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You are currently reading a thread in /sci/ - Science & Math

For questions that do not deserve their own threads. There isnt one at the moment, so Im starting a new one.

Given that X,Y have joint density f(x,y)=2 for 0<y<x<1, find P(X-Y > z). What I did is sketch the area over which I should be integrating, and found the following bounds: z<x<1 and 0<y<x-z. Integrating 2dydx using these bounds results in (z-1)^2, while according to my book the answer should be 1/2 * (z-1)^2. What went wrong?
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>>6573905
>joint density f(x,y)=2 for 0<y<x<1
The overall probability should stay 1 so you should renormalize it and there's your 1/2 factor
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>>6573910
>0<y<x<1
as
>0<y<1,0<x<1
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So I was going through past exam papers to revise for my maths next week and came across this.
Why isn't this \sqrt[4]{16}*cis(pi/4) for the first root?
I thought the way to find roots was \sqrt[n]{R}*cis((\alpha + 2kpi)/n).
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>>6573932
The fuck is cis?
Anyway, they give you an example of how to find the roots with the length being 1, with 16 the angles will be the same but the vector length different.
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I'm trying to settle a debate with a friend and I need some help. It's getting our heads in a twist.

"Someone who has pushed someone who has pushed Abdul won't be pushed by Abdul unless they have pushed Abdul."

Write this in symbolic form where Pushed(x,y) means x pushed y. So for example: there exists someone who has pushed Abdul would be: "(backwards E)x(Pushed(x,A)"
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>>6573932
It is, it's just that they're using the example z^4=-1 but you have to do z^4=-16
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>>6573936
cis(theta) = cos(theta) + isin(theta)
It's for the polar form of complex numbers.
cos(theta) is the real component while isin(theta) is the imaginary.
>>6573939
Find it a bit weird since they did it like that on both questions from past exams but okay.
Guess they just wanted to show the general method so if you didn't actually study they'd know.
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>>6573954
I use complex numbers on practically everyday basis in my field of study (purely mathematical) and have never encountered this notation in 8 years. I've read textbooks and articles in various languages. Is it an engineering thing? How does it differ from exp(i theta)?
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>>6573961
I see it all the time, a lecturer of mine commented on the fact that this notation is pretty regional and not used everywhere (context:australia)
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>>6573961

It's elitist notation that doesn't exactly see use in anywhere else but introductory trig books and UIL Math tests in high school.
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>>6573966 (You)

Speaking of UIL Math, I remember when I was on my school's math team back in high school and there was a technique I remember vaguely for getting approximate forms of high-powered numbers. For example:

Calculate 9283745^{1239840932} rounded to three significant digits.

Obviously in standard scientific calculators, this would yield an overflow error. But we learned some technique to give an approximation. It involved taking a logarithm, just taking the decimal end of the logarithm or something and I don't remember much after that.

I can't find my notes on the technique and I can't find it on Google. Does anyone know what technique this is and how to do it?
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>>6573961
See
>>6573965
> (context:australia)
I guess it's an Australian thing since that's where I'm going to uni.
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>>6573976
m8
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>>6573976
>I guess it's an Australian thing since that's where I'm going to uni.
It's a retarded thing that they introduced for only the "specialist mathematics" subject. Once you pass that one subject, you learn e^{i\theta} which is exactly the same thing as cis (try it on your calculator) yet they don't use it for some reason.
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If we take the debroglie wavelength of a particle in a 1D box and square it, that gives us the probability of finding it anywhere in that box. As we increase the mass of the particle, the frequency of the wave increases, increasing the number of troughs in the wave and the number of locations where the probability of finding the particle is 0%. So with each "mass increase" we add another location where the particle can't exist, while keeping the same number of locations where it can exist. So anywhere the curve is non-zero, the particle can exist, but as it gets heavier, there arise more locations where it can't exist.

So, my question:
What's the difference between existing and having a probability of being found?
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>>6573993
>So anywhere the curve is non-zero, the particle can exist, but as it gets heavier, there arise more locations where it can't exist.
You are confusing probability of being found with being 1/total possible places.
>What's the difference between existing and having a probability of being found?
I don't see how you can get these confused with the first point.
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>>6573987
So Niel taught the Algebra parts last year instead of the Calculus portion.
Dunno why they'd switch it around.
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>>6574002
he actually taught both subjects for me, I think they just do it on whose available. It's probably all the same to them.
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>>6573998
>You are confusing probability of being found with being 1/total possible places.

The length of the box stays the same, so when we add more mass to the particle, we add another point where the debroglie wavelength crosses the x-axis, in turn, the probability of finding the particle there is 0.

I don't think I'm confusing the two. I understand that the probability of finding that particle anywhere on the number line is given by psi^2.

I'm concerned with the fact that a particle has less and less locations to occupy as it gets heavier.
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>>6574012
Here's my take.
> when we add more mass to the particle, we add another point where the debroglie wavelength crosses the x-axis,
And since the number of such crosses is at most countable\zero measure, the number of places it can be in is still continuum. I think your confusion might come from the following points:
>a subset of an infinite set can be as large as the initial set, in particular if you get rid of a countable set there are still as much points left on the real line
>squared wavefunction gives probability density and not probability mass function
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>>6573988
Probably because high school students don't take complex analysis courses which justify the use of "exp" notation let alone writing e to the power of complex numbers.
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>>6574032
But introducing an additional notation would confuse them even more then. I don't follow.
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>>6574032
>Probably because high school students don't take complex analysis courses which justify the use of "exp" notation let alone writing e to the power of complex numbers.

I didn't take complex analysis, yet I still know of exp(i) over cis. And the high school curriculum uses the same properties like "multiplying complex is the same as adding the theta in the cis function". I would assume that if you use e^itheta notation it would be much more obvious (you add the theta because when you multiply exponents, you add the exponent term)
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>>6574045
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>>6574024
>probability density
What's the difference between that and probability? In class we talked about the concept and used the words interchangeably.

This video came to mind when I asked myself that question: https://www.youtube.com/watch?v=aIggWlKr41w

>a subset of an infinite set can be as large as the initial set, in particular if you get rid of a countable set there are still as much points left on the real line

I don't understand why that's true though. I think that bit of math was developed before quantum mechanics, and later the two were reconciled.

I understand that I have to trust the math and its rigor. Okay, but what about physical significance? To me this implies that motion isn't smooth, but increments of smooth motion.

I can't say I'm confused, because I don't have two or more competing ideas, but I do want these questions answered. As an undergrad I feel like I'm in over my head.
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>>6574050
>What's the difference between that and probability?
1. Imagine a 6-sided dice. What's the probability of each number? 1/6. Total probability? 6*(1/6)=1.
2. Imagine your (squared) wavefunction. What's the probability of getting x=5? "0.12", you say, looking at the graph. Well, how many points are there? Infinitely many. If you multiply a finite probability over an infinite number you'll get infinity, not 1. What do?

This is the motivation of introducing "density" instead of "discrete" probability. In short, the value of the density at a point doesn't tell you shit - you need to integrate it over an interval to get the probability of the values in that interval. It worked alright when we had 6 dice sides, but now when we have infinitely many (your interval) we have to mess around a bit.
The related field is "Measure Theory", but I think you'll be ok with just reading wiki on probability density.
>I don't understand why that's true though
This is the basics of the set theory in general or real analysis in particular. You can read wiki or some introductory book (for instance, the Hausdorff's one), it's actually a rather easy concept to grasp.

I'll probably watch the video and post some comments on it+your other concerns a bit later, unless some other kind anons do that
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>>6573932
>cis
>>>/pol/
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>>6574065
>>>/hm/
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>>6574045
cis is an abbreviation, so it's not like it's that confusing.
>>6574046
It might be more obvious, but it sort of trivialises the intricate justification for why we can write e^i*theta = cos(theta)+i*sin(theta) and that properties of exponentials hold for complex numbers when appropriately defined. It often causes issues with people studying math further down the line.
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>>6573975
Ahhhh shit you posted that in the last thread too, would love to know the answer to this too, remember doing something like this in the preparation for the school maths team too.
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I need to find the speed of vibrating glass in a video. I need to assume the frequency and wavelength. 10, 000 Hz sounds like a good frequency (it is really thick glass), but I have no clue on what wavelength would make sense here.
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im thinking about changing my major from computer science to mathematics. i have my general requirements
am i making a good choice?
what are all (or most) of the concepts that will be covered?
how useful is a BS in mathematics?
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>>6573975
Do you need a computer-assisted method or a "by hand" method? It's easy enough to solve with a computer but I wouldn't know how to solve it by hand.

To solve it with a computer:

Step 1: rewrite as a power of 10: 10^x for some x that you have a closed form of.
Step 2: find the fractional of x with decent precision and call it y (you don't need that many digits, any computer can do that for instance with wolframalpha).
Step 3: The three significant digits of 10^x are the same as those of 10^y, and you have a decent estimate of y. Compute 10^y, keep the first three digits.
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>>6575312
(cont)
In your case, x would be 1239840932 * log10(9283745), which is around 8.6 billion, and y would be 0.95435...

