/sci/ - Science & Math

E=mc2

10/10

euler lagrange is also crucial for quantum field theory and all other sorts of theories that require delta action=0.. i'd give it a 9.5/10 meaning and a 8/10 good looking

>>8402093
>E=mc2
>not E=mc^2

>>8402109
he took the derivative with respect to c

topological index = analytical index

good looking: 2/10
meaning: 11/10

>>8402110
but then the LHS should be dE/dc and it's not really useful; i know you wanted to make a joke, idc.
>>8402085
can't really decide on my favorite equation. i really like the einsteinian field equations or the field-strength tensor formulation of the maxwell equations

>>8402120
good looking 6.5/10
meaning 10/10

>>8402112
Please explain

>>8402120

that's such an ugly fucking equation

0 /10 aesthetics

[math] \bar{\mathbb{R}[x]} \cong \mathbb{c} [/math]

r8 pls

>>8402143
[math] \overline{\mathbb{R}[x]} \cong \mathbb{C} [/math]

damn \bar

The best one of course!
1 = 0.999...

>>8402132
Atiyah-Singer index theorem, the highlight of spin geometry. It's a statement about elliptic differential operators on manifolds, yielding well known theorems such as Riemann-Roch or Chern-Gauss-Bonnet as special cases.

>>8402085
Cahn–Hilliard equation
[math]\frac{\partial c}{\partial t} = D\nabla^2\left(c^3-c-\gamma\nabla^2 c\right)[/math]
>Aesthetics
7/10
>Meaning
6/10

Diffusion equation based on the chemical potential of a mixture. The transient phases of any mixture system can be described with it. It will eventually have more important engineering significance when phase equilibria models become more accurate.

I'm not all smart like you guys, but I really appreciate polar conversion equations since rectangular coordinates are so pre-multicore cpu.

>>8402150
good looking 9.999.../10
meaning -1/12

>>8402162
seems like you continue into the imaginary

The Yoneda lemma. It's at the heart of modern geometry, homotopy, and things like Tannaka duality.

I guess some MHD equations are cool too, but this one is just so useful

>>8402120
That one looks like ass

>>8402085
pi=2e+0

Ties together three of the most important constants in mathematics

>>8402085
[math]f'=f[/math]
>>8402346
it relates four constants. You forgot about the pi/tau conversion constant.

>>8402192
why does everyone fawn over this?
it's intuitive and trivial once you understand natural transformations and hom functors

[math]H |\Psi \left>=E |\Psi\left>[/math]

>>8402369
Of course it is, but it leads immediately to nontrivial results and if it was not formalized it would be unwieldy.

Aesthetics: 5/10
Usefulness: 100/10

>>8402770
What is that?

>>8402774
the inequality that is going to make mochizuki solve the riemann hypothesis

Plebs

[math][\frac{\partial }{{\partial t}}{g_{ij}} = - 2{R_{ij}}[/math]

[math]\int_{\partial\omega}\omega = \int \mathbf{d}\omega [/math]

>>8402852
Meaning 7/10
Good looking 10/10

Let [math] M [/math] be a compact two-dimensional manifold without boundary. Then:

[math] \displaystyle \int_M K \ dA = 2 \pi \chi(M) [/math]

Where [math] K [/math] is the Gaussian curvature of [math] M [/math] and [math] \chi(M) [/math] represents its Euler characteristic. This is beautiful because it says that the total curvature of a surface is topologically invariant despite the fact that [math] K [/math] varies locally according to local deformations of the surface.

Y = C + I + G + NX

Good looking: 10/10
Meaning: 7/10

I've never met a research-level mathematician or physicist who has a "favourite equation", but engineers all seem to have one. Is this what a low IQ does to you? You start subjectively ranking equations? How unfortunate.

>>8403008

>>8403008
You're right; proofs > equations.

Favorite proofs? I vote for the proof that there is no differentiable space-filling curve.

