bluuebunny/crossref_metadata_2025 huggingface dataset:
On the Partially Ordered Set of Prime Ideals of a Distributive Lattice
Authors: Balbes Raymond
Published: December 2010

For a distributive lattice L, let denote the poset of all prime ideals of L together with ∅ and L. This paper is concerned with the following type of problem. Given a class of distributive lattices, characterize all posets P for which for some . Such a poset P will be called representable over. For example, if is the class of all relatively complemented distributive lattices, then P is representable over if and only if P is a totally unordered poset with 0, 1 adjoined. One of our main results is a complete characterization of those posets P which are representable over the class of distributive lattices which are generated by their meet irreducible elements. The problem of determining which posets P are representable over the class of all distributive lattices appears to be very difficult.
On Weighted Norm Inequalities for Fractional and Singular Integrals
Authors: Walsh T.
Published: December 2010

In a recent paper [12] Muckenhoupt and Wheeden have established necessary and sufficient conditions for the validity of norm inequalities of the form ‖ |x|αTƒ ‖q ≦ C‖ |x|αƒ ‖p, where Tƒ denotes a Calderón and Zygmund singular integral of ƒ or a fractional integral with variable kernel. The purpose of the present paper is to prove, by somewhat different methods, similar inequalities for more general weight functions.

In what follows, for p ≧ 1, p′ is the exponent conjugate to p, given by l/p + l/p′ = 1. Ω will always denote a locally integrable function on Rn which is homogeneous of degree 0, Ω∼ will denote a measurable function on Rn × Rn such that for each x ∈ Rn, Ω∼(x, .) is locally integrable and homogeneous of degree 0.
Isomorphic Subgroups of Finite p-Groups. I
Authors: Glauberman George
Published: December 2010

Let p be a prime and P be a p-subgroup of a finite group G. Suppose that g ∈ G and that P ∩ Pg has index p in P. In [4], we assumed that g normalizes no non-identity normal subgroup of P. We obtained some bounds on the order of P and some applications to the case in which p = 2 and P is a Sylow 2-subgroup of 〈P, Pg〉. In this paper, we examine this situation further by considering the isomorphism ϕ of P ∩ Pg-l onto P ∩ Pg given by ϕ(x) = xg. We actually consider arbitrary isomorphisms ϕ between two subgroups of index p in P. However, an easy argument (Lemma 2.3) shows that every such ϕ can be obtained as above for some G and some g. We obtain some results concerning the nilpotence class rather than the order of P.
On Unitary Polarities of Finite Projective Planes
Authors: Kantor William M.
Published: December 2010

A unitary polarity of a finite projective plane of order q2 is a polarity θ having q3 + 1 absolute points and such that each nonabsolute line contains precisely q + 1 absolute points. Let G(θ) be the group of collineations of centralizing θ. In [15] and [16], A. Hoffer considered restrictions on G(θ) which force to be desarguesian. The present paper is a continuation of Hoffer's work. The following are our main results.

THEOREM I. Let θ be a unitary polarity of a finite projective planeof order q2. Suppose that Γ is a subgroup of G(θ) transitive on the pairs x, X, with x an absolute point and X a nonabsolute line containing x. Thenis desarguesian and Γ contains PSU(3, q).
Injective Hulls of Torsion Free Modules
Authors: Zelmanowitz J.
Published: December 2010

In § 1, we begin with a basic theorem which describes a convenient embedding of a nonsingular left R-module into a complete direct product of copies of the left injective hull of R (Theorem 2). Several applications follow immediately. Notably, the injective hull of a finitely generated nonsingular left R-module is isomorphic to a direct sum of injective hulls of closed left ideals of R (Corollary 4). In particular, when R is left self-injective, every finitely generated nonsingular left R-module is isomorphic to a finite direct sum of injective left ideals (Corollary 6).

In § 2, where it is assumed for the first time that rings have identity elements, we investigate more generally the class of left R-modules which are embeddable in direct products of copies of the left injective hull Q of R. Such modules are called torsion free, and can also be characterized by the property that no nonzero element is annihilated by a dense left ideal of R (Proposition 12).
Immersions and Embeddings Up to Cobordism
Authors: Brown Richard L. W.
Published: December 2010

In 1944 Whitney proved that any differentiable n-manifold (n ≧ 2) can be (differentiably) immersed in R2n–1[15] and embedded in R2n [14]. Whitney's results are best possible when n = 2r. (One uses a simple argument involving the dual Stiefel-Whitney classes of real projective space Pn. See [9, pp. 14, 15].) However, there is a widely known conjecture that any R-manifold (n ≧ 2) immerses in R2n–α(n) and embeds in R2n–α(n)+1. Here, α(n) denotes the number of ones in the binary expansion of n. We prove (Theorem 5.1) that every compact manifold is cobordant to a manifold that immerses in (2n – α(n))-space and embeds in (2n – α(n) + 1)-space. (See § 1 for the definition of cobordant manifolds.) It is well known that if the conjecture were true it would be the best possible result.
Topics in Direct Differential Geometry
Authors: Park Ralph
Published: December 2010

In the theory of curves, one often makes differentiability assumptions in order that analytic methods can be used. Then one tries to weaken these assumptions as much as possible. The theory of curves which is presented here uses geometric methods, such as central projection, rather than analysis. In this way, no analytic assumptions are needed and a purely geometric theory results. Since this theory is not so well known as the analytic one, I have tried to make the treatment as self-contained as possible. It is hoped that this paper will form a quick introduction for a reader who has had no previous acquaintance with the subject.

