bluuebunny/crossref_metadata_2025 huggingface dataset:
The Analytic Continuation of the Riemannliouville Integral in the Hyperbolic Case
Authors: Riesz Marcel
Published: December 2010

In 1949 I published in the Acta Mathematica (vol. 81) a rather long paper: “L'intégrale de Riemann-Liouville et le problème de Cauchy.” This work will be quoted in the sequel as Acta paper. Only minor local references to this paper will be made here, and knowledge of it is not required for the reading of the present article. The notations used here are slightly different from those used in my former paper.

In the Acta paper I introduce multiple integrals and of the Riemann- Liouville type depending on a parameter α and converging for sufficiently large values of α. I give the solution of the Cauchy problem for the wave equation in a unique formula, the same for space-time of odd or even dimensions, implying an analytic continuation with respect to the parameter α.
The Groups of Regular Complex Polygons
Authors: Crowe D. W.
Published: December 2010

The two-dimensional unitary space, U2, is a complex vector space of points (x, y) = (x1 + ix2, y1 + iy2), for which the distance between (x, y) and (x', y') is defined by . A unitary transformation is a linear transformation which preserves distance. A line is the set of points (x, y) satisfying some complex equation ax + by = c. A unitary transformation is a (unitary) reflection if it is of finite period n > 1 and leaves a line pointwise invariant. Thus à unitary matrix represents a reflection if its two characteristic roots are 1 and a complex nth root (n > 1) of 1.
On Convex Fundamental Regions for a Lattice
Authors: Macbeath A. M.
Published: December 2010

Let ∧ be a lattice in Euclidean n-space, that is, ∧ is a set of points ε1a1+ … + εnanwherea1… ,anare linearly independent vectors and the ε run over all integers. Letμdenote the Lebesgue measure. A closed convex setFis called afundamental regionfor ∧ if the setsF+x (x∈∧) cover the whole space without overlapping; that is, ifF0is the interior ofF,and 0 ≠x∈ ∧, thenF0∩(F0+x) =ϕ.
Longest Increasing and Decreasing Subsequences
Authors: Schensted C.
Published: December 2010

This paper deals with finite sequences of integers. Typical of the problems we shall treat is the determination of the number of sequences of length n, consisting of the integers 1, 2, … , m, which have a longest increasing subsequence of length α. Throughout the first part of the paper we will deal only with sequences in which no numbers are repeated. In the second part we will extend the results to include the possibility of repetition. Our results will be stated in terms of standard Young tableaux.
Widths and Heights of (0,1) -Matrices
Authors: Fulkerson D. R. , Ryser H. J.
Published: December 2010

A number of combinatorial problems may be regarded as particular instances of the following rather general situation. Given a setXcomposed ofnelements x1, x2, ..., xn, andmsubsets X1X2, … ,XmofX, find a minimal system of representatives forX1, X2, … ,Xm. That is, single out a subset X* ofXsuch thatXi∩ X* is non-empty fori = 1,2, … ,m, and no subset ofXcontaining fewer elements than X* has this property. To illustrate, each of the following can be thought of in these terms.
Distributive Sublattices of a Free Lattice
Authors: Galvin Fred , Jónsson Bjarni
Published: December 2010

The purpose of this note is to characterize those distributive lattices that can be isomorphically embedded in free lattices. If it is known (cf. (2)) that in a free lattice every element is either additively or multiplicatively irreducible, and consequently every sublattice of a free lattice must also have this property. We therefore begin by studying the class of all those distributive lattices in which this condition is satisfied.
Integration of Subspaces Derived from a Linear Transformation Field
Authors: Kobayashi Edward T.
Published: December 2010

The problem we study is a generalization of a problem first solved by Tonolo (6), then generalized successively by Schouten (5), Nijenhuis (4), Haantjes (3), and Nijenhuis-Frölicher (2). The Tonolo- Schouten approach is distinct from that of Nijenhuis-Haantjes-Frölicher in the sense that the former consider the problem on a Riemannian space, while the latter consider it on a manifold without any further structure.

The object of investigation is the integrability of the distribution θ of vector subspaces θP of the tangent space Tp to a manifold M, when θP is intrinsically related to a given field h on M, of linear transformations hp on Tv. The research has so far been restricted to certain types of h.
On the Structure of Semi-Prime Rings and their Rings of Quotients
Authors: Lambek Joachim
Published: December 2010

We are mainly interested in the study of prime and semi-prime rings and their rings of quotients. However, our argument proceeds largely in the category of modules (§ 1 to 4) and bimodules (§ 5 to 7).

After a brief description of the generalized rings of quotients introduced recently by Johnson, Utumi, and Findlay and the present author, we study a closure operation on the lattice of submodules of a module. For the lattice of left ideals of a ring, the concept of closed submodules reduces to the If-ideals of Utumi. The lattice of closed submodules of a module is always a complete modular lattice. We are specially interested in the case when it is a complemented lattice. This happens, in particular, when the singular submodule of Johnson and Wong vanishes. We consider the lattice of closed right ideals of a prime ring S and determine the maximal ring of right quotients of S in the case when this lattice has atoms.
Sur La Representation Integrale Des Certaines Operations Lineaires. IV Operations Lineaires Sur L'espace Lap
Authors: Dinculeanu N. , Foias C.
Published: December 2010

La première partie de cet article a un caractère introductif; on reprend l'étude des mesures vectorielles sur un espace localement compact, présenté dans (8) pour les espaces compacts.

