bluuebunny/crossref_metadata_2025_split huggingface dataset:
Some Theorems About pr(n)
Authors: Newman Morris 
Published: December 2010

If n is a non-negative integer, define
pr(n) as the coefficient of xn in;otherwise define pr(n) as 0. In a recent paper (1) the author has
proved that if r has any of the values 2, 4, 6, 8, 10, 14,
26 and p is a prime > 3 such that
r(p + 1) ≡ 0 (mod 24), then1, ,where n is an arbitrary integer.
A Generalization of the Cauchy Principal Value
Authors: Fox Charles 
Published: December 2010

If a < u < b and n
> 0 then(1)is a so-called improper integral owing to the infinity in the integrand at
x = u. When n = 0 we
have associated with (1) the well-known Cauchy principal value, namely(2).Hadamard (1, p. 117 et seq.) derives from an improper integral an expression
which he calls its finite part and which, as he shows, possesses many
important properties.
Asymptotic Expansions II
Authors: Moser Leo , Wyman Max 
Published: December 2010

In a previous paper (1) the authors considered the problem of finding an
asymptotic formula for numbers or functions Bn,m whose generating
function is of the form(1.1),where Pm(x) is a polynomial of degree
m in x given by(1.2), am≠0.The above-mentioned paper contained the restriction that ak ≥
0.
On a Diophantine Problem
Authors: Roberts J. B. 
Published: December 2010

If a1, a2, … , ak are relatively prime
positive integers then the equationalways has solutions in non-negative xi for n
sufficiently large. Sylvester called the number of non-negative solutions of
(1) the denumerant of the equation. We shall denote the
denumerant by n(a1, … , ak).
Factorization Rings
Authors: Maranda J.-M. 
Published: December 2010

Let o be an integral domain with as field of
quotients. W. Krull has shown (3 ; 4) that the following three conditions on
o are equivalent:There is a set of rank 1, discrete valuations of such that for each non-null element for all except a
finite number, and such that for all if and only
if for all
i ∊ I.
Schlicht Dirichlet Series
Authors: Robertson M. S. 
Published: December 2010

For power series
(1.1)

for which
(1.2)

, it has been known for four decades (1) that ƒ(z) is regular
and univalent or schlicht in |z| &lt; 1. This theorem, due to J. W. Alexander,
has more recently been studied by Remak (5) who has shown that
w =
ƒ(z), under the
hypothesis (1.2), maps |z| &lt; 1 onto a star-like
region, and if (1.2) is not satisfied=(z) need not be
univalent in |z| &lt; 1 for a proper choice of the amplitudes of the
coefficients an.
On Some Relations Between Partial and Ordinary Differential Equations
Authors: Kreyszig Erwin 
Published: December 2010

The theory of solutions of partial differential equations
(1.1)

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

More than sixty years ago, Sylvester (13) proposed the following problem: Let
n given points have the property that the straight line joining any two of them
passes through a third point of the set. Must the n points all lie on one
line?
An alleged solution (not by Sylvester) advanced at the time proved to be
fallacious and the problem remained unsolved until about 1933 when it was revived
by Erdös (7) and others.
On Some Theorens of Doetsch
Authors: Rooney P. G. 
Published: December 2010

The spaces

P
(ω), 1 ⩽ p ⩽ ∞, ω
real are defined to consist of those analytic functions ƒ(s), regular for Re s
&gt; ω and for which μ
P
(ƒ;x) is bounded for x &gt; ω where
(1.1)

, 1 ⩽ p ⩽ ∞and
(1.2)
On the Inversion of the Gauss Transformation, II
Authors: Rooney P. G. 
Published: December 2010

In an earlier paper (5) we studied the inversion theory of the Gauss
transformation defined by
(1.1)

. Operational methods indicated that

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

If a group contains two subgroups A and B such that every element of the group is either in A or can be represented uniquely in the form aba', a, a’ in A, b ≠ 1 in B, we shall call the group an independent ABA-group. In this paper we shall investigate the structure of independent ABA -groups of finite order.A simple example of such a group is the group G of one-dimensional affine transformations over a finite field K. In fact, if we denote by a the transformation x’ = ωx, where ω is a primitive element of K, and by b the transformation x’ = —x + 1, it is easy to see that G is an independent ABA -group with respect to the cyclic subgroups A, B generated by a and b respectively.Since G admits a faithful representation on m letters (m = number of elements in K) as a transitive permutation group in which no permutation other than the identity leaves two letters fixed, and in which there is at least one permutation leaving exactly one letter fixed, G is an example of a Frobenius group.
Some Division Algebras
Authors: Zemmer Joseph L. 
Published: December 2010

