Man, school must own for people who like making beautiful notes.
i feel like the only thing that's pedagogically hard is the conjugate (light-blue) surface. imagine like a kid's ball with a flashy annoying LED in the center. if you took a long-exposure camera shot, of it rolling all around a surface, the pic would look like the light-blue surface.

in the 2nd set, maybe the kid's got my weird barbell w/ two diff colored LEDs in each ball. the pic would then have two, parallel planes. all i noticed is that wherever a plane "kinks" ( like- the surface is non-differentiable, it takes a hard turn ), the solid body loses a degree of freedom. both the manifold loses a DOF and the kinematic solid body does.

exact constraint theory is small but mighty. "kinematic coupling" = "exact constraint" = the graph is logarithmic, it is accurate to the diameter of virus capsids:

edit: the classic kinematic coupling is the maxwell 2-2-2, invented by james clerk maxwell, who wrote the equivalent of a physics poem next to it in marginalia: "No two coincide; no three are in one plane, and either meet in a point or in parallel; no four are in one plane, or meet in a point, or are parallel, or, more generally, belong to the same system of generators of an hyperboloid of one sheet. The conditions for five... and for six, are more complicated"

we can use it to manufacture tools to meet the needs of our class. i think part of cadre work will involve developing natural-scientific as well as social-scientific theory, to build the machines that will form the basis of a new communist society.

Edited by toyotathon ()

insta_gramsci can you explain cosets, conjugates and normal subgroups to me?
*undead ken voice* you should talk about... set theory...
i cold-emailed my drawings/explanations to a settler kinematician, who fwd'd it to other settler kinematicians, and they were like, 'very interesting!' and 'probably correct!' and 'neato!' but also, 'i have no time to look at this!'.

do any zoneheads in the academy know how to 1) structure a solo research project, 2) structure a modern monograph? all the monographs i've seen are like from 1880 on archive dot org and they look the part. i don't think i've written anything over 10 pages since high school... or shit, anybody (not FBI) want to study kinematics w/ me? plz PM me? let's advance the material condition, don't rely on imperialism to do it for us.

i have a little more work to do on the fundamentals (identification of the 'instant center' in non-symmetrical solid bodies, i dunno if blanding's method is general), but the rest is hopefully okay imo. i am testing the theory out w/ some fever dream experiments... anyway thx rhizzone cuz yall are the only intellectuals/postdocs/immortal scientists i know

Edited by toyotathon ()

id like to learn about math but i think i'd need to go all the way back to algebra or pre-calc as a starting place. is there a free way to do this?
I've been working through some of the free texts in this reddit post
read settler kinematicians dot org

graphicalUSSRinterface posted:

insta_gramsci can you explain cosets, conjugates and normal subgroups to me?

Sure! Cosets first:

So if G is a group, H is a subgroup of G, and g is any element (member) of G, then the set gH = { gh | h is in H } is a left coset of H in G. (The formation of right cosets is analogous). Intuitively it's the set of all "multiples" of elements of H by the group element g. For different choices of group elements g you may or may not get different cosets--for example, if g = h for some element h in h, then gH = H (the same if g is set to be the identity element of G).

Are you familiar with equivalence relations? They're essentially a generalization of the notion of equality. For an arbitrary set S with elements x,y, and z, an equivalence relation, denoted by ~ (the tilde can read verbatim as "is related to") is some conditional statement such the following conditions hold:

1.) Symmetry: x ~ x for all x in S
2.) Reflexivity: If x ~ y, then y ~ x for all x, y in S
3.) Transitivity: If x ~ y and y ~ z, then x ~ z for all x, y, z in S

We can define a relation ~ in G by "x ~ y if and only if there exists h in H such that x = yh". In other words, two elements x and y are "equal" provided that they lie in the same coset of H. Let's make sure it's an equivalence relation:

We first assume that x is some "h-multiple" of y--i.e. there exists h in H such that x = yh.

1.) Since any subgroup contains the identity 1, setting h = 1 gives x = x(1) = x, or x ~ x;

2.) Since x = yh, and subgroups are closed under taking inverses, it follows that y = xh^{-1}, so that y ~ x ;

3.) This one is especially boring to type but it follows from 1.) and 2.).

So our relation ~ is an equivalence relation of G--we're saying that we can think of two group elements x and y as "equal", or as belonging to the same coset, provided that one is an h-multiple of the other.

