bluuebunny/crossref_metadata_2025_split huggingface dataset:
On a Theorem of Hayman Concerning Quasi-Bounded Functions
Authors: Kennedy P. B. 
Published: December 2010

If f(z) is regular in |z| < 1, the expressionis called the characteristic of f(z). This is the notation of Nevanlinna (4) for the special case of regular functions; in this note it will not be necessary to discuss meromorphic functions. If m(r,f) is bounded for 0 < r < 1, then f(z) is called quasi-bounded in |z| < 1. In particular, every bounded function is quasibounded. The class Q of quasi-bounded functions is important because, for instance, a “Fatou theorem” holds for such functions (4, p. 134).
On the Class Number of Representations of an Order
Authors: Reiner Irving 
Published: December 2010

We shall use the following notation throughout:R = Dedekind ring (5).u = multiplicative group of units in R.h = class number of R.K = quotient field of R.p = prime ideal in R.Rp = ring of p-adic integers in K.We assume that h is finite, and that for each prime ideal p, the index (R:p) is finite.Let A be a finite-dimensional separable algebra over K, with an identity element e (4, p. 115). Let G be an R-ordev in A, that is, G is a subring of A satisfying(i)e ∈ G,(ii)G contains a i∈-basis of A,(iii)G is a finitely-generated i?-module.By a G-module we shall mean a left G-module which is a finitely-generated torsion-free i∈-module, on which e acts as identity operator.
Power Series Representing Certain Rational Functions
Authors: Melzak Z. A. 
Published: December 2010

Let denote the set of functions of a complex variable z, regular at z = 0, and let I denote the set of non-negative integers. For f ∈ putFor a given subset 0 of there arises the problem of characterizing the admissible gap sets If of functions f in 0. When 0 is the set R of rational functions a complete solution in given by the following theorem:(A) Let f ∈ R and let If be infinite. Then there exist integers L, L1, L2… , L3 such that 0 ≤ L1 < L2 … < Ls < L, and If = {n|n ∈ I, n ≡ Lj (mod L), j =1, … , s} U I\ where V is a Unite exceptional set.
On Matrix Commutators
Authors: Marcus Marvin , Khan Nisar A. 
Published: December 2010

Let A, B, and X be n-square matrices over an algebraically closed field F of characteristic 0. Let [A, B] = AB — BA and set (A, B) = [A, [A, B]]. Recently several proofs (1; 3; 5) of the following result have appeared: if det (AB) ≠ 0 and (A,B) = 0 then A-1B-1AB - I is nilpotent. In (2) McCoy determined the general form of any X satisfying1.1in the case that A has a single elementary divisor corresponding to each eigenvalue, that is, A is non-derogatory. In Theorem 1 we determine the structure of any matrix X satisfying (1.1) and also give a formula for the dimension of the linear space of all such X in terms of the degrees of the elementary divisors of A.
Perturbation of the Continuous Spectrum of Even Order Differential Operators
Authors: Butler John B. 
Published: December 2010

Let L0 be a differential operator of even order n = 2v on the half open interval 0 ≤ t < ∞ which is formally self adjoint and satisfies the conditions of Kodaira (5, p. 503). We consider a perturbed operator of the form L∈ = Lo + ∈q where q(t) is a real-valued bounded function and ∈ is a real parameter. The object of this paper is to set up conditions on the operator L0 and the function q(t) such that L∈ determines a self-adjoint operator H∈ and such that the spectral resolution operator E∈(Δ) corresponding to H∈ is analytic in a neighbourhood of ∈ = 0, where Δ is a closed bounded interval.Our conditions are a natural generalization of conditions considered by Moser for the case n = 2(6). Moser has given a number of examples showing that when his conditions do not hold E∈(Δ) need not be analytic.
Sums of Functions of Digits
Authors: Stewart B. M. 
Published: December 2010

We generalize in several directions a paper by Porges (2) who considered the integer F(A) obtained from the positive integer .1 by taking the sum of the squares of the digits of A. Porges showed that if A > 99, then F(A) < A, so that under iteration of F(A) all the positive integers are divided into a finite number of classes, called orbits in the terminology of Isaacs (1), each containing a finite cycle. For his F(A) Porges showed there are only two orbits: one with the 1-cycle: 1 → 1 ; and the other with the interesting 8-cycle: 4 → 16 → 37 → 58 → 89 → 145 → 42 → 20 → 4.Consider the set Z of non-negative integers and choose as a base of enumeration any desired integer B ≧ 2 (not necessarily B = 10). Then only the “digits” 0, 1, 2, … , B — 1 are needed, in suitable multiplicity, to represent any A of Z.
Arithmetical Inversion Formulas
Authors: Cohen Eckford 
Published: December 2010

