bluuebunny/crossref_metadata_2025 huggingface dataset:
A Generalization of Difference Sets
Authors: McEliece Robert J.
Published: December 2010

A(v, k,λ)difference set Dis a set ofkdistinct residues{a1, a2,… ,ak} modulovsuch that every residueb ≢0 (modv)can be expressed in exactly λ ways in the formb≡ai— aj(modv).With each difference set we may associate a binary periodic sequence (s1, s2, …) withsi= 1 ifi(mod v) is inD,andsi= 0 otherwise. Since this sequence is periodic of periodv,we need only consider one cycle from the sequence. Such cycles we agree to call (binary)difference cycles.Difference cycles (equivalently, difference sets) have been studied intensively (2, 4). They have important applications to digital communications, mainly because they have2-level autocorrelation.In this paper we shall point out certain other (equivalent) properties of difference cycles which seem susceptible to immediate generalization, but show that these generalizations are vacuous.
On Partitions of an Equilateral Triangle
Authors: Graham R. L.
Published: December 2010

Let T denote a closed unit equilateral triangle. For a fixed integer n, let dn denote the infimum of all those x for which it is possible to partition T into n subsets, each subset having a diameter not exceeding x. We recall that the diameter of a plane set A is given by

where ρ (a, b) is the Euclidean distance between a and b.

In this note we determined dn for some small values of n. Typical values of dn are given in Table I. These values were obtained by three methods. As would be expected, as the value of n increases, the complexity of the argument needed to obtain dn also increases. We begin with the simplest case.
On the Number of Spheres Which can Hide a Given Sphere
Authors: Heppes Aladár
Published: December 2010

Several years ago H. Hornich suggested the following problem: find the minimal number of unit spheres which can hide a unit sphere in the sense that each ray emanating from the centre of that sphere meets at least one of the hiding spheres, with no two of the spheres overlapping. We shall call any set of spheres which hide a given unit sphere acloud.

The first result concerning this and related questions can be found in a paper of Fejes Tóth (4 ; see also 5, 7, 8, 6, and 1 ). With respect to the original problem, Fejes Tóth has given a lower estimate for the minimal numberNof the spheres of a cloud.
Extensions of I-Bisimple Semigroups
Authors: Warne R. J.
Published: December 2010

A bisimple semigroup S is called I-bisimple if Es, the set of idempotents of S, with its natural order is order-isomorphic to I, the set of integers, under the reverse of the usual order. In (9), the author completely determined the structure of I-bisimple semigroups mod groups; in this paper, he also gave an isomorphism theorem, a homomorphism theorem, an explicit determination of the maximal group homomorphic image, and a complete determination of the congruences for these semigroups.
The Hilbert Transform on Rearrangement-Invariant Spaces
Authors: Boyd D. W.
Published: December 2010

The purpose of this paper is to investigate conditions under which the Hilbert transform defines a bounded linear operator from a given function space into itself. The spaces with which we deal have the property of rearrangement-invariance which is defined in §1. This class of spaces includes the Lebesgue, Orlicz, and Lorentz spaces.
Common Fixed Points of Commuting Monotone Mappings
Authors: Wong James S. W.
Published: December 2010

We are concerned here with the existence of fixed or common fixed points of commuting monotone self-mappings of a partially ordered set into itself. Let X be a partially ordered set. A self-mapping ƒ of X into itself is called an isotone mapping if x ⩾ y implies ƒ(x) ⩾ ƒ(y). Similarly, a self-mapping ƒ of X into itself is called an antitone mapping if x ⩾ y implies ƒ(x) ⩽ ƒ(y). An element X0 ∈ X is called well-ordered complete if every well-ordered subset with x0 as its first element has a supremum. An element x0 ∈ X is called chain-complete if every non-empty chain C ⊆ X such that x ⩾ x0 for all x ∈ C, has a supremum. X is called a well-ordered-complete semi-lattice if every non-empty well-ordered subset has a supremum. X is called a complete semi-lattice if every non-empty subset of X has a supremum.
Lower Bounds for Solutions of Parabolic Differential Inequalities
Authors: Ogawa Hajimu
Published: December 2010

Let P be the parabolic differential operator

where E is a linear elliptic operator of second order on D × [0, ∞), D being a bounded domain in Rn. The asymptotic behaviour of solutions u(x, t) of differential inequalities of the form

1

has been investigated by Protter (4). He found conditions on the functions ƒ and g under which solutions of (1), vanishing on the boundary of D and tending to zero with sufficient rapidity as t → ∞, vanish identically for all t ⩾ 0. Similar results have been found by Lees (1) for parabolic differential inequalities in Hilbert space.
On Homotopic Harmonic Maps
Authors: Hartman Philip
Published: December 2010

Let M, M′ be C∞ Riemann manifolds such that

(1.0) M is compact;

(1.1) M′ is complete and its sectional curvatures are non-positive.

In terms of local coordinates x = (x1, … , xn) on M and y = (y1, … , ym) on M′, let the respective Riemann elements of arc-length be

and Γijk, Γ′αβγ be the corresponding Christoffel symbols. When there is no danger of confusion, x (or y) will represent a point of M (or M′) or its coordinates in some local coordinate system.
On Meromorphic Operators, II
Authors: Caradus S. R.
Published: December 2010

This paper forms a continuation of (1), extending the concept of a meromorphic operator to not necessarily bounded, closed linear operators in complex Banach space. Let T denote such an operator with range and domain in Banach space X. We shall study the class of such operators T where λ = 0 and λ = ∞ are the only allowable points of accumulation of σ(T) and every isolated point of σ(T) is a pole of Rλ(T). We shall write (0, ∞) to represent the class of such operators. If λ = 0 (λ = ∞) is the only allowable point of accumulation of σ(T), we shall write (0) ((∞)) to denote the corresponding class of operators.
Direct Products of Normed Linear Spaces
Authors: Jones William B.
Published: December 2010

