@Alice No problem! :)

@PedroTamaroff

@leo Shakalaka, BAM!

:-)

What's up?¡

Lou Reed's dead :(

@FernandoMartin Never listened to his music, but I know he was a great guy.

Music-wise.

If a ball is bounded in a metric space, is it necessarily closed?

Hi @PedroTamaroff

@DonLarynx You have open balls and closed balls. That name is not coincidence, Don! =)

Look what it says for bounded. proofwiki.org/wiki/Definition:Bounded_(Metric_Space)

Doesn't "$\leq$" imply closed?

Because $<$ implies open.

@DonLarynx Yes, what is your confusion?

I didn't understand your message. What do you mean?

@DonLarynx First, balls are always bounded.

They are contained in themselves.

So they are, by definition, bounded.

oh

im sorry

i used the wrong quantifier

@Don Let $B(\epsilon, x)$ be an open ball of radius $\epsilon$ about $x$. It's bounded by $k\epsilon$ for $k \gt 0$, ie $diam(B) \lt k\epsilon$

Correct question: If a ball is bounded by an element less than or equal to all of the elements in the set, is it necessarily closed?

how do you define $\leq$ here?

between elements of a vector space?

@EnjoysMath Uh?

We're dealing with $d(x,y)$ which is in $\Bbb R$.

@EnjoysMath no

and yes

wt...

I mean your sentence is weird, needs correction

it can be elements of anything

it doesn't matter in the context of this question.

Can anyone help me with linear programs and strong duality

Tangential question: I don't understand the significance of open/closed sets in topology. Why are they so important?

I posted my work and question here:

your sentence doesn't make sense: "If a ball is bounded by an element less than or equal to all the elements in the set, ... " what's "the set"?

@PedroTamaroff When a question says to "find all solutions $x$ for the system of linear congruences", is there still only one solution, assuming the moduli are coprime in pairs?

@Helen interesting. I had a question about duality just a few minutes ago.

Open sets is just a another useful definition used for making theorems just like compactness.

@MartinSvanberg They're definitions.

For example, the question reads as follows: "Find all integers $x$ which have the following property: $x$ leaves remainder 6 when divided by 11, and $x$ leaves remainder 3 when divided by 7. I came up with $x\equiv 45\mod{77}$, but that's the only one I can see... Are there more?

Although that was more norm-based.

This problem is also interesting to me though.

@MartinSvanberg Because without them there wouldn't be topology in the first place.

in the set with the usual topology @EnjoysMath?

$45\equiv 1\mod 11$

@agent154 By CRT, you know that there is a unique solution $\mod 77$. Yes.

So what could it mean by "find all integers $x$"?

@Zibadawa Get an 11-hour clock and go around it 4 times. What is the number obtained at the first hour in $\mathbb{Z}$?

they are any one mod-77 solution plus multiples of 77

Right, I understand this. I guess my main problem is, what is the difference between an open and a closed set? it might seem like a trivial question but there's something that just isn't clicking for me

@DonLarynx What? 45 = 4*11+1.

Correct @Zibadawa!

Closed sets are complements of open sets.

Yes, I already knew this, Don.

I was responding to @agent154

@Zibadawa yes, that's the definition. is there something else i can think of to make it more intuitive?

@Zibadawa OK, I see where I did some bad arithmetic... I added instead of subtracting in one step... but otherwise, my question was whether or not there are "multiple" solutions

@MartinSvanberg You can start by picking a book on introductory topology and reading it.

I recommend Mendelson's Introduction to Topology.

That is exactly what I'm doing!

Maybe I should pick a different book

Open sets are a way of declaring which things are "pretty close" to each other (neighborhoods, which you can think of in a fairly literal fashion without being too far from the point).

@MartinSvanberg Consider $A = [0, \sqrt{2})$ in $\mathbb{R}$. Take the neighborhood of $\sqrt{2}$, so $B = N_\epsilon(\sqrt{2})$ for some $\epsilon > 0$. Is the intersection $A \cap B$ empty?

@Zibadawa So the real solution is $x\equiv 17\mod{77}$

Why am I only seeing the latex code? How do I turn on math symbols?

Of course, since the entire space is an open set of necessity, that analogy sort of doesn't work, but usually you specify a topology through a "base" which consists of neighborhoods which are reasonably similar to the analogy.

