Mathematics - 2016-10-03 (page 1 of 3)

usukidoll

00:19

hihi

there's a truth table for those $2^{3}=8 $ cases

Ajeet Kljh

00:37

any general tips to prove a logical equivalence using other logical equivalences?

because i always end up stuck

Balarka Sen

03:31

@Danu You have to stop bothering about "all the details" to appreciate something.

The conversations are not really encoded. Also, how would you know the conversations are extremely interesting if they are encoded so you cannot understand?

You and Ted do have a lot of conversations which sound like they are encoded to me. I don't understand crap about what you're talking about on Kahler manifolds.

That frustrates me too, I guess, but I get over it.

Perturbative

Hey everyone, just a quick question. I'm reading through Baby Rudin, and I've noticed that Rudin defines the interior point of a subset, E, of a metric space in terms of the existence of a neighbourhood that is a subset of E

Is there any reason why Rudin doesn't just define an inter point of E to be an element of E?

*interior

Balarka Sen

Consider $[0, 1]$. $1/2$ is an interior point, $0$ isn't.

The whole point is that the boundary points are not allowed.

Perturbative

Ah okay I see, 0 would be a limiting point in that case

arctic tern

1/2 is also a limiting point...

0 and 1 are boundary points

usukidoll

hihi

arctic tern

03:36

hiya

usukidoll

can anyone give me some hints to this? prntscr.com/cp4xvf

arctic tern

which part?

Perturbative

Correct me if I'm wrong, but every point in $[0,1]$ would be a limiting point?

usukidoll

closure under addition, closure under multiplication, identity, additive inverse are the requirements for a subring

which part? hmm umm a?

arctic tern

yes, @Perturbative, it's closed

usukidoll

03:37

I'm like SO stuck ;_;

arctic tern

@usukidoll okay, let's talk about part (a). what do you think? do you think it's a subring, or not a subring?

usukidoll

if closure under addition/multiplication, identity, or additive inverse fails we don't have a subring

arctic tern

So do you think it's a subring? Yes or no.

usukidoll

so that polynomial must be closed under addition/multiplication, have an identity or additive inverse

I think it's a subring... because it's asking for $ n \geq 3$ for $x^n$ so we have $x^3$
$a_{0}+a_{1}x+a_{2}x^2+a_{3}x^3$

arctic tern

huh?

usukidoll

03:40

wait I screwed it up royally

I didn't read the 0 part

arctic tern

Read before that. It's the set of polynomials that look like ax^2+bx+c.

usukidoll

oh crud we just have this then
$a_{0}+a_{1}x+a_{2}x^2 =f(x)$ and then we got to prove if it's a subring or not sorry sorry I had lazy eyes for a sec

Perturbative

@arctictern Rudin doesn't technically define boundary points, (well at least not yet up to the point I've read), how would you define it rigorously?

Balarka Sen

@Perturbative Points of $S$ every nbhd of which intersects both $S$ and it's complement.

arctic tern

@Perturbative google

srsly

usukidoll

03:42

so then we need to prove that closure under addition/multiplication, identity, and additive inverse is satisfied for that polynomial

arctic tern

@usukidoll okay, (ax^2+bx+c) plus (a'x^2+b'x+c') makes (a+a')x^2+(b+b')x+(c+c').

usukidoll

doesn't Rudin have that real analysis book or something? That author sounds familar

Balarka Sen

yes

arctic tern

what happens if you multiply (ax^2+bx+c) and (dx^2+ex+f)?

Perturbative

@arctictern Okay okay :p, thanks for your help

usukidoll

03:43

YEAH! there we go and then supposedly ouch hold up

Perturbative

You too @BalarkaSen, thanks!

usukidoll

sorry the fan opened my soft cover book and all the contents hit my foot

$(ax^2+bx+c)(dx^2+ex+f) = adx^4+aex^3+afx^2+bdx^3+bex^2+bfx+cdx^2+cex+cf$
$adx^4 +aex^3+bdx^3+afx^2+cdx^2+bex^2 +bfx+cex+cf$

arctic tern

well, you didn't have to actually work it out, just notice that you get terms higher than x^2 in the result

in particular, x^2 is in the subset, but x^2 times x^2 isn't

usukidoll

yeah isn't that for x^3 the coefficents go to 0? why do we have letters in here?!

