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Almost all of the questions on the front page these days are homework questions or textbook exercises. I think I'll be spending a lot less time here.

I hereby authorize whoever it may concern to post my comments as answers if they wish, whether as community wiki or otherwise if you like imaginary internet points.


Mar
23
comment An indirect optimizing problem
If $x$ and $y$ are unconstrained, $z$ never has a minimum because you can send $x \to -\infty$, $y=\epsilon$ or vice versa. If $x$ and $y$ are nonnegative, the minimum of $z$ is always $0$ at $x=0$, $y=0$, so changing $z_1$ and $z_2$ makes no difference. In either case it's not clear what you're asking.
Mar
22
comment Weighted nonlinear least squares fitting of circles
Are you familiar with Newton's method? You can define a cost function which adds up the squared error of distance from each beacon, and find the location that minimizes the cost using Newton's method. That's essentially what NonlinearModelFit is doing under the hood.
Mar
21
comment Why don't taylor series represent the entire function?
The analogy is perfectly fine. If you know the position, velocity, acceleration, jerk, etc. of a particle at one moment in time, you have no idea whether some other force is going to come along in the future and push it around.
Mar
21
comment Folding in $\mathbb{R}^2$ - which area of modern geometry formalises this concept?
It's not true. Fold an isosceles right triangle along its symmetry axis and you get another isosceles right triangle.
Mar
21
comment Can every smooth path be parametrized?
(...with additional smoothness properties, of course)
Mar
21
comment Do four dimensional vectors have a cross product property?
See the previous questions Cross product in $\mathbb R^n$ and Is the vector cross product only defined for 3D?
Mar
21
comment Can every smooth path be parametrized?
How do you define a smooth path? Often it's defined precisely as a function from $[a,b]$ to $\mathbb R^n$.
Mar
21
comment What are some alternative ways of describing n-dimensional surfaces using control points other than Bezier surfaces?
However, the more commonly used Bézier splines are not polynomials, but rather piecewise polynomial functions.
Mar
20
comment Solve without convergance?
Previously: Four turtles/bugs puzzle
Mar
20
comment What happens if to postulate that complex numbers whose argument differs by $2 \pi$ are not equal?
I guess what you're thinking of is working in polar coordinates but with $(r_1,\theta_1)=(r_2,\theta_2)$ iff $r_1=r_2$ and $\theta_1=\theta_2$ instead of $\theta_1=\theta_2\pm2n\pi$. In this system, multiplication, square roots, etc. are nice, sure, but it's not clear how you would define addition.
Mar
19
comment Best closed convex surface fitting N points in 3D
Well, if the points come from a convex surface with a few outliers, then the Eigencrust algorithm is likely to yield the same convex surface. If you want to guarantee convexity you can take its convex hull.
Mar
18
comment Why is it that $\overline{a}\langle u,v\rangle + a \overline{\langle u,v\rangle} = 2\mathfrak{Re}(\overline{a}\langle u,v\rangle)?$
For future reference: use \langle and \rangle for angle brackets and \overline for a wide bar.
Mar
18
awarded  Popular Question
Mar
18
comment Scaling a big range of small numbers to a small range of big numbers
I'd try $90x^k$ for some exponent $k$ between $0$ and $1$. For example, if you want $0.03$ to map to $50$ then use $k \approx 0.17$.
Mar
17
comment Is Lewis Carroll's reasoning correct?
@Omnomnomnom Quod erat faciendum, "which had to be done". en.wikipedia.org/wiki/Q.E.D.#QEF
Mar
15
comment discrete math: is there a difference between $\subseteq$ to $\supseteq$
Yes to "is there a difference between $\subseteq$ to $\supseteq$" or yes to "Is $X \supseteq I$ just the same as $I \subseteq X$"? :)
Mar
10
revised How can I show that if $P \in L(V)$ is such that $P^2 =P$ and $\|Pv\| \leq \|v\|$ for every $v\in V$, then $P$ is an orthogonal projection?
deleted 1 characters in body; edited title
Mar
10
comment Can we say that a scalar is a special case of vector?
Yes, the real numbers are a one-dimensional vector space, so any real number can be considered a vector. However, you still need the concept of a scalar because in general you can't multiply two vectors, but you can multiply a vector with a scalar.
Mar
9
comment Can numbers exist outside of the number space?
Figure out what a number is, you say? We have a question for that.
Mar
8
comment Can I rotate divergence away? / Can get divergence from a rotation?
You have to define what you mean by rotating a vector field. I can imagine several possibilities: (i) Take the vector at location $\vec x$ and rotate it by $R$. (ii) Take the vector at $\vec x$ and put it at location $R\vec x$ without rotating it. (iii) Take the vector at $\vec x$, rotate it by $R$, and put it at location $R\vec x$. The first is what Shuhao assumes. The third is what corresponds to picking up the paper that the vector field is drawn on and rotating the whole thing.