/sci/ - Science & Math

Physics questions:
>>11086125 (Depends on how much light they emmit. Asymptotics and all that. Not that I know any of the physics behind it.)
>>11086761
>>11098852
>>11098910
>>11099105

CS questions:
>>11085671 (Discrete maths isn't really a topic either.)
>>11096347

Maths questions:
>>11090407
>>11090418
>>11092047 [Sorta stupid.]
>>11096151 [I made a suggestion here >>11097734 that it was probably a graph theory problem. A friend of mine later told me that it's the problem of finding Hamiltonian paths in a Cayley graph. Since the Lovasz conjecture is still open, I doubt S_n gives a counterexample.]
>>11098047 followed up by >>11098052
>>11098651 followed up by >>11098689
>>11098330

Medicine questions:
>>11085791

Stupid questions:
>>11083804
>>11084613 [Sort of answered by >>11084622, but I'm not sure it counts.]
>>11089245
>>11092223
>>11092933
>>11093078
>>11095798
>>11096467 (Obviously go for the fucking books.)
>>11098966
>>11100283

are you guys close with your families? i feel like im ignoring them too much because of my study, is it normal to give up familial relationships in pursuit of knowledge?

>>11100942
No. Hug your Dad asap <3

>>11100962
i really cant make the time to go home right now, the soonest would be thanksgiving

can someone please help me with this? i dont understand what the given basis is for. the output space? it doesnt look like the basis for the output space. i am lost

>>11100979
Looks like they mean for both the domain and the codomain
Why would it not look as a basis for the output space? What would you expect a basis for the output space to look like?

>>11101024
>codomain
thats the term i was looking for
>Why would it not look as a basis for the output space? What would you expect a basis for the output space to look like?
just looking at the transformation, it doesnt appear to be a basis for the codomain

>>11101042
B is a basis for [math] \mathbb{R}^{2} [/math], and that is entirely independent of the transformation T you are given. In this particular case I'm pretty sure the image of the basis is a basis too, so I'm not sure I'm understanding what the issue is

>>11101078
the issue is im retarded desu
ill try this one more time before i give up

>>11101084
Maybe explain what concept you have of bases and writing linear maps with respect to different bases, so we can tell what's wrong

>>11101097
a basis of a space is a set of linearly independent vectors such that every vector of the space can be expressed as a linear combination of the vectors in the set
i have no clue what it means to write maps with respect to bases

>>11100942
Yep, even after coming out.
>>11098852
What do you mean by "on the physical level"? Particles are by definition representations of gauge groups. There is literally nothing more to "particles with spin" than "irreps of [math]SU(2)[/math]".

>>11101106
Yes, the basis is linearly independent and it generates the vector space. As you can see it is unrelated to whatever linear transformation you define to/from it.

The latter means that the image of the basis vectors are written in terms of the new basis, which is possible because by definition the basis of the codomain generates every vector in it (so it generates said image too). For example, if the image of a vector under the linear transformation were (2,0), and I wanted to write it using the basis (1,1) and (-1,1), I could write it using the coefficients 1*(1,1)+(-1)(-1,1). Notice that the vector of coefficients of (2,0) with the canonical basis is still (2,0), since it can be written as 2*(1,0)+0*(0,1). But with respect to my new basis, it's vector of coefficients become (1,-1).
Since you want the matrix of the linear map with respect to the basis B, you have to first:
1. See how the basis vectors transform under T
2. See what the coefficients of the images of these vectors become in the basis B of the codomain
The matrix you want can be obtained with these coefficients.

>>11101173
alright, i think im understanding it now
ty, anon

Can any physicist explain how water is able to project this bottle of coke further as opposed to some solid surface? How would this apply to other liquids with different viscosity/density?
https://twitter.com/reiwa_chann/status/1188319616654688257?s=21

>>11100942
>>11100963
i stay pretty close by phone even though i'm so far away 10 months of the year
i recommend video calling so they can see your face and you can see theirs, feels really nice and refreshing for me

>>11101366
me thinks there is somekind of trickery. that doesn't make any sense.

Are the determinants of the two matrices the same? The LHS has an a^2, but the RHS only has 1 a.

>>11101562
[math] |kA|=k^{n} |A| [/math], where A is a nxn matrix.
So no, they are not the same

How do you pronounce this? Is this Hebrew letter or something?

>>11102391
Thats the greek letter rho, /varrho is latex

>>11100698
chemistry
The equilibrium constant (K) for the gas phase reaction 2 NH3 N2 + 3 H2 is 3.0 x 10-3 at some temperature. The reaction is started by placing a 0.040 bar sample of ammonia in an empty one-liter flask. When equilibrium is established how much N2 is present?

Question 3 options:

) 0.010 bar


) 0.016 bar


) 0.019 bar


) 0.013 bar

>>11100698

Anyone realize late into undergrad that they want to take pure maths more seriously? I have been developing this interest in computation mathematics, solving HackerRank Project Euler problems, and I want to definitely study more, especially since I have access to so many math classes and professors.

I would say my biggest weaknesses right now are a lack in any upper-division math. I'm definitely going to take a proof-based LinAlg or Abstract Alg class next semester. But, I kinda just want to read some anecdotes, if anyone has some to share, about going from just average in math to really practicing and getting good at it later in life.

brainlet here, how do I integrate (1-x^2)^4 without writing it out?

Can anyone help with or direct me to reading on track transition curves? I'm working on a matlab script to evaluate an arbitrary roller coaster and am struggling to find a good way to define its path spline. My first though was to use cubic polynomials, constraining first and second derivatives at the knots, but it seems that will require buying a package for no guarantee of success (or going full numerical analysis and implementing an algorithm myself). Now I'm looking at euler spirals but I've never worked with fresnel integrals before and don't really know how to solve them to endpoint constraints of position and curvature, nor how to implement them in 4 dimensions (x,y,z,theta). Is there something else I should consider? I'd go roller coaster tycoon/parkitect and just define pieces to be scaled as necessary but then I have jerk problems and I'm not trying to make Arrow or Vekoma coasters up in this.

>>11092047
>Curious mathlet here, is tgere a way to do pic related with a single equation?
Yes:
((x-0)^2 + (y-0)^2)((x-0)^2 + (y-1)^2)((x-0)^2 + (y-2)^2)((x-0)^2 + (y-3)^2)((x-0)^2 + (y-4)^2)((x-0)^2 + (y-5)^2)((x-1)^2 + (y-0)^2)((x-1)^2 + (y-1)^2)((x-1)^2 + (y-2)^2)((x-1)^2 + (y-3)^2)((x-1)^2 + (y-4)^2)((x-1)^2 + (y-5)^2)((x-2)^2 + (y-0)^2)((x-2)^2 + (y-1)^2)((x-2)^2 + (y-2)^2)((x-2)^2 + (y-3)^2)((x-2)^2 + (y-4)^2)((x-2)^2 + (y-5)^2)((x-3)^2 + (y-0)^2)((x-3)^2 + (y-1)^2)((x-3)^2 + (y-2)^2)((x-3)^2 + (y-3)^2)((x-3)^2 + (y-4)^2)((x-3)^2 + (y-5)^2)((x-4)^2 + (y-0)^2)((x-4)^2 + (y-1)^2)((x-4)^2 + (y-2)^2)((x-4)^2 + (y-3)^2)((x-4)^2 + (y-4)^2)((x-4)^2 + (y-5)^2)((x-5)^2 + (y-0)^2)((x-5)^2 + (y-1)^2)((x-5)^2 + (y-2)^2)((x-5)^2 + (y-3)^2)((x-5)^2 + (y-4)^2)((x-5)^2 + (y-5)^2)=0

>>11102551
ICE table

quadratic formula, basic stoichiometry.

This isn’t a hw thread

>>11102605
| means "divides".

>>11102570
You don’t like or care about pure maths you’re just an easily distracted midwit who enjoys intangible ideas and computing nonsense to pass the time. If you can read a pure maths text for over 8 hours without losing focus and spend days thinking over proofs while doing other things than math explicitly you like math. Lots of people become enamored with things when they have no exposure to their inner workings, and just as many without having to sink any time into developing the skills to do the actual work. Pure maths requires years of training to be competent at.

>>11102614
Trig substitution, and/or reduction formulas.

>>11102726
It does? I've literally never seen that notation. Oh well.

>>11102727

Interesting take!

How do I find the limit of this?

tradie here. im trying to come up with a way to markup materials in a consistent manner. Yet it can't be a consistent % because I can't markup a blower motor by 50% because a customer can just buy a new furnace at that point.

I was trying to remember what I know about functions. What I want is something that looks like f(x)=1/x so low cost items have higher markup, and high cost items have lower mark up.

So for items that are a dollar I might charge $10, but for a $1000 item I might only charge $1150. Items under $100 i usually charge 100% markup. Just to give you a few data points. How can i make a nice "rational" function so I can give my customers consistent (linear) markup? I've been playing around in excel but i don't exactly understand how to transform the graph properly.

>>11102989
For starters, can you see that it's the limit of
[math] \frac {1} {2} n^{1/n} [/math]?

>>11102925
You can just aggress in response to being insulted instead of passive aggressive sniping while in half-retreat, faggot.

I'm asked to find h(+infty) and prove that h([0, infty)) is superset of [0, 1], how can I do it? Can I do it with the limits at infinity and 0+?

>>11103059
I assume he wants to draw attention to his post without engaging in shit flinging.

>>11102570
I'm a bit confused what you like - pure math or hacker challenges?

>>11103036
Look into something like the logistic function (or more interesting 1 - the logistic function), you obviously need to scale it correctly.

>>11103067
hints

>>11100698
I've got a stupid question but don't want to make a whole thread for it.

It's about light speed communications and time dilation.

When would people on earth get your radio messages if you are traveling at some percentage of light speed if time is passing at different rates for both of you???????

>>11103067
>how can I do it? Can I do it with the limits at infinity and 0+?
Yes, after you computed both of these limits you can use that your function has a certain property on that interval (which?) and use it to show that if it attains some values, it also attains all values between them.

>>11103105
Say Bob is moving at speed
beta=v/c,
with v<c, relative to you.

When T=1 second has passed for him, then on your "now" plane,
T · gamma = T /sqrt(1-beta^2) > T
seconds will have passed for you.
He's at the position
x = T · gamma·v/c
away from you at this point.

If he sends a signal, then the signal will just travel at the speed of light towards you.

So if he's controlling sending away messages and does it at 1 second interval (on his clock), then from your perspective it will seems like he's taking more than 1 second to send each beep.

