/sci/ - Science & Math

>>9609872
>reading
not science or math

>>9609872
Just doing some trig derivative problem sets, not too engaging but better than higher derivatives.
I’m doing an independent study on approximating pi, possibly converting it to hex/binary. Any good books on pi?

>>9609872
Refreshing Geometric Algebra and Calculus.

I plan to use some of its formalism in my next publication.

It's fascinating because I tried just using a Clifford algebra and quickly ran into issues that made it obvious that a discrete exterior calculus or the geometric calculus would be much simpler

I'm excited.


What do researchers here feel is the more popular/standard field? I tend to prefer the exterior calculus since the ties to differential geometry is already well developed, but the notation of geometric algebra appears to be really easy to work with.

Thoughts?

>>9609872
>Math general
define "Math"

I was reading a parsing theory book and having trouble with a proof on (recursively defined) regular expressions' properties. Not sure if properly applying double induction.

>>9609132
>All you need is basic set theory to get started.
I formally doubt this.
>>9609151
> i find algebra to be more fun
You mean abstract algebra?

Who ENS here?

>>9610463
who is the terence tao of the ENS?

For those of you who ascribe to ZF(C), would you care if it was proved to be inconsistent, or just continue using it?

If you cared enough to switch, what system would you use instead?

>>9610450
>I formally doubt this.
It is true both formally and informally. It's better to start with topology rather than infecting your brain with stuff like "analysis" which you will ultimately have to purge if you wish to continue studying mathematics.

>>9610480
>mathematics
This is not well-defined.

>>9610482
Mathematics is the study of mathematical structures.

>>9610484
>mathematical structures
This is not well-defined.

@9610484
Don't respond to the spammer, retard.

>>9610486
>spammer
?

>>9610480
I have a feeling that you're trolling me so I will not follow your advice, sorry.

>>9610491
It being formally true is trivial. Just check any introductory book on topology. Some of the shittier ones use examples from "analysis", but that's about it.

>>9610485
But it is.

>>9610510
>But it is.
That remains to be shown.

>>9610450
Why do you doubt it? The only reason to """""need""""" anal before topo is that you can then say
>oh this is why it works instead of some ridiculous inequality farce i saw earlier
when you see claims familiar from anal proved in a more general sense. It follows that >>9610480 is correct. Have fun studying topo!

t. >>9609132

>>9609872
I took an applied math thing but it's all Lagrangian and Hamiltonian systems and I keep getting bogged down in the physicsy things I don't need to know about. I'm becoming one of those people who can regurgitate enough to solve problems but can't understand what's going on.

>>9610642
>I'm becoming one of those people who can regurgitate enough to solve problems but can't understand what's going on.
Congratulations! You are becoming a true applied "mathematician".

lmao category ""theorists"" eternally fuming out their arse because the analyst will get employed after their degree whilst they sit on 4chan all day crying. HAHAHAHAHAHHAA

>>9610785
>lmao category ""theorists"" eternally fuming out their arse because the analyst will get employed after their degree whilst they sit on 4chan all day crying. HAHAHAHAHAHHAA
cringe

>>9610785
>qê wants to be employed by the bourgeoisie
No wonder you can't handle the deep insights of category theorists.

>>9610790
I'm not a "qê "

Did you know category theorists were once banned(informally) from receiving NSF funding because they were huge cunts?

>>9610790
>bourgeoisie
This is not well-defined.

>>9610797
>NSF
Do you mean MSF? Why would they be funding math?

>>9610792
Sorry for misgendering.

>>9610798
>a sociologically defined class, especially in contemporary times, referring to people with a certain cultural and financial capital belonging to the middle or upper stratum of the middle class: the upper (haute), middle (moyenne), and petty (petite) bourgeoisie (which are collectively designated "the Bourgeoisie"); an affluent and often opulent stratum of the middle class (capitalist class) who stand opposite the proletariat class.
Yes it is.

>>9610806
>an affluent and often opulent stratum of the middle class (capitalist class) who stand opposite the proletariat class.
Not well-defined.

>>9610785
>because
Your reasoning doesn't seem to make sense here, I'm assuming that's just on the surface level. Could you elaborate how you came to this deep conclusion?

>>9610812
>The proletariat (/ˌproʊlJˈtɛəriət/ from Latin proletarius) is the class of wage-earners in a capitalist society whose only possession of significant material value is their labour-power (their ability to work). A member of such a class is a proletarian. In Marxist theory, a dictatorship of proletarians is for the proletariat, through the proletariat and by the proletariat. This, in Marxist theory, will lead to proletarian self abolition and, thus, communism.
Take the complement of this set.

>>9610816
>set
Not well-defined.

>>9610816
This is just a bunch of gibberish, not a well-defined set. Some authors have even gone as far as to call it a "fictional set".

>>9610831
>Some authors have even gone as far as to call it a "fictional set".
Is that like a "fictitious force" (i.e. actually not fictitious)?

>>9610844
>fictitious force
Please keep this discussion in the proper board: >>>/toy/

>>9610801
I meant the US National Science Foundation

>>9610824
Yes it is in anarcho-primitivist mathematics.

>>9610831
Is there not a set of humans? Is there not a set of people living in a country? Is there not a set of people living in a country and not owning the means of production?

>>9610854
The proper class of humans doesn't contain things which are fictional and not well-defined, i.e., terms/objects/entities such as "proletariat" and "means of production". It also doesn't contain "communists", but the only known proof of this uses "biology" which we don't discuss here.

>>9610869
The class of humans is in bijective correspondence with a subset of natural numbers, and thus a set. It follows that all those sets are well defined.

>>9610341
Read up on the Monte Carlo approximation method for approximating pi.

