Mathematics - 2018-10-28 (page 1 of 2)

user193319

00:49

Problem: Let $X$ be a compact Hausdorff space and $f : X \to X$ a continuous function. Show that there is some non-empty subset $A$ such that $f(A) = A$. Proof: Let $A_0 = X$, and let $A_n := f^n(X)$ for every $n \in \Bbb{N}$. Then $\{A_n\}_{n=0}^\infty$ is a sequence of non-empty, decreasing, compact sets. Hence $A := \bigcap_{n=0}^\infty A_n$ is non-empty....

I've been able to show that $f(A) \subseteq A$, but I'm having trouble showing $A \subseteq f(A)$. I could use some help.

Leaky Nun

01:08

> More generally, if f is monotonic, then the least fixpoint of f is the stationary limit of fα(0), taking α over the ordinals, where fα is defined by transfinite induction: fα+1 = f ( fα) and fγ for a limit ordinal γ is the least upper bound of the fβ for all β ordinals less than γ. The dual theorem holds for the greatest fixpoint.

Knaster–Tarski theorem

In the mathematical areas of order and lattice theory, the Knaster–Tarski theorem, named after Bronisław Knaster and Alfred Tarski, states the following: Let L be a complete lattice and let f : L → L be an order-preserving function. Then the set of fixed points of f in L is also a complete lattice.It was Tarski who stated the result in its most general form, and so the theorem is often known as Tarski's fixed point theorem. Some time earlier, Knaster and Tarski established the result for the special case where L is the lattice of subsets of a set, the power set lattice.The theorem has important...

@user193319 idk if this will work lmao

I give up

Fargle

02:32

Why are discrete spaces sometimes seen as pathological? They're perfectly normal.

:^)

The Great Duck

im having issues converting normals

help please

1

Q: How to convert the normal vector of a point in some polygonal surface transformed onto the surface of a triangle in space?

It's complicated to explain what the transformation I'm dealing with is but basically it is a transformation that takes a 3D model and maps it to the surface of one polygon on another model. The issue is that I'd prefer to not have to recompute the normal vectors of the post-transformation model ...

Alucard

04:12

if anyone reads this: have a A1 day :)

4

The Great Duck

@Alucard k

CaptainAmerica16

04:25

@Alucard Sure thing :D

Alucard

@Secret hi :)

Secret

hi

user2236

wazzup?

Alucard

5:30 here

Math geek

An example given in the textbook is given below.

Let $X=\{(x,y)\in \mathbb R^2:x=0 \text{or } x=\frac{1}{n}$ for some $n \in \mathbb N$, and $0\leq y\leq 1\}$. I know that equivalence class $[(0,0)]=\{(x,y)\in \mathbb R^2|(x,y)$ and $(0,0)$ in the same connected component$\}$. Why $[(0,0)]$ is not an open set? Under what topology it is not open? Seeing $(X,\tau)$ as a subspace topology of usual topology on $\mathbb R^2$? $[(0,0)]$ can not be written as the intersection of $X$ with any open ball in $\mathbb R^2$. am I correct?

Am I done a rigorous proof that $[0,0]$ is not open?

user2236

04:42

Tweeted by mathematicalsvs on October 28, 2018 at 4:10 AM

The Great Duck

@user2236 broken image but i can manually go to it

Alucard

@user2236 I can not see your picture :(

The Great Duck

might want to look at that

@Alucard click it

it's also a link to the image

Math geek

please help me.

Alucard

possibly a antihotlinkingfeature? @TheGreatDuck

The Great Duck

04:44

idk

i just know that if you click the image it takes you to twitter

and the original phot

Alucard

. ux.stackexchange.com/questions/121802/… you never know where you land on SE^^

Unknown MathMan

05:22

How many elements in $S_6$ has order $6$?

Can anyone help me to find this?

I think $6$-cycles has order $6$, and disjoint product of 2-cycles and 3-cycles has order 6.

But I am not getting a way so that I can be sure that I haven't missed any order 6 element

user1414

07:01

A mathematician named Klein
Thought the Mobius strip was divine
He said "If you glue
The edges of two
You can make a strange bottle like mine".

