bluuebunny/crossref_metadata_2025 huggingface dataset:
On a Theorem of Osima and Nagao
Authors: Frame J. S. , Robinson G. de B.
Published: December 2010

If we define the weight b of a Young diagram containing n nodes to be the number of removable p-hooks where n = a + bp, then three fundamental theorems stand out in the modular representation theory of the symmetric group Sn.
On the Symmetries of Spherical Harmonics
Authors: Meyer Burnett
Published: December 2010

Letbe a finite group of transformations of three-dimensional Euclidean space, such that the distance between any two points is preserved by all transformations of the group. Such a group is a group of orthogonal linear transformations of three variables, or, geometrically speaking, a group of rotations and rotatory inversions. Thirty-two groups of this type are important in crystallography and are known as the crystallographic classes.
Complexes in Abelian Groups
Authors: Scherk Peter , Kemperman J. H. B.
Published: December 2010

Let G be an abelian group of order [G] ≤ ∞. Let A = {a}, B = {b}, … denote non-empty finite complexes in G. Let [A] be the number of elements of A. Finally put

A + B = {a + b}.
On Sums of Sets of Integers
Authors: Kemperman J. H. B. , Scherk Peter
Published: December 2010

Small italics denote integers. Let A, B, … be sets of non-negative integers. Let A (h) be the number of positive integers in A that are not greater than h. Finally let A + B denote the set of all integers of the form a + b where a ⊂ A, b ⊂ B. The following result is implicitly contained in Mann's Proposition 11 (4):

Theorem 1. Let n > 0 and

(1.1) 0⊂4, 0⊂B, n⊄C = A + B.
On Cyclotomic Numbers of Order Sixteen
Authors: Lehmer Emma
Published: December 2010

It has been shown by Dickson (1) that if (i, j)8 is the number of solutions of

(mod p),

then 64(i,j)8 is expressible for each i,j, as a linear combination with integer
coefficients of p, x, y, a, and b where

,

and

a ≡ b ≡ 1 (mod 4),

while the sign of y and b depends on the choice of the primitive root g. There are actually four sets of such formulas depending on whether p is of the form 16n + 1 or 16n + 9 and whether 2 is a quartic residue or not.
Note on the Algebra of S-Functions
Authors: Duncan D. G.
Published: December 2010

Considerable advance has been made recently towards a systematic method of evaluating the “product” {μ} ꕕ {λ}, most notably in the methods of Robinson (3), Littlewood (2), and Todd (5) and the differential operator technique of H. O. Foulkes.
An Expansion Theorem for a Pair of Singular First Order Equations
Authors: Conte S. D. , Sangren W. C.
Published: December 2010

Titchmarsh (4) has shown how the classical method of
complex variables can be used to obtain expansion theorems for the singular
cases of the second order equation

(1) .

The purpose of this paper is to indicate how these results can be generalized
to the singular cases of the pair of first order equations

,
On Regions Omitted By Univalent Functions II
Authors: Goodman A. W. , Reich E.
Published: December 2010

1. Introduction. Let S denote the family of functions f(z) regular and univalent in ∣z∣ < 1, with the expansion f(z) = z + a2z2 + … about z = 0, and let Af denote the area of the intersection of the open circle ∣ω∣ < 1 with Df, the image of ∣z∣ < 1 under f(z). A few years ago one of the authors (1) proved that if

1

then

2.
Generalized Matrix Algebras
Authors: Brown W. P.
Published: December 2010

The algebras considered here arose in the investigation of an algebra connected with the orthogonal group. We consider an algebra of dimension mn over a field K of characteristic zero, and possessing a basis {eij} (1 ≤ i ≤ m; 1 ≤ j ≤ n) with the multiplication property

1,

The field elements ϕij form a matrix Φ = (ϕij) of order n × m. It will be called the multiplication matrix of the algebra relative to the basis {ϕij}.
Point-Finite And Locally Finite Coverings
Authors: Michael Ernest
Published: December 2010

1. Introduction. An interesting feature of recent topological developments is the increasingly important role played by locally finite coverings. Point-finite coverings, on the other hand, even though conceptually simpler, have received very little attention. And deservedly so, since they are much
less useful. Nevertheless, it sometimes happens (as it did to the author in (5)) that one is confronted by a covering which is known to be point-finite, but not necessarily locally finite.
The Propagation Of a Plane Shock Into a Quiet Atmosphere II
Authors: Martin M. H.
Published: December 2010

1. Introduction. In a previous paper (3) the study of one-dimensional, unsteady flows, isentropic or anisentropic, was reduced to the integration of a Monge-Ampère partial differential equation

(1)

For a polytropic gas, the specific volume

(2) ,

takes the form

,

once the entropy distribution function S = S(Ψ) is specified.
An Inhomogeneous Minimum For Non-Convex Star-Regions With Hexagonal Symmetry
Authors: Bambah R. P. , Rogers K.
Published: December 2010

1. Introduction. Several authors have proved theorems of the following
type:

Let x0, y0 be any real numbers. Then for certain functions f(x, y), there exist numbers x, y such that

1.1 x ≡ x0, y ≡ y0 (mod 1),

and

1.2 .

