/mg/ maths general

[Anonymous No.12166845]

Space-filling edition
Talk maths
Formerly >>12157556

[Anonymous No.12166850]

>>12166845
Is this the famous Hitler curve?

[Anonymous No.12166853]

>>12166845
Fuck off cunt I already posted one.
>>12166538

[Anonymous No.12166869]

>>12166853
You made a highly inappropriate threat. We cannot talk maths in such a thread.

[Anonymous No.12166881]

>>12166869
>t.nikolaj
kys asap

[Anonymous No.12166909]

>>12166881
>>12166853
Nikolaj is based. Fuck off.

[Anonymous No.12166920]

>>12166909
Die in agony <130 IQ normalfag

[Anonymous No.12166935]

>>12166909
And cute.

[Anonymous No.12166981]

>>12166935
So you're a necrophile then?

[On the Existence of Digital Ob(...) No.12166987]

/gmmg/ on this blessed Sunday.

>>12166853
>>12166869
>>12166881
>>12166909
>>12166920
>>12166935
>>12166981
Do you guys have so little to do that you start a thread with 10 posts without math? The gun pic isn't bad though.

Here's some random geometric series related claims to get the thread starting

[math] \sum_{k=0}^\infty\left(\dfrac{z-a}{1-a}\right)^k = \dfrac{1-a}{1-z} [/math]

[math] \sum_{k=0}^\infty\, (-1)^k \cdot (1 - \epsilon)^k = \dfrac{1}{2-\epsilon} [/math]

[math] \sum_{k=0}^\infty z^k = \prod_n\exp\left(\dfrac{z^n}{n}\right) [/math]

[Anonymous No.12166998]

>>12166987
Those are rather trivial, aren't they? You only need first year undergrad knowledge to see why they hold.

[Hille–Yosida No.12167075]

>>12166998
Yes that's true, you indeed just need the geometric series, logarithms and arithmetic.

Duality of man, there's not so many theorems that aren't well known landmark statements such that one wouldn't have to explain the notation to make them into something in the first place.
But okay, here a vaguely related but hard one:

[math]z[/math] a linear operator defined on a linear subspace [math]D(z)[/math] of a Banach space, [math]b[/math] real and [math]c > 0[/math].
Then [math]z[/math] generates a strongly continuous semigroup [math]Z[/math] (as in "[math]Z(t)=e^{tz}[/math]") with [math] ||Z(t)|| \le e^{b t} [/math]
iff
[math]z[/math] is closed and D(z) is dense in the space and every real [math]a > b[/math] belongs to the resolvent set of [math]z[/math] and for such [math]a[/math] and for all positive integers [math]n[/math],
[math] \|\dfrac{1}{(a\cdot I-z)^n}\|\leq\dfrac{c}{(a-b)^n} [/math]

[Anonymous No.12167095]

What is the most difficult obvious fact to prove, and why is it the Jordan Curve Theorem?

[Anonymous No.12167097]

>>12167095
https://www.maths.ed.ac.uk/~v1ranick/jordan/tverberg.pdf

[Anonymous No.12167101]

>>12166853
I'd like a burger with extra seethe, a dilate Cope and large fries.

[Anonymous No.12167154]

>>12167095
If statements about the reals, such as the Jordan Curve Theorem can count as obvious, then this is my candidate

>"If a set has more sets than another, it also has more subsets"

More formally,

[math] |Y| > |X| \implies |{\mathcal P}Y| > |{\mathcal P}X| [/math]

which happens to be independent of ZFC!

And afaik large cardinal axioms don't pin it down either.
(To be fair you said, difficult to prove, and not unprovable from sensible axioms. I tried to make the case that Jordan Curve Theorem might be strong too, but I suppose it's moderate. fyi it appears to be equivalent to the König's lemma (https://en.wikipedia.org/wiki/K%C5%91nig%27s_lemma)), which is on the level of weak dependent choice)

[Anonymous No.12167213]

>>12167154
You mean "if a set has more elements than another." I don't think that's actually obvious at all for infinite sets though. By comparison, the Jordan Curve theorem says that a closed loop on a plane has an inside and an outside. Your statement says that if two sets cannot be put into one-to-one correspondence, then neither can the sets of all their subsets.

I mean, it *is* obvious in the finite case, where you are just counting elements. And it is easy to prove for sufficiently small infinite sets. But I wouldn't call it obvious.

[Anonymous No.12167236]

>>12167154
Wait, I just reread your post and now I'm confused. This fact cannot be proved from naive set theory (or it can, but so can its negation), but it is trivially easy to prove in ZF.

[Anonymous No.12167248]

>>12167236
Sadly it can't, as will be clearer a few sentences down the line

>>12167213
>You mean "if a set has more elements than another."
Same thing if everything is a set, but yes that's a clearer formulation.

>And it is easy to prove for sufficiently small infinite sets.
It actually fails right away, in that there's you can force models with

[math] |{\mathbb N}|<|S|\ \ \ \land\ \ \ |{\mathbb R}| = |{\mathcal P}S| [/math]

This is what violates the continuum hypothesis.

>By comparison, the Jordan Curve theorem says that a closed loop on a plane has an inside and an outside.
While I agree with your general sentiment, you make it yourself easy by speaking of "a loop on a plane."
What's a plane here? Is it some set of pairs [math]R\times R[/math] such that all axioms we know don't bound the the quantity of those points by the cardinality of any ordinal (this is again related to the CH)?

In case we have to report that a plane and a curve really is, I think the Curve theorem and the statement about subsets are similarly obvious.
If we don't have to report what R is, we might as well pull a Tooker, define it ourselves and prove whatever millenium price problem we want :P

[Anonymous No.12167256]

[eqn]\displaystyle{\int f(x) dx = \sum_{n=1}^{\infty} f \left(x_{i}^{*}\right) \Delta x}[/eqn]

[eqn]\displaystyle{\frac{d}{dx}f(x) = lim_{h \rightarrow \infty} \frac{f(x+h) - f(x)}{h}}[/eqn]

[Anonymous No.12167261]

>>12167256
now that's what I call BASED

[Anonymous No.12167269]

[eqn]\displaystyle{\displaystyle{\int_{a}^{b} f(x) dx = \sum_{n=1}^{\infty} f \left(x_{i}^{*}\right) \Delta x}}[/eqn]

[eqn]\displaystyle{\displaystyle{\frac{d}{dx}f(x) = lim_{h \to 0} \frac{f(x+h) - f(x)}{h}}}[/eqn]

[Anonymous No.12167279]

>>12167256
>>12167269
Use \lim, brother.

[Anonymous No.12167281]

>>12167248
Yeah my first impression was correct. When I looked at the post again, I did a double-take because somehow I thought you had written down Cantor's theorem.

>What's a plane here?
It doesn't really matter. A statement is not "obvious" because of its formal construction. It is obvious because of its physical analogs. It is obvious that a closed loop on a plane divides it into two disjoint parts. In fact, it is so obvious, that any model of the plane for which the theorem is false is arguably not even a model of a plane.

For instance, it doesn't have to be [math]\mathbb{R}^{2}[/math]. It could just be the constructible plane. Of course in that case, we would have to restrict our definition of a "curve" to constructible curves as well. In this case though, the theorem is still not easy to prove.

[Anonymous No.12167282]

>>12167269
>riemann instead of lebesgue integral
>no backslash before lim

[Anonymous No.12167284]

[eqn]\displaystyle{\iint_{R} f(x,y) dx dy = \sum_{i=1}^{\infty} \sum_{j=1}^{\infty} f \left(x_{i}^{*}, y_{j}^{*}\right) \Delta A}[/eqn]

[eqn]\displaystyle{\frac{\partial}{\partial x} f(x,y) = \lim_{h \to 0} \frac{f(x+h,y) -f(x,y)}{h} }[/eqn]


[eqn]\displaystyle{\frac{\partial}{\partial y} f(x,y) = \lim_{h \to 0} \frac{f(x,y+h) -f(x,y)}{h} }[/eqn]

[Anonymous No.12167287]

>>12167282
redpill me on lebesgue

[Anonymous No.12167292]

>>12167287
he made a better integral

[Anonymous No.12167294]

>>12167095
How about proving that R^n and R^m are not the same topologically for any n \neq m?
Or maybe proving some simple fact like that a torus has more holes than a sphere?
There has to be some appropriately obvious fact which takes homology to prove.

[Anonymous No.12167296]

>>12167269
[math]{}_a\mathbb{D}^q_tf(t) = \frac{d^qf(t)}{d(t-a)^q} = \frac{1}{\Gamma(n-q)} \frac{d^n}{dt^n} \int_{a}^t (t-\tau)^{n-q-1}f(\tau)d\tau[/math]

[Anonymous No.12167306]

>>12167287
Lets you integrate more things because it slices vertically instead of horizontally. Instead of partitioning into intervals, you partition into "measurable sets." The measures of such sets replace Delta x in your sum. Sounds like a simple change but if you formulate things correctly (by studying piecewise constant functions with finite many values, then taking limits) it's not hard to check that the Lebesegue integral provides a norm which makes the vector space of integrable functions into a complete metric space. This is wonderful for all sorts of things.

[Anonymous No.12167309]

>>12167294
Those are obvious facts and do require a lot of math background, though I'm not sure I would go as far as to say they are harder to prove than the Jordan curve theorem. They are the sort of thing you would prove as a homework problem.

[Anonymous No.12167317]

>>12167281
>It doesn't really matter. A statement is not "obvious" because of its formal construction. It is obvious because of its physical analogs. It is obvious that a closed loop on a plane divides it into two disjoint parts. In fact, it is so obvious, that any model of the plane for which the theorem is false is arguably not even a model of a plane.
I'm with you there - we have some things we want to see validated by our mathematical models of the things and it shouldn't matter, for this, what [math]R\times R[/math] is. But with your quote here above, it's fair to argue that a set Y bigger than X should also have more subsets.
While I'm not an intuitionist in the Browerian sense (i.e. not post Heytings formalization of the philosophy into a mere syntatic framework), I too think that the curve theorem should not have chance of failing just from picking the wrong foundation. I.e. the geometric "obviousness" should be respected, wheras some statement about powersets of infinite sets in either direction would probably naturally not come as clear to anyone.
We're probably not gonna say more about it and I don't think I disagree with you.

