/sci/ - Science & Math

First for elliptic PDEs > elliptic curves.

>>10880521
Threadly reminder to work with philosophers

>>10880521
How can I see show that ω1 + 1 is compact with its order topology?

ω1 + 1 = ω1 U {Ω}

>>10880774
it's isomorphic to {1} U {1 - 1/n: n=1,2,...}

What is a "good" definition for a fixed point in a continuous dynamical system.

>>10880752
what a truly lost soul

I'm teaching myself some math, and I recently became interested in infinite series, i'm hoping some anons can answer some questions I have.
What operations can be done on a divergent infinite series which still preserve its information/strucutre? For example, can I take every other number out and create a new divergent series from that and still preserve information? What about re-aranging terms in the series? Is addition in a divergent series commutative? What about associativity, for example:
[math]\frac{1}{1-a} = 1 + a + a^2 + a^3 + a^4 \cdots[/math]
Can I then say that
[math]\frac{1}{1-a} -1 = a + a^2 + a^3 + a^4 \cdots[/math]

>>10881789
Why are you talking about divergent series as if they have a limit? Do you mean convergent?
>What about re-aranging terms in the series? Is addition in a divergent series commutative?
That doesn't even make sense, a divergent series doesn't have a limit.
If you mean convergent then see this.
https://en.wikipedia.org/wiki/Riemann_series_theorem
Your last example makes sense if the series converges, otherwise the right hand side is meaningless in both cases.

I'd suggest looking into the formal definition of a series if you want to understand why they behave the way they do.

>>10881026
it's an orbit

I used to be very into the natural sciences. I was slacking off in math however, and it greatly limited me.
What should I start with if I want to be able to gladly use math as the wonderful tool it is? I've forgotten most of the basics of calculus.
I currently need math to understand/apply/tweak transition matrices, various Markov models, wavelet functions, Fourier transforms, spiking neuron models, Bayesian stats.

>>10880774
isn't it homeomorphic to something like {1/n} union {0} ?

Anyone else fucking hate it when people try to push computer science or programming on you?

>>10881964
I think you might be confusing this set for something else. It cannot be homeomorphic to a countable space because it's uncountable

>>10881967
probably, I thought ω1 is natural numbers. I haven't seen set theory for ages.

>>10880760
>philosophers
You want me to take statistics for your next paper on interracial trans theory?

>>10881966
Not anti-cs just don’t want to deal with their autistic ideas, simple as

>>10881966

I really don't mind CS at all.

Programming isn't bad either. It is useful for computing sequences and drawing pictures.

>>10880752
what a truly enlightened soul

>>10881026
A fixed point is a point which occupies its own orbit, i.e., for a dynamical system f_t(x) it is a point y such that for any t, f_t(y) = y.

>>10881789
I think what you're interested in are the notions of "formal power series" and "generating functions." We don't really discuss convergence or divergence of these, well, we sort of do, but the idea is that as long as it converges on some open interval then it's useful to act like it's the same as what it converges to. In here are key notions of complex analysis as well.
So the basic answer is, well, sort of, but you have to be specific about what sort of structure you want to retain. If it's an algebraic, symbolic structure you're good. But if you want to talk too much about convergence you can get dumb bullshit like the sum of the naturals is -1/12.

>>10881966
>>10882087
>>10882334
Learn to code.

>>10881962
Sounds like you should just start with linear algebra. No need for calculus.
There's a post in the previous thread where I recommend Strang's OCW lectures, go take a look at that post.

>>10880774
Of course >>10881964 is wrong but maybe you can pop it into a closed interval as a closed set? I forget exactly how one constructs the first uncountable ordinal.
Another way might be a similar argument to the 1/n one, where you cover the point at the very end and hence cover a ton of other shit along with it. But I can't say I know the full argument.

what some /mg/ approved YouTube channels?

>>10882672
Ssethtzeentach.

>>10882672
https://www.youtube.com/user/XylyXylyX/videos

>>10881966
I get you not wanting to specialize in it, but in today's world if you can't program you are pretty retarded. You should be able to utilize the tools at your disposal

>>10882414
lol at you brainlet
>>10882411
eat my dick

>>10882411
Already learning sweaty
>>10882414
Strang is trash, read a real book on lin alg or don’t bother at all.

>why yes, I`ve already pre-ordered PDEs with applications to Super Mario 64, how did you know?

Is Baby Rudin a meme?
I want it for reference, I already did Real Analysis at uni

>>10882967
People who can tell me about the quality of international editions of his books also particularly welcomed

>>10880774
>>10882422 hit the nail on the head.
There is actually a connection with the [math]\{1/n, n > 0\}\cup \{0\}[/math] in that [math]\omega+1[/math] is homeomorphic to this subset of the real line (and order isomorphic to it for the reverse order on the reals).
Now the problem is to generalize the proof of this statement to larger ordinals. The general statement we get by induction is that all successor ordinals are compact.
The finite case is dumb, so we limit ourselves to infinite successor ordinals.
The case of [math]\omega + 1[/math] follows from the above isomorphism.
Now, let [math]\alpha > \omega+1[/math] be a successor ordinal and assume that all ordinals less than [math]\alpha[/math] are compact.
Then, by definition, we may write [math]\alpha = \beta+1 = \beta \sqcup \{\beta\}[/math] for some ordinal [math]\beta[/math].
If [math]\beta[/math] is a successor ordinal, then it is compact by induction and therefore so is [math]\alpha[/math].
Hence, we may assume that [math]\beta[/math] is a limit ordinal. Let [math](U_i)_{i \in I}[/math] be an open cover of [math]\alpha[/math]. There is some [math]i \in I[/math] such that [math]\beta \in U_i[/math].
By definition of the order topology, there is some [math]\gamma < \beta[/math] such that [math](\gamma, \beta] \subset U_i[/math]. Then, we get [math]\alpha \setminus U_i \subseteq \gamma+1[/math].
Since [math]\beta[/math] is a limit ordinal, we have [math]\gamma + 1 < \beta < \alpha[/math], hence by induction [math]\gamma + 1[/math] is compact, hence we may find a finite subset [math]J \subset I[/math] that covers [math]\gamma+1[/math] (and therefore [math]\alpha \setminus U_i[/math]).
Finally, we get [math]\alpha = U_i \cup \bigcup_{j \in J} U_j[/math], with J finite, which completes the proof.

>>10882672
FlammableMaths.

>>10882718
Of course Strang isn't very good, and ideally everyone would read Halmos, Hoffman & Kunze, or Axler. But those don't have lecture videos available, do they? Much easier for many people to learn from lectures.
Actually, I think Axler might have video powerpoint lectures, so maybe those are a better option. Not sure of their quality though.
In any case, Strang is easier for people who are just learning for the first time what real math is like.

>>10883140
Beautiful! All I needed for my idea was the induction. Wow, that's just so satisfying.
I really need to learn more about order topologies.

Need help
Suppose that [math] f [/math] is a continuous function with compact support on [math]\mathbb{R}[/math] and [math] (\varphi)_{n\ge N}[/math]
a series of continuous functions also with compact support on [math] \mathbb{R}[/math] such that [math] \forall n \in \mathbb{N},\: \int_{-\infty}^{\infty}\varphi_n(t)dt = 1[/math]
Consider the function [math]f*\varphi_n, \; x \mapsto \int_{-\infty}^{\infty}\varphi_n(t)f(x-t)dt[/math] show that it converges to [math]f[/math], you can use the fact that f is uniformly continuous
on [math] \mathbb{R}[/math]. Just the approach is enough

>>10883403
converges in what sense? Also, I suspect that you are missing some hypothesis on the functions [math] \phi_n [/math], since otherwise one could simply take the sequence [math] \phi_n [/math] to be constant, in which case you would not get convergence to f

>>10881966
if you think you might ever want to do anything in your career besides academic pencil-and-paper mathematics I suggest you reconsider this attitude

>>10883412
I got a question like this for my quals where \phi_n is qualified by being compactly supported.

>>10883403
You're missing half the hypothesis for the theorem.
Google "Dirac delta function", the proof of whatever you want should be on the wikipedia page.

