bluuebunny/crossref_metadata_2025 huggingface dataset:
Characteristic Polynomials
Authors: Schneider Hans
Published: December 2010

Let F be a field and let V be a finite
dimensional vector space over F which is also a module over
the ring F[a]. Here a may lie in any extension ring of
F. We do not assume, as yet, that V is
a faithful module, so that a need not be a linear transformation on
V. It is known that by means of a decomposition of
V into cyclic F[a]-modules we may
obtain a definition of the characteristic polynomial of a on
V which does not involve determinants.
Classes of Positive Definite Unimodular
Circulants
Authors: Newman Morris , Taussky Olga
Published: December 2010

All matrices considered here have rational integral elements. In particular
some circulants of this nature are investigated. An n ×
n circulant is of the form

The following result concerning positive definite unimodular circulants was
obtained recently (3 ; 4 ):

Let C be a unimodular n × n circulant and assume that C = AA' where
A is an n × n matrix and A' its transpose. Then it follows that C =
C1C1', where C1 is again a
circulant.
On The Zeros of the Fresnel Integrals
Authors: Kreyszig Erwin
Published: December 2010

This paper is concerned with the Fresnel integrals

1.1

in the complex domain.

Recent research work in different fields of physical and technical
applications of mathematics shows that an increasing number of problems
require a detailed knowledge of elementary and higher functions for complex
values of the argument.
A Mixed Problem for Normal Hyperbolic Linear Partial
Differential Equations of Second Order
Authors: Duff G. F. D.
Published: December 2010

In the theory of hyperbolic differential equations a mixed boundary value
problem involves two types of auxiliary conditions which may be described as
initial and boundary conditions respectively. The problem of Cauchy, in
which only initial conditions are present, has been studied in great detail,
starting with the early work of Riemann and Volterra, and the well-known
monograph of Hadamard (4). A modern treatment of great generality has been
given by Leray (7).
Sets and Subseries
Authors: Petersen G. M.
Published: December 2010

Suppose αn ≠ 0 for all n. By replacing a set of terms
from the series Σαn by 0's, we form a subseries of the original
series. These subseries can be put in one to one correspondence with the
non-terminating binary expansion of the points on the real line segment (0,
1).
Upper and Lower Bounds for The Area of a Triangle for Whose
Sides Two Symmetric Functions are Known
Authors: Frucht Robert
Published: December 2010

Improving on inequalities given by Gerretsen (2), Beatty (1) has proved that
for the area Δ of any plane triangle with sides a, b, c the
following inequalities hold:

(1.1)

where

(1.2);

the signs of equality in (1.1) only apply when the triangle is equilateral.
Beatty has also remarked that the second inequality in (1.1) is of no value
in case 5H ≥ 3K, since then the lower estimate which it gives for
Δ2 is not even positive.
The Minimum of a Definite Ternary Quadratic Form
Authors: Dempster A. P.
Published: December 2010

A purely algebraic solution for the problem of the extreme definite ternary
quadratic form has been given by Mordell (2). The equivalent geometric
problem of the critical lattice of a sphere was solved by Mrs. Ollerenshaw
(3) under a preliminary assumption that was said to be justified by “simple
considerations” (p. 297).
Equivalent Abelian Groups
Authors: De Groot J.
Published: December 2010

Throughout this note all groups areabelian, written additively. We refer to Kurosh (8; 9) for notation, terminology and theorems used without reference. We recall the notion of aserving subgroup(or pure subgroup)of a group. This is a subgroupin which for every natural numbernevery equationnx = s, s ∊can be solved provided that it can be solved in. Ifis torsion-free, “linearly closed” subgroups coincide with serving subgroups andis a serving subgroup if and only if/is torsionfree.
Transitivities in Projective Planes
Authors: Ostrom T. G.
Published: December 2010

A perspectivity in a projective plane is a collineation which leaves some
line pointwise fixed. Baer (4) has considered the coordinatization of planes
which admit certain groups of perspectivities. André (1;2;3) has made an
extensive study of the Veblen-Wedderburn plane in terms of its
perspectivities. The author has shown (7) that finite doubly transitive
planes are Desarguesian if the number of points on a line is
n + 1 where n is an odd non-square.
A Tauberian Theorem For The Riemann-Liouville Integral Of
Integer Order
Authors: Rajagopal C. T.
Published: December 2010

1. Notation. Let s(x) be a function integrable in every finite interval of x
≥ 0. Then the Riemann-Liouville integral of s(x), of order a > 0, is
defined for x ≥ 0 by

(1).

The object of this note is to prove a Tauberian theorem for sα(x)
in the case in which α is a positive integer p, employing
certain difference formulae due to Karamata (4, Lemma 2) and Bosanquet (1,
Theorem 1) used already for a broadly similar purpose in an earlier paper
(12) where a is any positive number.
On the Complementary Functions of the Fresnel
Integrals
Authors: Kreyszig Erwin
Published: December 2010

1. Introduction. As is well known, the functions

(1.1)

have various applications in theoretical physics and engineering. It is thus
worthwhile to study their behaviour for real and complex values of the
argument.
Graphs with Given Group and Given Graph-Theoretical
Properties
Authors: Sabidussi Gert
Published: December 2010

n 1938 Frucht (2) proved the following theorem:

(1.1). Theorem. Given any finite group G there exist infinitely many
nonisomorphic connected graphs X whose automorphism group is isomorphic
to G.
Congruences for the Coefficients of Modular forms and Some New
Congruences for the Partition Function
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 (2) the author
established the following congruence:

