bluuebunny/crossref_metadata_2025_split huggingface dataset:
On a Restricted Class of Block Design Games
Authors: Di Paola Jane W. 
Published: December 2010

Block design games have been developed by Richardson (12) and by Hoffman and Richardson (8), who proved a number of theorems concerning such games by studying the number of elements in a blocking coalition. Hoffman and Richardson listed as unsolved (except for PG(2, 3)) the following problem: What is the minimum number of elements in a blocking coalition of a block design game?This note considers blocking coalitions in those games that are dual to block designs having λ = 1 and r — k > 0. For such games certain blocking coalitions are shown to be related to sets of mutually disjoint blocks in the design to which the game is dual. In particular, for Steiner triple systems the largest odd-numbered set of mutually disjoint triples is shown to yield a minimum blocking coalition in the dual.
Normal Functions and Non-Tangential Boundary Arcs
Authors: Lappan P. , Rung D. C. 
Published: December 2010

Let D and C denote respectively the open unit disk and the unit circle in the complex plane. Further, γ = z(t), 0 ⩽ t ⩽ 1, will denote a simple continuous arc lying in D except for Ƭ = z(l) ∈ C, and we shall say that γ is a boundary arc at Ƭ.We use extensively the notions of non-Euclidean hyperbolic geometry in D and employ the usual metricwhere a and b are elements of D. For a ∈ D and r > 0 letFor details we refer the reader to (4).
Gap Series on Groups and Spheres
Authors: Rider Daniel 
Published: December 2010

Let G be a compact abelian group and E a subset of its dual group Γ. A function ƒ ∈ L1(G) is called an E-function if for all γ ∉ E wheredx is the Haar measure on G. A trigonometric polynomial that is also an E-function is called an E-polynomial.
Invariant Subspaces on Riemann Surfaces
Authors: Voichick Michael 
Published: December 2010

In this paper we generalize to Riemann surfaces a theorem of Helson and Lowdenslager in (2) describing the closed subspaces of L2(﹛|z| = 1﹜) that are invariant under multiplication by eiθ.Let R be a region on a Riemann surface with boundary Γ consisting of a finite number of disjoint simple closed analytic curves such that R ⋃ Γ is compact and R lies on one side of Γ. Let dμ be the harmonic measure on Γ with respect to a fixed point t0 on R. We shall consider the closed subspaces of L2(Γ, dμ) that are invariant under multiplication by functions in A (R) = ﹛F|F continuous on , analytic on R}.
Equivalent Formulations of the Borsuk-Ulam Theorem
Authors: Bacon Philip 
Published: December 2010

Let R° denote a space consisting of just one point and for each positive integer n let Rn denote euclidean n-space. For each non-negative integer n let Sn denote the n-sphereIn 1933 K. Borsuk published proofs of the following two theorems (2, p. 178).
Local Properties of the Embedding of a Graph in a Three-Manifold
Authors: McMillan D. R. 
Published: December 2010

LetGbe a finite graph topologically embedded in the interior of a 3-manifoldM.Doyle (4) and Debrunner and Fox (3) have noted that the following local homotopy condition at each pointp∈Gis necessary in order for the embedding ofGto be tame:For each sufficiently small open set U containing p, there is an open set V such that p∈V⊂U and if W is any connected open set such that p∈W⊂V, then the image under the inclusion homomorphism i*: π1(W — G) → π1(U — G) is a free group on n —1generators.
A Class of Loops with the Isotopy-Isomorphy Property
Authors: Wilson Eric L. 
Published: December 2010

A loop L has the isotopy-isomorphy property provided each loop isotopic to L is isomorphic to L. A familiar problem is that of characterizing those loops having this property.It is well known (1, p. 56) that the loop isotopes of (L, ·) are those loops L(a, b, *) defined by x * y = x/b·a\y for some a, b in L. In this paper we first show (Corollary to Theorem 1) that a loop L with identity element 1 has the isotopy-isomorphy property if L is isomorphic to 1,(1, x) and to L(x, 1) for each x in L. We then determine necessary and sufficient conditions (Theorems 2 and 3) for L to be isomorphic to these isotopes under translations (i.e. permutations of the form xv = cx or xv = xc for c fixed).
Extrema in Space-Time
Authors: Quintas Louis V. , Supnick Fred 
Published: December 2010

