bluuebunny/crossref_metadata_2025 huggingface dataset:
Real Subspaces of a Quaternion Vector Space
Authors: Dlab Vlastimil , Ringel Claus Michael
Published: December 2010

If UR is a real subspace of a finite dimensional vector space VC over the field C of complex numbers, then there exists a basis ﹛e1, … , en﹜ of VG such that
Groups and Complements of Knots
Authors: Feustel C. D. , Whitten Wilbur
Published: December 2010

We investigate the extent to which knot groups determine knot manifolds and knot types. Let Ki(i = 1, 2) denote a tame knot in S3, let Ci denote a Ki-knot manifold, and assume that Π1(C1) ≈ Π1(C2). The first named author recently showed (in [6]) that, if C1 has no essential annulus, then C1 ≅ C2, and so K1 and K2 are equivalent, if K1 has property P.
Quotient Spaces Without Bases in Nuclear Frechet Spaces
Authors: Dubinsky Ed , Mitiagin Boris
Published: December 2010

The first example of a nuclear Fréchet space without a basis was given by B. S. Mitiagin and N. M. Zobin [9; 10]. The question of existence of subspaces without bases in nuclear Fréchet spaces was recently settled in papers by P. Djakov and B. S. Mitiagin [2] and Ed Dubinsky [5]. In this paper we consider the analogous question for quotient spaces. As in the case of subspaces we obtain a complete solution to the problem.
Planar Sublattices of a Free Lattice. II
Authors: Rival Ivan , Sands Bill
Published: December 2010

In Planar sublattices of a free lattice, I [8] we verify Jonsson's conjecture for finite planar lattices; in particular we obtain a characterization of finite planar sublattices of a free lattice among all finite lattices. In the present paper we use arguments of a quite different flavour to obtain another characterization. Let

be the family of lattices illustrated in Figures 1, 2, 3, and 4. Our goal is to prove the following theorem: a finite lattice is a planar sublattice of a free lattice if andonly if it does not have a member of as a sublattice.
Rings Characterized by their Cyclic Modules
Authors: Smith P. F.
Published: December 2010

A ring R (with identity element) is called a right PCI-ring if and only if every proper cyclic right R-module is injective; that is, if C is a cyclic right R-module then either C ≌ R or C is injective. Faith [3, Theorems 14 and 17] (or see [2, Proposition 6.12 and Theorem 6.17]) proved that if a ring R is a right PCI-ring then R is semiprime Artinian or R is a simple right semihereditary right Ore domain. These latter rings we shall call simple rightPCI-domains. Examples of non-Artinian simple right PCI-domains were produced by Cozzens [1]. The object of this paper is to examine rings with similar properties and thus extend Faith's results.
Lifting the Commutant of a Subnormal Operator
Authors: Olin Robert F. , Thomson James E.
Published: December 2010

Let S be a subnormal operator on a Hilbert space ℋ and let N be its minimal normal extension on the Hilbert space ℋ. (We refer the reader to [5, 15] for the basic material on subnormal operators.) Denote the commutant and double commutant of an operator T by ﹛T﹜’ and ﹛T﹜”, respectively.
Some Radical Properties of Jordan Matrix Rings
Authors: Rich Michael
Published: December 2010

Let A be a ring (not necessarily associative) in which 2x = a has a unique solution for each a ∈ A. Then it is known that if A contains an identity element 1 and an involution j : x ↦ x and if Ja is the canonical involution on An determined by

where the ai al−l, 1 ≦ i ≦ n are symmetric elements in the nucleus of A then H(An, Ja), the set of symmetric elements of An, for n ≧ 3 is a Jordan ring if and only if either A is associative or n = 3 and A is an alternative ring whose symmetric elements lie in its nucleus [2, p. 127].
A Note on the Normal Moore Space Conjecture
Authors: Devlin Keith J. , Shelah Saharon
Published: December 2010

F. B. Jones (1937) conjectured that every normal Moore space is metrizable. He also denned a particular kind of topological space (now known as Jones' spaces), proved that they were all non-metrizable Moore spaces, but was unable to decide whether or not Jones’ spaces are normal. J. H. Silver (1967) proved that a positive solution to Jones’ conjecture was not possible, and W. Fleissner (1973) obtained an alternative proof by showing that it is not possible to prove the non-normality of Jones’ spaces. These results left open the possibility of resolving the questions from the GCH. In this paper we show that if CH be assumed, then Jones’ spaces are not normal (Devlin, Shelah, independently) and that the GCH does not lead to a positive solution to the Jones conjecture (Shelah). A brief survey of the progress on the problem to date is also included.
On Global Inverse Theorems of Szász and Baskakov Operators
Authors: Ditzian Z.
Published: December 2010

The Szász and Baskakov approximation operators are given by

1.1

1.2

respectively. For continuous functions on [0, ∞) with exponential growth (i.e. ‖ƒ‖A ≡ supx\ƒ(x)e–Ax\ < M) the modulus of continuity is defined by

1.3

where ƒ ∈ Lip* (∝, A) for some 0 < ∝ ≦ 2 if w2(ƒ, δ, A) ≦ Mδ∝ for all δ < 1. We shall find a necessary and sufficient condition on the rate of convergence of An(ƒ, x) (representing Sn(ƒ, x) or Vn(ƒ, x)) to ƒ(x) for ƒ(x) ∈ Lip* (∝, A). In a recent paper of M. Becker [1] such conditions were found for functions of polynomial growth (where (1 + \x\N)−1 replaced e–Ax in the above). M. Becker explained the difficulties in treating functions of exponential growth.
Compatible Tight Riesz Orders II
Authors: Glass A. M. W.
Published: December 2010