Let r = 4, 6, 8, 10, 14, 26. Let p be a
prime greater than 3 such that r(p + l) /
24 is an integer, and set Δ = r(p2 − l)/24.
On Cayley's Parameterization
Authors: Pearl M. H.
Published: December 2010

A matrix P with elements from an arbitrary field
is called a
cogredient automorph (c.a.) of a symmetric matrix
A if P'AP = A, where
P' is the transpose of P.
Some Remarks Concerning Categories and Subspaces
Authors: Isbell J. R.
Published: December 2010

This paper is primarily a brief elaboration on the axioms for a
bicategory introduced in (3). From this point of view,
the main aim is the development of the structure of certain systems of
topological and uniform spaces, and the present paper merely points out some
very general properties which follow from axioms so weak that they are
satisfied by any system likely to be considered.
Products and Plethysms of Characters with Orthogonal, Symplectic and Symmetric
Groups
Authors: Littlewood D. E.
Published: December 2010

Murnaghan (9) has proposed the following method of analyzing the Kronecker
product of two symmetric group representations. If (λ) = (λ1,
λ2, … , λi) is a partition of
p, the representation of the symmetric group on n
symbols corresponding to the partition (n —
p, λ1 , … ,
λi) is denoted by [λ] and is said to be of depth
p.

If [λ] is of depth p and [μ] of depth
q, then the terms in the Kronecker product [λ] X
[μ] of depth p +
q are terms which correspond to the terms in the
product of S-functions {λ} {μ}).
The Term Rank of a Matrix
Authors: Ryser H. J.
Published: December 2010

This paper continues a study appearing in (5) of the combinatorial properties
of a matrix A of m rows and
n columns, all of whose entries are 0's and 1's. Let the sum of row i of A be
denoted by r

i
and let the sum of column i of A be noted by st.
We call R = (r
1, … , rm) the row sum vector and
S = (s
1, … , s
n) the column sum vector of
A. The vectors R and
S determine a class consisting of all
(0, 1)-matrices of m rows and n columns, with row sum vector R and column sum
vector S. Simple arithmetic properties of R and S are necessary and sufficient for
the existence of a class (1 ; 5).
Certain Transformations of Nearly-Poised Bilateral Hypergeometric Series of
Special Type
Authors: Shukla H. S.
Published: December 2010

A few years ago Bailey (1) gave certain transformations of both terminating
and non-terminating nearly-poised hypergeometric series of the ordinary type and
later on he also deduced basic analogues of some of his transformations. Recently,
(3) I gave certain transformations of both ordinary and basic terminating
nearly-poised bilateral hypergeometric series which generalized Bailey's
results.
Comma-Free Codes
Authors: Golomb S. W. , Gordon Basil , Welch L. R.
Published: December 2010

Let n be a fixed positive integer, and consider an
alphabet consisting of the numbers 1, 2, … , n. With
this alphabet form all possible k-letter words
(a
1
a
2 … a

k
), where k is also fixed. There are
evidently n

k
such words in all.
A Note on Division Algorithms in Imaginary quadratic Number fields
Authors: Dubois D. W. , Steger A.
Published: December 2010

An integral domain E is said to be
Euclidean if there exists a non-negative,
integer-valued function g denned on the non-zero
elements of E such that for every non-zero
x and y in
E,

(1) g(xy) ⩾
g(x);

(2) (division algorithm) if x does not divide
y then there exists an element
q in E, depending on
x and y, with

.
The Betti Numbers of the Simple Lie Groups
Authors: Coleman A. J.
Published: December 2010

The purpose of the present paper1 is to simplify the calculation of the Betti
numbers of the simple compact Lie groups.

For the unimodular group and the orthogonal group on a space of odd dimension
the form of the Poincaré polynomial was correctly guessed by E. Cartan in 1929 (5,
p. 183). The proof of his conjecture and its extension to the four classes of
classical groups was given by L. Pontrjagin (13) using topological arguments and
then by R. Brauer (2) using algebraic methods.
A Convergence Theorem for Double L 2 Fourier Series
Authors: Gosselin Richard P.
Published: December 2010

Our aim in this paper is to extend a known theorem about the convergence of
subsequences of the partial sums of the Fourier series in one variable of class
L
2 to Fourier series in two variables of the same class, (1,
p. 396). The theorem asserts that for each function ƒ in
L
2, there is a sequence {m

V
} of positive integers of upper density one such that

Smv(X;ƒ)

converges to ƒ almost everywhere where
sm(x;f) denotes the mth partial
sum of the Fourier series of ƒ.
A Singular Boundary Value Problem for a Non-Self-Adjoint Differential
Operator
Authors: Kemp R. R. D.
Published: December 2010

If

the differential expression
l(y) = —
y” +
g(x)y
generates a closed operator L on
L
2(— ∞, ∞), with domain D consisting of those functions
y ∈ L
2 with absolutely continuous derivatives and such that
l(y) ∈
L
2. The case where
g(x) is real-valued has
been extensively investigated and yields an expansion of any
ƒ ∈ L
2 in terms of the characteristic functions of
L. We shall investigate the case where g is
complex-valued.
On the Medians of a Triangle in Hyperbolic Geometry
Authors: Bottema O.
Published: December 2010

In non-Euclidean geometry the three medians of a triangle
A
1A2A3 (each joining
a vertex A

i
with the internal midpoint G
i of the opposite side) are concurrent; their common point is
the centroid G. But the Euclidean theorem

' which depends on similarity, does not hold.
Well Distributed Sequences
Authors: Keogh F. R. , Lawton B. , Petersen G.M.
Published: December 2010

