bluuebunny/crossref_metadata_2025 huggingface dataset:
Irreducibility of Bernoulli Polynomials of Higher Order
Authors: McCarthy P. J.
Published: December 2010

The Bernoulli polynomials of order k, where k is a positive integer, are defined by

Bm(k)(x) is a polynomial of degree m with rational coefficients, and the constant term of Bm(k)(x) is the mth Bernoulli number of order k, Bm(k). In a previous paper (3) we obtained some conditions, in terms of k and m, which imply that Bm(k)(x) is irreducible (all references to irreducibility will be with respect to the field of rational numbers). In particular, we obtained the following two results.
On Non-Orientable Closed Surfaces in Euclidean Spaces
Authors: Yang C. T.
Published: December 2010

Let us begin with a simple result.

Proposition. Let X be a non-orientable closed surface differentiably imbedded into the euclidean 4-space R4. Then there is a line in R4 which intersects X at more than two points.

Proof. Let c be any point of R4 and let r be the smallest number such that X is contained in the closed 4-spheroid W of centre c and radius r. Clearly the boundary 3-sphere S of W intersects X.
A Strong Converse of the Wiener-Levy Theorem
Authors: Rudin Walter
Published: December 2010

Let A1 denote the class of all complex functions on the unit circle which are sums of absolutely convergent trigonometric series; that is to say, the class of all f of the form

Similarly, if 1 < p < ∞, let Ap be the class of all complex functions f on the circle whose Fourier coefficients

satisfy the condition
The Gibbs Phenomenon and Lebesgue Constants for Regular Sonnenschein Matrices
Authors: Sledd W. T.
Published: December 2010

Ifψ(x)is a real-valued function which has a jump discontinuity atx= ε and otherwise satisfies the Dirichlet conditions in a neighbourhood ofx= ε then{sn(x)}the sequence of partial sums of the Fourier series forψ(x)cannot converge uniformly atx =ε. Moreover, it can be shown that given τ in [ — π, π] then there is a sequence {tn} such thattn→ ε and

This behaviour of{sn(x)}is called the Gibbs phenomenon. If {σn(x)} is the transform of{sn(x)}by a summability methodT, and if {σn(x)} also has the property described then we say thatTpreserves the Gibbs phenomenon.
On Commutative Squares
Authors: Leicht Johann B.
Published: December 2010

The following elementary facts about certain commutative diagrams, called "squares," are stated and proved in terms of abelian groups and their homomorphisms. However, they are valid for arbitrary abelian categories and can be proved also for them. This does not need to be shown, since every abelian category can be embedded into the category of abelian groups with preservation of exact sequences according to a result due to S. Lubkin (1). Proofs are often omitted or given only for one half of a theorem, the other half being dual to the first. Generalizations to a larger class of diagrams containing all finite commutative diagrams are possible.
Minimal Term Rank of a Class of (0, 1)-Matrices
Authors: Haber Robert M.
Published: December 2010

Let be the class of m × n matrices all of whose entries are either 0 or 1 where every matrix A in the class satisfies the conditions that row i of A has ri ones, i = 1, 2, . . . , m; and column j of A has sj ones, j = 1, 2, . . . , n. We let R = (r1 . . . , rm), S = (s1, . . . , sn), and assume that r1 ≥ r2 ≥ . . . ≥ rm ≥ 0; s1 ≥ s2 ≥ . . . ≥ sn > 0. When this is the case we say the class is normalized.
On Singular Fiberings
Authors: Mahowald Mark
Published: December 2010

The purpose of this note is to discuss some properties of singular fibrations introduced by Montgomery and Samelson (4) and studied by Hu (2) and Conner and Dyer (1). This work was motivated by the last paper, where singular fibrations in which the fibre is a sphere were studied. Here more general fibres will be considered.
Monotone Functions on Linear Lattices
Authors: Ellis H. W. , Nakano Hidegorô
Published: December 2010

If R is a sequentially continuous linear lattice, a function f(x), denned on R+ = {x:0 ≤ x ∊ R} with 0 ≤ f(x) ≤ + ∞, will be called a monotone function if it satisfies
A Census of Planar Maps
Authors: Tutte W. T.
Published: December 2010

In the series of "Census" papers, of which this is the fourth, we attempt to lay the groundwork of an enumerative theory of planar maps (12, 13, 14). The maps concerned are rooted in the sense that some edge is fixed as the root, and a positive sense of description and right and left sides are specified for it. This device simplifies the theory by ruling out the possibility of a map being symmetrical.
Note on Some Low-Order Perfect Squared Squares
Authors: Federico P. J.
Published: December 2010

In the Mathematical Games department of the Scientific American for November 1958 (12), William H. Tutte gave an entertaining account of the researches of himself and three colleagues (C. A. B. Smith, A. H. Stone, and R. L. Brooks) concerning "squaring the square." The problem is to subdivide a square into a finite number of non-overlapping smaller squares of which no two are the same size. At first suggested as impossible, a number of different solutions have been published. The first one, by Sprague in 1939 (9), showed a square divided into 55 smaller unlike squares.
Characterization of Certain Binary Relations on Connected Ordered Spaces
Authors: L'Heureux James E.
Published: December 2010

In an earlier paper (2) reflexive transitive binary relations were considered on a connected ordered space. These relations were topologically restricted and their minimal sets were either an end point of the space or empty. It was shown that these relations could be characterized as one of the two orders of the space. Viewing the situation somewhat differently as suggested by I. S. Krule, one could say that this class of relations was characterized in terms of the identity function on the space. In this case the relations are considered in their natural setting, the product of the space with itself.
Isomorphisms of Simple Rings
Authors: Yen Ti
Published: December 2010

