/sci/ - Science & Math

math:
>>14741383 (guess he didnt make it)
>>14742944
>>14749503
>>14749782
>>14754985
>>14756238
>>14756949

physics:
>>14739374
>>14743347
>>14745483
>>14747012
>>14750645

chemistry:
>>14752436

/g/:
>>14746081
>>14748970

/scg/:
>>14750643

med:
>>14740940

wemi fuckin ignoring me:
>>14739947

greek letters:
>>14750921

if f:R->R does it imply the following statement
statement: if x in R then there exist integer n such that 2^n>f(x)

is the function f(x)=1/|sin(x)| not a f:R->R function because it has infinitely many infinity?

>>14759546
The first one is true because of
https://en.wikipedia.org/wiki/Archimedean_property

And for the second function. It involves division by 0 which is not allowed.

>>14759546
The second one is a matter of perspective. One perspective is that the domain of f is R, f is a partial function, and its natural domain is R\{nπ:n∈Z}. The other perspective is that partial functions don't exist, all functions are total, domain and natural domain are the same thing, and the domain of f is R\{nπ:n∈Z}. You just need to figure out which perspective your professor considers to be the uncontested truth. Much like whether or not 0∈N.

>>14759437
Fixed this for ya.

>>14759437
>stupid
How much silver do I have to eat to look like this?

How do field extensions work?
Let's say I have a field [math]F[/math] and a polynomial [math]f(x) \in F[x][/math] without any roots in [math]F[/math]. Let [math]g(x)[/math] be an irreducible factor of [math]f(x)[/math] of degree [math]d[/math].
[eqn]F[x]/(g(x)F[x]) = \left\{ \sum_{k=0}^{d-1} a_k x^k + g(x) F[x] \middle | a_0,a_1,\ldots ,a_{d-1} \in F \right\}[/eqn]
is supposed to be a bigger field in which [math]f(x)[/math] has a root.

It's a ring since it's a quotient ring. Since [math]g(x)[/math] is irreducible, [math]\gcd(g(x),\sum_{k=0}^{d-1} a_k x^k) = 1[/math] so Bezout gives every non-zero element of [math]F[x]/(g(x)F[x])[/math] a multiplicative inverse which makes it a field.
But which element of [math]F[x]/(g(x)F[x])[/math] is the root of [math]f(x)[/math]?

>>14760195
[math]x[/math] is.

>>14760195
[math]a_0, a_1, ..., a_{d-1}[/math] are the d roots of [math]g[/math]

>>14759546
f(x) is a constant with regard to n so of course for some n, 2^n is greater

>>14760190
You'd die from metal poisoning before that happened.

>>14760556
silver's inert tho

>>14760580
It's toxic in high enough doses. So is gold.

>>14759437
could i get a decent job with just an undergrad math degree or should i just go the engineering route(will flip a coin between electrical and mechanical as both interest me)?

>>14760624
The job market is not great for undergrad math degrees.
You definitely want EE or MechE

these are the kind of problems i suck at when it comes to physics.

Ill ask my question in my next post which has the solution to the question in pic

>>14761152
So in the solution, for part a and c. They multiply by (1/d - 1/2d) I dont really understand how they determined they needed to do that. How do they know its "1/d-1/2d" not d+2d. Im sure its simple but i really suck at these types of problems

>>14761169
[eqn]
\frac{F}{L} = \frac{\mu_0 I'_1I}{2 \pi r_1} + \frac{\mu_0 I'_2I}{2 \pi r_2} \\
I'_1 = I \; \; \; \; \; \; \; r_1 = d \\
I'_2 = -I \; \; \; \; \; \; \; r_1 = 2d \\
\frac{\mu_0 I'_1I}{2 \pi r_1} + \frac{\mu_0 I'_2I}{2 \pi r_2} = \frac{\mu_0 I^2}{2 \pi d} - \frac{\mu_0 I^2}{2 \pi 2d} = \frac{\mu_0 I^2}{2 \pi} \left( \frac{1}{d} - \frac{1}{2d} \right)
[/eqn]

>>14761196
That makes a lot more sense but i still dont understand where the "2d" comes from

>>14761210
[eqn] 2d >> 3d [/eqn]

>>14761217
Easy, d<0. How do you get that expression? There's no inequality and negative d doesn't really make sense in this context.

>>14761225
yeah that confuses me. I dont really know what he was trying to say with that

>>14761225
okay, follow closely now:
>>14761210 the 2d comes from d + d which is the distance from the top wire to the bottom, or vice versa
>>14761217 this is a joke about finding cartoons attractive

>>14761234
thanks

i'm going to take theory of linear models my first semester of this stupid program

i feel like i'm going to flunk out

>>14759452
mmmm... i piedi

ive been going nuts trying to solve for i (current) in this equation (just for fun). Symbolab doesnt seem to want to solve for it so im coming here to see how its done.

i ended up with:

(2B)^(2/3)(x^2+a^2)/(U*a^2)

>>14761695
am i having a stroke? why are you struggling so much with basic algebra?

>>14761722
i dont have a good reason

>>14761746
you move the top stuff to the bottom and the bottom stuff to the top. your answer is almost correct but you decided to do some retarded shit with the 3/2 exponent. why did you do manipulate the a^2 term correctly but (x^2+a^2)^(3/2) made you have a conniption?

Let
[eqn]\int_{a}^{b}\int_{a}^{b} K^2(x,t)dxdt=B^2[/eqn]
and
[eqn]\int_{a}^{b}\int_{a}^{b} K_n^2(x,t)dxdt=B_n^2[/eqn]
where [math]K_n(x,t)[/math] is the [math]n^{th}[/math] iterated kernel for the kernel [math]K(x,t)[/math] (*). Show that, if [math]B_2=B^2[/math], then for any [math]n[/math], [math]B_n=B^n[/math]

(*) [math]K_1(x,t)=K(x,t)[/math] and
[math]K_n(x,t) = \int_{a}^{b} K(x,z)K_{n-1}(z,t)dz[/math] for [math]n\geq2[/math]

Any hint on this one? Anyone?

>>14761152
Halliday and Resnick? Great textbook.

>>14761754
i sure if i was supposed to remove the exponent or if i could just multiply the entire bottom part. Its the little things that get me sometimes. Also its been a year since ive done math.

>>14761797
sorry i was so mean. i just thought it was weird for someone to be making mistakes like that in an E&M course. you got this anon, we're here for you.

I have taken over 90 courses at school

my average is 68

ask me anythig

>>14761877
liberal arts courses don't belong to /sci/
and if I'm wrong then answer me this >>14761762

Can someone recommend me youtube channels where they talk about new technology that's coming? I don't mean the consoomer marketing channels, I mean ones that actually go into detail about the tech. I already know Two Minute Papers, I only saw a few decent LTT videos but what else?

>>14761902
i've only taken math, chemistry, physics, biology, biochemistry, computer science courses

only probably 4 or 5 non STEM at most

>>14761912
I hear you, but I don't see an answer to my problem, anon.

>>14759437
What's the opposite of a sigmoid curve? Whereas it goes right then up then right what is the name of one that goes up then right then up?

>>14761917
i remember the squiggly means integrate, kernel is obviously something computer related. the B is something about magnetic field strength, looks like a recurrence relation with that Kn-1 and dz means with respect to z.

>>14761979
get outta here, clown

>>14761531
How do I unlearn italian?
>>14761762
>Any hint on this one?
Stop trying to solve it as an integral operator problem, it's a bounded linear operator on Hilbert space problem.

There was some property that said something along the lines of [math]\| A^2 | = \| A \|^2[/math] implies [math]A[/math] is normal or something to that effect but I don't really remember.

>>14761984
>Stop trying to solve it as an integral operator problem
yup, I realized that after hitting my head on the problem for a while.

>There was some property that said something along the lines of [math]∥A2|=∥A∥2∥A2|=∥A∥2\| A^2 | = \| A \|^2[/math] implies [math]A[/math] is normal or something
This makes much sense, any source for reference on this?

>>14761981
that's pretty rude, i told you my average is 68

i'd say my answer is on par with a 68. writing out all the info i know would get me part marks

>>14761993
>any source for reference on this?
Nothing that wouldn't cost you two hours looking.
If this is a book problem, go through the previous chapter, it's probably in there.

>>14762000
>If this is a book problem, go through the previous chapter, it's probably in there.
trust, it's not in the book where it is from.
Anyway, I'm skimming Debnath and Mikusinski on Hilbert spaces at the moment. Hopefully I get some more insight on this stuff.

>>14761801
its all good. I took a hiatus from school and now im getting back into the grind. I was able to get my prof's syllabus early and im just preparing ahead of time

>>14762015
i see. well keep at it and im sure the algebra will come back to you.

>>14761795
nah young and Friedman. but i do have a copy of haliday for extra practice problems

>>14761984
why would you want to

>>14759437
What is this "mathematical maturity" everyone talks about when recommending Baby Rudin?

>>14762099
experience solving math problems. And you can only get proper experience by solving problems, from easy ones and increasing difficulty over time. Basically that. And in order for this you need to spend time solving problems. Not to be a genius, but to have some 'toolbox' of math problems solving skills.

What am I supposed to do if the classes I need are full, and my financial aid is contingent on me being continuously enrolled full time and on track to graduate on time (i.e. I can't muck around and take some random bullshit for a semester and try to get better luck next time, nor can I just wait until next semester to do anything)

>>14762111
which shithole country are you from that they don't do multiple sections for a class specifically to solve this problem?

>>14762115
The US, and there are multiple sections of each class but they're all packed.

>>14762099
Never really understood why Rudin is hailed to be difficult. Most people who mess around with sequences and functions should be able to flip through it.

>>14762120
>The US
ouch
>there are multiple sections of each class but they're all packed.
kill or make sick some other guy to take his place.

>>14762120
quit college it's a money sink

>>14762123
Is rudin hailed as difficult? I thought it’s the standard because it’s easy.

>>14762164
I often see things like this >>14759560

>>14760614
So is water.

Let both U and W be T-invariant subspaces of V. Why isn't U ∪ W necessarily a T-invariant subspace of V?

>>14762233
U union W can contain elements not in U or W

>>14762233
>>14762249
Wait lol T(u+v) = T(u)+T(v). Who said it wasnt?

>>14762268
That's true, but according to my textbook the union is not the same thing.

>>14762278
Are you sure it isn't the span of U union W ?

>>14762278
Are they using set union? That's not even a subspace.

>>14762233
It’s not a T-invariant subspace because it’s not in general a subspace, T-invariance isn’t the problem here.