>>10883412
Sorry, the question was ambiguous so I missed certain properties on [math]\varphi_n [/math]:
[math]\varphi_n \ge 0 [/math] for all [math]n \ge 1[/math] and [math] \forall \delta > 0 [/math] we have [math]\lim_{n \to \infty} \Big( \int_{-\infty}^{-\delta}\varphi_n(t)dt + \int_{\delta}^{\infty}\varphi_n(t)dt \Big) = 0[/math]
we are asked to show that [math] f*\varphi_n[/math] uniformly converges on [math] \mathbb{R}[/math] also note that: [math] f*\varphi_n = \varphi_n*f[/math]
>>10883430
Sorry the concepts are similar but not equivalent

>>10883545
Just use the continuity of f (the definition) to bound its values by [f(x)- epsilon, f(x) + epsilon] around some neighborhood and then take upper and lower bounds on the integral of the convolution, finally decrease the bounds by reducing epsilon.

>>10882967
It's analysis with a measure theory perspective. If real anal was easy and you would like to understand it from a deeper perspective, then it is good. As a reference, what do you want? a bunch of theorems ordered in a convenient way? I can't tell you anything about a non english version.

>>10881966
I love them both, desu I do not see a difference at all. There really isn't a difference, computer science is the supercategory and concerned with the "how", mathematics is the "what" where we actually perform operations/computations on the defined list of symbols.
I used to think that logic > math > computer science and that logic was the 'underlying reality' that mathematics was based on and such, but now I don't think that way anymore, as I've learned more about categories and constructive logics.

>>10883780
>computer science is the super category
>logic is the super category
>philosophy is the super category
>metaphysics is the super category
>history is the super category
how convenient for the people uninterested in math and science

>>10883782
computer science is math so how can someone interested in computer science not be interested in math

>>10883798
>computer science is math
Other way around.

>>10883813
That's the point, math is the "what" in terms of the actual computation/algebra/manipulation "how" of defined symbols and rules. That's why I was saying in the other post that I no longer think of it like I used to. I've started learning more about constructive logics, algebraic logic, Heyting algebra's and category theory and it's greatly affected my perspective on this.

>>10882672
https://www.youtube.com/channel/UClI8OrDeDbsSgbYuZoZYLjw

If I don't want the value of a function to exceed n, do I just write x < n?

>>10881966
>it's 2020
>he still doesn't know how to code
Learn one OOP

>>10884336
F(x)<n

Strictly speaking, computers are counting machines and CPUs contain about 20 or so registers that just count integers within them, and another 4 or so that use integers to represent decimal numbers.
Nearly every major advancement in cs is due to math
I'm a programmer who wish I would have majored in math because it would have made my programming ability that much better

How do I prove that the set of strictly convex real functions defiend on [0,1], with f [math] \leq [/math] g defiend IFF f(x) ]math] \leq [/math] g(x) for all x in [0,1] is a meet semilattice but not a join semilattice? I'm learnign lattice theory on my own through gratzers lattice theory book but this exercise is confusion because it seems wrong.
please smart math anons i need help

>>10884572
For any f, g, we can define h(x)=min{f(x), g(x)}, which is a convex function.
Does this still work for sup?

>>10884672
Is min{f(x), g(x)} defined like here:
http://mathonline.wikidot.com/the-maximum-and-minimum-functions-of-two-functions
I've never seen this type of function before but I think I see how to use it

>>10884692
Yes.
I might have mixed stuff up, tho.

>>10881789
Bit late, hope you still see this

>For example, can I take every other number out and create a new divergent series from that and still preserve information?
No. Consider the sequence

1 + 1 + 1 + 1/4 + 1 + 1/9 + 1 + 1/16 + ...

The odd index numbers are just 1, the even index numbers are reciprocals of squares. Obviously 1 + 1 + 1 + ... diverges, and 1 + 1/4 + 1/9 + ... is pi^2/6.


> What about re-aranging terms in the series?
https://en.wikipedia.org/wiki/Riemann_series_theorem


>Is addition in a divergent series commutative? What about associativity, for example:
Not really, consider these two (same?) sequences

1 + (1 - 1) + (1 - 1) + (1 - 1) + ... = 1 + 0 + 0 + ... = 1
(1 + 1) - (1 + 1) - (1 + 1) - (1 + 1) - ... = 2 - 2 - 2 - 2 - 2 - 2 ... which diverges

Not sure if this is what you're asking for though, because you wanted a divergent series, and this series isn't really divergent or convergent. Also subtraction isn't commutative in the first place, so perhaps this was a bad example. But in general, you can't just sum infinite series in a different order and expect the same result.

>>10882967
Don't use Baby Rudin if it's your first time doing Analysis. I recommend Tao's books, they're free and easily accessible to starting math students. One complaint I have about Rudin is that many of his solutions to his exercises come out of nowhere and are unrelated to the chapter, I think it would only confuse beginners.


>there are people on /mg/ who can't solve pic related

>>10882967
>>10884726
Wait, I didn't see the part where you already took Real Analysis in uni. In that case, I think it would be worth reading through the book just to see how Rudin does his proofs. It would be a decent reference book, and I think you would pick up a bunch of new tricks by working through the problems.

Also, Understanding Analysis by Abbott is very good too.

>>10880521

Here is a problem which I encountered in practice:

Let M be a [math]2^n \times 2^n[/math] matrix which is lower triangular and invertible with entries in GF(2).

In how many ways can you choose [math]2^{n-1}[/math] of the rows such that when treated as a system of linear equations they form a system with 0 degrees of freedom?

The answer will depend on the choice of M. For an example of a matrix for which it is hard to determine this number for, consider the Sierpinski matrix:

[math]S_0 = [1][/math]
[eqn]S_{n+1} = \begin{matrix}
S_{n} & 0 \\
S_{n} & S_{n}
\end{matrix}
[/eqn]

>>10884761

if it's triangular then the rows/columns are linearly independent, and any selection of 2^(n-1) will also be linearly independent. perhaps i'm misunderstanding the problem.

>>10884761

if it's lower triangular and invertible then your rows/columns are all linearly independent because they'll all have a nonzero value along the diagonal, and of course any selection of 2^(n-1) will also be linearly independent. perhaps i'm misunderstanding the problem.

>>10880521
Is there a book on undergrad CA that has lots of exam like questions?

>>10884963
cont.

well, the columns and rows of any invertible matrix are linearly independent

>>10884963
>>10884977
Based not retarded anon.

>>10884984

if you remove the requirement that it be invertible, then you just count the number of non-zero entries on the diagonal to get the row/column rank

>>10884984
not exactly sure what >>10884761 is asking for though. a 4x4 example would be helpful

>>10884984
>>10884963
>>10884977

Here is a picture of two different selections of 8 rows from the 16x16 Sierpinski matrix. White or red pixels represent 1 and black pixels represent 0. The color red signifies that row is in the selection.

Both selections of rows (treated as vectors) span a vector space with the same dimension. But the problem is asking you to treat the rows as a system of linear equations, so the columns correspond to variables and the rows correspond to an equation.

In the top selection, we have 8 equations and 8 variables. In the bottom selection, we have 8 equations and 16 variables.

Thus, the top selection has 0 degrees of freedom and the bottom selection has 8 degrees of freedom.

>>10885063

>>10885063
Jesus Christ, it's even stupider than I thought.

>>10885063
>In the top selection, we have 8 equations and 8 variables.

so you don't consider zero entries as coefficients? or are you just not counting the zeros to the right of the diagonal?

>>10885086

The zero coefficients are not considered to contribute to the degrees of freedom in this case.

That would imply that you could take a given linear system and say that it depends on a bunch of variables which happen to not actually effect the value of any of the equations, thus make the concept of "degrees of freedom" meaningless


>>10885077

Go ahead and answer it then

>>10885094

it depends on the matrix, as you said. you can pick any number of rows with only one nonzero entry. the variables corresponding to those non-zero entries must be set to zero, so you can then consider additional rows which now effectively introduce only one free variable (if there are any), and so forth.

>>10885111
cont.

in other words, pick a row with only one non-zero entry, then zero out that column the entry is on, and repeat.

I want to get a chalkboard for my room, any recommendations?

