E: by convention the square root symbol means the positive root which I always forget so I think your identity is true for non-negative x

Edited by Belphegor ()

This is a cool math YouTube channel

dimashq posted:https://www.youtube.com/channel/UCYO_jab_esuFRV4b17AJtAwThis is a cool math YouTube channel

The best!

graphicalUSSRinterface posted:i'm reading pinter's abstract algebra book right now

i'm picking it up quickly, also using another book called 'visual group theory' alongside it. it's kewl. my problem with math/programming though is that i love reading dry technical stuff too much and i'm skewed towards that rather than practice which i need to work at i guess

Lenin posted:Such is the first cause of “physical” idealism. The reactionary attempts are engendered by the very progress of science. The great successes achieved by natural science, the approach to elements of matter so homogeneous and simple that their laws of motion can be treated mathematically, encouraged the mathematicians to overlook matter. “Matter disappears,” only equations remain. In the new stage of development and apparently in a new manner, we get the old Kantian idea: reason prescribes laws to nature. Hermann Cohen, who, as we have seen, rejoices over the idealist spirit of the new physics, goes so far as to advocate the introduction of higher mathematics in the schools—in order to imbue high-school students with the spirit of idealism, which is being extinguished in our materialistic age (F. A. Lange, Geschichte des Materialismus, 5. Auflage, 1896, Bd. II, S. xlix). This, of course, is the ridiculous dream of a reactionary and, in fact, there is and can be nothing here but a temporary infatuation with idealism on the part of a small number of specialists. But what is highly characteristic is the way the drowning man clutches at a straw, the subtle means whereby representatives of the educated bourgeoisie artificially attempt to preserve, or to find a place for, the fideism which is engendered among the masses of the people by their ignorance and their downtrodden condition, and by the wild absurdities of capitalist contradictions.

https://www.marxists.org/archive/lenin/works/1908/mec/five8.htm

from https://link.springer.com/article/10.1007/BF01946586

the feedback on problem sets, that labor, i feel is the real value of a $$$ education. the low-cost alternative model to the school, which comes w/ that feedback, isn't youtube lectures, it is student communal research. but i can't really find that. people want that sheepskin.

i feel strongly like this stuff, math and physics and engineering, is not window dressing to the struggle, and can be creatively applied (not just WUO wiring up bombs, or your local infoshop's mesh network). and that additionally these technical skills are comprehensible by all. like how the world became literate the past two centuries. our brains aren't lacking, the pedagogy's lacking.

gUSSRi, any insights into abstract algebra? what's going on there

toyotathon posted:but i don't have confidence in it, like i'd get w/ teacher feedback, from watching khan youtubes...

the feedback on problem sets, that labor, i feel is the real value of a $$$ education. the low-cost alternative model to the school, which comes w/ that feedback, isn't youtube lectures, it is student communal research. but i can't really find that. people want that sheepskin.

there are ways around this though; stackexchange is an amazing resource where everything you would have thought to ask has been asked pretty much, i got clarification on an obscure point there just yesterday where someone was confused exactly where i was. i don't think learning math has to be much different from learning anything else, i think thinking it is is ideology probably. you learn to reason about math and programming the way you become fluent in marxism, it's something you can have intuition and opinions about too, it's a language more than anything. also if you really need to i've found emailing professors gets you places, they like to chat and probably will email exhcnage with you happily

toyotathon posted:gUSSRi, any insights into abstract algebra? what's going on there

abstract algebra is all about symmetry and transformations/rearrangments of algebraic structures, i'm still focusing on groups. but a group is basically a set along with an operation that is associative (though not necessarily commutative), satsfies closure axioms, has an identity element and is invertible. it's a very vague concept that really only makes sense after looking at a bunch of examples and sort of solidifies itself in specific concrete applications. one example is modular arithmetic, like on a clock. one of the cool things i learned is about cycles of permutations and that every group is isometric to a group of permutations, which i promise is cool but hard for me to explain in a nontechnical way

edit: if i had to summarize abstract alegbra id say its about the way you can rearrange things while keeping them fundamentally the same.

