Using first isomorphism theorem, prove the following isomorphisms $\Bbb R/\Bbb Z\xrightarrow\sim S^1,\; $ $\Bbb C/\Bbb R\xrightarrow\sim \Bbb R,\; $ $\Bbb C^\times/\Bbb R^\times_+\xrightarrow\sim T,\;$ $\text{GL}_n(\Bbb C)/\text{SL}_n(\Bbb C)\xrightarrow\sim \Bbb C^\times,\; $ $\text{GL}_n(\Bbb...

First-time poster here. While doing some research on Waring's problem and the term $\{(3/2)^n\},$ I determined that the following recurrence relation holds for a certain sequence (here $n$ is a fixed, positive integer): $$r_{k}={n \choose k}+\frac{1}{2}r_{k-1}, r_0=1;\,0\leq k\leq n$$ Mathemati...

Proposal: Combine ceiling-function and floor-function, maybe into ceiling-and-floor-functions. This feels similar to limsup-and-liminf: yes, there are results that deal exclusively with one of the two (for instance, Fatou's Lemma and liminf; or questions about Waring's problem and floors), but th...