/sci/ - Science & Math

How the fuck do I find the 2020^(7^15)mod7? How do I tackle these kind of problems that I can't write the exponent in binary?

>added my books link
Feels good man

Unanswered questions:

Math questions:
>>11889425
>>11891571
>>11896135
>>11902149
>>11902195

/g/ questions:
>>11897457
>>11901399 [This is either machine learning or biology. Honestly looks more like machine learning.]
>>11903349
>>11903380

Physics questions:
>>11894247
>>11899927
>>11900405 [This guy is desperate, someone help him.]
>>11901629
>>11901706
>>11901743
>>11901901

Biology questions:
>>11886601
>>11890940 [What did he mean by this?]
>>11893142
>>11899615

Stupid questions:
>>11888113
>>11890176 (Patchouli)
>>11900695
>>11901617
>>11901625
>>11902157
>>11902859

>>11903363
Can you use fermat's little theorem or something like that to reduce the exponent mod 6?

>>11903580
Yes. I've gotten final result 4 btw.

If i have a function [math]f : \mathbb{R} \to \mathbb{R}[/math] that is "strictly increasing", is this function surjective?

Would my reasoning be good if i argued that it can be but only if the function is continuous?

>>11903910
No. [math]e^x[/math] is a counterexample. Did you mean injective instead?

>>11903934
No it said surjective

>>11903398
Thanks anon, I hope someone can answer this >>11899927 (even if I am beginning to think that it's an error by my professor, since I haven't found that graph anywhere else).

>>11901706
This is an open question, but I'd say that the tree analogy isn't really accurate. One proposed explanation to the wave function collapse is decoherence, which is basically what you are referring to: when a quantum system interacts with the classical environment, it loses coherence and becomes a statistical mixture of different states rather than a superposition of states. For example, in the Stern-Gerlach experiment you have a beam of electrons which are in a superposition of spin up and spin down; when they interact with a magnetic field, they lose coherence and become a statistical mixture. This means that before the magnetic field, each electron is in a state that is half up and half down, while after the magnetic field half of the electrons become up and half become down. As you can see, the moment when the wave function collapses is not when we observe which electrons have spin up and which have spin down, but rather the moment they interact with the classical environment, i.e. the magnetic field. This means that consciousness has (fortunately) nothing to do with the outcome of an experiment.

>>11903934
The arctangent is another counterexample which is strictly increasing, has [math]\mathcal{R}[/math] as a domain, is continuous and yet has [math]\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)[/math] as codomain.

I have another question now. Suppose that, as [math]n[/math] varies, I have a variable [math]x_{n+1}[/math] uniformly distributed between [math]kx_n[/math] and [math]x_n[/math], where [math]k<1[/math] is a constant and [math]x_0[/math] is given. How can I calculate the probability distribution for [math]\frac{x_n}{x_0}[/math]? I am sure it's pretty easy and I am just being a brainlet, I am not very good with statistics. Thanks anons.

>>11903986
Fuck, I used \mathcal instead of \mathbb for [math]\mathbb{R}[/math]. Oh well, you understand what I mean.

>>11894247
When changing the inertial frame, you can't simply add the velocities up like that, spacetime doesnt behave that way. You have to do Lorentztransformations. Its then easy to see that the velocities stay below c in any inertial frame.

>>11903630
How

I posted this on /mg/ but since it falls in the category of advice perhaps this is a better thread.
How important is the choice of university for undergrad? I have an offer to read maths at St. Andrews, as an international student, which means fees (study + living costs) of nearly 30k british pounds per year. I also have an offer for a uni in my country (third world latin american country), said uni which has a lot less 'prestige' than st andrews, but fees are ridiculously low (the whole undergrad costs less than one year at st andrews). Anyway, would you recommend to do undergrad home, try to do really good and then aim for post-graduate at somehing more prestigious? Or should I take the offer? Any chance somebody here studied at st andrews?

>>11904191
>Anyway, would you recommend to do undergrad home, try to do really good and then aim for post-graduate at something more prestigious?

Definitely yes. Unless you plan to do a master in the same university, in which case the admissions are gonna be about 20-30% more lax but that's not worth it. Just don't focus only on gpa, build a resume. That's your goal.

whats the best DEFENSE of quantum immortality theory?

>>11903986
>I have another question now.
Ok, it was just a log-normal. I haven't figured out why, but the simulation gave me some insight.

>>11904191
It does matter, but 30k pounds a year is a fucking LOT of debt to take on, and it's really not worth it. Your choice of school doesn't matter that much; as long as it's remotely respectable, your undergrad name isn't going to lock you out of postgrad opportunities. Plenty of foreigners from nowhere schools in nowhere countries immigrate to the US/UK for grad school each year.
Also, this might be a more difficult decision if you had an admission to Oxbridge or MIT or something, but St.Andrews isn't one of those global top-tier schools. It's quite good, but not "oh shit he went to St.Andrews" good.

>>11904191
>as an international student
It doesn't matter nearly as much then, at least not for undergrad. As long as you manage to keep decent results in your home country and make it to a more prestigious uni for post grad you are good to go

>>11904066
[math]2020^{7^{15}} \mod 7 = 4^{7^{15}} \mod 7[/math].
[math]7^{15} \mod 6 = 1^{15} \mod 6 = 1[/math].

>>11904235
That it literally changes nothing? You still have the same probability to die in any given scenario regardless if it's "true" or not.

HEY guys big double R retard here again, with a question like this, the solution goes
>augmented matrix form
>row reduction
one of the rows ends up being [math]0 0 0 (b_1 + \frac{(b_2 + b_3)}{3})[/math], so my solution is that this system is only consistent if [math]b_1 = -\frac{(b_2 + b_3)}{3}[/math], right?

Can a base for a topology include the empty set?

I'm growing small brown moles on my body pretty fast, in addition to just being itchy. Search results for this net me little to nothing. Do I have melanoma or something? What preceeded this was chest/heart aches.

>>11904767
>>11904826
Sure

>>11904767
Almost, but that should be /2, not /3.

>>11903346
If I’m around 41N and 71.4W would the comet be easily visible in the area? I don’t know enough about it’s path and the Internet is strangely vague about it

Working on my econ class which is basically just statistics. Would like some help with this problem since I am not at all confident with what p-scores really mean, especially in relation to the confidence interval. I feel like mentally I'm confusing myself because you're testing a null hypothesis, but the question is asking about testing the hypothesized mean against a two-sided alternative. Any help clarifying the questions/thought process, and proofing what I've came up with would be really appreciated. I can take screen shots of table A if it helps, too.

>>11904952
anon, you should see a doctor ASAP, I wouldn't worry about one or two after spending time in the sun, but if they're itchy and appearing quickly you should definitely get them checked out

Ive been admitted to 2 different phd programmes. One is with a very strong prof in a local (unknown) uni and the other is in a phd school which is much more reputed (but there I dont have a professor yet, I have to take classes the first year).What would you do?

The topic is pdes/analysis (pure).

>>11905241
>what would you do
I would have done physics desu.

>>11905241
People always say in these threads to invest in advisor coin, but I wouldn't know.

>>11905241
>What would you do?
Dunno man, I'm a brainlet so I'd never get to that position. I think if you've gotten that far on your own you should probably try to make this choice without letting other people influence you too much. Chances are though if you're smart some professor will scoop you up, guess it depends if you care more about learning from someone who has a lot to offer you or the degree itself

>>11905270
id like to have an academic career (postdocs...) preferably

>>11905196
Can disregard this question, my solutions were correct. This page helped clarify my confusion regarding the p-scores in relation to confidence intervals.
https://www.ucl.ac.uk/child-health/short-courses-events/about-statistical-courses/research-methods-and-statistics/chapter-6-content-2

>>11905241
how well known are we talking?
the reputation of PhD programs does not always go with the reputation of the school. MIT is only 10th in the world in my field, although many people think they'd be higher. having a well-connected and famous advisor in the field is 99/100 times more valuable
if the uni is truly unknown even in your subfield then I would say go to the other uni. you'll find an advisor regardless of if it's your first choice or not

>>11904985
you're right, that's what I had on paper actually.

last question, I totally don't understand this one.

>>11905241
Are there other factors? Will the living situation be nearly identical in both scenarios? Do you like one area more? etc, etc. You obviously want to go down the path of greatest robustness, least effort, and highest odds of exponential payoff via luck/exposure.

If I had to pick, I'd say the strong local prof: gives you a chance to stand out, get close and maybe get lucky, but at a bigger school your exposure to general good luck events and fate happenings may be greater overall. But best move is probably to find a prof at the other school first, and get a more accurate comparison.

>>11905322
first of all, im talking about europe, so many universities are completely unknown (regarding maths).

the professor im talking about solved a conjecture when he was young and won some important prizes. I fear that if I choose the unknown uni I may be almost alone doing analysis research (very few people do math phds there, ~4/5 graduate every year counting applied mathematicians).

