Mathematics - 2017-02-02 (page 2 of 3)

10:03 AM

@Secret can you prove that omega^2 = omega by splitting N into N subsets each of cardinality N

10:16 AM

So, the expression $(A\cup B)\cap (A\cup B')\cap (A'\cup B)$ is equivalent to the logic expression $\Phi:=(p\lor q)\land (p\lor \neg q)\land (\neg p\lor q)$, right?

From the truth table we get that this is equivalent to $p\land q$, so to the intersection $A\cap B$, right?

DHMO

10:32 AM

@MaryStar yes

Mary Star

@DHMO Ok!! Thank you!! :-)

Dattier

@DHMO math.stackexchange.com/questions/2125477/…

la réponse

DHMO

@Dattier dieu, c'est une bonne reponse

@MaryStar bitte

Dattier

10:57 AM

@DHMO : une idée pour celui là : math.stackexchange.com/questions/2125559/…

Ramanujan

$$\lim_{x\to0} \dfrac{\sin(\pi (\cos^2(x))}{x^2}$$ $\small{HOW to solve?}$

SteamyRoot

l'Hôpital?

Ramanujan

Without it

@SteamyRoot

user400188

While looking around stack exchange I noticed this: ∀x∈U:q(x)≡∀x:[x∈U→q(x)]
http://math.stackexchange.com/questions/2124099/simple-question-about-universal-quantifier
what is the meaning of the ":" in that formula?
they seem to replace the →

also there appears to be one after the ∀x on the right hand side

SteamyRoot

It's just like a separator meaning "holds that" or "we have that"

Ramanujan

11:11 AM

such that

user400188

I've heard the english definition before; but I want to know what its mathamatical definition is

for instance X is a subset of Y has a definition ∀a(a∈X $\rightarrow$a∈Y). Is there a similar thing for ":"?

SteamyRoot

The : is just a different way of writing brackets.

user400188

if that is the case; then how does it replace the implication in "∀x∈U:q(x)≡∀x:[x∈U→q(x)] " ?

Tobias Kildetoft

@user400188 It does not

user400188

can someone take me through the steps in expanding the LHS to the RHS then?

SteamyRoot

11:15 AM

the implication is hidden in the fact that you write "$\forall x \in U$"

user400188

but that line is on both sides. What am I missing?

actualy I should be asking how is it hidden?

I was told by another that ∀x∈U is just a shorthand way of writing ∀x (x∈U)

which may be a sorce of some confusion

SteamyRoot

Lolno

That would mean every object $x$, whatever it may be, belongs to some set $U$.

Tbh, Mauro's comments on that question is pretty much all I can say about this...

Also, @Ramanujan, you could probably use the answers to this question to solve your limit.

user400188

Reading Mauro's comments; I am left more confused. Is there some set of rules that allow us to take it to the abreviated form? Or is it literaly just a shorthand version that is not mathamaticly correct in itself, but stands for something that is?

Tobias Kildetoft

@user400188 Once we define it as an abbreviation, it automatically becomes mathematically correct

user400188

Ok then. So it is just an abrivation and not some simplification using existing rules.

Tobias Kildetoft

11:24 AM

right

user400188

I am somewhat surprised that abreviations like that exist. Most I am formiliar with using different symbols entirely. This one seems to use existing symbols. Reading out load the abriviation as if those sybols were used souds like a different statement entirely.

@SteamyRoot Reading back on your comments I noticed you wrote: "That would mean every object x, whatever it may be, belongs to some set U" for ∀x∈U . But is not that exactly what ∀x (x∈U) says?

SteamyRoot

Yes....?

user400188

didnt you say Lolno at the start??

SteamyRoot

$\forall x: (x \in U)$ means that every object $x$ belongs to some set $U$.

In particular, that statement will lead to a paradox (Russell's paradox)

$\forall x \in U$ is something very different

It means you have a predefined set $U$ and you only consider the elements $x$ of that set.

