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A classical result of K. L. Chung and W. Feller deals with the partial sums S k S_k arising in a fair coin-tossing game. If N n N_n is the number of “positive” terms among S 1 S_1 , S 2 S_2 , …, S n S_n then the quantity P ( N 2 n = 2 r ) P(N_{2n} = 2r) takes an elegant form. We lift the restriction on an even number of tosses and give a simple expression for P ( N 2 n + 1 = r ) P(N_{2n+1} = r) , r = 0 r = 0 , 1 1 , 2 2 , …, 2 n + 1 2n+1 . We get to this ansatz by adaptating the Feynman–Kac methodology.

Concerns have been voiced about the correctness of certain technical points in DiPerna’s paper (Comm. Math. Phys. 91 (1983), 1–30) related to the vacuum state. In this note, we provide clarifications. Our conclusion is that these concerns mainly arise from the statement of a lemma for constructing the viscous approximate solutions and some typos; however, the gap can be either fixed by correcting the statement of the lemma and the typos or bypassed by employing the finite difference methods. In [Di], DiPerna found a global entropy solution of the isentropic Euler equations for the following exponents in the equation of state for the pressure: γ = 1 + 2/(2m+ 1), m ≥ 2 integer. (1) He divided his arguments into the following two steps. 1. Compactness framework Assume that a sequence of approximate solutions (ρ (x, t),m (x, t)), 0 ≤ t ≤ T , satisfies: (i). There exists a constant C(T ) > 0, independent of > 0, such that 0 ≤ ρ (x, t) ≤ C, |m (x, t)/ρ (x, t)| ≤ C; (ii). For all weak entropy pairs (η, q) of the isentropic Euler equations, the measure sequence η(ρ ,m )t + q(ρ ,m )x is contained in a compact subset of H −1 loc (R× [0, T ]). If γ satisfies (1), then the sequence (ρ (x, t),m (x, t)) is compact in Lloc(R× [0, T ]). The reason for the restriction on the number γ is that, in such a case, any weak entropy function is a polynomial function of the Riemann invariants (w, z). This is the key step in DiPerna’s arguments and is also his main contribution to the compensated compactness method in this aspect. Received by the editors May 16, 1996. 1991 Mathematics Subject Classification. Primary 35K55, 35L65; Secondary 76N15, 35L60, 65M06.

In this short paper we prove that every sufficiently large odd integer can be written as a sum of one prime, two squares of primes and 83 83 powers of 2 2 .

The main purpose of this paper is to explore the structure of local and regular Dirichlet forms associated with symmetric one-dimensional diffusions, which are also called symmetric linear diffusions. Let ( E , F ) (\mathcal {E},\mathcal {F}) be a regular and local Dirichlet form on L 2 ( I , m ) L^2(I,m) , where I I is an interval and m m is a fully supported Radon measure on I I . We shall first present a complete representation for ( E , F ) (\mathcal {E},\mathcal {F}) , which shows that ( E , F ) (\mathcal {E},\mathcal {F}) lives on at most countable disjoint “effective" intervals with an “adapted" scale function on each interval, and any point outside these intervals is a trap of the one-dimensional diffusion. Furthermore, we shall give a necessary and sufficient condition for C c ∞ ( I ) C_c^\infty (I) being a special standard core of ( E , F ) (\mathcal {E},\mathcal {F}) and shall identify the closure of C c ∞ ( I ) C_c^\infty (I) in ( E , F ) (\mathcal {E},\mathcal {F}) when C c ∞ ( I ) C_c^\infty (I) is contained but not necessarily dense in F \mathcal {F} relative to the E 1 1 / 2 \mathcal {E}_1^{1/2} -norm. This paper is partly motivated by a result of Hamza’s that was stated in a theorem of Fukushima, Oshima, and Takeda’s and that provides a different point of view to this theorem. To illustrate our results, many examples are provided.

The study of the global structure of ultrafilters on the natural numbers with respect to the quasi-orders of Rudin-Keisler and Rudin-Blass reducibility was initiated in the 1970s by Blass, Keisler, Kunen, and Rudin. In a 1973 paper Blass studied the special class of P-points under the quasi-ordering of Rudin-Keisler reducibility. He asked what partially ordered sets can be embedded into the P-points when the P-points are equipped with this ordering. This question is of most interest under some hypothesis that guarantees the existence of many P-points, such as Martin’s axiom for σ \sigma -centered posets. In his 1973 paper he showed under this assumption that both ω 1 {\omega }_{1} and the reals can be embedded. Analogous results were obtained later for the coarser notion of Tukey reducibility. We prove in this paper that Martin’s axiom for σ \sigma -centered posets implies that the Boolean algebra P ( ω ) / FIN \mathcal {P}(\omega ) / \operatorname {FIN} equipped with its natural partial order can be embedded into the P-points both under Rudin-Keisler and Tukey reducibility. Consequently, the continuum hypothesis implies that every partial order of size at most continuum embeds into the P-points under both notions of reducibility.

Let $\Omega$ be either $\mathbb{R}^n$ or a strongly Lipschitz domain of $\mathbb{R}^n$, and $\omega\in A_{\infty}(\mathbb{R}^n)$ (the class of Muckenhoupt weights). Let $L$ be a second order divergence form elliptic operator on $L^2 (\Omega)$ with the Dirichlet or Neumann boundary condition, and assume that the heat semigroup generated by $L$ has the Gaussian property $(G_1)$ with the regularity of their kernels measured by $\mu\in(0,1]$. Let $\Phi$ be a continuous, strictly increasing, subadditive, positive and concave function on $(0,\infty)$ of critical lower type index $p_{\Phi}^-\in(0,1]$. In this paper, the authors introduce the "geometrical" weighted local Orlicz-Hardy spaces $h^{\Phi}_{\omega,\,r}(\Omega)$ and $h^{\Phi}_{\omega,\,z}(\Omega)$ via the weighted local Orlicz-Hardy spaces $h^{\Phi}_{\omega}(\mathbb{R}^n)$, and obtain their two equivalent characterizations in terms of the nontangential maximal function and the Lusin area function associated with the heat semigroup generated by $L$ when $p_{\Phi}^-\in(n/(n+\mu),1]$. As applications, the authors prove that the operators $\nabla^2{\mathbb G}_D$ are bounded from $h^{\Phi}_{\omega,\,r}(\Omega)$ to the weighted Orlicz space $L^{\Phi}_{\omega}(\Omega)$, and from $h^{\Phi}_{\omega,\,r}(\Omega)$ to itself when $\Omega$ is a bounded semiconvex domain in $\mathbb{R}^n$ and $p_{\Phi}^-\in(\frac{n}{n+1},1]$, and the operators $\nabla^2{\mathbb G}_N$ are bounded from $h^{\Phi}_{\omega,\,z}(\Omega)$ to $L^{\Phi}_{\omega}(\Omega)$, and from $h^{\Phi}_{\omega,\,z}(\Omega)$ to $h^{\Phi}_{\omega,\,r}(\Omega)$ when $\Omega$ is a bounded convex domain in $\mathbb{R}^n$ and $p_{\Phi}^-\in(\frac{n}{n+1},1]$, where ${\mathbb G}_D$ and ${\mathbb G}_N$ denote, respectively, the Dirichlet Green operator and the Neumann Green operator., Comment: This paper has been withdrawn by the authors

This paper complements a recent result of Dilworth, Ferenczi, Kutzarova and Odell regarding the ergodicity of strongly asymptotic ℓ p \ell _p spaces. We state this result in a more general form, involving domination relations, and we show that every asymptotically Hilbertian space which is not isomorphic to ℓ 2 \ell _2 is ergodic. In particular, every weak Hilbert space which is not isomorphic to ℓ 2 \ell _2 must be ergodic. Throughout the paper we construct explicitly the maps which establish the fact that the relation E 0 E_0 is Borel reducible to isomorphism between subspaces of the Banach spaces involved.

In this paper we give simple, sufficient conditions for the existence of the stochastic integral for vector-valued processes X with values in a Banach space E; namely, X is of class (LD), and the stochastic measure I X is bounded and strongly additive in L p E (in particular, if I X is bounded in L p E and c 0 ⊈ E) and has bounded semivariation. The result is then applied to martingales and processes with integrable variation or semivariation. For martingales the condition of being of class (LD) is superfluous. For a square-integrable martingale with values in a Hilbert space, all the conditions are superfluous. For processes with p-integrable semivariation or p-integrable variation, the conditions of I X to be bounded and have bounded semivariation are superfluous. For processes with 1-integrable variation, all conditions are superfluous. In a forthcoming paper, we shall extend these results to local summability. The extension needs additional nontrivial work.

