bluuebunny/crossref_metadata_2025 huggingface dataset:
Multiplicative Commutators of Operators
Authors: Brown Arlen , Pearcy Carl
Published: December 2010

An invertible operator T on a Hilbert space is a multiplicative commutator if there exist invertible operators A and B on such that T = ABA–1B–1. In this paper we discuss the question of which operators are, and which are not, multiplicative commutators. The analogous question for additive commutators (operators of the form AB — BA) has received considerable attention and has, in fact, been completely settled (2). The present results represent the information we have been able to obtain by carrying over to the multiplicative problem the techniques that proved efficacious in the additive situation. While these results remain incomplete, they suffice, for example, to enable us to determine precisely which normal operators are multiplicative commutators.
The Genus, Regional Number, and Betti Number of a Graph
Authors: Duke Richard A.
Published: December 2010

Let the genus of an orientable 2-manifold M be denoted by γ(M). The genus, γ(G), of a graph G is then the smallest of the numbers γ(N) for orientable 2-manifolds N in which G can be embedded. An embedding of G in M is called minimal if γ(G) = γ(M). When each component of the complement of G in M is an open 2-cell, the embedding of G in M is called a 2-cell embedding. In (3), J. W. T. Youngs has shown that each minimal embedding is a 2-cell embedding. It follows from the results of (3) that for each graph G, there is a number d(G), called the regional number, such that for any 2-cell embedding of G in an orientable 2-manifold, the number of (2-cell) complementary domains of G is ⩽d(G), with equality holding if and only if the embedding is minimal.
Infinite Packings of Disks
Authors: Melzak Z. A.
Published: December 2010

Let U be the closed disk in the plane, centred at the origin, and of unit radius. By a solid packing, or briefly a packing, C of U we shall understand a sequence ﹛Dn﹜, n = 1, 2, … , of open proper disjoint subdisks of U, such that the plane Lebesgue measures of U and of are the same. If rn is the radius of Dn and the complex number cn represents its centre, then the conditions for C to be a packing are

It was proved by Mergelyan (3) that for any packing the sum of the radii diverges:

1

Mergelyan's demonstration of (1) is somewhat involved and leans heavily on the machinery of functions of a complex variable. An elegant direct proof of (1) is given by Wesler (5), who uses the technique of projecting the boundaries of the disks of the packing on a diameter I of U.
On Unions of Two Convex Sets
Authors: McKinney Richard L.
Published: December 2010

Valentine (3) introduced the three-point convexity property P3 : a set S in En satisfies P3 if for each triple of points x, y, z in S at least one of the closed segments xy, yz, xz is in S. He proved, (3 or 1) that in the plane a closed connected set satisfying P3 is the union of some three convex subsets. The problem of characterizing those sets that are the union of two convex subsets was suggested. Stamey and Marr (2) have provided an answer for compact subsets of the plane. We present here a generalization of property P3 which characterizes closed sets in an arbitrary topological linear space which are the union of two convex subsets.
On Divisibility and Injectivity
Authors: Helzer Garry
Published: December 2010

In the category of abelian groups the concepts of divisible group and injective group coincide. In (4) this is generalized to modules over an integral domain and it is proved for a (commutative) integral domain that the concepts of divisible module and injective module coincide if and only if the ring is hereditary if and only if the ring is a Dedekind domain.

In (8) the assumption of commutativity is dropped and the ring is assumed to have a one-sided field of quotients. The result obtained is essentially the same as in the commutative case; see the theorem following 6.2. In (13) the requirement that the ring have no zero-divisors is also dropped and the ring is assumed to possess what we have called an Ore ring (see the definition following 6.2). The result obtained is the equivalent of parts (a) and (b) of our 6.13.
The Integral Extension of Isometries of Quadratic Forms Over Local Fields
Authors: Trojan Allan
Published: December 2010

Let F be a local field with ring of integers 0 and prime ideal π0. If V is a vector space over F, a lattice L in F is defined as an 0-module in the vector space V with the property that the elements of L have bounded denominators in the basis for V. If V is, in addition, a quadratic space, the lattice L then has a quadratic structure superimposed on it. Two lattices on V are then said to be isometric if there is an isometry of V that maps one onto the other.

In this paper, we consider the following problem: given two elements, v and w, of the lattice L over the regular quadratic space V, find necessary and sufficient conditions for the existence of an isometry on L that maps v onto w.
A Remark on Flat and Projective Modules
Authors: Jensen Chr. U.
Published: December 2010

It is the purpose of this note to give some characterizations of flat and projective modules, partly in ideal theoretical terms, partly in terms of the exterior product of a module (“puissance extérieure“); cf. (1).

We shall consider left modules over a ring R with identity element and without proper zero divisors. The left module M is called flat if X ⊗R M is an exact functor on the category of right R-modules X. If M is flat over a commutative domain R, M is necessarily torsion-free. Therefore when looking for flatness of a module M over a commutative domain, one may assume from the start that M is torsion-free.