Edited by graphicalUSSRinterface ()

lenochodek posted:

lenochodek posted:

this thread seems like your opportunity to make an argument instead of deploying "lenin" and "soviet" as buzzwords

graphicalUSSRinterface posted:every group is isometric to a group of permutations, which i promise is cool but hard for me to explain in a nontechnical way

Cayley's Theorem is beautiful, its proof too

It's not a practical result, but it's one of those results that's almost philosophical in what it says about the nature of groups

sometimes i like to kid myself that i got a nose for prime #s... this thread had 337 views as of a second ago, and guess who was fuckin correct, yeah that's right

amazing feats possible when your odds are better than 1 in 3, guessing the prime status of odd numbers below 1000

toyotathon posted:^^^ what's cayley's theorem? i never did group theory. i remember playing with cayley trees in undergrad but i assume that's unrelated

Let G be a group. Then G is isomorphic to a subgroup of a symmetric group. If G is finite, then G is isomorphic to a subgroup of S_{n}.

an isomorphism is a structure-preserving map between two groups; two groups that are isomorphic are actually the same underlying group, the elements are just labeled differently. for example, as groups, the set of permutations of the digits 1, 2, and 3 is the same as the set of rotations and reflections of a triangle. they're ontologically different but their algebraic structures are exactly the same.

Cayley's Theorem says that any group is either a subgroup of the infinite symmetric group or a subgroup of the symmetric group on n symbols, S_{n}.

but in a more philosophical or thematic sense what Cayley's Theorem says is that all groups are just abstract manifestations of symmetry

Edited by insta_gramsci ()

elemennop posted:yo cat admin, enable mathjax for the forums

cars posted:lenochodek posted:lenochodek posted:

this thread seems like your opportunity to make an argument instead of deploying "lenin" and "soviet" as buzzwords

I'm not sure if I have a point to make but here is my position in any case. (Also I'm not a mathematician, I'm active in the field of chemistry)

I think Fock truly stated his approach on how to handle interpretation of quantum mechanics in a very lucid and to the point way but I would put it like this:

in physics we could state the evolution of theories as follows

1. there is a contradiction between a prediction from the theory we use and the measurement of that which we predicted

2. a new speculative theory which theoretically enhances the old theory is proposed, and is empirically tested

3. in case the new speculative theory gives better predictions we adopt it until further enhancement is achieved

Described like that, it seems clear this is fully consistent with the dialectical materialist approach. However, when we look at the approach of some self-described Marxist-Leninists to the "New Physics" we see some issues arise. In my opinion most of those result from a confusion of on one hand incorrect idealist model interpretations with on the other hand models with weak predictive power. The latter is lethal for a theory but the former is an error of the interpreter and not of the theory/tool/model.

To give a concrete example, check out this snippet essay called "Marxism and Quantum Mechanics" from 1975:

The author objects to the Heisenberg Uncertainty Principle, which is a fundamental principle that indeed makes better and broader predictions than what preceded it. The objection seems to be not with the predictions, but with the fact that certain interpretations of this fundamental equation are idealist and/or incorrect, and indeed for didactic reasons sometimes an analogy with the observer effect is used, which is an interpretation which has no verifiable grounds in reality (and is actually imprecise and incorrect). In the end, because the new theory does not line up with the classical physical deterministic theory, the new theory is rejected in its useful form. The confusion that occurred here is that we are testing our new theories to the interpretation of the old theory instead of to material reality. Of course, this is an error, tautological, dogmatic, not empirical and at odds with dialectical materialism.

Some other examples of where I feel that similar errors are made is in case of the slow correction of Lysenkoism and in the soviet rejection of resonance structures because they are "Machist" ( see here (NB:article written from anti communist standpoint)), here too they reject an actual material enhancement (e.g. correct understanding of the abstraction that is resonance structures allows for more efficient planning of successful chemical synthesis of fine chemicals including medicines) on the grounds that some interpretations of the model are idealist, which is true, but does not affect the better result.

This ties into Lenin’s remarks on physical idealism and in my opinion his criticism is to an extent faulty. For example, the great mathematician Poincare worked on a broad range of topics in pure mathematics which includes original and fundamental studies in dynamical systems which at this stage in time have plenty of real-world applications. Essentially, his philosophy of science was at odds with Lenin's approach and yet were empirically more powerful e.g. his "conventionalist" stand that Euclidian geometry is not a priori true was essentially confirmed by later approaches to relativity and space-time. To call Poincare an empiriocritic does him injustice because because of some (imo) minor idealist aspects to his conventionalist philosophizing is of no importance compared to Poincare's actual body of mathematical work, which has many real life applications by now.

Lenin states:
*“A law of physics, properly speaking, is neither true nor false, but approximate"—writes Duhem (p. 274). And this “but” contains the beginning of the falsity, the beginning of the obliteration of the boundary between a scientific theory that approximately reflects the object, i.e., approaches objective truth, and an arbitrary, fantastic, or purely conventional theory, such as, for example, a religious theory or the theory of the game of chess.
*With some goodwill, Duhem's statement is actually correct - a law of physics is a tool and this tool has its limitations and is thus approximate. Enhancement of the tool results in enhancement of the result. When we use this tool, perhaps we will interpret it somehow and maybe make some idealist assumptions (atoms are localized particles, molecular orbitals exist, such stuff) which are abstractions with no ground in measurable reality but which are still, although not real, useful to deploy the tool successfully. A religious theory of course has no predictive value which is better than alternative theories, so I don't think this is a valid angle of criticism to Duhem. But what about "the theory of the game of chess" or more broadly, abstractions which are mathematically consistent but not at all connected with the matter of physical reality?