In this note we discuss some properties of well distributed sequences. We
take 0 ⩽ a < b ⩽ 1
and let I(x) denote the characteristic function of the interval
[a, b], so that

.
On Bounded Matrices with Non-Negative Elements
Authors: Putnam C. R.
Published: December 2010

It is known (Perron (10); Frobenius (5, 6)) that if
A = (a

ik
) is a finite matrix with elements aik ⩾ 0, then
A has a real, nonnegative eigenvalue μ, satisfying μ
=max|λ| where λ is in the spectrum of A, with a
corresponding eigenvector x =
(x
1, … , xn) for which
x
i≥ 0. Moreover if a
ik
> 0, then μ is a simple point of the
spectrum with an eigenvector x (unique, except for constant multiples) with
components xi ≥0. Much has been written on this and related
issues; cf., for example, the recent papers (4, 12) wherein are given several
references.
Orthogonal Polynomials and Hypergeometric Series
Authors: van der Sluis A.
Published: December 2010

In Part I of this paper we present a theory of Padé-approximants for Laurent
series, and discuss their relation to orthogonal polynomials. For earlier results
in this direction we may refer to (1 ; 7; 8). It is also indicated how this theory
can be extended, for example, to matrix polynomials.
Generalized Integrals with Respect to Functions of Bounded Variation
Authors: Jeffery R. L.
Published: December 2010

A considerable literature has grown up around the analysis of the structure
of a function in terms of its derivative, and the structure of functions
F(x) which are integrals
of various kinds. Some of this relates to derivatives and integrals of
F(x) with respect to functions of bounded variation
ω(x) (1-6) or, in the case of a paper by Ward (4),
with respect to a function of generalized bounded variation in the restricted
sense.
On Babinet's Principle
Authors: MacCamy R. C.
Published: December 2010

In acoustic and electromagnetic theory much use is made of what is called the
Principle of Babinet (1). The principle states that the problems of diffraction by
an aperture, S, in a plane screen and by a plane obstacle occupying the position
S, are equivalent. It is the intent of this paper to
indicate how this notion of equivalent boundary problems, for partial differential
equations, can be extended to other situations.
On Free Products of Cyclic Rotation Groups
Authors: Dekker TH. J.
Published: December 2010

We consider the group of rotations in three-dimensional Euclidean space, leaving the origin fixed. These rotations are represented by real orthogonal third-order matrices with positive determinant. It is known that this rotation group contains free non-abelian subgroups of continuous rank (see 1).

In this paper we shall prove the following conjectures of J. de Groot (1, pp. 261-262):

Theorem 1. Two rotations with equal rotation angles a and with arbitrary but different rotation axes are free generators of a free group, if cos α is transcendental.

Theorem 2. A free product of at most continuously many cyclic groups can be isomorphically represented by a rotation group.
On the Irreducibility of Convex Bodies
Authors: Woods A. C.
Published: December 2010

We select a Cartesian co-ordinate system in ndimensional Euclidean space Rn with origin 0 and employ the usual pointvector notation.

By a lattice Λ in Rn we mean the set of all rational integral combinations of n linearly independent points X1, X2, … , Xn of Rn. The points X1 X2, … , Xn are said to form a basis of Λ. Let {X1, X2, … , Xn) denote the determinant formed when the co-ordinates of Xi are taken in order as the ith row of the determinant for i = 1,2, … , n. The absolute value of this determinant is called the determinant d(Λ) of Λ. It is well known that d(Λ) is independent of the particular basis one takes for Λ.

A star body in Rn is a closed set of points K such that if X ∈ K then every point of the form tX where — 1 < t < 1 is an inner point of K.
Note on Generalized Witt Algebras
Authors: Ree Rimhak
Published: December 2010

Throughout this note K will denote a field of characteristic p > 0. Let I be the set {1,2, … , m}, and a finite additive group of functions on I with values in K. We assume that is total in the sense that, for any λ1, … , λm in K Σi=imλiσ(i) = 0 for all a in G implies all σi = 0. It is clear that is an elementary p-group. Let pn be the order of . A generalized Witt algebra is defined as an algebra over K with basis elements {e(σ, i)∣ σ ∈ , i ∈ I} and the multiplication table

is a simple Lie algebra except when p = 2, m = 1.
Non-Linear Recursive Sequences
Authors: Walker Elbert A.
Published: December 2010

The purpose of this paper is to investigate non-linear recursive sequences of maximum length with elements from GF(2). In particular, the question of whether or not a recursive sequence of maximum length can be equal to its dual is settled. This question, as far as the author knows, was originally asked by Rosser. Part I contains the necessary background for Part II, and in the main is a condensation of some unpublished work (1955) of W. A. Blankinship and R. P. Dilworth.
On a Paper of Maurice Sion
Authors: Mahowald Mark
Published: December 2010

Let M0 be the set of measures μ on the real line such that open sets are μ*-immeasurable. While attempting to find out whether a set μ*-measurable for all μ in Mo is mapped into a similar set by a continuous function of bounded variation, Maurice Sion develops a theory for what he calls variational measure (4). As an application of the theory, he gets conditions on a function f and a set of measures M in order that f map a set, which is μ*-measurable for all μ ∈ M, into a set of the same kind. In particular he proves for his class M2 (def. 2.5), the following theorem (4, § 8.11).
Complemented Modular Lattices
Authors: Amemiya Ichiro , Halperin Israel
Published: December 2010

1.1 This paper gives a lattice theoretic investigation of “finiteness“ and “continuity of the lattice operations” in a complemented modular lattice. Although we usually assume that the lattice is-complete for some infinite,3we do not require completeness and continuity, as von Neumann does in his classical memoir on continuous geometry (3); nor do we assume orthocomplementation as Kaplansky does in his remarkable paper (1).

