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Matthew Christopher Bartsh
Matthew Christopher Bartsh
6:00 PM
@PM2Ring I.e PM is the time to ring?
 
PM 2Ring
PM 2Ring
@MatthewChristopherBartsh Maybe... :)
 
Matthew Christopher Bartsh
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
PM 2Ring
@MatthewChristopherBartsh Not really. It was a while ago!
 
leslie townes
leslie townes
i don't remember the replies to a post i made last night. i live in the fast lane.
 
Matthew Christopher Bartsh
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
PM 2Ring
6:09 PM
@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
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
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
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
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
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
leslie townes
6:23 PM
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
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
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
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
Semiclassical
@leslietownes the integer formerly known as ten, you mean
 
leslie townes
leslie townes
precisely.
 
Matthew Christopher Bartsh
Matthew Christopher Bartsh
6:25 PM
*isn't very important to smoothly divide by three.
@Semiclassical lol
 
Semiclassical
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
PM 2Ring
@MatthewChristopherBartsh There's "trisect", but admittedly, it's a technical term, not part of the common vocabulary.
 
leslie townes
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
Semiclassical
with base-12 you can defer that conversation to when you start handling 1/5 and such
 
leslie townes
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
PM 2Ring
6:29 PM
@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
Semiclassical
right
 
leslie townes
leslie townes
you need a marked ruler.
 
Semiclassical
Semiclassical
yeah. so not that advanced, really, but hardly obvious how to use that
 
Matthew Christopher Bartsh
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
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
Ted Shifrin
6:31 PM
I'm not sure you can do it with perpendiculars.
 
Matthew Christopher Bartsh
Matthew Christopher Bartsh
I've never heard of trisection of a line.
 
PM 2Ring
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
leslie townes
you can definitely ruler-and-compass divide a segment into an integer number of parts.
 
Ted Shifrin
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
Matthew Christopher Bartsh
Where in maths is dividing by three seen?
 
leslie townes
leslie townes
6:33 PM
you just do another segment with as many parts as you want and draw a family of parallel lines.
 
Ted Shifrin
Ted Shifrin
See, @leslie, you are talking about something geometric. I'm stunned.
 
PM 2Ring
PM 2Ring
Yep. Ruler & compass let's you construct any quadratic number.
 
leslie townes
leslie townes
it's all professor wu. that's where i stole it from.
 
Ted Shifrin
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
Thorgott
it's probably not what you mean by dividing by three, but there's the famous paper Division by three
 
Semiclassical
Semiclassical
6:35 PM
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
leslie townes
haha, the set theory paper.
i love that
 
PM 2Ring
PM 2Ring
@Semiclassical I'm not sure whether Gauss would be pleased or perturbed by that statement. :)
 
leslie townes
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
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
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
Thorgott
6:39 PM
constructibility of the n-gon is algebra, not geometry :P
 
Ted Shifrin
Ted Shifrin
@Semiclassical You mean for suitable $n$?
 
Thorgott
Thorgott
awaits slap from Ted
 
leslie townes
leslie townes
i will give you $e^{1.5}$ slaps.
how the tables have turned.
 
Semiclassical
Semiclassical
right
 
Ted Shifrin
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
Thorgott
6:40 PM
didn't Gauss do that?
or am I misremembering
 
leslie townes
leslie townes
yeah but he was a psychopath
 
Ted Shifrin
Ted Shifrin
@leslietownes I have not yet designated you a legitimate smacker.
 
Semiclassical
Semiclassical
yeah. i mean, to bring 1/3 into it---cone:cylinder::1:3
 
leslie townes
leslie townes
i provisionally withdraw my potentially transcendental number of slaps
 
Matthew Christopher Bartsh
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
Semiclassical
6:42 PM
per my comment above, the formula for the volume of a cone would like to have a word with you :P
 
Ted Shifrin
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
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
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
Matthew Christopher Bartsh
I'd argue that dividing by sixteen is probably more important than dividing by three.
 
Thorgott
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
Ted Shifrin
6:43 PM
I disagree with that one.
 
leslie townes
leslie townes
user2103480, you had me until brownian motion. i was loving the semigroups and hilbert space stuffs.
 
Thorgott
Thorgott
the guy was Johann Gustav Hermes*
 
leslie townes
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
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
user2103480
whoa that was intense, had like 5 seconds left to find the LaTeX mistake in the last equation
 
Semiclassical
Semiclassical
6:46 PM
again, volume of a cone. $V=Ah/3$
which to my brain is not a small application
 
leslie townes
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
Ted Shifrin
@Semiclassical Well, the same formula works in $n$ dimensions with an $n$ denominator.
 
Matthew Christopher Bartsh
Matthew Christopher Bartsh
@Semiclassical Okay, cone to cylinder is 1:3. I forgot about that. Any more, though?
 
