Mathematics - 2019-10-13

4:43 AM

What functional is linear interpolation a minimizer of?

5:00 AM

Arclength?

Aladdin

7:24 AM

Can somebody explain me the limits of integration for calculating volume of ellipsoid of equation x^2/a^2+y^2/b^2+z^2/c^2=1

This is the region for which I an calculating volume

So I think the first thing we do is to constrct small rectangle in x-y plane

And find heights of them for them which will give the volume

Considering the region I would fix y, calculate limits of x and then find the height? Am I correct

Andrew Styles

Hello

What's so special about Fibonacci sequence?

Aladdin

7:39 AM

When do higher students learn about this sequence?

Mary Star

9:43 AM

Hello!! I want to draw $x^2-y^\geq 0$.

If we consider the equality $x^2-y^2=0 \Rightarrow x^2=y^2 \Rightarrow y=\pm x$. For that we draw the lines $y=x$ and $y=-x$.
Which points are then $x^2-y^\geq 0$ ? Could you give me a hint?

Mary Star

9:54 AM

I have seen the graph in Wolfram but I want to know how we get that

Do we do the following?
$x^2-y^2\geq 0 \Rightarrow (x-y)(x+y)\geq 0$

That means that $x-y\geq 0$ and $x+y\geq 0$ or $x-y\leq 0$ and $x+y\leq 0$.

Secret

10:54 AM

Toying with a new inference rule to bypass the principle of explosion without dropping or intro nor most forms of double negation

Inference rule 15

x

not x

y

Secret

(x and not x) and y

In particular, one can prove the following, letting D = (x and not x)

1. (not x and (x or y))=> D or (D and y) = D and (D or y)
2. (D and not D) => D

3. not D and (D or y) => D or ~D or y

Now D is not really that intractable, but the only way to remove D is to transform its constituents so it no longer contradictory e.g.

D

x => a

~x => b

x

~x

a

b

a and b

Of course, if neither x nor ~x contradicts with whatever the other member is transformed to, it is enough to remove D

Note how the principle of explosion is avoided because the or case that includes the y term cannot be completed if there is no sentence involving y can be proven first

Thus we now have a form of classical paraconsistent logic, where inference rule 15 is executed at a much higher priority than all other rules

I am not sure if the principle of explosion can still arise in some form here, however

Secret

11:23 AM

o wait, De morgan law will fail for D in this case

and all the results above will be wrong

2:44 PM

Is the composition of two rotations in 3-space another rotation? Would using the product rule for determinants suffice to prove this? (i.e. $D(AB)=D(A)D(B)=1 \cdot 1= 1$)

Ryan Unger

3:29 PM

@schn that's right

Mike Miller

3:49 PM

The product rule for determinants only shows that the product of two rotations is again determinant 1, but being a rotation is more than that. (The statement is nonetheless correct.)

@RyanUnger @MikeMiller Okay. Given two reflections in 3-space in a plane, how does one show that their composition is a rotation about the intersection of the planes and the angle of rotation?

4:16 PM

Well the determinant will be $(-1)(-1) = 1$ so it's definitely a rotation. The plane of the first reflection is left unchanged by that reflection and the plane of the second reflection is left unchanged by that reflection so their intersection is left unchanged by the composition of the reflections so it's left unchanged by the rotation and is therefore the axis of rotation.

Anush

if I have a mixture of two exponential distributions with prob $p_1$, $p_2$ and rates $\lambda_1$ and $\lambda_2$, what is the $P(x| p_1, p_2, \lambda_1, \lambda_2)$?

@Anush en.wikipedia.org/wiki/…

Anush

@user76284 oh that looks very simple!

@user76284 thank you

@user76284 thanks. How could one deduce the angle of rotation? Is there a standard reflection matrix for 3-space as for 2-space as given here (en.m.wikipedia.org/wiki/…)?

4:32 PM

The rotation angle has to do with the angle between the reflection planes.

You can try seeing it in 2D.

Hint: Choose any point that lies on the first reflection plane so it's left unchanged by that reflection. When it gets reflected by the second reflection what is the angle it's rotated by, as a function of the angle to the second reflection plane?

I should say: Choose any point on the first reflection plane that is not on the second reflection plane.