>>11103059
I got nothing to prove, man. I’m just here to learn.

>>11103070
I guess I’m also a little confused in that regard. I enjoy computational math challenges where you have to consider properties of the problem and derive a solution like, for example, if given a set of points find the maximum number of rectangles by calculating line segments that share x and y ranges.

At the same time, I’ve done a bit of discrete math and see a lot of the fun in going further into maths and have been doing more advanced lin alg recently because of an ML class. I guess I’m at that point where I want to dedicate time into going further into mathematics and have been actively doing so, just curious about where other people lie in the journey.

>>11103105
Say Bob is moving at speed
beta=v/c,
with v<c, relative to you.

When T=1 second has passed for him, then on your "now" plane,
t = T · gamma = T /sqrt(1-beta^2) > T
(e.g. t=1.25*T, see pic)
seconds will have passed for you.
He's at the position P,
x = T · gamma·v/c
away from you at this point.

If he sends a signal, then the signal will just travel at the speed of light towards you (45° lines in the diagram).

So if he's controlling the beeper, sending away messages and does it at 1 second interval (on his clock), then from your perspective it will seems like he's taking more than 1 second to send each beep.

>>11100698
ok guys please help ke out here. so you are reading a math textbook and look at a theorem then you think ‘hmm sounds interesting’ then its just one cimplicated ass stinkin calculation

or in general when you look at physics derivations or calculation examples. how tf are you supposed to study these portions of the text???

>>11103180
thanks anon

>>11103205
Look through it to make sure there aren't any tricks or important ideas involved. If there aren't, it's usually safe to file the proof away as "ugly routine calculation" and move on.
It's a good idea to occasionally force yourself to do calculations like this yourself to safety-check that they really are _routine_ calculations you're skipping.

Was trying to wrap your heads around differential equations particularly difficult for any of you guys?
Usually I can at least catch the gist of what's going on when it comes to math I haven't yet been introduced to. With diffeq all bets are off

I can't make head or tails of the notation, I can't understand what people are doing to solve them, hell, I'm not even sure if you're supposed to be able to visualize what the solution to a differential equation is supposed to look like like you can with a derivative/integral of a graph
I can't even really say I know what one is beyond the basic "Equation with one or more derivatives of a function in it". It all seems hopelessly opaque, none of the functions seem to be functions that I'm familiar with. I really, really want to learn about them, they're so damn interesting, especially dynamical systems/chaos theory and how they relate to analog computers and electronic circuits

is there some key insight I'm missing? Did anyone else here have this much trouble?

[math](\sqrt-25)(\sqrt-49)=-35[/math]
why wouldn't
[math](\sqrt-25)(\sqrt-49)\rightarrow\sqrt(-25*-49)[/math]
wouldn't that just give me 35? can i not do any operations with imaginary numbers without distributing out the -1? Sorry for the stupid question.

>>11103229
sorry, first time using the math editor

>>11103229
The problem is that -25 has two square roots; 5i, and -5i.
The equation [math]\sqrt{-25}\sqrt{-49} = \sqrt{(-25)(-49)}[/math] is only true if you pick the root with the negative coefficient for one of the terms and the positive root for the other, which would be odd since the radical sign is generally assumed to mean take the root corresponding to +i.

tl;dr no, you can't multiply under the radical as carelessly as you can with real variables.

>>11103245
I don't understand, but I know not to make this mistake in the future regardless.

>>11103245
>>11103272
Ah fuck I get it now. Fuck... thanks anon

>>11103217
thanks anon. will do

>>11103217
>>11103344
oh yeah but how do you know which tricks are important?

>>11102454
Thanks friend

Is eternal inflation truly the expected outcome of inflation theory. Or is inflation theory not understood enough to come to that conclusion?

>>11101491
What makes you think this? He tries it at multiple angles.

cheapest way to bleach bones without making them brittle?

>get an email telling me to sign up for a PhD program in maths
>applications literally close tomorrow
>lads shouldn't even have my personal email
>it doesn't even make sense to send me that shit
>the program is from a subdepartment of one of my country's top three unis
>insititutional email, too
Did one of you somehow dox me and sent that as a joke?
>>11100942
Yeah.
>>11103222
>is there some key insight that I'm missing
Yes, but it's hard to explain. The function just passes through some point (possibly with a certain speed, etc) and then "goes with the flow." That is, the necessity of satisfying the equation "drags" the function around, even though bifurcations and the like might show up. Visualizing it physically in terms of a particle attracted by various forces is very convenient and recommended.
For ODEs, that is. This intuition largely becomes inconvenient for PDEs, except in some stuff like the method of characteristics.
>>11103349
The non-trivial ones.

S aSb | aaS | Sbb | ε


How can I write this in set notation?

Brainlet math question:
You have a divergent series, you assign it to a variable x.
You multiply both sides by 2:
2x = 2 + x
Somehow x is supposed to equal 2, meaning the series sums to 2. However my brainlet equation tampering results in x = 1 - x/2.

>>11103205
Try reading Lehmann's book on many-body QM it's full of half-page computations dressed up as "lemmas".
>>11103784
Who knows, maybe the department is desperate. It does seem uncommon to go straight for the personal email and on such short notice.
If they're desperate then you can probably ask for an extension. These deadlines are never really set in stone (unless the central admin wants it to be).

>>11103088
Why the logistic function?

>>11104244
Because it has the properties you desire, correctly scaled it can model higher markups at lower prices and lower markups at higher prices.
Any sufficiently smooth, monotone function with values between 0 and 1 could be chosen as well, but the logistic function was the first which came to my mind.

>>11100698
Why is 90sin70 being subtracted?
Is it because the value of sin in the 4th quadrant is negative?

>>11103222
> I can't even really say I know what one is beyond the basic "Equation with one or more derivatives of a function in it".
But that's exactly what a differential equation is. No more, no less.

> is there some key insight I'm missing? Did anyone else here have this much trouble?
The concept is straightforward enough; it's an equation where the "variable" being solved for is a function. The difficulty comes in actually solving them, which largely comes down to memorising a bunch of cases with known solutions and recognising when your equation fits one of those cases.

>>11104865
Yes

>>11103636
because there is no physical reason that should be the case

>>11103222
It feels tacky answering questions in this thread with "watch the 3blue1brown video on it" but it really helps to have nice visualisations and they're pretty good here. I came to DEs from physics and found the physical intuition was really helpful. Maybe this bites you in the ass later but I never progressed past physical systems to wild nonsense. https://www.youtube.com/watch?v=p_di4Zn4wz4

Why are the vertical components being resolved with cos instead of sin?

Is the derivative of y=f(x) y'=f'(x)/2sqrt(f(x))?

Is the derivative of y=sqrt(f(x)) y'=f'(x)/2sqrt(f(x))?

>>11105510
>Why are the vertical components being resolved with cos instead of sin?
I remember you asking this problem some weeks ago. Just draw the triangles instead of using your memorized formulas.

>>11105525
But using the tip to toe method and sin rule feels like cheating

>>11105528
Ok, let's see if this explanation convinces you: cos(x)=sin(90º-x). If you consider the two non-right angles of a right triangle, the cosine of one is the sine of the other. That's why they use cos instead of sin.

Really stupid question but why and how does it work to translate a matrix into a vector.

Lets say I have 4 matrises that make up a basis in R4, I get that these have 4 dimensional vectors that are equivalent to each. But why can I just make a vector that consists of the entires on this from (M[1,1],M[1,2],M[2,1],M[2,2])^t ?

>>11105528
>using elementary geometry is cheating

>>11105562
Found it on wikipedia, nvm. Never saw it in my linear algebra book.

https://en.wikipedia.org/wiki/Vectorization_(mathematics)

alright guys, i'm 19, have adhd and my academic level is one of a 8 years old. I, somehow, seem to fail at everything, way below the average human but not deep enough to be considered autistic. how do i become a genius?

PLEASE HELP ME
a) Consider the following functions:
>pic related
>mit = with
Specify fi for each of the above functions. if it's injective and if it's surjective. For each non-injective function fi, enter two elements x, y ∈ Def(fi), so that x 6 ≠ y and fi(x) = fi(y) applies. For each non-surjective function fi, specify an element x from the image area, so that x ∉ picture (fi) applies. You do not need to justify your answers.

b) Be A, B and C any quantities and f : A B and g : B C any functions. We define the function h: A C as the concatenation of f and g through.

h(a):= g(f(a)) a)) f.a, a ∈ A

I) Show: If f and g are injective, then h is injective.
ii) Disprove: If h is surjective, then f and g are surjective

>>11105783
>b) Be A, B and C any sets

>pic related "und m ist ein primzahl = and m is a prime number"

>>11105744
You wont become a genius. Set more realistic goals for yourself or you are guaranteed to fail.

>>11103628
Bump

I'm quite lost at spin quantum mechanics, I get the procedure but I fail to see the big picture. We have a finite (say N) dimensional Hilbert space. We decompose it into eigenvectors of Jz. We assume there is a highest eigenvalue m, and using commutation relations we construct the whole ladder of quantum numbers (-m, -m+1, ... , m-1, m). That's the gist of it. But what I fail to see is what this finite-dimensional Hilbert space looks like. The J-matrices are still 2×2, and the vectors have 2 entries, so in what sense are states |j,m>, |j,m-1>, etc. different? In other words, what does such a state look like explicitly (not indexed by its quantum numbers)? Perhaps I'm rather blind to what the Hilbert space in the first place. How can I see that the Hilbert space is spanned by N states in the first place? I don't see how N vectors are needed to span the space when they have only 2 entries (naïveté would suggest you only need 2). Is there a degree of freedom in the theory that I am overlooking?

Anyone wanna help me with some algebra?
How do I solve for these two unknowns? I could do it if I were to seperate them as z = x + y where I solve for x and y, but they're being multiplied.

>>11105842
When you write down the Pauli matrices you're selecting a specific irrep of [math]\mathfrak{su}(2)[/math] labeled by a half-integer (for spins) or an integer (for orbitals) [math]j[/math]. Think about what this [math]j[/math] is when you say "each [math]J[/math]-matrices are [math]2\times 2[/math]".
Note that each irrep gives a homomorphism [math]SU(2) \xrightarrow{\rho_j} GL_{2j-1}(V)[/math] into a [math]2j+1[/math]-dimensional vector space because we have [math]2j+1[/math] roots distinguished by the Weyl group of [math]\mathfrak{su}(2)[/math]. This leads to, given a unique singular highest-weight vector [math]|0\rangle[/math] with [math]J_-|0 \rangle = 0[/math], [math]2j+1[/math] number of linearly-independent eigenvectors with weights [math]|m| \leq j[/math]. How many of these eigenvectors do you think you have when [math]j[/math] is such that "the [math]J[/math]-matrices are [math]2\times 2[/math]"?