Tobias Kildetoft

@SteamyRoot Actually, in many contexts, it will not lead to a paradox because the logic will only allow $x$ to be an element of a specified set anyway

user400188

11:36 AM

I was about to say that earlier actualy but I did see that it would lead to that paridox in some cases.
to be honest i had never thought to connect russles paradox to that statement

Tobias Kildetoft

@user400188 Note that $\forall x: x\in U$ is a full statement whereas $\forall x\in U$ is missing something be be an actual statement

user400188

I noted that too

I was surprised when Root said it actualy had a meaning

Junaid Aftab

I'm looking for a textbook that covers, at least for a large part, special relativity with tensors/a geometrical approach. Most textbooks I have found develop tensors for the purposes of GR; I'd like a book that goes through this exercise for the purposes of understanding SR. I'll then feel more comfortable moving on with the GR bit.

Ideally, the textbook/resource will first cover tensors, and then move on to discuss tensor analysis in SR.

Any suggestions?

SteamyRoot

It "means" something, I didn't say it was a statement...

DHMO

@user400188 "∀x∈U:q(x)≡∀x:[x∈U→q(x)]" means exactly the same thing as "[∀x∈U[q(x)]]≡[∀x[x∈U→q(x)]]"

user400188

11:41 AM

ah thank you.

DHMO

@Ramanujan taylor series

user400188

The : remind me of the dots Bertrand used in Principia Mathematica

the order of operations was set each time by the number of dots two statements had between each other

DHMO

11:58 AM

@Dattier avancement: j'ai decouvert que $\left(1+\sqrt2\right)^n$ est tres pres des entiers quand $n$ croit...

@Dattier Le polynôme minimal de $\left(1+\sqrt2\right)^{50}$ est $x^2 - 13765255184676885126 x + 1$...

Dattier

12:17 PM

Ivan a trouvé la solution

Ramanujan

@DHMO any good source to learn Tylor series?

DHMO

@Dattier merci

@Ramanujan I don't know

Dattier

avec plaisir

Secret

Last night dream: I have a white box of infinite size X (although physically it is onnly as large as a pencil case). I then took away countable number of styrofoam balls inside, and the box still as full as ever. I don't end up with $> 2^{X}$ things in total. I then suspect because the box only has $2^{X}$ possible components in it, thus I cannot take out more

I am however pondering what if I tried to take out that many things, what will happen:

$$X-2^{2^X}$$

s.harp

what?

DHMO

12:28 PM

@Secret maybe you should focus on your university instead

Secret

@DHMO o that is taken care of. I cannot do anything until 1 Marhc when it starts

that's why I am doing munkres and it just happened I go on a mad tangent to infnite sets because of its bizarre properties

DHMO

@Secret I'm still thinking how we can have 2^N cuttings of Q

Secret

12

Q: Uncountable chains

$P(\mathbb N)$ = power set of $\mathbb N$. $A \subset P(\mathbb N)$ is a chain if $a,b \in A \implies$ either $a \subseteq b$ or $ b \subseteq a$ That is we have something like this: $$\ldots a \subseteq b \subseteq c \subseteq\ldots$$ where $a,b,c \in A$ are distinct. We can show easy enough...

It's because of trees, you can make 2 branches for each node

(In the set theory room, I then ask why the tree proof fails for finite sets)

Here's a more visual representation of the proof:

Alessandro Codenotti

@DHMO that's because most of them are not cuts "at a rational" but cuts in a set with no sup and a set with no inf

Akiva Weinberger

What are you using for your images?

Secret

12:33 PM

@AkivaWeinberger just power point

I made all these images myself

(the latex is cropped from the ask question window in the main site when they render it there)

You mean something like this?
(... x x x x x | x x x ...)

But then naive intuition will say to us there will only be countably many possible spaces you can place the cut |

or do you mean something like this?
(... x x x x x ... | ... x x x x x ...)

(but naively it stil seemed like you can only put | at countable number of places...

o wait a sec.... what if...

(... x x x | x x x x ...)
(... x x x | x x x x ...)
(... x x x | x x x x ...)
(... x x x | x x x x ...)

what if all of these are actually 4 different cuts? (even if the cut is located between the same two rationals)

there are uncountably many reals between any two rationals after all, not matter how close they are...

Jack Lam

popping in with a completely unrelated question

Does anyone know if there is already a proof of either of the two

They aren't necessarily related

I found these numerically.

Equivalent integrals are also accepted

I just don't want to ask something that already exists....