[math]\delta \hat{S} = 0[/math]

>>8403374
I like the Spanish hotel theorem

[eqn]
\begin{cases}
\dfrac{\partial \mathbf{u}}{\partial t} + (\mathbf{u} \cdot \nabla ) \mathbf{u} = \dfrac{1}{\rho} \nabla p - \nu \Delta \mathbf{u} -\left(\frac{\xi}{\rho} -\frac{1}{3} \nu \right)\nabla (\nabla \cdot \mathbf{u}) + \mathbf{f} \\
\dfrac{\partial \rho}{\partial t} + \nabla \cdot (\rho \mathbf{u}) = 0
\end{cases}
[/eqn]

7/10 good looking
10/10 describes the general motion of a fluid, and is still not solved.
It's my favorite because my professor says so.

The incompressible case is much more A E S T H E T I C.

[eqn]
\begin{cases}
\dfrac{\partial \mathbf{u}}{\partial t} + (\mathbf{u} \cdot \nabla ) \mathbf{u} - \dfrac{1}{\rho} \nabla p + \nu \Delta \mathbf{u} = \mathbf{f} \\
\nabla \cdot \mathbf{u} = 0
\end{cases}
[/eqn]

>>8402946
Just cancel the stroked u's retard, H=E

Looks depend too much on notation and choice of symbols capturing other bulks of expressions

>>8403433
is that a pic of u

be my gf pls

Maxwell equations, especially Gauss's law
Looks:9/10
Meaning:10/10
Also using it for gravity, dayum

I know this thread is all just entry-level LaTeX wanking but I really like the Maxwell equations:
[eqn]\vec{\nabla}\cdot\vec{E}=\frac{\rho}{\varepsilon_{0}}[/eqn]
[eqn]\vec{\nabla}\cdot\vec{B}=0[/eqn]
[eqn]\vec{\nabla}\times\vec{E}=-\frac{\partial\vec{B}}{\partial t}[/eqn]
[eqn]\vec{\nabla}\times\vec{B}=\mu_{0}\vec{j}+\mu_{0}\epsilon_{0}\frac{\partial\vec{E}}{\partial t}[/eqn]

>>8403585
Are you me? I was gonna post the TeX but for some reason, my browser fucks up block equations.

>>8403433
>Hat

>>8403433
:^)

>>8403565
Thx obama

>>8403565
>stroked u's
Subtle.

>>8403008
>I've never met a research-level mathematician or physicist who has a "favourite equation"
Without a base rate for meeting mathematicians this statement is empty. Someone who wasn't merely shitposting would know this.

>>8402794
This

>>8403005
Kek

>>8402085

>>8403565
more like

hehehe

>>8402369
It has firepower. It has a lot of firepower. Yet, it is kinda like the abstraction of [math]Hom( \mathbb{Z} , G) \cong G[/math] for abelian groups. [math](^{*} \smile ^{*})[/math]

>>8402448
Oi m8, any progress?

>>8402085
[eqn]\Box A^{\alpha }=\mu _{0}J^{\alpha }[/eqn]
ayyyyyy

>>8403880
I've been doing lots of good stuff! Working on hybrid dynamical systems with some collaborators right now, and we are finding categorical ways for characterizing properties of attractors and reduction conditions.

On my own time, I have been looking into surreal analysis, total categories (and how they behave like manifolds, where their presheaf categories are like their tangent bundles), and how certain constructions from homotopy theory translate into the directed case (for (∞,1)-categories). Also, I'm trying to find ways to treat certain diagrams in a category as generalized objects, just like how we sometimes treat subsets of a set as generalized elements (with fixed "stages").

How are you? What are you working on?

>>8404965
Will we have the notion of an attractomorphism soon? I don't know much anything about attractors, but this is how I'd try constructing such maps in a way analogous to the homomorphisms between covering spaces: take the points around which the stuff spins and does what ever it does (let's call them attracting points for a while), and map each attracting point of an attractor A to an attracting point of an attractor B, and then somehow send the active part of A into the active part of B preserving whatever properties there are for them. These would be composed associatively, identities would satisfy these conditions, and it would also be necessary for A and B to have the same number of attracting points to be equivalent. I don't know if this makes any sense, though.