The first result of this type, but with replaced by min , was given by Barnes (3) for the case when the function is an indefinite binary quadratic form. A generalisation of this was proved by elementary
geometry by K. Rogers (6).
Systems Of Linear Congruences
Authors: Butson A. T. , Stewart B. M.
Published: December 2010

1. Introduction. On recent occasions papers have been presented concerned with the problem of solving a system of linear congruences. Apparently the authors were not aware that this problem was solved very neatly and completely a long time ago by H. J. S. Smith (5; 6).
On Partial Differential Equations in a Field of Prime Characteristic
Authors: Jaeger Arno
Published: December 2010

In classical analysis ordinary differential equations and partial differential equations are distinct concepts, and the transition from one derivation to several partial derivations changes some of their properties distinctly. On the other hand, the algebraic theories of modified ordinary and partial differential equations (5; 6), based on the differentiations in the sense of Hasse (2) and Schmidt (3) and the multidifferentiations in the sense of Jaeger (4), turn out to be strikingly similar in the case of fields of prime number characteristic.
The Normalizer Of Certain Modular Subgroups
Authors: Newman Morris
Published: December 2010

Introduction. Let G denote the multiplicative group of matrices

where a, b, c, d are integers and ad — bc = 1. G is one of the well-known modular groups. Let G0(n) denote the subgroup of G characterized by c ≡ 0 (mod n), where n is a positive integer. In this note we determine the normalizer
of G0(n) in G, denoted by .
On Unitary and Symmetric Matrices With Real Quaternion Elements
Authors: Wiegmann N. A.
Published: December 2010

1. Introduction. In general when symmetric matrices are considered, the elements of the matrix are taken at least in a principal ideal ring. It is interesting to determine what can be attained when the elements are not, in general, commutative and, to this end, the following is concerned with symmetric matrices with elements in the non-commutative field of real quaternions. At the same time some properties of real quaternion unitary matrices are obtained which involve symmetric matrices.
Submethods Of Regular Matrix Summability Methods
Authors: Goffman Casper , Petersen G. M.
Published: December 2010

By a submethod of a regular matrix method A we mean a method (see 1 or 3) whose matrix is obtained by deleting a set of rows from the matrix A. We establish a one-one correspondence between the submethods of A and the points of the interval 0 < ξ ≤We designate the submethod which corresponds to 𝞷 by A (𝞷) and are accordingly able to speak of sets of submethods of measure 0, of the first category, etc.
A Condition For Existence Of Moments Of Infinitely Divisible Distributions
Authors: Shapiro J. M.
Published: December 2010

Let F(x) be an infinitely divisible distribution and let ϕ(t) be its characteristic function. As is well known according to the formula of Lévy and Khintchine, ϕ(t) has the following representation:

1

where γ is a real constant and G(u) is a bounded nondecreasing function.
On Relationships Amongst Certain Spaces Of Sequences In An Arbitrary Banach Space
Authors: McArthur C. W.
Published: December 2010

1. Introduction. Let X be a Banach space (B-space). A sequence {s(i)} in X is unconditionally summable if and only if every rearrangement of the series Σis(i) is convergent. The set of unconditionally summable sequences in X will be written as U(X). In this paper several classes of summable sequences in X will be compared with one another. Each class to be considered is identical with U(X) when X has finite dimension.
Some Algebraic Properties Of Asymptotic Power Series
Authors: Hull T. E.
Published: December 2010

1. Introduction. Let us consider all power series of the form

It was shown first by Borel (1) that to each such series there corresponds a non-empty class of functions such that each function in the class has the given series as its asymptotic expansion about z = 0, the expansion being valid in a sector of the right half z-plane with vertex at the origin.
The Asymptotic Series For a Certain Class Of Permutation Problems
Authors: Mendelsohn N. S.
Published: December 2010

1. Introduction. This paper is concerned with problems connected with the permutations of the integers 1, 2, … , n subject to certain special restrictions. One such class of problems, the so-called “card matching” problems, deals with conditions of the type, “the number i is in the jth position,” “the number k is in the mth position,” etc.
Orthogonal Isomorphic Representations Of Free Groups
Authors: De Groot J.
Published: December 2010

1. Introduction. We consider the group of proper orthogonal transformations
(rotations) in three-dimensional Euclidean space, represented by real orthogonal matrices (aik) (i, k = 1,2,3) with determinant + 1 . It is known that this rotation group contains free (non-abelian) subgroups; in fact Hausdorff (5) showed how to find two rotations P and Q generating a group with only two non-trivial relations

P2 = Q3 = I.
On Commuting Rings Of Endomorphisms
Authors: Curtis C. W.
Published: December 2010

Various problems concerning the general theory of centralizers of modules which are not assumed to be completely reducible have been discussed by Fitting (3), Brauer (2), and Nakayama. In this paper we present a new approach to some of these questions, which has its origin in
Weyl's discussion (15) of the centralizer of a finite group of collineations.
On Some Polynomials Of Touchard
Authors: Wyman Max , Moser Leo
Published: December 2010

In the preceding paper Touchard considers a set of polynomials Qn(x) defined by

1.

Touchard uses (1) to compute Qn(x) for 0 ≤ n ≤ 9 and also finds Qn(-½).

He remarks however “l'expression générale des polynômes Qn(x) nous echappe.”

The object of this note is to derive an explicit expression for Qn(x).
Maschke Modules Over Dedekind Rings
Authors: Reiner Irving
Published: December 2010

We use the following notation throughout:

.
Standard and Accessible Rings
Authors: Kleinfeld Erwin
Published: December 2010

1. Introduction. A ring is defined to be standard (1) in case the following
two identities hold :

(1) (wx, y, z) + (xz, y, w) + (wz, y, x) = 0,

(2) (x, y, z) + (z, x, y) − (x, z, y) = 0,

where the associator (x, y, z) is defined by (x, y, z) = (xy)z − x(yz). Albert has determined the structure of finite-dimensional, standard algebras (1).