In a recent study of sign-balanced, labelled posets, Stanley introduced a new integral partition statistic s r a n k ( π ) = O ( π ) − O ( π ′ ) , \begin{equation*} \mathrm {srank}(\pi ) = {\mathcal O}(\pi ) - {\mathcal O}(\pi ’), \end{equation*} where O ( π ) {\mathcal O}(\pi ) denotes the number of odd parts of the partition π \pi and π ′ \pi ’ is the conjugate of π \pi . In a forthcoming paper, Andrews proved the following refinement of Ramanujan’s partition congruence mod 5 5 : p 0 ( 5 n + 4 ) a m p ; ≡ p 2 ( 5 n + 4 ) ≡ 0 ( mod 5 ) , p ( n ) a m p ; = p 0 ( n ) + p 2 ( n ) , \begin{align*} p_0(5n+4) &\equiv p_2(5n+4) \equiv 0 \pmod {5}, p(n) &= p_0(n) + p_2(n), \end{align*} where p i ( n ) p_i(n) ( i = 0 , 2 i=0,2 ) denotes the number of partitions of n n with s r a n k ≡ i ( mod 4 ) \mathrm {srank}\equiv i\pmod {4} and p ( n ) p(n) is the number of unrestricted partitions of n n . Andrews asked for a partition statistic that would divide the partitions enumerated by p i ( 5 n + 4 ) p_i(5n+4) ( i = 0 , 2 i=0,2 ) into five equinumerous classes. In this paper we discuss three such statistics: the ST-crank, the 2 2 -quotient-rank and the 5 5 -core-crank. The first one, while new, is intimately related to the Andrews-Garvan (1988) crank. The second one is in terms of the 2 2 -quotient of a partition. The third one was introduced by Garvan, Kim and Stanton in 1990. We use it in our combinatorial proof of the Andrews refinement. Remarkably, the Andrews result is a simple consequence of a stronger refinement of Ramanujan’s congruence mod 5 5 . This more general refinement uses a new partition statistic which we term the BG-rank. We employ the BG-rank to prove new partition congruences modulo 5 5 . Finally, we discuss some new formulas for partitions that are 5 5 -cores and discuss an intriguing relation between 3 3 -cores and the Andrews-Garvan crank.

The Rees algebra is the homogeneous coordinate ring of a blowing-up. The present paper gives a necessary and sufficient condition for a Noetherian local ring to have a Cohen-Macaulay Rees algebra: A Noetherian local ring has a Cohen-Macaulay Rees algebra if and only if it is unmixed and all the formal fibers of it are Cohen-Macaulay. As a consequence of it, we characterize a homomorphic image of a Cohen-Macaulay local ring. For non-local rings, this paper gives only a sufficient condition. By using it, however, we obtain the affirmative answer to Sharp’s conjecture. That is, a Noetherian ring having a dualizing complex is a homomorphic image of a finite-dimensional Gorenstein ring.

This paper completes the classification of Artin-Schelter regular algebras of global dimension three. For algebras generated by elements of degree one this has been achieved by Artin, Schelter, Tate and Van den Bergh. We are therefore concerned with algebras which are not generated in degree one. We show that there exist some exceptional algebras, each of which has geometric data consisting of an elliptic curve together with an automorphism, just as in the case where the algebras are assumed to be generated in degree one. In particular, we study the elliptic algebras A ( + ) A(+) , A ( − ) A(-) , and A ( a ) A({\mathbf {a}}) , where a ∈ P 2 {\mathbf {a}}\in \mathbb {P}^{2} , which were first defined in an earlier paper. We omit a set S ⊂ P 2 S\subset \mathbb {P}^2 consisting of 11 specified points where the algebras A ( a ) A({\mathbf {a}}) become too degenerate to be regular. Theorem. Let A A represent A ( + ) A(+) , A ( − ) A(-) or A ( a ) A({\mathbf {a}}) , where a ∈ P 2 ∖ S {\mathbf {a}} \in \mathbb {P}^2\setminus S . Then A A is an Artin-Schelter regular algebra of global dimension three. Moreover, A A is a Noetherian domain with the same Hilbert series as the (appropriately graded) commutative polynomial ring in three variables. This, combined with our earlier results, completes the classification.

In his notebooks, Ramanujan gave the values of over 100 class invariants which he had calculated. Many had been previously calculated by Heinrich Weber, but approximately half of them had not been heretofore determined. G. N. Watson wrote several papers devoted to the calculation of class invariants, but his methods were not entirely rigorous. Up until the past few years, eighteen of Ramanujan’s class invariants remained to be verified. Five were verified by the authors in a recent paper. For the remaining class invariants, in each case, the associated imaginary quadratic field has class number 8, and moreover there are two classes per genus. The authors devised three methods to calculate these thirteen class invariants. The first depends upon Kronecker’s limit formula, the second employs modular equations, and the third uses class field theory to make Watson’s “empirical method”rigorous.

Let F, G E C[x, y]. If the Jacobian determinant of F and G is 1, then G is said to be a Jacobian mate of F. If, in addition, G has degree less than that of F, then G is said to be a younger mate of F . In this paper, a necessary and sufficient condition is given for a polynomial to have a younger mate. This also gives rise to a formula for the younger mate if it exists. Furthermore, a conjecture concerning the existence of a younger mate is shown to be equivalent to the Jacobian conjecture. Throughout this paper, F and G will be polynomials in C[x, y] where C denotes the field of complex numbers. We say that F and G satisfy the Jacobian hypothesis if their Jacobian determinant is one, i.e., Fx Gy Fy Gx = 1 . In this case, we also say that G is a Jacobian mate of F. Furthermore, if the x-degree (resp. y-degree, total degree) of G is less than that of F, then G is said to be a younger mate of F relative to the x-degree (resp. y-degree, total degree). For instance, x + y has younger mates y and -x relative to the x-degree and the y-degree, respectively, but has no younger mate relative to the total degree. This paper was motivated by the Jacobian conjecture which asserts that if F has a Jacobian mate G, then (F, G) is an automorphism pair. In Section 1, it is shown that a younger mate is unique (up to an additive constant) and universal, i.e., if a Jacobian mate G of F exists, then any other mate of F can be expressed as G plus a polynomial in F. In Section 2, the problem of existence of a younger mate of F is reduced to the case where F is monic in both variables. In Section 3, a necessary and sufficient condition for the existence of a younger mate and a formula for a younger mate provided one exists are given. Finally, in Section 4, a conjecture concerning the existence of younger mates is formulated and shown to be equivalent to the Jacobian conjecture. Received by the editors July 6, 1993 and, in revised form, January 18, 1994. 1991 Mathematics Subject Classification. Primary 13B25, 13F20, 14E09, 16W20.

By using Deodhar's combinatorial setting and Bernstein-Gelfand projective functors, this paper provides some necessary and sufficient conditions for a highest weight category to have a Kazhdan-Lusztig theory. A consequence of these conditions is that in the semisimple Lie algebra case, the Kazhdan-Lusztig conjecture on the multiplicities of a Verma module implies the nonnegativity conjecture on the coefficients of Kazhdan-Lusztig polynomials. One of the central topics in representation theory in recent years is the socalled Kazhdan-Lusztig theory. The Kazhdan-Lusztig polynomials play a key role in this theory. These polynomials can be defined by using a distinguished basis of the Hecke algebra associated to a Coxeter group. In [KL1], there are two conjectures about these polynomials: (a) For any Coxeter group, the coefficients of these polynomials are nonnegative integers; (b) If the Coxeter group is the Weyl group of a complex semisimple Lie algebra, then the multiplicities of the composition series of a Verma module are given by the values of these polynomials at 1. Conjecture (b) is usually referred to as the Kazhdan-Lusztig conjecture and was proven in [BB] and [BK] shortly thereafter. Conjecture (a) is now known to be true for all crystallographic Coxeter groups (for a more upto-date reference on recent developments of Kazhdan-Lusztig theory, we refer to [DS]). It was shown in [D] that if the coefficients of the Kazhdan-Lusztig polynomials of a Coxeter group are nonnegative, then these polynomials can be defined by using certain sets derived from the elements of the Coxeter group. In fact, these sets give a closed formula for the Kazhdan-Lusztig polynomials under the nonnegativity assumption (see [D]). Since the Kazhdan-Lusztig polynomials are not easy to get at in general, the results in [D] give strong evidence for the importance of the nonnegativitiness. In an attempt to understand the results of [D], we observed that in the semisimple Lie algebra case, conjecture (b) implies conjecture (a). The connection is provided by some tensor functors called projective functors defined in [BG]. In this paper, we will give some necessary and sufficient conditions for the validity of the Kazhdan-Lusztig conjecture in certain special cases of the highest weight categories defined by CPS (see [CPS1] Received by the editors June 2, 1993; the contents of this paper have been presented to the Nineteenth Holiday Symposium held in December 1992 at New Mexico State University. 1991 Mathematics Subject Classification. Primary 22E47, 17B10; Secondary 22E46, 17B35.

It is well known that a generic compact Riemann surface of genus greater than two admits only the identity automorphism; however, examples of such Riemann surfaces with their defining algebraic equations have not appeared in the literature. In this paper we give the defining equations of a doubly infinite, two-parameter family of projective curves (Riemann surfaces if defined over the complex numbers), whose members admit only the identity automorphism. It is well known that a generic curve of genus greater than two admits only the identity automorphism. Although this result was probably known by the turn of the century, the first published proof was given by Bailey in 1961 [1]. To obtain the strongest results, Bailey's method is necessarily nonconstructive; it does not yield an example of a defining algebraic equation for a curve with no nontrivial automorphisms. Similarly a proof by Greenberg [4], using techniques of Teichmuller theory, does not yield an explicit example of a Riemann surface with a trivial automorphism group. Much of the subsequent work on automorphisms of Riemann surfaces, including the author's, has relied on the representation of a given Riemann surface as the upper half plane under the action of a Fuchsian group. This again has the disadvantage of rarely yielding a defining algebraic equation for the given Riemann surface. Indeed, in the preface to his book, The complex analytic theory of Teichmiiller spaces, Subhashis Nag exclaimed, "Almost every compact Riemann surface of genus g > 3 allows only the identity automorphism. (I don't know, though, of even a single explicit such algebraic curve whose automorphism group is demonstrably trivial!)." The author believes that examples pertinent to famous theorems should be readily at hand. Therefore, in this paper we give the defining equations of a doubly infinite, two-parameter family of curves which admit only the identity automorphism (see equation (1) below). The curves in this family have genus (n 1)(m 1)/2 for relatively prime integers m and n which satisfy n > m+ 1 > 3. Let C be a curve defined by (1) and let C' be a nonsingular projective model for C. The proof that C' admits only the trivial automorphism depends on the Received by the editors February 5, 1993. 1991 Mathematics Subject Classification. Primary 14E09, 14H55, 30F99. @ 1994 American Mathematical Society 0002-9939/94 $1.00 + $.25 per page

