bluuebunny/crossref_metadata_2025 huggingface dataset:
Remarks on an Arithmetic Derivative
Authors: Barbeau E. J.
Published: December 2010

Let D(n) denote a function of an integral variable n ≥ 0 such that

(1)

D(1) = D(0) = 0

(2)

D(p) = 1 for every prime p

(3)

D(n1n2) = n1D(n2) + n2D(n1) for every pair of non-negative integers n1, n2.
On the Problem of Steiner
Authors: Melzak Z.A.
Published: December 2010

There is a well-known elementary problem:

(S3) Given a triangle T with the vertices a1, a2, a3, to find in the plane of T the point p which minimize s the sum of the distances |pa1| + |pa2| + |pa3|.

p, called the Steiner point of T, is unique: if an angle of T is ≥ 2π/3 then p is its vertex, otherwise p lies inside T and the sides of T subtend at p the angle 2π/3. In the latter case p is called the S-point of T, and it can be found by the following simple construction: let a12 be the third vertex of the equilateral triangle whose other two vertices are a1 and a2, and whose interior does not overlap that of T, let C be the circle through a1, a2 a12; then p is the intersection of C and the straight segment a12a3. It is easily proved that any one of the three ellipses through p with two of the vertices of T as foci is tangent at p to the circle through p about the third vertex of T.
On the Colourings of Graphs
Authors: Tutte W.T.
Published: December 2010

A graph G is defined by a set V(G) of vertices, a set E(G) of edges, and a relation of incidence which associates with each edge two distinct vertices called its ends. We consider only the case in which V(G) and E(G) are both finite.

An n-colouring of G is usually defined as a mapping f of V(G) into the set of integers { 1, 2,…, n} which maps the two ends of any edge onto distinct integers. The integers 1 to n are the n "colours". Much work has been done on n-colourings in recent years, especially by G. A. Dirac.
The Reversibility of a Differentiable Mapping
Authors: Atkinson F. V.
Published: December 2010

Given n functions of n variables, in the real domain, by the equations

1

we have in various contexts to consider whether the equations are soluble for the xr when the yr are given. Such questions receive fairly complete answers in complex variable theory; a complex variable relation w = f(z) is of course brought under the heading of the real equations (1) by setting w = y1 + iy2, z = x1 + ix2. For example, if f(z) is a polynomial the fundamental theorem of algebra asserts that the equations are soluble, though not in general uniquely. Again, a basic theorem on conformal mapping gives conditions under which the equations are uniquely soluble, to the effect that a (1,1) mapping of the boundaries of domain and range implies a (1,1) mapping of the interiors.
A Note on the Tarry-Escott Problem
Authors: Melzak Z.A.
Published: December 2010

The Tarry-Escott problem in Diophantine analysis is the following: consider the system of equations

1

for the unknowns a1, …, ak, b1, …, bk, what is the smallest integer K = K(n) in the set of all k! s for which the system (1) possesses a non - trivial solution in integers ? By an integer we mean a rational integer and a solution is called trivial if the sets { a1, …, ak} and {b1, …, bk} are permutations of each other. By the Tarry-Escott problem one may also mean the problem of finding an actual solution of (1) or finding all such solutions but we shall be concerned with the bound K(n) only. However, the method given here is constructive, that is to say, with each estimate K for K it leads to finding actual non - trivial solutions of (1) with k = K1.
How to Program an Infinite Abacus
Authors: Lambek Joachim
Published: December 2010

This is an expository note to show how an “infinite abacus” (to be defined presently) can be programmed to compute any computable (recursive) function. Our method is probably not new, at any rate, it was suggested by the ingenious technique of Melzak [2] and may be regarded as a modification of the latter.

By an infinite abacus we shall understand a countably infinite set of locations (holes, wires etc.) together with an unlimited supply of counters (pebbles, beads etc.). The locations are distinguishable, the counters are not. The confirmed finitist need not worry about these two infinitudes: To compute any given computable function only a finite number of locations will be used, and this number does not depend on the argument (or arguments) of the function.
A Generalization of a Problem of L. Lorch and L. Moser
Authors: Gould H. W.
Published: December 2010

In connection with problem P33, Canad. Math. Bull. 3(1960), 189, I should like to offer the following solution and remarks which include a generalization.
Infinite Doubly Stochastic Matrices
Authors: Isbell J.R.
Published: December 2010

This note proves two propositions on infinite doubly stochastic matrices,
both of which already appear in the literature: one with an unnecessarily
sophisticated proof (Kendall [2]) and the other with the incorrect assertion
that the proof is trivial (Isbell [l]). Both are purely algebraic; so we
are, if you like, in the linear space of all real doubly infinite matrices A
= (aij).

Proposition 1. Every extreme point of the convex set of ail doubly
stochastic matrices is a permutation matrix.

Kendall's proof of this depends on an ingenious choice of a topology and the
Krein-Milman theorem for general locally convex spaces [2]. The following
proof depends on practically nothing: for example, not on the axiom of
choice.
On Approximation by Fejér Means to Periodic Functions Satisfying
a Lipschitz Condition
Authors: Lorch Lee
Published: December 2010

S. M. Nikolski [4, Theorem 1; cf. 3, esp. pp. 144 and 148] considered the
remainder term in the approximation by the n-th Fejér mean,
σn(x), to a function, f(x), of period 2π satisfying a Lipschitz
condition of order α, 0<α≤1. In this connection, he introduced the
quantity

1

where the maximum is taken over all x and the supremum is taken over all
functions of period 2π, bounded by 1 (a notational convenience only) and
satisfying a Laps chitz condition of order α.
On Maximal Rings of Right Quotients
Authors: Christensen Joanne
Published: December 2010

Utumi has shown [3, Claim 5.1] that for a certain class of rings the
associated maximal rings of right quotients are isomorphic to the
endomorphism rings of modules over division rings. We shall prove a
generalization of this theorem and then show how it is obtained as a
corollary. The following proofs do not depend on Utumi's paper; instead,
they make extensive use of results proved in [1]. The terminology and
notations employed here are the same as in [1].

