Mathematics - 2018-07-05 (page 1 of 2)

02:46

0

Q: How do Tensors are the generalisation of ordinary vectors?

Doubt:- What is the physical meaning of tensor? Tensors are $k-$ linear functions. Right?. I can see that $L_0(V)$ is a set of scalars. So I can understand tensors are the generalization of scalars. How do Tensors are the generalisation of ordinary vectors? Do tensors have direction? How do I...

nitsua60

@FailedtobeaMathematician Is there any particular reason you link that here?

Failed to be a Mathematician

@nitsua60 I didn't get the answer. If anyone in this group knows the answer. Please help me.

nitsua60

Okay. It's been 23 minutes. I suggest a little patience. Also, if you have a request for the people in the room go ahead and make it. Just dropping the link in here it's unclear if you want help answering it, you want help writing a coherent question, if you think someone in here you know might be interested in it, you think it should be closed....

Failed to be a Mathematician

@nitsua60 Can you help me to correct the errors in the question? Someone had downvoted my question :(

nitsua60

Probably not very much--I don't know if you've noticed my rep, but I've got virtually no experience around math.se. But I know chat, so felt free to chime in on a bald link dropped into an empty room.

Failed to be a Mathematician

02:59

okay

Piyush Divyanakar

05:01

Hey there guys this might be off topic but I have question on how to manage and handle notes?

Context: I am preparing for indian civil services examination and for it i have to study maths as a subject. Now there are about 14 topics ranging from all areas of mathematics and I will have study mutiple books and keep in my head large numbers of concepts ranging from linear algebra to modern algebra, real analysis, odes etc.

I want to make notes, but usually what happens is that notes tend to get more and more unorganzied as I do make more of them. I wanted to know if there is some system of organization that you are familiar with that i can use to manage the content.

Leaky Nun

06:27

For a topology with compact basis, an open set is compact iff it is a finite union of the basis.

Secret

07:12

What happens if the basis is noncompact?

i don't see how a finite union of them will fail to cause the resulting open set to be noncompact

Leaky Nun

just take the non-compact basis, and express it as the finite union of basis as such: it is the union of itself, and 1 is finite

Abcd

$$I = 2 \int_0^1 e^{-x^4}(1-4x^4) dx$$

I have tried integration by parts but that doesn't help here.

Leaky Nun

what does wolfram alpha say

Abcd

@LeakyNun integral calculator gives an answer: integral-calculator.com

Leaky Nun

"gives an answer" is not enough

it gives an answer in terms of gamma

Abcd

07:21

@LeakyNun See the "manually computed antiderivative" part

Leaky Nun

I see

then it answers your question

just click "show steps"

Abcd

@LeakyNun K

Secret

Hmm, so you mean for the noncompact open sets in the noncompact open basis, those will be the union of itself hence noncompact open with finite unions?

Leaky Nun

@Secret I mean if B is not compact then B is not compact

B is the union of {B}

the closure of S is:
1. the intersection of every closed set that contains S
2. points x such that every open set containing x intersects non-trivially with S

how on earth can it be described with closed sets and with open sets

@Secret exercise: prove that they are equivalent

Secret

08:58

Consider a point x in an open set U such that $S \cap U$ is empty. Then by definition of closed set $U^{\complement}=X-U$ is closed and contains $S$. Repeat for all those disjoint $U$s and take their arbitrary intersection. Then you end up with the smallest closed set containing $S$ and this is $cl(S)$

Now as for those x whose $U$ such that $S\cap U$ is nonempty ... (processing)

Secret

09:13

Ok for 2, the points x are limit points of S by definition. A set is closed if it contains all its limit points. 2 ensures the resulting set K contains S and all its limit points hence it is a closed set containing S, in fact the smallest such thus K=cl(S)

So they are equivalent

...

hmm...

can we generalise this further...?

Consider some structure X that is more general than a topolpgy such that you have some set A and a collection in X such that for each point m in each member M in X, $M \cap A$ is nonempty but there exists some points a in U in A such that $U \cap M$ is empty

Intuitively, this capture some asymmetric notion of closeness such that there are two points a and m such that a is in all neighbourhoods of m (hence m is a limit point of a), but m is not in all neighbourhoods of a (thus a is not a limit point of m)?

