bluuebunny/crossref_metadata_2025 huggingface dataset:
Some Properties of the Rank and Invariant Factors of Matrices*
Authors: Gomory R.E. , Hoffman A.J. , Hsu N.C.
Published: December 2010

Some years ago, G. Pall observed that the invariant factors of the incidence matrices of a certain pair of non-isomorphic projective planes of order 9 were different. With the aim of investigating such phenomena experimentally, we have constructed a code to calculate the invariant factors of rational integral matrices (actually, we compute the Smith's normal form of these matrices, as described, e.g., in MacDuffee [2; p. 41]), and this note is in the nature of a report on some preliminary experiments in the use of this code. In particular, we have computed the invariant factors of all (0, 1) matrices of order ≤ 8, with constant row and column sums, and these data are presented in the Appendix.
Generating Reflections For U(2, p2n). II, p =
2
Authors: Crowe D. W.
Published: December 2010

It is known [4] that the finite two- dimensional unitary group U(2,
p2n) is generated by two reflections if p ≠ 2. The present
note completes that result by giving two generating reflections for U(2,
22n), n > 1. As in [4] this implies that the points of the
"unit circle " in the
unitary plane over GF(22n), n > 1, are the vertices of a
"regular unitary polygon" whose group of auto- 2n morphisms is U(2,
22n).

The final section gives abstract definitions for the particular groups
U(2,24) and U(2,52) in terms of their generating
reflections.
Permanents of (0, 1)-Circulants
Authors: Minc Henryk
Published: December 2010

The permanent of an n-square matrix A = (aij) is defined by

where the summation extends over all permutations σ of the symmetric group
Sn. A matrix is said to be a (0, 1)-matrix if each of its
entries is either 0 or 1. A (0, 1)-matrix of n-1 the form , where θj = 0 or 1, j = 1,…, n, and
Pn is the n-square permutation matrix with ones in the (1, 2),
(2, 3),…, (n-1, n), (n, 1) positions, is called a (0, 1)-circulant. Denote
the (0, 1)-circulant . It has
been conjectured that

1
A Machine Program for Coset Enumeration
Authors: Trotter H. F.
Published: December 2010

We are concerned with an algorithm for determining the index (when it is finite) of a subgroup H of a group K when K is specified by a finite set of generators and relations and H is specified as generated by a finite set of words in the generators of K. A systematic computational attack on the problem was discovered by Coxeter and Todd [l], and has proved to be a very useful tool in problems involving generators and relations in groups [2]. Although the method was not completely formalized it was clearly possible to convert it into a computer program, and this has been done by a number of people. Leech has given a survey of this work in [3], where further references may be found.
A Quartic with 28 Real Bitangents
Authors: Blum R. , Guinand A. P.
Published: December 2010

According to Pluecker's equations a plane quartic can have at most 28 bitangents. These can be all real as has been shown by Pluecker himself with the help of the example (see [l]):
On Injective Sheaves
Authors: Kleisli H. , Wu Y.C.
Published: December 2010

A divisible abelian group D can be characterized by the following property: Every homomorphism from an abelian group A to D can be extended to every abelian group B containing A. This together with the result that every abelian group can be embedded in a divisible group is a crucial point in many investigations on abelian groups. It was Baer, [1], who extended this result to modules over an arbitrary ring, replacing divisible groups by injective modules, that is, modules with the property mentioned above. Another proof was found later by Eckmann and Schopf, [3]. This proof assumes the proposition to hold for abelian groups and transfers it in a very simple and elegant manner to modules. In the sequel, we shall refer to this proof as to the Eckmann-Schopf proof.
New Proofs for two Theorems of Capelli
Authors: Rowlinson Elizabeth
Published: December 2010

The following two theorems are due to Capelli.

THEOREM 1. Let g(x) and h(x) be polynomials over a field R of characteristic 0; let f(x) = g(h(x)). Then f(x) is irreducible over R if and only if

(i)

g(x) is irreducible over R

and

(ii)

h(x) - β is irreducible over R(β), where β is a root of g(x).
On Rellich's Theorem Concerning Infinitely Narrow Tubes
Authors: Clark Colin
Published: December 2010

Let G be a region in Eućlidean n-space En and consider the eigenvalue problem Δ2u = λu on G, with boundary conditions u = 0 on Γ, the boundary of G. (To be precise, we are considering the eigenvalue problem for the self-adjoint 2 realization L associated with the Laplacian -Δ2and zero boundary condition, acting in L2(G), cf Browder [2]). If G is bounded, the spectrum of this problem is discrete, but Rellich showed in 1952 [6] that the spectrum could also be discrete for certain unbounded regions which he introduced and called "infinitely narrow tubes".
On Lattice Paths with Diagonal Steps
Authors: Rohatgi V.K.
Published: December 2010

In [1] L. Moser and W. Zayachkowski considered lattice paths from (0, 0) to (x, y) where the possible moves are of three types: (1) a horizontal step, (2) a vertical step, and (3) a diagonal step. They obtained an expression for the number of paths from (0, 0) to (n, n) lying below the main diagonal except at the terminal points. In this note we extend their results to cover any point (m, n) lying below the main diagonal.
On the Number of Positive Entries in the Powers of a Non-Negative Matrix
Authors: Pullman N.
Published: December 2010

A real matrix A is said to be non-negative if and only if none of its entries is negative. Suppose A is an r by r non-negative matrix. We want to examine:

(A)

The first power of A to maximize the number of positive entries in An,

(B)

For each 1 ≤ i ≤ r the first power of A to maximize the number of positive entries in the i-th row of An.

