Recurrent and sweep-out sets for Markov processes

LetP be a Markov process on a probability space (X, ,m). Roughly speaking, a sweep-out set is a set which is reached with probability 1 under the action of the process form-almost all starting points. Obviously, in a finite state space no sweep-out sets of arbitrary small measure exist. The authors show that in general arbitrary small sweep-out sets exist, unless there is an invariant subset of the state space on which the process behaves as on a finite state space. Moreover, if there exist arbitrary small sweep-out sets and if is countably generated, then there exists an algebra of sweepout sets generating . The main tool to obtain these results is the use of embedded processes. Some properties of these processes are collected, and as a side-result a short and elementary proof of the decomposition theorem ofE. Hopf of the state space in a conservative and a dissipative part is given.     

Σ-Subsets of Natural Numbers

It is shown that the class of all possible families of -subsets of finite ordinals in admissible sets coincides with a class of all non-empty families closed under e-reducibility and . The construction presented has the property of being minimal under effective definability. Also, we describe the smallest (w.r.t. inclusion) classes of families of subsets of natural numbers, computable in hereditarily finite superstructures. A new series of examples is constructed in which admissible sets lack in universal -function. Furthermore, we show that some principles of classical computability theory (such as the existence of an infinite non-trivial enumerable subset, existence of an infinite computable subset, reduction principle, uniformization principle) are always satisfied for the classes of all -subsets of finite ordinals in admissible sets.     

Uniform boundedness of a family of set functions

Let be a ring of sets, X a normed space, : X ( ) a bounded family of triangular functions. The following generalized Nikodym theorem is established: the family {} is uniformly bounded on if and only if it is bounded on every sequence of pairwise disjoint sets of which the union is a part of some set in . An analogous criterion is established also for semiadditive functions. In addition, it is shown that uniform boundedness of a family of triangular functions is preserved in passing from a ring to the -ring it generates.Translated from Matematicheskie Zametki, Vol. 23, No. 6, pp. 855–861, June, 1978.     

On the Ershov Upper Semilattice

We find some links between -reducibility and T-reducibility. We prove that (1) if a quasirigid model is strongly -definable in a hereditarily finite admissible set over a locally constructivizable B-system, then it is constructivizable; (2) every abelian p-group and every Ershov algebra is locally constructivizable; (3) if an antisymmetric connected model is -definable in a hereditarily finite admissible set over a countable Ershov algebra then it is constructivizable.     

4-Dimensional compact projective planes with small nilradical

We consider 4-dimensional compact projective planes with a solvable 6-dimensional collineation group and with orbit type (2, 1), i.e. fixes a flagv W, acts transitively onL \{W} and fixes no point in the setW\{v}. We We prove a series of lemmas concerning the action of invariant subgroups of . These lemmas are applied to prove that the maximal connected nilpotent invariant subgroup of has dimension at least 4.Dedicated to Prof. H. Salzmann on the occasion of his 65th birthday     

The inverse hull of the free semigroup on a set X

In this paper the inverse hull of the free semigroup on a non-empty set X is characterized. It is shown that if F^O is a certain semilattice and (G,*) is the free group on X, then is isomorphic to a Rees factor semigroup of a subsemigroup of (F^O x G, #), where (A,w)#(B,u) = ((A*w B)*w^–1, w*u). Several properties of , including Green's relations on are also determined.     

Embedding a non-embeddable stable plane

In [4], K. Strambach describes a 2-dimensional stable plane admitting =SL₂ as a group of automorphisms such that there exists no -equivarient embedding into a 2-dimensional projective plane. R. Löwen [3] has given a 4-dimensional analogue , admitting =SL₂. He posed the question whether there are embeddings of Strambach's plane into . We show that such embeddings exist, in fact we determine all -equivariant embeddings of 2-dimensional stable planes admitting as atransitive group of automorphisms.     

Proper 2-Covers of PG(3,q), q Even

A k-cover of =PG(3q) is a set S of lines of such that every point is on exactly k lines of S. S is proper if it contains no spread. The existence of proper k-covers of is necessary for the existence of maximal partial packings of q ²+q+1–k spreads of . Here we give the first construction of proper 2-packings of PG(3,q) with q even; for q odd these have been constructed by Ebert.     

C k -B-splines with Square Support on a Three-Direction Mesh of the Plane

Let be the uniform triangulation generated by the usual three-directional mesh of the plane and let ₁ be the unit square consisting of two triangles of . We study the space of piecewise polynomial functions in C ^k (R ²) with support ₁ having a sufficiently high degree n, which are symmetrical with respect to the first diagonal of ₁. Such splines are called ₁-splines. We first compute the dimension of this space in function of n and k. Then, for any fixed k0, we prove the existence of ₁-splines of class C ^k and minimal degree. These splines are not unique. Finally, we describe an algorithm computing the Bernstein–Bézier coefficients of these splines, and we give an example.     

