well I'm pretty sure kids are taught equivalence of statements like 0.5 = 1/2 etc so why not go the extra mile to introduce infinite decimal expansions through geometric series
Another perhaps silly question. We have "Axiom. of extension. Two sets are equal if and only if they have the same elements." But what do we really mean by the same elements? Would it be more correct to write "Axiom. of extension. Two sets are equal if and only if every element of the first is an element of the second and vice versa" or should I be able to convince myself that "have the same elements" actually means something? I suspect my second writing is more correct since only
@TedShifrin My sister is taking calc 1 for the first time and she's turned a leaf on math (she actually enjoys it a lot now) but she had confusion with limits and approaching numbers from the left/right I don't think it would have been as confusing if she were taught infinite decimal expansions
I was thinking before 1900s like calculus probably had so many different forms. geometry was probably standardized since the ancient greeks. algebra probably somewhere in between
@Thorgott Cool problem: Prove that for any closed oriented surface $\Sigma$ of genus $g \geq 2$, there exists a pair of closed embedded curves $C_1, C_2$ on $\Sigma$ such that $\Sigma \setminus (C_1 \cup C_2)$ is a union of disks.
oh disregard my question: $k$ can be anything in the form $3s$ where $(3)^2 \nmid k$
Just to confirm, if $f(x) \in F[x]$ for a field $F$ and the degree of $f(x)$ is $5$ but $f(x)$ has no roots in $F$ and $f$ has no quadratic factors, we can say $f$ is irreducible right?
since you cannot express the sum of $5$ with numbers greater than 2
I feel so defeated in recitation for this algebra class lol. The TA is so smart and helpful but everyone has been complaining that the work is too hard and so now the expectations are so low
I wasn't complaining but I too had a hard time lol. It 100% wouldn't be a problem if we all dedicated more time to reviewing & studying but yeah, it is what it is
I always tried to encourage my students, but I never dropped standards. I taught one class years ago in which I gave no A's because no one earned one. The person with the highest B never spoke to me again. shrug
@TedShifrin I'm curious how you feel when students go up to you and ask if they can earn extra credit or plead for a better grade lol. After a while I'm guessing you get desensitized to it.
Yeah, that actually happened relatively rarely to me. I mean, I often had close to 10 office hours a week to work with and help students. And, in certain courses, I made it very clear that in order to pass one needed to demonstrate certain particular skills.
@Tedshifrin what was your favorite part about your job? Did you find it as fulfilling as doing research/working on higher level stuff or was it something to pay the bills lol
profs are like shepherds of knowledge but I'm sure it gets old answering the same questions a million times over, but maybe there's enough variance that it doesn't get old
In "How To Solve It", George Pólya writes:
"There was a seminar for advanced students in Zürich that I was teaching and von Neumann was in the class. I came to a certain
theorem, and I said it is not proved and it may be difficult. Von
Neumann didn't say anything but after five minutes h...
@Obliv For the most part, I absolutely loved teaching. A few students were annoying. When I taught future elementary school teachers, they hated me because I didn't give them As for effort — they actually told me that they were taught in their school of education classes that a sign of a good teacher is that all the students get high grades. In the end, I retired (thank goodness pre-Covid) because I was no longer motivating the majority of my students to work hard and do their best.
@user85795 For the record, I taught quite a few students who were exceptional and clearly more talented than I. It was great fun. I was not defensive about it at all.
@user85795 von Neumann said stuff like "If you say why not bomb them tomorrow, I say why not today? If you say today at five o' clock, I say why not one o' clock?" in reference to the atomic bombings in WW2 so I'd be pretty uncomfortable around someone like that as well.
@TedShifrin Yikes back. I do think that elementary school teachers have a big influence on kids though, so they're probably just worried that they'd damage their confidence if they reflected their actual abilities.
How do you know what they meant? They literally told me to my face that their teachers had taught them that I was a horrible teacher because they were not getting all As in my class. One young woman used profanity at me as I handed back her homework with a grade of 6/10 (approximately), yelling that she had spent a whole 40 minutes on her homework and how dare I.
It was amusing. There was a particularly bright young woman in the class who had been an engineering major at Georgia Tech before transferring and changing majors. She had had a few years of calculus, etc., and we were discussing how to understand arithmetic, things like multiplication and division of fractions. At any rate, she got it immediately.
