bluuebunny/crossref_metadata_2025 huggingface dataset:
On Ward's Perron-Stieltjes Integral
Authors: Henstock Ralph
Published: December 2010

In the paper (5), Ward defines an integral of Perron type of a finite
function f with respect to another finite function
g, where g need not be of bounded
variation. There arise two problems, (a) and
(b) below, that have not been dealt with in (5).

If f = j at a countable number of points
everywhere dense in (a, b), where
f and j are both integrable with
respect to g, then f — j
can be nonzero on a large set of points of (a,
b).
Short Proof of a Map-Colour Theorem
Authors: Dirac G. A.
Published: December 2010

Heawood (3) showed that for h > 1 the countries of any
map on a surface of connectivity h = 3 − x
can be coloured using at most

colours. In a previous paper (1)1 proved the following
Groups Generated by Unitary Reflections of Period
Two
Authors: Coxeter H. S. M.
Published: December 2010

In complex affine n-space with a unitary metric, a
reflection is a congruent transformation leaving
invariant all the points of a hyperplane. Thus the characteristic roots of a
unitary reflection of period p consist of a primitive
pth root of unity and n – 1 unities. A
group generated by n reflections is conveniently
represented by a graph having a node for each generator and a branch for
each pair of non-commutative generators.
Imbedding Conditions for Hermitian and Normal
Matrices
Authors: Fan Ky , Pall Gordon
Published: December 2010

Let A, B be two square matrices with
complex coefficients, of respective orders n and
m, where n ≥ m. We
shall say that B is imbeddable in
A if there exists a unitary matrix U of
order n such that U*AU contains
B a s a principal submatrix. In other words,
B is said to be imbeddable in A if
there exists a matrix V of type n ×
m such that V*V = IW (= the identity matrix of order m) and
V*AV = B.
Inequalities for Symmetric Functions and Hermitian
Matrices
Authors: Marcus M. , Lopes L.
Published: December 2010

The purpose of this paper is to present two concavity results for symmetric
functions and apply these to obtain inequalities connecting the
characteristic roots of the non-negative Hermitian (n.n.h.) matrices
A, B and A +
B.
Prime Power Representations of Finite Linear Groups
II
Authors: Steinberg Robert
Published: December 2010

The aim of this paper is two-fold: first, to extend the results of (4) to
the exceptional finite Lie groups recently discovered by Chevalley (1), and,
secondly, to give a construction which works simultaneously for the groups
An, Bn, Cn, Dn, En, F4 and G2 (in the usual Lie group notation), and which depends only on
intrinsic structural properties of these groups.
Combinatorial Properties of Matrices of Zeros and
Ones
Authors: Ryser H. J.
Published: December 2010

This paper is concerned with a matrix A of
m rows and n columns, all of whose
entries are 0's and l's. Let the sum of row i of A be
denoted by ri (i = 1, … , m) and let the sum of column i of
A be denoted by si (i = 1, … , n).
The Projection of a Linear Functional on the Manifold of
Integrals
Authors: Gordon H. , Lorch E. R.
Published: December 2010

If μ is a measure defined on a space and fn(x) is a sequence of
μ-integrable functions converging to
f(x), then under suitable conditions of
boundedness, one has the theorem of Lebesgue that

.
The Equivalence of Quadratic Forms
Authors: Watson G. L.
Published: December 2010

The main object of this paper is to find the number of classes in a genus of
indefinite quadratic forms, with integral coefficients, in
k ≥ 4 variables, distinguishing for even
k two cases, according as improper equivalence is or is
not admitted.
Geodesic Groups of Minimal Surfaces
Authors: Helfenstein H. G.
Published: December 2010

In a previous paper (6) we have studied those minimal surfaces which admit
geodesic mappings without isometries or similarities on another, not necessarily
minimal, surface. Here we determine all pairs of minimal surfaces which can be
geodesically mapped on each other. We find that two such surfaces are
either:

(i) similar Bonnet associates of each other, or

(ii) both Poisson surfaces (that is, isometric to a plane), or

(iii) both Scherk surfaces (2).
On the Convergence of Mean Values Over Lattices
Authors: Schmidt Wolfgang
Published: December 2010

Recently C. A.Rogers (2, Theorem 4) proved the following theorem which
applies to many problems in geometry of numbers:

Let
f(X1,X2, …
, Xk) be a non-negative B or el-measurable function in the
nk-dimensional space of points (X1,X2.
Further, let Λo be the fundamental lattice, Ω a linear
transformation of determinant 1, F a fundamental region in the space of linear
transformations of determinant 1, defined with respect to the subgroup of
unimodular transformations and μ(Ω) the invariant measure1 on the space of linear
transformations of determinant 1 in Rn.
Algebraic Approximation of Curves
Authors: Wallace A. H.
Published: December 2010

In his paper on the algebraic approximation of differentiable manifolds Nash
(1) introduced the concept of a sheet of a real algebraic variety (see the
definition in §16 below) and raised certain questions of a general nature. In
attempting to answer these questions it has been necessary to evolve some sort of
technique for manipulating curves on algebraic varieties, and, in particular, to
set up a criterion for the possibility of approximating a sequence of analytic
arcs (definition in §1) joined end to end by a single analytic arc.
On Certain Pairs of Matrices which Generate free Groups
Authors: Chang Bomshik , Jennings S. A. , Ree Rimhak
Published: December 2010

Denote by Fα,β the multiplicative group generated
by the two matrices

where α and β are complex numbers. Sanov (3) proved that
F
α,β is free, and Brenner (1) showed that
F
m,m is free if m ⩾ 2.
Groupoïdes Automorphes par le Groupe Géométrique et Quasіgroupes
“Endo”
Authors: Sade A.
Published: December 2010

