bluuebunny/crossref_metadata_2025 huggingface dataset:
On Effective Witt Decomposition and the Cartan–Dieudonné Theorem
Authors: Fukshansky Lenny
Published: December 2010

LetKbe a number field, and letFbe a symmetric bilinear form in 2Nvariables overK. LetZbe a subspace ofKN. A classical theorem of Witt states that the bilinear space (Z,F) can be decomposed into an orthogonal sum of hyperbolic planes and singular and anisotropic components. We prove the existence of such a decomposition of small height, where all bounds on height are explicit in terms of heights ofFandZ. We also prove a special version of Siegel's lemma for a bilinear space, which provides a small-height orthogonal decomposition into one-dimensional subspaces. Finally, we prove an effective version of the Cartan–Dieudonné theorem. Namely, we show that every isometry σ of a regular bilinear space (Z,F) can be represented as a product of reflections of bounded heights with an explicit bound on heights in terms of heights ofF,Z, and σ.
Strichartz Inequalities for the Wave Equation with the Full Laplacian on the Heisenberg Group
Authors: Furioli Giulia , Melzi Camillo , Veneruso Alessandro
Published: December 2010

We prove dispersive and Strichartz inequalities for the solution of the wave equation related to the full Laplacian on the Heisenberg group, by means of Besov spaces defined by a Littlewood–Paley decomposition related to the spectral resolution of the full Laplacian. This requires a careful analysis due also to the non-homogeneous nature of the full Laplacian. This result has to be compared to a previous one by Bahouri, Gérard and Xu concerning the solution of the wave equation related to the Kohn Laplacian.
On the Convergence of Successive Approximations in the Theory of Ordinary Differential Equations
Authors: Luxemburg W. A. J.
Published: December 2010

Let R denote the rectangle: |t-t0| ≤ a, | x-x0| ≤ b (a,b > 0) in the (t,x) plane and let f(t, x) be a function of two real variables t and x, defined and continuous on R. If I is the interval |t—t0| ≤ d with d = min(a,b/M), where M = max(|f(t, x)|, (t, x) ϵ R), then every solution x = x (t) of the differential equation x' = f(t, x) defined on I and which satisfies the initial condition x(t0) = x0, satisfies the integral equation

1.1

and conversely. In some cases, in order to prove the existence and uniqueness of the solutions of (1.1) on I, one forms the successive approximations

1.2
On Some Properties of Functions Regular in the Unit Circle
Authors: Rooney P.G.
Published: December 2010

The space Hp, 1 ≤ p ≤ ∞ consists of those analytic functions f(z) regular in the unit circle, for which Mp (f;r) is bounded for O ≤ r ≤ 1, where

These spaces have been extensively studied.

One well known result concerning these spaces is that if f(z) = Σ ∞n=0 anzn and {an} ɛ lp for some p, 1 ≤ p ≤ 2, then f ɛ Hq, where p-1+q-1 = 1, and conversely if f ɛ Hp, 1 ≤ p ≤ 2, then {an} ɛ lq. We propose to generalize this result to deal with functions f(z) = Σ ∞n=0 anzn with {n-λ an; n = 1, 2,...} ɛ lp, where λ ≥ 0. The resulting generalization is contained in the theorems below.

However, in order to make these generalizations we must first generalize the spaces Hp. To this end we make the following definition.
On a Normal Form of the Orthogonal Transformation I
Authors: Zassenhaus Hans
Published: December 2010

At the Edmonton Meeting of the Canadian Mathematical Congress E. Wigner asked me whether one knew something about the distribution of the characteristic roots of the linear transformations that leave invariant the quadratic form t2+x2-y2-z2, just as one knows that a Lorentz transformation has two complex conjugate characteristic roots and two real characteristic roots that are either inverse to one another or the numbers 1 and -1.

In this paper an answer to E. Wigner’s question will be obtained.

We are concerned with the pairs of matrices (X,A) with coefficients in a field of reference F such that the condition

0.1

is satisfied, where XT = (ξki) is the transpose of the matrix X = (ξki). It follows that both matrices are quadratic of the same degree d.
Yet Another Proof of the Minimax Theorem
Authors: Peck J.E.L.
Published: December 2010

There are so many proofs of this theorem in the literature, that an excuse is necessary before exhibiting another. Such may be found by examining the proof given below for the following: it uses no matrices, almost no topology and makes little use of the geometry of convex sets; it applies equally well to the case where only one of the pure strategy spaces is finite; also there is no assumption that the payoff function is bounded. Thus it can provide a short route to the more general forms of the theorem.
A Regular Quaternion Polygon
Authors: Crowe Donald W.
Published: December 2010

Repeated application of the unitary reflections

to the point (1,0) and line x + iy = 1 yields 24 points and 24 lines. These are the vertices and edges of the regular complex polygon 4 {3} 4 whose group has the abstract definition R4 = I, RSR = SRS [l]. The purpose of this note is to introduce the notion of regular quaternion polygon and give an example analogous to the preceding one.
On Some two way Classifications of Integers
Authors: Lambek J. , Moser L.
Published: December 2010

In this note we use the method of generating functions to show that there is a unique way of splitting the non-negative integers into two classes in such a way that the sums of pairs of distinct integers will be the same (with same multiplicities) for both classes. We prove a similar theorem for products of positive integers and consider some related problems.
Minkowski's Theorem with Curvature Limitations (I)
Authors: Melzak Z. A.
Published: December 2010

The well-known theorem of Minkowski, [1], [2], states that;

(M) a plane convex region, symmetric about the origin O, includes a lattice point other than O if its area is greater than 4.