Let n and r be integers, r positive, and define the coreγ(r) of r to be the product of the distinct prime factors of r (γ(1) = 1). Let f(n,r) be a complex-valued, arithmetical function of n and r. If for all n,f﹛n,r) = f((n,r), r) then f(n, r) is called an even function (mod r), and if f(n,r) = f(γ(n, r), r) for all n, γ(n, r) = γ((n, r)), then f(n, r) is said to be a primitive function (mod r). Clearly, both classes of functions are subclasses of the periodic functions (mod r), while the primitive functions form a subclass of the even functions (mod r).In a series of three papers (3; 5; 6) the author developed parallel, though interrelated, trigonometric and arithmetical theories of the even and primitive functions (mod r).
Modifications and Cobounding Manifolds
Authors: Wallace Andrew H. 
Published: December 2010

The object of this paper is to establish a simple connection between Thorn's theory of cobounding manifolds and the theory of modifications. The former theory is given in detail in (8) and sketched in (3), while the latter is worked out in (1). In particular in (1) it is shown that the only modifications which can transform one differentiable manifold into another are what I call below spherical modifications, which consist in taking out a sphere from the given manifold and replacing it by another. The main result is that manifolds cobound if and only if each is obtainable from the other by a finite sequence of spherical modifications.The technique consists in approximating the manifolds by pieces of algebraic varieties. Thus if M1 and M2 form the boundary of M, the last is taken to be part of an algebraic variety such that M1 and M2 are two members of a pencil of hyperplane sections.
On Continuous Regular Rings and Semisimple Self Injective Rings
Authors: Utumi Yuzo 
Published: December 2010

Brainerd and Lambek (2, Corollary 4) have proved recently that any complete Boolean ring is self-injective. It is easy to see that every complete Boolean ring is a continuous regular ring, that is, a regular ring of which the lattice of principal left ideals is continuous. This suggests that in a continuous regular ring it might be possible to prove the injectivity. However, a simple example (Example 3) shows that the conjecture is not true in general. Our main theorem is the following. Every continuous regular ring with no ideals of index 1 is (both left and right) self-injective (Theorem 3).It is known to Wolfson (13, Theorem 5.1) and Zelinsky (15) that the ring S of all linear transformations of a vector space of dimension ≥ 2 over a division ring is generated by idempotents and also by non-singular elements.
Reciprocal Convergence Classes for Fourier Series and Integrals
Authors: Guinand A. P. 
Published: December 2010

The classical result of Plancherel for Fourier cosine transforms of functions f(x) of the class L2(0, ∞) states that (see (7) for references)converges in mean square to a function g(x) which also belongs to L2(0, ∞), and furthermoreSome years ago in a series of papers (1; 2; 3) on summation formulae I showed that a similar symmetrical theory for narrower classes of functions and ordinary convergence of the integrals can also be developed.
Summability Methods on Matrix Spaces
Authors: Mitchell Josephine 
Published: December 2010

The matrix spaces under consideration are the four main types of irreducible bounded symmetric domains given by Cartan (5). Let z = (zjk) be a matrix of complex numbers, z' its transpose, z* its conjugate transpose and I = I(n) the identity matrix of order n. Then the first three types are defined by(1)where z is an n by m matrix (n ≤ m), a symmetric or a skew-symmetric matrix of order n (16). The fourth type is the set of complex spheres satisfying(2)where z is an n by 1 matrix. It is known that each of these domains possesses a distinguished boundary B which in the first three cases is given by(3)(In the case of skew symmetric matrices the distinguished boundary is given by (2) only if n is even.)
Arithmetic Linear Transformations and Abstract Prime Number Theorems
Authors: Amitsur S. A. 
Published: December 2010

Shapiro and Forman have presented in (4) an abstract formulation of prime number theorems which includes the various prime number theorems; for primes in arithmetic progressions, for prime ideals in ideal classes etc. The methods of proofs are “elementary” and follow closely Shapiro's proof for the primes in arithmetic progression (for reference see bibliography in (4)).The author has followed in (1) some ideas of Yamamoto (5) on arithmetic linear transformations to introduce a symbolic calculus in dealing with arithmetic functions. This calculus proved to be very useful in unifying many of the “elementary” proofs in the behaviour of arithmetic functions. In (6) Yamamoto has extended his theory to ideals in algebraic number fields, and with this extension the symbolic calculus of (1) can be extended to cover the abstract case of prime number theorem in countable free abelian groups as discussed in (4).
Graph Theory and Probability. II
Authors: Erdös P. 
Published: December 2010