In this paper we shall study properties of a locally convex space (l.c.s.) which guarantee that it is a direct product of normed linear spaces or Banach spaces. The conditions will be given both as properties of the original space itself and as properties of the dual, and will take the form of a completeness condition and the existence of sub-basic sets of pseudo-norms with certain properties (a set of pseudo-norms is basic if the set of unit balls of its members is a base of neighbourhoods of 0.
Rational Surfaces with Exceptional Unodes
Authors: Val Patrick du
Published: December 2010

Many years ago, I defined (8) three types of exceptional unode on an algebraic surface, which I called U*8, U*9, U*10, corresponding, on a non-singular model of the surface, to sets of six, seven, and eight rational curves, each of grade — 2, with the intersection patterns represented by the Coxeter-Dynkin graphs now usually known as E6, E7, E8:

where each dot represents a curve, and linked dots intersecting curves. In each case we shall denote the curves in the horizontal sequence by S1, s2, … from left to right, and the extra curve meeting s3 by s*.
On Order in a Plane
Authors: Forder H. G.
Published: December 2010

When a set of axioms is laid down as the basis of any mathematical doctrine, it must be proved that this set never leads to a contradiction. In this note we turn the question around. A set of axioms is given and we wish to adjoin an axiom of a specified type. How far does the demand of non-contradiction limit the choice of the new axiom?
Orthogonal Matrices with Zero Diagonal
Authors: Goethals J. M. , Seidel J. J.
Published: December 2010

The central problem in the present paper is the construction of symmetric and of skew-symmetric ( = skew) matricesCof orderv,with diagonal elements 0 and other elements + 1 or — 1, satisfying

The following necessary conditions are known:v≡ 2 (mod 4) and

aandbintegers, for symmetric matricesC(Belevitch (1, 2), Raghavarao (14)), andv =2 orv≡ 0 (mod 4) for skew matricesC.
Some Remarks on Semigroup Presentations
Authors: Neumann B. H.
Published: December 2010

Let a semigroupAbe given by generators a1, a2, … , adand defining relationsu1= v1, u2= v2, … ,ue=vebetween these generators, theui,vibeing words in the generators. We then have apresentationofA, and write

The same generators with the same relations can also be interpreted as the presentation of a group, for which we write
Collineations, Correlations, Polarities, and Conics
Authors: Rigby J. F.
Published: December 2010

It is well known that planes of characteristic 2 behave differently from other Pappian projective planes. For this reason their detailed properties are usually ignored in books on synthetic projective geometry, especially when conies are being discussed. This can give rise to the misleading impression that planes of characteristic 2 are more difficult to deal with, while a cursory introduction to conies in such planes (e.g. Theorem 4.3) may suggest that the notion of “pole and polar” no longer exists.
On Differentiable Arcs and Curves, VI: Singular Osculating Spaces of Curves of Order n + 1 in Projective n-Space
Authors: Scherk Peter
Published: December 2010

A closed curve Kn+1 of order n + 1 in real projective n-space Rn has a maximum number of n + 1 points in common with any (n — 1)-space. These curves are subjected to certain differentiability assumptions which make it possible to describe their singular points and to provide them with multiplicities in analogy with algebraic geometry.
On Cauchy's Lemma Concerning Convex Polygons
Authors: Schoenberg I. J. , Zaremba S. K.
Published: December 2010

One of the authors having just presented Cauchy's theorem on convex polyhedra to a class on “Convexity” at the University of Wisconsin, a discussion on Cauchy's lemma on convex polygons led to an exchange of letters which are here reproduced. The three letters are independently readable. Two new proofs of Cauchy's lemma are given, of which the second (§3) is very short.
Finite Incidence Structures with Orthogonality
Authors: Sherk F. A.
Published: December 2010

An incidence structure consists of two sets of elements, called points and blocks, together with a relation, called incidence, between elements of the two sets. Well-known examples are inversive planes, in which the blocks are circles, and projective and affine planes, in which the blocks are lines. Thus in various examples of incidence structures, the blocks may have various interpretations. Very shortly, however, we shall impose a condition (Axiom A) which ensures that the blocks behave like lines. In anticipation of this, we shall refer to the set of blocks as the set of lines. Also, we shall employ the usual terminology of incidence, such as “lies on,” “passes through,” “meet,” “join.” etc.
A Contribution to Chronogeometry
Authors: Alexandrov A. D.
Published: December 2010

This paper deals with systems of pairwise equal and parallel cones in the affine n-space En, i.e. cones of a system are obtained one from another by means of translations. This subject is closely connected with the geometrical interpretation of the theory of relativity, so one may say that it belongs to “elementary chronogeometry.” The term “chronogeometry,” which is due, it seems, to A. D. Fokker, means the relativistic theory of space-time.
Regular Skew Polyhedra in Hyperbolic Three-Space
Authors: Garner Cyril W. L.
Published: December 2010

The study of regular skew polyhedra was initiated in 1926 by Petrie's discovery of two infinite polyhedra in Euclidean three-space E3 which were free of false vertices; the only other regular skew polyhedron in E3 was found by Coxeter (1, pp. 33-34). The simplest of these is denoted {4, 6 | 4} and is derived from the space-filling of cubes by omitting half the faces.
On a Geometrical Theorem in Exterior Algebra
Authors: Pedoe Daniel
Published: December 2010

In this paper we shall give necessary and sufficient conditions for three lines, passing respectively through the vertices of a proper triangle PQR in the real Euclidean plane, to be concurrent. Of course, the theorem of Ceva deals with this problem, but it is useful to have a criterion which involves only vectors localized at a point O of the plane, and the exterior products of these vectors. Applications are made to theorems which are not easily proved by other methods.
Geometric Models of Certain Genetic Processes
Authors: Yaglom I. M.
Published: December 2010