I'm trying to find a formula for the number of elements in P1(Z/nZ)

The number of elements in P1(Z/pZ) is obviously p+1 as you have p elements of the form (1:k) and then 1 element (0:1)

It's also obvious that Pn(Z/pZ) has 1+p+...+p^n elements for the same reason as the above.

If I can get a formula for just P1(Z/p^kZ) then I'm done as I can use the Chinese Remainder Theorem to get the answer for all n, but I'm lost trying to get any formula for this.

Any help?

>>9609872
>modern algebra lecture
>professor asks class to define the characteristic of a ring
>classmate flips to the page in the book with the definition and paraphrases "It's the least positive integer n such that nx = 0 for all x in the ring."
>I chime in and say that nx is not necessarily the same thing as n · x, nx is multiplication while n · x is summing x with itself n times.
>professor says I'm being pedantic, multiplication is repeated addition. We move on.
>as I'm sitting through the rest of the lecture, I think of the ring [math]M_{2}(\mathbb{Z}_{2})[/math].
>you can sum the matrices in [math]M_{2}(\mathbb{Z}_{2})[/math], but you cannot multiply them by an integer.
>didn't ask professor about this case because too much of a pussy.
Am I being autistic and pedantic, or was the definition given by my classmate valid? I usually don't win when I disagree with professors, but I would like to make sure that lies are not being taught in class.

>>9611206
Every abelian group (G,+) is a Z-module with Z x G ---> G given by ng=g+...+g .

>>9611206
>nx is multiplication
In a ring this is usually written as x^n.
>n · x
This is just the [math]\mathbb{Z}[/math]-module action on your ring, usually it is written as nx.

>>9611034
Follow up to this:
it seems the formula is p^k+p^(k-1)
or p^k(1+1/p)

This has confused me somewhat as for Z/4Z I think I can make 9 distinct elements, not 6:
(1:0) (1:1) (1:2) (1:3) (0:1) (0:2) (2:0) (2:1) (2:2)

Any help?

>>9610640
>Implying topologists actually prove their claims

>>9611206
you can multiply the matrices in M2 by integers, that is scalar multilication. it's the same thing as adding a matrix to itself n times (or the additive inverse of that matrix)

>>9610416
Mathematics is (by definition) the study of TQFT and string theory and maybe some AQFT over exotic spacetimes.

>>9611260
seems like 0:1 and 0:2, 1:1 and 2:2 aren't distinct. 2:0 and 1:0 also but I don't know how you justify it

>>9611260
and maybe I just don't understand your terminology but if 2:2 != 1:1 aren't we talking about a cartesian product instead? i.e. Z X Zn

>>9611286
>>9611291
these are distinct because 2 is not a unit in Z/4Z

>>9611206
What scares me is that he is even talking about "multiplication" in a group theory context.

The operation nx of the ring isn't necessarily multiplication in the trivial sense.

That Prof is confusing his class.

What's the best software for making figures of combinatorial graphs?

>>9611310
python-networkx

Well-definiteness is not well-defined.

>>9611354
yes it is, it's often not well-explained though. well-defined things just fit the labels you use for them, e.g. checking if a map is well-defined is the same as checking if it's actually a map.

>>9611034
How do you define the projective line over a commutative ring that is not a field?

>>9611512
Proj(R[x,y])

>>9610478
Set theory student here. No one except philosophers and babies first logic class care about the consistency of ZF. The reason is that if ZF is inconsistent everything follows, so the only interesting case is when ZF is consistent. Thus that assumption is implicit when working ZF. Very few set theorists actually care or work with axiomatics, I took a grad set theory course with someone who did work with axiomatics and it was fucking painful. Hurr durr does this set exist well take the power set of the power set of the power set of this set and cut down with comprehension. I do work in effective theory so I could restrict ZF down to Kripke Platek (KP) set theory + omega and everything would carry through.

>>9611517
Ok, I see. The problem with the (algebro-)geometric definition is then that [math] \mathbb{P}^1(\mathbb{Z}/p) [/math] already has infinitely many elements - there are not only the points rational over [math]\mathbb{Z}/p[/math] but also other points. But if you restrict yourself to rational points, the question over [math]\mathbb{Z}/n[/math] for general n is somehow weird.

>>9611544
>infinitely many elements
it has p+1 elements

>>9611552
These are the rational points. But the Proj-construction generally gives more (closed) points (whose residue field is a proper extension field of the ground field).
But if you mean rational points - what do they mean over arbitrary Z/nZ.
Or do you mean something else by Proj? If you mod something out (set-theoretically) of some kind of R^2 you might get the correct object.

Does it matter where you get your undergrad degree? I go a decent public Uni and I am thinking of transferring to a more rigorous private school or an IV league.

>>9611608
some "decent" public universities have abysmal math programs, especially if you stay off the graduate track
this of course means transferring from such a school would be really hard, because everyone around you would have underlying knowledge that you lack

>>9611608

>>9611632 is right.
I go to what would be considered an ivy league school if it were in the states and the math program I'm in is a total joke. The honours stream is legit though and will actually prepare you for grad school. Research thoroughly the program you want to be admitted to regardless of the reputation of the uni.

>>9611608
I went to a pretty shitty undergrad institution but got into a Group I school for grad school. Get good letters of rec, and try to get your name on as many papers before you graduate as you can. I'd also recommend self-studying your dream school's curriculum if your current one isn't challenging you enough.

>>9611650
How did you cope? My courses are terrible so I self study the material from proper books, but having to do all this computational bullshit in the actual class is brutal.

>>9611632
I haven't started the math program at my school yet. I've completed all the Calc courses and am currently taking Differential Equations. At this point I'm a mathlet. Which is why I was wondering if I should transfer out before I even have the chance to be aware of how shit my school's curriculum is. I plan to self study anyway and to review Spitvaks Calculus book since my school uses Stewart. Could I still lack crucial underlying knowledge if I haven't even seen a real math course yet?

