A random map is a discrete time process in which one of a number of maps, 𝓜, is chosen at random at each stage and applied. In this note we study a random map, where 𝓜 is a set of piecewise linear Markov maps on [0, 1]. Sufficient conditions are presented which allow the determination of the unique absolutely continuous invariant measure of the process.

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).

In 1967, Powers [7] proved that there exists a one-parameter family of pairwise non-isomorphic hyperfinite factors of type III. Powers′ result on hyperfinite factors has been extended by Araki and Woods [1]. Connes [4], and Williams [11], with different proofs, showed that there exists a continuous family of mutually non-isomorphic non-hyperfinite factors of type III.

If G is any group then g ∊ G is called an involution if g ≠ 1 and g o g = 1. A group G is called bireflectional if every element in G is a product of two involutions. It is known that 2- dimensional, 3- dimensional, and some types of n-dimensional (n > 3) absolute geometries (in the sense of H. Kinder) are bireflectional. In this article the author proves the general result that every n-dimensional absolute geometry is bireflectional.

In the study of algorithms for the iterative solution of an arbitrary analytic equation f(z) = 0, acceleration procedures are of importance in practice and of considerable interest in the theory of the subject. Let

.

be an iteration formula which has a zero ξ of f(z) as attractive fixed point. An algorithm of this type is said to converge towards a root ξ of f(z) = 0 for all initial approximations z = z0 in a vicinity of ξ, of order k > 0, when

1.

In 1957, Mordell [3] proved

Theorem. If p is an odd prime there exist non-negative integers x, y ≤ A p3/4 log p, where A is a positive absolute constant, such that

(1.1)

provided (abc, p) = 1.

Recently Smith [5] has obtained a sharp asymptotic formula for the sum where r(n) denotes the number of representations of n as the sum of two squares.

The purpose of this paper is to study the rigidity and deformations of secondary characteristic classes for subfoliations.

Given any homogeneous complex manifold X = G/H, there exists a natural coset map π :G/H → G/K satisfying π (X1) = π (x2) if and only if dx(x1 x2) = 0, where dx denotes the Kobayashi pseudometric on X. Its typical fiber Z : = K/H is a connected complex submanifold of X. Also G/K has a (7-invariant complex structure, provided K satisfies a certain technical assumption (see Theorem 3). If Z is compact as well, then G/K is biholomorphic to a homogeneous bounded domain.

In this paper we shall give an affirmative solution to a problem, posed by Berger, Coburn and Lebow, for ${{C}^{*}}$-algebras on Hardy submodules.

We show that a locally compact semigroup S is topological left amenable iff a certain space of left uniformly continuous functions on the convolution semigroup of probability measures M0(S) on S is left amenable or equivalently iff the convolution semigroup M0(S) has the fixed point property for uniformly continuous affine actions on compact convex sets.

In this note we use functional analysis arguments to prove the existence of families of analytic solutions for the singular system of complex ordinary differential equations zW′ = h(z, W).

We obtain a Bogomolov type of result for the affine space defined over the algebraic closure of a function field of transcendence degree 1 over a finite field.

It is shown in this paper that an immersion of a connected, closed n-manifold into another connected n-manifold is a diffeomorphism if and only if the induced homomorphism between the fundamental groups is surjective at some point. This is proved as a consequence of a more general assertion about topological spaces.

We show that there exists a non-archimedean Fréchet-Montel space $W$ with a basis and with a continuous norm such that any non-archimedean Fréchet space of countable type is isomorphic to a quotient of $W$. We also prove that any non-archimedean nuclear Fréchet space is isomorphic to a quotient of some non-archimedean nuclear Fréchet space with a basis and with a continuous norm.

Define a group $G$ to be in the class $S$ if for any finitely generated subgroup $K$ of $G$ having the property that there is a positive integer $n$ such that ${{g}^{n\,}}\in \,K$ for all $g\,\in \,G,\,K$ has finite index in $G$. We show that a free product with amalgamation $A{{*}_{_{C}}}B$ and an $\text{HNN}$ group $A{{*}_{C}}$ belong to $S$, if $C$ is in $S$ and every subgroup of $C$ is finitely generated.

In this paper, we study the behaviour of singular isometries in orthogonal groups. These are isometries whose path is a singular subspace. We shall prove that the path of such a singular isometry is always even-dimensional. We shall use this result to show that the subgroup of the orthogonal group On(K, Q) which is generated by the singular isometries is the commutator subgroup Ωn(K, Q).