/sci/ - Science & Math

Unanswered questions from the previous thread:
Math questions:
>>11331621
>>11333561
>>11337135
>>11345178
>>11347448

/g/ questions:
>>11330997
>>11332510

Physics questions:
>>11331777
>>11332388
>>11335483
>>11335628
>>11337711
>>11344535
>>11347334

Chemistry questions:
>>11334823
>>11338301

Biology questions:
>>11347479

Stupid questions:
>>11332131
>>11332978
>>11336540
>>11341202
>>11347367
>>11347748

Note: Half of the serious questions should be in stupid.

I'm trying to understand Wannier states in full. I'm using "A short introduction to topological insulators", chapter 2. It is said that Wannier states [math]|w(j)\rangle[/math] are the Fourier transform of Bloch eigenstates [math]|\Psi(k)\rangle = |u(k)\rangle \otimes |k\rangle[/math], where the first factor is the internal part (eigenstate of the Fourier transformed Hamiltonian) and the second factor is the external part, plane waves. Noting the two Fourier transforms, I've tried making some sort of commutative diagram to tie it all together, but I had no success. Is something like this possible?

How do we kill all normalfags who browse 4chan?

I've been reading Graham Scott's essential ornithology in between studying maths and I'm finding it very comfy.
Does anyone have any recs of other books on ornithology or other subjects in science that you can study while relaxing?

How does dry climate START wildfires? (Not how it help spread it.)

Still on pic related, let me address the kind anon that replied:
>>11345947
I get this, but I don't see how this implies p divides q. Like p can be 3 and q 4, for example. 4 does not divide 3. Wait, I get it now. Clearly q is a factor of p because it's a factor of what p equals (p = q(a*p + ...). And yes, I know this is exactly what you said, but I already uploaded the pic.

I've been researching a lot of chemistry papers, following up on citations, writing down reactions, making note of conclusions and shieeeeeet, and OneNote just ain't cutting it for organizing information. I need something which can cross-reference data in a way more complex than a pitiful table, preferably something with chemistry features like molecule structures. Any help, please?

why the absolute fuck is the answer 0.009% and not 0.09%?
I know the percentage should decrease but my math keeps giving me 0.09

>>11348795
shit I forgot the pic

>>11348795
>>11348798
also heres what I did
please help, this is driving me nuts

nevermind I got it jesus FUCK

How the fuck are you supposed to deal with burnout?

>>11348998
>>11348998
Good boy.
>>11348740
Excel maybe?

>>11347954
answer to the 3. math question
let v1 be an eigenvector of A with eigenvalue j
then ABv1=BAv1=Bjv1=jBv1
therefor Bv1 is Eigenvector of A with eigenvalue j
let Bv1=v2 eigenvector of A
becuse v2, v1 have the same eigenvalue they are in the same eigenspace and therefore
k*v1=v2 k element in C
Bv1=v2=kv1
therefore v1 is eigenvector of B

>>11348602
Okay.
>>11349062
Hot blood and passion.
>>11349306
>the eigenspace is spanned by one vector

Why are larger nuclei only stable when they're composed of a mixture of proton and neutrons? Why can't the strong force hold together a group of, say, 10 neutrons by themselves, or 10 protons by themselves?

>>11348807
well, you added a zero on the second line
>5 * (0.0001/100)

>>11348807
and by the, you're retarded for LITERALLY writing out what is essentially 1*5

guys I need some help with a basic probability problem, it follows:
We have a chain of christmas lights consisting of 20 bulbs. For each bulb, the probability of it working after a 300 hour mark is 0.9. Calculate the probability of the event, that after 300 hours at most one bulb is not working.

when I was initially faced with this problem I instantly thought (0.9)^19 which is ~0.14, then I checked the answers and they said 0.406
my mind is kinda blown how the answer is not (0.9)^19, can someone explain why the answer is 0.406 and not 0.14?

>>11349136
>Excel maybe?
I'm looking for something more specific to chemistry. I mean, there has to be at least one program which lets you plan multiple routes to a final product by dragging-and-dropping substances in a flowchart, surely? It seems such an elementary (hurrrrr) idea, yet I can't find anything like that. Molecule and equation drawing software is common as muck, but don't let you see the big picture and multiple possibilities.