Semiclassical
Semiclassical
also, volume of sphere
4/3 pi R^3, after all
 
Ted Shifrin
Ted Shifrin
Hi, demonic @Alessandro.
 
Thorgott
Thorgott
6:49 PM
this seems like an even more pointless variant of arguing whether pi or tau is the better constant
 
Matthew Christopher Bartsh
Matthew Christopher Bartsh
@Semiclassical What if you use tau instead of pi?
tau = 2 pi
 
Semiclassical
Semiclassical
then it's 2/3 tau R^3
 
Alessandro Codenotti
Alessandro Codenotti
Hi Ted
 
PM 2Ring
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
Matthew Christopher Bartsh
Then the division by three is even more prominent. Darn.
Any others?
 
leslie townes
leslie townes
6:51 PM
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
Matthew Christopher Bartsh
Are these divisions by three a consequence of having three dimensions in a sphere and a cone ?
 
Semiclassical
Semiclassical
yeah
per Ted's comment above
 
PM 2Ring
PM 2Ring
@leslietownes Thanks. Maybe the OP just doesn't understand about accepting. They are pretty new.
 
Ted Shifrin
Ted Shifrin
Well, the bicylinder (region inside two orthogonal congruent cylinders) also gives $16a^3/3$. More for $3$.
 
Semiclassical
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
leslie townes
6:54 PM
i do think the upvoting process is not intuitive.
 
user2103480
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
leslie townes
accepting and upvoting seem to be separate, that's weird. maybe it's not.
 
Matthew Christopher Bartsh
Matthew Christopher Bartsh
@Semiclassical Does that support the case for using base twelve?
 
user2103480
user2103480
@leslietownes you'd love the stochastic integrals
 
leslie townes
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
PM 2Ring
7:00 PM
@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
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
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
Ted Shifrin
I always left mine for posterity (and for people to marvel at the multiple-color drawings).
 
Semiclassical
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
Ted Shifrin
Wiener
 
PM 2Ring
PM 2Ring
7:03 PM
@leslietownes Definitely separate. A 1 rep newbie can accept, but you need 15 points to upvote. And 125 to downvote (thank goodness).
 
geocalc33
geocalc33
what are applications of mappings that do not preserve distances between elements but preserve the order of the elements?
 
PM 2Ring
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
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
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
user2103480
$\Phi_s \circ \sqrt Q$ is hilbert-schmidt since Q is trace-class
 
Thorgott
Thorgott
7:09 PM
@user2103480 ah thanks, now everything makes sense
 
user2103480
user2103480
@Thorgott tHiS coNtAiNs aLl NeCesSaRy iNfO
 
Semiclassical
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
Ted Shifrin
It's cuz Mericans don't know the difference in pronunciation between ie and ei when they come in German names. :)
 
Semiclassical
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
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
Semiclassical
7:15 PM
hah
 
user2103480
user2103480
@Semiclassical I certainly do love that we have nice separable hilbert-spaces of operators
 
Semiclassical
Semiclassical
i think there's a way to view the above in QM as just "hurr, Riesz representation theorem"
 
Thorgott
Thorgott
of course
no need to state the obvious
 
user2103480
user2103480
@Semiclassical my crippled topologist's brain area was shortly confused about the cone of the positive hermitian operators
 
Semiclassical
Semiclassical
eh, nothing too strange there
 
user2103480
user2103480
7:18 PM
yeah yeah it's just linear combinations
 
Semiclassical
Semiclassical
convex combinations, but yes
(you don't want negative coefficients, after all)
 
Thorgott
Thorgott
that's what an algebraic cone be
 
user2103480
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
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
Ted Shifrin
Hard to see. Are we dropping an egg in a bottle?
 
Semiclassical
Semiclassical
7:24 PM
okay, yeah, that's better
that's a short bended straw with paper clips holding it closed (and providing weight)
 
Ted Shifrin
Ted Shifrin
bended? Good day for English :P
 
Semiclassical
Semiclassical
pfff
 
Ted Shifrin
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
Semiclassical
near the bottom, yes. but near the top it floats :)
 
Ted Shifrin
Ted Shifrin
But your stopper is closed at the top?
 
Semiclassical
Semiclassical
7:26 PM
yep
and yeah, it's a pressure thing
 
Ted Shifrin
Ted Shifrin
So partial vacuum ...
 
Semiclassical
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
Ted Shifrin
Ohhh ... There's water in there.
 
Semiclassical
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
Ted Shifrin
So the story of the moral is that water pressure is linear with depth.
 
PM 2Ring
PM 2Ring
7:29 PM
It's a Cartesian diver. I spent many hours making & playing with them when I was a kid.
 
Semiclassical
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
Ted Shifrin
Ah, good old flexible glass.
 
Semiclassical
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
Matthew Christopher Bartsh
It's a good physics lesson. So few people master buoyancy.
 
