Mathematics - 2021-03-08 (page 3 of 4)

Matthew Christopher Bartsh

18:00

@PM2Ring I.e PM is the time to ring?

PM 2Ring

@MatthewChristopherBartsh Maybe... :)

Matthew Christopher Bartsh

Do you remember the reaction of the other posters to the post you were replying to and to your reply to it, in that thread from years ago?

PM 2Ring

@MatthewChristopherBartsh Not really. It was a while ago!

leslie townes

i don't remember the replies to a post i made last night. i live in the fast lane.

Matthew Christopher Bartsh

@PM2Ring Everyone was hostile to the guy you replied to, except you. I was shocked. I realized that had I posted my idea there, I would have likely had the same response. I could barely believe my eyes. I was also shocked that no one on the rest of the internet quoted either of you. The discussion ended there.

@PM2Ring The pair of you combined had summarized my favorite idea, and no one was interested. There was only hostility. Mild, bored, dismissive hostility. No one even called him a crank. It was really a shock to me.

PM 2Ring

18:09

@MatthewChristopherBartsh Well, it's a bit like dozenal.org who promote using base 12 instead of decimal. Such people tend to be very enthusiastic about their cause, but the mainstream tend to regard them as cranks.

Similarly, we could have a more regular calendar, but the inertia opposing such a huge change for such a small benefit is enormous. But that doesn't stop calendar enthusiasts from spending time investigating the possibilities. :)

Matthew Christopher Bartsh

@PM2Ring I don't like the comparison with the base twelve enthusiasts. They are right about base twelve being clearly better than base ten (do you agree?), but it is not the best base. It must be a power of two. My unified naming scheme allows us to put off the difficult decision of which power of two to use because all of them use the one set of names. Thus we can try out several bases at the same time, without conflicts or excessive amounts of effort or memorization.

PM 2Ring

@MatthewChristopherBartsh Powers of two have certain benefits. But bases with a couple of small prime factors, like 12 and 60, have their benefits too. It's not just inertia that we still use the Babylonian base 60 scheme for time. And a few centuries ago, base 60 was still often used by mathematicians for fractions, especially for trig-related stuff, until it was finally out-competed by decimals.

Matthew Christopher Bartsh

The problem with promoting base eight and/or sixteen (hex) is that unlike base 12, which has everything base ten has and more, base eight's superiority, although much greater than base twelve, is also much subtler.

@PM2Ring As for the calendar, the year is inherently irregular, so I don't see how anyone could get excited about that.

leslie townes

there is a lot of interesting stuff in de morgan's budget of paradoxes about calendars.

i do love how so much useful mathematics grew out of people wanting to know when they should pray, or where mecca was. ideas can reap dividends.

PM 2Ring

@MatthewChristopherBartsh A good calendar could make it a little less irregular. We could even have a soli-lunar calendar that stayed in synch with the phases of the Moon, and in synch with the equinoxes & solstices. But it would mean that we'd have a whole leap month ever few years, not just a leap day. :)

leslie townes

18:23

i think we should have decimal years. i also think that daylight saving time should fall back every six months. no spring forward, that messes with my life. i don't care if it's noon in the nighttime, just don't make me get up earlier.

Semiclassical

@MatthewChristopherBartsh it's also hard to envision base-12 as a day-to-day matter because we don't have a built-in unit for 10,11

we would of course have those in a base-12 world

Matthew Christopher Bartsh

@PM2Ring What is the great importance of being able to smoothly divide by three, once or twice, or by fifteen? As I see it, being able to halve or double smoothly an infinite number of times is a big advantage. We have verbs to halve and quarter but no verb to divide by three. That indicated to me that it isn't very important.

leslie townes

the unit for 10 is the symbol that prince used when he was under legal distress, and the symbol for 11 is the cry-laugh emoji.

Semiclassical

@leslietownes the integer formerly known as ten, you mean

leslie townes

precisely.

Matthew Christopher Bartsh

18:25

*isn't very important to smoothly divide by three.

