would it be possible to grow crystals of pyrite in a home lab?

I've seen people grow those copper pentahydrate crystals at home and thought it would be cool to try and grow pyrite (iron sulfide) from a solution with a seed crystal. I'd expect the pyrite crystal to be a perfect cube, like in the picture.

I'd like to learn about continued fractions. What concepts should I have under my belt and through which texts should I work?

What concepts should I have under my belt and through which texts should I work?
I don't think you really need any particular knowledge beforehand, but try chapter 7 of pic related

what happens if i get a 0 or infinity for the radius of convergence for a power series? what do i write as the answer to the problem?

what happens if i get a 0 or infinity for the radius of convergence for a power series? what do i write as the answer to the problem?
0 or infinity, respectfully

do i put that as the answer if the question is asking for the interval of convergence and not the radius?

0 = series converges nowhere
inf = series converges everywhere

that's what i was thinking too but symbolab is telling me that if the radius is 0 then the series converges everywhere and if it's infinity then the series converges at only one point

symbolab.com/solver/power-series-calculator/\sum_{n=0}^{\infty} n!\left(x-1\right)^{n}

symbolab.com/solver/power-series-calculator/\sum_{n=0}^{\infty} \frac{\left(2x\right)^{n}}{\left(3n\right)!}

For that first one doesn't it say that it diverges for all x not 1? And then say it's a radius of 0?

oh yeah it does say that. i was thinking that the answer to the ratio test was the radius of convergence which was infinity.

There was a study on policy positions of americans and it mentioned 58% of latinos felt a certain way and 68% of black people felt another way, but omitted white opinion.
How do I find the missing percentage? I don't need a precise answer and asians and other races are negligible.

Anyone know of good exercise material for linear algebra? I'm a dumb fuck that's trying to (re)learn fundamental maths stuff, and I'd like to spend some time each day doing exercises on the things I learn. I jewgled for this stuff but it was mostly million page PDFs that were way over my head. I just want something where it gives me N instances of X problem to solve, repetition is key and whatnot.

Schaum's is good for that.
3,000 Solved Problems in Linear Algebra
amazon.com/000-Solved-Problems-Linear-Algebra/dp/0070380236

I took an Edx math course. I don't know if the one I took is still there but it had an interactive system sponsored by some college that introduced every concept with randomly generated problems in an order and would reintroduce concepts after a period of time.

I've heard it said that energy is quantized because of E=hf, but wouldn't it make more sense to say that frequency is quantized instead?
Esprcially since mass is continuous (afaik) and mass-energy equivalence is a thing.

Total energy is quantized in many situations with a potential (e.g. hydrogen atom, quantum harmonic oscillator), since there are discrete energy levels the particle can be at.

True, I guess what I'm really asking is for a satisfying interpretation of the energy-momentum relation (E^2 being the sum of a discrete and a continuous term seems to me like trying to add different types of things, though it's probably because I have no formal training in phyaics).

i made a B in physics 1 although I literally just plugged values randomly into equations i was given and hoped for the best. i made As in calc 1 - calc 3 so i swear im not a brainlet. i basically just cannot for the life of me understand torque and tension and rotational acceleration and momentum and friction. stuff like the ladder problem, my brain shuts down.

how will this come back to haunt me in physics 2 if at all?

(sage for mathematical masturbation and phoneposting, though I wouldn't mind knowing if there are sensible.interpretations for what I'm doing)

Maybe frequency f = c/lambda is discrete but the (imaginary) angular frequency w = 2pi*i*f makes it continuous, which allows pc = i*hbar*w to be meaningfully added to a continuous term.

The relation $p^2 + m^2 = 0$ is called the "on-shell" relation, which means that only physical momenta satisfy these relations, while the energy quantization relation holds for particles. The "continuous" 4-momenta (which includes the energy $E = p_0$) are attached to vertices of the Feynman diagrams arising from fields lying in the closure of the net of local observable algebras under the weak operator topology $\mathcal{A} = \operatorname{cl}\left(\bigcup_{U \subset \mathbb{M}} \overline{\mathcal{A}}(U)\right)$, while the "discrete" energy is assigned to the particle counterpart of these fields, namely the excitations on the states in the GNS Hilbert space $\mathcal{H} = \bigwedge_i \mathcal{H}_i/\{\Psi \in \mathcal{H}_i \mid \langle \Psi|\Psi\rangle \leq 0\}$ (or, for free fields, the Fock space $\mathcal{F}$). So the "continuous" notion of energy is associated only with the coordinates of the fields, which are operator-valued distributions on the symplectic vector space $(\mathbb{M}^2,dp^\mu dq_\mu)$ while the "discrete" notion of energy is associated with the eigenvalues of the Hamiltonian operator on states $\Psi \in\mathcal{H}$. These are in essence two different things, and technically only the "discrete" energy is observable.
I hope this clears up any misunderstanding.

would it be possible to grow crystals of pyrite in a home lab?

i swear im not a brainlet
stuff like the ladder problem melts my brain

wouldn't it make more sense to say that frequency is quantisized instead?

what grades did you make in calc 1, 2, and 3?

physics 2 is more about electricity and not mechanics so you're good

I haven't ever been to college. I'm 18 and barely old enough to post. I understand roughly 1/7 of the conversations on this board, and that's thanks to retards talking about global warming and /his/ shit leaking over, which I can understand. I love posting retard faces, of which I have more than fifty, especially in stupid questions threads.

I hope this clears up any misunderstanding
I'd be lying if I said I understood enough of that to clear up *all* misunderstandings, but from what I could glean it seems that the discrete-continuous dichotomy essentially boils down to doing physics in function space vs physical space, and manifests as 'wave-particle duality'.
Which is about as far as I can go given my current level of knowledge really.

Can I scientifically prove the existence of numbers?

Rephrasing the question, can the existence of numbers be proven by science or it is an abstract object that the existence is accepted based in the "certainty" that they exist?

I don't think so, but you can prove the existence of numerals.

It is sensible to interpret it that way, though I'd say that they're entirely distinct things, just named similarly. The mathematical statement of particle-wave duality is the existence of a unitary morphism taking boson fields $(\Gamma,A,\omega,v_0)$ to an $N$-particle Fock space $F$ that maps the vacuum field $v_0\in\Gamma$ to the cyclic vacuum state $|0\rangle$ in the Fock space. Since this statement doesn't directly concern energies I'd be hard-pressed to say that the continuous/discrete dichotomy is due to particle-wave duality.

would a gun shooting a 200 grain bullet have twice as much recoil as the same gun shooting a 100 grain bullet with the same charge?

I jack off to 2hu and have an interest in math.
How do I access my pineal gland to unleash the massive levels of 2hu math autism as you have consistenly shown through the last few months.