The aim of this paper is to present an example of a bounded balanced domain of holomorphy in C" (n > 3) with continuous Minkowski function that is not Kobayashi-finitely-compact. INTRODUCTION It is known [6] that if G c C" is a bounded Reinhardt-domain of holomorphy with 0 C G then G is finitely-compact with respect to (w.r.t.) the Carath6odorydistance cG, i.e., all cG-balls are relatively compact subsets of G w.r.t. the usual topology. In a more general case, if G = Gh = {z C Cn: h(z) < 1} is a bounded balanced domain of holomorphy with continuous Minkowski function h, then G is finitely-compact w.r.t. the Bergman-distance bG [4]. On the other hand, the continuity of h is a necessary condition for G = Gh to be finitely-compact w.r.t. the Kobayashi-distance kG [5, 1]. In this paper we give an example of a bounded balanced domain of holomorphy G = Gh C C3 with continuous h that is not kG-finitely compact and therefore, not cG-finitely compact. This answers a question formulated by J. Siciak in [7]. In particular, the example shows that, in general, there is no comparison of type bG < CkG for bounded balanced domains of holomorphy with continuous Minkowski function. DEFINITIONS AND STATEMENT We repeat some of the notions that will be needed in the sequel. Definition. A domain G c Cn is called balanced' iff whenever z C G and E C, 11?

The purpose of this paper is to introduce and investigate cardinal functions called pseudonet weight, weak net weight, and weak pseudonet weight. These are similar to but generally smaller than net weight. We look at how these cardinal functions relate to hereditary Lindelof degree, hereditary density, and spread, and we study their behavior under products. An important and useful cardinal function for a topological space is that of weight, namely, the minimum cardinal for a base of open sets. Net weight is similar to weight, except that "base" members need not be open. In this paper we look at three cardinal functions which are slight variations of net weight. Throughout this paper, k denotes an infinite cardinal number, and for sim- plicity, all cardinal functions will be infinite. The smallest (infinite) cardinal number k such that X is hereditarily /c-Lindelof (hereditarily «r-separable, resp.) is denoted by hl(X) ihdiX), resp.). The spread of X (equivalently, hereditary Souslin number) is denoted by siX). As usual, wiX) denotes the weight of X. The set of all real numbers is denoted by R. For notation and terminology not defined here, see (1). In § 1 we generalize the notion of nets (also called networks) by introducing «r-pseudonets. Some examples are given of (nonregular) Hausdorff spaces which have K-pseudonets but have no nets of cardinality < k . Theorem 1.9 shows that pseudonet weight coincides with net weight in regular spaces. We also examine weak net weight and weak pseudonet weight. In terms of definition, weak net weight is to net weight as weak pseudonet weight is to pseudonet weight. The main theorem in §2 is Theorem 2.3, which shows that the Cartesian product X x Y of a hereditarily K-Lindelof space X and a space F having a k-pseudonet is hereditarily «r-Lindelof. 1. Definitions, examples, and elementary relationships

A counterexample to a compactness embedding result of Nagy is provided. Let V and H be real separable Hilbert spaces with V densely and continuously embedded in H. Identifying H with its dual we write V c H c V' algebraically and topologically, where V' is the dual space to V. Given T > 0, let Y = L2(0, T; V), * = L2(0, T; H) and 7' = L2(0, T; V') denote the spaces of the square summable functions defined on the interval (0, T) with values in V, H and V', respectively. We define 2 = {v E 7: v' E 7'},whichwiththeusualnorm Ilullr = (I1uII2+I1u'II2,)1/2 is a Hilbert space (cf. [7], Theorem 25.4). The following remarks about the space 2f are in order. In the definition the derivative u' = du is understood in the dt weak sense. Nevertheless, when we view u as a V'-valued function, we know (see [8], Proposition 23.23) that it is absolutely continuous, so its derivative exists in the strong sense almost everywhere on (0, T). It is known (see, e.g., [7], Theorem 25.5) that the embedding (1) 7C C(0 T; H) is continuous. Here C(0, T; H) denotes the space of continuous functions from [0, T] into H, endowed with the supremum norm. Therefore every function in /l can be, after modification on the set of measure zero, considered as an element of C(0, T; H). We also know that 2 c X compactly (see [8]). The following is the main result of Nagy (see [3], Theorem 2). Theorem. Let V and H be infinite-dimensional separable Hilbert spaces such that the embedding V c H is dense, continuous and compact. Then the embedding (1) is also compact. The aim of this paper is to exhibit a simple example which shows that the embedding (1) cannot be compact, i.e., the above theorem is not true. It should also be noted here that this theorem was exploited in several recent papers in Received by the editors December 1, 1993. 1991 Mathematics Subject Classification. Primary 46E35.

Let $X$ be a $d\times d$ symmetric random matrix with independent but non-identically distributed Gaussian entries. It has been conjectured by Lata��{a} that the spectral norm of $X$ is always of the same order as the largest Euclidean norm of its rows. A positive resolution of this conjecture would provide a sharp understanding of the probabilistic mechanisms that control the spectral norm of inhomogeneous Gaussian random matrices. This paper establishes the conjecture up to a dimensional factor of order $\sqrt{\log\log d}$. Moreover, dimension-free bounds are developed that are optimal to leading order and that establish the conjecture in special cases. The proofs of these results shed significant light on the geometry of the underlying Gaussian processes., 18 pages, 1 figure; final version, to appear in Trans. Amer. Math. Soc

In this paper, we proved that for generic Tonelli Lagrangian, there always exists a residual set $\mathcal{G}\subset H^1(M,\mathbb{R})$ such that \[ \widetilde{\mathcal{M}}(c)=\widetilde{\mathcal{A}}(c)=\widetilde{\mathcal{N}}(c),\quad \forall c\in\mathcal{G} \] with $\widetilde{\mathcal{M}}(c)$ supports on a uniquely ergodic measure., Comment: 8 pages, 1 figure

Denote by B2(G) the Herz-Schur multiplier algebra of a locally compact group G and by B2x(G) the closure of the Fourier algebra in the topology of pointwise convergence boundedly in the norm of B2(G). G is said to be weakly amenable if B2X(G) = B2(G). We show that every amal- gamated product of a countable collection of locally compact amenable groups over a compact open subgroup is weakly amenable. This improves and extends previous results that hold for amalgams of compact groups. Let G be a locally compact group, A(G) its Fourier algebra and B(G) its Fourier-Stieltjes algebra. Denote by Bx(G) the closure of A(G) in the topol- ogy of uniform convergence on compact sets, boundedly in norm. Then G is amenable if and only if B^(G) = B(G). Moreover, if G is amenable, then B(G)=JtA{G) (the algebra of multipliers of A(G)). Recent papers on amenability have devoted a significant amount of attention to the Herz-Schur multiplier algebra B2(G) (He). It was noted in (BF) that B2(G) coincides with the algebra Jf0A(G) of completely bounded multipliers of A(G), introduced in (dCH), where many of its interesting properties were investigated. Amenability can be characterized in terms of B2(G) as follows. G is amenable if and only if B(G) = B2(G) ((Lo); for discrete groups, (Bo)). Ac- tually, for the purpose of studying amenability, the multiplier algebra B2(G) is better suited than J?A , because it has nicer functorial properties. For instance, the Herz-Schur multiplier constant AG (i.e., the infimum of the norms of all approximate identities in B2(G)) is a von Neumann algebra invariant ((Ha2); see (CoH) for a deep study of this invariant on simple Lie groups of real rank 1), whereas a similar statement does not hold for J(A. Above all, B2(G) is nicer in taking products. In fact, B2(GX)®B2(G2) c B2(GX x G2) (dCH), but the same statement does not hold for JfA . This paper makes use of B2(G) to study a property related to amalgamation for another type of products: free products with amalgamation. This property, studied in §2, can be naturally phrased in

We define a Cantor set C in R3 to be scrawny if for each p E C and each e > 0 there is a 3 > 0 such that for each map f: S1 -Int B(p , a)C there is a map F: D2 -IntB(p, e) such that FIaD2 = f and F-1(C) is finite. We show the existence and explore some of the properties of wild scrawny Cantor sets in R3. We prove, among other things, that wild scrawny Cantor sets in R3 are not definable by solid tori. 0. INTRODUCTION: HISTORICAL BACKGROUND This paper concerns the existence and properties of a class of wild Cantor sets in R3. Any perfect, uncountable, zero-dimensional compact metric space is called a Cantor set, and any two Cantor sets are topologically equivalent. Since the publication of Antoine's Necklace in 1921 [A], it has been known that there are inequivalent embeddings of Cantor sets in R3; that is, there are Cantor sets C1 and C2 in R3 such that, for any homeomorphism h: R3 -R3 h(C1) e h(C2). Definition. Let C be a Cantor set in R3. If there is a homeomorphism h: R3 R3 such that h(C) lies on a straight line then C is called tame. Otherwise, C is wild. Definition. Let C be a Cantor set in R3. Let {14'jn E N} be a sequence of finite collections On = {Mn ,kI < k < m(n)} of disjoint connected PL 3-manifolds-with-boundary Mn k C R3 such that (1) for each positive integer UAn+I C Int Uad; (2) A{Wu ln E N} = C. Then {J } is called a defining sequence for C. The following result is well known. Received by the editors May 30, 1990. 1991 Mathematics Subject Classification. Primary 57M30, 57M35. The contents of this paper were presented at the Fifth International Topology Conference in Dubrovnik, Yugoslavia, in June, 1990, sponsored by the Union of Mathematicians, Physicists, and Astronomers of Yugoslavia. A travel grant from the National Science Foundation and the Association for Women in Mathematics enabled the author to attend the conference. (?) 1992 American Mathematical Society 0002-9939/92 $1.00 + $.25 per page

One of the principal results of the paper is that each scalar-type spectral operator in the quasicomplete locally convex space X X is reflexive. The paper also studies in detail the relation between the theory of equicontinuous spectral measures in locally convex spaces and the order properties of equicontinuous Bade complete Boolean algebras of projections.

