bluuebunny/crossref_metadata_2025 huggingface dataset:
Logarithmic Capacity of Sets and Double Trigonometric Series
Authors: Shapiro V. L.
Published: December 2010

It is the purpose of this paper to establish a closer connection between the logarithmic capacity of sets and double trigonometric series. In (9), closed sets of logarithmic capacity zero were established as sets of uniqueness for a particular class of double trigonometric series under circular (C, 1) summability. By slightly changing this class of series but still maintaining closed sets of logarithmic capacity zero as sets of uniqueness, it is shown in this paper that closed sets of positive logarithmic capacity form sets of multiplicity.
Combinatorial Relations and Chromatic Graphs
Authors: Greenwood R. E. , Gleason A. M.
Published: December 2010

1. Introduction. The following elementary logical problem was a question in the William Lowell Putnam Mathematical Competition held in March 1953 (1): Six points are in general position in space (no three in a line, no four in a plane). The fifteen line segments joining them in pairs are drawn, and
then painted, some segments red, some blue. Prove that some triangle has all its sides the same color.
Remarks On Finite Groups Defined By Generating Relations
Authors: Frucht Robert
Published: December 2010

1. Introduction. After establishing aduplication principle(§2) which enables us to derive a group of order 2h with k + 1 involutory generators from any group of order h with k generators, we shall prove the followingcombination principle :

Letandbe two groups, each havingkgenerators, such that the rth generators ofandhave relatively prime periods for each r from 1 to k.
A Class Of Permutation Groups Of Prime Degree
Authors: Fryer K. D.
Published: December 2010

In (1), using the theory of group representations, Brauer studied groups of finite order g containing elements A of prime period p which commute only with their own powers Ai. If is a p-Sylow subgroup of , the normalizer can be generated by A and another element B such that

1.1,

where ϒ is a primitive root (mod p), and t and q are positive integers such
that

1.2 tq = p − 1.
Set-Transitive Permutation Groups
Authors: Beaumont R. A. , Peterson R. P.
Published: December 2010

The concept of an s-ply transitive (1 ≤ s ≤ n) permutation group on n symbols is of considerable importance in the classical theory of finite permutation groups, which was in the height of its development in the period around the turn of the century. The obvious generalization to a permutation group which is s set-transitive (i.e., a group which, for each pair of s-element unordered subsets S, T of the given n symbols, contains a permutation which carries S into T) seems to have received little attention.
The Existence of a Distribution Function for an Error Term Related to the Euler Function
Authors: Erdös Paul , Shapiro H. N.
Published: December 2010

The average order of the Euler function ϕ(n), the number of integers less than n which are relatively prime to n, raises many difficult and still unanswered questions. Thus, for

1.1,

and

1.2,

it is known that R(x) = O(x log x) and H(x) = O(log x). However, though these results are quite old, they were not improved until recently.
On Explicit Bounds In Schottky's Theorem
Authors: Jenkins J. A.
Published: December 2010

1. Introduction. To Schottky is due the theorem which states that a function F(Z), regular and not taking the values 0 and 1 in |Z| < 1 and for which F(0) = a0, is bounded in absolute value in |Z| ≤ r, 0 ≤ r < 1, by a number depending only on a0 and r. Let K(a0 r) denote the best possible bound in this result. Various authors have dealt with the problem of giving an explicit
estimate for this bound.
On An Inversion Formula For The Laplace Transformation
Authors: Rooney P. G.
Published: December 2010

1. Introduction. The literature of the Laplace transformation contains many examples of inversion operators. Particular attention has been given in this literature to the so-called “real” inversion operators, that is, those operators which make use of values of the generating function arising only from real values of the independent variable.
On The Potential Theory Of Coclosed Harmonic Forms
Authors: Duff G. F. D.
Published: December 2010

1. Introduction. The potential theory of real harmonic tensors, which was first studied by Hodge (5), offers a variety of problems by no means all of which have yet been examined. In the present paper there are formulated the solutions of some boundary value problems for the Poisson equations
associated with coclosed harmonic forms. These problems include as special cases a number of previous results on coclosed harmonic forms and harmonic fields.
The Group Ring Of a Class Of Infinite Nilpotent Groups
Authors: Jennings S. A.
Published: December 2010

Introduction. In this paper we study the (discrete) group ring Γ of a finitely generated torsion free nilpotent groupover a field of characteristic zero. We show that if Δ is the ideal of Γ spanned by all elements of the form G − 1, where G ∈, then

and the only element belonging to Δwfor all w is the zero element (cf. (4.3) below).
A Theorem Concerning Three Fields
Authors: Herstein I. N.
Published: December 2010

Several authors (1; 2; 3; 4; 5; 6) have recently studied the existence and non-existence of certain types of extensions of a given field. In this note we prove a theorem closely related to these results which, in a sense, contains essential portions of each of these.
Weak Compactness and Vector Measures
Authors: Bartle R. G. , Dunford N. , Schwartz J.
Published: December 2010

Introduction. It is the purpose of this paper to develop a Lebesgue theory of integration of scalar functions with respect to a countably additive measure whose values lie in a Banach space. The class of integrable functions reduces to the ordinary space of Lebesgue integrable functions if the measure is scalar valued. Convergence theorems of the Vitali and Lebesgue type are valid in the general situation. The desirability of such a theory is indicated by recent developments in spectral theory.
An Application Of Some Spaces Of Lorentz
Authors: Rooney P. G.
Published: December 2010