The stuff you are doing on your own sounds cool, too. You mean you would use a diagrammatic scheme or just the diagram? Since the scheme would give internal structure to these generalized elements!

I've been doing fine. I feel pretty energetic for some uncertain reason, and so I've basically written on paper my master's excluding the study of localization. I have some ideas regarding it I want to pursue the most. If there are these formal inverses, then there are arrows being functored into those in the original category; what if the ring of endomorphisms happened to be a localization of another ring, what could we conclude from this? I haven't had time to think about that stuff yet, though, and next week I'll be mostly unable to do anything related to it, too.

[math](^{-} \smile ^{-} )[/math]

Maxwells equations in relativistic tensor form.

>>8403810
is that a triple fucking integral?

>>8405009
That sounds really cool! It sounds like a case of the macrocosm principle; since abelian categories are sort of like categorified abelian groups, it would be really cool if localizations upstairs corresponded somehow to quotients of hom-groups, and localizations of endomorphism rings!

Regarding the hybrid systems stuff, I shouldn't say too much because I don't know how the professor would feel about me talking about unfinished results, but we are currently using a diagrammatic condition which is equivalent to a specific attractor condition. I am working on a way to talk about the attractor as a colimit over a carefully-constructed diagram of manifolds, and I think the group action property will simplify things greatly.

So, a total category is one who's Yoneda embedding has a left adjoint (which is therefore a localization!), and because of the Yoneda lemma and folklore regarding idempotent monads, it seems that the Yoneda embedding is a very moral categorification of the zero section to the tangent bundle. Furthermore, this modality is actually a major part of higher geometry, called the affine modality. It all fits quite nicely, but I am trying to define a vector field on a total category. I don't think a naïve section to the affine localization is quite right, since the Yoneda embedding itself carries more structure. I think the section should come with a natural transformation from the Yoneda embedding, but I will toy with some ideas more tomorrow. Thoughts?

>>8403810

what is this, a meme for ants?

>>8404965
>>8405009
>>8405072
Does this have anything to do with what you two are talking about?
http://philsci-archive.pitt.edu/1236/1/Axiomath.pdf

I was recently asked by a professor to start investigating this topic and I must say I'm a bit overwhelmed. Currently reading Category Theory for Scientists, but is there something else you'd recommend?

>>8405139
It looks like a lot of similar ideas are going on in that paper, mainly the exploitation of adjoint functors to model universal constructions (this is a very broad idea though, so there may not be much overlap). My collaborator's goal is to model concepts of bisimulation in the end, and this has been studied extensively in terms of coalgebras of comonads (which are very much related to adjoint functors).

I would recommend browsing around on the nLab to find accessible information l, and then you can work your way into the more niché topics that you will be using. It's a very comprehensive resource. Interesting stuff, good luck!

f(x)=Ax^(2)+Bx+C

>>8405072
Sorry, I fell asleep. I'll try to give a more thorough answer when I've digested this better, but let's throw some ideas in the air anyway.

Could you somehow construct a product of the tangent bundle and some space with internal structure corresponding to the data the Yoneda embedding carries, and then show such a construction would be the G(A) of this post >>8402192?

If such a construction was possible, you could define the projections from (let's call it a Yoneda bundle temporarily) a Yoneda bundle onto the manifold as the canonical projection of the product followed by the bundle projection. Then the sections would easily be augmented in a way that they would be not from the manifold into the tangent bundle, but from the manifold into the Yoneda bundle: set them constant in the other coordinate. The morphisms between Yoneda bundles would be products of tangent bundle morphisms and whatever morphisms would be the most suitable for the structure preserver.

>>8405139
nLab, Wikipedia (not even a joke), maybe Mac Lane's book available online, etc.