>>11349687
Either all of them work (chance 0.9^20) or a single one fails
(chance 20*(0.1* 0.9^19) because there are 20 possible choices of failing bulb, and that's the chance of each one failing alone).
This gives 0.391746998125 in Desmos.
I'm not sure if this is a rounding error. Can you post the full, original problem?

>>11349707
ok I think I get it now, the final probability is the sum of the probabilities of a) 1 bulb being broken and b) no bulbs being broken, which should be:
a) 20*(0.1*0.9^19)
b) (0.9)^20
I'd like to post the original problem but it's from my book which is not in English and I translated it
the solution in my book uses something called Bernoulli scheme, which I tried reading about but totally couldn't understand so I looked for a simpler solution, it probably is just a rounding error
is my understanding correct at this point?
I'd appreciate if anyone could explain the Bernoulli scheme kinda intuitively but I don't know if it can be done, it looks pretty complicated to me

how can somebody simplify the following

f(n) = \sum_{k=0}^{n} c^k
where c > 1, n out of the natural numbers

by simplifying I mean either approximate by some constant degree term or a fixed power series?polynomial interpolation

>>11349687
P(at most one fails)= P(one fails or 0 fails)= P(1 fail) + P(0 fails)- P(1 fail and 0 fail)= P(1 fail) + P(0 fails), since they are exclusive.
P(1 fail)= P(19 work, 1 fails)= (20 choose 19) 0.9^19 * 0.1^1= 20*0.1*0.9^19~~ 0,2702
P(0 fails)= P(20 work)= 0.9^20 ~~ 0,1216
adding both you get 0,3918 as >>11349707
said

>>11349726
>>11349753
this is exactly bernoulli trials. either 19 successes in 20 trials or 20 successes in 20 trials

>>11349600

The protons and neutrons themselves do not add up to "whole notes".

>>11349565
Thats the hole point:
v1 and v2 have the same corresponding eigenvalue which means that v1 and v2 are in the same eigenspace and
v1 and v2 are lineary dependent

>>11349759
ok but I imagine the bernoulli scheme could be used more simply for an arbitrary number of bulbs and fails
I now know how to solve this problem using my method, but now if we rephrased it into 5000 bulbs and at most 100 of them failing I would be forced to calculate 101 different probabilities which would be very tedious

>>11349762
But why does a nucleus of 10 neutrons have less binding energy than a nucleus of 5 neutrons and 5 protons? Is the strong force weaker due to the quark profile of the former being an uneven split between up quarks and down quarks?

>>11349726
for example, 19 bulbs work and 1 fails:
we need to multiply the probabillity that each one of 19 works and one of them fails, we assume they are independent events i suppose (but would that make sense in a physical situation when they are connected with one wire? i dont know). BUT we also need to note that the order is not important, we are only asking if one of them fails, doesnt matter if it was the first one or last one etc
(n choose k) gives you all the possible ways you can select k different objects from n different objects. So we need to count every possible order of trials that give k successes and n - k losses. in our example n=20, 19 successes and 1 fail. 20 choose 19 is just 20.

>>11349726
> it probably is just a rounding error
It's wrong by two digits, I think it might not be.
>is my understanding correct at this point?
If I've read the statement of the problem correctly, yes.
>it's from my book which is not in English
If it's in spanish, portuguese or italian then I can read it.
>>11349735
https://math.stackexchange.com/questions/971761/calculating-sum-of-consecutive-powers-of-a-number

>>11349735
or more general, where f(n) is the function of the fundamental series breaking up at n+1
sum_{k=0}^{n} {a_{k}}^k

>>11349781
[math]
sum_{k=0}^{n} {a_{k}}^k
[/math]

>>11349767
you just leave a correct indexed sum as an anwser then, and practically it would be in the computing abillity of most machines

>>11349778
yeah I now get it, thank you
>>11349787
is there really no way to do it manually in a decently fast manner?

Is there a resource anywhere for the flavor compounds in known substances?

Like sometimes i sense a flavor like laundry detergent smell in blue cheese, and i'd love to know what flavor compounds are doing that.

>>11349785
How can somebody approximate this function with a fixed term?

