bluuebunny/crossref_metadata_2025 huggingface dataset:
Sur Les Surfaces a Courbure Moyenne Isotherme
Authors: Lalan Victor
Published: December 2010

Nous nous occupons dans ce travail d'une classe de surfaces dont l'équation différentielle est du cinquième ordre, et qui jouissent de la propriété que, sur elles, les lignes d'égale courbure moyenne forment, avec leurs trajectoires orthogonales, un système isotherme. Dans cette classe rentrent les surfaces admettant un groupe de déplacements à un paramètre (cylindres, surfaces de révolution et hélicoïsian Bonnet).
Functionals of Bounded Frechet Variation
Authors: Morse Marston , Transue William
Published: December 2010

In a series of papers which will follow this paper the authors will present a theory of functionals which are bilinear over a product A × B of two normed vector spaces A and B. This theory will include a representation theory, a variational theory, and a spectral theory. The associated characteristic equations will include as special cases the Jacobi equations of the classical variational theory when n = 1, and self-adjoint integrodifferential equations of very general type. The bilinear theory is oriented by the needs of non-linear and non-bilinear analysis in the large.
On the Uniqueness of the Green's Function Associated with a Second-order Differential Equation
Authors: Titchmarsh E. C.
Published: December 2010

The Green's function G(x, ξ, λ) associated with the differential equation is of importance in the theory of the expansion of an arbitrary function in terms of the solutions of the differential equation. It is proved that this function is unique if q(x) ≧ — Ax2— B, where A and B are positive constants or zero. A similar theorem is proved for the Green's function G(x, y, ξ, η, λ) associated with the partial differential equation
On The Motion of Three Vortices
Authors: Synge J. L.
Published: December 2010

In a perfect incompressible fluid extending to infinity, the determination of the motion of N parallel rectilinear vortex filaments involves the solution of N non-linear differential equations, each of the first order. The method of Kirchhoff provides certain constants of the motion. If we describe the positions of the vortices by their point-traces on a plane perpendicular to them, the following facts follow from the theory of Kirchhoff:

(1.1) The mean centre of the system is fixed.
Angular Measure and Integral Curvature
Authors: Busemann Herbert
Published: December 2010

The Gauss-Bonnet Theorem leads through well known arguments to the fact that the integral curvature of a two-dimensional closed orientable manifold M of genus p equals 4π(1 — p). This implies, for instance, that the Gauss curvature K can neither be everywhere positive nor everywhere negative, if M is homeomorphic to a torus.
On a Theorem of Latimer and Macduffee
Authors: Taussky Olga
Published: December 2010

The matrix solutions of an irreducible algebraic equation with integral coefficients were studied by Latimer and MacDuffee. They considered matrices with rational integers as elements. If A is such a matrix, then all matrices of the “class” S1 AS will again be solutions if S is a matrix of determinant ± 1. On the other hand, in general all solutions cannot be derived in this way from one solution only. It was in fact shown that the number of classes of matrix solutions coincides with the number of different classes of ideals in the ring generated by an algebraic root of the same equation.
Congruence Relations Between the Traces of Matrix Powers
Authors: Frame J. S
Published: December 2010

Let A be a matrix of finite order n and finite degree d, whose characteristic roots are certain nth roots of unity a1, a2…, ad. We wish to prove a congruence (6) between the traces (tr) of certain powers of A, which is suggested by two somewhat simpler congruences (1) and (3). First, if tr (A) is a rational integer, it is easy to establish the familiar congruence

even though tr(Ap) may not itself be rational.
A Generalization of Epstein Zeta Functions
Authors: Minakshisundaram S.
Published: December 2010

§ 1. An Epstein zeta function (in its simplest form) is a function represented by the Dirichlet's series where a1… ak are real and n1, n2, … nk run through integral values. The properties of this function are well known and the simplest of them were proved by Epstein [2, 3]. The aim of this note is to define a general class of Dirichlet's series, of which the above can be viewed as an instance, and to discuss the problem of analytic continuation of such series.
On a Theorem of Liouville in Fields of Positive Characteristic
Authors: Mahler K.
Published: December 2010

A classical theorem of J. Liouville states that if z is a real algebraic number of degree n ≥ 2, then there exists a constant c > 0 such that for every pair of integers a, b with b ≠ 0.
On the Field of Origin Of an Ideal
Authors: Mann H. B.
Published: December 2010

In this paper we shall consider integral ideals in finite algebraic extensions(, …) of the field of rational numbers.

Two ideals in the same field are said to be equal if and only if they contain the same numbers.
The Bianchi Identities in the Generalized Theory of Gravitation
Authors: A. Einstein
Published: December 2010

1. General remarks. The heuristic strength of the general principle of relativity lies in the fact that it considerably reduces the number of imaginable sets of field equations; the field equations must be covariant with respect to all continuous transformations of the four coordinates. But the problem becomes mathematically well-defined only if we have postulated the dependent variables which are to occur in the equations, and their transformation properties (field-structure). But even if we have chosen the field-structure (in such a way that there exist sufficiently strong relativistic field-equations), the principle of relativity does not determine the field-equations uniquely. The principle of “logical simplicity” must be added (which, however, cannot be formulated in a non-arbitrary way). Only then do we have a definite theory whose physical validity can be tested a posteriori.
Elementarfunktionen Auf Riemannschen Flächen Als Hilfsmittel Für Die Funktionentheorie MEHRERER Veränderlichen
Authors: Heinrich Behnke  , Karl Stein
Published: December 2010

In the present paper, the authors extend the Cousin theorems and the continuity theorem, using some previous results on analytic functions connected with open Riemann surfaces.

