bluuebunny/crossref_metadata_2025 huggingface dataset:
Répartition Des Nombres Hautement Composés de
Ramanujan
Authors: Nicolas Jean-Louis
Published: December 2010

On dit qu'un nombre entier A est hautement composé si tout
nombre M plus petit que A a moins de
diviseurs que A. Si l'on définit d(n) =
nombre de diviseurs de n, on sait que, si la décomposition
en facteurs premiers de n est n = Πi
piai, on a:

La définition devient: A est hautement composé si et
seulement si:

1

Ramanujan [8] a défini et étudié les nombres hautement
composés, démontrant les propriétés suivantes:

Si A = 2a2 3a3 … pkapk est un nombre hautement composé, on a:

et à l'exception de A = 4 et A = 36, on a:
apk = 1 [8, §8].
Periodic Solutions of Non-Linear Evolution Equations in Banach
Spaces
Authors: Ton Bui An
Published: December 2010

In this paper the theory of Browder [2] and of Lions
[3] on periodic solutions of non-linear evolution equations
in Banach spaces is put in a more general framework so as to include the
Navier-Stokes equations and their variants.

An abstract existence theorem is proved in § 1. Applications are given in §
2. The existence of periodic solutions of the Navier-Stokes equations
without any restriction on the dimension of the space domain is established.
Application of the abstract theorem to the following problem is given:

1. Let H be a Hilbert space and (., .)H the inner product in H. Let V and
W be two reflexive separable Banach spaces with
W ⊂ V ⊂ H. W is dense
in V and V is dense in H.
Ramification Groups of Abelian Local Field Extensions
Authors: Marshall Murray A.
Published: December 2010

Let k be a local field; that is, a complete discrete-valued field having a perfect residue class field. If L is a finite Galois extension of k then L is also a local field. Let G denote the Galois group GL|k. Then the nth ramification group Gn is defined by

where OL, denotes the ring of integers of L, and PL is the prime ideal of OL. The ramification groups form a descending chain of invariant subgroups of G:

1

In this paper, an attempt is made to characterize (in terms of the arithmetic of k) the ramification filters (1) obtained from abelian extensions L\k.
Linearization of the Product of Jacobi Polynomials. III
Authors: Askey Richard , Gasper George
Published: December 2010

In a series of papers [1; 2; 3; 4] the operation of linearizing the product of two Jacobi polynomials Pn(α, β)(x), α, β > –1, has been investigated and the existence of a natural Banach algebra associated with the linearization coefficients has been proven. This was proven for α + β + 1 ≧ 0 in [3] and for a slightly larger region in [4]. It was shown in [4] that such a Banach algebra does not exist for . The method used in [1; 3; 4] was to prove the non-negativity of the expansion coefficients from which the existence of the Banach algebra easily follows. However, as shown in [4], the coefficients for a subset of can be negative infinitely often and so a different method must be used for these values of α and β.
On the Exponent of an Osculatory Packing
Authors: Boyd David W.
Published: December 2010

Suppose that U is an open set in Euclidean N-space which has a finite volume |U|. A complete packing of U is a sequence of disjoint N-spheres C = {Sn} which are contained in U and whose total volume equals that of U. In an osculatory packing, the spheres are chosen recursively so that for all n larger than a certain value m, Sn has the largest radius of all spheres contained in U\(S1– ∪ … ∪ Sn-1–) (S– is the closure of S). An osculatory packing is simple if m = 1. If rn denotes the radius of Sn, the exponent of the packing is defined by:

This quantity is of considerable interest since it measures the effectiveness of the packing of U by C.
A Relationship between Left Exact and Representable Functors
Authors: Stauffer H. B.
Published: December 2010

Our aim in this paper is to demonstrate a relationship between left exact and representable functors. More precisely, in the functor category whose objects are the additive functors from the dual of an abelian category 𝔄 to the category of abelian groups and whose morphisms are the natural transformations between them, the left exact functors can be characterized as those equivalent to a direct limit of representable functors taken over a directed class. The proof will proceed in the following manner. Lambek [3] and Ulmer [7] have shown that any functor T in can be expressed as a direct limit of representable functors taken over a comma category. When T is left exact, it is easily observed that this comma category is a filtered category. When T is left exact, it is easily observed that this comma category is a filtered category.
Partial Spreads and Replaceable Nets
Authors: Bruen A.
Published: December 2010

A blocking set S in a projective plane Π is a subset of the points of Π such that every line of Π contains at least one point of S and at least one point not in S. In previous papers [5; 6], we have shown that if Π is finite of order n, then n + √n + 1 ≦ |S| ≦ n2 – √n (see [6, Theorem 3.9]), where |S| stands for the number of points of S. This work is concerned with some applications of the above result to nets and partial spreads, and with some examples of partial spreads which give rise to unimbeddable nets of small deficiency.

In the next section we re-prove a well known result of Bruck which states that if N is a replaceable net of order n and degree k then k ≧ √n + 1, and show how this bound can be improved if n = 7, 8, or 11.
Biconnected Multifunctions of Trees which have an end Point as Fixed Point or Coincidence
Authors: Schirmer Helga
Published: December 2010

It was proved almost forty years ago that every mapping of a tree into itself has at least one fixed point, but not much is known so far about the structure of the possible fixed point sets. One topic related to this question, the study of homeomorphisms and monotone mappings of trees which leave an end point fixed, was first considered by G. E. Schweigert [6] and continued by L. E. Ward, Jr. [8] and others. One result by Schweigert and Ward is the following: any monotone mapping of a tree onto itself which leaves an end point fixed, also leaves at least one other point fixed.