Let \mathrm{c} \in \mathbb{R} \mathrm{c} > 1
[math]
f: \mathbb{N} -> \mathbb{R}
f(n) = \sum_{k=0}^{n} {c}^k
[math]


More general, what are the properties of such descrete series?
Can anybody give me a clue where to start searching for?
[math]
f: \mathbb{N} -> \mathbb{C}
f(n) = \sum_{k=0}^{n} {a_{k}}^k
[math]

How can somebody approximate this function with a fixed term?

[math]Let \mathrm{c} \in \mathbb{R} \mathrm{c} > 1/[math]
[math]f: \mathbb{N} -> \mathbb{R}/[math]
[math]f(n) = \sum_{k=0}^{n} {c}^k[/math]


More general, what are the properties of such descrete series?
Can anybody give me a clue where to start searching for?
[math]f: \mathbb{N} -> \mathbb{C}/[math]
[math]f(n) = \sum_{k=0}^{n} {a_{k}}^k/[math]

>>11349780
freaking thank you
fucking spammed the whole page here

>>11349792
there is also normal approximation. For large n, you can effectively approximate number of successes in n trails by normal distribution as pic related.
This is a case of a more general central limit theorem which is not really intuitive.

>>11349810
this should be an Taylor approximation
or better an Maclaurin approximation (Taylor approximation for x0=0

>>11349826
phi is standard normal cumulative distribution function

>>11349826
I think this is quite a bit above my level so I'll leave at that for now, thank you for the help nonetheless

>>11349827
Edit for the none fixed one
withe the ak

Is current economics and political science just applied statistics?

We start with a Hilbert space [math]\mathcal{H}[/math]. Then, we take the free associative complex algebra over the bras and kets and quotient out [math]\langle \phi | * | \psi \rangle \sim Id \langle \phi , \psi \rangle[/math] , where [math]*[/math] is the product in the algebra and the right side is just the inner product.
Is there a name for this construction? The "Dirac Bra-Ket algebra" or something?
Is it useful for anything other than reducing the abuse of notation in QM?

>>11349920
>quotient out the bras and kets
Small mistake. You also need to quotient out [math]\alpha \langle \phi | + \langle \psi | \sim \langle \alpha \phi + \psi |[/math] plus the equivalent for kets.

I have an idea, but i don't know enough detail about science to know whether it's bullshit.

I kept wanting to find a method of lowering the atmospheric pressure in a chamber to make it easier to evaporate salt water to turn it into fresh water, but every closed system i could think of would lose precious pressure when you opened it to get the fresh water out, or put salt water in. But then it occurred to me that if you passed a super fast jet of air over the salt water, it would also lower the effective pressure of that air.

I am suggesting we build a solar updraft tower, but its primary purpose is to pass jets of super fast air over salt water, which would lower the effective pressure in the air over that water, which would lower the evaporation temperature, and it would make desalination much faster.

This system would require no electricity. You can just shunt heat into the water with a heat sink, like a net of metal plates in direct exposure to the sun that would channel into poles in the water to heat it up. Then we use a solar updraft tower design to pass air over the salt water at super high speeds, and then just catch the air at the end of the chimney and collect it.

>>11349920
Actually, I think it might admit a full C*-Algebra structure.
The involution is swapping the bras with kets and inverting the order should be well defined).
Then [math]||x^* x||[/math] makes appropriate sense, and we can define [math]||x|| = \sqrt{||x^* x||}[/math]. The proof of [math]||xy|| \leq ||x||||y|| [/math] should just be a few successive applications of Cauchy-Schwartz.

Why do i think that intoxication will be a valuable experience when i cannot remember the experience later?

