Mathematics - 2012-05-04 (page 6 of 7)

Matt N.

20:01

But... do you think this makes my hint incorrect?

Martin Sleziak

afk brb

Matt N.

Meh. Fed up. I need a drink.

Gigili

Oh @Peter, I think you should use $v^2 - v_0^2 = 2 a \Delta x$

Peter Tamaroff

@Gigili In the problem? I'll try that.

Gigili

What's the initial velocity of the first block? 0?

Peter Tamaroff

20:05

@Gigili Yes, but that of the other is $2 m/s$

Gigili

It's $v^2-4 = 2a \Delta x$ for the second one

@PeterTamaroff Yes, that.

So I think you can describe it according to that formula.

Martin Sleziak

@MattN Well, I tried to think how I would avoid that problem and the only idea I've got was using decimal expansion.

Matt N.

@MartinSleziak I meant, if it's a hint it doesn't have to point out the problem he runs into.

Peter Tamaroff

@Gigili The damn proble is the curved axis! The professor didn't explain it very well, he just said "We should take this curved axis". It seems it has to do with Lagrangian mechanics, which I know nothing about. I can understand the idea intuitively but we never finished a problem like this yet, so some things are pretty unclear.

Martin Sleziak

@MattN I agree that it covers the basic idea.

BTW I was about to post similar comment, but I saw that Arturo had pointed it out already.

Matt N.

20:09

@MartinSleziak Good. Doesn't change the fact that I wasn't aware of something I actually knew. : ( Cheers!

Martin Sleziak

Cheers! I've opened a bottle of non-alcoholic beer, too.

Matt N.

No, I'm on wine today. Need the real deal to comfort me over writing this "hint". Haven't had drink in weeks.

Peter Tamaroff

@MartinSleziak I'll toast with coffee.

Sorry if I'm being pedantic, but any help here is appreciated. Anyone here knows physics? (dynamis and cinematic)

Martin Sleziak

Isn't knowing that the vectors |T|=|T'| sufficient to solve it?

Mariano Suárez-Alvarez

@PeterTamaroff, a midterm

Matt N.

20:18

Random fact of the day: an average Swiss supermarket chain offers you a choice of more than 13 different sorts of milk.

shrug

Martin Sleziak

@MartinSleziak I think this is basically the first equation in the solution which is posted there.

N3buchadnezzar

milk

David K.

Can someone help me with a rather simple question regarding automorphisms of cyclic groups? Suppose I have a cyclic group $C_{2}$ with 2 elements and a cyclic group $C_{4}$ with 4 elements. Is it true that $C_{2}\cong\operatorname{Aut}(C_{3})$ and that $C_{4}\cong\operatorname{Aut}(C_{5})$ ?

Martin Sleziak

Number of automorphisms of a cyclic group is exactly the number of generators.

Since a homomorphism is uniquely determined by the image of generator, isomorphism must map generator to generator.

$C_3$ has 2 generators, $C_5$ has 4 generators.

So $\operatorname{Aut}(C_5)$ has 4 elements. If it has a generator, it is cyclic.

It suffices to check whether you have an element of order 4 in $\operatorname{Aut}(C_5)$.

But I guess there's an easier way - maybe someone else will write a different solution.

I hope I did not miss some problem there.

@PeterTamaroff Ignore my comment about |T|=|T'|, that was non-sense. (It would be true only if the system were in equilibirum.

Mariano Suárez-Alvarez

the sentence «Since a homomorphism is uniquely determined by the image of generator, isomorphism must map generator to generator.» is wrong :)

Martin Sleziak

20:27

ok

what I meant was this

Peter Tamaroff

@MartinSleziak It is the case. The rope and pulleys are ideal

Mariano Suárez-Alvarez

you really want to say two things: an homomomprhism is determined by the image of a generator, and a automorphism must map a generator to a generator

there is no since involved :)

you do need to check, additionally, that a generator can be mapped to any other generator for your claim to be completely established

Martin Sleziak

I meant to say: We know that number of automorphisms = number of generators, if we use these two facts: 1. A homomorphism is uniquely deter.... 2. Isomorphism maps....

