Find an analytic formula for the recurrent sequence $$q_{n+1}=q_n(q_n+1)+1,\;\;q_0\in\mathbb N.$$ (The question was asked on 03.05.2018 by M. Pratsovytyi, see page 109 of Volume 1 of the Lviv Scottish Book).
Given a function $f(z)$ on the complex plane, define the Schwarzian derivative $S(f)$ to be the function $S(f) = \frac{f'''}{f'} - \frac{3}{2} (\frac{f''}{f'})^2$ Here is a somewhat more conceptual definition, which justifies the terminology. Define $[f, z, \epsilon]$ to be the cross ratio $[f...
Is there a heuristic argument behind the exponent in the circle problem? The problem that I am referring to is the following: Consider a circle of radius $R$ centered at the origin in the plane and let $N(R)$ denote the number of integer lattice points contained in the circle. Then it easy to ...
The tag lattices is a problem since a long time. Over the years there were varied discussions about this, but no clear resolution emerged, though some things were done to better the situation. This is an attempt at a fresh start of the discussion to find a long-term solution. The current situa...
Problem. Is the series $$\sum_{n=1}^\infty\frac{|\sin(n)|^n}n$$convergent? (The problem was posed on 22.06.2017 by Ph D students of H.Steinhaus Center of Wroclaw Polytechnica. The promised prize for solution is "butelka miodu pitnego", see page 37 of Volume 1 of the Lviv Scottish Book. To g...
Let us call a series $\sum_n x_n$ is a Banach space "good" if there exists a permutation $\sigma:\mathbb N\to\mathbb N$ such that the rearranged series $\sum_n x_{\sigma(n)}$ converges. Find a simple proof of the following theorem (which was proved by E.Steinitz in 1913 according to V.Kadets). ...
Problem. Is there a partition $\mathbb R^2=A\sqcup B$ of the Euclidean plane into two Lebesgue measurable sets such that for any disk $D$ of the unit radius we get $\lambda(A\cap D)=\lambda(B\cap D)=\frac12\lambda(D)$? (I.V.Protasov called such partitions kaleidoscopic). Observe that for ...
« first day (1893 days earlier) ← previous day next day → last day (1978 days later) »