I wish to thank Dr. B. Banaschewski for his suggestions and helpful
criticism.

LEMMA: If J is a left ideal with zero left annihilator in a ring R then a
maximal ring of right quotients of R is also a maximal ring of right
quotients of J.
On the Gram Matrix
Authors: Halperin Israel
Published: December 2010

The material sketched here is mostly well known and concerns the geometrical inter-relations of vectors in a Hilbert (that is, complete inner product) space. The discussion and references are (obviously) not exhaustive but I hope the reader will find here some interesting problems.

We shall permit the scalars to be real, complex or quaternionic numbers. The inner product of vectors x and y will be denoted (x|y).
Some Remarks on a Problem of E. L. Whitney
Authors: Kimura N.
Published: December 2010

We shall obtain a little more general formula than the one given in the problem P44.

Put

1

First of all we shall prove the identity

2

where Cmi are constants whose values will also be determined
A Remark on a Result of Marvin Marcus
Authors: Dulmage A.L. , Mendelsohn N.S.
Published: December 2010

Marcus [2] has proved the following theorem.

Suppose A is a non-negative normal matrix satisfying p(A) = 0 in which p(λ)
is a monic polynomial no two of whose non-zero roots have the same modulus.
Then there exists a permutation matrix P such that PAP* is a direct sum,
PAP* = A1 ⊕ A2 ⊕ … ⊕ Am, in which each
Ai is either O or primitive.

This note gives a generalisation of this result, dropping the non-negative
assumption and weakening the normality assumption.
Asymptotic Variational Formulae for Eigenvalues
Authors: Swanson C.A.
Published: December 2010

The eigenvalues of a second order self-adjoint elliptic differential
operator on Riemannian n-space R will be considered. Our purpose is to
obtain asymptotic variational formulae for the eigenvalues under the
topological deformations of (i) removing an ɛ -cell (and adjoining an
additional boundary condition on the boundary component thereby introduced);
and (ii) attaching an ɛ -handle, valid on a half-open interval 0 < ɛ ≤
ɛo. In particular the formulae will exhibit the non-analytic
nature of the variation. Similar variational problems for singular ordinary
differential operators have been considered by the writer in [3] and
[4].
Order in Affine and Projective Geometry
Authors: Lipman Joe
Published: December 2010

In this note we give a characterisation of ordered skew fields by certain
properties of the negative elements.

Properties of negative elements of a skew field K are then interpreted as
statements about "betweenness" in any affine geometry over K, or as
statements about "separation" in any projective geometry over K.

In this way, our axioms for ordered fields permit of immediate translation
into postulates of order for affine or projective geometry (over a field).
The postulates so obtained seem simple enough to be of interest, (cf. [4],
p. 22)
On the Sum of Consecutive Primes
Authors: Moser L.
Published: December 2010

Let A: a1 < a2 < … be a sequence of positive integers and let f(A:n) = f(n) denote the number of representations of n as the sum of one or more consecutive a's. Although f(n) will usually oscillate rapidly, an asymptotic estimate for its average value

can be obtained for many sequences A.
Insolvability of
Authors: Moser L.
Published: December 2010

A number of interesting Diophantine equations are of the a form f(n) = f(a) f(b). Thus the case f(a) = aa has been studied by C. Ko [2] and W. H. Mills [3] and a class of non-trivial solutions has been found, though whether these give all the solutions is still unsettled. The case f(n) = n! has been mentioned by W. Sierpinski. Here the situation is that besides the trivial solutions m! = m! 1! and (m!-1)! m! = (m!)! and the special solution 10! =7! 6! no other solutions are known, nor are they known not to exist. In the present note we show that the equation in the title has no solutions. A sketch of a somewhat different proof that this equation has at most a finite number of solutions was recently communicated to the author by P. Erdös.
On (L1)* for General Measure Spaces
Authors: Ellis H. W. , Snow D. O.
Published: December 2010

It is well known that certain results such as the Radon-Nikodym Theorem, which are valid in totally σ -finite measure spaces, do not extend to measure spaces in which μ is not totally σ -finite. (See §2 for notation.) Given an arbitrary measure space (X, S, μ) and a signed measure ν on (X, S), then if ν ≪ μ for X, ν ≪ μ when restricted to any e ∊ Sf and the classical finite Radon-Nikodym theorem produces a measurable function ge(x), vanishing outside e, with
A Model of the Real Numbers
Authors: Trott Stanton M.
Published: December 2010

The model of the real numbers described below was suggested by the fact that each irrational number ρ determines a linear ordering of J2, the additive group of ordered pairs of integers. To obtain the ordering, we define (m, n) ≤ (m', n') to mean that (m'- m)ρ ≤ n' - n. This order is invariant with group translations, and hence is called a "group linear ordering". It is completely determined by the set of its "positive" elements, in this case, by the set of integer pairs (m, n) such that (0, 0) ≤ (m, n), or, equivalently, mρ < n. The law of trichotomy for linear orderings dictates that only the zero of an ordered group can be both positive and negative.
Small Positive Values of Indefinite Quadratic Forms
Authors: Chalk J. H. H.
Published: December 2010

Let ∧ denote the lattice of points X = (x1, …, xn) with integral coordinates. A basis of ∧ is a set of n points X1, …, Xn of ∧ such that every point of ∧ is expressible in the form where ui are integers. It is easy to see that points X1, …, Xn of ∧ form a basis if, and only if

(1)

where Xr = (x1(r), …, x1(r)).
A General Turán Expression for the Zeta Function
Authors: Gould H. W.
Published: December 2010

In 1948 Gabor Szegő [9] gave four proofs of a remarkable inequality communicated to him by Paul Turán, who later published an original proof [10]. The Turán theorem states that if Pn(x) is the Le gendre polynomial, then

1.1

with equality holding only when |x| = 1.

