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/sci/ - Science & Math


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12283965 No.12283965 [Reply] [Original] [archived.moe]

Pure Mathematics edition
Talk maths
Prior >>12275236

>> No.12283975

>>12283965
>premature ejaculating a new thread fifteen posts too early

>> No.12283986 [DELETED] 
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12283986

>>12283975
Tell us how you really feel.

>> No.12284227
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12284227

I was gonna make the hyperglyph edition :{

Anyway, are there any fundamental properties that govern relations or algebras, that require more than 3 inputs? Associativity, order, commutativity, and equivalence are all 2 input. Transitivity is three. I see it relating to dimensions, with 2 dimensions you have a symmetry between objects, and with 3D, you have transitivity which lets you flesh out interactions throughout the space. But are there any other base level properties that require a fourth dimension?

>> No.12284233

>>12284227
Oh and distributive is 3 as well

>> No.12284305
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12284305

Can someone explain this to me, I'm having a hard time wrapping my head around the concept.

27 - -16 = 43

Why do the two negative signs here cancel each other out?

My brain is taking this to mean that I'm "subtracting" the negative sign from the value of -16?

>> No.12284334

>>12284227
Your question is not really clear because it's not obvious what should count as fundamental. People consider identities and relations involving arbitrarily many variables all the time.

>> No.12284337

>>12284305
A negative number is an anti. You're removing anti units. If negative is below sea level, you're removing depth, going the opposite direction. That means height.

>> No.12284340

>>12284334
Some property of reals or other simple, intuitive sets, that requires >3 elements to describe, that the set would look unusual without

>> No.12284355

How they know outliers are Q1-1.5 X IQR and Q3+ 1.5 x IQR?

>> No.12284357

>>12284340
Well you can't really take a property away from a structure, it either has it or it doesn't. Abelian groups are medial magmas, which means that they have (xy)(zw)=(xz)(yw)

>> No.12284358

>>12284355
It's the definition of outlier They chose it to be that way

>> No.12284367

>>12284357
>abelian groups
Thanks thats interesting. Although less powerful than others imo because it relies on position very strongly

>> No.12284375

>>12284357
But wait aren't they commutative so that's trivial?

>> No.12284378

>>12284337
That's a good way of conceptualizing it, thanks.

>> No.12284409

>>12284375
Well it's a thing that is true for any abelian group by an easy argument. If you don't know that you have an abelian group it can be useful though, see the Eckmann-Hilton argument.

>> No.12284449
File: 200 KB, 1024x931, Pacfc3.jpg [View same] [iqdb] [saucenao] [google] [report]
12284449

Any songs for the feel of pic related + writing symbols/definitions? Here's one example:
https://www.youtube.com/watch?v=TY5Hq_5FbT0

>> No.12284452

>>12284409
Thanks nonny

>> No.12284859

>>12284305
For any number x, -x is (by definition) the unique number y satisfying the property that x+y=0. But this defining property tells us that x = -y = -(-x).

>> No.12284895

Suppose there is an idempotent [math] f: M\to M[/math] such that [math] Im(f) = A[/math]I'm trying to prove that the map [math]A\bigoplus B \to M[/math] is an isomorphism. I would greatly appreciate a hint in the right direction. I can't seem to figure out why the induced homomorphism would have anything to do with the idempotent.

>> No.12285011

>>12284895
Can someone actually clarify what the notation MM means?

>> No.12285147

>>12285011
What do you mean by MM?

>> No.12285160

>>12284895
Also I should’ve mentioned these are modules. Let f’ denote the induced homomorphism. I’ve thought about it a little more and I think it comes down to coming up with a [math] g:M \to A \bigoplus B [/math] such that f’g = 1_M and [math] gf’ = 1_{A \bigoplus B} [/math]. Coming up with the right g is not obvious to me tho.

>> No.12285214
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12285214

Does someone know why Poincaré's inequality for a ball ends up with the constant in the upper bound scaling with the radius?

It seems like since you're doing the same COV in the last step, the Jacobians would cancel. I'm sure this is really trivial but it's driving me crazy.

>> No.12285220
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12285220

Solve this

>> No.12285261

>>12285214
Because along the way you use Holders inequality for the function on rhs and the characteristic function of the ball.

>> No.12285287

>>12285261
Are you speaking in general, or is that what's actually happening in the last step here in the screenshot? Thanks for the help.

>> No.12285308

>>12285147
Sorry, the entire f: m --> m, notation.

>> No.12285314

>>12285287
Just look at this:

https://ocw.mit.edu/courses/mathematics/18-152-introduction-to-partial-differential-equations-fall-2005/lecture-notes/lecture4.pdf

I know it is in 1 dimension and only for C^1 functions with 0 values on the boundary, but the general idea is similar, just more involved.

>> No.12285451

>>12285308
Function f from set M to set M.

>> No.12285457

>>12285451
Oh, and for the armchair cat theorists, I deliberately called it a function because that anon will have no use of knowing that it's a morphism or an arrow or a mapping or whatever.

>> No.12285465

How are you supposed to prove that 0 < 1 just from the ordered field axioms? He says you can prove it in my book but I think he might be forgoing to mention some assumptions or something

>> No.12285469

>>12285465
What are the axioms?

>> No.12285473

>>12285469
There exists operators + and * and an order relation < such that
>x+y=y+x
>x*y=y*x
>x(y+z)=x*y+x*z
>x*(y*z)=(x*y)*z
>x+(y+z)=(x+y)+z
>if x>y, x+z>y+z
>if x>y and z>0, x*z>y*z

>> No.12285478

>>12285469
Oh wait sorry there's more
>0 exists and x+0=x
>1 exists and x*1=x
>there exists y such that x+y=0
>there exists y such that x*y=1 unless x=0

>> No.12285498

>>12285473
>>12285478
Doesn't it specify that the order is total?

>> No.12285503

>>12285498
It has to be, i.e. the symbol < is misleading and he should have used [math]\leq[/math] instead.
As it stands, his axiomatization of field division doesn't even rule out 0=1.

>> No.12285504

Bros... Holy shit... I just invented something... Cyclical commutativity. Just arrange x y and z in a circuit like a triforce, and it goes so that (x*y)*z = x*(y*z) = x*)*y*(z* where it would loop around, and you put x in back as y*(z*x) because y is the new open position and is first in line in the cycle, and then this becomes (y*z)*x which gives you associativity for free!

>> No.12285506

>>12285498
>>12285503
What is a total order, he hasn't used that word. Also the symbol < is just to denote an arbitrary order relation in general

>> No.12285513

>>12285503
Yes I know, I was mentioning that to see if that was enough for him to come up with a proof.
>>12285506
It means that for all two members of the field it must be that x<y, x=y or x>y.

>> No.12285516
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12285516

>>12285504

>> No.12285520

>>12285513
Ah yes, his order relation is total although the = symbol only refers to x and y that are equivalent in the system of discourse

>> No.12285541

>>12285513
>It means that for all two members of the field it must be that x<y, x=y or x>y.
No, that's called something else, "connected" or "connex" or something like that (the constructivists may have yet another word).
Totality is specifically "x < y or y < x" and entails reflexivity (the "connex" relations are precisely those whose reflexive closure is total), which is why I suggested that writing the order relation symbol as < was misleading, as it suggests irreflexivity even though only the reflexive interpretation is correct (in the sense of being consistent with 0 < 1, because the axiomatization is satisfied by the field with one element).

>> No.12285599
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12285599

I'm gonna do it anyways, but will studying pure geometry have any carryover to university level mathematics?
It seems that nowadays most geometry is done analytically

>> No.12285613

>>12285599
It'll help you learn basic proof structure if you're new to maths. Other than that pure geometry is pretty niche.

>> No.12285616

>>12285613
I already understand proofs for the most part.
But I just really like geometry.
and by pure geometry I don't necessarily mean euclid only, I was studying projective geometry

>> No.12285622
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12285622

>>12285616
Just do it friend, maybe it will give you a new perspective that people don't get doing analytic geometry.

>> No.12285626

>>12285622
Thank you for the support bro :)

>> No.12285630
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12285630

>>12285599 (checked)
I am also curious about this. Geometry seems to be the most "controversial" branch of mathematics (at least according to my limited knowledge of the subject). There are a few people like Coxeter who try to keep pure synthetic geometry alive, and at the other extreme you get people like Dieudonne who reportedly said, "A bas Euclide! Mort aux triangles!" ("Down with Euclid! Death to triangles!").

Compared to other branches of modern mathematics synthetic geometry feels very much out of place. The other branches of math seems more or less unified whereas pure synthetic geometry seems to proceed in its own particular way. Maybe its because pure geometry is such an old subject. Do the big brains of /sci/ know how we ended up with this state of affairs? Does pure geometry have a future besides being a quaint curiosity taught in a high school course and then forgotten?