Let K* be an associative algebra over a field F with identity u, and let u, e1, e2, … , be a basis for K*. Denote by K the linear space, over F, spanned by the ei,i = 1, 2, … . Then for x, y in K, xy = αu + a, where a ∈ K. Define h(x, y) = α and x.y = a. With respect to the operation thus defined, K becomes an algebra over F satisfying1Further, the bilinear form h(x, y) is associative on K. Any algebra, over a field F, which possesses an associative bilinear form h(x, y) and satisfies (1) will be called a algebra. It is not difficult to show that any algebra K can be obtained from a unique associative algebra K* with identity by the process described above. The algebra K* will be called the associated associative algebra of K.
Characters of Cartesian Products of Algebras
Authors: Warner Seth 
Published: December 2010

Let R be a commutative ring with identity 1. A character of an R-algebra E is a homomorphism from E onto R, regarded as an algebra over itself. If (Eα) is a family of R-algebras indexed by a set A and ifthen for every β ∈ A and every character vβ of Eβ, vβ o prβ is a character of E where prβ is the projection homomorphism from E onto Eβ. Further if A is finite and if the only idempotents of R are 0 and 1 (equivalently, if R is not the direct sum of two proper ideals), it is easy to see that every character of E is of this form.
Concerning Binary Relations on Connected Ordered Spaces
Authors: Krule I. S. 
Published: December 2010

In a recent paper Mostert and Shields (4) showed that if a space homeomorphic to the non-negative real numbers is a certain type of topological semigroup, then the semigroup must be that of the non-negative real numbers with the usual multiplication. Somewhat earlier Faucett (2) showed that if a compact connected ordered space is a suitably restricted topological semigroup, then it must be both topologically and algebraically the same as the unit interval of real numbers with its usual multiplication.In studying certain binary relations on topological spaces there have become known (see, in particular, Wallace (5) and the author (3)) a number of properties analogous to those possessed by topological semigroups. Because of these analogous properties between relations and semigroups the author was motivated by the general nature of the Faucett and Mostert-Shields results (that is, that the multiplication assumed turned out to be the same as the usual multiplication) to feel that certain relations on a connected ordered space should turn out to be the same as the orders whose order topologies are the topology on the space.
Generating Functions for Bessel Functions
Authors: Weisner Louis 
Published: December 2010

On replacing the parameter n in Bessel's differential equation1.1by the operator y(∂/∂y), the partial differential equation Lu = 0 is constructed, where1.2This operator annuls u(x, y) = v(x)yn if, and only if, v(x) satisfies (1.1) and hence is a cylindrical function of order n. Thus every generating function of a set of cylindrical functions is a solution of Lu = 0.It is shown in § 2 that the partial differential equation Lu = 0 is invariant under a three-parameter Lie group. This group is then applied to the systematic determination of generating functions for Bessel functions, following the methods employed in two previous papers (4; 5).
On Connections Between Growth and Distribution of Zeros of Integral Functions
Authors: Rahman Q. I. 
Published: December 2010

The following theorem was proved by Paley and Wiener (4, p. 70; 1, p. 136).Theorem 1. If f(z) is a canonical product of order 1 with real zeros, and f(0) = 1, the conditionsandare equivalent. n(r) denotes the number of zeros of absolute value not exceeding r.Instead of assuming the zeros to be all real Pfluger assumed that the zeros are close to the real axis and proved the following theorem (5 or 1, p. 143).
On Finite Nilpotent Groups
Authors: Bachman G. 
Published: December 2010