Mariano Suárez-Alvarez

you need to add the third point to that list

Martin Sleziak

You're right of course.

@DavidK Did the above analysis help? (Now after Mariano's corrections.)

Peter Tamaroff

20:31

@MarianoSuárezAlvarez Do you happen to know where I can find examples of examenes libres for analysis and algebra in the CBC?

Mariano Suárez-Alvarez

no idea

luckily, I have absolutely no contact with CBC :)

David K.

@MartinSleziak @MarianoSuárezAlvarez Yes. Thank you very much.

Peter Tamaroff

@MarianoSuárezAlvarez Ha! I had some bad impressions of one professor that insists on the $\lim = \infty$ thing. But in general, I'm enjoying the other professor's expositions and rowing through the other's classes.

Matt N.

And we need toothpaste...

Peter Tamaroff

I'm really interested in topology now, rather than in the $\epsilon$-$\delta$ approach in analysis.

@MattN For what¿

Mariano Suárez-Alvarez

20:34

well, you need to know both things

it is not like "topology" can magically eliminate the need to do $\epsilon$-$\delta$

Matt N.

@PeterTamaroff Too long to explain : ) Just ignore. (to brush our teeth)

Mariano Suárez-Alvarez

when you want to actually show that something is contimuous, or compute a limitm or what not, you need to descend to that

Peter Tamaroff

@MarianoSuárezAlvarez Not even with the ngbhs and balls?

Mariano Suárez-Alvarez

no

that is just convenient language

David K.

Ok so now I have a slightly more complex question that arises from my last one. I have now established that $$G=\operatorname{Gal}(\Bbb{Q}(\zeta_{13\cdot 7})/\operatorname{Gal}(\Bbb{Q}))$$ contains the normal subgroup $$H=\operatorname{Gal}(\Bbb{Q}(\zeta_{3\cdot 5})/\Bbb{Q}).$$ Is there anything I can say about $G/H$ ? That is, I know that $G/H\cong C_{3}\times C_{3}$, but I'm wondering if it is possible to write $G/H$ as something more simple.

Peter Tamaroff

20:36

@MarianoSuárezAlvarez I see. But the thing that interests me in Topology is how it creates mathematical structures and analyses them...

Mariano Suárez-Alvarez

that sentence does not mean much to me, honestly :D

Peter Tamaroff

@MarianoSuárezAlvarez in what sense¿

Between the little i know about topology and the english, maybe I cant transmit my idea.

Mariano Suárez-Alvarez

in that "mathematical structure" is too vague a term; I have no idea what "creating a mathematical structure" means; and I don't know how topology analyses anything :)

your English es quite fine!

Peter Tamaroff

I mean: metric spaces, and a distance function. It creates an order between points or objects. It embeds a set with a structure.

I don't know if it is really so simple, but it is my first impression.

Mariano Suárez-Alvarez

don't try to go too fast

Peter Tamaroff

20:40

@MarianoSuárezAlvarez Yes, I have some IGCSE diplomas of my own, but english limits me sometimes.

Mariano Suárez-Alvarez

you are able to communicate just fine

:)

Matt N.

Looks as if Asaf is bored again.

Peter Tamaroff

@MarianoSuárezAlvarez Yes, I know. What I mean is that sometimes, with our mothertounge, we can be vaguer and transmit the same idea, use less words, more idioms....

Mariano Suárez-Alvarez

well, do not aim to be vaguer

it never helps :)

be precise

as Serre likes to say: «precise yet informal»

you'd be hard pressed to find many idioms in his writing though :)

Peter Tamaroff

Who is Serre?

Mariano Suárez-Alvarez

20:43

oh

kids these days!!

t.b.

:)

Mariano Suárez-Alvarez

:D

Jean-Pierre Serre

Jean-Pierre Serre (born 15 September 1926) is a French mathematician. He has made fundamental contributions to the fields of algebraic geometry, number theory, and topology. Biography Early years Born in Bages, Pyrénées-Orientales, France, to pharmacist parents, Serre was educated at the Lycée de Nîmes and then from 1945 to 1948 at the École Normale Supérieure in Paris. He was awarded his doctorate from the Sorbonne in 1951. From 1948 to 1954 he held positions at the Centre National de la Recherche Scientifique in Paris. In 1956 he was elected professor at the Collège de France, a positi...

kahen

Try looking at the video "How to write mathematics badly" if you want to see Serre give a useful and very accesible talk: modular.math.washington.edu/edu/basic/serre :)

Jordan

I am back and I want attention

Peter Tamaroff

@Jordan Is that a serious comment?

