In this paper we study lattice properties of operator algebras which are invariant under compact perturbations. It is easy to see that if and are two operator algebras with contained in , then the reverse inclusion holds for their lattices of invariant subspaces. We will show that in certain cases, the assumption thats is contained in , where is the ideal of compact operators, implies that the lattice of is “approximately” contained in the lattice of . In particular, supposed and are reflexive and have commutative subspace lattices containing “enough” finite dimensional elements. We show (Corollary 2.8) that if is unitarily equivalent to a subalgebra of , then there is a unitary operator which carries all “sufficiently large” subspaces in lat into lat .

Let X be any set and A be a uniformly closed algebra of bounded real valued functions on X which contains the constants and separates the points. For a lattice ℒ of subsets of X (we assume throughout that ∅ and X belong to ℒ), let MR(ℒ) denote the space of all finite, finitely additive,ℒ-regular measures defined on the field of sets generated by ℒ . Generalizing the notion of an integral representation, in [5] Kirk and Crenshaw define a standard representation of A*, the Banach dual of A, in MR(ℒ) to be a linear map I of A* into MR(ℒ) with the property that if 0 ≦ ϕ ∈ A*, then

for every W in ℒ. The space MR(ℒ) is said to represent A* if there exists a (unique) standard representation I of A* onto MR(ℒ) which is a Banach lattice isomorphism.

This paper is a contribution to applied stability theory. Our purpose is to investigate the complexity of lattices by determining the stability of their first order theories.

Stability measures the complexity of a theory T by counting the number of different “kinds” of elements in models of T. The notion of ω-stability was introduced by Morley [26] in 1965 and generalized by Shelah [31] in 1969. Shelah classified all first order theories according to their stability properties.

Stability and -categoricity are closely related (see [26] and [1]). In fact, the notions of stable, superstable and ω-stable can be regarded as successive approximations of -categorical. -categoricity is a very strong property while stability, superstability and ω-stability facilitate the classification of more “complex” theories.

The fundamental theorem of this paper is stated in Section 8. In this theorem, the stochastic differential equation dX = a(X)dZ is studied when Z is a *-dominated (cf. [15]) Banach space valued process and a is a predictable functional which is continuous for the uniform norm.

For such an equation, the existence of a “weak solution” is stated; actually, the notion of weak solution here considered is more precise than this one introduced by Strook and Varadhan (cf. [30], [31], [23]).

Namely, this weak solution is a probability, so-called “rule,” defined on (DH × Ω), DH being the classical Skorohod space of all the cadlag sample paths and Ω is the initial space which Z is defined on: the marginal distribution of R on Ω is the given probability P on Ω. This concept of rule is defined in Section 3.

In 1958 Gleason [6] proved the following :

THEOREM. In the category of compact Hausdorff spaces and continuous maps, the projective elements are precisely the extremally disconnected spaces.

The projective elements in many topological categories with perfect continuous functions as morphisms have been found since that time. For example: In the following categories the projective elements are precisely the extremally disconnected spaces:

• (i) The category of Tychonov spaces and perfect continuous functions. [4] [11].

• (ii) The category of regular spaces and perfect continuous functions. [4] [12].

• (iii) The category of Hausdorff spaces and perfect continuous functions. [10] [1].

• (iv) In the category of Hausdorff spaces and continuous k-maps the projective members are precisely the extremally disconnected k-spaces. [14].

In 1963 Iliadis [7] constructed for every Hausdorff space X the so called Iliadis absolute E[X], which is a maximal pre-image of X under irreducible θ-continuous maps.

Let be a valuation ring with unit element, i.e., is a commutative ring such that for any a and b in , either a divides b or b divides a. We assume 2 is a unit of . V is an n-ary nonsingular quadratic module over , O(V) or On(V) is the orthogonal group on V, and S is the set of symmetries in O(V). We define l(σ) to be the minimal number of factors in the expression of a of O(V) as a product of symmetries on V. For the case where is a field, l(σ) has been determined by P. Scherk [6] and J. Dieudonné [1]. In [3] I have generalized the results of Scherk to orthogonal groups over valuation domains. In the present paper I generalize my results of [3] to orthogonal groups over valuation rings.

Since is a valuation ring, it is a local ring with the maximal ideal A which consists of all nonunits of .

