Previous SQT thread maxed

Forget what I said... just help me answer question c if ya can

>>8438194
Here's a hint

AB dot BC equals the magnitude of AB times the magnitude of BC times the cosine of the angle between AB and BC

You can rearrange and solve for the angle

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anyone know how to do this? completely stumped

Is the infimum of the difference greater than the difference of the infimums?

>>8438193
Remember how to make a vector from to points?
AB will be <Bi-Ai,Bj-Aj,Bk-Ak> where the i, j and k represent the components in the x,y and z directions. Apply this to AB,AC and BC.

The magnitude of a vector is related to... you guessed it, the pythagoras theorem (at least in euclidean coordinates).

So IBCI will be equal to ((Bi-Ai)^2 + (Bj-Aj)^2 + (Bk-Ak)^2)^1/2

The last question can be answered using the definition of the dot product.
Sorry for the format, I'm on my phone.

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How hard is CS?

https://www.cpp.edu/~sci/computer-science/docs/SCI_Comp_Sci_2016-2017_VML.pdf

I transferred this semester from Computer Engineering, and I completely hated the intro courses to CE (Intro to Electrical Engineering).

http://www.cpp.edu/~engineering/ECE/documents/CPE_15-16.pdf

I looked at the future courses for CE and I just didn't care for them. I already completed the Math up to Differential Equations, but how much Math is involved in the CS courses? I want to work in cyber security.

>>8439218
>how much Math is involved in the CS courses
The most that ever comes up is a little algebra (maybe linear algebra if you're going to get real hardcore).

There's a reason people say computer science is a meme degree.

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Anyone know what the name of this shape is?

>>8439241
pyramid

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>>8439246
Oh of course

>>8438200
Start by calculating
n*T(n) - (n-1)*T(n-1)

The sums cancel except for the T(n-1) term, leaving you with
n*T(n) - (n-1)*T(n-1) = T(n-1) + 10*n - 1
=> T(n) - T(n-1) = 10 - 1/n
=> T(n) -> 10*n

IOW: c) f(n) = n

can any geology guys tell me what's the difference between "cleavage" and 'schistosity" on rocks? I don't think the two terms can be used interchangeably, and yet I can't figure out a difference other than the fact that schistosity only occurs on (duh) schists. both terms seem to indicate that the rock's structure has a planar orientation.

I have a test tomorrow with a really annoying teacher and I just want to have a good response in case he asks this

Matlab's acos function is return ArcCos values in the range [0, pi], which is kinda expected but I need it go from [0, 2pi]

Anyone have any ideas how I could tell whether a result should be in the range 0-pi or pi-2pi

>>8439278
Im sorry I dont quite follow could you expand on the working more? Its a bit too simplified for me to follow..

T(n) = 1/n * (T(0) + T(1) + T(2) + . . . + T(n-1)) + 5n

nT(n) = T(0) + T(1) + T(2) + . . . + T(n-1) + 5n^2

Where am I supposed to go from here?

>>8439427
cos(-x)=cos(2π-x)=cos(x)
For x in the range [0,2π], every value of cos(x) occurs twice. If that's the only information you have, there's no way to distinguish between the two possible angles.

>>8439449
n*T(n) = T(0)+T(1)+T(2)+...+T(n-2)+T(n-1) + 5n^2
(n-1)*T(n-1) = T(0)+T(1)+T(2)+...+T(n-2) + 5(n-1)^2

n*T(n) - (n-1)*T(n-1)=T(n-1) + 5n^2 - 5(n-1)^2
=T(n-1) + 5n^2 - 5(n^2-2n+1)
=T(n-1) + 5n^2 - (5n^2-10n+5)
=T(n-1) + 5n^2 - 5n^2 + 10n - 5
=T(n-1) + 10n - 5

=> n*T(n) - (n-1)*T(n-1) - T(n-1) = 10n - 5
=> n*T(n) - n*T(n-1) = 10n - 5
=> n*(T(n) - T(n-1)) = 10n - 5
=> T(n) - T(n-1) = 10 - 5/n
=> T(n) - T(n-1) -> 10 as n->infinity
=> T(n) -> 10n - k as n->infinity

>>8439493
Much clearer thanks, but what happened here?

=> n*T(n) - (n-1)*T(n-1) - T(n-1) = 10n - 5
=> n*T(n) - n*T(n-1) = 10n - 5

How does (n-1)*T(n-1) - T(n-1) simplify to
n*T(n-1)?

and

=> T(n) - T(n-1) -> 10 as n->infinity
=> T(n) -> 10n - k as n->infinity

Where did -T(n-1) go? T(n) - T(n-1) = T(n-1) right? how is 10"n" back on the last line

wot is a measurable function?
I've seen two definitions
1: a function of a measurable space (X,M) into a topological space (Y,τ)
such that the inverse of an open set from Y is a measurable set.

2.a Function on a measurable space Xinto another measurable set Y such that the inverse image of a measurable set from Y is a measurable set in X.

i see that the first definition seems to imply the second if we use the Borel σ-algebra, but idk if there are pathological examples that would make it impossible to get a topology on Y if Y is a measure space or something like that.

is there any way to find the sum of the first n squares, without resorting to pyramids or cubes?

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Wanting to check if I've done these correctly. I can't really remember how to do the first one - you substitute the supplied value for ψ(x) and take the second derivative of it (Will this always be (2mE/h)ψ?
I have a feeling I'm supposed to go ψ(x) = Asin(pix/a) + Bcos(pix/a) and ψ(-a) = ψ(a) but I'm not sure where to go for there

>>8439598
You mean [math]\sum_{k=1}^{n}k^2[/math]? That's just [math]\frac{n(n + 1)(2n + 1)}{6}[/math].

>>8439617
yes - thanks for the reply

Iv been searching around on this, and i have found the formula you posted on several pages and youtube videos, but i cannot understand how this is derived.

i have seen the proofs involving pyramids, its cute but i would rather find it through algebra

All of the algebra i have found just assumes we need a cubic function to solve it. but i dont understand why we need cubes to solve a function based on squares.

nor can i see how i can start with
>1^2 + 2^2 + 3^2 ... (n-1)^2 + n^2
and end up with a cubic

please forgive my stupidity and my lack of latex formatting,

If A and B are sets, is "A contains B" just another way of saying "B is a subset of A"?

>>8439636
Induction, my man.

>>8439636
I found this: https://proofwiki.org/wiki/Sum_of_Sequence_of_Squares

I'm sure that will satisfy you. The first proof is just proving that the formula is valid, but the other proofs provide ways of deriving it. They all involve a bit of machinery that you may or may not have yet, but they're still valid.

>>8439652
What's the context? Another possibility is "B is an element of A."

>>8439652
yes

A contains B means every element of B is an element of A

>>8439673
"Let K be a compact set and let U be an open set that contains K."
Pretty sure it's >>8439674
Thanks, guys.

>>8439541
> Much clearer thanks, but what happened here?
> How does (n-1)*T(n-1) - T(n-1) simplify to n*T(n-1)?
It doesn't. But /both/ terms are negative, and
- (n-1)*T(n-1) - T(n-1)
= (-n+1)*T(n-1) - T(n-1)
= -n*T(n-1)

> => T(n) - T(n-1) -> 10 as n->infinity
> => T(n) -> 10n - k as n->infinity
> Where did -T(n-1) go?
f(x)-f(x-1)=m => f(x)=m*x+c
If the difference between successive values is a constant, then the function is linear. This is the point where the recurrence relation is converted to an explicit form.

Well, it's not quite constant because of the 5/n term, but as n gets larger 5/n approaches zero, 10-5/n approaches 10, and T(n) approaches 10*n-k. Which is enough to say that it's O(n).

