/sci/ - Science & Math

Unanswered questions from the previous thread.
Maths questions:
>>11042932 (No.)
>>11043375 [The context is long, but he essentially wants a formula that returns even numbers that aren't multiples of 6.]
>>11043468 (Yes.)
>>11043473 (Yes.)
>>11044866
>>11046740 (Calculus and probability theory on finite sets, respectively)
>>11047845 (Yes.)
>>11054531
>>11055740 [I think he just wants a random tangent normalized vector, but I'm not sure.]
>>11055873
>>11056154 (Opposite angles.)

Physics questions:
>>11044795
>>11046366
>>11047074
>>11047904
>>11052450
>>11052456
>>11054387

Unclassified questions:
>>11043355

Biology questions:
>>11049388
>>11052454 plus >>11052636

Stupid questions:
>>11043411
>>11046019
>>11046578 (From God.)
>>11047512 [There is a reply, but I'm not quite sure it qualifies as an answer, because I have fuck all idea what it means.]
>>11055558 (Yes.)

I've decided to use parenthesis for when I'm answering and brackets for when I'm commenting.

>>11056713
>>11044866

Try professor leonard’s videos on limits in his Calculus 1 playlist, and make use of certain tricks like solving infinite*0 form limits with fractions. Strang’s Open Stax Calculus book (free) has a whole chapter on infinite limits and indeterminate forms which gives precisely the techniques you’re asking for without using L’Hopital’s Rule (not a good book but does a good job with that exact problem).

>>11056801
Small mistake replying to >>11056154.
As far as I know, there is no name for, given an angle A, A plus two right angles.

>>11056672
>>11056690
Thank you.

OK, how come come
[math]
c_{-1} = \frac{1}{2\pi}\int_{0}^{2\pi}\sin(x)e^{-(-1)ix} dx = \frac{1}{2}i
[/math]
when we know that Fourier coefficient number -1 of sin(x) should be -i/2? I feel like I'm missing something super obvious here, but I'm just using the formula:
[math]
c_{n} = \frac{1}{2\pi}\int_{0}^{2\pi} u(x) e^{-inx} dx
[/math]
for Fourier coefficient n of u(x).

>>11056801
my question didnt make it :c
i was counting on you!

How the fuck can I prove that the ideal generated by <x,4> is a primary ideal of the ring [math] \mathbb{Z} [x] [/math]?
We define a primary ideal I as an ideal such that if [math] ab \in I \land a \notin I \implies b^{n} \in I, \text{ for some } n \in \mathbb{Z}^{+} [/math]

Tried to go off just using the definition but I have no clue how to give an arbitrary element of the ideal the form of the product of two polynomials, while being able to get to the conclusion.

>>11056713
ugly 2hu post something better

how come stuff like stable orbits are possible if gravity isn't instantaneous. should't systems with multiple orbital bodies fall apart since the information of gravity lags behind in time

>>11057196
My bad.
Was it the differential equations one?

>>11057246
Primary ideals are those ideals where zero divisors are nilpotent
Z[x]/<x,4>=Z_4

Given a number in a given base (usually 10) and its representation in a different unknown radix, how can I find such radix?
[math]1100_{x} = 150_{10}[/math]

Specifically without using high-degree polynomials. I already know that I could do:
[math]r^3 + r^2 = 150[/math]

With larger numbers you would end up with octics and so on, and I do not want that.

>>11058348
in this case the solution is 5, by the way

>>11058348
Pretty sure the polynomials are unavoidable, but the solutions are integers, so you can solve it through the usual approximation techniques.

>>11052450
Not around the equator, but within the core of the earth. Basically coriolis forces and temperature differentials within the earth's core cause charges to rotate.
>dynamo theory

>>11057792
Im not sure why you think gravity acting instantaneously should be a requirement of that.

>>11057246
It's pretty easy to prove this directly from the definition too. If ab=cx+4d with a and b nonzero, the division algorithm tells you that one of them must be an integer and the other is at most a degree 1 polynomial. Then it's easy to argue that one of them must be in the radical of I simply by comparing coefficients.

How fucked am I if I don’t get an internship by the time I graduate?
I have a 3.5 GPA, but not many noteworthy projects/engineering experience. My last interview I fucked up because I’m an autistic fuck and haven’t had one since.

how do I find a basis of T such that the matrix of T according to that basis is upper triangular?

>>11058655
find standard bases s1, s2, ..., sn
let your new bases be e1=s1, e2=s1+s2, ..., en=s1+s2+...+sn

>>11058655
Write block matrix (A|E) where E is the identity matrix, perform row operations to make A into upper triangular. This will transform E into some other matrix and rows of this matrix is the desired base. I will explain why this works only if you're interested.

>>11058309
no, it was a stupid question asking for life advice, probably didnt belong on sci anyway
im not mad at you anymore, anon, and i appreciate what you do

Let me answer a couple of old ones:

>>11044795
>If the weak force is mediated by massive bosons does that mean the weak field doesn't change at the speed of light?
Exactly, and it speed will depend on the energy the boson carry. It's just an usual massive particle.

> If so how fast should it theoretically travel?
Like any massive particle, if you pump enough energy it can get close to c but never reach.

>>11052450
>if there is a magnetic field originating and ending at the earth's poles, does that mean there's an electric current circulating around the equator
You need a varying magnetic flux you create an electric current. Because Earth's magnetic field doesn't change (it changes, but very slowly) and the equator doesn't change its shape (it changes, but very slowly), then no.

>>11058619
Isn't the generated ideal supposed to have the form of [math]\langle x,4 \rangle = \{ p(x)x+q(x)4 : p(x),q(x) \in \mathbb{Z}[x] \} [/math]? So p and q could be any element from the ring, and then the product could have an arbitrary (finite of course) degree. That's why I was struggling factoring the polynomial at first

>>11058700
I already gave you a very easy solution.
The only nonzero zero divisor in Z_4 is 2, which is nilpotent. <x, 4> is primary.

>>11058723
Yeah thanks I'll probably end up doing it like that. I'm just trying to prove both of the statements in >>11058312 first.

>>11058670
please explain or link to proof

>>11058766
and is this what you want ? A is a matrix of some linear transformation

>>11058781
yes (i didn't try anything yet)

>>11058697
>You need a varying magnetic flux you create an electric current
It's pretty obvious he is asking about the orgin of this magnetic flux, anon.

>>11058796
okay, I probably fucked up and it doesn't work. I'll think about it for a while and reply again

>>11058805
>>11058796
okay. it's easy if you allow distinct bases for input and output. if you want those bases to be the same (that's how you formulated the question), then it's not even always possible. because upper triangular matrix implies that the first basis vector is an eigenvector and those don't always exist (over R at least). even if it is possible, you won't find it just by elementary means. you will need eigenvectors at least. actually the Jordan form gives you what you want.

Are there any anons who studied digital signals?
I'm looking for the equation for Frequency-shift keying (FSK) which can explain correlation between signal/noise power to the difference between the higher and lower frequency.

>>11058903
Signal power is dependant on the period of a continuous signal.

>>11056713
Why can't I browse the internet for 2 weeks without falling in love with some imaginary characters?
Why is Momiji so cute. Why are human creations so cute.

>>11059001
Psychologically based on Maslow’s Hierarchy of Needs, it’s because you lack intimate relationships and therefore project your feelings towards whatever reflects back any significance of warmth. In this case, it’s what is identified as close to human, but not human, but an imaginary character and as a result able to fulfill even the slightest bit of intimacy for yourself. Cute is desirable and desirable is good.

>>11059008
As much as it troubles me, I can't seem to get rid of my human nature.

>>11058969
Yes, but somehow I got worse signal/noise ratio with bigger frequency deviation, I read that it should be the opposite.

>>11059018
It’s alright, it’s like Plato’s cave, what is real isn’t always what you’ve only ever been exposed to, but what you’ll judge as real. This is life. This is human nature.

>>11056713
So if an atom is the smallest form an element can take. How many atoms would be present in say a cup of water? Is there an estimate at all? Trillions? Since water is ina liquid form, that means they slide past each other and all that, but still retain even at electronic levels the properties of water? Does that mean there’s quintillions of water atoms in the air?

>>11056713
An object falls from a height of h meters. In the last second before hitting the ground, it travels h/3 meters.
How do I even begin to approach this?

>>11059058
Isn’t that just a calculus problem

>>11059059
IIRC there is a non-calc solution, but I can't come up with either.
I can integrate acceleration w.r.t time, but not distance.

>>11059049
>How many atoms would be present in say a cup of water? Is there an estimate at all? Trillions?
Using the molar mass, you can get a number. The MM of water is 18g/mol.
Say you have 100mL water. Pure water is 1g/mL. Divide that by 18g/mol, you end up with 5.55 moles of water.
What is a mol? It's just a really big number. It's like saying a dozen, only instead of a dozen, it's 6x10^23. That's 600 billion trillion.
So there are 3.33 trillion trillions of water molecules in 100mL of water.
>Does that mean there’s quintillions of water atoms in the air?
Say air is 1% water, by mole. Using the eqn PV=nRT, at 1atm, 1L, 300K, and with R=0.082, there are n=0.04moles in 1L of air. Times 1%, there is ~0.0004moles of water. That's 240 million trillion water molecules, or 240 quintillion.

>>11059058
~150 meters

>>11056713
>>11056713
For particles with spin, the [math](2j+1)[/math]-multiplets of a spin-[math]j[/math] particle are given as representations of some [math]SU[/math] group (for example for spin-[math]1/2[/math] we have [math]SU(2)[/math]). I'm fairly confused as to if indeed these are representations of the group or its lie algebra. In the spin-1/2 case, the groups [math]SU(2)[/math] and [math]SO(3)[/math] have the same lie algrbra:
[math] [J_a, J_b] = i \epsilon _{abc} J_c [/math]

so elements of [math]SO(3)[/math] might as well satisfy it. How do i know if i'm representing [math]SU(2)[/math] or [math]SO(3)[/math]?