Jordan

20:45

nope

Peter Tamaroff

@MarianoSuárezAlvarez Oh I see. Now I would love to major in number theory and topology. (Let alone make fundamental contribution)

@Jordan Then go back and study man. You really need to.

Mariano Suárez-Alvarez

when I said don't try to go too fast, that included: do not pick a subject before you know a few

Jordan

I am studying

Peter Tamaroff

@Jordan You're chatting.

Jordan

I need an A on my final or I can't transfer

Yeah, well I am eating now while I reaad Arturo's comment

Mariano Suárez-Alvarez

20:47

it is clear that you are either unsuccessfully trying to be funny or trolling, @Jordan

Peter Tamaroff

@MarianoSuárezAlvarez Wha would you reccommend to have clear before reading this?

t.b.

@PeterTamaroff You're 17, right? You've got amply enough time before you decide even on the rough direction of your studies. Be open-minded, everything else would be a pity.

Mariano Suárez-Alvarez

I would recoment not reading that :D

Peter Tamaroff

@tb I'm 18

Mariano Suárez-Alvarez

18-17 = epsilon

t.b.

20:48

@PeterTamaroff Oh, sorry. Doesn't change the gist of my comment :)

Gigili

@PeterTamaroff I didn't notice your question got an answer.

Wow, 18?

Peter Tamaroff

@MarianoSuárezAlvarez But it starts with sets and subsets, set operations, then indexed families, then products of sets, functions, relations, composition and diagrams, inverses, extensions restrictions and then arbitrary products. It doesn't seem so hard. =(

@Gigili Yes, why?

@tb That is great advice.

Gigili

I thought you were 29, if not 30.

Mariano Suárez-Alvarez

a good part of most general topology books is more or less pointless

Jordan

@MarianoSuárezAlvarez why?

Matt N.

20:50

@PeterTamaroff It's a great book. I read it cover to cover. It's very nice to read.

Peter Tamaroff

@Gigili I don't know if I should be sad or happy of that. Why did you think I was older?

@MattN I feel the same about it. It is really clear.

Mariano Suárez-Alvarez

@Jordan, because there is no possible way a sensible person would insist with «I need an A on my final or I can't transfer» adter having been told repeatedly to stop that line of conversation

many times

Matt N.

@PeterTamaroff Oh, so you've already read it? I thought you were asking what you should know before you read it.

Peter Tamaroff

@MattN Not really. I'm scanning it.

But not properly reading or studying from it.

Gigili

@PeterTamaroff The way you talk and such, or perhaps it was your T-Ps. I'd assume that everyone is older than me!

Matt N.

20:53

@Gigili Tee-Pees?

t.b.

Theorems and Proofs...

Jordan

@MarianoSuárezAlvarez Well it is true, plus it is motivation for me to study

Gigili

@MattN Theorems-proofs, I guess

Matt N.

: )

Gigili

Umm, what Tee-Bee said.

Mariano Suárez-Alvarez

20:53

there are all sort of true things that are offtopic here

I will not continue this conversation with you

Gigili

What's your reference book for your exam @Peter?

Mariano Suárez-Alvarez

because we have been trhough this many times before

by now, you should have gotten the point

Peter Tamaroff

@Gigili You mean physics? We have a booklet, but I don't know if we have a book reference.

Gigili

There were several problems like the one you asked in my book, IIRC. It was many years ago.

Peter Tamaroff

I have Halliday Resnik Crane, but that is overkill for sure.

Gigili

20:56

I can take a look at it and see if I find a solution, if you want.

Peter Tamaroff

@Gigili That would be nice.

However, I mostly have everything clear

Matt N.

@Gigili Tee-Bee.