>>11347448
If [math]V[/math] is an inner product space, that's true only when Riesz's lemma [math]V\cong V^*[/math] holds and [math]T[/math] is Ferdholm. An inner product lets us decompose the space into orthogonal complements such that [math]0\rightarrow \operatorname{im}T \rightarrow V \rightarrow\operatorname{ker}T\rightarrow 0[/math] is exact in [math]({\bf Vect},\oplus)[/math]. Now as rank-nullity [math]\operatorname{ker}T^*= \operatorname{im}T = \operatorname{coker}T[/math] holds for Fredholm [math]T[/math], we can identify [math]\operatorname{coker}T \cong \operatorname{ker}T[/math] by Riesz [math]T\psi \mapsto (T\psi)^*[/math].
>>11348066
In the non-interacting scenario, a smooth Bloch Hamiltonian [math]H(k)[/math] defines as its image a finite dimensional Hilbert space [math]\mathcal{H}_k = \operatorname{im}H(k)[/math]. The external part can be taken to be the space [math]l^2(\widehat{\Lambda})[/math] of square summable sequences on the BZ [math]\widehat{\Lambda}[/math] as the Pontrjagyn dual of the lattice [math]\Lambda[/math], whence [math]\mathcal{H} = \left[\bigoplus \mathcal{H}_{k\in\widehat{\Lambda}}\right] \otimes l^2(\widehat{\Lambda))[/math]. Fourier transform is certainly an isometry here, and you can write down whatever diagram you want (though that wouldn't be very useful).
But the point is that, because of the underlying [math]U(1)[/math] (charge) or [math]\mathbb{Z}_2[/math] (particle-hole) symmetry of TIs/TSCs, the Chern number [math]c_1 = 0[/math] vanishes if you take as fibre your entire space [math]\mathcal{H}[/math] for your Bloch bundle; the topological property is only exhibited by the [math]filled[/math] Bloch states below the Fermi level. Projection [math]P_\mu[/math] onto these states may not commute with the Fourier transform.

>>11350482
Shit. I of course meant
[math]\mathcal{H}=\left[\bigoplus\mathcal{H}_{k\in\widehat{\Lambda}}\right]\otimesl^2(\widehat{\Lambda})
[/math]

>>11350484
AAAAAAAAAAAA FUCK
[math]\mathcal{H} = \left[\bigoplus_{k\in\widehat{\Lambda}}\mathcal{H}_k\right]\otimes l^2(\widehat{\Lambda})[/math]
>>11349920
You quotiented out a two-sided ideal that doesn't care about the complex structure [math]K[/math] in your original algebra. You also need to require that [math]K[/math] projects down to complex conjugation along your identification. This is what you need for a sesquilinear quadratic form.
To get anything useful you also need bilinearity (as you have posted >>11349927 but with [math]\overline{\alpha}[/math] for bras and [math]\alpha[/math] for kets) and the non-degenerate property of the [math]\ast[/math]-product, which gets you an inner product.
Now a sesquilinear inner product gets you Riesz on [math]\mathcal{H}[/math], and your projection defines an algebra homomorphism which commutes with taking duals, so you don't get anything new or useful (unless you want to sacrifice any of the properties of the inner product).

[eqn]L=\lim_{n \to \infty}\frac{|x_{n+1}|}{|x_{n}|}[/eqn]

exists and L<1

{X_n} is a sequence and X is in the real numbers. Show that {X_n} converges to X.

I'm teaching myself analysis. How do I go about doing this?

[math]L=\lim_{n \to \infty}\frac{|x_{n+1}-x|}{|x_{n}-x|}[/math]

exists and L<1

{X_n} is a sequence and X is in the real numbers. Show that {X_n} converges to X.

I'm teaching myself analysis. How do I go about doing this?

>>11350683
Let R be such that L < R < 1. You know that there is a point N after which (for all n >= N) we have |x_(n+1) - x|/|x_n - x| < R. Then |x_(n+1) - x| < R|x_n - x|.
Now, what can you say about |x_(N+k) - x| in comparison to |x_N - x|? Use the inequalities we've found for each n.

Is Tau Beta Pi worth joining?
I recieved an email saying I'm invited to join. I'm a senior right now, graduating next year in EE. Not sure if it's really worth it because I'm fairly busy with schoolwork and work. What do they even do?

I don't quite understand how introducing a neutron into the nucleus of a high proton/neutron ratio atom increases its stability. The neutron has no charge, so it doesn't offset the EM force in that way, and it only introduces a minuscule amount of distance between the protons, so I have a hard time believing it's because of that either. Does it have something to do with the types of quarks being even? What is it exactly that causes Helium 2 to throw a fit and refuse to exist?

>>11350501
Yeah, the sesquilinearity is the most finicky part of the construction.
But I was mostly hoping that it admitted an appropriate norm which gave it a Banach Algebra structure. Ideall, we'd have [math]|| ~ | \phi \rangle || = || \phi ||[/math] for kets, the same for bras, and operator norm for sums of elements like [math]| \psi \rangle \langle \phi |[/math].