The Cousin theorems, concerning the existence of analytic functions of several complex variables with prescribed poles and zeros in a given domain, have been generalized in various manners, but only in the case where the domain is schlicht. The authors proceed to the case where the given domain is the direct product of n open Riemann surfaces. They prove the following two theorems.
An Equation for the Degrees of the Absolutely Irreducible Representations of a Group of Finite Order
Authors: Zassenhaus Hans
Published: December 2010

If there is a nonsingular symmetric bilinear form defined on a distributive algebra A with basis elements C1, C2, … , Cr over
a field F such that and , then the
so called Casimir operator [1,2]
Minimal Solutions of Diophantine Equations
Authors: Holzer L.
Published: December 2010

Our aim is to prove: If a, b, c are positive integers, if ab > 1, if a, b, c are relatively prime in pairs, and all are free of squares, if — ab is a quadratic residue of c, bc of a, ca of b, and if F(x, y, z) = ax2 + by2 — cz2, we have non-trivial solutions of

with the inequalities
Approximations to the Area of an
n-Dimensional Ellipsoid
Authors: Lehmer D. H.
Published: December 2010

The fact that the perimeter S(a, b) of an ellipse is not an elementary function of its semiaxes a, b has led to many suggested approximations of S in finite form.
Généralisation Du Scalaire De Courbure Et Du Scalaire Principal D'un Espace Finslérien áNDimensions
Authors: Moór Arthur
Published: December 2010

Introduction. Dans ses mémoires “Über zweidimensionale allgemeine metrische Räume” [1] et “On Finsler and Cartan Geometries III,” [3] Berwald a définie deux invariants fondamentaux des espaces finslériens à deux dimensions. Dans le présent travail nous allons établir une formule pour le scalaire de courbure et une pour le scalaire principal, qui définissent un invariant de l'espace finslérien à n dimensions, et qui sont identiques, pour deux dimensions, aux invarients définis par L. Berwald. Notre idée fondamentale sera d'exprimer le vecteur normal (hi, hi)—qui n'est défini qu'à deux dimensions—par le vecteur d'Euler, déjà défini pour n dimensions.
Contribution a L'etude Du Probleme Des Timbres Poste
Authors: Touchard Jacques
Published: December 2010

On demande le nombre Sn de maniéres dont on peut replier une bande de n timbres-poste (TP) sur un seul timbre. Dans le graphique de Sainte Lagüe (graphique SL) [3, p. 39], les liaisons entre timbres ne doivent pas se couper (fig. 1). Nous supposerons toujours que le timbre 1 occupe la premiére place à gauche, car si Pn est, dans ce cas, le nombre de permutations bonnes, on aura Sn = nPn. Nous supposerons de plus que la liaison (1, 2) est au-dessus de l'axe. Nous appellerons arc pair ou couple pair un arc (a, a + 1) où a est pair; arc impair ou couple impair un arc (a, a + 1) où est impair. Nous appellerons aussi permutation inverse de la permutation .
On Representations as a Sum of Consecutive Integers
Authors: Leveque W. J.
Published: December 2010

1. Introduction. It is the object of this paper to investigate the function γ(m), the number of representations of m in the form

(1)

where . It is shown that γ(m) is always equal to the number of odd divisors of m, so that for example γ(2k) = 1, this representation being the number 2k itself. From this relationship the average order of γ(m) is deduced ; this result is given in Theorem 2. By a method due to Kac [2], it is shown in §3 that the number of positive integers for which γ(m) does not exceed a rather complicated function of n and ω, a real parameter, is asymptotically nD(ω), where D(ω) is the probability integral
Incidence Relations in Multicoherent Spaces II
Authors: Stone A. H.
Published: December 2010

Introduction. One standard method of studying the incidences of a system of sets A1, A2, .… , An is to consider the nerveof the system. However, this gives no direct information as to the numbers of components of the various intersections of the sets—information which would be desirable in several geometrical problems. The object of the present paper is to modify the definition of the nerve so that these numbers of components can be taken into account, and to study this modified nervefor systems of sets in a connected, locally connected, normal T1space S of a given degree of multicoherence r(S). The principal result (Theorem 6, 6.4) is a refinement of a theorem of Eilenberg [4, p. 107], and asserts that, ifthen under suitable hypotheses we have

(1)
On Exponential Sums Over an Algebraic Number Field
Authors: Hua Loo-Keng
Published: December 2010

Let K be an algebraic field of degree n over the rational field, and let b be
the ground ideal (différente) of the field. Let
Vorticity Averages
Authors: Truesdell C.
Published: December 2010

Recent studies of turbulent fluid motions have drawn attention to the
transfer of vorticity. There have been some attempts to study turbulence by plane
models, but these have been criticized justly for failing to reveal the true
nature of a phenomenon which depends essentially on threedimensional convection. I
have thought it worthwhile to eschew current conjectures regarding turbulence,
turning in preference to the method of a master of the theory of vortices and
applying it to the discovery of certain average properties of the continuous
rotational motion of any medium whatever. This memoir therefore makes no attempt
to deal with the problem of turbulence, but it is possible that the theorems
presented here, being exact consequences of the kinematical equations, may
nevertheless enjoy a certain relevance.
Green's Forms and Meromorphic Functions on Compact Analytic Varieties
Authors: Kodaira Kunihiko
Published: December 2010