So the cosets of H in G actually serve as a partition of G, in the same way that you can draw a circle and split it by drawing lines through it. I'll connect this to conjugation next and then to normal subgroups after.

Are you familiar with the idea of an abelian group? An abelian group is just one in which the group operation is commutative--A group G is said to be abelian if for all x and y in G, xy = yx.

Now, within the group G, two elements x and y are said to be conjugate to each other if there exists g in G such that gxg^{-1} = y.

Now, for an abelian group, xy = yx holds for any two elements x and y in G, which (by inversion) is equivalent to xyx^{-1} = y, so in an abelian group any two elements are conjugate.

So conjugating elements, as an "operation", is a generalization of commutativity. Abelian groups are very friendly to work with since you can take multiplication for granted--distinctions between left- and right- whatever are unnecessary. In other words multiplying elements doesn't really give you a lot of useful information about the group's structure. But for non-Abelian groups, conjugation is a useful tool to further study the group and it's the "closest we can get" to commutativity in a practical (or at least computational) sense.

That's conjugation as an "operation"--let's connect it to cosets and then to normal subgroups:

A coset is the set of all multiples of a subgroup H in G by a fixed group element, say g. So for some arbitrary h in H and g in G, the group element gh is a member of the left coset gH. Analogously for right cosets, hg would be a member of Hg.

Suppose that G is an abelian group. Then gh = hg, and since this holds for any h in H, it follows that gH = Hg. So in an abelian group the left and right cosets coincide, i.e. they give no new information about the group.

For a non-abelian group, however, it's not necessarily true that gH = Hg. As sets of objects gH and Hg could be completely distinct. But when they do coincide for every choice of g in G, we recover a certain amount of "abelian-ness" within a non-abelian group. Such subgroups, whose left and right cosets coincide, are said to be normal subgroups of G--

A subgroup H of G is said to be normal in G if gH = Hg for every g in G.

This definition is equivalent to this one, which is formulated more in the "conjugation as an 'operation'" sense from earlier:

A subgroup H of G is said to be normal in G if for every h in H and for every g in G, ghg^{-1} is in H.

Now, from here, normal subgroups become an extremely important tool in the classification of groups and also in the formation of new groups. But intuitively, or at least in a motivational sense, you can think of them as being an attempt to recover as much "abelian" behavior in a non-abelian group as possible, which might be one of the reasons they are called "normal".

can anybody make a stab at explaining Godel's incompleteness theorem to somebody with super rudimentary knowledge of formal logic. that person is me.
Within any axiomatic system governing something like basic arithmetic there will always be unanswerable questions

And any axiomatic system that purports to be able to demonstrate its own validity is actually
I understand that much, which is why I'm interested in understanding it on a more granular level because I don't want to pull a Badiou and start applying it metaphorically in all kinds of dumb situations. If you have recommendations for a easy to understand intro to logic (for someone without an analytic/math background) I'd be grateful
The book 'Godel's Proof' by Nagel and Newman is a self-contained explanation of the incompleteness theorem and it teaches you everything you need to know.
someone please help me work this one out

the bourgeoisie loves graphs like that, asks their scientists how to make the ball fall faster
what might make it clearer is if you draw a satanic Moneybags behind the most distant ball, lining his club up for a putt

looks like he'll sink it in one, but if it rolls wide he can still probably eagle it
if im reading this graph right, "stabilized earth" is markedly less "stable" than "hellhouse earth"
what could be more stable than the peaceful sterility of a lifeless rock?

ialdabaoth posted:

if im reading this graph right, "stabilized earth" is markedly less "stable" than "hellhouse earth"

What that means is that a huge amount of energy would be required to move the earth from the hothouse earth state. "Stability" doesn't mean stability of modern society or anything like that, it means that more energy is required to move Earth out of the Hothouse state than it takes to move Earth now, in the less stable state.

