bluuebunny/crossref_metadata_2025 huggingface dataset:
Disjoint Transversals of Subsets
Authors: Higgins P. J.
Published: December 2010

Let A1, A2,… , An be a finite collection of subsets (not necessarily distinct) of a set A. By a transversal of A1, A2,… , An we shall mean a set of n distinct elements a1, a2,… , an of A such that, for some permutation i1 i2, … , in of the integers 1, 2, … , n,

More generally, we shall say that the set {a1, a2, … , ar}, (r ≤ n) is a partial transversal oi A1, A2, … An of length r if (i) a1, a2, … , ar are distinct elements of A and (ii) there exists a set of distinct integers i1, i2… , irsuch that

A well-known theorem of P. Hall (2) states that the sets A1, A2… , An have a transversal (of length n) if, and only if, every k of them contain collectively at least k distinct elements (k = 1, 2, … , n).
Sur les Representations Unitaires des Groupes de Lie Nilpotents. IV
Authors: Dixmier Jacques
Published: December 2010

Soit n un entier ≥ 1. On notera Mn l'ensemble des matrices carrées d'ordre n à éléments réels, et Gn le groupe des x = (εJk) ∈ Mn tels que εJk = 0 pour 1 ≤ j ≤ k ≤ n, εJk; = 1 pour 1 εJkj εJkn. Le groupe Gn est un groupe de Lie nilpotent simplement connexe, dont l'algèbre de Lie s'identifie à l'ensemble Qn des x = (εJk) ∈ Mn tels que εJk = 0 pour 1 ≤ j ≤ k ≤ n. Nous allons déterminer:

(1°) le centre de l'algèbre enveloppante de gn;

(2°) la série “principale” de représentations unitaires irréductibles de Gn; (la recherche de toutes les représentations unitaires irréductibles de Gn ne semble pas facile) ;

(3°) la formule de Plancherel pour Gn ;

(4°) les caractères globaux (au sens de (5)) des représentations de la série principale.
A Note on Hyperconvexity in Riemannian Manifolds
Authors: Nijenhuis Albert
Published: December 2010

Let M denote a connected Riemannian manifold of class C3, with positive definite C2 metric. The curvature tensor then exists, and is continuous.

By a classical theorem of J. H. C. Whitehead (1), every point x of M has the property that all sufficiently small spherical neighbourhoods V of x are convex; that is, (i) to every y,z ∈ V there is one and only one geodesic segment yz in M which is the shortest path joining them:f:([0, 1]) → M,f(0) = y, f(1) = z; and (ii) this segment yz lies entirely in V:f([0, 1]) V; (iii) if f is parametrized proportional to arc length, then f(t) is a C2 function of y, t, and z.

Let V be a convex set in M; and let y1 y2, Z1, z2 ∈ V.
Some Principles Underlying The Construction of Measures
Authors: Hayes Charles A.
Published: December 2010

Measures may be obtained from suitable non-negative valued functions in a number of ways. It is the purpose of this paper to present an abstract formulation of certain principles which may be used to construct measures, and to show that the various methods most frequently encountered in the literature are in fact all special applications of these principles.

The basic requirements A1-A4 are set forth in § 2, and it is there shown that a measure can be denned when these are fulfilled. These requirements are satisfied in every case known to the writer. Usually, however, conditions stronger than A1-A4 hold, and it is these extra restrictions which yield information on the class of measurable sets and other matters. In §§ 3, 4, and 5 certain abstracts form of such restrictions are considered, and results are derived thereforom. The paper concludes with an analysis of how a number of measures occur as special cases of the theory.
Pencils of Null Polarities
Authors: Schuster Seymour
Published: December 2010

A theorem due to von Staudt states that a null polarity in complex projective space of three dimensions is determined by a selfpolar skew pentagon. By allowing an element of the self-polar pentagon to vary in a suitable manner we can arrive at a family of ∞1 null polarities, which we term a pencil of null polarities. Each polarity of the pencil distinguishes a linear complex as the class of self-polar lines. Thus, associated with the pencil is a family of ∞1 linear complexes, which we term a pencil of linear complexes.

It is the purpose of this paper to continue an earlier investigation of pencils of polarities (2), by applying analogous techniques to the study of pencils of null polarities and pencils of linear complexes.

Since it develops that the lines common to all linear complexes of a pencil are the lines of a linear congruence, the central question has been: How many of the different types of linear congruences can be achieved in this manner? Happily, it can be reported that the classification of pencils of null polarities yields all of the three types of linear congruences (4, pp. 140-141).
Sur Quelques Series de Lambert et de Dirichlet
Authors: Touchard Jacques
Published: December 2010

Nous nous occupons dans ce travail de séries dont voici le type le plus simple

et de leurs analogues lorsqu'on remplace xn par n-s et yn par n-s. Faute d'une
autre désignation, nous avons cru pouvoir appeler ces séries respectivement séries de Lambert et séries de Dirichlet. Une série plus générale que la précédente
est

où les ak sont des constantes et où g(x) est un polynôme dont les coefficients sont des symboles de Jacobi (1, pp. 132-40; 4 pp 361-9).

possédant la période 2ω. Les expressions

ont des propriétés multiplicatives analogues à celles des sommes de Ramanujan (6) d'où l'on peut déduire deux propositions concernant les racines de g(x).
A Cosine Functional Equation in Hilbert Space
Authors: Kurepa Svetozar
Published: December 2010

Throughout this paper R denotes the set of all real numbers, m(K) the Lebesgue measure of K ⊆ R, H a Hilbert space, L(H) the set of all linear continuous mappings of H into H, endowed with the usual structure of a Banach space.