In the following theorem, we shall not restrict ourselves to commutative rings R, but the modules concerned have to be torsion-free, which, of course, should mean that rm = 0 implies r = 0 or m = 0.
Sums of Fractions with Bounded Numerators
Authors: Stewart B. M. , Webb W. A.
Published: December 2010

The general problem considered in this paper is that of sums of a finite number of reduced fractions whose numerators are elements of a finite set S of integers, and whose denominators are distinct positive integers. Egyptian, or unit, fractions are merely the case S = ﹛1﹜. Problems concerning these fractions have been treated extensively. Another specific case S = ﹛1, — 1﹜ has been treated by Sierpinski (2).
Zero Tracts of Blaschke Products
Authors: Linden C. N. , Somadasa H.
Published: December 2010

Let ﹛an﹜ be a sequence of complex numbers such that

and

Then {an} is called a Blaschke sequence. For each Blaschke sequence {an} a Blaschke product is defined as

Thus a Blaschke product B(z, ﹛an﹜) is a function regular in the open unit disk D = {z: |z| < 1﹜ and having a zero at each point of the sequence ﹛an﹜.
The Hilbert-Schmidt Property for Embedding Maps between Sobolev Spaces
Authors: Clark Colin
Published: December 2010

Let H0m(Ω) denote the so-called Sobolev space consisting of functions denned on a region Ω in n-dimensional Euclidean space, which together with their generalized derivatives of all orders ⩽m belong to , and which vanish in a certain sense on the boundary ∂Ω. (Precise definitions are given in the next section.) For each pair m, k of non-negative integers the inclusion H0m+k(Ω) ⊂ H0m(Ω) defines a natural “embedding” map. For the case of a bounded region Ω it is well known that these maps are completely continuous, and even, for sufficiently large k, of Hilbert-Schmidt type. We have discussed complete continuity in the case of unbounded regions in an earlier paper; here we consider conditions on Ω which imply the Hilbert-Schmidt property for embeddings.
A Quasi-Linear Elliptic Boundary Value Problem
Authors: Adams R. A.
Published: December 2010

Let Ω be a bounded open set in Euclidean n-space, En. Let α = (α1, … , an) be an n-tuple of non-negative integers;

and denote by Qm the set ﹛α| 0 ⩽ |α| ⩽ m}. Denote by x = (x1, … , xn) a typical point in En and put

In this paper we establish, under certain circumstances, the existence of weak and classical solutions of the quasi-linear Dirichlet problem

1
Piecewise Analytic Solutions of Mixed Boundary Value Problems
Authors: Eisen M.
Published: December 2010

In a mixed problem one is required to find a solution of a system of partial differential equations when the values of certain combinations of the derivatives are given on two or more distinct intersecting surfaces. If the differential equations arise from some physical process, the correct boundary conditions are usually apparent. Many particular problems of this type have been solved by special methods such as separation of variables and the method of images. However, no general criterion has been given for what constitutes a correctly set mixed problem. In fact such problems have usually been formulated in connection with hyperbolic differential equations with data prescribed on two surfaces (called the initial and boundary surfaces).
Primary Ideals and Prime Power Ideals
Authors: Butts H. S. , Gilmer Robert W.
Published: December 2010

This paper is concerned with the ideal theory of a commutative ringR.We sayRhas Property (α) if each primary ideal inRis a power of its (prime) radical;Ris said to have Property (δ) provided every ideal inRis an intersection of a finite number of prime power ideals. In (2, Theorem 8, p. 33) it is shown that ifDis a Noetherian integral domain with identity and if there are no ideals properly between any maximal ideal and its square, thenDis a Dedekind domain. It follows from this that ifDhas Property (α) and is Noetherian (in which caseDhas Property (δ)), thenDis Dedekind.
Next-To-Interpolatory Approximation on Sets with Multiplicities
Authors: Motzkin T. S. , Sharma A.
Published: December 2010

It is known that given a set X of m (⩾n) distinct real numbers and a real-valued function f denned on X, there exists a unique polynomial pn-1,f,x of degree n — 1 or less which approximates best to f(x) on X, that is, which minimizes the deviation δ = δ(f, p) defined by the αth-power metric (α < 1) with positive weights, or by the positively weighted maximum of |f — p| on X; these deviations shall be denoted by δα and δβ. The polynomial pn-1,f,x has the property that f — pn-1,f,x has at least n strong sign changes; in other words, there are at least n + 1 points in X where the difference takes alternatingly positive and negative values.
On the “Essential Metrization” of Uniform Spaces
Authors: Zakon Elias
Published: December 2010

The notion of “essential metrization” was introduced in (8) where it was used to obtain some extensions of the theorems of Egoroff and Lusin on measurable functions. In the present note we shall further develop the theory of essential metrization, in its own right.

As is well known, sets of measure zero (“null sets“) may be disregarded in many problems of measure theory. Hence the usual topological prerequisites of such problems are actually too restrictive and may be replaced by what could be called “topology modulo null-sets,” imitating such notions as “approximate continuity,” “essential supremum,” etc. Thus we consider spaces in which certain topological properties (such as metrizability, separability, etc.) hold not in the usual sense but only “essentially,” i.e. to within some null sets, as defined below.
Additive Functionals on Lp Spaces
Authors: Friedman N. , Katz M.
Published: December 2010

In (1) a representation theorem was proved for a class of additive functionals defined on the continuous real-valued functions with domain S = [0, 1]. The theorem was extended to the case where S is an arbitrary compact metric space in (3). Our present purpose is to consider the corresponding class of additive functionals defined on Lp spaces, p > 0. In (4) Martin and Mizel have considered functionals defined on the class of bounded measurable functions which, however, satisfy a certain “stochastic” condition which we do not require.
Classification of Demushkin Groups
Authors: Labute John P.
Published: December 2010