Let be a compact
complex analytic variety of the complex dimension n
with a positive definite Kâhlerian metric [4] ; the local analytic coordinates on
will be denoted by
z = (z
1
z
2, … , zn). Now, suppose a meromorphic
function f(z) defined on
as given. Then the
poles and zero-points of
f(z) constitute an analytic
surface in consisting of a
finite number of irreducible closed analytic surfaces Γ1,
Γ2, … , Γk, each of which is a polar
or a zero-point variety of
f(z).
Sur Le Théoréme De Lebesgue-Nikodym (V)
Authors: Dieudonné Jean
Published: December 2010

Soient E un espace compact (par exemple l' intervalle et
μ une mesure de Radon sur E. Une fonction numérique
f intégrable pour μ définit une mesure
sur l'ensemble des
parties mesurables (pour μ) de E. On peut aussi
considérer la forme linéaire qu'elle définit
sur l'espace C des fonctions numériques continues dans E, et cette forme linéaire
est continue pour la topologie définie par la norme
. Si on appelle
encore “mesure” une forme linéaire continue sur C, le théoréme de Lebesgue-Nikodym
classique caractérise celles de ces “mesures” forme qui sont de la
. par une condition
de “continuité absolue” par rapport à μ.
Metrization of Topological Spaces
Authors: Bing R. H.
Published: December 2010

A single valued function D(x, y) is a metric for a topological space provided
that for points x, y, z of the space:

1. the equality
holding if and only if x = y,

2.
(symmetry),

3. (triangle
inequality),

4. x belongs to the closure of the set M if and only if P(x, m) (m element of
M) is not bounded from 0 (preserves limit points).
On Recursions Connected With Symmetric Groups I
Authors: Chowla S. , Herstein I. N. , Moore W. K.
Published: December 2010

Although the title of the paper suggests that the nature of the problem
considered is group theoretic, our methods are almost completely combinatorial and
number theoretic in nature, the group theory entering only insofar as it leads us
to various recursions that we study. Let Tn denote the number
of solutions of x2 = 1 in Sn the
symmetric group of degree n.
Normal Samples With Linear Constraints and Given Variances
Authors: Fraser D. A. S.
Published: December 2010

1. Summary. In Biometrika (1948) a paper [1] by H. L. Seal contained a
theorem applying to “n random variables normally
distributed about zero mean with unit variance, these variables being connected by
means of k linear relations.” Arising from this is the
question of how to obtain a set of normal variates connected by
k linear relations and such that each variate has
unit variance: or, more generally, connected by k
linear relations and such that each variate has a given variance.
Extreme Forms
Authors: Coxeter H. S. M.
Published: December 2010

The two ternary quadratic forms

are said to be reciprocal (to each other) since their coefficients form
inverse matrices :
Cyclic Incidence Matrices
Authors: Hall Marshall , Ryser H. J.
Published: December 2010

Let it be required to arrange v elements into v sets such that each set
contains exactly k distinct elements and such that each pair of sets has exactly λ
elements in common (0 < λ < k < v). This problem we refer to
as the v, k,λ combinatorial problem.
Analyse Et Synthese Harmoniques Dans Les Espaces De Distributions
Authors: Schwartz Laurent
Published: December 2010

Cet article supposera connue la théorie des distributions, pour laquelle nous
renverrons à notre livre Théorie des distributions (Paris, 1950). Nous noterons
par (D) cet ouvrage. Rappelons cependant les définitions des principaux espaces
utilisés, sur une variété indéfiniment différentiable Vn:

() = espace
des fonctions indéfiniment différentiables à support compact;

() = espace
des fonctions indéfiniment différentiables à support quelconque;

(') = espace
des distributions, dual topologique de ();

(') = espace
des distributions à support compact, dual de ().
The Homomorphic Mapping of Certain Matric Algebras onto Rings of Diagonal Matrices
Authors: Goldhaber J. K.
Published: December 2010

The problem of determining the conditions under which a finite set of matricesA1A2, … ,Akhas the property that their characteristic roots λ1j, λ2j, … , λki(j= 1, 2, …,n) may be so ordered that every polynomialf(A1A2… ,Ak) in these matrices has characteristic rootsf(λ1j, λ2j…,λki) (j= 1, 2, … ,n) was first considered by Frobenius [4]. He showed that a sufficient condition for the (Ai〉 to have this property is that they be commutative. It may be shown by an example that this condition is not necessary.

J. Williamson [9] considered this problem for two matrices under the restriction that one of them be non-derogatory. He then showed that a necessary and sufficient condition that these two matrices have the above property is that they satisfy a certain finite set of matric equations.
On the Geometry of Lineal Elements on
A Sphere, Euclidean Kinematics, and
Elliptic Geometry
Authors: Feld J. M.
Published: December 2010

The geometry of slides and turns of oriented lineal elements in the plane was first studied by Kasner [10]. Slides and turns generate whirls, which constitute a three-parameter group W3. The product of W3 and M3, the three-parameter group of Euclidean displacements in the plane, yields a sixparameter group of whirl-motionsG6. The geometry of turbines, and also of general series of lineal elements, under G6 was investigated by Kasner in [10] and, in subsequent papers, by Kasner and DeCicco, particularly in [3], [4], [11], [12].
On the Property С and A Problem of Hausdorff
Authors: Rothberger Fritz
Published: December 2010

In an earlier paper [3] I studied the property С and related properties С′ and С; but the principal problem, viz, to prove, with the axiom of choice only (without any other hypothesis), the existence of a nondenumerable set of property С, remains open.
On Simple Alternative Rings
Authors: Albert A. A.
Published: December 2010