The spaces Δ (α) and M(α) were defined by Lorentz (2) as follows. Let 0 < α < l , 0 < l ≤ ∞ ; let ϕ be measurable on (0,l), and, in case l = ∞, let the set where |ϕ(x)| > ∊ have finite measure for each positive ∊.
Pseudo-Regularity
Authors: Divinsky Nathan
Published: December 2010

Introduction. An element x is said to be right-quasi-regular (r.q.r.) if there exists an element y such that x + y + xy = 0. This concept had its inception in the fact that (for rings with unity) if 1 + x has an inverse, written as 1 + y, then (1 + x)(l + y) = 1, x + y + xy = 0. Thus in rings without unity elements it seemed (1; 3; 12) profitable to consider this latter equation.
On The Coordinatization Theorem Of J. Von Neumann
Authors: Fryer K. D. , Halperin Israel
Published: December 2010

A classical theorem of n-dimensional projective geometry asserts: if Desargues' theorem holds (in particular, if n ≥ 3) then the points of the geometry can be assigned homogeneous coordinates (α1, … , αn+l) with αi not all zero in a suitable division ring (2, p. 104; 3, Theorems 10, 11, ex. 19, p. 204).
Subspaces Of Riemannian Spaces
Authors: Blum Richard
Published: December 2010

In this paper, results obtained by the author for Riemannian Spaces Vn imbedded in Euclidean Spaces EN (3; 4) are extended to Vn imbedded in VN.

The first section is introductory. In §2 the general result is obtained. This is the establishment of a certain dependency among the three basic sets of equations of the Vn with respect to the VN, namely the equations of Gauss, Codazzi and Kuehne.
Induced and Produced Modules
Authors: Higman D. G.
Published: December 2010

We shall consider here two generalizations to rings of the concept of induced representation as it occurs in the representation theory of finite groups (6).

If A is a ring, S a subring of A, we shall associate with each S-module M an induced pair (I(M), K) consisting of an A -modulo I(M) and an S-homomorphism K: M → I(M).
Some Results On The Asymptotic Behavior Of Linear Systems
Authors: Marcus M.
Published: December 2010

1. Introduction. We consider first in §2 the asymptotic behavior as t → ∞ of the solutions of the vector-matrix differential equation

(1.1) ,

where A is a constant n-square complex matrix, B{t) a continuous complex
valued n-square matrix defined on [0, ∞ ), and x a complex n-vector.
Countable Paracompactness and Souslin's Problem
Authors: Ellen Rudin Mary
Published: December 2010

1. Introduction. A linearly ordered space S in which neighborhoods are
segments is called a Souslin space if

(i) S is not separable, but

(ii) every collection of disjoint segments of S is countable.

Whether a Souslin space exists is not known; this is the problem referred to in the title and was proposed by Souslin in (2).
Sur La Forme Hermitienne Canonique Des Espaces Homogenes Complexes
Authors: Koszul J. L.
Published: December 2010

Introduction. Soit G un groupe de Lie connexe et soit B un sous-groupe fermé de G. Supposons que la variété homogène G/B ou G opère transitivement à gauche soit munie d'une structure complexe invariante par les opérations de G, et qu'il existe sur G/B un volume invariant par G. Ce volume étant unique à un facteur constant près, la forme hermitienne de G/B qui est canoniquement associée au volume et à la structure complexe se trouve entièrement déterminée par la structure complexe de G/B et le groupe transitif G (5).
On Polynomial Extensions Of Rings
Authors: Yoshida Michio
Published: December 2010

Let A be a commutative ring with unit element, and let A [x] be a ring of polynomials in an indeterminate x with coefficients in A. There are a number of well-known properties which A shares with A [x]. We shall state one of them in the following.
A Theorem On Zariski Rings
Authors: Yoshida Michio
Published: December 2010

Given a Noetherian ring A with unit element and an ideal m of A such that

,

we may topologize A by adopting {mn; n = 1, 2, …} as a fundamental system
of neighborhoods of 0. This topologized ring is usually referred to as an m-adic ring, and is called a Zariski ring if its ideals are all closed.
Entire Solutions Of The Functional Equationf(f(z)) = g(z)
Authors: Thron W. J.
Published: December 2010

In this note it is proved that:the functional equation

(1)f(f(z)) = g(z),

where g{z) is an entire function of finite order, which is not a polynomial, and which takes on a certain value p only a finite number of times, does not have a solution f(z) which is an entire function.
On Disconjugate Differential Systems
Authors: Hartman Philip , Wintner Aurel
Published: December 2010

1. Introduction. Let F, G (and all capital letters to be used below) denote n by n matrices the elements of which are real-valued continuous functions on an interval a ≤t ≤ b. Correspondingly, by a solution x = x(t) of a differential system

1,

with det G(t) ≠ 0, or of a differential system

2,

will be meant a (vector) solution all n components of which are real-valued.
Harmonic and Analytic Functions Of Several Variables and The Maximal Theorem Of Hardy and Littlewood
Authors: Rauch H. E.
Published: December 2010

The present paper, an edited excerpt from my dissertation, arose from the suggestion of S. Bochner that I try to extend the maximal theorem of Hardy and Littlewood

(2) to functions analytic in the solid unit hypersphere
An Extension Problem For Functions With Monotonic Derivatives
Authors: Jurkat W. B.
Published: December 2010

Introduction. This paper deals with questions of the following type. Problem (A) : Let F(x) be the nth integral of a positive non-decreasing function for all large positive x, the problem is to find a function f(x), being the nth integral of a non-decreasing function for all x (— ∞ < x < ∞), with the property

A.
On a Quasi-Linear Equation
Authors: Bellman Richard
Published: December 2010

1. Introduction. The purpose of this note is to establish some limit theorems for the non-linear recurrence relations

1.1, i= 1,2, …, N; n ≥ 0

under certain assumptions concerning the initial values ci = xi(0), and the
coefficient matrices A (q) = (aij(q)).
Modified Boundary Value Problems For a Quasi-Linear Elliptic Equation
Authors: Duff G. F. D.
Published: December 2010

1. Introduction. The quasi-linear elliptic partial differential equation to be studied here has the form

(1.1) Δu = − F(P,u).