@Semiclassical lol

Semiclassical

one thing i will say for base-10: for better or worse, it forces you to git gud at handling repeating decimals

PM 2Ring

@MatthewChristopherBartsh There's "trisect", but admittedly, it's a technical term, not part of the common vocabulary.

leslie townes

i've always thought in terms of a number line. do you know the concept of a number symbol? some people visualize spirals with numbers on them to conceive of numbers. it's a rare but not uncommon thing.

Semiclassical

with base-12 you can defer that conversation to when you start handling 1/5 and such

leslie townes

people who have them don't know that other people don't have them.

Number form

This article refers to the neurological phenomenon. For Unicode numbers, see Number Forms. A number form is a mental map of numbers, which automatically and involuntarily appears whenever someone who experiences number-forms thinks of numbers. Numbers are mapped into distinct spatial locations and the mapping may be different across individuals. Number forms were first documented and named by Sir Francis Galton in his The Visions of Sane Persons (Galton 1881a). Later research has identified them as a type of synesthesia (Seron, Pesenti & Noël 1992; Sagiv et al. 2006). == Neural mechanisms == It...

PM 2Ring

18:29

@Semiclassical And with base 60, you can defer it to 1/7. Incidentally, in Australia we use 7-pointed stars on our flag, partly to symbolize the 6 states + territories, but also because the regular heptagon needs advanced mathematics to construct, it can't be done using a (finite) traditional Euclidean straightedge & compass construction.

Semiclassical

right

leslie townes

you need a marked ruler.

Semiclassical

yeah. so not that advanced, really, but hardly obvious how to use that

Matthew Christopher Bartsh

@PM2Ring I forgot about 'trisect'. But I've never heard it except in 'trisect an angle'. Which actually supports my point. We have a word for divide by three, but we never use it. I've never heard of a perpendicular trisection of a line. Have you?

leslie townes

i had a student once who had a number form. she drew it for me. i said, what the hell is that. she thought everybody had them. then i googled it and learned about the phenomenon.

Ted Shifrin

18:31

I'm not sure you can do it with perpendiculars.

Matthew Christopher Bartsh

I've never heard of trisection of a line.

PM 2Ring

And regarding trisection, one annoying thing about using 360° to the circle, is that an angle of 1° degree isn't constructible, since it requires a trisection.

leslie townes

you can definitely ruler-and-compass divide a segment into an integer number of parts.

Ted Shifrin

Yup.

You can construct any rational number, in fact. :)

And ... more.

Back to the Greeks (instead of Egyptians). I think.

Matthew Christopher Bartsh

Where in maths is dividing by three seen?

leslie townes

18:33

you just do another segment with as many parts as you want and draw a family of parallel lines.

Ted Shifrin

See, @leslie, you are talking about something geometric. I'm stunned.

PM 2Ring

Yep. Ruler & compass let's you construct any quadratic number.

leslie townes

it's all professor wu. that's where i stole it from.

Ted Shifrin

Well, good :)

I always had fun teaching this section of my algebra course, because the math education majors could chime in vocally about how to do the constructions.

Thorgott

it's probably not what you mean by dividing by three, but there's the famous paper Division by three

Semiclassical

18:35

the fact that being able to construct a regular n-sided polygon has to do with prime numbers is deeply weird to me

leslie townes

haha, the set theory paper.

i love that

PM 2Ring

@Semiclassical I'm not sure whether Gauss would be pleased or perturbed by that statement. :)

leslie townes

i do think the constructibility of the n-gon is a great example of non obviousness in mathematics. i answered a question a long time ago on math.SE about this. feynman notoriously said if you gave him a question he could provide its truth value without necessarily a method of getting to the truth. i don't think that's possible with the n-gon.

Matthew Christopher Bartsh

@PM2Ring We should be using turns, half turns, quarter turns, and so on, and base eight or some other power of two, and a base eight based metric system, (and maybe it could have some names that are more charming than the metric system - though I am not sure how to do that last bit).

leslie townes

constructibility itself is a weird notion to take as a primitive.

if you read euclid he never measures anything. it's fractions of a right angle. this and that make two right angles. we could learn from him.

i've outed myself as having read geometry. i'm crawling back in the closet.

