bluuebunny/crossref_metadata_2025 huggingface dataset:
Averaging Operators and C(X)-Spaces with the Separable Projection Property
Authors: Warren Baker John , Wolfe John
Published: December 2010

The Banach space of bounded continuous real or complexvalued functions on a topological space X is denoted C(X). An averaging operator for an onto continuous function ϕ : X → Y is a bounded linear projection of C(X) onto the subspace ﹛ƒ ∈ C(X) : f is constant on each set ϕ -1(y) for y ∈ Y﹜. The projection constant p(ϕ) for an onto continuous map ϕ is the lower bound for the norms of all averaging operators for ϕ ﹛p(ϕ) = ∞ if there is no averaging operator for ϕ).
Rings With Involution Whose Symmetric Units Commute
Authors: Lanski Charles
Published: December 2010

In the last few years many results have appeared which deal with questions of how various algebraic properties of the symmetric elements of a ring with involution, or the subring they generate, affect the structure of the whole ring. If the ring has an identity, similar questions may be posed by making assumptions about the symmetric units or subgroup they generate. Little seems to be known about the special units which exist in rings with involution, although several questions of importance have existed for some time. For example, given a simple ring with appropriate additional assumptions, is the unitary group essentially simple? Also, what can be said about the structure of subspaces invariant under conjugation by all unitary or symmetric units (see [7])?
Two More Hereditarily Separable Non-Lindelöf Spaces
Authors: Juhász I. , Kunen K. , Rudin M. E.
Published: December 2010

Our method usingCHis a blend of two earlier constructions (Hajnal-Juhász [2] and Ostaszewski [4]) of hereditarily separable(HS),regular, non-Lindelöf, first countable spaces. [4] produces a much better space than ours in § 1 ; it has all of our properties except that it is not realcompact (which is probably more interesting), and it is countably compact as well; however, the construction works only under ◇, which implies the continuum hypothesis(CH)but is not equivalent to it.
Amicable Orthogonal Designs-Existence
Authors: Wolfe Warren
Published: December 2010

Definition. An orthogonal design in order n and of type (u1, … , us) on the commuting variables x1, . . . , xs is an n X n matrix, X, with entries from the set ﹛0, ±x1, … , ±xs﹜ such that
Asymptotic Values Along Julia Rays
Authors: Gauthier P. M. , Hwang J. S.
Published: December 2010

Let ƒ be a function meromorphic in the finite complex plane C. If for some number θ, 0 ≦ θ < 2 π, the family, fr(z) = f(reθz), is not normal at z = 1, then the ray arg z = θ is called a Julia ray. Such a ray has the property that in every sector containing it, F assumes every value infinitely often with at most two exceptions. Many authors have taken this property as the definition of a Julia ray.
A Non-Standard Characterization of Perfect Mappings
Authors: Warren Button Robert
Published: December 2010

The central theorem of this note was proposed to me by I. Juhâsz on a napkin in Canter's restaurant in Pittsburgh, Pennsylvania. We agreed that if the proposition could be proven it would have some elegant applications and although it required a slight change before it could be proven, hopefully those expectations will be realized in this note.
Semi-Compactness with Respect to a Euclidean Cone
Authors: Wagner Daniel H.
Published: December 2010

Our motivation for this note originates with consideration of a subset A of Euclidean w-space, Rn, which contains only part of its boundary. The part contained is t h a t part of the closure of A which cannot be “bettered“ within A with respect to the preference associated with a fixed closed convex cone Γ. Here b is preferred to a if and only if a — b ∊ Γ; if, for instance, Γ is the non-negative orthant of Rn, this preference is ordinary vector inequality.
On Alexandroff Base Compactifications
Authors: Wasileski J. S.
Published: December 2010

In [13] we characterized the completely regular Hausdorff spaces as the class of spaces whose topology is generated by an Alexandrofï base. A space may have more than one Alexandrofï base and each such base determines a Hausdorff compactification .
On Homeomorphisms of a 3-Dimensional Handlebody
Authors: Suzuki Shin'ichi
Published: December 2010

By a 3-dimensional handlebody Vn of genus n, we mean an oriented 3-manifold which is a disk-sum of n copies of D2 X S1, where D2 is the unit disk and S1 is the boundary ∂D2; and by a surface Fn of genus n we mean the oriented boundary surface ∂ Vn.
Every Hausdorff Compactification of a Locally Compact Separable Space is a Ga Compactification
Authors: Van Mill J.
Published: December 2010

In [4], De Groot and Aarts constructed Hausdorff compactifications of topological spaces to obtain a new intrinsic characterization of complete regularity. These compactifications were called GA compactifications in [5] and [7]. A characterization of complete regularity was earlier given by Frink [3], by means of Wallman compactifications, a method which led to the intriguing problem of whether every Hausdorff compactification is a Wallman compactification.
Kronecker Products and Local Joins of Graphs
Authors: Farzan M. , Waller D. A.
Published: December 2010

When studying the category raph of finite graphs and their morphisms, Ave can exploit the fact that this category has products, [we define these ideas in detail in § 2]. This categorical product of graphs is usually called their Kronecker product, though it has been approached by various authors in various ways and under various names, including tensor product, cardinal product, conjunction and of course categorical product (see for example [6; 7; 11 ; 14; 17 and 23]).
Multiplicative Functionals on Frechet Algebras with Bases
Authors: Husain T. , J. Liang
Published: December 2010

