this is where dimensional analysis helps. mathematicians call this combination x^y^z (with ^ indicating the wedge product), or xyz for short, a volume form because it has dimensions of volume, simple as that. if you have a parallelepiped or cuboid (like a 3d parallelogram or rectangle respectively) or similar simple 3d shape, the product of its three spatial dimensions is its volume. it's actually not a separate degree of freedom, but it is a different way of measuring something: you cant change the volume of some shape without changing its spatial extent in at least one dimension. in the same way a cuboid can be uniquely identified only in terms of its three dimensions. for more complicated shapes you have you use a differential measure and then integrate: dx^dy^dz. this is the standard little symbol you put in a volume integral behind the integrand to indicate that you want to measure all three dimensions at once, that is, that you want to measure volume. time actually doesnt come into this at all, and again you can tell this just from looking at the dimensions of the quantities we are considering: all have dimensions of distance, so there is no way for time to enter unless we introduce something else to our theory. for example, every theory of wave propagation has at least one implicit timescale. this introduces a quantity with dimensions of time and you have a whole new set of things you could want to measure. unless the theory is quite simple (i.e. a linear theory), there are actually a ton of different things you could want to measure and you have to be careful and precise about what you're interested in, etc.

now, if you want to measure things in special or general relativity you introduce time as a coordinate with a different metric signature. then the total spacetime "volume form" has dimensions of volume times time.

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You're certainly right in that its very easy to hide energy is all sorts of ways and you cant tell how much energy there is in some volume of space just by looking at it, as opposed to something like momentum or electric current which can be immediately identified. If there's a net momentum to some collection of matter it will be immediately obvious since the whole thing will be moving. The fact that you cant immediately identify all of the operative modes of energy distribution in some region of spacetime is one of the things that makes physics hard, and is in part why its often understood by its practitioners as an exercise in modeling the world as opposed to an identical description. Most of the models in physics end up taking the form of some proposed energy function, along with very particular rules for how this energy function induces dynamics. An energy function could account for all of the things you mention: isotropic strain, torsion, rigid motion, heating, electric charge, etc. An important principle of practice when studying these systems is that "anything that is not forbidden is mandatory". What this means is that any type of interaction between different types of energy ought to be considered unless it is forbidden by some superseding principle (e.g. some type of symmetry, experimental setup, material conjecture, separation of scales, or the impatience of the person doing the math). That is to say, for each type of energy that is embedded in some body, there is usually a way for it to be converted to each other type (although this can be ignored if, say, it takes a much longer time than the duration of interest).

On the topic of volume, though, its important to note that its very possible to change shape without changing volume. This is what every incompressible fluid does when it moves around, and is also the type of deformation described by a "pure shear" of some solid body. It's important not to confused the derivative of the volume (dV/dt, a single number) with the total set of derivatives of the individual material elements (the set {dX_i/dt} of the derivative of each displacement vector X in each direction i, d*N numbers where d is the spatial dimension and N is the number of material elements in the solid).
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it doesnt seem to work since you have the unit the same as the quantity so in your algebra, for energy you would have E(nergy number) quantities of E(nergy unit) (E x E) rather than the current system which is E x J (or eV or whatever unit you are using) it makes it confusing and makes it even harder to understand that the units are arbitrary quantities of a concept and can be any value (imagine you had the following units of distance: 1) metre, 2) light year 3) distance; each one being a different distance in space)

sorry, i am immediately thinking of the teaching nightmare having the name of the concept and the name of one of the the units for the concept being the same!

edit: i am mad about the volts thing - having volts as a unit, and having "voltage" used as a catch all to refer to p.d./e.m.f. is tough to deal with

Edited by tears ()

I'm torn between tears pedagogical argument and the coolness of smoothing out the unit names
this is good, but is still in english.. which is bad. better to use esperanto as starting point.
| Nova Nomo |

| Energio |
| Potenco |
| Forto |
| Metro |
| Litro |
| Gram |
| Dua |
| Temp |
| Lumen |
| Stango |
| Premo |
| Turnu |
| eFluo |
| ePremo |
| Om |
| eMole |
| Mole |
| Persekundo |
| ? |
| ? |
| Zum |
some things to think about:

if you were simplifying units surely the first thing you would do is to strip out stuff like pascals and bar all together and just express it as N/m2 or MN/m2. it would help with understanding - same for watts as J/s

i dont understand why you would rename kilogram to gram when there is already a quantity of mass called a gram - i could understand if you wanted to switch the SI unit of mass to the quantity denoted by a gram - that would be cool, since the use of the kg is just some historical quirk. then kilogram would just be the unit + the prefix like everything else