L'ensemble des nombres ∈
Z/n, premiers avec
l'entier n, forme un groupe (le groupe géométrique) G,
par rapport à la multiplication. Etant donné un ensemble de nombres réels, M, un
groupoïde Q, formé d'éléments quelconques,
x, est automorphe par le groupe géométrique si (i)
pour tout élément x ∈ Q et
tout nombre m ∈ M, la
multiplication xm est définie; (ii) l'application
(x→xm) est un
automorphisme de Q.
On Integration of Vector-Valued Functions
Authors: Snow D. O.
Published: December 2010

Among the variety of integrals which have been devised for integrating
vector-valued functions the most widely used is that of Bochner (2), perhaps
because of the simplicity of its formulation. Other approaches, including one by
Birkhoff (1), have yielded more general integrals yet none of them seems to have
supplanted the Bochner integral to a significant extent.
Function which have Generalized Riemann Derivatives
Authors: Kassimatis C.
Published: December 2010

Letƒ(x) be a measurable function denned in the
interval (a, b), and let

If the limit of

exists and is finite at the point
x, as h → 0, it is called
the wth generalized Riemann derivative of ƒ(x) at the
point x,
Dnƒ(x).
Spectral Theory for the Differential Equation Lu= λ Mu
Authors: Brauer Fred
Published: December 2010

Let L and M be linear
ordinary differential operators defined on an interval
I, not necessarily bounded, of the real line. We
wish to consider the expansion of arbitrary functions in eigenfunctions of the
differential equation Lu =
λMu on I. The case where
M is the identity operator and
L has a self-adjoint realization as an operator in
the Hilbert space L
2(I) has been treated in various
ways by several authors; an extensive bibliography may be found in (4) or
(8).
On the Problème Des Ménages
Authors: Wyman Max , Moser Leo
Published: December 2010

The classical problème des
ménages asks for the number of ways of seating at a
circular table n married couples, husbands and wives
alternating, so that no husband is next to his own wife.

An outline of the history of the problem to 1946 was given by Kaplansky and
Riordan (11). They also presented a bibliography, which is augmented and brought
up to date in the bibliography of the present paper.
The Chords of the Non-Ruled Quadric In PG(3, 3)
Authors: Coxeter H. S. M.
Published: December 2010

In the preceding paper, Tutte described the forty-five chords of an
“ellipsoid” in the finite space PG(3, 3), showing that
they may be regarded as the edges of a remarkable graph whose group is the group
of automorphisms of the symmetric group . The object of
this sequel is to relate Tutte's idea to Sylvester's combinatorial investigation
of the fifteen “duads” and fifteen “synthemes” formed by six symbols, and to
Richmond's discussion of the figure of six points in a projective
4-space.
Coverings of Bipartite Graphs
Authors: Dulmage A. L. , Mendelsohn N. S.
Published: December 2010

For the purpose of analysing bipartite graphs (hereinafter called simply
graphs) the concept of an exterior covering is introduced. In terms of this
concept it is possible in a natural way to decompose any graph into two parts, an
inadmissible part and a core. It is also possible to decompose the core into
irreducible parts and thus obtain a canonical reduction of the graph. The concept
of irreducibility is very easily and naturally expressed in terms of exterior
coverings. The role of the inadmissible edges of a graph is to obstruct certain
natural coverings of the graph.
Oriented Flat Submanifolds in an Affine Space
Authors: Englefield M. J.
Published: December 2010

The simplest examples of figures which possess inner orientation are a sensed
line, and a plane in which a sense of rotation is specified. Suppose two sensed
lines, which intersect in a finite point, are given in a definite order. Then
there is only one way in which the first can be rotated to coincide with the
second without passing through the second. In this way, two ordered, sensed lines
determine a sense of rotation in the plane which contains them.
Exemple Effectif D'un Ensemble Transfiniment Non-Projectif
Authors: Rothberger F.
Published: December 2010

La notion d'ensemble projectif de classe finie P

n
peut être généralisée pour les classes transfinies
P

α
(α < Ω). Nous allons ici construire effectivement un ensemble
qui n'est pas projectif, même dans ce sens généralisé.
On the Denjoy Conjecture
Authors: Jenkins James A.
Published: December 2010

In recent years many of the properties of regular functions have been shown
to extend to quasiconformal mappings. (The latter term is here understood in the
sense defined in (5).) This is particularly true of those results which can be
proved by use of the method of the extremal metric. It is rather strange, then,
that a result which constitutes one of the first notable applications of this
method has not been so extended (at least to the author's knowledge).
F-Rings of Continuous Functions I
Authors: Brainerd Barron
Published: December 2010

It is well known (2, 4) that the ring of all real (complex) continuous functions on a compact Hausdorff space can be characterized algebraically as a Banach algebra which satisfies certain additional intrinsic conditions. It might be expected that rings of all continuous functions on other topological spaces also have algebraic characterizations. The main purpose of this note is to discuss two such characterizations. In both cases the characterizations are given in the terms of the theory of F-brings (1). In one case a characterization is given for the ring of all (real) continuous functions on a generalized P-space, that is, a zero-dimensional topological space in which the class of open-closed sets forms a σ-algebra. A Hausdorff generalized P-space is a P-space in the terminology of (3). In the other case a theorem of Sikorski (6) is employed to give a characterization of the ring of all (real) continuous functions on an upper X1-compact P-space.
Continued Fractions with Absolutely Convergent Even or Odd Part
Authors: Dawson David F.
Published: December 2010

