bluuebunny/crossref_metadata_2025_split huggingface dataset:
Some Results on Small-Deviation Probability Convergence Rates for Sums of Independent Random Variables
Authors: Heyde C. C. 
Published: December 2010

Let﹛Xj,j =1, 2, 3, …﹜ be a sequence of independent, non-degenerate random variables and writeUnder quite a diverse variety of conditions we may obtainasn→ ∞ for allx,— ∞ <x< ∞, and some realp⩾ 0. For example, suppose the ﹛Xj﹜ happen to be distributed identically and belong to the domain of normal attraction of a symmetric stable law with characteristic exponentα,0 <α⩽ < 2,α≠ 1.
The Kernel of the General-Sum Four-Person Game
Authors: Peleg B. 
Published: December 2010

In this paper we apply various results and methods of previous papers on the kernel to four-person games.Section 2 contains the basic definitions needed. In §3 we prove that the kernel of the general-sum four-person game consists of a line segment (which may shrink to a point). A method for classifying games according to their kernels is suggested in §4 and is used there to characterize all four-person games whose kernel consists of a non-degenerate interval. In the last section, §5, we offer a bargaining procedure, based on principles established in (1), which leads to the kernel in the case of a non-degenerate interval.
On the Periodicity of Compositions of Entire Functions
Authors: Gross Fred 
Published: December 2010

For two entire functions f(z) and g(z) the composition f(g(z)) may or may not be periodic even though g(z) is not periodic. For example, when f(u) = cos √u and g(z) = z2, or f(u) = eu and g(z) = p(z) + z, where p(z) is a periodic function of period 2πi, f(g(z)) will be periodic. On the other hand, for any polynomial Q(u) and any non-periodic entire function f(z) the composition Q(f(z)) is never periodic (2).The general problem of finding necessary and sufficient conditions for f(g(z)) to be periodic is a difficult one and we have not succeeded in solving it. However, we have found some interesting related results, which we present in this paper.
Infinite Packings of Disks
Authors: Melzak Z. A. 
Published: December 2010

Let U be the closed disk in the plane, centred at the origin, and of unit radius. By a solid packing, or briefly a packing, C of U we shall understand a sequence ﹛Dn﹜, n = 1, 2, … , of open proper disjoint subdisks of U, such that the plane Lebesgue measures of U and of are the same. If rn is the radius of Dn and the complex number cn represents its centre, then the conditions for C to be a packing areIt was proved by Mergelyan (3) that for any packing the sum of the radii diverges:1Mergelyan's demonstration of (1) is somewhat involved and leans heavily on the machinery of functions of a complex variable. An elegant direct proof of (1) is given by Wesler (5), who uses the technique of projecting the boundaries of the disks of the packing on a diameter I of U.
On Divisibility and Injectivity
Authors: Helzer Garry 
Published: December 2010

In the category of abelian groups the concepts of divisible group and injective group coincide. In (4) this is generalized to modules over an integral domain and it is proved for a (commutative) integral domain that the concepts of divisible module and injective module coincide if and only if the ring is hereditary if and only if the ring is a Dedekind domain.In (8) the assumption of commutativity is dropped and the ring is assumed to have a one-sided field of quotients. The result obtained is essentially the same as in the commutative case; see the theorem following 6.2. In (13) the requirement that the ring have no zero-divisors is also dropped and the ring is assumed to possess what we have called an Ore ring (see the definition following 6.2). The result obtained is the equivalent of parts (a) and (b) of our 6.13.
The Integral Extension of Isometries of Quadratic Forms Over Local Fields
Authors: Trojan Allan 
Published: December 2010

