bluuebunny/crossref_metadata_2025_split huggingface dataset:
A Sufficient Condition that an Operator Algebra be Self-Adjoint
Authors: Radjavi Heydar , Rosenthal Peter 
Published: December 2010

It is well-known, and easily verified, that each of the following assertions implies the preceding ones.
(i)Every operator has a non-trivial invariant subspace.(ii)Every commutative operator algebra has a non-trivial invariant subspace,(iii)Every operator other than a multiple of the identity has a non-trivial hyperinvariant subspace.(iv)The only transitive operator algebra on is Note. Operator means bounded linear operator on a complex Hilbert space , operator algebra means weakly closed algebra of operators containing the identity, subspace means closed linear manifold, a non-trivial subspace is a subspace other than {0} and , a. hyperinvariant subspace for A is a subspace invariant under every operator which commutes with A, a transitive operator algebra is one without any non-trivial invariant subspaces and denotes the algebra of all operators on .
Representations of Groups as Automorphisms on Orthomodular Lattices and Posets
Authors: Gudder Stanley P. 
Published: December 2010

In this paper we study the problem of representing groups as groups of automorphisms on an orthomodular lattice or poset. This problem not only has intrinsic mathematical interest but, as we shall see, also has applications to other fields of mathematics and also physics. For example, in the “quantum logic” approach to an axiomatic quantum mechanics, important parts of the theory can not be developed any further until a fairly complete study of the representations of physical symmetry groups on orthomodular lattices is accomplished [1].We will consider two main topics in this paper. The first is the analogue of Schur's lemma and its corollaries in this general setting and the second is a study of induced representations and systems of imprimitivity.
Multiplicative Integration of Infinite Products
Authors: Lovelady David Lowell 
Published: December 2010

Let G be a complete normed abelian group with norm N1. Let S be an interval (bounded or otherwise) of real numbers. We propose to study the Stieltjes integral equationwhere p is in G, a is in S, and each Fk is a function on S each value of which is a function from G to G. Our primary tools of investigation will be the works of J. S. MacNerney [6; 7] and their extensions by the author [4; 5]. Our main result, Theorem 4, will show that the equation above can be solved by a product integral of infinite products of solutions for the summands of the integrator. After obtaining our results, we shall specialize them to a linear situation and then show how this specialization allows us to obtain representations for analytic functions having only invertible values in a complex Banach algebra with identity.
Sphere Packings and Error-Correcting Codes
Authors: Leech John , Sloane N. J. A. 
Published: December 2010

Error-correcting codes are used in several constructions for packings of equal spheres in n-dimensional Euclidean spaces En. These include a systematic derivation of many of the best sphere packings known, and construction of new packings in dimensions 9-15, 36, 40, 48, 60, and 2m for m ≧ 6. Most of the new packings are nonlattice packings. These new packings increase the previously greatest known numbers of spheres which one sphere may touch, and, except in dimensions 9, 12, 14, 15, they include denser packings than any previously known. The density Δ of the packings in En for n = 2m satisfiesIn this paper we make systematic use of error-correcting codes to obtain sphere packings in En, including several of the densest packings known and several new packings.
Connectivity of Function Spaces
Authors: Borges Carlos R. 
Published: December 2010

Given two spaces X and Y with Y either an AE (metrizable) or ANE (metrizable), little is known with regard to when the function space (Yx, τ), for some topology τ, is an AE (metrizable) or ANE (metrizable) except when very strong separation properties are imposed on X and Y (see [5, pp. 186-189]). One of our tasks will be to eliminate most of these separation property requirements, therefore complementing or extending some of the results of [5]. We also attach an appendix, which contains some needful information, as well as the complete local analogue of [2, Theorem 5.1].Throughout, we use either the terminology and notation of [2] or of the appendix. We also let co stand for the compact-open topology and pc stand for the pointwise convergence topology of any function space
Varieties of Orthomodular Lattices
Authors: Bruns Günter , Kalmbach Gudrun 
Published: December 2010

In this paper we start investigating the lattice of varieties of orthomodular lattices. The varieties studied here are those generated by orthomodular lattices which are the horizontal sum of Boolean algebras. It turns out that these form a principal ideal in the lattice of all varieties of orthomodular lattices. We give a complete description of this ideal; in particular, we show that each variety in it is generated by its finite members. We furthermore show that each of these varieties is finitely based by exhibiting a (rather complicated) finite equational basis for each variety.Our methods rely heavily on B. Jonsson's fundamental results in [8]. This, however, could be avoided by starting out with the equations given in sections 3 and 4. Some of our arguments were suggested by Baker [1],
Quasi-Injectivité de Certains Modules sur Leur Anneau D'endomorphismes
Authors: Cailleau Annie 
Published: December 2010