>>9611639
>>9611650
Thank you.

this. someone recommend something better for linear algebra

>>9611666
If you're not being challenged by the material, use it as an opportunity to work learning a new skill into it.

I took a cryptography course at my old school that wasn't hard, but it was extremely computation heavy. So I taught myself how to code and wrote methods to do all the long calculations for me.

What's a physicist friendly book for differential geometry?

Plotting on how to overthrow my upcoming calculus classes.
Algebra 2 person here, freshman, the only freshman in my class of all sophomores. I already raised the pits of Satan himself upon the class by smoking everyone, which is fun until our teacher counts everything off for not showing work, but hey, I might consider studying pre calculus over the summer just got the heck of it, that way I'll send the class into oblivion.
Then I'll just enjoy senior year.
What am I doing.

>>9611383
>yes it is
That remains to be shown.
>well-defined things just fit the labels you use for them
This is not well-defined.

>>9611684
hoffman & kunze obviously come on now

>>9611713
you have to be 18 to post here I thought. If this isn't b8 tho just realize now that you aren't special and chances are you won't amount to anything like the overwhelming majority of the rest of us so try to understand that now so you don't end up being disappointed in the future.

>>9611713
Showing your work is what mathematics is. Don't be a brainlet by thinking you aren't one

>>9611785
>mathematics
This is not well-defined.

>>9611785
apparently he talks about high school math (algebra/precalculus), not actual mathematics

>>9611775
Thanks for the recommendation. I'm taking a class in this and this book is really confusing.

>>9611671
take a look at the GRE's, that should give you some idea what you are supposed to learn. analysis and algebra are typically 3 semesters each.

How did you guys learn numerical/computational math? Any good books? Youtube?

>>9612296
This is the mathematics general. If you would like to discuss engineering please direct yourself to the engineering thread(s) on /toy/ and elsewhere.

Hey, can someone here answer my question? (>>9612214)

Thank you!

>>9612303
>general
Not well defined.

>>9612325
Never mind, it's been answered. Thank you!

>>9611707
Start with Spivak's calc on manifolds and Schutz's geometry in physics.
Kuhnel's curves surfaces manifolds, Spivak's differential geometry vol 1-2, Lee's smooth manifolds, Milnor's differential topology, Guillemin and Pollack's differential topology, Schlichenmaier's Riemann surfaces algebraic curves moduli spaces, Naber's gauge fields, Nakahara's geometry topology, Baez's gauge fields knots gravity, Gockeler and Schucker's differential geometry

>>9611713
Isnt Algebra 2 and Trigonometry precalculus? Trig is very easy if you can remember and visualize the unit circle and understand it. Precalc algebra is harder than trig

>>9612519
>Schlichenmaier's Riemann surfaces algebraic curves moduli spaces
That book tries to cover way too much material and ends up not really giving any detail at all.

>>9611707
>>9612519
>>9612716
Try posting about this in the physical threads located at >>>/toy/.

How can I show that every ideal is maximal in the non-standard integers [math]\underline{\mathbb{Z}}^\infty[/math]?

>>9610797
>because no one understood their research

ftfy

>>9612918
>because they produced no tangible results

ftfy

>>9612918
No one in engineering you mean? Why are you posting about that here?
>>9612919
>tangible results
Try asking in the physics threads if you want "tangible" (i.e. understandable by the mentally impaired) results such as the existence of blacks holes and so on.

>>9611765
>>9611383
Generally it means checking that a relation is functional.

>>9612922
No need to get mad Anon.

I meant tangible in an abstract sense, as in lacking results within the field. How many useful connections have actually been made that weren't previously known though other techniques?

>>9612915
Show any ideal quotient defines a field

>>9612716
That's true. I included it because its short and gives a fair intro to complex geometry with good references. I thought about including Bott and Tu and Bredon as well but I think they're too involved.

Hey guys I'm a real brainlet when it comes to maths and am having some trouble trying to solve this problem I have. I want to find the coordinates of B and D in pic related. I have the coordinates of A and C currently and have calculated the length of each edge and the diagonal length.

The problem is it's a 3 dimensional object and it's on a reasonable angle such that A is much deeper than C.

If anyone could point me in the right direction of what they'd do next that would be super appreciated.

Whats a good book about tensors. I can't understand them.
t.physics brainlet

>>9613043
https://www.amazon.com/Tensor-Analysis-Manifolds-Dover-Mathematics/dp/0486640396
https://www.amazon.com/Tensors-Differential-Variational-Principles-Mathematics/dp/0486658406/
https://www.amazon.com/Vector-Tensor-Analysis-Applications-Mathematics/dp/0486638332
https://www.amazon.com/Elements-Tensor-Calculus-Dover-Mathematics/dp/0486805174/
https://www.amazon.com/Vector-Tensor-Analysis-Dover-Mathematics/dp/0486601099
https://www.amazon.com/Applications-Tensor-Analysis-Dover-Mathematics/dp/0486603733
http://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/20050175884.pdf

>>9610640
>>9610480
What Topology book would you recommend?

>>9612938
>as in lacking results within the field
That's understandable since it's not a field about "results" in the usual sense.

>How many useful connections have actually been made that weren't previously known though other techniques?
What is "useful" to you? Is pretty much the entirety of algebraic geometry, algebraic topology and homological algebra somehow not "useful" and "tangible"? In that case, maybe you should start using the physics threads over at >>>/toy/? Although even their kind would laugh at you for openly claiming such things. So maybe >>>/g/ would be a more suitable place?

>>9613043
>tensors
What do you mean? This isn't standard terminology.

>>9613052
Dugundji's book.

>>9613043
Rotman's book on homological algebra.

>>9613103
>Dugundji's
An old dusty Topology book. I knew that you were a troll, you can't do Topology without Analysis.

>>9613107
>you can't do Topology without Analysis
No. You in particular can't do topology (with or without an*lysis) and that's understandable given the limits of your brain.