A pro-p-group G is said to be a Demushkin group if

(1)

dimFp H1(G, Z/pZ) < ∞,

(2)

dimFp H2(G, Z/pZ) = 1,

(3)

the cup product H1(G, Z/pZ) × H1(G, Z/pZ) → H2(G, Z/pZ) is a non-degenerate bilinear form. Here FP denotes the field with p elements. If G is a Demushkin group, then G is a finitely generated topological group with n(G) = dim H1(G, Z/pZ) as the minimal number of topological generators; cf. §1.3. Condition (2) means that there is only one relation among a minimal system of generators for G; that is, G is isomorphic to a quotient F/(r), where F is a free pro-p-group of rank n = n(G) and (r) is the closed normal subgroup of F generated by an element r ∈ F9 (F, F); cf. §1.4.
On the Enumeration of Tree-Rooted Maps
Authors: Mullin R. C.
Published: December 2010

It is the purpose of this paper to show that many of the enumerative techniques available for counting rooted plane trees may be extended to tree-rooted maps, that is, rooted maps in which a spanning tree is distinguished as root tree. For example, tree-rooted maps are enumerated by partition, and the average number of trees in a rooted map with n edges is determined. An enumerative similarity between Hamiltonian rooted maps (that is, rooted maps with a distinguished Hamiltonian polygon) and tree-rooted maps is discussed. A 1-1 correspondence is established between treerooted maps with n edges and Hamiltonian rooted trivalent maps with 2n + 1 vertices in which the root vertex is exceptional, being divalent, both of which are in 1-1 correspondence with non-separable Hamiltonian-rooted triangularized digons with n internal vertices, where both the latter are as defined in (2).
Derivatives and Integrals with Respect to a Base Function of Generalized Bounded Variation
Authors: Ellis H. W. , Jeffery R. L.
Published: December 2010

In this paper we consider measures determined by arbitrary functions G(x) for which finite right and left limits exist everywhere and indicate how some of these measures permit the definition of generalized integrals of constructive or Denjoy type. These definitions are related to corresponding descriptive definitions based on the Perron approach as given by Ward (6) and Henstock (2). An exposition of the introductory theory is given in (1).
Extreme Points in H1(R)
Authors: Forelli Frank
Published: December 2010

Let R be an open Riemann surface. ƒ belongs to H1(R) if ƒ is holomorphic on R and if the subharmonic function |ƒ| has a harmonie majorant on R. Let p be in R and define ||ƒ|| to be the value at p of the least harmonic majorant of |ƒ|. ||ƒ|| is a norm on the linear space H1(R), and with this norm H1(R) is a Banach space (7). The unit ball of H1(R) is the closed convex set of all ƒ in H1(R) with ||ƒ|| ⩽ 1. Problem: What are the extreme points of the unit ball of H1(R)? de Leeuw and Rudin have given a complete solution to this problem where R is the open unit disk (1).
Proof of the Fixed Point Theorems of Poincaré and Birkhoff
Authors: Barrar Richard B.
Published: December 2010

In 1912, shortly before his death, Poincaré (8) conjectured the following theorem in his investigation of the restricted problem of three bodies.

Poincaré's Last Geometric Theorem. Given a ring 0 < a ⩽ r ⩽ b in the r, θ plane and a homeomorphic, area-preserving mapping T of the ring onto itself under which points on r = a advance and those on r = b regress, there will exist at least two points of the ring invariant under T.

Poincaré was able to prove this theorem in only a few special cases. Shortly thereafter, Birkhoff was able to give a complete proof in (2) and in, (3) he gave a generalization of the theorem, dropping the assumption that the transformation was area-preserving. Birkhoff's proofs were very ingenious; however, they did not use standard topological arguments.
Univalent α-Spiral Functions
Authors: Libera Richard J.
Published: December 2010

Supposeƒis regular in the open unit disk Δ, |z| < 1, and has a simple zero at the origin and no other zeros. Špaček (15) essentially showed thatƒis univalent in Δ if and only if

such that 0 <r< 1 and 0 <t2—t1⩽ 2π.
Triangulation of Fibre Bundles
Authors: Putz H.
Published: December 2010

In this paper we consider the following problem. Let (E, M, N, π) be a differentiable fibre bundle, whereEis the total space,Mthe base space,Nthe fibre, andπ: E→Mthe projection map. Then, given aCrtriangulation (ƒ, D) ofM,can one obtain aCrtriangulation (F, K) ofEsuch that the induced mapƒ–1πF:K→ D is linear? R. Lashof and M. Rothenberg (3) have obtained this result for vector bundles.

Using methods quite different from theirs, we obtain a solution in the general case. The methods we use are the geometric methods developed by J. H. C. Whitehead. (7) in his triangulation of differentiable manifolds, as found in (5). In fact, our solution consists of generalizing his techniques in a fibre bundle setting.
An Extremal Problem for Polygons Inscribed in a Convex Curve
Authors: Bollobás Béla
Published: December 2010

A. Zirakzadeh (1) has determined for n = 3 the minimal value of the perimeter length of a polygon A1 A2 … An, where A1, A2, … , An–1, and An divide the perimeter of a convex curve C, of perimeter length l, into n parts of equal length; further he has stated a conjecture concerning the general case. In the following a simpler proof for the case n = 3 is given; the minimum for even values of n, which confirms the conjecture of A. Zirakzadeh, is determined; and a fairly precise estimation for odd values of n, which refutes the conjecture of A. Zirakzadeh, is given. For n = 3 we have the following theorem.
Inertial Isomorphisms of V-Rings
Authors: Heerema N.
Published: December 2010

