Robert D. Berman

Cardiac and Pulmonary Complications in Duchenne's Progressive Muscular Dystrophy

IT IS the purpose of this paper to present a clinical and pathologic study of myocardial involvement in seven previously unreported eases of the Duchenne form of progressive muscular dystrophy. The clinical signs of cardiac involvement in 139 cases and the electrocardiograms taken on the majority of these patients are also discussed. The frequent occurrence of pneumonia has proved to be a serious complication in nonambulatory patients and the treatment employed in such cases is presented. Finally, the problem of the terminal illness and the immediate cause of death are discussed. The patients in this series have all been selected as cases of the Duehenne form of muscular dystrophy as defined by Walton and Nattrass.1 This corresponds to the childhood group of Tyler and Wintrobe.2 Walton and Nattrass define this group as (a) generally affecting male but rarely female subjects; (b) usually beginning in the first year of life but occasionally later; (e) transmitted as a sexlinked recessive eharacter; and (d) usually progressing rapidly, giving total disability and often death in adolescence but sometimes with survival to adult life.

A converse to the Lusin-Privalov radial uniqueness theorem

Let E be a subset of the unit circumference C. If for everv nonemptv open arc A of C. the set E is not both metnrcallv dense and of second category in A, then there exists a nonconstant analvtic function f on the open unit disk A such that 0. = E , where f* is the radial limit function of f Let A ={z I< 1} and let E be a subset of C= {jzj= 1}. For f an analytic function on A, denote by f* the radial limit function off Thusf*(Ti) limr. i f(rq) for each 7 in C where the limit exists (finite or infinite). According to a classical theorem of Fatou [2], a bounded analytic function f on A has radial limits almost everywhere in C. F. and M. Riesz [6] proved that if there exists a nonconstant bounded analytic function J on A such that f*(Ti) =0, ?7 C E, then E is of measure 0. A converse to the Riesz theorem was provided by Privalov [5, p. 214]. THEOREM 1. If E is of measure 0, then there exists a nonconstant bounded analytic function f on A such that f*(q) = 0, q E E. Here we are concerned with a converse to the following celebrated uniqueness theorem of Lusin and Privalov [3, pp. 187-189]. THEOREM 2. If there exists a nonconstant analytic function f on A such that f*(Ti) = O, T E E, then for every nonempty open arc A in C, the set E is not both metrically dense and of second category in A. By definition E is metrically dense in an open arc A if, for every nonempty open subarc B of A, the set E n B has positive outer measure. Also, E is nowhere dense if int E ? 0, that is, the interior of the closure of E is empty, and is of first (resp. second) category if it is (resp. is not) a countable union of nowhere dense sets. Lusin and Privalov [3, ??1 1 and 33-35] constructed nonconstant analytic functionsf and g such that f*(7) = 0 for a set of q in C of measure 2 7 and g*(q) = 0 for a set of q in C which is of measure 7 in the upper half-circle and of second category in the lower half-circle. These examples show that some nontrivial combination of the second category and metric density conditions in Theorem 2 is indeed necessary. In this paper, the full converse of Theorem 2 is proved. Received by the editors May 3, 1982. 1980 Mathematics Subject Classification. Primary 30D40. Ket words and phrases. Lusin-Privalov, radial uniqueness. ? 1983 Amencan Mathematical Society 0002-9939/82/0000-0404/

Activity and passive-avoidance learning in cobalt-injected rats

02.00

THE LEVEL SETS OF THE MODULI OF FUNCTIONS OF BOUNDED CHARACTERISTIC

A wide range of cognitive-behavioral sequelae, including memory deficits, results from hard metal disease in humans. Cobalt is a common component in the manufacture of hard metals and is a biologically active, toxic substance. This study examined the effects of cobalt exposure in rats. Results showed decreased exploratory behavior and a trend for higher-dose subjects to show decreased passive avoidance learning. No significant differences in active maze learning were found. These results indicate the value of further explorations of the cognitive-behavioral effects of cobalt exposure and suggest a number of methodological cautions.

Generalized derivatives and the radial growth of positive harmonic functions

For / a nonconstant meromorphic function on A = {|z| 0. When g is an arbitrary meromorphic function, the equality of two pairs of level sets implies that g = cf, where c # 0 and a £ (-oo, oo). In addition, an inner function can never share a level set of its modulus with an outer function. We also give examples to demonstrate the sharpness of the main results.

Some results concerning the boundary zero sets of general analytic functions

Let A and C denote the unit disk (1~1 in the complex plane. The classical Riesz-Herglotz theorem provides a representation of each positive harmonic function u on A as the PoissonStieltjes integral of a monotone nondecreasing function p. The Fatou radial-limit theorem establishes a basic connection between the radial behavior of u and symmetric derivatives of p. Generalized upper and lower symmetric derivatives of p with respect to functions o(t) such as P, 0 < c( < 1, or t log( l/t), are defined analogously to the standard symmetric derivatives with w(t) replacing t in the denominator. These derivatives compare the local concentration of mass of the measure associated with p to the function o. Samuelsson [ll] correlated the generalized symmetric derivatives of p with the radial growth of u to co. In Theorem 1.1, we collect his results along with an additional fact, the proof of which is based on a method of Mu1 [S]. Essentially, Theorem 1.1 asserts that the generalized upper symmetric derivative of p with respect to w at 0 is roughly the same size as the limit superior of the quantity [ (1 r)/o( 1 r)] U(re”) as Y -+ 1, and the corresponding lower symmetric derivative (up to a constant) is no larger than the limit inferior of this quantity. Stated another way, the theorem relates the radial growth of u(reis) to the function ~(1 r)/( 1 -r), the latter a function such as (1 -T)‘-‘, O<a< 1, or -log(l -r) which approaches cc as y--t 1. For example, if the generalized upper (or lower) symmetric derivative of p with respect to w at 8 is positive, then the value of u(re’“) is intermittently (or eventually) as large as some constant multiple of o( 1 r)/( 1 r) as Y + 1. Following Samuelsson, we consider next the size of the set of radii along 394 0022-247X/89

Coefficient inequalities for a subclass of starlike functions

3.00

Generalized variation and functions of slow growth

Two results concerning the boundary zero sets of analytic functions on the unit disk A are proved. First we consider nonconstant analytic functions f on i for which the radial limit function f * is defined at each point of the unit circumference C. We show that a subset E of C is the zero set of f * for some such function f if and only if it is a % that is not metrically dense in any open arc of C. We then give a precise version of an asymptotic radial uniqueness theorem and its converse. The constructions given in the proofs of each of these theorems employ an approximation theorem of Arakeljan.

The zero-sets of the radial-limit functions of inner functions

A function f(z) = z − ∑∞n = 2 anzn, an ⩾ 0, analytic and univalent in the unit disk, is said to be in the family T∗(a, b), a real and b ⩾ 0, if ¦(zf′f) − a¦ ⩽ b for all z in the unit disk. A complete characterization is found for T∗(a, b) when a ⩾ 1. Also, sharp coefficient bounds are determined for certain subclasses of T∗(a, b) when a < 1; however, examples are given to show that these bounds do not remain valid for the whole family.

Angular limits, maximum principles, and level domains in the MacLane class

Theoremes de representation integrale des fonctions analytiques a variation bornee et a croissance lente