>>9613107
>old dusty
Not an argument.
>i knew that you were a troll
How am I a troll?
>you can't do topology without anal
Yes you can. Anal serves as a source of examples of how topology can be applied to certain cases, but it is not needed for topology. You can't do linear algebra without analysis either because differential operators are linear?

>>9613113
This.

>>9613113
>an*lysis

>>9613144
You say this as a person who did analysis before topology or are you just speculating because you feel like your mathematical maturity didn't play a role because you fail to realize it's importance and just focus on the analysis-aspects? Or genuinely someone who did topology before analysis?

>>9613144
You say this as a person who did topology before analysis or are you just speculating because you feel like your mathematical maturity didn't play a role because you fail to realize it's importance and just focus on the analysis-aspects?

>>9613152
I'm sorry about your brain loss. It must have been painful, right?

>>9613184
Not him, but he's completely correct and I'm saying this as someone who has never touched any branch of engineering (including analysis).

>>9613043
>>9613083
an element in the tensor algebra

>>9613184
We used some anal stuff as examples when doing metric space stuff, for example Lipschitz continuity. Otherwise we had no use for anal.

>>9609872
>HOME-uh-toe-pee

>>9613323
Reddit frogs are not welcome here. Refer to >>>/v/ and >>>/r/eddit/.

>>9613323
Ignore the charlatans.

Reddit frogs are very welcome here, friend.

>>9613365
This but unironically

>>9613365
Reddit spacing and reddit frogs are not welcome here. Try using proper websites for your kind, one example would be >>>/r/eddit/.
>>9613368
>"unironically"
Refer to the message above .

>>9613368
Errrrrrrrrrrrrr, nope.

>>9613323
If you can't pronounce shit like "homotopy" using only the fact that it's derived from Greek, you are not welcome here.

>>9613372
>>>9613365
>Reddit spacing and reddit frogs are not welcome here. Try using proper websites for your kind, one example would be >>>/r/eddit/.
>>>9613368 (You)
>>"unironically"
>Refer to the message above .
>>9613375
>>9613375
>>>9613368 (You)
>Errrrrrrrrrrrrr, nope.
>>>9613323
>If you can't pronounce shit like "homotopy" using only the fact that it's derived from Greek, you are not welcome here.

>>9613345
No one gives a shit faggot. Go drink your soy milk you pathetic kike. You can play all the nintendo switch you want you limpwristed faggot no mudslime yuro or shitskin spic will ever accept you into their commie nigger club

>>9613450
any takers

>>9613492
This holds trivially, since any class which is a set is finite.

I'm interested in Logic, Number theory, Proof theory and to a certain degree Probability theory.
I'm a physics major.
Using the common division, Algebra, Analysis and Geometry/Topology. Would Algebra be the right way to go?

>>9613498
what?

>>9613549
What are you having trouble with in my post?

>>9613565
what's a class

>>9613529
>Would Algebra be the right way to go?
In what sense are you interested in number theory if you have to ask this?

>>9613578
what you missed because the question is trivial

>>9613492
Suppose epsilon is smaller than 1 (otherwise, the set is empty or has very few elements). Then there is by archimedean property of the reals, there is some natural number q such that [math]\frac \epsilon 2 q>1[/math]. Hence [math]\frac{1}q<\frac \epsilon 2[/math]. Now, how many natural numbers p(and hence rational) are there such that p/q is smaller than 1? Hint: it's finite

>>9613529
All the things you are interested in are completely unrelated, and hence they require literally all the fields you mentioned

Studying for a functional analysis test and reading Rick Miranda's Algebraic Curves and Riemann Surfaces along with Milnor's Characteristic Classes. Miranda's book is a gem.

Vilfredo Pareto’s Manual of Political Economy.

Not mathematical enough desu senpai. I liked Irving Fisher’s mathematical investigations a bit more

>>9613057
>Is pretty much the entirety of algebraic geometry, algebraic topology and homological algebra somehow not "useful" and "tangible"?
Of course those are not useful or tangible.

>>9613613
>In what sense are you interested in number theory if you have to ask this?
Presumably the analytic sense.

>>9613708
>All the things you are interested in are completely unrelated
This is false.

>>9613949
Perhaps to a "computer ""scientist""" such as yourself. Refer to the >>>/g/hetto/.

>>9613996
>"computer ""scientist"""
This is a meaningless notion.

>>9613956
>analytic
And why would he be posting about that in a mathematics thread? Presumably he has read the subject.

>>9611271
>t. literal drooling retard

>>9614001
Indeed. "Computer "scientists"" such as >>9613949 are meaningless notions.

>>9613949
True. They're too abstract and untangible for me. I'm a black person by the way.

>>9614009
>"Computer "scientists""
What do you mean?

>>9610482
Mathematics is that which is studied in Mathematics departments in non-profit academic institutions

kys

>>9613107
>old dusty
do you think theorems change over time, faggot-kun?
dust is also topologically invariant so just discard it

>>9614024
>faggot
Why the homophobia?

>this fucking thread
We should just call it /shit/. Do you autistic losers EVER discuss mathematics? The thread is literally solid shitposting from post 1 to here.

>>9613152
>>an*lysis

>>9614027
>Do you autistic losers EVER discuss mathematics? The thread is literally solid shitposting from post 1 to here.

>>9614027
See >>9613144

>>9614026
>>faggot
Why the homophobia, faggot-chan?

>>9614016
>I'm a black person by the way.
Why the racism?

>>9614033
What do you mean? How is me being black racism?

Hey guys. What are some good algebraic topology books for black people? I find the usual references too hard and abstract for some strange reason.

>>9614039
>What are some good algebraic topology books for black people?
It's not a book, but http://freevideolectures.com/Course/2721/Algebraic-Topology

>>9614035
>How is me being black racism?
How is it not?