R. N. Ball (unpublished) and G. E. Davis and C. D. Fox [1] established that if Ω is a doubly homogeneous totally ordered set, the l-group A (Ω) of all orderpreserving permutations of Ω endures a compatible tight Riesz order. Specifically T = {g ∈ A(Ω)+ : supp (g) is dense in Ω} is a compatible tight Riesz order for A(Ω). Using this fact, I inserted Theorem 3.7 into [2; MR 53 (1977), #13070] at the galley proof stage. (It was also included in MR 54 (1977), #7350 and [3; p. 472].) Theorem 3.7 stated: Let Ω be homogeneous. Then A(Ω) endures a compatible tight Reisz order if and only if Ω is dense. I stated that it was obvious that if Ω were homogeneous and discrete, A(Ω) could not endure a compatible tight Riesz order. This “obvious” is neither obvious nor true.
A Semilinear Dirichlet Problem
Authors: Castro Alfonso
Published: December 2010

Introduction and notations. Let Ω be a bounded region in Rn. In this note we discuss the existence of weak solutions (see [4, Section 2]) of the Dirichlet problem

(I)

where Δ is the Laplacian operator, g : Ω × R → R and f : Ω × Rn+1 → R are functions satisfying the Caratheodory condition (see [2, Section 3]), and ∇ is the gradient operator.

We let λ1 < λ2 ≦ … ≦ λm ≦ … denote the sequence of numbers for which the problem

(II)

has nontrivial weak solutions.
On Bernstein's Inequality
Authors: Giroux A. , Rahman Q. I. , Schmeisser G.
Published: December 2010

1. Introduction and statement of results. If pn(z) is a polynomial of degree at most n, then according to a famous result known as Bernstein's inequality (for references see [4])

(1)

Here equality holds if and only if pn(z) has all its zeros at the origin and so it is natural to seek for improvements under appropriate assumptions on the zeros of pn(z). Thus, for example, it was conjectured by P. Erdös and later proved by Lax [2] that if pn(z) does not vanish in │z│ < 1, then (1) can be replaced by

(2)

On the other hand, Turán [5] showed that if pn(z) is a polynomial of degree n having all its zeros in │z│ ≦ 1, then

(3)
Biplanar Surfaces of Order Three
Authors: Bisztriczky Tibor
Published: December 2010

0. Introduction. A surface of order three, F, in the real projective threespace P3 is met by every line, not in F, in at most three points. F is biplanar if it contains exactly one non-differentiable point v and the set of tangents of F at v is the union of two distinct planes, say τ1 and τ2. In the present paper, we classify and describe those biplanar F which contain the line τ1 ∩ τ2.

We describe a surface by determining the tangent plane sections of the surface at the differentiable points. This approach was introduced in [1] and it is based upon A. Marchaud's definition of “surfaces of order three” in [4].
Localization, Algebraic Loops and H-Spaces I
Authors: Shar Albert O.
Published: December 2010

If (Y, µ) is an H-Space (here all our spaces are assumed to be finitely generated) with homotopy associative multiplication µ. and X is a finite CW complex then [X, Y] has the structure of a nilpotent group. Using this and the relationship between the localizations of nilpotent groups and topological spaces one can demonstrate various properties of [X,Y] (see [1], [2], [6] for example). If µ is not homotopy associative then [X, Y] has the structure of a nilpotent loop [7], [9]. However this algebraic structure is not rich enough to reflect certain significant properties of [X, Y]. Indeed, we will show that there is no theory of localization for nilpotent loops which will correspond to topological localization or will restrict to the localization of nilpotent groups.
Ratio and Stochastic Ergodic Theorems for Superadditive Processes
Authors: Fong Humphrey
Published: December 2010

1. Introduction. Let (X, , m) be a σ-finite measure space and let T be a positive linear operator on L1 = L1(X, , m). T is called Markovian if

(1.1)

T is called sub-Markovian if

(1.2)

All sets and functions are assumed measurable; all relations and statements are assumed to hold modulo sets of measure zero.

For a sequence of L1+ functions (ƒ0, ƒ1, ƒ2, …), let

(ƒn) is called a super additive sequence or process, and (sn) a super additive sum relative to a positive linear operator T on L1 if

(1.3)

and

(1.4)
Quotients and Inverse Limits of Spaces of Orderings
Authors: Marshall Murray A.
Published: December 2010

A connection between the theory of quadratic forms defined over a given field F, and the space XF of all orderings of F is developed by A. Pfister in [12]. XF can be viewed as a set of characters acting on the group F×/ΣF×2, where ΣF×2 denotes the subgroup of F× consisting of sums of squares. Namely, each ordering P ∈ XF can be identified with the character

defined by

It follows from Pfister's result that the Witt ring of F modulo its radical is completely determined by the pair (XF, F×/ΣF×2).
Partial λ-Geometries and Generalized Hadamard Matrices Over Groups
Authors: Drake David A.
Published: December 2010

Section 1 of this paper contains all the work which deals exclusively with generalizations of Hadamard matrices. The non-existence theorem proven here (Theorem 1.10) generalizes a theorem of Hall and Paige [15] on the non-existence of complete mappings in certain groups.

In Sections 2 and 3, we consider the duals of (Hanani) transversal designs; these dual structures, which we call (s, r, µ)-nets, are a natural generalization of the much studied (Bruck) nets which in turn are equivalent to sets of mutually orthogonal Latin squares. An (s, r, µ)-netis a set ofs2µpoints together withrparallel classes of blocks. Each class consists ofsblocks of equal cardinality. Two non-parallel blocks meet in preciselyµpoints. It has been proven thatris always less than or equal to (s2µ– l) / (s– 1).
On Countability of Point-Finite Families of Sets
Authors: Junnila Heikki J. K.
Published: December 2010

It is well known that in a separable topological space every point-finite family of open subsets is countable. In the following we are going to show that both in σ-finite measure-spaces and in topological spaces satisfying the countable chain condition, point-finite families consisting of “large” subsets are countable.