Just bee yourself user.

any pro tips or guides?
Where do I start to unlock your subterranean autism?

$\textrm{ which touhou h-doujins do you prefer }$

can i multiply together 2 numbers with a variable as an exponent or do i need to leave them separate? example:
$(-1)^n (-2)^n$

it depends on what n is

for example
1 =
(1)^(1/2) !=
(-1)^(1/2)(-1)^(1/2) =
i*i =
-1

any pro tips or guides?
Can't tell you since I've never used any.
Ran is qt.
Where do I start to unlock your subterranean autism?
Get one of those IQ increasing drugs from Eirin or something lmao.
$\mathfrak{which}~\mathfrak{ touhou}~\mathfrak{ h}-\mathfrak{doujins}~\mathfrak{ do}~\mathfrak{ you}~\mathfrak{ prefer}$
I liked the one where Yukari and Yuyuko raped this one village boy.

i'm trying to test pic related for convergence and i don't know if i need to simplify this further or just do the alternating series test on it or something as-is

it is true for integers n, so you can simplify that to n^2

so it would simplify to pic related and then cancel?

respectfully

so it would simplify to pic related and then cancel?
yep

cool thanks

this brainlet meme is getting out of hand

well, at least you're honest. good luck when you make it to college. this stuff ain't easy.

You're very honest. Your honesty will be used against you soon.

Can / will you teach / tutor me?

$\frak{do\ you\ enjoy\ spacejin\ ?}$

not paying your respects to math
t. brainlet

The Veeky Forums I know and love

he's just a kid out of high school posting pictures he thinks are funny. there's no malice behind his posts. i'm one of the guys he replied but i have no desire to poke fun at him. he's probably a nice dude irl.

Can / will you teach / tutor me?
How much are you paying me?
And no I fucking hate spacejin.

$\textrm{I don't have a disposable income. Do you accept nudes, 2hu doujins, or have anything else you need? I can be your best friend! (I am a worthless NEET---go figure).}$

why are you trying to learn extremely niche math concepts as a NEET? get an engineering degree and you'll be swimming in dosh. this high level math is pointless to learn. there's no reason outside of academic settings to know it.

I have a Bsc. in physics from a garbage SJW university. I met some people along the way that did dynamics, chaos, and dynamical systems in neuroscience (this one fucking russian autist professor can do math like no other, he is my spirit animal). I just look up to them as a brainlet naturally would. Seeing 2hu and real math (I.E. topology, sets, real linear algebra) made me remember how happy I was in college. I also am a lonely NEET who wants to jack off to 2hu NTR doujins with someone else.

look user, instead of trying to find a jack off buddy you can hot glue a fumo with, why don't you try to find a job instead or at the very least get a more marketable degree?

maslow's heirarchy, user. get a job and meet that basic need. then , if you must, be a degenerate.

I might be a navy officer in Feb. of next year.

$\frak{can\ we\ at\ least\ be\ fap\ buddies}$

I also am a lonely NEET who wants to jack off to 2hu NTR doujins with someone else.
You fucking sick fuck

wants to join the navy
also wants to find a dude to jack off with
like pottery

Not gay just lonely. But I am glad you could get some entertainment.

Were all lonely autists, this is Veeky Forums after all

Just. Not all of us want to mutually masturbate with a jper

I didn't quite literally mean circlejerking. I know you understood what I meant since I am acoustic/statistic.

Anyone have an idea of how I would find a parametric equation to fit this curve. Pic is all information I was given.

Why is the notation of second order derivative like d^2/dx^2?
Why not d^2/d^2x?

when you take high order partial derivatives of multivariable functions you want it to be clear how many times you take a derivative of each variable i.e.
$\frac{\partial^3}{\partial y^2 \partial x}$

Say $f: A \rightarrow B$, does $f ^{-1}$ necessarily have to be a function?

d(dy/dx)/dx = d^2y/dx^2

if f^(-1) denotes the general pre-image, then no. if it's being used to denote the inverse function then by definition yes it's a function

fucked up my latex:
Say
$f: A\rightarrow B$, does $f^{-1}$ need to be a function, or can it just be a relation

thanks!

Which is more fundamental: logic, or the natural numbers?

Logic, obviously.

How do I solve the laplace transform of Piecewise[{{Sin[t], Sin[t] > 0}}, 0] ?
WolframAlpha gives e^(π s)/((e^(π s) - 1) (s^2 + 1)).

I tried verifying it by hand using the integral formula for periodic functions with f(t) = Sin(t) - Sin(t) * UnitStep(t- Pi) and with period T=2pi, but this gives a slightly different answer of e^(π s)/((e^(2 * π s) - 1) (s^2 + 1)). Why would the period be pi, not 2pi? I understand the integral can be simplified by interpreting 1 - UnitStep[t-pi] as halving the upper limit, but why would the denominator e ^ (sT) also change?

That's also what I would have said going by intuition, until I started writing up their definitions:

Natural numbers
0 is a natural number.
The successor of a natural number is a natural number.
(All subsequent operations can be defined in terms of these, qua Peano)

Logic
Absurdum is a proposition.
For any two propositions A and B, the implication "if A, then B" is a proposition.
(All subsequent operations can be defined in terms of these.
Using NAND alone doesn't work since you still need to assume the existence of at least one proposition P for the theory to be semantically non-vacuous.)

Or maybe this just means that the natural numbers are syntactically more fundamental (they're representable as tally marks, while you need parentheses for propositional logic as it has non-associative operators) while logic is semantically more fundamental (your domain of discourse only needs two values, qua Suszko, rather than infinitely many of them).
Unfortunately this is imprecise since I don't actually have a working definition of what it means to be "fundamental" (and I suspect that such a definition, if it existed, won't turn out to be fundamental at all).

nevermind, made a typo, and turns out to be the same thing after simplifying because of periodicity of e ^ z.

In terms of vector arithmetic, for an equation like:
4u-3v
would v be multiplied by 3 or -3?

What's the deal with the zero point energy field?
How can the vacuum be filled with energy?

they're the same thing because (-v) = (-1)v in vector spaces
proof literally trivial

proof literally trivial

Fuck I think I might be retarded, studying differential equations and the book says:
dy/dt = ay - b -> (dy/dt)/(y - b/a)

I don't understand how they get that, at most I can do (lets say dy/dt = x because I'm lazy):
x = ay - b -> x = (ay - b)/a -> x = (y - b/a) -> x/(y - b/a) = 1

I also just realized while writing this that I am retarded and somehow forgot that if I'm going to divide the right side by a, I have to divide the left side too (which then results in the correct equation). I am still posting this because I'm a fucking retard who deserves the shame.