>> No.12285649

>>12285630
set theory is the ultimate cause of our modern degeneracy. embrace type theory, embrace topos theory... embrace synthesis

>> No.12285651

>>12285630
I'm not entirely sure. I think a lot of it is derived from Hilbert iirc, when he started the shift from geometry as being the foundation of mathematics to logic, and geometry was kind of thrown to the side as being a lesser derivation from logic.
"Hilbert once remarked that instead of points, lines and planes one might just as well talk of tables, chairs and beer mugs"
I'm not quite sure if I really agree with that though.
Anyways, It still seems really strange to me to exclude pure geometry from modern mathematics since it still seems that much modern mathematics is derived from geometry, e.g. the real numbers which are essentially just an attempt to describe a line on which you can choose any point you would like, no?

>> No.12285654

>>12285649
Are these the salvation for would be pure geometers?
I recall seeing something about topos theory while reading about synthetic geometry.

>> No.12285657

>>12285649
>topos theory
No, no, don't embrace topos theory. It's like dialectical materialism, a fad claiming novelty when it's just old crap in disguise. Topoi are a failed attempt at type theory that was clinging too much to set theory.

> embrace synthesis

There is not only one synthetic approach. But yeah, please embrace types and computation.

>> No.12285661

>>12285651
>real numbers which are essentially just an attempt to describe a line on which you can choose any point you would like, no?

Euclidean geometry can get away with real numbers. You just need "constructible numbers" which is the closure of [math]\mathbb{Q}[/math] by taking square roots. It's a strict subset of algebraic numbers, and in particular it is a countable set with decidable equality.

>> No.12285676

>>12285651
Is it really Hilbert's fault? Ultimately Hilbert's system was merely a correction of Euclid's. That quote about tables and chairs was really meant to say that we should not accept any "intuitively obvious" fact about points, lines, and planes except for those explicitly stated in the axioms. I think Hilbert's work was ultimately a continuation of a long tradition of refinement in the foundations of geometry going back to von Staudt, Pasch, and others. Hilbert's being overly autistic about rigorous logic may be a turn off for most people interested in pure geometry rather than mathematical logic per se, but I think ultimately his work was in line with the geometric tradition up to that point, it was only that it took hairsplitting to a new level. (Which is not a bad thing in itself, so far as it is limited to courses in the foundations. Trying to implement Hilbert into an elementary textbook would likely lead to failure, as George Bruce Halsted exemplifies.)

I suspect that the big shift in geometry came when linear algebra came to the forefront as a possible foundation for geometry, and so the "relational" axioms of pure geometry were substituted with more "operational" (algebraic) axioms, as it is done in the definition of a vector space.

>> No.12285677

Does anyone know the following topological property? I noticed that it shouldnt be possible to decompose S^n into a disjoint union of arbitrarily many images of closed paths. Formally, can you find paths [math] \gamma_i: [0,1] \rightarrow X [/math] so that
[math] X = \bigcup_{i \in I} \gamma_i ([0,1]) [/math] is a disjoint union?
A space which has this property is clearly [math] [0,1]^2 [/math]. But for [math] S^n [/math] it seems impossible! It would be obvious to try the paths which go from a point in the boundary towards 0, but just intuitively I don't know how to resolve the situation at 0 then because all the paths interfere with eachother. It reminds me of the hairy ball theorem, as in that there must be a point where the paths circle or meet, both situations which would leabe that point not in the image of the paths...
Furthermore given such a collection of paths, one could factor [math]X[/math] into a family of subspaces
[math] A_t = {\gamma_i(t) : i \in I} [/math] which reminds me of the product measure, because there an integral in R^2 decomposes into integrals over lines...

>> No.12285679

>>12285616
Euclidean geometry has already been solved by Tarski-Seidenberg, so unless you're interested in the computational efficiencies of geometric constructions, or an acolyte of Wildberger, you won't find much theoretical insight from it anyway.

>>12285622
>maybe it will give you a new perspective that people don't get doing analytic geometry
I was going to say that this is the best reason to study geometry, aside from artistic purposes. But "gaining a fresh perspective" sounds pretty artistic as well.
And in the wake of the mechanization of geometry and automation of constructions, "artistic insights" may be the only way to advance the subject. I don't keep up with what they're doing at Erlangen, but Hilbert axiomatization (really, Godel's completeness theorem) didn't put an end to logic, nor did the Turing machine put an end to computability theory. If these are any indication, geometry will return to being an "applied math", with its development guided by advances in other subjects like cosmology, geography, and maybe even art.

>>12285541
Turns out the proof isn't quite as simple as I expected: to prove [math]1 < 0 \implies 0 < 1[/math], show that 0 is absorbing, then multiply both sides by -1, and then add 1.

>> No.12285680

>>12285661
Well yes, euclidean geometry can be done with just constructable numbers of course. I saw a little bit about geometry with field extensions of the rationals. But I was just trying to say that the real number system is an attempt to "fill in" the line, originally, right?
Like we can take the rational numbers plus some constructable irrationals, but then you have "numbers" like pi which obviously are not algebraic and not constructable, but there is some idea of its range. And then the real numbers say that we can continually approximate this an infinite number of times.
Not necessarily saying I feel the real numbers are necessary, just trying to say that the idea of real numbers doesn't really make sense without geometry as the foundation, I feel.

>> No.12285693

>>12285661
Sounds like something of a cop out. And you still have to deal with irrational numbers, even if it's a countable subset of them. And even the ancient Greeks weren't so strict about compass and straightedge. They came up with a lot of curves that can't be constructed just with compass and straightedge. Even something as simple as the conic sections can't be constructed that way in their entirety (you can only construct conics point-by-point with compass and straightedge, not a whole continuous curve). The obsession with compass and straightedge was just something of a curiosity and a desire to reduce geometric constructions to the simplest tools, but it's not like it was illegal for Greek mathematicians to work with curves which involved non-constructible numbers.

And Dedekind cuts really are a refinement of Eudoxus' theory of proportion as described in Euclid's fifth book, which arose as a nned to dela with irrational quantities after the Pythagoreans found out about incommensurable geometric quantities.

>> No.12285708

>>12285680
>Not necessarily saying I feel the real numbers are necessary, just trying to say that the idea of real numbers doesn't really make sense without geometry as the foundation, I feel.
I feel the same way, but there may be more to it. For example, Euler's number and the exponential function arise when dealing with compound interest, so we have a fairly concrete application of mathematics involving real numbers (even transcendental numbers) and yet pretty much unrelated to geometry. Then again, the underlying issue is the same: the assumption that space (in the case of geometry) and time (in the case of compound interest) are continuous. And this assumption may or may not be true, but it works well enough in practice, so it's a good enough approximation for most purposes.

>> No.12285760 [DELETED] 
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12285760

>>12285708
>pretty much unrelated to geometry
On the contrary, it demonstrates a fundamental dependence on the geometry of continuous-time, and this is a sticking point for mathematical finance, where the actual trading and information updating can only ever happen in discrete time steps.

>>12285677
Decompose into a disjoint union of spheres, and take the latitude lines for each?

>> No.12285814 [DELETED] 

>>12285677
I think you can skewer the sphere through its diameter, and then for each [math]r\in (0,1][/math] form the radius-r sphere (minus the two punctured points at the poles, which are part of the axis) out of its latitude lines (great circles).

>> No.12285878 [DELETED] 

>>12285677
Does [math]e^{2\pi it}[/math] count as a closed path for [math]S^1[/math]?
If it does, then you can skewer [math]S^2[/math] along any axis, and then decompose it as this axis plus a concentric series of radius-r spheres (minus the pair of antipodal "poles") for [math]r\in \left(0,1\right][/math]. Each sphere decomposes in turn into a series of [math]S^1[/math] "latitude lines", indexed by their angle with the north pole, and for which the decomposition is "known".
The case for general [math]S^n[/math] should follow inductively.

>> No.12286048

>>12283965

What do you call the idea that if you choose an infinite number of things from a finite set, some element gets chosen an infinite amount of times?

I know I've seen this before in Ramsey theory

>> No.12286052

>>12286048
Pigeonhole principle or (sometimes) the Dirichlet principle

>> No.12286053

>>12285214
never mind I'm retarded! This is why I shouldn't do math at 3am

>> No.12286141

>>12285677
>I noticed that it shouldnt be possible to decompose S^n into a disjoint union of arbitrarily many images of closed paths
It is possible, use the constant paths.

>> No.12286209

Have you guys ever printed off an entire textbook PDF? I'm thinking about starting with Lang's basic mathematics but I have a hard time focusing on a computer and I don't want to pay $60 for the book. I was thinking about just going somewhere to get it printed off and just throwing it into a binder. Has anyone done this before and how much did it cost?

