/sci/ - Science & Math

>>12410456
first for forcing

>>12410456
i need help with this triple integral.

SSSd(x^2 + y^2)dxdydz d: x^2 + y^2 + z^2 <= 1, z >= 0

the result is 4pi/15

but don't know how to get there.

>>12410615
also, it's my first time posting on /sci/, how active is this board?

>>12410615
>>12410619
Honestly not terribly active. Just at a glance and assuming I'm reading this right, have you considered the divergence theorem?

>>12410630
no, but we haven't studied that in class.

i think it's about transform the domain to spherical coordinates, and then integrate. but still i cannot reach that result.

>>12410634
ok so basically we're looking at the upper half ball. In spherical coords it has bounds [math] 0 \leq \theta \leq \frac{\pi}{2} [/math], [math] 0 \leq \varphi \leq 2\pi [/math] and [math] 0 \leq r \leq 1 [/math]. The jacobian is as usual [math] r^2 \sin\theta [/math].

We deduce [math] x^2 + y^2 = r^2 - z^2 [/math] , and as usual we have [math] z = r\cos\theta [/math]. Hence [math] x^2 + y^2 = r^2 - r^2\cos^2\theta = r^2\sin^2 \theta [/math]. Lastly, putting this all together. I will use [math] B [/math] to represent the domain.

[math] \iiint_B (x^2 + y^2) dxdydz = \int_0^{2\pi} \int_0^{\frac{\pi}{2}} \int_0^1 r^4\sin^3\theta dr d\theta d\varphi [/math].

Evaluating the integral on the right is simple and nets you the desired answer.

>>12410660
thanks a lot anon.

i was integrating by r^3 sin^2

>>12410660 thanks again, i have my exam in a few hours and you really help me. i was giving up with those kind of exercises because they always use x^2 + y^2

>>12410671
>>12410680
No worries. I'm studying for my own exam but if you have other questions you can ask here and I'll try to answer. Others might also but sometimes they get snobby about the kinds fo questions they'll answer. Here's my discord in case the thread dies and you want more help.
Candy Corn Enjoyer#2732

>>12410695
you are very kind anon, but it's almost 5 AM here, and my exam start at 1 PM, im gonna get some rest and see how it goes in a few hours.

>>12410711
Good luck

>>12410456
Best edition.

>[math]S^N[/math]
wtf is that?

>>12410774
the hypersphere embedded in [math] \mathbb{R}^{N+1} [/math]. So to be precise, I'll define [math] S^{N-1} [/math] as the following subset of [math] \mathbb{R}^{N} [/math];

[math] S^{N-1} := \{\mathbf{x} \in \mathbb{R}^N: |\mathbf{x}| = 1 \} [/math].

>>12410456
>. Show that any open connected subset of SNSN contains a rational point.
inverse stereographic projection centered at a point with all but one coordinates zero is a rational parametrization of S^N. the open connected subset in question clearly intersects at least one domain of such projection in an open subset.

>>12410456
Oh you like set theory huh. Name every set.

>>12410780
Oh ok, I'd probably think of that if you used lowercase "n".

>>12410456
Counterexample: the empty set is both connected and open but does not contain a rational point

>>12410990
A space is connected iff it is not a disjoint union of finitely many nonempty subspaces. Zero is finite, and the empty space is a disjoint union of zero nonempty subspaces.

>>12411010
>A space is connected iff it is not a disjoint union of finitely many nonempty subspaces
that's not what connected means, wtf

How do I solve this without recursive time steps:
>container filled with liquid A
>hole in bottom, draining out at constant rate
>liquid B is entering from above at constant rate
Find percentage of it thats liquid A at any given time

>>12410456
Hello /mg/ friends, Im having a bit of a crisis here, can someone pls throw some advice?

>Be EE student
>Start taking pure math classes for a minor (Bunch of math professors from my calc classes suggested I do so)
>End up liking mathematics a fuck ton more than EE
>Realizing I also have more talent for mathematics than for EE as ive been topping classes and having profs talk about me and so on. (This is so weird, I honestly think im retarded and I cant believe any of it. To put it simply I think my profs are too lenient and that I would get filterred the fuck out if I were studying in a harder uni)
>I ended up doing 2 minors (requisite for the math minor was taking 4 math classes, ive taken 6 at the moment and by the time im done with uni id have taken 8) but Id still need 5 more classes and a thesis (at the very least) in order to get the math degree
>Regardless I want to go into math gradschool in europe, for applied and computational math specifically
>I wont have a math degree tho
>Is it even possible for me to get admitted to a good masters program? Is my decision to go into math for the rest of my life retarded? I actually like the systems and signals part of EE but other than that its just so goddamn stale and boring

help. im having so many different thoughts and shit torment me I fucking cant

>>12411010
What the actual fuck

How do i solve those:
1) R^3 can be decomposed into disjoint union of unit circles
2) R^3 can be decomposed into a disjoint union of straight lines, none of which are parallel
should be some kind of transfinite induction but i don't see it

>>12410456
well known fact - the rational points are dense in unit circle
then, you can join solutions in the following manner:
if x^2 + y^2 = 1 and a^2 + b^2 = 1, then (ax)^2 + (ay)^2 + b^2 = 1
and fuck around with induction and probably you'll get that rational points are dense in S^N

alternatively, use the standard method for finding rational points of degree 2 polynomial equation

>>12410854
big brain solution, respect to you sir

>>12411561
>1) R^3 can be decomposed into disjoint union of unit circles
Trivial (by inspection).
>2) R^3 can be decomposed into a disjoint union of straight lines, none of which are parallel
Doubt this can be done.

>>12411570
I changed my mind. Both are actually impossible.

>>12410456
Reminder that constructive proofs are inherently superior.

>>12411570
>>12411573
They're both possible and you can find a solution by googling.

>>12411618
>relies on axiom of choice

>>12411573
A bunch of cylinders my nig nog

>>12411561
Straight lines hmm... What about some kind of interlocking staircases

>>12411742
>A bunch of cylinders my nig nog
That doesn't work.

Favourite books on combinatorics?
I'm math undergrad who so far took all his courses on analysis and algebra so my counting skills are whacky. Any books to help me improve on them?

