bluuebunny/crossref_metadata_2025 huggingface dataset:
Bilateral Testicular Epidermoid Cysts
Authors: Loberant Norman , Bhatt Shweta , Messing Edward , Dogra Vikram S.
Published: December 2010

Testicular epidermoid cysts are the most common benign tumors of the testes, but account for only 1-2% of all testicular tumors. In a young man presenting with a testicular mass, a high index of suspicion must be maintained for the malignant testicular germ cell tumor, which is 50-times more common than testicular epidermoid cyst. Bilateral testicular epidermoid cysts are a very rare condition, with only a few reports in the literature. It is extremely important in this condition to make a correct pre-operative diagnosis on imaging to enable a testis-sparing surgery.
Identification of Predictive Pathways for Non-Hodgkin Lymphoma Prognosis
Authors: Han Xuesong , Li Yang , Huang Jian , Zhang Yawei , Holford Theodore , Lan Qing , Rothman Nathaniel , Zheng Tongzhang , Kosorok Michael R. , Ma Shuangge
Published: December 2010

Despite decades of intensive research, NHL (non-Hodgkin lymphoma) still remains poorly understood and is largely incurable. Recent molecular studies suggest that genomic variants measured with SNPs (single nucleotide polymorphisms) in genes may have additional predictive power for NHL prognosis beyond clinical risk factors. We analyzed a genetic association study. The prognostic cohort consisted of 346 patients, among whom 138 had DLBCL (diffuse large B-cell lymphoma) and 101 had FL (follicular lymphoma). For DLBCL, we analyzed 1229 SNPs which represented 122 KEGG pathways. For FL, we analyzed 1228 SNPs which represented 122 KEGG pathways. Unlike in existing studies, we targeted at identifying pathways with significant additional predictive power beyond clinical factors. In addition, we accounted for the joint effects of multiple SNPs within pathways, whereas some existing studies drew pathway-level conclusions based on separate analysis of individual SNPs. For DLBCL, we identified four pathways, which, combined with the clinical factors, had medians of the prediction logrank statistics as 2.535, 2.220, 2.094, 2.453, and 2.512, respectively. As a comparison, the clinical factors had a median of the prediction logrank statistics around 0.552. For FL, we identified two pathways, which, combined with the clinical factors, had medians of the prediction logrank statistics as 4.320 and 3.532, respectively. As a comparison, the clinical factors had a median of the prediction logrank statistics around 1.212. For NHL overall, we identified three pathways, which, combined with the clinical factors, had medians of the prediction logrank statistics as 5.722, 5.314, and 5.441, respective. As a comparison, the clinical factors had a median of the prediction logrank statistics around 4.411. The identified pathways have sound biological bases. In addition, they are different from those identified using existing approaches. They may provide further insights into the biological mechanisms underlying the prognosis of NHL.
Mean-Continuous Integrals
Authors: Ellis H. W.
Published: December 2010

Descriptive definitions of Cesàro-Denjoy integrals (CD-integrals) equivalent to the Cesàro-Perron integrals (CP-integrals) introduced by J. C. Burkill [1, 2] have been given by Miss Sargent [6] (see §2). The CD§integrals are generalizations of the special Denjoy integral [5, p. 201]. They are somewhat complicated in that modifications of the definitions of continuity, generalized absolute continuity in the restricted sense (ACG*) [5, p. 231], and of derivatives are required for each order. In the present paper a scale of integrals is obtained which is based on the descriptive definition of the general Denjoy integral [5, p. 241]. The approximate derivative and a slightly modified definition of generalized absolute continuity (ACG) are used for all orders so that the only concept generalized for increasing orders is that of continuity.
Completely Indecomposable Modules
Authors: Snapper Ernst
Published: December 2010

The purpose of this paper is to investigate completely indecomposable modules. A completely indecomposable module is an additive abelian group with a ring A as operator domain, where the following four conditions are satisfied.

1-1. A is a commutative ring and has a unit element which is unit operator for .

1-2. The submodules of satisfy the ascending chain condition. (Submodule will always mean invariant submodule.)
On the Disjoint Product of Irreducible Representations of the Symmetric Group
Authors: de B. Robinson G.
Published: December 2010

The results of the present paper can be interpreted (a) in terms of the theory of the representations of the symmetric group, or (b) in terms of the corresponding theory of the full linear group. In the latter connection they give a solution to the problem of the expression of an invariant matrix of an invariant matrix as a sum of invariant matrices, in the sense of Schur's Dissertation. D. E. Littlewood has pointed out the significance of this problem for invariant theory and has attacked it via Schur functions, i.e. characters of the irreducible representations of the full linear group. We shall confine our attention here to the interpretation (a).
Subgroups of Finite Index in Free Groups
Authors: Hall Marshall
Published: December 2010

This paper has as its chief aim the establishment of two formulae associated with subgroups of finite index in free groups. The first of these (Theorem 3.1) gives an expression for the total length of the free generators of a subgroup U of the free group Fr with r generators. The second (Theorem 5.2) gives a recursion formula for calculating the number of distinct subgroups of index n in Fr.