10^0.95435 = 9.0022..., so the most significant digits are 900.
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will they allow minorities to colonize mars?
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>>6575340
If you're asking because you're worried they won't let you go there because you have Down's, don't worry, chances are you'll be retardedly obsessed about something else by the time we start sending people to Mars.
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>>6573905
How can I be an evil scientist?
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>>6575340
No, they will not allow the racist white minority to colonize Mars.
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What is it called when a reaction, once it starts, provides enough energy for it to keep going?

Like when a reaction becomes "Self sufficient" so to speak when it comes to energy, until the things to react with run out?
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Anyone know how to solve linear congruence relations with large numbers?
eg.
629d = 1 mod 3432

I can get my head around doing it with smaller numbers, and essentially using trial and error, but it seems like that approach would take me several hours to answer this one question.
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>>6575362

Study the effects of prolonged pain on kittens.
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>>6575599
If you want to find a 'd' for which this relation holds, just write down what "629d = 1 mod 3432" means mathematically and you got your answer.
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>>6575632
not sure what you mean by this.
I can basically work out that I could solve it by saying
629d=3432n, for some integers d, n

and then just trying to solve it with trial and error, but I'm sure that there's an actual method to solving this, and I just can't work out what it is.
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>>6575635
oops
>629d=3432n+1
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>>6575599
it isn't clear if there even exist a solution, like for example 2 d = 1 mod 2
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>>6575649
I guess that's sort of true. But I know that there is one in this case. and that its like 1997 or something. But obviously, I can't spend all day trying to find that answer like how I would with your example, just trying a few numbers.

Does it help if I mention that I'm doing RSA cryptography?
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>>6575599
Saying that linear congruence ax \equiv b \; (mod \,n) has a solution is equivalent to saying that GCD(a,n) \mid b which is the equivalent of saying that b = GCD(a,n)\cdot k
So let GCD(a,n)=d
Thus a solution is going to be
\frac{a}{d}x \equiv \frac{b}{d} \; \left( mod \, \frac{n}{d} \right)

In your case the gcd is 1, and obviously 1 divides n=3432, you will obliviously have that a/1 =a, and b/1 = b, and n/1 = n.

Now getting to actually solving it:
By Euler's theorem, you have that a^{\phi(n)} \equiv 1\;(mod\,n)
Where \phi(n) is Euler's function:
\phi(n) = \# \left\{ 1\leq x \leq n : GCD(x,n)=1 \right\}

and # is the cardinality of the set (ie, the cardinality of the set of the numbers relatively prime to n that are less than n).

In your case \phi(3432) = 960
So 629^{960} \equiv 1\;(mod\,3432)
So your case is solved here, but if b wasn't 1:
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>>6575682
Continuing:
So: a^{\phi(n)} \equiv 1 (mod\,n)
You also have that b \equiv b \; (mod\,n)
Now, by the following property, that if you have c,d,e,f such that c \equiv d \; (mod\,m) and e \equiv f \; (mod\,m), then c\cdot e \equiv d\cdot f \;(mod\,m)

Thus a^{\phi(n)} \cdot b \equiv 1 \cdot b\;(mod\,n)
But you also have that ax \equiv b \; (mod \,n) and since congruences are simmetric \left[a \equiv b \; (mod\,n) \rightarrow b \equiv a \;(mod\,n)\right] you have that b \equiv ax \; (mod \,n)

And now, by transitivity [if you have a \equiv b \; (mod\,n) and b \equiv c \; (mod\,n), then a \equiv c \; (mod\,n)]

a^{\phi(n)} \cdot b \equiv ax \;(mod\,n) \rightarrow a^{\phi(n)-1} \cdot b \equiv x \;(mod\,n)

\rightarrow n \mid a^{\phi(n)-1} \cdot b - x \rightarrow a^{\phi(n)-1} \cdot b - x = nk \rightarrow x = nk + a^{\phi(n)-1} \cdot b

S = \left\{x = nk + a^{\phi(n)-1} \cdot b : k \in \mathbb{Z} \right\}
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>>6575682
>So your case is solved here, but if b wasn't 1:
Oops, I put that on the wrong place, and wrote it badly.
and if b wasn't 1, I mean that a^{\phi(n)-1} \cdot b \equiv x \;(mod\,n) obviously becomes a^{\phi(n)-1} \equiv x \;(mod\,n)

So in your case you have 629^{959} \equiv x \; (mod\,3432)
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>>6575682
>>6575684
>>6575691
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>>6575682
>Thus a solution is going to be
My mistake, An equivalent congruence, with the same solution set, but that gets you into the hypothesis of Euler's theorem (the gcd(a/d,n/d)=1)

>>6575699
There are some useful properties of euler's function which should help have if you need to calculate it by hand too, such fermat's little theorem, which you get as corollary of euler's: \phi(p) = p-1 where p is a prime.
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>>6575682
isn't getting Euler's \varphi (n) pretty much as hard as getting the prime factor decomposition of that n?
that would make it infeasible to calculate for big numbers
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>>6575715
I don't have much experience with that, so I can't say how hard it is, but I can see how it could be.

Well, you also have that if ax \equiv b \; (mod \,n) has a solution, that is the equivalent to saying that the diophantine equation ax + ny = b
ie, if (x_0,y_0) is a solution to the diophantine equation, then x_0 is a solution to the congruence. \left[ ax_0 -b = n(-y_0) \right]
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I don't get all that arithmetics crap when http://en.wikipedia.org/wiki/Extended_Euclidean_algorithm is the basic arithmetic solution to that problem?
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I heard about one of Fermat's conjectures having to do with primes.
The conjecture states:
p = 4n + 1 \Rightarrow p = x^2 + y^2

Where n,x,y \in \mathbb{Z}
Does anyone know how to go about proving this?
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>>6574009
Well I fucked it all up.
Ended up needing to do [I-C\D] for a consumption matrix but then entirely forgot to actually subtract it from an identity and just subtracted it from a [1] 4x4 instead.
I also fucked up in the end of integrating tcos(t) since after doing it by parts twice and finishing it I said cos(pi) was 0.
This is why you should always sleep before exams.
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>>6576668
You probably want p to be prime, since n=4 \Rightarrow p=21 would be a counter example. This is one of the basic theorems from Girard or Fermat in number theory, there should be a lot of elementary proofs on the internet. See for example: http://en.wikipedia.org/wiki/Proofs_of_Fermat%27s_theorem_on_sums_of_two_squares
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>>6576724
you mean p = 17 right?
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>>6576807
sorry, i meant n= 5 \Rightarrow p=21, p = 17 would just be 4^2 + 1^2
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How would gravity act on an object the deeper it goes ?
If one managed to set an empty sphere at the very centre of the Earth and put an object into the centre of that sphere, would that object be in a state of weightlessness ?
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>>6577060
Newtons superb theorem/shell theorem. Look it up.
At that point the object would count as part of the sphere.
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>>6577060
>How would gravity act on an object the deeper it goes ?
The force of gravity would decrease

>If one managed to set an empty sphere at the very centre of the Earth and put an object into the centre of that sphere, would that object be in a state of weightlessness ?
Yes.

Though if the object is an empty sphere, a shell, then the force due to gravity would be 0 anywhere in it.

I'm just gonna take a page from halliday's book for this one, but sadly, it seems they don't go to much into the shell theorem.
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This ought to be easy but for some reason I can't get it:

"Two coherent microwave sources which are a distance a blah blah blah typical Young's Double Slit, what happens, in terms of intensity and position of the maxima, when one of the sources is has its phase reversed so there's a phase difference of 180 degrees between them?"

I'd say that because the waves are now in antiphase, there isn't any maxima left, so the intensity is now zero and the position doesn't really make scene as the waves cancel each other out.

But for some reason the answer is apparently "intensity unchanged, maxima move to positions of minima (and vice versa)", anyone able to clear that up for me? It ought to be simple.
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\cot{(x+y)}=\frac{\cot{x}\cot{y}-1}{\cot{x}+\cot{y}}
therefore:
\cot{(x-y)}=\cot{(x+(-y))}=\frac{ \cot{x} \cot{(-y)}-1}{\cot{x}+\cot{(-y)}}=\frac{-\cot{x}\cot{y}-1}{\cot{x}-\cot{y}}
but it should be:
\cot{(x-y)}=\frac{\cot{x}\cot{y}+1}{\cot{x}-\cot{y}}
What did I fuck up?
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\cot{(x+y)}=\frac{ \cot{x} \cot{y}-1}{ \cot{x}+ \cot{y}}
therefore:
\cot{(x-y)}= \cot{(x+(-y))}=\frac{ \cot{x} \cot{(-y)}-1}{ \cot{x}+ \cot{(-y)}}=\frac{- \cot{x} \cot{y}-1}{ \cot{x}- \cot{y}}
but it should be:
\cot{(x-y)}=\frac{ \cot{x} \cot{y}+1}{ \cot{x}- \cot{y}}
What did I fuck up?
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>>6573905
this is a stupid question:

when evaluating limits, why does factoring "plug in holes"? How does that work?

example (x^2 - 25)/)x-5
obviously x != 5, but if we want to evaluate the limit as x --> 5, why that it the same as evaluating the limit of (x+5) as x -->5?
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>>6579781
You got wrote the formula for cot(x-y) wrong, it should be:
\cot{(x-y)}=\frac{ \cot{x} \cot{y}+1}{ \cot{y}- \cot{x}}
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>>6579800
Yeah I assumed they fucked it up in the book I'm learning from (see pic related), but it says that here too http://www.analyzemath.com/trigonometry/trigonometric_formulas.html (at "9. Addition formulas"). Anyway, I'll test which one works.
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>>6579814
That site and your book got it wrong, you can check it, google it or something.
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>>6579800
>>6579822
Yeah, you were right, they fucked it up (pic related). But how fucking unlikely it is that it's fucked up the same way in a random Hungarian textbook and in the first result in Google?
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>>6579825
Well, at least for me, it's nowhere near the first result on google if I write it as cot(a+b) or cot(a-b), which was is normally used for trig functions, if you write it as x and y, then you get it as one of the first.