Consider an astronomer and his observation field, i. e., the set of observable (light or radio) signal-emitting loci of the universe. Let the observation field be ordered by attaching a date to each observable locus indicating the time in the history of the universe that the signal was emitted from its source. Whereas both the astronomer and his observation field age with time, the observations of the astronomer may trace a sequence of loci whose time labels proceed forward or backward in time (cf. Appendix).Consider now a finite set S of events in Ln,n-dimensional space-time (Riemannian n-space having the fundamental form
Measures and Tensors II
Authors: Gil de Lamadrid Jesús 
Published: December 2010

The present work is a sequel to our previous article (10) with the same title. The major theme remains the study of the relationship between tensor products of spaces of functions and vector-valued measures on a space S. In (10) S was a compact Hausdorff space. Here we extend our considerations to locally compact Hausdorff spaces. B(S) still stands for the Borel class of S.Three types of vector-valued measures m : B(5) → E, E a Banach space, are considered here (§3), namely, weak*, weak, and strong vector measures, but the concepts of weak and strong measures coincide. This result is due to Bartle, Dunford, and Schwartz (1) for abstract vector measures, i.e. defined on an abstract σ-algebra of sets.
A Permanental Inequality—The Case of Equality
Authors: Marcus Marvin , Minc Henryk 
Published: December 2010

In (3) we proved that if A is a complex n-square normal matrix with characteristic roots α1 … , αn,then1If A is positive semi-definite hermitian, the inequality (1) becomes2This inequality partially answers the problem of determining the maximum permanent of a positive semi-definite hermitian matrix with prescribed characteristic roots (6). In (1), Brualdi and Newman proved that (2) also holds when A is an n-square circulant with non-negative entries. In a recent conversation Dr. Newman raised the question of determining the cases of equality in (1). In the present note we answer this question.
Notes on Sphere Packings
Authors: Leech John 
Published: December 2010

These notes are to supplement my paper (4), and should be read in conjunction with it. Both are divided into three parts, and in these notes the section numbers have a further digit added; thus §1.41 here supplements §1.4 of (4). References by section numbers are always to (4) or to the present notes, but references to other papers are numbered independently.The principal results of these notes are the following. New sphere packings are given in [2m], m ⩾ 6, and in [24], which are twice as dense as those of §§1.6, 2.3. Others are given in [2m], m ⩾ 5, with the same density as those of §1.6, but in which each sphere touches fewer other spheres than in the earlier packings.
Gocommutative Hopf Algebras
Authors: Larson Richard G. 
Published: December 2010

A coalgebra over the field F is a vector space A over F, with maps δ: A → A ⊗ A and ∊: A → F such that1and2The notion of coalgebra is dual to the notion of algebra with unit, with δ as coproduct (equation (1) says that δ is associative) and ∊ as the unit map (equation (2) is just the statement that ∊ is a unit for the coproduct δ). If A is also an algebra with unit and δ and ∊ are algebra homomorphisms, A is a Hopf algebra.
On Arcs in a Finite Projective Plane
Authors: Martin G. E. 
Published: December 2010

The aim of this paper is to generalize and unify results of B. Qvist, B. Segre, M. Sce, and others concerning arcs in a finite projective plane. The method consists of applying completely elementary combinatorial arguments.To the usual axioms for a projective plane we add the condition that the number of points be finite. Thus there exists an integer n ⩾ 2, called the order of the plane, such that the number of points and the number of lines equal n2 + n + 1 and the number of points on a line and the number of lines through a point equal n + 1. In the following, n will always denote the order of a finite plane. Desarguesian planes of order n, formed by the analytic geometry with coefficients from the Galois field of order n, are examples of finite projective planes. We shall not assume that our planes are Desarguesian, however.
On Extensions of Topologies
Authors: Borges Carlos J. R. 
Published: December 2010

If (X, τ) is a topological space (with topology τ) and A is a subset of X, then the topology τ(A) = {U ⋃ (V ⋂ A)|U, V ∈ τ} is said to be a simple extension of τ. It seems that N. Levine introduced this concept in (4) and he proved, among other results, the following:(A) If (X, τ) is a regular (completely regular) space and A is a closed subset of X, then (X, τ(A)) is a regular (completely regular) space.(B) Let (X, τ) be a normal space, and A a closed subset of X. Then (X, τ(A)) is normal if and only if X — A is a normal subspace of (X, τ).(C) Let (X, τ) be a countably compact (compact or Lindelöf) and A ∉ τ.
On the Non-Existence of a Type of Regular Graphs of Girth 5
Authors: Brown William G. 
Published: December 2010