I also just realized while writing this that I am retarded and somehow forgot that if I'm going to divide the right side by a, I have to divide the left side too (which then results in the correct equation).
This phenomenon actually happens more often than you think, to the point that it's acquired the name of "rubber duck debugging".
wiki.c2.com/?RubberDucking probably explains it better than I can.

dy/dt = ay - b -> (dy/dt)/(y - b/a)
What the fuck is that supposed to mean?

lets say dy/dt = x
ever heard of y' ?

if you can make a vector u from a linear combination of vectors v and w, does that mean that u belongs to the span of vectors (v, w)?

If a large asteroid were to hit the moon today (think of the asteroid that made the Copernicus or Tycho crater), would you hear the impact from Earth or would it be silent?

Would you hear any sound from outside Earth if
it was loud enough and came from close enough?

That's kinda why I ask, 75% of the time I come up with the answer while typing the question myself. Second best way to learn, after trying to teach the stuff to others.

What the fuck is that supposed to mean?
That's me forgetting the = a at the end.

ever heard of y' ?
Good point but that's 2 key presses, a whole 100% increase over x.

No, sound waves require a medium to travel through, so they can't travel through space.

yeah man

I have a AxAx1xN array in matlab I wanna save for later, is there a format/method for saving it so that the AxAx1xN structure to it is preserved other than just writing it as a AxAN csv file and reshaping it myself after importing?

Nevermind I got it, shoulda just saved them as the variable it already is, thanks Matlab

why the fuck is
x^2+x+1 / x^2+x+2
NOT
1/2

Because 1/2 doesn't equal x^2+2x+2+x^(-2)

If my wife and I are sex addicts would this affect a baby in utero? Can you have too much sex, even without penetration?

yes you could get the baby pregnant if you penetrate too far

Does listening to a lecture with only one headphone in one ear any different than with headphones in both ears? If so does it matter which side?

is three fourths a half?

you can't cancel out the x's unless you can factor them out and you cant factor them out because the 1 and 2 don't have an x

But the pre-image is a function from B to the powerset of A :)

What if the baby comes out addicted to sex? I don't want a daughter to come out of the womb a slut.

will we ever find out what happened before the planck epoch?

ln e x = x
derivative of e^x = e^x
I guess I had a shitty precalc teacher but how does e do both of those things? How are they related?
Not important just curious

Because the logarithm is literally defined as the inverse of the exponential and the exponential is literally defined as the solution of f'=f , f(0)=1.

If the determinant of a system of vectors is 0, does that mean they don't span R^n?

If the determinant of a system of vectors is 0, does that mean they don't span R^n?
Yes

if I'm modeling compartmental drug delivery with a system of ODEs and my last compartment is irreversible and non-saturable I should be modeling the last equation as the rate of drug entering the cellular fluid multiplied by the concentration right? Or is there something I'm missing. Seems too simple.

Yes.
Also you should have written "If the determinant of a system of *n* vectors is 0"

If a system of equations has more unknowns than equations, the system will never have a unique solution.
What does it mean then to have more equations than unknowns?>=

take (1,0,0), (0,1,0), (1,1,0)
their determinant is 0
they span R^2
shiggy

(x-a)^2 + (y-b)^2 = 0
unique solution x = a, y = b
shiggy

how do into Evo-Biology and what are some good starter texts for uni students or undergrads i can snag? not looking for brianlet stuff, i'm willing to do math and think for myself and study. Please someone your fucking stupid wiki doesn't work at all and there is no evo-biology section.
t. auto-didact trying to get better at thinking

tfw you forgot the chain rule in your calc 3 final and fucked up a flux question

more unknowns than equations means you can have an infinite amount of solutions, if it's the other way around then it's trivial, and when you row reduce it, one or more of the equations should go entirely to 0.

forgot the chain rule
how the fuck can one forget that?

It's more so that I took the derivative wrong than forgot it. I was sloppy and what should've been a relatively easy flux ended up requiring me to use integration by parts twice.

and when you row reduce it, one or more of the equations should go entirely to 0.
Thanks. Does that then mean each of the column vectors making up the system are dependent?

faculty.bard.edu/belk/math213f12/LinearDependence.pdf
First theorem here seems to state it but yeah I'll ask him tomorrow.

I am assuming you mean linear system.
First of all, start with a homogenous m (equations) by n (unknowns) system.
It can be written as Ax=0.
Consider the linear transformation $f: \mathbb{R}^n \ni x \mapsto Ax \in \mathbb{R}^m$ .
{solutions of Ax=0} = ker f
This thing holds:
$\dim \mathbb{R}^n = \dim ker f + \dim Im f$
which can be rewritten to
$\dim ker f = n - k$ where $0 \leq k \leq m$

The system has a unique solution iff ker f={0} which is equivalent to dim ker f = 0.
For this to happen you need k to be equal to m, therefore m (equation) needs to be at least n (unknowns).

If m is greater than n you can't say much about how the solutions are.
See this:

en.wikipedia.org/wiki/Overdetermined_system

You have a vector space of dimension n.
And you have m vectors from this vector space where m>n.
Can those vectors be linearly independent?

0≤k≤m
0≤k≤min{m,n} , sorry

No.
Why does a row of 0s tell you it can't though?

Why does a row of 0s tell you it can't though?
what is "it" and what it "can't"?

what are some good books on Statics, I'm gonna be switching into MechE next term and for some reason the school will let me take dynamics before I've dealt with statics.

If a sine wave increases in frequency with time is it still a sine wave?

looks like the composition sin(f(x))

Depends on how you define "sine wave". If it is just a function of the form c*sin(ax+b) then no, cause the one you have in your picture is of the form sin(x^2) or something like that.

dumb question but when finding the distance between two objects i subtract the vectors correct? does it matter which one i put first?

dumb question but when finding the distance between two objects i subtract the vectors correct?
Yes, you subtract them and find the length of the resulting vector.

does it matter which one i put first?
If you substract them the other way, then you get the same vector but with opposite direction.
The length stays the same.

so when finding the distance to figure out the force of gravity i should find the distance vector and then find the magnitude correct? so it wouldnt matter which one i put first?

so it wouldnt matter which one i put first?
No, since ||-v|| = ||v|| for all v.
||u-w|| = ||-(w-u)|| = ||w-u||

if you just want to find the magnitude of the force then it doesn't matter, but if you want to find the force vector then it does

Appreciated
What would the decreasing function be?

- On the Origin of Species, Darwin (evolution from observational/logical viewpoint)
- Evolutionary Analysis, Freeman/Herron (evolution from genetic viewpoint)
- Evolution: What the fossils say and why it matters, Donald Prothero (evolution from fossil record viewpoint)
- The Blind Watchmaker, Dawkins (evolution from creationism-debunking viewpoint)

Pic: cave structure built by Neanderthals

the decreasing function
the what?

