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6h
comment Prove that for any given $c_1,c_2,c_3\in \mathbb{Z}$,the equations set has integral solution.
@lanse2pty The number $(c_1, c_2)$ is the smallest non-zero number that can be written as a linear combination of $c_1$ and $c_2$. All other numbers that can be written as a linear combination of $c_1$ and $c_2$ is a multiple of that. And any multiple of the gcd can be written as a linear combination. Since if $mc_1 + nc_2 = (c_1, c_2)$, then the multiple $r(c_1, c_2) = r(mc_1 + nc_2) = (rm)c_1 + (rn)c_2$.
9h
comment Prove that for any given $c_1,c_2,c_3\in \mathbb{Z}$,the equations set has integral solution.
This can be rewritten into vector cross product notation: $$ [a_1, a_2, a_3]\times [b_1, b_2, b_3] = [c_1, c_2, c_3] $$
12h
comment Prob. 1, Sec. 27 in Munkres' TOPOLOGY, 2nd ed: How to show that the compactness of every closed interval implies the least upper bound property?
You have to show $p \in U$ before you can conclude $[a, p] \in \mathscr L$, but that's just a matter of which order you write things. Otherwise it looks fine to me.
20h
comment gaming - How to calculate odds of roulette “Strategy”
Many roulette tables have a maximal possible bet to stop this exact behaviour.
20h
comment Prob. 1, Sec. 27 in Munkres' TOPOLOGY, 2nd ed: How to show that the compactness of every closed interval implies the least upper bound property?
Perhaps add a detail about how there is only one point $p$ and not a whole interval $[p_0,p_1]$. It's not very important, though.
21h
comment Find a unit vector that is parallel to $\nabla f(\cos\theta,\sin\theta)$
The gradient is at any given point orthogonal to level curves, and its length is proportional to the "distance" between the level curves at that point.
1d
revised Show $\langle u,v \rangle = -\frac{1}{2}$ when $u+v+w=0$ and $\|u\|=\|v\|=\|w\|=1$?
Use '\langle \rangle' for inner products.
1d
comment BMO1 2006/07 Question 4 Geometry Problem
You've got it backwards: $Q$ must be a tangent point, but $P$ doesn't have to be. The text says "let the tangent from $P$ to $T$ touch it at $Q$". If the circles aren't of the same size, then the line will intersect $S$ at two points: between $P$ and $Q$ if $S$ is the larger circle, and on the far side of $P$ if $T$ is the larger circle.
1d
comment BMO1 2006/07 Question 4 Geometry Problem
Note that $PQ$ is not a tangent to both circles, just one of them.
1d
comment Show that S is closed but not compact
Compact in $\Bbb R^3$ means closed and bounded. Since we're told that $S$ is closed, or only hope is that it's not bounded.
1d
comment Does this curiousity have any meaning?
There is nothing mathematically significant about $360^\circ$ degrees of a circle, other than that it's a round number.
1d
comment Trigonometry express $4\cos x+3\sin x$ in the form $R \cos (x+a)$.
Use your formula, and get, say, $a=2.6$. Then you know that $R\cos(x-2.6)$ is what you're looking for. Rewrite to $R\cos(x+(-2.6))$, and you have it on the form you were asked for.
1d
comment If a = 3i + 2j and b = -7i + 4j, find a + b as…
On this site there is a guide to formatting mathematics on math.SE. In short, writing $\vec{i}$ gives you $\vec{i}$. A longer example: $\vec{a} = 3\vec{i} + 2\vec{j}$ gives $\vec{a} = 3\vec{i} + 2\vec{j}$.
1d
comment Prob 12, Sec 26 in Munkres' TOPOLOGY, 2nd ed: How to show that the domain of a perfect map is compact if its range is compact?
No, I don't think we need it. Note that if $f$ is not surjective, then every $U(y)$ contains the complement of $f(X)$. Also, if we don't have surjectivity, we can just reduce to $Y'=f(X)$, which is closed and therefore compact. So it's probably there as a bonus that you would get almost for free anyway.
2d
comment Which of the following statements are false?
If you want to prove $B \setminus A \neq \emptyset$, then you should find some $x$ that is in $B$, but not in $A$.
2d
answered Prob 12, Sec 26 in Munkres' TOPOLOGY, 2nd ed: How to show that the domain of a perfect map is compact if its range is compact?
2d
comment Question about limit and dividing by zero
Note that in the function $$f(x)=\cases{1& if $x=0$\\0& otherwise} $$ we have $\lim_{x\to0}f(x)=0$, because that's how limits are defined to work.
2d
comment Question about limit and dividing by zero
But using your notation, we have $\lim_{n\to 0}\frac{1}{-n} =-\infty$, so $1/0$ is both positive and negative infinity at the same time. If that doesn't smell "undefined", then I don't know what does. Plus, we don't like to have $\infty$ as an actual function value. We put it on limits more as a shorthand for " grows indefinitely ", rather than as an actual value.
2d
comment Can a nonempty set ever equal its Cartesian product with another set?
If $S$ is finite, then $T$ must be a one-point set if you want $S$ and $S\times T$ to have the same size. If $S$ is infinite, then it's enough that $T$ is no bigger than $S$.
2d
comment Let $f$ be an odd meromorphic function , what can I deduce about $res (f,0)$
Just off the bat I'd say "nothing", since $z$ and $1/z$ both are odd and meromorphic. I might be missing something, though.