bluuebunny/crossref_metadata_2025 huggingface dataset:
A Dimension Theorem for Real Primes
Authors: Dubois D. , Efroymson G.
Published: December 2010

Let k be a real closed field (see § 2 for a definition). Let be an algebraic closure of k. An algebraic set denned over k is, as usual, a subset of (n some integer greater than 0) which is the set of zeros of some polynomials in k[X1, . . . , Xn]. A variety is denned to be an absolutely irreducible algebraic set. We define the real points of an algebraic set X to be the points in X ∩ kn. One can then define X to be real if I(X ∩ kn) = I(X). (I(X) = the polynomials in k[X1, . . . , Xn] which vanish on X.)
Rings with Involution in which Every Trace is Nilpotent or Regular
Authors: Montgomery Susan
Published: December 2010

A theorem of Marshall Osborn [15] states that a simple ring with involution of characteristic not 2 in which every non-zero symmetric element is invertible must be a division ring or the 2 × 2 matrices over a field. This result has been generalized in several directions. IfRis semi-simple and every symmetric element (or skew, or trace) is invertible or nilpotent, thenRmust be a division ring, the 2 × 2 matrices over a field, or the direct sum of a division ring and its opposite [6; 8; 13; 16].
Double Covers and Metastable Immersions of Spheres
Authors: Wells Robert
Published: December 2010

The real line will be R, Euclidean n-space will be Rn, the unit ball in Rn will be En, the unit sphere in Rn+1 will be Sn, and real projective n-space will be Pn. The canonical line bundle associated with the double cover Sn → Pn will be ηn. If γ is a vector bundle, E(γ) will be its associated cell bundle, S(γ) its associated sphere bundle, P(γ) its associated projective space bundle (P(γ) = S(γ) / (-1)) and T(γ) = E(γ)/S(γ) its Thorn space.
On Going Down in Polynomial Rings
Authors: Dawson Jeffrey , Dobbs David E.
Published: December 2010

Our main purpose is to enlarge upon the studies of McAdam [9; 10] on the property of going down (GD) for prime ideals in extensions of (commutative integral) domains. Unlike the investigations of McAdam and the earlier work of Krull [8] and Cohen-Seidenberg [4] on GD and the related property of going up (GU), this paper is not primarily concerned with integral extensions. Consideration of more general extensions of domains A ⊂ B is facilitated by the following basic definitions. A prime ideal P of A is unibranched in B if there exists exactly one prime ideal Q of B satisfying Q ∩ A = P.
An Improved Subgroup Theorem for HNN Groups with Some Applications
Authors: Karrass A. , Pietrowski A. , Solitar D.
Published: December 2010

In [4], a subgroup theorem for HNN groups was established. The theorem was proved by embedding the given HNN group in a free product with amalgamated subgroup and then applying the subgroup theorem of [3]. In this paper we obtain a sharper form of the subgroup theorem of [4] by applying the Reidemeister-Schreier method directly, using an appropriate Schreier system of coset representatives. Specifically, we prove (in Theorem 1) that if H is a subgroup of the HNN group

1
On Compact Perturbations of Operators
Authors: Anderson Joel
Published: December 2010

Recently R. G. Douglas showed [4] that if V is a nonunitary isometry and U is a unitary operator (both acting on a complex, separable, infinite dimensional Hilbert space ), then V — K is unitarily equivalent to V ⊕ U (acting on ⊕ ) where K is a compact operator of arbitrarily small norm. In this note we shall prove a much more general theorem which seems to indicate "why" Douglas' theorem holds (and which yields Douglas' theorem as a corollary).
Oscillation and Comparison Theorems for Second Order Linear Differential
Equations with Integrable Coefficients
Authors: Butler G. , Macki J. W.
Published: December 2010

The classical comparison and interlacing theorems of Sturm were originally
proved for the equations

under the assumption that all coefficients are real-valued, continuous, and
p > 0, P >
0. Atkinson [1, Chapter 8] has carried out the standard theory for eigenvalue
problems involving (1), under the more general hypothesis
Combinatorial Properties of Associated Zonotopes
Authors: Shephard G. C.
Published: December 2010

Let S
1 . . . ,Sr
be r line segments, each of non-zero
length, in n-dimensional euclidean space R
n
. If a polytope Z is defined as the vector
(Minkowski) sum

(1) Z = S
1 + . . . + Sr
,

then the segments Si
will be called the components of
Z. Since we do not wish to exclude the possibility
that some of the components may be parallel, the polytope
Z may be written in the form (1) in many different
ways. For this reason it is convenient to define a
zonotope to be the polytope
Z together with some specified set of components {S1
, . . . , Sr
}. Figures 1, 2 and 3 show some zonotopes of 1, 2 and 3 dimensions
with 4, 5 and 6 components.
Operators on the Fourier Algebra with Weakly Compact Extensions
Authors: Dunkl Charles F. , Ramirez Donald E.
Published: December 2010

We let G denote an infinite compact group, and
Ĝ its dual. We use the notation of our book [3,
Chapters 7 and 8]. Recall that
A(G) denotes the Fourier
algebra of G (an algebra of continuous functions on
G), and denotes its dual
space under the pairing 〈f, ϕ〉 ,
(f ∊ A
(G), ϕ ∊ ). Further, note
is identified
with the C*-algebra of bounded operators on
L
2(G) commuting with left
translation. The module action of A
(G) on is defined by the
following: for f ∊ A
(G), ϕ ∊ ,
f · ϕ ∊
by
Some Open Sets for Which the Heat Equation is Simplicial
Authors: Taylor Peter D.
Published: December 2010