@MartinSvanberg Having some experience with metric spaces helps. The leap from metric spaces to topological spaces is quite natural

So maybe the "answer" to this question is $x=17+77k$ where $k\in\mathbb{Z}$

@agent154 correct

@MartinSvanberg math.ucla.edu/~robjohn/math/mathjax.html

thanks!

there are so many topologies possible on a set they can mean very different things, or potentially have no real meaning at all. in metric spaces for example, open means every point has wiggle room and closed means you can't go outside via limits. more generally open sets can mean "semidecidable properties." ultimately, open sets and nbhds are used to formalize closeness, closed is a useful adjective to have for complement of open, and topologies in general are an artifact of formalization

I haven't bothered with jax'ing stuff. I just read the code directly. I've written enough in latex that it's fairly natural. Plus, if you try REALLY hard, it's like you're in an exceptionally nerdy Matrix, only this time without the godawful cave orgy.

@anon BAM.

Letting open sets be unions of ideals forms a ring

huh?

so the notion of open/closed in metric spaces is different from open/closed in topologies?

Little error there, enjoy.

*forms a topology of a ring

oops

The notion in general topologies is a generalization of the metric space case.

@MartinSvanberg in the same way that television is different from telecommunications

one is a special case of the other

If you feel like you understand open/close for metrics, then for a general topology they are meant to convey the exact same ideas and machinery, only without a metric.

a metric space is a well-behaved topology, i.e. it behaves like familiar spaces

@EnjoysMath What?

Well, metric spaces are normal and stuff. Type IV. Basically has every nice property imaginable.

Hm, I think I need to go back to reading. It'll make sense soon enough. Thanks everyone.

If you let the topo on $R$ be unions of ideals as open sets, then the topo on $R \times R$ is also unions of ideals of open sets, i.e. it coincides with the induced product topology

where $R$ is a ring

Exercise: prove whether or not $R$ is a topological ring

under that topo

$(x- y)^{-1}(I)$ seems tricky to calculate

$= \{ (x, y) \in R^2 s.t. x-y \in I \}$

$\implies x + I = y + I$

If I wanted to get in to grad school, would I be better off taking an REU in the summer or taking another set of courses in summer, and starting a DIS in fall (my last fall semester as an undergrad. I've been straight in school since summer '12)?

I feel much more comfortable taking courses in summer

@DonLarynx Graduate school?

ya

That comes after undergraduate studies.

yeah

@DonLarynx Well...?

In short, is it too late to start research the fall before you graduate?

I dont know about math, but iin physics definitely not

most people dont do independent research til senior year

@DonLarynx You're an undergraduate student?

he is

hes wondering about directed independet study

Yes @PedroTamaroff

@KevinDriscoll But you cannot expect to enter graduate school if you don't master undergraduate level mathematics.

@Pedro I dont see the relevance

Me either

@KevinDriscoll Err... how come?

Well hes gonna graduate with an undergrad degree in math presumably

and is wondeirng how ot best help his application to grad school

@DonLarynx In what year are you?

Last year, senior

"(my last fall semester as an undergrad"

@DonLarynx You're in your last year?

Yep, $(R, \tau)$ where $\tau = $ unions of ideals of ring $R$ is a topological ring.

Speaking of Graduate School.... I'm intending to (though not having as easy of a time with it) complete a joint Honors in Computer Science and Pure Math... My goal is to get into a Graduate program somewhere and focus on Cryptography... Does anybody have any advice on the differences of doing so on the Math vs the CS side? Or are there Joint Masters programs as well?

@agent154 "Joint Masters." Chuckles.

@PedroTamaroff! I'm in my second year of college. I finish all of my undergrad requirements in spring this year. Provided I don't slack off, I am planning on taking applied math courses like PDE and grad school courses in my third year

@agent154 In grad school, people dont usually have official joint programs, but what happens is you find a professor who is working in both fields

@DonLarynx So, two years of undergrad courses?

I am a pure math major atm.

Yeah

that's my spring schedule

@PedroTamaroff I just have two questions. To clarify my first one, is it too late to begin research in your last year of undergrad? 2) I see the requirements for a masters are 32 hours of coursework and passing qualifiers. If I pass some qualifiers, do I get to skip coursework?

@DonLarynx I wholeheartedly think you should strengthen up your bases before taking grad or masters courses.

you can't graduate until you master a joint

With what courses, @Pedro?

@DonLarynx I dont understand how you've managed to fulfill all of the math major requirements in just 2 years

even with summer classes

how many required calsses are there?

@DonLarynx That's for you to decide.

@KevinDriscoll I graduated high school with 45 credits, fulfilling most of my liberal arts requirements. I then proceeded to pursue a degree in biology but that wasn't fulfilling. I switched to math this summer. Now here I am.

@DonLarynx When was "this summer"?

@DonLarynx Ah okay. I also went ot a middle college program for high school, but my school didn't accept my transfer credits

Summer 13'

@DonLarynx I mean we have different seasons, I am down south.

oh sorry

@Pedro May-August

What month?