Notation and terminology. Let A be a set. The family consisting of all (finite) subsets of A is denoted by . Let be a family of subsets of A. The sets and are denoted by and , respectively. We say that the family is point-finite (disjoint) if for each a ∈ A , the family has at most finitely many members (at most one member).
Inverse Semigroups of Homeomorphisms are Hopfian
Authors: Baird Bridget B.
Published: December 2010

If X is a nonempty topological T1 space then the set of all homeomorphisms whose domains and ranges are closed subsets of X forms a semigroup under partial composition of functions. We call it IF(X). If, in a semigroup, every element a is matched with a unique element b such that aba = a and bab = b then the semigroup is an inverse semigroup (b is called the inverse of a and is denoted by a−1). We have that IF(X) is an inverse semigroup with the algebraic inverse of a map ƒ being just the inverse map ƒ-1. In this paper we examine epimorphisms from IF(X) onto IF(Y). The main theorem gives conditions under which an epimorphism must be an isomorphism.
Not Every K1-Embedded Subspace is K0-Embedded
Authors: Mill Jan van
Published: December 2010

All topological spaces under discussion are assumed to be Tychonoff.

For any topological space X let τ(X) denote the topology of X. If X ᑕ Y then a function κ : τ(X) ⟶ τ(Y) is called an extender provided that κ(U) ∩ X = U for all U ∊ τ(X). In addition, X is said to be Kn-embedded in Y (cf. [3]) provided there is an extender κ : τ(X) ⟶ τ(Y) such that

The extender κ is called a Kn-function (cf. [3]).

Eric van Douwen has asked whether there is a space X with a subspace Z which is Ki-embedded but not K0-embedded. The aim of this note is to answer this question.
Simple Algebras Over Rational Function Fields
Authors: Nyman T. , Whaples G.
Published: December 2010

The well-known Hasse-Brauer-Noether theorem states that a simple algebra with center a number field k splits over k (i.e., is a full matrix algebra) if and only if it splits over the completion of k at every rank one valuation of k. It is natural to ask whether this principle can be extended to a broader class of fields. In particular, we prove here the following extension.
Sets of Uniqueness for the Group of Integers of a p-Series Field
Authors: Wade William R.
Published: December 2010

Let G denote the group of integers of a p-series field, where p is a prime ≦ 2. Thus, any element can be represented as a sequence {xi }i = 0∞ with 0 ≦ xi < p for each i ≦ 0. Moreover, the dual group {Ψm}m = 0∞ of G can be described by the following process. If m is a non-negative integer with for each k , and if then

(1)

where for each integer k ≧ 0 and for each x = {xi} ∈ G the functions Φk are defined by

(2)

In the case that p = 2, the group G is the dyadic group introduced by Fine [1] and the functions are the Walsh-Paley functions. A detailed account of these groups and basic properties can be found in [4].
A Simple C*-Algebra Generated by Two Finite-Order Unitaries
Authors: Choi Man-Duen
Published: December 2010

We present an example which illustrates several peculiar phenomena that may occur in the theory of C*-algebras. In particular, we show that a C*-subalgebra of a nuclear (amenable) C*-algebra need not be nuclear (amenable).

The central object of this paper is a pair of abstract unitary matrices,

acting on a common Hilbert space. For an explicit construction, we may decompose an infinite-dimensional Hilbert space H into H = H0 ⴲ H1 , H1 = Hα ⴲ Hβ with dim H0 = dim H1 = dim Hα = dim Hβ, letting u, v Є B(H) be any two unitary operators such that

and u2 = 1, v3 = 1. Whereas many choices of u, v are possible, it might be surprising to see that C*(u, v), the C*-algebra generated by u and v, is algebraically unique; namely, if (u1,V1) is another pair of such unitaries, then C*(u, v) is canonically *-isomorphic with C*(u1, v1) (Theorem 2.6).
On Infinite-Difference Sets
Authors: Stewart C. L. , Tijdeman R.
Published: December 2010

1. Introduction. Let A be a sequence; throughout this paper sequences are understood to be infinite, strictly increasing and composed of non-negative integers. We define D, the infinite-difference set of A, to be the set of those non-negative integers which occur infinitely often as the difference of two terms of A. Plainly D has no positive terms if and only if ai+1 — ai → ∞ as i → ∞. Note that D contains zero. We shall be interested in the case when . Then D certainly contains more than one term. In fact, see Corollary 1, §2, in this case. Here and ḏ denote the (natural asymptotic) upper and lower density respectively.
Lie Algebras with Nilpotent Centralizers
Authors: Benkart G. M. , Isaacs I. M.
Published: December 2010

We consider finite dimensional Lie algebras over an algebraically closed field F of arbitrary characteristic. Such an algebra L will be called a centralizer nilpotent Lie algebra (abbreviated c.n.) provided that the centralizer C(x) is a nilpotent subalgebra of L for all nonzero x ∈ L.

For each algebraically closed F, there is a unique simple Lie algebra of dimension 3 over F which we shall denote S(F). This algebra has a basis e−1, e0, e1 such that [e−1e0] = e−1, [e−1e1] = e0 and [e0e1] = e1. (If char(F) ≠ 2, then S(F) ≅ sl2(F).) It is trivial to check that S(F) is a c.n. algebra for all F.

There are two other types of simple Lie algebras we consider. If char (F) = 3, construct the octonion (Cayley) algebra over F.
Almost Split Sequences whose Middle Term has at most Two Indecomposable Summands
Authors: Auslander M. , Bautista R. , Platzeck M. I. , Reiten I. , Smalø S. O.
Published: December 2010

Let Λ be an artin algebra, and denote by mod Λ the category of finitely generated Λ-modules. All modules we consider are finitely generated.