>>8439696
>= (-n+1)*T(n-1) - T(n-1)
>= -n*T(n-1)
Man... im not getting it how is this possible youre subtracting stuff against a thing thats being multiplied with something? wtf Ive never done anything like this in my life I just cant see for the life of me whats going on

>f(x)-f(x-1)=m => f(x)=m*x+c
Oh right haha

>>8439598
(x+1)^n = x^n+n*x^(n-1)+...
(x+1)^n-x^n = x^n+n*x^(n-1)+... - x^n
=n*x^(n-1)+...

I.e. the difference between successive terms of a degree-n polynomial is a degree-(n-1) polynomial.

And the difference between the sum to N+1 terms and the sum to N terms is just the N+1 term.

So the sum of successive values of a degree n polynomial is a degree-(n+1) polynomial.

From there, it's just a matter of working out the coefficients.

E.g. if you have f(x)=a*x^3+b*x^2+c*x+d, then
f(x+1)=a*(x+1)^3+b*(x+1)^2+c*(x+1)+d
= a*(x^3+3*x^2+3*x+1) + b*(x^2+2*x+1) + c*(x+1) + d
f(x+1)-f(x)=a*(3*x^2+3*x+1) + b*(2*x+1) + c
= (3*a)*x^2 + (3*a+2*b)*x + (a+b+c)

So if you want f(x+1)-f(x) = (x+1)^2 = x^2+2*x+1, then
3*a=1 => a=1/3
3*a+2*b=2 => b=1/2
a+b+c=1 => c=1/6
And f(0)=0 => d=0

So f(x)=(1/3)*x^3 + (1/2)*x^2 + (1/6)
= (2*x^3+3*x^2+x)/6
= x*(x+1)*(2*x+1)/6

Somewhat easy question here,

If I have a plane (for example -5x-7y+7z=35) then the normal vector is n=<-5, -7, 7>. Is it true that any vector on the plane x has to satisfy the condition x.n=35?

>>8439727
> youre subtracting stuff against a thing thats being multiplied with something?

- (n-1)*X - X
= - (n*X - 1*X) - X
= - n*X + X - X
= - n*X

E.g. if n was 8, n-1 would be 7, -7X-X=-8X.

>>8439752
Yes.
<x,y,z> (dot) <-5,-7,7> = 35
<=> -5x-7y+7z = 35

>>8439665
Awesome, thanks, ill digest what i can, have a great day!

>>8439761
Holy that's obvious bless u anon

>>8439752
Let x be a unit vector on the plane. Then [math]x \cdot n = -5 - 7 + 7 = -5 \neq 35.[/math]

>>8439761
This is not a proof.

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>>8439755
TYYYY

What are the first and second derivatives of (4-x)e^-x?

My calculator and Wolfram give different answers (opposite signs) for the first derivative, and Wolfram and DerivativeCalculator give answers with opposite signs for the second derivative.

Maybe my algebra is shitty, but I can see how it logically works both ways when I work it out on paper, and I know they aren't equivalent cause the graphs different.

******VERY IMPORTANT****


I have an interview for joining a lab tomorrow. This will be my first one, what should I expect for questions?

>>8439898
Same poster here:
By hand, I get:

>f'(x) = (4-x)'(e^-x)+(4-x)(e^-x)'
>f'(x) = (-1)(e^-x)+(4-x)(-e^-x)
>f'(x) = -e^-x + (-4e^-x) + xe^-x
>f'(x) = -e^-x - 4e^-x + xe^-x
>f'(x) = -e^-x (1 + 4 - x)
>f'(x) = -e^-x (5 - x)

According to Wolfram and derivativecalculator, it's positive, on derivative calculator it distributed -1 back into the parenthesis, but this changes the graph. The derivative function on my TI graphs what I got.

>>8439925
>derivative calculator it distributed -1 back into the parenthesis, but this changes the graph
No it doesn't. That's an equivalent function.

You're correct. This is how you go from your answer to Wolfram's answer:
[math]-e^{-x}(5 - x) = e^{-x}((-5) + x) = e^{-x}(x - 5).[/math]

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>>8439935
And this is what Wolfram gives me, for the record.

>>8438193
What's the purpose of accumulation points in analysis? why do we define limits, the derivative in particular, with them, as opposed to simply requiring the domain to contain an interval or neighborhood around the point(s) of interest?

Say I have an array with 16 million numbers in it and it is sorted. Then I add 10 unsorted onto the end, sorting with insertion sort would be fastest right? Since it would be Initial input * the new input which would work out to be 160m vs for example a quick sort which would be 16m*log16m=240m would that just be O(N)?

Can some retard give me a nice simple explanation of the branch of the logarithm/complex logarithm? I just can't grasp the fucking thing. Reading out of Complex Analysis by Bak and Newman, they seem to do a shit job of explaining it. I need some intuition behind it.

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I want to do either embedded systems or machine learning. Would it be smarter for me to do a BS in math then CS? I got the freedom to pick quite a few classes if I do a BS in math and so I will have the prerequisites for both. I do realize that there is a difference in difficulty. But I've never understood how somthing can be too hard to learn. Not sure if I'm just a delusional brainlet or nah.

So CS or math?

>>8438200
Try to rephrase the recurrence relation as a recursive function. Suppose you have computed T(n), how do you compute T(n+1) from that information? T(n+1) = [[T(n) - 5n]*n + T(n)]/(n+1)] + 5(n+1) = [(n+1)*T(n) - 5n^2)/(n+1) + 5(n+1)= T(n) + 5(n+1)^2/(n+1) - 5n^2/(n+1) = T(n) + 5*[(2n + 1)/(n+1)] = T(n) + (10n + 5)/(n+1) = T(n+1). For very large n, (10n + 5)/(n+1) is approximately equal to 10, so as n increments by 1, T(n) increments by 10, so choice C best represents this relationship (up to a constant).

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I know you can express any rational number as a continued fraction in two ways [a0;a1,… an−1,an] and [a0;a1,… an−1,(an−1),1]
I want to know how you prove these are the only two ways to do it?
I know the two expressions are ending with n > 1 i.e. [ ..... , n ] or ending with 1 i.e. replace the final n by (n-1) + 1/1 i.e. [ ..... , n-1 , 1 ]
Wondering how you prove there are no other expressions

>>8439427
Do you have the coordinates for your point?

>>8440216
What sort of explanation are you looking for?

The logarithm is the inverse of the exponential, i.e. y=e^x <=> x=log(y).

If x is complex, x=a+bi. And f(x)=e^(ix) is periodic with period 2π, i.e. e^(x+2πi)=e^x. So e^(a+bi+2nπi)=e^(a+bi) for all integer n.

Which means that log(e^(a+bi)) = log(e^(a+bi+2nπi)) = {a+bi+2nπi}, i.e. the logarithm of a complex number has infinitely many values which differ by integer multiples of 2πi.

Whereas the complex logarithm is multi-valued, a branch is a single-valued function which returns a value whose imaginary part lies in a specific interval of size 2π. A branch cut is a line in the plane between branches, where the branch is discontinuous: if a+bi lies on the branch cut, then Log(a+(b-ε)i) and Log(a+(b+ε)i) will differ by approximately 2πi (more precisely, the one-sided limits lim[ε->0-]Log(a+(b+ε)i) and lim[ε->0+]Log(a+(b+ε)i) will differ by 2πi).

>>8440435
if n is a positive integer, then 0<1/n<=1.

So to express a positive real number x as a continued fraction:
x = a0 + 1/r0
0<1/r0<=1 => (x-1/r0)<=a0<x
If x is an integer, r0=1 and a0=x-1, otherwise a0 is floor(x).
Solving for r0 gives
r0 = 1/(x-a0)
To continue the fraction:
r0 = a1 + 1/r1
Repeat until you end up with the left-hand side equal to an integer. At which point, you can either stop, or take one more step:
rp = an+1/rn
As rp and an are integers, rn must be 1 and an must be rp-1.

In short, all of the coefficients are dictated by the value which you're trying to represent as a continued fraction. The only choice available is whether to express the final integer term rp as rp or as (rp-1)+1/1.