Of some independent interest are two theorems used which do not involve any finiteness conditions. These are concerned with ways of determining a subgroup U of F.
Application of Combinatorial Formulae to Generalizations of Wilson's Theorem
Authors: Mendelsohn N. S.
Published: December 2010

Summary. By the use of finite difference operators a series of combinatorial formulae are obtained. Two applications of the formulae are given. First, the following explicit formulae for the number, ƒ(n), of non-isomorphic equivalence relations among n elements are obtained:
Representation by Quadratic Forms
Authors: Pall Gordon
Published: December 2010

1. Introduction. The elementary portions of the theory of integral representation of numbers or forms by quadratic forms will be somewhat simplified and generalized in this article. This indicates certain directions in which new applications can be made. The applications made here will be largely to the representation of numbers or binary quadratic forms by ternary quadratic forms. Particularly, we shall obtain the correct estimate (Theorem 10) needed to fill a lacuna in certain work of U. V. Linnik [1] on the representation of large numbers by ternary quadratic forms. Since Linnik applied his theorem on ternaries to prove [9] that every large number is a sum of at most seven positive cubes, a lacuna in this proof can now be regarded as filled.
Star Diagrams and the Symmetric Group
Authors: Staal R. A.
Published: December 2010

Introduction. The irreducible representations of the symmetric group Sn, were shown by A. Young to be in one-to-one correspondence with certain arrays of n nodes. E.g. for n = 12 and the partition λ = [4, 4, 3, 1] we have the array

which we call a “Young diagram.” The question arises as to the manner in which various properties of the representations are reflected in their corresponding Young diagrams.
Squaring the Square
Authors: W. T. Tutte
Published: December 2010

1. Introduction. It is the object of this paper to describe in more detail than has hitherto been done the general methods by which a square may be
dissected into smaller unequal non-overlapping squares. Examples of such
dissections are given
The Fréchet Variation, Sector Limits, and Left Decompositions
Authors: Morse Marston , Transue William
Published: December 2010

1. Introduction. The Fréchet variation of a function g defined over a 2-interval I2 was introduced by Fréchet to enable him to generalize Riesz's theorem on the representation of functionals linear over the space C [7]. Recently the authors have found this variation fundamental in the study of functionals bilinear over the Cartesian product A ⨯ B of two normed linear spaces with certain characteristic properties, and in the further use of this theory in spectral and variational analysis. The recent discovery by the authors of several new properties of the Fréchet variation has made it possible to to give new and natural tests for the convergence of multiple Fourier series generalizing the classical Jordan, de la Vallée Poussin, Dini, Young and Lebesgue tests under considerably less restrictive hypotheses than those now accepted.
The Distribution of the Maximum of Partial Sums of Independent Random Variables
Authors: Kac Mark , Pollard Harry
Published: December 2010

1. The problem. It has been shown [1] that if Xi, + X2, … are independent random variables each of density then

(1.1)
Lattices With a Given Abstract Group of Automorphisms
Authors: Frucht Robert
Published: December 2010

The problem of finding a lattice with a given abstract group of automorphisms has been solved by Garrett Birkhoff who proved that for any group of order g there exists a distributive lattice with at most elements. That this number can be somewhat reduced by modifications of Birkhoff's original procedure has already been shown by the author; it turns out, however, that it remains rather high for finite groups of relatively low order.
Equation De Hill Et Probleme De Störmer
Authors: De Vogelaere Rene
Published: December 2010

1. Introduction. Pour la détermination des orbites infiniment voisines de l'équateur, dans le probléme de Störmer, une équation de Hill est à résoudre. Les méthodes sont expliquées d'abord sur l'équation générale, puis appliquées au probléme de Störmer. Signalons les résultats suivants: les orbites équatoriales, considérées du point de vue de leur perturbation dans le plan méridien sont successivement stables, instables impaires, stables, instables paires et cela indéfiniment quand γ1 se rapproche de un; quelques orbites limites entre les zones de stabilité et d'instabilité sont obtenues avec une méthode qui permet n'importe quelle précision désirée.
Quasiconvex Sets
Authors: Green J. W. , Gustin W.
Published: December 2010

Introduction. Let I be the closed real number interval: Any subset Δ of I containing at least one number interior to I, will be called a quasiconvexity generating set. To each quasiconvexity generating set Δ we associate as follows a generalized notion of convexity, here called quasiconvexity or Δ convexity. Two numbers α and β, one of which belongs to Δ, the other being determined by the relation a α + β = 1, are called complementary ratios of Δ. A set Q in a real vector space is said to be A convex if for every pair of complementary ratios α and β in Δ and every pair of points a and b lying in Q the point αa +β b also lies in Q.
On Regular Surfaces Of Genus Three
Authors: Du Val Patrick
Published: December 2010

Enriques, in his posthumous magnum opus (1),
devotes a chapter to the canonical or (where the genus is small) bi- or
tricanonical models of regular surfaces, for various values of the genus
and of the linear
genus p(1). If, however, the cases he deals with are
tabulated as follows (* marking the surfaces described by Enriques):
The Duality Theorem for Curves of Order n in n-Space
Authors: Derry Douglas
Published: December 2010

LetCnbe a curve in real projectiven-space which is a continuous 1— 1 image of either the projective line or one of its closed segments. Consequently its points depend continuously on a real variablesfor which, with the understanding thats= 0 ands= 1 represent the same curve point in the case thatCnis the image of the complete projective line. The points ofCnwill be described by their corresponding real numberss.
A Theorem on Division Rings
Authors: Kaplansky Irving
Published: December 2010

The object of this note is to prove the following theorem.