Just throwing that out there.
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>>6579843
I googled cot(x-y), because I thought a and b are not used that often. Shit, had I googled cot(a-b), I wouldn't have wasted an hour with this shit.
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>>6579857
I think I remember doing something similar at the beginning of college but with sines or cosines, heh.
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recently got into mathematics and i have been wondering about something
assuming a matrix M has real entries, can i do this
M_{m,n} = f:\mathbb{N}_{m}\times\mathbb{N}_{n}\rightarrow\mathbb{R}
where
\mathbb{N}_x is the first x natural numbers

can i extend the domain to real numbers to get some sort of continuous matrix?
>>
Say there's a project which counts for 20% of the overall grade of an exam. I get full marks in this project. Then, say that in the written exam, which makes up the remaining 80%, I get 200/400 marks. What is my final percentage?
>>
>>6580125
could you rephrase your question a little? I'm not sure what you want to do.
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>>6580125
You mean to a function
f : \mathbb{R}^2 \mapsto \mathbb{R}

Sure, you can do that. But it has nothing more to do with a matrix and the whole matrix algebra wouldn't make sense anymore. A matrix is by definition just a "finite grid" with some values related to these gridpoints.
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>>6580125
Yeah, if v is a real vector - vector components v_i given as a map from a finite set of natural numbers to the reals - then matrix application
M:v\mapsto Mv
or in detail
M:v_i\mapsto\sum_jM_i^jv
is the linear mapping, i.e. the defining property is
M(aÂ·v+w)=aÂ·Mv+Mw

If you consider a vector v with components given as map from the reals to the reals, then this is just a real function v(t) and the linear mapping turns the sum to an integral
M:v(t)\mapsto\int M(t,s)v(s) ds
where you also can choose any integral bounds
Notice
\int M(t,s)(aÂ·v(s)+w(s)) ds=aÂ·\int M(t,s)v(s) ds+\int M(t,s)w(s) ds
So continous matrix multiplication is just an integral transformation.

For example, if you look at a fourier transformation
f(x)\mapsto Ff(k):=\int f(x)e^{-ikx}dx
then you just have
M(t,s):=e^{-iÂ·tÂ·s}

"Matrix multiplications" are called
http://en.wikipedia.org/wiki/Integral_transform
and you could/should view
http://en.wikipedia.org/wiki/Integral_equation
in that light.
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>>6580130
45%, hope your class is curved
>>
That one line should better read M:v_i\mapsto\sum_jM_{ij}v_j.

PS: you can think about how you can write each vector in R^3 as a function on R to R (tip: with delta-functions, cutting off most of the real domain) and then you soon see that you any inner product
vÂ·w:=sum_{i=1}^3v_iÂ·w_i
turns into an integral \int_Rv(t)Â·w(t)dt
and so on.
>>
weird.
For a "normal" function
f: \mathbb{R} \to \mathbb{R}
x \mapsto f(x)

what about functions like g(x,y) and functionals (functions of functions) like L(t,q(t),v(t) ) ?
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>>6580291
g:\mathbb{R}^2\to\mathbb{R}
which is
g:\mathbb{R}\times\mathbb{R}\to\mathbb{R}
which is also, via currying (look it up)
g:\mathbb{R}\to(\mathbb{R}\to\mathbb{R})

and

L:\mathbb{R}\times(\mathbb{R}\to\mathbb{R}^3)\times(\mathbb{R}\to\mathbb{R}^3)\to\mathbb{R}

Relevant for
>>6580125
will also be
http://en.wikipedia.org/wiki/Curry%E2%80%93Howard_correspondence
The "algebra" of function spaces is the same as intuitionistic logic.
E.g.
f:\mathbb{R}\times(\mathbb{C}\to\mathbb{N})\to\mathbb{Z}
if you drop the set meaning of R, C, N and Z,
means that from R and a proof of N being implied by C, Z follows. If you find a function with that type than you can convert it to a proof of the proposition.

It's dark magic.
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>>6580274
So you're saying that 1/5+1/2*4/5=6/10=3/5
equals 45%=9/20?
>>
x = cos(A)
y = cos(A - B)

What trig formula can I use to eliminate A? It is suggested to to use the double angle formulae but I cant get it to work
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>>6580130
0.2 + 0.8 * 1/2 = 0.2 + 0.4 = 0.6 = 60%
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>>6580319
pardon me just fixing your LaTeX
g: \mathbb{R} \times \mathbb{R}\to \mathbb{R}
g:\mathbb{R} \to( \mathbb{R} \to \mathbb{R})
L:\mathbb{R} \times( \mathbb{R} \to \ma thbb{R}^3) \times( \mathbb{R} \to \mathbb{R}^3) \to \mathbb{R}
f: \mathbb{R} \times( \mathbb{C} \to \mathbb{N}) \to \mathbb{Z}
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>>6580455
the only one I actually cared about...
L:\mathbb{R} \times( \mathbb{R} \to \mathbb{R}^3) \times( \mathbb{R} \to \mathbb{R}^3) \to \mathbb{R}
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>>6578354
This confused me for a bit too.
I think where you're looking at it wrong, is that you're imagining two sinusoidal waves and looking at them as either having no amplitude at the same time, or having constructive interference. Like
-_-^-_-^

-_-^-_-^

Doing that, then with the phase reversal, they only ever have destructive interference, so there arent any fringes, as you were saying before.

However, you should look at it like pic related. The dark fringes are where there is destructive interference -- imagining the two waves always in phase like you were before, there is never any destructive interference, so it isnt accurate. Looking at this picture it becomes more clear that when you do the phase reversal, when there was previously constructive interference there is now destructive and vice versa, so the pattern is simply shifted. Sorry for the terrible explanation but i tried
>>
There's a symmetry for velocity right?
r' = r+ u t
Is there a conservation law as well?

Or is there none, because some relativity magic?
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>>6580319
That's why I like type theory. Btw what a disgusting avatar?
>>
>>6575454
>What is it called when a reaction, once it starts, provides enough energy for it to keep going?

fire?
>>
If you severed someone's head with a laser 1 atom thick, would the person still die or would the cut be too small to have noticable effects?
>>
>>6580924
>There's a symmetry for velocity right?
>r' = r+ u t
The "symmetry for velocity" you are talking about is a Galilean transformation.

>Is there a conservation law as well?
Yup, but it's kind of trivial.
Basically, it just says that the center of momentum of a system is equal to the total mass times the center of mass.
http://en.wikipedia.org/wiki/Noether's_theorem#Example_2:_Conservation_of_center_of_momentum
>>
What creates neural waste?
I recall hearing that neural waste is cleaned out during sleep.
Is it related to creating neural connections by using one's cognition?
If one were to be void of sleep and have neural waste in one's brain, would using one's cognitive abilities be similar to the affects of exercising the same muscle groups multiple days in a row?
Can one over work one's brain in a day without boycotting sleep?
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>>6580515
>Sorry for the terrible explanation but i tried

Not that anon but I got your meaning.
>>
>>6575454
Self-sustaining / self-perpetuating
>>
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Suppose you have a polynomial of an even degree, with all imaginary roots, for example x^2+3 = f(x).
If you look for roots you get out x = \pm{i{sqrt}3}

But if you multiply by x x^3+3x = 0, to which x=0 is an obvious solution. Are you not allowed to multiply across by 0? I would have thought you could, because 0 = 0, but apparently not.