Define f(k, l) as the least integer so t h a t every graph having f(k, l) vertices contains either a complete graph of order k or a set of l independent vertices (a complete graph of order k is a graph of k vertices every two of which are connected by an edge, a set of I vertices is called independent if no two are connected by an edge).Throughout this paper c1, c2, … will denote positive absolute constants. It is known (1, 2) that(1)and in a previous paper (3) I stated that I can prove that for every ∈ > 0 and l > l(∈), f (3, l) > l2-∈. In the present paper I am going to prove that(2)
Matrix Commutators
Authors: Smiley M. F. 
Published: December 2010

A classical theorem states that if a square matrix B over an algebraically closed field F commutes with all matrices X over F which commute with a matrix A over F, then B must be a polynomial in A with coefficients in F (2). Recently Marcus and Khan (1) generalized this theorem to double commutators. Our purpose is to complete the generalization to commutators of any order.Let F be an algebraically closed field and let Fn be the ring of all n by n matrices with elements in F. We define ΔYZ — = [Z, Y] = ZY — YZ for all Y, Z in Fn.
Orthomorphisms of Groups and Orthogonal Latin Squares. I
Authors: Johnson Diane M. , Dulmage A. L. , Mendelsohn N. S. 
Published: December 2010

Euler (6) in 1782 first studied orthogonal latin squares. He showed the existence of a pair of orthogonal latin squares for all odd n and conjectured their non-existence for n = 2(2k + 1). MacNeish (8) in 1921 gave a construction of n — 1 mutually orthogonal latin squares for n = p with p prime and of n(v) mutually orthogonal squares of order v wherewith p1 p2, … , Pr being distinct primes and
Partition Rings of Cyclic Groups of Odd Prime Power Order1
Authors: Appel K. I. 
Published: December 2010

A ring R over a commutative ring K, that has a basis of elements g1, g2, … , gn forming a group G under multiplication, is called a group ring of G over K. Since all group rings of a given G over a given K are isomorphic, we may speak of the group ring KG of G over X.Let π be any partition of G into non-empty sets GA, GB, … . Any subring P of KG that has a basis of elementsis a partition ring of G over K.If P is a partition ring of G over Z, the ring of integers, then the basis A, B, … for P clearly serves as a basis for a partition ring P’ = Q ⊗ P of G over Q, the field of rationals.
On the Hausdorff and Trigonometric Moment Problems
Authors: Rooney P. G. 
Published: December 2010

Let K be a subset of BV(0, 1)—the space of functions of bounded variation on the closed interval [0, 1]. By the Hausdorff moment problem for K we shall mean the determination of necessary and sufficient conditions that corresponding to a given sequence μ = {μn|n = 0, 1, 2, …} there should be a function α ∈ K so that(1)For various collections K this problem has been solved—see (3, Chapter III)By the trigonometric moment problem for K we shall mean the determination of necessary and sufficient conditions that corresponding to a sequence c = {cn|n = 0, ± 1, ± 2, …} there should be a function α ∈ K so that(2)For various collections K this problem has also been solved—see, for example (4, Chapter IV, § 4). It is noteworthy that these two problems have been solved for essentially the same collections K.
Normal Operators on the Banach Space Lp(-∞,∞). Part I
Authors: Krabbe Gregers L. 
Published: December 2010

Let be the Boolean algebra of all finite unions of subcells of the plane. Denote by εpthe algebra of all linear bounded transformations of Lp(— ∞, ∞) into itself. Suppose for a moment that p = 2, and let Rp be an involutive abelian subalgebra of εp if Rp is also a Banach space and if Tp ∈ Rp, then:(i) The family of all homomorphic mappings of into the algebra Rp contains a member EPT such that(1)
Fourth Order Boundary Value Problems and Comparison Theorems
Authors: Barrett John H. 
Published: December 2010

This paper is primarily concerned with the existence of solutions of the fourth-order self-adjoint differential equation(1)(where r(x) > 0, q(x) ≥ 0, p(x) ≥ 0 and all three coefficients are continuous on [a, ∞)) and one of the two-point boundary conditions:(2)or(3)the subscript notation for any solution y(x) denoting:(4)
The Clifford Algebra and the Group of Similitudes
Authors: Wonenburger Maria J. 
Published: December 2010