Dans la seconde partie on donne un théorème de représentation intégrale d'une mesure vectorielle par rapport à sa variation (théorème 2), duquel on déduit une généralisation du théorème de Lebesgue-Nikodym (théorème 3) et une généralisation du théorème de Lebesgue sur la décomposition d'une mesure (théorème 5). Ces théorèmes ont été déjà démontrés par les auteurs sous des conditions un peu plus restrictives, (10) et (13), mais ici les démonstrations sont nouvelles, et l'ordre dont on les déduit l'un de l'autre, est inversée.
A Note on Induced Modules
Authors: Curtis Charles W.
Published: December 2010

In this paper, A denotes a ring with an identity element 1, and B a subring of A containing 1 such that B satisfies the left and right minimum conditions, and A is a finitely generated left and right B-module. The identity element 1 is required to act as the identity operator on all modules which we shall consider. For any left B-module V, there is a standard construction of a left A -module which is, roughly speaking, the smallest A -module containing V.
Real Analytic Functions on Product Spaces and Separate Analyticity
Authors: Browder Felix E.
Published: December 2010

Let f be a function on the product space V × W, where V and W are analytic manifolds, both either real or complex. The function f is said to be analytic (or bi-analytic) on V × W if it is analytic in the analytic structure induced on V × W by the corresponding structures on V and W. The function f is said to be separately analytic on V × W if, for each x in V, the function f(x, .) is analytic on W while, for each y in W, the function f (. ,y) is analytic on V. In the case of complex analytic manifolds, the classical theorem of Hartogs (3, chapter VII) states that the two notions of analyticity and separate analyticity are equivalent. For real analytic manifolds, it is known that such an equivalence does not hold, even if one adds the additional hypothesis that f is infinitely differentiate on V × W.
Uniform Continuity of Continuous Functions on Uniform Spaces
Authors: Atsuji Masahiko
Published: December 2010

Recently several topologists have called attention to the uniform structures (in most cases, the coarsest ones) under which every continuous real function is uniformly continuous (let us call the structures the [coarsest] uc-structures), and some important results have been found which closely relate, explicitly or implicitly, to the uc-structures, such as in the vS of Hewitt (3) and in the e-complete space of Shirota (7). Under these circumstances it will be natural to pose, as Hitotumatu did (4), the problem: which are the uniform spaces with the uc-structures? In (1 ; 2), we characterized the metric spaces with such structures, and in this paper we shall give a solution to the problem in uniform spaces (§ 1), together with some of its applications to normal uniform spaces and to the products of metric spaces (§ 2). It is evident that every continuous real function on a uniform space is uniformly continuous if and only if the uniform structure of the space is finer than the uniform structure defined by all continuous real functions on the space.
On Normed Algebras which Satisfy a Reality Condition
Authors: Aurora Silvio
Published: December 2010

Various results exist which permit real Banach algebras satisfying some sort of “reality condition” to be identified with the algebra of all continuous real-valued functions on a suitable compact space (or with the algebra of all continuous real-valued functions that * Vanish at infinity“ on a suitable non-compact, locally compact space in case the algebra has no unit). (Terminology follows (4), so that compact and locally compact spaces must be Hausdorff spaces, in addition to satisfying the usual requirements.) Kadison established such a result in (6, Theorem 6.6) for any Banach algebra with unit, if the algebra satisfies an appropriate reality condition and has a norm N such that N(x)2 — N(x2) for all x.
Study of Certain Similitudes
Authors: Wonenburger Maria J.
Published: December 2010

One important point in the determination of the automorphisms of the classical groups is the study of group-theoretic properties of the elements of order 2, that is, the involutions. The group of similitudes and the projective group of similitudes of a non-degenerate quadratic form Q are extensions of the orthogonal group and the projective orthogonal group, respectively. These extended groups may contain involutions which do not belong to the orthogonal group or the projective orthogonal group. To study the automorphisms of such groups, group-theoretic properties of some of these new involutions should be established.

The aim of this paper is to give some properties of a similitude T of ratio ρ whose square T2 is equal to the scalar multiplication by — p (it is always assumed that we are dealing with a field of characteristic ≠2). If ρ = — 1, T is an involution in the group of similitudes and for any ρ the coset of T in the projective group of similitudes is an involution.
Homotopy Theory in Abelian Categories
Authors: Kleisli Heinrich
Published: December 2010

The concept of homotopy for homomorphisms of modules, suggested by analogy with fundamental properties of the topological homotopy, was developed by Eckmann and Hilton (3; 9; 10). In the present paper, this concept of homotopy is generalized to additive categories with an additional structure, and the theory of homotopy, including in particular various exact homotopy sequences, is established in full detail. It may be helpful to the reader to realize that most of our theorems, concepts, and constructions have their analogues in the homotopy theory of topological spaces. Many of these analogues may be found in a recent paper by Eckmann and Hilton (4). Explicit references will be given at various places in the paper.
The Permanent Function
Authors: Marcus Marvin , May F. C.
Published: December 2010

Let X be an n-square matrix with elements in a field F. The permanent of X is defined by