>>14762233
It is T-invariant.
[eqn] x \in U \cup V \\
\Rightarrow x \in U \lor x \in V \\
\Rightarrow T(x) \in U \lor T(x) \in V \\
\Rightarrow T(x) \in U \cup V
[/eqn]

It doesn't have to be a subspace. Consider [eqn]V = \mathbb{R}^2 \\
U = \mathbb{R} \times \{0\} \\
W = \{0\} \times \mathbb{R} \\
e_1 \in U \\
e_2 \in V \\
e_1 + e_2 \notin U \cup W
[/eqn]

What does /sci/ think of data scientists?

what is the formula for the standard deviation of a variable C which is the product of two variables A and B with their own standard deviations?

the only eqn ive been able to find is that the cv(C) = sqrt(sum(cv(A)^2 + cv(B)^2))
i could then back-calculate the sd of C from its cv.

however when i attempt this in excel, the sd i calculate for C is different from the sd of the actual values of C (pic related)

if you want a reason why, imagine you had a material which had properties A, B, and C, (perhaps fabric with volume, porosity, and internal surface area) and C = A*B
you could experimentally find C, sd(C), A, and sd(A), and wanted to know sd(B) (how much the porosity of the material varies from one batch to another).

>>14762448

[math]\operatorname{var}(AB) = \left(\operatorname{cov}(A^2,B^2)+E[A^2]E[B^2]\right)
- \left(\operatorname{cov}(A,B)+E[A]E[B]\right)^2[/math]

>>14762471
i saw this online but im actually too stupid to know what these functions are.
could you define or just name var, cov, and E so i can google further? my bach was bio so i only took to calc 2

>>14759437
Is [math] F_2[/math] a field if: [math] 1 + 1 = 1[/math]?

how realistic is this https://lunarpedia.org/w/Rectenna ? Reading up on rectenna could it be possible to harvest nuclear radiation if we had a fast enough diode? Don't rectennas rely on wave mechanics of EM radiation and therefore won't work with nuclear radiation which is made up of particles?

>>14762673
Not if 1 stands for the neutral element of multiplications. One of the requirements of a field is that the neutral elements of addition and multiplications are different elements.
If you subtract 1 from the equation
1 + 1 = 1
you get that 1 is the neutral element of addition.

Why does it suffice, to prove that [math]( \vert A \vert \leq \vert B \vert ) \land ( \vert B \vert \leq \vert A \vert )\Rightarrow ( \vert A \vert = \vert B \vert )[/math], to show that [math]X\subset Y \subset Z[/math] and [math]\vert X \vert = \vert Z \vert[/math] imply [math]\vert X \vert = \vert Y \vert[/math]?

>>14763110
meant to say [math]X \supset Y\supset Z[/math]

>>14763110
[math]X\subset Y \subset Z[/math] implies that you can take the identity maps from [math]X[/math] to [math]Y[/math] and from [math]Y[/math] to [math]Z[/math] as injective maps. This tells you that [math]|X| \leq |Y|[/math] and [math]|Y| \leq |Z|[/math]. Now since [math]|X| = |Z|[/math] you have [math]|Y| \leq |X|[/math]. So you can use Cantor-Schröder-Bernstein to prove that [math]|X| = |Y|[/math]

Any tips for rewriting mathematical statements in pure formal logic? For instance, the axiom of union would become
[math]\forall x \exists y \forall z (z\in y \Leftrightarrow \exists \omega (z\in \omega \land \omega\in x))[/math]

>>14762749
It's physically impossible to make a diode fast enough to rectify gamma rays.

Is a phd in CS or EE worthwhile if I don't intend to become a professor/teach and would instead go into industry work?

>>14761762
no immediate idea, but maybe try working with Kn+ = (Kn(x,t)+Kn(t,x))/2 and Kn- = ((Kn(x,t)-Kn(t,x))/2
You should find some nice properties and reduce everything to Kn+, and try to work with scalar products and norms or something

>>14759437
How should I think about parallel transport in higher dimensions? Intuitively, it's what happens when you keep the direction of a vector constant when following a geodesic (a general curve being approximated by geodesics). This becomes equivalent to keeping the angle between the vector and the tangent vector constant in 2d (for a geodesic), but in higher dimensions (say 3d for simplicity) the vector can rotate, meaning we need additional information.

>>14759437
Are all the MUH CLOT SHOT lines still bullshit?

>>14763354
You're insisting on the Riemannian metric when you shouldn't. A covariant derivative tells you how to differentiate a vector field by a vector. Keeping the direction constant means that the derivative of the vector field by the velocity vector zeroes.

I want to learn about electrical engingeering and theory surrounding electricity in general. Are there any good resources you lot know of?

Could someone recommend me some good resources(books, videos, etc) on quaternions? More specifically a pathway to understanding it, starting from the basic necessary math required to the quartenions themselves. I feel like, having been out of school for a long time, my understanding of the basic stuff necessary to understand quaternions have fsimply disappeared from my memory.

please walk me through part b. Ive spent too much time on it.

the value for i= 3.67 amps

>>14762288
>>14762295
Check pic rel please.

>>14763761
All 4 sets are T-invariant.
[math]U \cup W[/math] is not a subspace in general. The others are all subspaces.

>>14763771
Then why is U∩W a subspace but U∪W isn't?

>>14762536
>>14762471
>>14762448

if anyone reads this the correct formula seems to be:

Var(XY) =
cov(X^2,Y^2) +
[var(X)+E(X)^2] *
[var(Y)+E(Y)^2] −
[cov(X,Y) + E(X)E(Y)]^2

var variance, cov covariance, E expected value (mean)

using this formula solves the problem from pic.

>>14761488
c-can anyone help me?

here is another problem which seems more solvable than that one

>>14763780
If x,y are in both U and W then x+y is in both U and W.
Since x,y are in U then x+y is U and since x,y are in W then x+y is in W

If x,y are in U or W then it could be x is in U and y is in W and then x+y is not necessarily in either

>>14763511
FLoyd books on electric circuits and electronic devices are a good starting point to begginers. Also Sadiku book.
Right after this you'll need the very basics on diff eqs, some linear algebra (kramer rule specifically) and the basics on complex analysis (but this one is normally already covered in circuit analysis books)

>>14761488
Just take the scalar product of [math]v_i[/math] with the equation [math]P_Sy = c_1 v_1 + \ldots + c_m v_m [/math] for [math]i=1,2,...,n[/math]

For the RHS you get
[eqn] \langle v_i, c_1 v_1 + \ldots + c_m v_m\rangle = c_1 \langle v_i, v_1 \rangle + \ldots + c_m \langle v_i, v_m \rangle [/eqn]
For the LHS you get
[eqn] \langle v_i, P_Sy \rangle = \langle v_i, y \rangle + \langle v_i, (P_S - I)y \rangle = \langle v_i, y \rangle[/eqn]

[math]P_S[/math] isn't actually defined in the problem but I assume it's the orthogonal projection onto the subsapce S so [math](P_S - I)y [/math] is orthogonal to all vectors in it including the [math]v_i[/math]

So with the equations above you have
[eqn] \langle y, v_i \rangle = c_1 \langle v_i, v_1 \rangle + \ldots + c_m \langle v_i, v_m \rangle [/eqn]
for every [math]i \in \{1,2,\ldots,m\}[/math]. All those equations together form the matrix from the problem.

>>14763938
Thank you anon!
I still will shill my problems here but that makes a lot of sense!

How do you study all this stuff anyways? It's not like real analysis, where there are pictures and it's neat...

anyone have the /scg/ OP?

Is there any quick way to see that for [math]a \in \mathbb{Z}[/math]
[eqn]a^2 + a +1 \not \equiv 0 \qquad \text{(mod 25)}[/eqn]
without checking all 25 cases?

I'm not sure if this question is gonna make sense, but here goes.
Is it possible that one's eyesight can be temporarily impacted by surroundings, like light?
I was having a conversation across 15meters or so, and i just realized how blurry the other person's face seemed.
Maybe my eyes have just gotten bad, and i only now noticed because i don't usually hold conversations at a distance? Or could it have been the lighting in the room messing with my eyes?

>>14764175
Not sure if this is quick, but it might be faster than checking 25 cases.
First, 2 doesn’t divide 25 so you can use quadratic formula. With this you can see that there’s a root of this polynomial only if b^2-4ac is a perfect square mod 25, this is just -3, which in turn is 22. Now -1 is a perfect square mod 25, since 25 is congruent to 1 mod 4. Thus, there’s a root of the original polynomial only if 3 is a perfect square mod 25.
Now if 3 is a perfect square mod 25, then it’s definitely a perfect square mod 5, but you can check by hand that this isn’t true. All this shows your equation has no roots

>>14764175
I’ll add to this, and say that you might be interested in reading about quadratic reciprocity. But it doesn’t quite apply here since 25 isn’t prime

>>14764290
not a medfag, but yes, muscles that control pupil aperture also fatigues as any other muscle. And the effort they do is related to the amount of light received by the eye.

>>14764291
Whoops, some of the steps were unnecessary. Once you see that 22 must be a square mod 25 for the original polynomial to have roots. You can just immediately say that 22 would have to be a square mod 5. Meaning 3 is a square mod 5.
In other words you can skip the step where I talk about -1 being a perfect square mod 25.

>>14764175
try mod 5. If none are 0, then they can't be 0 mod 25 either.

I'm thinking on getting a kindle, just to read pdfs and djvu articles/books. Is it a bad idea? the screen is 6 inches and I can get a bigger tablet for a bit more, but I'm worried about the eyestrain

\mathbb{R}

Does anyone here work at a nuclear power plant? How hard were the safety exams? I'm average IQ if that matters.

Give it to me plain and simple, /sci/. I'm not a retard but I'm not smart either, should i study mathematics or just go to the navy. I want to have a house and every person i know who lives from mathematics is pretty intelligent. I don't want to kms or live under a bridge.

>>14764643
Join the navy or study engineering. Math has no job prospects unless you go to grad school.

t. too brainlet for grad school but going anyways

I know math phd programs can be fully funded in the states, but how does it work in European countries?

>>14764643
figure out what you value more. Developing your brain skills on math or having money.
A math career will necessarily involve post grad school to achieve something worth the money you've spent studying it.

Now, once I told a guy asking a similar question but about studying physics, if he has family to support him on the long meanwhile he was studying, then go ahead.
Sadly, some careers are still that kind of elitist which the people with some kind of financial support can take and pursue. Unless you're a genius and can ace every class at first attempt.

