Mathematics - 2017-11-28 (page 1 of 4)

Leaky Nun

00:02

@identicon the first step should be $\|x^t A x\| \le \sqrt {\| x \| \cdot \| Ax \|}$ by Cauchy-Schwarz

but the rest looks good

identicon

@LeakyNun Thanks

Kasmir Khaan

hi handsome folks

MatheinBoulomenos

@MohammadAreebSiddiqui $b_1$ and $b_2$ will of course depend on the angle

Leaky Nun

edited @identicon to say $\|x\|$ instead of $\|x^t\|$

Kasmir Khaan

so order of G is 15

MatheinBoulomenos

00:04

Hey @Kasmir

Kasmir Khaan

so it has 5 sylow and 3 sylow

@MatheinBoulomenos Hey :D

Leaky Nun

@KasmirKhaan third sylow

Mohammad Areeb Siddiqui

@MatheinBoulomenos how?

Kasmir Khaan

let H_5 be generated by <a>

Mohammad Areeb Siddiqui

whats a sylow

Kasmir Khaan

00:05

and let H_3 be generated by <b>

Leaky Nun

@MohammadAreebSiddiqui it's group theory

Mohammad Areeb Siddiqui

dayum

Leaky Nun

Sylow theory is a powerful tool in group theory

Mohammad Areeb Siddiqui

are u all undergrads?

Leaky Nun

Sylow is a guy

pretty much yes

Kasmir Khaan

00:06

bab' = a^t

since H_5 is normal

Leaky Nun

what is t?

Kasmir Khaan

just some power

Mohammad Areeb Siddiqui

dayum

Leaky Nun

fair enough

Kasmir Khaan

the teacher said that that t^3 is congurent to 1 mod 5

I dont know where he got that

fermat?

MatheinBoulomenos

00:08

what happens if you evaluate bbbab'b'b'? @KasmirKhaan

Kasmir Khaan

I get a back

"a"

Leaky Nun

@MatheinBoulomenos nice

in other words H_3 act on H_5 by conjugation?

Kasmir Khaan

because b^3 = e

MATH ASKER

i'm having trouble finding the anti-derivative of this $∫ sin(√x)/(√x)) $

could anyone help me

Leaky Nun

@MATHASKER wolfram alpha first

to check if an elementary antiderivative exists

MATH ASKER

00:09

Should sin√x = u

wolfram?

MatheinBoulomenos

@KasmirKhaan correct, but what if you use identity bab'=a^t?

@LeakyNun yes

Mohammad Areeb Siddiqui

y cant we uv - v du are way through every integral?

MATH ASKER

oh nvm its a website

Leaky Nun

@MATHASKER I've checked it for you, it exists

MATH ASKER

what does elementary antiderivative mean

Leaky Nun

00:11

ignore it

it just means it's a question you can solve, for now it means that

as opposed to integrals like int sin(x)/x

try $u=\sqrt x$ @MATHASKER

Kasmir Khaan

@MatheinBoulomenos i dont see it

b^3 a b^-3 = a

that we know

and we know that the order of a is 5

Leaky Nun

@KasmirKhaan bbab'b' = ba^tb' = a^tt

Kasmir Khaan

aha

so inside that product

Leaky Nun

aha = h^t

Kasmir Khaan

i can do it 3 times and get exponent of a

on one hand that equals a

Leaky Nun

00:16

precisely

Kasmir Khaan

and other other hand it is equal to a^t^3

so they must be congurent mod the order

neat

@MatheinBoulomenos @LeakyNun thanks :)

Mohammad Areeb Siddiqui

@MatheinBoulomenos i checked by expanding again, b1 and b2 are in terms of c1 and c2 not the opposite like the book says

Kevin Driscoll

01:00

Can you have an inclusion map from the circle to the 2-torus thats like $f: S^1 \to T^2: p \mapsto (p,q)$ thinking of $T^2 = S^1 \times S^1$?

Daminark

@Mathein oh that is beautiful

MatheinBoulomenos

@Daminark what are you referring to?

