bluuebunny/crossref_metadata_2025 huggingface dataset:
Embedding Circle-Like Continua in the Plane
Authors: Bing R. H.
Published: December 2010

A finite sequence of open sets L1 L2, … , Ln is called a linear chain if each Li intersects only the L's adjacent to it in the sequence. The finite sequence is a circular chain if we also insist that the first and last links intersect each other. The 1-skeleton of the covering is an arc for a linear chain and a simple closed curve for a linear chain.

A compact metric continuum X is called snake-like if for each ∈ > 0, X can be covered by a linear chain of mesh less than ∈. Likewise, X is called circle-like if for each ∈ > 0, X can be irreducibly covered with a circular chain of mesh less than ∈. This definition is more restrictive than that given in (3, p. 210) for there a pseudo-arc is not circle-like but here it is. The present usage is in keeping with definitions of Burgess.
Some Results on Quadrics in Finite Projective Geometry Based on Galois Fields
Authors: Ray-Chaudhuri D. K.
Published: December 2010

In a paper (5) published in the Proceedings of the Cambridge Philosophical Society, Primrose obtained the formulae for the number of points contained in a non-degenerate quadric in PG(n, s), the finite projective geometry of n dimensions based on a Galois field GF(s). In § 3 of the present paper the formulae for the number of p-flats contained in a non-degenerate quadric in PG(n, s) are obtained. In § 4 an interesting property of a non-degenerate quadric in PG(2k, 2m) is proved. These properties of a quadric will be used in solving some combinatorial problems of statistical interest in a later paper.
Weak Containment and Induced Representations of Groups
Authors: Fell J. M. G.
Published: December 2010

Let G be a locally compact group and G† its dual space, that is, the set of all unitary equivalence classes of irreducible unitary representations of G. An important tool for investigating the group algebra of G is the so-called hull-kernel topology of G†, which is discussed in (3) as a special case of the relation of weak containment. The question arises: Given a group G, how do we determine G† and its topology? For many groups G, Mackey's theory of induced representations permits us to catalogue all the elements of G†. One suspects that by suitably supplementing this theory it should be possible to obtain the topology of G† at the same time. It is the purpose of this paper to explore this possibility. Unfortunately, we are not able to complete the programme at present.
The Construction of Representations of Lie Algebras of Characteristic Zero
Authors: Noonan B.
Published: December 2010

In this paper a procedure is given whereby, from a representation of an ideal contained in the radical, explicit representations of a Lie algebra by matrices can be constructed in an algebraically closed field of characteristic zero. The construction is sufficiently general to permit one arbitrary eigenvalue to be assigned to the representation of each basis element of the radical not in the ideal. The theorem of Ado is proved as an application of the construction. While Ado's theorem has several proofs (1; 3; 5; 6), the present one has a value in its explicitness and in the fact that the degree of the representation can be given.
Minimum and Conjugate Points in Symmetric Spaces
Authors: Crittenden Richard
Published: December 2010

The purpose of this paper is to discuss conjugate points in symmetric spaces. Although the results are neither surprising nor altogether unknown, the author does not know of their explicit occurrence in the literature.

Briefly, conjugate points in the tangent bundle to the tangent space at a point of a symmetric space are characterized in terms of the algebraic structure of the symmetric space. It is then shown that in the simply connected case the first conjugate locus coincides with the minimum (cut) locus. The interest in this last fact lies in its identification of a more or less locally and analytically defined set with one which includes all the topological interest of the space.
Some Results for the Generalized Lototsky Transform
Authors: Cowling V. F. , Miracle C. L.
Published: December 2010

Let A = (ank) and x = {sn} (n,k = 0,1,2, … ) be a matrix and a sequence of complex numbers, respectively. We write formally

(1.1)

and say that the sequence x is summable A to the sum t or that the A matrix sums the sequence x to the value t if the series in (1.1) converges and

exists and equals t. We say that the matrix A is regular provided it sums every convergent sequence to its limit.
Arithmetic Properties Of Freely α-Generated Lattices
Authors: Jónsson Bjarni
Published: December 2010

In § 1 we give a characterization of a lattice L that is freely α-generated by a given partially ordered set P. In § 2 we obtain a representation of an element of such a lattice as a sum (product) of additively (multiplicatively) irreducible elements which, although not unique, has some of the desirable features of the canonical representation, in Whitman (2), of an element of a free lattice. The usefulness of this representation is illustrated in § 3, where some further arithmetic properties of these lattices are derived.

We use + and . for the binary operations of lattice addition and multiplication, and Σ and II for the corresponding operations on arbitrary sets and sequences of lattice elements. In other respects the terminology will be the same as in Crawley and Dean (1).
Extremal Problems for Functions Starlike in the Exterior of the Unit Circle
Authors: Royster W. C.
Published: December 2010

Let Σ represent the class of analytic functions

(1)

which are regular, except for a simple pole at infinity, and univalent in |z| > 1 and map |z| > 1 onto a domain whose complement is starlike with respect to the origin. Further let Σ- 1 be the class of inverse functions of Σ which at w = ∞ have the expansion

(2).