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What is a limit of a function?

>>8440619
Oops; just realised that isn't quite enough; for 1/r<=1 you need r>=1. But if r<1, then e.g. r0=a1+1/r1 implies a1=0, which is invalid. If you allow zero coefficients, there would be infinitely many such expansions, as
p/q = 1/(0+q/p) = 1/(0+1/(0+p/q)) = ...

>>8440628
What f(x) looks like it would be if by looking at f(values near x).

Is there a similar document for linear algebra?

http://www.damtp.cam.ac.uk/user/fq201/DEsummary1.pdf
Found it from the sticky, also like, I found a pdf for linear algebra there, but it requires a password.
>Linear Algebra (PDF) by Jim Hefferon, Saint Michael’s College
>ftp://joshua.smcvt.edu/pub/hefferon/book/book.pdf

I'm taking DE this semester but my school doesn't require linear algebra for my major for some damn reason.

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How did this happen?

>>8440788
Variance is based upon the square of differences.
Var(x)=E[(x-m)^2]

If the mean of x is m, then the mean of x/n is m/n.
Var(x/n)=E[(x/n-m/n)^2]
=E[(x-m)^2/n^2]
=E[(x-m)^2]/n^2
=Var(x)/n^2

>>8440818
Thanks anon. I have a feeling that you're answering a lot of these questions ITT. How do you know so much? Are you a grad student doing maths?

What calc course do I learn differentiation under the integral sign in? It looks really cool

>>8440822
Math background: school, then a CS degree, then spending too much time on wikipedia and Project Euler.

>>8440834
Do you work as a programmer or something more mathsy?

>>8440837
Programmer. But work programming rarely involves much math; that's more of a hobby.

Dear resident category theory experts, in what way does a span represent a collection-valued attribute?

I can provide some context if this does not make sense on its own.

Why is it that people in my classes can be frequently disruptive (talking in class while the lecture is going) or just not show up to class frequently, and talk about having not prepared for exams or left assignments to the last minute, then pass the course without issue while I make a reasonable effort to show up to every class and do my work but I don't do very well? Is there some trick I'm not getting

>>8439231
Not really depends on your field. If you want to make games you need algebra, calculus, physics, and geometry

Are there any situations that arise in the universe where matter travels at a significant fraction of the speed of light?

I'm thinking something around a fraction of say 1/10 or maybe 1/5

>>8441103
en.wikipedia.org/wiki/Oh_my_god_particle

99.99999999999999999999951% of the speed of light.

>>8441024
>friends
>form group
>group studies together
>collection of minds with varying knowledge
>each fill each others gaps of misunderstanding
>all whilst enjoying witty banter
>many minds> one mind ???

My book defines the limit of a function at a point p as: Let f : D -> R, D a subset of R, and the point p in D an accumulation point of D. Then f has a limit as x approaches p iff there exists L such that for every epsilon>0 there exists delta>0, such that x in D and 0 < |x-p| < delta implies |f(x) - L| < epsilon.

I understand that p doesn't have to be in the domain for the limit to exist, and the motivation for defining it this way is clear to me. My confusion arises from the fact we don't require a deleted epsilon neighborhood around p (on at least one side) as a subset of the domain. This way we still wouldn't care what goes on with f at p, but could be ensured f is defined at all points leading up to p. Requiring only that p is an accumulation point in the domain allows for holes or gaps in the domain approaching p. any sort of insight or help would be most appreciated

>>8439231
My CS course had a fairly substantial chunk of discrete math: propositional logic, predicate logic, temporal logic, formal grammars, Turing machines, lambda calculus.

That's aside from the math-related courses (graphics, computational geometry, optimisation and linear programming).

>>8441082
not him but do we need actual geometry (like studying The Elements) or would the analytic geometry encountered in calculus plus some linear algebra suffice? Euclid was of course very axiomatic, which is awesome, but doesn't appear to me to be all that important if all you are wanting to do is make games.

Also, formally speaking, is analytic geometry a subset of euclidean geometry? or maybe rather an extension of it?

What is the point of making students prove trig identities during a test when we can just assume they were proven centuries ago?

Okay. cos(x)/(1+tan(x))=cos(x)-sin(x)

Having to prove this is tedious:
cos(x)/(1+(sinx/cosx))-sinx/(1+(cosx/sinx))

Okay, I'm already to bored to type the rest out. Do most professors make you do this shit or is she just a bitch? What is the point in the big picture of modern mathematics now?

>>8441136
> Requiring only that p is an accumulation point in the domain allows for holes or gaps in the domain approaching p.
Like how you can define an irrational number as the limit of a sequence of rationals?

>>8441140
Your Uni miles ahead of mine then (but sadly that isn't saying much). All we require of our CS undergrads math-wise is calculus 1 and 2, and their choice between discrete math or some watered down "equivalent" the CS department offers called data structures, or something like that. No linear algebra, no number theory or logic, and certainly no lambda calculus or optimisation.

>>8441155
Well, a lot of that was for optional courses (I tended to choose theoretical courses over the more vocational stuff).

Basic discrete math (mostly logic) was mandatory, calculus wasn't. But subjects which had some underlying mathematical theory would actually teach it (e.g. databases used relational algebra a lot, concurrent programming used temporal logic for expressing invariants, etc).

AFAICT, places where CS is offered by the math faculty tend to do more theory than where CS (or Software Engineering) is part of the engineering faculty.

>>8441136
>Requiring only that p is an accumulation point in the domain allows for holes or gaps in the domain approaching p.
This is exactly the point. Limiting behaviour does not have anything to do with whether you contain a complete interval of the metric space or not.

All you care about is whether the function itself is getting close to something. All that depends on is the values of the function. All _those_ depend on are the domain. Why would numbers outside the domain make a difference?

You ought to take some time to reread (or get some professor's help with) the earlier sections of your book on limit points of sets and limits of sequences (presumably your book does sequences before limits of functions). The fact that you got through those and think a limit point has anything to do with "gaplessness" indicates you didn't really understand those sections very well.

>>8441153
I'm in a babby's first analysis course and the instructor completely glossed over the construction of the reals, could you provide another hint or direct me to some requisite material?

>>8441175
This is precisely what I needed, thank you. Yes I sort of put this course on the back burner and ended up doing very poorly on the first exam, so i am trying to go back from the beginning and get caught up. thanks again

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>>8441176
>the instructor completely glossed over the construction of the reals
really makes you think

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Retard here. Just want to make sure I understand something.

Mythbusters did pic related a long time ago. They fired a ball out the back of a car with the same speed as the car was moving, cancelling the momentum. For clarification's sake, let's say the car was moving at 20mph and the ball was launched at 20mph to make this happen.

From the frame of reference of an observer not in the car, the ball's speed is 0, since the momentum was cancelled.

But from the frame of reference of someone in the car, the ball was launched at 20mph. From their point of view, the ball's behavior was no different at all than if it had been launched from a stationary car. But this observation is wrong because we know that the ball's speed is 0mph.

Now, the only reason we have these two frames of reference is because we have the background and context to figure that out. But let's say there was no way of figuring that out; let's say the car was in a featureless void and the only observers were located in the car. These observers have no clue how fast the car is moving, or if the car is moving at all.

If the ball were launched from this void-car at 20mph, how would the observers know the true speed of the ball? It would be moving 20mph away from the car, but is it the car that's moving, the ball, or both? To what degree? How could something like that be measured without an outside point of reference?

And to extrapolate from that, doesn't the same concept apply to measuring the speed of any given celestial body in the real world, since the only reference are other celestial bodies that we also don't know the speed of? Is there no such thing as true speed, but only relative speed instead?

And given all of this, how do we know what the actual speed of light is?

>>8441218
>But this observation is wrong because we know that the ball's speed is 0mph.
No. There is no privileged reference frame.