Dans tout ce qui suit, R désigne un anneau unitaire. Soit M R un R-module à droite; M R est muni canoniquement d'une structure de module à gauche sur l'anneau Λ de ses endomorphismes, pour laquelle il sera noté AM.On sait [3, Théorème 3.1] que si M R est quasi-injectif à sous-module singulier nul, Λ est un anneau régulier au sens de Von Neumann, auto-injectif à droite. Dans [5] B. Osofsky étudie l'auto-injectivité à gauche de Λ et pose les problèmes suivants.(1) Si MR est un module injectif à sous-module singulier nul, et si Λ est auto-injectif à gauche, ΛM est-il injectif?(2) Si Ton remplace l'hypothèse M R injectif par M R quasi-injectif, ΛM est-il quasi-injectif?On apporte ici une réponse affirmative à ces deux questions lorsque R est un anneau commutatif (Théorèmes 7 et 8).
A Characterization of Spin Representations
Authors: Brown Robert B. 
Published: December 2010

Associated with a non-degenerate symmetric bilinear form on a vector space is a Clifford algebra and various Clifford groups, which have spin representations on minimal right ideals of the Clifford algebra. Several invariants for these representations have been known for some time. In this paper the forms are assumed to be “split”, and several relations between the invariants are derived and promoted to the status of axioms. Then it is shown that any system satisfying the axioms comes from a minimal right ideal in a Clifford algebra and that the automorphism groups associated with the system are the Clifford groups. Hence, the axioms characterize spin representations.A description of split forms and spin representations is in section two. In section three the invariants and their properties are described.
The Free Product of Two Groups with a Malnormal Amalgamated Subgroup
Authors: Karrass A. , Solitar D. 
Published: December 2010

In [1], B. Baumslag defined a subgroup U of a group G to be malnormal in G if gug–1 ∈ U, 1 ≠ u ∈ U, implies that g ∈ U. Baumslag considered the class of amalgamated products (A * B; U) in which U is malnormal in both A and B. These amalgamated products play an important role in the investigations of B. B. Newman [13] of groups with one defining relation having torsion. In this paper, we shall be concerned primarily with a generalization of this class.Let U be a subgroup of a group G and let u ∈ U. Then the extended normalizer EG(u, U) of u relative to U in G is defined byif u ≠ 1, and by EG(u, U) = U if u = 1.
Fundamental Biorthogonal Sequences and K-Norms on ϕ
Authors: Crone L. , Fleming D. J. , Jessup P. 
Published: December 2010

A biorthogonal sequence is a double sequence (xi,fi) where each xi is from some locally convex space X, each fi is from X* and fi(xj) = δij. A biorthogonal sequence is called total if the functionals (fi) are total over X and is called fundamental if sp(xi) is dense in X. If a biorthogonal sequence is both total and fundamental we refer to it as a Markushivich basis or, more simply, an M-basis.If (xi,fi) is a total biorthogonal sequence for X, then X can be identified with the space of all scalar sequences (fi(x)) under the correspondence x ↔ (fi(x)). We refer to this space as the associated sequence space with respect to (xi, fi). With this correspondence, xi corresponds to and fi corresponds to Ei, the ith coordinate functional.
A Functional Analytic Description of Normal Spaces
Authors: Binz E. , Feldman W. 
Published: December 2010

Throughout the paper, X will denote a completely regular (Hausdorff) topological space and C(X) the R-algebra of all real-valued continuous functions on X. When this algebra carries the continuous convergence structure [1], we write CC(X). We note that CC(X)is a complete [5] convergence R-algebra [1].Our description of normality reads as follows. A completely regular topological space X is normal if and only if CC(X)/J (endowed with the obvious quotient structure; see § 1) is complete for every closed ideal J ⊂ CC(X).
Indecomposable Vector Bundles on the Projective Line
Authors: Roberts Leslie G. 
Published: December 2010