Throughout this paper R and Rn will denote v-rings, that is, complete discrete rank-one valuation rings of characteristic zero, having a common residue field k of characteristic p. R is assumed unramified and Rn has ramification index n. Let π be a prime element in Rn. Then Rn = R[π], where π is a root of an Eisenstein polynomial ƒ = xn + pƒn-1 xn-1 + … + pƒ0 with coefficients in R and ƒ0 a unit. Thus Rn is inertially isomorphic to R[[x]]/ƒR[(x)], that is, the rings are isomorphic by a mapping which induces the identity mapping on the common residue field. R[[x]] represents the power series ring in the indeterminate x over R. In this paper we identify Rn with R[[x]]/ƒR[[x]], R with its natural embedding in Rn and π with x + ƒR[[x]].
Intrinsic Functions on Semi-Simple Algebras
Authors: Hall C. A.
Published: December 2010

Rinehart (5) has introduced and motivated the study of the class of intrinsic functions on a linear associative algebra , with identity, over the real field R or the complex field C. In this paper we shall consider a semi-simple algebra = ⊕ … ⊕ over R or C with simple components . Let G be the group of all automorphisms or anti-automorphisms of which leave the ground field elementwise invariant, and let H be the subgroup of G such that Ω = (i = 1, 2, … , t) for each Ω in H.
Semi-Groups of Maps in a Locally Compact Space
Authors: Dorroh J. R.
Published: December 2010

Suppose that S is a locally compact Hausdorff space. A one-parameter semi-group of maps in S is a family {ϕt; t ⩾ 0} of continuous functions from S into S satisfying

(i)

ϕt0ϕu = ϕt+u for t, u ⩾ 0, where the circle denotes composition, and

(ii)

ϕ0 = e, the identity map on S.

A semi-group {ϕt} of maps in S is said to be

(iii)

of class (C0) if ϕt(x) → x as t → 0 for each x in S,

(iv)

separately continuous if the function t → ϕt(x) is continuous on [0, ∞) for each x in S, and

(v)

doubly continuous if the function (t, x) → (ϕt(x) is continuous on [0, ∞) x S.
On Meromorphic Operators, I
Authors: Caradus S. R.
Published: December 2010

If X is a complex Banach space and B(X) denotes the space of bounded linear operators on X, then the class of meromorphic operators consists of those T in B(X) such that the non-zero points of σ(T) are poles of the resolvent Rλ(T). If we also require that each non-zero eigenvalue of T have finite multiplicity, members of the class ⊆ so defined have been called operators of Riesz type. and have been studied in (2, 6, 7) and (1,4) respectively.
The Invariant Subspace Lattice of a Linear Transformation
Authors: Brickman L. , Fillmore P. A.
Published: December 2010

The purpose of this paper is to study the lattice of invariant subspaces of a linear transformation on a finite-dimensional vector space over an arbitrary field. Among the topics discussed are structure theorems for such lattices, implications between linear-algebraic properties and lattice-theoretic properties, nilpotent transformations, and the conditions for the isomorphism of two such lattices. These topics correspond roughly to §§2, 3, 4, and 5 respectively.
Soluble Artinian Groups
Authors: Baer Reinhold
Published: December 2010

The principal aim of this note is the proof of the following:

Main Theorem. The following properties of the group G are equivalent:

I. G is artinian and soluble.

II. (a) To every maximal subgroup S of G there exists a normal subgroup T of G with G = ST and S ∩ T = (S ∩ T)T.
A Variant of the Problem of the Thirteen Spheres
Authors: Tóth L. Fejes , Heppes A.
Published: December 2010

We use the termballsfor congruent, closed spheres no two of which have interior points in common. In Euclideann-space letNnbe the maximal number of balls which can touch a ball. Obviously,N2=6. R. Hoppe (see (1)) proved that N3= 12, settling thereby a famous point of controversy between Newton and David Gregory, known as the problem of the thirteen spheres (see (3)). Simpler proofs were given by Günter (6), Schütte and van der Waerden (10), and Leech (7).
Antisymmetrical Digraphs
Authors: Tutte W. T.
Published: December 2010

We call a digraph “antisymmetrical” if there is an automorphismθof its graph, of period 2, which reverses the direction of every edge and maps no edge or vertex onto itself. We construct a theory of flows invariant underθfor such a diagraph. This theory is analogous to the Max Flow Min Cut theory for ordinary flows in digraphs. It is found to include that part of the theory of undirected graphs which discusses the existence of spanning subgraphs with a specified valency at each vertex.
Extensions of Endomorphisms from the Higher Centres
Authors: Haimo Franklin
Published: December 2010

If 0 → A → C → B → 0 is an exact sequence of abelian groups, if ƒ is a 2-cocyle for this extension, if α ∈ End A, and if β ∈ End B, then a necessary and sufficient condition that α extend to an endomorphism γ of C which induces β is that (M) αƒ and ƒβ be cohomologous ; see Montgomery (2). We shall extend this result to the case where 1 → A → G → B → 1 is an exact sequence of groups and A is abelian.
On Almost Regular Homeomorphisms
Authors: Kaul S. K.
Published: December 2010

Let (X, d) be a metric space with metric d, and h be a homeomorphism of X onto itself. Any point y in X is called a regular point (2) under A if for any given ϵ > 0 there exists a δ > 0 such that d(x, y) < δ implies that d(hn(x), hn(y)) < ϵ for all integers n, where hn is the composition of h or h-1 with itself |n| times, depending upon whether n is positive or negative, and h0 is the identity on X. If y is not regular under h, then y is called irregular. We shall denote the set of regular points by R(h) and the set of irregular points by I(h).
On Integral Functions Having Prescribed Asymptotic Growth. II
Authors: Clunie J. , Kövari T.
Published: December 2010

One of the authors published in 1965 a paper with identical title (1), in which the following result was proved:

Theorem A. Let ϕ(r) be increasing and convex in log r, with
Four-Dimension Equivalences
Authors: Gard J. R. , Johnson R. D.
Published: December 2010