We recall from [6] that a nonsplit exact sequence in mod A is said to be almost split if A and C are indecomposable, and given a map h: X → C which is not an isomorphism and with X indecomposable, there is some t: X → B such that gt = h.

Almost split sequences have turned out to be useful in the study of representation theory of artin algebras. Given a nonprojective indecomposable Λ-module C (or an indecomposable noninjective Λ-module A), we know that

there exists a unique almost split sequence [6, Proposition 4.3], [5, Section 3].
Perturbations of AF-Algebras
Authors: Phillips John , Raeburn Iain
Published: December 2010

Let A and B be C*-algebras acting on a Hilbert space H, and let

where A1 is the unit ball in A and d(a, B1) denotes the distance of a from B1. We shall consider the following problem: if ‖A – B‖ is sufficiently small, does it follow that there is a unitary operator u such that uAu* = B?

Such questions were first considered by Kadison and Kastler in [9], and have received considerable attention. In particular in the case where A is an approximately finite-dimensional (or hyperfinite) von Neumann algebra, the question has an affirmative answer (cf [3], [8], [12]). We shall show that in the case where A and B are approximately finite-dimensional C*-algebras (AF-algebras) the problem also has a positive answer.
H-Finite Irreducible Representations of Simple Lie Algebras
Authors: Lemire F. , Pap M.
Published: December 2010

Let L denote a simple Lie algebra over the complex number field C with H a fixed Cartan subalgebra and C(L) the centralizer of H in the universal enveloping algebra U of L. It is known [cf. 2, 5] that one can construct from each algebra homomorphism ϕ:C(L) → C a unique algebraically irreducible representation of L which admits a weight space decomposition relative to H in which the weight space corresponding to ϕ ↓ H ∈ H* is one-dimensional. Conversely, if (ρ, V) is an algebraically irreducible representation of L admitting a one-dimensional weight space Vλ for some λ ∈ H*, then there exists a unique algebra homomorphism ϕ:C(L) → C which extends λ such that (ρ, V) is equivalent to the representation constructed from ϕ. Any such representation will be said to be pointed.
Defining Relations in Orthogonal Groups of Characteristic Two
Authors: Güunther Georg , Nolte Wolfgang
Published: December 2010

Introduction. It is well-known that a group is uniquely determined by a system of generators, and a set of defining relations on those generators. Clearly it is of interest to find relations that are as simple as possible. In this paper, this question is dealt with for certain orthogonal groups of characteristic 2, which are generated by involutions.

Let V be a vector space over a field K of characteristic 2 (we always exclude the prime field K = G F (2)). Let Q be a quadratic form over V, and let S be the set of orthogonal transformations of (V, Q) whose path is 1-dimensional and not contained in the radical of V. Letting O* be the group generated by S, we shall show that every relation among generators in S is a consequence of relations of length 2, 3, or 4.
Union of Realcompact Spaces and Lindelöf Spaces
Authors: Kato Akio
Published: December 2010

All spaces in this paper are completely regular Hausdorff and all maps are continuous onto, unless otherwise stated. The purpose of this paper is to investigate the realcompactness of a space X which contains a Lindelöf space L such that every zero-set Z (in X) disjoint from L is realcompact. We show in § 2 that such a space X is very close to being realcompact (Theorems I, II and III). But in general such a space fails to be realcompact. Indeed, in §§ 3 and 4 the following questions of Mrówka [18, 19] are answered, both in the negative:

(Q. 1) If X = L ∪ G where L is Lindelöf closed and G is E-compact, then is X E-compact?

(Q. 2) Suppose f:X → Y is a perfect map such that the set M(f) = {y ∊ Y| |f−l(y)| > 1} of multiple points of f is Lindelöf (especially, countable) closed. If X is E-compact, is Y also E-compact?
An Inequality Characterizes the Trace
Authors: Gardner L. Terrell
Published: December 2010

1. Introduction. While analogues of the Schwarz inequality have been much studied in the context of positive linear maps of operator algebras ([1], [2], [6], [7], [10]) the simpler triangle inequality |ϕ(x)| ≦ (|x|) has been neglected, outside of (possibly non-commutative) integration theory—perhaps partly because except for the important and familiar example of traces, scalar maps satisfying the triangle inequality are rarely encountered. In fact we here prove that they are never encountered: every such map is a trace.

For C*-algebras (norm-closed self-ad joint algebras of bounded operators on a Hilbert space) this means, for instance, that if the linear functional ϕ on the C*-algebra satisfies

(†)

then ϕ satisfies also the equivalent conditions

(i) ϕ(xy) = ϕ(yx) for all x, y in ;

(ii) ϕ(x*x) = ϕ(xx*) for all x in ;

(iii) ϕ(x) = ϕ(uxu*) for all x in and all unitary u in Ae, the C*-lgebra formed from by adjunction of a unit element.
The Saturation of a Product of Ideals
Authors: Wagon Stanley
Published: December 2010

In this note we discuss how the saturation I X J, where I, J are k-complete ideals on a regular uncountable cardinal K, depends on the saturation of I and J. We show that if 2k = k+ then the saturation of I X J is completely determined by the saturation of I and J . A consequence of a negative saturation result is that NSk X NSk is not k+-saturated, where NSk is the nonstationary ideal on k (even though it is still open whether NSk can be K+- saturated). We also discuss the preservation of precipitousness under certain products, obtaining a simple example of an ideal on k that is precipitous but not k+-saturated.
Some Structural Properties of the Set of Remote Points of a Metric Space
Authors: Gates Catherine L.
Published: December 2010

A remote point of a metric space X is a point in βX |Xnot in the βX-closure of any discrete subset of X. Remote points have been studied by Fine and Gillman [2], Plank [4], Robinson [6], Woods [9, 10], Van Douwen [7] and others. The main results concerning the existence of remote points are listed in Section 1. In this paper we determine some structural properties of the set of remote points of a metric space which has no isolated points. The notation of Gillman and Jerison [3] and Walker [8] will be used. All spaces are assumed to be Tychonoff.