The simple ones turn out to be either Jordan algebras or associative ones.
Constructions In Hyperbolic Geometry
Authors: Handest Frans
Published: December 2010

Introduction. In hyperbolic geometry we have three compasses, namely an ordinary compass for drawing ordinary circles with a given centre and a given radius, a hypercompass for drawing hypercycles with a given axis and a given radius, and a horocompass for drawing horocycles with a given diameter and passing through a given point.
Local Connectedness Of Extension Spaces
Authors: Banaschewski Bernhard
Published: December 2010

1. Introduction. An extensionE* of a topological spaceE(that is, a space containingEas a dense subspace) determines a family of filters(u) onE, given by the tracesU ∩ Eof the neighbourhoodsU ⊆ E*of each u ∈ E* −E.Many topological properties of an extensionE* of a given space E can be related to properties of these trace filters (as we shall call them) belonging toE*.
On the Basis Problem for Vector Valued Function Spaces
Authors: Ellis H. W.
Published: December 2010

1. Introduction. In a recent paper (2) Halperin and the author considered separable Banach spaces Lλ of real valued functions on general measure spaces and proved the existence of 1-regular (§2) Haar or σ-Haar bases when λ was the classical p-norm or any levelling length function (3) and, more generally, of K-regular Haar or σ-Haar bases when λ was a continuous length function satisfying certain additional conditions (2, Theorem 3.2).
Resolvents Of Certain Linear Groups In a Finite Field
Authors: Carlitz L.
Published: December 2010

Let Fq = GF(q) denote the finite field of order
q = pn, where p is a prime. Consider the group Γ of linear transformations

(1.1) x' = (ax + b)//(cx + d)

with coefficients a, b, c, d ∊ Fq and of determinant 1. The order of Γ is
½q(q2 − 1) or q(q2 − 1) according as q is odd or even, i.e., according as p > 2 or p = 2.
On Indefinite Ternary Quadratic Forms
Authors: Jones B. W. , Watson G. L.
Published: December 2010

1, Introduction. The first systematic study of equivalence of indefinite ternary quadratic forms seems to be that of A. Meyer (10) (see also Bachmann (1)). By methods which are often obscure he showed that the number of classes in a genus is a power of 2, the exact power depending on certain quadratic
characters associated with the form.
Some Global Theorems on Hypersurfaces
Authors: Hsiung Chuan-Chih
Published: December 2010

The purpose of this paper is to establish the following theorems, which were
obtained by Hopf and Voss in their joint paper (2) for the case where
n = 2.

THEOREM 1. Let Vn, V*n be two closed
orientable hyper surfaces twice differ entiably imbedded in a Euclidean
space En+l of dimension n + 1 ≥ 3.
Some Theorems About pr(n)
Authors: Newman Morris
Published: December 2010

If n is a non-negative integer, define
pr(n) as the coefficient of xn in

;

otherwise define pr(n) as 0. In a recent paper (1) the author has
proved that if r has any of the values 2, 4, 6, 8, 10, 14,
26 and p is a prime > 3 such that
r(p + 1) ≡ 0 (mod 24), then

1, ,

where n is an arbitrary integer.
A Generalization of the Cauchy Principal Value
Authors: Fox Charles
Published: December 2010

If a < u < b and n
> 0 then

(1)

is a so-called improper integral owing to the infinity in the integrand at
x = u. When n = 0 we
have associated with (1) the well-known Cauchy principal value, namely

(2).

Hadamard (1, p. 117 et seq.) derives from an improper integral an expression
which he calls its finite part and which, as he shows, possesses many
important properties.
Some Polynomials of Touchard Connected with the Bernoulli
Numbers
Authors: Carlitz Leonard
Published: December 2010

In a recent paper (4) Touchard has constructed a set of
polynomials Ωn(z) such that

(1),

where after expansion of the left member Bm is replaced by Bm,

,

and

(2).
Asymptotic Expansions II
Authors: Moser Leo , Wyman Max
Published: December 2010

In a previous paper (1) the authors considered the problem of finding an
asymptotic formula for numbers or functions Bn,m whose generating
function is of the form

(1.1),

where Pm(x) is a polynomial of degree
m in x given by

(1.2), am≠0.

The above-mentioned paper contained the restriction that ak ≥
0.
On a Diophantine Problem
Authors: Roberts J. B.
Published: December 2010

If a1, a2, … , ak are relatively prime
positive integers then the equation

always has solutions in non-negative xi for n
sufficiently large. Sylvester called the number of non-negative solutions of
(1) the denumerant of the equation. We shall denote the
denumerant by n(a1, … , ak).
Groupoïdes Automorphes Par Le Groupe Cyclique
Authors: Sade A.
Published: December 2010

Nous appellerons groupoïde un ensemble non vide,
G, muni d'une loi (×) faisant correspondre à tout couple
ordonné x, y ∊ G, au plus un élément z de
G, appelé produit de x par y, et satisfaisant à la loi
d'homogénéité (2).
Certain Classes of Ideals in Polynomial Rings
Authors: McCoy N. H.
Published: December 2010

The purpose of this paper is to establish some results, of a somewhat miscellaneous nature, concerning certain classes of ideals in polynomial rings. Although the results will be formulated for a ringR[x] of polynomials in one indeterminate over a given ringR, they can be easily extended to polynomial rings in any finite number of indeterminates.
A Class of Non-Desarguesian Projective Planes
Authors: Hughes D. R.
Published: December 2010