Since then many similar inequalities have been found for various special functions, particularly for the Legendre and Hermite polynomials. Reference may be had to the recent work of Danese [2] and Chatterjea [1]. Danese gives an extensive bibliography.
On the Consistency of the Two-Sample Empty Cell Test
Authors: Csorgo M. , Guttman Irwin
Published: December 2010

This paper considers the consistency of the two-sample empty cell test suggested by S. S. Wilks [2]. A description of this test is as follows: Let a sample of n1 independent observations be taken from a population whose cumulative distribution function F1(x) is continuous, but 1 otherwise unknown. Let X(1) < X(2) < … < X(n1) be their order statistics. Let a second sample of n2 observations be taken from a population whose cumulative distribution function is F2(x), assumed continuous, but otherwise unknown.
Computation of the Number of Score Sequences in Round-Robin Tournaments
Authors: Narayana T.V. , Bent D.H.
Published: December 2010

We consider round-robin tournaments of n players in which, at each encounter, the winner is awarded 1 point and the loser 0 (ties are excluded).

Let

1

be the n scores, ordered in a non-decreasing sequence. Clearly

2
On a Combinatorial Problem of Erdös and Hajnal
Authors: Abbott H. L. , Moser L.
Published: December 2010

In this note we consider some problems related to the following question:
What is the smallest integer m(n) for which there exists a family
Fn of sets A1, A2,…, Am(n)
with the following properties, (i) each member of Fn has n
elements and (ii) if S is a set which meets each member of Fn,
then S contains at least one member of Fn?

Erdős and Hajnal [1] observed that

and that m(l) = 1, m(2) = 3, m(3) = 7.
Monotony of the Osculating Circles of Arcs of Cyclic Order
Three
Authors: Lane N. D. , Singh K. D. , Scherk P.
Published: December 2010

It is well-known in elementary calculus that if a differentiable function
has a monotone increasing curvature, then its curvature is continuous and
the circles of curvature at distinct points have no points in common. In
particular, two one-sided osculating circles at distinct points of an arc
A3 of cyclic order three have no points in common; cf. [l],
[2], [3]. The conformai proof given here that any two general osculating
circles at distinct points of A3 are disjoint (Theorem 1), may be
of interest. We also prove that all but a countable number of points of
A3 are strongly conformally differentiable (Theorem 2).
A Generalization of Cauchy' s Double Alternant
Authors: Carlson David , Davis Chandler
Published: December 2010

The subject of alternants and alternating functions was widely studied
during the last century (cf. Muir [6]). One of the best-known alternants is
actually a double alternant (rows and columns) defined by Cauchy [2] in
1841. Cauchy's result may be stated as follows:If D = [dpq] p, q
= l, …, n, where dpq =
(xp+yq)-1, then

1.
The Solutions of Some Functional Equations
Authors: Howroyd T.D.
Published: December 2010

In this paper we are concerned with functional equations of the type

1.

in which x, y do not appear explicitly. J. Aczel [1] has given a method for
finding real solutions of some of these equations. We prove a theorem which
can sometimes be used to solve problems concerning the uniqueness of
solutions of such equations.
Finite Reciprocities
Authors: Artiaga Lucio
Published: December 2010

A. P. Guinand [1] has shown that for appropriate infinite sequences
an, αn,(an, αn real and
positive for all n) the relation

1.

implies that

2.
Some Useful Matrix Lemmas in Statistical Estimation
Theory*
Authors: Tiao George C. , Guttman Irwin
Published: December 2010

In this note, we present two matrix lemmas (one without proof) which have
interesting applications in statistical estimation theory.

LEMMA 1. Let A be a k X k positive definite matrix. Then for any k X 1
vector c, we have that

1.
A Decomposition of Orthogonal Transformations
Authors: Wonenburger María J.
Published: December 2010

The purpose of the present note is to give a partial answer to a question raised by Professor Coxeter, namely, if an orthogonal transformation is expressed as a product of orthogonal involutions, how many involutions do we need? Our answer is partial because we are going to consider only non-degenerate symmetric bilinear forms of index 0 and fields of characteristic ≠2. Under these conditions we prove that any orthogonal transformation is the product of at most two orthogonal involutions, which implies that we can write any orthogonal transformation as the product of two involutions.
Block Intersections in Balanced Incomplete Block Designs
Authors: Stanton R. G. , Sprott D. A.
Published: December 2010

One of the most interesting of the smaller BIBD's is the system (8, 14, 7, 4, 3), where we write the parameters in the standard order v, b, r, k, λ. One representation of a design with these parameters is 1248, 3567; 2358, 1467; 3468, 1257; 4578, 1236; 5618,2347; 6728,1345; 7138,2456. This particular design has the feature that every block B is paired with a complementary block B' consisting of all varieties not lying in B. Thus
Finite Projective Planes with Affine Subplanes
Authors: Ostrom T. G. , Sherk F. A.
Published: December 2010

A well-known theorem, due to R. H. Bruck ([4], p. 398), is the following:

If a finite projective plane of order n has a projective subplane of order m < n, then either n = m2 or n > m 2+ m.