Classical genetics studies the production of the descendants of a pair of parents with certain genetic characteristics. In the simplest case the genetic type of an individual is determined by a pair of genes each of which can be of one of two types, G and g. A given individual can have these genes in the combination GG (dominant person), or Gg (genetically not different from the combination gG; hybrid individual), or, finally, gg (recessive individual); GG- and Gg-individuals do not differ in appearance from one another, but a gg-individual is different from them.
On Quasi-Metrizability
Authors: Sion M. , Zelmer G.
Published: December 2010

The notion of a quasi-metric was introduced by Wilson (7) and has been studied by himself, Albert (1), and Ribeiro (6) among others. In this paper, we extend and unify some of their work, and connect it with results of Csaszar (2) and Pervin (4, 5) on quasi-uniformities.
Mixed Abelian Groups
Authors: Oppelt John A.
Published: December 2010

The difficulties encountered in the theory of mixed Abelian groups can become decidedly less complex, if it is possible to reduce the question to mixed groups whose torsion subgroup is 𝒫-primary. Call such a group a p-mixed group. In §1 we show that the splitting problem for a mixed group is reducible to the same problem for certain associated 𝒫-mixed groups. In §2 we look at groups which are a direct sum of 𝒫-mixed groups.
Two Term Conditions in π Exact Couples
Authors: Dyer Micheal N.
Published: December 2010

In achieving his celebrated results on the homology groups of fibre spaces, J. P. Serre used the exact couple of a fibring defined by J. Leray. One of his main tools was the so-called two-term condition on the E2 term of this exact couple, which, if satisfied, yielded exact sequences, such as those of Gysin and Wang (see (5), Chapter IX). H. Fédérer, in (3), defined an exact couple (X, F, v) on the mapping space M(X, Y) = {ƒ:X → Y|X, Y are spaces and ƒ is continuous} with the compact-open topology, where X is a locally finite CW complex and Y is arc-connected and n-simple for all n.
Commutativity for Matrices of Quaternions
Authors: Carlson R. E. , Cullen C. G.
Published: December 2010

For any ring we shall denote by the ring of all n × n matrices with elements from and by the set of all polynomials in x with coefficients from .

will denote the non-commutative four-dimensional division algebra of real quaternions with 1, i1, i2, i3 as generators
Axioms for Absolute Geometry
Authors: Rigby J. F.
Published: December 2010

The axioms of Euclidean geometry may be divided into four groups: the axioms of order, the axioms of congruence, the axiom of continuity, and the Euclidean axiom of parallelism (6). If we omit this last axiom, the remaining axioms give either Euclidean or hyperbolic geometry. Many important theorems can be proved if we assume only the axioms of order and congruence, and the name absolute geometry is given to geometry in which we assume only these axioms. In this paper we investigate what can be proved using congruence axioms that are weaker than those used previously.
Anneau De Fractions D'un J-Anneau
Authors: Djabali M.
Published: December 2010

Nous nous proposons d'étudier une question posée par L. Lesieur et R. Croisot (12). Un G-anneau possède un anneau de fractions à gauche semi-simple qui est son enveloppe injective. Un J-anneau possède-t-il un anneau de fractions qui soit un sous-anneau de son enveloppe injective ? Nous donnerons plusieurs exemples de J-anneaux qui ne possèdent pas d'anneau de fractions. Cependant nous donnerons une réponse positive dans un certain nombre de cas.
A Tauberian Theorem and Analogues of the Prime Number Theorem
Authors: Davison T. M. K.
Published: December 2010

In 1945 Ingham (3) proved the following Tauberian theorem: if ƒ is a non-decreasing, non-negative function on [1, ∞) and

1

then ƒ(x) ∼ cx. His proof is based on the non-vanishing of the Riemann zeta-function, ζ (s), on the line , and uses Pitt's form of Wiener's Tauberian theorem; (see, e.g., 5, Theorem 109, p. 211).
Semi-Homomorphisms of Groups
Authors: Herstein I. N.
Published: December 2010

A mapping ϕ from one group, G, into another, H, is said to be a semi-homomorphism of G if ϕ(aba) = ϕ(a) ϕ(b) ϕa) for all a, b ∊G. Clearly any homomorphism or anti-homomorphism is a semi-homomorphism; the converse, however, need not be true in general. It is perfectly clear what one intends by a semi-isomorphism or semi-automorphism.

Our purpose here is to show that for a rather general situation a semi-homomorphism turns out to be a homomorphism or an anti-homomorphism. In (2) we proved that any semi-automorphism of a simple group which contains an element of order 4 must automatically be either an automorphism or an anti-automorphism.
The Classification of Algebras by Dominant Dimension
Authors: Mueller Bruno J.
Published: December 2010

Nakayama proposed to classify finite-dimensional algebras R over a field according to how long an exact sequence

of projective and injective R-R-bimodules Xi they allow. He conjectured that if there exists an infinite sequence of this type, then R must be quasi-Frobenius; and he proved this when R is generalized uniserial (17).
Homology of Deleted Products of Contractible 2-Dimensional Polyhedra. I
Authors: Patty C. W.
Published: December 2010

If X is a space and , the kth deleted product space Xk* of X is the topological product X ⨯ X ⨯ . . . ⨯ X of copies of X minus the set of all points of the form (x, x, . . . , x), where x ∊ X. In (4), the author shows that the homology groups of Xk*, where X is a tree, produce as much information about trees as counting the orders of vertices.
Rings with Involution and Polynomial Identities
Authors: Baxter W. E. , Martindale III W. S.
Published: December 2010

