A Theory of some Multiple Decision Problems, I

I I Simultaneous Test Procedures - Some Theory of Multiple Comparisons

Some decision problems

Given a finite set of matrices with integer entries, consider the question of determining whether the semigroup they generated 1) is free, 2) contains the identity matrix, 3) contains the null matrix or 4) is a group. Even for matrices of dimension 3, questions 1) and 3) are undecidable. For dimension 2, they are still open as far as we know. Here we prove that problems 2) and 4) are decidable ...

Some Problems in Number Theory I: the Circle Problem

C.F. Gauss [Gaus1] [Gaus2] showed that the the number of lattice points in circles is the area, πt , up to a tolerance of order at most the circumference, i.e. the square root, O( √ t) . Sierpinski in 1904, following work of Voronoi [Vor] on the Dirichlet Divisor Problem, lowered the exponent in Gauss’s result to 1/3 [Sierp1] [Sierp2]. Further improvements that have been obtained will be descri...

Some Group Theory Problems

This is a survey of some problems in geometric group theory which I find interesting. The problems are from different areas of group theory. Each section is devoted to problems in one area. It contains an introduction where I give some necessary definitions and motivations, problems and some discussions of them. For each problem, I try to mention the author. If the author is not given, the prob...

Some Decision Problems Related To

The problem of deciding whether a monoid that is given through a nite presentation is a regular monoid as well as two closely related decision problems are considered. Since these problems are undecidable in general, they are investigated for certain restricted classes of nite presentations. Some new decidability results as well as some new undecidability results are obtained, which show that f...