We assume that our curves satisfy a condition, which we call direct differentiability. Roughly this condition is that, at each point of the curve, all the osculating spaces exist.
Basic Objects for an Algebraic Homotopy Theory
Authors: Cherenack Paul
Published: December 2010

The purposes of this paper are:

(A) To show (§§ 1, 3, 5) that some of the usual notions of homotopy theory (sums, quotients, suspensions, loop functors) exist in the category of affine k-schemes where the affine rings are countably generated.

(B) By example to demonstrate some of the more geometric relations between two objects of and their quotient or to study the algebraic suspension of one of them. See §§ 2.1, 2.2, 2.3, 3.

(C) To prove (§4) that the algebraic suspension (in R/) of the n-sphere is homeomorphic to the n + 1 sphere for the usual topologies.

(D) To show that the algebraic loop functor is right adjoint to the algebraic suspension functor (§5).

These results can be viewed as a precursor of definitions for an algebraic homotopy theory from a “geometric” point of view (rather than a more algebraic standpoint employing Galois theory [5]).
An Approximation Theorem for Extended Prime Spots
Authors: Brown Ron
Published: December 2010

We introduce here a generalization to arbitrary fields of the prime spots of algebraic number theory, essentially by allowing absolute values to take the value ∞. The set of “extended” prime spots of a field admits a natural topology, and an approximation theorem is given here for compact sets of extended prime spots. Among the corollaries of the approximation theorem are the weak approximation theorem for absolute values [13, p. 8], Ribenboim's generalization of the approximation theorem for independent valuations [14, p. 136], Stone's approximation theorem for type V topologies [16, p. 20], and an approximation theorem for Harrison primes.
Certain Artinian Rings are Noetherian
Authors: Shock Robert C.
Published: December 2010

Throughout this paper the word “ring” will mean an associative ring which
need not have an identity element. There are Artinian rings which are not
Noetherian, for example C(p∞) with zero
multiplication. These are the only such rings in that an Artinian ring
R is Noetherian if and only if R
contains no subgroups of type C(p∞)
[1, p. 285]. However, a certain class of Artinian rings is
Noetherian. A famous theorem of C. Hopkins states that an Artinian ring with
an identity element is Noetherian [3, p. 69]. The proofs of
these theorems involve the method of “factoring through the nilpotent
Jacobson radical of the ring”. In this paper we state necessary and
sufficient conditions for an Artinian ring (and an Artinian module) to be
Noetherian. Our proof avoids the concept of the Jacobson radical and depends
primarily upon the concept of the length of a composition series. As a
corollary we obtain the result of Hopkins.
Radicals of PID's and Dedekind Domains
Authors: Propes R. E.
Published: December 2010

The purpose of this paper is to characterize the radical ideals of principal
ideal domains and Dedekind domains. We show that if T is a
radical class and R is a PID, then T(R) is
an intersection of prime ideals of R. More specifically,
if

then T(R) = (p1p2 … pk), where p1,
p2, … , pk are distinct primes, and where (p1p2 … Pk) denotes the principal ideal of
R generated by p1p2 …
pk. We also characterize the radical ideals of
commutative principal ideal rings. For radical ideals of Dedekind domains we
obtain a characterization similar to the one given for PID's.
Nowhere Dense Subsets of Metric Spaces with Applications to
Stone-Cech Compactifications
Authors: Porter Jack R. , Woods R. Grant
Published: December 2010

Let X be a metric space. Assume either that
X is locally compact or that X has no
more than countably many isolated points. It is proved that if
F is a nowhere dense subset of X, then
it is regularly nowhere dense (in the sense of Katětov) and hence is
contained in the topological boundary of some regular-closed subset of
X. This result is used to obtain new properties of the
remote points of the Stone-Čech compactification of a metric space without
isolated points.

Let βX denote the Stone-Čech compactification of the
completely regular Hausdorff space X. Fine and Gillman
[3] define a point p of βX
to be remote if p is not in the βX-closure
of a discrete subset of X.
The Eigenvalues of Complementary Principal Submatrices of a
Positive Definite Matrix
Authors: Thompson R. C. , Therianos S.
Published: December 2010

Let C be an n-square Hermitian matrix,
presented in partitioned form as

where A is a-square and B
is b-square. Let denote the eigenvalues of C, A, B,
respectively. In a recent paper [10] the following inequality
was established:

1.1

if

1.2

This inequality is a simplification and a sharpening of an inequality
established earlier in [6], and is a wide generalization of an
inequality of Aronszajn [4].
Uniform Finite Generation of SU(2) and SL(2, R)
Authors: Lowenthal Franklin
Published: December 2010

A connected Lie group H is generated by a pair of
one-parameter subgroups if every element of H can be
written as a finite product of elements chosen alternately from the two
one-parameter subgroups, i.e., if and only if the subalgebra generated by
the corresponding pair of infinitesimal transformations is equal to the
whole Lie algebra h of H (observe that the
subgroup of all finite products is arcwise connected and hence, by Yamabe's
theorem [5], is a sub-Lie group). If, moreover, there exists a
positive integer n such that every element of
H possesses such a representation of length at most
n, then H is said to be uniformly
finitely generated by the pair of one-parameter subgroups. In this case,
define the order of generation of H as the least such
n ; otherwise define it as infinity.
Associated Prime Divisors in the Sense of Krull
Authors: Kuntz Richard A.
Published: December 2010

In a recent paper by Douglas Underwood [8] several definitions of “associated prime divisors” were discussed and shown to be unique. In this note we produce a fifth type, which is due to W. Krull, and is found in his classical paper [2] and further discussed by B. Banaschewski in[1].Historically this characterization considerably predates the other four definitions.