An involution * of a ring A is a one-one additive mapping of A onto itself such that (xy)* = y*x* and x** = x for all x, y ∊ A. If A is an algebra over a field Φ, one makes the additional requirement that (λx)* = λx* for all λ ∊ Φ, x ∊ A. S will generally denote the set of symmetric elements s* = s, K the set of skew elements , and Z the centre of A.
Closed Maps and Paracompact Spaces
Authors: Shapiro H. L.
Published: December 2010

Let ƒ be a map from a topological space X into a topological space F. We say that ƒ is proper in case ƒ is closed continuous and ƒ-1(y) is compact for all y ∊ Y. Proper maps have been extensively studied, see for example (3, Chapter I, §10) or (6). (The definition of a proper map given above is different from, but equivalent to, that given by Bourbaki in (3). In (6) only surjective proper maps are considered and these maps are called ƒitting maps.) It is known that if ƒ is a proper map, then X is compact if and only if ƒ(X) is compact, and X is paracompact if and only if ƒ(X) is paracompact.
Natural Partial Orders
Authors: Dean R. A. , Keller Gordon
Published: December 2010

Let n be an ordinal. A partial ordering P of the ordinals T = T(n) = {w: w < n} is called natural if x P y implies x ⩽ y.

A natural partial ordering, hereafter abbreviated NPO, of T(n) is thus a coarsening of the natural total ordering of the ordinals. Every partial ordering of a finite set 5 is isomorphic to a natural partial ordering. This is a consequence of the theorem of Szpielrajn (5) which states that every partial ordering of a set may be refined to a total ordering. In this paper we consider only natural partial orderings. In the first section we obtain theorems about the lattice of all NPO's of T(n).
On Polynomial Algebras and Free Algebras
Authors: Grätzer George
Published: December 2010

It is well known that given the polynomial algebra (for definitions, see §2), an algebra of type τ, and a sequence a of elements of , one can define a congruence relation θa of such that the factor algebra is isomorphic to the subalgebra of generated by a, and the isomorphism is given in a very simple way.
On The Determination of Sets by Sets of Sums of Fixed Order
Authors: Ewell John A.
Published: December 2010

The present investigation is based on two papers: “On the determination of numbers by their sums of a fixed order,” by J. L. Self ridge and E. G. Straus (4), and “On the determination of sets by the sets of sums of a certain order,” by B. Gordon, A. S. Fraenkel, and E. G. Straus (2).

First of all, we explain the terms implicit in the above titles. Throughout these considerations we use the term “set” to mean “a totality having possible multiplicities,” so that two sets will be counted as equal if, and only if, they have the same elements with identical multiplicities. In the most general sense the term “numbers” of (4) can be replaced by “elements of any given torsioniree Abelian group.”
Union and Extension of Arcs of Cyclic Order Three
Authors: Singh K. D. , Lane N. D.
Published: December 2010

In (2) Lane and Scherk discussed differentiate points of arcs in the conformai (inversive) plane. Arcs A3 of cyclic order three were discussed in (3; 4). In the present note we give necessary and sufficient conditions for the union of two A3's to be an A3 (Theorem 1), and for an A3 to be extensible to a larger one (Theorem 2). The related problem of extending arcs in projective n-space was dealt with by Haupt in (1) and Sauter in (5; 6).
Zur Darstellungstheorie Von Kranzprodukten
Authors: Kerber Adalbert
Published: December 2010

Einer Anregung I. Schurs folgend, untersuchte W. Specht bereits in seiner Dissertation (10) Gruppen, die unter den neueren Oberbegriff des Kranzproduktes zweier Gruppen fallen: die sogenannten Symmetrien . Allgemeine Kranzprodukte wurden von ihm in einer nachfolgenden Arbeit (11) behandelt; dort wird deren gewöhnliche Darstellungstheorie in aller Ausführlichkeit angegeben.
On the Diameter of a p-Cyclic Strongly Connected Digraph
Authors: Stuart Lynn M.
Published: December 2010

In this paper we follow the notation of (2). In (5), Luce showed, in other terminology, that if d is the diameter of a strongly connected digraph, D, on n vertices with m edges, then

1.1

this inequality being sharp; from (1.1) one may immediately derive sharp upper bounds for d in terms of m and n, this being a generalization of the obvious and well-known inequality

1.2
On Packings of Unequal Spheres in Rn
Authors: Larman D. G.
Published: December 2010

Suppose that a sequence of spheres is packed in order of decreasing diameters into the unit cube In of Rn. In recent work (2), I have shown that for n = 2, there exist positive constants K2, s ( = 0.97) such that the area of has an asymptotic lower bound K2(d(Sm))s. Although the methods used were complicated and possibly only viable in two dimensions, it is intuitively clear that such a result should also be true in higher dimensions.
A Study of Tensors which Characterize a Hypersurface of a Finsler Space
Authors: Brown Gillian M.
Published: December 2010

The literature on Finsler geometry contains more than one definition for the normal curvature vector of a hypersurface and for coefficients of the second fundamental form; see Berwald (1), Davies (3), and Rund (5). In the first case this situation has arisen from the basically different approach to the subject adopted by the authors; Davies, following the locally Euclidean school and Rund the locally Minkowskian theory. In both cases, a comparison of the definitions shows that they are linked by expressions in a vector Mαwhich was introduced in the paper by Rund (7).
On a Combinatorial Result of R. A. Brualdi and M. Newman
Authors: Marcus Marvin , Pierce Stephen
Published: December 2010

LetHbe a subgroup ofSnand letAbe ann-square matrix over a fieldƒ. Following Schur (7) we define the generalized matrix functiondH(A)by
Multiplication in Vector Lattices
Authors: Rice Norman M.
Published: December 2010