By a lattice point we shall understand a point in the plane, both of whose coordinates are rational integers. In connection with (M) a critical region is defined to be a convex symmetric region of area 4, which includes no lattice point other than O. One such region is the open square S={(x, y)| |x| < 1, |y|<1} an infinite set of critical regions is formed by the parallelograms bounded by the lines y = x+1, y = x-1, y-1 = k (x-l), y+1 = k(x+l), 0≤-k<∞. Finally, there is a critical hexagon H, bounded by the following six lines: y-1 = -(x-l), y+1 = -(x+1), y-1 = mx, y+1 = mx, y = (1/m)(x-1), y = (l/m)(x+l), where m = tan π/12. All the vertex angles of H are equal to 2π/3.
A Generalization of Wythoff's Game*
Authors: Connell Ian G.
Published: December 2010

W.A. Wythoff [1] in 1907 defined a modification of the game of Nim by the following rules:

(i)

there are two players who play alternately;

(ii)

initially there are two piles of matches, an arbitrary number in each pile;

(iii)

a player may take an arbitrary number of matches from one pile or an equal number from both piles but he must take at least one match;

(iv)

the player who takes the last match wins the game.

If, after his move, a player leaves one match in one pile and two in the other he can force a win; for if his opponent takes one match from the pile containing two he can take both remaining matches; and similarly for the other possibilities.
Sur la Structure d'Une Classe d'Anneaux
Authors: Thierrin Gabriel
Published: December 2010

L'objet de ce travail est de déterminer la structure d'une classe d'anneaux, celle des anneaux bipotents à droite.

1. Anneaux bipotents à droite. Un anneau A est dit bipotent à droite, si l'on a

pour tout a ϵ A, On voit facilement qu'un anneau A est bipotent à droite, si et seulement si pour tout couple a, b ϵ A, il existe x ϵ A tel que l'on ait ab = a2x.
A Minimal Cubic Graph of Girth Seven
Authors: McGee W. F.
Published: December 2010

A “cubic” graph is one with three edges incident on each vertex. Let v and e be the number of vertices and edges, respectively. Then 3v = 2e for a cubic graph. The girth of a graph is the smallest number of edges in any non-trivial polygon. A minimal graph is one with the smallest number of edges with its particular properties. The minimal cubic graphs of girths one to eight, excluding seven, are discussed in Tutte's paper [1]. A minimal cubic graph of girth seven is given here.
A Note on Hilbert's Tenth Problem
Authors: Melzak Z. A.
Published: December 2010

The tenth problem on Hilbert's well known list [1] is the following.

(H 10) For an arbitrary polynomial P = P(x1,x2,…,xn) with integer coefficients to determine whether or not the equation P = 0 has a solution in integers.

By 'integers' we always mean 'rational integers'. The problem (H 10) is still unsolved but it appears likely that no decision procedure exists; in this connection see [2]. It will be shown here that (H 10) is equivalent to deciding whether or not every member of a certain given countable sec of rational functions of a single variable x is absolutely monotonie. We recall that f(x) is absolutely monotonie in I if f(x) possesses non-negative derivatives of all orders at every x ∊ I.
On an Algorithm of G. Birkhoff Concerning Doubly Stochastic Matrices
Authors: Johnson Diane M. , Dulmage A. L. , Mendelsohn N. S.
Published: December 2010

In [1] G. Birkhoff stated an algorithm for expressing a doubly stochastic matrix as an average of permutation matrices. In this note we prove two graphical lemmas and use these to find an upper bound for the number of permutation matrices which the Birkhoff algorithm may use.

A doubly stochastic matrix is a matrix of non-negative elements with row and column sums equal to unity and is there - fore a square matrix. A permutation matrix is an n × n doubly stochastic matrix which has n2-n zeros and consequently has n ones, one in each row and one in each column. It has been shown by Birkhoff [1],Hoffman and Wielandt [5] and von Neumann [7] that the set of all doubly stochastic matrices, considered as a set of points in a space of n2 dimensions constitute the convex hull of permutation matrices.
Permutations with Confined Displacements
Authors: Mendelsohn N. S.
Published: December 2010

A fundamental problem in combinatorial analysis is the classification of the permutations of 1, 2, …, n which satisfy a system of constraints. Thus one may ask such questions as how many permutations are there which have exactly r k-cycles; how many have at least s cycles regardless of cycle length. Again, one may ask how many permutations are there in which k ascending sequences appear; or how many permutations are there in which specified numbers may not appear in specified places or at specified distances from other numbers. The literature on these problems is quite extensive. References [1,2,5,7,10,14,17] give an indication of the present, status of these problems.
On Certain Pairs of Matrices which do not Generate a Free Group
Authors: Ree Rimhak
Published: December 2010

A complex number λ will be said to be free if the multiplicative group Fλ generated by the two matrices

is a free group, and non-free, otherwise. Very little is known about the distribution of free and non-free numbers [1]. It is, for instance, unknown whether the domain

contains an open set which consists of only free points.
Notes on Numerical Analysis III
Authors: Schwerdtfeger Hans
Published: December 2010

This note is to complement the earlier paper on the same subject (Canad. Math. Bull. 3 (1960), 41–57) in two points. The first part presents a simpler proof of the minimum property (cf. 1. c. section 3) of the orthogonal functions Ψv(x) (cf. 1. c. p. 46). In the second part we introduce another orthogonal system of sectionally linear functions χo(x), …, χn(x) which leads to a particularly simple interpolation formula. These functions appeared, mutatis mutandis, in the author's study on sectionally linear functions over an infinite range about which a report will be given elsewhere.
On Continuous Regular Rings
Authors: Utumi Y.
Published: December 2010

A subset K of a lattice is said to be directed if for any a, b∊K there is c∊K with c ≥ a, b. A complete lattice L is called upper continuous if for every directed subset (aα) and every element b.