In this paper we prove an analagous theorem concerning affine subplanes of finite projective planes (Theorem 1). We then construct a number of examples; in particular we find all the finite Desarguesian projective planes containing affine subplanes of order 3 (Theorem 2).
Axiomatic Proof of J. Lambek's Homological Theorem
Authors: Leicht J. B.
Published: December 2010

A category with zero-maps is called "quasi-exact" in the sense of D. Puppe (see [4], page 8, 2. 4), if it satisfies the following axioms:

(Q1)

Every may f is a product f=με of an epimorphisrn εfollowed by a monomorphism μ

(Q2)

a) Every epimorphism ε has a kernel k = ker ε

b) Every monomorphism μ has a cokernel γ = Coker ε, where Ker and Coker are characterized by the familiar universality properties (see [3], page 252, (1. 10) and (1. 11)).
A Note on a Self Injective Ring
Authors: Koh Kwangil
Published: December 2010

A ring R with unity is called right (left) self injective if the right (left) R-module R is injective [7]. The purpose of this note is to prove the following: Let R be a prime ring with a maximal annihilator right (left) ideal. If R is right (left) self injective then R is a primitive ring with a minimal one-sided ideal. If R satisfies the maximum condition on annihilator right (left) ideals and R is right (left) self injective then R is a simple ring with the minimum condition on one-sided ideals.
Sur les Automorphismes de Certaines Structures Presque Cosymplectiques*
Authors: Akbar-Zadeh H.
Published: December 2010

Il s'agit d' étudier les automorphismes infinitésemaux (a. i.) de certaines structures presque cosymplectiques liées aux variétés finslériennes
A Note on a Prime Ring with a Maximal Annihilator Right Ideal
Authors: Koh Kwangil
Published: December 2010

A ring R is called a prime ring [1] if and only if a·R·b = 0 implies that a = 0 or b = 0 for all a, b ϵ R. Hence if R is a prime ring and a is a non-zero element of R, a·R ≠ 0 and R·a ≠ 0. In the present note we prove that a prime ring with a maximal annihilator right ideal has no non-zero nil right or left ideal.
The Algebra of Dimension-Linking Operators
Authors: Bruck R. H.
Published: December 2010

In the course of preparing a book on group theory [1] with special reference to the Restricted Burnside Problem and allied problems I stumbled upon the concept of a dimension-linking operator. Later, when I lectured to the Third Summer Institute of the Australian Mathematical Society [2], G. E. Wall raised the question whether the dimension-linking operators could be made into a ring by introduction of a suitable definition of multiplication. The answer was easily found to be affirmative; the result wasthat the theory of dimen sion-linking operators became exceedingly simple.
On Extensions of Monotone Functions from Linear SubLattices
Authors: Ellis H. W.
Published: December 2010

In this note real valued functions, defined on a linear sublattice S of a linear lattice R and satisfying the two order conditions (M1) and (M2), are studied from the point of view of the existence and uniqueness of extensions to R. The paper is partly expository and supplements and extends §3 of [4] where S was assumed to be an ℓ-ideal.
The Geometry of Finite Markov Chains
Authors: Pullman N.
Published: December 2010

The purpose of this paper is to present a geometric theorem which provides a proof of a fundamental theorem of finite Markov chains.

The theorem, stated in matrix theoretic terms, concerns the asymptotic behaviour of the powers of an n by n stochastic matrix, that is, a matrix of non-negative entries each of whose row sums is 1. The matrix might arise from a repeated physical process which goes from one of n possible states to another at each iteration and whose probability of going to a state depends only on the state it is in at present and not on its more distant history.
Some Properties of Compositions and their Application to the Ballot Problem
Authors: Mohanty S. G.
Published: December 2010

This paper is a continuation of two papers [4], [5] and brings out the solution of the ballot problem in its general form.

In [5], Narayana has considered a generalised occupancy problem which can be viewed as a problem in compositions of integers. In what follows, we use the definitions of [6].
A Characterization of Group Rings as a Special Class of Hopf Algebras
Authors: Takahashi Shuichi
Published: December 2010

By a group ring we mean in this paper a ring defined by a finite group G and an integral domain K:

such that A contains G and is freely generated by G over K, so that

The ring A = KG has a co-multiplication

defined by

so that A is a Hopf algebra.
A Generalization of Divisibility and Injectivity in Modules
Authors: Sanderson D. F.
Published: December 2010

Classically, there has been, for obvious reasons, an intimate relation between the concepts "rings of quotients" and "divisible modules". Recently, however, their generalizations have appeared to diverge.

For example, Hattori ([9]) and Levy ([15]) have generalized the concept of "divisibility" as follows: Hattori (respectively Levy) defines a left R-module M over a ring R to be divisible if and only if Ext1R(R/I, M)=0 for every principal left ideal I ⊂ R (respectively, every principal left ideal I ⊂ R which is generated by a regular element of R).
Stability on the basis of Orthogonal Trajectories
Authors: Burton T. A.
Published: December 2010

We consider a system of differential equations of second order given by

1

(' = d/dt) where P and Q have continuous first partial derivatives with respect to x and y in some open and simply connected set R containing O = (0, 0) which we assume to be the only singular point in R. In fact, let R be the whole plane; for if not then the following discussion can be modified.
Chromatic Number and Topological Complete Subgraphs
Authors: Dirac G. A.
Published: December 2010

A graph with m(>1) vertices, each pair of distinct vertices connected by an edge, and also a graph obtained from such a graph by the process of subdividing edges through the insertion of new vertices of valency 2, will be denoted by ≪m, o≫. A graph obtained from a graph with m(>2) vertices in which each pair of distinct vertices are connected by an edge, by deleting n (≤ m-1) edges incident with one and the same vertex, and also a graph obtained from such a graph by the process of subdividing edges through the insertion of new vertices of valency 2, will be denoted by ≪m, n≫.
Elementary Generalizations of Hilbert's Theorem 90
Authors: Connell Ian G.
Published: December 2010

