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seen Dec 16 at 17:01

Systems (or computational) biologist.


Nov
18
comment Finding an angle between side and a segment from specified point inside an equilateral triangle
It's simple: note that arc length = angle (in radians) when the radius of the circle is 1 (i.e. unit circle). So when the arc length is $u$, the chord length is $2\sin(u/2)$.
Nov
15
comment Find the line that intercepts the lines $r$ and $s$ and forms congruent angles to the coordinate axes
You're almost done. From the first part of your argument, if you set $\overrightarrow{AB}=(x,\,y,\,z)$, you get $|x|=|y|=|z|$. (The angle is not 45 degrees by the way, but the argument still stands.) Then you have four cases to consider: $x=y=z$, $x=-y=z$, $x=y=-z$, $x=-y=-z$.
Nov
15
comment Finding an angle between side and a segment from specified point inside an equilateral triangle
I used $\cos 3\theta=4\cos^3\theta-3\cos\theta$ and $\sin 2\theta=2\sin\theta\cos\theta$. So $\cos 3\theta=\sin 2\theta$ becomes $\cos\theta(4\cos^2\theta-3)=\cos\theta(2\sin\theta)$, and dividing by $\cos\theta$ (which is not zero) we get $4(1-\sin^2\theta)-3=2\sin\theta$.
Nov
14
comment Finding an angle between side and a segment from specified point inside an equilateral triangle
A proof of $d_3+d_5=d_9$: it suffices to show $\sin 54^{\circ}-\sin 18^{\circ}=\sin 30^{\circ}$, that is $2\cos 36^{\circ}\sin 18^{\circ}=1/2$. Using the fact that $\alpha=\sin 18^{\circ}$ satisfies $4(1-\alpha^2)-3=2\alpha$ (from $\cos 54^{\circ}=\sin 36^{\circ}$), we can show that $2(1-2\alpha^2)\alpha=1/2$.
Nov
14
comment Finding an angle between side and a segment from specified point inside an equilateral triangle
This lemma is stated in www-math.mit.edu/~poonen/papers/ngon.pdf (pages 4-5). I'll check if it is really true.
May
13
comment Prove $ |\vec{a_1}-\vec{b}|+ \cdots +|\vec{a_n}-\vec{b}| > n $
The condition given is $|\vec{b}|<1$, not $|\vec{b}|>1$.
May
1
comment Variance over two periods with known variances?
A clear explanation, easily generalizable to cases with more than two sets.
Apr
23
comment Geometric interpretation for sum of fourth powers
30 comes from the fact that the Bernoulli number $B_4=-1/30$. See "Faulhaber's formula" in Wikipedia.
Jan
17
comment The volume of the solid obtained by revolving the region bounded by $y^2=x$ and $x=2{y}$ about the $y$-axis.
The interval of integration is [0, 2], not [0, 1]; the result seems correct.
Jan
17
comment Let $a, b$ and $c$ be the lengths of the sides of an arbitrary triangle. Pick out the true statements.
What will the value of $x$ be if (i) $a=b=c$, or (ii) $a=b\gg c$?
Nov
20
comment Negative real parts and of the solution of a polynomial and stable matrices
This is called "the Routh-Hurwitz criterion". For a proof, see p.78 (Theorem 11) of this book.
Nov
18
comment Proof of Convergence: Babylonian Method $x_{n+1}=\frac{1}{2}(x_n + \frac{a}{x_n})$
@Clash en.wikipedia.org/wiki/…
Nov
15
comment Elegant way to prove this inequality
You're welcome.
Nov
12
comment What is the integral of $x(1-x)^8$?
Why $x=(x-1)+1$? Because I want to treat $x-1$ as sort of a unit, which makes the calculation easier.
Nov
12
comment What is the integral of $x(1-x)^8$?
$(x-1+1)(x-1)^8=(x-1)\cdot (x-1)^8+1\cdot (x-1)^8$ follows from the distributive law.
Nov
11
comment Finding pairs of integers such that $x^2+3y$ and $y^2+3x$ are both perfect squares
Symmetry ($x^2+3y$ and $y^2+3x$) suggests that without loss of generality we can assume $y\ge x$. Also notice that if $x\sim y$, $y^2+3x$ is not much larger than $y^2$, so it may be equal to $(y+1)^2$ or $(y+2)^2$... To discuss this rigorously you need to assume $y\ge x$.
Nov
10
comment Show that the value of a definite integral is unity
As Dinesh points out, $g(x)=\frac{f(x)}{f(x)+f(6-x)}$ satisfies $g(x)+g(6-x)=1$ (more generally, $g(x)+g(a-x)=b$ where $a$ and $b$ are constants); this is the condition where you can use this integration trick.
Nov
10
comment Is $ x^2+ax+a$ irreducible over ring $\mathbb{Z}$ of integers?
@pedja I think in that case, the same method as posted by Bruno Joyal works (i.e. think in $\mathbb{F}_2[x]$).
Nov
10
comment Must every event have a probability?
The probability that you can find life on Mars (or a Nessie in Loch Ness). I don't think it is "impossible to define any probability", but I think it is mathematically ill-defined (is it?).
Nov
10
comment Show that the value of a definite integral is unity
Because $y$ is a dummy variable; please see my edited comment.