The only known simple alternative rings which are not associative are the Cayley algebras. Every such algebra has a scalar extension which is isomorphic over its center F to the algebra where . The elements e11 and e00 are orthogonal idempotents an , for every xij of . The multiplication table of C is then completed by the relations
A Closure Criterion for Orthogonal
Functions
Authors: Graves Ross E.
Published: December 2010

In this paper we give a simple, necessary, and sufficient condition for a sequence of orthogonal functions to be closed in L2. In theory the question of closure is reduced to the evaluation of certain integrals and the summation of an infinite series whose terms depend only upon the index n. Our principal result is

Let p(t) be a function whose zeros and discontinuities have Jordan content zero, such that for each x ∊ (a, b), p(t) ∊ L2 on min (c, x) < t < max (c, x), where a ≤ c ≤ b. (a, b, and c may be infinite.)
A One-Regular Graph of Degree Three
Authors: Frucht Robert
Published: December 2010

Soon after the publication of Tutte's paper [5] on m-cages, H. S. M. Coxeter asked in a letter to the author whether one-regular graphs of degree 3 exist. The purpose of the following paper is to show by an example that the answer is in the affirmative.
Abelian Quadratic Forms
Authors: Ramanathan K. G.
Published: December 2010

Hermite [4] in the course of his investigations on the transformation
theory of abelian functions, introduced the notion of abelian quadratic
forms. They are quadratic forms whose matrices of orders 2n, satisfy

where k ≠ 0 is a real number, and is the unit matrix of order n. Laguerre [7] and, more systematically, Cotty [3] developed an arithmetical theory of abelian forms in four variables.
A New Representation and Inversion Theory for the Laplace Transformation
Authors: Rooney P. G.
Published: December 2010

In the literature, considerable attention has been devoted to the study of inversion operators for the Laplace transformation. In particular, much interest attaches to “real” inversion operators, i.e., operators which make use of the values of the generating function arising only from real values of the independent variable. Several of these operators are known (see for example Widder [3, chap. 7, §6; chap. 8, §25], Hirschman [2]).
The Propagation of a Plane Shock into a Quiet
Atmosphere
Authors: Martin M. H.
Published: December 2010

In this note the problem of determining the propagation of a plane shock
moving through a polytropic gas into an undisturbed body of the gas, leaving
a non-isentropic disturbance behind it, is reduced to the solution of the
problem of Cauchy for a Monge-Ampére partial differential equation of
special type. The application of this result to specific examples is being
studied with a view to later publication.
The Boolean Algebra of Regular Open Sets
Authors: Pierce R. S.
Published: December 2010

Let S be a completely regular topological space. Let
C(S) denote the set of bounded,
real-valued, continuous functions on 5. It is well known that
C(S) forms a distributive lattice under
the ordinary pointwise joins and meets. For any distributive lattice
L and any ideal I⊆L, a
quasi-ordering of L can be defined as follows :
f⊇g if, for all h ∈
L, f ∩ h ∈ I implies g
∩ h ∈ I. If equivalent elements under this
quasi-ordering are identified, a homomorphic image of L is obtained.
Continuous Solutions of the Functional Equation fn(x)
= f(x)
Authors: Ewing G. M. , Utz W. R.
Published: December 2010

In this note the authors find all continuous real functions defined on the
real axis and such that for an integer n > 2, and for each x,
Regular Surfaces of Genus Two: Part I
Authors: Du Val Patrick
Published: December 2010

The present paper is a sequel to one I published lately (3) on
regular surfaces of genus 3, and like it, is intended to fill up some of the
gaps in our detailed knowledge of the regular surfaces of moderately low
genus p = pg = pa and linear genus p(1) = n + 1 > 1. (The surfaces for which
p(1) = 1 form a rather separate field of study on which a good
deal of work has been done, and I shall not consider them.)
Summability Methods Defined by Riemann Sums
Authors: Hill J. D.
Published: December 2010

Let f(x) be real valued, bounded and, integrable in the sense of Riemann on
the interval X = (0 < x < 1), with the value of its integral over X
equal to one. For brevity we call such a function
admissible.The symbol Xnk will always denote the interval an arbitrarily chosen point of Xnk and δ any specified set of intermediate points
On the Curvature Tensor of Einstein's Generalized Theory of
Gravitation
Authors: Lamson K. W.
Published: December 2010

Einstein, in his Generalized Theory of Gravitation [I],
deals with a tensor

whose ninety-six independent components are complex-valued functions of four
real variables xα. Schouten [4, p. 261 (89)] has decomposed the
general relative tensor,
antisymmetric in α, β, γ, and in ρ,σ, into five irreducible parts. This
decomposition can be applied to the R of Einstein if we define the tensor v
by
Weighted Quadratic Partitions Over a Finite
Field
Authors: Carlitz Leonard
Published: December 2010

Using some known results on Gauss sums in a finite field, it is shown that
the sum (1.3) defined below can either be evaluated explicitly or expressed
in terms of a Kloosterman sum. The same result applies to the more general
sum S(α, λ, Q) defined in (5.1). The latter sum also satisfies the
reciprocity formula (5.5). Some related sums are discussed in §§6, 7.
On -Adic
Integral Representations of Finite Groups
Authors: Maranda Jean-Marie
Published: December 2010

It has been shown by Diederichsen [2] that for integral representations of a
finite group, the irreducible constituents in any complete reduction are not
necessarily unique up to order and unimodular equivalence. In this same
article, it is shown that for certain finite groups, such as the cyclic
group of order 4, there are infinitely many classes of indecomposable
representations under unimodular equivalence.
On Integers n Relatively Prime to
[αn]
Authors: Watson G. L.
Published: December 2010