What's a good reference for the Picard-Lindelof theorem and other results about solutions of odes?

>>12411811
Honestly just to hone your skills.
Combinatorial Problems and Exercises by Laszlo Lovasz is great. This is assuming you already have some proficiency and are just looking to git gud. It's hard.

>>12411811
>whacky
Alex?

>>12411911
Do they come with solutions?
>>12411920
Nigga who?

A huge part of algeraic topology isthe study of paths and maps of higher dimensional spheres into a space. We usually consider these paths under homotopy.
But is there any study of non-self intersecting paths? There seems to be some nontrivial stuff going on:
Consider a punctured disk with two basepoints in the boundary. Normally we might look at the homotopy classes of paths connecting these two points and see that they have the structure Z. You can wind around the puncture several times in each direction.
But if the paths have to be not self-intersecting then there should only be two homotopy classes - going around the puncture and the puncture this way and that way.

>>12412019
It's probably not very interesting because it doesn't give you a group (to compose two loops you need self-intersection). Even if you allow for self-intersection but only at the basepoint then it's also not homotopy invariant.
For example take the annulus, which is homotopic to a circle.

>>12412069
Well it doesn't do the same things as fundamental groups but I still think there are interesting questions to ask. Such as the following:
Consider [math] \mathbb{R}^2 \setminus \{x_1,...,x_n\} [/math], the plane with [math] n [/math] holes, and fix two additional points [math] a,b \in \mathbb{R}^2 [/math].
Then how many basepoint-fixed homotopy classes of injective paths from [math] a [/math] to [math] b [/math] are there? (There are different possible strengths of homotopy equivalence one could choose here, the homotopy could be arbitrary, injective as a path for every time step t, or completely injective.)

>>12412019
Isn't that what knot theory is about?

>>12412160
maybe
maybe knot

>>12410456
How do I get better at math for personal enjoyment because I probably have autism or something?
What books should I start off with, given that I do not have a very strong base currently?

I graduated HS with good grades months ago but now i'm a neet due to corona. If I want to study math for my own enjoyment and remain sharp where should I dig around?

>>12412019
>>12412120
What you're talking about is called isotopy, and yes it does come up in some fields, usually related to low-dimensional topology.

>>12412120
I am not very familiar with homotopy classes, but aren't two paths p1,p2 from a to b homotopic iff there is a continuous function mapping one to the other?

if so, isn't this just a question about combinatorics?
each hole [math] x_i [/math] allows a path to go around it in 2 ways, clockwise or counter-clockwise. Because the paths must be injective, each path cannot intersect itself and no hole can be circeled twice.
So we have [math] \prod_{i=1}^n 2^i = 2^{\frac{n(n+1)}{2}[/math] different homotopy classes?

I am not sure though. Could someone check this?

>>12412446
sorry, [math] \prod_{i=1}^n 2^i = 2^\frac{n(n+1)}{2} [math]

also, I made this litle drawing. Does this make sense?

>>12412449
is that Spa Francorchamps?

So why is infinity not treated as a number (it isn't anyway), but zero is? Both don't exist and don't make any sense, so why the discrimination?

>>12412449
so for n=1 you're saying there are 2 classes?
i think it's possible to also fuck around the origin points
this picture has 3 non-homotopic paths (and there's fourth which i didn't draw for clarity)

can someone recommend me a (older) book which is a compendium of math proofs and knowledge?

>>12412546
zero has better algebraic properties

>>12412548
actually there should be infinititely many by going in spirals around a and p1 before going to b....
>>12412348
Huh I've never heard of this before... sounds interesting. Would it also deal with something like self intersecting paths on a discrete lattice?
I asked this question in the first place because I read somewhere that non self-intersecting random walks are a very difficult problem in probability theory. And then I thought man I really don't have a good idea of the topological behaviour of such paths.

>>12412548
isn't the set of homotopy classes in your picture infinite? after all, we could always make one more round around p1 and this would not be homotopic to another path that makes a different number of rounds aroung p1, because we would need to move the path along p1, which is not continuous.

>>12410660

What are you using to achieve the formatted annotations?

>>12412546
Name a number that does exist

Explain this shit right fucking now math niggers.

>>12412802
-1^2 = -(1^2) by convention

>>12411594
They're fine as long as [math](P\wedge\lnot P)[/math] is true

>>12412802
Write [math](-1)^2[/math], if you mean that.

>>12412345
Get a book on proofs. Also Stewart's Calculus is the go-to for starting on Calculus.

>>12412808
>>12412805
Ah okay thank you guys (:

Have some cute jap girls is christmas costume as thanks

>>12412818
based newfag

>>12412818
we don't want cute japs here, we want your answer to the following question:
does 0.999... equal 1?

>>12412860
Obviously not, I am familiar with the meme saying otherwise. You can trust me on this btw.

>>12411060
That's exactly what connected means. Are you confusing "connected" with "path-connected"? They're not the same.

>>12412925
every space is connected according to your definition

>>12413004
*none, my bad

>>12410456
Consider higher order n...n^x as a set of higher dimensional Pascal's triangles.
11^×
.[1(n^2), 2(n^2),1(n^2)]=nn^2 [1(n^3),3(n^3),3(n^3),1(n^3)]=nn^3 ........

111^×
Etc

And that those can describe multiplication itself as a waveform

>>12411561
https://mathoverflow.net/questions/28647/is-it-possible-to-partition-mathbb-r3-into-unit-circles

>>12412988
>he isn't ironically posting some hard problem with kurisu, he's actually posting his homework
Anon, I...

I'm almost done Linear Algebra Done Right, I want to move on to multilinear/tensor algebra.

There's a chapter on the tensor product in Serge Lang's Algebra. Is there a multilinear algebra done right?

>>12413247
There's Greub's Multilinear Algebra, which is good, cute and funny.
I think it assumed some topology tho.

WHY I CAN'T REMEMBER HOW TO USE OR FORMULATE THE SPECTRAL THEOREM FOR UNBOUNDED SELF ADJOINT OPERATORS AAAAAAH IT COMES UP ALL THE TIME

>>12411811
>combinatorics
Intredasting, just wanted to make a post that I'm studying combinatorial geometry, or at least some part of it, right now. Ewald's book "Combinatorial Convexity and Algebraic Geometry" is really nice if you're interested in that topic.