It is further known that not only single-valued mappings, but also upper semi-continuous (use) and connected-valued multifunctions of trees have a fixed point [7], and that two use and biconnected multifunctions from one tree onto another have a coincidence [5].
Quasi-Splitting Exact Sequence
Authors: Hou Hsiang-Dah
Published: December 2010

Let R be a ring with 1 ≠ 0 and α, β, γ R-endomorphisms of R-modules A, B, and C respectively. We shall denote by M(R) the category of R-modules, and by End(R) the category of R-endomorphisms. For objects α and β of End(R) a morphism λ: α → β is an R-homomorphism such that λα = β λ. We shall denote by Idm(R) the full subcategory of End(R) whose objects are idempotents. Idm(R) is an abelian category, ker, coker and im are constructed in the naive way and hence

is exact in M(R) if and only if

is exact in Idm(R), where the domains of α,β, and γ are A, B, and C respectively. One sees that End (R) as well as Idm(R) is abelian.
On the Number of Prolongations of a Finite Rank Valuation
Authors: Wright Michael J.
Published: December 2010

A (non-archimedean) valuation v on a field K is said to be henselian if it has a unique prolongation to a valuation on Ka, the algebraic closure of K. A henselization (Kh, vh) of a valuated field (K, v) is a smallest separable extension of K containing a henselian prolongation vh of v. (Kh, vh) is unique up to K-isomorphism, and (Kh, vh) = (K, v) if and only if v is henselian. In this paper we confine ourselves to valuations of finite rank.

If v is a non-henselian rank one valuation on K, and if [Ks:K] = ∞, Ks being the separable closure of K, then it is known that v has infinitely many prolongations to Ka [1, (27.11)]. We shall see that this is no longer true if the rank of v is greater than one.
First Homology of Irreducible 3-Manifolds
Authors: Evans Benny
Published: December 2010

In [2], J. Gross provides an infinite collection of topologically distinct irreducible homology 3-spheres. In this paper, we construct for any finitely generated abelian group A, an infinite collection {Mi} of topologically distinct irreducible closed 3-manifolds such that H1(Mi) = A for each i.

The proof consists of first constructing a closed irreducible 3-manifold MA with H(MA) = A, and then providing a method for producing more such manifolds with the same first homology group.

All maps and spaces in this paper are assumed to be in the piecewise linear category, and all subspaces are assumed to be piecewise linear subspaces.

A 3-manifold M is irreducible if each 2-sphere in M bounds a 3-cell in M. A compact 2-manifold (or surface) F in a compact 3-manifold M is properly embedded in M if F ∩ bdM = bdF.
Tensor Products and the Splitting of Abelian Groups
Authors: Lawver D. A. , Toubassi E. H.
Published: December 2010

In [2], Irwin, Khabbaz, and Rayna discuss the splitting problem for abelian groups through the use of the tensor product. Throughout the paper they make a basic assumption, namely, that the torsion subgroup contains but one primary component. Under this restriction they introduce the concept of “splitting length”, which is a positive integer indicator of how far a group is from splitting. The results obtained along these lines may be extended to groups whose torsion subgroups contain any finite number of primary components by applying the work of Oppelt [4].

Irwin, Khabbaz, and Rayna [2] define the notion of a p-sequence and show that for groups A where T(A) is p-primary and A/T(A) has rank one, the existence of a torsion-free element with a p-sequence is sufficient for the group to split.
The Lattice of Equational Classes of Commutative Semigroups
Authors: Nelson Evelyn
Published: December 2010

There has been some interest lately in equational classes of commutative semigroups (see, for example, [2; 4; 7; 8]). The atoms of the lattice of equational classes of commutative semigroups have been known for some time [5]. Perkins [6] has shown that each equational class of commutative semigroups is finitely based. Recently, Schwabauer [7; 8] proved that the lattice is not modular, and described a distributive sublattice of the lattice.

The present paper describes a “skeleton” sublattice of the lattice, which is isomorphic to A × N+ with a unit adjoined, where A is the lattice of pairs (r, s) of non-negative integers with r ≦ s and s ≧ 1, ordered component-wise, and N+ is the natural numbers with division.
Weierstrass Points at the Cusps of Γ0(16p) and Hyperellipticity of Γ0(n)
Authors: Larcher H.
Published: December 2010

For a fixed positive integer n we consider the subgroup Γ0(n) of the modular group Γ(l). Γ0(n) consists of all linear fractional transformations L: z → (az + b)/(cz + d) with rational integers a, b, c, d, determinant ad – bc = 1, and c ≡ 0(mod n). If ℋ = {z|z = x + iy, x and y real and y > 0} is the upper half of the z-plane then S0 = S0(n) = ℋ/Γ0(n), properly compactified, is a compact Riemann surface whose genus we denote by g(n). A point P of a Riemann surface S of genus g is called a Weierstrass point if there exists a function on S that has a pole of order α ≦ g at P and is regular everywhere else on S.

Lehner and Newman started the search for Weierstrass points of S0 (or, loosely, of Γ0(n)).
Extreme Points in the Hardy Class H1 of a Riemann Surface
Authors: Pranger Walter
Published: December 2010

The theorems presented here extend known results on the set of extreme points of the unit ball of the Hardy class H1 of a disk to the situation of an arbitrary Riemann surface. Several new results are obtained. The initial motivation for this work was provided by the theorem of de Leeuw and Rudin [2, p. 471] characterizing the extreme points in the case ol a disk. Careful scrutiny of the proof of that theorem yields one necessary and one sufficient condition for being an extreme point in H1 of an arbitrary surface (Theorems 1 and 4 below). The material presented here on compact bordered surfaces is closely related to the beautiful results of Gamelin and Voichick [4] and the results of Forelli [3].