Akiva Weinberger

07:17

@user1414 Those are impressive! Did you make them?

user1414

nope

found them here @AkivaWeinberger

Zee

@AkivaWeinberger haven’t seen you here in a while

Not that I been active either

Secret

08:04

Why Beautiful Things Make us Happy – Beauty Explained

►

Isana Yashiro

math is beautiful sometimes but I'm not always happy with it

I'm both so excited and sad about how long will I really understand advanced topics in linear algebra.

Could anyone please help me about why linear transformation $T$ and its inverse relation $T^{-1}$ map(?) subspace to subspace?

Secret

Quaternions and 3d rotation, explained interactively

►

Isana Yashiro

i.e. why given $W$ a subspace of domain then $T(W)$ is subspace and $W'$ a subspace of codomain then $T^{-1}(W')$ is also subspace.

Secret

T(0)=0 otherwise you cannot invert T

Isana Yashiro

@Secret I eventually laughed when I heard about 32-ternion or something at the end

@Secret no, $T^{-1}$ is to denoted a inverse relation not necessarily function

Mary Star

08:16

Hey!! Let $K$ be a normal extension of $F$ and $f\in F[x]$ be irreducible over $F$.
Let $g_1, g_2$ be irreducible factors of $f$ in the ring $K[x]$. Could you give me a hint how we could show that there exists $\sigma \in G(K/F)$ such that $g_2=\sigma (g_1)$ ?

Isana Yashiro

bye my question, die, right?

Secret

Nope, otherwise why will I try to answer?

Ok, I think the first step is since T is linear, it follows it will map a linear combination of vectors to a corresponding linear combination

so linear conbinations are preserved

In particular, linear combinations that gives the zero vector will also be preserved

You cannot have T(0)=/=0 as T(x)=T(x+0)=T(x)+T(0) thus T(0) must also be an additive identity, hence can only be a zero vector in the image. Hence subspace structure is preserved

same reasoning applies to the inverse map

Isana Yashiro

you're right, let me think about it

@Secret when you said linear com. are preserved what do you mean? and should i pick a basis/spanning set to think about this?

Secret

if {x,y,z} are a basis, then {T(x),T(y),T(z)} will also be a basis if T is invertible

Isana Yashiro

I think the problem is that I haven't convinced myself that $T^{-1}(W')$ is a subspace

Secret

08:29

Recall that to check whether something is a subspace, check whether it contains the zero vector and check that it is closed under addition and multiplication

$T^{-1}(0)=0$ by the same logic as $T(0)$ as described below because $T^{-1}$ is also a linear map

Isana Yashiro

If, some elements in $W'$ are never hit/map by $T$, e.g.

no element $x\in W$ s.t. $T(x)=w'$ for some $w'\in W'$

why won't this affect that $T^{-1}(W')$ be a subspace?

this is what I can prove but never understand why

I randomly circle some elements in $W'$ to make them a subspace then something are never mapped back to domain and then I say: yeah, $T^{-1}$ keep subspace.

Secret

Hmm... if there is no $x$ such that $T(x)=w$ for some $w \in W'$, will that mean there is no $a,b,c$ such that $w=a+b+c$ such that $w = T(a+b+c) = T(a)+T(b)+T(c)$?

Isana Yashiro

ok i now know the simplified version of my doubt: if $W'$ is bigger than the range of $T$, why it necessarily $T^{-1}(W')$ a subspace of domain?

and there also cases where $Ran(T)-W'$ not empty right? how to deal with this case?

Secret

08:45

The domain of $T^{-1}$ is the range of $T$ be definition of being its inverse. That means, anything that is in $W' - \text{Ran}(T)$ lies outside the domain of $T^{-1}$. Thus the image of $T^{-1}$ is a subspace properly contained in $W'$

Isana Yashiro

I'm really sorry about my misleading symbol, which is from my stopid book, but $T^{-1}$ I meant something like $R^{-1}$ where $R$ is a relation not necessarily function.

so for example $T$ not one-one then $T^{-1}$ never a function

Secret

The conclusion is the same, because points in $W'-\text{Ran}(T)$ were never reached from mapping by $T$, thus the inverse relation $T^{-1}$ that gives the set of points that map to it is empty. This holds even if $T$ is not injective

Since $T$ is not surjective, the preimage of $W'-\text{Ran(T)}$ is empty

Isana Yashiro

it sounds like every $W'$ sub-slice(or something) a part of $Ran(T)$ still a subspace then mapped back to domain then still subspace?