In this paper we develop variational formulas for functions of the classes Σ and Σ- 1 and obtain certain properties of functions that extremalize some rather general functionals pertaining to these classes. In particular, we obtain precise upper bounds for |b2| and |b3|. Precise upper bounds for |b1|, |b2| and |b3| are given by Springer (8) for the general univalent case, provided b0 =0.
Lattice Coverings ofn-Space By Spheres
Authors: Bleicher M. N.
Published: December 2010

Alattice∧ in euclideann-space,Enis a group of vectors under vector addition generated bynindependent vectors,X1, X2… ,Xn, called abasisfor the lattice. The absolute value of then×ndeterminant the rows of which are the co-ordinates of a basis is called thedeterminant of the latticeand is denoted byd(∧). For any lattice ∧ there is a unique minimal positive numberrsuch that, if spheres of radiusrare placed with centres at all points of ∧, the entire space is covered. Thedensityof this covering may be defined as(Jnrn)/(d(∧))whereJnis the volume of the unit sphere inn-dimensional euclidean space. This density will be denoted by θn(∧). Thedensity of the most efficient lattice coveringofn-space by spheres,θnis the absolute minimum of θn(∧) considered as a function from the space of all lattices to the real numbers.
Subnormal Subgroups of Division Rings
Authors: Herstein I. N. , Scott W. R.
Published: December 2010

Let K be a division ring. A subgroup H of the multiplicative group K′ of K is subnormal if there is a finite sequence (H = A0, A1, . . . , An = K′) of subgroups of K′ such that each Ai is a normal subgroup of Ai+1. It is known (2, 3) that if H is a subdivision ring of K such that H′ is subnormal in K′, then either H = K or H is in the centre Z(K) of K.
Sur la Cohomologie Non Abelienne, II
Authors: Dedecker Paul
Published: December 2010

Cet exposé fait suite à des travaux antérieurs et notamment à mon article (4). Il a fait l'objet d'une communication au Colloque de géométrie différentielle et de topologie algébrique de Zürich en juin 1960. Dans une première partie on aborde, par les méthodes cohomologiques, le problème de la restriction du groupe structural G d'un espace fibre à un sous-groupe N, ce qui, lorsque N n'est pas normal, conduit à une "cohomologie à coefficients dans un faisceau d'espaces homogènes." Dans le second on donne une interprétation cohomologique des espaces fibres en théorie simpliciale, une fonction tordante étant interprétée comme un cocycle.
The Divided Central Differences of Zero
Authors: Carlitz L. , Riordan John
Published: December 2010

Put

In a recent paper (4), Lohne showed that
Torsion-Free and Divisible Modules Over Non-Integral-Domains
Authors: Levy Lawrence
Published: December 2010

In trying to extend the concept of torsion to rings more general than commutative integral domains the first thing that we notice is that if the definition is carried over word for word, integral domains are the only rings with torsion-free modules. Thus, if m is an element of any right module M over a ring containing a pair of non-zero elements x and y such that xy = 0, then either mx = 0 or (mx)y = 0. A second difficulty arises in the non-commutative case: Does the set of torsion elements of M form a submodule? The answer to this question will not even be "yes" for arbitrary non-commutative integral domains.
The Number of Solutions of Congruences in Incomplete Residue Systems
Authors: Chalk J. H. H.
Published: December 2010

Let denote the set of points x = (x1, . . . , xn) with integral co-ordinates in Euclidean n-space. For any fixed integer m ≥ 2, let C = C(m) be the set of such points in the cube 0 ≤ xi < m (i = 1, 2, . . . , n) and let be any subset of C. Suppose that f(x) is any single-valued, integral-valued function, defined for all x ∊ . We consider solutions x ∊ of the congruence
On Certain Remarkable Triangles
Authors: Bottema O.
Published: December 2010

In his fine and stimulating book Introduction to geometry (1), Coxeter considers three mutually tangent circles C1 , C2, and C3 (Fig. 1) and calculates the radii of the two circles E1 and E2 that touch all these three. If the centres of Ci are the vertices of a triangle ABC their radii are s — a, s — b, and s — c; α, β, and γ are defined by
Width Sequences for Special Classes of (0, 1)-Matrices
Authors: Fulkerson D. R. , Ryser H. J.
Published: December 2010

The α-width ∊A(α) of a (0, 1)-matrix A is the minimal number of columns that can be selected from A in such a way that all row sums of the resulting submatrix of A are at least α. This notion was introduced in (2) and further studied in (3). In these papers the major emphasis was on the minimal α-width sequence for the class of (0, 1)-matrices generated from an arbitrary A by interchanges:
Radial and Angular Limits of Meromorphic Functions
Authors: Cargo G. T.
Published: December 2010

Let us say that a function denned in the open unit disk D has the Montel property if the set of those points eiθ on the unit circle C where the radial limit exists coincides with the set where the angular limit exists. By a classical theorem of Montel (4), every bounded holomorphic function has this property. Meromorphic functions omitting at least three values and, more generally, the normal functions recently introduced by Lehto and Virtanen (3) also enjoy the Montel property (also see 1).
On the Character Rings of Finite Groups
Authors: Banaschewski B.
Published: December 2010

The characters of the representations of a finite groupGover a fieldKof characteristic zero generate a ring oK(G) of functions onG, theK-character ring of G, which is readily seen to be Zϕ1+ . . . + Zϕn, where Z is the ring of rational integers andϕ1, . . . ,ϕnare the characters of the different irreducible representations ofGoverK. The theorem that every irreducible representation ofGover an algebraically closed field Ω of characteristic zero is equivalent to a representation ofGover the subfield of Ω which is generated by the g0th roots of unity (g0the exponent ofG) was proved by Brauer (4) via the theorems that

(1) OΩ(G) is additively generated by the induced characters of representations of elementary subgroups of G, and

(2) the irreducible representations over 12 of any elementary group are induced by one-dimensional subgroup representations(3).
On Some Tactical Configurations
Authors: Hanani Haim
Published: December 2010

Given a set E of v elements, and given positive integers k, l (l ≤ k ≤ v), and λ, we understand by a tactical configurationC[k, l, λ, v] (briefly, configuration) a system of subsets of E, having k elements each, such that every subset of E having l elements is contained in exactly λ sets of the system.