The object of this paper is to consider a problem suggested by Dr. K. F.
Roth, on the distribution of integers n that are relatively prime to the
integral part of an, a being a fixed real number. He conjectured that the
number of positive integers up to x with this property is asymptotic to
(or in
other words that they have the density ), for
irrational a. I prove this and rather more in the following
A Tensor Equation of Elliptic Type
Authors: Duff G. F. D.
Published: December 2010

The theory of the systems of partial differential equations which arise in
connection with the invariant differential operators on a Riemannian
manifold may be developed by methods based on those of potential theory. It
is therefore natural to consider in the same context the theory of elliptic
differential equations, in particular those which are self-adjoint. Some
results for a tensor equation in which appears, in addition to the operator
Δ of tensor theory, a matrix or double tensor field defined on the manifold,
are here presented. The equation may be written

in a notation explained below.
Function Spaces Determined by A Levelling Length
Function
Authors: Ellis H. W. , Halperin Israel
Published: December 2010

In this paper we introduce the function spaces Lx and Lx(5) which generalize
the classical Lp and Lp(B) spaces respectively. For
those λ which possess what we call the levelling property
we give a discussion of the conjugate space to Lλ(B); our
treatment applies, in particular, to the L(W)p(B) and M(W)q(B) spaces defined in [7, §2]. (See also the note at end
of this paper.)
Core-Consistency and Total Inclusion for Methods of Summability
Authors: Lorentz G. G. , Robinson A.
Published: December 2010

We shall consider methods of summation A, B, … defined by matrices of real elements (amn), (bmn), (m, n = 1, 2, …) which are regular, that is, have the three well-known properties of Toeplitz (4, p. 43). A method A is said to be core-consistent with the methodBfor bounded sequences if the A-core (3, p. 137; and 4, p. 55) of each real bounded sequence is contained in its B-core. B is totally included in A, B ≪A, if each real sequence which is B-summable to a definite limit (this limit may be finite or infinite of a definite sign) is also A-summable to the same limit.
An Embedding Theorem for Balanced Incomplete Block Designs
Authors: Hall Marshall , Connor W. S.
Published: December 2010

From a symmetric balanced incomplete block design we may construct a derived design by deleting a block and its varieties. But a design with the parameters of a derived design may not be embeddable in a symmetric design. Bhattacharya (1) has such an example with λ = 3 . When λ = 1, the derived design is a finite Euclidean plane and this can always be embedded in a corresponding symmetric design which will be a finite projective plane.
On Linear Partial Differential Equations of the Second Order Having Geodesic Solutions
Authors: Duff G. F. D.
Published: December 2010

The coefficients of the second derivatives in an elliptic or hyperbolic differential equation of the second order determine a Riemannian metric on the space of the independent variables. A Riemannian space has been called harmonic if the Laplace equation corresponding to it has a solution which is a function only of geodesic distance in that space. Harmonic spaces have been studied in some detail (1; 3). In this note are examined the circumstances under which a non-parabolic second order linear equation may have a “geodesic” solution of the type described. It will be shown that the equation must be self-adjoint, that the Riemann space corresponding to the equation must be harmonic and that the coefficient of the dependent variable must be a constant. Conversely, if these conditions are satisfied, the equation has two geodesic solutions, one of which is an elementary solution. In the case of elliptic equations, the second solution is connected with a certain mean value property which is valid in harmonic spaces.
An Inversion and Representation Theory for the Laplace Integral of Abstractly-Valued Functions
Authors: Rooney P. G.
Published: December 2010

The theory of the Laplace integral of abstractly-valued functions of a real or complex variable has been developed, in the last few years, to an extent that it is almost approaching the degree of completeness enjoyed by the classical theory of the Laplace integral of numerically-valued functions. In certain respects, however, there are still large gaps. One of the gaps occurs in representation theory.
Les Isomorphismes Exceptionnels Entre Les Groupes Classiques Finis
Authors: Dieudonné Jean
Published: December 2010

Les groupes classiques de petite dimension présentent entre eux des isomorphismes que l'on peut qualifier de “génériques”, c'est-à-dire que ces isomorphismes ne dépendent que de la dimension et du type de groupe considéré, et non du corps de base de l'espace vectoriel où opère le groupe.
A Note on Balanced Incomplete Block Designs
Authors: Sprott D. A.
Published: December 2010

A balanced incomplete block design is defined as an arrangement of v objects in b blocks, each block containing k objects all different, so that there are r blocks containing a given object and λ blocks containing any two given objects. Such designs have been studied for their combinatorial interest, as in (3), and also for their application to statistics, where the objects are usually varieties.
The Sixteenth Power Residue Character of 2
Authors: Whiteman A. L.
Published: December 2010

The problem of giving a criterion for the eth power residue character of 2 has long interested number theorists. This paper is primarilyconcerned with the cases e = 4, 8 and 16. Gauss (8) proved that 2 is a biquadratic residue of a prime p of the form 4n + 1 if and only if p is representable as x2 + 64y2.
The Relaxation Method for Linear Inequalities
Authors: Motzkin T. S. , Schoenberg I. J.
Published: December 2010

Let A be a closed set of points in the n-dimensional euclidean space En. If p and p1 are points of En such that

1.1

then p1 is said to be point-wise closer than p to the set A. If p is such that there is no point p1 which is point-wise closer than p to A, then p is called a closest point to the set A.
A Two-Point Boundary Problem for Ordinary Self-Adjoint Differential Equations of Fourth Order
Authors: Sternberg H. M. , Sternberg R. L.
Published: December 2010

