Mathematics - 2018-12-15 (page 2 of 2)

@TedShifrin I know, my mom freaked out because the doctor kept mentioning I might be legally blind. I can see 20/20 with glasses though, so not that bad at the moment.

Ted Shifrin

Yeah, that sounds like legally blind.

Daminark

Astigmatism and my vision kept decreasing badly until I was about... 15?

Ted Shifrin

I still need glasses for driving and movie theaters, but I'm fine otherwise.

Akiva Weinberger

Are your glasses really heavy?

CaptainAmerica16

Astigmatism as well, I'm also near-sighted.

@AkivaWeinberger No, but they are large.

Daminark

7:07 PM

Once I started wearing antiglare glasses it stopped plummeting

Ted Shifrin

Oh, I have that (as well as progressive sunglasses) because of my cataracts.

Akiva Weinberger

I might need to invest in those

On occasion I can see my eye's reflection in my glasses

Ted Shifrin

DogAteMy: I think they now use light plastics rather than glass, so glasses aren't heavy any more.

Akiva Weinberger

Interesting

I'm no chemist

At some point I should look up why glass is transparent, on a molecular level

Ted Shifrin

Crystal structure? Hmm ... I dunno.

Akiva Weinberger

7:10 PM

I still don't know where color comes from tbh

There's a blue pigment called techelet whose formula had been lost for two thousand years

CaptainAmerica16

Isn't it obvious? You can't see through opaque objects. The only other option was clear.

Ted Shifrin

Certain things absorb or transmit only certain frequencies of light, I guess.

Akiva Weinberger

(Or tekhelet, however you want to spell it)

Ted Shifrin

So it's all about wavelengths.

Akiva Weinberger

(תכלת)

Anonymous

7:11 PM

@AkivaWeinberger There's a nice thread on that here

CaptainAmerica16

That reminds me - I saw a rainbow on the ground today.

Akiva Weinberger

Neat

Had it rained recently?

Ted Shifrin

Probably oil, CaptainAmerica.

Anonymous

@CaptainAmerica16 Well, there's translucent

CaptainAmerica16

Yeah, it rained all night.

Akiva Weinberger

7:12 PM

There's a water fountain in the park near my house that I used to have fun making rainbows with

CaptainAmerica16

@Blue Oof, I didn't think of that

Akiva Weinberger

'cause if you put your finger in it in the right way, it makes this massive spray of water

Ultradark

during a storm I saw lightning go across a rainbow

Akiva Weinberger

and then the sun hits it and makes a rainbow

CaptainAmerica16

@Ultradark That sounds intense.

Akiva Weinberger

7:13 PM

Zeus got angry with Iris?

Ultradark

Yeah the gods were angry that day

Akiva Weinberger

There's a neat thing with light

where, technically it's this infinite dimensional space of frequencies, you can add together as many different frequencies as you want,

CaptainAmerica16

Really?

Akiva Weinberger

but our eyes can only see a three-dimensional subspace of that 'cause we only have red, green, and blue cones

(or maybe fewer if you're color blind)

So as a result there are types of light that look the same to us but our actually different

and they'll refract off of materials differently

I forget the word for this

Animals with more types of cones than us can distinguish more light than we can, as a result

There's also the question of how sensitive to change each individual cone is, but that's a separate thing

CaptainAmerica16

Animals must see some freaky stuff.

Akiva Weinberger

7:18 PM

Computer screens, for example, only make red, green, and blue light

because they're built for humans

so they can't make yellow, they can only make something that's indistinguishable from yellow for humans (by combining R, G, and B)

but different animals would be able to tell that it's not the same

CaptainAmerica16

The aquaponics lab I work in has lights that only give off red and blue, so that when you back into normal light, everything looks green.

Akiva Weinberger

Cool

(This is why the color space on computers is RGB. It's entirely to do with human anatomy, rather than a property of the universe)

CaptainAmerica16

@Ted The Spivak guilt got to me, I'm back in ch. 1

Ted Shifrin

I'm sick of chapter 1.

