nah, representation theory of finite groups (the relevant Serre text is Linear representations of finite groups)

ah nice :)

are you gonna attend that seminar?

I definitely want to. The preliminary meeting is in two weeks and I'm not sure how much interest there is, but hopefully I'll get a spot

It sounds like a really cool topic

I need a hint: if $K/F$ is an algebraic extension, then for any non-zero $f\in K[x]$, you can find $g\in K[x]$ such that $fg\in F[x]$.

I feel like I saw that question on main just today

!

Main chat, or as a question on the site?

Oh I found the question.

Interesting.

@Thorgott nice man, I hope you get a spot!

thanks :)

I'm probably gonna attend 2 seminars this semester

or rather next semester

one on algebraic number theory and the other on non-commutative algebra

sounds neat (and like a lot of work)

yeah it will be lol

Hi @TedShifrin!

I guess I am seeing things. ;)

Merriam–Webster gives "pulverize" as a synonym of "extinguish"

so I'mma use it like that from now on

"Hi fire department could you pulverize a fire"

That is with regards to the definition "to annihilate", I bet.

> The Game may have been created in 1977 by members of the Cambridge University Science Fiction Society when attempting to create a game that did not fit in with game theory.

- Wikipedia

(I lost the game)

Contraction idea

It is possible -> It's possible

Is it possible -> Is't possible

What's the quickest way to break a number up into the form (2^s)*m where m is odd? Can logarithms be used?

@northerner divide by 2 until it's odd

@LeakyNun ok. But is there a faster way that isn't brute force?

You could also try dividing by a larger power of 2 e.g. 4 to accelerate the process

you can tell whether a number is divisible by 2^s by looking at the last s digits

so keep scanning the digits from right to left one at a time

you can even do it inductively

let's say the last s digits is divisible by 2^s

if the last s digits is divisible by 2^(s+1), then the last (s+1) digits is divisible by 2^(s+1) iff the (s+1)-th digit is even;

if the last s digits is not divisible by 2^(s+1), then the last (s+1) digits is divisible by 2^(s+1) iff the (s+1)-th digit is odd;

(obviously this isn't the most efficient way to do it in a program, but only by hand)

@northerner do you need an example?

yes please

@LeakyNun

so take 1923847218579874819752

What is meant by "we need the next digit to be odd"? We need it to be odd for what? @LeakyNun

so if the last two digits were e.g. '...62' then 2^2 would fit in, is that what you're saying?

if it were ....62 then it wouldn't be divisible by 2^2

@LeakyNun wouldn't 202 be a counter example since 202=(2^1)121 even though it's last 3 digits are even, giving 2^3

@LeakyNun actually nvm I think I get what you're saying now.

cool

So with your method do you always need to inspect each digit or can you stop early sometimes?

you can stop when it is no longer divisible by the next power of 2

@northerner I almost threw up when I realised how many factorization algorithms that are computationally more efficient that brute force, I mean I'll dig up the references for what I'm reading at the moment if it will help?

Having said that, if you have got onto something that is explained in vague way or not at all, what ive assumed is that you have to appreciate that the most current and efficient carry a proprietary value that's lost if the creator explicitly announces to the public precisely how the algorithm works, so kind of understandable really

@Rithaniel I had an awesome one that I sadly lost that was so undersold with it's title being "Statistics for Engineers" but yeah depends you may be a lot more advanced I didn't go past first year in stats, but yeah I didn't put any effort into the year I did do so it got me a good result considering how little time I spent

anyway good day ppl

@Adam I would be interested

Yeah, free probability is a different kind of topic than normal probability. It's probability where random variables aren't necessarily real numbers

Such as integer valued probabilities, off the top of my head

Waddup

Not much, Meg, hbu?

Just in the train on my way to Heidelberg :) how’s it going @Rithaniel

It's okay, getting geared up for working on homework tonight

Trying to figure out how to make a short exact sequence where the middle is isomorphic to the direct product of the two ends, but the sequence doesn't split

This is as $R-$modules

What is i and j here?

All combination of i and j that is possible to make of N?

So (0,0), (0,1), (1,0), (1,1), (2,0), (2,1), (2,2), (3,0) ...

?

So $$B={1,2,4,5,8,10,16,20,30,32,...}$$

Right?

Sebastian please use $\mathbb N$ if you are referrng to the natural numbers ive already found these American flip cards that have made me grumpy I don't need that

Adam, that is an image from my textbook lol.

I am reading "Introduction to Languages and the Theory of Computation".