1.1

where σ runs over the symmetric group of permutations on 1, 2, … , n. This function makes its appearance in certain combinatorial applications (13), and is involved in a conjecture of van der Waerden (6; 9). Certain formal properties of per (X) are known (1), and an old paper of Pólya (12) shows that for n > 2 one cannot multiply the elements of X by constants in any uniform way so as to convert the permanent into the determinant. In a subsequent paper we intend to investigate this problem for more general operations on X.
Representations Subduced on an Ideal of a Lie Algebra
Authors: Noonan B.
Published: December 2010

This paper considers the properties of the representation of a Lie algebra when restricted to an ideal, the subduced* representation of the ideal. This point of view leads to new forms for irreducible representations of Lie algebras, once the concept of matrices of invariance is developed. This concept permits us to show that irreducible representations of a Lie algebra, over an algebraically closed field, can be expressed as a Lie-Kronecker product whose factors are associated with the representation subduced on an ideal. Conversely, if one has such factors, it is shown that they can be put together to give an irreducible representation of the Lie algebra. A valuable guide to this work was supplied by a paper of Clifford (1).
On the Measure of Sets of Parallel Linear Subspaces in Affine Space
Authors: Santaló L. A.
Published: December 2010

LetEnbe then-dimensional euclidean real space andthe group of unimodular affine transformations which operates on it. It is known that the sets of linearh-spacesLh(0 <h < n) have no invariant measure with respect tou(5). We wish now to consider sets of elements

1.1

composed byqparallel subspaces of dimensionsh1, h2, … ,hqwhich transform transitively by.
An Integral Over Function Space
Authors: Eberlein W. F.
Published: December 2010

Real functions

may be identified with elements x = (x0, x1, x2, … ) of the sequence space l1 Since the unit sphere S∞ of l1 is compact under the weak* topology = topology of co-ordinatewise convergence, a countably additive measure on S∞ is induced by a positive linear functional E (integral) on C(S∞), the weak* continuous real-valued functions on S∞. There exists a natural integral over S∞ reducing to

when f is a function of x0 alone. The partial sums Sn = Sn(x) of the power series for x(t) then form a martingale and zero-or-one phenomena appear. In particular, if R(x) is the radius of convergence of the series and e is the base of the natural logarithms, it turns out that R(x) = e for almost all x in S∞. Applications of the integral to the theory of numerical integration, the original motivation, will appear in a later paper.
A Census of Hamiltonian Polygons
Authors: Tutte W. T.
Published: December 2010

In this paper we deal with trivalent planar maps in which the boundary of each country (or “face“) is a simple closed curve. One vertex is distinguished as the root and its three incident edges are distinguished as the first, second, and third major edges. We determine the average number of Hamiltonian polygons, passing through the first and second major edges, in such a “rooted map” of 2n vertices. Next we consider the corresponding problem for 3-connected rooted maps. In this case we obtain a functional equation from which the average can be computed for small values of n.
Prime Ideals in Vector Lattices
Authors: Johnson D. G. , Kist J. E.
Published: December 2010

Projectors, spectral functions, carriers, and collections of these objects are some of the tools which have been used to study vector lattices. One of our objectives in this paper is to show that these various approaches are not essentially different. We do this by proving that each of the above-mentioned objects can be identified with a collection of prime ideals.
On a Class of Finitely Presented Groups
Authors: Macdonald I. D.
Published: December 2010

The groups in question are generated by elements A and B subject to the relations

in which α and β are fixed integers. We prove:

Theorem. Each group of the class just presented is finite when neither α nor β equals 1, and is nilpotent. Its order is a factor of 27 (α — 1) (β — l)∈8where ∈ is the greatest common divisor of α — 1 and β — 1, and its nilpotency class is at most 8.

We denote the commutator X-1Y-1XY by [X, Y], and Y-1XY by XY. If X1, X2, … , Xr are elements of some group then {X1, X2, … , Xr} will mean the subgroup which they generate.
Structure Theory of Complemented Banach Algebras
Authors: Tomiuk Bohdan J.
Published: December 2010

If A is an H*-algebra, then the orthogonal complement of a closed right (left) ideal I is a closed right (left) ideal P. Saworotnow (7) considered Banach algebras which are Hilbert spaces and in which the closed right ideals satisfy the complementation property of an H*-algebra. In our right complemented Banach algebras we drop the requirement of the existence of an inner product and only assume that for every closed right ideal I there is a closed right ideal IP which behaves like the orthogonal complement in a Hilbert space (Definition 1). Thus our algebras may be considered as a generalization of Saworotnow's right complemented algebras.
Fractional Integration and Dual Integral Equations
Authors: Erdélyi A. , Sneddon I. N.
Published: December 2010

In the analysis of mixed boundary value problems by the use of Hankel transforms we often encounter pairs of dual integral equations which can be written in the symmetrical form

(1.1)

Equations of this type seem to have been formulated first by Weber in his paper (1) in which he derives (by inspection) the solution for the case in which α — β = ½, v = 0, F ≡ 1, G ≡ 0.

The first direct solution of a pair of equations of this type was given by Beltrami (2) for the same values of α— β and v with G(p) ≡ 0 but with F(ρ) arbitrary.
A Census of Slicings
Authors: Tutte W. T.
Published: December 2010

A band is a closed connected set in the 2-sphere, bounded by one or more disjoint simple closed curves.

