/mg/ math general -- bernoulli numbers edition

[Anonymous No.9442329]

talk maths fagets

[Anonymous No.9442397]

[Anonymous No.9442420]

Why is an integral sign used as a symbol for a sumation? Also the notation is not well defined.

[Anonymous No.9442426]

>>9442420
>Why is an integral sign used as a symbol for a sumation?
Can you give an example? I haven't seen it used that way.

[Anonymous No.9442428]

>>9442426
The fucking OP...

[Anonymous No.9442435]

>>9442428
>The fucking OP...
Do you need to swear?

[Anonymous No.9442437]

>>9442420
go ask bernoulli

[Anonymous No.9442438]

>>9442420
What could that giant "S" possibly stand for?

[Anonymous No.9442473]

>>9442426
>Why is a large S used as the symbol for summation
The absolute state of American education.

[Anonymous No.9442503]

>>9442329
I dislike that every textbook hides the relation between sums and the power rule.
Whereas in this picture it is obvious

[Anonymous No.9442590]

>>9442329
Is it possible to survive half a year of Galois theory with acceptable grades if you struggle a lot even with Linear algebra proofs?

[Anonymous No.9442650]

>>9442590
Why would you want to ?

[Anonymous No.9442653]

>>9442650
I'll have to pass that course if I decide to continue studying maths.

[Anonymous No.9442654]

>>9442590
>Galois theory
This is a maths thread.

[Anonymous No.9442661]

>>9442653
Well you don't really need to know a lot to understand Galois theory but it's going to be a rough ride if you struggle with linear algebra

[Anonymous No.9442666]

>>9442661
yeah that's what I'm talking about
They more or less allow us to choose what courses we want to take from the 5th semester, but I'm not sure if I can make it through 3 more, let alone pass the upcoming exams

[Anonymous No.9442710]

>>9442590
Maybe if you change your study habits. How did you study for linear algebra?

[Anonymous No.9442726]

>>9442710
during the semester? Not much so I can't really say. It sounds like I'm a failure, and maybe I am, but I'm just not used to working.
Currently studying, Two weeks before the test. Although not very efficient.
Basically doing all the homework we were given over again, even though I haven't really done the last few sheets, so just doing homework. Before that I took some time to write down all the relevant theorems and definitions. Wanted to do the previous exams next.

[Anonymous No.9442733]

>>9442590
galois theory relies on linear algebra

[Anonymous No.9442736]

>>9442733
>>9442654
>>9442661
it's not just that specific subject / course, I was just wondering in general - about how, and if I'm even able to continue with maths.

[Anonymous No.9442832]

>>9442329
> fagets
Why the homophobia?

[Anonymous No.9442839]

>>9442832
Cuz they are dumb fucks who cant do math and deserve to die

[Anonymous No.9443360]

[math]\mathbb{F}_1[/math] for everyone
Oliver Lorscheid
https://arxiv.org/pdf/1801.05337.pdf

>This text serves as an introduction to [math]\mathbb{F}_1[/math]-geometry for the general mathematician. We explain the initial motivations for [math]\mathbb{F}_1[/math]-geometry in detail, provide an overview of the different approaches to [math]\mathbb{F}_1[/math] and describe the main achievements of the field.

[Anonymous No.9443361]

>>9442736
linear algebra is fundamental. you're supposed to struggle with it, if it's taught right. it means you're learning. so carry on, and work hard

>>9442832
>look mom I did it again hahahahahaha

[Anonymous No.9443373]

Let A_1, ..., A_m be subsets of {1, ..., n} such that for all i, j, A_i is not a subset of A_j.

Show that
[math]\sum_{i=1}^m \frac{1}{\binom{n}{|A_i|}} \leq 1[/math]

Any idea ?
I think I could prove it combinatorially when the sets all have the same size, but that would be ugly and incomplete. There should be a clever solution.

[Anonymous No.9443377]

Do you guys know about inverse problems?

[Anonymous No.9443380]

>>9443360
Every approach to this subject feels wrong.

[Anonymous No.9443439]

How do I prove that, if x and y are real numbers, then |xy| ≤ |x||y|?

[Anonymous No.9443440]

>>9443439
>How do I prove that, if x and y are real numbers, then |xy| ≤ |x||y|?
Use the fact that |xy|=|x||y|.

[Anonymous No.9443444]

>>9443440
how do I prove that without doing "for x>0, |x| = x and for y<0, |y| = -y and then |x||y| = xy..." etc
sorry, I'm a literal brainlet :(

[Anonymous No.9443453]

>>9443444
you don't, that's exactly how you have to do it

[Anonymous No.9443454]

>>9443444
that's a definition

[Anonymous No.9443676]

>>9443444
Define the positive and negative parts of x to be x+ = (x+|x|)/2 and x- = (|x|-x)/2, so that x = x+ - x- and |x| = x+ + x-. Show case-by-case that either x+ = |x| and x- = 0 or x- = |x| and x+ = 0. Then expand out xy = (x+ - x-)(y+ - y-) = x+y+ - x-y+ - x+y- + x-y-. Out of these four different combinations, only one can be nonzero, and will then equal |x||y|.

[Anonymous No.9444176]

Has anyone here read Gunnar Carlsson's paper on Topological Data Analysis? I am looking to figure out the prerequisites from algebraic topology.

https://math.stackexchange.com/questions/2574902/prerequisites-for-gunnar-carlssons-topology-and-data

[Anonymous No.9444190]

Hey guys, brainlet here checking in. Soft question: how important is this crap? I'm taking my intro proofwriting class and although I'm not really struggling with it or anything, basic symbolic logic manipulation just seems trivial in comparison to the real math proofs I see. I know you have to crawl before you walk, but I need to know, is this babby-tier symbolic logic stuff a meme?

[Anonymous No.9444196]

>>9442438
>>9442473
It's not a "giant S" but a large ſ.
https://en.m.wikipedia.org/wiki/Long_s

[Anonymous No.9444200]

>>9444190
Yes and no.
Very low likelihood you will ever see symbolic logic notation again unless you seek out a logic textbook.
But manipulating propositions in that manner will occur in literally every theoretical course you take so it's important to internalize the results themselves.

[Anonymous No.9444213]

>>9444190
Boolean algebra is kind of useless, the real deal will be when you get to natural deduction / sequent calculus. Then you'll define what a proof actually is.

[Anonymous No.9444269]

How can i model rough jagged manfiolds evolving into frsgmented pieces and recombining in a rigorous way? I need a tool to describe this for properly modeling turbulent systems.

[Anonymous No.9444306]

>>9442329
I still find it amazing that the odd bernoulli numbers turn out to be zero.
Also, who the fuck decided B_1 might be convenient as +1/2?

[Anonymous No.9444340]

>>9444200
>Yes and no.
What did he mean by this?

[Anonymous No.9444344]

>>9444190
>I'm taking my intro proofwriting class
Which school for brainlets do you go to?

[Anonymous No.9444355]

>>9444340
It is a contradiction. It means zů can introduce a negation and cancel the assumption leading to yes and no. Zů will learn this stuff on zįr introductory course on logic, assuming zů hasn't already done it in hs.

