bluuebunny/crossref_metadata_2025_split huggingface dataset:
On Asymmetrical Derivates of Non-Differentiable Functions
Authors: Garg K. M. 
Published: December 2010

Let ƒ(x) be a non-differentiable function, i.e. a realvalued continuous function denned on a linear interval which has nowhere a finite or infinite derivative. We shall say that ƒ(x) has symmetrical derivates at a point x if the four Dini derivates of ƒ(x) at x satisfy the relationsand otherwise we shall say that ƒ(x) has asymmetrical dérivâtes at x.
Free Pq-Algebras
Authors: Grillet Pierre Antoine 
Published: December 2010

By a PQ (product-quotients) algebra, we mean a non-empty set together with three single-valued and not necessarily associative operations ., / , \ that we shall treat as product, right quotient, and left quotient although we require no relation between them. The theory of binary systems provides the following examples:A. is the set of all subsets of a groupoid G (which may be a semigroup) with the operations defined by:
A Class of Polynomials in Self-Adjoint Operators in Spaces with an Indefinite Metric
Authors: Lo C.-Y. 
Published: December 2010

Let H be a Hilbert space with the usual product [x, y] and with an indefinite inner product (x, y) which, for some orthogonal decompositionin H, is defined bywhereand dim H1 = κ, a fixed positive integer.
Commutative Non-Associative Algebras and Identities of Degree Four
Authors: Marshall Osborn J. 
Published: December 2010

The main result of this paper is the following.Theorem 1. Let A be a simple, commutative, finite-dimensional algebra containing an idempotent over a field of characteristic 0, and let the algebra A' obtained from A by adjoining a unity element satisfy an identity of degree ≦ 4 not implied by commutativity. Then either A is a Jordan algebra or A is two-dimensional over an appropriate field E.
The Coarseness of the Complete Graph
Authors: Guy Richard K. , Beineke Lowell W. 
Published: December 2010

The coarseness, c(G), of a graph G is the maximum number of edge-disjoint, non-planar subgraphs of G. We consider only the complete graph, Kp, on p vertices here. For p = 3r, Erdös conjectured that the coarseness was , but it has been shown (1) that1where square brackets denote integer part.
Convergence of Continued Fractions
Authors: Jones William B. , Thron W. J. 
Published: December 2010

Let {sn(z)} be a given sequence of linear fractional transformations (or simply l.f.t.'s) of the form1.1and let1.2The sequence of l.f.t.'s {Sn(z)} is called a continued fraction generating sequence (or simply a c.f.g. sequence).
Non-Stationary Processes and Spectrum
Authors: Nagabhushanam K. , Bhagavan C. S. K. 
Published: December 2010

In 1964, L. J. Herbst (3) introduced the generalized spectral density Function1for a non-stationary process {X(t)} denned by1where {η(t)} is a real Gaussian stationary process of discrete parameter and independent variates, the (a;)'s and (σj)'s being constants, the latter, which are ordered in time, having their moduli less than a positive number M.
Non-Local Elliptic Boundary-Value Problems
Authors: An Ton Bui 
Published: December 2010

Let G be a bounded open set of Rn with a smooth boundary ∂G. We consider the following elliptic boundary-value problem:where A and Bj are, respectively singular integro-differential operators on G and on ∂G, of orders 2m and rj with rj < 2m; Ck are boundary differential operators, and Ljk are linear operators, bounded in a sense to be specified.
Complete Regularity as a Separation Axiom
Authors: de Groot J. , Aarts J. M. 
Published: December 2010

Although the axiom of complete regularity ought to be a separation axiom, in none of its usual forms does it look like an intrinsic separation axiom. Our aim in this paper is to establish such characterizations of complete regularity which naturally fit in between regularity and normality and which already have proved to be fundamental and useful. This can simply be achieved by replacing the family of all open sets (as used in the formulation of the separation axioms) by some suitable (sub)base of open sets. For the sake of simplicity, we assume all our spaces to be T1 and we shall operate with (sub)bases of closed sets (instead of open sets). It is appropriate to define the notion of a screening.Two subsets A and B of a set X are said to be screened by the pair (C, D) if C ∪ D = X, A ∩ D = ∅ and C ∩ B = 0. (Consequently, A ⊂ C and B ⊂ D.)
A Characterization of Intrinsic Functions on
Authors: Carlson R. E. , Cullen C. G. 
Published: December 2010