It is well known that if (n, ϕ(n)) = 1, where ϕ(n) denotes the Euler ϕ function, then the only group of order n is the cyclic group. This is a special case of a more general result due to Dickson (2, p. 201); namely, ifwhere the pi are distinct primes and each αi > 0, the necessary and sufficient conditions that the only groups of order n are abelian are (1) each αi ≤ 2 and (2) nois divisible by any p1 … , ps.We wish to establish a theorem which includes these two results. We let G(n) equal the number of groups of order n whereand we seek necessary and sufficient conditions on n so thatClearly, this problem is equivalent to finding necessary and sufficient conditions on n so that all existing groups of order n be nilpotent.
Finite Groups Which Admit An Automorphism With Few Orbits
Authors: Gorenstein Daniel 
Published: December 2010

In the course of investigating the structure of finite groups which have a representation in the form ABA, for suitable subgroups A and B, we have been forced to study groups G which admit an automorphism ϕ such that every element of G lies in at least one of the orbits under ϕ of the elements g, gϕr(g), gϕrϕ(g)ϕ2r(g), gϕr(g)ϕr2r(g)ϕ3r(g), etc., where g is a fixed element of G and r is a fixed integer.In a previous paper on ABA-groups written jointly with I. N. Herstein (4), we have treated the special case r = 0 (in which case every element of G can be expressed in the form ϕi(gj)), and have shown that if the orders of ϕ and g are relatively prime, then G is either Abelian or the direct product of an Abelian group of odd order and the quaternion group of order 8.
Sur La Cohomologie Non Abelienne I (Dimension Deux)
Authors: Dedecker Paul 
Published: December 2010

On sait que la définition de la cohomologie d'un espace X à coefficients dans un faisceau de groupes non abéliens est liée à la classification de certains types d'espaces fibres principaux de base X (1; 8; 10). Cette cohomologie se définit assez facilement en dimensions zéro et un, mais pour certains problèmes, il serait utile de pouvoir la définir en dimension deux également.J'ai abordé la question dans (3; 6) mais un certain nombre de difficultés subsistaient qu'il semble utile de chercher à éliminer, ce qui est l'objet de la présente note.Remarquons d'abord que la définition de H°(X, ), groupe des sections de , est triviale, aucune difficulté ne résultant du caractère non abélien de . En ce qui concerne H1(X, ) une difficulté naît du fait que cet ensemble n'est plus un groupe, même non abélien; il possède toutefois un élément neutre privilégié.
Sur Les Representations Unitaires des Groupes de Lie Nilpotents. VI
Authors: Dixmier Jacques 
Published: December 2010

Soient Γ un groupe localement compact, l'ensemble des classes de représentations unitaires irréductibles de Γ. Dans (4), Godement a défini une topologie sur que nous appellerons "topologie canonique." Cette topologie est facile à étudier lorsque Γ est abélien ou compact, mais fort mal connue en général. Dans un article récent (3), Fell a repris l'étude de la topologie canonique de , et réussi à la calculer lorsque Γ est le groupe spécial linéaire complexe à n variables. L'espace est alors "presque" séparé, et, par suite, "presque" localement compact.Il semble nécessaire d'étudier complètement quelques cas particuliers avant d'entreprendre une théorie générale de .
On Relatively Invariant Measures
Authors: Mahowald Mark 
Published: December 2010

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

This paper deals with the discrete groups of rigid motions of the hyperbolic plane. It is known (12) that the finitely generated, orientation-preserving groups have the following presentations:Generators: .Defining relations: where km = ambmam-1bm-1. We shall denote this group by F(p; n1, … , nd; r).In particular, the finitely generated free groups are contained among these. Indeed, one purpose of this paper is to indicate some geometrical methods for investigating free groups.
The Gibbs Phenomenon for Taylor Means and for [F, Dn] Means
Authors: Miracle Chester L. 
Published: December 2010

The Gibbs phenomenon may be described, quite generally, as follows. Let a sequence {fn(x)} (n= 0, 1, 2, … ,) converge to a functionf(x)for x in the interval x0<x <x0+ h. We say that {fn(x)} displays the Gibbs phenomenon in a right-hand neighbourhood of the point X0, ifA similar definition holds for a left-hand neighbourhood. If {fn(x)} displays the Gibbs phenomenon at both sides of x0, we say simply that {fn(x)} displays Gibbs phenomenon at the point X0.
Properties of the Coefficients of Orthonormal Sequences
Authors: Bullen P. S. 
Published: December 2010

