Mathematics - 2021-03-15 (page 1 of 2)

schn

12:03 AM

Well, if I also have $\int_0^{10} \frac{d^2 \phi}{d y^2}dy=\int_0^{10} 0 dy$.

Then $ \frac{d \phi}{d y}=C $, right?

Thorgott

no, not at all

anakhro

@KarimMansour One thing I found effective with high school students was motivating with a problem. So for my euclidean geometry lecture, it was "can every isometry be inverted?". The question can be phrased in plain english (they know examples of isometries like translations, reflections, rotations) but the answer is hard to place a finger on due to the problem of not having rigorous enough notation/definitions and because it's genuinely not an easy question.

I then spent the rest of the lecture developing the ideas, always referring back the same problem.

schn

How come? I'm solving Laplace equation $\Delta \phi(x,y)=0$ with the boundary conditions $\phi(x,0)=0$ and $\phi(x,10)=V_0$ by the way.

Karim Mansour

@anakhro That is good way to teach in general. I motivate myself that way to read certain books or research papers.

schn

The RHS is 0.

anakhro

12:06 AM

@KarimMansour Indeed! I do the same. :)

Karim Mansour

@anakhro One thing is also try to link every connection. Try to find something in nature that resembles what your teaching also works.

schn

Then the LHS also needs to be 0.

Karim Mansour

@anakhro what do you study ?

schn

So either $\phi'$ is 0 or a constant, no?

anakhro

I am in the middle of switching fields somewhat. I kind of do symplectic stuff. I see you do algebraic geometry stuff?

Thorgott

12:08 AM

integration is an averaging process

it loses almost all information about a function

knowing the integral of a function along just some interval tells you almost nothing about the function

if you have a differential equation, go solve it, a single integral on the interval [0,10] won't give you the answer

Karim Mansour

@anakhro Yeah I am in intersection of complex geometry and algebraic geometry.

anakhro

@KarimMansour Very nice! So do you see a lot of Kahler stuff?

Karim Mansour

Yeah and Chow groups.

variants of Chow groups and variants of Hodge conjecture.

Ted Shifrin

Most grad students and faculty are not good teachers, but you really expect even a good undergraduate math major to have the skills to be a good teacher magicallly? @Karim The lower level you teach students, the more experience it takes to be successful.

anakhro

@KarimMansour Nice! I don't know what a Chow group is. Is it related to the Chow lemma at all?

Karim Mansour

12:16 AM

Chow groups are just homology group in the category of algebraic varieties. Though I deal so far with smooth varieties.

@TedShifrin Yeah I agree. But I think that option should be there. But I guess you could enrol in school earlier if you want that option.

anakhro

@TedShifrin who is thinking they are a good teacher magically?

@KarimMansour Nice. Is there a reason they call them Chow groups instead of homology groups?

Karim Mansour

@anakhro Probably to honour Chow.

Wei-Liang Chow

Chow Wei-Liang (simplified Chinese: 周炜良; traditional Chinese: 周煒良; pinyin: Zhōu Wěiliáng; Wade–Giles: Chou Weiliang; October 1, 1911, Shanghai – August 10, 1995, Baltimore) was a Chinese mathematician born in Shanghai, known for his work in algebraic geometry. == Biography == Chow was a student in the USA, graduating from the University of Chicago in 1931. In 1932 he attended the University of Göttingen, then transferred to the Leipzig University where he worked with van der Waerden. They produced a series of joint papers on intersection theory, introducing in particular the use of what are now...

anakhro

Is there anything that sets them apart from homology groups in general?

Karim Mansour

they are very rigid

and if you look at higher version of them then they are the motivic cohomology theory.

@anakhro This is also very important

Chow group

In algebraic geometry, the Chow groups (named after Wei-Liang Chow by Claude Chevalley (1958)) of an algebraic variety over any field are algebro-geometric analogs of the homology of a topological space. The elements of the Chow group are formed out of subvarieties (so-called algebraic cycles) in a similar way to how simplicial or cellular homology groups are formed out of subcomplexes. When the variety is smooth, the Chow groups can be interpreted as cohomology groups (compare Poincaré duality) and have a multiplication called the intersection product. The Chow groups carry rich information...

relations to $K_0$

you can calculate $K_0$ by using Chow groups.

anakhro

Very cool!