*ask for something

DHMO

12:50 PM

@AlessandroCodenotti sure, but they're still cuts of the rationals

Ramanujan

@DHMO can we do it without Tylor series because we haven't introduced to it and it was given in exam today

Secret

@JackLam just a small check, is cot-1 inverse cotan or reciprocal cotan (hence tan)?

@DHMO No that actually make quite a difference (at least for rationals, I have no idea how one can make sense of that in naturals as consecutive naturals there are no naturals there). Since any two rationals, no matter how close, will always have countable number of rationals within, two different cuts can be put arbitrarily close to each other, thus approaching continuum instead of some fixed separation

DHMO

$\displaystyle \lim_{x\to0} \frac{\sin(\pi (\cos^2(x))}{x^2}$
$= \displaystyle \lim_{x\to0} \frac{\sin(\pi (1 - \cos^2(x))}{x^2}$
$= \displaystyle \lim_{x\to0} \frac{\sin(\pi (\sin^2(x))}{x^2}$
$= \displaystyle \lim_{x\to0} \frac{\sin(\pi (\sin^2(x))}{\sin^2(x)} \frac{\sin^2(x)}{x^2}$
$= \displaystyle \lim_{x\to0} \frac{\sin(\pi (\sin^2(x))}{\sin^2(x)} \frac{\sin(x)}{x} \frac{\sin(x)}{x}$
$= 1\times1\times1$
$=1$

Ramanujan

@DHMO what??

We can multiplying and divides simply sin ?

DHMO

@Ramanujan I don't understand your question

and I made a mistake

$\displaystyle \quad \lim_{x\to0} \frac{\sin(\pi (\cos^2(x))}{x^2}$
$= \displaystyle \lim_{x\to0} \frac{\sin(\pi (1 - \cos^2(x))}{x^2}$
$= \displaystyle \lim_{x\to0} \frac{\sin(\pi (\sin^2(x))}{x^2}$
$= \displaystyle \lim_{x\to0} \pi \frac{\sin(\pi (\sin^2(x))}{\pi\sin^2(x)} \frac{\sin^2(x)}{x^2}$
$= \displaystyle \lim_{x\to0} \pi \frac{\sin(\pi (\sin^2(x))}{\pi\sin^2(x)} \frac{\sin(x)}{x} \frac{\sin(x)}{x}$
$= \pi\times1\times1\times1$
$= \pi$

Ramanujan

1:04 PM

$\dfrac 1x =\dfrac{\sin(1)}{\sin(x)}$ ?

DHMO

@Ramanujan no, where did I say that?

Let $y = \csc^{-1} \left(\sqrt{1 + \cot \theta}\right)$.
$\csc y = \left(\sqrt{1 + \cot \theta}\right)$
$\csc^2 y = 1 + \cot \theta$
$\csc^2 y - 1= \cot \theta$
$\cot^2 y = \cot \theta$
That's the farthest I've gone to.

Ramanujan

Hehe,nice but still mistake cos^2 x = 1 - sin^2 x

DHMO

@Ramanujan sin(pi(cos^2 x)) = sin(pi - pi(cos^2 x)) = sin(pi(1-cos^2 x))

Jack Lam

@Secret It's inverse, the function would be impossible to express in elementry terms (or so I would reason to believe)

DHMO

@JackLam it is possible, but you would need complex numbers.

Jack Lam

1:09 PM

well Ihave my dounts on the tan of a csc, but if you choose to believe that...

*doubts

anyway, it's standard inverse notation, I didn't think I'd be required to elaborate on that.

DHMO

It wasn't me who asked

Jack Lam

nested trig functions of the same type (forward XOR inverse) as an integrand are usually horrible

unless they have something extra to make them trivial

DHMO

@JackLam Actually, I think I can continue.

Jack Lam

...I'm confused....

Secret

For the first integral, wolfram alpha's results got a 6 decimal number that matches the first 6 decimals of $\frac{\pi^2}{12}$. There are no numbers followed after that but there is for $\frac{\pi^2}{12}$, so it is not exactly $\frac{\pi^2}{12}$ but close up to 6 d.p to $\frac{\pi^2}{12}$

Second integral still computing

Jack Lam

1:16 PM

I did state these results were obtained numerically.