One day before class, she and I exchanged a bit of witty banter. I don't remember about what. On my end-of-semester evaluations, a handful of them wrote a paragraph complaining how inappropriate it was for me to have flirted with a student in the class. So many eye-rolls.
And I actually learned some good stuff teaching that class. Division actually has two distinct meanings. How many groups of $b$ in $a$, or ... if I want $b$ groups, how many should be in each group?
I certainly was not taught it as conceptually as I tried to teach these future teachers (based on a wonderful textbook written by one of my colleagues).
Here are a few questions from one of my exams in that class.
Bobby tells Susie that $2\div\frac34$ is $2\frac12$. Is he correct? If so, explain why. If not, explain his misconception and show him (using the {\bf meaning} of division) how to fix his answer. Diagrams are recommended.
A pizza parlor offers ten toppings from which Tiffany may choose. How many different pizzas with exactly {\bf three} different toppings might she order? Explain your answer is such a way that someone who is not in this class will understand.
Consider the multiplication problem $\dfrac34\times\dfrac25$. (a) Make up a story problem or real-world example where this problem might occur. (b) Using the {\bf meaning} of multiplication (and {\bf not} a formula for multiplying fractions), explain how you might calculate $\dfrac34\times\dfrac25$.
Extremely obstinate, as in He's stubborn as a mule about wearing a suit and tie. This simile evokes the proverbial stubbornness of mules, whose use as draft animals was once so common that the reputation for obstinacy can hardly be as warranted as the term indicates. [ Early 1800s]
I think that plenty of math majors would learn a great deal from a course like this, actually. Yes, of course, we work with the rational numbers as the field of fractions of an integral domain, but conceptually that is far from what most people can handle. And appropriately so.
Average grade 2.54/4 (so C+/B- border). 6 As, 10 Bs, 9 Cs, 2 Ds, 1 F.
Interestingly, I remember that one of my students, who got a B and was the child of two teachers, subsequently asked me for a letter of recommendation.
@Soumik The students do not know what grades are given, unless they talk to all their friends in the class.
@Jakobian For only one guy or for the D? In the US it's almost all women who teach elementary school, very few men. At the high school level, most of the men who do it do it because they want to coach sports.
more or less. I think i've worked out the math on paper, but there's some error somewhere. trying to figure it out. my radius formula for the circle has a square root in it and the numbers im getting plotting are negative inside the square root
so i've got some problem somewhere. trying to decide whether its the numbers im using or a poor derivation
well, it depends. in my old model, when i set the intial leg lengths, they didn't satisfy the triangle inequality so i got undefined errors. nothing was wrong in the derivation, but i put in bad parameter values. the same could be true here
I mean, the final equation I used was the simplifed form from the wolfram page but the same stuff. I can use the one you have instead and try it.
I could have also entered it wrong into geogebra. its very clunky
I also did verify that biomechanically its fine to use $\mathbf{u} = \mathbf n \times (0,0,1)$ for the basis vectors, so I did go with that for my first basis vector in the plane and then the cross product $\mathbf v = \mathbf n \times \mathbf u$.
So in $n$ dimensions, if you have an extrema, you can't solve in terms of any other variable and get an extrema, or it doesn't generalize that strongly?
Something I find amusing is when you find people whose jobs profit from climate change. For example in Germany there is a practical explosion in places you can grow wine, and the quality of wine from old slopes has drastically improved in the last decade.
These people are acutely aware that climate change is having a real and strong impact on the world right now -- in a way most people are not. But due to luck the change is helpful for their ventures.
@TedShifrin yeah but the implications are huge. its not the math that matters. its the implication of what you can use it for. im excited for the medical insights
@TedShifrin I'm thinking of mapping to $\mathbb{R}$. So like $z = f(x, y, a, b, c, ...)$ has at least $1$ max or min. Does it follow that $b = f(x, y, z, a, c, ...)$ does not have any?
@AlessandroCodenotti hm wait but these notes say that the converse of the theorem is not true. but making the additional statement $a \in U$ makes the theorem an if and only if, no?
oh yes as corrected they would be identical. but the notes (from my prof from which the screenshot is taken) explicitly says the converse of thm 2.0.11 is not true, which makes me think they intentionally left out the statement $a \in U$, else the converse would definitely be true (as is the case for the corrected theorem)
although i should try to construct a counter example for the uncorrected theorem itself. i also emailed them about it :P
I think I see the problem. I'm using inverse function in higher dimensions imprecisely. I don't mean all the inputs have to be recoverable from the output (as in matrix inversion). I just mean solving something like $z = 3x - 2y + sin(a)$ for $x$ in terms of $y$, $z$, and $a$ and then looking for extrema of $x$. I'm wondering if it's the case that for any multivariable function union the set of multivariable functions obtainable by swapping the output variable for one of the input...
variables, at most one function in that set can have any extrema.