The purpose of this paper is to give conditions under which the absolute convergence of the subsequence of odd or of even approximants to a continued fraction implies convergence of the continued fraction. In § 2 we consider the problem in general, and in § 3 we impose a condition which gives absolute convergence of the odd or of the even part of the continued fraction and state conditions which imply convergence of the continued fraction.
Series of Products of Bessel Polynomials
Authors: Ragab F. M.
Published: December 2010

The Bessel polynomials, which arise as solution of the classical wave equation in spherical co-ordinates, are defined by Krall and Frink (3) by the equation

1

The purpose of this paper is to present some series of products of these polynomials when the two arguments are different as in the case of Legendre and Hermite polynomials. Such an explanation was given by Brafman (2), namely :

2

These series will be stated and proved in § 2.
On the Representation of Functions as Fourier Transforms
Authors: Rooney P. G.
Published: December 2010

If f ∈ Lp (— ∞, ∞), 1 < p ≤ 2, then f has a Fourier-Plancherel transform F ∈ Lq (— ∞, ∞) where p-1 + q-1 = 1. Also if ∣x∣1-2/qf(x) ∈ Lq (— ∞, ∞), q ≥ 2, then / has a Fourier-Plancherel transform F ∈ Lq (— ∞, ∞). These results can be found in (2, Theorems 74 and 79). In neither case, however, does the collection of transforms cover Lq, except when p = q = 2, and in neither case, with the same exception, has the collection of transforms been characterized.

Further, if f ∈ Lp, (— ∞, ∞), 1 < p ≤ 2, then its transform F has the property |x|1-2/pF(x) ∈ Lp (— ∞, ∞) (see 2, Theorem 80) but, except when p = 2, the collection of transforms does not cover the set of functions with this property, and again, except when p = 2, the collection of transforms has not been characterized.
The Elliptic Integrals of the Third Kind
Authors: Neville E. H.
Published: December 2010

This paper develops a case for adopting as the standard elliptic integrals of the third kind the function IIs(u, a) defined by

and the three functions lIs (u, a + Kc), Πs(u, a + Kn), Πss(u, a+ Kn) where KC, Kn, Kn are the three quarter-periods of the Jacobian system. The function Πs(u, a) is the same function whether qs u is cs u, ns u, or ds u.

The origin of the paper was a wish to understand how it has come about that the integrals commonly accepted as standard are not related symmetrically to the theta functions in terms of which they are expressed.
An Improved Result Concerning Singular Manifolds of Difference Polynomials
Authors: Cohn Richard M.
Published: December 2010

Let be a difference field of characteristic 0, m an irreducible manifold of effective order n over {y}, and F an algebraically irreducible difference polynomial in {y} of effective order n + k, k > 0, which vanishes on 3 m. In an earlier paper (2, p. 447) I gave necessary conditions, restated below as (a), (b), and (c) of the main theorem, for m to be an essential singular manifold of F. These conditions are analogous to the low power criterion of Ritt (1, p. 65) for the corresponding problem of differential algebra. Like that criterion they depend, in the special case that m is the manifold of y, only on which power products appear effectively in F. Unlike the low power criterion, however, conditions (a), (b), and (c) are only necessary, not sufficient.
Subspaces of a Generalized Metric Space
Authors: Eliopoulos H. A.
Published: December 2010

In a paper published in 1956, Rund (4) developed the differential geometry of a hypersurface of n— 1 dimensions imbedded in a Finsler space of n dimensions, considered as locally Minkowskian.

The purpose of the present paper is to provide an extension of the results of (4) and thus develop a theory for the case of m-dimensional subspaces imbedded in a generalized (Finsler) metric space.

We consider an n-dimensional differentiable manifold Xn and we restrict our attention to a suitably chosen co-ordinate neighbourhood of Xn in which a co-ordinate system xi (i= 1, 2, … , n), is defined. A system of equations of the type xi = xi(t) defines a curve C of Xn the tangent vector dxi/dt of which is denoted by xi.
Dense Subgraphs and Connectivity
Authors: Nettleton R. E. , Goldberg K. , Green M. S.
Published: December 2010

A proper subgraph of a connected linear graph is said to disconnect the graph if removing it leaves a disconnected graph. In this paper we characterize, in the following sense, the disconnecting subgraphs of a fixed connected graph. We define two distinct types of disconnecting subgraphs (isthmuses and articulators) which are minimal in the sense that no proper subgraph of either type can disconnect the graph. We then show that any disconnecting subgraph must contain either an isthmus or an articulator. We also define a set of subgraphs (called dense) which form a lattice. We show that the union of the minimal dense subgraphs contains all isthmuses and articulators. In terms of these subgraphs we investigate some of the consequences of assuming that a disconnecting subgraph must contain at least m points.
The Term and Stochastic Ranks of a Matrix
Authors: Mendelsohn N. S. , Dulmage A. L.
Published: December 2010

The term rank p of a matrix is the order of the largest minor which has a non-zero term in the expansion of its determinant. In a recent paper (1), the authors made the following conjecture. If S is the sum of all the entries in a square matrix of non-negative real numbers and if M is the maximum row or column sum, then the term rank p of the matrix is greater than or equal to the least integer which is greater than or equal to S/M. A generalization of this conjecture is proved in § 2.