Let be an associative algebra over the field and let be the group of all automorphisms and anti-automorphisms of which leave elementwise invariant. A function Fwith domain and range contained in is called an intrinsic functionon if (i) for each Ω in and (ii) F(ΩZ) = ΩF(Z) for every Z in .Rinehart (5) has introduced and motivated the study of the class of intrinsic functions on , and has characterized these functions for the cases in which is the algebra of real quaternions, the algebra of n × ncomplex matrices, or the algebra of n× nreal matrices (5; 6). The algebras listed above, along with the algebra of n × nquaternion matrices, constitute the full list of possibilities for the simple direct summands of any semi-simple algebra over or ; see (2).
Absolute Convergence Factors for Hp Series
Authors: Caveny James 
Published: December 2010

A famous theorem of Hardy asserts that if f ∊ H1, then the sequence of Fourier coefficients satisfies . For this reason we say that the sequence (1, 1/2, 1/3, …) belongs to the multiplier class (H1, l1). In this paper, we investigate the multiplier classes (Hp, l1) for 1 ≧ p ≧ ∞. Our observations are based on the fact that a sequence (λ(0), λ(l), …) belongs to (Hp, l1) independent of the arguments of its terms. We also show that (Hp, l1) may be thought of as the conjugate space of a certain Banach space.1. Preliminaries.Lp denotes the space of complex-valued Lebesgue measurable functions f defined on the circle |z| = 1 such thatis finite.
The Schwarzian Derivative and Disconjugacy of nth order Linear Differential Equations
Authors: Lavie Meira 
Published: December 2010

In this paper we deal with the number of zeros of a solution of the nth order linear differential equation1.1where the functions pj(z) (j = 0, 1, …, n – 2) are assumed to be regular in a given domain D of the complex plane. The differential equation (1.1) is called disconjugate in D, if no (non-trivial) solution of (1.1) has more than (n – 1) zeros in D. (The zeros are counted by their multiplicity.)The ideas of this paper are related to those of Nehari (7; 9) on second order differential equations. In (7), he pointed out the following basic relationship. The function1.2where y1(z) and y2(z) are two linearly independent solutions of1.3is univalent in D, if and only if no solution of equation(1.3) has more than one zero in D, i.e., if and only if(1.3) is disconjugate in D.
Collineation Groups of Generalized André Planes
Authors: Foulser David A. 
Published: December 2010

In a previous paper (5), I constructed a class of translation planes, called generalized André planes or λ-planes, and discussed the associated autotopism collineation groups. The main question unanswered in (5) is whether or not there exists a collineation η of a λ-plane Π which moves the two axes of Π but does not interchange them.The answer to this question is “no”, except if Π is a Hall plane (or possibly if the order n of Π is 34) (Corollary 2.8). This result makes it possible to determine the isomorphisms between λ-planes. More specifically, let Π and Π′ be two λ-planes of order n coordinatized by λ-systems Qand Q′, respectively. Then, except possibly if n = 34, Π and Π′ are isomorphic if and only if Q and Q′ are isotopic or anti-isotopic (Corollary 2.13). In particular, Π is an André plane if and only if Q is an André system (Corollary 2.14).
Cercles de Remplissage and Asymptotic Behaviour
Authors: Gauthier Paul 
Published: December 2010

In 1908, Lindelöf showed that if w = f(x) is a bounded holomorphic function in a sector S: |arg z| < θ1, and if f(z) has an asymptotic value w0 as z tends to ∞ along a half-ray in S; then f(z) tends uniformly to w0 as z tends to ∞o within any sector |arg z| ≦ θ, 0 ≦ θ < θ1. Montel (8) later replaced the condition that f(z) be bounded by the condition that f(z) be meromorphic and omitted three values. The following is an immediate consequence of the Lindelöf-Montel theorem.THEOREM 1. Let w = f(x) be a function meromorphic in the sector | arg z| < θ1, and let f(z) tend to a value w0 as z → ∞ along the positive real axis.
Lower Radicals in Associative Rings
Authors: Watters J. F. 
Published: December 2010

Given a homomorphically closed class of (not necessarily associative) rings , the lower radical property determined by is the least radical property for which all rings in are radical. Recently (7) a process of constructing the lower radical property from a class of associative rings has been given which terminates after a countable number of steps. In this process, an ascending chain of classesis obtained and the property of being a ring in the class is the lower radical property determined by . In Theorem 1 we give another characterization of the rings in the class , λ ∈ {1, 2, …, omega;0}, and a procedure for constructing the lower radical determined by in an arbitrary associative ring is given.
On the Behaviour of a Series Associated with the Conjugate Series of a Fourier Series
Authors: Mohanty R. , Ray B. K. 
Published: December 2010