Let K, k be fields and K|k a finite galois extension with galois group G. The multiplicative group K* of K is a G-module, that is, a module over the integral group ring ZG, the module action of an element σ ϵ G being its effect as an automorphism. It is shown in [2, p. 158] that the first cohomology group vanishes:

1
On a Problem by R.A. Hirschfeld: Counterexample
Authors: Edelstein M.
Published: December 2010

The following question was asked by R. A. Hirschfeld in [1]: "E and F are Banach spaces, F reflexive, D is a subset of E and T: D→F a nonlinear contraction, i.e.,

Can T be extended to a contraction (for E = F = Hilbert space the answer is yes)."
Curves on Surfaces of Constant Width
Authors: Armstrong William W.
Published: December 2010

A surface S of constant width is the boundary of a convex set K of constant width in euclidean 3-dimensional space E3. (See [l] pp. 127–139. )

Our first result concerns the interdependence of five properties which a curve on such a surface may possess. Let S be a surface of constant width D > 0 which satisfies the smoothness condition that it be a 2-dimensional submanifold of E3 of class C2.
Topology of Complex Manifolds
Authors: Srinivasacharyulu K.
Published: December 2010

The aim of this paper is to collect some interesting unsolved problems in the topology of compact complex manifolds which are scattered in literature; needless to remark, some of them are known to specialists. Besides, the readermay consult the well-known list of problems by F. Hirzebruch [ 5 ] which has stimulated remarkable progress in this direction. But the topology of complex manifolds has always baffled workers in this field and is still far from satisfactory. Of course, there is remarkable progress in the case of homogeneous compact complex manifolds. We start with the following result.
Note on Generalized Schreier Extensions of Groups
Authors: Kuyk Willem
Published: December 2010

By a (generalized) Schreier extension we mean a group G decomposed into a subinvariant series Gn ↣ Gn-1 ↣ Gn-2 … ↣ G1 ↣ G0 = Gn is anti-invariant in G, i. e. the only subgroup of G which is normal in Gn is the trivial one. ( “ ↣ ” denotes a group monomorphism, i. e. an injection homomorphism. ) As is well known, such groups G can be embedded into the repeated wreath product where Fi ≅ Gi / Gi+1 (cf. [ 2 ], notation of M. Hall [ 1 ], p. 81).
On the Boundary and Tensor Product of Function Algebras
Authors: Fox A. S.
Published: December 2010

Let be an arbitrary family of continuous complex-valued functions defined on a compact Hausdorff space X. A closed subset B ⊆ X is called a boundary for if every attains its maximum modulus at some point of B. A boundary, B, is said to be minimal if there exists no boundary for properly contained in B. It can be shown that minimal boundaries exist regardless of the algebraic structure which may possess. Under certain conditions on the family , it can be shown that a unique minimal boundary for exists. In particular, this is the case if is a subalgebra or subspace of C(X) where X is compact and Hausdorff (see for example [2]). This unique minimal boundary for an algebra of functions is called the Silov boundary of .
On Sums of Valencies in Planar Graphs
Authors: Bowen Robert
Published: December 2010

Planarity in graphs implies relatively small valencies and numbers of edges. In this note we find the maximum sum of valencies and the maximum number of incident edges for a set of n vertices in a planar graph with v vertices. Graphs considered are without multiple edges or loops.
Some Inequalities Related to the Wald-Wolfowitz-Noether Condition*
Authors: Mehra K. L. , Wong J. S. W.
Published: December 2010

If {avα: or = 1, 2, …, N }, with Nv → ∞ as v → ∞, is a double sequence of real numbers with the property that , then

1.1

is known in statistical literature as the Wald- Wolfowitz- Noether condition and it plays an important role in the proofs of certain types of central limit theorems (see e. g., [ l ], [2] ).
The Densest Packing of Six Spheres in a Cube
Authors: Schaer J.
Published: December 2010

This packing problem is obviously equivalent to the problem of locating six points Pi(l ≤ i ≤ 6) in a- closed unit cube C such that is as large as possible, where d(Pi, Pj) denotes the distance between Pi and Pj. We shall prove that this minimum distance cannot exceed (= m, say), and that 4 it attains this value only if the points form a configuration which is congruent to the one of the points Ri(l≤i≤6) shown in fig. 1. Note that , and so the six points are the vertices of a regular octahedron.
On a Lemma of M. Abramson
Authors: Church C.A.
Published: December 2010

Kaplansky's Lemma [3] states: the number of k-combinations of 1, 2,…, n with no two consecutive integers in any selection is Using this, Abramson [l; lemma 3] solves the problem: find the, number of k-combinations so that no two integers i and i+2 appear in any selection. (This is generalized by Abramson in [2].) An interesting solution, also using Kaplansky's lemma, is obtained as follows.

If n = 2m, we choose s from among the m even integer s, no two consecutive, and k- s from among the m odd integers, no two consecutive.
On Axioms for Semi-Lattices
Authors: Padmanabhan R.
Published: December 2010

By a semi-lattice we mean a system  where L is a set and. is a binary operation in L that is idempotent, commutative and associative. In a recent article [2] D.H. Potts considers the problem of reducing the number of axioms for a semi-lattice. His result was that the following two axioms viz. (1) xx=x, (2) (uv)((wx)(yz)) = ((uv)(xw))(zy) are sufficient to give a semi-lattice. But the second identity contains six elements instead of the original three. In the following we give a set of two simple identities with just three elements for a semi-lattice. This improves the above mentioned result.
On Carathéodory's Theorem
Authors: Scherk P.
Published: December 2010

The following proof of Caratheodory's Theorem, while not essentially new, seems to be natural and therefore of interest.