Here's something easy I'm probably missing
The equation is: 10^x + 1 = 0.8(10^x + 2)
How to simplify that to 10^x = 3?

>>11105880
Just label 10^x something else, like a, and it should be obvious enough.

>>11105893
10^x + 1 = 0.8(10^x) + 1.6
10^x = 0.8(10^x) + .6
10^x - .8(10^x) = .6
.2(10^x) = .6
10^x = 3

Yep. I feel like an idiot.

>>11105796
my only realistic goal could be becoming assistant manager in a McDonald

>>11105876
I lied. That was pretty easy.
My real question is this.

>>11105879
Thanks, I've got several questions.
>When you write down the Pauli matrices you're selecting a specific irrep of su(2)
There is one represenation that's automatically obtained through the similarity transformation, the others are obtained by adding a half-integer m to the exponents. Then we can undo the similarity transformation and carry this integer back to the group SU(2) which acts on generic states. Is this a correct way to think about it? I'm trying to make the quantum number manifest in the original problem.

>we have 2j+1 roots distinguished by the Weyl group of su(2)
This is simply a more neat way of identifying the size of the ladder, right? I've worked with roots and weights before, actually. Yes indeed, there are 2j+1 orthogonal vectors for a given quantum number j. But I still don't know what they look like explicitly, it seems to contradict the idea that the vectors have 2 entries.

I believe it's so that we need the different representations (the different values of m) in order to decompose our reducible matrix into irreps. But said reducible matrix is a Pauli matrix, right? Seems like there is little to reduce, especially if we consider that we have an infinite number of irreps for it to decompose into.

>>11105976
>Is this a correct way to think about it?
It's [math]one[/math] way to think about it, though not essentially correct. What's correct from an a priori way is to characterize the irreps with the root system and Weyl groups, then construct singular vectors. This is what ladders are doing, then you find the matrix elements from the action of the ladders on the singular vectors.
>But I still don't know what they look like explicitly
Again, given the action of the ladders on the [math]2j+1[/math] vectors, you can write down the matrix elements. The former characterizes the latter, not the other way around.
>it seems to contradict the idea that the vectors have 2 entries.
For which [math]j[/math]? If [math]j = 1[/math], say, are the eigenvectors still 2-component (as in, are the Pauli matrices still [math]2\times 2[/math])?
>different representations (the different values of m)
Irreps are labeled by [math]j[/math], not [math]m[/math]. The [math]m[/math]'s label the singular vectors in a given irrep.

>>11105580
>Never saw it in my linear algebra book.
because it's not really interesting (at least not in pure math). you just write numbers into a long column instead of a box.

How do I show that [math]1-x^n=(1-x)(1+x+x^2+...+x^{n-1})[/math]

i don't understand proofs at all

>>11105994
>What's correct from an a priori way is to characterize the irreps with the root system and Weyl groups, then construct singular vectors
Is a singular vector just a state |j,m>?

>Again, given the action of the ladders on the 2j+1 vectors, you can write down the matrix elements.
I suppose I should repeat the last question in my previous post, lest I risk going in circles: how come our 2×2 matrix group is being decomposed into a direct sum of an arbitrary (infinite?) number of 1×1 irreps? The |j,m> vectors are a decomposition of our original generic state (or what else is their relation?), but I don't see how this works. Suddenly our 2×2 Jz matrix has become (2j+1)×(2j+1).

>Irreps are labeled by j, not m. The m's label the singular vectors in a given irrep.
I may have been confused by my book (Jones). It suggests that "irreps are labelled by the integers [math]D^{(m)}(\phi) = e^{- i m \phi}[/math]" (this is on SO(3) but all the same). Maybe it is tacitly assumed there is a maximum j, and |m| <= j, because he proceeds calculating [math]\langle \chi^{(m)}, \chi^{(m') = \delta_{m,m'}}. I'm unsure about this.

>>11106092
I screwed up the latex.
[math]\langle \chi^{(m)}, \chi^{(m') = \delta_{m,m'}}[/math]

>>11106092
I screwed up the latex again.
[math]\langle \chi^{(m)}, \chi^{(m')}\rangle = \delta_{m,m'}}

>>11106092
Fuck me.
[math]\langle \chi^{(m)}, \chi^{(m')}\rangle = \delta_{m,m'}[/math]

>>11106064
Induction dumbass

>>11106064
but that isn't a true statement?

>>11106064
Sum over all x^k from k=0 to 0 is 1

Sum over all x^k from k=0 to n+1 is the sum over all x^k from k=0 to n with x^{n+1} added.
This is
(1-x^n)/(1-x) + x^{n+1} = (1-x^n+(1-x)*x^{n+1})/(1-x) = (1-x^{n+1})/(1-x)

>>11106092
>Is a singular vector just a state |j,m>?
Yeah.
>Suddenly our 2×2 Jz matrix has become (2j+1)×(2j+1).
Please don't conflate the rank of the matrices in the Lie group with the rank of the irreps of the Lie algebra. You have already picked a specific [math]j[/math] (labeling the latter) when you say the Pauli matrices are [math]2\times 2[/math]. I don't know why you're so hung up on the rank-2 case, perhaps you've never seen Pauli matrices for higher spins? Spin-1 particles have rank-3 Pauli matrices because the [math]SU(2)[/math] irrep space you're considering has rank 3, [math]not[/math] that you're considering [math]SU(3)[/math] or any other Lie group.
> It suggests that "irreps are labelled by the integers
Different irreps of [math]\mathfrak{su}(2)[/math], and hence that of [math]SU(2)[/math] itself, are characterized by the dimension of the representation space. In other words, as the Wigner matrices [math]D_{j,m}[/math] are irreps of [math]SU(2)[/math] (the argument in the [math]\exp[/math] are irreps of [math]\mathfrak{su}(2)[/math]), its rank stays constant within the same irrep. Since the rank of [math]D_{j,m}[/math] does not change as [math]m[/math] varies, this should suggest to you that [math]D_{j,m}[/math] lies in the same irrep of [math]SU(2)[/math] for fixed [math]j[/math] but different [math]m[/math]. Again, this means that [math]m[/math] labels not the irreps but elements of the "singular vector basis" therein.
The [math]\chi[/math]'s are group characters, which are also labeled by the irrep [math]j[/math]. We don't really need it for this discussion though since they can only get you the ranks of the irrep spaces, not details like the singular vectors.

>>11106064
All of the answers so far are way too complicated for this simple property. All you need to do is to distribute:
(1-x)(1+x+..+x^(n-1))=
1+ x + x^2 + ... + x^(n-1)
- ( x + x^2 + .... + x^(n-1) + x^n)
=1 - x^n.

A way you could have discovered this property yourself, which actually also generalizes to many other problems is the following:
First, notice that with x=1, 1-x^n becomes 0.So 1- x^n has 1 as a root. That means it's divisible by (1-x). Now use the long division to find that the divisor is 1+x+...+x^n-1. Hence you get your property again. Same method can be used to factorize, say x^n + x^(n-1) - 2. You notice that x=1 is a root, and use long division to divide it by (x-1). Try finding a nice formula for it yourself and then prove that it's true.

>>11106169
I think we're getting down to my real issues here
>Spin-1 particles have rank-3 Pauli matrices because the SU(2) irrep space you're considering has rank 3, not that you're considering SU(3) or any other Lie group.
You got me, I was thinking of higher spin Pauli matrices as belonging to another group. We have established they don't, they are precisely the higher j's. I've all this time been thinking about j=1/2, where the Pauli matrices are 2×2 and the rank of the irreps is also 2. A large portion of my confusion has been cleared up. But SU(2) is still a group of 2×2 matrices, so I still don't see why we are now considering larger matrices. You've probably explained it already; I'll take some time to think about it for myself. Thanks for your patient responses so far, I need to leave but I'll be back at a later time.

>>11106064
Just expand the RHS then simplify. (a-b)*c = a*c-b*c.

(1-x)(1+x+x^2+...+x^{n-2}+x^{n-1})
= (1+x+x^2+...+x^{n-1}) - x(1+x+x^2+...+x^{n-2}+x^{n-1})
= (1+x+x^2+...+x^{n-1}) - (x+x^2+x^3+...+x^{n-1}+x^n)
= (1 + (x+x^2+...+x^{n-1})) - ((x+x^2+x^3+...+x^{n-1}) + x^n)
= 1-x^n

>>11106064
Look at what (1-x) does to some term in [math]1+x+x^2+x^3...x^{n-1}[/math]

In particular consider that
[math](1-x) x^k = x^k - x^{x^k+1}[/math]
[math](1-x) x^{k+1} = x^{k+1} - x^{x^k+2}[/math]

The second term from the first expression cancels the first term of the second expression.
actually all terms will cancel like this. Except for the very first and very last, because they don't have anyone to pair up with and cancel.

>>11106064
Look at what (1-x) does to some term in [math]1+x+x^2+x^3...x^{n−1}[/math]
In particular consider that
[math](1−x)x^k=x^k−x^{k+1}[/math]
[math](1−x)x^{k+1}=x^{k+1}−x^{k+2}[/math]
The second term from the first expression cancels the first term of the second expression.
Actually all terms will cancel like this. Except for the very first and very last, because they don't have anyone to pair up with and cancel.

>>11106332
>because they don't have anyone to pair up with and cancel

too real

I submitted a resume, cover letter, and a transcript recently for this student position working with microwaves.
They reached out to me recently and now want a writing sample. What exactly do I send? Like a lab report?

>>11106345
tell them about your favourite microwave meal.

how do I do a bibliographic revision? I need to do it for an animal

>>11106345
They just emailed me to send a Map Report or an essay if I don't have one.
What the fuck is a map report? Is it worth trying to make one, or should I just type up an essay?

Let's say you have a loop that has a p_1 chance to stop at each iteration and each iteration takes randomly between t_1 and t_2 sec. Then X ={chance of stopping at the n-th iteration} follows a geometric distribution with p=p_1 and the average expected execution time until halting is (1/p_1)*(t_1+t_2)/2. Now if we extend the problem to also have a p_2 chance of sleeping for an extra t_3 on each iteration my intuition tells me the average expected time before halting will be(1/p_1)*(t_1+t_2)/2 + (1/p_2)*t_3. Is this correct?