B. Z. Vulih has shown (13) how an essentially unique intrinsic multiplication can be defined in a Dedekind complete vector lattice L having a weak order unit. Since this work is available only in Russian, a brief outline is given in § 2 (cf. also the review by E. Hewitt (4), and for details, consult (13) or (11)).
Isomorphisms of Multiplier Algebras
Authors: Gaudry G. I.
Published: December 2010

Suppose that G1 and G2 are two locally compact Hausdorff groups with identity elements e and e’ and with respective left Haar measures dx and dy. Let 1 ≦ p ≦ ∞, and Lp(Gi) be the usual Lebesgue space over Gi formed relative to left Haar measure on Gi. We denote by M(Gi) the space of Radon measures, and by Mbd(Gi) the space of bounded Radon measures on Gi. If a ϵ Gi we write ϵa for the Dirac measure at the point a. Cc(Gi) will denote the space of continuous, complex-valued functions on Gi with compact supports, whilst Cc+ (Gi) will denote that subset of Cc(Gi) consisting of those functions which are real-valued and non-negative.
On the Periodicity of Compositions of Entire Functions. II
Authors: Gross Fred
Published: December 2010

In (1) the author suggested the following research problem. Does there exist a non-periodic entire function ƒ such that ƒƒ is periodic? My aim in this note is to give a partial answer to this question and, more generally, to give a partial solution to the following problem: if ƒ and g are entire functions and ƒ(g) is periodic, what can one say about g? These results also extend a previous result of mine; for details, see (2, Theorem 4). We begin with some simple lemmas.
An Inequality Between Numerical Homotopy Invariants
Authors: Priddis M. J. M.
Published: December 2010

In (1), Berstein and Ganea denned the nilpotency class of a based topological space. For a based topological space X we write nil X for the nilpotency class of the group ΩX in the category of based topological spaces and based homotopy classes. Hilton, in (3), defined the nilpotency class, nil class K of a based semi-simplicial (s.-s.) complex; actually, the restriction of connectedness can be removed. Hence, by using the total singular complex functor S, an invariant (nil class SX) can be defined for a based topological space X.
Fixed Points and Characters in Groups with Non-Coprime Operator Groups
Authors: Isaacs I. M.
Published: December 2010

Let A and H be finite groups with A acting on H, i.e., there is a given, fixed homomorphism A → Aut(H). In this situation, A acts on the set of conjugacy classes of H and also on the set of irreducible characters of H. If A is cyclic, it follows from a lemma of Brauer (see, for instance, 1, 12.1) that the number of fixed points in these two actions are equal and therefore one can conclude that for any A, the number of orbits in the two actions are equal.
Some Tauberian Theorems for the Logarithmic Method of Summability
Authors: Kwee B.
Published: December 2010

The series is said to be summable (L) to s if the sequence {sn}, where sn = a0 + a1 + … + an, is L-convergent to s, i.e., if

If the sequence {sn} is l-convergent to s, i.e., if the sequence {tn}, where

1
The Number of -Coloured Graphs
Authors: Klarner David A.
Published: December 2010

In this paper we describe an algorithm for finding the number of non-isomorphic -coloured graphs with n nodes and e edges. We use Pόlya's fundamental enumeration theorem (in a form similar to that given by de Bruijn (see 1)) which reduces the problem to finding the cycle index for a certain permutation group. Harary (3) followed this same program for bi-coloured graphs, but failed to find the cycle index of the relevant group for general -coloured graphs.
Lifting Disjoint Sets in Vector Lattices
Authors: Conrad Paul
Published: December 2010

A subset {s α| α ϵ △} of a lattice-ordered group (l-group) is disjoint if Sα Λ Sβ= 0 for all α≠ β in △. An l-group G has the lifting property if for each l-ideal S of G and each countable disjoint subset X1, X2, … of G/S one can choose elements 0 ≦ xi ϵ Xi so that x1, x2, … is a disjoint subset of G. In (2) Topping showed by an example, that uncountable sets of disjoint elements cannot necessarily be lifted and asserted (Theorem 8) that each vector lattice has the lifting property. His proof is valid for finite disjoint subsets of G/S, but we show by an example that this is, in general, all that one can establish.
Polytopes, Valuations, and the Euler Relation
Authors: Sallee G. T.
Published: December 2010

By a d-polytope we shall mean a d-dimensional convex polytope. We shall denote a j-dimensional face (or j-face) of a polytope by Fj. Every d-polytope P has proper j-faces for 0 ≦j ≦d — 1 and we shall also say that P is a d-face of itself. Observe that every face of a polytope is again a polytope. The collection of all convex polytopes shall be denoted by .
Groups with Relatively Few Non-Linear Irreducible Characters
Authors: Isaacs I. M. , Passman D. S.
Published: December 2010

In (4), Seitz characterized those finite groups which have exactly one non-linear irreducible character (over the complex numbers). In this paper we are concerned with the general question of what can be deduced about a finite group G if the number of its non-linear irreducible characters m(G) is given. In particular, does the assumption that m(G) is in some sense small when compared with the order |G| impose any restrictions on the structure of G?
Structure Theory for Montgomery-Samelson Fiberings between Manifolds, II
Authors: Antonelli Peter L.
Published: December 2010

Let f: Mn→ Npbe the projection map of an MS-fibering of manifolds β with finite non-empty singular set Aand simply connected total space (see 1). Results of Timourian (10) imply that (n, p) = (4, 3), (8, 5) or (16, 9), while a theorem of Conner (2) yields that #(A), the cardinality of the singular set, is equal to the Euler characteristic of Mn. We give an elementary proof of this fact and, in addition, prove that #(A) is actually determined by bn/2(Mn), the middle betti number of Mn, or what is the same, by bn/2(Np – f(A)). It is then shown that β is topologically the suspension of a (Hopf) sphere bundle when Np is a sphere and bn/2(Mn) = 0. It follows as a corollary that β must also be a suspension when Mn is n/4-connected with vanishing bn/2. Examples where bn/2 is not zero are constructed and we state a couple of conjectures concerning the classification of such objects.
Hereditary Radicals and Derivations of Algebras
Authors: Anderson T.
Published: December 2010

In the Amitsur-Kurosch theory of radicals in rings (2), an important problem is to determine the relationship between the radical of a ring and the radical of each of its ideals. The first result on this problem was by Amitsur who proved that if β is a hereditary radical in the sense that ideals of β-radical rings are β-radical, then for each associative ring R and ideal I of R, β(I) = I ∩ β(R), where β(R) denotes the β-radical of R; see (2).