Let A be a simple ring with minimum condition, and B1, B2, and C be regular subrings of A such that Bi > C, i = 1, 2. A pair of isomorphisms σi of Bi into A such that σi|C is the identity, and that Biσi are regular subrings of A (i = 1, 2), is called compatible if σ1|B1 ∩ B2 = σ2|B1 ∩ B2. Here σ|X means the restriction of σ to X. Bialynicki-Birula has proved some necessary and sufficient conditions that every compatible pair (σ1, σ2) has a common extention to an automorphism σ of A (1 ). When A is a division ring, he shows that the linear disjointness of the division subrings B1 and B2 is necessary and almost sufficient for the existence of a common extension of any compatible pair.
Some Regular [F, dn] Matrices with Complex Elements
Authors: Miracle Chester L.
Published: December 2010

Let A = (ank) and {sn} (n, k = 0, 1, 2, . . .) be a matrix and a sequence of complex numbers, respectively. Let the members of the sequence {σn} be defined by

then we say {σn} is the A -transform of {sn}. The matrix A = (ank) is called regular if
Point-Like, Simplicial Mappings of a 3-Sphere
Authors: Finney Ross
Published: December 2010

A decomposition of a topological space X is a partitioning of X into non-empty, disjoint sets called elements of the decomposition. An element of a decomposition is non-degenerate if it contains more than one point. Associated with each decomposition D of X is a topological space D*, called the hyperspace of the decomposition. A classical problem on decompositions of topological spaces is to find conditions under which D* is homeomorphic to X. Often decompositions arise from mappings: if g is a mapping of a space X onto a space Y, then D = {g-1(y) |y ∊ Y} is a decomposition of X.
On Measurability for Vector-Valued Functions
Authors: Snow D. O.
Published: December 2010

The problem of developing an abstract integration theory has been approached from many angles (6). The most general of several definitions based on the norm topology is that of Birkhoff (1), which includes the well-known and widely used Bochner integral (3).

The original Birkhoff formulation was based on the notion of unconditional convergence of an infinite series of elements in a Banach space and the closed convex extensions of certain approximating sums.
Integral Representations of the Direct Product of Groups
Authors: Jones Alfredo
Published: December 2010

Let G be a finite group and R a Dedekind domain with quotient field K. We denote by RG the group ring of formal linear combinations of elements of G with coefficients in R. By an RG-module we understand a unital left RG-module which is finitely generated and torsion-free as R-module. In particular, if R is a principal ideal domain this is equivalent to considering representations of G by matrices with entries in R.
Dual Integral Equations
Authors: Love E. R.
Published: December 2010

Erdélyi and Sneddon (4) have reduced the dual integral equations (4, (1.4))

where Ψ is unknown, to a single Fredholm integral equation (4, (4.4)), from the solution of which Ψ is explicitly obtainable. Their work extended and clarified an investigation by Cooke (1), placing it in a context of standard integral transforms. Cooke's reduction was obtained after consideration of the Fredholm integral equation obtained by Love (8) in discussing Nicholson's problem of the electrostatic field of two equal circular coaxial conducting disks (9).
Subrings of the Maximal Ring of Quotients Associated With Closure Operations
Authors: Murdoch D. C.
Published: December 2010

This paper contains a number of results that grew out of an attempt to solve the following problem : Given a non-commutative ring R with suitable ascending chain condition, and a prime ideal P in R, to construct a corresponding local ring RP in which the extension P′ of P is a unique maximal prime, and to prove, if possible, that the intersection of the powers of P′ is zero. The present investigation is at best a preliminary attack on this problem since the contribution to the complete solution is comparatively small and the central problem of the intersection of the powers of P′ has not been touched.
Ideals in Topological Rings
Authors: Yood Bertram
Published: December 2010

We present here an investigation of the theory of one-sided ideals in a topological ring R. One of our aims is to discuss the question of "left" properties versus "right" properties. A problem of this sort is to decide if (a) all the modular maximal right ideals of R are closed if and only if all the modular maximal left ideals of R are closed. It is shown that this is the case if R is a quasi-Q-ring, that is, if R is bicontinuously isomorphic to a dense subring of a Q-ring (for the notion of a Q-ring see (6) or §2). All normed algebras are quasi-Q-rings. Also (a) holds if R is a semisimple ring with dense socle.
Collineations of Affine Moulton Planes
Authors: Pierce William A.
Published: December 2010

In a recent article on Moulton planes (8), I have generalized the non-Desarguesian planes introduced by F. R. Moulton (6) and Pickert (7, pp. 93-94). Let F = (0, 1; a, b, . . . , x, y, . . .} denote a given field, having P as a multiplicative subgroup of index 2. Define x > 0 or x < 0 according as x lies in P or in the other coset of non-zero elements. The function ϕ is order-preserving, and can be assumed to satisfy ϕ(0) = 0, ϕ(1) = 1. For any n < 0, the maps x —> ϕ(x) and x —> ϕ(x) — nx both carry F onto itself. The elements of F form a Cartesian group, {+,0} being defined so that
A Note on Cayley Loops
Authors: Ginzburg A.
Published: December 2010