Let X be a space. The general remote points of X, denoted TX, are those points in βX |X which are not in the βX-closure of any nowhere dense sets of X.
S-Subgroups of the Real Hyperbolic Groups
Authors: O'Malley Thomas J.
Published: December 2010

IfHis a closed subgroup of a locally compact groupG, withG/Hhaving finiteG-invariant measure, then, as observed by Atle Selberg [8], for any neighborhoodUof the identity inGand any elementginG, there is an integern >0 such thatgnis inU·H·U.A subgroup satisfying this latter condition is said to be anS-sub group,or satisfiesproperty (S).IfGis a solvable Lie group, then the converse of Selberg's result has been proved by S. P. Wang [10]: IfHis a closedS-subgroup ofG,thenG/His compact. Property(S)has been used by A. Borel in the important “density theorem” (see Section 2 or [1]).
Pairs of Rings with the Same Prime Ideals
Authors: Anderson David F. , Dobbs David E.
Published: December 2010

There are numerous instances in which the partners in an extension of commutative rings R ⊂ T have the same prime ideals, i.e., in which Spec(R) = Spec(T). Although this equality is intended to be taken set-theoretically, the identification easily extends to the corresponding spaces endowed with their Zariski topologies (see Proposition 3.5(a)), but of course need not extend to an identification of Spec(R) and Spec(T) as affine schemes. Perhaps the most striking recent illustration of this phenomenon arises from the work of Hedstrom and Houston [14] in which R is a pseudo-valuation domain and T is a suitable valuation overring. Other examples may be found by means of the D + M construction, either in its traditional form [12, p. 560] or in the generalized situation introduced by Brewer and Rutter [5].
On Asymptotic Centers and Fixed Points of Nonexpansive Mappings
Authors: Lim Teck-Cheong
Published: December 2010

Let X be a Banach space and B a bounded subset of X. For each x ∈ X, define R(x) = sup{‖x – y‖ : y ∈ B}. If C is a nonempty subset of X, we call the number R = inƒ{R(x) : x ∈ C} the Chebyshev radius of B in C and the set the Chebyshev center of B in C. It is well known that if C is weakly compact and convex, then and if, in addition, X is uniformly convex, then the Chebyshev center is unique; see e.g., [9].

Let {Bα : α ∈ ∧} be a decreasing net of bounded subsets of X. For each x ∈ X and each α ∈ ∧, define
Groupings of Metabelian Groups and Extension Categories
Authors: Roggenkamp K. W.
Published: December 2010

Let G be a metabelian group and R an integral domain of characteristic zero, such that no rational prime divisor of │G│ is invertible in R. By RG we denote the group ring of G over R. In this note we shall prove

THEOREM. If RG ≌ RH as R-algebras, then G ≌ H

The question whether this result holds was posed to me by S. K. Sehgal. The result for R = Z is contained in G. Higman's thesis, and he apparently also proved a more general result. At any rate, I think that the methods of the proof are interesting eo ipso, since they establish a “Noether-Deuring theorem” for extension categories.

In proving the above result, it is necessary to study closely the category of extensions (ℊs, S), where the objects are short exact sequences of SG-modules
A Generalization of Degree Two Simple Finite-Dimensional Noncommutative Jordan Algebras
Authors: Conlon Mary Ellen
Published: December 2010

Let be an algebra over a field . For x, y, z in , write (x, y, z) = (xy)z – x(yz) and x-y = xy + yx. The attached algebra is the same vector space as , but the product of x and y is x · y. We aim to prove the following result.

THEOREM 1. Let be a finite-dimensional, power-associative, simple algebra of degree two over a field of prime characteristic greater than five. For all x, y, z in , suppose

1

Then is noncommutative Jordan.

The proof of Theorem 1 falls into three main sections. In § 3 we establish some multiplication properties for elements of the subspace in the Peirce decomposition . In §4 we construct an ideal of which we then use to show that the nilpotent elements of form a subalgebra of for i = 0, 1.
Semigroups with Quasi-Zeroes
Authors: Rankin S. A. , Reis C. M.
Published: December 2010

Let S be a semigroup. An element a ∈ S is said to be a left quasi-zero if x ⋂  ≠ ∅ for all x ∈ S, where  denotes the cyclic sub-semigroup of S generated by a. In a recent study [6] of semigroups with a maximum right congruence, such elements proved to be useful in providing characterizations of these semigroups. Left quasizeroes have appeared in the literature under different names in a variety of situations. In the context of semigroup radicals, left quasi-zeroes are called right quasi-regular elements, where an element is defined to be right quasi-regular if it is not a left identity for any right congruence other than the universal congruence (see [4], [5], [2], [7], and [8]).
Uniqueness of Massey Products on the Stable Homotopy of Spheres
Authors: Kochman Stanley O.
Published: December 2010

The product on the stable homotopy ring of spheres π*scan be defined by composing, smashing or joining maps. Each of these three points of view is used in Section 2 to define Massey products on π*s. In fact we define composition and smash Massey products (x1, … , xt)where X1, … ,xt-1∈ π*s, xt∈ π*(E) and E is a spectrum. In Theorem 3.2, we prove that these three types of Massey products are equal. Consequently, a theorem which is easy to prove for one of these Massey products is also valid for the other two. For example, [3, Theorem 8.1] which relates algebraic Massey products in the Adams spectral sequence to Massey smash products in π*s is now also valid for Massey composition products in π*s
Boxes in Rn —A ‘Fractional’ Theorem
Authors: Katchalski Meir
Published: December 2010

