/sci/ - Science & Math

What's it like doing research in mathematics?
Do you still solve problems from textbooks sometimes?
How do you plan your research, and the ideas to use in your research?
Do you find it fun?

Nobody proved this is smooth yet.
Over 100 years.
1 million bucks on the line and a millennium prize.

>>12238015
I don't think Genzen's proof (involving ϵ0) requires any set theoretical axiom. You can discuss order types in theories of arithmetics without ever invoking ∈ or rules about it.
Actually, in my head ordinals aren't even a set theoretical notion. The total order notion as such is less loaded with axioms than ∈, so it's primary or at least orthogonal to it. Basically we could speak about any binary predicate and ask about a collection of totally orders w.r.t. it.
But I guess you can (or even should?) formally define your terms to have the word "ordinal" reserved for transitive _sets_.

I asked in the other thread but no one answered. Let [math]W[/math] be a set. Let [math] A \subset \mathcal{P}(W) [/math]. If A is closed under differences and contains [math]W[/math], is A necessarily a [math]\sigma[/math]-algebra? I proved that it is an algebra but I don't know how to prove it is a [math]\sigma[/math]-algebra if it is one.

>>12238784
Do your own homework.

[math]ABCD[/math] is a square of side [math]a[/math] and [math]A^{\prime}B^{\prime}C^{\prime}D^{\prime}[/math] is a similarly situated concentric square of side [math]b[/math]. If [math]a=\left( 1+\sqrt{2} \right) b[/math], prove that any general point [math]P[/math] of [math]AB[/math] is one vertex of a quadrangle [math]PQRS[/math] such that [math]P,Q,R,S[/math] lie respectively on [math]AB, BC, CD, DA[/math] and [math]PQ, QR, RS, SP[/math] pass respectively through [math]B^{\prime},C^{\prime},D^{\prime},A^{\prime}[/math].

>>12238767
Can you not post something so ugly in /mg/ please, thank you very much

Say that I have a function [math]f(x,y)[/math] from some spaces into positive reals. When can I conclude that [eqn]\min_{x \in X} \max_{y \in Y} \log f(x,y) = \log \min_{x \in X} \max_{y \in Y} f(x,y)[/eqn] for some convex sets [math]X,Y[/math]? Do I need to assume anything about f?

>>12238750
I work in mathematical physics.
>What's it like doing research in mathematics?
You kinda just think about shit and then write it up.
>Do you still solve problems from textbooks sometimes?
Sure, when I want to learn something new or when I need to remind myself how some basic concept that I learned in undergrad but already forgot works.
>How do you plan your research, and the ideas to use in your research?
I just kinda wing it
>Do you find it fun?
Why else would I be doing it? It's not like it pays well.

Real dumb question,

for vectors x, a
If [math] \lim_{x \to a} f(x) = L [/math]
then is
[math] L= f(a) [/math]

>>12238670
>what do we mean by "infinitesmally small" in differentials?
>>inb4 limits
>anything else?

Mostly just limits, you can also interpret it in terms of smooth infinitesimal analysis or non-standard analysis and it'll give valid results if you're careful but most mathematicians won't understand it.

>>12238875
No, just choose any discontinuous function f.

What is /mg/'s favorite axiom system

>>12238855
Cool, thanks for the answer! Did you find it difficult to get your research position?

>>12238945
I'm still a postdoc so not very difficult, the difficult part comes next.

>>12238943
ZFC is the highest IQ one.

>>12238784
kill yourself nigger

>>12238750
>What's it like doing research in mathematics?
Ecstatic or extremely depressing, depending on how it is proceeding. Basically, you just read a lot of related stuff (or seemingly unrelated if you want to find a connection) and try to fit that into the framework of what you are doing yourself. Or that's how it goes with me.
>Do you still solve problems from textbooks sometimes?
Yes. If there is something new I need to learn quickly, the textbook problems are good for intuition or for the actual problems I need to face.
>How do you plan your research, and the ideas to use in your research?
I don't. I just do stuff.
>Do you find it fun?
At the moment yes, a week ago no.

>>12238943
> axiom system

You know it's not the 30's anymore, right? You can throw predicate calculus in the bin of outdated foundational systems and embrace the Curry-Howard correspondence, maybe? Like, instead of positing a list of 10 or so random axioms, maybe root your logic in a principled approach that enjoys good proof-theoretical properties?

>>12238943
A better known Set Theory poster got triggered recently by the appearance of this paper
https://arxiv.org/abs/2010.03664
which formulates a spooky theory of (something like) functions and it interprets ZF. They call it fluents. It's somehow objects that have all terms in the domain of discourse (let's call it V) as domain and return stuff. And there's one (the "0") mapping everything to 0 and one (identity) that maps everything to itself. And generally, all terms x have x(x)=x.

https://mathoverflow.net/questions/373700/is-the-theory-flow-actually-consistent
I'm not interested in the framework, but I'll follow what comes from this episode

>>12238980
Cope.

Is this still accurate?

>>12239088
No it has been debunked on Snopes.

What are some of the topics in advanced probabilty theory?

>>12238839
COPE.
So, solve the problem and make it more beautiful.

>>12239129
odds of losing virginity relative to time spent on 4chan

>>12239154
Positive correlation between odds of getting laid and increased time spent on 4chan. Next question.

>>12239154
>>12239166
Not a virgin

>>12239129
I might add that I'm currently taking an intermediate course in probability (so it's called) and another one in stochastic processes.
First one deals with the transformation theorem of random variables, some "advanced" conditioning, probabilty/moment generating functions and characteristic functions, order statistics, distributions with a random parameter, sums with a random parameter, branching processes, multivariate normal distribution, and a shit ton on convergence of all sorts

How do I integrate [math]\sqrt[x]{x}[/math] from 0 to any finite n?

