12:00 AM
You told me you had $a$ and $b$ earlier. What are they?

a=8+11n
b=-5-7n

Give me specific ones (as small as possible).

n=0 gives a=8 and b=-5

You can use the Euclidean algorithm, and there are smaller ones.

n=-1 gives a=-3 and b=2

12:03 AM
But ok. So if $-55\equiv 1\pmod 7$, what is $\equiv 2$?

Chatjax keeps stopping

-3*7+2*11=-21+22 = 1

Oh, you're fine, sorry. ok, so use $22$ instead of $-55$ and answer me.

11b is 22-77(n+1)

12:05 AM
Stop. Think. How do you get from $1$ to $2$?

Can someone help me find the eigenvectors for the 2x2 matrix ( (1 0) (0 4)) ?

@TedShifrin what are you trying to do, form b in terms of n differently? Dare I say add 1?

How 'bout multiply by $2$?

Where do we have 1? 11b=22-77(n+1) 's RHS is divisible by 11, so 22-77(n+1)$\equiv$0 mod 11

I give up. I've typed it three times.

12:09 AM
We have $11b=22-77(n+1)$ where do you see $1$? Or are you talking about $7a+11b=1$ then $14a+22b=2$?

What are the eigenvectors for $\begin{bmatrix} 1 & 0\\ 0 & 4 \\ \end{bmatrix}$
?

@JakeShellman solve the characteristic equation

I already did....the eigenvals are 1, 4
@Alec

$\text{det}(A-\lambda I)=0$

@AlecTeal I keep getting 0,0 when I use them though

12:11 AM
Right, so you are looking for $Av=\lambda v$
Or $Av-\lambda v=0$

@JakeShellman The matrix is diagonal. What are the eigenvectors of a diagonal matrix?

Yah...I know its diagonal...but I have to show it using the characteristic equation not the short cut

@JakeShellman you will always get 0,0 as one, that is the trivial solution, if v=0, then Av=$\lambda v$
You need to choose a non-trivial one
So at some point say "x=1" then bung that in to your solution to get a y.
You could say "x=2" or "x=\frac{-1}{3}" choose whatever looks nicest.

@DanielFischer @AlecTeal If you use 1, then the matrix becomes \begin{bmatrix}
0 & 0\\
0 & 3 \\
\end{bmatrix}

@JakeShellman And the kernel of that matrix is spanned by?

12:13 AM
Okay, you need a vector, (x,y) where (0,3y)=(x,y)
Oh I see what you mean.
Wait, no, @JakeShellman your eigenvalues are wrong

exactly @AlecTeal .... the eigenvalues of $\begin{bmatrix} 1 &0\\ 0 & 4 \\ \end{bmatrix}$ are 1 and 4

So where has 0,0,0,3 come from?

A - 1I = 0, 0, 0, 3

That matrix (call it A) times (x,y) as a column is (x,4y)
You need to find (x,4y)=(x,y).
x=1, y=0
x=90, y=0
x=$-\pi$, y=0
All those vectors define the same line

Wait dont you solve eigenvectors by doing (A-rI)x = 0, and x is the eigenvector for the eigenvalue r?

12:16 AM
An eigenvector is a vector that doesn't change direction after you apply a transform to it.
Like... draw a circle of r=2 and r=3, take the vector (t,t) and plot A timees it, that is the vector (t,4t)
Draw this for all t on your picture, but only where r>=3
For 0<=r<=2 draw the vector (t,t)
Then join where they end in between your two circles, see how it moves a lot!

But how do you mathematically solve for an eigenvector?

Now use the vector that just points along the x axis (either direction) that line doesn't move, it may stretch a bit (it doesn't as the value is 1) or change direction wrt 2, but the space between your two circles, r=2 and r=3 that it goes though is the same line
You've just done that

I still dont get it lol

12:45 AM
@Charlie
:'(

1:16 AM
lol @skullpatrol

1:50 AM
Can someone please help me understand how to find the horizontal asymptotes of a rational function? I know the degree of the numerator has to be equal to the degree of the denominator OR the degree of the denominator has to be greater than the degree of the numerator but I don't know and cant figure out how to find the actual number(s)?