THEOREM. Let A be a division ring with centre Z, and suppose
that for every x in A,
some power (depending on x) is in . Then A is
commutative.
On a Theorem of Aubry-Thue
Authors: Brauer Alfred , Reynolds R. L.
Published: December 2010

In 1913 L. Aubry [1] proved the following theorem:

If a and m are relatively prime, m > 0, and if
is not an
integer, then it is always possible to find integers x and y not both zero such
that
On Products of Sets of Group Elements
Authors: Mann Henry B.
Published: December 2010

Let = {A1, … , As}, = {B1 … , Bt} be sets of elements of a group of finite order g. We define

By (), (), … we shall denote the number of elements in , , … respectively and by , , … the sets of elements of not in ,, ….
A Symmetric Proof of the Riemann-Roch Theorem, and a New Form of the Unit Theorem
Authors: Beatty S. , Lane N. D.
Published: December 2010

LetF(z,u) denote

1

whereF1(z),… ,Fn(z) are rational functions ofzwith complex coefficients. We shall speak ofF(z, u) = 0 as the fundamental algebraic equation and shall adoptzas the independent variable and u as the dependent, except in § 4, where we usexandyinstead of them, and where it is understood thatxandyare connected birationally withzandu.
Quelques Remarques Sur La Généralisation Du Scalaire De Courbure Et Du Scalaire Principal
Authors: Moór Arthur
Published: December 2010

Dans un espace finslérien à n dimensions rapporté à un système quelconque de coordonnéesx1,x2…,xnla distance de deux points infiniment voisins est donnée par la formule

1
Isoperimetric Problems in the Calculus of Variations
Authors: Karush William
Published: December 2010

We are concerned with establishing sufficiency theorems for minima of simple integrals of the parametric type in a class of curves with variable end points and satisfying isoperimetric side conditions. The results which are obtained involve no explicit assumptions of normality. Such results can be derived by transforming our problem to a problem of Bolza and using the latest developments in the theory of that problem. More recently [6] an indirect method of proof has been published. Our object is to present a direct method of proof without transformation of the problem which is based upon a generalization of the classical theory of fields.
Cyclic Affine Planes
Authors: Hoffman A. J.
Published: December 2010

Let Π be an affine plane which admits a collineation τ such that the cyclic group generated by τ leaves one point (say X) fixed, and is transitive on the set of all other points of Π. Such “cyclic affine planes” have been previously studied, especially in India, and the principal result relevant to the present discussion is the following theorem of Bose [2]: every finite Desarguesian affine plane is cyclic. The converse seems quite likely true, but no proof exists. In what follows, we shall prove several properties of cyclic affine planes which will imply that for an infinite number of values of n there is no such plane with n points on a line.
On the Number of Solutions of Some General Types of Equations in a Finite Field
Authors: Faircloth Olin B.
Published: December 2010

The conditional equation f(x1, … , xs) = 0, where f is a polynomial in the x´s with coefficients in a finite field F(pn), is connected with many well-known developments in number theory and algebra, such as: Waring's problem, the arithmetical theory of quadratic forms, the Riemann hypothesis for function fields, Fermat's Last Theorem, cyclotomy, and the theory of congruences in commutative rings.
Multiply Subadditive Functions
Authors: Lorentz G. G.
Published: December 2010

Let S denote a Boolean ring with elements e, that is, a distributive, relatively complemented lattice with zero element 0 [2, p. 153]. In this paper we study real-valued functions which have a representation of the form

1.1
Map-Colour Theorems
Authors: Dirac G. A.
Published: December 2010

A map wil be calledk-chromatic or said to have chromatic numberkifkis the least positive integer having the property that the countries of the map can be divided intokmutually disjoint (colour) classes in such a way that no two countries which have a common frontier line are in the same (colour) class.Heawood [4] proved that forh> 1 the chromatic number of a map on a surface of connectivityhis at mostnh, where
The Coordinate Conditions and the Equations of
Motion
Authors: Infeld L.
Published: December 2010

The problem of the field equations and the equations of motion in general
relativity theory is now sufficiently clarified. The equations of motion can
be deduced from pure field equations by treating matter as singularities,
[2; 3], or from field equations with the energy momentum tensor [4].
Recently two papers appeared in which the problem of the coordinate system
was considered [5; 8]. The two papers are in general agreement as far as the
role of the coordinate system is concerned. Yet there are some differences
which require clarification.
Proof of a Conjecture of Schoenberg on the Generating Function
of a Totally Positive Sequence
Authors: Edrei Albert
Published: December 2010

Let

be a sequence of real terms with which we associate the generating power
series

We consider the following definition due to Schoenberg [7, p. 362]:
Integral Functions With Negative Zeros
Authors: Boas R. P.
Published: December 2010

If f(z) is an integral function of non-integral order with only real
negative zeros, there is a close connection between the rates of growth of
the function and of n (r), the number of zeros of absolute value not
exceeding r. The best known theorem is that of Valiron [12], which may be
stated as follows.
On the Commutativity of Certain Division Rings
Authors: Nakayama Tadasi
Published: December 2010

Formerly Hua [1] proved that if A is a
division ring with centre Z and if there exists a natural
number n such that an ∈ Z for every a ∈
A, then A is commutative; this
generalizes Wedderburn's theorem on finite division rings. Another
generalization of Wedderburn's theorem, due to Jacobson [3],
asserts that every algebraic division algebra over a finite field is
commutative. On the other hand, a theorem of Noether and Jacboson
[3] states that every noncommutative algebraic division
algebra contains an element which is not contained in the centre
Z and is separable over Z.
On Non-Averaging Sets of Integers
Authors: Moser Leo
Published: December 2010