Secret

Isana Yashiro

Cool I like it

Secret

08:55

Any subset of a subspace containing a basis set of size the same as W or smaller and their linear combination is also a subspace

so for example, a line passing through the origin contained on a plane passing the origin is a subspace

Isana Yashiro

but in your picture it seems like $Ran(T)\subset W'$, what if it's not?

if $W'$ just contain a part of $Ran(T)$?

how can I confirm that the intersection $W'\cap Ran(T)$ must be a subspace?

ohhhhh

wait

the intersection of two subspaces must be subspace!?

oh man, I think I just understand a little part of math which God thought it as garbage, but I'm so happy

Secret

well they must at least contain the subspace $\{0\}$

so yes

Isana Yashiro

@Secret Thank you so much

I've decided to use your image as my desktop picture when I study linear algebra

LoveWithMaths

Anybody knows how to find coordinate of point D. In my textboook its given as -4,-3

Textbook says since its a parallelogram, AB and CD have same length and same slope.

But i need to know how. I know the length is 5 and all sides are same. But how to find point d ?

Secret

$\vec{DA}=\vec{AC}-\vec{BA}$

LoveWithMaths

09:08

what is vec ?

Secret

do you have latex enabled?

LoveWithMaths

how to do that ?

Secret

tinyurl.com/cfqcvpc

LoveWithMaths

I clicked on start chat jax. Still no luck

@Secret can you please explain what it means ?

what is vec ?

Secret

vector DA = vector AC - vector BA

LoveWithMaths

09:13

for example vector AC means length of AC ?

Secret

have your course gone through vector calculations yet?

LoveWithMaths

No thats why its alien to me.

is there simple way of calc the same

Secret

ah no wonder...

hmm... then it will be a bit hard to explain... let me think...

LoveWithMaths

Textbook says since ABCD is parallelogram, line segments AB and CD have same length and same slope. There fore point D is (-4,-3)

Dint understood above statement

how to calculate the same.

Secret

Well, at least for this example, you can use symmetry to get the coordinates of D, note how D is a reflection of B about the diagonal AC

The textbook solution, is basically saying that since the lengths are the same, you have:
$$(-7-d_1)^2+(-7-d_2)^2=(-3-0)^2+(-4-0)^2$$

where coordinates of $D=(d_1,d_2)$

LoveWithMaths

09:19

How do i enable latex, cannot understand anything

Secret

meanwhile, because they have the same slope, you also have the equation:

$$\frac{-7-d_2}{-7-d_1}=\frac{-4-0}{-3-0}$$

This gives you two equations in two unknowns to solve for $D$

LoveWithMaths

Great above is easy to solve ! Cheers

I am getting $4d1-3d2=-7$

Is that correct ?

by the way above equation is no where considering the length which is 5. Then how can we ever get correct coordinates ?

Secret

09:39

You have the distance equation above, but you also have the slope equation below. Combine the two to get a system of two equations in two unknowns

Prashin Jeevaganth

09:58

Anyone here is willing to confirm answers with me for Q5a,b?

I got 9/80
229/1200

And I did it by a super tedious method, not sure if there's a simpler way

Alucard

@PrashinJeevaganth i'll look into it

@PrashinJeevaganth i get 9/80 for a too

for b) it's too much of a hassle sorry :D

ouch misread ;)

Prashin Jeevaganth

10:40

@Alucard thanks for your attention, this question is just a slaughter

Alucard

@PrashinJeevaganth if you write down the way of your solution I'll look it. :)

Prashin Jeevaganth

@Alucard I think I got a careless mistake somewhere ... because I couldn't solve part c... It's handwritten do u still want it?

