Joseph Miles

On the growth of a meromorphic function and its derivatives

The relative rates of growth of a function F meromorphic in the complex plane and its q derivative F (q) are studied via the Nevanlinna Characteristics T(r.F)and T(r.F (q)) and It is shown that lim inf T(r.F)/T(r.F (q)) ≤ 3ethat for all meromorphic functions. A lower bound on the size of the set {r>1 T(r.F) T(r.F (q))for K 1 is obtained. The upper bounds established for T(r.F)/T(r.F′)justify in a weakened form an old conjecture of Nevanlinna.

On the growth of solutions of certain linear differential equations

A box for containing an article having a relatively stiff planar portion positioned flush against the front or back wall of the box and an internal tuck closure flap hinged to the front or back wall at an end of the box includes a tab hinged to the wall of the box opposite the closure flap hinge that folds inwardly so that when the closure flap is tucked inside the box to close off an end of the box the closure flap overlaps the tab. The tab prevents the article from hanging up on the edge of the closure flap or interfering with the tucking of the closure flap inside the box.

Quotient representations of meromorphic functions

Let f be a meromorphic function in [z I 0. It is implicit in the method of proof that for any B > 1 there is a corresponding A for which the desired representation holds for all f . We show that in general B cannot be chosen to be 1 by giving an example of a meromorphicfsuch that if f = fl]f2 where f l and f2 are entire then T(r,f2) # O(T(r,f)). Rubel and Taylor have obtained the above theorem for special classes of meromorphic functions. In particular it is shown in I-5] that such a representation exists for any meromorphic f such that either sup T(2r,f)]T(r,f) _l or such that log T(r,f) is a convex function of log r. Although the representation is not obtained in [-5] for all meromorphic functions, the general result is shown to be equivalent to a seemingly more elementary proposition concerning sequences of complex numbers. The contribution of this paper is to prove the result concerning sequences of complex numbers and to provide an example showing the theorem is sharp. Results in this direction for functions of several complex variables appear

Pay now or pay later: aging and the role of boundary salience in self-regulation of conceptual integration in sentence processing.

Previous research has suggested that older readers may self-regulate input during reading differently from the way younger readers do, so as to accommodate age-graded change in processing capacity. For example, older adults may pause more frequently for conceptual integration. Presumably, such an allocation policy would enable older readers to manage the cognitive demands of constructing a semantic representation of the text by off-loading the products of intermediate computations to long-term memory, thus decreasing memory demands as conceptual load increases. This was explicitly tested in 2 experiments measuring word-by-word reading time for sentences in which boundary salience was manipulated but in which semantic content was controlled. With both a computer-based moving-window paradigm that permits only forward eye movements, and an eye-tracking paradigm that allows measurement of regressive eye movements, we found evidence for the proposed tradeoff between early and late wrap-up. Across the 2 experiments, age groups were more similar than different in regulating processing time. However, older adults showed evidence of exaggerated early wrap-up in both experiments. These data are consistent with the notion that readers opportunistically regulate effort and that older readers can use this to good advantage to maintain comprehension.

Adult age differences in self-regulated learning from reading sentences.

We examined age differences in the heuristic used to allocate effort in learning information from sentences. Younger and older adults read and reread sentences varying in propositional density for recall, making judgments of learning before producing recall. The allocation of effort in rereading items that were less well learned on the first reading was optimized for sentences of intermediate complexity, especially for older adults. These data support a model of self-regulated learning in which readers reduce the discrepancy between current and optimal states of learning. However, self-regulation, which may be procedure based or rely on an implicit representation of the current state of learning, may be particularly efficient for older adults within a region of proximal learning.

On a theorem of hayman and stewart

Suppose f id a meromorphic function on the complex plane. Let n(r, a) be the number of solutions of f(z=a in |z| ≤ r counting multiplicity, and let n 1(r) = sup a n(r, a). If A(r, f) denotes the average of n(r, a) with respect to surface area measure as a varies over the Riemann sphere, it is shown that exists μ < e independent of f such that , thereby establishing strict inequality in an earlier result of Hayman nad Stewart.