>>9613052
Mendelson's and Jänich's

>>9614072
How is black people (me) simply existing somehow racism?

>>9610816
the complement of the middle class is not the proletariat
unless you consider people like Gates, Trump, and Weinstein to be proles
you know what, fine, i like this definition
leftists are so funny sometimes

>>9610854
>Is there not a set of humans?
no. there are only sets of sets. formally all that exists are sets. your entire premise is ill-conditionned

>>9614090
>formally

>>9610872
>The class of humans is in bijective correspondence with a subset of natural numbers, and thus a set.
Dunning-Kruger: the post

>>9613052
This >>9614078
Also check out Lee's "Introduction to Topological Manifolds"

>>9611305
it's in a ring context; they're talking about characteristics of rings
it's honestly perfectly fine

>>9614090
>all that exists are sets

>>9614082
>How is black people (me) simply existing somehow racism?
Blackness IS racism

>>9614085
>proletariat
>proles
This is a meaningless notion.

>>9614082
>>9614102
The Black Body is an ONTOLOGICAL condition. Ontology here is not in the Heideggerian sense of Dasein, but a social formation in which Being, as such, is determined by the system which produces it. Being, then, is defined by the outside. Who you are, or who you think you are, is irrelevant against the condition of Blackness. Blackness is the condition of a body which is labelled as inhuman, uncivilized, etc, and is the figure by which Civil Society is defined against. This opposition of Civil Society being everything which is NOT the Black Body is the way racism perpetuates itself in the status quo, and comes from a long tradition of slavery.

>>9614085
Middle class = petty bourgeoisie, high class or w/e it's called in English are the capitalists. Combine those and you get the non-proletariat.

>>9614090
Are we not sets of atoms in the end? Are atoms not sets of smaller stuff? Eventually we get down to some primitive notion.

>>9614094
What do you mean? There is an injective class function from the class of humans to the set of natural numbers.

>>9614108
As a Black Body, this is too abstract for me.

>>9614111
>petty bourgeoisie
>high class
>non-proletariat
all ill-defined

>ESL
oh that explains it, you're just retarded.

>>9614111
>Are we not sets of atoms in the end?
no, faggot
of course the literal communist posting in the math general has never been exposed to the most fundamental branch of mathematics
kindly kys

>>9614111
>There is an injective class function from the class of humans to the set of natural numbers.
no there isn't. if you'd studied even the most rudimentary set theory or type theory you'd know this to not be the case. instead, you're content to copy and paste buzzwords from wikipedia to impress anonymous faggots on the worst thread on the worst board on 4chan

>>9614099
Hey friend, are you the one that makes the topological friends images?

>>9614111
>the set of natural numbers.
The natural numbers object in "[math]\mathbf{Set}[/math]" can be shown to be a proper class.
>>9614119
He is actually correct if you assume that LEM is false.

>>9613107
> you can't do topology without analysis.
this is kind of true in a sense, but really it's more like they're both intertwined.
think of the Jordan curve theorem for example. most of complex analysis is virtually impossible without it but you have to prove it topologically.

>>9614111
>atoms
Please go back to >>>/toy/

>>9614122
I wish. They are by @omnisucker (twitter). These three are the only ones which exist as far as I know.

>>9614119
>no there isn't. if you'd studied even the most rudimentary set theory or type theory you'd know this to not be the case. instead, you're content to copy and paste buzzwords from wikipedia to impress anonymous faggots on the worst thread on the worst board on 4chan
cringe

>>9614114
>oh that explains it, you're just retarded
Speaking the same language as British island monkeys and Amerimutts makes you the retard here.

>>9614116
>faggot
Why the homophobia?
>of course the literal communist posting in the math general has never been exposed to the most fundamental branch of mathematics
You mean TQFT? That is not mathematics.

>>9614119
How is there not an injective class function? Collect all humans in a line and label them with consequent positive integers. What you end up is a function mapping every human to a unique natural number, and thus a function from a class to a set. Since it is injective, the class of humans must be a set.

>>9614125
>he
Did I permit you to assume my gender?

>>9614126
>this is kind of true in a sense, but really it's more like they're both intertwined
False. Anal is an application of algebra and topology.

>>9614128
The category of [math]\mathbb{R}\text{world}[/math] can be used as an example. Read Bourbaki's critique on Proudhon.

>>9614125
>He is actually correct if you assume that LEM is false.
no he is not, as the "class" of humans is not a class of any type, neither a proper class nor a set

>>9614125
>He is actually correct if you assume that LEM is false.
I'm not a "he".

>>9614126
Literally this. Actually, you can even prove a meta-theorem that the definition of an exact sequence depends in a crucial way on analysis.

>>9613057
>pretty much the entirety of algebraic geometry
nice meme

>>9614135
>Speaking the same language as British island monkeys and Amerimutts makes you the retard here.
he said, in English.

>>9614135
>Collect all humans in a line and label them
hahahahahHAHAHAHAHHAHAHAHHAHAHAAAAAAAAAAA

>>9614141
By no means an argument.

>>9614143
>th-things aren't an a-a-argument when YOU do them!!
he winged, in English.

>>9614145
Nice damage control, mutt cuck.

>>9614136
It's provably a type in the topos [math]D(\mathbf{Ho}(\infty\mathbf{Type}))[/math] which is definable if and only if LEM is provably false.

>>9614142
Exactly what a retard would say.

>>9614139
>, you can even prove a meta-theorem that the definition of an exact sequence depends in a crucial way on analysis.
What do you mean?

>>9614160
What exactly isn't clear?

>>9614164
>What exactly isn't clear?
What the "meta-theorem" precisely states

>>9614133
Thank you friend :3

I am not reading anything. I am doing a really tedious partial derivative Webassign before Spring Break.