In (7), Veblen and Wedclerburn gave an example of a non-Desarguesian
projective plane of order 9; we shall show that this plane is self-dual and
can be characterized by a collineation group of order 78, somewhat like the
planes associated with difference sets. Furthermore, the technique used in
(7) will be generalized and we will construct a new non-Desarguesian plane
of order p2n for every positive integer n and every odd
prime p.
Dedekind Completeness and a Fixed-Point Theorem
Authors: Wolk E. S.
Published: December 2010

McShane (5, 6) has introduced the concept of “Dedekind completeness” for
partially ordered sets, which seems to be a natural generalization of the
usual concept of completeness for lattices. It is the purpose of this paper
to discuss some of the properties of Dedekind completeness, particularly
with respect to a rather natural class of partially ordered sets which we
call “uniform.”
The Closed Ideals in an Algebra of Analytic
Functions
Authors: Rudin Walter
Published: December 2010

Let K and C be the closure and boundary,
respectively, of the open unit disc U in the complex plane.
Let be the
Banach algebra whose elements are those continuous complex functions on
K which are analytic in U, with
norm

(f ∊ ).
Games On a Compact Set
Authors: Peck J. E. L. , Dulmage A. L.
Published: December 2010

Von Neumann's fundamental theorem of the theory of games has been extended
by various authors and recently in two different directions by Kneser (8)
and Nikaidô (9). We present here a form of the theorem, which is more
general than that of both these authors. We develop some consequences of
this theorem, which make it easier to decide whether certain classes of
games have a value and we give several illustrative examples
On a Metric that Characterizes Dimension
Authors: De Groot J.
Published: December 2010

Sometimes it is possible to characterize topological properties of a
metrizable space M by claiming that a certain
(topologypreserving) metric ρ can be introduced in
M.
Factorization Rings
Authors: Maranda J.-M.
Published: December 2010

Let o be an integral domain with as field of
quotients. W. Krull has shown (3 ; 4) that the following three conditions on
o are equivalent:

There is a set of rank 1, discrete valuations of such that for each non-null element for all except a
finite number, and such that for all if and only
if for all
i ∊ I.
Goursats Theorem and the Zassenhaus Lemma
Authors: Lambek Joachim
Published: December 2010

In this paper we study generalized homomorphisms between two algebras, namely
the binary relations whose graphs are subalgebras of the direct product of the
given algebras. In 1897 Goursat proved that every subgroup of the
direct product of two groups is determined by an isomorphism between factor
groups of subgroups of the given groups (10, §§11, 12; 25, pp.
15, 16). A like result is here shown for a general class of algebras, including
loops and quasigroups, by a method due to Riguet (22). This result is used to
obtain general forms of the Zassenhaus lemma and the Jordan- Hälder-Schreier
theorem for normal series (26, §9).
On the Random Disorientation of two Cubes
Authors: Handscomb D. C.
Published: December 2010

We are given two identical symmetrical bodies (e.g., cubes) with independent
random orientations; then we can always, in several ways, turn one of these bodies
about some axis through its centre of gravity, so as to bring it into the same
orientation as the other body. The smallest angle of rotation needed will be
called the disorientation, d, of the two bodies, and we
shall be concerned with the distribution of d under these conditions.
Mixed Problems for Linear Systems of first Order Equations
Authors: Duff G. F. D.
Published: December 2010

A mixed problem in the theory of partial differential equations is an
auxiliary data problem wherein conditions are assigned on two distinct surfaces
having an intersection of lower dimension. Such problems have usually been
formulated in connection with hyperbolic differential equations, with initial and
boundary conditions prescribed. In this paper a study is made of the conditions
appropriate to a system of R linear partial
differential equations of first order, in R dependent
and N independent variables.
Schlicht Dirichlet Series
Authors: Robertson M. S.
Published: December 2010

For power series

(1.1)

for which

(1.2)

, it has been known for four decades (1) that ƒ(z) is regular
and univalent or schlicht in |z| < 1. This theorem, due to J. W. Alexander,
has more recently been studied by Remak (5) who has shown that
w =
ƒ(z), under the
hypothesis (1.2), maps |z| < 1 onto a star-like
region, and if (1.2) is not satisfied=(z) need not be
univalent in |z| < 1 for a proper choice of the amplitudes of the
coefficients an.
Laplace Transforms and Generalized Laguerre Polynomials
Authors: Rooney P. G.
Published: December 2010

Various sets of necessary and sufficient conditions are known in order that a
function ƒ(s), analytic for
Re s > 0, be represented as the Laplace
transform of a function in L
p(0,∞), 1 < p ⩽ ∞ . Most of
these theories are based on the properties of some inversion operator for the
transformation— see, for example, (7, chap. 7). However in the case
p = 2 a number of representation theorems of a much
simpler type are available.
On Some Relations Between Partial and Ordinary Differential Equations
Authors: Kreyszig Erwin
Published: December 2010

The theory of solutions of partial differential equations

(1.1)

with analytic coefficients can be based upon the theory of
analytic functions of a complex variable; the basic tool in this approach is
integral operators which map the set of solutions of (1.1) onto the algebra of
analytic functions. For certain classes of operators this mapping which is first
defined in the small, can be continued to the large, cf. Bergman (3).
On the Number of Ordinary Lines Determined by n Points
Authors: Kelly L. M. , Moser W. O. J.
Published: December 2010

More than sixty years ago, Sylvester (13) proposed the following problem: Let
n given points have the property that the straight line joining any two of them
passes through a third point of the set. Must the n points all lie on one
line?