Sorry if i'm being a bit naive, just starting to grasp how reps fit into the picture (not pic related, thats unrelated).

>>11059049
>How many atoms would be present in say a cup of water? Is there an estimate at all?
Yes, there is a rough estimate, although as you might expect measuring an amount of objects that absurdly small makes it extremely hard to be too accurate, you just estimate it from the volume of water, the size of the molecules and their density. A quick google search says it's 8.36 x10^24 molecules, that's about 8,360,000,000,000 times your estimate of a trillion. So yeah, molecules are stupidly small.

>but still retain even at electronic levels the properties of water?
What do you mean by properties of water? A lot of them are consequences of their properties at a molecular level, so I guess in that sense it's true, but it's hard to identify those properties at such a scale.
>Does that mean there’s quintillions of water atoms in the air?
There's probably much more than that, but yeah there's a fuckload of water molecules in the air as well.

Also, formally speaking "water atoms" aren't a thing, they are water molecules which are basically a bundle of atoms of other basic elements (oxygen and hydrogen of course).

I like what you're doing anon, keep it up!

>>11059271
what a fat hand.

>>11059272
whats ur problem lol?

>>11059058
>>11059068
lol

>>11058821
retarded faggot

>>11059289
good comment anon

>>11059058
>>11059068
Alright. You have ye ole [math]\Delta s=v_0 t+a t^2[/math]. Plus, you have [math]v_f ^2=v_0 ^2 + 2 a \Delta s[/math], which is Torricelli.
We use Torricelli to determine its speed after falling two thirds of the fall, [math]v_1 ^2= 20 \frac{2}{3} h[/math], and then the ole trusty to finish the job [math]\frac{h}{3}= v_1 1 +10[/math].
My bad for formatting errors, I'm on my phone.
>>11059271
Anon who?

op anon

>>11059104
First of all, all spins labeled by [math]j\in \frac{1}{2}\mathbb{Z}[/math] are irreps of [math]SU(2)[/math]. You're not representing anything else for higher spin particles.
Second of all, the very fact that [math]j[/math] can take half-integer values is due to the rep theory on [math]SU(2)[/math]. As a manifold, [math]SU(2)[/math] double covers [math]SO(3)[/math] so all reps of [math]SO(3)[/math] factor through the central extension [math]1 \rightarrow \mathbb{Z}_2 \rightarrow SU(2)\rightarrow SO(3)\rightarrow 1[/math]. You can check that irreps of [math]SO(3)[/math] are labeled by integers by, for instance, computing eigenvalues of the Casimir [math]L^2[/math] and using Schur's lemma. This is why the [math]j[/math]'s for the spherical harmonics are integers.

>>11059089
Wow, looks like >>11059049 has some really good estimation skills.

I killed a bug in my room and it fell behind a hard to reach area
I'm creeped out that its body is still there. how long do bugs take to decompose? It was a cricket

>>11060650
>It was a cricket
You're fucked.

>>11059917
thanks yukari!

Im a first semester math student and I dont know how to learn math really. I have a final appointment for my ADHD diagnosis in 4 weeks. Apart from that I check many marks for Aspergers too imo (very bad with social stuff as in i dont understand, sensory issues, bad at keeping friends due to neglecting friendships even though i like them, generally feeling like being on a different wavelength than most other people).
During highschool I generally got good grades without doing anything, so I never developed good habits to learn stuff.
That being said, based on my grades in highschool I dont feel I do have a solid foundation in math. Im not very confident.

I do have some stimulants on hand which are very helpful, but I dont know how to make use of that time. I can sit down for hours, but at the end of the day I feel I havent learned anything new. I have no structure on how to approach this. Ideally I want to follow some sort of algorithm that I can repeat for every new concept I have to learn.

In Germany (probably every other decent math program too) you do Linear algebra and Analysis in the first semester.
We are provided with a script, book recommendations and weekly practice problems you have to submit.
I go to the lectures and copy whatever the professor writes on the board. Then at home I go trough them and try to do the problems from the weekly sheet. This went semi well in the first week. While my understanding of these subjects is greater than before the first day of university, I doubt I will get a good grade at the end of the semester if I get better at the current speed. I also didnt manage to complete all problems. I feel I sometimes get so lost in the problems that I spend 45 minutes without really progressing and then only going further with help from an online source. Then even after "completing" the problem I dont feel I really grasped it.

>>11061013
Our Analysis course is structured like this: Proof by induction -> axiomatic characterization of real numbers -> sequences, limits, Supremum, Infimum -> ...
Linear Algebra: R^n -> vectors -> linear equations -> ....

Pretty basic like most other universitys, everything with proofs and so on.

Idk I feel scared and my confidence is really low. Im anxious and I dont know how to approach this with a clear head. I refuse to give up so early though.

So do you have some advice? Maybe some sort of algorithm on how I should approach all these topics from the lectures. Im lost

I need help with basic thermodynamics, this is what our notes say:
we know the differential of the internal energy: [eqn]dU=TdS-PdV+\mu_idn_i[/eqn]
and that the internal energy is a linear homogeneous function. If [math]\ f(x_{1}, . . ., x_{n})[/math] is linear homogeneous then:
[eqn]\sum_{i=1}^{n} \frac{\partial f}{\partial x_{i}} x_{i} =f(x)[/eqn]
so we calculate the partial derivatives [math]\frac{\partial U}{\partial S}=T[/math], [math]\frac{\partial U}{\partial V}=-P[/math], [math]\frac{\partial U}{\partial n_i}=\mu_i[/math] and this lets us express U as:
[eqn]U=TS-PV+\mu_in_i[/eqn]
Now how does this make sense? U is a function of T,S,P,V and n_i, so how can it be correct to just ignore the dependency on T and P and slam it back together like that?

>>11061179
U is only actually a function of S, V, and n_i. You can think of T and P as defined by dU/dS and -dU/dV respectively, partial derivatives of course

>>11061013
are you me?

>>11061263
Likewise, the mu_i's are defined as dU/dn_i. Enthalpy is instead defined as a function of S, P, and Ni, for which V is defined as dH/dP

is there a website for plotting on an argand diagram in the [math] e^{i\theta}[/math] notation?

it's just for checking answers to roots of unity questions

>>11061483
you can always convert to rectangular by Euler's formula, then plot it as you would in R^2

>>11060164
hi tuba jam

If the Loerntz factor used in relativity, [math]\gamma=\frac{1}{\sqrt{1-\frac{v^2}{c^2}}}[/math], can be rewritten as [math]\gamma=\frac{c}
{ \sqrt{c^2-v^2} }[/math]:
Is it equally valid to say
>[math]\gamma=\frac{c}
{ \sqrt[4] {(c+v)(c-v)(c+iv)(c-iv)} }[/math]
?

>>11060164
Yeah, that blew me away too. I thought he was just pulling numbers out of his ass, but lo and behold.

>>11061634
Screwed the formatting:
Is [math]\gamma=\frac{c}{\sqrt[4]{(c+v)(c-v)(c+iv)(c-iv)}}[/math] equivalent to the Lorentz factor?

>>11061634
No. [eqn]\sqrt[4]{(c+v)(c-v)(c+iv)(c-iv)} = \sqrt[4]{c^4-v^4} \neq \sqrt{c^2-v^2}[/eqn]

how do you do a three layer fraction (don't know the term for it) in latex

>>11061712
Like [math]\frac{1}{1-\frac{1}{x}}[/math]?
>>11061015
In my experience, my understanding of linear algebra was extremely static and simple from inception, while my comprehension of analysis grew gradually more geometric and linear algebraic.
So I'd tell you to just focus on the linear algebra and build off the analysis from it.
>>11059444
Oh, anon me.
Thanks.

If X^23=10000 then how can I calculate X? I.e. if a number has to be raised by a certain power to result in a certain number, how do I find that number?

>>11061740
No like x/x/x

>>11061179
dU is a "total exact differential". This means that it's only function of S, V, and n_i (here).

>>11061740
>it shows up correctly in Tex
Fuck. Let's try again.
[math]\frac{1}{1- \frac{1}{x}}[/math]
>>11061818
https://tex.stackexchange.com/questions/202017/a-fraction-with-three-stories-and-two-bars-of-equal-length
>>11061817
By hand? Newton's method.
In a computer, just ask it to calculate 10000
^(1/23)

>>11061848
>In a computer, just ask it to calculate 10000
>^(1/23)
Yeah it's for a program, this is great thanks!

>>11061817
x^23 = 10000
=> log(x^23) = log(10000)
=> 23*log(x) = log(10000)
=> log(x) = log(10000)/23
=> x = e^(log(10000)/23)
Log base 10 would also work, and is simpler in this particular case: x=10^(4/23)=1.492495545.

>>11061263
>>11061820
Alright thanks. So, just for clarity, writing this would be correct?
[eqn]dU(S,V,n_i)=T(S,V,n_i)dS-P(S,V,n_i)dV+\mu_i(S,V,n_i)dn_i[/eqn]

>>11061848
thanks for the fraction thing

>>11056801
>>11043468 # (Yes.)
Alright, which one?

