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Avv
Avv
00:20
For cosine distance, it does not satisfy the four properties of a metric over Euclidean space. Why we need these 4 metrics and why they chose those 4 metrics over others please? Among these metrics is triangle inequality and another one is being indiscernible .
leslie townes
leslie townes
avv: the axioms of a metric offer one level of abstraction away from an intuitive notion of 'distance' as premised on our lived experience with the euclidean distance in R^2 and R^3. i don't know that we "need" these axioms in any philosophical sense, but they are useful because when present they are usually simple to verify and allow a whole bunch of things as consequences.
Avv
Avv
@leslietownes. Thank you
Very useful
@leslietownes. Why the triangle inequality is important to be there for example?
to validate a metric?
leslie townes
leslie townes
00:38
i dunno what you mean by "important." if you look at any book that works at the level of metric spaces, you will find that it is used a lot. so it is "important" in the sense that if you assume it, it can be used to prove a lot of generally useful things. this is the sense in which a lot of common axioms are "important." nobody's out there insisting that they have to be satisfied, but they are, good for you, here are some useful consequences
Avv
Avv
Thanks
 
1 hour later…
Ted Shifrin
Ted Shifrin
01:51
@Avv Analysis is to a great extent about estimates. If you ever have a sum (or difference) as your upper bound, then you immediately need the triangle inequality to get the efficient bound.
robjohn
robjohn
02:09
@TedShifrin that's approximately correct, except in sum obtuse cases.
Ted Shifrin
Ted Shifrin
02:24
@robjohn Acute remark.
Cotton Headed Ninnymuggins
Cotton Headed Ninnymuggins
why is $(x)_2$ interpreted as a rising factorial in Wolfram Alpha and how do I get a falling factorial?
Unknown x
Unknown x
user image
How do I find the direction and value(positive/negative) of the curl of the vector field using the figure?
How do I find the value(positive/negative) of thedivergence of the vector field using the figure?
user image
This was the actual question.
Could you suggest reference for learning this?
Ted Shifrin
Ted Shifrin
This tests your understanding of the divergence theorem and Stokes’s Theorem. Take little circles or squares around the appropriate point and find the flux across or work around the curve.
Unknown x
Unknown x
02:39
user image
user image
How do I find the work done using small circle, if the formula of the field is not explicitly given?
I understood, how work done help me to to find the value(positive/ negative) of the curl.
Ted Shifrin
Ted Shifrin
This is flux, not work, according to your post. You estimate geometrically.
Unknown x
Unknown x
@TedShifrin okay
Ted Shifrin
Ted Shifrin
They are asking about the curl at that point, not divergence. Why did you post about flux/divergence?
D.C. the III
D.C. the III
Never knew the double integral was a completely separate notion from the iterated integral...
Ted Shifrin
Ted Shifrin
Well, you should have.
Unknown x
Unknown x
02:54
@TedShifrin in the option D, they ask about the divergence as well. If I able to solve for just (7,8), I can apply to other points.
D.C. the III
D.C. the III
😥😥😥...I blame the teaching.
Unknown x
Unknown x
That is why I asked about the divergence and curl for the point (7,8).
sorry for the confusion.
when I drew a small cirlcle around (7,8), I can see that three arrows goes in and three arrows goes out.
Ted Shifrin
Ted Shifrin
This is a bad question, Unknown, as both div snd curl are very close to 0 at that point. Usually questions like this are far more decisive. You have to estimate whether the force spins the little wheel counter/clockwise and whether the next flux across is positive/negative. This one is hard to see.
Unknown x
Unknown x
okay.
Ted Shifrin
Ted Shifrin
You have to estimate the net tangential/normal effects.
Unknown x
Unknown x
02:58
okay sir.
Ted Shifrin
Ted Shifrin
Try looking at some different problems where it will be clearer.
Unknown x
Unknown x
okay
Thank you
UnderMathUate
UnderMathUate
03:12
@TedShifrin I'm back. I went to class earlier, then had dinner, then got distracted...
As far as what you said:

I guess I could just label $S_k = \sum\limits_{n=1}^{\infty} (x_{2n} + x_{2n+1})$. Since this subsequence starts from $x_2$ rather than $x_1$, we have something like
$$S_k \leq x_n$$
So it must be that $\sum\limits_{n=1}^{\infty} S_k$ converges since $ \sum\limits_{n=1}^{\infty} x_n$ does.
Cotton Headed Ninnymuggins
Cotton Headed Ninnymuggins
What are these relationships between the Stirling numbers called and where can I find a proof of them?: $\sum_{k\ge 0}c(n,k)(x)_k=x^n$

$\sum_{k\ge 0}S(n,k)x^k=(x)_n$
robjohn
robjohn
@TedShifrin you're so right.