In § 1 the existence of two non-isotopic loops satisfying the same system of relations is proved: C1 is isomorphic to the well-known Cayley loop (see 1 or 6); C2 seems not to have been mentioned in the literature. In § 2 a characterization of the second type is given in terms of Tamari's generalized normal multiplication table (7; 8) as a complete symmetric quasiregular partition, i.e., a quasi-associative loop (4, 5), or a so-called near-group (9). In § 3 these distinctions are followed by comparing the corresponding algebras.
Determining a Set from the Cardinalities of its Intersections with Other Sets
Authors: Cantor David G.
Published: December 2010

Let n be a positive integer and put N = {1, 2, . . . , n}. A collection {S1, S2, . . . , St} of subsets of N is called determining if, for any T ⊂ N, the cardinalities of the t intersections T ∩ Sj determine T uniquely. Let €1, €2, . . . , €n be n variables with range {0, 1}. It is clear that a determining collection {Sj) has the property that the sums
The Current Theory of Analytic Sets
Authors: Bressler D. W. , Sion M.
Published: December 2010

In this paper we describe the outlines of the theory of analytic sets from the point of view of recent work on the subject. Our aim is to present the concepts and some of the principal results in a setting useful to workers in analysis, especially those workers not familiar with the field or its current developments. No attempt has been made to include all the results concerning analytic sets—not even in a particular category. There are some excellent monographs(12; 18)as well as chapters in books(2, 8, 9, 15, 17)where the subject is treated extensively. These, however, do not contain the recent results and consider only metric spaces. Our emphasis is on a general topological setting.
Bernstein Power Series
Authors: Cheney E. W. , Sharma A.
Published: December 2010

In Bernstein's proof of the Weierstrass Approximation Theorem, the polynomials

are constructed in correspondence with a function f ∊ C [0, 1] and are shown to converge uniformly to f. These Bernstein polynomials have been the starting point of many investigations, and a number of generalizations of them have appeared. It is our purpose here to consider several generalizations in the form of infinite series and to establish some of their properties.
Groups with Representations of Bounded Degree
Authors: Isaacs I. M. , Passman D. S.
Published: December 2010

Let G be a discrete group with group algebra C[G] over the complex numbers C. In (5) Kaplansky essentially proves that if G has a normal abelian subgroup of finite index n, then all irreducible representations of C[G] have degree ≤n. Our main theorem is a converse of Kaplansky's result. In fact we show that if all irreducible representations of C[G] have degree ≤n, then G has an abelian subgroup of index not greater than some function of n. (The degree of a representation of C[G] for arbitrary G is defined precisely in § 3.)
Pairs of Consecutive Power Residues or Non-Residues
Authors: Jordan J. H.
Published: December 2010

For a positive integer k and a prime p ≡ 1 (mod k), there is a proper subgroup, R, of the multiplicative group (mod p) consisting of the kth power residues (mod p). A necessary and sufficient condition that an integer t be an element of R is that the congruence xk ≡ t (mod p) be solvable. The cosets, not R, formed with respect to R are called classes of kth power nonresidues, and form with R a cyclic group of order k. Let ρ be a primitive kth root of unity and let S be a class of non-residues that is a generator of this cyclic group. There is a kth power character X (mod p) such that
A Generalization of the Hardy Spaces
Authors: Rooney P. G.
Published: December 2010

The Hardy spaces for right half-planes, , σ real, 1 ≤ p ≤ ∞, are defined to consist of all those functions f(s), holomorphic for Re s > σ, for which μp(f, x) exists and is bounded for x > σ, where

These spaces have been studied extensively (see, for example, 3, Chapter 8, and 2, §19.1).
On Subplanes of Free Planes
Authors: Sandler Reuben
Published: December 2010

In 1945, L. I. Kopejkina (4), at the suggestion of A. G. Kurosh, began a programme of studying the properties of free projective planes and the analogies between free planes and free groups. In this paper, this study will be extended by proving a tool theorem and several of its consequences. The theorem deals with the existence of "minimal free generators" for subplanes of free planes.

A set of points and lines and an incidence relation are said to form a projective plane if the following three axioms are satisfied.
On the Positive Semidefinite Nature of a Certain Matrix Expression
Authors: Wigner Eugene P. , Yanase Mutsuo M.
Published: December 2010

By "positive definite matrices" or, briefly, definite matrices, we mean in this note self-adjoint matrices all the characteristic values of which are positive. Alternatively, they can be defined as matrices, all the hermitian quadratic forms of which are real and positive. It is well known that these two definitions are equivalent and it is the existence of two equivalent definitions which renders the subject particularly interesting. Thus, it follows from the first definition that A-1 is positive definite for positive definite A ; it follows from the second definition that the sum of two positive definite matrices is also positive definite. Similarly, we shall call a matrix positive semidefinite or, briefly, semidefinite if it is self-adjoint and none of its characteristic values are negative, or if all of its hermitian quadratic forms are non-negative. A matrix which is definite is also semidefinite but a semidefinite matrix is, of course, not necessarily definite.
Concerning Metrizability of Pointwise Paracompact Moore Spaces
Authors: Traylor D. R.
Published: December 2010

Although it is known that there exists a pointwise paracompact Moore space which is not metrizable (1), very little seems to be known about the metrizability of pointwise paracompact Moore spaces. This paper is devoted to determining some of the conditions under which a pointwise paracompact Moore space is metrizable.

