@Semiclassical if you plant to enter this into a computer program, find out the language, then match the Multi-dim array ordering with your math notation
Same here. The first thing you want to derive is how natural transformations are represented as string diagrams, then show how adjoint unit / counit form a diagram where unfolding the string yields identity, then from there on it gets even more interesting. But the unfolding the string is nice, it's a simpler proof of unit/counit properties
and then use the failure of set-theoretic models to work as categories (b/c you have isomorphisms instead of equations) to motivate natural transformations
From that natural (natural transformation ie!) isomorphism (it's really a family of isomorphism maps), you can let $B = F(A)$ etc. That gives you $\text{id}_{F(A)} \in \text{Hom}_C(F(A), F(A))$ and from there you can derive the unit of the adjunction
I want to prove the following statement:
Suppose that $S$ is a non-empty subset of $\mathbb R^n$. Suppose that $f:S\to \mathbb R^m$ is differentiable at an interior point $c$ of $S$. Suppose that $f=(f_1,...,f_m)$, where $f_i:S\to \mathbb R$ for every $i\in \{1,2,...,m\}$. Then $f$ is differentia...
It has only two parts. To prove statement A holds iff statement B holds. The post contains: ($Rightarrow$) that is, proof for statement A $\implies $ statement B and then proof for converse.
@PenAndPaperMathematics I think I haven't been introduced to these so far in multivariable calculus.
I want to quickly say that if $f:S\to R$ is differentiable at c in S, where S is an open set and f attains maxima (local) at c, then $f'(c)=0$. I say: suppose that $f'(c)\ne 0$ then there is an $e_k$ (standard unit vector) such that $\nabla f(c). e_k\ne 0$. $f(c+he_k)-f(c)=h\nabla f(c).e_k+o(h)\leq 0$ (because LHS is always negative) in a small neighborhood of $c$). Sign of RHS depends only on the first term. If $\nabla f(c).e_k$ is positive then positive $h$ gives contradiction and if negative then negative h gives contradiction. So the only posibility is the gradient term must be zero.
95% are just cyberbullying each other online and the rest, i.e. the mathematicians, teachers, engineers, etc. of tomorrow, are learning math from ted's viral tiktok dances
But the fact that spin-1/2 fermions like positrons exists has to do with the spinor representation $\mathrm{Spin}(2n) \to U(2^n)$ decomposing into two irreducible representations of equal dimension each.
One of them is "electron", the other of them is "positron"
you wanna use \operatorname{Spin} or whatever the new syntax is for that. it won't space right as mathrm. at least if characters might be placed in front of it. if that ever happens.
But if you're into popular music and know a lot about history of popular music I highly recommend the book.
Or if you're into lefty politics and looking for a narrative that coherently answers why there's a "longing for old times" or why there's an "anxiety in the air" that you might often feel about modern technological world.
Hey guys I don't know if this is a stupid question, but I can't see an error in my work right now.
So let $f(x)$ be a differentiable function in $[a, b]$, and let's subdivide the interval using the following equally spaced points $\nu_0 = a, \nu_1, \nu_2, \cdots, \nu_n = b$. This means that as $n \to \infty$ then the length of the sub-intervals approaches $0$. Furthermore, assume that $f^{'}(x) < M$ for $x\in[a,b]$.
Focusing on the sub-interval $[\nu_{n-1}, \nu_n]$, we can write the following expression using the mean value theorem:
Balarka: I didn't know about ISI Kolkata/Bangalore till my class 12. I knew only of IITs. Had I known, I would have tried to get admission in one of those. :) I ended up taking engineering after class 12. :)
yes, most US systems have something called 12th grade which is the last year of high school.
in canada they invented 'grade 13' and they can't be trusted for at least that reason.
i figured out why the cat hangs out in my daughter's bedroom at night. we put her to bed, she immediately turns on the light and then begins throwing toys for the cat to chase. it went on for 20 minutes.
@Koro i know someone in tifr who quit midway through their masters, did a job for quite a few years, and now recently joined the phd program in math again
one reason is they students are not aware of various options available after highschool. I didn't have access to internet till my class 12 (year 2014) and I didn't know about ISI Kolkata/Bangalore etc. I knew only of IITs and that's because after the end of exercises in some books there was a special section: Questions from varous entrance exams.