>>12167296
oof.

>>12167292
did he, though?

[Anonymous No.12167318]

WOAH DUDE

ANOTHER MATH GENERAL??? SO FRIGGIN EPIC!!!!

MATH IS THE MOST BEAUTIFUL LANGUAGE IN THE UNIVERSE

SO HECKIN EPIC

[Anonymous No.12167319]

>>12167287
In a certain sense, the Lebesgue integral is the best you can do for any given measure. That said, there are sometimes still reasons to assign values (like the Cauchy principal value) to improper integrals which otherwise won't have any value (as either a Riemann or Lebesgue integral). For instance, the standard Cauchy distribution has no expectation, but the Cauchy principal value of its expectation is 0, which reflects the fact that the distribution is symmetric around x=0.

[Anonymous No.12167330]

>>12167319
>>12167306
>>12167292

why is it overshadowed by riemann's then, i don't remember seeing it in undergrad calculus, is this some faggy elitism thing

[Anonymous No.12167343]

>>12167330
because the reimann integral is fairly east to define. The lebesgue integral is not hard to conceptualize either, for most (riemann integrable) functions.
Really the lesbegue integral extends the family of integrable functions, like the characteristic function of the rationals. to use its full power you need the lesbegue measure, which is probably a bit too sophisticated for most hs students/ndergrads to develop properly

[Anonymous No.12167344]

>>12167330
>why do you first learn sin and cos via triangles while unit circle is clearly superior definition

[Anonymous No.12167348]

>>12167309
Oh certainly, I was just adding to the discussion. Jordan curve is definitely harder (though maybe not as obvious as the sphere and torus holes).
>>12167317
>philosophy shit
>thinks picking foundations is how people's intuition works instead of geometry
>lebesgue integral is somehow worse (presumably due to nonmeasurable sets)
fuck off

[Anonymous No.12167351]

>>12167330
Riemann integration is much simpler I guess, and for the usual measure on the real numbers, it will always have the same value (if it has one at all). The Riemann integral can be understood even if you only have a handwavy understanding of limits, while the Lebesgue integral requires a lot more rigor in defining things like limit points, limits of functions, limits superior and inferior, and so on. In other words, it needs at least a little real analysis to get a footing. Probably the most complex part of the definition of a Lebesgue integral is the sigma algebra (a collection of sets satisfying certain axioms that allows us to call them "measurable"), but I do think that can be handwaved away easily enough unless you need to get down to brass tacks.

[Anonymous No.12167353]

>>12167348
>thinks picking foundations is how people's intuition works instead of geometry
Pretty sure I said nothing in that direction

>lebesgue integral is somehow worse (presumably due to nonmeasurable sets)
I didn't make a call, but it seems like you'd be generalizing yourself into a corner if you dropped the other

>>12167330
There's an MO question along those lines
https://mathoverflow.net/questions/52708/why-should-one-still-teach-riemann-integration
but I agree with >>12167343.

[Anonymous No.12167359]

>>12167330
Because defining the Lebesgue measure which you use to talk about partitions is very, very hard. People in calculus classes always use Riemann. If you get to a second course in real analysis, or a measure theory course, you will learn Lebesgue and then no one ever goes back. Every single analyst means the Lebesgue integral when they write down an integral over the reals (or an integral similarly taken with respect to some other Borel measure). It is not overshadowed by the Riemann integral, it is universally used in mathematics.

[Anonymous No.12167364]

>>12167348
True. One thing that shocked me was how difficult 1+1=2 and other basic arithmetical facts were to prove in Russel and Whiteman's Principia. I haven't read the book, but I have seen their notation and axioms, and they seem really cumbersome. Starting with ZF on the other hand, we get the natural numbers for free and can immediately define things like addition with hardly any work. Even so, basic properties like association take a bit of cleverness to prove. It is fun to ask students about this sort of thing, because most have never even considered the notion that these needed to be proved, or that they could be.

[Anonymous No.12167367]

>>12167353
You can't escape from the humiliation of basing your mathematical worldview around foundations by merely arguing semantics with me.

[Anonymous No.12167374]

>>12167348
You got it exactly backwards. That poster was saying that we pick our foundations to suit our intuitions (geometric or otherwise), not the other way around. And that's clearly true. We have defined a circle and a square in formal terms, and we pick our definitions in such a way to match our physical intuitions about what circles and squares are. If our definition implied that squares had five sides, we would not use that definition, or at least not for squares.

[Anonymous No.12167378]

>>12167351
>while the Lebesgue integral requires a lot more rigor in defining things like limit points, limits of functions, limits superior and inferior, and so on
Doesn't it require a whole lot more than that? You need something as strong as powerset on the reals to define a sigma algebra, wheras for theorems of the Riemann integral you only need a metric. (The integral is computable in Weihrauchs analysis, so it's in a way even better behaved than differentiation)

[Anonymous No.12167377]

>>12167364
Yes, it's fun to poison young minds with trivialities which no man was ever intended to ponder. Sounds like a lot of fun to me. Hahaha! We're having FUN formulating Peano arithmetic in various collections of set theory axioms! Fun fun fun!!!

[Anonymous No.12167387]

>>12167378
Integrals are far better behaved than derivatives in every way except closed formulas. Integrals are smoothing and provide bounded linear operators.

[Anonymous No.12167396]

>>12167378
Not really. The hardest thing is to find a collection of measurable sets. As I said in that post, if you handwave that the way a first year calc course handwaves the Riemann definition of a limit, then defining a Lebesgue integral is really not that hard. In other words, if we accept that there is some measurable space with R as the underlying set, the rest follows. Granted, for integrals over certain extremely complicated sets, you won't be able to tell for sure if they have a value (or what it is) without something deeper, but these sets will never be constructible in the first place, so it's not a practical problem in a first year real analysis class.

>Weihrauchs analysis
I don't know what that is.

[Anonymous No.12167411]

Inspired by the anon who is posting infantile calculus definitions, I'm curious: What do you consider to be the most beautiful definition in mathematics?
For me, it's the definition of the abstract C* algebra. An abstract C* algebra is a complex Banach *-algebra where the identity [math] \|x^* x\| = \|x^*\|\|x\| [/math]. It is such a simple additional restriction on such a large class of objects, and yet it turns out any such algebra is a subalgebra of bounded operators on a Hilbert space. And you can construct the representation in a universal manner! It's incredible.

[Anonymous No.12167418]

>>12167396
Well, checking that the measure actually works on that space of measureable sets is not easy (that countable additivity holds with the outer measure). But that proof can also be handwaved.

[Anonymous No.12167423]

>>12167396
>I don't know what that is.
Computability theory but with streams instead of encodings into finite object. I.e. you can speak about effective results of real number computation that go a tad beyond encodings or real numbers into integers. It's sort of a hard subject imho
https://en.wikipedia.org/wiki/Computable_analysis

[Anonymous No.12167491]

>>12167418
True, this is actually what my professor did. It's a little awkward, but from the student's perspective, you don't really lose much. I was already familiar with sigma algebras going in to some extent, and I was mildly disappointed that they didn't come up, but it's not too surprising for an undergrad course.

On the other hand, we did do (ε,δ)-definitions in high school, and I did think they were worthwhile, so idk.

[Anonymous No.12167493]

>>12167423
This seems like a fascinating and relatively recent field. Do many people care about it or is it very niche? What sort of "flavor" is the research (in terms of techniques and such)?

[Anonymous No.12167502]

>https://en.wikipedia.org/wiki/Computable_analysis
>The differentiation operator over real valued functions is not computable, but over complex functions is computable.
I should expect shit like this and it makes perfect sense but it still screws with my head every time.

[Anonymous No.12167505]

>>12167423
>https://en.wikipedia.org/wiki/Computable_analysis
>there are continuous and computable functions over the computable real numbers which are total, but which map some closed intervals to unbounded open intervals
OK, now THIS is hard to picture. How can such a function be continuous?

[Anonymous No.12167524]

>>12167505
Not that guy but you can pick some uncomputable real number a and talk about, say, x maps to 1/(x - a). This is continuous over the computables but maps [0, 2a] intersect the computables to an unbounded set.

[Anonymous No.12167548]

>>12167359
There are Riemann integrable functions which are not Lesbegue integrable though.

[Anonymous No.12167551]

>One of the basic results of computable analysis is that every computable function from [math]\mathbb {R}[/math] to [math]\mathbb {R}[/math] is continuous (Weihrauch 2000, p. 6).
>the Kronecker delta function is either not computable or continuous

[Anonymous No.12167560]

>>12167551
Well as a function of real variables it wouldn't be computable

[Anonymous No.12167561]

>>12167548
Only improper integrals. I guess it's just a question of semantics whether an improper integral is technically a Riemann integral or not. Arguably, improper integrals are just an extension to Riemann integrals in the same way the Cauchy p.v. is.

[Anonymous No.12167575]

>>12167560
I get that if I print out a number's infinite decimal expansion you'll hardly be able to tell me if it's zero all the way through or not, but this still feels dumb.

[Anonymous No.12167605]

Does plane with Manhattan metric count as Euclidean geometry?

[Anonymous No.12167628]

>>12167493
>This seems like a fascinating and relatively recent field. Do many people care about it or is it very niche? What sort of "flavor" is the research (in terms of techniques and such)?
I suppose the programme is only a few decades old, but it borders work in descriptive set theory, realizability and such. Work by Polish mathematicans, Kleene, and those guys.
The answer is I think that it looks close to realizability theory, which looks like pic relatated, a paper by Andrej Bauer. Who you can easily find online, wrote his PhD thesis on realizibility and he shills this stuff quite a bit.

It's computability where (instead of demanding real r numbers to be, upfront, if possible at all, encoded into objects n_r that would e.g. fit into memory) you can think of it as having machines that read in streams of such objects. The difference is then that to compute something does not mean the completed process of input->output, but instead it's demanding that for every new input on the stream, you compute an output. E.g. the machine popping out a stream of digits is effective computing in this model.
The two notion are orthogonal.