Let A denote a complex (or real) Fréchet algebra (i.e. a complete metrizable locally m-convex algebra, see [2] or [3]). It is known [2] that the topology of such an algebra can be defined by an increasing sequence [qn] (i.e. qn(x) ≦ qn+i(x) for all x £ A and n ≧ 1) of submultiplicative (i.e. qn(xy) ≦ qn(x)qn(y) for all x, 3/ Ç ^4 and for each n ≧ I) seminorms.
A Lebesgue Decomposition for Vector Valued Additive Set Functions Defined on a Lattice
Authors: Dence Thomas P.
Published: December 2010

Our aim is to establish the Lebesgue decomposition for s-bounded vector valued additive functions defined on lattices of sets in both the finitely and countably additive setting. Strongly bounded (s-bounded) set functions were first studied by Rickart [15], and then by Rao [14], Brooks [1] and Darst [5; 6]. In 1963 Darst [6] established a result giving the decomposition of s-bounded elements in a normed Abelian group with respect to an algebra of projection operators.
An Analog of Nagata's Theorem for Modular LCM Domains
Authors: Beauregard Raymond A.
Published: December 2010

The theorem referred to in the title asserts that for an atomic commutative integral domain R, if S is a submonoid of R* (the monoid of nonzero elements of R) generated by primes such that the quotient ring RS-1 is a UFD (unique factorization domain) then R is also a UFD [8]. Recently several definitions of a noncommutative UFD have been proposed (see the summary in [6]).
On Transformation and Oscillation of Linear Differential System
Authors: Mary Donald F. St.
Published: December 2010

In this paper we study second order linear differential systems. We examine the relationship between oscillation of n-dimensional systems and certain associated n-dimensional systems, where m ≧ n. Several theorems are presented which unify and encompass in the linear case a number of results from the literature. In particular, we present a transformation which extends an oscillation theorem due to Allegretto and Erbe [1], and a comparison theorem due to Kreith [9], and explains some work of Howard [7].
Finite Quotients of the Automorphism Group of a Free Group
Authors: Gilman Robert
Published: December 2010

Let G and F be groups. A G-defining subgroup of F is a normal subgroup N of F such that F/N is isomorphic to G. The automorphism group Aut (F) acts on the set of G-defining subgroups of F. If G is finite and F is finitely generated, one obtains a finite permutation representation of Out (F), the outer automorphism group of F. We study these representations in the case that F is a free group.
Matrices Doubly Stochastic by Blocks
Authors: Fischer Pal , Holbrook John A. R.
Published: December 2010

The present work stems from the following classical result, due to G. H. Hardy, J. E. Littlewood, G. Pólya [7], and R. Rado [10].

THEOREM 1. Concerning a pair of n-tuples x, y ϵ Rn, the following four statementsare equivalent:

(a) for every continuous, convex function f : R → R
Formulas for the Nehari Coefficients of Bounded Univalent Functions
Authors: De Temple Duane W. , Oulton David B.
Published: December 2010

The Grunsky inequalities [6] and their generalizations (e.g., [5; 14; 17]) have become an increasingly important tool for the study of the coefficients of normalized univalent functions defined on the unit disc. In particular, proofs based upon the Grunsky inequalities have now settled the Bieberbach conjecture for the fifth [15] and sixth [13] coefficients. For bounded univalent functions the situation is similar, although the Grunsky inequalities go over to those of Nehari [11].
A Canonical Factorization for Graph Homomorphisms
Authors: Fawcett Barry
Published: December 2010

The graphs are undirected, without loops or multiple edges. The edge set E(X) of a graph X is a set of certain unordered pairs [x, y] of distinct elements of the vertex set V(X). For x ϵ V(X) we denote by E(x; X) the edges of X incident with x. A (homo)morphism ϕ : X ⟶ Y is a function from V(X) to V(Y) which preserves edges; thus it induces ϕ# : E(X) ⟶ E(Y) by ϕ# [x, x’] = [ϕx, ϕx’].
Changes of Variables Preserving Fourier-Stieltjes Transforms on Simply Connected Nilpotent Lie Groups
Authors: Drury S. W.
Published: December 2010

In [1] Beurling and Helson prove the following theorem.

THEOREM. Let φ: R → R be a continuous map such that

Then φ is affine. (Here denotes the space of Fourier-Stieltjes transforms on R.)
Cyclic Transformations of Polygons and the Generalized Inverse
Authors: Davis Philip J.
Published: December 2010

By a polygon with n vertices (an n-gon) we shall mean an ordered n-tuple of complex numbers (z1, z2, … , zn). The numbers zi will be called the vertices of the polygon. The zi may or may not be distinct, and it shall not matter whether a “conventional” polygon can be drawn with the zi as vertices. We shall operate in the spirit of the book of Bachmann and Schmidt [2] though not at the same level of algebraic generality inasmuch as these authors allow zi to belong to a general field.
The Caratheodory Metric and its Majorant Metrics
Authors: Burbea Jacob
Published: December 2010

One of the main purposes of the present paper is to provide a proof for the following statement:

Theorem A. Let M be a complex manifold of a complex dimension n. Let be a fixed point in M such that there exists a square integrable holomorphic n-form a(z) on M with .
The Dichromate and Orientations of a Graph
Authors: Berman Gerald
Published: December 2010

Internal and external activities are defined for any orientation of a graph relative to a fixed labelling of its edges. It is shown that the number of such orientations of having internal activity r and external activity s is 2r+sXrs where Xrs is the coefficient of xrys in the dichromate It follows that the number of orientations of in which the resulting digraph is acyclic is given by , where is the chromatic polynomial associated with .
Even Whitehead Squares are Not Projective
Authors: James Milgram R. , Zvengrowski Peter
Published: December 2010