ƒ(k, 5) is defined to be the smallest integer n for which there exists a regular graph of valency k and girth 5, having n vertices. In (3) it was shown that1.1Hoffman and Singleton proved in (4) that equality holds in the lower bound of (1.1) only for k = 2, 3, 7, and possibly 57. Robertson showed in (6) that ƒ(4, 5) = 19 and constructed the unique minimal graph.
Pairs of Consecutive Residues of Polynomials
Authors: Williams Kenneth S. 
Published: December 2010

Let p be a large prime and let ƒ(x) be a polynomial of fixed degree d ⩾ 4 with integral coefficients, say,1.1Recently Mordell (8) has considered the problem of estimating the least positive residue of ƒ(x) (mod p), that is, the unique integer l (0 ⩽ l ⩽ p — 1) such that the congruence1.2is soluble for r = l but not for r = 0, 1, … , l — 1.
Cell Growth Problems
Authors: Klarner David A. 
Published: December 2010

Thesquare latticeis the set of all points of the plane whose Cartesian coordinates are integers. Acellof the square lattice is a point-set consisting of the boundary and interior points of a unit square having its vertices at lattice points. Ann-ominois a union ofncells which is connected and has no finite cut set.The set of alln-ominoes,Rnis an infinite set for eachn; however, we are interested in the elements of two finite sets of equivalence classes,SnandTn, which are defined on the elements ofRnas follows: Two elements ofRnbelong to the same equivalence class (i) inSn, or (ii) inTn, if one can be transformed into the other by (i) a translation or (ii) by a translation, rotation, and reflection of the plane.
The Number of Very Reduced 4 × n Latin Rectangles
Authors: Moser W. O. J. 
Published: December 2010

Two permutations (displayed in the two rows)of the integers 1, 2, … , n are called discordant if ai ≠ bi, i = 1, 2, …, n. Let v(4, n), n ⩾ 4, be the number of permutations discordant with the three Permutations
Differences, Derivatives, and Decreasing Rearrangements
Authors: Duff G. F. D. 
Published: December 2010

The decreasing rearrangement of a finite sequence a1, a2, … , an of real numbers is a second sequence aπ(1), aπ(2), … , aπ(n), where π(l), π(2), … , π(n) is a permutation of 1, 2, … , n and(1, p. 260). The kth term of the rearranged sequence will be denoted by . Thus the terms of the rearranged sequence correspond to and are equal to those of the given sequence ak, but are arranged in descending (non-increasing) order.
Generating Functions for Ultraspherical Functions
Authors: Viswanathan B. 
Published: December 2010

The ultraspherical function1.1for |1 — x| < 2 is a solution of the differential equation1.2This equation has two independent solutions; of the two, only Pn(λ)(x) is analytic at x = 1, aside for some special values of λ, which we shall not consider.
A Representation Theorem for Relatively Complemented Distributive Lattices
Authors: Nanzetta Philip 
Published: December 2010

In this note, we are concerned with the following generalization of a wellknown theorem of M. H. Stone; see (2, 8.2).Theorem 1. Let L be a relatively complemented distributive lattice.(I) If L has no least element, then L is isomorphic to the lattice of non-empty compact-open subsets of an anti-Hausdorff, nearly-Hausdorff, T1-space with a base of open sets consisting of compact-open sets.(II) (3, Theorem 1) If L has a least element, then L is isomorphic to the lattice of all compact-open subsets of a locally compact totally disconnected space. Moreover, the spaces of (I) and (II) are compact if and only if L has a greatest element.The space in question is the space of prime ideals of L with the hull-kernel topology.The author is indebted to M. G. Stanley for several conversations concerning this note.
Operators of the form PAQ-QAP
Authors: Brown Arlen , Pearcy Carl 
Published: December 2010

In this note the Hilbert spaces under consideration are complex, and the operators referred to are bounded, linear operators. If is a Hilbert space, then the algebra of all operators on is denoted by .It is known (1) that if is any Hilbert space, then the class of commutators on , i.e., the class of all operators that can be written in the form PQ — QP for some , can be exactly described. A similar problem is that of characterizing all operators on that can be written in the form for some .
Topological Homotheties on Compact Metrizable Spaces
Authors: Janos Ludvik 
Published: December 2010

Notation and definitions.Definition 1. Let (X, ρ) be a metric space and ϕ: X → X a continuous self-mapping of X. We shall call ϕ and α-contraction on (X, ρ) if and only if α ϵ [0,1) and . We shall call ϕ an α-homothety on (X, ρ) if and only if α > 0 and .
Unitary Representations of Generalized Symmetric Groups
Authors: Puttaswamaiah B. M. 
Published: December 2010