I feel like I'm breaking some obvious rule, but is there a reason I can't do this?
>>
>>6583413
Oh well, I'm a dumb guy using Latex
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>>6583413
could you elaborate on what you mean by "multiply across by 0"?
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>>6583454
Well, if you are allowed to multiply x^2+3=0 by x, you get x(x^2+3)=x(0), so x=0 is a solution. Therefore, multiplying both sides by x is the same as multiplying both sides by 0, no?
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>>6583458
I think I see what you're getting at.
Multiplying by 0 is always fine, because
a = b => 0*a = 0*b => 0 = 0
which is true, but the implication only goes one way, as you can't divide by 0 to get from 0 = 0 to a = b.
When you take f(x) = 0 and multiply by x to get x*f(x) = 0, you are creating the solution x = 0. Nothing illegal is happening, but nothing interesting is happening either.
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>>6582343
"In fact, the restorative nature of sleep appears to be the result of the active clearance of the by-products of neural activity that accumulate during wakefulness."
http://www.huffingtonpost.com/2013/10/17/brain-waste-removal-sleep_n_4109670.html
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>>6583893
What about the over working one's brain thing?
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>>6581479
http://en.wikipedia.org/wiki/Noether's_theorem#Example_2:_Conservation_of_center_of_momentum
I don't have quite enough math to follow that wiki page.
(Anyone else think that page is pretty terrible considering how important it is? It just seems erratic)
pic related is my proof for my notes.
What exactly are generator functions?
Is there a name for Q in my proof? Like current or w/e.
If it's not current what is the current?
It's the time derivative of the conserved quantity right?
>>
How do you determine the distance of one arcsecond at a given latitude
>>
Is a Poisson random variable times a constant still Poisson? It's not homework I swear.
>>
How do I prove that a multivariable function xy^2/x^2+y^2 is differentiable at 0,0
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>>6584235
I could determine this by creating a right triangle of height latitude and theta of one second
right?
>>
>>6584268
use the definition of differentiability at a point
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>>6584284
From defintion we know it is differentiable if:
$\lim_{h\to 0}\frac{F(X+h)-F(X)-c*h}{|h|}$ exists, where $c$ is the gradient of the function.

Is the following correct:

If I have to prove differntiablity at $(a,a)$

$\lim_{h\to 0}\frac{F((x+h),(a))-F(x,a)-c*h}{|h|}$ That is take $y$ as constant, and show that the x limit exists and then do vice versa.

Is this the correct way of trying to do this?
>>
>>6584290
That's
lim h->0 (f(a+h)-f(a)-c*h)/h

Is it correct to show
first

f(x+h,y)-f(x,y) - c_x*h_x / h_c

then the other side? Is that the correct process
>>
>>6573905

Okay, I have a question that's actually not homework.

I'm self-learning topology.
I keep reading about "closed" surfaces. But these "closed" surfaces seem like they should be open sets, if we take the Euclidean topology.

So are "closed surfaces embedded in 3-space" not actually "closed sets with the Euclidean topology of 3-space"? This is really annoying me.
>>
>>6584089
bump
>>
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How do I solve these questions?
Pic related.
>>
I roll a die. If a 1 comes up, I discard it, but otherwise I write down the result and then I may choose to roll the die again. I can stop this process at any time, and then my score for this round will be the total of my dice, but if a 1 ever comes up, my score for this round of rolling dice is 0.

How many times should I roll the die, such that my average score will be the highest over the course of 10 rounds?
>>
>>6584296
They are not actually open. Being open would prevent them from being surfaces, since they'd need to contain 3D neighborhoods around each of their points. I do see what you mean though: "without boundary" looks like the same thing as "open" depending on how you understand it:
- An open set S is without boundaries in the sense that the boundary B(S) are outside of the set: B(S) \cap S = \emptyset. The converse doesn't hold, though, obviously: closed surfaces like a sphere (which is the boundary of a ball, so it's not a volume but a surface living in a 3D world) have boundary \emptyset, so obviously their intersection with their boundaries is empty.
- "without boundary" can also mean what I just mentioned about closed surfaces: the boundary B(S) is empty (not just disjoint from the set S). That doesn't mean S is open: S open implies B(S) disjoint from S, but B(S) disjoint from S (or even B(S) empty) doesn't imply S open.

Those closed surfaces are actually closed sets too. If a sequence of points from a sphere has a limit, then that limit is on the sphere too (not only because the sphere is not open as you thought it was, but also don't be too fast thinking that a set that isn't open is closed or vice versa: there are sets that are neither and sets that are both).
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>>6585760
I'm sorry to tell you this, but your question assumes that you are going to use a bad strategy. The best strategy isn't to stop after a fixed number of rolls, but to stop after reaching a fixed value (or encountering a 1). If you state the problem that way, computing that value becomes relatively easy. Proving that it's the optimal way to do things is slightly harder than finding the value (it doesn't take many lines to do the proofs but it requires understanding why the strategy is actually optimal).
>>
What is the difference between a regular map and a rational map between algebraic varieties. I've been looking at the definitions but I'm not seeing it. And what about a morphism? I get that morphisms are supposed to be morphisms in the categorical sense (structure preserving), but how is it different from a rational function that doesn't blow up at a point?
>>
>>6585845
>Being open would prevent them from being surfaces, since they'd need to contain 3D neighborhoods around each of their points.
Yes, that's what I was thinking. So an "open surface" in 3-space isn't actually open.

And "closed surfaces" like the 2-sphere are actually closed (in 3-space), since they contain all of their limit points (all of its points are limit points).

So in an n-dimensional Euclidean space with the usual topology, any open set must also be n-dimensional?
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>>6585866
>So in an n-dimensional Euclidean space with the usual topology, any open set must also be n-dimensional?
Yes, (or empty).
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>>6585866
> So in an n -dimensional Euclidean space with the usual topology, any open set must also be n -dimensional?
Yes. I'm not sure if it always holds true for other topologies (in particular anything discrete), but in the Euclidean case, the neighborhood around a point is n-dimensional, so if you contain at least one point and you're open, you're n-dimensional. Of course the empty set is open and 0-dimensional, but that's the only exception.
>>
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What physical force keeps cells together?
>>
How probable is it that there is another galaxy in our universe called the Milky Way galaxy by humans on a planet on one of its arms and being virtually the same as our planet?
>>
>>6577637
Halliday a shit
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>>6585862
A a regular map is given by a polynomial in each variable. A morphism of varieties is the same thing as a regular map; it preserves the algebraic structure of the variety. Morphisms come from/induce (depends how you view it) ring homomorphisms on the coordinate rings of the varieties, but going in the opposite direction. The rational maps instead relate to the field of fractions of the coordinate ring, called the function field of the variety.
>>
>>6585873
http://en.wikipedia.org/wiki/Fundamental_interaction
You have 4 fundamental interactions in physics:
- gravitational
- electromagnetic
- strong nuclear
- weak nuclear
You should be able to guess which ones happen at the nuclear level by reading the names :)
>>
>>6585845

>>6585868
>>6585870
>...and the empty set

Thanks a lot, I feel like I understand the basics of topology much better now.
>>
>>6585882
So a morphism induces a contravariant functor from the category of varieties to the category of comm rings, but the morphisms in the target category are called rational maps? How does one try to imagine rational maps concretely? The definition with equivalence classes of morphisms seem too esoteric to grasp at once.
>>
>>6585886
cells=/=atoms
http://en.wikipedia.org/wiki/Cell_(biology)
>>
What makes a (co)homology theory?

Like what are the philosophical cornerstones that must be present in order to get that title? There are so many out there and I have such little exposure that I just want to try to understand the main ideas first and then dive into some.
>>
>>6585892
My bad, I'm not a native English speaker, I misread that. I guess electromagnetism is a big part of what holds a molecule together, what holds molecules together to form a cell is probably a mixture of everything except for gravity, but I'm not sure about that.
>>
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can you determine if that formula is evaluated true or false?

I don't even know what this is. I found this on my exercises. I have a test about this tomorrow and i have no idea what this is, all i know is:

it has something to do with first order logic.

my professor may (very probably) be retarded, and may not be using properly the notation...

any idea on how to solve this?
>>
>>6585948
If it's supposed to mean
"for all x, there exists y and z such that sin(xy)<=cos(z/y)", then it's pretty obvious. If x is not 0, then just pick y=pi/x so that sin(xy)=sin(pi)=0, and pick z=y*pi/2 so that cos(z/y)=cos(pi/2)=1. If x is 0, just pick y=1 and z=pi/2 for the same result.

Not sure why those symbols are used for "forall" and "there exists" though when there's \forall and \exists instead.
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>>6585958
oh, that makes sense... 3rd world sucks man, i am pretty sure my professor doesnt know how to write the "forall" and "there exists" symbols...
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>>6585960
Well algebraically, they form the same kind of structure as \vee and \wedge do, but yeah it's pretty weird. Maybe in makes more sense in context, but with just the image you posted, I'd definitely not use those notations.
>>
How do drugs expire?
>>
How does bacteria survive if it just sits on a doorknob or something?
It has no energy.
>>
>>6585911
The main force that holds a cell together is the electromagnetic force. The strong nuclear force only has relevant influence at the scales of the size of an atomic nucleus. And the weak force is essentially irelivant.
>>
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Can someone explain to me the steps that I've crudely highlighted in pic related?

http://en.wikipedia.org/wiki/Lambert_W_function
>>
>>6586090

The first step is x=exp(ln(x)) with x=1/c.

The second step is W(z)=exp(W(z))/z (when z is non-zero),

The second step is to observe that if z=a*exp(a), then W(z)=a by definition of W. Replace a with ln(1/c).

Then apply exp on both sides.

Then notice that W(b)*exp(W(b)) = b, thus exp(W(b)) = b/W(b), and replace b with -ln(z).

Every step is atomic and either the relationship between exp and ln, or straight forward from the definition of the Lambert W function.
>>
>>6586242
Thanks!
>>
>>6585891
Can anyone clarify this?
>>
How do you determine step or chain growth polymers experimentally?
>>
>>6573905
What does integral without an range mean?
>>
>>6583413
You are picking up what is called an "extraneous root".