>>11105510
Oh god this shit confused the fuck out of me throughout physics I. They explain it so horribly.
It's because you're trying to find the components with respect to the given angle, not with respect to the x axis.
In cases like that, you need to dispose of your sin=y cos=x intuition, and instead return to the pythagorean definition of sin=O/H and cos=AH.
Pic related.
Even though the *red* side of the triangle is aligned horizontally, it's the *opposite* side of the angle you're using to find the length-along-horizontal-axis of the hypotenuse. Thus, it's OPPOSITE/HYPOTENUSE=sin
Even though the *green* side of the triangle is aligned vertically, it's the *opposite* side of the angle you're using to find the length-along-vertical-axis of the hypotenuse. Thus, it's ADJACENT/HYPOTENUSE=cos.

>>11105510
Oh god this shit confused the fuck out of me throughout physics I. They explain it so horribly.
It's because you're trying to find the components with respect to the given angle, not with respect to the x axis.
In cases like that, you need to dispose of your sin=y cos=x intuition, and instead return to the pythagorean definition of sin=O/H and cos=A/H.
Pic related.
Even though the *red* side of the triangle is aligned horizontally, it's the *opposite* side of the angle you're using to find the length-along-horizontal-axis of the hypotenuse. Thus, it's OPPOSITE/HYPOTENUSE=sin
Even though the *green* side of the triangle is aligned vertically, it's the *adjacent* side of the angle you're using to find the length-along-vertical-axis of the hypotenuse. Thus, it's ADJACENT/HYPOTENUSE=cos.

why do sequences have to be defined only for integers? Likewise with infinite sums?
What is stopping someone from setting the step of the series/sequence to, say, epsilon so that it is continuous everywhere?

>>11106865
How do you perform 2.5 addition operations? Or have a 2.5th member of a sequence?
Whatever you may choose to label each term of a sum or member of a sequence, it must be possible to count and iterate through the labels for a sum or sequence to make sense.

>>11106943
the same could be said for non-integer/irrational exponents though
[math]x^{\pi }[/math]
how do we multiply something by itself π times? But there's no quandary with that
I don't mean to come off combative, I'm just really confused

>>11106286
I spoke a little too soon back there. On a more formal level, the unitary algebra [math]\mathfrak{u}(N)[/math] as well as its oriented counterpart [math]\mathfrak{su}(N)[/math] are thought of as an algebra on [math]N^2[/math],[math]N^2-1[/math] generators satisfying certain commutation (Clifford) relations (of course, this was not the historic approach). The algebra is completely characterized by these relations (the universal property), regardless of the rank of the representations you have in mind.
Now when you're thinking explicitly of matrices to describe [math]\mathfrak{su}(N)[/math], you're already thinking of its "minimal" irrep which has rank [math]N[/math] (you can prove this for general Clifford groups in fact, that you can represent [math]\operatorname{Cliff}_{0,N}[/math] with a rank-[math]N[/math] matrix algebra). We can certainly find matrix algebras of rank larger than [math]N[/math] that both satisfy the Clifford relations as well as fixing no subspace, hence they nevertheless form irreps.
>Thanks for your patient responses so far
You're very welcome dear.

If [math](X,Y)[/math] are independent random variables that follow the bivariate standard normal distribution, [math]X, Y \sim N(0,1)[/math], how do I find the probability that the 2-dimensional vector [math](X,Y)[/math] has a length the exceeds 3, i.e. [math]P(X^2+Y^2>3)[/math]?
I understand that in order to calculate probabilities for bivariate distributions, you need to doubly integrate the pdf with the proper limits, but I am struggling to find what the limits of integration are for [math]P(X^2+Y^2>3)[/math]. Is there something to do with the properties of [math]P(Y|X)[/math] or [math]\mathbf{E} (X^2+Y^2)[/math]?

>>11106960
To find irrational exponents, we can take, for example, 2^π by making a series of 2 raised to the power of rational numbers. To 1 significant digit, 2^3 is perfectly known to be 8. To 2 significant digits, 2^3.1 would be written as 2^(31/10), which is now the 10th root of 2 to the 31st power, which is 8.57. We can continue in this way, finding a sequence that will go something like
8, 8.574, 8.815, 8.827, ... and eventually, the series does converge to 2^π, or at least arbitrarily close, to about the same precision that we know pi to be.

The same cannot be said for sequences. They are meant to be discrete. If you had a sequence defined as y = N, the sequences goes 1, 2, 3, and so on. It does not get half-terms, because the point is that it's discrete. The reason there's a distinction at all between sequences and functions is because they are useful for different things.

>>11106865
You mean as in the indexes? Why indexes for sequences/series are integers?
A sequence is really just a function [math] f: \mathbb{N} \longrightarrow X [/math], where X is just the space where the elements of the sequence reside. You could very well do something like [math]g: \mathbb{Z} \longrightarrow X [/math] or even [math]h : \mathbb{Q} \longrightarrow X [/math]. Why don't we do that? Well, think about why we define sequences in the first place. The naturals in the domain are really just indexes for each element, they don't mark a particular "length step", since the elements [math] x_n \in X [/math] are already free to be whatever you want them to be or however far apart you want them to be.
Both Z and Q are countable sets. That means that there exists a bijection between N and Z, and also between N and Q, so using a composition between those bijections and the functions g,h defined above, defining a sequence using Q and Z is exactly the same as using N. The distance between the points (assuming the space has a metric to begin with) wouldn't even change, because all you are doing is changing their indexes, basically just renaming them. Why N, then? Because it's simply the "easiest" countable set to work with, we can use tools like mathematical induction, and in general the structure of the naturals is pretty well known.
Even if we used your idea, using steps of epsilon length as indexes, you could see that the first step would be one epsilon apart, the second two, and so on. In other words, [math] \mathcal{E} : \mathbb{N} \longrightarrow \varepsilon \mathbb{N} \text{ such that } n \mapsto \varepsilon n [/math] would just be a bijection, and if we have a sequence with epsilon "length steps" [math] f: \varepsilon \mathbb{N} \longrightarrow X [/math], then using the composition [math] f \circ \mathcal{E} : \mathbb{N} \longrightarrow X [/math] we'd arrive again at a sequence that has N as its domain, which is just much easier to work with.

Can you use rref to get an inverse of a 2x2 matrix or is that only for 3x3 and above ?

Can someone give me the steps for the rref to get the inverse of
[4 7
1 2]
I know how to get it using determinant but rref idk

>>11107084
nvm i got it
But FUCK LINEAR ALGEBRA

>>11107084
You do know rref and inverse are not even close to the same thing, right?

>>11107120
Where did I say that they are? Retard
You can use rref to get the inverse

Does anyone have any recommendations for texts or websites to understand probability and intro statistics? Such as series, stirlings aprox, binomial distribution, gaussian, intro to probability theory, whatever.
I know it is really easy but I want to truly understand it.
Also is there a way to improve focus to what it used to be? Will stopping use of social media (4chan, youtube) help? I really hope it's that and not permanent brain damage =(

How do I solve this matrix?

How can we stop thee fact that in a closed system the entropy always increases so that we can become immortals?

>>11107273
kys

>>11106865
look up nets, this is exactly what you're talking about. you can index a net by any poset and then talk about nets converging in topological spaces. nets are the correct way to generalize sequences from metric spaces (continuity = maintains convergence of nets, compactness = every net has a convergent subnet, etc etc)

>>11106865
>Likewise with infinite sums?
An infinite sum over an arbitrary set is usually called an "Integral".
An infinite sum if just a special case of the integral.

>>11107273
The question is about as sane as asking why Santa Claus isn't real.

How do I find the stoichiometric factors of this reaction experimentally using mass balance:

0,500 mL 10 mM Ascorbic acid is added to a solution of 10 mL 1,6 mM K3Fe(CN)6.

The absorbance of the solution is measured to 0,664 A, which means the concentration of K3Fe(CN)6 is 0,618 mM after total oxidization of Ascorbic Acid.

0,618 * 0,0105 L / 1000 = 6,4*10^-6 moles

0,5 mL * 10 mM / 1000 = 5,0*10^-6 moles

6,4*10^6 / 5,0*10^-6 = 6,4 / 5 = 1,28

The stoichiometric factor of this reaction should be 2,0, so what am I doing wrong?

>>11107028
You can just switch the integral to polar, and it's r>sqrt(3)

Is there a sequence of transpositions ( permutations switching 2 elements) such that o_1, o_1 o_2, o_1 o_2 o_3, etc make up all the permutations of S_n, without repetition?

Are there any relations between a function being even/odd and subjective/injective?

>>11107718
Even functions can't be injective, since for each nonzero x in the domain, -x is a distinct element guaranteed to map to the same f(x).
Odd functions aren't necessarily injective (x^3 is, but sin x isn't).
Even and odd functions may or may not be surjective. sinx and cosx aren't surjective, but tan x (odd) and ln(abs(x)) (even) are.

>>11107812
I guess technically a function defined only at 0 is both even and injective, so there's that. I don't think any other cases are possible.

>>11107162
>calling someone retard
>can't do basic linear algebra

>>11107162
> You can use rref to get the inverse
Concatenating an invertible matrix A with the identity matrix and transforming the augmented matrix [A|I] to rref gives you [I|A^-1]. The rref of A itself is just the identity matrix.

>>11107814
Every function is even over a ring of characteristic 2.

In pic related,what's the limit of 2^- and 2^+? the furthest point or the closest?

the image is 1, right?