Koro
Koro
03:44
O(n) is a deformation retract of GL_n(R).
leslie townes
leslie townes
hot take
Koro
Koro
I think that this is because if I take A in GL_n(R), then by Gram Schmidt, this can be changed to a matrix A' in O(n).
And this happens continuously. We also note that if we started with A as orthogonal, then A'=A.
But I don't know how to write it out for an exam.
leslie townes
leslie townes
"by gram schmidt" is covering a lot of ground. if you are implicitly defining a map there, i would be careful to check whether and how you are making any arbitrary choices. you'll need to do that to define a map.
unless you've proved properties of "is a deformation retract of" that let you step away from that a little.
Ted Shifrin
Ted Shifrin
@UnderMathUate sloppy. Get the details right!
leslie townes
leslie townes
04:17
phbbhpbphth
Semiclassical
Semiclassical
04:50
@CottonHeadedNinnymuggins the latter at least looks like a generating function for Stirling numbers. (Or rather, a family of such g.f.s indexed by n)
Indeed, WP claims that this is how the Stirling numbers of the first kind were initially defined…but this has a [citation needed] so I dunno
The main upshot is that your proof will depend on how you’re defining Stirling numbers in the first place
leslie townes
leslie townes
"depends on your definitions" is a very mathy answer for a physicist
Semiclassical
Semiclassical
Eh. It’s the one you have to give for families of special functions all the time
leslie townes
leslie townes
if you say so, "physicist"
Semiclassical
Semiclassical
eg there’s a number of equivalent ways to define Legendre polynomials.
leslie townes
leslie townes
preaching to the choir, my dude
Semiclassical
Semiclassical
04:59
Lol
I did have this same problem in a different context recently
What’s the best way to justify Euler’s formula for complex exponentials?
leslie townes
leslie townes
take them as a definition? (no idea what those are)
Semiclassical
Semiclassical
Ugh, mobile typo (fixed)
leslie townes
leslie townes
oh haha euler?
yeah i dunno, it's too intertwined with the definition
Semiclassical
Semiclassical
Ya. In the context of trying to explain it to a non-calculus audience
leslie townes
leslie townes
yeah i literally don't know
there are ten thousand routes to complex numbers, and i don't know why anyone would prefer any one of them
Semiclassical
Semiclassical
05:03
I almost feel like defining it as true for imaginary exponentials is a sufficient approach. Then show that this at least preserves angle addition as motivation
This isn’t truly rigorous, but without calculus I dunno if you can do better
leslie townes
leslie townes
i'm a big fan of "if thing X is true, then [zillions of things] are also true"
Semiclassical
Semiclassical
Yes
“It lets us prove high school trig identities easily” is a good argument in my book
It also does depend a bit by what “non-calculus” means. Derivatives/integrals are definitely out
But what about “consider what happens if we do something small many times”
Eg $(1+1:n)^n\to e$ as $n\to \infty$
That’s a limit and thus does involve “calculus” in a strict sense, but doesn’t require differentiation
Plus you can point to a calculator and say “hey, 1.001^1000 is already a decent approximation to $e$
 
3 hours later…
copper.hat
copper.hat
07:46
whoa, what is going on here?
yet another unexplained downvote. depressing.
Mouse
Mouse
@copper.hat I think down vote doesn't matter much for you at least.-3 rep only.
but I can give you up vote.
Although I am no match for your old brain.
robjohn
robjohn
08:13
@Semiclassical I consider limits precalculus, but differentiation and Riemann integration are based on limits.
Silly Goose
Silly Goose
08:45
Hello friends i have been having great trouble understanding some mathematical definitions from a paper (Locality from the Spectrum by Cotler et al.)
I will paste the relevant definitions and then state my question
user image
user image
I understand the definition of a TPS. We are saying two isomorphisms $T$ and $T'$ should be called equivalent if the codomains (the tensor product factorization of the hilbert space) either have different basis WITHIN individual tensor factors or if isomorphic factors in the tensor product factorization induced by $T$ are permuted in $T'$
PNDas
PNDas
09:00
I forgot how they defined tangent line in school before calculus course. Like in conic sections such as parabola, ellipse etc.
how did they find tangents to the these curves
Mouse
Mouse
Mathematical curve is sexy the less the radii of curvature the more sexy?
robjohn
robjohn
@PNDas lines that touch only at one point, perhaps.
PNDas
PNDas
Okay they were just putting $y=mx+c$ in the equation and find $m,c$
@robjohn Hmm, I was really confused when we first found tangent to $\sin x$.