1.1. Definition. Let λ ≡ λ(ω) be continuous, differentiable, and monotonic increasing in (0, ∞) and let it tend to infinity as ω → ∞. Suppose that ∑ an (we write ∑ for throughout the present paper) is a given infinite series and letThe series ∑ an is said to be summable |R, λ, r|, where r > 0, ifwhere A is a fixed positive number (6, Definition B). Now, for r > 0, m < ω < m + 1,Hence, ∑ an is summable |R, λ, r|, where r > 0, if
On the Growth of Blaschke Products
Authors: MacLane G. R. , Rubel L. A. 
Published: December 2010

It is well known that the distribution of the zeros of an analytic function affects its rate of growth. The literature is too extensive to indicate here. We only point out (1, p. 27; 2; 3; 5), where the angular distribution of the zeros plays a role, as it will in this paper. In private communication, A. Zygmund has raised the following related question, which is the subject of our investigation here.Let {zn}, n = 1, 2, 3, …, be a sequence of non-zero complex numbers of modulus less than 1, such that ∑(1 – |zn|) < ∞, and consider the Blaschke product1Let2What are the sequences {zn} for which I(r) is a bounded function of r?
Biorthogonal Systems in lp-Spaces
Authors: Keown R. , Conatser C. 
Published: December 2010

Our aim in this paper is to generalize certain ideas and results of Bary (1) on biorthogonal systems in separable Hilbert spaces to their counterparts in separable lp-spaces, 1 < p.The main result of Bary is to characterize a natural generalization of the concept of orthonormal basis for a Hilbert space. That of this paper is to characterize the concept of a Bary basis which is a generalization of the idea of standard basis of an lp-space. The result is interesting for lp-spaces because of the paucity of standard bases in these spaces.Before summarizing our results, we shall introduce some notation and recall a few pertinent definitions and facts. The symbols and denote mutually conjugate lp-spaces, where is the space lt and the space lswith 1 < r <2 and 2 < s = r/(r – 1).
Systems of Ideals
Authors: Cohn Richard M. 
Published: December 2010

Some well-known theorems on the representation of an ideal in a commutative ring as an intersection of ideals of a specified type remain valid when attention is restricted to certain subclasses of the ideals of the ring. For example, a homogeneous ideal in a graded Noetherian ring is the intersection of homogeneous primary ideals, and Seidenberg (16) has recently proved a similar result for differential rings. Other examples are provided by theorems on the representation of certain differential and difference ideals as intersections of prime differential or difference ideals.Kolchin (6) began the development of a general theory applicable to such phenomena. In this paper, I continue the development of that theory.In §§ 1 and 2, I have followed Kolchin's definition and discussion of divisible systems with minor modifications and additions.
On the Lower Range of Perron's Modular Function
Authors: Kinney J. R. , Pitcher T. S. 
Published: December 2010

The modular function Mwas introduced by Perron in (6). M(ξ) (for irrational ξ) is denned by the property that the inequalityis satisfied by an infinity of relatively prime pairs (p, q)for positive d,but by at most a finite number of such pairs for negative d.We will writefor the continued fraction expansion of ξ ∈ (0, 1) and for any finite collection y1,…, ykof positive integers we will writeIt is known (see 6) thatWhere
On Simply Transitive Primitive Permutation Groups
Authors: Bercov R. D. 
Published: December 2010

In (1) we considered finite primitive permutation groups G with regular abelian subgroups H satisfying the following hypothesis:(*) H = A × B × C, where A is cyclic of prime power order pα ≠ 4, B has exponent pβ < pα, and C has order prime to p.We remark that an abelian group fails to satisfy (*) (apart from the minor exception associated with the prime 2) if and only if it is the direct product of two subgroups of the same exponent.We showed in (1) that such a group G is doubly transitive unless it is the direct product of two or more subgroups each of the same order greater than 2. This was done by showing that (in the terminology of (3)) the existence of a non-trivial primitive Schur ring over H implies such a direct decomposition of H.
Homotopy Groups of Transformation Groups
Authors: Rhodes F. 
Published: December 2010