[Anonymous No.9444356]

>>9444355
>zů

[Anonymous No.9444357]

>>9444356
Yes, and?

[Anonymous No.9444360]

>>9444357
Why does zįr have zůcchini on xhr face?

[Anonymous No.9444364]

>>9444360
I don't discuss such matters with """""people""""" for pronouns are too hard to use. And you are one of them. Literally vomit inducing. (Vomit = food^*)

[Anonymous No.9444369]

>>9444364
For whom*, your retardation of mental sort is contagious.

[Anonymous No.9444373]

>>9444364
>>9444369
Remark: poo = food_*

[Anonymous No.9444385]

>>9444355
>It is a contradiction.
Which he proved with no assumptions.

[Anonymous No.9444389]

>>9444269
What you want to do only works for [math]n[/math]-manifolds with [math]n\geq5[/math].

[Anonymous No.9444398]

>>9444364
>>>/tumblr/

[Anonymous No.9444451]

>>9443373
[math]
\frac{1}{\binom{n}{|A_i|}} = \frac{|A_i|!(n-|A_i|)!}{n!}
[/math]
So, you want to show that [math] \sum\limits_{i=1}^{m} |A_i|!(n-|A_i|)!\leq n! [/math] (1) .
By noticing that [math] |A_i^{c}|=n-|A_i| [/math], (1) becomes [math] \sum\limits_{i=1}^{m} |A_i|!|A_i^c|!\leq n! [/math]

Maybe you can get it from there, dunno.

[Anonymous No.9444542]

>>9444190
You are going to be using these stuff literally in every proof. Sometimes they are going to be trivial, so you won't be needing to think about them symbolically, but you'll eventually find yourself checking logical stuff symbolically so that you are sure you don't fuck up.
For example:
A function is not continuous at [math] x_0 [/math] means:
[math]
\neg (\forall \varepsilon >0 : \exists \delta >0 : \forall x : (|x-x_0|<\delta \implies |f(x)-f(x_0)|<\varepsilon))
[/math]

which is logically equivalent to
[math]
\exists \varepsilon >0 : \forall \delta >0 : \exists x : \neg\{ \neg [ |x-x_0|<\delta \land \neg (|f(x)-f(x_0)|<\varepsilon) ] \}
[/math]

equivalent to
[math]
\exists \varepsilon >0 : \forall \delta >0 : \exists x : \neg\{ |x-x_0| \geq \delta \lor |f(x)-f(x_0)|<\varepsilon \}
[/math]

equivalent to
[math]
\exists \varepsilon >0 : \forall \delta >0 : \exists x : ( |x-x_0| < \delta \land |f(x)-f(x_0)| \geq \varepsilon )
[/math]

You can definitely derive the last thing without symbolic manipulation, but you might fuck up.
From it you can for example derive that there exists a sequence [math] x_n \to x_0 [/math] , but [math] \neg ( f(x_n) \to f(x_0) ) [/math] .
So for if for all sequences [math] x_n [/math] you have that [math] x_n \to x \implies f(x_n) \to x [/math] , then f must be continuous at x_0.

[Anonymous No.9444552]

>>9444542
> [math] x_n \to x \implies f(x_n) \to x [/math]
meant x_n \to x_0 \implies f(x_n) \to f(x_0)

[Anonymous No.9444576]

How do i into odes/pdes? Ive heard arnolds book is good for odes. I know baby rudin analysis

[Anonymous No.9444595]

>>9444542
>but you might fuck up
Only if you are a brainlet.

[Anonymous No.9444601]

>>9444576
Evans PDE. The appendix will direct you to any material you need to learn.

[Anonymous No.9444707]

if + is a function, what kind of notation rule allows one to represent an addition as
>x + y
when it should be
>+(x, y)
context: i am playing with peano's axioms.

[Anonymous No.9444709]

>>9444707
that's exactly the notation rule
a + b is defined as +(a,b)

[Anonymous No.9444722]

>>9444709
fugg. either high-school sucked or i didn't pay attention back then. thanks.
https://en.wikipedia.org/wiki/Function_(mathematics)#Binary_operations
https://en.wikipedia.org/wiki/Infix_notation

[Anonymous No.9444870]

>>9444398
I had a user there. Sadly I don't remember the log in stuff.

[Anonymous No.9445465]

Looking for a book on graph theory. Anyone have recommendations?

[Anonymous No.9445508]

>>9445465
Bondy & Murty.

[Anonymous No.9445522]

>>9445508
Much appreciated, /mg/ is one of the most reliable places for good math recommendations.

[Anonymous No.9445525]

>>9445522
I'm not a "place".

[Anonymous No.9445533]

>>9445525
I complemented you. Why respond like this anon?

[Anonymous No.9445558]

>>9442329
anybody want to collaborate on Number Theory stuff? I like trying to solve open problems and just dicking around with primes and various tests and stuff like that. Any undergrad math majors or the like want to join a Discord and work on stuff together? Looking for 1-3 people

[Anonymous No.9445598]

>>9445558
I am down for that. I do number theory every day. Quick disclaimer though, I only know elementary number theory and the first 5 chapters of Apostol's analytic number theory. I can also solve IMO/Putnam tier problems some of the time. Going from the putnam scale, I can solve #1's 100% of the time, #2-4 70% of the time, #5-6 10% of the time.

[Anonymous No.9445607]

>>9445558
sure

[Anonymous No.9445683]

>>9444451
Yes, almost there.

n! is the total number of permutations of [math]\{1, \ldots, n\}[/math].
[math]|A_i|! |A_i^c|![/math] is the number of such permutations that start with elements of [math]A_i[/math] and then elements of [math]A_i^c[/math] (when I say "starts", I view a permutation as an ordered list [math]\sigma(1), \ldots, \sigma(n)[/math])

Then if these sets of permutations are disjoint, we're done.
Suppose we have a permutation that both begins by all the elements of A_i and by all the elements of A_j, taking the least of those prefixes then one of them is included in the other, which is a contradiction.

[Anonymous No.9445689]

>>9445598
ok, sounds like you have a bit more experience than me but that's good. My knowledge comes mostly from wikipedia but I've tried solving some Putnam problems and it's fun. I mess around with Mathematica and Matlab mostly. Join quick and I'll delete this once we have everybody

https://discord.gg/hTSFbn

[Anonymous No.9445700]

>>9445558
>>9445598
I actually have Apostol Analytic Number theory open right now on my computer.

[Anonymous No.9445702]

>>9445689
I need to get home. Can't download the discord app on my phone.

[Anonymous No.9445705]

>>9445702
kk I'll wait

[Anonymous No.9445750]

>>9445465
Did you check the wikia?
http://4chan-science.wikia.com/wiki/Mathematics#Introduction_to_Graph_Theory

[Anonymous No.9445767]

>>9445750
thank you for the link. i also like to ask what people here have read as well.

[Anonymous No.9445872]

>>9444344
your mom's

[Anonymous No.9445895]

is ramsey theory considered mememath?

https://en.wikipedia.org/wiki/Ramsey_theory

[Anonymous No.9445903]

>>9444344
techniques and introduction to writing proof is offered at many universities, including top 10 schools like UC Berkeley. If you do not have such a class, you are probably a brainlet at a shitty school.

sorry f@m.