oh so x^2 is in the subset but x^4 isn't because our original polynomial only went up to x^2. Oh this fails under multiplication

because for additive inverse
$a_{0}+a_{1}x+a_{2}x^2 + (-a_{0})+(-a_{1}x)+(-a_{2}x^2)$
$a_{0}+a_{1}x+a_{2}x^2 -a_{0}-a_{1}x-a_{2}x^2$

$a_{0}-a_{0}+a_{1}x-a_{1}x+a_{2}x^2 -a_{2}x^2$ all the terms cancel

Balarka Sen

I have to run for school now, cruds.

usukidoll

03:50

for identity that's just $a \cdot 1_{R}=a$\\
so
$a_{0}+a_{1}x+a_{2}x^2 \cdot 1 =a_{0}+a_{1}x+a_{2}x^2$

this isn't a subring because closure of multiplication just flops due to the higher exponents of $x^3$ and $x^4$

oh I see and then part b is a small version of a

Mike Miller

@BalarkaSen He was talking about a conversation I had with Andrew where we used initials instead of names and referenced a PDF we linked to each other via email.

Balarka Sen

Oh, I see the objection. I thought the mathematics sounded cryptic (which happens for me so often). Sorry about that, @Danu.

usukidoll

04:14

Hmm how does $f(x) = a_{0}$ be seen as a subring?
unless this fails under addition?

Closure under addition \\
$a_{0} +a'_{0}$\\
Closure under multiplication \\
$(a_{0})(a'_{0})$\\
Identity\\
$a_{0} \cdot 1 a_{0}$\\
Additive inverse \\
$a_{0}+(-a_{0}) = a_{0}-a_{0}=0$\\
Ok I can see the last two clearer than the first two :/

arctic tern

if you add two rational numbers, you get a rational number. same for multiplying. you know that @usukidoll

Balarka Sen

The HNQ skpetic.SE post is amazing

usukidoll

if I had a number x 1 I'll just have that number
ex. 2 x 1 = 2
then for additive inverse if I had a number +(-number) it goes to 0
2+(-2)=0
2-2=0
0=0
so addition will look something like
1+2=3
multiplication 1(2) = 2
so I'll just get a single number as the result without any variables.. . so for the closure under addition and multiplication I would just get something in the form$a_{0}$

I'm thinking that as long as it's in the form $a_{0}$ it would be a subring. If we end up with a variable x or above while doing the requirements for a subring then it isn't a subring

Dysanix Official

04:32

Hey guys, I have a math question I am unable to solve, could someone help me?

A studio sells photographs and prints. It costs $20 to purchase each photograph and it takes 2 hours to frame it. It costs $25 to purchase each print and it takes 5 hours to frame it. The store has at most $400 to spend and at most 60 hours to frame. The studio makes $30 on each photograph and $50 profit on each print.

>> Find the number that the studio should purchase to maximize profits.

usukidoll

something is wrong with your latex

Dysanix Official

Latex?

arctic tern

lol

we're using tools that turn latex code into math equations in the chatroom

the code is activated by putting dollar signs around it

but when you use the dollar sign as an actual dollar sign multiple times in a normal sentence, it turns all the text in between the signs into italics with no spaces

if you haven't seen "LaTeX in chat" and don't have it enabled, you do not experience this

Dysanix Official

Aahhh, should I resend the question with dollars converted to euros?

Lmao

A studio sells photographs and prints. It costs €20 to purchase each photograph and it takes 2 hours to frame it. It costs €25 to purchase each print and it takes 5 hours to frame it. The store has at most €400 to spend and at most 60 hours to frame. The studio makes €30 on each photograph and €50 profit on each print.

>> Find the number that the studio should purchase to maximize profits.

arctic tern

you can put ` around things to make it look like code: $. if you do this around dollar signs, it will negate the latex thing.

well, I think it will. test: $ 1+1 $ versus $ 1+1 $

yep, negates the effect

usukidoll

04:42

so.. if my answers are in the form $a_{0}$ then we have a subring?

arctic tern

"if my answers are in the form" is not what you say. what you say is "so... if my subset is all the polynomials which are constants then it's a subring?"

and yes

usukidoll

ok... :3

arctic tern

sorry if english isn't your first, not trying to be rude.

usukidoll

english is my first.. math_english is 2nd

lol

Dysanix Official

Can you guys help me then?

usukidoll

04:45

Since the subset is all the polynomials which are constants, we have a subring.