We shall call the former first power the index of A and the latter the i-th row index of A (index (i, A)).
Goursat's Theorem and Homological Algebra
Authors: Lambek Joachim
Published: December 2010

This expository note consists of two parts: In the first we present a homological formulation of Goursat's theorem. In the second we indicate how some rudimentary concepts of homological algebra can be introduced with the help of this form of Goursat's theorem. The second part is addressed to those readers with an algebraic background who wish to be initiated into homological algebra as painlessly as possible. The ideas developed here are close to the spirit of [5, Chapter II, § 6], where further references may be found. In fact, these ideas are very much in the air, and any originality in the present note is purely coincidental.
On the Exactness of the Homotopy Sequence
Authors: Scherk P.
Published: December 2010

This note is based on Hu's Homotopy Theory (New York, 1959), pp. 115 f.

The theorem that the homotopy sequence is exact splits into six statements. The fifth in Hu's exposition can be stated as follows: Let f:In, In-1, Jn-1→X, A, x0,

1

Then there exists a map

2

such that f~j g: In, In-1, Jn-1→X, A, x0.
The Isomorphism between Graphs and their Adjoint Graphs
Authors: Menon V. V.
Published: December 2010

A graph G is defined as a set X = {x1, …, xn} of elements xi called vertices, and a collection Γ of (not necessarily distinct) unordered pairs of distinct vertices, called edges. An edge (xi, xj) is said to be incident to xi and xj which are its end-vertices.
On a Class of Complete Simple Sets
Authors: McLaughlin T. G.
Published: December 2010

Since the solution of Post' s Problem by Friedberg and Muchnik, a good deal of abstract knowledge about the semilattice of recursively enumerable degrees has been developed, especially in recent papers of G. E. Sacks. In this note, we show that one specific class of simple sets contains only sets of degree 0'; no contribution to the general theory is claimed. One of the sets belonging to the special class considered is the original simpie-but-not-hypersimple S of Post [1]. According to information received from Sacks, J. R. Myhill gave a proof, in 1953, of the completeness of S. However, as far as we know, Myhill' s proof (presented in a seminar) does not exist in published form.
On Certain Simultaneous Functional Equations
Authors: Howroyd T. D.
Published: December 2010

G. de Rham [1] has shown that certain remarkable curves, defined by limits of sequences of inscribed polygons, are solutions of functional equations of the form

If the functions FP (z) (p = 1,2) are contraction mappings of the complex plane into itself, i.e. for fixed r < 1
Remark on Co-Null Matrices
Authors: Macphail M. S.
Published: December 2010

A well-known theorem of Copping [2] states that a conservative matrix with a bounded left inverse cannot evaluate a bounded divergent sequence. (Definitions are given in the next paragraph.) A proof was given by Parameswaran [3, Theorem 6. 1], using only the simplest Banach-space ideas. This proof, however, is valid only for co-regular methods; it was stated in [3, Theorem 6. 2] that a co-null matrix cannot have a bounded left inverse, but the proof there given is incorrect, as it uses for co-null methods a theorem established only for co-regular. It would be desirable to have a short independent proof of this known result, which excludes co-null matrices from consideration in Copping' s theorem. This is furnished by the slightly more general result given below.
A General Hamilton-Jacobi Equation and Associated Problem of Lagrange
Authors: McKiernan M. A.
Published: December 2010

It is well known [1] that the variational problem of minimizing

1

where F is positive homogeneous of degree one in (henceforth abbreviated to "plus-one" in ) leads to a Hamiltonian H{xi, pi} and corresponding Hamilton-Jacobi equation

2
An Example Concerning Exposed Points
Authors: Edelstein M.
Published: December 2010

In his paper [1] V. L. Klee gave an example of a smooth bounded convex body C, in En, with the property that extC s closed and extC - expC is dense in extC. As in [1] extC and expC denote the sets of extreme and exposed points of C respectively. It is the purpose of this note to exhibit a similar example in a general separable Banach space using a direct construction which involves Minkowski summation of convex sets.
A Theorem on Harmonic Homologies
Authors: Marsden J. E.
Published: December 2010

A collineation is a one-one mapping of a projective plane onto itself, taking points into points, lines into lines and preserving incidence, ([1], p. 247). A perspective collineation (sometimes called a central collineation) is a collineation which leaves invariant all points on a line h called the axis, and all lines through a point H called the centre. The perspective collineation is an elation if H is incident with h; otherwise it is a homology.
A Differential Geometry Associated with Dissipative Systems
Authors: McKiernan M. A.
Published: December 2010

Consider the following problem of Lagrange in the calculus of variations: relative to differentiable curves xi(t) satisfying xi(t0) = xi0 and xi(t1) = xi1 find a curve minimizing

1
Some General Instability Criteria for Ordinary Differential Equations
Authors: Burton T. A.
Published: December 2010

We consider a system of differential equations

1

where 0 = (o, o) is an isolated singular point. Thus, there exists B > o such that S(0, B) contains only one singular point. Here, S(0, B) denotes a sphere centered at 0 with radius B. We shall denote the boundary of S(0, B) by ∂ S(0, B).
A New Proof of a Theorem of Dirac
Authors: Brown William G.
Published: December 2010

In [1] Dirac has determined the structure of all 3-connected graphs which do not contain two independent (i.e., disjoint) circuits. We shall here provide a short proof of this theorem by applying Tutte' s theory of 3-connected graphs [2].
The Maximum Number of Strongly Connected Subtournaments*
Authors: Beineke Lowell W. , Harary Frank
Published: December 2010