And I solved those questions and got stuck at those tagged with IIT JEE.
So I thought that meant -International Institute of technology
Until one day when my teacher told me actual full form of IIT.
and that's when I decided to study at one of those. This happened around 2011.
@leslietownes this was the part i felt was the problem as well, but aren't all of the $f^{'}(\xi_n)$ contained in the range of the function? So i thought no matter what the value of $f^{'}(\xi_n)$ is i thought the fact that the length of the sub-intervals tending to zero would eventually make $f^{'}(\xi_n)(\nu_n-\nu_{n-1})$ tend to 0
Balarka, you will notice that a lot from engineering physics, Biotechnology etc. I have friends from engineering physics and Biotechnology who switched to Maths after their undergraduation.
But rare to see the switching from electrical engineering, civil engineering.
david; it will, but you're summing a number of terms of those that is growing with each n. the finite "limit law" that distributes the thing applies to fixed finite numbers of sequences, not unbounded, n-dependent numbers of them.
but anyway you can try to opt to do a masters in math in ISI. you have to prepare for the admission test, but that can only be a gain since you dont have a formal training in math at a more advanced level, you can pick it up as you prepare
the masters entrance is better than the undergrad one, less about competitive tricks and more about solid foundations
if a_{m,n} = 1/n for each m and n then a_{1,n} + a_{2,n} + ... + a_{n,n} = 1 for all n and hence lim_n (a_{1,n} + a_{2,n} + ... + a_{n,n}) = 1 also. and this holds despite the fact that lim_n a_{k,n} = 0 holds for every fixed k.
I think one reason is -not many companies visit for recruitment from engineering physics or Bio technology. At my college (IITG), I don't think any company visited to recruit from engineering physics or Biotech. They went for higher studies or took up IT jobs.
@BalarkaSen I understood that after I wrote the entrance in 2020. I didn't know group theory at all back then, it was new to me. I knew some linear algebra but not through operators;rather through matrices. I have worked on those things -group theory, linear algebra etc. I wish to appear in that entrance exam again. :)
david but note in the example lim_n (a_{1,n} + ... + a_{n,n}) is not lim_n a_{1,n} (which is 0) + lim_n a_{2,n} (also 0) + "..." (kinda meaningless because the number of terms is not controlled and the value of every one of them changes with n) + lim_n a_{n,n} (also 0)
I say that $f=o(g)$ iff there exists $\alpha (x)$ such that $f(x)=\alpha(x) g(x)$ and that $\alpha (x)\to 0$ as x tends to something. Isn't that very formal?
no detail is obscured I think. $f$ is diff. at c (talking one variable) then I say $\lim_{x\to c}\frac{f(x)-f(c)}{x-c}$ exists or equivalently $f(x)-f(c)=f'(c) (x-c)+o(x-c)$ as $x\to c$.
Note: I myself am not a math educator, though I plan to be one someday.
In this letter, Donald Knuth suggests an alternate way of teaching calculus, based on big-O (introduced via a related big-A notation). He says "it would be a pleasure for both students and teacher if calculus were taught...
I don't think the temp of the water for the yeast matters. I did everything by "Hey, Googling" with a device my dad developed at a company, but I could also wing all of the dough recipe from experience
@DavidChoi I know leslie has commented, but as an example consider $$\lim_{n\to\infty}\underbrace{\frac1n+\dots+\frac1n}_{n\text{ terms}}\overset{\text{?}}=\underbrace{\lim_{n\to\infty}\frac1n+\dots+\lim_{n\to\infty}\frac1n}_{n\text{ terms}}$$ which would imply that $1=0$. Note that the right hand side doesn't really make sense anyway, since $n$ in the limit is undefined.
@robjohn are there known formulae/estimates for functions like $\nu_p(n)$ (largest power of $p$ that divides $n$) and $\sigma_p(n)$ (sum of digits of base-$p$ expansion of $n$)?
Note: I myself am not a math educator, though I plan to be one someday.
In this letter, Donald Knuth suggests an alternate way of teaching calculus, based on big-O (introduced via a related big-A notation). He says "it would be a pleasure for both students and teacher if calculus were taught...