For a subharmonic function u, which has a harmonic majorant on the Riemann surface R, Mu will denote the least harmonic majorant of u.
A Class of Non-Central E-Functors
Authors: Chapman G. R.
Published: December 2010

We refer the reader to [1, Chapters 1 and 2] for the notions of E-functor and centrality. Let R1 ⊆ R be the integral group rings of the groups G1⊆ G. Butler and Horrocks [1, Chapter 26] have shown that on the category of left, unitary R-modules the Hochschild E-functor determined by R1 is central. There are no examples of non-central Hochschild E-functors, and our purpose is to establish the existence of a class of such E-functors.

Take G to be finite, non-abelian and let 5 be the centre of R. Denote by φ the Hochschild E-functor determined by S. We obtain a necessary condition for the centrality of φ in terms of the group structure of G. Let G* denote the subgroup of G generated by the commutators of G together with the set ﹛gh(g): g £ G﹜, where h(g) is the class number of g in G.
Generalization of Watson's Lemma
Authors: Wong R. , Wyman M.
Published: December 2010

Many functions, F(z), have integral representations of the form

1.1

the so-called Laplace transform of f(t). When f(t) satisfies certain regularity conditions, it is possible to use Cauchy's theorem to deform the contour so that F(z) has the integral representation

1.2

where 7 is a fixed real number, and the path of integration is the straight line joining t = 0 to t = ∞ etr, small indentations in the path of integration being allowed to avoid singularities of f(t) where necessary. When the two functions defined by (1.1) and (1.2) are not the same, (1.2) provides the more general situation for a theoretical discussion of the properties of F(z).
On Inflation and Torsion of Amitsur Cohomology
Authors: Dobbs David E.
Published: December 2010

Studies of torsion [2] and inflation [11] of Amitsur cohomology have primarily been concerned with module-finite faithful projective algebras. In this paper, our goal is to consider these topics for more general algebras. The fundamental tool, in case R is a domain with quotient field K, is the functor UK/U (defined in § 1), together with the monomorphic connecting map of Amitsur cohomology H1(S/R, UK/U) → H2(S/R, U) arising from a (commutative) flat R-algebra S and the unit functor U. This map was first considered in [10, Chapter IV, Theorem 1.6], where it was proved to be an isomorphism for certain étale faithfully flat algebras S in case R is an algebraic number ring with trivial Brauer group. In Corollary 1.5 it is shown, in the case of a module-finite faithful projective S over a regular domain R, that the connecting map is the kernel of the canonical homomorphism [8] from H2(S/R, U) to the split Brauer group B(S/R). As the latter map is often injective [8, Corollary 7.7], one might expect instances of vanishing of H1(S/R, UK/U) without assuming both R Noetherian and S module-finite R-projective. Much of § 1 (viz. (1.7)-(1.9)) is devoted to such examples.
Extensions of One Primitive Inverse Semigroup by Another
Authors: Ault Janet E.
Published: December 2010

Every inverse semigroup containing a primitive idempotent is an ideal extension of a primitive inverse semigroup by another inverse semigroup. Consequently, in developing the theory of inverse semigroups, it is natural to study ideal extensions of primitive inverse semigroups (cf. [3; 7]). Since the structure of any primitive inverse semigroup is known, an obvious type of ideal extension to consider is that of one primitive inverse semigroup by another. In this paper, we will construct all such extensions and give an abstract characterization of the resulting semigroup.

The problem of extending one primitive inverse semigroup by another can be essentially reduced to that of extending one Brandt semigroup by another Brandt semigroup. The latter problem has been solved by Lallement and Petrich in [3] in case the first Brandt semigroup has only a finite number of idempotents.
Two R-Closed Spaces
Authors: Stephenson R. M.
Published: December 2010

Throughout this paper all hypothesized spaces are T1. A regular space is called R-closed[11](regular-closed [7] or, equivalently, regular-complete [2]) provided that it is a closed subset of any regular space in which it can be embedded. A regular space (X, ℐ) is called minimal regular [2; 4] if there exists no regular topology on X which is strictly weaker than J. We shall call a regular space X strongly minimal regular provided that each point x ∈ X has a fundamental system of neighbourhoods such that for every V ∈ , X\V is an R-closed space.

In §2 we note that a strongly minimal regular space is minimal regular, but we do not know if the converse holds. M. P. Berri and R. H. Sorgenfrey [4] proved that a minimal regular space is R-closed, and Horst Herrlich [7] gave an example of an R-closed space that is not minimal regular.
Varieties of Orthomodular Lattices. II
Authors: Bruns Günter , Kalmbach Gudrun
Published: December 2010

In this paper we continue the study of equationally defined classes of orthomodular lattices started in [1].

