Hi! I came across an instance where the combination of a variable and a constant (e.g. 5+$x$) is also called a variable. I've never heard of the combination being called a variable before. While 5+$x$ will vary depending on the value of $x$, is it correct to call it a variable?
(I didn't even really finish 1.2, kinda skipped the last bit on bases and discriminants but I felt I probably wasn't gonna need it and I'm trying to make a decent amount of progress in... about a week or two)
@LeakyNun Oh ok. In this instance, a variable is defined as a 'symbol which can be given different values, and it represented by letters.' This is for grade 8 so they're just learning about variables and constants for the first time.
@Alessandro rip me, right after that he proves that rings of integers are Noetherian by referencing the fact that the integral closure of a PID in finite extensions of the field of fractions are finitely generated modules
Wait I'm not sure how to parse the statement "they are given as the quotient of an element of $A$ by the determinant $\det(Tr_{L\mid K} (\alpha_i\alpha_j)) = d$"
Does that mean those elements of $A$ are given as $\frac{x}{d}$ for $x\in A$?
Because if so I have no idea why that's the case
Oh oh wait hmm
I misread
I read $Tr_{L\mid K}(\alpha_i\alpha_j)$ and thought that the "they" was referring to those guys
When problem-solving, I keep making this same mistake: I think of a possible solution, I discard it thinking it won't work, then, after an hour of struggle, I find out the solution I thought of to begin with worked. Does anybody have the same experience? Does anybody have any advice regarding this issue?
Integral calculator gives something like $$\frac{\tan^{-1}\left(\frac{x}{2(\sqrt{2}+1)}\right)-\tan^{-1}\left(\frac{x}{2(\sqrt{2}-1)}\right)}{4}+c$$ :||
In everyday work who'd ever encounter that in everyday work? why would anyone want to integrate that by hand rather than just plugging it into WA or some other such service :P
If you guys have time , could you please look at my question ? Here is the link : https://bit.ly/2Nxp2Uu It has been on for 12 days and till now has no answers . Help would be appreciated . Thank you .
It seems to me that $\prod_{i=1}^\infty F_q$ has the cardinality of the continuum for any fixed $q$, where $F_q$ is a field of $q$ elements, but I have two questions I'm less certain about. What about $\prod_{i=1}^\infty F_{p^i}$? Then, I have the same questions for products of countable fields. Can any of these have cardinality different than that of the continuum?
This is just about sets, really. Pick a surjection from each set $X_n$ to {0,1}; using choice, $\prod X_n \to \{0,1\}^{\Bbb N} = \mathcal P(\Bbb N)$ is surjective.
On the other hand, $\Bbb N^{\Bbb N}$ has the same cardinality. (You can inject it into the reals using continued fractions.)
The actual Number Munchers was a bit too old for my time. We had the slightly more graphically advanced Math Munchers, as well as Math Blaster.
Of course, the name "Math Munchers" set against the name "Number Munchers" just makes me think about a game like, "Alright kids, get ready to blast the aliens using the fundamental theorem of finitely generated abelian groups!"
@Semiclassic was right when he said this is a homogeneous differential equation. Let $y/x = z$, or $y = xz$. Then $y' = xz'+z = z + \sqrt{1-z^2}$, so this is separable. You get $dz/\sqrt{1-z^2} = dx/x$. @Jasmine
@Semiclassical I haven't found the solution yet . But I was thinking of converting x and y into polar coordinates . x=rcosθ and y=rsinθ. Then we can write xdy-ydx=r^2 dθ and then proceed from there .
I have answered probably a dozen of such questions on multivariable limits. People are very fond of polar coordinates, but I disparage that unless it's absolutely necessary.
@TedShifrin not understanding...Okay leave it ... we dont have multivariable calculus in our syllabus...i just saw this problem and found it interesting...