The object of this paper is to establish the equivalence of four functionrelated dimension concepts in arbitrary topological spaces. These concepts involve stability of functions (3, p. 74), the modification of covering dimension involving basic covers (1, p. 243) (which is equivalent to Yu. M. Smirnov's definition using normal covers), the definition involving essential mappings (2, p. 496), and a modification of the closed set separation characterization of dimension in (3, p. 35).
A Riemann-Type Integral of Lebesgue Power
Authors: Henstock Ralph
Published: December 2010

The introduction of a mathematics student to formal integration theory usually follows the lines laid down by Riemann and Darboux. Later a change of ideas is necessary if he tackles Lebesgue's more powerful theory, and connections between the two are laboriously constructed. On the other hand, the commonest method of evaluating an integral is through the operation inverse to differentiation (the indefinite integral taken between limits). We refer to this as the calculus integral; few realize that this can succeed in cases where even the Lebesgue integral does not exist, let alone the Riemann one. An example is given later.
Convex Sets of Non-Negative Matrices
Authors: Brualdi R. A.
Published: December 2010

In(8)M. V. Menon investigates the diagonal equivalence of a non-negative matrixAto one with prescribed row and column sums and shows that this equivalence holds provided there exists at least one non-negative matrix with these row and column sums and with zeros in exactly the same positionsAhas zeros. However, he leaves open the question of when such a matrix exists. W. B. Jurkat and H.J. Ryser in (7) study the convex set of allm×nnon-negative matrices having given row and column sums.
Commutators of Matrices with Prescribed Determinant
Authors: Thompson R. C.
Published: December 2010

Let K be a commutative field, let GL(n, K) be the multiplicative group of all non-singular n × n matrices with elements from K, and let SL(n, K) be the subgroup of GL(n, K) consisting of all matrices in GL(n, K) with determinant one. We denote the determinant of matrix A by |A|, the identity matrix by In, the companion matrix of polynomial p(λ) by C(p(λ)), and the transpose of A by AT. The multiplicative group of nonzero elements in K is denoted by K*. We let GF(pn) denote the finite field having pn elements.
Monotone and E-Schauder Bases of Subspaces
Authors: Russo John P.
Published: December 2010

The notions of monotone bases and bases of subspaces are well known in a normed linear space setting and have obvious extensions to pseudo-metrizable linear topological spaces. In this paper, these notions are extended to arbitrary linear topological spaces. The principal result gives a list of properties that are equivalent to a sequence (Mi) of complete subspaces being an e-Schauder basis of subspaces for the closed linear span of . A corollary of this theorem is the fact that an e-Schauder basis for a dense subspace of a linear topological space is an e-Schauder basis for the whole space.
Maximal or Greatest Homomorphic Images of Given Type
Authors: Tamura Takayuki
Published: December 2010

Let Q be a quasi-ordered set with respect to ⩽ ; that is, the order ⩽ is reflexive and transitive. An element a of Q is called maximal (minimal) if

a is called greatest (smallest) if

Obviously a greatest (smallest) element is maximal (minimal). A greatest (smallest) element in a partially ordered set is unique, but it is not necessarily unique in a quasi-ordered set.
Problems on Measure Algebras
Authors: Kaufman Robert
Published: December 2010

Suppose that G is a locally compact abelian group, u an element of infinite order, and w a complex number of modulus 1. It is a familiar fact that there is a complex homomorphism Ψ of the measure algebra M of G, which maps ϵu (the unit mass concentrated at u) to w. Beyond this, one may specify an element μ of M, and require a homomorphism Ψ which does not annihilate μ. The resolution of this problem leads to an abstract lemma on measurable transformations, derived in some generality in the first section.
Standard Representations of Simple Lie Algebras
Authors: Bouwer I. Z.
Published: December 2010

Let L be any simple finite-dimensional Lie algebra (defined over the field K of complex numbers). Cartan's theory of weights is used to define sets of (algebraic) representations of L that can be characterized in terms of left ideals of the universal enveloping algebra of L. These representations, called standard, generalize irreducible representations that possess a dominant weight. The newly obtained representations are all infinite-dimensional. Their study is initiated here by obtaining a partial solution to the problem of characterizing them by means of sequences of elements in K.
A Note on Certain Subalgebras of C()
Authors: Hager Anthony W. , Johnson Donald G.
Published: December 2010

Let be a completely regular Hausdorff space and C() the algebra of continuous real-valued functions on . In attempts to characterize abstractly those algebras that are isomorphic to C() for some , one produces subalgebras of C() which: (a) contain the constant functions, (b) separate points and closed sets in , (c) are closed under uniform convergence, and (d) are closed under inversion in C() (see, for example, (2; 5)).
Approximation in Bounded Summability Fields
Authors: Hill J. D. , Sledd W. T.
Published: December 2010

This paper deals with several related properties of bounded summability fields of regular, real matrices. For a matrix A = (ank) and a sequence x = {xn}, we write formally

We denote by m the space of bounded real sequences, and by A* the bounded summability field

of A. The strong summability field of A is the set
The Size of the Unit Sphere
Authors: Whitley Robert
Published: December 2010

Banach (1, pp. 242-243) defines, for two Banach spaces X and Y, a number (X, Y) = inf (log (‖L‖ ‖L-1‖)), where the infimum is taken over all isomorphisms L of X onto F. He says that the spaces X and Y are nearly isometric if (X, Y) = 0 and asks whether the concepts of near isometry and isometry are the same; in particular, whether the spaces c and c0, which are not isometric, are nearly isometric. In a recent paper (2) Michael Cambern shows not only that c and c0 are not nearly isometric but obtains the elegant result that for the class of Banach spaces of continuous functions vanishing at infinity on a first countable locally compact Hausdorff space, the notions of isometry and near isometry coincide.
The Hereditary Property in the Lower Radical Construction
Authors: Armendariz E. P. , Leavitt W. G.
Published: December 2010