Ted Shifrin
Ted Shifrin
I always assigned Archimedes law as homework after we'd done the divergence theorem.
 
Semiclassical
Semiclassical
7:36 PM
it's pretty quick if i remember right
 
Matthew Christopher Bartsh
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
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
Semiclassical
yeah
 
Ted Shifrin
Ted Shifrin
You need the vector form of the divergence theorem.
 
Matthew Christopher Bartsh
Matthew Christopher Bartsh
The cause is the pressure gradient, in case you didn't know.
 
Ted Shifrin
Ted Shifrin
7:37 PM
This comes after I've dropped someone through a tunnel from one point on the earth to another.
 
Semiclassical
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
Ted Shifrin
The issue is about weight, though, not just volume.
The mathematical volume has net weight.
 
Semiclassical
Semiclassical
yeah. it's a volume of fluid, after all
 
Ted Shifrin
Ted Shifrin
No, it's displacing a volume of fluid.
I'm not just being an ass here.
 
Semiclassical
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
Ted Shifrin
7:39 PM
If the mathematical volume has no mass, forget about it.
It has to be partially submerged.
 
Semiclassical
Semiclassical
no. otherwise we couldn't apply archimedes principle to bubbles
 
Ted Shifrin
Ted Shifrin
I do not follow. Maybe our statements of Archimedes' principle are just different.
 
Semiclassical
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
Ted Shifrin
Be careful. Only the portion of $\partial V$ below water level.
 
Semiclassical
Semiclassical
ah, yes. i was taking for granted that $V$ was entirely below the water level
 
Ted Shifrin
Ted Shifrin
7:43 PM
If it's entirely below, then it's going to sink in a minute.
 
Semiclassical
Semiclassical
(since everything above isn't actually displacing fluid)
uh
 
Ted Shifrin
Ted Shifrin
It may sink.
Once we introduce mass.
 
Semiclassical
Semiclassical
right. (or at least it can)
 
Ted Shifrin
Ted Shifrin
I always explained it with partially-submerged objects. It could just be barely at equilibrium with its top at the surface.
 
Semiclassical
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
Ted Shifrin
7:44 PM
Well, you had zero mass to this point. Part of my bitch.
 
Semiclassical
Semiclassical
how could i have zero mass? a fluid isn't massless
 
Ted Shifrin
Ted Shifrin
That's where we started the whole thing.
 
leslie townes
leslie townes
i'm a massless fluid
 
Ted Shifrin
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
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
leslie townes
7:47 PM
i'm filled with the same water
who is contradicting me
 
Semiclassical
Semiclassical
but the fluid within that boundary still has volume and weight
 
Ted Shifrin
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
leslie townes
ugh. some of my homework for today was to write a one-page contract. no thanks.
 
Semiclassical
Semiclassical
"ugly giant bags of mostly water"
@TedShifrin of the displacing object, yeah
 
leslie townes
leslie townes
that is very much it. that is humanity. thank you, star trek
 
Ted Shifrin
Ted Shifrin
7:48 PM
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
Semiclassical
meh. divide it into two parts, one lying above the water and one lying below.
 
Ted Shifrin
Ted Shifrin
Yes, that's what I suggest.
 
Semiclassical
Semiclassical
as a tangent, there is a simple physics question i love to pose in this vein
 
Ted Shifrin
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
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
Ted Shifrin
7:52 PM
Yeah, it's unnecessary.
 
geocalc33
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
Semiclassical
now come back 15 minutes later when all of the ice has melted. what has happened to the water level?
 
Ted Shifrin
Ted Shifrin
Oh yeah. That's a nice puzzle, @Semiclassic. Decades and decades ago, I thought about it.
 
PM 2Ring
PM 2Ring
Oddly, we only have 3 Cartesian diver questions on Physics.SE physics.stackexchange.com/search?q=cartesian+diver
 
Ted Shifrin
Ted Shifrin
@geocalc No.
 
Semiclassical
Semiclassical
7:53 PM
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
PM 2Ring
@Semiclassical You could do it with a normal ice cube, floating in deuterium water. :) Deuterium ice sinks in normal water.
 
Semiclassical
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
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
geocalc33
@ted I think the main reason because the non-isometric map will not preserve addition
 
Ted Shifrin
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
PM 2Ring
7:57 PM
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
Ted Shifrin
LOL ... @PM2Ring seems to have a radioactive bent.
 
Semiclassical
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
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
Ted Shifrin
Right @Semiclassic. So if it's buoyant, the ice has to be denser.
 
Semiclassical
Semiclassical
less dense. $V_w<V_i,$ so $\rho_w>\rho_i$
 
Mike Miller
Mike Miller
7:59 PM
This clarifies why it's important :)
 
Ted Shifrin
Ted Shifrin
Oh, right, good, because that's what I remembered. Your notation confused me.
 
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