LEMMA 1. Let P denote a supporting hyperplane of the convex polytope K. Then P∩K is a convex polytope whose vertices are vertices of K.
Eisenstein's Criteria for Absolute Irreducibility Over a Finite Field
Authors: Williams Kenneth S.
Published: December 2010

Let p denote a prime and n a positive integer. Write q = pn and let kq denote the Galois field with q elements. The unique factorization domain of polynomials in m(≤ 2) indeterminâtes x1,…, xq with coefficients in k is denoted by kq [x,…, xm. It is the purpose of this note to prove the foliowing generalization of Eisenstein's irreducibility criteria and to point out some of its consequences.
A Type of Quasi-Frobenius Ring
Authors: Feller Edmund H.
Published: December 2010

In [3], the author proved that a ring R with identity is right noetherian and right injective if and only if R is a direct sum of a finite number of uniform right ideals, which are completely primary in the sense of that paper. In this paper, we shall determine the structure of such rings in the case where the sum of the isomorphic uniform components are twosided ideals. The ring is found to be a direct sum of total matrix rings over local rings.
On the Summability of a Class of Derived Fourier Series
Authors: Govil Narendra K.
Published: December 2010

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

1.1

be its Fourier series. The series

1.2

obtained by term by term differentiation of the series (1.1) is called the derived Fourier series of f.
On Uniform Approximations of Abstract Functions
Authors: Zakon Elias
Published: December 2010

As is well known, every real function is the pointwise (uniform) limit of a sequence of functions with a finite (countable) range of values. Monna [5] and Kvačko [4] suggested some extensions of this theorem to functions with values in a separable metric space. In the present note we give some further generalizations, with an emphasis on uniform approximations which have many applications in the generalized theory of measure and integration. In particular, we consider measurable abstract functions (mappings).
Integrals of the Differential Equations ẍ + f ( s ) x = 0
Authors: Datko R.
Published: December 2010

In this note we consider a relatively ancient stability problem: the behaviour of solutions of the second order differential equation ẍ + f(s) x = 0, where f(s) tends to plus infinity as s tends to plus infinity. An extensive survey of the literature concerning this problem and a resume of results may be found in [ l ]. More recently McShane et a l. [2] have shown that the additional assumption f(s) ≥ 0 is not sufficient to guarantee that all solutions tend to zero as s tends to infinity. Our aim is to demonstrate a new criterion for which all solutions do have the above property. This criterion overlaps many of the cases heretofore considered.
On Complex Homogeneous Manifolds
Authors: Srinivasacharyulu K.
Published: December 2010

Compact complex homogeneous manifolds have been studied in great detail by Borel, Goto, Remmert and Wang (cf., (13)): it was shown that every compact, connected complex homogeneous manifold M is a holomorphic fiber bundle over a projective algebraic homogeneous manifold B with a connected, complex parallelizable fiber F.
An Extension of Fermat's Theorem
Authors: Chinthayamma  , Gandhi J. M.
Published: December 2010

In [1[ Trypanis has proved the following extension of Fermat's theorem: If a is any integer, paprime, , then

1

This result is to be understood in the sense that a(p-1)/pn-1=p1/pnα where α is an algebraic integer.
On the Steiner Problem
Authors: Cockayne E. J.
Published: December 2010

Let M be a metric space with metric ρ which has the following properties.

1. M is finitely compact.

2. There exists a geodesic in M joining each two points of M.

3. For all a, b∈M, ρ(a, b) is equal to the length of a geodesic joining a and b.
A Note on Affine Pappus Conditions
Authors: Lane N. D.
Published: December 2010

Let ℓ, m, n be three mutually distinct lines in the projective plane. The (ℓ, m, n)-Pappus condition can be described as follows.

Let A, B, C, A', B', C' be any six mutually distinct points such that A, B, C lie on ℓ; A', B', C' lie on m; and none of these points lies on ℓ∩m, m∩n, or n∩ℓ. If the points AB'∩BA' and BC'∩CB' both lie on n, then the point AC'∩CA' also lies on n.
Exact Distribution of the Quotient of Independent Generalized Gamma Variables
Authors: Malik Henrick John
Published: December 2010

Let X be a random variable whose frequency function is

1.1

Form (1.1) is Stacy′s [3] generalization of the gamma distribution. The familiar gamma, chi, chi-squared, exponential and Weibull variâtes are special cases, as are certain functions of normal variate - viz., its positive even powers, its modulus, and all positive powers of its modulus.
On the Möbius Ladders
Authors: Guy Richard K. , Harary Frank
Published: December 2010

Consider the graph Mn, where n = 2r ≥ 6, consisting of a polygon
of length n and all n/2 chords joining opposite pairs of vertices. This
graph has 2r vertices which we denote by 1, 2, 3,..., 2r, and the 3r
(undirected) edges

We call Mn the n-ladder, defined thus far only for n even. The
three smallest n-ladders with n even are shown in Figure 1.
The Maximum Order of the Group of a Tournament
Authors: Dixon John D.
Published: December 2010

To each tournament Tn with n nodes n there corresponds the
automorphism group G(Tn) consisting n of all dominance preserving
permutations of the set of nodes. In a recent paper [3], Myron Goldberg and
J. W. Moon consider the maximum order g(n) which the group of a tournament
with n nodes may have. Among other results they prove that