1.2. Our exposition is elementary in the sense that it can be read without reference to the literature. Our brief preliminary § 2 should enable the reader to read this paper independently.

1.3. Von Neumann's theory of independence (3, Part I, Chapter II) leans heavily on the assumption that the lattice is continuous, or at least upper continuous.
On a Theorem of Hayman Concerning Quasi-Bounded Functions
Authors: Kennedy P. B.
Published: December 2010

If f(z) is regular in |z| < 1, the expression

is called the characteristic of f(z). This is the notation of Nevanlinna (4) for the special case of regular functions; in this note it will not be necessary to discuss meromorphic functions. If m(r,f) is bounded for 0 < r < 1, then f(z) is called quasi-bounded in |z| < 1. In particular, every bounded function is quasibounded. The class Q of quasi-bounded functions is important because, for instance, a “Fatou theorem” holds for such functions (4, p. 134).
Quadrics Over GF(2) and their Relevance for the Cubic Surface Group
Authors: Edge W. L.
Published: December 2010

About half the following pages are concerned with the “cubic surface group” G, of order 51840, and the geometry cognate to a certain representation thereof. The literature of this group, with its subgroup of order 25920, is already voluminous; the addition of these few pages to it will not, it is hoped, be regarded askance as a perverse and misdirected indulgence in archaism.
On the Class Number of Representations of an Order
Authors: Reiner Irving
Published: December 2010

We shall use the following notation throughout:

R = Dedekind ring (5).

u = multiplicative group of units in R.

h = class number of R.

K = quotient field of R.

p = prime ideal in R.

Rp = ring of p-adic integers in K.

We assume that h is finite, and that for each prime ideal p, the index (R:p) is finite.

Let A be a finite-dimensional separable algebra over K, with an identity element e (4, p. 115). Let G be an R-ordev in A, that is, G is a subring of A satisfying

(i)

e ∈ G,

(ii)

G contains a i∈-basis of A,

(iii)

G is a finitely-generated i?-module.

By a G-module we shall mean a left G-module which is a finitely-generated torsion-free i∈-module, on which e acts as identity operator.
Power Series Representing Certain Rational Functions
Authors: Melzak Z. A.
Published: December 2010

Let denote the set of functions of a complex variable z, regular at z = 0, and let I denote the set of non-negative integers. For f ∈ put

For a given subset 0 of there arises the problem of characterizing the admissible gap sets If of functions f in 0. When 0 is the set R of rational functions a complete solution in given by the following theorem:

(A) Let f ∈ R and let If be infinite. Then there exist integers L, L1, L2… , L3 such that 0 ≤ L1 < L2 … < Ls < L, and If = {n|n ∈ I, n ≡ Lj (mod L), j =1, … , s} U I\ where V is a Unite exceptional set.
Sheets of Real Analytic Varieties
Authors: Wallace Andrew H.
Published: December 2010

In a previous paper (4) the author worked out some results on the analytic connectivity properties of real algebraic varieties, that is to say, properties associated with the joining of points of the variety by analytic arcs lying on the variety. It is natural to ask whether these properties can be carried over to analytic varieties, since the proofs in the algebraic case depend mainly on local properties. But although this generalization can be carried out to a large extent, there are, nevertheless, difficulties in the analytic case, owing mainly to the fact (cf. 2, § 11) that a real analytic variety may not be definable by means of a set of global equations. Thus, although the general idea of the treatment given here is the same as in (4), some variation in the details of the method has proved to be necessary, and some of the final results are slightly weaker in form.
Sur le Radical Corpoïdal D'un Anneau
Authors: Thierrin G.
Published: December 2010

Le radical d'un anneau, dans le sens général de N. Jacobson (1; 2), peut être caractérisé de plusieurs manières. On peut, par exemple, le définir comme l'intersection de tous les idéaux primitifs de l'anneau envisagé. Dans ce travail, nous considérons une classe particulière d'idéaux primitifs, la classe des idéaux corpoïdaux: un idéal est dit corpoïdal si et seulement si l'anneauquotient correspondant est un corps. L'objet principal de ce travail est de donner quelques caractérisations de l'intersection C de tous les idéaux corpoïdaux d'un anneau A, intersection appelée le radical corpoïdal de l'anneau A. Si C est distinct de A, l'anneau-quotient A/C est isomorphe à une somme sous-directe de corps Dans le cas d'un anneau commutatif, tout idéal primitif est corpoïdal, et par conséquent son radical corpoïdal coïncide avec son radical.
Loops with Adjoints
Authors: Cowell W. R.
Published: December 2010

It is shown in (6) how to represent certain sets of orthogonal Latin squares as a group together with a set of permutations of the group elements. The correspondence between 3-nets and loops is well known; for example, see (8). We shall consider a loop G together with a certain set of permutations on the elements of G and shall interpret such a structure as an incidence system in which the 3-net of the loop is embedded. Specifically, the permutations or “adjoints” will give rise to lines which may be adjoined to the 3-net of G in the sense of (3). The group of autotopisms of the loop determines a group of automorphisms of its 3-net analogous to collineations in an affine plane. We shall study the problem of extending these incidence preserving mappings to the adjoined lines. By analogy with the study of loops with operators, we shall consider homomorphisms of loops with adjoints and examine geometric consequences. Particular attention will be paid to the case where G has the inverse property and the adjoints are “linear.” The special case in which G is an abelian group is of geometric interest in that the corresponding incidence systems include the Veblen-Wedderburn affine planes.
On Homeomorphisms between Extension Spaces
Authors: Banaschewski Bernhard
Published: December 2010

