Mathematics - 2018-03-16 (page 2 of 2)

Silent

13:02

Please someone look at this

Secret

I am really starting to suspect the new age is an age of analysis, given the huge spike in analysis questions here

Xander Henderson

@AkivaWeinberger I love all the anal: real anal, complex anal, funky anal, harm anal. I just love anal SO MUCH!

Secret

Age of Analysis: confirmed

aka The Age of Anal

Secret

13:34

O they dried out...

quallenjäger

@XanderHenderson Is this true that one should avoid passive voice in the paper?

Xander Henderson

The passive voice should be avoided.

Slereah

@XanderHenderson *You should avoid the passive voice

Secret

sniped

Xander Henderson

@Slereah That sentence which was said was the sentence which was meant.

Slereah

13:43

Although I think "you" is also to be avoided in papers

The author should avoid the passive voice

Xander Henderson

"You" is terrible.

Slereah

Rude

Xander Henderson

(or is it "'You' are terrible"?)

Slereah

depends how rude you're feeling

Xander Henderson

We prefer to use "we" in papers. Never "I" and certainly not "you". "You" is to be avoided.

Slereah

13:44

Perhaps "One should avoid the passive voice"

Same

which is weird because I write it alone

quallenjäger

Really? For example, if I want to express something like "The following expression can be brought into the following closed form".

Slereah

So I suppose it's the royal we

Xander Henderson

It is the "mathematical we".

quallenjäger

How can I rewrite it in active voice?

Slereah

But I thought there was no royal way to mathematics

Xander Henderson

13:45

Not the royal we, the mathematical we.

quallenjäger

Sometimes, it does appears to me that passive voice cannot be avoided.

Xander Henderson

The way that I have always interpreted it is that once a result is proved, it is indisputable fact.

Therefore all of us know that it is true, therefore we see how it must be.

Slereah

*proven

Xander Henderson

I also have nothing against the passive voice; I was trying to make that clear by using the passive voice to admonish against it.

@Slereah Yuck. No. Proved.

Slereah

although I beware of "proven results", but maybe that's my physicist ways :p

Xander Henderson

13:46

Regularize them verbs!

Slereah

There's a lot of fishy proofs around

quallenjäger

Using too much passive voice won't lead to the rejection of the article right?

Xander Henderson

@quallenjäger No idea. But I would imagine that is WAY down the list.

Over use of the passive voice can be fixed with editing. Uninteresting results or incorrect proofs likely cannot.

quallenjäger

I have a horrible style of writing. My supervisor criticized me many times.

Slereah

I'd say until you have the actual proof you're wanting to write, writing style should be pretty low on your priority :p

It's easier to change phrasing than proving theorem

quallenjäger

13:51

Is it common that error occurs on published paper?

Slereah

In physics certainly :p

Dunno about math

quallenjäger

I am currently reading a published paper. Something appears not right to me

I don't know if I am not smart enough to see it or they really did a mistake.

Slereah

well you can ask here

Xander Henderson

Errors occur from time to time, but I think that they are reasonably rare in mathematics (at least compared to fields where one might have to gasp experiment).

That being said, one of the first things I did as a graduate student was find a counter-example to a theorem in one of my masters advisor's papers.

Slereah

Oh I don't even mean experimental things

just that theoretical physicists aren't necessarily big on rigor

Xander Henderson

13:54

Yes, I know. We make fun of them all the time.

I mean, what the fuck is a Dirac function? That's nonsense, yo!

Slereah

to be fair they're usually awful bad unphysical counterexample :p

Well technically it is a function

Xander Henderson

The Dirac distribution, on the other hand... :P

Slereah

A function on the space of test functions!

Xander Henderson

There are so many words that are better than function, in that case: continuous linear form, bounded linear functional, distribution, etc

And it certainly isn't a function on the underlying space, which is how I see physicists using it. :P

Secret

I use the word maps when the domain or the range is something exotic that is not based on the reals

Slereah

13:57

But on the other hand there is an embedding of distributions into a space where they are functions!