Karim Mansour

12:25 AM

Yeah it is very cool stuff.

ery cerious matherfunker

12:39 AM

@TedShifrin It is odd since they need to learn prolog.

leslie townes

i find it helps to start with Logo

love_sodam

12:54 AM

anyone can handle my question?

leslie townes

i've got one two three four five, senses working over time

love, where's the question, i scrolled back and couldn't find it. which isn't to say i know anything, which i definitely don't. but i wouldn't object if you repeated it.

love_sodam

2

Q: If $\phi:X'\to X$ is a homotopy equivalence then an induced map $C_{f\circ\phi}\to C_f$ is a homotopy equivalence

If $\phi:X'\to X$ and $f:X\to Y$ are maps, define an induced map $F:C_{f\circ\phi}\to C_f$. If $\phi$ is a homotopy equivalence then show that $F$ is a homotopy equivalence. Abusing the notation, I first tried to show $F:M_{f\circ\phi}\to M_f$ is a homotopy equivalence. My attempt is basically ...

user2103480

2:09 AM

@Thorgott agree

Mike Miller

2:45 AM

@love_sodam You are trying to prove the theorem in the textbook, with the proof written below? I don't follow. The proof is there.

love_sodam

@MikeMiller No I'm trying the prove the problem which (I think) use similar idea of the proof given in the textbook

The proof I wrote in the question is imitating the proof in the situation given in the problem.

$f:X\to Y$ a map.
Textbook: $\phi: Y\to Y'$ a homotopy equivalence then $M_f\to M_{\phi\circ f}$ is a homotopy equivalence
Problem: $\phi: X'\to X$ a homotopy equivalence then $M_{f\circ\phi}\to M_f$ is a homotopy equivalence

$M_f$ is a mapping cylinder

Actually the original question is $C_{f\circ\phi}\to C_f$ is a homotopy equivalence but I think I should prove the mapping cylinder case first

Akiva Weinberger

3:28 AM

Hello and welcome to another episode of: I'm asking simple questions on Math SE that I'm too lazy and/or tired of answering myself, because I know it'll give me reputation points

1

Q: Why does the equation of the circumference of a circle in spherical and hyperbolic space satisfy $C''=-KC$?

In a space of constant curvature $K$, the function for $C(r)$ where $C$ is the circumference of a circle of radius $r$ satisfies: $C''=−KC$, with initial conditions $C(0)=0$ and $C'(0)=2\pi$. (Units check: Gaussian curvature $K$ is units $1/\rm length^2$, $C$ is units $\rm length$, and $C''$ is u...

ordinary-differential-equations riemannian-geometry circles curvature pi

leslie townes

crap, i totally have notes from a class that would answer this. they are decomposing in my garage. the electronic version is on a computer that does not have a monitor connected to it. my apologies.

i don't even remember the name of the person who taught that class. he was a postdoc or assistant professor. almost incapable of speaking english but very clear.

Akiva Weinberger

Damn

Let me know if you remember

leslie townes

he brought plum wine to a thanksgiving dinner that a professor organized for people who didn't go home for thanksgiving. does that help?

i might even remember the brand name of the plum wine. that's how my memory works.

Akiva Weinberger

That's nice

leslie townes

very appropriate for pi day, in any case.

i was annoyed by pi day when i worked in math, now i bother my coworkers with "happy pi day!!!!"

Akiva Weinberger

3:46 AM

I memorized pi through music

vocaroo.com/1oLKn5dquVQo

(40 digits)

leslie townes

the pitches are correlated with the digits?

that's pretty good

Akiva Weinberger

Yeah: 1 through 8 are the major scale

and 0 is the note below 1, 9 is the note above 8

love_sodam

Maybe my question is harder than I first thought

so maybe I'm not a complete idiot!

Leonhard Euler

4:33 AM

ah no

happy $\pi+0.01$ day!

Note: The $\pi$ here is the engineer's pi

leslie townes

in celebration of pi day, i ate some curried chickpeas.

love_sodam

5:06 AM

How about $H((x,s),t) = fH(x,2s+t) for 2s+t\geq 1, H^{-1}(x,t(2s+t-1)/(2s), s2+t-1/(t+1)) for 2s+t\leq 1$

at 2s+t = 1 then pasting lemma condition satisfied in $M_f$ and

H(-,0) = h and $H(-,1) = \phi\psi$

Think it's ok i think all continuous

copper.hat

6:45 AM

${22 \over 7}$ is close enough.

Jakobian

7:18 AM

Ok so yesterday some people tried to help me/come up with solution

They didn't really get to the point of my issue but I didn't completely describe my attempt so I'll do that ig

Assume not. Take compact set of positive measure contained in {g<0}

Moreover take ball B so that K is contained in it.

We can find continuous non-negative functions u_k with support contained in B which converge to u = -1_K g in L^1

The problem now is whetever u_k g converges to ug in L^1

If it does, then we'll get a contradiction

This is the issue

robjohn

@copper.hat So are you saying that in Europe, July 22 is $\pi$ day?