The following is a slight improvement of [4; Anmerkung 1. 11, p. 11].
Some Elementary Properties of Bilinear Forms
Authors: Swanson Jonathan
Published: December 2010

The purpose of the present note is to help popularize a section of Artin' s “Geometric Algebra” (chapter 1,4; Interscience, New York (1957) ) by elaborating on its contents. The author will have succeeded when the reader discovers that his results are either presented more simply in Artin' s book or that they are trivial corollaries of its theorems, in particular of theorem 1.11.
Definition of Affine Geometry by a Group of Transformations
Authors: Lipman Joe
Published: December 2010

In his Geometric Algebra (New York, 1957) E. Artin poses the problem of co-ordinatizing an affine plane in the following terms.

How little do we have to assume, from a geometric point of view, about an affine plane, in order to be able to describe its points by pairs of elements of a field, and its lines by linear equations?
A Family of Regular Maps of Type {6, 6}
Authors: Sherk F.A.
Published: December 2010

In (4), pp. 25−27, Coxeter divided the regular maps on a surface of genus 1
into three infinite families. They are:

(i)

Maps of type {4, 4}.

(ii)

Maps of type {6, 3}.

(iii)

Maps of type {3, 6} (the duals of (ii)).

We consider the family (iii). By adjoining an element to the group of any
map in (iii) we shall derive the group of a regular map of type {6, 6}. Thus
we produce a 1−1 correspondence between the members of the family (iii) and
of the new family. Corresponding members in the two families have certain
properties in common, the most interesting of which is the property of
reflexibility. Our results are summarized in Theorems 1 and 2.
A Note on a Theorem of Moser and Whitney
Authors: Narayana T. V. , Pettigrew H. M.
Published: December 2010

In a recent paper [1], L. Moser and E.L. Whitney have proved the
following.

Theorem: The number of compositions of n into parts=1, 2, 4 or 5 (mod 6) and
involving an even number of parts=4 or 5 (mod 6) exceeds by n the number of
compositions of n into parts=1, 2, 4 or 5 (mod 6) and involving an odd
number of parts=4 or 5 (mod 6).

Their method of proof utilizes the notion of weighted compositions and the
method of generating series. They remark that they have not been able to
find a direct combinatorial proof. The purpose of this note is to give a
direct proof of a more general result.
Note on the Structure of Graphs
Authors: Dirac G.A.
Published: December 2010

This paper is concerned with undirected graphs which may be infinite and may contain multiple edges. The Axiom of Choice is assumed. The terms path, infinite path and circuit are used in the same sense as Weg, unendlicher Weg and Kreis, respectively, are used in D. Konig's book [1]. The valency of a vertex is the number of edges incident with it.The length of a path is the number of edges in it. The following theorem is a generalization of the well known fact that if a vertex of a graph is not a cut-vertex (Artikulation [2]) and has valency ≧2, then the graph contains at least one circuit to which the vertex belongs.
A Pair of Generators for the Unimodular Group
Authors: Trott Stanton M.
Published: December 2010

It is well known that ℳn, the multiplicative group n consisting of n-rowed square matrices with integer entries and determinant equal to ±1 can be generated by:

It is also known that U4 can be generated by U1, U2, and U3 (cf.[2] p.85).

However, by a construction which is much simpler than the one just mentioned for U4, it is possible to generate U3 by just U2 and U4. Since U2 and U4 affect only the first
Congruence Relationships for Integral Recurrences
Authors: Mendelsohn N. S.
Published: December 2010

A sequence {un}, n=0, 1, 2, 3,… is said to be an integral recurrence of order r if the terms satisfy the equation

for n=r+1, r+2,…, and a1, a2,…, ar are integers, ar≠0. In this case we will say that {un} satisfies the relation [a1, a2,…, ar]. The sequence {un} is uniquely determined when u1, u2,…, ur are given specified values. If u1, u2,…, ur are integers all the terms of {un} are integers. The generating function f(t)=u1t + u2t2+… takes on the form where Q(t) depends on the values of u1, u2,…, ur and R(t)=tr-a1tr-1-a2tr-2-…-ar. We will refer to R(t) as the characteristic polynomial of the recurrence.
Special Orthogonal Latin Squares of Order 10
Authors: Weisner Louis
Published: December 2010

The orthogonal latin squares displayed in [1] and [2] have the property that
their row permutations are transformed amongst themselves by a permutation
of order 7. In this note I present three examples of orthogonal latin
squares of order 10 whose row permutations are transformed amongst
themselves by a permutation of order 9.
Indefinite Green's Functions and Elementary
Solutions
Authors: Duff G. F. D. , Ross R. A.
Published: December 2010

Linear differential equations both ordinary and partial are often studied by
means of Green's functions. One reason for this is that linearity permits
superposition of solutions. A Green's function describes the "effect" of a
point source, and the description of line, surface, or volume sources is
achieved by superposing, that is to say, integrating, this function over the
source distribution.