i dont like the term electron pressure to refer to p.d. because it brings in all sorts of misconceptions - electricity is one of the hardest things to teach and i have used plenty of water flow metaphors, but they tend to come back and bite you - since volts are just joules per coulomb i think that is a far better way to unit it - since it is very descriptive as energy per packet of charge -> i also find it helps with visulisation in terms of energy transfer. The problem does build with this since converting derived units back to SI units is pretty hard when you get to things like ohms - V/A, which would then be (J/C)/A or J/(A2 x s) or whatever which quickly gets complicated - expressing units in terms of other units and back to SI units tends to get left to A-level physics here

i also dont like the term eFlow for rate of flow of charge, just because it uses an e, which tends to get associated with electrons, and charge flow is charge flow not necessarily electron flow - i know that e is used for elementary charge but theres a big difference between teaching elementary charge at A-level and teaching flow of charge to 12 year olds who cant write good.

i can see problems with relating quantity of charge to moles in the eMole for coulombs and moles for amount, unless you were to redefine one or the other so that the mole part refereed to the same number of particles and elementary charge multiples - which you could do - it would be quite nice to standardise

im not against renaming units, i just think that ease of learning is paramount
my feelings about units are that the metric is fine except that celsius is worse than fahrenheit for everyday use, and if youre not talking about everyday use you can just use kelvin
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*gesturing at my unit*
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i dont think there's too much to be gained really from overhauling the names for orders of magnitude. people who need to deal with them a lot can just use powers of whatever base and units are necessary and otherwise the only important thing is having a name that you can use to communicate effectively, so i dont see what you really get with this.
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There's actually a pretty intuitive reason why most infinite series would sum to irrational numbers, and its simply that there are infinitely more of them than there are rational numbers!
dear math thread posters,

please advise on how to build a no-kill mouse trap. the mice have outsmarted all my attempts so far. they are cute but are becoming overly confident and comfortable. preferably construction plans only use household items, no cnc here.

thanks in advance.
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Gssh posted:

dear math thread posters,

please advise on how to build a no-kill mouse trap. the mice have outsmarted all my attempts so far. they are cute but are becoming overly confident and comfortable. preferably construction plans only use household items, no cnc here.

thanks in advance.

mice are notoriously bad at maths, so if you write some simple algebra on a piece of paper in a box the mouse will feel compelled to go into the box and try to solve it, which will take him hours.

keep all your algebra books at least 10 inches off the floor, otherwise they will learn and the trap will be useless
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nice work! the reason the curve is approx n^2 is because the number of iterations you have to do to calculate the sum of row n and all of its predecessors is f(n) = n(n+1)/2 = (n^2+n)/2 (which incidentally is (n+1,2) lol).

there are some minor optimizations you could make to your code. my intuition says not to allocate vectors as frequently and avoid clever math tricks that feel good but probably don't translate well to code (like the dot product thing for summing the vector's elements!). you could also translate this code into a language that uses your CPU's native floating point arithmetic instead of using the arbitrary precision decimal numbers that i assume scad uses by default... but 64-bit floating point numbers are only large enough to calculate like 100 rows of this calculation before running out of precision.

anyway all this is small potatoes because you'll only be reducing the coefficient of your time function (e.g. 4n^2 vs. n^2). those sorts of gains are overshadowed infinitesimally by n^2 as n gets real big. so if you want major speed improvement you'll need to exploit some sort of other identity.. probably something related to summing reciprocals. if you were calculating just the sum of the sums of each row in the triangle, you could just use the triangle's identity sum(n)=2^n to reduce the algorithmic complexity down from O(n^2) to just O(n)! but obviously you need to sum reciprocals of elements not just the elements... idk!
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btw toyot if you find yourself lacking computing power on your laptop, you could always just set up an amazon lightsail server for pennies on the hour to crunch numbers for you. they don't charge you when the server is offline too. for example, i run their cheapest tier 24/7 to host a web server and a dns-only openvpn tunnel w/ pihole to block ads & tracking on my phone when i'm off wifi, and it comes out to $3.50/month. so if you used one to run a program like that for a few hours at a time it'd probably cost you like 15 cents.

e: (i think amazon ec2 also has micro instances that might be better for your use-case that are just.. free -- but i'm unsure about that works)

Edited by aerdil ()

hah i wasn't thinking geometrically but yeah n(n+1)/2 is just the sum of all numbers 1..n, which is also the formula for the triangular number. yeah you can parallelize the calculation by splitting up the triangle as you mentioned, you just have to keep in mind the quadratic growth. so for balanced utilization of threads you'll have to divide the triangle up by area and not just e.g. top 1000 rows & bottom 1000 rows.