Spaazmaster

8:41 AM

hello

porridgemathematics

9:02 AM

in an infinite dimensional CW complex $X$, do the $n$-skeleta necessarily embed into $X$? I can see its true in the finite dimensional case, but not totally sure about the infinite dimensional case

via the inclusion map

porridgemathematics

10:14 AM

oh, they do

ery cerious matherfunker

10:25 AM

Hi anyone here from logic?

Simone

10:51 AM

@eryceriousmatherfunker there's a Logic chat room

2

porridgemathematics

11:10 AM

im pretty confused about CW complexes, I can grasp $n$-skeletons fine, but im not sure what a union of $n$-skeletons for all $n$ is really supposed to mean, apparently we need to build an increasing sequence of $n$-skeletons of the form $X_0 \subset X_1 \subset X_2 \subset ... $ but im not sure how this really works, because $X_{n+1} = X_n \cup_{f} \coprod_{\alpha} D^{n+1}_{\alpha}$ but this doesn't entail $X_n \subset X_{n+1}$

at best we need to reidentify $X_n$ with its homeomorphic image in $X_{n+1}$ every time we build a higher dimensional skeleton, so I don't see how we can just take a union of increasing sets, since we need to redefine the head of the sequence everytime we build a higher dimensional skeleton, and this isn't explained at all in hatcher

and I know 0 category theory so i'm not sure how I can wrap my head around what $X = \cup_{n \geq 1} X_n$ really is :/

Thorgott

11:23 AM

@Jakobian dominated convergence?

or just monotone convergence even

you can pick the u_k to converge monotonously from above to u

Mike Miller

@porridgemathematics I don't understand what it means to say we need to redefine the head of the sequence every time. In an inductive definition of CW complexes $X_{n+1}$ is defined to contain $X_n$; it's defined as a quotient of $X_n \sqcup_\alpha e^{n+1}_\alpha$, the disjoint union of $X_n$ and a bunch of new $(n+1)$-cells, where the quotient is given by pasting them to $X_n$ by their attaching maps.

$X_n$ sits inside the disjoint union. It then sits inside the quotient.

Do you also object to the notation $\Bbb R \subset \Bbb C$?

Thorgott

There's a canonical map $f_n\colon X_n\rightarrow X_n\sqcup\coprod_{\alpha}D_{\alpha}^{n+1}\rightarrow X_n\cup_f\coprod_{\alpha}D_{\alpha}^{n+1}=X_{n+1}$. An alternative way to interpret this union is as a direct limit. Explicitly, you take the disjoint union $\coprod_nX_n$ and equip it with the equivalence generation generated by $x_n\sim f_n(x_{n+1})$ for all $x_n\in X_n$ and $n\in\mathbb{N}$.
This does precisely identify $X_n$ with its homeomorphic image in $X_{n+1}$, but on all stages at once. The topology on this, after the identifications, is precisely the CW topology (open iff inters…

porridgemathematics

sure, but as a set $X_n$ isn't a subset of $X_{n+1}$, rather there is a (closed) embedding $X_n \rightarrow X_{n+1}$, $X_n$ looks like a bunch of equivalence classes inside $X_{n+1}$ but topologically is the same thing as $X_n$

Mike Miller

You're making this obscure when it doesn't have to be.

But since you're really worried about this issue, did you ever learn about the Prufer groups in algebra?

porridgemathematics

i didn't :/

Mike Miller

11:31 AM

OK, then step back. Did you ever learn about the set-theoretic construction of the natural numbers?

porridgemathematics

@Thorgott oh I see

@MikeMiller yes I did

Mike Miller

Nevermind, you're going to be more satisfied with Thorgott's point. I'll step out.

ery cerious matherfunker

@Simone thanks for info

Simone

np

Thorgott

11:52 AM

No one objects to that, but there is a subtlety here. When you have an embedding $K_0\rightarrow K_1$, you can identify $K_0$ with its image in $K_1$ no problem, but you can't just do this inductively on a sequence of embeddings $K_0\rightarrow K_1\rightarrow K_2\rightarrow\dots$; you effectively need a space in which all these spaces embed simultaneously to simultaneously carry out the necessary identifications.
This is a subtlety when it comes to constructing algebraic closures that most texts don't adequately address (otherwise, there's little stopping you from just saying "take union of…

Mike Miller

"At least once" should not be the first pass.

I agree that one cannot take a literal union of $\{1, \cdots, n\}$ if your model for that is a subset of $\{n\} \times \Bbb R \subset \Bbb Z \times \Bbb R$.