>>12167505
>>12167551
It helps if you got a good hold on [math] {\mathbb B} := {\mathbb N}^{\mathbb N} [/math] with the product topology.
(i.e. https://en.wikipedia.org/wiki/Baire_space_(set_theory))

The function [math]{\mathbb R} \to {\mathbb R}[/math] are already all continuous in e.g. the works of Brouwer and Markov Jr.

[Anonymous No.12167629]

>>12167605
no

[Anonymous No.12167644]

>>12167575
It sounds like the whole point of distinguishing computable functions over R from computable functions over computable numbers is to reject functions exactly like this.

[Anonymous No.12167649]

>>12167605
There is a reason they call [math]d(x,y) = \sqrt{x^2+y^2}[/math] the Euclidean metric.

[Anonymous No.12167652]

>>12167649
*Euclidean norm

[Anonymous No.12167663]

>>12167605
Only if 0=1

[Anonymous No.12167683]

>>12167628
(Where w.r.t. the last sentence, R can not be taken to be the Dedekind reals with the order topology. It's very easy to get confused here without a lot of distinguishing notation.)

[Anonymous No.12167688]

>>12167663
That's only a metric space on [math]\mathbb{R}^0[/math]

[Anonymous No.12167763]

>>12167548
>implying that there's any reason one should consider sin(x)/x integrable
>implying that conditional convergence is anything but an unfrortunate consequence of poorly chosen definitions

[Anonymous No.12167771]

>>12167628
Ah, so it sounds to be much more of a "computability" flavor than an "analysis" one. Oh well. I like computability theory but I need to be doing analysis and really just analysis if I'm proving something.

[Anonymous No.12167775]

Fun puzzle for the mathlets

[Anonymous No.12167778]

>>12167775
>5 KB

[Anonymous No.12167784]

>>12167778
Yes, what's the problem?

[Anonymous No.12167786]

>>12167775
this is an imo problem from the most recent competition
please don't bully the mathlets

[Anonymous No.12167788]

How do I write a mixed fraction in mathjax?

[Anonymous No.12167794]

>>12167788
Like [math]\dfrac{3}{2} = 1 \dfrac{1}{2}[/math]?

[Anonymous No.12167802]

>>12167786
Fine then, how about an easier question?
Determine the value of the following expression:
67kg-82kg+34kg

[Anonymous No.12167806]

>>12167794
Yes but with the fractional part written smaller so it all fits on one line.

[Anonymous No.12167809]

>>12167806
[math]\frac{3}{2} = 1 \frac{1}{2}[/math]?

[Anonymous No.12167818]

>>12167784

[Anonymous No.12167821]

>>12167802
I determine its value is worthless.

[Anonymous No.12167835]

>>12167775
https://artofproblemsolving.com/community/c6h2278647p17821569
what a boring, gross problem

[Anonymous No.12167970]

>>12167835
>what a boring, gross problem
this

[Anonymous No.12167982]

>>12167970
Here's a fun problem for everyone to enjoy: find a closed form for [math]\\int \frac{dx}{x^8+1}[/math] in terms of trigonometric and logarithmic functions. Fun math puzzle!

[Anonymous No.12167986]

>>12167982
Why didn't it come out right?
[math]\int \frac{dx}{x^8+1}[/math]

[Anonymous No.12167994]

>>12167982
just do partial fraction decomposition lmao

[Anonymous No.12168013]

>>12167994
That's the joke. The solution to the IMO problem above ("fun puzzle for the mathlets") requires you to just multiply out polynomials, collect like terms, and compare all 16 coefficients. It's annoying to solve and not satisfying when you do. Kind of like an 8th degree partial fraction decomposition.

[Anonymous No.12168019]

>>12167771
>if I'm proving something
Well you're proving something, obviously. It has more algorithms and less choice functions, if that's what you mean.

[Anonymous No.12168026]

>>12167809
Thanks

[Anonymous No.12168035]

>>12168013
no, most people are too dumb to solve the imo problem

[Anonymous No.12168049]

>>12168035
It's not even that they're too stupid, it's just that most people don't know the inequality of weighted arithmetic and geometric means. But knowing that, the problem reduces to pure grunt work.

[Anonymous No.12168061]

>>12168019
I was just saying I really only like doing analysis, so if the proofs in computable analysis are more like computability theory proofs than analysis proofs, I'm probably not going to enjoy doing computable analysis very much.

[Anonymous No.12168067]

>>12168035
not knowing an obscure version of jensen's inequality [math] \neq [/math] being "too dumb"

[Anonymous No.12168068]

i solved the rot.

you can suck my cock, fuck my mouth, or eat my asshole for my +1.

thank you,

jerel canty

[Anonymous No.12168098]

>>12168068
i want to rim your asshole, please contact me

[Anonymous No.12168125]

What's a good way to input Unicode math symbols that's faster than just googling them and copying and pasting? For instance, if I want to type Σaₙxn×yn+2 (i.e. [math]\Sigma a_{n} \mathbf{x}^{n} \times \mathbf{y}^{n+2}[/math]) or whatever, my best option right now is to use google or charmap to find the characters and paste them into the comment. Is there anything more convenient?

[Anonymous No.12168131]

>>12168125
Ah, looks like unicode superscripts don't even work on 4chan for whatever reason. Strange that the subscript does.

i0231456789+-=()n0123456789+-=()ₐₑₒₓₔₕₖₗₘₙₚₛₜ

[Anonymous No.12168144]

>>12168067
Why does it seem like so many IMO problems are designed around knowing these esoteric methods/theorems?

[Anonymous No.12168150]

>>12168144
I have a feeling the people who design the problems were themselves into competition math in school. It's incestuous. So now it's its own thing with its own special trivia you have to know to compete at the top level.

[Anonymous No.12168161]

>>12168035
Why yes, I'm too dumb to solve it, how could you tell?

[Anonymous No.12168168]

>>12167982
You put in two \s at the beginning.
If anything it's weird how it didn't line break.
[math]a \\ b[/math]

[Anonymous No.12168173]

>>12167344
because k12 teachers dont know anything and learning math as a system of floating ungrounded facts is a joke unless you 're still living in 1800

[Anonymous No.12168176]

>>12166869
Nikolaj what's the best breakfast have before doing maths?

[Anonymous No.12168177]

>>12168144
How else do you ask remotely difficult questions to high schoolers with no real mathematics knowledge?

[Anonymous No.12168197]

>>12166845
space filling curves are bs, convergence doesn't count. An ACTUAL space filling curve would be a continuous surjection [math] [0,1] \longrightarrow [0,1]^2 [/math]

[Anonymous No.12168206]

>>12168197
>An ACTUAL space filling curve would be a continuous surjection [0,1]⟶[0,1]2
that's what space filling curve is

[Anonymous No.12168214]

>>12168176
Stop moving Discord conversations to sci pls, nobody gains from that

[Anonymous No.12168221]

>>12168173
It's the other way around. Secondary school teachers want to leverage the intuition about right triangles to define a potentially confusing function to kids. And that they want to use the Pythagorean Theorem to demonstrate why these values make sense. They use the right triangle functions to motivate the circle functions. Going straight to the circle off the bat without explaining the motivation would be the mistake you seem to think they are making right now.

[Anonymous No.12168227]

>>12168176
How did I get to /b/ by mistake?

[Anonymous No.12168240]

>>12168197
>convergence doesn't count
The space-filling curve is the pointwise limit of the sequence of curves. But it's still a single function. It maps each point in [0,1] to a point in [0,1]×[0,1]. And it's a curve, so it's continuous. And it's space-filling, so it's a surjection. The picture in the OP just doesn't show the actual Peano curve, because its image is the entire unit square. It shows a couple of the iterations, the pointwise limit of all of whom is the true Peano curve.

[Anonymous No.12168293]

>>12166850
That’s literally the nazi ideology permeating all the space.

[Anonymous No.12168327]

I have a finite sequence of numbers. I'm interested in how often the partial sum equals zero.
Do permutations affect this? What abou char > 0?

[Anonymous No.12168339]

>>12168327
have you tried a single example? i doubt that

[Anonymous No.12168341]

>>12168327
>Do permutations affect this
Sure. Consider the partial sequence (1, -1, 0). This reaches 0 twice, on the second and third partial sums. But then consider (1, 0, -1). This reaches 0 only once, on the third partial sum. Finding the number of places where a partial sum is exactly zero sounds like a challenging problem.

[Anonymous No.12168380]

>>12168341
exclude 0 from the possible values. this was something i had in my head but failed to specify

[Anonymous No.12168406]

>>12168380
OK. Consider the sequence (1, -1, 1). Now think about its permutations. It's not hard to see that permutations matter.

[Anonymous No.12168412]

>>12168380
-1 -1 1 1

[Anonymous No.12168455]

>>12168125
Anyone?

[Anonymous No.12168486]

>>12168068
worship these feet

[Anonymous No.12168487]

>>12168455
Some kind of an input method. There's a LaTeX input method available for ibus

[Anonymous No.12168503]

Hey, I'm studying topology with Engelking, loving it so far but I've noticed he uses a lot of times AC or a weaker version for his proofs. Is there a topology book with enphasis on the theorems equivalent or using versions of AC?

[Anonymous No.12168559]

How the hell do you guys deal with burnout? None of this stuff is interesting anymore and it's been this way for months. I used to have an unhealthy obsession with math, I want to go back to those days.

[Anonymous No.12168568]

>>12168559
Is it possible to take a break for a while?

[Anonymous No.12168572]

>>12168559
By taking a break from trying to think about it all the time. Occupy yourself with another hobby and then eventually the other hobby will get boring and you'll have the drive to spend that time on math again.

[Anonymous No.12168584]

>>12168559
Learn a new language.

[Anonymous No.12168598]

>>12168568
From university? Unfortunately no.

>>12168572
I have picked up other hobbies and have been waiting for the spark to reignite but I get these short bursts of interest that last like an hour a couple times a week if I am lucky. I have no desire to apply to graduate school anymore, I just hope I don't end up doing something that I regret.