Anyone willing to solve a Laplace IVP HW problem? Am stuck need help, thank you.

What is so special in number 9232?

>>12414501
it's the smallest number that can be written as a product of 1 and 9232

>>12414490
I'm too lazy right now to solve it, sorry, but I can tell you what I would do. Consider your eqn first on [math][0,4)[/math]. With the additional initial conditions, you need to solve [math]y''-4y=1[/math]. Multiply by [math]y'[/math], integrate, integrate again and you should have the moste general solution for [math]y[/math] depending on two parameters. Then do the same for [math][4,\infty)[/math] and find the fitting two parameters.
This is "by hand", there's probably an easier method using Fourier stuff iirc, but cannot recall it in detail right now.

>>12414501
9232 = (((((((((1214/2)*3+1)/2)*3+1)/2)*3+1)/2)*3+1)/2)*3+1
As such, 9232 is unusually frequent peak value in Collatz conjecture. See http://oeis.org/A095387

>>12414531
Might seem like a foolish question to ask but: There's two answers right? A y=f(t) for t greater then t and another for when it's between 0 and 4?

what do people mean when they say "model of ZF" or "model of X"

>>12414636
>what do people mean when they say "model of ZF" or "model of X"
Some mathematical structure in which the axioms of ZF (resp. X) hold.

is there such a thing as a proof that something cannot/ can be proven?

>>12414726
Halting problem

>>12414726
I believe the continuum hypothesis.

>>12414726
The incompleteness theorems?

>>12414636
a model is a structure or example that satisfies the axioms of a theory.
for example integers under addition are a model of a group.

one form of consistency proof is creating a model.

You can get autistic and define what structures are in terms of set theory (bourbaki structure) but that's a formality.

>>12412988
find an onto homo that has the ideal as its kernel

>>12415063
interesting
is { { [math] \emptyset, P(\emptyset), P(P(\emptyset)), ... [/math] } , [math] \in [/math] } a model of ZF?

>>12415134
>, ...

>>12415138
you know what I mean silly boy

>>12415144
How would I know what you mean?
Let X = {1,4,12455,-pi, 123, ...}. Do you know what I mean by X?

>>12415151
more like let X = {0, 1, 2, 3, ...}
I think you know what set X is

>>12415156
Is it {0,1,2,3,5}?

>>, ...

>>12415158
no
keep guessing

>>12415165
Why don't you simply write down what you actually mean instead of playing a game where others have to guess, retard?

>>12415170
context makes it clear in 99% of cases.

>>12415134
No, because it doesn't satisfy the axiom of infinity.

>>12415170
>>12415180
There are two kinds of mathematicians in this world.

Are there any good reasons to think ZFC is consistent apart from the fact that we haven't found a contradiction so far?

>>12415212
yes, those who get work done and those who squabble pedantically all day achieving nothing.

>>12415134
Yes.

>>12415275
No, ZFC proves that set exists, which would contradict the second incompleteness theorem. That set is a model of ZFC minus the axiom of infinity plus the negation of the axiom of infinity. In fact, that model is elementarily equivalent to the standard model of the natural numbers.

>>12415183
Why do you think so?
[math]\mathcal{P}(\emptyset)=\{\emptyset\}\\
\mathcal{P}(\mathcal{P}(\emptyset))=\{\emptyset,\{\emptyset\}\}\\
\dots[/math]
So we have [math]\emptyset \in S[/math] and [math]\forall_{X \in S} X \cup \{X\} \in S[/math].
The "axiom of infinity" is satistfied.

>>12411811
taking a grad level combinatorial analysis course next semester
what am I in for?

>>12410615
Use cylindrical coordinates.
[math] x = r\cos(\theta),\quad y = r\sin(\theta),\quad z = z [/math]
and
[math] dx dy dz=r dr d\theta dz [/math]

>>12415267
>Are there any good reasons to think ZFC is consistent
sure, we've been working with it long enough and it seems to work just fine
>inb4 is there a proof
you can only prove something inconsistent

>>12414490
>>12414531
Use unit step functions. [math] U(t-c)=0 [/math] when [math] t<c [/math] and [math] U(t-c)=1 [/math] if [math] t\geq c[/math]
so
[math]
f(t) = U(t) - 8 U(t-4)
[/math]
using unit step functions.
The Laplace transformation for unit step function is known
[math]
/mathcal{L}[U(t-c)] = \frac{e^{cs}}{s}
[/math]
You don't have to analyze it piecewise

>>12415294
>That set is a model of ZFC minus the axiom of infinity plus the negation of the axiom of infinity
Wrong. He's obviously enumerating over all ordinals, so his set satisfies the axiom of infinity.

>>12415366
Sorry, the last equation had the wrong slash
[math]
\mathcal{L} [ U(t-c)] = \frac{e^{cs}}{s}
[/math]

>>12415306
The underlying set of the model may be infinite but every element of the model is finite. Think about the natural numbers, the set of natural numbers is infinite but every natural number is finite.

>>12415350
Read the post before responding to it, dumbass.

>>12415267
is there any good reason to think that peano axioms are consistent apart from the fact that they're the simplest fucking thing imaginable, and for the last 5000 years we haven't found a contradiction so far?

for the last 50 years set theorists have been proving stuff such as "if ZFC is consistent then ZFC+retarded_axiom_no_123 is consistent", i would say that "zfc is very weak because all kinds of shit can happen in the models" is a more valid complaint
and in some sense "axioms are very weak" and "axioms are self-contradictory" are on the opposite sides of a spectrum

>>12415375
>every element of the model is finite
Are you retarded? At the step [math]\omega+1 [/math] it generates an infinite set which contains all the natural numbers, so obviously is satisfies the axiom of infinity.

>>12415375
ZF doesn't require a model to have an element with infinitely many elements.
Just as you argued: [math]\infty \notin \mathbb{N}[/math]

>>12415388
You are clearly interating over the natural numbers the way you have written things out. Why don't you use transfinite recursion like a big boy and stop being obtuse.

>>12415401
>You are clearly interating over the natural numbers
??? How was it clear. This is set theory so obviously it's going over all the ordinals, just like in constructions of V and L.
>Why don't you use transfinite recursion like a big boy and stop being obtuse
The transfinite induction is clear from the context. That's what ... stands for.

