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Aug
27
comment Online tool to check if number is rational or irrational?
Any number you can type in by entering a finite number of digits is always going to be rational. I guess you mean you want to enter expressions like $\sqrt{7}$, $\pi/5$, and so on, and see if they are rational. But this is a surprisingly hard problem in general: for example, nobody knows whether $\pi + e$ is rational or irrational.
Aug
27
revised Finding a logarithmic function from a graph
deleted 3 characters in body
Aug
27
comment Finding a logarithmic function from a graph
It should probably be $-x+1$ in the argument of $\log$. When $x=0$, you want to take $\log 1$, not $\log(-1)$.
Aug
26
comment Can you approximate a vector field?
Anything you can do to a scalar field, you can do to the components of a vector field... Well, not quite: anything linear in the scalar values can be applied to the components of a vector field, and the result will be sensible under change of basis of the vector values. So you can do bilinear interpolation, spline interpolation, componentwise Fourier transforms, and so on...
Aug
26
comment A way to identify a unique number '$a_1$' has performed operations on '$x$'
Do you get to pick the numbers $x$, $a_1$, $a_2$, and so on, or are they all given to you and you only get to choose the operation?
Aug
26
comment Why do $r=\cos[2\theta]$ and $r=\frac{1}{2}$ have 8 intersections?
More formally: The curves are $(x,y)=(\cos2\theta\cos\theta,\cos2\theta\sin\theta)$ and $x^2+y^2=1/4$, which intersect when $\cos^22\theta=1/4$, or $\cos2\theta=\pm1/2$.
Aug
25
comment Yet another differential equation
This is a damped harmonic oscillator (cont'd.).
Aug
25
comment uniform sampling of the sphere w.r.t. inner products other than the usual one
If I understand correctly, you want to sample the set $S = \{v: v^TAv=1\}$ in such a way that the distribution is uniform when $S$ is linearly mapped to a sphere? Then it should be $v^TAv=\|f(v)\|_2^2$, given that the left-hand side is quadratic in $v$, right? Sorry, I don't have an answer to your actual question.
Aug
25
comment What's the thing with $\sqrt{-1} = i$
Editing this answer seems to be a yearly ritual for you. :)
Aug
25
comment linearity and nonlinearity of the PDEs below
By $ux$ do you mean $u_x$, i.e. $du/dx$? If so, have you expanding out the linearity conditions for (c) like you did for (a) and (b)?
Aug
24
comment Rectangular WZYX inside the rectangular ACDB
It may not be possible to keep both $Y$ and $Z$ on the edges of $ABCD$ when you move $X$ around.
Aug
24
comment Learning to differentiate, with respect to $f_{\overline{z}}$
You differentiate a function to find its derivative. It's confusing, I know.
Aug
24
comment what is the geometric idea of this theorem?
You go from $(x,y)$ to $(x,y+t)$ applying the mean value theorem on $t$. Then you go from $(x,y+t)$ to $(x+s,y+t)$ applying the mean value theorem on $s$. The latter doesn't quite fit because $u_x$ is evaluated at $(x+s^*,y+t^*)$ and not $(x+s^*,y+t)$; are you sure that's not a typo?
Aug
24
revised Comparison of norms
edited title
Aug
24
comment How to precisely distinguish vectors and points?
I'm not sure I understand what you are looking for when you say "define precisely the distinction". It might help to explain how the answers to the previous question on the distinction between vectors and points are not what you are looking for.
Aug
24
comment Can all Integration and differentiation of a real function form be determined?
The question is unclear, but perhaps you mean How can you prove that a function has no closed form integral?
Aug
24
answered Rounding of complementary percentages
Aug
24
comment What are the Eigenvectors of the curl operator?
This paper "Eigenfunctions of the Curl Operator..." (Moses, 1971) turned up in a quick Google search, but given that you're considering $\nabla$ to be an eigenvector I'm not at all sure it's what you're looking for...
Aug
24
comment How to find the equation for a graph of a polynomial function?
You turned $a_1$ into $2a_1$ in the expression for $f'(x)$ going from step 2 to step 3.
Aug
24
comment Solving an optimization problem involving reciprocals
@Guy: Yes, $v_i^{-2} = d_i/\lambda$ would be the usual way to write it, though it makes no difference to the solution. The functions $v_i^{-1}$ are convex as can be seen from their second derivatives; $\sum v_i^{-1}$ is a sum of convex functions and is therefore convex; $\sum v_i^{-1} \le 1$ is the sublevel set of a convex function, so it is a convex set. Although on second thought it doesn't really matter, because the objective is linear and so cannot have an optimum in the interior in the first place.