bluuebunny/crossref_metadata_2025 huggingface dataset:
Finite Linear Groups of Degree Seven. I
Authors: Wales David B.
Published: December 2010

1. 1. This paper is the second in a series of papers discussing linear groups of prime degree, the first being (8). In this paper we discuss only linear groups of degree 7. Thus, G is a finite group with a faithful irreducible complex representation Xof degree 7 which is unimodular and primitive. The character of Xis x- The notation of (8) is used except here p= 7. Thus Pis a 7-Sylow group of G.In §§ 2 and 3 some general theorems about the 3-Sylow group and 5-Sylow group are given. In § 4 the statement of the results when Ghas a non-abelian 7-Sylow group is given. This corresponds to the case |P| =73 or |P|= 74. The proof is given in §§ 5 and 6. In a subsequent paper the results when Pis abelian will be given.
Projective Ideals of Finite Type
Authors: Smith William W.
Published: December 2010

The main results in this paper relate the concepts of flatness and projectiveness for finitely generated ideals in a commutative ring with unity. In this discussion the idea of a multiplicative ideal is used.

Definition.An ideal Jis multiplicative if and only if whenever I is an ideal with I ⊂ J there exists an ideal Csuch that I = JC.

Throughout this paper Rwill denote a commutative ring with unity. If I and Jare ideals of R,then I: J = {x| xJ ⊂ I}. By “prime ideal” we will mean “proper prime ideal” and Specie will denote this set of ideals. Ris called a local ring if it has a unique maximal ideal (the ring need not be Noetherian). If P is in Spec R then RPis the quotient ring formed using the complement of P.
Transformations Affines Agissant Sur Un Ensemble Compact Convexe
Authors: Dubuc Serge
Published: December 2010

Soit Eun espace vectoriel topologique localement convexe séparé et soit K un sous-ensemble convexe compact de E, le problème que nous voulons considérer est le suivant : nous partons d'une famille A de transformations affines de Kdans lui-même, pour chaque point xde K, nous considérons la fermeture de l'enveloppe convexe de {A(x)| A∊ A}, ce que nous appelions A(x). Si nous voulons que l'application x → A(x) vérifie certaines propriétés, quelles conditions devons-nous imposer à A ou à K? La solution de ce problème est reliée à plusieurs théorèmes de point fixe (en particulier au théorème de Markov-Kakutani (2) et de Ryll-Nardzewski (3)), ainsi qu'au lemme de Choquet-Deny sur les solutions bornées de l'équation de convolution μ * σ = μ. Le principal résultat de cet article est que l'application identité est un élément extrême de l'ensemble des applications affines d'un convexe borné en lui-même.
Automorphism Groups of Unary Algebras on Groups
Authors: Wenzel G. H.
Published: December 2010

This paper presents a systematic study of the automorphism groups of those unary (universal) algebras whose carrier set G is the carrier set of some group ) and whose automorphism set contains the right translations of the latter group. These algebras appear, apart from the known classical contexts, repeatedly in characterization theorems of endomorphism semigroups (End) and automorphism groups (Aut) of algebras due to Grätzer (3; 4; 5), Makkai (7), Armbrust and Schmidt (1), Birkhoff (2), and others.

Our main result (Theorem 1) constitutes an essential strengthening of a theorem of Birkhoff and represents the automorphism group of a unary algebra (where F is contained in the set of left translations of the group as wreath product of two groups that are easily determined from F and G.
An Inequality Concerning Analytic Functions with a Positive Real Part
Authors: MacGregor Thomas H.
Published: December 2010

This paper contains an inequality about functions which are analytic and have a positive real part in the unit disk. A first consequence of the inequality is the fact that if is analytic for |z| < 1 and has values lying in a strip of width δ. This result is known and was first proved by Tammi (1).

Our second theorem is a generalization of this. Namely, ifis analytic for |z| < 1 and satisfies Re{zmf(m>(z)} ≧ A and

thenconverges.

Another application of our fundamental inequality is the following. Let be analytic for |z| < 1 and satisfy Re p(z) ≧ 0 and set and .
The Reflection Principle for Banach Space-Valued Analytic Functions
Authors: Finkelstein Mark
Published: December 2010

We give sufficient conditions for the continuation of an analytic function with values in a Branch space. For analytic functions taking complex numbers as values, the principle is known as the Schwarz Reflection Principle.

A function defined on a domain of the complex plane with values in a Banach space X is analytic if it possesses at each point Z0 of the domain a convergent power series in z, with coefficients in X.

THEOREM. Let D be a domain in the upper half-plane, and E a regular subset of the boundary of D. Suppose that E is an interval of the real axis (a,b). Let f be an analytic function defined on D, continuous up to E, taking values in a Banach space X. Let the image of D under f be Ω, and let Γ be the part of the boundary of Ω which is the image of E under f.
Abstract Quadratic Forms
Authors: Kaplansky Irving , Shaker Richard J.
Published: December 2010

We shall be studying the following structure, which we shall call a V-form (“Vector-valued form”). Let G and W be additive abelian groups with every element of order 2 (i.e. vector spaces over the field GF(2) of two elements). Let there be given a symmetric bilinear map from G × G to W; we shall write it simply as a product ab. We define an equivalence relation on unordered n-ples of G. For n = 2: (a, b) ~ (c, d) if a + b = c + d and ab = cd. For n > 2 we define equivalence “piecewise”: there is to be a chain from (a1, …, an) to (b1, …, bn) where at each step only two elements are changed in accordance with the equivalence just defined for n = 2.
On Plane Sections and Projections of Convex Sets
Authors: Groemer H.
Published: December 2010

Let K be a three-dimensional convex body. It has been conjectured (cf. 3) that one can always find a plane H such that the intersection K ∩ H is, in a certain sense, fairly circular. Instead of the plane section K ∩ H one can also consider the orthogonal projection of K onto H. Our aim in this paper is to prove some results concerning this type of problems. It appears that John has found similar theorems (cf. the remarks of Behrend, 1, p. 717). His proof of the first inequality of our Theorem 1 has been published (6). It is based on a property of the ellipse of inertia which will not be used in the present paper.