Now, if you feel discouraged about what I wrote, I was studying EE back then and wanted to switch to Math majoring. But I was poor enough to not be able to pay for applying to that change of school and all the process. So I stayed and graduated in EE, and now I study math, and whatever I want on my own (the perks of not having gf, dependents and not many friends, so my individual case is being autistic) without the stupid pressure of exams, deadlines and shit. And that way I enjoy when I study instead of feeling like courses are something to tame, pass, and go for the next.
Best of luck anon, reevaluate your priorities and consider what resources are at your disposal.

>>14759565
I read it as x being chosen after n. In other words the proposition was: exists n. for all x, 2^n > f(x). And that's false for f(x)=x by the same principle you used. It looks vaguely like a definition for nice function, though the bound should depend on x.

>>14764511
>Is it a bad idea?
I had a bootleg kindle thingy once.
It was nice but I don't particularly miss it. Lower brightness on your pc screen, turn on the lights and reading is fine.

>>14763239
read up on propositional logic, especially the foundations and motivations. Just writing mathematical statements with symbols instead of words has neither the readability of writing complete sentences nor the rigor of true sentential (or further, first order) logic. That is, there's no point in writing it as shorthand except for yourself, in which case there's no need to be strict about it. If you want to do it formally you need to read a book

do you guys think a university is more likely to accept my dismal gpa while applying for grad school if it's my alma mater? Will I get any alumni appreciation? Also, I'd like any general tips people have for getting into grad school (for math) with my bad gpa. I promise I'm not too dumb!

>>14765065
How bad is bad? I got into grad school with a 3.1. Of course, I have to fund it, and I feel likely to not pass the first semester of courses...

>>14759437
Not a question, but a statement rather.

Here's the 411.

You do not exist. I do not exist. No one exists. Nothing exists. Therefore, there is no point in doing anything. Since knowledge also does not exist there is no point on pursuing it. Since you do not exist, there is no point in eating, drinking, breathing, living, dying, etc. Ideally, you do not have any kids and you wait to starve to death to be taken from this illusion of being, though, it may be eternal. I feel stupid making this post because there was no point since I don't exist and neither do you. Kinda pointless even thinking about it. But also pointless to not think about it.

Anyway, starving is taking too long so maybe an hero? Either way, your life is a lie and your goals and dreams are useless. Goodbye, /sci/, you fat fucks. Goodbye forever. Goodbye.

Goodbye.

>>14765103
Goodbye.

>>14765097
it's 2.7, although my math courses by themselves are a 3 by being half As (classes with low homework grades) and half Cs (lots of homework grades). I'm fairly certain I can perform well in grad school, I'm in a much better position to put effort in, I just need to find the opportunity. which will probably require a lot of pleading with the admissions guys.
>Of course, I have to fund it, and I feel likely to not pass the first semester of courses...
you're in your first year now? Is it more difficult than you thought it would be?

>>14765097
Can someone explain how funding works. Is it just contingent on grades? Research hours? Teaching hours?

>>14765011
Thank you. I know that formal logic isn't necessarily short-hand. My analysis book has a little introductory chapter on that (well, rather just the zf axioms and talks about formal logic a tiny bit in the end) and includes exercises such as writing the condition of injectivity in formal logic. It just piqued my interest and I've found it somewhat difficult to do these exercises (or rather know I'm doing them right) since it doesn't really introduce propositional logic and basically has you figure it out on your own by comparing the zf axioms to their propositional form.
I will read up on propositional logic, thank you.

>>14765011
>>14765182
actually, I have just checked: the text doesn't really even introduce propositional logic. The exercise does and more or less only says "we shall only deal with sets" and "logicians usually denote all sets by uppercase letters" on the matter. In any case, thank you!

>>14765182
>>14765196
oh, I see. Are you interested in learning more of the notation? Try the first chapter of this book https://archive.org/details/StevenR.LayAnalysisWithAnIntroductionToProof2013PearsonEducationLimited/mode/2up
I think it does a good cursory job

>>14765218
Yes, thank you very much.

>>14759437
Feasibility of heating our homes with bioreactors?
>all biodegradable waste goes into an isolated tank under the house
>resulting combustible gas is either used domestically, connected to the grid or collected by the community
>resulting heat rises to the house
>depleted waste is used domestically as gardening soil (?) or collected

What do you think and know about this?
Calculations?
Bacteria strains?
Experiments?

When i measure/observe a particle which had its wave function intact, must the entire wavefunction collapse, or does only the part of it involved in what I measured collapse, or is it more complicated than either of those?

related question: when i collapse a particle's wavefunction, does it collapse completely, or does it only collapse insofar as I have the ability to measure? if it's the latter, has this been proven to be the case, or is it merely suspected or unfalsifiable?

third question: what explains the apparent lack of quantum effects at the macro scale? are all the particles that make up macro objects effectively observed by virtue of interacting with their surroundings? if so, is the limit/transition zone for this known? has it been narrowed down in any meaningful way?

>>14765643
the entire wavefunction collapses.

>>14765652
same answer though it may be clearer if you think of it as the wavefunction is now localised at the position/value it was measured at and then begins to disperse.

>>14765663
because any particle interaction is the same thing as a measurement. it's kind of hard to have macroscopic objects without many, many particles interacting. this is sometimes knows as quantum decoherence.

>>14765462
I dont get why we don't make biofuels from sewage. It's weird the obsession with electric cars instead of closed systems with less waste

Do waves collapse simply because, when observed, they must pick a position. Ie they exist as math, but when seen in order to be seen they must be at a location, so it causes collapsing into that location

>>14766042
The argument about if the wavefunction collapse represents reality or is just an artefact of the mathematics of quantum mechanics is still being argued about today. The answer is that no one knows and we currently have no way to prove it either way.

Is it viable to do a bachelors in geology and then a masters in geophysics? How should one go about it?

Does this work or is it just some schizo shit? https://www.researchgate.net/publication/255709759_The_Pyramid_Electric_Generator

>>14766091
everything on the site is schizo shit

>>14766038
Yeah... The infrastructure is already there for sewage. Some places make gas for public transportation from collected household food scraps but I doubt the generated heat is put to use and scarcely populated places won't much use for it. Better having a steel tank under the house with a proper monitoring system and sleek design for easy maintenance.
I guess one should weight the investment cost and materials against the output. Maybe it won't ever be worth the work on such a small scale.

>>14766110
>Some places
Scandinavia
https://www.foodunfolded.com/article/fuel-made-from-food-waste

>>14766057
The collapse of the wavefunction is the empirically verifiable fact that repeated measurements produce the same result.

>>14766091
found his presentation of it lol https://youtu.be/0chbsMejdR0

>>14766125
That doesn't prove there is a wavefunction.

Basically the statement of the exercise is before the line.

After the line is what I did. I know it's right but I'm sure it's presented in the wrong way. What way do I have to present my solution for it to be considered right?

I have no idea what you did but the trick here is to use the binomial theorem on
[eqn]5^n = (1 + 4)^n = \sum_{k=0}^n {n \choose k} 4^k[/eqn]
You split of the k=0 term and then factor the 4 out of the sum.

[eqn]6 \cdot 5^n - 30 = 6 \sum_{k=0}^n {n \choose k} 4^k - 30 = 24 \left( \sum_{k=1}^n {n \choose k} 4^{k-1} - 1\right)
[/eqn]

What is the route of study I should take if I have just finished with calc 3 and would like to understand what the Monster Group is and all the stuff about symmetry?

>>14765462
There isn't enough biomass. If you heat with wood you need a couple cords per winter. That's many times bigger than your septic tank and kitchen waste.

>>14766267
>I know it's right but I'm sure it's presented in the wrong way.
Use words. I stared at your proof until I figured out it was an induction proof, but I had to actually think to do so.
You can be as terse as you want, just explain what is it you're doing and what results you're using.
>>14766336
He's using 6x5 mod 24 = 6 mod 25 to induct.

>>14766653
> I stared at your proof until I figured out it was an induction proof
Oh sorry, I should have specified.

So my demonstration is correct?

>>14766770
Yes.

If you professor marked it as incorrect he's still in the right because shit's hard to read but the proof is correct.

>>14766778
>he's still in the right because shit's hard to read
this is exactly why i dont think relativity is correct and black holes are fake.

>>14764511
For reading fiction, it's fine, but for reading texts, it is extremely preferable for it to be fast, so just go for the iPad, since you can switch between pages back and forth quickly, and the fact that you can look up other relevant materials like help, on the web in the same device is just a bonus. Kindles are VERY slow. There isn't much eye strain for me. Apple screens are much easier on the eye than your standard LCD screen especially with the absolute pile of shit that is ClearType.

When a single photon is released how does EM wave spread? Is it only one direction like on the pic or does it spread as an expanding sphere?

>>14766935
One direction. A photon is not a source of EM, it carries no charge, so it does not generate a radial field.

>>14759437
Why does the graph of complex functions have gay colors in it?

>>14767214
So if we consider any point outside of x axis there won’t ever be any EM vectors? When EM wave travels there will only be EM vectors at certain point on x axis at one moment?
Also I realize photon is not a source but when there are for example chemical reactions in which it is said that a photon is released it actually means EM wave is created?

>>14767325
> So if we consider any point outside of x axis there won’t ever be any EM vectors?
Your confusion seems to arise around the fact you can treat EM as a wave or a particle depending on the situation. For example it's useful to use waves when calculating something like interference but that does not mean those waves have an amplitude (height) you can measure along the y or z-axes.

Similarly it may be beneficial to think of something like bioluminescence as continuously emitting multiple photons is all directions.

>>14767399
>Your confusion seems to arise around the fact you can treat EM as a wave or a particle depending on the situation
This brings even more confusion desu. My previous understanding was that light’s wave property is associated with wave-like fluctuations of EM vectors along x axis and now you’re saying that wave property is associated with interference created by many multiple photons emitted which doesn’t have amplitude?

>>14767499
That would mean single photon or its EM wave equivalent doesn’t have wave property which doesn’t make sense. That’s why i want to simplify this by only considering how a single photon behaves

>>14767499
>>14767526
A full and rigorous answer would require quantum field theory, specifically QED, and the fact when there are lots of photons you can treat it like a classical field but fundamentally it's a quantum object.

>>14759437
Pre-Laplace transform algebra

I managed to do the polynomial division. I have the quotient and the remainder but I'm stumped when it comes to writing it in its final form

G(s) = s + 2 + ( s+3 )/(s+1)(s+2)

>>14767655
Ok i think i have it now, i had the identity for the division but it needed plugged back into G(s) and then cancel everything out.

Feels like my algebra is still a hack, the s + 3 should be in brackets...?

Is the mass of excited hydrogen atom exactly the same as the mass of ground state hydrogen atom? Or is there any, no matter how indiscernible difference?