>>12410456
brainlet from /lit/ here, i'd like to ask about your thoughts on abstract objects like numbers. Do they exist? Is it empirically observed or mentally constructed? Are you a neoplatonist or a complete nominalist?

>>12415414
>Do they exist?
Only the small natural numbers and simple rational numbers that we can compute with, like 2999, or 12356122/51920312. Other structures also exist like Gaussian integers (very fascinating structure) or p-adic numbers.
"Real numbers" don't exist.

>>12415404
No it isnt', doing one step of the induction at the base case is not clear. The construction of ordinal indexed levels of the cumulative hierarchy depends on which ordinal you want to stop at or continue to. If you said consider the set S that contains the empty set, if x_alpha in S then P(x_alpha) in S and if lambda is a limit ordinal then x_lambda=Ux_alpha in S. That would be proper classification of V.

>>12415430
>The construction of ordinal indexed levels of the cumulative hierarchy depends on which ordinal you want to stop at or continue to.
You don't stop. It's the union over all the ordinals. Obviously that's not a set but it's still a model of ZF.
>if lambda is a limit ordinal then x_lambda=Ux_alpha in S.
This was implicit in the way it was written down. You always take the union when you're doing a cumulative hierarchy and reach a limit ordinal.

>>12415429
interesting, do you study metaphysical implications of this in an academic setting?

>>12415437
My point was you can stop. The poster in >>12415134 put set brackets around his iterations of the power set. This means it is a set. So I assumed he stopped at omega since there was not transfinite inductive set. That is why I am saying you have to be clear and use transfinite recursion.

>>12415414
there's no meaning to this question, it only seems "deep" because you throw around some unclear words and leave it to the interrogated party to clear up what the fucking question actually means
once you clear up what does it mean "numbers exist" or "numbers are empirically observed" or "numbers are mentally constructed" the question becomes so dumb that you realize it's not worth asking

can I walk out of my house and trip over a number because it's lying on a pavement? no
can I count my fingers? yes there's ten of them

Directed here from /mfg/. I'm a highschooler aiming for IMO. Any tips?

>>12415456
>This means it is a set
No it doesn't retard, obviously it's not a set because it's bigger than any ordinal. Classes are regularly denoted by brackets in set theory.
>So I assumed he stopped at omega
Why would you assume he stops at such an arbitrary place?
>That is why I am saying you have to be clear and use transfinite recursion
Your pedantic quibbling is not helping anyone. The transfinite recursion is completely obvious from the context.

U can't like prove infinity exists cuz like you can't count to infinity on your fingers and toes

>>12415456
Even if it stops at [math]\omega [/math] that's still reliant on transfinite induction. He didn't specify that it stops at [math]\omega[/math] so it's implicit that he doesn't, i.e. he goes through all the ordinals.

>>12415502
You have to reading comprehension, and I said why I assumed he stopped at omega, because there was no transfinite inductive step in his definition. V_omega, the set I thought he wrote down, is a very important set, it is a model of ZFC without infinity plus the negative infinity. It is used to show that the axiom of infinity is independent of the rest of the ZFC axioms. It comes up in exercises in set theory all the time, I even had a question about it on my set theory final. If you think being clear is pedantic quibbling youre ngmi as a mathematician.

>>12415414
Abstract objects are a tool of the mind to process sensory data and develop long-term behavioral strategies.

>>12410456
Stereographic projection is a rational transformation with integer coefficients (for a reasonable choice of hyperplane). It is easy to check that the property holds in a hyperplane of R^{n+1}, so it will also hold on the sphere

>>12415537
>You have to reading comprehension
...
>because there was no transfinite inductive step in his definition
The transfinite inductive step is obviously implicit, just take the union like you do in all such constructions.
>V_omega, the set I thought he wrote down,
Why would you think he wrote down V_omega? That's a baseless assumption.
>it is a model of ZFC without infinity plus the negative infinity
Ok but who cares? He was talking about models of ZF not models of ZF-infinity so obviously he didn't mean V_omega (and not just because he didn't specify that his induction stops at omega).
>If you think being clear is pedantic quibbling youre ngmi as a mathematician
Just because you come up with your own retarded misinterpretations of what he wrote and make petty arguments about it doesn't make you smart or interesting.

>>12415553
>the recursive step is implicit
lol, keep coping

>>12415559
That's what "..." means, retard.

There are x cards in a deck.
y of these cards lay on the board.
What's the possibility of encountering the card on the same space two games in a row, provided that the positioning of the cards is always the same, and the chance of encountering any of the cards is the same for each

>>12415490
Michael Penn is probably a good YouTuber for you.

>>12415662
Thank you

>>12415663
Don't thank me without having watched a video or two.
Otherwise you won't know, if to thank or to curse me.

>>12415672
I don't thank you for your recommendation, but for your attempt. I already know him

what jobs are actually interesting as a maths graduate

hardmode: no academia

>>12415854
nascar driver

>>12415854
work at a hedge fund, earn money, retire and then forget about all the numbers and math shit and actually start doing something interesting

>>12415854
skiing instructor

>>12415854
successor

>>12415985
why is Ted so fucking based bros

>>12415854
courier

How do I formulate this? It's a finite sum, which is basically a geometric progression sum, but it uses an increasing multiple (n), which is an arithmetic progression.

>>12416269
It's simple anon
[math]
\product_{n=2}^{k} \frac{n^2}{n-1} = \frac{(k!)^2}{(k-1)!} = k(k!) [/math]

>>12416341
[math]
\prod_{n=2}^k\frac{n^2}{n-1} = k(k!)
[/math]

>>12416349
thanks!

>>12415854
Independent Gentleman.

>>12415854
Take some stats courses and don't be a fucking asspie and you can get plenty of job options. Also lie that you know how to program and just try to look like a buisness type. Most people hire based on their guts and even ask you to take personality fb tests lmao and it is all bullshit. They just need to think you are not "walled" by academic nonsense and you can get the work done (knowing how search for libraries and bullshiting your shit well). There's a reason fucking data sciene, a fucking meme non profession is now in such hype, because basically they managed to sell the idea that someone that can use python r and julia is in some sense an expert in data analysis and presentation. Obviously it doesn't work everywhere, data encryption and protection at high levels will require you to know some shit, but what they actually use can be learnt fairly easily with a good background on math.

