bluuebunny/crossref_metadata_2025 huggingface dataset:
Group Algebra Modules. II
Authors: Gulick S. L. , Liu T. S. , Van Rooij A. C. M.
Published: December 2010

The present paper began as a natural outgrowth of our first paper, where we characterized the module homomorphisms from group algebras into a fairly restrictive class of group algebra modules. We now investigate module homomorphisms from group algebras into a more general class of group algebra modules. Although the two papers are thus related, they can be read quite independently.

Section 2 contains our extension, Theorem 2.1, of P. J. Cohen's theorem on factorization in Banach algebras (1). Our extension is to Banach modules over Banach algebras equipped with an approximate identity. We should mention first that J.-K. Wang observed the existence of such a generalization, and secondly, that our proof requires no ideas different from those in Cohen's proof. Nevertheless, we include a proof that condenses the original proof considerably.
Concerning Non-Planar Circle-Like Continua
Authors: Ingram W. T.
Published: December 2010

In this paper it is proved that if a circle-like continuum M cannot be embedded in the plane, then M is not a continuous image of any plane continuum (Theorem 5).

Suppose that (S, ρ) is a metric space. A finite sequence of domains L1, L2, … , Ln is called a linear chain provided Li intersects Lj if and only if |i — j| ⩽ 1. If, in addition, there is a positive number ∊ such that, for each i, the diameter of Li is less than ∊, then the linear chain is called a linear ∊-chain. If for each positive number ∊ the continuum M can be covered by a linear ∊-chain, then M is said to be chainable (or snake-like) (2).
Sets of Disjoint Lines in PG(3, q)
Authors: Mesner Dale M.
Published: December 2010

Let ∑ be a projective space PG(3, q) of dimension 3 and finite order q. Then ∑ contains (q + 1)(q2 + 1) points and an equal number of planes, and (q2 + 1) (q2 + q + 1) lines. It will be convenient to consider lines and planes as sets of points and to treat the incidence relation as set inclusion. Each plane contains q2 + q + 1 points and an equal number of lines. Each line contains q + 1 points and is contained in an equal number of planes. Each point is contained in q2 + q + 1 planes and an equal number of lines.

A spread of lines of ∑ is a set of q2 + 1 lines of ∑ which are pairwise disjoint, or skew; it can also be defined as a set of lines such that each point (or each plane) is incident with exactly one of the lines.
Embedding -Like Compacta in Manifolds
Authors: McCord Michael C.
Published: December 2010

A compactum is a compact, metrizable space. A continuum is a connected compactum. All polyhedra will be finitely triangulable spaces. If a is an open cover of a compactum X, a map of X onto a compactum Y is called an α-map provided that the inverse image of each point in Y is contained in some member of α.

If is a class of polyhedra, then, following Mardešić and Segal (10), we say a compactum X is -like provided that for each open cover α of X there exists an a-map of X onto some member of .
A Decomposition Theorem for Matrices
Authors: Pearl Martin H.
Published: December 2010

According to a classical theorem originally proved by L. Autonne (1; 3) in 1915, every m × n matrix of rank r with entries from the complex field can be decomposed as

where U1 and U2 are unitary matrices of order m and n respectively and D is an m × n matrix having the form

1

where Δ is a non-singular diagonal matrix whose rank is r. If r = m, then the row of zero matrices of (1) does not actually appear. If r = n, then the column of zero matrices of (1) does not appear. The main purpose of this paper is to give a necessary and sufficient condition under which both U1 and U2 may be chosen to be real orthogonal matrices.
On a Kind of Homotopy Manifold
Authors: Akasaki T.
Published: December 2010

In a recent paper (6), S. T. Hu investigated the initial projection from the mth enveloping space of a topological space X into X and proved that, under some local conditions on X, the initial projection is a fibering. In a subsequent paper (7), Hu showed that the terminal projection from the mth enveloping space is a fibering without assuming the local conditions on X and in (8) he used the terminal projection from the second enveloping space in his topological immersion theorem.
Notes on Hypersurfaces in a Riemannian Manifold
Authors: Yano Kentaro
Published: December 2010

H. Liebmann (3) and W. Süss (7) proved

Theorem A. The only convex closed hypersurface with constant mean curvature in a Euclidean space is a sphere.

Y. Katsurada (1; 2) gave the following generalization.

Theorem B. Let M be an orientable Einstein space which admits a proper conformai Killing vector field, that is, a vector field generating a local one-parameter group of conformai transformations which is not that of isometries, and S a closed orientable hypersurface in M whose first mean curvature is constant. If the inner product of the conformai Killing vector field and the normal to the hypersurface has fixed sign on S, then every point of S is umbilical.
An Extension of the Concept of the Order Dual of a Riesz Space
Authors: Luxemburg W. A. J. , Masterson J. J.
Published: December 2010

Let L be a σ-Dedekind complete Riesz space. In (8), H. Nakano uses an extension of the multiplication operator on a Riesz space into itself (analagous to the closed operator on a Hilbert space) to obtain a representation space for the Riesz space L. He calls such an operator a “dilatator operator on L.” More specifically, he shows that the set of all dilatator operators , when suitable operations are defined, is a Dedekind complete Riesz space which is isomorphic to the space of all functions defined and continuous on an open dense subset of some fixed totally disconnected Hausdorff space. The embedding of L in the function space is then obtained by showing that L is isomorphic to a Riesz subspace of . Moreover, when L is Dedekind complete, it is an ideal in , and the topological space is extremally disconnected.
Differentiable Manifolds with an Area Measure
Authors: Brickell F.
Published: December 2010

In this section we fix some notations and give a definition of an area measure on a differentiate manifold, where throughout the paper the word differentiable implies differentiability of class C∞. LetMdenote a differentiate manifold of dimensionnand call a set ofmlinearly independent vectors{e1,… ,em} at a point ofManm-frame ofM.The setE′of all suchm-frames can be given the structure of a differentiable fibre bundle overMand we denote the projection ofE'ontoMbyπ′.
On Unsolvable Groups of Degree p = 4q + 1, p and q Primes
Authors: Appel K. I. , Parker E. T.
Published: December 2010

This paper presents two results. They are:

Theorem 1. Let G be a doubly transitive permutation group of degree nq + 1 where a is a prime and n < g. If G is neither alternating nor symmetric, then G has Sylow q-subgroup of order only q.

