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Apr
10
awarded  Yearling
Apr
8
revised Computing the inverse error function
added 6 characters in body; added 2 characters in body
Apr
8
revised Computing the inverse error function
added 147 characters in body
Apr
8
answered Computing the inverse error function
Apr
2
answered Uniqueness proof for $\forall A\in\mathcal{P}(U)\exists!B\in\mathcal{P}(U)(C\setminus A=C\cap B)$
Feb
26
awarded  Popular Question
Jan
20
revised Prove that the preimage of a prime ideal is also prime.
I REALLY hate the 6 character limit on edits, especially on a math site, where each character matters.
Jan
20
suggested approved edit on Prove that the preimage of a prime ideal is also prime.
Jan
19
comment Binary operation commutative, associative, and distributive over multiplication
Nice, but the question wasn't for it to distribute over itself, but to distribute over regular multiplication. (Which this might do, I haven't checked.)
Dec
16
comment Lebesgue Measure - positive measure sets not containing intervals
In part 2., the $\alpha$ is fixed. You can have a positive measure set $A$, such that $m(A \cap I)<1$ for all intervals, but as you take smaller and smaller intervals $I_n$ "zooming in" on a part of $A$, then $m(A \cap I_n)$ arbitrarily approaches 1. Part 2. works because the $\alpha$ bounds this measure away from 1 (by some tiny amount).
Dec
4
comment Minimize the Frobenius norm of the difference of two matrices with respect to matrix: $\underset{B} {\mathrm{argmin}} \left\| A- B \right\|_F$
This is VERY nice, but I must admit I have no idea how you did it. I see what you're doing, but not how. In your very second line, how do you differentiate the Frobenius inner product without first converting it to some other notation and then trying to figure out how to convert back?
Dec
3
comment Minimize the Frobenius norm of the difference of two matrices with respect to matrix: $\underset{B} {\mathrm{argmin}} \left\| A- B \right\|_F$
@nullgeppetto Those diagonal elements will contribute the smallest square sum when their average is 0 (when they are "on average" as close to zero as they can get). Thus taking $b = \mathop{avg}\limits_{i} \{d_A(i)\}$, so that $B$ subtracts off the average of the diagonal of $A$, will minimize this case.
Dec
3
comment Minimize the Frobenius norm of the difference of two matrices with respect to matrix: $\underset{B} {\mathrm{argmin}} \left\| A- B \right\|_F$
@nullgeppetto Picture the graph of a function $d_A(i)$, where this is the $i^{th}$ element on the diagonal of $A$. Since your $B$ is diagonal, no matter how it changes, it won't affect the off diagonal elements. Since the Frobenius norm squares and then sums each element, you want $B$ to modify the diagonal of $A$ so they contribute as little as possible. When $B$ was any diagonal matrix, taking $d_B(i):=d_A(i)$ gave $d_A(i)-d_B(i)=0$, so they contributed nothing. When $d_B(i)=b$, $d_A(i)-d_B(i)$ will slide the diagonal elements of $A$ up and down equally.
Dec
3
revised Minimizing Frobenius norm of a special type
added 580 characters in body
Dec
3
answered Minimizing Frobenius norm of a special type
Dec
2
comment Is “product” of Borel sigma algebras the Borel sigma algebra of the “product” of underlying topologies?
+Brian I don't mean to nitpick, but for clarity for later readers, the standard product $\sigma$-algebra (generated by cylinders) and the "box" $\sigma$-algebra (generated by rectangles) are still the same for countably infinite products (because you can intersect countably many cylinders to get the rectangles). If you're indexing uncountably many dimensions, every cylinder is an "infinite" rectangle, but you can not intersect uncountably many cylinders to get the smaller rectangles anymore, so the product $\sigma$-algebra is properly contained in the "box" $\sigma$-algebra.
Nov
16
comment Solving the recurrence $a_{n+2} = 3a_{n+1} - 2a_n, a_0 = 1, a_1 = 3$ using generating functions
Some of us just want hints, and only look for explicit results after failing to be able to work it out. Yes they can find this online, but providing it takes away that choice.
Nov
16
comment Solving the recurrence $a_{n+2} = 3a_{n+1} - 2a_n, a_0 = 1, a_1 = 3$ using generating functions
Since that's the simplest power series there is, I don't think it should be given out on homework problems. It's easy to compute.
Nov
16
answered Solving the recurrence $a_{n+2} = 3a_{n+1} - 2a_n, a_0 = 1, a_1 = 3$ using generating functions
Nov
16
comment Solving the recurrence $a_{n+2} = 3a_{n+1} - 2a_n, a_0 = 1, a_1 = 3$ using generating functions
It's been a long time since I've worked with generating functions, but it seems you should be able to figure out the power series for $1/(1-z)$ and $1/(1-2z)$, figure out the constants $C$ and $D$, and then you have $A(z)$ as a linear combination of two other series, so combine them term by term.