The statement that 5 is a Moore space means that there exists a sequence of collections of regions in 5 satisfying Axiom 0 and the first three parts of Axiom 1 of (2). A Moore space is complete if and only if it satisfies all of Axiom 1 of (2).
Finite Groups with All Maximal Subgroups of Prime or Prime Square Index
Authors: Kohler Joseph
Published: December 2010

In this paper finite groups with the property M, that every maximal subgroup has prime or prime square index, are investigated. A short but ingenious argument was given by P. Hall which showed that such groups are solvable.

B. Huppert showed that a finite group with the property M, that every maximal subgroup has prime index, is supersolvable, i.e. the chief factors are of prime order. We prove here, as a corollary of a more precise result, that if G has property M and is of odd order, then the chief factors of G are of prime or prime square order. The even-order case is different. For every odd prime p and positive integer m we shall construct a group of order 2apb with property M which has a chief factor of order larger than m.
An Algorithmic Solution for a Word Problem in Group Theory
Authors: Mendelsohn N. S.
Published: December 2010

This paper describes a systematic procedure which yields in a finite number of steps a solution to the following problem. Let G be a group generated by a finite set of generators g1, g2, g3, . . . , gr and defined by a finite set of relations R1 = R2 = . . . = Rk = I, where I is the unit element of G and R1R2, . . . , Rk are words in the gi and gi-1. Let H be a subgroup of G, known to be of finite index, and generated by a finite set of words, W1, W2, . . . , Wt. Let W be any word in G. Our problem is the following. Can we find a new set of generators for H, together with a set of representatives h1 = 1, h2, . . . , hu of the right cosets of H (i.e. G = H1 + Hh2 + . . . + Hhu) such that W can be expressed in the form W = Uhp, where U is a word in .
The Maximum Number of Disjoint Permutations Contained in a Matrix of Zeros and Ones
Authors: Fulkerson D. R.
Published: December 2010

A well-known consequence of the König theorem on maximum matchings and minimum covers in bipartite graphs (5) or of the P. Hall theorem on systems of distinct representatives for sets (4) asserts that an n by n (0, 1)-matrix A having precisely p ones in each row and column can be written as a sum of p permutation matrices:

Our main objective is a generalization of (1.1) along the following lines. Let A be an arbitrary m by n (0, 1)-matrix. Call an m by n (0, 1)-matrix P a permutation matrix if PPT = I, where PT is the transpose of P and I is the identity matrix of order m.
Generalized Hadamard Matrices and Orthogonal Arrays of Strength Two
Authors: Shrikhande S. S.
Published: December 2010

The purpose of this note is to point out some connexions between generalized Hadamard matrices (4, 5) and various tactical configurations such as group divisible designs (3), affine resolvable balanced incomplete block designs (1), and orthogonal arrays of strength two (2). Some constructions for these arrays are also indicated.

A balanced incomplete block design (BIBD) with parameters v, b, r, k, λ is an arrangement of v symbols called treatments into b subsets called blocks of k < v distinct treatments such that each treatment occurs in r blocks and any pair of treatments occurs in λ blocks.
Nil Rings Satisfying Certain Chain Conditions
Authors: Herstein I. N. , Small Lance
Published: December 2010

In general, given the fact that every element in a ring is nilpotent, one cannot conclude that the ring itself is nilpotent. However, there are theorems which do assert that, in the presence of certain side conditions, nil implies nilpotent. We shall prove some theorems of this nature here; among them they contain or subsume many of the earlier known theorems of this sort.
Semigroups in Which all Subsemigroups are Left Ideals
Authors: Kimura Naoki , Tamura Takayuki , Merkel Rudolph
Published: December 2010

A semigroup S is a λ- [ρ-, σ-] semigroup if and only if each subsemigroup of S is a left [right, two-sided] ideal of S. Since the concept of ρ-semigroup is the dual of that of λ-semigroup, the results for ρ-semigroups are generally not stated explicitly, σ-semigroups are treated as a special case of λ-semigroups ; in fact, a semigroup S is a σ-semigroup if and only if it is a λ-semigroup and a ρ-semigroup. The purpose of this paper is to determine the structure of λ- [ρ-, σ-] semigroups.
Some Configurations in Finite Projective Spaces and Partially Balanced Incomplete Block Designs
Authors: Ray-Chaudhuri D. K.
Published: December 2010

Using the methods developed in (2 and 3), in this paper we study some properties of the configuration of generators and points of a cone in an w-dimensional finite projective space. The configuration of secants and external points of a quadric in a finite plane of even characteristic is also studied. I t is shown that these configurations lead to several series of partially balanced incomplete block (PBIB) designs. PBIB designs are defined in Bose and Shimamoto (1). A PBIB design with m associate classes is an arrangement of v treatments in b blocks such that.
On Generalized Powers of the Difference Operator
Authors: Unni K. R.
Published: December 2010

Sumner (3) discussed for arbitrary real λ and h, where the averaging operator ∇h is defined by

(1.1)

when f(z) is an entire function of exponential type <2π/|h|. Boas (2) gave an alternative definition of ∇h which gave Sumner's results quickly and showed that his definition is equivalent to that of Sumner.
Fubini Theorem For a Certain Type of Integral
Authors: Hayes Charles A.
Published: December 2010

In (1), the writer defined a process of integration that leads to a kind of Riemann integral under certain rather general conditions. The purpose of this paper is to show how it is possible to use the process of integration of (1) to obtain integrals in a product space that satisfy a Fubini theorem. In this connection, we define a class of integrands that are the analogues of continuous functions in the product space, establish some of their properties, and finally arrive at a Fubini theorem for this class.
The Kernel of m-Quota Games
Authors: Peleg Bezalel
Published: December 2010