The projectivity of the Whitehead square wn = [in, in] in π2n-1(SN) has been studied by Randall [6] who proved that if wn is projective then n must be a power of 2 or one less than a power of 2. Here we solve the question in the even case, proving by means of bo homology:

Theorem. if and only if n = 1, 2, 4.
Banach Spaces that are Uniformly Rotund in Weakly Compact Sets of Directions
Authors: Smith Mark A.
Published: December 2010

In a Banach space, the directional modulus of rotundity, δ (ϵ, z), measures the minimum depth at which the midpoints of all chords of the unit ball which are parallel to z and of length at least ϵ are buried beneath the surface. A Banach space is uniformly rotund in every direction (URED) if δ (ϵ, z) is positive for every positive ϵ and every nonzero element z. This concept of directionalized uniform rotundity was introduced by Garkavi [6] to characterize those Banach spaces in which every bounded subset has at most one Čebyšev center.
Enlargement of σ-Algebras and Compactness of Time Changes
Authors: Baxter J. R. , Chacon R. V.
Published: December 2010

Given a stochastic process adapted to an increasing family of right-continuous σ-algebras, it is often useful for many purposes to enlarge the a-algebras. In the present paper we shall consider enlargements which involve embedding the process in a larger probability space. The first question investigated is what kinds of enlargements it might be useful to consider. To study stopping times, the least requirement needed to have a complete theory is that convergent sequences of stopping times converge to a function which is also a stopping time, and for this it is necessary to make the enlargement right continuous.
A Failure of Stability Under Complex Interpolation
Authors: Connett William C. , Schwartz Alan L.
Published: December 2010

Let denote the spaces of Bessel potentials as denned and discussed in [6]. When 1 < q < ∞ and α is an integer , the Sobolev space which consists of functions Fin Lq with α derivatives in Lq and with norm

If now we make the change of variables x = ev, f(x) = F (In x) it is easily seen that the ratio
Compactification of Hereditarily Locally Connected Spaces
Authors: Tymchatyn E. D.
Published: December 2010

All spaces considered in this paper are completely regular and T1. A continuum is a compact, connected, Hausdorff space. A continuum is hereditarily locally connected if each of its subcontinua is locally connected. The reader may consult Whyburn [5] or Kuratowski [2] for a discussion on hereditarily locally connected metric continua. Nishiura and Tymchatyn [3] recently obtained some metric characterizations of connected subsets of hereditarily locally connected metric continua.
Extensions of Closure Spaces
Authors: Chattopadhyay K. C. , Thron W. J.
Published: December 2010

Extension theory has been intensively studied for completely regular spaces and is fairly well developed for T0-topological spaces. (See, for example, [1] and [5]). However, except for definitions of some of the basic concepts in [4] and results on embedding of closure spaces in cubes in [2] and [7], ours is the first study of the general theory of extensions of G0-closure spaces. (Definitions will be given following these introductory paragraphs).
On the Growth of Entire Functions Bounded on Large Sets
Authors: Hansen Lowell J.
Published: December 2010

There have been many indications of a relationship between the rate of growth of an entire function and the “size” of the set, E(c), where the modulus of the function is larger than the constant, c. Theorems of this type include the classical theorem of Wiman on functions of bounded minimum modulus, the Phragmén-Lindelöf Theorem, the Denjoy-Carleman-Ahlfors Theorem, and its many subsequent improvements. These theorems can all be understood as quantitative versions of the statement that if ƒ is an entire function such that, for some c > 0, the set E(c) is ‘'small”, then the maximum modulus function M(R, f) is forced to grow rapidly with R.
Geometry on the Unit Ball of a Complex Hilbert Space
Authors: Hahn Kyong T.
Published: December 2010

Furnishing the open unit ball of a complex Hilbert space with the
Carathéodory-differential metric, we construct a model which plays a
similar role as that of the Poincaré model for the hyperbolic geometry.

In this note we study the question whether or not through a point in the
model not lying on a
given line there exists a unique perpendicular, and give a necessary and
sufficient condition for the existence of a unique perpendicular. This enables us
to divide a triangle into two right triangles. Many trigonometric identities in a
general triangle are easy consequences of various identities which hold on a right
triangle.
A Generalization of Gasper's Kernel for Hahn Polynomials: Application to
Pollaczek Polynomials
Authors: Rahman Mizan
Published: December 2010

In this paper we consider a generalization of the discrete Poisson kernel for
the Hahn polynomials obtained recently by Gasper [6]. The
Hahn polynomials of degree n are defined by

and are known to be orthogonal on the set of non-negative integers
x = 0, 1, . . . , N
provided Re α, β > - l or Re α, β < -N [7; 8].
Topologies Extending Valuations
Authors: Rigo Thomas , Warner Seth
Published: December 2010

Let K be a field complete for a proper valuation
(absolute value) v. It is classic that a
finite-dimensional K-vector space
E admits a unique Hausdorff topology making it a
topological K-vector space, and that that topology is
the “cartesian product topology” in the sense that for any basis
c1
…, cn
of E,
is a topological
isomorphism from K

n
to E [1, Chap. I, §
2, no. 3; 2, Chap. VI, § 5, no.
2]. It follows readily that any multilinear mapping from
E
m to a Hausdorff topological
K-vector space is continuous. In particular, any
multiplication on E making it a K-algebra is continuous
in both variables. If for some such multiplication E is
a field extension of K, then by valuation theory the
unique Hausdorff topology of E is given by a valuation
(absolute value) extending v.
On Certain Sequences of Plus and Minus Ones
Authors: Borwein D. , Gawronski W.
Published: December 2010