We consider the mapping F of the set R into L(H) such that

holds for all x, y ∈ R. In (2) we have solved this equation under the assumption that H is of finite dimension. In this paper we prove that a weak measurability of F implies its weak continuity in the case of separable Hilbert space. In Theorem 2 we prove that every weakly continuous solution of (1) in the set of normal transformations has the form F(x) = cos (xN), where the normal transformation N does not depend on x.
On Quadruple Systems
Authors: Hanani Haim
Published: December 2010

Given a set E of n elements we denote by S(l, m, n), (l ≤ m ≤ n) a system of subsets of E, having m elements each, such that every subset of E having l elements is contained in exactly one set of the system S (l, m, n).

It is clear (3), that a necessary condition for the existence of S (l, m, n) is that

1

is the number of elements of S(l, m, n) and

is the number of those elements of S (l, m, n) which contain h fixed elements of E.

It is known that condition (1) is not sufficient for S(l, m, n) to exist. It has been proved that no finite projective geometry exists with 7 points on every line. This implies non-existence of S(2, 7, 43).
On the Composition of Balanced Incomplete Block Designs
Authors: Bose R. C. , Shrikhande S. S.
Published: December 2010

The object of this paper is to develop a method of constructing balanced incomplete block designs. It consists in utilizing the existence of two balanced incomplete block designs to obtain another such design by what may be called the method of composition.

1. Preliminary results on orthogonal arrays and balanced incomplete block designs. Consider a matrixA =(aij) of k rows andNcolumns, where eachaijrepresents one of the integers 1, 2, … ,s. Consider all t-rowed submatrices ofNcolumns, which can be formed from this array,t≤k.Each column of any Crowed submatrix can be regarded as an orderedt-plet.
On Null-Recurrent Markov Chains
Authors: Lamperti John
Published: December 2010

Throughout this paper, the symbol P = [Pij] will represent the transition probability matrix of an irreducible, null-recurrent Markov process in discrete time. Explanation of this terminology and basic facts about such chains may be found in (6, ch. 15). It is known (3) that for each such matrix P there is a unique (except for a positive scalar multiple) positive vector Q = {qi} such that QP = Q, or

1

this vector is often called the "invariant measure" of the Markov chain.

The first problem to be considered in this paper is that of determining for which vectors U(0) = {μi(0)} the vectors U(n) converge, or are summable, to the invariant measure Q, where U(n) = U(0)Pn has components

2

In § 2, this problem is attacked for general P. The main result is a negative one, and shows how to form U(0) for which U(n) will not be (termwise) Abel summable.
Lattice Octahedra
Authors: Mordell L. J.
Published: December 2010

Let Ai, A2, … , An be n linearly independent points in n-dimensional Euclidean space of a lattice Λ. The points ± A1, ±A2, . . , ±An define a closed n-dimensional octahedron (or “cross poly tope“) K with centre at the origin O. Our problem is to find a basis for the lattices Λ which have no points in K except ±A1, ±A2, … , ±An.

Let the position of a point P in space be defined vectorially by

1

where the p are real numbers. We have the following results.

When n = 2, it is well known that a basis is

2

When n = 3, Minkowski (1) proved that there are two types of lattices, with respective bases

3

When n = 4, there are six essentially different bases typified by A1, A2, A3 and one of

4

In all expressions of this kind, the signs are independent of each other and of any other signs. This result is a restatement of a result by Brunngraber (2) and a proof is given by Wolff (3).
A Note on Continued Fractions
Authors: Oppenheim A.
Published: December 2010

Any real number y leads to a continued fraction of the type

(1)

where ai, bi are integers which satisfy the inequalities

(2)

by means of the algorithm

(3)

the a's being assigned positive integers. The process terminates for rational y; the last denominator bk satisfying bk ≥ ak + 1. For irrational y, the process does not terminate. For a preassigned set of numerators ai ≥ 1, this C.F. development of y is unique; its value being y.

Bankier and Leighton (1) call such fractions (1), which satisfy (2), proper continued fractions. Among other questions, they studied the problem of expanding quadratic surds in periodic continued fractions. They state that “it is well-known that not only does every periodic regular continued fraction represent a quadratic irrational, but the regular continued fraction expansion of a quadratic irrational is periodic.
The Equivalence of Two Extremum Problems
Authors: Kiefer J. , Wolfowitz J.
Published: December 2010

Let f1 , …, fk be linearly independent real functions on a space X, such that the range R of (f1, …, fk) is a compact set in k dimensional Euclidean space. (This will happen, for example, if the fi are continuous and X is a compact topological space.) Let S be any Borel field of subsets of X which includes X and all sets which consist of a finite number of points, and let C = {ε} be any class of probability measures on S which includes all probability measures with finite support (that is, which assign probability one to a set consisting of a finite number of points), and which are such that

is defined. In all that follows we consider only probability measures ε which are in C.
Traces of Matrices of Zeros and Ones
Authors: Ryser H. J.
Published: December 2010

This paper continues the study appearing in (9) and (10) of the combinatorial properties of a matrix A of m rows and n columns, all of whose entries are 0's and l's. Let the sum of row i of A be denoted by ri and let the sum of column j of A be denoted by Sj. We call R = (r1, … , rm) the row sum vector and S = (s1 . . , sn) the column sum vector of A. The vectors R and S determine a class