All rings considered are associative. We show that if a homomorphically closed class P1 of rings is hereditary in the sense that every ideal of a ring in P1 is also in P1, then the lower Kurosh radical construction terminates at P3. This is an improvement on the result of Anderson, Divinsky, and Sulinski (3) showing that the lower radical construction terminates at P2 provided P1 is homomorphically closed, hereditary, and contains all zero rings. Examples are given to show that the third step is actually attained in some constructions.
On S2-Groups and Groups of Moebius Transformations
Authors: Lorimer P. J.
Published: December 2010

Let PGL(2, ƒ) denote the group of all Moebius transformations

over a field F. The object of this paper is to prove the following theorem.

Theorem 1. G is an S2-group and the centre of G is trivial if and only if G is isomorphic to a group PGL(2, ƒ), char ƒ ≠ 2.

This theorem was proved for finite groups in (1). The present paper extends the result to infinite groups and also improves the method of proof used in that paper. Many of the theorems given there were proved for infinite groups and are used here with appropriate references.
On Factorization of Elliptic Functions
Authors: Gross Fred
Published: December 2010

In this paper we shall be concerned with the following problem: If h is an elliptic function and h(z) = ƒ(g(z)), what can be said about the functions ƒ and g? In order to simplify the discussion we introduce some basic definitions.

Definition 1. A meromorphic function h(z) = ƒ(g(z)) is said to have ƒ(z) and g(z) as left and right factors, respectively, provided that either ƒ(z) is non-linear and meromorphic and g(z) is non-linear and entire or f(z) is rational and g(z) is meromorphic.
Catégories Semi-Simples et Catégories Primitives
Authors: Leduc P.-Y.
Published: December 2010

Le but de cet article est d'établir une théorie des catégories additives axée sur l'idée de structure. Etant donné ce que nous connaissons aujourd'hui des catégories additives, il était tout naturel qu'une telle étude fut entreprise, et ce, dans l'optique d'une généralisation de la théorie classique (Wedderburn-Artin-Jacobson) des anneaux non-commutatifs avec élément unité. C'est dans “On the radical of a category” de G. M. Kelly (2) que l'on retrouve une amorce à l'étude des catégories additives, telle que nous la concevons dans cet article.
A Theorem Concerning Partitions and its Consequence in the Theory of Lie Algebras
Authors: Hughes J. W. B.
Published: December 2010

In the first part of this paper we state and prove a theorem concerning the partition (j; l, i) of an integer j into at most l integers , none of which exceed i; l and i being themselves integers, (j; l, i) is thus the number of distinct solutions of the equations

1.1

where the satisfy the inequalities

1.2
On Metrizability of Topological Spaces
Authors: Borges Carlos J. R.
Published: December 2010

Our present work is divided into three sections. In §2 we study the metrizability of spaces with a Gδ-diagonal (see Definition 2.1). In §3 we study the metrization of topological spaces by means of collections of (not necessarily continuous) real-valued functions on a topological space. Our efforts, in §§2 and 3, are directed toward answering the following question: “Is every normal, metacompact (see Definition 2.4) Moore space a metrizable space?” which still remains unsolved. (However, Theorems 2.12 through 2.15 and Theorem 3.1 may be helpful in answering the preceding question.) In §4 we prove an apparently new necessary and sufficient condition for the metrizability of the Stone-Čech compactification of a metrizable space and hence for the compactness of a metric space.
The Use of S-Functions in Combinatorial Analysis
Authors: Read Ronald C.
Published: December 2010

The aim of this paper is to present a unified treatment of certain theorems in Combinatorial Analysis (particularly in enumerative graph theory), and their relations to various results concerning symmetric functions and the characters of the symmetric groups. In particular, it treats of the simplification that is achieved by working with S-functions in preference to other symmetric functions when dealing with combinatorial problems. In this way it helps to draw closer together the two subjects of Combinatorial Analysis and the theory of Finite Groups. The paper is mainly expository; it contains little that is really new, though it displays several old results in a new setting.
Homology of Deleted Products of Contractible 2-Dimensional Polyhedra. II
Authors: Patty C. W.
Published: December 2010

The deleted product space X* of a space X is X × X — △. In (4), I computed the homology groups of the deleted product of a polyhedron in a subcollection (see §2 of this paper for the definition of ) of the finite, contractible, 2-dimensional polyhedra. In the present paper, I show that there is an infinite subcollection ( of such that the deleted product of each member of has the homotopy type of the 2-sphere.
Direct Sums of Partial Algebras and Final Algebraic Structures
Authors: Schmidt Jürgen
Published: December 2010

Słomiński (9), as well as the author (8), gave a descriptive, i.e., noncategory-theoretic, definition of the direct sum of partial algebras, i.e., the co-product in the category of partial algebras (A,ƒ), where ƒ = (ƒi)i∈I, ƒi: dom ƒi ⟶ A, dom ƒi ⊂ AKi, of fixed type A = (Ki)i∈I.
Rings Satisfying (x, y, z) = (y, z, x)
Authors: Sterling Nicholas J.
Published: December 2010