[Anonymous No.9445913]

>you will never be Kolmogorov

[Anonymous No.9445914]

>>9445903
>techniques and introduction to writing proof is offered at many universities, including top 10 schools like UC Berkeley. If you do not have such a class, you are probably a brainlet at a shitty school.
When people ask me if I went to a school for brainlets with an intro to proofs class, I tell them, "No, I went to math classes."

[Anonymous No.9445921]

>>9445895
Absolutely not. The fact that you would think you need to link the wikipedia like you just discovered some obscure topic is embarrassing and I suggest you delete your post to save face.

[Anonymous No.9445924]

asking in a very broadly way, is there any mathematical model, or sub-field that can be used to talk about self-reference? - be it from systems or just an abstract way - I've read something about Operads and first-order logic,etc.

[Anonymous No.9445927]

>>9445921
>need to link the wikipedia like you just discovered some obscure topic

[Anonymous No.9445933]

>>9445924
You might like the links between differential equations and recurrence relations, and generating functions

[Anonymous No.9446043]

What's the mathematical definition of an "economy"?

[Anonymous No.9446088]

I made a discord for discussing number theory. Anyone interested or knowledgeable in the subject is welcome to join.
https://discord.gg/hTSFbn

[Anonymous No.9446101]

>>9446043
Mathematical means it's well defined, so define it shithead

[Anonymous No.9446107]

>>9446101
>Mathematical means it's well defined, so define it shithead
What do you mean?

[Anonymous No.9446112]

>>9446107
Ask your question more specifically

[Anonymous No.9446119]

>>9446112
>Ask your question more specifically
What do you mean by "Mathematical means it's well defined, so define it shithead"?

[Anonymous No.9446151]

>>9445525
>I'm not a "place".
Then what are you?

[Anonymous No.9446152]

Give me one good reason to care about HoTT.

What's the simplest non-trivial result worth knowing about it?

[Anonymous No.9446163]

>>9446119
You want to play stupid games, or you want an answer to your well-defined question?

[Anonymous No.9446165]

>>9446163
>You want to play stupid games, or you want an answer to your well-defined question?
The latter, obviously.

[Anonymous No.9446167]

>>9446165
Oops, well looks like you forgot to include your well-defined question

[Anonymous No.9446172]

>>9446167
>Oops, well looks like you forgot to include your well-defined question
What do you mean?

[Anonymous No.9446179]

>>9446151
Not szïr, but Fritz von Bothmer said he is not a meaningless point in the universe. This is the antroposophical basis for pointless topology, and similarly this pointless approach makes it reasonable to assume people are not places, but instead a concentration of aether twirls on some area. This is why one should start their day by practicing yoga with electrodes attached all over the body in order to find the optimal aether flow through one's corporeal manifestation. This was taught to me by mr. Kauko Nieminen.

[Anonymous No.9446458]

ok /sci/ help this brainlet out.
I was using linear regression but I soon realized that my model has a bidimensional output variable (let's say I am trying to find a predictor [math]\hat{f}~:~\mathbb{R}^2\rightarrow\mathbb{R}^n[/math]).

Usually everyone is using it to fit a scalar function to whatever number of parameters ([math]\hat{f}~:~\mathbb{R}\rightarrow\mathbb{R}^n[/math]), but what if my output space has dimension >1? Can't find anything on google it seems... multiple linear regression is apparently only for the input variable being more than 1 dimensional, not the output variable...

[Anonymous No.9446470]

>>9443380
maybe your brain is wrong.

[Anonymous No.9446538]

>>9445914
>No, I went to math classes.
What would be some examples of these math classes?

[Anonymous No.9446568]

>>9446538
>What would be some examples of these math classes?
The usual applied repertoire: machine learning, numerical methods, PDEs, etc.

[Anonymous No.9446569]

>>9446538
Xô is obviously referring to high school math.

[Anonymous No.9446574]

>>9446568
Lol
Your idea of applied math is "shit similar to math for engineers"

[Anonymous No.9446575]

>>9446569
>Xô

[Anonymous No.9446577]

>>9446574
>math for engineers
This is a meaningless notion.

[Anonymous No.9446579]

>>9446575
Yes, and?

[Anonymous No.9446615]

>>9446574
>applied math
No such thing.

[Anonymous No.9446902]

What course am I supposed to take after Linear Algebra? Uni says Number Theory + Analysis I

[Anonymous No.9446911]

>>9446902
real analysis + linear algebra is the basics
then abstract algebra and more analysis
number theory is fine

[Anonymous No.9446912]

>>9446902
>Number Theory + Analysis I
Just take any engineering course since that's what you seem to be studying if they teach that shit in your school.

[Anonymous No.9446945]

>>9446912
What? I’m a first year mathematics major. What about those courses makes me out to be an engineer?

[Anonymous No.9446977]

>>9446902
>>9446945
Learn Linear Algebra, Calculus/Analysis and Probability extremely well.
That's all you need to graduate with ease.

[Anonymous No.9446983]

>>9446902
>What course am I supposed to take after Linear Algebra?
Another course on algebra like group theory or ring theory

Also multivariable calculus so you can use linear algebra there

[Anonymous No.9447004]

>>9446977
shut up

[Anonymous No.9447006]

>>9447004
no u

[Anonymous No.9447013]

>>9447004
>shut up
Do you need to swear?

[Anonymous No.9447016]

>>9446902
Analysis is pivotally important to take soon.

Number theory is a wonky subject because there are so many techniques you can apply to the integers than "number theory" is meaningless as a course label. The one for undergraduates won't use any linear algebra at all and probably won't even use calculus.
It's kind of a fluff course in most math departments.

[Anonymous No.9447044]

>>9447016
>Number theory is a wonky subject
>It's kind of a fluff course in most math departments.
The same can be said about analysis.

[Anonymous No.9447046]

>>9447044
what are you talking about? the analysis sequence is very standard and streamlined at this point

[Anonymous No.9447057]

Common sense tells me x3=10 and x1 and x2 = 0
with the minimum value being -10
this is worth 14% of a paper though so am I missing a chunk of the problem?
Also how to formally state my reasoning?

[Anonymous No.9447060]

>See unfamiliar greek/hebrew/alien symbol in a proof
>Don't know how to pronounce it in my head
>Don't even know what to search since no idea what it's called
>It's used throughout the entire publication

[Anonymous No.9447063]

>>9447060
Why do you need to know how to pronounce it?

[Anonymous No.9447068]

>>9447060
>>Don't even know what to search since no idea what it's called
http://detexify.kirelabs.org/classify.html

[Anonymous No.9447075]

Did /sci/ suffer a fucking head injury since I last visited? Is it because a new semester started? What the fuck is going on.

[Anonymous No.9447140]

>>9447063
I pronounce things vocally in my mind as I read through text in order to enhance short term memory

[Anonymous No.9447144]

>>9447060
just call it epsilon! you can pronounce any greek letter "epsilon"

[Anonymous No.9447209]

>>9447144
what if there's too many

[Anonymous No.9447225]

>>9447057
that's what i got too.

as for actually going through the reasoning, use lagrange multipliers

[Anonymous No.9447427]

>>9447209
>baby's first arithmetic functions
What's the issue?

