This chapter presents the stratigraphic distribution of 154 diatom species in a composite upper Eocene to lower Miocene section recovered in cores from Holes 511 and 513A. The section is divided into 12 biostratigraphic zones based on the highest and lowest occurrences of 11 species. Abundance curves for several diatom species in the vicinity of the Eocene/Oligocene and Oligocene/Miocene boundaries are presented; they suggest that abundance changes may have utility as stratigraphic tools, at least on a local scale. Taxonomic notes on all species studied are presented and eight new species are described. Ranges of Zonal Species Diatom Zones D. hustedtii/D. lauta/Vjate Miocenes early Miocene Coscinodiscus rhombicus / Rocella gelida Triceratium groningensis Rocella vigilans Kozloviella minor Pyxilla prolongata group Coscinodiscus superbus group Rhizosolenia gravida Brightwellia spiralis Melosira architectural is Asterolampra insignis Ry/andsia inaequiradiata Age late Oligocene early Oligocene late Eocene Ranges of Secondary Marker Species ll σ>

Let F be a family of real functions, F ⊆ R R. In the paper we will examine the following question. For which families F ⊆ R R does there exist g : R → R such that f + g ∈ F for all f ∈ F ? More precisely, we will study a cardinal function A(F) defined as the smallest cardinality of a family F ⊆ R R for which there is no such g. We will prove that A(Ext) = A(PR) = c + and A(PC) = 2 c , where Ext, PR and PC stand for the classes of extendable functions, functions with perfect road and peripherally continuous functions from R into R, respectively. In particular, the equation A(Ext) = c + immediately implies that every real function is a sum of two extendable functions. This solves a problem of Gibson [6]. We will also study the multiplicative analogue M(F) of the function A(F) and we prove that M(Ext) = M(PR) = 2 and A(PC) = c. This article is a continuation of papers [10, 3, 12] in which functions A(F) and M(F) has been studied for the classes of almost continuous, connectivity and Darboux functions.

Let A stand for the class of all almost continuous functions from R to R and let A(A) be the smallest cardinality of a family F ⊆ R R for which there is no g: R → R with the property that f + g ∈ A for all f ∈ F . We define cardinal number A(D) for the class D of all real functions with the Darboux property similarly. It is known, that c < A(A) ≤ 2 c [10]. We will generalize this result by showing that the cofinality of A(A) is greater that c. Moreover, we will show that it is pretty much all that can be said about A(A) in ZFC, by showing that A(A) can be equal to any regular cardinal between c + and 2 c and that it can be equal to 2 c independently of the cofinality of 2 c . This solves a problem of T. Natkaniec [10, Problem 6.1, p. 495].We will also show that A(D) = A(A) and give a combinatorial characterization of this number. This solves another problem of Natkaniec. (Private communication.)

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