Let F be a local field with ring of integers 0 and prime ideal π0. If V is a vector space over F, a lattice L in F is defined as an 0-module in the vector space V with the property that the elements of L have bounded denominators in the basis for V. If V is, in addition, a quadratic space, the lattice L then has a quadratic structure superimposed on it. Two lattices on V are then said to be isometric if there is an isometry of V that maps one onto the other.In this paper, we consider the following problem: given two elements, v and w, of the lattice L over the regular quadratic space V, find necessary and sufficient conditions for the existence of an isometry on L that maps v onto w.
A Remark on Flat and Projective Modules
Authors: Jensen Chr. U. 
Published: December 2010

It is the purpose of this note to give some characterizations of flat and projective modules, partly in ideal theoretical terms, partly in terms of the exterior product of a module (“puissance extérieure“); cf. (1).We shall consider left modules over a ring R with identity element and without proper zero divisors. The left module M is called flat if X ⊗R M is an exact functor on the category of right R-modules X. If M is flat over a commutative domain R, M is necessarily torsion-free. Therefore when looking for flatness of a module M over a commutative domain, one may assume from the start that M is torsion-free.In the following theorem, we shall not restrict ourselves to commutative rings R, but the modules concerned have to be torsion-free, which, of course, should mean that rm = 0 implies r = 0 or m = 0.
Zero Tracts of Blaschke Products
Authors: Linden C. N. , Somadasa H. 
Published: December 2010

Let ﹛an﹜ be a sequence of complex numbers such thatandThen {an} is called a Blaschke sequence. For each Blaschke sequence {an} a Blaschke product is defined asThus a Blaschke product B(z, ﹛an﹜) is a function regular in the open unit disk D = {z: |z| < 1﹜ and having a zero at each point of the sequence ﹛an﹜.
A Quasi-Linear Elliptic Boundary Value Problem
Authors: Adams R. A. 
Published: December 2010

Let Ω be a bounded open set in Euclidean n-space, En. Let α = (α1, … , an) be an n-tuple of non-negative integers;and denote by Qm the set ﹛α| 0 ⩽ |α| ⩽ m}. Denote by x = (x1, … , xn) a typical point in En and putIn this paper we establish, under certain circumstances, the existence of weak and classical solutions of the quasi-linear Dirichlet problem1
On the “Essential Metrization” of Uniform Spaces
Authors: Zakon Elias 
Published: December 2010

The notion of “essential metrization” was introduced in (8) where it was used to obtain some extensions of the theorems of Egoroff and Lusin on measurable functions. In the present note we shall further develop the theory of essential metrization, in its own right.As is well known, sets of measure zero (“null sets“) may be disregarded in many problems of measure theory. Hence the usual topological prerequisites of such problems are actually too restrictive and may be replaced by what could be called “topology modulo null-sets,” imitating such notions as “approximate continuity,” “essential supremum,” etc. Thus we consider spaces in which certain topological properties (such as metrizability, separability, etc.) hold not in the usual sense but only “essentially,” i.e. to within some null sets, as defined below.
Classification of Demushkin Groups
Authors: Labute John P. 
Published: December 2010

A pro-p-group G is said to be a Demushkin group if(1)dimFp H1(G, Z/pZ) < ∞,(2)dimFp H2(G, Z/pZ) = 1,(3)the cup product H1(G, Z/pZ) × H1(G, Z/pZ) → H2(G, Z/pZ) is a non-degenerate bilinear form. Here FP denotes the field with p elements. If G is a Demushkin group, then G is a finitely generated topological group with n(G) = dim H1(G, Z/pZ) as the minimal number of topological generators; cf. §1.3. Condition (2) means that there is only one relation among a minimal system of generators for G; that is, G is isomorphic to a quotient F/(r), where F is a free pro-p-group of rank n = n(G) and (r) is the closed normal subgroup of F generated by an element r ∈ F9 (F, F); cf. §1.4.
Proof of the Fixed Point Theorems of Poincaré and Birkhoff
Authors: Barrar Richard B. 
Published: December 2010