How does x-5 vanish instead of the denominator ending up as (x-5)(7-5x)?

Oh nvm I see what you mean.
Well I wanna say sin(1/x^2) but it doesn't seem like it. Nor sin(sqrt(x)) or stuff like ln(x). It seems like it has to be a rational function.

If increasing frequency is sin(x^2) then what would decreasing frequency be?
Pls no bully my retardation

$f \circ g (x) = f(g(x)) = \frac{9}{g(x)-5} = \frac{9}{\frac{7}{x}-5} = \frac{9x}{7-5x}$

Thanks for the reply, it points me in the right direction

sin(x^2) has decreasing frequency as x increases.
Decreasing frequency would just be sin(f(x)) where f(x)/x is increasing.

???
sin(sqrt(x)) works fine

Thank for spoonfeeding

For a prime number p, and a,b integers less than p, can I have a*b divides p?

Numbers don't divides numbers. People divides numbers.

No, unless a=b=1 or a*b=p. Prime numbers by definition have 1 and themselves as their only factors.

I'm confused about modular arithmetic notation. So I have [xy]*z equivalent to x*[yz], meaning the equivalence class of xy * integer z equals integer x * equivalence class of yz, modulo n. Is there something that lets me bring the integers inside the equivalence class? To get [xyz] = [xyz] modulo n

[xy] is a member of Z_n and z is a member of Z. technically you cannot multiply elements of two different rings. how do we make a sense of this? in abelian groups you can automatically multiply by integers: you define k*g = g + .. + g k-times. (this is to say that every abelian group is a module over Z). so [xy]*z really stands for

[xy] + .. + [xy] z-times

which, by the laws of modular arithmetics, is

[xy + .. + xy] z-times

which clearly equals

[xyz]

Is anyone know linear algebra. I need some clarification regarding similarity.
If A~B, then $A = SBS^{-1}$
But then I also see it written as $A = S^{-1}SB$
Is there any logic on why it's written differently? Doe sit matter?

Fucking Hausdorff spaces man, how do they work? Pic related is an example given in Jänich's Topology of how quotient spaces of Hausdorff spaces may or may not be Hausdorff. The solid lines depict equivalence classes. If p is the projection from the original space to the quotient space (sending a point to its equivalence class), a set U in the quotient space is open iff the inverse image of U under p is open.

Now I don't get how even one of these examples can be Hausdorff. For example, consider the space to the left and in that, consider any two distinct squiggly curves. Any open set (in the original space) containing any of the squiggly curves must have an open neighborhood around the end points of the curve, so it must also contain some points from the straight line equivalence classes. So any open set in the quotient space containing the equivalence class of the squiggly line will have to contain the equivalence classes of infinitely many straight lines as well, right? So two open sets containing two squiggly line equivalence classes will always have the straight line equivalence classes as common points, and so they cannot be disjoint? The same reasoning applies to the space on the right, so even that one can't be Hausdorff, right? Where did this reasoning go wrong?

Tldr: Can you help a brainlet identify which of these quotient spaces is Hausdorff?

What did you guys use to learn about forcing? I'm slowly coming to grasp with Boolean valued forcing from Jech (which I prefer to Kunen), but it still seems very mysterious. It feels like I'm studying a huge machine with thousands of tiny parts that miraculously work together to produce the generic extension.

What end point are you talking about? The curves extend asymptotically to infinity.

Quotient spaces have coinduced topology.

the similarity relation tells you this: two matrices are similar if and only if they represent the same linear transformation but expressed in different bases. If $A$ represent $f$ in $\alpha$ and $B$ represent $f$ in $\beta$, then $S$ is either the transition matrix $\alpha \to \beta$ or the transition matrix $\beta \to \alpha$, this depends on which of the two definitions you use, and obviously it doesn't change any logic of what's going on. So it doesn't matter.

You can always multiply them together, you cannot always pull them apart

Ahh now that I look closer at the picture it does appear that there are no end points. So the quotient space to the left will be Hausdorff correct?

Yes.
Same thing. The two notions are each other's dual.

Thanks a lot, that was really helpful.

I am trying to prove that, with the standard topology, a subset A of $\mathbb{R}$ is compact if every sequence in it has a convergent subsequence.

The characterization of compactness I am using is "A set if compact if and only if it is closed and bounded". I already proved that A must be bounded so now I am trying to prove A is closed. My argument is as follows:

A is closed if and only if $A^c$ is open. Suppose that $A^c$ is not open. This means that there exists an $x \in A^c$ such that for all $\epsilon > 0$, the $\epsilon-neighbourhood$ around x is not contained in $A^c$.

Notation: The epsilon neighbourhood around x is $V_{\epsilon} (x)$.

If $V_{\epsilon} (x)$ is not contained in $A^c$ This must be because the intersection $V_{\epsilon} (x) \cap A$ is not empty.

Consider the family of sets $B_n = V_{\frac{1}{n}} (x) \cap A$ and let $x_n$ be an arbitrary element of $B_n$

The sequence $x_n$ is a sequence of points in $A$ which means it has a subsequence that converges to a point $x_0 \in A$.

But because the way the sequence is defined, it is always true that $|x_n - x| < \frac{1}{n}$ so the sequence of $x_n$ converge to $x$.

If a sequence converges to a limit, then every convergent subsequence of that sequence must converge to that same limit. This means that $x_0 = x$ so $x \in A$ and this contradicts the fact that $x \in A^c$.

This is the contradiction I needed to find, which arises because I assumed $A^c$ is not open, therefore it must be open. And by definition $A$ must be closed.

My question is now, is this proof rigorous? Is it really okay to choose an arbitrary element of $B_n$ ?

well, it's much easier. take a sequence on A that converges. if it converged outside of A, then every subsequence of it would converge outside of A.

but since you asked about your proof: yes, it's rigorous. it's okay to choose an arbitrary element. it may happen that you repeat elements a lot, but it satisfies convergence anyway.

It's perfectly OK if you're aware that you're using AC to make an infinite number of choices - one for each n. Even countable choice is sufficient for this.

Okay, I see.
Even countable choice is sufficient for this.
Could you explain this?

As you're arbitrarily choosing one point x_n for each n, you only need to make a countable number of choices, so your proof will be OK even if you assume the axiom of countable choice instead of the more general axiom of choice.

you are taking the set {B_n, n \in N} which is a set
you know there is an element x_n in B_n \cap A for each n
how do you know the (x_n) are a sequence, which for examples implies that {x_n} is a set? which of the axioms of ZFC are you using? (hint: it's the axiom of choice, but it's enough to have a weaker version that works in countable sets)

to illustrate why you don't need AC to say the (B_n) form a sequence, they are the image of a function:

f : N \to P(R)
n \mapsto {x \in R / d(x,0) < 1/n }

Okay. Could this be proven without the axiom of choice or even countable choice?

