The h Bar

12:00 AM

@ACuriousMind Agreed. And also uncertainty about what may or may not have changed in interim.

@dmckee Why should the question "nod in the direction of the solution"? It sort of makes sense in this case when I'm cleaning the question up, but not for a genuine stumper.

dmckee

@rob Hm. Sorry. Not in the direction of the solution to the question being asked here. In the direction of the solution that Levine provides—I consider that part of making the question self-contained, is all.

Something along the lines of "by conserving energy you can obtain the simple harmonic equation of motion" or "the system is equivalent to a pendulum so it is a simple harmonic oscillator for small oscillations".

0celo7

12:16 AM

@ACuriousMind Nonsense

Can you give a reference?

rob

@dmckee Yeah, that makes sense. With guidance from in here I feel more comfortable just re-opening. Thanks, all.

dmckee

@rob Nicely done.

David Z

@heather for the record, the license allows you to import questions from PSE to other sites, as long as the attribution guidelines (and, in general, the terms of the license) are followed. You don't technically need to ask for permission; you already have permission, and we do not legally have the right to revoke that permission.

Bernard Meurer

Sup people

dmckee

@BernardMeurer Supper is still cooking. I can't yet.

But don't ask me why I interpreted that as a verb. I have no clue. It just happened.

Bernard Meurer

12:27 AM

@dmckee Lol, how are you man?

rob

dmckee

I'm behind on grading and going short on sleep because of it. But being short on sleep makes it hard to stay on task...

@rob But sup was a verb a long time ago. It means to eat. Particularly to eat the main meal of the day.

0celo7

@BernardMeurer you need to teach Reb python

Bernard Meurer

"Maybe we can make language a complete impediment to understanding" And then VHDL was created

@0celo7 Skype

I'll teach you guys

I taught this shit back in Rio

rob

@dmckee Verbing is as old as English, at least. Shakespeare does it for sure; I'm not well-read further back than that.

Bernard Meurer

12:32 AM

@dmckee I need VHDL help

dmckee

Hmmm. So we need to know if supper predates sup or vice versa. This is where I miss the OED in the graduate reading room in the NMSU physics building.

I'm sure the university library has one, but that's too far to go. In Gardner hall it was just sitting on a podium right next to the door of the reading room.

rob

@dmckee dict.org suggests that sup, sip, and soup are all closely related, and that perhaps "supper" started off as the adjective: "supper time", "supper bell", etc.

@dmckee oed claims "sup" was present in Old English, but "supper" dates to the 1300s and then in the liturgical context.

Also, holy cow, I've forgotten how great the OED is.

vzn

@Secret violation of bells inequality in fluid mechanics / anderson, brady

0celo7

12:55 AM

@ACuriousMind Hey, can you give an example of a wave function not in $L^2_c=L^2\cap C_c$?

Is that space not a Hilbert space or something stupid?

ACuriousMind

@0celo7 According to Wiki most people and in particular Coxeter (reference given there) agree with me

0celo7

@ACuriousMind Coxeter is also the one who says the boundary of the $n$-ball is the $n$-sphere.

His terminology and definitions are worthless.

ACuriousMind

@0celo7 I can, but we've done this at least twice already, so I won't do it again for you.

0celo7

@ACuriousMind Sigh.

Not a function in $L^2_c$.

A wave function.

Specifically, a solution to the SE.

ACuriousMind

A wave function is by definition a function in $L^2$.

0celo7

12:59 AM

No, a wave function is a solution of the SE.

ACuriousMind

@0celo7 Pick any function $\psi$ in $L^2 - L^2_c$. Then $\mathrm{e}^{-\mathrm{i}Ht}\psi(x)$ is a solution to the Schrödinger equation. I don't know what you want from me.

0celo7

@ACuriousMind That's cheating.

I don't think your $\psi$ should be allowable initial data.

bolbteppa

MO post appears here on an awesome topic

ACuriousMind

@0celo7 Maybe, but it's true.

@0celo7 That's not my problem.

0celo7

Sorry for aggravating you

Bernard Meurer

1:04 AM

Someone stop VHDL from being a thing

@ACuriousMind Help me

0celo7

1:17 AM

@ACuriousMind How would you distinguish a nuclear reactor from a nuclear explosive?

Bernard Meurer

@0celo7 I was about to make a racist joke, but I held myself

0celo7

Feel free

dmckee

Freedom of speech means that you can make such jokes. It doesn't mean that you should.

Though of course, someone has to do it occasionally just to make the point, which always makes me uncomfortable.

Bernard Meurer

@dmckee Meh, it's pointless, ArtOfCode will come and ban me

dmckee

Well, you don't experience unlimited freedom of speech in a SE chat-room.

