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May
21
comment Chi-Square Computations
stats.stackexchange.com/questions/121662/…
Mar
13
comment Evaluating the integral $\int \frac{1}{x+ \text{ ln }(x)}dx$
wolframalpha.com/input/…
Mar
13
comment The formula of Eclidean distance to a hyperplane.
If you consider $H$ as a hyperplane in $\mathbb{R}^{mn}$ and $X$ and $Y$ are two coordinates (i.e. $X,\,Y\in\mathbb{R}$), then $mn$ real numbers, not just $X$ and $Y$, are necessary to specify a point in $\mathbb{R}^{mn}$.
Mar
12
comment Weaker version of the A.M.-G.M. inequality
Related: mathoverflow.net/questions/122411/…
Mar
12
comment Positive numbers inequality
The above answer can be rescued if you show $\sum iy_i\ge \sum i\cdot\frac1n$ when $0< y_1\le y_2\le \cdots \le y_n$ and $y_1+\cdots+y_n=1$.
Mar
11
comment Positive numbers inequality
Where did you use $y_1\le y_2\le \cdots \le y_n$? and why is $\left(\frac2{n(n+1)}\sum y_i\right)^{\log_2 3}\ge \frac1{n^2}$ (the last inequality)?
Feb
17
comment prove that $f_n = 37111111…111$ is never prime
Try Factor[371], Factor[3711], Factor[37111] etc. (up to $n=10$) with WolframAlpha.
Feb
13
comment Model a chemical phenomenom
Yes, periodic boundary conditions are often used in mathematical / physical modeling and are a good starting point.
Feb
12
comment Model a chemical phenomenom
I think the lattice should reflect the actual shape of the receptor, but you could start with a rectangle (if it's 2D) or a cuboid (if it's 3D).
Feb
10
comment Express $\binom{n+2}{k}$ according to $\binom{n}{k}$
Similar question: math.stackexchange.com/questions/649567/…
Feb
5
comment Hard inequality with conditions
Then you can construct a counterexample easily, following the above argument. (such as in the case $b_2/g_2=r_2>1$ and $b_3/g_3=r_3<1$)
Feb
5
comment Find one solution for system of inequalities (if exits)
Thank you, I corrected the note.
Feb
4
comment Find one solution for system of inequalities (if exits)
$A(1)=0.020302010304655=20302010304655/10^{15}$ and not equal to $326/160723$ etc. I don't see why you want to convert a finite decimal to a different rational number (circulating decimal).
Feb
4
comment Find one solution for system of inequalities (if exits)
Really? Even though $0.112843485160579$ was changed to $0.112843485160578$? You might need to refresh the page.
Feb
4
comment Find one solution for system of inequalities (if exits)
Thank you, I changed $A(8)$ to satisfy $A(1)+\cdots+A(8)=1$ and updated the results accordingly.
Feb
4
comment Find one solution for system of inequalities (if exits)
@user64494 I'll check, but I think that can be dealt with without affecting inequality signs.
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 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.