulus of Normal Meromorphic Functions and Applications to Value Distribution
Authors: Gauthier Paul
Published: December 2010

Let f(z) be a function meromorphic in the unit disc D = (|z| < 1). We consider the maximum modulus

and the minimum modulus

When no confusion is likely, we shall write M(r) and m(r) in place of M(r,f) and m(r,f).

Since every normal holomorphic function belongs to an invariant normal family, a theorem of Hayman [6, Theorem 6.8] yields the following result.

THEOREM 1. If f(z) is a normal holomorphic function in the unit disc D, then

(1)

This means that for normal holomorphic functions, M(r) cannot grow too rapidly. The main result of this paper (Theorem 5, also due to Hayman, but unpublished) is that a similar situation holds for normal meromorphic functions.
Unsolvable Problems in Groups With Solvable Word Problem
Authors: McCool James
Published: December 2010

Let G be a finitely presented group with solvable word problem. It is of some interest to ask which other decision problems must necessarily be solvable for such a group. Thus it is easy to see that there exist effective procedures to determine whether or not such a group is trivial, or nilpotent of a given class. On the other hand, the conjugacy problem need not be solvable for such a group, for Fridman [5] has shown that the word problem is solvable for the group with unsolvable conjugacy problem given by Novikov [9].
Simple Quotients of Euclidean Lie Algebras
Authors: Moody Robert V.
Published: December 2010

In [2], we considered a class of Lie algebras generalizing the classical simple Lie algebras. Using a field Φ of characteristic zero and a square matrix (Aij) of integers with the properties (1) Aii = 2, (2) Aij ≦ 0 if i ≠ j, (3) Aij = 0 if and only if Ajt = 0, and (4) is symmetric for some appropriate non-zero rational a Lie algebra E = E((Aij)) over Φ can be constructed, together with the usual accoutrements: a root system, invariant bilinear form, and Weyl group.

For indecomposable (A ij), E is simple except when (Aij) is singular and removal of any row and corresponding column of (Aij) leaves a Cartan matrix. The non-simple Es, Euclidean Lie algebras, were our object of study in [3] as well as in the present paper. They are infinite-dimensional, have ascending chain condition on ideals, and proper ideals are of finite codimension.
On the Classification of Lie Pseudo-Algebras
Authors: van Quê Ngö
Published: December 2010

For every ( differentiable) bundle E over a manifold M, Jk(E) denotes the set of all k-jets of local (differentiable) sections of the bundle E. Jk(E) is a bundle over M such that if X is a section of E, then

is a (differentiable) section of Jk(E). If E is a vector bundle, Jk(E) is a vector bundle and we have the canonical exact sequence of vector bundles

where Sk(T*) is the symmetric Whitney tensor product of the cotangent vector bundle T* of M. and π is the canonical morphism which associates to each k-jet of section its jet of inferior order.
Lines and Hyperplanes associated with Families of Closed and Bounded Sets in Conjugate Banach Spaces
Authors: Edelstein M.
Published: December 2010

Let be a family of sets in a linear space X. A hyperplane π is called a k-secant of if π intersects exactly k members of . The existence of k-secants for families of compact sets in linear topological spaces has been discussed in a number of recent papers (cf. [3–7]). For X normed (and a finite family of two or more disjoint non-empty compact sets) it was proved [5] that if the union of all members of is an infinite set which is not contained in any straight line of X, then has a 2-secant. This result and related ones concerning intersections of members of by straight lines have since been extended in [4] to the more general setting of a Hausdorff locally convex space.
Incidence Systems associated with Non-Planar Nearfields
Authors: Davis Elwyn H.
Published: December 2010

In [2] Sandler studied generalizations of projective planes called pseudo planes. These structures gave rise to ternary operations for which addition and multiplication are loop operations. Our aim in this paper is to investigate the pseudo planes for which the operations of addition and multiplication give rise to nearfields. The pseudo planes for which this is true will be called π-systems in this paper, and a list of postulates for such systems is given in § 2. In that section some results on collineations of π-systems are given which are stronger than analogous results for general pseudo planes.
Minimal Relations for Certain Wreath Products of Groups
Authors: Johnson D. L.
Published: December 2010