[Anonymous No.9447448]

>>9447060
>tfw Greek
>tfw writting down math symbols was never a problem (except aleph, fuck aleph)
Seriously though. How hard can it be? I learned the Russian alphabet in like 15 minutes. Just open wikipedia and read.

[Anonymous No.9447639]

Taking topology without real analysis, am I boned?

[Anonymous No.9447642]

>>9447639
>Taking topology without real analysis, am I boned?
No.

[Anonymous No.9447645]

>>9447639
That's how any non-brainlet does it.

[Anonymous No.9447653]

Is MathOverflow a circlejerk?

[Anonymous No.9447684]

>>9443360
>>9443380
We'll know it's right when it proves the Riemann Hypothesis.

[Anonymous No.9447689]

>>9445924
mathematical logic & model theory

[Anonymous No.9447693]

>>9447639
you'll have some issues with intuition compared to someone who did, but it should be ok

[Anonymous No.9447694]

>>9447639
The only difficulty you may have is that your professor will probably assume everyone knows analysis and keep using examples from it all semester
The subject itself is completely self-contained

[Anonymous No.9447714]

>>9442420
It stands for funky sum.

[Anonymous No.9447751]

>>9447639
real analysis is completely useless

[Anonymous No.9447776]

What classes aren’t completely useless, then?

[Anonymous No.9447858]

>>9447776
complex analysis

[Anonymous No.9447863]

>>9447858
>take complex analysis
>it requires a lot of material from real analysis
>don't learn shit and drop it because real analysis is useless

[Anonymous No.9447942]

>>9447863
C=R+iR

what did you expect?

[Anonymous No.9447957]

hello, I have a question about Asymptotic Notation, specifically proving Θ() when given a table / graph.

Lets say I'm given a graph of n and time(ms), that would look like this:

n Time (ms) T(n)/T(n-1)
5 10
6 30 3
7 230 7.7
8 250 1.1
9 1287 5.1
10 1810 1.4
...

and so on and so on, when graphed it is an exponential function.

How do I determine if the runtime has the form THETA( 2^n ) ? I get that to be bigtheta is has to be bigO (upper bound) and bigOmega(lower bound), but when given a table with multiple columns (time and T(n)/T(n-1) ) do you prove bigTheta for each column? Or just once?

Sorry I know its kindof a complicated question, i just cant find that many good online resources for proving bigtheta based on a graph / table

[Anonymous No.9447962]

>>9447942
I know right? What a muppet.

[Anonymous No.9447963]

>>9447957
fuck my photo screwed up the formatting of the table:


n - Time (ms) - T(n)/T(n-1)
5 - 10
6 - 30 - 3
7 - 230 - 7.7
8 - 250 - 1.1
9 - 1287 - 5.1
10 - 1810 - 1.4
...

[Anonymous No.9448629]

>Consider E = {1, {2, 3} , {2, 4}}
>the subset {2, 3} is in E but 2 is not.
then how come 2 is in R?

[Anonymous No.9448658]

>>9448629
That's not a subset.

[Anonymous No.9448671]

>>9448658
it is a set within E, that makes it a subset of E, although one could simply call it a set but for the sake of clarity, i am calling it a subset so you don't think i am defining a new one, namely {2, 3}.

[Anonymous No.9448675]

>>9448671
No. That makes it an element. {{2, 3}} would be a subset.

[Anonymous No.9448688]

>>9448675
That makes N an element of Z, Z an element of Q, but N is not an element of Q.

[Anonymous No.9448695]

>>9448688
You are disabled in a psychological way.

[Anonymous No.9448700]

>>9448695
literally book of proof. page 4, end of third paragraph
http://www.people.vcu.edu/~rhammack/BookOfProof/Sets.pdf

[Anonymous No.9448702]

>>9448700
>book of proof
Waste of money.

[Anonymous No.9448760]

>>9448629
>>9448671
>>9448688
Embarrassing.

[Anonymous No.9448804]

>>9448629
Because {2,3} is an element of E, not a subset. Similar to how the UN can be regarded as a set of nations, and nations can be regarded as a set of the people who live in them.

So you and I are not member of the UN, although the countries we are part of are.

[Anonymous No.9448810]

>>9448804
{2,3} is clearly a subset of {{2,3},4,3,7,2} so it's a subset of E

[Anonymous No.9448819]

>>9448810
I'm talking about E in the link. If you set E = {{2,3},4,3,7,2} then {2,3} would be a subset of E, yes.

[Anonymous No.9448922]

>>9448804
>Similar to how the UN can be regarded as a set of nations
The UN does not form a set, since it would have to include itself as an element.

[Anonymous No.9448936]

>>9448629

Hey man, I saw you struggling and just thought I should come over an help.

No disrepect or anything, bro. I think you're doing your best. Hell, I even think you're doing a great job.

But here's the thing, man. And again, I don't mean any disrespect. But, I think (and this is just from my own experience), that you might be a little off.

If E = {1,{2,3},{2,4}}, then {2,3} isn't a subset of E. Sorry. It's just an element of E. A subset of E is a set that contains elements of E. And since {2,3} is a set that contains 2, and 2 isn't an element of E, it isn't a subset.

But that's just my 2 cents, bro.

[Anonymous No.9449100]

>>9448700
>All of mathematics can be described with sets.
>The theory of sets is a language that is perfectly suited to describing and explaining all types of mathematical structures

[Anonymous No.9449184]

>>9449100
This reminded me of a thing I heard this week. A few people were discussing how the inconsistency of ZFC would bring down all of category theory.

[Anonymous No.9449204]

What are your thoughts on amateur mathematicians? I used to like discovering infinite series but stopped because I didn't really have anywhere to discuss my results because not in college.

[Anonymous No.9449208]

>>9449204
Discuss them here.

[Anonymous No.9449220]

>>9448936
Sorry for asking, please don't think I'm trying to disrespect you or anything. I just have one very quick question. Would it then be correct to say that {{2,3}} is a subset of E? Sorry for asking.

[Anonymous No.9449222]

>>9449204
I'm a memegineer doing amateur maths myself, my engineering education actually helped me a lot since it gives the practical calculations tools as well as a bit of proofs/mathematical cultures to jump into high-level books without being too much at a loss

>>9449208
Here's a funny problem if you're into them
Find [math]\alpha[/math] and [math]\beta[/math], complex numbers, such as :

[eqn]a = & \frac{\alpha + \beta}{1 + \alpha^2} \\
b = i * \frac{\alpha - \beta}{1 + \alpha^2} \\
c = \frac{1 - \alpha^2}{1 + \alpha^2}[/eqn]

Are all simultaneously real ; what is the set of solution ? What is the relation between [math]\alpha[/math] and [math]\beta[/math] ?
The textbook I'm reading is old as fuck and has a ton of geometry-oriented problems like this one

[Anonymous No.9449225]

>>9449222
Fug it worked on the preview
[eqn]
a = \frac{\alpha + \beta}{1 + \alpha^2} \\ b = i * \frac{\alpha - \beta}{1 + \alpha^2} \\ c = \frac{1 - \alpha^2}{1 + \alpha^2}
[/eqn]

[Anonymous No.9449276]

>>9448700
>literally book of proof. page 4, end of third paragraph
>http://www.people.vcu.edu/~rhammack/BookOfProof/Sets.pdf
Why are you reading a memebook?