Suppose throughout that c is a fixed positive
integer, that

and that
Non-Isomorphic Burnside Groups of Exponent p 2
Authors: Hickin K. K. , Phillips R. E.
Published: December 2010

In the recent paper [8] Phillips has shown that for each prime
p there are 2
ℵ0 non-isomorphic 2-generatecl
p-groups. This same result was obtained
independently by S. Jeanes and J. S, Wilson (unpublished) who show that the groups
constructed in [1] have 2
ℵ0 non-isomorphic images. The groups in both
of these proofs all have infinite exponent. In this paper we show that, for large
enough primes p, there are 2
ℵ0 non-isomorphic 2-generated groups of
exponent p2.
An Extrapolation Theorem for Positive Operators
Authors: Miller H. D. B.
Published: December 2010

Denote by S and M
respectively the complex vector spaces of simple and measurable complex valued
functions defined on the finite measure space X. Let
T be a positive linear map from
S to M such that for each
p, 1 < p <
∞, sup {||T f||p:
f ∈ S, ||f||P
≦ 1} is finit. finite. T then has an extension to a bounded
transformation of every LP(X), 1
< p < ∞ , and these extensions are "consistent". The norm of
T as a transformation of Lp is denoted
||T||P. The aim of this note is to prove the following
theorem.
Differential Operators with Abstract Boundary Conditions
Authors: Brown R. C.
Published: December 2010

Suppose F is a topological vector space. Let
ACm
≡ ACm[a, b] be the
absolutely continuous m-dimensional vector valued
functions y on the compact interval [a,
b] with essentially bounded components. Consider the boundary
value problem

where A0, A are respectively... operator with range in
F.
Invariant Subgroups in Rings with Involution
Authors: Montgomery Susan
Published: December 2010

Let R be a ring with involution*. In this paper,
we study additive subgroups A of
R which are invariant under all mappings of the form ϕ
x
: a → xax*.
That is, xAx* ⊆ A for all x ∈
R. Obvious examples of such subgroups
A are ideals of R, the
set of symmetric elements, and the set of skew-symmetric elements. We will prove
that when R is *-prime, these examples are essentially
the only ones.
Higher Derivations and Tensor Products of Commutative Rings
Authors: Brown W. C.
Published: December 2010

The genesis of this paper is the following well known result in field theory:
Let R denote a field of characteristic p
≠ 0, and let denote a subfield
of R such that for some
e sufficiently large. Then
R is isomorphic to the tensor product (over
) of primitive
extensions of if and only if
there exists a finite set Γ of -higher
derivations on R such that is the field of
constants of Γ. A proof of this theorem can be found in
[6].
The Embedding of Compact Convex Sets in Locally Convex Spaces
Authors: Roberts James W.
Published: December 2010

In studying compact convex sets it is usually assumed that the compact convex
set X is contained in a Hausdorff topological vector
space L where the topology on
X is the relative topology. Usually one assumes that
L is locally convex. The reason for this is that
most of the major theorems such as the Krein-Milman, Choquet-Bishop-de Leeuw, and
most of the fixed point theorems require that there be enough continuous affine
functions on X to separate points.
A Hardy-Davies-Petersen Inequality for a Class of Matrices
Authors: Johnson P. D. , Mohapatra R. N.
Published: December 2010

Let ω be the set of all real sequences a = ﹛an﹜
n ≧0. Unless otherwise indicated operations on sequences will
be coordinatewise. If any component of a has the entry oo the corresponding
component of a-1
has entry zero. The convolution of two
sequences s and q is given
by s * q
. The
Toeplitz martix associated with sequence
s is the lower triangular matrix
defined by
tnk = sn-k (n ≧ k),
tnk = 0 (n < k). It can be seen that Ts(q) =
s * q for each sequence q
and that Ts is invertible if and only if s0 ≠ 0. We shall
denote a diagonal matrix with diagonal sequence s by
Ds.
Functionals on Real C(S)
Authors: Farnum Nicholas , Whitley Robert
Published: December 2010

The maximal ideals in a commutative Banach algebra with identity have been
elegantly characterized [5; 6] as
those subspaces of codimension one which do not contain invertible elements. Also,
see [1]. For a function algebra
A, a closed separating subalgebra with constants of
the algebra of complex-valued continuous functions on the spectrum of
A, a compact Hausdorff space, this characterization
can be restated: Let F be a linear functional on
A with the property:

(*) For each ƒ in A there is a point
s, which may depend on f,
for which F(f) = f(s).
The Decomposition of the Module of n-th Order Differentials in Arbitrary
Characteristic
Authors: Fischer Klaus G.
Published: December 2010

Throughout this paper, it is assumed that A is the
complete, equicharacteristic, local ring of an algebraic curve at a one-branch
singularity whose residue field is algebraically
closed and contained in A. Hence, the domain
A is dominated by only one valuation ring in its
quotient held F, and if t is a uniformizing parameter,
then the integral closure of A in
F, denoted by Ā, is
[[t]].
Automatic Continuity for Linear Functions Intertwining Continuous Linear
Operators on Frechet Spaces
Authors: Thomas Marc P.
Published: December 2010

Many results concerning the automatic continuity of linear functions
intertwining continuous linear operators on Banach spaces have been obtained,
chiefly by B. E. Johnson and A. M. Sinclair [1;
2; 3; 5].
The purpose of this paper is essentially to extend this automatic continuity
theory to the situation of Fréchet spaces. Our motive is partly to be able to
handle the more general situation, since for example, questions about Fréchet
spaces and LF spaces arise in connection with the
functional calculus.
Discrete Copies of Rings of Sets in Groups and Orlicz-Pettis Theorems
Authors: Labuda Iwo
Published: December 2010

