Month: September 2016
Ralf Schiffler Awarded Simons Visiting Professorship
Professor Ralf Schiffler has been awarded a Simons Visiting Professorship by the Mathematisches Forschungsinstitut Oberwolfach in Germany. This program annually supports up to 40 Simons Visiting Professors, distinguished scientists from outside of Europe, who wish to combine an existing invitation to an Oberwolfach Workshop with a research visit to a European university of up to two weeks.
Professor Schiffler will use this award to work on several existing research projects as well as create new research collaborations during a two week visit to Germany. He will stay at the University Bielefeld and participate in the workshop Representation Theory of Quivers and Finite Dimensional Algebras in Oberwolfach.
Fall 2016 Challenge Problem #1
A fairly standard problem in the study of sequences is to make rigorous sense of the expression
$$
\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{2+\cdots}}}}
$$
and find its value. In this instance, one looks at the sequence of “partial” expressions obtained by cutting the given expression off after only finitely many radicals and $2$s. It is easy to see that the sequence of these is increasing and bounded above by $2$, so must converge to a limit $L$. It is straightforward that then $L$ must satisfy the equation $L = \sqrt{2+L}$, from which it $L = 2$.
There have been variations of this problem among our past challenge problems, and our current offering is another one.
Problem. Make rigorous sense of the expression
$$
L = \sqrt{1 – \sqrt{2- \sqrt{3-\sqrt{4-\sqrt{5-\cdots}}}}}
$$
and determine whether $L$ has a numerical value. If it does, you are not required to evaluate it.
Remark. Real numbers only!
Solutions are due by email by 3:00 p.m. on Thursday 22 September 2016.