1

2
On the Convergence of Product Moments
Authors: Chan L. K.
Published: December 2010

Let Ⓗ be an abstract space and for every positive integer n let Fn,
θ(x, y), θ ∈ Ⓗ, be a family of distribution n, 0 function s of
random variables (Xn, Yn)θ, θ ∈ Ⓗ. For
every θ ∈ Ⓗ, Eθg (Xn, Yn) will denote the
expected value of the function g of (Xn, Yn
)θ. The following proposition is proved.
Some Interpolators Properties of Laguerre
Polynomials*
Authors: Prasad J. , Saxena R. B.
Published: December 2010

In 1955, J. Suranyi and P. Turán [8] introduced the nomenclature (0,
2)-interpolation for the problem of finding polynomials of degree ≤ 2n-1
whose values and second derivatives are prescribed in certain given nodes.
In a series of papers ([2], [3], [8]) Professor Turán and his associates
discussed the problems of existence, uniqueness, explicit representation and
convergence of such interpolatory 2 polynomials when the nodes are the zeros
of (1-x2) P′n- 1(x), Pn- 1(x) being the
Legendre polynomial of degree n - 1.
On General Polynomials
Authors: Williams Kenneth S.
Published: December 2010

Let d denote a fixed integer > 1 and let GF(q) denote the finite field of
q = pn elements. We consider q fixed ≥ A(d), where A(d) is a
(large) constant depending only on d. Let

1

where each aiεGF(q). Let nr(r = 2, 3, …, d) denote the
number of solutions in GF(q) of

for which x1, x2, …, xr are all
different.
Lattice Paths in E3With Diagonal Steps
Authors: Stocks D.R.
Published: December 2010

Moser and Zayachkowski [1] have discussed certain planar lattice paths from (0, 0) to (m, n). In this note we consider analogous paths in three dimensional space. For basic definitions see reference [2]. Throughout this note each of m, n and k is a positive integer and if S is a finite set, |S| will denote the number of elements of S.

Each path under consideration here is such that each of its steps is of one of the following types:
On the Relation Between Boundedness and Oscillation of Differential Equations of Second Order
Authors: Kartsatos A. G.
Published: December 2010

In this paper we are dealing with differential equations of the forms:

Ei

where the functions pi are positive.

By a solution of an equation of the above forms, we mean a function x(t) ∈ C2 [c, + ∞) where c is a non-negative constant, which satisfies the corresponding equation on the whole interval [c, +∞). By an oscillatory solution of (Ei), we mean a solution with arbitrarily large zeros.
A Note on Fourier Transforms and Imbedding Theorems
Authors: Adams Robert A.
Published: December 2010

It is well known that Sobolev′s Lemma on the continuity of functions possessing L2 distributional derivatives of sufficiently high order is a simple consequence of elementary properties of the Fourier transform in L2 (e.g. [1, p. 174]). (In fact this statement remains true if 2 is replaced by p, 1 ≤ p ≤ 2). In this note we show that imbedding theorems of the type Wm, p ⊂Lq can also be obtained using Fourier transforms and an elementary lemma which reduces the cases p > 2 to the case p = 2. The simplicity of this approach is obtained at the expense of a slight loss of generality in the imbedding theorem.
Footnote to a Formula of Gioia and Subbarao
Authors: Segal S. L.
Published: December 2010

Recently Gioia and Subbarao [2] studied essentially the following problem: If g(n) is an arithmetic function, and , then what is the behaviour of H(a, n) defined for each fixed integer a ≥ 2 by

1

By using Vaidyanathaswamy′s formula [e.g., 1], they obtain an explicit formula for H(a, n) in case g(n) is positive and completely multiplicative (Formula 2.2 of [2]). However, Vaidyanathaswamy′s formula is unnecessary to the proof of this result, which indeed follows more simply without its use, by exploiting a simple idea used earlier by Subbarao [3] (referred to also in the course of [2]).
On the Distribution of Integer Solutions of f(x, y) = z2 for a Definite Binary Quadratic Form f
Authors: Nobusawa Nobuo
Published: December 2010

Let f be a positive definite binary quadratic form with rational coefficients. We shall call a point (x, y) in E2 with integers x and y a Pythagorean point of f when f(x, y) = z2 is satisfied with some integer z, and shall prove the following theorem.
A Note on a Fully Ordered Ring
Authors: Koh Kwangil
Published: December 2010

A ring R (associative ring) is said to be fully ordered provided that R is a linearly ordered set under a relation such that for any a, b and c in R, implies that and if c ε 0 then and . We say a subset K of R is convex provided that if a, b ε K such that then the interval [a, b] is a subset of K. Obviously an additive subgroup K of R is convex if and only if b ε K and b > 0 implies [a, b] ⊆ K.
On Operators and Distributions
Authors: Struble Raimond A.
Published: December 2010

Mikusinski [1] has extended the operational calculus by methods which are essentially algebraic. He considers the family C of continuous complex valued functions on the half-line [0,∞). Under addition and convolution C becomes a commutative ring. Titchmarsh's theorem [2] shows that the ring has no divisors of zero and, hence, that it may be imbedded in its quotient field Q whose elements are then called operators. Included in the field are the integral, differential and translational operators of analysis as well as certain generalized functions, such as the Dirac delta function. An alternate approach [3] yields a rather interesting result which we shall now describe briefly.
A Note on Homogeneous Dendrites
Authors: Melzak Z. A.
Published: December 2010

In graph - theoretic terms a homogeneous p-dendrite, p ≥ 2, is defined as a finite singly-rooted tree in which the root has valency 1 while every other vertex has valency 1 or p. More descriptively, a homogeneous p-dendrite may be imagined to start from its root as the main, or 0th order, branch which proceeds to the first - order branch point where it gives rise top first - order branches. Each of these either terminates at its other end (which is a second-order branch point) or it splits there again into p branches (which are of third order), and so on. The order of the dendrite is the highest order of a branch present in it. For completeness, a 0-th order dendrite is also allowed, this consists of the 0-th order branch alone.
On Real Almost Hermitian Structures Subordinate to Almost Tangent Structures
Authors: Closs Mike P.
Published: December 2010