A non-empty compact convex set 5 which is contained in some plane of euclidean three-dimensional space E3 will be called a convex domain.
On a Theorem of Herstein
Authors: Chacron M.
Published: December 2010

Throughout this paper, Z is the ring of integers, ƒ*(t) (ƒ(t)) is an integer monic (co-monic) polynomial in the indeterminate t (i.e., each coefficient of ƒ* (ƒ) is in Z and its highest (lowest) coefficient is 1 (5, p. 121, Definition) and M* (M) is the multiplicative semigroup of all integer monic (co-monic) polynomials ƒ* (ƒ) having no constant term. In (3, Theorem 2), Herstein proved that if R is a division ring with centre C such that

1

then R = C. In this paper we seek a generalization of Herstein's result to semi-simple rings. We also study the following condition:

(1)*

Our results are quite complete for a semi-simple ring R in which there exists a bound for the codegree ofƒ (ƒ*) (i.e., the degree of the lowest monomial of ƒ(ƒ*)) appearing in the left-hand side of (1) ((1)*).
Some Remarks Concerning (f, dn) and [F, dn] Summability Methods
Authors: Koch C. F.
Published: December 2010

In this paper we note a simple connection between the (ƒ, dn) method of summability (defined by Smith (8)) and a composition of [F, dn] (defined by Jakimovski (4)) and Sonnenschein methods (9; 10). This connection is then used to supply some sufficient conditions for the regularity of (ƒ, dn) methods by using known regularity conditions for various [F, dn] and Sonnenschein methods. Finally, the connection is further exploited to obtain information about the Lebesgue constants for a certain class of [F, dn] methods by investigating related (ƒ, dn) methods.

2. Definitions and the regularity theorem. The (ƒ, dn) method of summability is defined by Smith (8) essentially as follows. Letƒ(z) be a nonconstant entire function satisfying ƒ (l) = 1 and let {dn} be a sequence of complex numbers satisfying dt ≠ – 1, di ≠ – ƒ(0) (i ≧ 1).
Right Bol Quasi-Fields
Authors: Kallaher Michael J.
Published: December 2010

We shall consider quasi-fields which satisfy the multiplicative Identity

1.1

(1.1) will be called the right Bol law and a quasi-field satisfying it will be called a right Bol quasi-held. Moufang quasi-fields, i.e., those satisfying the Moufang identity

1.2

were studied in (5). Quasi-fields satisfying the left Bol identity

1.3

were studied by Burn (3) and the author (6). Such quasi-fields are called Bol quasi-fields.

Our investigation will parallel the investigations in (5; 6). In § 2 we derive necessary and sufficient conditions for a right Bol quasi-field to be an alternative division ring and also criteria for it to be a near-field. With this information we derive in §§ 3 and 4 new characterizations of Moufang planes similar to those in (5; 6).

Loops satisfying (1.1) have been studied by Robinson (10). He calls such loops Bol loops.
Implications between Conditions on l-Groups
Authors: Pedersen Franklin D.
Published: December 2010

The study of lattice-ordered groups (l-groups) in recent years has yielded several publications concerning the lattice of all convex l-subgroups (denoted by Γ(G)). As for example, in [3], Conrad has shown a direct connection between one possible structure of Γ(G) as a lattice and certain finite conditions on the elements of the group G. In the process of developing variations of this approach, there has been imposed several different conditions on G and Γ(G). For instance, it was proposed by Conrad that if all the minimal prime subgroups have non-zero polars, then the group elements would all be finite-valued. In this paper we show that in general this is not true.
The Width of a Module
Authors: Wichman Michael
Published: December 2010

An R-module N is said to have finite width n if n is the smallest integer such that for any set of n + 1 elements of N, at least one of the elements is in the submodule generated by the remaining n. The width of N over R will be denoted by W(R, N).

The notion of width was introduced by Brameret [2, p. 3605]. However, Cohen [3] investigated rings of finite rank, which, in the case that R is a local Noetherian domain, is equivalent to width (Proposition 1.6). He showed that finite width of R was both equivalent to R having Krull dimension one, and to R having the restricted minimum condition (Theorem 1.12).
On Multiple Integral Geometric Integrals and Their Applications to Probability Theory
Authors: Streit Franz
Published: December 2010

It has been pointed out repeatedly in the literature that the methods of integral geometry (a mathematical theory founded by Wilhelm Blaschke and considerably extended by several mathematicians) provide highly suitable means for the solution of problems concerning “geometrical probabilities“ [2; 6; 12; 15]. The possibilities for the application of these integral geometric results to the evaluation of probabilities, satisfying certain conditions of invariance with respect to a group of transformations which acts on the probability space, are obviously not yet exhausted. In this article, such applications are presented. First, some concepts and notation are introduced (§1). In the next section we derive some integral geometric relations (§ 2). These results are generalizations of known systems of formulae and they are valid in the k-dimensional Euclidean space. In § 3, we determine mean-value formulae for the fundamental characteristics of point-sets, generated by randomly placed convex bodies.
Galois Extensions as Modules over the Group Ring
Authors: Garfinkel Gerald , Orzech Morris
Published: December 2010