The Orlicz-Pettis type theorems are essentially statements about the equivalence of strong and weak subseries convergence of series in topological groups. The corresponding problem of the equivalence of subfamily summability for summable families was examined for instance in [2] and (16). We give here another uncountable generalization of the Orlicz-Pettis theorem in the sense that we assume less than subfamily summability and obtain straightforward generalizations of the classical case, which include or imply in a trivial manner—at least for polar topologies on groups—all known theorems “from weak to strong σ-additivity”.
A Logarithmic Property for Exponents of Partially Ordered Sets
Authors: Duffus Dwight , Rival Ivan
Published: December 2010

In an effort to unify the arithmetic of cardinal and ordinal numbers, Garrett Birkhoff [2; 3; 4; 5] (cf. [6]) defined several operations on partially ordered sets of which at least one, (cardinal) exponentiation, is of considerable independent interest: for partially ordered sets P and Q let PQ denote the set of all order-preserving maps of Q to P partially ordered by f ≦ g if and only if f(x) ≦ g(x) for each x ∈ Q.
Frobenius Algebras and Their Quivers
Authors: Green Edward L.
Published: December 2010

This paper studies the construction of Frobenius algebras. We begin with a description of when a graded -algebra has a Frobenius algebra as a homomorphic image. We then turn to the question of actual constructions of Frobenius algebras. We give a, method for constructing Frobenius algebras as factor rings of special tensor algebras. Since the representation theory of special tensor algebras has been studied intensively ([6], see also [2; 3; 4]), our results permit the construction of Frobenius algebras which have representations with prescribed properties. Such constructions were successfully used in [9].
Baer Endomorphism Rings and Closure Operators
Authors: Soumaya M. Khuri
Published: December 2010

A Baer ring is a ring in which every right (and left) annihilator ideal is generated by an idempotent. Generalizing quite naturally from the fact that the endomorphism ring of a vector space is a Baer ring, Wolfson [5; 6] investigated questions such as when the endomorphism ring of a free module is a Baer ring, and when the ring of continuous linear transformations on a pair of dual vector spaces is a Baer ring. A further generalization was made in [7], where the question of when the endomorphism ring of a torsion-free module over a semiprime left Goldie ring is a Baer ring was treated.
Projective Representations of Minimum Degree of Group Extensions
Authors: Feit Walter , Tits Jacques
Published: December 2010

Let G be a finite simple group and let F be an algebraically closed field. A faithful projective F-representation of G of smallest possible degree often cannot be lifted to an ordinary representation of G, though it can of course be lifted to an ordinary representation of some central extension of G. It is a natural question to ask whether by considering non-central extensions, it is possible in some cases to decrease the smallest degree of a faithful projective representation.
A Cellular Constraint in Supercompact Hausdorff Spaces
Authors: Bell Murray G.
Published: December 2010

In this paper we prove a cardinal inequality for supercompact Uausdorff spaces which gives insight into the cellular structure of such spaces and yields new examples of compact Uausdorff non-supercompact spaces.
Index Theory for Perturbations of Direct Sums of Normal Operators and Weighted Shifts
Authors: Berg I. D.
Published: December 2010

In this paper we consider the construction of a norm-continuous index theory for unitary equivalence up to commutative normality of operators on a separable Jlilhert space, and establish some essentially best possible results in this direction for operators which can be written as direct sums of weighted shifts and normal operators. Indeed, for such operators we establish both a normcontinuous analogue and a norm-continuous extension of the elegant results of L. Brown, R. Douglas and P. Fillmore [4].
Planar Sublattices of a Free Lattice. I
Authors: Rival Ivan , Sands Bill
Published: December 2010

There are three lattice-theoretic properties that are generally used to open a discussion on sublattices of a free lattice:

(W) for all a, b, c, d, a Λ b ≦ c V d implies a A b ≦ c, a Λ b g rf, a ≦cVd,orb ≦ c V d;

(SDv) for all a, b, c, a V b = a V c implies a V b = a V (b Λ c ) ;

(SDΛ) for all a, b, c, a Λ b = a Λ c implies a Λ b = a Λ (b V c).
Discrete Subsets of Proximity Spaces
Authors: Mattson Don A.
Published: December 2010

The distinct Hausdorff compactifications δX of a completely regular (Hausdorff) space X are in one-one correspondence with the admissible proximity relations δ on X, or alternatively, with the admissible totally bounded uniform structures for X. (See [1], [2].) Thus, δX is the Smirnov compactification of (X, δ). Generalized uniform structures for X will be described by means of pseudometrics on X (cf. [5], [7], [13]). Let where is in the proximity class π(δ) associated with (X, δ). Then a subset S of X is σ-discrete of gauge if , for all x, y ∈ S where x ≠ y.
The Density of Zeros of Dirichlet's L-Functions
Authors: Heath-Brown D. R.
Published: December 2010

Let L(s, χ) be a Dirichlet L-function and let N(σ, T, χ) denote the number of zeros ρ of L(s, χ), counted according to multiplicity, in the rectangle σ ≦ Re(ρ) ≦ 1, |Im(ρ)| ≦ T, (T ≦ 1). In this paper we shall prove several new estimates for the sum

where Σ* denotes summation over primitive characters only. These estimates will all be of the type

(1)

where denotes any fixed positive quantity.
On Certain Polynomials of Gaussian Type
Authors: Reich Daniel
Published: December 2010

Introduction. We shall consider functions of the form

where {ri} and {si} are sets of positive integers. Such functions were studied by E. Grosswald in [2], who took {si} to be pairwise relatively prime, and asked the following two questions:

(a) When is ƒ(t) a polynomial?