In this note, conditions are obtained which will ensure that two topological spaces are homeomorphic when they have homeomorphic extension spaces of a certain kind. To discuss this topic in suitably general terms, an unspecified extension procedure, assumed to be applicable to some class of topological spaces, is considered first, and it is shown that simple conditions imposed on the extension procedure and its domain of operation easily lead to a condition of the desired kind. After the general result has been established it is shown to be applicable to a number of particular extensions, such as the Stone-Čech compactification and the Hewitt Q-extension of a completely regular Hausdorff space, Katětov's maximal Hausdorff-closed extension of a Hausdorff space, the maximal zero-dimensional compactification of a zero-dimensional space, the maximal Hausdorff-minimal extension of a semi-regular space, and Freudenthal's compactification of a rim-compact space. The case of the Hewitt Q-extension was first discussed by Heider (6).
The Commutativity of a Special Class of Rings
Authors: Martindale Wallace S.
Published: December 2010

A well-known theorem of Jacobson (1) states that if every element x of a ring R satisfies xn(x) = x where n(x) > 1 is an integer, then R is commutative. A series of generalizations of this theorem have been proved by Herstein (2; 3; 4; 5; 6), his last result in this direction (6) being that a ring R is commutative provided every commutator u of R satisfies un(u) = u. We now define a γ-ring to be a ring R in which un(u) — u is central for every commutator u of R (where n(u) > 1 is an integer). In the present paper we verify the following conjecture of Herstein: every commutator of a γ-ring is central.
On Matrix Commutators
Authors: Marcus Marvin , Khan Nisar A.
Published: December 2010

Let A, B, and X be n-square matrices over an algebraically closed field F of characteristic 0. Let [A, B] = AB — BA and set (A, B) = [A, [A, B]]. Recently several proofs (1; 3; 5) of the following result have appeared: if det (AB) ≠ 0 and (A,B) = 0 then A-1B-1AB - I is nilpotent. In (2) McCoy determined the general form of any X satisfying

1.1

in the case that A has a single elementary divisor corresponding to each eigenvalue, that is, A is non-derogatory. In Theorem 1 we determine the structure of any matrix X satisfying (1.1) and also give a formula for the dimension of the linear space of all such X in terms of the degrees of the elementary divisors of A.
Perturbation of the Continuous Spectrum of Even Order Differential Operators
Authors: Butler John B.
Published: December 2010

Let L0 be a differential operator of even order n = 2v on the half open interval 0 ≤ t < ∞ which is formally self adjoint and satisfies the conditions of Kodaira (5, p. 503). We consider a perturbed operator of the form L∈ = Lo + ∈q where q(t) is a real-valued bounded function and ∈ is a real parameter. The object of this paper is to set up conditions on the operator L0 and the function q(t) such that L∈ determines a self-adjoint operator H∈ and such that the spectral resolution operator E∈(Δ) corresponding to H∈ is analytic in a neighbourhood of ∈ = 0, where Δ is a closed bounded interval.

Our conditions are a natural generalization of conditions considered by Moser for the case n = 2(6). Moser has given a number of examples showing that when his conditions do not hold E∈(Δ) need not be analytic.
Sums of Functions of Digits
Authors: Stewart B. M.
Published: December 2010

We generalize in several directions a paper by Porges (2) who considered the integer F(A) obtained from the positive integer .1 by taking the sum of the squares of the digits of A. Porges showed that if A > 99, then F(A) < A, so that under iteration of F(A) all the positive integers are divided into a finite number of classes, called orbits in the terminology of Isaacs (1), each containing a finite cycle. For his F(A) Porges showed there are only two orbits: one with the 1-cycle: 1 → 1 ; and the other with the interesting 8-cycle: 4 → 16 → 37 → 58 → 89 → 145 → 42 → 20 → 4.

Consider the set Z of non-negative integers and choose as a base of enumeration any desired integer B ≧ 2 (not necessarily B = 10). Then only the “digits” 0, 1, 2, … , B — 1 are needed, in suitable multiplicity, to represent any A of Z.
Arithmetical Inversion Formulas
Authors: Cohen Eckford
Published: December 2010

Let n and r be integers, r positive, and define the coreγ(r) of r to be the product of the distinct prime factors of r (γ(1) = 1). Let f(n,r) be a complex-valued, arithmetical function of n and r. If for all n,f﹛n,r) = f((n,r), r) then f(n, r) is called an even function (mod r), and if f(n,r) = f(γ(n, r), r) for all n, γ(n, r) = γ((n, r)), then f(n, r) is said to be a primitive function (mod r). Clearly, both classes of functions are subclasses of the periodic functions (mod r), while the primitive functions form a subclass of the even functions (mod r).

In a series of three papers (3; 5; 6) the author developed parallel, though interrelated, trigonometric and arithmetical theories of the even and primitive functions (mod r).
Canonical Forms for Certain Matrices Under Unitary Congruence
Authors: Stander J W. , Wiegmann N. A.
Published: December 2010

If A is a matrix with complex elements and if A = AT (where AT denotes the transpose of A), there exists a non-singular matrix P such that PAPT = D is a diagonal matrix (see (3), for example). It is also true (see the principal result of (5)) that for such an A there exists a unitary matrix U such that UAUT = D is a real diagonal matrix with nonnegative elements which is a canonical form for A relative to the given U, UT transformation.
Modifications and Cobounding Manifolds
Authors: Wallace Andrew H.
Published: December 2010

The object of this paper is to establish a simple connection between Thorn's theory of cobounding manifolds and the theory of modifications. The former theory is given in detail in (8) and sketched in (3), while the latter is worked out in (1). In particular in (1) it is shown that the only modifications which can transform one differentiable manifold into another are what I call below spherical modifications, which consist in taking out a sphere from the given manifold and replacing it by another. The main result is that manifolds cobound if and only if each is obtainable from the other by a finite sequence of spherical modifications.