If you want to solve f(x)=0, and you multiply by g=g(x) and solve g(x)f(x)=0 instead, then you need to check your solutions to make sure they aren't just solutions to g(x)=0.
>>
Is it possible to condition my body to associate learning with extreme pleasure?
If so, how are some ways I'd be able to do that?
>>
How intelligent can a human be before his or her brain and or skull can't hold the intelligence?
Is there a limit to possible human intelligence?
>>
>>6587198
This should be at the top of the medical R&D lists...
>>
What is a lemma and how does it differ from an axiom.
>>
>>6587212
>>
My father is a rather intelligent guy. My mother, not so much or maybe she's average, it seems rather low to me.
How would I be able to tell just how much my mother (genetically) set back my intelligence? Is it even possible to figure it out?
>>
>>6587214
So, are they really different than Theorems? Because they don't seem so... How did the authors decide that a Lemma was different than a Theorem, was it based solely on the fact that a Lemma is used in other Theorems?
>>
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>>6587229
The difference between lemmas and theorems is different from author to author.

I use lemmas as a stepping stone for more important, named theorems.
>>
>>6587229
>So, are they really different than Theorems?
No, it's an arbitrary distinction of the 'value' of the result
>>
>>6587249
>>6587250
Thank you
>>
Michio Kaku says we're approximately 50 years from having his lightsaber.
What would an accurate estimate be for having force field batons?
>>
I'm trying to make physics in my game more realistic, but I can't understand how to derive acceleration from engine power and resistance force.
I can compute air and road friction, I can compute energy produced by engine at the given time frame, but I have no idea what to do with them to get acceleration.
Any clues?
>>
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>>6587294
Play with DC motor modelling.
>>
>>6586620
An integral like \displaystyle \int f(x) dx is called an anti-derivative. It is the inverse operation of the derivative.

Let \displaystyle \int f(x) dx = F(x) + c where c is a constant.

Then \displaystyle \frac{d}{dx} \left( F(x) + c \right) = f(x) .

So \displaystyle \frac{d}{dx} \left( \int f(x) dx \right) = f(x) .

And similarly \displaystyle \int \left( \frac{d}{dx} \int f(x) dx \right) dx = f(x) .
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>>6587368
>And similarly \displaystyle \int \left( \frac{d}{dx} \int f(x) dx \right) dx = f(x) .

That should be \displaystyle \int \left( \frac{d}{dx} f(x) \right) dx = f(x) .
>>
>>6587368
>>
>>6587376
I'm 2fast4u
>>
>>6587294

If the engine produces a force, I don't see a problem.

If the engine produces energy, then what you want to do is:

delta-energy = engine power*dt - F*ds

note that ds = v*dt, so

delta-energy = power*dt - F*v*dt

this tells you the amount of energy added/substracted from the object in a timeframe dt. This you can obviously translate to a speed and vice versa
>>
>>6587373
>That should be \displaystyle \int \left( \frac{d}{dx} f(x) \right) dx = f(x)

\displaystyle \int \left( \frac{d}{dx} f(x) \right) dx = f(x) + c

Shit.
>>
>>6587373
>>6587368
thats not true, its inverse upto the constant c,
and
\frac{d}{dx} \int f(x) dx = \int \frac{d}{dx} f(x) dx
is only true for uniformly continous functions :(
>>
How does one geometrically interpret duality theorems? For example, is there a visualization of something like Poincare duality?
>>
>>6587393
>even sci forgets the + c
>>
>>6587401
For all intensive purposes it's the inverse.
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>>6587549
* for all intents and purposes
>>
How do you prove \zeta(1/2) \not= 0?
>>
>>6587582
Are you too stupid to verify these yourself? For God's sake, (a) and (b) can be confirmed near instantly on a computer, while (c) would just require you to look at the fucking table that's certainly in your book/notes.
>>
>>6587549
>intensive purposes

As opposed to non-intensive purposes?
>>
Is an IQ of approximately 130 and studying habits of approximately 2 hours a day adequate to contribute to multiple fields i.e. computer science, neuroscience, physics, music, visual arts, etcetera in a relatively meaningful way?
>>
Is it true that the strings in string theory aren't made of anything? I cannot understand this. Are they 1 dimensional energy?
>>
>>6588099
If you know and attach importance to your IQ, you're unlikely to make a meaningful contribution to any serious field. I don't think there's an inherent problem with doing research only 2 hours a day, but I think it's unrealistic to think that:
- you'll be able to do anything working only 2 hours a day (doing research 2 hours a day might be fine, but you still have a lot of things not directly related to research that you have to do as a successful academic, unfortunately),
- you can have contributions to many fields without being a sufficiently hard worker to be knowledgeable about the state of the arts of all of them, which doesn't necessarily require more intelligence than being knowledgeable about just 1, but does require much more time.
>>
What does it mean to glue topological spaces "along open sets" of a certain kind? Like what does that even mean intuitionally?
>>
>>6587726
>extensive purposes
>>
>>6573905

Are there any proofs of the uncountability of the real numbers besides Cantor's diagonal argument?
>>
>>6588228
Well, "gluing" is pretty much the intuition behind it. You take a certain number of points, and you say "those points are now the same point". Formally, you define an equivalence relation in which any points that are supposed to be glued together are equivalent, and you consider the quotient of your original space by that equivalence relation.

One trivial example is that if you use the equality as your equivalence relation, then no point will be glued to a distinct point, so you won't glue anything. Indeed, this matches the fact that taking the quotient of a space by the equality relation yields a space that is homeomorphic (actually unless you really want to be extra formal, it's the same space you started with: if you want to be extra formal, if your original space was S, the result is the space {{x} : x in S}).

Now if you take a line segment, let's say the line segment from 0 to 1, and you glue 0 and 1 together by using an equivalence relation in which x is equivalent to y if only if (x=y) or (x=1 and y=0) or (x=0 and y=1), you obtain a new topological space that instead resembles a circle. If you use the same equivalence relation but your initial space isn't a line segment, and instead it's the real line, you obtain a new space which looks like if you'd imagine the real line as a long string, and you'd make a loop to glue 0 and 1 together: left of 0, it's a half line, right of 1, it's a half line, between them there's a circle connecting them. If you use again that same equivalence relation that glues 0 to 1 but in the complex plane, it'd look like what happens when you take a large bed sheet, pick two points and put them next to each other. It's really "gluing".
>>
>>6588263
> Extensive purposes
Shit... That makes almost more sense than "intensive purposes"...
>>
lim n->infinity [(n+1)^6 - (n-1)^6] / [(n+1)^5 + (n-1)^5]

Wat do?
>>
>>6588564
(n+1)^6 = n^6 + 6n^5 +o(n^5)
(n-1)^6 = n^6 - 6n^5 +o(n^5)

Thus (n+1)^6 - (n-1)^6 = 12n^5 + o(n^5).

Furthermore, (n+1)^5 + (n-1)^5 = 2n^5 + o(n^5).

Thus the quotient is [12n^5 + o(n^5)] / [2n^5 + o(n^5)] and its limit is 6.
>>
>>6588589
Thanks mate, really appreciated.
So, can I use the o(...) notation each time that i want to indicate something which is "negligible" in respect to the other terms? I used to use it only when I applied the Taylor expansion...
>>
>>6588639
There's a few things to be careful about in terms of how to combine them in expressions more complex than simple polynomials and rational fractions, but yeah. The formal definition is http://en.wikipedia.org/wiki/Big_O_notation#Little-o_notation

The important things to realize is that sometimes they aren't all-powerful. For instance:
Example 1: (n+2)/(n+1) = (n+o(n))/(n+o(n)), thus the limit when n goes to infinity is 1.
but
Example 2: exp(n+2) / exp(n+1) = exp(n+o(n)) / exp(n+o(n)), and yet the limit when goes to infinity is clearly e and not 1.

Basically you have to be careful when you apply a function to an o(...) or O(...).
>>
>>6588378
But specifically, what does it mean to glue along sets/morphisms? I knew about that kind of glueing before, but I'm reading algebraic geometry now where the formalism is sort of handwaved in that regard.
>>
I'm doing the coursera bioinformatics course. My knowledge of genetics isn't really up to par, what does 5', 2', 3', etc. mean in the context of

(5') CGCTATAGCGTTT (3')

It's a coding strand for who the fuck knows
>>
>>6588778
I assume those sets are the non-singleton equivalence classes of the equivalence relation you're supposed to glue with, and those morphisms are the morphism from the original space to the quotient space, but I've not seen the formalism before so I can't tell for sure. Can you post a few concrete examples?
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>>6588885
Well general schemes are colloquially defined as objects created by "gluing together" affine schemes, which are locally ringed spaces isomorphic to the spectrum of some ring. I'm not sure how to imagine gluing such objects together.
>>
>>6588783
Each strand has polarity, such that the 5'-hydroxyl (or 5'-phospho) group of the first nucleotide begins the strand and the 3'-hydroxyl group of the final nucleotide ends the strand; accordingly, we say that this strand runs 5' to 3' ("Five prime to three prime") . It is also essential to know that the two strands of DNA run antiparallel such that one strand runs 5' -> 3' while the other one runs 3' -> 5'. At each nucleotide residue along the double-stranded DNA molecule, the nucleotides are complementary. That is, A forms two hydrogen-bonds with T; C forms three hydrogen bonds with G. In most cases the two-stranded, antiparallel, complementary DNA molecule folds to form a helical structure which resembles a spiral staircase. This is the reason why DNA has been referred to as the "Double Helix".
>>
>>6588134

I don't know my IQ, approximately 130 was an estimate.
So, basically, In order to do what I want to do, I need to dedicate a lot more time than I currently am.
About that IQ thing. Is there some bias to that? One's ability to obtain and use knowledge, and solve problems will surly have an impact on one's ability to do anything.
If you have two identical people on two identical planets with identical societies, etcetera, the one with the higher IQ who dedicates the same exact amount of time will be relatively more well off, despite how (as I've heard) IQ is not a valid measure of intelligence.