Result 2. There is no unsolvable transitive permutation group of degree p = 29, 53, 149, 173, 269, 293, or 317 properly contained in the alternating group of degree p.

Result 2 was demonstrated by a computation on the Illiac II computer at the University of Illinois.
Some Generalized Hardy Spaces
Authors: Meeker L. D.
Published: December 2010

This paper is concerned with generalizations of the classical Hardy spaces (8, p. 39) and the question of boundary values for functions of these various spaces. The general setting is the “big disk” Δ discussed by Arens and Singer in (1, 2) and by Hoffman in (7). Analytic functions are defined in (1). Classes of such functions corresponding to the Hardy Hp spaces are considered and shown to possess boundary values in (2). Contrary to the classical case, such functions do not form a Banach space; hence they are not the functional analytic analogue of the classical spaces. In (3) quasi-analytic functions are defined while in (4) Hardy spaces of such functions are considered and are shown to have boundary values and to form a Banach space.
On Partially Ordering Operator Algebras
Authors: Demarr Ralph
Published: December 2010

In this paper, we consider linear spaces and algebras with real scalars. It is well known that if X is a Banach space and is the set of all bounded linear operators which map X into itself, then is a Banach algebra. In this paper we shall show that can be partially ordered so that it becomes a partially ordered algebra in which norm convergence is equivalent to order convergence. This motivates a study of Banach algebras of operators in which one uses the order structure to obtain various results. In addition, it encourages a study of partially ordered algebras in general, since our result shows that among such algebras one finds all real Banach algebras of operators. Of course, there are many other real algebras which are naturally partially ordered and which have been studied from that point of view.
Directed Covers and Paracompact Spaces
Authors: Mack John
Published: December 2010

A cover is directed (is well-ordered, totally ordered) if it is directed (is well-ordered, totally ordered, respectively) by set inclusion. Specifically is directed if, given U, V in , there exists W in such that U ⋃ V ⊂ W.

Directed covers have been used implicitly in numerous theorems dealing with paracompactness and related properties. In particular, characterizations of paracompactness in terms of product spaces seem to require the use of directed covers (3; 11, §2; or 12). However, the utility of directed covers has not been fully exploited.
Representation of Groups by Generalized Normal Multiplication Tables
Authors: Ginzburg A.
Published: December 2010

G will denote a finite (or infinite) group of order n. In a normal multiplication table (n.m.t.) of G (7, 8, 9, 12) all entries in one diagonal are equal to e (the identity of G), and if the entry on the intersection of the ith column and jth row is gi,j ∈ G, then

The following is a n.m.t. of Z6 = {0, 1, 2, 3, 4, 5}:

Remark. The cyclic groups Zn will always be written in additive notation. The table is uniquely defined by every one of its columns, in particular by the first.
On Indecomposable Graphs
Authors: Harary Frank , Plummer Michael D.
Published: December 2010

A set of points M of a graph G is a point cover if each line of G is incident with at least one point of M. A minimum cover (abbreviated m.c.) for G is a point cover with a minimum number of points. The point covering number α(G) is the number of points in any minimum cover of G. Let [V1, V2, … , Vr], r > 1 be a partition of V(G), the set of points of G. Let Gi be the subgraph of G spanned by Vi for i = 1, 2, … , r.
Phragmèn-Lindelöf and Comparison Theorems for Elliptic-Parabolic Differential Equations
Authors: Oddson J. K.
Published: December 2010

Theorems of Phragmèn-Lindelöf type and other related results for solutions of elliptic-parabolic equations have been given by numerous authors in recent years. Many of these results are based upon the maximum principle and the use of auxiliary comparison functions which are constructed as supersolutions of the equations under various conditions on the coefficients. In this paper we present an axiomatized treatment of these topics, replacing specific hypotheses on the nature of the coefficients of the equations by a single assumption concerning the maximum principle and another concerning the existence of positive supersolutions, in terms of which the theorems are stated.
On Some Applications of Graph Theory to Geometry
Authors: Erdös P.
Published: December 2010

Let [Pn(k)] be the class of all subsets Pn(k) of the k-dimensional Euclidean space consisting of n distinct points and having diameter 1. Denote by dk(n, r) the maximum number of times a given distance r can occur among points of a set Pn(k).Put

In other words Dk(n) denotes the maximum number of times the same distance can occur between n suitably chosen points in k-dimensional space.
Axes of Points of Certain Linear Systems of Polarities
Authors: Schuster Seymour
Published: December 2010

In this issue honouring Professor Coxeter, I am pleased to present some results of an investigation that was prompted by questions which he himself raised over a decade ago.