CaptainAmerica16

I felt bad for skipping so many problems.

So that's what I was doing all night.

Anonymous

7:22 PM

@AkivaWeinberger Isn't that a bit of an oversimplification? :/ Red, blue and green aren't three specific frequencies any way and there's sufficient overlap between those colors in the visual spectrum.

Akiva Weinberger

That graph describes the map to the 3D subspace

CaptainAmerica16

@Ted I'm done with ch. 1 by the end of Dec. no matter how I feel.

Anonymous

@AkivaWeinberger Which graph?

Ted Shifrin

It really would be better if you got to more interesting stuff, @CaptainAmerica.

Akiva Weinberger

This one

Jasper Loy

7:25 PM

@TedShifrin That's why once we are done with all of math, we can get on to more interesting stuff. =)

CaptainAmerica16

@TedShifrin This is it. The final stretch, then I'm seriously going to move on.

Ted Shifrin

Thank goodness for transfinite induction, Jasper.

Akiva Weinberger

Based on that graph, I'd guess 500nm light would look the same as the right combination of 450nm and 550nm light

give or take

CaptainAmerica16

I plan on finishing math when I go to grad school. Can't wait to see what classes come after that.

Akiva Weinberger

$\sin(500x)$ and $\sin(450x)+\sin(550x)$ are different functions though

Ted Shifrin

7:26 PM

DogAteMy: You get serious beats if you combine 450 and 550.

CaptainAmerica16

I hear MATH2 is intense.

Akiva Weinberger

@TedShifrin Is that a thing that happens with light??

I mean I guess it must

but I dunno what that would mean

Ted Shifrin

You get pulsing, I presume.

user616128

@AkivaWeinberger Why are you calling this a 3D subspace?

Ted Shifrin

I need to ask my physicist friends.

Akiva Weinberger

7:27 PM

Er, that's probably actually $\sin(x/450)+\sin(x/550)$

'cause it's wavelength, not frequency

@user616128 One dimension per cone cell

Color, as seen by humans, can be described as an amount of red, an amount of green, and an amount of blue

This is what the RGB encoding is on computers

Jasper Loy

@CaptainAmerica16 What is MATH2?

CaptainAmerica16

@JasperLoy It's the math you do after you finish math.

Akiva Weinberger

Color as it actually exists in the universe is an arbitrary linear combination of waves

user616128

"One dimension per cone cell" makes no sense at all

Akiva Weinberger

It's the configuration space of the colors

not an actual geometric shape

You need three numbers to describe it, is what I mean

user616128

7:30 PM

The different types of cones have different responsivity to different wavelength within the visible spectrum

Akiva Weinberger

Yeah, and we can describe a color by three numbers, describing how activated each type of cone is

Jasper Loy

@CaptainAmerica16 Whether it's grad classes or undergrad classes, they only cover a fraction of a good book, and a good book only covers a fraction of the important stuff in a topic. So classes are just introductions to topics to enable you to get a paper qualification along the way.

user616128

You need one number to describe visible colour, something like a weighted average of the frequencies of the EM-waves corresponding to the photons

Why are you saying there are three numbers?

Akiva Weinberger

Wait sorry I misread

user616128

The picture was responsivity of each of the three types of cones

Jasper Loy

7:32 PM

@AkivaWeinberger On Windows 10 I can specify three numbers in RGB to get a colour.

Akiva Weinberger

It's not the average of the frequencies

CaptainAmerica16

@JasperLoy I was kidding, but that's exactly why I'm going to study calc in more depth at the end of this school year.

user616128

@AkivaWeinberger You're going to have trouble perceiving a single photon

Akiva Weinberger

No monochromatic light is pink, for example

CaptainAmerica16

@Ted Do I need a lot of linear algebra to really understand affine geometry?