In the ranking of a collection of p objects by the method of paired comparisons, a measure of consistency is provided by the relative number of transitive (or consistent) triples and cyclic (or inconsistent) triples. This point of view was introduced by Kendall and Babington Smith [4]. They found a formula for the maximum number of cyclic triples, thereby determining the greatest inconsistency possible. The purpose of this note is to extend the result to obtain the maximum number of "strongly connected" collections of n objects among the given p objects.
An Application of Ramsay's Theorem to a Problem of Erdos and Hajnal
Authors: Abbott H. L.
Published: December 2010

A family of sets is said to possess property if there exists a set such that and F ⊄ B for each In [1], P. Erdos and A. Hajnal ask the following question: Does there exist for every positive integer k a finite family of finite sets satisfying

(i)

|F|=k for each

(ii)

| F∩ G| ≤ 1 for each F, , F ≠ G

(iii)

does not possess property ?
On Abelian Permutation Groups
Authors: Bercov R. , Moser L.
Published: December 2010

The principal object of this note is to determine the maximal order of Abelian subgroups of the symmetric group sn of degree n. We also discuss some related results and problems.

A largest Abelian subgroup of sn has order f(n) where
Some Remarks on Transformations in Metric Spaces
Authors: Wong James S. W.
Published: December 2010

Recently A. Haimovici [1] has proved a general fixed point theorem of transformations in metric spaces from which he obtained existence theorems for certain types of ordinary and partial differential equations. However, both the result and the proof are given for a rather special case. One of the purposes of this present note is to put his result on a more concrete basis and give a stronger characterization of the kind of transformations used in [l]. (Theorem 3).
Note on the Hardy-Landau Summation Formula
Authors: Pearson T. L.
Published: December 2010

Broadly speaking, the Hardy-Landau summation formula * is given by

where r(n) is the number of integer solutions of the Diophantine equation x2 + y2 = n, and f(x) and g(x) are transforms with respect to the Watson kernel that is:

and
Weak Families of Maps
Authors: Taylor J. C.
Published: December 2010

Let Ω be an index set and for each αϵΩ let fα:X → Xα be a function where X and Xα are sets. Assume that, for each α, a topology is given for Xα. Then, as is well-known, the functions fα and the topologies determine a topology for X. This is the so-called weak or initial topology, which is generated by .
On the Number of Complete Subgraphs of a Graph
Authors: Moon J. W.
Published: December 2010

A graph Gn consists of n nodes some pairs of which are joined by a single edge. A complete k-graph has k nodes and edges. Erdos [1] proved that if a graph Gn has edges, then it contains at least complete 3-graphs if it contains any at all. The main object of this note is to extend this result to complete k-graphs.
Extensions of Sylvester's Theorem
Authors: Balomenos Richard H. , Bonnice William E. , Silverman Robert J.
Published: December 2010

Sylvester [7] proposed the following question in 1893. If a finite set of points in a plane is such that on the line determined by any two points of the set there is always a third point of the set, is the set collinear? Equivalently, given a finite planar set of non-collinear points, does there exist a line containing exactly two of the points?
A Class of Prime Rings
Authors: Koh Kwangil , Mewborn A. C.
Published: December 2010

If R is a ring and I is a right ideal of R then I is called faithful if R - I is a faithful right R-module, i.e. if { r ∊ R: Rr⊆ I} = (0). I is called irreducible [ 1 ] provided that if J1 and J2 are right ideals such that J1 ∩ J2 = I, then J1 or J2 = I. Let N(I){ r ∊ R: rI⊆ I} and [ I: a ] = { r ∊ R: ar⊆ I} for a ∊ R. We write (a)r for [(0): a ].
A Proof that Souslin Souslin H ⊂ Souslin H
Authors: Simons S.
Published: December 2010

We write ω for the set of natural numbers (including zero) andfor the set of all finite sequences of natural numbers. If n ∊ ω we write. If x is a function which takes its values in cu and whose domain of definition containsthen we writefor the element (x(0), …, x(n)) of.
Sets of Convergence for Series Defined by Iteration1
Authors: Brauer George
Published: December 2010

Let f(x) be a real-valued function defined on an interval Ia: [ 0, a ]. For each point x in Ia we form the series , where u0 and un+1 = f(un) for n ≥ 0. If the series converges, x will be called a point of convergence; if this series diverges, x will be called a point of divergence. La this note several properties of sets of convergence2 will be obtained.
Sequences of Euler Graphs
Authors: Sabidussi Gert
Published: December 2010

A graph is called Euler if it has no isolated vertices and every vertex has even or infinite degree. The graphs we consider may have multiple edges but no loops. Loosely speaking, we will be concerned with the conditional compactness of the set of all countable Euler subgraphs of a given graph.
A Generating Function for Jacobi Polynomials
Authors: Saxena R. K.
Published: December 2010

The following notations will be employed throughout this note.

The object of the present note is to obtain a new generating function for the Jacobi polynomials defined by [4, page 268]
On Graphs that do not Contain a Thomsen Graph
Authors: Brown W. G.
Published: December 2010

A Thomsen graph [2, p. 22] consists of six vertices partitioned into two classes of three each, with every vertex in one class connected to every vertex in the other; it is the graph of the “gas, water, and electricity” problem [1, p. 206]. (All graphs considered in this paper will be undirected, having neither loops nor multiple edges.)
Integration of Non-Measurable Functions
Authors: Zakon Elias
Published: December 2010

This is primarily an expository paper based on (and generalizing) some ideas of J. Pierpont [6], W.H. Young [8], R. L. Jeffery [4] and S. C. Fan [2]. Our aim is to give a simple and easily applicable theory of integration for arbitrary extended - real functions over arbitrary sets in a measure space. This will be achieved by using a generalized version of Pierpont1 s upper and lower integrals (with the upper integral playing the main role), and by appropriately defining the operations in the extended real number system, henceforth denoted by E*, so as to make it a commutative semigroup under addition and multiplication.
Unbounded Vector Measures
Authors: Byers William
Published: December 2010