Throughout this note,Rdenotes a commutative ring with unity, and all ideals and elements are assumed to be in such a ring. We shall let upper case letters, most frequently the beginning of the alphabet, denote ideals and lower case letters, elements ofR.On the whole, our terminology will be that of [9]. We do, however, take exception with [9] in two instances, viz.
Descendingly Incomplete Ultrafilters and the Cardinality of
Ultrapowers
Authors: Adler Andrew , Jorgensen Murray
Published: December 2010

Let D be an ultrafilter on I, and let k be
a cardinal. D is said to be k-descendingly
incomplete (k-d.i.) if there exists a chain
Xα : α < k of elements of
D such that α < β →
Xα ⊆ Xβ and Xα = ϕ. Such a chain will be called a
k-chain for D. The notion of
k-descending incompleteness is due to Chang [3].

In this paper we explore the relationship between the cardinality of the
ultrapower kI/D and the existence of certain
chains on D. Since we deal so much with questions of size,
we do not ordinarily make a notational distinction between a set and its
cardinality. Where such a distinction is useful, the cardinality of a set
A will be denoted by |A|.
Analytic Toeplitz and Composition Operators
Authors: Deddens James A.
Published: December 2010

This paper is a continuation of [1] where we began the study of
intertwining analytic Toeplitz operators. Recall that X
intertwines two operators A and
B if XA = BX. Let H2 be the Hilbert space of analytic functions in the open unit disk
D for which the functions fr(θ) =
f(reiθ) are bounded in the L2 norm, and H∞ be the set of bounded functions in H2.
For φ ∊ Hφ, Tφ (or Tφ(z)) is the analytic Toeplitz
operator defined on H2 by the relation (Tφf)(z) = φ(z)f(z). For φ
∊ H∞, we shall denote {φ(z):
|z| < 1} by Range (φ) or
φ(D). Then where
and
σ(Tφ) = Closure(φ(D))
[1]. If φ ∊ H∞ maps D into D,
then we define the composition operator Cφ on H2 by the relation (Cφf) (z) = f(φ(z)). J. Ryff has shown
[11, Theorem 1] that Cφ, is a bounded linear operator on H2.
Residue Free Differentials and the Cartier Operator for
Algebraic Function Fields of one Variable
Authors: Kodama Tetsuo
Published: December 2010

Let K be a field of characteristic p >
0 and let A be a separably generated algebraic function
field of one variable with K as its exact constant field.
Throughout this paper we shall use the following notations to classify
differentials of A/K:

D(A) : the K-module of all differentials,

G(A) : the K-module of all differentials of the first
kind,

R(A) : the K-module of all residue free differentials in
the sense of Chevalley [2, p. 48],

E*(A) : the K-module of all pseudo-exact
differentials in the sense of Lamprecht [7, p. 363], (compare
the definition with our Lemma 8).
The Continuity of the Visibility Function on a Starshaped
Set
Authors: Beer Gerald
Published: December 2010

The visibility function assigns to each point x of a fixed
measurable set E in a Euclidean space En the Lebesgue outer measure of
S(x), the set {y : rx + (1 — r)y ∊ E
for every r in [0, 1]}.

The purpose of this paper is to determine sufficient conditions for the
continuity of the function on the interor of a starshaped set.
Cohomology Theorems for Borel-Like Solvable Lie Algebras in Arbitrary Characteristic
Authors: Leger G. , Luks E.
Published: December 2010

This paper develops some techniques for the study of derivation algebras and cohomology groups of Lie algebras. We are especially concerned with solvable algebras over arbitrary fields with structural properties like those of the Borel subalgebras of complex semi-simple Lie algebras. In particular, these algebras are semi-direct sums of nilpotent ideals and abelian subalgebras which act on the ideals in a semi-simple fashion. We make strong use, in our discussion, of a cohomology theorem of Hochschild-Serre. This result is stated herein (§ 2) in a modified form which allows us to omit the original hypothesis that the base field have characteristic 0.
Some Properties of Anosov Flows
Authors: Byers W.
Published: December 2010

Anosov flows are a generalization of geodesic flows on the unit tangent bundles of compact manifolds of negative sectional curvature. They were introduced and are dealt with at length by Anosov in [2]. Moreover they form an important class of examples of flows satisfying Smale's axioms A and B (see [15]). In that paper Smale poses the problem of determining which manifolds admit Anosov flows. In this paper we obtain information about the fundamental groups of such manifolds. These generalize results which have been obtained for the fundamental groups of manifolds of negative curvature (see Preissmann [13], Byers [6]).
Group Rings with Only Trivial Units of Finite Order
Authors: Hughes Ian , Wei Chou-Hsiang
Published: December 2010

We denote by ZG the integral group ring of the finite group G. S.D. Berman [1] showed that every unit of finite order μ in G is trivial (i.e., μ = ±g for some g in G) if and only if either G is abelian or G is a Hamiltonian 2-group. In this note, we give a new and shorter proof for the “only if” part.
On Kaehler Immersions
Authors: Ogiue Koichi
Published: December 2010

Let be an (n + p)-dimensional Kaehler manifold of constant holomorphic sectional curvature . B. O'Neill [3] proved the following result.

PROPOSITION A. Let M be an n-dimensional complex submanifold immersed in . If and if the holomorphic sectional curvature of M with respect to the induced Kaehler metric is constant, then M is totally geodesic.

He also gave the following example: There is a Kaehler imbedding of an w-dimensional complex projective space of constant holomorphic sectional curvature ½ into an -dimensional complex projective space of constant holomorphic sectional curvature 1. This shows that Proposition A is the best possible.
Distributional Watson Transforms
Authors: Hsu Hsing-Yuan
Published: December 2010

All our notation is as denned in [2] with the restriction to n = 1. However, for our purposes, we introduce a sequence of norms by

in It is not difficult to see that turns out to be a fundamental space.