>If the ball were launched from this void-car at 20mph, how would the observers know the true speed of the ball?
There is no such thing as "true speed".

>And given all of this, how do we know what the actual speed of light is?
The speed of light is constant in all reference frames and is an exception to the rule.

>>8438193
Are animals aware they have organs?

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So far I have 2....4,5....7,8....10,11....13,14....16,17

sort of a pattern maybe.

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How the hell do I do this?

>>8441218
>It would be moving 20mph away from the car, but is it the car that's moving, the ball, or both?
depends on how you define your reference frame.

>To what degree? How could something like that be measured without an outside point of reference?
it can't, speed is only a meaningful concept with some reference frame. in a total void there's no physical difference between it being the car moving or the ball moving

>And to extrapolate from that, doesn't the same concept apply to measuring the speed of any given celestial body in the real world, since the only reference are other celestial bodies that we also don't know the speed of?
yes
>Is there no such thing as true speed, but only relative speed instead?
yes
>And given all of this, how do we know what the actual speed of light is?
You can calculate it experimentally, it also falls out of the equations of E&M as the speed of electromagnetic radiation in a vacuum.
I'm sure it falls out of the relativity equations as well, but I've only had a few brief lectures on that and cbf to self study it.

If I have a block of wood that is partialy sumerged in both water and oil (the oil being on top lf the water) of length 10, what would be the pressure at the bottom of the block?

>>8441686
Find the derivative of the curve at an arbitrary point construct the general equation of the tangent line at point p and equate it to to thag point. Solve the system. Considering that the point must be on the curve.

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>>8441713
What about this guy? I got the first part right but I can't get the cos(x) correct. I tried doing a*b=|a||b|cos(x) where a and b are the values at the derivative and it doesn't work

>>8441757
What other value of t you got?

>>8441767
None

>>8441767
Oh, I got 3 and -7

Can you maintain a geosynchronous orbit at points other than the equator and what's the difference exactly between geosynchronous and geostationary?

>>8441149
Depending on your course, yes. If the course aims to teach proof techniques, then it's obvious that you would be expected to use those techniques on the test.

The point is developing your skills.

>>8441788
Geosynchronous means the time of orbit is 1 day. This is possible for an infinite amount of orbits, it does not have to be "parallel" to the equator it just has to be at the right distance away.

Geostationary means the satellite remains above the same spot on earth. You're still moving with some orbital speed, but that orbital speed matches the speed of the earths rotation. This is only possible if you're in geosynchronous orbit above the equator moving in the direction of earths orbit.

>>8440628
It's having an input that gives an undefined output. Like 1÷x and x=infinity. The function is indeterminant when x is zero, but we can see it approaches zero. y = 0 is the limit. But I'm just now taking calc I. I'm no math major.

>>8441774
For the tangent vectors, I get [-20,0,-10] and [20,0,110] with magnitudes 10*sqrt(5) and 50*sqrt(5), and dot-product -1500. So cos(a)=-1500/2500 = -3/5.

>>8439278
>>8440244
not solving via DE
plebs

Should I even practice algebra and calculus word problems? Or will that come naturally in my Electrical Engineering studies?

>>8439017
Yes.

X >= m, Y >= M => X - Y >= m - M, so inf(X - Y) >= m - M

>>8442250
word problems?

>>8439575
The second definition is the correct one. They should coincide if you use the Borel σ-algebra.

>>8441012
https://golem.ph.utexas.edu/category/2008/12/groupoidification_made_easy.html

Spans of groupoids represent a binary relation where, instead of just having two possibilities (yes = related, no = unrelated), you can have a groupoid (the intersection of the preimages of the two objects) witnessing the relation. This groupoid in turn can be turned into a nonnegative rational number. You can also just use sets, then you'll just get a set for each pair.

>>8438193
How do smart people organize their work? Why do I feel like I don't want to do ANYTHING these weekends? Is that because I wanted to do so much?

Why do I feel urges to do the most stupid things and desperately want to talk-talk-talk-and-talk to people? wtf is happening

any recommendations for introductory Quantum M textbooks? better if it is fit for self study

>>8442472
How proficient are you in math?

>>8441149
The point is to teach you how to be good at algebra/logic, which will help you with proofs later on.

You should be grateful that your course is being taught rigorously and use the opportunity to distinguish yourself among your peers.

>>8442476
decent, I wouldn't mind going through a math prelude tho

Does anyone know the easiest way to solve this proof:
-a = -b ≡ a = b
Starting with the LHS -a = -b?

>>8442718
multiply both sides by -1

>>8442726
if only... not allowed to use that axiom though

>>8442732
why dont you tell us which axioms you're allowed to use then instead of making us guess?

>>8442736
>>8442718
Sorry, my bad. These are what I'm allowed to use.

(a + b) + c = a + (b + c)
(a · b) · c = a · (b · c)
a + b = b + a
a · b = b · a
0 + a = a
1 · a = a
a · (b + c) = a · b + a · c
a + b = a + c ≡ b = c
a · 0 = 0
a + z = a ≡ z = 0
a + (-a) = 0
a - b = a + (-b)
x + a = 0 ≡ x = -a

>>8442745
-a=-b
-a+b=0 (last axiom)
b+(-a)=0 (third axiom)
b=a (last axiom)

>>8442753
thank you

>>8440228
math with a cs minor, more marketable than straight math, more smarterer than straight CS

>>8441902
Thanks anon.

when you make y = 0 you get the roots but what happens when you make x = 0?

>>8443519
I know this is SQT, but this has to take the cake. Do you even read what a function was?

>>8443519
you get the y-intercept

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Has anyone heard of a kilotap? If so, what is it and why are there so few instances of it being used?

Is there a textbook/video series that has lots of problem sets of trig identities (with full working out). I want to study it because I was never taught them and I'm scared of calc II.

>>8443519
y-intercept. Unless it's an exponential or cosecant. Then you get a discontinuity.

Testing ignore me pls

[eqn] i\hbar \frac{\partial}{\partial t} \left | \psi \right \rangle = \mathit{H} \left | \psi \right \rangle [/eqn]

>>8441149
Did your professor ask you to prove this exact identity? It's trivially incorrect.

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How did this happen?

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What. How does f(x) = x not have a function for 0 < x < 1? Why is the maximum not x = 0.(9)?

>>8444076
>x = 0.(9)?
because 0.(9) = 1 and 1 isn't in the domain

>>8444090
So then no unbounded convex function has a maximum?

What's the difference between the dot product between x and y; and x^Ty where [math]x,y\in\mathbf{R}^n[/math]?

>>8438193

are quaternions just 3d imaginary numbers?

why is their 4 (ijk + a real number) and not 3 (one for each axis)?

What other theorems besides Cauchy-Swartz inequality are very popular for proving stuff?

>>8444103
No, counter example:
fx) = -x^2 + 1 defined on the domain (-0.5,1)

>>8444133
how is that a convex function

Are KhanAcademy and PatrickJMT a meme, or would those sites alone be enough to get my math skills up to at least high school level, maybe slightly beyond that?

l'm planning on majoring in medical physics, which has an analysis course in the first semester, which I don't think I'd be able to handle yet. I'm switching from pharm school, where we only had a statistics course.

>>8444142
khan definitely beyond high school

>>8441149
Because A. being able to convert an expression into various forms can come handy when you have to do actual analysis (ie. estimate quantities), as various forms of the same quantity tell you different things and B. because it's important to be able perform calculations neatly and efficiently
Math is not just "light up and talk about concepts". In many situations you have to sit down and do unappetizing things

>>8444140
how is it not?

>>8444149
by being a (non-affine) concave function

Can an injective continuous map [math]f : S^2 \to R^2[/math] NOT be a homeomorphism?