A necessary condition for the existence of a configuration C[k, l, λ, v] is known (6) to be
Interval Functions and Non-Decreasing Functions
Authors: Appling William D. L.
Published: December 2010

In a previous paper the author (1) has shown the following theorem.

Theorem A. If each of H and K is a real-valued bounded function of subintervals of the number interval [a, b] and m is a real-valued non-decreasing function on [a} b] such that each of the integrals

exists, then the integral

exists.
Integral Bases in Kummer Extensions of Dedekind Fields
Authors: McCulloh Leon R.
Published: December 2010

Let J be a Dedekind ring, F its quotient field, F′ a finite separable extension of F, and J′ the integral closure of J in F′. It has been shown by Artin (1) that a necessary and sufficient condition that J′ have an integral basis over J is that a certain ideal of F (namely, √(D/Δ), where D is the discriminant of the extension and Δ is the discriminant of an arbitrary basis of the extension) should be principal. More generally, he showed that if is an ideal of F′, then a necessary and sufficient condition that have a module basis over J is that N()√(D/Δ) should be principal.
Embedding Theorems for Abelian Groups
Authors: Chehata C. G.
Published: December 2010

Given an abelian group G and a mapping θ that maps a subgroup A of G homomorphically onto another subgroup B of G, then it is known (3) that there always exists an embedding group G* ⊇ G which is abelian and possesses an endomorphism θ* which coincides with θ on A, i.e. aθ = aθ* whenever aθ is defined. θ is called a partial endomorphism of G and θη a total endomorphism or simply an endomorphism that extends or continues θ.
From Matrices to Graphs
Authors: Tutte W. T.
Published: December 2010

All the matrices considered in this paper have their elements in the field of residues mod 2.

Two non-singular matrices are equivalent if each row of either matrix is a linear combination of rows of the other. The matrices then have equal numbers of rows and equal numbers of columns.

A nodal matrix is a non-singular matrix in which no column has more than two l's. A graphic matrix is a non-singular matrix equivalent to a nodal matrix.
Free Lattices Generated by Partially Ordered Sets and Preserving Bounds
Authors: Dean R. A.
Published: December 2010

A construction of the free lattice generated by a partially ordered set P and preserving every least upper bound (lub) and greatest lower bound (glb) of pairs of elements existing in P has been given by Dilworth (2, pp. 124-129) and, when P is finite, by Gluhov (5).

The results presented here construct the free lattice FL generated by the partially ordered set P and preserving

(1) the ordering of P

(2) those lub's of a family of finite subsets of P which possess lub's in P
Complemented Banach Algebras
Authors: Olubummo A.
Published: December 2010

Let A be a complex Banach algebra and Lr (Ll) be the lattice of all closed right (left) ideals in A. Following Tomiuk (5), we say that A is a right complemented algebra if there exists a mapping I —> IP of Lτ into Lr such that if I ∊ Lr, then I ∩ Ip = (0), (Ip)p = I, I ⴲ Ip = A and if I1, I2 ∊ Lr with I1 ⊆ I2 then .

If in a Banach algebra A every proper closed right ideal has a non-zero left annihilator, then A is called a left annihilator algebra. If, in addition, the corresponding statement holds for every proper closed left ideal and r(A) = (0) = l(A), A is called an annihilator algebra (1).
Sums of the Divisor Function
Authors: Gordon B. , Rogers K.
Published: December 2010

Shapiro and Warga (2) have proved in an elementary way that all large integers are expressible as the sum of at most 20 primes. In so doing, they proved that

as x —> ∞, where u is a positive square-free integer,
The k-Extremally Disconnected Spaces as Projectives
Authors: Cohen Henry B.
Published: December 2010

The letter k denotes an infinite cardinal. A space is a compact Hausdorff space unless otherwise indicated. A space is called extremally disconnected (k-extremally disconnected) if it is the Stone space for a complete (k-complete) Boolean algebra. A map is a continuous function from one space into another. A map f:X —> Y is called minimal if f is onto, but f(M) is properly contained in Y for each closed proper subset M of X. A space F is called free if F has a dense subset X such that every space-valued function on X extends to a map on all of F or, equivalently, if F is the Stone-Cech compactification of some discrete topological space X.
Complete Reductibility of Infinite Groups
Authors: Dixon John D.
Published: December 2010

The theorems of the present paper deal with conditions which are necessary and sufficient in order that a solvable or nilpotent infinite group should have a completely reducible matrix representation over a given algebraically closed field.

It is known (17) that a locally nilpotent group of matrices is always solvable. Thus the first theorem of the present paper is a partial generalization of Theorem 1 of (16), which states:

If G is a locally nilpotent subgroup of the full linear group GL(n, P) over a perfect field P, then G is completely reducible if and only if each matrix of G is diagonizable (by a similarity transformation over some extension field of P).
Randomly Packed and Solidly Packed Spheres
Authors: Gilbert E. N.
Published: December 2010

In the classical packing problem unit spheres are placed without overlapping in D-dimensional space. When D = 2, the densest packing is a familiar regular arrangement of circles, each circle touching six others. In this packing, circles cover a fraction π/√12 = 0.9069 . . . of the area of the plane. The densest packing is not known for D ≥ 3.