However, it should still be visually clear what is meant.

I think the picture I just outlined is the entire content of the picture here. Yes, yes, you need a directed system to formalize this.

Thorgott

Yeah, I agree

The way I think about this is also in the picture that you have outlined, I'm not advising to think of a CW complex as a direct limit at any time

but from the formal point of view, this seems like a slight inaccuracy, so it's important to figure out once what the correct formal notion is and why this way of interpreting it is a non-issue

if you're comfortable with identifications to begin with, this is fine to ignore, but if you're worried about the identifications like porridge was, I think the right way to deal with it is formalize it once to make sure all is well

Mike Miller

OK, that's fair, but I think you should explain the informal point as well :)

Thorgott

well, you already did

my explanation should be taken as complementing yours, not disagreeing with it

Szabolcs

I have a tentative graph theory question, not yet ready to be posted on the main site.

Consider a graph in which every edge is part of some cycle. Take a minimum weight cycle basis of the graph. Often, a subset of this basis is sufficient to cover all edges.

Question: Is it true that for any minimum weight cycle basis of a given graph, the size of the smallest subset that will cover all edges will be the same?

Generally, I am looking for results about what subset of a minimum weight cycle basis is necessary to cover all edges, and in what way this might be related to the planarity (or genus) of the graph.

anakhro

1:34 PM

What's an easy 1st year example of a property of functions (e.g. continuous) such that an f satisfying said property and being bijective is still not enough to guarantee the inverse has the property?

I want something a little easier than continuity to give examples for.

But I can't think of any.

I guess differentiable is somewhat okay with x^3.

Thorgott

I don't think differentiability is easier than continuity, but ymmv

measurability is the only other example coming to mind immediately and clearly much less instructive than those other two

anakhro

You don't think f(x) = x^3 is an easy example for differentiability?

It's at least easily inspected from the graph (of the cube root).

I don't know many 1st years who know anything about measurability.

Oh you said much less instructive, my bad.

Thorgott

no, x^3 is certainly a good example for differentiability

I just think of continuity as more fundamental than differentiability

anakhro

Oh. Yeah, indeed. The examples for continuity are a pain to convey to first years, though.

The best one I can think of is mapping a bit of the interval to S^1 but to explain continuity in this case is harder than explaining differentiability in the case of x^3.

Thorgott

I think both of these are very instructive examples for a first year to understand

anakhro

1:50 PM

How would you go about explaining the continuity one to a first year student who has taken only calculus?

I'd think you'd have to explain continuity all over again, basically. The easiest version for this being the open neighbourhood definition.

Thorgott

the point of continuity is that "close points get mapped to close points" (putting the subtle issues of how to quantify this aside), which clearly is the case for I->S^1, but continuity if the inverse then means "close points come from close points" and this gets violated precisely because the circle thus described is the result of joining the two ends of an interval together (it's precisely the quotient space [0,1]/0~1, but they don't need to know that).
So there is a way a "point can move in the codomain", namely traversing that gluing point, which is "invisible" in the domain and that's …

anakhro

Seems pretty wordy, tbh

You would have to get them across two bridges, the first being that this notion of continuity accurately represents their "don't lift the pencil" definition of continuity, and the second being that the inverse fails this criteria.

I would think they would be still mystified about the first bridge by the time they came close to the second one.

Thorgott

is the audience not acquainted with the notion of continuity at all?

cause I definitely think understanding continuity first should precede trying to understand such a counter-example

anakhro

1st year calculus courses do not really drive home anything about continuity other than the limit definition, I think. And at that, it's usually just in the graphical sense.

At least, here.

Not sure about where you are from.

Limit definition being "if the left and right limits exist and agree, then it is continuous there".

Thorgott

that's enough, the limit definition is the rigorous equivalent of "not lifting the pencil" (which should hopefully be clear enough if you have an intuition for limits), then what I'm saying is just that e^{2\pi t}->1=e^{2\pi 0} for t->1=/=0. You can travel along the circle and come back to where you started without lifting the pencil, but you have to lift the pencil to jump from one end of an interval to the other.

anakhro

2:01 PM

I dunno, Thorgott, seems pretty hefty still.

Thorgott

I'm afraid I don't have anything better to offer. Continuity is a non-trivial thing to understand at first, after all.

robjohn

2:24 PM

A friend sent this to me yesterday.

jeea

2:40 PM

what is limit of the sum $\frac{1^n+2^n+... N^n}{N^{n+1}}$ as $N \to \infty$ is it $\frac{1}{n+1}$ and how it can be showed?