Bernard Meurer

1:28 AM

@dmckee Also, it's pointless, I'm not drunk so I won't even find it funny

dmckee

::chuckles::

0celo7

@BernardMeurer Text me the joke.

God I hope she doesn't win.

You're right @bolbteppa

dmckee

If there is any actual humor in them I find them funny anyway. Then I feel a little guilty about it.

Bernard Meurer

@0celo7 Smart

0celo7

@dmckee I'll tell you the joke if it's not too bad.

Bernard Meurer

1:30 AM

@dmckee I never feel guilty, but I do try and avoid offending people who didn't ask for it

I save the effort to those who deserve my wrath

0celo7

Are you gonna text me or not

Bernard Meurer

@0celo7 It wasn't a funny joke

But sure

0celo7

jesus

you'd get banned in a second

Bernard Meurer

@0celo7 I know lol

@0celo7 Did you giggle?

0celo7

No, it's not funny

Bernard Meurer

1:37 AM

@0celo7 Told you

0celo7

you're a bad person

Bernard Meurer

@0celo7 I'm aware

user228700

Hi everyone :-) I was wondering if anybody could help me with some very basic geometry? Basically, I'm trying to derive the value of the eccentricity of the standard ellipse, $x^2/a^2+y^2/b^2$.

user228700

$e$ is defined as the ratio of the distance of any point on the conic from a fixed point, the focus (in this case, focii) to its distance from a fixed line, the directrix.
And I'm having trouble with this because although I know the coordinates of the focii, I don't know the equation of the directrices.

0celo7

1:56 AM

@BernardMeurer how many times?

Bernard Meurer

@0celo7 12

0celo7

Bingo bango bongo

Bernard Meurer

Hm?

user228700

2:13 AM

OK, I've boiled it down to one small question. How is it that the distance b/w the point $(0,b)$ to the focii $(±c,0)$ is $a$ in the following diagram:

user228700

(Pls excuse my drawing skills...its lack thereof :-P)

Secret

Last night dream is a complex one: Basically it talked about one of my professor have published an electromagnetism model in a open access journal called Hecco. Similar to the multipole expansion, the electromagnetic field was divided into many terms in a series. This is represented as a figure as follows:

The 0th term is an irregular shaped blob in the middle, the 1st order term are some fragments hovering above the blob. The 2nd order terms are represented by even sparsely distributed fragments which are surface bundles.

While whatever use of potrygian duality mentioned in the dream is nonsense, the basic idea I believed have been used in various video game rendering: Render only as much you need to know given a scale, so as to conserve computation resources

For this example here, according to the dream, the fragments will become more numerous as you move closer to the object

Bernard Meurer

2:30 AM

@Secret What the hell is that drawing?

Secret

A very rough (my drawing skills kinda sucks) illustration of the main object I saw in that journal webpage in last night dream

Bernard Meurer

@Secret Your dreams are too hardcore for me

I dream of like, getting the kernel to compile, women and free beer

mostly free beer

also I dream that I forgot to file in my Digital Systems assignment

So I wake up at 5AM to check :)

Secret

within the dream, it is said to be the potrygian dual (a group theoric version of fourier transform) of the electromagnetic field, pretreated in some unknown way

I do have slice of life dreams like those you mentioned, and in fact they are the more common ones compared to these

Bernard Meurer

I actually had a very peculiar dream this week

I bought this super stinky cheese (real life)

but it's really good

and (dream) because it was so stinky my neighbor came to complain with me, the Hungarian, not the Slovenian

Secret

what type of cheese?

Bernard Meurer

2:36 AM

Goat cheese, really soft inside, cream-like

And she opened my fridge and got my cheese, and I was so angry that we got in a fight

I don't know what happens next, but I know it ends up with we having sex

Which is very weird to me because I'd never have sex with someone who doesn't enjoy good cheese

But yeah, go figure, weird dreams

Secret

Yeah, dreams are very good at continuing whatever stories happening in waking life. I often have more extreme versions of this when I am on the plane and there is basically no boundary between waking and dreaming. For example, I had one where I fell asleep in my chair (real life), then heard some guy next to me mumbling. I then wake up and ask my friends they said the guy in question never said a thing

Some dreams reflects what you subconscious desires or complaints. If you remember or log down enough of them, you started to see the pattern

Bernard Meurer

@dmckee You still around prof?

dmckee

Sorta.