With respect to a linear system of polarities in complex projective three-space, the polars of a fixed point Q form an axial pencil of planes. The axis of the pencil is called the axis of point Q with respect to the linear system of polarities. Since there are ∞3 axes and ∞4 lines in the space, not every line is an axis. The following discussion answers the questions of how many and which lines are axes with respect to the linear systems of polarities that have a fixed self-polar tetrahedron.
Zur Verallgemeinerung der Neperschen Regel inr-Dimensionalen Riemannschen Räumen Konstanter Krümmung
Authors: Böhm J.
Published: December 2010

Bei der geometrischen Interpretation von gewissen Aufgaben z.B. aus dem Bereiche der Statistik einerseits oder der Matrizenalgebra andererseits treten Simplexe in Riemannschen Räumen konstanter Krümmung auf. Ein spezieller Typus eines solchen Simplexes ist das Orthoschem. Für dieses ergibt sich da nun die Aufgabe, anzugeben, in welcher Weise die Masse der Kanten und der verschiedensten Winkel eines Orthoschems miteinander zusammenhängen.
Polytopes with Centrally Symmetric Faces
Authors: Shephard G. C.
Published: December 2010

If a convex polytope P is centrally symmetric, and has the property that all its faces (of every dimension) are centrally symmetric, then P is called a zonotope. Zonotopes have many interesting properties which have been investigated by Coxeter and other authors (see (4, §2.8 and §13.8) and (5) which contains a useful bibliography). In particular, it is known (5, §3) that a zonotope is completely characterized by the fact that all its two-dimensional faces are centrally symmetric.
Solvability of Ordinary Differential Equations Near Singular Points: An Analytic Case
Authors: Ellis Homer G.
Published: December 2010

The question of solvability of the differential equation

1

with x ranging over an interval (0, a], and with the boundary condition ƒ(0+) = 0, can be investigated as an initial-value problem at 0, which may be a singular point for the equation.
On The Generalized Cauchy Equation
Authors: Monroe Itrel
Published: December 2010

It is the purpose of this note to prove the following theorem: Let ƒ: G → R be a non-constant continuous function with G a locally compact connected topological group and with R the real numbers. Let C = ƒ(G) and suppose that F: C × C → C is a junction such that

Then ƒ is monotone and open and F is continuous.
Cycles and Connectivity in Graphs
Authors: Watkins M. E. , Mesner D. M.
Published: December 2010

In this note, G will denote a finite undirected graph without multiple edges, and V = V(G) will denote its vertex set. The largest integer n for which G is n-vertex connected is the vertex-connectivity of G and will be denoted by λ = λ(G). One defines ζ to be the largest integer z not exceeding |V| such that for any set U ⊂ V with |U| = z, there is a cycle in G which contains U. The symbol i(U) will denote the component index of U. As a standard reference for this and other terminology, the authors recommend O. Ore (3).
Lifting Inductive and Projective Limits
Authors: Sonner Johann
Published: December 2010

In this paper, which is the fourth in a series of articles (11, 12, 13) on universal solutions in categories, a relationship between inductive limits and final structures (or projective limits and initial structures) is studied. The problems to be encountered are illustrated by the following example.
Free Vector Lattices
Authors: Baker Kirby A.
Published: December 2010

This note presents a useful explicit characterization of the free vector lattice FVL(ℵ) on ℵ generators as a vector lattice of piecewise linear, continuous functions on Rℵ, where ℵ is any cardinal and R is the set of real numbers. A transfinite construction of FVL(ℵ) has been given by Weinberg (14) and simplified by Holland (13, § 5). Weinberg's construction yields the fact that FVL(ℵ) is semi-simple; the present characterization is obtained by combining this fact with a theorem from universal algebra due to Garrett Birkhoff.
Ramanujan Congruences for p-k(n)
Authors: Atkin A. O. L.
Published: December 2010

Let

1

2

Thus p-1(n) = p(n) is just the partition function, for which Ramanujan (4) found congruence properties modulo powers of 5, 7, and 11. Ramanathan (3) considers the generalization of these congruences modulo powers of 5 and 7 for all ; unfortunately his results are incorrect, because of an error in his Lemma 4 on which his main theorems depend. This error is essentially a misquotation of the results of Watson (5), which one may readily understand in view of Watson's formidable notation.
On Asymmetrical Derivates of Non-Differentiable Functions
Authors: Garg K. M.
Published: December 2010

Let ƒ(x) be a non-differentiable function, i.e. a realvalued continuous function denned on a linear interval which has nowhere a finite or infinite derivative. We shall say that ƒ(x) has symmetrical derivates at a point x if the four Dini derivates of ƒ(x) at x satisfy the relations

and otherwise we shall say that ƒ(x) has asymmetrical dérivâtes at x.
On Invariant Means which are Not Inverse Invariant
Authors: Rajagopalan M. , Witz K. G.
Published: December 2010

In (1) R. G. Douglas says: “For a finite abelian group there exists a unique invariant mean which must be inversion invariant. For an infinite torsion abelian group it is not clear what the situation is.” It is not hard to see that if every element of an abelian group G is of order 2, then every invariant mean on G is also inversion invariant (see 1, remark 4). In this note we prove the following theorem (Theorem 1 below): An abelian torsion group G has an invariant mean which is not inverse invariant if, and only if, 2G is infinite. This result, together with the theorems of Douglas, answers completely the question of the existence (on an arbitrary abelian group) of invariant means which are not inverse invariant.
On the Adams Spectral Sequence
Authors: Mahowald Mark
Published: December 2010