LetC(M, Q)be the Clifford algebra of an even dimensional vector spaceMrelative to a quadratic formQ. WhenQis non-degenerate, it is well known that there exists an isomorphism of the orthogonal groupO(Q)onto the group of those automorphisms ofC(M, Q)which leave invariant the spaceM⊂C(M, Q). These automorphisms are inner and the group of invertible elements ofC(M, Q)which define such inner automorphisms is called the Clifford group.If instead of the groupO(Q)we take the group of similitudesγ(Q)or even the group of semi-similitudesΓγ(Q), it is possible to associate in a natural way with any element of these groups an automorphism or semi-automorphism, respectively, of the subalgebra of even elementsC+(M, Q)⊂C(M, Q). Each one of the automorphisms ofC+(M, Q)so defined can be extended, as it is shown here (Theorem 2), to an inner automorphism ofC(M, Q), although the extension is not unique.
A Generalization of Finsler Geometry
Authors: Vanstone J. R. 
Published: December 2010

Modern differential geometry may be said to date from Riemann's famous lecture of 1854 (9), in which a distance function of the form F(xi, dxi) = (γij(x)dxidxj½ was proposed. The applications of the consequent geometry were many and varied. Examples are Synge's geometrization of mechanics (15), Riesz’ approach to linear elliptic partial differential equations (10), and the well-known general theory of relativity of Einstein.Meanwhile the results of Caratheodory (4) in the calculus of variations led Finsler in 1918 to introduce a generalization of the Riemannian metric function (6). The geometry which arose was more fully developed by Berwald (2) and Synge (14) about 1925 and later by Cartan (5), Busemann, and Rund. It was then possible to extend the applications of Riemannian geometry.
Some Formulas Involving Ramanujan Sums
Authors: Nicol C. A. 
Published: December 2010

The purpose of this note is to establish an identity involving the cyclotomic polynomial and a function of the Ramanujan sums. Some consequences are then derived from this identity.For the reader desiring a background in cyclotomy, (2) is mentioned. Also, (4) is intimately connected with the following discussion and should be consulted.The cyclotomic polynomial Fn(x) is defined as the monic polynomial whose roots are the primitive nth roots of unity. It is well known that2.1For the proof of Corollary 3.2 it is mentioned that Fn(0) = 1 if n > 1 and that Fn(x) > 0 if |x| < 1 and 1 < n.The Ramanujan sums are defined by2.2where the sum is taken over all positive integers r less than or equal to n and relatively prime to n. It is also well known that2.3where the sum is taken over all positive divisors d common to n and k.
Angular and Tangential Limits of Blaschke Products and their Successive Derivatives
Authors: Cargo G. T. 
Published: December 2010

In this paper, we shall be concerned with bounded, holomorphic functions of the formwhere(1)(2)and(3)B(z{an}) is called a Blaschke product, and any sequence {an} which satisfies (2) and (3) is called a Blaschke sequence. For a general discussion of the properties of Blaschke products, see (18, pp. 271-285) or (14, pp. 49-52).According to a theorem due to Riesz (15), a Blaschke product has radial limits of modulus one almost everywhere on C = {z: |z| = 1}. Moreover, it is common knowledge that, if a Blaschke product has a radial limit at a point, then it also has an angular limit at the point (see 14, p. 19 and 6, p. 457).
Perturbation of the Continuous Spectrum of Systems of Ordinary Differential Operators
Authors: Butler John B. 
Published: December 2010

Letbe an ordinary differential operator of order h whose coefficients are (η, η) matrices defined on the interval 0 ≤ x < ∞, hη = n = 2v. Let the operator L0 be formally self adjoint and let v boundary conditions be given at x = 0 such that the eigenvalue problem(1.1)has no non-trivial square integrable solution. This paper deals with the perturbed operator L∈ = L0 + ∈q where ∈ is a real parameter and q(x) is a bounded positive (η, η) matrix operator with piecewise continuous elements 0 ≤ x < ∞. Sufficient conditions involving L0, q are given such that L∈ determines a selfadjoint operator H∈ and such that the spectral measure E∈(Δ′) corresponding to H∈ is an analytic function of ∈, where Δ′ is a subset of a fixed bounded interval Δ = [α, β]. The results include and improve results obtained for scalar differential operators in an earlier paper (3).
Irreducibility of Bernoulli Polynomials of Higher Order
Authors: McCarthy P. J. 
Published: December 2010

