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For convex d-polytope P let f t {P) equal the number of faces of P of dimension i, 0< i < d -1. f(P) = (f 0 (P) 9 . . . , fd^QP)) is called the f vector of P An important combinatorial problem is the characterization of the class of all /-vectors of polytopes, and in particular of simplicial polytopes (i.e. those for which each facet is a simplex). McMuUen in [5] conjectures a set of necessary and sufficient conditions for (/ 0 , . • . 9 f d~i ) to be the/-vector of a simplicial d-polytope and proves this conjecture in the case of polytopes with few vertices. We sketch here a proof of the sufficiency 3 of these conditions, and derive in a related way a general solution to an upper bound problem posed by Klee.The /-vectors of simplicial d let with the convention that £_ x = 1 and f f = 0 for / < -1 or i > d -1. We note here that these relations are invertible, allowing us to express the f (

We describe here the notion of generalized stress on simplicial complexes, which serves several purposes: it establishes a link between two proofs oftbe Lower Bound Theorem for simplicial convex polytopes; elucidates some connections between the algebraic tools and the geometric properties of polytopes; leads to an associated natural generalization of infinitesimal motions; behaves well with respect to bistellar operations in the same way that the face ring of a simplicial complex coordinates well with shelling operations, giving rise to a new proof that p.1.-spheres are Cohen-Macaulay; and is dual to the notion of McMullen's weights on simple polytopes which he used to give a simpler, more geometric proof of the g-theorem.

Prospective teachers are expected to construct, emphasise, integrate, and make use of mathematical connections; in doing so, they acquire an understanding of mathematics that is fluid, supple, and interconnected (Evitts Dissertation Abstracts International, 65(12), 4500, 2005). Given the importance of mathematical connection making, an exploratory study was conducted to consider the ability of prospective middle-grades teachers to make mathematical connections while engaging in card-sorting activities. Twenty-eight prospective middle-grades teachers participated in both an open and closed card sort. Data were analysed using constant comparative methods to extract meta themes to describe the types of connections made. Findings indicate that these prospective teachers tended to make more procedural-and categorical-type mathematical connections and far fewer derivational or curricular mathematical connections.