Peter Tamaroff

I'm just missing how to represent $x(t)$. Tomorrow I'm studying with some friends, so maybe I'll clear it out. Let's keep physcs out for now!

@MattN They are talking about these TPs

Gigili

@PeterTamaroff O-kay.

Peter Tamaroff

@Gigili BTW; how old are you?

Gigili

20:58

25.

Matt N.

@PeterTamaroff I see : )

Peter Tamaroff

Is Dedekinds cuts a modern topic? I mean, is it studied nowadays?

Gigili

I'm off to bed, good night.

t.b.

@Peter: I would really recommend that you get a firm grasp of (linear) algebra and real analysis first before you seriously endeavor entering general topology. You'll encounter enough point-set topology while doing that to cover all your needs for a long time. (I'm not saying you shouldn't do topology, but don't waste too much time with it right now).

Dylan Moreland

I agree with@tb. I spent a fair amount of time reading Munkres and so on and it wasn't useless, but it's not a secret weapon like some other things are.

Matt N.

21:02

@Gigili Sleep well!

Peter Tamaroff

@Gigili It is only 6PM here. What time is it there?

@tb I see. Good to know.

Matt N.

Around 11 pm.

Dylan Moreland

On the other hand, it isn't that hard.

t.b.

@PeterTamaroff I think it's long past midnight where she is.

Matt N.

Never mind : )

t.b.

21:05

@MattN I don't actually know.

Dylan Moreland

The stuff that is hard (the freaky counterexamples and so on), I've never really had any use for.

Obviously this is just an opinion.

kahen

Well... you kind of have to have them. Otherwise you might think that regular and completely regular is the same

t.b.

99% of mathematicians don't need to know there's a difference :)

(I'm estimating)

Peter Tamaroff

@kahen What are you talking about?

Dylan Moreland

I guess I'm not often in that position, is what I'm saying.

kahen

21:07

But of course one can always look it up in Steen & Seebach

Matt N.

@JM Why?

Peter Tamaroff

@MattN To get a new Noether?

Dylan Moreland

A lot of my spaces are in the Zariski topology anyway, so regular is usually out.

t.b.

I don't recall ever seriously working with any beast between $T_2$ and $T_{3\, 1/2}$ and most of the time $> T_4$ is granted for me.

Mariano Suárez-Alvarez

Peter, topology

t.b.

21:11

(I like the examples, though)

Hey, @leo: did you manage to convince Asaf of the measure-theoretic proof of uncountability of the reals? :)

Mariano Suárez-Alvarez

T_{3\tfrac12} is a more common way to write it :)

t.b.

Yeah, I was too lazy. Sorry

Mariano Suárez-Alvarez

you added the \, though!

:D

kahen

I'd do \usepackage{xspace}\def\tychonoff{\ensuremath{T_{3\tfrac12}}\xspace}

Peter Tamaroff

@MarianoSuárezAlvarez All this conversation is out my league. I'll go to main for a while.

Mariano Suárez-Alvarez

21:14

the many mysteries of separation properties!

don't worry, it'll get to you eventually, Peter :D

Matt N.

@JonasTeuwen Ello?

kahen

There are also the really crazy ones that don't show up in Munkres. Such as $T_{2\tfrac12}$. en.wikipedia.org/wiki/Urysohn_space

Matt N.

Belsa-Mosquitoes alert.

Dylan Moreland

@MattN What do you mean?

leo

@tb No, I think.

t.b.

21:18

@leo don't worry then. Wast of effort and time. You should have told him that you assume choice (countable dependent one to be specific) per default, and all his arguments would have evaporated...

@DylanMoreland see here and put on ignore.

Matt N.

@DylanMoreland There was a guy called "Wolfgang Mückenheim" (Mücke=mosquito) who asked a ton of questions about infinity and uncountability (?) and basically said Cantor had it all wrong. (iirc)

I didn't read all his stuff. But I did look him up and he's got a degree in maths from Göttingen.

Dylan Moreland

Ah, well, I don't subscribe to math.GM.

Matt N.

.

t.b.