Let 5 be a set of positive integers no three of which are in arithmetical
progression, i.e., if A, B, C are distinct elements of S, A + B ≠ 2C. We
call such a set a non-averaging set. Let v(n) denote the maximum number of
elements not exceeding n in any non-averaging set. The problem of finding
bounds for v(n) has been treated by several authors [1, 3, 5, 6, 7]. The
question first arose in connection with a theorem of van der Waerden
[8].
A Relation Between Ultraspherical and Jacobi Polynomial
Sets
Authors: Brafman Fred
Published: December 2010

The Jacobi polynomials may be defined by

where (a)n = a (a + 1) … (a + n — 1). Putting β = α gives the
ultraspherical polynomials which have
as a special case the Legendre polynomials .
A Theorem of Glaisher
Authors: Carlitz Leonard
Published: December 2010

Let

Then if p is a prime > 3, Glaisher [4] proved
Unitary Groups Generated by Reflections
Authors: Shephard G. C.
Published: December 2010

A reflection in Euclidean n-dimensional space is a
particular type of congruent transformation which is of period two and
leaves a prime (i.e., hyperplane) invariant. Groups generated by a number of
these reflections have been extensively studied [5, pp. 187-212]. They are
of interest since, with very few exceptions, the symmetry groups of uniform
polytopes are of this type. Coxeter has also shown [4] that it is possible,
by Wythoff's construction, to derive a number of uniform polytopes from any
group generated by reflections. His discussion of this construction is
elegantly illustrated by the use of a graphical notation [4, p. 328; 5, p.
84] whereby the properties of the polytopes can be read off from a simple
graph of nodes, branches, and rings.
An Extreme Duodenary Form
Authors: Coxeter H. S. M. , Todd J. A.
Published: December 2010

Let f(x1, … , xn) be a positive definite quadratic
form of determinant Δ; let M be its minimum value for integers
x1, … , xn not all zero; and let 2s be the
number of times this minimum is attained, i.e., the number of solutions of
the Diophantine equation
Numerical Integration of Functions of Several
Variables
Authors: Tyler G. W.
Published: December 2010

Methods of mechanical quadrature of functions of more than one variable
apparently have received little systematic investigation and the few
available results are widely scattered in the literature. In this paper a
systematic approach to this problem is given and a number of formulae are
derived which may prove to be useful.
The Non-Existence of Certain Affine Resolvable Balanced
Incomplete Block Designs
Authors: Shrikhande S. S.
Published: December 2010

A method of proving the impossibility of certain Affine Resolvable Balanced
Incomplete Block Designs (A.R.B.I.B.D.) has been given by the author
elsewhere [9]. More complete results in the same direction are obtained here
using the ideas of a paper by Connor [4].
On Residue Difference Sets
Authors: Lehmer Emma
Published: December 2010

In recent years the subject of difference sets has attracted a considerable
amount of attention in connection with problems in finite geometries [4].
Difference sets arising from higher power residues were first discussed by
Chowla [1], who proved that biquadratic residues modulo p form a difference
set if (p — l )/4 is an odd square. In this paper we shall prove a similar
result for octic residues and develop some necessary conditions which will
eliminate all odd power residue difference sets and many others. We also
prove that a perfect residue difference set (that is, one in which every
difference appears exactly once) contains all the powers of 2 modulo p.
On the Waring-Siegel Theorem
Authors: Ayoub R. G.
Published: December 2010

The Waring problem deals with the decomposition of integers into sums of kth
powers. Consider
On the Number of Primitive Lattice Points in a
Parallelogram
Authors: Estermann Theodor
Published: December 2010

1. Let a be any irrational real number, and let F(u) denote the number of
those positive integers for which
(n, [nα]) = 1. Watson proved in the preceding paper that
Postulates for Boolean Algebras
Authors: Sholander Marlow
Published: December 2010

The independence of postulates for well-known systems is a question of
general interest. A closely related question is whether or not, by altering
one or more of the postulates in an independent set, the set remains
independent. From this standpoint the best sets of postulates are those
which involve, first, the fewest postulates and, next, the least number of
variables. As a rule progress is made in this direction at the sacrifice of
simplicity of postulates. In this paper, in counting postulates, we ignore
properties such as closure under the operations and count only identities or
those stating that one equation implies another.
General Transformations of Bilateral Cognate Trigonometrical
Series of Ordinary Hypergeometric Type
Authors: Agarwal R. P.
Published: December 2010

Whipple [6] was the first to consider transformations connecting well-poised
hypergeometric series as particular cases of relations between cognate
trigonometrical series. He used contour integrals of the Barnes type to
deduce such transformations. Later Sears [3] gave a systematic theory of
general and well-poised transformations of trigonometrical series of any
order which included Whipple's results as particular cases.
Linear Combinations of Bernstein Polynomials
Authors: Butzer P. L.
Published: December 2010

If f(x) is denned on [0, 1], then its corresponding Bernstein polynomial

approaches f(x) uniformly on [0, 1], if f(x) is continuous on [0, 1]. If
f(x) is bounded on [0, 1], then at every point x where the second derivative
exists
(Voronowskaja [7], see also [5])
A Contribution to the Theory of Chromatic Polynomials
Authors: Tutte W. T.
Published: December 2010

Two polynomials θ(G, n) and ϕ(G, n) connected with the colourings of a graph G or of associated maps are discussed. A result believed to be new is proved for the lesser-known polynomial ϕ(G, n). Attention is called to some unsolved problems concerning ϕ(G, n) which are natural generalizations of the Four Colour Problem from planar graphs to general graphs. A polynomial χ(G, x, y) in two variables x and y, which can be regarded as generalizing both θ(G, n) and ϕ(G, n) is studied. For a connected graph χ(G, x, y) is defined in terms of the “spanning” trees of G (which include every vertex) and in terms of a fixed enumeration of the edges.
The Steinitz-Gross Theorem on Sums of Vectors
Authors: Behrend F. A.
Published: December 2010

α1α2, …, αpare n-dimensional vectors,

,;

they are arranged to form a closed polygon

.