>Now a sesquilinear inner product gets you Riesz on H
Speaking of, you can run the entire construction on an arbitrary complex vector space [math]V[/math] by taking the free associative algebra over [math]V[/math] and [math]V^*[/math], quotienting out the conjugate-linearity and quotienting out evaluation. Does that one have a name?
>an algebra homomorphism
An algebra homomorphism with what?

>>11348081
Start with yourself

I'm a little confused, just wondering if you guys can help. Resistivity is directly associated with temperature, so an increase in temperature means higher resistance. So if you did an experiment and measured voltages at the resistors in a short circuit, the voltage should be lower, correct? Compared to mathematically working out the voltages at the same resistors in the same shorted circuit, using mesh analysis or whatever.

>>11349801
Yep
http://www.thegoodscentscompany.com/allodor.html
http://www.thegoodscentscompany.com/allflavor.html
it's "bleu" btw

How should each equation be labeled in a lab paper? 1, 2, 3, 4, etc?

>>11347448
We let [math]a, b[/math] be two orthonormal generators for [math]\mathbb{C}^2[/math].
Let [math]A: \mathbb{C} ^2 \rightarrow \mathbb{C} ^2 [/math] be given by [math]A(a)=A(b)=a[/math], and extend to the rest of the domain by linearity. Clearly, the subspace generated by [math]a[/math] is invariant.
Then we have that [math]1 = \langle a, a \rangle = \langle Ab , a \rangle = \langle b, A^* a \rangle[/math], which implies that the subspace generated by [math]a[/math] isn't invariant under [math]A^*[/math]

Can someone explain horizontal graph shifting in simple logic? So [math]f(x+h)[/math] for example.

My own understanding behind it is that the reason it "shifts" to the left (when we add+h) is because in the original function x would get paired to some value f(x), but in this new function, x gets paired with f(x+h), ie. it's getting paired with the f(x) value of some x that would be ahead of it on the x axis in the original function, effectively shifting it to the left. Is there some other way to look at it?

>>11351211
bump

>>11351417
well you are stealing 3 from the x of the original function and lets say you want both of them to be 0 at some point x. so it has to shift to compensate. there is not much to add other than what you have already said, anything beyond that is overthinking it.

>>11351211
what do you mean by short circuiting resistors? exceeding their power rating? they will melt and burn and that may result in an actual short circuit, i.e. 0 ohm.

>>11351211
First:
>resistors in short circuit
I don't think that makes sense. Short circuits have no resistance, by definition. You would never intentionally build a short circuit in a basic physics/circuits lab.
>Resistivity is directly associated with temperature, so an increase in temperature means higher resistance
Not necessarily true. Ceramics and semiconductors have a negative temperature coefficients (decreased resistivity at higher temps.) while metals all have positive coefficients (more resistive at higher temp.). Thermistors are resistors that are specifically designed to exploit this phenomenon and are used in sensors, and can have negative or positive coeffs.

Experiment:
Let's say we have a simple circuit with a 9 volt battery in series with 1 kohm ceramic resistor. If we measure the current at the spec. temperature for the resistor, we get a current of about 9 mA. US Resistor, the manufacturer of our resistor, rates the thing with a temp. coefficient of about [math] \alpha=-0.0002\ ^\circ\text{C}^{-1}[/math]. http://www.usresistor.com/index.php/materials/ceramic-resistors
The exact equation that models this is
[eqn] \frac{\text
{d}R}{R}=\alpha\text{ d}T [/eqn]
However, usually we can assume that [math] \alpha [/math] is a constant and [math] R [/math] doesn't change very much with temperature. Someone comfortable truncating a Taylor expansion after the first term would get the following approximation. Say we remove and heat the resistor by 200°C in an oven before replacing:
[eqn] \Delta R=R_0\alpha\Delta T=(1\text{ k}\Omega)(-0.0002\ ^\circ\text{C}^{-1})(200\ ^\circ\text{C})=-40\ \Omega [/eqn]
So our new resistance is [math] R'=1000+\Delta R=960\ \Omega [/math] and the new current in your circuit is [math]9/960=9.375\text{ mA}[/math] which is greater than before. The voltage is exactly the same because our power source did not change, but the resistance and current did.
>>11351511
You've got to be patient, lover.