The term doubly stochastic has been used to describe a matrix of nonnegative entries in which the row and column sums are all equal to one. In this paper, by a doubly stochastic matrix, the, authors mean a matrix of non-negative entries in which the row and column sums are all equal to the same real number T.
Separation and Approximation in Topological Vector Lattices
Authors: Leader Solomon
Published: December 2010

Spectral theory in its lattice-theoretic setting proves abstractly that the indicators of measurable sets generate the space L of Lebesgue-integrable functions on an interval. We are concerned here with abstractions suggested by the fact that indicators of intervals suffice to generate L. Our results show that the approximation of arbitrary elements of a topological vector lattice rests upon the ability to separate disjoint elements/ and g by an operation that behaves in the limit like a projection annihilating/ and leaving g invariant.

The introduction of this concept of separation together with the notion of limit unit leads (via the Fundamental Lemma) to abstract generalizations of the Radon-Nikodym Theorem (Theorem 1) and the Stone-Weierstrass Theorem (Theorem 3).
Supersoluble Immersion
Authors: Baer Reinhold
Published: December 2010

Supersoluble immersion of a normal subgroup K of a finite group G shall be defined by the following property:

If σ is a homomorphism of G, and if the minimal normal subgroup J of Gσ is part of Kσ then J is cyclic (of order a prime).

Our principal aim in the present investigation is the proof of the equivalence of the following three properties of the normal subgroup K of the finite group G:

(i)

K is supersolubly immersed in G.

(ii)

K/ϕK is supersolubly immersed in G/ϕK.

(iii)

If θ is the group of automorphisms induced in the p-subgroup U of K by elements in the normalizer of U in G, then θ' θp-1 is a p-subgroup of θ.

Though most of our discussion is concerned with the proof of this theorem, some of our concepts and results are of independent interest.
Extremal Properties of Hermitian Matrices. II
Authors: Marcus M. , Moyls B. N. , Westwick R.
Published: December 2010

Let H be an n-square Hermitian matrix with eigenvalues h1 ≥ h2 ≥ . . . ≥ hn. Fan (2) showed that

1

k — 1,2, … , n, where the max and min are taken over all sets of k orthonormal (o.n.) vectors in unitary w-space Vn. Marcus and McGregor (3) have generalized this result in the case that H is non-negative Hermitian. For vectors X1, … , xr r ≤ n, in Vn, let X1 Λ X2 Λ … Λ xT denote the Grassmann exterior product of the xt; it is a vector in Vm,, where
Prime Dual Ideals in Boolean Algebras
Authors: Heider L. J.
Published: December 2010

Let denote an arbitrary Boolean algebra. Let Latin letters a, b, … denote general elements of while the symbols 0, 1 denote the special smallest and largest elements. Let Greek letters α, β, … denote various prime dual ideals of elements of . It is recalled that a prime dual ideal of is a proper subset of closed under finite intersections of its elements and maximal with respect to those properties. Every prime dual ideal includes the element 1 and for each element a of includes either a or ā (complement of ain ) but not both. Occasional reference will be made to principal dual ideals of . These are subsets of composed of all elements of majorizing some fixed non-zero element of . Finally, let X() denote the collection of all prime dual ideals of .
On a Type Problem
Authors: Jenkins James A.
Published: December 2010

Considerable interest has attached to the problem of determining the type of a Riemann surface obtained by performing an identification between the edges of a strip or a half-strip (1, 2, 4, 5, 8). A fairly thorough analysis was made in 1946 by Volkovyskii (6) who gave various sufficient conditions for parabolic and hyperbolic type. The object of the present paper is to show that his principal sufficient condition for hyperbolic type can be substantially improved.
A Network-Flow Feasibility Theorem and Combinatorial Applications
Authors: Fulkerson D. R.
Published: December 2010

There are a number of interesting theorems, relative to capacitated networks, that give necessary and sufficient conditions for the existence of flows satisfying constraints of various kinds. Typical of these are the supply-demand theorem due to Gale (4), which states a condition for the existence of a flow satisfying demands at certain nodes from supplies at other nodes, and the Hoffman circulation theorem (received by the present author in private communication), which states a condition for the existence of a circulatory flow in a network in which each arc has associated with it not only an upper bound for the arc flow, but a lower bound as well. If the constraints on flows are integral (for example, if the bounds on arc flows for the circulation theorem are integers), it is also true that integral flows meeting the requirements exist provided any flow does so.
Sur un Theorème de G. Szekeres, Concernant les Fonctions Monotones et Convexes
Authors: Marcus S.
Published: December 2010

Dans une note récente (1), Szekeres a donné le théorème suivant:

Sif est une fonction réelle, définie, strictement croissante et ayant la deuxième dérivée continue sur (a, b) ( — ∞ ≤ a < b ≤ ≤) alors il existe une fonction réelle ϕ, définie et strictement croissante sur (a, b), et une fonction réelle ψ définie et strictement croissante sur (ϕ(a), ϕ(b)), telles que ϕ˝,(x) ≥ 0 pour a < x < b, ψ ˝(u) < 0 pour ϕ(a) < u < ϕ(b) et f(x) = ψ(ϕ(x)) pour a < x < b.

