ha, i didn't realize math.SE gave 'test' questions on votes to close. i was informed that i had decided correctly on a vote to leave something open to see if i was 'paying attention.'

spooky but actually a really good site feature.

adding another layer of tinfoil to my hat though.

@Leslie: This was the question: If $\sum \alpha_i \beta_i$ converges for every sequence $\{\beta_i\}$ such that $\beta_i \to 0$ as $i\to\infty$, prove that $\sum |\alpha_i| < \infty$. I suggested applying Hahn-Banach to $\ell^\infty$, proceeding by contradiction. What thinkest you?

Did you mean thinketh?

isn't that way too complicated

I never knowest.

@Thor Well, @robjohn had an elementary argument, but epsilon said it was in the Banach spaces chapter of his text.

mm, in banach space language it seems to be an attempt to identify [a portion of] the dual of c_0(N)? am i missing something?

Correct.

i.e. this math.stackexchange.com/questions/678911/…

I'm thinking of $c_0\subset\ell^\infty$.

yeah, that makes sense.

the hahn banach idea is an interesting one.

yeah, I saw his nice argument. I was trying to do something similar when the question was posed, but couldn't quite get it right. it's certainly elementary, but I wouldn't know how to interpret it correctly in the Banach space context, so I should probably keep quiet.

That's why I pang you.

I'm the antipodal point of a functional analyst, but I was curious.

hi

hi satan

What is the inverse fourier transform of $$\dfrac{1}{a+iw}$$ ?

no

w

my book's answer is contradicting wolfram's answer

i am reluctant to apply hahn banach whenever possible because even on classical spaces a black box application tends to invoke the axiom of choice. and maybe you can be more direct here.

my book claims its $u(x)e^{-ax}$, which makes sense I believe

ell^infty is a complete chaos of a space. i try to stay away from it. but the idea ought to work.

I concede to your reluctance, but I'm curious if what I'm thinking works, @Leslie.

@shintuku did you figured the circle functions out?

but wolfram claims its $u(-x) e^{ax}$

My point is that if there were a linear functional on $c_0$ other than pairing with $\ell^1$, it would extend to give something on $\ell^\infty$ we know cannot exist. Or am I full of it?

@satan: I am not thinking about your question (it's been 35 years since I have thought about Fourier transforms), but what is $u(x)$?

unit step function

1 for x>=0, 0 otherwise

Oh, Heavyside function.

i guess it isn't clear that a hahn banach extension would be governed by the same formula, or really any formula, when applied to sequences other than ones that happen to belong in c_0. you just get some linear extension.

Agreed, @leslie, but it's still an element of $(\ell^\infty)^*$.

my only zoom meeting of the day just got canceled, so as a treat i'm ordering pizza for all of you.

Well, send the pizza quickly. I have to be at the bridge club in an hour.

bridge club. lah-di-dah.

ok, i actually think you're right.

we need to stop doing this.

Well, I wanted the functional analyst to critique my idea, as I didn't think it through very far.

My days of thinking about this stuff are long gone.

i actually got off of my a-se and imported some books from boxes in the garage into bookcases into the house. i think this might be in hille and phillips.

Since we're speaking of functional analysis what's a simple example of a Rosenthal space? (That is a Banach space not containing a copy of $\ell^1$)

the UPS man

R^1 doesn't contain a copy of ell^1. or does it?

Is this supposed to be an isometric copy?

i think i met rosenthal once.

i'll email him and let you know. :)

I find it interesting that video-cameras or an insurance work, but not both together. but maybe only with trash insurances.

(against theft)

Sure but an infinite dimensional one is the interesting case

because if you caught the thief on tape, the insurance says "get the money/ware from him)

@TedShifrin yes (for separable spaces that's the same as a linear isometric copy)

insurance companies also use video cameras to deny coverage to policyholders. they will send goons out to your house and tape you to see if you are as sick as you say you are.

you get this blurry stuff like candid camera. they don't use good equipment. usually no sound.

but you can always make a picture sharper, i mean that's what CSI Miami says!

is ell^2 a rosenthal space? subspaces of reflexive spaces are reflexive.

c_0(N) might also be an example.

i havent thought this through carefully for approximately 10 years.

c_0 has a separable dual, namely, and recently, ell^1. i don't think something with a copy of ell^1 in it can do that.

Dumb question: Why $ax^2 + 2hxy + by^2 + 2gx + 2fy + c=0$ does not contain $d$ or $e$?

notational torture.

Ah yeah that's a good example, of course

what's that quote about some guy. he would give lectures with proofs where similarly situated variables would have names like epsilon_1 and c_0 and alpha_30.

i arguably see a reason for e being excluded from that kind of setup, it's used for the eccentricity of an ellipse. d may have some similar significance for conic sections.