>>11351546
>voltage is exactly the same
Sure thats because you considered a single resistor. He may have asked about multiple resistors which would make a voltage divider, so if one of them gets hot that would slightly increase its resistance and therefore the voltage drop over it. But in practice thats close to zero or it just burns out if the current is too high, maybe thats what he meant by a "short circuit" but I am not sure what he is asking honestly.

>>11349899
bump

>>11351584
>He may have asked about multiple resistors which would make a voltage divider
Maybe. If he's got a battery in series with two equal resistors, he's going to have an equal voltage drop over each. If he heats one of them up in the oven, its resistance will decrease (assuming negative coeff.), and the voltage over that resistor in particular will decrease.

how do i go from 6sin(2x)+8cos(2x) to a formula using only one trigonometric funtion?
alternatively it could say 6sin(2x)+8sin(2x+pi/2) still dont find the way

>>11349953
I'm not sure if cooling towers achieve "super high speeds"

>>11351546
>I don't think that makes sense. Short circuits have no resistance, by definition. You would never intentionally build a short circuit in a basic physics/circuits lab.
The resistors in a short will be causing no resistance to the flow of current?

>>11351661
Draw a picture of what you think a "short" circuit is. Where I come from, a short circuit contains no resistors. Your question makes no sense.

>>11351643
anybody???

>>11351669
We started with the open, then shorted it. Measured voltages across each resistor with a voltmeter.

>>11351653

Seems like getting it up to "super high" is just a matter of building a very large platform to heat the air in the tubes.

>>11351643
Sine and cosine are linearly independent, anon.

>>11349953
>this system requires no electricity
If you are evaporating water into air in a tower, by the time the air reaches the spout, it will be quite cold. To condense the water and reclaim pure H2O, you need to decrease its entropy. To do this, you need to draw heat out of it. The only way to cool something that is already quite cool, you need something colder. This means refrigeration, which definitely needs a lot of power.
Unless you mean that the "super high speed" air is coming from the top of the spout, in which case your idea makes less sense.
The science of the hotness and wetness of air and how it draws in moisture is called "psychrometrics," by the way. Research that to learn more.

>>11351743
But you can use
[math]6 sin y + 8 \sqrt{1- sin^2 y}[/math], where y=2x.

>>11351706
The circuit on the bottom is definitely short-er than the one on the top, but a short circuit it is not. Did I answer your question?
>>11351716
>>11349953
>this system requires no electricity
If you are evaporating water into air in a tower, by the time the air reaches the spout, it will be quite cold. To condense the water and reclaim pure H2O, you need to decrease its entropy. To do this, you need to draw heat out of it. To cool something that is already quite cool, you need something colder. This means refrigeration, which definitely needs a lot of power.
Unless you mean that the "super high speed" air is coming from the top of the spout, in which case your idea makes less sense.
The science of the hotness and wetness of air and how it draws in moisture is called "psychrometrics," by the way. Research that to learn more. If you want more help, you will have to formulate a more specific question.

>>11351761

Are you familiar with solar updraft towers?

>>11351771

Whoever designed solar updraft towers designed it to generate electricity. I just want to take advantage of the fast moving air it would create.

>>11351785
>Are you familiar with solar updraft towers?
A SUT is just a cooling tower with turbines at the base. I know a thing or two about evaporative cooling.

>>11351794

Okay, well don't care about the temperature of the air, so much as i care about the speed of the air. I plan to heat the water by letting the sun hit metal plates which attach to poles that go down into the water, but the goal isn't to get the water to 212, it's to get it to some parboil temperature. And we make up the rest of the temperature with the high speed air lowering the pressure.

It wouldn't be necessary to forcibly cool the water-laden air. If you just stick a net at the end of tower and blow past it, some of the water will cool onto the net and then we can collect it. A lot of the water would just fly past. But that's fine.

The draw of this system is that it requires no power, it would just lens the same system that generates fresh water normally - the sun and clouds.