I've verified them to hundreds of decimal places.

DHMO

Let $t = \cot \theta$. Then, $t^2 = \cot^2 y$.
$\mathrm dt = - \csc^2 \theta \ \mathrm d\theta = -\sqrt{t^2+1} \ \mathrm d\theta$
$\displaystyle \quad \int_0^{\frac\pi2} \csc^{-1} \left(\sqrt{1 + \cot \theta}\right)\ \mathrm d\theta$
$\displaystyle = \int_0^{\frac\pi2} y \ \mathrm d\theta$
$\displaystyle = \int_0^\infty \frac y {\sqrt{t^2+1}} \ \mathrm dt$
... and then I'm stuck.

@JackLam never mind, I can't continue.

Secret

The second integral has a very scary looking closed form involving a polylogarithm Li$_2$, and other terms in arcsin and arctan. It is nonelementary

the first integral when indefinite wolfram just give up. Now checking MSE for similar integrals

DHMO

@Secret there must br a clever way to evaluate the definite integral

Secret

@DHMO try sub $\cot \theta$ with $-\cos ^2 \theta$, and $\csc \theta$ with $\cot ^2 \theta$. See if anything nice pop out...

I need to double check how one get csc^2 from sin^2+cos^2=1 identity

DHMO

@Secret cot^2 + 1 = csc^2 but I don't know what you are trying to do

Secret

1:28 PM

I am trying to make a substitution so that the resulting term in the squareroot will become a trig function that is the inverse of the inverse trig function, and hopefully nothing nasty pops up in the $d\theta$

The two integrals are nonelementary, but perhaps they might have some symmetry that allow them to express in terms of other nonelementary integrals (preferably one which is some constant multiple of it) that are easier to check results

DHMO

I need a way to create $\pi^2$

Jack Lam

1:41 PM

I briefly considered fourier expansions but I gave up after a moment's thought

but maybe someone more familiarized can work with it

aquire

2:00 PM

may be $\int_0^a f(x)dx=\int_0^a f(a-x)dx$ would help

DHMO

@aquire somehow I've never tried it... let me try

trilolil

Hello everyone

I am goiung to ask an extremely stupid question

but it got me stumped

$2 = \frac{10}{8-x}$

how do you get x out of there o.O ?

DHMO

$\displaystyle 2 = \frac {10} {8 - x}$
$\displaystyle 2 (8 - x) = \frac {10 ( 8 - x)} {8 - x}$
$\displaystyle 2 (8 - x) = 10$
$\displaystyle 16 - 2 x = 10$
$\displaystyle 16 - 2 x + 2x = 10 + 2x$
$\displaystyle 16= 10 + 2x$
$\displaystyle 16 - 10 = 10 + 2x - 10$
$\displaystyle 6 = 2 x$
$\displaystyle \frac 6 2 = \frac {2 x} 2$
$\displaystyle 3 = x$
$\displaystyle x = 3$

trilolil

@DHMO Wow I feel so stupid, thx.

BTW why overcomplicate?

Akiva Weinberger

@trilolil Multiply left and right sides by $(8-x)$

DHMO

2:05 PM

@trilolil you are welcome

trilolil

@DHMO I mean no matter what case that seems totally useless to me. Or not?

DHMO

@trilolil because i don't know how much you know

Akiva Weinberger

Alternatively, you could do $10$ divided by each side, giving $5=8-x$.

(Note that $10\div\left(\frac{10}{8-x}\right)=8-x$)

aquire

taking inverses of both sides

Jack Lam

inspect the denominator

clearly, it must be equal to 5

Akiva Weinberger

2:08 PM

Give it to Wolfram Alpha

trilolil

@aquire nice one.