In (1), M. Davis and M. Maschler define the kernel K of a characteristic-function game; they also prove, among other theorems, that K is a subset of the bargaining set M1(i) and that it is never void, i.e. that for each coalition structure b there exists a payoff vector x such that the payoff configuration (x, b) belongs to K. The main advantage of the kernel, as it seems to us, is that it is easier to compute in many cases than the bargaining set M1(i).
Tauberian Estimates Concerning the Regular Hausdorff and [J, f(x)] Transformations
Authors: Meir A.
Published: December 2010

Denote by {t(x)} some linear transform of the sequence

of the form

where x attains continuous or only integer values. The problem of estimating |t(x) — sm| as x and m tend to ∞ with some connection between them was considered first by H.
T2-Groups And a Characterization of the Finite Groups of Moebius Transformations
Authors: Lorimer P. J.
Published: December 2010

In recent years a number of algebraic characterizations of the groups of Moebius transformations over finite fields have been given in the literature; see (1, 3, 6). H. W. E. Schwerdtfeger has noticed (4) that the group G of Moebius transformations over the real, complex, and certain other fields has the property:
On a Theorem of Beurling and Livingston
Authors: Browder Felix E.
Published: December 2010

In their paper (1), Beurling and Livingston established a generalization of the Riesz-Fischer theorem for Fourier series in Lp using a theorem on duality mappings of a Banach space B into its conjugate space B*. It is our purpose in the present paper to give another proof of this theorem by deriving it from a more general result concerning monotone mappings related to recent results on non-linear functional equations in Banach spaces obtained by the writer (2, 3, 4, 5) and G. J. Minty (6).
Entire and Meromorphic Functions with Asymptotically Prescribed Characteristic
Authors: Edrei Albert , Fuchs Wolfgang H. J.
Published: December 2010

If f(z) is a non-constant, entire function, then Hadamard's three-circles theorem asserts that

is a convex, increasing function of log r. Hence, by well-known properties of logarithmically convex functions,
The Generation of the Lower Central Series
Authors: Gallagher P. X.
Published: December 2010

Let G be a finite group with commutator subgroup G′. In an earlier paper (4) it was shown that each element of G′ is a product of n commutators, if 4n ≥ |G′|. The object of this paper is to improve this result in two directions:

Theorem 1a. If (n + 2)!n! > 2|G′| — 2, then each element of G′ is a product of n commutators.

Theorem 1b. If G is a p-group, with |G′| = pa, and if n(n + 1) > a, then each element of G′ is a product of n commutators.
Lattice-Ordered Rings of Quotients
Authors: Anderson F. W.
Published: December 2010

R. E. Johnson (10), Utumi (18), and Findlayand Lambek (7) have defined for each ring R a unique maximal "ring of right quotients" Q. When R is a commutative integral domain (in this paper an integral domain need not be commutative) or an Ore domain, then Q is the usual division ring of quotients of R. Moreover, it is well known that in these special cases, if R is totally ordered, then so is Q.

The main purpose of this paper is to study the ring of quotients Q, and in particular its order properties, for certain lattice-ordered rings R.
Hyperbolic Convolution Operators
Authors: Kakita Takao
Published: December 2010

Hyperbolic differential operators with constant coefficients introduced and studied systematically by Gårding (4), were characterized by the existence of the fundamental solution with some cone condition, according to Hörmander (6). Recently Ehrenpreis, extending the notion of hyperbolicity due to Gårding, has defined hyperbolic operators for distributions with compact support in the convolution sense. Under the hypothesis that the operator is invertible as a distribution, he has established a theorem analogous to the theorem of Hörmander mentioned above (3).
A New Group Algebra for Locally Compact Groups II
Authors: Ernest John
Published: December 2010

In an earlier work, we defined and described a new group algebra , which is a von Neumann algebra containing the group G (3). In this paper we continue this study be relating the lattice of normal subgroups of the group G to the lattice of central projections of the group algebra . More precisely, we shall exhibit a one-to-one mapping ϕ of the lattice of closed normal subgroups of G into the lattice of central projections of , having the property that if N1 ⊂ N2, then ϕ(N2) ≤ ϕ(N1).
Some Relationships Between Bers and Beltrami Systems and Linear Elliptic Systems of Partial Differential Equations
Authors: Caldwell W. V.
Published: December 2010

Much work has been done in the investigation of the properties of solutions of linear elliptic systems of partial differential equations. Among these systems, the class of Beltrami systems has been studied for many years and has been shown to be of fundamental importance. Another class, perhaps of equal importance, is the class defined by Bers (1), which the author has taken the liberty of calling Bers systems. Solutions of these systems will be called Beltrami and Bers functions respectively.
Invariant Subspaces of Continuous Functions
Authors: Hasumi Morisuke , Srinivasan T. P.
Published: December 2010

Since Beurling (1 ), the study of closed invariant subspaces of Hp(X) and Lp(X) on the unit circle X = {z:|z| = 1} has been done extensively and culminated into a very fine theory of generalized analytic functions ; cf. Hoffman (4), Srinivasan (7). Here we say that a space E of complex-valued functions on X is invariant if zE ⊆ E. Little is known, however, about the structure of closed invariant subspaces E of the space C(X) of continouus functions on X.
On the line Graph of a Finite Affine Plane
Authors: Hoffman A. J. , Ray-Chaudhuri D. K.
Published: December 2010

