Mathematics

Annamalai Sriram

1:08 AM

21 hours ago, by Annamalai Sriram

in JEE Maths Zone, 2 days ago, by Annamalai Sriram

This is the solution provided to the problem

If anyone explain the first part of solution that's enough.

copper.hat

1:21 AM

solve y' = y/x

not sure why that didn't occur to me yesterday when you asked.

Ted Shifrin

The first two lines come from working out the algebra they give you with the tangent and normal lines.

monoidaltransform

1:59 AM

hi

If $X_n\in \mathcal{L}^1$ are random variables, how do I show that for each $n$, there exists $c_n$ such that $\mathbb{P}(|\frac{X}{c_n}|\geq \frac{1}{n})\leq \frac{1}{2^n}$

?

epic_math

2:35 AM

I, the god of mathematics, came here to see how modern people do mathematics.

Which book on analytic number theory should I read after Apostol?

copper.hat

3:28 AM

@monoidaltransform it is equivalent to showing that $\lim_{c \to \infty} P[|X| \ge c \epsilon ] = 0$ for any $\epsilon>0$. this follows from continuity of measure.

Mike

Hello, can anyone take a look at my post regarding mutually orthogonal probability measures? Thank you.

0

Q: Show that two orthogonal probability measures have the following representations for disjoint intervals on $[0,1]$

Problem: $\alpha$ and $\beta$ are two mutually orthogonal probability measures in the sense that for some Borel subset $E\subset[0,1]$, $\alpha(E)=0$ and $\beta(E)=1$. If $F(x)=\alpha\{[0,x]\}$ and $G(x)=\beta\{[0,x]\}$ for $0<x\leq 1$, show that for any $\varepsilon>0$, there is finite collectio...

user 6232128

3:44 AM

@AlessandroCodenotti shouldn't all historical events be starred , in this chatroom, for future reference?

Ted Shifrin

@user6232128 Certainly not the jokes.

user 6232128

:p

Rithaniel

I'd actually disagree. Jokes deserve to be immortalized the same as any other thought. (Within reason, of course)

user 6232128

when is your next math club zoom presentation?

Rithaniel

We have them on Fridays. Different speaker each week

user 6232128

3:51 AM

thanks

Rithaniel

So, my next presentation would probably be next semester

If you want, I can actually talk to the organizer and see if we can invite people from off campus

user 6232128

nah, I just like the idea in general

Rithaniel

Yeah, I still haven't watched the video of my presentation back

There's just something nerve wracking about it

user 6232128

you did a fine job

Rithaniel

Well, I'm glad about that

In other news, it's the last week before finals where I am, and I made a game-assignment for my kids to compete against each other while practicing math. However, it looks like it might not be played because I over-estimated the enthusiasm of Freshmen in a math course

I'm thinking about spreading it around to others so that it doesn't go to waste

user 6232128

4:10 AM

yeah, providing games so close to the final exams is being a bit overly optimistic :P

Rithaniel

Well, that's a thing for me to remember for next time

user 6232128

Right now, most of them, are scrambling to find practice final exams.

Ted Shifrin

4:41 AM

Heya @Rithaniel

Leonhard Euler

Hello everyone

Leonhard Euler

5:35 AM

0

Q: Evaluating the sum $\sum_{n=1}^{\infty}\frac{1}{F_n}$

Let $F_n$ be the $n^\text{th}$ Fibonacci number. I wanted to calculate $$\sum_{n\ge1}\frac{1}{F_n}$$ I simplified it to $$\sqrt{5}\sum_{n\ge1}\frac{1}{\varphi^n-\phi^n}$$ But this didn't seem to help. The value is approximately $3.35988566624317755317201130291892...$. The OEIS entry of this cons...

sequences-and-series fibonacci-numbers

Does someone know the answer?

Annamalai Sriram

5:51 AM

@TedShifrin Thanks I came up with equations but I have a small doubt that is if we consider the equation of the tangent y = mx + c , the equation comes up only when y = f(x) but don't the y in tangents equation represent the trajectory of tangent

Ted Shifrin

I don't understand. Write the equation of the tangent line to $y=f(x)$ at $x=a$. Same for $g$. The equation for them to intersect on the $y$-axis is an equation with $f$, $g$, $f’$, $g’$ all evaluated at $a$. They then rewrote it with $x$ instead of $a$.

Annamalai Sriram

6:08 AM

but I think only the slope "m" comes from the y = f(x)

at x=a

copper.hat

6:23 AM

the equation of the tangent line at a is given by f(a)+f'(a)(x-a), so the value of the y-axis intercept is when x=0, so that gives f(a)-af'(a)

hence f(x)-xf'(x)=g(x)-xg'(x)

Ted Shifrin

@copper That first one is not an equation.

feynhat

@TedShifrin How much longer do you plan to stay up?