Let A be a commutative ring, and let Proj A[t0, ti]. By a vector bundle on X we mean a locally free sheaf of finite rank on X. Set t = t1/to. Then X is made up of two affine pieces U1 = Spec A[t], and U2 = Spec A[t-1]. Let P(R) denote the category of finitely generated projective modules over the ring R. The category of vector bundles on X is equivalent to the category of triples (P1,f1, P2), where P1 ∊ 𝓅 (A[t]), P2 ∊ 𝓅(A[t-1]), andis an A[t, t-1] -isomorphism. In [2], the category of vector bundles on is denned directly in this manner, without first defining (so that one could work over a non-commutative ring). We prove that if A is a Krull ring (or a Noetherian ring with connected spec) of dimension > 0, then there is an indecomposable vector bundle of rank n on X, for every positive integer n.
Sums and Products of Normal Functions
Authors: Bash David W. 
Published: December 2010

Let D be the unit disk in the complex plane. Let p(z, z’) denote the hyperbolic distance between z and z’ in ((1 + u)/ (1 — u)) = tanh-1 u, [6, chapter 15]). Let W be the Riemann sphere with the chordal metric. A complex valued function F(Z) in D is a normal function if lor each pair of sequences {zn} and {zn’} of points in D such that the convergence of {(fzn)} to a value α in W implies the convergence of {f(zn’)} to α. Two sequences {zn} and {zn’} of points in D are called close sequences if ρ(zn, zn’) → 0. (There are several equivalent definitions of normality if the functions are meromorphic.) The definition of a normal function implies that a normal function is continuous at each point of D when using the Euclidean metric in the domain and the chordal metric in the range.We wish to study the sums and products of normal functions. Some functions, such as a function in a Hardy p-class, p > 0, (or really any function of bounded characteristic) can be written as a sum or product of two normal functions, but sums and products of normal functions need not be normal (see Lappan [7]).
Uniformities on a Product
Authors: Hager Anthony W. 
Published: December 2010

All topological spaces shall be uniformizable (completely regular Hausdorff). A uniformity on X shall be viewed as a collection μ of coverings of X, via the manner of Tukey [20] and Isbell [16], and the associated uniform space denoted μX. Given the uniformizable topological space X, we shall be concerned with compatible uniformities as follows (discussed more carefully in § 1). The fine uniformity α (finest compatible with the topology); the “cardinal reflections“ αm of α (m an infinite cardinal number) ; αc, the weak uniformity generated by the real-valued continuous functions.With μ standing, generically, for one of these uniformities, we consider the question: when is μ(X × Y) = μX × μY For μ = αℵ0 (the finest compatible precompact uniformity), the problem is equivalent to that of whenβ(X × Y) = βX × βY,β denoting Stone-Cech compactification; this is answered by the theorem of Glicksberg [9]. For μ = α, we have Isbell's generalization [16, VI1.32].
The Torsion Submodule of A Cyclic Module Splits Off
Authors: Teply Mark L. 
Published: December 2010

A prominent question in the study of modules over an integral domain has been: “When is the torsion submodule t(A) of a module A a direct summand of A?” A module is said to split when its torsion module is a direct summand. Clearly, every cyclic module over an integral domain splits. Interesting splitting problems have been explored by Kaplansky [14; 15], Rotman [20], Chase [4], and others.Recently, many concepts of torsion have been proposed for modules over arbitrary associative rings with identity. Two of the most important of these concepts are Goldie's torsion theory (see [1; 12; 22]) and the simple torsion theory (see [5; 6; 8; 9; 23], and their references).
Projective Homotopy Classes of Stiefel Manifolds
Authors: Strutt Joseph 
Published: December 2010

Given a homotopy class [f] in πn(X), we say that [f] is projective if and only if there is a homotopy commutative factorizationwhere v is the standard double covering. We then denote by the subset of projective homotopy classes in πn(X).The notion of projective homotopy classes was studied in the author's thesis [5], and the projective homotopy classes for spheres in the stable range, up through the 3-stem were calculated in [6].
Intersections of Primary Ideals in Rings of Continuous Functions
Authors: Williams R. Douglas 
Published: December 2010

Let C be the ring of all real valued continuous functions on a completely regular topological space. This paper is an investigation of the ideals of C that are intersections of prime or of primary ideals.C. W. Kohls has analyzed the prime ideals of C in [3 ; 4] and the primary ideals of C in [5]. He showed that these ideals are absolutely convex. (An ideal I of C is called absolutely convex if |f| ≦ |g| and g ∈ I imply that f ∈ I.) It follows that any intersection of prime or of primary ideals is absolutely convex. We consider here the problem of finding a necessary and sufficient condition for an absolutely convex ideal I of C to be an intersection of prime ideals and the problem of finding a necessary and sufficient condition for I to be an intersection of primary ideals.
Riemann Surfaces as Orbit Spaces of Fuchsian
Groups
Authors: Moore M. J. 
Published: December 2010