Denote by R(α1,α2, …,αp) the radius of the smallest circumscribed hypersphere with centre at 0 ; by R(α1,α2, …,αp) the minimum of

(α1,α2, …,αp)

for all possible reorderings

ofα2, …,αp−1and by cnthe least possible constant such that

for all possible choices ofpandα1α2, … ,αp.
A Theory of Convergence
Authors: McShane E. J.
Published: December 2010

The literature already contains several theories of limits which have great generality (10; 12; 1; 5; 6; 13; 3; 14; 4, p. 34). Nevertheless, the intrinsic importance and frequent use of the concept may justify the publication of another variant, provided that it has advantages in ease of application without
sacrifice of generality. The theory of convergence studied in this note includes the other theories and their applications in a smooth way, without artifice.
On a Generalization of the First Curvature of a Curve in a Hypersurface of a Riemannian Space
Authors: Pan T. K.
Published: December 2010

The unit tangent vector at a point of a curve in a hypersurface of a Riemannian space has two derived vectors along the curve, one with respect to the Riemannian space in which the hypersurface is imbedded and one with respect to the hypersurface itself. When the former vector is
decomposed along the directions normal and tangent to the hypersurface, its tangential component, which is called the first curvature vector of the curve at the point in the hypersurface, is exactly the latter vector.
Brouwerian Geometry
Authors: Nordhaus E. A. , Lapidus Leo
Published: December 2010

From time to time attention has been directed to the study of spaces in which the “distances” are selected from algebraic structures other than the real or complex numbers. For example, Menger (9) and Taussky (11) have considered spaces in which the distances are taken from a group. More recently, Ellis (4), Blumenthal (2), and Elliott (3), have investigated spaces in which the distances are elements of a lattice.
On Homomorphic Images of Special Jordan Algebras
Authors: Cohn P. M.
Published: December 2010

A linear algebra is called a Jordan algebra if it satisfies the
identities

(1) ab = ba, (a2b) a = a2(ba).

It is well known that a linear algebra S over a field of characteristic different from two is a Jordan algebra if there is an isomorphism a → a of the vector-space underlying S into the vector-space of some associative algebra A such that

1,

where the dot denotes the multiplication in A. Such an algebra S is called a
special Jordan algebra.
On Bessel Polynomials
Authors: Agarwal R. P.
Published: December 2010

Recently a number of papers have been written on Bessel
polynomials which arise as the solutions of the classical wave equation in
spherical coordinates. Krall and Frink (5) studied in some detail the properties
of these polynomials yn(x, a, b) defined as

(1) .
On the Continuous Spectra of Singular Boundary Value Problems
Authors: Putnam C. R.
Published: December 2010

Suppose that p(t) > 0, that both p(t) and f(t) are continuous functions on the half-line 0 ≤ t < ∞, and that λ denotes a real parameter.
Only real-valued functions will be considered in this paper. Let the differential equation

,

be of the limit-point type (3, p. 238), so that (1) and a linear homogeneous boundary condition

, 0 ≤ α < π,

determine a boundary value problem on 0 ≤ t < ∞ for every fixed α.
On Integral Closure
Authors: Butts Hubert , Hall Marshall , Mann H. B.
Published: December 2010

Let J be an integral domain (i.e., a commutative ring without divisors of zero) with unit element, F its quotient field and J[x] the integral domain of polynomials with coefficients from J . The domain J is called integrally closed if every root of a monic polynomial over J which is in F also is in J.
On Integers n Relatively Prime To ƒ(n)
Authors: Lambek Joachim , Moser Leo
Published: December 2010

1. Introduction. If m and n are two integers chosen at random, the probability that they are relatively prime (2, p. 267) is 6π-2. This result may still hold when m and n are functionally related. Thus, Watson (3) recently proved that for α irrational, the positive integers n for which (n, [αn]) = 1, have density 6π-2. A different proof of a slightly more general result was given by Estermann (1).
A Theorem On Fixed Points Of Involutions In S3
Authors: Montgomery Deane , Samelson Hans
Published: December 2010

1. Introduction. Let T be an orientation-preserving homeomorphism of period two of the 3-sphere S3 onto itself; further let T be different from the identity and have at least one fixed point. It has been shown by Smith (8, p. 162) that the set F of all fixed points of T is a simple closed curve. However, very little is known about the position of F in S3.
Reducible Diophantine Equations and Their Parametric Representations
Authors: Rosenthall E.
Published: December 2010

1. Reducible diophantine equations. The present paper will provide a general method for obtaining the complete parametric representation for the rational integer solutions of the multiplicative diophantine equation

1.1

for some specified range of k, where the aijk,bijk are non-negative integers and the fki, hki are decomposable forms, that is to say they are integral irreducible homogeneous polynomials over the rational field R of degree k in k variables which can be written as the product of k linear forms.
Ovals In a Finite Projective Plane
Authors: Segre Beniamino
Published: December 2010

1. Let be a finite projective plane (8, §17), i.e. a projective space of dimension 2 over a Galois field γ. We suppose that γ has characteristic p ≠ 2, hence order q = pn, where p is an odd prime and h is a positive integer. It is well known that every straight line and every non-singular conic of then contains q + 1 points exactly.
On Fourier-Stieltjes Transforms
Authors: Calderón A. P. , Devinatz A.
Published: December 2010

Let be the class of bounded non-decreasing functions defined on the real line which are normalized by the conditions ϕ(− ∞) = 0 , ϕ(t + 0) = ϕ(t).