The only atom in the lattice of varieties of orthomodular lattices is the variety of all Boolean algebras. Every nontrivial variety contains it. It follows from B. Jónsson [4, Corollary 3.2] that the variety [MO2] generated by the orthomodular lattice MO2 of Figure 1 covers the variety of all Boolean algebras. I t was first shown by R. J. Greechie (oral communication) and is not difficult to see that every variety not consisting of Boolean algebras only contains [MO2]. Again it follows from the result of Jónsson's mentioned above that the varieties generated by one of the orthomodular lattices of Figures 2 to 5 cover [MO2]. The Figures 4 and 5 are to be understood in such a way that the orthocomplement of every element is on the vertical line through this element.
Higher Monotonicity Properties of Certain Sturm-Liouville Functions. IV
Authors: Lorch Lee , Muldoon Martin E. , Szego Peter
Published: December 2010

The Sturm-Liouville functions considered in this instalment are real (as are all other quantities discussed here) non-trivial solutions of the differential equation

1.1

Higher monotonicity properties, as defined in § 2, are investigated for a number of sequences (finite or infinite) associated with these functions. One such sequence, discussed in detail later, has the kth term

1.2

where the constant X > — 1 (to assure convergence of each integral), W(x) possesses higher monotonicity properties and, moreover, is such that, again, each integral converges, and X1, X2, … is a sequence (finite or infinite) of consecutive zeros of a solution of (1.1), which may or may not be linearly independent of y(x), in the interval of definition of the functions under consideration.
On Strong Nörlund Summability Fields
Authors: Kuttner Brian , Thorpe Brian
Published: December 2010

Let p denote the sequence {pn} and set wherever this series converges. (Where no limits are stated, sums are throughout to be taken from n = 0 to n = ∞.) We use a similar notation with other letters in place of p. Given two sequences p, q, the convolution p*q is defined by

it is familiar, and easily verified, that the operation of convolution is commutative and associative. We write Pn = (p*l)n (where 1 denotes the sequence {1} ), and take P-1 to mean 0. If, for all n ≦ 0, Pn ≠ 0, then we define the Nörlund mean (N, p) of the sequence s as σn, where

and (σ-1 = 0. If σn → λ as n → ∞, then 5 is said to be limitable (N, p) to the number λ
Positive Linear Maps on C*-Algebras
Authors: Choi Man-Duen
Published: December 2010

The objective of this paper is to give some concrete distinctions between positive linear maps and completely positive linear maps on C*-algebras of operators.

Herein, C*-algebras possess an identity and are written in German type . Capital letters A, B, C stand for operators, script letters for vector spaces, small letters x, y, z for vectors. Capital Greek letters Φ, Ψ stand for linear maps on C*-algebras, small Greek letters α, β, γ for complex numbers.

We denote by the collection of all n × n complex matrices. () = ⊗ is the C*-algebra of n × n matrices over .
A Note on Complementary Subspaces in c0
Authors: Berg I. D.
Published: December 2010

A well known result of A. Pelcynski [2] states that each subspace of c0 which is isomorphic to c0 and of infinite deficiency has a complementary subspace which is itself isomorphic to c0. We are concerned here with the question of when there exists R, a subset of the integers, such that the complementary subspace X can actually be taken to be C0(R). That is, we are concerned with determining when the basis vectors for X can be chosen as a subset of the usual basis vectors for c0. If T: C0 → C0 is norm increasing and ‖T‖ < 2, it is not hard to see, as we shall show, that Tco admits a complement of the form C0(R). However, this bound cannot be improved; indeed, it is possible to construct norm increasing T: C0 → C0 such that ‖T‖ = 2 and yet Tc0 admits no such complement. The construction of such a T is the main point of this note. This construction also enables us to dispose of a speculation of ours in [1].
Rings with Finite Norm Property
Authors: Levitz Kathleen B. , Mott Joe L.
Published: December 2010

A ring A has finite norm properly,
abbreviated FNP, if each proper homomorphic image of
A is finite. In [3], Chew and Lawn
described some of the structural properties of FNP rings
with identity, which they called residually finite rings. The twofold aim of
this paper is to extend the results of [3] to arbitrary rings
with FNP and to give characterizations of
FNP rings independent of the results of
[3].

If A is a ring, let A+ denote A regarded as an abelian group. In the
first section of this paper, we explore the effects of FNP
upon the structure of A+. The following theorem
is typical of the results in this section.
Galois Theory of Essential Extensions of Modules
Authors: Wiegand Sylvia
Published: December 2010

The purpose of this paper is to exploit an analogy between algebraic
extensions of fields and essential extensions of modules, in which the role
of the algebraic closure of a field F is played by the
injective hull H(M) of a unitary left
R-module M. (The notion of * ‘algebraic’
extensions of general algebraic systems has been studied by Shoda; see, for
example [5].)

In this analogy, the role of a polynomial p(x) is played by
a homomorphism of R-modules

(1)

which will be called an ideal homomorphism into
M. The process of solving the equation
p(x) = 0 in F, or in an algebraic
extension of F, will be replaced by the process of
extending an ideal homomorphism (1) to a homomorphism
F* from R into M, or into an
essential extension of M.
Normed Lie Algebras
Authors: Vasilescu F.-H.
Published: December 2010

In this paper we attempt to define the notion of normed Lie algebra by
endowing the corresponding algebraic concept with topologicalmetric
properties. More precisely, we define normed Lie algebras as being normed
spaces possessing a Lie product, the latter satisfying a compatibility
relation. It turns out that any normed algebra, in particular the algebra of
continuous linear operators on a normed space, is a normed Lie algebra in
the sense denned below, with the usual Lie product given by the additive
commutator.