Later, Suliński, Divinsky, and the author proved that if β is any radical and R is an associative or alternative ring, then β(I) ⊆ I∩ β(R) for each ideal I of R; see (3). Since every hereditary radical β has the property β(I) ⊇ I ∩ β(R), this result provided another proof of Amitsur's theorem and extended that theorem to alternative rings. Of course, this raises the question of whether Amitsur's theorem is true for Lie or Jordan rings.
Generalized Frattini Subgroups of Finite Groups. II
Authors: Beidleman James C.
Published: December 2010

The theory of generalized Frattini subgroups of a finite group is continued in this paper. Several equivalent conditions are given for a proper normal subgroup H of a finite group G to be a generalized Frattini subgroup of G. One such condition on H is that K is nilpotent for each normal subgroup K of G such that K/H is nilpotent. From this result, it follows that the weakly hyper-central normal subgroups of a finite non-nilpotent group G are generalized Frattini subgroups of G.

Let H be a generalized Frattini subgroup of G and let K be a subnormal subgroup of G which properly contains H. Then H is a generalized Frattini subgroup of K.

Let ϕ(G) be the Frattini subgroup of G. Suppose that G/ϕ(G) is nonnilpotent, but every proper subgroup of G/ϕ(G) is nilpotent. Then ϕ(G) is the unique maximal generalized Frattini subgroup of G.
Homotopical Nilpotency of Loop-Spaces
Authors: Hoo C S.
Published: December 2010

In this paper we shall work in the category of countable CW-complexes with base point and base point preserving maps. All homotopies shall also respect base points. For simplicity, we shall frequently use the same symbol for a map and its homotopy class. Given spaces X, Y, we denote the set of homotopy classes of maps from X to Y by [X, Y]. We have an isomorphism τ: [∑X, Y] → [X, Ω Y] taking each map to its adjoint, where ∑ is the suspension functor and Ω is the loop functor. We shall denote τ(1 ∑x) by e′ and τ-1(1Ωx) by e.
Maximal Groups on Which the Permanent is Multiplicative
Authors: Beasley Leroy B.
Published: December 2010

Let Δn be the set of all n × n, non-singular matrices of the form PD, where P is a permutation matrix and D is a diagonal matrix with complex entries. In (1, conjecture 12), Marcus and Mine asked: Is Δn a maximal group on which the permanent function is multiplicative? (that is, per AB = per A per B). The field over which the entries range was not mentioned in the conjecture; however, we assume that the complex number field was intended. Corollary 1 answers this in the affirmative. In fact, Δn is the only maximal group (or semigroup) on which the permanent is multiplicative. Let ρi be the set of all non-zero entries in the ith row and let λj be the set of all non-zero entries in the jth column.
On the Ideal Theory of the Kronecker
Function Ring and the Domain D(X)
Authors: Arnold Jimmy T.
Published: December 2010

Let D be an integrally closed domain with identity having quotient field L. If {Vα} is the set of valuation overrings of D and if A is an ideal of D, then Ã = ∪ αAVα is an ideal of D called the completion of A. If X is an indeterminate over D and f ∈ D[X], then we denote by Af the ideal of D generated by the coefficients of f. The Kronecker function ring DK of D is defined by DK = {f/g| f, g ∈ D[X], Ãf ⊆ Ag} (4, p. 558); and the domain D(X) is defined by D(X) = {f/g| f, g ∈ D[X], Ag = D} (5, p. 17). In this paper we wish to relate the ideal theory of D to that of DK and D(X) for the case in which D is a Prüfer domain, a Dedekind domain, or an almost Dedekind domain.
A Note on the Caradus Class of Bounded
Linear Operators on a Complex Banach Space
Authors: Ruston A. F.
Published: December 2010

1. In a recent paper (1) on meromorphic operators, Caradus introduced the class of bounded linear operators on a complex Banach space X. A bounded linear operator T is put in the class if and only if its spectrum consists of a finite number of poles of the resolvent of T. Equivalently, T is in if and only if it has a rational resolvent (8, p. 314).

Some ten years ago (in May, 1957), I discovered a property of the class g which may be of interest in connection with Caradus' work, and is the subject of the present note.

2. THEOREM. Let X be a complex Banach space. If T belongs to the class, and the linear operator S commutes with every bounded linear operator which commutes with T, then there is a polynomial p such that S = p(T).
A Class of Reflexive Symmetric Bk-Spaces
Authors: Garling D. J. H.
Published: December 2010

We denote by ω the linear space of all sequences of real or complex numbers. A linear subspace of ω is called a sequence space. A sequence space E is a BK-space (9) if it is equipped with a norm under which: first, E is a Banach space and second, each of the coordinate maps x → xi is continuous. Let ∑ be the group of all permutations of Z+ = {1, 2, 3, …}. If x ∈ ω and σ ∈ ∑, the sequence xσ is defined by (xσ)i = xσ(i)). A sequence space E is symmetric if xσ ∈ E whenever x ∈ E and σ ∈ ∑. Accounts of symmetric sequence spaces occur in (3; 7; 8).
A Generalization of Global Class Field Theory
Authors: Seo Tae Kun , Whaples G.
Published: December 2010