>>9614168
It states that the notion of an exact sequence in an abelian category is undefinable without having the deep prior notion of an "integral".

>>9614172
>I am doing a really tedious partial derivative Webassign before Spring Break.
Why did you decide to tell us about your non-mathematical activities?

>>9609872
spinors all day everyday

What is cardinality of a set which contains all of the cardinalities?

>>9614218
Burali Forti paradox

>>9614218
>contains all of the cardinalities
What do you mean?

I'm currently teaching myself abstract algebra.
Is it just me or do all of the interesting results in this field come from other, superior fields? number theory, polynomial algebra, combinatorics, etc.

>>9614397
>superior fields
>number theory
>polynomial algebra
>combinatorics
Nice b8, m8. I r8 8/8.

>>9614397
>number theory
That's a branch of algebra.
>polynomial algebra, combinatorics, etc.
Haven't heard of those. Not an expert in engineering.

>>9614397
What are these interesting "results" you speak of?

>>9614410
>That's a branch of algebra.
This is false.

>>9610806
who else /petitbougie/ af here?

>>9614421
>This is false.
This is false.

>>9614428
>petitbougie
This is provably not well-defined.

>>9614434
>This is provably not well-defined.
The burden of proof is on you.

>>9614466
>burden of proof
Refer to >>>/lit/ for the proper place to discuss such topics.

>>9614474
>Refer to >>>/lit/ for the proper place to discuss such topics.
Refer to >>>/x/ if you are unable to provide proof of your claims as per the burden of proof.

>>9614476
Is this a simulation?

>>9614420
most of the less-obvious stuff to me, at least so far. e.g. the structure of a cyclic group is essentially derived from number theory. it's obvious that every group is isomorphic to a permutation group, but the features of permutation groups (e.g. parity) are much less obvious and are proved with more outside math.
compared to those kinds of results, stuff like lagrange's theorem seems really basic
I know I'm still in baby land but I wanna know if this pattern continues -- does most of the insight in abstract algebra stem from concrete examples?

>>9614497
Do you know what makes abstract algebra abstract? You abstract away redundancies like numbers.

>>9614485
>Is this a simulation?
Is what a simulation?

>>9614497
>structure of a cyclic group
>lagrange's theorem
>features of permutation groups (e.g. parity)
>""results""
It's funny how high-schoolers always seem to be the ones writing such laughable posts and yet with a serious tone. I wanna know if this pattern continues -- do most of these people ever become self-aware enough to realize their own retardation?

>>9614534
Am I in a simulation right now? I demand to be briefed.

>>9614497
not every group is isomorphic to a permutation group
every group *embeds in* a permutation group
consider: the order of a permutation group on a set of cardinality n is n!, but there are groups of each prime order

>>9614555
>I demand to be briefed.
We use "we", not "I".

>>9614593
Permutation groups are by definition the subgroups of symmetric groups.

>>9614604
oh. my bad, probably shouldnt've inferred definitions

>>9614497
don't listen to the other anons. Group theory is trash. Better speed up to get to the juicy ring/module theory

>>9614099
Seconding the Lee recommendation, gr8 book altho it doesn't cover some areas of non-directly-manifold-related point-set topology, if you care about that)

>>9614615
>Group theory is trash. Better speed up to get to the juicy ring/module theory
Oh, the irony...

Pure maff student here --any tips for an aspiring hs math teacher? I know it sounds gay, but I want to give kids the education I never received.

Where(which book) can I find levi civita tensors demonstrations in vectors? Like Ax(B.C), determinant, trace, det(I + kA), something like that, not the definition of index.

>>9614183
the fuck are you talking about, it's completely algebraic

>>9614627
Not really mg material, but don't be that "cool" teacher. That shit killed my desire to show up for geometry. I might be biased though since I was a loner.

>>9614615
>speed up
What do you mean? Are you implying that finite group theory is somehow a prerequisite for studying rings and modules?

>>9614670
>don't be that "cool" teacher.
Kek, that won't be an issue.

>>9614627
I've often felt the same way but if I'm being honest with myself most people probably wouldn't appreciate the kind of education that I wish I'd had (i.e., most people are not autistic)

>>9614684
>finite group theory
As opposed to what?

>>9614735
In what sense?

>>9613052
The books the topology guys are recommending you are find, though I would say they're probably going to be too advanced if you're fresh off Spivak (it's not there there's a formal prerequisite for lots of this, it's just you want more mathematical maturity first). It also makes some sense to do a good analysis book first, because such books will do some point set topology in a more simple concrete context, which helps when you generalize later when studying topology properly.
But before you do all that, it's really crucial you do some linear algebra next. It just shows up everywhere and it's a good way to build more mathematical maturity. Hoffman and Kunze is the best linear algebra book in my opinion, but there are lots of good options out there.
After this, for analysis, I think Pugh's book is excellent. Rudin is a more terse alternative with no pictures, so I like it less for self study (it does have good problems though). Since you mentioned Tao, Analysis I covers very similar material to Spivak, so you'd really want to start on Analysis II. After any of these you'd be more than ready to read a point set topology book if you want (Mendelson or Munkres for example).
There's lots of options out there for algebra, like Aluffi, Artin, or Dummit and Foote. I have no strong feelings on these, maybe someone else will.

>>9614747
>I would say they're probably going to be too advanced if you're fresh off Spivak
Post disregarded. Try recruiting people into your organization elsewhere.

>>9614741
>In what sense?
As opposed to a term that is not redundant.

>>9614747
>I have no strong feelings on these
You don't have any strong feelings on actual math and yet you seem to be an expert in "undergrad" "analysis" as you guys call it. Maybe you shouldn't be giving advice?

What are some applications of vector spaces which do not have a basis?