Some of the most important G-structures of the first kind (1) are those defined by linear operators satisfying algebraic relations. Let J be a linear operator acting on the complexified space of a differentiable manifold V, and satisfying a relation of the form

where λ is a complex constant and I is the identity operator. In the case λ ≠ 0 the manifold has an almost product structure (2) which in the case λ = i reduces to an almost complex structure (3). In the remaining case, λ = 0, the manifold has an almost tangent structure (4).
A Room Design of Order 14
Authors: O'Shaughnessy C D.
Published: December 2010

A Room design of order 2n, where n is a positive integer, is an arrangement of 2n objects in a square array of side 2n - 1, such that each of the (2n - 1)2 cells of the array is either empty or contains exactly two distinct objects; each of the 2n objects appears exactly once in each row and column; and each (unordered) pair of objects occurs in exactly one cell. A Room design of order 2n is said to be cyclic if the entries in the (i + l) th row are obtained by moving the entries in the i th row one column to the right (with entries in the (2n - l)th column being moved to the first column), and increasing the entries in each occupied cell by l(mod 2n - 1), except that the digit 0 remains unchanged.
On Partitioning Planar Graphs
Authors: Hedetniemi Stephen
Published: December 2010

In 1879 Kempe [5] presented what has become the most famous of all incorrect proofs of the Four Colour Conjecture, but even though his proof was erroneous his method has become quite useful. In 1890 Heawood [4] was able to modify Kempe's method to establish the Five Colour Theorem for planar graphs. In this article we show that other modifications of Kempe's method can be made which enable one to establish more results about planar graphs. By this process we obtain upper bounds for several parameters which involve partitioning the point set of a graph. In particular, we show that the point set of any planar graph can be partitioned into four or less subsets such that the subgraph induced by each subset is either disconnected or trivial (consists of a single point). We also show that the point set of any planar graph can be partitioned into three or less subsets such that the subgraph induced by each subset contains no cycles.
Moment Problems and Quasi-Hausdorff Transformations
Authors: Leviatan Dany
Published: December 2010

The sequence to sequence quasi - Hausdorff transformations were defined by Hardy [1] 1 1. 19 p. 277 as follows. For a given sequence {μn} (n ≥ 0) of real or complex numbers, define the operator Δ by for k > l. {tm} (m ≥ 0) is called the sequence to sequence quasi-Hausdorff transform by means of {μn} (or, in short, the [QH, μn] transform) of {sn} (n ≥ 0) if if , provided that the sums on the right-hand side converge for all m ≥ 0. Ramanujan in [11] and [12] has defined the series to series quasi-Hausdorff transformation s and has proved necessary and sufficient conditions for the regularity of the two kinds of transformations.
On Polynomials with Real Zeros
Authors: Ahmad Manzoor
Published: December 2010

Let

be a polynomial of degree n with real and non-negative zeros x1 ≤ x2 ≤ … xn. The zeros xj. will be said to have extent 1 if

Let ξ1 ≤ ξ2 ≤ … ξn-1 be the zeros of the derived polynomial p'(x). The zeros ξ1, ξ2, …, ξn-1 are real and non-negative, and moreover their extent can be at most equal to the extent of the zeros x1, x2, …, xn. The two can indeed be equal. For if the extent of the zeros xj. is 1 and 1 is a multiple zero of p'(x) then ξn-1 = 1. However it is not quite clear how small ξn-1 = 1 can be if Xn = 1. The extent ξn-1 = 1. of the zeros of p'(x) is less than 1 only if 1 is not a multiple zero of p(x). So let us suppose that p(x) has a simple zero at x = 1. Consequently xn-1 is the largest zero of p(x)/(x-l) or equivalently the largest zero of p(x) in 0 < x < 1 and it follows by Rolle's theorem that p'(x) n as a zero in the interval (xn-1,1). Thus the extent ξn-1 of the zeros of p'(x) is greater than xn-1 and it remains to see how small it can be.
Gap Formulae for the Weierstrass Transforms
Authors: Ditzian Z.
Published: December 2010

A gap formula for a transform

(1)

is an operator J on f(x)

(2)

Such operators are known for Laplace transform [1; p.91] and [4; pp. 296–299], Stieltjes transform [4; pp. 351–353], a class of convolution transforms [2] and others. Gap formulae for the Weierstrass transform

(3)

and the Weierstrass-Stieltjes transform

(4)

will be proved in the following two theorems.
On Exceptional Polynomials
Authors: Williams Kenneth S.
Published: December 2010

Let f(x) be a polynomial of degree d ≥ 2 defined over the finite field kq with q = pn elements. Let

If f*(x, y) has no irreducible factor over kq which is absolutely irreducible, f is called an exceptional polynomial [1]. Davenport and Lewis have noted that when d is small compared with p, a permutation (substitution) polynomial is necessarily an exceptional polynomial. It is the purpose of this paper to prove the converse; that is, we will show the existence of a constant a(d), depending only on d, such that if f(x.) is an exceptional polynomial over kq, where p ≥ a(d), then f(x) is a permutation polynomial.
On the Cube of a Graph
Authors: Karaganis Jerome J.
Published: December 2010

The nth power Gn of a connected graph G is the graph with the same point set as G and where two points u and v are adjacent in Gn if and only if the distance between u and v in G is at most n. The graph G2 is called the square of G while G3 is referred to as the cube of G.