arctic tern

@DysanixOfficial ask on main, don't feel like learning about optimization enough to help you right before bed

Dysanix Official

/help

oops

I thought there might be commands

Where do I find main? I'm new to this chat heh

usukidoll

so the last one the set of polynomials
$f(x) = Q[x]$ all the coefficients are in the set of all integers
how do I apply the requirements of the subring into this?

that means that closure under add/multiply, identity, and additive inverse must be satisfied in f(x) = Q[x] with the coefficients in the set of integers

@DysanixOfficial post at math.stackexchange.com and use latex.

arctic tern

@DysanixOfficial math.stackexchange.com. click the logo in the lower right corner

usukidoll

I think accounts still get downvoted for not using latex :S

arctic tern

04:48

lol @ irc commands btw

usukidoll

chatstackexchange $ \neq $ irc

arctic tern

do they have bots for those in other chatrooms on the SE network?

Dysanix Official

Let me try some stuff I don't know how to work this but I just enabled latex

$ 1 + 1 $

usukidoll

might wanna search latex commands :S

Dysanix Official

But I am not sure which parts of my question to turn into latex

since it's kind of just a bunch of text

usukidoll

04:50

the numbers and math stuffz

pfft if $f(x) = Q[x]$ and $R = Q[x]$ then $f(x) = R$ WHATT

Dysanix Official

A studio sells photographs and prints. It costs € $ 20 $ to purchase each photograph and it takes $ 2 $ hours to frame it. It costs € $ 25 $ to purchase each print and it takes $ 5 $ hours to frame it. The store has at most € $ 400 $ to spend and at most $ 60 $ hours to frame. The studio makes € $ 30 $ on each photograph and € $ 50 $ profit on each print.
>> Find the number that the studio should purchase to maximize profits.

so like this?

I dont really see the point

lmfao

arctic tern

don't bother with latex for writing it on the mainsite, just make sure dollar signs don't screw things up

Dysanix Official

A studio sells photographs and prints. It costs $€$ $20$ to purchase each photograph and it takes $ 2 $ hours to frame it. It costs $€$ $25$ to purchase each print and it takes $ 5 $ hours to frame it. The store has at most $€$ $400$ to spend and at most $ 60 $ hours to frame. The studio makes $€$ $30$ on each photograph and $€$ $50$ profit on each print.
>> Find the number that the studio should purchase to maximize profits.

arctic tern

latex is for more complicated equations that would give people headaches to read in ascii

Dysanix Official

I latex'ed the euros

Yeah I was about to say

what the hell

usukidoll

04:55

usually with just $2 I just type like that w/0 fancy symbols

it's x^2 that gives headaches
$x^2$

prntscr.com/cp5exw now if only I can apply the requirements of the subring in here . There's like no polynomial... it's just says the f(x) is in Q[X] where the coefficients are in the set of integers

Dysanix Official

0

Q: Find the number that the studio should purchase to maximize profits

A studio sells photographs and prints. It costs $€$ $20$ to purchase each photograph and it takes $ 2 $ hours to frame it. It costs $€$ $25$ to purchase each print and it takes $ 5 $ hours to frame it. The store has at most $€$ $400$ to spend and at most $ 60 $ hours to frame. The studio m...

inequality

I did it! Thanks guys

usukidoll

woohoo

Jessy Cat

Does there exist a nonempty, bounded subset of the Euclidean plane $\mathbb{E}^{2}$ with an isometry group of order 5?

user228700

Hi everyone :-)

usukidoll

hi

user228700

05:03

I suspect that I may have asked this before but I'm not quite sure if anybody replied...

user228700

If the discriminant of a quadratic equation in $x$ is greater than 0, does that imply that the expression gives a positive number for every real $x$?

user228700

I need this piece of information to solve inequalities of the form:

user228700

$x^2-12x+30>0$, for real $x$.

arctic tern

@JessyCat sure, draw rays from the origin to split the plane into five parts, place an asymmetric blob in one of the parts, then take its rotated image in the other four parts and take the union

Jessy Cat

I was going to say, "what's an aymmetirc blob"

blog

arctic tern

05:09

@KaumudiHarikumar no, a discriminant of x greater than 0 implies it has two real roots, which means the parabola intersects the x-axis twice, which means the x-axis cuts the parabola into a top part and a lower part, which means the quadratic function takes both positive and negative values

user228700

@arctictern Damn; I know this but I was hoping that there is some piece of information that I've missed :/

arctic tern

@KaumudiHarikumar x^2-12x+30 = (x-6)^2 - 6

user228700

Then how does one solve inequalities of the above kind, where the quadratic expression in $x$ is not factorizable..?

arctic tern

completing the square

Jessy Cat

@arctictern, how would I rigorously say that?

arctic tern

05:11

rigorously say the equality I just said? it's a direct equality, it's already rigorous.