>> No.12286226

>>12286048
>>12286052
is this contradiction - m elements in set, N = max{n|n= times element e_i chosen i=1,...,m}
then finite mN+1 > inifinity choices
or is that the pigeonhole

>> No.12286235

Just lost my 4.0 because the shittily made chink spyware that my Uni forces everyone to install decided to crash in the middle of a proctored test.

>> No.12286249

>>12286209
I did this for the first courses. I think it was around 6 dollars with binding included for each, but I'm not american.

>> No.12286276
File: 2.39 MB, 2277x1867, ClXItsR.png [View same] [iqdb] [saucenao] [google] [report]
12286276

>>12286249
6 dollars would be amazing. I'm looking at printing it at my local staples and it's 5 cents for each page if you print over a thousand pages which isn't too bad. But If I have to print at least 1000 pages I'll have to print multiple textbooks but I don't really know which ones to get. Should I just follow the main texts on this meme chart?

>> No.12286284

>>12286276
Just print off Napkin and Lang, that'll be over 1000.

>> No.12286290

>>12286276
Book of Proof and Spivak are high quality textbooks, too.

>> No.12286304

>>12286048
Common sense.

>> No.12286326

>>12286284
>>12286290
By napkin do you mean guesstimation? So I should go Lang's -> proofs -> napkin -> Spivak?

>> No.12286335

>>12286326
Evan Chen's Napkin. Look it up.

>> No.12286339

>>12286209
That's somewhat standard practice in my country. Can't tell you about prices but just to let you know it's not that out there as an idea.
>>12286276
I never noticed the background on that image, hah.
I'm noone to say, but personally for "precalc" stuff I just kept some pdfs or googled when I had forgotten something while reading other courses. For proofs I just kinda went by feeling/learning what the book used, but maybe a book on that isn't bad.
Spivak is a neat book, I didn't complete it. But it's clear and very readable. Cool exercises.
Shilov I couldn't quite get through. I used Friedberg and I loved it.
D&F always gets criticized for being more of a reference than anything, but I'll let someone else comment on that.
>>12286326
He means Evan Chen's Napkin Project: https://web.evanchen.cc/napkin.html

>> No.12286351

>>12286326
The infinitely large napkin, by Evan Chen https://venhance.github.io/napkin/Napkin.pdf

You could also print off Fuchs Fomenko, another top tier book

>> No.12286357

>>12286326
He means the napkin by Evan Chen.

>> No.12286361

>>12286326
I think he means Evan Chen's napkin. You can read it at https://venhance.github.io/napkin/Napkin.pdf

>> No.12286369

>>12286335
>>12286339
Jesus christ that napkin project looks seriously impressive. Do I just start that after I'm done with calc?

>> No.12286379
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12286379

>>12286335
>Evan Chen's Napkin. Look it up.
>>12286339
>He means Evan Chen's Napkin Project: https://web.evanchen.cc/napkin.html
>>12286351
>The infinitely large napkin, by Evan Chen https://venhance.github.io/napkin/Napkin.pdf
>>12286357
>He means the napkin by Evan Chen
>>12286361
>I think he means Evan Chen's napkin. You can read it at https://venhance.github.io/napkin/Napkin.pdf

>> No.12286424

>>12286369
In all seriousness tho, don't actually print off Napkin (unless you want to ). Use it during your summers (I assume you're actually a student, and not a hobbyist. If you are read it whenever you want to find a new topic to learn) to wet your appetite for future topics that you'll find interesting

>> No.12286428
File: 10 KB, 149x533, sq.png [View same] [iqdb] [saucenao] [google] [report]
12286428

I simplified this square root without calculator

>> No.12286455

>>12286424
I am a student but the reason I'm considering printing it out is because I have ADD so it's hard for me to study at my computer.
>>12286379
Kek

>> No.12286496

>>12286369
I mostly used it to fuck around and learn neat things
Some of it you don't even need calc for
>>12286455
Napkin has a graph inside of chapter dependence
You could print something that appeals to you and its related chapters

>> No.12286636
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12286636

I'm a mathlet. Can I pass Calculus 1 by memorizing problem types? It's the only thing that works for me and I end up reverting back to it. For example I'm learning logarithms right now so I just memorize what certain problems look like and I copy the steps on homework/quizzes/exams.

>> No.12286646

>>12286235
Email the professor. Heavily suggest in the email that you will contact the dean if they do not permit you to retake the exam.

>> No.12286649

>>12286636
it's easier just to do the practice.

>> No.12286658

>>12286428
Impressive

>> No.12286673

Feeling a bit disappointed lads. Is all math really just abstract nonsense? I thought it was supposed to describe reality.

>> No.12286674

>>12286649
No that's what I'm saying, I practice and I end up just mentally categorizing everything into types of problems because that's the only way I learn/complete my work. It's the only way I ""learn"" math, I don't understand jack shit otherwise.

>> No.12286710

>>12286428
>>12286658
You know the trick for when something is a multiple of 3 right?

>> No.12286723

>>12286674
doomed to fail the moment you encounter a problem that isn't in one of these categories (which will happen).
what's so difficult about reading the definitions and theorems?

>> No.12286752

Why do we focus so much on arbitrary categorizations of objects and existence theorems? Why don't mathematicians work more on computational structures and methods, on making structures that represent physics, and of finding their numerical/elemental properties? I know some theorems can come to be useful but the majority of them seem in the wrong direction.

>> No.12286756

>>12286674
That's not the "only way you learn". There's no way you can't understand calculus, except if you're actually retarded.
I am 100% sure that you jump straight into solving problems without having even looked up the theory.
That just won't work.
Read the theory to understand what you're doing.

>> No.12286759

>>12286752
because abstract, in time, leads to the tangible.

>> No.12286763

>>12286752
the answer is because most of them want to intentionally avoid that so that they don't feel like watered down physicists

>> No.12286769

>>12286710
What trick?

>> No.12286774

>>12286752
Because mathematics is not engineering. It's truth-seeking.

>> No.12286775

>>12286769
Add the digits. If what you get is divisible by 3, the number is divisible by 3.

>> No.12286792

>>12286775
Ah pretty neat
https://math.stackexchange.com/questions/1435339/for-numbers-divisible-by-three-why-is-the-sum-of-their-digits-able-to-be-divide

>> No.12286797

>>12286792
Yep. Another slightly less useful one is that if a number is divisible by 3 and by 2, it's divisible by 6.

>> No.12286799

>>12283975
How else was he gonna get the thread for some basedboy Nazi LARPer that decided to die young in the Eastern Front rather than doing mathematics?

>> No.12286813

I have a family of operator [math]C(\dot):[0,T]\times\mathcal{S}\to\mathcal{S}[/math] with [math]\mathcal{S}[/math] some Schwartz space such that for a fixed [math]f\in\mathcal{S},\; C(t)f[/math] is smooth in [math]t[/math]. I managed to find the estimate [math]\lvert\lvert C(t)f\rvert\rvert_2\leq e^{Kt}||f||_2[/math] in [math]L^2[/math]. Is the unique continuous extension also smooth or at least continuous in [math]t[/math]?

>> No.12286833

>>12286813
[math]K[/math] is a positive constant just to clarify.

>> No.12286844
File: 104 KB, 647x478, 1595188917036.png [View same] [iqdb] [saucenao] [google] [report]
12286844

>the one guy in class who asks 'are we using the axiom of choice here'?

>> No.12286861

>>12286752
Many of them do. I think because we start off learning "pure" math as undergrads and many people here then either graduate and stop doing math or go on to do pure math as postgrads, we tend to underestimate how many applied mathematicians there actually are.
At my uni, the applied math department is several times larger than the pure math department. They are the ones bringing the money in

>> No.12286863

>>12286844
Do we have a choice?

>> No.12286866

>>12284895
Splitting lemma

>> No.12286874

>>12286710
>>12286775
I had no idea about this. I checked many divisors manually with long division.

>> No.12286880

2+2=4
6*8=48
32-1=31.


Kneel faggots

>> No.12286898

>>12286880
I KNEEL

>> No.12286926

>>12286874
Oof, hope it wasn't too tedious.

>> No.12286936

>>12286880
wtf
how did you discover this?

>> No.12286937

>>12286141
no constant paths, I forgot to mention.

>> No.12286956

>>12286844
I am that guy lol. What about it?

>> No.12286958

>>12286880
>6*8=48
prove it

>> No.12286976
File: 295 KB, 1080x1099, 1600130676116.jpg [View same] [iqdb] [saucenao] [google] [report]
12286976

>>12286844
>the one guy in class who asks 'are we using the Zorn's Lemma here'?