Dans la démonstration de ce théorème, Szekeres utilise une expression où la dérivée f‘(x) apparaît au dénominateur, ce qui exige qu'on ait f'(x) ≠ 0 pour a < x < b, condition qui ne figure pas dans l'énoncé du théorème. Il est naturel alors de poser le problème si le théorème ci-dessus reste en vigueur si cette condition n'est pas remplie. Il y a lieu aussi de se demander si les autres conditions, par exemple l'existence et la continuité def“(x) sont nécessaires pour qu'une fonction strictement croissante puisse être représentée comme une superposition de deux fonctions croissantes, l'une convexe, l'autre concave.
The Asymptotic Behaviour of the Laurent Coefficients
Authors: Wyman Max
Published: December 2010

LetG(z)be a function of a complex variable, regular in the annulus 0 ≤ a ≤ |z| <b≤ ∞. We shall assume there exists a curve within the annulus for which

providedzis restricted to be a point of this curve. Under these restrictionsG (z)has a Laurent expansion of the form

1.1

where the Laurent coefficientsanhave the integral representation

1.2

andCcan be any contour, within the domain of regularity, that enclosesz= 0. We shall also assume that theanare all real numbers. Using the usual complex conjugate notation, we can, therefore, write

1.3

The problem of determining the asymptotic behaviour ofanasn —> co is very old in mathematical literature and appears in many forms and disguises.
On the Structure of Half-Groups
Authors: Whittaker James V.
Published: December 2010

Furstenberg (1) and the author (4), among others, have exhibited postulate systems for groups in terms of the operation x — y = x + ( — y).Furstenberg also investigated a system obtained by removing one of his postulates which defines what he called a half-group. A structure theorem for half-groups was given in (1). In the present paper, we prove another structure theorem for half-groups. A more restricted entity, called a pseudo-group, is introduced, and its structure, together with that of a half-group satisfying the left and right cancellation laws, is studied. Finally, some topological questions concerning their structure are also considered.
Further Results on the Construction of Mutually Orthogonal Latin Squares and the Falsity of Euler's Conjecture
Authors: Bose R. C. , Shrikhande S. S. , Parker E. T.
Published: December 2010

If

is the prime power decomposition of an integer v, and we define the arithmetic function n(v) by

then it is known, MacNeish (10) and Mann (11), that there exists a set of at least n(v) mutually orthogonal Latin squares (m.o.l.s.) of order v. We shall denote by N(v) the maximum possible number of mutually orthogonal Latin squares of order v. Then the Mann-MacNeish theorem can be stated as

MacNeish conjectured that the actual value of N(v) is n(v).
On the Maximal Number of Pairwise Orthogonal Latin Squares of a Given Order
Authors: Chowla S. , Erdös P. , Straus E. G.
Published: December 2010

In the preceding paper Bose, Shrikhande, and Parker give their important discovery of the disproof of Euler's conjecture on Latin squares. In this paper we show that their results can be strengthened to imply that N(n), the maximal number of pairwise orthogonal Latin squares of order n, tends to infinity with n. In fact there exists a positive constant c, such that N(n) > nc for all sufficiently large n.

Our proof involves no new combinatorial insights, but is based entirely on a number-theoretical investigation of the following inequality due to Bose and Shrikhande.
Limits of Lattices in a Compactly Generated Group
Authors: Macbeath A. M. , Świerczkowski S.
Published: December 2010

Let G be a locally compact and (σ-compact topological group and let H be a discrete subgroup of G. We shall use G/H to denote the space of right cosets Hx of H with the usual topology (cf. (8, pp. 26-28)). Let μ be the left Haar measure in G. μ induces a measure in the space G/H3; this measure will, without ambiguity in this paper, also be denoted by μ. If μ(G/H) is finite, the group H is called a lattice. If the space G/H is compact, then H is certainly a lattice and is called a bounded lattice. These terms are an extension of the usage of the Geometry of Numbers, where G is the real n-dimensional vector space Rn. In this case any lattice is generated by n linearly independent vectors, all lattices are bounded, and the whole family of lattices is permuted transitively by the automorphisms of G (which are the non-singular linear transformations).
A Theorem on Pure Submodules
Authors: Kolettis George
Published: December 2010

In (1) Baer studied the following problem: If a torsion-free abelian group G is a direct sum of groups of rank one, is every direct summand of G also a direct sum of groups of rank one? For groups satisfying a certain chain condition, Baer gave a solution. Kulikov, in (3), supplied an affirmative answer, assuming only that G is countable. In a recent paper (2), Kaplansky settles the issue by reducing the general case to the countable case where Kulikov's solution is applicable. As usual, the result extends to modules over a principal ideal ring R (commutative with unit, no divisors of zero, every ideal principal).

The object of this paper is to carry out a similar investigation for pure submodules, a somewhat larger class of submodules than the class of direct summands. We ask: if the torsion-free i?-module M is a direct sum of modules of rank one, is every pure submodule N of M also a direct sum of modules of rank one? Unlike the situation for direct summands, here the answer depends heavily on the ring R.
Generalized Lie Elements
Authors: Ree Rimhak
Published: December 2010

Let λ(ij),i,j =1, 2, … ,m, bem2elements in a fieldKof characteristic zero such thatλ(ij)λ(ji) = 1 for alliandj, and X1,x2, … , xmnon-commutative associative indeterminates overK. Define the elements [xi1Xi2…xin] inductively by [xi] =xiand

Any linear combination of the elements

with coefficients inKwill be called ageneralized Lie elememt. Generalized Lie elements reduce to ordinary Lie elements ifλ(ij) = 1 for alliandj.