Most of the packings to be considered here use spheres of many different sizes. In this way greater densities are obtainable; small spheres can fill up some of the space left over after large spheres have been packed.
Criteria for Solvability of Certain Congruences
Authors: Muskat Joseph B.
Published: December 2010

Lower case italics will denote rational integers, while lower case Greek letters will denote algebraic integers. The Law of Quadratic Reciprocity can be formulated as follows:

If p and a are distinct odd primes, then a is a quadratic residue(modp)if and only if ( — 1)(p-1)/2p is a quadratic residue(modq).

LetXpdenote the non-principal quadratic character (modp) and letζpbe a primitivepth. root of unity. Then
One-Dimensional Cell Complexes with Homeotopy Group Equal to Zero
Authors: Quintas Louis V.
Published: December 2010

Let K denote a connected finite 1-dimensional cell complex (1, p. 95), G(K) its group of homeomorphisms, and D(K) the group of homeomorphisms of K which are isotopic to the identity. The group (K) = G(K)/D(K) is a topological invariant of K and is called the homeotopy group ofK (4). K may be thought of as a linear graph (connected finite 1- dimensional simplicial complex) extended to admit loops and multiple edges and (K) as the topological analogue of the automorphism group A(L), (the permutations of vertices which preserve edge incidence relations) of a linear graph L.
Trace Formulas for Powers of a Sturm-Liouville Operator
Authors: Gilbert Richard C. , Kramer Vernon A.
Published: December 2010

Let H0 be the mth power (m a positive integer) of the self-adjoint operator defined in the Hilbert space L2(0, π) by the differential operator — (d2/dx2) and the boundary conditions u(0) = u(π) = 0. The eigenvalues of H0 are μn = n2m and the corresponding eigenfunctions are ϕn = (2/π)1/2 sin nx, n = 1 , 2 , . . ..

Let p be a (2m — 2)-times continuously differentiate real valued function defined over the interval [0, π] satisfying the conditions p(j)(0) = p(j)(π) = 0 for j odd and less than 2m — 4.
Equivalent Conditions for a Ring to Be a Multiplication Ring
Authors: Mott Joe Leonard
Published: December 2010

In this paper a ring will always mean a commutative ring with identity element. Furthermore, a ringRis called a multiplication ring if, wheneverAandBare ideals ofRandAis contained inB, there is an idealCsuch thatA=BC. Noetherian multiplication rings have been studied by Asano(1), Krull(4, 5), and Mori(6, 7). Krull also studied non-Noetherian multiplication rings(3). In(8, 9), Mori studied non-Noetherian multiplication rings which did not necessarily contain an identity element.
On the Decomposition of A Class of Functions of Bounded Variation
Authors: Laha R. G.
Published: December 2010

Let F1(x) and F2(x) be two distribution functions, that is, non-decreasing, right-continuous functions such that Fj(— ∞) = 0 and Fj(+ ∞) = 1 (j = 1, 2). We denote their convolution by F(x) so that

the above integrals being defined as the Lebesgue-Stieltjes integrals. Then it is easy to verify (2, p. 189) that F(x) is a distribution function. Let f1(t), f2(t), and f(t) be the corresponding characteristic functions, that is,
Linear Homogeneous Equations Over Finite Rings
Authors: Stevens Harlan
Published: December 2010

The intent of this paper is to apply the following theorem in several particular instances:

Theorem 1. For any finite ring of q elements, let {} be a collection of s subsets of , each containing hi (i = 1, 2, . . . , s) members, and let denote the set of all differences d′ — d″ with d′ and d″ from including d′ = d″. Furthermore, suppose that
Relations Between the Digits of Numbers and Equal Sums of Like Powers
Authors: Roberts J. B.
Published: December 2010

I t is straightforward, but tedious, to write down the integers whose representations in a given base do not have particular digits in certain positions. In the first section of this paper we give a computational scheme that enables us to carry out such operations in a rapid and simple fashion.

In the second section of the paper we derive a general identity involving the digits of integers in arbitrary Cantor systems of notation.

In the third section we apply this identity and deduce a number of results concerned with the splitting of integers into classes with equal power sums. The computational scheme of the first section leads us to an algorithm for the determination of such splittings.
Some Sphere Packings in Higher Space
Authors: Leech John
Published: December 2010

This paper is concerned with the packing of equal spheres in Euclidean spaces [n] of n > 8 dimensions. To be precise, a packing is a distribution of spheres any two of which have at most a point of contact in common. If the centres of the spheres form a lattice, the packing is said to be a lattice packing. The densest lattice packings are known for spaces of up to eight dimensions (1, 2), but not for any space of more than eight dimensions. Further, although non-lattice packings are known in [3] and [5] which have the same density as the densest lattice packings, none is known which has greater density than the densest lattice packings in any space of up to eight dimensions, neither, for any space of more than two dimensions, has it been shown that they do not exist.
The Isometries of Hp
Authors: Forelli Frank
Published: December 2010

Let a be the Lebesgue measure on the unit circle |z| = 1 with

and let Lp be the space of complex-valued σ-measurable functions f such that

is finite. Hp is the closure in Lp of the algebra of analytic polynomials
Derivative of Singular Set-Functions
Authors: Anvari Morteza
Published: December 2010

The purpose of this paper is to prove that the general derivative of a completely additive singular set-function defined on certain measurable subsets of an abstract measure space is zero almost everywhere. As a corollary the celebrated Lebesgue decomposition theorem has been sharpened.