>>11351771
>The circuit on the bottom is definitely short-er than the one on the top, but a short circuit it is not. Did I answer your question?
We were told that we were shorting the original circuit, on the breadboard

>>11351790
You certainly can use that fast moving air to evaporate a pool that is on the outside of the tower, and it would certainly evaporate faster than normal. But I'm not sure how this helps you. Once the water is evaporated, you are left with a pile of salt on one hand and a bunch of wet air on the air. The vapor in the air-vapor mixture then needs to be condensed to get liquid water. I already explained how to do this, but I wasn't clear on why this is a problem.
It is a fact that you will need more power to cool down this water than the turbines can produce. The reason is the second law of thermodynamics, specifically the Kelvin-Planck statement of this law:
>It is impossible to devise a cyclically operating device, the sole effect of which is to absorb energy in the form of heat from a single thermal reservoir and to deliver an equivalent amount of work.
Your system is the tower, the dry air that enters it, the salty water that enters it, the distilled liquid water that is returned, and the dry air that remains after condensation. However, the only energy source is the sun. You have no heat sink for this giant heat engine thing.

>>11351831

I don't know why you think we need to forcibly cool the air. I mean, you could do that to get *more* water out of the air. But you don't *have* to do it, and i'm not suggesting we do it.

>>11351815
You are shorting R4, not the circuit. This is a pretty big distinction.
>>11351803
It's pretty important that the air be hot and with low specific humidity if you want water to evaporate into it.
>>11351837
>>11351803
>net
Water vapor doesn't magically stick to nets. Water vapor is NOT little droplets of water floating in air, it is literally a gas. The way to get this gas to condense is to cool it. If you plan on letting it cool passively, then basically you just have a big solar-powered cloud-making machine. It's kinda cool, but it doesn't do what you think it does.

What do i have to study to work in sony's developers team?? Serious question, i'm already looking for a japanese course at the embassy.

>>11351853
>You are shorting R4, not the circuit. This is a pretty big distinction.
Sorry, you are right, I should have been more exact with my language.

Okay, so ODEs are equations where the terms are derivatives, and the solution is an integral? PDEs are the same thing, but with multiple variables?

I'm having trouble differentiating between P/ODEs and functions. For example, a particular ODE might be df/dt = a + b + c (a,b,c being a derivative-term), how is that different than a single derivative with multiple terms f(x) = df/dt = a + b + c?

I'm a service-industry brainlet, but am committed to learning math to an undergrad level before I'm 50. I got 20 years left, so it should be possible. thanks for your help.

>>11351768
oh i forgot, but suspected that. thanks anyways

>>11351937
Function f is a function of one variable, so f=f(x)
Function g is a function of many variables, so g=g(x,y,z)
An ODE can only contain f(x), x, and any possible derivative of f w.r.t. x.
A PDE can possibly contain g(x,y,z,...), x, y, z, ..., and the partial derivative of g w.r.t. any of its arguments. It's pretty simple.

Quick
statistics or pure math?
statistics: guaranteed internship and well paying job
pure math: more interesting
honorary choice: EE, unfortunately engineering doesn't attract me because it's way too applied for my taste.

I asked a stupid question a few threads ago about a patrickjmt equivalent for physics. I think I see how it's a dummy question. Is it that the solutions to most mechanics questions are just plug and chug into the correct formula? I don't need a detailed video showing me how to go basic algebra.
Additional resources for university mechanics 1 would still be appreciated though.

>>11351937
>f(x) = df/dt
what?
nigger just ignore the retarded names. ODE is a relation of differentials of a one vriable function and the variable itself.
the soulutions are the posible formulas of the function

>>11351025
You can just consider crossed products [math]V\otimes_\mathbb{C} V^*\subset \bigoplus_n V^{\otimes n}[/math] where [math]\mathbb{C}[/math] acts by skew-conjugation. This gets you a sesquiinear product which you can just evaluate into [math]\mathbb{C}[/math] and then quotient out the entire thing [math]\bigoplus_n V^{\otimes n}[/math] by the ideal generated by its preimage (i.e. elements in [math]V\otimes_\mathbb{C} V^*[/math] with well-defined inner products). As long as this product structure inherits Cauchy-Schwarz and polarization identity then it will automatically induce a norm and orthogonal projectors.
>Does this have a name
Don't know but just know that you're working [math]a~priori[/math] with [math]V[/math] this algebra coincides with its own *-irrep in [math]V[/math] (which induces a Hilbert structure), and no representation of it will be faithful on any proper subspace.
>algebra homomorphism
I was referring to the quptienting projection.

>>11351896
what are you asking then

>>11352207
>spent ten minutes doing calculations
>the norm induced by the adaptaded inner product on the algebra seems to actually coincide with the operator norm for an operator [math]|a \rangle \langle b| + | c \rangle \langle d |[/math]
Fucking nice.
Thanks.