We present a new free resolution for k as an G-module, where G is an associative augmented algebra over a field k. The resolution reflects the combinatorial properties of G. Introduction. Let k be a field and let G be an associative augmented k-algebra. For many purposes one wishes to have a projective resolution of k as a G-module. The bar resolution is always easy to define, but it is often too large to use in practice. At the other extreme, minimal resolutions may exist, but they are often hard to write down in a way that is amenable to calculations. The main theorem of this paper presents a compromise resolution. Though rarely minimal, it is small enough to offer some bounds but explicit enough to facilitate calculations. As it relies heavily upon combinatorial constructions, it is best suited for analyzing otherwise tricky algebras given via generators and relations. Since several results we get as consequences of the main theorem have been obtained before through other means, this paper may be viewed as generalizing and unifying several seemingly unrelated ideas. In particular, we are generalizing Priddy's results on Koszul algebras [12], extending homology computations by Govorov [9] and Backelin [3], and complementing Bergman's methods regarding the diamond lemma [6]. Three results may be of interest. The homology of the mod p Steenrod algebra is given in terms of the homology of a new chain complex smaller than the A-algebra in Theorem 3.5. Formula (16) offers an efficient algorithm for the determination of Hilbert series, and Theorem 4.2 asserts the existence of new bounds on the torsion groups of commutative graded rings. 1. Definitions and the main theorem. Throughout this paper, k denotes any field and G is an associative k-algebra with unity. The field k embeds in G via 77: k -*G and we suppose that G has an augmentation, i.e., a k-algebra map s: G -? k for which 77 is a right inverse. S denotes a set of generators for G as a k-algebra and k(S) is the free associative k-algebra with unity on S. There is a canonical surjection f: k(S) -? G once S is chosen, and the augmentation E is determined once we know s(x) for each x E S. In particular, this means that k(S) may be augmented such that f becomes a map of augmented algebras. To S we associate a function e: S -? Z+ called a grading. In the absence of a more compelling choice we often take e to be grading by length, i.e., e(x) = 1 for Received by the editors May 23, 1983 and, in revised form, February 22, 1984. This paper was the subject of an invited one-hour address in Boulder, Colorado, during the week-long AMS summer program on Combinatorics and Algebra, June 1983. 1980 Mathematics Subject Classification (1985 Revision). Primary 16A62; Secondary 13D03, 55S10. (?)1986 American Mathematical Society 0002-9947/86 $1.00 + $.25 per page

A set-theoretic lemma is introduced and various applications are given, including: (1) a result of Erdos and Hajnal on the coloring number of a graph; (2) game characterizations of the coloring number of a graph; (3) K. Alster's result that a point-countable collection of open, compact scattered spaces has a point-finite clopen refinement; (4) normal, locally compact, metacompact spaces which are scattered of finite height are paracompact. 1. Introduction. In this paper, a purely set-theoretic lemma is presented which appears to be the "common part" of K. Alster's result (A) that a point-countable collection of compact scattered spaces has a point-finite clopen (open and closed) refinement, and a result of Erdos and Hajnal (EH) on the coloring number of a graph. The author has found this lemma to be quite useful (see (G2j, for example). In this paper, we will show how the lemma can be used to prove the above results, and we will also give some other applications. One such application is a "game characterization" of the coloring number of a graph, from which the Erdos-Hajnal result easily follows. Other applications are partial answers to the question of whether metalindelkf spaces are preserved by closed maps, and Tall's question of whether normal, locally compact, metacompact spaces are paracompact. 2. The lemma. In this section, we state and prove the main lemma. In the lemma, 2M is the set of all subsets of M, and Aa / A means that, for some ordinal K

This paper introduces the notion of weak subordination: If F F and G G are meromorphic in the unit disc U \mathcal {U} , then F F is weakly subordinate to G G , written F > G F > G , provided there exist analytic functions ϕ \phi and ω : U → U \omega :\mathcal {U} \to \mathcal {U} , with ϕ \phi an inner function, so that F ∘ ϕ = G ∘ ω F \circ \phi = G \circ \omega . A class X \mathcal {X} of meromorphic functions is termed stable if F ≺ w G F \stackrel {w}{\prec } G and G ∈ X ⇒ F ∈ X G \in \mathcal {X} \Rightarrow F \in \mathcal {X} . The motivation is recent work of Burkholder which relates the growth of a function with its range and boundary values. Assume F F and G G are meromorphic and G G has nontangential limits, a.e. Assume further that F ( U ) ∩ G ( U ) ≠ ∅ F(\mathcal {U}) \cap G(\mathcal {U}) \ne \emptyset and G ( e i θ ) ∉ F ( U ) G({e^{i\theta }}) \notin F(\mathcal {U}) , a.e. This is denoted by F > G F > G . Burkholder proved for several classes X \mathcal {X} that ( ( ∗ ) ) F > G and G ∈ X ⇒ F ∈ X . \begin{equation}\tag {$(\ast )$}F > G \qquad {\text {and}}\quad G \in \mathcal {X} \Rightarrow F \in \mathcal {X}.\end{equation} The main result of this paper is the Theorem: F > G ⇒ F ≺ w G F > G \Rightarrow F{ \prec ^w}G . In particular, implication (*) holds for all stable classes X \mathcal {X} . The paper goes on to study various stable classes, which include BMOA, H p {H^p} , 0 > p ⩽ ∞ 0 > p \leqslant \infty , N ∗ {N_{\ast }} , the space of functions of bounded characteristic, and the M Φ {M^\Phi } spaces introduced by Burkholder. VMOA and the Bloch functions are examples of classes which are not stable.

The strong lower limit and the weak upper limit of a net of von Neumann subalgebras on which the conditional expectations exist with respect to a fixed faithful normal state are defined. The limits coincide if and only if the corresponding conditional expectations converge strongly. 1. Preliminaries. Let M be a a-finite von Neumann algebra and (p a faithful normal state on M. By the GNS construction it can be considered that M is acting on a Hilbert space H and there exists a cyclic separating vector 'D E= H with (p(x) = (D I xD) for every x E M. Denote by M* the space of all a-weakly continuous linear functionals on M. That is, M* is the predual of M. For a von Neumann subalgebra N of M, if there exists a projection 8 of norm one from M onto N with (p o 8 = (p ? is called the conditional expectation onto N [3, 6]. 1?. The conditional expectation onto N exists if and only if at(N) = N for every t E R, where { a, } is the modular automorphism group on M with respect to (P. 2?. If the conditional expectation 8 onto N exists, then ?(x)?D = PxD for every x E M, where P is the orthogonal projection of H onto ND. Throughout this paper we fix a net { Na } of von Neumann subalgebras of M and assume that the conditional expectation 8a onto Na exists for each a. The orthogonal projection of H onto Ha = NaD is denoted by Pa. In the recent paper [5] we proved that if { Na } is increasing (resp. decreasing), then the conditional expectation 8 onto VaNa (resp. laNa) exists and 8a(X) --8(x) strongly for every x E M and f 0 -?a f ? E, in norm for everyf E M*. In this paper we shall introduce the notion of the strong limit of { Na } and show that the limit exists if and only if the corresponding {fea } converge strongly. The following are elementary but will be useful below. 30. For any uniformly bounded net { xy } in M and x E M, xy -x strongly (resp. weakly) if and only if x yxD strongly (resp. weakly) in H. 40. Let { Py } be a net of orthogonal projections of H, and P an orthogonal projection of H. For any t E H, if -y* Pt weakly, then it does strongly. Received by the editors January 23, 1984 and, in revised form, July 16, 1984. 1980 Mathena(dtics Subject Classification . Primary 46L10.

In this paper a necessary and sufficient condition for a composition operator CT on L2[0, 1] to be a Fredholm operator is given. In addition, all Fredholm composition operators on 12(N) are characterized. 1. Preliminaries. Let (X, S, X) be a a-finite measure space and T be a measurable nonsingular (AT-'(E) = 0 whenever X(E) = 0) transformation from X into itself. Then a composition transformation CT on L2(X, S, X) is defined as CTJ = f o T for everyf EL2(X, ,X ). In case CT is a bounded operator with range in L2(X, , A), we call it a composition operator induced by T. The main purpose of this paper is to study Fredhohm composition operators on L2(X, S, X) (briefly written as L2(X)), where X is the unit interval, S is the a-algebra of all Borel subsets of X, and X is the Lebesgue measure on S. A criterion for a composition operator to be Fredholm on 12(N) is also given here. Let B(L2(X)), R(CT)1 and [x,y, z,... ] denote the Banach algebra of all bounded linear operators on L2(X), the orthogonal complement of the range of CT and the closed linear span of the vectors x, y, z, . . . respectively. DEFINITION. An operator A on a Hilbert space H is called a Fredholm operator if the range of A is closed and if the dimensions of the kernel and the cokernel are finite. 2. Fredholm composition opertors. A characterization of Fredholm composition operators on H2(D) is given by J. Cima, J. Thomson and W. Wogen in [2] where they proved that a composition operator CT is Fredholm if and only if T is a conformal automorphism of the disc. The following theorem gives an analogous characterization of Fredholm composition operators on L2(X) = L2[0, 1]. THEOREM 1. Let CT E B(L2(X)). Then CT is a Fredholm operator if and only if it is invertible. PROOF. If CT is invertible, then clearly CT is a Fredholm operator. It is known from [5, p. 82] that CTCT = Mf0, where Mf is the multiplication operator induced by fo = dAT-/dA. Since X is nonatomic [3, pp. 171-174] and Ker CT = Ker CTCT = Ker Mf= L2(Xo), where X0 = {x: f0(x) = 0), it follows that the dimension of the kernel of CT of either zero or infinite. Received by the editors May 3, 1977 and, in revised form, June 18, 1979. AMS (MOS) subject classifications (1970). Primary 47B30; Secondary 47B30.