It seems that some algebraic features have within the framework of normed
Lie algebras the natural topological extensions. We mention, for instance,
the convergence of the well-known Campbell-Hausdorff formula. Let us also
mention the occurence of a variant of the Kleinecke-Sirokov theorem,
obtained via universal enveloping algebra, which might be unknown in this
context.
Restrictive Semigroups of Continuous Functions on 0-Dimensional
Spaces
Authors: Hofer Robert D.
Published: December 2010

Let X be a topological space and Y a
nonempty subspace of X. Γ(X, Y) denotes
the semigroup under composition of all closed self maps of
X which carry Y into
Y, and is referred to as a restrictive semigroup of closed
functions. Similarly, S(X, Y) is the analogous semigroup of
continuous selfmaps of X, and is referred to as a
restrictive semigroup of continuous functions. It is immediate that each
homeomorphism from X onto U which carries
the subspace Y of X onto the subspace
V of U induces an isomorphism between
Γ(X, Y) and Γ(U, V), and also an
isomorphism between S(X, Y) and S(U, V).
Indeed, one need only map f onto h o f o
h-1. An isomorphism of this form is called
representable. In [5, Theorem (3.1), p. 1223] it was shown that
in most cases, each isomorphism from Γ(X, Y) onto
Γ(U, V) is representable. The analogous problem was
discussed for the semigroup S(X, Y) and it was pointed out
by means of an example that one could not hope to obtain the same result for
these semigroups without some further restrictions.
Abelian Groups Quasi-Injective Over their Endomorphism
Rings
Authors: Poole George D. , Reid James D.
Published: December 2010

L. Fuchs has posed the problem of identifying those abelian
groups that can serve as the additive structure of an injective module over
some ring [1, p. 179], and in particular of identifying those
abelian groups which are injective as modules over their endomorphism rings
[1, p. 112]. Richman and Walker have recently answered the
latter question, generalized in a non-trivial way [7], and have
shown that the groups in question are of a rather restricted structure.

In this paper we consider abelian groups which are quasi-injective over
their endomorphism rings. We show that divisible groups are quasi-injective
as are direct sums of cyclic p-groups. Quasi-injectivity of
certain direct sums (products) is characterized in terms of the summands
(factors). In general it seems that the answer to the question of whether or
not a group G is quasinjective over its endomorphism ring
E depends on how big HomE(H,
G) is, with H a fully invariant subgroup of
G.
On the Ring of Quotients at a Prime Ideal of a Right Noetherian
Ring
Authors: Heinicke A. G.
Published: December 2010

J. Lambek and G. Michler [3] have initiated the study of a ring
of quotients RP associated with a two-sided prime ideal P in a
right noetherian ring R. The ring RP is the quotient ring (in the sense of [1]) associated
with the hereditary torsion class τ consisting of all right R-modules
M for which HomR(M, ER(R/P)) = 0, where ER(X) is the injective hull of the R-module
X.

In the present paper, we shall study further the properties of the ring
RP. The main results are Theorems 4.3 and
4.6. Theorem 4.3 gives necessary and sufficient conditions for the torsion
class associated with P to have property
(T), as well as some properties of RP when these conditions are indeed satisfied, while Theorem 4.6
gives necessary and sufficient conditions for R to satisfy
the right Ore condition with respect to (P).
A Note on Division Algebras
Authors: Herstein I. N. , Ramer A.
Published: December 2010

In this note we prove some results on the intersection properties of maximal subfields of division algebras which are finite dimensional over their centers. These results indicate that we can get very small intersections with any subalgebra if we use the appropriate maximal subfields. As a consequence of our first theorem, we obtain some theorems which are known and some which can be obtained from these known theorems (see, for instance, Theorem 3, Chapter VII in [3]). The proofs of these known results given here are very elementary and are quite different from the ones in the literature.
Right-Ordered Groups
Authors: Rhemtulla A. H.
Published: December 2010

A group G is right-ordered if it can be totally ordered so
that for any a, b, c in G, a <
b implies that ac < bc.
Right-ordered groups, considered as order preserving automorphisms of
ordered sets, were studied by Cohn in [4]; but the first
systematic study of the structure of these groups was made by Conrad in
[5] where he gave several natural characterizations of
right-ordered groups. We mention here that the class of right-ordered groups
is precisely the subgroup closure of the class of lattice ordered groups
(see [6], [7], [9] or
[10]).
Super-Reflexive Banach Spaces
Authors: James Robert C.
Published: December 2010

A super-reflexive Banach space is defined to be a Banach space
B which has the property that no non-reflexive Banach
space is finitely representable in B. Super-reflexivity is
invariant under isomorphisms; a Banach space B is
super-reflexive if and only if B* is super-reflexive. This concept has many equivalent formulations,
some of which have been studied previously.
An Embedding Theorem for Ordered Groups
Authors: Minassian Donald P.
Published: December 2010

An O*-group is a group wherein every partial
order can be extended to some full order.

THEOREM. Suppose the group G has a normal chain G = G1 ⊇ G2 ⊇ … such that

and each G/Gi is locally nilpotent and torsion-free. Then
G can be embedded in thecomplete direct product G’ of divisible
O*-groups.
Extensions of Homomorphisms of Parabolic
Subgroups
Authors: Wilson Edward N.
Published: December 2010

If one wishes to construct a homomorphism of a group, an obvious approach is
to begin with a homomorphism of a subgroup and attempt to extend it in some
way. Any such extended homomorphism is determined by its action on elements
of the group which, together with the subgroup, generate the whole group.
The values assigned to such generators must then satisfy various functional
equations. By a governing list for the extension problem,
we shall mean a list of functional equations whose solutions describe all
possible homomorphisms extending a given subgroup homomorphism.
An Lp Saturation Theorem for Splines
Authors: Butler G. J. , Richards F. B.
Published: December 2010

Let 1 be a
subdivision of [0, 1], and let denote the
class of functions whose restriction to each sub-interval is a polynomial of degree at most
k. Gaier [1] has shown that for uniform
subdivisions △n (that is, subdivisions for which

if and only if f is a polynomial of degree at most
k. Here, and subsequently, denotes the usual norm in Lp[0,
1], 1 ≦ p ≦ ∞, and we should
emphasize that functions differing only on a set of Lebesgue measure zero
are identified.
On Open Projections of GCR Algebras
Authors: Digernes Trond , Halpern Herbert
Published: December 2010