Will figure this out later. I need a lot of pen and paper

(the aforementioned asymmetric closeness is very commonly encountered in Escher-esq themed games, where walking pass some one way doorway and you will be sent far away from the starting point)

Hmmm... maybe a quasiuniform space can model that

60

Q: What concept does an open set axiomatise?

In the context of metric (and in general first-countable) topologies, it's reasonably clear what a closed set is: a set $F$ is closed if and only if every convergent sequence of points in $F$ converges to a point also in $F$. This naturally generalises to the definition of a closed set in an arbi...

general-topology math-history

A computational perspective of topology

0

Q: Is the intersection of two non-open/non-measurable sets non-open/non-measurable?

Very simple question. If I have two sets A and B that are both non-open in some topological space, is their intersection necessarily non-open? And similarly, is the intersection of two non-measurable sets non-measurable? I think the answer is that it could go either way. But I can't find any go...

general-topology measure-theory

More exotic topologies

Jasmine

10:33

2

Q: Need help with this question.

I was trying to solve this question but got stuck. If we further solve it we get $I'(e)=0 $ which does no help to find the value of the integral. I know an alternate way to take $7$ as variable but I want to know if the process I used can help in anyway to evaluate this integral. How will we fin...

definite-integrals

I need help with this.

mick

11:24

0

Q: How to solve $ f(\sqrt[3](1 - z^3))^2 = 1 - f(z)^2 $?

How to Find analytic $f(z)$ such that $$ f(\sqrt[3](1 - z^3))^2 = 1 - f(z)^2 $$ Koenigs function can not be used here So I am stuck. How does the riemann surface look like ?

dynamical-systems functional-equations riemann-surfaces complex-dynamics

Sonu Lamba

14:04

***Question:-*** Let $V$ denotes the vector space of all sequences $ \boldsymbol{a}=(a_{1},a_{2},...)$ of real numbers such that $\sum{2^{n}\left|{a_{n}}\right|}$ converges.

Define $\|.\| : V \rightarrow \mathbb{R}$ by $\|\boldsymbol{a}\|=\sum{2^{n}\left|{a_{n}}\right|}$. Which of the following ***are*** true?

1. $V$ contains only the sequence $(0,0,...)$.
2. $V$ is finite dimensional.
3. $V$ has a countable linear basis.
4. $V$ is a complete normed space.

***My attempt:-*** As we know $$\|\boldsymbol{a}\|=0 \Leftrightarrow \boldsymbol{a} = 0$$

Silent

@LeakyNun, here, do we measure angle t from TC to CP or from CP to TC? In particular, I am finding very hard to digest that coordinates of vector CP are (-a sin t,-acos t), which is iscussed a couple minutes later

Failed to be a Mathematician

14:25

in CSIR-TIFR-ISI-NBHM, 5 mins ago, by Failed to be a Mathematician

for a cauchy sequence in the space $a_n$, for $\epsilon>0$,$\exists N;n,m \ge N||a_m-a_n||<\epsilon$
$a_{n}=(a_{n1},a_{n2},...,)$
$a_{m}=(a_{m1},a_{m2},...,)$
$||a_n-a_m||=\sum 2^k|a_{nk}-a_{mk}|$
$\implies$ $\{a_{nk}\}$ is cauchy sequence of real numbers. Hence, convergent.
$|a_{nk}-a_{mk}|\leq 2^k|a_{nk}-a_{mk}|\leq ||a_m-a_n||<\epsilon$
let it converges to $a_{0k}$.
$a_{n1}\to a_{01}$
$a_{n2}\to a_{02}$
...
Let $a_0=(a_{01},a_{02},...)$
Now we need to prove $a_0 \in V$
We need $\sum 2^k |a_{0k}|$ convergent.

I hope this solution is correct.

Sonu Lamba

Yes. It is right. Thanks @FailedtobeaMathematician

Failed to be a Mathematician

mmm

Failed to be a Mathematician

15:48

How does (3.7) coming?

How $\sigma (\sum_{\tau \in S_k} (sgn \tau)(\tau f) \tens g)$ transformed?

How $\sigma (\sum_{\tau \in S_k} (sgn \tau)(\tau f) \otimes g)$ transformed?