They are found by looking at what happens for very large values of x.
Also what happens for very large negative values.

for example $(5+x)/(5-x)$
sorry $f(x) = (5+x)/(5-x)$

Sub in larger and and larger values of x and look at what happens to the value of f(x).
Then do the same for negative values of x.

the book tells me the horizontal asymptote is -1 but I cant understand how this is a definite answer?
to be exact it says the answer is y=-1

as x gets huge 5 + x ---> x and 5 - x ---> -x.
try it for x = 10, 100, 1000 ...
basically as the x value becomes larger, the 5 becomes insignificant, right?

2:04 AM
yes
but how does that make y -1

look at what happens with 5 - x
4 mins ago, by skullpatrol
try it for x = 10, 100, 1000 ...

-101000995

5 - x ---> -x
because of the subtraction, the result becomes a large negative number, right?

yes =)

so you're left with x/(-x)

2:09 AM
okay im still a little confused, can we try with a different problem

Hi all

before that ill ask, if there CAN BE a horizontal asymptote will it always be -1?

first clear up the confusion with yjis one :)

@Brittany If you want to be somewhat formal about it, let $r$ be an arbitrary real number and calculate $r-f(x)$. Then $r$ is a horizontal asymptote of $f$ if and only if $0$ is a horizontal asymptote of $r-f(x)$.

0

What are the best methods to accurately generate random integers distributed according to a power law? The probability of getting $k$ ($k=1,2,\ldots$) should be equal to $p_k = k^{-\gamma} / \zeta(\gamma)$ and the method should work well for any $\gamma > 1$. I can see two naive approaches: C...

2:11 AM
@Brittany Not always, it depends on your expression

It's applied math ... but I won't be surprised if people here find it boring.

@Brittany what do you mean by "CAN BE"?

@skullpatrol I mean is there IS a HA (ill use that to abbreviate horizontal asymptote from now on) Because is N > D there is no HA, correct?

correct.

@KarlKronenfeld I am slightly confused by this seemingly elementary concept because I dont know how to solve $r-f(x)$
@skullpatrol Do you know any examples of then the HA will NOT be -1?

2:19 AM
I'll leave that to your textbook :)

Im still confused hahha

@Brittany ok, $$r-f(x)=r-\frac{5+x}{5-x}=\frac{r(5-x)}{5-x}-\frac{5+x}{5-x}=\frac{r(5-x)-(5+x)‌​}{5-x}$$

Maybe I need to rephrase what Im asking, Is HA something you solve for, like solving a quadratic by making $f(x)=0$

I'm just trying to give you a quick numerical way to approach this :)

@Brittany You can, in some sense, solve for it. For instance, once you simplify my expression, there will be a unique $r$ such that the numerator is constant.
Then the HA for $f$ will be $r$.
@Brittany For HA not equal to $-1$ try $\frac{x+1}{6x-1}$.

2:31 AM
@KarlKronenfeld $f(x)=\frac{3x^2+1}{x^2+x+9}$ somehow has the HA of $y=3$
I just dont know the steps taken to find the HA =/ Whya re there no steps?

@Brittany There is a recipe for doing this formally somewhat like what I said above, but no one actually does that. You may find this annoying. Instead, we use "trick": namely ignore all terms but the terms of highest degree in the numerator and denominator.

divide the numerator and the denominator by 1/(x^2)

x being what

the variable

lol

2:38 AM
@Brittany So you forget $1$ in the numerator, and $x+9$ in the denominator, leaving you with $\frac{3x^2}{x^2}=3$.

ahhhhhhhh okay
thats nice and simple

The trick works, in the sense that you actually can compute the horizontal asymptote this way and be right

@KarlKronenfeld I like the trick, thank you
for $f(0)=\frac{x^2-1}{x^2-2x-3}$ I get Vertical asymptotes 3,-1 but the book only lists x=3. Do you know why?

numerator also goes to 0 at x=-1
I was doing something else, I can explain better now.
@Brittany Factorize both the numerator and denominator completely.