The technique consists in approximating the manifolds by pieces of algebraic varieties. Thus if M1 and M2 form the boundary of M, the last is taken to be part of an algebraic variety such that M1 and M2 are two members of a pencil of hyperplane sections.
A Semimodular Imbedding of Lattices
Authors: Finkbeiner D. T.
Published: December 2010

The study of structural or arithmetic properties of a general lattice often can be facilitated by imbedding as a sublattice of a lattice of a more restricted type whose properties are known. However, if is too restricted, a general imbedding is impossible; for example, cannot be modular because , as a sublattice of , would then have to be modular. One of the best results of this nature has been given by Dilworth in an unpublished work in which he shows that any finite dimensional lattice is isomorphic to a sublattice of a semi-modular point lattice (1, pp. 105 and 110). In the present paper Dilworth's imbedding process is modified to obtain a sharper result: Any finite dimensional lattice is isometrically isomorphic to a sublattice of a semi-modular lattice which has the same number of points as and which preserves basic properties of the join-irreducible arithmetic of .
On Continuous Regular Rings and Semisimple Self Injective Rings
Authors: Utumi Yuzo
Published: December 2010

Brainerd and Lambek (2, Corollary 4) have proved recently that any complete Boolean ring is self-injective. It is easy to see that every complete Boolean ring is a continuous regular ring, that is, a regular ring of which the lattice of principal left ideals is continuous. This suggests that in a continuous regular ring it might be possible to prove the injectivity. However, a simple example (Example 3) shows that the conjecture is not true in general. Our main theorem is the following. Every continuous regular ring with no ideals of index 1 is (both left and right) self-injective (Theorem 3).

It is known to Wolfson (13, Theorem 5.1) and Zelinsky (15) that the ring S of all linear transformations of a vector space of dimension ≥ 2 over a division ring is generated by idempotents and also by non-singular elements.
On a Class of Non-Self-Adjoint Differential Operators
Authors: Kemp R. R. D.
Published: December 2010

The problem of spectral analysis of non-self-adjoint (and non-normal) operators has received considerable attention recently. Livšic (5), and more recently Brodskii and Livšic (1) have considered operators on Hilbert space with completely continuous imaginary parts. Dunford (3) has generalized the notion of spectral measure and defined a class of spectral operators on Hilbert and Banach space. Schwartz (8) and Rota (7) have investigated conditions under which a differential operator will be spectral. The work of Naimark (6) and the author (4) on non-self-adjoint differential operators leads to an expansion theorem which implicitly defines a type of spectral measure. However the projections involved in this will not in general be bounded, much less uniformly bounded.
Reciprocal Convergence Classes for Fourier Series and Integrals
Authors: Guinand A. P.
Published: December 2010

The classical result of Plancherel for Fourier cosine transforms of functions f(x) of the class L2(0, ∞) states that (see (7) for references)

converges in mean square to a function g(x) which also belongs to L2(0, ∞), and furthermore

Some years ago in a series of papers (1; 2; 3) on summation formulae I showed that a similar symmetrical theory for narrower classes of functions and ordinary convergence of the integrals can also be developed.
Summability Methods on Matrix Spaces
Authors: Mitchell Josephine
Published: December 2010

The matrix spaces under consideration are the four main types of irreducible bounded symmetric domains given by Cartan (5). Let z = (zjk) be a matrix of complex numbers, z' its transpose, z* its conjugate transpose and I = I(n) the identity matrix of order n. Then the first three types are defined by

(1)

where z is an n by m matrix (n ≤ m), a symmetric or a skew-symmetric matrix of order n (16). The fourth type is the set of complex spheres satisfying

(2)

where z is an n by 1 matrix. It is known that each of these domains possesses a distinguished boundary B which in the first three cases is given by

(3)

(In the case of skew symmetric matrices the distinguished boundary is given by (2) only if n is even.)
Arithmetic Linear Transformations and Abstract Prime Number Theorems
Authors: Amitsur S. A.
Published: December 2010

Shapiro and Forman have presented in (4) an abstract formulation of prime number theorems which includes the various prime number theorems; for primes in arithmetic progressions, for prime ideals in ideal classes etc. The methods of proofs are “elementary” and follow closely Shapiro's proof for the primes in arithmetic progression (for reference see bibliography in (4)).