>>12238789
This is not homework. I'm solving practice problems for an exam. I would appreciate some help as I've been stuck on this for quite a while.

>>12239267
complex analysis

>>12239046
Seems cool I'll probably give it a read

Is the space of all degree-2 complex polynomials in two variables [math]ax^2+bxy+cy^2+dx+ey+f[/math] for real integer coefficients [math]a,b,c,d,e,f[/math] completely solved or are there any remaining questions about it?

>>12239267
https://www.wolframalpha.com/input/?i=integrate+x%5E%281%2Fx%29

>>12239288
cool, report back.
I only skimmed it.

>>12239299
one of the remaining questions is whether the space of all degree-2 complex polynomials in two variables with real integer coefficients is completely solved or whether there are any remaining questions about it

>>12239327

reposting this here cuz why not
what do we mean by "infinitesmally small" in differentials?
>inb4 limits, epsilon
anything else?

reposting this here cuz why not
what do we mean by "infinitesmally small" in differentials?
>inb4 limits, epsilon
anything else?

>>12238784
No it's not. This is a trivially easy problem and you must be low IQ to ask this.
Take W= {1,2,3,4...} and A be the the set of subsets of W that are either finite or cofinite.
W is obviously not a sigma algebra because it doesn't contain the even naturals.

>>12239327
Based.

>>12238799
but it isn't true

>>12239320
Got any pics of your sisters bare soles?

>>12239392
proof?

>>12239358
>>12238885

>>12239414
Yes.

I finally did it /mg/
after 5 years of forcing myself to sit down and work problems for 8 hours a day, I'm finally starting to enjoy math

too bad I got kicked out my PHD program

bros... i didn't do maths today...

>>12239374
Yeah you're right I should've seen that. I actually thought about that as a counterexample but for some reason I wrote it off that it wouldn't work.

>>12239451
Took me three years, but it was worth it. Good job, anon-sama uwu

>mediocre grades
>no relationships with professors
>no research experience
>parents starting to ask me when I'm applying to grad school

How do I explain this to them, that just going to an "elite" school for undergrad doesn't mean shit for grad school applications if you don't perform well or get involved with math beyond the classroom?

I think Maths, for me, is a cope. I'll just go back to just semen slurping.

>>12239600
pics?

>>12239598
By telling them exactly what you just told us?

>>12239672
but i did that and they think as long as my GREs are good I'll be fine

>>12239675
Sucks being American, huh.

>>12239685
Yes though I'm not sure what you mean in this context

>>12239088
No, a non mathematician made it and it shows. It is stupid.

I'm trying to find the base-3 representation of 21,127 by hand but I think I've done something wrong. I've got 221,222,111 but wolfram alpha tells me I've fucked up and got the wrong number.
The thing is, I've manually subtracted 6,691 from 21,127 three times and it gives a negative number so I know that it has to be a nine digit number and not a ten digit number. But even 222,222,222 in base 3 is smaller than 21,127 according to wolfram.
Is Wolfram Alpha wrong?

>>12239810
Anon, 3^8 isn't 6,691. It's 6,561. Furthermore, 3^9 is 19,683, which is also less than 21,127.
Try manually calculating each power of 3 again, making sure you don't fuck up the addition this time.

>>12239810
Why would you do that by hand?

>>12239833
Seconded, why the fuck would anybody do that?

is this book good, just bought it

>>12239875
It's not great but you can manage.

>>12239875
Depends on what you're looking for.

>>12239875
I fon't get the book buying thing. Doesn't your university give you the eBook versions for free? And why would you want a book in the first place? Don't you get your materials from your profs?

>>12239906
i just want a basic understanding of calc rn
prob gonna look at more advanced stuff later after i finish the book

>>12239833
The first exercise in my textbook is to find the Base-2, Base-3, Base-8, Base-10, Base-16 and Base-36 representations of 21,127.
I'm not good at math past basic algebra and I was suggested to start here.
I managed to find the Base-2 representation with no problems and used Wolfram Alpha to confirm that I got the correct answer so I'm pretty happy with it so far.
>>12239817
Thanks, I've since tried that and got the correct answer which is 1,001,222,111.

>>12239906
A simple introduction to Clifford algebra

>>12239913
i literally cant read ebooks, im one to click away and play games eventually.

>>12239919
Do you want to get better at math or just get better at calculating? Cause if you want to get better at math this won't do shit.

>>12239921
Assuming you're serous, Porteous has a good book.

>>12239987
Thanks, I was being serious just not that anon.

>>12239919
lmao what a shitty textbook

>>12239088
I have done everything above serious math and nothing below it
Seems accurate
t. engineer

>>12240347
Based and engpilled.

>>12239088
>Poly-Dimensional Topology at the bottom, Topology barely in serious math
Does this brainlet think ordinary topology isn't poly-dimensional? Like we just study 1-dimensional spaces?

What is the difference between "Peano arithmetic", "Peano Axioms", "categorical arithmetic", "full second-order arithmetic"?

I understand that Peano axioms, unlike Peano arithmetic, is a theory of second order. There is some proof that if a theory is monomorphic (i.e. categoric), therefore it is complete (syntax-complete, which is proven being the same as semantic-complete for categoric systems).

Which one of the aforementioned systems is the categoric and thus, complete? For which one does the (what order?) incompleteness theorem apply?