In a previous paper (2) I defined the fundamental group σ(X, x0, G) of a group Gof homeomorphisms of a space X, and showed that if the transformation group admits a family of preferred paths, then σ(X, x0, G) can be represented as a group extension of π1(X, x0) by G. In this paper the homotopy groups of a transformation group are defined. The nth absolute homotopy group of a transformation group which admits a family of preferred paths is shown to be representable as a split extension of the nth absolute torus homotopy group τn(X, x0) by G.In § 6 it is shown that the action of G on X induces a homomorphism of Ginto a quotient group of a subgroup of the group of automorphisms of τn(X, x0). This homomorphism is used to obtain a necessary condition for the embedding of one transformation group in another, in particular, for the embedding of a discrete flow in a continuous flow with the same phase space.
On Operator Algebras and Invariant Subspaces
Authors: Davis Chandler , Radjavi Heydar , Rosenthal Peter 
Published: December 2010

If is a collection of operators on the complex Hilbert space , then the lattice of all subspaces of which are invariant under every operator in is denoted by Lat . An algebra of operators on is defined (3; 4) to be reflexive if for every operator B on the inclusion Lat ⊆ Lat B implies .Arveson (1) has proved the following theorem. (The abbreviation “m.a.s.a.” stands for “maximal abelian self-adjoint algebra”.)ARVESON's THEOREM. Ifis a weakly closed algebra which contains an m.a.s.a.y and if Lat, then is the algebra of all operators on .A generalization of Arveson's Theorem was given in (3). Another generalization is Theorem 2 below, an equivalent form of which is Corollary 3. This theorem was motivated by the following very elementary proof of a special case of Arveson's Theorem.
Unbounded Positive Definite Functions
Authors: Stewart James 
Published: December 2010

Let G be an abelian group, written additively. A complexvalued function ƒ, defined on G, is said to be positive definite if the inequality1holds for every choice of complex numbers C1, …, cn and S1, …, sn in G. It follows directly from (1) that every positive definite function is bounded. Weil (9, p. 122) and Raïkov (5) proved that every continuous positive definite function on a locally compact abelian group is the Fourier-Stieltjes transform of a bounded positive measure, thus generalizing theorems of Herglotz (4) (G = Z, the integers) and Bochner (1) (G = R, the real numbers).If ƒ is a continuous function, then condition (1) is equivalent to the condition that2
Groups Generated by two Parabolic Linear Fractional Transformations
Authors: Lyndon R. C. , Ullman J. L. 
Published: December 2010

We are interested in the structure of a group G of linear fractional transformations of the extended complex plane that is generated by two parabolic elements A and B, and, particularly, in the question of when such a group G is free. We shall, as usual, represent elements of G by matrices with determinant 1, which are determined up to change of sign. Two such groups G will be conjugate in the full linear fractional group, and hence isomorphic, provided they have, up to a change of sign, the same value of the invariant τ = Trace(AB) – 2. We put aside the trivial case that τ = 0, where G is abelian. In the study of these groups, two normalizations have proved convenient. Sanov (17) and Brenner (3) took the generators in the formwhile Chang, Jennings, and Ree (4) took them in the form
Linearly Stable Polytopes
Authors: McMullen P. 
Published: December 2010

Our aim in this paper is to describe a new class of convex polytopes, which will be called linearly stable. These have properties analogous to those of projectively stable polytopes (called projectively unique by Grunbaum (1, exercise 4.8.30)), which were first investigated early in 1966 by Grunbaum and Pedes. Although many particular examples of projectively stable polytopes have been found, at present no general criteria for projective stability are known.The main result of this paper is a theorem which enables us to classify linearly stable polytopes completely.I wish to thank Professor G. C. Shephard for his many helpful suggestions for improvements to this paper. During the period of this research I held a research studentship, for which I would like to thank the Science Research Council.
Generalized Casimir Operators
Authors: Douglas A. J. 
Published: December 2010

Throughout this paper, S will be a ring (not necessarily commutative) with an identity element ls ≠ 0s. We shall use R to denote a second ring, and ϕ: S→ R will be a fixed ring homomorphism for which ϕ1S = 1R.In (7), Higman generalized the Casimir operator of classical theory and used his generalization to characterize relatively projective and injective modules. As a special case, he obtained a theorem which contains results of Eckmann (3) and of Higman himself (5), and which also includes Gaschütz's generalization (4) of Maschke's theorem. (For a discussion of some of the developments of Maschke's idea of averaging over a finite group, we refer the reader to (2, Chapter IX).) In the present paper, we define the Casimir operator of a family of S-homomorphisms of one R-module into another, and we again use this operator to characterize relatively projective and injective modules.
On Balanced Incomplete Block Designs with Large Number of Elements
Authors: Hanani Haim 
Published: December 2010