Alucard

@PrashinJeevaganth yeah why not, if you got a scanner or camera :) If you have a diagram of the probabilities just mark what you added for b) ;)

Prashin Jeevaganth

@Alucard Alright a moment

@Alucard Well here it is ... not sure if it's illegible

My head is not thinking straight after doing this tedious work for 20minutes

Alucard

10:59

@PrashinJeevaganth why not this way?

Prashin Jeevaganth

Is this much easier to see? I don't know ...

Alucard

@PrashinJeevaganth your works should be on a din A4 site, but actually it could be 1/2 a site

maybe even less

(if you are allowed to use a calculator, that is)

"your workings should not exceed one A4 page"

Prashin Jeevaganth

@Alucard Well I think they meant just show that u wrote something tedious

As a plagiarism check haha

I don't think anyone will question my work the moment they see that disgusting chunk of expansion

How can this even be done on half a page ... someone must be able to expand this without any difficulty

Alucard

nr for example means Not Rainy ;)

Prashin Jeevaganth

@Alucard hmm neat way of writing

I will try that

but your answer still doesn't work on my part (c) ...

damn haha

Alucard

11:15

it works, i'll add that

oh well, too much time passed i forgot how you do that, dependant probability or such thing, i'll look it up

@PrashinJeevaganth i think this is what you look for: mathsisfun.com/data/probability-events-conditional.html

Prashin Jeevaganth

@Alucard hmm I mean using my old part

I'm trying to use the tree method now ...

Hold on why do I get 1 ...

Alucard

probably because you summed somehow all possible eventchains together^^

Prashin Jeevaganth

isn't P(A|Q) = P(A ^ Q) / P(Q) ... and P(Q) is the earlier answer

@Alucard Could I even work with the same tree for (c)?

I realised my method for attaining (b) and (c) is the same ... something's wrong with my logic

Alucard

@PrashinJeevaganth that is what I'm not sure about

Prashin Jeevaganth

11:30

haha great :)

gosh problems everywhere haha

@Alucard do you know of any good recommendations for probability and combinatorics practice problems with solutions? I'm in need of them right now.

To be honest, I have been doing probability using the Venn diagram for the most of my life. Didn't know the life hack for the tree minutes ago

Alucard

@PrashinJeevaganth a statistic course will help you a lot. i don't know any good book on it, but i guess there are plenty who teach the basics

Prashin Jeevaganth

@Alucard I'm taking a Discrete Mathematics course now, until then I need some reference guides and practice resources as combinatorics is taking a huge component

the other 2 being graph theory and set theory too

Alucard

@PrashinJeevaganth to understand c) i would probably first write down a simpler problem

it rains 20% of the days, and while it's raining i am 30% late, if it's not raining i am 10% late

now, given i am too late, what is the chance that it is a raining day?

if you can solve this you can solve c) ;)

Prashin Jeevaganth

11:54

@Alucard By this question, are you saying that you found out that the answer for (c) is not 1?

Alucard

@PrashinJeevaganth there are cases where i am late, but it didn't rain, so yes, the answer to c) can't be 1. that's sure.

Prashin Jeevaganth

Hmm is the answer 3/7 to this new problem?

Just divided the 1 long branch where (late and raining) / 2 cases where I was late

Alucard

@PrashinJeevaganth the way is right, i think!

Prashin Jeevaganth

haha ok

Alucard

but i don't gurantee that :P

maybe wait for someone who actually got a clue XD

Prashin Jeevaganth

11:59

This time of the week it seems nobody is chatting here

haha

this day*

Alucard

@PrashinJeevaganth you can always ask on the mainsite, as long as your question meets the standards :) I'm interested too

Secret

[Random]

\begin{align}
& 0\\
& 10\\
& 01\\
& 0010\\
& 020\\
& 00010\\
& 0110\\
& 000010\\
& 030\\
& 0020\\
& 01010\\
& 0000010\\
& 0210\\
& 00000010\\
& 010010\\
& 00110\\
& 040\\
& 000000010\\
& 0120\\
& 0000000010\\
& 02010\\
& 001010\\
& 0100010\\
& 00000000010\\
& 0310\\
& 00020\\
& 01000010\\
& 0030\\
& 020010\\
& 000000000010\\
& 01110\\
& 0000000000010\\
& 050\\
& 0010010\\
& 010000010\\
& 000110\\
& 0220\\
& 00000000000010\\
& 0100000010\\
& 00100010\\
& 03010\\
& LONG\\
& 011010\\
& LONG\\
& 0200010\\

nope, absolutely no pattern at all

hmm...

