We define, as in [2], a random knock-out tournament with n players as a vector (m1, m2, …, mk) of positive integers satisfying

1

On the first round of the tournament 2m1 players, chosen at random, are paired off randomly; the remaining n — 2m1 players have a "bye". The m1 losers are knocked out, leaving a tournament n—m1 players with vector (m2, m3, …, mk).

Much current research is concerned with the fixed points of contractive mappings (mappings which shrink distance in some manner) from a metric space into itself. In this remark we shall point out that most mappings treated in the literature are very special in the sense that all these mappings satisfy a condition which is rather severe: every periodic point must necessarily be a fixed point.

A (meet-) semilattice is an algebra with one binary operation ∧, which is associative, commutative and idempotent. Throughout this paper we are working in the category of semilattices. All categorical or general algebraic notions are to be understood in this category. In every semilattice S the relation

defines a partial ordering of S. The symbol "∨" denotes least upper bounds under this partial ordering. If it is not clear from the context in which partially ordered set a least upper bound is taken, we add this set as an index to the symbol; for example, ∨AX denotes the least upper bound of X in the partially ordered set A.

A well known theorem of Hasse [1] says that every quadratic form in at least 5 variables over the field Qp of p-adic numbers has a nontrivial zero. This fact has led Artin to make the conjecture

(C): "Every form over Qp of degree d in n > d2 variables has a non-trivial zero." However, a counterexample has been provided by Terjanian [2] in the case d=4.

The case d=2 is distinguished by the fact that every quadratic form may be "diagonalized", i.e., assumed to be of the type Σ aiX2i. One is therefore led to the weaker conjecture

(C): "Every form f= Σ aiXdi over Qp in n > d2 variables has a nontrivial zero in Qp,"

which still generalizes Hasse's theorem.

In [1] it was shown that a continuous function of bounded variation on the real line determined a Method II outer measure for which the Borel sets were measurable and the measure of an open interval was equal to the total variation of f over the interval. The monotone property of measures implied that if an open interval I on which f was not of bounded variation contained subintervals on which f was of finite but arbitrarily large total variation then the measure of I was infinite. Since there are continuous functions that are not of bounded variation over any interval (e.g. the Weierstrasse nondifferentiable function) the general case was not resolved.

We are concerned with the existence proof of solution of the Falkner-Skan equation

1

subject to boundary conditions

2

The first existence and uniqueness proof based on a fixed point theorem was given by Weyl [4] in 1942, with the added assumption that f' > 0. In 1960, Coppel [1] proved the existence (and uniqueness with the assumption 0 < f' < 1) by considering trajectories in the three-dimensional phase space.

This note is intended as a commentary on a part of the paper [1] by Gulliksen, Ribenboim, and Viswanathan. It takes its inspiration from a colloquium talk by P. Ribenboim.

Our aim is a partial generalization of the theorem of Cohen-Seidenberg (cf. [2], IX. 1, Propositions 9 and 10) to noncommutative algebras.

Definition. Let A be a ring with 1 in the center of another ring B (with the same 1). B is said to be integral over A, if each element b ∊ B satisfies an integral equation

It is shown that no pair of orthogonal Steiner triple systems of order 9 exists.

If p is a polynomial in one real variable and p(x) = p(-x) then p has only even powers of x and is thus a polynomial in x2. If p is a polynomial in n variables and p(x1, …, xn) = p(y1, …, yn) when x12 + … + xn2 = y12+ … + yn2 then p is a polynomial in q where q(x1, …, xn) = x12 + … + xn2.

The theorem stated below is due to R. Balbes. The present proof is direct; it uses only the following two well-known facts: (i) Let K be a category of algebras, and let free algebras exist in K; then an algebra is projective if and only if it is a retract of a free algebra, (ii) Let F be a free distributive lattice with basis {xi | i ∊ I}; then ∧(xi | i ∊ J0) ≤ ∨(xi | i ∊ J1) implies J0∩J1≠ϕ. Note that (ii) implies (iii): If for J0 ⊆ I, a, b ∊ F, ∧(xi | i ∊ J0)≤a ∨ b, then ∧ (xi | i ∊ J0)≤ a or b.

Consider the linear ordinary differential equation

1

where x ∊ En, the n-dimensional Euclidean space and A is an n × n constant matrix. Using a matrix result of Sylvester and a stability result of Perron, Lyapunov [4] established the following theorem which is basic in the stability theory of ordinary differential equations:

Theorem (Lyapunov). The following three statements are equivalent:

(I) The spectrum σ(A) of A lies in the negative half plane.

(II) Equation (1) is exponentially stable, i.e. there exist μ, K>0 such that every solution x(t) of (1) satisfies

2

where ∥ ∥ denotes the Euclidean norm.

(III) There exists a positive definite symmetric matrix Q, i.e. Q=Q* and there exist q1,q2>0 such that

3

satisfying

4

where I is the identity matrix.

By an r-graph, we mean a finite set V of elements called vertices and a collection of some of the r-subsets of V called edges with the property that each vertex is incident with at least one edge. An A-chromatic r-graph is an r-graph all of whose edges are coloured A.

Theorem. Let G1, …, Gt denote r-graphs. There exists a nonempty class of r-graphs such that for each if the edges of G are painted arbitrarily in t colours A1, …, At, then for at least one i in {1, …, t}, G has an Ai-chromatic r-subgraph which is isomorphic to Gi.

Of special interest in summability theory are those conservative matrices possessing the "mean-value property". If cA={x: Ax ∊ c} denotes the convergence domain of a conservative matrix A, then A has the mean-value property in case, for each x in cA, there exists M = M(A, x) > 0 such that

1

This property has been considered by many writers and has been shown, among other things, to be equivalent to the requirement that the matrix be associative, i.e., for each x in cA

2

R is a ring and M is a right R-module for which Rl = {m ∊ M | mR = 0} is the zero submodule. Let and be the injective hull and the quasi-injective hull of M respectively. Then where [1]. The ring plays an important role, in many cases, in the studying of R especially when D is a division ring. For x ∊ M, we denote the annihilator of x in R by xγ = {r ∊ R | xr=0}, whereas xγl={m ∊ M | mxγ=0}. If N is a submodule of M and x ∊ M, x-1(N) is the right ideal in R consisting of elements r in R where xr ∊ N.

Let X1, X2, …, Xn be n independent and identically distributed random variables having the positive binomial probability function

1

where 0 < p < 1, and T = {1, 2, …, N}. Define their sum as Y=X1 + X2 + … +Xn. The distribution of the random variable Y has been obtained by Malik [2] using the inversion formula for characteristic functions. It appears that his result needs some correction. The purpose of this note is to give an alternative derivation of the distribution of Y by applying one of the results, established by Patil [3], for the generalized power series distribution.