The purpose of this paper is to generalize to the generalized Lie elements the following: a theorem of Friedrichs, a theorem of Dynkin-Specht-Wever (2), and the Witt formula on the dimension of the space spanned by homogeneous Lie elements of a fixed degree. The set of all generalized Lie elements will be made into an algebra which generalizes the ordinary free Lie algebra. This algebra turns out to be free in a certain sense. We shall also generalize the algebra associated with shuffles in (2).
On Immersion of Manifolds
Authors: Samelson Hans
Published: December 2010

In (3) R. Lashof and S. Smale proved among other things the following theorem. If the compact oriented manifold M is immersed into the oriented manifold M', with dim M' ≥ dim M + 2, then the normal degree of the immersion is equal to the Euler-Poincaré characteristic x of M reduced module the characteristic x’ of M'. If M’ is not compact, x' is replaced by 0. “Manifold” always means C∞-manifold. An immersion is a differentiable (that is, C∞) map f whose differential df is non-singular throughout. The normal degree is defined in a certain fashion using the normal bundle of M in M', derived from f, and injecting it into the tangent bundle of M'

It is our purpose to give an elementary proof, using vector fields, of this theorem, and at the same time to identify the homology class that represents the normal degree (Theorem I), and to give a proof, using the theory of Morse, for the special case M’ = Euclidean space (Theorem II).
On Orientations, Connectivity and Odd-Vertex-Pairings in Finite Graphs
Authors: Nash-Williams C. ST. J. A.
Published: December 2010

The integer part of a non-negative real number p will be denoted by [p]. For any integer n, n* will denote the greatest even integer less than or equal to n, that is, n* = n or n — 1 according as n is even or odd respectively.

The order of a set A, denoted by |A|, is the number of elements in A. The set whose elements are a1, a2, … , an will be denoted by {a1, a2 … , an. The empty set will be denoted by Λ. A set will be said to include each of its elements. A set separates two elements if it includes one but not both of them.

An unoriented graph U consists of two disjoint sets V(U), E(U), the elements of V(U) being called vertices of U and the elements of V(U) being called edges of U, together with a relationship whereby with each edge is associated an unordered pair of distinct vertices which the edge is said to join.
On the Projective Centres of Convex Curves
Authors: Kelly Paul , Straus E. G.
Published: December 2010

We consider a closed curve C in the projective plane and the projective involutions which map C into itself. Any such mapping γ, other than the identity, is a harmonic homology whose axis η we call a projective axis of C and whose centre p we call an interior or exterior projective centre according as it is inside or outside C. The involutions are the generators of a group Γ, and the set of centres and the set of axes are invariant under Γ. The present paper is concerned with the type of centre sets which can exist and with the relationship between the nature of C and its centre set.

If C is a conic, then every point which is not on C is a projective centre. Conversely, it was shown by Kojima (4) that if C has a chord of interior centres, or a full line of exterior centres, then C is a conic.
Foundations of the Theory of Dynamical Systems of Infinitely Many Degrees of Freedom, II
Authors: Segal I. E.
Published: December 2010

The notion of quantum field remains at this time still rather elusive from a rigorous standpoint. In conventional physical theory such a field is defined in essentially the same way as in the original work of Heisenberg and Pauli (1) by a function ϕ(x, y, z, t) on space-time whose values are operators. It was recognized very early, however, by Bohr and Rosenfeld (2) that, even in the case of a free field, no physical meaning could be attached to the values of the field at a particular point—only the suitably smoothed averages over finite space-time regions had such a meaning. This physical result has a mathematical counterpart in the impossibility of formulating ϕ(x, y, z, t) as a bona fide operator for even the simplest fields (in any fashion satisfying the most elementary non-trivial theoretical desiderata), while on the other hand for suitable functions f, the integral ∫ϕ(x, y, zy t)f(x, y, z, t)dxdydzdt could be so formulated.
A Note on Non-Negative Matrices
Authors: Putnam C. R.
Published: December 2010

This note can be regarded as an addendum to the paper (4). On the complex Hilbert space of vectors x = (x1, x2, … ,) a matrix A is said to be bounded if there exists a constant M such that ||Ax|| ≦ M||x|| whenever ||x||2 = Σ|xk|2 < ∞ ; the least such M is denoted by ||A||. Only bounded matrices A and vectors x satisfying ||x|| < ∞ will be considered in the sequel. The spectrum of A, denoted by sp(A), is the set of values for which the resolvent R(λ) = (A — λI)-1 fails to be bounded. The notation A ≥ 0 or A > 0, where A = (aij), means that, for all i and j , aij ≥ 0 or aij 0 respectively.
Homotopy and Isotopy Properties of Topological Spaces
Authors: Hu Sze-Tsen
Published: December 2010

The most important notion in topology is that of ahomeomorphism f: X→Yfrom a topological spaceXonto a topological spaceY. If a homeomorphism f:X→Yexists, then the topological spaces X andFare said to behomeomorphic(ortopologically equivalent), in symbols,

X ≡ Y.

The relation ≡ among topological spaces is obviously reflexive, symmetric, and transitive; hence it is an equivalence relation. For an arbitrary familyFof topological spaces, this equivalence relation ≡ divides /Mnto disjoint equivalence classes called thetopology typesof the familyF. Then, the main problem in topology is the topological classification problem formulated as follows.

The topological classification problem:Given a familyF oftopological spaces, find an effective enumeration of the topology types of the familyFand exhibit a representative space in each of these topology types.
Some Problems for Typically Real Functions
Authors: Jenkins James A.
Published: December 2010

Many extremal properties of the class of normalized univalent functions are shared by the class of typically real functions each considered in the unit circle. By the class T of typically real functions we mean those functions f(z), regular for |z| < 1 with f(0) = 0, f'(0) = 1, and such that f(z) > 0 for z > 0, (z) < 0 for > 0. This class was first studied by Rogosinski (4) who proved various simple properties for it. Later Robertson (3) took up the study and proved the following important representation result.
Properties of Solutions of Parabolic Equations and Inequalities
Authors: Protter M. H.
Published: December 2010

In this paper we shall be concerned with two problems: (i) the asymptotic behavior of solutions of parabolic inequalities and (ii) the uniqueness of the Cauchy problem for such inequalities when the data are prescribed on a portion of a time-like surface. The unifying feature of these rather separate problems is the employment of integral estimates of the same type in both cases.

