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Clash Royale CLAN TAG #URR8PPP up vote 1 down vote favorite 1 Let $c in mathbbC$ and $f(z)=z^2+c$. Then define a sequence: $z_0=0$, $z_n=f(z_n-1)$ for all $n in mathbbN$. Show that $|c|>2 Rightarrow z_n rightarrow infty$. I had no clue which theorem I could use in this case in order to show this. I'd be thankful for any advice. complex-analysis share | cite | improve this question edited Aug 22 at 11:26 Did 243k 23 208 443 asked Aug 22 at 11:23 Thesinus 226 2 9 (Terrible title.) No need for a "theorem" here, just find a lower bound for $|z_n|$ in terms of $|z_n-1|$ and iterate. – Did Aug 22 at 11:26 2 Please refer to "escape criterion" here . – Ng Chung Tak Aug 22 at 18:25 add a comment  |  up vote 1 down vote favorite 1 Let $c in mathbbC$ and $f(z)=z^2+c$. Then define a sequence: $z_0=0$, $z_n=f(z_n-1)$ f...

Clash Royale CLAN TAG #URR8PPP up vote 2 down vote favorite I want to count the number of $ell_1timesell_2$ matrices with entries in $0,1,2$ and prescribed sum of entries for each row and column. For example, there are three $2times 2$ matrices with row and column sums equal to 2: $$beginpmatrix2&0\0&2endpmatrix,beginpmatrix0&2\2&0endpmatrix,beginpmatrix1&1\1&1endpmatrix$$ There are 21 $3times 3$ matrices with row and column sums equal to 2 - not listed here, but you can check in GAP using the following command: gap> R:=Filtered(Tuples([0,1,2],3),r->Sum(r)=2); gap> Q:=Filtered(Tuples(R,3),M->ForAll(TransposedMat(M),r->Sum(r)=2)); gap> Size(Q); I also found a paper by Wang and Zhang (1997) for the number of matrices with entries in $0,1$ with prescribed row and column sums, so I am wondering how I can extend it to $0,1,2$. combinatorics matrices share | cite | improve this question asked Aug 22 at 11:28 Alvin ...

Clash Royale CLAN TAG #URR8PPP up vote 2 down vote favorite 1 $$lim_ntoinftysumlimits_0leqslantkleqslant2n frackk+n^2$$ I can't figure out the "right" function for this limit. The previous problem was very similar except there was $k^2$ in the denominator, and it wasn't difficult to recognize the Riemann sum for $f(x)=fracxx^2+1$. Given the limits, this integral sum can be written as $sumf(frac2kn)cdotfrac2n$. limits definite-integrals riemann-sum share | cite | improve this question edited Aug 22 at 11:20 Did 243k 23 208 443 asked Aug 22 at 9:52 Timur Sharapov 23 4 "Given the limits, this integral sum can be written as $sumf(frac2kn)cdotfrac2n$." Well, no. – Did Aug 22 at 11:20 Sorry, but why not? – Timur Sharapov Aug 22 at 11:22 Which function $f$ would yield this identity? Not $f(x)=frac xx^2+1$ in any case... – D...