>>12168584
I actually am doing that, it's been pretty fun so far despite not being able to say anything yet.

[Anonymous No.12168702]

>>12167154
A statement must first be true before it can be obvious imo

[Anonymous No.12168726]

Jason needs to buy new carpet tiles for his kitchen. A plan of the kitchen is shown as 5m, 5m, 7m and 3m. The tiles Jason has chosen are 0.5m2, how many tiles will he need to cover the entire kitchen floor?
Jason must store crates measuring 6m wide, 8m long and 4m high. What is the max number of crates Jack can fit into the storeroom?

[Anonymous No.12168733]

>>12168726
Are Jack and Jason dating?

[Anonymous No.12168734]

>>12168559
>I used to have an unhealthy obsession with math, I want to go back to those days.
Sounds unhealthy

[Anonymous No.12168851]

How many hours are you guys studying a day? I can barely do 4 hours, am I not destined to be a good mathematician with just 4 hours?
t. second year undergrad

[Anonymous No.12168875]

>>12168851
Very variable and hard to say just based off time. Some people can finish their problem set in an hour and others it takes a few days. Neither is any better or worse it's just what works best for you. As long as you find you're performing well and understanding the material then you should be fine. Do all the exercises and seek to understand.

[Anonymous No.12168898]

>>12168851
I haven't done anything for maybe 3 weeks now.

[Anonymous No.12168951]

>>12168702
Fair enough, but the relation of this perspective is not so straight forward.
Unless we agree to make provability from the ZFC axiom the arbiter of truth.

[Anonymous No.12168954]

>>12168702
Fair enough, but the relation of this perspective to the statement is not so straight forward. Unless we agree to make provability from the ZFC axiom the arbiter of truth. Without that, one may argue that the claim is obvious and probably true, just that ZFC is an insufficient framework for set theory.

[Anonymous No.12168990]

>>12168177
Well something obviously at a higher level than hs? The amount of time they spend studying could be used to learn high level undergrad shit.

[Anonymous No.12168992]

>>12167248
My attempt at “pulling a Tooker”:

A disproof of the Jordan Curve Theorem

Define R^2 as (-inf,inf)x(-inf,inf)

Claim that there exists a curve whose coordinates are R-hat numbers with out an inside or outside in the neighborhood of infinity.

Claim that somehow this curve is also in R^2.

Q.E.D.

[Anonymous No.12169062]

>>12168992
You found the counterexample [math]\hat{\text{c}} \hat{\text{u}} \hat{\text{r}} \hat{\text{v}} \hat{\text{e}}[/math]. Congratz.

[Anonymous No.12169102]

Lads, I've always been fascinated with void/nothingness and the apparent paradoxes. For the past week since I started learning set theory, I've been working towards actually formalizing the concept of void. Maybe we can work together! Here are some thoughts I had today:

In the vein of a vacuous truth, suppose x^2 < 0 in R. What this does is suppose reality is in a state which cannot occur. This places our analysis in the void. There are multiple ways to be in the void, one can set a contradictory/impossible state of truth values, but let all statements exist, and one can make all statements nonexistent. If we assume the former, then if we ask "does x^2 < 0 imply x=23 in R", we are looking for the truth-state of "x = 23" in a realm where at least one truth-value doesn't agree with reality's truth value (namely that x^2<0). Since the vacuous statement is placed in the void due to analysis using the standard implications of statements, "x = 23" still contains its implication that "x^2<0" is false. Either we hold that contradictions can't occur, and x=23 must be false, and its negation is false, thus making eachother both true as well, which leads to contradiction again, so we hold that "x=23" has to have a value, and since both possible truth values of it are contradictory, contradictions are allowed. Contradictions might be allowed universally, letting the void assume all possible values.

However, other ways of analyzing this give interesting results too. If we say something else that's vacuous such as "x^2<0 implies x is real", and allow all statements to exist in the void, just with potentially non-legitimate truth values (as before), we aren't forced to allow contradiction because we can have x be non real. If complex numbers have not been defined yet, this could be construed to generate a new class of numbers. If we accept limitating all x to R, we might non contradictorily accept the vacuous state of x is not in R, since we are already in the void.

[Anonymous No.12169111]

>>12168898
Go watch some mindyourdecisions videos, at least

[Anonymous No.12169124]

>>12165621
>Also, what is your notetaking methodology?
What's yours?

[Anonymous No.12169143]

>>12169111
I think it's because of having to give a presentation (maybe) in the beginning of the week. If I fail, I'll be kicked out of the program, which will most likely not happen, but it's just bugging me a lot. Also, there is a little bit of stress related to that because the other assessor isn't replying to my emails when I'm asking him when he would like to do it, while the other one was happy to share his preferred times with me. At least I have the presentation ready and I've presented it to my stuffed dog a few times for practice.

[Anonymous No.12169182]

>>12168990
Oh, it certainly could. But it isn't. Which is stupid.

[Anonymous No.12169191]

>>12167802
>mixed fraction
Are you a literal toddler?
Are you making baking recipes?

[Anonymous No.12169198]

>>12169191
That will be the newest book on category theory, I assume.

[Anonymous No.12169211]

>>12169198
Lel someone shop this onto a textbook cover

[Anonymous No.12169220]

>>12169102
>Cont'd (2000 characters is far too short for true thoughts...)
Lastly, I want to ponder on the situation where we generate a void state as before, with x^2<0, and we don't assume that all statements exist in that void. Given x^2<0, x=23, but we note that no such x fulfills that property, so in that state, there are no x's. Then when we ask if x is 23 or not, we are asking if a nonexistent object, the Void Element (V), is an R equal to 23 or not. It inherits properties of all nonthings in the void (simply that of being in the void). If an element of the Void is an R, then it would also be in Reality, which is contradictory because then it would be boolean and not be allowed to be in Void and Reality. So it must have those equivalences be false about it, or it must be equivalent to both so that it is contradictory and never enters reality. What about traits that describe but do not set it equal to an element in Reality, such as "V is green"? This would not place V in a set if we only allow extant elements with a given trait to enter sets. So V is allowed to have all traits, but does it truly have all or any traits yet no equivalence? For this, we have to understand what constitutes an element of reality, and what combination of traits sets it equal to an element. Reality is relative, so an element is real so long as it can operate on another element considered real. Operations can be defined for any trait. Thus V must have all traits false, or all traits and their negations as true. Having a trait as false can be cast into another trait that is true (x has the trait (x's color is green is false)), so the latter must be the true.

For some V that has the traits of not x and x, this is not contradictory, because V can generate a third class of state, v, so it is v x, thought of as equivalent to all things at once.

[Anonymous No.12169224]

>>12169143
What's the presentation on?

[Anonymous No.12169230]

>>12169102
>>12169220

>This places our analysis in the void. There are multiple ways to be in the void, one can set a contradictory/impossible state of truth values, but let all statements exist, and one can make all statements nonexistent.
This is a bit too funky for my taste, but I've seen things along those lines being studied.

If you take a space [math]X[/math] and a topology given as a collection of open sets [math]O(X)\subset {\mathcal P}X[/math] and work exactly the aspects of it that are algebraic in nature (the topology axioms are already about certain operations, albeit potentially infinite ones ones), you can get a lattice theoretical characterization of much of it. The definition of continuity for functions [math]f()[/math] is something about inverse "backwards" images [math]f^{-1}[][/math] of open sets.
Now in the algebraic setting, if you dualize again, you got a "forward" theory on [math]{\mathcal P}X[/math] where you can do a lot of weird shit because the theory actually stopped caring about the point notion of [math] X[/math], just about subsets.
That's a handwavy sketch of it anyway.

But don't get into that, only topos theory autists and maybe some far out-there algebraic geometers look at that.

[Anonymous No.12169232]

>>12169143
Get off of 4chan(nel), lad.

[Anonymous No.12169233]

>>12169220
Er sorry, it is contradictory, but allowable because it's not in the domain where contradiction is disallowed. But it is equivalent to all things at once.

[Anonymous No.12169258]

>>12169230
>locale has no points but contains nontrivial open subspaces
I wonder what this could mean

[Anonymous No.12169284]

>>12169232
Actually, I have hardly been on this site. I've just been unable to concentrate on anything. Of course, subconscious is working at the same time, but I feel stuck with all work. Have you had any nice insights on anything?

Anyway, /gnmg/.

[Anonymous No.12169289]

>>12169284
Night, lad.

[Anonymous No.12169291]

>>12169284
Night.

[Anonymous No.12169292]

Does anyone have that image of a letter to the editor in a newspaper where someone is complaining about government funding for science and he mentions complex analysis and says something like "for that much money, it had better be complex!"

[Anonymous No.12169308]

>>12169258
I think that bit isn't all that bad if you allow yourself to step out of the frame of Descartes and Cantor, where everything is a set, including the things in geometry.

E.g. if you look at the late 19'th century work by Hilbert on geometry, it's a typed theory of points, lines and planes and not a Cartesian setup on [math] {\mathbb R}^n [/math].
E.g. look at the triangle construction in pic related, expressing the fact that in a triangle, given two lines 'uy' and 'vx' inside it. They will intersect in a point 'a'.
In formal logic, this reads

[math] ({\mathrm B}\,xuz \land {\mathrm B}\,yvz) \rightarrow \exists a\, ({\mathrm B}\,uay \land {\mathrm B}\,vax) [/math]

where [math] {\mathrm B}xyz [/math] is the tertiary predicate saying that y is on the line xz.

We're reminded that even if most math can be modeled by [math] \in [/math] with it's 10 axioms, we can also do fully formal math without it.
100 years ago I'm sure there were people for whom it would have been heresy to postulate the thing that we call "continuum" is actually a sort of spectrum of points and not a continuum at all. From that more typed angle, I think it's fair to conceive of lines that have segments of other lines in it, but not points.

[Anonymous No.12169319]

[math] {\mathrm B} [/math] stands for between btw.

So [math] {\mathrm B} uvw [/math] means v is between u and w.

Points are endpoints of lines and points can be between other points, but B is a predicate of only points here.