>>11905345
>least effort
I meant least resistance, i.e go to the uni that lets you live with your parents so you don't have to worry about paying rent (just an example)

>>11905346
I see that it comes down to two things:
1. is the prof well-known enough in the field
2. where will your work have more impact

only you can really evaluate 1. or, I recommend talking to faculty at your current uni about their impressions. people in the field often have better perspectives
regarding 2., you should talk to the guy you want to work with. ask him what type of work he envisions you doing, how involved he wants to be or if he'd rather be a hands-off advisor. at the same time you should get in contact with some of the people you'd want to work with at the bigger uni, partly to answer the same questions and partly to see whether or not they're even taking students.

>>11905358
its hard to talk with profs (summer + covid). I talked with my msc thesis advisor and he said very good things about the prof (he did his phd with him).

>>11905389
it's good that he likes him. don't take that for granted, some advisors suck donkey dick. did he say anything about the program at the other school?
also yeah it's hard to talk with them in person but even a simple email correspondence will do. I emailed 15 potential advisors and got like 12 responses. ended up working with one because of our continued conversation following the email

>>11905326
a)
[ 1 -5 4 ]
[ 0 1 -6 ]
b) Yes (the matrix has rank 2, i.e. it has two linearly-independent rows). No (a mapping can't be one-to-one if the number of columns is greater than the rank).

Where and how do I learn those advanced integration techniques (Feynman, tangent substitution, countour, etc.) like in those autistic yet kino math videos? Do they get taught in a class?

>>11905465
these pop up naturally as you progress in math. if you don't know them yet I feel like you're still in high school.
literally just pick any calculus textbook and read it.

>>11905475
I'm not. I'm going on to my second year of uni, and I finished calculus 3 this year. It never came up in any of my classes, so I was just wondering when I could expect to learn them, or even better how I could self-study them. To be clear, I already know about the fundamental ones: by parts, u substitution, and partial fractions.

>>11905494
>finished calc III without contour integrals
you absolutely must have. or you called them something else. trig substitution is a calc I or II topic.
>how to self-study them
like I said, if you take any calculus book it will include sections on these. they are just ways to solve specific problems.
Feynman/Leibniz: differentiating the integral so that it becomes an easier integral to solve.
"tangent" trig substitution: replacing a complicated square-root integral via trigonometric relationships
contour: fancy term for an integral whose path follows a certain contour, such that it is bounded on the countour. often used in physics.

>>11905475
retarded faggot
>>11905494
you can find a few of them in Spivak's chapter on primitives in his book 'Calculus'. Look up Keith Conrad, differentiating under the integral sign, and check out Vladimir Zorich's Mathematical Analysis volume I specifically the chapter on primitives (I believe its the end of chapter 5) where he gives a compact but rather robust tutorial on integrating. For contour integrals I'm assuming a book on mathematical methods for physicists or complex variables would help.

>>11905508
what makes me retarded? I learned about these topics in high school/1st year of uni. ANY non-shit multivariable calculus textbook will discuss them.

>>11905512
no they wont
people dont care about dumb integral tricks like your teacher did
its not even useful so Anon should just not worry about it anyway

>>11905517
???
through any basic fucking calc class you learn trig substitution as a way of evaluating certain integrals. did you even take calc? am I being memed? they don't say: "this is what an integral is" and then move on to a different topic. you apply it in different situations, in different problems, etc. and all of these techniques show up in one way or another

>>11905517
>>11905512
Integrating is a g loaded activity, it is a good way to pass the time and legitimately useful in applied math and physics. While you waste time with GAY combinatorics, number theory, algebra homoshit, brain chads solve differential equations and integrals. Sorry!

>>11905505
i think when he says contour he means computing real integrals by considering a contour in the complex plane and using the residue theorem / cauchy integral formula

>>11905524
yeah, you learn easy trig sub in a calc class, and then you dont use it after that, hardly even in diff eq
you dont need Feynman for anything
contours ill admit are useful but thats it
>>11905526
>brain chads solve differential equations and integrals.
you fucking retards prove its impossible to solve in elementary functions and then plug it into the computer, you dont do jack shit

>>11905575
I highly doubt it when contour integrals are a thing that are more similar in scope to the other ideas he's talking about.
>>11905576
>you're a retard you don't learn any of these dumb tricks in math
>oh shit yeah you actually learn 2/3 of these topics in math
can you pick a side and stick with it jfc. I don't even understand the point you're making, you're somehow now trying to argue the usefulness of certain topics when that wasn't what the original question was asking?

>>11905581
if youre a dumb enough stump to do the integrals by hand, then yeah 2 of them are kinda sometimes useful
but because we get computers to solve integrals, its a waste of fucking time to worry about the tricks to begin with
there is never ever a case where people prefer to solve an integral or de by hand rather than just get the computer to do it

>>11905601
now you're literally arguing a brand new 3rd point holy shit what is going on

>>11905608
my point the entire time has been "computers do it better, so its worthless"
i didnt start by mentioning computers, because everyone whose ever done DE already knows to use a fucking computer for this
youre the one dumb enough to talk about doing this shit by hand, so yeah i backed up and lied about "contours are useful" so that you wouldnt get upset that i insulted your undergrad toy
i guess i shouldnt have even bothered
a computer can replace you easier than a truck driver, so try and prove me wrong and drive off a cliff

>>11905633
the original question was "where and how to learn these integration techniques"
I responded with where they show up in the course of math curriculum
now you're autistically arguing a new point that "they aren't needed" which is fine if you feel that way but nobody asked. go be retarded somewhere else
>i guess i shouldnt have even bothered
now you're starting to get it

>>11905638
>I responded with where they show up in the course of math curriculum
>>u use AdVaNcEd integration techniques in calc
yes, and what an insightful answer it was
in every part of the math curriculum, you never use integration techniques outside of baby calc
because in adult calc, the computers do it for you, and outside of calc, you never mention them
dont let knowing a few integration techniques bloat your head, or else the rest of the useless competition math will hit you on the way out

is this sensible/coherent?

>>11903346
Hey pls help with finding the complex fourier coefficients, I think im integrating over the wrong part of the period but im this point im broken. can find anything too usefull either no matter where I look.

the goal is to find the complex fourier coefficients [math]X_n[/math] for a signal [math]x(t)[/math] thats given by:
[eqn]x(t) = \begin{cases} A \cos(\omega_0 t) & Acos(\omega_0 t) > 0 \\ 0 & else \end{cases}[/eqn]

The fourier coefficients are given by:
[eqn]X_n = \frac{1}{T_0} \int_{t_0}^{t_0 + T_0} x(t)e^{-jn \omega_0 t} dt[/eqn]

Im pretty sure what im doing is correct, but the aproximation doesnt seem to work, at this moment my result is:
[eqn]\dfrac{A \cos \big( \frac{\pi}{2} n \big)}{\pi (1- n^2)}[/eqn]

im just stumped, pls help

>>11905465
Feynman integration is probably learned in some physics classes. Not sure what tangent integration is. That might be trig sub, which is learned in a second semester of calculus. Contour integration is learned in complex analysis. But integration tricks are best learned through doing many problems that use them.

https://www.springer.com/us/book/9783030437879

>>11905759
My result is the same as yours apart from a factor of [math](-1)^n[/math].

>>11905806
Can you please state your full result?
Does the function produced by it match the desired outcome?

>>11889425
You need continuity or PL-smoothness for Fourier inversion.
>makes no use of the provided hint
Are you sure?
>>11901629
Microstates vs macrostates, hun.
>>11903986
Decoherence is not at all an open question sweetie.

>>11905820
>you need continuity or PL-smoothness for fourier inversion
>implying

>>11905813
Period is [math]\frac{2\pi}{\omega_0}[/math], so I just used [math]0 \leq t \leq \frac{2\pi}{\omega_0}[/math] for integrating.
Assuming [math]A[/math] is positive, then the signal [math]x(t)[/math] is nonzero for [math]0 \leq t \leq \frac{\pi}{2\omega_0}[/math] and for [math]\frac{3\pi}{2\omega_0} \leq t \leq \frac{2\pi}{\omega_0}[/math].
Quickly just plugged in the results into Mathematica and got
[eqn]X_n = \frac{(-1)^n A \cos \left( \frac{\pi n}{2}\right)}{ \pi(1-n^2) }[/eqn].

>>11905836
OK, something's not right with this approach, answer gives [math]\frac{0}{0}[/math] for [math]n=\pm1[/math]. Plugging in [math]n=\pm1[/math] into the integrals before evaluating gives [math]A_{\pm1} = \frac{A}{4}[/math].
I thought I remembered my Fourier Analysis. Guess I'm rusty.

>>11905749
bump

>>11905749
Some nitpicks over LaTeX but overall makes sense to me.

>>11905836
>>11905854
for n=1 the integral doesnt seem to exist bro.
Im integrating over [math]\Big[ -\frac{\pi}{2 \omega_0 , \frac{pi}{2 \omega_0}[/math] should I be integrating elsewhere? my reasoning is that:
[eqn]x(t) = \begin{cases} 0 & t \in \Big[ -\frac{\pi}{\omega_0} , \frac{\pi}{2\omega_0} \Big) \\ A \cos(\omega_0 t) & t \in \Big[ -\frac{\pi}{2\omega_0} , \frac{\pi}{2\omega_0} \Big] \\ 0 & t \in \Big( \frac{\pi}{2\omega_0} , \frac{\pi}{\omega_0} \Big] \end{cases}[/eqn]

ffs why doesnt this work? what im I missing?