Let n be a finite affine plane with n points on a line. We denote by G(II) the graph whose vertices are all points and lines of II, with two vertices adjacent if and only if one is a point, the other is a line, and the point and line are incident. Let L(11) denote the line graph of G(II), i.e., the vertices of L(II) are the edges of G(II), and two vertices of L(II) are adjacent if the corresponding edges of G(II) are adjacent. It is clear that L(II) is a regular, connected graph with n2(n + 1) vertices and valence 2n — 1.
Trees and Tree-Equivalent Graphs
Authors: Ramanujacharyulu C.
Published: December 2010

As is well known in the theory of graphs a tree is a connected graph without cycles. Many characterizing properties of trees are known (1), for example the cyclomatic number is equal to zero, which is also equal to p — 1, where p is the number of connected components of the graph. The graphs with cyclomatic number equal to p — 1 are defined here as tree-equivalent graphs. A tree is always a tree-equivalent graph but not conversely. The properties of tree-equivalent graphs are studied here.
Convexity Properties for Weak Solutions of Some Differential Equations in Hilbert Spaces
Authors: Zaidman S.
Published: December 2010

In this work we obtain a simultaneous extension of Theorems 1.6 and 1.7 in Agmon and Nirenberg (1), together with a partial extension of the result on backward unicity for parabolic equations by Lions and Malgrange (4).

Let H be a Hilbert space. (·) and | | are the notations for the scalar product and the norm in this space. Consider in H a family B(t), 0 ≤ t ≤ T, of closed linear operators with dense domain DB(t) (varying) with t. Let L2(0, T, H) be the space of Bochner square-integrable vector-valued functions with values in H. Our main result is the following
A Characterization of the Hughes Planes
Authors: Ostrom T. G.
Published: December 2010

Baer (1) introduced the term "(p,L)-collineation" to denote a central collineation with centre p and axis L. We shall find it convenient to use a modification of the related notion of "(p, L)-transitivity."

Definition. Let π0 be a subplane of the projective plane π. Let L be a fixed line of π0, and let p be a fixed point of π0. Let r and s be any two points of π0 that are collinear with p, distinct from p, and not on L. If, for each such choice of r and s, there is a (p, L)-collineation of π that (1) carries π0 into itself and (2) carries r into s, we shall say that π is (p, L, π0)-transitive.
Domain Perturbations of the Biharmonic Operator
Authors: Swanson C. A.
Published: December 2010

Eigenvalue problems for the biharmonic operator (iterated Laplacian)L= ΔΔ will be studied on bounded plane domains. Our purpose is to obtain asymptotic variational formulae for eigenvalues and eigenfunctions under the deformation of removing an ∊-disk and adjoining additional boundary conditions on the new boundary component thereby introduced, valid on a positive interval 0 < ∊ < ∊0. Eigenvalue problems can be considered in connection with each of the following sets of homogeneous boundary conditions.
A Fourier Formula for Series Summable (C, 1)
Authors: Cross George
Published: December 2010

Recently Bullen (1) developed a J3-integral that gives a Fourier representation for trigonometric series when both the series and its conjugate are summable (C, 1) everywhere. Earlier Cross (4) had shown that the C2P-integral (3) was strong enough to integrate such series and, indeed, that any trigonometric series for which the series and its conjugate were summable (C, k) were Fourier series where the integral used in the formula for the coefficients was the Ck+iP-integral (3). In (1) Bullen defines a J4-integral and says, without proof, that a Fourier formula in terms of this integral may be obtained for series for which the series and its conjugate are summable (C, 2) and the coefficients o(n).
On the Average Number of Trees in Certain Maps
Authors: Mullin R. C.
Published: December 2010

For a formal definition of “map” the reader is referred to (7, §2). The maps in this paper are rooted by specifying an orientation for one of the edges. This also specifies a root vertex, the negative end of the root, and a root face, the face on the left of the root edge. Counting is, as usual, defined on isomorphism classes.

Regular maps of even valence have been enumerated in a recent paper by Tutte. In this paper we determine the average number of trees in such maps, and include similar results for regular tri valent maps, that is, maps with three edges incident on every vertex. In the development for the latter, a formula for the number of trivalent maps with 2t vertices is produced.
A Further Note on Lototsky-Type Transformations
Authors: Meir A.
Published: December 2010

In two recent papers by V. F. Cowling and C. L. Miracle (1; 2), the regularity of generalized Lototsky transformations, as well as their application to the geometric series, has been investigated. The main interest of the papers centres around (1, Theorems 3.1 and 4.1) and (2, Theorem A). In (1, Theorem 3.1) and in (2, Theorem A), the authors stated and proved a set of sufficient conditions for the regularity of Lototsky-type transformations. In (1, Theorem 4.1), they proved under certain additional conditions that these transformations sum the geometric series ∑zn to (1 — z)–l if Re z < 1.
Distributive Extensions and Quasi-Framal Algebras
Authors: Hu Tah-Kai
Published: December 2010

In (2; 3; 4), A. L. Foster denned Boolean extensions of framal algebras and bounded Boolean extensions of framal-in-the-small algebras. Foster proved that the class of Boolean (of bounded Boolean) extensions of a framal (a framal-in-the-small) algebra A is coextensive up to isomorphism with a certain class of subdirect powers of A, namely, the class of normal (of bounded normal) subdirect powers of A. His proofs apply, however, to considerably more general situations. Indeed, as remarked in (2), the construction of Boolean extensions may be carried out for an arbitrary universal algebra with finitary operations; this is done, in fact, in (4).
Nil Rings Satisfying Certain Chain Conditions: An Addendum
Authors: Herstein I. N. , Small Lance
Published: December 2010