Let us associate to each open set U ⊂
R

n+1 the space
HU
of real functions f which are twice
continuously differentiable in x
1 . . . xn
and once continuously differentiable in
x

n+1 and which satisfy the heat equation:
Δf =
∂f/∂x

n+1 where Then we have what
in the axiomatic of Bauer is called a strong harmonic
space [2, p. 61]. We will call functions of
HU

harmonic in U.
Non-Isomorphic Tensor Products of Von Neumann Algebras
Authors: Williams J. J.
Published: December 2010

This paper investigates special conditions under which the tensor product of
two von Neumann algebras will be non-isomorphic to the tensor product of two
others. The main tools are the algebraic invariants property Λ
x
(x ≧ 0) (first defined by Powers [18])
and the r
∞ and ρ sets (defined by Araki and
Woods [3]).
Hall Higman Type Theorems, I
Authors: Berger T. R.
Published: December 2010

Suppose Q is a q-group for a prime q and C ≦ Aut(Q) is cyclic of order pe for a prime p ≠ q. Let k be a splitting field for CQ, the semidirect product of C by Q, of characteristic r ≠ q. Let V be a faithful irreducible k[CQ]-module. The k[CQ]-module V has been widely studied. When r = p the situation outlined above is similar to the situation occurring in the proof of Theorem B of Hall and Higman [3]. When r ≠ p it is similar to the corresponding theorem of Shult [5]. In all but restricted cases V|C has a regular k[C]-direct summand.
Operations in Grothendieck Rings and the Symmetric Group
Authors: Burroughs John
Published: December 2010

In [1] Atiyah described how to use the complex representations of the symmetric group, Sn, to define and investigate operations in complex topological K-theory. In this paper operations for more general Grothendieck groups are described in terms of the integral representations of Sn using the representations directly without passing to the dual as Atiyah did. The principal tool, which is proved in the first section, is the theorem that the direct sum of the Grothendieck groups of finite integral representations of Sn form a bialgebra isomorphic to a polynomial ring with a sequence of divided powers. A consequence of this theorem is that the only operations that can be constructed from the symmetric groups will be polynomials in the symmetric powers.
On Implicational Completeness
Authors: Ahmad Shafaat
Published: December 2010

A class of (universal) algebras [4; 5] of the same type or species τ is called equationally complete if the variety [4] Var() generated by has exactly two subvarieties —namely Var() itself and the class of all one element algebras. I t follows that equationally complete varieties are the atoms in the complete lattice of all varieties of a given type of algebras. J. Kalicki, D. Scott [7; 8; 9; 10] and others [2; 3; 6; 12] have considered several questions about equational completeness. A good many of these results have appeared in books. (See, for instance, [4, 5], to which we also refer the reader for more extensive bibliography on subjects related to equational completeness.)
Determining Subgroups of a Given Finite Index in a Finitely Presented Group
Authors: Dietze Anke , Schaps Mary
Published: December 2010

The use of computers to investigate groups has mainly been restricted to finite groups. In this work, a method is given for finding all subgroups of finite index in a given group, which works equally well for finite and for infinite groups. The basic object of study is the finite set of cosets. §2 reviews briefly the representation of a subgroup by permutations of its cosets, introduces the concept of normal coset numbering, due independently to M. Schaps and C. Sims, and describes a version of the Todd-Coxeter algorithm. §3 contains a version due to A. Dietze of a process which was communicated to J. Neubuser by C. Sims, as well as a proof that the process solves the problem stated in the title. A second such process, developed independently by M. Schaps, is described in §4. §5 gives a method for classifying the subgroups by conjugacy, and §6, a suggestion for generalization of the methods to permutation and matrix groups.
Strict Topologies for Vector-Valued Functions
Authors: Fontenot Robert A.
Published: December 2010

This paper is motivated by work in two fields, the theory of strict topologies and topological measure theory. In [1], R. C. Buck began the study of the strict topology for the algebra C*(S) of continuous, bounded real-valued functions on a locally compact Hausdorff space S and showed that the topological vector space C*(S) with the strict topology has many of the same topological vector space properties as C0(S), the sup norm algebra of continuous realvalued functions vanishing at infinity. Buck showed that as a class, the algebras C*(S) for S locally compact and C*(X), for X compact, were very much alike. Many papers on the strict topology for C*(S), where S is locally compact, followed Buck's; e.g., see [2; 3].
The SP-hull of a Lattice-Ordered Group
Authors: Bleier Roger D.
Published: December 2010

There have been several recent papers on the subject of the P-hull and the SP-hull of an l-group (lattice-ordered group). The most natural formulation of the concepts was given by P. Conrad in [6]. T. Speed studied P-groups extensively in [11]; his work was motivated by earlier work by H. Nakano and I. Amemiya in a vector lattice setting. A. Vecksler [12] produced the SP-hull for f-rings. The ortho-completion of S. Bernau [2] is a related concept.
The Group of Extensions and Splitting Length
Authors: Toubassi E. H.
Published: December 2010

This paper is concerned with the internal structure of Ext(Q, T) where Q is the group of rationals and T a reduced p-primary group of unbounded order. In [1] Irwin, Khabbaz, and Rayna define the splitting length of an arbitrary abelian group A, written l(A), to be the least positive integer n, otherwise infinity, such that A ⊗ . . . ⊗ A (n factors) splits. The concept of splitting length has been induced on Ext(Q, T), see [2; 5].
N-Series and Filtrations of the Augmentation Ideal
Authors: Losey Gerald
Published: December 2010

Let G be a group. Denote by ZG the group ring of G over the integers and by Δ = Δ(G) the augmentation ideal of ZG, that is, the kernel of the augmentation map ϵ : ZG → Z defined by . Then Δ is a free abelian group with a free basis . A filtration of Δ is a sequence
Some Counterexamples in Link Theory
Authors: Rolfsen Dale
Published: December 2010