>>12239419
Wanna post them here as a prank? haha

>>12238875
if f is continuous at a

This thread is shit

>>12240434
I mean did you google your terms?
Sounds like
https://en.wikipedia.org/wiki/Complete_theory
answers your last question right away.
See also
https://en.wikipedia.org/wiki/Categorical_theory

For a short history, Frege came up with quantifiers etc. in 1879. Around that time, or shortly after, people like Dedekind used this fully formal language to write down math. Around that time, Cantor came up with transfinite ordinals (to deal with function theory) and this was all fully formalized in the decades to come. Logicans like Russel did programs to capture math via logic and all this history.
So regarding the first question, Peanos original formulation was second order, meaning you have not just quantifiers over one domain of discourse, but also over collections of those. See
https://en.wikipedia.org/wiki/Second-order_arithmetic

But if you say Peano axioms, I'd say it's not strictly tied to second-order languages.

If you interpret quantifiers to range over just all there is of their respective sort, then second order logic doesn't have the nice metamathematical properties that first order logic has. E.g. completeness (the syntax is in some sense faithful to the models for a theory - at least for those things that all models share. This is not necessarily related to incompletness theorems or decidability).
But second order PA is categorical.
Meanwhile, PA in first order logic doesn't capture any particular model. If you look at the syntax of PA and prove theorems, you actually never prove statements that are just true for "the first infinite amount" of numbers.
After all numbers "N" we know, one can attach a dense (like Q) collection of infinite elements (like Z). Roughly, there's models like "M=N+Q·Z" and whenever you use first-order logic to prove any statement, it can also be understood to be a statement about this weird model.

>>12240486
Get an image from fractalforums to open the next thread then dipshit

>>12240501
>But second order PA is categorical.
>>Every categorical theory is complete
>>>>PA is complete.

What happened? What did i miss?
You can find easily that >> theorem, for instance: https://modeltheory.fandom.com/wiki/Categoricity

>> theorem
huh?

which notion of "complete" is being discussed here?
https://en.wikipedia.org/wiki/Complete_theory
?

Can someone who was born to parents who didn't beat them please explain what I'm doing wrong?

What does "value" mean here?

>>12240621
>Completeness of axioms system T
>For every relevant proposition (i.e. contains only subjects and predicates already occuring in T).
>For every relevant proposition H in T, either H or ~H follows from T

>>12240637
There are six ways to do this
Division, then multiplication, then addition
Division, then addition, then multiplication
Multiplication then division then addition
Multiplication then addition then division
Addition then division then multiplication
Addition then multiplication then division
You have to guess which order is correct

Okay /mg/, I have a question for you. Do psychoactive drugs like LSD improve your mathematical ability? I've done it a couple times in my life and both times I noted a marked improvement in my skills in the successive weeks.

>>12239374
>W={1,2,3,4,...}
>W ... doesn't contain the even naturals
Huh?

>>12240650
The solution wolfram is stating indicates it wants me to express the division in fraction form first, then perform a bunch of computations to arrive at 10/3

>>12240665
He meant A. Take the even numbers as singletons. Then the union of them is the even numbers. The odd numbers are their complement. Neither are finite.

mathlet that wants to get in to logic and discrete math for a non-math field

do i really have to get that good at calculus to even start with discrete math? can i just jump in to a discrete math text book with basic algebra skills?

So, this right here.

Express 7/3 in fraction form, right? Done. Then multiply that by two?

But why just the numerator?

>>12240702

>>12239088
This image is so retarded

>>12240669
Discrete math and calculus are totally different subjects without much overlap. Calculus deals with continuous functions, while discrete math...I won’t insult your intelligence. Discrete math is easy and can be fun, I say go for it anon

>"mathematical maturity"

>>12240730
What about it?

>mathematical immaturity

>>12239598
why go to grad school if you don't want to go to grad school?

>>12239088
Increased level of abstraction doesn't equal more difficult, multi-variable calculus is harder than a lot of the topics in 'serious math' in my opinion despite being less abstract. And how the fuck is Chaos Theory not in serious math? I didn't do that until my last year.

>>12240730
https://mathvault.ca/math-glossary/#maturity

>>12240729
no no, i get the difference between continuous and discrete math (to the degree a mathlet can...) but i was really wondering why every fucking university math track has like 4 semesters of calculus as a pre-req to any discrete math

>mathematical pedophilia

>>12240807
Someone else is probably better qualified to answer, but it might be because calculus is the first place that most students encounter theorems, proofs, and derivations, so they want you to have that experience before taking discrete. Again, not too sure about that.

If you wanna get into logic and discrete math, it’s fortunately very easy to teach yourself with a book or by using online resources. Start out by learning Boolean algebra, propositional logic, and how logic gates operate. Then just take it from there as your interest grows

>>12240846
i found a pdf of a textbook that looks like it's used by a number of universities and the intro chapter says it only really expects a basic grounding in algebra, so i'll work out of that and watch youtube videos as I need. I'm mostly in it to get to formal language theory and a strong base in mathematical logic

>>12239810
https://www.rapidtables.com/convert/number/base-converter.html

>>12240637
4/6*8+9
2/3*8+9
16/3+9
16/3+27/3
43/3

Does anybody here edit Wikipedia
On the page for LEM it says LEM is equivalent to the law of non contradiction by demorgans laws. But even though lem implies no contradiction, non contradiction doesn't imply lem. In reality if you do demorgans laws on -(P and -P) you would get -P v -(-P). And you can only do double negation elimination if you already have lem, so if you're trying to prove lem that cancellation is invalid

>>12240854
That’s a great idea. I wish you luck in your studies brother

>>12240764
Because a much larger (and higher-paying) variety of jobs are available to someone with a graduate degree in math than to someone with just a bachelor's. Most people in the latter category seem to end up in software engineering or data science.
I like math enough that I want it to be a core part of what I do for a living, but I also highly doubt I have what it takes to be a successful academic. If I apply to grad school it would most likely be for a Master's.