>>10885125
cont.

so specifically, it depends on the number of zeros below the diagonal. if there are none, you will have no choice of the next row to pick, and you'd have to start with the first, so you must pick the first n rows. more zeros does not necessarily mean more choices though, and fewer zeros does not necessarily mean fewer choices. is the sierpinski matrix a worst case for this?

>>10884692
Just to be clear, that sort of min and max is how you typically define a lattice structure on spaces of real valued functions (or perhaps any kind of function with values in another lattice - not sure about that actually).
It turns out that this "pointwise max/min" is very useful in a lot of places.

>>10885151

The highest number of possible choices is the identity because all choices work

>>10884736
Thanks, I'll take a look.
Do they both (Rudin and Abbott) cover similar material at similar levels?

I'm basically looking for analysis text to help me get to grips with stuff for PhD next year.

Do you (or anyone else who might be reading) have experience with Stein's series?

>>10883768
>As a reference, what do you want? a bunch of theorems ordered in a convenient way?
Yeah, basically. I guess I'm looking to fill in any gaps as I never took any 3rd/4th year analysis (so only really know up to Munkres manifolds).

I don't mean a non-English version - I mean the one with this cover, that's marketed as an 'International Edition'.
I know the physical quality of some international editions, especially ones for the south Asian market, can be dire.

>>10885150
That really doesn't sound healthy if it's where you sleep. Is a whiteboard no good?

>>10885301
Stein and Shakarchi are delightful books. Rudin is fine but if you're looking to prep for PhD I'd go with S&S. Alternatively, you could check out Pugh if what you want is problems that will twist your brain and teach you how to think like a mathematician. But S&S is a well rounded book and the whole series is just exquisite.
I would recommend against Abbott, that guy probably thought you didn't understand analysis the first time through. Abbott is great for first time learners, it's not a reference.
And of course Rudin will always work as a reference. It's just so bland and flavorless, and the later chapters are shit, after chapter 7. No reason to learn measure theory from rudin.

>>10860247
I am having a hard time understanding how to prove trigonometric identities.
For example, I don't understand:
[math]
cos(x-y) - cos(x+y) = 2sin(x)sin(y)
[/math]
I know that I can use difference identities to turn cos(x-y) to cos(x)cos(y) - sin(x)sin(y). But I don't know where to go from there.
I've read my university's books, but they're McGraw Hill. I tried a few youtube videos over the last few weeks, and websites... but I have my exam next week and while I'll pass the class, I don't want to leave this class without understanding this topic. I checked out 'purplemath' and all that on the front page of my search engine.
So, can anyone either explain it or send me in the right direction?

i need some ideas for innovative concepts in graph theory (research) any ideas?

>>10885337
Why yes, I'll just hand over my research projects for you to do, why didn't I think of that?
>>10885332
Just plug in the identity for cos(x+y), it should be in your book.
Also, why did you quote sqt and post here?

>>10883403
Nigga this is not true and you should be ashamed for believing that.

>>10885332
Well you have a formula for cos(x-y) (which is actually cos(x)cos(y) _+_ sin(x)sin(y)), from there you deduce a formula for cos(x+y), substract the two and Bob's your uncle.
Now the real interesting question is how you prove that formula for cos(x+y).
You can find a nice geometric proof of this starting from pic related and using some projections and the pythagorean theorem

>>10885368
>Also, why did you quote sqt and post here?
I meant to actually post in there. I figured it was a stupid question.
>>10885393
pic related is where I'm lost now. I thought I understood it, but now I don't know.

>>10885430
You're fucking up the signs.
Make sure to find the identities on your book and copy them down again. Then do it again, carefully.

>>10885430

do you not see that the bold terms are negatives of each other hence add to zero

>>10885436
Oh wow. Thank you for pointing that out. I had swapped the sum/diff identities for cos/sin when I was writing my flashcards.

>>10885306
Even if I don't keep it near my bed? If it's not healthy, I could get a whiteboard. I just prefer chalk.

I like messing around with simple math stuff in my free time. I don't really know the notation or vocabulary beyond like Algebra 2, so I don't know what I should be using to describe a lot of stuff. And I kind of came up with an interesting little puzzle that I wanted to share. So please bear with me while I try to explain the problem. My background is programming so I might use some terminology from there for lack of a better option.
So let's say there's this function [math]p[/math] that takes a repeating decimal and returns how many digits repeat. For example, [math]\frac{1}{81} = 0.\overline{0123456789}[/math], so [math]p(x) = 9[/math]. (WolframAlpha refers to this as the period so that's why I'm calling it p here)
[math]p[/math] does not count any inital digits in the decimal that don't repeat. For example, [math]\frac{1}{36} = 0.02\overline{77}[/math], [math] p(1/36) = 1[/math], since it doesn't care about the initial 02, only the 7's that repeat every 1 digit.
What I found is that for all values [math]x > 0[/math] and [math]n > 1[/math],
[math]\frac{p(\frac{1}{3^{xn}})}{p(\frac{1}{3^{x(n-x)}})} = 3^x[/math], and for some reason, this seems to only work for perfect squares of 3. So I kind of rearrange it and get [math]\log_{y}(\frac{p(\frac{1}{y^{xn}})}{p(\frac{1}{y^{x(n-x)}})}) = x [/math] if [math]x > 0, n > 1, [/math]and [math]log_{3}(y)[/math] is a positive whole number.
So I wanted to try and understand why that is, if there are any numbers besides [math]3^x[/math] that would work, or if this already has a proof of some kind. Unfortunately after looking around I can't find anything. Also, just for my own personal benefit, I was hoping someone could help me understand how to describe this problem in a way that's friendlier to math conventions.
Also please point out where I made a mistake / missed something obvious, because I'm sure I did. Typing this all out on a phone at work where I've been writing this stuff on some scratch paper.

Is anyone intimidated by math and it holds them back?

>>10885746
There's nothing to be intimidated about, it's just a bunch of ideas that are built on simple rules. If you struggle to understand something, it's because you're lacking some comprehension at a lower level, whether you realize it or not.
What are you intimidated by specifically?

>>10885094
/mg/ has such a consistently shit intuition for lin alg, why is this?
>>10885454
>flashcards
don’t do this

>>10885888
They believed the Axler meme.

Does anyone know the name of the Riemann rearrangement's theorem analogue for Cauchy principal values?
I don't know the exact statement, but it should be something like:
For any function f that has a Cauchy V.P., but isn't in L^1, and any real a, there's a series of characteristic functions, whose sum converges in measure in every bounded set to the real set's characteristic function, such that the sum of the integrals of f times the characteristic functions converges to a.

>>10885888
>/mg/ has such a consistently shit intuition for lin alg, why is this?
pretty sure
>>10885111
answers his question, or at least you can construct a valid selection of 2^(n-1) vectors rather easily. not sure if there's a more efficient procedure for counting the number of possible selections unless you know something about the structure of the matrix.

Hey friends, what's a nice introductory book about differential equations? I'm about to start a course on stochastic DE and we're supposed to leveraged our knowledge of regular DE, but I remember nothing and idk which book to read.

I'm struggling to find any content more difficult than just algebra or calculus 1 material publicly related to physics, anyone got sources?

>>10886319

>>10885746
I feel intimidated, for sure. I'm gonna have to take real analysis soon and I'm sure its gonna be rape.

>>10885932
Funny enough, I liked Treil's Linear Algebra Done Wrong better than Axler.

>>10886363
maybe something not thaat introductory?

>>10886319
Differential Equations George Simmons

hey. I'm an undergrad who doesn't know shit about shit. I'm transferring from a community college to an out of town university this fall, so I don't really have any professors to go to right now, but I think I disproved the Riemann hypothesis. How do I go about getting this checked out by the community to decide if I actually did something cool or if I just missed something?

>>10883140
>The finite case is dumb
laughed

How many of you here are pursuing a Ph.D? What do you expect to do after it? Are professorships still possible to get?

>>10885932
>>10886404
absolutely and overwhelmingly based

Are there any good jobs for a major in maths that doesn't involve research or finance/economics?