This note is concerned specifically with links of two (disjoint) n-spheres in an (n + 2)-manifold M, i.e. embeddings L:Sn + Sn —> M. The links L0 and L1 are isotopic if they are t he ends of a continuous family Li:Sn + Sn —> M (0 ≦ t ≦ 1) of links. They are ambient isotopic or equivalent if there is a continuous family of self-homeomorphisms ht: M—> M (0 ≦ t ≦ 1) such that h0 = identity and h1◯L0 = L1. Ambient isotopic links are isotopic, but not conversely. For example, an isotopy can tie and untie little knots (as in Figure 3) in the components of a link, thus changing the original link into one which is inequivalent to the original.
Topological Rings of Quotients
Authors: Schelter William
Published: December 2010

We investigate here the notion of a topological ring of quotients of a topological ring with respect to an arbitrary Gabriel (idempotent) filter of right ideals. We describe the topological ring of quotients first as a subring of the algebraic ring of quotients, and then show it is a topological bicommutator of a topological injective R-module. Unlike R. L. Johnson in [6] and F. Eckstein in [2] we do not always make the ring an open subring of its ring of quotients. This would exclude examples such as C(X), the ring of continuous real-valued functions on a compact space, and its ring of quotients as described in Fine, Gillman and Lambek [3].
On Algebras Generated by Composition Operators
Authors: Cima J. A. , Wogen W. R.
Published: December 2010

Let Δ be the open unit disk in the complex plane and let be the group of automorphisms of Δ onto Δ, define by

The Banach spaces Hp = Hp(Δ), 1 ≦ p < ∞, are the Hardy spaces of functions analytic in Δ with their integral p means bounded,
On Embedding Essential Annuli in M3
Authors: Feustel C. D.
Published: December 2010

In [7; 8; 11] it is shown that an appropriate map of a planar surface into a 3-manifold can be replaced by an embedding. In [1 ; 4 ; 6 ; 7 ; 9 ; 10] conditions are given so that a "non-trivial" map of a planar surface (2-sphere) can be replaced by a non-trivial embedding of a planar surface (2-sphere). In this paper we give conditions on an annular map which guarantee the existence of a non-trivial embedding of an annulus. It is reported that F. Waldhausen has proved a similar but stronger "annulus theorem".
Semi-Simplicity Relative to Kernel Functors
Authors: Rubin Robert A.
Published: December 2010

Let Λ be a ring and σ a kernel functor (left exact preradical) on the category of left Λ-modules. A left Λ-module M is called σ-semi-simple if whenever N is a submodule of M with M/Nσ-torsion, N is a direct summand of M. In Section 1 we consider alternative characterizations and properties of σ-semi-simplicity for modules. In Section 2 conditions equivalent to the σ-semi-simplicity of the ring are obtained. Section 3 is devoted to the condition, which frequently arises in Section 2, that every σ-torsion module be semisimple.
Products of Transvections
Authors: Phadke B. B.
Published: December 2010

This paper is concerned with the presentation of certain elements of the groupSL(n,K) as products of a minimal number of transvections. To explain the terminology, letVbe ann-dimensional left vector space over a (not necessarily commutative) fieldK. The group of all non-singular linear transformations ofVontoV(i.e. the group of all collineations ofV) is the groupGL(n,K). This group is generated by collineations leaving a hyperplane pointwise fixed. Whenn= 2 these collineations are called axial collineations and the invariant hyperplane (line) is then called an axis.
On Some Complex Submanifolds in Kaehler Manifolds
Authors: Kon Masahiro
Published: December 2010

The purpose of this paper is to give some conditions for complex submanifolds in a Kaehler manifold of constant holomorphic sectional curvature to be Einstein.

For a complex hypersurface which is Einstein, Smyth [8] has obtained its classification and Chern [2] has proved the corresponding local result. Moreover, Takahashi [9] and Nomizu-Smyth [3] generalized this to a complex hypersurface with parallel Ricci tensor. We shall consider a condition weaker than the requirement that the Ricci tensor be parallel, that is we shall consider a complex submanifold with commuting curvature and Ricci operator, which condition was treated by Bishop-Goldberg [1].
Green's Relations for Regular Elements of Semigroups of Endomorphisms
Authors: Magill K. D. , Subbiah S.
Published: December 2010

X is a set and End X is a semigroup, under composition, of functions, which map X into X. We characterize those elements of End X which are regular and then we completely determine Green's relations for these elements. The conditions we place on End X are sufficiently mild to permit such semigroups as S(X), the semigroup of all continuous self maps of a topological space X and L(V), the semigroup of all linear transformations on a vector space V, to be regarded as special cases.
A Characteristic Subgroup of π-Stable Groups
Authors: Arad Zvi
Published: December 2010

All groups in this paper are assumed to be finite.

Let G be a group with op(G) ≠ 1 which is p-constrained and p-stable, p odd. If P is an Sp-subgroup of G, then by Glaubermaris Theorem, [3, 8.2.11],

G = Op′(G)NG(ZJ(P)).
Absolute Purity
Authors: Fieldhouse David J.
Published: December 2010

Throughout this paper we use the Bourbaki [1] conventions for rings and modules: all rings are associative but not necessarily commutative and have a 1; all modules are unital.

Our purpose is to extend and simplify some recent results of Maddox [7], Megibben [8], Enochs [3], and the author [5] on absolutely pure modules by introducing several new dimensions, and using the absolutely pure dimension introduced by the author in [6], This completes some work on character modules and dimension in [5] and [6].