if we read the 2nd digit of ever second row, we get:

Vrouvrou

hello

plaise if for a function $u:\Omega\subset \mathbb{R}^n\to \mathbb{R}$ we know that the positive part $u^+\neq 0$ why $|\Omega_+|>0$ where $\Omega_+=\{x\in \Omega, u(x)>0\}$

Secret

0,1,2,1,3,1,2,1,4,1,2,1,3,1,2,1,5,1,2,1,3,1,2,1,4,1,2,1,3,1,2,1,6,1,2,1,3,1,2,1,‌4,1,2,1,3,1,2,1,4,1,2,...

Vrouvrou

@LeakyNun hello

Secret

12:09

Or more concisely:

0,2,3,2,4,2,3,2,5,2,3,2,4,2,3,2,6,2,3,2,4,2,3,2,4,2,...

Or even more concisely:

0,2,3,2,4,2,3,2,5,2,3,2,4,2,3,2,6,2,3,2,4,2,3,2,4,2,..‌.

$$0,1,{\color{red}2},1,{\color{yellow}3},1,{\color{red}2},1,{\color{blue}4},1,
{\color{red}2},1,{\color{yellow}3},1,{\color{red}2},1,5,1,{\color{red}2},1,{\color{yellow}3},1,{\color{red}2},1,{\color{blue}4},1,{\color{red}2},1,{\color{yellow}3},1,{\color{red}2},1,6,1,{\color{red}2},1,{\color{yellow}3},1,{\color{red}2},1,‌{\color{blue}4},1,{\color{red}2},1,{\color{yellow}3},1,{\color{red}2},1,{\color{blue}4},1,{\color{red}2},...$$

nope, 4 breaks down

It is easy to check whether 2 eventually will break down by computing:

e.g.

Prashin Jeevaganth

@Alucard math.stackexchange.com/questions/2974604/…

I asked my question

Secret

05130
010000010...10
010...10
012000000110
0300000010...10
01010...10
02101110
010...10
040...10
0110...10
02010...10

Alucard

@PrashinJeevaganth one upvote you got :D

Prashin Jeevaganth

12:25

@Alucard ah haha thanks

Secret

So yes, it does breaks down: because 010...10

We expect the same thing will happen for 3, though we are not too bothered to calculate it

1 seemed to show no signs of breaking down though. Could that be our silver bullet?

Highest tried:

Unknown MathMan

Let $\phi:G_1\to G_2$ is a onto homomorphism. If $G_1$ is abelian then can we say $G_2$ is also abelian?

Secret

$1_13_21_51_{16}1_{\#[84881507]}$

MatheinBoulomenos

@UnknownMathMan yes

Unknown MathMan

@MatheinBoulomenos I have another query, can we say that $G$, with order $260$ has exactly 12 elements with order 13?

mercio

12:39

o..O

Unknown MathMan

I tihnk yes, as $\phi(13)=12$ there are 12 elements that has order 13, but I am not sure if there are any other element with that order

mercio

are you saying that every group has 12 elements with order 13 ?

MatheinBoulomenos

yes, you can say that $G$ of order 260 has exactly 12 elements of order 13, but you need more of an argument than what you gave

it follows from Sylow theory

Unknown MathMan

Ok, can you give any hint?

MatheinBoulomenos

try to work out how many $13$-Sylow subgroups there are in a group of order 260

Unknown MathMan

12:45

1?

MatheinBoulomenos

yeah

Unknown MathMan

As 13 is prime, the group is cyclic

So, by previous argument there are exactly 12 of them

Right?

MatheinBoulomenos

right

Unknown MathMan

Thanks :)

MatheinBoulomenos

np

Vrouvrou

12:59

Someone have an idea please :math.stackexchange.com/questions/2974594/…

someone knows something about measure theory ?