Let R be a field of rational functions of one variable over a field of constants R0. Dock Sang Rim (6) has proved that the global reciprocity law in exactly the usual sense holds whenever R0 is an absolutely algebraic quasi-fini te field of characteristic not equal to 0: this was known before only when R0 was a finite field. We shall give another proof of Rim's result by means of a noteworthy generalization of the usual global reciprocity law. Namely, let R0 be a finite field and let F be the set of all fields k contained in some fixed Ralg.clos. and of finite degree over R. The reciprocity law states that there exists a family {fk}, k ∈ F, of functions fk: Ck → G(kabel.clos./k) (where Ck is the idèle class group of k) enjoying certain properties such as the norm transfer law.
Groups with Equal Uniformities
Authors: Ramsay R. T.
Published: December 2010

If G = (G, τ) is a topological group with topology τ, then there is a smallest topology τ* ⊇ τ such that G* = (G, τ*) is a topological group with equal left and right uniformities (1). Bagley and Wu introduced this topology in (1), and studied the relationship between Gand G*. In this paper we prove some additional results concerning G* and groups with equal uniformities in general. The structure of locally compact groups with equal uniformities has been studied extensively. If G is a locally compact connected group, then G has equal uniformities if and only if G ≅ V× K,where F is a vector group and Kis a compact group (5). More generally, every locally compact group with equal uniformities has an open normal subgroup of the form V× K(4).
The mod ℭ Suspension Theorem
Authors: Brown Benson Samuel
Published: December 2010

Our aim in this paper is to prove the general mod ℭ suspension theorem: Suppose that X and Y are CW-complexes,ℭ is a class offinite abelian groups, and that

(i) πi(Y) ∈ℭfor all i < n,

(ii) H*(X; Z) is finitely generated,

(iii) Hi(X;Z) ∈ℭfor all i > k.

Then the suspension homomorphism

is a(mod ℭ) monomorphism for 2 ≦ r ≦ 2n – k – 2 (when r= 1, ker E is a finite group of order d, where Zd∈ ℭ and is a (mod ℭ) epimorphism for 2 ≦ r ≦ 2n – k – 2

The proof is basically the same as the proof of the regular suspension theorem. It depends essentially on (mod ℭ) versions of the Serre exact sequence and of the Whitehead theorem.
A Spectral Sequence for Cohomotopy
Authors: Brown Benson Samuel
Published: December 2010

For a prime number p let be the class of finite abelian groups whose orders are prime to p. For a finitely generated abelian group G, let Gp be the sum of the free and p-primary components of G. Our aim in this paper is to prove the following theorem.

Theorem. Suppose that

(i) Hi(X;Z) = 0 for i > k,

(ii) for i > k – d

Then there exists a spectral sequence with

and the differential is given by
A One-Parameter Subsemigroup which meets many Regular -Classes
Authors: Tymchatyn E. D.
Published: December 2010

We give an example which answers affirmatively the following problem posed by Hofmann and Mostert (1, p. 200):

P. 10. Let S be a compact semigroup with identity 1, zero 0, and totally ordered -class space . Suppose that there is a one-parameter semigroup containing 0 and 1. Can S have any regular -classes aside from D(0) and D(1)?

I am indebted to Professor K. H. Hofmann and Professor P. S. Mostert for their many helpful suggestions.

1. We first give a method of embedding a given semigroup S in a semigroup R so that in the commutative diagram

the bottom map is an isomorphism of partially ordered spaces and some non-regular -class of S is embedded into a regular -class of R.
A Tauberian Theorem for Borel-Type Methods of Summability
Authors: Borwein D.
Published: December 2010

Suppose throughout that α >0, β is real, and Nis a non-negative integer such that αN+ β> 0. A series of complex terms is said to be summable (B, α,β) to l if, as x→ ∞,

where sn= a0 + a1 + … + an.The Borel-type summability method (B, α, β) is regular, i.e., all convergent series are summable (B, α,β) to their natural sums; and (B,1, 1) is the standard Borel exponential method B.

Our aim in this paper is to prove the following Tauberian theorem.

THEOREM. Iƒ

(i) p ≧ – ½, an = o(np), and

(ii) is summable (B, α,β) to l, then the series is summable by the Cesaro method(C, 2p + 1) to l.
Remetrization in Strongly Countabledimensional Spaces
Authors: B. R. Wenner
Published: December 2010

Although the Lebesgue dimension function is topologically invariant, the dimension-theoretic properties of a metric space can sometimes be made clearer by the introduction of a new, topologically equivalent metric. A considerable amount of effort has been devoted to the problem of constructing such metrics; one example of the fruits of this research is the following theorem by Nagata (2, Theorem 5).

In order that dim R ≦ n for a metrizable space R it is necessary and sufficient to be able to define a metric p(x, y) agreeing with the topology of R such that for every ∊ > 0 and for every point x oƒ R,

imply

A metric ρ which satisfies the condition of this theorem is called Nagata's metric (this term was introduced, to the best of the author's knowledge, by Nagami (1, Definition 9.3)).
On a Transformation Group
Authors: Kaul S. K.
Published: December 2010

0. Let Γ denote a group of real linear fractional transformations (the constants defining any element of Γ are real numbers); see (3, § 2, p. 10). Then it is known that Γ is discontinuous if and only if it is discrete (3, Theorem 2F, p. 13).

Now Γ may also be regarded, equivalently, as a group of homeomorphisms of a disc D onto itself; and if Γ is discrete, then, except for elements of finite order, each element of Γ is either of type 1 or type 2 (see Definitions 0.1 and 0.2 below).

