Suppose you have a map $\sigma$ from a $k$-simplex to a space $X$. Now, partitioning the $k$-simplex to $n$ other $k$-simplices, we can restrict the $\sigma$ onto each of the simplices to get $n$ maps $\sigma_1, \cdots, \sigma_n$. The formal sums $\sigma$ and $\sigma_1 + \cdots + \sigma_n$ are formally different, but how to show the belong to same homology class, i.e $\sigma - \sigma_1 - \cdots - \sigma_n$ is homologous to $0$ ?
A rectangle has property that if you place two copies of it side by side the resulting rectangle of twice the area of the original has sides in the same ratio. Show that this ratio is $\sqrt{2}$.
but according to my calculation it is 1:2 and not same
but book says it is same and it did this
steps
suppose the longer side is $x$ and the shorter is $1$. Then the new rectangle has side 2 and $x$. Hence if these are in same ratio then $x/2=1/x$ so $x^2=2$
it is odd that it is not x/2:x/1 when you arrange the rectangle as book told me to do
So ? I don't understand where you're getting In your question you said " according to my calculation it is 1:2 " but your calculation gives a ratio $x=\sqrt2$, not $x=2$
@user791811 Since the new rectangle is a $2\times x$ rectangle, assuming $x<2$, then the ratio big/small side is $2/x$, which has to be equal to the original ratio $x/1$, right ?
Hello!! Is it possible to have a natural cubic spline, but at one of the intervals we have a polynomial of degree $2$ ? (All the other conditions are satisfied.)
@MaryStar (not an expert but) isn't the natural cubic spline a complete linear system ? (as in, n equations and n unknowns) ? As such, requiring an interval to be of order two would make it a n equations & n-1 unknowns system wouldn't it ?
Or do you relax some conditions on the degree 2 interval ?
@MaryStar But if we do so, it becomes trivial that a 2nd order solution exists since in particular a linear - 2nd order - linear solution can be created with four points
In several notes I see just that these have to be cubic polynomials of the form $a_0+a_1x+a_2x^2+a_3x^3$ but I haven't seen the restriction that $a_3\neq 0$.
Yes. Practical example: Your index set is the set of all your employees and the surjection maps each employee to their salary (think of this as, say, an excel table where you list how much you pay all your employees). Of course, two employees can get the same salary.
Let $f(x)=x^3-1$. To approximate the root $x^{\star}=1$, we consider the sequence $(x_n)$ that we get if we apply Newton's method with $x_0>0$. Show that the sequence converges to $1$.
I used $x_0=0,5$ and applied the method and in that way we see that the sequence converges to $1$.
Is that correct?
An other way could be: From Newton's method we get $x_{n+1}=\frac{1}{3x_n^2}$ and we have to show that this sequence converges to $1$, or not?
I have a question about the algebraic group $A^1$ of units of reduced norm one in a quaternion algebra $A$ over a number field $F$. Let $E/F$ be a splitting field of $A$ so that $A \otimes_F E \cong M_2(E)$. Is it possible to characterize the elements $a \in A^1$ which have diagonalizable images in $M_2(E)$ purely in the language of algebraic groups?
... something like "(regular) semisimple" elements, maybe?
@Abdullah: He's removed all the functions $\varphi$ that are nonzero on $C$, so all the remaining functions are zero on $C$. And they add up to at most $1$ at all the points of $A-C$.
This is a partition of unity, after all, but some of the functions that are nonzero on $C$ might also be nonzero at points of $A-C$, so perhaps the sum is a bit less than $1$.
Well, one fellow who was annoying me got to the point where I stopped answering his questions in comments, so he emailed me. I went back and answered and he asked an inane question, so I suggested he stop to think. He posted the question on main. Good riddance :P
Spivak is not the easiest book to read, so, if you don't know about them, I also recommend my YouTube lectures on multivariable analysis. They might come in handy from time to time, @Abdullah.
@TedShifrin next time I won't consider reading a book just because it's about 140 pages lol. To be frank I haven't even took an analysis course so reading this book is a huge challenge for me
You definitely need single-variable analysis experience (as well as a solid knowledge of multivariable calculus). You might check out my book (which integrates linear algebra in throughout the course).
@TedShifrin I read the book Thomas calculus and linear algebra by William goodie cover to cover I think I understood it well so I thought it's time to be rigorous because I need to study general relativity.
Spivak is not a great choice. Thomas's Calculus is an engineering text. If you don't want to look at (or spend money on or steal) mine, try C.H. Edwards's Advanced Calculus.
Reading cover to cover doesn't suffice for math, anyhow. You have to work lots of exercises, and with proofs you'll generally need someone to criticize what you've done.
You can get the link in my profile. There is no legal .pdf available, but the book seems to be at the usual illegal source of books that you all know about.
The other thing I noticed is that some Yale student uploaded there his transcription of my video lectures. His handwriting or pictures are not remotely as good as mine. I wonder why such things are uploadable?
@Abdullah: I'm not asking you to spend the money. I am telling you that if you go to libgen (which most students know about) you can currently find it there, so take advantage. I hope my publisher gets it removed eventually.
But the videos are totally free and pretty much free-standing. But you miss out on all my excellent exercises :P
I've also seen that some people upload what they allege are solutions to various of my differential geometry problems. Again, of dubious quality, and they also certainly never asked if I minded.