[Anonymous No.12169359]

And I don't know the first thing about locale theory, but I think similar to what that paraconsistent guy (>>12169102 >>12169102) seem to do, you can work with predicates in themselves a lot, if you don't axiomize them away.

E.g. the ZF Axiom of Separation says that given a set [math] A [/math] and a predicate [math] P [/math], the set [math] B=\{x\in A\mid P(x)\} [/math] exists. More formally, it really rather says that for all [math] A [/math], there is a [math] B [/math], such that
[math] \forall x. ((x\in B)\iff (x\in A \land P(x))) [/math]

Now if you assume excluded middle, then P is either [math]\top[/math] (True) or [math]\bot[/math] (False) and the rules for [math]\land[/math] (And) immediately kill the predicate in favour of sets. I.e. [math]Q\land \top[/math] becomes [math]Q[/math] and [math]Q\land \bot[/math] becomes [math]\bot[/math].
But if you don't adopt LEM, the predicates survive and the comprehension axiom (saying that for any predicate, there's a set), makes for a whole lot of sets.
So e.g. in constructive set theory, the collection of subsets of the singleton set [math]\{0\}[/math] is so big (or unknown), it's not even a set. (But that's not a bad thing,)

>>12169220
Btw. have you looked at
https://en.wikipedia.org/wiki/Paraconsistent_logic
?
Might be another angle at your game.

[Anonymous No.12169371]

My point being he does formal manipulation of expressions "x^2 < 0" that have no evident "material" (set theoretical) realization, but that's not unheard of, since you can work with predicates in this sense themselves.

[Anonymous No.12169372]

holy shit stop talking about fucking logic this is the math thread

[Anonymous No.12169379]

>>12169372
this

[Anonymous No.12169437]

When does math get fun?

[Anonymous No.12169444]

>>12169437
Analysis

[Anonymous No.12169477]

It's known that if two permutations are disjoint, they commute. I'm wondering if there's any further condition that can be assumed that makes the converse hold true, i.e. if two permutations commute, then they are disjoint.

[Anonymous No.12169490]

I provided the entire proof for the sake of context but really only one part matters.
Why can we write the division with |r| Is less than c/2
I understand that the Euclidean algorithm shows that there's r with absolute value less than c, so how can we just make this assertion

[Anonymous No.12169566]

>>12169437
when you start getting good at it

[Anonymous No.12169922]

>>12167788
What is a mixed fraction?
t. Math grad

[Anonymous No.12169960]

>>12169922
[math]1 \frac{3}{4} = \frac{7}{4}[/math]

[Anonymous No.12170120]

>>12169490
>>12166714

[Anonymous No.12170316]

>>12169477
(1,2) and (2,1) commute but are not disjoint.The same is true of (1,2,3,4) and (1,3,2,4). A necessary and sufficient condition for permutations to commute is given here: https://math.stackexchange.com/questions/440568/are-there-any-conditions-such-that-2-permutations-in-s-4-are-commutative/440605..

[Anonymous No.12170389]

>>12170316
>(1,2) and (2, 1) are two different permutations

[Anonymous No.12170401]

>>12170316
>(1,2) and (2,1) commute but are not disjoint
That is very true, indeed. Any element commutes with itself.

[Anonymous No.12170430]

>>12169292
Bumping for this

[Anonymous No.12170434]

>>12170389
If you prefer, (1,2) and (1,2) commute. The point is the same.

[Anonymous No.12170787]

[math] \exp(x) \approx \dfrac{1 + {\frac{1}{2}}x + {\frac{1}{12}}x^2}{1 - {\frac{1}{2}}x + {\frac{1}{12}}x^2} [/math]

[Anonymous No.12170826]

>>12169230
>But don't get into that, only topos theory autists and maybe some far out-there algebraic geometers look at that.
huh, you cant do serious topology without locale theory if you want to go beyond undergrad FOL logic.

[Anonymous No.12170831]

>>12169258
>>>locale has no points but contains nontrivial open subspaces
>I wonder what this could mean
it means there is no locale map 1->X, ie frame map omega[X]->omega[1]

1 is the singleton

[Anonymous No.12170833]

>>12170826
but is it non-autistic?

[Anonymous No.12170887]

>>12169258
>>12169230
For all the locale stuff, Vickers is pedagogical
https://www.cs.bham.ac.uk/~sjv/

the first main article is
LOCALES AND TOPOSES AS SPACES

https://www.cs.bham.ac.uk/~sjv/LocTopSpaces.pdf

[Anonymous No.12170889]

>>12170887
Introduction
Mac Lane and Moerdijk, 1992, in their thorough introduction to topos
theory, start their Prologue by saying –
A startling aspect of topos theory is that it unifies two seemingly
wholly distinct mathematical subjects: on the one hand, topology and
algebraic geometry, and on the other hand, logic and set theory. In-
deed, a topos can be considered both as a “generalized space” and as a
“generalized universe of sets”.
This dual nature of topos theory is of great importance, and one can
quite reasonably understand Grothendieck’s name “topos” as meaning
“that of which topology is the study”.
Mac Lane and Moerdijk are
unquestionably masters of the spatial nature of toposes, yet one could
easily read through their book without grasping it. The mathematical
technology is so firmly expressed in the set theory and the logic that the
spatiality is obscured.
The aim in this chapter is to provide a reader’s guide to the spatial
content of the major texts. Those texts can also provide a more detailed
account of original sources and other applications than has been possible
here.
We have on the one hand, the logic and set theory, and, on the other,
the topology. In a nutshell, the topos connection between them is that
the topos acts like a “Lindenbaum algebra” (of formulae modulo equiv-
alence) for a logical theory whose models are the points of a space.
2
The prototype is Stone’s Representation Theorem for Boolean alge-
bras, which relates propositional logic to Hausdorff, totally disconnected
topology. However, it takes some work to develop the idea to its full
generality.
First, the logic is not at all ordinary classical logic.
It is
an infinitary positive logic known as geometric logic. Second, we are in
general talking about predicate theories, and for these the appropriate
notion of Lindenbaum algebra is not straightforward. It is really the
“category of sets generated by the theory”.

[Anonymous No.12170891]

>>12170889
Lastly, “space” of points is
not an ordinary topological space – it is a real generalization.
However, the propositional fragment of the predicate logic does cor-
respond more or less to ordinary topological spaces. As a rough picture
of the correspondence, in the propositional case we find –
space ∼ logical theory
point ∼ model of the theory
open set ∼ propositional formula
sheaf ∼ predicate formula
continuous map ∼ transformation of models that is definable within
geometric logic
These “propositional toposes” are called localic, or (with slight abuse
of language) locales. They are equivalent to the locales introduced in –
say – Johnstone, 1982 or Vickers, 1989.
Now the topos theorists discovered some deep facts about the inter-
action between continuous maps and the logic and set theory of toposes.
A map f : X Y gives a geometric morphism between the correspond-
ing toposes of sheaves. A topos is sufficiently like the category of sets
that a kind of set theory can be modelled in it. Roughly speaking, in
sheaves over X it is set theory “continuously parametrized by a variable
point of X”. The map f then comes to be seen as a “generalized” point
of Y , parametrized by a point of X, and this is a point of Y in the
non-standard set theory of sheaves over X. So by allowing topological
reasoning to take place in toposes instead of in the category of ordinary
sets, one gains a simple way to reason about the generalized points of Y
– in other words the maps into Y .
However, to make this trick work one has to reason constructively
because the internal logic of toposes is not in general classical.
And
constructive topology does not work well unless one replaces topological
spaces with locales. For instance, the Tychonoff theorem and the Heine-
Borel theorem hold constructively for locales but not for topological
spaces.

[Anonymous No.12170895]

>>12170891
Locales and toposes as spaces
3
Now there is a well known drawback to locales. They do not in general
have enough points and for this reason are normally treated with an
opaque “point-free” style of argument. However, they do have enough
generalized points.
Since constructive reasoning gives easy access to
these, it also allows locales to be discussed in a spatial way in terms of
their points. We in fact get a cohesive package of mathematical deals.
1 Constructive reasoning allows maps to be treated as generalized
points.
2 Locales give a better constructive topology (better results hold)
than ordinary spaces.
3 The constructive reasoning makes it possible to deal with locales
as though they were spaces of points.
What’s more, the more stringent geometric constructivism has an in-
trinsic continuity – one might almost say it is the logical essence of con-
tinuity. The effect of this is that constructions described in conformity
with its disciplines are automatically continuous.
The prime aim of this chapter is to explain how this deep connection
between logic and topology works out. However, as a spinoff we find
that “generalized spaces” corresponding to toposes become more acces-
sible. They are spaces in which the opens are insufficient to define the
topological structure, and sheaves have to be used instead.
These ideas are not essentially new. They have been a hidden part of
topos theory from the start. Some writers, such as Wraith, 1979, have
made quite explicit use of the virtues of geometric logic. Our aim here
is to make them less hidden. At the same time we shall also stress a
peculiarity of geometric logic, namely that it embodies a geometric type
theory. This provides a more naturally mathematical mode of working
in geometric logic.
For further reading as a standard text on topos theory, we particularly
recommend Mac Lane and Moerdijk, 1992.

[Anonymous No.12170898]

>>12170895

The standard reference text
(Johnstone, 2002a, Johnstone, 2002b) is much more complete and ulti-
mately indispensible. In particular, it treats in some depth the notion
of “geometric type construct” that is very important for us. However,
it can be impenetrable for beginners.
Though the chapter is so closely linked to toposes, many of its tech-
niques can also be used in other (and distinct) constructive foundations
such as formal topology in predicative type theory (Sambin, 1987). We
refer to Vickers, 2006 and Vickers, 2005 for some of the connections.

So like he says here
>However, to make this trick work one has to reason constructively because the internal logic of toposes is not in general classical.
And constructive topology does not work well unless one replaces topological spaces with locales. For instance, the Tychonoff theorem and the Heine-Borel theorem hold constructively for locales but not for topological spaces.

