Abstract: The problem of characterizing derivatives is "old and celebrated" to quote a recent Monthly article on the subject, but also has been vexing. Characterizations have been neither transparent nor particularly useful. However, derivatives are both pointwise limits of continuous functions (Baire Class 1) and satisfy the intermediate value property (Darboux), and there are many quite elegant and very useful characterizations of the class DB₁ of Baire one, Darboux functions. A good starting point for the study of DB₁ is Chapter 2 of Andy Bruckner's book, Differentiation of Real Functions where one finds a list of eight statements each of which is equivalent to the notion of Darboux for functions of the first Baire class. An example of one of the equivalences was published in 1907 by W. H. Young, and is known as Young's Theorem; condition 2 in the theorem below is referred to as Young's Criterion. Theorem [Young, 1907]. Let f:R→R be a Baire 1 function. Then the following are equivalent:

1. f is a Darboux function.
2. For each x, there exist sequences x_n↑x and x_n^*↓ x such that f(x)=lim_n→∞f(x_n)=lim_n→∞f(x_n^*).

The situation in higher dimensions is considerably more delicate. First, there are several "natural" ways to generalize the notion of Darboux to functions f:Rⁿ→R and they are not equivalent. To confound matters further, each of the aforementioned equivalences for functions f:Rⁿ→R gives rise to one or more equally "natural" generalizations to functions f:Rⁿ→R, and these too are not equivalent, in many cases not even comparable. However, in 1997, Malý showed that gradients of differentiable functions map closed convex sets with non-empty interior to connected sets; as a consequence, partial derivatives of differentiable functions map closed convex sets with non-empty interior to intervals. This lends significant credibility to the claim that for real valued functions of several variables, Darboux functions should be defined as those that map closed convex sets with non-empty interior to intervals. The purpose of this paper is to lend a little more weight to that argument by showing that a higher dimensional analogue of Young's Criterion is equivalent to this condition.

Abstract: Rising carbon dioxide levels should increase crop yields. But what is their effect on the nutritional value of our food? It is known that elevating the level of carbon dioxide can significantly reduce the leaf nitrogen content and hence leaf mites' reproduction and renders pesticide unnecessary in greenhouse vegetable production. This raises the question of how resource quality impacts the population dynamics in general. Mathematical biologists have built on variants of the Lotka-Volterra equations and in almost all cases have adopted the physical science's single-currency (energy) approach to understand population dynamics. However, biomass production is essentially a mass transfer process that requires more than just energy. It is crucially dependent on the chemical compositions of both the consumer species and food resources. In this talk, we explore how depicting organisms as built of more than one thing, for example, C to represent energy, and an important nutrient, such as P (or N), to represent quality, results in qualitatively different and realistic predictions about the resulting dynamics.