Let p be a rational prime, G a non-trivial finite p group, and K the field of p elements, regarded as a trivial G-module according to context; then we define:

d(G) = dimKH1(G, K), the minimal number of generators of G,

r(G) = dimKH2(G, K),

r′(G) = the minimal number of relations required to define G,

where, in the last equation, it is sufficient to take the minimum over those presentations of G with d(G) generators. It is well known (see § 2) that the following inequalities hold:

We shall consider only finite p-groups, so that the class of groups with r = d coincides with that consisting of those groups whose Schur multiplicator is trivial.
k-Degenerate Graphs
Authors: Lick Don R. , White Arthur T.
Published: December 2010

Graphs possessing a certain property are often characterized in terms of a type of configuration or subgraph which they cannot possess. For example, a graph is totally disconnected (or, has chromatic number one) if and only if it contains no lines; a graph is a forest (or, has point-arboricity one) if and only if it contains no cycles. Chartrand, Geller, and Hedetniemi [2] defined a graph to have property Pn if it contains no subgraph homeomorphic from the complete graph Kn+1 or the complete bipartite graph

For the first four natural numbers n, the graphs with property Pn are exactly the totally disconnected graphs, forests, outerplanar and planar graphs, respectively. This unification suggested the extension of many results known to hold for one of the above four classes of graphs to one or more of the remaining classes.
Algebraic Extensions of Commutative Regular
Rings
Authors: Raphael R. M.
Published: December 2010

In this paper we study algebraic closures for commutative semiprime rings.
The main interest, however, is with rings which are regular in the sense of
von Neumann. These play the same role with respect to semiprime rings as
fields do with respect to integral domains. Two generally distinct notions
are defined: “algebraic” and “weak-algebraic” extensions. Each has the
transitivity property and yields a closure which is unique up to isomorphism
and is “universal”. Both coincide in fields.

The extensions here called “algebraic” were studied independently by Enochs
[5] and myself. Our results on these extensions proceed from
a different point of view, and allow us to answer a question posed by
Enochs. Furthermore, these results are required (and were developed) in
order to obtain the weak-algebraic closure, which was the original closure
sought. The motivation for the weak-algebraic extensions is found in the
work of Shoda [14, p. 134, no. 1].
On Quasi-Essential Subgroups of Primary Abelian
Groups
Authors: Benabdallah Khalid , Irwin John M.
Published: December 2010

All groups considered in this paper are abelian. A subgroup
N of a group G is said to be a
quasi-essential subgroup of G if G =
〈H, K〉 whenever H is an
N-high subgroup of G and
K is a pure subgroup of G containing
N. We started the study of such subgroups in
[5]; in particular, we characterized subsocles of a primary
group which were both quasi-essential and centres of purity. In this paper
we show that quasi-essential subsocles of a primary group are necessarily
centres of purity answering thus in the affirmative a question raised in
[5].

We obtain the following theorem: A subsocle S of a p-group G is
quasi-essential if and only if either S ⊂ G1or (pnG)[p] ⊃ S ⊃
(pn+1G)[p] for some non-negative integer n. The
notation is that of [1]. If G is a group,
then

where p is a prime integer.
Some Addition Theorems of Group Theory with Applications to
Graph Theory
Authors: Yap H. P.
Published: December 2010

Let G be an additive group with nonempty subsets
S and T. Let S ±
T = {s ± t;
s ∊ S, t ∊ T}, let
S be the set complement of S in
G, and let |S| be the cardinality of
S. We abbreviate {f} where
f ∊ G to f. If
S + S and S have no
element in common, then we say that S is a sum-free set in
G or that S is sum-free in
G. If S is a sum-free set in
G and if for every sum-free set T in
G, |S| ≧ |T|, then
S is said to be a maximal sum-free set in
G. We denote by λ(G) the cardinality of
a maximal sum-free set in G. We say that S
is in arithmetic progression having difference d if
S = {s, s +
d, …, s + nd} for some
s, d ∊ G and some integer
n ≧ 0.
A Difference-Differential Basis Theorem
Authors: Cohn Richard M.
Published: December 2010

Our aim in this paper is to extend to difference-differential rings the
beautiful theorem of Kolchin [5, Theorem 3] for the
differential case. The necessity portion of Kolchin's result is not
obtained.