>>12240871
you too, appreciate it

can someone explain to me why the map f: A --> B where A and B are the set of even and odd permutations on a set I_n = {1, 2, ... n}, respectively, defined as f (a) = (1, 2) (a) is a bijection? Injectivity is obvious but why is it surjective?

>>12240910
same person. I'm obviously misunderstanding something, but how can (1 2) * (a) cover all the odd permutations when not every odd permutation has a (1 2) in the products that make up the cycle. e.g. (1 4) (2 3) (5 6) for I_6

>>12240920
a is an even permutation and f(a) is supposed to take you to all odd permutations. If you want f(a) to be an odd thing with no (1 2) factor, just pick some a that starts with (1 2) then they cancel out and you still have an odd number of

Can anyone help a sperg with my midterm?

I need to find the GCD of (34,58), which is 2 (Im not that retarded)

But then I need to do the extended Euclidian algo finding integer s and t such that 2 = (34)s + (58)t, I got the answer (12, -7) but I need to show work. I cant seem to get the math right on the 3rd line going back up, can someone show me how to get to s and t?

58 - 34(1) = 24
34 - 24(1) = 10
24 - 10(2) = 4
10 - 4(2) = 2

>>12239451
>>too bad I got kicked out my PHD program
what was the topic

>>12240949
thank you, i always suspected i was retarded

>>12240958
10 - 4(2) = 2
replace 4 with (24 - 10(2)) and and then 10 with (34-24(1)) and so on

>>12240637
The value is always 0
t. Nietzsche

>>12238739
Is there a book that hold most of the math knowledge discovered to this point? Kinda like a encyclopedia of math.

>>12241028
no. math is way to massive a subject

>>12241028
The internet.

>>12241028
You know encyclopedias aren't just one book right

>>12241039
>to

>>12239088
>Counting is at the top
If you only knew....

>>12241028
The Princeton Companion to Mathematics is the closest thing I can think of, and even it has gaping holes larger then your mom on a lonely night

>>12241088
>and even it has gaping holes larger then your mom on a lonely night
Is it bad that this made me laugh out loud?

>>12241088
Well that is good start I guess. Is there other books I can complement with?

>>12241102
Principa Mathematica

>>12240997

Thanks for this anon, for the next one I would replace 24 and 10 as: (58-34(1)) - (34-24(1))(2)

according to one of online calculators, the numbers I need for the third level are -2 * 24 + 5 * 10, but I cant see how to get them

thanks again

Any good math books where the entire goal of the book is to master proving things with inequalities?

t.Can't into Delta/Epsilon thinking

>>12241129
If you can't into epsilon delta proofs, your issue is probably with quantified logic and not the actual inequalities

>>12241106
Hey, I found this
https://www.amazon.ca/dp/1770857931/ref=cm_sw_r_cp_apa_i_GHNIFbG2SH3XV
Do you anything about it? Is worth looking?

>>12241129
analysis is fake math

>>12241152
>Do you have anything to say*
Shit, and now writing nonsense. Probably will go to sleep soon

>>12241156
What is real math, then?

>>12239046
>fluents
Looks like an ad hoc, informally-specified, bug-ridden, implementation of half of the lambda calculus.

>>12239177
Foundations of Modern Probability - Kallenberg

Very dense, but very nice.

>>12241202
analysis

>>12241102
there a encyclopedia for physics too

>>12241285
name?

>>12240910
because f o f = id
so its bijective and f^-1 = f

Ok /mg/, tell me your favourite trick in Mathematics

>>12241529
2+2=5

>>12241533

>>12241529
Multiplying by 1 and adding 0 in clever ways.

>>12241538
just go addition mod 1 desu

Can somebody recommend a book in statistics that can take me from knowing nothing about statistics to the end of the first year of a statistics degree?

>>12241538
what's the ,5 there after mod 2? Seems unnecessary

>>12241605
Identity element

>>12241623
I see. Well if it says "+ mod 2" and not a generic operation symbol, it's already implied you inherit the structure from addition on N, I'd say.

What's a good textbook for stochastic PDEs? Also, if anyone has taken a stochastic PDE class, how do you think I would fare in a grad level stoch PDE course with some graduate courses in probability and an undergrad course in diffeq?

>>12241597
This is tricky - that would include calculus and linear algebra (and some more basic math) as well. But assuming you already know the math, someone I know learned statistics from nothing using a book called 'All of Statistics' by Wasserman. That book would take you way further than end of year 1 if you study it seriously, but you should also be able to skip chapters and jump back and forth without losing out on anything essential.

Why can't I wrap my head around variations of the power series for e^x?
e.g e^(x - 2)

Anyone else not trust themselves with even the most basic calculations?
>10/2 = 5
>Better check on the calculator anyway

I see Joel David Hamkins (set theorist and guy with the highest MO rating) just started a series on the philosphy of mathematics.
Might be basic, though, but I see he already has one or two talks of his stuff online too.

https://www.youtube.com/channel/UCeMZeXYIhdxnQvZP360uoBg

>>12241785
[math] {\mathrm e}^{x+d} = c\,{\mathrm e}^x [/math]

How to solve this for all real numbers? I saw this on /b/.

