@amWhy Of course. No objections from me.

@XanderHenderson And Kant was around then, too! But the key point in your choice is the "if I could choose to be born into an academic track..."

@amWhy Exactly.

@Xander Your daughter for president!!!

But I think that if you ask the question "If you could choose to be born at any moment in history (prior to today), into a random family taken from among the population of the world (or even a chosen geographical region), when (and where) would you choose to be born?" the only possible answer is "Now-ish".

@amWhy I mean, I would support her, but I would not wish a life in politics on anyone.

@XanderHenderson Yeah, I'm with you on that!!

Katja found a "Super Nanny" clip on YouTube, and is shouting at me that the narrator is the same narrator as from The Great British Bake Off. It is not, but they both have British accents.

@XanderHenderson As Plato argued, we need a Philosopher Queen to lead.

I guess all British men are the same...?

@amWhy Angela Merkel gets close, no?

@XanderHenderson Oh, I thought all men are the same?? :-)

@amWhy hash tag all men?

@XanderHenderson True!

She has a physics degree? right...?

"Quantum Chemistry"

Anywho, time to go make dinners.

when I become president, we'll have a philosopher (and mathematician) queen!

@XanderHenderson Anywhy, go eat!

@Arjun firebolt is wayyyyyyyyy better

@amWhy i'd decide after looking into the mirror of erised. if i see myself holding a pair of woolen socks i'd choose to be myself

@amWhy planning on running for elections? (i'd ask my cousins in the US to vote for you)

How was the interview @XanderHenderson ?

@amWhy Yay!

@sai-kartik I think it went well.

Best question they asked: "What is a logarithm?"

@XanderHenderson for someone with a doctorate, i guess you must have paused for a second and thought: "okay. Where to start?"

No, because it is a teaching position.

So they are looking for a particular kind of answer.

I.e. one which could be presented to a student.

@XanderHenderson ahh yes of course..

But the logarithm is one of two functions which I have thought quite deeply about with respect to instruction (the other (really two) is the (co)sine function(s)).

What a logarithm "is" very much depends on who you are teaching.

@XanderHenderson oh sweet! really tested out your teaching skills then!

In a precalculus class, the emphasis is on coming to terms with functions. What are functions, how do they behave, etc. In that context, we typically define the exponential via a little bit of handwaving (but based on the intuition of $x^q$ for rational $q$), then define the logarithm as the inverse of the exponential.

@XanderHenderson hmm true. i saw a question here on mse asking how to explain a logarithm to 5 yo and there was answer that used cookies (and probably the cookie monster)

In a calculus class, I think that it is reasonable to pretend that logarithms don't exist until you get to integration. The logarithm them becomes the antiderivative of $1/x$. Geometrically, the fact that $\log(ab) = \log(a) + \log(b)$ pops out in a nice way, and the inverse function theorem becomes interesting for relating $\log$ and $\exp$.

And in an abstract algebra course, one might happily decree that the logarithm is an isomorphism of groups from the multiplicative group of positive real numbers to the additive group of all real numbers.

And there are other approaches, as well.

so which definition have you given?

I gave those three.

In that context.

ohh nice..

For me, however, the "right" definition is geometric: $\log(a)$ is the area under the graph of $1/x$ between $1$ and $a$.

@XanderHenderson so they must have thought you know what you're talking about..

@XanderHenderson i was taught the inverse power definition back in my 9th grade.. so i still stick with that

@sai-kartik You shouldn't just "stick with" a definition. Since all of the definitions are equivalent, you use the one which best helps to understand a given problem.

@XanderHenderson yes..that's not what I intended but most of the time, I don't have to use the other two definitions (i've never even heard the 3rd until you've mentioned it) so using the inverse-power is kind of second nature to me

@sai-kartik From that, I would guess you to be a second or third year undergraduate, at most.

That is, entering into your second or third year.

@XanderHenderson ive just finished my 12th grade :P xD

So, I'm not off by much.