3:00 AM
@Brittany the numerator is the difference of two squares

yessssss
so x+1 will cancel out?

@Brittany For any $x\ne -1$ it will cancel.
If $x=-1$, then the expression is just undefined. One calls it a "hole".
If you graph the function, it appears to be the graph of $\frac{x-1}{x-3}$. So, indeed, $x=-1$ is not a vertical asymptote.

@Brittany when you divide the the numerator and denominator by x +1 they will cancel out

i see
thanks you guys!!

3:04 AM
see you guys later

later

bye
Hi @Mike

hey

3:21 AM
yo

Hi
Could anyone help me with a physics problem?
I'm having trouble finding the equation of the overall moment of a system

I have that system
How would I find the equation of the moment?
It can be left in terms of variables of course.
I have the equations of forces already but this eludes me
@skullpatrol Any idea of how I may get it?
I know it is a static ssytem so the moment is zero

What is the definition of "the moment"?

It is the torque
Just another term for it

4:01 AM
link you've been gone for a while

@David Yolo.
@Chris'sis $$\sum_{d|n} \tau(d)^3 = \left [ \sum_{d|n} \tau(d) \right ]^2$$
@robjohn Got some TNT problems for you to sit on.
@Karl Keep on topologizing.
@Mike Starting algebraic geometry from D&F. Do you think I need lots of module algebras?
@BalarkaSen (Taking the limit points in the inlusion too...)
Yeah, now I am closed =D

possibly

I gave a new name for lacunatic functions today.
"Lunatic functions"
@skullpatrol Look what you did to the convex set.

4:27 AM

4:40 AM
@Sawarnik Here's something for you to do : Derive a combinaotiral proof of $$3^n = \sum_{0 \leq i \leq j \leq n} \binom{n}{i}\binom{i}{j}$$
Have fun!
I am off now, pip-pip!

5:11 AM
Aw, someone flagged a bad word.

was it yolo

sighs

build character

@Mike It was one of the most amazing bad words. Pretty sure he made it up.

aw man
now I want to know it

5:15 AM
Definitely worth typing it into a chat, yolo.

well, you're the one with the ability to do so

So can chat mods.

good point
on the other hand, no mods no masters

I approved the flag simply because it looked potentially trollish since it was out of the blue and I couldn't see any recent activity from the user in here. I would have dismissed it if it was just random language from a regular

I don't think Karl is mocking your decision
He's just being silly

5:18 AM
ah, k, I don't care either way, just figured I'd comment on it
besides, who could be dissin on the math?

just for the record, he used to be a chat regular, approx 2 mos ago.

yeah, I wasn't sure about it since I saw he had a fair bit of rep on Math

(hence the 2 mos ago piece of the message)

I wish I had enough time to check out the Math site occasionally, but there are just too many good sites on SE

I'm not too fond mse myself. Too much clutter, so to say.

5:21 AM
(computer scientist here, otherwise known as a math major with a job)

I see they're going at it again, except now it's crystal @Mike

@Karl Yop
They've got one incredibly overleveled 'mon and the rest are pretty low
Because they're incapable of getting someone else as the starter

@AJHenderson I used to enjoy programming, but computer science is a bit too far in that direction for my tastes.

@KarlKronenfeld yeah, if you can get a software architecture job or certain algorithm design jobs then it's more mathy
I actually much prefer the logic/mathy parts of my job to the programing parts
the fun of it is solving the problem, not getting the computer to know how to follow your algorithm

agreed

5:26 AM
but I went to a high end comp sci school that was all theory, and it was a regular statement from first and second year professors to tell students they were great programmers but not computer scientists
which also probably explains the 65% attrition rate we had in the major

fun
@KarlKronenfeld Why don't kleptomaniacs get jokes?

@Mike because they never knock?

Because they always take things literally

ouch

looks back at screen after admiring product tag collection I don't get it.