Sorry, I messed up horribly :))

(and got the too late to edit notification)

leo

@tb :-) Thanks for the bullets. I was thinking in add some thoughts in my answer, but I didn't because the OP is asking for straightforward proofs

t.b.

21:23

@leo It's way over the top, I'd say, as is the one using Baire.

leo

it is no hard, but it is tedious

Matt N.

@DylanMoreland And here is his MO account.

t.b.

@leo yeah, but it is much more work than diagonalization or other proofs and the fact that the outer measure of an interval is its length is tricky.

Matt N.

Thanks for pointing this out, I hadn't seen this answer. Too cool! He gets plus one from me for using my favourite theorem.

leo

@tb Yes. I already have a post that implies that the outer measure of an interval is its length. You have an answer in there as well

t.b.

21:26

@leo I know :)

@MattN heh! I like the pimped nested intervals principle, too :)

Matt N.

@tb soulmates : )

leo

see you! all

Matt N.

Byee!

t.b.

Bye, leo

ymar

@DylanMoreland May I take a bit of your time?

t.b.

21:32

@MattN I wish I'd be better at applying it, though. It's one of the hardest things for me to use properly.

Matt N.

@tb I've been meaning to post a question about it. I want to see demonstrations of its alleged power. Just haven't gotten around to writing up the question. The side effect I am hoping for is that I get an idea of how to apply it.

t.b.

@MattN That'd be really nice. My favorite exercise is: if $f:(0,\infty) \to \mathbb{R}$ is continuous and $f(nx) \xrightarrow{n\to\infty} 0$ for all $x \in (0,\infty)$ then $f(x) \xrightarrow{x\to\infty} 0$.

Matt N.

@tb I know, you've told me once before : ) But I forgot about it, so thanks for saying it again : )

Peter Tamaroff

@tb What would the steps be? Or hints? It find it intuitively true, but I'd like to take a shot.

Matt N.

@PeterTamaroff Hint: use the Baire category theorem.

Peter Tamaroff

21:37

@MattN Oh... bummer. It is topology related?

Matt N.

@tb Unless you would like to post this as an answer I could include it in the question. (along with some other application I had in mind)

t.b.

@MattN Sure, go ahead. You could link to Gowers's take on it, too, to save yourself some work. Another nice application is the uncountability of dimension of infinite-dimensional Banach spaces and of course the standard facts in basic Futile Attempts.

@PeterTamaroff Suppose that $f(x+n) \xrightarrow{n \to \infty} 0$ for all $x \in (0,\infty)$. Is the result still true?

Matt N.

@tb Thank you! I'll see. I don't want to save too much work because the process of writing the question is already intended to help me learn some things.

(exactly some of those standard facts)

t.b.

@MattN But to answer more directly: feel free to include that one.

Matt N.

@tb I understood. "Sure, go ahead" is pretty direct. : )

t.b.

21:43

@MattN Not too coherent today, am I? :/ I'm sure I've linked you to the Sokal thingie already.

(not directly linked to your query in that it avoids Baire for uniform boundedness)

Peter Tamaroff

@tb Mmmh. I'm trying to think of a counterexample.

Matt N.

@tb Not that I remember! Thank you. I've been incoherent since Wednesday night. I have a hint of a headache since then and for some reason it causes me to be unable to focus. No idea what it is but I hope it'll go away soon. Hope you don't have the same.

(not that I'm much more coherent normally...)

N3buchadnezzar

My place is full of drunk and high people sigh...

skullpatrol

@N3buchadnezzar Make sure your door is locked ;)

N3buchadnezzar

Indeed, I just took in my rice and curry from the hallway.

t.b.

21:48

@MattN Oh, too bad. I guess it's too darn hot... No, I'm fine headache-wise.

Peter Tamaroff

@tb Maybe something with $\left\lceil {x + n} \right\rceil $. I really don't know.

Dylan Moreland

@ymar Doubtful, but what's up?

t.b.

@PeterTamaroff remember that you want it to be continuous. I wouldn't have asked if it were true, of course.

Peter Tamaroff

@tb Oh, right! I'm thinking.

user19161

@N3buchadnezzar You take curry there?

t.b.

21:50

@DylanMoreland ymar's left the building a few minutes ago.

