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I'm in the math PhD program at UCLA, focusing on convex optimization.


1d
comment Find the conditions required for the values of a, b, and c that make the following matrix symmetric.
@Kaj_H The post had been vandalized.
1d
comment Advanced Calculus of Several Variables
Two good, popular books are Analysis on Manifolds by Munkres and Introduction to Smooth Manifolds by Lee.
2d
comment Mysterious Proof about Induced Norms (was: Uniqueness of SVD)
But, $|c| + |s| > 1$.
Apr
12
comment Recommended Textbook/Resources
Check out artofproblemsolving.com.
Apr
11
comment What is the definition of a positive integer?
I think one thing you might like to do is find a modern introductory number theory textbook that presents the axioms for the integers as a starting point, and builds the entire theory on that. Also, you could take a look at the answers to this question on math.stackexchange and see if it sheds any light on the issue.
Apr
11
comment Is a least squares method a cost function?
Hmm, there are (as you say) many ways to fit a line to data. One method (least squares) is to pick the line that minimizes the sum of the squares of the errors. Another method (not least squares) would be to pick the line that minimizes the sum of the absolute values of the errors. Other methods might not involve minimizing a cost function at all -- for example you could just pick two data points at random and use the line that connects them. The "sum of squared errors" cost function could be used to evaluate any of these lines, but that's only one way to measure how good a line is.
Apr
11
comment Is a least squares method a cost function?
When you fit a line to data using least squares, you are finding the line that minimizes the sum of the squares of the errors. That's actually what least squares does. Here's a picture to show the idea.
Apr
11
comment Prove that if m|a and m|b then m|a+b
Your proof that if $m|a$ and $m|b$ then $m|a + b$ is good. (Though perhaps you should mention $k,\ell \in \mathbb Z$.) Are you sure you can't prove the other one?
Apr
10
comment How do you prove n(n-1) by induction?
One issue here is that it doesn't make sense to "prove $n(n+1)/2$", because the expression $n(n+1)/2$ by itself isn't a statement. What Khan proved is that if $n$ is a positive integer, then $1 + 2 + \cdots + n = n(n+1)/2$.
Apr
10
comment Can I get a better image from two images that focused on different objects?
This is the kind of image processing problem that is often studied by applied mathematicians, so I think the question is on topic here. For example, the applied mathematician Stan Osher is an author on this article entitled "Shape from defocus via diffusion".
Apr
9
comment Visually stunning math concepts which are easy to explain
Along the lines of @MaximUmansky's comment, a two dimensional picture seems unnecessarily complicated. You could just draw a line segment, cut it in half, cut one of the remaining pieces in half, etc.
Apr
9
comment Comprehensive, rigorous calculus book with a small number of exercises?
You could just do the homework problems for one of the Calculus courses on MIT Opencourseware.
Apr
9
comment Help needed with Masters' Thesis
I also noticed that Karen has the Informed badge, which is a nice touch.
Apr
9
comment Could the fourth root of $1$ be $i$?
$i$ is a fourth root of $1$, yes.
Apr
9
comment How did Newton and Leibniz actually do calculus?
@user7530 well, I think you can find examples of arguments based on manipulating infinitesimals that lead to incorrect results. (But I bet a careful mathematician, even without $\epsilon - \delta$ proofs, would usually be able to recognize that those arguments are somehow not valid.)
Apr
8
comment The range of the controllability matrix
It helps to know that $Cx$ is a linear combination of the columns of $C$. More explicitly, if the $i$th component of $x$ is $x_i$ and the $i$th column of $C$ is $c_i$, then $Cx = \sum_i x_i c_i$.
Apr
8
comment How can I write an SDE in Matlab?
You can "divide through by $dt$" first.
Apr
7
comment Definition of tangent
@user72694 Probably I should have said that it's impossible for two distinct points on a curve in $\mathbb R^2$ to be infinitely close to each other, and noted that one can make sense of such ideas using nonstandard analysis (as you explain in your answer). I haven't studied nonstandard analysis but I'd like to learn more about it.
Apr
6
comment Definition of tangent
If $f$ is differentiable at $x$, then the tangent line to the graph of $f$ at $(x,f(x))$ is, technically, defined to be the line through $(x,f(x))$ whose slope is $f'(x)$. Note that it's impossible for two distinct points to be "infinitely close" to each other, so the definition you mentioned doesn't make sense.
Apr
6
comment What is the definition of a positive integer?
I think the answer to this depends on what approach you take to developing various number systems. If your starting point is a set of axioms for the integers, then the term "integer" may be left as an undefined term.