5:34 AM
they always take things, literally

Ah, too much of my silliness for one conversation?

wha
kleptomaniacs always take things, literally
but if someone always takes things literally then jokes are never funny

I actually did get the joke, but I was being silly..

damn you
I guess I'm a bit of a klepto myself

Well there are other kinds of criminals who don't get jokes. For instance, murderers never get jokes because they kill the joke teller before the punchline.

5:40 AM
I have broken every law
Including perjury
Coincidentally, I just took the oath

6:17 AM
@Complexanaylsis Where did you get the floor question from?

anyone here?

@robjohn are you there?
@robjohn my reputation keeps dropping over night but no change shows on the reputations tab.
@robjohn it was 16,691 two days ago then yesterday it suddenly was 16,689 and today it's 16,679.

@MattN. Fewtimes happens with me to, drops by 2 or 3 for no reason.

6:36 AM
0

Prove $(A \cup B) \cap C$ = $(A \cap C) \cup (B \cap C)$ Starting from the left side, $(A \cup B) \cap C =$ By distributive law, ( distributing the $\cap C$), we have $(A \cap C ) \cup (B \cap C) =$ Therefore, $(A \cap C ) \cup (B \cap C) = (A \cap C) \cup (B \cap C)$ If I start ...

6:50 AM
is it really one line or do I have to prove the distributive law in its entirety

7:12 AM
Does anyone know how to show if a function has no inverse in elementary functions?

I forgot D:"

@Sawarnik. I don't think there is a general method. Can you post the function

$x\cot(\pi/x)$ for $x \geq 3$.

out of curiosity, why do you care

See his 4 questions.

7:20 AM
i see where you found the question, but not why you care
:p

lawl

@Mike Its a good question still.

it's a question, to be sure

Pretty sure it's not ADD that is making it so hard for me to think about that problem for more than 10 seconds.

7:25 AM
@Mike sounds good, not kidding.

oh crap. linking that was, like, super illegal i think.
nobody click it.

@boywholived Do you have any idea for that question?

oh, cool.
it's Just War Theory In A Post-Cold War World, by J. Bryan Hehir in the Journal of Religious Ethics
Vol. 20, No. 2 (Fall, 1992), pp. 237-257
@KarlKronenfeld what's your email? I don't have it
you have some solid pseudonyms dude

7:45 AM
who is to say that's not my real name?

I think you'd be more careful as to not give away your information to me, and thus my legion of assassins

Yes, I may come off as cautious. E.g. I used 'dot' and 'at' as if the inbox was not packed full of spam.

7:58 AM
lol

Who dat? flagged? wat?

it's because you said yolo
in the presence of children

Yeah.
it was yolo that was flagged?

huh?

8:05 AM
anyone wanna help me disprove $(A+B) \cap C$ = $(A \cap C) +(B \cap C)$

@BalarkaSen huh

+?

-.- symmetric difference
maybe I should post on main and then you can see the whole thing @Mike

that's not standard notation bro

@KarlKronenfeld what was flagged, actually?

8:06 AM
oh that's odd because that bs was on my exam

@Mike $\Delta$

-_- but my prof used + oh I don't know wth to believe anymore XD

@BalarkaSen nah
I've seen that notation for symmetric difference before.
Think it was algebraic graph theory

@KarlKronenfeld Was it "yolo"? What's it to flag about?

@BalarkaSen nah

8:08 AM
it's probably used when they want to emphasize the ring structure

@KarlKronenfeld Then what was that flag about? Flagged what?

@BalarkaSen nah

@KarlKronenfeld Shut yer mouth.

@BalarkaSen nah
2

8:09 AM
Shall I?

Sure.

@BalarkaSen nah
2

Cut it out, common.
Was yolo flagged or not? yeah or nah?

@BalarkaSen nah
2

@Mike then what was flagged? Mike?

8:11 AM
hello; please i have a function f(t,u(t)), how to express the first derivativ of f with respect to u at 0 using limites ?

Do flagged things usually automatically remove?

with the definition of the derivative

$lim \frac{f(t,u(t))-f(t,0)}{u-0}$ ?

Ah, here's a mod.
@MarianoSuárez-Alvarez

@Mike ???

8:13 AM
1 min ago, by Balarka Sen
Do flagged things usually automatically remove?