Thorgott

18:39

constructibility of the n-gon is algebra, not geometry :P

Ted Shifrin

@Semiclassical You mean for suitable $n$?

Thorgott

awaits slap from Ted

leslie townes

i will give you $e^{1.5}$ slaps.

how the tables have turned.

Semiclassical

right

Ted Shifrin

@Thor: I can't argue too strongly. I mean, how the hell do you discover the construction of the 17-gon just geometrically? The pentagon is doable. (Of course, the Greeks were way better at this stuff than we are.)

Thorgott

18:40

didn't Gauss do that?

or am I misremembering

leslie townes

yeah but he was a psychopath

Ted Shifrin

@leslietownes I have not yet designated you a legitimate smacker.

Semiclassical

yeah. i mean, to bring 1/3 into it---cone:cylinder::1:3

leslie townes

i provisionally withdraw my potentially transcendental number of slaps

Matthew Christopher Bartsh

@leslietownes Again, it supports my point. It's possible to do it, but it serves little purpose, and so it is so rarely done. Dividing by three, as I see it, is barely more important than dividing by seven or eleven.

Semiclassical

18:42

per my comment above, the formula for the volume of a cone would like to have a word with you :P

Ted Shifrin

Yup, I was wrong. The Greeks could only do 5 and 15. It took Gauss at the age of 19. Stewart said this turned him into a mathematician.

leslie townes

my immediate family consists of three people. i think about dividing by three all the time. it does help that my daughter mostly wants to eat waffles. that does turn it into dividing by two and then figuring out where to get some waffles.

user2103480

@MikeMiller The setting is that we have a generator $A \colon H \rightarrow H$ of a $C_0$-semigroup $(T_t)_{t \geq 0}$ (on some Hilbert space), a measurable disturbance $B \colon H \rightarrow H$, and an operator-valued thingy $C$ that governs the magnitude of our noise, induced by some brownian motion $W_t$. This last thing you can imagine as time-dependent noise at every point in space.

We want to solve the stochastic PDE $$\mathrm d X_t = (AX_t + B(t,X_t)) \mathrm d t+ C(t,X_t) \mathrm d W_t$$ in a "mild" sense, which means that we search for an $H$-valued time-dependent process $X$ wi…

Matthew Christopher Bartsh

I'd argue that dividing by sixteen is probably more important than dividing by three.

Thorgott

some guy spent 10 years at the end of the 19th century writing a 200 page paper with an explicit construction of the 65537 gon

Ted Shifrin

18:43

I disagree with that one.

leslie townes

user2103480, you had me until brownian motion. i was loving the semigroups and hilbert space stuffs.

Thorgott

the guy was Johann Gustav Hermes*

leslie townes

that's a weird mix of names and nationalities. hi, i might be german, and now i'm greek. figure out who you are, dude.

Matthew Christopher Bartsh

@leslietownes That's about the only time dividing by three comes up: when arbitrarily there are exactly three people, or three nations or whatever. But it could just as easily have been five or seven people or nations.

user2103480

whoa that was intense, had like 5 seconds left to find the LaTeX mistake in the last equation

Semiclassical

18:46

again, volume of a cone. $V=Ah/3$

which to my brain is not a small application

leslie townes

i don't think it's arbitrary that there are exactly three people in my house. what do you want, for me to have 15 more kids?

or 14 more kids, if i'm included in the total

i'm obviously kidding. i just don't have that kind of energy. one was enough

Ted Shifrin

@Semiclassical Well, the same formula works in $n$ dimensions with an $n$ denominator.

Matthew Christopher Bartsh

@Semiclassical Okay, cone to cylinder is 1:3. I forgot about that. Any more, though?