Bernard Meurer

@dmckee Could you give the cover letter I'm writing for applying to a job at GitHub a read when you have time? It'd mean a lot to me :)

(I know you're busy with grades, so saying no is totally fine, no stress)

dmckee

I can give it a once-over.

bolbteppa

2:42 AM

@Kaumudi (Using my notes) an ellipse is the set of all points whose distance from two fixed points (called focal points, or foci) is constant. Taking $F = (c,0), F' = (-c,0)$ as the foci we see the distance from any point $P = (x,y)$ to $F$ and $F'$ is constant: $PF + PF' = 2a$. Why $2a$? Well, if it was a circle it'd be $PF + PF' = a + a = 2a$ so by convention just use $2a$ I guess.

@Kaumudi Now, drawing $F,F',P$ you immediately see $PF = \sqrt{(x-c)^2+y^2}$ and $PF' = \sqrt{(x+c)^2+y^2}$. Now, squaring $\sqrt{(x-c)^2+y^2} + \sqrt{(x+c)^2+y^2} = 2a$ you end up with $(a^2-c^2)x^2+a^2y^2=(a^2-c^2)a^2$ or $b^2x^2+a^2y^2 = a^2 b^2$ so that $a$ and $c$ distinguish between ellipses.

Bernard Meurer

@dmckee Sounds good to me :). What's your email (it's on Google Docs so I can share it)

bolbteppa

@Kaumudi Scaling $a$ and $c$ by $\lambda$ sends $x^2/a^2 + y^2/b^2 = 1$ into a new $(x')^2/a^2 + (y')^2/b^2 = 1$ of the same shape just a different size, so that $a/c$ or $c/a$ characterize an ellipse invariantly. Choosing $c/a = e < 1$ as our invariant, different values determine how 'eccentric' our ellipse is.

Obliv

@secret My friend had a dream a couple days ago where he was playing videogames with me as per usual and everything seemed normal (as if he wasn't sleeping) then he decided it was time to sleep. He woke up as soon as he fell asleep in his dream lol

dmckee

@BernardMeurer Have you stalked me enough to know where I work?

Bernard Meurer

@dmckee You bet

dmckee

2:43 AM

mckee-d @ <that place>

bolbteppa

@Kaumudi (Write $b^2 = a^2 - c^2$ then scale $a$ and $c$ to get the new $x'$ and $y'$ if it's not clear)

Bernard Meurer

not dmckee@<theotherplacewithk>? :P

Obliv

@secret I also had a dream a couple weeks ago where I was dreaming about having an inception dream. I woke up in my dream to tell my sister about the inception dream, then eventually I woke up because things weren't adding up and realized that inception dream was 2 dreams deep.

Bernard Meurer

@dmckee Sent

bolbteppa

(ellipse is the set of points such that the sum of the distances from two foci is constant)

dmckee

2:45 AM

@BernardMeurer No, I don't get that one anymore. They cut me off about one year after I separated.

Secret

@Obliv I have something like this before, but it is more common for me to actually sleep through to the next day' (' mean the time is based on the dream's own history)

Obliv

she added cockroaches in the pot of rice she was cooking so that was a red flag for me to wake up lol

Bernard Meurer

@dmckee Edit your inspireheap profile then I'd say, you got'em both there :)

Man this cheese really stinks holy shit

It's like I'm living inside a cheese

And I keep that shit inside a box, inside a fridge, inside a closet

Secret

@Obliv My highest record dream is 5 layers. But I have dreams more extreme than that. I have dreams where I did something in layer 1, went to sleep and end up in layer 2, did something and wakoe up back in layer 1, go to sleep to the next day' and then finally woke up

Obliv

oh my god LOL @secret yeah this is not a common occurrence for me like it is for you.

bolbteppa

2:49 AM

@Secret Lunacy, I like it

Bernard Meurer

My dreams are a drawing made by an unartistic 4 year old

Obliv

your home smells like cheese even though it's in the fridge? o_o @bernard

that is strong stuff

Bernard Meurer

@Obliv Dude I'm telling you this cheese has 3 containment levels but I still can't even smell my own fart over the stink of the cheese

dmckee

@BernardMeurer Uhg. I think I have to make an ORCID ID before I can fix that.

Obliv

just imagine what your farts would smell like after eating the cheese @bernard

Bernard Meurer

2:52 AM

@Obliv I just ate a fifth of the cheese lol

@dmckee Damn :/

@heather What time zone (relative to UTC) are you in? You're only here when it's 3AM for me :P

@Obliv And 2 beers

Secret

I also have dreams where real dates and dream dates became scrambled in memory and the dream scenes, and having brothers or sisters I don't really have yet I will not recall they don't exists until I wake up

The most common format of my dreams based on analysing my 5 years worth of dream logs is the following:

1. Every dream has its own continuity, thus I can have memory of past events that actually don't exist. It sometimes get more interesting when I realised that within the dream. Sometimes real life memories will play along in the dream in a smooth fashion

bolbteppa

Laplace-Runge-Lenz vector, derivation from thin air? Comes from crystallizing the notion of orientation? Landau has some absolutely mental classical mechanics degeneracy thing underlying it

Bernard Meurer

@dmckee Email sent btw

bolbteppa

Can get it taking a time derivative and it magically arising, but still, wtf

Obliv

@secret If only I could be in your head for a night. Would be more wild than taking LSD probably.