The aim of this paper is to extend the idea of a measure which takes on values in Euclidean n-space so as to allow it to assume infinite values while preserving its countable additivity over a given σ-ring. It is shown that in order to do this it is necessary to restrict the range of the measure to one infinite value.
A Theorem on Nets
Authors: Shimrat M.
Published: December 2010

It is well-known that Tychonoff's theorem on the product of compact spaces may be proved, for the special case of a countable number of metric spaces X1, X2…, Xn,…, in the following simple manner.
Semi-Hausdorff Spaces
Authors: Murdeshwar M.G. , Naimpally S.A.
Published: December 2010

It is well-known that in a Hausdorff space, a sequence has at most one limit, but that the converse is not true. The condition that every sequence have at most one limit will be called the semi-Hausdorff condition. We will prove that the semi-Hausdorff condition is strictly stronger than the T1 -axiom and is thus between the T1 and T2 axioms. In this note, we investigate into some properties of the spaces satisfying the semi-Hausdorff condition.
Order in Finite Affine Planes
Authors: Beattie Mary , Marsden J.E. , Sharpe R. W.
Published: December 2010

An ordered plane [1, p. 177 ff.] is a set of undefined entities called points, together with an undefined relation of intermediacy, satisfying the following axioms: (The symbol [ABC] means ″B is between A and C″.)
On Semigroups of Transformations Acting Transitively on a Set
Authors: Tully E.J.
Published: December 2010

We call a semigroup S transitive if S is isomorphic to a semigroup T of transformations of some set M into itself, where T acts on M transitively, that is in such a manner that for all x, y ∊ M we have Xπ = y for some transformation π∊T. In [4] the author showed that S is transitive if and only if there exists a right congruence σ (i.e., an equivalence relation for which a σ b always implies ac σ bc for all c ∊ S) on S, satisfying:

(1)There exists a left identity modulo σ, that is an element e such that ea σ a for all a ∊ S .

(2)Each σ-class meets each right ideal, or, equivalently, for all a, b ∊ S we have ac σ b for some c ∊ S .

(3)The relation σ contains ( i. e. , is less fine than) no left congruence except the identity relation (in which each class consists of a single element).
On Some Special Factorizations of (1-xn)/(1-x)
Authors: Carlitz L. , Moser L.
Published: December 2010

Let A: a1 < a2 < … < ak be a set of non-negative integers We call the corresponding polynomial the characteristic polynomial, or briefly, the c-polynomial of A. Any polynomial of such a form we call a c-polynomial and any factorization of a c-polynomial into others of the kind we call a c-factorization. If a c-polynomial cannot be factored in this way we call it c-irreducible. In this note we will determine all c-factorizations of the polynomial 1 + x + x2 + … + xn-1, and will find under what circumstances the c-irreducible factors are also irreducible in the usual sense, i.e., irreducible over the field of rationals.
Linear Normed Spaces with Extension Property
Authors: Elliott George , Halperin Israel
Published: December 2010

In this paper we shall say “E has the (F, G) (extension) property” to mean the following: F is a subspace of the real normed linear space G, E is a real normed linear space, and any bounded linear mapping F→E has a linear extension G→E with the same bound (equivalently, every linear mapping F→E of bound 1 has a linear extension G→E with bound 1).
A Characterization of Deviation from Normality Under Certain Moment Assumptions
Authors: McGillivray W.R. , Kaller C.L.
Published: December 2010

If Fn is the distribution function of a distribution n with moments up to order n equal to those of the standard normal distribution, then from Kendall and Stuart [1, p.87],
A Combinatorial Theorem
Authors: Bassett Paul G.
Published: December 2010

Let n be an arbitrary but fixed positive integer. Let Tn be the set of all monotone - increasing n-tuples of positive integers:

1

Define

2

In this note we prove that ϕ is a 1–1 mapping from Tn onto {1, 2, 3,…}.
On Closed, Totally Bounded Sets
Authors: Murdeshwar M. G.
Published: December 2010

C. Goffman asserts that "… in a metric space X a set S is compact if and only if it is closed and totally bounded." [1] and "… every totally bounded sequence in a metric space has convergent subsequence." [2].

The statements (incidentally, equivalent to each other) are both wrong, as the following counter-example shows. Take the set of all reals in the open interval (0, 1) with the usual metric. This space is closed and totally bounded, but not compact.
Demi-Groupes Separateurs
Authors: Derdérian J.-C. , Thierrin G.
Published: December 2010

Un demi-groupe d'applications d'un ensemble est dit séparateur si, pour tout triple d'éléments x, y, z de l'ensemble, avec x ≠ y, il existe une application α du demi-groupe telle que x.α ≠ z et y α ≠ z. Un tel demi-groupe d'applications est évidemment transitif. Tout demi-groupe isomorphe à un demigroupe séparateur d'applications est dit un demi-groupe séparateur.
Note on Almost Product Manifolds and their Tangent Bundles
Authors: Houh Chorng Shi
Published: December 2010

Let Mn be an n-dimensional manifold of differentiability class C∞ with an almost product structure . Let have eigenvalue +1 of multiplicity p and eigenvalue -1 of multiplicity q where p+q = n and p≧1, q≧1. Let T(Mn) be the tangent bundle of M. T(Mn) is a 2n dimensional manifold of class C∞. Let xi be the local coordinates of a point P of Mn. The local coordinates of T(Mn) can be expressed by 2n variables (xi, yi) where xi are coordinates of the point P and yi are components of a tangent vector at P with respect to the natural frame constituted by the vectior ∂/∂xi at P.
On Disjoint Connected Subsets of a Square Containing Pairs of Antipodal Points
Authors: Morrison D. , Kiang M. , Wright J.
Published: December 2010