There's a clear difference in intelligence between people who score 100 and people who score 180+, there must be some merit...
>>
Memories seem to be encoded relatively easily when it registers on an emotional level with the amygdala.
If everything were to register on an emotional level, would one be able to learn more easily?
Should there be a new teaching style that attaches very emotional stimuli to the things being learned?
>>
>>6588999
IQ is a not a measure of an individual's abilities. Just like BMI, it's a measure that was intended to be used to compare large populations.

If you have 2 populations of 10000 individuals take IQ tests, a the first scores an average of 100 while the second scores an average of 110, that is a pretty good indicator that the second population is more intelligent than the first.

If you take two guys and make them take an IQ test, and one scores 100 while the other scores 110, you can't draw any conclusion. It's slightly more likely that the second is more intelligent than the first, but that's all you can say. Too many factors are involved (one's luck, one's strategy, one's training at doing IQ tests well, one's willingness to do well on that test, whether one is in a poor physical or mental state during the test, etc etc). You can eliminate some of these factors by making those 2 guys take many IQ tests and averaging their results, but then you create new factors (do you value the speed at which they learn how to solve IQ tests more than their initial ability? where do you put the tradeoff between the two?).

(...)
>>
>>6589057
(...)

Take another example. Imagine you want to judge someone's ability to play soccer. To do that, you give them a basketball and ask them to try 3-pointers until they succeed. You write down a score which depends on how many attempts they needed to score their first 3-pointer.

Clearly, people better at soccer are more likely to be athletic, to like sports games etc, so they will in average score better than people bad at soccer, like nerds who can't even reach the basket, so on large and sufficiently "random" populations, the test will work decently.

However, for two individuals, not only if you do just one test, one might be lucky and the other unlucky, but also, one's ability to score 3-pointers might be very high even if he's bad at soccer (let's say a decent basketball shooter living in the US who has never played soccer in his life and doesn't even know the rules), and someone might be a one-armed guy that is in a paralympics soccer team or whatever, and he can't even throw the basketball but he's actually pretty good at soccer.

TL;DR: Use IQ to compare the intelligence of large populations, not to estimate the intelligence of an individual.
>>
>>6588983
This is so foreign to me that I'm not even sure it makes sense. Sorry, >>6588228, I can't answer your question then.
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>>6589060
Okay.
>>
Are plants in the oceans producing oxygen and is that oxygen contributing more to our air than the oxygen produced by plants on land?
>>
>>6589253
yes.
1) phytoplancton is the firdt provider of O2 on earth
2) Also, CO2 is captured by the ocean's water (solubilization). It then incorporates itself in the limestone which accumlulates for very long period of time on the ocean's ground.

"Forest are the earth's lungs" is a common but untrue urban legend.
>>
>>6589313
Thanks.
>>
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limit(sqrt(tan^2*x+tan(x))-tan(x), tan = -(1/2)*Pi)

The answer is +infinity when approaching from the positive (right) side and 1/2 from the negative (left) side. I know little about limits and the first time I did this question I thought the limit was 1/2 but it's actually indefined since the right side limit=/=left side limit. What I want to know is: How do I find out the positive side limit?
>>
A topological space is an ordered pair (X,t) where X is the underlying set and t is the topology. The topology consists of open sets.

>my question:
Does the topology of a topological space contain ALL open sets in the space?
>>
>>6589063
Ah well, thanks for trying. Perhaps an algebraic geometer will show up who can answer.
>>
Where can I find information on defusing bombs?
>>
Can a 3x3 matrix have more than one inverse?
>>
Why can't we determine fraction values for the Riemann Zeta function? Wouldn't it just be like a limit and get really close to something?
>>
>>6589854
No, assume the 3x3 matrix A has the inverses B and C then

B = B(AC) = (BA)C = C
>>
>>6589865
>Wouldn't it just be like a limit and get really close to something?
Every real number can be written as a limit of rational numbers. Yet there are uncomputable real numbers.
>>
>>6589868
fugg. Been looking for a fault in my working for fucking too long
>>
>>6589869
Well there's values for 1, -1 and 0, why isn't there a value for 1/2?
>>
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What is the answer to this and why?
>>
>>6589533
what did you try?

Usually you solve these kind pf problem by using taylor series.
for instance, tan(x)=x+1/3x^3+o(x^4) for x~0

and since you're at x=-Pi/2, just switch the variable (X=x+Pi/2 -> 0 so you can apply the formula with X).
>>6589558
No, it depends of the definition.
Sometimes T (collection of open sets) may be defined to be the closed sets rather than (all) the open sets.

>>6589847
I would have thought of >>>/k/ or maybe >>>/diy/ for this kind of question, but anyway:
-official manual released by aussie gvt: http://www.afp.gov.au/~/media/afp/pdf/b/bombs-defusing-the-threat.ashx
-thise one is interesting too. But expensive as shit, so try to dl it and if you can't, let it go: http://www.amazon.fr/Defusing-Workplace-Time-Bombs-Robert-Fitzpatrick-ebook/dp/B0049H9F6A
and just for the fun, this very short one: http://www.amazon.co.uk/Defuse-Bomb-Collins-Shorts-Book-ebook/dp/B008ZU6FF6

But again, for more accurate answer, I'm not sure /sci/ is the best place

>>6589886
Are you aware that this exact same "riddle" is posted every 1 or 2 weeks?
pro-tip: the content of this board is archived here: archive.foolz.us/sci/ You can eszily find your way (old interesting threads for instance), by looking up in google with a query like:
[concise description of the thread] inurl:archive.foolz.us/sci/ (removeing [])
>>
>>6589886
rolling
>>
>>6575454
The term you might be looking for is a spontaneous process or reaction
>>
How the fuck do I solve the indefinite integral of

1/(x^(5)+x)

tried to rewrite it as 1/x*(x^(4) + 1) and substitute t=x^4 + 1 but had no luck so far.
Sorry if "homework" tier, I'm actually a self learner and I'm struggling through calculus.
>>
>>6590308
1/(x^5+x)=1/x(x^2+1)(x+1)(x-1)

Partial fractions. You'll get a few logs and an arctan for the (x^2+1)^-1 term or somthing like that

Protip for possibly faster calculations : integral (x^2-1)dx=arctanh(x)
>>
>>6590314
That was meant to be
\int\frac{dx}{1-x^2}=\textup{arctanh}(x)
>>
>>6590314
But you don't get (x^5+x) from x(x^2+1)(x+1)(x-1), you get (x^5 - x). I used that method sometimes but I thought it can't be done here given that x^4 + 1 has only complex solutions. Am I wrong?
>>
>>6590320
Oh yea wait you're right oops

All polynomials can be factored down into linear and quadractic factors however

in this case, after messing around with roots of unity I get somthing like x^4+1=(x^2-sqrt(2)x+1)(x^2+sqrt(2)x+1)

Now do your arctanh arctan stuff by partial fractions and completing the square and doing some substitutions

Also that equation I was trying to post before was
\int\frac{dx}{1-x^2}=arctanh(x)
>>
This sentence has __ letters in it.

What number would be in the blank, spelled out (example: twenty five), for the sentence to be true?

Also, was my grammar correct?
>>
>>6589558
>Does the topology of a topological space contain ALL open sets in the space?
Yes. To define a topological space, you say here is the underlying set and here are ALL the open sets.

>>6589854
Inverses are unique in any group

>>6589533
hmmm this question is difficult but
sqrt(tan^2(x)+tan(x))-tan(x)
=sqrt(tan^2(x)(1+1/tan(x)))-tan(x)
=|tan(x)|*sqrt(1+1/tan(x))-tan(x)

As you approach -pi/2 from the right,
|tan(x)| gets arbitrarily large and positive
-tan(x) gets arbitrarily large and positive
sqrt(1+1/tan(x)) approaches 1

So therefore the function is unbounded
>>
>>6590379
>Does the topology of a topological space contain ALL open sets in the space?
>Does the topology
>topology
A topology on a set is the collection of ALL open sets. The questioner wanted this clarification. YOU need to check your definitions again about what a "topology" is as opposed to say a "Neighborhood function". Also it is extremely clear that here we are using the open set definition of a topological space.

>wat.
>f(X):=X*(1+1/X)-X
>left X goes to infinity when X->inf.
>right X goes to infinity when X->inf
>1+1/X goes to 1 when X->inf
>--> same hypos, so "therefore" f(X) should be unbounded. But >f(X)=1...

