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Jun
27
revised Capital and small letter function notations
bold --> capital
Jun
26
comment Is There a $R^2 \rightarrow R^2$ Linear Transformation to Make XOR Problem Separable?
Neural networks aren't linear either; the activation function is typically sigmoidal. In any case, as pzf's answer shows, if neural networks were linear, they wouldn't be able to solve the XOR problem no matter how high-dimensional you made the feature space.
Jun
26
comment Comparing and contrasting equations and functions
An equation is something that has an equals sign in it. A function is something like a monkey.
Jun
21
comment Tables and histories of methods of finding $\int\sec x\,dx$?
Why is the integral in \displaystyle? It looks conspicuous on the front page.
Jun
21
comment An elementary (?) minimization problem
Consider the segment of the path from one of the given points to the straight line. If the segment is not straight, you can replace it with a straight line segment while keeping its endpoints fixed. This reduces the length of the path, so the original path cannot be optimal.
Jun
20
comment Minimize $\| ACE \|$ by geometrical means
@robjohn: Heh. I guess this question is simple enough that the Anna Karenina principle applies: All correct answers are alike; every incorrect answer would be incorrect in its own way.
Jun
20
revised Minimize $\| ACE \|$ by geometrical means
added note about Fermat's principle
Jun
20
comment Minimize $\| ACE \|$ by geometrical means
@robjohn, it certainly feels like a question that must have been asked before already.
Jun
20
comment Minimize $\| ACE \|$ by geometrical means
I tried to check whether this question had a duplicate before posting this answer, but I couldn't find one.
Jun
20
answered Minimize $\| ACE \|$ by geometrical means
Jun
19
comment Conditions stronger than differentiability or weaker than integrability
Apropos Leonid's comment, the relevant Wikipedia article: Smooth functions and differentiability classes.
Jun
17
comment $0 \leq a^2 + b^2 - abc \leq c \implies a^2 + b^2 - abc$ is a perfect square
Regarding my previous comment, make that at most two values of $c$. I had checked that the length of the range was at most $1$ and jumped to a conclusion, but you can have $a=1$, $b=1$, $c\in\{1,2\}$. I still think for $(a,b)\ne(1,1)$ there can't be more than one solution, though.
Jun
17
comment Why is it hard to prove whether $\pi+e$ is an irrational number?
@Qiaochu, I agree that the obvious answer to the stated question is "why not?", but negative results about what kinds of techniques cannot possibly work can still give much insight. For example, there are several results regarding what classes of proofs are insufficiently powerful to resolve P vs. NP. I've upvoted this question because I guess I'm hoping to learn something similar here.
Jun
17
comment Why is it hard to prove whether $\pi+e$ is an irrational number?
I don't think this is precisely a duplicate of the other question, as this one asks for references and discussion about why previous techniques are insufficient to resolve the problem. (I've edited the title to match.) This can be more illuminating than a simple yes/no answer, which is what the previous question received.
Jun
17
revised Why is it hard to prove whether $\pi+e$ is an irrational number?
direct link to subsection on Wikipedia page; edited title
Jun
17
comment $0 \leq a^2 + b^2 - abc \leq c \implies a^2 + b^2 - abc$ is a perfect square
The inequalities imply that $c$ must lie between $(a^2+b^2)/(1+ab)$ and $(a^2+b^2)/ab$. For any $a, b \ge 1$, there is at most one $c$ that can satisfy those conditions. I'm not sure if that leads anywhere.
Jun
17
answered Confusion regarding convex and affine set
Jun
17
comment Intuition behind two functions being equal up to order $n$ at $a$
Actually, if two functions agree to first order, then the more you zoom in, the less you can tell the difference between them! They have the same slope at $a$, but they bend in different ways, and you have to "zoom out" a bit to see that. (I'm assuming that by "zooming" what we mean is that we're looking at the graphs of $y = f(x)$ and $y = g(x)$ and scaling both $x$ and $y$ by the same amount, preserving the aspect ratio as if zooming into a picture.)
Jun
17
comment Dimension in mathematics and physics
Related (possible duplicate): What is a physical “dimension” - in the sense of “dimensional” analysis?
Jun
17
comment Product rule for the derivative of a dot product.
@Arturo: Sometimes I don't even do the former when I'm feeling lazy, so I'm not going to say anything! :)