:^^^^^^^^^^^^^^^^^^^^^^)

>>8405594
Anon, I believe I failed in communicating what I meant: I am looking at the Yoneda lemma as the zero section to a tangent bundle projection, and am trying to transport differential geometric constructions into the language of categories. I think categories have too much data for things like representables to be translated into the language of manifolds, although a more fleshed out theory may allow some things to carry over.

Regarding G(A), that is just a bare set, as G is a presheaf. The Yoneda lemma as I posted there is a natural isomorphism in G and A. I'm going to toy around with the concepts today during my classes and get back to you this evening with results.

a=a

>>8403810
>triple integral
I'm not a phd yet senpai, keep that shit away

>>8405699
Sorry, my bad. I should have waited until I was fully awake, but no, I decided to post while still in bed.

It starts to make sense now. I used the pic of the isomorphims just to point to express my idea with less words. The functor G would have been a functor taking a space X to the underlying set of the Yoneda bundle, but I don't have a clue whether this would be fruitful or not.

If you want to decrease the amount of information, then how about an approach of the following sort?

Take a category [math]\textbf{C}[/math] with its objects satisfying some properties [math]P, Q[/math] and all morphisms combatible with both the properties, and a category [math]\mathbf{C}'[/math] with its objects satisfying just [math]P[/math], and for which [math]\mathbf{C}[/math] is a subcategory. If the properties are suitable, the original category is not a full subcategory, and then you can add the required morphisms to get a full subcategory with necessarily less info. Restricting the Yoneda embedding to this full subcategory would then embed a less informative version of the original category (assuming it is locally small).

I hope you reach some conclusion.

>>8402794
not writing it
[math]e^{i\pi}=-1[/math]

[math] pV=nRT [/math]

>>8405699
>I am looking at the Yoneda lemma as the zero section to a tangent bundle projection

What is what here?
If you write
p:E->B
what's p for you?

>>8402085
[eqn] \frac{ \partial L }{ \partial \mathbf{q} } [/eqn]
disgusting

>>8406554
That was the question I had, and my solution was to ask that B be a total category so that p is the left adjoint to the Yoneda embedding.

>>8405854
Hmm, I like your construction, but it seems like we should only be able to specify if objects have properties by specifying the faithful functor. That is, if C is some category, then a subcategory of objects of C satisfying some property is just given by a faithful functor into C, and then we can use the language of factorizations in Cat to manipulate it from there.

>>8406587
Maybe the fibred category theory in Jacob B's work (categorical logic and type theory) gives some ideas - although it's probably completely different.

Regarding
>>8406367
how do you decide who action costs what? And what's the point of this phase space anyway, other than making the necessary stuff the cheapest. What's the goal.

>>8406608
The goal is to highlight the analogy to physics, where a system is described by a Langrangian and the dynamics are determined by minimizing it. My end goal would be to have some sort of left adjoint functor which would assign a system with Lagrangian its dynamics, and the right adjoint would assign a dynamical system the smallest Lagrangian that describes it. It's just a sort of romantic idea to me, that free constructions would govern physical dynamics. Also, I have this gut feeling that it's all tied up intimately with entropy and that entropy of a system measures the difference between the known Lagrangian and whatever the right adjoint assigns to the system. I'm just musing, please don't murder me if this is all crackpottery.

>>8406594
You mean you would somehow attack the category in [math]\mathbf{Cat}[/math] or the faithful functor? If the functor, then how about taking its factorization category?

>>8402282
aesthetics: -1000/10
usefulness: 4/10

>>8402794
aesthetics: 7/10
usefulness: popsci/10

>>8406682
I meant the functor. Every functor admits an essentially unique three-part factorization into a 0-surjective (surjective) part, a 1-surjective (full) part, and a 2-surjective (faithful) part. It's essentially the Moore-Postnikov decomposition generalized from groupoids to categories. What exactly do we want to do with this functor?

