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Jan
26
comment How to solve the Few Scientists Problem (big word problem) in its general form?
This doesn't answer the intended question, which was even written in bold in the original post: "what I really need is how to solve this in its general form."
Jan
26
comment “Convex” polynomials
I guess you are only considering the restriction of $f$ to $[0,1]$, and you mean $p(x)=4x-4x^2$.
Jan
26
revised Postive-semidefiniteness of matrix with entries $1/(a_i+a_j)$
edited title
Jan
25
comment “Convex” polynomials
Wait, $f(x)=\frac12x$ is in $\mathcal C$ but $f^{-1}(\{0,1\})) = \{0,2\} \not\subseteq \{0,1\}$. Also, $p(x)=4x^2-4x$ maps $[0,1]$ to $[-1,0]$ not $[0,1]$. Am I missing something?
Jan
25
comment Number of optimas of product of convex functions
For multivariable functions you can more than two local minima, for example when $f(x,y)=10x^2+(y-1)^2$ and $g(x,y)=(x-1)^2+10y^2$ then $f(x,y)g(x,y)$ has minima near $(0,1)$, $(1,0)$, and $(0,0)$. Probably lots of local minima are possible if $f$ and $g$ can be less than zero.
Jan
25
comment What's so special about involute curves??
I assume you've seen this? "Although many tooth shapes are possible for which a mating tooth could be designed to satisfy the fundamental law, only two are in general use: the cycloidal and involute profiles. The involute has important advantages -- it is easy to manufacture and the center distance between a pair of involute gears can be varied without changing the velocity ratio."
Jan
24
comment Number of optimas of product of convex functions
Suppose $f(x)=e^x$ and $g(x)=e^{-x}$, so that $h(x)=1$. Now consider replacing $f(x)$ with $(1+\epsilon\sin x)e^x$...
Jan
24
comment Do standard gradient descent methods work on complex variables
Is $f:\mathbb C\to\mathbb R$ or $f:\mathbb C\to\mathbb C$? If the former, then there is no problem, you can treat it like a function $\mathbb R^2\to\mathbb R$. If the latter, then it is not clear what it means to optimize $f$.
Jan
22
comment What equation would give me a graph like this?
math.stackexchange.com/a/321157/856
Jan
20
comment Is there any attempt to explain irrational numbers from a geometrical point of view?
"no matter how big you choose a and b to be, you never get a result which is not irrational"... as long as $a$ and $b$ are both rational.
Jan
19
comment geometric meaning of blowing up the affine space at a line
Is it not the same as the product of the blow-ups of the normal spaces at each point $p\in\ell$?
Jan
19
comment geometric meaning of blowing up the affine space at a line
Here's a related question about visualizing the blow-up of a plane at a point. Maybe it helps.
Jan
18
comment Laplace, Legendre, Fourier, Hankel, Mellin, Hilbert, Borel, Z…: unified treatment of transforms?
@Cameron: What would you like me to do about it?
Jan
16
comment Is $G^{(X, Y)} = (G^X)^Y?$ ($A^B$ just means that $B$ is mapped to $A$)
I think you're misunderstanding either what the notation $A^B$ means or what it means to map a codomain to a domain. Nevertheless, it is true that $G^{X\times Y}=(G^X)^Y$, this is called currying.
Jan
15
comment Functions Mapping Integers to Zero?
@David: That gives no greater generality because any $f(x)=h(x)g(x)$ obtained through your procedure can also be obtained as $\sin(\pi x)\left(\frac{h(x)}{\sin(\pi x)}g(x)\right)$ in Théophile's procedure.
Jan
15
comment What is the point of finding a limit? Does limit give us a real/exact value?
Alternatively, there is only one line passing through $(a, a^2)$ such that the parabola is entirely on one side of the line.
Jan
13
comment How to find the corners of a shape given 4 inequalities?
Considering every possible pair of inequalities gives you $\binom42 = 6$ candidate vertices. Discard the ones that do not satisfy the other two inequalities.
Jan
13
comment How does $(1+\sin x)\cos x =\cos x+\sin x \cos x$? I have a feeling i missed a basic fact
$(a+b)c=ac+bc$, y'know?
Jan
13
comment Function for this 3D curve
Try $x + y + z + a (x y + y z + z x) = 1$ for some $a>0$.
Jan
12
comment Generalizing the Cantor Set to the $n$-dimensional plane
See Cantor dust, and perhaps also Sierpinski carpet.