[Anonymous No.9449301]

>>9449184
Disprove that then

[Anonymous No.9449312]

What is the best book on linear algebra and why? I'm looking for proof-heavy, rigorous books, and keeping applications to a minimum.

>>9442420
The integral symbol stands for 'infinite sum'. So, the integral of f(x) is the infinite sum of f(xi)dx

[Anonymous No.9449322]

>>9449301

[Anonymous No.9449445]

>>9449312
Look into Linear Algebra Done Wrong

[Anonymous No.9449452]

>>9449312
>What is the best book on linear algebra and why? I'm looking for proof-heavy, rigorous books, and keeping applications to a minimum.
Hoffman and Kunze is canonical

[Anonymous No.9449460]

Algebraic geometry is the mathematician's mathematics

[Anonymous No.9449461]

>>9449460
>Algebraic geometry is the mathematician's mathematics
Actually, it's hardly "mathematics" at all.

[Anonymous No.9449486]

https://terrytao.wordpress.com/2018/01/19/the-de-bruijn-newman-constant-is-non-negativ/

>Brad Rodgers and I have uploaded to the arXiv our paper “The De Bruijn-Newman constant is non-negative“. This paper affirms a conjecture of Newman regarding to the extent to which the Riemann hypothesis, if true, is only “barely so”.

[Anonymous No.9449487]

>>9449460
Oops it seens that you've conflated trash plebian geometry with godtier topology, anon. Rookie freshman mistake, I forgive you.

[Anonymous No.9449491]

>>9449322
Not a proof, a contradiction in zfc could be translated to topos.

[Anonymous No.9449590]

>>9449487
Why is it trash?

[Anonymous No.9449594]

>>9449590
because he doesn't understand it

[Anonymous No.9449601]

>>9449594
>because he doesn't understand it
I'm not a "he".

[Anonymous No.9449603]

>>9449594
I'm not a "he".

[Anonymous No.9449606]

>>9449601
>>9449603
yeah you are faggot

[Anonymous No.9449610]

>>9449606
I'm not a "you". Please learn my pronouns before opening your mouth next time.

[Anonymous No.9449616]

>>9449606
>yeah you are faggot
Why the homophobia?

[Anonymous No.9449618]

>>9449610
ill get right on that faggot

[Anonymous No.9449623]

>>9449618
>faggot
Why the homophobia? I'll let you answer this after you're back from learning my pronouns.

[Anonymous No.9449628]

>>9449616
>>9449623
because faggots are faggots

[Anonymous No.9449687]

>>9449491
Could be translated to topos =/= would wreck topos theory per se.

[Anonymous No.9449716]

>>9449276
i want to learn how to write mathematical proofs in my spare time. that is the go to recommendation, no?

[Anonymous No.9449721]

>>9449716
You learn how to write mathematical proofs by actually reading and writing real proofs, not by using a toy book.

[Anonymous No.9449746]

>>9449623
the only pronoun you need is "that annoying faggot"

[Anonymous No.9449749]

>>9449721
tried that, i have seen techniques such as mathematical induction, i don't see how adding one to a variable makes it a valid proof, but certainly shows that i am missing common techniques used.

[Anonymous No.9449760]

>>9449749
You're retarded.

[Anonymous No.9449771]

>>9449760
Get cucked, cunt

>>9449749
Just read the proof book. Who gives a shit, as long as you enjoy it.

[Anonymous No.9449777]

>>9449716
>i want to learn how to write mathematical proofs in my spare time. that is the go to recommendation, no?
No, read actual math books.

[Anonymous No.9449788]

>>9449687
Not an argument in favor of your point

[Anonymous No.9449789]

>>9449749
> techniques such as mathematical induction
It's not really a technique, it's a result saying that it suffices to prove two "simpler" statements when proving something about an arbitrary natural number.
>adding one to a variable
This is engineer-level garbage. No wonder people such as yourself gravitate towards those kinds of books.

[Anonymous No.9449792]

>>9449749
The variable parametizes the cases, induction shows its true in all cases

[Anonymous No.9449794]

>>9449788
See if I care.

[Anonymous No.9449795]

>>9449771
>Who gives a shit, as long as you enjoy it.
He might give a shit about wasting time on something useless.

[Anonymous No.9449797]

>>9449788
Why is that?

[Anonymous No.9449801]

>>9449792

If you're most worried about sparing us from wasting our time, then you should quit making us suffer your advice.

[Anonymous No.9449808]

>>9449771
Fucking retard samefag.

[Anonymous No.9449809]

>>9449777
ok

[Anonymous No.9449812]

>>9449792

Oops! That catty remark was meant for

>>9449795

[Anonymous No.9449813]

>>9449797
Doesnt disprove what the person said earlier that was replied to naively

[Anonymous No.9450219]

>>9442329
If [math]P[/math] is a p-group, is it true that every proper subgroup [math]Q[/math] of order [math]p^k[/math] is contained in a subgroup of order [math]p^{k+1}[/math]?

[Anonymous No.9450277]

>>9444389
elaborate

[Anonymous No.9450688]

>>9449486
analytic number theory providing some results once again. start studying hard analysis right now you n-category virgins.

[Anonymous No.9450902]

What's ba tensor. I know its a k-linear function and I know its an element of the algebra formed by the freest bilinear operator over a vector space and its dual and I know how it transforms based on its type.

[Anonymous No.9450938]

>>9450902
so what exactly is your question ?

[Anonymous No.9450961]

>>9450902
>bilinear
Multilinear*.

[Anonymous No.9451001]

>>9447057
easier to see:
[math]2x_1 + 2x_2 - x_3 = 2(x_1 + x_2 + x_3) - 3x_3 = 20 - 3x_3[/math]
the conditions give [math]x_i \in [0,10][/math] and the minimum of the monotone function is taken on the border, namely for [math]x_3 = 10[/math]

[Anonymous No.9451056]

>>9450219
Yes because all maximal subgroups of any p-group has index p. It's a standard result.

[Anonymous No.9451348]

>>9450688
>start studying hard analysis right now
I'm not subhuman enough to do this.

[Anonymous No.9451385]

>>9442329

what's {-13/10} ?

am i correct in saying

-1.3 - [-1.3] = -1.3 - (-2) = 0.7

[ ] defaults to the floor function

[Anonymous No.9451389]

>>9451385
>what's {-13/10} ?
In what context? It's the fractional part but there are different ways to define it for negative numbers

https://en.wikipedia.org/wiki/Fractional_part#For_negative_numbers

i.e. depending on your definition it could be several different things (0.7, 0.3, -0.3...)

[Anonymous No.9451404]

Anyone interested in a grad or above level fourier series problem?