Jack Lam

thus, x=3

aquire

for things like ${p\over q}=\frac{a+x}{c-x}$ you can do $\frac{p}{p+q}=\frac{a+x}{(a+x)+(c-x)}$

Secret

Take $1+\csc \theta=\cot ^2 u$ get scary integral:

$$\int_0^{\frac{\pi}{2}} \cot^{-1}\left(\sqrt{1+csc(x)}\right)dx=4\int_{-\infty}^{\infty}(u+n\pi)\left(\frac{du}{\sin (2u)(\cot^2 u-1)\sqrt{1-(\cot^2 u-1)^2}}\right)$$

where $n \in \mathbb{Z}$

whether there are anymore symmetries I can find here I have no idea

Actually, I need to doubel check my limits...

aquire

@DHMO $\tan^{-1}x=\frac{\pi}{2}-\cot^{-1}x$

Jack Lam

2:20 PM

remember

with a border symmetry flip, you can add the integrals in pairs and they reduce to something trivial

1/6+1/12 = 1/4

1/8+1/8=1/4

DHMO

@JackLam easier said than done...

Jack Lam

Oh, I'm just giving a trivial result

evaluating the integrals independently is much more difficult

I'm well aware of that :P

oh

wait

for the pi^2/8

those 2

it MAY be possible to prove they are equal integrals

without direct knowledge of their values

then use the trivial result to obtain the value

and thus both of them are exactly pi^2/8

DHMO

@JackLam how do you use numerical methods to find a hundred digits?

Secret

Correction of typo:
$$\int_0^{\frac{\pi}{2}} \cot^{-1}\left(\sqrt{1+csc(x)}\right)dx=4\int_{\infty}^{\cot^{-1}\sqrt{2}}(u+n\pi)\left(\frac{du}{\sin (2u)(\cot^2 u-1)\sqrt{1-(\cot^2 u-1)^2}}\right)$$

Now trying acquire's method instead... please wait

Jack Lam

@DHMO Mathematica

DHMO

2:35 PM

@Secret @AlessandroCodenotti I'm thinking that if we order the naturals using the inversed dictionary order, where we first compare the units and then the tens and then so on, we can construct similar Dedekind cuts...

Secret

NB: Sorry that scary integral above is wrong, I made an error when chanigng variables, bleh, I am going to do that later...

@DHMO But is this still a linear order?

DHMO

@Secret what is a linear order?

Secret

otherwise known as total order, where $a \leq b$ or $b \leq a$

DHMO

@Secret of course it is

Secret

right, in that case it will work. Alkiva's proof using dynadic rationals showed similar things, where as we move down the chain of subsets, the dynadic rationals have larger denominators, and hence few of them for a give subset

DHMO

2:40 PM

@Secret they're essentially... p-adics with finite support

@Secret but one would need a new metric...

@Secret essentially the terminating decimals...

Balarka Sen

Hi

DHMO

namaskar

Secret

I really felt like I have clogged up the set theory room. Hopefully my countable number of questions can be killed in finite time else it will be a problem

DHMO

@Secret what the hell...

Secret

(there's a joke here, finite is also a type of countable, it is just not infnite)

N1ng

3:47 PM

I don't know how to apply the answer in my question about Darboux sum.

Thanks @Martin R

Martin Sleziak

@N1ng You know using @username syntax in chat does not at all ping the user who commented on the question, right? See here for some details on comment reply.

In fact MartinR is at the moment not pingable in chat - only users who were in the room during the past week are. An exception is a direct reply to a message in chat.

N1ng

Sorry...

Martin Sleziak

4:13 PM

@N1ng I am not sure to which extent it helps, but I have tried to reply to your comment from the main site in the analysis chat room.

trilolil

Hi

could somebody please explain this to me

I was looking at a formula about the compressibility of something:

$k=-\frac{1}{V}\{\Delta V}{\Delta p}$

turns out:

${\Delta V}{\Delta p} = {\partial V}{\partial p}$

How is this possible? To me they both express something different.

to me $\Delta$ just expresses a change while $\partial$ expresses a rate of change.

Am I misunderstanding something here?

$k=- \frac{1}{V} \frac{\Delta V}{\Delta p}$

Balarka Sen

$\Delta x$ represents small change, while $dx$ represents infinitesimal change. In physics one interchangeably goes from one to the other.

DHMO

@trilolil $\partial$ is not a rate of change.

trilolil

@DHMO oh...

but it is a derivative

partial, but still.

Balarka Sen

No

DHMO

4:17 PM

$\dfrac{\partial y}{\partial x}$ is a rate of change because it is a small change in $y$ divided by a small change in $x$

Balarka Sen

It's a differential, not derivative

trilolil

Sorry guys!

made a typo

Balarka Sen

@DHMO Well, you'd want $x$ to be the time variable most of the time.