What do you call it when you multiply some algebraic term by a fraction which is equal to 1 in order to factor/simplify whatever?

e.g. Multiplying by [math]\frac{x}{x}[/math]

>>11061179
In the context of geometrothermodynamics, the internal energy is encoded in the Gibbs contact 1-form [math]\Phi = dU - \sum_i I_idE_i[/math], where [math]I_i, E_i[/math] are the symplectic coordinates in phase space denoting intensive and extensive quantities. Physical trajectories are characterized by curves in phase space on which [math]\Phi = 0[/math]. Now in order to express [math]U[/math] purely in terms of partial derivatives along [math]I,E[/math], we must be able to integrate [math]\Phi[/math] (i.e. find a contact 0-form [math]\theta[/math] for which [math]\Phi = d\theta[/math]) along physical trajectories.
Globally speaking (i.e. not just on physical trajectories), by treating [math]T,S,P,V[/math] as (orthogonal) coordinate functions and [math]dS,dV[/math] their differentials, this can be accomplished locally by an application of Poincare lemma, which is what your notes had done. As long as the physical trajectories are at least [math]C^1[/math] and Legendrian, Poincare lemma can be applied. Obstructions are characterized by [math]H^1[/math] of the physical trajectory.

>>11062033
Euclid's Elements not only introduced great mathematicians, such as Euler and Gauss to proofs, it introduced humanity to them. There is absolutely no better text.
Kiddos nowadays just use the Book of Proof, tho. It's available for free on the internet.
>>11062076
We usually say "multiply above and below" where I'm from.

I'm not sure if I'm the worst brainlet ever for this question, or if it's something poorly explained by schools. Some background, I'm definitely no scientist, but I've had a renewed interest in chemistry (mainly as a consequence of interest in geology and minerals). My question however is about physics...

What in the ever loving fug does it mean that every action has an equal and opposite reaction?This idea is utterly insane to me. It's always glossed over or stated as intuitive, but how can it be true (or is it really true) that if you apply a force the exact same amount of force pushes back? How would things ever move? Wouldn't everything either be completely static or bouncing perfectly off each other with nothing ever being able to push anything?

>>11062302
when you punch someone really hard, his jaw breaks, but also your hand hurts.

>>11061977
Yeah

>>11062302
The key is that the acceleration, which is how the motion changes, is F/m. So with the same force, a more massive object has a smaller change in motion than a smaller mass.

>>11062302
Just what is force? Apparently it has something to do with field theory.

>>11062306
That's the kind of sentence equated to the equal and opposing rule that just seemed more confusing to my grug brain.
>>11062315
Thanks lad. That actually helps clarify things.

>>11062302
When you just move your hand and there's nothing in front of it, it just moves freely, right?
But when there's a wall, your hand stops. By Newton's first law, if it stops, then there's a force. The third law essentially just specifies the magnitude of this force.

>>11059442
how do i get a wife like marisa?

>>11062422
You don't.
Alimari or nothing.

>>11063172
Alice is too good for her

If G is a finite group such that g^4=1 for all g in G, prove that the center has a nontrivial element.
I can assume that G has an element of order 4, else each element squares to 1 and then it can be shown that G is abelian. I want to claim that g^2 commutes with every element of G, but I don't know how to show it.

>>11063456
Forgot to mention that g is the element of order 4.

>>11056713
Why does sleep help with learning?

>>11056713
Good-day /sqt/. I'm on an engineering mathematics module which so far I've found easy as its essentially a refresher of GCSE/ A level content. This seems to have gotten hard though as I've gotten to vectors. For some reason no matter how hard I study I can never visualize problems at all, its like I have actual brain damage. There was a point where I could imagine some problems by picturing Osaka from Azumanga Daioh being acted upon by whirlwinds, or rowing a boat through a harsh current, or operating a pulley etc. etc. but now I seem unable to get even the basics right.

With that polava out of the way, how do you complete these questions? The worksheet mentions using the triangle rule which I did for the first question, though it only appears to mention the cos rule afterwards for problems that clearly do not use the cos rule. Have I simply imagined the forces acting in the wrong ways? How do you visualize this type of stuff, when something is due in X direction I picture it as an object being acted ON a stationary point rather than said point travelling in a direction WITH the force. Its such a simple thing and yet so complicated.

>inb4 not science of math
"A rational consumer has 10 monetary units at his disposal, that he can use to buy two goods: A, or B. If he successively consumed six units of good A, the consumer would feel the following "satisfactions": 10, 8, 6, 5, 2, and 1, and if he did the same for good B, he would feel 20, 15, 12.5, 10, 5 and 2.5 satisfaction.
How much of each good will he buy?"

>>11063990
Forgot to mention that one of good A costs 2 monetary units, and one of good B costs 2.5 monetary units.

>>11063835
>this exercise can be done using vertical and horizontal components
>>11063990
>>11063997
3 units of B, one unit of A.
Just calculate the ratio of utilities and compare it with the price ratio.
In other words, maximize the utility price ratio.
Once you have that, just keep choosing until his money runs out.

if there exists a rational number between two real numbers, how is Q countable?

>Lang - Basic Mathematics
>Enderton - A Mathematical Introduction to Logic
>Enderton - Elements of Set Theory
>Bertsekas and Tsitsiklis - Introduction to Probablity
>Andrews - Number theory
>Spivak - Calculus
Is this a good study plan? Also, what about geometry? I'm thinking about maybe doing Kiselev before Spivak but I don't really know.

>>11064014
But he can afford to buy 4 of B, so why doesn't he just do that if B makes him the happiest? Is it because 1A + 3B = 9.5 and he would still have money to spare?

>>11064090
Once he has consumed three B, he has 2.5 dollars, and buying either A or B would give him 10 utility.
He can purchase either one and obtain the same utility, so both options are correct, 4 B or 3 B and one A. I just picked the one that keeps 50 cents.
>>11064072
No.
Just read Spivak all the way through.
And get a linear algebra text.

>>11064115
>Just read Spivak all the way through.
But anon, don't I need to have some foreknowledge before I try that?

>>11064045
why should it be ?

>>11064124
No, studying babby set theory and logic is indistinguishable from wasting time.

>>11064045
Nice observation. It can seem like a paradox, though it's not. Q is countable because it's really easy to inject it in the naturals.

>>11064155
Not really, it's just that it's much easier/"funnier" to learn it in the context of calc.

>>11064045
The construction of Q (and the countability arguments) make no appeal to R (in fact, you need to construct Q to construct R).
However, constructing R (via Dedekind's cuts) requires it to be uncountable and fundamentally different in nature. You can take two infinite subsets of Q with no upper limit on the "lower" one and no lower limit on the "upper" one, but you can't do that with R.

>>11064045
The same rational number is between lots and lots of pairs or real numbers (uncountably many of them)

Is 74 Numberwang™?

>>11064312
no

>>11064335
thanks

When writing the solution of a system of equations, how do I write it as a span of vectors? It is confusing

Is circadian ryhthm actually real or is it just an observable pattern?

Can I learn trig alongside Calculus if my algebra skills are up to par?

>>11064579
Are observable patterns real?

>>11064793
By the time you do calc you should already be very comfortable with trig functions, logarithms, and exponentials.

>>11063835
By triangle rule I think they meant connecting the two vectors head-to-butt. The resulting vector, when adding, is then the vector that completes the triangle.

>>11064793
Yes, the very basics of calculus only require arithmetic

>>11064793
Its going to become absolutely necessary for integral and multivariable calc so yes.

I've cured my cumbrain (by myself, it was hard). Now if you tell me how to cure my massive laziness and brutal procrastination, i will forever love you! (no homo)

>>11064434
You have Ax=b.
So if you have Az=b, for some z, then every solution can be written as z+ ker A, where the kernel is the set of solutions of Ax=0.
>>11064987
Get really angry at niggas for knowing more than you do until studying hard becomes a habit.

Is 487411 Numberwang?

I'm having some trouble understanding the disk and washer method for finding volume.
My most pressing question is are there two different ones for just disk and disk AND washer? Or are they both the same thing? My prof never really went into depth about it.

I feel like I understand the way to find volume using shells pretty okay. But for some reason trying to do disk/disk and washer method is giving me trouble.
To piggy-back on that, how do you know whether using shells or disk/washer would be easier for a volume problem? How can you tell which method would be the most straightforward/easy for a given problem?

is the breakdown of IQ into spacial, verbal, etc a valid categorization?

When talking about vector spaces (or more generally modules), I've seen that even if we are talking about a linearly dependent set whose cardinality we don't know (so could be infinite), we still always take finite sums to test for linear dependency. Does the definition of linear independence always require that the sum we evaluate (where the scalars have to 0 iff the sum is 0) be always finite? Or am I just reading cases where the set S is finite?

how will the third equation here help me to get H

>>11065133
https://en.wikipedia.org/wiki/Cavalieri%27s_principle
>>11065835
This suffices when the dimension of the vector space is finite.
>>11065909
Integrate term by term?

>>11065929
A finite-dimensional vector space can't have an infinite linearly dependent set though
I'm confused cause the cardinality S is never specified, but the fact that they use an arbitrary indexed set S with indices [math] i \in I [/math] leads me to believe it could be infinite. In that case I don't know why the linear dependence sum should be restricted to a finite sum regardless of the set S' cardinality.

>>11065108
No.
>>11065835
>finite
Yes. Infinite sums aren't even defined for pure vector spaces, they only show up when you add a metric or a topology.
>>11065909
Can't you integrate a polynomial?

>>11065942
>Can't you integrate a polynomial?
i dont know why lambda is there

>>11065945
>why lambda is there
Functional calculus. Essentially, write out the Riemann integral, but take the sums as matrices.
The integral up to T is most likely a typo and should have been up to |T|.
I say most likely because I could be wrong.

>>11065941
>A finite-dimensional vector space can't have an infinite linearly dependent set though
Why not? Any infinite set of spanning vectors will be linearly dependent on any finite-dim vector space.

>>11065979
>why not
Tip: it's finite dimensional.

>>11065979
I'm sorry I meant to say independent, my bad. The set S I'm talking about is a set whose linear dependency we still don't know and want to test, so what I'm trying to say is that I don't think we could assume the vector space is finite-dimensional, because in that case if S is infinite then it will necessarily be linearly dependent

>>11065990
Wait, he said linearly dependent, not independent.
What the fuck even is the point of saying that.