One of the really significant advances in stable homotopy theory has been the Adams spectral sequence (see (1) for a general discussion). To date there has been no useful general way to obtain differentials in this spectral sequence. There is a general feeling that these differentials come about because of some geometric fact which forces a difference between the ring structure imposed on π*(S0), the stable homotopy ring, by the spectral sequence and its own natural one.
A Generalized Fredholm Theory for Certain Maps in the Regular Representations of an Algebra
Authors: Barnes Bruce Alan
Published: December 2010

Given an algebra A, the elements of A induce linear operators on A by left and right multiplication. Various authors have studied Banach algebras A with the property that some or all of these multiplication maps are completely continuous operators on A ; see (1-5). In (3)1. Kaplansky defined an element u of a Banach algebra A to be completely continuous if the maps a ⟶ ua and a ⟶ au, a ∊ A, are completely continuous linear operators.
Graph-Colouring and Combinatorial Numbers
Authors: Sobczyk Andrew
Published: December 2010

We shall use the termn-configurationfor the complete graph havingnvertices, which we visualize as a regular convex polygon. The sides and diagonals will be referred to asedges. A complete sub-configuration ofpvertices (subset ofpvertices with all interconnecting edges) will be called ap-tuple. A 3-tuple will be called a triple or triangle, a 4-tuple a quadruple, etc.
Free Pq-Algebras
Authors: Grillet Pierre Antoine
Published: December 2010

By a PQ (product-quotients) algebra, we mean a non-empty set together with three single-valued and not necessarily associative operations ., / , \ that we shall treat as product, right quotient, and left quotient although we require no relation between them. The theory of binary systems provides the following examples:

A. is the set of all subsets of a groupoid G (which may be a semigroup) with the operations defined by:
A Geometrical Theory of Multiple Integral Problems in the Calculus of Variations
Authors: Rund H.
Published: December 2010

Let Xn denote an n-dimensional differentiate manifold referred to local coordinates xi. An m-dimensional subspace Cm (m < n) of Xn can be represented parametrically in the form

1.1
On the Invariance of the Spectrum in Locally m-Convex Algebras
Authors: Brooks R. M.
Published: December 2010

In this paper we consider two closely related problems concerning a complete locally m-convex (LMC) algebra A with identity. Let a be a fixed element of A, and let P(a) be the smallest closed subalgebra containing a and 1. If B is any subalgebra containing a and 1, we let σ(a; B) denote the spectrum of a as an element of B. (I) Describe the set σ(a; P(a)) in terms of σ(a; A). (II) Give necessary and sufficient conditions in order that σ (a; B) = σ(a; A) for every closed subalgebra B of A which contains a and 1.
A Class of Polynomials in Self-Adjoint Operators in Spaces with an Indefinite Metric
Authors: Lo C.-Y.
Published: December 2010

Let H be a Hilbert space with the usual product [x, y] and with an indefinite inner product (x, y) which, for some orthogonal decomposition

in H, is defined by

where

and dim H1 = κ, a fixed positive integer.
Finite Rotation Groups in Low Dimensions
Authors: Friesen Donald K.
Published: December 2010

Let F be a vector space of dimension two, three, or four over a field of characteristic not two, and let V have a non-singular orthogonal metric. The problem discussed in this paper is the determination of all finite groups that can occur as subgroups of the rotation group of V.
Matrix Transformations in an Incomplete Space
Authors: Maddox I. J.
Published: December 2010

Let X = (X, p) be a seminormed complex linear space with zero θ. Natural definitions of convergent sequence, Cauchy sequence, absolutely convergent series, etc., can be given in terms of the seminorm p. Let us write C = C(X) for the set of all convergent sequences for the set of Cauchy sequences; and L∞ for the set of all bounded sequences.
Commutative Non-Associative Algebras and Identities of Degree Four
Authors: Marshall Osborn J.
Published: December 2010

The main result of this paper is the following.

Theorem 1. Let A be a simple, commutative, finite-dimensional algebra containing an idempotent over a field of characteristic 0, and let the algebra A' obtained from A by adjoining a unity element satisfy an identity of degree ≦ 4 not implied by commutativity. Then either A is a Jordan algebra or A is two-dimensional over an appropriate field E.
The Coarseness of the Complete Graph
Authors: Guy Richard K. , Beineke Lowell W.
Published: December 2010

The coarseness, c(G), of a graph G is the maximum number of edge-disjoint, non-planar subgraphs of G. We consider only the complete graph, Kp, on p vertices here. For p = 3r, Erdös conjectured that the coarseness was , but it has been shown (1) that

1

where square brackets denote integer part.
Endomorphism Rings of Quasi-Injective Modules
Authors: Osofsky B. L.
Published: December 2010

Y. Utumi (14 and 15) obtained some interesting results on self-injective rings. He showed that, if R is right self-injective, then so is R/J, where J is the Jacobson radical of R. Also, if R is right self-injective and regular, then R is left self-injective for any set of orthogonal idempotents is an essential extension of . This note extends these results to endomorphism rings of quasi-injective modules.
Extreme Points in Spaces of Analytic Functions
Authors: Gamelin T. W. , Voichick M.
Published: December 2010

Our aim in this paper is to obtain some theorems concerning spaces of analytic functions on a finite open Riemann surface R which extend known results for the disc △ = {|z| < 1}. Suppose that R has a smooth boundary bR consisting of t closed curves, and that the interior genus of R is s. The first Betti number of R is then r = 2s + t — 1.
On Some Methods for Construction of B.I.B. Designs
Authors: Majindar Kulendra N.
Published: December 2010