Suppose that R is a commutative ring and G is a finite abelian group. In § 2 we review the definition of E(R, G) (T(R, G)), the group of all (commutative) Galois extensions S of R with Galois group G. We discuss the properties of these groups as functors of G and give an example which exhibits some of the pathological properties of the functor E(R, – ). In § 3 we display a homomorphism from E(R, G) to Pic (R(G)); we use this homomorphism to prove that if S is commutative, G has exponent m, and R(G) has Serre dimension 0 or 1, then a direct sum of m copies of S is isomorphic as a G-module to a direct sum of m copies of R(G). (This result is related to [5, Theorem 4.2], where it is shown that if S is a free R-module and G is any finite group with n elements, then Sn is isomorphic to R(G)n as G-modules.) We also give some examples of Galois extensions without normal bases.
Polytopes with an Axis Of Symmetry
Authors: McMullen P. , Shephard G. C.
Published: December 2010

During the last few years, Branko Grünbaum, Micha Perles, and others have made extensive use of Gale transforms and Gale diagrams in investigating the properties of convex polytopes. Up to the present, this technique has been applied almost entirely in connection with combinatorial and enumeration problems. In this paper we begin by showing that Gale transforms are also useful in investigating properties of an essentially metrical nature, namely the symmetries of a convex polytope. Our main result here (Theorem (10)) is that, in a manner that will be made precise later, the symmetry group of a polytope can be represented faithfully by the symmetry group of a Gale transform of its vertices. If a d-polytope P ⊂ Ed has an axis of symmetry A (that is, A is a linear subspace of Ed such that the reflection in A is a symmetry of P), then it is called axi-symmetric. Using Gale transforms we are able to determine, in a simple manner, the possible numbers and dimensions of axes of symmetry of axi-symmetric polytopes.
On the Simplex of Completely Monotonic Functions on a Commutative Semigroup
Authors: Fine N. J. , Maserick P. H.
Published: December 2010

Bernstein's classical integral representation theorem for completely monotonie functions can be proved most elegantly, on a commutative semigroup with identity, by the integral version of the Kreĭn-Milman theorem [2]. The key to this approach is the identification (as exponentials) of the extremal points of the normalized completely monotonie functions. Alternate proofs of this identification are given in § 1. The first (Corollary 1.3) is based on the Kreĭn-Milman theorem and the second (see remarks following Corollary 1.5) is derived from elementary analytic techniques. Other interesting facts about completely monotonie functions are mentioned in passing. For example, we observe that the normalized completely monotonie functions form a simplex (Corollary 1.4). In Corollary 1.6 we note that the product of completely monotonie functions corresponds to the convolution of their representing measures. Thus the normalized completely monotonie functions form an affine semigroup [3].
Homotopy of Natural Transformations
Authors: Hardie K. A.
Published: December 2010

Let C be a full subcategory of T, the category of based topological spaces and based maps, and let Cn be the corresponding category of n-tuples. Let S, T: Tn → T be covariant functors which respect homotopy classes and let u, v: S → T be natural transformations, u and v are homotopic inC, denoted u ≃ v(C), if uX ≃ vX: SX → TX (X ∈ Cn), that is to say, for every X ∈ C, uX and vX are homotopic (all homotopies are required to respect base points), u and v are naturally homotopic inC, denoted u ≃n v; (C), if there exist morphisms

such that, for every X ∈ C, utX is a homotopy from uX to vX and such that, for every t ∈ I, ut:S → T is a natural transformation.
A Generalization of Commutative and Alternative Rings
Authors: Kleinfeld Erwin , Kleinfeld Margaret Humm , Kosier Frank
Published: December 2010

In [3] Schafer has defined generalized standard rings as rings satisfying the identities

(1)

(2)

(3)

and observed that these identities imply (y,y, (x, z)) = 0 and if the characteristic is not three, (x, y, x2) = 0. Schafer determined the structure of simple, finite-dimensional generalized standard algebras of characteristic not two or three by showing that they must be either commutative, Jordan, or alternative.

Previously one of us [2] had studied accessible rings, which are defined by the identities (x,y,z) + (z,x,y) – (x,z,y) = 0 and ((w,x), y,z) = 0.
On the Representation of Mappings of Compact Metrizable Spaces as Restrictions of Linear Transformations
Authors: Edelstein Michael
Published: December 2010

Let f: X → X be a continuous mapping of the compact metrizable space X into itself with a singleton. In [3] Janos proved that for any λ, 0 < λ < 1, a metric ρ compatible with the topology of X exists such that ρ(f(x), f(y)) ≦ λρ(x, y) for all x, y ∈X. More recently, Janos [4] has shown that if, in addition, f is one-to-one, then a Hilbert space H and a homeomorphism μ: X → H exist such that μfμ-1 is the restriction to μ[X] of the transformation sending y ∈ H into λy. Our aim in this note is to show that in both cases a homeomorphism h of X into l2 exists such that hfh-1 is the restriction of a linear transformation.
Cercles De Remplissage and Asymptotic Behaviour along Circuitous Paths
Authors: Gauthier P. M.
Published: December 2010

In this paper we consider the value distribution of a meromorphic function whose behaviour is prescribed along a spiral. The existence of extremely wild holomorphic functions is established. Indeed a very weak form of one of our results would be that there are holomorphic functions (in the unit disc or the plane) for which every curve “tending to the boundary” is a Julia curve.

