Strong Properties Research Articles

Sufficiency, generalized likelihood function and exponential family

The problem of characterizing an exponential family by sufficiency of certain statistics is considered. In distinction to most of the papers on this subject we do not want to characterize an exponential family of order less than or equal to k by the existence of a-dimensional sufficient statis tics in.GepenoenX' oi inc sample size. Since such a characterization is only valid under regularity assumptions, which is shown in the paper, we consider a stronger property of an exponential family which turns out to be a characteristic one. At this the concept of generalized likelihood functions will play an important role.

Left context precedence grammars

Left context precendence grammars which are defined in this paper are a proper subclass of the precedence grammars and contain properly the simple precedence grammars. Left context precedence grammars need not be uniquely invertible. The parsing algorithm developed for left context precedence languages works with linear time and has the viable prefix property which is a stronger property than the correct prefix property.

Stable Range in AW ∗ Algebras

We show all finite A W* algebras (and somewhat more general C* algebras) satisfy a stronger property than having I in their stable range. I. A ring R has 1 in its stable range if for all pairs a,b of elements of R, aR + bR = R implies there is an element c in R such that a + bc is invertible (e.g. [1, p. 231]). We show finite A W* algebras (and somewhat more general C* algebras) satisfy an even stronger property, that the c may be chosen to be unitary. Some related properties are also studied. LEMMA 1. Let S1S2' ... s, S be positive elements of a C* algebra R. If there is a projection p in R such that E ', s1R = pR, then (Xs1)R = pR. In particular, if p = 1, E: s, is invertible in R. PROOF. Since psi = si, each s, commutes with p, and since p = E: siai = X2(psip)aip = X2(psip)(paip) for some collection of elements (a,), by reducing to the ring pRp, we may assume p = 1. It now suffices to show s = E: s, is invertible, and we may assume s i , zz* (proved by induction on m: consider xx*, where x = 1 zi) zm; by varying the x, 3m2 may be improved to 2110g2(M 1)J+ 1 -1), we obtain 1 Si2, and thus 1 S 3m-2 KY si, whence I si is invertible. EJ (The preceding proof is an elaboration of the referee's quick proof for the case m = 2, and is simpler than my originally intended proof.) LEMMA 2 (A SLIGHT GENERALIZATION OF [7, p. 122]). If R is a finite A W* algebra, then for all a in R, there exists a unitary u such that au is positive. Received by the editors October 20, 1976 and, in revised form, October 26, 1977. AMS (MOS) subject classifications (1970). Primary 46L10; Secondary 16A54.

Finite central height in linear groups

J.D. Dixon has characterized those pairs ( n, F), where n is a positive integer and F a field, for which every locally nilpotent subgroup of GL( n, F) is nilpotent. He showed further that these pairs ( n, F) have the stronger property that there is a bound on the nilpotency class of the nilpotent subgroups of GL( n, F). In this note we show that these pairs ( n, F) have the still stronger property that every subgroup of GL( n, F) has finite bounded central height. Our main result generalizes to groups of automorphisms of Noetherian modules over commutative rings.

Baire*\ $1$, Darboux functions

It is well known that a function $f:[0,1] \to R$ is Baire 1 if and only if in any closed set C there is a point ${x_0}$ at which the restricted function $f|C$ is continuous. Functions will be called Baire$^\ast$ 1 if they satisfy the following stronger property: For every closed set C there is an open interval (a, b) with $(a,b) \cap C \ne \emptyset$ such that $f|C$ is continuous on (a, b). Functions which are both Baire$^\ast$ 1 and Darboux are examined. It is known that approximately derivable functions are Baire$^\ast$ 1. Among other things it is shown here that ${L_p}$-smooth functions are Baire$^\ast$ 1. A new result about the ${L_p}$-differentiability of ${L_p}$-smooth, Darboux functions is shown to follow immediately from the main properties of Baire$^\ast$ 1, Darboux functions.