A box in Euclidean 𝓀-dimensional space Rk is a set of the type

A family of boxes, unless stated otherwise, is finite.
The Largest Class of Hereditary Systems Defining a C0 Semigroup on the Product Space
Authors: Delfour M. C.
Published: December 2010

The object of this paper is to characterize the largest class of autonomous linear hereditary differential systems which generates a strongly continuous semigroup of class C0 on the product space Mp = Rn × Lp(-h, 0), 1 ≦ p < ∞, 0 < h ≦ + ∞ (R is the field of real numbers and Lp(– h, 0) is the space of equivalence classes of Lebesgue measurable maps x:[ – h, 0] ⌒ R → Rn which are p-integrable in [ –h, 0] ⌒R.) Our results extend and complete those of [4] and [15], [16] for linear hereditary differential equations possessing “finite memory” (h < + ∞ ) and those of [14], [5] and [6] in the “infinite memory case (h = + ∞ )”.

Consider the autonomous linear hereditary differential equation

(1.1)

where x(t) ∊ Rn, x:[–h, 0] ⌒ R → Rn is defined as xt(θ) = x(t + θ), C(–h, 0) is the space of bounded continuous functions [–h, 0] ⌒ R → Rn and L:C(–h, 0) → Rn is a continuous linear map.
On the Automorphism Group of a Finite p-Group with a Small Central Quotient
Authors: Davitt Richard M.
Published: December 2010

In recent years there has been considerable interest in the conjecture that |G| divides |Aut G| for all finite non-cyclic p-groups G of order greater than p2. In particular, the conjecture has been established for a considerable number of (not necessarily distinct) classes of finite p-groups ([6], [7], [8], [9], [15], [16]); additionally, results have been obtained, often using homological methods, which permit reductions in any attempt to establish the overall conjecture ([5], [10], [13], [15]). In the former case, the p-groups G have generally been regular p-groups (see, for example, [6]) and the prime p = 2 has either been excluded (see, for example, [8]) or treated as a special case (as in [9]).It is the purpose of this paper to establish the conjecture for the class of all p-groups G where |G: Z(G)| ≦ p4 with no restrictions on the prime p.
Cover Set Lattices
Authors: Adams M. E. , Sichler J.
Published: December 2010

The proof of a main result in [1] concerning (0,1)-endomorphisms of finite lattices is based on properties of lattices A(G) derived from the system of independent sets of an undirected loop-free graph G. For a number of questions naturally arising from [1] and [2], however, constructions employing only graph-induced complementation and properties of the lattices A (G) associated with these are no longer adequate. The present paper introduces cover set lattices (a generalization of the lattices A(G)) to deal with some of these questions. A special case of the main result presented here states that for every (0, 1)-lattice L and any monoid homomorphism φ:M → End0,1(L) there exists a lattice K containing L as a (0, 1)-sublattice in such a way that the monoid End0,1(K) of all (0, 1)-endomorphisms of K is isomorphic to M, and the restriction to L of every (0, 1)-endomorphism m of K is the (0, 1)-endomorphism φ(m) of L.
Capitulation in Class Field Extensions of Type (p, p)
Authors: Chang S. M. , Foote R.
Published: December 2010

Let K be a number field, K(1) its Hilbert class field, that is, the maximal abelian unramified extension of K, let K(2) be the Hilbert class field of K(1), and let G = Gal(K(2)/K) (alternatively, for p a prime the first and second p class fields enjoy properties analogous to those of the respective class fields discussed in this introduction; the particulars may be found surrounding Lemma 2). Since G/G’ is the largest abelian quotient of G, G/G′ = Gal (K(1))/K) and so G’ is the abelian group Gal(K(2)K(1)); moreover, class field theory provides (Artin) maps φK, φK(1) which are isomorphisms of the class groups Ck, Ck(1) onto G/G′, G′ respectively. In the remarkable paper [1] E. Artin computed the composition VG′

where e is the homomorphism induced on the class groups by extending ideals of K to ideals of K(1), and he gave a formula for computing VG′, the now familiar transfer (Verlagerung) homomorphism, in terms of the group G alone (see Lemma 1).
Arithmetic Invariants of Simplicial Complexes
Authors: Brown M. , Wasserman A. G.
Published: December 2010

What invariants of a finite simplicial complex K can be computed solely from the values v0(K), V1(K), …, vi(K), … where Vi(K) is the number of i-simplexes of K? The Euler chracteristic χ(K) = Σ i (– 1)ivi(K) is a subdivision invariant and a homotopy invariant while the dimension of K is a subdivision invariant and homeomorphism invariant. In [3], Wall has shown that the Euler chracteristic is the only linear function to the integers that is a subdivision invariant. In this paper we show that the only subdivision invariants (linear or not) of K are the Euler characteristic and the dimension. More precisely we prove the following theorem.
Uniform Approximation on the Graph of a Smooth Map in Cn
Authors: Weinstock Barnet M.
Published: December 2010

Let X be a compact set in Cn, and let ƒ1, …, ƒm, m ≧ n, be continuous, complex-valued functions on X which have C1 extensions to some neighborhood of X. We wish to describe the algebra A of continuous complex-valued functions on X which can be approximated uniformly by polynomials in the functions z1 …, zn, ƒ1 …, ƒm. For this purpose we introduce the sets

and
On the Number of Automorphisms of a Finite p-group
Authors: Exarchakos Theodoros
Published: December 2010