Jessy Cat

@arctictern er...the thing about the five parts, aymmetric blog, etc.

arctic tern

sorry, mixed you up with Kau @Jessy

user228700

@arctictern OK. Is there any rule to follow when taking square root on both sides of an inequality..?

Jessy Cat

@arctictern I know

arctic tern

@JessyCat well, you'll have to figure out what asymmetric figure you want to use

(if c is nonnegative and u is an algebraic expression):
u^2>c is equivalent to (u>sqrt(c) or u<-sqrt(c))
u^2<c is equivalent to (u<sqrt(c) and u>-sqrt(c))

user228700

05:13

Wait, no, never mind. I think I got it. Thank you! :-)

Jessy Cat

@arctictern, how about a bean shape tilted to the left a bit?

arctic tern

sure.

user228700

@arctictern Oh, crap. Sorry that I made you type all that :/ Thanks! :-D

arctic tern

wasn't much to type. have it basically memorized since I have to teach it to hundreds of students.

user228700

@arctictern Oh, OK then. U're a teacher/professor?

arctic tern

05:15

student teacher, yeah

user228700

@arctictern Cool :-)

Jessy Cat

@arctictern, and I guess rotate it by 360/5 degrees?

arctic tern

yes

well, revolve it around the origin

Jessy Cat

And that's an isometry group?

arctic tern

no, it's a subset of E^2.

its isometry group is the group of five rotations

Jessy Cat

05:19

So the way the question is worded is "Does there exist a nonempty bounded subset of the Euclidean plane $\mathbb{E}^{2}$ with the isometry group of order 5?"

Now, I'm assuming the reason my prof wrote "the isometry group" rather than "an isometry group" is a lack of proficiency with the English language.

Unless there is something called "the isometry group of order 5"...

arctic tern

nah, you're right

Jessy Cat

But if you could construct one so easily like that, why would he be asking us this? Do there not exist some nonempty bounded subsets of the Euclidean plane with isometry groups of certain orders?

arctic tern

no, you can do the same thing with any order

Jessy Cat

Then, what's the deal with the question?

arctic tern

you say how easy the construction is, yet you came here to ask about the question, hence the question was nontrivial for you

sometimes solutions that are simple and easy to understand still require time and enlightenment to find in the first place

Jessy Cat

05:25

I suppose.

Out of curiousity, what is the appropriate word for quadrants, but when there are 5 of them. quintents?

arctic tern

shrug

Jessy Cat

Well, thank you for all your help, @arctictern

arctic tern

mmhmm

user116211

08:07

Why should the last term of $\cos n\theta$ is $(-1)^{\frac{n-1}{2}}\binom n{n-1}\cos\theta\sin^{n-1}\theta$ when $n$ is odd; and $(-1)^\frac n2 \sin^n\theta$ when $n$ is even?

user116211

Not getting that.

Tobias Kildetoft

@MAFIA36790 What do you mean the last term?

user116211

@TobiasKildetoft I mean the last term in the expansion of $\cos n\theta$ using De Moivre's Theorem: $$\cos n\theta= \cos^n\theta - \binom n2 \cos^{n-2}\theta\sin^2\theta+ \binom n4\cos^{n-4}\theta\sin^4\theta - \ldots$$

Szabolcs

08:20

Hello everyone

I am looking for ideas for showing off some (computational) graph theory package.

user116211

Any insight on this; how to approach, @TobiasKildetoft?

Szabolcs

I am trying to find problems that can be mapped to subgraph finding (induced or otherwise) or coloured graph isomorphism. I am looking for problems that would be interesting for anyone with some mathematical aptitude, even if they don't know graph theory. E.g. engineers.

A nice example would be my solution to the soma cube posted here (scroll down), based on mapping it to a clique finding problem.

usukidoll

08:52

any abstract algebra people?

Tobias Kildetoft

@usukidoll I am an algebra person. But I am fairly concrete (I mean, I exist in the real world and everything).

usukidoll

-.- just c prntscr.com/cp7cl6 something to jump start besides the function is in Q[x] and coefficents are in the set of integers

user116211

@tobias, got an answer; let me read...

usukidoll

how do I put the def. of the subring in here?!