Let R be a ring satisfying the identity

1

for all x, y, z ϵ R, where (x,y,z) = (xy)z — x(yz). If R also satisfies the identity (x, x, x) = 0 for all x ϵ R, then R is alternative. It is known that if R satisfies (1), it need not be an alternative (see 6). Thus, the class of rings satisfying (1) is a non-trivial extension of the class of alternative rings. P. Jordan remarked that (x, x, x)2 = 0 is an identity in R (see 9).
On Subsemigroups of the Projective Group on the Line
Authors: Lowenthal Franklin
Published: December 2010

The subsemigroups of the projective group on the line that are described in this paper are those that can be generated by a pair of infinitesimal transformations. One denotes by G the connected component of the identity of this group; Theorem 1 gives a necessary and sufficient condition for a pair of infinitesimal transformations to generate a subsemigroup which is equal to G (and hence is actually a group). This condition is reformulated in a geometric manner in Theorem 1*.
Simultaneous Unitary Invariants for Sets of Matrices
Authors: Radjavi Heydar
Published: December 2010

It is our aim in this paper to give an elementary solution to the problem of simultaneous unitary equivalence of two finite sets of matrices, i.e., given two ordered sets {Aj} and {Bj} of n × n matrices, j = 1, 2, … , m, we wish to determine whether there exists a unitary matrix U such that Bj = U*AjU for all j. A special case of this problem is that of unitary equivalence of two arbitrary matrices.
Extreme Points and Linear Isometries of the Banach Space of Lipschitz Functions
Authors: Roy Ashoke K.
Published: December 2010

Let X be a compact metric space with metric d. A complex-valued function ƒ on X is said to satisfy a Lipschitz condition if, for all points x and y of X, there exists a constant K such that

The smallest constant for which the above inequality holds is called the Lipschitz constant for ƒ and is denoted by ||ƒ||d, that is,
Restrictive Semigroups of Closed Functions
Authors: Magill Kenneth D.
Published: December 2010

It is assumed that all topological spaces discussed in this paper are T1 spaces. A function ƒ mapping a topological space X into itself is a closed function if ƒ[H] is closed for each closed subset H of S. The semigroup, under composition, of all closed functions mapping X into X is denoted by Γ(X). These were among the semigroups under consideration in (4).
Subordinate and Pseudo-Subordinate Semi-Algebras. II
Authors: Barbeau Edward J.
Published: December 2010

This paper is a sequel to (1), to which the reader is referred for definitions and known results. As before, £ is a compact Hausdorff space and C+(E) is the semi-algebra of all continuous non-negative functions defined on E.
Finite Groups with Normal Normalizers
Authors: Hobby C.
Published: December 2010

We say that a finite group G has property N if the normalizer of every subgroup of G is normal in G. Such groups are nilpotent since every Sylow subgroup is normal (the normalizer of a Sylow subgroup is its own normalizer). Thus it is sufficient to study p-groups which have property N. Note that property N is inherited by subgroups and factor groups.
Concerning Upper Semi-Continuous Decompositions of En Whose Non-Degenerate Elements are Polyhedral Arcs or Star-Like Continua
Authors: Treybig L. B.
Published: December 2010

In (1) Armentrout raised the question “Is there a monotone decomposition of E3 into arcs?” The analogous question for E2 was answered negatively by Roberts in (8). Our aim in this paper is to give a partial answer to Armentrout's question by proving the following theorem.
Engel Congruences in Groups of Prime-Power Exponent
Authors: Gupta N. D. , Newman M. F.
Published: December 2010

It is a well-known result of Sanov (5) that groups of exponent pk (p prime) satisfy the th Engel congruence (definition below). Recently, an alternative proof of this has been given by Glauberman, Krause, and Struik (3). Bruck (2) has conjectured that such groups satisfy the th Engel congruence. In this note we go some way towards proving this.
On Spectral Properties of Matrices with Positive Characteristic Vectors
Authors: Majindar Kulendra N.
Published: December 2010

Unless stated otherwise, all our matrices (denoted by capital letters) are square matrices of size n × n and composed of real numbers. A' denotes the transpose of A. The characteristic or eigenvectors of matrices are written as column vectors having n coordinates. If ζ is a vector, ζ’ denotes its transpose.
Correspondences of Characters for Relatively Prime Operator Groups
Authors: Glauberman George
Published: December 2010

Let G be a finite group and let A be a finite solvable operator group on G. Suppose that A and G have relatively prime orders. Let T be the fixed-point subgroup of G with respect to A. We say that A fixes a complex character ζ of G if ζ (gα) = ζ (g) for all g ∈ G and α ϵ A. Our aim in this paper is to define a one-to-one correspondence between the irreducible characters of T and those irreducible characters of G that are fixed by A, and to prove some properties of this correspondence that were mentioned in (8).
On the Lower Derivate of a Set Function
Authors: Pfeffer W. F.
Published: December 2010

In (5), the following theorem was proved in a very general setting:

(1) An additive set function is non-negative whenever its lower derivative is non-negative.

For a continuous additive function of intervals, theorem (1) can be improved as follows:

(2) A continuous additive set function is non-negative whenever its lower derivative is non-negative except, perhaps, on a countable set.
The Categorical Product of Graphs
Authors: Miller Donald J.
Published: December 2010

Undirected graphs and graph homomorphisms as introduced by Sabidussi (6, p. 386), form a category that admits a categorical product. For the category of graphs and full graph homomorphisms, the categorical product was introduced by Čulik (1) under the name cardinal product. It was independently defined by Weichsel (8) who called it the Kronecker product and investigated the connectedness of products of finitely many factors. Hedetniemi (4) was the first to make use of the fact that the cardinal product is categorical.
On Projective Modules and Automorphisms of Central Separable Algebras
Authors: Childs L. N.
Published: December 2010

This paper developed from, and complements, the paper by F. R. DeMeyer (see 6).