>>8444158
*[math]\mathbb{R}^2[/math]

>>8444158
*Homeomorphism onto its image, of course

>>8444158
Nope, it's automatically open because the domain is compact (write it down)

>>8444103
unbounded functions have no maximum by definition. but I think you mean something else.
continous real functions on compact sets always have a max and min

>>8444068
Just add the thing on the left into a whole fraction and you will gey your answer.

>>8444107
IIRC quaternions are not a field. unlike C or R, they are another algebraic structure, can't remember its name tho.
you need to real part so that quotients will always be well defined. i never studied quats tho
>>8444158
what do you mean by S square?

>>8444189
>what do you mean by S square?
Oops, what's going on with me today.
I wanted [math]S^1[/math], a circle, of course.

So the question:
Can an injective continuous map [math]f : S^1 \to \mathbb{R^2}[/math] NOT be a homeomorphism onto its image.

>>8444107
4d imaginary numbers if you want, yes
They're 4-dimensional because it turns out that you cannot construct a 3-dimensional analog of the complex numbers (more accurately that's not really the reason they are 4d but rather the reason why they are used instead of some 3d analog of the complex numbers, there just isn't one).
They do have some relation to 3d geometry though, because the "purely imaginary quaternions" (ie. the linear combinations of i, j and k) form a 3 dimensional vector space over which quaternions act as rotations.
tldr: there are no 3d complex numbers (whatever that means) but quaternions are related to 3d rotations

>>8444189
>can't remember its name tho.
division ring
>>8444107
ask hamilton.
Quaternions were created to describe rotations in 3d like complex numbers do in 2d.

>>8444185
>add the thing on the left into a whole fractio
what

>>8444180
Yeah sorry I meant functions on open sets, not unbounded

>>8444280
Nope, assume there were a local maximum at an interior point x of your domain D. Then for small enough [math]r > 0[/math] that [math]B(x,r) \subset D[/math], you could write [math]f(x+h) < f(x)[/math] for each h such that [math]0 < ||h|| < r[/math].
But if h is such that [math]0 < ||h|| < r[/math], we can write [math]f(x) \le \frac{f(x+h) + f(x-h)}{2} < f(x)[/math], contradiction.

What's the easiest way to prove that a space minus homeomorphic image of an interval is homotopy equivalent to the same space minus one point? Or how to prove their homology groups are isomorphic?

>>8444179
>>8444218
Yep, this works, thank you. All the closed sets in a sphere are compact since the domain is, vice versa, continuous image of a compact is a compact, hence the map is closed, which is enough for homeo- in this case.

>>8444068
(x+c)(x-c)=x^2-c^2

x+c=(x+c)*((x-c)/(x-c))
=((x+c)*(x-c))/(x-c)
=(x^2-c^2)/(x-c)
=x^2/(x-c)-c^2/(x-c)

x^2/(x-c)=x+c+c^2/(x-c)

>>8444107
> are quaternions just 3d imaginary numbers?
No, they're 4D imaginary numbers.

3D rotations are often expressed as unit quaternions (versors), i.e. quaternions of norm one. The unit-norm constraint reduces the dimensionality of the space from 4 to 3.

This is similar to expressing an angle (1D) as a 2D unit vector or as a complex number of magnitude one, or a direction in 3D (which could be represented using two scalars e.g. latitude and longitude) as a 3D unit vector.

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How do I do these by induction?

>>8444474
You don't need induction. name the first finite sum S and then compute r *S subtract the latter from the former and divde. Theb compute the limit for the above one.

>>8444106
nothing

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Why is thermal diffusivity a property of the material if it depends on its dimensions?

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2 workers need to work 8 hours daily to finish the job in 15 days. How many days would it take 3 workers if they worked for 10 hours daily?

This is an exercise for 7th graders and I can't solve this after having done no math in couple of months since I focused on learning the language of the country I want to move to in order to begin university. It's truly defeating that I can forget so much in such a short period of time. And I planned on getting a degree in mechanical engineering... I fucked up, I shouldn't be pursuing such a degree, I'm a fucking brainlet...

>>8444561
Thermal diffusivity doesn't depend upon the dimensions.

For that experiment, the time depends upon thermal diffusivity and thickness.

>>8444647
Total effort is proportional to all three, i.e. 2*8*15= 240 man-hours.

With 3 workers doing 10 hours a day, that's 30 man-hours per day so it should take 240/(3*10) = 8 days.

IOW: 2*8*15=3*10*X => X=(2*8*15)/(3*10) = 8

>>8444106
The dot product is only defined for vectors, and always yields a scalar. If they're column vectors, then it's x^T.y, if they're row-vectors it's x.y^T.

OTOH, x^T.y is defined for any combination of (n×a) and (n×b) matrices, yielding an a×b matrix.

>>8444474
Assume [math]1+r + \dots + r^n = \frac{r^{n+1}-1}{r-1}[/math]. Rewrite [math]1+ r + \dots + r^n + r^{n+1}[/math] using the previous identity, conclude.

Can anybody recommend websites or books other than Khan Academy or PJMT for beginner level math? Because I can't be assed to watch 15 minutes of video for a topic I could read in five. The books recommended in the /sci/ guide are still too high level for me, though.

>>8444269
>Quaternions were created to describe rotations in 3d like complex numbers do in 2d.

no they weren't

>>8444140
Give the pair of points which segment contradicts he definition of convex.

>>8444787
What do you know already?

>>8445032
High school level statistics, integrals/differentials, vectors.
...Basically, assume that I'm a 10th grader.
I forgot a lot, but reading it brings it back instantly.

>>8445051
What do you want to learn?

>>8445060
Group Theory, to prepare myself for quantum chemistry. I heard linear algebra also helps for that.

>>8445080
Nigga, you are far from close. Get a more realistic goal. Group theory is fucking crazy.

>>8445087
I hear you. I just have no idea what's based on what in mathematics, since I only had one course in the first semester, and it was mostly statistics. I'll try to look into other shit first.

>>8445098
I mean, if you are talking about quantum chemistry and haven't taken linear algebra you need to slow down. Group theory is amazing, but you need to have a pretty good understanding of other subjects.

>>8445101
Sounds like a plan. I'll leave it for another semester.
Do I need anything but my basic bitch school algebra to get started on that?

>>8444694
Holy fuck, I did the same thing, but I guess I erred somewhere somewhere along the calculation and divided 240 with 20 for two people, instead of 30 for 3 people, and got 12 days instead of 8 like you did. Thanks a lot!

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Which one is it?

What are some good textbooks on organic chemistry?

>>8445324
(A∩B)∪C or A∩(B∪C)?

They aren't the same.

>>8445510
what is the convention in boolean algebra?
i think it is A ^(B v C)

>>8445518
The convention is to use parentheses.

>>8445542
REKT

Have all problems in NP been shown to be equivalent or are there some problems which we know are in NP but haven't worked out how to reduce an NP-complete problem to?

>>8439575
Like the other anon said, the second definition is the correct one. If you only require the codomain to be a topological space, then you won't be able to compose measurable functions (because not all topological spaces can be measure spaces, and so won't be able to serve as the domain of a measurable function). Relatedly, definition 1 does not allow for functions that map into measurable spaces whose sigma-algebra does not arise the Borel algebra of a topology.

>>8445552
>>8445542
>>8445518
>>8445510
no parenthesis

what is the solution?

One infinite integer sequence is [math]3 \rightarrow 5 \rightarrow 7 \rightarrow 9 \rightarrow ...[/math] which is trivially generated by [math]S(0) = 3, S(n) = S(n-1)+2[/math].

Question 1: How would I mathematically represent the finite integer sequence [math]3 \rightarrow 5 \rightarrow 7 \rightarrow 9[/math]?

Question 2: Is there always a set of non-conditional rules which generates a finite integer sequence [math]S[/math]? If so, how can I find this set of rules?

Question 3: What branch of mathematics are devoted to these types of questions?

