Mathematics - 2018-10-23 (page 2 of 2)

Semiclassical

7:00 PM

actually, the Abel Conference for Langlands going to be here at the UMN in November

definitely going to try to sit in on some of those

and Langlands himself will be a participant

so yeah, hopefully I don't forget about that :/

Leaky Nun

@Liad @Semiclassical It seems to be saying this: Let $k:[0,1]\to[0,\infty)$ be a non-negative measurable function. Let $X$ be the set of equivalence classes of functions for which that integral [1] exists, modulo a.e. equality. (a) show that $X$ is a vector space. (b) show that the following formula [2] fulfills all the requirements of an inner product, except that $f \ne 0$ might occur even when $(f,f) = 0$.

[1]: $\displaystyle \int_0^1 |f(x)|^2 k(x) \ \mathrm dx$

[2]: $(f,g) = \displaystyle \int_0^1 f(x) \overline{g(x)} k(x) \ \mathrm dx$

Mike Miller

@Semiclassical i don't know what any of that stuff means

i'm content with it

Semiclassical

the Abel Prize is one of the big yearly awards

and there's a conference that goes with it

This year, Langland's got it, so the conference is in his honor

Leaky Nun

@Liad so the answer is, just take $k=0$.

Semiclassical

and it's being hosted at the University of Minnesota

Abcd

7:08 PM

Semiclassical

@LeakyNun ...lol

that's underwhelming

Abcd

Can someone please give this question a look?

Leaky Nun

40 mins ago, by Liad

let $k:[0,1] \to [0\infty)$ be measurable and let $X$ be all the function (module equality a.e) that $\int_0^1 |(fx)|^2 k(x) dx \lt \infty$. i want to find $f \ne 0$ s.t $\int_0^1 |(f(x))|^2 k(x)dx =0$, someone can help?

so @Semiclassical, to be fair, I think we just misinterpreted the quantifiers

Alessandro Codenotti

@MikeMiller what are you referring to?

Leaky Nun

that's rather subtle though

Semiclassical

7:09 PM

right

Leaky Nun

that's why the original question is important

even though it takes me 25 minutes to decipher

Semiclassical

@LeakyNun and liable to be lost in translation, yeah

Eric Silva

@Semi do u have a classic minnesota accent

Alucard

@Semiclassical the only thing I found is math.stackexchange.com/questions/2517415/…

Semiclassical

ya shuure u betcha

(honestly, I don't think I really do)

Eric Silva

7:11 PM

lmao

everyone ive met from minnesota hasnt had one

Semiclassical

i mean, there's probably some accent

but nothing anywhere as strong as is stereotyped

Mike Miller

@AlessandroCodenotti the link I posted a message afterwards :P

are you a minnesota native

Semiclassical

ya

the apple hasn't fallen far from the tree

Mike Miller

i haven't ever lived outside of california for more than 6 months

Semiclassical

same with me in minnesota

Eric Silva

7:18 PM

is there a california accent

Daminark

Usually I think of either the kinda "valley girl" thing or surfers

"There's some hella waves bruh"

Eric Silva

my gf is from california (not from socal tho) and she says hella a lot

but Mike lived there for his whole life and he doesnt sound like a valley girl

so where's the lie

Daminark

Is this what mathematical logic studies? :thonk:

Mike Miller

valley girls are mostly a myth

its one of those invented accents people actually take on sometrimes fi they identify with the picture

we do say hella though

Liad

@LeakyNun Ah? k is given

Mike Miller

7:22 PM

and "yeah no" and "no yeah" mean the opposite

Eric Silva

@MikeMiller holy shit i picked this AND hella up

uh oh

Leaky Nun

@Liad yes. but it just means, for any given $k$, it is possible for $f$ to be non-zero despite $(f,f)$ being zero.

Daminark

Wait that's a California thing? I think I use it like that too

Mike Miller

also everyone says "i get it" constantly

Leaky Nun

@Liad it doesn't mean that for any given $k$ there needs to be such an $f$.