I apologize for pissing off the category theory haters in the chat, but I am looking for a few examples of spanning classes in triangulated categories. Except the trivial case where you take the triangulated category of vector spaces, and the spanning class is just the 0 vector space, what are other interesting examples of spanning classes? For instance in the homotopy category?

sayan, for the record, i only pretend to hate category theory and made fruitful use of categorical language in my dissertation. i am thinking about homotopy classes but my brain is 10-year-old mush.

i'm thinking.

category theory is like inception for math for me.

I like category theory, I Just don't know anything

i think of it as a language and not really as its own math. which might offend people who do it. i think it is very helpful to see different instantiations of it in various fields, but, the study of itself seems frankly a little onanastic.

I like the kind of category theory I am doing right now. So we have this two semester long reading project on Fourier Mukai transforms in Algebraic geometry, so we are prepping up with some amount of triangulated categories

i guess that's my basepoint. i think you use it with a goal in mind. people who are just doing it, some of whom are my closest friends, seem to be goofing off to no effect.

i can see the power of it, but it is beyond my intellectual grasp to use it in any other way than following someone else's proof.

Hi

Can we say that $(\mathbb{Z}_5 \times \mathbb{Z}_5) \rtimes \mathbb{Z}_3$ has a faithful representation as a group of matrices?

@SayanChattopadhyay What about spheres

@leslietownes isnt this sort of what happened to noncommutative geometry though? it started off with a pretty incredible premise, but now it just doesn't seem to be going anywhere

i dont know nearly enough but that is my inference

you'd know better than i

no we entirely agree on that. a handful of good ideas, a new field, handful of cool results, then a swarm of ants onto the syrup of grant funding, and then basically nothing.

i'm not saying don't learn those results, but it's not a fountain of youth or a blueprint for the future. it's just some results in papers, same as everyone else did.

i don't know how it works elsewhere; in the USA it seems easier to obtain grant funding from the few organizations that fund pure research in mathematics if you are in an early field. nobody knows what is or isn't difficult yet, so you can basically say anything and deliver anything and it's all amazing.

well maybe would be a blueprint for the future if there were clear programmes on where to go next. part of the issues with noncommutative algebra seems to me to be similar to category theory; there's an incredible amount of formal insight needed to understand what things for actual spaces exactly goes through for noncommutative spaces

even if from a mathematical point of view it might not be any different, in terms of what it provides to future researches, from what others are doing.

but i bet NC owes many of its success to physics

physics is toxic. i used to review papers for a physics journal. everyone was quoting other papers with the same buzzwords and announcing 'results' that were linear algebra from 1800-1920.

nobody was reading papers without their own buzzwords in the title. it was nonsense.

i think maybe they got the virus from mathematicians

i dont think physics is per se toxic. modern physics is a lot of buzzwords yes

i think we are a little better about the fundamentals but you do have a point.

i think physics is a good context for a lot of mathematics. its just easy to get lost in the thick of it if the subject is formally technical without any guiding light of experiments to back up your insights.

the experiments need not be as people understood experiments in the 1850s or whatever. im saying, things have to make sense

a lot of NC makes sense because QM

i think there is a profitable trade in people who write stuff, where the physicists think "this must be interesting to mathematicians" (when it isn't), and the mathematicians think "this must be interesting to physicists" (when it isn't), and a lot of crap gets through.

I do not know enough homotopy theory to be able to answer that @BalarkaSen What's your intuition for that?

@Sayan $[S^n, X] = 0$ for all $n$ implies $X$ is contractible. Whitehead's theorem

Work in the homotopy category of CW complexes or something

i am cynical but i think the natural hesitance to comment too harshly on something out of someone's expertise can create a kind of additive noise where suddenly nobody really understands what a paper is for, but it gets accepted for publication.

Isn't that the definition of spanning objects in triangulated cats

I know a bunch of NC-geometers who got tons of grants and now seem to be trying to generalize pontryagin duality for quantum groups

@leslietownes makes sense

if you work in something that is just one thing, it's harder, because the people your paper gets sent to will actually know everything that you are talking about.

@BalarkaSen Oh yeah that is, with the shift operation being the cone of the chain map

Neat man

chain map?

the shift operator on hoCW is just suspension

No so you have $X$, $Y$ as complexes, say CW ones, then anything is a triangle iff its isomorphic to something of the form $X \rightarrow Y \rightarrow \text{cone}(f) \rightarrow X[1]$ for some map $f X \to Y$

So?