It is a well-known fact that the Watson transform and the Mellin transform are connected by the fact that

and

if and only if K(s)K(l — s) = 1, where K(s) is the Mellin transform of k(x). Further, the Hankel transform and Hilbert transform can be considered as special cases of Watson transforms.
Finite p-groups with Isomorphic Subgroups
Authors: Currano John J.
Published: December 2010

Throughout this paper, p will denote a prime and t an integer greater than 1. We consider those finite p-groups P, which satisfy the following conditions:

(1.1) P has subgroups R and Q, both of index p, and there is an isomorphism φ of R onto Q which does not fix any non-identity subgroup of R setwise.
The Strong Perfect Graph Conjecture for Planar Graphs
Authors: Tucker Alan
Published: December 2010

A graph G is called γ-perfect if ƛ (H) = γ(H) for every vertex-generated subgraph H of G. Here, ƛ(H) is the clique number of H (the size of the largest clique of H) and γ(H) is the chromatic number of H (the minimum number of independent sets of vertices that cover all vertices of H). A graph G is called α-perfect if α(H) = θ(H) for every vertex-generated subgraph H of G, where α (H) is the stability number of H (the size of the largest independent set of H) and θ(H) is the partition number of H (the minimum number of cliques that cover all vertices of H).
On the Uniform Approximation of Smooth Functions by Jacobi Polynomials
Authors: Prasad J. , Hayashi H.
Published: December 2010

Let ωn(x) denote the Jacobi polynomials with the weight Function

If we denote the corresponding normalized Jacobi polynomials by we Have

(1.1)

Now let

be the nth partial sum of the Fourier series of Jacobi polynomials of a function f(x).
QF - 1 Rings of Global Dimension ≦ 2
Authors: Ringel Claus Michael
Published: December 2010

R. M. Thrall [10] introduced QF — 1, QF — 2 and QF — 3 rings as generalizations of quasi-Frobenius rings. (For definitions, see section 1. It should be noted that all rings considered are assumed to be left and right artinian.) He proved that QF — 2 rings are QF — 3 and asked whether all QF — 1 rings are QF — 2, or, at least, QF — 3. In [9] we have shown that QF — 1 rings are very similar to QF — 3 rings. On the other hand, K. Morita [6] gave two examples of QF — 1 rings, one of them not QF — 2 and therefore not QF — 3, the other one QF — 3, but not QF — 2.
Three Remarkable Graphs
Authors: Biggs Norman
Published: December 2010

In the development of any mathematical theory it is often advisable to test the known theoretical results in particular cases, for in this way we are able to judge the state of the subject, its strengths and its weaknesses. It is the purpose of this paper to put forward three remarkable graphs which can be used to indicate the progress being made in research in graph theory. We shall undertake three tasks - the construction of the graphs themselves, the investigation of their properties (insofar as the theory allows), and the indication of possible lines for further enquiry.
Majorization-Subordination Theorems for Locally Univalent Functions, II
Authors: Campbell Douglas Michael
Published: December 2010

Let denote the set of all normalized analytic univalent functions in the open unit disc D. Let f(z), F(z) and φ(z) be analytic in |z| < r. We say that f(z) is majorized by F(z) in we say that f(z) is subordinate to F(z) in where .

Let be the set of all locally univalent (f’(z) ≠ 0) analytic functions in D with order ≦α which are of the form f(z) = z +… . The family is known as the universal linear invariant family of order α [6]. A concise summary of and introduction to properties of linear invariant families which relate to the following material is contained in [1]. The present paper contains the proofs of some of the results announced in [1]
Chromatic Sums for Rooted Planar Triangulations: The Cases λ = 1 and λ = 2
Authors: Tutte W. T.
Published: December 2010

In this paper we derive a functional equation whose solution would give the sum of the chromatic polynomialP(M,λ) over certain classes of rooted planar mapsMcalled “ triangulations” and “near-triangulations”. For an integral colour-number λ this sum is the number of λ-coloured rooted maps of the kind considered, but the sum can also be discussed for non-integral λ.
A Class of Normal (0, 1)-Matrices
Authors: Bridges W. G.
Published: December 2010

If A is a real normal matrix At (the transpose of A) is a real polynomial in A. We study here those normal (0, 1)-matrices A with constant row sums which have At a polynomial of degree two in A.
Quotient Rings of a Class of Lattice-Ordered-Rings
Authors: Steinberg Stuart A.
Published: December 2010

An f-ring R with zero right annihilator is called a qf-ring if its Utumi maximal left quotient ring Q = Q(R) can be made into and f-ring extension of R. F. W. Anderson [2, Theorem 3.1] has characterized unital qf-rings with the following conditions: For each q ∈ Q and for each pair d1, d2 ∈ R+ such that diq ∈ R

(i) (d1q)+ Λ (d2q)- = 0, and

(ii) d1 Λ d2 = 0 implies (d1q)+ Λ d2 = 0.

