For k-step Markov chains, factors generated by finite length codes split off with Bernoulli complement when maximal in entropy. Those not maximal are relatively finite in another factor which generates or splits off.

These results extend to random Markov chains with finite expected step size, implying that random Markov chains with finite expected step size can have only finitely many ergodic components, each of which is isomorphic to a finite rotation, a Bernoulli shift, or a direct product of a Bernoulli shift with a finite rotation. This result limits the type of zero entropy factors which occur in random Markov chains with finite expected step size, providing a counterpoint to the work of Kalikow, Katznelson, and Weiss, who have shown that each zero entropy process can be embedded in some random Markov chain.

Extending Rudolph and Schwarz, random Markov chains with finite expected step size are limits in of their canonical Markov approximants. The -closure of the class is the Bernoulli cross Generalized Von Neuman processes.

Finitary isomorphism of aperiodic ergodic random Markov chains with finite expected step size is considered.

Applications are made to a class of generalized baker's transformations.

Examples of finite complexes are given whose self-homotopy equivalences group is isomorphic to the group of integers.

Suppose that a subset of a finite dimensional lattice has the property that there are many orthogonal tensor products that are almost 1 on the set, then the set is forced to have unusual concentrations of points in small cartesian products.

The main purpose of this paper is to investigate isotropic immersions with low codimensions into a real space form.

If G is a group such that every infinite subset of G contains a commuting pair of elements then G is centre-by-finite. This result is due to B. H. Neumann. From this it can be shown that if G is infinite and such that for every pair X, Y of infinite subsets of G there is some x in X and some y in Y that commute, then G is abelian. It is natural to ask if results of this type would hold with other properties replacing commutativity. It may well be that group axioms are restrictive enough to provide meaningful affirmative results for most of the properties. We prove the following result which is of similar nature.

If G is a group such that for each positive integer n and for every n infinite subset X1,...,Xn of G there exist elements xi of Xii = 1,... ,n, such that the subgroup generated by {x1,... ,xn} is finite, then G is locally finite.

Jean-Philippe Anker made an interesting conjecture in [2] about the growth of the heat kernel on symmetric spaces of noncompact type. For any symmetric space of noncompact type, we can write

where ϕ0 is the Legendre function and q, "the dimension at infinity", is chosen such that limt—>∞Vt(x) = 1 for all x. Anker's conjecture can be stated as follows: there exists a constant C > 0 such that

where is the set of positive indivisible roots. The behaviour of the function ϕ0 is well known (see [1]).

The main goal of this paper is to establish the conjecture for the spaces SU*(2n)/ Sp(n).

Among the eight-dimensional stable planes, the compact connected generalized Hughes planes and the geometries induced on the outer points are characterized by the property that these planes admit an effective action of the group SL3 ℂ.

Known results for strictly stable motions as finiteness of moments and local boundednessof sample-path variation are generalized to self-affine processes, i.e., self-similar processes with stationary increments. The proofs are new, even for stable motions, and are obtained by applying the ergodic theorem to powers of the (one-sided) increments.

In this note we are going to give a geometric proof, using the method of the extremal metric, of the following result of Beurling. For any analytic function f(z) in the unit disc Δ of the plane with a bounded Dirichlet integral, the set E on the boundary of the unit disc where the nontangential limits of f(z) do not exist has logarithmic capacity zero. Also, using an unpublished result of Beurling, we will prove different results on boundary values for different classes of functions.

The range of the transformation, defined by

is characterized on the spaces Lμ,p defined by the norm

for

Let n ∊ ℤ+ and R be a ring which possesses a unit element, a left ideal J, and a derivation d such that dn(J) ≠ 0 and dn(r) is 0 or invertible, for all r ∊ J. We prove that either R is primitive, in which case R is Di with 1 ≤ i ≤ n+ 1, where Di is the ring of i × i matrices over a division ring D, or else there exist positive integers i, l and p with p prime and 2 ≤ ipl ≤ n + 1, such that R is where D is a division ring with characteristic p, and furthermore there is a derivation f of Di and a1, a2,..,al ∊ ZDi., the center of Di, such that a ∊ Di then

and

for all 2 ≤ j ≤ l

We present a proof of a theorem announced by van Douwen concerning existences of universal spaces for certain closed images of σ-discrete metric spaces.

We show that a restriction on the angular derivative of the inducing map does not determine compact composition operators on large weighted Hardy spaces, thus answering in the negative a question posed by T. Kriete.

Let F be a field containing a primitive p-th root of unity, let K / F be a cyclic extension with group 〈σ〉 of order pn, and choose a in K. This paper shows how the Galois group of the normal closure of K(a1/p) over F can be determined by computations within K. The key is to define a sequence by applying the operation x ↦ σ(x)/x repeatedly to a. The first appearance of a p-th power determines the degree of the extension and restricts the Galois group to one or two possibilities. A certain expression involving that p-th root and the terms in the sequence up to that point is a p-th root of unity, and the group is finally determined by testing whether that root is 1. When (σ(a)/a G Kp, the results reduce to a theorem of A. A. Albert on cyclic extensions.

We give a simple proof that, for any ॉ > 0, there is an outer measure μ* on a finite set X such that, for any measure Thus there is a non-zero outer (finitely subadditive) measure v* on the clopen subsets of the Cantor set such that, if v ≤ v* is a finitely additive measure on the clopen subsets of the Cantor set, then v ≡ 0.

T. Ding has shown that a topologically transitive flow on the torus given by a real analytic vector field is orbitally equivalent to a Kronecker flow on the torus, modified so as to have a finite number of fixed points, provided the original flow had only a finite number of fixed points. In this paper it is shown that the assumption that there are only finitely many fixed points is unnecessary.

We prove that with some weak chain conditions, left SF-rings are semisimple Artinian or regular. We also prove that MERT left SF-rings are really regular.

We propose an analogue of the Banach contraction principle for connected compact Hausdorff spaces. We define a J-contraction of a connected compact Hausdorff space. We show that every contraction of a compact metric space is a J-contraction and that any J-contraction of a compact metrizable space is a contraction for some admissible metric. We show that every J-contraction has a unique fixed point and that the orbit of each point converges to this fixed point.

Let RM be a nonsingular module such that B = EndR(M) is left nonsingular and has as its maximal left quotient ring, where is the injective hull of RM. Then it is shown that there is a lattice isomorphism between the lattice C(M) of all complement submodules of RM and the lattice C(B) of all complement left ideals of B, and that RM is a CS module if and only if B is a left CS ring. In particular, this is the case if RM is nonsingular and retractable.

A relation between the zeros of the partial sums and the zeros of the corresponding tails of the Maclaurin series for ez is established. This allows an asymptotic estimation of a quantity which came up in the theory of the Riemann zeta-function. Some new properties of the tails of ez are also provided.