In this paper some methods for the construction of balanced incomplete block (b.i.b. for conciseness) designs are given. In the last section it is established that the existence of an affine resolvable b.i.b. design implies the existence of two other b.i.b. designs; §§ 6 and 7 are independent of §§ 3, 4, and 5.
Projective Systems on Trees and Valuation Theory
Authors: Laudal Olav Arnfinn
Published: December 2010

It is our aim in this note to introduce methods from homological algebra in the study of some problems in valuation theory. In particular, we will use such methods to give a new, and, in some respect, simpler proof of a well-known theorem of Krull and Ribenboim; see (2). We shall also show that the same methods can be used to prove the Riemann-Roch theorem for algebraic curves and the Weierstrass product theorem.
Convergence of Continued Fractions
Authors: Jones William B. , Thron W. J.
Published: December 2010

Let {sn(z)} be a given sequence of linear fractional transformations (or simply l.f.t.'s) of the form

1.1

and let

1.2

The sequence of l.f.t.'s {Sn(z)} is called a continued fraction generating sequence (or simply a c.f.g. sequence).
The Number of Graphs with a given Automorphism Group
Authors: Sheehan J.
Published: December 2010

In this paper, the graphs under consideration may have multiple edges but they do not have loops. We enumerate the number N[H: n, p] of topologically distinct graphs with n vertices and p edges whose automorphism group is the permutation group H. As in (5), this enumeration is considered in the context of the theory of permutation representations of finite groups. We begin with some definitions and notation.
Generation of Local Integral Orthogonal Groups in Characteristic 2
Authors: Pollak Barth
Published: December 2010

In two previous papers (see4;5) O. T. O'Meara and I investigated the problem of generating the integral orthogonal group of a quadratic form by symmetries in the case where the underlying ring of integers was the integers of a dyadic local field of characteristic not 2. In this paper, the investigation is concerned with a local field of characteristic 2. As in (5), only the unimodular case is treated. As in (4) and (5), groupsS(L), Xh(L), andO(L) are introduced for a unimodular latticeLand the relationship betweenS(L) andO(L) studied. As in the previously cited papers, generation by symmetries means thatS(L) =O(L). The following result is obtained.
Non-Stationary Processes and Spectrum
Authors: Nagabhushanam K. , Bhagavan C. S. K.
Published: December 2010

In 1964, L. J. Herbst (3) introduced the generalized spectral density Function

1

for a non-stationary process {X(t)} denned by

1

where {η(t)} is a real Gaussian stationary process of discrete parameter and independent variates, the (a;)'s and (σj)'s being constants, the latter, which are ordered in time, having their moduli less than a positive number M.
Some Open Questions on Minimal Primes of a Krull Domain
Authors: Eakin Paul M. , Heinzer William J.
Published: December 2010

Let A be an integral domain and K its quotient field. A is called a Krull domain if there is a set {Vα} of rank one discrete valuation rings such that A = ∩αVα and such that each non-zero element of A is a non-unit in only finitely many of the Vα. The structure of these rings was first investigated by Krull, who called them endliche discrete Hauptordungen (4 or 5, p. 104).
Special Abelian Group Difference Sets
Authors: Johnsen E. C.
Published: December 2010

A abelian group difference set (abbreviated AGDS) (G, D) is a -subset D = {di}1k taken from an abelian group G of order v such that each element different from the identity e in G appears exactly λ times in the set of differences {didj-1}, where . Combinatorially, AGDS is equivalent to a design having an abelian collineation group which is transitive and regular on the elements and on the blocks of the design (1).
Complete Sets of Observables and Pure States
Authors: Gudder Stanley P.
Published: December 2010

It was shown in (1) that a complete set of bounded observables is metrically complete. However, an extra axiom was needed to prove this result (1, footnote, p. 436). In this note we prove the above-mentioned result without the extra axiom. We also show that there is an abundance of pure states if M is closed in the weak topology and give a necessary and sufficient condition for the latter to be the case.
Non-Local Elliptic Boundary-Value Problems
Authors: An Ton Bui
Published: December 2010

Let G be a bounded open set of Rn with a smooth boundary ∂G. We consider the following elliptic boundary-value problem:

where A and Bj are, respectively singular integro-differential operators on G and on ∂G, of orders 2m and rj with rj < 2m; Ck are boundary differential operators, and Ljk are linear operators, bounded in a sense to be specified.
Linearization of a Contractive Homeomorphism
Authors: Janos Ludvik
Published: December 2010

Let X be a topological space and ϕ: X ⟶ X a continuous self-mapping of X. We say that ϕ is linearized in L by Φ if there exists a topological embedding μ: X ⟶ L of the space X into the linear topological vector space L such that for all x ϵ X, μ (ϕ (x)) = Φ (μ (x)), where ϕ is a continuous linear operator on L.
Free Planes and Collineations
Authors: Alltop W. O.
Published: December 2010

Our aim in this paper is to consolidate and extend some of the notions in (1; 2; 5; 6) concerning free planes in order to facilitate the study of their collineation groups. An upper bound mn for the orders of the finite subroups of Gn will be established, where Gn is the collineation group of the free plane ƒn of rank n + 4. In the process, a result of (6) will be generalized. Indeed, mn will be shown to be the best upper bound for all n ≠ 5.
Some Algebraic Structure in the Dual of a Compact Group
Authors: Iltis Richard
Published: December 2010