The author has followed in (1) some ideas of Yamamoto (5) on arithmetic linear transformations to introduce a symbolic calculus in dealing with arithmetic functions. This calculus proved to be very useful in unifying many of the “elementary” proofs in the behaviour of arithmetic functions. In (6) Yamamoto has extended his theory to ideals in algebraic number fields, and with this extension the symbolic calculus of (1) can be extended to cover the abstract case of prime number theorem in countable free abelian groups as discussed in (4).
An Enumeration Problem Related to the Number of Labelled Bi-Coloured Graphs
Authors: Lee C. Y.
Published: December 2010

We will consider the following enumeration problem. Let A and B be finite sets with α and β elements in each set respectively. Let n be some positive integer such that n ≦ αβ. A subset S of the product set A × B of exactly n distinct ordered pairs (ai, bj) is said to be admissible if given any a ∈ A and b ∈ B, there exist elements (ai, bj) and (ak, bl) (they may be the same) in S such that ai = a and bl = b. We shall find here a generating function for the number N(α, β n) of distinct admissible subsets of A × B and from this generating function, an explicit expression for N(α, β n). In obtaining this result, the idea of a cut probability is used. This approach in a problem of enumeration may be of interest.
Sublattices of a Free Lattice
Authors: Jónsson Bjarni
Published: December 2010

Professor R. A. Dean has proved (1, Theorem 3) that a completely free lattice generated by a countable partially ordered set is isomorphic to a sublattice of a free lattice. In particular, it follows that a free product of countably many countable chains can be isomorphically embedded in a free lattice. Generalizing this we show (2.1) that the class of all lattices that can be isomorphically embedded in free lattices is closed under the operation of forming free lattice-products with arbitrarily many factors. We also prove (2.4) that this class is closed under the operation of forming simply ordered sums with denumerably many summands. Finally we show (2.7) that every finite dimensional sublattice of a free lattice is finite.
On a Class of Singular Differential Operators
Authors: Kemp R. R. D.
Published: December 2010

In the considerable literature on linear operators in L2 or Lp arising from ordinary differential operators it has always been assumed that the coefficient of the highest order derivative appearing does not vanish in the interior of the interval under consideration. If this coefficient vanishes at one or both endpoints of the interval, or if one or both of the endpoints is infinite the differential operator is said to be singular. In this paper we shall allow this leading coefficient to vanish in the interior of the interval, and show that the theory of such operators can sometimes be reduced to a consideration of several operators of the well-known type. We shall also indicate how those which cannot be so reduced should be dealt with.
Graph Theory and Probability. II
Authors: Erdös P.
Published: December 2010

Define f(k, l) as the least integer so t h a t every graph having f(k, l) vertices contains either a complete graph of order k or a set of l independent vertices (a complete graph of order k is a graph of k vertices every two of which are connected by an edge, a set of I vertices is called independent if no two are connected by an edge).

Throughout this paper c1, c2, … will denote positive absolute constants. It is known (1, 2) that

(1)

and in a previous paper (3) I stated that I can prove that for every ∈ > 0 and l > l(∈), f (3, l) > l2-∈. In the present paper I am going to prove that

(2)
Matrix Commutators
Authors: Smiley M. F.
Published: December 2010

A classical theorem states that if a square matrix B over an algebraically closed field F commutes with all matrices X over F which commute with a matrix A over F, then B must be a polynomial in A with coefficients in F (2). Recently Marcus and Khan (1) generalized this theorem to double commutators. Our purpose is to complete the generalization to commutators of any order.

Let F be an algebraically closed field and let Fn be the ring of all n by n matrices with elements in F. We define ΔYZ — = [Z, Y] = ZY — YZ for all Y, Z in Fn.
Orthomorphisms of Groups and Orthogonal Latin Squares. I
Authors: Johnson Diane M. , Dulmage A. L. , Mendelsohn N. S.
Published: December 2010

Euler (6) in 1782 first studied orthogonal latin squares. He showed the existence of a pair of orthogonal latin squares for all odd n and conjectured their non-existence for n = 2(2k + 1). MacNeish (8) in 1921 gave a construction of n — 1 mutually orthogonal latin squares for n = p with p prime and of n(v) mutually orthogonal squares of order v where

with p1 p2, … , Pr being distinct primes and
Partition Rings of Cyclic Groups of Odd Prime Power Order1
Authors: Appel K. I.
Published: December 2010

A ring R over a commutative ring K, that has a basis of elements g1, g2, … , gn forming a group G under multiplication, is called a group ring of G over K. Since all group rings of a given G over a given K are isomorphic, we may speak of the group ring KG of G over X.

Let π be any partition of G into non-empty sets GA, GB, … . Any subring P of KG that has a basis of elements

is a partition ring of G over K.

If P is a partition ring of G over Z, the ring of integers, then the basis A, B, … for P clearly serves as a basis for a partition ring P’ = Q ⊗ P of G over Q, the field of rationals.
On the Hausdorff and Trigonometric Moment Problems
Authors: Rooney P. G.
Published: December 2010

Let K be a subset of BV(0, 1)—the space of functions of bounded variation on the closed interval [0, 1]. By the Hausdorff moment problem for K we shall mean the determination of necessary and sufficient conditions that corresponding to a given sequence μ = {μn|n = 0, 1, 2, …} there should be a function α ∈ K so that

(1)

For various collections K this problem has been solved—see (3, Chapter III)

By the trigonometric moment problem for K we shall mean the determination of necessary and sufficient conditions that corresponding to a sequence c = {cn|n = 0, ± 1, ± 2, …} there should be a function α ∈ K so that

(2)

For various collections K this problem has also been solved—see, for example (4, Chapter IV, § 4). It is noteworthy that these two problems have been solved for essentially the same collections K.
Normal Operators on the Banach Space Lp(-∞,∞). Part I
Authors: Krabbe Gregers L.
Published: December 2010

Let be the Boolean algebra of all finite unions of subcells of the plane. Denote by εpthe algebra of all linear bounded transformations of Lp(— ∞, ∞) into itself. Suppose for a moment that p = 2, and let Rp be an involutive abelian subalgebra of εp if Rp is also a Banach space and if Tp ∈ Rp, then:

(i) The family of all homomorphic mappings of into the algebra Rp contains a member EPT such that

(1)
Homogeneous Continua Which are Almost Chainable1
Authors: Burgess C. E.
Published: December 2010