>>11107557
>>11107079
Not quite an integral, what I'm trying to get at and where this all stems from is my professor telling me there's no such thing as a derivative or integral for the graph of a series since it's not continuous, it's just points defined at the integers.
Yeah, that's true, but the way my (admittedly naive) brain sees it, I don't know why we can't use a limiting process such that there is a point defined at every real number, like the dirichlet function. Then, since there are infinitely many points, the line would essentially be made continuous (I think)

like with this graph of a series, obviously there's a... curve that it follows, similar in form to the natural log or square root. What I'm thinking is, why can't we just sample more and more points to the point where it's a continuous graph? With it being discrete, there is no way to take a derivative or integral of the series, even though there is clearly a rate of change between the points and an area bounded by the envelope of the curve. I'm aware of discrete differences, but I don't know much about them

>>11108131
>there's no such thing as a derivative or integral for the graph of a series since it's not continuous, it's just points defined at the integers.
That's not really quite correct. Sequences ARE continuous functions, and in fact every function defined on a discrete domain is a continuous function. The reason the derivative doesn't exist is simply that the points aren't limit points (cause the set is discrete), which is a necessary condition for the usual definition of the derivative. And an integral could technically be defined, too. At least a Lebesgue integral.
>Yeah, that's true, but the way my (admittedly naive) brain sees it, I don't know why we can't use a limiting process such that there is a point defined at every real number
That exists, and it is... just a regular function. See >>11107079, in short a (real) sequence defined on the naturals is just a function [math] f: \mathbb{N} \longrightarrow \mathbb{R} [/math], and the only thing that makes it stand out when viewed like that is its domain, and of course its special notation. If you change it from the naturals to the reals, you'd have [math] f: \mathbb{R} \longrightarrow \mathbb{R} [/math]. But at that point that is just entirely indistinguishable from a normal function. Obviously, depending on how you define the function, it will have properties like differentiability, integrability, continuity, etc.
>like with this graph of a series, obviously there's a... curve that it follows, similar in form to the natural log or square root. What I'm thinking is, why can't we just sample more and more points to the point where it's a continuous graph?
We can. It's [math] f(x) = \log_{10} (x) [/math] (if I'm not mistaken reading the graph). And that's really it. It has the properties you want. And really the only difference with a sequence is the domain, it changed from the naturals to the reals. As a sequence it would still just be [math] x_{n} = \log_{10} (n) [/math].

>>11108131
A quick follow up from >>11108199. Keep in mind that there are some properties that are particularly useful about sequences and can't be retrieved from just replacing the domain with any random set. If you want a more strict generalization, see >>11107333. If we are talking about sequences like [math]f: \mathbb{N} \longrightarrow \mathbb{R}^{2}[/math], for example, or sequences of functions, then just changing the domain from the naturals can't really be applied, and even if you can study functions too (to look for differentiability, integrability, w.e.), there are some properties of sequences that are particularly useful and can be only generalized if you use nets that have to be defined from directed sets instead. As you can guess from the name, having a sense of direction, e.g going some steps forward or backwards, is really useful to study some properties. [math] \mathbb{R} [/math] is just as always a particularly nice example.

how do I stop feeling so inadequate
it seems like all my peers are 200+IQ geniuses with 300+ internships under their belts and 4.0 GPAs, while here I am struggling to just stay in the program
I'm even having difficulty with a shitty freshman-level gened that I would've laughed my way through two years ago
it's fucking killing me
is this what it's like to have a breakdown?

>>11108291
Just stop paying attention to those faggots

>>11107189
since X is a 2x2 matrix, call the entries a,b,c,d and just multiply it out. solve for each letter with the system of equations it creates

for hensel's lifting lemma, what operation is []^-1 in pic related, surrounding f'(x)?

>>11108462
from the looks of it, multiplicative inverse mod 7

>>11108462
probably means the equivalence class mod 7

>>11108291
Where do you study?

>>11108477
>>11108488
oh I see. so [4]^-1 mod 7 is the same as
4x ≡ 1 mod 7
it's stupid how no source writes that out, but now it's obvious. thanks.

>>11107595
right, i'm an idiot. thanks.

does this remain true if the first x term has a coefficient? do you just ignore it if it does?

>>11108780
If the first term has a coefficient you can just divide the whole polynomial by that coefficient and the roots will not change, since by definition they are the points where the polynomial takes the value 0 and dividing 0 by any coefficient will still result in 0.

>>11108795
but it would change the alpha beta thing since you add them together

who was that math nigga who was seminal in set theory and proved there were no "quadratic" formulae for polynomials greater than degree four but then went full SJW and got himself killed in a petty duel during the french revolution

>>11100698
So if the majority of mass that makes up plants are from carbon dioxide, what purposes do nutrient play? And to what extent how important are they? Is there such a thing as good soil that allows plants to grow quickly? How can that be when carbon dioxide is the mass and not nutrients in the soil?

>>11108821
galois, based on abel's stuff

>>11108820
>>11108824
damn it it was group theory
I knew it was either set or group theory
thank you

>>11108819
You do divide the polynomial by said coefficient first. To apply that formula the coefficients have to be 1, although that isn't really a problem since you can always do that.

>>11108821
Good soil doesn't necessarily mean your plants will grow faster. A plant needs elements like hydrogen and nitrogen and magnesium for things like chlorophyll, for example. Phosphorus is present in cell membranes and some lipids.

>>11108834
okay thank you

>>11108081
any help?

>>11108081
>>11108876
I don't know what you mean by farthest or closest (to what, the image at 2?). At any rate, the left limit and the right limit both coincide at 4. Limits don't care about what the image of the point is because it may not even exist, and the graph approaches 4 from both sides around 2, before it jumps down to 1 just for that value.

I need recs of books about teaching high school and middle school maths.
Essentially, my little sister has been having issues with the usual high school stuff and shaky foundations.
Thanks in advance.
>>11104192
>desperate
Probably.
I really just don't know where it is they got my email.
>>11105518
I think so. Check with wolfram.
>>11105783
i is neither surjective nor injective. 1 x 2 = 2 x 1. 17 isn't in the image.
ii is injective, but not surjective. {1, 3} isn't in the image.
iii isn't injective or surjective. [math]f_3(2, 1)=f(1, 3)[/math] . 1 isn't in the image.
iv isn't surjective. 1 isn't in the image. I'm pretty sure it's also not injective, but no immediate counterexample comes to mind.

I)trivial and left to the reader as an exercise.
II) [math]{0} \rightarrow [0, 1] \rightarrow {0}[/math], where the first map is the identity, and the second map is constant zero.
>>11106514
>randomly between t_1 and t_2 sec
Be specific.
>>11107651
Yes.
See: http://www.sciencedirect.com/science/article/pii/0012365X95000725/pdf?md5=8ec3598f1533fd0e2fed6477a8426d0a&pid=1-s2.0-0012365X95000725-main.pdf , theorem 18.
And thus concludes my short foray into graph theory. Hopefully never to happen again.

>>11101366
>physicist
This is high school physics, didn't you learn babby optics?
https://en.wikipedia.org/wiki/Snell's_law
>>11106359
Pick up an Aristotle text and immitate the beginning.

why would someone define sine in terms of cosine like in pic related? the book is Miller Puckett's Theory and Technique of Electronic Music

also, tangentially, captchas are not ethically approvable; i hate this shit but don't want to pay $20/year to shitpost, but I must spend at least 2 hours per year solving captchas so technically it's probably worth it

>>11109234
nvm, I think I get it now: cos is just an example of a function within the sinusoid family.. soz for retard

>>11109234
nvm, I think I get it now: cos is just an example of a function within the sinusoid family.. soz for retard, i sorta skimmed it initially and thought sine was being defined here

edit: captchas suck

So I completed my applied math degree 6 months ago. I don't feel qualified for any job nor do I even know what applied math majors do. How stupid would it be if I went back to school and got like an associates for accounting or something?

>>11109309
>stuck with school for 4 years
>can't work more than 6 months without giving up
How is this possible?

>>11109322
They're very different things. One you're working toward a clear goal with a clear path to achieving it and the other you're working toward nothing specific and you don't believe it'll work anyway.

>>11109309
You probably aren't actually qualified to do any specific job but neither is anyone else who's just graduated. The best you can hope for is to feel like you can learn to do the jobs you're applying for. Also apply for graduate programs at larger businesses - they're specifically designed to help you through that.

Is there any way of solving equations like sin(x) + cos(2x) = 0 other than by looking at the unit circle and finding an applicable value?

>>11109724
"Where is the negative x value of sin equal to two times the x value of cosine"

>>11109724
>>11109856
in other words, no there isn't really a way to do this algebraically

>>11109724
Taylor series for sine/cosine

>>11109724
Double-angle and Pythagorean identities.
cos^2(x)=1-sin^2(x)
cos(2x)=cos^2(x)-sin^2(x)
=1-2*sin^2(x)
So:
cos(2x)+sin(x)=0
=> 1-2*sin^2(x)+sin(x)=0
=> 2*sin^2(x)-sin(x)-1=0
This is a quadratic in sin(x) which can be solved by the quadratic formula. Or by inspection; clearly sin(x)=1 is a solution, polynomial division gives the other:
=> (sin(x)-1)*(2*sin(x)+1)=0
=> sin(x)=1 or sin(x)=-1/2
=> x=π/2 or x=-π/6 or x=-5π/6

A similar approach can be applied to higher multiples or even rational multiples of x. But apart from the simplest cases you'll end up with a high-degree polynomial which has no closed-form solution in radicals.

>>11109859
>>11109856
>>11109874
jesus

>>11109724
write cos(2x) = cos^2(x) - sin^2(x) and then write write cos^2(x) = 1 - sin^2(x). you'll obtain an equation which contains only sines in power at most 2. solve as a quadratic equation.

>>11109144
Thanks!
t. symmetric transpo guy

>Tried to study every single day for 7 to 9 hours
>Got "burned out" and now I just don't want to do anything while my grades are plummeting to point of no return
I know that I shouldn't be in /sci/ and am a brainlet compared to anyone here, but I really wanted to get As on my all shitty compsci/math courses, but I started to not enjoy anything because of the amount of work I need to put into to get a decent grade. Anyone had this happen and care to give advice so this doesn't repeat again? I put two and a half years into this, so I can't change majors, just need to finish two more extra years or more so I can be done with this.

complete brainlet question here so i apologize but when you make a polynomial equal to zero and graph it it's two horizontal lines. what's that about?

why do we always make it equal to zero if that's what it looks like

>>11108821
https://en.wikipedia.org/wiki/Liebig%27s_law_of_the_minimum

>>11110364
seems like you're trying to do too much

try 2 3-hour sessions but make sure you're clear of all distraction, coffee, adderall, etc.

>>11106359
1 search literature on said animal
2 find literature on said animal
3 read literature said animal
4 compare literature on said animal

>>11110574
Setting a polynomial equal to zero lets you solve for the x values where the y value is zero. Whatever your solutions are, say x=-1 and x=1, you can graph these as vertical lines which intersect the polynomial at y=0. I'm not sure what you mean by "graphing its horizontal lines."