Semiclassical

also, volume of sphere

4/3 pi R^3, after all

Ted Shifrin

Hi, demonic @Alessandro.

Thorgott

18:49

this seems like an even more pointless variant of arguing whether pi or tau is the better constant

Matthew Christopher Bartsh

@Semiclassical What if you use tau instead of pi?

tau = 2 pi

Semiclassical

then it's 2/3 tau R^3

Alessandro Codenotti

Hi Ted

PM 2Ring

Speaking of angle bisection, a couple of weeks ago I answered a question about using bisection to calculate pi, math.stackexchange.com/a/4034914/207316

Matthew Christopher Bartsh

Then the division by three is even more prominent. Darn.

Any others?

leslie townes

18:51

that's a quality answer, @PM2Ring, i don't know why it wasn't accepted.

i mean i do, it's just pearls before swine, all of this is. nothing means anything. why don't i just yell at people in the street.

but the more restrained version is, that is a good answer.

Matthew Christopher Bartsh

Are these divisions by three a consequence of having three dimensions in a sphere and a cone ?

Semiclassical

yeah

per Ted's comment above

PM 2Ring

@leslietownes Thanks. Maybe the OP just doesn't understand about accepting. They are pretty new.

Ted Shifrin

Well, the bicylinder (region inside two orthogonal congruent cylinders) also gives $16a^3/3$. More for $3$.

Semiclassical

setting aside that, though, in pedagogy i see three as important as the first fraction where there's two possible cases (1/3 vs. 2/3)

leslie townes

18:54

i do think the upvoting process is not intuitive.

user2103480

@MikeMiller I missed a dependence on $t$ there. So $\mathcal K_1(H)(t) = ...$ and if we restrict both the process $H$ (with random paths) to a small enough initial segment (so its paths are defined on $[0,t_0]$ for small enough $t_0$), and apply the functional in that domain, we obtain a contraction.

leslie townes

accepting and upvoting seem to be separate, that's weird. maybe it's not.

Matthew Christopher Bartsh

@Semiclassical Does that support the case for using base twelve?

user2103480

@leslietownes you'd love the stochastic integrals

leslie townes

i don't know that i would. i saw them on the chalkboard in the rooms i had to teach in. i loved erasing them. it gave me a cool vibe.

PM 2Ring

19:00

@leslietownes I sometimes explain accepting to newbies, even on questions that I haven't answered. But on that particular question, it's a tricky choice. Joshua posted the first answer, which directly answers the OP's question. My answer was originally just going to be a comment responding to other comments, but it got out of control. :)

Claude's answer is pretty groovy: he's a wizard at finding approximations with great convergence. His answer's probably a bit too technical for the OP, but I'm sure it will be of great interest to some future readers.

Ted Shifrin

A lot of faculty who taught before me started erasing their blackboards, because I sometimes would comment about something they'd left on the board :D

leslie townes

one thing i really like about math.SE is the plurality of perspectives on a problem. nobody needs only my perspective. i'm deranged. don't follow me anywhere.

Ted Shifrin

I always left mine for posterity (and for people to marvel at the multiple-color drawings).

Semiclassical

i'd love to be able to say "hah, physics is where stochastic integrals rule" but I think Weiner and Kac have pride of place there

Ted Shifrin

Wiener

PM 2Ring

19:03

@leslietownes Definitely separate. A 1 rep newbie can accept, but you need 15 points to upvote. And 125 to downvote (thank goodness).

geocalc33

what are applications of mappings that do not preserve distances between elements but preserve the order of the elements?