For equations with constant coefficients the use of integral transforms
permits the calculation of such source functions in the form of integrals.
Only in the simplest cases is explicit evaluation by elementary functions
possible, and this has perforce led to the use of asymptotic estimates,
which so thoroughly pervade the domain of applied mathematics.
Some Results Concerning the Structure of Graphs
Authors: Dirac G.A.
Published: December 2010

The object of this paper is to present results concerning the structure of 3-connected graphs and of 5-chromatic and 6-chromatic graphs and also a theorem on contraction and a theorem of Turan type. The Axiom of Choice is assumed.
Partition Functions and Spiralling in Plane Random Walk
Authors: Melzak Z.A.
Published: December 2010

Consider the plane symmetric random walk on a square lattice: a particle is initially at the origin in the xy-plane, it makes n consecutive steps of unit length, and each step is made with the probability 1/4 in each one of the four directions parallel to the axes. We call the path of the particle a spiral if the following conditions are met: a) the particle never occupies the same position twice, b) the path of the particle, whenever it turns, either turns always clockwise or always counter-clockwise throughout the path, and c) for every m > n the given n-step path can be continued in atleast one way to give an m-step path meeting the conditions.
Variational Aspects of the Abel and Schroder Functional Equations
Authors: McKiernan M.A. , Rényi A.
Published: December 2010

Given an analytic function f, the successive iterates of f are defined by

f[0](z) = z, f[n+1](z) = f{f[n](z)} for every z.

In particular f[1](z) = {f[0](z)} f(z). Extensive study has been given [1] to the problem of generalizing the iterates f[n], for integer n, to f[t] for arbitrary t, where the iterative character of f[t] is to be preserved by the conditions,

f[0](z) = z and f[s]{f[t](z)} = f[s+t](z) for arbitrary s and t.
Generalization of a Bracket Function Formula of L. Moser
Authors: Gould H. W.
Published: December 2010

In Problem P 60 L. Moser has proposed the formula

1

This can be expressed in the more elegant form

2

The formula has been given in the literature a number of times. For example, it was proved by Bouniakovsky [1].
On Subgraphs of the Complete Bipartite Graph
Authors: Erdös P. , Moon J. W.
Published: December 2010

G(n) denotes a graph of n vertices and Ḡ(n) denotes its complementary graph. In a complete graph every two distinct vertices are joined by an edge. Let Ck(G(n)) denote the number of complete subgraphs of k vertices contained in G(n). Recently it was proved [1] that for every k

1

where the minimum is over all graphs G(n).
A Distribution Function of Cantor-Vitali Type
Authors: Sumner D. B.
Published: December 2010

In his 1922 article [l] on functions of bounded variation, Vitali gave a method for constructing monotone non-absolutely continuous functions, generalizing ideas from the ternary set introduced in another connection by Cantor. In [2], Hille and Tamarkin gave a full account of the "middle-third" function, showing it to be a singular distribution function, and finding its characteristic function. In [3], Evans obtained a generalization of the middle - third function by discarding middle intervals of length other than one-third, and obtained algorithms by which the moments of his function could be calculated. Invarious papers, among them [4], Wintner studied infinite convolutions of symmetric Bernoulli distributions, finding a great variety of distributions whose characteristic functions were of the form
A New Proof of an Inequality of Heinz
Authors: Bullen P. S.
Published: December 2010

In a recent paper, [l], Dixmier has proved Heinz' inequality by deducing it from a lemma due to Thorin. In this note it is proved directly from a convexity theorem.

Let(M(k), ℳ(k), μ(k)), k = 0, …, n, be measure spaces and Lq(k) (M(k), ℳ(k), μ(k)) be all the functions on M(k) such that
Remarks on Quasi-Hermite-Fejér Interpolation
Authors: Sharma A.
Published: December 2010

Let

1

be n+2 distinct points on the real line and let us denote the corresponding real numbers, which are at the moment arbitrary, by

2

The problem of Hermite-Fejér interpolation is to construct the polynomials which take the values (2) at the abscissas (1) and have preassigned derivatives at these points. This idea has recently been exploited in a very interesting manner by P. Szasz [1] who has termed qua si-Hermite-Fejér interpolation to be that process wherein the derivatives are only prescribed at the points x1, x2, …, xn and the points -1, +1 are left out, while the values are prescribed at all the abscissas (1).
Some Remarks on the Mathieu Groups
Authors: Garbe Dietmar , Mennicke Jens L.
Published: December 2010

In the present note we shall study some properties of the Mathieu groups.

We shall give an invariant characterisation of the 2-Sylow subgroups. The 2-Sylow subgroup of M24is the holomorph of the elementary abelian group of type (1, 1, 1, 1), and for the 2-Sylow subgroups of the other Mathieu groups there are similar characterisations.
Partitions of the Natural Numbers
Authors: Angel Myer
Published: December 2010

We obtain in this article some results concerning partitions of the natural
numbers, the most important of which is a generalization of that quoted
immediately below. Some intuitive material is included.

In 1954, J. Lambek and L. Moser [l] showed that "Two non-decreasing
sequences f and g (of non-negative integers) are inverses if and only if the
corresponding sets F and G of positive integers, defined by F(m) = the mth
element of F = f(m) + m and G(n) = g(n) + n are complementary."
A Module is Flat if and Only if its Character Module is
Injective
Authors: Lambek J.
Published: December 2010

The purpose of this expository note is to establish the fact mentioned in
the title. While this is not difficult and requires no new ideas, it seems
worth doing, as such a simple characterization does not appear explicitly in
a recent treatise on the subject of flat modules [Bourbaki XXVII, Chapter
1].
On Abel's Binomial Identity
Authors: Majindar Kulendra N.
Published: December 2010

The identity of Abel [1] [2] we deal with here can be stated in the
following form: If n is a positive integer,

.

(In order that all terms be defined we require a ≠ 0, b ≠ n.)