May

Hello MSE. Question: if $A$ is a compact linear bounded operator (on a Hilbert space $H=l^2(\mathbb{Z}_+$) then is it true that $\overline{ \text{range } A} = H$?

@agent154 You will probably need to pay a lot of attention to the faculty of the CS and Math departments at the schools you are looking at. There may not be much of a crypto presence in the math department, at least not specifically so (algebraic geometers and number theorists could both lay claim to being relevant, at least).

@DonLarynx You started studying math three months ago Don. You cannot expect to get into a graduate course next year. You need to keep studying!

Why not? I did that.

I think @pedro is probably right. You have probably taken many engineering/science-level math courses, but there should be lots of math major courses left to take

@Zibadawa I'm a ways off from having to worry about anything yet (aside from maintaining as good grades as possible for the time being) but I was told that McGill in Montreal has a strong Crypto program... I don't know if it's CS or Math leaning though. The University I'm at presently (Memorial University) has a lot of algorithm complexity stuff, and is supposedly great for that... but not quite how I want to focus.

@PedroTamaroff Time and intelligence are not linearly correlated.

And to start research in your last year will probably be too much, unless you've already filled most/all of your degree requirements. It'll consume time. Plus you'd have to find a project you could actually complete in a year, or otherwise not leave the project dead and hanging from your sudden absence.

@KevinDriscoll The only courses left are applied mathematics courses.

@DonLarynx Sure. But you still have a lot to learn.

Certainly not @DonLarynx and neither @Pedor nor I know exactly what courses you're taken.

We all do.

@KevinDriscoll I posted my schedule a few posts back

@Zibadawa Who are you talking to?

@PedroTamaroff How many hours do you study a day?

@DonLarynx I don't know. But I read and study everyday.

That's not enough, I do that as well.

does anyone know how to take the derivative of this :$ [1/2 \ ||y-z||_2^2 + u^Tz$ and find z?

In fact I go to the loo reading math

@agent154 That was to @DonLarynx.

When you're far enough along, you'll go to the loo doing math.

Indeed.

I don't understand how to do separate z after taking the derivative.

@DonLarynx Taking one or two applied courses is good. Rounds you out better. You can never be too sure that you'll never have need for it in a research career. You may be working on something very "abstract", and suddenly a formula pops out that turns out to be something from the "applied" side.

@DonLarynx How many classes in real and ocmplex analysis have you had?

If I consider a graph where

the vertices are denoted by a series of bits

@KevinDriscoll I'm planning on taking complex in the summer and real analysis 2 in the spring. I really wanna take measure theory next year...

and an edge exists between two vertices if the difference in bits is 1

@DonLarynx which school do you go to?

The school with the best QB in the nation. F! L! O! R! I! D! A! S! T! A! T! E!

hmm.

so there would be an edge between 001 and 000, but no edge between 000 and 011

What is a QB?

BWAHAHAHAH!!!!! Florida State best QB in the nation............lololololololol

How would you go about showing your graph is connected if every bit combination is shown

You'd like ot think so playing in the ACC

every bit combination exists as an edge

@Helen Quarterback, talking about football

@KevinDriscoll Well we will have the full data at the end of the season.

@DonLarynx I may have to eat those words

He very well might be the best, I'm just so far not entirely convinced given the subpar opponents

They're subpar by definition, @KevinDriscoll....

Because they play in the ACC of course

It was Jeff Driskell, but he got hurt

so I dont know either now

Never mind. I got it.

@KevinDriscoll I see.

@DonLarynx I go to Georgia tech, but when it comes to football I'm a GAMECOCK

@Zibadawa i know right? you never know when you might need it. it makes you well-versed enough.

just enough and not more!

@AlexMardikian Sounds like your graph is a cube

@KevinDriscoll are you a grad student?

@DonLarynx It seems that graduate schools in math are much more flexible when it comes to acceptable undergraduate courses. IN physics, there are something like 7 or 8 courses that every grad school requires

@Helen Yes

@KevinDriscoll phd or masters?

@KevinDriscoll. I have quite a few friends there actually.

@Helen PhD in physics

Well then, I shall take each of your advices. I shall do another load of courses in summer!

@KevinDriscoll I see. my friends are doing a phd in computer science. Much like me.

I don't go to Georgia Tech though

@Helen Ah okay. I dont have any contact with CS folks

@KevinDriscoll makes sense.

aren't bases subcovers as well?

bases are covers of their space

@DonLarynx Subcovers of what?

if $X$ is compact, then there's a finite subcover of the basis covering $X$

hm

@KarlKronenfeld Yao.

@PedroTamaroff Hello

@KarlKronenfeld How's your sunday?

@PedroTamaroff Not bad. The Sun peeked out for the first time in ages.