In particular I have the following sequence [math]0 \rightarrow 1 \rightarrow 2 \rightarrow 3 \rightarrow 0 \rightarrow 2[/math] but I have no clue how to go on about this without using conditionals for the 5th and 6th term.

I am sorry if I use confusing notation, I have never worked with this kind of math before :)

>>8441218
Welcome to (special) relativity.
There is no center point in the universe, so there is no absolute point to measure anything that needs space AND time to measure (and even this is arguably only an interpretation)

>somehow get into a top university for my master's
>i'm several weeks in and i'm getting completely wrecked
this was quite a blow to my ego. i thought i was worth something, now i feel like i should just pack up and fuck off. i thought my maths was alright but now i think i should kill myself.

i'll still go forward and try to push myself because there's nothing else out there for me, but feels like shit and very demotivational (apparently not a word)

>x is a geometric random variable
>what is the expectation of x^3?
Not asking you to do my homework, just give some pointers, because I'm completely lost.
Thanks!

>>8445653
not, then and, then or
but anyone on their right mind would use parentheses

>>8446108
In general, the expected value of a variable X is
sum[-∞<k<∞](P(X=k)*k).

So to get the expected value of f(X), you'd use
sum[-∞<k<∞](P(X=k)*f(k))

If X is geometric, then you can change the lower bound from -∞ to 0 or 1 (depending upon which definition of "geometric distribution" you're using).

Then it's just boils down to calculating the limit of the resulting series.

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I have absolutely no idea how to approach this question. How do you prove things like this? My professor sucks, please help me.

Also, L_k is an elimination matrix.

How can I __really__ strictly prove that something is a fibre bundle? For example [math]SO(n)\to S^{n-1}[/math] or Hopf fibration?

What's given is a definition (as in wiki) and homotopy lifting property. Anyone can see that they are kiiind of or """obviously""" fibre bundles one a fiber [math]SO(n-1)[/math] and [math]S^1[/math] respectively. But really, I want to know how to actually prove it.

Reading this book that tells me that some probability distributions are "parameteric" because they have adaptive parameters. What's that? I know that Gaussian has mean and variance, but why are they adaptive? What distributions are non-parametric?

I need to prove something really obvious and I'm not sure how to because I haven't written a proof in years

say I have 5 3 9 8 11 and I need to add them together at each step fibonacci style

so in this example 5 8 17 25 36
I need to prove that the lowest average of this line (5 8 17 25 36 average will be 91/5) will be obtained by putting the initial digits in ascending order ie

3 5 8 9 11
3 8 16 25 36 average = 88/5

this feels like a really self-evident thing but I have to prove it as part of something bigger and I'm just hung up here because I don't really remember how to prove stuff

it'd be for a general case that was just an example I made up

How do I write the cardinality of the union of two power sets as a function of the cardinality of the sets? Basically I need to find f such that

[eqn] f(|A| , |B|) = | P(A) \cup P(B) | [/eqn]

>>8446389
whats wrong with what you wrote down already?

>>8446407
I haven't found a formula for the RHS, only stated that I need one. To put it as plainly as possible,

What I've got:

[math] |A| [/math]
[math] |B| [/math]
[math] |A \cup B| [/math]

What I need to find:

[math] | P(A) \cup P(B) | [/math]

What I don't have:

[math]f(|A|, |B|, |A \cup B|)[/math] that equals [math] | P(A) \cup P(B) | [/math]

(I forgot to say I know the cardinality of the union of A and B in my first post)

>>8446383
Probably by induction. Assume that you have a sequence of N-1 numbers in ascending order, and you add an Nth number. Derive an expression for the increase in the average for a given placement. Show that the smallest increase occurs when you place it so as to retain the ascending order.

Hey can anyone explain or point me toward more information about non-isolated fixed points in the context of non-linear systems. Apparently trace(a)<0 implies stability and trace(a)>0 implies instability. Also I need a better explanation of when this fails due to small non-linear terms and how to analyze it if it does.

>>8446389
>>8446462
The cardinality of the power set of A is the number of possible subsets of A. So
[eqn]|P(A)| = \sum_{n=0}^{|A|} C(|A|, n)[/eqn].
Of course, by C(n, k) I mean [math]\frac{n!}{k! (n - k)!}[/math]. The cardinality of the power set of B is similar. The cardinality of the unions of the two power sets is just the sum.

>>8446688
>|P(A)|=∑n=0|A|C(|A|,n)
theres a much easier way to write this, it's just 2^|A|

>The cardinality of the unions of the two power sets is just the sum.
no, you're forgetting the sets can have intersection

>>8446695
>tfw btfo

Thanks for keeping me in check family

>>8439636
[math]\text{Define}\ \Delta_n f(n):=f(n+1)-f(n). \text{Then}\ \sum\limits_{n=0}^{m-1}\Delta f(n)=f(m)-f(0)\ \text{and}\ \\ \Delta_m\sum\limits_{n=0}^{m-1}f(n)=\Delta_m(f(0)+f(1)+f(2)\ ...+f(m-1)=\\ (f(0)+f(1)+f(2)\ ...+f(m))-(f(0)+f(1)+f(2)\ ...f(m-1))=f(m)\\
\text{what I'm getting at is that}\ \Delta\ \text{and}\ \sum\ \text{are inverse operations analogous to taking}\\ \text{the derivative and the antiderivative. It should be clear that for a polynomial}\ P(n)\\ \text{of degree k}\ \Delta P(n)\ \text{is a polynomial of degree k-1 (to convince yourself just}\\ \text{show it for}\ n^k. \text{and since}\ \sum\ \text{is the inverse of}\ \Delta,\ \sum P(n)\ \text{is a polynomial}\\ \text{of degree k+1}[/math]

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How would I show that the covectors defined here are linearly independent?

Is the intersection of an infinite amount of compact sets also compact? I think it is compact, I can't seem to think of a counterexample where this is not the case.

>>8446882
I forgot to mention that those compact sets are all contained in R^n

>>8446882
>>8446885
Pretty trivial consequence of Heiny-Borel

>>8446860
suppose v=sum of c_ie^i =0. then v(e_j)=c_i delta_{ij} implies c_i=0 for all i

>>8446882
>>8446885
yes, use heine-borel and show infinite intersection of bounded sets is bounded

>>8446882
it is true even if the sets aren't on Rn
closed subsets of compact sets are compact
compact sets are closed
arbitrary intersctions of closed sets are closed.
it follows

>>8446860
Just use the definition. Write out a linear combination of them and assume it's the zero map. Use the fact that [math]\{ \mathbf{e}_i \}[/math] is a basis for every input [math]\mathbf{x}[/math]. Everything follows pretty naturally from the linearity of the maps.

>>8446901
>compact sets are closed
But that is only true in R^n

>>8446860
>>8446860

for each covector in the set there is a subspace of elements in V which they will not send to zero, but every other covector in the set will, I think that is sufficient......I hope

>>8446901
what

are you trying to imply closed = compact? the result isn't true outside of R^n

>>8446909
>>8446908
compact implies closed, the converse is true for R^n
let c be a limit point of E, E compact
assume c dosnt lie on E. take as an open cover all open balls around every single point x of E, with radius equal to one half the distance between x and c
no finite subcover of this cover can cover E. therefore E isn't compact. contradiction. it follows that c belongs to E, and E is closed

>>8446917
>compact implies closed, the converse is true for R^n
no, not even for n=1, go back to calculus class please

>>8446924
closed and bounded, jeez
show me a compact sunset of a metric space that isn't closed

>>8444076

There's a difference between a maximum and a supremum.

>>8446928
>adding extra hypotheses after the fact
yeah no

>>8446924
Are you stupid? Heine Borel theorem: In R^n, compact is equivalent to closed and bounded.

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>>8446917
>no finite subcover of this cover can cover E.
[proof needed]

>>8446943
it is trivial
gotta admit tho, it may not hold for general topological spaces.