Throughout this paper will denote
a C*-algebra and will denote
its second dual, which is identified with the enveloping von Neumann algebra
of . A
projection E in is said to
be open if it supports a left ideal in , that is, if E =
- for some
left ideal in
. Here the
bar - means the stong closure. When has a unit, this definition coincides with the
definition of Akemann [1, Definition II. 1]. In the sequel, we
shall solely be concerned with two-sided ideals, and consequently central
projections [4, I, § 3, Corollary 3 of Theorem 2]. Our aim is to show that
is CCR if
and only if the open central projections are strongly dense in the set of
central projections on .
Graphical Regular Representations of Non-Abelian Groups, II
Authors: Nowitz Lewis A. , Watkins Mark E.
Published: December 2010

The present paper is a sequel to the previous paper bearing the same title by the same authors [3] and which will be hereafter designated by the bold-face Roman numeral I. Further results are obtained in determining whether a given finite non-abelian group G has a graphical regular representation. In particular, an affirmative answer will be given if (|G|, 6) = 1.

Inasmuch as much of the machinery of I will be required in the proofs to be presented and a perusal of I is strongly recommended to set the stage and provide motivation for this paper, an independent and redundant introduction will be omitted in the interest of economy.
𝔪-Stone Lattices
Authors: Davey B. A.
Published: December 2010

A Stone lattice is a distributive, pseudo-complemented lattice L such that a* V a** = 1, for all a in L; or equivalently, a bounded distributive lattice L in which, for all a in L, the annihilator a⊥ = {b ∊ L|a ∧ b = 0} is a principal ideal generated by an element of the centre of L, namely a*.

Thus it is natural to define an 𝔪-Stone lattice to be a bounded distributive lattice L in which, for each subset A of cardinality less than or equal to m, the annihilator A⊥ = {b ∊ L|a ∧ b = 0, for all a ∊ A} is a principal ideal generated by an element of the centre of L.

In this paper we characterize 𝔪-Stone lattices, and show, by considering the lattice of global sections of an appropriate sheaf, that any bounded distributive lattice can be embedded in an 𝔪-Stone lattice, the embedding being a left adjoint to the forgetful functor.
The Friedberg-Muchnik Theorem Re-Examined
Authors: Soare Robert I.
Published: December 2010

In the well-known solution to Post's problem, Friedberg [1] and Muchnik [13] each constructed a pair of incomparable recursively enumerable (r.e.) degrees a and b. Subsequently, Sacks [15, p. 81] constructed r.e. degrees c and d such that c ∪ d = 0’ and C’ = d’ = 0'. Lachlan showed [7, p. 69] that such degrees c, d could have no greatest lower bound in the upper semilattice of r.e. degrees. We show that the original Friedberg-Muchnik degrees a, b automatically satisfy Sack's conditions and hence witness that the upper semi-lattice of r.e. degrees is not a lattice.
Sequentially Relatively Uniformly Complete Riesz Spaces and Vulikh Algebras
Authors: Tucker C. T.
Published: December 2010

Throughout this paper V will denote an Archimedean Riesz space with a weak unit e and a zero element θ. A sequence f1,f2,f3, … of points of V is said to converge relatively uniformly to a point f (with regulator the point g of V) if, for each ∈ > 0, there is a number N such that, if n is a positive integer and n > N, then |f — fn| < ∈g. In an Archimedean Riesz space a relatively uniformly convergent sequence has a unique limit. The sequence f1, f2, f3, … is called a relatively uniform Cauchy sequence (with regulator g) if, for each ∈ > 0, there is a number N such that if n and m are positive integers and n, m > N, then |fn — fm| < eg. A subset M of V is said to be sequentially relatively uniformly complete, written s.r.u.-complete, whenever every relatively uniform Cauchy sequence of points of M (with regulator in V) converges to a point of M. This property was defined by Luxemburg and Moore in [4] and some related conditions were derived.
A Note on Open Extension of Maps
Authors: Kohli J. K.
Published: December 2010

In recent years there has been some interest in trying to improve the behaviour of maps by extending their domains (see Whyburn [10], Baur [3], Krolevec [8], Dickman [5], Franklin and Kohli [6]). It was shown in [6] that every map can be extended to an open map so that certain properties of the domain and range are preserved in the new domain. In [6] and [7] we also related the topological properties of the domain and range of the mapping with the new domain; also these results were then used to obtain analogues and improvements of recent theorems of Arhangelskii, Čoban, Hodel, Keesling, Nagami, Okuyama, and Proizvolov. In this note we give a method of unifying the domain and range of a mapping so as to yield a meaningful open extension.
Defining Families for Integral Domains of Real Finite Character
Authors: Heinzer William , Ohm Jack
Published: December 2010

Throughout this paper R and D will denote integral domains with the same quotient field K. A set of integral domains {Di} i∊I with quotient field K will be said to have FC (“finite character” or “finiteness condition“) if 0 ≠ ξ ∊ K implies ξ is a unit of Di for all but finitely many i. If ∩i∊IDi also has quotient field K, then {Di} has FC if and only if every non-zero element in ∩i∊IDi is a non-unit in at most finitely many Di. A non-empty set {Vi}i∊:I of rank one valuation rings with quotient field K will be called a defining family of real R-representativesfor D if {Vi} i∊:I has FC, R (⊄ ∩i∊IVi, and D = R∩ (∩i∊I Vi).
Normed Right Alternative Algebras Over the Reals
Authors: Nieto José I.
Published: December 2010

One of the most interesting results on real normed division algebras says that every real normed associative division algebra is finite dimensional [6, Theorem 1.7.6], and hence by a classical theorem of Frobenius either isomorphic to the real field, the complex field, or the algebra of quaternions. Thus the dimension of the algebra can only be either 1, 2 or 4.
A Theorem on Isometries and the Application of it to the Isometries of Hp(S) for 2 < p < ∞
Authors: Forelli Frank
Published: December 2010

Let X and Y be sets, let λ be a bounded positive measure on X, and let μ be a bounded positive measure on Y. Furthermore let M be a subalgebra of L∞(λ), let p ∈ (0, ∞), and let A be a linear transformation of M into Lp(μ) such that

for all f in M.