You found me a mere child at the University, knowing nothing. I don't know how it happened, but I cannot be satisfied any more with analytic functions and Taylor series ... it happened about a year ago. ... To see the misery of people, to see the torment of life, to wend my way home from a mathematical meeting ... where, shivering in the cold, some women stand waiting in vain for dinner purchased with horror - this is an unbearable sight. It is unbearable, having seen this, to calmly study (in fact to enjoy) science. After that I could not study only mathematics, and I wanted to transfer to the medical school.

- Nikolai Luzin

What was going through his mind bros? I want to find out more about Luzin.

>>12241205
>Looks like an ad hoc, informally-specified, bug-ridden, implementation of half of the lambda calculus.
You sir, are a man of culture.

>>12241205
>bug-ridden
how so?

Any good resource to selfstudy linear algebra and discrete math? Just Khan Academy? I have done calc I, II and III.

Can you prove the fundamental theorem of calculus?

>>12241913
Watch the essence of algebra by 3blue1brown
And get algebra for dummies

>>12238847
I assume you have topologies on X,Y and X,Y are compact. Then you can assume continuity of f, It will suffice.

Hello guys, I want to take the international SAT exam since my grades from the romanian bacalaureat aren't as good as they could've been to enter a nicer college.(we cannot retake the exam, only if we had a bad grade or we didn't pass). Can you tell me what manuals I could use or what type of math is needed?
Thank you in advance.

>>12239875
>spending money on "Calculus for dummies"
If the book serves you by all means use it, but don't waste money on it. Better use it to buy Spivak or something of quality.

>>12241887
Most integrals don't have any simple formula, but you can compute them numerically with standard methods. 1/x! oscillates strongly on the negative half line, so it won't have an integral from -infinity to 0, but it does have one from 0 to +infinity.

>>12241935
I know that, but I don’t know how to get the 0 to infinity integral without shoving it into a calculator then adding a very small amount <1 to account for the rest to infinity

>>12241943
Yeah, there's not really much of a better way to do it than that. If you want rigorous upper and lower bounds you can use [math]0 \lt 1/n! \lt k^{-n}\text{ for }n \gt k[/math] to get [math]0 \lt \int_k^{\infty} 1/n! dn \lt \int_k^{\infty} 1/k^{-n} dn = \frac{k^{-k}}{ln(k)} [/math] to handle the tails.

>>12241028
Well for some stuff there's Mathematics: It's Methods, Content and Meaning

>>12241913
I've been going through Friedberg's Linear Algebra and it's nice so far.
If you done that much calc maybe stuff like Hubbard&Hubbard would be more appealling, but I haven't read that one

>>12241887
I like this question.

Seems finite, but I wasn't able to simplify it.
I looked for formulas for 1/Gamm(x+1), but didn't see anything useful. Didn't try the reflection formula, but this might just make it more complicated. I also didn't see a regularization alla
1/Gamm(x+1) = g(x) + series
with g integrable in 0 to inf.
Those were my ideas.

>>12242024
>Seems finite, but I wasn't able to simplify it.
It caps around 4? I think

>>12238739

getting letters of reccomendation for grad school applications is fucking exhausting.

>>12242027
Scratch that, it caps at 3

>>12242043
Scratch that again, it caps at e. Why does it cap at e?

>>12242027
>>12242043

Hello everyone, I would like some feedback on this real analysis proof.

Let [math]a_n[/math] by a sequence that is convergent to [math]0[/math] and [math]a_n\geq a_{n+1}\geq ...[/math] for any [math] n \in \mathbb{N}[/math].

Prove that

[math]\sum_{n=1}^{\infty}a_{n}(-1)^{n+1}[/math]

converges, by showing that the partial sums [math]s_n[/math] are Cauchy.

So I have that [math]a_n[/math] is decreasing and convergent to [math]0[/math], so it is Cauchy and I can make [math]|a_n-a_m|[/math] arbitrarily small.
Let [math] \epsilon > 0 [/math]. I choose [math]N \in \mathbb{N}[/math] such that for any [math]m,n \geq N[/math]:

[math]|a_{n+1}-a_{n+2}| < \frac{\epsilon}{m-n-1}[/math]
[math]|a_{n+3}-a_{n+4}| < \frac{\epsilon}{m-n-1}[/math]
...
[math]|a_{m-1}-a_{m}| < \frac{\epsilon}{m-n-1}[/math]

Now I have that by triangle inequality

[math]|s_m-s_n| = \bigg| \sum_{j=n+1}^{m}a_j(-1)^{j+1} \bigg| \leq \sum_{j=n+1}^{m-1}|a_j-a_{j+1}| = \epsilon [/math]

So I have shown that the partial sums are Cauchy. What is wrong/inaccurate with this?

>>12242047
Nvm I’m a fucking retard.

>>12242059
Sorry, last equality is actually [math]<\epsilon [/math]

fuck algebra

>>12242063
[math] \int \dfrac{1}{\Gamma(x+1)} \sim \sum_{n} \dfrac{1}{n!} = {\mathrm e}^1 [/math]

btw. I just found
https://en.wikipedia.org/wiki/Frans%C3%A9n%E2%80%93Robinson_constant
(where x is off by 1 w.r.t. this integral)

>>12241923
Sorry, I'm retarded it is always true, it follows from continuity of log, you dont need any assumptions on X and Y.

>>12242032
Most of my professors just offered to write one so I avoided having to ask.

>>12241925
Take the sat but also take the sat math 2 subject test, it's balls easy so you can get a perfect score but barely anyone takes it so it looks really good for colleges

>>12238943
>>12238971
ETCS is the non-brainlet answer

>>12238739
Given f(g(t)) = [...] where g is a curve g:R->C in C and f:C-> C, is it valid to apply the chain rule to find the derivative?