An A -module will be called an FFR-module if and only if it has a resolution by finitely generated free A -modules.
The Ascending and Descending Varietal Chains of a Variety
Authors: Jónsson B. , McNulty G. , Quackenbush R.
Published: December 2010

Let F be a variety (equational class) of algebras. For n ≧ 0, Vn is the variety generated by the F-free algebra on n free generators while Vn is the variety of all algebras satisfying each identity of V which has no more than n variables. (Equivalently, Vn is the class of all algebras, , such that every n-generated subalgebra of is in V.) Note that unless nullary operation symbols are specified by the similarity type of V, V0 is the variety of all one element algebras while V° is the variety of all algebras.
Cardan Motion in Elliptic Geometry
Authors: Bottema Oene
Published: December 2010

Cardan motion in Euclidean geometry may be defined as the motion of a plane Г1 with respect to a coinciding plane Г such that two points A1, A2 of Г1 move along two orthogonal lines ai, a2 of Г1 The properties of this classical motion are well-known: the path of a point of Г1 is in general an ellipse with its center at the intersection o of a1 and a2; there are ∞ l points of Г1 (their locus being the circle C1 with A1, A2 = 2d as diameter) the paths of which are line segments. The moving polhode is the circle Ci, the fixed polhode is the circle (o; 2d). We investigate here Cardan motion-defined in the same way-in the elliptic plane.
Unit Preserving Isometries are Homomorphisms in Certain Lp
Authors: Schneider Robert
Published: December 2010

(a) (ᓂ1 and ᓂ2 will always denote positive bounded measures of equal mass defined on sets X and F respectively. Lp(ᓂ1) and Lp(ᓂ2) will always be complex Lp spaces.

(b) M C L∞(ᓂ1) will always denote a subalgebra of L∞(ᓂ1) containing constants.

(c) Let be a 1 inear map of ikf into Lp(ᓂ2). We shall say that T is a linear isometry in LP norm if

We shall prove the following:

THEOREM B. If 2 < p < ∞ and is a linear isometry in the Lp norm with T(l) = 1 then T is a homomorphism on M; that is
Norm Inequalities for Ultraspherical and Hankel Conjugate Functions
Authors: Andersen Kenneth F.
Published: December 2010

The notion of conjugate functions associated with ultraspherical expansions and their continuous analogues, the Hankel transforms, was introduced by Muckenhoupt and Stein [14], to which we refer the reader for general background and an excellent discussion of the motivation underlying these notions. The operation of passing from a given function to its conjugate is in many ways analogous to the passage from a function to its Hilbert transform, indeed, Muckenhoupt and Stein proved, among other things, that these operations acting on appropriate weighted Lebesgue spaces, Lp(𝝁), satisfy inequalities of M. Riesz type analogous to those satisfied by the Hilbert transform on the usual Lebesgue spaces, Lv( — ∞, ∞).
Some Results on Totally Isotropic Subspaces and Five-Dimensional Quadratic Forms Over GF(q)
Authors: Cordes Craig M.
Published: December 2010

In [5] Pall denned a partitioning of a quadratic space over a field of characteristic not 2 to be a collection of disjoint (except for ﹛0﹜ ) maximal totally isotropic subspaces whose union formed the set of isotropic vectors. Clearly isometric quadratic spaces simultaneously do or do not have partitionings. Pall exhibited the existence of partitionings for the spaces associated with

over formally real fields for n = 1, 2, 4, 8 and over Z/(p), p prime, for n = 1, 2. Using the latter, he was able to find a new proof for Jacobi's formula for the number of representations of a positive integer as the sum of four integral squares.
Variation Reducing Properties of Decreasing Rearrangements
Authors: Chong Kong-Ming
Published: December 2010

One well-established characteristic of the operation of decreasing rearrangement is its variation reducing property. A systematic study of this property has been made in considerable detail by G.F.D. Duff in [5] and [6]. He proved some inequalities related to the operation of rearrangement in decreasing order showing that the total variation of a sequence or an absolutely continuous function is in general diminished by such rearrangement. He also showed that the Lp norm of the difference sequence (or the derivative function) is diminished by this rearrangement operation unless the given sequence (or absolutely continuous function) is already monotonie (or equal to a monotonie function almost everywhere).
A Variational Technique for Bounded Starlike Functions
Authors: Barnard R. W.
Published: December 2010

Let KM = {z : \z\ < M}, 1 ≦ M < ∞ and K = K1. Let S denote the collection of functions f(z) = z + a2z2 + a3s3 + … that are regular and univalent in K. We write, for 1 < M < ∞,

S(M) = ﹛f : f ∞ S, f (K) ⊂ KM ﹜,

S*(M) = ﹛f : f ∞ S(M), f(K) is starlike with respect to the origin﹜.

In this paper we develop a variational technique for slit domains and give some applications with respect to finding the

and the

for any nonconstant entire function ᶲ(w) and a given z ∊ K.
A Minimax Equality Related to the Longest Directed Path in an Acyclic Graph
Authors: Vidyasankar K. , Younger D. H.
Published: December 2010

As an analog of a recently established minimax equality for directed graphs [1], I. Simon has suggested that the following be investigated.

1.1. For a finite acyclic directed graph G, a minimum collection of directed coboundaries whose union is the edge set of G has cardinality equal to that of a maximum strong matching of G.

This minimax equality is here proved, using a characterization of a maximum strong matching of an acyclic graph as the set of edges of a longest directed path in the graph.