A (tame) knot kn in S3 is said to have period n if there exists a homeomorphism ϕ: S3 → S3, necessarily orientationpreserving, such that

• (i) the fixed point set of ϕ is a circle disjoint from kn

• (ii) ϕ(kn) = kn;

• (iii) ϕ has order n.

Several necessary conditions for a knot to have period n have already been established in the literature; see [3] [11] [14] [18]. Here we establish the following further condition, involving the signature σ(kn) of (kn).

A topological space X is called developable if it has a development, i.e., a sequence of open covers of X such that for each x ∈ X the collection is a neighbourhood base of x, where

This class of spaces has turned out to be one of the most natural and useful generalizations of metrizable spaces [23]. In [4] it was shown that some well known results in metrization theory have counterparts in the theory of developable spaces (i.e., Urysohn's metrization theorem, the Nagata-Smirnov theorem, and Nagata's “double sequence theorem”). Moreover, in [3] it was pointed out that subspaces of products of developable spaces (i.e., D-completely regular spaces) can be characterized in much the same way as subspaces of products of metrizable spaces (i.e., completely regular T1-spaces).

1. It was proved by A. A. Markov [3] that if is a polynomial of degree at most n and |pn(x)| ≦ 1 in the interval –1 ≦ x ≦ 1, then in the same interval

(1)

The problem was proposed by the chemist Mendeleev who knew the answer for polynomials of degree 2. For a historical background of the problem see [1].

A. A. Markov's younger brother W. A. Markov considered the problem of determining exact bounds for the j–th derivative of pn(x) at a given point X0 in [ – 1, 1]. His results appeared in a Russian journal in the year 1892; a German version of his remarkable paper was later published in [4]. Amongst other things he proved the following two theorems.

In this paper we consider the problem of determining when an algebra of formal power series over a commutative ring R is the homomorphic image of a reduced incidence algebra P(R, ∽). The question of when two such algebras are isomorphic is answered in Section 8 of [1]. A slight generalization of their notion of full binomial type is introduced here.

Section 1 contains background material together writh a summary of the results of [1]. In Section 2 we present the desired characterization, and to conclude an application appears in Section 3. In Section 3 the tools of Section 2 are used to derive an equation of R. W. Robinson and R. P. Stanley which counts labelled, acyclic digraphs.

If x0,x1, … xn are homogeneous coordinates in [n], projective space of n dimensions, the prime (to use the standard name for a hyperplane)

osculates, as θ varies, the rational normal curve C whose parametric form is [2, p. 347]

Take a set of n + 2 points on C for which θ = ηjζ where ζ is any complex number and

so that the ηj, for 0 ≦ j < n + 2, are the (n + 2)th roots of unity. The n + 2 primes osculating C at these points bound an (n + 2)-hedron H which varies with η, and H is polar for all the quadrics

(1.1)

in the sense that the polar of any vertex, common to n of its n + 2 bounding primes, contains the opposite [n + 2] common to the residual pair.

We let Z denote the ring of rational integers, Q the field of rational numbers, R the field of real numbers, and C the field of complex numbers.

For elements e and f of a Lie algebra, [e,f] denotes the bracket of e and f. A generalized Cartan matrix C = (cij) is a square matrix of integers satisfying cii = 2, cij ≦ 0 if i ≠ j, cij = 0 if and only if cji = 0. For any generalized Cartan matrix C = (cij) of size l × l and for any field F of characteristic zero, denotes the Lie algebra over F generated by 3l generators e1, …, el, h1, …, hl, f1, …, fl with the defining relations

for all i, j,

for distinct i, j.

A semigroup which is a union of groups is said to be completely regular. If in addition the idempotents form a subsemigroup, the semigroup is said to be orthodox and is called an orthogroup. A completely regular semigroup S is provided in a natural way with a unary operation of inverse by letting a-l for a ∈ S be the group inverse of a in the maximal subgroup of S to which a belongs. This unary operation satisfies the identities

(1)

(2)

(3)

In fact a completely regular semigroup can be defined as a unary semigroup (a semigroup with an added unary operation) satisfying these identities. An orthogroup can be characterized as a completely regular semigroup satisfying the additional identity

(4)

The concept of θ-closure was introduced by Velicko to study H-closed spaces and to generalize Taimanov's extension theorem [11], [12]. More recently, this notion has been used by Dickman and Porter [1] to characterize those Hausdorff spaces in which the Fomin H-closed extension operator commutes with the projective cover (absolute) operator and [2] to study extentions of functions. If X is a topological space and A ⊂ X, we let Σ(A) and Γ(A) represent, respectively, the family of open subsets which contain A and closed subsets which contain some element of Σ(A). The θ-closure of A ⊂ X, denoted by clθ(A) (clθ(v) if A = {v}), is {x ∈ X: each V ∈ Γ(x) satisfies V ∩ A ≠ ∅} and A is called θ-closed in case clθ(A) = A.