We remark that this characterization holds even when R does not have an identity element.
Chromatic Sums for Rooted Planar Triangulations II: The Case λ = τ + 1
Authors: Tutte W. T.
Published: December 2010

In an earlier paper [2] we denned the chromatic sums g, q, l and h. We determined the derivatives of these sums with respect to the colour-number λ at the special values λ = 1 and λ = 2. In the present paper we find parametric equations for h, l and q in the case λ = τ + 1, where τ is the golden ratio. We obtain h, l and the basic indeterminate z explicitly in terms of the parameter u, but for q we exhibit only a cubic equation with coefficients depending on u. We obtain an exact formula for the coefficients in h by applying Lagrange's theorem to the parametric equations.
Graphs with 6-Ways
Authors: Leonard John L.
Published: December 2010

In a finite graph with no loops nor multiple edges, two points a and b are said to be connected by an r-way, or more explicitly, by a line r-way a — b if there are r paths, no two of which have lines in common (although they may share common points), which join a to b. In this note we demonstrate that any graph with n points and 3n — 2 or more lines must contain a pair of points joined by a 6-way, and that 3n — 2 is the minimum number of lines which guarantees the presence of a 6-way in a graph of n points.

In the language of [3], this minimum number of lines needed to guarantee a 6-way is denoted U(n). For the background of this problem, the reader is referred to [3].
Rings Over Which Every Simple Module is Rationally Complete
Authors: Brown S. H.
Published: December 2010

In 1958, G. D. Findlay and J. Lambek defined a relationship between three R-modules, A ≦ B(C), to mean that A ⫅ B and every R-homomorphism from A into C can be uniquely extended to an irreducible partial homomorphism from B into C. If A ≦ B(B), then B is called a rational extension of A and in [5] it is shown that every module has a maximal rational extension in its injective hull which is unique up to isomorphism. A module is called rationally complete provided it has no proper rational extension.
On Maximal Torsion Radicals
Authors: Beachy John A.
Published: December 2010

Let R be an associative ring with identity, and let denote the category of unital left R-modules. It is known that if R is a commutative, Noetherian ring, then the maximal torsion radicals of correspond to the minimal prime ideals of R. In fact, Nӑstӑsescu and Popescu [15, Proposition 2.7] have given a more general result valid for arbitrary commutative rings. This paper investigates maximal torsion radicals over rings not necessarily commutative.
Génératrices Extrémales d'un Cône de Fonctionnelles Linéaires Positives Invariantes
Authors: Dubois Jacques
Published: December 2010

Si S est un espace topologique compact séparé et si ϕ : S → S est une fonction continue, l'opérateur: A : C(S) → C(S), défini par Ag = g o ϕ est linéaire positif tel que Ae = e (sur C(S) nous considérons le cône usuel et e désigne la fonction identiquement 1 sur S).
Kasteleyn's Theorem and Arbitrary Graphs
Authors: Little Charles H. C.
Published: December 2010

Familiarity with the basic notions of graph theory is assumed. Loops and multiple edges are not permitted. An orientation of an edge e of a graph G is a designation of one of the ends of e as the positive end and the other as the negative end. We say that e is oriented from the positive end to the negative end. If e joins vertex v to vertex w and v is the positive end of e, we write e = (v, w). An orientation of G is a set of orientations, one for each edge of G; a graph with an orientation is called a directed graph.
Frobenius Symbols and the Groups SsGL(n), O(n) and Sp(n)
Authors: Abramsky Y. J. , Jahn H. A. , King R. C.
Published: December 2010

Frobenius [2; 3] introduced the symbols

to specify partitions and the corresponding irreducible representations of the symmetric group Ss.
On the Isomorphisms Between Certain Congruence Groups, II
Authors: Solazzi Robert
Published: December 2010

For integral domains of characteristic not 2, we prove here that the symplectic and unitary congruence groups are not isomorphic if the Witt indices are at least 3. This is Theorem 2.1; Theorem 3.3 describes the isomorphisms of unitary congruence groups.
A Generalization of Čaplygin's Inequality with Applications to Singular Boundary Value Problems
Authors: Willett D.
Published: December 2010

Let

(1.1)

where pk ∈ C(α, β) and - ∞ ≦ α < β ≦ ∞ . A solution of (1.1) is a nontrivial function y ∈ Cn(α, β), a neighborhood of β is an interval of the form (γ, β), α ≦ γ < β, and a neighborhood of α is an interval of the form (α, γ), α < γ ≦ β.
The Dual of Frobenius' Reciprocity Theorem
Authors: Robinson G. de B.
Published: December 2010

In two preceding papers [2; 3] the author has studied the algebras of the irreducible representations λ and the classes Ci of a finite group G. Integral representations {λ} and {Ci} of these algebras are derivable from the appropriate multiplication tables [4]. It should be emphasized, however, that the symmetry properties of the two sets of structure constants are not the same, and this leads to somewhat greater complexity in the formulae relating to classes as compared to representations.
A Theorem on Permutations of a Finite Field
Authors: Bruen A. , Levinger B.
Published: December 2010

The purpose of this note is to give a new proof of a theorem of L. Carlitz [2] and R. McConnel [5]. The theorem is as follows:

THEOREM 1. Let F = GF(pn) be the finite field of order q = pn and let
K — {x ∈ F|xd = 1} for some proper divisor d of q — 1.
A Technique for Studying the Boundedness and Extendability of Certain Types of Operators
Authors: Rooney P. G.
Published: December 2010

For 1 ≦ p < ∞, μ real, let Lμ, p denote the collection of functions f, Lebesgue measurable on (0, ∞ ), and such that ‖ f ‖μP < ∞ , where

(1.1)

Also, if X and Y are Banach spaces, denote by [X, Y] the collection of bounded linear operators from X to Y; [X, X] denote by [X].
On the Structure of Finite T0 + T5 Spaces
Authors: Das Shawpawn Kumar
Published: December 2010

The object of this paper is to study some structural aspects of finite T0 + T4 and T0 + T5 spaces in order to establish certain recursion relations that can be used to obtain the number of (labelled as well as unlabelled) T0 + T5 topologies on a finite set.Here, as in [2], a topology 𝒥 is a T4(T5) space provided for any pair of disjoint closed sets A and B (separated sets A and B = A ∩ closure B = B ∩ closure A = 0) there exist disjoint open sets 0A and 0B 𝒥 such that A ⊆ 0A and B ⊆ 0B. An almost immediate consequence of these investigations is that the inherent simplicity of the connected T0 + T5 topologies ensures that they are reconstructable.
On the Commutant of Certain Automorphism Groups
Authors: Tam P. K.
Published: December 2010

Let be a W*-algebra, A ( ) the group of all automorphisms of . In this paper we have determined the commutant G’ of a subgroup G of A () for certain classes of G and . The main results are as follows.