>>11905854
The Fourier transform is a continuous function of [math]n\in \mathbb{R},[/math] so to compute the coefficient for [math]n = \pm1[/math], just take the limit as [math]n \to 1[/math] and apply l'Hopital. Doing so, you get the coefficients. But I will say, that my computation of the coefficients doesn't get the factor of [math](-1)^n[/math].

>>11905957
I get a perfectly well defined integral. Where does it go wrong?
[eqn]\frac{A\omega_0}{2\pi}\int_{-\frac{\pi}{2\omega_0}}^{\frac{\pi}{2\omega_0}} \cos(\omega_0 t) e^{-i\omega_0 t} dt = \frac{A}{2\pi} \frac{\pi}{2}.[/eqn]

>>11905957
OK, using this approach, I'm not getting the factor of [math](-1)^n[/math]. Now I'm remembering just how important it is to specify what the domain of periodicity is.
Also the integral does exist for [math]n=\pm 1[/math], I get exactly what >>11905978
, and it gives me the same result as using >>11905961 (l'Hopital worked perfectly fine btw)

>>11905978
shit youre right.

im trying to aproximante by using the cosine series and I get aomething not even remotely close to the desired product.
[eqn]x(t) = X_0 + 2 \sum_{n=1}^{\infty}||X_n|| \cos(n \omega_0 t) \\ = \frac{A}{2} + \frac{2A}{\pi} \sum_{n=1}^{\infty} \Bigg| \frac{\cos \Big( \frac{pi}{2} n \Big)}{1-n^2} \Bigg| \cos(n \omega_o t)[/eqn]

>>11903363
2020 mod 7 = 4 mod 7.
now observe that 4^2 = 2, 4^3 = 1 mod 7.
7^15 mod 3 = 1 mod 3.
therefore,
4^(7^15) = 4^(1) = 4 mod 7

>>11903346
hey uh doesn't pic related mean sqrt(x) isn't well defined because it gives two answers(sqrt(64) = 0 +- 8)?

positive_sqrt and negative_sqrt are defined, but sqrt which would give 8 and -8, would imply that 8 = -8, and therefore be irrational, wouldn't it?

>>11906231
[math]\sqrt{64}=8[/math]. That's it. It is not also -8, by definition.

>>11906243
oh, i thought sqrt(x) is the c such that c^2 = x, which would mean sqrt(64) = 8 and sqrt(64) = -8 would both be true, but 8 obv cant = -8

Looking to to a text mining project with machine learning using recipes and ingredients as datasey, to determine what recipe best matches the input ingredients. Would this even count as machine learning if you consider the recipes as classes, ingredients as features and set of ingredients/input as a feature vector?

It seems overkill to me to use ml algorithms for such a trivial task but it still counts as ML, right?

>>11906250
Nope, if we are talking about real roots then there is the condition [math]c\geq0[/math]. Complex roots instead are usually defined (or at least we defined them like this when I took the course) to take all the values, i.e. [math]\sqrt[n]{x}=c, c^n=x[/math]. This means that they aren't functions.

wtf does the highlighted part mean?

>>11906269
j=1, 2, 3
k=1, 2, 3
Therefore, you have 3x3=9 possible combinations of the two indices.

>>11906269
[eqn]v_1 = \epsilon _{1, 1, 1} \omega _1 r_1 +\epsilon _{1, 1, 2} \omega _1 r_2 + \epsilon _{1, 1, 3} \omega _1 r_3 + \epsilon _{1, 2, 1} \omega _2 r_1 + \epsilon _{1, 2, 2} \omega _2 r_2 + \epsilon _{1, 2, 3} \omega _2 r_3 + \epsilon _{1, 3, 1} \omega _3 r_1 + \epsilon _{1, 3, 2} \omega _3 r_2 + \epsilon _{1, 3, 3} \omega _3 r_3[/eqn]

>>11905989
So no one knows wtf it should result in?

>>11905759
[math]x : [0, T_0] \rightarrow \mathbb{R}[/math]
[math]x(t) = \max [ A \cos ( \omega t), 0][/math]
[math]y : [0, T_0] \rightarrow \mathbb{R}[/math]
[math]y(t) = \min [A \cos ( \omega t), 0] [/math]
[math]y(t) = - x(t + T_0 / 2)[/math]
[math]x(t) + y(t) = A \cos ( \omega t)[/math]
Try to use these relations to compute it.

>>11906250
A real square root refers to the positive root. A complex square root may refer to the principal root (the one closest to the positive real axis) or to "some" root (e.g. the roots of a cubic polynomial are a+∛u+∛v for u,v being the roots of a quadratic), but you have to use "matching pairs" of cube roots to get the 3 actual roots).

If you're using the principal root, you need to bear in mind that √(a.b)=√a.√b doesn't necessarily hold (this is used in the troll proof of 1=-1).

>>11905434
>a)
SHOW YOUR WORK!!!!

>>11907287
There is literally no work to show.

>>11907291
so the standard matrix is basically just a matrix of the coefficients, where R1 is the first equation, R2 is the second? And so in this case the resultant matrix is 2 rows by 3 columns?

>>11907362
>so the standard matrix is basically just a matrix of the coefficients, where R1 is the first equation, R2 is the second?
Pretty much.
>2 rows by 3 columns?
Yeah.

>>11907362
I think you mean array of numbers

do you guys have anything on information as a quantity in physics? Wikipedia is hopeless

>>11907447
did you look at the "entropy and information" page? maybe that's what you're looking for

I'm looking for a table of refractive indices of light of various wavelengths in water. Google doesn't give me anything useful.

>>11907549
https://en.wikipedia.org/wiki/Optical_properties_of_water_and_ice#Refractive_index_(real_and_imaginary_parts)_for_liquid_water

>>11907362
You need to read a book or google stuff. You won't be able to do your homework if you don't know what anything means. The standard matrix of your T is the matrix that represents T with respect to the standard bases of [math]\mathbb{R}^3[/math] and [math]\mathbb{R}^2[/math].

aside from being born, where did i go wrong here?

>>11907528
thanks

>>11907565
Thank you!

>>11907575
Did you actually compute the product of the two matrices or did you just flip the sign on entry 1, 1?

>>11907575
>>11907588
oops just realize i only rotated pi/6, got it now

did i do this right

>>11907755
Yeah, but you probably want to write it out like this:
[math]T(v) = T(5 e_1 - 3 e_2) = 5 T(e_1) - 3 T(e_2) = (13, 7)[/math].
Pretty much formalities tho.

>>11907863
thanks

>>11907863
take 2

>>11907901
Looks good.

>>11907901
>>11907915
At a second look, the alignment on the formatting is kinda funny. Otherwise good.

I'm supposed to find the shortest method to make that product using reagants and only those two reactants. I can tell there's sulfonation but how the hell do you make/add that hexane chain?

>>11908116
grignard?

>>11908116
I think somehow the benzene ring needs to electrophilically grab that pi-bond, leaving a carbocation. If you count the hydrogens, it's apparent there's a [1,2] H-shift involved. Presumably, the now-benzylic carbocation will grab another pi-bond, forming the hexane chain.
I have no idea what the reagents would be, or how that last cation would get dealt with.

Can someone explain to me in this graph showing the correlation between sun radiation and temperature how exactly they are explaining the measures of sun radiation before 1970?

I recall there being a colonization meme here, can anyone remind me?
Not the colonize Venus one, I think it was an abstract concept. "When will we colonize love" or something.

>>11908505
Pyranometers were invented at the end of the 19th century.

>>11908578
The start on that chart is before that though.

I'm supposed to find the flux through the surface of this prism in a uniform electric field pointing along +x. I redrew the prism as a flat shape and added some colors here.
Flux through purple, green, and yellow should be zero because their normals are perpendicular to the field. Flux through blue should be [math]-EWH[/math], and flux through red should be [math]\frac{1}{2}EW\sqrt{H^2+B^2}[/math].
Why isn't the total flux zero? It's a closed surface with no charge inside. I'm imagining an infinite charge plate to the left of the prism generating the uniform electric field. It wouldn't matter what closed surface you rope off; there would be no charge inside it, so the total flux electric flux be zero.
The field also has zero divergence, so there's no charge density anywhere.

I got the labels wrong on my previous post

bros
whats a fuckin skyrmion

Feeling like a brainlet. How the fuck would you prove that

[math]\frac{1+||y||^2}{1+||x||^2}\leq \left(1+||x+y||\right)^2\quad x,y\in\mathbb{R}^N[/math]

what do they mean by this? how do i even eliminate the constant terms in all equations?

>>11908838
r1+3*r2, or r2+r3, or r1-3*r3, i.e. use weights which cause the constants to cancel. Each of those gives you 5x+4z+4w=0 (up to a scale factor).

Can someone explain to me what this even means? Not what the variables are just the maths behind it.

>>11909167
You know notation is not universal right? What is the context of that symbol?

>>11908689
physicists just make things up sometimes

>>11909171
Something to do with heuristics. This formula is literally labelled as "heuristics".

>>11909071
so there is only 1 equation left?