Throughout this paper, G will denote a compact (Hausdorff) topological group with identity e. When G is abelian, there is no difficulty in relating the group multiplication in G to the multiplication in the dual of G since characters are homomorphisms with respect to pointwise multiplication and pointwise multiplication of characters yields another character. However, in the non-abelian case, there are two multiplications associated with the dual of G: (1) representations are homomorphisms with respect to composition multiplication, and (2) the tensor product of representations yields another representation.
The Fredholm Elements of a Ring
Authors: Barnes Bruce Alan
Published: December 2010

In (1), Atkinson characterized the set of Fredholm operators on a Banach space X as those bounded operators invertible modulo the two-sided ideal of compact operators on X. It follows from this characterization that the Fredholm operators can also be described as those bounded operators which are invertible modulo the two-sided ideal of bounded operators on X which have finite-dimensional range. This ideal is the socle of the algebra of all bounded operators on X. Now, if A is any ring with no nilpotent left or right ideals, then the concept of socle makes sense (the socle of A in this case is the algebraic sum of the minimal left ideals of A, or 0 if A has no minimal left ideals). Also, in this case, the socle is a two-sided ideal of A. In this paper we study the elements in a ring A which are invertible modulo the socle. We call these elements the Fredholm elements of A.
Complete Regularity as a Separation Axiom
Authors: de Groot J. , Aarts J. M.
Published: December 2010

Although the axiom of complete regularity ought to be a separation axiom, in none of its usual forms does it look like an intrinsic separation axiom. Our aim in this paper is to establish such characterizations of complete regularity which naturally fit in between regularity and normality and which already have proved to be fundamental and useful. This can simply be achieved by replacing the family of all open sets (as used in the formulation of the separation axioms) by some suitable (sub)base of open sets. For the sake of simplicity, we assume all our spaces to be T1 and we shall operate with (sub)bases of closed sets (instead of open sets). It is appropriate to define the notion of a screening.

Two subsets A and B of a set X are said to be screened by the pair (C, D) if C ∪ D = X, A ∩ D = ∅ and C ∩ B = 0. (Consequently, A ⊂ C and B ⊂ D.)
A Characterization of Intrinsic Functions on
Authors: Carlson R. E. , Cullen C. G.
Published: December 2010

Let be an associative algebra over the field and let be the group of all automorphisms and anti-automorphisms of which leave elementwise invariant. A function Fwith domain and range contained in is called an intrinsic functionon if (i) for each Ω in and (ii) F(ΩZ) = ΩF(Z) for every Z in .

Rinehart (5) has introduced and motivated the study of the class of intrinsic functions on , and has characterized these functions for the cases in which is the algebra of real quaternions, the algebra of n × ncomplex matrices, or the algebra of n× nreal matrices (5; 6). The algebras listed above, along with the algebra of n × nquaternion matrices, constitute the full list of possibilities for the simple direct summands of any semi-simple algebra over or ; see (2).
Structure Theory for Montgomery-Samelson Fiberings between Manifolds. I
Authors: Antonelli Peter L.
Published: December 2010

In (12), Montgomery and Samelson conjectured that an MS-fibering of polyhedra with total space an n-sphere must have a homology sphere as its singular set. Mahowald (11) has shown that, indeed, an orientable fibering with n ≧ 4 must have a Z2-cohomology sphere as its singular set, while Conner and Dyer (4) have shown this for n arbitrary provided the fiber itself is a Z2-cohomology sphere. We show that if the singular set is tame, then it is a Z-homology sphere if the fiber is also one. This result together with those of Stallings (15), Gluck (7), and Newman and Connell (13) are applied in the case where the singular sets are locally flat and tame. It is shown (Theorem 5.2) that MS-fiberings of spheres on spheres, with closed connected manifold fibers and singular sets, are topologically just suspensions of (Hopf) sphere bundles. In a subsequent publication, the case where the singular sets are finite shall be considered. The reader is invited to consult (3) and (18) in this case.
Absolute Convergence Factors for Hp Series
Authors: Caveny James
Published: December 2010

A famous theorem of Hardy asserts that if f ∊ H1, then the sequence of Fourier coefficients satisfies . For this reason we say that the sequence (1, 1/2, 1/3, …) belongs to the multiplier class (H1, l1). In this paper, we investigate the multiplier classes (Hp, l1) for 1 ≧ p ≧ ∞. Our observations are based on the fact that a sequence (λ(0), λ(l), …) belongs to (Hp, l1) independent of the arguments of its terms. We also show that (Hp, l1) may be thought of as the conjugate space of a certain Banach space.

1. Preliminaries.Lp denotes the space of complex-valued Lebesgue measurable functions f defined on the circle |z| = 1 such that

is finite.
The Schwarzian Derivative and Disconjugacy of nth order Linear Differential Equations
Authors: Lavie Meira
Published: December 2010

In this paper we deal with the number of zeros of a solution of the nth order linear differential equation

1.1

where the functions pj(z) (j = 0, 1, …, n – 2) are assumed to be regular in a given domain D of the complex plane. The differential equation (1.1) is called disconjugate in D, if no (non-trivial) solution of (1.1) has more than (n – 1) zeros in D. (The zeros are counted by their multiplicity.)