Tobias Kildetoft

@usukidoll You do not put the definition anywhere. You check whether this thing satisfies the definition

usukidoll

08:59

what the?!

so I have to check whether $f(x) = Q[x]$ satisfies the subring definiton?

Tobias Kildetoft

no

usukidoll

:/

Tobias Kildetoft

You have to check whether the given subset of $\mathbb{Q}[x]$ satisfies the subring definition

usukidoll

rational ring?!

Tobias Kildetoft

you are given a subset of $\mathbb{Q}[x]$ and you are asked whether it is a subring. So you check whether is satisfies the definition.

usukidoll

09:01

$Q [a+b]$ $Q[ab]$ $Q[x] \cdot 1 = Q[x]$ $Q[x]+(-Q[x])=0$?!

Tobias Kildetoft

none of those things make sense

usukidoll

I know x.x

could you give me an example of how to determine if a subset is a subring

I know the requirements... I somehow can't apply this x!x

Tobias Kildetoft

So given two polynomials of degree $3$ or less, what can be the degree of their sum?

usukidoll

is the polynomial like $ax^2+bx+c$?

Tobias Kildetoft

They are just some polynomials with degree at most $3$, we don't care what the coefficients are labelled

usukidoll

09:07

wouldn't taking the sum be?
$ax^2+bx+c +dx^2+ex+f = x^2(a+d)+x(b+e)+c+f$

I feel like I'm doing a again.. it's not a subring because under multiplication we end up with $x^4$

P Vanchinathan

11:05

Is the terminology "cyclic group* appropriate the the group of integers?

Tobias Kildetoft

@PVanchinathan Yes, since the term is defined in such a way as to include the integers

Balarka Sen

Hi @Tobias

Tobias Kildetoft

@BalarkaSen Hi

P Vanchinathan

Take an intege $m$, the set $m+m, m+m+m,\ldots$ never repeats itself. Whereas for a root of unity $\zeta$ the seta $\zeta^2, \zeta^3,\ldots$ repats itself exhibiting cyclicity (periodicity)

Tobias Kildetoft

@PVanchinathan Why is the root of unity thing relevant? The definition of cyclic is that it is generated by a single element.

P Vanchinathan

11:12

That is precisely my objection: abuse of the word "cyclicity" for being generated by a single element.

Tobias Kildetoft

I don't see the issue

P Vanchinathan

The seasons change cyclically means, if summer ends now it will come again. Price changes are cyclic means the price that has gone up will come back. If a mechanical system goes through many stages cyclically means it returns to the first stage and so on. The word "cyclic" has the same meaning in English across so many technical disciplines. Whereas in algebra it has been assigned to mean something else.

Balarka Sen

@Tobias How goes it?

Tobias Kildetoft

@BalarkaSen Good. Trying to do some calculations in type $B_6$, but they are taking a long time

Balarka Sen

@PVanchinathan It's an "infinite" version of cyclicity.

Hence, "infinite cyclic".

P Vanchinathan

11:18

Infinite cyclic is as illuminating as rational irrational number.

Tobias Kildetoft

@PVanchinathan except one makes sense, the other does not

requiring that everyday words have their intuitive meaning in math would just mean that we had to invent a new word for every single thing we want to define

Balarka Sen

@TobiasKildetoft Nice

P Vanchinathan

The only algebraic property common between the groups Z/n and Z is being monogenic, not the cyclic property

Tobias Kildetoft

@PVanchinathan monogenic is not a mathematical term (at least not one people use)

and they are all cyclic, as per the definition of cyclic

Also, all cyclic groups share a very nice property: Maps from them are uniquely determined by the value on a specified element. This is a common feature.

Balarka Sen

This sounds like a nonmathematical issue you're having anyway. It's pretty clear why one says Z is cyclic (Z/n's are all generated by a single element and vice versa restricted to finite groups, and Z is the only infinite group satisfying that so why not include that too), so I don't think there's any point being pedantic about this terminology.

P Vanchinathan

11:22

I am aware that monogenic is not the word people use. I am very much aware that Z/n and Z should be part of one single family deserving a single name. The question is what is the most appropriate name?

Tobias Kildetoft

@PVanchinathan the most appropriate name is the one everyone already uses.