In this paper we consider complete orthonormal sequences defined on the interval [0, 1] and satisfying an inequality of the type(1)for allnand some sequence {Fn}. Such sequences were first considered by Zygmund and Marcinkiewicz (8). They extended the well-known results of Hausdorff-Young and Paley, originally proved for the casev= ∞,Fn=Mfor alln(12). We will consider cases of equality in t he Hausdorff-Young theorems and certain limiting cases of the Paley theorems. Application of these results and the results in (8) will be made to functions harmonic in the unit α-sphere.
Polar Means of Convex Bodies and a Dual to the Brunn-Minkowski Theorem
Authors: Firey William J. 
Published: December 2010

This paper deals with processes of combining convex bodies in Euclidean n-space which are, in a sense, dual to the process of Minkowski addition and some of its generalizations.All the convex bodies considered will have a common interior point Q. Variables x and y denote vectors drawn from Q; we shall speak of their terminal points as the points x and y. Unit vectors will be denoted by u; ||x|| signifies the length of x. Convex bodies will be symbolized by K with distinguishing marks. ∂K means the boundary of K. λK will mean the image of K under a homothetic transformation in the ratio λ : 1. The centre of the homothety will always be Q.
On Nilpotent Products of Cyclic Groups. II
Authors: Struik Ruth Rebekka 
Published: December 2010

In a previous paper (18), G = F/Fn was studied for F a free product of a finite number of cyclic groups, and Fn the normal subgroup generated by commutators of weight n. In that paper the following cases were completely treated:(a) F a free product of cyclic groups of order pαi, p a prime, αi positive integers, and n = 4, 5, … , p + 1.(b) F a free product of cyclic groups of order 2αi, and n = 4.In this paper, the following case is completely treated:(c) F a free product of cyclic groups of order pαi p a prime, αi positive integers, and n = p + 2.(Note that n = 2 is well known, and n — 3 was studied by Golovin (2).) By ‘'completely treated” is meant: a unique representation of elements of the group is given, and the order of the group is indicated. In the case of n = 4, a multiplication table was given.
An Inversion Formula for the Weierstrass Transform
Authors: Bilodeau G. G. 
Published: December 2010

The Weierstrass transform f(x) of a function ϕ(y) is defined by1.1wherewhenever this integral exists (7, p. 174). It is also known as the Gauss transform (11; 12). Its basic properties have been developed and studied in (7) and in particular it has been shown that the symbolic operatorwill invert this transform under suitable assumptions and with certain definitions of this operator. We propose to study the definitionfor f(x) in C∞. This formula seems to have been first examined by Pollard (9) and later by Rooney (12). In so far as convergence of (1.2) is concerned, we will considerably improve the results (12).
Transcendental Elements in Continuous Rings
Authors: Halperin Israel 
Published: December 2010

In (2), John von Neumann introduced the concept of a continuous ring as a generalization to the infinite limiting case of the total matric algebras over a division ring. Von Neumann sketched a theory of arithmetic for such continuous rings and asserted :(⋆) every continuous ring contains purely transcendental elements c.This means: for every polynomial p(t) = tm + Z1tm-1 + . . . + zm {m ≥ 1) which has all coefficients zi in the centre of , the element p(c) has a reciprocal in , that is, (p(c))-l exists such that p(c).(p(c))-l = (p(c))-l.p(c) = 1.
Embedding Circle-Like Continua in the Plane
Authors: Bing R. H. 
Published: December 2010

A finite sequence of open sets L1 L2, … , Ln is called a linear chain if each Li intersects only the L's adjacent to it in the sequence. The finite sequence is a circular chain if we also insist that the first and last links intersect each other. The 1-skeleton of the covering is an arc for a linear chain and a simple closed curve for a linear chain.A compact metric continuum X is called snake-like if for each ∈ > 0, X can be covered by a linear chain of mesh less than ∈. Likewise, X is called circle-like if for each ∈ > 0, X can be irreducibly covered with a circular chain of mesh less than ∈. This definition is more restrictive than that given in (3, p. 210) for there a pseudo-arc is not circle-like but here it is. The present usage is in keeping with definitions of Burgess.
Minimum and Conjugate Points in Symmetric Spaces
Authors: Crittenden Richard 
Published: December 2010