>>12242248
f(g(t)) will decompose into a(t)+ib(t) and you can go from there.
I think the result will be the same as considering g as a vector field R^2->R^2, looking at the Jacobian and it's normal rules, but only be interested in the partial derivative along the t-axis.
I don't think you can envoke the more restricted notion of the complex derivative and it's chain rule (the derivative which amounts to those Jacobians who's matrix fulfills the Riemann equations)

>>12242248
Yep.

>>12241919
i hate that faggot and his 'original visualizations' that distract you more than 'teach' you
i actually thought he was a pajeets

>>12242290
that said, his linear algebra vids are not bads

>>12242059
how do you have these inequalities? you cant bound a_n - a_m by something which depends on m and n.

[math]\bullet[/math] What textbooks, papers or text did you read today?
[math]\bullet[/math] What non-textbooks did you read today?
[math]\bullet[/math] Did you write something today?
[math]\bullet[/math] Did you do some programming today?
[math]\bullet[/math] Did you build something today?
[math]\bullet[/math] Did you plan/clean something today?
[math]\bullet[/math] Did you manage to work off some bureaucracy/paper work today?
[math]\bullet[/math] Did you practiced any skills today? If so, which?
[math]\bullet[/math] Did you do sports/cardio/weightlifting today?
[math]\bullet[/math] What were you eating today?
[math]\bullet[/math] How much sleep did you get yesterday?

>>12241129
Inequalities book by littlewood and Hardy
>>12241145
>>12241156
shut the fuck up faggot

>>12241129
Cauchy-Schwarz masterclass?

Real question, do you math guys feel inadequate compared to physics majors?

>>12242416
No, it's the other way around.

>>12242417
really, I knew one of you had to be the top guy I wasnt sure if it was you guys or the physics guys.

>>12242416
Nope. Only compared to other mathemagicians.

>>12242417
lol

>>12238971
Set theory is absolutely nigger-tier

Fact: the only math job you can get with a bachelor's degree is actuary

I'm personally hoping to just steal a code monkey job from CS brainlets if I don't get funding for grad school

>>12242380
[math]a_n[/math] is decreasing and bounded by [math]a_1[/math] and 0 (by definition) so the differences decrease as [math]N[/math] grows large

>>12242212
Thanks a lot! :)

>>12240434
>>12240501
>>12240621
Well, i found what happened. The original Peano system was stated without the axioms for the recursive definition of addition and multiplication. Whithout this axioms the second order theory is categoric and therefore complete. No surprise the first-order Presburguer arithmetic with addition but without multiplication is also complete. Does it follow that first-order Peano arithmetic again without addition nor multiplication is also complete? I heard that this first order Peano has infinite models and therefore cannot be complete, then how is Presburguer's complete?

>>12242484
I'd say call PA the theory with + and * and then things are clear?
Probably all those theories without * are complete in the sense that all statements can be decided (or at least incompletness can't be prove). I'm guessing here because I don't have my definitions down so much either.

>>12242432
It's nigger repellent.

>>12238943
>axiom system
has been debunked

>>12242554
[citation needed]

>>12242565
>[citation needed]
https://web.math.princeton.edu/~nelson/papers/warn.pdf

do most people that get a pure maths PhD want to work in academia? I have a friend in a top applied maths program who said that basically everyone he knows is looking to return to industry after.

Is geometric group theory a good field to specialize in?

>>12242637
What else is left for us? When you chose mathematics, you have already chosen academia, those who go to the industry are the ones who failed.

>>12242651
Hum, not really, AG and AT are the trendy fields these days, no one really cares about classic abstract algebra or classic analysis anymore outside of a small niche

>>12242637
I mean, what else are you going to do with a pure maths PhD?

>>12242659
I don't really care about what are the most prestigious fields. As long as it's a sufficiently lively field and has reasonably good prospects.

>>12242059
You cannot bound your inequalities by m or n. Also, you never used the fact that [math]a_n[/math] is decreasing. Although your sequence is convergent, there is more information than that, namely that is convergent to 0.

>>12242637
I'm getting a PhD for fun, but I doubt I could use it for anything else than academia.

>>12238739
Nonners I did it, I finally wrote down my first own created equation in applied mathematics

∃: ∀λᵃ/∀λᵇ = J1 = Pᵃ = Sᵃ | ∀β = A> ^ ∀β ᵛ ∀β + ∀λ >B

||

R̾ = ∀β>Is|...|Ip>∀γ (Is = "∀β"{<T̾1ᴀ}, Sᵃ. ∀γ‖...‖∀β. Ip = "∀β"{<T̾2ᴀ})

I used a gilgamesh of wikipedia notations and Instagram font generators so it looks like the bastard mutt son of like 20 disciplines so I will most likely have to clean everything but still feel very proud of popping my math cherry

>>12238971
>not even finitely axiomatizable
Sad!

>>12242743
Do you come from a wealthy background?

The more time you spend on pure logic and set theory, the more you grow dissatisfied and suspicious of everything. A definite step forward is never taken; instead, the doubts and undecidable questions pile up.
Here >>12242570 we have a cautionary tale of an old man who spent his life studying logic and axiomatic systems, in the end he learned nothing and ended up having doubts about natural numbers.

>>12242828
No. I've just never had to pay (essentially) anything for the studies.

>>12242809
Nice, I guess.

>>12242570
Reading this paper was like licking the diarrhea running down an intuitionists leg.