The primary purpose of this paper is to show that every valuation on the field of meromorphic functions of bounded type on a finitely sheeted unlimited covering Riemann surface is a point valuation if and only if the same is true on its base Riemann surface. The result is then applied to concrete examples and some related results are obtained. Any valuation on the field M(W) of single valued meromorphic functions on a Riemann surface W is a point valuation [9]. What happens to valuations on subfields of M(W)? An especially interesting subfield in this context is the field M??(W) of meromorphic functions of bounded type on W (cf. [2]). We are thus concerned with the following question in this paper: When is it true that any valuation on nontrivial M??(W) is a point valuation? The paper consists of five parts. ?1 covers preliminaries. The main part, ?2, concerns the covering stability. We say that a Riemann surface W is stable if M??(W) is nontrivial and any valuation on M??(W) is a point valuation. Then it is shown in this section as the main theorem of this paper that a finitely sheeted unlimited covering surface R of a Riemann surface S is stable if and only if S is stable. In ?3 on stable surfaces, an example due to Forelli [4] of stable surfaces is given among others. In ?4 the relation between H?-maximality and stability of a Riemann surface W and in particular of a plane region W is discussed. We relax the definition of stability in ?5 to obtain the notion of weak stability of a Riemann surface W. Here a simple but powerful device of what we call H?-barrier is introduced, which is used to exhibit a weakly stable plane region of infinite connectivity. 1. Preliminaries. 1.1. Fields F we consider in this paper are all assumed to be extensions of the complex number field C. We denote by F* the multiplicative group consisting of nonzero elements in F, i.e. F* = F\ {O}. By a valuation v on F we mean a discrete valuation v on F, i.e. a group homomorphism of the multiplicative group F* into the additive group Z of integers such that (1.1) v(f + g) > min(v(f), v(g)) (f, g E F*) where we make the convention that v(O) = +oo. Received by the editors June 16, 1987. 1980 Mathematics Subject Classification. Primary 30H50; Secondary 30F99, 30D50, 46J15. To complete this work the author was supported in part by Grant-in-Aid for Scientific Research, No. 61540094, Japanese Ministry of Education, Science and Culture. ?1988 American Mathematical Society 0002-9947/88 $1.00 + $.25 per page

A Galois correspondence between cylindric set algebras and permutation groups is presented in this paper. Moreover, the Galois connection is used to help establish two important algebraic properties for certain classes of finite-dimensional cylindric algebras, namely the amalgamation property and the property that epimorphisms are surjective. The importance of the amalgamation property (AP for short) in algebraic logic has been recognized for a long time. In [16] the connection with the interpolation property of first-order logic is discussed. The positive amalgamation results from the author's thesis [2] and their extensions announced in [31 are cited in ?2.4 of [16]. These results are established, with a slight improvement, in ?4 below. The key to this study is a lattice anti-isomorphism between subgroups of the symmetric group S, and subalgebras of the full set algebra % (a, ,t) when 0 ,u > a + 1. In particular, this applies to Theorems 2.2, 2.8, 2.9, 3.7, 3.8(2), Corollaries 3.10, 5.5 and Lemma 5.2(2). Essentially the only result where the value of ,u is not known to be the best possible is in Theorem 2.5. Received by the editors July 6, 1983. Some of the contents of this paper were presented at the 74th Annual meeting of the American Mathematical Society in San Francisco, California, January 23-27, 1968. 1980 Mathematics Subject Classification. Primary 03G15; Secondary 06A15, 08B25.

The countable box product of ordinals is examined in the paper for normality and paracompactness. The continuum hypothesis is used to prove that the box product of countably many a-compact ordinals is paracompact and that the box product of another class of ordinals is normal. A third class trivially has a nonnormal product. Because I have found a countable box product of ordinals useful in the past [1], this class of spaces particularly interests me. The purpose of this paper is to tell what I know about which of these spaces is paracompact or normal. In [2] I prove that the continuum hypothesis implies the box product of countably many a-compact, locally compact, metric spaces is paracompact. I prove here that the continuum hypothesis implies the box product of countably many a-compact ordinals is paracompact (Theorem 1) and the box product of another class of ordinals is normal (Theorem 2). The proof of Theorems 1 and 2 is a quite messy join of the techniques of [1] and [2] which raises some doubt in my mind as to whether these theorems are worth proving. Because I care, because I think these spaces are set theoretically interesting and topologically useful, because I think these theorems are best possible, the theorems are worth the mess to me. A. If {XXAOA)\ is a family of topological spaces, a box in HIAeA XA is a set TheA UA where each UA is open in XA. The box product of {XAIA A iS HAA XA topologized by using the set of all boxes in it as a basis. Throughout the paper the following notation is used. An ordinal a is the set of all ordinals less than a and a is topologized by the interval topology. The statement that a is a cardinal means that a is an ordinal and no smaller ordinal has the same cardinality as a. The notation IAnA BAA is used to mean the ordinary Cartesian product of the f,3's and never the cardinal or ordinal arithmetic product. Similarly a#9 means the set of all functions from ,B into a rather than an arithmetic operation. If a is an ordinal, let cf(a) denote the cofinality of a; that is cf(a) is the smallest ordinal 8 such that there is a subset A of a, order isomorphic with 8, such that /3 < a implies there is a y El A with /3 < y. Observe that a is a a-compact ordinal if and only if a is compact or cf(a) =w. Received by the editors December 10, 1971 and, in revised form, October 1, 1972. AMS (MOS) subject classiflcations (1970). Primary 54B10, 54A25, 54D15, 54D20, 54D30, 02K25.

In this paper it is shown that every nondiscrete LCA group contains a compact independent Helson set which disobeys spectral synthesis. In [3] T. W. Korner has constructed an independent compact H 1-set of type M. This result solves negatively the long-standing problem whether or not every Helson set obeys spectral synthesis. K6rner's construction of such a set is, however, very complicated, and R. Kaufman [2] has simplified it (see also [6]. In this paper, we modify Kaufman's method to prove that every nondiscrete metrizable LCA group contains an independent compact Helson set of type M. Consequently it is shown that every nondiscrete LCA group contains a Helson set which disobeys spectral synthesis. Let G be a LCA group with dual G. We denote by A(G) and PM(G) the Fourier algebra on G and the conjugate space of A(G), respectively. Each element of PM(G) is called a pseudo-measure on G. For / e A(G) and P e PM(G), we define (/, p) =( * Pi)(1)= f X (x)P(X )dx, where f and P denote the functions in L (G) and in L(G) whose (inverse) Fourier transforms are / and P, respectively. If t e M(G) and -(x) = ftX(x) dji(X) (x e G), then we denote by jP the pseudo-measure on G defined by the requirement (WP)= ,u * P. It is well known that if P e PM(G) has compact support, then P can be chosen from the space C(G); we will always do this. For such a P, (-, P) = (f * P)(1) (G e M(G)) is well defined, and we have (X, yP) = P(Xly 1j (XX y e G). A pseudo-function on G is any pseudo-measure whose Fourier transform is a continuous function on G which vanishes at infinity. The space of all pseudo-functions is denoted by PF(G). Received by the editors July 25, 1973. AMS (MOS) subject classifications (1970). Primary 43A45; Secondary 43A20.

In this paper we determine all the weakly compact derivations of a nest algebra. We also obtain necessary and sufficient conditions in order that a nest algebra admit compact derivations. Finally we prove that every compact derivation of a nest algebra d is the norm limit of finite-rank derivations. 1. Let _ be a Banach algebra and let X be a Banach s-module. By an X-valued derivation of _ we mean a linear mapping 8: sV-X with the property 8(ab) = a8(b) + 8(a)b for all a E :, b E .@. The derivation 8 is called compact if 8 is a compact operator between the Banach space s and X, and weakly compact if 8 is a weakly compact operator from s to X (i.e. 8(s) is relatively weakly compact in X, where SV is the unit ball of sV [4]). Let H be a complex Hilbert space, B(H) the algebra of all bounded operators on H and Y(H) = X the set of compact operators on H. In [7] Johnson and Parrott investigated derivations of a von Neumann subalgebra of B(H) with range contained in Y. They proved that in most cases such derivations are implemented by a compact operator. The general result was recently obtained by Popa [8] who proved that this is the case for all von Neumann subalgebras of B(H). Such derivations are known to be weakly compact [1]. On the other hand, in a series of papers [1, 9, 10], C. A. Akemann, S. K. Tsui, and S. Wright have determined the structure of all compact and weakly compact s-valued derivations of a C*-algebra s, and of all compact B(H) valued derivations of a C *-subalgebra of B(H). In this note we determine the structure of all s-valued compact and weakly compact derivations of a nest algebra S. In particular we prove that every compact derivation of a nest algebra s is the norm limit of finite rank derivations. We need the following result. LEMMA 1 [6]. Let H be an infinite dimensional Hilbert space. If 8 is a compact derivation of B(H) then 8 0. Received by the editors January 16, 1984 and, in revised form, July 1, 1985. 1980 Mathenmatics Subject Classification. Primary 47D25; Secondary 47B05.

