In Lee (submitted), the GW-integral (the generalized Riemann integral using Wiener measure) is defined. The object of this article is to define stochastic integral in the set up given in Lee (submitted). We also investigate the connection between the stochastic integral defined with the Legesgue counter part, the Paley-Wiener-Zygmund integral in Paley, Weiner and Zygmund (1933). Applications of the stochastic integral will be explained elsewhere.

Consider a (real) projective plane which is topologically locally flatly embedded in S4. It is known that it always admits a 2-disk bundle neighborhood, whose boundary is homeomorphic to the quaternion space Q, the total space of the nonorientable S1-bundle over RP2 with Euler number ± 2, with fundamental group isomorphic to the quaternion group of order eight. Conversely let f: Q → S4 be an arbitrary locally flat topological embedding. Then we show that the closure of each connected component of S4 − f(Q) is always homeomorphic to the exterior of a topologically locally flatly embedded projective plane in S4. We also show that, for a large class of embedded projective planes in S4, a pair of exteriors of such embedded projective planes is always realized as the closures of the connected components of S4 − f(Q) for some locally flat topological embedding f: Q → S4.

We prove that every (locally) contractible topological group is (L)EC and apply these results to homeomorphism groups, free topological groups, reduced products and symmetric products. Our main results are: The free topological group of a θ-contractible space is equiconnected. A paracompact and weakly locally contractible space is locally equiconnected if and only if it has a local mixer. There exist compact metric contractible spaces X whose reduced (symmetric) products are not retracts of the Graev free topological groups F(X) (A(X)) (thus correcting results we published ibidem).

In [4], Hanson has obtained necessary conditions and sufficient conditions for optimality of a program in stochastic systems. However, in many cases, especially in a general treatment, a program satisfying these conditions cannot be determined explicitly, so that the question of existence of an optimal program in such systems is significant. In this paper, we obtain conditions sufficient for existence of an optimal program by applying the direct methods of the calculus of variations [9], [6] and the theory of optimal control [7], [5].

Consider a set x together with a σ-algebra B of subsets of x. Let G be a family of B-measurable transformations on x, let p(X) be the convex set of all prbability measures on B and let I be the convex set of all G-invariant probablity measures in p(X). For μµ p(X) we define Bµ = {A ∈ B: µ(g A Δ A) = 0 for all g ∈ G} and we define B0 = {A ∈B: gA = A for all g ∈ G}. Then B0 ⊆ Bµ and both are σ-subalgebras of B. G is said to act transitively on X if for x ∈ X, y ∈ X, gx = y for some g ∈ G.

Under certain general conditions an n x n stochastic matrix P = (pij), where

is known to possess the “ergodic” property. This means that there exists a finite matrix E such that

Let G be a second countable locally compact group possessing a normal subgroup N with G/N abelian. We prove that if G/N is discrete then G has T1 primitive ideal space if and only if the G-quasiorbits in Prim N are closed. This condition on G-quasiorbits arose in Pukanzky's work on connected and simply connected solvable Lie groups where it is equivalent to the condition of Auslander and Moore that G be type R on N (-nilradical). Using an abstract version of Pukanzky's arguments due to Green and Pedersen we establish that if G is a connected and simply connected Lie group then Prim G is T1 whenever G-quasiorbits in [G, G] are closed.

A group G is called a c-group if each of its subnormal subgroups is characteristic in G. It is the object of this note to give a characterization of finite solvable c-groups.

A two-term Edgeworth expansion for the distribution of an M-estimator of a simple linear regression parameter is obtained without assuming any Cramér-type conditions. As an application, it is shown that certain modification of the naive bootstrap procedure is second order correct even when the error variables have a lattice distribution. This is in marked contrast with the results of Singh on the sample mean of independent and identically distributed random variables.

Let t, m > 2 and p > 2 be positive integers and denote by N(t, m, p) the largest integer for which there exists a t-uniform hypergraph with N (not necessarily distinct) edges and having no independent set of edges of size m and no vertex of degree exceeding p. In this paper we complete the determination of N(t, m, 3) and obtain some new bounds on N(t, 2, p).

A tight Riesz group G is a partially ordered group G that satisfies a strengthened form of the Riesz interpolation property. The term “tight” was introduced by Miller in (1970) and the tight interpolation property has been considered by Fuchs (1965), Miller (1973), (to appear), (preprint), Loy and Miller (1972) and Wirth (1973). If G is free of elements called pseudozeros then G is a non-discrete Hausdorff topological group with respect to the open interval topology U. Moreover the closure P of the cone P of the given order is the cone of an associated order on G. This allows an interesting interplay between the associated order, the tight Riesz order and the topology U. Loy and Miller found of particular interest the case in which the associated partial order is a lattice order. This situation was considered in reverse by Reilly (1973) and Wirth (1973), who investigated the circumstances under which a lattice ordered group, and indeed a partially ordered group, permits the existence of a tight Riesz order for which the initial order is the associated order. These tight Riesz orders were then called compatible tight Riesz orders. In Section one we relate these ideas to the topologies denned on partially ordered groups by means of topological identities, as described by Banaschewski (1957), and show that the topologies obtained from topological identities are precisely the open interval topologies from compatible tight Riesz orders.