Xander Henderson

@Slereah As you say, they are functions on the space of test functions, but that it not the way that I see people using the word "function" in general.

Slereah

digitalcommons.calpoly.edu/cgi/…

^Dirac distributions as functions :p

Secret

actually, is there a difference between maps and functions?

Slereah

Functions on ${}^*\mathbb{R}$

Xander Henderson

@Slereah "generalized functions"

which is not the same thing :P

Secret

13:59

I mean, we use linear maps instead of linear functions when referring to matrices, for example

Slereah

It's just $\mathbb R$ with extra numbers!

Xander Henderson

or, according to their abstract "asymptoic functions"

linear maps are a superset of linear functionals

and between the two sits linear operators

(or map is a synonym for operator, depending on who you talk to)

Secret

eh? I always thought linear operators and linear maps are the same thing, bleh, those sloppy quantum physicists...

Slereah

I never really understood why Todorov uses asymptotic numbers rather than the hyperreal directly

Silent

@AkivaWeinberger, will you please look at this?

Secret

14:02

wait a minute, Slereah knew heaps of anal, thus he might be able to help out, whether he wants to is another story

in particular, he knew hilbert space stuff much MUCH better than I am

Slereah

@Secret What kind of slander is this

Secret

@Slereah well, xanderson has a habit of referring analysis as anal and that becomes an in-joke of math chat

thus I just go with the trend and refer that as anal

Akiva Weinberger

Relevant

Princess Luna

@Secret I'll take care of it

Secret

cool thanks

Slereah: And based on the h bar and maths chat data, I am pretty sure you knew at least a lot of formalism about hilbert spaces, distribution thoeries, linear operators, complex analysis and some real analysis, so I pretty sure you know most analysis in general, thus I don't think I am slandering you

Akiva Weinberger

14:10

I imagine you're not too familiar with the subject

Princess Luna

@Secret I went back to the Java backend, connected it to the neural network through Flask. So the bot loaded an old database in which it's still in this room

Slereah

I'm terrible at analysis

Secret

Princessluna: Ah I see

Slereah: Ok

Princess Luna

It should be gone, and I'm pingable here for 7 days. If Alisha rejoins, ping me and I'll unsummon her again and make sure it gets saved.

Secret

ok

Silent

14:28

@AkivaWeinberger, thanks for that answer. :)

Secret

Discrete mathematics

Discrete mathematics is the study of mathematical structures that are fundamentally discrete rather than continuous. In contrast to real numbers that have the property of varying "smoothly", the objects studied in discrete mathematics – such as integers, graphs, and statements in logic – do not vary smoothly in this way, but have distinct, separated values. Discrete mathematics therefore excludes topics in "continuous mathematics" such as calculus and analysis. Discrete objects can often be enumerated by integers. More formally, discrete mathematics has been characterized as the branch of mathematics...

hmm...

Akiva Weinberger

@Silent Incidentally, I think the answer would be different if we change "bounded" to "totally bounded".

Note that $\Bbb R_{()}$ from your answer is not totally bounded, since you can't cover it with a finite number of balls of radius $\frac12$.

Secret

If potential infinity can be ditched since we don't have ideal computers, perhaps it will be a good idea to start studying about discrete analysis and be familiar with the world where things are chunky

Akiva Weinberger

(A set is totally bounded, if, for every $\epsilon$, the set can be covered with finitely many balls of radius $\epsilon$)

Secret

is that basically the analysis version of compactness in general topology?

Akiva Weinberger

14:32

Er, I guess; a metric space is compact iff it's compete and totally bounded.

It's strange that neither completeness nor total boundedness is a topological property and yet their conjunction is

[Looks up to double-check] Conjunction is "and", disjunction is "or"

Secret

/thonking/

Secret

15:15

O they dried out...

Leyla Alkan

17:03

Hi guys!

Secret

I found it interesting that the logic room is a lot more populated than the main chat atm

Leyla Alkan

Oh really

Is its name just Logic?