This identity and deductions from it have been very useful in many problems,
for instance in mathematical statistics [3]. Usually this identity is
established by means of the Lagrange - Bűrman theorem [4]. Here we will
derive it very simply.
Explicit Expressions for a Class of Permutation Problems
Authors: Abramson Morton
Published: December 2010

The purpose of this paper is to give some explicit formulae for probability problems such as those dealing with maximum runs upon tossing a coin n times in succession. Although the actual computation involved is quite lengthy, and generating functions can be used for approximations, the lemmas obtained may be of some interest in themselves.
A Problem on Tournaments
Authors: Erdös P. , Moser L.
Published: December 2010

By a tournament we mean the outcome of a round-robin tournament in which there are no draws. Such a tournament may be represented by a graph in which the n players are represented by vertices labelled 1, 2, …, n, and the outcomes of the games are represented by directed edges so that every pair of vertices is joined by one directed edge. We call such a graph a complete directed graph. One can also represent such a tournament by an nXn matrix T=(tij) in which tij is 1 if i beats j, and 0 otherwise, so that T is a (0, 1) matrix with tij + tji = 1 for i≠j and (by definition) tii = 0.
Some Lower Bounds for Density of Multiple Packing
Authors: Blundon W.J.
Published: December 2010

A system of circles is said to form a k-fold packing of the plane if every point of the plane is an interior point of at most k circles. In this paper we consider only lattice packings, that is, we suppose that the centers of the circles form a lattice. We further restrict our consideration to circles of equal radius. Without loss of generality, the circles may have unit radius.
The Densest Packing of 9 Circles in a Square
Authors: Schaer J.
Published: December 2010

Packing problems of this kind are obviously equivalent to the problems of placing k (here 9) points in a unit square such that the minimum distance between any two of them be as large as possible. The solutions of these problems are known for 2 ≤ k ≤ 9. The largest possible minimum distances mk are given in table 1, and the corresponding "best" configurations shown in figure 1.
The Series Σ∞1 f(n)/n for Periodic f
Authors: Livingston Arthur E.
Published: December 2010

We are here concerned with the problem of deciding when Σ∞n=1 f(n)/n ≠ 0, given that f is periodic and the series convergent. In particular, we consider

Conjecture A. Let p be a positive integer and f a (real-or complex-valued) number-theoretic function with period p.
Inequalities Associated with the Triangle
Authors: Blundon W. J.
Published: December 2010

Let R, r, s represent respectively the circumradius, the inradius and the semiperimeter of a triangle with sides a, b, c. Let f(R, r) and F(R, r) be homogeneous real functions. Let q(R, r) and Q(R, r) be real quadratic forms. The latter functions are thus a special case of the former. Our main result is to derive the strongest possible inequalities of the form

1

with equality only for the equilateral triangle.
On the Summability of a Class of the Derived Conjugate Series of a Fourier Series
Authors: Govil Narendra K.
Published: December 2010

Let f(t) be integrable L(-π, π) and periodic with period 2π, and let

1.1

be its Fourier series. The series

1.2

obtained by the term by term differentiation of the series (1. 1), is called the Derived Fourier series of (1. 1). The series conjugate to (1. 2),

1.3

is called the Derived conjugate series of the Fourier series of f.
On Simple Rings with Maximal Annihilator Right Ideals
Authors: Koh Kwangil
Published: December 2010

If R is a simple ring with 1 which contains a maximal annihilator right ideal then R is the endomorphism ring of a unital torsion-free module over an integral domain.

We first prove the following:

Let R be a ring with 1. If a ϵ R such that (a)r = {r ϵ R|ar = 0} is a maximal annihilator right ideal then HomR(aR, aR) is an integral domain.
The Meaning of Mono and EPI in Some Familiar Categories
Authors: Burgess W.
Published: December 2010

This expository note was prompted by some questions asked by Professor P. Hilton during his lectures "Catégories non-abétiennes" at the University of Montréal, July 1964.

The descriptions of set functions as one to one and as onto can be characterized in terms of set function composition. A set function is one to one iff it has the left cancellation property, that is, f · g = f · h implies g = h.
On Random Variables which have the same Distribution as their Reciprocals
Authors: Seshadri V.
Published: December 2010

The motivation for this paper lies in the following remarkable property of certain probability distributions. The distribution law of the r. v. (random variable) X is exactly the same as that of 1/ X, and in the case of a r. v. with p. d. f. (probability density function) f(x; a, b) where a, b are parameters, the p. d. f. of 1/X is f(x; b, a). In the latter case the p. d. f. of the reciprocal is obtained from the p. d. f. of X by merely switching the parameters. The existence of random variables with this property is perhaps familiar to statisticians, as is evidenced by the use of the classical 'F' distribution. The Cauchy law is yet another example which illustrates this property. It seems, therefore, reasonable to characterize this class of random variables by means of this rather interesting property.
On the Square of a Homological Monoid
Authors: Bonyun Rosemary
Published: December 2010

Homological monoids, as first defined by Hilton and Ledermann [ l ], are a generalization of abelian categories. It is known that if is an abelian category, so is here we prove the more general theorem that if is a homological monoid, so is . Our definition differs from that originally given by Hilton and Ledermann by the addition of a uniqueness condition in Axiom 1.
Automorphisms of the Semigroup of all Relations on a Set
Authors: Magill Kenneth D.
Published: December 2010

An automorphism Φ of a semigroup S is said to be an inner automorphism if there exists a unit u in S such that

for each a in S.
On Convergence of Projections in Locally Convex Spaces
Authors: Simpson J. E.
Published: December 2010