Consider a band B with bounding curves J1, J2, … , Jk. On each curve Ji let there be chosen mi ≥ 0 points to be called vertices, with the restriction that the sum of the k integers mi is to be even. Write

(1)

Next consider a set of n disjoint open arcs in the interior of B which join the 2n vertices in pairs and partition the remainder of the interior of B into simply connected domains. We call the resulting dissection of B a slicing with respect to the given set of vertices. The arcs are the internal edges of the slicing and the simply connected domains are its internal faces, or slices.
The Distribution of Sequences and Summability
Authors: Dowidar A. F. , Petersen G. M.
Published: December 2010

We suppose that 0 <sn≤ 1 for everyn, and denote byn(α, β) the number ofS0, S1,S2, . . . ,Snwhich fall in the interval 0 ≤ α <x≤ β ≤ 1. If there exists a functiong(t), 0 ≤t≤ 1, such that

for every interval (α, β] with 0 ≤ β — α ≤ 1, the sequence (Sn) is said to have a distribution functiong(t), 0 ≤t≤ 1, in the interval [0, 1], (see 9, p. 87).
Relations Among Generalized Hadamard Matrices, Relative Difference Sets, and Maximal Length Linear Recurring Sequences
Authors: Butson A. T.
Published: December 2010

It was established in (5) that the existence of a Hadamard matrix of order 4t is equivalent to the existence of a symmetrical balanced incomplete block design with parameters v = 4t — 1, k = 2t — 1, and λ = t — 1. A block design is completely characterized by its so-called incidence matrix. The existence of a block design with parameters v, k, and λ such that the corresponding incidence matrix is cyclic was shown in (3) to be equivalent to the existence of a cyclic difference set with parameters v, k, and λ. For certain values of the parameters, Hadamard matrices, block designs, and difference sets do coexist.
Union Curves in a Subspace Vn of a Riemannian Vm
Authors: Singal M. K.
Published: December 2010

A union curve on a surface in a euclidean 3-space, relative to a given congruence is characterized by the property that its osculating plane at each point contains the ray of the congruence through that point. Springer (2) and Pan (1) have studied union curves in a hypersurface Vn of a Riemannian Vn+1. In the present paper we proceed to obtain the equations of union curves in a subspace Vn of a Riemannian Vm.
Multipliers of Difference Sets
Authors: Newman Morris
Published: December 2010

Let λ, K, v be integers such that 0 < λ < k < v. Then the set of integers

is a difference set with parameters v, k, λ if each non-zero residue modulo v occurs precisely λ times and zero occurs precisely k times among the k2 numbers
On the "Edge of the Wedge" Theorem
Authors: Browder Felix E.
Published: December 2010

In the mathematical justification of the formal calculations of axiomatic quantum field theory and the theory of dispersion relations, a strategic role is played by a theorem on analytic functions of several complex variables which has been given the euphonious name of the edge of the wedge theorem. The statement of the theorem seems to be due originally to N. Bogoliubov (cf. 3, Mathematical Appendix, pp. 654-673) but no complete proof which is fully satisfactory from the mathematical point of view has yet appeared in the literature.
Properties of Harmonic Functions of Three Real Variables Given by Bergman-Whittaker Operators
Authors: Mitchell Josephine
Published: December 2010

Let be a closed rectifiable curve, not going through the origin, which bounds a domain Ω in the complex ζ-plane. Let X = (x, y, z) be a point in three-dimensional euclidean space E3 and set

The Bergman-Whittaker operator defined by
Pairs of Consecutive Power Residues
Authors: Lehmer D. H. , Lehmer Emma , Mills W. H.
Published: December 2010

Until recently none of the numerous papers on the distribution of quadratic and higher power residues was concerned with questions of the following sort: Let k and m be positive integers. According to a theorem of Brauer (1), for every sufficiently large prime p there exist m consecutive positive integers r, r + 1 , . . . , r + m — 1, each of which is a kth power residue of p. Let r(k, m, p) denote the least such r.
Enumeration of Bicolourable Graphs
Authors: Harary Frank , Prins Geert
Published: December 2010

In a previous paper (2), one of us has derived a formula for the counting series for bicoloured graphs.2 These are graphs each of whose points has been coloured with exactly one of two colours in such a way that every two adjacent points have different colours.

In this paper we first enumerate bicoloured graphs without isolated points and connected bicoloured graphs. This leads us to corresponding problems for bicolourable graphs. Such a graph has the property that its points can be coloured with two colours so as to obtain a bicoloured graph.
Entropies of Sets of Functions of Bounded Variation
Authors: Clements G. F.
Published: December 2010

In this paper the entropies of several sets of functions of bounded variation are calculated. The entropy of a metric set, a notion first introduced by Kolmogorov in (2), is a measure of its size in terms of the minimal number of sets of diameter not exceeding 2∊ necessary to cover it. Using this notion, Kolmogorov (4; p. 357) and Vituškin (7) have shown that not all functions of n variables can be represented by functions of fewer variables if only functions satisfying certain smoothness conditions are allowed.
Tauberian Theorems for Integrals
Authors: Henstock Ralph
Published: December 2010

When, for the generalized summation of series, we use A and B methods, giving A and B sums, respectively, we say that the A method is included in the B method, A ⊂ B, if the B sum exists and is equal to the A sum whenever the latter exists. A theorem proving such a result is called an Abelian theorem. For example, there is an Abelian theorem stating that if the A and B sums are the first Cesàro mean and the Abel mean, respectively, then A ⊂ B. If A ⊂ B and B ⊂ A, we say that A and B are equivalent, A = B. For example, the nth Hölder and nth. Cesàro means are equivalent.
Boolean Averages
Authors: Wright Fred B.
Published: December 2010