Here Δ is the Laplacian while F(P,u) is a continuous function of a point P
and the dependent variable u. We shall study the Dirichlet problem for (1.1) and will find that the usual formulation must be modified by the inclusion of a parameter in the data or the differential equation, together with a further numerical condition on the solution.
Asymptotic Expansions
Authors: Moser Leo , Wyman Max
Published: December 2010

1. Introduction. Let a1 a2, …, am be a set of real non-negative numbers and let

1.1 P(x) = a1x + a2x2 + … + amxm (am ≠ 0).

Many combinatorial problems can be reduced to the study of numbers Bn generated by

1.2.

Some problems of this type were treated by Touchard (7), Jacobsthal (3), Chowla, Herstein, Moore and Scott (1; 2), and the present authors (4).
Nombres Exponentiels Et Nombres De Bernoulli
Authors: Touchard Jacques
Published: December 2010

Introduction. Les nombres entiers positifs a0, a1, … , an, … définis par la fonction génératrice

et que l'on appelle nombres exponentiels jouent, pour la sommation de certaines séries, un rôle qui rappelle le rôle sommatoire des nombres de Bernoulli. Nous avons rassemblé ici les principales propriétés des nombres an dont plusieurs sont, croyons nous, nouvelles.
On the Theory of Ring-Logics
Authors: Yaqub Adil
Published: December 2010

Boolean rings (B, ✗, + ) and Boolean logics ( = Boolean algebras) (B, ∩, *) are equationally interdefinable in a familiar way (6). Foster's theory of ring-logics (1; 2; 3) raises this interdefinability and indeed the entire Boolean theory to a more general level. In this theory a ring (or an algebra) R is studied modulo K, where K is an arbitrary transformation group (or “Coordinate transformations”) in R.
Convex Hulls Of Simple Space Curves
Authors: Derry Douglas
Published: December 2010

The convex hull of an arbitrary setMin real Euclideann-space is known to consist of all the points within ther-simplexes withr+ 1 vertices fromM, r ≤ n.This note shows that ifMis specialized to be a curveAnof real ordern, then its convex hull consists of all the points within ther-simplexes withr+ 1 vertices onAn, n = 2r + 1 or n = 2r.
Note On a Stieltjes Type Of Inversion
Authors: Porcelli Pasquale
Published: December 2010

If F(z) is an analytic function for z ∉ [− ∞, −l],g(t) of bounded variation and real valued for 0 ≤ t ≤ 1 and

then the Stieltjes type of inversion between F(z) and g(t) (cf. 1, p. 339, Theorem 7a) is

where 0 ≤ v < u ≤ 1, ImF(z) is the imaginary part of F(z) and z = − t−l+iy.
On The Zeros Of Solutions Of Second-Order Linear Differential Equations
Authors: Beesack P. R. , Schwarz Binyamin
Published: December 2010

Introduction. In §1 of this paper we consider the complex differential equation

1, |z| < 1,

where q(z) is a regular function in the open unit circle. We shall give a lower bound for the non-Euclidean distance of any pair of zeros of any non-trivial (i.e., not identically zero) solution u(z) of (1).
Relative Cohomology
Authors: Higman D. G.
Published: December 2010

It is our purpose in this paper to present certain aspects of a cohomology
theory of a ring R relative to a subring
S, basing the theory on the notions of induced and produced
pairs of our earlier paper (2), but making the paper self-contained except
for references to a few specific results of (2). The cohomology groups
introduced occur in dual pairs. Generic cocycles are defined, and the groups
are related to the protractions and retractions of
R-modules.
A Minimum-Maximum Problem for Differential
Expressions
Authors: Carter D. S.
Published: December 2010

In the study of approximate methods for solving ordinary differential
equations, an interesting question arises. To state it roughly for a single
first order expression, let y0(t) be the solution of the equation

(1.1)

which satisfies the initial condition y(a)
= na. Let nb be an approximation to the value of
y0 at a later time, t = b.
Hyperbolic Mixed Problems for Harmonic Tensors
Authors: Duff G. F. D.
Published: December 2010

This paper may be regarded as a sequel to (1), where the initial value or
Cauchy problem for harmonic tensors on a normal hyperbolic Riemann space was
treated. The mixed problems to be studied here involve boundary conditions
on a timelike boundary surface in addition to the Cauchy data on a spacelike
initial manifold. The components of a harmonic tensor satisfy a system of
wave equations with similar principal part, and we assign two initial
conditions and one boundary condition for each component.
Some Generating Functions for Laguerre and Hermite
Polynomials
Authors: Brafman Fred
Published: December 2010

The Laguerre polynomials Ln(α)(x) have the following hypergeometric and Rodrigues
representation (3, pp. 188, 189):

1,

2.
A Simple Algorithm for Finding Maximal Network Flows and an
Application to the Hitchcock Problem
Authors: Ford L. R. , Fulkerson D. R.
Published: December 2010

The network-flow problem, originally posed by T. Harris of
the Rand Corporation, has been discussed from various viewpoints in
(1; 2; 7; 16). The problem arises naturally in the study of
transportation networks; it may be stated in the following way. One is given
a network of directed arcs and nodes with two distinguished nodes, called
source and sink, respectively. All other nodes are called
intermediate. Each directed arc in the network has
associated with it a nonnegative integer, its flow capacity. Source arcs may
be assumed to be directed away from the source, sink arcs into the sink.
Subject to the conditions that the flow in an arc is in the direction of the
arc and does not exceed its capacity, and that the total flow into any
intermediate node is equal to the flow out of it, it is desired to find a
maximal flow from source to sink in the network, i.e., a flow which
maximizes the sum of the flows in source (or sink) arcs.