Locale theory is really the best candidate for topology in a constructive sense

[Anonymous No.12170902]

Anonymous 14 seconds ago No.12170898

>>12170895#

The standard reference text
(Johnstone, 2002a, Johnstone, 2002b) is much more complete and ulti-
mately indispensible. In particular, it treats in some depth the notion
of “geometric type construct” that is very important for us. However,
it can be impenetrable for beginners.
Though the chapter is so closely linked to toposes, many of its tech-
niques can also be used in other (and distinct) constructive foundations
such as formal topology in predicative type theory (Sambin, 1987). We
refer to Vickers, 2006 and Vickers, 2005 for some of the connections.

So like he says here
>However, to make this trick work one has to reason constructively because the internal logic of toposes is not in general classical. And constructive topology does not work well unless one replaces topological spaces with locales. For instance, the Tychonoff theorem and the Heine-Borel theorem hold constructively for locales but not for topological spaces.

Locale theory is really the best candidate for topology in a constructive sense.

[Anonymous No.12170928]

>>12170787
Where does this approximation come from?

[Anonymous No.12170940]

Where can I read on the proof of the laws of exponents using supremum/sequences and the definition of logarithm explained in the same way?

[Anonymous No.12171016]

>>12170940
rudin maybe

[Anonymous No.12171277]

/gamg/

>The Midwest Topology Seminar in Fall 2020 will be held in an online format. Three 45-minute talks will be separated by two 30-minute coffee breaks. The breaks will be structured to provide an opportunity for small group conversation in breakout rooms.
>Thursday October 8 (all times eastern)
>12:45-1:30pm, Maria Yakerson, ETH Zürich
>1:30-2:00pm: Coffee break
>2:00-2:45pm: J. D. Quigley, Cornell University
>2:45-3:15pm: Coffee break
>3:15-4:00pm: Inbar Klang, Columbia University
https://wayne-edu.zoom.us/j/94838246637?pwd=ZE45ZnpmUy9CbE90RkwzYkRHVm8vZz09
>Meeting ID: 948 3824 6637
>Passcode: MTS

[Anonymous No.12171293]

>>12171277
Afternoon, lad.

[Anonymous No.12171306]

>>12170787
But that's wrong

[Anonymous No.12171325]

>>12171306
It's pretty accurate for [math]x \in [-2,2][/math].

[Anonymous No.12171328]

>>12170928
It's the Pade approximation (the 3,3 one).

Take a function [math]f[/math] and compute the two polynomials p, q of degree n, m making
[math]p(x)\cdot f(x)=q(x)[/math]
as close to zero as possible in some sense or on some subset.

If [math]p(x)=1[/math], this is just the Taylor series. In either case, [math]\frac{p(x)}{q(x)}\approx f(x)[/math].

>>12171306
It expands as [math] \sum_{k=0}^n \frac{1}{k!} x^k [/math] where [math] n=4 [/math], followed by alternating terms.
So around [math]x=0 [/math]it can't be very wrong anyway.

[Anonymous No.12171332]

[math] p(x)·f(x) - q(x) [/math] close to zero, that is

[Anonymous No.12171340]

>>12170928
It's the Pade approximation (the 3,3 one).

Take a function f and compute the two polynomials p, q of degree n, m making
p(x)⋅f(x) - q(x)
as close to zero as possible in some sense or on some subset.

If p(x)=1, this is just the Taylor series. In either case, q(x)/p(x)≈f(x).

>>12171306
It expands as x^k/k! up to n=4, followed by alternating terms.
So around x=0it can't be very wrong anyway.

[Anonymous No.12171359]

>>12171340
Why did you delete your post and then repost if but with no Latex?

[Anonymous No.12171363]

>>12171277
>The breaks will be structured to provide an opportunity for small group conversation in breakout rooms.
oof so this one is not for me

[Anonymous No.12171432]

>>12171293
Cheers.

>>12171363
I'm quite sure you need not join those. They are meant to provide opportunities, not to force people into doing anything.

[Anonymous No.12171532]

>>12171359
I flipped p and q but didn't save the tex code in ram. Was too lazy to to lazy it up again, sorry

[Anonymous No.12171633]

>>12167095
>>12167294
Topological arguments are harder because our intuition is really on that level. Most mathematicians before Weierstrass didn't believe there could be a continuous function that was differentiable nowhere. These things are usually easy to prove in some smooth category. I would say tit is in no way "obvious" in general in terms of how we picture continuous shit.

[Anonymous No.12171649]

>>12171277
Whose all talking there

[Anonymous No.12171651]

>>12171633
our intuition isn't really on that level*

[best hot take No.12171657]

>>12171633
People today are just myriads smarter than any mathematician before 1900. It's probably sugars or something. Yes, those people couldn't imagine that, but they also had no internet = no imagination. It's different times now. Dedegrumpy and Gaussi could go home now facing any Indian school boy. Math also got a lot harder. Finance wasn't even a thing back then.

[Anonymous No.12171660]

>>12171657
what the fuck are you talking about retard

[Anonymous No.12171689]

>>12171649
I have literally no idea what you are actually trying to ask. All I know is presented in that post. Those are the names and times, but I have no idea about the topics.

[Anonymous No.12171701]

If the fourier series are about orthogonal cos/sin and the taylor series are orthogonal polynomials, then it is basically they same approach, approximating a function with some other functions, generally speaking?

[Anonymous No.12171754]

>>12171701
>taylor series are orthogonal polynomials
You should have payed more attention in calc anon

[Anonymous No.12171785]

How can I find all the asymptotes of a function? Teacher pretty much skips this and I didn't find any good online material.

[Anonymous No.12171794]

>>12167330
one is for doing computations, the other is for proofs.

[Anonymous No.12171809]

>>12171701
Yes, that's fair to say. The difference being that Taylor series don't relate to the circle group.

[Anonymous No.12171833]

>>12171701
That would be fair to say, except it's completely incorrect.

[Anonymous No.12171851]

Is there any good book on non-classical logics?

[Anonymous No.12171877]

>>12171657
>smarter
no
>have the opportunity to memorise results that others have found for them
I hope you meant this one instead

[Anonymous No.12171895]

>>12166845
I see it

[Anonymous No.12171898]

>>12171851
Don't know if it's good but there's that one books called Introduction to non-classical logic

[Anonymous No.12171938]

Who would win in a fight to the death between tooker and nikolaj, I think it would be tooker since he has absolutely nothing to lose

[Anonymous No.12171946]

>>12171938
Anime tranny

[Anonymous No.12171978]

>Applying topoi theory to music theory
>More than 1000 pages

Has anyone read that book? Is it bullshit or does it really use topoi to study music in a non trivial way?

[Anonymous No.12172070]

>>12171946
Anime tranny isn't even in the fight.
Are you implying that anime tranny and Nikolaj are the same person?

[Anonymous No.12172079]

>>12171978
i member i did not need it to learn topos theory

[Anonymous No.12172082]

>>12171833
why is it fair to say then. i see a contradiction.

[Anonymous No.12172102]

>>12171851>>12171898

>non-classical logics

On libgen, You can try Anita Wasilewska - Logics for Computer Science_ Classical and Non-Classical-Springer (2018).pdf

and any book by troelstra

[Anonymous No.12172116]

These threads suck.
I'm leaving until they no longer suck.
If you need me, you can find me in /wsr/.
That is all.

[Anonymous No.12172126]

>>12172116
Take care.

[Anonymous No.12172136]

>>12171978
It's just someone who knows topos theory and music theory and mashes them together because he likes writing about it.
At least I'm 98% confident that that's the situation here.

[Anonymous No.12172185]

What's the best notation for introducing new notation?

[Anonymous No.12172200]

>>12172116
Where in /wsr/

[Anonymous No.12172209]

>>12171978
I have yet to see anything to do with category theory applied to music theory to any result other than making the author look like they're trying way too hard to be fashionable.

[Anonymous No.12172211]

If there were buttons and shining stuff in textbooks I would probably know a lot more maths than I do now.

[Anonymous No.12172216]

Is it possible for me to self teach myself Analysis barely knowing Calculus or proofs?

[Anonymous No.12172218]

>>12172211
Sorry, I don't get it. What kind of buttons and shining stuff do you mean?

[Anonymous No.12172221]

>>12172211
>they can't see the buttons and shining stuff

[Anonymous No.12172222]

>>12172216
I think so.

[Anonymous No.12172231]

>>12172218
Like interactive stuff. You press on a proposition or a theorem and the proof appears. Or the diagrams would move in interesting ways. Or the pages be made in interesting texture and smells. Things like that.

[Anonymous No.12172258]

>>12172185
The word let

[Anonymous No.12172276]

>>12172222
The thing i'm more afraid of it are the proofs, i'm using the textbook "Understanding Analysis"

[Anonymous No.12172302]

>>12172231
The first two ideas sound nice but the third is just weird. I'm sure something in that general direction could be done as software though, but I'm worried that it would be regarded as 'not serious' and so nobody would buy it. Most textbooks are written either for super specialized researchers, or with the intention to get professors to adopt them for their classes. What you're thinking about would only be a self-study tool

[Anonymous No.12172314]

>>12172276
Are you understanding "Understanding Analysis"?

[Anonymous No.12172331]

Hello friends. Today I have proved that the power set of any set with N elements contains 2^N elements. I have not seen a proof for it before so I did it all from scratch, other than knowing the end result to look for!

First, we need a lemma. The lemma is pic related which shows that the nth layer of the magic pyramid (counting from layer 0), has a sum of terms equal to 2^n.

Now, we show that for a set of size N, it has Ki subsets of size i, for each i in layer n of the magic pyramid, giving it 2^n subsets. We can show this recursively, starting from layer n = 1, which can easily be checked that it holds. For a set of N = 2, we see that it contains the empty set, so it has 1 subset of size 0, agreeing with the magic pyramid for K0.
Next we prove that for any set, its subsets of size i>0 equal the number of subsets of size i>0 in any set size N-1, plus the number of subsets of size i-1 in set N-1. Take a given set, and remove an element from it. All the subsets of size i in the lesser set are also in the greater set. Also, all Ki-1 subsets of size i-1 in the lesser set can be united with the element removed from the greater set, creating Ki-1 additional subsets of size i. We must show that these are the only subsets of size i in the greater set. This can be seen by the fact that if a subset is solely made of elements from the lesser set, it is in the former group, and if it contains any elements outside the lesser set, it can only be one element (since that is the difference of sets), leaving the other i-1 in the lesser set. So, if a subset of size i or i-1 is in lesser, a corresponding set is in greater, and if it is in greater, it is in or corresponds to from a subset in lesser (with no overlap), leading to Ki on the new layer being sum of Ki-1 and Ki on the preceding layer of the pyramid.