Vanishing Theorems for Flag Manifolds

Introduction. In this paper, I prove some vanishing theorems for the cohomology groups of invertible sheaves on flag manifolds. Most of these vanishing theorems follow from Kodaira's vanishing theorem in characteristic zero. I prove these by induction. This induction involves certain subvarieties of the flag manifold, which I call Schubert manifolds. These vanishing theorems are used to deduce the Cohen-Macaulayness of Schubert varieties in Grassmannians and cones over them. They were recently proved to be Cohen-Macaulay by Eagon, Hochster and Laksov [3, 5, and 8] by more algebraic methods. I prove that the Schubert varieties have rational resolutions. This is a stronger property than Cohen-Macaulayness. The original idea for this proof was to generalize the results in [6]. The vanishing theorem has been applied by T. Svanes to show the rigidity of some Schubert varieties [12].

The isomorphism property in nonstandard analysis and its use in the theory of Banach spaces

The basic setting of nonstandard analysis consists of a set-theoretical structure together with a map * from into another structure * of the same sort. The function * is taken to be an elementary embedding (in an appropriate sense) and is generally assumed to make * into an enlargement of [13]. The structures and * may be type-hierarchies as in [11] and [13] or they may be cumulative structures with ω levels as in [14]. The assumption that * is an enlargement of has been found to be the weakest hypothesis which allows for the familiar applications of nonstandard analysis in calculus, elementary topology, etc. Indeed, practice has shown that a smooth and useful theory can be achieved only by assuming also that * has some stronger properties such as the saturation properties first introduced in nonstandard analysis by Luxemburg [11].This paper concerns an entirely new family of properties, stronger than the saturation properties. For each cardinal number κ, * satisfies the κ-isomorphism property (as an enlargement of ) if the following condition holds:For each first order language L with fewer than κ nonlogical symbols, if and are elementarily equivalent structures for L whose domains, relations and functions are all internal (relative to * and ), then and are isomorphic.

Locally 𝑒-fine measurable spaces

Hyper-Baire sets and hyper-cozero sets in a uniform space are introduced, and it is shown that for metric-fine spaces the property “every hypercozero set is a cozero set” is equivalent to several much stronger properties like being locally e-fine (defined in §1), or having locally determined precompact part (introduced in §2). The metric-fine spaces with these additional properties form a coreflective subcategory of uniform spaces; the coreflection is explicitly described. The theory is applied to measurable uniform spaces. It is shown that measurable spaces with the additional properties mentioned above are coreflective and the coreflection is explicitly described. The two coreflections are not metrically determined.

A note on commutative Baer rings III

If R is a commutative semiprime ring with identity Kist [4], [5] has shown that R can be embedded into a commutative Baer ring B(R), and has given some properties of this embedding. More recently Mewborn [7] has given a construction which embeds R into a commutative Baer ring with the stronger property that every annihilator is generated by an idempotent. Both of these constructions involve a representation of R as a ring of global sections of a sheaf over a Boolean space.

Tables for unbiased confidence intervals for ratios of variances when sampling from two independent normal distributions

Tables are given for the upper tail values for unbiased confidence intervals for ratios of variances when sampling from two independent normal distributions for sample sizes (N 1-l) = 1(1)20(5)30(30)60(60)120 and (N 2−l) = 1(1)30(5)40(20)60(60)120, and alpha levels of 0.10,0.05,0.01 (where the alpha level is the sum of the upper and lower tail areas). These confidence intervals are logarithmically shortest confidence intervals, in the sense of Scheffe. The tables can also be used for a two-sided test of the equality of variances, which is unbiased and which also possesses the stronger property of monotonicity. The computation of the tables is discussed, and an example comparing the standard equal tail area procedure with the unbiased procedure is given.