In this paper all representations are over the complex field K. The generalized symmetric group S(n, m) of order n!mn is isomorphic to the semi-direct product of the group of n × n diagonal matrices whose rath powers are the unit matrix by the group of all n × n permutation matrices over K. As a permutation group, S(n, m) consists of all permutations of the mn symbols {1, 2, …, mn} which commute withObviously, S (1, m) is a cyclic group of order m, while S(n, 1) is the symmetric group of order n!. If ci = (i, n+ i, …, (m – 1)n+ i) andthen {c1, c2, …, cn} generate a normal subgroup Q(n) of order mn and {s1, s2, …, sn…1} generate a subgroup S(n) isomorphic to S(n, 1).
Characterizations of Finite Projective and Affine Spaces
Authors: Kantor William M. 
Published: December 2010

A well-known result of Dembowski and Wagner (4) characterizes the designs of points and hyperplanes of finite projective spaces among all symmetric designs. By passing to a dual situation and approaching this idea from a different direction, we shall obtain common characterizations of finite projective and affine spaces. Our principal result is the following.Theorem 1.A finite incidence structure is isomorphic to the design of points and hyperplanes of a finite projective or affine space of dimension greater than or equal to4if and only if there are positive integers v, k, and y, with μ> 1and(μ– l)(v — k) ≠ (k—μ)2such that the following assumptions hold.(I)Every block is on k points, and every two intersecting blocks are on μ common points.(II)Given a point and two distinct blocks, there is a block containing both the point and the intersection of the blocks.(III)Given two distinct points p and q, there is a block on p but not on q.(IV)There are v points, and v– 2 ≧k>μ.
Generalization of Hölder's Theorem to Ordered Modules
Authors: Viswanathan T. M. 
Published: December 2010

Hölder's theorem on archimedean groups states:An ordered (abelian) group G is order isomorphic to an ordered subgroup of the ordered group R of real numbers if and only if it is archimedean.We comprehend this theorem in the following setting: G is a Z-module and Ris the completion with respect to the open interval topology of the ordered field Q; Qitself is the ordered quotient field of the ordered domain Z.Rephrasing the situation, we raise the following question: We start with a fully ordered domain A,let Kbe its ordered quotient field. We endow Kwith the open interval topology and consider , the topological completion of K. Is it possible to impose a compatible order structure on and if this can be done, when can we say that an ordered A-module Mis order isomorphic to an ordered A-submodule of ? In Theorem 3.1, we obtain a set of necessary and sufficient conditions for this isomorphism to hold.
Topologies on Generalized Inner Product Spaces
Authors: Prugovečki Eduard 
Published: December 2010

In the present note we introduce a straightforward algebraic generalization of inner product spaces, which we appropriately name generalized inner product (GIP) spaces. In the same fashion in which different topologies :an be introduced in inner product spaces, adequate topologies can be introduced in GIP spaces in such a manner that topological vector spaces are obtained. We enumerate and derive some fundamental properties of different topologies in GIP spaces, having primarily in mind their possible later application to quantum physics.The desirability of having in quantum physics more general structures than Hilbert spaces (in which quantum mechanics is usually formulated) is suggested by Dirac's formalism (2), which deals with “unnormalizable” vectors. Unfortunately, although this formalism is very elegant from the point of view of the facili ty of dealing with its symbolism, it completely lacks in mathematical rigour.
On Certain Discrete Inequalities involving Partial Sums
Authors: Beesack Paul R. 
Published: December 2010

Our aim in this paper is to prove inequalities of the form1or2for all real values of the parameters α, β and all non-negative (in some cases all positive) xi. Obviously, an is finite in all cases, and we shall show that An is finite if α and α + β are both non-negative. In all cases, we obtain sharp values of the constants an, An (when finite), as well as bounds for these constants, and their behaviour as n → ∞. In case a < 0, we naturally consider only positive xi, otherwise the xi may be non-negative. Although we always write xi ≧ 0 in the following, this should be read as xi > 0 in case α < 0; similar remarks apply to the parameter t introduced below.
Idempotent Ideals in Perfect Rings
Authors: Michler Gerhard 
Published: December 2010