>>12412988
Literally just the quadratic formula

pic related, it's my problem solver

>>12412988
Ring structure on R[x] or vector space structure?

>Garden of Eden theorem
wtf this is literal pottery

>>12416851
Watch out, the Lemma of Indecency is ahead of you.

>>12416985
>not a real lemma

>>12416851
>tfw studying this type of shit for my undergrad research

how do you effectively search to make sure you're not re-treading ground covered by some bored dude centuries ago
do you just need to jam through the 10k unrelated search results on an academic search to make sure your shit's original?

A well known result is that you can measure all whole number weights between 1 and 40 on a balance scale with only 4 different weights. You just have to make the weights 1, 3, 9 and 27 and combine them on both sides of the scale to get every possible weight. I have found that you can actually make the weights 2, 5, 15, 45 and get even better results, since you can now measure all the way to 67. What would be the max amount of consecutive natural number weights which could be measured with only 4 weights?

>>12417649
can the weights be repeated?
If so, just use 1. You can now measure all the way up to infinity.

>>12417649
>I have found that you can actually make the weights 2, 5, 15, 45
How do you measure 1 with such weights?

>>12415414
I think it's empirically motivated. The foundations weren't set a priori. You develop a theory of counting and numerosity, which is absorbed by a more general theory of integers. Lots of math is in essence just casting existing theories as special cases of more general theories.
Are numbers real? They describe real things. They're as real as language.

>>12417659
They can be repeated, but then you will have 4 weights of 1 and I'm not sure how would you measure above 4 with those.
>>12417663
Well, the main difference between the original and mine is that I allow the scale to tell you if the desired object had a weight less, equal or more than whatever you're measuring. This makes sense in real life and it makes it so that I don't have to actually generate all the weights between 1 and 67. So to measure 1, you just have to measure less than 2.

>>12417649
>>12417670
I guess I might as well fully explain how mine works and hopefully someone can come up with an optimization or something:

1)Notice that the weights 5, 15 and 45 can measure any multiple of 5 in the range.
2)This means our object of weight x could be bound between two multiples of 5, a and a+5.
3)If x is a multiple of 5, we could directly measure it.
4)Otherwise there are only 4 possibilities:
a)x = a+2
b)x = (a+5)-2
c) x < a+2, therefore x = a+1
d) x > (a+5)-2 , therefore x = a+4

All of this could be measured when using the 4th weight that weights 2.

How do I put this into one equation:
>x^2+y^2+z^2=1
>y=z
The intersection

>>12417717
If you know that y=z, then you just turn the z^2 into y^2.

>>12417746
That turns into an ellipse-cylinder

>>12417759
And? [math]\{ (x, y, z) \in \mathbb{R}^3\ |\ x^2 + y^2 + z^2 = 1\} \cap \{ (x, y, z) \in \mathbb{R}^3\ |\ y=z\} = \{ (x, y, y)\in \mathbb{R}^3\ |\ x^2 + 2y^2 = 1 \}[/math]

>>12417786
I want one equation for xyz coordinates to use in computers

>>12417717
If you mean one equation to plot them together then take
[math]x^2+y^2+z^2=1\to x^2+y^2+z^2-1=0[/math]
and
[math]y=z\to y-z=0[/math]
then their product
[math](y-z)(x^2+y^2+z^2-1)=0[/math]
is them together since the product is zero if and only if one of the two factors is zero

So it seems that I will write my undergrad thesis in a topic in homological algebra.. I am going to be an undergrad category theorist

>>12411010
wait, isn't every space connected by that def? since the space itself is a trivial subspace, and clearly said subspace will intersect every other subspace since it contains all of them...

>>12418443
>isn't every space connected by that def?
actually no space is connected by this def. he's talking out of his ass

>>12418283
Have you already come out to your family? As a category theorist?

>>12415485
you are incredibly dumb.

>>12411214
Holy shit bro I’m literally in the same situation as you, 3rd year EE major

Thankfully uni allows late applications for double majoring and I’ve done enough courses to wrap it up in 1 extra year . Have the same worries as you regarding admissions in case they don’t accept the double major

>>12418458
sorry, i am an undergrad and have never seen this def before (studying calc iii rn). could you explain why no spaces are connected?

>>12418460
I'm sure they already know from all the thigh-high socks I own

>>12418552

Those are programming socks, they think you're an indie game developer.

>>12418461
uhm sweetie you are the dumbdumb here

>>12418539
a space [math]X[/math] is disconnected if there exist non-empty open sets [math]A,B \subseteq X[/math] such that [math]A \cap B = \emptyset[/math]
and [math]X = A \cup B[/math]. a space is connected if it's not disconnected.
his definition is wrong

what are some math redpills?

if I want to make a mathematical model that describe the relation between energy demand and the covid situation such as lockdown etc, what mathematical model can I use? my knowledge on mathematical modelling is very limited

>>12418849
buy an econometrics book

>>12418695
Math is currently under attack by the SJW types. They claim that

Math is elitist, because it excludes those who can't do it.
Math is racist, because of crime statistics.
Math is white supremacist, because many mathematicians were white.
Math is a social construct, thus anyone can claim a 'personal' math which is correct regardless of logic.

Many mathematicians have bought into this rhetoric.
Stockpile Springer books. Download books and pdfs. Store papers and lecture notes. Protect them and hide them all. They will soon come to burn them.

>>12417150
Well, then create it yourself.
You already got the name, now find the content.

>>12418875
fuck

>>12417875
He looks for a equation for the intersection, not the union.

>>12418875
>They will soon come to burn them.
You are saying that as a joke, but don't underestimate them.
They will unironically do that.

>>12418875
>Math is elitist, because it excludes those who can't do it.
When you think about it throughout history, that is not entirely false. Rather than "those who can't do it", it would be, "those who couldn't afford it".
>Math is racist, because of crime statistics.
Haven't seen anyone make that claim.
>Math is white supremacist, because many mathematicians were white.
Again, some parts of it are true. Indian and Chinese mathematicians are often times not appreciated enough in favor of some westerner. Just look at the "Fibonacci" sequence.
>Math is a social construct, thus anyone can claim a 'personal' math which is correct regardless of logic.
I saw someone with a math phd make this claim on Twitter and I was at awe at the stupidity of it. According to him, anyone could claim that 2+2=5, even without any context, and other people should be the ones to ask the claimer which definitions and context were used to get such a result. This is of course really stupid and defeats the purpose of having language in the first place.