The theorems in this paper generalize results of Gavrilov [7], Lange [9], and Seidel [11].

I wish to express my thanks to Professor W. Seidel for his guidance and encouragement.

2. Preliminaries. For the most part we will be dealing with the metric space (D, ρ) where D is the unit disc, |z| < 1, and ρ is the non-Euclidean hyperbolic metric on D. The chordal metric on the Riemann sphere will be denoted by x.
Normal and Canonical Representations in Free Products of Lattices†
Authors: Lakser H.
Published: December 2010

In solving the word problem for free lattices, Whitman [4] showed that free lattices admit canonical representations, that is, of all polynomials over the generating set representing an element of the lattice, the polynomial of shortest length is unique up to commutativity and associativity. These well-defined shortest polynomials have proved very important in analyzing the internal structure of free lattices in detail; see, e.g., [5].

Sorkin [3] proved that the free product of chains also admits canonical representations; these were exploited by Rolf [2]. In the above-mentioned paper, Sorkin also suggested that the free product of two copies of 22 does not admit canonical representations.
The Dimensions of Irreducible Representations of Linear Groups
Authors: King R. C.
Published: December 2010

The theory of the relationship between the symmetric group on a symbols, Σa, and the general linear group in n-dimensions, GL(n), was greatly developed by Weyl [4] who, in this connection, made use of tensor representations of GL(n). The set of mixed tensors

forms the basis of a representation of GL(n) if all the indices may take the values 1, 2, …, n, and if the linear transformation

is associated with every non-singular n × n matrix A. The representation is irreducible if the tensors are traceless and if the sets of covariant indices (α)a and contra variant indices (β)b themselves form the bases of irreducible representations (IRs) of Σa and Σb, respectively. These IRs of Σa and Σb may be specified by Young tableaux [μ]a and [v]b in the usual way [4].
Remark to a Theorem Due to R. Sandler
Authors: Iden Oddvar
Published: December 2010

To prove [2, Theorem 1], it is enough to remark that if x is a point or line of F4 which is fixed under H, then x is fixed under each subgroup of H. By Sylow's theorem, H has a subgroup H’ of order three. By [1, p. 420, Lemma 2.2], it follows that x is in the subplane of F4 generated by those elements of {A, B, C, D} which are fixed under H'. This subplane consists of a point only. Hence, x is in {A, B, C, D}, which is impossible. This argument can be used in many other cases.
Surfaces Embedded in M2 × S1
Authors: Jaco William
Published: December 2010

In this paper we study incompressible and injective (see § 2 for definitions) surfaces embedded in M2 × S1, where M2 is a surface and S1 is the 1-sphere. We are able to characterize embeddings which are incompressible in M2 × S1 when M2 is closed and orientable. Namely, a necessary and sufficient condition for the closed surface F to be incompressible in M2 × S1, where M2is closed and orientable, is that there exists an ambient isotopy ht, 0 ≦ t ≦ 1, of M2 × S1onto itself so that either

(i) there is a non-trivial simple closed curve J ⊂ M2 and h1(F) = J × S1, or

(ii) p\h1(F) is a covering projection of h1(F) onto M2, where p is the natural projection of M2 × S1onto M2.
Sublattices and Initial Segments of the Degrees of Unsolvability
Authors: Thomason S. K.
Published: December 2010

In this paper we shall prove that every finite lattice is isomorphic to a sublattice of the degrees of unsolvability, and that every one of a certain class of finite lattices is isomorphic to an initial segment of degrees.

Acknowledgment. I am grateful to Ralph McKenzie for his assistance in matters of lattice theory.

1. Representation of lattices. The equivalence lattice of the set S consists of all equivalence relations on S, ordered by setting θ ≦ θ’ if for all a and b in S, a θ b ⇒ a θ’ b. The least upper bound and greatest lower bound in are given by the ⋃ and ⋂ operations:
Some Results on the Schur Index of a Representation of a Finite Group
Authors: Ford Charles
Published: December 2010

Let ℭ be a finite group with a representation as an irreducible group of linear transformations on a finite-dimensional complex vector space. Every choice of a basis for the space gives the representing transformations the form of a particular group of matrices. If for some choice of a basis the resulting group of matrices has entries which all lie in a subfield K of the complex field, we say that the representation can be realized in K. It is well known that every representation of ℭ can be realized in some algebraic number field, a finitedimensional extension of the rational field Q.
Subgroups of Conjugate Classes in Extensions
Authors: Burroughs John E. , Schafer James A.
Published: December 2010

Often in various mathematical problems one encounters an extension B of the group G by the group π in which one wishes to extract certain information about B from information given in terms of G, π, the action of π on G, and the class of the extension in H2(π, centre G). An example of this type of problem is to determine some intrinsically defined subgroup of B, for instance the centre of B, given knowledge of the corresponding subgroup for G and π, and, of course, the usual information concerning the extension.

In this paper we shall use the fact that any extension is congruent to a crossed product extension [2] to investigate a class of subgroups which naturally generalizes the notion of the centre. The definition of this class appears in § 3.

1. Let

by an extension of G by π.
On a Theorem of Heilbronn Concerning the Fractional Part of θn2
Authors: Liu Ming-Chit
Published: December 2010

1. In 1948 Heilbronn [4] proved the following theorem.

THEOREM H. For every real θ and every positive integer N, there is an integer n satisfying

(1.1)

whereis an arbitrarily small number,depends only on, and ‖t‖ means the distance from t to the nearest integer.