Thus, if we let P1 be the source, Pn the sink, we are required to find xij (i,j =1, . . . ,
w) which maximize
Abstract Definitions for Reflection Groups
Authors: Shephard G. C.
Published: December 2010

This paper is an appendix to “Finite unitary reflection groups” by J. A.
Todd and the author (3) in which the irreducible finite groups generated by
reflections in unitary space (2, p. 364) are enumerated and abstract
definitions are given for all those n-dimensional groups
which may be generated by n reflections.
On the Maximum Principle of Ky Fan
Authors: Marcus M. , Moyls B. N.
Published: December 2010

In 1951 Fan (1) proved the following interesting extreme value result:
Let A1, … , Am be completely continuous
operators on a Hilbert space . For
be the
characteristic roots of Aσ* Aσ. Then, for any positive integer
k,

(1),

(2),

where both maxima are taken over all unitary operators
U1, … , Um and all sets of k orthonormal (o.n.)
vectors.
Certain Groups of Orthonormal Step Functions
Authors: Price J. J.
Published: December 2010

It was first pointed out by Fine (2), that the Walsh functions are
essentially the characters of a certain compact abelian group, namely the
countable direct product of groups of order two. Later Chrestenson (1)
considered characters of the direct product of cyclic groups of order α (α =
2, 3, …). In general, his results show that the analytic properties of these
generalized Walsh functions are basically the same as those of the ordinary
Walsh functions.
On Countably Paracompact Normal Spaces
Authors: Mansfield M. J.
Published: December 2010

A. H. Stone (9), E. Michael (3, 4), J. L. Kelley and J. S. Griff en (2) have
established many necessary and sufficient conditions that a regular
Hausdorff space be paracompact. It is the purpose of this note to show that
if the word “countable” is inserted in the appropriate places in the
above-mentioned conditions they become, in general, necessary and sufficient
conditions that a normal space be countably paracompact.
A Condition for the Commutativity of Rings
Authors: Herstein I. N.
Published: December 2010

A well-known theorem of Jacobson (1) asserts that if every
element a of a ring A satisfies a relation
an(a) = a where n(a) > 1 is
an integer, then A is a commutative ring. Thus the condition used in
Jacobson's theorem is a sufficient condition for commutativity. However the
condition is by no means a necessary one, as it is satisfied by a very
restricted class of commutative rings.
On the Structure of Frobenius Groups
Authors: Feit Walter
Published: December 2010

Let G be a group which has a faithful representation as a
transitive permutation group on m letters in which no permutation other than
the identity leaves two letters unaltered, and there is at least one
permutation leaving exactly one letter fixed. It is easily seen that if
G has order mh, a necessary and sufficient condition for
G to have such a representation is that
G contains a subgroup H of order
h which is its own normalizer in G and
is disjoint from all its conjugates. Such a group G is
called a Frobenius group of type (h, m).
Network Flow and Systems of Representatives
Authors: Ford L. R. , Fulkerson D. R.
Published: December 2010

The theory developed for the study of flows in networks (2; 3; 4; 5; 6; 7)
sometimes provides a useful tool for dealing with certain kinds of combinatorial
problems, as has been previously indicated in (3; 4; 6; 7). In particular,
Hall-type theorems for the existence of systems of distinct representatives which
contain a prescribed set of marginal elements (10; 11), or, more generally, whose
intersection with each member of a given partition of the fundamental set has a
cardinality between prescribed lower and upper bounds (9), can be obtained in this
way (7).
On the Existence of the Burkill Integral
Authors: Kober H.
Published: December 2010

The main problem of the present paper is the existence of the Burkill
integral of an interval function
ƒ(I) which is not
supposed to be continuous. Little is known about this case, though otherwise the
theory of the integral can be considered as complete: we may refer to Ringenberg's
comprehensive paper (2) in which further references are given.
On Generalized Averaging Operators
Authors: Boas R. P.
Published: December 2010

Let

Sumner (4) has discussed

for arbitrary real λ and h,
whereƒ(Z) is an entire function of exponential type

I shall show that in this case an alternative definition of
∇λλ, which leads to Sumner's results more quickly, is
equivalent to Sumner's. (However, Sumner's definition is, in principle, applicable
to a wider class of functions.)
A New Proof of a Theorem of Lehmer
Authors: Roberts J. B.
Published: December 2010

In 1851 Prouhet (2) stated that any b
k+1 consecutive positive integers
(b a positive integer ⩾ 2) can be separated into
b sets
o, …,
b-1 each with b

k
members in such a way that
where
designates the
sum of the tth powers of the numbers in
j. In 1947 Lehmer (1) generalized this result.
Central Limit Theorems for Interchangeable Processes
Authors: Blum J. R. , Chernoff H. , Rosenblatt M. , Teicher H.
Published: December 2010

Let {Xn} (n = 1, 2 , …)
be a stochastic process. The random variables comprising it or the process itself
will be said to be interchangeable if, for any choice of distinct positive
integers i
1, i
2, H
3 … , ik, the joint distribution
of

depends merely on k and is independent of the
integers i
1, i
2, … , i
k. It was shown by De Finetti (3) that the probability
measure for any interchangeable process is a mixture of probability measures of
processes each consisting of independent and identically distributed random
variables.
Some Generalizations of the Problem of Distinct Representatives
Authors: Mendelsohn N. S. , Dulmage A. L.
Published: December 2010