@XanderHenderson yep

You haven't taken any upper-division, proofs-based courses.

nope

Once you start proving things, it becomes important to be able to think about things from a lot of different angles.

we dont have such options for high school in india

@XanderHenderson yeah.. mse makes that point really clear for me

In any event, I should go to bed.

It is late.

what time zone btw?

(Well, it is 8pm, but I'm, like, an old man...)

ah ok

12hr diff

ill remember.

good night!!

@amWhy, today is friday ;-), just saw some portion of the movie Into the wild.

i gotta say it is better then most of the crap that's circulating

@XanderHenderson Eeks. That's not the right definition for me. It doesn't even work for the complex logarithm. Might as well just say it's the path integral rather than area..

I personally prefer the inverse of exp definition so we can get straight to using it algebraically. But talking about the path integral, what's the easiest way you know to motivate it and prove its equivalence to the inverse definition?

Happy Friday @amWhy

Thanks @Arjun and @sai-kartik !

@amWhy what'd you plan for today?

@user21820 Again, depends on context.

I should also note that I am not really thinking of it as a path integral; I am thinking of it is as an area under a curve, corresponding to a real integral. This is, perhaps, a quibble, but given that I am thinking of this in the context of a calculus (or maybe real analysis) class, it is an important distinction.

To get from $\log(a) = \int_{1}^{a} 1/x \,\mathrm{d}x$ to the logarithm being the inverse of the exponential is not too difficult: you get the important properties of the logarithm without too much difficulty (e.g. $\log(ab) = \log(a) + \log(b)$), which makes it possible to prove that $\log(a^n) = n \log(a)$ by induction.

This extends to rational powers in the "obvious" manner, and can be extended to $\log(a^x) = x \log(a)$ by a continuity argument.

The simplest trick to get that logs and exponentials are inverses at this point is maybe an application of the chain rule... I'll have to think about that for a second...

Or maybe the inverse function system (as I originally claimed)...

Upon further thought (and I'll note that I am still brainstorming), I think it is going to take a few more steps. Define $\exp$ to be the inverse of $\log$. Show that $\exp$ has all of the nice properties of $x \mapsto \mathrm{e}^x$. Conclude that the logarithm is the inverse of the exponential.

Oh, wait... it is easier...

Define $\exp(x) = \log^{-1}(x)$. Then $$\frac{\mathrm{d}}{\mathrm{d}x} \exp(x) = (\text{inverse function theorem}) = \exp(x). $$

@XanderHenderson: I meant the path integral for the complex log.

Give that an appropriate initial condition, and you are done (so, define $\mathrm{e}$ as the solution to $\log(\mathrm{e}) = 1$, then $\exp$ solves $u'=u$ with $u(1) = \mathrm{e}$).

I know it's relatively easy to prove equivalence for real log definitions.

But how to motivate the path integral for complex log?

What's the easiest way?

I don't know. I haven't thought about that sequence of steps.

In complex analysis, I doubt that I would ever try to define the logarithm as an integral in the first place.

Rather, that is a property which should pop out from a more useful definition in that context.

But many people do in fact define the complex log via the path integral that does not cross the negative real line.

@user21820 Interesting. I believe it, but I haven't every taught complex analysis, so I haven't thought about this in any great detail.

The first thing that comes to mind is to prove everything on $(0,\infty)$, then apply the identity theorem.

But this seems to be in the wrong spirit.

That's why I asked, because I don't see an easy way to motivate it.

I know one can use the Cauchy theorem to make sure the definition is well-defined.

But why should we think that it produces the inverse of exp?

@user21820 I don't think that there is any "obvious" or "natural" reason to think that it does. We can certainly prove it, but that doesn't mean it is pedagogically correct.

Well there is one particular good reason to do it like this, because there is one particular important usage of this path integral, when we want to define a branch-cut using a simple continuous path from the origin to infinity.

And the only way to do it is via path integral; inverse of exp seems to be a terrible idea even if we can get it to work.

@user21820 Indeed. This is a situation where good understanding of the underlying theory is important.

One day I should sit down and think carefully about it.

Right now I'm thinking about perfect-information abstract 2-player games with infinite plays.