[Anonymous No.9451414]

>>9451389

{x} := x - [x]

[Anonymous No.9451454]

>>9450961
Ah yeah.
>>9450938
What's a tensor

[Anonymous No.9451463]

>>9451454
>What's a tensor
A (p,q)-tensor is an element of [math]{V^{ \otimes p}} \otimes {V^*}^{ \otimes q}[/math]

[Anonymous No.9451476]

how do i write latex here? \phi does nothing

[Anonymous No.9451480]

>>9451476
http://boards.4chan.org/sci/thread/5942502/

[Anonymous No.9451482]

>>9451463
I know that already

[Anonymous No.9451483]

>>9451482
Then what are you asking

[Anonymous No.9451488]

>>9451480
okay, im just wondering why we choose

[math]\hat{r} = \rho'\hat{\rho}[/math]

and not,

[math]\hat{r} = \rho'\hat{\rho} + \phi'\hat{\phi}[/math]

and take the limit from 0 to 2pi and 0 to a ???

[Anonymous No.9451494]

>>9451488
both [math]\hat{r}[/math] how be [math]\hat{r}'[/math]

[Anonymous No.9451640]

>>9451483
What is a tensor

[Anonymous No.9451649]

>>9451640
It's an undefined primitive concept.

[Anonymous No.9451655]

>>9451640
I just told you

[Anonymous No.9451661]

What's the best concise introduction to mathematical (read: rigorous) statistics?

I want something that's like baby rudin. Small, thin book that is essentially just a bunch of results and their proofs. I don't care if rudin is a meme, I love terse books like that

[Anonymous No.9451672]

>>9451661
there's komolgorov's books for probability
I don't think there is one for statistics specifically. statistics is usually done in a pedagogical way

[Anonymous No.9451701]

>>9450688
hard analysis is such an ugly field
>juggle twelve epsilons to get an upper bound of 14 bajillion
>for the next 3 years a horde of analysts twiddle your epsilons around to get the lower bound down to 2.12e^1.00001
>some guy comes along with a respectable proof method and proves it's zero

[Anonymous No.9451731]

>>9451649
>>9451655
Tensors are k-linear functions. They are elements of the freest algebra over the cartesian product. They are elements of the tensor product of the tensor product of the r tensor algebra of a vector space V and the s tensor algebra of the dual vector space V*. Its the building block of the symmetric, exterior, and geometric algebra (transitively). Its an element of the tensor bundle of a manifold. But what the FRICK is it.

[Anonymous No.9451734]

>>9451731
You wouldn't understand, even if I were to explain it.

[Anonymous No.9451736]

>>9451734
What do you mean

[Anonymous No.9451787]

>>9450688
I tried and attended a class in analytic number theory. It kicked my ass after 3 weeks desu and I stopped going.
It's sad because I actually thought I liked analysis. I guess it's more fun to work on that stuff by yourself at your own speed than watch someone rush through pages and pages of computations

[Anonymous No.9451818]

>>9451787
>It's sad because I actually thought I liked analysis.
Why not do actual analysis instead of some dumbshit related to number theory?

[Anonymous No.9451824]

>>9451731
Oh ok, you're just a faggot.

>inb4 why the homophobia

[Anonymous No.9451832]

>>9451787
I'm the opposite. I didn't really like analysis but I enjoyed analytic number theory and it helped motivate me to study analysis more.

[Anonymous No.9451848]

>>9442329
Eulers number is awesome

https://www.youtube.com/watch?v=Z00Zn7ictIY

[Anonymous No.9451854]

>>9451832
>I enjoyed analytic number theory

[Anonymous No.9451920]

>>9451854
>anime reaction image

[Anonymous No.9451923]

>>9451920
>>>/r/eddit/

[Anonymous No.9451946]

>>9451920
anime website

[Anonymous No.9451950]

>>9451920
>spongebob reaction image
>calls someone else mentally retarded

[Anonymous No.9451961]

>>9451950
Assuming anything that is not proven (except axioms lol) cannot yield a proof. Any mathematician thinking otherwise is an idiot.

[Anonymous No.9451969]

>>9451731
Tensors are like a matrix with more dimensions. Only physicists use this crap.

[Anonymous No.9451980]

>>9451969
No. Sounds like what people would gather from skimming Griffith. Go read an actual QM book like Townsend, Sakurai or Landau-Lifshitz.

[Anonymous No.9451989]

>>9451980
I won't touch that shit. I only do real math, like calculus and linear algebra.

[Anonymous No.9451990]

>>9451946
>/sci/ - Science and Math

[Anonymous No.9451999]

>>9451989
>can't understand basic quantum mechanics
>wants to study string theory
>>9451990
A good place to start is the Landau-Lifshitz series.

[Anonymous No.9452019]

>>9451999
>implying you need babby QM to understand string theory
That's what Mr Shekelberg wants you believe, so you buy his expensive textbooks.

[Anonymous No.9452026]

>>9452019
The cobordism hypothesis can be proved, so where are the proofs of this homotopy "hypothesis"?

[Anonymous No.9452061]

>>9451488
>>9451494
Read Baez's Guage Fields, Knots and Gravity.

[Anonymous No.9453152]

[eqn]v_{\xi \eta} + \frac{1}{(2x+1)^2}v_{\xi} - \frac{1}{(2x+1)^2}v_{\eta} =
0\\
where\\
\xi(x,y)=y-x^2-x, \thinspace \eta(x,y)=y+x^2+x[/eqn]
Just what the fuck am I supposed to do here?
>inb4 no homework
It's not homework, I'm just preparing for my upcoming exam and this appeared two years ago.

[Anonymous No.9453211]

>>9453152
What is the problem asking for?

[Anonymous No.9453232]

>>9453211
For explicit solutions. And I'm pretty sure that's the equation I have to solve (they gave me another one and that's its canonical representation) because more than one person has asked me to help them and showed me that bullshit, which is also what I arrived at.
This kinds of problems usually simplify quite well and you just integrate over [eqn]\xi[/eqn] and then over [eqn]\eta[/eta] (or the other way around) and that's it, but I don't know what's the deal with this one.

[Anonymous No.9453235]

>>9453232
Oops, fucked that [math]\eta[/math] up real badly and forgot to use /math, sorry.

[Anonymous No.9453283]

can someone give me some intuition behind filters and ultra filters on a set?

[Anonymous No.9453338]

>>9453283
these are quite general notions so its hard to say something satysfying
filter: a family of all sets which are 'large' in some sense, for example: all neighbourhoods of a given point in a topological space
all subsets of R containing 0
all subsets of R which have finite complement
ultrafilter: i mostly encountered these in various nonconstructive proofs of existence of some retarded object
'maximal filter' doesn't say much, it is perhaps easier to think about the following equivalent property of ultrafilters: for any set A, either A is in the ultrafilter, or A is not in the ultrafilter and then [math]A \cap M = \emptyset [/math] for some M from ultrafilter
or: for any set A, either A is in the ultrafilter, or X\A is in the ultrafilter

[Anonymous No.9453500]

>>9444707
infix notation btw

[Anonymous No.9453541]

>>9453338
what confuses me is this notion of "largeness".

[Anonymous No.9453757]

>>9449460
Algebraic geometry is cool because Grothendieck MADE it cool.

[Anonymous No.9453762]

>>9451701
Honestly their proof is kinda ridiculous too though. One day algebra will subsume hard analysis and then we'll have a good proof.