Secret

3:00 AM

One thing I am pretty sure about is that dreams are based on the waking life. This is because before I first read about tesseracts, I never had any higher spatial dimension dreams, similarly for time travel

I also differ from most of my friends in that I cannot really incubate dreams, because there seemed to be a dream element selection rule that selects the thing that I am least aware of in real life unless it shocked me enough

user228700

@bolbteppa

user228700

> "so that $a/c$ or $c/a$ characterize an ellipse invariantly. Choosing $c/a = e < 1$ as our invariant, different values determine how 'eccentric' our ellipse is."

user228700

What dyou mean by "$c/a$ is invariant"..? How so?

vzn

@Obliv cf zhuangzis butterfly dream en.wikiquote.org/wiki/Zhuangzi

dmckee

@BernardMeurer Couple of comments: You've got one paragraph doubled. And it sounds like you are answering the question "Why do you want to work at GitHub?" without answering the question "Why does GitHub want to hire you."

In most cases you should be addressing the latter as much or more than the former.

Even if you can't show a lot of credentials you have answers to the question about why they want you. You have a broad enough interest in technology to put linux on a calculator.

bolbteppa

3:09 AM

@Kaumudi Given $x^2/a^2 + y^2/b^2 = x^2/a^2 + y^2/(a^2-c^2) = 1$, you see the ellipse only depends on $a$ and $c$, and scaling $a \mapsto \lambda a, c \mapsto \lambda c$ by the same value $\lambda$ you see we have the same ellipse $x^2/(\lambda a)^2 + y^2/(\lambda ^2 a^2 - \lambda^2 c^2) = (x')^2/a^2 + (y')^2/b^2 = 1$ but sent $(x,y)$ to $(x',y')$

dmckee

You are interested in assembly and other low level topics. This gives you insight into the way really grand sounding high-level ideas can hit below-the-waterline snags.

Bernard Meurer

@dmckee Oops, deleted the duplicate, I was doing some editing and I like having two versions :). The reason I'm writing the letter in this tone is because of this:

vzn

@Secret you might look into jungian theory/ analysis sometime...

Bernard Meurer

@dmckee I got the idea from that that I should just say why I wanted to work there.

dmckee

Ah. Yep. Do they give you a space later to answer the question of why they want you?

Secret

3:11 AM

@vzn It is revealed that dream analysis is highly dependent on the person. Only the person itself can decipher one's own dreams

dmckee

If not you might sneak some in here.

vzn

@Secret it is revealed where? [citation needed] ... agreed in general but there are different schools of thought on that

Bernard Meurer

@dmckee Nope. All they have is an additional information field

dmckee

BTW-- If you are going to end up getting involved in legal you have to be willing to take elaborate structures on their internal merits even when, seen from the outside, they are totally bonkers.

I hope you know what you're getting into.

@BernardMeurer Hmm ... they expect you to do the convincing in the resume.

vzn

Bernard as a lawyer lol

reminds me of lawrence lessig...

Bernard Meurer

3:15 AM

@dmckee I thought a little about that. I really want to get this, I feel like my experience so far (and the way it's headed) is really monotone into "LETS PROGRAM HOW THE ELECTRONS MOVE IN THE IC ONE BY ONE", and I want to diversify that

If I could manage something like this I guess in the future it could stand to show that I am also part human

You know, makes me more employable since right now it looks like my original life plan won't work at all and I need employability

And since I love privacy policies and user rights and so on I thought this was a good opportunity

Job description

bolbteppa

@Kaumudi From $x^2/\lambda^2 a^2 + y^2/[\lambda^2 (a^2 - c^2)] = (x/\lambda)^2/a^2 + (y/\lambda)^2/(a^2 - c^2) = (x')^2/a^2 + (y')^2/b^2 = 1$ and the fact that $a,c$ determine the ellipse, if we take $(\lambda a, \lambda c)$ we have the same ellipse, but a point $(x,y)$ is sent to the point $(x',y')$, so because $a/c = \lambda a / \lambda c$ we see $a/c$ (or $c/a$) is a scale-invariant way to represent the ellipse, i.e. it throws away the redundancy of $a/c = \lambda a / \lambda c$.