>> No.12286986

>>12286937
Do the paths have to be injective? If they can be a line that goes back and forth along itself then you can do it. Otherwise I think it's impossible because if you remove finitely many paths you get a bunch of disks and annuli, possibly some things like annuli with multiple holes, but there's always at least two disks and the intersection of an infinite descending sequence of disks is a point that can't be covered. Hmm, unless the diameters din't converge to zero... but the intersection should still be simply connected... but so is R3 and you can cover that with disjoint circles... hmm

>> No.12287047

>>12286797
that's true in general
if n is divisible by r and s, and r and s are relatively prime (i.e. they have no common divisors besides 1, i.e. their gcd is 1), then n is divisible by rs as well.

>> No.12287098

>>12285677
by closed you mean gamma(0) = gamma(1) ?

>> No.12287124
File: 8 KB, 247x250, 1603125119147s.jpg [View same] [iqdb] [saucenao] [google] [report]
12287124

Can somebody help me with the proof of this?

Show that every quotient space of a locally path connected space is locally path connected

>> No.12287183
File: 143 KB, 1224x1316, help.png [View same] [iqdb] [saucenao] [google] [report]
12287183

can somebody explain to this mega-brainlet what i need to learn in order to understand this?
i took a machine learning class because i thought it would be fun but i'm starting to think i made a mistake since i'm shit at math

>> No.12287194

[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 clean up something today?
[math]\bullet[/math]Did you plan 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?

>> No.12287215

>>12285504
>Just arrange x y and z in a circuit like a triforce, and it goes so that (x*y)*z = x*(y*z) = x*)*y*(z* where it would loop around, and you put x in back as y*(z*x) because y is the new open position and is first in line in the cycle, and then this becomes (y*z)*x which gives you associativity for free!
I tried to read this 3 times but I don't get it. Write better sentences

>> No.12287242

>>12285676
Where do you take these historical facts and references to rather obscure mathematicans?
I'm not saying that's a bad thing, but I'm surprised where this comes from. In partcilar since we talked about synthetic geometry a few threads ago.

>> No.12287249

>>12287194
what i read today
-The Real Projective Plane
-Apollonius' Conics

>> No.12287251

I've been having fun doing mathematics

>> No.12287278

>>12287124
What have you tried?

>> No.12287283

>>12287183
This book is for people like you: https://mml-book.github.io/
You mostly just need linear algebra and vector calculus.

>> No.12287311

has /mg/ ever worked through textbooks together? if not would it be something of interest?

>> No.12287315
File: 83 KB, 1491x1118, 0.jpg [View same] [iqdb] [saucenao] [google] [report]
12287315

>>12287278
This is what I have so far. I'm not sure if my approach is correct

>> No.12287338

>>12287315
I think your approach will work but I find it a weird way to think about things.
Here's my intuition: the quotient space is more connected since we can attach things but not disconnect them. Paths that worked before will still work. Open sets that worked before will still work.
All you need to do is formalise this idea by talking about the quotient topology.

>> No.12287347

>>12287315
>>12287124
>>12287278
Retard here, wouldn't the entire [math]\mathbb{R}^2[/math] plane be a quotient group of itself?

>> No.12287358
File: 7 KB, 436x179, Capture2.png [View same] [iqdb] [saucenao] [google] [report]
12287358

>>12283965
i dont even know what to do with this
i know cos(0)= pi/2 but after that is where i get lost

>> No.12287363

>>12287347
You mean under the trivial quotient where you don't change anything? In that case yes.

>> No.12287365

>>12287358
Solve the equation cos(x)=0, then find what makes e^y+1.2 =x for some y

>>12287363
Elaborate?

>> No.12287373

>>12287365
why would i solve for y now? im super confused

>> No.12287376

>>12287365
Elaborate?
Let ~ be the trivial equivalence relation where x~y if and only if x=y. Then [math]\mathbf{R}^2/\tilde \cong \mathbf{R}^2 [/math]. Or if you want to think of [math]\mathbf{R}^2[/math] as a group then consider the quotient group [math] \mathbf{R}/0 \cong \mathbf{R} [/math]. This is a trivial statement that is true for any topological space (or group).

>> No.12287383

>>12287376
Oh that notation go really messed up, I mean [math] \mathbb{R}^2/~ \cong \mathbb{R}^2 [/math]

>> No.12287392

>>12287376
I made the mistake of reading the wikipedia article, what's a good source on this?

>>12287373
I used y the second time to show that it's a different variable than x

>> No.12287432

>>12287392
https://www.youtube.com/watch?v=aO9l7zteKlE
Any introductory topology course will cover the quotient topology construction. If you haven't seen topology before then make sure you understand metric space analysis first (for example definitions of continuity, compactness, connectedness, etc in terms of open sets). Otherwise everything will seem really abstract and unmotivated.

>> No.12287453

>>12286844
I hope it's you.

Stop living in the past, make a contribution. Get rid of Choice and the lie of excluded middle.

>> No.12287460

>>12287249
>Apollonius' Conics
nice

the Burger always keeps talking about him

>> No.12287464

>>12287311

https://discord.gg/95pAyAM

>> No.12287466

>>12286976
>Zorn's Lemma
>not Kuratowski's Lemma
fucking plebs

>> No.12287468

>>12287283
thanks anon, going through it right now

>> No.12287476

What do you call a sequence f(i):N-> S where S is some finite set that is defined by a set of rules for what symbol can come next?

For example,

S = {A,B,C}
B can only come after C
C can only come after A

some sequences would be:

AAAAABC...
AAAABA...
AAAAAAA...
ABABABC...

>> No.12287484

>>12287476

whatever i got the rules rong fuck it doe

>> No.12287487

>>12287476
finite automata?

>> No.12287490

>>12287476
regular expression.

>> No.12287517

>>12287392
got it
ln((pi/2)-1.2))

could you walk me through this last problem?

>> No.12287519
File: 47 KB, 1280x496, Capture3.png [View same] [iqdb] [saucenao] [google] [report]
12287519

>>12287517
forgot pic

>> No.12287532

Please rate this curriculum:
1st Semester:
>Introduction to Calculus
>Math Lab 1 (problem solving)
>Arithmetic Fundamentals
>Quantitative Geometry 1

2nd
>Calculus 1
>Quantitative Geometry
>Analytic Geometry
>Introduction to Combinatorics and Probability

3rd
>Physics 1
>Calculus 2
>Math Lab 2
>Linear Algebra 1
>Seminars

4th
>Physics 2
>Calculus 3
>Linear Algebra 2
>Linear Analysis
>Algebra 1

5th
>Probability and Statistics
>Algebra 2
>Introduction to Topology
>Ordinary Differential Equations
>Computational Math Lab

6th
>Physics 3
>Calculus 4
>Analysis 1
>Differential Geometry
>Numerical Methods

7th
>Physics 4
>Topology
>Complex Variables
>Thesis 1

8th
>Introduction to Galois Theory
>Computational Science
>Partial Differential Equations
>Thesis 2

>> No.12287535

>>12287532
cringe

>> No.12287633

>>12287532
Forgot Analysis 2 on the 7th

>> No.12287644

>>12287466
>Kuratowski's Lemma
>not well ordering theorem
super pleb

>> No.12287649

>just introduced Taylor series in my math class
holy smokes. am i gonna be okay?

>> No.12287677

>>12287649
They're fun

>> No.12287797

>>12287649
taylor series are your friend.

>> No.12287811

>>12285473
Use the second to last property, let z be 0, x be 1, and y be 0.

>> No.12287936
File: 103 KB, 1280x720, trump_dab_election_night.jpg [View same] [iqdb] [saucenao] [google] [report]
12287936

why do elementary divisibility proofs trip me up so much number theory bros

also

>masturbating
>start thinking about number theory problem
>become interested in problem and not in boobs
>stop masturbating
>solve problem
>no longer feel like masturbating

send help

>> No.12287971

Physical books all the way right?

>> No.12287991

>>12283965
What's the fastest way to multiply two n digit numbers w/o a forrier transform?

>> No.12287995

>>12287936
/sci/ - science and math

>> No.12288014

Can /sci/ help me with this gay ass fucking stats problem? Don't know where I'm going wrong.

Suppose that Θ is a random variable that follows a gamma distribution with parameters λ and α, where α is an integer, and suppose that, conditional on Θ, X follows a Poisson distribution with parameter Θ. Find the unconditional distribution of Y = α + X.

So I try and solve for the mgf of X because I can use that to get the mgf of Y so I can fucking be done with this. Since I want the mgf of Y, I'm gonna use the iterated expectation formula, which relies on me getting th expectation of e^{Xt}. That is:
[eqn]M_X(t) = E[e^{tX}] = E[E[e^{tX} \ | \ \Theta]]= \int_{-\infty}^\infty e^{xt} \frac{\theta^x e^{-\Theta}}{x!}dx [/eqn] but this fucking integral doesn't give me anything... wolfram alpha can't even evaluate it. Am I fucking up somewhere?

>> No.12288018

>>12288014
sorry, that integral is [math] E[e^{tX} \ | \ \Theta] [/math], not the expectation of this.