Mike Miller

7:23 PM

jk @Eric just trolling

Daminark

"No yeah that's totally right!"
"Yeah, no"

Eric Silva

uh oh ok

i got it once u said the i get it thing

Leaky Nun

@MikeMiller what does each of them mean?

Mike Miller

"yeah no" means "uhhhh... no."

Eric Silva

"yeah no" means "no, also ur an idiot"

Mike Miller

7:24 PM

"no, yeah" means "that's right"; the no signifying that there's no reason to think the opposite

like "no you were right"

Liad

@LeakyNun i dont follow, k is given and they asked to show that there might be $f\ne 0 .. $ , what do you mean?

Mike Miller

but thats a little fuzzier

Will Njundong

what's up with this nonsense wording? :/ imgur.com/laK8ntm

Leaky Nun

@Liad it means, you don't need to verify the requirement that says "$(f,f)=0 \implies f=0$" because it might not be true.

Will Njundong

It says my mistake is selecting that the given function is "not onto"

"f(x) is onto: For every y, there is an x such that f(x)=y."

i see y's there that have nothing mapped to them ...

Liad

7:26 PM

Ahhhhhhhh. @LeakyNun that's a really tricky wording ...

Leaky Nun

@Liad just bear in mind that my only knowledge of Hebrew is its alphabet and some basic words

so I can recognize "function" and "integral" and "modulo" but for other words I relied on Google Translate

Liad

how do you know basic words in Hebrew?

Leaky Nun

@WillNjundong because the x that maps to such y are outside the picture

@Liad general interest in languages

Liad

and im not the only student who understood this question like that ^^

Leaky Nun

oh

Will Njundong

7:29 PM

...oh @LeakyNun :(

thank you

Leaky Nun

treat pictures with caution :P

Liad

@LeakyNun can you see why $tr(AB^*) = tr(BA^*) $? $A,B \in M_n(\Bbb C)$

the star for adjoint matrix

Mike Miller

@TedShifrin A bi-invariant metric on $G$ of volume 1 is unique up to isometry. What can we say about 'bi-invariant metrics' on orbits? Precisely, this means that it is invariant under the left action of $G/H$ as well as the right action of $W(H) = N_G(H)/H$.

It works fine in the case I need.

Leaky Nun

@Liad $\operatorname{tr}(AB^\ast) = \operatorname{tr}(A^{\ast\ast}B^\ast) = \operatorname{tr}((BA^\ast)^\ast) \ne \operatorname{tr}(BA^\ast)$

Liad

wait

$tr(AB^*) = \overline {tr(BA^*)}$ @LeakyNun forgot the conjugation

Leaky Nun

7:34 PM

$\operatorname{tr}(AB^\ast) = \operatorname{tr}(A^{\ast\ast}B^\ast) = \operatorname{tr}((BA^\ast)^\ast) = \overline{\operatorname{tr}(BA^\ast)}$

Liad

why the last equation hold?

Rithaniel

@Alucard There should exist a bijection between the unit circle and the unit square.

In fact, I think they're probably homeomorphic.

Alucard

@Rithaniel yeah but hard to come with one up alone

Rithaniel

Well, maybe try and find the opposite direction. Unit square to unit circle.

Leaky Nun

@Liad $\operatorname{tr}(M^\ast) = \operatorname{tr}(\overline{M^T}) = \overline{\operatorname{tr}(M^T)} = \overline{\operatorname{tr}(M)}$

Liad

7:40 PM

$A^* $ is the adjoint of $A$ , yes?

because there is another defintion for that.

Adjugate matrix

In linear algebra, the adjugate, classical adjoint, or adjunct of a square matrix is the transpose of its cofactor matrix.The adjugate has sometimes been called the "adjoint", but today the "adjoint" of a matrix normally refers to its corresponding adjoint operator, which is its conjugate transpose. == Definition == The adjugate of A is the transpose of the cofactor matrix C of A, adj ⁡ ( A ) = C T ...