I suppose is giving me a hint but I don't quite get it.

a set A is closed in R iff every sequence in A that converges in R converges in A
this equivalence probably needs AC too lmao. you can't do analysis without AC

Okay. Could this be proven without the axiom of choice or even countable choice?
I don't think so, but don't worry AT ALL about axiom of choice, especially when doing analysis.

You could also use nested interval theorem.

It depends on the limit you get from the ratio test. If your limit is zero, when related to one, that means it always converges since that satisfies the condition of convergence for all x.

If you get an infinity for the limit, it immediately fails since the sequence is fundamentally divergent. Hope this helps.

but don't worry AT ALL about axiom of choice
Using unnecessary assumptions is a sign of a true brainlet.

if you're inside a car moving down the highway at 50mph. And you reach your arm outside the window and fire a bullet from from a pistol pointed the same direction the car is moving.

does the bullet travel its regular speed, or its regular speed + the speed of the moving car?

Ladder problems are just equilibrium. So torque net has to be zero. Write out the equation and solve for the value you want.

It depends on where you're measuring the speed from. If you measure from the car, it's just its regular speed. If you measure from the ground, it will be the added speeds.

Now I'll you need to do is identify the 4 other people like you who are ruining this board. Together we can round them up and make their lives better. (Or kill them)

Hey Veeky Forums. Found myself in a bit of a pickle working on matrices.

I've proven the transposition of a matrix product $(AB)^T = B^T A^T$ via induction but I can't figure out how to prove the general case (i.e. $(A_1 A_2 ... A_n)^T = A^T_n ... A^T_2 A^T_1$ ).

Naturally, it is assumed all the matrices are of the same type and their product is defined. Any ideas, hints, tips? Seems like it should be simple, but I don't know how to get rid of the transposition step by step.

Is Introduction to Mechanics by kleppner and kolenkow regarded as difficult? I can hardly do any of the problems. But am managing University Physics Young Freedman problems fine as well as my college assignements. Am I just a brainlet or is Kleppner more of a 2nd year undergrad text?

Do you know how to do induction?

obvious. use associativity of matrix multiplication

you mean the opposite thing, right?

I'm actually retarded. Thank you, user.

I've proven the transposition of a matrix product (AB)T=BTAT(AB)T=BTAT via induction
I doubt it.

As you should, my bad, not induction, wouldn't make any sense. I rewrote the post once or twice, and left induction where it shouldn't be because I knew the general case was to be proven via induction and I must've left it there while I was rewriting. Good catch.

I have to write a piss easy 20 page essay, but i can't seem to find the motivation to start.
How do i do it?
It used to help to listen to ASMR but that doesn't help anymore because i overused it.
It also doesn't help that i don't have a fixed time frame to finish the essay. The deadline is basically "this semester"

Why humans are so rare? Just one species capable of building and improving shit overtime? Something is not right.

Why us? Why monkeys dont build skyscapers or fly into space?

You can always multiply them together, you cannot always pull them apart
But the post you quoted has an example where you can't multiply them together

Why birds are so rare? Just one species capable of flying and migrating places overtime? Something is not right.

Why us? Why monkeys dont grow wings or fly into sky?

what are flying insects

plenty of shit have wings and is able to fly yo but none of it builds skyscrapers

ok but why can parrots talk and insects are dumb and fly into a burning hot lightbulb

a better question is why do we have brains when they're so energy intensive (20% of our calories go to the brain), create dangerous birth conditions (brain must squeeze though birth canal), are fragile, and require a long development (10+ year childhood)

No? The "normal speed" is being defined as the speed of a shot bullet from the reference frame it was shot in.

Prove that if the general case is true for n=k, then it's true for n=k+1. This means that if it's true for n=2 (which you already showed), then it's true for n=3, then n=4, and so on (proof by induction).

is there any 'fast track' to general relativity for a student of pure math? i took intro physics in my first year undergrad, but didn't take a course since.

What is the inheritability of intelligence? Does it mean that intelligent ppl will usually have intelligent children and unintelligent ppl the opposite? If that is the case, were the 20th century eugenicists right?

alright, what now

IQ is inheritable to an extent, but that's not the only factor in determining it. If a society's goal is to increase the population's intelligence, then yes, the eugenicists were probably right. I believe that the brand of eugenics they practiced required making people infertile and treating people without beneficial genes as lesser, which is why society considers the practice unethical, though there could be other forms of eugenics which work around these problems.
Change in enthalpy is equal to the heat at a constant pressure.

I just had this question on an exam.
For what values of x does the function f(x) = sin(x) * ln(sin(x)) have horizontal tangent lines on the interval [0, π]?
I got π/2 for one of my answers, it said there were two.

f(x) = sin(x) * ln(sin(x))
that's a preddy cool function

Not sure how I solve that without using a graphing calculator, unfortunately.

take derivative
set to zero

i count three
you want to solve for x in $f'(x)=0$. that is
\begin{align}0&=\cos(x)\log(\sin(x))+\sin(x)\frac{\cos(x)}{\sin(x)}\\&=\cos(x)\log(\sin(x))+\cos(x)\\&=\cos(x)(\log(\sin(x))+1)\end{align}

so $\cos(x)=0$, in which case $x=\pi/2$, or $\log(\sin(x))+1=0$ in which case
$\sin(x)=e^{-1}$ and $x=\arcsin(e^{-1})$ in [0,pi] whatever that works out to be

f'(x) = cos(x) + ln(sin(x)) * cos(x) = 0
I knew that if cos(x) was 0 then the derivative would be 0, so π/2.
Other than that I don't know how to solve it.

plugging $x=\arcsin(e^{-1})$ works out. The more interesting question is how to get that third point. Clearly something to do with the periodicity, but how could you reason that?

Factor out a cosine in your calculation to get: cos(x) ( 1 + ln(sin(x)) = 0 . Now it's two cases, cos(x) = 0 , and 1+ln(sin(x)) = 0 .

the solutions to that are found like they are to cos(x)=0: by $x=\arcsin(e^{-1})+2n\pi,\quad n\in\mathbb{Z}$ where x is in [0,pi]

plugging x=arcsin(e−1) works out. The more interesting question is how to get that third point.
for any a in (0,1), sin(x)=a has two solutions in [0,pi]

Oh I actually missed factoring that out. I may be retarded. Thanks anons.

ah yes, seem to be a bit of a brianlet myself

Other than that I don't know how to solve it.
Learn basic algebra before you go to calculus.