In [1], p. 132, it is shown that there exist two disjoint subsets of a square each connecting a pair of opposite vertices. Our interest in obtaining covers by infinite collections of such sets was prompted by a question posed by R.B. Reed in [3], We are grateful also to Dr. M. Edelstein for his guidance and encouragement of our work on the subject.
Translativity for Strong Borel Summability
Authors: Lorch Lee
Published: December 2010

It is an obvious property of convergence that implies that sn+k exists and equals s for k = -1 (left translativity ) and for k = 1 (right translativity). Not so for sumrnability.
Stability of Interpolation on an Infinite Interval
Authors: Saxena R.B.
Published: December 2010

In 1958, Egerváry and Turán [3] proposed and solved the problem of finding a stable interpolation process of minimal degree on a finite interval. Later [4] they investigated the same problem for an infinite interval with a suitable modification of the definition of stability. For the interval (-∞, ∞) their definition naturally differs from the one for the semi-infinite interval.
An Isoperimetric Inequality for Tetrahedra
Authors: Melzak Z.A.
Published: December 2010

Let T be a tetrahedron and let V(T) and L(T) denote its volume and the sum of its edge-lengths. In this note we prove

Theorem 1.

with equality if and only if the tetrahedron T is regular.
Identities for Multiplicative Functions
Authors: Subbarao M. V. , Gioia A. A.
Published: December 2010

Throughout this paper the arithmetic functions L(n) and w(n) denote respectively the number and product of the distinct prime divisors of the integer n > 1, with L(1) = 0 and w(1) = 1. Also let

We recall that an arithmetic function f(n) is said to be multiplicative if f(1) = 1 and f(mn) = f(m)f(n) whenever (m, n) = 1, where (m, n) denotes as usual the greatest common divisor of m and n.
Note on Pointwise Convergence on the Choquet Boundary
Authors: Grossman Marvin W.
Published: December 2010

In [6] J. Rainwater obtained the following theorem.

Theorem. Let N be a normed linear space, {xn} a bounded sequence of elements in N and X ∊ N. for each extreme point f of the unit ball of N✶, then {xn} converges weakly to x.

Now let X be a compact Hausdorff space and H a linear subspace of C(X) (all real-valued continuous functions on X ) which separates the points of X and contains the constant functions. If x∊X, then MX(H) denotes the set of positive linear functionals μ on C(X) such that μ(h) = h(x) for all h in H.
An Extremal Problem in Number Theory
Authors: Abbott H. L. , Gardner B.
Published: December 2010

Let n and k be integers with n ≥ k ≥ 3. Denote by f(n, k) the largest positive integer for which there exists a set S of f (n, k) integers satisfying (i) and (ii) no k members of S have pairwise the same greatest common divisor. The problem of determining f(n, k) appears to be difficult. Erdős [2[ proved that there is an absolute constant c > 1 such that for every ∈ > 0 and every fixed k

1

provided n > no (k, ∈). In [l[ it i s proved that for every ∈ > 0 and every fixed k
On Characterizing Classes of Functions in Terms of Associated Sets
Authors: Bruckner A. M.
Published: December 2010

Let κ be a class of real valued functions defined on an interval which need not be bounded. The class κ is said to be characterized in terms of associated sets if there exists a family of sets of real numbers P such that f ∈ κ if and only if for every real number α the sets a remembers of P. Many classes of functions have been characterized in terms of associated sets. The chart below summarizes a few such characterizations.
The Injective Envelope of S-Sets
Authors: Berthiaume P.
Published: December 2010

If S is a semigroup, then an S-set AS is a set A together with a representation of S by mappings of A into itself. In this article, the theory of injective envelopes is carried from R-modules to S-sets. These results are known to hold in every Grothendieck category, but the category EnsS of (right) S-sets is not even additive.
On a Theorem of Privaloff
Authors: Bullen P. S.
Published: December 2010

It is the object of this note to extend to general harmonic structures a theorem due to Privaloff [2] concerning the definition of harmonic functions. The notation is that of [8, 9, 10], where many of the definitions not given here will be found.
Open Disk Packings of a Disk
Authors: Wilker John B.
Published: December 2010

It is an old problem to find how a collection of congruent plane figures should be arranged without overlapping to cover the largest possible fraction of the plane or some region of the plane. If similar figures of arbitrary different sizes are permitted, Vitali's theorem ([7] p. 109) guarantees that packings which cover almost all points are possible. It is natural to study the diameters of figures used in such a packing and we will investigate this for the case of a closed disk packed with smaller open disks.
Coincidence Producing Maps Onto Trees
Authors: Schirmer Helga
Published: December 2010

Holsztyński [1] called a map f : X → Y f r om a space X into a space Y ‘universal for all maps of X into Y f if for all maps g : X → Y there exists a point x ∊ X such that f(x) = g(x) , i. e., if f has a coincidence with all maps from X into Y. As the word ‘universal’ is already widely used with different meanings, we prefer the more precise term ‘coincidence producing’ for these maps. Such maps must clearly be surjective.
A Remark on R1 Spaces
Authors: Rao B. V.
Published: December 2010

A topological space is an R1 - space if and only if x, y ε X, and such that U∩V = ϕ and x ε U, y ε V, where Ā denotes closure of the set A. Let be an R1 - space.
A Note on p-Cyclic Matrices and Digraphs
Authors: Heap B. R. , Lynn M. S
Published: December 2010