The person teaching the course at Yale resorted to my videos because of the pandemic. I think he plans to use them again next year. He didn't respond when I gently castigated him for not getting to forms and the variants of Stokes's Theorem. I consider that dereliction of duty, but not surprising when a combinatorist teaches such a course. I gather he's a good teacher, though, regardless.
I've seen students offer money for solutions to test questions in real time as they take the test. We had all sorts of fun when I was associate department head at UGA.
Ok suppose i break my unit interval into 0, a, 1 then the positive linear maps are $L_1(s)= as$ and $L_2(s)= (1-t)a +t $ so my $f_1(s)=f(as)$ and $f_2(s)=f( (1-t)a+t) $ so now $f_1*f_2= f(2as) , f(-2sa+2s-1) $ and now i need a homotopy between f and f1*f2
@Abdullah: Before I retired, I occasionally caught some of my own students posting homework questions (from my course!) on MSE. Suffice to say there was a stern speech in class the next day.
Those guys are way grumpier than this chat, I think, so waving online hi's are not usually part of the culture.
It's fine, he will never remember I was the guy who rambled about Postnikov towers in a random internet chat room when I get the Fields medal 20 years from now.
Oh yeah Dennis Sullivan wrote a 1 line answer below my long explanation of why S^1 x S^2 is not homotopy equivalent to S^1 v S^2 v S^3 just saying "oh yeah note that if you suspend once they become homotopy equivalent"
@TedShifrin I've been studying math for a while now but after some time I forget the proof of a theorem partially or completely but I use the theorem anyway is this okay from a mathematician point of view
If you understood the proof at one point, that's fantastic. I don't remember most of the proofs in my own papers. In my books, I think I do mostly because I've taught things so much.
you don't need to understand the proof in order to apply a theorem, but you most likely need to understand the proof in order to understand the theorem
@TedShifrin the reason for choosing Spivak's calculus on manifolds is that it had a fewer pages_which Is good compared to Harvard book on advanced calculus. and in the preface it was only assumed that the reader had a respected linear algebra and one variable calculus courses. Plus time is a major factor since I'm originally a physics major and I still have to study electrodynamics, thermodynamics,.. Etc
to me, understanding a theorem includes an understanding of why the theorem is true and I couldn't tell you why any of these theorems ought to be true without at least recounting the basic idea behind their proof
Citing the wikipedia intro: "Although the JCT is one of the best known topological theorems, there are many, even among professional mathematicians, who have never read a proof of it."
@TedShifrin i found the lectures but I have to wait till I Get back to the town since the network connection barely makes it possible to browse the web
@Abdullah: I think you won't really learn satisfactorily from that book. When I was a Berkeley grad student and faculty taught the Honors multivariable course out of it, I had to spend hours every week helping the undergraduates (and that was with a teacher). Sometimes a few more pages and examples are far better.
@Thorgott: Even in calculus, many proofs give you no insight into why the thing really works. My rule of thumb when I taught multivariable calculus for science/engineering (not pure math) was that I would only do proofs when the proof gave that insight. It was a handful of the theorems in the course.
Well, the reason it's difficult is because Jordan curves can be a lot more pathological than our intuition permits. I wouldn't call it being intuitively plausible a good reason for it working out mathematically.
By the way, a great book for you to look at because of your physics interest, @Abdullah, is by Bamberg and Sternberg (from Harvard): A course in mathematics for students of physics.
Yes, I think that in elementary number theory that is probably right. And things in algebra like understanding why $\Bbb Z$ is a PID.
And for what it's worth, anything we think about intuitively will be a piecewise smooth curve and the theorem is easier for those curves, the reason being that they locally divide the plane - and this can be turned into a proof.
I think the theorems in calculus that have genuine content also have insightful proofs. The ones whose proofs generate little insight tend to be the technical lemmas that themselves aren't very insightful, in my experience.
I might agree with Thorgott for something like Poincaré-Hopf or the Hopf Degree Theorem.
The problem is that the way we teach calculus in the US (i.e., we do not teach all college students an analysis course), there are black boxes everywhere — limits, maximum value theorem, intermediate value theorem. But I would still prove that at a local extremum the derivative must vanish, Fundamental Theorems of Calculus, and a few others.
The fundamental fact for understanding most of these theorems from calculus I did emphasize when teaching the theoretical course: If a function is continuous and positive at a point, it must be positive in a neighborhood of that point. If you actually have a rigorous definition of limit, this is crucial.
Yes, I'm agreeing those are insightful. Same thing with fundamental theorem of calculus for line integrals, Green's Theorem (for a rectangle), and a few others.
About 460, but a fair portion is linear algebra that, presumably, you already know, although perhaps you don't know it as well or as geometrically as you might.
Anyhow, I'm not trying to twist your arm. But I provide a lot more intuition for things than Spivak, and it's not quite so technical.
I am being whiny about calculating, of course. I can write something down here: $4 f_{z_p \bar{z}_q} = (f_{x_p x_q} + f_{y_p y_q}) + i(f_{x_q y_p} - f_{x_p y_q})$
I don't know. So you want an $n\times n$ thing out of the $2n\times 2n$ thing. If the real Hessian is $\begin{bmatrix} A|B \\ \hline C|D\end{bmatrix}$, then you want $A + D + i (B-C)$.