@KarlKronenfeld Do you happen to live in the north or south pole?

@PedroTamaroff Maybe you mean hemisphere, I live in the north hemisphere.

@KarlKronenfeld I was being a little more dramatic.

Bad weather? Clouds, storms?

:)

@PedroTamaroff Mainly rain if anything, but it's been insanely cloudy.

@PedroTamaroff I used to love looking for fixed points like this.

@KarlKronenfeld Heh, with a program?

@PedroTamaroff A hand calculator.

@KarlKronenfeld Hehe. I remember "discovering" the number $e$ when meddling with my calculator.

I cannot recall what I did, but started calculating some quotients of logs and obtained 2.718281828

@PedroTamaroff I guess I also had a graphing calculator, since I realized that you wouldn't find all of them this way (fixed points are the intersections of the graph of $f(x)$ with the graph of the identity function).

@PedroTamaroff Ah, cool.

I mean, I took several different quotients and $e$ always showed up.

In fact, it is easy to remember 2.718281828459045

Right, so the finite subcover covers the basis covering $x$. gotcha

@DonLarynx You don't usually cover collections of sets, Don. You usually cover sets.

@AlexYoucis

Nitpicking!

how do i put plus/minus sign?

@Omnitic $\pm$?

ty

I wouldn't call that nitpicking. For me, a field is Archimedean with respect to some norm if and only if the integers are unbounded.

@AlexYoucis Aha. Sounds reasonble.

So, to each his own I guess haha

@AlexYoucis Could you explain the "respect to some norm" part?

Unboundedness, by definition, basically means you can measure size. So, you must be talking about some field F along with a norm |.|:F--->R

@AlexYoucis I am talking about order in my answer. Which now has a downvote ¬¬

Two downvotes?

Maybe I'm just having a brain fart, but... how do I calculate $x\equiv 2^{61}\mod{125}$?

I mean, this works here because you have transported to R via the usual norm. This seems a little contrived because you're already in R, but this is the right context in which to view the question.

@AlexYoucis Transported what to $\Bbb R$?

I mean, yes, in our case our field is R, and so we have a well-defined ordering. But, if we are in a general field, the right generalization, we need to transport to R by some absolute value so that we can talk about order. Ordered fields are a little too rare for this to be the right generlization (e.g. C isn't even orderable).

@AlexYoucis Aha. But I didn't aim to generalize anything. =P

@PedroTamaroff Right, which is fine. But I was trying to explain why Robert Israels' comment makes sense.

@AlexYoucis Yeah, of course it does.

and why it isn't nitpicking

I didn't mean to be disrespecful with the nitpicking. Sometimes I am a little too playful with my words. =)

I have to go now, be back in 20'

Haha, I didn't perceive any offense.

See ya

hello everyone

hi

i posted a question to physics SE, but i wondered if it might be better suited for the mathematics site

perhaps someone has an opinion about it: physics.stackexchange.com/questions/82379/…

Part of Sylow's theorem states that for a group $G$ and a prime $p$, any two Sylow-p subgroups are conjugate in $G$. Does this just mean that for any $P,P' \in Syl_p(G)$, there exists $g,g' \in G$ such that $P \leq g'P'g^{-1}'$ and $P' \leq gPg^{-1}$?

or is that not what it means

@AlexMardikian Yes.

@hadsed Even if the content of the question is really mathematical, I really doubt many people here would be able to make the translation. I have no idea what 60% of those words mean.

ah okay, that makes sense

i wish i could translate it a little better... alas i am but a nub

@AlexYoucis not be, but for sure there are people here in math.se who yes

@AlexMardikian Another thought process, one which is useful for one version of the proof, is the fact that G acts on Syl_p(G) by conjugation. The content of that part of Sylow's theorem is then that the action is transitive.

@leo I didn't say that there was no one, I just think that honestly you'd have a hard time finding people. The only students of mathematics I know who would have a fair chance of understanding that are those who are very, very on the side of geometry, bordering on mathematical physics. If my graduating class of 22 is any indication at my school, I believe only 1 would be able to help. That's just my opinion though. @hadsed i more than welcome to try.

*is more than welcome

Anyone: Should I take undergrad PDE in summer, then grad PDE 1 in fall, then grad PDE 2 in spring?

@DonLarynx It's hard to give course advice to a name and a picture of a spiral :)

ha ha

@AlexYoucis I am planning on taking undergrad topology in spring and grad topology 1 in the fall. I am a pure math major but i think the undergrad pde in summer, and grad sequence of pde in next year will be helpful. Would it be a waste of time?

I want to do a lot of analysis.

also brb.

I'd appreciate an answer :)

but it's ture I guess

ha ha

but it's ture I guess