Rings in which certain subsets satisfy polynomial identities

Let R be an associative ring and Y' a set of polynomials (each in some finite number of noncommuting indeterminates) with integer coefficients. R will be called an IF-ring if and only if for every finite subset S of R there is anf in F which vanishes identically on S. Thus certain kinds of rings can be characterized as 3Frings for appropriate choices of Y: nil rings, with F the set of polynomials xn for all positive integers n; locally nilpotent rings, with 3 the set of monomials x1x2 ... xn in n indeterminates for all n; nilpotent rings, with s the single monomial x1x2... xn for some n; rings satisfying a polynomial identity, with Y finite (or equivalently, with Y containing a single polynomial); and so forth. In addition to Y-rings we shall consider rings satisfying a somewhat stronger property. R is called an F+-ring if and only if for every finitely generated additive subgroup S of R there is anf in F which vanishes identically on S. Clearly every S+-ring is an Y-ring. Also all subrings and homomorphic images of F-rings are F-rings, and similarly for F + -rings. This paper is concerned with determining conditions on , which will ensure that every Y-ring (or F+-ring) has nil commutator ideal (the ideal generated by all commutators [a, b] = ab ba). This intent is similar to that of Drazin in [5] insofar as both are concerned with conditions under which, in a given class of rings, the fact that the commutator ideal is contained in the Jacobson radical implies that the commutator ideal is nil. Drazin's work is self-contained (depending only on Zorn's lemma) but requires that every ring in the class be an f-ring (cf. [5]), whereas here, rather than imposing conditions on our rings individually, we instead confine our attention to classes defined in special ways, i.e., as complete classes of Yor F+-rings, and make use of some relatively special properties of the Jacobson radical (whereas Drazin's arguments would apply equally well to any radical not containing nonzero idempotents). The commutator ideal of R is denoted C(R) and the Jacobson radical J(R). The ring of n x n matrices over R is denoted Rn. R is said to be of characteristic 0, written char (R) =0, if and only if mxA 0 for every nonzero integer m and every nonzero x in R. A linear polynomial means a polynomial which is linear in each of its indeterminates; F will be called linear if and only if each member of F is linear. We consider only sets Y of polynomials each of which is homogeneous in each of its indeterminates. One final assumption about F: Note that if all coeffi-

On the extension property for compact operators

1. Introduction. The main theorem of this note shows the equiv­ alence of several extension properties for operators and contains also some geometrical characterizations of the spaces having these proper­ ties. In particular a characterization of such spaces is given in terms of intersection properties of their cells, similar to that given by Nachbin for (Pi spaces [8], Our theorem extends previous results of Grothendieck [4]. Some applications are given, among them a new character­ ization of C{K) spaces. In this connection some problems raised by Aronszajn and Panitchpakdi [l], Grothendieck [4] and Nachbin [8; 9] are solved. Notations. All Banach spaces are assumed to be over the reals. Sx(xo, r) denotes the cell {x;xGl, ||x — xo Sr}. A Banach space X is called a (Px space if from any Z containing X there is a projection on X with norm ^X (see Day [2, pp. 94-96]). We say that a Banach space has the metric approximation property (M.A.P.) if for every compact subset K of X and every e>0 there is a compact operator from X into itself such that || T = 1 and || Tx-x ue for x£K. A (possibly) stronger property was introduced by Grothendieck [3, pp. 187-191]. He proved that the common Banach spaces have the M.A.P. It is an open problem whether there exists a Banach space which does not have the M.A.P. 2. The main results. We state now the main result of this note (the equivalencies l<-*2<-»3<-»5 are due to Grothendieck [4]). In the extension properties stated below F, Z and V will be arbitrary Ba­ nach spaces satisfying Z~2)Y, TO^and the indicated restrictions (if any).