The terms employed in the above theorem are defined as follows. Let G be a finite directed graph with vertex set VG and edge set eG
Uniform Finite Generation of Threedimensional Linear Lie Groups
Authors: Koch R. M. , Lowenthal Franklin
Published: December 2010

A connected Lie group G is generated by one-parameter subgroups exp(tX1), … , exp(tXk) if every element of G can be written as a finite product of elements chosen from these subgroups. This happens just in case the Lie algebra of G is generated by the corresponding infinitesimal transformations X1, … , Xk ; indeed the set of all such finite products is an arcwise connected subgroup of G, and hence a Lie subgroup by Yamabe's theorem [9]. If there is a positive integer n such that every element of G possesses such a representation of length at most n, G is said to be uniformly finitely generated by the one-parameter subgroups.
The n-Dimensional Distributional Hankel Transformation
Authors: Koh E. L.
Published: December 2010

The Hankel transformation was extended to certain generalized functions of one dimension [1; 2; 3]. In this paper, we develop the n-dimensional case corresponding to [1]. The procedure in [1] is briefly as follows:

A test function space Hμ is constructed on which the μth order Hankel transformation hμ defined by

is an automorphism whenever μ ≧ —1/2. The generalized transformation hμ' is then defined on the dual Hμ' as the adjoint of hμ through a Parseval relation, i.e.
Handles, Semihandles, and Destabilization of Isotopies
Authors: Webster Dallas E.
Published: December 2010

Kirby and Siebenmann showed that handles of index other than 3 can be straightened, and then went on to prove an abundance of immensely important results such as triangulation theorems, Hauptvermutungen, and classification and structure theorems. (See [7; 6; 2; 3; 8; 4; 5].) Their pr∞fs relied on some nonsimply connected surgery techniques and computations due to Wall, Hsiang, and Shaneson [10; 11; 12; 1], which at the time seemed so difficult as to be practically inaccessible to the large number of mathematicians interested in the consequences. The main result presented here, viz., that a stably straightened handle can be straightened, was obtained by the author some time ago in an attempt to prove the stable homeomorphism theorem without the assistance of Wall, Hsiang and Shaneson (cf. [13]).
On a Theorem of Arhangel'skiĭ Concerning Lindelöf P-Spaces
Authors: Hodel R. E.
Published: December 2010

1. Introduction. In [4] Arhangel'skiĭ proved the remarkable result that every regular space which is hereditarily a Lindelöf p-space has a countable base. As a consequence of the main theorem in this paper, we obtain an analogue of Arhangel'skiĭs result, namely that every regular space which is hereditarily an ℵi-compact strong ∑-space has a countable net. Under the assumption of the generalized continuum hypothesis (GCH), the main theorem also yields an affirmative answer to Problem 2 in Arhangel'skiĭs paper.

In § 3 we introduce and study a new cardinal function called the discreteness character of a space. The definition is based on a property first studied by Aquaro in [1], and for the class of T1 spaces it extends the concept of Kicompactness to higher cardinals.
A Note on Certain Spaces with Bases (mod K)
Authors: Bennett Harold R. , Martin Harold W.
Published: December 2010

In this note all spaces are assumed to be regular T1 spaces and all undefined terms and notations may be found in [8], In particular let cl(A) denote the closure of the set A and let Z+ denote the set of natural numbers.

Definition 1. Let X be a topological space and a covering of X by compact sets. An open covering of X is said to be a basis (mod K) if whenever and an open set V contains Kx, then there exists such that . In such a case X is written as the ordered triple .
The Convergence of Series for Various Choices of Sign in Banach Spaces
Authors: Shirey James
Published: December 2010

1. Let (xn, Xn) denote a basis for a Banach space (X, ∥ • ∥) of measurable functions in (0, 1).

It is shown in [2] and [9] that the equivalence of the norms

and ∥ • ∥ is equivalent to the unconditionality of the basis (xn, Xn). In [8] a weaker relationship between these norms is exploited to establish the existence of an element of L1(E) for each E ⊂ (0, 1), |£| > 0, whose Haar series expansion is conditionally convergent in the norm of L\(E).

In this note, a Lemma of Orlicz [7] is generalized to provide a relationship between , and the changes in sign that are tolerated in without disruption of norm convergence.
A Remark on Colimits
Authors: Osofsky Barbara L.
Published: December 2010

Let MR be a right module over the associative ring R (with 1). Assume one has an expression for M as a colimit (direct limit) of a system

over the (directed) poset D. A natural way to get M as a colimit of the family ﹛F∞ —” Fβ|∞, fβ ∈ E﹜ for some subset £ of D is to take E cofinal in D. However, if one is concerned about the cardinality of the set E, cofinal subsets may be too large. Let us look at a specific example. Lazard [3] has shown that any flat MR is a direct limit of finitely generated free R-modules. The cardinality of his indexing set depends on the cardinality of M.
Cotorsion Theories and Colocalization
Authors: McMaster R. J.
Published: December 2010

Let R be an associative ring with unit element. Mod-R and R-Mod will denote the categories of unitary right and left R-modules, respectively, and all modules are assumed to be in Mod-R unless otherwise specified. For all M, N ϵ Mod-R, HomR(M, N) will usually be abbreviated as [M, N]. For the definitions of basic terms, and an exposition on torsion theories in Mod-R, the reader is referred to Lambek [6]. Jans [5] has called a class of modules which is closed under submodules, direct products, homomorphic images, group extensions, and isomorphic images a TTF (torsion-torsionfree) class.
On Infinite Systems of Linear Differential Equations
Authors: McClure J. P. , Wong R.
Published: December 2010

Let A = [αtj] (i,j = 1, 2, …) be an infinite matrix with complex entries, and let z = (ζj) (j = 1, 2, …) be a sequence of complex numbers. In this paper we wish to investigate the existence, uniqueness and asymptotic behavior of solutions to the infinite system of linear differential equations

with the initial conditions
Redfield's Theorems and Multilinear Algebra
Authors: White Dennis E.
Published: December 2010