@Secret

$\sigma (\sum_{\tau \in S_k} (sgn \tau)(\tau f) \otimes g)=\sigma (\sum_{\tau \in S_k} (sgn \tau)\tau (f \otimes g)$

Is $\sigma$ linear?

Failed to be a Mathematician

16:33

Hai @Silent

Jasmine

16:46

Hello! For a,b be real positive numbers what is wrong with $a^2+b^2>or=0$ then a+b is > or = (2ab)^(1/2)(why isnt it in accordance with Am Gm inequality i know that minimum value of a+b is (4ab)^(1/2)

Abcd

@Jasmine $\ge$ (\ge)

@Jasmine a can be negative, same for b. AM-GM is for positive numbers.

@Jasmine Sum of squares is always positive.

@Jasmine There's nothing to laugh about it.

FrownyFrog

How to calculate the probability of getting each outcome at least once from n attempts? Like if I have coins it’s just 1 − 2/2ⁿ

Jasmine

@Abcd yes actually that wasn't my question it's little different I interpreted it differently

FrownyFrog

17:11

nevermind, I got it

GFauxPas

hi friends

math.nyu.edu/student_resources/wwiki/index.php/… in the first solution, how do we know that the lower limit of integration should be $0$?

GFauxPas

17:30

nm I see it

Silent

17:43

I get to know from here that angle $\theta$ here is measured clockwise. Then why is $\theta$ positive?

(The picture gives derivation of parametric equations)

@GFauxPas, do you have any idea?

Semiclassical

because as the circle moves to the right, it'll rotate clockwise as well

Silent

@Semiclassical I can see why angle is measured is clockwise. But I was taught that if you measure angles counter-clockwise, then they are positive but if you measure them clockwise, they are considered negative. But here , even though clockwise, $\theta$ is cosidered positive.

Semiclassical

positive angles being CCW is a convention, not a necessity

Silent

So, we can violate that as and when we want?

Semiclassical

if you're clear about it, sure. same as any coordinate system

Silent

17:54

So, if i stick to that convention, will i get same result?

Semiclassical

you should, yes. in fact, i'd say it's easier to do it this way than the other

Silent

ok, thank you

Semiclassical

if you do it the other way, you'll need to $s=-a\theta$. that still works, but it's a good deal more tedious

alternatively, you could have the disk roll to the left and get a CCW rotation

then it'd match the convention

GFauxPas

math.nyu.edu/student_resources/wwiki/index.php/… now I dont understand the very last line. How can we throw away $\delta$ like that?

Semi, that would depend if there's any room behind the circle for you and your phone camera

:P

Semiclassical

idgi

GFauxPas

17:59

if it's rotating clockwise, to get it to rotate ccw you have to step behind it

maybe there's no room

wuz joke

Semiclassical

lol

or, y'know, you could just stop it and push it the other direction :P

GFauxPas

can you take a look at the last line of the solution I linked to Semi, please

the question's phrased very sloppily

"find an equivalent"

but the question is, how to absorb $\delta$ into a constant multiple

Secret

@FailedtobeaMathematician I am too sleepy and afking to look at the question (as well I am not that familar with k vectors). you might need to ask someone else

Semiclassical

I probably can't say anything. My recollection of real analysis is very old, and I"m not sure I remember dominated convergence

GFauxPas

no not that part

just the very last line, it's one of those "after some algebra" cheats

Semiclassical

18:02

ah

GFauxPas

which I dislike

Semiclassical

glancing it at it quickly, I imagine it's something like: In this manipulated form, we see that it'd have been more convenient had we defined $a=b+1$ not $a=b$. So we'll redefine it like that

the mathjax isn't loading correctly for me tho

GFauxPas

try in firefox

Semiclassical

or maybe it's just loading v e r y s l o w l y

GFauxPas

it doesnt work in Chrome well for me for osme reason

Semiclassical

18:05

that'd do it

user193319

Let $F_n$ denote the free group on $n$ generators, and let $g_1,...,g_\ell \in F_n$ be distinct elements. Then $\langle g_1,...,g_\ell \rangle$ is a free subgroup of $F_n$. Suppose that $\pi : \langle g_1,...,g_\ell \rangle \to \mathcal{U}(M_k(\Bbb{C}))$ be some unitary representation. Is it possible to extend $\pi$ to all of $F_n$?