@Vrouvrou i've said all i shall

just tel me if it's right ?

it's not.

@BalarkaSen you speak to me >
?

@Vrouvrou Yes.

8:16 AM
@KarlKronenfeld After the author stops talking about the actual problems with a modern war for the JWT he begins to ramble and it's hard to see what his point is

so how to do please ?

i've said all i shall.

pffff

it seems to be essentially 'it's a waste of time to talk about the technical issues with JWT in the modern era; the politics are more important'

@Mike What was flagged?
And who posted the word that was flagged?
Silence is something I do not prefer.

8:25 AM
nah

@Karl
4 mins ago, by Balarka Sen
And who posted the word that was flagged?
"Nah" is not a proper answer here.

nah
5

And now you're just trolling.
That's exactly the reason I am not visiting chat often.

nah
5

Better than that, Sodium Hydride

8:31 AM
0

Prove or disprove $(A + B) \cap C = (A \cap C) +(B \cap C)$ I want to disprove this statement. $(A+B)$ is the symmetric difference and has the form of $(A \cup B) \backslash (A \cap B)$ I am starting on the left which is $(A + B) \cap C$ If I take the complement definition of $(A+B)$, I wou...

@Mike So I presume yolo is not the word

@BalarkaSen @Mike why you say that is not right ??

@Mike Can you suggest a good book for algebraic topology?
I figured geometric topology is not quite my thing.

@BalarkaSen the book of Hatcher

@Vrouvrou The diff geo perspective of usual books are killing me. I want one that has a slight notch towards algebra.
And also, one that starts from the basics of homology

8:43 AM
i don't know

@Vrouvrou Have you learned topo?
@Mike Is D&F ideal for homo and cohomology?

rawr'

no, but i'ts my subject of master
so i trying to understand it

@Vrouvrou Integrated Circuit

???

8:52 AM
@Karl
12 mins ago, by Balarka Sen
@Mike Can you suggest a good book for algebraic topology?
No one's talking here.
Just that stupid "nah"

@BalarkaSen nah
5

I think I will leave
starpanelsfilledwithnahs
@Karl @Mike Stop nahing.

@BalarkaSen nah
5

Then stop starring.

@BalarkaSen nah
5

8:59 AM
Boy o boy, it's a series of nahs I see at my right. O_o
@Mike @Karl You two are complete trolls.

@BalarkaSen nah
5

@KarlKronenfeld what I got out of the article is that the author thinks there's no further value for the classical theory of JWT, but that a modern one that changes focus might still have untapped value

@Mike nah.
What's JWT?
Something starting with Jordan?
W is the Weierstrass, I guess
Am I right?
No "nah", so I am gonna take that as a yes.

@Mike Mm, my impression is not far from the opposite.
@BalarkaSen Nah

@KarlKronenfeld I'd like to hear it
I honestly had trouble getting much out of the article after his background section (though that did take up half the paper)

9:13 AM
@Mike I thought that the author was saying: (1) yeah sovereignty is crumbling in the modern times but battles aren't fought among them anyway, (2) the larger blocs like the UN will have to consider the questions implicitly posed by the JWT eventually, (3) the existence of nuclear weapons does not fundamentally change warfare since they are unlikely to be used (we're using other high-tech weapons)

I definitely see that point
I guess the biggest thing I would use for my POV is the section bridging pages 253 and 254

how to disprove?!
http://math.stackexchange.com/questions/697399/prove-or-disprove-a-b-cap-c-a-cap-c-b-cap-c/697408?noredirect=1#697408

Yeah, you are definitely right in that the author wants to perhaps shift focus, but I disagree about whether the claim is "there is no further value for the classical theory"

Yeah, you're right
I definitely overphrased that - the parts that he objects to most are essentially the requirements of proportionality and sovereignity perhaps need to be re-examined or roughly ignored in favor of the other topics, simply because it's hard to not fail those immediately
And he's not willing to consider war trivially immoral, but rather that if one's going to be making the argument that it is, it should be more subtle
immoral's not the right word, but you get my meaning