nice proof!! i'm not familiar with sum(1/(n,k)) = 1 + 1/n.

i made my way through most of the exercises in book of proof last summer and that stuff is really rewarding. also rewarding are the exercises on project euler, if you're not familiar with that site and are looking for more coding practice
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i think i did this right. i'm also a terrible very beginner coder, and i'm not sure if i understand your math completely, but here's a python script that i believe does the same thing as your openscad code, but is substantially faster. i'm still trying to figure out list comprehensions and what not, so please excuse the inefficient mess:

import numpy as np

def sumtworows(hardstop):
thesum = (1/6+1/5)
uprow = [5, 10]
while hardstop != 0:
k = len(uprow) - 2
midodd = 2 * uprow[k+1]
oddrow = [uprow[0]+1]
for n in range(k + 1):
oddrow.append(uprow[n] + uprow[n + 1])
evnrow = [oddrow[0]+1]
for n in range(k + 2):
evnrow.append(oddrow[n] + oddrow [n + 1])
which = [i for i in range(1, k+3)]
oddsum = 2 * sum([1 / oddrow[i] for i in which]) * np.prod([1 for i in which])
evnsum = 2 * sum([1 / evnrow[i] for i in which]) * np.prod([1 for i in which])
uprow = evnrow
thesum = oddsum - 1/midodd + evnsum + thesum
hardstop = hardstop - 1
return thesum
hardstop = int(input("Enter a hardstop value to calculate to: "))
print (sumtworows(hardstop))

Edited by aerdil ()

cjeck this https://sci-hub.tw/https://www.jstor.org/stable/2690456?seq=1
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Here's the python code I was using. It will show the graph that toyot claims converges to 3/2. This could be made to run faster but I tried to make it the mos usable code.

from fractions import Fraction
from scipy.special import binom # program this yourself if you don't have scipy
import matplotlib.pyplot as plt

# calculate desired reciprocals down to the row n_rows
reciprocals = lambda n_rows: [[Fraction(1,int(binom(n,k))) for k in range(2, n-1)] for n in range(4, n_rows + 4)]

# sum all the values in reciprocals(n_rows)
sum_of_reciprocals = lambda n_rows: float(sum([sum(row) for row in reciprocals(n_rows)]))

# display sum_of_reciprocals in graph
plt.plot([sum_of_reciprocals(i) for i in range(1,50)])

Edited by Acdtrux ()

had fun going through donald knuth's intro to conway's numbers, called SURREAL NUMBERS : how two ex-students turned on to pure mathematics and found total happiness, a mathematical novelette by D. E. Knuth. recommended for those nostalgic about being israeli hippies in goa. it opens

A. Bill, do you think you've found yourself?

B. What?

A. I mean -here we are on the edge of the Indian Ocean, miles away from civilization. It's been months since we ran off to avoid getting swept up in the system, and "to find ourselves." I'm just wondering if you think we've done it.

B. Actually, Alice, I've been thinking about the same thing. These past months together have been really great-we're completely free, we know each other, and we feel like real people again instead of like machines. But lately I'm afraid I've been missing some of the things we've "escaped" from. You know, I've got this fantastic craving for a book to read - any book, even a textbook, even a math textbook. It sounds crazy, but I've been lying here wishing I had a crossword puzzle to work on.

A. Oh, c'mon, not a crossword puzzle; that's what your parents like to do

this line cut too close. i never got good at crossword. RIP Conway, a lovely man. fuck this corona shit


toyot posted:

sum(1/(n,k)) = 1 + 1/n, for 0 < k < n

i dont think this is true? try n = 3: then you have 1/(3,0)+1/(3,1)+1/(3,2)+1/(3,3) = 1+1/3+1/3+1 = 8/3 != 1+1/3

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I think that's also incorrect. I get sum_{k to infinity} 1/((n + k) choose k) = n/(n-1) which matches that for n=3 but not for n=4, for example
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you can prove it by noting that sum_{k to infinity} 1/((n + k) choose k) = n! sum_{k to inf} k!/(n+k)! = n! sum_{k to inf} 1/(k)_n where (k)_n is the falling factorial and using the identity in Eq 19 in this link. note that our "n" is their "p" and their "n" is the maximal value of the k that the sum is taken to, so you have to take the limit of their expression as n goes to infinity and then exchange p for n. Also they use the Gamma function notation instead of factorials so you have to remember that Gamma(n) = (n-1)!