Andrew M. Bruckner

Generalized convex kernels

The notion of the convex kernel of a setD is generalized to that of then-th order kernel ofD. Such kernels are studied for compact, simply connected subsets of the Euclidean plane. In particular, it is shown that under certain circumstances [see Theorem 4 and also section 5], these kernels have rather simple structures.

ON THE DIFFERENTIABILITY STRUCTURE OF REAL FUNCTIONS

In the process of obtaining the results mentioned above, we obtain several other results. In particular, in ?2, we obtain a lemma which is of fundamental importance in the proof of the main theorem, but which appears to be of independent interest. This lemma states, roughly, that if every interval contiguous to a perfect nowhere dense set P is colored either black or white, and the endpoints of those intervals colored black form a set somewhere dense in P, then there exists a perfect set Q contained in P such that if x and y are arbitrary points of Q, the longest of the colored intervals contained in the interval determined by x and y is black. We use this lemma in the proof of the main theorem, but it can be used in other ways as well. For example, we use it to obtain a relatively simple proof of a theorem of Filipczak [4], according to which a function continuous on a perfect set P is

Conditional Correlation Phenomena with Applications to University Admission Strategies

This paper considers the situation (as in college admissions) where one is given two attributes,X and Y, which one uses to predict a third attribute, Z, by some function Ẑ ofX andY. However, one only retains values ofX, Y, andZ for which Ẑ is large. A thorough discussion, under fairly general conditions on the distributions, is given of how the correlation coefficients of X, Y, andZ are affected by this restriction of the range of values. In the case of the normal distribution, where linear prediction is optimal, the role of suppressor variables is discussed.

A category analogue of a theorem on metric density

The main result of this paper is to prove that for a subsetB of the euclidean plane possessing the property of Baire, each point outside of some first category subset ofB is a point of categorical directional density in all directions except, perhaps, a first category set of directions. This result is the category analogue of a result on metric directional densities derived by Bruckner and Rosenfeld and provides an affirmative answer to a question raised by Ceder in the setting of set theory.

POROSITY AND APPROXIMATE DERIVATIVES

Etude de la porosite et des derivees approchees, en relation avec les resultats de A. Khintchine et G.H. Sindalovskii