What might well be called the Ritt basis theorem states that if a
commutative ring R with identity is finitely generated over
a subring R0, then the ascending chain condition for radical ideals of
R0 implies the ascending chain condition for radical ideals of
R. (This is indeed a basis theorem. If we define a basis
for a radical ideal A to be a finite set B
such that then every
radical ideal of a ring R has a basis if and only if the
ascending chain condition for radical ideals holds in R.)
It is the Ritt basis theorem rather than the Hilbert basis theorem which has
appropriate generalizations in differential and difference algebra, where in
fact it originated.
Higher Monotonicity Properties of Certain Sturm-Liouville
Functions. III
Authors: Lorch Lee , Muldoon M. E. , Szego Peter
Published: December 2010

A Sturm-Liouville function is simply a non-trivial solution of the
Sturm-Liouville differential equation

(1.1)

considered, together with everything else in this study, in the real domain.
The associated quantities whose higher monotonicity properties are
determined here are defined, for fixed λ > –1, to be

(1.2)

where y(x) is an arbitrary (non-trivial) solution of (1.1)
and x1, x2, … is any finite or infinite sequence of consecutive zeros of
any non-trivial solution z(x) of (1.1) which may or may not
be linearly independent of y(x). The condition
λ > –1 is required to assure convergence of the
integral defining Mk, and the function W(x) is taken subject to the
same restriction.
Vertex Subgroups of Irreducible Representations of Solvable
Groups
Authors: Malzan J.
Published: December 2010

If ρ(G) is a finite, real, orthogonal
group of matrices acting on the real vector space V, then
there is defined [5], by the action of
ρ(G), a convex subset of the unit
sphere in V called a fundamental region.
When the unit sphere is covered by the images under
ρ(G) of a fundamental region, we obtain
a semi-regular figure.

The group-theoretical problem in this kind of geometry is to find when the
fundamental region is unique. In this paper we examine the subgroups,
ρ(H), of
ρ(G) with a view of finding what
subspace, W of V consists of vectors held
fixed by all the matrices of ρ(H). Any
such subspace lies between two copies of a fundamental region and so
contributes to a boundary of both. If enough of these boundaries might be
found, the fundamental region would be completely described.
Pentagon-Generated Trivalent Graphs with Girth 5
Authors: Robertson Neil
Published: December 2010

The terminology of [1] will be assumed in what follows. LetPb(G) stand for the set of pentagons in the graphG.Call a graphpentagongeneratedwhen it is the union of its contained pentagons. LetP5,3be the class of connected trivalent pentagon-generated graphs with girth 5. These graphs form a family including the Petersen graph and the graph of the dodecahedron. They are studied here and completely classified in terms of a decomposition which all but some specifically determined indecomposable graphs admit.

Assume henceforth thatH∈P5,3. LetEk(H) be the set of edges in exactlyk∈ 0 pentagons ofH. ClearlyEk(H) = 0 ifk≠ 1, 2, 3, 4 and |E1(H) ∩E(P)r ≦ 2, for allP∈P5(H).P∈P5(H) issingularwhen |E1(H) ∩E(P)r = 2,.
On Expanding Locally Finite Collections
Authors: Krajewski Lawrence L.
Published: December 2010