PM 2Ring

@TedShifrin One of my highschool maths teachers was ambidextrous. When one hand got tired from writing on the blackboard, he'd just switch hands. Sometimes we'd have to tell him to slow down and give us a break.

user2103480

They integrate over operator-valued random stochastic processes $\Phi_s \in L(U,H)$ with respect to $U$-valued brownian motion $W_t$ whose covariance is a positive definite trace class operator $Q$ andthen we have identities like $$\Bbb E \left[ \left \| \int_0^t \Phi_s \, \mathrm d W_s \right \|_H^2 \right] = \Bbb E \left[ \int_0^t \|\Phi_s \circ \sqrt{Q} \|^2_{L_2} \, \mathrm{d}s \right]$$

So any functional analyst should revel in this @leslietownes

Ted Shifrin

@PM2Ring LOL. I can imagine. I wrote rather too fast for most of my students (and neatly so!), but I often stopped to talk and entertain conversations with the students :P

user2103480

$\Phi_s \circ \sqrt Q$ is hilbert-schmidt since Q is trace-class

Thorgott

19:09

@user2103480 ah thanks, now everything makes sense

user2103480

@Thorgott tHiS coNtAiNs aLl NeCesSaRy iNfO

Semiclassical

@TedShifrin shoot, i knew it was going to do that wrong

@user2103480 on that note, the notion of Hilbert-Schmidt norm in QM is something I plan to do some research on

Ted Shifrin

It's cuz Mericans don't know the difference in pronunciation between ie and ei when they come in German names. :)

Semiclassical

at some level it's just trivial, of course

QM people love hilbert spaces of state vectors, so they should love the Hilbert space of operators on that space

(modulo annoying conversations about bounded operators that makes a typical physicist want to gag)

I do know a little about htat story already, though. like, there's the whole notion of a density matrix in QM

which basically amounts to "take the cone of positive Hermitian operators and restrict to those with trace 1"

user2103480

@Thorgott it's just an isometry from square-integrable $H$-martingales to $L^2(\Omega \times [0,t];L_2(\sqrt Q(U), H))$

@Semiclassical I wouldn't know, physics is an enigma to me

Semiclassical

19:15

hah

user2103480

@Semiclassical I certainly do love that we have nice separable hilbert-spaces of operators

Semiclassical

i think there's a way to view the above in QM as just "hurr, Riesz representation theorem"

Thorgott

of course

no need to state the obvious

user2103480

@Semiclassical my crippled topologist's brain area was shortly confused about the cone of the positive hermitian operators

Semiclassical

eh, nothing too strange there

user2103480

19:18

yeah yeah it's just linear combinations

Semiclassical

convex combinations, but yes

(you don't want negative coefficients, after all)

Thorgott

that's what an algebraic cone be

user2103480

I just wanted to show that I know he doesn't mean taking the product with $[0,1]$ and collapsing the top, not post a whole definition

Semiclassical

ah yeah

i like how i'm having this abstract conversation while at the same time i'm working on my middle-school level demo for tomorrow's lab

hrm, that second one didn't come out right

Ted Shifrin

Hard to see. Are we dropping an egg in a bottle?

Semiclassical

19:24

okay, yeah, that's better

that's a short bended straw with paper clips holding it closed (and providing weight)

Ted Shifrin

bended? Good day for English :P

Semiclassical

pfff

Ted Shifrin

I do not understand what the demo is.

The straw falls because of the weight?

Is there something to do with air pressure?

Semiclassical

near the bottom, yes. but near the top it floats :)

Ted Shifrin

But your stopper is closed at the top?

Semiclassical

19:26

yep

and yeah, it's a pressure thing

Ted Shifrin

So partial vacuum ...

Semiclassical

when the diver, as they call it, is near the top, the air bubble is displacing enough water to keep it bouyant

Ted Shifrin

Ohhh ... There's water in there.

Semiclassical

yeah

hard to show this with pictures :/

(esp. my crappy webcam)

but near the bottom, the additional hydrostatic pressure compresses the air bubble just enough that the diver will sink instead

so there's a point of neutral bouyancy about halfway along the bottle

above that, it rises. below that, it sinks

(it's an unstable equilibrium, so you can't really balance it there)

Ted Shifrin

So the story of the moral is that water pressure is linear with depth.

PM 2Ring

19:29

It's a Cartesian diver. I spent many hours making & playing with them when I was a kid.