 
3 hours later…
123
123
11:56
Hi All...
I have a question about Euclidean geometry. According to SMSG axioms postulate-1 line uniqueness says given two distinct points there is exactly one line.
I am confused with the idea of coincidence and collinearity in the context of postulate-1. First of all pls clear do we consider points, lines and plane are different objects or line and plane is considered as set of points?
 
2 hours later…
leslie townes
leslie townes
14:27
phbhbbphtht
shintuku
shintuku
14:51
ah, yes, phbtphbtphbt, or polyhydroxybutyratepolyhydroxybutyratepolyhydroxybutyrate
leslie townes
leslie townes
... ok
Koro
Koro
15:18
haha
one potato two potato
one potato two potato
15:34
peep
robjohn
robjohn
15:57
@PNDas If you want to compute the slope of the tangent to $\sin(x)$, you need to use calculus, or at least imitate the computation of the derivative.
ILikeMathematics
ILikeMathematics
@robjohn By the way, do you buy physical or digital copies of books? Or do you lend them out the university library?
冥王 Hades
冥王 Hades
16:18
user image
Hades vs Seiya
^my mind whenever I'm debating myself
PNDas
PNDas
16:58
@robjohn I was saying, the first time I found tangent lines of $\sin x$, I got confused because at that time I thought tangent line touches the curve only ONCE.
冥王 Hades
冥王 Hades
Imagine learning physics without calculus 💀
user4539917
user4539917
A tangent line does touch a curve only once; that is, as close to "once" as you want.
shintuku
shintuku
i think they mean that a tangent line of any point of a sine curve cuts the curve at least twice
and there are tangent lines which are tangent to infinitely many points
user4539917
user4539917
But "once" is the whole idea behind "instantaneous" velocity.
shintuku
shintuku
it is also eleven in spanish
Thorgott
Thorgott
17:10
once locally
user4539917
user4539917
^
Exactly once.
shintuku
shintuku
locally
and not with all curves
robjohn
robjohn
17:26
@PNDas and the tangent of $\sin(x)$ at $x=0$ is not even on one side of the curve.
user4539917
user4539917
Also at x=180°.
PNDas
PNDas
@robjohn another good example is tangent line of $x^2\sin(\frac1x)$ at $x=0$.
There is a book called "Understanding Real analysis". It explained everything nicely. How to see continuity, Uniform continuity, derivatives, sequence convergence etc.
It really helped me whe I was in B.Sc.
robjohn
robjohn
@ILikeMathematics I buy physical versions of most books.
copper.hat
copper.hat
i prefer real paper not pdfs
@Mouse Thanks, its not the rep, its the why
Ted Shifrin
Ted Shifrin
17:55
@PNDas Please remember to give authors, not just titles.
Rithaniel
Rithaniel
(I find myself in a position where wish that I had already taken a PDE course)
Ted Shifrin
Ted Shifrin
A parenthetical PDE course?
ILikeMathematics
ILikeMathematics
@robjohn Oh, alright, do you buy them used?
Rithaniel
Rithaniel
@TedShifrin Lol, well, what would the physical interpretation of a parenthetical? My assumption would be a course that was attended without being enrolled
Shinrin-Yoku
Shinrin-Yoku
Wrong a tangent can tough a curve infinitely many times
it is Wrong inuition
dont spread it
locally is also wrong
There are many monsters out there
Limit of secants is the correctest def of tangwnts
It leads to wrong thinking
ok?
dont spreas
d
Xander Henderson
Xander Henderson
18:12
@Shinrin-Yoku Disagree. The intuition is that a line is tangent to a curve if it only intersects that curve once. When you try to make that intuition rigorous, you learn that this isn't the right definition, but it is a perfectly reasonable intuition.
Limits of secants is a way of making that intuition rigorous.
Another approach is to notice that if $p$ is a quadratic polynomial, then there is a unique linear function $\ell$ (e.g. $\ell(x) = mx + k$) such that $p-\ell$ has a unique zero (of multiplicity 2), i.e. the line and the parabola intersect exactly once. This means that $(p-\ell)(x) = b(x-a)^2$ for some $b$ ($a$ is the $x$-coordinate of the intersection). It is possible to solve for $m$ and $k$; the derivative is $k$.
This approach extends to any polynomial $p$: $\ell$ is tangent to $p$ at $(a, p(a))$ if $(p-\ell)(x) = q(x)(x-a)^2$, where $q$ is a polynomial of degree 2 less than $p$.
