Existence theorems for sum and product integrals

ON CONVERGENCE THEOREMS FOR FUZZY HENSTOCK INTEGRALS

The main purpose of this paper is to establish different types of convergence theorems for fuzzy Henstock integrable functions, introduced by  Wu and Gong cite{wu:hiff}. In fact, we have proved fuzzy uniform convergence theorem, convergence theorem for fuzzy uniform Henstock integrable functions and fuzzy monotone convergence theorem. Finally, a necessary and sufficient condition under which th...

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...

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

Existence and Differentiability Theorems for the Solutions of Variational Problems for Multiple Integrals

1. The object of the research [ l ] with which this address is concerned is twofold : first, to demonstrate, by direct methods, the existence of solutions, perhaps in some generalized sense, for a wide class of variational problems for multiple integrals, and second, to investigate further differentiability properties of the generalized solutions thus obtained. We consider only problems in nonp...