Anyways, I don't think they'll come very far, but is definitely something to keep an eye on.They want to get a Trojan horse inside, the exterior looking like reasonable points, and the interior being absolute nonsense. I think some of their exterior points are valid, just need to keep the inside out.

>>12418888
Quads of doom

>>12419313
>They want to get a Trojan horse inside, the exterior looking like reasonable points, and the interior being absolute nonsense.
My guess would be an attempt to make maths more accessible to "unprivileged" students.
They killed orthography in my area with a "writing by listening"-concept. Now there are several years of children that couldn't write a single sentence without mistake to protect their life.
Maybe it will be a "counting by feeling"-concept for maths.

>>12419313
>anyone could claim that 2+2=5, even without any context, and other people should be the ones to ask the claimer which definitions and context were used to get such a result

>tfw you're a six year old learning addition in 2040 and you need to attach a disclaimer to your homework explaining that you utilize the symbol of addition with full understanding of the historical origin of contemporary basic arithmetics and its current presentation with indo-arabic numerals and symbols and that other cultures might have a different understanding of addition, and that they might prefer to use different symbols to denote addition

>>12419495
>tfw you're a six year old learning addition in 2040 and you need to attach a disclaimer to your homework explaining that you utilize the symbol of addition with full understanding of the historical origin of contemporary basic arithmetics and its current presentation with indo-arabic numerals and symbols and that other cultures might have a different understanding of addition, and that they might prefer to use different symbols to denote addition
so according to you, being a racist asshole is a better alternative?

>>12419616
>a racist asshole
LOL it's called being a decent human being.

>>12419622
I think you should leave this board right now. Here on /mg/ we do not tolerate nazis.

>>12419647
lmao

>>12418875
Twitter retardation has nothing to with actual math lmao. You guys have no fucking idea of the state of academia.

>>12419647

>Statistically speaking, you are probably taking up room that should go to someone else. If you are a white cis man (meaning you identify as male and you were assigned male at birth) you almost certainly should resign from your position of power. That’s right, please quit.

Where do I start with automata theory?

>>12419683
http://www.theliberatedmathematician.com/wp-content/uploads/2015/11/PiperThesisPostPrint.pdf
>this is what it takes to get a PhD at Princeton
I wasn't even thinking about applying there but I'm a lot more confident now

>>12419651
My department just established an antiracism committee. There's a lot of pressure coming from the younger cohort. It's only a matter of time.

Funniest part is how clueless the boomer professors in my department are to the cultural shift happening before their eyes. They try to be supportive of the grad students and younger professors organizing the whole thing only to find themselves being criticized and forced to write apology emails for not doing enough or daring to say that we are primarily math researchers not therapists.

How much linear algebra do I need for representation theory? I was basically retarded and didn’t pay that much attention during my linear algebra course.

>>12419971
>How much linear algebra do I need for representation theory?
All of it lol.

>>12419990

it's been enough times since I last checked /mg/ that there's no elliptic curves homo

>>12419809
You'd better have mad connections or you won't be allowed to blow off a PhD like that.

Any finite group [math]G[/math] has an order in the natural numbers, say [math]n[/math],

Any such integer has a prime factorization, so there exists [math]p[/math] a prime such that [math]p|n[/math],

By Cauchy's Theorem, (a partial converse to Lagrange's theorem), if [math]p| n [/math] , for a group [math]G[/math] such that [math]|G|=n[/math], then there exists a subgroup [math]H[/math] of [math]G[/math] such that [math]|H|=p[/math]

Since [math]H[/math] has the order of a prime [math]p[/math], it must be cyclic, and since all cyclic groups are abelian, [math]H[/math] is abelian.

So for any finite group [math]G[/math] must have an abelian, nontrivial subgroup [math]H[/math]. But this implies all groups have a nontrivial center [math]Z(G)[/math], which is not true (the symmetric group [math]S_n[/math] has a trivial center for [math]n>2[/math].)

Where am I fucking up here?

>>12419687
It's not supposed to be difficult, just look up what you need to know. It's not important which resource you use, but if you want to study a textbook I can recommend Kozen. His book is very easy and pedagogical, and has a good problem set. But really you can google any lecture notes about the subject, this is not something very difficult that should require some kind of "just right" material to learn efficiently.

>>12420088
>having a non-trivial abelian subgroup implies it has a non-trivial center
Did your brain rot and fall off through your ears?

>>12420103
whoops, you're right. I think I was thinking that [math]H \leq G[/math] implies the existence of [math]H \subseteq G[/math] and that

[math]{id_G} \subset H \subseteq Z(G) \subseteq G[/math]

I get that that's NOT true, though it isn't super clear why. But I guess asking why things aren't true is pretty silly

Let's be honest, nobody really likes studying maths except a handful of turboautists

Yes I have a test on monday and yes I am utterly fucked

>>12410456
>Number Theory Edition
Hi /mg/. I've come to ask the tired question of how do I into math. I'm interested in number theory, I found this book http://math.gordon.edu/ntic/ntic.pdf which seems neat.
Thing is tho I've only ever taken up to Calc I (lol) and Linear Algebra. Should I study more Calculus/Analysis? Or is there something else neat and beginner friendly, like abstract algebra or graph theory?

>>12419651
Just look timnit gebru case, research on Ethics AI in Google
>She wrote one paper
>Reviewers just say no good for publish
>She demand getting identity of anonymous review
>She angry respond "Publish or I resign"
>Google: goodbye
>Mass media and SJW twitter Evil Google fires black woman
https://twitter.com/timnitgebru

She repost pic

Top Research on Google comment about case.
https://docs.google.com/document/d/1f2kYWDXwhzYnq8ebVtuk9CqQqz7ScqxhSIxeYGrWjK0/preview?pru=AAABdlNwLxs*PKCOHN-Ks0PI5nFrljenMg

>>12419687
Don't directly dive into automata.
Start with regular expressions, context-free languages, the Chomsky hierarchy.