If S
1, S
2, S
3, … , S
n are subsets of a set M then it is known that a necessary
and sufficient condition that it is possible to choose representatives at such
that ai is in S
i for (i = 1, 2, 3, … ,
n) and such that ai ≠
aj for i ≠ j , is that for k =
1, 2, 3, … , n, the union of any k of the
sets S1, S
2, … , S
n, contains at least k elements. The theorem has a number of
consequences amongst which we list the following.
Dimension of Ideals in Polynomial Rings
Authors: Auslander Maurice , Rosenberg Alex
Published: December 2010

A well-known theorem asserts that if K is a field,
a prime ideal in
the polynomial ring S =
K[X1, …
Xn] and d the transcendence degree of
S / over K

n = rank +
d.

In the first half of this paper we extend this result to the case of
arbitrary commutative noetherian K, as well as giving a
purely homological proof of the classical theorem. In the second half we use our
first result to compute the analogue of the dimension of the product and
intersection of two affine varieties when K is a
Dedekind ring.
On Conjugates in Division Rings
Authors: Faith Carl C.
Published: December 2010

Let D be a non-commutative division ring with
centre C, and let Δ be a proper division subring not
contained in C. In (4) Cartan raised the question: is
it ever possible for each inner automorphism of D to
induce an automorphism of Δ? As is well-known, Cartan (4, Théorème 4) with the aid
of his Galois Theory answered this negatively in case D
is a finite dimensional division algebra. Later Brauer (3), and Hua (8), using
elegant, elementary methods, extended Cartan's theorem to arbitrary division
rings.
Congruence Representations in Algebraic Number Fields II. Simultaneous Linear
and Quadratic Congruences
Authors: Cohen Eckford
Published: December 2010

Let ƒ and λ be positive integers and p a positive
odd prime. Suppose further that P is an ideal of norm
p
f in a finite extension F of the rational field. In (2),
which will also be referred to as I in the present paper, we obtained the number
of solutions Ns(m) of the
quadratic congruence,
Analytic Equivalence of Algebroid Curves
Authors: Wallace Andrew H.
Published: December 2010

Let k be an algebraically closed field and let x1, x2, . . . , xn be indeterminates. Denote by Rn the ring k[[x1, x2, … , xn]] of power series in the xi With coefficients in the field k. Let and be two ideals in this ring. Then and will be said to be analytically equivalent if there is an automorphism T of Rn such that T() = . and will be called analytically equivalent under T.

The above situation can be described geometrically as follows. The ideals and can be regarded as defining algebroid varieties V and V' in (x1, x2, … , xn)-space, and these varieties will be said to be analytically equivalent under T.

The automorphism T can be expressed by means of equations of the form :

where the determinant is not zero and the fi are power series of order not less than two (that is to say, containing terms of degree two or more only).
Involutions Associated with the Burkhardt Configuration in [4]
Authors: Horadam A. F.
Published: December 2010

Horadam (11) has established the existence of a locus L in [8] (projective 8-space) having order 45 and dimension 4, which is invariant under a group of order 51840 X 81 (the Clifford similarity transform group CT). Associated with CT are two other groups, the Clifford collineation group CG of order 81, and the Clifford substitution group CS of order 51840. Furthermore, CS may be regarded as either a subgroup of CT, or a symplectic group of index matrices of size 4. Among the matrices of size 9 which perform the operations of CT, there is a set of 81 involutory, symmetric, orthogonal matrices JW. As collineation matrices in [8], these produce 81 pairs of invariant spaces Σ, Π of dimensions 3 and 4 respectively. These [4]'s give rise to a configuration C invariant under the operations of CT, consisting of 360 points, 1080 lines, 120 Jacobian planes, and 81 [4]'s, and their various inter-relationships.
Graph Theory and Probability
Authors: Erdös P.
Published: December 2010

A well-known theorem of Ramsay (8; 9) states that to every n there exists a smallest integer g(n) so that every graph of g(n) vertices contains either a set of n independent points or a complete graph of order n, but there exists a graph of g(n) — 1 vertices which does not contain a complete subgraph of n vertices and also does not contain a set of n independent points. (A graph is called complete if every two of its vertices are connected by an edge; a set of points is called independent if no two of its points are connected by an edge.) The determination of g(n) seems a very difficult problem; the best inequalities for g(n) are (3)

It is not even known that g(n)1/n tends to a limit. The lower bound in (1) has been obtained by combinatorial and probabilistic arguments without an explicit construction.
A Further Extension of Cayley's Parameterization
Authors: Pearl Martin
Published: December 2010

In a recent paper (3)* the following theorem was proved for real matrices.

Theorem 1. If A is a symmetric matrix and Q is a skew-symmetric matrix such that A + Q is non-singular, then

1

is a cogredient automorph (c.a.) of A whose determinant is + 1 and having theproperty that A and I + P span the same row space.

Conversely, if P is a c.a. of A whose determinant is + 1 and if P has theproperty that I + P and A span the same row space, then there exists a skew symmetricmatrix Q such that P is given by equation (1).