>>11909180
That is not a mathematically defined concept, from what subject it is from(Don't say heuristics, what is the title of the book and from what chapter is it)? Could you put a screenshot of the whole page?

>>11909188
Its from "Regularized Predictive Control Framework for
Robust Dynamic Legged Locomotion"
Chapter 3.3
The screenshot was from the appendix, the actual formula is used to calculate something else so I'm trying to work out what that is.

>>11909198
It is just a function of those variables, I don't understand your confusion

>>11909217
I was thinking that because it said heuristics that it was just going to be something as simple as an xyz plot.

>>11909230
Wasn't

>>11909217
The paper states these as just "heuristics" not a heuristics function.

>>11909266
For that equation to make sense, you must be adding two vectors of the same dimension, so that must be a function of those variables that has as an ouput a vector as the same dimension as [math]\chi[/math]. If you look at the examples at the apendix, that is exactly what it is. It is just a generic function mathematically, I suppose "heruistic" to this concept
https://en.wikipedia.org/wiki/Heuristic_(computer_science)#:~:text=A%20heuristic%20function%2C%20also%20called,may%20approximate%20the%20exact%20solution.
Which again, is not a mathematical concept, but defined in the context of human problem solving. I'm not really sure though, but that question goes beyond just the "math" of it. Mathematically it is just a vector valued function.

>Blacks holes are locations in the universe where so much mass is concentrated in such a small space,the gravity is so strong that not even light can escape.

Physics says that black holes grow in size.
To grow it means to start occupying the space around the object.
Which means ,there has to be a limit on how much mass can be put in a point in space.

To solve the conundrum the naive way of approaching the problem of whats the smallest size of space we need to observe a star,know its parameters of mass,radious,and see what we get when it collapses into a black hole.

i dont have questions just wanted to know if my observations make sense.

Maybe I am missing something completely obvious but how do I calculate numerically stable the exponent x^(a/b) where x is a rational number and (a/b) is a small rational number but with very large numerator and denominator. Since a and b are coprime, I fail to see how I can avoid very large/small numbers since I either have to take the power or the n-th root first.
One approach uses x^r = exp(r*ln(x)) where exp and ln are approximated using the Taylor series. Are there other ways as well?

What is dimension or 4D rotation group?

>>11909535
Dimension of [math]SO(n)[/math] is [math]\frac{n(n-1)}{2}[/math]

>>11909286
Thank

Relativityfag here. A metric measures distance, right? Two points in spacetime are lightlike connected if their distance equals 0. How should I understand that their distance is 0? Or am I committing a conceptual fallacy?

>>11909709
Maybe I should be asking for clarity on an even more basic concept: Take the 2d Euclidean metric. In this case the inner product corresponds to the dot product, and the inner product of vectors (0,1) and (1,0) is zero. How do I see this as a distance? Clearly the points are separated right? What's the difference between the Euclidean distance squared as understood by high schoolers (it is sqrt(2) in this case) and the inner product of the two vectors?

>>11909728
Edit: it is 2, not sqrt(2) which is the norm.

>>11909728
>How do I see this as a distance?
The euclidean norm of x is the square root of <x,x>.

>Clearly the points are separated right?
Their distance is the norm of their difference. Clearly their distance is the square root of 2.

>What's the difference between the Euclidean distance squared as understood by high schoolers (it is sqrt(2) in this case) and the inner product of the two vectors?
The distance between x and y is ||x-y|| or, equivalently the square root of <x-y,x-y>.

>>11909729
>it is 2, not sqrt(2
Please look at the definition of the euclidean distance before attempting to understand relativity.

>>11909737
>norm of their difference
Thanks, I did not realize this. That basically means translating the system to put one of the points at the origin and then measuring the norm squared of the other point. That stills leaves me to ask what the interpretation of the inner product between the two is. What is the "physical" meaning of the number that I get when I give a metric two vectors? For Euclidean metric it can be seen as a projection but I'd like to know if there's a way to understand this for more nontrivial metrics.

>>11908729
Assume that [math]||y|| \geq ||x||[/math] for convenience. We have that, [math]|| y + x || \geq ||y|| - ||x||[/math]. This inequality is, in fact, achieved, so our problem becomes, at the worst case scenario:
[eqn]\frac{1 + a^2}{a + b^2} \leq (1+ a - b)^2[/eqn].
Where [math]a = ||y||[/math] and [math]b = ||x||[/math].
We set [math]a = b + c[/math], and rewrite the problem as
[eqn]\frac{1 + b^2 + c^2 + 2bc}{1 + b^2} \leq 1 + 2c + c^2[/eqn]
[eqn]1 + b^2 + c^2 + 2bc \leq 1 + 2c + c^2 + b^2 + 2b^2c + b^2c^2[/eqn]
[eqn]2bc \leq 2c + 2 b^2 c + b^2c^2[/eqn]
If [math]b \leq 1[/math], [math]2bc \leq 2c[/math], and we're done, because the other terms are nonnegative. If [math]b > 1[/math], [math]2bc \leq 2 b^2 c[/math], and we're also done.

>>11909747
Fucking typos.
We don't assume that [math]||y|| \geq ||x||[/math] for convenience, we do it because that case is trivial.
And the first inequality is [eqn]\frac{1+a^2}{1+b^2} \leq (1+a-b) ^2[/eqn]

>>11909742
>measuring the norm squared
No, just the norm.

>That stills leaves me to ask what the interpretation of the inner product between the two is.
Something like an angle. E.g. if they are orthogonal to each other their scalar product is zero.

>What is the "physical" meaning of the number that I get when I give a metric two vectors?
Stop throwing terms around. A metric is not a dot product. And for a general metric there is no geometric interpretation.

>For Euclidean metric it can be seen as a projection
I do not get what this means. A projection has to be onto something, you can define a projection onto something, through a metric, in general.

>if there's a way to understand this for more nontrivial metrics.
This basically means nothing. This is like asking "what is the geometric intuition behind a chair". The question does not make sense if you understand the terms involved.
For any given metric on some metric space there might be an intuitive way to understand it...

I don't know math, but how does pic related work?
>(2-1) + (2-2), (3-1) + (3-2), (4-1) + (4-2), (5-1) + (5-2)
1 3 5 7
>supposed fib number from 2 to 5
1 2 3 5

>>11909763
The "n-1" and "n-2" is referring to the index. You start with F_1 = 1 and F_2 = 1 and then you apply the formula, i.e.
F_3 = F_(3-1) + F_(3-2) = F_2 + F_1 = 2
F_4 = F_(4-1) + F_(4-2) = F_3 + F_2 = 2 + 1
and so on

>>11909749
>No, just the norm.
Excuse me.

>Stop throwing terms around. A metric is not a dot product. And for a general metric there is no geometric interpretation.
A metric is a rank (0,2) tensor, it takes two vectors and returns a scalar. For a Euclidean metric it works identically like a dot product, although this is only the familiar picture that I am trying to generalize from.

>I do not get what this means.
I meant projecting the one vector on the other, which is a picture of the dot product that is commonly taught to physicists. To be clear, I am not a freshman on this topic, just someone who is trying to grasp procedures that I have followed for a long time without questioning the deeper implications of the mathematical structure of the theory. In GR one works with metrics. What does a metric do? It is used to measure distance (among other things). If giving the same vector twice to a metric yields the norm, how do I understand the number that I get when giving it for two different vectors? I know from dot products its akin to an angle, perhaps it generalizes.

>This basically means nothing. This is like asking "what is the geometric intuition behind a chair".
A metric is quintessentially a geometric object, of course I can ask what the geometric relevance is of some calculation involving a metric. Or if not please explain me why, it just means I am all the worse mistaken and need to be corrected.

>>11909773
>A metric is a rank (0,2) tensor, it takes two vectors and returns a scalar.
That does NOT make it a tensor. A metric is NOT a rank (0,2) tensor. Please just Google what a metric is.
In some geometric setting you can locally describe a metric through tensors, but OBVIOUSLY the euclidean metric is not a tensor, as it is always positive, for example.

>For a Euclidean metric it works identically like a dot product
Absolutely not, except that the dotproduct, which is a multi linear mapping, so essentially a tensor, induces a metric. Which in no way makes a metric a tensor.

>I meant projecting the one vector on the other
What the fuck does that mean? Do you just mean the difference?

>which is a picture of the dot product that is commonly taught to physicists.
I highly doubt that.

>To be clear, I am not a freshman on this topic,
Indeed. It seems you have absolutely no understanding at all, I am sorry but you repeatedly equate "metric" and "dot product" which are VERY different things.

>It is used to measure distance (among other things).
A metric is the generalization of distance. It is a mapping from a product of vector spaces to the real numbers which obeys some axioms which intuitively any notion of distance should.

>If giving the same vector twice to a metric yields the nor
What????
How did you get to this, is this the first time you encounter a metric?
If you plug in the same 2 vectors into a metric you get ZERO. That is literally part of the definition.

>how do I understand the number that I get when giving it for two different vectors? I know from dot products its akin to an angle, perhaps it generalizes.
One idea you can apply generally if you want to understand a METRIC (WHICH IS NOT A DOT PRODUCT) is to draw the unit circle in that metric space, this can, for example, make the difference between the p-norms quite clear.

