The following was proved in Proposition 7.1.11(c) (in Tao's Analysis): 7.1.11(c) (Substitution, part I) If $X$ is a finite set, $f:X\to \mathbb{R}$ is a function, and $f:Y\to X$ is a bijection, then $$ \sum_{x\in X} f(x) = \sum_{y\in Y} f(g(y))$$ Later in proof of Lemma 7.1.13 Tao sir uses this...

I am trying to come up with an example of a group $G$ which has 2 elements (maybe the group itself has more elements) s.t. $g,h\in G$, and $o(h)=o(g)=3$, but $o(gh)=2$. Clearly it is not an Abelian group. I am trying to avoid using group presentations and relations, just trying to find one concre...

I’m interested in the following question. Consider two identical, rigid cylinders (in 2D, two circles) of radius $r$, with centers separated by a distance $s$. A point particle is released above with a vertical downward velocity, its initial horizontal offset measured by the distance $\xi $ from ...