>>11065997
See, I knew something was weird.
First of all, does [math]V[/math] have a norm equipped so you can talk about convergence? Or is it formally arbitrary-ranked? If it's the latter then testing linear independence is by definition the same as testing linear independence on every finite subset.

>>11066012
>>11065942
Yeah I guess in an arbitrary vector space the convergence of the series may not make much sense then. Thanks

At an introductory level, how "hard" (in terms of mathematical rigor and non-triviality of proofs/theorems) is abstract algebra compared to real analysis?

>>11066352
much easier

>>11065941
the point is not about "why don't we allow infinite sums?", but rather "we don't know what an infinite sum is". the axioms of vector space define addition as a binary operation, it easily extends to any finite number of summands by associativity. but we don't know what an infinite sum is, we need some kind of limit and convergence available and this is not a priori the case in a general vector space or module.

>>11066395
second

I'm pretty much completely lost here, I've no idea how to relate taylors theorem to epsilons and deltas

>>11066352
im taking both this semester and breezing through abstract algebra, real analysis is hard though (my question)

Is this a legit good recommendation for physical chemistry or is this one of the /sci/ memes?

>>11066352
The difficulty in learning analysis comes from the fact that you have an intuition for the objects under study (from calculus or something) that's wrong and needs to be corrected. Whereas in algebra, you often have no intuition and need to learn what intuition to have. So it really depends on the individual which of these scenarios is more difficult.

I know it sounds fucking stupid, but what is the equivalent of Italian "Elettrotecnica" for english courses?
It seems to be related to Circuit Analysis, but the programs diverge pretty hard, and I don't have much time left to study
I ask because the material given to me has the most annoying bug, it's 56 .pptx with misaligned OLE objects
It's giving me a solid headache trying to decipher some of them

Is there any way I can do math problems on my computer? I'm too messy to use paper and I always fuck something up and have to scratch it.

>>11067858
Latex
Math Fonts
Math Software

Given a set X and a metric d on it such that forall x,y in X d(x,y) = 0 if x = y and d(x,y) = 1 if x != y.
Is it true that for any point x in X, a sphere centered in x with a radius 1 is equal to X\{x}?

>>11067898
Yes.
>>11066590
The hint is immediate from Taylor, just pass x^6 to the other side and throw some Landau's.
I'm extremely confused about what you should do with the hint, tho.
I'm honestly getting the impression there's a typo somewhere. Why would f grow faster?
By the by, the hint gives that the function is bell-like around zero, so local minimum.

Brainlet here just trying to exercise my jaw to make it more robust according to Wolff's law, I'm having trouble interpeting https://www.sciencedirect.com/science/article/abs/pii/0021929084900034?via%3Dihub , what would be a static or dynamic loading? Is pushing against my mandible in a resisted closing a static or dynamic force?

>>11068034
pushing my mandible in a resisted opening

Here is what i have problem with, we have 2 function:
[math]
$f(x) = f(x-1)+3.0*f(x-1) * (1-f(x-1))$
$g(x)=4.0*g(x-1)-3.0*g(x-1)^2$
[/math]

I need to compare float/double performance on both function and explain difference.

So second function is obviously expansion of first function, so why results are different? I mean in second function difference between double and floats results are smaller than in first function, it means second function is better, why?

I still dont understand why in recursive function, first few elements of series is similiar in float/double (difference is 10^-5 in general), but in later elements of series there could difference of 10^-1 or even fucking 10^0, whole 1 of difference!
But machine epsilon of float is 10^-7 it means first 7 digits of representation should be exact, and machine epsilon of double is 10^-16 it means first 16 digits should be exact, so first 7 digits of float result should be same as in double result, whats going on? Why there is such a large error in later results?

>>11065990
By the way,what anime is this from?

>>11068205
>I still dont understand why in recursive function, first few elements of series is similiar in float/double (difference is 10^-5 in general), but in later elements of series there could difference of 10^-1 or even fucking 10^0, whole 1 of difference!

accumulated roundoff errors.

>>11068205
>why results different
Floats and doubles do that.
>first sixteen digits should be corrects
No, no, no. The computer doesn't round at the end. It rounds before and after every step.
>why is there such a large error in later results
It calculates and rounds, then calculates and rounds, iteratively. Every rounding accumulates more and more error.
f gets it wrong more because it rounds
more. The path of least error is the path of least rounding.
>>11068245
Touhou.

>>11068483
Ok, that makes sense. What i dont get why this function:
[math]
f(x)=f(x−1)+3.0∗f(x−1)∗(1−f(x−1))
[/math]
is "better" than this function
[math]
g(x)=4.0*g(x-1) - 3.0*g(x-1)*g(x-1)
[/math]

By "better" i mean: i compare 1: difference between float and double result of first function with 2: difference between float and double result of second function. In general error of second function is smaller by aprox 10^-7 and smaller. Why are those difference not equal?
My ideas:
1: compiler finds that g(x-1)*g(x-1) is just g(x-1)^2 so there is only one call squared, and not 2, less calls smaller error. I work with iterative version so probably not
2: you cant really compare two double values, you can just check if difference is <= epsilon. Problem with this idea: double epsilon is 10^-16 so difference between those differences should be much smaller right?

>>11068205
> f(x) = f(x-1)+3.0*f(x-1) * (1-f(x-1))
> g(x)=4.0*g(x-1)-3.0*g(x-1)^2

> So second function is obviously expansion of first function, so why results are different? I mean in second function difference between double and floats results are smaller than in first function, it means second function is better, why?
Both functions have 4 arithmetic operations: f() has 2 multiplies, 1 addition, 1 subtraction, g() has 3 multiplies and one subtraction. But the largest source of error in most calculations is additions or subtractions which almost cancel, i.e. where the magnitude of the result is less than that of the operands.

Consider 1.234578-1.234567 = 0.000011. The operands have 7 sig.figs. of precision, the result only has 2, so the relative error is magnified by 10^5. Multiplication just sums the relative error: if a and b have relative errors of e_a and e_b, their product is a(1+e_a)*b(1+e_b) = a*b*(1+e_a+e_b+e_a*e_b), the absolute error is a*b*(e_a+e_b+e_a*e_b) and the relative error is e_a+e_b+e_a*e_b (and the e_a*e_b term is negligible compared to the total).

f() has one addition and one subtraction, g() has one subtraction. So all other things being equal, f() has twice the scope for cancellation error. If this is the cause, you would see the error grow at twice the rate for f compared to g, i.e. f(n) would have roughly the same error as g(2*n).

>>11068613
thank you a lot.
According to my calculation difference between f(3) error and g(6) error is about 2*10^-6. I guess with such a small numbers i dont know what "roughly the same means". However my difference function is returning double which has 10^-16 epsilon, it means difference should be smaller than 10^-6,
however difference function takes float and double as parameter, but now i am passing 2 double values to it as in:
difference(difference(a,b), difference(a,b))
so that could be cause of such large error as well

>>11056713
Would you be more likely to buy a math text book if it had big round slopes of breasts on the cover instead of the normal pic of a bridge with repeated stone arches,

How come if you look up chemical compounds (lsd for example) it shows tons of different pictures? can anyone link a good video talking about how chemical structures are made and explains them?

>>11068483
Thanks a lot mate.

I have a genetics question.

Is there greater heritability of hair colour between a mother and daughter than between a mother and son? Also father to son?

Let's say we have two white people. If a blonde haired mother has offspring with a brown haired father, is it more likely that the daughters will be blonde, and the sons get brown hair, or is the chances of the sons and daughters getting one colour or the other the same?

I understand that a dominant allele will override a recessive one but is there a degree of sex-specific insulation to that effect, and the (as I understand) dominant brown hair of the father won't impose on the daughters as much as it would the sons?

I noticed that effect because I know a couple where the mother is very blonde from Finland while the father is German and brown haired, and they have 4 blonde daughters. But that could just be chance.

Why does almond milk make me pee so much?

>>11056713
How do I download books from #bookz using irssi?

Anyone got a ressource on how to prove transvections generate the linear group ?
I'm not sure if I understood the proof in the textbook.

>>11071025
Nevermind, I figured it out.

>>11069703
>Is there greater heritability of hair colour between a mother and daughter than between a mother and son? Also father to son?
no

>>11056713
if thermal energy is lost in blackbody radiation, and light has wave properties, couldn't out of phase blackbody radiation waves cancel out, destroying the energy?

>>11071393
Phase only matters for coherent monochromatic waves. Black bodies sources are not coherent (the photons emitted vary widely in polarisation) and aren't monochromatic.
Two lasers can interfere because lasers are coherent monochromatic sources.

Need some verification on this meme problem.
Graphed on Desmos: https://www.desmos.com/calculator/qevzqjrg4k
Pretty much what I did was split the yellow area into 2 areas where the lower function would change form [math]y=1/2(1-2(-x^{2}-x)^{.5})[/math] to [math]y=\sqrt{1-x^{2}}[/math] at [math]x=-\frac{5}{8}-\frac{\sqrt{7}}{8}[/math] and did the usual integration business of [math]A_{intersect}=\int _a^bf(x)dx-\int _a^bg(x)dx[/math] where [math]f(x)>g(x), x\in[a,b][/math].
The smaller area being
[math]A_{small}=\int _{-1}^{-\frac{5}{8}-\frac{\sqrt{7}}{8}} 1/2(2(-x^2-x)^{.5}+1)dx -\int _{-1}^{-\frac{5}{8}-\frac{\sqrt{7}}{8}}1/2(1-2(-x^2-x)^{.5})dx\approx 0.012257759872[/math]
The bigger being
[math]A_{larger}=\int_{-\frac{5}{8}-\frac{\sqrt{7}}{8}}^{\sqrt7/8 - 5/8}1/2(2(-x^2-x)^{.5}+1)dx-\int_{-\frac{5}{8}-\frac{\sqrt{7}}{8}}^{\sqrt7/8 - 5/8}\sqrt{1-x^{2}}dx\approx 0.134123499657[/math]
Total gives [math]0.146381259530[/math]

Is there a way to do this problem purely from a geometric standpoint using only the area formulas for a square and circle?