I have used this in many occassions but dont know is it correct or not :D

Thorgott

yes, the sum $1^n+\dots+N^n$ for fixed $n$ is a polynomial in $N$ whose degree is $n+1$ and whose leading coefficient is $1/(n+1)$, which implies the limit

I think you can prove this by induction

there's an explicit formula for this polynomial too, but it's unwieldy and its full strength should not be needed for this

jeea

@Thorgott can we write $\frac{1^n+2^n+... N^n}{N^{n}}$ as $\frac{N}{n+1}$ for large $N$, like $N=100$

not quite, error is quite big, Is there a good approximation for this

Thorgott

2:57 PM

asymptotically, yes, that follows from what I said above

porridgemathematics

3:09 PM

hi, im not sure why this implies that $A$ is a CW complex with the induced topology imgur.com/a/p9fcnNf , I can see that the image of each cells 'characteristic map' (not attaching map like it says in the screenshot) is contained in $A$ and so $A$ is of the form $\cup_{\alpha} \Phi_{\alpha}(D^n_{\alpha})$, but I don't see how to realize it via building skeletons

Thorgott

you build it the exact same way you build X, but only use the cells in A

leslie townes

i am also a pi day truther. wake up, sheeple

porridgemathematics

oh okay, i think i see it

Wolgwang

3:28 PM

Can someone justify this construction?

leslie townes

is the construction of PT and PM? i'm piecing this together.

Wolgwang

Steps of construction.

. Take a point P outside the circle and draw a secant PAB, intersecting the

circle at A and B.

iii. Produce AP to C such that AP = CP.

iv. Draw a semi-circle with CB as diameter.

v. Draw PD perpendicular to CB, intersecting the semi-circle at D.

vi. With P as centre and PD as radius draw arcs to intersect the given circle

at T and M.

vii. Join PT and PM.

viii. Then PT and PM are the required tangents.

@leslietownes Yes

leslie townes

OK, P is given.

that answers my question.

leslie townes

3:51 PM

this is a weird construction. normally people work with the segment OP. i'm trying to relate the usual construction to this one.

Wolgwang

@leslietownes Actually the question is "Draw tangents to a circle without using the centre." :-/

leslie townes

yeah, what kind of madman would want that.

ugh, this is so taking me back to 10th grade. it's 1994, i'm in a prefabricated classroom which was intended to be temporary but will eventually become a permanent part of my school. my teacher knows nothing. i'm trying to figure out what is in the solution book.

thank you for these negative vibes. happy monday, everyone

Nicolás Castellanos

4:25 PM

Hello

how i can express log(y) = log(0,25e^2x) as y=x?

is that y=, not log(y)=

did you understood me

?

robjohn

@Wolgwang [Cheating]: Find a pair of antipodal points using the center, then without using the center, draw a diameter between these two antipodal points and draw a perpendicular to the diameter at one of the points.

leslie townes

if y is a multiple of e^(2x), it is not possible to express y as a linear function of x. it's possible to express log y as a linear function of x.

linear, affine, something like that.

robjohn

@Wolgwang If you are not given a point at which to draw the tangent and were only asked to "draw a tangent to the circle", then take any two points on the circle, bisect the line between them and construct the perpendicular to the line at the midpoint. You can probably take it from there.

Now, extend that method to construct a tangent at a pre-specified point.

Think isosceles triangles

@leslietownes it's not that bad, if you think about it a bit.

leslie townes

that's a big "if." i'm too busy drinking green beer in advance of st. someone's day.

your comments do make it clearer than it was to me. i think i see it now.

leslie townes

4:53 PM

it's weird how many constructions become simpler if you forget a piece of information, or introduce a new piece of information

Wolgwang

@robjohn That line would pass through the centre...Unable to realise what you want me to :-(

Semiclassical

5:13 PM

Hilbert space question. If I'm given a Hilbert space $\mathcal{H}$, I can form the Hilbert space $B_2(\mathcal{H})$ of Hilbert-Schmidt operators on $\mathcal{H}$. Is there a meaningful way to invert this? E.g. given a Hilbert space $\mathcal{H}_2$, can I find $\mathcal{H}$ such that $\mathcal{H}_2=B_2(\mathcal{H})$?

leslie townes

given a family of operators on something, shoehorn them into acting on a hilbert space and being hilbert schmidt?

Semiclassical

well, it's not apparent from $\mathcal{H}_2$ alone that it's the set of operators on anything.