Fuck brahs I have a calc I final in a few days and I'm completely lost. What's the most streamlined way for me to learn to take limits, take derivatives, solve related rates and optimization problems, solve EVT, MVT, etc. problems, and solve problems related to the fundamental theorem of calculus? Note I already have a vague sense of these concepts I just need a really efficient way to consolidate them to get a good grade on this final, and I don't really know how to study this shit efficiently. Help appreciated.

My mother and I discovered we have the same 4 digit PIN

What are the chances of that happening?

What you ask is extremely vague. In general, practice makes perfect, however if you want anything more specific you'll have to post problems that you struggle on and we might help you.

Is the set of language {<M> | L(M) is decidable} semidecidable?

in that case the function can be reduced to 2^n, something like 3^n * 2^n can be simplified as 6^n.

What's the best way for me to review? What problems, that if I knew how to solve, would prepare me fully for a standard calc I final?

Can I still use the divergence theorem if the orientation is downwards?

in that case the function can be reduced to 2^n
not for all n

If you understand the theorems and know the definitions, you will be fine. If you want problems, sift through this tutorial.math.lamar.edu/Classes/CalcI/CalcI.aspx , and look at the examples.

All the possible PINs are 10^4=10000.
Construct a 10000 by 10000 table listing all the possible pairs pairs of (your PIN, your mom's PIN). They are 10000*10000 in number.
The pairs in the diagonal are the desired ones. They are 10000 in number.
Assuming that all PINs are equally likely (which they aren't, but whatever) the probability is 10000/(10000*10000) = 1/10000 = 0.01%.

Do you guys doing hard science college/uni courses work while you study? I may have to start working 20-30 hours a week while doing medical sciences and I'm starting to doubt whether I can do it without failing.

I worked 20 hours a week for two years when I started my math major. All I can say is that it is a huge burden.

At the beginning you may not even notice it, but the problem is the domino effect that happens when anything starts going bad. If something bad happens in your job that puts you behind schedule and causes stress then that stress goes into your university life, causing you to also go behind schedule on your uni work and then that causes more stress and you do even worse at your job, and that continuous until something big happens, and there are only two big things that can happen

1) Your problems suddenly disappear (can happen but rare)
2) You fail big and get an F in uni or get shit on your job by your boss

The last time I faced this domino effect I decided I needed to force 1) to happen so i just quit my job.

Not quite the problems I was expecting. I assume I just won't have enough time to do assignments and get on top of the content. Plus when I start doing practicals I probably won't have time to go to work. This is all topped off by a lack of actual jobs, let alone ones that can actually fit my uni hours and pay my fucking rent and perhaps food.

brainlet here

i have a 2d "random variable", how i should calculate variance of only of its "dimensions" ?
should i just take marginal distribution as pr. values ?

Have any of you taken astronomy courses in college? How were they? Worth a minor?

If I want to be a scientist (phd) and I want to be a drag racer, what science field am I going for, what degree? Thank you, sincere question.

PLS HALP DUE IN 1 MINUTE

Niggaroni pepperoni you are being spoonfed what you have to do. Find the normal vector using the literal formula you are given and then the normal vector determines the plane.

I found the normal vector. When I plug that into [z-component of normal vector](z - [initial z]) = [x-component of normal vector](x - [initial x]) + [y-component of normal vector](y - [initial y]), it says WRONG FUCKING ANSWER, BUP BOY

At this point the only thing to do is get a book of problems and start trying to solve them. Then look up what you don't understand, or as a final resort ask here.

Differential geometry: Schutz - Geometrical Methods of Mathematical Physics
GR intro: Wald - Space Time and Gravity
GR advanced: Wald - General Relativity

Cool. Well with the human population being what it is it's bound to happen once or twice.

I have to show that $\sum_{n=0}^{\infty}\frac{1}{(z-1)^n}$ converges uniformly on all closed disks contained in $\{z\in\mathbb{C}|\lVert z-1 \rVert > 1\} := A$. I tried a method, but it's not what the professor usually uses, so I don't know if it's right...

Let $z_{0}\in A$ and $r>0$ such that $\overline{D(z_{0},r)}\subset A$. $\frac{1}{(z-1)^n}$ is holomorphic in the disc (so continuous in the boundry) because $1\notin A$ so by the maximum modoulus principle $\exists z^*\in \partial\overline{D(z_{0},r)}$ such that $\forall z \in \overline{D(z_{0},r)}$ $\frac{1}{\lVert z-1\rVert^n}\leq \frac{1}{\lVert z^*-1\rVert^n}=\left(\frac{1}{\lVert z^*-1\rVert}\right)^n$. Because $z^* \in A, \frac{1}{\lVert z^*-1\rVert}<1$ then $\sum_{n=0}^{\infty}\frac{1}{\lVert z^*-1\rVert^n}$ converges and then just use Weirstrass M test...

Do maths degree they said, do well on assessments all year, barely pass end of year exams everytime (bar once). Leave with a pass grade.

So what are my options with a worthless grade maths degree?
the assessment questions tended to be more difficult but you had time

doing a math degree is like taking an official public IQ tests, your diploma is just a proof of your low IQ.

Kunen, have never done Boolean-valued models, but I really should, apparently its easier. Learning the machinery of forcing is the worst part of forcing. But once you get past that it should be ok. Im trying to learn Gandy forcing right now for a research paper, god I have no idea whats going on. I really like the idea of effective theory but it gets really tedious at times. What subfield are you interested in?

Mechanics (Dover Books on Physics) by Hartog

Dawkins

kill yourself.

Is there a biological reason why I find the act of removing objects from bodily orifices stimulating.(earwax/tick/botfly/tonsil stone/blackhead ?removal)

But i've got a 144 IQ on mensa.

Bump. Just go into the compsci meme? Got a 2:1 on applied probability at least and have an idea where i could go with it.

How is drawing curves in a solid using circles called ? like I need to mimic the curve on the inseam of a pair of jeans but I can't find any books or tutorials on how to determine the radius of a circle for said curvature etc

I know a CAD software would help but I'm looking for a book or a resource that explains it

Sorry if I'm unclear and sorry if this is /wsr/ I'm a brainlet

Can someone help me understand pic ?
Is $\delta_{k,0}$ the kronecker delta or the delta distrubtion ? Why is it N, shouldnt it be 0 since all term would cancel each other ?

at last I found an illustration

Am I reading the science on this wrong, or can most high level nuclear waste be thrown into breeder reactors to be turned into more nuclear fuel where it'll eventually decay into non-radioactive stuff?

Why are we not doing this as much as we can instead of trying to find holes to shove it underground?