Might as well argue that we are putting too many requirements on being a group, since a everyday terms, a group is just a collection of objects, so all sets should be groups.

Balarka Sen

Terminology is the one which is the most efficient, not necessarily the most accurate or appropriate. Everyone uses infinite cyclic to denote Z, and everyone always ever will

Doesn't matter if you come up with a new terminology

P Vanchinathan

I agree that a word widely used for many years gains acceptance. Of course miles and feet were used in Physics books, but they started using metres and kilograms

Tobias Kildetoft

@PVanchinathan that is not a change in terminology. That is a change is choice of unit

It does happen that terminology changes, but it takes more than "it doesn't correspond to people's intuition". Usually it takes overlap between things.

Balarka Sen

Good luck trying to bring revolution to the terminology of cyclic groups.

P Vanchinathan

11:32

@BalarkaSen: I have no illusion that I will get into bringin such revolution. But this awareness in the general meaning and technical meaning has helped me as a teacher when a student gets coloured by the usual meaning of the word cyclic and struggles.

Balarka Sen

That's fine. You asked whether calling Z cyclic is appropriate, we answered (which are of course entirely our opinions). If you want to give it a new name when introducing them to students and if you think that will help them, for sure. But I think, since most texts in group theory use that terminology, they will face trouble in the future. But again, feel free to teach however you want to.

paul23

11:50

Hey, I got a question about mathjax: how can I make function "split" into multiple lines (ie to describe F(x) = {1 if x < 0 \n 0 if x > 0 )

Tobias Kildetoft

@paul23 I think mathjax supports \cases

Test : $$F(x) = \begin{cases} 1 & blah \\ 2 & blah\end{cases}$$

so \begin{cases}

paul23

kk thanks

Lozansky

what do you guys do if your professor doesn't recommend exercises from the textbook?

SteamyRoot

Does he disrecommend them?

Or recommend other exercises?

paul23

Nothing? What you mean actually: he says the exercises are bad, or he just doesn't give advice on what to do?

Lozansky

12:02

no he doesn't have any recommended exercises is all

doesnt give advice

i just know what chapters to go through

Tobias Kildetoft

@Lozansky Did you ask for recommendations?

Lozansky

it

paul23

Well that's actually up to the university(section) rules: some enforce these, others don't.

SteamyRoot

If the textbook is something you follow during lectures, then most likely the exercises within will supplement the course well

paul23

But at the basics a prof only has to tell you what topics he wishes you to learn: in later years (masters) you don't even get textbooks anymore.

Lozansky

12:04

exercises within? we just go through the theory during lectures

paul23

You have to take your own initiative.

SteamyRoot

Well, but the theory is from a textbook, and that textbook has exercises?

Or am I misunderstanding this?

Lozansky

it does but there are like hundreds of exercises per chapter

so which ones do i choose to do?

i could just do the ones i do not understand

Tobias Kildetoft

@Lozansky Did you in fact ask for recommendations?

paul23

Just skim & pick them based on what looks difficult?

Lozansky

12:05

but even that'd be really time consuming

SteamyRoot

Usually they get more difficult the further you're in. Make an easy one, see how difficult you found it

If it was way too easy, skip like 10-20 exercises and make another one

Lozansky

i can try but when i asked him when we would get our mid terms exams back he just said "you'll get then when you get them"

paul23

Btw what field of mathematics (I take it is this) is the problem?

Lozansky

complex analysis

SteamyRoot

Easy -> Skip a bunch; Hard but doable -> make more around this one; Way too hard -> Return a bit

That was my strategy

Lozansky

12:06

yeah that sounds reasonable

Tobias Kildetoft

@Lozansky Why would that be in any way related to this? Nobody likes to be pinned down to havig to correct exams within some time frame

paul23

Also old examns & ask older students; there are always facebook groups

SteamyRoot

Or, if the textbook is well-known: some professors post homework or exercise recommendations online

Lozansky

@TobiasKildetoft Maybe you are right, I can at least shoot him an email and ask I guess

I just really prefer when you have a list of recommended exercises so you know exactly what to do

paul23

@SteamyRoot TBF: given his topic is "complex analysis" I doubt it's a well known textbook: most universities have complex numbers as part of the linear algebra or calculus courses.

What textbook you use?