1. Introduction. The remarkable 1927 paper by J. H. Redfield [13] which anticipated many recent combinatorial results in Polya counting theory and, in fact, predated Polya's theorem by ten years has been discussed at length by Harary and Palmer [8], Foulkes [5; 6], Sheehan [15; 16] and Read [12], not to mention de Bruijn [3] and others. We shall, in this paper, demonstrate how multilinear techniques may be used in this context. The Redfield superposition theorem and decomposition theorem turn out to be statements about a group acting on finite function spaces, and may thus be dealt with in multilinear terms. We shall prove Redfield's results and an extension due to Foulkes [5].
Synthese Harmonique Des Sous-Groupes Discrets
Authors: Lohoué N.
Published: December 2010

Dans [4] C. S. Herz soulève la question suivante:

Soit G = SL(2, R) le groupe des matrices réelles d'order deux, de déterminant un; soit H = SL(2, Z) le sous-groupe discret de G, formé des matrices à coefficients entiers. Existe-t-il une constante absolue c > 0 telle que pour toute function f de At(H) à support fini, on puisse trouver une function de AP(G), à support compact dont la restriction à H coïncide a vec f et
Modules Over Hereditary Noetherian Prime Rings
Authors: Singh Surjeet
Published: December 2010

Quasi-injective and quasi-projective modules over hereditary noetherian prime rings ((hnp)-rings) were studied in [17]. In the present paper we give some applications of the results established in [17]. Kulikov, Kertesz, Prufer, Szele had made basic contributions to the problem of decomposability of abelian p-groups (Fuchs [4]). Kaplansky [9] studied analogous problems for modules over (commutative) Dedekind domains. Let R be an (hnp)-r'mg, which is not right primitive. Using the structure of an indecomposable infective torsion R-module, established in [17, Theorem 4], some of the basic concepts and results on the decomposability of a torsion abelian group are generalized in Section 2, to modules over R.
A Non-Hausdorff Multifunction Ascoli Theorem for 𝓴3-Spaces
Authors: Morales Pedro
Published: December 2010

A non-Hausdorff Ascoli theorem for continuous functions was established in [6]. The present purpose is to extend this result to point-compact continuous multifunction, using Levine's generalization for closed subsets [12]. The paper is organized as follows: the object of section 2 is to establish the necessary multifunction lemmas and to introduce the notion of a Tychonoff set; section 3 generalizes to multifunction context the partial exponential law of R. H. Fox [9, p. 430], and establishes a special exponential law for multifunctions; section 4 concerns the crucial properties of even continuity for multifunctions, introduced in [8]; the main theorem of the paper is established in section 5.
Edgeconvex Circuits and the Traveling Salesman Problem
Authors: Kalmanson Kenneth
Published: December 2010

This paper will continue certain investigations into the geometric nature of the well-known traveling salesman problem: that of determining the extreme Hamiltonian circuits (H-circuits) of a graph.
Generalizations of the Simple Torsion Class and the Splitting Properties
Authors: Teply Mark L.
Published: December 2010

In this paper all rings R are associative rings with identity and all modules are members of R-mod, the category of unital left R-modules, unless the contrary is specifically stated.
A General Asymptotic Result for Partitions
Authors: Richmond Bruce
Published: December 2010

In this paper we are concerned with partition functions pϒ(n) that have generating functions of the form

where γ(n) ≧ 0. We shall obtain an asymptotic relation for pϒ(n) under suitable restrictions on ϒ (see Theorem 1.1). These restrictions are weaker than those of Brigham [2] who considered this problem previously.
Modularity vs. Separability for Field Extensions
Authors: Kreimer H. F. , Heerema N.
Published: December 2010

In this paper we compare the properties separability and modularity for field extensions. Let be fields of characteristic . K is separable over if K and are linearly disjoint over . K is modular over if K and are linearly disjoint over their intersection for all n > 0. The latter definition is due to Sweedler [12] and is important particularly for Galois theories of purely inseparable extensions [2; 3; 4; 7].
Oscillation of Elliptic Equations in General Domains
Authors: Noussair E. S.
Published: December 2010

Oscillation criteria will be obtained for the linear elliptic partial differential equation

in an unbounded domain G of general type in n-dimensional Euclidean space En. The differential operator D is defined as usual by

where each α (i), i = 1, … , n, is a non-negative integer.
A General Construction of Spaces of the Type of R. C. James
Authors: Lohman Robert H. , Casazza Peter G.
Published: December 2010

In 1950, R. C. James [7] exhibited a nonreflexive Banach space with a basis that is of finite codimension in its second dual. This space is the first example of a quasi-reflexive space. General results on quasi-reflexive spaces have been obtained by P. Civin and B. Yood [3], and quasi-reflexive spaces with bases have been studied by D. Dean, B. L. Lin, and I. Singer [4 ; 12].
Pseudo-Confluent Mappings and a Classification of Continua
Authors: Lelek A. , Tymchatyn E. D.
Published: December 2010

In this paper we introduce a new class of mappings and apply it to study some local properties of continua. A solution is obtained to a problem raised in [14] by the first author (see 4.4 below). By a mapping we always mean a continuous function.
Initially Structured Categories and Cartesian Closedness
Authors: Nel L. D.
Published: December 2010