And the same idea extends to rational functions fairly simply, and to analytic functions via power series expansions (so just define all of transcendental functions in terms of their power series).
user4539917
user4539917
@冥王Hades there are intro courses on non-calculus based physics.
Imagine that :P
Xander Henderson
Xander Henderson
Another approach which starts from the basic idea of "tangent means unique point of intersection" is Descartes method of normals. I don't quite know all the details, but the basic idea is that, on a circle, radii are perpendicular to tangents. So build a circle which is tangent to a curve (in the "touches only once" sense), and the tangent line to the curve is the perpendicular to the radius of that circle.
Shinrin-Yoku
Shinrin-Yoku
18:30
It is true that the history of tangents was developed that was, but the historical approach is not always the correct one. My best definition of tangent is that it is the best linear approximation to the function.
And from a physics point of view I find it hard to motivate the definition a tangent only touches once. But by F=ma best linear appproximation is crystal
@XanderHenderson
PNDas
PNDas
@TedShifrin Understanding Real Analysis by Paul Zorn
shintuku
shintuku
Paul Wrath
Shinrin-Yoku
Shinrin-Yoku
Zorn?! I thought it was Max Zorn of Zorn lemma fans.
*fame
PNDas
PNDas
He is born in India but doesn't look like Indian. Anyway he is MAA president.
Shinrin-Yoku
Shinrin-Yoku
I think Max Zorn must be one of the few people in the 20 th century who proved a basically trivial theorem(given the machinery available) and became so famously known
PNDas
PNDas
18:35
@Shinrin-Yoku Who is spreading wrong idea? Nobody said that tangent has to touch the curve only once.
Shinrin-Yoku
Shinrin-Yoku
Really you may want to scroll and read comments
PNDas
PNDas
@Shinrin-Yoku I asked about how they defined tangent line in school before calculus course and Robjohn said "lines that touch only at one point, perhaps."
Xander Henderson
Xander Henderson
@Shinrin-Yoku I made no claim about "best" approach. Nor, really, did I make a claim about the historiocity of any approach. All I said is that there is nothing wrong with the intuition that a tangent should touch a curve only once. It is an entirely reasonable intuition, which can be made rigorous, by, for example, taking limits of secant lines (because a secant line touchs a curve at two explicit points, and we reduce that to one point in the limit).
Regarding the intuition of "best linear approximation": how is that intuitive for a beginning calculus student?
D.C. the III
D.C. the III
@冥王Hades If you just let your Ikki side dominate there would be no confrontations...
shintuku
shintuku
the only way i can make sense of best linear approximation is least squares
Shinrin-Yoku
Shinrin-Yoku
18:42
Imagine a force acting on a particle the tangent is where the object would travel if suddenly all the force was removed @XanderHenderson
Xander Henderson
Xander Henderson
@Shinrin-Yoku You are showing a tremendous bias for physics. Most beginning calc students don't know any physics. Why would I want to appeal to that intuition?
Shinrin-Yoku
Shinrin-Yoku
it is quite intuitive and absolutely correct by Sir Issac newtons second law
Xander Henderson
Xander Henderson
Also, I feel like you are not really reading what I've written, because I have never said that you are wrong for noting that there are other intuitions. I think that you are wrong to say that the "touches only once" intuition is wrong.
Shinrin-Yoku
Shinrin-Yoku
But it is wrong!
Xander Henderson
Xander Henderson
I disagree.
And I have outlined several ways in which that intuition can be made rigorous.
Shinrin-Yoku
Shinrin-Yoku
18:45
Maybe it is a “lie for children” but whether you consider that wrong or not is a matter of taste so we do actually agree @XanderHenderson
But you must agree that as a blanket statement it is wrong. As shown by the over of Bartle and Sherberts analysis book
Xander Henderson
Xander Henderson
@Shinrin-Yoku Which "blanket statement"?
冥王 Hades
冥王 Hades
@user4539917 Wait what? Who are those for? 3rd graders?
Shinrin-Yoku
Shinrin-Yoku
i mean that the tangent to a curve only intersects onxe
冥王 Hades
冥王 Hades
@D.C.theIII Ikki is a joke. Got thrashed by Saga
Shinrin-Yoku
Shinrin-Yoku
locally
user4539917
user4539917
18:47
@冥王Hades freshman physics for Arts majors
Xander Henderson
Xander Henderson
@Shinrin-Yoku I don't think that anyone has every argued otherwise. But I also don't think that anyone is telling that to students.
Thorgott
Thorgott
you just have to add a single adjective to avoid all confusion, as I said
Shinrin-Yoku
Shinrin-Yoku
Wait you someone should add it to the lies to children math educators thread
Xander Henderson
Xander Henderson
@Thorgott Indeed. As I keep telling my students, the derivative is local.