Secret

indeed, and you seemed to found it

Leyla Alkan

It's hard to use chat on phone :D

Semiclassical

17:44

hi @ted

Ted Shifrin

Heya @Semiclassic.

Semiclassical

Think I have a better way to explain what I was saying the other day re: the gamma function

Ted Shifrin

You need to focus on your other problem:)

Semiclassical

If we start from the integral definition $\Gamma(z)=\int_0^\infty u^{z-1}e^{-u}\,du$, and make the substitution $u=ze^{t}$, then it becomes $$\Gamma(z)=z^z e^{-z}\int_{-\infty}^\infty e^{-z(e^t-t-1)}\,dt$$

Secret

hey ted, FYI and unrelated, we have preliminary evidence that the chat have entered an Age of Analysis this year, after the Age of Number Theory in 2017, Age of Algebra in 2016-2017 and the Integral Age circa <2015

Ted Shifrin

17:48

There's not that much analysis. I would argue there's been a shift to more geometry (from more topology). Not that I'm biased.

Semiclassical

The exponent in that case has an obvious saddle point at $t=0$ which gives the usual $\sqrt{2\pi /z}$ factor.

But it also has an infinity of other saddle points along the imaginary axis.

Ted Shifrin

@Semiclassic: As you know, I no longer have my notes [oh, maybe I did this in the applied math class], but this seems like what I did following Bender/Orszag.

PrithiviRaj

Hi ! @LeylaAlkan

Semiclassical

For most purposes, these other saddle don't matter: You're only integrating along the real axis, so why should you care about these other saddle points?

However, if you want to do analytic continuation in $z$, then you want to know what happens to this integral as you deform the contour.

Ted Shifrin

Right, fair enough.

Secret

17:51

Ted: Hmm, as you mentioned that, it seems the topology discussion had massively decreased ever since Balarka gone chemistry and I don't see Eric, Balarka and Mike on at the same time that often

Semiclassical

And therefore it shouldn't be too surprising that things are rather more involved if you take $z=iy$ with $y\gg 0$.

Ted Shifrin

Mike is around way less, and Balarka has been studying a lot of geometry. So has Akiva. And 0celo is doing geometric analysis, as is Eric.

Fair enough, @Semiclassic. Thanks for thinking about that.

Semiclassical

well, I'm doing it more because my advisor asked me to think about it :)

Ted Shifrin

Ah ...

Semiclassical

But I explained it so awkwardly before that I wanted to correct the point

Secret

17:53

hmm, interesting... should continue to observe closely then

Semiclassical

By comparison, the full asymptotics for $\Gamma(iy)$ (which i'm stealing from another source) are
$$
\Gamma(i y)\sim \sqrt{2\pi}(i y)^{iy-1/2}e^{-i y}\left(1+\frac{1}{12iy}-\frac{1}{288y^2}+\cdots \right)(1-e^{-2\pi y})^{-1/2}$$

Ted Shifrin

Oh, @Semiclassic, I found it in my applied math lecture notes. Interesting, I also got Stirling out of the difference equation.

Semiclassical

the stuff in front is just what you get from stirling's formula. the stuff in the middle is the usual asymptotic series

and what's left is there because the reflection formula should give $|\Gamma(iy)|=\sqrt{\dfrac{\pi}{y\sinh \pi y}}$ for $y>0$.

the upshot, as I understand it, is that the effect of all those additional saddle points at $z=2\pi i k$ along the imaginary axis is to give that factor of $(1-e^{-2\pi y})^{-1/2}=1+\frac12 e^{-2\pi y}+\mathcal{O}(e^{-4\pi y})$

(there's a thesis out there which does this in a lot more detail)

Ted Shifrin

I see. That shouldn't be too hard to check. I'm looking back at my asymptotics notes from that class.