That really looks like one of those troll papers.

is this textbook lying to me?

see d.
(90 - z)
(90 - 19)
71
yes

is holomorphic in the disc
Circular. Prove that it is holomorphic in a non-empty open subset of $A$ first.

Kronecker.
Why is it N, shouldnt it be 0 since all term would cancel each other ?
What?

Is the determinant of the bordered Hessian in a Lagrangian function always a linear transformation of the constraint?

Am I reading the science on this wrong, or can most high level nuclear waste be thrown into breeder reactors to be turned into more nuclear fuel
The only stuff you can turn into fuel is heavy elements which absorb neutrons to produce fissionable isotopes. Most of the research has concentrated on Uranium-238 or isotopes of Thorium.

Why are we not doing this as much as we can instead of trying to find holes to shove it underground?
Producing fuel using breeder reactors is more expensive than just mining fresh Uranium or using Plutonium from reprocessing spent fuel.

And it doesn't do much to solve the waste problem.
Most of the radioactivity in spent fuel comes from fission fragments (mainly Caesium-137 and Strontium-90), which aren't useful as fuel (and you need to remove them before
you do much else with the spent fuel).

Not really, the definition of holomorphic we got was that it complex diferentiable (and that that's equivalent to C-R, and being diferentiable as a function from R^2 to R^2).

Given that X is a continuous random variable uniformly distributed on the interval $-1<x<1$, find the
probability density function of $Y = 4 - x^2$.
Attempted solution: If X is uniformly distributed we can write the PDF of X as:
$f(x) = \frac{1}{2}$ if $x \in (-1,1)$ and $f(x)=0$ otherwise.
Now, we can define $G(Y) = Prob(Y \leq y) = Prob(4-x^2 \leq y) = Prob(x \geq +-\sqrt{(4-y))}$
Alright, so what should I do now? How do I use the integral of the PDF of X to find this? I tried using both
negative and positive value for the square root inside the Prob, but both returned me screwed up intervals for y that
won't hold for the interval of x given.
Thanks for the help!

I don't know if this is the right place to ask, but I'd appreciate advice. I fucked up this semester and failed two of my courses, guys. Fell into deep depression because of untimely deaths in my family and isolated myself from my peers. How do I get back on my feet? Got prescription for Trintellix and trying to study everyday to prepare for my next semester.

Is δk,0 the kronecker delta or the delta distrubtion ?
Maybe δk,0 = 1 iff k=0 and δk,0=0 when k != 0?

Prob(4−x2≤y)=Prob(x≥+−(4−y))−−−−−−−√
Goddammit, learn some basic fucking algebra before you go to higher subjects.
First of all, $R_Y = [3,4]$ .
$P(Y \leq y) = P(4-x^2 \leq y) = P(x^2 \geq 4-y) = P(x \geq \sqrt{4-y} \lor x \leq - \sqrt{4-y} ) = P(x > \sqrt{4-y}) + P(x \leq -\sqrt{4-y}) = 1-P(x \leq \sqrt{4-y}) + P(x \leq -\sqrt{4-y})$ .
The rest do them yourself.

How did you come up with that interval?

What's the image of [-1,1] under 4-x^2?

Ok. I got it.
I'm having some problems with this subject because my professor gave us a really shitty material to study with. Now I understand what he meant on his notes.
Thanks. And yes, that algebra thing was ridiculous, though I'm usually not that retarded.

I thought RMS*sqrt2 = Vp. So Vp/sqrt2 = RMS
Why is this right? I tried an online RMS calculator and I got 340 as well

Really stupid quest but are the various small islands in the middle of the ocean basically peaks of underwater mountains?

they've clearly mentioned that 480 is the peak to peak voltage

Yes it is.

how do i enable viewing math equations as LaTex here?

im using 4chanx btw

the average Veeky Forums browser can't solve this

How do you answer one-sided limit problems on an exam? The way it's explained in youtube videos seems very imprecise/"unmathematical" to me, given how autistic all the other memes are.

$\forall \varepsilon >0 : \exists \delta > 0 : \forall x : a<x<a+\delta \implies |f(x)-l|<\varepsilon$
If this is true, then you write $\lim\limits_{x \to a^+} f(x) = l$

$\forall \varepsilon >0 : \exists \delta > 0 : \forall x : a-\delta<x<a \implies |f(x)-l|<\varepsilon$
If this is true, then you write $\lim\limits_{x \to a^+} f(x) = l$

Meant $\lim\limits_{x \to a^-} f(x) = l$ at the second part.

trying to understand the physics of musical instruments and all that.

when you pluck a guitar string why does that produce its resonant frequency? I thought to get a string to resonate you had to apply a driving frequency to it at just the right frequency at its natural resonant frequency. Why is it that we can just pluck strings and they somehow magically resonate without having to hit it with the right frequency?

How do I solve this using partial fractions?

Ahhh

This is a brainlet response, but I remember hearing that frequencies that aren't an integer multiple of the resonant one will end up destructively interfering.
Maybe you could start with some fairly random initial condition as your pluck (say a delta function in the middle), make the fourier series and attach time dependence and see what happens. Also I wonder if adding damping will cause high frequencies to die faster than low frequencies (that's typically the case with damping). So then if you start with an admixture of frequencies, the non-integer multiples of the resonant effectively die by destructive interference and the integer multiples die by some sort of damping mechanism as damping affects high frequencies more.

That Valenza Linear Algebra book is actually quite brainlet friendly and fun.Thanks to the user that recommend it me a while a go.

Why doesnt pic rel form a racemic mixture? I know all the reactions but im still getting confused as to why some form racemic mixtures and some dont

are you using noscript or something similar? the latex is displayed using javascript code loaded from a third party site

When you twist the C3 and C4 sigma bond by 180 degrees, you can see that there exists a mirror plane. Compounds with stereoactive centers but have mirror planes are called meso compounds. Since they have mirror planes, they are optically inactive and non racemic.

Whether or not the reaction results in a racemic mixture depends on both the substrate and reaction type.

Resonance doesn't require that the excitation is a pure sine wave, only that it contains some component at the resonant frequency. If you calculate the Fourier transform of an impulse (Dirac delta), you find that it contains all frequencies (convolution with an impulse is the identity function).

Another way of looking at it is that the solution to d^x/dt^2+(w^2)*x=0 is a sine wave, whose angular frequency is w. The initial conditions affect the amplitude and phase, but not the frequency.