An alleged solution (not by Sylvester) advanced at the time proved to be
fallacious and the problem remained unsolved until about 1933 when it was revived
by Erdös (7) and others.
On the Number of Sides of a Petrie Polygon
Authors: Steinberg Robert
Published: December 2010

Let {p, q, r} be the regular 4-dimensional poly
tope for which each face is a {p, q} and each vertex
figure is a {q, r}, where {p,
q}, for example, is the regular polyhedron with p-gonal faces, q
at each vertex. A Petrie polygon of {p, q} is a skew
polygon made up of edges of {p, q} such that every two
consecutive sides belong to the same face, but no three consecutive sides do. Then
a Petrie polygon of {p, g, r} is defined by the
property that every three consecutive sides belong to a Petrie polygon of a
bounding {p, q}, but no four do. Let h
Pqr be the number of sides of such a polygon, and
g

p,q,r
the order of the group of symmetries of {p, g, r}.
Retraceable Sets
Authors: Dekker J. C. E. , Myhill J.
Published: December 2010

Let us compare two properties of sets of non-negative integers: (1) the set a
has property Γ, if there exists an effective procedure
which when applied to any element of a different from its maximum (which
α does not necessarily possess) yields the next
larger element of α; (2) the set
α has property Δ, if
there exists an effective procedure which when applied to any element of
α different from its minimum yields the next smaller
element of α. It is readily seen that every recursive
set has both properties.
On Some Theorens of Doetsch
Authors: Rooney P. G.
Published: December 2010

The spaces

P
(ω), 1 ⩽ p ⩽ ∞, ω
real are defined to consist of those analytic functions ƒ(s), regular for Re s
> ω and for which μ
P
(ƒ;x) is bounded for x > ω where

(1.1)

, 1 ⩽ p ⩽ ∞and

(1.2)
On the Product of two Kummer Series
Authors: Henrici Peter
Published: December 2010

Let a, β, μ, ν,
z be complex numbers such that 2μ and
2ν are not negative integers. Using the notation of (4)
for generalized hypergeometric series, we set
The Chords of the Non-Ruled Quadric In PG(3, 3)
Authors: Tutte W. T.
Published: December 2010

The 8-cage (3) may be defined as the simplest cubic graph having no circuit
of fewer than eight edges. To construct it we first observe that it must contain a
tree T whose vertices are of degrees 1 and 3 and in
which each vertex of degree 1 is separated by an arc of just three edges from a
central edge AB. These properties fix the structure of
T uniquely.
On the Inversion of the Gauss Transformation, II
Authors: Rooney P. G.
Published: December 2010

In an earlier paper (5) we studied the inversion theory of the Gauss
transformation defined by

(1.1)

. Operational methods indicated that

and we showed that in certain circumstances this equation was
true if exp (— D
2) f(x)
was interpreted as the sum of the series
A Class of Frobenius Groups
Authors: Gorknstein Daniel
Published: December 2010

If a group contains two subgroups A and B such that every element of the group is either in A or can be represented uniquely in the form aba', a, a’ in A, b ≠ 1 in B, we shall call the group an independent ABA-group. In this paper we shall investigate the structure of independent ABA -groups of finite order.

A simple example of such a group is the group G of one-dimensional affine transformations over a finite field K. In fact, if we denote by a the transformation x’ = ωx, where ω is a primitive element of K, and by b the transformation x’ = —x + 1, it is easy to see that G is an independent ABA -group with respect to the cyclic subgroups A, B generated by a and b respectively.

Since G admits a faithful representation on m letters (m = number of elements in K) as a transitive permutation group in which no permutation other than the identity leaves two letters fixed, and in which there is at least one permutation leaving exactly one letter fixed, G is an example of a Frobenius group.
Some Division Algebras
Authors: Zemmer Joseph L.
Published: December 2010

Let K* be an associative algebra over a field F with identity u, and let u, e1, e2, … , be a basis for K*. Denote by K the linear space, over F, spanned by the ei,i = 1, 2, … . Then for x, y in K, xy = αu + a, where a ∈ K. Define h(x, y) = α and x.y = a. With respect to the operation thus defined, K becomes an algebra over F satisfying

1

Further, the bilinear form h(x, y) is associative on K. Any algebra, over a field F, which possesses an associative bilinear form h(x, y) and satisfies (1) will be called a algebra. It is not difficult to show that any algebra K can be obtained from a unique associative algebra K* with identity by the process described above. The algebra K* will be called the associated associative algebra of K.
Characters of Cartesian Products of Algebras
Authors: Warner Seth
Published: December 2010

Let R be a commutative ring with identity 1. A character of an R-algebra E is a homomorphism from E onto R, regarded as an algebra over itself. If (Eα) is a family of R-algebras indexed by a set A and if

then for every β ∈ A and every character vβ of Eβ, vβ o prβ is a character of E where prβ is the projection homomorphism from E onto Eβ. Further if A is finite and if the only idempotents of R are 0 and 1 (equivalently, if R is not the direct sum of two proper ideals), it is easy to see that every character of E is of this form.
Concerning Binary Relations on Connected Ordered Spaces
Authors: Krule I. S.
Published: December 2010

In a recent paper Mostert and Shields (4) showed that if a space homeomorphic to the non-negative real numbers is a certain type of topological semigroup, then the semigroup must be that of the non-negative real numbers with the usual multiplication. Somewhat earlier Faucett (2) showed that if a compact connected ordered space is a suitably restricted topological semigroup, then it must be both topologically and algebraically the same as the unit interval of real numbers with its usual multiplication.