Anonymous

I'm a bit confused about this terminology: My textbook (Axler) calls $\Bbb{F}^{n}$ a set of functions from $\{1,2,3,...,n\}$ to $\Bbb{F}$. It also says that the elements of $\Bbb{F}^{n}$ are a list of numbers. I'm not sure how a list of numbers can be function from $\{1,2,3,...,n\}$ to $\Bbb{F}$.

Leaky Nun

18:00

@Blue simple: the k-th element is just f(k)

Semiclassical

@TedShifrin it's this thesis, for reference: mathematics.ceu.edu/sites/mathematics.ceu.hu/files/attachment/….

Leaky Nun

(0,1,0,3) is precisely the function that sends 1 to 0; 2 to 1; 3 to 0; 4 to 3

Semiclassical

if nothing else, the figure on page 103 (page 127 in the pdf file) is worth glancing at

Ted Shifrin

By a list of numbers, you mean an ordered $n$-tuple $(x_1,x_2,\dots, x_n)$, so, as Leaky said, $x_k = f(k)$.

Leaky Nun

@TedShifrin sniped :P

Ted Shifrin

18:01

I was typing more info than you, Leaky.

Anonymous

Oh. It's a set of functions from $\{1,2,3,..,n\}\to\Bbb{F}$. I missed that. Thanks!

Semiclassical

I'm trying to avoid looking at their thesis too much because there's so much detail that I fear going blind :P

Daminark

Hello

Ted Shifrin

Hi Demonark.

Daminark

How's it going?

Semiclassical

18:03

The main tedious bit of all of this is that there's Stokes phenomenon going on along the imaginary axis for $\Gamma(z)$

Ted Shifrin

I know Gibbs phenomenon. Who's Stokes phenomenon?

0celo7

@Semiclassical jeez

so many equations

Semiclassical

ikr

short description: if you look at the coefficients of an asymptotic series, you'll often find that they need to be different in different sectors of the complex plane

Eric Silva

oi

0celo7

@BalarkaSen I'm on a ski trip, and while tumbling down the slopes and almost dying I dreamed up two more Hodge theorem proofs for you, I think.

Ted Shifrin

18:06

hi Eric.

Probably you should be paying attention to the skiing and not the almost dying, 0celo.

Semiclassical

and unless one handles the error terms very carefully, you'll incorrectly conclude that those coefficients must change discontinuously across the imaginary axis

0celo7

Well I was paying attention to the skiing but when you're on the ground in pain, elliptic PDE comes naturally

Semiclassical

For a more quantitative description:

Daminark

@Eric done with finals?

Eric Silva

just finished my Foucault paper and now i am free

Ted Shifrin

18:07

0celo, it probably should be parabolic PDE.

hi @Sha

0celo7

@TedShifrin I wasn't flying through the air

just falling to the ground

Sha Vuklia

omg hi @Ted! do you know any logic by chance:p

Semiclassical

As $|z|\to \infty$ with $|\theta|<\pi$ where $\theta=\text{arg }z$,
$$\Gamma(z)\sim \sqrt{2\pi}z^{-1/2}e^{-z}\left(1+\frac{1}{12z}+\frac{288z^2}+\cdots +S(\theta)e^{\pm 2\pi i z}\right)$$

0celo7

and certainly not at the speed of light, so not hyperbolic either

Ted Shifrin

Very little, @Sha. You should ask @Alessandro. He's our resident logician.

Semiclassical

18:09

where $S(\theta)=0$ if $|\theta|<\pi/2$, $S(\pm \pi/2)=1/2$, and $S(\theta)=1$ otherwise.

Sha Vuklia

alright, if he comes around, I'll do that

Semiclassical

also, it's the plus sign in the exponent if im(z)>0

Ted Shifrin

But parabolic eqns are more painful than elliptic, 0celo.

0celo7

@BalarkaSen So there's a very nice proof where you derive a p-form Poincare inequality for forms orthogonal to the kernel of the Laplacian. This gives you nice control for an existence theorem.

Semiclassical

The main point is that $S(\theta)$ changes discontinously to this order of approximation.