1.1

consisting of all (0, 1)-matrices of m rows and n columns, with row sum vector R and column sum vector S. The majorization concept yields simple necessary and sufficient conditions on R and S in order that the class 21 be non-empty (4; 9). Generalizations of this result and a critical survey of a wide variety of related problems are available in (6).
A New Type of Characteristic Subgroup of Prime-Power Groups
Authors: Brahana H. R.
Published: December 2010

In a recent paper (1) the fifty-eight metabelian groups of order p11 that are generated by five elements and have all their elements of order p were determined and characterized in terms independent of any particular selection of the generating elements. In dealing with fifty-seven of these groups there was no occasion to distinguish between one odd prime and another, except that in exhibiting canonical forms it was necessary to select irreducible polynomials and these, of course, depended on p. The fifty-eighth group was described in two ways in terms that were independent of p, but the proof of uniqueness could not be made without taking into account properties of p. These properties distribute the primes into classes, and the properties are reflected in the groups of order p11 in characteristic subgroups some of which exist for one prime and not for another. It may be that examination of the groups of isomorphisms of some of the fifty-seven groups would produce characteristic subgroups for one p that would not exist for another, but the writer considers it doubtful.
Some Generalized Theorems on Connectivity
Authors: Nettleton R. E.
Published: December 2010

The “k-dense” subgraphs of a connected graph G are connected and contain neighbours of all but at most k-1 points. We consider necessary and sufficient conditions that a point be in Γk, the union of the minimal k-dense subgraphs. It is shown that Γk contains all the [m, k]-isthmuses” and [m, k]-articulators“— minimal subgraphs which disconnect the graph into at least k + 1 disjoint graphs—and that an [m, k]-isthmus or [m, k]-articulator of Γk disconnects G. We define “central points,” “degree” of a point, and “chromatic number” and examine the relationship of these concepts to connectivity. Many theorems contain theorems previously proved (1) as special cases.
A Metrical Theorem in Diophantine Approximation
Authors: Schmidt Wolfgang
Published: December 2010

In this paper we prove a sharpening and generalization of the following Theorem of Khintchine (4):

Let ψ1(q), …, ψnq) be n non-negative junctions of the positive integer q and assume

is monotonically decreasing. Then the set of inequalities

1

has an infinity of integer solutions q > 0 and p1, … , pn for almost all or no sets of numbers θ1, … , θ2, according as Σψ(q) diverges or converges.
A Local Property of Measurable Sets
Authors: Eames W.
Published: December 2010

Let Ω be a metric space with metric ρ, let C be a class of closed sets from Ω and let τ be a non-negative real-valued set function on C. We assume that the empty set ϕ is in C and that τ(I)= 0 if and only if I = ϕ. For each set A in Ω, we define φ(A), 0 ≤ φ(A) ≤ ∞ by:

where the infimum is taken for all possible countable collections of sets I(n) from C such that:

and the diameter of I(n), d(I(n)), is less than ∈ for every n.
On a Discriminant Inequality
Authors: Mordell L. J.
Published: December 2010

The following result has been conjectured by Dr. Birch. Let z1 z2, . . . , zn be any n complex numbers such that

(1)

Then

(2)

attains its greatest value when the z are at the vertices of a regular n-sided polygon inscribed in the circle |z| =1.

It seems to be difficult to prove this but Dr. Birch informs me that some work by Mullholland shows that the result is false for large n. I can, however, prove that the result is true for n = 3, and then Δ ≤ 27. The suggested general result would be Δ ≤ nn.
Construction of Primitives of Generalized Derivatives with Applications to Trigonometric Series
Authors: Bullen P. S.
Published: December 2010

This paper is an extension of the ideas discussed in (3, §§ 14-16); the extension consisting of the use of the third and fourth symmetric Riemann derivative instead of the Schwarz or second symmetric Riemann derivative.

The J2integral, due to James (1), is defined in (3) as follows. Let f(x) be measurable on [a, b] and finite at each point; if there exists a continuous function F(x) such that D2F = f everywhere on (a, b),

then

1.1

The definition is unique since if F(x) and G(x) are continuous and D2F =D2G everywhere then
Simple Algebras of Type (1,1) are Associative
Authors: Kleinfeld Erwin
Published: December 2010

In the classification of almost alternative algebras relative quasiequivalence by Albert two new classes of algebras of type (γ, δ) were introduced, namely those of type (1, 1) and ( —1, 0) (1, equations (34), (35), and Theorem 6). Since rings of type (1, 1) and ( — 1, 0) are anti-isomorphic it suffices to consider those of type (1, 1). They may be defined as rings satisfying

1

and

2

for all elements x, y, and z of the ring, where the associator (a, b, c) is given by (a, b, c) = (ab)c — a (bc).

Actually the identity

2'

together with (1) implies (2) whenever the characteristic of the ring is different from 2. This may readily be verified by linearizing (2’). Consequently we may use (1) and (2’) as the defining relations for a ring of type (1, 1).
Decomposition of Finite Graphs into Open Chains
Authors: Nash-Williams C. ST. J. A.
Published: December 2010

If m, n are integers, “m = n” will mean “m = n (mod 2 ).“ The cardinal number of a set A will be denoted by |A|. The set whose elements are a1 a2, … , an will be denoted by {a1, a2, … , an}. The empty set will be denoted by Ø. If A, B, C are sets, A — B will denote the set of those elements of A which do not belong to B, and A — B — C will denote (A — B)— C. The expression Σε∈Af(ε) will be denoted by f. A. The statements “f = g on A,” “f ≡ g on A” will mean that f(ε) = g(ε) or f(ε) ≡ g(ε) respectively for every ε ∈ A.
On the Derivation Algebras of Lie Algebras
Authors: Tôgô Shigeaki
Published: December 2010

LetLbe a Lie algebra over a field of characteristic 0 and letD(L)be the derivation algebra ofL, that is, the Lie algebra of all derivations ofL. Then it is natural to ask the following questions: What is the structure ofD(L)?What are the relations of the structures ofD(L)andL? It is the main purpose of this paper to present some results onD(L)as the answers to these questions in simple cases.