This result is well known for set-functions defined on measurable subsets of an n-dimensional Euclidean space (2, p. 119). The proof in this setting depends on two things: Vitali's covering theorem and the fact that for every measurable set A there exists an open set O which contains A and the images of O and A under the set-function can be made arbitrarily close. Here the covering theorem is due to Trjitzinsky and the open set is replaced by an envelope, an entirely measure-theoretic concept.
A General Perron Integral
Authors: Bullen P. S.
Published: December 2010

In this paper integrals are considered from the point of view of inverting differential operators. In order to do this it is necessary to introduce integrals more general than the Lebesgue integral and these integrals turn out to have other interesting properties (6, 7, 12). The integral introduced here is defined in the setting of axiomatic potential theory (2, 4). By defining it as generally as possible it not only includes the James P2-integral but inverts many of the standard second-order differential operators.
A Class of Translation Lattices
Authors: Taylor John C.
Published: December 2010

Let E be a completely regular space, let C* denote the Banach algebra of continuous bounded real-valued functions on E, and let A* denote a Banach subalgebra which contains the constant function 1. Let

The purpose of this note is to discuss some properties of the subsets L(A*) of the ring C of continuous real-valued functions on E. They arose out of an unsuccessful attempt to distinguish C* from its Banach subalgebras.
Uniform and Equicontinuous Schauder Bases of Subspaces
Authors: McArthur Charles W. , Retherford James R.
Published: December 2010

A sequence {Mi} of non-trivial subspaces of a linear topological space X is a basis of subspaces for X if and only if corresponding to each x ∊ X there is a unique sequence {xi}, xi ∊ Mi, such that

Corresponding to a basis of subspaces {Mi} for X is a sequence of orthogonal projections {Ei} (Ei2 = Ei and EiEj = 0 if i ≠ j) defined by Ei(x) = xi if
On the Degenerate Cauchy Problem
Authors: Carroll R. W. , Wang C. L.
Published: December 2010

The problem treated here is an abstract version of the Cauchy problem for an equation of mixed type in the hyperbolic region with initial data on the parabolic line (cf. 2, 3, 5, 11, 13, 14, 15, 16, 21, 27). A more complete bibliography may be found in (3, 5, 18). We begin with the equation (6)

(1.1)
Refinement-Unbounded Interval Functions and Absolute Continuity
Authors: Appling William D. L.
Published: December 2010

In this paper we prove the following characterization theorem (Section 3) :

Theorem 1. If each of g and m is a real-valued non-decreasing function on the number interval [a, b], then the following two statements are equivalent: (1) If R is a real-valued, refinement-unbounded (Section 3) function of subintervals of [a, b], then the integral (Section 2)

exists and is equal to g (b) — g (a), and

(2) g is absolutely continuous with respect to m.
-Homogeneity of Projective Planes and Polarities
Authors: Jónsson W.
Published: December 2010

In (1) Baer introduced the concept of (C, γ)-transitivity and (C, γ)- homogeneity. A projective plane (see (5) for the requisite definitions and axioms) is (C, γ)-transitive if, given an ordered pair (P1, P2) of points collinear with C but distinct from C and not on γ, there is a collineation which maps P1 into P2 and leaves fixed every point on γ as well as every line through C. A projective plane is (C, γ)-homogeneous for a non-incident point-line-pair if it is (C, γ) transitive and there is a correlation which maps every line through C into its intersection with γ and every point on γ into its join with C.
On Independent Circuits Contained in a Graph
Authors: Erdös p. , Pósa L.
Published: December 2010

A family of circuits of a graphGis said to beindependentif no two of the circuits have a common vertex; it is callededge-independentif no two of them have an edge in common. A set of vertices will be called arepresentingset for the circuits (for the sake of brevity we shall call it a representing set), if every circuit ofGpasses through at least one vertex of the representing set. Denote byI(G) =kthe maximum number of circuits in an independent family and byR(G) the minimum number of vertices of a representing set. Dirac and Gallai asked whether there is any relation betweenI(G) andR(G) (triviallyR(G) ≥I(G)).
The Double Transitivity of a Class of Permutation Groups
Authors: Bercov Ronald D.
Published: December 2010

Certain finite groups H do not occur as a regular subgroup of a uniprimitive (primitive but not doubly transitive) group G. If such a group H occurs as a regular subgroup of a primitive group G, it follows that G is doubly transitive. Such groups H are called B-groups (8) since the first example was given by Burnside (1, p. 343), who showed that a cyclic p-group of order greater than p has this property (and is therefore a B-group in our terminology).

Burnside conjectured that all abelian groups are B-groups. A class of counterexamples to this conjecture due to W. A. Manning was given by Dorothy Manning in 1936 (3).
The k-Normal Completion of Function Lattices
Authors: Cohen Henry B.
Published: December 2010

A subset G of a non-empty partially ordered set C is called normal if it coincides with the set of all upper bounds of the set of lower bounds of G. This is equivalent to stipulating that G be the set of all upper bounds of some subset of C called a set of generators for G. When ordered by inclusion, the family of all normal subsets of C forms a complete lattice with maximum C and minimum empty or singleton. The meet operation is simply point set intersection; whence, the meet of a family Gi of normal subsets is the set of upper bounds of ∪ Fi where Fi generates Gi for each i. A normal subset is called proper if it is neither void nor C, and the proper normal subsets. of C form a boundedly complete lattice.
A Theorem on k-Saturated Graphs
Authors: Hajnal A.
Published: December 2010