Bernard Meurer

@bolbteppa Lamba should be a character, I don't blame you

bolbteppa

Happens so many times I agree, but then it's a slippery slope to Simba as a symbol of anything but childhood

Secret

@vzn Ok I think I have a citation issue here, because I read about that in some magazines lately. But a related notion that dream is based on what you saw in waking life, I saw that in newscientist (which they sometims have journal references there)

Bernard Meurer

@bolbteppa Lol

Secret

3:21 AM

This website also mention simialr things, in fact there are a lot thus there must be osme underlying school of thought that they all based on but not cited

bolbteppa

This seems to be the Freudian view that dreams are about wish fulfillment

Usually things that happen in your day trigger unconscious desires that your dreams can fulfill unbridled

and math dreams definitely fit into that pattern

vzn

@Secret fine but why do so many people have similar dream themes? falling, naked, in school/ class, etc

dmckee

@BernardMeurer That's a great internship for showing future employers that you can work the human/organizational side of things as well as the purely technical side. Good luck.

Bernard Meurer

Hm, the hungarian is kind of hot

bolbteppa

Unconscious fears played out unbridled

Bernard Meurer

3:24 AM

@dmckee Glad my thinking makes sense (not an usual sight :P). Thanks a lot prof!

user228700

@bolbteppa Thank you :-) But, this seems like a justification of why we are allowed to take $c/a=e$...

Secret

Speaking about maths, my dream logs told me the following mechanism:

5. Intellectual amplification: If mentally tasking activity that are not repetitive is being carried out on the day before the dream (such as thinking and computing about science), the dreams that result has a higher probability of becoming vivid and even surreal with an increase in structural complexity.

vzn

@Secret lol this line: Dream dictionaries are generally not considered scientifically viable by those within the psychology community (citation needed)... dreams themselves are not considered scientifically viable by many in the scientific community :P

Secret

Or in short: doing maths increases dream complexity

user228700

Under the definition of $e$ being the ratio of the distance of any point on the conic from a fixed point to its distance from a fixed line and all, how dies this make sense..?

Secret

3:27 AM

@vzn Well I have not read that site in detail actually. After all I have 5 years worth of raw data for analysis

Bernard Meurer

@vzn Come on, I'd be a great lawyer

vzn

@Secret its not bad

Bernard Meurer

@vzn "Fuck you judge, you're wrong, you're going to jail you twat"

Secret

I do however, have dream characters that seemed to play the role of the anima and animus

Bernard Meurer

Ah this reminds me. The other day I head the word prat and it got me thinking if some brit made it by combining prick and twat

vzn

3:28 AM

@BernardMeurer lol... anyway maybe read a little on lawrence lessig, interesting character, wrote a book on how law/ "code" are related...

Secret

The animuses are not a single person: They are usually black business suit men that gave advices or professors

The anima, however is the same person

Bernard Meurer

$$\text{prick }+\text{ twat }=\text{ prat}$$

dmckee

@vzn Several such books.

Secret

It is not entirely sure if they are really anima and animuses, but they do act simialrly

bolbteppa

@Kaumudi Most of what I said is here people.richland.edu/james/lecture/m116/conics/elldef.html but the fact about choosing $c/a = e$ is not there. I have given you a mathematical reason for even defining such a notion, the fact that it remains invariant when you scale the two terms characterizing an ellipse, so the next step is to link this invariant to something else geometrically invariant on the ellipse, and what invariants to we have?

We have the distance between the foci, $2c$, and we have $2a$, what is their ratio?

vzn

3:30 AM

@Secret interesting, was just reading it a little, hadnt heard before, think very similar to ancient yin-yang theory... which also shows up somewhat in hinduism etc

Bernard Meurer

Man do I miss Swartz

vzn

@BernardMeurer ?

bolbteppa

@Kaumudi 'The eccentricity of an ellipse, usually denoted by ε or e, is the ratio of the distance between the two foci, to the length of the major axis or e = 2f/2a = f/a.' en.wikipedia.org/wiki/Ellipse#Elements_of_an_ellipse

Bernard Meurer

@vzn That kid is Aaron Swartz. He's a big deal to me. He is dead.

bolbteppa

Is that Larry Lessig?