Theorem 1. If G is a locally compact abelian group acting by translation on the W*-algebra L∞(G), then the commutant of a dense subgroup of G is G itself.
Kinetic Theory and Boundary Conditions for Fluids
Authors: Schnute Jon , Shinbrot Marvin
Published: December 2010

A rigorous derivation of the molecular theory of a confined, deterministic gas is given. Then, a molecular reflection law is presented with the property that the corresponding fluid does not slip at the boundary. It is also shown that, within a certain reasonable class of reflection laws, the one we give is the only one that leads to no-slip.
The SCnP-Integral and the Pn+1-Integral
Authors: Bullen P. S. , Lee C. M.
Published: December 2010

In [2], we have briefly described, as examples of the general theory developed there, a scale of symmetric Cesaro-Perron integrals, namely SCnP-integral for n = 1, 2, 3, … . The purpose of this paper is to consider the integrals in a greater detail.

As a preliminary, we prove some lemmas, which are also interesting for their own sake, concerning the de la Vallée Poussin derivatives in Section 1, and we also state two deep theorems concerning the n-convex functions in Section 2.
The Banach-Saks Theorem in C(S)
Authors: Farnum Nicholas R.
Published: December 2010

A Banach space X has the Banach-Saks property if every sequence (xn) in X converging weakly to x has a subsequence (xnk) with (1/p)Σk=1xnk converging in norm to x. Originally, Banach and Saks [2] proved that the spaces Lp (p > 1) have this property. Kakutani [4] generalized their result by proving this for every uniformly convex Banach space, and in [9] Szlenk proved that the space L1 also has this property.
Some Results on Semi-Perfect Group Rings
Authors: Woods S. M.
Published: December 2010

The aim of this paper is to find necessary and sufficient conditions on a group G and a ring A for the group ring AG to be semi-perfect. A complete answer is given in the commutative case, in terms of the polynomial ring A[X] (Theorem 5.8). In the general case examples are given which indicate a very strong interaction between the properties of A and those of G. Partial answers to the question are given in Theorem 3.2, Proposition 4.2 and Corollary 4.3.
Chains of Varieties
Authors: Gupta Narain , Levin Frank , Rhemtulla Akbar
Published: December 2010

If is a variety of groups that can be denned by n-variable laws and (m) is the variety all of whose m-generator groups are in then there corresponds the chain: (1) ≧ (2) ≧ . . . ≧ (n) = . In this paper such chains are investigated to determine which of the inclusions are proper for certain varieties . In particular the inclusions are shown to be all proper for the varieties where is the variety of nilpotent-of-class-c groups, is the abelian variety and is the variety of centre-bymetabelian groups. For the inclusions are likewise proper but for the corresponding chain is:
Some Good Sequences of Interpolatory Polynomials
Authors: Freud G. , Sharma A.
Published: December 2010

In 1963, P. L. Butzer [4, p. 180] asked whether it was possible to prove Jackson's theorem by means of an operator which is "almost" interpolatory in the sense that it is based on the values of the approximee at a finite number of nodes. In answer to this question, G. Freud introduced [4] a sequence of operators which led to an independent proof of Jackson's theorem. Strictly speaking these operators are not interpolatory but they are "almost" interpolatory in the above sense.
Some Subfields of Qp, and their Non-Standard Analogues
Authors: Dubrovsky Diana L.
Published: December 2010

The desire to study constructive properties of given mathematical structures
goes back many years; we can perhaps mention L. Kronecker and B. L. van der
Waerden, two pioneers in this field. With the development of recursion theory it
was possible to make precise the notion of "effectively carrying out" the
operations in a given algebraic structure. Thus, A. Frölich and J. C. Shepherdson
[7] and M. O . Rabin [13] studied
computable algebraic structures, i.e. structures whose operations can be viewed as
recursive number theoretic relations. A. Robinson [18] and
E. W. Madison [11] used the concepts of computable and
arithmetically definable structures in order to establish the existence of what
can be called non-standard analogues (in a sense that will be specified later) of
certain subfields of R and C, the
standard models for the theories of real closed and algebraically closed fields
respectively.
Ordering Uniform Completions of Partially Ordered Sets
Authors: Redfield R. H.
Published: December 2010

Let (P, ) be a (nearly) uniform ordered space. Let (P, ) be the uniform completion of (P, ) at . Several partial orders for P are introduced and discussed. One of these orders provides an adjoint to the functor which embeds the category of uniformly complete uniform ordered spaces in the category of uniform ordered spaces, both categories with uniformly continuous order-preserving functions. When P is a join-semilattice with uniformly continuous join, two of these orders coalesce to the unique partial order with respect to which P is a join-semilattice, P is a join-subsemilattice of P, and the join on P is continuous.
Regular Categories and Regular Functors
Authors: Herrlich Horst
Published: December 2010

Let be a category with nice factorization-properties. If a functor G: —> which has a left-adjoint behaves nice with respect to factorizations then it can be shown quite easily that G behaves well in many other respects, especially that it lifts nice properties from into .
The G-Function of Macrae
Authors: Rush David E.
Published: December 2010

Let R be a commutative ring with identity. A finitely generated R-module M is called a torsion module if the annihilator of M contains a non zero-divisor. In [18] MacRae proved the following

Theorem. If R is a noetherian ring then there is a map G with the following properties from the class of torsion R-modules of finite homological dimension to the set of integral invertible ideals of R.