>>8406685
>aesthetics: 7/10
Tsundere score

>>8406545
>/vp/ = NTR

>>8406635
I don't get what you mean by "assigning the dynamics". Naming the Lagrangian are constraints on the (a priori bigger) state space that implicitly specifies the legal configurations. Then initial conditions determine solutions within that space (and you're not gonna find a procedure to solve some super general Lagrangian system explicitly). So given a Lagrangian, what does the left-right construction give you?
And if you move through some space of Lagrangians that descibe the same system, that basically sounds like renormalizaion theory.

>>8406708
Yes, I just wanted to make sure. We would like to forget something. We wouldn't want to forget everything, but more like we taking a ring or a module to its abelian group, though this would not be an embedding without requiring only one object per an underlying object. Those examples would still give a faithful functor, I think. Could a similar trick be used in the general case?

>>8406731
The left adjoint would give you a trajectory in phase space, or maybe you have to assign this to a Lagrangian paired with initial conditions. Whatever needs to be satisfied to facilitate a functorial handling of it all, my hypothesis is simply that this functor is a left adjoint to a faithful functor (doesn't this then imply it is full and essentially surjective?), so we can immediately deduce that this preserves colimits. This intuitively makes sense for coproducts, as disjoint systems should exhibit disjoint dynamics. It would also mean that we can glue systems together by gluing together their Lagrangians if they have somehow compatible boundary conditions.

>>8402085
Dirac equation
>>8405618
this guy gets it

>>8402192
yoneda lemma gives me big cums

>>8402085
>Literally just a mathematical reformulation of F = ma except with linearly independent coordinates
You know it fails whenever solutions are path-dependent right?

>>8402946
holy shit I was literally just complaining about this after my TA meeting

why are undergrads so fucking fascinated by an eigenvalue equation

>>8403585
>>8403592
>maxwell's equations
can high school leave please
[eqn]\psi \mapsto e^{i\theta}\psi\\A_\mu \mapsto A_\mu + \partial_\mu\theta\\\Box A^\mu = \bar{\psi}\gamma^\mu\psi[/eqn]

>>8406912
sorry should be [math]\overline{\psi}[/math]

>>8406743
Yes, the forgetful functor taking a ring to its underlying abelian group is faithful (the functor forgets structure, so it is going to be faithful, and I think it is also essentially surjective).

>>8406912
Fuck off with your 4 vectors, they're superfluous anyway.

GR a shit

>>8406928
Could we generalize this idea? Would this reduce the information stored in the category enough for you to define your manifoldoids?

I'm very shy. Be nice, please.

>>8406933
?
this is qed, and only the last line anyway
the first two are literally just local and global U(1) invariance
you know, the things that make maxwell's equations work
well no, you actually don't know, huh

>>8406953
I know Maxwells equations as I have a Physics degree

I just don't know your retarded-ass oversimplified notation
Also fuck off with your social signalling bullshit. Don't be one of those wispy bearded faggots trying to get one over on people by proving you know more than them

>>8406685
>usefulness: popsci/10

That's why you use the general form

[math]e^{i\theta} = cos(\theta)+i sin(\theta)[/math]

aesthetics: 6/10 not quite as nice looking
usefulness: 10/10 it's fucking everywhere

>>8406635
That sounds like total fucking garbage. Keep your masturbatory abstract bullshit to yourself and out of physics.

who /numerical analysis/ here?

being an engineer is dope, you just brute force solutions to everything with computers.

p=ρRT

appearance: 9.5/10 it's like ideal gas law but better
usefulness: 10/10 i use this literally every day
confusion: 10/10 why would you put p and ρ in the same equation

>>8402085
Not really math (you sort of add), but I think the Friedel-Crafts Acylation is the best equation in organic chemistry.

It's mostly because I like the way I pronounce acyl (ass-ill).

>>8404970
Accurate

>>8407432
Needs catalysis via lewis acids therefore into the trash. Diels-alder a best

It's kinda ugly.

>>8403577
mugi is for friendly love only!

>>8407432
it is math, the reaction can be represented with a graph transformation.