The only known examples of nondegenerate homogeneous plane continua are the simple closed curve, the circle of pseudo-arcs (6), and the pseudo-arc (1; 13). Another example, called the pseudo-circle, has been suggested by Bing (2), but it has not been proved to be homogeneous. (Definitions of some of these terms and a history of results on homogeneous plane continua can be found in (6).) Of the three known examples, the pseudo-arc is both linearly chainable and circularly chainable, and the simple closed curve and the circle of pseudo-arcs are circularly chainable but not linearly chainable. It is not known whether every homogeneous plane continuum is either linearly chainable or circularly chainable. Bing has shown that a homogeneous continuum is a pseudo-arc provided it is linearly chainable (4).
Fourth Order Boundary Value Problems and Comparison Theorems
Authors: Barrett John H.
Published: December 2010

This paper is primarily concerned with the existence of solutions of the fourth-order self-adjoint differential equation

(1)

(where r(x) > 0, q(x) ≥ 0, p(x) ≥ 0 and all three coefficients are continuous on [a, ∞)) and one of the two-point boundary conditions:

(2)

or

(3)

the subscript notation for any solution y(x) denoting:

(4)
The Clifford Algebra and the Group of Similitudes
Authors: Wonenburger Maria J.
Published: December 2010

LetC(M, Q)be the Clifford algebra of an even dimensional vector spaceMrelative to a quadratic formQ. WhenQis non-degenerate, it is well known that there exists an isomorphism of the orthogonal groupO(Q)onto the group of those automorphisms ofC(M, Q)which leave invariant the spaceM⊂C(M, Q). These automorphisms are inner and the group of invertible elements ofC(M, Q)which define such inner automorphisms is called the Clifford group.

If instead of the groupO(Q)we take the group of similitudesγ(Q)or even the group of semi-similitudesΓγ(Q), it is possible to associate in a natural way with any element of these groups an automorphism or semi-automorphism, respectively, of the subalgebra of even elementsC+(M, Q)⊂C(M, Q). Each one of the automorphisms ofC+(M, Q)so defined can be extended, as it is shown here (Theorem 2), to an inner automorphism ofC(M, Q), although the extension is not unique.
A Generalization of Finsler Geometry
Authors: Vanstone J. R.
Published: December 2010

Modern differential geometry may be said to date from Riemann's famous lecture of 1854 (9), in which a distance function of the form F(xi, dxi) = (γij(x)dxidxj½ was proposed. The applications of the consequent geometry were many and varied. Examples are Synge's geometrization of mechanics (15), Riesz’ approach to linear elliptic partial differential equations (10), and the well-known general theory of relativity of Einstein.

Meanwhile the results of Caratheodory (4) in the calculus of variations led Finsler in 1918 to introduce a generalization of the Riemannian metric function (6). The geometry which arose was more fully developed by Berwald (2) and Synge (14) about 1925 and later by Cartan (5), Busemann, and Rund. It was then possible to extend the applications of Riemannian geometry.
Some Formulas Involving Ramanujan Sums
Authors: Nicol C. A.
Published: December 2010

The purpose of this note is to establish an identity involving the cyclotomic polynomial and a function of the Ramanujan sums. Some consequences are then derived from this identity.

For the reader desiring a background in cyclotomy, (2) is mentioned. Also, (4) is intimately connected with the following discussion and should be consulted.

The cyclotomic polynomial Fn(x) is defined as the monic polynomial whose roots are the primitive nth roots of unity. It is well known that

2.1

For the proof of Corollary 3.2 it is mentioned that Fn(0) = 1 if n > 1 and that Fn(x) > 0 if |x| < 1 and 1 < n.

The Ramanujan sums are defined by

2.2

where the sum is taken over all positive integers r less than or equal to n and relatively prime to n. It is also well known that

2.3

where the sum is taken over all positive divisors d common to n and k.
Angular and Tangential Limits of Blaschke Products and their Successive Derivatives
Authors: Cargo G. T.
Published: December 2010

In this paper, we shall be concerned with bounded, holomorphic functions of the form

where

(1)

(2)

and

(3)

B(z{an}) is called a Blaschke product, and any sequence {an} which satisfies (2) and (3) is called a Blaschke sequence. For a general discussion of the properties of Blaschke products, see (18, pp. 271-285) or (14, pp. 49-52).

According to a theorem due to Riesz (15), a Blaschke product has radial limits of modulus one almost everywhere on C = {z: |z| = 1}. Moreover, it is common knowledge that, if a Blaschke product has a radial limit at a point, then it also has an angular limit at the point (see 14, p. 19 and 6, p. 457).
Perturbation of the Continuous Spectrum of Systems of Ordinary Differential Operators
Authors: Butler John B.
Published: December 2010

Let

be an ordinary differential operator of order h whose coefficients are (η, η) matrices defined on the interval 0 ≤ x < ∞, hη = n = 2v. Let the operator L0 be formally self adjoint and let v boundary conditions be given at x = 0 such that the eigenvalue problem

(1.1)

has no non-trivial square integrable solution. This paper deals with the perturbed operator L∈ = L0 + ∈q where ∈ is a real parameter and q(x) is a bounded positive (η, η) matrix operator with piecewise continuous elements 0 ≤ x < ∞. Sufficient conditions involving L0, q are given such that L∈ determines a selfadjoint operator H∈ and such that the spectral measure E∈(Δ′) corresponding to H∈ is an analytic function of ∈, where Δ′ is a subset of a fixed bounded interval Δ = [α, β]. The results include and improve results obtained for scalar differential operators in an earlier paper (3).