>>14765118
If your math courses are a 3.0, then you *might* have a chance. Don't expect to be funded though.

>>14765139
It usually depends on grades, and you apply for a GTA or a GRA position once you get into grad school. I got neither, but am still going.

>>14767782
More energy means more mass in a system. It's just like how most of the mass in a proton or neutron comes from the strong force binding energy not the higgs field. But for hydrogen the ionisation energy is 13.6 eV but the ground state mass of hydrogen is ~9.4 x 10*8 eV, so yes the difference is very, very small.

>>14767719
It's not strictly necessary for the s+3 to be in brackets, but it can make it clearer.

You don't need to be able to factor anything for polynomial division. It's usually easier to leave it unfactored (until you get to the next step, which is partial fraction decomposition).

s^3+5s^2+9s+7 / s^2+3s+2 = s + ... (s^3/s^2 = s)

s^3+5s^2+9s+7 - s(s^2+3s+2)
= s^3+5s^2+9s+7 - (s^3+3s^2+2s)
= 2s^2+7s+7

2s^2+7s+7 / s^2+3s+2 = 2 (2s^2/s^2 = 2)

2s^2+7s+7 - 2(s^2+3s+2)
= 2s^2+7s+7 - (2s^2+6s+4)
= s+3

s^3+5s^2+9s+7 / s^2+3s+2 = (s+2) / s^2+3s+2 + (s+3)

This example is pathologically easy because all of the coefficients remain positive at every step. In practice, you'll often have negative coefficients.

The process is identical to numerical long division except that each step cancels the leading monomial rather than the leading digit. It's arguably simpler than numerical division because you don't have to deal with carries.

What are some good YouTube channels to (re)learn some basic chemistry? I only learned it in school and never cared much for it or think I'll realistically ever need it in my life, but thought it'd be an interesting subject to have some basic understanding of.

You guys know how to get the max amplitude of pic related from the equation V(t)=A*cos(wt + r)?

suggestions on easiest way to prove this? i really suck at analysis.:(

im so glad im not a math major

>>14768127
Differentiate and set slope to zero

>>14768127
> from the equation V(t)=A*cos(wt + r)?
The amplitude is A.

How to get it from the graph? Use the zero-crossing times to get w and r.
V(t0)=0 => A*cos(w.t0 + r) = 0 => w.t0 + r = -π/2
V(t1)=0 => A*cos(w.t1 + r) = 0 => w.t1 + r = π/2
Subtracting
=> w(t1-t0)=π/2-(-π/2)=π => w=π/(t1-t0)
w.t1 + r = π/2
=> π.t1/(t1-t0) + r = π/2
=> r = π/2 - π.t1/(t1-t0) = π(1/2-t1/(t1-t0)) = -π(t0+t1)/2(t1-t0)

At t=0, V(0)=A*cos(r) => A=V(0)/cos(r). You're given V(0) and you've just found r.

>>14768127
>>14768242
FWIW, I get w=1000, r=π/5, A=2 (to 3 sig. fig.).

>>14768242
>>14768250
Cool. Thanks.
Why w=1000 though? isn't the formula for angular frequency just 2pi/T and T is also 2pi? wouldn't that just be 1?

>>14768242
>π/2
How do you know it's zero here when there's a phase shift?

>>14763737
halp

>>14768586
post the equation you used for a)

Can anyone explain my pic?

>>14768589

>>14768598
no, this one >>14761695
at the center of the coil the field is
[math] \displaystyle
B_{center} = \frac{\mu_0 I a^2}{2(0^2+a^2)^{\frac{3}{2}}} = \frac{\mu_0 I}{2a}
[/math]
you want to find the distance [math]x[/math] at which the field is half of that.
[math] \displaystyle
B_x = \frac{B_{center}}{2} = \frac{\mu_0 I}{4a} = \frac{\mu_0 I a^2}{2(x^2+a^2)^{\frac{3}{2}}}
[/math]
solve for x.

>>14768260
T is ~2π milliseconds, so w is 1 rad/ms = 1000 rad/s.

>>14768610
here's the solution i found but i genuinely cannot follow the math for part b.

>>14768629
wait nvm. the format was so weird that i didnt see they were cross multiplying

>>14768629
they did it in a kinda gay way.
>[math] \displaystyle
\frac{\mu_0 N I a^2}{2(x^2 + a^2)^{\frac{3}{2}}} = \left( \frac{\mu_0 N I}{2a} \right) \left( \frac{a^3}{2(x^2 + a^2)^{\frac{3}{2}}} \right)
[/math]
they factored out the constants, and then factored out an extra a which made that first term equal [math]B_c[/math], but since there wasnt actually an a to factor out they added one to the numerator.

>>14768652
oof, they fucked me hard on that one
>[math] \displaystyle \frac{\mu_0 N I a^2}{2(x^2 + a^2)^{\frac{3}{2}}} = \left( \frac{\mu_0 N I}{2a} \right) \left( \frac{a^3}{2(x^2 + a^2)^{\frac{3}{2}}} \right) [/math]

>>14765663
look up the de broglie wavelength, it describes how matter can act as a wave, and also describes how anything with mass above a small molecule simply has too small of a wavelength for it to have any effect

>>14768655
>>14768652

i dont fully understand what they did with the factored part. it just seems to disappear

How do you measure temperature in electronovolts???

>>14768805
simply multiply the temperature in Kelvin by the Boltzmann constant k_B

If [math] \forall i \in \mathbb N \; A_i [/math] denotes a nonempty set bounded from above, I have proved that: [math] \forall n \in \mathbb N \; \bigcap_{i=1}^n \operatorname{ub} A_i \neq \emptyset [/math], where [math] \operatorname{ub} A_i [/math] denotes the set of upper bounds of [math] A_i [/math]. How do I show that this is not true for: [math] \bigcap_{i \in \mathbb N} A_i [/math]? [math] A_i = \{ i \} [/math] is a clear counterexample.

>>14768655
Holy shit! You can green text math??

>>14769174
Never mind, a counter example is a disproof. How the fuck did I not think of that?

>>14769174
>>14769177
Actually no. It's not a counterexample until I show that the example is in fact, empty. So how do I do that?

>>14769181
Assume a common element [math] m[/math]. But then, [math] \neg \left( m \in \operatorname{ub} A_{m+1} \right) [/math]

>>14766500
>If you heat with wood you need a couple cords per winter.
If you burn it.
Self-sustaining bacteria metabolizing the wood would not let heat and nutrients go up in smoke.

>>14759437
Good morning /sci/entists!

I have come from the Dra/g/on Maid board to ask all of you some questions.

>Do you know any good books for learning and then (hopefully) getting pretty good at lambda calculus?

An anon in the LISP thread called my project a lambda calculus implementation and I want to see if he's right and also if there are ideas I can take from it that maybe I haven't used yet. I did study Lambda Calculus at one point, but it was in excess of a decade ago so I forgot almost everything.

I prefer old books as physical hardcover copies, but I'll read anything from any time period if it helps me play science more.

>Is there a latex template I can use to make reports for people who donate anime maids my project? Something that tracks where I put the picture and has a small thank you comment area?

Finally

>what's the most fun science textbook?

I like the antique computer kind, but I'll read anything fun. A good intro to general physics would be nice but I am not picky on the topic.

Also, I like books that have a lot of pictures.

>tl:dr; suggest lambda calculus books.

>>14769270
Taylor's classical mechanics is really fun introduction to physics for me as a non physics major.

>>14769359
>Classical Mechanics by John Taylor
Thanks for the suggestion fren. I pulled a PDF of it to take a peek and if I like it I will buy a physical one.

The other day i saw a good tiktok that involved pouring concrete into golf holes since they got an exemption from the hosepipe ban in the UK. I was wondering if any of you have any ideas that would give that same eco-terrorism vibe. I know about Kudzu bombs and the like, any ideas?

>>14769521
Consider doing something constructive instead of destructive? Maybe grow some edible plants and teach others to do the same?

Being an eco-tarderist is just going to make everyone but the most obnoxious and insufferable activists hate you, and you can't feed anyone or make anything good with a cement filled golf hole.

>>14769521
What's the point? It's not like blocking the hole stops them watering the grass.

>>14769664
Boring, if that worked this wouldnt have even crossed my mind. It's not that i want to cause harm its more of a taking on those who are abjectly against any kind of environmental efforts. Most of these things are not ways that will harm, more ways that will inconvinience those in charge. I dont plan on doing stupid shit like gluing my hand to a road. Its more just like greening the side of a public building, planting protected species in private areas, things that will cause a fuss if theyre removed. Eco terrorism is a bit strong of a word for what i want but im not sure how to articulate it better. Also i already have an excelent vegetable patch. :)

What does analytic mean in math? I see it often: analytic function, analytic continuation, analytic geometry, etc.

>>14769821
That a specific set of allowed operations and functions are used / available. It's very similar to a closed form expression with a few extra special functions on top.

https://en.wikipedia.org/wiki/Closed-form_expression#Analytic_expression

>>14769821
>What does analytic mean in math?
Most usages of analytic eventually go back to the usage of analytic in analytic function, including all examples you gave, so long as you take care to note that "analytic continuation" refers to continuation by a complex analytic function and analytic geometry typically refers to real analytic functions.
Tbqh I don't entirely remember if analytic geometry usually refers to real manifolds equipped with analytic structures or to complex analytic sets.

>>14769821
>>14769875
Wait do you mean analytic geometry as in coordinate geometry?
I think the analytic there comes from the same analytic used in analytic philosophy, so it's meaningless for all practical purposes.

why is there an anomalocariid on the catalog now?

>>14761762
Here's the meme:
[math]\displaystyle\iiiint \Bigl[K(x,y)K(z,t)-K(x,z)K(y,t)\Bigr]^2\,dx\,dy\,dz\,dt = 2B^4-2B_2^2 = 0[/math],
so [math]K(x,y)K(z,t) = K(x,z)K(y,t)[/math] for all x,y,z,t.
From here you get [math]K(x,y)K_n(z,t) = K(x,z)K_n(y,t)[/math] by expanding K_n to its integral. With that established you can do straightforward induction:
[math]\displaystyle B_{n+1}^2 = \iiiint K(x,y)K(x,z)K_n(y,t)K_n(z,t)\,dx\,dy\,dz\,dt=\iiiint K^2(x,y)K_n^2(z,t)\,dx\,dy\,dz\,dt = B^2\cdot B_n^2 [/math].

>>14770074
Not sure if this legit or you're trolling. Anyway I'll give it a good read.
And thanks a lot for taking a little time for this anon. Really appreciate it.