The ideas of this paper are related to those of Nehari (7; 9) on second order differential equations. In (7), he pointed out the following basic relationship. The function

1.2

where y1(z) and y2(z) are two linearly independent solutions of

1.3

is univalent in D, if and only if no solution of equation(1.3) has more than one zero in D, i.e., if and only if(1.3) is disconjugate in D.
Polynomials with Certain Prescribed Conditions on the Galois Group
Authors: Rowlinson Elizabeth , Schwerdtfeger Hans
Published: December 2010

In this paper, some contributions are made to the theory of algebraic equations over the rational field with conditions imposed on the Galois group. In § 1, for a given abstract group G all faithful permutation representations Ḡ are obtained, and it is shown that if one of them is the group of some equation with splitting field K, then any of them is the group of some equation, also with splitting field K. The method of proof enables us to construct an equation having as group a given faithful permutation representation Ḡ of a prescribed group G if we are given an equation having as group some faithful representation of G. In § 2, equations having nilpotent group are considered, non-normal extension fields are discussed, and a canonical form is obtained for the roots of non-normal irreducible equations; this form is used to characterize fields and equations with nilpotent groups.
Rings of Meromorphic Functions on Non-Compact Riemann Surfaces
Authors: Kelleher James
Published: December 2010

In this paper we shall be concerned with the algebraic structure of certain rings of functions meromorphic on a non-compact (connected) Riemann surface Ω. In this setting, A = A(Ω) and K= K(Ω) denote (respectively) the ring of all complex-valued functions analytic on Ω and its field of quotients, the field of functions meromorphic on Ω. The rings considered here are those subrings of K containing A,which we term A-rings of K. Most of the results given here were previously announced without proof (15) and are contained in the author's doctoral dissertation (16), completed at the University of Illinois under the direction of Professor M. Heins, whose encouragement and advice are gratefully acknowledged.
Partition Relations for Ordinal Numbers
Authors: Milner E. C.
Published: December 2010

Capital letters denote sets and the cardinal of A is |A|. Greek letters always denote ordinal numbers and, unless stated otherwise, small latin letters denote non-negative integers. The symbol [A]r is used to denote the set {X: X ⊂ A; |X| = r} of all subsets of A with relements. If A is a simply ordered set with the order relation <, then the order type of A with this ordering is written as tp <A or simply as tp A when there is no ambiguity about the intended order relation. A graph G = (A, E) is an ordered pair with A as the set of vertices and E ⊂ [A]2 as the set of edges. In particular, if A is simply ordered, we call G a graph of type tp A. A complete subgraph of G = (A, E)is a set B ⊂ A such that [B]2 ⊂ E; a set C⊂ A is independent if [C]2 ∩ E = ∅.
Collineation Groups of Generalized André Planes
Authors: Foulser David A.
Published: December 2010

In a previous paper (5), I constructed a class of translation planes, called generalized André planes or λ-planes, and discussed the associated autotopism collineation groups. The main question unanswered in (5) is whether or not there exists a collineation η of a λ-plane Π which moves the two axes of Π but does not interchange them.

The answer to this question is “no”, except if Π is a Hall plane (or possibly if the order n of Π is 34) (Corollary 2.8). This result makes it possible to determine the isomorphisms between λ-planes. More specifically, let Π and Π′ be two λ-planes of order n coordinatized by λ-systems Qand Q′, respectively. Then, except possibly if n = 34, Π and Π′ are isomorphic if and only if Q and Q′ are isotopic or anti-isotopic (Corollary 2.13). In particular, Π is an André plane if and only if Q is an André system (Corollary 2.14).
Cercles de Remplissage and Asymptotic Behaviour
Authors: Gauthier Paul
Published: December 2010

In 1908, Lindelöf showed that if w = f(x) is a bounded holomorphic function in a sector S: |arg z| < θ1, and if f(z) has an asymptotic value w0 as z tends to ∞ along a half-ray in S; then f(z) tends uniformly to w0 as z tends to ∞o within any sector |arg z| ≦ θ, 0 ≦ θ < θ1. Montel (8) later replaced the condition that f(z) be bounded by the condition that f(z) be meromorphic and omitted three values. The following is an immediate consequence of the Lindelöf-Montel theorem.

THEOREM 1. Let w = f(x) be a function meromorphic in the sector | arg z| < θ1, and let f(z) tend to a value w0 as z → ∞ along the positive real axis.
Lower Radicals in Associative Rings
Authors: Watters J. F.
Published: December 2010

Given a homomorphically closed class of (not necessarily associative) rings , the lower radical property determined by is the least radical property for which all rings in are radical. Recently (7) a process of constructing the lower radical property from a class of associative rings has been given which terminates after a countable number of steps. In this process, an ascending chain of classes

is obtained and the property of being a ring in the class is the lower radical property determined by . In Theorem 1 we give another characterization of the rings in the class , λ ∈ {1, 2, …, omega;0}, and a procedure for constructing the lower radical determined by in an arbitrary associative ring is given.
On the Behaviour of a Series Associated with the Conjugate Series of a Fourier Series
Authors: Mohanty R. , Ray B. K.
Published: December 2010

1.1. Definition. Let λ ≡ λ(ω) be continuous, differentiable, and monotonic increasing in (0, ∞) and let it tend to infinity as ω → ∞. Suppose that ∑ an (we write ∑ for throughout the present paper) is a given infinite series and let

The series ∑ an is said to be summable |R, λ, r|, where r > 0, if

where A is a fixed positive number (6, Definition B). Now, for r > 0, m < ω < m + 1,

Hence, ∑ an is summable |R, λ, r|, where r > 0, if
On the Growth of Blaschke Products
Authors: MacLane G. R. , Rubel L. A.
Published: December 2010

It is well known that the distribution of the zeros of an analytic function affects its rate of growth. The literature is too extensive to indicate here. We only point out (1, p. 27; 2; 3; 5), where the angular distribution of the zeros plays a role, as it will in this paper. In private communication, A. Zygmund has raised the following related question, which is the subject of our investigation here.