A balanced incomplete block design (BIBD) B[k, λ; v] is an arrangement of v distinct elements into blocks each containing exactly k distinct elements such that each pair of elements occurs together in exactly λ blocks.The following is a well-known theorem [5, p. 248].THEOREM 1. A necessary condition for the existence of a BIBD B[k, λ,v] is that(1)It is also well known [5] that condition (1) is not sufficient for the existence of B[k, λ; v].There is an old conjecture that for any given k and λ condition (1) may be sufficient for the existence of a BIBD B[k, λ; v] if v is sufficiently large. It is attempted here to prove this conjecture in some specific cases.There is an old conjecture that for any given k and X condition (1) may be sufficient for the existence of a BIBD B[k, λ; v] if v is sufficiently large. It is attempted here to prove this conjecture in some specific cases.
On Convex Univalent Functions
Authors: Basgoze T. , Frank J. L. , Keogh F. R. 
Published: December 2010

In what follows, we suppose that ƒ(z) = Σ0∞anzn is regular for |z| < 1. LetandThen (see, for example, [6, pp. 235-236]), for 0 ≦ r < ρ < 1, we have:The following results are well known.
On the Hyperplane Sections Through two Given Points of an Algebraic Variety
Authors: Kuan Wei-Eihn 
Published: December 2010

1. Let k be an infinite field and let V/k be an irreducible variety of dimension ≧ 2 in a projective n-space Pn over k. Let P and Q be two k-rational points on V In this paper, we describe ideal-theoretically the generic hyperplane section of V through P and Q (Theorem 1) and prove that the section is almost always an absolutely irreducible variety over k1/pe if V/k is absolutely irreducible (Theorem 3). As an application (Theorem 4), we give a new simple proof of an important special case of the existence of a curve connecting two rational points of an absolutely irreducible variety [4], namely any two k-rational points on V/k can be connected by an irreducible curve.I wish to thank Professor A. Seidenberg for his continued advice and encouragement on my thesis research.
The Spectra of Semi-Normal Singular Integral Operators
Authors: Putnam C. R. 
Published: December 2010

Suppose that(1.1)and define the bounded self-adjoint operators H and J on the Hilbert space L2(0, 1) by(1.2)the integral being a Cauchy principal valueIt is seen that(1.3)or, equivalently,(1.4)Since (Cƒ, ƒ) = π–1|(ƒ, ϕ)|2 ≧ 0, A is semi-normal. (For a discussion of such operators, see [4].)
Linearization of the Product of Jacobi Polynomials. I
Authors: Gasper George 
Published: December 2010

Let Pn(α,β) be the Jacobi polynomial of degree n, order (α,β), α,β > – 1, defined by[9, p. 67], and let Rn(α,β)(x) = Pn(α,β)(x)/Pn(αβ)(1). Then for n ≧ m,whereSince Rn(α, β)(l) = 1, it follows that(1)It is known that if (the ultraspherical case) or if α = β + 1, then α = β + 1, then g(k, n, m) ≧ 0.
A Characterization of the Approximately Continuous Denjoy Integral
Authors: Kubota Yöto 
Published: December 2010

The approximately continuous integral which includes the Lebesgue integral has been considered by Burkill [1] and Ridder [3; 4]. I [2] also defined the AD-integral of this kind which is more general than the AP-integral of Burkill [1] and the Denjoy integral in the wide sense. But this integral is equivalent to the β-integral of Ridder.Our aim in this paper is to characterize the AD-integral in the following way: The AD-integral is the least general approximately continuous integral (Definition 1) which includes the Lebesgue integral and fulfils the Cauchy and Harnack conditions (Definition 2).
Un Théorème De Régularité Pour Une Équation Différentielle Abstraite
Authors: Arminjon Paul 
Published: December 2010

Soit Y un espace localement convexe complet, la topologie étant définie par une famille de semi-normes {pα}. Soit A un opérateur fermé de domaine D(A) dense dans Y. Nous considérons dans cet article des fonctions u de la variable réelle t à valeurs dans Y. On dit que u(t) ∈ Cl[(a, b); Y] est une solution au sens classique de l'équation différentielle abstraite (ou opérationnelle)(1)si pour tout t ∈ (a, b), u(t) ∈ D(A), et satisfait la relationélément de Y, f étant une fonction continue à valeurs dans Y.Dans leur important mémoire [1], Agmon et Nirenberg ont étudié en détail les propriétés des solutions d'une telle équation dans le cas où Y est un espace de Banach.