That's Stokes phenomenon.

0celo7

18:11

@Semiclassical does this have anything to do with the stokes layer in fluid dyanmics

Ted Shifrin

Ah, interesting how we label "phenomenon" all these sorts of discontinuity/overshoot things.

Semiclassical

No idea. It wouldn't terribly surprise me, given how boundary layers in fluid mechanics are related to the method of multiple scales (which is just a version of asymptotic analysis)

But then it could just reflect the fact that Stokes did a lot of stuff

0celo7

yeah, true

Oh, @TedShifrin, my advisor gave me a copy of this Wu book on Bochner stuff

You mentioned it some years ago

Ted Shifrin

Yes, Wu's an old, old friend of mine. Great writer.

0celo7

Turns out they did a 2017 reprint and he picked up two of them

Semiclassical

18:13

@TedShifrin the main novelty about Stokes phenomenon in the last decades is that one doesn't have to settle for S(theta) being discontinuous

Ted Shifrin

Oh cool. Well, Wu's still alive and kicking.

0celo7

It's nicely written, and also why I have a newfound interest in Hodge's theorem.

Ted Shifrin

Yes, he's a great teacher and expositor.

Semiclassical

One can get a smooth description of what happens across the imaginary axis, it just requires more work

0celo7

I don't like the one I usually think about (from Gilkey)

Warner's is very long

Ted Shifrin

18:14

Warner's tries to use minimal amounts of analysis.

0celo7

Jost's is...Jost. I don't think it's quite right.

Semiclassical

I'm not really interested in the rigorous bounds on the error term (which is where much of the equations in that thesis come from)

Ted Shifrin

I think Griffiths pretty much stole Warner's argument for their treatment in Griffiths/Harris. I've forgotten now.

Semiclassical

But there's also just a lot of careful manipulation going on

0celo7

@TedShifrin Well he does this stuff with Fourier but if you're going to do that, might as well do pseudos a la Gilkey or Lawson--Michelson.

Ted Shifrin

18:15

@Semiclassic: Does your adviser have a good reason for having you look at this?

Wells has a proof using PsiDO.

0celo7

Yes, that's true.

Semiclassical

He's curious about it as a warm-up example for stuff involving the parabolic cylinder function

0celo7

But I found his exposition less clear and certainly less detailed than Gilkey.

Semiclassical

the main point there being that it's got a similar-looking integral representation

Ted Shifrin

I no longer have any of these books. Oh well.

And I'm about to have to move in a few weeks, and I'm regretting the number of books I do still have.

Semiclassical

18:16

So it'd be nice to understand how the story for the gamma function works as a warm-up for the parabolic cylinder function

Ted Shifrin

I see.

0celo7

@TedShifrin Anyway, I think I can do a proof by mimicking the proof that $\int f=0\implies \Delta u=f$ for some $u$

You just have to work a bit harder because the kernel of $\Delta$ on $\Omega^p$ is more complicated than on $\Omega^0$

And this requires very little analysis.

Semiclassical

There's also a temptation on my part to get really into this stuff since this kind of 'resurgent analysis' of integral representations is something I've had an interest in for a while now.

Ted Shifrin

Semiclassic: You really are a mathematician at heart.

Semiclassical

I'm a certain kind of mathematician, at any rate.

proving rigorous error bounds isn't something that excites me, so I'm probably not an analyst

Ted Shifrin

18:20

Right, that makes you more a physicist.

Semiclassical

Yeah.

I live on the line :P

Ted Shifrin

Like the Maginot line ...

0celo7

Error bounds don't really excite us, it's the fact that we can bound errors that does...?

Semiclassical

Yeah, great precedent there :P

@0celo7 That's fair.

Daminark

@Ted where are you heading?

Ted Shifrin

18:21

To the building next door, Demonark. But it's still a royal pain.

0celo7

@Semiclassical It's just a part of the life I guess

Semiclassical

In that spirit, I do find it interesting when there are neat things to be said about how fast the error goes down.

