1
vote
0answers
35 views

Solving a system of equations with fractional parts and a system with round parts

I have the following two systems of equations: $a = x_{11} - \{x_{11} + \frac{4 - \sqrt{2}}{7}b + \frac{4 - \sqrt{2}}{7}c + \frac{2\sqrt{2} - 1}{7}d\}$ $b = x_{12} - \{x_{12} + \frac{4 - ...
0
votes
1answer
75 views

A combinatorics problem refer to this problem?

If i define $f(m,n)=$ $$\sum_{1\leq k\leq mn}\left\{ \frac{k}{m}\right\} \left\{ \frac{k}{n}\right\} .$$ Then prove $$f(m+n,n) - f(m,n) =\frac{n^2-n}{4}$$ for all $m$ and $n$. This question came ...
17
votes
3answers
403 views

A sum of fractional parts.

I am looking to evaluate the sum $$\sum_{1\leq k\leq mn}\left\{ \frac{k}{m}\right\} \left\{ \frac{k}{n}\right\} .$$ Using matlab, and experimenting around, it seems to be $\frac{(m-1)(n-1)}{4}$ when ...