>>11071502
Yes.
My bad for the image's rotation, my phone refuses to cooperate.
Anyhow, calculate the area of the arc O'AB (in the large circle), and subtract the area of the triangles O'OB and O'OA. This gives the area of the OAB thingy.
OACB is just an arc, and OACB-OAB is the desired area.
Also, pretend I attached an image of a cute lass. I tried to scribble one on the side of the image, but it didn't work.

So I'm trying to design a monitor mount that I can 3D print, and have it still be adjustable as well.

Thinking of just having the monitor hook onto the stand, and just using a string to keep it tight.

Any idea if it would work or would the monitor still be too heavy and just pull it forward unless I made the feet really long?

A normal monitor might have a lot of weight in the mechanism in the back, and the 3D print would be pretty light.

Any dentists here?

I have some questions about my two broken upper frontal incisors

>>11069703
For the heritability of a gene to be affected by sex it would need to be located on the X or Y chromosome. The most well researched hair colour gene is MC1R which is found on chromosome 16. There are over a dozen other genes associated with hair colour (mostly involved in the melanin production pathways) and the only one located on a sex chromosome is GPR143 which is on the X chromosome (meaning females would have two copies of this gene whereas males would only have one). The protein produced by GPR143 is mostly found in the eyes and skin. It seems to have more involvement in eyesight than hair colour.

Sex of the children would have little affect on hair colour.

>>11069180
>tons of different pictures?
Chemical compounds are represented in various ways. Usually the way someone draws a compound is based on what they're trying to show or explain. Most will be line drawings where lines represent bonds between atoms. Since carbon is very common people tend to assume when two lines meet and no atom is specified, it's carbon.

Some drawings show the three dimensional shapes of molecules. Usually this is done to emphasize that certain atoms are farther apart (or closer to each other) than what a 2D drawing seems to imply. Molecules are not flat though most are drawn as if they are because their shape sometimes isn't important.

Instead of lines sticks and balls are used to show the relative size of the various atoms and their "spheres" of influence. Again usually only helpful if there is a reason to point these things out.

Of course with LSD you'll get people drawing more artistic version of the chemical which are meant just to look cool because drugs man.

Here's some introduction to how elements bond to each other and form compounds and the various way people show bonds.
http://employees.csbsju.edu/cschaller/Principles%20Chem/molecules/index%20to%20molecs.htm

I actually hacked this on /sci/ like two years ago, a fun midwit problem which admits of both a calculus-based and non-calculus based solution (if you take as given formulae for circular sectors/segments). Another user made a rather clunky numerial picture with the solution which has been reposted frequently (his numerical solution was correct) but I preferred to keep things exact.

Your terms and values look about right. To your final, most substantive question (since the body of your post is you thinking through the calc approach and sounding like you've got those details right), I refer you back to the circular sector/circular segment stuff I mentioned above. In a certain strict sense, "the area formula for a circle" itself depends on calc.

When you actually do the thing as a closed form it's a frightful inverse trig thing with like 8-12 terms using the non-calc approach, which turns out (it had better be) to be the same as the calc approach once some simplifying is done.

Being on this board really makes me want to do some math again. I really enjoyed my courses in U but elected to study something more libby

Recommend me some Calc materials please. I'm probably back at basic limit shit again.

>>11072621
Pick one: (Spivak, Apostol, Courant) and stick with it.

>>11072658
Apostol it is. Thank you

>>11072677
Imagine being so obsessed that you have to pick an inferior book cause the author used some terms literally decades before the issue they are associated with was even relevant.

>>11072717
Guy had a better academic career on paper. Of course Congrats, you got DABBED ON

>>11072621
Just buy some used textbook for like $10

>>11072816
Yeah nah I got this one already

>>11072840
I like real books better

>>11072851
Well I got delivered in 2 minutes

I'll get the hard copy later down the line

>>11072840
>62MB
Who's the monkey who made that pdf?

Any crystallography and crystal chemistry book recs?

>>11073137
First chapter of Kittel for crystallo

>>11072677

What's a master's seminar?

>>11068613
>>11068665

i wonder if squaring a number is generally more accurate than taking the product of two different numbers.

I have N boxes and N apples. I select one box at random, regardless of how many it already contains, and put one apple in it. Repeat until all apples are in a box. I then sort the boxes in increasing order by number of apples they contain.

What distribution results?

On average, what fraction of the total number of boxes remain empty, as a function of N, and what does this value approach as N goes to infinity? On average, how many apples will be in the box with the most apples, as a function of N, and is this function unbounded as N goes to infinity?

>>11056713
Hi /sci/, I started learning arithmetics and group theory and I'm having trouble with the types of proof that require you to build/find an isomorphism.
For example how could I find a subgroup of [math](\mathcal{S}_5, \circ) [/math] to which [math] ((\mathbb{Z}/6) \backslash \{0, 3\}, +_6) [/math] is isomorphic? ([math]S_5[/math] is the permutations of integers up to 5).

Likewise, how could I prove that
[math] ((\mathbb{Z}/4), +_4) is not isomorphic to the subgroup [math]D_6\backslash\{ e, r^2, r^4\}[/math] where [math]D_6[/math] is the group of symmetries of a hexagon.

>>11074389
>how do I show two groups are isomorphic
It's a two step program:
>uniquely characterize one of them
>show that the other satisfies said characterization
For example, [math]Z_4[/math] is the only cyclic group of order 4. If you want to show the other group is isomorphic to it, find an element which generates it and has order four.

I get the impression that in the first problem you've posted, the second group collapses to either Z_2 or Z_3, neither of which is hard to embed in S_5.

When is it appropriate to sum a function with its integral? Isn't this basically what taylor series do? At least, regarding sin/cos/e^x, or any function made of [math]\frac{x^n}{n!}[/math]

And for the Riemann series, shouldn't it really be [math]\zeta(-1)=\frac{1}{1^{-1}}+c_1+\frac{1}{2^{-1}}+c_2+\frac{1}{3^{-1}}+c_3+... [/math]
or something of that line of thought, since each integration produces an unknown constant "c"?

>>11074533
>sum a function with its integral
what do you mean by this?

The difference between sum and integral are given by

https://en.wikipedia.org/wiki/Euler–Maclaurin_formula

The sum has no c's.

>>11074560
>what do you mean by this?
I guess what I mean is, when is it appropriate to write a differential equation? How do you justify, or explain a function interacting with its derivative, rather than being linearly independent?

>The sum has no c's.
Assuming the zeta's series may be a sum of integrals (parent function + 1st integral + 2nd integral), wouldn't that mean that each integration generates an indeterminate number "c?" Since [math]\frac{d}{dx}f(x)+c=\int{f'(x)dx}[/math], then the opposite should hold true.

>>11074588
It's not clear what you're asking. Sum and integral are different but related things. Nice functions admit representations in various ways.
>interacting with its derivative
Not sure what you mean, but if the derivative of a function exist (some limit) then it does and if it doesn't it doesn't.

>Since f' = int f'
Something fishy's going on there.

>>11074593
It's clearly a typo. If a series is produced by repeated integration, why is the constant "c" not considered?

>>11074606
The constant is for indefinite integrals.

>>11074588
I'm not sure you fully understand the concepts you are mentioning or if you just struggle putting the idea into words, so I don't know how much this will clear things up. But the Taylor series is not a differential equation. Sure, it's construction depends on the values the derivative takes at a given point x_0, but by then you are already evaluating said derivatives and finding a sum that becomes independent of them. The construction uses them, but the final result doesn't anymore, which is why it is such an important property.
>How do you justify, or explain a function interacting with its derivative, rather than being linearly independent?
Again I don't get what you wanna say, especially with the linear independence part. I assume you mean to ask how is it justified to write a function in terms of the values of its derivatives at certain points, and in that case you should check out the proof of Taylor's theorem.
For a differential equation, you can just define them how you need, and it's solution will by definition meet the relation with it's derivatives you want it to meet, assuming it exists to begin with. Proofs of existence and uniqueness of the solutions are helpful if your differential equation meets the properties that are required to apply those.

Also by the way, if you have several integration constants in your result, you can group them all up in a single constant since they are all arbitrary so the sum is arbitrary too. That's not the case for the Zeta function, but in case it happened then you don't need to have a lot of separate constants if you can just sum them. I don't see where they should be supposed to show up from, though.

>>11074635
does a range of [math]-\infty[/math] to [math]\infty[/math] count as indefinite?

>>11074654
No.

>>11074643
>the Taylor series is not a differential equation.
Isn't the presence of factorials in the series for e^x a dead giveaway that it's a differential equation?

I've got an imbecilic question.

Suppose the earth weighs more than we think it does. Or perhaps mass curves space more than we think it does.

How much higher would the earth's mass(same size) need to be for the slowdown from being in the gravity well to make the apparent speed of rotation of distance galaxies make sense given the amount of matter we actually see in them?

>>11074667
A differential equation, in this case an ordinary one, is usually defined as a function [math] F(t,x(t),x'(t),...,x^{(n)}(t)) = 0 [/math]. Each derivative has to be in a direct relation with it's derivatives up to the n-th derivative. Note that when I mean the derivatives of x, that means with respect to a different variable, so the function (or dependent variable) x depends on an independent variable t. So you don't just pick the variable x and try to give other expressions the form of any of its derivatives.
So for example, if I have the affine function x+1=0, I can't say that it is a differential equation just cause x'=1 and then x+x'=0. I imagine that's where you get the idea that the factorials somehow imply that each term is a derivative of the next one and that would make up a differential equation, which is not true. The Taylor series of a function doesn't put the derivatives of a dependent variable in relation to one another.