In the first section of this paper we note a correspondence between projective modules of a central separable R-algebra A and the two-sided ideals of central separable algebras in the same class as A in the Brauer group of R. When R has the property that rank one projective A-modules are free, this correspondence yields a bijection between isomorphism types of indecomposable projective A-modules and the isomorphism types of algebras in the Brauer class of A which are the analogue of division algebra components in the field case. This bijection was remarked on without proof by DeMeyer in (6).

Pursuing the ideas behind this correspondence, we consider the situation for a separable order A in a central simple algebra A over an algebraic number field, and obtain, by means of results involving the reduced norm, a generalization of DeMeyer's remark except when the division algebra component of A is a totally definite quaternion algebra (Theorem 3.3).
An Analogue of a Problem of J. Balázs and P. Turán
Authors: Prasad J. , Varma A. K.
Published: December 2010

In 1955, J. Surányi and P. Turán (8) initiated the problem of existence and uniqueness of interpolatory polynomials of degrees less than or equal to 2n — 1 when their values and second derivatives are prescribed on n given nodes. This kind of interpolation was termed (0, 2)-interpolation. Later, Balázs and Turán (1) gave the explicit representation of the interpolatory polynomials for the case when the n given nodes (n even) are taken to be the zeros of πn(x) = (1 — x2)Pn′(x), where Pn–i(x) is the Legendre polynomial of degree n — 1. In this case the explicit representation of interpolatory polynomials turns out to be simple and elegant.

Balázs and Turán (2) proved the convergence of these polynomials when f(x) has a continuous first derivative satisfying certain conditions of modulus of continuity. They noted (1) that a significant application of lacunary interpolation could possibly be given in the theory of a differential equation of the form y′ + A(x)y= 0.
Hjelmslev Planes Derived from Modular Lattices
Authors: Artmann Benno
Published: December 2010

In several papers, W. Klingenberg has elaborated the connections between Hjelmslev planes and a class of rings, called H-rings (4; 5; 6), which are rings of coordinates for the corresponding Hjelmslev planes. Certain homomorphic images of valuation rings are examples of H-rings. In these examples, the lattice of (right) ideals of the ring, say R,is a chain, and the coordinatization of the corresponding Hjelmslev plane yields a natural embedding of the plane in the lattice L(R3) of (right) submodules of the module R3. Now, L(R3) is a modular lattice with a homogeneous basis of order 3 given by the submodules a1 = (1, 0, 0)R, a2 = (0, 1, 0)R, a2 = (0, 0, 1)R, and the sublattices L(N, ai) of elements less than or equal to ai are chains. Forgetting about the ring, we find ourselves in the situation of a problem suggested by Skornyakov (7, Problem 23, p. 166), namely, to study modular lattices with a homogeneous basis of chains. Baer (2) and Inaba (3) investigated lattices of this kind with Desarguesian properties and assuming that the chains L(N, ai) were finite. Representations of the lattices by means of certain rings can be found in both articles.
A Note on Non-Distributive Sublattices of Degrees and Hyperdegrees
Authors: Thomason S. K.
Published: December 2010

In (1, §§ 2.3 and 2.4) we proved that certain distributive lattices are simultaneously lattice-embeddable in the degrees of recursive unsolvability and in the hyperdegrees. Let ℒ be the non-distributive lattice {0,1, a0, a1,…}, where ai ∪ aj = 1 and ai ∩ aj = 1 whenever i ≠ j. We shall prove the following theorem.

THEOREM. The lattice ℒ is simultaneously lattice-embeddable in the degrees and hyperdegrees.

For A ⊆ N, let deg(A) and hyp(A) be the degree and hyperdegree of A, respectively. To prove the theorem we must construct hyperarithmetically incomparable sets A0, A1, … such that for Δ = deg, hypand for all distinct i, j:

1

2

Now, if each 〈Ai, Aj〉 were a generic pair in the sense of (1), then (2) would hold.
On Finite Collineation Groups of F5
Authors: Sandler Reuben
Published: December 2010

Our aim in this paper is to fill one gap left in (1) and to prove that if H is a finite collineation group of F5, the free plane generated by a finite open configuration of rank 9, then |H | ≦ 12. Alltop has shown that |H | ≦ 24 and that there exist finite collineation groups of F5 which have order 12, so that the argument in this paper shows that |H| ≦ 12 is the best estimate which can be given. In (1), Alltop has completely settled the question for Fn, n ≠ 5. The notation of this paper will generally be that of (1).

Most of the arguments used here will consist of case analyses of degenerate planes of ranks 7 and 8 and will be sketched rather than given in detail. The one theorem of some interest in this paper, other than the main result, is the following theorem which yields some information about the collineation group of F4 (π2 in the notation of (4)).
On the Criterion of Stasheff
Authors: Hoo C. S.
Published: December 2010

In (9), Stasheff showed that a loop space ΩX is homotopy-commutative if and only if the map e∇ : ΣΩX ∨ ΣΩX → X may be extended to ΣΩX × ΣΩX, where ∇ is the folding map and e:ΣΩX → X is the map whose adjoint is the identity map of ΩX. In (5), Ganea, Hilton, and Peterson showed that this criterion does not dualize. Our aim in this paper is to give a reformulation of Stasheff's criterion, which is equivalent to it, but in a form which does dualize. In the course of the paper, we shall discuss why Stasheff's criterion, in its original form, does not dualize. In (5), the authors have, of course, given an explicit counterexample to the dual of StashefFs criterion in its original form.