We consider parabolic operators in self-adjoint form

(1)

as well as the non-self-adjoint operator

(2)

where the coefficients aij(x, t) = aij (x1, x2... , xn, t) are C1 functions of x and t and the bij= bij(x, t) are C2 functions of x and t.
Moulton Planes
Authors: Pierce William A.
Published: December 2010

In 1902, F. R. Moulton (12) gave an early example of a non-Desarguesian plane. Its ‘points” are ordered pairs (x, y) of real numbers. Its “lines” coincide with lines of the real affine plane except that lines of negative slope are “bent” on the x-axis, line {y = b + mx}, for negative m, being replaced by {y = b + mx if y ≤ 0, y = [m/2]. [x + (b/m)] if y > 0}. A certain Desarguesian configuration in the classical plane is shifted just enough to vitiate Desargues’ Theorem for Moulton's geometry. The plane is neither a translation plane (“Veblen-Wedderburn” in the sense of Hall (7), p. 364) nor even the dual of one (Veblen and Wedderburn (17). It is natural to ask if the same construction is feasible when real numbers are replaced by elements from an arbitrary field.
Tetraspheres. I
Authors: De Majo A.
Published: December 2010

Les propriétés anallagmatiques de groupes de sphères ont été étudiées dans des contextes divers ces dernières années (voir (4), (5)). Dans la note qui suit nous étudions les propriétés anallagmatiques de certains groupes de 4 sphères, nous plaçant à un point de vue élémentaire.

A fin de ne pas alourdir la rédaction nous omettrons de spécifier chaque fois que les points, droites ou sphères considérés sont toujours supposés être dans la position relative la plus générale possible compatible avec les définitions données.

L'index i pourra toujours prendre les valeurs i = 1, 2, 3, 4.
On Stability in the Large for Systems of Ordinary Differential Equations
Authors: Hartman Philip
Published: December 2010

Autonomous systems. This note concerns the stability of systems of (real) differential equations in the large on Euclidean space En and on certain Riemannian manifolds Mn. The results will be refinements of those of Krasovski (3), (4), (5) and of Markus and Yamabe (8) and will make clear the role of the various assumptions in the type of theorems under consideration.

In this section, the main theorems are stated for autonomous systems

(1)

Their proofs are given in § 2, 3, 4. In § 5, 6, 7, generalizations to non-autonomous systems are made.
Some Properties of Noetherian Domains of Dimension One
Authors: Matlis Eben
Published: December 2010

Throughout this discussion R will be an integral domain with quotient field Q and K = Q/R ≠ 0. If A is an R-module, then A is said to be torsion-free (resp. divisible), if for every r ≠ 0 ∈ R the endomorphism of A defined by x → rx, x ∈ A, is a monomorphism (resp. epimorphism). If A is torsion-free, the rank of A is defined to be the dimension over Q of the vector space A ⊗R Q; (we note that a torsion-free R-module of rank one is the same thing as a non-zero R-submodule of Q). A will be said to be indecomposable, if A has no proper, non-zero, direct summands. We shall say that A has D.C.C., if A satisfies the descending chain condition for submodules. By dim R we shall mean the maximal length of a chain of prime ideals in R.
General Radicals that Coincide with the Classical Radical on Rings with D.C.C.
Authors: Divinsky N.
Published: December 2010

General radical theories were obtained by Amitsur (1; 2; 3) and Kurosh (6). Following Kurosh we say that a property of rings is a radical property if:

(a) Every homomorphic image of an -ring is an -ring;

(b) Every ring R contains an -ideal S which contains every other -ideal of R;

(c) The factor ring R/S is-semi-simple (that is, has no non-zero -ideals).
Counting Coloured Graphs
Authors: Wright E. M.
Published: December 2010

A graph on n labelled nodes is a set of n objects called “nodes”, distinguishable from each other, and a set (possibly empty) of “edges,” that is, pairs of nodes. Each edge is said to join its pair of nodes, at most one edge joins any two nodes and no edge joins a node to itself. By a k-colouring of such a graph we mean a mapping of the nodes of the graph onto a set of k distinct colours, such that no two nodes joined by an edge are mapped onto the same colour. We take k > 1.
A Normal form for a Matrix under the Unitary Congruence Group
Authors: Youla D. C.
Published: December 2010

Let C be a square matrix with complex elements. If C = C' (C' denotes the transpose of C) there exists a unitary matrix U such that

(1)

where the μ's are the non-negative square roots of the eigenvalues μ12, μ22, … , μn2 of C*C (C* is the adjoint of C) (2). If C is skew-symmetric, that is, C= — C”, there exists a unitary matrix U such that

(2)

where

(3)

and the α's are the positive square roots of the non-zero eigenvalues α12, α22, … , αk2 of C*C (1). Clearly rank C = 2k and the number of zeros appearing in (2) is n — 2k. Both (1) and (2) are classical. In a recent paper (3) Stander and Wiegman, apparently unaware of (1), give an alternative derivation of (2) with its appropriate generalization to quaternions.
A Census of Planar Triangulations
Authors: Tutte W. T.
Published: December 2010

Let P be a closed region in the plane bounded by a simple closed curve, and let S be a simplicial dissection of P. We may say that S is a dissection of P into a finite number α of triangles so that no vertex of any one triangle is an interior point of an edge of another. The triangles are ‘'topological” triangles and their edges are closed arcs which need not be straight segments. No two distinct edges of the dissection join the same two vertices, and no two triangles have more than two vertices in common.
Permanent Preservers on the Space of Doubly Stochastic Matrices
Authors: Moyls B. N. , Marcus Marvin , Minc Henryk
Published: December 2010