The purpose of this paper is to discuss conjugate points in symmetric spaces. Although the results are neither surprising nor altogether unknown, the author does not know of their explicit occurrence in the literature.Briefly, conjugate points in the tangent bundle to the tangent space at a point of a symmetric space are characterized in terms of the algebraic structure of the symmetric space. It is then shown that in the simply connected case the first conjugate locus coincides with the minimum (cut) locus. The interest in this last fact lies in its identification of a more or less locally and analytically defined set with one which includes all the topological interest of the space.
Some Results for the Generalized Lototsky Transform
Authors: Cowling V. F. , Miracle C. L. 
Published: December 2010

Let A = (ank) and x = {sn} (n,k = 0,1,2, … ) be a matrix and a sequence of complex numbers, respectively. We write formally(1.1)and say that the sequence x is summable A to the sum t or that the A matrix sums the sequence x to the value t if the series in (1.1) converges andexists and equals t. We say that the matrix A is regular provided it sums every convergent sequence to its limit.
Arithmetic Properties Of Freely α-Generated Lattices
Authors: Jónsson Bjarni 
Published: December 2010

In § 1 we give a characterization of a lattice L that is freely α-generated by a given partially ordered set P. In § 2 we obtain a representation of an element of such a lattice as a sum (product) of additively (multiplicatively) irreducible elements which, although not unique, has some of the desirable features of the canonical representation, in Whitman (2), of an element of a free lattice. The usefulness of this representation is illustrated in § 3, where some further arithmetic properties of these lattices are derived.We use + and . for the binary operations of lattice addition and multiplication, and Σ and II for the corresponding operations on arbitrary sets and sequences of lattice elements. In other respects the terminology will be the same as in Crawley and Dean (1).
Extremal Problems for Functions Starlike in the Exterior of the Unit Circle
Authors: Royster W. C. 
Published: December 2010

Let Σ represent the class of analytic functions(1)which are regular, except for a simple pole at infinity, and univalent in |z| > 1 and map |z| > 1 onto a domain whose complement is starlike with respect to the origin. Further let Σ- 1 be the class of inverse functions of Σ which at w = ∞ have the expansion(2).In this paper we develop variational formulas for functions of the classes Σ and Σ- 1 and obtain certain properties of functions that extremalize some rather general functionals pertaining to these classes. In particular, we obtain precise upper bounds for |b2| and |b3|. Precise upper bounds for |b1|, |b2| and |b3| are given by Springer (8) for the general univalent case, provided b0 =0.
On the Character Rings of Finite Groups
Authors: Banaschewski B. 
Published: December 2010

The characters of the representations of a finite groupGover a fieldKof characteristic zero generate a ring oK(G) of functions onG, theK-character ring of G, which is readily seen to be Zϕ1+ . . . + Zϕn, where Z is the ring of rational integers andϕ1, . . . ,ϕnare the characters of the different irreducible representations ofGoverK. The theorem that every irreducible representation ofGover an algebraically closed field Ω of characteristic zero is equivalent to a representation ofGover the subfield of Ω which is generated by the g0th roots of unity (g0the exponent ofG) was proved by Brauer (4) via the theorems that(1) OΩ(G) is additively generated by the induced characters of representations of elementary subgroups of G, and(2) the irreducible representations over 12 of any elementary group are induced by one-dimensional subgroup representations(3).
On Some Tactical Configurations
Authors: Hanani Haim 
Published: December 2010

Given a set E of v elements, and given positive integers k, l (l ≤ k ≤ v), and λ, we understand by a tactical configurationC[k, l, λ, v] (briefly, configuration) a system of subsets of E, having k elements each, such that every subset of E having l elements is contained in exactly λ sets of the system.A necessary condition for the existence of a configuration C[k, l, λ, v] is known (6) to be
Interval Functions and Non-Decreasing Functions
Authors: Appling William D. L. 
Published: December 2010

In a previous paper the author (1) has shown the following theorem.Theorem A. If each of H and K is a real-valued bounded function of subintervals of the number interval [a, b] and m is a real-valued non-decreasing function on [a} b] such that each of the integralsexists, then the integralexists.