Akiva Weinberger

7:33 PM

Pink can only be obtained by combining lights of different frequencies

It doesn't look the same as the average frequency

Jasper Loy

@CaptainAmerica16 After you read Spivak's Calculus, you can read Shifrin's Multivariable Mathematics, which is essentially multivariable calculus with linear algebra.

Akiva Weinberger

Think of it this way: Sound is a wave, just like light. If I play more than one note at once, it doesn't sound like one note.

It sounds like a chord.

Ted Shifrin

@CaptainAmerica: You need linear maps/matrices.

Akiva Weinberger

But our ears are better than our eyes, because our ears don't have just three types of sound sensors.

Ted Shifrin

Unless you're tone-deaf, DogAteMy :D

Akiva Weinberger

7:34 PM

If they did, then every noise would sound like some combination of C, E, and G (for example)

@TedShifrin True

Jasper Loy

When I was in the choir some singers were tone deaf.

user616128

@AkivaWeinberger I don't understand how this refutes what I said?

Akiva Weinberger

You wouldn't be able to tell apart different instruments, even

Jasper Loy

Tone deaf people can play the piano, but they can sing the wrong tone even after you play it to them.

user616128

Monochromatic light is taking a stream of photons with the same associated EM frequency

Akiva Weinberger

7:35 PM

@user616128 You said "You need one number to describe visible colour"

No one number can describe pink

CaptainAmerica16

@JasperLoy That sounds like a plan.

Something I feel like I agreed to before - it's been so long.

user616128

I said "weighted average of the frequencies"

Not frequency

Akiva Weinberger

I'm not sure I understand you

If you take the weighted average of frequencies you end up with a single number

user616128

Yes, and each of your photons in monochromatic light has the same frequency

Akiva Weinberger

Sure, but pink is a color and you can't get monochromatic pink

user616128

7:38 PM

But in the case that you are not perceiving monochromatic light, each of your cones has a different responsivity to different wavelengths

(which was normalised out in your picture above)

Akiva Weinberger

and you have three cones, resulting in three numbers describing what you're seeing

Do you know what the RGB color space is, on computers?

You give the computer three numbers and it gives you a color

user616128

And when looking at something that is pink, you have a massive number of photons being absorbed, with non-constant frequency

Akiva Weinberger

@user616128 OK

Doesn't mean pink isn't a color

user616128

@AkivaWeinberger You realise how computer monitors use RGB do you not?

You know what a pixel actually looks like?

Akiva Weinberger

Yes

Three subpixels

user616128

7:41 PM

Yes

And if you use a magnifying glass you can clearly observe them

The photons aren't activating cones in an extremely local region

When you observe the screen with the unaided eye, photons are indeed activating cones in the same region

Tell me what white is in your three dimensional space?

Akiva Weinberger

(100%,100%,100%)

Anonymous

Biology SE has a very nice discussion on the human color vision model. It's a bit more complicated than the RGB model we use for computers. The point here is that the L, M, S cone cell frequencies have considerable regions of overlap whereas the RGB model does not - so you can describe it as three independent coordinates (what Akiva calls a 3D subspace), whereas that simplistic model does not apply to cone cells but approximates it rather well.

Akiva Weinberger

#FFFFFF, in RGB

Anonymous

Life would have been simpler if we could have described our vision in terms of "activations" of RGB cone cells with values ranging from 0 to 1 for each coordinate. Unfortunately, real physics sucks. :)

Akiva Weinberger

Hey I found a Wikipedia thing

> A "physical color" is a combination of pure spectral colors (in the visible range). Since there are, in principle, infinitely many distinct spectral colors, the set of all physical colors may be thought of as an infinite-dimensional vector space, in fact a Hilbert space. We call this space $H_{\rm color}$.