(b) When does ƒ(t) have positive coefficients?

These questions arise naturally from the work of Allday and Halperin, who show in [1] that under suitable circumstance ƒ(t) will be the Poincare polynomial of the orbit space of a certain Lie group action. Grosswald gives a complete answer to (a), but (b) is a much harder question, and a complete answer is provided only for the case m = 2. His treatment involves the representation of the coefficients of ƒ(t) by partition functions, and uses a classical description by Sylvester of the semigroup generated by {si}.
Mean Values of Character Sums
Authors: Montgomery H. L. , Vaughan R. C.
Published: December 2010

For a non-principal Dirichlet character χ modulo q,

Let the Pólya-Vingradov inequality asserts that M(x) < q1/2 log q see [7]. in the opposite direction it is a trivial consequence of lemma 1 below and 1. Parseval's identity that if χ is primitive modulo q, then

We show that on average the latter of these estimates is the more precise.

THEOREM 1. For any real k > 0

where the summation is over all non-principal characters modulo q.

THEOREM 2. For any k > 0,
Noether Lattices Representable as Quotients of the Lattice of Monomially Generated Ideals of Polynomial Rings
Authors: Anderson D. D. , Johnson E. W. , Johnson J. A.
Published: December 2010

Noether lattices were introduced by R. P. Dilworth in [5] and constitute a natural abstraction of the lattice of ideals of a Noetherian ring. In his definitive work, Dilworth showed that a minimal prime of an element generated by n principal elements has rank ≦ n. Following standard ring theoretical terminology, a local Noether lattice with (unique) maximal element M is said to be regular if M has rank n and can be generated by n principal elements.

In [3], K. P. Bogart showed that a distributive regular local Noether lattice of Krull dimension n is isomorphic to RLn, the sublattice of all ideals generated by monomials of any polynomial ring (k a field). In a later paper [4], Bogart extended his results on distributive regular local Noether lattices by showing that any distributive local Noether lattice is the image of a multiplicative map θ which preserves joins, and can in fact be thought of as the related congruence lattice.
Localization, Algebraic Loops and H-Spaces II
Authors: Shar Albert O.
Published: December 2010

In a previous work [6] it was shown that by imposing certain finiteness conditions on a nilpotent loop certain algebraic results yielded properties about [X, Y] where X is finite CW and Y is an H-Space. In this sequel we further restrict the category of nilpotent loops to a full subcategory called H-loops which still contains all loops of the form [X, Y], We prove that on this category there is a unique and universal P-localization if P ≠ ∅ which corresponds to topological localization. We also show that if the H-loop is a group then the two concepts of localization agree.

The first section of this paper is devoted to the definition and basic properties of H-loops. In the second section we develop the localization construction and prove uniqueness. Finally, in the third section we consider the topological and group theoretic situations.
Strict Topology on Spaces of Continuous Vector-Valued Functions
Authors: Choo Seki A.
Published: December 2010

In this paper, X denotes a completely regular Hausdorff space, Cb(X) all real-valued bounded continuous functions on X, E a Hausforff locally convex space over reals R, Cb(X, E) all bounded continuous functions from X into E, Cb(X) ⴲ E the tensor product of Cb(X) and E. For locally convex spaces E and F, E ⴲ, F denotes the tensor product with the topology of uniform convergence on sets of the form S X T where S and T are equicontinuous subsets of E′, F′ the topological duals of E, F respectively ([11], p. 96). For a locally convex space G , G ′ will denote its topological dual.
Combinatorial Oriented Maps
Authors: Tutte W. T.
Published: December 2010

An orientable map is often presented as a realization of a finite connected graphGin an orientable surface so that the complementary domains ofG, the “faces” of the map are topological open discs. This is not the definition to be used in the paper. But let us contemplate it for a while.

On each edge ofGwe can recognize two opposite directed edges, or “darts”. Letθbe the permutation of the dart-setSthat interchanges each dart with its opposite. The darts radiating from a vertexvoccur in a definite cyclic order, fixed by a chosen positive sense of rotation on the surface. The cyclic orders at the various vertices are the cycles of a permutationPofS. The choice ofPrather thanP–l, which corresponds to the other sense of rotation, makes the map “oriented”.
On a Uniqueness Theorem
Authors: Niimura Mikio
Published: December 2010

The classical uniqueness theorems of Riesz and Koebe show an important characteristic of boundary behavior of analytic functions in the unit disc. The purpose of this note is to discuss these uniqueness theorems on hyperbolic Riemann surfaces. It will be necessary to give additional hypotheses because Riemann surfaces can be very nasty. So, in this note the Wiener compactification will be used as ideal boundary of Riemann surfaces. The Theorem, Corollaries 1, 2 and 3 are of Riesz type, Riesz-Nevanlinna type, Koebe type and Koebe-Nevanlinna type respectively. Corollaries 4 and 5 are general forms of Corollaries 2 and 3 respectively.