High school teaching doesn't count

>>10887018
>How many of you here are pursuing a Ph.D?
I am
>What do you expect to do after it?
I dunno, depends on how my thesis goes. If not research maybe teach (I already have a certification), maybe become a quant (I have connections in several banks), maybe fuck off to the woods idk
>Are professorships still possible to get?
Yes, but you have to be actually good, lucky and willing to put in the time. There are so few spots that even some very good candidates do not get them the first time around (or after the first post doc).
In my country, unless you are Fields-medalist tier, you are not getting a spot until maybe two or three post-docs. For about 30 jobs country-wide (all areas of pure math included), there are just too many people already waiting in line, and the inevitable 4-5 geniuses.

>>10882672
People's Veto

What's an example for when the Newton method (or a variation of it) is used in practice? Bonus points for needing to solve algebraic problems?

>>10883780
>computer science is the supercategory and concerned with the "how", mathematics is the "what" where we actually perform operations/computations on the defined list of symbols.
I used to think that logic > math > computer science and that logic was the 'underlying reality' that mathematics was based on and such, but now I don't
It's a simplification to imply that all of computer science is about the how - all the language and type theory is pretty much designing systems and classifying them as if it was topology.

I think the distinctions between the majors are pretty off - even a lot of (theoretical) physicists aren't really distinguishable from mathematicians, it's just that number theories only pass on to a new frameworks after decades while physicists stick to one for fewer years.
For a CS guy using adga to formalize proofs, any math paper in differential geometry is a holey sketchbook and no rigor in sigh - the same way mathematicans would speak about physicist. It's all le same.

>>10887112
>the Newton method
Do you mean Newton Raphson or interpolating polynomials? Or another method?

Newton Raphson gets used a lot everywhere.

>>10887127
Yes, Newton-Raphson.

Can you give me an attractive example where a 3th or higher polynomial comes up that's solved with Newton-Raphson?

>>10883813
lol

>>10887112
Newton is not really used in practice because it does not scale well.

Gradient decent is the most common method used, with various rules for step sizes.

>>10887018
>How many of you here are pursuing a Ph.D?
I finished my PhD in 2018 and have since started part time finance and data science orientated things just to keep busy and try to plug CV gaps.

>What do you expect to do after it?
Well I expected at least some kind of work. I had published good research papers in good journals including a breakthrough that got me invited to talk at several conferences and universities. Unfortunately my adviser did not have the right network. I made friends at conferences, but I wasn't ever let into the inner circles of other networks because of the money involved in the field. My university has a great ranking, but it is not known for my field (after my previous adviser left academia for the data consulting meme I was the only one still working in the field in the whole department).

In any case after applying to around 20 or so postdocs without success I started applying to hundreds of industry positions. I figured that my coding experience (not related to my PhD, I've been coding as a hobby since I was 12, have a really active GitHub profile and even some paid freelance experience) might get me hired in a starting software position. I would not be shocked if I wrote over 1000 cover letters custom tailored. I never got a single interview.

>Are professorships still possible to get?
In theory yes. Statistically 5 in every 100 PhD candidates should make it. I don't know how good those 5 are though. But I can guarantee you they have h-indices of at least 20, but probably double that, by the time they graduate.


It's gloomy, but many of my friends made it. Some got really good positions at banks with high pay. I'm not sure how they landed those initial internships, as I said never got to the interview round for even after resorting to paying agencies to help taylor my CV and cover letters. Another friend of mine who dropped out is now thoroughly established in the ML job market etc. No one I know is still in academia though.

>>10887131
Why would you solve a third degree polynomial with Newton-Raphson?

>>10887152
But gradient decent finds the minimum, which is only the zero of a linear equation, no?

>>10887183
Or higher.

>>10887194
>But gradient decent finds the minimum
Yes.

>, which is only the zero of a linear equation, no?
No it isn't, the minimum of a linear equation is unbounded.


All equation solvers are really just optimisation routines in practice, you turn an optimisation problem into a root finding problem by simply taking the absolute value of any function. The only exception is systems of linear equations which is usually solved with decomposition methods or even good old Gaussian elimination.

>>10887177
Where do you live?
I did my PhD in physics in the Netherlands and it was a shitshow when it comes to results and papers but from about 6 jobs I ever applied to in my life, I got 4. Working for big-tech for an American company now (Netherland office).
For what it's worth, some of the jobs I did were 1 month internship (while you can do something else too - even if it's private projects in your free time) which always end up in them hiring you anyway - you already know the stuff and the people know you. So that might be considered a "strategy".

>>10887201
Okay thanks. Is there some sort of overview of different common algorithms. Like a cheat sheet?

>No it isn't, the minimum of a linear equation is unbounded.
I meant the linear equations Ax-b=0 are framed as minimum of f(v):=vAv-bv

>>10887202
I live in Europe, but I studied in another country. That was a big mistake (the project was ideal for what I wanted to do and I thought it would be an adventure to leave my home country). The university had a good reputation and ranking, I thought it would be ok.

And yeah Europe has a great market for its Physics and Math graduates in general. It seems to be very respected there, they just don't seem to care for any degrees obtained outside the EU unless it's MIT/Caltech/Standford.

As I said the picture isn't all gloomy, it can just happen to some people like me that you fall into some "gotcha" and destroy your career before it even begins. I think one important point is to just apply to the best possible school straight after your Masters. Prestigious unis actually take in a lot of students. Many of my friends got in even without publications and worse grades than I had.

>For what it's worth, some of the jobs I did were 1 month internship
Anon I would kill for a 1 month internship. I've applied to those, but as I've said I've never made the interview state. I wish I could do another PhD in Germany/Netherlands/Belgium, but I don't think anyone in their right mind would give a position to me anymore, even at a lower ranked university.

>>10887213
Sorry to hear, but if you can god and got education, I can't imagine it will take too long to get something sensible. My point with the internships was that after you're at a company for a month and they know you do your job, they are likely to hire you.

I sometimes hear that people with PhD are considered overqualified and not taken for that reason, but till now I didn't encounter that, thankfully

>>10887204
>Is there some sort of overview of different common algorithms.

I don't know any good concise theoretical resources, I think your best bet is this page:
https://scipy-lectures.org/intro/scipy.html#finding-the-roots-of-a-scalar-function

SciPy uses a lot of common routines from many different FORTAN and C++ libraries. Plus it's open source so you can actually see what a modern root finding algorithm looks like under the hood. You'll notice a lot of edge cases etc.

The API of the root function:
https://docs.scipy.org/doc/scipy/reference/generated/scipy.optimize.root.html

Lists a bunch of different methods for root finding

>I meant the linear equations Ax-b=0 are framed as minimum of f(v):=vAv-bv
Ah yes, solving a linear system of equations can be done very efficiently and therefore uses different algorithms from root finding in systems of polynomial equations.

https://docs.scipy.org/doc/scipy-0.14.0/reference/generated/scipy.linalg.solve.html


For a general overview of numerical routines and the different problems you could encounter check out:
https://docs.scipy.org/doc/scipy/reference/optimize.html

>>10887220
Thanks that's nice of you to say Anon.

Do you mind if ask you what your stats were when you got those offers? Number of publications, (rough) h-index etc?

I don't know if I'm considered extremely unqualified, maybe I don't pad my CV with enough minor things like awards, undergrad internships etc. or maybe I'm padding too much. I don't have articles in Nature or anything, but I do have two in the highest impact journal in my field. I don't know if that might be why I never got anything.

>>10887235
Awesome, thanks.

>>10886368
It's just proving stuff from calculus dude, it's fine. Don't worry. It will be alright.

>>10886923
You just missed something.

>>10887245
Maybe the academic credentials aren't that important and it's more about coming across confident, sympathetic and with skills carved out for the position at hand

Has the "get a gf" problem been formulated yet?

>>10887358
uncomputable function

>>10887349
Ah yes for industry especially I absolutely degree. I meant for post docs.

I remember back when I used to have confidence, before I got worn down by the rejection letters...excuse me Anons.

>>10887358
Pretty sure you can solve that with algorithms.

>>10887501
GET OFFF MY FUCKING IMAGE BOARD GET OUT REEEEEEEEEEEEEEEEEEEEEEEE

> a/b(c+d)
Implied multiplication is idiotic. You remove the parentheses after adding c+d.