>> No.12288024
File: 689 KB, 2300x1618, 20201028_045105.jpg [View same] [iqdb] [saucenao] [google] [report]
12288024

>>12287532
>calculus 4

>> No.12288064

>>12286866
I'm not particularly sure how that helps me here. Are you saying I can construct a SES using my idempotent f?

>> No.12288072 [DELETED] 
File: 42 KB, 300x421, 1601379061509.jpg [View same] [iqdb] [saucenao] [google] [report]
12288072

>>12287532
How's UFSC treating ya?

>> No.12288130

>>12287315
you write like a bitch, let me fuck your girlfriend

>> No.12288142

I saw your post
It's treating me well
I saw the archives and apparently it has a 90% drop rate, is this true?

>> No.12288145

>>12288018
>Am I fucking up somewhere?
Yes.
>have expression with [math]x![/math] term
>decide to integrate it rather than summing*
>starting from the lower bound [math]x = -\infty[/math]
Bluntly put, you should be thinking about what you're calculating. In fairness to you, this is not something that comes naturally to people, it has to be honed by practicing problems.

* For the puremathfags: in this case, the urge to generalize by writing [math]\Gamma(x+1)[/math] everywhere instead of x!, ends up working against you by obscuring the error of integrating against the wrong measure.

>> No.12288156

>>12288142
No idea. Probably? Wouldn't surprise me.

>> No.12288191

>>12288156
Why so? Apparently it kinda sucks? It's hard to get information with this pandemic so I'm not sure about transfering. I'm pleasantly surprised someone in /mg/ recognized the curriculum though

>> No.12288192

>>12288145
thanks homie. bluntly put, it's a stupid mistake that reflects badly on my competence of the material. then again, this is an online class that meets once a week about my least favorite subject. I do needa learn stats eventually, but I'll do this over winter break when i don't have any class.

>> No.12288244

>>12288192
>competence of the material
This isn't incompetence, it's carelessness resulting from unfamiliarity. Being incompetent would be if you were unable to rectify the mistake, even after it was pointed out to you.

Protip: Redefine your probability distributions to be multiplied by a {0,1}-valued indicator function (or "Iverson bracket") that explicitly marks their support, e.g. [eqn]p_X = \mathbf{1}\{ x\in \mathbb{N} \} \cdot \theta^x \cdot e^{-\theta}/x![/eqn]
or [eqn]f_X = \mathbf{1}\{ x\in\mathbb{R}, x > 0 \} \cdot x^{a-1} \cdot e^{-x/\lambda} \cdot \lambda^{-a} / \Gamma(a)[/eqn]
Then you no longer have to keep track of the integration bounds, since the indicator variables do that for you. They combine according to Boolean logic, e.g. [math] \mathbf{1}\{ x\in A \} \cdot \mathbf{1}\{ x \in B \} = \mathbf{1}\{ x \in A \text{ and }x \in B \} = \mathbf{1}\{ x \in A \cap B \}[/math], which is how you calculate the support of your final distribution.

>> No.12288261

>>12288191
It does suck, yeah, and the first two semesters are filled with mem shit. IIRC it didn't cover much less than, say, USP, tho, so there's no big deal.
>transfering
A friend of mine once considered transferring to maths. I specifically told him not to because he'd get tired within the first two semesters and drop out. I stand by my opinion.

>> No.12288263

>>12287649
He’ll fucking yes. Most interesting part of calculus. Super satisfying when you prove Taylor’s thm. One of my favorite parts of intro analysis.

Speaking of, what’s everyone’s favorite result or technique from analysis?

>> No.12288276

>>12287936
>>12287995
I once stopped having sex with a girl to work on a problem relating to idempotent functions. It was really awkward

>> No.12288283

>>12287936
This sounds like the origin to a very strange math related fetish.

>> No.12288290

>>12287811
>if 1>0, 1+0>0+0
>if 0>1, 0+0>1+0
I dont get it

>> No.12288326

>>12288261
Thanks man, I really wanted a opinion on this matter. Guess I'll just do CS and self study higher math

>> No.12288557

>>12288276
Just have sex and think about math at the same time. It's not that hard, especially if you get the other person to do all the work.

>> No.12288560

>>12287242
>Where do you take these historical facts and references to rather obscure mathematicans?
There is plenty of books on the history of mathematics. You'll pick up some of these facts if you read a few of them.

>> No.12288563

>>12288276
more like impotent functions

>> No.12288567
File: 122 KB, 720x720, 1481730293586.jpg [View same] [iqdb] [saucenao] [google] [report]
12288567

>>12288563
Savage.

>> No.12288576
File: 210 KB, 645x968, 1503241345483.jpg [View same] [iqdb] [saucenao] [google] [report]
12288576

>Consider the subgroup [math]G=\left\langle \sigma,\pi \right\rangle [/math] of [math]S_{4}[/math]...
can someone clarify what this notation means?

>> No.12288578

>>12288563
kek'd

>>12288576
The smallest (sub)group containing [math] \sigma [/math] and [math] \pi [/math]. ie you're going to need the identity, inverses, and anything you can generate with the group operation.

>> No.12288582
File: 32 KB, 530x492, CF-k9JAUEAAhvqg.jpg [View same] [iqdb] [saucenao] [google] [report]
12288582

>>12288578
thanks

>> No.12288657
File: 207 KB, 1110x1600, 1569958429044.png [View same] [iqdb] [saucenao] [google] [report]
12288657

>>12288263
>what’s everyone’s favorite result or technique from analysis?
Intermediate value theorem and >>12286048

The triangle inequality may look like analysis but is really algebraic, as evidenced by its appearance in discrete math.

>> No.12288672

>>12288657
>apperance in discrete math
Which theorem? I've seen it in linear algebra but it was with inner products and norms, kind of definitional

>> No.12288684

>>12284895
Just Karoubi complete, bro

>> No.12288693

>>12285220
No, I'm lazy. But you square it first, multiply by the two, and then divide 8 by the resulting number.

The two counts as part of the P step of PEMDAS. If they wanted 8 divided by the 2, they would've written it as a fraction, or put parenthesis around it.

>> No.12288700

>>12288672
It's in the name: the triangle inequality is a theorem of Euclidean geometry, which is assuredly non-analytic as it predates the conception of [math]\pi[/math].

In other words, the definitional character of the triangle inequality (as you've observed) is the evidence that it did not originate [math]from[/math] analysis; rather, it is being applied [math]to[/math] analysis.

>> No.12288765

Undergrad here. What course should I take next sem lads, an advanced PDEs course (that focuses on the theoretical stuff not just the solving bullshit) or combinatorial design theory? Leaning towards PDEs.

>> No.12288867
File: 118 KB, 994x340, Screenshot from 2020-10-30 07-59-43.png [View same] [iqdb] [saucenao] [google] [report]
12288867

Can anyone take a quick look at this proof? I believe it is fine but would like to be sure.

>> No.12288900

Stupid anon here.

I have a question about probability theory.
Say, there are a soldier and tank. We know only that the probability to shoot an enemy for the first one is 0.7 and 0.4 for the second one.

So, if we take these two together the probability to shoot an enemy EITHER by soldier OR tank is 1 - q1*q2 = 1 - 0.3 * 0.6 = 0.82.

Is is right to say that tank gives additional credits of success to soldier = 0.82 - 0.7 = +0.17 and soldier to tank = 0.82 - 0.4 = +0.42? And how are these additions called officially?

I try to understand the effect of support without any data about their real cooperation.

>> No.12288984

>>12288900
>Is is right to say that tank gives additional credits of success to soldier = 0.82 - 0.7 = +0.17 and soldier to tank = 0.82 - 0.4 = +0.42? And how are these additions called officially?
I suspect you want the probability ratios 0.82/0.7 and 0.82/0.4 (or equivalently, the log-probabilities log(0.82)-log(0.7) and log(0.82)-log(0.4)) instead. Those are called "(log) odds" or "odds ratios".
>without any data about their real cooperation.
No, your analysis is only valid if you explicitly assume that they are acting independently. That can be your null hypothesis though.

>>12288867
I don't see any problem either.

>> No.12288985

>>12288263
Harmonic analysis as a whole.

>> No.12288994

Today I struggled with something for about 2 hours and it basically boiled down to the definition of the matrix representation of a linear transformation. I’m in the third year of my undergrad. Which method of suicide do you recommend?

>> No.12289024

>>12288984
Thank you for your answer.

>No, your analysis is only valid if you explicitly assume that they are acting independently. That can be your null hypothesis though.
Is there any valid way to make a reasonable assumption about what will be the support of each other if they are NOT acting independently?