>>11074730
>can't say that it is a differential equation just cause x'=1 and then x+x'=0
That sounds perfectly reasonable though. Sure, it's a trivial example, but it's not wrong.
> factorials somehow imply that each term is a derivative of the next one
Why else would the factorial function exist? Repeatedly integrating polynomials is the only method I'm aware of that generates them

>>11074782
>That sounds perfectly reasonable though. Sure, it's a trivial example, but it's not wrong.
But it is wrong. From the expression x+x'=0 I should be able to just so this
x=-x'
x'=-x''
And then by transitivity x=-x''. But then that would imply x=0, which of course doesn't make sense in my equation. x in the differential equation is a function, x in my algebraic equation is just an independent variable. The solution to the differential equation that does meet the conditions is c*e^(-x).
>Why else would the factorial function exist? Repeatedly integrating polynomials is the only method I'm aware of that generates them
And it is true that the factorials come up from using derivatives for the construction of the Taylor series. But they are not the derivative functions anymore. They are already just constants that were evaluated at a point x_0, around which the Taylor series was expanded. The Taylor expansion in fact does not include any of the derivative functions of f of any order. Refer to the proof of Taylor's theorem if you want more details.

>>11074675

Like, by just pretending for a moment that particles have more mass to them than we think they do, and we can't tell because it all looks the same to us in our relatively flat spacetime from our perspective on the ground, and mostly just out to low earth orbit(i'm aware satellites must carefully consider the curvature of our gravity well to remain properly synced to our time, thus proving that gravity wells do exist and we have at LEAST usable if not perfectly accurate math that successfully describes them), but what if that continues for much of the solar system, and we can't really see it because not much shit has gotten beyond our solar system yet and we've massively underweighed ourselves because we can't tell how much of our mass is inherent and how much of it is just apparent mass due to the mass moving through the curvature of space due to itself. If a spacetime is compressed at all, the matter moving through it has more energy in it(because it's moving faster through the same amount of space, and moving objects inherently have more energy and therefore effective mass). That could be interfering with our own measurements of our mass. We don't have a control group outside the solar system.

What if either the gravity well is much bigger than we think it is, or we're much further into it than we think we are. In any event, what if we're wrong about the pace which the whole rest of the universe is moving at?

If we were deeper into a gravity well than we think we are, and we weigh a lot more than we think we do, then it's possible that the rest of the universe appears to be moving way faster than it actually is, just because of mild time dilation.

So what would earth's mass have to be to create the level of time dilation required to explain the apparent rotation speed we see in distant galaxies unexplained by their apparent mass?

>>11074867

Also yes, i realize that if that did turn out to be right it would like, require reworking like 2/3rds of physics. But just humor me, somebody please check and see if there is some speed the whole universe could actually be going at that makes perfect sense for the amount of apparent mass - that we for some reason or another just aren't perceiving as the speed it's actually moving at?

>>11074831
>x'=-x''
>differentiating both sides
You sure you can just do that? I know squares and squareroots can cause a lot of trouble that way, it's not something you can take for granted.
>but they are not the derivative functions anymore. They are already just constants that were evaluated at a point x_0, around which the Taylor series was expanded
That doesn't make sense. Even if the taylor series is just an approximation, the derivates are being used for a reason. In the context of the equation your trying to approximate, sure, the derivatives might be meaningless. But for the taylor series itself, I don't think you can dismiss the connection.
I'm working through the proof rn, but I don't see your justification.
https://www.math.cuhk.edu.hk/course_builder/1516/math1010c/Taylor.pdf

>>11074675
>How much higher would the earth's mass(same size) need to be...to make the apparent speed of rotation of distance galaxies make sense given the amount of matter we actually see in them?
Any small error on our side would be orders of magnitude higher for galaxies, especially since the error seems to concentrate around massive bodies. Basically, it's their problem, not ours.

>>11074950

Perhaps that is the case. Perhaps we do have a small error on our side and a massive error on their side, but in any event, how much would we need to weigh, like ballpark for me here, how many orders of magnitude more would we need to weight to create a time dilation effect that would explain the apparent matter in the galaxy and the speed at which galaxies are appearing to rotate?

>>11074950
>>11074958

Like, i don't really feel like we need a shit load of time dilation to be really really off about how fast galaxies look like they're rotating. I don't think it needs to be an extreme amount, just a lot more than our current belief about our own mass would suggest is possible.

And by the way, i feel like this shit is testable. We just need to send a probe into as deep a space as we can get it to, then slow down to as close a standstill relative to earth as we can get, then see if there are significant differences in our observations of distant planets and galaxies.

>>11074950
It's true that you can't do that for every case. I'm abusing a property of linear ODEs with constant coefficients, since their solutions happen to be infinitely differentiable in all of R. I don't expect you to know that (although maybe you already learned it, don't know), but I'm just using that to get to the core of the counterexample: My algebraic equation can't be converted arbitrarily into an ODE.

The reason I'm trying to make the difference between a dependent and an independent variable is the following: In the case I gave you, x is independent. x=1 isn't a function. You could maybe see it as a constant function if you defined a variable and a domain for the "function" x, but as is x is just a point on the line. Now, assume x were actually a function with respect to a variable t, defined on some subset [math]S \subset \mathbb{R} [/math]. Then x(t)=1, which IS a function. If we try evaluating that in that ODE, see what happens:
x(t)=-x'(t), where x(t) is the constant function, and then
1=0, which is obviously wrong.
In the general form >>11074730 I gave for an ODE earlier, what you are doing is trying to define an ODE using the derivatives of t. But t isn't even a function, just cause you can make it take a value doesn't turn it into one.

>But for the taylor series itself, I don't think you can dismiss the connection.
But I can. And that's what the Taylor series actually does. A function f'(x) and the point [math]f'(x_{0})[/math] are fundamentally very different things. Can I find one using the other? Sure. But one is a function, defined on a domain to a codomain, and the other is still just a point in space.
See the construction of the Taylor series you are reading. For it to be a differential equation, you would need expressions using the derivative of x, [math]\forall x \in S \subset \mathbb{R} [/math]. That never happens, as you only use VALUES at a singular point, not FUNCTIONS that are n-th derivatives of f(x).

>>11074950
And just as a short follow up from >>11075005 since I ran out of space. To be able to talk about a derivative, I need a variable I take the derivative with respect to. Which is what we usually talk about when we write [math] \frac{dx}{dt} [/math]. In what I said about x-1=0 and then trying to write an ODE out of it, think about what it would mean to say that if I take x as a function, then x'=1. What am I even differentiating with respect to?

what does it mean for the loss tangent to be pi/2 radians, I know it's supposed to mean the material is a good conductor, but I don't understand why.

>>11074971
I don't know how to answer your question, but is this what you're trying to ask?
"What would the mass of the earth have to be for the real velocity at point A (red line) to resemble the apparent velocity (blue line)? For point B? For C?"
You're gonna get different values, that's for sure

>>11073837
>I have N boxes and N apples. I select one box at random, regardless of how many it already contains, and put one apple in it. Repeat until all apples are in a box. I then sort the boxes in increasing order by number of apples they contain.
So I wrote a short program that does exactly this

>On average, what fraction of the total number of boxes remain empty
On average, 37% of the boxes remain empty. The experiment was performed 10,000 times.

> as a function of N, and what does this value approach as N goes to infinity?
Seems like it's 37%.
N = 100, results in 36.5%
N = 5000 results in 36.7%

>On average, how many apples will be in the box with the most apples
N = 100, approximately 4 apples on average in the biggest box
N = 1000, approximately 5 apples on average in the biggest box
N = 10000, approximately 6 apples on average in the biggest box


Not sure if code tags work here


import random

def run_experiment_empty(n):
boxes = [0] * n
for ii in range(0,n):
boxes[random.randint(0,n-1)] += 1
return (boxes.count(0) / len(boxes))

def run_experiment_max(n):
boxes = [0] * n
for ii in range(0,n):
boxes[random.randint(0,n-1)] += 1
return (max(boxes))

def get_average_empty(repeat, n):
results = []
for ii in range(0,repeat):
results.append(run_experiment_empty(n))
return (sum(results) / len(results))

def get_average_max(repeat, n):
results = []
for ii in range(0,repeat):
results.append(run_experiment_max(n))
return (sum(results) / len(results))

print(get_average_empty(1000, 100)) #N = 100, repeat 1,000 times
print(get_average_max(1000, 100))


>>11073837
I'm guessing you get some kind of power law in the limit.

so (0.999...) isn't contained within the real interval (0,1), right? (if not is it an integer?)
idk why but it's annoying me a little bit

>>11075179
>N = 100, approximately 4 apples on average in the biggest box
>N = 1000, approximately 5 apples on average in the biggest box
>N = 10000, approximately 6 apples on average in the biggest box
That's sick!

>>11070574
bump

I'm about discretising some PDE, is the value for the variables still in vector or in scalar?

>>11056713
If f=ma then why does it hurt when I run into a wall at a constant velocity?

>>11075579
Because force=change in momentum/change in time.
Your momentum changes very suddenly. Therefore, force is large.

>>11075478
No it is not. It is an integer, because it is equal to 1.

How does a computer read a transistor without the state of the transistor being changed?
As in: How does the CPU receive a meaningful value when asked to read memory address X, and not change the bit at memory address X?

>>11075616
Pretty sure the transistor is just doing math based on what's in it's memory.

just go watch computerphile or something though

>>11075579
you accelerate because your speed changes (really, you decelerate)
you go from moving forward to stopped, that requires force
the force is you hitting the wall, and the wall produces an equal and opposite reactive force against you. which is what hurts and what stops you.