In studying certain binary relations on topological spaces there have become known (see, in particular, Wallace (5) and the author (3)) a number of properties analogous to those possessed by topological semigroups. Because of these analogous properties between relations and semigroups the author was motivated by the general nature of the Faucett and Mostert-Shields results (that is, that the multiplication assumed turned out to be the same as the usual multiplication) to feel that certain relations on a connected ordered space should turn out to be the same as the orders whose order topologies are the topology on the space.
Generating Functions for Bessel Functions
Authors: Weisner Louis
Published: December 2010

On replacing the parameter n in Bessel's differential equation

1.1

by the operator y(∂/∂y), the partial differential equation Lu = 0 is constructed, where

1.2

This operator annuls u(x, y) = v(x)yn if, and only if, v(x) satisfies (1.1) and hence is a cylindrical function of order n. Thus every generating function of a set of cylindrical functions is a solution of Lu = 0.

It is shown in § 2 that the partial differential equation Lu = 0 is invariant under a three-parameter Lie group. This group is then applied to the systematic determination of generating functions for Bessel functions, following the methods employed in two previous papers (4; 5).
Some Results on Uniqueness and Successive Approximations
Authors: Brauer Fred
Published: December 2010

In the theory of ordinary differential equations, there is a strange relationship between uniqueness of solutions and convergence of the successive approxi mations. There are examples of differential equations with unique solutions for which the successive approximations do not converge (8) and of differential equations with non-unique solutions for which the successive approximations do converge (2). However, in spite of the known logical independence of these two properties, almost all conditions which assure uniqueness also imply the convergence of the successive approximations. For example, the hypotheses of Kamke's general uniqueness theorem (5), have been shown by Coddington and Levinson to suffice for the convergence of successive approximations, after the addition of one simple monotonicity condition (4). There is one counterexample to this “principle,” a generalization of Kamke's result, to which another condition in addition to a monotonicity assumption must be added before convergence of the successive approximations can be proved (2).
The a-Local Operator Problem
Authors: de Branges Louis
Published: December 2010

Let be the Fourier transform of a Borel measure of finite total variation. The formula

can be justified if the integral on the right converges absolutely. For

where

Now let h → 0 in both sides of this equation and use the Lebesgue dominated convergence theorem.
Certain Bilateral Hypergeometric Identities Of Cayley And Orr Type
Authors: Agarwal Nirmala
Published: December 2010

Recently I (1) gave some new basic hypergeometric identities of the Cayley and Orr type with the help of a certain basic differential operator. The present paper deals with some bilateral generalizations of those identities together with certain new identities of the same type. In § 4 is indicated how the generalizations of Orr's identities given recently by Shukla (8, Theorems I, II) may be connected with each other. Later in § 5 certain general expansions of hypergeometric functions are deduced.
On Connections Between Growth and Distribution of Zeros of Integral Functions
Authors: Rahman Q. I.
Published: December 2010

The following theorem was proved by Paley and Wiener (4, p. 70; 1, p. 136).

Theorem 1. If f(z) is a canonical product of order 1 with real zeros, and f(0) = 1, the conditions

and

are equivalent. n(r) denotes the number of zeros of absolute value not exceeding r.

Instead of assuming the zeros to be all real Pfluger assumed that the zeros are close to the real axis and proved the following theorem (5 or 1, p. 143).
On Finite Nilpotent Groups
Authors: Bachman G.
Published: December 2010

It is well known that if (n, ϕ(n)) = 1, where ϕ(n) denotes the Euler ϕ function, then the only group of order n is the cyclic group. This is a special case of a more general result due to Dickson (2, p. 201); namely, if

where the pi are distinct primes and each αi > 0, the necessary and sufficient conditions that the only groups of order n are abelian are (1) each αi ≤ 2 and (2) no

is divisible by any p1 … , ps.

We wish to establish a theorem which includes these two results. We let G(n) equal the number of groups of order n where

and we seek necessary and sufficient conditions on n so that

Clearly, this problem is equivalent to finding necessary and sufficient conditions on n so that all existing groups of order n be nilpotent.
Finite Groups Which Admit An Automorphism With Few Orbits
Authors: Gorenstein Daniel
Published: December 2010

In the course of investigating the structure of finite groups which have a representation in the form ABA, for suitable subgroups A and B, we have been forced to study groups G which admit an automorphism ϕ such that every element of G lies in at least one of the orbits under ϕ of the elements g, gϕr(g), gϕrϕ(g)ϕ2r(g), gϕr(g)ϕr2r(g)ϕ3r(g), etc., where g is a fixed element of G and r is a fixed integer.

In a previous paper on ABA-groups written jointly with I. N. Herstein (4), we have treated the special case r = 0 (in which case every element of G can be expressed in the form ϕi(gj)), and have shown that if the orders of ϕ and g are relatively prime, then G is either Abelian or the direct product of an Abelian group of odd order and the quaternion group of order 8.
Integral p-adic Normal Matrices Satisfying the Incidence Equation
Authors: Goldhaber J. K.
Published: December 2010

The problem of arrangingvelements intovsets in such a way that every set contains exactlykdistinct elements and that every pair of sets has exactly λ =k(k — l)/(v —1) elements in common, where 0 < » <k<v, is equivalent to finding a normal integralvbyvmatrixAsuch thatATA = B, whereBis thevbyvmatrix havingkin every position on the main diagonal and λ in all other positions (10). Utilizing the fact that for the existence of a λ,k, vdesign it is necessary that I (thevbyvidentity matrix) representBrationally, (2) and (3) have proved the non-existence of certain λ,k, vdesigns. Neither of the proofs utilize the fact that it is necessary thatAbe normal. However, Albert (1) for the projective plane case and Hall and Ryser (5) for the general design proved that if there exists a rationalAsuch thatATA = Bthen there exists a normal rational matrix satisfying the same equation. Thus the requirement of normality does not exclude any λ,k, vwhich were not previously excluded.
Sur La Cohomologie Non Abelienne I (Dimension Deux)
Authors: Dedecker Paul
Published: December 2010

On sait que la définition de la cohomologie d'un espace X à coefficients dans un faisceau de groupes non abéliens est liée à la classification de certains types d'espaces fibres principaux de base X (1; 8; 10). Cette cohomologie se définit assez facilement en dimensions zéro et un, mais pour certains problèmes, il serait utile de pouvoir la définir en dimension deux également.