Balarka Sen

@0celo7 Write up all those Hodge proofs for me and I'll look into it three weeks from now

Ted Shifrin

Oh, look, it's a Balarka.

Semiclassical

An example being the rates at which Fourier series coefficients decrease based on how smooth the function is.

Ted Shifrin

18:27

Sure, @Semiclassic, you can be sure I taught that and emphasized it in my applied class.

Semiclassical

My favorite example in that vein being stuff like $f(x)=\sqrt{1-m \sin^2 x}$.

That's analytic in the neighborhood of the real line, so the Fourier coefficients decrease like $c_n\sim a^{-n}$ for some $a>1$.

on the other hand, when $m\to 1^{-}$ there's a kink at $x=\pi$.

Hence there's some interesting stuff going on for how the coefficients behave asymptotically as $m\to 1^-$

0celo7

ok, time to go back out

cheerio

Ted Shifrin

Don't break too many legs, 0celo.

Semiclassical

"too many"

Daminark

See you Ocelot

Ted Shifrin

18:30

@Semiclassic: I'm going back to geometry. :)

Semiclassical

kk

0celo7

@TedShifrin Morwen Thistlethwaite (maybe you know him) said when he was at Cambridge, all the rich kids when to Switzerland and broke their legs

Semiclassical

I haven't done a lot with that matrix stuff I was doing this week, since the main prof I talk to about it is out of town

0celo7

I'm not rich, and not in Switzerland, so maybe I'll be safe

Semiclassical

But I think he'll like the use of the spectral factorization once I get a chance to talk with him again

Daminark

18:31

@0celo7 but you broke both conditions so it's like a double negative

More Anonymous

Is this observation trivial?

0celo7

:( crap

More Anonymous

#math.stackexchange.com/questions/2694045/…

Ted Shifrin

I've met him, 0celo.

Eric Silva

@Daminark double negatives either make him safer or in danger 50/50 chance

Daminark

18:43

(I wonder who in the world had time and decided to make that, but wow does it come in handy)

More Anonymous

WOW! SUPER COOL

who knew the sun had a face :P

Eric Silva

@Daminark i dont understand what u mean, that's 100% real footage of the actual sun

Daminark

oshit

Astyx

18:59

@Daminark I'm bookmarking this

Alessandro Codenotti

Hi everyone

Astyx

Hi

Alessandro Codenotti

@ShaVuklia I heard you have a logic question

nbro

Feel free to add more intuitive answers (i.e. interpretations of the value of the p.d.f., which changes from r.v. to r.v.) to the following question.

8

Q: What does the value of a probability density function (PDF) at some x indicate?

I understand that the probability mass function of a discrete random-variable X is $y=g(x)$. This means $P(X=x_0) = g(x_0)$. Now, a probability density function of of a continuous random variable X is $y=f(x)$. Wikipedia defines this function $y$ to mean In probability theory, a probability...

Alessandro Codenotti

I can't seem to remember the names in probability, cdf, pdf, density, they're all too similar

Daminark

19:09

@Astyx it's excellent

Astyx

pdf is probability density function I think

Mike Miller

portable digital file

Leaky Nun

@Astyx cimer

c'est tres tilu

:P

Astyx

"tilu" isn't something you'd say :p

And that kind of language should only be used with people you are very familiar with

Leaky Nun

i'm in a frenzy now

Astyx

19:11

Else you look like a thug or something

Leaky Nun

je verlan tous

Astyx

Your french seems to have improved at least

Alessandro Codenotti

@Astyx the one for absolutely continuous variables that you integrate to get the probability?

Leaky Nun

cepar que je lanver?

Astyx

Yeah I think

Eric Silva

19:12

@MikeMiller portable document format?