A Fuchsian group is a discrete subgroup of the hyperbolic
group, L.F. (2, R), of linear fractional
transformationseach such transformation mapping the complex upper half plane
D into itself. If Γ is a Fuchsian group, the orbit space
D/Γ has an analytic structure such that the projection
map p: D → D/Γ, given by
p(z) = Γz, is holomorphic
and D/Γ is then a Riemann surface.If N is a normal subgroup of a Fuchsian group Γ, then
N is a Fuchsian group and S = D/N is a
Riemann surface. The factor group, G =
Γ/N, acts as a group of automorphisms (biholomorphic
self-transformations) of S for, if γ ∊ Γ and z ∊
D, then γN ∊ G, Nz ∊ S, and
(γN) (Nz) =
Nγz. This is easily seen to be independent of
the choice of γ in its N-coset and the
choice of z in its N-orbit.Conversely, if S is a compact Riemann surface, of genus at
least two, then S can be identified with
D/K, where K is a Fuchsian group acting
without fixed points in D.
A Characterization of the Tits' Simple Group
Authors: Parrott David 
Published: December 2010

In [6], J. Tits has shown that the Ree group 2F4(2) is not simple but possesses a
simple subgroup of index 2.
In this paper we prove the following theorem:THEOREM. Let G be a finite group of even order and let z be an
involution contained in G. Suppose H = CG(z) has the following
properties:(i) J = O2(H) has order 29and is of class at least 3.(ii) H/J is isomorphic to the Frobenius group of order
20.(iii) If P is a Sylow 5-subgroup of H, then
Cj(P) ⊆ Z(J).Then G = H • O(G) or G ≊
, the
simple group of Tits, as defined in [6].For the remainder of the paper, G will denote a finite
group which satisfies the hypotheses of the theorem as well as G ≠
H • O(G). Thus Glauberman's theorem
[1] can be applied to G and we have that
〈z〉 is not weakly closed in H (with
respect to G). The other notation is standard (see
[2], for example).
Rational Tensor Representations of Hom(V, V)
and an Extension of an Inequality of I. Schur.
Authors: Marcus Marvin , Gordon William Robert 
Published: December 2010

Let V be an n-dimensional vector space
over the complex numbers equipped with an inner product (x,
y), and let (P, μ) be a symmetry class in the
mth tensor product of V associated with a
permutation group G and a character χ (see
below). Then for each T ∊ Hom (V, V) the
function φ which sends each m-tuple
(v1, … , vm) of elements of
V to the tensor μ(TV1, … ,
Tvm) is symmetric with respect to
G and x, and so there is a unique linear map
K(T) from P to P such that φ =
K(T)μ.It is easily checked that K: Hom(V, V) →
Hom(P, P) is a rational representation of the
multiplicative semi-group in Hom(V, V): for any two linear
operators S and T on VK(ST) = K(S)K(T).Moreover, if T is normal then, with respect to the inner
product induced on P by the inner product on V (see below),
K(T) is normal.
On Scalar Dependent Algebras
Authors: Coughlin Raymond , Rich Michael 
Published: December 2010

The intent of this paper is to study a class of algebras which do not
necessarily obey the association law but instead obey a law which bears a
marked resemblance to associativity. For lack of a better name we call this
class the class of scalar dependent algebras. Specifically, an algebra
A over a field F is called
scalar dependent if there is a map g: A
× A × A → F such that (xy)z = g(x,
y, z)x(yz), for all
x, y, z in A. To obtain our results we
shall assume throughout that A is a scalar dependent
algebra with an identity element e over a field of
characteristic not 2 satisfying(I) (x, x, x) = 0.As usual, the associator (x,y,z) is defined by
(x,y,z) = (xy)z —
x(yz). An example is given to show that
(I) is not implied by scalar dependency.
On the β-Construction in
K-Theory
Authors: Naylor C. M. 
Published: December 2010