This note is concerned with the extension to locally convex spaces of a theorem of J. Y. Barry [ 1 ]. The basic assumptions are as follows. E is a separated locally convex topological vector space, henceforth assumed to be barreled. E' is its strong dual. For any subset A of E, we denote by w(A) the closure of A in the σ-(E, E')-topology. See [ 2 ] for further information about locally convex spaces. By a projection we shall mean a continuous linear mapping of E into itself which is idempotent.
Total Digraphs
Authors: Chartrand Gary , James Stewart M.
Published: December 2010

The line - graph of an ordinary graph G is that graph whose points can be put in one-to-one correspondence with the lines of G in such a way that two points of are adjacent if and only if the corresponding lines of G are adjacent. This concept originated with Whitney [ 5 ], has the property that its (point) chromatic number equals the line chromatic number of G, where the point (line) chromatic number of graph is the minimum number of colors required to color the points (lines) of the graph such that adjacent points (lines) are colored differently.
On the Quotients of Indecomposable Injective Modules
Authors: Tiwary A.K.
Published: December 2010

It is well known that Z(p∞) is isomorphic to each of its non-zero homomorphic images [3]. The aim of the present note is to generalize this fact about Z(p∞) to indecomposable infective modules over rings more general than the ring of integers which will include Dedekind domains as a special case.
Some Remarks on Angular Derivatives and Julia's Lemma
Authors: Jackson H. L.
Published: December 2010

Let w = f(z) be holomorphic on the unit disk D = { z: | z | < 1}, with the additional restrictions that | f ( z ) | < l and , where denotes the (outer) angular limit of f (z) at z = 1. Let us now define and then focus our attention on the behaviour of g(z) in an arbitrary angular neighbourhood of z = 1. Whenever exists, this limit is commonly referred to as the angular derivative of f(z) at z = 1.
On the Densest Packing of Spheres in a Cube
Authors: Schaer J.
Published: December 2010

How many spheres of given diameter can be packed in a cube of given size? Or: What is the maximum diameter of k identical spheres if they can be packed in a cube of given size? These questions are obviously equivalent to the following problem:

Let d(Pi, Pj) denote the distance between the points Pi and Pj, and Γk the set of all configurations of k points Pi (1 ≤ i < j ≤ k) in a closed unit cube C.
The Error Term for the Squarefree Integers
Authors: Moser L. , MacLeod R.A.
Published: December 2010

Let Q(x) denote the number of squarefree integers ≤ x. Recently K. Rogers [ l ] has shown that Q(x) ≥ 53x / 88 for all x, with equality only at x = 176. Define R(x) to be Q(x) - 6/п2 X. (We observe that and ) Our objective will be to examine R(x). In particular, we show that for all x and observe that for x ≥ 8.
A Property of Entire Functions of Exponential Type
Authors: Rahman Q.I.
Published: December 2010

We prove the following

THEOREM 1. Let f1(z), f2(z) be entire functions of exponential type τ1, τ2; respectively. Suppose that for certain constants K1, K2,

on the real line.
On the Functional Equations f(x+iy)| = |f(x)+f(iy)| and |f(x+iy)| = |f(x)-f(iy)| and on Ivory's Theorem
Authors: Haruki Hiroshi
Published: December 2010

Considering Cauchy's functional equation

f(z1+z2)=f(z1)+ f(z2),

where f(z) is an entire function of z, we have the following functional equation:

(1) |f(x+iy)|=|f(x)+f(iy)|,

where x and y are real.
Fatigue Spaces in Electromagnetic-Gravitational Theory
Authors: McKiernan M.A.
Published: December 2010

In an n-dimensional manifold Vn, coordinates xi for i = 1, …, n, let each curve x(t) for t0≤t≤t1 of class C1 define a corresponding λ by means of the integral equation

1
An Informal Arithmetical Approach to Computability and Computation, III
Authors: Melzak Z.A.
Published: December 2010

It is assumed that the reader is acquainted with the first two parts of the present paper, [1] and [2], in which there was developed informally the theory of a certain class of hypothetical computing devices, the Q-machines. In the present part of the paper we develop a way of describing Q-programs and Q-computations; then, following the theorem in [2], we obtain some further connections between the arithmetical nature of a number x and the computational schémas for generating sequences of rational numbers converging to x; and finally, by means of the usual device of Goedel numbering we begin to examine some properties of programs which operate on programs.
A Note on Combinations
Authors: Abramson M. , Moser W.
Published: December 2010

We call k integers x1 < x2 … < xk chosen from 1, 2, …, n} a k-choice (combination) from n. With 1, 2, …, n arranged in a circle, so that 1 and n are consecutive, we have a circular k-choice from n. A part of a k-choice from n is a sequence of consecutive integers not contained in a longer one. Let denote the number of circular k-choices from n with exactly r parts all ≤ w.
A Note on the Borsuk Conjecture
Authors: Melzak Z.A.
Published: December 2010

According to the still unproved conjecture of Borsuk [1] a bounded subset A of the Euclidean n-space En is a union of n + 1 sets of diameters less than the diameter D of A. Since A can be imbedded in a set of constant width D, [2], it may be assumed that A is already of constant width. If in addition A is smooth, i. e., if through every point of its boundary ∂A there passes one and only one support plane of A, then the truth of Borsuk′s conjecture can be proved very easily [3]. The question arises whether Borsuk′s conjecture holds also for arbitrary smooth convex bodies, not merely for those of constant width. Since it is not known whether a smooth convex body K can be imbedded in a smooth set of constant width D, the answer is not immediate. In this note we show that the answer is affirmative.
Categorical Fibrations
Authors: Seip Ulrich
Published: December 2010