6/2(5+10)
6/2*15
3*15
45

Anyone who defends implied multiplication is a moron.

>>10887543
No one fucking cares. No one writes division that way in real math unless you're writing a fractional exponent inline. Stop shitting up our thread.

>>10887026
programming

While I was studying complex manifolds, I remember having a small headache. Once I was trying to sleep, my head was blasting, ceaselessly repeating autistic results about sheaves of holomorphic vector fields and deformations and I couldn't fucking sleep, but then I did, and I woke up 2 in the morning, complex manifolds full blast, not even recalling any results, just complex manifolds, and I ended up vomiting.
It gradually subsided, I didn't manage to catch some sleep, I went to classes, and once I got back home I rushed through the remainder of Kodaira's book, and never studied complex manifolds again.
This happened a couple months ago, but I kept forgetting I should tell you lads.

>>10885673

> Also, just for my own personal benefit, I was hoping someone could help me understand how to describe this problem in a way that's friendlier to math conventions.

1. You are choosing to work in base 10. Most mathematicians would prefer binary or perhaps the 2-adics.

2. I don't understand what you accomplish by taking the logarithm base y. Are you trying to generalize your identity by replacing 3 with y? You haven't proven your identity in the first place.

3. Your identity is hard to understand. Why don't you try to actually prove (or atleast explain) what you think the numerator and denominator are? Are they both powers of three at least?

> all x>0 and n>1

What exactly is x allowed to be?

> This only seems to work for perfect squares of 3

The only perfect square of 3 is 9.

Are you saying "I conjecture that the base 10 period function satisfies [that identity] when n is an even power of 3?"

Any tips on taking notes? Most of the time I don't take any, and just do exercises. And that's worked decently due to topics being highly connected and building on top of each other but recently I've had to remind myself some things I hadn't used in a while and I feel like it would've gone much smoother if I had made notes.

>>10883403
I know this post is old but this is the notion you should be looking into
https://en.wikipedia.org/wiki/Summability_kernel

>>10885673
I don't know how much you know about algebra but hopefully you can understand this. Here [math]k[/math] is an integer, for the sake of not dealing with annoying cases, let's say it's coprime with 10. You should be able to google or solve any of these.
- The value [math]p(1/k)[/math] is equal to the smallest positive integer [math]n[/math] such that [math]10^n \equiv 1 \pmod{k}[/math]. In other words, [math]p(1/k)[/math] is the order of 10 in the multiplicative group modulo k.
- So if you know anything from group theory you can instantly infer this: for any k, the value p(1/k) divides [math]\phi(k)[/math] (this is euler totient)
- From here I think it's kind of painful to progress for arbitrary k, so one can consider the case when k is a perfect power of a prime.
- So basically what you're interested (in this case) in is the value [math]p(1/q^j)= [/math] order of 10 modulo [math]q^j[/math]. This can be dealt with by induction. For example I can prove the following theorem: [math]p(1/7^j) = 6*7^{j-1}[/math].

>>10887931
Taking notes in class is a waste of time IMO. It's better to read the book before class to get some exposure, listen to the professor's take on it and ask questions, then read the book again/do exercises to really make it stick. I TeX up my notes at the end of the week once I've had enough time to let the details sink in.

>>10887630
So you're telling me, don't study complex manfiolds?

>>10884336
[math]g(x):= \min(f(x),n-\epsilon)[/math]

>>10887624
is it easy to get a job in that despite only having a major in Phys and Math? I've done programming but not sure if that's enough

>>10888181
You'll be just fine then

>>10888072
Nah, go ahead. Great subject.

Someone please help me, I think I want to completely change paths in life, go back to school, and start majoring in math despite only having experience up to high school algebra 2
Can someone help convince me I'm wrong and that I should keep working at Del Taco, I'm sure I'd be making a mistake

>>10883311
I am convinced he posts on 4chan

>>10887630
>>10888072
>>10888508
Kek, copypasta tier.

>>10885337
Try to prove that all bipartite graphs of girth 5 are 1 factorable

>>10887018
My fiance will be finishing up his clinical psych PhD a year after I finish mine so if I can't get a postdoc after that I can just be his trophy husband.

What is a Hidden Markov Model?

>>10887630
excellent post

>>10889129
Good, then you can post here even more.

>>10889129
That's pretty gay, anon.

What's the state of mathematics in Australia? What are the topics that are the most active in research?

>>10889401
>gay
Why the homophobia?

>>10889549
Bumping for this, I'm curious too.

I'm taking calc 2 next semester. I have less than 5 months to get ready. I haven't taken a math class, except for statistics, in about 13 years. I picked up an algebra review book. I'm scared

>>10889558
>>Why the homophobia
Why so you associate the word "gay" with homophobia? Ya fag.

>>10888540
what do you want to do after the major ? do you want to have a better job or do you want to be a mathematician ? do you have any vague idea what is actual mathematics about ?

>>10883311
Stop dude, we know it's you

>>10889818
I'm just interested in the idea of Number Theory and proofs. Never really cared about what job I have.
From where I'm sitting, mathematics at the college / professional level involves a lot of proofs, which sounds interesting to me. But no, beyond that I'm not sure what actual mathematics is about, besides solving problems and justifying your solutions.
So yeah I'm sure I'm way off but I don't really know any better.

>>10890319
well, it's not that you're way off. it's just that becoming a professional mathematician takes time and effort and requires some serious dedication. it's definitely not something you find yourself doing because "yeah well why not". you're right that there are a lot of proofs, but that's similar to saying that professional fotball involves kicking a ball.
I suggest you try to self-study linear algebra (including proofs) in your spare time and see where it takes you. I don't have any suggestion for a book though.

guise im taking undergrad analysis and algebra at a high tier university and working 20 hours a week, what am i in for.

>>10890486
any other classes you're taking ?

>>10890488
no just those two. and its also the quarter system.

>>10890492
you're fine. algebra is mostly easy if you're not a total brainlet, analysis can be pain in the ass, but it's not like you need to study 8 hours each day for it.

>>10890502
just for reference of how much of a brainlet i am, i got a b in combinatorics thanks to a stupid jew teacher where a 63 in the class was curved up to a b and i got an a in graph theory and complex anal. i guess im being a weenie i just need to take these classes and stfu.

Requesting methods for proving the irreductibility of a polynomial in the rationals that is not either of the Gauss's lemmas nor the Eisenstein criterion nor the fact that if [tex]a\in R[\tex] is a unit of a unital, conmutative, asociative ring [tex]R[\tex] with [tex]b\in R[\tex] then [tex]p(x)[\tex] is reductible iff [tex]p(ax+b)[\tex] is too

Requesting methods for proving the irreductibility of a polynomial in the rationals that is not either of the Gauss's lemmas nor the Eisenstein criterion nor the fact that if [math]a\in R[/math] is a unit of a unital, conmutative, asociative ring [math]R[/math] with [math]b\in R[/math] then [math]p(x)[/math] is reductible iff [math]p(ax+b)[/math] is too

>>10890580
Did you try multiplying it so that the polynomial is in Z, and using exhaustion plus geometric arguments?

>>10880774
UωU

>>10890593
ugh, are there any irreductibility methods non-mentioned that aren't geometrical?

>>10880521

When is it okay to treat dy over dx as a fraction? Considering how that is done in solving homogenous differential equations, it has to be okay most of the time, right? Also, what area of mathematics covers this question, is it in galois theory or abstract algebra?

>>10890995
All the time, this is a basic result of mathematical physics.

>>10890995
all of the time

real analysis+analysis on manifolds, differential geometry

>>10890995
you've probably heard that "dx" represents "an INFINITESIMAL displacement" in physics, so this is a thing in maths about infinitesimals (a.k.a non-standard analysis)

N=n-g*w
Rearrange the variables to find g

Shouldn't it be: g=N+n/-w?
Wtf

>>10890580
What do you need them for? Because for all exam problems and such, usually those suffice. Plus maybe rational roots theorem for degree 2 and 3 polys.

>>10891141
no. check your signs.

>>10891265
I was being dumb and was getting rid of -g by multiplying -1/-1 instead of -1/1. Thanks

>>10891233
because i think there should be other methods

>>10891265
>he used a name for a shitpost and forgot to remove it when posting again
Relatable.