In § 2 of this paper we will prove the following theorem.
R-isomorphisms of Transformation Groups and Prolongations
Authors: King Larry
Published: December 2010

In [8] the notion of a reparameterizing isomorphism of transformation groups, henceforth called an R-isomorphism, is introduced generalizing Ura's type-2 isomorphism (see [13]). We have shown [8] that an R-isomorphism is weaker than a transformation group isomorphism. For example, although R-isomorphisms preserve pointwise almost periodicity and minimality they do not preserve the existence of slices [7] or almost periodicity. This suggests that R-isomorphisms might be a useful classification tool in topological dynamics.
Testing Categories and Strong Universality
Authors: Sichler J.
Published: December 2010

A category A is binding (or universal) if any full category of algebras is isomorphic to a full subcategory of A. There are many binding categories: the category of all commutative rings with unit and all unit-preserving homomorphisms [1], the category of bounded lattices [2], the category of semigroups [3], the category A(1, 1) of all algebras with two unary fundamental operations and the category of directed graphs [4], the category of all commutative groupoids [11] and many others.
On the Wedderburn Theorem
Authors: Szeto George
Published: December 2010

In [6], Pierce studied the modules over a commutative regular ring R by using the representation of R as the global sections of a sheaf which we call the Pierce sheaf. When the stalks of the Pierce sheaf are regular, Magid gave a Galois theory and some properties for a central separable R-algebra [4, (2.4), (2.5), (2.6) and (2.7)]. When the stalks of the Pierce sheaf are semi-local, DeMeyer presented a Galois theory for a central separable R-algebra [3, sections 2 and 3] and the author characterized the finitely generated and projective modules over a central separable R-algebra in terms of the R-modules in [7] and [8].
Generalizations of Noshiro's Theorem and Their Applications
Authors: Yoshida Hidenobu
Published: December 2010

Meier [8, Hauptsatz] proved a remarkable theorem concerning the boundary behavior of functions meromorphic in the upper half plane; but his techniques are very complicated. So Noshiro [10, p. 72-73] proved an analogous (but somewhat weaker) result to Meier's by a simple method using the theorem of Gross and Iversen.

In this paper, we sharpen and generalize Noshiro's theorem in some directions by making use of the notion “porosity”, and we state some applications.
A 1-ALG Simple Closed Curve in E3 is Tame
Authors: Boyd W. S. , Wright A. H.
Published: December 2010

Let J be a simple closed curve in a 3-manifold M3. We say M — J is 1-ALG at p ∈ J (or has locally abelian fundamental group at p) if and only if for each sufficiently small open set U containing p, there is an open set V such that p ∈ V ⊂ U and each loop in V — J which bounds in U — J is contractible to a point in U — J.
A Note on Root Decision Problems in Groups
Authors: Lipschutz Seymour , Lipschutz Martin
Published: December 2010

Consider a positive integer r > 1. We say that the rth root problem is solvable for a group G if we can decide for any W ॉ G whether or not W has an rth root, i.e. whether or not there exists V ॉ G such that W = Vr.

Baumslag, Boone and Neumann [1] proved that there exists a finitely presented group with all root problems unsolvable. Here we are concerned with the relationship between the different root problems.
Sum Theorems for Countably Paracompact Spaces
Authors: Potoczny Henry
Published: December 2010

In this paper, we extend the class of spaces to which the Σ and β theorems of Hodel apply, as well as the sum and subset theorems of [2]. Instead of the open cover definition of countable paracompactness, we utilize an equivalent formulation of countable paracompactness, due to Ishikawa [3].
Embedding Circle-Like Continua in E3
Authors: Ball B. J. , Sher R. B.
Published: December 2010

A space X is locally planar if each point of X has a neighborhood which is embeddable in the plane. If X is a closed, locally planar subset of E3, we will say that X is locally tame if each point of X has a neighborhood in X which lies on a tame disk in E3; if every cell-like subset of X has such a neighborhood, we say that X is strongly locally tame.
Infinite Quasi-Normal Matrices
Authors: Wiegmann N. A.
Published: December 2010

If A is a finite matrix with complex elements, and if A = AT (where AT denotes the transpose of A ), it is known (see [8] ) that there exists a unitary matrix U such that UA UT = D is a real diagonal matrix with non-negative elements which is a canonical form for A relative to the given U, UT transformation.
Metrizability of M-Spaces
Authors: Martin Harold W.
Published: December 2010

An unsolved problem in metrization theory is whether every Hausdorff M-space with a Gg-diagonal is metrizable. There are several recent results which have a bearing upon this question. In [9], P. Zenor showed that an M-space is metrizable if and only if it has a regular Gδ-diagonal; in [1], Borges showed that a regular meta-Lindelöf M-space is metrizable if and only if it has a -diagonal.
Algebras Over Dedekind Domains
Authors: Wehlen Joseph A.
Published: December 2010

The purpose of this paper is two-fold : first, to show that Dedekind domains satisfy a generalization of the Wedderburn-Mal'cev Theorem and, secondly, to classify certain types of finitely generated projective algebras over a Dedekind domain.