>> No.12289037

>>12288994
Watch 3blue1brown

>> No.12289045

>>12288867
I love mathscr so much bros

>> No.12289076

>>12286799
contributed more to mathematics than you

>> No.12289100
File: 83 KB, 900x900, no.jpg [View same] [iqdb] [saucenao] [google] [report]
12289100

>>12289024
Is there any valid way to draw a reasonable map of a territory if you do NOT know what it looks like?

(And do you actually know that they are not acting independently, or is that just an assumption? In certain situations, knowing that some event will occur with positive probability can be useful, but this is unlikely to be what you're looking for.)

>> No.12289122

>>12289100
You have a point.

Thank you again.

>> No.12289124

How to show?

Let [math] [x] [/math] denote the equivalence class of [math]x \in X[/math]. This is called the path-component of [math]x[/math]and consists of all points that you can connect to [math]x[/math] by a path in [math]X[/math]. Let [math]\pi_0(X) = X /\sim[/math] denote the set of path components of [math]X[/math]. Give [math]\pi_0(X)[/math] the natural quotient topology and write the quotient map [math]q_X:X \w \pi_0(X), q_X(x) = [x][/math]. Show that if [math]f:X \w Y[/math] is a continuous function, then [math]f_0: \pi_0(X) \w \pi_0(Y), f_0([x]) = [f(x)] [/math]is well defined and a continuous function.

I think I can do it through showing the composition is continuous but I'm not sure?

>> No.12289127

>>12289124
sorry \w is a shitty macro for \longrightarrow

>> No.12289130

>>12289045
I too love it when half my letters look the same.

>> No.12289264

>>12289127
Just use \to

>> No.12289293

Why are eigenvectors of AT * A and, conversely, singular vectors of A orthonormal? I understand why they are orthogonal (AT * A is symmetric) but I don't see how they are unit length.

>> No.12289297

>>12289293
You can just choose them to have unit length. Do you understand what an eigenspace is?

>> No.12289298

>>12289293
Eigenvectors can be made any length you want.

>> No.12289300

>>12289297
>>12289298
Yep, posted this and realized what an idiot I am, we don't care about their norm, we only care about direction, so you can just choose a unit vector in this "eigendirection". Silly me.

>> No.12289356

Guys. How come "if anon is a faggot, anon watches anime" is true?
Also, listen to Tinariwen:
https://www.youtube.com/watch?v=H1YIgwPsX5Q

>> No.12289389

>>12289124
>>12289124
>>I think I can do it through showing the composition is continuous but I'm not sure?
yeah you don't have much else at your disposition

as usual with eq classes, just be sure your result is independant of the representant.

>> No.12289391

In the future the low IQ populations will look at us mathematicians and think it's all just a very elaborate meme or that we're all just pretending to be smart to boost our egos. This process will be accelerated by low tier diversity hire mathematicians similar to Piper Harron who will constantly publicly complain about the arrogance, the exclusivity of mathematicians, how they obscure their thoughts with obscure jargon, how they take pride in not being understood by PoCs and common people who want to get into the field, how they fail to take the historical mathematical achievements of various african tribes seriously. These voices will be amplified by the media on all fronts, continually diminishing the respect people hold for mathematics.
There will be multiple websites like /r/iamverysmart dedicated to posting mathematical papers, excerpts from textbooks and our faces. The typical comments will be "yup, this guy needs to leave his room once in a while", "I bet he's fun at parties", the classic "Yikes", "Someone likes the thesaurus too much".

>> No.12289406

>>12283965
Seeking advice here. What should I orient my graduate education if my goals are to learn useful things for my job (predictive modelling) but also specialize in an area where I can sit comfy writing interesting papers.

I know analysis and stochastic analysis are good but is there anything more specific? Like specific classes that would aid? My goal is to become competent enough to produce independent research but I also want to become better at my job.

>> No.12289424

>>12289391
>continually diminishing the respect people hold for mathematics.
nobody respects mathematicians, even before the birth of nihilism. Also mathematicians are very few, all over the world. Cartier estimates the number at 10k.

>> No.12289552

>>12288563
Kek nice one, I got btfo

>>12288576
The angle bracket section is what's called a "group presentation", and S4 is the 4th permutation group

>> No.12289566

>>12289552
That's not a presentation you dweeb, there are no relations. And finite symmetric groups don't have free subgroups.

>> No.12289579

>>12287532
>intro calculus
>calculus 1
>calculus 2
>calculus 3
>calculus 4
>"math lab"
>quantitative/analytic geometry (high school math?)
>arithmetic fundamentals (middle school math?)
>physics classes just have numbers, no names
>called complex "variables" instead of complex analysis
>implying the differential geometry class is real differential geometry if taken simultaneously with analysis 1
>intro to topology and topology
can you just start at 4th semester, cut out the bloat, and take grad classes for 2 years? i mean come on.

>> No.12289580

>>12289566
Elaborate?

>> No.12289585

>>12288765
advanced pde is fun

>> No.12289596

>>12288263
>result
banach fixed point theorem
>technique
summing things with increasingly bad properties /2^n to get something with really awful properties. simple example would be weierstrass functions.

>> No.12289599
File: 345 KB, 1280x720, Screenshot_20201030-174929.jpg [View same] [iqdb] [saucenao] [google] [report]
12289599

>>12283965
>messes up sentence
>loudly exhales
>repeats sentence

Why does he always do this?

>> No.12289606

Is there a direct proof of the fact that there's no integer between 0 and 1? All proofs I've read use contradiction

>> No.12289611

>>12286209
Printed off Pressley's Elementary Differential Geometry yesterday; school library for cost of tuition

>> No.12289613

>>12289580
It's the subgroup generated by those two elements, not a presentation. If it was a presentation it would be the free group on two generators, which would be infinite. Do you know what the subgroup generated by a set is?

>> No.12289679

>>12289606
Not sure if you can directly prove a negative, but perhaps you could show that [math]x \mapsto 2^x[/math] is an order-isomorphism from [math]\left\{ x\in\mathbb{Z} \mid 0 \leq x \leq 1 \right\}[/math] to the powerset of {0}.

>> No.12289693

>>12288244
I came back to this problem and I still can't fucking crack it... Taking the sum in place of the integral does simplify, and then I'm left with
[eqn]M_X(t) = E\left[\exp(\theta(e^t - 1))\right] [/eqn]
as the mgf of X, but this is horseshit still. [math] \theta [/math] follows a gamma distribution, and so the expected value of that garbage is a mess (again, wolfram alpha doesn't evaluate it), so I'm at another fucking wall

>> No.12289696

>>12289679
Wait no, [math]x\mapsto 2^x[/math] makes no sense.
But [math]\mathrm{card}: \mathcal{P}(\{0\}) \to \left\{ x\in\mathbb{Z} \mid 0 \leq x \leq 1 \right\}[/math] seems like it should work.

>> No.12289712

>>12289693
that's just a compound Poisson distribution bro

>> No.12289719

>>12289712
I don't even know what that means familia, please elaborate.

>> No.12289724

>>12289606
The integers are all R who are + or - by 1 from 1, and no others. 1-1 is 0. Any int that's +1 to 1 or -1 from 0 is greater than 1 or less than 0. So none of them fit 0 < x < 1. And of course 1 and 0 don't fit either.

>> No.12289738

>>12289719
Let [math]\theta[/math] be Gamma-distributed. Now let [math]N \sim \textrm{Poisson}(\theta)[/math]. Then the variable
[eqn]X = \sum_{k=1}^{N} Y_k,[/eqn]
where the variables [math]Y_k[/math] are iid, is compound Poisson distributed. Calculating the moment generating function of this variable would be a fun exercise for you and what you get is going to look like the function you have in >>12289693
(Hint: the variables Y are allowed to be deterministically a constant)

>> No.12289752

>>12286209
>Have you guys ever printed off an entire textbook PDF?
lol yes, i printed loads, but from a lab and from a computer belonging to someone else, in order not to have my counter going up


all printing jobs are tracked.

>> No.12289755

>>12289738
>would be a fun exercise
for someone who likes stats maybe
>and what you get is going to look like the function you have in
And I'm stumped here. This compound variable stuff hasn't been discussed yet, so I'm unsure how to proceed after this expectation.

>> No.12289765

>>12289606
I'd use proof by contradiction

>>12289613
I know what subgroups are, admittedly I'm still learning and permutation groups have been a thorn in my side

>> No.12289796

>>12289755
You don't need any compound shit. Literally look up the MGF of a Poisson distribution and compare with what you have.

>> No.12289805

>>12289738
>>12289755
Alright, so
[eqn]\exp(\theta(e^t - 1))[/eqn] is the mgf of a Poisson random variable w/parameter [math]\theta[/math], so the expected value of this is just
[eqn]E[\exp(\theta(e^t - 1))] = E[E[e^{Xt}]] = E[e^{Xt}] =...[/eqn]wtf

>> No.12289809

>>12289606
It follows from the usual construction of the real numbers and the definition of the naturals as a subset. What you have to understand is that mathematicians don't prove obvious shit because it isn't obvious, but to see if their particular axiomatic system works. There are many ways to construct the real numbers, and so you have to specify how you are defining the natural numbers in the real numbers. I'm going to assume you are using field axioms and so >>12289724 Is tthe usual definition because you only know you have special elements 1 and 0.