>>14764150
pls halp

99% sure I just failed my PhD qualifying exam. How do I break the news to my advisor I'm getting kicked out?

>>14770498
If you want to do a PhD and can't answer that yourself there's a reason you failed.

how do you prove this?

>>14770616
Suppose that
[eqn]\lim_{n \to \infty} x_n = L [/eqn]
Then for every [math]\varepsilon > 0[/math] there is a [math]N \in \mathbb{N} [/math] such that for all [math]n \geq N[/math]
[eqn]|x_n - L| < \varepsilon[/eqn]

For every [math]\varepsilon > 0[/math] set [math]K=N[/math] then for every [math]x \geq K[/math]
[eqn]|f(x) - L| < \varepsilon[/eqn]
So
[eqn]\lim_{x \to \infty} f(x) = L[/eqn]

Same for the other direction except you have to choose [math]N = \lceil K \rceil[/math].

What's a kugilblitz

>>14761762
>>14770135
You're still stuck on this?
We use the properties in https://en.wikipedia.org/wiki/Hilbert%E2%80%93Schmidt_operator#Properties
First of all [math]\| K^2 \|_{HS} \leq \| K \| \| K \|_{HS}[/math]
Since [math]\| K^2 \|_{HS} = \| K \|_{HS}^2[/math] we have [math]\| K \| _{HS} \leq \| K \|[/math]. Because [math]\| K \| \leq \| K \|_{HS}[/math] is always true we have [math]\| K \| = \| K \|_{HS}[/math]
The definition then trivially implies [math]K[/math] is a one-dimensional operator and the result follows.

>>14770717
>You're still stuck on this?
kind of. I'm working out the hint in >>14770074 right now. Since this not an exercise from a FA I'm trying to use what I know. That's the reason Hilbert spaces stuff sounds obscure to me. (yeah call me brainlet, whatever). I'll give a look at those properties you mention tho.

>>14770728
>I'm working out the hint in >>14770074 right now.
I'm not saying it's baloney, I'm saying it filtered me and I don't get the proof even in principle.
Especially the first line. [math]\displaystyle \iiiint 2 K(x, y) K(y, z) K(z, t) K(y, t) = 2B_2^2[/math]. I don't get it.

>>14770672
much appreciated sir/madam

>>14761762
>>14770074
>>14770717
>>14770777
Ok I think I got a sketch of a proof thing.

[eqn]B^2 = \int_a^b \int_a^b K^2(x,t)dxdt = \int_a^b \int_a^b K^2(y,z) dydz [/eqn]
then: [eqn]B^4 = \bigg[\int_a^b \int_a^b K^2(x,t)dxdt \bigg]\bigg[ \int_a^b \int_a^b K^2(y,z) dydz\bigg] = \int_a^b \int_a^b \int_a^b \int_a^b K^2(x,t) K^2(y,z) dxdtdydz[/eqn]

and

[eqn]B_2^2 = \int_a^b \int_a^b K_2^2(x,t)dxdt = \int_a^b \int_a^b K_2^2(y,z) dydz [/eqn]
where
[math]K_2(x,t) = \int_a^b K(x,z) K_1(z,t) dz =\int_a^b K(x,z) K(z,t) dz [/math]
and also [math]K_2(y,z) = \int_a^b K(y,t) K(t,z) dt =\int_a^b K(y,x) K(x,t) dx [/math]

By hypothesis [math]B_2=B^2[/math] then [math]B_2^2 = B^4 \implies B^4 -B_2^2=0[/math], and using above expressions:

[eqn] \int_a^b \int_a^b \int_a^b \int_a^b K^2(x,t) K^2(y,z) dxdtdydz - \int_a^b \int_a^b K_2^2(y,z) dydz = 0 [/eqn]
[eqn] \int_a^b \int_a^b \int_a^b \int_a^b K^2(x,t) K^2(y,z) dxdtdydz - \int_a^b \int_a^b \bigg[ \int_a^b K(y,x) K(x,t) dx \bigg]\bigg[ \int_a^b K(y,t) K(t,z) dt \bigg] dydz = 0 [/eqn]
[eqn] \int_a^b \int_a^b \int_a^b \int_a^b K^2(x,t) K^2(y,z) dxdtdydz - \int_a^b \int_a^b\int_a^b \int_a^b K(y,x)K(x,z)K(y,t)K(t,z) dxdtdydz = 0 [/eqn]
making the following variable interchanges (because the integral doesn't depende on this): [math]y \leftarrow x, x \leftarrow t , t \leftarrow y[/math] we come up with:
[eqn] \int_a^b \int_a^b \int_a^b \int_a^b K^2(x,t) K^2(y,z) dxdtdydz - \int_a^b \int_a^b\int_a^b \int_a^b K(x,t)K(t,z)K(x,y)K(y,z) dxdtdydz = 0 [/eqn]
[eqn] \int_a^b \int_a^b \int_a^b \int_a^b K^2(x,t) K^2(y,z) - K(x,t)K(t,z)K(x,y)K(y,z) dxdtdydz = 0 [/eqn]
So the whole thing under the integral sign must vanish:
[eqn] K^2(x,t) K^2(y,z) - K(x,t)K(t,z)K(x,y)K(y,z) = 0[/eqn]
Given that [math]K(*,*) \not\equiv 0[/math] (this is by definition), then:
[eqn] K(x,t) K(y,z) - K(t,z)K(x,y) = 0[/eqn]
[eqn]\therefore K(x,t) K(y,z) = K(t,z)K(x,y)[/eqn]

>continue in next post

>>14770916
>[math]\displaystyle \int_a^b \int_a^b K_2^2(y,z) dydz = \int_a^b \int_a^b \bigg[ \int_a^b K(y,x) K(x,t) dx \bigg]\bigg[ \int_a^b K(y,t) K(t,z) dt \bigg] dydz[/math]
Excuse me but what?

>>14770916
>cont.
>here the reasoning becomes blury.

[eqn]B_n^2 = \int_a^b \int_a^b K_n^2(x,t)dxdt [/eqn]
But
[eqn]K_n(x,t) = \int_a^b K(x,z)K_{n-1}(z,t) dz[/eqn]
Then:
[eqn]B_{n+1}^2 = \int_a^b \int_a^b K_{n+1}^2(x,t)dxdt = \int_a^b \int_a^b \bigg[ \int_a^b K(x,z)K_n(z,t) dz \bigg]\bigg[ \int_a^b K(x,z)K_n(z,t) dz \bigg] dxdt[/eqn]
[eqn]B_{n+1}^2 = \bigg[ \int_a^b \int_a^b K(x,z)K_n(z,t) dxdz \bigg]\bigg[ \int_a^b \int_a^b K(x,z)K_n(z,t) dtdz \bigg] [/eqn]
[eqn]B_{n+1}^2 =\int_a^b\int_a^b\int_a^b\int_a^b K(x,z)K(x,z) K_n(z,t) K_n(z,t) dtdxdzdz [/eqn]
[eqn]B_{n+1}^2 =\int_a^b\int_a^b\int_a^b\int_a^b K^2(x,z) K_n^2(z,t) dtdxdzdz [/eqn]
[eqn]B_{n+1}^2 =\bigg[ \int_a^b\int_a^b K^2(x,z) dxdz \bigg] \bigg[\int_a^b\int_a^b K_n^2(z,t) dzdt \bigg] [/eqn]
[eqn]\therefore B_{n+1}^2 =B^2 B_n^2[/eqn]
then
[eqn]B_{n+1} = B B_n[/eqn]
Finally, given that [math]B_1^2 = \int_a^b \int_a^b K_1^2(x,t)dxdt = \int_a^b \int_a^b K^2(x,t)dxdt = B^2 \implies B_1=B[/math]
so
[eqn]B_2 = B B_1 = B B = B^2[/eqn]
[eqn]B_3 = B B_2 = B B^2 = B^3[/eqn]
etc....
[eqn]B_n = B^n[/eqn]
As were supposed to be.
I know this has some holes to be filled but I guess is enough as a draft. i would appreciate some feedback if possible, specially to fill the gaps and horrors that I (probably) commited.

>>14770963
[math]K_2(x,t) = \int_a^b K(x,z) K_1(z,t) dz =\int_a^b K(x,z) K(z,t) dz [/math]
change variables, multiply them, double integrate both sides from a to b and dydz

>>14770981
I don't follow.
[math]\displaystyle \int_a^b \int_a^b K_2^2(y,z) dydz = \int_a^b \int_a^b \left[ \int_a^b K(y, x) K(x, z) dx \right] \left[ \int_a^b K(y, t) K(t, z) \right] dydz[/math]
Explain how to get from there to the thing.

>>14771012
sure.
[math]\displaystyle \int_a^b \int_a^b K_2^2(y,z) dydz = \int_a^b \int_a^b \left[ \int_a^b K(y, x) K(x, z) dx \right] \left[ \int_a^b K(y, t) K(t, z) dz \right] dydz[/math]
[math] = \int_a^b \int_a^b \left[ \int_a^b K(y, x) K_1(x, z) dx \right] \left[ \int_a^b K(y, t) K_1(t, z) dz \right] dydz[/math]

By definition of [math]K_n(*,*)[/math] we have [math]\int_a^b K(y, x) K_1(x, z) dx = K_2(y,z)[/math] and [math]\int_a^b K(y, t) K_1(t, z) dx = K_2(y,z)[/math]
Then
[math]\displaystyle \int_a^b \int_a^b K_2^2(y,z) dydz = \int_a^b \int_a^b \left[ K_2(y,z) \right] \left[ K_2(y,z) \right] dydz[/math]

>>14771012
sure. (sorry, deleted previous reply bc a typo)
[math]\displaystyle \int_a^b \int_a^b K_2^2(y,z) dydz = \int_a^b \int_a^b \left[ \int_a^b K(y, x) K(x, z) dx \right] \left[ \int_a^b K(y, t) K(t, z) dt \right] dydz[/math]
[math] = \int_a^b \int_a^b \left[ \int_a^b K(y, x) K_1(x, z) dx \right] \left[ \int_a^b K(y, t) K_1(t, z) dz \right] dydz[/math]

By definition of [math]K_n(*,*)[/math] we have [math]\int_a^b K(y, x) K_1(x, z) dx = K_2(y,z)[/math] and [math]\int_a^b K(y, t) K_1(t, z) dx = K_2(y,z)[/math]
Then
[math]\displaystyle \int_a^b \int_a^b K_2^2(y,z) dydz = \int_a^b \int_a^b \left[ K_2(y,z) \right] \left[ K_2(y,z) \right] dydz[/math]

>>14771024
You seem lost, but I figured out you just made a typo and didn't carry it on to the next equation.
>inb4 what typo
Reread >>14770963 and compare to >>14771012

>So the whole thing under the integral sign must vanish:
Elaborate.
The other anon was doing a "squares are positive argument".
[math]K^2(x,t) K^2(y,z) - K(x,t)K(t,z)K(x,y)K(y,z)[/math] isn't a square, so you can't do that.