So, let's say we're putting in X trillion BTU equivalents today to warming the Earth. It's relatively more easy to change its state today than in Hothouse Earth. Assuming that society sticks around, we might need 3*X (for example) trillion BTU equivalents to get Earth back to where it is today, once we've reached Hothouse Earth.


ialdabaoth posted:

if im reading this graph right, "stabilized earth" is markedly less "stable" than "hellhouse earth"

theres literally an axis labeled "stability"

been reading From Natural Numbers to Quaternions by jurg kramer. i've had good libgen luck picking translated books. if it's good enough to translate, it must be distinguishable in other ways yeah?

starts with the Peano Axioms to derive the set of natural numbers, and the rules of arithmetic... lots of proofs... i never did proofs before... a 3D CAD program i'm working on, i used quaternions w/o knowing anything except that they didn't tweak out when the euler transformations did. and i remember a bug where the order of multiplication for quats affected the answer. i now know this is a big quaternion thing. at this rate i will understand them by 2019.

it goes rational numbers -> imaginary -> quaternions -> octonions and some weirdos think the octonions underlie the standard model, like how quaternions can express both GR and QM?

edit-- never saw a proof for the fundamental theorem of arithmetic b4 and i'm gonna try to 'translate' it. it's basically that, you can prove that a natural number (1,2,3,4...) has a single prime factor. a = prime * b. then you induce, since b's a natural number, b also's got a prime factor, b = prime * c. so, a = prime * prime * c, and you keep going until you hit primes all the way down. they have a proof that the set of prime factors for any number is unique, which they prove by assuming it's NOT unique, then watching the logic lead to a contradiction. i don't get this second proof very well tho.

Edited by toyotathon ()

i'm trying to determine why degrees of freedom in kinematics are in the set of natural numbers, and the answer at first blush is, because spacial dimensions are in the set of natural numbers. but then why is that so
what is a black hole toyotathon?

Edited by toyotathon ()

looks like time can escape a forum after all,
through the tiny hole in my post

toyotathon posted:

i'm trying to determine why degrees of freedom in kinematics are in the set of natural numbers, and the answer at first blush is, because spacial dimensions are in the set of natural numbers. but then why is that so

this is true but to ask "why?" can be misleading. a more practical question might be "what would it mean for a 'space' to have a dimension not in the natural numbers?" there are a few different ways of approaching this, fractals can be said to have non-integer dimensions which has to do with their self-similarity and scaling under transformations (effectively a fractal embedded in some number of dimensions can have fewer "effective" dimensions arising from its self-similarity). there are also spaces with an continuously infinite number of dimensions, partial differential equations have solutions in spaces like these but each of these dimensions corresponds to a specific point in physical space so it might not be what you have in mind. generally, in mathematics, you can always generalize the concepts you're working with in any number of ways, the trick is to find generalizations that are productive and useful.

cool+thanks, i will look into those things... it's good to know, in certain cases the spacial dimensions are differentiable? where i was coming from is, well the perspective is, in kinematics, there is a physical point of contact, theoretically at a perfect point, a ball rolling down a simple surface touches the surface exactly once. 1 point of contact to create 5 DOF, 2 for 4, etc. if you hooked up an ohm-meter to the two solid bodies, contact is unambiguous, it's where resistance drops to zero (you know what i mean). but these are single points of contact, which you can count w/ natural numbers. what does it mean to smear contact over a surface, like in a pressure field?

the application i had in mind is a spindle bearing, which could receive force in an arbitrary direction (let's call it 3 o'clock), and instead of how spindle bearings today return it equal-and-opposite (thanks tears + newton) at 9 o'clock, it could return two equal forces from 11 and 7, per exact constraint. the other 3 assume are constrained, 2 in a second spindle bearing, and the last in a thrust bearing, leaving 6-5=1 rotational DOF. spindles today chase manufacturing perfection, ya know, perfect ball bearings pre-loaded on perfect spindles in perfect housings, and they don't wear in, they wear out. they have to do this, because they in truth only have 4 constraints, and try to reduce the last, unwanted DOF down as far as they can. you could probably achieve this with a really elaborate hydrostatic manifold, and get theoretically infinite stiffness (true-zero displacement) in any wear state. whereas a spindle today, ya know, maybe you can make it x diameter, and its housing x + 1 micron, the moment it leaves the factory (and you leave less $xx,xxx). and it's still 1 micron greater inaccuracy than could be achieved with an exact constraint spindle.

but this is no longer a kinematic solid body acting on another, it's a hydrostatic field of fluid pressure. so what's that mean for the DOF analysis. before you could unambiguously count contact points, 1, 2, 3..., but now one whole body is wet! the pressure fields i know are continuous/differentiable, but constraints (via blanding) are not, they jump w/ the natural numbers, mirroring our spacial dimensions.