With respect to the first problem, E. C. Ingraham has shown that a Dedekind domain R is an inertial coefficient ring (IC-ring) if and only if R has zero radical or R is a local Hensel ring.
An Essential Ring Which is Not A v-Multiplication Ring
Authors: Heinzer William , Ohm Jack
Published: December 2010

An integral domain D is called an essential ring if D = ∩αVα where the Vα are valuation rings which are quotient rings of D. D is called a v-multiplication ring if the finite divisorial ideals of D form a group. Griffin [2, pp. 717-718] has observed that every v-multiplication ring is essential and that an essential ring having a defining family of valuation rings {Vα} which is of finite character (i.e. every nonzero element of D is a non-unit in at most finitely many Vα) is necessarily a v-multiplication ring; but he conjectures that, in general, there exists an essential ring which is not a v-multiplication ring.
On C0-Sufficiency of Complex Jets
Authors: Chang S. H. , Lu Y. C.
Published: December 2010

In this paper we shall study the sufficiency of complex jets. Let A (Cn, C) be the set of all analytic functions f : Cn → C with f (0) = 0. We call two functions f and g of A (Cn, C) equivalent of order r at 0 if, at 0, their Taylor expansions up to and including the terms of degree ≦ r are identical.
On Hereditary and Cohereditary Modules
Authors: Shrikhande M. S.
Published: December 2010

A recent paper by Goro Azumaya on M-projective and M-injective modules [1] suggests a generalization of the concept of hereditary rings to modules which is also capable of dualization. Section 2 is devoted to preliminaries on M-projective and M-infective modules.

In section 3, we introduce the notion of hereditary and cohereditary modules. An R-module is called hereditary if every R-submodule of it is projective. Cohereditary modules are defined dually.
A Tauberian Theorem for a Scale of Logarithmic Methods of Summation
Authors: Phillips R.
Published: December 2010

We suppose throughout that p is a non-negative integer, and use the following notations:

where (n = 0 , 1 , 2 , … );
The Radius of Convexity of a Linear Combination of Functions in or uα
Authors: Campbell Douglas Michael
Published: December 2010

Labelle and Rahman [4] showed that if f , g ∈ , the normalized convex functions in the unit disc D, then has a radius of convexity at least as large as the smallest root of 1 – 3r + 2r2 — 2r3 = 0. Their method requires neither the properties of the arithmetic mean nor the strong geometric properties of ; indeed, the procedure works for a linear combination of functions from any linear invariant family of finite order.
Centers of Infinite Bounded Sets in a Normed Space
Authors: Calder J. R. , Coleman W. P. , Harris R. L.
Published: December 2010

Čebyšev centers have been studied extensively. In this paper an alternate concept of center for infinite bounded point sets is introduced. Some of the results in this paper for this new type of center are similar to previous results for Čebyšev centers.
T-Orthogonality and Nonlinear Functionals on Topological Vector Spaces
Authors: Sundaresan K. , Kapoor O. P.
Published: December 2010

In recent years the problem of concretely representing a class of nonlinear functionals on Banach spaces has received considerable attention. Suppose B is a Banach space equipped with an orthogonality relation ⊥ ⊂ B X B.
The Sphericity of Higher Dimensional Knots
Authors: Dyer Eldon , Vasquez A. T.
Published: December 2010

In 1956 CD. Papakyriakopoulos showed [5] that the complement C of a 1-sphere S1 tamely imbedded in a 3-sphere S3 is aspherical; that is, that for all i ≧ 2, πi(C) = 0. In this note we show that for n ≧ 2 the complement C of an n-sphere Sn smoothly imbedded in Sn+2 is aspherical only if the fundamental group of C is infinite cyclic. Combined with results of J. Stallings [6] or of J. Levine [3], this implies that if the complement of an Sn smoothly imbedded in Sn+2 is aspherical, n ≥ 4 , then Sn is topologically unknotted in Sn+2.
Non-Abelian Torsion Theories
Authors: Barr Michael
Published: December 2010

Torsion theories have proved a very useful tool in the theory of abelian categories; for example, in one proof of Mitchell's embedding theorem (Bucur and Deleanu [3]) and in ring theory (Lambek [5]). It is the purpose of this paper to initiate an analogous theory for non-abelian categories. Originally we had hoped to prove the non-abelian analogue of Mitchell's theorem this way (Barr, [2, Theorem III (1.3)]), but so far this had not been possible. Nonetheless an interesting variety of examples fit into this theory.
The Primitive Ideal Space of a C*-Algebra
Authors: Dauns John
Published: December 2010

The commutative Gelfand-Naimark Theorem says that any commutative C*-algebra A is isomorphic to the ring C0(M, C) of all continuous complex-valued functions tending to zero outside of compact sets of a locally compact Hausdorff space M. A very important part of this theorem is an intrinsic and also a complete characterization of M as exactly the primitive ideal space of A in the hull-kernel (or weak-star) topology. In the non-commutative case, A ≌ Γ0(M, E)—the ring of sections tending to zero outside of compact subsets of a locally compact Hausdorff space M with values in the stalks or fibers E.
On the Splitting of Modules and Abelian Groups
Authors: Hill Paul
Published: December 2010

In a fundamental paper on torsion-free abelian groups, R. Baer [1] proved that the group P of all sequences of integers with respect to componentwise addition is not free. This means precisely that P is not a direct sum of infinite cyclic groups. However, E. Specker proved in [9] that P has the property that any countable subgroup is free. Since an infinite abelian group G is called -free if each subgroup of rank less than is free, these results are equivalent to: P is -free but not free.