The purpose of this note is to investigate the properties of a mapping α, of a Boolean algebra A into itself, which satisfies the functional equation α(p·aq) = ap·aq, where the multiplication is infimum. This is the so-called averaging identity of Kampé de Fériét. Garrett Birkhoff has noted (2) the occurrence of this identity both in the statistical theory of turbulence and in mathematical logic. That this is more than coincidence is shown by the existence of an explicit connection between the notion of quantification in logic and certain non-linear "extremal operators" in function algebras (16). Certain linear operators are associated in a natural way with these extremal operators.
Parabolic Differentiation
Authors: Lane N. D.
Published: December 2010

The purpose of this paper is the study of parabolically differentiable points of arcs in the real affine plane. In Section 2, two different definitions of convergence of a family of parabolas are given and it is observed (Theorem 1) that these are equivalent. In Section 3, tangent parabolas at a point p of an arc A are discussed and it is proved (Theorem 2) that all the non-degenerate non-tangent parabolas of A through p intersect A at p or that all of them support. In Section 4, osculating parabolas are introduced and the condition that an arc be twice parabolically differentiable at a point p is stated.
On Groups of Order p3
Authors: Cohn James A. , Livingstone Donald
Published: December 2010

The simplest example of two non-isomorphic groups with the same character tables is provided by the non-abelian groups of order p3, p ≠ 2. Let G1 be the one of exponent p and let G2 be the other. If Q denotes the field of rational numbers, then Berman (2) has shown that QG1 ≈ QG2, where QGi denotes the rational group algebra. In this note we shall show that the corresponding statement is false for ZGi where Z is the ring of rational integers. More explicitly we shall show that ZG1 does not contain a unit of order p2 so that it is impossible to embed ZG2 in ZG1.
The Number of Hexagons and the Simplicity of Geodesics on Certain Polyhedra
Authors: Grünbaum B. , Motzkin T. S.
Published: December 2010

The problem of determining the possible morphological types of convex polyhedra in three-dimensional Euclidean space E3 is well known to be quite hopeless. We lack not only any general way of determining whether there exists a convex polyhedron having as faces ƒ3 triangles, ƒ4 quadrangles, . . . , and ƒnn-gons, but even much more special questions of this kind seem to be rather elusive.
On Outwardly Simple Line Families
Authors: Ceder Jack
Published: December 2010

In (5) Hammer and Sobczyk defined an outwardly simple line family in the plane as a family of lines in the plane having the property that each point outside some given circle lies on exactly one line of the family, and they characterized planar outwardly simple line families as follows (5): the extended diameters of a convex body, whose boundary has no pair of parallel line segments in it, form an outwardly simple line family; moreover, each outwardly simple line family is the family of extended diameters of a convex body having constant width.
On Covering Systems
Authors: Mallory Donald J. , Sion Maurice
Published: December 2010

Differentiation of a set function μ with respect to another v at a point x involves taking the limit of the ratio μA/vA as A "converges" to x in some sense which requires that A belong to some family of sets N(x) (e.g. spheres with centre x). For the development of a reasonable theory of differentiation certain restrictions must be placed on the families N(x). The best-known restriction is that they form a Vitali system. However, other systems have been considered.
Counting Coloured Graphs II
Authors: Wright E. M.
Published: December 2010

As in my earlier paper (2), a graph on n labelled nodes is a set of n objects called "nodes" distinguishable from each other and a set (possibly empty) of "edges," i.e. pairs of distinct nodes. Each edge is said to join its pair of nodes and no edge joins a node to itself. By a k-colouring of the nodes of such a graph we mean a mapping of the nodes onto a set of k distinct colours, such that no two nodes joined by an edge are mapped onto the same colour. By a colouring of the edges of such a graph we mean a mapping of the edges onto a set of colours.
Motions of Matrix Rings
Authors: Keller Roy F.
Published: December 2010

Metric spaces in which the distances are not real numbers have been studied by several people (2, 3, 4, 7, 9). Any ringRtogether with a mapping,X—>ϕ(X), ofRinto a lattice A with 0 and 1 satisfying

is called a "lattice-valued ring," where the operations union, ∪, and intersection, ∩, are the usual lattice operations. The mapping ϕ is called a "valuation" andAis a 'Valuation lattice." IfRis a lattice-valued ring and a mappingdis defined by
Almost-Bounded Holomorphic Functions with Prescribed Ambiguous Points
Authors: Cargo G. T.
Published: December 2010

Let f be a function mapping the open unit disk D into the extended complex plane. A point ζ on the unit circle C is called an ambiguous point of f if there exist two Jordan arcs J1 and J2, each having an endpoint at ζ and lying, except for ζ, in D, such that
Extension Of Finite Projective Planes I. Uniform Hjelmslev Planes
Authors: Craig Robert T.
Published: December 2010

In his paper (3) on finite H-planes, Kleinfeld has defined invariants s and t for an H-plane π as follows: let P and k be any point and line of π such that P is incident with k, then let s be the number of non-neighbour points to P on k and let t be the number of neighbour points to P on k. He has shown that s and t are independent of the choice of P and k, t divides s, π has s2 + st + t2 points (lines), π* has order s/t, and, if t ≠ 1, s ≤ t2. In the case s = t2, π is called uniform and has the property that each pair of neighbour lines (points) has exactly t points (lines) in common.
Convolutions as Bilinear and Linear Operators
Authors: Edwards R. E.
Published: December 2010

Throughout this paper X denotes a fixed Hausdorff locally compact group with left Haar measure dx. Various spaces of functions and measures on X will recur in the discussion, so we name and describe them forthwith. All functions and measures on X will be scalarvalued, though it matters little whether the scalars are real or complex.