The element a is in the center of the Banach algebra A modulo its radical if and only if there is an upper bound for the diameters of the spectra of a -ta-I for u invertible. Applications of this result are given to general Banach algebras and to the essential spectrum of operators on a Hilbert Space. In this paper we relate the diameters of the spectra of elements of a Banach algebra A to commutativity properties of A and to the Jacobson radical, rad(A). For a in A we let 8(a) be the diameter of the spectrum of A, and p(a) be the spectral radius of A. Also, we let Z(A) be the center of A modulo its radical; that is, a belongs to Z(A) if and only if ar ra E rad(A) for all r in A. For the algebras that usually occur in applications, like C*-algebras, group algebras, the Calkin algebra, etc., the radical is (O}, so that Z(A) is actually the center of A. Our results about the spectral diameter are similar to results of Aupetit [1] and Zemanek [10] about the spectral radius, and in fact, are generalizations of these results (see Corollary (1.3) below). Our main result, Theorem (1.1), gives a technical characterization of Z(A). In ?2, we obtain a variety of consequences of this technical characterization. For instance, we show in Corollary (2.2), that if the Hilbert space operator T is not the sum of a compact operator and a scalar, then there is a quasinilpotent operator Q for which T + Q has more than one point in its essential spectrum. 1. The basic result. Essentially all the results in this paper are easy consequences of the following Theorem. THEOREM (1.1). Suppose that A is a Banach algebra with identity and that a belongs to A. Then a belongs to Z(A), the center of A modulo its Jacobson radical, if and only if sup 8(a uau) < oo, where the sup is taken over the invertible elements u of A. Notice that if a belongs to Z(A), then a uau-' E rad(A) for all invertible u, so that 8(a uau-') is always 0. In fact, for all our results characterizing commutativity modulo the radical, the only nontrivial part of the characterization is to show that the given property implies commutativity modulo the radical. Received by the editors February 22, 1983. The contents of this paper were presented to the annual meeting in Denver, January, 1983. 1980 Mathemnatics Subject Classificcation. Primary 46H05; Secondary 47A10. Ke! vi'ords and phrases. Spectral diameter, radical, center, commutativity. 'Partially supported by NSF Grant MCS-8002923. CN1984 American Mathematical Society 0002-9939/84 $1.00 + $.25 per page

Using a technical integral inequality, J. Moser proved a sharp result on exponential integrability of a certain space of Sobolev functions. In this paper, we show that the integral inequality holds in a general setting using nonincreasing functions and a certain class of convex functions. We then apply the integral inequality to extend the above result by J. Moser to other spaces of Sobolev functions. A second application is given generalizing some different results by M. Jodeit. 1. The integral inequality (Theorem 3) presented in this paper has been of interest ever since a simple version of it was proven in 1971 by J. Moser. Subsequent improvements were made by M. Jodeit in 1972, B. F. Jones in 1979 and C. J. Neugebauerin 1980. Let 1 ^ 0 be locally integrable, and define t(x)=\(X ?(y)dy\/P and F{x) = ff(y)$(y) dy. /o J •'o Let $ be a nonincreasing function on [ 0, oo). In this paper we investigate for which real-valued functions A(x) we get the inequality / 4>{N[ + {x)] -N[F(x)]} dN[4,(x)] and / if and only if N(x) is a certain type of convex function called a C*-convex function. Finally in §§5 and 6 we improve upon the applications given by J. Moser and M. Jodeit, respectively. 2. Definition 1. A continuous function p: [ 0, oo) -» [ 0, oo) will be called a C*-function provided there exists a constant Cp C(d) and 0 < s < oo. Received by the editors March 26, 1982 and, in revised form, March 21, 1983. 1980 Mathematics Subject Classification. Primary 42A96. ©1984 American Mathematical Society 0025-5726/84 $1.00 + $.25 per page 157 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use

In this paper several results on analytic diameters and analytic centers are obtained. We show that the extremal function for analytic diameter is unique and that there exist compact sets with many analytic centers. We answer negatively several problems posed by F. Miinsker. Introduction. In the theory of functions in the complex plane, many capacitytype set functions are defined in order to measure the size of sets. The notions of analytic diameter and analytic center were first introduced by Vitushkin [13], [14] in the way of developing his beautiful theory of rational approximation. Recently F. Minsker [9] obtained their several properties and posed in the last part of his paper [9, p. 93] open problems on them. In this paper we investigate certain properties of the extremal function for the problem of analytic diameters and construct instructive examples, most of which offer negative answers to Minsker's problems. In ? 1 we list the definitions and the notation which we use throughout this paper. In ?2 we state preliminary known results as a series of lemmas. In ?3 we derive a property of the extremal function, and as a corollary we show that the extremal function is unique. In ?4 we consider the case where the compact set consists of finitely many continua. In ?5 we give examples, most of which serve as counterexamples to Minsker's problems. 1. Definition and notation. Let K be a compact set in the complex plane, and let D = D(K) denote the unbounded component of the complement KC of K with respect to the Riemann sphere. By AB(D) we denote the Banach space of all bounded analytic functions in D, normed by the uniform norm 1 in D, that is 0110 = sup{ff(z)1: z E D} (1) for anyf E AB(D). Let eg(K) = {f E AB(D): IItIOO < l,f(oo) = 0). Any function which belongs to Gi(K) is often called an admissible function for K. The analytic capacity y(K) of K is defined by y(K) = sup{If'(oo)I:f E zie(K)} (2) It is well known that there exists a unique extremal function fo J e6(K) with fo(oo) = y(K), which is called the Ahlfors function of K [1]-[4], [7], [8]. Received by the editors January 11, 1980. A MS (MOS) subject classifications (1970). Primary 30A34, 30A44. ( 1981 American Mathematical Society 0002-9947/81/0000-041 4/$03. 50

An operator is compact relative to a semifinite von Neumann algebra, i.e., belongs to the two-sided closed ideal generated by the finite projections relative to the algebra, if and only if it maps vector sequences converging weakly relative to the algebra into strongly converging ones (generalized Hilbert condition). The generalized Wolf condition characterizes the class of almost Fredhoim operators. Introduction. The elements of the two-sided closed ideal i generated by the projections finite relative to a von Neumann algebra 6, are called compact operators of C, and it has been shown (see [7, 4]) that they satisfy many of the properties of the compact operators on a Hilbert space. The weak convergence of vectors plays an important role in classical operator theory. The aim of this paper is to study a relative weak (RW) convergence that could play an analogous role in the operator theory relative to a semifinite von Neumann algebra. A bounded sequence of vectors x,, is defined to converge RW to x if Px -> Px for every projection P finite relative to ,. This convergence is shown to be not equivalent (apart from trivial cases) to weak or strong convergence. The classical Hilbert characterization is extended to the compact operators of 6C and a generalized Wolf property (see [8]) is used to characterize a new class Ci+ called almost left-Fredholm [5]. We remark that the RW convergence can be used, in analogy with Calkin's construction (see [2]), to obtain a representation of the generalized Calkin algebra C, / i. Finally a RW topology is defined on the unit ball of the predual of 6, and both the generalized Hilbert and Wolf properties are reformulated in a space-free setting. The author wishes to thank M. Sonis, L. Brown and P. Fillmore for valuable suggestions. 1. The relative weak convergence. Let H be a Hilbert space, let L(H) be the algebra of all bounded linear operators on H, let 6i be a semifinite von Neumann algebra on H and P (C) be the set of the projections of (B. Let i be the ideal of compact operators (relative to ei), i.e., the norm closed two-sided ideal of a, generated by the finite projections of ,. It is known that i is proper iff C, is Received by the editors April 28, 1980. Some of the results in this paper were presented to the American Mathematical Society at the 85th Annual Meeting at Biloxi, Mississippi, January 1979. 1980 Mathematics Subject Classification. Primary 46L10, 47C15; Secondary 47A53, 47B05.

n The collection of all closed linear subspaces of H invariant under A (i.e. invariant under every a E A) is denoted by Lat A. A weakly closed algebra A c B(H) is reductive [8] if 1 E A, and Lat A = Lat MA. A linear subspace K c H is paraclosed [4] if there exist a Hilbert space HO and a bounded linear operator Q: Ho+H such that QHo = K. The collection of all paraclosed subspaces of H, invariant under A is denoted Latl/2A. A weakly closed algebra A c B(H) will be called parareductive if 1 E A, and Latl/2A = Latl/2MA. In this paper (Theorem 2.1) we show that if A is a parareductive algebra then A is a von Neumann algebra. In order to prove this result, in' ? 1, we give some new results on paraclosed operators. Finally, in ?3 we prove a result, announced (without proof) in [7]. Since the paper was written a proof of Theorem 2.1 in the separable case has appeared [1]. Our approach covering the possibly nonseparable case is entirely different and in fact simplifies Azoff's proof. I am indebted to the referee for calling my attention to Azoff's paper and for the suggestion that Theorem 2.1 can be formulated in this general form.