Concerning the questions above, we give an example showing that there exist non-isomorphic Lie algebras whose derivation algebras are isomorphic (Example 3 in § 5). Therefore the structure of a Lie algebraLis not completely determined by the structure ofD(L). However, there is still some intimate connection between the structure ofD(L)and that ofL.
Programmes in Paired Spaces
Authors: Kretschmer K. S.
Published: December 2010

The basic problem of linear programming is to minimize (or maximize) a linear function of a finite number of variables constrained by a finite number of linear inequalities. From a mathematical point of view the subject may be regarded as being divided into two areas. One is primarily analytical and deals with certain questions of duality and consistency. The other is algorithmic and is concerned with computational questions and methods.
The Lebesgue Constants for Regular Hausdorff Methods
Authors: Lorch Lee , Newman Donald J.
Published: December 2010

The unboundedness of the sequence of Lebesgue constants (norms), at a point, of certain transforms implies, as is well known, that there exist (i) a continuous function whose transform fails to converge to the function at the point in question (the du Bois-Reymond singularity), and (ii) another such function whose transform, while converging everywhere to the function, does not do so uniformly in any neighbourhood of the stipulated point (the Lebesgue singularity). The converses also hold in our case.
The Expansion Problem with Boundary Conditions at a Finite Set of Points
Authors: Cole Randal H.
Published: December 2010

The problem of expanding an arbitrary function in a series of characteristic solutions of the ordinary differential equation

(1.1)

and the boundary relations

(1.2)

is well known. The various discussions are distinguished by the manner in which a parameter λ appears in the differential system and by the number of points at which the boundary conditions apply. The case in which the boundary conditions apply at intermediate as well as at the end points of a fundamental interval has been considered by Wilder (3).
Asymptotic Solutions of Equations in Banach Space
Authors: Swanson C. A. , Schulzer M.
Published: December 2010

The equationPx = yin Banach spaces has aroused considerable interest, particularly in view of the various situations in applied analysis which it encompasses, and consequently it has been the topic of numerous investigations (2; 9; 10; 12). Detailed references may be found in (10). The equation is of special interest because of its interpretation as an integral equation; and in turn, many problems related to differential equations can be reformulated as integral equations (5; 7; 13).
Transformation Algebras
Authors: LeBlanc Leon
Published: December 2010

The purpose of this paper is to show that most results concerning polyadic algebras can be generalized to transformation algebras. The results of this paper will clearly indicate that a great deal can be done in polyadic algebras without ever mentioning the quantifier structure (for instance, terms and operations can be characterized without the help of the quantifier structure, at least in the case where an equality is present). In § 1, we develop the elementary theory; in § 2, we study the different ways of extending a (locally finite) transformation algebra (of infinite degree) to a polyadic algebra; in § 3, we study equality transformation algebras; finally, in § 4, we show how terms and operations can be defined in equality transformation algebras.
Anticommuting Linear Transformations
Authors: Kestelman H.
Published: December 2010

It is well known that any set of four anticommuting involutions (see §2) in a four-dimensional vector space can be represented by the Dirac matrices

(1)

where the B1,r are the Pauli matrices

(2)

(See (1) for a general exposition with applications to Quantum Mechanics.) One formulation, which we shall call the Dirac-Pauli theorem (2; 3; 1), is

Theorem 1. If M1,M2, M3, M4 are 4 X 4 matrices satisfying

then there is a matrix T such that

and T is unique apart from an arbitrary numerical multiplier.
The Analyticity of Kernels
Authors: De Barros-Neto J. , Browder F. E.
Published: December 2010

Let V be a paracompact real analytic manifold of dimension n ≥ 1. Following the terminology of the theory of distributions of Schwartz (4), is the linear space of infinitely differentiable functions with compact support in V with the appropriate inductive limit topology, is the Frechet space of infinitely differentiable functions on V, is the dual space of consisting of the distributions on V, the dual space of consisting of the distributions with compact support on V. Let 𝒰(V) be the linear space of real analytic functions on V.
On Normed Algebras Whose Norms Satisfy Polynomial Identities
Authors: Aurora Silvio
Published: December 2010

In this note we are concerned with normed algebras over a non-discrete field with absolute value. The norm, N, of a normed algebra A is said to satisfy a polynomial identity on a subset B if there is a polynomial such that P(N(x1), … , N(xr)) = N(P(x1 … , xr)) whenever x1, … , xr are in B, where the polynomial has rational integer coefficients, degree greater than 1, constant term zero, and non-negative coefficients for each term of highest degree. It is shown in Theorem 1, following a method of proof used by Kadison in (6, § 7), that if the norm of a normed algebra satisfies a polynomial identity on the entire algebra, then the norm is power multiplicative. (That is, then N(x)2 = N(x2) for all x.)
Symmetric Exterior Differentiation and Flat Forms
Authors: Shapiro Victor L.
Published: December 2010