Suppose that n competitors compete in r races and in each race they are awarded placings l, 2, 3, …, n – 1, n. After the r races each competitor has a result consisting of his r placings. Let such a result be written (αj)1≦j≦r where for convenience the positive integers αj are arranged in ascending order. For example, if n = 4 and r = 6 a typical result is (1, 2, 2, 3, 4, 4).

A final ranking of the n competitors will be determined in all circumstances if a strict ordering is assigned to all possible results. In the next section it is shown that the number or results is . Of course only some strict orderings of this set of results will be feasible in that there are criteria which a final ranking should satisfy. For example, (1, 1, …, 1) should clearly be ranked first in any feasible ranking.

Our work was motivated by attempts to find a criterion for the existence of a classical quotient ring, for a noetherian ring, in analogy with the various known criteria for the existence of an artinian classical quotient ring ([9], [10], [13], [2]).

We have restricted our attention to Krull symmetric noetherian rings R, and we make heavy use of the fact that all their Krull composition factors are non-singular (Proposition 7). The collection Kprime R of the associated primes of the Krull composition factors of R plays a central role, taking the place of the collection of the associated primes of R.

A Steiner system S(t, k, v) is a collection of k-subsets, called blocks, of a v-set of points with the property that any t-subset of points is contained in a unique block. We assume 1 < t < k < v. A Steiner system is called block-schematic if the blocks form an association scheme with the relations determined by size of intersection. Ito and Patton [3] proved that if S(4, 5, v) is block-schematic, then v = 11. The purpose of this paper is to extend this result, and we prove the following theorem.

THEOREM. A Steiner system S(t, t + 1, v) is block-schematic if and only if one of the following holds: (i) t = 2, (ii) t = 3, v = 8, (iii) t = 4, v = 11, (iv) t = 5, v = 12.

Let (X, ) be a Steiner triple system on v = |X| points, and suppose that is a partial parallel class (transversal, clear set, set of pairwise disjoint blocks) of maximum size . We want to derive a bound on . (I conjecture that in fact r is bounded, e.g., r ≦ 4 – 4 is attained for the Fano plane, but all that has been proved so far (cf. [1], [2]) are bounds r < C.v for some C. Here we shall prove r < 5v2/3.)

Define a sequence of positive real numbers by q0 = Q · r2/v, , where l is determined by ql ≧ 6, , i.e.,

(The constant Q will be chosen later.) Define inductively sets Ai, Ki and collections as follows. Let

This paper addresses some flatness properties of an (n – 1)-sphere Σ in Euclidean n-space En resulting from the presence of round balls in En tangent to Σ. The notion of tangency used here is geometric rather than differentiable, for a round n-cell Bp (that is, the set of points whose distance, in the standard metric, from some center point is less than or equal to a fixed positive number) is said to be tangent to the (n – 1)-sphere Σ in En at a point p ∈ Σ if p ∈ Bp and Int Bp ⊂ Σ = ∅. The ball Bp is called an interior tangent ball at p if Int Bp ⊂ Int Σ; otherwise, it is called an exterior tangent ball at p.

Let n ≧ 1 be an integer and let Rn be the usual n-dimensional real vector space, considered together with all its usual structure. The usual n-dimensional Lebesgue measure on Rn is denoted by λn. The positive cone of Rn is Rn+ and the interior of Rn + is Pn. Hence Pn is the set of vectors with strictly positive coordinates. A subset of Rn is called an interval if it is the cartesian product of one dimensional bounded intervals. If a, b ∊ Rn then [a, b] denotes the interval {u|a ≦ u ≦ b|. The closure of any interval I is of the form [a, b]; the initial point of I will be defined as the vector a. The class of all intervals contained in Rn+ is denoted by . Also, for each u ∊ Pn, let be the set of all intervals that are contained in the interval [0, u] and that have non-empty interiors. Finally let en ∊ Pn be the vector with all coordinates equal to 1.