A space X is in-expandable, where
m is an infinite cardinal, if for every locally finite
collection {Hα| α ∈ A} of subsets of
X with |A| ≦
m(cardinality of A ≦ m)
there exists a locally finite collection of open subsets
{Gα| α ∈ A} such that
Hα ⊆ Gα for every α ∈ A.
X is expandable if it is
m-expandable for every cardinal m. The
notion of expandability is closely related to that of collection wise
normality introduced by Bing [1], X is
collectionwise normal if for every discrete collection
of subsets {Hα|α ∈ A} there is a discrete
collection of open subsets {Gα|α ∈ A} such that
Hα ⊆ Gα for every α ∈ A. Expandable
spaces share many of the properties possessed by collectionwise normal
spaces. For example, an expandable developable space is metrizable and an
expandable metacompact space is paracompact.
Commutative Endomorphism Rings
Authors: Zelmanowitz J.
Published: December 2010

The problem of classifying the torsion-free abelian groups with commutative
endomorphism rings appears as Fuchs’ problems in [4, Problems
46 and 47]. They are far from solved, and the obstacles to a solution appear
formidable (see [4; 5]). It is, however, easy to see that the
only dualizable abelian group with a commutative endomorphism ring is the
infinite cyclic group. (An R-module Miscalled dualizable if HomR(M, R) ≠ 0.) Motivated by this, we study the class
of prime rings R which possess a dualizable module
M with a commutative endomorphism ring. A
characterization of such rings is obtained in § 6, which as would be
expected, places stringent restrictions on the ring and the module.

Throughout we will write homomorphisms of modules on the side opposite to
the scalar action. Rings will not be assumed to contain identity elements
unless otherwise indicated.
Finite-to-One Open Mappings
Authors: Duda Edwin , Haynsworth W. Hugh
Published: December 2010

The class of finite-to-one open mappings on manifolds contains some
important subclasses. Any non-constant analytic function from a bounded
region in its domain of definition is finite-to-one. Church [2]
showed that any light strongly open Cn map f: Rn → Rn is discrete. A number of papers concerning discrete open mappings on
manifolds have been published; see [1-6; 8-9; 11-14].

A result of Černavskiĭ [1] (see also [13]) shows
that for any discrete strongly open mapping f :
Mn → Nn of an n-manifold into an
n-manifold, the branch set of f has
dimension less than n – 1. If f is also a
closed map, then N(f) is finite and the set of points
x for which N(x, f) =
N(f) is an open dense connected subset of
Mn.
On the Smallest Degrees of Projective Representations of the
Groups PSL(n, q)
Authors: Harris Morton E. , Hering Christoph
Published: December 2010

In this paper, we obtain information about the minimal degree
δ of any non-trivial projective representation of the
group PSL(n, q) with n ≧ 2 over an
arbitrary given field K. Our main results for the groups
PSL(n, q) (Theorems 4.2, 4.3, and 4.4)
state that, apart from certain exceptional cases with small
n, we have the following rather surprising situation: if
q = pf (where p is a prime integer) and char K =
p, then δ = n, but if
q = pf and char K ≠ p, then
δ is of a considerably higher order of magnitude,
namely, δ is at least qn–l – 1 or if
n = 2 and q is odd. Note that for
n = 2, this lower bound for δ is the
best possible. However, for n ≧ 3, this lower bound can
conceivably be improved.
Grothendieck Groups of Bass Orders
Authors: Roggenkamp Klaus W.
Published: December 2010

Commutative Bass rings, which form a special class of Gorenstein rings, have
been thoroughly investigated by Bass [1]. The definitions do
not carry over to non-commutative rings. However, in case one deals with
orders in separable algebras over fields, Bass orders can be defined. Drozd,
Kiricenko, and Roïter [3] and Roïter [6] have
clarified the structure of Bass orders, and they have classified them. These
Bass orders play a key role in the question of the finiteness of the
non-isomorphic indecomposable lattices over orders (cf. [2;
8]). We shall use the results of Drozd, Kiricenko, and Roïter
[3] to compute the Grothendieck groups of Bass orders
locally. Locally, the Grothendieck group of a Bass order (with the exception
of one class of Bass orders) is the epimorphic image of the direct sum of
the Grothendieck groups of the maximal orders containing it.
Transitive Orientation of Graphs and Identification of
Permutation Graphs
Authors: Pnueli A. , Lempel A. , Even S.
Published: December 2010