if there are two return pressure maxima, are there two constraints? where does one constraint become two. for a pressure field, is the kinematic equivalent the scalar field maxima (which will follow the natural numbers), or do we need to invoke something like, 1.5216 DOFs? lucky me, that ppl have been thinking about fields for 200 years, so the math's out there, if that is the math that's applicable. sorry if this is rambling, i'm not a scientist or math person if it's not obvious lol, just a communist workin on The Means

Edited by toyotathon ()

You cant imagine how intimidating this thread is to me

Caesura109 posted:

You cant imagine how intimidating this thread is to me

seems like you are in a great life situation to learn mathematics with some of the world's best math lecturers and make that part of your education!!!! i wish i appreciated math class when i had it instead of cheating off my roommate in linear algebra

Math is great and i dont think that much more effort to learn than a foreign language or a musical instrument. You can definitely learn to do math very well more or less just by practicing a lot. Being at a university is probably the best environment to do the practice you need, since the classes have a pretty organized structure and there are usually many people available to help with conceptual difficulties. My personal experience is not so much that math is "more difficult" than the humanities so much as it requires a bit more personal motivation to get into, since the humanities are mostly concerned with interpreting things like texts in plain language or historical accounts, things which require less specialized knowledge to understand the possibility of significance.

The short version of that i guess is that i think math is probably harder to self study so getting into it in a structured environment like a university course is great. i was incredibly average at math until college where i had to sit and do it over and over again.

Edited by c_man ()

kitchen math

experimenting adding gluten to my flour for bread. turned 10% gluten AP flour into 15%. so far, the bread texture is a lot better, going to keep upping the gluten 1% each batch and see what happens.

i hope this 'gluten-free' fad keeps going cuz, right now there's a gluten glut. my pet theory for the mass gluten allergy is, since white people have been eating bread for millenia, to just now diagnose a mass allergy, that's probably not real. BUT-- maybe the chemical processing step that removes gluten (dissolving the starch in water, removing sediment incl gluten, then drying the starch) might be removing the actual causative agent, like a new pesticide or something. someone was tellin me that their friend with celiacs was fine eating european-wheat breads, maybe cuz of different pesticide regimes in europe and US.

the formula to figure out how much of something to add is:

x = b*(c - a) / (1 - c)

x = the amount to add (weight)
a = the original %/100 (from 0 to 1)
b = total original weight
c = target %/100 (from 0 to 1)

if you want to stick with percents you can turn that '1' into '100'.

so if you have 51oz of 10% gluten AP flour, and want to turn it into 15% gluten bread flour,

x = 51oz * (0.15 - 0.10) / (1 - 0.15) = 3oz

if you don't believe the formula: 10% of 51oz is 5.1oz gluten, 5.1+3=8.1, 8.1/(51+3)=15%

started with semigroups/monoids/groups in this number theory book that's supposed to get to the integers eventually. i am very interested in group theory if anybody wants to PM chat cuz of the euclidean group and its relation to kinematics/exact constraint, and why point-symmetry in solid bodies makes 3-DOF, line-symmetry makes 5-DOF, and plane-symmetries and higher give 6-DOF (?). or honestly geometry. vvvvvv shit TG that gives me an idea.

Edited by toyotathon ()


Caesura109 posted:

You cant imagine how intimidating this thread is to me

i volunteer at the library tutoring a woman for her ged. we recently got to the math part and when we made it to algebra - basic, 2x + 3 = 7 type stuff - and i was fumbling through it like an idiot. i never learned calculus so i will be even more useless going forward. so yes, i can imagine

trying to learn maths but having some problems when u do a "subtraction" where does the bit u take away go to? example: 13 - 3, where does the 3 go to?
the 3 and the -3 collide and obliterate each other
ok that makes sense; so with 10 - 3 its three minuses hitting 3/10ths of ten positives? like wave neutralisation?
like this?

+ + + + + + + + + +
- - -
+ + + + + + + | (+-) (+-) (+-)
If a socialist should understand anything it is the taking from one group to give to another

taxation is theft vote Ron Poal
ok so i get it now, +1 + -1 = 0 and some gamma radiation, or is it photons, i cant work this maths stuff out