Leaky Nun

@Alucard observe that every straight ray from the origin meets the circle and the square at exactly one point respectively

@Liad yes

Liad

so why $A^* = \overline{A^T}$ ?

Semiclassical

because that's the definition of adjoint?

in this context, anyways

Adam

If a question you have posted has "grown" into a series of quite similar questions, but none the less, more than one mathematics problem that requires rigorous proof, is it better that A) I continue to update the original question, by including all the material involved and how it relates to the original to the best of my ability at the present time or B) make a new question for each individual problem, and link them in that feature provided

Semiclassical

i think the question is how distinct it is from the initial question

if the scenario were "this is my initial question, but here's the question I should have asked" then I'd lean towards keeping the question

on the other hand, if it's a new question that's motivated by the old one then I'd create a new one

but I'd still try to keep the new question as self-contained as possible. referencing the old question as context is fine, but typically you want the question to make sense by itself

Adam

7:55 PM

Ok well that in itself is most definiately the problem, but I go thru phases of which particular problem I want the reader to see as central or the most significant, is there anyway I can have a webspace where by it is available to the people I know will provide the right type of criticism necessary,

and then once I have their approval of the content, post the question on SE officially? it's just that atm I am doing choice A, but editing and updating my current position on a public post is a little embarrassing in that its just exposing the lunatic I actually am logic wise, and so it would feel a lot better if I could get help with writing it by having it somewhere number theory people can see, then post a polished "end product"

Semiclassical

maybe this?

124

Q: Sandbox for drafts of long, complex posts

This sandbox is intended for saving drafts of long, complex posts, especially posts whose composition takes a long time. It serves to localize to one thread the front-page "bumps" caused by edits to drafts of such posts, so that they may be easily ignored. Also, it helps to guard against losing l...

Adam

ah yes ive been put onto this before I was trying to remember it ok ill take a look

ty

Semiclassical

np

Will Njundong

what should i expect from
$$
f (f^-1 (x)))
$$

how to turn into math syntax?

Liad

@Semiclassical the def. im working with is this the one given here:
https://en.wikipedia.org/wiki/Adjugate_matrix

how is it the same as $\overline{A^T}$

Semiclassical

8:09 PM

then you're using the wrong definition

the adjugate matrix is defined so that $AA^* =(\det A)I$

it arises in relations to cofactors and the like

Rithaniel

Having a little bit of difficulty interpreting this statement: "Let $\tau$ be a topology on $\mathbb{N}$ with the property that each open cover of $\mathbb{N}$ has a subcover of no more than 2 sets."

Semiclassical

the adjoint of A as an operator, by contrast, is $\overline{A^\top}$

Rithaniel

Could I petition someone to rephrase, perhaps not using the word "cover?"

Leaky Nun

@Rithaniel if $\{U_i\}_{i \in I}$ is a family of open sets with $\bigcup U_i = \Bbb N$ then there is $i, j \in I$ with $U_i \cup U_j = \Bbb N$ or there is $i \in I$ with $U_i = \Bbb N$

Rithaniel

Okay, so every point "splits" $\mathbb{N}$ into two open sets, and dividing these sets cannot possibly cover $\mathbb{N}$ (therefore, splitting the sets in the subcover must result in at least one non-open set?)

Adam

8:18 PM

If a question you ask requires both the use of set builder notation and expressions of the fractional part, (traditionally having the notation of curly brackets as used in set builder notation) is it necessary to re-express things in terms of the ${\{x}\}=x-\lfloor x \rfloor$ or something of this nature?

also @Semiclassical do I just post the question as a comment on the page you linked to add it to the "sand box"?

Mike Miller

@Rithaniel Part of the job of learning this is learning how to translate between words and their definitions - which is what Leaky did.