GFauxPas

well here we're making $x < \delta$ so that the error of a limit term is less than epsilon

so we can always make $\delta$ smaller

still dont see how that helps

Semiclassical

it does seem a bit handwavey

GFauxPas

indeed it does, and now I have to unwave the hand

Semiclassical

good luck with that :P

GFauxPas

18:08

ugh

Semiclassical

One thing I do find weird about that description of it: As $\epsilon\to 0$, one has $\log(1+\delta^2/\epsilon^2)\approx \log(\delta^2/\epsilon^2)=2\log (1/\epsilon)+2\log \delta$

which means that the $2\log \delta$ term is not going to be part of the $C$ term in front

it'll be an additive constant, not a multiplicative one.

GFauxPas

that's exactly my problem

not to mention I don't get the squiggly equality either

Semiclassical

well, if $\epsilon$ is small (compared with $\delta)$, then $\delta^2/\epsilon\gg 1$

GFauxPas

sure

ah, I see

$\delta \in (0..1)$ so we can't just set it to $1$

in this case

$\epsilon$ is arbitarily small and $\delta$ is strictly between 0 and `1

Semiclassical

I think the thing to bear in mind is what 'arbitrarily small' actually means

GFauxPas

18:14

maybe it's a mistake

hm?

Semiclassical

0.01 is small compared to 1, but not compared to 0.001

GFauxPas

right

Silent

@Semiclassical Thank you so much for this. So, if $\theta$ negative, $x$ co-ordinate will be $-a\theta+\sin\theta$, right?

GFauxPas

just flip the horizontal axis

Semiclassical

hmm

GFauxPas

18:16

positive on the left and negative on the right

Semiclassical

you may need to have it be $\sin(-\theta)=-\sin \theta$ as well

GFauxPas

but I think it's less confusing to just have the wheel roll the other way

to the left

Semiclassical

yeah

I always get a bit mixed up geometrically with this business if I'm honest

there's a reason I stick with stuff like "better make sure s and \theta have the same sign"

GFauxPas

so Semi do you think it's a mistake or is the answer salvagable and I just have to unwave the hand

Semiclassical

i think it's just a matter of being clear

GFauxPas

18:19

but I think it should be $+2 \log \delta$, where $\delta \in (0..1)$

Semiclassical

sure, which (following their tack) you could equally well write as $-2\log(1/\delta)$

with the advantage that the log is then positive

GFauxPas

good point

Semiclassical

I'd also point out that $\delta$ doesn't actually appear in the problem statement

so its definition is inherently arbitrary

GFauxPas

well, it is between zero and one

so it's an error term

Semiclassical

as such, you could redefine $\delta \to \delta \epsilon$

GFauxPas

18:23

because $\delta \epsilon < \delta$ and $\delta$ was arbitrary

okay, then what

Semiclassical

then $\delta/\epsilon\to 1/\epsilon$

more formally, I'm doing $\delta'=\delta/\epsilon$

Leaky Nun

math.stackexchange.com/questions/tagged/galois-theory

GFauxPas

oooh

Leaky Nun

last question posted 10 hours before :(

GFauxPas

that goes back to your first guess

Semiclassical

18:24

ya

though, actually, something's off

yeah, that doesn't make sense. $\delta/\epsilon \to (\delta \epsilon)/\epsilon = \delta$ under that prescription

i dunno

GFauxPas

well $\epsilon < \delta \epsilon$ as well

err other way

$\delta \epsilon < \epsilon$

Semiclassical

i'm not convinced that helps

if one wanted $\delta/\epsilon\to 1/\epsilon$, one would do $\epsilon\to \delta \epsilon$

i.e. redefine $\epsilon$

this feels goofy

GFauxPas

the whole question is silly

"find an equivalent"

Semiclassical

it reminds me of physics tbh---figure out the asymptotic behavior of a given integral

GFauxPas

well the first part does that

first half of the question

actually that's not helpful

user193319

18:30

I think it might be possible using the universal property of a free group, though I am not certain.