I'm thinking it might be false, though. We should have $B_2(\mathcal{H})\cong \mathcal{H}^* \otimes \mathcal{H}$

which is problematic if $\mathcal{H}_2$ is odd-dimensional

(I could well understand the answer being "Yes if H2 is even-dimensional, but not in any interesting way.")

leslie townes

i don't fully understand the question, but it's an interesting thought. to hilbert-schmidt-ize a collection of, if you don't want to call them operators, elements of some algebra.

this is reminding me of an unpublished paper of my advisor and myself which as far as i know only exists on a computer under my desk. we had a construction, from an algebra, to a hilbert space formed out of that algebra, in which the elements of the algebra had decay properties.

i hope there's somebody smarter here who can answer your question. i'm an idiot.

Semiclassical

Yeah. Given an arbitrary Hilbert space H, can you construct another Hilbert space H0 such that H behaves as operators on H0

and again, a plausible answer is "yes if H is even-dimensional, but in a trivial way"

leslie townes

there's no such thing as finite dimensions to me.

finite dimensional stuff, just go and type into matlab and get the answer. i was thinking infinite dimensional.

Semiclassical

5:21 PM

i do QM, everything is finite dimensional /s

leslie townes

is it finite dimensional "over" something infinite dimensional? throw me a bone here.

Semiclassical

tbh, in most cases it's really just a fancy way of saying "hey, let's do finite-dimensional matrices"

leslie townes

i'm fine with finite matrices, but the elements of the matrices might act on anything. that's what i'm about. it would surprise me if physicists got particular about this.

Semiclassical

the typical case being a qubit. when you distill it down, you're just doing operators on $\mathbb{C}^2$

leslie townes

right around when i quit academia, i refereed a paper where someone who had grown up on 'quantum information theory' thought it appropriate to announce results from approximately 1905 as new results because he didn't know his history and had never read a paper without 'quantum' in the title.

Semiclassical

5:24 PM

and even there you're really just interested in rays. so in that sense it really just reduces to $\mathbb{C}$

leslie townes

i didn't recommend rejecting, i said, around theorem 2, please cite this paper from 100 years ago which establishes this result. the editor ended up rejecting.

Semiclassical

@leslietownes yeah, i mean, there's an honest way to do that: "This result is known in mathematics, but in QI it's not known and it should be."

it's not a matter of "I did it first" but "I'm the first person to realize that it matters to my community"

leslie townes

i do think it's helpful to translate known stuff into new stuff. the only thing that bothered me was the presentation of the thing as not-known stuff.

Semiclassical

right.

it can be genuinely hard to get a handle on what's known, though

like, a result I came up with and was proud of? after a lot of digging, I realized that it first shows up in a paper from 1896

leslie townes

i do get it. and i don't think "does this logically follow from published papers" is an appropriate test for preventing publication.

Semiclassical

5:28 PM

that said, once I did dig back that far, I made sure to highlight and credit said initial source

leslie townes

all mathematics logically follows from other mathematics. i'm personally terrible at remembering who established what. i just didn't try to make it sound like i was the guy who realized something.

my thesis cited papers from the 1940s which were basically what i did in less generality. i was frank about this. they gave me the PhD anyway.

Semiclassical

The Arnold Principle: If a notion bears a personal name, then this name is not the name of the discoverer.
The Berry Principle: The Arnold Principle is applicable to itself.

leslie townes

i saw a talk by arnold once. he was wonderful.

the soviet system is a complete catalogue of toxicity, but it produced very good results.

Semiclassical

I do consider myself an archivist to no small extent. I'm as interested in finding old, useful tools as I am of creating new ones.

leslie townes

you just need to look over the anti-semitism and misogyny and the elitism and the, oh, i don't know, twenty thousand other things.

Semiclassical

5:32 PM

you're not wrong

it's easy to forget that when looking at math papers, of course

But that "find old useful stuff" is one reason why, while I do have certain preferences when it comes to QM interpretations, I don't find myself inclined to search for my "preferred interpretation". I find the QM formalism as it stands to be perfectly worthy of study.

leslie townes

there's some very funny stuff from chinese math. you have journals that begin with a love letter to chairman mao, and sometimes introductions to articles that suggest that mao brought forth the observations to come. it's sad, i guess, and not funny.

Semiclassical

It does make me fond of Bohm's interpretation if simply because of the way the extra structure naturally adds in, but by that token I'm not so interested in strict modifications of QM.

leslie townes

my phd advisor was obsessed with the fundamentals of QM although he never published anything about it.

Semiclassical

I prefer substructure which naturally supports the existing formalism rather than substructure which insists that it's wrong, if only because the former has so much more experimental support

(The main problem with Bohm being that, while it naturally supports non-relativistic QM, it really doesn't mesh nicely with special relativity. you can make it work if you insist on a preferred reference frame and absolutely simultaneity, but that's a heck of a price.)

leslie townes

you are an orthodoxer and not a paradoxer.

i am a paradoxer.