>A metric is quintessentially a geometric object
Each individual metric might be understood geometrically.

>>11908729
Look at x=0
Then look at x > 0 and see if that makes it easier or tougher for the inequality to hold
If tougher, look at x -> inf

>>11909854
>That does NOT make it a tensor. A metric is NOT a rank (0,2) tensor. Please just Google what a metric is.
Ah, I think we're at a misunderstanding: I am, like always in relativity theory, talking about a /metric tensor/ which is often abbreviated to "metric". (E.g. "Minkowski metric", "Fubini–Study metric", I did not make this up) I thought this was clear but apparently not. Lazy physicists' notation, doesn't matter, let's continue.

>Absolutely not, except that the dotproduct, which is a multi linear mapping, so essentially a tensor, induces a metric. Which in no way makes a metric a tensor.
I suppose the above clears this misunderstanding up as well. Even Wikipedia calls it a generalization of the dot product.

>What the fuck does that mean? Do you just mean the difference?
https://en.wikipedia.org/wiki/Vector_projection

>I highly doubt that.
In fact, projection of vectors is defined through the dot product, see the link above. From my own textbook by Giancoli:
>The scalar product can be interpeted as the magnitude of one vector (...) times the projection of the other vector..." (ellipses omit denotation)

So, now that that's cleared up: what geometric meaning one give to a metric tensor being given two distinct vectors, knowing that it returns the norm for two identical vectors, and that we understand the dot product to give the project length of one vector onto the other?

is the issue with blacks racial? arguing with a race realist right now and he brought africa as evidence that its in their genetics.

Could trilobites roll up into a ball?

>>11909854

Understand the importance of context. It sounds like you are talking about metrics as presented in the theory of metric spaces from Real Analysis. But the other guy, when he is talking about a metric as discussed in General Relativity, isn't talking about that exactly. Even in the math department, say in a Riemannian Geometry course, people use the term "metric" when they really mean a field of positive definite inner products. And Physics Guy probably doesn't even mean "positive definite" when he says "metric" !

So for this I need to find the probability that X is in A, and that will be the pmf for Y correct?

>>11905465
When you take math 1208 you will learn the answer is horseshoe-mathematics

>>11909418
There are other ways to compute logs, but x^y is typically computed as exp(y*log(x)). Calculating x^(a/b) as (x^a)^(1/b) is possible but slow for large a, although you might use it if you want a relatively simple algorithm with easily-understood error bounds. x^a is an integer and it's straightforward to find increasingly-accurate rational y s.t. y^b ~= x^a. (x^(1/b))^a doesn't offer any advantage over exp(log(x)*a/b) and typically requires vastly more internal precision to get the same overall accuracy (x^(1/b) is going to be 1.00000..0000<something> for large b).

>>11909978
>is the issue with blacks racial?
No single person on this earth things otherwise.
It is true pretty much by definition.

> evidence that its in their genetics
If you do not believe in evolution you can safely disregard that argument. Otherwise you are faced with the fact that literally every single human trait has to vary between population groups.

If you do believe in evolution, then your starting assumption has to be that differences between groups are most likely genetic, at least to a certain extent.
Just imagine, what would be the chances that there is no genetic variation in running ability between Africans and Europeans?
Of you want to tell me that the reason that blacks outperform whites in a large area of running contests is because of some mythical „black privilege“, or because of „institutional discrimination against whites“ I would call you insane.

>>1190990
Sorry for the misunderstanding.
I think the best way to imagine this is by comparing a sphere to the R^n.
On the R^n you have your usual dot product, which allows you to measure distances and „angles“.

Now imagine a sphere, as you can surely imagine on a sphere angles and distances behave differently, the distance, for example, is not the length of the line through the points but what you would get by following a path on the sphere.

The metric tensor allows you to pretend like you are in the R^n with the dot product *at a single point* and by e.g. integrating in some way you can actually measure distances on the sphere.

> In fact, projection of vectors is defined through the dot product
You can actually define projections without a dot product.

> and that we understand the dot product to give the project length of one vector onto the other?
This isn’t *exactly true*, but the point is that the metric tensor lets you pretend that you are actually working in R^n, while operating on complex geometric objects. It is something like a dot product, for any given point.

is it true that if I sleep sitting down i will
A)require less sleep naturally
B) have more vivid,or lucid,dreams

>>11909747
>>11909748
Great proof. Can't believe I didn't try reverse triangle myself.

>>11910421
No, but you'll have more lower back problems

>>11909763
This is probably the dumbest question I've seen in these threads.

>>11910457
I Reject your answer

>>11910234
Yes, the probability [math]Y=1[/math] is the same probability that [math]X \in A.[/math]

>>11910376
Yes I do believe in evolution and differences between races, his argument though had to do more with differences in the brain caused by nature and mine was about nurture (their family, the influences of culture, the amount of money they receive and other factors that plays out). The brain is complex and people shouldn't come to conclusions so quickly but the dude spams fbi statistics and mocks me.

why is the formula of the perimeter of a circle known as [math]2*r* \pi[/math] and not just as [math]d * \pi[/math] ?

Where is the smart questions thread?

>>11910609
Because it's the radius that is important when constructing a circle. A circle is by definition a set of points at fixed distance from some point, and that fixed distance is the radius.

>>11910609
2πr is more useful mathematically, as the formulae used to describe a circle normally use the radius (x^2+y^2=r^2, or x=r*cos(a), y=r*sin(a)). In engineering, it's common to use πd (or πd^2/4 for the area) as it's easier to measure the diameter of a physical cylinder than to measure the radius.

>>11910626
Ironically this one. Don't even bother to check the catalog of this hellhole of a board. Don't bother to make your own thread of you'll get memed hard.

Does anybody know something about the SISSA PhD school (Trieste, Italy)?

What do you think of it? Is it a good place to pursue a PhD? (Pure maths)

Hey smarties gentooman here. I need help with some math for my website.

Please give me all x and y where x and y are integers and
360 - 2x
------------
y
is also an integer. Thanks anons

>>11910725
>its easier to measure the diameter than the radius
How does one justify this

>>11911089
for example x = 30 and y = 5 is one combination. I could brute force this myself, but if there is some easy way to find it that you could do I appreciate it.

>>11910451
Thank you very much.
By the way, I noticed a bit later that an anon posted a proof that's pretty similar in /mg/.
>>11910626
Theoretically, you can just make a new thread for smart questions.
>>11911089
Unless I'm misunderstanding, you basically want to enumerate the divisors of all numbers smaller than 180 (because [math]360-2x = 2 (180-x)[/math], so you just repeat the list with every element multiplied by 2).
I doubt that there is some trickery solution to this, it is going to be some form or another of brute force.
Most efficient solution is probably going to be some Erathostenes sieve copy.

>>11911089
Call the whole thing z so you have a triple (x,y,z) with all three variables integer. Then zy = 360 - 2x. The right side is always an even number, so zy must also be even. What I'd suggest is take combinations of y and z that are even and then solve for x. There are three ways to make yz even: (y,z) = (even,even), (even,odd) and (odd,even); consider them all and you will exhaust every possibility.

>>11911188
But if x lies in exclusively in the range [0, 180] it might be better to use it as input rather than to fix it based on values of y and z so >>11911172 might have the better idea.

>>11911172
Alright. I could do it in max 360*360 iterations which should be instant in c++ though.

Consider the PDE Uxx=Ut with x in [0,1] and t>0 and U=U(x,t). (I think it's the heat equation). Using finite differences with deltaT=0.05 and deltaX=0.25 (that is the increment per step of t is 0.05 and the increment per step on x is 0.25) I need to estimate U(0.25;0.2)

I'm getting 0.1296, but apparently it's wrong. Could anyone compue it and tell me what they get? I tried writing a soultion in python but I'm still learning the basics.

>>11911229
Forgot to add initial and boundary conditions.
U(x,0)=sin(2*pi*x) for x [0,1] and U(0,t)=U(1,t)=0 for t>0.

If you got rid of all the atoms, particles, protons, quarks, whatever, everything out of like a mile cubed of space, what would be left? Would it still be black in colour? Is that the fabric of space? I feel like a retard kek

>>11910886
selfbump!

Has anyone here seen this atrocity before?

I am trying to wrap my head around complex analysis and i seem to be a brainlet. Could someone explain to me the Cauchy integral and why they caculate with circles? I don't get the circles especially

>>11911602
Theres like 7 different Cauchy Integral related things in Complex Analysis, you should be more specific
But the reason is that Circles are easy, and Closed Paths have an integral of 0 over a holomorphic function
And we can just use that calculation on a circle to another Closed path, just by moving around a bit
So we can prove stuff for all Closed Paths just by focusing on Circles and then extrapolating

>>11911602
[math]\frac{f(z) - f(a)}{z - a}[/math] is practically holomorphic, so it's integral around a circle zeroes.
This lets you equal [math]\frac{1}{2 \pi i} \int \frac{f(z)}{z-a} = \frac{1}{2 \pi i} \int \frac{f(a)}{z-a} = f(a)[/math], where the last equality follows from literally just computing it.