Shinrin-Yoku
Shinrin-Yoku
nooooo @Thorgott locally also false :(
It is possible for tangent to keep intersecting locally
it is sad but true. I was very distraught when I learn it
Xander Henderson
Xander Henderson
18:50
@Shinrin-Yoku Again, who is telling students "a tangent line crosses a curve only once"?
Shinrin-Yoku
Shinrin-Yoku
You were saying loclally
it only intersects once
Xander Henderson
Xander Henderson
:63111215 Imagine a function which oscillates very quickly near a point, which such that the oscillation damps down quickly enough that the function will be differentiable.
Something like $x^2 \sin(1/x)$ at zero.
Shinrin-Yoku
Shinrin-Yoku
one needs local assumptions of convexity for the statement to hold. Locally
冥王 Hades
冥王 Hades
@user4539917 what business do they have learning physics in college?
robjohn
robjohn
@ILikeMathematics not usually
Xander Henderson
Xander Henderson
18:52
In any event, the usual spiel is "Hey, kiddos! A tangent to a circle touches the circle at only one point. This doesn't really work for general curves, but we get to something similar by taking limits of secant lines (in the same way that you might take limits of chords in a circle)."
冥王 Hades
冥王 Hades
@XanderHenderson I have the urge to eat that same hotdog again
Am I supposed to overcook it?
Shinrin-Yoku
Shinrin-Yoku
The fact it does not hold for general curves must be slowly revealed
or it can be very damaging
Xander Henderson
Xander Henderson
@Shinrin-Yoku "Slowly revealed"? No---you reveal it almost instantly.
Shinrin-Yoku
Shinrin-Yoku
Then student will ask give me curve which locally tangent intersects infinitely many times
what would you say?
Xander Henderson
Xander Henderson
We have an intuition: the tangent to a circle or a parabola (or to any non-degenerate conic section) intersects that curve exactly once. How do we generalize that notion?
Shinrin-Yoku
Shinrin-Yoku
18:56
it is better for students to discover it themselves, when they learn analysis
Xander Henderson
Xander Henderson
One approach is via limits of secant lines. Almost immediately, we learn that "intersects only once" fails. Take nearly any cubic, and consider a tangent line at a local maximum.
Shinrin-Yoku
Shinrin-Yoku
Highly oscialling curves will not come up in calc
Xander Henderson
Xander Henderson
I don't need to talk about highly oscillatory functions to find a counter-example.
Thorgott
Thorgott
just look at a line
Shinrin-Yoku
Shinrin-Yoku
I am talking about local ally non line functions pfcpirse
user4539917
user4539917
18:57
y=mx+b
Shinrin-Yoku
Shinrin-Yoku
of cpufse
user4539917
user4539917
ok. f(x)=mx+b
Xander Henderson
Xander Henderson
Don't be rude.
user4539917
user4539917
y=f(x)
yes?
Shinrin-Yoku
Shinrin-Yoku
He is being rude by saying trivial to things to me and making statements such as “prove it” it is very demeaning for me.
user4539917
user4539917
18:59
sorry
冥王 Hades
冥王 Hades
New to internet?
Shinrin-Yoku
Shinrin-Yoku
No problem.
Sine of the Time
Sine of the Time
hi chat
user4539917
user4539917
I want to start from first principles.
hi
Xander Henderson
Xander Henderson
user image
user4539917
user4539917
19:02
:(
冥王 Hades
冥王 Hades
Is the arrow necessary?
Shinrin-Yoku
Shinrin-Yoku
No need to
Thorgott
Thorgott
i think it adds charme
Xander Henderson
Xander Henderson
@user4539917 Sorry. I shouldn't have used your account as an example.
Shinrin-Yoku
Shinrin-Yoku
take extreme mesaure
冥王 Hades
冥王 Hades
19:03
@XanderHenderson no. Its perfect
Shinrin-Yoku
Shinrin-Yoku
just communicate to user
Xander Henderson
Xander Henderson
@Shinrin-Yoku Sure, if you are polite about it.
Sha Vuklia
Sha Vuklia
user image
user image
I don't see how to prove the if-direction in (ii) in Theorem 3 (see screenshot). To check that $f:a\to b$ is monic, we consider a parallel pair of arrows $x,x':c\to a$ such that $fx=fx'$, and we need to show that $x=x'$. Since $fx=fx'$ implies in particular $fx\equiv fx'$, it follows that $x\equiv x'$. This means that there exists epics $u,v:d\to c$ such that $xu=x'v$. However, how to arrive at $x=x'$?