In 1912, shortly before his death, Poincaré (8) conjectured the following theorem in his investigation of the restricted problem of three bodies.Poincaré's Last Geometric Theorem. Given a ring 0 < a ⩽ r ⩽ b in the r, θ plane and a homeomorphic, area-preserving mapping T of the ring onto itself under which points on r = a advance and those on r = b regress, there will exist at least two points of the ring invariant under T.Poincaré was able to prove this theorem in only a few special cases. Shortly thereafter, Birkhoff was able to give a complete proof in (2) and in, (3) he gave a generalization of the theorem, dropping the assumption that the transformation was area-preserving. Birkhoff's proofs were very ingenious; however, they did not use standard topological arguments.
Univalent α-Spiral Functions
Authors: Libera Richard J. 
Published: December 2010

Supposeƒis regular in the open unit disk Δ, |z| < 1, and has a simple zero at the origin and no other zeros. Špaček (15) essentially showed thatƒis univalent in Δ if and only ifsuch that 0 <r< 1 and 0 <t2—t1⩽ 2π.
Triangulation of Fibre Bundles
Authors: Putz H. 
Published: December 2010

In this paper we consider the following problem. Let (E, M, N, π) be a differentiable fibre bundle, whereEis the total space,Mthe base space,Nthe fibre, andπ: E→Mthe projection map. Then, given aCrtriangulation (ƒ, D) ofM,can one obtain aCrtriangulation (F, K) ofEsuch that the induced mapƒ–1πF:K→ D is linear? R. Lashof and M. Rothenberg (3) have obtained this result for vector bundles.Using methods quite different from theirs, we obtain a solution in the general case. The methods we use are the geometric methods developed by J. H. C. Whitehead. (7) in his triangulation of differentiable manifolds, as found in (5). In fact, our solution consists of generalizing his techniques in a fibre bundle setting.
Semi-Groups of Maps in a Locally Compact Space
Authors: Dorroh J. R. 
Published: December 2010

Suppose that S is a locally compact Hausdorff space. A one-parameter semi-group of maps in S is a family {ϕt; t ⩾ 0} of continuous functions from S into S satisfying(i)ϕt0ϕu = ϕt+u for t, u ⩾ 0, where the circle denotes composition, and(ii)ϕ0 = e, the identity map on S.A semi-group {ϕt} of maps in S is said to be(iii)of class (C0) if ϕt(x) → x as t → 0 for each x in S,(iv)separately continuous if the function t → ϕt(x) is continuous on [0, ∞) for each x in S, and(v)doubly continuous if the function (t, x) → (ϕt(x) is continuous on [0, ∞) x S.
Soluble Artinian Groups
Authors: Baer Reinhold 
Published: December 2010

The principal aim of this note is the proof of the following:Main Theorem. The following properties of the group G are equivalent:I. G is artinian and soluble.II. (a) To every maximal subgroup S of G there exists a normal subgroup T of G with G = ST and S ∩ T = (S ∩ T)T.
Antisymmetrical Digraphs
Authors: Tutte W. T. 
Published: December 2010

SummaryWe call a digraph “antisymmetrical” if there is an automorphismθof its graph, of period 2, which reverses the direction of every edge and maps no edge or vertex onto itself. We construct a theory of flows invariant underθfor such a diagraph. This theory is analogous to the Max Flow Min Cut theory for ordinary flows in digraphs. It is found to include that part of the theory of undirected graphs which discusses the existence of spanning subgraphs with a specified valency at each vertex.
Maximal or Greatest Homomorphic Images of Given Type
Authors: Tamura Takayuki 
Published: December 2010

Let Q be a quasi-ordered set with respect to ⩽ ; that is, the order ⩽ is reflexive and transitive. An element a of Q is called maximal (minimal) ifa is called greatest (smallest) ifObviously a greatest (smallest) element is maximal (minimal). A greatest (smallest) element in a partially ordered set is unique, but it is not necessarily unique in a quasi-ordered set.
Approximation in Bounded Summability Fields
Authors: Hill J. D. , Sledd W. T. 
Published: December 2010