[Anonymous No.12172433]

>>12172302
You can imagine that people in a very advanced society would have put a lot of effort into teaching math in a very engaging way from an early age. Greg Egan's novel Diaspora has a depiction of something along these lines.

[Anonymous No.12172444]

>>12172331
Just use binary strings yo.

[Anonymous No.12172451]

>>12172211
I have had similar ideas anon! One is a website with a wide array of visualization tools, and the other is a smartglasses software that recognizes mathematical statements and displays alternative simplifications/formulations of them instantly, in a list which you can browse or next to the dominant. There are many more ideas like this.

[Anonymous No.12172511]

>>12172331
Do you want to know the easy way to do that where you need no counting like that?

[Anonymous No.12172571]

[math] 1, 1, 1, 2, 3, 6, 12, 25, 52, 113, 247, 548, 1226, 2770, 6299, 14426, \dots [/math]

[Anonymous No.12172602]

>>12172314
DIdn't begin my self-teaching journey yet

[Anonymous No.12172620]

What do you prefer, algebra, analysis, topology, number theory, or combinatorics?

[Anonymous No.12172624]

>>12172511
I sure do anon. Could you also tell me how the following metric satisfies the triangle inequality?
>norm(f(x),g(x) a<=x<=b) = max (|f(x)-g(x)| for all x on that interval)

[Anonymous No.12172630]

>>12172620
Algebra CHAD reporting in.

[Anonymous No.12172646]

>>12172620
each of those is based, except for topology which i like less (probably because i didn't force myself to learn it as well as the rest)

[Anonymous No.12172650]

>>12172620
Algebra

[Anonymous No.12172670]

>>12172646
Are there any parts of mathematics that you consider cringe? Either the material or the average person who specializes in it.

[Anonymous No.12172690]

>>12172571
It's like a recursive 2x with an extra term that gets stronger, 2x+f(x)

[Anonymous No.12172717]

>>12172624
We may prove the powerset claim inductively. Start with the empty set, so that N=0. Then there is precisely one subset, so [math]|\mathscr{P}(\emptyset)| = 1 = 2^0[/math], as desired. Next, suppose the claim holds up to some [math]n\ge0[/math] and consider a set [math]A = \{ a_i \ |\ 1\le i\le n+1\}[/math] (we may label the elements like that because it is a finite set). What are the subsets of this set? They are all either of the form [math]B \subseteq A \setminus \{ a_{n+1}\}[/math] or [math]B\cup \{ a_{n+1}\}[/math]. By assumption, we have [math]|\mathscr{P} (A\setminus \{ a_{n+1}\})| = 2^n[/math] such subsets [math]B[/math], so we have [math]2^n + 2^n = 2\cdot2^n = 2^{n+1}[/math] subsets of [math]A[/math], as claimed.

The metric thing is based on this: [math]|f(x) - g(x)| = |f(x) -h(x) +h(x) -g(x)| \le |f(x)-h(x)| + |h(x) - g(x)|[/math] (for any x in the interval). Then the result follows by taking maxima on both sides, as we will then have [math]\max_{x\in [a, b]} |f(x) - g(x)| \le \max_{x\in [a, b]}(|f(x) - h(x)| + |h(x) - g(x)|) \le \max_{x\in [a, b]} |f(x)-h(x)| + \max_{x\in [a, b]} |h(x) - g(x)|[/math].

>>12172314
If not, then the anon should analyse the problems in understanding Understanding Analysis.

>>12172620
Algebra first, topology second.

[Anonymous No.12172725]

N E W N I K O L A J V I D E O
>>12171978
Alain Connes said it's a meme.

[Anonymous No.12172732]

>>12172690
>not just looking in the encyclopedia

[Anonymous No.12172745]

>>12172620
Topology first, algebra second.

[Anonymous No.12172754]

>>12172725
Yeah thanks for the shoutout, I just want to get the topos stuff over with. Wednesday I try to get the second Heyting algebra 101 piece done.

>Alain Connes said it's a meme.
What did he say exactly?
Speaking of which, I just came across this 2h recording of Grothendieck.
Sadly I don't speak French.

https://youtu.be/ZW9JpZXwGXc

[Anonymous No.12172764]

>>12172717
Ohh so basically what I did without the autistic refractallized counting! Thank you anon! Arigatogozaimashita! Arigatogozaimasu!

Adding 0 truly is a powerful tool... What's the deal with Triangle Inequality anyway? Why do people care so much about making sure it's satisfied? Why aren't triangle-inequality-free "metrics" studied?

[Ouch No.12172767]

>>12172620
geometry
>>12172630
faggot

[Anonymous No.12172782]

>>12172571
From 3 the extra term after doubling is
0, 0, 1, 2, 9, 21, 54, 130, 268, 759

[Anonymous No.12172783]

>>12172754
>What did he say exactly?
https://youtu.be/ggOzrB8a1nc?t=3353
>"the great danger is that there are a bunch of scammers/frauds, and for example I read a book called "The Topos of music", but it's incredibly stupid, and in general these people will try to impress others by using an obscure mathematic language without understanding it"
also learn french and read EGAs.
Also it's a shame you can't listen to grothendieck's conference, it's great.

[Anonymous No.12172814]

>>12172783
Life's tough, but I got my Asterix and Gaston comix.

[Anonymous No.12172823]

>>12172764
Intuitively, if you can choose between the straight path from A to B or going through a point C that is not on the path, you would want this A->C->B path to be at least as long as the direct A->B path. One wants the metric to be a reasonable abstraction of what we would call distance, and that's why there are those conditions. No negative distance, detours will not shorten the path, the length is independent of direction and zero distance if and only if the same spot. There is, however, at least this kind of generalisation (called pseudometric): non-negativity & triangle inequality & symmetry & x=y implies d(x, y)=0. An example of such would be points on the plane and then you take the absolute value of the difference of the first (or second) coordinates. This will satisfy all of those, but d((0, 0), (0, 1))=0 so it is only a pseudometric. Even in this generalised version, one wants detours not to shorten the travel.

[Anonymous No.12172981]

>>12172620
I have sex with very attractive women quite often so naturally I prefer number theory.

[Anonymous No.12173142]

calculus is just try hard sudoku

[Anonymous No.12173228]

>>12173142
How will number geeks ever recover from this severe burn?

[Anonymous No.12173257]

>>12173228
allow me to correct myself. graph theory is just try hard sudoku
now you may question the existence of your whole number thingy

[Anonymous No.12173268]

>>12172981
can you imagine fucking a girl for free. literally making her cum and not even wanting to be paid for it.
Can you imagine literally spending your own money to accommodate a woman and making her feel at home, having a lavish life full of sex and pleasures.

I wonder what would happen to society if men stopped putting women on a pedestal

[Anonymous No.12173287]

>>12172620
analysis (but topology is close behind)

[Anonymous No.12173289]

>>12172823
Why is your example a pseudometric? It doesn't agree with standard Euclidean but it follows the axioms so isn't it just a different metric?
As far as your intuition, that would assume that the shortest path is the straight path, and that a straight path always exists, and that the metric on any two points is a function for finding this shortest/straight path. Hmm... What is straight?....

[Anonymous No.12173291]

>>12173142
sudoku is already quite difficult at the most advanced level. if anyone here hasn't checked out really advanced sudoku, i highly recommend it. there's some beautiful stuff going on there.

[Anonymous No.12173304]

>>12168293
holy based

[Anonymous No.12173334]

>>12173291
for sure. even without going into the merit of difficulty, one the best things that's happened to me was finding out about sudoku variants

[Anonymous No.12173337]

>>12172981
I applaud your sexual popularity and success. I too, enjoy some number theory, but my favourite area of mathematics out of the ones listed is probably algebra.

[Anonymous No.12173339]

>>12173289
It fails to satisfy the axiom that says d(x, y)=0 iff x=y. It satisfies the weaker one "if x=y, then d(x, y)=0". The straightness of the shortest path is also a good way (in this case) to see why it is called the triangle inequality. Given any points x and y, distinct or otherwise it will be degenerate. If d(x, y) = d(x, z) + d(z, x), then z has to be on the line segment with x and y as its endpoints. If we now move z away from that, we obtain a triangle with vertices x, y and z, and d(x, y) < d(x, z)+d(z, y). For shortest paths between two points in general, check geodesics. You shouldn't necessarily think about them when you think about metrics, though.

[Anonymous No.12173351]

>>12172981
based

[Anonymous No.12173352]

>>12173142
https://en.wikipedia.org/wiki/Mathematics_of_Sudoku

[Anonymous No.12173408]

>>12173352
wait, it's formalized by group theory? i thought sudoku would be graph theory. huh. but then again i'm a mathlet and don't know what group theory is

[Anonymous No.12173416]

The Miracle. Normal Sudoku rules apply (numbers 1-9 in every column, row, and box). Cells a king's move or a knight's move away from one another cannot contain the same number. Orthogonally adjacent cells may not contain consecutive numbers.
There is a unique solution. Find it.

[Anonymous No.12173447]

Did a janny just delete the post about "the miracle" sudoku puzzle? What gives?

[Anonymous No.12173449]

>>12173416
have you solved this, anon?

[Anonymous No.12173459]

>>12173449
Yes, it took quite a bit to get through the first third but afterwards it snowballs. The grid is incredibly restricted. I highly, highly recommend trying to make as much progress as you can. It's beautiful.
If you give up here's an expert solving it blind: https://www.youtube.com/watch?v=yKf9aUIxdb4

[Anonymous No.12173461]

>>12173408
Yes, the relation between algebra and Sudoku is quite fascinating.