In this paper we consider finite graphs without loops and multiple edges. A graph is considered to be an ordered pair 〈G, *〉 where G is a finite set the elements of which are called the vertices of while * is a subset of [G]2 (where [G]2 is the set of all subsets of two elements of G). The elements of * are called the edges of . If {P, Q} ∊ *, we say that Q is adjacent to P. The degree of a vertex is the number of vertices adjacent to it. Let k be an integer. We say that is the complete k-graph if G has k elements and * = [G]2.
A Filter Description of the Homomorphisms of H∞
Authors: Kerr-Lawson A.
Published: December 2010

The algebra of bounded analytic functions on the open unit disk D, usually written H∞ i is a commutative Banach algebra under the supremum norm. Since its compact maximal ideal space M (space of complex homomorphisms) is an extension space of the unit disk, there must be a continuous mapping form βD, the Stone-Čech compactification of D, onto M. R. C. Buck has remarked (4), that this mapping fails to be one-one, in the light of a classical theorem of Pick. If the points of βD are represented by filters of subsets of D, we can identify those filters which are sufficiently close in terms of the hyperbolic metric on D in an attempt to get a one-one correspondence between filters and points of M.
Topological H-Surfaces
Authors: Helfenstein H. G.
Published: December 2010

An H-space is a topological space T for which it is possible to define a continuous binary composition

with the following properties: there exists a homotopy unit, i.e. an element u ∊ T such that
Immersions of Metric Spaces into Euclidean Spaces
Authors: Akasaki Takeo
Published: December 2010

In a recent paper on isotopy invariants (1), S. T. Hu denned the enveloping space Em(X) of any given topological space X for each integer m > 1. By an application of the Smith theory to the singular cohomology of the enveloping space Em(X), he obtained his immersion classes for every n = 1, 2, 3, . . . and proved (3) the main theorem that a necessary condition for a compact metric space X to be immersible into the ^-dimensional Euclidean space Rn is . This theorem was proved earlier by W. T. Wu (4) for finitely triangulable spaces X using purely combinatorial methods.
Majorants in Variational Integration
Authors: Henstock Ralph
Published: December 2010

In Perron integration, majorants are usually functions of points. If the domain of definition is a Euclidean space ofndimensions, we can define a finitely additiven-dimensional majorant rectangle function by taking suitable differences of the majorant point function with respect to each of thencoordinates. The way is then open to a generalization, in that we need only suppose that the majorant rectangle function is finitely superadditive. Similarly, we need only suppose that a minorant rectangle function is finitely subadditive. These kinds of rectangle functions were used by J. Mařík (5) to prove the Fubini theorem for Perron integrals in Euclidean space ofm+ndimensions. He also proved that for a function that is Perron, and absolutely Perron, integrable, the majorant and minorant rectangle functions can be taken to be finitely additive. As a result he posed the following problem.
Arithmetic Invariants of Subdivision of Complexes
Authors: Wall C. T. C.
Published: December 2010

The following problem was raised by M. Brown. Let K be a finite simplicial complex, of dimension n, with αi(K) simplexes of dimension i. Which of the linear combinations have the property that they are unaltered by all stellar subdivisions of K? The most obvious invariant is the Euler characteristic; there are also some identities that hold for manifolds (2), so, if K is a manifold, they remain true on subdivision. We shall see that no other expressions are ever invariant, but if K resembles a manifold in codimensions ⩽2r (in a sense defined below) that r of the relations continue to hold.
Term Rank of the Direct Product of Matrices
Authors: Brualdi Richard A.
Published: December 2010

Let A = [aij] be a matrix of 0's and 1's or a (0, 1)-matrix of size m by m′. The term rank of A is denned as the maximal number of 1's of A with no two of the 1's on the same row or colunn. A theorem due to D. König (3, Theorem 5.1, p. 55) asserts that the term rank of A is also equal to the minimal number of rows and columns of A that collectively contain all the 1's. The term rank of A will be denoted by ρ(A). Obviously it is invariant under arbitrary permutations of the rows and columns of A. We assume without loss of generality that all matrices considered have no rows or columns consisting entirely of 0's.
On the Group of a Directed Graph
Authors: Hemminger Robert L.
Published: December 2010

In 1938, Frucht (2) proved that for any given finite group G there exists a finite symmetric graph X such that G(X) is abstractly isomorphic to G. Since G(X) is a permutation group, it is natural to ask the following related question : If P is a given finite permutation group, does there exist a symmetric (and more generally a directed) graph X such that G(X) and P are isomorphic (see Convention below) as permutation groups? The answer for the symmetric case is negative as seen in (3) and more recently in (1). It is the purpose of this paper to deal with this problem further, especially in the directed case. In §3, we supplement Kagno's results (3, pp. 516-520) for symmetric graphs by giving the corresponding results for directed graphs.
Sieve-Generated Sequences
Authors: Wunderlich M. C.
Published: December 2010

We shall consider a generalization of the sieve process introduced by W. E. Briggs (1) in 1963. Let A(1) be the sequence {ak(1)}, where ak(1) = k + 1, so that A(1) = {2, 3, 4, … }. Suppose inductively that A(1), A(2), … , A(n) has been defined. 4(n+1) will be defined from A(n) = {a1(n), a2(n), a3(n), …} in the following manner: For each integer t ⩾ 0, choose an arbitrary element αt(n) from the set where an = an(n), and delete the elements αt(n) from A(n) to form A(n+1). The sequence A is defined to be the sequence {an}. It is also the set-theoretic intersection of all the sequences A(n), n — 1, 2, … . Let be the class of all sequences that can be generated by this sieve process.
Elementary Factorization in π-Regular Rings
Authors: Steger Arthur
Published: December 2010