@northerner ok first one ive got (theres only two for the subject im not great at reading I should really water boarded for it or I don't know) " Pipeline Architecture for Factoring Large Integers with the Quadratic Sieve Algorithm" by Carl Pomerance

A* Pipeline Architecture sorry

And the second one is on another kind of quadratic sieve algorithm by Scott Patrick Contini under Carl's direction but ill be lucky if I properly digest the first one, and yeah I assume that would be pretty tough to get thru such is life really its not a popular subject you either love it or hate I guess

anyway im pretty sure I'll get the wooden spoon award as far as computational efficiency of algorithms is concerned well I know my first one ever did not fare well all all lol

Hey @AlessandroCodenotti let me know when you'll be online, just wanted to chat about Bonn again.

I mean I literally only learnt about these because a few of the algorithms I wrote that lead to some of the problems I put up as questions on here spit the dummy at a certain point tell me I don't have the memory to handle the computations, and I was very fortunate in that the error that maple returned when it did crash very often mentioned a "quadratic sieve"

so yeah I naturally decided not knowing what that was maybe an issue with things

so I just wanted to make the point that I was not seriously proposing water boarding as a disciplinary measure for bad students, I have a warped sense of humour

@Perturbative I'm here now

@AlessandroCodenotti If I may ask, through what means are you funding your stay in Germany?

My parents are paying until I graduate

any smart way to prove $\log_{6}(3)\log_{15}(30)+\log_{6}(5)\log_{15}(2)=1$?

@vesii Just write everything in terms of log (one fixed base) and simplify.

Is there a name for vector spaces equipped with metrics?

Kind of like how ones equipped with topologies are called topological vector spaces and those with norms are normed vector spaces

What's the difference between a norm and a metric in a vector space? Doesn't either one immediately give you the other?

wait are all metrics in vector spaces induced by norms?

not the discrete metric i imagine

Oh, no, duh. Apologies.

math.stackexchange.com/questions/2841855/… covers some of it but doesn't answer your question, I think.

Hey guys, to disprove that $ f(C - D) \subseteq f(C) - f(D)$, if $f: A \rightarrow B$, and $C, D \subseteq A$, I let $f = x^2$, C = {$1$}, and D = {$-1$}. Would this be a valid counter-example to disprove this claim?

I would choose C={-1,1} just so that you have D as a proper subset of C.

@StevenStadnicki Ok, thanks!

@Thorgott Ah, gotcha. Thanks

@Abwatts your definition of the image, as a set, should have no free variables in it.

So, the image set should be fixed, without any conditionals?

of course, think about the definition

I see. So, I'm not quite sure how would I be able to describe the image of this function? It always changes depending if $x$ is even or odd..

The image is the union of the (singleton) sets of values that $f$ takes over its entire domain.

e.g. if $f(x):\mathbb{N}\mapsto\mathbb{N}$ is $f(x)=x+1$, then the image of $f()$ is $\{2, 3, \ldots\}$.

A set-theoretic way of writing the definition of the image $I(f)$ of a function $f:D\mapsto E$ is that $I(f)\subseteq E = \{e\in E: \exists d\in D. f(d)=e\}$.

Ahh, that makes sense! We are basically using the defintion of an image as a guideline. So, in the case of the function I mentioned above, its image would be $f(\mathbb{N}) = \{1, 2, 3, \ldots\}$ up to infinity (since there isn't a value that would make it undefined, etc)?

@user193319 Isn't that the literal definition of $p_K(z)$? — oh wait, not quite, because the definition has an infimum. But I believe you should be able to factor your argument easily through the infimum operation.

Yeah, the definition is $p_{K}(x) := \inf \{\lambda > 0 \mid x \in \lambda K\}$

It's true that $z\in p\cdot K$ for all $p\gt p_K(z)$, though (again, by infimum). So if $p_K(z)\lt 1$, then...

(The proof that $p\leq q\implies pK\subseteq qK$ is pretty straightforward but you should probably run through it at least in your head if not on paper.)

@Steven Stadnicki Would it be also valid to say that $f(\mathbb{N}) = \mathbb{N} \cap \mathbb{R}$ for the function I listed above?

@Abwatts No, because $\mathbb{N}\cap\mathbb{R}=\mathbb{N}$. Is there a $n\in \mathbb{N}$ with $f(n)=2$?

Oh ok.. So, the image would only be the set $\{-1,1\}$ since these are the only possible outputs of the function, and $\{-1,1\} \subseteq \mathbb{R}$?

Correct!

@StevenStadnicki Awesome, thanks!

