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Aug
16
comment Why is any number divided by 0 is infinite?
Actually, the non-continuity of $\frac{1}{x}$ at $0$ isn't the main reason that we don't usually have a number called $\infty$ and define $\frac{1}{0} = \infty$. The main reason for not doing this is IMHO that this would lead to $\frac{1}{0} = \frac{2}{0} = \infty$, meaning you still wouldn't know how to define $0\cdot\infty$. In other words, if you add the special number $\infty$ the resulting rules for algebraic manupulation have to contain more special cases and exceptions, not fewer...
Aug
16
comment Why do we use “congruent to” instead of equal to?
If you use $\textrm{mod}$ to denote the binary modulo operation, you shouldn't IMHO use \mod in $\TeX$, but instead write \textrm{mod}. The increases whitespace on the left-hand side produced by \mod makes things look weird if used as a binary operation, and blurries the distribution between the binary operator $\textrm{mod}$, and the clause $\mod n$ which indicates that an equality holds in $\mathbb{Z}_n$, not $\mathbb{Z}$.
Aug
3
answered K is the least positive integer divisible by all positive integers less than or equal to 10. Find the total number of factors of K?
Jul
4
comment Random variable with all higher order moments zero?
If there is, it's characteristic function would need to be a second-degree polynomial, i.e. of the form $\varphi(t) = a\cdot t^2 + b\cdot t + c$. This follows from $\mathbb{E}(X^k) = (-i)^k \varphi_X^{(k)}(0)$.
Jul
4
comment Why can't you add apples and oranges, but you can multiply and divide them?
@mike4ty4 True, but putting it that way pre-supposes that we must be able to express quantities as a scalar times a unit. This is true, of course, and also intimately connected to scale invariance, but it's nevertheless that something I'd have had to argue.
Jun
30
answered What are the most prominent uses of transfinite induction outside of set theory?
Jun
10
awarded  Nice Answer
May
25
comment System of linear equations: get approximate solution with non-negative coefficients
To be able to formulate an algorithm, you first need to formalize your notion of closeness. You seem to want to minimize $d(\vec{m}_3, x_1\cdot\vec{a}+ x_2\cdot\vec{b},+x_3\cdot\vec{c})$ according to some distance function $d$, but you didn't specify how that distance is defined.. A possible choice would be $d(u,v) = \|u-v\|$, i.e. the euclidean distance, but there are other options. (And btw, your notation seems to be inconsistent. In the first equation, the vectors are $\vec{a}$,$\vec{c}$, $\vec{w}$, but in the second they are $\vec{a}$,$\vec{b}$, $\vec{c}$)
May
25
comment derivative of $\ln(4)$
The derivative of a constant is always zero, so in particular the derivative of $\ln(4)$ is zero.
May
25
comment Isomorphism between vector spaces of linear transformations
What have you tried?
May
25
revised Invertible , bounded linear operator on a Hilbert space
added 164 characters in body
May
25
comment Invertible , bounded linear operator on a Hilbert space
@NateEldredge On second thought, my operator is bounded, but since its also compact, it can't be invertible. It's definitely injective, but surjectivity fails I believe - there's no pre-image of $(\frac{1}{i^2})_{i\in\mathbb{N}}$, since the only candidate would be $(\frac{1}{i})_{i\in\mathbb{N}}$ but that doesn't lie in $H$. So I guess this is not what the OP was looking for (or I now managed to utterly confuse myself - it's late here...)
May
25
revised Invertible , bounded linear operator on a Hilbert space
Fixed notation
May
25
comment Invertible , bounded linear operator on a Hilbert space
@NateEldredge Yeah, I wanted to write $\ell^2(\mathbb{N})$ but couldn't figure out how to get the small script "l". Thanks to your comment I now know that the correct command is \ell. Will fix.
May
25
answered Invertible , bounded linear operator on a Hilbert space
May
25
revised Proving that a function is discontinuous
added 442 characters in body
May
25
answered Proving that a function is discontinuous
May
23
comment How to check that the function is not absolute continuous
Do you have a specific function in mind?
May
20
comment How to compare dispersion of data?
There are different measured of dispersion. The standard deviation is one of them, and it can be used sensibly for a large class of non-normal distributions. But, as you have observed, a few very large outliers can greatly influence it. If this makes it unsuitable for your application, then you'll have to explain what exactly you are trying to achieve better so that people can suggest alternatives.
Mar
27
answered Why is a function space considered to be a “vector” space when its elements are not vectors?