N3buchadnezzar

@JasperLoy Huh? Its mine! I just store it outside because my room is so small.

Dylan Moreland

Ah, good timing by me

user19161

@N3buchadnezzar What kind of curry?

N3buchadnezzar

And because of all the drunk people, I prefer not to wake up with the bathroom and toilet flooded with curry and rice.

Peter Tamaroff

@tb Give me any hint.

N3buchadnezzar

21:52

Oh, and the curry is a nowegian blend. It is sort of spicy, but I guess not in the eyes of any indian.

Drunk logic: "We have this large toilet here, and curry... Duuude we should totally make food like in the toilet, that would be so cool dude"

skullpatrol

The smell of curry drives drunks wild :)

t.b.

@PeterTamaroff Take a function whose graph is a triangle of height $1$ in each interval $[n,n+1]$ and make the base of the triangle smaller and smaller.

Jonas Teuwen

@N3buchadnezzar You could also tell little kids: "If you don't stfu I'll rip off your scrotum and eat your balls. I'm Maori/Scandinavian you know! We eat balls"

user19161

@JonasTeuwen They really eat balls?

N3buchadnezzar

@JonasTeuwen We also like to worship Satan in deep forests, and eat goat heads.

Jonas Teuwen

21:56

@JasperLoy I don't know, I've just made that up.

But Scandinavians probably do eat balls.

user19161

@JonasTeuwen I like Swedish meatballs. I tried them at Ikea.

Jonas Teuwen

@N3buchadnezzar And drink the blood?

skullpatrol

stfu?

Jonas Teuwen

@JasperLoy Yes, those are actually balls. But of deer, not humans.

Peter Tamaroff

@skullpatrol Shut the --------- up

N3buchadnezzar

21:57

@JonasTeuwen No, just the eyes and so forth.

Actually I am not joking about the goat head...

skullpatrol

@PeterTamaroff Thanks:)

Peter Tamaroff

@tb Dont you mean larger and larger? Where does $x$ come in?

skullpatrol

Hi @ZhenLin

t.b.

@PeterTamaroff No, smaller and smaller is what I mean. Remember that you want $f(x+n) \xrightarrow{n\to\infty} 0$ for all $x \in (0,\infty)$.

Peter Tamaroff

@tb But you just said the base is always $1$. ($n+1-n$)

Maybe I'm not understanding the first idea of the function.

Should it look like a sawtooth?

t.b.

22:00

@PeterTamaroff No, the height is $1$. As base you could take the interval $\left[n-\frac{1}{n},n\right]$

It is sort of a saw-tooth that gets spikier and spikier.

ymar

@DylanMoreland I'm having trouble determining whether $\mathbb R[x,x^{-1}]$ is a Euclidean domain.

Peter Tamaroff

OH! Yes! I understood the idea wrongly.

Matt N.

▒

N3buchadnezzar

I should party even harder than them, all by myself!

Peter Tamaroff

@tb You first said, in each interval $[n,n+1]$, that's why.

t.b.

22:04

@PeterTamaroff yeah, I tried not to give too much away. But you see that $f(n - \frac{1}{2n}) = 1$ and doesn't converge to zero as $n \to \infty$.

(if the triangle is isosceles for each $n$)

While for every fixed $x \in (0,\infty)$ you have $f(x+n) = 0$ eventually.

Peter Tamaroff

@tb Right. Somewhat like $\sin^{(2n)} x$ goes to $0$ everywhere but in special multiples of $\pi$ where it is always $1$.

N3buchadnezzar

@JonasTeuwen People from Norway are peaceful, and like to take slow walks in the forest

t.b.

@PeterTamaroff a bit. Yes. I only meant to say: $f(x+n) \xrightarrow{n\to\infty} 0$ for all $x$ is not enough to conclude that $f(x) \xrightarrow{x\to\infty}0$, while $f(nx)$ for all $x$ does suffice

Peter Tamaroff

@tb I see.

t.b.

so it is not that immediately obvious why it should hold, after all.

Peter Tamaroff

22:09

@tb Intuitively $n$ is pulled by $x$ and viceverse in the product, but the sum seems to keep them unrelated.

t.b.