Let ω be a continuous differential r-form defined in a bounded domain R of Euclidean n-space, En, where n ≧ 1 and 0 ≦ r ≦ n — 1. ω is called a flat form in R, (3, p. 263), if there exists a constant N such that for every (r + 1)-simplex σ contained in R, where |σ| designates the (r + 1)-volume of σ. For n = 1 and ω a zero form, flatness is the same thing as the usual Lip 1 condition. As is well known, a necessary and sufficient condition that a continuous real-valued function of one variable f(x) satisfy a Lip 1 condition over an interval (a, b) is that its upper and lower symmetric derivatives be bounded in (a, b) (4, pp. 22, 327). We intend to prove the analogue of this result for r-forms.
Generators for Simple Groups
Authors: Steinberg Robert
Published: December 2010

The list of known finite simple groups other than the cyclic, alternating, and Mathieu groups consists of the classical groups which are (projective) unimodular, orthogonal, symplectic, and unitary groups, the exceptional groups which are the direct analogues of the exceptional Lie groups, and certain twisted types which are constructed with the aid of Lie theory (see §§3 and 4 below). In this article, it is proved that each of these groups is generated by two of its elements. It is possible that one of the generators can be chosen of order 2, as is the case for the projective unimodular group (1), or even that one of the generators can be chosen as an arbitrary element other than the identity, as is the case for the alternating groups. Either of these results, if true, would quite likely require methods much more detailed than those used here.
On Finite Groups with an Abelian Sylow Group
Authors: Brauer Richard , Leonard Henry S.
Published: December 2010

We shall consider finite groups of order of g which satisfy the following condition:

(*) There exists a prime p dividing g such that if P ≠ 1 is an element of p-Sylow group ofthen the centralizer(P) of P incoincides with the centralizer() of in.

This assumption is satisfied for a number of important classes of groups. It also plays a role in discussing finite collineation groups in a given number of dimensions.

Of course (*) implies that is abelian. It is possible to obtain rather detailed information about the irreducible characters of groups in this class (§ 4).
The Lattice of Subalgebras of a Boolean Algebra
Authors: Sachs David
Published: December 2010

It is well known (1, p. 162) that the lattice of subalgebras of a finite Boolean algebra is dually isomorphic to a finite partition lattice. In this paper we study the lattice of subalgebras of an arbitrary Boolean algebra. One of our main results is that the lattice of subalgebras characterizes the Boolean algebra. In order to prove this result we introduce some notions which enable us to give a characterization and representation of the lattices of subalgebras of a Boolean algebra in terms of a closure operator on the lattice of partitions of the Boolean space associated with the Boolean algebra. Our theory then has some analogy to that of the lattice theory of topological vector spaces. Of some interest is the problem of classification of Boolean algebras in terms of the properties of their lattice of subalgebras, and we obtain some results in this direction.
On Certain Onto Maps
Authors: Namioka Isaac
Published: December 2010

Let Δn (n > 0) denote the subset of the Euclidean (n + 1)-dimensional space defined by

A subset σ of Δn is called a face if there exists a sequence 0 ≤ i1 ≤ i2 ≤ … < im ≤ n such that

and the dimension of σ is defined to be (n — m). Let denote the union of all faces of Δn of dimensions less than n. A topological space Y is called solid if any continuous map on a closed subspace A of a normal space X into Y can be extended to a map on X into Y. By Tietz's extension theorem, each face of Δn is solid. The present paper is concerned with a generalization of the following theorem which seems well known.
Finite Sublattices of a Free Lattice
Authors: Jónsson Bjarni , Kiefer James E.
Published: December 2010

It is known that every sublattice A of a free lattice satisfies the following conditions:

(W) For all a, b, c, d ∈ A, if ab ≤ c + d, then ab ≤ c or ab ≤ d or a ≤ c+dor b ≤ c + d.

(SD) For all u, a, b, c ∈ A, if u = a + b = a + c, then u = a + bc.

(SD′) For all u, a, b, c ∈ A, if u = ab = ac, then u = a(b + c).

In fact, (W) is one of the four conditions used in Whitman (4) to characterize free lattices, and in Jónsson (3) it was shown that (SD) and (SD′) follow from Whitman's canonical representations of elements of a free lattice.

This note is concerned with lattices that satisfy one or more of the above conditions, and especially with finite lattices that satisfy all three conditions.
An Algorithm for a Minimum Cover of an Abstract Complex
Authors: Ray-Chaudhuri D. K.
Published: December 2010

Let X = {x1, x2,...,xm} be a finite set of m points and = {A1, A2, . . . , An] be a class of n subsets of X. Such a system of points and sets is called a complex (X, ). If every set of the class contains two points, the complex is a graph with m points x1x2, . . . , xm and n edges A1, A2, . . . , An. A complex (X, ) in which every set has the same number of points is called regular.
A New Proof of Certain Metrization Theorems
Authors: Smith Marion B.
Published: December 2010

A well-known problem in topology is the so-called metrization problem. This consists of asking for the topological conditions that are necessary and sufficient in order to guarantee that a topological space be metrizable. The first solution of this problem was given in 1923 by P. Alexandroff and P. Urysohn (1).
Triangles in an Ordinary Graph
Authors: Nordhaus E. A. , Stewart B. M.
Published: December 2010

An ordinary graph is a finite linear graph which contains no loops or multiple edges, and in which all edges are undirected. In such a graph G, let N, L, and T denote respectively the number of nodes, edges, and triangles. One problem, suggested by P. Erdös (1), is to determine the minimum number of triangles when the number of edges is specified, subject to suitable restrictions.
The Embedding Theorems of Malcev and Lambek
Authors: Bush George C.
Published: December 2010

A given semigroup is said to be embeddable in a group if there exists a group which contains a subsemigroup isomorphic to .