Let f be a mapping from an open Riemann surface R into a Riemann surface R′.
Local Spaces with Three Cells as H-Spaces
Authors: Hagelgans Nancy L.
Published: December 2010

The question of which finite CW-complexes are if-spaces has been studied for many years. Since a finite CW-complex is an H-space if and only if its localization at each prime p is an H-space [21], an examination of finite local cell complexes as H-spaces yields results concerning CW-complexes. On the other hand, if it is known that a particular CW-complex is not an H-space, one would like to know for which primes p its localization at p fails to be an H-space. The main result of this paper gives a condition equivalent to a three cell local CW-complex's being an H-space for a prime p > 3.
Bounding the Valency of Polygonal Graphs With Odd Girth
Authors: Perkel Manley
Published: December 2010

In this paper we investigate the action of finite groups G on finite polygonal graphs. The notion of a polygonal graph was introduced in [17]: A polygonal graph is a pair (, ) consisting of a graph which is regular, connected and has girth m for some m ≧ 3, and a set of m-gons of such that every 2-claw of is contained in an unique element of (See Section 2 for the définitions of the terms used here.) If is the set of all m-gons of , so that there is in an unique m-gon on every one of its 2-claws, then we write for (, ) and call a strict polygonal graph. If we wish to emphasize the integer m, then we call (, ) an m-gon-graph (respectively, a strict m-gon-graph).
Algebras Intertwining Normal and Decomposable Operators
Authors: Jafarian Ali A.
Published: December 2010

The celebrated result of Lomonosov [6] on the existence of invariant subspaces for operators commuting with a compact operator have been generalized in different directions (for example see [2], [7], [8], [9]). The main result of [9] (see also [7]) is: If is a norm closed algebra of (bounded) operators on an infinite dimensional (complex) Banach space , if K is a nonzero compact operator on , and if then has a non-trivial (closed) invariant subspace. In [7], it is mentioned that the above result holds if instead of compactness for K we assume that K is a non-invertible injective operator with a non-zero eigenvalue belonging to the class of decomposable, hyponormal, or subspectral operators.
Complete Decomposability in the Exterior Algebra of a Free Module
Authors: Boratynski M. , Davis E. D. , Geramita A. V.
Published: December 2010

Recall the classical criterion for the complete decomposability of exterior vectors: the completely decomposable vectors in ∧pRn, R a field, are precisely the “Plücker vectors,” i.e. those whose coordinates (relative to the standard bases for ∧pRn) satisfy the Plücker equations. For R an arbitrary commutative ring, completely decomposable exterior vectors are still Plücker vectors, but the converse is not generally true. Rings for which the converse is true (for all 1 ≤ p ≤ n) are called Towber rings. Noetherian Towber rings are regular and, in fact, are characterized by the property that every Plücker vector in ∧2R4 is completely decomposable. (See [10] for these two results as well as for the above mentioned facts.) The present note develops a new characterization of Towber rings, combining it with results of Kleiner [9] and Estes-Matijevic [5] in (1) below.
Equivalence of Topologically-Algebraic and Semi-Topological Functors
Authors: Herrlich H. , Nakagawa R. , Strecker G. E. , Titcomb T.
Published: December 2010

Throughout let be faithful.

1.1. A U-morphism with domain X is a pair (e, A), where e ∈ Hom (X, UA). A [U-morphism (e, A) is called U-epi ( = generating) provided that r, s ∈ Hom (A, A’) and (Ur)e = (Us)e imply that r = s.

1.2. A U-source is a pair (X, (fi,Ai)I), (written more simply (X, (fi,Ai)I),, where (fi,Ai)I 7 is a family of U-morphisms each with domain X.

1.3. A factorization of a U-source (X,fi,Ai)I is a triple (e,A,gi)I such that (e, A) is a U-morphisms with domain X and for each i ∈ I gi ∈ Hom (A,Ai) and (Ugi)e = fi.
Modular Forms from Codes
Authors: Maher David P.
Published: December 2010

In this paper we construct modular forms from combinatorial designs, and codes over finite fields. We construct codes from designs, and lattices from codes. Then we use the combinatorial properties of the designs and the weight (or shape) structures of the codes to study the theta functions of the associated lattices. These theta functions are shown to be modular forms for the modular group or for various congruence subgroups. The levels of the forms we examine are determined by the dimensions of the codes and the characteristics of the fields. Using the Lee polynomial of the codes we can write the theta functions as homogeneous polynomials in modified Jacobi theta functions. By extending the underlying combinatorial structure, a modular form of half-integral weight is associated with a modular form of integral weight.
On Unions of Metrizable Subspaces
Authors: Van Douwen E. K. , Lutzer D. J. , Pelant J. , Reed G. M.
Published: December 2010

In this paper we study the question “which generalized metric spaces can be written as the union of k (closed) metrizable subspaces, where k is a cardinal number with k ≤ c?“ Questions of this type first arose in [16] where J. Nagata asked for examples of certain generalized metric spaces which could not be written as the union of countably many closed metrizable subspaces. Using Baire Category arguments, Fitzpatrick provided the required examples in [12]. We begin this paper by sharpening Fitzpatrick's examples, showing in Section 2 that there is a Moore space which is not the union of countably many metrizable subspaces of any kind. Then in Section 3 we present a positive result, proving that any cr-space, and a fortiori any Moore space, can be written as the union of c = 2ω0 closed metrizable subspaces.
Convergence of Classes of Amarts Indexed by Directed Sets
Authors: Millet Annie , Sucheston Louis
Published: December 2010