=D

Heh.

@XanderHenderson: I finally recall the theorem name: open-mapping theorem. If I recall correctly the path integral construction of log was needed to construct a sqrt function with such arbitrary path to infinity branch-cut, which in turn was needed to prove that theorem.

Wait... that's not the one.

It's Riemann mapping theorem.

@user21820 That's the one that asserts that all open sets (other than $\mathbb{C}$ and the emptyset) are equivalent up to bi-holomorphic mappings, right?

Yea.

Simply connected open sets.

Sure. Yes.

Yeah, I can see how a path-integral log would be useful there.

Yea, and the not-all-of-C is where the path to infinity comes in.

Ah, right. Okay, that makes sense. I'll have to think about that in my spare time...

It's not my area, so maybe there are better proofs, but that's the one I learnt back in undergraduate days.

@XanderHenderson Who has spare time these days!?! ;D

@amWhy Only on fridays!

=D

@user21820 Ha!

@amWhy :P

Obviously you, @Xander, @user21820, even @quid, and I, have enough free-time to chat once in a while. So I have my answer!

Heh.

That's right!

You too. =)

@user21820 Indeed, hence the "and I"...

Oops I missed that on first reading. =P

@user21820 (I edited in @quid).

Hi, @AloizioMacedo! And all!

@amWhy, I have a new book for you

The room on the roof

Written by Ruskin Bond, published when he was only 17!

It is set in the post colonial rule in India, about the life of an Anglo Indian teen

@Arjun Thanks! I'll look for it!

@user21820 @XanderHenderson I don't know if this is in line with what you want, but one way to motivate the path integral is by proceeding with the following ansatz-like approach: We want some $f$ for which $\exp(f(z))=z$. By the chain rule, this will imply $\exp(f(z))f'(z)=1$, and thus $zf'(z)=1$, $f'(z)=1/z$. This suggests the definition via the path integral due to the FTC.

@amWhy Hello!

@AloizioMacedo Yes that's the usual way I do it. Even at the lower level, if we have variables x,y such that y = exp(x), then dy/dx = exp(x) = y ≠ 0, so dx/dy = 1/y.

The question though is how to motivate the idea that you must cut the plane when trying to get the complex log.

I think a pedagogical approach is to rationalize that this is where the ansatz-like approach must proceed carefully: If we define it as the integral and just leave at that, then it will not be a well-defined value. Now it depends on whether it is already intuitive that simply-connected domains allow for that or not. If it is, then things close up nicely. If it is not, I'd suggest to build that intuition beforehand, for example via Stokes' theorem ideas which the student are probably familiar w/.

@quid, Happy Friday! I hope the day has not been too busy for you!

@Arjun I want to read it so bad! you have a link in hand right now?

@Arjun where i can read it for free of course (typical indian middle class)

@sai-kartik ,as you are an indian. I'm sure you will love it, 1 min for link please

@XanderHenderson If i didn't have my chatjax on, i wouldn't have seen it in its splendor..

@XanderHenderson You going to do that again, next Friday, July 3rd? ;-) Very nice, by the way!!

@amWhy Probably not. That took too much work.

@amWhy the G and F stand out with the grey bg when i hover over my/your reply

@XanderHenderson :-) Nice job.

Also, what is "Fry Day"?

I seem to remember that, back in the before times, there were distinct days.

@XanderHenderson french fry day?

And something called "Weak Ends".

@XanderHenderson Fish Fry Friday, abbreviated Fry day during Lent!

But then the 'rona came.

And all the days bleed into one.

i have a tshirt that says "Its <french fry.jpg> day!"

@sai-kartik, sorry to say but I could not find any link

I recommend you to buy it

@Arjun don't worry! i'll eventually find it!

@sai-kartik, btw I got this book signed from him

@Arjun OH MY GOD!!!

you lucky, lucky guy

btw have you read the harry potter series?

I think you have ..

@Arjun .....

speechless

@Arjun ahh i see you are a man of culture as well

@sai-kartik ,this is nothing in comparison to the time I spent with him

@sai-kartik , :-)

@Arjun stop.making.me.jealous.