Bernard Meurer

3:32 AM

I think that's the right surname

@bolbteppa On the right, yes

vzn

@BernardMeurer oh right. heard about him. kinda a variation on assange. (feel that also may not end well...) which reminds me did anyone see the stone/ snowden movie? just saw it ~1½ wk ago, amazing, highly recommend it... riveting/ emotional etc

bolbteppa

I have to admit, this ellipse, parabola, hyperbola stuff drove me absolutely crazy

Old ways of defining them, new ways, 5 ways, matrix ways

user228700

@bolbteppa That's weird :/ I don't understand how $e$ is defined differently for different conic sections. The value of $e$ is supposed to be different, but not the definition itself..?

vzn

bitcoin is another recent remarkable/ interesting intersection of law/ code... vzn1.wordpress.com/category/bitcoin

Secret

About conic sections we are never taught about the existence of the directrix back in my high school, which is why attempt to answer these questiosn will mean spending the next hour in wikipedia

bolbteppa

3:36 AM

@Kaumudi What do you mean? You have an ellipse $x^2/a^2 + y^2/b^2 = x^2/a^2 + y^2/(a^2 - c^2) = 1$, you can play around with $a$ and $c$ and still get an ellipse so long as $c/a = e < 1$, that is $c^2/a^2 = e^2 < 1$, that is $c^2 < a^2$, that is $0 < a^2 - c^2$ which we have in the denominator.

See, I can't remember where the directrix is in what I said, but it's there somewhere

user228700

3:48 AM

@Secret WHAT?! 😱

user228700

@bolbteppa I feel that I am trying to solve the classic chicken-and-egg paradox. Yes, defining the eccentricity like this:

user228700

18 mins ago, by bolbteppa

@Kaumudi 'The eccentricity of an ellipse, usually denoted by ε or e, is the ratio of the distance between the two foci, to the length of the major axis or e = 2f/2a = f/a.' https://en.wikipedia.org/wiki/Ellipse#Elements_of_an_ellipse

bolbteppa

There are basically two ways to look at eccentricity: en.wikipedia.org/wiki/… I gave one

(Flattening? Forget that, non-standard)

Secret

@Kaumudi Indeed, your questions first let me to realise there is so much I don't know about parabolas alone

user228700

makes sense and all, but only because that ratio gives me $e$ because I know that the distance b/w the two focii is $2ae$ and the length of the major axis is $2a$.

user228700

3:52 AM

But I dunno how we got that in the first place; how the focii are $(±ae,0)$ and the vertices are (±a,0)$

user228700

^ Chicken-and-egg. See, I was under the impression that the excentricity for all conics is given by:

bolbteppa

I defined the foci to be at $F: (c,0)$ and $F'(-c,0)$ and then the distance between them is $2c$ so that $2c/2a = c/a$ holds. Then I gave you a reason why we even care about $c/a$ by analyzing the ellipse.

user228700

> an ellipse may also be defined as the set of points such that the ratio of the distance of each point on the curve from a given point (called a focus or focal point) to the distance from that same point on the curve to a given line (called the directrix) is a constant. This ratio is called the eccentricity of the ellipse.

bolbteppa

The mathematical reason for even considering $c/a$ is motivated by scaling them in the equation for an ellipse, finding you get the same ellipse, so their ratio is scale invariant. Thus we have a scale invariant representation of the ellipse in terms of the parameters defining the ellipse.

user228700

@bolbteppa Yes, thank you very much :-) I do understand this, but I feel like that was a justification, rather than a definition. 'Cause conic sections are defined like that ^ (my last message)

user228700

3:56 AM

@Secret I see...

Secret

AI encryption not understable to humans

user218912

@SirCumference hax

bolbteppa

@Kaumudi I guess it's a justification for even defining the concept of eccentricity. But you can take a geometric definition of it using the directrix, and I haven't tried to motivate the directrix yet so there is a gap to that definition which I guess is the issue

user228700

@bolbteppa Yes, that is the issue ._.

bolbteppa

I have to say even finding that perspective of eccentricity took me f***ing ages, absolutely ages, so much little nonsense on conic sections you don't even think of and different books do different things

'Any conic section can be defined as the locus of points whose distances to a point (the focus) and a line (the directrix) are in a constant ratio'

So basically, you can define conic sections using (point,point) or (point,line)

This is reminiscent of projective geometry duality I still can't make full sense of

user228700

4:09 AM

@bolbteppa Yes, and I don't understand the def. of the eccentricity of an ellipse in this context.

user228700

> (point, point) or (point, line)

user228700

?

DanielSank

Anyone know why you can always find a Hamiltonian (Lagrangian) to describe a physical system?

bolbteppa

Okay, so if you take the (point,point) perspective of conic sections, I have done almost everything, apart from a final step to derive the directrix. However, if you take the (point,line) perspective of conic sections, we should be able to directly to a similar calculation and end up with the eccentricity in terms of the directrix, I think I seen this ages ago

@DanielSank What do you mean?

DanielSank

i.e. is it the case that any differential equation (with some particular properties) admits a Hamiltonian description?

user228700

4:11 AM

@bolbteppa The latter is what I'm going for...

DanielSank

@bolbteppa I think I mean, why can any (some?) differential equations be expressed as Hamiltonian equations of motion?