>>11075623
No, a single transistor that holds a single bit in memory. Not a circuit.

How does the transistors state
A) be read to be on or off 0 or 1
B) how does this not effect the state of the transistor

>>11075647
https://www.youtube.com/watch?v=UvI-AMAtrvE

https://www.youtube.com/watch?v=XETZoRYdtkw

>>11075616
you don't know what a transistor is
a transistor is a gate. if the lock is on, energy doesn't flow through. if it's off, then energy can get through.
the lock is controlled by a third wire.
this lets you make logical decisions.
one thing which you can build out of transistors (quite a few of them) is called a flip flop. there are a couple different types, but the one that's relevant here is the d-q flip flop. a d-q flip flop has one input line and one output line, as well as a control line. whenever the control line is turned on, the flip flop looks at the input line and stores whatever is there. when the control line is turned off, the flip flop ignores whatever is happening to the input line and just keeps whatever it was last time the control was on. so it's a memory cell. at all times, the output line is showing what's in the flip flop.
this is how a computer reads a single bit of memory, through a flipflop which is only allowed to change when a separate control line is on.
but you have to remember that a flip flop is not just one transistor, it is a bunch of them wired together in a clever way (really, they're locking and unlocking each other somehow!) a single transistor cannot remember a thing.
there's the dq flipflop in the image.

>>11075647
the transistor doesn't hold memory idiot. jesus christ.
read >>11075657 and ignore the memey numberphile shitter who doesn't know what he's talking about

>>11075659
I didn't say a transistor holds memory. I said the transistor was 0 or 1. My question was how does this state be read and how does the reading process not effect the state of the transistor
>>11075657
Ty

>>11075657
So this is in essence the same thing as 2 not gates?

>>11075666
oh. well the reading process and the control of the state happen on different wires entirely. the control wire "opens up" or "closes off" the path between the input wire and the output wire by chemically changing the composition of the metal in the transistor, i.e. by tugging out / feeding in electrons which make/fill in holes in the actual metal atoms. when there are holes, electrons may freely pass through them. when it's full, there's a lot of resistance and the electrons don't get through. this is a very very physical process.

>>11075666
Pretty sure it dumps the solution to memory and deals with it then

>>11075678
So, because the memory cell is always outputting, the reading has no actual change happening to the cell, only writing?

>>11075673
eh, it's closer to 2 latches,
think about this first, it's called an RS latch. the two logic gates are NORs. what happens when you turn on S and then turn it off? what happens when you turn on R and then turn it off?
we can build a proper DQ out of these little memory elements.

>>11075692
shit forgot picture

Does liquid chromatography separation needs specialized silica from lab supplier or can i just use D.E (i heard it's 90% silica) with similar results?

>>11075692
>>11075697
This has helped a lot, thanks man. I gotta go to sleep but appreciate the answers.

>>11075692
>>11075697
now, we'll send D to R and not D to S. but we don't want to always write from D, we only want to when the control (the clock) says to. so we need to do an and first.
pretend in this circuit that the not Q line is on, i.e. the RS latch contains a 0 bit. suppose D has been 1 for a few seconds already. we quickly flash the clock on and off. what happens to the thing stored in the latch? follow all the wires and see where it stabilizes. write down the AND and NOR truth tables if you need to. when you get this, it all clicks.
then you can do the same if D is off (so not D is on) and you flash the clock.
notice that reading from the flipflop only happens on the right, and electricity flows in the left end of a logic gate and out the right end. this corresponds to the left end being the control line of a transistor or two (NOT the input line!), and the right end being the output line of the transistors. that way, electricity from the "read" or "Q" side can never go back in and screw with what's stored in our SR latch - and the ANDs make it so that electricity from the input side can never screw with our SR latch unless we let it in with the CLK.
attach 32 of these together and you get a memory register!

Read or watch cardcaptor sakura?
I’m 6 episodes in, watch like an episode a everyday or every other day. Pretty comfy, don’t know if the manga is better though.

>>11075712
You'll need a certain mesh size, I think flash chromatography is usually "60," in whatever units.
Dry column vaccuum chromatography uses something much finer, see: https://curlyarrow.blogspot.com/2006/10/dry-column-vacuum-chromatography.html
Otherwise, if you just buy coarse shit from the store you may have to use these bigass gravity columns in pic related.
>liquid chromatography
You wouldn't, by chance, mean HPLC? Good luck getting shit out of a ~4" column. Hope you didn't need it in reverse phase.

>>11075838
Thanks for the answer. My bad, i mean a standard column one, not HPLC. I guess i have to buy alumina online then. Shit from merck are way too expensive for my simple experiment like seperating leaf extract and black ink.
My column is a puny 20x300mm i bought for like $14

>>11075876
Yeah, with 7' of 60 mesh silica you'd be able to purify maybe 100mg at a time. I think alumina is supposed to be slightly lower resolution.
If you use a polarity gradient, from low to high, you can hack a pretty respectable separation even with a coarse solid phase.

I swear Im not trying to be inflammatory. Here goes. Would UBI work? Part two I saw a thing were it said if we did it china could dump there dollar reserves and rekt us.

>>11075917
*their pardon me

What's the best method of studying multiple subjects? Like say I want to learn japanese, drawing, and writing a good essay.
Do I study each one 1 hour a day for 3 hours a day?

Or do I study one subject 3 hours a day and switch subjects every day?

And finally, when studying, is the best method 20 minute sessions of pure study and 5 minutes rest in between each set the best for memory retention?
or is there a better method?

>>11075924
I'm afraid you'll have to try for yourself. also try 2+2 hours each day, but rotate the subjects.

Congrats, you faggots baited me into reading Baby Rudin. is my proof of the irrationality of sqrt(12) correct?

[math]\sqrt{12} = 2\sqrt{3}[/math]. If [math]\sqrt{3} \in \mathbb{Q}[/math], then [math]\sqrt{3} = \frac{a}{b}[/math] for some coprime [math]a,b \in \mathbb{N}[/math].
[math]3b^{2} = a^{2} \implies a[/math] is divisible by [math]3[/math]. Let [math]a = 3k[/math], then [math]3b^{2} = 3k^{2}[/math], but [math]a[/math] and [math]b[/math] cannot have common factors. Thus [math]\sqrt{3}[/math] is irrational.

Next, we prove that rational * irrational is irrational.
Let [math]m, \in \mathbb{Q}, n \in \mathbb{Q'}[/math] with [math]m[/math] nonzero.
[math][/math]If [math]mn[/math] were rational, then [math]\frac{mn}{m}[/math] would be rational. It isn't. QED

I wasted so much time on this.

>>11076059
I don't see anything wrong with it but I'm honestly retarded, so who knows.

>>11076059
you're correct

>I wasted so much time on this.
Focus on later chapters, this constructon of real numbers stuff is not really that important.

rehearsing complex number theory for fourier analysis

why is e^(iw) = e^(-iw) ?

>>11076125
> why is e^(iw) = e^(-iw) ?
It isn't. The two are conjugates (same real parts, imaginary parts have the same magnitude but opposing signs).
e^ix = cos(x)+i.sin(x)
e^-ix = cos(-x)+i.sin(-x) = cos(x)-i.sin(x)

I really want to derive the commutation relations of the generators of the Lorentz group [math]SO(3,1)[/math]:
[eqn][ I^{\mu \nu} , I^{\rho \sigma} ] = i \left( g^{\nu \rho} I^{\mu \sigma} + g^{\mu \sigma} I^{\nu \rho} - g^{\nu \sigma} I^{\mu \rho} - g^{\mu \rho}I^{\nu \sigma} \right) [/eqn]

where we defined the generators to be: [math]\left( I^{\rho \sigma} \right) ^{\mu} _{\nu} = g^{\mu \rho} \delta ^{\sigma} _{\nu} - g^{\mu \sigma} \delta ^{\rho} _{\nu}[/math]. I know that changing basis one can break [math]\mathfrak{so(3,1)}[/math] into two [math]\mathfrak{su(2)}[/math] subalgebras but that doesn't help derive the relation. Is there a way to get it other than "close your eyes, put everything in the commutator and calculate"?. Any help appreciated.

>>11076292
yeah thanks, unit circle and e^(iw) = cosw + isinw helped me understand. e^(-i*pi) = e^(i*pi)

>>11076372
Usually you try to look for symmetry under switching of indices to save work, been a while since I've done this type of thing though.

>>11076372
Did you try computing over the quaternions?
>>11076380
It's a bit simpler to think of it in terms of [math]e^{iw} e^{=iw} = e^0 = 1[/math], and thus [math]e^{iw}= \overline{e^{-iw}}[/math]

>>11076436
>Did you try computing over the quaternions?
well i need this for a particle physics class, we don't cover quaternions.

What is the proper way to cite a proceedings?
For instance, if I want to cite a paper from POPL, should I write "POPL", or "Principles of Programming Languages", or "Principles of Programming Languages (POPL)", or "ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages", or "ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages (POPL)", or "nth ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages 2007 (POPL'2007)", or something else???

How is the inverse laplacetransform of dirac(s-2i) = cos(2t) ? can't find any info on this

>>11076724
Because it's the Dirac delta, it's only got a single frequency.
Computing it yourself isn't hard or anything.

>>11076733
anon please
I can only laplace transform dirac functions, not inverse transform them?

>>11076733
>>11076742
even wolfram doesn't know what to do

What 3d graph plotting software is suitable for finding exact values of graph points after plotting? I use gnuplot and OriginPro to plot the data set, but as far as I can tell, neither can do that.

>>11076763
ti-nspire

>>11076724
> How is the inverse laplace transform of dirac(s-2i) = cos(2t) ?
It isn't. L{cos(2t)}=s/(s^2+4).