Semiclassical

5:38 PM

I like the phrase of "shut up while you calculate"

as contrasted with "shut up and calculate"

leslie townes

i used to tell my students, "it's calculus, not thinkulus."

Semiclassical

old anecdote:

Neumann, to a physicist seeking help with a difficult problem: "Simple. This can be solved by using the method of characteristics."
Physicist: "I'm afraid I don't understand the method of characteristics."
Neumann: "In mathematics you don't understand things. You just get used to them."

(it is a rather dubious quote, but it's a useful story nonetheless)

leslie townes

i used that line a lot in class. don't set the bar at like, isaac newton level. just use the cheat codes. who cares if anybody knows if it's right.

robjohn

@Wolgwang You are not using the center. This gives you a diameter without reference to the center.

leslie townes

to put it in modern terms, if i watch a video of someone beating a video game, i can beat the video game. just do it.

robjohn

5:41 PM

@Wolgwang Once you have a diameter, you just need to construct a perpendicular at the point on the circle.

Semiclassical

@leslietownes unless it's a tool-assisted speedrun and you don't know it, lol

leslie townes

silvanus thompson had a good book on this, calculus made easy.

robjohn

@leslietownes I have that book.

Semiclassical

(though the point still stands. if you see a genuine demonstration of video game play, then that's proof that it can be replicated.)

leslie townes

it's a happy spirit to take into mathematics. grab your tools from wherever.

people do shady things on speed runs. i have yet to adapt the techniques i have seen to my video game of preference. but the principle stands.

Semiclassical

5:44 PM

depends what you mean by shady

the old question of glitch vs. exploit comes to mind.

when are you breaking the game, and when are you merely using the tools the game gave you in a way that the devs didn't anticipate?

leslie townes

we run into this a lot in my law practice, where it is not a video game but the same question exists.

the devs [legislators] did not anticipate.

my copy of calculus made easy was dematerialized and sent into the ether by the post office when i moved from iowa to massachusetts. thankfully public domain copies are still available.

Semiclassical

to have a good competition one needs to put down boundaries, but those boundaries are inherently artificial. that's not the same as being subjective---if you have declared rules, then that's objective---but it's socially constructed

Wolgwang

@robjohn Ohk

Semiclassical

ultimately the question of "why is this a rule in this speedrun" comes down to "we thought it was a healthy rule"

leslie townes

i've been thinking about when is the correct age to introduce my daughter to video games. she already receives stimuli from screens, although they are stimuli that she cannot control. i don't see this as better than playing actual games.

Semiclassical

5:49 PM

it's such a different world now, yeah

i group up near computers, but as a kid I didn't grow up using computers. it wasn't until i was a teen that i really engaged with them

and i certainly didn't grow up with "i've got the internet in my pocket"

now it feels unimaginable to go back to that

leslie townes

i'm trying to think of the first time i touched a general purpose computer. it would have been in the 1980s. some guy in my town decided that it would be good to retrofit a bus from the 1950s with terminals so that we could all do Logo at the behest of instructors.

they parked this bus on the playground. that may have been the most significant thing for me. a bus in the playground?? what's next, a talking banana?

that's a simpsons reference.

Anakhand

Sorry to interrupt the conversation. Don't you find it irksome that we have a standard name for the interior, exterior, boundary, etc. of a set A in a metric/topological space, but we don't have a name for the set of isolated points? What would you think of the name "Seurat set of A", after pointillist painter Georges Seurat?

leslie townes

Vote in favor

Jakobian

6:09 PM

@Thorgott Thanks, I was thinking of convergence in L^1 constantly but better strategy is just to take bump functions of neighbourhoods of K and do as you say.

copper.hat

6:21 PM

@robjohn approximately :-) albeit since the day is written first it does not quite have the same association.

robjohn

@copper.hat In some regions, the date format is dd/mm/yyyy (which makes it confusing in some instances without knowing the person's origin).

I prefer yyyymmdd

it makes it lexicographic

leslie townes

pi day was invented to sell meat pies in jurisdictions that don't even put the month before the day.

and still have meat pies.

jeea

8:42 PM

German tank problem

In the statistical theory of estimation, the German tank problem consists of estimating the maximum of a discrete uniform distribution from sampling without replacement. In simple terms, suppose there exists an unknown number of items which are sequentially numbered from 1 to N. A random sample of these items is taken and their sequence numbers observed; the problem is to estimate N from these observed numbers. The problem can be approached using either frequentist inference or Bayesian inference, leading to different results. Estimating the population maximum based on a single sample yiel...