Theorem 1 reduces to the well-known Cayley parameterization in the case where A is non-singular. A similar and somewhat simpler result (Theorem 4) was given for the case when the underlying field is the complex field. It was also shown that the second part of the theorem (in either form) is false when the characteristic of the underlying field is 2. The purpose of this paper is to simplify the proof of Theorem 1 and at the same time, to extend these results to matrices over an arbitrary field of characteristic ≠ 2.
A Remark on the Orthogonality Relations in the Representation Theory of Finite Groups
Authors: Nagao Hirosi
Published: December 2010

Let G be a finite group of order g, and

be an absolutely irreducible representation of degree fμ over a field of characteristic zero. As is well known, by using Schur's lemma (1), we can prove the following orthogonality relations for the coefficients :

1

It is easy to conclude from (1) the following orthogonality relations for characters:

where

and is 1 or 0 according as t and s are conjugate in G or not, and n(t) is the order of the normalize of t.
On Certain Extensions of Function Rings
Authors: Banaschewski Bernhard
Published: December 2010

The present note is concerned with the existence and properties of certain types of extensions of Banach algebras which allow a faithful representation as the normed ring C(E) of all bounded continuous real functions on some topological space E. These Banach algebras can be characterized intrinsically in various ways (1); they will be called function rings here. A function ring E will be called a normal extension of a function ring G if E is directly indecomposable, contains C as a Banach subalgebra and possesses a group G of automorphisms for which C is the ring of invariants, that is, the set of all elements fixed under G. G will then be called a group of automorphisms of E over C. If E is a normal extension of C with precisely one group of automorphisms over C, which is then the invariance group of C in E, then E will be called a Galois extension of C. Such an extension will be called finite if its group is finite.
Lattice Theory of Generalized Partitions
Authors: Hartmanis Juris
Published: December 2010

In (1) the lattice of all equivalence relations on a set S was studied and many important properties were established. In (2) and (3) the lattice of all geometries on a set S was studied and it was shown to be a universal lattice which shares many properties with the lattice of equivalence relations on S. In this paper we shall give the definition of a partition of type n and investigate the lattice formed by all partitions of type n on a fixed set S. It will be seen that a partition of type one on S can be considered as an equivalence relation on S and similarly a partition of type two on S can be considered as a geometry on S as defined in (2). Thus we shall obtain a unified theory of lattices of equivalence relations, lattices of geometries and partition lattices of higher types.
On A Class of Operators Occurring in the Theory of Chains of Infinite Order
Authors: Ionescu Tulcea C.
Published: December 2010

Let T, E be two sets and I ⊂ β(T),l, ⊂ β (E) two tribes. For every n ∈ N* denote by En the product E{1....,n} and by the tribe. For every x ∈ E let ux be a mapping of T into T. For x = (x1,… , xn) ∈ ∈En define ux = uxo … oxl and suppose that {(t, xi1, …, xn) ∣u(x1.....,xn)(t) ∈ A} .

Let m be the Banach space of functions defined on T, real-valued, bounded and I measurable with norm
Typically-Real Functions
Authors: Brown Richard K.
Published: December 2010

A function

an real, is called typically-real of order one in the closed region ∣z∣ ≦ R if it satisfies the following conditions (6).

(1) f(z) is regular in ∣z∣ ≦ R

(2) ∮{f(z)} > 0 if and only if ∮{z} >0.

The same function is called typically-real of order p, p a positive integer greater than one, if it satisfies condition (1) above and in addition the followingcondition (4; 5):

(2’) there exists a constant ρ, 0 < p < R, such that on every circle ∣z∣ = r, ρ, 0 < p < R, ∮{f(z)} changes sign exactly 2p times.
Generating Functions for Hermite Functions
Authors: Weisner Louis
Published: December 2010

Hermite's function Hn(x) is denned for all complex values of x and n by

where F (α; γ; x) is Kummer's function with the customary indices omitted. It satisfies the differential equation

1.1

of which

is a second solution. Every solution of (1.1) is an entire function.
Mixed Problems for Hyperbolic Equations of General Order
Authors: Duff G. F. D.
Published: December 2010

The object of this paper is the extension to linear partial differential equations of order m in N independent variables, of the existence theorems for mixed initial and boundary value problems which have been established for systems of first order equations in (3). In such mixed problems an initial surface S and a boundary surface T are the carriers of the two types of data, and the number of datum functions to be assigned on T depends on the configuration of the characteristic surfaces relative to S and T.

For the first part of the paper (§§ 1-5) the coefficients in the differential equation, the initial and boundary surfaces, and the data prescribed are all taken to be real analytic in the variables x1 … xN. In this “analytic” case an existence theorem is established for boundary conditions of considerable generality. We assume that the differential equation is regularly hyperbolic with respect to 5 and T, a notion which is stated precisely in § 1, and is weaker than the usual regular hyperbolic condition.
Tensor Products of Banach Algebras
Authors: Gelbaum Bernard R.
Published: December 2010

This paper is concerned with a generalization of some recent theorems of Hausner (1) and Johnson (4; 5). Their result can be summarized as follows: Let G be a locally compact abelian group, A a commutative Banach algebra, B1 = Bl(G,A) the (commutative Banach) algebra of A-valued, Bochner integrable junctions on G, 3m1the maximal ideal space of A, m2the maximal ideal space of L1(G) [the [commutative Banach] algebra of complex-valued, Haar integrable functions on G, m3the maximal ideal space of B1. Then m3and the Cartesian product m1 X m2are homeomorphic when the spaces mi, i = 1, 2, 3, are given their weak* topologies. Furthermore, the association between m3and m1 X m2is such as to permit a description of any epimorphism E3: B1 → B1/m3 in terms of related epimorphisms E1: A → A/M1 and E2:L1(G) → Ll(G)/M2, where M1 is in mi i = 1, 2, 3.
A Class of Solvable Groups
Authors: Gorenstein Daniel , Herstein I. N.
Published: December 2010

Numerous studies have been made of groups, especially of finite groups, G which have a representation in the form AB, where A and B are subgroups of G. The form of these results is to determine various grouptheoretic properties of G, for example, solvability, from other group-theoretic properties of the subgroups A and B.