Let be a probability space, J a directed set filtering to the right. (X t) t∈J is a family of random variables adapted to an increasing family of σ-algebras . Vitali conditions V and V' on the σ-algebras, abstracting classical assumptions in Lebesgue's derivation theory, were made to insure essential convergence of martingales and submartingales (under proper boundedness assumptions). In reality these conditions, guaranteeing the existence of certain disjoint and properly measurable sets Bi, are better suited for study of amarts, since the sets Bi are a natural habitat and breeding ground for stopping times, thriving, as well known, precisely on disjoint and properly measurable sets. Thus K. Astbury [1] showed that the condition V, proved by K. Krickeberg to be sufficient for convergence of martingales (see [20] or Neveu [26], p. 98) is both necessary and sufficient for convergence of amarts. (We follow Neveu denoting by V the condition Krickeberg denotes by V+∞.) The Vitali condition V’, proved by Krickeberg [21] to be sufficient for convergence of submartingales, is shown here to be both necessary and sufficient for convergence of ordered amarts, defined similarly to amarts, except that the stopping times are ordered. We also introduce the controlled Vitali condition Ve, properly weaker than V’, and show that Ve is sufficient for convergence of controlled amarts, including submartingales.
A Characterization of Sharply Transferable Lattices
Authors: Grätzer G. , Platt C. R.
Published: December 2010

A lattice L is called transferable if and only if, whenever L can be embedded in the ideal lattice I(K) of a lattice K, L can be embedded in K. L is called sharply transferable if and only if, for every lattice embedding ψ(x) , there exists an embedding such that for x, y ϵ L, if and only if x ≤ y. Finite sharply transferable lattices were characterized in [3]. In this paper we extend the characterization to the infinite case. We begin by revising some of the terminology of [3].

1.1. Definition, (a) Let 〈P; ≧〉 be a poset and X, Y ⊆ P. Then X dominates Y (written X Dom Y) if and only if, for every y ∈ F, there exists x ∈ X such that y ∈ x. Dually, X supports Y (written X Spp Y) if and only if, for every y ∈ Y, there exists x ∈ X such that x ≦ y.
SL(2,5) and Frobenius Galois Groups Over Q
Authors: Sonn Jack
Published: December 2010

A finite transitive permutation group G is called a Frobenius group if every element of G other than 1 leaves at most one letter fixed, and some element of G other than 1 leaves some letter fixed. It is proved in [20] (and sketched below) that if k is a number field such that SL(2, 5) and one other nonsolvable group Ŝ5 of order 240 are realizable as Galois groups over k, then every Frobenius group is realizable over k. It was also proved in [20] that there exists a quadratic (imaginary) field over which these two groups are realizable. In this paper we prove that they are realizable over the rationals Q, hence we Obtain

THEOREM 1. Every Frobenius group is realizable as the Galois group of an extension of the rational numbersQ.
Rational Approximation to xn II
Authors: Newman D. J. , Reddy A. R.
Published: December 2010

Introduction. In 1858 Chebyshev showed that xn+l can be approximated uniformly on [–1, 1] by polynomials of degree at most n with an error 2–n. Let 0 ≦ σ ≦ (n + l)tan2(π/2n + 2). In 1868 Zolotarev established that x n + 1σxn can be approximated uniformly on [ –1, 1] by polynomials of degree at most (n – 1) with an error 2–n(l + σ/n + l)n+1. It is interesting to note that for the case σ = 0, Zolotarev's result includes Chebyshev's result. Achieser ([1], p. 279) proved the following analogue for rational approximation. Let a0 ≠ 0, a1, a2, a3, …, an be any given real numbers. Then for every N > n,

where λ is numerically the smallest root of the polynomial

with
Pure Unital Local Principal Ideal Domains in Local Fields
Authors: Benabdallah K. , Roggenkamp K. W.
Published: December 2010

The main purpose of this note is to give a characterization of p-pure unital subrings of the p-adic completion of the ring of integers R of an algebraic number field K localized at a maximal ideal p. This yields a characterization of the valued subfields of the p-adic field. In this context there turn up valuations of rational function fields in many indeterminates which seem to be new. The proof that the underlying function is indeed a valuation is quite easy here, however direct computations would involve a large amount of combinatorics. Our approach seems to fit well with Kronecker's, apparently forgotten, approach to ideal theory in rings of algebraic integers [3]. The concept of p-pure unital subrings arose from a study by the first author and A. Laroche of quasi-p-pure-injective (q.p.p.i.) abelian groups ([1], p. 582).
Generalized Riemann Integration and an Intrinsic Topology
Authors: Henstock Ralph
Published: December 2010

In generalized Riemann integration theory it is becoming increasingly clear that a particular collection of sets has some properties of a topology; it is a useful topology when general requirements hold, and the present paper examines the background. Thomson [23, 24] altered my original theory of the variation and Riemann-type integration that has Lebesgue properties, defining the variation of a function of interval-point pairs over the whole of a space T by using partial divisions of T instead of divisions covering T entirely, and also defining a Lebesgue-type integral. His reason might have been that a decomposable division space seems impossible in a general compact or locally compact space. McGill mentioned this to me, and in [15] connected Thomson's setting with topological measure and Topsøfe [25], giving an interesting theorem on the variation of the limit of a monotone increasing generalized sequence of open sets.
On a Class of One-Relator Groups
Authors: Brunner A. M.
Published: December 2010

In this paper, we consider the class of groups G(l, m; k) which are defined by the presentation

where k, l, m are integers, and |l| > m > 0, k > 0. Groups in this class possess many properties which seem unusual, especially for one-relator groups. The basis for the results obtained below is the determination of endomorphisms.

For certain of the groups, we are able to calculate their automorphism groups. One consequence of this is to produce examples of one-relator groups with infinitely generated automorphism groups. This answers a question raised by G. Baumslag (in a colloquium lecture at the University of Waterloo). Our examples are, perhaps, the simplest possible; J. Lewin [10] has found an example of a finitely presented group with an infinitely generated automorphism group.