>not constructing $\mathbb{F}_1$
>mfw

[Anonymous No.9453782]

>>9453283
Filters are extremely intuitive!
(btw, don't read the wikipedia article because you will be left with zero intuition on the subject)

A filter represents a collection of consistent assertions, beliefs, or pieces of information.

If you believe P and you are rational then you must believe all logical consequences of P (upward closure).
If you believe P and you believe Q and you are rational then you must believe P & Q (intersection).

An ultrafilter is a maximal collection of consistent beliefs, so it's "complete information." If you take all properties of x (subsets containing the point x) then you get an ultrafilter representing x (principal ultrafilter). The tricky thing is that an ultrafilter may not specify a particular x.

[Anonymous No.9453786]

>>9442329
Where do odd perfect numbers live?
Aldi Nestor de Souza
https://arxiv.org/pdf/1801.06182.pdf

>The existence of a perfect odd number is an old open problem of number theory. An Euler's theorem states that if an odd integer n is perfect, then n is written as n=p^rm^2, where r,m are odd numbers, p is a prime number of the form 4k+1 and (p,m)=1, where (x,y) denotes the greatest common divisor of x and y. In this article we show that the exponent r, of p, in this equation, is necessarily equal to 1. That is, if n is an odd perfect number, then n is written as n=pm^2.

[Anonymous No.9453789]

>>9453786
Big if true

[Anonymous No.9453836]

>>9453786

[Anonymous No.9453858]

>>9453786
>https://arxiv.org/abs/1202.0569
>A Constructive Proof of the Cycle Double Cover Conjecture
>Comments: Due to a flaw in Lemma 9, the paper has been withdrawn

Red flag much?

[Anonymous No.9453860]

>>9453858
Different author.

[Anonymous No.9453865]

>>9453860
oops

[Anonymous No.9454611]

>statistics and actuary is for brainlets
>my countries best living mathematician has made several contributions to stats and actuaryshit
:thinking:

[Anonymous No.9454628]

>>9454611
>statistics and actuary
Not science or math.

[Anonymous No.9454638]

>>9454628
http://staff.math.su.se/lindskog/manuscriptELL.pdf
Looks pretty mathy to me

[Anonymous No.9454639]

Any good courses for Differential Equations online?
MIT one is good but the video lectures are so crappy

[Anonymous No.9454641]

>>9454638
>pretty mathy
Seems like buzzwords to me.

[Anonymous No.9455090]

>>9454611
ACTUAL Statistics is one of the hardest math/science subjects.

[Anonymous No.9455359]

babbys first proofs here

can I take P implies Q AND R and just use P implies R for a proof if I have no interest in Q?

ie
P -> Q ^ R
means
P -> Q
and
P -> R

[Anonymous No.9455361]

>>9455359
try to read your question in colloquial language and you'll realize it's obvious

while you're learning the formalism, try not to lose track of what the symbols mean and the intuition behind them

[Anonymous No.9455399]

>>9454611
Do you live in a country of brainlets?

[Anonymous No.9455409]

>>9455359
What are you trying to prove?

[Anonymous No.9455412]

>>9455359
Write a truth table for P -> (Q^R), P -> Q, and P -> R. This will give you your answer.

Don't ever forget that valid statements in these kinds of proofs (and really, all proofs) rely on tautologies.

[Anonymous No.9455428]

C = speed of light
N = newtons
G = universal gravitational constant = 6.66*10^-8
M = gravitational mass and inertial mass
E = energy
K = coulombs law constant
Q = consciousness
i = square root of -1
r^2 = vector direction of r

Newtons formula of universal gravitation transformed and stated as cgs units:

f = -G M1 N2 / r^2 r^0 = (i G^1/2 M1) (i G^1/2 M2) / r^2 r^0

Q1 = i G^1/2 M1
Q2 = i G^1/2 M2

Thus the universal gravitational pull between M1 and M2 is equal to the gravitational pull between Q1 and Q2, given that the quantity of virtual charge is considered to be consciousness the conclusion is that the cause of gravity or attractive force lies in the consciousness that a body possesses. Thus when matter M is destroyed through the collapse of gravity without being converted into energy by Einsteins E = MC^2 it turns out that it is converted into consciousness of the order of Q = G^1/2 M, it is then possible for the conscious energy to be converted into real energy in which case we require that the energy E when it is converted into real energy through consciousness Q and law of conservation energy than the conversion coefficient for consciousness Q and real energy E can be summarised as:

E = MC^2 = KQ
= KG^1/2 M

K = MC^2 / C^1/2 M = C^2 / C^1/2
f = Q1 Q2 / r^2 r^0

[Anonymous No.9455439]

>>9450277
get a manifold and set n to at least 5

[Anonymous No.9455555]

>>9453836
s^(r-1)+...+1 is an integer
therefore (1-b1/a1)s^r is an integer
therefore a1 divides s^r

[Anonymous No.9455600]

>>9455555
>therefore (1-b1/a1)s^r is an integer
>therefore a1 divides s^r
The claim is that a1 divides s.

[Anonymous No.9455723]

>>9455399
>per martin-löf
>brainlet

[Anonymous No.9455731]

>>9455723
In a way, he is.

[Anonymous No.9455848]

Starting uni this year, I want to specialize in Analysis, what are some books that'll help me get to that throughout the career? I'm thinking of
>Tao's Analysis
>Rudin
>Spivak

[Anonymous No.9455868]

>>9455848
Ahlfors Complex Analysis

[Anonymous No.9455876]

>>9455848
>I want to specialize in Analysis

[Anonymous No.9455894]

>>9449461
don't be a pussy because you miss number

[Anonymous No.9455897]

>Suck ass at math since 5th grade
>Get to college
>Start liking it and genuinely considering changing my major to a math focused one
what the fuck happened

[Anonymous No.9455941]

>>9443360
Oliver stop promoting your shitty paper on 4chan and continue marking your undergrad exams

love

your advisor

[Anonymous No.9455945]

>>9444176
>https://math.stackexchange.com/questions/2574902/prerequisites-for-gunnar-carlssons-topology-and-data
Hatcher

eat that shit up

[Anonymous No.9455950]

>>9455897
Any math you do before Calc isn't even close to real math. Hell, Calc isn't even real math. Now that you're being exposed to actual math, you're liking it.

[Anonymous No.9455987]

Suppose there is a matrix M, as well as a vector v such that M(k)v(t) approximately equals v(t+k) for any value of t, and the approximation becomes more and more accurate as k decreases. Would the relation also be true if k is infinitesimal ?

[Anonymous No.9456488]

>>9454641
>buzzwords
Seems like buzzwords to me.

[Anonymous No.9456501]

f : N → N

f ◦ f(x) + f(x) = 2x + 15 ∀x ∈ N

How the fuck do I prove that this shit is injective?

[Anonymous No.9456598]

>linear algebra professor told us today that it won't be anything like calc and some classes will be us grinding out row reduced matrices.
Does this seem right? If so, will integrals/derivatives mix with what we learn here in later classes?

[Anonymous No.9456617]

>>9456598
>grinding out row reduced matrices
That's engineering garbage, not linear algebra.