The purpose of this note is to establish Theorem A below for the two-point homogeneous vector boundary problem

where the Pi(x) are given real m × m symmetric matrix functions of x with P0(x) positive definite and Pi(x) of class C2−i on an infinite interval [a, ∞), and where by a solution of (1.1) — (1.2) for a ≤ x1 < x2 < ∞ we understand a real m-dimensional column vector u = u(x) of class C2 on [a, ∞) which is such that Pi(x)u(2−i) is of class C2−i on [a, ∞) and which satisfies (1.1) — (1.2) with the former a vector identity on [a, ∞).
On a Double Integral Variational Problem
Authors: Garabedian P. R. , Schiffer M.
Published: December 2010

The results presented here were motivated by a desire to give a simple treatment of the non-linear elliptic partial differential equation governing the steady irrotational subsonic flow of an ideal compressible fluid. For two independent space variables x and y, the stream function Ψ of such a flow satisfies an equation

,

where is an analytic, increasing, convex function of whose explicit form depends on the equation of state of the fluid in question.
A Generalization of the Young Diagram
Authors: Burrow M. D.
Published: December 2010

The method of A. Young for finding the set of primitive idempotents of the group algebra of the symmetric group is classical; it was first given by Frobenius (4) using results of Young (10 and 11). A concise account can be found in (9) and a very detailed treatment in (6).
Some Remarks on the Characters of the Symmetric Group, II
Authors: Osima Masaru
Published: December 2010

Let p be a fixed prime number. We denote by k(n) the number of partitions of n. As is well known, the number of ordinary irreducible characters of the symmetric group Sn is k(n). We set k(0)=1.
A Short Proof of the Cartwright-Littlewood Fixed Point Theorem
Authors: Hamilton O. H.
Published: December 2010

The purpose of this paper is to give a short proof of the Cartwright-Littlewood
fixed point theorem (2, p. 3, Theorem A).

Theorem A. If T is a (1-1) continuous and orientation preserving transformation of the Euclidean plane E onto itself which leaves a bounded continuum M invariant and if M does not separate E, then some point of M is left fixed by T.
The Cauchy Problem for a Hyperbolic Second Order Equation with Data on the Parabolic Line
Authors: Protter M. H.
Published: December 2010

In this paper we consider the Cauchy problem for the
equation

(1) h(x, y) K(y) vxx − vyy + a(x, y) vx + b(x, y) vy + c(x, y) v + f(x, y) = 0

with initial values prescribed on a segment of the x-axis. The coefficients in (1) are assumed to possess two continuous derivatives with respect to x and one continuous derivative with respect to y in the closure of the domain under consideration.
On Linear Perturbation of Non-Linear Differential Equations
Authors: Atkinson F. V.
Published: December 2010

In the theory of the asymptotic solution or stability of ordinary differential equations most attention has been given to linear or nearly-linear cases. Investigations in this field, starting primarily with those of Kneser(7)on the equation y″+f(x)y= 0, have by now mostly been summed up in results on the vector-matrix system dy/dx =Ay +f(y,x), whereyandfdenoten-vectors of functions, andAann-by-nmatrix, frequently assumed constant.
Convolution Transforms Related To Non-Harmonic Fourier Series
Authors: Sumner D. B.
Published: December 2010

1. Introduction. Widder has pointed out (2, p. 219) in connection with Wiener's fundamental work on the operational calculus (1, pp. 557-584), that the convolution transform

(1.1)

will be inverted by the operator DE (D), where D = d/dx, and

,

where a suitable interpretation must be found for E(D).
Asymptotic Behaviour Of The Inverse Of a Laplace Transform
Authors: Hull T. E. , Froese C.
Published: December 2010

1. Introduction. There are many problems, particularly in circuit theory, where the inverse of a Laplace transform is required but only for large values of the independent variable. For example (2, p. 66) the Laplace transform

of the current function for a semi-infinite cable under certain conditions turns
out to be

.
On Selberg's Lemma For Algebraic Fields
Authors: Ayoub R. G.
Published: December 2010

1. Introduction. Recently two Japanese authors (1) gave a beautifully simple proof of Selberg's fundamental lemma in the theory of distribution of primes. The proof is based on a curious twist in the Möbius inversion formula. The object of this note is to show that their proof may be extended to a proof of the result for algebraic fields corresponding to Selberg's lemma. Shapiro (2) has already derived this result using Selberg's methods and deduced as a consequence the prime ideal theorem.
Two Remarks On Extreme Forms
Authors: Kneser Martin
Published: December 2010

1. Introduction. The following remarks concern two different parts of the theory of extreme quadratic forms. In §2, I shall give a new proof for the theorem of Voronoï (4), which asserts that a form is extreme if and only if it is both perfect and eutactic. (For the definitions see, e.g., Coxeter (2) or the text below.) There is indeed a comparatively simple proof in Bachmann's Zahlentheorie, IV, 2. For two reasons, however, it may not be useless to communicate another proof here.
Note On Extreme Forms
Authors: Barnes E. S.
Published: December 2010

Letƒ(x1, … ,xn) = Σaijxixjbe a positive definite quadratic form of determinantD= |aij|, and letMbe the minimum offfor integralx1, … ,xnnot all zero. The formƒis said to beextremeif the ratioMn/Ddoes not increase when the coefficients aijoffsuffer any sufficiently small variation.
Some Theorems On Matrices With Real Quaternion Elements
Authors: Wiegmann N. A.
Published: December 2010