This paper extends the results of A. L. Foster (1) on elementary factorization in Boolean-like rings to commutative π-regular rings. After proving some preliminary lemmas we proceed to the partition of the set of non-units of a π-regular ring into irreducible and composite elements. Finally, we prove a number of theorems concerning factorization rings, weakly unique factorization rings, principal ideal rings, etc. The principal result is that a π-regular ring is a weakly unique factorization ring if and only if it is a principal ideal ring.
A Characterization of Machine Mappings
Authors: Ginsburg Seymour , Rose Gene F.
Published: December 2010

A generalized sequential machine (abbreviated gsm) is a 6-tuple (K, ∑, Δ, δ, λ, p1), where K, ∑, Δ are finite non-empty sets (of “states,” “inputs,” and “outputs” respectively), δ (the “next state” function) is a mapping of K X ∑ into K, λ (the “output” function) is a mapping of K X ∑ into Δ*, and p1 (the “start“ state) is a distinguished element of K.
Structure of Semigroups
Authors: Hoehnke Hans-Jürgen
Published: December 2010

The treatment of semigroups given in a previous paper (3) is based upon representations of a semigroup by means of transformations of a set (cf. also 12). In this paper we try to remove the assumption of the existence of a zero element proposed in (3). In accordance with our general programme explained at the beginning of (3) we utilize certain minimum conditions in order to gain more information on the structure of semigroups.

Our main results are structure theorems on primitive semigroups which have irreducible right ideals generated by idempotents (§§15-17). As we have shown in (5), these theorems permit the explicit construction of primitive semigroups.
Inversion Theorems for the Laplace-Stieltjes Transform
Authors: Ditzian Z. , Jakimovski A.
Published: December 2010

The Laplace-Stieltjes transform f(x) of the function α(t) is denned by

1

where α(t) is a function of bounded variation in each closed interval [0, R] (R > 0) and the right-hand side of (1) is supposed to be convergent for some x = X0. The Laplace transform f(x) of the function ϕ(t) is defined by

2

where ϕ(t) ∈ L1(0, R) for each R > 0 and the right-hand side of (2) is supposed to be convergent for some x = x0.
On the Integral Part of a Linear form with Prime Variables
Authors: Danicic I.
Published: December 2010

The object of this paper is to prove the following:

Theorem. Suppose that λ, μ are real non-zero numbers, not both negative, λ is irrational, and k is a positive integer. Then there exist infinitely many primes p and pairs of primes p1, p2 such that

In particular [λp1 + μp2] represents infinitely many primes.

Here [x] denotes the greatest integer not exceeding x.
Invariant Sub-Bundles of the Tangent Bundle of a Homogeneous Space
Authors: Tondeur Philippe
Published: December 2010

Let M = G/H be the homogeneous space of a Lie group G and a closed subgroup H. Denote by p : G → G/H the canonical projection, e ∈ G the identity and x0 = p(e). Let W be a subspace of the tangent space Tx0(M).

Definition. A lift W* of W is a subspace of the Lie algebra of G satisfying ∩ W* = ﹛0﹜ and p*W* = W, where p* : → Tx0(M) denotes the tangent map of p at e.

Consider a G-invariant sub-bundle of the tangent bundle of M (4), i.e., a field of vector subspaces x ⊂ Tx(M) for every x ∈ M satisfying

1

Here μg : M → M denotes the diffeomorphism defined by g ∈ G and (μg)*x : Tx → Tμg(x) the induced tangent map at x.
Series Expansions of Generalized Temperature Functions in N Dimensions
Authors: Haimo Deborah Tepper
Published: December 2010

The generalized heat equation is given by

1.1

where Δxf(x) = f″(x) + (2v/x)f′(x), v a fixed positive number. In a recent paper (5), the author established criteria for representing solutions of (1.1) in either the form

1.2

or

1.3

where Pn,v(x, t) is t he polynomial solution of (1.1) given explicitly by

1.4

and Wn,v(x, t) is its Appell transform; cf. (1). Our object is to generalize these results by extending them to higher dimensions. D. V. Widder (8) studied the problem for the ordinary heat equation.
Semi-Prime Modules
Authors: Feller E. H. , Swokowski E. W.
Published: December 2010

Properties and characterizations for prime and semiprime rings have been provided by A. W. Goldie (2, 3). In a previous paper (1), the authors used the results of (2) to characterize prime and uniform prime modules. It is the aim of the present paper to generalize Goldie's work on semi-prime rings (3) to modules. In this setting certain new properties will appear.

Notationally, in the work to follow, the symbol R always denotes a ring and all R-modules will be right R-modules.

In the theory of rings an ideal C is said to be prime if and only if whenever AB ⊆ C for ideals A and B, then either A ⊆ C or B ⊆ C. A ring is prime if the zero ideal is prime.
Enumeration of Locally Restricted Digraphs
Authors: Harary Frank , Palmer Ed
Published: December 2010

Among the unsolved problems in graphical enumeration listed in (4) is included the determination of the number of graphs and digraphs with a given partition. Parthasarathy (9) has developed a formulation for counting graphs with a given partition by making a suitable modification of the method given in (2) for the enumeration of graphs. We present here an analogous modification that leads to a formula for the number of digraphs with a given partition. Not surprisingly, the main combinatorial device for this purpose is provided by the classical theorem due to Pólya.
Construction of Principal Functions by Orthogonal Projection
Authors: Nakai Mitsuru , Sario Leo
Published: December 2010