@user76284 It depends on the angle between the planes, doesn’t it?

That's right.

By how much?

At most $\pi/2$, or?

I mean what is F in rotation angle = F(angle between planes)?

Draw a diagram in 2D with a top down view of the planes (so they're represented as lines).

Okay, so there is an explicit function F?

4:47 PM

Of course

Why wouldn't there be?

Sure.

Did you draw a diagram?

Yes. Trying to move the second plane closer and closer to the first.

Just draw two lines crossing each other at some angle, and a point on one of the lines. How far does it end up rotating around the point of intersection when reflected by the second line (plane), as a function of that angle?

2 times the angle between the planes.

4:50 PM

Bingo.

Thanks!

No problem

5:07 PM

Given a finite set of samples from some unknown distribution, the distribution that maximizes the likelihood of those samples is (I think) a mixture of delta functions at those samples. Does KDE maximize the likelihood of those samples for some set of constraints? If so, what are they?

If you constrain the distribution to be both smooth and nonzero everywhere in the desired region you get increasingly narrow Gaussians around each sample, which is not what we're looking for.

Perhaps an entropy constraint?

5:28 PM

Hi @BalarkaSen

Hi @Alessandro!

Oh and @loch

Have you been doing any interesting math recently?

Sort of

In which area?

Well, I have been learning some algebra (algebraic geometry/commutative algebra/homological algebra), but I am also trying to understand some pure topology

Gordon-Luecke theorem in particular: knot complements are homeomorphic implies knots are equivalent

5:37 PM

I see

Akiva Weinberger

Apr 3 '18 at 19:42, by Akiva Weinberger

Thing I'm adding to my reading list but in all likelyhood will never actually read: https://a16z.com/2018/02/10/crypto-readings-resources/

Can confirm: never read

Hi @Ted @Mathei @Erico @Akiva

HI @Alessandro

Érico Melo Silva

hi

@MatheinBoulomenos Ready for the new semester?

5:50 PM

@Alessandro yeah

Which classes are you going to take?

Topics in algebraic geometry - Introduction to moduli theory
Intro Japanese
Affine Algebraic Groups
Adic Spaces

and some applied courses I'd really rather not take ...

and you?

@MatheinBoulomenos lol the famous two courses you need to finish your bachelor?

yes

I'm doing global analysis I, models of set theory II, a set theory seminar, introduction to higher category theory and semigroups and applications to hyperbolic PDEs

(don't ask about the last one)

5:57 PM

nice, higher categories

I'm still not 100% sure about that course

I don't need to do it for credits, so I might not do the exam or stop going, but it seems interesting so far

oh right, Bonn starts a week earlier

Yep, we started last Monday

oh and I'm TAing intro abstract algebra

so, did you start with simplicial sets?

yep

6:08 PM

cool stuff

6:23 PM

Can someone here take a look at this question? It's getting next to no attention and I am waiting for an answer so I can continue a project. math.stackexchange.com/questions/3388423/…

Mike Miller

@BalarkaSen I don't know how the proofgoes

Ultradark

howdy demonic @Alessandro, @MikeM, a @Balarka, @Mathein

Why is it "a balarka"? Are there other Balarkas I missed?

hi @Ted

6:29 PM

Balarka is multiplicitous.

@MikeMiller The proof in Josh Greene's notes span several lectures. At some point he uses integer programming!

I can tell you after I have understood the ideas

@Brendon: I can't understand what you're talking about in the post. You need to give a PRECISE and CLEAR definition of what it means for one point to affect another.

The main theorem is that a nontrivial surgery along a nontrivial knot in S^3 is never S^3

I had a lot of fun understanding a specific example: Take a knot K in S^3 cabling around a trefoil knot with slope 1/3. Then surgering along tubular neighborhood of K such that the curves parallel to K on it's boundary are glued to meridian disks will give you a 3-manifold which has a L(3, 1) as a connected sum

I like how this can be read off from pure combinatorics

Oh also hi @Ted

Semiclassical

6:45 PM

@BalarkaSen ooo, integer programming

Q: Projecting a stretched grid to the unit disk?

yeah very cool stuff

Semiclassical

in other news: if there's one part of working in intro physics which drives me up a wall, it's figuring out lab report grades

part of it is that there's a lot of room for judgment on my part, including figuring out how to divy up points according to the rubric

and another part is that I feel like categories on the rubric overlap

i've been stuck in a loop on that for days

Ultradark

7:32 PM

1

What is happening mathematically in this image? How does one represent the grid on the ball mathematically? I've observed that there seem to be four points at infinity in which grid lines meet. I would also say I think this is in the scope of spherical geometry. Q: If you projected the grid ...