Annamalai Sriram

Ah! Yes, I got it. Generally in the equation of tangent y is a variable but at point x=a the "y" of the tangent is equal to that of "y = f(a)" of curve so it is equal to f(a)

Ted Shifrin

No, that's not right. But use the point-slope formula for lines.

Why? @feynhat

feynhat

I had a couple of questions.

Sections of line bundles and stuff.

Annamalai Sriram

6:33 AM

yes it comes with slope formula y = mx +c as you said I've misunderstood that point

Ted Shifrin

Around for a half hour or so.

feynhat

Okay. Give me a minute.

Annamalai Sriram

@TedShifrin @copper.hat Thank you very much.

copper.hat

@TedShifrin sry, i meant the function t(x)=f(x)+f'(x)(x-a)

@AnnamalaiSriram it might help to draw a little diagram. my research advisor used to call such things as 'proof by picture'.

Ted Shifrin

Better to write the equation since we want to set the $y$-intercepts equal.

feynhat

6:42 AM

Let $F(x, y) = x^2 - y^2$. I want to show that the section of $\gamma_{\mathbb{CP}^1}^{\ast\otimes 2} \to \mathbb{CP}^1$ (ie. the second tensor power of the dual of the tautological line bundle. I believe its also sometimes written as $\mathcal{O}(-2)$) induced by $F$ is transverse to the zero section.

Lets denote the induced section by $s_F$. I need to check that at the points where $F=0$, $s_F$ is a submersion. The point in $\mathbb{CP}^1$ where this happens is $[1, 1]$.

What does the induced section look like in charts? Let $U_i$ denote the standar atlas of $\mathbb{CP}^1$. In $U_0$, the polynomial is just the dehomogenization $f(z) = F(1, z)$. Now, the target of the section is $\gamma_{\mathbb{CP}^1}^{\ast\otimes 2}$ which has a standard trivialization over $U_0$.

So, the section can be thought of as a map $U_0 \to U_0 \times \mathbb{C}$. I will ignore the the first output because its just identity. So, basically, this is just $U_0 \to \mathbb{C}$. What is this map? Is it just $F(1, z) = 1 - z^2$?

Ted Shifrin

No, it’s written $\mathscr O(2)$.

There are two zeroes, of course.

feynhat

right the other being [1,-1].

Ted Shifrin

Right.

Yes, that's the dehomogenization.

And the derivative is non-zero at $z=\pm 1$.

feynhat

Thanks. It seems a bit hacky to me. Like I missing some detail.

Ted Shifrin

Hacky?

Amin Idelhaj

6:53 AM

Hey Ted!

Ted Shifrin

Hi Demonark

Amin Idelhaj

How are you doing?

feynhat

So, if I want to check if a section induced by a homogeneous polynomial is transverse to the zero section, I just need to see if look at the roots of its dehomogenization and see that all of those roots are simple (derivative doesn't vanish at those roots.)

Ted Shifrin

Still alive.

Amin Idelhaj

That's good, same here

Ted Shifrin

6:55 AM

In general, what does it mean for a section to be transverse to the zero section?

For any line (vector) bundle?

feynhat

At the points where they intersect, the images of the derivatives should span the tangent space at those points.

Ted Shifrin

So if the section is given locally by a function, the graph if that function should be transverse to the graph of the zero function. Just go back to first principles.

feynhat

Hmm... so locally, the sections are given by dehomogenization of the polynomial, so my assertion about roots being simple is true.

Ted Shifrin

Yes

Rithaniel

Heya Ted. Sorry about the delay in response
(I find myself on a nocturnal schedule once again)

Ted Shifrin

7:05 AM

No problem. Balarka breaks everyone’s sleep.

Rithaniel

Hehe, fair enough. I break my own sleep, though. Of course, I'm thirty now and it's becoming more difficult to switch between different sleep schedules

Ted Shifrin

Old man!

Amin Idelhaj

Now that I think about it my sleep schedule was better before I met Balarka

10

Correlation \implies causation and all that

Ted Shifrin

You've been around here since late your first year in college, I think.

Amin Idelhaj

Maybe mid second year, I think I already finished a quarter of analysis by then?

Ted Shifrin

7:09 AM

Approximately.

feynhat

7:55 AM

@SayanChattopadhyay Hi.

Have you returned to Mohali?

Leonhard Euler

8:47 AM

2

liked it?

Alessandro Codenotti

10:23 AM

@Balarka I noticed only now but King Gizzard put out a new album a couple of weeks ago

Abhigyan Chattopadhyay

12:57 PM

Hi all! I'm currently a 2nd year EE student. I plan on taking an introductory Graph Theory course in my next semester and wanted some insights from people who possibly have more experience in that field.

Would anyone say that Graph Theory is, in general, a heavy course for someone who has only done Multivariable Calculus and Partial Differential Equations, in terms of mathematical maturity?

Any and all insights are appreciated

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$Mathematics$ Mathematics