@ACuriousMind

user228700

@DanielSank: Greetings :-)

bolbteppa

@DanielSank Do you mean this en.wikipedia.org/wiki/Inverse_problem_for_Lagrangian_mechanics

Some derivation like this for (point,line) conic sections: math.stackexchange.com/q/656746/82615

I don't think I'll be happy with this until I motivate directrices from what I posted so far tbh

user228700

4:41 AM

@bolbteppa Yeah, me neither :/

user228700

Anyway, thanks for all ur help :-)

Secret

5:23 AM

0

Q: If space is quantized, what cause them to attract each other?

According to Newton's law of motion, object moves in a straight line unless it is intercepted by another force, and the geodesic path the object takes in a distorted space time due to the presence of another yet massive object looks like its falling. My question is if space is quantized what happ...

LQG expert needed

user507974

5:56 AM

@BernardMeurer I spy Lawrence Lessig

also Aaron Swartz was such a shame, such great potential

user228700

7:03 AM

Hi again. Is anybody familiar with ellipses?

user228700

in Mathematics, 45 mins ago, by Kaumudi

The equation of tangent to the ellipse $x^2/a^2+y^2/b^2=1$ is given by $y=mx ± \sqrt{a^2m^2+b^2}$

user228700

in Mathematics, 44 mins ago, by Kaumudi

Where $m$ is the slope of the tangent. Clearly, for every slope, there are two such tangents, with different y-intercepts.

user228700

in Mathematics, 44 mins ago, by Kaumudi

These two tangents are clearly parallel to each other.

user228700

in Mathematics, 43 mins ago, by Kaumudi

My textbook has given a result related to this and it starts off with "Let this tangent pass through a point $(h,k)$"

user228700

in Mathematics, 41 mins ago, by Kaumudi

My textbook has then gone on and written $k=mh ± \sqrt{a^2m^2+b^2}$ by substituting this point on the equation of both tangents!

user228700

7:06 AM

in Mathematics, 42 mins ago, by Kaumudi

How is it that these two parallel tangents will pass through a common point $(h,k)$?!

user228700

in Mathematics, 40 mins ago, by Kaumudi

Or have I misunderstood? 'Cause my textbook says, at the end "If we know what $(h,k)$ is, we can solve the obtained quadratic equation for two values of $m$ "

user228700

Um..?

user228700

U know, maybe I am right, in that the two tangents don't intersect.

user228700

Perhaps one of these tangents passes through the point $(h,k)$ but another tangent that isn't parallel to the first one also passes through this point!

user228700

But the only problem is that there is no evidence of the existence of such a tangent.

user228700

7:13 AM

I got the equation for the tangent as $y=mx ± \sqrt{a^2m^2+b^2}$ by solving the equation of a line $y=mx+c$ with the parabola $x^2/a^2+y^2/b^2=1$ and equating the discriminant of the obtained equation to zero to get that $c^2=a^2m^2+b^2$

user228700

Wait, this equation can be interpreted another way, too!

user228700

If we forget all about the fact that the two tangents are supposed to be parallel and all, then this equation can be seen as the equation of the two tangents passing through the point! The point is the hero, no the slope!

user228700

Okay, thank you, my rubber ducks! :-D

user228700

(:-P)

Mew

7:43 AM

@Kaumudi, the two tangents don't intersect

sorry they do intersect, but they don't intersect on the ellipse

Draw an elippse

then pick any x value

and draw a vertical line that passes through that x value

it intersects the ellipse in two places

thus there are two tangents for this x-value. One at the first intersection, and another at the second intersection.

user116211

8:04 AM

I can't read so long transcript T__T

Dvij

@Slereah Thanks for your explanation regarding coordinate conditions. I think I understand now how it works.

So the thing is that from the curvature tensor, one doesn’t get the complete metric components. One get them up to a “gauge” which disappears when we calculate the curvature tensor from the metric. It is this gauge that ultimately decides my coordinate and thus, my symbols.

So in a way, even if I start computing the components of the curvature tensor from the EFEs, I am not really opting a particular frame but rather I am opting a class of frames and then when I would choose the gauge then I would be choosing the particular coordinate system.

user228700

@Mew No, this isn't what I was getting at...I think.

user228700

I wasn't talking about a point on the ellipse, I was talking about a point outside the ellipse, from which two tangents are drawn to the ellipse. Since this point lies on one of these two parallel tangents, I substitute $(h,k)$ and get an quadratic in $m$.

user228700

And since we square both sides, it turns out that it doesn't matter which of the two parallel tangents we pick.

user228700

Am I making sense?