Matrices with real quaternion elements have been dealt with in earlier papers by Wolf (10) and Lee (4). In the former, an elementary divisor theory was developed for such matrices by using an isomorphism between n×n real quaternion matrices and 2n×2n matrices with complex elements. In the latter, further results were obtained (including, mainly, the transforming of a quaternion matrix into a triangular form under a unitary similarity transformation) by using a different isomorphism.
Full Individual and Class Differentiation Theorems In Their Relations To Halo and Vitali Properties
Authors: Hayes C. A. , Pauc C. Y.
Published: December 2010

Introduction. In his article (22), de Possel laid the foundations for anabstract theory of differentiation of set functions, the term “abstract” being meant in the sense of Fréchet-Nikodym, that is, without reference to a euclidean, metric, or topological background. In 1.1, we adopt, substantially, his notion ofderivation basis.De Possel considered two Vitali properties for a derivation basis.
On Dynamical Systems With One Degree Of Freedom
Authors: Putnam C. R.
Published: December 2010

1. Introduction. Consider the (vector, n-component) system of differential equations

1,

where f(x) is of class C1. Let Ω denote a set of points, x, consisting of unrestricted solution paths x(t), so that the x(t) exist and lie in Ω for — ∞ < t < ∞.
The Distribution Of Totatives
Authors: Lehmer D. H.
Published: December 2010

This paper is concerned with the numbers which are relatively prime to a given positive integer

where the p's are the distinct prime factors of n. Since these numbers recur periodically with period n, it suffices to study the ϕ(n) numbers ≤n and relatively prime to n.
On the Modular Representation of the Symmetric Group
Authors: Robinson G. De B.
Published: December 2010

1. Introduction. It has been observed (2) that the number of p-regular classes of Sn, i.e. the number of classes of order prime to p, is equal to the number of partitions (λ) of n in which no summand is repeated p or more times. For this relation to hold it is essential that p be prime. It seems natural to call the Young diagram [λ] associated with (λ) p-regular if no p of its rows are of equal length, otherwise p-singular.
On Orders In Separable Algebras
Authors: Higman D. G.
Published: December 2010

Introduction. The present note began from the observation that the arguments produced by J-M. Maranda in developing his very interesting theory of representations of groups by automorphisms of modules over Dedekind rings (4, 5) were applicable without essential change to arbitrary orders, instead of just group rings, provided that a suitable generalization of Theorem 1 of (4) could be supplied.
On The Equivalence Of Representations Of Finite Groups By Groups Of Automorphisms Of Modules Over Dedekind Rings
Authors: Maranda J.-M.
Published: December 2010

Let t be a Dedekind ring whose quotient field we denote by K. If is a prime ideal of t, let o denote the ring of all -regular elements of K. If is a torsion free i-module, let K denote the smallest K-module into which can be embedded and let o denote the o-submodule of K generated by .
A Note On The Separability Of An Ordered Space
Authors: Ball B. J.
Published: December 2010

An open interval of a simply ordered set S is a subset I of S such that either

(1) for some ,

(2) for some , or

(3) for some .

A simply ordered set with its interval topology (i.e., the topology in which “neighborhood of x” means “open interval containing x”) will be called an ordered space.
A Class Of Abelian Groups
Authors: Tutte W. T.
Published: December 2010

1. Introduction. If M is any finite set we define a chain on M as a mapping f of M into the set of ordinary integers. If a ∈ M then f(a) is the coefficient of a in the chain f. The set of all a ∈ M such that f(a) ≠ 0 is the domain |f| of f. If |f| is null, that is if f(a) = 0 for all a, then f is the zero chain on M. If M is null it is convenient to say that there is just one chain, a zero chain, on M.
On An Inversion Formula For The Laplace Transformation, II
Authors: Rooney P. G.
Published: December 2010

In an earlier paper (3) we discussed at some length a certain inversion
operator for the Laplace transformation. If

I

then the inversion operator is given by

II ,

and we showed that if v > — 1, then under certain conditions

.
A Regular Singular Functional
Authors: Martin A. D.
Published: December 2010

1. Introduction. In a joint paper with Leighton (2), the author considered
quadratic functionals of the type

1.1 (0 < a < b)

in which x = 0 is a singular point of the functional which is otherwise regular on [0, b]. The hypothesis on a regular functional includes the assumption that r is continuous and positive on a closed interval [0, b].
On a Theorem Of Le Roux
Authors: Diaz J. B. , Ludford G. S. S.
Published: December 2010

1. The Theorem of Le Roux. Let U(x, y, α) be any solution of the linear hyperbolic differential equation

1

containing a parameter α. J. Le Roux (5) has shown that the function u(x, y) defined by

2 α0 = const.
Uniqueness In Boundary Value Problems For The Second Order Hyperbolic Equation
Authors: Duff G. F. D.
Published: December 2010

Introduction. We study linear normal hyperbolic partial differential equations of the second order, with one dependent variable u, and N independent variables xi (i = 1, … , N). The uniqueness theorem connected with the Cauchy problem for this type of equation is well known and in effect states that if u and its first normal derivatives vanish on a spacelike initial surface S then u vanishes in a certain conical region which contains S (1, p. 379).
A Generalization Of An Inequality Of Hardy and Littlewood
Authors: Smith K. T.
Published: December 2010

1. Introduction. A well-known inequality of Hardy-Littlewood reads as follows (4): if p > 1 and f > 0, then

,

where is defined as the supremum of the numbers

the constant depends on p only. The statement obtained by putting p = 1 is false; its substitute reads:

the constants depend on p but not on f.
On a Theorem Of Baer and Higman
Authors: Tobin Seân
Published: December 2010