>>11911630
>>11911622
I still don't get the circle stuff. Why do we use circles? How do i relate circles to a complex number?

How would I go about answering this problem? My book isn't really clear
>Pay attention to both conditions for inference in the following box that summarizes the confidence interval: we must, as usual, be willing to regard the sample as an SRS from the population, and the sample must have both enough successes and enough failures. The condition on successes and failures ensures that the sample size is large enough to use the Normal approximation without knowing p.
How would I even go about defining these different measurements as failures or successes?

>>11911730
Because a circle loops once around its center.
Any other curve that loops once works just as well, but proving things for a circle is usually easier than for an arbitrary curve.
Because circles are neat and clean in polar coordinates.

>>11911730
in most cases you just use circles cause they are the easiest closed paths to compute

>>11911754
You can immediately rule out the second, third and fourth for being nonsense.
I'd then go with the first one, but it's fifty fifty. To be entirely honest, if I was writing an answer to the question, I'd say that you have to remove the outlier, and that it looks asymmetric so you have to test for that.

>>11911754
I'm guessing the reason I can't figure out how to define the measurements as a success/failure is because I don't know the population mean.

>>11911172
was getting connection issues before, but wanted to tell you that's a good proof because you deserve to hear it!

>>11911784
Thanks for the recommendations. It really doesn't seem to match the chapter, since the chapter is devoted to inferences about population proportions. Throws me off too that it mentions the chi-square distribution which isn't even part of the course (it is included in later chapters, but not part of the material we cover in this class). Makes me think there's just some obvious rule I'm meant to know to apply but reading the chapter doesn't seem to help much. Might just have to guess.

>>11911797
Apparently there's hints available for questions, which I didn't realize (the professor never told us, and this is literally the last question of my last assignment). The hint reads:
>When estimating a population variance, if the sample is drawn from a population that is normally distributed, the sample variance follows a chi‑square distribution and a chi‑square distribution is used to develop the confidence interval estimate.
So looks like the answer is 3 and it literally had nothing to do with the chapter, not sure how it got included.

>>11911814
So the hint was a trick. Fuckers. 1 was the right answer. Why would they consider that bullshit a hint?! Oh well. Guess I deserved to miss that one.

>>11911814
>three mentions chi-square
>hint mentions chi-square
>three is right
Bro I literally told you it was nonsense.

>>11911830
Yeah, thanks for trying to help, but you didn't tell me why it was nonsense so I assumed the hint was actually a hint instead of a misdirection. Guess I should be glad I didn't try to use those hints earlier. It isn't a big deal, I have a 98.44%. Thanks again.

>>11911844
in my country if you addressed a stranger who tried to help you in such a low minded and arrogant manner you would be torn limb from limb and the pieces of your body fed to pigs.

>>11911872
Thank goodness I'm not from your country. I said thank you, which part did you take insult in, that I didn't trust your help, or that it didn't really matter anyway?

>>11911886
>Continues to insult the person that helped him
>Socially retarded and misattributes identity
Just lol

>>11911905
Nigga, you say you'd tear me up and throw me to the pigs and expect me to be courteous? Get over yourself. It would be one thing if you actually had explained anything to me, but just telling me you know some answers are bogus and a best guess? Like I said, yeah thanks I appreciate the help, but didn't exactly inspire me with confidence, and ultimately it wouldn't matter because I'm finishing the class with an A and picking your answer based on your best guess wouldn't teach me anything. At least by getting the answer wrong I got to read the explanation for the solution, so I get to learn something.

>>11911919
Unbelievable that your mother didn't beat you every day for this kind of insolent attitude, make sure to punch her in the gut for me the next time you see her.

>>11911937
Thankfully my parents are loving and nurturing. But you can feel free to think about me the next time you're beating on a woman, for whatever that's worth. Or when you're feeding whoever hurts your thin skin to the pigs.

>>11911754
>>11911784
>>11911797
>>11911814
>>11911826
>>11911830
>>11911844
>>11911872
>>11911886
>>11911905
>>11911919
>>11911937
>>11911955
the absolute state of /sci/ anons lul

>>11911998
Kek. I got in contact with my professor and he told me that the question wasn't meant to be in there, so at least there's that.

I'll say this just once: people giving children the attention is no better. Don't ever engage flamefags again.

fuck online classes I really can't be helped for this one

>>11912208
Find the mass flow rate

I was applying eye drops minutes ago, sadly I miscalculated and actually two ended up running down my cheekbone into my lips, I don't know if it got in or not, am I fucked?
Felt a knot on my throat minutes ago and did some gargling
It's made of
chondroitin sulfate
sodium hyaluronate
excipient csp
HELP HELP HELP help

>>11912004
He was being rude and needed to be told that he is benefiting from the distance and enabling factors of online, in the real world there are consequences for being rude.

>>11912373
You're fine bro if it is okay for your eyes it won't hurt you, remember your eyes absorb stuff too

What's the best way to learn physics on my own? I want to try it before I sign up for uni because I don't know if I am big brained enough. I am especially interested in relativity theory and everything that is related and also quantum mechanics. I started with reading wikipedia articles and looking up stuff that I don't understand but I am not sure if this is the best way to learn.

>>11912471
Halliday and Resnick or Hugh D. Young's University Physics with Modern Physics. Try MIT Open Courseware's lecture series on Physics 8.0.1-8.0.3 with Walter Lewin its quite good. When you finish that you can move on to their lecture series on Quantum Mechanics. Check out the link in the sticky thread or in the first post in this thread for textbook recommendations and check out libgen.is for a way of acquiring the books.

>>11911548
That's just an n-dimensional ball isn't it? Set of points with distance smaller than 1 to the origin?

>>11911844
Bro, bro, I'm not angry at you for not listeniong to me or for getting the question wrong. If you went and marked five, that'd be fine.
I'm angry at you for asking for help on the internet, receiving help, and then ignoring help because of nonsense reasons.

>>11911756
>>11911755
I still don't get how i manage to turn complex numbers into circles

>>11912521
Yes, the notation tho.

>>11912875
You're confused about the notation? It says: the set of all x, such that (:) x is an element of n-dimensional real space, and (,) the absolute value of x is smaller than 1.

>>11912856
The thing about integration in the complex plane is that it doesn't matter what sort of path you take. If the path is closed the value of the integral is given by the enclosed residues. For convenience we will therefore always envision the path as a circle, even though it's precise form doesn't matter and we can deform it freely provided the deformation won't cross over any poles.

As to why one might be interested in calculating a loop integral in complex space: for example there exist functions that we wish to integrate over the real numbers. We can close the path in the complex plane and push the part that is off the real line out to infinity, where its contributions become infinitely small. I don't mean what kind of problem you're referring to when you say "turn complex numbers into circles".

>>11912919
>I don't mean
I don't see

>>11903346
Let's say I have a sink full of water. I then take a cup and completely submerge it in the water and point the bottom of the cup upwards. As I pull the cup out of the water, there's a bit of resistance but the water then falls out and the cup is now as light as a feather.

What is this effect called and how does it work?

>>11913146
> What is this effect called
Atmospheric pressure.

If you hold an empty mug upside-down, you have air on both sides of the base of the cup and the atmospheric pressure equalises. If you do the same with the cup of water, the pressure on the inside will be equal to the atmospheric pressure at the surface of the water in the sink, minus the weight of the water divided by the area. So you have a pressure differential. The net result is as if the weight of the water inside the cup is pulling the cup down. The water itself can't exert a downward force on the cup; that comes from the external air pressure. But the magnitude of the force is equal to the weight of the water above the surface.

If you did the same experiment in a vacuum, as you pull the cup out the water inside the cup would remain level with the water outside, leaving a vacuum at the top of the cup; as there's also a vacuum outside, there'd be no "pull". To get a vacuum inside the cup at atmospheric pressure the cup would need to be over 10 metres tall (1 atm = ~10332 mm H20).

>>11912727
That's okay, sorry to have made you angry. But obviously the question was over my head and outside the scope of the class, and the help didn't help me to understand the question at all. Thanks again for trying, but silver lining is I got to read a pretty detailed explanation (a full page of text) explaining the answer, so getting the question wrong was better for me in the long term anyway.

>>11913177
Interesting. So the water is able to stay inside the cup due to the pressure differential? One would normally expect to fall out with little issue but that's not the case.

Wait, so if the pressure in the cup would be the same at the surface of the water minus the weight and divided by the area, does that mean the pressure in the cup is lower which creates a kind of "sucking" force? It reminded me of how airplane wings use the low pressure area under the wings to generate lift.

>>11913273
Pressure increases with depth. For a given depth, the water inside the cup will be at the same pressure as the water outside. At the level of the surface of the (outside) water, the pressure will be equal to atmospheric pressure. So the water above that level (all of which is inside the cup) will be below atmospheric pressure.

It doesn't fall out because the atmospheric pressure pushing down on the (external) water exceeds the weight of the water inside the cup. Once the cup is removed from the water, air can bubble up into the cup and displace the water.

>>11913498
Ooohhhh that makes sense. Thank you for the information. It helped a lot.

How would you say something (Y) is X^2 greater than X, like 64 is 4^3 but 256 is 4^4, an exponent greater?