It can easily be proved that cancellation is a necessary condition for embeddability. It can also be shown that we can adjoin an identity to a semigroup without identity in such a way that the new semigroup is embeddable if and only if the original was embeddable. Therefore, we can, without loss of generality, restrict our attention to cancellation semigroups with identity, whenever this is convenient.
On the Number of Disjoint Edges in a Graph
Authors: Weinstein Joseph M.
Published: December 2010

In what follows we prove that a finite graph of n nodes in which each node has degree ≥ 1 but ≤ possesses a set of at least n/(1 + ) pairwise disjoint edges. Our principal theorem states an analogue of this result for the case when each node has degree ≥2: we show that in this case the graph possesses a set of at least 2n/(2 + max(4, )) mutually disjoint edges.
Unions of Two Convex Sets
Authors: Stamey William L. , Marr J. M.
Published: December 2010

Valentine (1, Theorems 2 and 3) has defined a three-point property which he called P3 and has shown that a closed subset of the euclidean plane possessing this property is expressible as the union of at most three convex sets. He also showed that if the number of isolated points of local non-convexity of such a set is one, finite and even, or infinite, the set is the union of two convex sets. In this paper we give properties which, together with Valentine's results, characterize those subsets of a plane which may be represented as a union of two closed, convex sets.
Regent Results in Comma-Free Codes
Authors: Jiggs B. H.
Published: December 2010

A set D of k-letter words is called a comma-free dictionary (2), if whenever (a1a2 . . . ak) and (b1b2 . . . bk) are in D, the "overlaps" (a2a3 . . . akb1), (a3a4 . . . akb1b2), . . . , (akb1 . . . bk-1) are not in D. We say that two k-letter words are in the same equivalence class if one is a cyclic permutation of the other. An equivalence class is called complete if it contains k distinct members. Comma-freedom is violated if we choose words from incomplete equivalence classes, or if more than one word is chosen from the same complete class.
Rational Points on Linear Subspaces. Representation of an Integer as a Sum of Squares with Accessory Conditions
Authors: Motzkin T. S.
Published: December 2010

The present study was motivated by an investigation of algebraic conjugates in the complex plane (cf. 4 for one of the results) where some of its concepts are extended and applied.

Let F be a flat (linear subspace) in real affine n-space. The points z = (ζ1, . . . , ζn) on F for which the least common denominator of the coordinates ζv is minimum form a grid G, the main grid of F, studied in § 3.
Principal Right Ideal Rings
Authors: Johnson R. E.
Published: December 2010

This paper is concerned with the recent work of A. W. Goldie (1 ) on principal right ideal rings (p.r.i. rings). We shall prove some of his main structure theorems using the methods of (3) and (4), and in so doing shall weaken some of his hypotheses.
The Automorphisms of the Group of Rotations and its Projective Group Corresponding to Quadratic Forms of any Index
Authors: Wonenburger María J.
Published: December 2010

Let M be a vector space of dimension n over a field K of characteristic ≠2 and f a non-degenerate quadratic form on M. The automorphisms of the orthogonal group On(K,f), the rotation group On+(K,f), and their corresponding projective groups POn(K,f), POn+(K,f), were determined by J. Dieudonné for n sufficiently large under the condition that the index of f is greater than zero (see 2).
On Weakly Positive Matrices
Authors: Wigner Eugene P.
Published: December 2010

A matrix is said to be positive definite if it is hermitian and if all of its characteristic values are positive. It is well known, and easy to prove, that the necessary and sufficient condition for a matrix P to be positive definite is that its hermitian quadratic form

with any vector v ≠ 0 be positive. (This will imply, in the present article, that it is real.) It is easy to see from (1) that if P1 and P2 are positive definite, the same holds of a1P1 + a2P2 if a1 and a2 are positive numbers.
On Unitary Equivalence of Matrices over the Ring of Continuous Complex-Valued Functions on a Stonian Space
Authors: Pearcy Carl
Published: December 2010

This paper is a continuation of the earlier papers (1, 5) in which the author studied matrices with entries from the algebra C() of all continuous, complex-valued functions on an extremely disconnected, compact Hausdorff space . (Such spaces are sometimes called Stonian, after M. H. Stone, who first considered them in (8). They arise naturally as maximal ideal spaces of abelian W*-algebras.) In this note, three theorems are proved.
The Asymptotic Behaviour of the Hermite Polynomials
Authors: Wyman Max
Published: December 2010

In a recent paper, Olver (2) obtains a set of formulae that completely determine the asymptotic behaviour of the Hermite polynomials, Hn(z), as n —> ∞ and z is unrestricted. His proof depends on a technique that he has developed for discussing the asymptotics of solutions of second-order, linear, homogeneous differential equations satisfying certain conditions. We believe it fair to say that Olver's work follows the tradition of most of the major theorems of classical asymptotics. The results contained in theorems such as Watson's lemma and Perron's proof of the Method of Laplace are based on an acceptance, on an a priori basis, of the Poincaré type expansion.
Intrinsic Functions on Matrices of Real Quaternions
Authors: Cullen C. G.
Published: December 2010

It is well known that any semi-simple algebra over the real field R, or over the complex field C, is a direct sum (unique except for order) of simple algebras, and that a finite-dimensional simple algebra over a field is a total matrix algebra over a division algebra, or equivalently, a direct product of a division algebra over and a total matrix algebra over (1). The only finite division algebras over R are R, C, and , the algebra of real quaternions, while the only finite division algebra over C is C.
Decomposition of Graphs into Two-Way Infinite Paths
Authors: Nash-Williams C. ST. J. A.
Published: December 2010