GFauxPas

that would tell you $h(\epsilon) \sim 0$ for $\epsilon$ small

not helpful when integrating over a shrinking interval

Semiclassical

Well, let's be specific for a moment and take $f(x)=x$

as the simplest example

GFauxPas

sure

Semiclassical

then $\displaystyle \int_0^1 \frac{x}{x^2+\epsilon^2}\,dx = \frac12 \log(1+1/\epsilon^2)$

GFauxPas

right

Semiclassical

18:32

I think I know why this question is bothering me. In the physics context, this integral doesn't make sense unless $x,\epsilon$ are already pure numbers

since the range is $x=0$ to $1$

GFauxPas

pure? meaning, constants?

Semiclassical

right. i.e. dimensionless

in physics, you'd instead have something like $\displaystyle \int_0^\Lambda \frac{x}{x^2+\epsilon^2}\,dx$

GFauxPas

$\Lambda$ is?

Semiclassical

and $\epsilon$ being small/large would be $\epsilon\ll \Lambda$ and $\epsilon\gg \Lambda$ respectively

some quantity with the same dimensions as $x$ and $\epsilon$

You'd need that for the dimensions to work out

GFauxPas

oh, okay

well let's not worry about that

Semiclassical

18:35

I'm not sure that's a crucial difference tho.

Agreed.

GFauxPas

so here instead of $x$ we have $f(x)$ where $f(0) = 0$, $f'(0) \ne 0$, and $f$ is continuous on a compact set

Semiclassical

Right

I dunno. That last line does just feel sloppy

GFauxPas

I don't even know if it's correct

Semiclassical

"absorb it into the constant"

which constant?

GFauxPas

this isn't an official page, it's a wiki made by students

Semiclassical

18:38

yeah

GFauxPas

so it might be wrong

Semiclassical

I don't think it's 'wrong,' just not well stated

I mean, the $-2\log(1/\delta)$ is an additive constant after all

it won't be relevant as $\epsilon\to 0$

GFauxPas

uh

Semiclassical

derp

GFauxPas

yeah

okay better

Semiclassical

18:40

though it seems like the key point there is that $\delta$ is independent of $\epsilon$

GFauxPas

right, all we assumed about it was that $0 < \delta < 1$

we didn't assume anything else

Semiclassical

Right. and the quantity being bounded doesn't depend on $\epsilon$

so the required value of $\delta$ is also not $\epsilon$-dependent

GFauxPas

so it will be negligible compared to the $\epsilon$ term, as the $\epsilon$ terms grow without bound. slowly, but they do

Semiclassical

I guess the reason they want that factor of 1/2 in said bound is to have $\frac12 \log(1/\epsilon^2)=\log(1/\epsilon)$

eh, just do $\epsilon$ in reciprocal powers of 10 :P

GFauxPas

the 1/2 factor was convenient earlier

Semiclassical

18:43

in physics, btw, the terminology is that the integral has a 'logarithmic divergence'

GFauxPas

I like that

so yeah, the 1/2 was from earlier, it was $1/c$ where $c >1$

Semiclassical

more precisely, you'll have integrals of the form $\int_0^1 \frac{f(x)}{x^2}\,dx$

and you argue that this shows up because the setup of the calculation wasn't sufficiently careful

GFauxPas

I didnt understand that last line

Semiclassical

yeah, I'm handwaving a bit

it's more or less a point about how you have to regularize certain badly-defined integrals in QFT

ah, the discussion here is pretty much identical to the $f(x)=x$ case given earlier: hitoshi.berkeley.edu/230A/regularization.pdf

(with $d^2 p = 2\pi p \,dp$ upon going to polar coordinates with radial coordinate $p$ and integrating over angle)

(also, $dp^2=d(p^2)=2p\,dp$ on the second page. i have no idea why they did that)

GFauxPas

thanks for your help Semi

Semiclassical

19:09

np

Abcd

19:21

What is the maximum value of:

$$f(x)= \int_0^1 t\sin(x+\pi t)dt$$

I don't understand why Leibniz Rule isn't helping in this problem .

I differentiated and set the resultant integral to zero which is when $x= \dfrac{\pi}{2}- \pi t$

So we get maximum value $= \dfrac{1}{2}$

Which is not even remotely close to the right answer.