Mew

8:17 AM

no you're not because the tangents are not parallel

DanielSank

@MAFIA36790 I was making the same edit :P

user228700

Yes, they are! The equation $y=mx ± \sqrt{a^2m^2+b^2}$ gives the equation of two parallel tangents to an ellipse (Notice, same slope!) with diff. y-intercepts.

user116211

It would have completed faster had I not used \prime.

user116211

@DanielSank o/

Mew

@Kaumudi, the equation is wrong

user228700

8:22 AM

No, it's not. Try to solve a line $y=mx + c$ with the ellipse $x^2/a^2+y^2/b^2=1$

user228700

And tell me what u get for the line to be a tangent to the ellipse.

Mew

parabola?

user228700

Why do u think it's wrong?

DanielSank

@MAFIA36790 \o

user228700

@Mew Sorry, I've just come from finishing parabola.

DanielSank

8:23 AM

@Kaumudi Greetings and salutations.

user228700

:-)

user228700

@Mew..?

Mew

@Kaumudi, it's wrong

show me your calculation and i'll find ur error

user228700

What dyou mean "it's wrong"? Tell me why it's wrong!

Mew

because for a given x-value

the tangents of an elipse are not parallel

just draw it

user228700

8:27 AM

Is the given $x$ on the ellipse or outside it?

Mew

parabola?

user228700

Ugh, sorry.

Mew

for a given x-value on the elipse, there are 2 tangents that are clearly not parallel

in fact the gradient of one is the negative gradient of the other

and the y-intercept of one is the negative of the y-intercept of the other (provided origin is at the centre)

user228700

Hang on, one second.

Mew

see for a given x-value, there are two tangents

that are claerly not parallel

but they intersect outside of the elipse

their gradients are negative of each other

user228700

8:33 AM

I'm saying that for any m, there will be two tangents to the ellipse (which makes the two tangents parallel, of course) But you're saying that for a given x value, no two tangents to the ellipse can be parallel.

Mew

true dat

user228700

Yeah, so clearly, we're not even talking about the same thing.

Mew

yeah

should have posted an image bro

user228700

Yes, shoulda :-P

user228700

Sorry.

user228700

8:36 AM

But I understand what u're saying, thanks :-)

Mew

no worries dawg

I thought you were getting confused

John Rennie

@Kaumudi: so what are you asking? I've got lost with the multiple posts?

user228700

Yeah, I was :-P

user228700

@Mew: How did u figure out that their gradients are negative of each other?

user228700

@JohnRennie Hi :-) I'm asking about the tangent to an ellipse, basically.

John Rennie

8:44 AM

@Kaumudi that's ... not amazingly specific :-)

@Kaumudi the ellipse is symmetric about the x axis

user228700

:-P I know. Dyou want me to get into the nitty gritty details again?

John Rennie

@Kaumudi well I've finished work for the day so I'm chilling and drinking coffee. If you want me to have a look at the problem I'm happy to.

Is it the problem starting here:

2 hours ago, by Kaumudi

Hi again. Is anybody familiar with ellipses?

user228700

I have many problems related to ellipses lol :-P

user228700

Yeah.

John Rennie

The dictionary says:
elliptical, adjective, using or involving ellipsis, especially so as to be difficult to understand.

Difficult to understand eh? :-)

Mew

8:49 AM

How can a oval be complicated?

user228700

@JohnRennie What, the dictionary?!

John Rennie

@Mew Spolier: it's referring to ellipsis not ellipses

user228700

Ah :-P Well, that's not what I'm going for.

John Rennie

But it's the same adjective elliptical in both cases.

user228700

Oh! Interesting.

John Rennie

8:51 AM

I just thought it was mildly amusing that in English elliptical can mean hard to understand

OK, I'll shut up now :-)

Mew

lol that's a good maths joke

John Rennie

@Kaumudi anyhow, if you have problems with ellipses (not ellipsis) please ask

user228700

@JohnRennie Indeed :-)

user228700

I'm still trying to figure out how the gradients of those two tangents are negative of each other.

user228700

Like, by solving the equations I have at disposal ._.

John Rennie

8:57 AM

Which two, the ones Mew drew?

Mew

@Kaumudi, gradient = rise/run

user228700

Yeah.

Mew

Clearly the Run is the same for both

And the Rise of one = -Rise of the other

John Rennie

If so the symmetry guarantees that

Because the symmetry maps dy to -dy that means dy/dx maps to -dy/dx

user228700

Yeah, but I'm trying to do it by solving the equations I have to get $m_1m_2=-1$

Mew

8:59 AM

that would be the case for perpendicular lines

but clearly the tangents in the picture are not perpendicular (at right angles) to one another

John Rennie

Don't you mean $m_1/m_2 = -1$ ?

user228700

Oh, wait.

Transcript for