Shinrin-Yoku
Shinrin-Yoku
I am not native English apeaker
user4539917
user4539917
neither am i
Thorgott
Thorgott
19:05
Sha you'll have to do some notation explaining
Shinrin-Yoku
Shinrin-Yoku
For me I just wanted to communicate the message no sharp edges …
Sha Vuklia
Sha Vuklia
hm? the first screenshot explains the notation $\in_m$ and $\equiv$
user4539917
user4539917
@Shinrin-Yoku then just say that :-)
Shinrin-Yoku
Shinrin-Yoku
Should mathematicians go to industry instead of acedemia? Does it have better pay?
Sha Vuklia
Sha Vuklia
I guess I could have added that by definition of $\in_m$, we have $x,x'\in_m a$
Xander Henderson
Xander Henderson
19:09
@Shinrin-Yoku That's a loaded question...
Shinrin-Yoku
Shinrin-Yoku
What does loaded mean in this context?
Xander Henderson
Xander Henderson
@Shinrin-Yoku It means that you have phrased the question in such a way that it invites one answer, and discourages others.
Shinrin-Yoku
Shinrin-Yoku
I don’t understand why anyone would go to acedemia when the they can just study math in their free time
Xander Henderson
Xander Henderson
@Shinrin-Yoku What free time?
Also, what do you have against academia?
user4539917
user4539917
Loaded question
A loaded question is a form of complex question that contains a controversial assumption (e.g., a presumption of guilt).Such questions may be used as a rhetorical tool: the question attempts to limit direct replies to be those that serve the questioner's agenda. The traditional example is the question "Have you stopped beating your wife?" Whether the respondent answers yes or no, they will admit to having a wife and having beaten her at some time in the past. Thus, these facts are presupposed by the question, and in this case an entrapment, because it narrows the respondent to a single answer...
Ted Shifrin
Ted Shifrin
19:12
@Shinrin-Yoku Generally, much better pay. Most of us in academia take lower pay because of research independence and/or the love of teaching .
Xander Henderson
Xander Henderson
@TedShifrin This.
Shinrin-Yoku
Shinrin-Yoku
I don’t have anything against it. But it seems it is quite hard in acedemia, and there is not enough reward
Ted Shifrin
Ted Shifrin
not that university administrations reward dedication to teaching, in general.
Sha Vuklia
Sha Vuklia
hm, I have a counter example to the implication: $u,v:d\to c$ epic such that $xu=x'v$ implies that $x=x'$, so I should take another route
Xander Henderson
Xander Henderson
I will also add that I see teaching at a community college (at the fringes of academia) to be a public service.
Shinrin-Yoku
Shinrin-Yoku
19:13
For example if a phd is hell bet on acedemia is it easy to get a permanent job?
Ted Shifrin
Ted Shifrin
I thought teaching at a state university was as well.
Sha Vuklia
Sha Vuklia
though I guess it could still be true that $x=\phi x'$ with $\phi$ an iso
Xander Henderson
Xander Henderson
In exchange for getting paid laughably little compared to my industry counterparts, I have very good job security, the option to take summers off (to research, for example), the opportunity to take off a semester ever couple of years to pursue my own interests, the opportunity to teach, and the opportunity to serve my community.
Sha Vuklia
Sha Vuklia
which means $f$ is essentially monic or sth
Xander Henderson
Xander Henderson
@Shinrin-Yoku This depends very much on the field.
Right now, the academic job market in mathematics is extremely competitive. A recent math PhD, hell bent on a tenure-track position, will pretty much be required to spend three to five years in post-doc positions. Even then, they are going to face stiff competition for nearly every position out there.
Sha Vuklia
Sha Vuklia
19:16
$x'\phi$ i mean
Xander Henderson
Xander Henderson
Community college positions are much less competitive, and I am reasonably certain that any new math PhD who wants to teach at a community college will find a job. Though it might not be where they want to be (geographically).
user4539917
user4539917
@冥王Hades as part of their science requirement
Shinrin-Yoku
Shinrin-Yoku
another interesting question is how easy it is to bail into industry, having only mathematics knowledge….
Xander Henderson
Xander Henderson
On the other hand, a math PhD with a background in topological data analysis would likely be easily hired at a number of institutions (it is a sexy field right now, and a lot of places are trying to build departments around it).