>>12242850
How do you not pay for phd
Are you euro (peeing)

>>12242859
he's the finnish tranny. he lives on goverment gibs and probably autism bux as well.

I have a maths degree and I have never studied abstract algebra, topology or complex analysis

>>12242859
>How do you not pay for phd
They pay me for doing it.
>Are you euro (peeing)
Yes.

>>12242876
False.

>>12242876
Sounds based

>>12242878
Would they pay US immigrants as well
I want gibs

>mfw I realize Pascal's rule is simply the summing thing you do in Pascal's triangle
Such a fucking brainlet, intuition is key people!

>>12238739
Alright, let's see how good /mg/ really is...

Develop a model that describes how a girls emotions change over time. More specifically suppose she is initially in love with you, describe what happens when you have no contact (without any indication of future contact) for a long period of time. How long before she stops loving you? How long before she starts being in love with someone else? If you answer these it will be considered succesful but try to generalise for different contact functions and try to go one step further deacribing loss of interest in the case you are married (and have daily/continuus contact). It goes without saying that the best model is the one with the most accurate predictions for as many situations as possible. Try it for fun and post results/ideas, you don't have to get it right. You can use any math you like

>>12242743
What will you do when you eventually get refused a job everywhere? I think that will be the time I'll finally end myself for real.

What is the connection between graph theory and counting?

>>12242877
You learned high school maths in your degree then? Lmao

>>12242922
he can just get early retirement handouts for autism.

>>12242898
I don't know, probably not. You should check the government's website for example, perhaps. I've never received any other than 450 a month of student support back when I lived in Mongolia, so I'm no expert of any sort.

>>12242922
I don't know. I have no plans. I used to be an incoming goods inspector in a warehouse, so maybe something like that again. It was fun, strong recommendations for a job like that. Sounds boring, but it actually requires using your brain a bit.

>>12242938
How old are you lad?

>>12242943
27.

>>12242955
Are you a bottom

>>12242964
S-sometimes

>>12242966
based

>>12242964
Yes. Why?

this shit any good?
http://www-math.mit.edu/~rstan/papers/comb.pdf

>>12242978
I see Richard P. Stanley... I say it's good

>>12242955
How old will you be when you finish your PhD?

>>12242983
If in schedule, then a bit under 30.

>>12242989
Are you cute

>>12242989
You're finished then

>>12243010
he (no longer) has a penis but do not be fooled.

>>12242849
I both agree and disagree with this.

I wouldn't recommend learning logic, at least not at any dept. On the other hand, the way you formulate this requires to have a good outlook on meaning of life otherwise. Basically, you can say "well that was not worth it either" to pretty much anything.
Nelson was the logic prof of Tao and Tao certainly rose to greater fame. Tao is likely more smart too. But Tao also did nothing interesting imho, despite being famous. He wouldn't even be as famous if he wasn't as extravertedly engaged on various fronts.
So what is the "cautionary tale", really? What I can grant you is that it's more than just a gamble to go into a field where the fruits are all high hanging. The lecture is to choose well or have people around you that can help you choose.
I can't judge how Nelsons life was. He had some influence. His photos have /fa/ aestetic too.

How many PhD students are at this moment sent down to "rigorously" formalize Luries tombs? Will their lifes be well spent? They work hard on making Luries legacy bigger than it now already is. Maybe 5 of them will become well known beyond the specific topic and maybe one or two will eventually win a price and end up with 50 lines in the Times.

>>12242974
lets have sex and spend the holidays toghether <3

tomes

>>12243049
Why is Lolitamoko holding a rifle in pijamas
Why would you take a picture of that

>>12243053
It's like >>12239320, I digged out an old hard drive with a lot of artsy shit

Blessed be the setting sun
https://www.youtube.com/watch?v=rFjcYwU69r8

>>12242849
Correct.
>>12243010
He is cute, yes.

>just assume a normal distribution

>>12242989
You wouldn't happen to be from Oulu, would you?

>>12242416
Honestly I would but since Physics has a huge $$$ barrier into research outside of Theoretical Physics, I think its an acceptable tradeoff

>>12243013
By definition, yes.

>>12243093
Nope, not blessed with the presence of his ghost. Do you like it up there?

>>12243098
It's nice. Biggest complaint is that the weather sucks, but that's Finland as a whole I suppose.

/mg/ finnfag meetup fucking when

perkele saatana juma lauta yksi kaksi kolme

A linear function requires no recognition of the state of a function, but does require memory of the system. For each input, drop the preordained value into the output bucket. To know the increase, you don't need to know the system, but to know the total, you need the rest of the bucket to measure.

For exponentials, both the function and the total require the system, but it is simple still. They cannot drop a value into the bucket but they can transform the bucket each time.

For x^2 functions, it is like layered linear. To go between any distance, you simply need to know one extra piece of data, the directly prior history of the function's increase. Then you add the preordained base addition, and drop in the bucket. For each ^n greater than 2, you need one extra prior term.

How about more complex curves, like sigmoids? You can obviously always just follow the explicit function, and drop into the bucket based on arbitrary inputs. But what if you want the reflexive function, how to transform the bucket for each consecutive action/input? Do you need to know a certain number of derivatives, can you locally approximate with an x^n and use that, or are there other methods?

>>12243106
How many of us are there? Finland always seems underrepresented on 4chan(nel).

Hate = love from the inverse time direction. Murder is revival. God loves satan and Satan hates god, the two are equivalent. No things nonexist and Every thing exists.

>>12243121
Seben :DDDD

>>12243102
At least in the Homo City, the winters sucked a few years ago because it would freeze all the water during the night and melt it during the day, and then you would kill yourself by trying to walk on the streets.