Astyx

@Leaky Non et cette phrase était juste incompréhensible à vrai dire

Mike Miller

whatever

Leaky Nun

lmao

that's the goal, verlan until tu peux pas prendcom

Eric Silva

lol

Astyx

PDF (disambiguation)

PDF often refers to the Portable Document Format in computing. PDF or PdF may also refer to: == In mathematics, science, and technology == === In biology and medicine === Parkinson's Disease Foundation, in medicine Pigment dispersing factor, in biology === In computing and telecommunications === Package Definition File, System Center Configuration Manager Page Description File, used in variable data publishing Pop Directional Formatting, Unicode character U+202C, a formatting character that cancels a previous bi-directional formatting character Printer Description File, describing cap...

Balarka Sen

19:14

3

Leaky Nun

@Astyx one should edit a french wiki page to verlan every mot

Eric Silva

i hate this meme

Leaky Nun

@EricSilva I'll keep using it; CHANGE MY MIND :P

Eric Silva

you can do w.e. u want my dude

Mike Miller

I liked it originally

"I drink piss" was a classic

Balarka Sen

19:15

you won't change my mind

change my mind

@MikeMiller pee is stored in the balls

that's a 50/10

Alessandro Codenotti

How is the pushforward measure of the random variable called in English? Distribution or density?

Balarka Sen

hard to not love

its just such a perfect summary of Steve Crowdy boi

Semiclassical

19:35

Poisson summation is cool

the content is sorta obvious---Fourier transform of a periodic function should be related to the Fourier series coefficients---but it's still really powerful.

Sha Vuklia

20:07

@AlessandroCodenotti hey thanks for getting back at me, but the question is solved!

Alessandro Codenotti

I see, what kind of logic are you dealing with?

Sha Vuklia

we've just been introduced to Stone duality

and we are also doing stuff with interpretations

like, those are two separate topics we cover

Alessandro Codenotti

I'm glad the question is solved because I don't know anything about Stone duality :P it's a topic I should read something about though

Sha Vuklia

o lol, yea well my question had to do with atoms, and it was actually something very simple, but it took me hours

I wanted to show that each finite BA has an atom, but I forgot a very simple thing, which resulted in me complicating a lot

Daminark

@Alessandro I think it's the distribution

And that should be the CDF

The density is gonna be the Radon-Nikodym derivative of the distribution wrt the Lebesgue measure, and that's sometimes called PDF

If I remember this right

Alessandro Codenotti

21:48

@Daminark makes sense, so you need an absolutely continuous variable to have a PDF

user193319

Suppose that $V$ is a vector space with some involution $*$ (e.g., matrices with conjugate transpose). If $\phi : V \to \Bbb{C}$ is some linear functional, is it true that $\overline{\phi(v)} = \phi(v^*)$?

Alessandro Codenotti

Is the involution also linear?

user193319

@AlessandroCodenotti Yes.

Drew Brady

Anyone able to answer this quick question? Suppose A and B are symmetric matrices. What are necessary and sufficient conditions for Tr(AB) = 0?

Let me restate the problem as follows: Let A be a symmetric, square real matrix. For which rank-1 positive (non-negative) definite matrices B is it true that Tr(AB) = 0?

Alessandro Codenotti

22:07

If I'm not mistaken $\Bbb C^2$ as a $\Bbb C$ vector space with $*(x,y)=(y,x)$ and $V((x,y))=x$ is a counterexample

The Testosterone Fanatic

23:21

hey @AlessandroCodenotti

ah he's gone

Tuki

Hello

The Testosterone Fanatic

hello

Tuki

How's it going ?

The Testosterone Fanatic

I'm good, just doing my homework and wasting time on stackexchange

Tuki

Homework on what topic if i may ask ?

The Testosterone Fanatic

23:31

measure theory

you?

Tuki

Well i am trying to practise for entrance exams for uni right now

The Testosterone Fanatic

good luck

Tuki

thanks

Xander Henderson

23:54

I would once again like to invite anyone with better harm anal knowledge than myself to have a look at this question.

According to the exercise the OP posted, I am an idiot (which is an entirely reasonable conclusion), but I would like to know in what specific way I am an idiot.

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$Mathematics$ Mathematics