We use the terminology of [1]. Let D be a strongly connected digraph on n
points and containing m lines, and let A = A(D) be the corresponding
adjacency matrix, so that A is an n x n 0-1 matrix containing m unit
elements. We recall that A and D are said to be p-cyclic if p is the
greatest common divisor of the lengths of all directed cycles of D. Clearly,
the larger the value of p, the smaller the value of m must be; in this note
we make the latter and related statements precise.
On Irregular Fixed Points
Authors: Kaul S.K.
Published: December 2010

Throughout this paper (X, d) will be a metric space with metric d, and h a
homeomorphism of X onto itself. For any real number r > 0, and p ∊ X,
U(p, r) will denote the open r - sphere about p. Any point p ∊ X is called
regular [3] if for any given ∊ > 0 there exists a δ > 0 such that d(p,
y) <δ implies d(hn(p), hn(y)) < ∊ for all
integers n, where hn denotes the iterates of h for n > 0, of
h-1 for n < 0, and h0 is the identity. Any point
of X which is not a regular point i s called an irregular point. Let I(h)
denote the set of all the irregular points of X and R(h) = X-I(h). Lim inf
and Lim sup are defined as in [4].
On A Problem of P. Turán on Lacunary Interpolation*
Authors: Varma A. K.
Published: December 2010

In 1955, Suranyi and P. Turán [8] considered the problem of existence and
uniqueness of interpolatory polynomials of degree ≤ 2n-1 when their values
and second derivatives are prescribed on n given nodes. Around this kind of
interpolation - aptly termed (0, 2) interpolation - considerable literature
has grown up since then. For more complete bibliography on this subject we
refer to J. Balazs [3], Later we considered [10] the problem of modified (0,
2) interpolation when 2 the abscissas are the zeros of (1-x2)
Tn(x), where Tn(x) is the Tchebycheff polynomial of
the first kind (Tn(x) = cos n θ, x = cos θ).
On Extremal Polynomials
Authors: Williams Kenneth S.
Published: December 2010

Let p denote a prime number and let κp denote the finite field of
p elements. Let f(x) ∊ κp[x] be of fixed degree d ≥ 2. We suppose
that p is also fixed, large compared with d, say, p ≥ p0(d). By
V(f) we denote the number of distinct values of f(x), x ∊ κp. We
call f maximal if V(f) = p and quasi-maximal if V(f) = p + O(1). Clearly a
maximal polynomial is quasi-maximal but it is not known under what
conditions the converse holds.
Formal Contraction of the N-Simplex
Authors: Peterson Bruce B.
Published: December 2010

If K is a finite geometric (i.e. admitting a rectilinear triangulation) n-complex and σn is an n - simplex of K which is n -1 not a face of any n + 1 simplex of K, and if σn-1 is an n-1 face of σn which is not a face of any other n - simplex in K, then the complex K - σn - σn-1 (the complex whose simplexes are those of K except for σn and σn-1) is called an elementary contraction of K of order n. The correspondence K → K - σn - σn-1 will also be called an elementary contraction, there being no possibility of confusion.
On Stability of Solutions of Certain Differential Equations of the Third Order
Authors: Lalli B.S.
Published: December 2010

The purpose of this paper is to obtain a set of sufficient conditions for “global asymptotic stability” of the trivial solution x = 0 of the differential equation

1.1

using a Lyapunov function which is substantially different from similar functions used in [2], [3] and [4], for similar differential equations. The functions f1, f2 and f3 are real - valued and are smooth enough to ensure the existence of the solutions of (1.1) on [0, ∞). The dot indicates differentiation with respect to t. We are taking a and b to be some positive parameters.
On the Monotonicity of Certain Functionals in the Theory of Analytic Functions
Authors: Rahman Q.I.
Published: December 2010

Let f(z) be a function regular for | z | < R, and denote by L(r) the length of the curve Γ onto which the circle | z | = r is mapped by f(z), i. e.

If Dr is the image of | z | ≤ r on the Riemann surface of f(z) then its area S(r) is given by
Rellich′s Embedding Theorem for a “Spiny Urchin”
Authors: Clark Colin
Published: December 2010

From the plane R2 we remove the union of the sets Sk (k = 1, 2, …) defined as follows (using the notation z = x + iy):

Sk = {z: arg z = nπ2-k for some integer n; |z|≥k}.

The remaining connected open set Ω we call the spiny urchin.
Note on an Application of the δ-Function in the Representation of Solutions of Algebraic Equations
Authors: Sabat J. B.
Published: December 2010

The “function” δ(x - xo) is known as the Dirac Delta function and may be defined as zero everywhere except at xo, where it is infinite in such a way that

1

having property that for every continuous function φ(x) on (a, b)

2

It is well known [2] δ(x-xo) can be approximated as a limit of a sequence of piecewise continuous functions, and there is an abundance of such sequences.
Group Theory and the Principle of Duality
Authors: Lightstone A.H.
Published: December 2010

In [1] the Principle of Duality of mathematical logic (which states that the duals of equivalent well-formed formulas are also equivalent) is established by first showing that certain transforms related to the duality transform, also possess this property. The transforms involved are ,, , and which are defined as follows.
Note on Pairs of Consecutive Residues of Polynomials
Authors: Williams Kenneth S.
Published: December 2010

Let f(x) be a polynomial of degree d ≥ 3 with integral coefficients, 'say,

In a previous paper [6] I deduced, from a deep result of Lang and Weil [2], that there is a constant k1(d), depending only on d, such that for all primes p ≥ k1(d), p ⫮ ad, f(x) has a pair of consecutive residues (mod p), that i s, there exists an integer r(0 ≤r ≤ p-1) with the property that
Non-Existence of Estimates of Prescribed Accuracy in Fixed Sample Size
Authors: Singh Rajinder
Published: December 2010