(i) If M is a finitely generated torsion R-module with homological dimension ≦ 1 then G(M) = F(M), the first Fitting ideal of M.

(ii) If S is a multiplicative subset of R then G(Ms) = G(M) s.

(iii) If 0 → L → M → N → 0 is an exact sequence of torsion modules of finite homological dimension then G(M) = G(L)G(N).
Chromatic Sums for Rooted Planar Triangulations, V: Special Equations
Authors: Tutte W. T.
Published: December 2010

In I we obtained an equation, called the "chromatic equation", for the generating function g(x, y, z, λ). In II and III we obtained special equations, valid in the cases λ = τ + 1 and λ = 3 respectively, for the generating function l(y, z, λ), defined as the coefficient of x2 in g(x, y, z, λ). The argument was independent of that in I and no attempt was made to derive the new formulae from the chromatic equation.
An Extension of M. Riesz's Mean Value Theorem for Infinite Integrals
Authors: Choudhary B. , Kumar Ashok
Published: December 2010

Isaacs [4] has proved the following theorem.

Theorem A. If 0 < α < 1 and

is convergent, then for u < w,

1.1
The Maximum Genus of Cartesian Products of Graphs
Authors: Zaks Joseph
Published: December 2010

The maximum genus γM(G) of a connected graph G has been defined in [2] as the maximum g for which there exists an embedding h : G —> S(g), where S(g) is a compact orientable 2-manifold of genus g, such that each one of the connected components of S(g) — h(G) is homeomorphic to an open disk; such an embedding is called cellular. If G is cellularly embedded in S(g), having V vertices, E edges and F faces, then by Euler's formula

V-E + F = 2-2g.
The Translational Hull of an Inverse Semigroup
Authors: Reilly N. R.
Published: December 2010

Let S be a semigroup. A function λ(ρ) on S is a left (right) translation of S if, for all x, y ∊ S, λ(xy) = λ(x)y ((xy)ρ= x(yρ)). A left translation λ and a right translation ρ are said to be linked if x(λy) = (xρ)y, for all x,y ∊ S,and then the ordered pair (λ, ρ) is called a bitranslation. Clearly the set Λ(S) (P(S)) of all left (right) translations is a semigroup with respect to composition of functions. The set of bitranslations forms a subsemigroup of the direct product Λ(S) × P(S)which is called the translational hull, Ω(S), of S. A valuable survey of results relating to Ω(S) and its importance in relation to semigroup extensions will be found in Petrich's review [6], to which the reader is referred for basic results on translational hulls.
Localization of Right Noetherian Rings at Semiprime Ideals
Authors: Lambek J. , Michler G.
Published: December 2010

In [11] and [12] we investigated the process of localization of right Noetherian rings R at prime ideals. We shall now extend these investigations to semiprime ideals N of R.

In Section 2 we show that localizing at the injective right R-module E(R/N) is the same as localizing with respect to the multiplicative set
A Generalization of the Mapping Degree
Authors: Bourgin D. G.
Published: December 2010

For the single-valued case the notion of degree has been given recent expression by papers of Dold [5] for the finite dimensional case, and by Leray-Schauder [8] for the locally convex linear topological space. Klee [7] has removed this restriction by use of shrinkable in place of convex neighborhoods with the central role filled by a form of (2.15) below. For set-valued maps a modern formulation is, for instance, to be found in Gorniewicz-Granas [6]. These contributions relate the degree to the Lefschetz number, and the set-valued maps are required to map points into acyclic sets; that is to say, into "swollen points".
A Local Ergodic Theorem on Lp
Authors: Baxter J. R. , Chacon R. V.
Published: December 2010

Two general types of pointwise ergodic theorems have been studied: those as t approaches infinity, and those as t approaches zero. This paper deals with the latter case, which is referred to as the local case.

Let (X, , μ) be a complete, σ-finite measure space. Let {Tt} be a strongly continuous one-parameter semi-group of contractions on , defined for t ≧ 0. For Tt positive, it was shown independently in [2] and [5] that

1.1

almost everywhere on X, for any f ∊ L1. The same result was obtained in [1], with the continuity assumption weakened to having it hold for t > 0.
A Note on Quadratic Forms and the u-invariant
Authors: Ware Roger
Published: December 2010

The u-invariant of a field F, u = u(F), is defined to be the maximum of the dimensions of anisotropic quadratic forms over F. If F is a non-formally real field with a finite number q of square classes then it is known that u ≦ q. The purpose of this note is to give some necessary and sufficient conditions for equality in terms of the structure of the Witt ring of F.
Regularity of Richardson's Compactification
Authors: Gazik R. J.
Published: December 2010

In [7] Richardson constructed a Stone-Čech type compactification R(E) of a Hausdorff convergence space E. Two questions arise in this regard. First, when is R(E) homeomorphic to β(E), β(E) the topological Stone-Čech compactification of E, for a Tychonoff topological space E? Second, if E is a regular convergence space, when is R(E) regular? The last question is motivated by the study of regular compactifications in [6]. In section 2 it will be shown that a necessary and sufficient condition in answer to both questions, is that α = cl(α) for each nonconvergent ultrafilter α on E.
On the Weak Basis Theorem in F-spaces
Authors: Shapiro Joel H.
Published: December 2010