All rings considered in this note have an identity element, and all R-modules are unitary.Bass (2) defined a left perfect ring as a ring R satisfying the minimum condition on principal right ideals. A commutative ring R is perfect if and only if R is a direct sum of finitely many local rings whose radicals are T-nilpotent. Therefore, the commutative perfect rings with finite global projective dimension are just the direct sums of finitely many commutative fields, and hence they trivially satisfy the minimum condition for all ideals. However, in the non-commutative case, even hereditary perfect rings are not necessarily right or left artinian (cf. Example 3.4).Each left perfect ring R has only finitely many idempotent (two-sided) ideals (Corollary 2.3), where the ideal X of R is called idempotent, if X = X2. Hence, it makes sense to consider minimal idempotent idealsof the left perfect ring R, i.e., ideals of R which are minimal in the set of all idemponent ideals of R.
On Planar Continuous Families of Curves
Authors: Zamfirescu Tudor 
Published: December 2010

In a recent paper (3), Grünbaum has found a general and unifying setting for a number of properties of some special lines associated with a planar convex body. Besides various interesting results, two conjectures are stated and two kinds of convexity and polygonal connectedness are introduced.In the present paper, we shall prove one of Grünbaum's conjectures (§ 3, Theorem 1); we consider the other in § 4 and establish some related results in §§ 5 and 6. Six-partite problems are studied in this general setting (§ 7) and a question raised by Ceder (2) is answered. We give a generalization of the notion of a continuous family of curves in § 8, and discuss some new kinds of connectedness in § 9.
A Lower Bound for the Scholz-Brauer Problem
Authors: B. Stolarsky Kenneth 
Published: December 2010

In (6)Scholz asked if the inequality1.1held for all positive integers q, where l(n)is the number of multiplications required to raise xto the nth power (a precise definition of l(n)in terms of addition chains is given in § 2). Soon afterwards, Brauer (2) showed, among other things, that l(n) ∼(log n)/(log2). This suggests the problem of Calculating1.2It can be deduced from (2) that θ≦ 1. If θ <1, (1.1) follows immediately for infinitely many q.My main result,Theorem 5 of § 4, merely shows that θ is slightly larger than ⅓.Actually, I know of no case where (1.1) is not in fact an equality; a tedious calculation verifies this for 1 ≦ q≦ 8.
A First Approximation to {X, Y}
Authors: Samuel Brown Benson 
Published: December 2010

If ℭ and ℭ′ are classes of finite abelian groups, we write ℭ + ℭ′ for the smallest class containing the groups of ℭ and of ℭ′. For any positive number r, ℭ < r is the smallest class of abelian groups which contains the groups Zp for all primes p less than r.Our aim in this paper is to prove the following theorem.THEOREM. Iƒ ℭ is a class of finite abelian groups and(i) πi(Y) ∈ℭ for i < n,(ii) H*(X; Z) is finitely generated,(iii) Hi(X;Z)∈ ℭ for i > n + k,ThenThis statement contains many of the classical results of homotopy theory: the Hurewicz and Hopf theorems, Serre's (mod ℭ) version of these theorems, and Eilenberg's classification theorem. In fact, these are all contained in the case k = 0.
Remarque Sur La Moyenne Arithmetique de Fonctions Univalentes Convexes
Authors: Labelle G. , Rahman Q. I. 
Published: December 2010

Soient f(z) et g (z) deux fonctions univalentes normalisées convexes dans |z|. Nous avons démontré récemment (3) que leur moyenne géométrique F = (fg)1/2 applique le disque sur un domaine convexe et l'estimation est précise.Nous voulons ici considérer le même problème pour la moyenne arithmétique des deux fonctions f(z) et g (z).THÉORÈME. Soient f(z) et g(z) deux fonctions univalentes normalisées convexes dans |z|. Si K représente la plus petite racine positive de l'équation1alors la moyenne arithmétique applique |z| < κ sur un domaine convexe.Il nous faut, pour la preuve du théorème, le lemme suivant.LEMME. Si la function
The Asymptotic behaviour Of μ(z, β, α)
Authors: Wyman M. , R.Wong  
Published: December 2010

The function μ(z, β, α) defined bywhereplays an important role in Volterra's theory of convolution-logarithms, and also in the Paley-Wiener inversion formula for the Laplace transformation. These and other properties of pi are briefly described in (3).Our aim in this paper is to find the asymptotic behaviour of μ as |z| < ∞, a result which, as far as we are aware, has not been obtained. It is of interest to note that the procedure to be used also gives, with some minor modification, the asymptotic behaviour of μ as z → 0, a result that is well known (3, p. 219). When this behaviour becomes known, it becomes possible to write a significant generalization of Watson's Lemma.In 1906, Barnes (1) published a significant paper containing many asymptotic results of major importance. It is of passing interest to note that this paper contains a general theorem from which the result of Watson's Lemma, published in 1918, can easily be obtained. It would seem that this latter result is misnamed in mathematical literature, and might well be called Barnes’ Lemma.