On the Order Structure of the Set of Sufficient Subfields

In [5], the concept of statistical sufficiency is studied within a general probability setting. The study is continued here. The notation and definitions of [5] are used. Here we give an example of sufficient statistics $t_1$ and $t_2$ such that the pair $(t_1, t_2)$ is not sufficient. The example also has the property that, in a sense to be made precise, no smallest sufficient statistic containing $t_1$ and $t_2$ exists. In Example 4 of [5], sufficient subfields $\mathbf{A}_1$ and $\mathbf{A}_2$ are exhibited such that $\mathbf{A}_1 \vee \mathbf{A}_2$, the smallest subfield containing $\mathbf{A}_1 and \mathbf{A}_2$, is not sufficient. Such an example is given here with the even stronger property that no smallest sufficient subfield containing $\mathbf{A}_1$ and $\mathbf{A}_2$ exists. Let $(X, \mathbf{A}, P)$ be the probability structure under consideration. Here $X$ is a set, $\mathbf{A}$ is a $\sigma$-field of subsets of $X$, and $P$ is a family of probability measures $p$ on $\mathbf{A}$. Let $\mathbf{N}$ be the smallest $\sigma$-field containing the $P$-null sets and let $K$ be the collection of sufficient subfields of $A$ containing $N$. (Restricting attention to sufficient subfields containing $N$ is technically convenient. Note that any sufficient subfield is equivalent, in the usual sense, to one containing $N$.) Some of the properties of $K$ can be described in the language of lattice theory as follows. Let $L$ be the set of subfields (= sub-$\sigma$-fields) of $\mathbf{A}$. Then $L$, partially ordered by inclusion, is a complete lattice. (Our terminology is essentially that of Birkhoff [4].) Example 4 of [5], mentioned above, shows that $K$ is not always a sublattice of $L$. The example given below shows more: The set $K$, partially ordered by inclusion, is not always a lattice in its own right. Note, however, that if $H$ is a finite, or even countable, subset of $K$, then the greatest lower bound of $H$ relative to $L$ exists and is in $K$ ([5], Corollary 2). The difficulty is with the least upper bound. There is less difficulty if $\mathbf{A}$ is separable. Corollaries 2 and 4 of [5] indicate that if $A$ is separable, then $K$ is a $\sigma$-complete sublattice of $L$. This is about as strong a result as could be expected here. For even if $\mathbf{A}$ is separable, $\mathbf{K}$ is sometimes neither complete nor conditionally complete: Each of the nonsufficient subfields exhibited in Example 1 of [5] is easily seen to be both the greatest lower bound of a subset of $K$ and the least upper bound of a subset of $K$. There is no difficulty if $P$ is dominated. If $P$ is dominated, then $K$ is a complete sublattice of $L$. This follows easily from the existence in this case (Bahadur [2], Theorems 6.2 and 6.4; Loeve [6], Section 24.4) of a subfield $\mathbf{A}_0$ in $K$ such that $K = \{\mathbf{B} \mid \mathbf{B} \epsilon L, \mathbf{A}_0 \subset \mathbf{B}\}$.

Remote Points in βR

The proof turns out to be considerably more difficult than anticipated. If we assume the continuum hypothesis (designated [CH]), then we can find such a point p (2.5); however, we do not know whether the continuum hypothesis is necessary. The result is obtained as a consequence of a more general theorem: for a suitably restricted class of spaces X, a point with a somewhat stronger property exists if and only if X admits an unbounded continuous function (2.3). A byproduct is: [CH] there exists a countable, completely regular space without isolated points, one of whose points is not a limit point of any discrete set (2.6).

Gamma-positivity of variations of Eulerian polynomials

An identity of Chung, Graham and Knuth involving binomial coefficients and Eulerian numbers motivates our study of a class of polynomials that we call binomial-Eulerian polynomials. These polynomials share several properties with the Eulerian polynomials. For one thing, they are $h$-polynomials of simplicial polytopes, which gives a geometric interpretation of the fact that they are palindromic and unimodal. A formula of Foata and Schutzenberger shows that the Eulerian polynomials have a stronger property, namely $\gamma$-positivity, and a formula of Postnikov, Reiner and Williams does the same for the binomial-Eulerian polynomials. We obtain $q$-analogs of both the Foata-Schutzenberger formula and an alternative to the Postnikov-Reiner-Williams formula, and we show that these $q$-analogs are specializations of analogous symmetric function identities. Algebro-geometric interpretations of these symmetric function analogs are presented.