Let X be a random variable whose density (or distribution if discrete) f(x; θ) depends on an unknown parameter θ, real or vector-valued. By making observations on X we want to know whether there exist estimates of prescribed accuracy for the real-valued parametric function g(θ). By an estimate of prescribed accuracy for g(θ) we mean a confidence interval of prescribed length and confidence coefficient or a point estimate with prescribed expected loss W. In the following our loss functions W will always satisfy the requirement that W(δ, θ) = V(|δ - θ|), where V is a strictly increasing function of its argument. The class of such loss functions includes among others the squared error loss.
The Circle Problem in an Arithmetic Progression
Authors: Smith R.A.
Published: December 2010

In following a suggestion of S. Chowla to apply a method of C. Hooley [3] to obtain an asymptotic formula for the sum ∑ r(n)r(n+a), where r(n) denotes the number of representations of n≤x

n as the sum of two squares and is positive integer, we have had to obtain non-trivial estimates for the error term in the asymptotic expansion of

1
On a Modification of Romberg Quadrature
Authors: Manohar R. , Turnbull C.
Published: December 2010

Meir and Sharma [1] have suggested a modification of Romberg quadrature using Newton-Cotes and, in particular, Simpson sums in place of trapezoidal sums. By comparing the error term with that obtained by Bulirsch [3] for trapezoidal sums, they concluded that the use of Simpson sums would lead to an improvement of the results. The procedure adopted by Meir and Sharma [1] permits them to obtain an expression for the error in the numerical quadrature. However, for the purpose of numerical computation, this procedure appears to be less suitable. In section 3, we give an alternative formulation which would enable us to carry out the computation, using Simpson sums, in the same wasy as is done in the case of Romberg quadrature with trapezoidal sums. Some numerical results are discussed in section 4.
A New Characterization of Finite Prime Fields
Authors: Maxson Carlton J.
Published: December 2010

Let N ≡  be a (right) near-ring with 1 (we say N is a unitary near-ring)[1] and recall that a near-field is a unitary near-ring in which  is a multiplicative group. In [2], Beidelman characterizes near-fields as those unitary near-rings without non-trivial N-subgroups. We show that in the finite case this absence of non-trivial N-subgroups is equivalent to the absence of non-trivial left ideals.
A Functional Equation for the Cosine
Authors: Kannappan PL
Published: December 2010

It is known [3], [5] that, the complex-valued solutions of

(B)

(apart from the trivial solution f(x)≡0) are of the form

(C)

(D)

In case f is a measurable solution of (B), then f is continuous [2], [3] and the corresponding ϕ in (C) is also continuous and ϕ is of the form [1],

(E)

In this paper, the functional equation

(P)

where f is a complex-valued, measurable function of the real variable and A≠0 is a real constant, is considered. It is shown that f is continuous and that (apart from the trivial solutions f ≡ 0, 1), the only functions which satisfy (P) are the cosine functions cos ax and - cos bx, where a and b belong to a denumerable set of real numbers.
On Compact Prime Rings and their Rings of Quotients
Authors: Koh Kwangil
Published: December 2010

In [10], it is defined that a right (or left) ideal I of a ring R is very large if the cardinality of R/I is finite. It is also proven in [10, Theorem 3.4] that if R is a prime ring with 1 such that its characteristic is zero, then R is a right order in a simple ring with the minimum condition on one sided ideals if every large right ideal of R is very large. In the present note, we shall prove that if R is a prime ring with 1 such that its characteristic is zero and R is also a compact topological ring, then R is a right and left order in a simple ring with the minimum condition on one sided ideals, which is also a non-discrete locally compact topological ring if and only if every large right ideal of R is open. In particular, if R is an integral domain with 1 (not necessarily commutative) such that its characteristic is zero, then R is openly embeddable [13, p. 58] in a locally compact (topological) division ring if and only if every large right ideal of R is open. Following S. Warner [13], we shall say R is openly embeddable in a quotient ring Q(R) if there is a topology on Q(R) which is compatible with its structure, which induces on R its given topology and for which R is an open subset.
Convergence of the Hausdorff Means of
Double Fourier Series*
Authors: Ustina Fred
Published: December 2010

In this paper we prove that if {sm, n(x, y)} is the sequence of partial sums of the Fourier series of a function f(x, y), which is periodic in each variable and of bounded variation in the sense of Hardy-Krause in the period rectangle, then {sm, n(x, y)} converges uniformly to f(x, y) in any closed region D in which this function is continuous at every point. This result is then used to prove that the regular Hausdorff means of the Fourier series of such a function also converge uniformly in such a region.
A Note on an Oscillation Criterion for an
Equation with a Functional Argument
Authors: Waltman Paul
Published: December 2010

It might be thought that, as far as the oscillation of solutions is concerned, the behaviour of

and

would be the same as long as t - α(t) → ∞ as t→∞. To motivate the theorem presented in this note, we show first that this is not the case. Consider the above equation with α(t) = 3t/4, a(t) = l/2t2 i.e.

