Ranks of collinear Morse forms

Morse Theory of Harmonic Forms

We consider the problem of whether it is possible to improve the Novikov inequalities for closed 1-forms, or any other inequalities of a similar nature , if we assume, additionally, that the given 1-form is harmonic with respect to some Riemannian metric. We show that, under suitable assumptions, it is impossible. We use a theorem of E.Calabi C], characterizing 1-forms which are harmonic with r...

Dyson’s Ranks and Maass Forms

which were defined by Ramanujan and Watson decades ago. In his last letter to Hardy dated January 1920 (see pages 127-131 of [27]), Ramanujan lists 17 such functions, and he gives 2 more in his “Lost Notebook” [27]. In his paper “The final problem: An account of the mock theta functions” [32], Watson defines 3 further functions. Surprisingly, much remains unknown about these enigmatic series. R...

Binary Forms with Three Different Relative Ranks

Suppose f(x, y) is a binary form of degree d with coefficients in a field K ⊆ C. The K-rank of f is the smallest number of d-th powers of linear forms over K of which f is a K-linear combination. We prove that for d ≥ 5, there always exists a form of degree d with at least three different ranks over various fields. The K-rank of a form f (such as x3y2) may depend on whether -1 is a sum of two s...

Mock Theta Functions, Ranks, and Maass Forms

Recent results on p - ranks and Smith normal forms of some 2 − ( v , k , λ ) designs

We survey recent results on p-ranks and Smith normal forms of some 2 − (v, k, λ) designs. In particular, we give a description of the recent work in [11] on the Smith normal forms of the 2-designs arising from projective and affine spaces over Fq .