The graphs considered in this paper are assumed to be finite, with no edge
joining a vertex to itself and with no two distinct edges joining the same
pair of vertices. An undirected graph will be denoted by G
or (V, E), where V is the set of vertices
and E is the set of edges. An edge joining the vertices
i,j ∊ V will be denoted by the
unordered pair (i,j).

An orientation of G = (V,
E) is an assignment of a unique direction i →
j or j → i to every
edge (i,j) ∊ E. The resulting directed
image of G will be denoted by G→ or (V, E→), where
E→ is now a set of ordered pairs E→ =
{[i,j]| (i,j) ∊ E and
i → j}. Notice the difference in
notation (brackets versus parentheses) for ordered and unordered pairs.
Approximation by Unimodular Functions
Authors: Fisher Stephen
Published: December 2010

The theorems in this paper are all concerned with either pointwise or uniform approximation by functions which have unit modulus or by convex combinations of such functions. The results are related to, and are outgrowths of, the theorems in [4; 5; 10].

In § 1, we show that a function bounded by 1, which is analytic in the open unit disc Δ and continuous on may be approximated uniformly on the set where it has modulus 1 (subject to certain restrictions; see Theorem 1) by a finite Blaschke product; that is, by a function of the form

*

where |λ| = 1 and |αi| < 1, i = 1, …, N. In § 1 we also discuss pointwise approximation by Blaschke products with restricted zeros.
On the Differentials of Certain Matrix Functions
Authors: Powers David L.
Published: December 2010

In [5], Rinehart showed that if X is an n × n complex matrix with distinct eigenvalues, then a suitably defined diagonalizing matrix P and the diagonal matrix Λ of eigenvalues in P–lXP = Λ are both Hausdorff differentiate functions in an open set containing X. Furthermore, if the scalar function ƒ(z) is analytic at the eigenvalues of X, then the primary matrix function ƒ(X) is Hausdorff differentiable, and its differential may be represented in terms of the differentials of P and Λ [4]. Rinehart noted that the actual computation of differentials was difficult and ad hoc. This difficulty clearly arises because of the definition given for the diagonalizing matrix. Therefore, our aim in this note is to give a different definition of the diagonalizing matrix, one which simplifies the computations.
On the Point-Arboricity of a Graph and its Complement
Authors: Mitchem John
Published: December 2010

The point-arboricity ρ (G) of a graph G is defined as the minimum number of subsets into which the point set V(G) of G may be partitioned so that each subset induces an acyclic subgraph. Equivalently, the point-arboricity of G may be defined as the least number of colours needed to colour the points of G so that no cycle of G has all of its points coloured the same. This term was introduced by Chartrand, Geller, and Hedetniemi [1], although the concept was first considered by Motzkin [4].

As with the chromatic number of a graph G, which we denote by χ(G), there is no explicit formula for the point-arboricity of a graph. However, Nordhaus and Gaddum [5] have shown that if G is a graph with p points, then
On Certain Functional Identities in EN
Authors: Mercer A. McD.
Published: December 2010

1. If f is a real-valued function possessing a Taylor series convergent in (a — R, a + R), then it satisfies the following operational identity

1.1

in which D2 = d2/du2. Furthermore, when g is a solution of y″ + λ2y = 0 in (a – R, a + R), then g is such a function and (1.1) specializes to

1.2

In this note we generalize these results to the real Euclidean space EN, our conclusions being Theorems 1 and 2 below. Clearly, (1.2) is a special case of (1.1) but in higher-dimensional space it is of interest to allow g, now a solution of

