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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.