We introduce a method for constructing Boolean algebras from trees which preserves some of the trees' properties. The method is used to produce a very simple construction for rigid Boolean algebras and to construct a weakly homogeneous Boolean algebra without homogeneous factors. 0. Introduction. In this paper we present two constructions: a weakly homogeneous Boolean algebra that answers a question of Rubin, and what we feel is the simplest construction of a rigid Boolean algebra. A rigid Boolean algebra is one which has no nonidentity automorphism. We begin our construction of a rigid Boolean algebra by taking a tree T satisfying (a) the height of T is w; (b) T has a single root; (c) for all distinct a, ,3 E T, the number of immediate successors to a is different from the number of immediate successors to ,8; and (d) the number of immediate successors is always regular. We form " wedges". Sa = (/3 E T: a < ,B}, for each a E T, and close the set of wedges under finite unions and complements. The result is our algebra. Many constructions of rigid Boolean, algebras have been given. In [vDMR] there is a short history. van Douwen, Monk and Rubin ask for a "natural" construction of a rigid Boolean algebra. We offer our construction as a candidate. van Douwen, in [vD] has a similar construction. He begins with a tree T satisfying (a) and (c) and with the property that for all a E T, icK, the number of immediate successors to a, satisfies KNO = K.. He then topologizes the tree by taking certain infinite unions of wedges. The result is a rigid 0-dimensional compact space. The algebra of closed-and-open sets is the desired rigid Boolean algebra. A Boolean algebra B is homogeneous if for any nonzero b E B, the set (a E B: a < b} viewed as a Boolean algebra is isomorphic to B. It is weakly homogeneous if for all distinct nonzero a, b E B there exist nonzero b, < b, a, < a such that the algebras consisting of {c: c < b,} and (c: c < a,} are isomorphic. We answer a question of Rubin in [R] by constructing a weakly homogeneous Boolean algebra that has no homogeneous factors. Received by the editors January 27, 1982 and, in revised form, October 25, 1982. 1980 Mathematics Subject Classificatiotn. Primary 06E99. 'These results are contained in Chapter 2 of the author's Ph. D. dissertation, prepared under the direction of J. D. Monk at the University of Colorado. The author expresses his gratitude to Professor Monk. 2This paper is dedicated to Merry Havens. ((1983 American Mathenatical Socicty 0002-9939/82/(XXX)-0385/$02.(X)

In a recent paper F. Treves studied a model of complexes of pseudodifferential operators in an open set of R", establishing necessary and sufficient conditions for its semiglobal solvability. In the present paper, the authors give necessary and sufficient conditions for the global solvability of an analogous complex defined on an orientable, compact smooth manifold without boundary. 1. Introduction and statement of the theorem. Let B be a ^-dimensional (v > 1), compact, connected, orientable C°° manifold without boundary. We will denote by A a linear selfadjoint operator, densely defined in a complex Hilbert space H, which is unbounded, positive and has a bounded invsise^-1. (We may think of A as being, for instance, (1 - Ax)s on R", or a selfadjoint extension of \DX\.) We will use the scale of "Sobolev spaces" Hs (for s E R), defined by A : if s > 0, Hs is the space of elements u of H such that Asu E H, equipped with the norm \\u\\s = M^mIIo. where || ||q denotes the norm in H = H°; if s < 0, Hs is the completion of H for the norm \\u\\s = WA'uWq. The inner product in Hs will be denoted by ( , )s. Whenever s E R, m E R, Am is an isomorphism (for the Hilbert space structures) of Hs onto Hs~m. By H°° we denote the intersection of the spaces Hs, equipped with the projective limit topology, and by H~°° their union, with the inductive limit topology. Since for each s E R, Hs and H~s can be regarded as the dual of each other, so can i/00 and H~°° : with their topologies, they are the strong dual of each other. We denote by C°°(fi; H°°) the space of C°° functions in fi valued in H00. It is the intersection of the spaces C7(s;// ) (of the ^/'-continuously differen- tiable functions defined in fi and valued in H^ as the nonnegative integers y, k, tend to +00. We equip C°°(fi; Hx) with its natural C°° topology. We will denote by

Primary ideals in the ring of all analytic functions on a noncompact Riemann surface are analyzed with the aid of classical valuation theory. Let A be the ring of all analytic functions on a noncompact (connected) Riemann surface X. The ideal theory of A has been studied since 1940. (See [1] for references.) The main piece of analysis needed is the general Weierstrass (product) theorem and the Mittag-Leffler theorem. The finitely generated ideals of A are all principal. Each principal maximal ideal M of A is of the form }/E A: /(x) = 0}, for some xE X. All other maximal ideals are creatures of the axiom of choice. Maximal and prime ideals of A have been well understood for 20 years. Recently R. Douglas Williams [4] analyzed the primary ideals Q of A. While reviewing Williams' paper, the author found it convenient to cast the definitions and proofs into purely valuation theoretic terms, and found that this seemed to make the ideas of Williams much more transparent and elementary. The purpose of this paper is to elaborate and to expose the results the author found before he wrote his review for Math Reviews. I1 is devoted to the analysis of primary ideals in an abstract valuation ring. It is pure valuation theory that could have been done at any time over the last 40 years. Having failed to find these results in the literature, they have been included here. In 42 results of ?1 are applied to the ring A to obtain new proofs of some of Williams' results as well as giving some new results. 1. Primary ideals in a valuation ring. Let B be a (proper) valuation ring in a field K, V its valuation, and G its value group. Given a (commutative) ring R, let 9(R) denote the set of all proper ideals of R, ordered under inclusion, and let IR(R), C(R), and 9(R) be the set of all maximal, prime, and primary ideals of R, respectively. A subset S of G+ -= Ige G: g > 0} will be called an upper interval if sE S and te G+ such that s < t, implies t e S. Let il(G+) be the set of all proper, upper intervals of G+. Received by the editors May 2, 1974 and, in revised form, October 18, 1974. AMS (MOS) subject classifications (1970). Primary 46E25, 30A98; Secondary 13A15.

It is shown that not every Boolean algebra is the Boolean part of a nondiscrete relation or cylindric algebra, but every nonatomless Boolean algebra is. Solutions of Tarski's Cube Problem for nondiscrete relation and cylindric algebras are given. Introduction. Relation algebras (RAs) and cylindric algebras (CAs) are Boolean algebras (BAs) endowed with additional structure. The question naturally arises: Which BAs are Boolean parts (i.e. Boolean reducts) of RAs or CAs? This question can be considered as a representation problem for BAs, too: Which BA can be represented as a BA of some binary relations on a set U closed under composition, inversion, and containing the identity relation on U (this is the RA case); and which BA can be represented as a BA of n-ary relations closed under, roughly, first-order definability, i.e. containing with every relation Rj,..., Rk the relations first-order definable (with n variables) from them, too? RA and CA,a denote the classes of RAs and CAs of dimension a respectively. There is a trivial way of turning every BA into a RA or a CAa; these RAs and CAs are called "Boolean" and "discrete" ones (cf. [7, p. 276 and 5, 1.3.11]). By [5, 1.2.14], every BA is the Boolean part of a nondiscrete CA1. J. D. Monk in 1981 and independently R. Laver in 1984 asked the question: Which BAs are Boolean parts (reducts) of nondiscrete CA ,s, a( > 2? Theorems 1 and 2 give partial answers. To CAa5s there correspond another BA, the so-called zero-dimensional part 3b W of a CA, W. By [5, 2.4.35], every BA is the zero-dimensional part of a hereditarily nondiscrete CA a (for all a > 0). In the last part of the paper we use this fact (together with Ketonen's solution for BAs) to give a full solution of Tarski's Cube Problem for CA,as as well as for RAs (of course we mean nondiscrete CA,as and RAs here). See Theorems 3 and 4. This solves that part of Problem 2.4 of [5] which was still left open in [6, p. 127]. The definitions of RA and CA.a, both originating with Tarski, can be found e.g. in [5, 7, 8, 10] and in [5,6] respectively. Below we recall that part of their definitions that will be needed in the present paper. Received by the editors April 2, 1985 and, in revised form, August 19, 1985 and February 11, 1986. 1980 Mathematics Subject Classification (1985 Revision). Primary 03G05, 03G25; Secondary 03E15, 06E99. C1987 American Mathematical Society 0002-9939/87 $1.00 + $.25 per page

The theory of abstract Hammerstein operators is applied to obtain existence theorems for Urysohn's integral equation. Urysohn's integral equation is of the form (*) u(s) + f 1(s, t, u(t))dt = 0. Usually one assumes that Q is a subset of R', and that V(s, t, u) is a function of three variables s, t E Q, u e R, satisfying the so-called Caratheodory conditions. Urysohn's equation has been discussed by Urysohn [6], Kolomy [4], Krasnosel'skil [5] and others. Attempts have been made to apply the theory of monotone operators to get existence theorems for (*). In this paper we apply the theory of abstract Hammerstein operators to obtain existence theorems for ( *) with rather simple conditions on the function (D. We define a linear operator A: L2(Q x ) -L2(Q x Q) with range in L2(Q) and a nonlinear operator F: L2(Q) L2(Q x Q) as follows: (1) [Au](s) = u(s, t)dt, (2) [Fu](s, t) = F(s, t, u(t)). In all our considerations in this paper, Q will be a set of finite measure in Rn and (3) L 2(Q) = u: IQ u(t)| 2 dt < } (4) L2(Qx Q)={u: fflU(s t)|2dtds

In an earlier paper we studied real laminated lattices (or Z {\mathbf {Z}} -modules) Λ n {\Lambda _n} , where Λ 1 {\Lambda _1} is the lattice of even integers, and Λ n {\Lambda _n} is obtained by stacking layers of a suitable ( n − 1 ) (n - 1) -dimensional lattice Λ n − 1 {\Lambda _{n - 1}} as densely as possible, keeping the same minimal norm. The present paper considers two generalizations: (a) complex and quaternionic lattices, obtained by replacing Z {\mathbf {Z}} -module by J J -module, where J J may be the Eisenstein, Gaussian or Hurwitzian integers, etc., and (b) integral laminated lattices, in which Λ n {\Lambda _n} is required to be an integral lattice with the prescribed minimal norm. This enables us to give a partial answer to a question of J. G. Thompson on integral lattices, and to explain some of the computer-generated results of Plesken and Pohst on this problem. Also a number of familiar lattices now arise in a canonical way. For example the Coxeter-Todd lattice is the 6 6 -dimensional integral laminated lattice over Z [ ω ] {\mathbf {Z}}[ \omega ] of minimal norm 2 2 . The paper includes tables of the best real integral lattices in up to 24 24 dimensions.