>>11914250
[math]Y[/math] is [math]X^2[/math] greater than [math]X[/math] if and only if there is are integers [math]m[/math] and [math]n[/math] such that [math]X = m^n[/math] and [math]Y= m^{n+1}[/math]?

>>11914332
So theres no word equivalent of this, like
Something is twenty times greater than something else ; Y = 20x. ?

Given an arbitrary derivable f(t) function, is there an h(x,t) function so that h=(dh/dt) - (dh/dx)(x-f(t))?

>>11914454
Same anon, should be (dh/dt) divided by a constant c.

>>11914417
Never heard of anything tbqh.

>>11914250
an order of magnitude greater?

Why doesn't someone just build the world's tallest skyscraper in a huge ravine, attached to the sides for support? Since apparently the limiting factor is foundation strength, you could spread the load over the cliff face. And it'd help block wind.

f(x, y)=x/y is a homogenous function of order 0, right?

>>11914929
because there's no motivation to build a gigantic billion-dollar pole in the middle of absolute nowhere wilderness

Consider a graph with [math]n[/math] vertices. Suppose we are guaranteed a particular structure of the edges: the maximal clique size is [math]d[/math], and furthermore there are [math]k[/math] unique maximal cliques in the graph. I want to cover the graph with as few such cliques as a polytime algorithm will allow me.

I know that the minimal clique cover is NP-complete and has a good number of heuristics for approximately solving. However, so far none of them work well enough for my purposes. I was wondering if there are any algorithms which can take advantage of additional known structure in the graph, like what I described above.

>>11912911
Are you a literary autistic my guy? it's a strange notation that's all he wanted to say

>>11903346
Hello /sci/, i'm doing a presentation on stochastic quantum mechanics and i can't for the life of me figure out how the path integral representation of the propagator of QM calculates Brownian paths. I was expecting to find a Gaussian distribution somewhere in the functional measure of integration but its not clear to me.. any help appreciated.

>>11915043
Yah.

>>11903346
Anybody knows statics here? I need someone to check if I got the reaction forces right.

This angled support is the problem I am not sure if I got it right and STRIAN doesnt allow for angled support simulation

Can anyone remember who it was who spilled what was at the time the country's (UK I think?) entire stockpile of a certain element (something radioactive iirc) onto a lab bench and then had to recover it by cutting up the lab bench and ashing the section with the spill on it?
Google is failing me.

Uk anon here, which degree has a higher pay ceiling. Electronic engineering or mathematics. Planning to move university after doing my year one in EE due to personal crap. I enjoy learning the physics part of the degree but I dont think I'd actually enjoy doing that as a job and main reason I did engineering was for maths. Looking at going to Brighton which has both departments but mathematics is such a open subject is there actual money to be made?

For my PChem/Quantum bros:
I remember working on a proof of sorts for electrons/subatomic particles being able to exist in two places at once.
It involved taking and integral of a wave function (I think) from -infinity to infinity to define the bounds as “the universe.”
The end result was supposed to be that the probability of finding the particle in both places was 1, or something like that.
Can someone explain what the probability you’re actually calculating is? I can’t put it into words since it’s been so long, and realized I forgot what the proof is even saying.

>>11915910
I don't get it. For a wave function, one of the basic axioms is [math]\int _{\Omega} | \psi |^2 d \mu = 1[/math], which should mean that it can't be in two different places at once.
It can have non-zero probability of being in two different places, tho.

>>11915951
My memory is sketchy, that’s why I don’t remember what exactly was being integrated, only that a probability was found, and found to be “1”. I was wondering what process would have led up to that, and what that actual probability signifies.

My assignment says

"Graph a level surface of your own design in GeoGebra. Do something interesting! It doesn’tneed to be completely crazy but see if you can come up with an equation in three variablesthat is at least more interesting than one of the six standard level surfaces we’ve covered inthis class. What equation did you settle on? Write it here!"

Can you think of an interesting function? We were told to replace the phrase "level surface" with just "surface".

>>11903346
Anybody know how to prepare myself for going back to UNI for prerequisites on math and physics after working for 2 years?

The probability of choosing a number, say 1, between 0 and 2 (all real numbers between 0 and 2) is 0. But, we could still theoretically get 1 from randomly picking, so shouldn't the probability be non zero? How does this work?

Why won't Geogebra graph this?
(sin^(2)(x))/(2)+(cos^(2)(y))/(2)+(z^(2))/(9)=9

>>11903398
>>11886601
Hoping my gene modding question gets answered

>>11916101
The axiom of probability you are referring to is that for disjoint sets [math]E_1, E_2, \dots,[/math], we get
[eqn]\mathbb{P}\left(\bigcup_{n=1}^\infty E_n\right) = \sum_{n=1}^\infty \mathbb{P}(E_n).[/eqn] The issue comes that probability splits up between countably many subdivisions, when the set of real numbers between 0 and 2 is an uncountable set, so this property doesn't apply.

>>11916101
https://en.wikipedia.org/wiki/Almost_never

>>11916141
Does it require sin(x)^2 etc rather than sin^2(x)?

>>11915126
Shitty idea as follows:
If you manage to find [math]k[/math] non-adjacent vertices, you know by default that each of them belongs to its own maximal clique that's disjoint from the remainder. I have no idea how fast you can do that, tbqh.
From there, you can construct the clique around each of those vertices by adding in the vertices that aren't adjacent to the other cliques (which can take you this far or that far depending on luck), and progressively delete cliques once they reach the element cap, until you need to finish it off the old way.

>>11916276
I'm having trouble getting it graph anything that isn't a basic quadric desu

>>11916276
for example, I can't graph a torus either. I'm not really familiar with Geogebra but this class requires it

>>11916101
a point has volume zero, but it's still a non-empty set. exactly the same thing.

>>11916414
>torus
https://en.wikipedia.org/wiki/Torus#Geometry

I'm reading this book on special relativity for beginners and stumbled upon this principle at the beginning see pic related.

How does this work with black holes? Does it mean that black holes aren't inertial references frames? Because it can't be confirmed that black holes are completely equivalent in physics considering no information about what actually happens in black holes exists

>>11916498
It's not just a situation of "there aren't black holes in special relativity", there isn't even gravity in special relativity.
You're asking "how does the model fit this in", and the answer is "it doesn't, it fails miserably."

>>11915484
just insert more resolutions of the identity bro lol

>>11916444
I know what the equation is ya dingus

i need a gradient vector for
[math]\frac{x^2}{3^2}+\frac{y^2}{3^2} - \frac{z^2}{5^2} = -1[/math]
So as a function this is
[math]z=f(x,y)=5\frac{\sqrt{x^2+y^2+9}}{3}[/math]

So the gradiant vector is just the partial derivative of x, y and z, right? Do I take the derivatives from the first equation of the quadric, or the second where x,y are variables of f? I believe gradiants are supposed to be perpendicular to the surface but I'm having trouble achieving this.

I want to look at amber insect inclusions up close. Can anyone recommend a suitable low-power microscope? It’s not something I would use a whole lot, so I’m looking for something that’s cheap and just good enough to use.

>>11916517
no

>>11916600
So I think I would do this as
[math]F(x,y,z)=\frac{x^2}{9} + \frac{y^2}{9} - \frac{z^2}{25} - 1[/math]
And then take the derivatives for x, y and for z I would take the derivative of z in my quadric surface as -2(f(x,y)/9.. So I have that for my gradient vector, but it's not perpendicular to the point on the surface.

>>11916634
your surface function is
[math]F(x,y,z)=\frac{x^2}{9}+\frac{y^2}{9}-\frac{z^2}{25}[/math]
the gradient is
[math]\nabla F=\frac{2x}{9} \hat{x}+\frac{2y}{9} \hat{y} - \frac{2z}{25} \hat{z}[/math]

applying to some phd programs next year. im ee/physics for bachelors and mostly thinking about applying for ee or physics for optics, photonics, lasers, etc research which seem to be mostly in physics or ee programs based on the university or the type of research, but there are also some universities like in UArizona or URochester that have a whole program for an Optics phd. would one of those be better or just a normal physics or ee program?

>>11916736
this is a good question to ask someone closer to those programs. I'd just email profs or the dept. at those schools because they'll advise you which way to go.

Heeyyyyyyyy I've got a question for y'all scientists:

h-index & similar metrics: based or cringe?

>>11917172
cringe. read their actual work and if you aren't competent or knowledgeable enough to judge its merit no artificial metric of relevance is going to change how worthless your opinion is. Especially cringe in fields like biology where cronyism is rampant.

>>11904430
Actually the American dollar translates into Euros quite well. He might end up paying 21-24k a year.