>>12420287
>Top Research on Google comment about case.
Do you have a pastebin of that?

>>12419906
Those departmemts are political memes. Professors are if anything overly protected by unis (in general I know there are bs cases it doesnt mean it signals a tendency)
>>12420287
Google pseudo academic shit is not a representation of academia.

You really don't know what you are talking about.

>>12420302
Google document by Jeff dean
https://docs.google.com/document/d/1f2kYWDXwhzYnq8ebVtuk9CqQqz7ScqxhSIxeYGrWjK0/preview?pru=AAABdlNwLxs*PKCOHN-Ks0PI5nFrljenMg

What will be the Great Reset of mathematics?

>>12420328
So there is no pastebin. Okay.

>>12420334
rational trig

>>12410456
For one fucking masturbated latex you're circle jerk again, what's S^N?

>>12420412
The N-sphere, obviously.
Your question was already asked,
>>12410774
>>12410780
So read the thread before posing such a question :)

>>12420367
Anon was being retarded.
Not to worry tho.
https://pastebin.com/ANr6xfVL

>>12420334
Conitnuum hypothesis

>>12420334
Tookerism

>>12420435
Thank You.

>Pastebin’s SMART filters have detected potentially offensive or questionable content ahead.
topkek

>>12420485
Those filters are really bad.

>>12420334
Martin-Löf type theory

the irish phenotype

>>12419906
>from the younger cohort.
Lol no, the pressure comes from lobby groups.

>>12420557

>While attending Trinity College, Hamilton proposed to his friend's sister, who rejected him.[11] Hamilton, being a sensitive young man, became sick and depressed, and almost committed suicide.

imagine being the f*male responsable for killing hamilton

>>12420596
>think finally I have found a successful mathematician who is a bigger loser than myself
>look up his wiki page
>married, 3 kids
fuck my life

Brainlet filter:

What does GL mean in mathematics? If you don't know this you should give up on maths.

anyone here like probability theory? what's the natural progression after something like Durrett?

>>12420130
A lot of number theory is really elementary. Unfortunately the highest levels are locked behind algebra and analysis, but you can learn an awful lot to develop a taste without them.

>>12420805
>chingon

>>12420617
at least you will never pussy pedestal yourself to death like galois

>>12420920
That's cool then, thanks anon. Spent the afternoon reading, ita fun so far. What about Graph Theory and Algebra?

how would I go about finding n from a matrix n x n size, given a determinant, and multiple of the determinant?
just want to pointed in the right direction, cant seem to figure out what to google for this type of problem

>>12421006
If you're given a matrix, you also know it's size.

>>12421026
maybe I miscommunicated what i meant was that I have matrix A, All I know is that A is n x n size, and that det(A) = 5 and det(5A) = 75, how to find 'n'

Ive got it now, though i'd still be interested in how to explain this question to google since it didn't turn up any results

>>12421084
[math]det(Ak)=k^{n}det(A)[/math]

>>12421084
You probably could build a reasonable search term but that's a "were you paying attention" kinda question

>>12421084
>>12421112
>All I know is that A is n x n size [...] how to find 'n'
Even if n pops up in the determinant relation, the question is extremely contrived.
You can't have A without knowing n.

>>12421146
technically you know the size is n but not which specific n

>>12410695
Neck yourself discord tranny. Stop hand holding retards.

>>12412777
Sci has LaTeX support vis JSmath I think.
Put your formulas between [math*] [/math*] tags but without the asterisks.

>>12410864
S = {X| X is a Set}

>>12410456
Brehs I'm somehow in year 3 of financial maths with good grades and still don't understand any maths lmao.

Have exams in metric spaces and stochastic models in a couple weeks. Shitting bricks rn.

Could someone recommend me a book on basic geometry? Like where they are proving basic shit. Feel like I should start from scratch.

>>12410565
Set theory is a forced meme.

>>12415005
The Continuum Theorem was proven by Prof. Uuno Saarnio in the 1968 article "Eine konstruktive Darstellung für die Richtigkeit der Kontinuumhypothese", published in the highly respected journal Mathematische Annalen, volume 178.

Cohen's independence proof only tells us that ZF set theory is an insufficient axiomatisation of Platonic set theory.

>>12422185
Here is the article btw.
https://gdz.sub.uni-goettingen.de/id/PPN235181684_0178?tify={%22pages%22:[353],%22view%22:%22info%22}

If you do not know German, then no worries, the proof is also available in Finnish in the book "Mitä tiedämme äärettömästä?", chapter VI.

>>12422106
You shouldn't start from scratch, but if you are intent on ignoring this advice then "Basic Geometry" by Birkhoff & Beatley is my recommendation.

>>12422106
Geometry revisited
>>12422199
I don't know anon, that all seems really fishy.

Speaking of set theory, did you know that Cantor was a serial plagiarist? His famous indexing of the algebraic numbers is due to Dedekind, but Cantor gives him no credit in "Ueber eine Eigenschaft des Inbegriffes aller reellen algebraischen Zahlen". Moreover, one of his most famous techniques, the method of diagonalisation, is clearly inspired from du Bois-Reymond (whose work Cantor was knowingly aware of), who applied it to a different problem (orders of growth of real functions). Really makes you rethink Cantor, huh?

Is it true that the square root of an integer is either irrational or an integer?

>>12422106
Euclid's elements

>>12422302
yes it's true sir

>>12422302
Yes (for non-negative integers obviously). The square root of a non-perfect square is irrational.

Also Euclid's Elements is a bad recommendation.

>>12422324
Au contraire, anon. The Elements is an excellent book on basic Geometry, but it is emphatically not a book for the basic Geometer.

>>12422324
Elements is the only recommendation. Don't tell me you're one of those who fell for the "it's not rigorous" meme.

>>12422328
(cont'd)
I would not recommend someone learn "Geometry" as their first step anyway. Like any normal person, you should learn calculus (first by the manipulation of algebraic expressions) and then algebra (i.e. matrices).

>>12422330
The Elements is a very hard book to actually read unless you are already smart enough to know roughly what it is trying to say

>>12422335
Well, yes, you need to brush up on Platonic philosophy before you can attempt to read the Elements. In this sense it is not an elementary book, suitable for an entirely from-scratch approach.

