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12h
comment How to prove that $\det A=|\lambda|^n$
You are right that $\dim V_0<\dim V$ implies that $A$ is singular.
12h
comment Intersection of two circles giving reversed answer
I don't see any coordinates reversed in the graphics on the referenced mathworld page
12h
comment Why isn't the number of cosets equal to cardinalities of the groups?
@mavavilj Yes. For example, if $x\in H$ (which happens for $|H|$ different $x$) we have $xH=H$.
21h
comment Proving Injectivity $x + \sin(x)$
@ArturodonJuan That iniquality holds for all real $x\ne 0$. Even with the geometrical definition of sine, $\sin x$ is the straight line distance from the $x$-axis to a point on the unit circle, whereas $x$ is the not-so-straight length of the arc along the unit circle from the $x$-axis to that point
1d
comment Let $\mathcal S$ be the collection of all straight lines in the plane $\mathbb R^2$. If $\mathcal S$ is a subbasis for a topology …
@user290425 Of course, see also the meanwhile existing answer.
1d
comment If $c$ is critical point and $x_{n}\to c$ then $f''(c)=0.$
Can we assume that $f$ is twice differentiable? Or only once and the existence of $f''(c)$ is to be shown?
1d
comment Let $\mathcal S$ be the collection of all straight lines in the plane $\mathbb R^2$. If $\mathcal S$ is a subbasis for a topology …
Actually, this new topology is also Hausdorff ...
1d
comment In $\mathbb{Z}[\omega]$, if $a^3+b^3+c^3=0$ then $1-\omega$ divides at least one of $a,b,c$
@ThomasAndrews Your finding leads to $a\equiv b\equiv c\pmod{1-\omega}$. In case $\equiv 0$, we are done. In case $\equiv 2$, just mutiply everything with $2$ to arrive in the remaining case $a\equiv b\equiv c\equiv 1\pmod{1-\omega}$. Then $a^3+b^3+c^3\equiv 3\pmod 9$ by the useful result.
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 $g(0)\ge 0\ge g(1)$
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
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
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 $Y$
2d
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 $\{\,(a,b)\in\omega\times\omega\mid a=0\lor b=0\,\}$?
2d
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 ...
2d
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
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.