This paper deals with several related properties of bounded summability fields of regular, real matrices. For a matrix A = (ank) and a sequence x = {xn}, we write formallyWe denote by m the space of bounded real sequences, and by A* the bounded summability fieldof A. The strong summability field of A is the set
On S2-Groups and Groups of Moebius Transformations
Authors: Lorimer P. J. 
Published: December 2010

Let PGL(2, ƒ) denote the group of all Moebius transformationsover a field F. The object of this paper is to prove the following theorem.Theorem 1. G is an S2-group and the centre of G is trivial if and only if G is isomorphic to a group PGL(2, ƒ), char ƒ ≠ 2.This theorem was proved for finite groups in (1). The present paper extends the result to infinite groups and also improves the method of proof used in that paper. Many of the theorems given there were proved for infinite groups and are used here with appropriate references.
On Factorization of Elliptic Functions
Authors: Gross Fred 
Published: December 2010

In this paper we shall be concerned with the following problem: If h is an elliptic function and h(z) = ƒ(g(z)), what can be said about the functions ƒ and g? In order to simplify the discussion we introduce some basic definitions.Definition 1. A meromorphic function h(z) = ƒ(g(z)) is said to have ƒ(z) and g(z) as left and right factors, respectively, provided that either ƒ(z) is non-linear and meromorphic and g(z) is non-linear and entire or f(z) is rational and g(z) is meromorphic.
A Theorem Concerning Partitions and its Consequence in the Theory of Lie Algebras
Authors: Hughes J. W. B. 
Published: December 2010

In the first part of this paper we state and prove a theorem concerning the partition (j; l, i) of an integer j into at most l integers , none of which exceed i; l and i being themselves integers, (j; l, i) is thus the number of distinct solutions of the equations1.1where the satisfy the inequalities1.2
Rings Satisfying (x, y, z) = (y, z, x)
Authors: Sterling Nicholas J. 
Published: December 2010

Let R be a ring satisfying the identity1for all x, y, z ϵ R, where (x,y,z) = (xy)z — x(yz). If R also satisfies the identity (x, x, x) = 0 for all x ϵ R, then R is alternative. It is known that if R satisfies (1), it need not be an alternative (see 6). Thus, the class of rings satisfying (1) is a non-trivial extension of the class of alternative rings. P. Jordan remarked that (x, x, x)2 = 0 is an identity in R (see 9).
Extreme Points and Linear Isometries of the Banach Space of Lipschitz Functions
Authors: Roy Ashoke K. 
Published: December 2010

Let X be a compact metric space with metric d. A complex-valued function ƒ on X is said to satisfy a Lipschitz condition if, for all points x and y of X, there exists a constant K such thatThe smallest constant for which the above inequality holds is called the Lipschitz constant for ƒ and is denoted by ||ƒ||d, that is,
On the Lower Derivate of a Set Function
Authors: Pfeffer W. F. 
Published: December 2010

In (5), the following theorem was proved in a very general setting:(1) An additive set function is non-negative whenever its lower derivative is non-negative.For a continuous additive function of intervals, theorem (1) can be improved as follows:(2) A continuous additive set function is non-negative whenever its lower derivative is non-negative except, perhaps, on a countable set.
On Projective Modules and Automorphisms of Central Separable Algebras
Authors: Childs L. N. 
Published: December 2010

This paper developed from, and complements, the paper by F. R. DeMeyer (see 6).In the first section of this paper we note a correspondence between projective modules of a central separable R-algebra A and the two-sided ideals of central separable algebras in the same class as A in the Brauer group of R. When R has the property that rank one projective A-modules are free, this correspondence yields a bijection between isomorphism types of indecomposable projective A-modules and the isomorphism types of algebras in the Brauer class of A which are the analogue of division algebra components in the field case. This bijection was remarked on without proof by DeMeyer in (6).Pursuing the ideas behind this correspondence, we consider the situation for a separable order A in a central simple algebra A over an algebraic number field, and obtain, by means of results involving the reduced norm, a generalization of DeMeyer's remark except when the division algebra component of A is a totally definite quaternion algebra (Theorem 3.3).
An Analogue of a Problem of J. Balázs and P. Turán
Authors: Prasad J. , Varma A. K. 
Published: December 2010