In this paper we study groups with Černikov conjugacy classes which are nilpotent-by-Černikov groups, giving full characterizations of them and applying the results obtained to some related areas.

A new version of Erdös-Turán's inequality is described. The purpose of the present paper is to show that the inequality provides better upper bounds for the discrepancies of some sequences than usual Erdös-Turán's inequality.

We extend the Burger–Mozes theory of closed, nondiscrete, locally quasiprimitive automorphism groups of locally finite, connected graphs to the semiprimitive case, and develop a generalization of Burger–Mozes universal groups acting on the regular tree T_{d}$T_{d}$ of degree d\in \mathbb {N}_{\ge 3}$d\in \mathbb {N}_{\ge 3}$. Three applications are given. First, we characterize the automorphism types that the quasicentre of a nondiscrete subgroup of \operatorname {\mathrm {Aut}}(T_{d})$\operatorname {\mathrm {Aut}}(T_{d})$ may feature in terms of the group’s local action. In doing so, we explicitly construct closed, nondiscrete, compactly generated subgroups of \operatorname {\mathrm {Aut}}(T_{d})$\operatorname {\mathrm {Aut}}(T_{d})$ with nontrivial quasicentre, and see that the Burger–Mozes theory does not extend further to the transitive case. We then characterize the (P_{k})$(P_{k})$-closures of locally transitive subgroups of \operatorname {\mathrm {Aut}}(T_{d})$\operatorname {\mathrm {Aut}}(T_{d})$ containing an involutive inversion, and thereby partially answer two questions by Banks et al. [‘Simple groups of automorphisms of trees determined by their actions on finite subtrees’, J. Group Theory 18(2) (2015), 235–261]. Finally, we offer a new view on the Weiss conjecture.

The associativity of the regular semidirect product of existence varieties introduced by Jones and Trotter was proved under certain condition by Reilly and Zhang. Here we estabilsh associativity in many new cases. Moreover, we prove that the regular semidirect product is right distributive with respect to the join operation. In particular, both associativity and right distriutivity yiel within the varieties of completely simple semigroups. Analogous results are obtainedj for e-pseudovarieties of finite regular semigroups.

Let f be a Cr (r ≥ 0) map from the interval [0, 1] into itself and m be a positive integer. This paper gives a sufficient and necessary condition under which the set of periodic points of period m disappears after a certain small Cr-perturbation.

Hashimoto (1952; Theorems 8.3 and 8.5) proved the following theorems: Theorem A. If L, is a distributive lattice, then there exists a generalized Boolean alegbra Lr and an isomorphism from the lattice of all congruence relations of L onto the lattice of all congruence relations of Lr.

In the study of locally finite varieties of groups it has often been illuminating to consider when a group A is a factor (i.e. quotient of a subgroup) of a group B. We write A ⋨ B to express this and say that A is involved in B. It follows from elementary isomorphism theorems that the relation ⋨ is a partial order on any set of finite groups. The conjecture that we consider in this paper (and to which we only give the beginning of an answer) is the following:

We show that the family of all holomorphic functions f$f$ in a domain D$D$ satisfying

\begin{eqnarray}\frac{|f^{(k)}|}{1+|f|}(z)\leq C\quad \text{for all }z\in D\end{eqnarray} $$\begin{eqnarray}\frac{|f^{(k)}|}{1+|f|}(z)\leq C\quad \text{for all }z\in D\end{eqnarray}$$

k$k$

C>0$C>0$

\unicode[STIX]{x1D6FC}>1$\unicode[STIX]{x1D6FC}>1$

k\geq 2$k\geq 2$

\begin{eqnarray}\frac{|f^{(k)}|}{1+|f|^{\unicode[STIX]{x1D6FC}}}(z)\leq C\quad \text{for all }z\in D\end{eqnarray} $$\begin{eqnarray}\frac{|f^{(k)}|}{1+|f|^{\unicode[STIX]{x1D6FC}}}(z)\leq C\quad \text{for all }z\in D\end{eqnarray}$$

An exponential polynomial is a finite linear combination of terms unea: t→tneat where n is any non-negative integer and a is any complex number. The set X of exponetial polynomials is clearly a vector space over the field of complex numbers C and this set is identical with the set of solutions to all homogeneous linear ordinary differential equations with constant coefficients.