@PeterTamaroff yes, that's one of the observations to make towards the solution: that intervals around $x$ are blown up in length if you multiply them with $n$.

Peter Tamaroff

@tb Right. Off.topic: how do you code the \case environment in math.SE?

if i want to write something like

$f(x)=1 \text{if} P(x)$ and $f(x) = 0;\text{otherwise}$ in only one block.

t.b.

use \begin{cases} ... \end{cases}

like so:

kahen

$f(x) = \begin{cases} 1 & \text{for } x \in \mathbb Q \cr 0 & \text{for } x \in \mathbb R \setminus \mathbb Q\end{cases}$

Matt N.

Good night!

t.b.

22:13

Good night, Matt! Hope your headache is going to get better!

Matt N.

Thanks : )

t.b.

@PeterTamaroff It is a hard problem. As I mentioned, I think Gowers's take on it is really nice.

(It starts a bit above the middle of the post, in case you're impatient)

Jordan

what does it mean if I have the limit as x goes to infinity of f(x) = L?

Jonas Teuwen

There! Booked a ticket to London.

@N3buchadnezzar Immortal. Such fools.

@MattN Kopfschmerz? Good night!

Jordan

a function won't have a limit under certain conditions, but a limit will always exist on a specific interval of that function wont it?

Peter Tamaroff

22:24

@Jordan Not really. It depends on your function.

Jordan

I mean if a function approaches infinity it still has a general limit but it is is approaching infinity from the left and negative infinity from the right then it would have to be a one sided limit

this guide online is saying that if the function approaches infinity it is considered to not have a limit

Peter Tamaroff

@Jordan That is right.

Jordan

weird

but the limit is infinity

Peter Tamaroff

@Jordan Not really, if you read the definition of infinity with care.

The limit of a function has to be a real number.

What you're talking about is a special case which people like to introduce.

Jordan

so if I get infinity on a test I where I am suppose to find the limit I can just right Does Not Exist

kahen

22:26

What do you mean exactly by your question "what does it mean if I have the limit as x goes to infinity of f(x) = L"? Are you talking about the definition of what it means for that to exist and equal L?

Peter Tamaroff

The idea there is that $f(x) \to \infty$ if for every number $M$ we choose there is an $x_0$ such taht for $x \geq x_0$, $f(x) > M$

Jordan

I am not sure @kahen

I am reviewing limit laws and they are quite confusing, I dont remember them at all

skullpatrol

@Jordan What is your definition of infinity?

Jordan

I am not sure

N3buchadnezzar

Bigger than two!

Jordan

22:34

infinity is just a large value?

skullpatrol

@Jordan not quite

t.b.

@PeterTamaroff: for your latest question: I recommend that you draw Venn diagrams for figuring out 2. and 3.

arete

@t.b. hey, I'm still struggling with that problem I had yesterday...

skullpatrol

@Jordan It is "large"

t.b.

@arete I forgot what it was. That the union of rational translates of the Cantor set is uncountable in every interval?

Jordan

22:36

I do not know how to define it without using the word infinity

t.b.

Or something to that effect.

arete

Yeah, sorry to bug you, but I'm attempting to set up a proof for it and I still feel like it's rickety haha.'

skullpatrol

@Jordan But you do know it is large right?

t.b.

@arete What have you got so far?

(and no problem re: bugging)

Jordan

@skullpatrol or small for negative

arete

22:39

Basically given an interval (a,b) show that that there is always a point in that interval with due to the density of $\mathbb{Q} \in \mathbb{R}$ This gives us the set $C_q$ will always be nonempty.

skullpatrol

@Jordan Not "small" for negative but "large" in the negative direction, right?

Jordan

yes

kahen

@Jordan the definition of $\lim_{x\to \infty} f(x) = L$ is that $\forall \epsilon>0 \exists K\in \mathbb R: x > K \implies \lvert f(x)-L\rvert < \epsilon$. One way of thinking about this to think of what it means that $\lim_{x \to c} f(x) = L$: For all $\epsilon>0$ there exists a $\delta>0$ such that $0 < |x-c| < \delta$ implies that $|f(x) - L| < \epsilon$. Then think about the notion of "$x\to\infty$" means: that $x$ "escapes away from all finite bounds" and you get the definition above.

t.b.