I would call those theories "internally true abstractions" and in fact the theoretical toolkit which we can use when we want to build the tools that learn us how to engage with reality properly. This set of internally true abstractions essentially contains all mathematics without applications. Concretely, we can look at the famous example of "complex numbers" - for many years since their formal introduction in the 16th century they were viewed as senseless folly, fictions for bored math people until the work of Euler and Gauss and by now they have many concrete applications via quantum mechanics, electromagnetism etc. When we present theoretical work as reality, we become idealists. But when we apply these abstractions for concrete enhancements of theories for grasping material reality, they are objectively useful. With this empirical knowledge in hand, we know that developing and working on "internally true abstractions" on the long run provides us with ingredients to develop our concrete tools and material theories and this evidence should convince us to entertain these abstractions and not reject them a priori on philosophical grounds.

Of course, Lenin himself does not argue for that explicitly and he in fact welcomed the fusion of mathematics and the new physics of his time. That said, I think based on my statements Lenin would probably classify me as a "physical" idealist.

exact constraint is a graphical kinematic method that reasons, there are 6 degrees of freedom in this universe, and so to make a body go in N degrees of freedom, it must be constrained in 6 - N.

"When an object touches a 2D surface, the object begins to orient itself in 3D space. Let me use this example: Imagine four balls on the edge of a cliff. Say a first ball starts at the top of a cliff and rolls down it. After a time delta-t, the formerly first ball becomes the second, a second rolls down, and then a third, and then a fourth rolls down the cliff."

Edited by toyotathon ()

toyotathon posted:not really math strictly but i'm trying to learn kinematics, and drew this to convince myself of an explanation for Exact Constraint

exact constraint is a graphical kinematic method that reasons, there are 6 degrees of freedom in this universe, and so to make a body go in N degrees of freedom, it must be constrained in 6 - N.

"When an object touches a surface, the object begins to orient itself. Let me use this example: Imagine four balls on the edge of a cliff. Say a first ball starts at the top of a cliff and rolls down it. After a time delta-t, the formerly first ball becomes the second, a second rolls down, and then a third, and then a fourth rolls down the cliff."

bozhe moi

*all*directions, the 1D line drops a dimension down to a 0D point, and the ball is stationary.

edit: another way to say it, for a given manifold, take the surface normal of r length, where r is equal to the radius of a ball. this makes a second manifold, say M2. where the surface of M2 is non-differentiable in 1 dimension, the ball is constrained to a line. M2 loops back on itself, intersects itself, and this makes a line on the top point of the loop. the ball travels along this line of self-intersection. where it's non-differentiable in 2 dimensions on M2, the line collapses to a point: ball is stuck, like at t=3. M2 also loops back on itself here in another place. where the line of self-intersection pierces this new piece of the manifold looping back on itself, is the point where the ball rests.

Edited by toyotathon ()

Susskind isn't perfect, but I do genuinely feel like I get the math of what's going on now, and a visceral understanding of the fact that it is just

*a mathematical description of observed effects*. Like what lenochodek mentioned about the observer effect being a groundless interpretation -- almost everything commonly "known" about QM is complete masturbation, just handwaving about some mathematical relationships that sounds cool. I thought i understood that before, but i dropped out of college before i bothered learning the specific details and it is kind of a shock to learn that these toothpicks of math are what is holding up the entire pop sci edifice of quantum bullshit.

not that i am an expert in QM now, of course, but i am a pretty practiced bullshit recognizer

lenochodek posted:

ty.

toyotathon posted:another way to say it, for a given manifold, take the surface normal of r length, where r is equal to the radius of a ball. this makes a second manifold, say M2. where the surface of M2 is non-differentiable in 1 dimension, the ball is constrained to a line. M2 loops back on itself, intersects itself, and this makes a line on the top point of the loop. the ball travels along this line of self-intersection. where it's non-differentiable in 2 dimensions on M2, the line collapses to a point: ball is stuck, like at t=3. M2 also loops back on itself here in another place. where the line of self-intersection pierces this new piece of the manifold looping back on itself, is the point where the ball rests.

a picture's worth 1000:

sorry about the graphical errors i don't have liquid paper. i'm kind of excited tho cuz this is a generalization of the science of exact constraint that preserves its major observations: the constraint as complement of degrees of freedom, the instant center of rotation (which naturally comes out of this manifold-based analysis). the intersection of a 2D manifold with another 2D manifold is a 1D line, so whenever this happens, the object is constrained along a 1D line. and the intersection of a 1D line with a 2D manifold is a 0D point, so whenever this happens, the object is rendered motionless, trapped at the point. this corresponds to blanding's DOF analysis method, but could potentially generalize it (haven't thought about the 3 rotational constraints, just ball bearings rollin down surfaces).

what math do i need to study, that'll tell me what happens when a manifold self-intersects?