>>8407276
You're LaTeX offends me
[math] \exp(i x) = \cos x + i \sin x [/math]

>>8406928
Another idea: Grothendieck topology. This would concretize the notion of continuous partial differentioids on your manifoldoids in the sense that you wouldn't have to invent your own concept of continuity, but could have them continuous iff continuous with respect to the Grand Wizard's topology.

Some essential questions now would be
>would the Yoneda embedding be continuous with respect to this topology?
If you would use it as a zero section, then it would be from a point to a point (right?), and depend on
>is the site Hausdorff or even [math]T_1[/math]?
>what would its basis be like?
>would it be locally euclidean?
I'm on my way to these topologies, so I can't answer them on my own yet. Thus, I will outsource them to you or someone else interested in this project.

>>8407649
peasant/10

>>8402085
I liked the variational principles of some important equations when I learned about them, but cannot find them
From what I remember:
The heat equation becomes something like:
[eqn] \delta \int_A \left[ \left( \frac{\partial u}{\partial t} \right)^2 + \sum_i \left( \frac{\partial u}{\partial x_i} \right)^2 \, \mathrm{d} A\right] = 0[/eqn]
and the wave equation becomes
[eqn] \delta \int_A \left[ \left( \frac{\partial u}{\partial t} \right)^2 - \sum_i \left( \frac{\partial u}{\partial x_i} \right)^2 \, \mathrm{d} A\right] = 0[/eqn]

>>8407483
Where's this from?

>>8407746
The standard model of particle physics.

>>8406532
But that way you dismiss the possibility of including both, 0 and 1, as the neutral elements of addition and multiplication in the equation. In my third week physics, when my calc Prof introduced said equation, he named it "the most beatuful equation in all of mathematics"

>>8402369
Undergrads, thats why

>>8402872
Nice

>>8407570
>you're
Disgusting

>>8406934
Well, the idea is that the presheaf category is a tangent bundle with respect to functors (functoriality is our continuity). However, you raise a good point: choosing categories such that functors between them correspond to maps of spaces (for example, with locales) would allow us to compare the categorical tangent bundle of a space to its geometric tangent bundle.

>>8407343
Wow, you sure are sensitive to new ideas. That's okay, you will see one day that this idea is useful.

>>8407842
Nice. How would you/we define the partial derivatives now, or are we interested in manifolds in general instead of differential manifolds? One way to define being differentiable n times would be that the functor can be factored through n categories without any functor in the factorization being an identity. I just don't know if this would be analogous enough to the ordinary differentiability, or even preserved by sufficiently many functors for this to be a sensible approach.

>>8407884
Interesting! I will work on this more later. Good idea!

>>8407894
Awesome! Please do and keep me informed!

[math]^{o} \smile ^{o}[/math]

>>8402282
Holy fuck my eyes.

>>8406969
Its not oversimplified.
dont be mad please

>>8402339
>>8402134
>not liking tensor notation
how's first year?

>>8402085
>All of classical mechanics can be summarized to this equation
All of classical mechanics can be summarized in newton's law too, that's not really saying much

>>8403968
>d'alembertian
not even once

>>8405618
what's this?

>>8408130
https://en.wikipedia.org/wiki/Dirac_equation#Covariant_form_and_relativistic_invariance

>>8408161
cheers
>puts away GR assignment
>reads about particle physics for hours

>>8408175
It's amazing how they came from the Klein-Gordon/Schrödinger to the Dirac Equation. And from that on to the electroweak model and how by breaking a symetry it allows particles to have mass.

>>8406761
Not that I have a good mental picture of what you cook up, but I'd assume a trajectory is an element of one of your objects, how could a left/right adjoint functor give you one?

>>8407649
>favourite equation
>posts inequality

>>8403374
I like the Lévy construction of the Wiener process.

>>8407894
I'll give you my construction for this idea. [math]\mathbf{Cat}[/math] shall stand for the category of categories and functors.