>>14771034
>You seem lost
>The other anon was doing a "squares are positive argument"... isn't a square, so you can't do that.
ugh my lifepoints, lol.
>you just made a typo and didn't carry it on to the next equation.
Lol, yes, sorry for that, noticed again after replying the second time.

>Elaborate
With [math]f(x) \geq 0[/math] and integrable on interval [math][a,b][/math] if [math]\int_a^b f(x) dx = 0 \implies f(x) = 0[/math]

>>14771066
>With f(x)≥0
That's precisely the issue, why do you think [math]K^2(x,t) K^2(y,z) - K(x,t)K(t,z)K(x,y)K(y,z) \geq 0[/math]?

>>14771012
Seeing a surprisingly large amount of struggling with my solution.
The first part is just straight algebra; you have to get comfortable with multivariable integration like this to see it.
Because [math]K_2(y,z) = \int_a^b K(y,x)K(x,z) dx[/math], the square is a product of two integrals:
[math]K_2^2(y,z) = \int_a^b K(y,x)K(x,z) dx \cdot \int_a^b K(y,t)K(t,z) dt[/math].
Make sure to integrate with respect to different variable names, so that we can bring the constants inside the integrals:
[math]K_2^2(y,z) = \int_a^b K(y,x)K(x,z) \left[\int_a^b K(y,t)K(t,z) dt \right] dx = \int_a^b \int_a^b K(y,x)K(x,z)K(y,t)K(t,z) dt dx[/math].
This is the expression we integrate of y and z to get the result.

The second step of the argument is that if f is a real integrable function and f^2 integrates to zero, then f must be zero in the region we integrate.
Otherwise, f^2 would be positive around some point, and the integral would become positive rather than exactly zero.
Proving this rigorously for all integrable functions is tricky (and isn't even strictly true), but if we assume f is at least piecewise continuous the proof is basically what I just wrote.

>>14771068
Yeah, I know what you're pointing out. I'm trying to recollect the hypothesis of all this stuff.
The [math]K^2[/math] terms are non-negative, bc [math]K[/math] is a real and integrable function on the rectangle [math]R=\{a \leq x \leq b, a \leq t \leq b\}[/math]

I'm trying to think how to justify the positiveness on the 2nd portion, or how to modify the previous equations. My brain hurts at the moment.

>>14771082
Are you the person who originally posted that solution?
See >>14770777
>>14771083
Relax.
You can always give up and use the other solution.

>>14771097
>Relax.
>You can always give up and use the other solution.
I'll go for a walk. Hopefully something clicks on my mind on the meantime.

Kind of a stupid question, but is it weird for an undergrad to do research outside their home university?

>>14771234
depends on many factors. if it's because you don't have the resources available at your uni then no. if it's a summer research program then no. basically the only time it is kind of weird is if you're doing research at a lower-tier program than the one at your own uni, unless the projects are significantly different.
I'm at a top uni and many students work here from other places, it's not uncommon at all

>>14771247
I might be doing it at a much lower tier. I was thinking that doing research at another university might give me a bit of an in come admissions for PhD programs, that and it would be nice to explore other parts of the country (my uni is in the midwest, so I was exploring unis like UNC Chapel Hill or Duke to do some Embedded or Distributed Systems research at). I'm not exactly sure what I want to do come next summer, but I was thinking of applying to both internships and other universities, as I'm not quite sure whether I want to go to grad school, since my stats are somewhat lacking.

>>14771265
Sorry i should have clarified. I might be doing research at a much lower tier uni in my home city if I can't get anything elsewhere/if circumstances force me to get work in the local area (since my parents are pretty old, though it shouldn't be a problem).

Why are popular color spaces dominated by green? Seriously, look at it. Blue and Red get scrunched up into their little corners, and look woefully inadequate next to giant bands of magenta, purple, orange, yellow, and cyan. Then, for some stupid reason, green gets a whole 27% or more of the entire color space all to itself, even though it looks entirely the same on one end as the other. It's just fucking green. The tiny, tiny portion dedicated to Blue goes through a great transformation, whereas the fucking ocean of Green is spread so thin that it doesn't even look like it's any different. What's the big deal? What's the point of having a color space which includes so many shades that human green and red cones can't tell the difference between? Why represent it on all the professional color spaces? Why?

you see it too, right? RIGHT?

>>14770693
https://www.heroesfire.com/hots/wiki/talents/kugelblitz

>>14770693
it's a black hole but instead of matter it's light or heat

>>14769821
Roughly "analytic" tends to refer to precise distances and functions in some way.

In real and complex analysis analytic functions are "those which can be expanded in power series", but its clearer to understand analytic functions as having nice derivatives which then results in taylor series.

There are smooth real functions which don't have taylor series. This is because the error term in the taylor series proof is based on the nth derivative not growing faster than n!. So analytic functions in real analysis are smooth with derivatives that don't grow too fast.

In complex analysis it turns out being once differentiable implies twice differentiability (because complex derivatives are stronger real ones) which implies infinite differentiability and also having a taylor series so everything kind of collapses.

>>14771097
ok, I think I managed well enough to go in detail with this >>14770074 and fill the gaps. If this thread is still up tomorrow I'll post it. For now I'm done. Good night.

>>14771466
>In real and complex analysis analytic functions are "those which can be expanded in power series
positive powers must be required, so no terms like [math]\frac{1}{x^n}[/math] appears in the series

>>14771474
Ye my intention. I really do not care for the power series definition of analytic. Its the reason students are surprised when a smooth function does have a matching taylor series.

>>14771477
>Its the reason students are surprised when a smooth function does have a matching taylor series.
Agreed. But I'm curious. Can you an example of analytic function with no power series? Only if it's not much of a hassle.

>>14771480
Analytic functions have taylor series. That's what an analytic function is.

For a smooth function without a taylor series just imagine any smooth thing where the derivatives diverge fast. e^(-1/x^2) is common.

>>14771477
*doesn't

>>14771302
>>14771320
Because of human evolution our visual systems is more sensitive to green that any other color and can discern the difference between more shades.

Need someone who is good with probabilities to help me with this one:
>Exercise 1.7.14. Suppose that we have a tennis tournament with 32 players. Players are matched in a completely random fashion, and we assume that each player always has probability 1/2 to win a match. What is the probability that two given players meet each other during the tournament.
So i solved this exercise, in my solution the probability is [math]\frac{15}{496}[/math] (see below for details)
However checking the back of the book gives [math]\frac{1}{16}[/math] as an answer
Where did my solution go wrong?

My solution:
Since there are 32 players, that means there are 5 matches (32 players in the first match, 16 in the second, 8 in the third, 4 in the fourth and 2 players in the fifth and final match)
Denote the event that two given players meet in the [math]k[/math]-th match as [math]M_k[/math], now note that the events [math]M_k[/math] are pairwise disjoint, therefore we can sum their probabilities as follows:
Probability of two players meeting on any match = [math]P(M) = P(M_1 \cup M_2 \cup M_3 \cup M_4 \cup M_5) = P(M_1) + P(M_2) + P(M_3) + P(M_4) + P(M_5) [/math]
Suppose we pick some player, before the first match this player has equal probability of playing a game (and thus meeting) with any of the other 31 players, therefore the probability that this player meets with any other particular player on the first match is [math]P(M_1) = \frac{1}{31}[/math], now what it is the probability that those two players meet on the second match? in order for these two given players to meet on the second match they must've not met in the first match and each of them must've won and then they can meet with probability [math]\frac{1}{15}[/math] since there are 16 players now, the probability that both of them win is [math]\frac{1}{4}[/math] and the probability that they don't meet in the first match is [math]\frac{30}{31}[/math] therefore [math]P(M_1) = \frac{30}{31}\frac{1}{4}\frac{1}{15}[/math]

(1/2)

Applying the reasoning above to all those events we have:
[eqn]P(M_1)=\frac{1}{31}\\P(M_2)=\frac{30}{31}\frac{1}{4}\frac{1}{15}=\frac{1}{31}\frac{1}{2}\\P(M_3)=\frac{30}{31}\frac{1}{4}\frac{14}{15}\frac{1}{4}\frac{1}{7}=\frac{1}{31}\frac{1}{4}\\P(M_4)=\frac{30}{31}\frac{1}{4}\frac{14}{15}\frac{1}{4}\frac{6}{7}\frac{1}{4}\frac{1}{3}=\frac{1}{31}\frac{1}{8}\\P(M_5)=\frac{30}{31}\frac{1}{4}\frac{14}{15}\frac{1}{4}\frac{6}{7}\frac{1}{4}\frac{2}{3}\frac{1}{4}\frac{1}{1}=\frac{1}{31}\frac{1}{16}[/eqn]
And summing these we get the probability i had given earlier:
[eqn]\frac{1}{31}\frac{1}{2}+\frac{1}{31}\frac{1}{4}+\frac{1}{31}\frac{1}{8}+\frac{1}{31}\frac{1}{16} = \frac{1}{31}\left(\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}\right)=\frac{1}{31}\left(\frac{15}{16}\right)=\frac{15}{496}[/eqn]

(2/2)

>>14771645
>>14771646
nvm i just figured out that i'm a blind idiot because i forgot to sum the [math]\frac{1}{31}[/math] term on the far left lol
the solution is now correct after summing this missing term
[eqn]\frac{1}{31}+\frac{15}{496}=\frac{1}{16}[/eqn]

I guess the moral of the story here is solve your problems on a clean piece of paper so you can see your terms clearly or you'll end up in my situation
I'm leaving this problem here and the solution because it was an interesting one

How come you can move through time without moving through space but you can't move through space without moving through time?

>>14771666
That question currently has no answer. We still don't even know what time actually *is*.

>>14771666
The following is just speculation by me and not an answer, i'd like someone knowledgable to confirm or deny the following:
If you consider time to be a dimension just like the other three dimensions then everything is moving at the speed of light
If you don't move through space then you're moving at the speed of light in the direction of time
If you're moving at the speed of light in the any of the three direction of space then you're not moving in the direction of time
Generally if you're moving in space at a speed of A then you're moving through time at a speed of sqrt(c^2 - A^2)
Since time is considered another spacial dimension in this model, we can measure distances in time using meters, in fact the units of time and distance become the same, in this case 1 second is equivalent to moving 299,792,458 meters in the direction of time.