>>10891342
That's silly. If one method suffices for all practical cases, why would you need more?

>>10891342
There are others, like using tower law in field theory, you can also try checking in math overflow, I think there was a post about it last time I also googled irreducibility criteria.

Can you self-learn math and become a mathematician that way?

>>10882679
true and good

>>10889549
bump

>>10891852
I don't know of any examples of this in the last 100 years. One stabling block would simply be to not have a real reason to send in works to math journals and have them published, and being a mathematician (in that sense) also generally means working on problems that that are of interest to contemporaries.
So apart from math being tedious to learn without mentors, few put polished works online and nobody self taught publishes correct work on hot topics

If a textbook has prefaces to a series of editions, do you read them chronologically or as printed (reverse order)?

>>10891852
>>10892589

You can learn all of undergraduate and a lot of graduate level maths at the very least

>>10892603
The thing is no one would take you seriously if you 1) don't have a math degree 2) haven't published.

>>10892608
>at the very least
as in what you said is true although if one were inclined to just learn the mathematics taught to most math undergrads it is entirely feasible though the amount of effort and stress involved in self teaching that much material might dissuade many otherwise motivated and bright people. you can’t really do research in any stem subject without access to the correct peers, equipment, funding and mentors

>>10892603
I'm sure a smart guy can easily learn graduate math too, but a mathematician is someone who publishes math papers. If a theoretical physicist learns enough math to enable him to easily learn new post-doc mathematics, this doesn't make him a "mathematician". There's more to the profession than math knowledge. Being an actor doesn't just mean you can act a role from scratch well, you also need to do films and be involved with various professionals in the fields, and know what directors are like and how a movie set functions.
People who are self taught barely think "hey let's now sit here at home and put ours of work into writing a paper for the lolz". Some people who are not mathematicians still actually do that, but it's - if I scan my mind for examples - never on topics that other mathematicians are interested in and you're not in their community.

>>10892593
Don't read them

How do I learn proofs? I just picked up a calculus book and I can understand everything but when I get to the exercises I pretty much get stuck. So I think about it for a few minutes and then look at the solution. Should I just struggle for like 15 minutes or will this come naturally as I do more and more. I feel like a brainlet

>>10892709
Try to follow how the author does their proofs.

>>10892589
>to not have a real reason to send in works
Self-taught autist here. I've stumbled across a couple problems or conjectures that I ended up taking a few days to solve on my own, but I've never bothered to try to derive further results that could be useful or write anything of the like that could be published. Largely because I don't think it's interesting, and because I don't want to bother people by making them check if my work is correct.

>>10892758
What are the most advanced topics you've learned? How long have you been self-studying math?

>>10884410
OOP is POO
learn haskell

>>10892763
Symplectic topology, Riemannian geometry, C*-Algebras, ergodic theory.
Around 2 years now. I don't have to do well in tests, so I can just go through a subject, feel like I've understood it well enough and move on. I do return relatively often, tho.
If I spontaneously come up with a problem, it's usually topology.

>>10892709

imo, it's often helpful to consider why something must be true and then formalize that intuition. doesn't always help but it might save you some time. a lot of proofs in introductory books don't usually require a big leap from the basic axioms and definitions of whatever topic they cover, for example the triangle law for analysis, axler's definition of a matrix of linear transformation, basic group/field properties, etc. if you get really stumped then it can't hurt to look for an errata list, as exercises seem to be where authors make the most mistakes.

>>10892589
Kurt Heegner is an example. Whose work is now the basis for a lot of results related to BSD.

>>10889129
You flamming faggot. I hope you and your faggot boyfriend both die from aids.

CompSci/Programming has so many comfy YT-videos of only sort off technical topics (Defcon, Blackhat, randos describing their projects, etc.). Any similar stuff but for maths (except for Numberphile)? Most math stuff seems to be college lecture-esque, but often I just want to be comfy with some maths.

ok I am mad
what the fuck drives people to propose stuff like this
how brainwashed do you have to be

what is a commutative diagram used for

>>10889129
you'll definitely get a good postdoc but may have a 2 body problem ......

>>10889129
Sounds fun! Honestly just a win-win for you. Good luck with that degree!

>>10893734

to illustrate identities involving composition of functions

i guess it's time to learn CT for real
lost it at vertical/horizontal composition

>>10893816
wtf

>>10889549
Bump

>>10893789
Thank you.

AAAAAAAAAAAAAA
I'LL HAVE TO TAKE FUCKING ANALYTIC GEOMETRY SINCE THEY ADDED IT OUT OF NOWHERE TO THE CURRICULUM AND AREN'T OFFERING A TEST TO DODGE IT
WHAT THE FUCK

can I put equal signs inside a function or do I have to list them individually to not be confusing

Will paypal $15 if you can prove this, but I don't think it's true and there's probably an easy counterexample:

Suppose,
[math]f(x)\leq \int^x_0 g(z,f(z))\;dz[/math]
Where [math]f,g>0[/math], [math]g[/math] is decreasing in the first argument and increasing and concave in the second. And the derivative [math]g_2(x,y)[/math] is bounded above by some constant whenever [math]x>0[/math].

Show that [math]f(x)=0[/math] for any [math]x>0[/math].

>>10880521
What are the uses of imaginary numbers?
I only learned about them in high school as i^2 = -1

Like that was it, high school math here is "just grind problems and put in the forumla"

>>10894449
The complex numbers are the algebraic closure of the reals. This means they are the "smallest" field (https://en.wikipedia.org/wiki/Field_(mathematics)) which contains the reals where every polynomial has roots. They were first motivated by the fact that the polynomial [math]x^2+1[/math] had no roots in the reals. This property of algebraic closure makes lots of different theory much easier, for example describing solutions to polynomials in algebraic geometry. They also share a lot of other "nice" properties with the reals (inner products, metrics, completeness, etc).
In many cases the algebraic results we can prove with complex numbers can then tell us things we want to know about the reals. As a simple example if a polynomial's roots in the complex numbers are complex, then it has no factorisation in the reals. There are real functions that are easier to integrate using stuff frmo complex analysis.
Complex numbers are also interesting as an object of study in their own right however. They can be used to describe physical phenomena in areas like quantum mechanics for example.

>>10894471
> They were first motivated by the fact that the polynomial [math]x^2+1[/math] had no roots in the reals.
Actually, the original motivation was in finding the roots of cubic polynomials (Cardano, ca 1545).

If you have a cubic polynomial with real coefficients and three real roots (the /casus irreducibilis/), the expression for the roots has intermediate terms which are complex.

In the case where you only have one real root, the two complex roots involve intermediate terms which are complex. But that's to be expected. The sole real root can be found using only real arithmetic.

Briefly, each root is the sum of the cube roots of the two roots of a quadratic polynomial. If the quadratic has two real roots, then each has a real cube root and their sum is real. If the quadratic has complex roots, then they form a conjugate pair (a+bi and a-bi for some a,b). Their cube roots are complex and form three conjugate pairs. The sum of a conjugate pair is always real ((a+bi)+(a-bi)=2a), leading to three real roots for the cubic polynomial.

>tfw have to start TAing soon
How do you anons make it through the slog?

>>10894578
>If you have a cubic polynomial with real coefficients and three real roots (the /casus irreducibilis/), the expression for the roots has intermediate terms which are complex.
Neat.

>>10894585
TAing is fun. It pays really well here too.

>>10894293
g(x,y) = -x + ln(y+1)
f(x) = x

>>10894293
let g be constant

>>10891852
>>10892589
>>10892603
>>10892647
>>10892608
>The thing is no one would take you seriously if you 1) don't have a math degree 2) haven't published.

This. Especially 1. I studied engineering because I got a bursary for it and I wouldn't have been able to afford going to university without it. I've always wanted to be a mathematician and so I had self studied math and took extra math courses whenever I could and I eventually published an article with an important result in a pure math journal (as the sole author, it was related to previous applied work).

However, because I don't have a math degree, no professor would like twice at my gradschool applications. In one case this was despite the fact that I published in the exact same journal he published his only 2 papers in that year, but whatever. Math professors are extremely snobbish and have an artificial protection of their profession, much like medicine.