The interest of the result (1.1) is that the inequality is uniform in θ, and is therefore analogous to the classical inequality of Dirichlet for the fractional part of θn. In this paper we shall prove the following theorem.

THEOREM. For every real θ and every positive integer N, there is an integer n satisfying

(1.2)

where A is an absolute constant and.
A Maximum Principle for Bounded Harmonic Functions on Riemannian Spaces
Authors: Kwon Y. K. , Sario L.
Published: December 2010

Harmonic functions with certain boundedness properties on a given open Riemann surfaceRattain their maxima and minima on the harmonic boundaryΔBofR.The significance of such maximum principles lies in the fact that the classification theory of Riemann surfaces related to harmonic functions reduces to a study of topological properties of Δ(cf. [11; 8; 3; 12].

For the corresponding problem in higher dimensions we shall first show that the complement of ΔRwith respect to the Royden boundary ΓRof a Riemannian N-spaceRis harmonically negligible: given any non-empty compact subsetEof ΓR– ΔRthere exists an Evans superharmonic functionv,i.e., a positive continuous function onR* = R∪ΓR,superharmonic onR,withv= 0 on ΔR, v≡ ∞ onE,and with a finite Dirichlet integral overR.
On Positive Definite Functions over a Locally Compact Group
Authors: Price J. F.
Published: December 2010

In this note we are concerned with several questions on positive definite functions over a Hausdorff locally compact group. The main result, Theorem A, gives some necessary and sufficient conditions for to be a positive definite function when μ is a (complex Radon) measure. In particular, is a positive definite function if and only if μ ∊ L2, and Theorem B then follows by giving a complete characterization of functions of the type , where f ∊ L2. Perhaps the most interesting aspect of these results is that they provide further examples of results over a non-abelian, non-compact group, which otherwise are simple consequences (with μ, a bounded measure in Theorem A) of the theorems of Plancherel and Bochner.

Unless otherwise specified, all notation and definitions will follow [1;2]. The underlying group will always be G, a Hausdorff locally compact group with identity e, and with left Haar measure dx.
A Theorem on Steiner Systems
Authors: Mendelsohn N. S.
Published: December 2010

1. Definitions and notation. A generalized Steiner system (t-design, tactical configuration) with parameters t, λt, k, v is a system (T, B), where T is a set of v elements, B is a set of blocks each of which is a k-subset of T (but note that blocks bi and bj may be the same k-subset of T) and such that every set of t elements of T belongs to exactly λt of the blocks. If we put λt = u we denote by Su(t, k, v) the collection of all systems with these parameters. Thus Q ∈ Su(t, k, v) means Q = (T, B) is a system with the given parameters. If λt = u = 1, we write S(t, k, v) instead of S1(t, k, v) and refer to the system as a Steiner system. If t = 2, the system is called a balanced incomplete block design.
Perfect Mappings and Spaces of Countable Type
Authors: Vaughan J. E.
Published: December 2010

In [1, p. 41, Theorem 3.10] Arhangel'skiï proved that the
perfect image of a completely regular space of countable type is of
countable type, and he asked [1, p. 60, problem 4] if a similar
result held for regular or Hausdorff spaces. In this paper, it is proved
that the perfect image of a space of countable type is of countable type,
provided that the image is Hausdorff or regular. An affirmative answer to
both of Arhangel'skiï's questions follows immediately from this.
Arhangel'skiï made use of the Stone-Čech compactification in the proof of
his result, but the proofs below are of a different nature.

Let X be a topological space and let K ⊂
X. A collection of open sets
is called a base at K provided that for every open set
W ⊃ K there exists such that K ⊂ U ⊂
W. Clearly, we may assume that every member of
contains K.
On the Nilpotency of Nil Subrings
Authors: Fisher Joe W.
Published: December 2010

A famous theorem of Levitzki states that in a left Noetherian ring each nil
left ideal is nilpotent. Lanski [5] has extended Levitzki's
theorem by proving that in a left Goldie ring each nil subring is nilpotent.
Another important theorem in this area which is due to Herstein and Small
[3] states that if a ring satisfies the ascending chain
condition on both left and right annihilators, then each nil subring is
nilpotent. We give a short proof of a theorem (Theorem 1.6) which yields
both Lanski's theorem and Herstein- Small's theorem. We make use of the
ascending chain condition on principal left annihilators in order to obtain,
at an intermediate step, a theorem (Theorem 1.1) which produces sufficient
conditions for a nil subring to be left T-nilpotent.
Geometric Mappings on Geometric Lattices
Authors: Sachs David
Published: December 2010

It is a classical result of mathematics that there is an intimate connection
between linear algebra and projective or affine geometry. Thus, many
algebraic results can be given a geometric interpretation, and geometric
theorems can quite often be proved more easily by algebraic methods. In this
paper we apply topological ideas to geometric lattices, structures which
provide the framework for the study of abstract linear independence, and
obtain affine geometry from the mappings that preserve the closure operator
that is associated with these lattices. These mappings are closely connected
with semi-linear transformations on a vector space, and thus linear algebra
and affine geometry are derived from the study of a certain closure operator
and mappings which preserve it, even if the “space” is finite.
On Purifiable Subsocles of a Primary Abelian
Group
Authors: Irwin John , Swanek James
Published: December 2010

