Hagen von Eitzen
Reputation
397/400 score
 1d answered Is direction of logical consequence in FOL arbitrary? 1d comment Under which additional hypothesis are open maps locally injective The constant map to a one-point space is open, but locally injective only at isolated points ... 1d comment Let $I=[0,1]$ and $f:[0,1]\to[0,1]$ be a continuous function. Then exist $x\in I$ such that $f(x)=x$. Consider $g(x)=f(x)-x$. Then $g(0)\ge 0\ge g(1)$ 1d answered Why is $\varphi(X_i) = X_i + b_i$ an automorphism of $K[X_1,\dots,X_n]$? 2d comment Does there exist a function with following properties? I just see that after much reformulating my first attempt, this ends up as a filled-out version of @Siminore's answer - so maybe all credit should go there 2d answered Does there exist a function with following properties? 2d answered Find A and B for $A(x^{2}+3)+B(x^{3}+2)=5x^{4}+9x^{2}+4x$ 2d comment Irreducibility of a Polynomial after a substitution +1 So here's how to show a polynomial is irreducible by successfully factoring it :) 2d comment Rational Question for $a + b$ and Irrationality of $a^2 + b^2$ ... where $\pi$ can be replaced with any irrational (except square roots of a rational) between $0$ and $\frac{10}3$ ... 2d answered Number of ways to pick 16 people and place them in 11 non-empty groups. Details follow… 2d comment Finding the Limit of a Sequence (No L'Hopital's Rule) +1 You even only need that $\lim_{h\to 0}(1+h)^{1/h}$ exists at all, not that it $e$. 2d comment Counting the number of “distinct” permutations of two sets? @Finalfire All possible $Y$ can be obtained as follows: Pick an arbitrary bijection $f\colon A\to B$. This induces a homomorphism $\phi\colon S_A\to S_B$, $\sigma\mapsto f\circ \sigma\circ f^{-1}$. Note that by this $(p,q)$ is equivalent to $(\sigma\circ p, \phi(\sigma)\circ q)$. Now pick an arbitrary map $g\colon S_A\to S_A$. Let $Y=Y_g=\{\,(g(p)\circ p,\phi(p))\mid p\in S_A\,\}$. - Hence there are $n!^{n!}$ possible $Y$ Apr 26 comment Real numbers as element of a universe Why (in defining $\Bbb Z$ the usual way) do we need $-$ to write down $(a,b)\sim (c,d)\iff a+d=c+b$? This does use only $+$. - In contrast, idon't we need $-$ in order to define $+$ for $\{\,(a,b)\in\omega\times\omega\mid a=0\lor b=0\,\}$? Apr 26 comment Prove: If $\ker(A)\cap \mbox{Im}(A)\not = \{0\}$ then $A$ is a singular matrix But already $\ker(A)\ne\{0\}$ implies that $A$ is singulkar ... Apr 26 answered Is the 2011th term odd for this sequence and why so? Apr 26 comment Prove $P(X=k \mid X+Y=n) = \frac1{n+1}$ Maybe as a guidance, the heuristic is: $X,Y$ can be viewed as "number of tails before first head" for a biased coin. We are looking for "number of tails before first head, given that there are $n$ tails before second head" or "given that exactly one out of $n+1$ flips is head, at which position does it occur?" - The latter variant is clearly symmetric in the $n+1$ positions, hence uniform. Apr 26 awarded Good Answer Apr 25 answered An algorithm to find a subgroup generated by a subset of a finite group Apr 25 comment How to prove that Bezier(t) polynomial lies in convex hull of points (i/n,ai) for i from 1 to n "and i found that Bezier curve lies in convex hull of it's control points" - then what are the points $(i/n,a_i)$ if they are not the control points? Apr 25 comment Showing that the powers of a root of the p-th cyclotomic polynomial are distinct roots thereof. @Peter You do not use the (unstated by the OP) fact that $p$ is prime. For $p=4$, we might have $\gamma=-1$ and then fail because $\gamma^2$ is not a root of $x^3+x^2+x+1$ and because $\gamma^3$ and $\gamma^1$ are equal.