DHMO

@BalarkaSen the folly of physics

trilolil

it is: $\frac{\Delta V}{\Delta P} = \frac{\partial V}{ \partial P}$

Sorry not used to my current keyboard.

Balarka Sen

4:19 PM

Well, only approximately but yeah

As you make $\Delta P \to 0$ it is that

trilolil

but they both express different things I think

One is absolute change while the other is rate of change.

DHMO

@trilolil why?

Balarka Sen

@trilolil What is "absolute change"?

DHMO

@trilolil not really. absolute change / absolute change = rate of change

just like how you find the slope on a graph

trilolil

@BalarkaSen I mean by that just the difference.

Balarka Sen

4:20 PM

No! $\Delta x$ represents absolute change. Not $\Delta y/\Delta x$.

trilolil

@DHMO true.

Actually it is just the geometrical definition of a derivative.

Right?

Sure

@trilolil yes it is

Thank you! :)

4:38 PM

As a follow up question in @DHMO that case, do you think something like $\frac{\partial \Delta p}{\partial x}$

makes any sense?

DHMO

@trilolil well you need to define it so that it makes sense

N3buchadnezzar

Heya

This might seem weird, but I am looking for a definition of functions in $h^1$. Where I think it is meant the space of harmonic functions.

trilolil

@DHMO sorry, what do you mean? Do you mean the contex?

if yes: imgur.com/a/xfsiE

Balarka Sen

$\Delta p$ is not a function, so it doesn't make sense.

trilolil

first line below Newton's second law.

DHMO

4:41 PM

N3buchadnezzar

Is this simply $h^1 := \{f \colon\nabla^2f\}$? Not sure where the 1 comes in tbh.

trilolil

@DHMO does this make more sense to you now? Looking at the previous explanations about the difference between $\Delta$ and $\partial$ $\frac{partial \Delta p}{\partial x}$ doesn't make much sense to me logically speaking.

Semiclassical

hi chat

DHMO

@trilolil I have no idea. You should ask @Semiclassical

Balarka Sen

I already answered you up there

trilolil

4:50 PM

@BalarkaSen oh didn't notice. But it is a university teacher with a PhD who wrote this, so it must mean something.

Semiclassical

Well, one way it could make sense is if $\Delta p$ means $p(x+\Delta x)-p(x)$.

In that case $\Delta p$ would indeed be a function of $x$.

Balarka Sen

That's not a function of $x$ either...

Semiclassical

Sure it is, if $\Delta x$ is just some fixed constant.

And if it's not constant, well---it's a function of both $x$ and $\Delta x$, just as $p(x)-p(y)$ would be a function of $x,y$.

trilolil

So @Semiclassical what would be the correct way to interprete $\frac{\partial \Delta p}{\partial x}$ ? the rate of change of the pressure p? If yes, I am affraid I don't understand what this is supposed to mean. Some sort of second derivative?

Balarka Sen

@Semiclassical That's just strange. This is exactly the kind of physics I'm inclined to call nonsense.

Semiclassical

4:54 PM

The way I would understand it is this. I pick a point $x$ along the tube, and then another point which is $\Delta x$ farther than that. There will be a difference in pressure from one end of the tube to the other.

trilolil

that would fit some of the explanation

Semiclassical

And then that pressure difference can depend on how far along the tube I am.

Now, if you've got uniform flow at a constant height, I believe that said pressure differential doesn't change.

trilolil

@Semiclassical I think you are entirely correct, except (I think) it is: "that pressure difference can depend on the wave traveling the rod".

Semiclassical

Sounds right.

trilolil

What you are seeing is the displacement of a rod after it being hit by a hammer.

Mike Miller

4:56 PM

"This is just as well, as I’m thereby able to concentrate on the trickle of information coming in from the wicked world beyond the fence. Lately I’ve been getting garbled reports of hoverboards, as well as some sort of new fascist movement that could conceivably take control of the White House this year, though I find it difficult to believe that the boards actually float like the ones from the movie." - journalist Barrett Brown, from prison, Feb. 2, 2016.

Balarka Sen

Why not let it be a function of $x$ and $y$ instead of $x$ and $\Delta x$?