We prove here a result closely related both in spirit and technique to those obtained in our paper (1). We also take this opportunity to present a counter example, due to E. Sąsiada, to the conjecture we made there. Finally, we extend a result of Wedderburn (3) from algebras over fields to algebras over commutative rings; the result follows easily from the paper by Posner (2) although it does not appear there explicitly.
A Relationship between Arbitrary Positive Matrices and Stochastic Matrices
Authors: Sinkhorn Richard
Published: December 2010

The author (2) has shown that corresponding to each positive square matrix A (i.e. every aij > 0) is a unique doubly stochastic matrix of the form D1AD2, where the Di are diagonal matrices with positive diagonals. This doubly stochastic matrix can be obtained as the limit of the iteration defined by alternately normalizing the rows and columns of A.

In this paper, it is shown that with a sacrifice of one diagonal D it is still possible to obtain a stochastic matrix. Of course, it is necessary to modify the iteration somewhat. More precisely, it is shown that corresponding to each positive square matrix A is a unique stochastic matrix of the form DAD where D is a diagonal matrix with a positive diagonal. It is shown further how this stochastic matrix can be obtained as a limit to an iteration on A.
A Stieltjes–Volterra Integral Equation Theory
Authors: Hinton D. B.
Published: December 2010

Suppose S = [a, b] is a number interval and F is a function from S X S to a normed algebraic ring N with multiplicative identity I. We consider the problem of finding, for appropriate conditions on F, a function M from S X S to N such that for all t and x,

where the integral is a Cauchy-left integral.
Boundary Regularity in the Sobolev Imbedding Theorems
Authors: Hurd A. E.
Published: December 2010

In (6) (see also 7), Sobolev introduced a class of function spaces Wm,p(Ω) (m a non-negative integer, 1 < p < ∞) defined on open subsets Ω of Euclidean space En, which have important applications in partial differential equations. They are defined as follows. For each n-tuple α = (α1, … αn) of non-negative integers let
A Martingale Convergence Theorem in Vector Lattices
Authors: DeMarr Ralph
Published: December 2010

The martingale convergence theorem was first proved by Doob (3) who considered a sequence of real-valued random variables. Since various collections of real-valued random variables can be regarded as vector lattices, it seems of interest to prove the martingale convergence theorem in an arbitrary vector lattice. In doing so we use the concept of order convergence that is related to convergence almost everywhere, the type of convergence used in Doob's theorem.
On the Smoothness of General Kernels
Authors: Barros-Neto Jose
Published: December 2010

In (3, §2), the writer and F. E. Browder stated briefly, without proof, some results concerning general distribution kernels. It is our aim here to prove and complete those results.

The terminology and notations are introduced in §1.

In §2 we define the notion of domain of dependence with respect to the kernel Kx,y (Definition 1) as well as the notion of smoothness of a distribution kernel at a point (Definition 2). Theorem 1 states that the set of points, where the distribution kernel is smooth, is open and the kernel is a smooth function in this set. Theorems 2 and 3 are the converse of Theorem 1.
Uniqueness Classes for Difference Functionals
Authors: DeMar Richard F.
Published: December 2010

If ﹛Ln﹜ is a sequence of linear functionals on a linear space C of functions to the complex numbers, then a subspace C1 ⊂ C is a uniqueness class for ﹛Ln﹜ if a function f in C1 is uniquely determined by the sequence ﹛Ln(f)﹜ of complex numbers; i.e., if f ∈ C1 and Ln(f) = 0, n = 0, 1, 2, … , implies f = 0. For example, the class of all functions f analytic at the origin is a uniqueness class for the sequence ﹛f(n)(0)﹜ of linear functionals. Gontcharoff (9) asked the following question: Suppose, instead of ﹛f(n)(0)﹜, we use ﹛f(n)(an)﹜.
Replaceable Nets, Net Collineations, and Net Extensions
Authors: Ostrom T. G.
Published: December 2010

A net of degree k and order n is a set of n2 points and nk designated sets of points, called lines, such that

(1) The lines fall into k disjoint parallel classes, i.e. each line occurs in exactly one parallel class.

(2) Lines in the same parallel class have no points in common; lines in different parallel classes have exactly one point in common.

(3) Each point lies on exactly one line of each parallel class.
Euler Lines in Infinite Directed Graphs
Authors: Nash-Williams C. ST. J. A.
Published: December 2010

While the contents of the author's doctoral thesis (4) have, owing to their lengthy nature, been published only in small part (5, §2; 6; 7), the absence from the literature of graph theory of any characterization of infinite directed graphs with Euler lines seems to constitute a definite gap that prompts the publication in the present paper of some further material from (4). The main results characterizing such directed graphs will be obtained in §§2 and 3. In §4, we shall indicate an alternative (and perhaps better) formulation of one of these results, some extensions obtained in (4), and some comparisons between parallel results for undirected and directed graphs. A familiarity with the definitions and results of (7) will be assumed in §4, but not before.
The Reconstruction of a Tree from its Maximal Subtrees
Authors: Harary Frank , Palmer Ed
Published: December 2010

One of the many interesting conjectures proposed by S. M. Ulam in (5) can be stated as follows:

If G and H are two graphs with p points vi and ui respectively (p ⩾ 3) such that for all i, G — vi is isomorphic with H — ui then G and H are themselves isomorphic.