Let be the class of Fourier-Stieltjes transforms of elements of i.e. the elements of and are connected by the relation

where ϕ ∊ and Φ ∊ .It is well known, and easy to verify that this mapping from to is one to one (1, p. 67, Satz 18).
Asymptotic Solution Of Differential Equations In a Domain Containing a Regular Singular Point
Authors: Kazarinoff N. D. , McKelvey R.
Published: December 2010

1. Introduction. In this paper we study the asymptotic behavior in λ of the solutions about the origin in the z-plane of the differential equation

.

Both the variable z and the parameter λ are complex. The coefficient P(z, λ) is assumed to be analytic and single-valued in λ at infinity and in z throughout a bounded, closed, simply connected domain D containing z = 0.
Pencils Of Polarities In Projective Space
Authors: Schuster Seymour
Published: December 2010

1. Introduction. A polarity in complex projective space of two dimensions (S2) is completely determined by a self-polar triangle ABC, and a pair of corresponding elements: a point P and its polar line p. We denote the polarity by (ABC) (Pp). We follow Coxeter (2) in denning a pencil of polarities as the ∞1 polarities (ABC) (Pp) where A, B, C, P are fixed while p varies in a pencil of lines.
On Connections Of Cartan
Authors: Kobayashi Shôshichi
Published: December 2010

Introduction. Consider a differentiable manifold M and the tangent bundle T(M) over M, the structure group of which is usually the general linear group G'. Let P' be the principal fibre bundle associated with T(M). Consider the fibre F of T(M) as an affine space, then we have acting on F the affine transformation group G, which contains G' as the isotropic subgroup.
On a Class Of Almost Alternative Algebras
Authors: Kokoris L. A.
Published: December 2010

Introduction. In the study of almost alternative algebras (2) relative to quasiequivalence an important class called algebras of (γ, δ) type arises. An algebra of (γ, δ) type is a finite dimensional algebra over a field satisfying the identities

1,

and

2

where γ and δ are elements of satisfying γ2 − δ2 + δ = 1.
Radicals Of Polynomial Rings
Authors: Amitsur S. A.
Published: December 2010

Introduction. Let R be a ring and let R[x] be the ring of all polynomials
in a commutative indeterminate x over R. Let J(R) denote the Jacobson radical (5) of the ring R and let L(R) be the lower radical (4) of R. The main object of the present note is to determine the radicals J(R[x]) and L(R[x]). The Jacobson radical J(R[x]) is shown to be a polynomial ring N[x] over a nil ideal N of R and the lower radical L(R[x]) is the polynomial ring L(R)[x].
Conics and Orthogonal Projectivities In a Finite Plane
Authors: Edge W. L.
Published: December 2010

1. Introduction. The ternary orthogonal group of projectivities over a finite field leaves a non-singular conic ✗ invariant, but the geometry consequent thereupon does not appear to have been investigated. The group is isomorphic to a binary group of fractional substitutions over the same field
and this fact may, since these binary groups and their subgroups are so well known, have inhibited projects to embark on a detailed description of the geometry of the ternary group.
Implications Of Hadamard's Conditions For Elastic Stability With Respect To Uniqueness Theorems
Authors: Ericksen J. L. , Toupin R. A.
Published: December 2010

Introduction. The purpose of this paper is to discuss implications of Hadamard's condition for elastic stability (2, §269) with respect to uniqueness of solutions of boundary value problems in the theory of small deformations superimposed on large. We show that a slightly refined form of his condition implies a uniqueness theorem for displacement boundary value problems. We construct a counter-example showing that his condition does not imply uniqueness of solutions for one type of stress boundary value problem.
Hypersurfaces Of a Finsler Space
Authors: Rund Hanno
Published: December 2010

Introduction. Certain aspects of the theory of subspaces of a Finsler space had been treated by the present author in earlier papers (7). These developments were based on an approach essentially different from the classical theory of Cartan (2) and subsequent writers, whose use of the element of support enables one to introduce the so-called “euclidean connection,” which effects the vanishing of the covariant derivative of the metric tensor.
New Characterizations of Polyhedral Cones
Authors: Mirkil H.
Published: December 2010

A pyramid clearly has all its projections closed, even when the line
segments from vertex to base are extended to infinite half-lines. Not so a
circular cone. For if the cone is on its side and supported by the
(x, y) plane in such a way that its infinite half-line
of support coincides with the positive x axis, then its
horizontal projection on the (y, z) plane is the open upper
half-plane y > 0, together with the single point (0,
0).
Spaces Of Dimension Zero
Authors: Banaschewski Bernhard
Published: December 2010

1. Introduction. In a recent paper (1) it was remarked that the theory of
zero-dimensional spaces is exactly that part of general topology which can
be described in terms of equivalence relations. Here, it will be shown how
this idea can be used to obtain the following characterizations of certain
types of zero-dimensional spaces:

Any compact zero-dimensional space which has a denumerable basis for
its open sets and is dense in itself is homeomorphic to the space of
2-adic integers.
Matrices with Elements in a Boolean Ring
Authors: Butson A. T.
Published: December 2010