In this paper we shall investigate an interesting connection between the
structure of G/S and G, where
S is a purifiable subsocle of G. The
results are interesting in the light of a counterexample by Dieudonné
[3, p. 142] who exhibits a primary abelian group
G, where G/S is a direct sum of cyclic
groups, but G is not a direct sum of cyclic groups.
Surprisingly, the assumption of the purifiability of S
allows G to inherit the structure of G/S.
In particular, we show that if G/S is a direct sum of
cyclic groups and S supports a pure subgroup
H, then G is a direct sum of cyclic
groups and if is a direct summand of G which is of course a
direct sum of cyclic groups. It is also shown that if G/S
is a direct sum of torsion-complete groups and S supports a
pure subgroup H, then G is a direct sum of
torsion-complete groups and H is a direct summand of
G, and is also a direct sum of torsion-complete
groups.
Representation of certain Linear Operators in Hilbert
Space
Authors: Harvey Bernard Niel
Published: December 2010

In this paper we represent certain linear operators in a space with
indefinite metric. Such a space may be a pair (H, B), where
H is a separable Hilbert space, B is a
bilinear functional on H given by
B(x, y) = [Jx, y], [,
] is the Hilbert inner product in H, and J
is a bounded linear operator such that J = J*
and J2 = I. If T is a linear operator
in H, then ‖T‖ is the usual operator norm.
The operator J above has two eigenspaces corresponding to
the eigenvalues + 1 and –1.

In case the eigenspace in which J induces a positive
operator has finite dimension k, a general spectral theory
is known and has been developed principally by Pontrjagin [25],
Iohvidov and Kreĭn [13], Naĭmark [20], and
others.
On Groups with Chain Conditions
Authors: Amberg Bernhard
Published: December 2010

Our aim in this note is to generalize results of Baer in [3;
5]. In § 1 an arbitrary formation n is considered,
the key result being Proposition 1.5. This is applied in § 2 to characterize
various finiteness conditions, for example the classes of groups with
maximum [minimum] condition on subgroups, subnormal subgroups, and normal
subgroups respectively, or the class of (not necessarily soluble)
polyminimax groups (see Theorems 2.1 and 2.6). These results may also be
regarded as generalizations of the well-known theorem of Malcev-Baer that a
radical group satisfies the maximum condition [is a polyminimax group] if
all its abelian subgroups satisfy the maximum condition [are minimax
groups].
A Remark on the Proof of a Theorem of Laufer and Tomber
Authors: Myung Hyo Chul
Published: December 2010

In this note we give a correction to the proof of the following theorem [1, Theorem 2].

THEOREM. Letbe a flexible, power-associative algebra, over an arbitrary algebraically closed field Ω of characteristic 0. Ifis a simple Lie algebra, thenis a simple Lie algebra isomorphic to.

Step (i) of the proof, which proves that the Cartan subalgebra of is a nil subalgebra of , is incomplete. Assuming that is not a nil subalgebra of , there exists an idempotent e ≠ 0 in .
On Reductive Lie Admissible Algebras
Authors: Sagle Arthur A.
Published: December 2010

A Lie admissible algebra is a non-associative algebra A such that A− is a Lie algebra where A− denotes the anti-commutative algebra with vector space A and with commutation [X, Y] = XY – YX as multiplication; see [1; 2; 5]. Next let L−(X): A− → A−: Y → [X, Y] and H = {L−(X): X ∊ A−}; then, since A− is a Lie algebra, we see that H is contained in the derivation algebra of A− and consequently the direct sum g = A − ⊕ H can be naturally made into a Lie algebra with multiplication [PQ] given by: P = X + L−(U), Q = Y + L−(V) ∊ g, then

and note that for any P, [PP] = 0 so that [PQ] = −[QP] and the Jacobi identity for g follows from the fact that A− is Lie.
Maximal Abelian Subgroups of the Symmetric Groups
Authors: Dixon John D.
Published: December 2010

Our aim is to present some global results about the set of maximal abelian subgroups of the symmetric group Sn. We shall show that certain properties are true for “almost all” subgroups of this set in the sense that the proportion of subgroups which have these properties tends to 1 as n → ∞. In this context we consider the order and the number of orbits of a maximal abelian subgroup and the number of generators which the group requires.

Earlier results of this kind may be found in the papers [1; 2; 3; 4; 5]; the papers of Erdös and Turán deal with properties of the set of elements of Sn. The present work arose out of a conversation with Professor Turán who posed the general problem: given a specific class of subgroups (e.g., the abelian subgroups or the solvable subgroups) of Sn, what kind of properties hold for almost all subgroups of the class?
On Additive Operators
Authors: Friedman N. A. , Tong A. E.
Published: December 2010

Representation theorems for additive functional have been obtained in [2, 4; 6-8; 10-13]. Our aim in this paper is to study the representation of additive operators.

Let S be a compact Hausdorff space and let C(S) be the space of real-valued continuous functions defined on S. Let X be an arbitrary Banach space and let T be an additive operator (see § 2) mapping C(S) into X. We will show (see Lemma 3.4) that additive operators may be represented in terms of a family of “measures” {μh} which take their values in X**. If X is weakly sequentially complete, then {μh} can be shown to take their values in X and are vector-valued measures (i.e., countably additive in the norm) (see Lemma 3.7). And, if X* is separable in the weak-* topology, T may be represented in terms of a kernel representation satisfying the Carathéordory conditions (see [9; 11; §4]):
Homeomorphic Sets of Remote Points
Authors: Woods R. Grant
Published: December 2010