C = C(X) is the space of all continuous functions on X, Cc = Cc(X) its subspace formed of functions with compact supports. M = M(X) denotes the space of all (Radon) measures on X, Mc = MC(X) the subspace formed of those measures with compact supports. In general we denote the support of a function or a measure ξ by [ξ].
Linear Relations Between Higher Matrix Commutators
Authors: Khan Nisar A.
Published: December 2010

Let Mn denote the space of all n-square matrices over an algebraically closed field F. For A, B ∊ Mn, let

define the iterated commutators of A and B. Recently several research papers (1, 2, 4, and 5) have appeared on these commutators. In (1), Kato and Taussky have proved that for n = 2 the iterated commutators of A and B satisfy the linear relation
Arcs of Parabolic Order Four
Authors: Lane N. D.
Published: December 2010

This paper is concerned with some of the properties of arcs in the real affine plane which are met by every parabola at not more than four points. Many of the properties of arcs of parabolic order four which we consider here are analogous to the corresponding properties of arcs of cyclic order three in the conformai plane which are described in (1). The paper (2), on parabolic differentiation, provides the background for the present discussion.

In Section 2, general tangent, osculating, and superosculating parabolas are introduced. The concept of strong differentiability is introduced in Section 3; cf. Theorem 1. Section 4 deals with arcs of finite parabolic order, and it is proved (Theorem 2) that an end point p of an arc A of finite parabolic order is twice parabolically differentiable.
On Two Problems Concerning the Generalized Lototsky Transforms
Authors: Meir Amram
Published: December 2010

The generalized Lototsky transform (or the [F, dn] transform) of a sequence {sn} into a sequence {tn) was defined in (2) in the following way: Let {dn) (dn ≠ —1) be a fixed complex sequence and let

then the sequence {tn} is defined by

In a recent paper by V. F. Cowling and C. L. Miracle (1) on these transformations, the following two problems were left open.
On Anti-Commutative Algebras with an Invariant Form
Authors: Sagle Arthur A.
Published: December 2010

In this paper we consider anti-commutative algebras with an invariant form, that is, an algebra A over a field F such that

and A possesses a symmetric bilinear form f(x, y) such that

Lie and Malcev algebras (2, 3) are examples of such algebras and we shall consider generalizations of these algebras obtained by introducing commutation, x o y = xy — yx, as a new multiplicative operation in the non-commutative Jordan algebras of (1).
Tournaments With a Given Automorphism Group
Authors: Moon J. W.
Published: December 2010

The set of all adjacency-preserving automorphisms of the vertex set of a graph form a group which is called the (automorphism) group of the graph. In 1938 Frucht (2) showed that every finite group is isomorphic to the group of some graph. Since then Frucht, Izbicki, and Sabidussi have considered various other properties that a graph having a given group may possess. (For pertinent references and definitions not given here see Ore (4).) The object in this paper is to treat by similar methods a corresponding problem for a class of oriented graphs. It will be shown that a finite group is isomorphic to the group of some complete oriented graph if and only if it has an odd number of elements.
Construction of Certain Semi-Simple Groups*
Authors: Ree Rimhak
Published: December 2010

In (3), Chevalley constructed, for every field K and every semi-simple Lie algebra g over the complex number field, a group GK(g) by using a system of root vectors Xr which satisfies a certain condition (more precisely, (1.2) in Section 1). The main point of Chevalley's construction lies in the fact that the above-mentioned root vectors furnish a basis of g such that, for every root r, exp(t ad Xr) is represented by a matrix (Aij(t)) whose entries Aij(t) are polynomials in t with integral coefficients.
Homotopy and Isotopy Properties of Topological Spaces
Authors: Gottlieb Daniel H.
Published: December 2010

In 1961, S. T. Hu published a paper (1) in which he discussed the desirability of discovering those topological properties which are preserved under homotopy and isotopy equivalences. In that paper he gave general tests in terms of weakly hereditary and hereditary topological properties for homotopy and isotopy properties.

In this paper, general tests for homotopy and isotopy properties in terms of weakly hereditary properties and of a class of properties which the author calls open properties are given. In the last sections, we shall show the strong role played by the notions of dimension and separating subsets in forming isotopy properties.
The Solution of an Integral Equation
Authors: Fox Charles
Published: December 2010

This paper deals with the problem of finding the solutions of the integral equation

where the constant a and the function giipc) are both given and both assumed to be real, while the function f1(x) is to be determined.

On writing

equation (1) simplifies to
On Covering of Balanced Incomplete Block Designs
Authors: Hanani Haim
Published: December 2010

Given a set E of v elements and given positive integers k (k ≤ v) and λ, we understand by balanced incomplete block design (BIBD) B [k, λ, v] a system of blocks (subsets of E) having k elements each such that every pair of elements of E is contained in exactly λ blocks.