Here if the tanks are captured with replacement, that is captured and let away, then will the formula for $N$ remain similar?

Because I have derived and using that I am getting $74.375$ much close to the wikipedia page where they have not replaced the tank

Semiclassical

@jeea This blog post seems like it may help. It treats both cases (with/without replacement) felixxiao.github.io/2017/10/german-tank-problem

Bottom line seems to be that the with-replacement case has a closed-form solution whereas without-replacement doesn't.

jeea

@Semiclassical Thanks a lot for the link! I think you mean the opposite!

Semiclassical

woops, yes

jeea

Yes I was approximating the with replacement case

using this formula

Faulhaber's formula

In mathematics, Faulhaber's formula, named after Johann Faulhaber, expresses the sum of the p-th powers of the first n positive integers ∑ k = 1 n k p = 1 p + 2 p + 3 p + ⋯...

Semiclassical

ah, Faulhaber. I can never manage to remember how to derive that

(the generic formula in terms of Bernoulli numbers, that is)

robjohn

9:00 PM

@Wolgwang I finally saw the diagram you were trying to understand. Let me know if you did not figure it out.

Hey @Ted!

Ted Shifrin

Hi @robjohn!

robjohn

I am proctoring an exam via Zoom. Really tough.

Ted Shifrin

You can't stop cheating via Zoom.

robjohn

It ends in 20 minutes.

Ted Shifrin

Modern-day realities.

robjohn

9:11 PM

@TedShifrin The software I wrote helps some

Ted Shifrin

Oh really?

Does it stop them from using their phones, too?

robjohn

We can't cut off the browser and email like we an at the university, but they do have to sign an honor statement.

Ted Shifrin

One of my friends who teaches at UCSD just posted on FB that he caught a Ph.D. student plagiarizing a research paper on a homework assignment for a grad course.

robjohn

and phones cannot be collected on the way in

Ted Shifrin

Right. So basically it's hopeless.

robjohn

9:12 PM

yeah

Ted Shifrin

I used to be very trusting of my students (unless I taught a low-level course or course for future teachers, ironically).

robjohn

The checks are mainly to keep someone outside of the classroom from doing the work for someone in the classroom

Ted Shifrin

Now they're at home, though, right?

robjohn

yep, so it's not so helpful

Ted Shifrin

Yeah, and even if you manage to check what's going on at the IP address they're logged in on, you can't stop phones or laptops. I guess if they have to have their video on, you might notice something super suspicious.

jeea

9:16 PM

hello dear professor, as a student of my college I can say with confidence, that in my college cheating has gone up by remarkable percent. In offline there was no scope of cheating in final exams, but now in online all assignments and exams are copied and everyone cheats

robjohn

Yeah, that's the benefit of in classroom teaching. You have eyes on

jeea

Even with video on it is not possible to stop

Ted Shifrin

Yeah, I'm very glad I retired. I want to teach and help students learn, not be cheated on.

The only possible solution I can think of is oral exams, and that takes huge effort/time and even then you cannot stop cheating.

jeea

in my college we had one oral exam , but students used to keep friend on the call who would tell the answer, or would write it in message service

like whatsapp

Ted Shifrin

I think I could change things as we go along so that the student would be shown up for that.

Long pauses and lack of comprehension.

But, yeah, no fun for anyone.

robjohn

9:21 PM

With FOL, it is harder to cheat with a phone because the symbols can get involved and the parens need to be exact. I am not saying it is impossible, just harder than a multiple choice exam.

Ted Shifrin

@robjohn Someone just posted an interesting question. If I know (say a series expansion to approximate) $(I+A/2)^{-1}$, how do I use my answer to approximate $(I+A)^{-1}$.

@robjohn: What if the students and their accomplices know LaTeX? :)

robjohn

squaring comes to mind as a first order approximation

Ted Shifrin

OK, the question was actually with $(I+cA)^{-1}$ with an arbitrary $0<c<1$. :)

I did think of that for the 1/2 case.

I think they want a good numerical solution.

(Scaling $A$ to make the eigenvalues $<1$.)

robjohn

The program itself supplies the symbols, and does not use LaTeX, so they would have to translate back and forth. As I say, not impossible, just more difficult.

Karim Mansour

9:35 PM

@TedShifrin Hi

Hi everyone

Ted Shifrin

Howdy

Semiclassical

9:53 PM

o/

@TedShifrin isn't that a resolvent thing

also i know i've run into it with QM. (mostly in the context of $c$ as a perturbation parameter)

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$Mathematics$ Mathematics