1.3

to possess singularities at isolated points away from the origin. It is then necessary to consider not only a neighbourhood of the origin but annular regions also.
Boolean Algebra Retracts
Authors: Cramer Timothy
Published: December 2010

A Boolean algebra B is a retract of an algebra A if there exist homomorphisms ƒ: B → A and g: A → B such that gƒ is the identity map B. Some important properties of retracts of Boolean algebras are stated in [3, §§ 30, 31, 32]. If A and B are a-complete, and A is α-generated by B, Dwinger [1, p. 145, Theorem 2.4] proved necessary and sufficient conditions for the existence of an α-homomorphism g: A → B such that g is the identity map on B. Note that if a is not an infinite cardinal, B must be equal to A. The dual problem was treated by Wright [6]; he assumed that A and B are σ-algebras, and that g: A → B is a σ-homomorphism, and gave conditions for the existence of a homomorphism ƒ:B → A such that gƒ is the identity map.
Actions of a Locally Compact Group with Zero
Authors: Hanson T. H. McH.
Published: December 2010

In [2] we find the definition of a locally compact group with zero as a locally compact Hausdorff topological semigroup, S, which contains a non-isolated point, 0, such that G = S – {0} is a group. Hofmann shows in [2] that 0 is indeed a zero for S, G is a locally compact topological group, and the unit, 1, of G is the unit of S. We are to study actions of S and G on spaces, and the reader is referred to [4] for the terminology of actions.

If X is a space (all are assumed Hausdorff) and A ⊂ X, A* denotes the closure of A. If {xρ} is a net in X, we say limρxρ = ∞ in X if {xρ} has no subnet which converges in X.
Some Imbedding Theorems for Sobolev Spaces
Authors: Adams R. A. , Fournier John
Published: December 2010

We shall be concerned throughout this paper with the Sobolev space Wm,p(G) and the existence and compactness (or lack of it) of its imbeddings (i.e. continuous inclusions) into various LP spaces over G, where G is an open, not necessarily bounded subset of n-dimensional Euclidean space En. For each positive integer m and each real p ≧ 1 the space Wm,p(G) consists of all u in LP(G) whose distributional partial derivatives of all orders up to and including m are also in LP(G). With respect to the norm

1.1

Wm,p(G) is a Banach space. It has been shown by Meyers and Serrin [9] that the set of functions in Cm(G) which, together with their partial derivatives of orders up to and including m, are in LP(G) forms a dense subspace of Wm,p(G).
On C-Matrices of Arbitrary Powers
Authors: Turyn Richard J.
Published: December 2010

A C-matrix is a square matrix of order m + 1 which is 0 on the main diagonal, has ±1 entries elsewhere and satisfies . Thus, if , I + C is an Hadamard matrix of skew type [3; 6] and, if , iI + C is a (symmetric) complex Hadamard matrix [4]. For m > 1, we must have . Such matrices arise from the quadratic character χ in a finite field, when m is an odd prime power, as [χ(ai – aj)] suitably bordered, and also from some other constructions, in particular those of skew type Hadamard matrices. (For we must have m = a2 + b2, a, b integers.)
On the Generality of the AP-Integral
Authors: Cross G. E.
Published: December 2010

In 1955 Taylor [6] constructed an AP-integral sufficiently strong to integrate Abel summable series with coefficients o(n). He showed that the AP-integral includes the special Denjoy integral and further that, when applied to trigonometric series, the AP-integral is more powerful than the SCP-integral of Burkill [1] and the P2-integral of James [3]. The present paper shows that the AP-integral includes the SCP-integral, and, under natural assumptions, the P2-integral.

After completing this manuscript I was advised by Skvorcov that he had shown [5] under more general conditions that the P2-integral is included in the AP-integral. The proof in the present paper seems to have some value in its own right and is considerably shorter.

Since the definition of the AP-integral is essentially for a function defined in (0, 2π] and elsewhere by 2π-periodicity, we shall consider SCP-integrable and P2-integrable functions defined similarly.