Lozansky

12:08

Otherwise it just feels overwhelming

Fundamentals of complex analysis 3rd edition by Saff & Snider

SteamyRoot

I used "Function theory of one complex variable" by Greene & Krantz

Lozansky

This is more applied to engineering I think

paul23

Haven't heard of it, sorry. (Though as said, all studies I know simply use stewart's calculus or one of hte linear algebra books for it).

Sawarnik

@BalarkaSen You changed your dp?

paul23

The integral of a probability density function over the full domain has to be "1" right?

Parth Kohli

12:25

Yeah.

sourav

12:58

anyone please have a look math.stackexchange.com/questions/1951889/…

user228700

Hi everyone :-) Quick question: a quadratic equation have unreal/complex coefficients, no..?

Starfall

@KaumudiHarikumar Sure.

user228700

@Starfall OK, thanks!

Lozansky

13:22

How do you show that $e^x(cosy+isiny)$ maps $\mathbf{C}\\ \{ 0 \}$ for $x,y \in \mathbf{R}$?

user228700

Also, it's given that any quadratic equation can be written in the form $a(x-\alpha)(x-\beta)=0$ where $\alpha$ and $\beta$ are the roots of this equation. Is this somehow a consequence of the factor theorem or is it directly derivable?

Lozansky

Did I just break the chat...

Starfall

@KaumudiHarikumar It is a consequence of the fact that your ring is factorial.

Granted, it is true generally that if $ P(\alpha) = 0 $ for a polynomial $ P $, then $ X - \alpha $ divides $ P $, regardless of whether the polynomial ring is factorial or not.

Lozansky

How do you show that $e^x(cosy+isiny)$ maps $\mathbf{C}\setminus\{0\}$ for $x,y \in \mathbf{R}$

Does it render correctly for anyone?

Starfall

What does it mean to "map $ \mathbf C \\ \{ 0 \} $"...?

oh, well

Lozansky

13:29

Its supposed to say $\mathbf{C}$ without $0$

Starfall

What does it mean, though?

Lozansky

All of C with the exception of the point 0

Starfall

I don't see how "some function mapping some set" means anything sensible.

That is not what I am asking.

What exactly do you want to prove about this function?

Lozansky

Ok how do you show that the function $f(z) = e^x(cosy+isiny)$ has the range stated above?

Does that make sense?

Starfall

Yes, that makes more sense.

Clearly, that function cannot map to zero, since its norm is always positive.

Lozansky

13:32

Yeah

Starfall

Conversely, you have to show that it does map to everything else, and to do this you note that you can write any $ z = a + bi $ as $ z = |z|(a/|z| + b/|z| i) $, with $ (a/|z|, b/|z|) $ being a point on the unit circle.

Since $ \sin $ and $ \cos $ parametrize the unit circle, you see that said point must be $ (\cos(y), \sin(y)) $ for some $ y $, and then you take $ x = \log(|z|) $, which makes sense because $ |z| > 0 $.

user228700

@Starfall I have no idea what "polynomial ring" is, sorry :-P

Lozansky

Oh okay, so it's like $(a/|z| + b/|z|i$ has the unit circle as its range and then $|z|$ is sort of a scale factor so we can describe any circle in the plane?

Starfall

@KaumudiHarikumar A factorial ring is a ring which behaves like the integers in its prime factorization - for example, if I tell you that an integer is divisible by both $ 5 $ and $ 7 $, you could deduce that it is divisible by $ 35 $, correct?

Likewise, if I tell you that a polynomial is divisible by $ X - \alpha $ and $ X - \beta $, you can deduce that it is divisible by $ (X - \alpha)(X - \beta) $.

@Lozansky Yes.

Lozansky

Awesome, thanks

user228700

13:55

@Starfall Ohh, okay...

user228700

Thanks sir/ma'am :-)

Daniel Cortild

14:59

Hello everyone

Sir Cumference

Say, anyone here good at computer science?

Daniel Cortild

Not really... Anyone good at inequalities/number theory? @SirCumference, what is the question?

Starfall

@DanielCortild Ask your question.

If nobody here can answer it, you can always ask it as a post on the main site.

Daniel Cortild

I have to prove that $\dfrac{a_1}{1^2}+\dfrac{a_2}{2^2}+\cdots+\dfrac{a_n}{n^2}\geq\dfrac{1}{1}+\frac {1}{2}+\dfrac{1}{3}+\cdots+\dfrac{1}{n}$ for $a_i$ whole different integers...

And the $a_i$ have to be positive!