Let {zn}, n = 1, 2, 3, …, be a sequence of non-zero complex numbers of modulus less than 1, such that ∑(1 – |zn|) < ∞, and consider the Blaschke product

1

Let

2

What are the sequences {zn} for which I(r) is a bounded function of r?
Biorthogonal Systems in lp-Spaces
Authors: Keown R. , Conatser C.
Published: December 2010

Our aim in this paper is to generalize certain ideas and results of Bary (1) on biorthogonal systems in separable Hilbert spaces to their counterparts in separable lp-spaces, 1 < p.The main result of Bary is to characterize a natural generalization of the concept of orthonormal basis for a Hilbert space. That of this paper is to characterize the concept of a Bary basis which is a generalization of the idea of standard basis of an lp-space. The result is interesting for lp-spaces because of the paucity of standard bases in these spaces.

Before summarizing our results, we shall introduce some notation and recall a few pertinent definitions and facts. The symbols and denote mutually conjugate lp-spaces, where is the space lt and the space lswith 1 < r <2 and 2 < s = r/(r – 1).
Some Linear Operators in the Lp Spaces
Authors: Peng-Yee Lee
Published: December 2010

Let Lp,1 ≦ p< ∞, denote the space of all functions ƒ (real or complex) such that ƒ and |ƒ|pare variationally integrable (see 2, p. 40, for definition) with respect to a pair h of interval functions in an elementary set E. In what follows, we fix both h and E and assume that h = {hl ht}is variationally integrable and hs ≧ 0 (s = l, r)in E. Further, let L∞denote the space of all functions ƒ (real or complex) which are h-measurable (cf. 2, p. 95) and bounded almost everywhere, i.e. except in a set X satisfying (cf. 2, p. 47) V(h; E; X)= 0.
Maps with Discrete Branch Sets Between Manifolds of Codimension One
Authors: Timourian J. G.
Published: December 2010

Let M n and Np be separable manifolds of dimensions n and p, respectively, with n ≧ p, and without boundary unless otherwise indicated. A mapƒ: M → N is proper if, for each compact set K ⊂ N, f –l(K) is compact. It is topologically equivalent to g: X → Y if there exist homeomorphisms α of M onto X and β of N onto Y such that βƒα–1 = g. At x ∈ M, ƒ is locally topologically equivalent to g if, for every neighbourhood W ⊂ M of x, there exist neighbourhoods U ⊂ W of x and V of ƒ(x) such that ƒ | U: U → V is topologically equivalent to g.
Generalization of Schwarz-Pick Lemma to Invariant Volume
Authors: Hahn K. T. , Mitchell Josephine
Published: December 2010

In this paper we give an extension of (6, Theorem 1), using a similar method of proof, to every homogeneous Siegel domain of second kind which can be mapped biholomorphically into a Kâhler manifold of a certain class (Theorem 1). Then by a well-known result of Vinberg, Gindikin, and Pjateckiï-Sapiro (10) that every bounded homogeneous domain D,contained in a complex euclidean space CN,can be mapped biholomorphically onto an affinely homogeneous Siegel domain of second kind, the theorem follows for D(Theorem 2). (6, Theorem 1) is a generalization of the Ahlfors version of the Schwarz-Pick lemma in C1(1) to invariant volume for a star-like homogeneous bounded domain in CN;see also (4). In § 3 we give the inequality for a special non-symmetric Siegel domain of second kind using an explicit form of TD(z, )due to Lu (7).
A Characterization of the Algebra of Functions Vanishing at Infinity
Authors: Mullins Robert E.
Published: December 2010

1. In this paper, X will always denote a locally compact Hausdorff space, C0(X) the algebra of all complex-valued continuous functions vanishing at infinity on X and B(X) the algebra of all bounded continuous complex-valued functions defined on X. If X is compact, C0(X) is identical to B (X) and all the results of this paper are obvious. Therefore, we will assume at the outset that X is not compact. If A represents an algebra of functions, AR will denote the algebra of all real-valued functions in A.
On the Lattice of σ-Algebras
Authors: Rayburn Marlon C.
Published: December 2010

In this paper I consider the relations between the lattice of topologies on a fixed space and the lattice of σ-algebras on that space. It is found that the intersection of these two lattices is the lattice of complete Boolean algebras, and that this lattice is anti-atomically generated. Some sufficient conditions for a topology to contain a maximal σ-algebra are noted.
Systems of Ideals
Authors: Cohn Richard M.
Published: December 2010

Some well-known theorems on the representation of an ideal in a commutative ring as an intersection of ideals of a specified type remain valid when attention is restricted to certain subclasses of the ideals of the ring. For example, a homogeneous ideal in a graded Noetherian ring is the intersection of homogeneous primary ideals, and Seidenberg (16) has recently proved a similar result for differential rings. Other examples are provided by theorems on the representation of certain differential and difference ideals as intersections of prime differential or difference ideals.

Kolchin (6) began the development of a general theory applicable to such phenomena. In this paper, I continue the development of that theory.

In §§ 1 and 2, I have followed Kolchin's definition and discussion of divisible systems with minor modifications and additions.
On the Lower Range of Perron's Modular Function
Authors: Kinney J. R. , Pitcher T. S.
Published: December 2010

The modular function Mwas introduced by Perron in (6). M(ξ) (for irrational ξ) is denned by the property that the inequality

is satisfied by an infinity of relatively prime pairs (p, q)for positive d,but by at most a finite number of such pairs for negative d.We will write

for the continued fraction expansion of ξ ∈ (0, 1) and for any finite collection y1,…, ykof positive integers we will write

It is known (see 6) that

Where