Leaky Nun

@KevinDriscoll what is q?

if you mean $p \mapsto (p,p)$, then this is the 1-1 knot (which is homotopic to a Villarceau circle)

Daminark

The 336 one

ALannister

0

Q: Calculate $E[XY]$ for $(X,Y)\sim N(\mu_{1},\mu_{2},\sigma_{1}^{2},\sigma_{2}^{2}, \rho)$

I need to calculate $E[XY]$ for $(X,Y) \sim N(\mu_{1},\mu_{2},\sigma_{1}^{2}, \sigma_{2}^{2}, \rho)$ by using integration and then determine the correlation coefficient afterwards. Now, when $X \sim N(\mu_{1},\sigma_{1}^{2})$ and $Y \sim N(\mu_{2}, \sigma_{2}^{2})$, the probability density funct...

MatheinBoulomenos

01:17

@Daminark ah I see, I forgot about that one

Kasmir Khaan

well hi again handsome peeps

I got an other Q about sylow

so in working abstractly on a Group of order n

sometimes n_p is just one value

sometimes many values are possible

so without any extra info how does one elliminate the other possiblities?

Leaky Nun

@KasmirKhaan one does not eliminate anything without info

Kasmir Khaan

hmm

If we take |G| = 90

one find many possiblies of n_2 , n_3 , n_5

kasmir shall return after gathering few data

._.

@LeakyNun here? :)

HiHello

Anyone here know a website I can find creative math problems to solve?

Leaky Nun

@KasmirKhaan ?

Kasmir Khaan

01:31

@HiHello searth for olympiade math questions

they are handsome Q's

@LeakyNun if we have 7 , 3sylow subgroups does that mean we have 18 elements of order 3 ?

based on that, we can make a counting arguemnt

Leaky Nun

14 elements

Kasmir Khaan

so even if many p-sylows are possibles some combinations will overflow no

yes sorry

14 elemnts of order 3

7*2

Leaky Nun

you can't have 7, 3 sylow subgroups if |G| = 90

HiHello

@KasmirKhaan yes. When I was on high school I have done some times. haha. But a person told me about a website where I can find creative problems and he does not remember the name.

I need to ocupy my brain. hahaha

Kasmir Khaan

@MatheinBoulomenos here? :)

MatheinBoulomenos

01:36

yeah

Kasmir Khaan

@LeakyNun was different problem leaky ><

Mathei :D

I igot exam question on sylow

G has order 90

a) suppose that G has more than one 3-sylow subgroup. and the intersection of any two 3 sylow is trivial

prove that G is not simple

let me tell you what I got

since we have more than one 3 sylow

means that n_3 = 10

Mohammad Areeb Siddiqui

if a < b

is

Kasmir Khaan

and we know that normal subgroup is a uninon of conjugacy classes

Mohammad Areeb Siddiqui

e^a < e^b

MatheinBoulomenos

@MohammadAreebSiddiqui yes, the exponential function is strictly monotonously increasing

@KasmirKhaan I don't think that criterion for normal subgroups is going to help you here

Kasmir Khaan

01:40

grrr okay ><

let me Think again

the goal is to get a non trivial normal subgroup

so i need to have some p sylow

to be unique

it cant be the 3-sylow

MatheinBoulomenos

not necessarily

a group can have normal subgroups that are not sylow subgroups

Kasmir Khaan

hmm

nice did not Think of that

but let me keep my experiment

n_5 is either 1 or 6

at some Point i need to get 90 elements

grrr

MatheinBoulomenos

okay, is n_5=1 possible?

Kasmir Khaan

1 is allways possible

MatheinBoulomenos

but, is it possible if the group is simple?