Semiclassical

right

the other cool part of this: if you compress the bottle one way, the diver spontaneously rises (even if it's on the bottom)

if you compress it another, it falls

Ted Shifrin

Ah, good old flexible glass.

Semiclassical

lol, i'm too cheap for that

it's just a Simply orange plastic bottle

which basically comes down to: if you squeeze a square along its diagonal, its cross-sectional area increases. if you compress it along the sides, the area decreases

which means you can either increase or decrease the volume of the air bubble at the top of the bottle. air bubble gets bigger, pressure of air goes down, therefore pressure throughout bottle goes down. hence, less pressure on air bubble and it expands. hence it displaces more water and rises

it's pretty finicky to get the balance, though

you want the point of neutral bouyancy in the middle, and that implies a pretty delicate range for how much air is trapped in the straw

Matthew Christopher Bartsh

It's a good physics lesson. So few people master buoyancy.

Ted Shifrin

I always assigned Archimedes law as homework after we'd done the divergence theorem.

Semiclassical

19:36

it's pretty quick if i remember right

Matthew Christopher Bartsh

Almost no one understands the cause of the fluid upthrust. They only parrot that it is less dense than water.

Ted Shifrin

Along with an exercise to prove that for any closed surface in $\Bbb R^3$, $\int_S \vec n\,d\sigma = 0$ ($\vec n$ being the unit outward normal).

Semiclassical

yeah

Ted Shifrin

You need the vector form of the divergence theorem.

Matthew Christopher Bartsh

The cause is the pressure gradient, in case you didn't know.

Ted Shifrin

19:37

This comes after I've dropped someone through a tunnel from one point on the earth to another.

Semiclassical

archimedes' principle (at least at the macro scale) boils down to just "if you replace a mathematical volume $V$ with a real volume $V$, the forces on the surface can't change"

@TedShifrin ah, that old chestnut

Ted Shifrin

The issue is about weight, though, not just volume.

The mathematical volume has net weight.

Semiclassical

yeah. it's a volume of fluid, after all

Ted Shifrin

No, it's displacing a volume of fluid.

I'm not just being an ass here.

Semiclassical

after i put the object in, yes. but if you have a mathematical volume in a fluid, you still have to have zero net force on that fluid volume

Ted Shifrin

19:39

If the mathematical volume has no mass, forget about it.

It has to be partially submerged.

Semiclassical

no. otherwise we couldn't apply archimedes principle to bubbles

Ted Shifrin

I do not follow. Maybe our statements of Archimedes' principle are just different.

Semiclassical

well, my point is like this: if I have a volume region $V$ with boundary $\partial V$ in my original fluid---no object introduced yet---then there's a certain distribution of hydrostatic force along $\partial V$

in particular, that distribution has to be net zero. otherwise, the fluid contained within that volume wouldn't be in hydrostatic equilibrium

Ted Shifrin

Be careful. Only the portion of $\partial V$ below water level.

Semiclassical

ah, yes. i was taking for granted that $V$ was entirely below the water level

Ted Shifrin

19:43

If it's entirely below, then it's going to sink in a minute.

Semiclassical

(since everything above isn't actually displacing fluid)

uh

Ted Shifrin

It may sink.

Once we introduce mass.

Semiclassical

right. (or at least it can)

Ted Shifrin

I always explained it with partially-submerged objects. It could just be barely at equilibrium with its top at the surface.

Semiclassical

that said, i did say something wrong and dumb above

the net hydrostatic force over the (submerged) surface will still be $mg$.

Ted Shifrin

19:44

Well, you had zero mass to this point. Part of my bitch.

Semiclassical

how could i have zero mass? a fluid isn't massless

Ted Shifrin

That's where we started the whole thing.

leslie townes

i'm a massless fluid

Ted Shifrin

The confusion in this discussion arises from the conflation of two things: A closed object with inherent mass. A "mathematical" object consisting of the portion of that object under water and filled with the same water.