There is no chain map there are spaces not chain complexes

What are you talking about

The homotopy category of an additive category $A$ is the category of complexes of objects in A, upto homotopy classes of morphisms of complexes

Hence chain maps

Oh, that homotopy category. I was giving the stable homotopy category as my example, which is triangulated, shift operator is suspension

Not hoCh

Sorry my bad, I should have mentioned

hoCh has obvious spanning objects, no? Just take the complex which is Z concentrated in degree n for various n? Hom(Z, A) = 0 implies A = 0

Any suggestions on this please

@SayanChattopadhyay it's a good idea to have a more solid foundation in classical homological algebra (which seems lacking imo, judging by the question you asked) before jumping into triangulated categories

@Astyx Bon noir!

Bon sois

hi

hi

Hi, @Astyx

hi Balarka

howdy

is anyone here familiar with some Local Group Theory (LGT)?

You can just ask your question, if someone feels like answering it they will

i think i will wait for the experts to arrive

hi @LeakyNun

do you know some Local Group Theory (LGT)?

@Wolgwang as mentioned $e$ is a number, and maybe $d$ not, because deriviatives?

so much topology, category theory, geometry -- but no one knows group theory??? pity

Has it already been asked if phi (the golden ratio) is (or is not) a normal number? I've tried to search around but haven't found any specific question for phi.

I believe such things are widely open

Darn, most common materials I've seen talk about pi, e, root(2) being unknown so I wasn't sure if phi was also unknown

Doesn't the divergence and ratio test say about the same thing? But for some reason we are told we cannot use the divergence test to prove convergence, but we can use the ratio test to prove convergence. The divergence test states that if $ \lim_{n \rightarrow \infty} a_n \not= 0$ then clearly $\sum a_n$ will not converge. But we are told the converse of this statement is not true.That is, if $\lim_{n \rightarrow \infty} a_n = 0$ then we cannot infer from this that $\sum a_n$ is convergent.

However, the ratio test says something that is identical to this in my opinion. That is, if $\lim_{n\rightarrow\infty} \frac{a_{n+1}}{{a_n}} = L$ then $\sum a_n$ will converge if $L<1$. That is because for each successive term in the $a_n$ sequence, it approaches $0$. And if $L>1$ each successive term is getting larger hence it will diverge. So, the way I see it, the divergence test says the same thing but more concisely so what can't we use that to prove convergence?

How do you suggest to prove something converges using the divergence test?

@Thorgott $1/(\sum a_n)$ converges if $\sum a_n$ diverges

Hello. I need some help regarding a Block cipher attack question. May I ask it here?

if $ \lim_{n \rightarrow \infty} a_n = 0$, wouldn't that be sufficient to conclude that $\sum a_n$ converges? If not, doesn't the ratio test say about the same thing?

consider $\sum 1/n$

@finitegronpnombertheorist no

@finitegronpnombertheorist yes

$1/\sum(1/n)$ does converge

but $\sum n$ diverges

so i don't think you're right @Thorgott

consider $\sum0$

Oh, I see. So, the terms of $\sum \frac{1}{n}$ approach 0 for large $n$, but it's divergent. So the test would fail in that case. thanks

This question in 3 steps is asking literally only for which is counterfit, but we don't have to exactly pinpoint that fake coin, is that correct please?

i don't see the distinction you are drawing although maybe i am missing something. the question does want the identity of the fake coin.

@leslietownes. Thanks! Yeah I got the trick behind these questions! I was confused to believe that I want to know the coin exactly! 2 hours spent on this question and similar questions

:/

All of them also ask to do that in at most 2 to 3 steps using trees!

the key insight is you don't need to put all unknown coins on the scale at the first step, or at any step.

you can learn information about a coin that you don't weigh. that cuts things down.

great problem.

@leslietownes Huh? I can get being able to determine how many weighings are necessary without the weight of all the coins, but you can't know the weight of the last coin without weighing it directly in some way.

you only need to know that one coin weighs less than the others. you can determine that without weighing it directly. if you take 6 of the 7, put 3 on each side of the balance scale, and get a balance, then the one you didn't weigh is the fake.

or if it's eight coins (i now realize the statement above says seven once and eight once), compare six in groups of 3, if you get a balance you have one more weighing between two. if not, from the lighter group of 3, reserve one and weigh the others. you get the information.

there are generalizations of this problem where you don't know if the fake coin is lighter or heavier, or if coins have 'true' weights and false weights and the scale is numerical. someone must have written a good paper on this.

i dunno what abstract slot of research this would fit into. maybe information theory. the number of pairwise comparisons needed to identify something when you control what you compare seems to have that flavor.

AMDG i like how your avatar is more or less the complete opposite of mine. sacred heart and immaculate heart, guy drowned in kiddie pool.

opposite isn't really the right word. contrasting.