Edited by toyotathon ()

used the same method as before, making conjugate manifolds, but for the two instant centers of this weird barbell (orange).

first example is a 3-constraint system. the large ball (orange) is nested in 2 constraints (light blue), the floor and the wall (dark blue), w/ nesting force in +Y/-Z (unpictured). the small ball (orange), constrained by the same (dark blue) surface makes a shorter, ~parallel conjugate plane (light blue). same observations as before: the wall+floor constraint (the red line) is formed along the non-differentiable line created by the self-intersection of a conjugate manifold. dimensionality drops from a 2D plane to a 1D line, loss of 1 DOF here. remaining DOFs are translation along the +X/-X constraint-path (in red), and 2 Z-theta rotations, about either instant center. at first i thought this was pretty boring but i realized something, the fact that there are "rotational" DOFs at all in this system is weird. cuz where are the circles? supposedly all the kinematic information, all possible locations, are in the two conjugate surfaces, plus the line-of-torque separating instant centers. interestingly tho: given a line of length L (the line-of-torque, the distance between instant centers), and two parallel planes separated < L (like in this system), if you fix one end of L in one plane, the solution for where it intersects the other plane

**is a perfect circle**. so that's where the rotational paths come out of. it makes a cone. one thing Blanding did in exact constraint was to collapse all DOFs into rotational constraints. translational DOFs had an infinite line-of-torque. like if you zoom really far into the edge of a circle, it starts to look like a line.

(NB this "local linearity" in calculus is the thing marx was looking at above, the transition where algebra becomes calculus, when you zoom in really really far into the edge of a curve. to where the limit definition of the derivative matches its algebra. dx (the distance between the edges of the screen you're using to zoom in on the curve) becomes zero)

blanding's trick was to say well maybe translation is really a rotation, if the line of torque is on the other end of the universe. maybe this is a transitive way to do the same, collapsing rotational paths into translations. rotational paths as the set of intersections, in the surface-surface-line system.

okay in the second set, i added a channel for the small ball, to constrain it along a manifold intersection (path is in red). it is a standard circle-vee. if the line-of-torque is long enough, there are two potential constraints for the large ball (2nd row, right: the two red dots). same observation as before, a 1D line intersecting a 1D line makes a 0D point, which coincides with a loss of 1 DOF. the constraint is located where the vee channel's red line, and the floor-wall's red line, are distance L apart. there are two solutions, so either creates the 4th constraint. if you look at the little vertical loop, in red, between the two points of constraint, i'm curious if this is a conic section -- hyperbolic? the only way to travel the little curve between constraints is to change the nesting force vector to +Y/+Z. this fits the observation about the nesting force vector's permissible orientation: that it is also a function of the conjugate surfaces.

in the last set, the small ball is translationally exactly constrained in 3 places, in its little cup, and the large ball, constrained in one place, is free to form a cone. this is a better corresponding illustration imo to the observation about the surface-surface-line system. it also sort of suggests another way to define exact constraint. constraints are locations where 1 spacial DOF is lost, either a surface intersects itself to make a line, a line intersects a surface to make a point, a line intersects another line at a point, etc. every intersection hard-kinks the kinematic paths, makes them non-differentiable. so an object is exactly constrained, when all instant centers have had all their kinematic paths, from these intersections, reduced down to a point. it makes intuitive sense, b/c like of course exact constraint means something is stuck at a point in space. but it also explains why a system like this has maximum 5 constraints, and the ball bearing 3.

i think the 3-ball system would be pretty pedagogically boring, so i'm going to skip it, unless a new context arises. i picture it sliding down a track, bumping into walls, and landing in the classic maxwell 2-2-2 coupling. the 4-ball system could be interesting, and could demonstrate why, thru the creation of a 4th conjugate plane, overconstraint results in play or binding, as the system switches between planes/lines/points, depending on the nesting force vector.

Edited by toyotathon ()

toyotathon posted:i think the 3-ball system would be pretty pedagogically boring, so i'm going to skip it, unless a new context arises. i picture it sliding down a track, bumping into walls, and landing in the classic maxwell 2-2-2 coupling. the 4-ball system could be interesting