Sha Vuklia
Sha Vuklia
I guess the $\implies$ direction is slightly more relevant for chasing diagrams anyways
Thorgott
Thorgott
19:20
@ShaVuklia sorry, I missed that one
have you done (i), cause (ii) should be an easy consequence
Sha Vuklia
Sha Vuklia
oh shoot, I did (i) but I didn't think to use it
thanks leme see
user4539917
user4539917
formula for momentum
►
Ted Shifrin
Ted Shifrin
@Shinrin-Yoku Computer skills are highly recommended. I started telling my undergraduate advisees that 25 years ago.
Sha Vuklia
Sha Vuklia
@Thorgott I see now that I went too fast when solving (i). I'm stuck at showing that if $x\equiv 0$ then $x=0$ (which is what I think I need)
Thorgott
Thorgott
that should follow from the UP of an epi
Sha Vuklia
Sha Vuklia
19:34
but I have two epis
that's the problem
if I can show that one epi suffices, then I'm done indeed by the UP
user4539917
user4539917
Fun fact: 11,083 messages found in the math room with the word "formula." While the Physics room has only 2,970 messages found.
Sha Vuklia
Sha Vuklia
oh
I think I see it
yea, got it now, thanks
btw, this result is pretty cool (there are further properties), because now I can do element chasing in abelian categories without invoking the Mitchell embedding theorem
user4539917
user4539917
19:54
"What is the most important thing a scientist should cultivate in himself?
One should get rid of excessive ambition. One should not think that only a
genius can be happy. One must learn to appreciate even a small achievement,
to rejoice in it, and never overestimate oneself. One has to cultivate a love
for work. One has to understand and cultivate the joy of learning, which
is almost the same as the joy of life. Happiness is when your life’s work is
needed."
Sobolev, Sergei Lvovich
Ted Shifrin
Ted Shifrin
@ShaVuklia What happened to the Sha I used to know?
Sha Vuklia
Sha Vuklia
well, for one, half of my courses use categorical language, so I better have my cat thy in order xD
Ted Shifrin
Ted Shifrin
Ugh.
Thorgott
Thorgott
20:14
it's a good thing
Ted Shifrin
Ted Shifrin
Ugh.
I'll stick with the feline sort of cat.
冥王 Hades
冥王 Hades
@TedShifrin One of my dad's friend works at Google, despite only having a Masters in Mathematics
He got a job after developing his coding skills
Ted Shifrin
Ted Shifrin
"only" isn't the right word there. Sure.
冥王 Hades
冥王 Hades
Not having good computer skills could put you at a serious disadvantage in the industry
Ted Shifrin
Ted Shifrin
Did you ever explain your "easy" proof of that trapezoid problem? My proof was totally wrong.
冥王 Hades
冥王 Hades
20:25
@TedShifrin well, "only", compared to yours
Ted Shifrin
Ted Shifrin
Why did cyclic quadrilaterals with angles of equal measure allow you to deduce similarity?
冥王 Hades
冥王 Hades
@TedShifrin No I didn't. I'll look into it again once I get home
Ted Shifrin
Ted Shifrin
OK, if you have a good proof, you should answer the question.
I feel bad because one person removed his proof after I asked if it wasn't just the same as mine. I can't unremove it :P
冥王 Hades
冥王 Hades
Its painfully obvious that $P$ is the dead center of that square. Just need to prove it
Ted Shifrin
Ted Shifrin
Overusing the letter $P$ isn't helpful. Yes, that's true.
@Rithaniel As opposed to an official audit? :D
Koro
Koro
20:29
Does removing boundary circle of a Mobius strip still keep the remaining space compact?
冥王 Hades
冥王 Hades
I've actually seen this question before now that I remember, however, it had additional information about the ratio of the corresponding lengths passing through P. This question omits that
Ted Shifrin
Ted Shifrin
The person who caught the gap in my proof has a proof posted. I haven't checked it carefully. He has something sounding like a continuity argument as a heuristic, but I don't think that's essential to his proof.
Koro
Koro
No, it's not compact.
I found the solution manual :-).
Julius
Julius
Hi. Let $r_i$, $i=1,\dots,n$, be known $k\times 1$ vectors and $R=(r_1,\dots,r_n)'$ an $n\times k$ matrix. Any ideas on how to characterize the cardinality $M(C) = |\{r_i'C: i=1,\dots,n\}|$? Fwiw, $r_i$ consist of -1, 0, and 1, and the coordinates of $C$ are distinct. I'm particularly interested in lower bounding $M(C)$. Experimentally I see that $2n-5 \leq M(C)$, and I'm able to find $C$ such that $M(C)=2n-5$, but I wonder how to show that $M(C) < 2n-5$ is not possible.
Novice
Novice
Anybody familiar with convolution of probability measures on $(\mathbb Z, 2^{\mathbb Z})$?
