/mg/ - Math general

>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.

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>mathematics
This is not well-defined.

Mathematics is the study of mathematical structures.

>mathematical structures
This is not well-defined.

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

>spammer
?

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

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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.

But it is.

>But it is.
That remains to be shown.

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 is correct. Have fun studying topo!

t.

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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.

>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".

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

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

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

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I'm not a "qê "

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

>bourgeoisie
This is not well-defined.

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

Sorry for misgendering.

>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.

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

>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?

>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.

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>set
Not well-defined.

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".

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>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)?

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

I meant the US National Science Foundation

Yes it is in anarcho-primitivist mathematics.

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?

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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.

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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.

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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?

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>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.

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

>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.

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?

>Implying topologists actually prove their claims

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)

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

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

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

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

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?

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python-networkx

Well-definiteness is not well-defined.

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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.

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

Proj(R[x,y])

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.

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.

>infinitely many elements
it has p+1 elements

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.

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

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.

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.

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.

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?

Thank you.

this. someone recommend something better for linear algebra

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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.

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

hoffman & kunze obviously come on now

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.

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

>mathematics
This is not well-defined.

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

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

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?

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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? ()

Thank you!

>general
Not well defined.

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

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

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

>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.

Try posting about this in the physical threads located at .

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

>because no one understood their research

ftfy

>because they produced no tangible results

ftfy

No one in engineering you mean? Why are you posting about that here?
>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.

Generally it means checking that a relation is functional.

No need to get mad user.

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?

Show any ideal quotient defines a field

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.