More recently the structure of finite groups G which have a representation in the form ABA, where A and B are subgroups of G, has been investigated. In an unpublished paper, Herstein and Kaplansky (2) have shown that if A and B are both cyclic, and at least one of them is of prime order, then G is solvable. Also Gorenstein (1) has completely characterized ABA groups in which every element is either in A or has a unique representation in the form aba', where a, a’ are in A, and b ≠ 1 is in B.
On Generalized Morse-Transue Function Spaces
Authors: Ellis H. W.
Published: December 2010

Marston Morse and William Transue (6, 8) have introduced and studied function spaces, called MT-spaces, for which the elements of the topological dual are of integral type. Their theory does not admit certain classical Banach function spaces including spaces of bounded functions and spaces. The theory of function spaces determined by a length function (λ-spaces) (4, 5), which depends on a fixed measure, admits many of the maximal MT-spaces, the spaces and spaces of locally integrable functions but does not admit certain maximal MT-spaces including the space of complex continuous functions with compact supports.
On Some Properties of Functions Analytic in a Half-Plane
Authors: Rooney P. G.
Published: December 2010

The spaces , w real, 1 ≤ p < ∞, consist of those functions f(s), analytic for Re s > w, and such that μp(f;x) is bounded for x > w, where

1.1

Doetsch (1) has shown that if e-wtϕ(t) ∈ Lp (0, ∞), 1 < p ≤ 2, and f is the Laplace transform of ϕ, that is,

then f ∈ , where

1.2

and that conversely if f ∈ , 1 < p ≤ 2, then there is a function ϕ, with e-wtϕ(t) ∈ Lq (0, ∞), such that f is the Laplace transform of ϕ.
The Regular Maps on a Surface of Genus Three
Authors: Sherk F. A.
Published: December 2010

A considerable volume of research on the theory of regular maps is now in existence. Systematic enumerations of regular maps on the surfaces of genus 1 and 2 were begun by Brahana (1; 2) and completed by Coxeter (6; 7, p. 141). In addition Coxeter enumerated the regular maps on the simplest non-orientable surfaces (7, pp. 116, 139), and constructed tables of some interesting families of regular maps (3; 7, p. 140).

Most of the regular maps on a surface of genus 3 have appeared in these papers, but no systematic enumeration of them seems to have been attempted. The ultimate goal of this paper is a complete list of these regular maps. However, the families of maps {j.p,q}and {j.p,j.q} which are defined in § 4 and listed in Tables I and II are of considerable interest in themselves. Also of some importance is the complete list of regular maps of type {p,3} with six or fewer faces (§ 5 and Table III).
Convex Structures and Continuous Selections
Authors: Michael Ernest
Published: December 2010

This paper continues the study of continuous selections begun in (13; 14; 15) and the expository paper (12). The purpose of these papers, which is described in detail in the introduction to (13), can be summarized here as follows. If X and Y are topological spaces, and ϕ a function (called a carrier) from X to the space 2Y of non-empty subsets of F, then a selection for ϕ is a continuous f: X → Y such that f(x) ∈ ϕ(x) for every x ∈ X. For reasons which are explained in (13), we restrict our attention to carriers which are lower semi-continuous (l.s.c), in the sense that, whenever U is open in Y, then is open in X. Our purpose in these papers is to find conditions for the existence and extendability of selections.

The principal purpose of this paper is to generalize the following result, which is half of the principal theorem (Theorem 3.2˝) of (13) (and is repeated as Theorem I of (12)).
Polytopes Over GF(2) and their Relevance for the Cubic Surface Group
Authors: Coxeter H. S. M.
Published: December 2010

In the preceding paper, Edge represented the celebrated “cubic surface group” of order 72.6! = 51840 as the group of automorphisms of a senary quadratic form over the field of residue-classes mod 2. The object of this sequel is to compare Edge's finite space with a real space, thus identifying his non-ruled quadric in PG(5, 2) with a modular counterpart of the semi-regular polytope 221 which was discovered by Gosset in 1897.
On Representations of Orders Over Dedekind Domains
Authors: Higman D. G.
Published: December 2010

We study representations of o-orders, that is, of o-regular -algebras, in the case that o is a Dedekind domain. Our main concern is with those -modules, called -representation modules, which are regular as o-modules. For any -module M we denote by D(M) the ideal consisting of the elements x ∈ o such that x.Ext1(M, N) = 0 for all -modules N, where Ext = Ext(,0) is the relative functor of Hochschild (5). To compute D(M) we need the small amount of homological algebra presented in § 1. In § 2 we show that the -representation modules with rational hulls isomorphic to direct sums of right ideal components of the rational hull A of , called principal-modules, are characterized by the property that D(M) ≠ 0. The (, o)-projective -modules are those with D(M) = 0. We observe that D(M) divides the ideal I() of (2) for every M , and give another proof of the fact that I() ≠ 0 if and only if A is separable. Up to this point, o can be taken to be an arbitrary integral domain.
A Simple Closed Curve is the Only Homogeneous Bounded Plane Continuum that Contains an Arc
Authors: Bing R. H.
Published: December 2010

One of the unsolved problems of plane topology is the following:

Question. What are the homogeneous bounded plane continua?

A search for the answer has been punctuated by some erroneous results. For a history of the problem see (6).