> A humanly perceived color may be modeled as three numbers: the extents to which each of the 3 types of cones is stimulated. Thus a humanly perceived color may be thought of as a point in 3-dimensional Euclidean space. We call this space $\Bbb R^3_{\rm color}$.

Ted Shifrin

7:48 PM

This is where spectral decompositiion of operators on Hilbert (Banach) spaces comes from :P

user616128

I have to go now, thanks for the discussion :-)

Akiva Weinberger

The map $H_{\rm color}\mapsto\Bbb R_{\rm color}^3$ has a kernel, resulting in two points with the same image

Ted Shifrin

Um, an infinite-dimensional (affine sub)space of points with the same image?

Akiva Weinberger

Right yeah

Here's the spectrum of a lemon

The spectrum of a computer image of a lemon would be mostly zero except for three spikes

Anonymous

@TedShifrin Lol, I knew there had to be a connection :P

Anonymous

7:52 PM

Spectral decomposition...spectroscopy.....phew

Akiva Weinberger

Finally found the name of the thing!

Metamerism (color)

In colorimetry, metamerism is a perceived matching of the colors with different (nonmatching) spectral power distributions. Colors that match this way are called metamers. A spectral power distribution describes the proportion of total light given off (emitted, transmitted, or reflected) by a color sample at each visible wavelength; it defines the complete information about the light coming from the sample. However, the human eye contains only three color receptors (three types of cone cells), which means that all colors are reduced to three sensory quantities, called the tristimulus value...

Metamers.

Or metametric colors.

Ted Shifrin

Wow.

Daminark

Almost done with grad apps, just have 3 left

Ted Shifrin

Quite the ordeal, Demonark. Remember that some of us had to do this with typewriters and pens.

Akiva Weinberger

> The term illuminant metameric failure or illuminant metamerism is sometimes used to describe situations where two material samples match when viewed under one light source but not another. Most types of fluorescent lights produce an irregular or peaky spectral emittance curve, so that two materials under fluorescent light might not match, even though they are a metameric match to an incandescent "white" light source with a nearly flat or smooth emittance curve.

Ted Shifrin

7:58 PM

Not to mention the pain of sending in recommendations on paper for years.

Daminark

Lack of autofill... How many schools did people apply to then?

Akiva Weinberger

> Normally, material attributes such as translucency, gloss or surface texture are not considered in color matching. However geometric metameric failure or geometric metamerism can occur when two samples match when viewed from one angle, but then fail to match when viewed from a different angle. A common example is the color variation that appears in pearlescent automobile finishes or "metallic" paper

Ted Shifrin

Probably 6-15.

It cost money then, too.

Daminark

I ended up with 15

But wow doing 15 on paper is a nightmare

Ted Shifrin

I was cocky and I think I applied only to 6.

I was confident in a few safety schools.

Plus I was super lucky and got an NSF graduate fellowship.

I still remember typing almost 100 job applications individually.

Daminark

8:02 PM

Seems to have worked out well enough in any event. I had 3-4 backups in there, and 4 wishful thinking. The remaining 7 I classify as rough but hopefully I have a shot

Maybe 2 of the 7 are a bit less confident

And oof, 100?

Ted Shifrin

Well, your list that you showed me looked reasonable to me in the end.

Yup.

Now there's MathJobs and people just post things electronically. Or maybe the Russians post them.

Daminark

Yeah, I ended up adding Notre Dame to it, I had a fee waiver because of the workshop I went to some time back and at least one guy there knows and seems to like me

Ted Shifrin

I know several people on the faculty there. It was also one of my multiple job interviews back in 1981. :P

Semiclassical

hi chat

Ted Shifrin

hi Semiclassic

Daminark

8:04 PM

So I'd like to think it's safe, and they have Behrens and Putnam who seem really interesting. Also two number theorists, though they're assistant professors so less sure bets

Ted Shifrin

OK, lunchtime for me and then I need to take a walk.