Ted Shifrin
Ted Shifrin
20:35
Of course it's not compact.
Ask the question about removing the boundary circle of a closed disk. Same issue.
Julius
Julius
(I meant $2k-5$ rather than $2n-5$ in my question.)
Rithaniel
Rithaniel
@TedShifrin $\{\text{Official Audit}\}\subseteq\{\text{Classes Observed}:\text{Not Student}\}$
Ted Shifrin
Ted Shifrin
Actually, at UGA students could actually register as auditors. For some graduate courses where we needed 5 registered students for the class to run, this was important. I suppose.
Rithaniel
Rithaniel
Ah, well, fair enough. So there isn't necessarily a containment
But, like, you could "attend a class" by watching videos of it on YT
Ted Shifrin
Ted Shifrin
I am suspicious of that. Watch 1 minute and claim to have attended a course?
Rithaniel
Rithaniel
20:47
Well, that would be "claiming to have attended a course." The same could be done by a chronically absent student. In order to actually attend, you have to actually pay attention and do the work involved
D.C. the III
D.C. the III
21:42
@XanderHenderson Literally just read a SIAM newsletter sent to my email about this. LOL
Silly Goose
Silly Goose
21:53
would it be accurate to say that compactness is a property inherent to a metric space itself as opposed to a property endowed to a metric space by a space it is embedded in?
Ted Shifrin
Ted Shifrin
What a convoluted question.
leslie townes
leslie townes
silly, if you make that inquiry slightly more precise, you might have something that has an answer.
Ted Shifrin
Ted Shifrin
Compactness is a property of a subset of a topological space.
leslie townes
leslie townes
what do you propose to do with an answer to that question?
Ted Shifrin
Ted Shifrin
It depends on the topology (in your case, coming from a metric) and on the subset.
Yeah, good question from Leslie
Thorgott
Thorgott
22:04
compactness is a property intrinsic to a metric (or topological) space
it is, in that sense, different from properties like openness/closedness, which are properties relative to an ambient metric (or topological) space
this is how I interpret the question
Silly Goose
Silly Goose
user image
i intend to understand what rudin means by the paragraph below the theorem heh
Thorgott
Thorgott
Rudin explains this stuff horribly
Silly Goose
Silly Goose
right so you don't need to care about any embedding space to make the claim that metric space (X, d) is compact?
because no matter what embedding space you look at (in particular you can look at (X,d) embedded in X itself) whether $(X,d)$ is compact or not does not change
Thorgott
Thorgott
yes
the entire notion of "compact relative to" is a red herring that is absolutely awful pedagogically
Silly Goose
Silly Goose
and this is in general what we mean by an intrinsic property of some set with structure? that we do not need to consider any embedding to make a claim about said intrinsic property
i am asking this question now because i was told that diffe g is about thinking about particular sets with structure without embedding them into anything and seeing how much information you can extract from there
D.C. the III
D.C. the III
22:12
WHat is Baby Rudin good for? All I ever read and see is vitriol for it?
Once considered the paragon of Real ANalysis it has become despised.....
Silly Goose
Silly Goose
I get Kunze and Hoffman vibes from baby Rudin :P. i tried with great effort to work through K&H's Linear Algebra only to reflect later that much of my effort was spent on understanding notation or strangeness not having to do with Linear Algebra :P
shintuku
shintuku
@D.C.theIII since when
D.C. the III
D.C. the III
before our time Shintuku, this was in the years of yonder of when Ted, Leslie, Copper, Xander and the others were just swash buckling youths with a dream and the endeavour to pursue it.
Ted Shifrin
Ted Shifrin
@SillyGoose Was this supposed to be differential geometry? In general, whether we're talking about geometry or topology, we can define topological (and geometric) structures on spaces intrinsically, without requiring that the spaces sit inside — say — Euclidean space. Compactness is an example. But there are lots of others.
Why does Rudin even use this language? I guess he's covering $K$ by open sets in the ambient space without just talking about the subspace topology. I don't remember.
Thorgott
Thorgott
22:29
yes, Ted, I think that's it
 
1 hour later…
mick
mick
23:43
i understand little about julia sets
confused
0
Q: Number of connected components for the filled julia set of $z^2 + c z^5$

mickFor any polynomial map $f$ we can define the filled Julia $K$ to be closure of the complement of $ \Omega$ in $\mathbb{C}$ of the basin of infinity $$\Omega = \{z \in \mathbb{C}; f^{\circ n}(z)\rightarrow \infty \}.$$ Thus the boundary of $K$ is what is often called the Julia set. Now there are e...

polynomials dynamical-systems closed-form fractals complex-dynamics

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