>>8446942
are you stupid or is english your second language? the word bounded is not in the post i relied to

>>8446953
if it's trivial then i'm sure you can prove it, you might even get a fields medal for strengthening the heine-borel theorem by removing this supposedly unnecessary hypothesis

>>8446960
you are a retard, m8. honestly
all im saying is that compact subsets of metric spaces are closed. this is easy to see, and has nothing to do with the Heine borel theorem.

>>8446977
>all im saying is that compact subsets of metric spaces are closed.
no, you said
>compact implies closed, the converse is true for R^n
>the converse is true for R^n
if you're still saying these are the same thing then you're implying compact iff closed which is false, what are you trying to get at?

>>8446988
you keep clinging to that. it has nothing to do with this discussion. I forgot to add "bounded" there.
this was my original post
>>8446901
please tell me what's wrong about it. I have to add, tho that im talking about metric spaces

>>8447017
if you add the hypothesis that you're working in a metric space the post is fine, but this is a big stretch from a general topological space, where the statement
>compact sets are closed
is not true

Chemistry question.

So i took 6mol HCl and some fresh, stranded copper wire and water. I waited until it turned pickle juice color, i believe it made Cu(ii)Cl2 in solution. Then i took a square, iron pipe, probably cheap steel, and cleaned the rust off. I attached a 2A car battery charger to one end and placed the other, unrusted end of the pipe into the solution. Then i put the other electric terminal in the solution. The pipe and terminal immediately started producing small bubbles. When i turned the charger off after 30 seconds and removed the pipe from solution. To my surprise, it was half plated in copper, but only around the corners of the pipe.

Why wasn't the entire submerged section of the pipe also plated in copper?

Im sorry i dont have pics, i was half drunk in my dads garage proving a point.

>>8446906
He's wrong, but compact does imply closed and bounded

>>8447051
>but only around the corners of the pipe.
Sounds like charge concentration on sharp profile. Can't remember exactly why that happens, but it does.

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How to prove pic, using only those rules http://imgur.com/a/6YDzA ? I feel like dump shit. but this is the only exercise that I haven't done.

>>8447228
Can you use the fact that multiplication of positive numbers preserve order?

>>8447228
a<b
a.a<a.b
ab<bb => aa<bb
because a >0 and b>0
if you aren't allowed to use that rule then il be fucked

>>8447140
I was thinking that the area at the corners have more grain boundries/smaller grain sizes, so the free electrons flow better through there, but its just a hunch since Im my material science class hasnt covered that yet.

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Having trouble with this one.

>>8447244
Print a bunch of copies and cut it until you meet your 6x6 frame. Brute force is allowed when the problem is simple like this.

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>>8447244
2EZ m8

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>>8447244

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How does the forth line become fifth line?

>>8447621
Nevermind. I figured it out by using Brownian Motion definition.

So I'm trying to learn moments and center of mass in calculus.

Everywhere online, I see this:
[math]\bar x={M_y\over M} \qquad \bar y={M_x\over M}[/math]

But my textbook, as well as earlier exams (with answers) say the opposite.
Like this:
[math]\bar x={M_x\over M} \qquad \bar y={M_y\over M}[/math]

Can anyone tell me which I should use, and why it's right? Or are both right in a certain way?

>>8447872
Oh, and also - it's calculated the same way. The textbook just renames Mx as My, and vice versa.

So anyone know what's up?

>>8447873
Sorry for tripleposting, but I noticed that the textbook says "Mx=0" and "My=0"

Does that mean it is y at x=0 and x at y=0?

>>8441743
something to do with measuring all of them at once

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How I find the weight of an object in a pulley system, where one object (mass 70g) is hanging off a table and accelerating downward at 1.970 M/S^2 (so with a force of 0.130 N), and the other is on a frictionless surface moving across a tabletop?

Can you parameterise certain space-filling curves?

>>8447936
Maybe you just have to believe.

>>8447936
If you can't parameterise it, then I'm not sure you should even be calling it a curve.

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for r(x+1), why is it not just
"if [math]-17\leq x \leq \frac{\pi}{3}[\math]"
why does he add the -1?

>>8447921
Note that the magnitude of the acceleration is the same for both objects.

The force applied to the object on the table is equal to the tension, the force applied to the hanging object is its weight minus the tension.

>>8448055
r is only defined for arguments up to pi/3.

r(x) is only defined for -17<=x<=pi/3
r(x+1) is only defined for -17<=x+1<=pi/3, i.e. for -18<=x<=pi/3-1
So an equation involving both r(x) and r(x+1) is only defined for the intersection of those two ranges, i.e. for -17<=x<=pi/3-1.

>>8448061
thank you

>>8438193
Is it true that
[eqn]\bigcup_{A,\ B\ \in \ \text{Obj}(\mathcal{C})} \text{Hom}_\mathcal{C}(A, B) = \text{Mor}(\mathcal{C})[/eqn]

My intro to analysis text defines an open set S as a subset of R such that for all x in S there exists a neighborhood N(x) such that N(x) is a subset of S.

Would it then follow the set S = (0,1)∪ {2}∪ (3,4) is not open, since no N(2) can be a subset of S?

Why define open sets this way? The definition for closed set is inky slightly more intuitive for me, so any insight of either would be most appreciated.

>>8448125
>Would it then follow the set S = (0,1)∪ {2}∪ (3,4) is not open, since no N(2) can be a subset of S?
yes

>>8448095
If the category is locally small.

>>8448125

Try some examples in R^2 to get an idea

>>8448164
What is Mor(C) if C is not locally small?

>>8448230
if C isn't locally small then the hom's aren't sets so you cant take their union

>>8448127
Thanks

>>8448171
Awesome, I'll try this. Our metric in R^2 is Pythagoras' theorem extended to vectors, right? So our neighborhood would then be a circle of radius epsilon? Sorry, we're only doing one variable on this class.

>>8448265
Yes, your neighborhood are now disks without the border.

Would it be possible for someone to perform a blood choke on themselves using only their own hands while sitting at a desk, and position it to where they wouldn't collapse or lose the choke after they passed out?

>>8448372
You could cut off your hands and use the material to make a noose and hang yourself.

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What did they mean by this?

>>8448395
How is that at all difficult to understand? Give an example where P(B|A) + P(B|~A) = 1 and an example where P(B|A) + P(B|~A) =/= 1.

If I have two linearly dependent rows of a square matrix is the determinant 0?

>>8448424
yes

>>8448427
Thanks m8 I thought so

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>>8448409
For the first part...it does not work

>>8448443
holy shit are you in middle school? how do you screw up baby's first probability question that badly??

>>8448449
I didn't do anything wrong. Is my method wrong for the question asked?

>>8448427
To see why this is the case, consider a 2x2 matrix of the form
[a*x a*y]
[b*x b*y]

Its determinant is
(a*x)*(b*y)-(a*y)*(b*x)
=a*b*x*y-a*b*x*y
=0

Now consider how the determinant can be defined recursively in terms of submatrices. If all of the submatrices formed from a given set of rows have zero determinants, all of the submatrices formed from any superset of those rows must also have zero determinants, and so the overall matrix must have a zero determinant.

>>8448241
fuk dis abstract nonsense

>>8448443
You've managed to give an example where P(B|A)+P(B|~A) =/= 1 (i.e. part b).

Now you need to come up with an example where it is equal to 1.

Also, it might be a good idea to create examples where the probability of neither A nor B is non-zero, so that it doesn't look like you're assuming that.

If you come from a line of people who've lived in the place where gravity is the strongest. But then you move to a place where gravity is the weakest and have a babby. Will he grow taller then if he would have stayed? //Curiousmanlet

>>8448482
No

>>8448482
yes

>>8448443
kys brainlet

My pocket calc thinks sqrt(2)*sqrt(2) is 1.999961. What's going on?

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>>8448449
>>8448518
Leave him alone

>>8448533
That's the correct answer.