J'ai abordé la question dans (3; 6) mais un certain nombre de difficultés subsistaient qu'il semble utile de chercher à éliminer, ce qui est l'objet de la présente note.

Remarquons d'abord que la définition de H°(X, ), groupe des sections de , est triviale, aucune difficulté ne résultant du caractère non abélien de . En ce qui concerne H1(X, ) une difficulté naît du fait que cet ensemble n'est plus un groupe, même non abélien; il possède toutefois un élément neutre privilégié.
On Maximal Abelian Subrings of Factors of Type II
Authors: Pukánszky Lajos
Published: December 2010

Although we possess a fairly complete knowledge of the abelian subrings of rings of operators in a Hilbert space which are algebraically isomorphic to the ring of all bounded operators of a finite or infinite dimensional unitary space, that is of factors of Type I, very little is known of abelian subrings of factors of Type II1. In (1), Dixmier investigated several properties of maximal abelian subrings of factors of Type II. It turned out that their structure differs essentially from that of maximal abelian subrings of factors of Type I. He showed the existence of maximal abelian subrings in approximately finite factors, possessing the property that every inner*-automorphism carrying this subring into itself is necessarily implemented by a unitary operator of this subring. These maximal abelian subrings are called singular. In addition, he constructed a IIi factor containing two singular abelian subrings which cannot be connected by an inner *automorphism of this ring.
Sur Les Representations Unitaires des Groupes de Lie Nilpotents. VI
Authors: Dixmier Jacques
Published: December 2010

Soient Γ un groupe localement compact, l'ensemble des classes de représentations unitaires irréductibles de Γ. Dans (4), Godement a défini une topologie sur que nous appellerons "topologie canonique." Cette topologie est facile à étudier lorsque Γ est abélien ou compact, mais fort mal connue en général. Dans un article récent (3), Fell a repris l'étude de la topologie canonique de , et réussi à la calculer lorsque Γ est le groupe spécial linéaire complexe à n variables. L'espace est alors "presque" séparé, et, par suite, "presque" localement compact.

Il semble nécessaire d'étudier complètement quelques cas particuliers avant d'entreprendre une théorie générale de .
On Relatively Invariant Measures
Authors: Mahowald Mark
Published: December 2010

In this note we will discuss the question of the measurability of the multiplier function of a relatively invariant measure on a group. That is, for a group G, σ-ring S, and a measure μ defined on the sets of S, we assume: E in S, x in G implies xE is in S and μ(XE) = σ(x)μ(E) and study the measurability of the function σ(x).

The problem was discussed by Halmos (1, p. 265), on locally compact groups and there the situation proved to be as nice as it could be, that is, if the measure is a non-trivial, relatively invariant Baire measure then the multiplier function is continuous. We prove two theorems for groups in which no topology is assumed. In the first theorem we assume a shearing condition and answer the question completely. The second theorem places a condition on the measure and weakens the shearing assumption. Its proof is complicated and occupies the major portion of this paper.
Discrete Groups of Motions
Authors: Greenberg Leon
Published: December 2010

This paper deals with the discrete groups of rigid motions of the hyperbolic plane. It is known (12) that the finitely generated, orientation-preserving groups have the following presentations:

Generators: .

Defining relations:

where km = ambmam-1bm-1. We shall denote this group by F(p; n1, … , nd; r).

In particular, the finitely generated free groups are contained among these. Indeed, one purpose of this paper is to indicate some geometrical methods for investigating free groups.
The Gibbs Phenomenon for Taylor Means and for [F, Dn] Means
Authors: Miracle Chester L.
Published: December 2010

The Gibbs phenomenon may be described, quite generally, as follows. Let a sequence {fn(x)} (n= 0, 1, 2, … ,) converge to a functionf(x)for x in the interval x0<x <x0+ h. We say that {fn(x)} displays the Gibbs phenomenon in a right-hand neighbourhood of the point X0, if

A similar definition holds for a left-hand neighbourhood. If {fn(x)} displays the Gibbs phenomenon at both sides of x0, we say simply that {fn(x)} displays Gibbs phenomenon at the point X0.
Block Design Games
Authors: Hoffman A. J. , Richardson Moses
Published: December 2010

In this paper, we define and begin the study of an extensive family of simple n-person games based in a natural way on block designs, and hitherto for the most part unexplored except for the finite projective games (13). They should serve at least as a proving ground for conjectures about simple games. It is shown that many of these games are not strong and that many do not possess main simple solutions. In other cases, it is shown that they have no equitable main simple solution, that is, one in which the main simple vector has equal components. On the other hand, the even-dimensional finite projective games PG(2s, pn) with s > 1 possess equitable main simple solutions, although they are not strong either. These results are obtained by means of the study of the possible blocking coalitions. Interpretations in terms of graph theory, network flows, and linear programming are discusssed, as well as k-stability, automorphism groups, and some unsolved problems.
Sylow Theory for a Certain Class of Operator Groups
Authors: Ayoub Christine W.
Published: December 2010