Should running a brain similarion and deleting it considered unethical? Want to know your opinion

are there any matlab pros here? i have a figure with 11 subplots. i'd like to retrieve the y-data set for the first subplot. googling and trying different shit i'm only able to retrieve the 11th subplot's data set.

is y'(x) the same as y'

>>11107009
In case you're still here, I've got several questions and observations. It seems we are interested not so much in SU(2) as its algebra. Therefore we look at any matrix satisfying this algebra. How does this work, why are we looking for so much more than simply the faithful SU(2)? Next, I was under the impression that representations of a matrix would always be smaller (this because I studied decoposition of finite representations). But we aren't decomposing SU(2), and we are infact looking for bigger matrices. How do we know that every matrix satisfying the SU(2) algebra is indeed irreducible?

>>11110678
If the only two variables involved are x and y, then yes.
If there's other variables than x and y, you shouldn't write y' without specifying that y is a function of x and not those other variables.

>>11110683
thanks

>>11107651
Yes, the inductive proof is actually quite straight forward.
Page 28 of https://www.math.uni-bielefeld.de/~grigor/kurosh-higher-algebra.pdf

I can't focus at all, what should i do?

I'm considering going for a PhD in nuclear engineering when I finish my bachelor's in two years. Is a PhD in nuclear engineering valuable? What sort of stuff can I do with it? Is waiting until you're thirty to get a real job worth it?

>>11109234
>something something captchas
Post less.
>>11110313
No problem.
>>11110752
Coffee.

t= 1/sqrt27 is an answer but t= -1/sqrt27 isn't part of the answer. the negative version is also on the graph of the derivative, but if the question asked to find all local and absolute min/max, would the negative version then be part of the answer?

>>11111438
Yes, if you want local maxima then the negative would be a correct answer. Along with the absolute maximum in that interval, of course.

>>11111450
but without graphing the derivative, how would you know whether or not the negative number isn't part of the answer?

>>11111487
You can analyze monotony of the function using the first derivative (no need to graph, just evaluate values between the zeros/discontinuity points since it's continuous there). If you know where it increases/decreases and until where, you can find which are the other candidates.

Is this proof incorrect? Suppose for example S = {1,3,5} a 3-combination of some larger set X. Then f(S) = {1,2,3} which clearly does not belong to B.

>>11111727
f(S) doesn't need to meet the no consecutive integer condition. B doesn't have that condition in its definition.

>>11090418
Assuming that ac is denoting the cardinality of the cartesian product of two sets with cardinality a, and c, the implication does not hold for the same reason it does not hold in general for regular numbers.

>>11111732
Ah, ok. I missed that bit. Thanks.

>>11108821
>purposes do nutrient play
stuff like this

>>11110625
>Should running a brain similarion
Will never exist and if it did it wouldn't be "alive" in any meaningful way so there is no ethical issue.

what do the arrows mean?

If there's a balloon and I'm a hunk man and I punch that balloon with incredible force... Can the balloon still respond with the same force? Does that mean that a balloon can exert any amount of force? (Because it always has to match an aggressor) So basically any object can exert any amount of force?

>>11100698
Artfag here, so the question must appear extra dumb.

How far could one see from a platform around 300m tall, assuming good weather conditions, from my research it was around 65km, but these distance is if the object is a ground level, no? What if it's something massive? Would it be possible to see a mountain around 1km tall over a distance of 200km from the 300m platform? If so, how large would it appear?

Is it bad to put a class project on a resume?
I have an antenna project we did for lab under projects on my resume, but it’s technically a project, right? Even though we just followed a predetermined procedure, but results did vary among students in regards to antenna length, how well you could fabricate it, and gain.

how difficult would it be to make a simple calculator app where you input a number, it goes through a function, and the result is then displayed? basically a calculator where you only need to do one calculation, but with different x vaules.

>>11111949
it's trying to help yu understand 3d structure. N is pulled in to the Co cation

Supposedly the farther you go into space, the more you travel back in time.
So what direction do you need to travel for this to actually occur?
Does this mean that from some other distance, earth is in the past relative to some position but also in the present relative to some other position?
I have no idea how this works. Shouldn't time be consistent and we are in the present regardless of location in space?

>>11112645
>Supposedly the farther you go into space, the more you travel back in time.
The statement you actually heard was probably that we observe distant objects as they appeared years ago because that's how long it took their light to reach us.

>>11112530
>app
Something between trivial and a 3 hour project, depending on you meaning of "app".
A simple command line application can be written in almost any programming language in a few minutes.
If you mean a smartphone app that will probably take quite a while to set up and get going, but it should be quite easily doable.

If I lift an object fast, I transfer high kinetic energy to the object. If I lif the object slow, the KE is lower. However at the end when I hold the object up, the same work has been done on the object because the difference in KE is 0 and also the potential energy counters the gravitational energy. So what happened to the higher KE during the motion? Why is there no more work done when lifting faster?

>>11112892
You aren't applying the same force continually. If you did, when you eventually stop applying the force the object would continue upward due to its momentum. Instead, as the object approaches the desired height you reduce the force allowing the object to decelerate under gravity so it stops at the desired height.

If you initially apply more force so that the object moves faster, it will have more momentum and need longer to decelerate, so you reduce the force earlier. The result is that the total energy is the same.

>>11112367
If you have absolutely nothing else you could write down then it's not going to get you in trouble or anything, but it's a very weak thing to list on a resume unless you're like, a freshman looking for summer internships.

>>11113082
But what if I continue to apply high force and just forcefully stop it at the end? Is is that just that I'm applying "negative work"? So in absolute sense I'm doing more work, but it just cancels out because of different directions?

>>11112046
Assuming the Earth is a sphere with radius R, that the tower has heigh h, and ignoring relativity, we have R * arccos (R+h)/h.
>>11112530
Depends on the language, I think.
Stuff like R is already capable of acting as a calculator, so all you do is make a frontend.
>>11112892
The potential gravitation energies are equal. If "lifting faster" implies it's faster when you stop pushing it upwards, then it obviously has more energy.

>>11113371
>arccos (R+h)/h
I meant arccos R/(R+h)
No idea what came over me.

>>11113371
>>11113456
>arccos
Oh good lord, what even is arccos? So basically this arccos thing * 6371/(6371+0,3)
Jesus, I should've just downloaded Google Earth.

>>11112046
> How far could one see from a platform around 300m tall
Straight-line distance to the horizon: d=√(h(h+2R)), where h is height above surface level, R is planetary radius (~6378 km). Assuming that h is much smaller than R, you can just use d=√(2Rh); the difference is ~40ppm for the kind of heights you're talking about. So ~62 km.

> What if it's something massive? Would it be possible to see a mountain around 1km tall over a distance of 200km from the 300m platform? If so, how large would it appear?
If you can just see the peak from the tower, then there is some point on the horizon which is visible from both the peak and the tower. From a height of 1 km, the horizon distance is ~113 km. So if the mountain peak and the top of the tower are more than 175 km apart, the earth's curvature will block line-of-sight.

Even if it didn't, it's doubtful you could see through 200 km of air even in really clear weather.

All the solutions for this problem just seem to create an arbitrary function. I don't know where you would even start. "let f(x) = sinx - x" This is in the mean value theorem chapter

>>11113620
can you show that the slope of sin(x) is smaller than the slope of (x) for all x in that region? and their value at x=0 is the same so sin(x) never catches up?

>>11113507
invserse cosine

cos(x)=y if and only if arccos(y)=x

>>11113580
>Even if it didn't, it's doubtful you could see through 200 km of air even in really clear weather.
Wouldn't the attitude counteract it, since air is denser on the ground and all?

Anyway, thank you, anon!

>>11113371
>Stuff like R is already capable of acting as a calculator, so all you do is make a frontend.
Name a single serious language that isn't capable of acting as a calculator.

>>11113653
So use cosx? The slope of x will always be 1 won't it?

What could I do with a degree in physics? What is the actual landscape looking like? Is it a growing field with abundant opportunities or is getting lucky just a high school teacher's position? What are good subsections of the field? Other options than positions like nuclear?

I’m at the point where I need to settle on a career trajectory. I’m finishing up a math/cs degree with some projects on github. I love math, but i’m probably too dumb too be a mathematician (plus the pay sucks) and being a plain ol dev sounds boring. Getting into finance sounds great, but i’m not an Ivy League type.

I’d like meaningful, challenging work. I don’t want to go into the trades or military. I want benefits and retirement. I’d love to run my own business, esp in software. But that hasn’t happened yet, so in the meantime i guess i’ll just go for grad school..

>>11111987
Yes. This is literally just newton's third law.

>>11113719
no, if you use cos(x) then the slope between zero and 2pi will start at one but decrease after that and only start turning around much later

at a more basic level, you know that cos(x) is curvy right? if it had a constant slope then it couldn't be curvy

>>11113768
every now and then i start to doubt my whole life and wallow in the meaninglessness of it all, and i always revert back to ‘finish degree, get into grad school and figure the remainder out then’, so i guess that’s what i’ll do. i know i won’t be satisfied in life if i don’t go ham right now.

hopefully i can nab a phd and get self employed

>>11100698
any medfags on? Anyone in neurology in particular?

How come when a limb falls asleep I never have the experience where I can feel it but not move it? There have been tons of times where I can't feel or move it, and several when I can move but not feel it, but never where I can feel but not move it. Are sensory neurons more susceptible to getting cut off, or attached to motor neurons in a way that prohibits this? What is it?

Why will phosphorylation on some proteins increase activity and on some proteins decrease activity? Is there some sort of unifying general explanation?

>>11112530
>>>/g/

>>11112762
but really this answer

>>11113620
Is this Analysis? How constrained are you when it comes to pointing out the obvious? sin(x) has a global peak at pi/2, which is greater than 1, and it's never downward concave before then. fuckin done, right?

>>11113620
Suppose there exists [math]x_0 \in (0,2\pi)[/math] with [math]\sin x_0 > x_0[/math]. By the mean value theorem there exists a number [math]a \in [0,x_0][/math] with [math]\cos a = \tfrac{\sin x_0}{x_0}[/math]. But [math]\tfrac{\sin x_0}{x_0} > 1[/math], contradiction. Maybe you need to take some care about the endpoints or absolute value or whatever, but this is the idea.

>>11113787
he was talking about repls so any compiled lang

>>11113677
> Wouldn't the attitude counteract it, since air is denser on the ground and all?
At an altitude of 1 km, atmospheric density is still 90% of its sea-level value.

Is mass cause of time? More mass near = faster time
No mass near = slower time

>>11114151
Mass bends space time

>>11100698
Anyone know how studying in sweden is for foreigners?
Is it easy to get in? How demanding are unis? Etc
Im in latin america but have a european citizenship/passport and ive been looking for options in europe for my EE masters degree studies and so far the best looking option is sweden by far.