>>14771702
>you're moving at the speed of light in the direction of time
that part is correct but it's just another way of stating >>14771666, it does not explain why that is true.

everything after that is nonsense.

>>14771645
The tourney consists of [math]31[/math] games. Number them all from [math]1[/math] to [math]31[/math]. Let [math]A_i[/math] be the event the two chosen players meet each other in the game number [math]i[/math]. Clearly the events [math]A_i[/math] are all disjoint as two players can't meet other more than once in the tourney.
There are [math]{32 \choose 2} = 496[/math] possible ways to select 2 of the 32 players and all are equally likely so for each game we have the probability [math]P(A_i) = \frac{1}{496}[/math].

[eqn]P \left( \bigcup_{i=1}^{31} A_i \right) = \sum_{i=1}^{31} P(A_i) = \frac{31}{496} = \frac{1}{16}[/eqn]

>>14771754
This is certainly more concise and less messy than my solution, Thanks!

>>14771471
>If this thread is still up tomorrow I'll post it.
I'm anxious to see it.

Are Sauropsida and Diapsida synonyms?

Is there any reason [math]s+\frac{2-s^2 }{3s}[/math] is chosen instead of some arbitrary other number?

>>14771514
are you implying that i lack green cones or something? the majority of these green "shades" in these popular color spaces are not discernable from one another, which is not how the other colors are represented.

>>14771666
>you can't move through space without moving through time
This is a matter of perspective. As an example, massless particles (such as light/forms of radiation which are literally light) do not experience time. There is also spooky action at a distance.
>move through time without moving through space
this is impossible while in a gravity well.
>>14771709
his post isn't nonsense he's just worded it in a confusing way, you really shouldn't get filtered by it

>>14772260
Sounds like you just have a shitty monitor.

>>14771666
There is a maximum speed of causality. I have no idea why or if it can even be deduced. Spooky action moves faster, but it is not causal.

>>14772379
>can it be deduced
it's c
>why
because there aren't any particles which have free/bonus time. They ran out of it. there is nothing more to be traded for extra speed, no matter how much energy is dumped into it.
>spooky action is faster
yep, all evidence points to it being instant.

>>14772431
By deduced is meant why is there a limit in the first place?

>>14772431
Relativity is deduced by the speed of causality. The results from relativity cannot be used to deduce why there is a limit.

Can the consequences of COVID be considered part of natural selection?
Overheard some colleagues arguing about it, where one said that the people who survived are literally the definition of "survival of the fittest" and that the others were going to die anyway.
The other said that it's not how natural selection works.
I frankly forgot most of my biology classes so I can't say which one is right desu.

>>14772612
anything that stops an organism from reproducing can be considered selection.

>>14772716
But covid kills people too old to reproduce.

>>14772721
men are never too old to reproduce.

>>14772721
Correct but it does kill some younger people too, so you could argue it's filtering out the genes for those rare few with a predisposition.

>>14772156
>>14770074
>>14761762
Ok. Here it is.

First, we pose the non-negative function
[eqn]D^2=\left[ K(x,y)K(z,t)-K(x,z)K(y,t) \right]^2 = K^2(x,y)K^2(z,t)+K^2(x,z)K^2(y,t) - 2K(x,y)K(z,t)K(x,z)K(y,t)[/eqn]
Then, integrating both sides wrt each variable from [math]a[/math] to b:
[eqn]\int_a^b\int_a^b\int_a^b\int_a^b D^2 dxdydzdt = \int_a^b\int_a^b\int_a^b\int_a^b K^2(x,y)K^2(z,t)+K^2(x,z)K^2(y,t) dxdydzdt - 2 \int_a^b\int_a^b\int_a^b\int_a^b K(x,y)K(z,t)K(x,z)K(y,t) dxdydzdt[/eqn]
[eqn]\int_a^b\int_a^b\int_a^b\int_a^b \left[ K(x,y)K(z,t) \right]^2 + \left[K(x,z)K(y,t)\right]^2 dxdydzdt - 2 \int_a^b\int_a^b \left[\int_a^b K(x,y)K(y,t)dy\right] \left[\int_a^b K(x,z)K(z,t) dz \right] dxdt[/eqn]
[eqn]\left[ \int_a^b\int_a^b K^2(x,y) dxdy \right] \left[ \int_a^b\int_a^b K^2(z,t) dzdt \right] + \left[ \int_a^b\int_a^b K^2(x,z) dxdz \right] \left[ \int_a^b\int_a^b K^2(y,t) dydt \right] - 2 \int_a^b\int_a^b \left[\int_a^b K(x,y)K_1(y,t)dy\right] \left[\int_a^b K(x,z)K_1(z,t) dz \right] dxdt[/eqn]
[eqn]\left[ \int_a^b\int_a^b K^2(x,y) dxdy \right] \left[ \int_a^b\int_a^b K^2(z,t) dzdt \right] + \left[ \int_a^b\int_a^b K^2(x,z) dxdz \right] \left[ \int_a^b\int_a^b K^2(y,t) dydt \right] - 2 \int_a^b\int_a^b \left[ K_2(x,t) \right] \left[K_2(x,t) \right] dxdt[/eqn]
[eqn]B^2 \cdot B^2 + B^2 \cdot B^2 - 2 \int_a^b\int_a^b K_2^2(x,t) dxdt [/eqn]
[eqn]2B^4-2B_2^2[/eqn]
By hypothesis, [math]B_2=B^2 \implies B_2^2=(B^2)^2 \implies B_2^2 = B^4 \implies 2B^4-2B_2^2 = 0[/math]
Therefore:
[eqn]\therefore \int_a^b\int_a^b\int_a^b\int_a^b D^2 dxdydzdt = 2B^4-2B_2^2 = 0[/eqn]
and given that [math]D^2 \geq 0 [/math] then [math]D^2 = 0 \therefore D=0[/math]
which means [math]B_2=B^2 \implies K(x,y)K(z,t)-K(x,z)K(y,t) = 0[/math] and then [math] K(x,y)K(z,t)=K(x,z)K(y,t)[/math]

>cont. in next post.

>>14772901
>cont.

Second. We take the relation [math] K(x,y)K(z,t)=K(x,z)K(y,t)[/math] and substitute [math]t[/math] by an auxiliar variable [math]s[/math]
[eqn] K(x,y)K(z,s)=K(x,z)K(y,s)[/eqn]
Multiplying both sides by [math]K(s,t)[/math] and integrating [math]\int_a^b ds[/math]
[eqn] K(x,y)\int_a^b K(z,s)K(s,t)ds=K(x,z)\int_a^b K(y,s)K(s,t) ds[/eqn]
[eqn] K(x,y)K_2(z,t)=K(x,z) K_2(y,t)[/eqn]
Proceeding likewise we come up with:
[eqn] K(x,y)K_n(z,t)=K(x,z) K_n(y,t)[/eqn]
Multiplying both sides by [math]K(x,z) K_n(y,t)[/math]
[eqn] K(x,y)K_n(z,t)K(x,z) K_n(y,t)=K^2(x,z) K_n^2(y,t)[/eqn]
Integrating both sides [math]\int_a^b\int_a^b\int_a^b\int_a^b dxdydzdt[/math]

[eqn] \int_a^b\int_a^b\int_a^b\int_a^b K(x,y)K(x,z) K_n(y,t)K_n(z,t) dxdydzdt =\int_a^b\int_a^b\int_a^b\int_a^b K^2(x,z) K_n^2(y,t) dxdydzdt =\left[\int_a^b\int_a^b K^2(x,z) dxdz \right] \left[\int_a^b\int_a^b K_n^2(y,t) dydt \right] [/eqn]
[eqn] \int_a^b\int_a^b\int_a^b\int_a^b K(x,y)K(x,z) K_n(y,t)K_n(z,t) dxdydzdt = [B^2] \cdot [B_n^2] = B^2 B_n^2 [/eqn]

On the other hand:
[eqn]\int_a^b\int_a^b\int_a^b\int_a^b K(x,y)K(x,z) K_n(y,t)K_n(z,t) dxdydzdt = \int_a^b\int_a^b\int_a^b\int_a^b K(x,y)K_n(y,t) K(x,z)K_n(z,t) dxdydzdt[/eqn]
[eqn]= \int_a^b\int_a^b \left[ \int_a^b K(x,y)K_n(y,t) dy \right] \left[ \int_a^b K(x,z)K_n(z,t) dz \right] dxdt[/eqn]
[eqn]= \int_a^b\int_a^b \left[ K_{n+1}(x,t) \right] \left[ K_{n+1}(x,t) \right] dxdt = \int_a^b\int_a^b K_{n+1}^2(x,t) dxdt = B_{n+1}^2[/eqn]

This way we have:
[eqn]B_{n+1}^2 = B^2 B_n^2 \implies B_{n+1} = B B_n[/eqn]
Given that
[eqn]B_1^2=\int_a^b\int_a^b K_1^2(x,t) dxdt = \int_a^b\int_a^b K^2(x,t) dxdt = B^2 \implies B_1=B [/eqn]
By using the previous recurrent relation:
[math]B_2 = B \cdot B_1 = B \cdot B = B^2[/math]
[math]B_3 = B \cdot B_2 = B \cdot B^2 = B^3[/math]
[math]B_n = B \cdot B_{n-1} = B \cdot B^{n-1} = B^n[/math]
[eqn]\therefore B_n=B^n[/eqn]

>>14772721
Dead grandparents can't help raise grandchildren, which might lead to the parents having fewer.

>>14772734
That's true and what I had thought as well.
I don't like to think like this but couldn't everyone who died to covid be considered "unfit" and were weeded out?

>>14772947
Isn't that a social thing? Would that also fit whitin natural selection?

>>14772451
I want to _______ Revy.

>>14773082
file jointly with?

gonna ask a stupid question that might piss people off but im genuinely asking to get a better understanding.

so mag flux is essentially the total magnetic field passing through an area. But gauss's law states that the total mag field out side of a closed loop is zero since mag fields create a closed loops on itself.

This confuses me since you could put neodymium magnet inside of a small copper cylinder but it would still attract something outside of the cylinder. Which means the magnetic field is coming out of the cylinder. so where am i misunderstanding this rule?

>>14773082
support and cherish

>>14773194
rewrite it and youll notice this: electric field is proportional to the surface electric charge. this way it is much more intuitive

>>14772901
>>14772904
Seems correct.
Very nice no-tricks proof.

>>14773206
so the mag field inside the cylinder is affecting the charge distribution on the surface outside of the cylinder and this in turn causes an electric field?