I can prove all of the above from the definitions but want to ask a "softer" question
Is there some useful (in your opinion) way to think about the asymmetry between the image and preimage functions in terms of how they commute with e.g. intersections
(The set-valued functions alluded to here are defined on the powersets of $X,Y$ and take subsets into subsets based on the standard definition of image and preimage)
a perspective i like is the following: the datum of a subset of a given set $X$ is equivalent to a function $X\rightarrow\{0,1\}$. the subset $A\subseteq X$ is identified with its "characteristic function" $1_A\colon X\rightarrow\{0,1\}$ that maps points of $A$ to $1$ and other points to $0$. the inverse is given by taking the preimage of $1$ under such a function. this is to say that subsets are just a type of inverse image, so naturally they behave well with respect to other inverse images, yet not necessarily with images (you can always "pull back" a function out of space, you can't alwa…
For image you look only at your point(s) in the domain in question. For preimage, you must examine $f$ not just near one point but everywhere in the domain.
Hmm, I will mark that down in my notes, thanks Thorgott. What does datum mean in this case though?
I've never heard "datum of a subset"
I follow Ted. Roughly why would you expect "globality" to imply good behavior when commuting with these various set operations?
Also Thorgott, I see you are from Germany so I am led to ask (since this is from Analysis I by Amann Escher). Have you read Amann Escher/do you have any opinions either way?
in a way, there's two natural statements (preimage commutes with intersection, image commutes with union) and two unnatural statements (preimage commutes with union, image commutes with intersections)
in a sense, it's surprising that precisely one of the unnatural statements is true and the other is not
I don’t want formalism. I want a concrete (non-constant) function (even on finite sets, but with pictures) where $f(A)\cap f(B)$ is bigger than $f(A\cap B)$. Why does it happen?
@EE18 well, alright. I'm not sure if this is proper English, but I'm not going to argue on that. I just thought you're interchanging e.g. with the word example
the inclusion in one direction is obvious. for any element of the image of an intersection, that element is the "output" of some input element common to both sets. thus, when we decide to first use $f$ and then intersect, that given input domain element is in sets and thus both images have the given output. So what goes wrong the other way is that we could have an element of the intersection of the images which we arrive at from two parts of the input sets which do not overlap
This is very 21st century but i do not have pencil and paper on me right now. is there good math software for drawing these things?
@Thorgott The point at which it keeps meeting is a tangency. Or not, depending on your "generification" procedure. I also mean an immersion with clean double point intersections -- those are the generic immersions
You can also use inkscape to do the drawings. It has an option to export the results so that they can be included in LaTeX and the text in them is rendered by LaTeX
@Thorgott The two embedded curves can be found explicitly, but I like the following: if you pick a homeomorphism $f : \Sigma_g \to \Sigma_g$ "randomly", choose any embedded curve $C_1 \subset \Sigma_g$ and another curve $C_2$ intersecting $C_1$ homotopically nontrivially, then $f^{\circ n}(C_1)$ and $C_2$ will do the job, for $n$ large enough
do it inductively. in the induction step, take a small disk around an intersection point where the curves look like coordinate axes, cut out that disk and glue in a punctured torus with meridian and longitude so that the curves match up
Here's the easy way. Take a pants decomposition of $\Sigma_g$. Take $C$ to be one of the seam curves. Let $C'$ be your favorite curve which algebraically intersects $C$ once. Then Dehn twist $C'$ about all the seams once.
@EE18 a meta-exercise would be: find an example online of a software generated graph that already has the properties you want (e.g. maybe you have to add in highlighting indicating what the sets A, B whatever are but the graph itself is pre-made)
In "How To Solve It", George Pólya writes:
"There was a seminar for advanced students in Zürich that I was teaching and von Neumann was in the class. I came to a certain
theorem, and I said it is not proved and it may be difficult. Von
Neumann didn't say anything but after five minutes h...