Thoughts?

7:50 PM

@BrendonShaw Are you asking whether $D$'s boundary touches $A$'s boundary?

i.e. whether their Voronoi boundaries overlap?

By "affect" I guess you meant that any change in $D$'s position, however small, will change $A$'s Voronoi region?

@Ultradark What's going on in the photo?

It's a glass orb in front of a grid in front of a candle?

Is the orb touching the grid behind it?

8:11 PM

@user76284 Yep! I want to check for overlapping boundaries. I am currently overhauling the question so that it actually makes sense.

Tanuj

I need to ask a doubt regarding a C program , can someone link the room where I can ask it ?

What is the limit of $xye^-{x^2+y^2}$ as (x,y) recedes to infinity? What is the strategy for finding it? Switching to polar coordinates?

@Tanuj chat.stackoverflow.com/rooms/54304/c

Tanuj

@user76284 Thanks

@user76284 @TedShifrin I've edited the question to make it clearer.

8:21 PM

I suggest editing the question and its title to just "Determining whether two Voronoi regions are adjacent".

Look up Delaunay triangulation

9:24 PM

@user76284 I just want an solution that will work in the case given in the question, not a completely different method for generating a Voronoi diagram.

10:00 PM

Three slots with two possible outcomes for each yield how many combinations in total?

10:13 PM

@Brendon: Start by editing to define what "mid-segment of a point" or whatever it is you're working with. Your diagram is of no help to me. In my 50+ years in mathematics, I have never seen language such as in your post.

@schn: Why are you using the word combinations? What do you think the answer is?

10:30 PM

Hi @loch @Eric

Hey @Ted

Hi @ÍgjøgnumMeg. How is your cycle route?

@Ted it's good! The semester starts tomorrow, which I'm excited for

I think my rucksack is too small though lol

Too small for you to curl up when you need a nap?

hahaha, too small to contain my laptop, my notebooks, my books, my clothes, my shoes, and my toiletries

10:38 PM

Oh, it's your overnight bag?

I cycle in lycra with road shoes and I need to shower when I arrive lol

I have an algebraic number theory lecture and a topology seminar tomorrow, very much excited for the semester to start finally

Yup, you've been waiting for months!

Indeed :) Perhaps a bad idea to attend a party tonight

But I'm irresponsible so

10:57 PM

Hey, guys! Are we allowed to ask questions here about calculus?

Of course

Awesome! I am just stuck with proving that if the limit of f(x) as x approaches 0 is 5, then the limit of sin(x)f(x) as x approaches 0 must be 0. I know that if you plug in 0 into the function sin(x), then you get 0, but I am not sure how to formally state it.

math.wikia.org/wiki/Algebra_of_limits

Thank you! I'll definitely take a look at this. I am just unsure whether I really need to use the epsilon-delta definition of a limit here?

Well it depends what is expected lol

I mean if you know that sin is continuous at 0 and that the limit of a product is the product of limits and you aren't expected to prove these facts then it's obvious

11:09 PM

I see. I think a justification of why it must be true should be sufficient (I hope so at least lol)

In any case, thanks!

No worrieeees

YOUSEFY

11:26 PM

people who are interested in the foundation of mathematics, I would like them to answer: In this article, Scientific American explained Godel's incompleteness theorem in simple words. Then they put the following unanswered question in the last of this paragraph:
[Strictly speaking, his proof does not show that mathematics is incomplete. More precisely, it shows that individual formal axiomatic mathematical theories fail to prove the true numerical statement "This statement is unprovable." These theories therefore cannot be "theories of everything" for mathematics. Is this an isolated phenom…

I mean: Godel shows that there are true statements that are unprovable. Now, What else we want to know?

scientificamerican.com/article/what-is-goumldels-proof