We consider undirected graphs in which two vertices may be joined by more than one edge and in which a vertex may be joined to itself by one or more edges. G will always denote a graph. The set of vertices [edges] of G will be denoted by V(G) [E(G)]. G is finite or infinite according as V(G) ∪ E(G) is finite or infinite. The degree, d(ξ) or dG(ξ), of a vertex ξ of G is 2a + b, where a is the number of edges joining ξ to itself and b is the number of those joining ξ to other vertices.
On Conjugate Functions
Authors: Chen Yung-Ming
Published: December 2010

In a previous paper (3) generalizations of M. Riesz's theorem by the method of asymptotic approximation have been given. The present paper is concerned with further generalizations for even and odd functions. In Section 3, we consider a generalization of a theorem due to Zygmund: If (10, p. 254). The results in this paper include generalizations of results by K. K. Chen (2) and T. M. Flett (4, Theorems 1, 2) as important special cases.
A Bipartitional Function Arising in Hall's Algebra
Authors: Davies I. J.
Published: December 2010

Hall's algebra (3) is an algebra over the field V(p) of rational functions in the indeterminate p with coefficients in the field V of complex numbers. The basis of the algebra consists of elements Gλ which are in one-one correspondence with the set of all partitions (λ) and whose multiplication "constants" are the "Hall polynomials" , i-e.
On the Group Ring
Authors: Connell Ian G.
Published: December 2010

LetRbe the discrete group ring of the groupGover the ringA. In this paper we attempt to find necessary and sufficient conditions onGandAso thatRwill have some standard ring-theoretic property ; among the properties considered are those of being artinian, regular, self-injective, and semi-prime.

The contents of this paper form essentially the author's doctoral thesis. The author would like to thank his supervisor Dr. J. Lambek for his generous encouragement and continued interest.
Variation-Diminishing Transformations and General Orthogonal Polynomials
Authors: Hirschman I. I.
Published: December 2010

Letα(dx) be a finite measure defined on the Borel subsets of [—1, 1], the spectrum of which is infinite. Letbe the family of orthonormal polynomials associated with a, so that

Theare uniquely determined by this and by the condition
A Homomorphism in Exterior Algebra
Authors: Kostant B. , Novikoff A.
Published: December 2010

In the following, V is a vector space over an arbitrary field F, dimFV = n. Let {e1, . . . , en} be a basis for V, and {f1, . . . , fn} be the dual basis for and , then the operators *(u) and i(g) (exterior and inner multiplication by u and g respectively) set up an equivalence between the ideal = range of ∊(u) and the sub-algebra = range of i(g) considered as vector spaces. That is, ∊(u)i(g) is the identity on , i(g) ∊(u) is the identity on .
Lattice Tetrahedra
Authors: White G. K.
Published: December 2010

A class of problems in the geometry of numbers, for which there are but fragmentary results, may be expressed in general terms, as follows. Let 5 be a given point-set and let G be a given discrete point-set, both in Euclidean n-space. Suppose that Λ is a lattice which contains G but no point of S not in G. Such lattices, if they exist, will be said to be admissible for S with respect to G, and the general problem is to investigate their properties and, if possible, classify them.
Compounds of Skew-Symmetric Matrices
Authors: Marcus Marvin , Yaqub Adil
Published: December 2010

In a recent interesting paper (3) H. Schwerdtfeger answered a question of W. R. Utz (4) on the structure of the real solutions A of A* = B, where A is skew-symmetric. (Utz and Schwerdtfeger call A* the "adjugate" of A ; A* is the n-square matrix whose (i, j) entry is (—1)i+j times the determinant of the (n — 1)-square matrix obtained by deleting row i and column j of A. The word "adjugate," however, is more usually applied to the matrix (AT)*, where AT denotes the transposed matrix of A ; cf. (1, 2).)

The object of the present paper is to find all real n-square skew-symmetric solutions A to the equation
Induced Representations and Alternating Groups
Authors: Puttaswamaiah B. M. , ROBINSON G. de B.
Published: December 2010

This paper is based on part of the thesis of one of the authors (5), submitted at the University of Toronto in 1963. In the first part of the paper a result on induced representations (2, 4, 9) is generalized slightly and a number of corollaries are derived. In the rest of the paper a special case of this result is applied to put the representation theory of the alternating group on a par with that of the symmetric group. A knowledge of the representation theory of Sn (7) on the part of the reader is assumed.
Diameters of Polyhedral Graphs
Authors: Klee Victor
Published: December 2010

The distance between two vertices of a connected finite graph is the smallest number of edges forming a path that joins the two vertices, and the diameter of the graph is the largest integer which is realized as the distance between two vertices of the graph. We are concerned here with the diameters of two graphs associated with a d-dimensional convex polytope P (called henceforth a d-polytope). The graph Γ(P) of P is the 1- complex formed by the vertices and edges of P , and the polar graph II (P) of P is the 1-complex whose vertices correspond to the (d — 1)-faces of P, with two vertices joined by an edge in II (P) if and only if the corresponding (d — 1)- faces intersect in a (d — 2)-face of P. The diameters of Γ(P) and II (P) will be denoted respectively by δ(P) (called the diameter of P) and ϕ(Ps) (called the face-diameter of P ).
A Formal Solution of Σ∞i=1AieBix = X
Authors: Raney George N.
Published: December 2010