>>9614771
>applications
Please go to the physics and engineering threads
>>>/toy/

>>9614747
>it's just you want more mathematical maturity first
And how exactly is someone supposed to develop it by doing engineering problems?
>It also makes some sense to do a good analysis book first, because such books will do some point set topology in a more simple concrete context, which helps when you generalize later when studying topology properly.
He didn't state that he was a brainlet who requires crutches to understand simple concepts.
>But before you do all that, it's really crucial you do some linear algebra next.
Linear algebra (the mathematical kind) requires basic ring theory and module theory, so it's advisable to learn those first.
>After any of these you'd be more than ready to read a point set topology book if you want
He would be more than ready even without reading any of the poisonous garbage you recommended.

>>9614825
>anime poster
Not even worth reading your shitty post. Why don't you fuck off to /a/ and stay there?

>>9614832
>Not even worth reading your shitty post. Why don't you fuck off to /a/ and stay there?
Do you need to swear?

>>9614832
Your kind is not wanted here, simpleton. Proceed to >>>/r/eddit/ if you dislike the fact that this website is based on anime discussion.

>>9614684
Not necessarily, but if he's taking the usual abstract algebra sequence, then they always start with group theory

>>9614771
none, which is why you want a basis so badly

>>9614832
So basically you're admitting that you're a redditor then?

if you havent read all 9 volumes of dieudonne's treaties on analysis what exactly have you been doing with your time? hint: wasting it

>>9613052
someone mentioned that pic was good. pdf on author's website.

>>9614852

>>9614844
>none, which is why you want a basis so badly
But what good is assuming you have a basis if you can't do anything else except assume it exists?

>>9614860
Every vector space has a basis by the AoC, I don't know what your problem is? you want an explicit basis?

>>9614880
You don't need AC to prove something which holds trivially.

>>9614884
it doesnt hold trivially for infinite dimensional vector spaces

>>9614880
>Every vector space has a basis by the AoC, I don't know what your problem is?
So every vector space has a basis only if you assume it has a basis. What's the point in making such an assumption?

>>9614891
it is a useful one. Turns a "suppose you have a basis", to "let this set be a basis"

>>9614895
>it is a useful one. Turns a "suppose you have a basis", to "let this set be a basis"
Useful for what? "suppose you have a basis" and "let this set be a basis" are functionally equivalent if you can only say "let this set be a basis" by assuming a basis exists.

>>9614890
>infinite dimensional vector spaces
This is redundant.

>>9614902
>redundant
This is not well-defined.

>>9614904
>well-defined
This is not well-defined.

>>9614906
>This is not well-defined.
This is false.

>>9614910
>This is false.
The burden of proof is on you.

>>9614890
It holds by definition of "vector space" for arbitrary vector spaces.

>>9614915
>arbitrary vector spaces
This is a meaningless notion.

>>9614902
Another poster interrupting - you might define "finite dimensional" without a basis: just take "generated by some finite subset" as your definition. Then definite "infinite dimensional" as "not finite dimensional".

>>9614919
>Another poster interrupting - you might define "finite dimensional" without a basis: just take "generated by some finite subset" as your definition. Then definite "infinite dimensional" as "not finite dimensional".
I am not sure what you are trying to imply here.

All vector spaces are infinite dimensional which is why "infinite dimensional vector spaces" is redundant, and "generated by some finite subset" is not remotely equivalent to the definition of a basis.

>>9614922
Ok, I misunderstood your statement.
Can you give a proof that all vector spaces are infinite dimensional?

>>9614916
How so?

>>9614931
>proof
Refer to >>>/lit/ for the proper place to discuss such topics. I am sure you can find one in any of the standard references (e.g. proofwiki, vixra).

>>9614938
>How so?
What meaning does it have?

>>9614931
The proof can't be shown to exist constructively.

>>9614952
>constructively
This is a meaningless notion.

>>9614959
>This is a meaningless notion
This is not well-defined.

>>9614961
>well-defined
What do you mean?

Can anything which is not "well-defined" be lifted to something "well-defined"? For example the function [math]f: \mathbb{Q} \to \mathbb{Z} [/math] defined by [math] f(\frac{a}{b})=a [/math] which is not "well-defined" since [math]\frac{1}{2}=\frac{2}{4}[/math] but [math]f(\frac{1}{2}) = 1[/math] and [math]f(\frac{2}{4})=2[/math], but can be lifted to a "well-defined" function [math] \tilde{f}: \{ \frac{a}{b} \mid a\in \mathbb{Z}, b\in \mathbb{Z}\backslash \{0\} \} \to \mathbb{Z}[/math].defined by [math] \tilde{f}(\frac{a}{b})=a [/math].

>>9613323
Clearly homotopy should have the same cadence as homology.

>>9614988
What do you mean by "well-defined"?

>>9614988
The formula you gave does define a relation, just not a function. What you can do is show that given an appropriate into the "domain" of the relation, the relational composition of that function with your relation is itself a function; this is your "lift". The function you'd compose with in this case would be (a, b) |-> a/b

>>9615095
s/given an appropriate into/given an appropriate function into/

>>9615095
wait never mind I have no fucking idea what I was thinking, pls ignore this

>>9610642
Maybe look into applied math about something you care about? I am an electrical engayneer who likes machine learning so taking advanced linear algebra with someone who researched applied problems in signal processing/control theory was pretty cool. Of course if you don't give a shit about anything then pure math is cool too.

>>9615086
hoe moe loe gee

Honest question: why is this general so cancerous?

>>9615270
>why is this general so cancerous?
What do you mean?

>>9615278
Things like this:
>>9610416
>>9610482
>>9610485
>>9610480
>>9610498
>>9610513
>>9610640
>>9610785
>>9610824
>>9612918
>>9612919
>>9612922
>>9613113
>>9614004
>>9614016
>>9614024
>>9614031
This is just from the first half the of the thread and I definitely skipped some. Basically, lots of trolling and elitism about branches of mathematics (inb4 somebody replies in a way that implies analysis isn't "true math"). I just wanna talk about math without 5 trolls responding to every post.