Secret

Initiate hopping algorithm

0,0,1,0,2,0,1,0,3,0,1,0,2,0,1,0,4,0,1,0,2,0,1,0,3,0,1,0,2,0,1,0,5,0,1,0,2,0,1,0,‌3,0,1,0,2,0,1,0,4,0,1,0,2,0,1,0,3,0,1,0,2,0,1,0,6,0,1,0,2,0,1,0,3,0,1,0,2,0,1,‌0,‌4,0,1,0,2,0,1,0,3,0,1,0,2,0,1,0,4,0,1,0,2,...

Pick 37[18]

Hop: 37[18]->111[55]

Hop: 111[55]->112[56]

ReadHop: [56]

LessSq: [56]=112<n^2 => n=11

LessSq2: 2^k < 121 => k =6

Decomp 112[56][2] <= 6

112 = 040010

Hop: 040010=>00010=7

112[56] -> 7[3]

Prashin Jeevaganth

13:28

@Alucard that was a terrible meme haha

we both missed it

Secret

summary: 18->55->**56**->3->10->11->5->16->17->...

hmm...

Alucard

@PrashinJeevaganth what you mean, still your question? :)

Prashin Jeevaganth

yes haha

the one about the probability tree

Alucard

@PrashinJeevaganth the funny thing is, i learned such things at school but forgot the important parts lol

Prashin Jeevaganth

@Alucard Well I think the biggest takeaway for this is I can stop using Venn diagrams ... I'm too used to doing Boolean algebra and Venn diagrams and didn't really know about the probability tree.

Are there scenarios where a probability tree can't be used ?

Alucard

13:34

@PrashinJeevaganth I don't know, surely they get ugly if too many branches are there, but exercises are usually made such that you can do it this way

a perfect venn diagram could be nice

such that the area is directly proportional to the probability

Vrouvrou

someone help me on measure theory please?

Secret

Odd $n\to 3n+1 \to$
Even $3n+2$
Odd $(3n+2)/2^{m < \lfloor 3n+2\rfloor_{2^x}} = \text{odd}$

Odd[even] $n \to 3n+1 \to$

Even[odd] $3n+1 \to 3n+2$

Odd[even] $n \to 3n +1 \to$

Odd[odd] $3n+1 \to 3n+2$

Prashin Jeevaganth

@Alucard well no one would have the time to draw them to scale...

Secret

Even[even] $3n+2 \to (3n+2)/2^{m < \lceil 3n+2 \rceil_{2^x}}$

$\to$ Odd[odd]

Vrouvrou

@AlessandroCodenotti hello

Secret

13:46

Odd[Odd] $n \to 3n +1 \to$

Odd[even] $3n+1 \to 3n + 2$

Even[odd] $3n +2 \to (3n+2)/2 = \text{odd[Even]}$

$2(2m+1)[2m+1] \to 2m+1[m] \to 3(2m+1)+1[]$

ERROR. CALCULATON TOOK TOO LONG. OPERATION TERMINATED

Alucard

@Secret :-D

Secret

F888 prime numbers

Alucard

15:10

@PrashinJeevaganth with what you created the graphic?

Prashin Jeevaganth

@Alucard I think you're talking to the wrong person

Alucard

@PrashinJeevaganth no, i mean your tree diagram :)

Prashin Jeevaganth

@Alucard Oh, just MS powerpoint: arrows and textboxes, I did it hastily...

Leyla Alkan

15:23

Hi. I am not a native speaker of English. So, the bold part in the following sentence seems odd to me a bit. "$ \vec{a} $ satisfies $\varphi \Leftrightarrow \psi $ iff either $ \vec{a} $ satisfies both $\varphi $ and $ \psi $ or $\textbf{$ \vec{\textbf a} $ doesn't satisfy either $ \varphi $ or $ \psi $}$"
It doesn't seem correct to me. What do you think?

I mean it doesn't seem to be equivalent to the phrase "$ \vec{a} $ satisfies neither $ \varphi $ nor $ \psi $ "

Mike Miller

I do not see the difference between the last sentence and your sentence, but I understand your confusion. In any case, your last sentence is what is meant there.