The β-construction assigns to each complex representation
φ of the compact Lie group G a unique
element β(φ) in (G). For the details of this
construction the reader is referred to [1] or [5].
The purpose of the present paper is to determine some of the properties of
the element β(φ) in terms of the invariants of the
representation φ. More precisely, we consider the following
question. Let G be a simple, simply-connected compact Lie group and let
f : S3 →G be a Lie group homomorphism. Then (S3) ⋍ Z with generator
x = β(φ1), φ1 the fundamental representation of S3 , so that if
φ is a representation of
G,f*(φ) = n(φ)x, where
n(φ) is an integer depending on φ and
f . The problem is to determine n(φ).Since G is simple and simply-connected we may assume that
ch2, the component of the Chern character in dimension 4 takes
its values in H4(SG,Z)≅Z. Let u be a generator of
H4(SG,Z) so that ch2(β (φ)) = m(φ)u, m(φ)
an integer depending on φ.
Multiplications on a Space with Finitely Many Non-Vanishing Homotopy Groups
Authors: Cheng Chao-Kun 
Published: December 2010

Copeland [1] proved that if X is a CW complex then the set of homotopy classes of multiplications is in 1-1 correspondence with the loop [X ∧ X, X], But in general [X ∧ X, X] is very hard to compute.Here we study the problem of finding how many different (up to homotopy) multiplications can be put on a space with finitely many non-vanishing homotopy groups. We reduce this problem to computing quotients of certain cohomology groups and to determining the primitive elements (Theorem 3.6). Our approach to this problem uses Postnikov decompositions and multiplier arguments.
On a Theorem of Cohen and Lyndon About Free Bases for Normal Subgroups
Authors: Karrass A. , Solitar D. 
Published: December 2010

Let S(≠1) be a subgroup of a group G. We consider the question: when are the conjugates of S “as independent as possible“? Specifically, suppose SG (the normal subgroup generated by S in G) is the free product II*S0α where and gα ranges over a subset J of G. Then J must be part of a (left) coset representative system for G mod SG. N where N is the normalizer of S in G. (For, g ∊ SGgαN implies Sg is conjugate to Sgα in SG; however, distinct non-trivial free factors of a free product are never conjugate.)We say that SG is the free product of maximally many conjugates of S in G if SG = II*Sgα where gα ranges over a (complete) left coset representative system for G mod SGN (or equivalently, gα ranges over a double coset representative system for G mod (SG, N)); in this case we say briefly that S has the fpmmc property in G.
Presentations of the Groups SL(2,m) And PSL(2,m)
Authors: Sunday J. G. 
Published: December 2010

In this paper, we refine the presentations of Behr and Mennicke [1] for SL(2,m)and PSL(2,m)wheremis odd. The group SL(2,m)is first shown to be presented by the following system of generators and relations:1.1The group PSL(2,m)appears as the factor group1.2This simplification then permits us to use the results of Schur [3] to establish three-relation presentations for these groups.
A General Hewitt-Yosida Decomposition
Authors: Traynor Tim 
Published: December 2010

In 1952, E. Hewitt and K. Yosida [3] proved that a bounded, finitely additive real-valued set function has a unique representation as the sum of a countably additive function and a “purely finitely additive” function.Below, using a variation of the Carathéodory process we give a suitable generalization to s-bounded vector-valued set functions. In fact, since the methods do not rely on scalar multiplication, we give the result for commutative Hausdorff topological groups.
A Theorem for Enumerating Certain Types of Collections
Authors: Osterweil Leon 
Published: December 2010

In this paper, we are concerned with proving a formula for the computation of what is variously called the pattern inventory (e.g., see De Bruijn [2]) or the configuration counting series (e.g., see Harary [3]). Rather than redeveloping a large number of definitions, we shall assume the reader is already familiar with the terminology used by De Bruijn [2].Polya, in a celebrated paper [4], proved a formula for computing the pattern inventory for all functions f defined on a set D (where D is acted on by a permutation group G), and mapping into a set R (which is called the store) for which the “store enumerator” in known.
Isometries of Hp(Un)
Authors: Schneider R. B. 
Published: December 2010

Forelli in [1] has described the isometries of Hp(U) into Hp(U) for p≠2, 0 < p < ∞. We shall extend his methods to characterize the isometries of Hp(Un) onto Hp(Un).The notation we shall use can be found in Rudin [3].Let II represent a permutation that induces a map on functions of n complex variables byClearly II is an isometry of Hp(Un) onto Hp(Un).
The Algebra of Differentials of Infinite Rank
Authors: Brown W. C. 
Published: December 2010

Let k and A denote commutative rings with identity and assume that A is a k-algebra. A qth order k-derivation δ of A into an A -module V is an element of Homk(A, V) such that for any q + 1 elements a0, … , aq of A, the following identity holds:Thus, a 1st-order derivation is just an ordinary derivation of A into V.