Is anyone going to reply?

nevermind, Ill ask on main then

Semiclassical

19:40

@Abcd What integral did you get upon differentiation? The $t$ in front doesn't go away, so I don't think it's any easier than the integral you started with

more to the point, one simple approach is by doing f(x) by parts and only then differentiate

Imago

Is there a definition for continuous functions between a set and a topology? Usually I see continuity only defined between two topological spaces.

Leaky Nun

no, there isn't.

Abcd

@Semiclassical Wont we partial differentiate wrt x?

$f'(x)= \displaystyle\int _0^1 t\cos (x+\pi t) dt$.

Semiclassical

Sure, but how do you do the resulting integral?

All you've done so far is trade sin for cos

Abcd

@Semiclassical For maxima f'(x)=0

so x= $\pi/2 - \pi t $

Semiclassical

19:47

...yes, and? You need $\int_0^1 t\cos(x+\pi t)\,dt=0$

Why, exactly?

Abcd

@Semiclassical cos pi/2 is zero

Thats why x = pi/2 - pi*t

Semiclassical

So? You've got $t$ ranging from 0 to 1

and you need to pick one value of x

at most, you can make the integrand vanish at one point

but vanishing at one point doesn't mean it vanishes everywhere

Abcd

oh

so what do I do?

Semiclassical

I think I already said what one could do.

Abdullah UYU

20:49

Hey, I'm not officially introduced to matrices but I've got a rough understanding for it. mathworld.wolfram.com/RotationMatrix.html Here there is the line $v' = \mathbb{R}_\theta\,v_0$. Is it the "matrix multiplication" going on that line or I'm wrong?

My concern is that $v_0$ is not a matrix but a vector.

Abcd

What;s the simplest way to say (prove/ciaim ) this^ ?

Semiclassical

@AbdullahUYU if you have matrices A,B which are j-by-k and k-by-l respectively, then the matrix multiplication AB is well-formed and gives a j-by-l matrix

Abcd

Do you have any idea about my question?

Semiclassical

In particular, if l = 1, then B is k-by-1 ie a column vector

So multiplying a column vector on the left by a matrix of appropriate size gives another column vector

In particular, one has v’=Av with matrix sizes n-by-1, n-by-n, and n-by-1 respectively

Abdullah UYU

Hmm, so a vector is not really much different from a matrix other than the dimension it has.

Considering them solely as data types.

Semiclassical

21:07

Yep

GFauxPas

well, yes, but it's often helpful to view rhem

view them as objects matrices act upon

Semiclassical

It should be noted that, in most linear algebra contexts, that ‘vectors’ are taken to be column vectors

Abdullah UYU

I don't have a motivation yet but yes, intuitively, the matrices are inclusive.

GFauxPas

though it's trivial to show that a space of row vectors is isomorphic to a space of column vectors of the same length

Semiclassical

Huzzah for the transpose

GFauxPas

21:10

trivial, but you should prove it to yourself for practice

we generally think of (rectangular) matrices as maps that act on objects called vectors

rather than thinking of column vectors as matrices in their own right. depends on context

but (column) vectors are nx1 matrices

Abdullah UYU

alright, thanks all

GFauxPas

@AbdullahUYU , are you comfortable with proofs by induction?

Abdullah UYU

Yes, i think so.

GFauxPas

alright then

Ted Shifrin

@Abcd: That's immediate if you use the addition formula for $\tan(\alpha-x)$ and do the little bit of algebra.

Abcd

21:23

@TedShifrin Not really.

Ted Shifrin

I just did it.

Abcd

Is an integral with variable limits always differentiable?

Ted Shifrin

That's a vague question. What do you mean?

Abcd

@TedShifrin let me show the question

Ted Shifrin

The answer is no.

Do you know a statement of the fundamental theorem of calculus (there are two parts)?

Abcd

21:25

@TedShifrin cant recall. But I know them

GFauxPas

if you can't recall them, review them

it's pretty ... fundamental

Nick

Is this generalization wrong?

Abcd

@GFauxPas Both those theorems were quite obvious I felt.

GFauxPas

so obvious you don't remember them

?

Ted Shifrin

@Abcd, "quite obvious" — again, do you understand what's going on? Is there a hypothesis in there you're forgetting that you need to answer your question?