A necessary condition for the existence of a design B [k, λ, v] is known to be (4)
Collineations of Projective Moulton Planes
Authors: Pierce William A.
Published: December 2010

In the article "Moulton Planes" (10), I studied F. R. Moulton's construction over any field containing a multiplicative subgroup of index 2. In "Collineations of Affine Moulton Planes" (11), I determined the collineations between two arbitrary affine Moulton planes.

The purpose now is to describe the collineations between two projective Moulton planes. Since the affine collineations are known from (11), we are concerned with collineations mapping ideal lines onto ordinary lines. Notations and conventions of (10) and (11) are retained.
Projective Geometry in the One-Dimensional Affine Group
Authors: Schwerdtfeger Hans
Published: December 2010

The idea of considering the set of the elements of a group as a space, provided with a topology, measure, or metric, connected somehow with the group operation, has been used often in the work of E. Cartan and others. In the present paper we shall study a very special group whose space can be embedded naturally into a projective plane and where the straight lines have a very simple group-theoretical interpretation. It remains to be seen whether this geometrical embedding in a projective space can be extended to other classes of groups and whether the method could become an instrument of geometrical investigation, like co-ordinates or reflections. In the final section it is shown how a geometrical theorem may lead to relations within the group.
Semigroups Co-Ordinatizing Orthomodular Geometries
Authors: Foulis D. J.
Published: December 2010

In (2, 3, 4, and 5), the author has established a connection between orthomodular lattices and Baer *-semigroups. In brief, the connection is as follows. The lattice of closed projections of any Baer *-semigroup forms an orthomodular lattice. Conversely, if L is any orthomodular lattice, there exists a Baer *-semigroup S which co-ordinatizes L in the sense that L is isomorphic to the lattice of closed projections in S. In this note we shall assume that the reader is familiar with the results and the notation of the quoted papers.
Identities of Non-Associative Algebras
Authors: Marshall Osborn J.
Published: December 2010

In the first part of this paper we define a partial ordering on the set of all homogeneous identities and find necessary and sufficient conditions that an identity does not imply any identity lower than it in the partial ordering (we call such an identity irreducible). Perhaps the most interesting property established for irreducible identities is that they are skew-symmetric in any two variables of the same odd degree and symmetric in any two variables of the same even degree. The results of the first section are applied to commutative algebras in the remainder of the paper.
Completions of Quadrangles in Projective Planes II
Authors: Killgrove Raymond B.
Published: December 2010

This viewpoint of studying projective planes was given in my previous paper (12). It is discussed in other papers: Hall (4, 6, 7), Maisano (16), Lombardo-Radice (14, 15), Wagner (19). In particular, we consider how to make identifications in the free plane, or how identifications are forced when one begins with a non-degenerate quadrangle and makes free extensions of this quadrangle of a known plane. We shall continue to develop this topic using the notations and definitions of the previous paper (12). We consider the number of subplanes of certain planes, finding exact values in the cases of the known order-nine planes, and deriving a lower bound in a general case. We prove a theorem concerning the structure of all singly generated planes. We give an example to show that this structure is not definitive. Finally, we prove that a specific Walker-Knuth plane is singly generated.
A Distortion Theorem for Analytic Maps of Annuli
Authors: Castonguay C. E. , Helfenstein H. G.
Published: December 2010

Every abstract open Riemann surface can be made "concrete" (in the terminology of (1)) by considering it as a covering surface (in general branched) of the complex plane by means of a suitable projection map p. Since this covering map is not unique, it seems natural to single out some such maps by an extremal property. The use of Riemannian metrics compatible with the conformai structure on the given surface for the study of $1 is well known ; from the point of view of differential geometry it suggests an investigation of the distortion caused by p between such a metric ds^ and the Euclidean metric of .
Almost Multiplication Rings
Authors: Butts H. S. , Phillips R. C.
Published: December 2010

It is well known that an idealAin a. Dedekind domain has a prime radical if and only ifAis a power of a prime ideal. The purpose of this paper is to determine necessary and sufficient conditions in order that a commutative ring with unit element have this property and to study the ideal theory in such rings. Domains with unit element having the above property possess many of the characteristics of Dedekind domains (however, they need not be Noetherian) and will be referred to in this paper as "almost Dedekind domains"—these domains are considered in Section 1.
The Cancellation Law for Ideals in a Commutative Ring
Authors: Gilmer Robert W.
Published: December 2010

We say that therestricted cancellation lawfor ideals (RCL) holds in the commutative ringRif from the equationAB=AC, whereA,B, andCare ideals ofRandAB≠ (0), it follows thatB=C. RCL is a weakened form of thecancellation lawfor ideals (CL) : IfA,B, andCare ideals ofRsuch thatAB=ACandA≠ (0), thenB=C. A ring in which CL holds is an integral domain and in an integral domain, RCL is equivalent to CL.
The Structure of Powers of Non-Negative Matrices
Authors: Dulmage A. L. , Mendelsohn N. S.
Published: December 2010

The theory of non-negative matrices was initiated by Perron (7) and Frobenius (4). Wielandt in (11) gives an elegant exposition of the subject.

It is well known to workers in the field that if a matrixAhas all its entries non-negative real numbers, then the pattern of zeros and non-zeros ofAcompletely determines the pattern of zeros and non-zeros in every power ofA. Ptak in (8) and Ptak and Sedlacek in (9) describe this behaviour in terms of some combinatorial constructs.
On Integral Functions Having Prescribed Asymptotic Growth
Authors: Clunie J.
Published: December 2010

In this paper I shall prove the following theorem:

Theorem. Let ϕ(r) be increasing and convex in log r with

(This condition is imposed to exclude certain trivial cases.) Then there is an integral function f(z) such that
Paths, Trees, and Flowers
Authors: Edmonds Jack
Published: December 2010

A graph G for purposes here is a finite set of elements called vertices and a finite set of elements called edges such that each edge meets exactly two vertices, called the end-points of the edge. An edge is said to join its end-points.