The following examples of bounded homogeneous plane continua are known : a point; a simple closed curve; a pseudo arc (2, 12); and a circle of pseudo arcs (6). Are there others?

The only one of the above examples that contains an arc is a simple closed curve. In this paper we show that there are no other such examples. We list some previous results that point in this direction. Mazurkiewicz showed (11) that the simple closed curve is the only non-degenerate homogeneous bounded plane continuum that is locally connected. Cohen showed (8) that the simple closed curve is the only homogeneous bounded plane continuum that contains a simple closed curve.
A Linear Diophantine Problem
Authors: Johnson S. M.
Published: December 2010

Let a1, a2, … , at be a set of groupwise relatively prime positive integers. Several authors, (2; 3; 5; 6), have determined bounds for the function F(a1, …, at) defined by the property that the equation

1

has a solution in positive integers X1, …, xt for n > F(a1, ..., at). If F(a1, …, at) is a function of this type, it is easy to see that

2

is the corresponding function for the solvability of (1) in non-negative x's.

It is well known that a1a2 is the best bound for F(a1, a2) and a1a2 — a,1 — a2 for G(a1, a2). Otherwise only in very special cases have the best bounds been found, even for t = 3.

In the present paper a symmetric expression is developed for the best bound for F(a1, a2, a3) which solves that problem and gives insight on the general problem for larger values of t. In addition, some relations are developed which may be of interest in themselves.
On Nilpotent Products of Cyclic Groups
Authors: Struik Ruth Rebekka
Published: December 2010

In this paperG=F/Fnis studied forFa free product of a finite number of cyclic groups, andFnthe normal subgroup generated by commutators of weightn. The case ofn =4 is completely treated(F/F2is well known;F/F3is completely treated in (2)); special cases ofn >4 are studied; a partial conjecture is offered in regard to the unsolved cases. Forn= 4 a multiplication table and other properties are given.

The problem arose from Golovin's work on nilpotent products ((1), (2), (3)) which are of interest because they are generalizations of the free and direct product of groups: all nilpotent groups are factor groups of nilpotent products in the same sense that all groups are factor groups of free products, and all Abelian groups are factor groups of direct products. In particular (as is well known) every finite Abelian group is a direct product of cyclic groups. Hence it becomes of interest to investigate nilpotent products of finite cyclic groups.
The Enumeration of Point Labelled Chromatic Graphs and Trees
Authors: Austin T. L.
Published: December 2010

Given n points with c1 of one colour, c2 of another colour, up to k colours, linear graphs are formed with the restriction that no line connects points of the same colour. Following fairly standard terminology, coloured graphs with this restriction will be called point chromatic graphs. Giving the points numerical labels running from 1 to ci for points of the ith colour (i = 1, 2, … , k) forms point labelled chromatic graphs. Note that this description is slightly different from assigning a label and a colour independently to each point.
Intersection Irreducible Ideals of a Non-Commutative Principal Ideal Domain
Authors: Feller Edmund H.
Published: December 2010

Let R always denote a fixed non-commutative principal ideal domain. A right (left) ideal aR (Ra) is termed right (left) ∩ irreducible provided it is not the intersection of two right (left) ideals that properly include it. In this case, the element a is called right (left) ∩ irreducible.

Since R satisfies the A.C.C. for right ideals every right ideal aR can be written in the form aR = a,1R ∩ a2R ∩ … ∩ anR, where atR properly include aR and is right ∩ irreducible, i = 1,2, … ,n. We shall investigate properties (including primary properties) of right ∩ irreducible one-sided and two-sided ideals of R. These properties will depend on the results given in (1) and (2, chapter III).

An element a is irreducible if it is not zero or a unit and has no proper factors. In this case aR (Ra) is a maximal right (left) ideal.
Continuous Translation of Hölder and Lipschitz Functions
Authors: Mirkil H.
Published: December 2010

All functions will be complex, periodic, integrable (on [0,2π]) functions of a real variablex. Moreover, we shall require that every function have mean zero on [0,2π], so that in particular non-zero constants are excluded.

1. Plessner's characterization of absolutely continuous functions. An old theorem of Plessner (4), generalized to arbitrary compact groups by Bochner (1), can be taken as our starting point. Consider the functions f ofbounded variationon [0,2π]. These f form a Banach spaceFwhen each f is normed by its total variation on [0,2π]. And translations define a natural one-parameter group of isometries onF.
On Lie Semi-Groups
Authors: Langlands R. P.
Published: December 2010

Suppose we have a semi-group structure defined on

a subset of real Euclidean n-space, En, by (p, q) → F (p, q) = poq. In this note we shall be concerned with a representation T(.) of π as a semi-group of bounded linear operators on a Banach space 𝒳. More particularly, we suppose that postulates P1, P2, P3, P5 and P6 of chapter 25 of (2) are satisfied so that, by Theorem 25.3.1 of that book, there is a continuous function, f(.), defined on π such that f((ρ + σ)a) = f(ρa)o f(σa) for a ∈ π, ρ,σ ≥ 0; that the representation is strongly continuous in a neighbourhood of the origin and that T(0) = I.
The Expression of Trigonometrical Series in Fourier Form
Authors: Cross George
Published: December 2010

In a paper published in 1936 Burkill (2) proved that, if the trigonometrical series

1.1

is bounded except on a countable set and if the series obtained by integrating series (1.1) once converges everywhere, then the coefficients can be written in Fourier form using the C1P-integral. In §3 of this paper an analogous result is shown to be true when (1.1) is bounded (C, k), k < 0. The proof of this depends on generalizations of theorems by Verblunsky and Zygmund and both of these generalizations are obtained in §2.