>>11915910
It's called a poor attempt to teach quantum mechanics by using tenuous classical intuition about silly things like "positions". It seems all you're recalling is the fact that a pure quantum state can be represented by a Hilbert space element with unit norm. The probabilistic interpretation is that I can rewrite [math]\langle \psi | \psi \rangle = 1[/math] by inserting a resolution of the identity (excuse my physics notation here, I'm too lazy to be mathematically precise): [eqn]\langle \psi | \psi \rangle = \langle \psi | \int | \xi \rangle \! \langle \xi | \, d\xi \, | \psi \rangle = \int | \langle \xi | \psi \rangle |^2.[/eqn] Hence [math]| \langle \xi | \psi \rangle |^2[/math] is a probability distribution wrt the measure [math]d\xi[/math]. The physicality of this is that the spectral theorem guarantees that every self-adjoint operator [math]A[/math] (which are the physical observables of quantum theory) can be written as [eqn]A = \int \lambda_{\xi} | \xi \rangle \! \langle \xi | \, d\xi,[/eqn] where the integral is taken over the spectrum of [math]A[/math]. For the physicists who don't care for the technicality of infinite dimensional unbounded operators, it essentially says that every observable has a complete orthonormal eigenbasis, and projectors onto that eigenbasis are precisely what describe projective, "Copenhagen" measurements with respect to measuring that observable.

What I'm trying to get at here is that the fact that you might have been taught about wavefunctions as taking [math]A[/math] to be the position operator and [math]\xi = x[/math] is largely irrelevant. It's a basis-dependent view of quantum mechanics and I can very well choose any other basis. It all depends on what observable you want to measure. The real crux of quantum mechanics is that states are self-adjoint, trace-class operators on a Hilbert space. The fact that we normalize their trace to 1 is because it's convenient and allows us to converse with experimentalists.

>>11917248
Not him, what's the best introduction to QM for someone with more of a math than physics background? Not all too familiar with functional analysis, but I have some good background in Lin Alg with proofs and discussion of the spectral theorem, Real Analysis, basic ODE's and Vector Calc. Looking for something erring more towards physical intuition but I can handle heavier doses of math if that's a requirement.

Alright guys Google isn't helping. Consider this your warming: this is a stupid question (pls no bully).

Can vaccines cause sterility? When the corona vaccine is out I'm 100% gonna get it (I'm not an antivaxer; I ended up getting the h1n1 vaccine) but I'm just a lil worried about the long term effects. I'm genuinely considering freezing some sperm and paying for sperm storage but if there's nothing to worry about wrt sterility then I won't freeze em.

>>11917274
https://www.youtube.com/playlist?list=PLPH7f_7ZlzxQVx5jRjbfRGEzWY_upS5K6
For books, Reed and Simon is the classic text but it's more of a reference than a pedagogical text. (Also it's old and expensive but just libgen it.) Maybe try Quantum Theory for Mathematicians by Hall (I enjoyed his Lie groups text, at least): https://link.springer.com/book/10.1007/978-1-4614-7116-5

Where do animal skeletons go (provided they aren't eaten whole, obviously)? I realise that flesh gets consumed by creatures which feed on carrion, or broken down by micro-organisms, but what happens to the skeletons? Why aren't forests and woodlands piled high with animal bones?

>>11917427
fungi and microorganisms eat the collagen. for the remaining calcium phosphate, soil is fairly acidic, so they break down over time just in the dirt.

>>11917461

Cool thanks bud

>>11917306
No, they generally just cause autism

Physics 1 material: I know you're suppose to integrate twice to get x(t) but I'm lost after that. I don't know how to solve for k when I don't even know t.

>>11917582
are you accounting for the fact its velocity is 0 when it reaches d=500m

>>11917592
Yes, but I don't see how that fits in.
>a(t) = kt^2
>v(t) = (kt^3)/3 + vi
>x(t) = (kt^4)/12 + vit + xi

>>11917624
you know vi and vf=0. isolate k in the 2nd equation and then solve for t in the 3rd eq, then plug in to get k. xi=0

>>11917582
So, the structure equations are:
[math]x(0) = 0[/math]
[math]x(f) = 500[/math]
[math]\frac{dx}{dt} (0) = 24[/math]
[math]\frac{dx}{dt} (f) = 0[/math]
[math]\frac{d^2 x}{dt^2} (t) = kt^2[/math].
Where [math]f[/math] is the final time.
From these we derive
[math]\frac{dx}{dt} (t) = 24 + \frac{kt^3}{3}[/math]
[math]x(t) = 24 t + \frac{k t^4}{12}[/math].
Using the equation for the velocity and the final velocity we get
[math]\frac{dx}{dt} (f) = 24 + \frac{kf^3}{3} = 0[/math] which we rearrange as [math]\frac{kf^3}{12} = -6[/math], which we plug into the position equation to get [math]x(f) = f(24 + \frac{kt^3}{12}) = f(24 - 6) = 18f = 500[/math].
The remainder is left as an exercise to the reader.

What's the difference between inverse trigonometric functions and reciprocal?

for example if [math]sin^-1 = csc[/math] aren't reciprocals = inverse?

>>11917670
no
arcsin is the inverse of sin
where "inverse" means "function inverse"

you know how sqrt is the "inverse" of x^2, its the same thing here, putting one inside the other cancels it out
sqrt(x^2) = x
and arcsin(sin(x)) = x
but sqrt(x) isnt the reciprocal of x^2

reciprocal is canceling out mutliplication wise
functional inverse is cancelling out input wise

and cosecant isnt arcsin, they look totally different pic related

>>11917686
thanks man

>>11917631
>>11917637
thank you thank you. oc pic unrelated.

>>11917306
Vaccines generally can't have any side-effects that the disease itself lacks, given that they're essentially crippled versions of the pathogen. Side-effects usually boil down to an immune response which is stronger than desired. AFAIK, sterility isn't a known side-effect of Covid (although admittedly we don't really know much about its long-term effects yet).

>>11917670
The inverse of a function "undoes" the original function. y=f(x) <=> x=f^-1(y), where f^-1 denotes the inverse of x. E.g. y=sin(x) <=> x=sin^-1(y) (alternative notations x=arcsin(y) or x=asin(y)).

The reciprocal is the multiplicative inverse. a*x=b <=> b*(1/x)=a, so 1/x is the multiplicative inverse of x.

>>11917686
arcsin isn't an inverse function

>>11917960
it literally is, stop being retarded

>>11917960
im not gonna explain about domain, range, injectivty and surjectivity to someone asking if inverse trig functions are the same as the reciprocal, you dumb autist

I've been given two definitions of field extensions: K is an extension of F if [math]F \subset K[/math], and K is an extension of F if there exists a monomorphism [math]\sigma : F \longrightarrow K[/math]. I have to prove the definitions are equivalent.

Getting from the first definition to the second one is obviously trivial, the inclusion map is injective so that's pretty much it. But I don't really understand the converse that much. If I know such a monomorphism exists, then obviously K doesn't necessarily have to be defined with elements from F and they could be disjoint sets such that F is just isomorphic to its image in K. So how do I go about constructing a field that is also a superset of F to get to the first definition? I assume the field I'm supposed to construct is isomorphic to K but also has elements from F, but I don't know how to get started with building these remaining elements

>>11918115
I would say you've done sufficient work. The field you would construct, which no one bothers to do, has elements in K but, the elements in the image, [math]\sigma(F)[/math] are replaced with their corresponding elements of F
And then you would need to define the operations depending on whether or not the elements are in K or F, and its really just a huge pain
you've got a subfield isomorphic to F, which is totally good enough

>>11918156
So I can just write that field extension as [math]F \cup (K \setminus \sigma (F))[/math] then? After defining the operations of course.
The idea seemed a bit trivial to me but honestly I don't see how else I would define it, so I guess that'll work then. Thanks

>>11917971
No it isn’t.
>>11918055
You’re lying to them though which is disgraceful.

>>11915655
bump

>>11917686
>arcsin(sin(x)) = x
wrong

>>11918409
You don't know whether it's wrong as the domain of x hasn't been specified. If the domain is [-π,π] (or any subset thereof) then it's correct.

Are there too many mathematicians or not enough?

>>11916498
>Does it mean that black holes aren't inertial references frames?

Yes. Even the surface of the earth isn’t. You feel the acceleration of gravity.

>>11916101
“Randomly picking” has no mathematical meaning.

>>11903346
Pls help with this

>>11918551

It's a question about how to memorise strings of letters where the letters are differentiated by upper and lower case.
(For example, in a YouTube URL.)

>>11918593
unspecified domain always means the largest domain for which the used expressions make sense

>>11918593
>If the domain is [-π,π] (or any subset thereof) then it's correct.
wrong

I absolutely love the taste of monster energy drinks but I'm very sensitive to caffeine.
Since Monster is run by complete troglodytes, is there a way to extract/get rid of only the caffeine and retain everything else?

>>11918965
Monster (and all other energy drinks) taste revolting, just drink them for long enough.
I am at a point where the caffeine does nothing for me and drinking the ones with sugar just makes me sleepy...

if i have [math]sin( \alpha )[/math] and i do [math]sin^{-1} (sin( \alpha )[/math] will i obtain the value of [math] \alpha [/math] ?

>>11919176
nope

>>11919176
Not necessarily. The thing is, sin doesn't actually have an inverse as it's not injective; there are multiple distinct x with the same value for sin(x).

sin^-1(x) typically means "the angle with the smallest absolute magnitude whose sine is x". For that definition, if α∈[-π/2,π/2] then sin^-1(sin(α))=α. Otherwise you'll get 2πk+α or 2πk+π-α for some integer k. But note that sin(sin^-1(x))=x is always true.