Semiclassical

It's the same thing. Just $\Delta x=y-x$.

That may or not be wise notation, but it's not ill-defined.

trilolil

Thank you very much for your explanation!

Balarka Sen

Yeah, it's crap notation.

Semiclassical

I think the sensibility of it here goes like this. If I hit the rod with a hammer, then there'll be a travelling shock front running down the rod.

Balarka Sen

5:00 PM

$\Delta x$ usually means difference of two consecutive values for a discrete variable $x$. Making that $y - x$ without explicitly knowing the variable $y$ is confuzzling

Mats Granvik

Has anyone ever computed the zeta zero counting formula by Andrew Guinand?

Semiclassical

If I now pick two points which are close together, I can compute the pressure difference between them. That pressure difference will initially be constant: the shock front won't pass through the two points until enough time has passed for it to reach them.

And if I wait long enough, then the shock should pass through both points and therefore the pressure difference will again be constant in time.

However, I will find that the pressure difference will vary rapidly in the moment where the shock is passing between the two points.

trilolil

I see.

Semiclassical

Now, that's a description in time. But I can similarly do a description in space: I pick points $x,x+\Delta x$ for $\Delta x$ small, and see how the pressure difference varies along the rod at any instant.

I'll find that it's uniform in the non-shock regions, but changes rapidly at the shock front.

So in that case it makes a lot of sense to study at what rate $p(x+\Delta x)-p(x)$ changes along the rod.

That said, it still leaves open the question of what $\Delta x$ is supposed to be.

And I'm not familiar enough with engineering mechanics to say any more, unfortunately.

DHMO

anyone have any idea?

symmetry might be involved.

trilolil

5:08 PM

ok no worries @Semiclassical thank you!

Semiclassical

I saw those earlier. Kinda goofy integrals.

DHMO

oh wait, symmetry solves it.

Semiclassical

Neat.

DHMO

@JackLam @Secret symmetry solves it. The lower integral when added to itself will give a constant $\dfrac\pi2$.

the upper integral still has to wait

Astyx

Hi chat

@DHMO What are cot and csc ?

Semiclassical

5:13 PM

Cotangent, cosecant.

DHMO

@Astyx cotangent and cosecant

Astyx

Right, I'll give this some thoughts

Semiclassical

@DHMO Note that the first integrand is symmetric under $\theta\mapsto \pi-\theta$.

DHMO

@Semiclassical yes it is

Semiclassical

So that probably tells us something...

It may also make things more obvious to shift the definition of $\theta$ by $\pi/2$, so that the symmetry is at the origin.

Astyx

5:29 PM

Does mathematica compute it ?

DHMO

@Astyx the first hundred digits match, according to OP

Astyx

I see

Sha Vuklia

hee guys

would someone mind giving a look at one question I posted?

1

Q: Central limit theorem; exercise

A fair die is thrown 12,000 times. Use the central limit theorem to find values of $a$ and $b$ such that $$ \mathbb P(1900<S<2200)\approx\int_a^b\frac{1}{\sqrt{2\pi}}e^{-\frac{1}{2}x^2}\,\mathrm dx, $$ where $S$ is the total number of sixes thrown. Right, so the central limit theorem goe...

central-limit-theorem

I got answer $2\sqrt{5}$, and the solution set says it's $2\sqrt{6}$.

Astyx

@DHMO perhaps make the variable change $t = \sin \theta$, then integrate by parts ?

Sha Vuklia

oh i see my mistake! i thought the variance was p, but apparently it's pq

Astyx

5:42 PM

not sure it will work

Semiclassical

@Astyx Trying it in Mathematica, it looks like the boundary term in that integration by parts would be singular at the origin.

Astyx

I didn't quite get how the lower one was solved : Why is the lower integrand $\pi\over 4$ ?

Semiclassical

(at least with the way I'm splitting the integrand to do integration by parts.)

Oh, no, I'm being silly.

Astyx

Perhaps Taylor series help ?