Presentability of Groups of Σ-Definable Permutations over Admissible Sets

It is proved that a group of all -permutations of an arbitrary locally countable, recursively listed, admissible set is not -presentable over that set. An example is constructed showing that this group could be presentable provided that the condition of being locally countable is dropped.     

Fourier series of additive vector measures and their term-by-term differentiation

On a measurable space (T, , ) we choose an additive measure: Z (Z is a Banach space) with the following property: for alle , we have ; this measure defines an indefinite integral over the measure onL ² (T, ,). We prove that if { _n (t)} _n ^=1/ is an orthonormal basis inL ² and _n (e)=e _n (t) d, then any additive measure: Z whose Radon-Nikodým derivatived/d belongs toL ² is uniquely expandable in a series(e)= _n ^=1/ _n n(e) that converges to(e) uniformly with respect toe can be differentiated term-by-term, and satisfies _n ^=1/ _n ^/2 <. In the caseL ²[0,2],Z=, the Fourier series of a 2-periodic absolutely continuous functionF(t) such thatF'(t) L ²[0, 2] is superuniformly convergent toF(t).Translated fromMatematicheskie Zametki, Vol. 64, No. 2, pp. 180–184, August, 1998.     

Classification de certains feuilletages associés à un cusp

We study singularities of foliations given by R. Moussu closely related to a conjecture of R. Thom. We give their analytic classification and prove their topological rigidity.

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Notations , ,F 0 - H 0, III - H ₁,H _p I - H-cusp,H _r -cusp I-3 - H _p I-3, III-2 - -cusp II-1 - _r -cusp III-1     

Spreads admitting net generating regulizations

We say a spread S carries a regulization , if is a collection of reguli contained in S and if each element of S, except at most two lines, is contained either in exactly one regulus of or in all reguli of . Replacement of each regulus of by its complementary regulus (exceptional lines remain unchanged) yields the complementary congruence S^c of S with respect to . If S^c is a hyperbolic or parabolic or elliptic linear congruence of lines, then is called a net generating, in particular, a hyperbolic or parabolic or elliptic regulization, respectively. For hyperbolic and parabolic regulizations we also give other geometric characterizations.     

Über beschränkte Mengen in induktiven Limiten topologischer Vektorräume

In topological linear spaces convex hulls of bounded sets are, in general, not bounded. The question arises whether there is at least for every bounded set B a sequence {|} of strictly positive numbers such that the set { _l ⁿ _v B|n} is bounded. When this obtains, bounded sets share several of the properties known in locally convex spaces. The main result of this note is an example of a countable inductive limit of complete metrizable topological linear spaces which is neither regular nor sequentially complete and also fails to have the above bounded summability property.     

Estimation of the Multivariate Normal Precision Matrix under the Entropy Loss

Let X ₁, , X _n (n > p) be a random sample from multivariate normal distribution N _p (, ), where R ^p and is a positive definite matrix, both and being unknown. We consider the problem of estimating the precision matrix ^–1. In this paper it is shown that for the entropy loss, the best lower-triangular affine equivariant minimax estimator of ^–1 is inadmissible and an improved estimator is explicitly constructed. Note that our improved estimator is obtained from the class of lower-triangular scale equivariant estimators.     

The Symmetry of 2-Flat Homogeneous Spaces

A Riemannian manifold M is called 2-flat homogeneous if every geodesic is contained in some 2-flat , and if the group of isometries of M acts transitively on the set of pairs (p, ) with p . By a 2-flat we mean a closed, connected, flat, totally geodesic, 2-dimensional submanifold of M. It is proved in the paper that 2-flat homogeneous spaces are symmetric.     

Four-dimensional compact projective planes with doubly transitive action on the fixed pencil

We consider a four-dimensional compact projective plane =( , ) whose collineation group is six-dimensional and solvable with a nilradical N isomorphic to Nil × R, where Nil denotes the three-dimensional, simply connected, non-Abelian, nilpotent Lie group. We assume that fixes a flag pW, acts transitively on _p \{W}, and fixes no point in the set W{p}. We study the actions of and N on and on the pencil _p \{W}, in the case that does not contain a three-dimensional elation group. In the special situation that acts doubly transitively on _p {W}, we will determine all possible planes . There are exactly two series of such planes.     

Eine Klasse von achtdimensionalen lokalkompakten Translationsebenen mit großen Scherungsgruppen

This paper is one of the final steps in a classification program to determine all eight-dimensional, locally compact translation planes having large collineation groups. Here, we describe all such planes whose collineation group contains a semidirect product ·N, whereN is an at least 3-dimensional normal subgroup consisting of shears with fixed axis, and is isomorphic to SO₃ ().     

A class of nonseparable dynamic programming problems

A solution procedure is proposed for a class of deterministic sequential decision problems whose objective functions are of the form f _n(x _n)+(g _n(x _n)) where is differentiable and either concave or convex. The procedure calls for the collaboration between dynamic programming and c-programming, and is demonstrated in our treatment of a minimum variance type problem.