>>12411109
That type of system can be solved with a differential, not sure if that's helpful

>>12422343
Reading number theory in the elements is a fools errand anon, you will be hopelessly lost trying to turn it into modern notation.
It *is* correct of course, and perfectly well set out, but the modern student will be quite lost with his talk of integers "measuring" and etc.

>>12422348
Well, it's important to see that arithmetic is founded on geometry.

Have you ever met a mexican mathematician?

>>12422356
Oh definitely, it's a very enlightening book, but you can't really expect someone to appreciate it until they are pretty good at maths

>>12422365
Somehow, the Elements was suitable as an introduction for hundreds of years until it was suddenly discovered to not be by moderns.

>>12422328
>>12422330
It's not a good recommendation. I never said whether or not the book itself is any good. Only a fool would say the proper way of teaching someone geometry is through the Elements. Perhaps if we had infinite time, but it's a waste to not use algebra when discussing geometry, just as it's waste to not use geometry when discussing algebra.

>>12422388
I was making a joke. Elements is very difficult for the modern reader, but it is an excellent book for what it is.

>>12421084
>>12421146
just look at diagonal matrices with 5 at position 1,1 and 1s for the rest of the main diagonal
construct examples and actually try the question

>>12422357
No. Why?

>>12421084
det(cA) = c^n*det(A)
So in your case it becomes:
75 = 5^n*5
or
15 = 5^n where n is a positive integer. No such integer exists, so your question seems strange to me.

>>12422593
ummm it should be 25 not 15

>>12422624
75/5 = 15

>>12422636
you are right i am retarded

>>12422640
You really are...

>>12422642
Don't bully him

>>12410615
change coord to cylindrical, then use fubini.

>>12422593
Yeah, I was thinking that last night. I hope he just got the wrong numbers.

start with the soviets

>>12410456
Does anyone here know discrete Morse theory, who can give a condensed overview of it's applications in graph theory?

All i could find was a thesis by some Norde, but i don't know enough discrete Morse theory myself to understand it.

I have a square, self-adjoint matrix [math]A[/math] and some projector [math]P[/math]. Is it always true that [eqn]\left |\mathrm{trace}(PA)\right | + \left |\mathrm{trace}((I-P)A)\right | \leq \sum_i |\lambda_i |[/eqn] where [math]\lambda_i[/math] are the eigenvalues of A?

It seems to me that this should be the case: A has only real eigenvalues, so there exists a projection [math]P[/math] such that equality holds in the equation (project onto the span of eigenvectors with positive eigenvalues). But I don't see a simple argument that inequality will hold for a sub-optimal choice of a projector. Is this obvious?

Hey /mg/, would you guys have any recommendations for the sorts of textbooks that would cover concepts like the score function and fisher information )or at least the name of the "field" or subject)? Intro Stats textbooks don't go into things like that, they usually just settle at hypothesis testing.

>>12422378
If you know modern math then it's abundantly clear why The Elements are no longer a good introduction. It's like you're complaining that children aren't taught how to ride horses any more.

I'm looking at the proof that the countably infinite cartesion product P = {0,1}^w is an uncountable set, and I need a bit of clarification.
>take any function F:Z+ > P
>choose an element p of P such that its the same at all values of some F(z), except at the zth position
>thus p is not in the image of F
But why can't another element of Z+ map to p? Is it because P has yet another element that would be different from z2, creating an inductive "chasing a tail and never finding it" situation?

>>12423885
>if you have calculators why should you learn arithmetic lmao

>>12423885
I know modern math quite well, I am a Lie theorist.
The debasement of synthetic geometry in education has been a catastrophe. As has been the debasement of Latin. No, don't blabber to me about "practicality" and "business needs".

>>12424013
It looks basically like cantor's diagonalization argument but in binary. Look it up, maybe it'll make more sense then.

>>12424026
You mean du Bois-Reymond diagonalisation. Cf. >>12422281

>>12424029
I really don't care. People refer to it as cantor's diagonalization argument, and so will i.

>>12424034
You should get ready. Cantor will be removed from his coveted place in the history books in the near future when people lose their faith in the completed infinite.

>>12424026
Ohhhh I see. It's finding a new element where each coordinate is different from the corresponding coordinate of every other element.

>>12424015
Reading the elements is like practicing multiplying six-digit numbers in your head, not doing basic arithmetic.

>>12424025
>Lie theorist
Nice touch lol.

>>12423819
If i didn't fuck up, it's true:
WLOG (since the trace and eigenvalues are preserved by any change of basis, and the notion of "being self-adjoint" is preserved by a unitary change of basis) we can choose an orthonormal e_1... e_n such that P is projection onto some of the coordinates, then I-P is the projection onto the remanining coordinates. Then trace(PA) is sum of some diagonal terms of A, and trace((I-P)A) is the sum of remaining diagonal terms.
So it is sufficient to prove, for a self-adjoint matrix A, that
[eqn]\sum_{i=1}^n |a_{ii}| \leq \sum |\lambda_i|.[/eqn]
From here, I googled around for various eigenvalue inequalities and I found this:
1) There is a "schur majorization theorem" https://en.wikipedia.org/wiki/Schur%E2%80%93Horn_theorem which tells you that the sequence of eigenvalues of a hermitian matrix majorizes the sequence of its diagonal elements
2) From this majorization, our condition follows if you fuck around a bit. (put t=0 into the last bullet point of https://en.wikipedia.org/wiki/Majorization#Equivalent_conditions))

there's also this tao blog https://terrytao.wordpress.com/2010/01/12/254a-notes-3a-eigenvalues-and-sums-of-hermitian-matrices/ which is related but i'm too lazy to read it

/mg/ trannies, did you study your dilation theorems today?
https://en.wikipedia.org/wiki/Dilation_theorem

>>12424133
https://en.wikipedia.org/wiki/Tsuruko_Haraguchi

>>12424177
kek

>>12424177
Kek.

>>12424177
Telling a tranny to study analysis is transphobia.

>>12424177
>tfw no bf that would dilate your boypussy

>>12424218
>>12424218

>>12424200
L O N D O N