Choose any categories C and D, and consider the class of functors [math]H=Hom(C, D)[/math]. Define the [math]\textit{decomposition}[/math] of [math]F \in H[/math] to be a sequence [math](F_i )_{i \in \textbf{n} }[/math] of functors [math]F_i \colon C_{i-1} \to C_i[/math], where [math]\textbf{n} = \{ 1, \dots , n \} [/math], [math]C_0 = C[/math] and [math]C_n = D[/math], and [math]F = F_n F_{n-1} \cdots F_1[/math]. Call the decomposition [math]\textit{proper}[/math] if [math]F_{i+1} F_i[/math] is not naturally equivalent to the identity functor [math]1_{C_i} [/math] for any [math]0 < i < n[/math]. If [math]n[/math] is the greatest number for which the decompositions of a functor [math]F[/math] can be proper, then call [math]F \textit{n times differentiable}[/math].

Let then [math]\partial _n[/math] be the category with its objects all n times differentiable functors and morphisms ordered n-tuples [math]\tau = (\tau _i )_{i \in \textbf{n} }[/math], where [math]\tau _i \colon F_i \to G_i[/math] is a natural transformation for every [math]i[/math]. If [math]m < n[/math], then [math]\partial _n[/math] is a subcategory of \partial _m[/math].

Hopefully I made no typos in Latex...

>>8408416
Sorry, my mom called me in the middle of writing my post and broke my concentration. Anyway, we can construct a category like this.

>>8408416
>>8408459
Reformulating the post now:
[math]\textbf{Cat}[/math] is the category of categories and functors. Taking any categories C, D, define [math]H=Hom_{\textbf{Cat} } (C, D)[/math]. For any [math]F \in H[/math], define a decomposition of [math]F[/math] to be an ordered n-tuple [math](F_i )_{1 \le i \le n }[/math], where [math]F_i \colon C_{i-1} \to C_i[/math] is a functor for all [math]i[/math], [math]C_0=C[/math] and [math]C_n=D[/math], and [math]F=F_n F_{n-1} \cdots F_1[/math]. Call a decomposition proper if [math]F_{i+1} F_i[/math] is not naturally equivalent to the identity functor [math]1_{C_{i-1} }[math] for any [math]1 \le i < n[/math]. If [math]n \ge 1[/math] is such that [math](F_i)_{1 \le i \le n}[/math] can be a proper decomposition, call the functor [math]n-1[/math] times differentiable.

Let [math]\partial _n[/math] be the category with its objects all [math]n[/math] times differentiable functors, and as morphisms ordered [math]n[/math]-tuples [math]\tau = (\tau _i)_{1 \le i \le n} \colon (F_i)_{1 \le i \le n} \to G_{1 \le i \le n}[/math], where [math]\tau _i \colon F_i \to G_i [/math] is a natural transformation for each [math]i[/math]. Clearly, [math]m < n[/math] implies that [math]\partial _n[/math] is a subcategory of [math]\partial _m[/math].

>>8408416
>>8408459
Reformulating the post now:
[math]\textbf{Cat}[/math] is the category of categories and functors. Taking any categories C, D, define [math]H=Hom_{\textbf{Cat} } (C, D)[/math]. For any [math]F \in H[/math], define a decomposition of [math]F[/math] to be an ordered n-tuple [math](F_i )_{1 \le i \le n }[/math], where [math]F_i \colon C_{i-1} \to C_i[/math] is a functor for all [math]i[/math], [math]C_0=C[/math] and [math]C_n=D[/math], and [math]F=F_n F_{n-1} \cdots F_1[/math]. Call a decomposition proper if [math]F_{i+1} F_i[/math] is not naturally equivalent to the identity functor [math]1_{C_{i-1} }[/math] for any [math]1 \le i < n[/math]. If [math]n \ge 1[/math] is such that [math](F_i)_{1 \le i \le n}[/math] can be a proper decomposition, call the functor [math]n-1[/math] times differentiable.