The Bernoulli polynomials of order k, where k is a positive integer, are defined byBm(k)(x) is a polynomial of degree m with rational coefficients, and the constant term of Bm(k)(x) is the mth Bernoulli number of order k, Bm(k). In a previous paper (3) we obtained some conditions, in terms of k and m, which imply that Bm(k)(x) is irreducible (all references to irreducibility will be with respect to the field of rational numbers). In particular, we obtained the following two results.
On Non-Orientable Closed Surfaces in Euclidean Spaces
Authors: Yang C. T. 
Published: December 2010

Let us begin with a simple result.Proposition. Let X be a non-orientable closed surface differentiably imbedded into the euclidean 4-space R4. Then there is a line in R4 which intersects X at more than two points.Proof. Let c be any point of R4 and let r be the smallest number such that X is contained in the closed 4-spheroid W of centre c and radius r. Clearly the boundary 3-sphere S of W intersects X.
A Strong Converse of the Wiener-Levy Theorem
Authors: Rudin Walter 
Published: December 2010

Let A1 denote the class of all complex functions on the unit circle which are sums of absolutely convergent trigonometric series; that is to say, the class of all f of the formSimilarly, if 1 < p < ∞, let Ap be the class of all complex functions f on the circle whose Fourier coefficientssatisfy the condition
The Gibbs Phenomenon and Lebesgue Constants for Regular Sonnenschein Matrices
Authors: Sledd W. T. 
Published: December 2010

Ifψ(x)is a real-valued function which has a jump discontinuity atx= ε and otherwise satisfies the Dirichlet conditions in a neighbourhood ofx= ε then{sn(x)}the sequence of partial sums of the Fourier series forψ(x)cannot converge uniformly atx =ε. Moreover, it can be shown that given τ in [ — π, π] then there is a sequence {tn} such thattn→ ε andThis behaviour of{sn(x)}is called the Gibbs phenomenon. If {σn(x)} is the transform of{sn(x)}by a summability methodT, and if {σn(x)} also has the property described then we say thatTpreserves the Gibbs phenomenon.
Some Regular [F, dn] Matrices with Complex Elements
Authors: Miracle Chester L. 
Published: December 2010

Let A = (ank) and {sn} (n, k = 0, 1, 2, . . .) be a matrix and a sequence of complex numbers, respectively. Let the members of the sequence {σn} be defined bythen we say {σn} is the A -transform of {sn}. The matrix A = (ank) is called regular if
On Measurability for Vector-Valued Functions
Authors: Snow D. O. 
Published: December 2010

The problem of developing an abstract integration theory has been approached from many angles (6). The most general of several definitions based on the norm topology is that of Birkhoff (1), which includes the well-known and widely used Bochner integral (3).The original Birkhoff formulation was based on the notion of unconditional convergence of an infinite series of elements in a Banach space and the closed convex extensions of certain approximating sums.
Dual Integral Equations
Authors: Love E. R. 
Published: December 2010

Erdélyi and Sneddon (4) have reduced the dual integral equations (4, (1.4))where Ψ is unknown, to a single Fredholm integral equation (4, (4.4)), from the solution of which Ψ is explicitly obtainable. Their work extended and clarified an investigation by Cooke (1), placing it in a context of standard integral transforms. Cooke's reduction was obtained after consideration of the Fredholm integral equation obtained by Love (8) in discussing Nicholson's problem of the electrostatic field of two equal circular coaxial conducting disks (9).
Bernstein Power Series
Authors: Cheney E. W. , Sharma A. 
Published: December 2010

In Bernstein's proof of the Weierstrass Approximation Theorem, the polynomialsare constructed in correspondence with a function f ∊ C [0, 1] and are shown to converge uniformly to f. These Bernstein polynomials have been the starting point of many investigations, and a number of generalizations of them have appeared. It is our purpose here to consider several generalizations in the form of infinite series and to establish some of their properties.
A Generalization of the Hardy Spaces
Authors: Rooney P. G. 
Published: December 2010

The Hardy spaces for right half-planes, , σ real, 1 ≤ p ≤ ∞, are defined to consist of all those functions f(s), holomorphic for Re s > σ, for which μp(f, x) exists and is bounded for x > σ, whereThese spaces have been studied extensively (see, for example, 3, Chapter 8, and 2, §19.1).
On Subplanes of Free Planes
Authors: Sandler Reuben 
Published: December 2010

In 1945, L. I. Kopejkina (4), at the suggestion of A. G. Kurosh, began a programme of studying the properties of free projective planes and the analogies between free planes and free groups. In this paper, this study will be extended by proving a tool theorem and several of its consequences. The theorem deals with the existence of "minimal free generators" for subplanes of free planes.A set of points and lines and an incidence relation are said to form a projective plane if the following three axioms are satisfied.