In this paper we find a new bound for the function g(h), for which |A(G)|p ≧ pn whenever |G| ≧ pg(h) G a finite p-group. The existence of such a function was first conjectured by W. R. Scott in 1954, who proved that g (2) = 3. In 1956 Ledermann and Neumann proved that in the general case of finite groups g(h) ≦ (h – 1)3.-pn−1 + h[10]. Since then, J. A. Green, J. C. Howarth and K. H. Hyde have reduced this bound considerably. The best (least) bound to date for finite p-groups was obtained by K. H. Hyde [9]. He proved that for h ≧ 5 and g(h) = h + 1 for h ≦ 4. For finite non-abelian p-groups, we improve this bound to: for h ≧ 13, g(h) = 2h – 5 for 5 < h ≦ 8, g(h) = h for h ≦ 5 and for 8 < h ≦ 12 we prove that g(9) = 14, g(10) = 17, g(11) = 20, g(12) = 23.
Congruence Subgroups of the Picard Group
Authors: Fine Benjamin
Published: December 2010

The Picard group Γ = PSL2 (Z [i]) is the group of linear fractional transformations

with ad – bc = ± 1 and a, b, c, d Gaussian integers.

Γ is of interest as an abstract group and in automorphic function theory. In an earlier paper [1], a decomposition of Γ as a free product with amalgamated subgroup was given and this was utilized to investigate Fuchsian subgroups. Karrass and Solitar used a similar decomposition to characterize abelian and nilpotent subgroups. Maskit [6], Mennicke [7] and Fine [2], used Γ to generate faithful representations of Fundamental Groups of Riemann Surfaces while more recently Wielenberg [10] represented certain knot and link groups as subgroups of Γ. In this paper, we will examine the structure of the congruence subgroups of Γ. Our technique will be to use the decomposition cited above [1], together with the Karrass-Solitar subgroup structure theory for free products with amalgamations [3]. Finally, we give a conjecture and some results concerning Fuchsian subgroups which are contained in congruence subgroups.
Injective Sheaves of Abelian Groups: A Counterexample
Authors: Banaschewski B.
Published: December 2010

It has been claimed that a sheaf of abelian groups on a Hausdorff space in which the compact open sets form a basis is injective in the category of all such sheaves whenever its group of global elements is divisible (Dobbs [1]). The purpose of this note is to present an optimal counterexample to this by showing, more generally, that on any nondiscrete T0-space there exists a sheaf of the type in question which is not injective.

Recall that a sheaf A of abelian groups on a space X assigns to each open set U in X an abelian group AU and to each pair U, V of open sets in X such that V ⊆ U a group homomorphism, denoted s ⟿ s|V, satisfying the familiar sheaf conditions ([3, p. 246]) which make A a special type of contravariant functor from the category given by the inclusion relation between the open sets of X into the category Ab of abelian groups, and that a map between sheaves A and B of abelian groups is a natural transformation h:A → B, with component homomorphisms hu:AU → BU. In the following, AbShX will be the category with these A as objects and these h:A → B as maps (= morphisms).
The Generalisation of Tutte's Result for Chromatic Trees, by Lagrangian Methods
Authors: Jackson D. M. , Goulden I. P.
Published: December 2010

A K-coloured rooted tree t is said to have colour partitionL if L is a K × ∞ matrix with elements lij equal to the number of non-root vertices of colour i and degree j. If adjacent vertices are of different colours then t is called a chromatic tree and L a chromatic partition. The tree has edge partitionD where D is a K × K matrix with elements dij equal to the number of edges, directed away from the root, from a vertex of colour i to a vertex of colour j.

In this paper we consider a method for enumerating trees with respect to colour and degree information. The method makes use of elementary decompositions of trees, and the functional equations which are induced. A number of new results are obtained by this means. More specifically, we consider (Section 3) the enumeration of rooted plane X-coloured trees with given colour and edge partitions.
Isometries of sp(α)
Authors: Fleming R. J. , Jamison J. E.
Published: December 2010

Let l < p < ∞, p ≠ 2 and α > 0. In what follows sp(α) will denote the space of all real or complex sequences for which

(1.1)

In this paper we show that the spaces sp(α) are Banach spaces under the natural norm and in fact share many properties that the usual lp spaces have. Our main results give characterizations of the surjective isometries of sp(α). These turn out to be quite different than the results for lp. For example, we show that for α ≠ 1, an operator T is a surjective isometry if and only if T is a modulus one multiple of the identity. The methods used are valid for both real and complex scalars. They involve the use of a disjoint support condition together with a property of semi inner products. In the complex case the information on isometries allows us to give complete descriptions of the Hermitian operators as well as the adjoint abelian operators.
Any 2-Sphere in E3 with Uniform Interior Tangent Balls is Flat
Authors: Daverman R. J. , Loveland L. D.
Published: December 2010

This paper addresses some flatness properties of an (n – 1)-sphere Σ in Euclidean n-space En resulting from the presence of round balls in En tangent to Σ. The notion of tangency used here is geometric rather than differentiable, for a round n-cell Bp (that is, the set of points whose distance, in the standard metric, from some center point is less than or equal to a fixed positive number) is said to be tangent to the (n – 1)-sphere Σ in En at a point p ∈ Σ if p ∈ Bp and Int Bp ⊂ Σ = ∅. The ball Bp is called an interior tangent ball at p if Int Bp ⊂ Int Σ; otherwise, it is called an exterior tangent ball at p.
σ4-Actions On Homotopy Spheres
Authors: Liang Chao-Chu
Published: December 2010

Let σ4 denote the group of all permutations of {a, b, c, d}. It has 24 elements, partitioned into five conjugacy classes: (1) the identity 1; (2) 6 transpositions: (ab), …, (cd); (3) 8 elements of order 3: (abc), …, (bcd); (4) 6 elements of order 4: (abcd), …, (adcb); (5) 3 elements of order 2: x = (ab)(cd), y = (ac)(bd), z = (ad)(bc).