Let [math]\partial _n[/math] be the category with its objects all [math]n[/math] times differentiable functors, and as morphisms ordered [math]n[/math]-tuples [math]\tau = (\tau _i)_{1 \le i \le n} \colon (F_i)_{1 \le i \le n} \to G_{1 \le i \le n}[/math], where [math]\tau _i \colon F_i \to G_i [/math] is a natural transformation for each [math]i[/math]. Clearly, [math]m < n[/math] implies that [math]\partial _n[/math] is a subcategory of [math]\partial _m[/math].

Constantly making typos.

>>8402144
someone pls explain the bar on the top for me pls

I know that the real polynomials mod x^2+1 are isomorphic to the complex numbers, but this is new to me

>>8408974
It means the algebraic closure of the polynomial ring, that is, a maximal algebraic extension of [math]\mathbb{R}[x][/math]. These are unique up to isomorphism, and [math]\mathbb{C}[/math] happens to be one of them.

>>8408907
>functors and anime

Holy shit, I've never seen a real life autist before...what's next, sonic and affine varieties?

>>8408974
It means an algebraic closure of [math]\mathbb{R}[x][/math].

>>8408998
I have seen atleast one. Not the fedora and tweed suit type, nor brony, as these are just weirdos with a chance of Asperger's. This autist I've seen was the real deal. He was swinging back and forth while sitting, mumbling his own stuff by himself, laughing his silent chuckle, and so on. I remember when he was chosen to prove something on the blackboard. He took the wiper and said something like "This is this", and then just started to laugh while wiping the board. He was nice, though.

But, as you clearly are implying I am autistic, well, I may be a bit aspie. I don't see how either of those would imply autism. I watch brainless moe series to give my brains some rest. There's a guy always asking why a "genius" like me (he uses that word) would waste his time with stuff like that instead of the "deep" shows. The answer is simple and comes in two parts: I want to balance my thinking with something that doesn't require brain activity that much, and I'm bad at following the plots and implications on the more philosophical shows.

>>8409024
How sad does your life have to fucking be that you get to the point where you spend your time shopping dover books onto anime drawings so you can use them to avatarfag

>>8408999
I see, very thanks cute trips anime!

>>8402085
>>8402085
Sin x = o/h

Stupidly basic, which makes it so, so satisfying.

>>8408999
[math]\mathbb{R}\left[ x \right][/math] isn't a field bro. I think you just mean [math]\mathbb{R}[/math].

>>8409050
Dunno. Nevertheless, I feel happier than for ages. This is still irrelevant, and I won't take part in derailing this thread any more than these few posts have done. Your fanmail has been read, though.

>>8409060
Cheers, m8!

>>8409097
Ah, yes ofcourse. My bad.

>>8409097
The algebraic closure of R[x] is tho

>>8409107
Algebraic closure is only defined for fields. The algebraic closure of R is C.

R[x] is not a field and thus algebraic closure isn't something defined. Now you could put some topology on the ring (like an I-adic topology) and take the topological closure, but that is very different.

>>8409220
ah gotcha. my bad.
still new.
> Now you could put some topology on the ring (like an I-adic topology) and take the topological closure, but that is very different.
couldn't you just use the field of rational functions?

>>8409227
Yes, R(x) is a field and does have algebraic closure. But that doesn't have anything to do with topology.

>>8407395
>appearance: 9.5/10
>has both ρ and p in it
nah

>>8403474
I second this answer. Used this a lot in school

s' = s + v * t + a/2 * t^2

>>8403469
>splitting into cases

how more pig disgusting can you get?

E=m

>>8407459
Was just about to post this.

D I E L S
A L D E R

While we're on the topic of Organic Chemistry, this is the most aesthetic lecture of all time:

https://www.youtube.com/watch?v=YvEB05xdAy4

The banter in it is top notch as well.

Someone please explain Nashes Mixed strategy algorithm to me please, I've looked at countless books and videos, I've always picked up even hard stuff very easily but this has me stumped