Given a point set E on an open Riemann surface V we denote by H(E) the space of functions u harmonic in open sets O(u) containing E. Let V0 be a regular region of V with border α, and consider restrictions f to α of functions in H(α). For V1 = V — ⊽0, an operator L from H(α) to H(⊽1) is, by definition, normal if

1

2

3

4

5
Extensions of Pseudometrics
Authors: Shapiro H. L.
Published: December 2010

If γ is an infinite cardinal number, a subset S of a topological space X is said to be Pγ-embedded in X if every γ-separable continuous pseudometric on S can be extended to a γ-separable continuous pseudometric on X. (A pseudometric d on X is γ-separable if there exists a subset G of X such that |G| ⩽ 7 and such that G is dense in X relative to the pseudometric topology A pseudometric d is continuous if d is continuous relative to the product topology on X × X.) We say that S is P-embedded in X if every continuous pseudometric on S can be extended to a continuous pseudometric on X.
Primary Ideals in Prüfer Domains
Authors: Ohm Jack
Published: December 2010

A Prüfer domain is an integral domainDwith the property that for every proper prime idealPofDthe quotient ringDPis a valuation ring. Examples of such domains are valuation rings and Dedekind domains, a Dedekind domain being merely a noetherian Prüfer domain. The integral closure of the integers in an infinite algebraic extension of the rationals is another example of a Prüfer domain (5, p. 555, Theorem 8). This third example has been studied initially by Krull (4) and then by Nakano (8).
Hermitian Varieties in a Finite Projective Space PG(N, q2)
Authors: Bose R. C. , Chakravarti I. M.
Published: December 2010

The geometry of quadric varieties (hypersurfaces) in finite projective spaces of N dimensions has been studied by Primrose (12) and Ray-Chaudhuri (13). In this paper we study the geometry of another class of varieties, which we call Hermitian varieties and which have many properties analogous to quadrics. Hermitian varieties are defined only for finite projective spaces for which the ground (Galois field) GF(q2) has order q2, where q is the power of a prime. If h is any element of GF(q2), then = hq is defined to be conjugate to h.
Half-Transitive Automorphism Groups
Authors: Isaacs I. M. , Passman D. S.
Published: December 2010

Let G be a finite group and A a group of automorphisms of G. Clearly A acts as a permutation group on G#, the set of non-identity elements of G. We assume that this permutation representation is half transitive, that is all the orbits have the same size. A special case of this occurs when A acts fixed point free on G. In this paper we study the remaining or non-fixed point free cases. We show first that G must be an elementary abelian g-group for some prime q and that A acts irreducibly on G. Then we classify all such occurrences in which A is a p-group.
Summability of the Heine and Neumann Series of Legendre Polynomials
Authors: Jakimovski Amnon
Published: December 2010

With a holomorphic function f(z) defined in a domain H which includes the closed interval [—1, 1] we associate the Neumann series

1

where Pn(z), Qn(t) are, respectively, the nth Legendre polynomials of the first and second kind and γ is a closed and rectifiable Jordan curve which includes [— 1, 1] in its interior and is included, together with its interior, in H.
Bounds for Solutions of a System of Linear Partial Differential Equations on Domains with Bergman-Silov Boundary
Authors: Mitchell Josephine
Published: December 2010

The method of integral operators has been used by Bergman and others (4; 6; 7; 10; 12) to obtain many properties of solutions of linear partial differential equations. In the case of equations in two variables with entire coefficients various integral operators have been introduced which transform holomorphic functions of one complex variable into solutions of the equation. This approach has been extended to differential equations in more variables and systems of differential equations. Recently Bergman (6; 4) obtained an integral operator transforming certain combinations of holomorphic functions of two complex variables into functions of four real variables which are the real parts of solutions of the system

1

where z1, z1*, z2, z2* are independent complex variables and the functions Fj (J = 1, 2) are entire functions of the indicated variables.
On the Construction of Sequence Spaces that have Schauder Bases
Authors: Ruckle William
Published: December 2010

It is known that every Banach space which possesses a Schauder basis is essentially a space of sequences (6, Section 11.4). The primary objectives of this paper are: (1) to illustrate the close connection between sectionally bounded BK spaces and Banach spaces which have a Schauder basis, and (2) to consider some results in these theories in such a way as to render them easy and natural. In order to reach the largest number of readers we shall use (6) as the sole basis of our discussion. References to other authors are made in order to direct the reader to the original source of a theorem or to a related discussion.
Laplace Transformations of Distributions
Authors: Cooper J. L. B.
Published: December 2010

In a previous paper (2), I discussed conditions in order that a holomorphic function f(w) be a Laplace transform of a function F(t) such that, for some a, F(t)e-at ∈ Lp(0, ∞). These conditions involved (a) the behaviour of integral transforms of the values of the function on a vertical line w = const, and (b) conditions involving the order of magnitude of the function at infinity. The present article discusses the corresponding question for Laplace transforms of distributions.
Complexes Over a Complete Algebra of Quotients
Authors: Tewari Krishna
Published: December 2010

Let R be a commutative ring with unit and A be a unitary commutative R-algebra. Let As be a generalized algebra of quotients of A with respect to a multiplicatively closed subset S of A. If (A) and (As) denote the categories of complexes and their homomorphisms over A and As respectively, then one easily sees that there exists a covariant functor T:(A) → (AS) such that T is onto and T(X, d) is universal over As whenever (X, d) is universal over A. Actually the category (AS) is equivalent to a subcategory of (A) where contains all those complexes (X, d) over A such that for each s in S, the module homomorphism ϕs: x → sx of Xn into itself is one-one and onto for each n ⩾ 1.