>> No.12289828

>>12289809
All constructions of [math]\mathbb{R}[/math] I know start from the rational numbers (Cauchy sequences, Dedekind cuts, decimal expansions), and the rationals, as far as I know, are built from the natural numbers.

>> No.12289838

>>12289828
You can start with an unspecified field with a total order and then afterwards construct all the subsets you know. Is the approach done in spivak and used by people who don't like set theoretic shit.

>> No.12289852

>>12289838
I know, but that still doesn't answer my question. The proof you mention is only valid if [math]\mathbb{R}[/math] exist, which (as far as I know) can only be proven by the construction of the natural numbers. Then the proof would be redundant

>> No.12289892

>>12289852
Again, the real numbers as defined through the axioms, not as a set theoretic construction. The real numbers "exist".

>> No.12289906
File: 28 KB, 480x480, anons second wife.jpg [View same] [iqdb] [saucenao] [google] [report]
12289906

>>12289809
We wants to prove
[math]\neg \exists n. \big((0<n) \land \big(n<1)\big)[/math]

where, I suppose,

[math]x<y := \exists k. (x+k=y)\land \neg(k=0)[/math]

In natural deduction style, assuming, for a contradiction,
[math]\exists n. \big((0<n) \land (n<1)\big)[/math]
instantiate
[math]n, i, j[/math]
with the four properties

[math]0+i=n[/math],
[math]\neg(i=0)[/math],
[math]n+j=1[/math],
and
[math]\neg(j=0)[/math]
using +0 being dropped by the PA axioms on either side, this is

[math]\neg(n=0)[/math],
[math]\neg(j=0)[/math]
[math]n+j=1[/math],
The question is thus reduced to deriving that the above 3 relations imply
[math](n=0)\lor (j=0)[/math]
formally in PA.
Seems obvious enough, although I'm too tired to put 1 and 1 together now, pun intended

>> No.12289919

To spell it out in words

If [math]n[/math] and [math]j[/math] are two nonzero numbers, use the Peano axioms to derive contradiction (i.e. derive [math](0=1)[/math]) from the claim
[math]n+j=1[/math].

>> No.12289931

Why is all math seemingly reduced to functions of two things rather than operators on >2 things?

>> No.12289934

>>12289931
Think.

>> No.12289942

For the love of God someone please kill me. Posting again because fuck me.

Suppose that Θ is a random variable that follows a gamma distribution with parameters λ and α, where α is an integer, and suppose that, conditional on Θ, X follows a Poisson distribution with parameter Θ. Find the unconditional distribution of α + X (Hint : Find the mgf by using iterated conditional expectations.

>> No.12289948

I suppose since 1 doesn't have more axioms than being singled out as the multiplicative unit, one must multiply (n+j)·x=x

>>12289931
It's called ternary operation fyi

>>12289934
I thought for a second but can't give a moral justification desu. Other than that it's easier.

One can of course say all functions are only ever really functions of a single arguments and 2-ary arguments are 1-ary one's in disguise. But I suppose the underlying question is whether there's a reason 3-ary notation is rarely the setup in mathematical theories.

I recall a SE thread on this.

And there's also of course counterexamples, e.g. the associator or what's it called, where you study
x·(y·z)-(x·y)·z in its own right.
But it's rare.

>> No.12289953

If x and y are two nonzero naturals, does x+y=1 lead to a contradiction?

Why, formally?

>> No.12289963
File: 305 KB, 1048x1584, 1588079118725.png [View same] [iqdb] [saucenao] [google] [report]
12289963

>logic and foundations

>> No.12289964

>>12289931
What about tensors, multilinear maps, determinants?

>> No.12289972
File: 132 KB, 800x1200, nicola_cavanis4.jpg [View same] [iqdb] [saucenao] [google] [report]
12289972

>>12289948
found the thread I mentioned

https://math.stackexchange.com/questions/94690/is-anybody-researching-ternary-groups

>>12289963
we're all deciding to live in our own personal hell

>> No.12289985

>>12289972
>nicola cavanis
i wonder how many men nicola cavanis picked up for anal sex
wonder also how much easy is nicola cavanis life
wonder if nicola cavanis is a feminist and claims she has a hard life

>> No.12289995

>>12289948
Redpill me on the associator

>> No.12289996

>>12284305
Think of it this way, subtraction shows us the amount of steps between two numbers
>27 - 16 = 11
There are 11 steps between 27 and 16
But how many steps are there between 27 and -16? The answer is 27 + 16
Or just think of it this way, you have 27 apples. Now take 16 apples. You just grabbed some apples. But if you were to take -16 apples, what would you do? The opposite

>> No.12290002

>>12289934
I would argue that it's because binary operations are the simplest form of interactions

>> No.12290053

>>12289995
measures associativity

>> No.12290055 [DELETED] 
File: 1.18 MB, 1612x1186, brafe.png [View same] [iqdb] [saucenao] [google] [report]
12290055

>>12289995
https://en.wikipedia.org/wiki/Associator

I only came across it in relation to the Star/Moyal product.
Don't ask me to redpill you about the Star product.

>>12289985
she brave woman

>> No.12290059 [DELETED] 

Although admittedly it was a shit example, since it's probably never studied abstractly (but instead is defined just in terms of a binary operation)

>> No.12290063
File: 861 KB, 1548x830, b.png [View same] [iqdb] [saucenao] [google] [report]
12290063

>>12289995
https://en.wikipedia.org/wiki/Associator
(Although admittedly it was a shit example, since it's probably never studied abstractly (but instead is defined just in terms of a binary operation)
)

I only came across it in relation to the Star/Moyal product.
Don't ask me to redpill you about the Star product.

>>12289985
she brave woman

>> No.12290119

>>12289942
Oh hey, I see you're still at it.
You've managed to get the MGF in >>12289693. I don't know why Wolfram doesn't recognize it, because that's exactly the expression for the MGF of [math]\theta[/math], but evaluated at [math]e^t-1[/math] instead of t. I.e.
[eqn]M_X(t) = \left(1 - \lambda(e^t-1) \right)^{-\alpha}/\mathbf{1}\{ e^t-1 < 1/\lambda \}[/eqn]
So all that remains is for you to recognize this MGF, for which it helps to rewrite the support as a radius of convergence [math]\mathbf{1}\{ t < \log( \frac{1+\lambda}{\lambda})\}[/math].
This suggests the parameter transform [math]p = \frac{\lambda}{1 + \lambda}[/math], i.e. [math]\lambda=\frac{1}{1-p}-1[/math], so that after substituting
[math]M_X(t) = \left(1 - (\frac{1}{1-p} - 1)(e^t-1)\right)^{-\alpha}/\mathbf{1}\{ t < -\log p \}[/math]
you can expect the numerator to simplify nicely (and you can probably already guess what the final distribution of X will be).

>> No.12290192

>>12290119
I can tell by the way you talk that you have a big cock. Thank you very much for continuing to help me on this stupid problem anon.
> that's exactly the expression for the MGF of [math]\theta[/math]
the MGF of a gamma distribution is the expected value of the MGF of a poisson distribution? Evaluating the expectation w.r.t. [math] \theta [/math] gets
[eqn]E[\exp(\theta(e^t - 1))] = \int_0^\infty \exp(\theta(e^t - 1)) \frac{\lambda^\alpha \theta^{\alpha - 1}}{\Gamma(\alpha)} e^{-\lambda \theta} \ d\theta = \int_0^\infty \frac{\lambda^\alpha}{\Gamma(\alpha)} \int_0^\infty \exp(\theta(e^t - 1))\theta^{\alpha - 1} e^{-\lambda \theta} \ d\theta [/eqn]
This part is incredibly confusing to me.

>> No.12290196

>>12290192
ugh ignore the first integral in the third expression, why can't we edit posts on 4chin?

>> No.12290226

Friendly reminder that nobody has proved the smoothness of the Navier-Stokes equations. Over 100 years old and you get one 1 million bucks for the proof. Get going. I mean, you are as smart as you say you are on 4chan, right?

>> No.12290236

>>12289972
Emil Artin wrote a 150 page book on n-ary group theory in 1940, initiating the discipline. It's still actively studied. People on StackExchange are so excited to answer even when they don't work in the area.