Wow way to be a massive dick when you're absolutely and utterly wrong
I think this wiki page might help you be less of a failure at maths
http://en.wikipedia.org/wiki/Absolute_value
>>
>>6585902
Cohomology is a very board term.
In algebraic topology, we have the Eilenbergâ€“Steenrod axioms, which tell you what is and is not a (co)homology theory. You can think of cohomology as a functor from the category of chain complexes of some kind, but I've also heard it been described as "determining the connected components of an infinity topos" or some such higher category madness. At the most basic level, it is just (co)cycles/(co)boundaries.
I'd say the most common definition is that of a homological functor, which gives rise to the modern theory of derived categories.
>>
>>6573905
Is it really that bad for US undergrads in the 18-20 age bracket to drink even though its legal in other developed nations? Ignoring any legal/ academic consequences, does drinking at this age cause any adverse long term side effects?
>>
>>6590339
37
pretty much
>>
>>6589684
Glueing along open sets basically means the same thing for schemes as it does for any other topological space. You take an open set in an affine scheme and glue it to an open in another scheme. The technical definition would be that a scheme must have an open covering U_i such that O_X restricted to each U_i is an affine scheme.
Gluing along morphisms is more or less the same as it is in Top (i.e pushouts, as the other guy described), but with extra stuff going on. See here: http://math.stackexchange.com/questions/422922/gluing-schemes-hartshorne-example.
Note that pushouts in Top are also quotenionts of the disjoint union, just as they are here.
>>
>>6573905
What are some subjects which could be studied that have some or all of these criteria checked?
>Making and synthesising new drugs or general invention
>Possibility for secret/Manhattan Project-like jobs
>Prestige
I applied for loads of different courses but I don't know what to do.
>>
>>6590393
>A topology on a set is the collection of ALL open sets
http://mathworld.wolfram.com/TopologicalSpace.html

You guys should stop arguing about definitions...
>>
>>6590424
If you only drink moderately, absolutely not. The risks of alcohol for someone with an adult's body are:
- Drinking FAR too much at once: ethylic coma.
- Drinking too much at once, getting drunk and doing something dangerous, or doing something stupid that has consequences that will make your life harder (getting caught pissing on the police station, randomly getting pregnant or getting a girl pregnant, etc).
- Drinking a bit too much at once, and driving.
- Not necessarily drinking a lot at once, but drinking really often so that you damage your liver, but it's a long-term effect and I don't think it matters whether you start doing that at 18 or at 21.
- Being in contact with other people who also drink and therefore at being the mercy of their own drunken stupidity, but honestly, just don't go to creepy parties and you're fine.
>>
>>6589886
Nobody has the answer for this?
>>
>>6590561
Yes, so? His definition agrees with that one.
>>
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>>6589886
>>6590601
pic related
>>
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Can I get a quick check on some basic clinical lab math? The last problem is the only one I'm a little unsure about.
>>
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>>6590658
>>
I have not decided what i want to do with my life. I wanna discover things through math but i want to be able to have a good paycheck and have a life. Is majoring in Comp Sci and Math a good idea?
>>
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>>6590654
>no tumblr redirect
>just plainly, and elegantly, CSFag

It's very rare for me to actually find an appreciable guide. Thanks for this.
>>
Are there any templates or games online (free) where one can practice applied deductive reasoning similarly to Sherlock Holmes?
>>
>>6590763
What is a life without more math in it?
>>
Does anyone have instructions for teaching plebs how to play Nash?
>>
>>6573905
What's the difference between an ordered list, an ordered tuple, a vector and a sequence?
>>
>>6591160
The vector lives in a vector field. The tuple or ordered list doesn't necessarily live in something with any kind of structure. The sequence is usually infinite.
>>
>>6591102
Not really a game but more so a riddle
Check out notpron
>>
>>6591160
>What's the difference between an ordered list, an ordered tuple, a vector and a sequence?

A vector is specifically an element of a vector \, space:
http://en.wikipedia.org/wiki/Vector_space

Vector spaces have a built-in mathematical structure which is different from tuples, ordered lists, and sequences.

>>6591215
>The vector lives in a vector field.
>vector field
Perhaps you meant "vector space."
>>
>>6591215
>>6591263
and the difference between tuple and list?
>>
>>6591160
A vector is an algebraic object that lives in a vector space. That means that it obeys certain axioms and is part of the structure of the space.

A tuple is an ordered set. You generally say n-tuple to specify that the ordered set has n elements. Also, a 2-tuple is commonly called an ordered pair and a 3-tuple is commonly called an ordered triple. Look up the construction on wiki because I'm sure I'll fuck it up. The 2-tuple is something like this.
(a,b) = {{a},{a,b}}
The idea is that since order doesn't matter in sets you can make a special set where containing your original set and the element that goes at the beginning. Then you use some sort of nesting to generalize this to n-tuples of any positive integer n. You can even define an infinite tuple which you usually denote with some notation like.
[a,b,c,d,...,h,0,0,...)
This is useful for defining the ring of polynomials.

Any way, sets are used as the basis to construct a ton of things. Sometimes (but definitely not always) you'll see vectors appear as ordered sets. Sometimes you'll see other algebraic objects represented the same way. At times you'll even see axioms and definitions written with ordered sets. Think of them as a kind of like a fundamental data structure that you can use to construct other mathematics with.
>>
What is a mild palace, how does it work, and is it bullshit?
>>
>>6591258
I'll check that out but there has to be more than that out there.
>>
Is anyone aware of some things on the internet that aids one in learning how to judge a person's height, body type, and body weight?
Perhaps pictures of people from various distances and information on their height, body type, and body weight.
>>
>>6591278
>[spoiler][/spoiler] Think of them as a kind of like a fundamental data structure that you can use to construct other mathematics with.
Thanks I do.

Do sequences have to either diverge or converge? Do members of a sequence have to be members of a metric space?
>>
>>6591263
>Perhaps you meant "vector space."
Yeah, my bad. Non-native speaker, sometimes I get things wrong.
>>
>>6591312
Get a job at a gas station. Their doors have a built in ruler that normal people don't notice. It's useful in case the place is robbed so that police can figure out the height of the robber. I'm sure if you sit behind the counter watching people walk past a ruler all day you'll start to get a feel for their heights.
>>
>>6591369

I have a bit of social anxiety.
One time when I was getting ready to go outside for a jog, I had to defecate at least 5 times.
I decided to exercise inside my hallway instead.
>>
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This should be just simple calculus but I'm not too familiar with numerical analysis:

I have an infinite series j(z) = \sum_{n=-1}^\infty c(m)q^m where q=exp(2*pi*i*z) (the j-invariant https://en.wikipedia.org/wiki/J-invariant to be precise). I have an upper bound for the coefficients c(m). I'm trying to find how small delta has to be and how large K has to be as functions of epsilon so that |z-z'|<\delta[\math] implies |j^{[k]}(z)-j^{[k]}(z')|<\epsilon[\math] where j^{[k]}(z)=\sum_{n=-1}^K c(m)q^m[\math] is a truncation of j(z). I think I can do this without the truncation but I'm not sure how to deal with getting a bound on delta and K simultaneously since many distinct pairs of delta and K should work.
>>
>>6591452
Woops, messed up the math tags on that one:

This should be just simple calculus but I'm not too familiar with this brand of numerical analysis:

I have an infinite series j(z) = \sum_{n=-1}^\infty c(m)q^m (the j-invariant https://en.wikipedia.org/wiki/J-invariant to be precise) where q=exp(2*pi*i*z). I have an upper bound for the coefficients c(m). I'm trying to find how small delta has to be and how large K has to be as functions of epsilon so that |z-z'|<\delta implies |j^{[k]}(z)-j^{[k]}(z')|<\epsilon where j^{[k]}(z)=\sum_{n=-1}^K c(m)q^m is a truncation of j(z). I think I can do this without the truncation but how to get bounds on delta and K simultaneously? There should be multiple pairs (delta,K) that will satisfy this.
>>
>>6591369
I kinda wonder. Once at a gas station the guy behind the counter was easily over 2m tall even though he was skinny like a stick. I suspected the area behind the counter to be raised but I couldn't check it.

Is this something they do to deter robbery?
>>
>>6573905
How do chemists, etc learn about molecular structure of all the shit in the world.
Can the be inferred through some general rule even for unknown things or is there some sort of measurement device or technique that requires study of each element and the compounds they form with other elements?

This structure shit has been around for ages too, so it's not a very recent thing so if it's a technology thing then that tech must be old.

How's it done? Google is kinda shitty for it.
>>
How do I find absolute max and min of a function in a given interval?

I have this f(x) = |x^2 - 2x| - |2x - 1|, and I have to find absolute maximum and minimum in [-1,3]
I solved the absolute value, so that f(x) is divided into 4 cases:
x^2 - 1 if x<=0
-x^2 + 4x -1 if 0<x<1/2
-x^2+1 if 1/2 =< x < 2
x^2 - 4x + 1 if x>=2

Should I do the derivative for each part, and then see when it's equal to zero?
>>
Anyone have a chart comparing how much scientists have contributed to our way of life and non-scientists?
>>
>>6592126
> Should I do the derivative for each part
Yes.