1. Introduction. Let be a Boolean
ring of at least two elements containing a unit 1. Form the set
of matrices A, B,
… of order n having entries aiJ, bij, … (i, j = 1, 2, …, n), which are members of . A matrix U of is called unimodular if there
exists a matrix V of such that
VU= I, the identity matrix. Two matrices
A and B are said to be
left-associates if there exists a unimodular matrix
U satisfying UA = B.
Simultaneous Pairs of Linear and Quadratic Equations In a Galois
Field
Authors: Cohen Eckford
Published: December 2010

Let F denote the Galois field GF(pr) with
pr elements, where p is an
odd prime and r is a positive integer.
Suppose further that m and n are arbitrary
elements of F and that αi, βi pt
(i = 1, …, s) are
nonzero elements of F. The purpose of
this paper is to evaluate the function Ns(m, n), defined, for an arbitrary positive integer
s, to be the number of simultaneous solutions in
F of the equations

1.1.
The Set of all Generalized Limits of Bounded
Sequences
Authors: Jerison Meyer
Published: December 2010

Let M be the normed linear space whose general element,
x, is a bounded sequence

of real numbers, and ‖x‖ = l.u.b. |ξn|. Let T denote the linear operation (of norm 1)
defined by Tx = (ξ2, ξ3, … , ξn+1,…). A generalized limit is a
linear functional ϕ on M which satisfies
the conditions

.
On Touchard Polynomials
Authors: Brafman Fred
Published: December 2010

Wyman and Moser (5) have recently given an explicit form for some
polynomials of Touchard (4). Equation (6) of this note will give a simpler
expression for these polynomials. Subsequently, generating functions (9) and
(12) will be obtained.
The Perfect and Extreme Senary Forms
Authors: Barnes E. S.
Published: December 2010

This paper is devoted partly to a summary of some results which will be published in detail elsewhere, partly to a description of a new extreme senary form, ϕ6, and an n-dimensional generalization of it.
Some Generalizations Of Burnsides Theorem
Authors: Wiegmann N. A.
Published: December 2010

1. Introduction. Burnside's Theorem in the theory of group representations
states that a necessary and sufficient condition that a semigroup of
matrices of degree n over the complex field be irreducible
is that the semigroup contain n2 linearly independent matrices. In the course of dealing with
sets of matrices with coefficients in a division ring, Brauer (1) obtained a
generalization of this theorem which concerned irreducible semigroups with
elements in a division ring.
On Binomial Coefficient Residues
Authors: Roberts J . B.
Published: December 2010

The number of binomial coefficients , which are
congruent to j , 0 ≤ j ≤
p − 1, modulo the prime number p is
denoted by θj(n). In this paper we give systems
of simultaneous linear difference equations with constant coefficients whose
solutions would yield the quantities θj(n)
explicitly.
Cyclically Ordered Sets
Authors: Aissen M. I.
Published: December 2010

The classical separation theorems regarding the relative position of the
zeros of two consecutive orthogonal polynomials, or of the positive zeros of
two consecutive Bessel Functions, are well known. Less well known is a
theorem of Bôcher and Porter (1, p. 237), which states that the positive
zeros of three consecutive Bessel Functions occur in cyclic order.
Representation of Banach Algebras with an
Involution
Authors: Schatz J. A.
Published: December 2010

In 1943 Gelfand and Neumark (3) characterized uniformly closed self-adjoint
algebras of bounded operators on a Hilbert space as Banach algebras with an
involution (a conjugate linear anti-isomorphism of period two) satisfying
several additional conditions. The main purpose of this paper is to point
out that if we consider algebras of bounded operators on complex Banach
spaces more general than Hilbert space, then we can represent a larger class
of algebras by essentially the same methods.
On the Inversion of the Gauss Transformation
Authors: Rooney P. G.
Published: December 2010

The inversion theory of the Gauss transformation has been the subject of
recent work by several authors. If the transformation is defined by

1.1,

then operational methods indicate that

,

under a suitable definition of the differential operator.
Completeness in Semi-Lattices
Authors: Ward L. E.
Published: December 2010

Let (X, ≤) be a partially ordered set, that is,
X is a set and ≤ is a reflexive, anti-symmetric,
transitive, binary relation on X.

We write

,

for each x ∈ X. If, moreover,

exists for each x and y in
X, then (X, ≤) is said to be a
semi-lattice.
The Inner Plethysm of S-Functions
Authors: Littlewood D. E.
Published: December 2010

In a previous paper (2), the inner product of two S-functions
{λ}.{μ} was defined for
(λ),(μ) partitions of the same integer n. Briefly,
the ordinary product {λ}.{μ} of two S-functions
corresponds to the analysis of the direct product of two corresponding
representations of the full linear group, while the inner product
{λ}.{μ} corresponds to the analysis of the direct
product of two representations of the corresponding symmetric group.
A Special Formula for the Lie Character
Authors: Davis Robert L.
Published: December 2010

In 1942 Thrall (3) introduced what he called the Lie
character to study the structure of the free Lie ring. These
characters are defined by associative representations of the full linear group
GL
q(C). Their importance is seen in
the fact that, for each n, the splitting of the
submodule of Lie forms of degree n into irreducible
invariant subspaces is completely specified by the Lie character
ζn.
The Representation Type of Algebras and Subalgebras
Authors: Jans J. P.
Published: December 2010