We can use the Mean Value Theorem:
$$
\frac{\tan^{-1}(x)-\tan^{-1}(0)}{x-0}=\frac1{1+\xi^2}
$$
for some $\xi\in(0,x)$. That is
$$
\frac1{1+x^2}\lt\frac{\tan^{-1}(x)}{x}\lt1
$$
@Semiclassic: Well, projectivizing isn't going to be helpful. Better to think about removing $z=0$ and what's left deformation retracts to $|z|=\epsilon$, which is a $3$-dimensional cylinder in $\Bbb C^2 = \Bbb R^4$.
Make sure you play up your computing skills. You should look into consulting stuff. Of course, a lot of this is in the world of investment/finance, which may come crashing down soon.
Since abandoning academia, I've worked for a math education software company, a 360-degree video camera company, and now a robotics company. It's been great
self-driving car company is a job to aspire to, but I don't know how long they're going to last, or at least how long the current ones are going to last
You can do a ton of stuff with algorithms based on the SVD
I listened a guy at PyCon talk about search algorithms based on SVD, and I was convinced he didn't understand the first thing about what the SVD does, but he understood what it accomplished in searches
for context, in quantum information theory one likes to talk about quantum correlation matrices
which basically amounts to being able to factorize such a matrix as $C=A^\top B$
i.e. a matrix whose entries are inner products
(in this case the inner product is just the Euclidean dot product between real vectors. You get to QM by using a generic Hilbert space)
Anyways. You can imagine that the SVD is helpful for the above factorization problem
for better or worse, though, the above has mostly pushed me to learning about more linear algebra/stats than it has python. so not the best project for that. (good thing I have a different one)
@TedShifrin the resume is a big stumbling block for me. I drafted one a while back, but didn't format it
lol Big Data/ ML/ Deep learning etc are far more than fads, but they do have overhyped elements... just have to separate baby/ bathwater... or signal vs noise :) :P ps recently got repl.it to do scikit-learn code, it wasnt super hard, thought it was awesome :)
The following is stated in the book Analysis on Manifolds by James Munkres
Just as is the case with linear transformations, a multilinear transformation
is entirely determined once one knows its values on basis elements. That
we now prove.
And then he gives the following lemma.
Lemm...
I am reading a paper and they have the following: $\sum_{m=0}^\infty \left( 3^n b_n + (4 b_n + 2 n c_n) m^{n-1} \right) e^{-\pi m^2}$
n is a natural number and b_n and c_n are just some constants m is the radius of an n-ball now they go from this line to the following line $\le \alpha \sum_{m=0}^\infty \left( 3^n b_n + (4 b_n + 2 n c_n) \right) e^{-\pi m^2/2}$
I understand the exponential /2 part but somehow they pull that m out as a constant, but the sum is dependent on m. How is that even possible?
You can let $x= k t$ to find
$$ I(k) \equiv\int \limits_0^\infty \frac{-k \ln(x)}{k^2+x^2} \, \mathrm{d} x = \int \limits_0^\infty \frac{-\ln(t)}{1+t^2} \, \mathrm{d} t + \int \limits_0^\infty \frac{-\ln(k)}{1+t^2} \, \mathrm{d} t \, .$$
Your substitution $t \to \frac{1}{t}$ shows that the first ...
Wrt this answer, If I try to solve it using Leibniz Rule , I am getting $dI/dk$ to be divergent (after putting x = kt)
So is it not possible to use that method in this question. Any comment?
How do I find
$$\lim_{(x,y) \rightarrow (0,0)} \frac{x^2y}{x^2 + y^2}$$
We can use the Mean Value Theorem:
$$
\frac{\tan^{-1}(x)-\tan^{-1}(0)}{x-0}=\frac1{1+\xi^2}
$$
for some $\xi\in(0,x)$. That is
$$
\frac1{1+x^2}\lt\frac{\tan^{-1}(x)}{x}\lt1
$$
The following is stated in the book Analysis on Manifolds by James Munkres
Just as is the case with linear transformations, a multilinear transformation
is entirely determined once one knows its values on basis elements. That
we now prove.
And then he gives the following lemma.
Lemm...