@sai-kartik , sorry. I'll stop here

^_^

@Arjun yes

thanks XD

What else have u read?

dan brown

have you read?

@sai-kartik , heard of him

da vinci code

Never read tho

must read^^

the suspense and the thrill

@sai-kartik , it's struck

the ending will never be what you expected

I ordered it, Amazon cannot deliver

@Arjun sorry?

@Arjun ohh

try getting a kindle version

you can read it on your ta

That was few months back

tab*

@Arjun ahh then the 'rona came

@sai-kartik , I already wear glasses

@sai-kartik , how do u know I have a tab?, Link?

@Arjun so do I. But that doesn't stop me ;P

@Arjun your screenshots match those of a samsung tab if im not wrong

s4?

I have -6, on one eye and -6.5 on other

@Arjun yikes. i have a -3

@sai-kartik , sorry to disappoint, it's Lenovo m 10

@Arjun dammit

Do u read fantasy?

@Arjun im open to all formats of novels

i just LOVE reading books

Jeffrey Archer

ohh

Bestest short storyteller

cool will check out

heard his name before

@sai-kartik , amateur 😂

@Arjun i could say the same for dan brown xD

Hey, do u know about EVP?

evp?

@sai-kartik , 😑😑

@sai-kartik yep

im sorry could you expand

Extreme value ...(something)

nope 🤐

😬

(okay someone's really disappointed xD)

I'm glad the two of you are bonding. But lets try to be more inclusive in this chat room? It is public, afterall.

@amWhy anyone is most welcome to comment about novels/classic literature. I'm sorry if that intention wasn't clear 😬

@sai-kartik No problem!

@amWhy :)

what bout you, @amWhy what books do you really like?

I have had very little brainspace for pleasure reading recently.

:\

The last thing I read for "fun" was, I think, NK Jemison's Broken Earth trilogy, which was incredible.

@XanderHenderson you'll probably get more time when you're a Professor, (X) :P

@sai-kartik , nah I was looking for it in books. sorry for leaving

@sai-kartik I don't read as often as I wish I could. But I like novels by Hermann Hesse, some Steinbeck, some Shakespeare (especially Hamlet), May Sarton, Ursula Legun... Oh, I can't begin to name them all!! Plus I like non-fiction, too.

@sai-kartik Doubtful.

@amWhy Ursula LeGuin was a master.

@amWhy have you heard of sydney sheldon?

I can't wait until my daughter is old enough to get into Earthsea.

@XanderHenderson I know!! At both short stories, and novels.

i think he was famous at around the 2005-ish time cus that's what all my grandpa's books of his are dated..

@Arjun thats alright. btw what is evp?

@sai-kartik I think he wrote a lot of mysteries?

@sai-kartik , couldn't find

@amWhy yeahh

@sai-kartik I think he was famous earlier, because my mom died in 1996, (too young), but she read a lot of his works.

@amWhy , we'll take care of that in future

@amWhy I did not enjoy her later, more mainstream, work as much, but The Word for World is Forest was mindblowing to my 10 year-old self.

@amWhy i've read are you afraid of the dark by him

@amWhy sorry bout that :/

And it is short story collections by LeGuin and Zenna Henderson (no relation at alll) which got me into sf in the first place.

@XanderHenderson Absolutely. I also enjoyed The disposseed? I've already posted a link to her mid seventies "the Ones Who Walk Away from Omelas" (Short Story) here in this chat.

@amWhy could you please re-share?

@amWhy The Dispossed was quite good. I haven't read "The ones who walk away from Omelas".

also ,people, if you remember some other literature that you want to share, you can do so in my chatroom: chat.stackexchange.com/rooms/109808/life-on-earth. It's specifically built for things like this..

@XanderHenderson The Ones Who Walk Away from Omelas:

sites.asiasociety.org/asia21summit/wp-content/uploads/2011/02/…

@XanderHenderson Actually, it seems that I have read it (it is collected in *The Wind's Twelve Quarters"). I just don't remember it at all.