What happens if you give a Rubik's cube Pro an unsolvable Rubik's cube?

you're right. turns out umatrix was the culprit

en.wikipedia.org/wiki/Partial_fraction_decomposition
Since, $s^2-2s+2$ is irreducible in $\mathbb{R}$ you do this:
[eqn] \frac{a}{s} + \frac{bx+c}{s^2-2s+2} = \frac{1}{s(s^2-2s+2) [/eqn]

[eqn] \frac{a}{s} + \frac{bs+c}{s^2-2s+2} = \frac{1}{s(s^2-2s+2)} [/eqn]

My notes give this result
[eqn]lim_{\eta\rightarrow0^+}\int_0^\infty e^{(i\omega-\eta)s}\text{d}s=\pi\delta(\omega)+i\mathcal{P}(1/\omega)[/eqn]where the curly P indicates the Caouchy principle value.

I tried this myself and I get the delta function but I don't get why the imaginary term has to be wrapped in the principal value. Can anyone help?

Let's try that again

My notes give this result

[eqn]lim_{\eta\rightarrow0^+}\int_0^\infty e^{(i\omega-\eta)s}\text{d}s=\pi\delta(\omega)+i\mathcal{P}(1/\omega)[/eqn]
where the curly P indicates the Caouchy principle value.

I tried this myself and I get the delta function but I don't get why the imaginary term has to be wrapped in the principal value. Can anyone help?

one more time

My notes give this result

$lim_{\eta\rightarrow0^+}\int_0^\infty e^{(i\omega-\eta)s}\text{d}s=\pi\delta(\omega)+i\mathcal{P}(1/\omega)$

where the curly P indicates the Caouchy principle value.

I tried this myself and I get the delta function but I don't get why the imaginary term has to be wrapped in the principal value. Can anyone help?

well fuck knows why Latex isn't working here

there's a tex previewer

They worked in the preview

Here, just use this image instead

I'm curious, how do you solve it?
I wrote down Ae1= ..... , Ae2= .... , etc. and only found that if you add them you get e1+...+en which makes it an eigenvector of eigenvalue 1. Can't think of any other tricks do to with those equations.
Then I tried to represent the matrix as a triangular one under some basis, but I can't find any such basis.

Holy shit this was available all this time? I didn't even think it was clickable. I was going too quicklatex.com to check if I wrote things down properly..........

Checkes, also idk if there's a more elegant method but just use induction and use laplace formula for the determinant and expand using rows. When expanding though the first row, everything is 0 except the last term and the minor will then have two non 0 terms and so on.

Really stupid question: what is the correct way of solving 2^logn-2 (base2).
Does it become: n^-2 or what?

Type it out in latex so someone might understand what you're trying to ask

is the function y=x^2+x a parabola and why?

yes, because it's a polynomial degree 2

When expanding though the first row, everything is 0 except the last term
Are you talking about det(A)? We gotta find the eigenvalues, which are the roots of det(A-xI).
Btw $\det(A)=(-1)^n \prod\limits_{i=1}^n a_{ii}$ straight from the permutation formula.

Yes, but I'm telling you to use laplace formula.

But how do you expand that? There are two non zero entries in every row and column.

oops it is not $(-1)^n$, it's:
$(-1)^{\frac{n}{2})$ , if n is even
$(-1)^{\frac{n-1}{2}$ , if n is odd

cause the permutation is (1,n) (2,n-1) (3,n-3) ...

Can someone explain the Taylor and Maclaurin Series to me like I'm a retard?
I'm looking at my notes and I feel like the equations for them make it seem more complicated than they are.

Heres a practice problem I looked at and got lost

"If f^n)(0) = (n + 1)!
for
n = 0, 1, 2, ,
find the Maclaurin series for f. "

When someone asks to explaina broad and branching conceot with many different things to it, it makes him look that he has no idea whatsoever. Could you ask a more direct question? What do you not understand?

So the two series are used to take any function and rewrite them as series, opposed to just functions that look like geometric series. It makes sense when I word it like this but the actual process looks like a clusterfuck.

I have this for the Maclaurin Series
[eqn]C_n = (f^n(0)/n!)x^n [/eqn]

And This for the Taylor Series
[eqn](Sigma)C_n(x-a)^n[/eqn]

I think it made slight sense during the lecture but looking at it now I don't understand how to use it or where to start.
This is an example I understand

"Find the Taylor Series for f(x)=e^x at a=2
Which is equal to the sum of [eqn](e^2/n!)(x-2)^2 [/eqn]"

What I don't understand is the problems like

"Find The Taylor Series for centered at 7
[eqn]f^n (7) = ((-1)^n n!)/3^n (n+2)[/eqn]"

I'm really just looking for the typical approach to these problems.

Read the wikipedia articles. They're good.
en.wikipedia.org/wiki/Taylor_series
en.wikipedia.org/wiki/Taylor's_theorem

Brainlet here.

How the fuck is random variable multiplication defined? I can figure out the summation by considering the events from the event space that get mapped to the different numbers and those number sum up - BUT WHAT THE FUCK IS EVEN MULTIPLICATION IN THIS CASE?

dumm frogposter
Yes I am pls halp me T.T

Confusion leading to "i must have messed something up let me start over"

"Cubers" just memorize rubicks-cube-solving algorithms and repeat them by rote as fast as possible

I can figure out the Multiplication by considering the events from the event space that get mapped to the different numbers and those number Multiply up

bump

Anyone in here have experience with R? I'm trying to generate a side by side histogram but can't figure out how to do it. I have two vectors that I'm trying to compare in one graph. My code looks like;

vector1 <- c(raw data)
vector2 <- c(raw data)

I've tried hist(vector 1, vector2, beside=TRUE) but that doesn't work. Any guidance/links would be much appreciated.

Use uniqueness of the maclaurin series to write f(x) as a sum of n-th derivatives, evaluated at 0 (since it's maclaurin series) divided by n!. This is the coefficient of x^n.

They will end up with a state which shouldn't be possible, at which point it's obvious that it's unsolvable.

For a valid cube, making a wrong move will simply result in a cube that's not as solved as it ought to be. There will still be a sequence of moves to solve it, and anyone familiar with solving will know what that sequence is. A state for which there is no correct sequence means that the cube is invalid.

-
Thanks

How do I generate the average amount of time for a single event given a set of data?
So in my program, 350 words are checked in a table, where the size of the table, N, increases in every 5 test samples.
What I'm doing at the moment is:
Sum all 5 times to add 350 words to a table sized N
Divide by 5 to get average
Divide by 350 to get time for one
Is this how to do it? I'm getting like 0.0002808ms for it which seems a bit small.

I don't understand how to parameterize and solve b)
I remember learning it in an earlier math class but forgot

Divide by 350 to get time for one
small mistake here, you're supposed to divide 350 by the average time, not the other way around

Divide by 350 to get time for one

Small mistake here, you're supposed to divide 350 by the average time, not the other way around