Longshot question: does anyone here know if the 4d analog of origami has been looked at, from a 'constructibility' perspective? e.g., what regular and semiregular polyhedra can be constructed (which I would think would mean all faces and all solid angles realizable?) from a unit cube, given suitable analogs of the possible 'fold' operations (find the plane that reflects one line onto another, etc)...

can someone ping me

trying to test the notification

@sevdaicmis

ok, allowing notifications doesn't make any difference then, i suppose

Hello

yes, the supremum always exists (possibly infinite)

@Thorgott Hi thor

Am ready for that hint now

haha

@TedShifrin Hello Ted

Hello @Jacksoja.

Are you familiar with big O notation ?

Are you up to no good, as usual?

Yes, big O and little o.

haha I was hoping that you would not find out

Can you please recommand some readings?

I have the defintion but it is not doing good for me with limits

O just means "on the same order as" and o means "small compared to," making those both rigorous.

It's pretty intuitive.

What can you not understand?

Are you doing $x\to 0$ or $x\to\infty$?

We are doing big O in my book

So I assume we want to say a function is bounded by another

OK, answer my last question.

am I supposed to find the best such function ? or any one would do ?

No, by a fixed constant times another.

x going to infinity

Come on, don't make me ask three times.

Thank you.

So, when I write $f(x) = x^3+4x^2 + x^{10/7}$, what term dominates?

As $x \to\infty$?

x^3

So $f(x) = O(x^3)$.

This means that for large $x$, $f(x)\le 2x^3$, for example.

Do we say that f= O(x^3) ?

Well, I would write $f(x)$ if I'm writing $x$'s.

I could write $f=O(g)$, where $g(x)=x^3$.

That is comfusing to me

What about $f(x) = e^x + x^{10000} + e^{-2x}$.

why are we writing both

f = O (e^x)

because that would dominate after a while right?

If your textbook writes it the way you did, that's OK. I don't like putting $x$ on one side and not on the other.

Yup, that's right.

Give me the actual official definition in your text.

ah no sorry, they do it like this f(x) = O (g(x))

OK, so they agree with me :)

It's important, because I might want to say $f(x^2) = O(x^3)$ if $f(x) = x^{3/2} + 7x$.

if the limit of f /g exists as x approches infinity

we say f(x) = O(g(x)

Hmm, that's not my usual definition.

it is equivalant to this at least?

So they couldn't say $f(x) = x\sin x$ is $O(x)$, even though $|f(x)|\le x$.

that limit does not exist

My definition is that there exists $C>0$ so that $f(x)\le Cg(x)$ for all large $x$.

Right, I'm showing you where our two definitions diverge.

I see

Obviously, you should use your text (course).

I don't think you're confused. I think you're fine.

We are not doing formal complextity theory i think that is why

thanks Ted!

@LeakyNun i remember you were writing proofs in some formal proof system like coq or something, don't remember which language it was

anyway, i wasn't aware of them at the time, now i'm so interested in them

I only know coq au vin.

started to write some trivial proofs a few weeks age

it's acceptable too, i presume :)

sometimes i'm mixing also and too, and it embarrasses me

:D

f(x)=O(g(x)) is one of my least favourite abuses of notation ever

ayw

aye*

also hi

Hi, EE.

Why abuse of notation, @Thorgott?

btw, these -proof assistant is what they call- are really cool @TedShifrin do you really didn't know about them?

Most people won't write $f=O(g)$ and need to define the functions separately.

Nope, @sevdaicmis — have no idea what you're talking about. But I do love coq au vin.

I'm gonna see lots of $\mathcal{O}(\log \log \log \log \log \sqrt{x}$) next semester

With maybe a $1+\varepsilon$ exponent.

lol probably

as if I know anything about it

f=O(g) isn't much better. The equality sign is the abuse. For a) O(g) is not a function, but a class of functions and b) it's a highly asymmetric relation.

Meh. I have no issues with it. I rarely taught with o and O, but to say $f(x)\in O(g(x))$ is too pedantic for me.

but @Ted, isn't there really a conceptual problem there, two normally not equal functions will be equal with that notation

Well, but the definition (at least mine) is an inequality, not an equality. But to me it's notation that is intuitive and I have no issue with it. I'm not the world's most pedantic mathematician.

I've always been really bad at reading o notation when it's used more loosely. Once I was proofreading a friends thesis and at one point, he used the symbol O(...) twice to represent a sequence with that asymptotic behavior, but the two occurrences of the symbol referred to different such sequences (whose existence was implicitly postulated by the proposition). It took me like an hour to sort that out conceptually. In fact, I complained so much about it that he changed it in the final version.

Well, I'm not defending the mathematical writings of all authors, students, or professors.

Lots of people do write garbage.

It's certainly not well-defined to define a sequence by a class of sequences. I mean, if what's you're saying holds for every element of that class, I suppose you should make it clear that you're not working with a specific sequence.

yeah