[Anonymous No.12173482]

>>12173459
I really like watching that guys' vids. he's both good at it and quite entertaining

[Anonymous No.12173512]

Here's another great Sudoku. It is impressive that a grid of digits is capable of achieving the sublime.
https://www.youtube.com/watch?v=LwkNChSO2yE

[Anonymous No.12173536]

My teacher groomed me and I liked it

[Anonymous No.12173544]

>>12173536
was he old

[Anonymous No.12173565]

>>12173544
Very.

[Anonymous No.12173568]

>>12173565
h-hot...

[Anonymous No.12173590]

>>12173536
Nothing wrong with this. Islam is trendy precisely because it is the complete package to people who want to be groomed.

[Anonymous No.12173851]

>>12173339
can any space with a pseudometric have equivalence classes (y in [x] when d(x,y)=0) =put onto the space to make a real metric?

[Anonymous No.12173895]

>>12173851
Yes. I think that works. Sorry for the flow of thought post, I'm thinking as I'm writing. We need reflexivity, but that is guaranteed by [math]x=y \Rightarrow d(x, y)=0[/math], we need symmetry, but that is also given by the axioms of a pseudometric, [math]d(x, y) = d(y, x)[/math], and transitivity. Now this thing uses the triangle inequality. If [math]d(x, y) = 0 = d(y, z)[/math], then [math]0 \le d(x, z) \le d(x, y) + d(y, z) = 0 + 0 = 0[/math], so defining [math]x \sim y \Leftrightarrow d(x, y) = 0[/math] does indeed give an equivalence relation. Now, this gives quotient space [math]X/\sim[/math] with all the equivalence classes as its points. And yes, we do get an induced metric [math]\tilde{d} \colon X/\sim \times X/\sim \to [0, \infty)[/math] which will be given by [math]\tilde{d}([x], [y]) = d(x, y)[/math] for some representatives of the classes. Is it well-defined? If [math]x, x'\in [x], y, y'\in [y][/math], then [math]\tilde{d}([x], [y]) = d(x, y) \le d(x, x') + d(x', y') + d(y', y) = 0 + d(x', y') + 0 = d(x', y') = \tilde{d}([x], [y])[/math], so yes it is! Very good. The answer is yes.

[Anonymous No.12173939]

>>12173408
Did an undergrad project about applying Burnside's lemma to sudoku.

I was based on the work of some guy who had answered the question "What is the minimal number of hints in a well defined sudoku problem", where by "well defined" we mean that the problem has a unique solution.

He showed that the answer is 17. And in order to do that, he had a super computer exhaust the possibilities.
Only you have an obvious group of symmetries that don't change the problem. Like, if you just rotate a grid, or swap all the 2s and the 9s, that is not going to change the minimal number of hints for a sudoku problem to have your grid as its unique solution.
So he applied Burnside's lemma and worked only with one element of each orbite.

It still took almost a year for the supercomputer to complete the exhaustion though.

As an undergrad, I reproduced the work for 4x4 sudoku grids, and after applying Burnside, I had reduced the problem to 2 essentially different grids (which also proved, as one may imagine, that 4x4 sudoku is absolutely uninteresting). If you care, the minimal amount of hints for a 4x4 sudoku problem is 4.

[Anonymous No.12173987]

Meta-mathematically speaking, why is convexity so ubiquitous?

[Anonymous No.12174585]

https://youtu.be/66ko_cWSHBU

F(x)= me hoopin on yo ass

[Anonymous No.12174622]

>>12173987
Non-convex stuff breaks more easily (in more convex stuff)

[Anonymous No.12175099]

>>12169284
I have a bad "sampling" habit where I'll start to read through concepts but I won't digest enough of them to truly unlock the power of them.

[Anonymous No.12175177]

Fuck normalfags.

[Anonymous No.12175191]

>>12167294
For lower dimensions this is pretty easy, but I remember for higher dimensions the proofs weren't so easy/intuitive.

[Anonymous No.12175210]

>>12175099
Same problem. Is there any hope for us? What sort of concepts do you like?

>>12175177
[math]F \trianglelefteq G[/math] where [math]F[/math] is free abelian?

[Anonymous No.12175211]

>>12175177
why this all of a sudden

[Anonymous No.12175568]

reading Lectures on the Hyperreals and i think i've been lost somewhere

given [math]a=[0,1,0,1,0,1,\ldots][/math] and
[math]b=[1,0,1,0,1,0,\ldots][/math]. It is clear [math]a\ne b[/math] since [math]a_n\ne b_n[/math] for all [math]n[/math].
Where I'm confused is which of [math]a>b[/math], or [math]b>a[/math] is true, or are [math]a[/math] or [math]b[/math] malformed somehow?
Both [math]a_n<b_n[/math] and [math]b_n<a_n[/math] are violated infinitely often, so both should be false, right? but they're also not equal?

[Anonymous No.12175627]

>>12175568
When you choose a nonprincipal ultrafilter you have to choose for each infinite set of indices with infinite complement which one you want in your ultrafilter.

[Anonymous No.12175649]

>>12175627
so it's a matter of choice of ultrafilter which is greater?
if [math]S=\{2,4,6,8,\ldots\}[/math] is in our ultrafilter [math]\mathcal{F}[/math] then [math]b<a[/math] and [math]a<b[/math] if [math]S^c\in\mathcal{F}[/math] instead?

[Anonymous No.12175710]

>>12175649
Yes, that's right.

[Anonymous No.12175898]

Four symbols
IIII
And three
III
You note a nonmatch
AND when you have
TWO SYMBOLS
43
You also notice a nonmatch
In your visual pixels
And what separates them? Solely the forced connection between Truthvoice and vision giving them the word, NICHT GLEICH
In math, we simply follow the Truthvoice as it pertains to things which it deems numerical, discrete, or others likewise

[Anonymous No.12175975]

Should I take a Symbolic Logic class this quarter while I review mathematics for my next quarter Calculus 1 class? Will I see benefit or should I choose something easier so I can focus on reviewing for Calculus 1? Is Symbolic Logic a kind of foundational subject from which I will benefit everywhere even now, or some tangential philosophy subject?
They use
>A Concise Introduction to Logic by Hurley & Watson (13th edition)
but it uses the Cengage Mindtap assignment garbage which is $100 while the PDF is freely available on libgen- very annoying, so I do not want to spend the funds if it is not necessary at least for now.

[Anonymous No.12176023]

>>12175975
It's not necessary but studying logic as early as possible makes the rest of math a lot easier. Without formal logic in the background a lot of your thinking is just trash which might be good enough for eating and fucking but does not work when you're trying to do math.

[Anonymous No.12176077]

what's /mg/ opinion on topological data analysis?

[Anonymous No.12176079]

Is there anything to do with measure theory and metric spaces? Specifically I'm wondering if there exists a path (which is a continuous ordered set) that describes the deformation between any two elements in a metric space, whose measure equals the metric.This would mean this continuous deformation has the lowest measure of all continuous deformations between A and B (assuming that you can generate all deformations from the variety of choices of intermediate paths), which gives a nicer intuition to metrics - if summing Met(x,y) and Met(y,z) doesn't give the value for something that "links" x and z (such as a distance from x to y and y to z links x and z in R^2), then the analogy of a triangle doesn't hold up well. A deformation as a set is a generalization of a path in space, and holds the property that it starts at one point and ends at the other, and that it goes through the specified middle point for "triangle deformations".

I noticed that you can always define a "discrete path" which is an ordered set containing only the elements you choose to be on the path, and it coincides with the metric nicely, but this is not as interesting, although satisfying enough I suppose.

[Anonymous No.12176087]

Calc 2 is making me want to blow my fucking brains out. I dont give a shit about these gay little magic tricks for solving integrals. I spent 30 minutes working on a problem, the final answer was .00004517, I got marked wrong because the faggot website actually wanted me to write "≈ 0.00"
What kind of retard makes these?

[Anonymous No.12176102]

>>12176077
The technique has some holes, or at least I think so...

[Anonymous No.12176118]

>>12176087
They're not really that hard just memorize them and bedone or learn the magic behind em

[Anonymous No.12176214]

>>12176118
It's just some calculation, man.

[Anonymous No.12176235]

>>12176077
it's a meme

[Anonymous No.12176240]

>>12176079
Wait, I think this might not work
>smallest continuous deformation on a set
If only one such exists, then the standard measure produces one result for any pair of elements. But, if there can be different metrics defined, they won't equate, unless we have different measures associated to each metric.

[Anonymous No.12176241]

110 iq brainlet here
Can someone explain how multiplying pi by the radius squared gives the area of a circle

[Anonymous No.12176242]

>>12176235
At least neural networks have a solid mathematical basis and aren't just fashionable nonsense.

[Anonymous No.12176246]

>>12176144

[Anonymous No.12176247]

>>12176242
>At least neural networks have a solid mathematical basis and aren't just fashionable nonsense.
fairly certan people have no fucking clue how they work so well

[Anonymous No.12176249]

>>12176241
Do you know any calculus or do you want the ancient informal reasoning?

[Anonymous No.12176258]

>>12176247
thatsthejoke.aux

[Anonymous No.12176269]

>>12176249
Ancient informal reason

[Anonymous No.12176601]

>Geometry by Brannan, Esplen, & Gray
>Geometry: A Comprehensive Course by Dan Pedoe
>Introduction to Geometry by Coxeter
How do these books compare? Which one do you recommend?

[Anonymous No.12177160]

>>12176077
took a few informal lectures on it, very neat stuff. It's shown practical use, but don't be fooled to think that PCA isn't still king for high dimensional data. TDA is niche and secondary to more immediate and traditional techniques in my eyes.

[Anonymous No.12177191]

>>12177160
>TDA is niche
Not for much longer.

[Anonymous No.12177198]

>>12177191
did something neat happen?
last I checked for practical (data crunching) purposes it was a ton of computation for usually not telling you much more than other methods could tell you with less. It does have a nice visial flair to it i guess, but you can get similar stuff with PCA if you know what you're doing..
Or did /g/ just hear about it now and is making it their next obsession or something?

[Anonymous No.12177571]

just enrolled in a maths msc with 0 research experience and no idea what to write my dissertation about but it's gonna be fine, right?