>>12243106
Never.

>Bully: Math this, nerd!
>

>>12243157
He can still punch your nose because he said or not xor

>>12243157
>proof by intimidation

>>12243157
>>12243160
but no. he seems to use disjunction syllogism having rules out the punching

not to destroy your joke but still

>>12238739
>Need to study Stokes Theorem
>Need to study Diff Eq.
>Need to study physics
>Need to learn LaTeX until the end of the year
>Need to git gud at the piano
>Need to study french everyday
>Need to play vidya consistently at night instead of shitposting on 4chan
Holy fuck, anyone else has procrastinated so much that you can't finish in time all the things you wanted this year? I'm completely fucked.

>>12243190
Probably everybody.
If you manage to reach your stretch goals, you're planning is lacking.

>>12243190
Simplement ouvre jusque'un page d'wikipedie, toujours, et lire quand tu veux.

>>12243078
How do you know each other?

>>12243227
Finland is a small place.

>>12238971
>Have bag of infinite marbles
>Pick one out somehow
Yay, I'm doing analysis!

>>12243184
*punches ur nose*

>>12243221
Est-tu fou? Comme c'est utile?

>>12243236
How is this a paradox? Just pick the top one. The same way there can be infinite real numbers and you can still reference the number 3994.2

>>12243239
Immersive pratique. J'utilise pour l'allemande.

>>12243229
Which one of you is GoatseHitler?

Should I?

>>12243221
>>12243239
>>12243247
I haven't studied French in years, but the fact that I can still read and understand this is kind of neat. Maybe I should pick it back up.

>>12243247
J'utilise le Twitter pour pratique le français, c'est très bon, merci pour l'aide.

>>12243190
>need to learn latex by the end of the year
First year math majors at my uni are expected to know latex before their first day of class

>>12243236
>Have bag of infinite marbles
>Pick one out somehow
>Spill bag
>Oh fuck oh shit
>Room is filling up with marbles
>Cry out 'Help me axiom of choice excluded bros!'
>They can't hear me over the bag of infinite marbles spilling out onto the floor
>House bursts apart, Marbles are how invading the US and no wall can stop them
>Hours later the world is beneath a layer of marbles
>Few years What used to be the sun gets sucked into the Marble hell that is now Earth
>Due to heat, mass and fusion, a few moments later the Marble collective collapses into a Black Hole
>Eons later, Aliens show up and analyze the Black Hole to find that its gaining mass faster then what's being put in, breaking Alien Physicists brains
>All because you picked one marble out of a bag of infinite marbles somehow

>>12243260
Omlette du fromage. Sesse bonjour! Grothendieck bien bien.

>have infinite marbles
>buy slingshot and throw them at w*men

>>12243270
based

>>12243043
i was exaggerating a bit for comedic value, clearly nelson is a respectable mathematician
>>12243270
would you throw them at trans women too?

>>12243277
No.

>>12243237
Ouch

>>12243263
>>12243263
Wow, congratulations to you then, want a prize or something?

>>12243277
Depends
I would make them a trans test to prove they are real Trans
They have to name 3 transformers that don't include decepticucks

>>12243290
>want a prize or something?

>>12243292
'ight, pick one

>>12243298
I'll take the monkey with the grimacing smile.

>>12243298
>no pony plushies
Shit stall

>>12243256
Are you cute?

>>12243319
I'm tiddy goth dog

>>12243247
You need a complement for "utilise"
>>12243256
Yes absolutely, why is that a question
>>12243259
>Maybe I should pick it back up.
Yes
>>12243264
Kek

SHUT UP!!!
JUST SHUT THE FUCK UP RETARDS!!!!

>>12243322
Prove it.

>>12243323
>You need a complement for "utilise"
why?

What are the tiddy goth doggie proofs formulations

>>12243323
Best resources to get back into it?

>>12243190
>trying to learn math
>trying to learn piano
>trying to learn japanese language
that's me, and i'm fucking up too bro...

>>12243333
You can start by posting pictures of your bare soles. Also, checked.

set me straight on the Axion of Choice

>>12243340
>trying to learn japanese language
It's one of, if not the hardest language to learn for a native English speaker. You'll never learn it.

>>12243323
Is just that the Leggings look kind of silly
I think I'll settle for fractals
Just as autistic yet more subtle and classy
>>12243342
Flattering, but I have a bf already silly . Only he can see those

new thread
>>12243381

Neu: >>12243380

>>12243392
>>12243385
Which one is better?
This one:
>>12243381

>>12243340
After french I'll focus on japanese, I already know the kanas, 300 kanjis and basic grammar.

>>12243380
>>12243380
>>12243380
>>12243380

Please wipe your feet on the doormat before entering!

How do you know if you're too dumb for math
I'm slightly struggling with Calculus (first time studying it, using Apostol, mostly autodidact) but I'm doing quite well in Linear Algebra

>>12243332
This isn't /int/ so I'm gonna keep it quick. Some verbs need a direct complement or an indirect complement. So <<J'utilise souvent>> doesn't make sense. What do you use often? If we know because of context use a pronom, like <<Je l'utilise souvent>>.
>>12243333
Nice trips. What >>12243342 said
>>12243337
??? No idea desu. Good luck. If you actually want to be able to use it you have to speak it regularly, so make some friends.
>>12243355
Languages aren't that hard.
>>12243358
Oooh those are nice

>>12243444
Trips of truth show that you are fine

>>12243455
Anon you are right. J'utilise is like saying "I use often", rather than "I use it often".