A matching in G is a subset of its edges such that no two meet the same vertex. We describe an efficient algorithm for finding in a given graph a matching of maximum cardinality. This problem was posed and partly solved by C. Berge; see Sections 3.7 and 3.8.
On the (f,dn)-Method of Summability
Authors: Smith Gaston
Published: December 2010

Letf(z) be a non-constant entire function and let {dn} be a sequence of complex numbers such that

The set of equations

(1.1)
Maxima for Graphs and a New Proof of a Theorem of Turán
Authors: Motzkin T. S. , Straus E. G.
Published: December 2010

Maximum of a square-free quadratic form on a simplex. The following question was suggested by a problem of J. E. MacDonald Jr. (1):

Given a graph G with vertices 1, 2, . . . , n. Let S be the simplex in En given by xi ≥ 0, Σxi = 1. What is
On Multipliers of Difference Sets
Authors: McFarland Robert , Mann H. B.
Published: December 2010

Multipliers are useful in constructing difference sets as well as in impossibility proofs. The existence of a fixed set facilitates their application, so that the results of this paper should prove to be useful in numerical work concerning difference sets.

Let G be a group which we shall write multiplicatively. For any set A of elements of G, we define

(1)
On Anti-Commutative Algebras and Analytic Loops
Authors: Sagle Arthur A.
Published: December 2010

In (4) Malcev generalizes the notion of the Lie algebra of a Lie group to that of an anti-commutative "tangent algebra" of an analytic loop. In this paper we shall discuss these concepts briefly and modify them to the situation where the cancellation laws in the loop are replaced by a unique two-sided inverse. Thus we shall have a set H with a binary operation xy defined on it having the algebraic properties

(1.1) H contains a two-sided identity element e;

(1.2) for every x ∊ H, there exists a unique element x-1 ∊ H such that xx-1 = x-1x = e;
Sur L'Indice de Certains Réseaux de R4 Permis Pour un Octaèdre
Authors: Bantegnie Robert
Published: December 2010

Si un réseau G de Rn contient Zn et est permis pour l'«octaèdre généralisé» Ωn d'équation

l'indice de Zn dans G est borné supérieurement par n! d'après les théorèmes généraux de Minkowski.
Some Refinements of Lyapunov's Second Method
Authors: Brauer Fred
Published: December 2010

Lyapunov's second method is a well-known and powerful tool for studying the behaviour of solutions of a system of differential equations. One approach to the theory is the comparison method developed by Corduneanu (4). This approach has the advantage that it also leads to other results on asymptotic behaviour which originally appeared to be unrelated to Lyapunov's method. Some of these results have been obtained by the author in (2). The purpose of this paper is to make use of the comparison method to obtain some refinements of Lyapunov's theory.
Finite Groups in Which Sylow 2-Subgroups are Abelian and Centralizers of Involutions are Solvable
Authors: Gorenstein Daniel
Published: December 2010

The purpose of this paper is to establish the following theorem :

Theorem 1. Let be a finite group with abelian Sylow 2-subgroups in which the centralizer of every involution is solvable. Then either is solvable or else /O is isomorphic to a subgroup of PΓL(2, q) containing PSL(2, q), where either q = 3 or 5 (mod 8), q ≥ 5, or q = 2n, n ≥ 2.
Isotropic and Kähler Immersions
Authors: O'Neill Barrett
Published: December 2010

Let Md and be Riemannian manifolds. We shall say that an isometric immersion ϕ: Md —> is isotropic provided that all its normal curvature vectors have the same length. The class of such immersions is closed under compositions and Cartesian products. Umbilic immersions (e.g. Sd ⊂ Rd+1) are isotropic, but the converse does not hold. If M and are Kähler manifolds of constant holomorphic curvature, then any Kähler immersion of M in is automatically isotropic (Lemma 6). We shall find the smallest co-dimension for which there exist non-trivial immersions of this type, and obtain similar results in the real constant-curvature case.
Undirected Graphs Realizable as Graphs of Modular Lattices
Authors: Alvarez Laurence R.
Published: December 2010

If (L, ≥) is a lattice or partial order we may think of its Hesse diagram as a directed graph, G, containing the single edge E(c, d) if and only if c covers d in (L, ≥). This graph we shall call the graph of (L, ≥). Strictly speaking it is the basis graph of (L, ≥) with the loops at each vertex removed; see (3, p. 170).

We shall say that an undirected graph Gu can be realized as the graph of a (modular) (distributive) lattice if and only if there is some (modular) (distributive) lattice whose graph has Gu as its associated undirected graph.
Formally Normal Operators Having no Normal Extensions
Authors: Coddington Earl A.
Published: December 2010

The domain and null space of an operator A in a Hilbert space will be denoted by and , respectively. A formally normal operatorN in is a densely defined closed (linear) operator such that , and for all A normal operator in is a formally normal operator N satisfying 35 . A study of the possibility of extending a formally normal operator N to a normal operator in the given , or in a larger Hilbert space, was made in (1).