For A an associative algebra with identity over a field
K, [A :
K] < ∞, and d an
integer, we define g
Λ(d) to be the number of
inequivalent indecomposable Λ-modules of degree d over
K. Following (6), we define Λ to be of finite
representation type if . Λis said to be of
bounded representation type if there exists d
0 such that g
Λ(d) = 0 for d ⩾
d
0; Λ is of unbounded representation type if not of bounded
type.
On the Hasse-Minkowski Invariant of the Kronecker Product of Matrices
Authors: Vartak Manohar N.
Published: December 2010

Let R = (rij) be an m × n
matrix and let S =
(sik
) be dip × q matrix denned over a field F. The Kronecker product
R × S of
R and S is denned as
follows:

Definition 1.1. The Kronecker product R ×
S of the matrices R and
S is given by

where rij S; i = 1, 2, … , m; j = 1, 2,
… , n, is itself a p × q matrix (1, 69-70).
A Family of Difference Sets
Authors: Stanton R. G. , Sprott D. A.
Published: December 2010

A difference set (,D) is defined in (2) as a subset D of k elements
in a group of order υ with
the following properties :

(1) if x ∊ ,
x ≠ 1, there are exactly λ distinct ordered pairs
(d
1
d
2) of elements of D such that
x = 1
d
2;

(2) if x ∊ ,
x ≠ 1, there are exactly λ distinct ordered pairs
(3
d
4) of elements of D such that x =
3
d
4−1.
Spectral Theory for a Class of Nonnormal Operators II
Authors: Gonshor Harry
Published: December 2010

In a previous paper (2) we have developed a spectral theory and a unitary
equivalence theory for a certain class of non-normal operators. We dealt primarily
with operators which were called J
2 operators. At present we are interested in studying the
uniformly closed rings generated by such operators.

The notation will be the same as in (2).
Sur Les Représentations Unitaires Des Groupes De Lie Nilpotents. III
Authors: Dixmier J.
Published: December 2010

Dans un article antérieur (3), j'ai donné quelques théorèmes généraux sur les
représentations unitaires des groupes de Lie nilpotents simplement connexes. On
va, dans le présent article, construire explicitement les représentations
unitaires irréductibles et la formule de Plancherel des groupes de Lie nilpotents
simplement connexes de dimension ⩽ 5.
On the MT*- and λ-Conjugates of Spaces
Authors: Ellis H. W.
Published: December 2010

Marston Morse and William Transue (9, 10), motivated by their theory of
bilinear functions, introduced and studied vector function spaces called
MT-spaces for which each element of the dual is
represented by an integral with respect to a suitable
(C) measure. In this paper the definition of real
MT-spaces is generalized to give spaces, called
MT*-spaces, for which part but not all of the dual
is of integral type and this part is called the
MT*-conjugate of the
space.
An S-Configuration in Euclidean and Elliptic n-space
Authors: Mandan Ram Sahib
Published: December 2010

“The remarkable analogies which exist between the complete quadrilateral and
the desmic system of points suggest that it may be possible to extend the
properties considered above to spaces of higher dimensions”, remarks Prof. N. A.
Court at the end of his paper (2). Here is an attempt in that direction in
Euclidean as well as in elliptic 4-space, suggesting extensions in higher spaces.
The corresponding figure, called an S-configuration, is
discussed.
On the Number of Dissimilar Graphs Between a Given Graph-Subgraph Pair
Authors: Harary Frank
Published: December 2010

The purpose of this paper is to integrate the theorems on enumerating
subgraphs and supergraphs in (2) and (3) respectively by generalizing to a result
which includes both of these as special cases. In this process we again utilize
the powerful enumeration method of Pólya (4).
On the Lattice of Topologies
Authors: Hartmanis Juris
Published: December 2010

In many cases Lattice Theory has proven itself to be useful in the study of
the totality of mathematical systems of a given type. In this paper we shall
continue one of such studies by investigating further the lattice of all
topologies on a given set S. A considerable amount of research has been done in
this field (1; 2; 3; 5; 6). This research, besides satisfying the intrinsic
interest in the lattice theoretic properties of this lattice, has aided the study
of interconnections of different properties of point set topologies.
Linear Transformations on Algebras of Matrices
Authors: Marcus Marvin , Moyls B. N.
Published: December 2010

Let Mn denote the algebra of n-square matrices over the complex numbers; and let Un, Hn, and Rk denote respectively the unimodular group, the set of Hermitian matrices, and the set of matrices of rank k, in Mn. Let ev(A) be the set of n eigenvalues of A counting multiplicities. We consider the problem of determining the structure of any linear transformation (l.t.) T of Mn into Mn having one or more of the following properties:

(a)

T(Rk) ⊆ for k = 1, …, n.

(b)

T(Un) ⊆ Un

(c)

det T(A) = det A for all A ∈ Hn.

(d)

ev(T(A)) = ev(A) for all A ∈ Hn.

We remark that we are not in general assuming that T is a multiplicative homomorphism; more precisely, T is a mapping of Mn into itself, satisfying

T(aA + bB) = aT(A) + bT(B)

for all A, B in Mn and all complex numbers a, b.
Some Remarks on Prime Factors of Integers
Authors: Erdös P.
Published: December 2010

Let 1 < a1 < a2 < … be a sequence of integers and let N(x) denote the number of a's not exceeding x. If N(x)/x tends to a limit as x tends to infinity we say that the a's have a density. Often one calls it the asymptotic density to distinguish it from the Schnirelmann or arithmetical density. The statement that almost all integers have a certain property will mean that the integers which do not have this property have density 0. Throughout this paper p, q, r will denote primes.

I conjectured for a long time that, if e > 0 is any given number, then almost all integers n have two divisors d1 and d2 satisfying

1