> and then see when it's equal to zero?
Yes, but not only. Checking where the derivative in each part is 0 will tell you where the local extrema are in each OPEN segment. But it's also possible (and likely) that there are extrema at the border between two segments. So once you've found where the derivatives are 0 and computed the values of the function there, also compute the value of the function at 0, 1/2 and 2, and find the highest and lowest of all of those.

As an example, consider |x| on [-1,1]. The derivative is -1 on [-1,0) and 1 on (0,1], so it's never 0. However, there are 3 global extrema: 1 at -1, 0 at 0 and 1 at 1.
>>
>>6592175
Okay man, thanks. Now, what about the extremes of the interval? {-1, 3} in the function that I wrote. Should I check them too? I suppose not, since they're not "natural" borders like those pointed out by the absolute value, but just arbitrary boundaries chosen for this particular exercise.
>>
>>6591258
I've been on level 5 for quite a while now...
>>
>>6592219
Yes, sorry of course. I don't know why I forgot to mention them and then added them in the example. Yes, every bound of every interval can be an extremum, even the outer ones.
>>
>>6590654
uh.. I can't read Math very well.. can someone dumb it down for me please?
>>
>>6592506
Let's say there are 10 answers, which are:

a, a, a, b, b, c, d, e, f, f.

a is a potentially valid answer if a=3/10
b is a potentially valid answer if b=2/10
c is a potentially valid answer if c=1/10
d is a potentially valid answer if d=1/10
e is a potentially valid answer if e=1/10
f is a potentially valid answer if f=2/10

There is a paradox if more than one of the above is valid at once. There is no paradox and there is actually a valid answer if exactly one of the above is a potentially valid answer.

Otherwise if none is a potentially valid answer, the probability to pick correctly is 0. Then if 0 is in the list of answers, there is a paradox, and if 0 isn't in the list of answers, it's a question which isn't paradoxical itself but its answer is just missing from the proposed list.
>>
How hard is Linear Algebra and Real Analysis? Taking both next semester, the former is not the baby applied shit intro to matrices and what not, I took that this semester, the one I'm taking next semester is proof heavy and theoretical.

Budding math major here.
>>
I have a dumb question. What is the graph of the exponential function (-2)^x or really any exponential function (-a)^x where a> 0. I don't really understand. What would the graph even look like?
>>
>>6573905
I have a water bottle with frozen water that sits approximately two inches from my laptop's exhaust thingy.
Does the water bottle affect the rate at which my laptop cools?
>>
>>6592659
I would say yes, but only marginally so. Then again I only took freshman year physics so wait for a more knowledgable response maybe.
>>
Given a Hamiltonian H, and the function describing a state at time t, how do you find the probability of energy being in a certain range? Do you calculate eigenvector of the Hamiltonian for the energy that can be found on that range, the proceed as usual with |<?|?t>|^2, where ? is the Hamiltonian eigenvector and ?t is the state function?
>>
>>6592694
yep. That's exactly what you do.
>>
>>6573905
Physics question

By what mechanism is the ionosphere always charged with the same electrical polarity when the solar wind is neutral?
>>
Is the pollution in the air more concentrated in large cities, does the air in large cities (due to pollution) cause negative measurable effects to human health and how bad are those effects?
>>
Can I use de L'Hopital rule in composite functions?
I mean, if I have lim x->0 arctg [ (sin2x) / (x^2 + x) ]^(1/3), can I use de L'Hopital just for the inner function (i.e. the argument of arctg)?
I know that if I have a sum or product of different functions then I can use it because of these lim f(x) * g(x) = lim f(x) * lim g(x) or lim f(x) + g(x) = lim f(x) + lim g(x), but I'm not sure if this applies even in the composite case. I can't find it anywhere...
TN5 I summon you, pls ;_;
>>
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Ive got a few questions because this book just keeps wrecking my shit.

A) Questions 1 and 2 (attached)
B) Question 3 http://i.imgur.com/XFmIbXz.jpg
C) 4 http://i.imgur.com/uAjL20L.jpg
D) 5 http://i.imgur.com/9kjTnru.jpg

1) Why do they integrate from 0 to z if it's a density function, not a distribution function?

2) Why do they integrate from z-1 to z, and where does f_{k-1}(y) come from?

3) The books says "When n=1, this reduces to the exponential(\lambda)". If I fill in n=1,
Im left with \lambda e^{-x}, but Ive learned that exponential(\lambda) = \lambda e^{-\lambda x}. Which is wrong?
Also, theorem 6.1 confuses me: the exponent of lambda is n+1, while just above that it says it should be\lambda^{n}.
Those cant be equal per definition, right?

4) What happened in the last step, E{XY-YEX-XEY+EXEY}=EXY-EXEY?

5) Is p the chance that a red is drawn? If so, if it's drawn k times without replacement, shouldnt
p vary for each X_{i}? Because if we draw one ball, the chance of it being red is \frac{m}{n}, if we draw one again,
the chance of it being red too is either \frac{m-1}{n-1} or \frac{m}{n-1}, right?
>>
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I've been learning pieces for the guitar since I was in elementary school.
By learning those pieces, was I developing my ability to learn, learn pattern, and patterns in relation to the fret board or was I developing my ability to learn, learn patterns, learn patterns in relation to the fret board then just patterns in relation to the fret board without developing much of the other two?
>>
>>6582343

Is braining while sleep deprived like working the same muscle twice in a day or what?
>>
>>6593895
1) Why do they integrate from 0 to z if it's a density function, not a distribution function?
The integral from 0 to z could just be written
f_{X+Y} (z) = \int_{x\in \{0,1\} }\sum_{y\in \{0,1\} :x+y=z} f_X(x)f_Y(y) \mathrm{d}x = \int_{x\in \{0,z\}}f_X(x)f_Y(z-x) \mathrm{d}x
because for 0<=x<=z, there is only one value of y to sum over and it's y=z-x, and for x>z, the sum is empty.
>>
>>6594145
*two days in a row
>>
>>6593864
pls guys respond
>>
Exactly what is the meaning of "diffuse" in the diffuse extragalactic background radiation?
>>
When I turn on my bedroom light, does light emit from the light source then goes in all possible directions then bounce off of everything in the room forever?
>>
>>6594161
Ah, yes, thank you. Do you know the rest of the answers, too? It's kind of an obstacle for me at the moment.
>>
>>6594283
Yes, you can use l'HÃ´pital. Note that because only the argument of arctan depends on x, you can rewrite lim arctan(f) to arctan(lim f). If you do that, you get the indeterminite form 0/0, so you can go on your merry way.
>>
How do scientists know exactly how old the earth and the universe is?
>>
>>6592646
It would fluctuate, like so. The reason for this becomes apparant when you take a good look at the rules for fractional exponents, and the definition of the complex numbers.
>>
>>6595492
Im going to wait for a physicist to answer this, in the meanwhile, take a look at this page:
>>
>>6592631
Linear algebra was perfectly managable to me, as long as youre set on abstract thinking. Real analysis wrecked my shit, because I lack the ability to find creative solutions. Im going to have to retake analysis.
>>
Is it true that if every human in the world farted at the exact same time, that the temperature would rise?
>>
What would be the best composite for a tube exposed to a 100 bars pressure from the inside ? It also have to be resistant to vibrations, but it would be stored in controlled atmosphere.

I'm looking for an answer in terms of matrix, fibers, orientation of fibers and production methods.
>>
>>6593206
The solar wind ejects the electrons from the atoms by photoelectric effect, Compton and Thomson scattering also happen but don't result in ionization.
>>
>>6595500
Age of earth is determined by radioactive dating. Not carbon, and I have had classes on that in HS but can't remember the isotope used.

For the universe, it's mainly a supposition based on the cosmic microwave background and the current universe expansion.
>>
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I plead for not having questions, if they are interesting, in "questions that deserve their own thread" threads.

They slow the board down and most of the things get overlooked by most people.
>>
>>6595548
I did not mean to give off the idea that it was the answer he was looking for, just an interesting read.
>>
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>>6595552
>>
>>6595490
thank you kind anon
>>
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anyone able to tell me how they got pi/12 from sin6?
>>
>>6595584
anyone?
>>
>>6595600
sin 6(1+a) probably means sin[6(1+a)]

2 = 4 sin[6(1+a)] - 2 =>
4 = 4 sin[6(1+a)] =>
1 = sin[6(1+a)] =>
6(1+a) = pi/2 =>
1+a = pi/12
>>
>>6595600
Gonna call alpha x:

2 = 4 sin 6(1+x) - 2
4 = 4 sin6(1+x)
1 = sin 6 (1+x)

sin(y) = 1 implies y = pi/2 (unit circle)

Thus 6(1+x) = pi/2 and 1 + x = pi/12
>>
>>6595602
>>6595603

Mathmind
>>
>>6595602
>>6595603
ah so arcsin of 1 which gives you pi/2 and then you remove then 6 and so on and so forth

cheers fellas
>>
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>>6595543
Anyone has an idea ? Also, currently aramid fibers with a polymer matrix are used, but I'm looking for ideas for a better replacement.
>>
>>6595626
Last polite bump.