In this paper, we study the differentiate actions of σ4 on odd-dimensional homotopy spheres modelled on the linear actions, with the fixed point set of each transposition a codimension two homotopy sphere.

A simple (2n – l)-knot is a differentiate embedding of a homotopy sphere K2n–l into a homotopy sphere Σ2n+1 such that πj(Σ – K) = πj(S1) for j < n.
Diamond Principles, Ideals and the Normal Moore Space Problem
Authors: Taylor Alan D.
Published: December 2010

If is a topological space then a sequence (Cα:α < λ) of subsets of is said to be normalized if for every H ⊆ λ there exist disjoint open sets and such that

The sequence (Cα:α < λ) is said to be separated if there exists a sequence of pairwise disjoint open sets such that for each α < λ. As is customary, we adopt the convention that all sequences (Cα:α < λ) considered are assumed to be relatively discrete as defined in [18, p. 21]: if x ∈ Cα then there exists a neighborhood about x that intersects no Cβ for β ≠ α.
Expansions of Arbitrary Analytic Functions in Series of Exponentials
Authors: Dickson D. G.
Published: December 2010

Let ϕ ≠ 0 be an entire function of one complex variable and of exponential type. Let B denote the set of all monomial exponentials of the form zneζ where ζ is a zero of ϕ of order greater than h. If R is a simply connected plane region and H(R) denotes the space of functions analytic in R with the topology of uniform convergence on compacta, then ϕ can be considered as an element of the topological dual H′(R) if the Borel transform of ϕ is analytic on , the complement of R. The duality is given by

where C is a simple closed curve in the common region of analyticity of ƒ and , and C winds once around the complement of a set in which is analytic.
Birkhoff Interpolation at the nth Roots of Unity: Convergence
Authors: Riemenschneider S. D. , Sharma A.
Published: December 2010

The first investigations on this type of problem were carried out by O. Kiš [2]. Kiš considered the problem of interpolating a function and its second derivative at the nth roots of unity (the (0, 2) problem) by 2n – 1 degree polynomials, and the convergence of such approximates. Later, Sharma [8], [9] extended the existence and uniqueness results to (0, m) interpolation, and essentially to (0, m1, m2) interpolation. In the latter case, Sharma established the convergence results for (0, 2, 3), (0, 1, 3) and (0, 1, 4) interpolation as well. Although some further special cases were considered [10] these were the essential results until very recently. Now Cavaretta, Sharma and Varga [1] have established the existence and uniqueness theorem for all possible interpolations of this type (see Theorem 2.1 below). Motivated by the work of Cavaretta, Sharma and Varga and the earlier work of Sharma [9], a different proof of this result is provided, and this proof is used to establish a convergence theorem in the general case.
Wallman Compactification and Representation
Authors: Hegde Shankar
Published: December 2010

Let X be any set and A be a uniformly closed algebra of bounded real valued functions on X which contains the constants and separates the points. For a lattice ℒ of subsets of X (we assume throughout that ∅ and X belong to ℒ), let MR(ℒ) denote the space of all finite, finitely additive,ℒ-regular measures defined on the field of sets generated by ℒ . Generalizing the notion of an integral representation, in [5] Kirk and Crenshaw define a standard representation of A*, the Banach dual of A, in MR(ℒ) to be a linear map I of A* into MR(ℒ) with the property that if 0 ≦ ϕ ∈ A*, then

for every W in ℒ. The space MR(ℒ) is said to represent A* if there exists a (unique) standard representation I of A* onto MR(ℒ) which is a Banach lattice isomorphism.
An Essential Integral Domain with a Non-Essential Localization
Authors: Heinzer William
Published: December 2010

An integral domain D is said to be an essential domain if D is an intersection of valuation rings that are localizations of D. D is called a v-multiplication ring if the finite divisorial ideals of D form a group. Griffin has shown [2, pp. 717-718] that every v-multiplication ring is an essential domain, and that an essential domain 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. It is noted in [4, p. 860] that any localization of a v-multiplication ring is again a v-multiplication ring. In this vein, Joe Mott has asked whether a localization of an essential domain must again be an essential domain. An example of an essential domain that is not a v-multiplication ring is given in [4], however it can be seen for this example that each localization is again an essential domain [6].
Nilpotent Partition-Inducing Automorphism Groups
Authors: Pettet Martin R.
Published: December 2010

If A is a group acting on a set X and x ∈ X, we denote the stabilizer of x in A by CA(x) and let Γ(x) be the set of elements of X fixed by CA(x). We shall say the action of A is partitive if the distinct subsets Γ(x), x ∈ X, partition X. A special example of this phenomenon is the case of a semiregular action (when CA (x) = 1 for every x ∈ X so the induced partition is a trivial one). Our concern here is with the case that A is a group of automorphisms of a finite group G and X = G#, the set of non-identity elements of G. We shall prove that if A is nilpotent, then except in a very restricted situation, partitivity implies semiregularity.
Counting Coloured Graphs of High Connectivity
Authors: Bollobás Béla
Published: December 2010

Find exact or asymptotic formulae for the number of labelled graphs of order n having a certain property. The property we are interested in in this note is that of being k-coloured and having connectivity at least l. Special cases of this problem have been tackled by many authors; in particular Gilbert [6], Read [9] and Robinson [11] found exact formulae, and Read and Wright [10], and Wright [12], [13] found asymptotic expressions (for many other examples see [7]). Recently Harary and Robinson [8] counted labelled bipartite blocks, that is 2-connected bipartite graphs. (For terms not defined here and general background in graph theory see [1].) Our present investigations have been prompted by [8]; in particular, as a very special case of our results, we shall prove the conjecture published in [8].

The exact formulae appearing in the enumeration of labelled graphs in general, and in the enumeration of k-coloured labelled graphs in particular, tend to be very pleasing, especially because of the functional equations relating them.