@XanderHenderson Indeed, that's where it first appeared. It has since been included in many anthologies of literature, philosophy.

I have a signed, first edition copy of a printing of "Nine Lives" running around somewhere.

It was a very formative short story for me, thanks to a teacher who knew I'd appreciate it.

@XanderHenderson Wow!

That is one of my favorites.

I don't have a signed copy of any book :/ just hope i get one someday..

I have a book, Conversations with Ursula K. Le Guin (Edited by Carl Freedman) which is quite delightful. Signed by Ursula! And I have a signed copy of William Sororyan's "The Human Comedy". (Apologize for any mis-spellings.)

Copyright, first edition, 1943. I found it at a used book store, and had the signature authenticated.

@amWhy whoaaa nicee

Can i ask a doubt?

@Arjun Sure!

Dammait, it's wrong orientation

2 nd question, just a hint would be appreciated

@amWhy Wow!

@amWhy I love Le Guin. I was devastated when she passed away.

@XanderHenderson I know. But I can't read it again, because despite having it under glass, it had already accumulated a lot of dust, prior to acquiring. So after my first read, which was tough enough (I'm extremely allergic to dust!!), I keep it because I loved the book, but I can't really read it without suffering severe sneezing and an asthma attack!

@EddieKal Me too. But I think she lived a very full and rewarding life. That's all any of can ask for.

@amWhy That sucks. :(

@EddieKal You've changed your name!! :-)

@Arjun take ln both sides to begin with

@sai-kartik , did that, next

@Arjun race material or module?

Module

@Arjun wait a min

@amWhy Have I? I thought I always sported this name on most sites. What was it before?

@EddieKal Ohhh, I'm sorry! I thought your were sai-kartik. I didn't look closely enough at the gravatars! I'm very glad you enjoy Ursula Le Guin's work! Thanks for stopping by!

(I was looking only at the tiny squares in front of comments, not at the larger gravatars to the right.)

@amWhy looks like you've confused Miles' hoodie with a roof :P

@sai-kartik lol. My avatar used to be in black and white. Someone from the ELL chatroom colored it for me.

It's a millhouse

@EddieKal ohh nice!

@EddieKal yeah.. neat coloring tho..

Thanks!

@EddieKal :)

@sai-kartik You said Miles, so that's Miles Morales?

@EddieKal yep. My fave spiderman

@sai-kartik I have only seen the 2018 film's trailer. Need to catch up

Neat avatar!

@EddieKal thanks! its an amazing film btw

@amWhy Editor of that book, this guy? lsu.edu/hss/english/faculty/faculty/cfreedman.php

I have been meaning to read more sci-fi criticism, sci fi studies. Havent seen many good books on sci-fi

@EddieKal Yes, I believe so. Have you ever read the classic Ray Bradbury?

Sci Fi didn't originally mean Star Trek and such. It was meant to be a projection into the future of trends in current society, and meant more as a commentary on current society, than on fantasy. There is definitely an overlap between Sci Fi and Fantasy, though.

Friday to @quid ... Signal Friday to @quid :-) Stop in if and when you can. Nothing urgent. Just, it's FRIDAY! ;-)

@amWhy Only Fahrenheit 451. Dystopian is my fav genre :)

I haven't been following the sci fi community since, um.. Sad Puppies

Sad Puppies scared me.

@EddieKal Ugh... those trolls.

Best to just forget them.

Well, if @quid is too busy to drop in here, so am I, :-) Good afternoon, good night, good morning, folks!.

In fact, @quid and Co., I don't think any user should put any more time into this site than an elected mod puts in. You might argue that your time is twice or three times more valuable, spend on this site, as a mod, but I don't buy it. I'd like to see how the mods handle regular users not putting in more time on this site than you do, because it seems, as a mod, you put in the most time here, relative to mods. But you couldn't function without the dedication, @quid, of other users who log in

...far more hours than you do, and hence, than any other mod does. There. I've said it. I think mods actually take advantage of all the loyal users on this site logging in more hours than they do.

Happy Saturday, @quid!