Abcd

21:31

Leaky Nun

@TedShifrin hi

geocalc33

I wish

Abcd

Leaky Nun

@TedShifrin you've been gone quite a while; I wanted to ask you a question

Abcd

I assumed that the integral is differentiable so applied L Hospital and got the right answer

@TedShifrin What exactly did you convert it to?

GFauxPas

21:33

so, is it differentiable?

Leaky Nun

@Abcd it's intuitive maybe (that's controversial), but it's anything but obvious

Ted Shifrin

Notice that the theorem assumes continuity of $f$ to get differentiability of $F$.

Leaky Nun

@Ted my question is, what is your opinion on this paper? (The Non-Existent Complex 6-Sphere, Michael Atiyah, Differential Geometry)

Ted Shifrin

And your question has continuity as a hypothesis. But if $f$ fails to be continuous, $F$ may fail to be differentiable.

@Leaky: I have consulted with experts on this and it's not believed to have any validity.

I myself am not qualified to judge it

Leaky Nun

isn't you yourself an expert

Ted Shifrin

21:35

Not on the stuff he's using, no, and it's totally vague and inexplicit.

Leaky Nun

isn't -> aren't

Abcd

@GFauxPas yes because $\sin (t^2)$ is differentiable

Ted Shifrin

@Abcd: There's nothing tricky going on. The denominator in $\tan(\alpha-x)$ cancels out and you're left with 4 terms to add up, 2 of which cancel. And then you use $1+\tan^2\alpha = \sec^2\alpha$.

Leaky Nun

lol at the references, everything is citing himself

geocalc33

Who's been watching the world cup?

Abcd

21:37

@LeakyNun I think I saw on English.SE that "how is you" is also acceptable in place of "how are you"...just telling

Leaky Nun

@Ted what is Spec(2^N) where 2 is the field of two elements? According to theorems, it should be in bijection to N, compact, hausdorff, but I have no idea what it is

Ted Shifrin

I don't know, @Leaky. I have a question on main I have to write up.

Leaky Nun

ok

Abcd

@TedShifrin I want to die now. I was using hellish things and it wassss so simple!

Leaky Nun

don't die

Ted Shifrin

21:40

I told you how to do it directly and, of course, you contradicted me without trying it. You get annoying!

I'm going to try to write up a multivariable calculus answer now.

Abcd

@TedShifrin Oh sorry about that.

geocalc33

Hey guys

Can someone tell me what type of graph this is? I posted a question about it but didn't get an answer

Abcd

$$\int_0^{\pi/2}\cos^{10}x (\sin 12x )dx$$

Maximus

Hi @TedShifrin @LeakyNun

Leaky Nun

hi

Maximus

21:48

Does anyone know by what theorem or property I can estimate an upper bound for a sum with its integral? i.e. I want to claim $\sum_{x=0}^\infty e^{-\frac{1}{2} \pi x^2/r^2} \le \int_{-\infty}^{+\infty}e^{-\frac{1}{2} \pi x^2/r^2}dx$

geocalc33

the summation is only summing for positive values of x and the integral is summing over negative values and positive values of x

Maximus

I mean bounding a summation by its integral. I remember there were some small details such as monotonicity from what I recall

geocalc33

I don't know, @LeakyNun @Rudi_Birnbaum do you guys know the answer to Maximus's question?

Rudi_Birnbaum

Hi guys!

geocalc33

Hi :)

Leaky Nun

21:56

7

Q: When is infinite sum bounded by an integral?

What conditions does $f(i)$ have to have for the inequality $\sum_{i=1}^{\infty} f(i) \leq \int_{i=0}^{\infty} f(i)$ to hold? My guess is that $f(i)$ has to be monotone and that the sum and integral should both exist. Is it right that you don't need to worry about whether $f$ is increasing or ...

inequality integration summation

Maximus

@LeakyNun Thank you so much

Rudi_Birnbaum

by the integral bound theorem?

Abcd

@LeakyNun Are you interested in that integral?

Rudi_Birnbaum

I mean some integrals really are upper bounds to series

Leaky Nun

@Abcd not really

Abcd

21:58

I see.

Integration is just tricks, tricks and tricks. Lot many tricks!

Transcript for

$Mathematics$ Mathematics