Let Mn be the linear space of n-square matrices with real elements. For a matrix X = (xij) ∈ Mn the permanent is defined by

where σ runs over all permutations of 1, 2, …, n. In (2) Marcus and May determine the nature of all linear transformations T of Mn into itself such that per T(X) = per X for all X ∈ Mn. For such a permanent preserver T, and for n < 3, there exist permutation matrices P, Q, and diagonal matrices D, L in Mn, such that per DL = 1 and either

or

Here X′ denotes the transpose of X. In the case n = 2, a different type of transformation is also possible.
On Finite Groups of the Form ABA
Authors: Gorenstein Daniel
Published: December 2010

The class of finite groups G of the form ABA, where A and B are subgroups of G, is of interest since it includes the finite doubly transitive groups, which admit such a representation with A the subgroup fixing a letter and B of order 2. It is natural to ask for conditions on A and B which will imply the solvability of G. It is known that a group of the form AB is solvable if A and B are nilpotent. However, no such general result can be expected for ABA -groups, since the simple groups PSL(2,2n) admit such a representation with A cyclic of order 2n + 1 and B elementary abelian of order 2n. Thus G need not be solvable even if A and B are abelian.

In (3) Herstein and the author have shown that G is solvable if A and B are cyclic of relatively prime orders; and in (2) we have shown that G is solvable if A and B are cyclic and A possesses a normal complement in G.
Some Lie Admissible Algebras
Authors: Laufer P. J. , Tomber M. L.
Published: December 2010

Several studies have been made to obtain larger classes of non-associative algebras from classes of algebras with a known structure. Thus, we have right alternative algebras (2)* and non-commutative Jordan algebras (6), (7), (8), and (9). These algebras are defined by a subset of the set of identities of the algebras from which they derive their names. Also, Albert (1), among others has studied Jordan admissible algebras. This paper is concerned with algebras which are related to Lie algebras in that they satisfy some of the identities of a Lie algebra and are Lie admissible. Theorem 2 answers a question raised by Albert in (1).
Analytic Hopf Surfaces
Authors: Helfenstein H. G.
Published: December 2010

The topological concept of H-space (7) has an analytic counterpart which so far has not been considered in the literature. We define: A complex-analytic manifold S will be called an analytic H-space if it is capable of carrying a continuous binary composition

with the following properties (i) and (ii).
A Geometrical Approach to the Second-Order Linear Differential Equation
Authors: Petty C. M. , Barry J. E.
Published: December 2010

In this paper various concepts intrinsically defined by the differential equation

1.1

are interpreted geometrically by concepts analogous to those in the Minkowski plane. This is carried out in § 2. The point of such a development is that one may apply the techniques or transfer known results in the theory of curves (in particular, convex curves) to (1.1), thereby gaining an additional tool in the investigation of this equation. For an application of a result obtained in this way, namely (3.12), see (4).

Throughout this paper,R(t)is a real-valued, continuous function ofton the real line (— ∞ < t < + ∞) and only the real solutions of (1.1) are considered.
Projections in Certain Continuous Function Spaces C(H) and Subspaces of C(H) Isomorphic with C(H)
Authors: Dean David W.
Published: December 2010

If A and B are sets then A — B = {x| x £A, x ∉ B}. This notation is also used if A and B are linear spaces. If X and Y are Banach spaces an embedding of X into Y is a continuous linear mapping u of X onto a closed subspace of F which is 1 — 1. In this case X is said to be embedded in Y. If |ux| = |x| for every x ∈ X (| … | stands for norm), then u embeds Xisometrically into Y. If u is onto then X and Y are isomorphic and if, in addition, |ux| = |x| for every x ∈ X, then X and Y are isometric. Then an embedding u has a continuous inverse u-1 (4, p. 36) defined on uX and this fact is used below without further reference. The conjugate space of X is denoted by X′. Unless otherwise noted, all topological spaces considered are Hausdorff spaces.
Approximate Continuity and Differentiation
Authors: Sion Maurice
Published: December 2010

The relation between the notions of measurability and continuity of a function has received a great deal of attention. The best-known result in this connection is the Vitali-Lusin theorem. Various versions of it can be found in (2; 3; 5; 8; 9). We prove one in this paper (Theorem 3.5) under very weak assumptions and state the classical one in Corollary 3.6. Functions satisfying the property stated in the theorem are frequently called quasicontinuous.

Here we are interested in the notion of approximate continuity, which we call μ-continuity for short and which is closely connected to differentiation. It was first introduced by Lebesgue (4) and it has so far required knowledge of density theorems in order to prove its relation to measurability. In this paper, making use of a certain type of Vitali property, we prove first the relation between μ-measurable functions and those μ-continuous almost everywhere (Theorems 3.8 and 3.9). Then, in §4, with Theorem 3.8 as the main tool, we derive several theorems about density and differentiation which extend results found in (1; 3; 7; 8).
Euler Graphs on Labelled Nodes
Authors: Read Ronald C.
Published: December 2010

In this paper we shall derive a concise formula for the number of Euler graphs on n labelled nodes and k edges. An Euler graph is a connected graph in which every node has even valency, where by the valency of a node is meant the number of edges which are incident with that node. Throughout most of the paper we shall be dealing with graphs whose nodes have even valencies but which may or may not be connected. For convenience we shall refer to these graphs as Euler graphs, although the usage is not, strictly speaking, correct. We shall impose the condition of connectedness in § 4.