>>11111111

>>11114501
>university courses
>demanding

Hey, does anyone happen to know if nicotine gum can be put into a coffee maker with the grinds to get some of the nicotine into the coffee (assuming it sliced into tiny pieces/etc). My older sister can't chew the nicotine gum because their teeth are garbage, and won't use the patch because of skin issues/hates having smg stuck to their body. So we've considered dicing up a few of the pieces of gum and putting it in with the ground beans when making coffee. I don't think it would chemically react with anything, and the heat should be fine since it survives burning cigarettes too. Does anyone know if this actually works or will it just waste the gum?

>>11114513
>Hurr durr im vety shmart
Major cringe, pal

>>11114526
Nicotine is not the addictive element of tobacco smoke.
Supplying tobacco addicts with nicotine has been proven to be unable to stop tobacco smoking.

>>11114527
>asks subjective question
>gets mad at subjective answer
I expect that the courses will quite demanding for you.

>>11114545
>Gets mad
What makes you think im mad?
Im just laughing at your cringe posting.
You sound like a freshman that just got an A in his first calc 1 quiz bro.

>>11114134
>so any compiled lang
Lang is a meme.

>>11114550
K tard

>>11114537
Oh dang, that's pretty hard to hear. Thank you for the information. Is there anything that's really proven to work besides cold turkey + heavy emotional support and distractions? They've considered vaping but it doesn't really stratch the same itch

>>11114561
>Doest anything work
Imo a postulate of the human condition is that no one will ever do something they dont want to do.
If your sisters wants to quit, she will, if sje doesnt want to, she wont.
[math] Knowing [/math] that she needs to quit is a very different thing from her actually [math] wanting [/math] to quit.

>Vaping
Vaping is seriously worse from smoking cigarettes because:
>Its absurdly convenient, you can vape at anytime, anywhere with minimal effort, you just need to pull a little pendrive frok your pocket and puff, no one will even be bothered by your smoke
>Smoke taste is very customizable and is generally much more pleasant that cigarette taste
>As the smoke solution is very customizable, it means people come up with all sorts of dangerous shit like 80% nicotine content smoke and pretty much anything you can imagine
>Smoke itself is generally much denser than cigarette smoke

Theres people dying from juuling already bro.

>>11114574
Thank you again for the info, she's got other things going in her life so quitting cigarettes is probably not her top priority. She's said one of the reasons she wants to stop is to actually save money, and we try and do the chores/make sure she doesn't have do deal with much when she's trying to quit, but she's only ever made it 2 days before.
It makes me upset that even if we don't include smoking, she spends the most money in the house by a good margin. Like she genuinely can't save money at all, which is it's own problem. If it comes to it I'm probably going to have to move on, and just leave her to her own devices, but it's hard knowing she has these quirks, and just refuses to stop spending, even when rent is overdue.

Thank you again for your advice, this is significantly better that a lot of threads

>>11114593
Yeah man, I have sisters too so i know it can be tough sometimes.
Hope everything work outs for, if it doesnt, dont beat yourself up as its in her to become a better person and its not your responsibility.

Is the fact that a change in amplitude doesn't affect the period related to the fact that a change in mass doesn't affect the time it takes for an object to fall? Is it related to conservation or what's going on?

How do you get the second inequality? (With the assumption that [math]x, y\ge 4[/math]

I'm trying some random ass home chemistry experiments and need a (preferably somewhat cheap) source of cold. I bought some peltier coolers, but they just don't get cold enough. I am thinking of trying dry ice next, but is there some other method? Can small amounts of liquid nitrogen be bought over the counter anywhere? Any tips/tricks for achieving deep cold?

>>11114504
what was >>11111111 anyway?

how to solve this what do i need to read?

>>11114792
forgot pic

>>11114791
>>/sci/thread/S11110566#p11111111

>>11114665
I assume you're talking about a simple pendulum. Yeah, it's related. Have you seen the derivation?

>>11114805
No

What is the most likely object that is in our solar system affecting everything's gravity but has yet to be discovered? I've seen planet and black hole hypothesized but I'm too brainlet to understand why it couldn't just be more asteroids we haven't mapped yet.

>>11114792
>>11114794
Just need to learn binomial theorem.

A few months back, someone posted a link to a paper claiming to have found a quantum interaction that was not quantum computable. Did anyone save that, or was it too much bullshit?

When talking about a union of linear subspaces, is there an easy way to know if the resulting set is a linear subspace?

>>11114915
A sufficient condition is that one is contained into the other.
Not really the most exciting result but otherwise unless you consider uncountable unions it's very rarely true that the union of linear subspaces is a subspace.

Can someone explain me or give me some resources to help me understand rigid body Dynamics and kinematics? I just fail to understand even what's even going on.

For instance, when you have various rods attached to each other (A B C). I've been taught this formula

VB = VA + w x AB

I'm talking about problems like this one: https://www.coursehero.com/qa/attachment/3207411/

I can't even picture how the rods move. Can someone whether it be one of you nice anons, or a PatrickJMT equivalent for physics (aka a god) explain it to me? That would be much appreciated

>>11114915
>>11114929
That's an if and only if goyim. The union of two vector spaces is a vector space if and only if one is contained in the other.

>>11114792
>>11114794
Binomial expansion.
[eqn]
(x+y)^n = \sum_{k=0}^n {n \choose k} x^{n-k} y^k
[/eqn]
Substitute x=1, y=-1, simplify 1^(n-k)=1, and you have the expansion for (1-1)^n = 0^n = 0.

A more explicit approach is to note that [math]{n \choose n-k}={n \choose k}[/math] so the sum must be zero for odd n (the k and n-k terms have the same coefficient but opposite signs). For even n, note that [math]{n \choose k}={n-1 \choose k-1}+{n-1 \choose k}[/math] for 0<k<n which results in a telescoping series leading to 1-1=0.

>>11115033
Yeah, I didn't know how to word it for arbitrary unions instead of just two, since you can still get a vector space even if not all of them are contained into one in the uncountable case.
I guess saying it's sufficient that at least one of them contains all the others would have been more accurate.

>>11115028
> Can someone explain me or give me some resources to help me understand rigid body Dynamics and kinematics?
https://en.wikipedia.org/wiki/Rigid_body_dynamics

In short, you have F=ma applied to a set of points, coupled with the constraint that the distance between any pair of points never changes (that's the nature of rigidity), meaning that the velocity of A relative to B must always be perpendicular to the position of A relative to B. IOW, (A'-B')·(A-B)=0. Everything follows from that.

> I can't even picture how the rods move.
In the absence of external forces, everything rotates about the centre of gravity, which moves in a straight line.

>>11114857
>>11115046
thanks

I figured this might be more appropriate for /sci/ than /g/.

How do you solve a recurrence if you aren't given a base case? I'm trying to get my CS homework done but I have no idea how you're supposed to solve it without any kind of base case(s).

The instructions are "Solve the following recurrences, but without computing multiplicative constants (Ci). Thus, in this exercise the base cases are not needed"
One of the problems is T(n) = 4T(n-1) - 5T(n-2) + 2T(n-3) + n - 3 + (5n^2)*2^n
And that's all the information I am given. I've looked through the chapter and every example they give a base case.

can someone help me understand these medical terms? I need to know where these take place and easy ways to think about them practically.
Primary hemostasis:
Secondary hemostasis: which includes
-intrinsic coagulation cascade
-extrinsic coagulation cascade

>>11115153
The base case only determines the constants. So if they don't give base cases, you end up with a formula involving symbolic constants.

In general, linear recurrence relationships give a formula consisting of a sum of terms of the form c*n^k*r^n. The relationship determines the r and k for each term, the base cases determine the c.

The usual methods for solving them find the k and r from the relationship. You then evaluate the expression for n=0,1,2,..., and equate the resulting expression (involving only the c) to the corresponding base case. That gives you a system of linear equations which are solved to find the c.

In this case, you don't have to do the last step.

FWIW, I get:
T(n) = (a0+(370/3)*n-30*n^2+(20/3)*n^3)*2^n + b0+b1*n-(1/6)*n^3

If you had base cases, you could use them to solve for a0, b0, b1.

>>11115535
Wait, so is it just asking for me to put it into the form of the characteristic equation and not find what the variables to it are?
For example, if it was T(n) = 2T(n-1)
it only wants me to write T(n) = C1*2^n ? (where C1 is the unknown constant)

How do ships and boats "tack"? The wind is coming at them from 45 degrees to the front, but how is this converted to forward movement instead of just pushing the boat back anyway? I know it has something to do with the keel but I don't get it.

>>11113787
I dunno, do I look like I browse /g/?
R is the only language I know.
>>11113994
>is this analysis
Yes.
>>11114545
That wasn't an answer, that was a comment.
An answer would've been "Studying anywhere is easy" or something along those lines.
>>11114684
[math]\sqrt{2(x^2+y^2})+2= \sqrt{2}\sqrt{(x^2+y^2} + 2[/math], and the inequality is equivalent to [math](2- \sqrt{2}) \sqrt{x^2+y^2} > 2[/math].
Since [math]2- \sqrt{2} > \frac{1}{2}[/math], we have that [math] (2 - \sqrt{2}) \sqrt {x^2 + y^2} \geq \frac{1}{2} * \sqrt{32} \geq 2[/math].
I give the inequality a weak 2/10. Are you sure that you have x, y>4? The inequality should be strict.
I've checked it at least five times now, but I still get the impression it's wrong.

>>11115551
Yes. That's what it means by "without computing multiplicative constants". You can't (in general) compute them without the base cases.

>>11115560
The force generated by the wind on the sail is sideways relative to the sail. This can be up to ±90° from the direction the wind is blowing towards. The keel prevents the boat from moving sideways, so the net force on the boat can be up to ±90° from the force provided by the sail. So in theory, you could get a net force that's almost opposite to the wind direction. In practice, the force gets weaker the greater the angle (in wind-sail and sail-keel), and the keel doesn't entirely prevent sideways motion, it just reduces it significantly.

You can't sail directly into wind, but you can sail maybe within 40° of that, and you can alternate from one side to the other so the long-term course is into wind.

>>11115789
Thanks. I assumed it meant without using the characteristic equation when it said do it without multiplicative constants.
CS textbooks are worded so weirdly.

I wanna prove that d(x,C) =inf{d(x,y) | y in C} where C is a closed set is equal to min{d(x,y) | y in C}. But if do proof by contradiction, I'm concerned that the minimum might not exist making my proof pointless.