If one considers the theories of Hurewicz, - Serre - or other fibrations in the categories of topological or pointed topological spaces, one can see that many of the fundamental theorems can be formulated and proven in the general case of categories for which certain functors and natural transformations are given. And, since fibrations may be defined either by a cylinder or a path space construction, we shall give in an analogous way two different definitions of fibrations in the general case.
On the Residues of a Cubic Polynomial (Mod p )
Authors: McCann K. , Williams K.S.
Published: December 2010

If f(x) is a polynomial with integral coefficients then the integer r is said to be a residue of f(x) modulo an integer m if the congruence

is soluble for x; otherwise r is termed a non-residue. When m is a prime p, Mord ell [4] has shown that the least nonnegative residue ℓ of f(x) (mod p) satisfies
On the Location of Zeros of Polynomials
Authors: Joyal A. , Labelle G. , Rahman Q.I.
Published: December 2010

The different results proved in this paper do not have very much in common. Since they all deal with the location of the zeros of a polynomial, we have decided to put them in one place. Improving upon a classical result of Cauchy we obtain in § 2 a circle containing all the zeros of a polynomial. In § 3 we obtain an extension of the well known theorem of Enestrőm and Kakeya concerning the zeros of a polynomial whose coefficients are non-negative and monotonie.
Polynomials with some Prescribed Zeros
Authors: Rahman Q. I. , Ali Khan Mohd.
Published: December 2010

In connection with various problems concerning polynomials

on the unit interval, the Tchebycheff polynomial

is known to play a very important role [11, problem 34].
On Some New Generalizations of the Functional Equation of Cauchy
Authors: Fischer P. , Muszély Gy.
Published: December 2010

Examining certain problems in physics M. Hosszu [l] obtained the functional equation

(1)

where x, y, f are real.

In another paper M. Hosszu [2] proved that the equation (1) is equivalent to the functional equation of Cauchy; i. e., to the equation

(1)

under the assumption that x is real and f is real and continuous.
The Hausdorff Means for Double Sequences
Authors: Ustina F.
Published: December 2010

The basic theory of the Hausdorff means for double sequences was developed some thirty - three years ago by C.R. Adams [1], and independently by F. Hallenbach [3], Yet today, many of the properties of these means remain largely uninvestigated. The calculations here, although clearly more complex, for the most part break down into obvious modifications of the calculations in the one dimensional case.
Maximal Subgroups of Infinite Symmetric Groups
Authors: Richman Fred
Published: December 2010

The purpose of this paper is to extend results of Ball [1] concerning maximal subgroups of the group S(X) of all permutations of the infinite set X. The basic idea is to consider S(X) as a group of operators on objects more complicated than X. The objects we consider here are subspaces of the Stone-Čech compactification of the discrete space X and the Boolean algebra of “big setoids” of X.
An Estimate for the order of Rational Matrices
Authors: Putz Randee
Published: December 2010

In this note we prove what we believe to be a new result concerning matrices, namely, that if an n × n matrix with rational entries has a finite order then this order is bounded. We also give an estimate for this bound and an application.
On A Theorem of Niven
Authors: Williams Kenneth S.
Published: December 2010

In 1940, I. Niven [2] proved that the gaussian integer z = x + iy is the sum
of two squares of gaussian integers if, and only if, y is even and not both
of 1/2x and 1/2y are rational odd integers. In this note we calculate the
total number g2(z) of representations of z in this form.

1

where a, b, c, d are rational integers, if and only if

2
A Note On Arc-Preserving Functions For Manifolds1
Authors: Charlton H. J.
Published: December 2010

Hall and Puckett [2] have shown that an arc - preserving function defined on
a locally connected continuum having no local separating points is a
homeomorphism if its total image is not an arc or point. This note shows
that their results can be extended to non-compact manifolds.
On Dependence in Matroids
Authors: Welsh D.J.A.
Published: December 2010

In this note we study dependence in matroids as an exercise in combinational
algebra. Because the work seems to have little connection with graph theory
we will not use Tutte′s approach (1) which uses dual concepts. To define a
matroid we use Edmunds (2).
Structure Theorems of a Module Over a Ring with a Bilinear Mapping
Authors: Nobusawa Nobuo
Published: December 2010

Let R be a ring with an identity 1, and R′ a ring anti - isomorphic to R. Let V be an R-module as well as an R′-module. We assume that 1 a = a for all elements a in V and that V satisfies the minimum condition for R-submodules. Elements of R will be denoted by α, β, …, and those of V by a, b, … Elements of R′ will be a α′, β′, …, where α′ fcorresponds to α by the anti - isomorphism. A mapping f of V x V to R is called a bilinear mapping of V to R if it satisfies the following.
A Note on the Homotopy-Commutativity of Suspensions
Authors: Hoo C.S.
Published: December 2010

Let A and X be spaces. Then as is wellknown, [∑A, X] is a group where ∑ denotes the suspension. We wish to find conditions on A which will imply that this group is abelian for all spaces X, that is, ∑A is homotopy-commutative. This is equivalent to saying that conii A≤ 1 (see [2] for definition). Our results contain relations between conil A and the generalised Whitehead product of [1]. We work in the category of complexes with base points.
Applying the Cauchy Integral
Authors: Taylor J. C
Published: December 2010

It could perhaps be reasonably maintained that for most students of calculus the definite integral is in fact the Cauchy Integral. That is to say, where F is a primitive of f.