(I imagine you parsed it as "a fails to satisfy at least one of phi, psi". This is not what they mean to say, but I also see how one can read ir that way.)

Akiva Weinberger

15:41

"A doesn't do B or C" is the same as "A does neither B nor C" in English

In fact, I think the former is more common

Vrouvrou

@AkivaWeinberger hello

how are you

Akiva Weinberger

"Neither" and "nor" are pretty rare, I think

or, at least, they're more formal

@Vrouvrou I'm good

Vrouvrou

please can you helpp me on measure

Akiva Weinberger

Now's not a really good time, but you can post your question anyway

Vrouvrou

i post it but I'm sure about the answer:math.stackexchange.com/questions/2974594/…

if for a function $u:\Omega\subset \mathbb{R}^n\to \mathbb{R}$ we know that the positive part $u^+\neq 0$ why $|\Omega_+|>0$ where $\Omega_+=\{x\in \Omega, u(x)>0\}$

Leyla Alkan

15:47

Oh okay, thanks guys!

Vrouvrou

@AkivaWeinberger you see my question ?

Alessandro Codenotti

@Vrouvrou I suppose $u$ is continuous?

Vrouvrou

yes

Alessandro Codenotti

If $u(x_0)>0$ then $u(x)>0$ on an open set around $x_0$, which is a set of positive measure

Vrouvrou

and if u is measurable ?

Alucard

15:59

I'd aprecciate if someone could break down exercise c) here: math.stackexchange.com/questions/2974604/…

Topologicalife

If $f:(-\infty, b) \to \mathbb{R}$ and I have, for some $k\in\mathbb{R}$ that $\displaystyle \lim_{x\to b^+} f'(x) = k$, is this a contradiction?

I think so because $f$ is not defined in $b$.

Alessandro Codenotti

@Topologicalife $f(x)=kx$?

@Vrouvrou This is not enough, since you can have $u$ be nonzero on a set of zero measure then

Topologicalife

I don't have info about $f$, @AlessandroCodenotti

Only that $f$ is continuous.

Alessandro Codenotti

That's an example of a function $(-\infty,b)\to\Bbb R$, compute its derivative

Topologicalife

(and sorry, forget to mention it)

@AlessandroCodenotti I can see that $\displaystyle \lim_{x\to b} f'(x) = k$ in your example, but I think $\displaystyle \lim_{x\to b^+} f'(x) $ is not defined.

Alessandro Codenotti

16:11

Ah, didn't notice the small $+$. I agree it's not defined then, but you can definitely talk about the limit of $f$ at $b$ approaching it from below

And this one can be finite with no issues

Topologicalife

I think, again, that limit shouldn't be defined.

Err sec.

What about if I have $f,g : \mathbb{R} \to \mathbb{R}$ with both continuous and with $g$ bounded, and $\phi : (-\infty, b) \to \mathbb{R}$. If I compute $\displaystyle \lim_{x\to b^+} f(x)g(\phi(x))$. My notes say $\displaystyle \lim_{x\to b^+} f(x)g(\phi(x)) = f(b)k $ since $g\leq k$ (bounded).

I think it shouldn't be defined.

Uhm does $\lim_{x\to +b}$ mean we approach $b$ from the left?

Vrouvrou

16:29

merci @AkivaWeinberger

Fargle

The sign by a limit indicates where you're approaching from, @Topologicalife. You would be approaching $b$ from the right in that instance.

Topologicalife

I see, thanks. Then I I think that limit shouldn't exist.

Fargle

Agreed. You'd need the function to be defined in at least a small open set in that direction (though not necessarily $b$ itself).

Topologicalife

The condition $g$ is bounded it has nothing to do with that limit, right?

Fargle

Well, other than allowing the left-hand limit at $b$ to exist, no.

The function could asymptote at $b$ if it were continuous but unbounded.