Kasmir Khaan

01:43

oh :D

smart !

we have to exclude that :D

n_2 is 45

to make this into 90 elements

90=1+20+24+45

hmm

well since 1+20+24 divide 90

MatheinBoulomenos

there are more possibilities for n_2 then 45, even if G is simple

Kasmir Khaan

but then we wont have 90 elements no?

since both n_3 and n_5 are foced to be 10 , 6 respectivly

those add up to 44+1

n_2 has to be 45

MatheinBoulomenos

hmm, I don't see how you arrive at 44+1

Kasmir Khaan

10*2 for the 3 sylow

and 6*4 =24 for six 5sylow

+1 idenity

MatheinBoulomenos

How many elements are contained in each 3-sylow?

Leaky Nun

01:47

90 = 2 x 3^2 x 5

Kasmir Khaan

2 non trivial elements

MatheinBoulomenos

no

Kasmir Khaan

oh

8

grrrrrrr

let me try again

Leaky Nun

@KasmirKhaan you don't know if they intersect trivially

@AkivaWeinberger hi

Akiva Weinberger

hi

MatheinBoulomenos

01:48

@LeakyNun he's allowed to assume that

Leaky Nun

@MatheinBoulomenos why?

Kasmir Khaan

it was in the question leaky

assumed that way

Akiva Weinberger

What's going on?

Leaky Nun

oh, ok

@AkivaWeinberger I taught Mathein the basics of model theory

Akiva Weinberger

Cool

Kasmir Khaan

01:49

aha it is done then

Akiva Weinberger

Was your explanation consistent and complete?

Kasmir Khaan

n_5 is forced to be 1

hence non trivial normal subgroup

because 80 elements of order 3

Leaky Nun

@AkivaWeinberger yes, it is even recursively enumerable and can encode PA

Kasmir Khaan

if n_5 = 6

we get 24 more eleemnts thus more than 90

@MatheinBoulomenos ? :DDD

MatheinBoulomenos

they can be of order 9 as well, technically, but this is the right idea

Kasmir Khaan

01:50

hmm what do you mean ?

Akiva Weinberger

Gödel's incompleteness theorem: No discussion is complete without mentioning Gödel.

2

Kasmir Khaan

oh yeah yeah

subgroup of order 9

Akiva Weinberger

@BalarkaSen O_O

Kasmir Khaan

we have 80 nontrivial elements in 3sylow subgroup

MatheinBoulomenos

@LeakyNun can we express algebraicity in first-order?

Akiva Weinberger

01:51

... O_O

blink blink

O_O

Leaky Nun

@MatheinBoulomenos let's go to logic room to discuss this?

Kasmir Khaan

okay part b) is wierd

MatheinBoulomenos

oh yeah right

Kasmir Khaan

mathein

wait :D

suppose that G has at least two distinct 3 sylow subgroups whose intersection is non trivial. prove that G is not simple

by intersection being non trivial here means that they are the same subgroup no ?

Leaky Nun

@KasmirKhaan no, the intersection can have 3 elements

Kasmir Khaan

01:53

or wait ._. the order is 9 so they might be of order 3

Leaky Nun

you see, you answered it 10 seconds later

Kasmir Khaan

ill text you guys later, ill keep trying

Akiva Weinberger

This is cool

Kasmir Khaan

haha=p am stressed out leaky , exam in 1 day

Akiva Weinberger

MatheinBoulomenos

01:54

@KasmirKhaan you need to look at the normalizer of the intersection

that's a bit tricky

Kasmir Khaan

@MatheinBoulomenos hmm I know what normalizer is , but what does it have to do here?

MatheinBoulomenos

it can help you prove that the group is not simple

Kasmir Khaan

normalizer of any subgroup is normal in G?

oh well ._. I need to keep thinking on this. I ll be back in few minutes

I hope you guys dont leave :D

Kevin Driscoll

@LeakyNun I meant $q$ to be any point in the '2nd circle' of $S^1 \times S^1$

Leaky Nun

@KevinDriscoll then it is still yes

it is one of the vertical circles

(or horizontal depending on how you define the product)

Kevin Driscoll

01:58

That was my intuition, just wanted to make sure I was right

I was thinking about the same idea but applied to $T^2 \to T^3$

Leaky Nun

trust the algebra

3

john

hi

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