In recent papers Horst Herrlich [4; 5] has demonstrated the usefulness of topological categories for applications to a large variety of special structures. A particularly striking result is his characterization of cartesian closedness for topological categories (see [5]). Spaces satisfying a separation axiom usually cannot form a topological category in Herrlich's sense however and some interesting special cases, e.g. Hausdorff C-spaces, remain excluded from his theory despite having many analogous properties. It therefore seems worthwhile to undertake a similar study in a wider setting. To this end we relax one of the axioms for a topological category and show that in the resulting initially structured categories a significant selection of results can still be proved, including the characterization of cartesian closedness.
Some Results on the Countable Compactness and Pseudocompactness of Hyperspaces
Authors: Ginsburg John
Published: December 2010

Let X be a Hausdorff space. Let 2X denote the set of all non-empty closed subsets of X. For a subset A of X, we set 2A = {F 6 2X : F ⊆ A}. Recall that the finite topology on 2X is that topology having as a sub-basis the family {2G : G is open in X} U }2X — 2F : F is closed in X). When endowed with this topology, 2X is referred to as the hyper space of X. For the fundamental properties of hyperspaces, we refer the reader to [6; 7]. Following [6], we adopt the following notation: If A0, A1, … , An are subsets of X, we set and for all .
An Application of Logic to Analysis
Authors: Becker Joseph , Lipshitz Leonard
Published: December 2010

Let F be a complex analytic subvariety of an open subset of Cn and p ϵ V let be the germs at p of holomorphic, weakly holomorphic, infinitely differentiable, and k times continuously differentiate functions respectively. Spallek [15] has shown that for any p £ V there exists an integer such that , generalizing the result of Malgrange [12] that .In [14], Siu proved Spallek's result from a more sheaf theoretic point of view and showed the minimal integer function is bounded on compact sets. Bloom [7] reproved Malgrange's result by using differential operators on varieties.
Possibility of Uniform Rational Approximation in the Spherical Metric
Authors: Gauthier P. M. , Roth A. , Walsh J. L.
Published: December 2010

Let ƒ b e a mapping defined on a compact subset K of the finite complex plane C and taking its values on the extended plane C ⋃ ﹛ ∞﹜. For x a metric on the extended plane, we consider the possibility of approximating ƒ x-uniformly on K by rational functions. Since all metrics on C ⋃ ﹛oo ﹜ are equivalent, we shall consider that x is the chordal metric on the Riemann sphere of diameter one resting on a finite plane at the origin.
Minimal Plat Representations of Prime Knots and Links are not Unique
Authors: Montesinos José M.
Published: December 2010

Letdenote the 2-fold cyclic covering space branched over a linkLin S3. We wish to describe an infinite family of prime knots and links in which each memberLexhibits two minimal 6-plat representations, where the associated Heegaard splittings ofare minimal and inequivalent. Thus each knot or link of that family admits at least two equivalence classes of 6-plat representations which are minimal.
Subinvariance in Solvable Lie Algebras
Authors: Chao Chong-Yun , Stitzinger Ernest L.
Published: December 2010

In a recent paper, Wielandt has continued his investigation of subnormal subgroups. Since the analogous concept is also of interest in Lie algebras, this note considers the Lie algebra counterparts to Wielandt's results. Generally the results do not carry over to all Lie algebras, but do hold in the solvable case. In order to state the main results, several definitions are needed and consequently we begin by listing some of the consequences. All Lie algebras considered here are finite dimensional over a field.
Compatible Tight Riesz Orders
Authors: Glass A. M. W.
Published: December 2010

N. R. Reilly has obtained an algebraic characterization of the compatible tight Riesz orders that can be supported by certain partially ordered groups [13; 14]. The purpose of this paper is to give a “geometric“ characterization by the use of ordered permutation groups. Our restrictions on the partially ordered groups will likewise be geometric rather than algebraic. Davis and Bolz [3] have done some work on groups of all order-preserving permutations of a totally ordered field; from our more general theorems, we will be able to recapture their results.
Pull-Backs in Homotopy Theory
Authors: Mather Michael
Published: December 2010

The (based) homotopy category consists of (based) topological spaces and (based) homotopy classes of maps. In these categories, pull-backs and push-outs do not generally exist. For example, no essential map between Eilenberg-MacLane spaces of different dimensions has a kernel. In this paper we define homotopy pull-backs and push-outs, which do exist and which behave like pull-backs and push-outs, and we give some of their properties. Applications may be found in [3; 5; 6 and 14].
A Note on the Dirichlet Condition for Second-Order Differential Expressions
Authors: Everitt W. N.
Published: December 2010

Let M denote the formally symmetric, second-order differential expression given by, for suitably differentiable complex-valued functions ƒ,

The coefficients p and q are real-valued, Lebesgue measurable on the halfclosed, half-open interval [a, b) of the real line, with - ∞ < a < b ≦ ∞, and satisfy the basic conditions:
A Classification of 2-Varieties
Authors: Anderson Tim , Kleinfeld Erwin
Published: December 2010

The purpose of this paper is to give a classification of those varieties of power-associative algebras over a field F which satisfy the condition

(1.1) For each A in and each ideal I of A, I2 is an ideal of A.
Split Subdirect Products and Piecewise Domains
Authors: Fuelberth John , Kuzmanovich James
Published: December 2010

Throughout this paper all rings will have unity and all modules will be unital.

If X ⊂ R, then r(X) (respectively, l(X)) denotes the right (left) annihilator of x.