Semiclassical

i think i reached the limit of my polytopes stuff, or at least the limit of what I can get a computer to do for me

Ted Shifrin

Putnam or Putman?

Guess you need a mainframe, Semiclassic.

Daminark

Uh oh, thought it was Putnam. I hope I didn't misspell his name in my SoP

Ted Shifrin

Oh oh.

The geometry guy I know is Putman.

Semiclassical

8:06 PM

when one of your steps takes your computer an hour and results in a convex hull of half a million vertices in 63 dimensions

that's a sign that you probably should stop asking your computer to do it

Daminark

You're right and I did misspell it. :'(

Ted Shifrin

Presumably only relatively few of those vertices are actually relevant, @Semiclassic.

I'm sure you're not the first to do so, Demonark.

Semiclassical

yep. once you apply an appropriate mapping from RR^63 down to RR^3, you end up with a convex hull of just 68 points

Ted Shifrin

Is this irrevocably posted without possibility of correction?

Why are you RRing, Semiclassic?

Semiclassical

because I'm used to how sage writes rings

Daminark

8:08 PM

Sadly, yes. Hopefully whoever is reading it is also dyslexic.

Semiclassical

QQ = rational field, RR = real field etc

Ted Shifrin

Oh.

Semiclassical

(it's actually working over rationals, since I want exact results)

Ted Shifrin

I'm sure that slows things down HUMONGOUSLY.

Semiclassical

yuuup

To make sure I was actually making progress, I split up the process to create the half-million vertices into 63 steps (one for each coordinate---you just need to ensure that each of them is nonnegative)

Ted Shifrin

8:11 PM

Ugh ... well, anyhow, I need to get going.

Semiclassical

the first 50 steps take tenths of a second each

the next 10 about half a second each

next is about a second, then 15 seconds, then 90 seconds

Daminark

Yeah flight is about to take off so I should go airplane mode

Semiclassical

that just leaves the last step...which takes an hour :3

Daminark

Heading to Texas for winter break

Akiva Weinberger

8:29 PM

@Daminark Have fun

Joe Shmo

8:57 PM

what are you trying to do @Semiclassical?

Balarka Sen

9:20 PM

@AkivaWeinberger That sounds right. The complement has countably many components.

Mary Star

10:05 PM

How can we show that the diophantine equation $x^3+5=11+y^3 \Rightarrow x^3-y^3=6$ has no integral solution?

If we calculate that modulo $6$ we get that $x^3\equiv y^3\bmod 6$. Does this help us?

Akiva Weinberger

10:47 PM

@MaryStar You need two cubes that are six away from each other

Think about the sequence of cubes (1, 8, 27, etc)

CaptainAmerica16

In Spivak, is 0 considered part of the positive numbers?

Nevermind, I'm pretty sure it isn't.

0 isn't a positive number anywhere 😑

Leyla Alkan

11:12 PM

Shouldn't have the underlined part been $[F(a,b):F]\leq mn$ ?

Nicholas Roberts

11:29 PM

Hi all, a quick question. Suppose I know that $\int f \leq M$ for some $M$. Can anything be said about the integral $\int f \cdot g$ for some integrable function g?

Like can I put any bound on the integral of the product using M

Mike Miller

@NicholasRoberts "Holder inequality"

Nicholas Roberts

Ive heard of Holder's ineq when studying LP spaces. Does it apply here?

Mike Miller

You'll want to assume either f is non-negative or replace that by a bound on the integral of $|f|$

Nicholas Roberts

Actually, the absolute value should have been there. I meant to write L1 norm. So thats good

But for holders, dont we need 1/p + 1/q = 1?

Mike Miller

11:46 PM

Sure, so figure out what p and q are appropriate here

Nicholas Roberts

Thanks for the tip, Mike! Btw i saw your talk at stonybrook last spring, was very good!

Ultradark

Given several points, that if connected would form a nonlinear, concave arc, find a function that passes through the points

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$Mathematics$ Mathematics