This equation has the non-oscillatory solution y(t) = t1/2 although all solutions of

are oscillatory [1, p. 121].
Another Proof of the Contraction Mapping Principle
Authors: Boyd D.W. , Wong J. S. W.
Published: December 2010

In a recent note of Kolodner [2], the Cantor Intersection Theorem is used to give an alternative proof of the well known Contraction Mapping Principle. Kolodner applied Cantor's theorem first to a bounded metric space and then reduced the general case to this special case. Sometime ago, we found a somewhat different proof of the Contraction Mapping Principle using Cantor's theorem. Since our proof seems somewhat more direct we propose to present it here.
On a Certain Set of Linear Inequalities
Authors: Kalbfleisch J.G. , Stanton R. G.
Published: December 2010

In this paper we shall discuss the following set of n + 1 linear inequalities:

If we let , and Z = (zi) (i = 0, 1,…, n) be (n+1)-dimensional column vectors, and define the n+1 by n+1 tridiagonal matrix Dn(ϕ) by

the set of inequalities (1) may be written

where An= Dn(1) and zi≥ 0 (i = 0, 1,…, n). In sections 2 and 3, we
Some Elementary Converse Problems in Ordinary Differential Equations*
Authors: Seminar D. E.
Published: December 2010

In studying differential equations, the usual task is to determine properties of the solutions of such equations from a knowledge of the coefficient functions. The converse question, namely, of determining the coefficient functions from properties of solutions, also has significance. It has been studied especially in the case of Sturm-Liouville equations.

A discussion of the inverse Sturm-Liouville problem can be found in [8, Chapter 8], where references are given to the work of W.A. Ambarzumiam, G. Borg, I.M. Gelfand, M.G. Krein, B.M. Levitan, N. Levinson and W.A. Marchenko on this problem. Work of a quite different character, but dealing also with questions of a converse type arising from Sturm-Liouville equations, has been done by O. Boruvka and his colleagues and students [2].
Note on T-Minimal Complete Bipartite Graphs
Authors: Bouwer I. Z. , Broere I.
Published: December 2010

The thickness of a graph G is the smallest natural number t such that G is the union of t planar subgraphs. A graph G is t-minimal if its thickness is t and if every proper subgraph of G has thickness < t. (These terms were introduced by Tutte in [3]. In [1, p. 51] Beineke employs the term t-critical instead of t-minimal.) The complete bipartite graph K(m, n) consists of m 'dark1 points, n 'light' points, and the mn lines joining points of different types.
On Polyhedral Realizability of Certain Sequences
Authors: Jucovič E.
Published: December 2010

A finite sequence (pk) = (p3, p4,…) of non-negative integers shall be called realizable provided there exists a 3-valent 3-polytope P which has pi. i-gonal faces for every i. P is called a realization of (pk).

For realizability of a sequence (pk), from Euler's formula follows

(*) as a necessary condition.
A Presentation of the Mathieu Group M12
Authors: Leech John
Published: December 2010

In the course of work on factor groups of the modular group, A. O. L. Atkin (private communication) obtained the permutations

which satisfy the relations

(1)

for T = T1, T2. It was evident that neither pair S, T generates either A12 or LF(2, 11), so he suggested that they probably generate the Mathieu group M2.
Solution to a Matroid Problem Posed by D. J. A. Welsh
Authors: Bean D. T.
Published: December 2010

The pair (S, M) is a matroid if S is a finite set and M a collection of subsets of S such that (1) every subset of a set of M is in M, and (2) all maximal sets in M have a common cardinality. The span of a set A ⊂ S is Γ(A) where y ∈ Γ (A) if and only if y ∈ A or there is A' ⊂ A, A' ∈ M and {y} ∪ A' ∉ M. A maximal set in M is called a base.
The Equality of a Manifold's Rank and Dimension
Authors: Thomas R. S. D.
Published: December 2010

T. J. Willmore has shown that if a differentiable manifold's rank (the maximum number of everywhere linearly independent commuting vector fields definable on it) equals the manifold's dimension, then the manifold is a torus of the appropriate dimension [1]. This theorem is proved more simply and without any differentiability hypothesis in the present note.
Hamiltonian Systems with Spin
Authors: Marsden J. E.
Published: December 2010

In this note we give a brief exposition of the mathematical foundations of the theory of spin for both classical and quantum mechanical systems on oriented Riemannian manifolds. We shall use freely the notations and theory developed in Abraham [1] and Marsden [2, 3], From the physical point of view nothing new appears. The whole purpose of the note is to explain how the theory fits in the spirit of [1].
On a Class of Semigroups
Authors: Djoković D. Ž.
Published: December 2010

Let Sn, r be the semigroup generated by two elements a and b with two defining relations

(1)

By symmetry, we may assume that n ≤ r. In [l] and [2] it was shown that the semigroups

and
On a Theorem of Cutler
Authors: Megibben Charles K.
Published: December 2010

In [1] Cutler proved the following theorem.

Theorem. If G and K are abelian groups such that nG ≅ nK for some positive integer n, then there are abelian groups U and V such that U ⊕ G ≅ V ⊕ K and nU = 0 = nV.

Cutler's proof is long and fairly involved. Walker [3] obtains the theorem rather elegantly as a corollary of his results on n-extensions. We give here a proof that is extremely simple both in conception and execution. Our proof relies on the notion of p-basic subgroups introduced by Fuchs in [2]. Therefore we shall first recall certain pertinent facts from [2].
Boolean Near-Rings
Authors: Clay James R. , Lawver Donald A.
Published: December 2010

In this paper we introduce the concept of Boolean near-rings. Using any Boolean ring with identity, we construct a class of Boolean near-rings, called special, and determine left ideals, ideals, factor near-rings which are Boolean rings, isomorphism classes, and ideals which are near-ring direct summands for these special Boolean near-rings.

Blackett [6] discusses the near-ring of affine transformations on a vector space where the near-ring has a unique maximal ideal. Gonshor [10] defines abstract affine near-rings and completely determines the lattice of ideals for these near-rings. The near-ring of differentiable transformations is seen to be simple in [7], For near-rings with geometric interpretations, see [1] or [2].