Let X be a completely regular Hausdorff space, and let βX denote the Stone-Čech compactification of X. A point p ∈ βX is called a remote point of βX if p does not belong to the βX-closure of any discrete subspace of X. Remote points were first defined and studied by Fine and Gillman, who proved that if the continuum hypothesis is assumed then the set of remote points of βR((βQ) is dense in βR – R(βQ – Q ) (R denotes the space of reals, Q the space of rationals). Assuming the continuum hypothesis, Plank has proved that if X is a locally compact, non-compact, separable metric space without isolated points, then βX has a set of remote points that is dense in βX – X. Robinson has extended this result by dropping the assumption that X is separable.
A Best Possible Tauberian Theorem for the Collective Continuous Hausdorff Summability Method
Authors: Peterson G. E.
Published: December 2010

The purpose of this paper is to prove that o(l/x) is the best possible Tauberian condition for the collective continuous Hausdorff method of summation. The analogue of this result for the collective (discrete) Hausdorff method is known [1, pp. 229, ff.; 7, p. 318; 8, p. 254]. Our method involves generalizing a well-known Abelian theorem of Agnew [2] to locally compact spaces and then applying the analogue for integrals of a result Lorentz obtained for series [6, Theorem 1].

Let T and X denote locally compact, non compact, σ-compact Hausdorff spaces. Let T′ = T ∪ (∞) and X′ = X ∪ (∞) denote the onepoint compactifications of T and X, respectively. Let B(T) denote the set of locally bounded, complex valued Borel functions on T and let B∞(T) denote the bounded functions in B(T).
Characterizations of Modules
Authors: Fieldhouse David J.
Published: December 2010

In this paper we use the Bourbaki [2] conventions for rings and modules. All rings are associative but not necessarily commutative and have a 1; all modules are unital.

Bass [1] calls a ring A left perfect if and only if every left A -module has a projective cover, which he shows is equivalent to every flat left A -module being projective. Bass calls a ring A semi-perfect if and only if every finitely generated module has a projective cover and shows that this concept is leftright symmetric.

We will define a ring A to be quasi-perfect if and only if every finitely generated flat left A -module is projective.

An exercise [6, Exercise 10, p. 136] is given by Lambek to show that every semi-perfect ring is quasi-perfect.
On Primitive Solvable Linear Groups
Authors: Wright C. R. B.
Published: December 2010

Let V be a vector space over the field K. A group G of K-linear transformations of V onto itself is primitive in case no proper nontrivial subspace of V is G-invariant and V cannot be written as a direct sum of proper subspaces permuted among themselves by G. Equivalently, G is primitive on V in case G is irreducible and is not induced from a proper subgroup.

Suprunenko showed [3, Theorem 12, p. 28] that the n-dimensional general linear group GL(n, K) has a solvable primitive subgroup only if

(1) there is a divisor, m, of n such that K has an extension field of degree m containing a primitive p-th root of 1 for each prime p dividing n/m.

The main result of this note is the converse fact.
Multilinear Functions of Row Stochastic Matrices
Authors: Pierce Stephen
Published: December 2010

In the study of inequalities, the cases of equality are often the most difficult and interesting part. The case of equality is, in some sense, a measure of the tightness of the inequality. In this paper, we generalize two inequalities of Brualdi and Newman [1, Theorems 3, 4], but the instances of equality are probably more interesting because of the variety of cases which can occur.

Let A = (aij) be an n × n matrix. Define the permanent of A by

We say that A is row stochastic if all entries are non-negative and all row sums are 1. In [1], several inequalities involving permanents of row stochastic matrices were proved. In two of these results, the case of equality was not determined. We will generalize both of these results to a class of functions which includes the permanent, and determine all cases of equality.
A Dual View of the Clifford Theory of Characters of Finite Groups
Authors: Roth Richard L.
Published: December 2010

Let G be a finite group, K a normal subgroup of G, χ an irreducible complex character of G. In the usual decomposition of χ|κ, using Clifford's theorems, G/K is seen to operate by conjugation on the irreducible characters of K and if σ is an irreducible component of χ|κ, then I(σ) the inertial group of σ, plays an essential role as an appropriate intermediate subgroup for the analysis. In this paper we consider the case where G/K is abelian and study the action of the dual group (G/K)^ (of linear characters of G/K) on the irreducible characters of G effected by multiplication. This action appears to be related in a dual way to the action of G/K on the characters of K. We define a subgroup J(χ) of G which plays a role similar to that of I (σ) and which we call the dual inertial group of χ.
Obstructions to Liftings in Commutative Squares
Authors: Pressman Irwin S.
Published: December 2010

A commutative square (1) of morphisms is said to have a lifting if there is a morphism λ: B1 → A2 such that λϕ1 = α and ϕ2λ = β

1

Let us assume that we are working in a fixed abelian category . Therefore, ϕi will have a kernel “Ki” and a cokernel “Ci” for i = 1, 2. Let k : K1 → K2 and c: C1 → C2 denote the canonical morphisms induced by α and β.

We shall construct a short exact sequence (s.e.s.)

2

using the data of (1). We shall prove that (1) has a lifting if and only if k = 0, c = 0, and (2) represents the zero class in Ext1(C1, K2). Furthermore, if (1) has one lifting, then the liftings will be in one-to-one correspondence with the elements of the set |Hom(G1, K2)|.
Isomorphic Subgroups of Finite p-Groups. II
Authors: Glauberman George
Published: December 2010

Suppose that we are given an isomorphism ϕ between two subgroups of index p in a finite p-group P. Let N(ϕ) be the largest subgroup of P fixed by ϕ. By a result of Sims [2, Proposition 2.1], n(ϕ) is a normal subgroup of P. In [2], we showed that P/N(ϕ) has nilpotence class at most two if p = 2, and at most three if p is odd. We then applied this result to investigate certain cases of the following question. Suppose that P is contained in a finite group G and that some subgroup of index p in P is a normal subgroup of G. Let α be an automorphism of P. Then, does α fix some nonidentity normal subgroup of P that is normal in G?