Or with the sum sign, $\sum_{k=1}^n\frac{a_k}{k^2} \geq \sum_{k=1}^n\frac{1}{k}.$

Sir Cumference

@DanielCortild This.

127

Q: Super Mario Galaxy problem

Suppose Mario is walking on the surface of a planet. If he starts walking from a known location, in a fixed direction, for a predetermined distance, how quickly can we determine where he will stop? More formally, suppose we are given a convex polytope $P$ in 3-space, a starting point $s$ on t...

ds.algorithms cg.comp-geom

Daniel Cortild

15:14

Yea.... So you know where he starts and how long he is walking... So the problem is that the planet is round or what?

Sir Cumference

@DanielCortild We want to know how quickly we can determine which facet of $P$ Mario will stop inside.

Daniel Cortild

Ohh ok... Sorry I don't think I can help you a lot...

Sir Cumference

All right, thanks

Starfall

@DanielCortild Are you trying to prove that this holds for any positive sequence $ a_i $?

Oh, my bad, you said "different".

In that case, you can use the rearrangement inequality.

Daniel Cortild

15:30

@Starfall But how do I come to that inequality? I know the rearrengement inequality, but how do I use it?

Starfall

Given any choice of $ a_i $, you note that the minimum value will be attained when the $ a_i $ are in ascending order.

After that, since $ a_i $ is an increasing sequence of distinct positive integers, you can say that $ a_i \geq i $ for all $ i $.

Daniel Cortild

Ohh I see!!

Because the minimum of the RHS will be when $a_i={1,2,3,...,n}$ and after simplification, it is the same as the RHS! So it is always true and with equality iff $a_i={1,2,3,...,n}$

Does that seem correct?

Starfall

That is correct.

Daniel Cortild

But how do I prove that the minimum is ${1,2,3,4,...,n}$ and not any bigger sequence?

Starfall

Read what I said.

I said something about how big a term of your sequence has to be to satisfy the condition of being an increasing sequence of positive integers.

Daniel Cortild

15:36

Ohh yea... Bigger or equal, and equality of the original equation occurs when all $a_i=i$?

Starfall

Yes.

Daniel Cortild

Ohh thank you soo much!!

I have another problem; Find all functions that satisfie $x \cdot f\left(\frac{x}{2}\right) - f\left(\frac{2}{x}\right) = 1 $

Semiclassical

morning chat

Starfall

@DanielCortild Try substituting $ x \to 4/x $ and see what happens.

Semiclassical

@TedShifrin Got more questions for you when you have a chance

Daniel Cortild

15:46

@Starfall Well that gives me $(x+1)f(\frac{x}{2})=\frac{4+x}{x}f(\frac{2}{x})$

Semiclassical

what happened to the 1 on the RHS?

Starfall

@DanielCortild Take a sheet of paper, write the functional equation you are given, and directly below it, write what you get when you substitute $ x \to 4/x $ into that equation.

After that, take a good look at your sheet of paper.

You should notice something.

Daniel Cortild

@Semiclassical By substituting $x->4/x$, I get $\frac{4}{x}f(\frac{2}{x})-f(\frac{x}{2})=1$ So I can get the equality proposed earlier

Semiclassical

the equation you just said (which is valid) isn't equivalent to what you said earlier.

Daniel Cortild

@Starfall What sould I notice? I just see what I mentioned earlier, but not more...

Starfall

15:50

@DanielCortild Did you do what I told you to do?

Daniel Cortild

Yes I wrote it down but don't see any link

Semiclassical

@DanielCortild it'll help, though, to multiply both sides of this by $-x$

Starfall

You wrote both equations down, correct?

And you still do not see anything?

Daniel Cortild

Yep

Well they are boith equal to 1

Starfall

@DanielCortild You should be looking at a system of two linear equations with two unknowns.

If that is not what you are looking at, something probably went wrong somewhere.

Daniel Cortild

15:52

But can't I say $\frac{4}{x}f(\frac{2}{x})-f(\frac{x}{2})=1=xf(\frac{x}{2})-f(\frac{2}{x})$?

Semiclassical

being able to and it being the most useful thing to do are different things.

Daniel Cortild

Ohh... So there is some more efficient thing to do?

Semiclassical

you've got both $f(x/2)$ and $f(2/x)$ in that equation. it'd be better to come up with an equation that only contains one or the other.

Daniel Cortild

I will think about it, comming back later...

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