We wish to generalize the result quoted above in purely topological terms. Thus, throughout this paper we denote by X a compact metric space with metric d, and by G a topological transformation group on X each element of which, except the identity e, is either of type 1 or type 2. Let L = ﹛a ∈ X: g(a) = a for some g in G — e﹜, and . We assume furthermore that 0 is non-empty.
Some Results Relating the Behaviour of Fourier Transforms Near the Origin and at Infinity
Authors: Nasim C.
Published: December 2010

It is known that under special conditions, Fourier sine transforms and Fourier cosine transforms behave asymptotically like a power of x, either as x → 0 or as x → ∞ or both. For example (3),

where f(x) = x–αϕ(x), 0 < α < 1, and ϕ(x) is of bounded variation in (0, ∞) and Fc(x) is the Fourier cosine transform of f(x). This suggests that other results connecting the behaviour of a function at infinity with the behaviour of its Fourier or Watson transform near the origin might exist. In this paper wre derive various such results. For example, a special case of these results is

where f(x) is the Fourier sine transform of g(x). It should be noted that the Fourier inversion formula fails to give f(+0) directly in this case. Some applications of these results to show the relationships between various forms of known summation formulae are given.
A Summation Formula Involving σk(n), k > 1
Authors: Nasim C.
Published: December 2010

The existence of certain formulae analogous to Poisson's summation formula (9, pp. 60-64),

where αβ = 2π, α > 0, and Fc(x) is the Fourier cosine transform of f(x), but involving number-theoretic functions as coefficients, was first demonstrated by Voronoï (10) in 1904. He proved that

where r(n) is an arithmetic function,/(x) is continuous in (a, b) and a(x) and i?(x) are analytic functions dependent on τ(n). Later, numerous papers were published by various authors giving formulae of this type involving d(n), the number of divisors of n (3), and rp(n), the number of ways of expressing n as the sum of p squares of integers (8).

In 1937, Ferrar (4) developed a general theory of summation formulae, using complex analysis. Around that time, Guinand (5) also published papers where he developed the general theory from a different point of view. He applied the theory of mean convergence for the transforms of class L2(0, ∞ ). Later in 1950, Bochner (1) gave a general summation formula.
Finite Groups whose Powers have no Countably Infinite Factor Groups
Authors: Billis M. J.
Published: December 2010

Let P be the class of all finite groups G whose powers GI have no countably infinite factor groups. Neumann and Yamamuro (1) proved that if G is a finite non-Abelian simple group, then G ∈ P. We generalize this result by proving the following theorem.

THEOREM. A finite group G ∈ P if and only if G is perfect

2. Inheritance properties ofP.

P1. If G ∈ P and N is normal in G, then G/N ∈ P.

Proof. Since (G/N)I is isomorphic to GI/NI it is clear that factor groups of (G/N)I are isomorphic to factor groups of GI, and hence finite or uncountable.

P2. If G = HK, where H ∈ P and K ∈ P, then G ∈ P.

Proof. We show that homomorphic images of GI are either finite or uncountable. Let ϕ be a homomorphism of GI. Then GIϕ = (HK)Iϕ = (HI/KI)ϕ = (HIϕ) (KIϕ). Since HIϕ and KIϕ must be finite or uncountable, the conclusion follows.
On the Irreducible Lattices of Orders
Authors: Roggenkamp Klaus W.
Published: December 2010

We shall use the following notation:

R = Dedekind domain;

K = quotient field of R;

Rp = ring of p-adic integers in K, p being a prime ideal in R;

A = finite-dimensional separable k-algebra;

G – R-order in A (for the definition cf. (3)).

All modules that occur are assumed to be finitely generated unitary left modules, unless otherwise specified. By a G-lattice we mean a G-module which is torsion-free as R-module. A G-lattice is called irreducible if it does not contain a proper G-submodule of smaller R-rank. If p is a prime ideal in R we shall write Gp = Rp ⊗RG; Mp = RP⊗RM for a G-lattice M, and KM = K ⊗R M. Two G-lattices M and N are said to lie in the same genus (notation M ∨ N) if Mp ≌ Np for every prime ideal p in R.

For any A-module L, let S(L) be the collection of G-lattices M, for which KM ≌ L. Suppose that S(L) splits into rg(L) genera, and into ri(L) classes under G-isomorphism. Maranda (6) has shown: If L is an absolutely irreducible A-module, then
Mean Value and Limit Theorems for Generalized Matrix Functions
Authors: Nikolai Paul J.
Published: December 2010

Let A = [aij] denote an n-square matrix with entries in the field of complex numbers. Denote by H a subgroup of Sn, the symmetric group on the integers 1, …, n, and by a character of degree 1 on H. Then

is the generalized matrix function of A associated with H and x; e.g., if H = Sn and χ = 1, then the permanent function. If the sequences ω = (ω1, …, ωm) and ϒ = (ϒ1, …, ϒm) are m-selections, m ≦ w, of integers 1, …, n, then A [ω| ϒ] denotes the m-square generalized submatrix [aωiϒj], i, j = 1, …, m, of the n-square matrix A. If ω is an increasing m-combination, then A [ω|ω] is an m-square principal submatrix of A.
Archimedean Closures in Lattice-Ordered Groups
Authors: Byrd Richard D.
Published: December 2010

Conrad (10) and Wolfenstein (15; 16) have introduced the notion of an archimedean extension (a-extension) of a lattice-ordered group (l-group). In this note the class of l-groups that possess a plenary subset of regular subgroups which are normal in the convex l-subgroups that cover them are studied. It is shown in § 3 (Corollary 3.4) that the class is closed with respect to a-extensions and (Corollary 3.7) that each member of the class has an a-closure. This extends (6, p. 324, Corollary II; 10, Theorems 3.2 and 4.2; 15, Theorem 1) and gives a partial answer to (10, p. 159, Question 1). The key to proving both of these results is Theorem 3.3, which asserts that if a regular subgroup is normal in the convex l-subgroup that covers it, then this property is preserved by a-extensions.