[Anonymous No.9456672]

>>9456598
no, you probably won't see an integral or derivative for a while if you are doing actual linear algebra.

at a brainlet american uni, expect to spend 2+ hours per day showing the steps of marching various matrices towards Reduced Row Echelon form.

[Anonymous No.9456693]

>>9456501
By simple arithmetic? You probably proved the identities you need already

2x+15=2y+15 means 2x=2y and x=y

[Anonymous No.9456700]

>>9456598
>Does this seem right
It's a matrix algebra class.

>will be us grinding out row reduced matrices
If he's nice, he will do it over Z/pZ with p=2, 3, or 5.

>will integrals/derivatives mix with what we learn here in later classes?
They always cover solving system of differential equations by diagonalization in matrix algebra and/or DEs.

[Anonymous No.9456815]

>>9455848
http://4chan-science.wikia.com/wiki/Mathematics#Analysis

[Anonymous No.9457600]

>>9456693
Sorry, i meant that I need to prove that f is injective, not 2x+15.

[Anonymous No.9457736]

Does learning functional analysis require a good understanding on linear algebra or can I go with my analysis knowledge?

[Anonymous No.9457780]

>>9457736
you need linear algebra almost everywhere in advanced mathematics including functional analysis or any analysis for that matter

[Anonymous No.9458065]

>>9457736
Functional Analysis is literally Linear Algebra on infinite dimensional spaces who are equipped with a norm.
If you don't have a good understanding on linear algebra you are going to struggle on functional analysis.
Actually, if you don't have a good understanding on linear algebra, you are going to struggle in pretty much everything.

[Anonymous No.9458420]

>>9455723
he's pretty badass, he gets a pass

[Anonymous No.9458426]

>>9455950
That's what happened to me

[Anonymous No.9458429]

>>9456501
Suppose f(x) = f(y) and then see if you can prove x = y.

[Anonymous No.9458472]

I cannot for the life of me figure out how they got this fucking answer, can anybody please alleviate my idiocy for a second and explain it to me?

I thought I was supposed to use Lorentz transformation, or Spacetime inerval, but I couldn't figure out how to apply either...

[Anonymous No.9458483]

>>9442329
Can somebody help me to solve

[math]\int^1_0 \ldots \int^1_0 \frac{- \prod_{i=1}^n (x_i)}{log(\prod_{i=1}^n (x_i))} dx_1 \ldots dx_n[/math]


wolfram says that for i=2 answer is

[math]\frac{1}{t+1}[/math]

[Anonymous No.9458495]

>>9458483
Fuck, it should be

[math]\int_0^1 \ldots \int_0^1 \frac{−(\prod_{i=1}^n(x_i))^t}{\prod_{i=1}^n(log(x_i))} dx_1 \ldots dx_n[/math]

[Anonymous No.9458500]

>>9458495
Now I broke the denominator, great

[Anonymous No.9458522]

>>9458500
Dumb bitch.

[Anonymous No.9458551]

>>9458429
How would I do that since I don't know anything about f(x)? Can i just compare 2x+5 = 2y+5 and it would also prove that it holds for f(x)?

[Anonymous No.9458683]

I have been looking up for a definition of degrees of freedom, but never found something rigorous.
Is this one I came up with correct?

"Degrees of freedom" is defined only for linear transformations of random variables, with the "Degrees of freedom of the transformation" being the rank of the trasformation.

Like, for example, let
[math]
\vec{X}=(X_1, \ldots , X_n)'
[/math]

Consider the transformation
[math] \vec{\bar{X}}= (\bar{X}, \ldots , \bar{X}) = \frac{1}{n}J\vec{X} [/math]
where J is the matrix filled with 1's
[math] \frac{1}{n}J [/math] has rank 1, so "the sample mean has has 1 degree of freedom".

Then consider the transformation
[math] \vec{X}-\vec{\bar{X}} = (I-\frac{1}{n}J)\vec{X} [/math]
[math] I - \frac{1}{n}J [/math] has rank n-1 , so "the centralization around the sample mean has n-1 degrees of freedom".

[Anonymous No.9458744]

>>9458683
Oh and if P is an orthogonal projection (1/n J and I-1/n J are) and X is a vector with [math] E(X)=0 [/math] and [math] V(X)=\sigma^2I [/math] (e.g. a vector of i.i.d. random variables) , then
[math]E(\lVert PX \rVert ^2) = E(X'PX)= tr(AV(X)) + E(X)'PE(X) = σ^2 rank(P) = \sigma ^2 \times d.f. (P) [/math]
From which one finds unbiased estimators of the sample variance: [math] \frac{1}{d.f.(P)} \lVert PX \rVert ^2 [/math] . When [math] P = I - \frac{1}{n}J [/math] we get the usual variance estimator.
And if X follows standard multivariable normal , then [math] \lVert PX \rVert ^2 = X'PX = [/math] sum of d.f.(P) i.i.d standard normals (by diagonalizing P) which follows chi-squared distribution with d.f.(P) Degrees of Freedom.

[Anonymous No.9458890]

>>9458551
Well, if f(x)=f(y), then f(f(x))=f(f(y)), isn't it?

[Anonymous No.9458956]

Is it possible to prove [math] \exists x, P(x) [/math] without giving an example of such an [math]x[/math], using the included middle in a proof by contradiction or [math]P(x)[/math] being trivially true for all [math]x[/math]?

[Anonymous No.9458965]

>>9458956
*excluded middle

[Anonymous No.9459004]

>>9458956
Literally anything invoking the axiom of choice.
- for any partial order where every chain is bounded above, there exists a maximal element
- for any vector space, there exists a basis
- etc.

[Anonymous No.9459009]

Should I take computer sci/Software engi courses or should I go all in pure maths for my degree?

[Anonymous No.9459063]

>>9459004
>for any vector space, there exists a basis
That's not true.

[Anonymous No.9459175]

>>9458956
>the included middle in a proof by contradiction
They are classically equivalent, so using proof by contradiction (double negation elimination) suffices.
For an example not using AC, Brouwer's fixed point theorem.

[Anonymous No.9459254]

>>9459009
I'd say take either compsci or stats.

[Anonymous No.9459287]

>>9459063
>rejecting axiom of choice

[Anonymous No.9459289]

>>9459009
Pure math

t. a pure math major

[Anonymous No.9459303]

>>9459063
It is true by definition of "vector space".

[Anonymous No.9459337]

>>9459303
>It is true by definition of "vector space".
Wrong.

[Anonymous No.9459339]

>>9459287
Assuming what you're trying to prove is an absurdity.

[Anonymous No.9459426]

>>9459303
this isn't right

with the standard vector space axioms, axiom of choice is required to prove every vector space has a basis, and the axiom "every vector space has a basis" can be used to recover the axiom of choice, so they're often said to be equivalent

[Anonymous No.9459486]

>>9459337
Right.
>>9459426
>with the standard vector space axioms
Which are the axioms for free modules over a field.

[Anonymous No.9459606]

>>9459486
>Which are the axioms for free modules over a field.
Wrong.

[Anonymous No.9459618]

>>9459606
>Wrong.
Wrong.