In 1955, J. Surányi and P. Turán (8) initiated the problem of existence and uniqueness of interpolatory polynomials of degrees less than or equal to 2n — 1 when their values and second derivatives are prescribed on n given nodes. This kind of interpolation was termed (0, 2)-interpolation. Later, Balázs and Turán (1) gave the explicit representation of the interpolatory polynomials for the case when the n given nodes (n even) are taken to be the zeros of πn(x) = (1 — x2)Pn′(x), where Pn–i(x) is the Legendre polynomial of degree n — 1. In this case the explicit representation of interpolatory polynomials turns out to be simple and elegant.Balázs and Turán (2) proved the convergence of these polynomials when f(x) has a continuous first derivative satisfying certain conditions of modulus of continuity. They noted (1) that a significant application of lacunary interpolation could possibly be given in the theory of a differential equation of the form y′ + A(x)y= 0.
A Note on Non-Distributive Sublattices of Degrees and Hyperdegrees
Authors: Thomason S. K. 
Published: December 2010

In (1, §§ 2.3 and 2.4) we proved that certain distributive lattices are simultaneously lattice-embeddable in the degrees of recursive unsolvability and in the hyperdegrees. Let ℒ be the non-distributive lattice {0,1, a0, a1,…}, where ai ∪ aj = 1 and ai ∩ aj = 1 whenever i ≠ j. We shall prove the following theorem.THEOREM. The lattice ℒ is simultaneously lattice-embeddable in the degrees and hyperdegrees.For A ⊆ N, let deg(A) and hyp(A) be the degree and hyperdegree of A, respectively. To prove the theorem we must construct hyperarithmetically incomparable sets A0, A1, … such that for Δ = deg, hypand for all distinct i, j:12Now, if each 〈Ai, Aj〉 were a generic pair in the sense of (1), then (2) would hold.
On Finite Collineation Groups of F5
Authors: Sandler Reuben 
Published: December 2010

Our aim in this paper is to fill one gap left in (1) and to prove that if H is a finite collineation group of F5, the free plane generated by a finite open configuration of rank 9, then |H | ≦ 12. Alltop has shown that |H | ≦ 24 and that there exist finite collineation groups of F5 which have order 12, so that the argument in this paper shows that |H| ≦ 12 is the best estimate which can be given. In (1), Alltop has completely settled the question for Fn, n ≠ 5. The notation of this paper will generally be that of (1).Most of the arguments used here will consist of case analyses of degenerate planes of ranks 7 and 8 and will be sketched rather than given in detail. The one theorem of some interest in this paper, other than the main result, is the following theorem which yields some information about the collineation group of F4 (π2 in the notation of (4)).
The Maximum Term and the Rank of an Entire Function
Authors: Sreenivasulu V. 
Published: December 2010

1. For an entire function , let M(r, f), μ(r, f), and v(r, f) denote the maximum modulus, the maximum term, and the rank for |z\ = r, respectively. Also, letand λ(r) the lower proximate order relative to log M(r, f). For the properties of the lower proximate order, we refer the reader to the paper by Shah (1).2. We prove the following theorems.THEOREM 1. For an entire functionwhere μ(r, f1) and M(r, f1) correspond to fl(z), the derivative of f(z), provided (n + l)Rn > nRn+1, when L(f) > 1.
A Presentation of the Groups PSL(2, p) with Three Defining Relations
Authors: Zassenhaus Hans J. 
Published: December 2010