Perhaps not

Semiclassical

shrug

Akiva Weinberger

6:19 PM

Prepare for February 7th

https://xkcd.com/843/

Vrouvrou

6:33 PM

ihave these two conditions: $(f_1)$: $f\in C([0,\infty[\times\mathbb{R})$ and for some $2<p<2^*, c_0>0$ $$|f(r,u)|\leq c_0(|u|+|u|^{p-1})$$

can i deduce from this

that

$(f_3)$ $$f(r,u)=o(|u|),|u|\rightarrow 0,~\text{uniformly on } \mathbb{R}^+$$

someone have an idea ?

JessyunBourne

I just put a bounty on this question: math.stackexchange.com/questions/2120233/…

I'm having a little difficulty in getting touch with the one person who posted an answer in order to ask follow up questions.

Samuel Yusim

6:54 PM

waiting for responses on grad school applications is so stressful

I just want to get in somewhere warm

Secret

7

Q: Countability of disjoint intervals

According this problem/solution set from an MIT class (http://ocw.mit.edu/courses/mathematics/18-100c-analysis-i-spring-2006/exams/exam1_sol.pdf), the assertion: "Every collection of disjoint intervals in R is countable." is True, because "every interval contains a rational number", and the rat...

real-analysis analysis elementary-set-theory

JessyunBourne

@SamuelYusim University of Hawaii at Manoa? Alooooha!

Samuel Yusim

I ended up not applying to hawaii

Aaron Hall

Do you guys understand Hindley Milner, by any chance? :)

JessyunBourne

I applied, got in, and then chose not to go.

Samuel Yusim

7:00 PM

my biggest hope is UCLA

Secret

Ok, so there are countable number of rationals, but they are arbitrarily near to the next one. So we practically have a continuum formed by all these infitesimal spaces between any two rationals...

Now to figure out how we can do the same thing in naturals to generate the dense order that is needed to contruct the uncountable chain...

JessyunBourne

UCLA is ranked significantly higher than UH

Samuel Yusim

yeah I know

rankings are rankings, but weather is forever

JessyunBourne

LOLZ!

Except for that whole global warming thingy...

Samuel Yusim

true

JessyunBourne

7:02 PM

One of these days, Canada and Sweden might be the only habitable places on earth.

4

Samuel Yusim

good thing I'm canadian then!

JessyunBourne

<3

You'll have to move to Yellowknife though, if you don't live there already.

Samuel Yusim

I do not

JessyunBourne

I am so stuck on that question :(

Semiclassical

I meant to ask, how did the PDE final go? @JessyunBourne

JessyunBourne

7:08 PM

There were a couple of problems I kind of freaked out about.

And for one of them, I didn't have time to calculate the Fourier coefficients, but I basically explained how to get them.

I don't know what my actual grade on the final was, but I wound up getting an A in the course.

Thanks for helping me whenever I had questions, @Semiclassical

You're really good at explaining stuff!

2

Q: $X$ is a basis for free abelian group $A_{n}$ if and only if $det (M) = \pm 1$

This question is related to another problem I asked a question about here. In fact, it is part (b) of a problem whose part (a) was Let $X = \{ x_{1}, x_{2}, \cdots, x_{n}\}$ be a set of elements of the free abelian group $A_{n}$. Let $M$ be the $(n\times n)$-matrix of coordinates of elements...

abstract-algebra matrices group-theory abelian-groups free-abelian-group

Mike Miller

@SamuelY good luck yo

Samuel Yusim

yeah, here's hoping my chances are good

@MikeMiller tell Igor Pak that I'm good

Semiclassical

7:29 PM

heh, i get a lot of practice.

Bernardo Meurer

Attention people

I have passed my Real Analysis course

I'd like to thank everyone who helped me during my studies

Cheers!

Mike Miller

8:06 PM

@SamuelY He doesn't know me

Nisman

Hey

user3026388

Hello guys!

Can someone explain to me the logic behind this solution?

Nisman

how can I use Lagrange remainder to calculate $e^(0.1)$ with an error $< 0.00005$ ??

user3026388

2

Q: Best formula for this case?

I have a programming assignment that asks me to create a block that is built off of components a, b, c, d, and e. The ideal block will have: Equals amounts of "a" and "b". Equal amounts of "c" and "d". When "c" and "e" are added together, they will equal "a". I must create a function that for...

functions programming

Samuel Yusim

8:23 PM

@MikeM well darn

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