In this paper we consider again the group-theoretic configuration studied in (1) and (2). Let G be an additive group (not necessarily abelian), let M be a system of operators for G, and let ϕ be a family of admissible subgroups which form a complete lattice relative to intersection and compositum. Under these circumstances we call G an M — ϕ group. In (1) we studied the normal chains for an M — ϕ group and the relation between certain normal chains. In (2) we considered the possibility of representing an M — ϕ group as the direct sum of certain of its subgroups, and proved that with suitable restrictions on the M — ϕ group the analogue of the following theorem for finite groups holds: A group is the direct product of its Sylow subgroups if and only if it is nilpotent. Here we show that under suitable hypotheses (hypotheses (I), (II), and (III) stated at the beginning of §3) it is possible to generalize to M — ϕ groups many of the Sylow theorems of classical group theorem.
Properties of the Coefficients of Orthonormal Sequences
Authors: Bullen P. S.
Published: December 2010

In this paper we consider complete orthonormal sequences defined on the interval [0, 1] and satisfying an inequality of the type

(1)

for allnand some sequence {Fn}. Such sequences were first considered by Zygmund and Marcinkiewicz (8). They extended the well-known results of Hausdorff-Young and Paley, originally proved for the casev= ∞,Fn=Mfor alln(12). We will consider cases of equality in t he Hausdorff-Young theorems and certain limiting cases of the Paley theorems. Application of these results and the results in (8) will be made to functions harmonic in the unit α-sphere.
Some Two-Dimensional Unitary Groups Generated by Three Reflections
Authors: Crowe D. W.
Published: December 2010

Shephard and Todd (5) give generators for the finite primitive irreducible groups generated by two unitary reflections in U2. It is the purpose of the present paper to give generating reflections, and defining relations in terms of these reflections, for the seven such groups requiring three generating reflections, that is, for their nos. 7, 11, 12, 13, 15, 19, 22. The reflections are chosen whenever possible so that their product has the property suggested by Theorem 5.4 of (5). That is, except for no. 15, the period of the product of the three generating reflections is h = m2 + 1, and the characteristic roots of this product are 2πim1/h and 2πim2/h, where m1 and m2 are the “exponents“ (5, p. 282) of the group. The reason for the impossibility of such a choice for no. 15 is given in § 4. In § 5 the homomorphisms between these groups and certain groups of motions in elliptic 3-space are determined.
Polar Means of Convex Bodies and a Dual to the Brunn-Minkowski Theorem
Authors: Firey William J.
Published: December 2010

This paper deals with processes of combining convex bodies in Euclidean n-space which are, in a sense, dual to the process of Minkowski addition and some of its generalizations.

All the convex bodies considered will have a common interior point Q. Variables x and y denote vectors drawn from Q; we shall speak of their terminal points as the points x and y. Unit vectors will be denoted by u; ||x|| signifies the length of x. Convex bodies will be symbolized by K with distinguishing marks. ∂K means the boundary of K. λK will mean the image of K under a homothetic transformation in the ratio λ : 1. The centre of the homothety will always be Q.
On Nilpotent Products of Cyclic Groups. II
Authors: Struik Ruth Rebekka
Published: December 2010

In a previous paper (18), G = F/Fn was studied for F a free product of a finite number of cyclic groups, and Fn the normal subgroup generated by commutators of weight n. In that paper the following cases were completely treated:

(a) F a free product of cyclic groups of order pαi, p a prime, αi positive integers, and n = 4, 5, … , p + 1.

(b) F a free product of cyclic groups of order 2αi, and n = 4.

In this paper, the following case is completely treated:

(c) F a free product of cyclic groups of order pαi p a prime, αi positive integers, and n = p + 2.

(Note that n = 2 is well known, and n — 3 was studied by Golovin (2).) By ‘'completely treated” is meant: a unique representation of elements of the group is given, and the order of the group is indicated. In the case of n = 4, a multiplication table was given.
An Inversion Formula for the Weierstrass Transform
Authors: Bilodeau G. G.
Published: December 2010

The Weierstrass transform f(x) of a function ϕ(y) is defined by

1.1

where

whenever this integral exists (7, p. 174). It is also known as the Gauss transform (11; 12). Its basic properties have been developed and studied in (7) and in particular it has been shown that the symbolic operator

will invert this transform under suitable assumptions and with certain definitions of this operator. We propose to study the definition

for f(x) in C∞. This formula seems to have been first examined by Pollard (9) and later by Rooney (12). In so far as convergence of (1.2) is concerned, we will considerably improve the results (12).
Contributions to the Cell Growth Problem
Authors: Read R. C.
Published: December 2010

The cell growth problem is a combinatorial problem which may be stated as follows: A plane animal is made up of cells, each of which is a square of unit area. It starts as a single cell, and grows by adding cells one at a time in such a way that the new cell has at least one side in contact with a side of a cell already present in the animal. The problem is to find the number of different animals of area n, it being understood that animals which can be transformed into each other by reflections or rotations of the plane will be regarded as the same animal.
Transcendental Elements in Continuous Rings
Authors: Halperin Israel
Published: December 2010

In (2), John von Neumann introduced the concept of a continuous ring as a generalization to the infinite limiting case of the total matric algebras over a division ring. Von Neumann sketched a theory of arithmetic for such continuous rings and asserted :

(⋆) every continuous ring contains purely transcendental elements c.

This means: for every polynomial p(t) = tm + Z1tm-1 + . . . + zm {m ≥ 1) which has all coefficients zi in the centre of , the element p(c) has a reciprocal in , that is, (p(c))-l exists such that p(c).(p(c))-l = (p(c))-l.p(c) = 1.