It is well-known that every weak basis in a Fréchet space is actually a basis. This result, called the weak basis theorem was first given for Banach spaces in 1932 by Banach [1, p. 238], and extended to Fréchet spaces by Bessaga and Petczynski [3]. McArthur [12] proved an analogue for bases of subspaces in Fréchet spaces, and recently W. J. Stiles [18, Corollary 4.5, p. 413] showed that the theorem fails in the non-locally convex spaces lp (0 < p < 1). The purpose of this paper is to prove the following generalization of Stiles' result.
Coefficients of Symmetric Functions of Bounded Boundary Rotation
Authors: Leach Ronald J.
Published: December 2010

Let denote the family of all functions of the form

that are analytic in the unit disc U, f′(z) ≠ 0 in U and f maps U onto a domain of boundary rotation at most . Recently Brannan, Clunie and Kirwan [2] and Aharonov and Friedland [1] have solved the problem of estimating |amp+1| for all , provided m = 1.
Growth Conditions and Decomposable Operators
Authors: Radjabalipour Mehdi
Published: December 2010

Throughout this paper T will denote a bounded linear operator which is defined on a Banach space and whose spectrum lies on a rectifiable Jordan curve J .

The operators having some growth conditions on their resolvents have been the subject of discussion for a long time. Many sufficient conditions have been found to ensure that such operators have invariant subspaces [2 ; 3 ; 7 ; 8 ; 12 ; 13; 14; 21; 27; 28; 29], are S-operators [14], are quasidecomposable [9], are decomposable [4 ; 11], are spectral [7 ; 10 ; 15 ; 17], are similar to normal operators [16 ; 23 ; 25 ; 26], or are normal [15 ; 18 ; 22]. In this line we are going to show that many such operators are decomposable.
Korovkin Theorems for Integral Operators with Kernels of Finite Oscillation
Authors: Marsden M. J. , Riemenschneider S. D.
Published: December 2010

There has been considerable interest recently in the investigation of "Korovkin sets". Briefly, for X a Banach space and a family of linear operators on X, a subset K ⊂ X is a Korovkin set relative to if for any bounded sequence {Tn} ⊂ , Tnk → k in X for each k ∊ K implies Tnx → x for each x ∊ X. A large portion of these investigations have been carried out for X being one of the spaces C(S), S compact Hausdorff, the usual Lp spaces of functions on some finite measure space, or some Banach lattice; while is one of the classes +-positive operators, 1-contractions (i.e., ||T|| 1), or + ⋂1
Ranges of Products of Operators
Authors: Grabiner Sandy
Published: December 2010

Suppose that T and A are bounded linear operators. In this paper we examine the relation between the ranges of A and TA, under various additional hypotheses on T and A. We also consider the dual problem of the relation between the null-spaces of T and AT; and we consider some cases where T or A are only closed operators. Our major results about ranges of bounded operators are summarized in the following theorem.

Theorem 1. Suppose that T is a bounded operator on a Banach space E and that A is a non-zero bounded operator from some Banach space to E.
A Note on Associative Polyverbal Operations On Groups
Authors: Macedońska-Nosalska O. N.
Published: December 2010

In his paper [2] O. N. Golovin introduced the notion of a neutral polyverbal operation on groups, of which Moran's verbal operation [6], and Gruenberg's and Šmel'kin's operations [3 ; 7] are special cases. (Bronštein [1] proved, more generally, that every regular operation for which MacLane's postulate (see [2]) holds and which is invariant under addition of trivial factors, is a neutral polyverbal operation.)
An Arithmetical Difference System with Application to B.I.B. Designs
Authors: Majindar Kulendra N.
Published: December 2010

In this paper we have established the existence of an arithmetical difference system by a constructive method. Our arithmetical difference systems are a generalization of cyclic difference sets.
The Translational Hull of a Semilattice of Weakly Reductive Semigroups
Authors: Petrich Mario
Published: December 2010

The translational hull is of central importance in the construction of ideal extensions and the study of densely embedded ideals particularly for weakly reductive semigroups (see [4, Chapter III]). The translational hull of semigroups belonging to a few special classes is known in an explicit form, and for some other classes of semigroups, certain properties of their translational hulls have been established (see [4, Chapter V]). We have generalized in [5] the concept of an inverse limit of groups in order to give a construction of the translational hull of a semigroup which is a semilattice of groups.
Transferring Results From Rings of Continuous Functions to Rings of Analytic Functions
Authors: Adler Andrew , Douglas Williams R.
Published: December 2010

Let C(X) be the ring of all real-valued continuous functions on a completely regular topological space X, and let A﹛Y) be the ring of all functions analytic on a connected non-compact Riemann surface F. The ideal theories of these two function rings have been extensively studied since the fundamental papers of E. Hewitt on C﹛X)[12] and of M. Henriksen on the ring of entire functions [10; 11]. Despite the obvious differences between these two rings, it has turned out that there are striking similarities between their ideal theories. For instance, non-maximal prime ideals of A (F) [2; 11] behave very much like prime ideals of C﹛X)[13; 14], and primary ideals of A(Y) which are not powers of maximal ideals [19] resemble primary ideals of C(X) [15]. In this paper we show that there are very good reasons for these similarities. It turns out that much of the ideal theory of A (Y) is a special case of the ideal theory of rings of continuous functions. We develop machinery that enables one almost automatically to derive results about the ideal theory of A(Y) from corresponding known results of ideal theory for rings of continuous functions.