Topologicalife

16:36

Yeah

This question is related to this post: math.stackexchange.com/questions/2973575/…

Akiva Weinberger

16:55

@Vrouvrou When they say $u^+\ne0$ do they mean they're not equal anywhere, or just that they're not equal almost everywhere?

tatan

17:59

If a,b,c are positive integers such that 2^a+4^b+8^c=328 then find (a+2b+3c)/abc

Is there any reasonable method to do it rather that finding (a,b,c) by trial and error?

aitía

Hello, is someone here available to help me with my multiple choice questions?

user2236

Askaway

Alucard

^

user2236

^^

aitía

Assume there are two communities in this network: {A, B, C, D, G} and {E, F, H}. Which of the following statements is(are) True? Select all that apply.
- The Common Neighbour Soundarajan-Hopcroft score of node C and node D is 2.
- The Common Neighbour Soundarajan-Hopcroft score of node A and node G is 4.
- The Resource Allocation Soundarajan-Hopcroft score of node E and node F is 0.
- The Resource Allocation Soundarajan-Hopcroft score of node A and node G is 0.7.

Mary Star

18:47

Let $K$ be a normal extension of $F$ and $f\in F[x]$ be irreducible over $F$.
Let $g_1, g_2$ be irreducible factors of $f$ in the ring $K[x]$. Show that there exists $\sigma \in G(K/F)$ such that $g_2=\sigma (g_1)$.

Could you give me a hint how we could show that?

ninja hatori

19:09

0

Q: About tensor product in inner product space

I know that for finitely generated space surjectivity implies injectivity. Only problem is that we need to show $\hat b$ is surjective map. So can somebody explain me that tensor product calculation?

linear-algebra commutative-algebra tensor-products

Akiva Weinberger

@Semiclassical The angle-addition formula for tanh is the same as the formula for adding velocities in relativity

I guess it makes sense (from the graph of tanh) but I never noticed it before

user193319

In the "polynomial" proof of the Cauchy-Schwarz Inequality, why does the discriminant of the polynomial have to be less than or equal to $0$? I never understood that...

I.e., the polynoimial $q(t) = \langle v - tu, v - tu \rangle = t^2 ||u||^2 - 2t \langle v,u \rangle + ||v||^2$.

user451552

19:38

What makes a person a great mathmatician?

Nick

Being like @TheSimpliFire and keep a question from last September open by commenting with valuable data in next year's October Now, that's dedication to a problem.

Tobias Kildetoft

@user451552 Great results :)

user451552

Agree @TobiasKildetoft

ninja hatori

What is field of fraction for Z localized at p

Tobias Kildetoft

@ninjahatori Localizing an integral domain anywhere does not change the field of fractions.

ninja hatori

19:50

It is Q then right

Tobias Kildetoft

yes

ninja hatori

Thanks

Nick

How do you define a fraction field?

finite set with abelian properties?

ninja hatori

@TobiasKildetoft any short proof?

Nick

Field (mathematics)

In mathematics, a field is a set on which addition, subtraction, multiplication, and division are defined, and behave as the corresponding operations on rational and real numbers do. A field is thus a fundamental algebraic structure, which is widely used in algebra, number theory and many other areas of mathematics. The best known fields are the field of rational numbers, the field of real numbers and the field of complex numbers. Many other fields, such as fields of rational functions, algebraic function fields, algebraic number fields, and p-adic fields are commonly used and studied in...

Tobias Kildetoft

19:53

@ninjahatori It is basically trivial from what the terms mean.

Nick

Oh, rationals. That makes sense.

P-adic number

In mathematics, the p-adic number system for any prime number p extends the ordinary arithmetic of the rational numbers in a different way from the extension of the rational number system to the real and complex number systems. The extension is achieved by an alternative interpretation of the concept of "closeness" or absolute value. In particular, p-adic numbers are considered to be close when their difference is divisible by a high power of p: the higher the power, the closer they are. This property enables p-adic numbers to encode congruence information in a way that turns out to have powerful...

This is neat.

Alessandro Codenotti

@ninjahatori note that for an integral domain $A$ every localization is a subring of $\operatorname{Frac}(A)$

Nick

ding ding ding xD

user451552

God knows everything => God knows infinity

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$Mathematics$ Mathematics