1.1 Baer has shown (1) that if the fact that the exponent of a group is m (that is, m is the least common multiple of the periods of the elements) implies a limitation on the class of the group, then m must be a prime. Graham Higman has extended this result by proving (3) that for any given integer M there are at most a finite number of prime powers q other than primes, such
that the fact that a group has exponent q implies a limitation on the class of the Mth derived subgroup.
Maximal Flow Through a Network
Authors: Ford L. R. , Fulkerson D. R.
Published: December 2010

Introduction. The problem discussed in this paper was formulated by T. Harris as follows:

“Consider a rail network connecting two cities by way of a number of intermediate cities, where each link of the network has a number assigned to it representing its capacity. Assuming a steady state condition, find a maximal flow from one given city to the other.”
Enumeration Of Labelled Graphs
Authors: Gilbert E. N.
Published: December 2010

1. Introduction. The number of connected linear graphs having V vertices labelled 1, … , V and λ (unlabelled) lines is found below. Similar formulas are found for graphs in which slings, lines “in parallel,” or both are allowed and for directed graphs with or without slings or parallel lines. Some of these graphs are also counted when the lines are labelled and the vertices are unlabelled.
Certain Infinite Zero-Sum Two-Person Games
Authors: Dulmage A. L. , Peck J. E. L.
Published: December 2010

1. Introduction. The theorem of von Neumann, that every finite, zero-sum two-person game has a value, has been extended in various ways to infinite games. In particular Wald (6) has shown that any bounded game in which one player has finitely many pure strategies, has a value. Our interest was aroused by the infinite analogue of the game of “hide and seek” as described
by von Neumann (5), which does not appear to fit any of the known cases, unless the matrix is bounded.
On Explicit Bounds In Landau's Theorem
Authors: Jenkins J. A.
Published: December 2010

1. The theorem of Landau in question may be stated in the form that if the function F(Z) is regular for |Z| < 1 and does not take the values 0 and 1, while

F(Z) = a0 + a1Z + …

is its Taylor expansion about Z = 0, then |a1| has a bound depending only on a0. In fact |a1| has a bound depending only on |a0| and Hayman (1) gave the explicit bound
The Cauchy Problem For Linear Partial Differential Equations With Restricted Boundary Conditions
Authors: Miles E. P. , Williams Ernest
Published: December 2010

We shall discuss solutions of linear partial differential equations of the form

1

where Ψ is an ordinary differential operator of order s with respect to t. Our first theorem gives a solution of (1) for the Cauchy data;

2

j = 1,2, ߪ,s − 1,

whenever the function P is annihilated by a finite iteration of the operator Φ.
Spectral Theory For a Class Of Non-Normal Operators
Authors: Gonshor Harry
Published: December 2010

1. Introduction. As is well known, the spectral theorem plays an important part in mathematics because of its many applications. Unfortunately, the theorem is valid for normal operators only. In view of this, attempts have been made by several mathematicians to obtain a theorem about a more general class of operators, which will reduce to the ordinary spectral theorem if the operator is normal. Brown (1) has developed a unitary equivalence theory for a certain class of operators.
Groups Of Positive Operators
Authors: Dye H. A. , Phillips R. S.
Published: December 2010

1. Introduction. Semi-groups of bounded positive operators on certain function spaces enter the theory of stochastic processes of the diffusion type in an essential way. It is a matter of experience that these semi-groups cannot be imbedded in groups of positive operators, or, in more special terms, that the solution of a diffusion equation does not define a one-parameter group of positive operators on the natural function space.
Extremal Properties Of Hermitian Matrices
Authors: Marcus M. , McGregor J. L.
Published: December 2010

In (1) Fan showed that if A is a Hermitian matrix with eigenvalues λ1 ≤ … ≤ λn then, for k ≤ n,

,

,

where X1, … , xk run over all sets of k orthonormal (o.n.) vectors in unitary n-space V.
Double Transitivity In Finite Projective Planes
Authors: Ostrom T. G
Published: December 2010

A projective plane is characterized to a certain extent by the amount of transitivity it possesses. This amounts essentially to saying that the plane is characterized by its group of collineations. The transitive planes that have been most thoroughly studied are the cyclic planes (8; 10; 12; 13; 14). It is believed that all finite cyclic planes are Desarguesian, but it has not been proved that this is the case.
Prime Power Representations Of Finite Linear Groups
Authors: Steinberg Robert
Published: December 2010

1. Introduction. There are five well-known, two-parameter families of simple finite groups: the unimodular projective group, the symplectic group,1 the unitary group,2 and the first and second orthogonal groups, each group acting on a vector space of a finite number of elements (2; 3).
Some Remarks on Noetherian Rings
Authors: Yoshida Michio
Published: December 2010

In his lecture at the University of Kyoto on September 23, 1955, Professor
Artin gave an important theorem on Noetherian rings, which seems to have not
a few interesting consequences. It is the purpose of our present note to
point out one of them. We begin by quoting a special case of the
theorem.
A Note on Asymptotic Series
Authors: Davis Harry F.
Published: December 2010

We extend some observations of Popken (2) on the algebraic foundations of
the theory of asymptotic series. The main result is the theorem in §5 which
characterizes, for a particular function space, a class of linear
functionals defined in §4. In §3 we discuss another class of linear
functionals related to asymptotic series. In the first two paragraphs we
give definitions which render this note self-contained.