>>9615297
>I just wanna talk about math
Like what?

The set [math]\{(a^2,a^3,a^4) | a \in \mathbb C\}[/math] is algebraic since it's the vanishing of [math] (x^3 - y^2, x^2 - z) [/math], right? It's an exercise to find the Zariski closure of this set, and based on how far into the section it is it feels like it shouldn't be this obvious.

>>9615314
Ok, let's try this and see what happens.
I'm taking a geometric measure theory seminar this coming semester. Does anyone have any book recommendations, and have there been any recent interesting results to give me some idea of what's being worked on right now?

>>9615327
That's not math

>>9615337
wow I'm so shocked

>>9615327
>Does anyone have any book recommendations
The most boring kind of question, are Amazon reviews not enough? Especially when you can download nearly any book instantly

>>9614432
>what is analytic number theory
>what are elliptic curves
the absolute state of algebraists

>>9614902
brain damage: the post

>>9614915
the definition of a vector space nowhere implies the existence of a basis
go retake undergraduate linear algebra for the third time; maybe you'll learn something this time

>>9614922
>All vector spaces are infinite dimensional
every day i browse this site i more quickly converge towards suicide

>>9615375
>brain damage: the post
Wrong: the post.

>>9615327
>geometric measure theory
sorry, i thought you wanted to talk about math?

anyways guys what's you're favourite functor/co-functor pair?

https://mathoverflow.net/questions/66084/open-problems-with-monetary-rewards

>>9615382
>all vector spaces are infinite-dimensional

>>9615270
Because of a failed jewish biologist spamming. He doesn't even understand what people are talking about here, but tries to act cool by "knowing" abstract non-sense.

>>9614988
Yeah, just take the equivalence class of being a certain number and project it down to the quotient, but don't expect it to be a homomorphism or anything

>>9615327
Federer. If you can tackle it, you'll be literally god

>>9615270
because we are on 4chan

So is this the homework help thread?

Because I need some help with a statistics question.

I have to use the maximum likelihood rule to determine which classification and object belongs too.

There are two classes of objects, and each object has two measured features.

The question asks you to use the "Naïve Bayes classifier" which is no problem, but all of the examples I have for applying this have the constraint:

Objects are equally likely to be in class 1 or class 2 and are normally distributed.

For this question though, objects are still normally distributed, but objects in class 1 are twice as likely to appear as class 2.

my set of data looks something like:

class 1:
object 1: x=3.4 y=2.3
object 2: x=4.5 y=2.1
...
object 10 ...

class 2:
object 1: x=2.4 y=1.3
object 2: x=3.5 y=1.1
...
object 10 ...

where both sets are the same size. My question is just what do I need to adjust in my calculation to reflect the increased likelihood of being in class 1 vs class 2 if the data sets are the same size?

>>9616030
your reddit spacing and retarded way of explaining things makes it hard to understand what you want

I have never done anything like this before so you should have clarified some more

>Objects are equally likely to be in class 1 or class 2
>My question is just what do I need to adjust in my calculation to reflect the increased likelihood
my guess is you simply multiply with the probability

what exactly is the formula you use? I dont know what naive bayes means. so when you have a new object, to guess its class you compare the probability that it belongs tto class 1 or class 2. now if twice as many appear in class 2, you simply add the respective probability multipliers 1/3 for class 1 and 2/3 for class 2 and then pick the one with the higher probability for maximum likelihood

when both are equally likely both are multiplied by 1/2 and since it is the same you can ignore this factor

>>9616077
> class 1 are twice as likely to appear as class 2
read that wrong
>ultipliers 1/3 for class 1 and 2/3 for class 2
should obviously be switched, 2/3 for class 1 and 1/3 for class 2

Ok lads, I wanna be ready for the start of my PhD next September on algebraic number theory/geometry. Recommend me books or a certain progression to be absolutely prepared for anything by the time I get there. Assume I'll work all day starting June.

>>9616030
For Naive Bayes with Gaussian class conditional you need

a) priors for all classes given by >>9616079
b) Estimate the class conditional for each class by estimating the mean of the feature vectors. Since its Naive Bayer you don't need to estimate a full covariance for each class but only the variances for each feature in each class

so [math]dx[/math] is a purely symbolic representation and not an actual division of differentials?

Why does this work then?
[math]\dfrac{df}{dg} \times \dfrac{dg}{dx} = \dfrac{df}{dx} [/math]

Why does this also appear to work?
[math]f(x) = y[/math]
[math]f(x+dx) = y+dy[/math]
[math]f(x+dx) - y = dy[/math]
[math]f(x+dx) - f(x) = dy[/math]
[math]\dfrac{f(x+dx) - f(x)}{dx} = \dfrac{dy}{dx}[/math]

>>9616309
You need to think of it as an arbitrary change in value locally then all that makes sense

>>9616309
https://en.m.wikipedia.org/wiki/Chain_rule
https://en.m.wikipedia.org/wiki/Taylor%27s_theorem (first order)

>>9616309
you can turn it into an actual division if you work in non-standard analysis
ultrapowers are neat

how hard would it be to learn multivariable calc on my own? ive just finished integration at uni.

>>9616079
>>9616142
>>9616142

Yeah that was pretty much it, I missed on the formula that the calculation was:

P(X|C1) P(C1) where C1 obviously reflects the probability of that class, so it will just not be 0.5 this time but 2/3.

So I have another question now. Is it valid if I get back a number greater than 1 from the Gaussian function in pic related. I've tried examples where I know the correct answer and the example matches so I'm not entering anything wrong, then I've tried numbers where I calculated the mean and variance (which I am certain are correct as I've calculated them in two different ways using the calculator and excel) and in one case I get a number greater than 1.