But we're going to arrive at the same proof eventually. :P

Semiclassic is mumbling that mathematicians shouldn't teach physics :D

Semiclassical

what's mathematical there is only the fact that, before you put the object in, there's no real boundary between one part of the fluid and another

leslie townes

19:47

i'm filled with the same water

who is contradicting me

Semiclassical

but the fluid within that boundary still has volume and weight

Ted Shifrin

sends leslie to write a 100-page contract

Of course, @Semiclassic, not to be confused with the weight of the original object.

leslie townes

ugh. some of my homework for today was to write a one-page contract. no thanks.

Semiclassical

"ugly giant bags of mostly water"

@TedShifrin of the displacing object, yeah

leslie townes

that is very much it. that is humanity. thank you, star trek

Ted Shifrin

19:48

In most real-world applications, the object is partly floating, and so it really is important not to consider it just submerged.

Anyhow, divergence theorem. Done.

Semiclassical

meh. divide it into two parts, one lying above the water and one lying below.

Ted Shifrin

Yes, that's what I suggest.

Semiclassical

as a tangent, there is a simple physics question i love to pose in this vein

Ted Shifrin

The exercise in my book assumes constant density fluid. I'm not sure why I did that.

Seems unnecessary to me at the moment.

Semiclassical

suppose you put an ice cube in a glass of water. once it stops bobbing around, the water will be at some consistent level

Ted Shifrin

19:52

Yeah, it's unnecessary.

geocalc33

when you non-isometrically map from R^2 to R^2_+, is there any reason to believe that a series in R_2 will behave well in R^2_+?

Semiclassical

now come back 15 minutes later when all of the ice has melted. what has happened to the water level?

Ted Shifrin

Oh yeah. That's a nice puzzle, @Semiclassic. Decades and decades ago, I thought about it.

PM 2Ring

Oddly, we only have 3 Cartesian diver questions on Physics.SE physics.stackexchange.com/search?q=cartesian+diver

Ted Shifrin

@geocalc No.

Semiclassical

19:53

crucially, i'm assuming that the water the ice melts into has the same density as the water in the glass

so it's different than, say, an iceberg of frozen freshwater melting into the Atlantic ocean

PM 2Ring

@Semiclassical You could do it with a normal ice cube, floating in deuterium water. :) Deuterium ice sinks in normal water.

Semiclassical

i've never been fully satisfied by the answer, though. not at a macroscopic level, to be clear: Archimedes' principle has an unambiguous answer here

@PM2Ring yeah, but then it's boring: all of the ice is below water, and ice expands when melted, so the mixture expands

Ted Shifrin

I'm confused now, thinking about it, because then the whole thing is that frozen water has different density than the corresponding melted water.

geocalc33

@ted I think the main reason because the non-isometric map will not preserve addition

Ted Shifrin

Even if it did, you wouldn't know anything about convergence of the series.

It could send very small things off to very large things.

PM 2Ring

19:57

Another substance with the solid less dense than the liquid is plutonium. But it's not easy to do demos with molten plutonium at the typical high school. ;)

Ted Shifrin

LOL ... @PM2Ring seems to have a radioactive bent.

Semiclassical

so, let the volume of ice be $V_i$ with $V_w$ of it lying below the water line. the bouyant force of the ice is $\rho_w g V_w$ and the weight of the ice is $\rho_i g V_i$, so we have $\rho_w V_w=\rho_i V_i$

Mike Miller

@user2103480 I don't follow all of the notation but I get your point. You get a contraction mapping property on spaces like $L^p \cap C^0$ where you use both norms, and the one that really contributes in an important way is the $L^p$ norm. (I was surprirsed you care about $L^p$ solutions but I see now they're continuous solutions that are also $L^p$.)

Ted Shifrin

Right @Semiclassic. So if it's buoyant, the ice has to be denser.

Semiclassical

less dense. $V_w<V_i,$ so $\rho_w>\rho_i$

Mike Miller

19:59

This clarifies why it's important :)

Ted Shifrin

Oh, right, good, because that's what I remembered. Your notation confused me.

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$Mathematics$ Mathematics