An element d of R is called right (left) regular if r(d) = 0 (1(d) = 0). An element which is both right and left regular is called regular.
Normality in Elementary Subgroups of Chevalley Groups Over Rings
Authors: Hurley James F.
Published: December 2010

In [6] we have constructed certain normal subgroups G7 of the elementary subgroup GR of the Chevalley group G(L, R) over R corresponding to a finite dimensional simple Lie algebra L over the complex field, where R is a commutative ring with identity. The method employed was to augment somewhat the generators of the elementary subgroup EI of G corresponding to an ideal I of the underlying Chevalley algebra LR;EI is thus the group generated by all xr(t) in G having the property that ter ⊂ I. In [6, § 5] we noted that in general EI actually had to be enlarged for a normal subgroup of GR to be obtained.
Plethysm of S-Functions
Authors: Usher A. O.
Published: December 2010

The S-function , where is the ‘new multiplication' or plethysm of D. E. Littlewood [1], corresponds, in the sense defined below in (1), to the character afforded by a representation of the symmetric group Slm induced from a representation of the subgroup . The aim of this paper is to define the latter representation and deduce its character using a somewhat different approach from that in [3].
The Operator Theory of Generalized Boundary Value Problems
Authors: Brown R. C.
Published: December 2010

In this paper we develop a theory of maximal and minimal operators and their duals associated with the system
On the Genus of Strong Tensor Products of Graphs
Authors: Garman B. L. , Ringeisen R. D. , White A. T.
Published: December 2010

The genus parameter for graphs has been studied extensively in recent years, with impetus given primarily by the Ringel-Youngs solution to the Heawood Map-coloring Problem [15]. This solution involved the determination of 𝛄(Kn), the genus of the complete graph Kn.
The Structure of Continuous {0, 1}-Valued Functions on a Topological Product
Authors: Broverman S.
Published: December 2010

In this paper we investigate the question of which continuous ﹛0, 1﹜-valued functions on a product space admit continuous extensions to where βXα is the Stone-Čech compactification of Xa and ﹛0, 1﹜ denotes the two point discrete space. This problem is clearly equivalent to determining which clopen subsets of have clopen closures in .
On a Structural Property of the Groups of Alternating Links
Authors: Mayland E. J. , Murasugi Kunio
Published: December 2010

In this paper, we will prove, as a consequence of the main theorem,

THEOREM A. (See Corollary 2.6). The group of an alternating knot, for which the leading coefficient of the knot polynomial is a prime power, is residually finite and solvable.
Dimension of a Topological Transformation Group
Authors: Ku Hsu-Tung , Ku Mei-Chin
Published: December 2010

Throughout this paper, the Alexander-Spanier cohomology with compact supports will be used. Suppose X is a compact connected topological ra-manifold which admits an effective action of a compact connected Lie group G (m ≧ 19).
Localization in Non-Commutative Noetherian Rings
Authors: Müller Bruno J.
Published: December 2010

To construct a well behaved localization of a noetherian ringRat a semiprime ideal S, it seems necessary to assume that the set(S)of moduloSregular elements satisfies the Ore condition ; and it is convenient to require the Artin Rees property for the Jacobson radical of the quotient ringRsin addition: one calls such 5classical.To determine the classical semiprime ideals is no easy matter; it happens frequently that a prime ideal fails to be classical itself, but is minimal over a suitable classical semiprime ideal.
Metric Spaces Without Large Closed Discrete Sets
Authors: Comfort W. W. , Hager Anthony W.
Published: December 2010

We investigate the structure of those non-separable metric spaces X, and their Stone-Čech compactifications, for which X has no closed discrete subspace of power equal to the weight of X. (Throughout this paper we denote the weight of X—the smallest power of a base for the topology of X—by the symbol wX.)
Fundamental, Picard, and Class Groups of Rings of Invariants
Authors: Magid Andy R.
Published: December 2010

Let G be a n affine algebraic group over the algebraically closed field k, and let V be an affine, normal algebraic variety over k on which G acts. Suppose that the ring of invariants k [F]G is finitely generated over k, and let W be the affine variety with k[W] = k[V]G. The purpose of this paper is to show that the induced homomorphism from the étale fundamental group of V to that of W is surjective, and to examine the consequences of this observation in terms of the relations between the Picard and divisor class groups of k[V] and k[W],
Fourier-Young Coefficients of a Function of Wiener's Class Vp
Authors: Siddiqi Rafat N.
Published: December 2010

N. Wiener [12] introduced the idea of the class Vp. A 2 π-periodic function ƒ is said to have bounded p-variation Vp(f)(1 ≦ p < ∞), or to belong to the class Vp, if
Multiplication Ideals, Multiplication Rings, and the Ring R(X)
Authors: Anderson D. D.
Published: December 2010

Let R be a commutative ring with an identity. An ideal A of R is called a multiplication ideal if for every ideal B ⊆ A there exists an ideal C such that B = AC. A ring R is called a multiplication ring if all its ideals are multiplication ideals. A ring R is called an almost multiplication ring if RM is a multiplication ring for every maximal ideal M of R. Multiplication rings and almost multiplication rings have been extensively studied—for example, see [4; 8; 9; 11; 12; 15; and 16].
Value Distribution of Biaxially Symmetric Harmonic Polynomials
Authors: D'Archangelo J. , McCoy P. A.
Published: December 2010

Consider the biaxially symmetric potential equation

where α, β > — 1/2. If 2α + 1 and 2 β + 1 are non-negative integers and if X corresponds to the hypercircle

then the biaxisymmetric Laplace equation in E2(α+ β+2),

and (1.1) are equivalent.
A Minimal Regular Space that is Not Strongly Minimal Regular
Authors: Pettey Dix H.
Published: December 2010

A regular T1 space is said to be R-closed if there is no regular T1 space in which it can be embedded as a nonclosed subspace. A regular T1 space is said to be minimal regular if no regular T1 topology on the underlying set is strictly weaker than the given topology. It is known (see [1, Theorem 4, p. 455]) that every minimal regular space is R-closed. An R-closed space, however, need not be minimal regular [3, Example 2, p. 288].