I'm assuming that δ(s-2i) comes from substituting s=iω into the Fourier transform (which is δ(ω-2)). But that equivalence only holds if the region of convergence of the Laplace transform includes the imaginary axis. But the RoC for cos(wt) is Re(s)>0, i.e. it doesn't include the imaginary axis.

>>11073837
>On average, what fraction of the total number of boxes remain empty, as a function of N
[eqn] \left( \frac{N-1}{N} \right)^N [/eqn]
>and what does this value approach as N goes to infinity?
[eqn] \frac{1}{e} [/eqn]

>>11076912
I'm referring to pic related (exam studies, not homework, before you throw a fit)
as you say one part of the conversion applies for Re(s) > 0, but the part I mentioned doesn't.

For mathematical morphology/image processing
is there a name for or operation that does what hit-or-miss does (binary)
but instead of the transformed image only having 1's where the origin of the structuring element leads to a match, there are 1s in all the places the structuring element coincides with the image?

so performing the operation using a structuring element shaped like a tetris-L padded with zeros should result in an image consisting only of isolated tetris-Ls.

>>11072658
What are the differences between Apostol and Courant?
Which one is better for a physics major?
Which one is better for a applied math major?

>>11073837
Let e(N) be the number of empty boxes on average, and m(N) the greatest number of apples in one box on average. Doing it the brainlet way and working out the first few case by hand:
[eqn]
\begin{tabular}{l|l|l}
& e(N) & m(N) \\
\hline
1 & 0 & 1 \\
2 & $\frac{1}{2}$ & $\frac{3}{2}$ \\
3 & $\frac{8}{9} = \frac{2^3}{3^2}$ & $\frac{17}{9}$ \\
4 & $\frac{81}{64} = \frac{3^4}{4^3}$ & $\frac{136}{64}$ \\
\vdots & & \\
N & $\frac{(N-1)^N}{N^{(N-1)}}$ & ??? \\
\end{tabular}
[/eqn]

e(N) suggests a simple pattern which seems to agree with >>11076931 (the fraction being e(N)/N). m(N) initially appears to be one more than e(N), but that rule fails starting at N=4

>>11076975

>>11077684
Goddamnit

Let [math]A[/math] be a symmetric matrix, and let [math]A[1,1][/math] be [math]A[/math] without the 1st column and row. Let [math]p[/math] be the characteristic polynomial of [math]A[/math] and similarly [math]g[/math] for [math]A[1,1][/math]. How do I go about showing that [eqn]\frac{p}{g} = ((Ix - A)^{-1})_{1,1}[/eqn]

>>11077834
Nvm, was simple application of cramer's rule

Bump.

>get to the polynomial section of Axler's Precalc text
>expect to learn to factor higher-degree polynomials since I see them show up every once in a while in Wolfram Alpha's problem generator
>Basically says if it's not something you can quickly factor like x^2 - 9 or something you can solve with the quadratic formula for its zeroes then fuck it and use a calculator

uh i'm not missing anything important right

>>11076744
Sure you're talking about Laplace transformations, buddy? Try taking the forward Laplace transformation of cos(2t)
[eqn]F(s) = \int_{0}^\infty dt \cos(2t)e^{-st} = \frac{s}{s^2 + 4} [/eqn],
which looks nothing like a dirac delta function. Now if we were talking about Fourier transformations, which are a bit similar, you'd be getting a lot closer but you'd need a sum of shifted Dirac delta's to find an actual cosine instead of a pure plane wave.

>>11079572
Based advice.
Factoring high degree polynomials by hand is a waste of time. Very few high-degree polynomials you encounter outside of textbooks are going to be solvable by hand, and of those that are, you could be spending your time doing something much more productive or enlightening than attempts at synthetic division.
Modern computer algorithms for polynomial factorization are very efficient. Use them.

>>11079572
For the purpose of say, analytic tests, my professors used to make sure you could find solutions by hand. One of the most useful ways, if you suspect there to be solutions like this, is Horner's method. In general cases the roots will be ugly and for [math]n > 2[/math] could just be calculated by computer.

>>11079661
Consider the polynomial [math]P(x) = x^3 + 6x^2 - x -30[/math]. Using Horner you can rapidly find at least one of the roots, then using Euclidian division you can reduce it to a polynomial of degree 2 and use the standard formula to break it down completely.
https://en.wikipedia.org/wiki/Horner%27s_method

>>11079676
Forgot to add 10 and 15 are also possible divisors of 30 so, err, it really depends on how much legwork you're willing to do.

>>11068034
Static means the force is constant, dynamic means it varies

>>11079644
apparently I was talking about an inverse laplace transform of an analytical distribution? (how the fuck do one keep track of all of these). pic related is the relation I was looking for.

>>11080089
https://en.wikipedia.org/wiki/Inverse_Laplace_transform
I'm not going to even pretend to be rigorous right now but I guess you could write something like
[eqn]\mathcal{L}^{-1}(\delta)(t) = \frac{1}{2\pi i} \text{lim}_{T \rightarrow \infty} \int_{\gamma - i T}^{\gamma + i T} e^{st} \delta (s) ds \sim \frac{1}{2\pi i}\int_{\gamma - i \infty}^{\gamma + i \infty} e^{st} \delta (s) \sim \frac{1}{2\pi i}2\pi i = 1 [/eqn]
using contour integration and assuming the pole behaves well enough. Again, this is about as far from exactly correct as it gets.

>>11080441
it's a dumb course anyways and we're provided with formula sheets for the exam but I appreciate your insight anyway anon

So, I got a question...
If physicists use dx/dy like a fraction, how do mathematicians use them if they get headaches everytime a physicist uses them like this? I mean, how do the methods of a mathematician differ from those of physicists?

>>11080450
you can treat x'(t)=dx/dt as a fraction without any consequences so long as you are not dealing with partial differentials or an order higher than 1. In reality, this is just an application of the chain rule. Real analysists just use the chain rule and the definition of the derivative/limit.

>>11080450
mathematicians don't solve differential equations in the first place, but if they do, they also treat it as a fraction. they just like being dicks about it.

>>11080450
You generally get a long long way by just knowing that a derivative fulfills the chain rule and the Jacobi identity - physicists and mathematicans usually manipulate algebraically.
Most people stop having to compute actual limits - unless you're in analysis and do proofs there.

Besides, there's axiomatizations of analysis where the differentials are numbers, which justifies the fractions perspectives in many cases
https://en.wikipedia.org/wiki/Non-standard_analysis

how the heck do you do this

>>11056713
Am I missing something here, just to validate?
>Use IPAT equation
I < 1
I = 1.07P x 1.045A x yT (let 'y' be constant).
I / (1.07Px1.045A) = yT
Yet it asks for rates, so am I purely putting the rates a la
I = 1.07 x 1.045 x T
I swear I'm not usually this retarded but the phrasing baffles me, why are they asking this? I mean they'd have to go backwards to not use more natural resources so what are they saying?

>>11080717
Nvm!

>>11080745
1.07/1.045 -1 lad.

>>11056713
>>11056713
How do I know if given A and B that belong to R^(3x3), C and D that belong to R^(3x3) and that det(D)=!0 det(C)=!0 exist so that: A=C*B*D
Or asked in other words, how do i know if two matrices represent the same linear transformation in diffeent bases?
Thank you

>how do I know two matrices represent the same transformation in different bases
iirc if and only if they have the same Jordan form.

>>11080894
Mhh, they hadn't taught me that when they gave me this exericise. They also make me say what bases are those... Maybe i am supposed to make it trying?? I think alpha has to be 1 or -1 so that the third row is a linel combinaton of the prevoius.

>>11080859
Easiest way is to reduce them to their Jordan canonical form. You technically have to find matrices B and D such that they are similar to a matrix J (the Jordan matrix) too, but the specific method for doing that is pretty straightforward. If they share the same canonical form then they're similar.

>>11081236
Well, didn't get to see >>11080950 since I had only seen the original post had no linked replies. If you can't compute the canonical form then what tools are you given? What have you learned so far that we can use? Since you know the specific transformations, at the very least you may be able to compute a matrix with arbitrary entries, it's inverse (so you'll have to force the condition det=0) and then compute the matrix products to get a fuckhuge system of equations. Sounds annoying though so maybe you already know other stuff that can be used.

>>11081236
>>11081261
thanks for the replies, I will ask the teacher what am i supposed to do with this.

>>11080894
>>11081236
not exactly.

jordan form is about classifying matrices up to the SAME basis at the input and at the input.

OP is free to have two distinct bases. as such, the matrices are classified just by its rank really. (any matrix can be reduced into an identity matrix in the top left corner and zeroes everywhere).

Consider the cyclic group G ={a, a, a... a^12=u}, u is identity, and its subgroup G'={a^2,a^4,a^6....a^12}. What is the element mapping for a^12 from G if the mapping is a^n to a^2n? It doesn't seem obvious to me that every element in G has an image in G' either.

>>11082068
Wait, what?
Let me see if I get this right.
You have the cyclic group with order 12 and generator a.
You define f: G -> G by f(x)=x^2.
G' = Im f.
f(a^12)=f(e)=e^2=e.
If you wanted the solution to f(x)=a^12=e, then we can have either x=a^6 or x=e.

Thks, slipped my mind that e times e is still the identity. Any answer for this part
>doesn't seem obvious to me that every element in G has an image in G' either.
Like eg. an a^10 becomes a^20 which won't be in the subgroup. Does it mean G can't have a^k with k more than 6 then

What is the answer to this limit? I am retarded and I got 1.

What a coincidence, I got 1 too

>>11082188
a^20=a^12a^8
=ua^8=a^8, which is in the subgroup

>>11082188
a^20=(a^12)(a^8)=a^8
Likewise, a^9 -> a^18 = (a^12)(a^6)=a^6

>>11082234
OK thx