P. J. Kelly (3) has shown this to be true for trees. The conjecture is, of course, not true for p = 2, but Kelly has verified by exhaustion that it holds for all of the other graphs with at most six points. Harary and Palmer (2) found the same to be true of the seven-point graphs.

In (1) Harary reformulated the conjecture as a problem of reconstructing G from its subgraphs G — vi and derived several of the invariants of G from the collection G — vi.
Parallel Lines Associated with Two Sets
Authors: Robkin Eugene , Valentine F. A.
Published: December 2010

What conditions determine when a collection of points A lies on a collection of parallel lines each member of which intersects a set B? In order to describe these conditions the following notations and definitions are used. Also for earlier results see Robkin and Valentine (2).
On the Basis and Chromatic Number of a Graph
Authors: Everett C. J.
Published: December 2010

The basis theorem for directed graphs is, in effect, a result on weakly ordered sets, and, in §1, a proof is given, based on Zorn's lemma, that generalizes, and perhaps clarifies the exposition in (1, Chapter 2). In §2, a graph G* is defined, on an arbitrary collectionQof non-void subsets of a setX(which includes all its one-element subsets), in such a way that the partitions ofXintoQ-sets correspond to the kernels ofG*.Applied to the collectionQof non-null internally stable subsets of a graphGwithout loops, this identifies the chromatic number ofGwith the least cardinal number of any kernel ofG*.
The Second Mean Values of Entire Functions
Authors: Rahman Q. I.
Published: December 2010

Let f(z) be an entire function of the complex variable z = x + iy defined by the everywhere absolutely convergent Dirichlet series

1.1

If

then log m(x,f) is an increasing convex function of x (2), and

is called the Ritt order of f(z).
Bicontinuous Isomorphisms between Two Closed Left Ideals of a Compact Dual Ring
Authors: Wu Ling-Erl E. T.
Published: December 2010

A quasi-Frobenius ring is a ring with minimum condition satisfying the conditions r(l)H)) = H and l(r(L)) = L for right ideals H and left ideals L where r(S) (l(S)) denotes the right (left) annihilator of a subset S of the ring. Nakayama first defined and studied such rings (8; 9) and they have been studied by a number of authors (2; 3; 4; 6). A dual ring is a topological ring satisfying the conditions r(l)H)) = H and l(r)H)) = L for closed right ideals H and closed left ideals L. Baer (1) and Kaplansky (7) introduced the notion of such rings, which is a natural generalization of that of quaso-Frobenius rings. Numakura studied the analogy between dual rings and quasi-Frobenius rings in (10).
On Rigid Undirected Graphs
Authors: Hedrlín Z. , Pultr A.
Published: December 2010

By an undirected graph we mean a couple (X, R), where X is a set and R is a subset of X × X such that (x, y) ∈ R implies (y, x) ∈ R. The cardinal of X, denoted by |X|, will be called the cardinal of the graph.

A mapping f:X → X is called an endomorphism of (X, R) if (x, y) ∈ R implies that (f(x), f(y)) ∈ R for all x, y ∈ R.

An undirected graph (X, R) is called rigid if there is only one endomorphism of (X, R), namely the identity mapping of X.

P. Erdös communicated orally that, using probability methods, it is possible to prove that almost all finite undirected graphs are rigid.
Connectivity in Matroids
Authors: Tutte W. T.
Published: December 2010

An edge of a 3-connected graph G is called essential if the 3-connection of G is destroyed both when the edge is deleted and when it is contracted to a single vertex. It is known (1) that the only 3-connected graphs in which every edge is essential are the “wheel-graphs.” A wheel-graph of order n, where n is an integer ⩾3, is constructed from an n-gon called its “rim” by adding one new vertex, called the “hub,” and n new edges, or “spokes” joining the new vertex to the n vertices of the rim; see Figure 4A.

A matroid can be regarded as a generalized graph. One way of developing the theory of matroids is therefore to generalize known theorems about graphs. In the present paper we do this with the theorem stated above.
Double Series of Isols
Authors: Barback Joseph
Published: December 2010

It is assumed that the reader is familiar with the following notions: regressive function, regressive set, regressive isol, infinite series of isols, the minimum of two regressive isols, combinatorial function, and canonical extension. We shall use the slightly more general definition of a regressive function introduced in (3). The next three notions are defined in (2), the fifth in (3), and the last two in (7 and 8).
Signal Metrics
Authors: Darsow William F.
Published: December 2010

The purpose of this paper is to introduce a generalization of metric space that arises naturally out of the notion of signal function as it occurs, for example, in (5). In §§ 2-5, the basic definitions and motivation are given. In §§ 6 and 7 several elementary topological properties are proved, and in §§ 8 and 9 an important example from special relativity is developed.
New Approximations for Wiener Integrals, with Error Estimates
Authors: Finlayson Henry C.
Published: December 2010

The principal theorem of this paper, a generalization of a theorem given by R. H. Cameron (2), provides a means of approximating certain Wiener integrals to any desired degree of accuracy by an (n + 1)-fold Riemann integral with sufficiently large n. The generalization is in the use of a general complete orthonormal set of functions, whereas Cameron's paper used only the odd harmonic set.

Let C′ be the class of real-valued functions x(t) defined on [0, 1] and such that x(0) = 0 and which are continuous except perhaps for one left continuous jump. Let C be the class of continuous members of C′.