So if you want to break in you NEED a math degree and to kiss some old man's ass for 3 years

The irony of it all is that a year later my paper has a couple of citations while his papers are largely ignored.

>>10881966
thank god I only had two programmings in my undergrad

>>10882672
3blue1brown

>>10880521
I'm starting college soon. Is there any good way to judge how good I'll be at upper level maths based on my high school performance?

>>10894293
How easily can Harvard/MIT undergrads solve this?

Uh this is my first time writing something outside of an intro to proofs course. Can I cut down the size of a proof by removing an almost obvious lemma and just assuming the truth of the statement and leaving it up to the reader to prove? My bad if this question makes no sense.

>>10894926
meme channel
>>10894928
no

>>10894959
are you writing a homework? then usually no.
are you writing a paper? then usually yes.

>>10894968
The latter, thanks.

Math peeps, thoughts?

I found this on a LaTeX stack exchange. Does anybody know what this signifying/signified thing is or what book this pic is from?

>>10894968
I'm writing a post-postmodernish kind of novel.

>>10895035
Cold reasoning and social skills work in entirely different ways? No shit really

>>10895039
>Does anybody know what this signifying/signified thing is
this signified and signifier terminology is part of the foundation in semiotics. i dont really get what the indermediate vertices are supposed to represent, but of course we are lacking context and im not an expert in semiotics either.

>>10889549
bump

>>10889549
>>10895052
ok since nobody is answering:

1) you can just look up universities and look through the staff, usually all professors have their research field listed
2) australia is not that small, so there will be a decent variety of fields
3) I don't think the precise field of mathematics you are working on is very important, for most people it just sort of develops organically (at least that is how it was for me and most other people I know)

That said, the only group I personally know of in Australia that is very strong is Shparlinskis group at university of new south wales i think. He's mostly famous for publishing an incredible amount of papers on exponential sums, but he also does LFSRs, general finite fields stuff and additive combinatorics.

>>10895039
The image is a fully connected neural network
i.e. a framework for (deep) machine learning in its simplest form.
On the left you have the input and on the right a processed output

Signifier / signified are terms from semiotics in philosphy of language, at least I associtate it with Lacan (a 60's Frenchy post-Freudian phychologist)

https://en.wikipedia.org/wiki/Neural_network
https://en.wikipedia.org/wiki/Sign_(semiotics)

>>10895091
I knew the semiotics thing but I didn't understand what it had to do with math. ty anon

What's go go with wolfram's notes on maths equations and expressions?
Particularly between distant objects that my have relevance?

Like for argument, the periodic table and a sting.
Is it being expressed in the maths as a whole.
Or as a segment is expressed but not purely just notation is existent?
Asking for a friend.

>>10895099
np
https://youtu.be/aircAruvnKk
https://youtu.be/z2aq21lMw40

>>10895101
??

I'm doing a physics problem where I end up having to antidifferentiate this expression: [math]x'=dx'/dx[/math]
The left side is easy enough, the antiderivative is simply 1 and the interval is 0 to x. But what do I do with the right side?

>>10895184
We set y=x'.
Then y=dy/dx, and y=ae^x, for some number a.
Making sense of whatever the fuck he's doing is left as an exercise for the reader.

>>10895114
Well a no souce figure equation. If it were calculated to exteame. And wolframalpha said.at given points that one is a 183 element table. And at a extreme it is a sting.
That'd be because of sheer numbers expressed. But not anything wort looking in to further, right?
Like there's no way that bastardised chi square explaining a region from sting to the stable periodic table in our reality.
Surely if that were to occur. It's physicists who designed the calculator taking the piss?

Because I'm not aware of how those two points are defined or referenced for.
But the equation used that prompted these both, was some what aimed that way.
But no where near as ready as thought to get a 1/^nth hat far.
However the argument used isn't inherently wrong from my knowledge.
But I was using wolfram for obvious case. I'm not that trained.
Proofs usually visible in my neck of the woods.

The table was interesting.to find,
But to get to a string?
That was so fucked off possibility it may have became fucked in.
But I never had to proofs, such things. Where would toy start.

I am out of town and I don't have my laptop but I want to calculate a table of a complicated binomial coefficient sum:

Table[Sum[(Binomial[2^(n - 1), k]*Binomial[k, 2^(n - 1) - k]), {k, 2^(n - 2), 2^(n - 1)}],5]

Wolfram alpha refuses to do what I want it to. Any wolfram Chads know what's wrong with my "code"?

hey guys i'm finishing hs how do I keep improving at maths without doing a degree in it?

>>10895407
get off /sci/, lurk on stackexchange and refer to the guide on which textbooks to go through. it's very important to get a strong background in algebra (not hs-level, but abstract algebra, particularly a good understanding of group theory as studying structure) since many things end up using it eventually. then an intro to analysis, pugh's real mathematical analysis is good for the exercises. keep in mind that post-hs maths is pretty much nothing like hs maths, daniel velleman's book how to prove it is a great exercise-led guide to how to approach and write proofs. much of the discussion here and online around mathematics is heavily skewed towards pure maths, but look into applied stuff too - things like game theory and statistics can be fun, you get to play around a lot more with computational models too.

there are also excellent series of lectures online, so whatever topic you're studying (say, galois theory) in the textbook, it always helps to look it up and watch a few lectures. if you have the advantage of a library, once you go through 2-3 textbooks you'll have an idea of what kind of style of mathematical writing you like, so you'll be able to pick out a few books on a topic, flick through a chapter and see which ones suit you.

>>10880521
who here has actually read through Any of Hormander's Analysis of Linear Partial Differential Operators I - IV?

>>10895456
>get off /sci/
Based advice.
>>10895467
>any of
I've read half of the first volume to review distributions. Good shit.
The remainder is in the backlog. Said backlog being:
Regularization>Elliptic PDEs>Hyperbolic PDEs>Parabolic PDEs>That other set on nonlinear fun anal, except the one on mathematical physics>Hormander.
Unless it changes, I can come back to you in a year.

>>10894867
Forgot to mention that lim g(•,y) =0 as y->0, but this example doesn't even satisfy g>0

>>10894585
I love it honestly. I do tutorial 4 hours a week, it’s a nice distraction. The prof does the actual teaching and writes the problem sets; all I have to do is select a few problems from the set each week, make a recap of the lecture and assign problems to the students, then have them solve them or do it myself.
It only takes about an hour of preparation out of my week (to read the material and do the problems), and I like interacting with the students.

how do i learn category theory

>>10895622
Don't. Analysis > abstract faggotry.

>>10880521
Any books that:
1) Are discursive in style
2) Great motivating definitions and showing their limitations (such as failures of the riemman integral)
3) Don't sacrifice mathematical rigour

>>10895645
Most books by Godement are exactly like this. I suggest his Analyse Mathématique I-IV (I think it was translated in English), as well as his Introduction to the Theory of Lie Groups.
You can also try anything by Hatcher, Milnor, Dieudonné, (Michael) Artin, Eisenbud, Shafarevich...

>>10895765
Thanks!

>>10895765
Any other authors like Godement?

>>10896343
Yukarifag.
Most authors of grad texts tbqhwyf, I'm still confused about why you're asking for something so common. Unless you've been going through lecture notes and advanced texts in mathematics, in which case you've completely fucked up.

>>10896360
Not true at all in my experience...

>>10895645
Noncommutative Geometry by Alain Connes. If you have any trouble with that you're a lost cause

>>10896413
Whatever! Just forget it!

Can I get a job doing type theory? I have to imagine there are tech companies that are doing programming languages research and shit, right?

>>10896764
Lol

>>10896764
Yes, become a fucking researcher. Even FaceBook has opened a lab on PLT / Type theory.

Anyone have any amusing examples of hideous commutative diagrams?
I've been referencing a book on Hopf algebras lately and it always makes me chuckle when I flip a page and half the next page is a gorillion arrows.

>find out my professor has written a book on the subject
>it's pretty good, skips out the bloat, has nice examples
>it also has small mistakes out the ass
Amazing.
>>10896921
I've never read cubical homotopy theory, but based on the premise it's probably absolutely horrifying.