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>>8448596
fug :DDDDDD

>>8448596
My other calc gives 2. The first calculator says 1/9*9 is 0.999999. The second says 1. (No bully). Is it a floating point round off error? Do calculators and computers do computation the same way? Just want to into numerical analysis.

>>8448620
>Is it a floating point round off error?
Yeah pretty much. Your calculator finds the decimal approximation of sqrt(2), and then multiplies it by itself. Since the calculator can only store a fixed number of digits in memory, sqrt(2) is rounded off to however many decimal places, so there will be an error. Same deal with 1/9*9; it divides 1 by 9, getting a decimal approximation, and then multiplies that by 9. Your other calculator probably does some fancier trickery to recognize that these expressions can by algebraically reduced, before going through with numerical calculations.

>Do calculators and computers do computation the same way?
Yes.

>>8448636
Thanks, anon. I'm really interested by this area.

How do I get from the expression up top to the simplified version?

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>>8448692
forgot picture

>>8448533
> My pocket calc thinks sqrt(2)*sqrt(2) is 1.999961. What's going on?
That's a pretty shitty calculator. Rounding error is to be expected, particularly for irrational numbers, but that amount of error corresponds to a precision of 16 binary digits or 5 decimal digits.

For comparison, single-precision floating point has 24 binary digits while double precision has 53.

>>8448620
Do calculators and computers do computation the same way?

Some calculators use decimal arithmetic. Which is somewhat slower, but if you're typing in numbers manually, it there's no difference between taking a microsecond or taking a millisecond.

Computers normally use binary, and tend to follow IEEE-754, which means that with a bit of effort you can get consistent results between different systems. There will still be rounding error, but it will at least be predictable.

The main exception is that mainframes often have hardware support for decimal arithmetic, because it's consistent with several centuries of accounting practice, some of which has been enshrined in law. While an error of one part per million may be trivial, lawyers will argue over anything.

A calculator could be using an 8-bit microcontroller with even primitive operations (addition, multiplication) implemented as software subroutines, so accuracy and rounding would be whatever the programmer decides to make them.

>>8448719
Alright I'm starting to see the merits of weighing three versus three to start.

You'll either have:
Case 1:
RRR CCC

Case 2: Unbalanced
RRC CCR


So you know the heavier side either has three counterfeits or two counterfeits.

>>8448728
One thing to consider:
There are 6C3 = 6!/(3!*3!)=20 combinations of 3 real and 3 counterfeit coins.

If each weighing has 3 possible results, then there are 3*3*3=27 possible result triples. If one of the weighings only has 2 possible results (i.e. can't balance), then you only have 2*3*3=18 result triples, which isn't enough to distinguish 20 cases.

And 3-vs-3 can't balance.

>>8441743
>>8447892
>>8448728
Alright I think I'm close to the solution.Can anyone help me out.

Weighing 1: Three Versus Three. R=Real C=Counterfeit

Case 1: RRR CCC

Case 2: RRC CCR


So you either have two or three counterfeits in the heavier stack.

How to figure out which case we have. Look at the lighter stacks.

In case 1: RRR. For weighing two compare the first two coins. If they are balanced they are both real. For the final weighing compare one of the real coins to the last. If they balance all of them are real and the three counterfeits are in the heavier stack.


Here's where it goes wrong.
Case 2: RRC

Weigh any two coins in this stack. If they balanced they are real, if they aren't the heavier one is the fake. We could do similar to what we did above and find the counterfeit, but there'd be no way to identify which two coins in our original stack were the counterfeits which is a problem since I've used all my weighings.

>>8448736
Aw so I don't start with three versus three. That explains why I was having trouble.

>>8448745
3x 1-vs-1 weighings:

0>1 2>3 3<4 CRCRCR
0>1 2>3 3=4 CRCRRC
0>1 2=3 3>4 CRCCRR
0>1 2=3 3<4 CRRRCC
0>1 2<3 3=4 CRRCCR
0>1 2<3 3>4 CRRCRC
0=1 0>2 3>4 CCRCRR
0=1 0>2 3=4 CCRRRC
0=1 0>2 3<4 CCRRCR
0=1 0=2 2>3 CCCRRR
0=1 0=2 2<3 RRRCCC
0=1 0<2 3>4 RRCCRC
0=1 0<2 3=4 RRCCCR
0=1 0<2 3<4 RRCRCC
0<1 2>3 3<4 RCCRCR
0<1 2>3 3=4 RCCRRC
0<1 2=3 3>4 RCCCRR
0<1 2=3 3<4 RCRRCC
0<1 2<3 3=4 RCRCCR
0<1 2<3 3>4 RCRCRC

>>8448768
In case that's too cryptic:

Coins are numbered 0 through 5. First weigh 0 against 1. If they balance, weight 0 against 2, otherwise weigh 2 against 3. If both weighings balanced, weigh 2 against 3, otherwise weigh 3 against 4.

That gives you 3 result triples, each of which uniquely identifies a combination. There are 7 "can't happen" cases, i.e. all three weighings balanced (one case), or the first one being unbalanced and the last two going the same way (2x3=6 cases).

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Can someone pls help me with this?

>>8448768
>>8448780
Oh wow. I'll have to mess with this to understand it better, but thanks. I kind of thought I might want to try 1v1 weighings, but I just figured it would be impossible.

>>8448781
I solved the first one: it's 2m to the right

Can anons help me with one with stability of the glasses?

>>8448781
>>8448786
2, 3, 1

Rank them by ratio of center of mass height to base radius. High center of mass is less stable, but you need the base size to be able to compare them.

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uh isnt c = 0?

or am i just retarded

How to find out area of the grey zector?
I think that : S(of the sector)=(Q2-Q1)/360 * pi(r1^2-r2^2)
Am I right?

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>>8448816
?

>>8448815
If (1,-1) is a point on the line then yes, c=0.

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Anyone want to help me with this? I am pretty sure you use the residue theorem.

If I toast bread long enough can I turn it into charcoal and increase the energy I can efficiency get out of it?

>>8448890
Yes

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So anyway, I understand and can visualize how composition of shift and rotation would always have a fixed point, but how can it remain fixed, when you take a homothety of < 1 after that, unless of course you specifically choose it's center to be that fixed point?

I've seen some proof on stackexchange, but it included linear algebra which is obviously unacceptable, since it's a FUCKING ANALYSIS book - intro chapter of the first volume in fact - and you're clearly supposed to be able to solve it without linear algebra.

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>>8448816
>>8448843
Forgot about the picture, that a retard.

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>>8448395
I-is this right, senpais?

Would it be possible for some sort of catastrophic event to wipe out all life on Earth, assuming it doesn't actually physically destroy the planet itself?

I can see something wiping out most of the life on Earth, even 99.9% of it, but I feel like there is always going to be something left over that can potentially breed and seed the planet back to life again. There's just too much diversity of life and spread too far for it to all get killed by one specific thing.

Though I'm speaking of more sudden events, not something like "the atmosphere gradually drifts off into space over a period of a million years" or something.

>>8448695
Thanks

>>8449220
Both of those sum to one.
(1/5)/(1/2)+(3/10)/(1/2)=2/5+3/5=5/5=1

>>8448913
>So anyway, I understand and can visualize how composition of shift and rotation would always have a fixed point, but how can it remain fixed, when you take a homothety of < 1 after that, unless of course you specifically choose it's center to be that fixed point?
the full composition has a fixed point, but the fixed point after the shift and rotation doesn't have to be the fixed point of the shift, rotation and homothety

>
I've seen some proof on stackexchange, but it included linear algebra which is obviously unacceptable, since it's a FUCKING ANALYSIS book - intro chapter of the first volume in fact - and you're clearly supposed to be able to solve it without linear algebra.
what's wrong with linear algebra?

>>8441660
Think of your integers in terms of modulo 3, try expressing mode 1 as a series with your first term being 17 (don't forget if you can get 2 you can also get 1).