Generalized product and sum theorems for Whitehead torsion

Generalized bipolar product and sum

Aggregation functions on [0, 1] with annihilator 0 can be seen as a generalized product on [0, 1]. We study the generalized product on the bipolar scale [−1, 1], stressing the axiomatic point of view ( compare also [9]). Based on newly introduced bipolar properties, such as the bipolar increasingness, bipolar unit element, bipolar idempotent element, several kinds of generalized bipolar product...

Sum-product Theorems and Incidence Geometry

In this paper we prove the following theorems in incidence geometry. 1. There is δ > 0 such that for any P1, · · · , P4, and Q1, · · · , Qn ∈ C, if there are ≤ n 1+δ 2 many distinct lines between Pi and Qj for all i, j, then P1, · · · , P4 are collinear. If the number of the distinct lines is < cn 1 2 , then the cross ratio of the four points is algebraic. 2. Given c > 0, there is δ > 0 such th...

Generalized Product Theorems for Torsion Invariants with Applications to Flat Bundles

This note announces generalizations of the product theorems for Wall invariants and Whitehead torsions due to Gersten [5], Siebenmann [7, Chapter VII], and Kwun and Szczarba [6], and applies these theorems to study torsion invariants of the total space of a flat bundle. The generalized product theorems are described in §§1 and 2. The applications are found in §3. These theorems were discovered ...

Some Problems Related to Sum-product Theorems

The study of sum-product phenomena and product phenomena is an emerging research direction in combinatorial number theory that has already produced several striking results. Many related problems are not yet fully understood, or are far from being resolved. In what follows we propose several questions where progress can be expected and should lead to advances in this general area. For two finit...

Sum-product Theorems in Algebraic Number Fields