We consider a combinatorial enumeration problem involving certain collections of labelled rooted trees having coloured nodes and edges. Notations and definitions are introduced in §§2 and 3, and the problem is described in §4. We give recursion formulas for its solution in §5. Then, by using a modification of a method of Priifer, we obtain a direct solution in terms of multinomial coefficients and power products in §§6 and 7. These results are combined in §8. Working in a formal power series algebra we find a formal solution of the equation

which expresses the unknown X as a multiple power series in the Ai and the Bi.
Infinite Euler Graphs
Authors: Sabidussi Gert
Published: December 2010

It is well known that for finite connected graphs the following are equivalent:

(i)

X is Euler (i.e., every vertex of X has positive even degree);

(ii)

X is traceable (i.e., the edges of X can be arranged in a sequence e1, . . . ,en such that ei ≠ ej if i ≠ j, and ei, ei+1 are adjacent, i = 1, . . . , n, subscripts considered mod n) ;

(iii)

X is cyclically coverable (i.e., X contains a family of non-overlapping circuits whose union is X).
A Census of Finite Automata
Authors: Harrison Michael A.
Published: December 2010

A finite automaton may be thought of as a possible abstraction of a digital computer. Imagine a tape or sequence of letters from some alphabet being fed into a device with a finite number of internal states. When the device is in a particular state and receives an input letter, the system passes to another internal state and prints a letter on an output tape. This operation is continued until the entire input sequence has been processed.

Both Harary (6) and Ginsburg (4) have focused attention on the previously unsolved problem of counting the number of equivalence classes of finite automata. In the present paper, this problem is solved completely by proving a variety of theorems about the enumeration of functions.
The Distribution of Irreducible Polynomials in Several Indeterminates II
Authors: Carlitz L.
Published: December 2010

It is well known that the number of normalized irreducible polynomials of degree m in a single indeterminate, with coefficients in GF(q), is given by

(1.1)

where μ(r) is the Möbius function. It follows from (1) that

(12)
Kurosh Radicals of Rings with Operators
Authors: Divinsky N. , Suliński A.
Published: December 2010

If A is an algebra over a commutative ring with unity, Φ, then the Jacobson radical of the algebra A is equal to the Jacobson radical of A, thought of as a ring (1, p. 18, Theorem 1). The present note extends this result to all radical properties (in the sense of Kurosh 2) and allows ϕ to be any set of operators on A.

If A is a ring and Φ is an arbitrary set, we say that Φ is a set of operators for A if for any α in Φ and any x in A, the composition ax is defined and is an element of A, and if this composition satisfies the following two conditions:
Enumeration of Quadrangular Dissections of the Disk
Authors: Brown William G.
Published: December 2010

A dissection of the disk will be a cell complex (1, p. 39) K with polyhedron the closed disk B2. It will further be required that:

(a) every edge of K be incident with two distinct vertices (called its ends) ;

(b) no two edges have the same ends ; and

(b) no two edges have the same ends ; and
Some Homological Pathology in Vector Lattices
Authors: Topping David M.
Published: December 2010

The purpose of this paper is to point out a number of curious phenomena in the category of (real) vector lattices and linear lattice homomorphisms. Birkhoff (3, p. 221, Ex. 2 and Problem 96) called attention to the question of constructing models of the free objects with more than one generator in this category, a problem recently solved by E. C. Weinberg (9). In §6 we construct a more manageable class of (non-free) projective vector lattices. Here, however, there is a countability restriction which suggests strong connections with free and projective Boolean algebras (in the category of Boolean algebras and their homomorphisms, such algebras must satisfy the countable chain condition (6)).
The Decomposition of Graphs into a Finite Number of Paths
Authors: Rothschild Bruce
Published: December 2010

In (3) Ore poses two problems concerning the decomposition of graphs into edge-disjoint paths. The first is to find the conditions on a graph so that it can be decomposed into a finite number k of edge-disjoint, two-way infinite paths and no fewer. In (2) Nash-Williams solves this problem. The results of (2) are used here to solve the second problem, to find conditions on a graph so that it can be decomposed into a finite number k of edge-disjoint paths (finite, one-way infinite, and two-way infinite) and no fewer.
On the Structure of Group Algebras, I
Authors: Cohn James A. , Livingstone Donald
Published: December 2010

With this paper we begin a study of the structure of the group algebra RG of a finite group G over the ring of algebraic integers R in an algebraic number field k. The basic question is whether non-isomorphic groups can have isomorphic algebras over R. We shall show that this is impossible if G is

(a) abelian,

(b) Hamiltonian,

(c) one of a special class of p-groups.
On Some Diophantine Inequalities Involving the Exponential Function
Authors: Baker A.
Published: December 2010

It is well known that for any real number θ there are infinitely many positive integersnsuch that

Here ||a|| denotes the distance of a from the nearest integer, taken positively. Indeed, since ||a|| < 1, this implies more generally that ifθ1,θ2, . . . ,θkare any real numbers, then there are infinitely many positive integersnsuch that
On the Location of Singularities of a Class of Elliptic Partial Differential Equations in Four Variables
Authors: Gilbert R. P.
Published: December 2010

In this paper we shall investigate the singular behaviour of the solutions to the elliptic equation

(1.1)

whereA(r2),C(r2) are entire functions of the complex variable
Formal Categories
Authors: Maranda J.-M.
Published: December 2010

A category is made up of its class of objects , of a function M (or ) which assigns to each (A,B) ∊ × the class M(A, B) of all morphisms from A to B, of an operation μ which is a class of maps

A, B, C ∊ , and of a family of maps