@arete You do agree that $C \cap [0,\varepsilon)$ is uncountable for every $\varepsilon \gt 0$, right?

Jordan

what does the upside down A mean?

kahen

22:41

$\forall$ = "for all", and $\exists$ = "there exists"

Jordan

epsillon? as in epsilon delta?

skullpatrol

@Jordan "Small" would be infinitesimally tiny, right?

arete

Every element of $C$ is a cluster point and c is uncountable (due to it being perfect), so intersecting that set with $[0, \epsilon)$ gives us an uncountable. Right?

Jordan

@skullpatrol yes

t.b.

@arete Right. I would say that there is $n$ such that $\frac{1}{3^n} \lt \varepsilon$. Then $x \mapsto 3^n x$ will send $[0,\varepsilon) \cap C$ onto $C$.

Jordan

22:45

does 0/0 exist? because 0 divide by anything is 0 but you can't divide by zero so which rule takes precedence?

Peter Tamaroff

@Jordan For that matters, surf math.SE. There are a lot of explanations of why it is undefined.

skullpatrol

@Jordan Do you see the arrow ----> between the x and the symbol for infinity, this means x can "approach" infinity but never reach it, right?

kahen

Depends... are you working in the en.wikipedia.org/wiki/Zero_ring ? hides

Jordan

ok

arete

@t.b. I don't follow that last bit.

t.b.

22:46

@arete You are working with the ternary Cantor set, are you?

arete

Aye

Peter Tamaroff

@Jordan Here!

Jordan

@PeterTamaroff Thanks that makes sense

t.b.

@arete At some point in the Cantor set's construction in $[0,1]$ you'll have the interval $I_n = [0,3^{-n}]$ from which you remove the middle third. If you perform the Cantor set's construction starting with $I_n$ and blow everything up by $3^{n}$ you get the usual Cantor set in $[0,1]$ back. Self similarity is the keyword here.

Jordan

a lot of answers to things in math aren't answers but just demonstrations of why the opposite can not be true, or similar scenarios

skullpatrol

22:50

@Jordan That arrow "--->" gives you a hint how to define infinity as something that is bigger than any preassigned value, understand?

Peter Tamaroff

@Jordan Proof by contradiction is a powerful tool.

Jordan

@skullpatrol yes, so infinity can not be computed as a specific value, just a larger value than other specific values?

I think proof by contradiction can be wrong sometimes

Peter Tamaroff

@Jordan If the premises are correct, it can't happen.

kahen

no, it's not a value at all (unless you're working in the extended real numbers, but those aren't a field...)

Jordan

it does not always prove that something is wrong, just that something is wrong in a specific case, so the statement could also be correct in specific cases just no the one that proved it wrong

arete

22:52

So is every interval of the cantor set uncountable?

Peter Tamaroff

@Jordan Proofs by contradiction assume general premises. For example, that there is a finite set of primes $\{p_n\}=\{ p :p \text{ is prime } \}$.

t.b.

@arete Every interval around a point in the Cantor set contains uncountably many points of $C$, yes.

Jordan

I guess a simple example then is that if you have $$2^3^x$$ than it can't be $2^3x$ because if x is zero than it doesn't work

but if x is not zero it can work

arete

I'm just trying to wrap my head around that because we know that the Cantor set is perfect and therefore uncountable. Isn't it the case that any interval of an uncountable set inherits the sets uncountability?

t.b.

@arete I wasn't saying that what you wrote was wrong. I was just saying that I consider it a bit of an overkill because you can "see" it

Peter Tamaroff

22:56

@Jordan A simple example of what¿

Jordan

proof by contradiction will sometimes only work with a specific scenario

crap I wrote it out wrong

Peter Tamaroff

@Jordan I don't think so. Proof by contradiction is quite far reaching.

Jordan

crap I just failed my chapter review test by a lot, the book doesnt offer answer so I am going to ask some in here

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$Mathematics$ Mathematics