In this paper, we consider characteristic subgroups of P rather than normal subgroups.
On Screenability and Metrizability of Moore Spaces
Authors: Reed G. M.
Published: December 2010

After showing that each screenable Moore space is pointwise paracompact and that the converse is not true, Heath in [4] asked for a necessary and sufficient condition for a pointwise paracompact Moore space to be screenable. In [12], Traylor asked for a necessary and sufficient condition for a pointwise paracompact Moore space to be metrizable. It is the purpose of this paper to provide such conditions, and to establish relationships between those conditions and metrization problems in Moore spaces.

A Moore space S is a space (all spaces are T1) in which there exists a sequence G = (G1, G2, …) of open coverings of S, called a development, which satisfies the first three parts of Axiom I in [7]. The statement that a collection H of subsets of the space S is point finite (point countable) means that no point of S belongs to infinitely (uncountably) many elements of H.
Compact Induced Representations
Authors: Busby Robert C. , Schochetman Irwin
Published: December 2010

In [15; 16; 17], Horst Leptin introduced what he called generalized group algebras. These Banach *-algebras are formed by letting a locally compact group G act on a Banach *-algebra A both by *-automorphisms and by a cocycle with values in the multiplier algebra, M (A ), of A. We will review the precise construction later, but for now we remark that examples include the group algebra of a group extension, the covariance algebras of quantum field theory, the “projective group algebras” of a group G (that is, for each complex-valued cocycle λ, called a multiplier in the literature, the Banach *-algebra whose nondegenerate *-representations are in bijective correspondence with the λ-projective representations of G), and the twisted group algebras of Edwards and Lewis [8; 9].
Conjugate p-Subgroups of Finite Groups
Authors: Currano John J.
Published: December 2010

Throughout this paper, let p be a prime, P be a p-group of order pt , and ϕ be an isomorphism of a subgroup R of P of index p onto a subgroup Q which fixes no non-identity subgroup of P, setwise. In [2, Lemma 2.2], Glauberman shows that P can be embedded in a finite group G such that ϕ is effected by conjugation by some element g of G. We assume that P is thus embedded. Then Q = P ∩ Pg. Let H = 〈P,Pg〉 and V = [H,Z(Q)], so Q ⊲ H and V ⊲ H.

Let E(p) be the non-abelian group of order p3 which is generated by two elements of order p. Then E(p) is dihedral if p = 2 and has exponent p if p is odd. If p is odd, then E* (p) is defined in § 2 to be a particular group of order p6 and nilpotence class three.
Extending Jordan Ideals and Jordan Homomorphisms of Symmetric Elements in a Ring with Involution
Authors: Smith Kirby C.
Published: December 2010

In this work, we show how the ideas in [3, pp. 6-12] can be used to give conditions under which Jordan ideals in the set of symmetric elements in an associative ring R with involution extend to associative ideals of R in a natural way. We also give conditions under which a Jordan homomorphism of the set of symmetric elements will extend to an associative homomorphism of R. Such work has been done on matrix rings with involution in [5; 6]. An abstract definition of a Jordan ring may be found in [3] as well as other background information.
Perturbation Theorems for Relative Spectral Problems
Authors: Hughes Edward
Published: December 2010

Eigenvalue problems of the form Af = λBf, where λ is a complex parameter and A and B are operators on a Hilbert Space, have been considered by a number of authors (e.g., [1; 3; 5; 7; 10]). In this paper, we shall be concerned with the existence and nature of eigenfunction expansions associated with such problems, with no assumptions of self-adjointness. The form of the theorems to be given here is: if the system (A, B) is spectral and complete (definitions below), and F and G are operators satisfying certain “smallness” conditions, then (A + F, B + G) is also spectral and complete. The hypotheses for these theorems are chosen with an eye to applying the results to boundary-value problems on a compact interval. Such applications, together with an examination of circumstances under which the system (Dn, Dm) (D denoting differentiation) is spectral and complete under a broad class of boundary conditions, will be made in a later paper.
Injectivity in Equational Classes of Algebras
Authors: Day Alan
Published: December 2010

The concept of injectivity in classes of algebras can be traced back to Baer's initial results for Abelian groups and modules in [1]. The first results in non-module types of algebras appeared when Halmos [14] described the injective Boolean algebras using Sikorski's lemma on extensions of Boolean homomorphisms [19]. In recent years, there have been several results (see references) describing the injective algebras in other particular equational classes of algebras.

In [10], Eckmann and Schopf introduced the fundamental notion of essential extension and gave the basic relations that this concept had with injectivity in the equational class of all modules over a given ring. They developed the notion of an injective hull (or envelope) which provided every module with a minimal injective extension or equivalently, a maximal essential extension. In [6] and [9], it was noted that these relationships hold in any equational class with enough injectives.
Limit Point Criteria for Differential Equations
Authors: Hinton Don
Published: December 2010

For certain ordinary differential operators L of order 2n, this paper considers the problem of determining the number of linearly independent solutions of class L2[a, ∞) of the equation L(y) = λy. Of central importance is the operator

0.1

where the coefficients pi are real. For this L, classical results give that the number m of linearly independent L2[a, ∞) solutions of L(y) = λy is the same for all non-real λ, and is at least n [10, Chapter V]. When m = n, the operator L is said to be in the limit-point condition at infinity. We consider here conditions on the coefficients pi of L which imply m = n. These conditions are in the form of limitations on the growth of the coefficients.