H. Behr and J. Mennicke (1) have proven that the group PSL(2, p) can be presented by the following system of generators and relations:1From this presentation, it follows that the three relations2for the same generators S and T suffice if p > 3, p ≠ 17. If p = 3, it is well known that the relations S3 = 1, T2 = 1, and (ST)3 = 1 define PSL(2, 3). For p = 2, the relations S3 = 1, T2 = 1, and (ST)2 = 1 define PSL(2, 2). For p = 17, the three relations3will suffice.Indeed, the group G, with generators S, T and defining relations (2), contains the subgroup 〈Sp〉 in its centre.
Lower Bounds for the Essential Spectrum of Fourth-Order Differential Operators
Authors: Kreith Kurt 
Published: December 2010

In this paper, we seek to determine the greatest lower bound of the essential spectrum of self-adjoint singular differential operators of the form1where 0 ≦ x < ∞. In the event that this bound is + ∞, our results will yield criteria for the discreteness of the spectrum of (1).Such bounds have been established by Friedrichs (3) for Sturm-Liouville operators of the formand our techniques will be closely related to those of (3). However, instead of studying the solutions of2directly, we shall exploit the intimate connection between the infimum of the essential spectrum of (1) and the oscillation properties of (2).
A Double-Centralizer Theorem for Simple Associative Algebras
Authors: Werner W. L. 
Published: December 2010

Consider the following result.PROPOSITION. Let D be a finite-dimensional central division algebra over a field F, and let Dn be the algebra (over F) of all n × n matrices with entries in D. Let A and B be in Dn, and suppose that BX = XB for every X in Dn such that XA = AX. Then B is a polynomial in A with coefficients in F.The case D = F is a well-known classical result. Recently, the particular case where D is the algebra of real quaternions was established by Cullen and Carlson (2). In this note, the general proposition is proved by reduction to the classical case by way of tensor products.
Some Lemmas on Interpolating Blaschke Products and a Correction
Authors: Kerr-Lawson A. 
Published: December 2010

A Blaschke product on the unit disc,where |c|= 1 and kis a non-negative integer, is said to be interpolatingif the
conditionCis satisfied for a constant δ independent of m.A Blaschke product always belongs to the set I of inner functions; it has norm 1 and radial limits of modulus 1 almost everywhere. The most general inner function can be uniquely factored into a product BS,where Bis a Blaschke product andfor some positive singular measure μ(θ) on the unit circle. The discussion will be carried out in terms of the hyperbolic geometry on the open unit disc D,its metricand its neighbourhoods N(x, ∈) = ﹛z′ ∈ D: Ψ(z, z′) < ∈ ﹜
On the Absolute Cesaro Summability of Negative Order of a Series Associated with the Conjugate Series of a Fourier Series
Authors: Mohanty R. , Ray B. K. 
Published: December 2010

1. Definition. Let λ ≡ λ(ω) be continuous, differentiable, and monotonie increasing in (0, ∞) and let it tend to infinity as ω → ∞. A series an is summable |R, λ, r|, where r > 0, ifwhere A is a fixed positive number (6, Definition B).Let f(t) be a periodic function with period 2π and Lebesgue integrable over (–π, π) and let1.1The series conjugate to (1.1), at t = x, is1.2
Conilpotency and Weak Category
Authors: Hoo C. S. 
Published: December 2010

Let ƒ : X → Y be a map and let e′: Y → ΩΣY be the usual embedding. Then we prove the following results.Theorem 1. cat ƒ = cat(e′ƒ), w cat ƒ = w cat(e′ƒ) if Y is an H-space.Theorem 2. conil ƒ = w Σ cat(e′ƒ) ≦ Σ w cat(e′ƒ) ≦ w cat(e′ƒ), where Σ the suspension functor. If we take X = Y and ƒ= lx, this result yields conil X ≦ w cat e', a result due to Ganea, Hilton, and Peterson(4).Theorem 3. Suppose that Y is (m– 1)-connected andThen conil ƒ = w Σ cat (e′ƒ) = Σ w cat(e′ƒ) = w cat(e′ƒ).