Given the Fourier coefficients of an even continuous function, we find a necessary and sufficient condition such that their arithmetic means are the Fourier coefficients of an odd continuous function. A similar result is shown for those Lipschitz classes whose elements are automatically equivalent to continuous functions. Introduction. Let I' la"cos nx and ' lb,, sin nx be the Fourier series of fc and fS, respectively. Hardy showed that X= J(a, + * + a")/n]cos nx and . Xs [(b1 + + bn)/n]sin nx [2, p. 51] will also be Fourier series-let THJC and THfJ, respectively, represent these two series. It was shown by M. Kinukawa and S. Igari [5, Theorem 2] that fc E LI? = THfc E L??. In this paper we show that if fc E C, then fc(O) = O X= THfC E C. A. Konyushkov [6, Theorem 15] showed that for 0 < a < 1/p < l,fc E AP X THfc E AP. In this paper we consider the case 0 < I/p < a < 1, and show that iffc Ef AP, then I' Ian = 0= THfc E AP. All the notations, unless otherwise mentioned, are taken from [9]. Every function is supposed to be defined a.e., periodic with period 21r and integrable on [T, ir]. We shall not distinguish between equivalent functions. Given a function f, by f(t) we would mean limn,o an (t; f)-whenever it exists. For convenience we shall write the Lipschitz classes AP and Aa as A(a, p) and A(a), respectively. Definition. For f and E E L1, we define Tsf as the function representing the Fourier series 21'1[Sn(0; f)/n]sin nt. We can easily check the following: (i) TSfC = THJC. (ii) For 0 < oa < 1 < p < oo and ap # 1, Tsfc E A(a, p) X= THfc E A(a, p). (iii) For 0 < Il/p < a < 1, by [4, Theorem 5 (ii)]

Given a holomorphic fibering with fibre P, we compare the cohomological Brauer group of the base to that of the total space. This allows us to prove that the geometric Brauer group of any symmetric product of a Riemann surface coincides with the cohomological one. Grothendieck has introduced [GROT] the notion of a geometric Brauer group. Given a variety X, this group Br(X) classifies, roughly speaking, Pa-bundles over X modulo those bundles of the form P(E) for some vector bundle E. In this paper we work in the category of complex manifolds (Grothendieck's setting was, of course, scheme-theoretic). Given any manifold Z, we put Br'(Z):= H 2 ( Z, Oz)tors (torsion part of H 2(Z, (*)). The Brauer group Br(Z) can be identified with a subgroup of Br'(Z) and a fundamental question is: does the equality Br(Z) = Br'(Z) hold? The aim of this paper is to give a positive answer to this question in the following cases (Theorems 1 and 2, respectively): (1) When Z is a Pa-bundle over a manifold X with the property Br(X) = Br'(X); (2) When Z is a symmetric product C(n) of a compact Riemann surface. ?1 is devoted to some preliminary material. In ??2 and 3 we prove the two theorems mentioned above. It is my pleasant duty to thank A. Fujiki for his judicious comments and the friendly interest he took in this work. 1. We work in the category of holomorphic manifolds. A Pr _-bundle over the manifold X is a holomorphic submersion 7r: P -* X such that all fibres of ST are isomorphic to Pr_1. Such a map is automatically a locally trivial fibration with structure group PGL(r, C). Hence the set of isomorphism classes of such Pr l-bundles, denoted by Projectr1( X), is in natural bijection with H'( X, PGL(r, ()). We have an exact sequence of sheaves of (noncommutative) groups on X, 1 -C9* -> GL(r, -x) PGL(r, -x) 1, Received by the editors June 11, 1984 and, in revised form, September 26, 1984. 1980 Mathenmatics Subject Classification. Primary 14C30; Secondary 32L05, 14F25, 14H15. ?)1985 American Mathematical Society 0002-9939/85 $1.00 + $.25 per page

Introduction. Let H be a compact Hausdorff space, let E and F be locally convex topological vector spaces over the real or complex field where Fis Hausdorff. Let C(H, E) be the space of continuous functions from H into E with the topology of uniform convergence. The general problem is to find an integral representation theorem for a continuous linear transformation T from C(H, E) into F. The well-known Riesz representation theorem [3] gives a Stieltjes integral representation for T when H is a closed interval and E and F are the real numbers. Tucker [6] gives a Stieltjes' integral type representation for T in the case of H, a closed interval, and E and F, linear normed spaces. This paper was generalized by Uherka [7] to the case of H, a compact space, and E and F, linear normed spaces. Also see Swong's paper [5] in which H is a compact Hausdorff space and E and F are locally convex topological vector spaces. This last paper takes a fundamentally different approach from that of Tucker and Uherka, and Swong writes y'T as an integral where y' is in the topological dual F' of F. However, Swong is able to write T as an integral under various additional assumptions on E, F or T. The approach taken in this paper more closely follows that of Tucker and Uherka where a much smaller class of sets is employed in the definition of integral. The result is that the integral converges in the s00 topology, as compared to the weak topology in Swong's paper. And T is written as an integral.

1. Summary. Given below is a brief description of the main results of this paper: ?3. It is shown that the study of a Markov process X,(Q , v) where Q is a general state space, E a separable a-field of subsets of Q, and v a a-finite stationary measure for the process, can essentially be reduced to the study of a real-valued Markov process Yn(K, A, A) where K is a bounded interval or the entire real line, A the Borel sets of K, and A Lebesgue measure, stationary for the process. To show this a notion of isomorphism of processes is introduced. The main tool is the geometric-isomorphism theorem of Halmos and von Neumann. ?4. Processes Xn(R, A, A) are discussed where R is the real line, and A and A are as defined in ? 3. The selection of R rather than a bounded interval K is made because we are particularly interested in infinite stationary measures and the infinite case introduces a few technical difficulties. Treatment of the finite case is practically identical to that of the infinite case except that it is simpler. The basic result is that the process Xn(R, A, A) may be approximated in a certain way by processes (called k-processes) which are essentially Markov chains on a countable state space. The nature of the approximation is "weak" convergence of measures on function space. Such an approximation is useful because certain results difficult to prove directly for processes on a continuous state space may be easy to prove for Markov chains, and then carried over to the general process by using the k-processes. (Similar approaches have been used to approximate continuous time by discrete time processes; in this paper it is the state space rather than the time parameter which is discretized-time is discrete throughout.) Next, if the process is conservative and ergodic (that is, the shift T is; see [9] and Halmos' Lectures on ergodic theory) we derive a few conclusions about the nature of the k-processes. ?5. As an application of ?4, we give a probabilistic proof of Birkhoff's ergodic theorem. ?6. Using the isomorphism of ?3, we extend the results of the paper to processes Xn(Q, , v) for E separable.

The object of this paper is to study the groups formed by the residue classes of a certain type of Kronecker modular system and some closely related generalizations of well-known theorems in number theory. The type of modular system to be studied is of the form 9) = (mn, mn-1, , Mnl, m). Here m, defined by (ml, m2, ***, Mk), is an ideal in the algebraic domain Q of degree k. Each term mi, i = 1, 2, *.., n, belongs to the domain of integrity of Qi = ( Q, xl, x2, * , xi), and is defined by the fundamental system ((1t/, t'2j), ***, i,t')). The various 1(i), j = 1, 2, *.., jI, are rational integral functions of xi with coefficients that are in turn rational integral functions of xl, x2, ... , xj_i, with coefficients that are algebraic integers in Q . In every case the coefficient of the highest power of xi in each of the 't') shall be equal to 1. We shall see later that the developments of this paper also apply to modular systems where the last restriction here cited is omitted, being replaced by another admitting more systems, these new systems in every case being equivalent to a system in the standard form as here defined. Any expression that fulfills all of the conditions placed upon each {() with the possible exception of the last one, we shall call a polynomial, and no other expression shall be so designated. This definition includes all of the algebraic integers of Q. Throughout this paper we shall deal exclusively with polynomials as here defined. The first part of this paper will contain the introduction with the necessary definitions and a discussion concerning the factoring of the system 9). The second section will then be devoted to setting up necessary and sufficient conditions that a set of residue classes belonging to 9) form a group when taken modulo 91. In the third section we shall study the structure of such a group with respect to groups belonging to certain modular factors of 9), besides

Let f be a real-valued function defined and finite on sets from a family F \mathcal {F} of bounded measurable subsets of Euclidean n-space such that if T ∈ F T \in \mathcal {F} , the measure of T is equal to the measure of the closure of T. An earlier paper [Trans. Amer. Math. Soc. 145 (1969), 439-454] considered the questions of finiteness and boundedness of the upper and lower regular derivates of f and of the existence of a unique finite derivative. The present paper is an extension of the earlier paper and considers the summability of the derivates. Necessary and sufficient conditions are given for each of the upper and lower derivates to be summable on a measurable set of finite measure. A characterization of the integral of the upper derivate is given in terms of the sums of the values of the function over finite collections of mutually disjoint sets from the family.