>>14773216
Thanks anon. And thanks a fuckin lot to the >>14770074 anon too (are you the same btw)?
My only doubt is with the very first line where [math]D^2=\left[ K(x,y)K(z,t)-K(x,z)K(y,t) \right]^2[/math] is defined. To me it looks something taken from under the sleeve, and still all the magic comes from that very first statement. is there any previous thing I should have known? or it's only intuition?
Once I faced a similar doubt working with very specific rational substitutions to solve a particular integral.
Anyway. Thanks again to the anons that pointed suggestions and gave feedback. I had this problem pending for a while and finally I can mark it as solved.
>feelsgoodman

>>14773252
>(are you the same btw)?
No.
>To me it looks something taken from under the sleeve, and still all the magic comes from that very first statement.
I think it's equivalent to the operator-is-one-dimensional thing.

Define [math]f(x) = K(c, x)[/math] where [math]c \in [a, b][/math]
Then [math]\int_a^b K(x, y) f(y) dy = \int_a^b K(x, y) K(c, y) dy = K(c, x) \int_a^b K(y, y) dy[/math]
I'm not really in the mood for a complete proof.

>>14773635
>I'm not really in the mood for a complete proof.
Understandable. Any reference on this?

>>14773082
WHAT WAS THAT?

Practically, what difference does a left ordering (∀a,b ∈ M)(∃c ∈ M)[(cRa)Ʌ(cRb)] or a right ordering (∀a,b ∈ M)(∃c ∈ M)[(aRc)Ʌ(bRc)] make? What are some examples?

Why doesn't Cantor's diagonalisation argument work for the bijective map from [math] \Bbb R[/math] to itself? Why doesn't it work for the surjective map from [math] \Bbb R[/math] to [math] \{1\}[/math]

Are you expected to TA as a master's student?

>>14774837
I have never heard of any master's student teaching. I think it's only for PhD students, and even in the latter case it's not compulsory in many situations.

>>14774886
huh? all my labs and practicals were overseen by master's students and some bachelor students who had completed the course earlier. never saw the professor in any of them.

>>14774886
huh? most of my labs and practicals were taught by master's students and some by bachelor students. never saw the professor in any of them.

>>14774837
>>14774908
I guess if you're American, then yes you have to teach, since Bachelor's is more valued there, being 4 years.

>>14763865
A lot of it is just knowing definitions anon. Do you know when to "columns" (meaning vectors) are linearly dependent? When x_0 = k x_1, for some scalar k. Here the "column matrix" is the matrix X such that Y =X\beta +\epsi, i.e. X =[1 x].
For b you now have two vectors and are asked to find the dimension of their span, do you know what this is?
For c you are asked when these vectors are orthogonal, i.e. when their dot product is 0. You know 1•x=sum(x), so the column has to be 'centered', i.e. have mean 0.
>>14764150
You get better probably when you take a proper linear algebra class, one that properly defines projections in Hilbert spaces and their uniqueness &c. For me that was called functional analysis, so maybe look at that.

Why does Nested Interval Property fail for rationals?

>>14773640
>Any reference on this?
Lax for functional analysis and Halmos for integral operators.
If you meant the specific result, no.
>>14775138
Because you can eenie meenie an irrational number [math]x[/math] and consider the nested intervals [math][x - 1/n, x + 1/n] \cap \mathbb{Q}[/math]

How to prove this?

[eqn]\sum_{k=0}^{r}2^{2(r-k)}\binom{n}{k}\binom{n-k}{2(r-k)}{\binom{2n}{2r}}^{-1}=1[/eqn]

>>14773810
I want to gently headpat Revy while she sleeps.

>>14775455
>consider the nested intervals
What about it?

>>14775455
>>14775610
Nevermind, I got it.

>>14775555
forgot to mention that 2r < n

>>14775555
Multiply with [math]{2n \choose 2r}[/math].

[eqn]\sum_{k=0}^{r}2^{2(r-k)}\binom{n}{k}\binom{n-k}{2(r-k)}={\binom{2n}{2r}}[/eqn]

Both sides count the number of ways to choose a subset with [math]2r[/math] elements from a set with [math]2n[/math] elements. For the RHS it should be obvious.
For the LHS group the elements into [math]n[/math] pairs. From each pair you can take either no, one or both elements. If you take both elements from [math]k[/math] pairs then you will need to take one elements from [math]2(r-k)[/math] of the [math]n-k[/math] remaining pairs as [math]2k + 2(r-k) = 2r[/math].
For each [math]k[/math] there are [math]{n \choose k}[/math] many ways to choose pairs from which you take both elements and [math]{n - k \choose 2(r-k)}[/math] many ways to choose pairs from which you take one elements. For each of the latter pairs you could either take the first or the second element so you have to muliply by 2 for each chosen one. Sum it all up and you get
[eqn]\sum_{k=0}^{r}2^{2(r-k)}\binom{n}{k}\binom{n-k}{2(r-k)}[/eqn]
as needed.

>>14775455
>If you meant the specific result, no.
no no, this is enough. Thanks anon.

>>14775805
I was hoping there was a more algebraic way to solve this but this is indeed interesting, thank you!

>>14774164
>Why doesn't Cantor's diagonalisation argument work for the bijective map from [math] \Bbb R[/math] to itself?
There isn't a bijection between R and the set of functions from R to R like there is with R and the set of functions from N to a finite set.

>Why doesn't it work for the surjective map from [math] \Bbb R[/math] to [math] \{1\}[/math]
The "diagonal" is a single element that can only be 1, and by being 1 makes it a surjection

>>14759437
Dear /sci/,

Please help. I am deeply troubled by something and I don't know if its psychological, or emotional or what but I'm wanting to understand something. Perhaps you can offer some explanation or theories on the principle of what this is about.

Ok, my question is about the feelings or impulsed a viewer gest when watching a selfie-type of social media video.
Like pic related here for example.

Here's my general reaction to it-

At first, I'm absolutely sickened and repulsed by the narcissisism implied by these constant social media selfies, women grooming and preening themselves in the mirror while filming themselves for the world

but then i become subconsciously uncontrollably drawn in and mesmerized by it at the same time. a morbid unhealthy fascination i think.

what is this?
why is it repulsing, and then mesmerizing at the same time? serious question. i really want to know what effect this is having on people as individuals but also our society as a whole. I think its going to be detrimental but the whole world seems to just be fine with it.

Can anyone help me understand which of these two setups make the most sense for providing the most water pressure and most even pressure to all the nozzles in the system?

I don't understand anything about water. I'm stupid. But I need a system to water my herb garden.

>>14775948
what's so mesmerizing in seeing other people having a better life/work conditions than the average joe? My guess is you are projecting yourself (non ironically) in this.
To me it's as interesting as watching paint dry. It has no effect on me. Just like some products or services ads.
Let me ask you something. Do you consider your purchasing behavior can be affected by ads? Maybe there's a correlation with that.
>what is this?
no idea.

>>14776024
literally the same

>>14776033
they can't literally be the same
imagine the side nozzle isn't there and there is only enough water pressure for 2 to spray then in the left picture the rightmost nozzle doesn't and in the left picture the middle nozzle doesn't

>>14776051
no, they would all spray at roughly two thirds capacity. the downstream nozzles might spray less if the pipes are super small and the pressure is super high, but pressure generally distributes itself equally as long as the fluid isnt moving too fast inside the pipes.

>>14775562
Why didn't you say so sooner?
>>14775555
>>14775925
If you work through the sum chapters in Concrete Math by Knuth and do the exercises I swear on my life you will ANNIHILATE combinatoric summations. He even introduces hypergeometric sums which is a sledgehammer among nuts.

>>14759437
What are the prerequisites for stochastic calculus? and can anyone recommend books to read to get there for someone with a solid foundation of regular calculus and linear algebra?

>>14776031
>Do you consider your purchasing behavior can be affected by ads?
no. not me far from it. im a brick wall to advertisement. dont respond to ads and i don't consume stuff.
and i dont view enough of this trash to be exposed to any ads that might be associated with them.
and i don't use social media at all. only see these clips posted on 4chan occasionally is the only reason im even aware of them.

the only clips i bother top watch is the ones of females of course. could be any subconscious enchantment i feel from them is an attraction to the young attractive women posing enticingly for the camera. and the false illusion they are posing "for me" even though obviously they are not. maybe that's all there is to it.

i dont follow social media i certainly don't know who these people are or follow them in any way. just see the occasional random clip of a female doing this and that is my only exposure to it

>>14776062
what if it doesn't provide enough flow to get past the lower nozzle

>>14776086
Thank you for the recommendation anon, i have always found sums with binomial coefficients very intimidating, i will definitely check out this book

>>14776031
>better life/work conditions than the average joe?
I'd say whatever she's showing in that clip is about average. doesn't look particularly special. and it looks quite shallow and probably temporary, considering "Big Tech" has been making announcements about getting ready to lay off the useless excess fat form their staff

>>14776200
>lower
i interpreted the picture as a top-down view of the situation, but if the whole structure is standing upright then that stuff i said about pressure distributing equally doesnt fully apply; the pressure would increase with depth.
in the case that all points in the pipe are the same height, then as long as the resistance of the nuzzles >> the resistance of the pipe, then theres no way for the fluid not to be able to "get past" the first nozzle.

is the axiom of foundation important? I've seen several books leave it out when discussing ZFC

>>14759437
How long until android waifus?

>>14776200
>>14776301

After I went to the hardware store and actually built this set up, and because I used transparent vinyl tubing, I could see what was happening in the tubes. What actually happened was:

1. All of the nozzles seemed to spray maximal amount of water.

2. In the side of the circuit with no nozzles, there was no flow. It was partially filled with bubbles of air and water, and bubbles weren't moving.

3. Putting in more nozzles allowed water to flow up to where the nozzle was, but the water never completely filled the tube.

>>14778022
>I went to the hardware store and actually built this set up
This being setup B.

>>14776024
>>14776301
>in the case that all points in the pipe are the same height, then as long as the resistance of the nuzzles >> the resistance
They make pressure compensating drippers. At high pressure the housing expands and shrinks the holes where the water comes out, making flow constant between say 30 and 80psi. Then the cheapest solution might use tubing one size smaller than the one with negligible resistance.

In other words, pressure compensating allows 30 4L/h inline drippers on 50ft of 1/4 inch tubing to be even. The tubing pressure drop isn't negligible. Before you needed 50ft (or less) of 1/2” to feed drippers on short lengths of 1/4” tubing. I'd much rather assemble the first system but the second is cheaper to pump and won't clog as dramatically.

>>14778022
Rotate the tee/Swap it's two outlets of the tee to get flow in the dead leg.