Rithaniel

Yeah, I'm always trying to translate from words to a mental image, and definitions are simply the tool to help along the way. :P

Also, thank you for clarifying @LeakyNun

Mike Miller

@LeakyNun No need to say the second - the second implies the first. ;)

Leaky Nun

aha

Adam

@Wi

@WillNjundong what do you mean by math syntax?

Will Njundong

8:40 PM

oh i figured it out, I was using { instead of $

to show fancy math symbols and style etc

Semiclassical

for reference, if you want to have {} show up in math mode, you'd do \{ \}

Yuriy S

I really dislike the word "plugging" when it comes to mathematics. Is it just me or every time someone says "I tried plugging [some value] into [some function]" it means they don't quite understand what they are doing? Maybe it's just the language barrier for me, I hope I don't offend anyone

user280247

Different numeric systems (decimal, dozens, binary etc) woudn't change anything about calculus, but are there some numeric systems which actually do it?

user280247

For example, numeric system with different lenghts between numbers 0,1,2,3

user280247

Actually, I think that's what a function does..

Isana Yashiro

8:50 PM

@santimirandarp measure theory? (not sure)

Fargle

@santimirandarp Don't quote me on this, but I believe that you can do something like calculus on the p-adic numbers, which have a different notion of distance than the standard Euclidean one.

user280247

hmm interesting, I suppose anyone'd answer. But isn't what I said a possible definition of a function?

user280247

For example, if we plot x axis and below a y axis; which is f(x)

Will Njundong

@YuriyS what would you rather used? inserting? substituting?

user280247

I find interesting that part of maths...

Will Njundong

8:53 PM

given the context, substitution action consists of "plugging" a certain element to take the place of something before it, yea?

Alucard

is there a prove that a bijective function exists between two things, but such a function noone has figured out yet?

Yuriy S

Not sure. As I said, it's likely just the language barrier. Technically, I would use "evaluating the function at the point", but again, I have learned in Russian, so what do I know

user280247

Maybe @YuriyS is referring to a pattern: usually the word appears when less understanding of maths is observed...

Will Njundong

with a function where there is a common f(x) result for more than one x, can i call it well defined?

Yuriy S

@santimirandarp, yes, it's likely that, from a few years on Math.SE

Thank you all for humoring me inbetween your deep discussions :)

user280247

9:00 PM

@YuriyS :)

user193319

1

Q: Extraneous Condition in the Hypothesis?

If $\{f_n\}$ is a sequence of continuously differentiable functions on $[a,b]$, $f_n \to f$ uniformly on $[a,b[$, and there is a function $g : [a,b] \to \Bbb{R}$ such that $f'_n \to g$ uniformly on $[a,b]$, the $f$ is continuous differentiable and $f'=g$. So, when I first approached this pr...

real-analysis indefinite-integrals uniform-convergence uniform-continuity riemann-integration

math.stackexchange.com/questions/2968164/…

user525966

In propositional logic, are well-formed formula, sentence, expression, statement, and proposition are all synonyms of each other?

Will Njundong

9:16 PM

when asked "What is the domain of g ο f?" do i take the range of f, or domain of f?

Leaky Nun

domain of f

Will Njundong

thank you

Monolite

10:45 PM

0

Q: On computing a conditional expectation $E[Y|X]$ where $Y$ is not known.

Preface: This question is based on the answer to this question given in the comments. The problem: Consider the Lebesgue probability space on the interval $[0,1)$. (I.e. the state space is $Ω = [0, 1)$, the $\sigma$-field is the set of Lebesgue measurable sets and the measure is the Lebesgue...

probability probability-theory expected-value

Could some probability theorist answer this question by me

Adam

11:44 PM

Does anyone else feel like they go thru an infinite loop of sulking because no one seems interested in the problems they are working on, then realising everyone has their own choices and how bad it would be to force everyone to be interested in one persons problems, then somehow ending up sulking again because no one gives a #$#$ about your specific math problem on this day?

it probably is just me there actually

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