>> No.12290239

>>12290226
tao is on the job

>> No.12290279

>>12290192
No, I meant that the expression is of the form [math]M_X(t) = E_\theta[e^{u\theta}][/math] for [math]u=e^t-1[/math] (or to be pedantic, [math]u(t)=e^t-1[/math]).
Protip 2: Subscript your expectation operators, so that you can see explicitly what you're integrating over. In this case, [math] E_\theta[e^{u\theta}][/math] is the telltale pattern of an MGF, and [math]\theta[/math] is a random variable whose distribution you already know. This should prompt you to write out the functional form of
[math]M_X(t) = M_\theta(u(t)) = M_\theta(e^t-1)[/math]
which has already been given.

>> No.12290285

>>12290239
Is he? Is that one of the problems he's trying to solve?

>> No.12290306

Do you guys use any made up symbols as you work? I made a symbol thats like a V with an | through it, like a trident. I use it to compare two objects before making a statement about them, it seems helpful. I call it comparator

>>12289948
The reason I asked the question is because I was counting an octagon, and I said 22 22 in my head and it instantly went to 8. But something felt off, as though it was disallowed to add 4 things at once or even recognize 4-ness as a distinct rationale without counting up by units. But then I wondered why going 2 4 6 8 is any better. Is recognizing 4 in the brain possible, or does the brain and reality split it into units at the lowest level? I believe n-ary is possible.

>> No.12290314

>>12290285
Yes.

>> No.12290318

Bros. I just discovered one fundamental difference between how normal life thinking differs from mathematical thinking
>almost all "if statements" irl, are in fact "iff statements", at least partially
This is especially true in biological, natural, and social things

>> No.12290321

>>12290226
Get us started then. Post your progress.

>> No.12290347

What are some practical applications of pure magmas?

>> No.12290403

Cons cells in Lisp, I guess.
It seems like anything you'd want from a pure magma [math]M:\Omega \to \Omega^\Omega[/math] can also be done by the more general action [math]S:\Sigma \to \Omega^\Omega[/math], which has both theoretical and practical uses.

>> No.12290408

>>12290347
Smelting iron.

>> No.12290416

>>12290403
Consing is not a binary operation.

>> No.12290425

>>12290347
protecting your virginity.

>> No.12290537

https://www.youtube.com/watch?v=8OkpRK2_gVs
Can someone post more songs like this?

>> No.12290662

>>12290537
https://www.youtube.com/watch?v=bq4SGw_L-c4

>> No.12290663

>>12283965
Any good resources for learning about series? I'm taking calc II and have an A so far but my professor really isn't very good and it's catching up. Am I supposed to be able to always be able to find what they converge to? Are partial sums always obvious? The convergence/divergence tests are fairly straight forward, we've learned up until the limit comparison test.

>> No.12290724

>>12290662
Anon you're gay.

>> No.12290734
File: 22 KB, 680x375, ElMCKAGXIAM1fJQ.jpg [View same] [iqdb] [saucenao] [google] [report]
12290734

Asked this in /sqt/ to no avail.

This being the case, what is the difference between the "p-adics" and "base p" ?

>> No.12290748

>>12290734
p-adics can have infinitely many digits left of the decimal point, but not right of it.

>> No.12290750

>>12290734
They're completely different things. One is an algebraic structure and the other is just a notation for natural numbers.

>> No.12290902

>let this axis represent n-1 dimensions

>> No.12291070

What's the consensus on the best measure theory book, cu/mg/uzzlers?

>> No.12291092
File: 33 KB, 334x334, alexandre-grothendieck.jpg [View same] [iqdb] [saucenao] [google] [report]
12291092

>>12291070
You're supposed to rederive it yourself using your intuition anon.

>> No.12291122

>>12291092
p-please anon, spare me a crumb of reference to the written assistance...

>> No.12291147

>>12290663
Sort of on the same boat. Is this your first time taking this subject?
What book are recommended by your professors? Meme books like Apostol, Spivak should be fine if you are proving stuff (I know this is /mg/ but sometimes non math majors come here to ask stuff).
>Am I supposed to be able to always be able to find what they converge to?
Kind of a tricky question but no, mostly just if they do. Also, if you're at power series, for what values of x they converge.
>Are partial sums always obvious?
No, not really. They require some work, there are techniques though.

>> No.12291157

>>12291147
Also Professor Leonard's videos are really good. No shame in watching them, particularly if your prof sucks

>> No.12291192
File: 18 KB, 333x499, 41bnIUsE2uL._SX331_BO1,204,203,200_.jpg [View same] [iqdb] [saucenao] [google] [report]
12291192

kino has arrived.

>> No.12291396

>>12285487
You misunderstood the implication, it's context-dependant.
If you don't understand common induction applications, then you don't really understand the concept. You barely have to think on how to apply on common examples, if you already understand the principle.

>> No.12291431

Do you guys set a benchmark of pages/problems to be finish for each block of time (hour/day/week/month/whatever) spent studying a book or paper? Why? Does it work?

>> No.12291458

>>12291431
-No
-I'm not THAT autistic
-yes

>> No.12291478
File: 18 KB, 416x133, asasas.png [View same] [iqdb] [saucenao] [google] [report]
12291478

Brainlet here.
Pls explain what this means and how to solve it. I know how to solve a separable diff equation (the chapter this is from) and I know what the initial value problem is
But I'm not being able to connect the dots.
Thanks.

>> No.12291480

>>12291396
>common
So you admit that understanding the induction principle need not imply the ability to solve induction problems, unless they are "common"?
I can buy that, but only if you make it clear what distinguishes a "common" application from an "uncommon" one.

>> No.12291500

>>12291478
Not my thing so take it with a grain of salt, but it'll probably be obvious if you differentiate both sides

>> No.12291502

>>12291478
Differentiate both sides and figure out a differential equation. This is why nobody talks about integral equations, because they can be transformed into differential equations

>> No.12291507

>>12291502
>>12291500
That is what I did but I thought I was missing something.
What do I do with the y(2) that shows up? Just treat it like a constant?

>> No.12291512

>>12291507
https://en.wikipedia.org/wiki/Fundamental_theorem_of_calculus#First_part

>> No.12291532

>>12291512
Now I have a family of differential equations. What does that "hint" part mean?

>> No.12291542
File: 37 KB, 361x500, think with full brain.jpg [View same] [iqdb] [saucenao] [google] [report]
12291542

>>12291532
bruh, use your brain
[math]y'(x)=\frac{d}{dx}\int_2^x f(t)dt=f(x) [/math]

>> No.12291549

>>12291157
Thank you anon, I'm doing alright so far, no missed marks on my homework, but I don't want to fall behind. I think I'll just diversify my resources.

>> No.12291556

What's a good book for finding the closed
form of the summations?

>> No.12291577

>>12291542
But that is not the equation on the picture:
[math]
y(x) = 2 + \int_2^x [t - ty(t)]dt
[/math]

[math]
y'(x) = x - xy(x) + C
[/math]

What does it mean to "use an initial condition obtained from the integral equation"?
If I had the value of y(a) = b then I could use it to find C and thus one equation from the family. Do I use a = 2? If so why?

Thanks for the help.

>> No.12291589

>>12291577
Let t-ty(t)=f(x). Use >>12291542. Notice how there is no +C there, since [math]\int_2^x f(t)=F(x)-F(2)[/math] and F(2) is a constant, so when you differentiate, it goes away

>> No.12291615
File: 1.45 MB, 900x972, 1558264300732.png [View same] [iqdb] [saucenao] [google] [report]
12291615

>>12291589
Thanks femanon.
>>12291542
Thanks femanon.

>> No.12291636
File: 39 KB, 564x640, 1591541389150.jpg [View same] [iqdb] [saucenao] [google] [report]
12291636

>>12291615
>>>12291589 (You)
>Thanks femanon.
>>>12291542 (You)
>Thanks femanon.
Yeah np

>> No.12291642

>>12291577
a=2 is a good choice because the integral of anything from 2 to 2 is 0

>> No.12291654
File: 89 KB, 700x827, sppjoztm7vu51.jpg [View same] [iqdb] [saucenao] [google] [report]
12291654

>>12286209
>>12283965
Looking at buying this book. Reviews are saying there are errors throughout the book. Is it still worth it? On Amazon you can open it to random pages to look through the book, and on a random page I saw a problem where the value -(-2) randomly got converted into 4... This book sounds great but random errors sounds like bullshit. Can anyone please let me know? Thanks!

>> No.12291671

>>12291654
Lang is a meme.

>> No.12291678

>>12291654
Download the pdf fag.

>> No.12291679

>>12291671
Yeah I think I'm gonna cancel the order. I saw grammatical errors and more math errors. Really lame.

>> No.12291680

>>12291678
will do ma'am.

>> No.12291745

>>12290236
*Emil Post

>> No.12291778

Successor >>12291777

>> No.12291779

>>12284305
Try drawing this on a number line.

>> No.12292654

>>12284305
Imagine someone removing (subtracting) your debt (negative money). That increases your net worth.

>>
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