>>10896764
Learn to code.

>>10896891
Can you get a position somewhere like that without being well known / well connected?

>>10896949
I know how2code. I don't know how2get cool job.

>>10896921

>>10895467
>partial differential equations
Try >>>/lgbt/

>>10897379
name a serious branch of math that doesn't build on linear pde.

>>10897384
algebraic topology

>>10897395
What is the Atiyah-Singer index theorem?

Whenever you compute cohomology of a Riemannian manifold, you are computing spaces of solutions to an elliptic linear pde.

>>10897384
redpill me on this? what the fuck do differential equations have to do with anything? to me it's just that lame class I had to take with engineers and memorize a bunch of shit

>>10897737
>>10897737
you aren't a mathematician, you are an undergrad or at best a second year grad student who has seen absolutely no deep mathematics. all of geometry and analysis is founded on things which change as a function of themselves.

>>10897834
How much of a faggot do you have to be to instantly go on the offensive even though I clearly expressed that I don't know much about it?

>>10897737
Not that guy and not a geometer but here are two examples that I recall from my studies.
In many situations that arise in geometry, you might want to construct diffeomorphisms with certain specific properties (say you want to move shit around, enlarge a hole, reduce a hole, push stuff down, etc.)
A simple and neat way to do that is to build a vector field that "indicates how to do it" and then use the theory of flows (a direct application of the Cauchy-Lipschitz/Picard-Lindelof theorem) to construct a diffeomorphism that follows the vector field.
Another completely different example comes from complex analysis. Now this is not going to come as a surprise since holomorphy is defined by a system of PDEs (the Cauchy-Riemann equations), but the existence of sections of vector bundles (eg. functions, differential forms etc.) on Riemann surfaces follows from general results on PDE. Therefore most results about vector bundles (especially Serre duality and Riemann-Roch) rest on analytic results (Fredholm theory and elliptic PDE iirc).
These existence results are crucial because many geometric arguments and constructions about Riemann surfaces rely on the existence of meromorphic functions or meromorphic differentials for example.

>>10897737
how the fuck do you think any process in nature is modelled ? has it never occured to you that

F = ma

is just a differential equation ?

/lit/ mathlet here
What the fuck did he mean by this?

How do I get better at analysis proofs? Is reading The Cauchy-Schwarz Master Class good? Any tips?

>>10897406
>the virgin mathematician that uses Atiyah-Singer to calculate the analytical index
>the chad shitposter that uses Atiyah-Singer to calculate the topological index

>>10897737
>what the fuck do differential equations have to do with anything?
Aside from being the most important object that seems to describe the real world they have applications nearly every where in mathematics.

The only real answer to that question is "everything".

>>10889129
>gay
>being in a relationship with someone who has a degree in bring mentally ill (unsurprising given the fiance)
>speeging out, trying to impress undergraduates on 4channel
Are these the risks of getting a math PhD? Am I missing anything?

>>10895546
Not him, but any tips?
Gonna be a TA next semester for Linear algebra..

>>10894959
If it's homework just write "it is obvious that ...", that also works if you have no idea why it is true.

>>10897834
>not answering the question
>>10898056
>answering with physics
>>10898234
>also not answering the question

SEETHING analysts.

>>10889129
positively based

>>10898139
kek

But Atiyah was hopeful the index theorem had applications to even the homotopy groups of spheres. For example, the index mod Z of the Dirac operator on a Spin manifold with boundary is a Spin-cobordism invariant. Evaluating this on all Spin-cobordism classes gives a map to Q/Z which reproduces the image of the J-homomorphism.

>>10898060
go back

>>10898246
I was also TAing for linear algebra. Honestly if you are comfortable with the material and enjoy what you are doing, I don’t think you can really go wrong.
As I said, I would usually start by asking if there were any questions about the previous lecture. Usually there wouldn’t be any, but of course it does not mean they understood everything so I would then expand a bit on the lecture (give a recap, review some examples, give some intuition etc.).
After that I would assign problems to be solved by the end of the session.
The hard part really is the timing and knowing what to do during idle time. I had two sessions of two hours on the same material. I would usually spend somewhere between 15 and 30 minutes on the recap, then the rest doing problems.
During that problem-solving time, I never really knew what to do. I am a bit shy so I do not really like to go snooping among the students, but I think I am going to start doing that next semester.
It gives you something to do and it allows for more interaction with the students (which makes things less boring for everyone). Plus, you have an easier time telling when students have solved a certain problem and therefore when to give a solution and move to the next one.
But these are the sorts of things that you will figure out as you go. Don’t sweat it, just go in and try to show them some interesting math, which begins by showing them that you find it interesting.

>>10897982
>most results about vector bundles (especially Serre duality and Riemann-Roch) rest on analytic results (Fredholm theory and elliptic PDE iirc)
It's the other way around really. Riemann-Roch doesn't require any anal beyond [math]\overline{\paritla}[/math]-Dolbeault diff alg's, while it can say something about the rank of holomorphic sections. Things like existence and index problems in anal can be tackled generally with alg top/geo.
An extreme case is Gromov-Witten, where point-counting in a Schubert variety tells us the rank of the [math]D[/math]-module of some exactly solvable DE, including KdV and NLSE.

>>10897384
Mate nobody cares about your engineering courses. Go try looking for cock on >>>/lgbt/

>>10898538
Why the homophobia?

>>10880752
You need tenure and two algebraist's wives

>>10898281
>also not answering the question
It's like you asking "what is adding for", there is no answer that could satisfy someone who knows nothing about adding.

>>10898281
what's wrong with answering with physics ?

>>10898571
First and foremost he needs to have sex, just like me.

>>10898571
Relax, I already have those.

>>10898454
Thanks for the answer and encouragement. Recaps seem like a good idea to structure the class, filling the time meaningfully is what I am most concerned about, but you are right that probably is something which you can get through experience.

How does one prove this

>>10898665
Match units on both sides like a monkey.

>>10898670
So you find x and y using the end result?

Did that already and got
y = 2
x = -2

Hey guys know any good YouTube channels that cover music maths up through college level shit? Possibly beyond too?

>>10898685
Basic* maths sorry typo

>>10898680
You messed up matching like monkey, should have x=-1/2 and k=1/2.

>>10898685
3blue1brown

>>10898685
wtf is "music math"??

>>10898504
Hey you.
If I were trying to proof the circumstance in how matter came to exist.
What check points would I need to hit as validation of the equation?

I'm asking for your answer to this. Specifically you.

As it's seemingly is that you're the only person here, or most places to have an actual grasp of physics.
I may have hit on the answer to this, I'm getting things in solve, but not more than a few key items.
I have to know which ducks and in what order they should be found.
That way I can get the logic in order. And I'm sure you have a better sub atomic to macro interaction list in mind now, than I'd have in the time spend imploded into a organic web crawler.
I need the sum root of each increment property. Not a property that has a later or over complex (larger particle/functions out of place for causality) equation to explain it.
But do note it in the list should you feel that the concept is right, but the maths is fuzzy for it.

All I get else where is name calling or dumbfukardness.
Can't say what it is that's got this potential. Other than a logical uncertainty. If you're really clicked on as I think you might.
You may even take it and run. I've been saying it for a while. Just thought the uncertainty would be that easy to isolate.
I ran it, and it was the one. I thought there had to be more nuance. But it's in the orders of operation that that's gained.

Hope you can give a hand. Won't go forgotten.

>>10898504
>>10899745
yukaricuck and the schizo: a match made in heaven

>>10899807
Two people whom share no qualm to posing things yet disproven, or shame if challenging contrived fact as fallacy?
And when give cause for pause, they have an answer in counter.
Why is this a issue?
Do you think that progression in knowledge has a master theist?
A single man or group. Whom hold absolute power as to what is allowed to be questioned, or thought about?

I can't remember such a thing existing apart from some place called the Vatican.

>>10899934
>why is this an issue?
it's not. the issue is that these two people wont leave us the fuck alone.

>>10899745
yikes

>>10900108
So instead of challenging facts, and striving to discover. Ignorance is what you wish to live in?
Why are you here, rather then some meme aggregate or tutor service?