What do? ;(

>>11115909
Prove the existence by contradiction if you want. Existence is the most important part of the proof, the equality itself is kinda trivial if the minimum exists.

>>11115648
Thanks for the answer.
>I give the inequality a weak 2/10. Are you sure that you have x, y>4? The inequality should be strict.
That was what confused me as well. I was just reading this from a lecture note, written based on a paper.
Anyway, they were trying to prove that the approximation ([math]x+y+2[/math]) is within a factor of 2 from OPT ([math]2\sqrt(x^2+y^2)[/math]). So in this case, even if the inequality is not tight, it's actually fine. The reason why the choose 2*OPT still bugs me though, especially when they could make it tight.

What is the idea behind free objects? I know their universal property, but I want to understand why they are of any interest to begin with. I've read that they are supposed to be a generalization of the concept of basis in vector spaces which is pretty obvious in the case of modules, but not so much for groups or rings, since I guess having no ring of scalars makes translating the concept of linear independence for groups a bit more difficult? Not too sure how to relate that
That said, I've heard that another way to see it is as if free objects were the most "generic" possible objects, but I don't really get what is meant with generic in this context. In this case I don't even get how being "generic" is something that's somehow related to the concept of bases in vector spaces which we were trying to generalize earlier.

How can I make sense of these intuitions?

Physics here
If a rocket goes straight up with an acceleration of a=2,25 m/s^2 and the engines go off when it hits a height of 525m, how would I go about calculating its max height?

Seems pretty basic but I'm blanking right now

>>11116313
think of the typical free object as the mathematical object you're talking about which is generated by a certain set of generators, which are governed by no relations besides the ones inherent to that object.
So a free group on a set of generators wouldn't have any relations on its elements to restrict it beyond just being the largest group you can make out of those generators: just multiplying them however you like in finite products, and cancelling anytime an element is next to its inverse.
How about the free abelian group over a set of generators then? Well, we have sums of the generators, and each generator can be in the sum an integer number of times. So you're ending up with integer linear combinations of the generators. There's no restrictions to quotient by, that's it.
You might be able to see why the generators are like the basis for a vector space in the commutative case, because you can separate them all out nicely.
It might also make it more obvious what's going on with the universal property if you have this intuition.
Think of some other objects you know about and what the free object on a nice (maybe finite) set of generators looks like.

>>11116773
let the engines go out at t=0. From basic kinematics, you know the velocity vi at t=0 and you also obviously know the height at t=0. The acceleration is 9.81 m/s/s downward. Now, you can apply some other kinematic equation to get the height where vf=0.

>>11116839
>set of generators wouldn't have any relations on its elements to restrict it
This made me remember the group presentation notation I had come across by accident a while ago, and all of a sudden it all just clicked and made sense. So no relations is basically how we generalize how the basis of a vector space is linearly independent and thus can't have a (non-zero) linear combination that equals 0, right?
And also all other groups can be "created" out of the free groups if you just quotient the free group by the subgroup generated by the relations, so the relations equal the identity as equivalence classes, then?

Damn I think I can see all the stuff I was missing now. Thanks a lot

Stony Brook or NYU for grad level math?

>>11111949
Coordination. Not quite covalent, not quite ionic either, but there's orbitals at play. Common with metals.
>>11114694
rock salt and ice gets about 10F colder than ice. Ice bath is standard. If you can find dry ice, dry ice/acetone bath are all you'll ever need.
I think dry ice and liquid N2 can be found at restaurant and welding suppliers, but they might look at you funny

>A method of de-bacterializing a 40%-alcohol beverage (from, specifically, streptococcus).
>The ("electricity potential") difference between a potentiometer and what the "more powerful" option is.
>How pure the product of electrolysis from within plastic is.

what would the sum of all numbers between 0 and 1 be? Or would it diverge?

>>11117946
A sum of all such numbers would have to be bigger than the harmonic series (1+1/2+1/3+...), since it contains all those elements and even more positive ones. The harmonic series already diverges though, so a bigger sum would diverge too. In fact a countably infinite sum only converges if almost every element but a countably infinite amount of them are zero. So even if you were to sum a tiny interval it would still diverge.

>>11118001
That's simpler than I thought, thanks for explaining.

I got half a report on the line. I've been at it for four hours, it should take about 3 more.
It's 3 am, i'm exhausted, my jead is fucking spinning and i have to have it ready by 11am.
Should i
>Brute force it now
>Sleep until 8am and do it then
>Nap for two hours and do it then
Please help

>>11118115
do it NOW. get off 4chan.

>>11116158
That's the thing, really. Shitty estimates are to be expected, mixing up strict and non-strict inequalities is really weird.
>>11116773
Work is force times distance, T=mad, where T is work, a is acceleration and d is distance through which the force is applied.
Potential kinectic energy is P=mgh.
Set T=P and solve for h, that is, m*2,25*525=m*9.81*h. Slash m on both sides.
>>11116842
Also works and actually more intuitive, but shittier to calculate by hand.
>>11118115
>my head is fucking spinning
If you're actually feeling unwell, just go to sleep, lad.
And make sure to drink some water.

>>11118129
>energy methods to solve a kinematics problem
cringe

>>11118141
It's either energy or two (2) Torricellis.
You solve for its speed at 525 meters with Torricelli, and then use it again for zero speed.

>>11100698
Is it possible to factor something like [math]x^3-216[/math]
without memorizing the difference of cubes formula: [math](x-y)(x^2+xy+y^2)[/math]

Help a brainlet out, is there some spooky shit I should know that lets me avoid memorizing yet another formula?

>>11118227
Polynomial division, if you are able to find a root.

>>11118256
What would be the first step in using polynomial division in the above expression? x-6/x^2-36?

Are there finite element simulation software (for peasants like us) out there that are capable of simulation gravity (in order to reproduce gravity waves)?

>>11118267
The first root you're likely to notice is ∛216=6 => x-6 is a factor, so:
(x^3-216)/(x-6)

>>11118279
That appears to have done the trick. Polynomial division is bizarre and doesn't feel like real math. But factoring felt the same way until I "got good" at it.

>>11118227
Don't waste your time with polynomial long division, it isn't exactly necessary for finding the factors. You start with your rational zeroes test, you know, the factors of the leading coefficient and the factors of the constant make a set of rational numbers. Try them until you get a zero. Then use synthetic division for the rest. It's a really easy mechanical method you can pick up after a 5 minute video on youtube.

I need some guidance on this problem. Unsure how to proceed. I know that an open set U in the p topology would be such that any f in U has the property that there exists a delta where we can put a p-metric ball around f and have it contained in U.

>>11118746
come on. if two functions are close to one another everywhere, i.e. d(f, g) < epsilon, then how big can the integral p(f, g) be?
now, if two functions have close integral p(f,g), does that necessarily mean they're close EVERYWHERE?
draw a fucking picture.

>>11118769
If d(f,g) < epsilon then p(f,g) < epsilon(b-a) right? Sorry I'm just very slow at this stuff.

I was going about this a different way. I was going to show that the function p : X x X --> R is continuous on the topology induced by metric space d, in which case the topology of metric space d would be finer than that of p.

>>11118818
nice numbers

>>11118405
>tfw synthetic division sometimes fails
what the fuck

I think I am retarded and I don't understand isomery. Does either of these compounds exist?

>>11119106
Yeah, if I'm not mistaken the first one is 2-butanol and the second one is 2-methylpropan-2-ol
What don't you understand about isomery?

does the dirichlet function pass the vertical line test

>>11119189
If it didn't it would mean there's a real number x such that f(x) can take the values 0 and 1, and by the definition of the function it would only happen if x is both rational and irrational. The irrationals are by definition just the non-rationals though, so it wouldn't make sense to be both at once.

>>11119185
I was just wondering if they are properly drawn. The exercise was something along the lines of "Write isomers of the saturated monohydric alcohol containing 4 carbon atoms".
Could the second one also be called "2 methyl 2 propanol"?

>>11119235
>Could the second one also be called "2 methyl 2 propanol"?
Yeah, in fact it would have been more consistent with the notation I used for the first case, guess I didn't give it enough thought. Rest seems fine to me.

How do I find an equation for a line between two points given the points and exclusion zones (with radius)?

>>11119348
What kind of line do you want? Cause if you literally mean a (straight) line then it could very well be impossible in euclidean spaces. The line between two points is unique, so you can find it as usual and just check if it intersects your exclusion zones. If it does then there is no line between both points.
If by line you just mean any curve or path that unites both points, then you can just define an arbitrary polygonal chain (so a bunch of connected segments) that goes around your exclusion zones. E.g in your pic you could just pick the union of the segments (2,1) to (2.5,1), then from (2.5,1) to (2.5,6), and lastly from (2.5,6) to (6,6). Obviously writing it as a function and much more elegantly than I did, but the point is that you can construct it like that. If you do it like that then the line may not be described by a single equation, but it should still be a well defined function.
Now if you ask for even more conditions (like making it differentiable, or constructing it out of elementary functions) then it becomes much harder.

>>11119491
>What kind of line do you want? Cause if you literally mean a (straight) line
I mean the line doesn't have to be straight
>Now if you ask for even more conditions (like making it differentiable, or constructing it out of elementary functions) then it becomes much harder.
Is there a process for coming up with such an equation other than just trying random stuff until it works?

>>11119500
As far as I'm aware it is extremely dependent on the set of your desired exclusion zones, so there's not a single process that could guide you through. If those exclusion zones we're simple enough maybe you could get away with using a convenient polynomial interpolation. Also with polynomials, you can find a continuous curve (like the chain I mentioned earlier), and approximate it through Bernstein polynomials, although the computation could get harder the moment you start working with more complex cases. Of course there are a lot of restrictions too, like if the two points are in a single vertical axis, you'd need to work with rotations of the graph to at least hope to get a solution.
For simple cases, just trial and error (assuming you at least know how most elementary functions look and behave like) would be less of a headache.

Need help trying to calculate my grade
HW = 10% (100pts)
2 Midterms (30% each) = 60% (100pts each)
Final = 30% (100pts)
X(.1)+Y(.6)+Z(.3)
My professor said he would add 5 points to the overall grade if I do all the HW, which I'm pretty sure I'll do.
So how do I fit points into this equation?

can sexual dimorphism be seen as a type of disruptive selection?

Any of you guys know a book or code or whatever on P&ID symbols? I have a diagram with some weird symbols I haven’t seen before and google isn’t helping.