If done heuristically, the discussion of the Riemann integral serves two purposes. Firstly, it indicates that every continuous function has a primitive.
Concerning the Minimum Function of a Stochastic Matrix
Authors: Sinkhorn Richard
Published: December 2010

A square matrix is said to be stochastic if its elements are non-negative and if each of its row sums is equal to one. Thus λ = 1 is always an eigenvalue of a stochastic matrix.
A Note on Lower Radical Constructions for Associative Rings
Authors: Heinicke A.G.
Published: December 2010

In [2], a construction for the lower radical class R∘ (η) with respect to a class η of rings was given as the union of an inductively defined ascending transfinite chain of classes of rings. It was shown there that this construction terminates, for associative rings, at ω∘, the first infinite ordinal, in the sense that if {ηα: α an ordinal} is the chain, then R∘ (η) =ηω∘. Also, examples of classes η for which R∘ (η) = η1, η2, η3 were given.
A Class of Positive Linear Operators
Authors: King J. P.
Published: December 2010

Let F[a, b] be the linear space of all real valued functions defined on [a, b]. A linear operator L: C[a, b] → F[a, b] is called positive (and hence monotone) on C[a, b] if L(f)≥0 whenever f≥0. There has been a considerable amount of research concerned with the convergence of sequences of the form {Ln(f)} to f where {Ln} is a sequence of positive linear operators on C[a, b].
The Nucleus of a Set
Authors: Strang Gilbert
Published: December 2010

Consider the subset containing those functions for which

One never attempts to visualize ; it is just a compact blur in the infinite-dimensional space . Nevertheless, we want to establish that it shares with several other sets an odd but rather remarkable "geometric" property: it is overwhelmingly concentrated around a single element. This element we call the nucleus of .
On a Discrete Analogue of Inequalities of Opial and Yang
Authors: Lee Cheng-Ming
Published: December 2010

Let be a non-decreasing sequence of non-negative numbers, and let U∘=0. Then we have

Yang [3] proved the following integral inequality:

If y(x) is absolutely continuous on a≤x≤X, with y(a) = 0, then
On Detours in Graphs1
Authors: Kapoor S. F. , Kronk H.V. , Lick D.R.
Published: December 2010

A path of maximum length in a graph G is referred to as a detour path of G and the length of such a path is called the detour number of G. It is not surprising that the study of detour paths is closely associated with the problem of investigating hamiltonian paths in graphs. Evidently few results have been obtained in this area, although Ore [3] has shown that any two detour paths intersect. It is the purpose of this article to further investigate these concepts. In particular, we obtain bounds for several graph theoretic parameters in terms of the detour number and also present formulae for the detour numbers of several important classes of graphs.
A Transformation Connecting Products of Generalised Basic Hypergeometric Functions
Authors: Verma Arun
Published: December 2010

Darling [2] in 1932 gave two types (equations 11 and 18) of transformations connecting generalised hyper geometric functions. The first was studied by Bailey [1] and extended by Sears [4] to a transformation connecting products of basic hyper geometric functions of the type r+1ϕr × r+1ϕr. In a number of papers [6, 7, 8] the author has extended these results to both unilateral and bilateral series with bases q and q1/2. The second type of transformation by Darling for a product 0F1 × 3F2 was extended by Bailey [1] to a transformation between 1F0 × r+1Fr. In the same paper Bailey mentioned the transformation of a 0ϕ1 × 3ϕ2 without proof.
Note on a Theorem on Singular Matrices
Authors: Djoković D. Ž.
Published: December 2010

J. A. Erdös proved recently [1] that every singular matrix over a field F is a product of idempotent matrices. He gave two proofs, one valid for matrices which are similar to triangular matrices and the other valid in general. We shall give a simple geometric proof of the above result. Instead of matrices we use linear operators. Moreover we get an explicit factorization in terms of projectors (idempotent operators).
Diameter of a (0, 1)-Matrix*
Authors: Murty U.S.R.
Published: December 2010

Let A be an m×n (0, l )-matrix. Let C1, C2, …, Cn denote its columns. A sequence of distinct columns is said to form a chain if the inner product of and (for 1 ≤ t ≤ k-l) is at least one. k-1 is called the length of the chain and this chain is said to connect are said to be connected. As can be easily seen, connectedness is an equivalence relation on the set of columns. A matrix is called connected if all its columns belong to the same equivalence class. If Ci and Cj belong to the same equivalence class, then s(Ci, Cj) will denote the length of the shortest chain between Ci and Cj We define the distance between any two columns Ci and Cj to be denoted by d(Ci, Cj), in the following manner.
A Remark on the Gibbs Phenomenon and Lebesgue Constants for a Summability Method of Melikov
Authors: Ustina Fred
Published: December 2010

Let u = ∑ukbe a given series and letMelikov [4] has defined the n-th σ- transform of u by

where ε and θ are assumed to be non-negative. This is easily shown to be equivalent to

The method is a generalization of a method used by Kaufman [l], and of another one used by Melikov [5]. It reduces to the (n-l)th (C;l) mean when θ = 0 and ε = 0, and to the n-th (C;l) mean when θ = 1 and ε = 0.
A Construction for Partitions Which Avoid Long Arithmetic Progressions
Authors: Berlekamp E.R.
Published: December 2010

For k ≥2, t ≥2, let W(k, t) denote the least integer m such that in every partition of m consecutive integers into k sets, atleast one set contains an arithmetic progression of t+1 terms. This paper presents a construction which improves the best previously known lower bounds on W(k, t) for small k and large t.
Circumscribing an Ellipsoid about the Intersection of Two Ellipsoids
Authors: Kahan W.
Published: December 2010

An ellipsoid G is associated uniquely with a positive definite matrix A via

Note that all ellipsoids discussed here are centred at 0. Given G1, and G2 we seek another ellipsoid circumscribed about G1 ∩ G2. It is easy to see that

if and only if x'hx ≤ maxi x'aix for all vectors x.