>>11352322
Wait, nope.
My mistake.
Ah, well. No real loss.

>>11352148
hold on, I think I've got it: is a differential essentially the same thing as an infinitesimal? a derivative is the relationship between two infinitesimals aka differentials?

>>11352409
yes and yes. where and infinitesimal is the limit of the variation of a funtion in one point
>>11351768
isnt it other method?
for instance i know allways any linear combination of sin ans cos will give me a curve that looks exactly as a sin or cos function
ex:
sinx+cosx=sqrt(2)sin(x+0,7854)
i could not figure out what number is 0,7854 but i am pretty sure there must be a way.
same because nce i relized cos^2x was similar to cos x and i wanst wrong... (later learned the identity)

>>11352409
>I think I've got it: is a differential essentially the same thing as an infinitesimal
Crudely speaking, yes. Don't bother trying to understand differential equations unless you first know calculus.

>>11352419
>i could not figure out what number is 0,7854 but i am pretty sure there must be a way.
yes. it is pi/4

>>11351768
and 6sinx+8cosx=10sin(x+0,9273)
and 6sin(2x)+8cos(2x)=10sin(2x+0,46365)
etc etc

>>11351643
[math]\sqrt{6^2+8^2}=10[/math]
[math]\cos \phi = \frac{6}{10}, \,\sin \phi = \frac{8}{10}\rightarrow \phi=.927\,\text{(radians)}[/math]
[math]6\sin (2x)+8\cos(2x)=10\sin(2x+.927)[/math]

>>11352473
thank you. i fountd it in my folder lmao i thought i was gonna discover something bout relized this.
of course w1=w2 or nothing of this stands

>>11352420
>>11352422
no it isn't you fucking retards

Say you had a magical chamber where particles inside always acted like waves and never had to "decide" on a single position or state. What happens if you put a cat in there for 1 minute, then deactivate the chamber?

>>11352499
You can treat first order, ordinary differentials as small, infinitesimal quantities without any negative consequence. You are trying too hard to sound smart.

>>11352521
The cat would die.

>>11352521
Let [math]H = H_E \otimes I + I \otimes H_B + \epsilon H_\text{int}[/math] be the Hamiltonian describing an interaction with range [math]R<\infty[/math] between the quantum system [math]B[/math] and the environment [math]E[/math] where [math]N = \operatorname{dim}E \gg \operatorname{dim}B[/math]. Suppose [math]\rho = \rho_E\otimes \rho_B[/math] is the initial density matrix of the coupled system with [math]\rho_E[/math] pure, then the thermalization hypothesis implies that there exists [math]T\in\mathbb{R}_{>0}[/math] and a pure state [math]\rho_0[/math] on [math]B[/math] such that the Heisenberg dynamics [math]\tau_\ast^H[/math] drives [eqn]||\operatorname{tr}_E(\tau_T^H(\rho)) - \rho_0|| \leq \epsilon O(N^{-\infty}).[/eqn]
What this means is that, given a macroscopic classical environment [math]E[/math], its massive number of degrees of freedom will eventually "drown out" the quantum properties of states on [math]B[/math]. In other words, putting the cat in a box will decohere all the quantum particles in the box. After 1 min all particles will behave as classically as the cat initially did.
>deactivate the chamber
I don't know what that means.

What prevents you from building a quantum computer using waves of water rather than wave-like particles?

So I am kinda stuck with Apostol.
This is part B of excercise 10 4.7
Prove by induction that for all n>= 1 we have:
[math]\sum_{k=n+1}^{2n} \frac{1}{k} = \sum_{m=1}^{2n}\frac{(-1)^{m+1}}{m}[/math]
for Induction lets assume the equality holds when n=x, so let's see if it also holds for n=x+1. Now I have searched for a solution in the internet (Pic related), now I understand the process, and the reason why I couldn't get to the answer is because I was adding a extra term to the equation, but I don't understand why I'm wrong. the summation on the left, since it's n = j+1, shouldn't the bottom side be k = j + 2, due to this in the equation I added the term [math]-/frac{1}{x+1}[/math] but this ruins the equality.
I have checked and the equality holds when you don't change the bottom part as they do in the image (it doesn't when you change it), so does this mean that the equality is untrue? or the bottom part is supposed to remain unchanged?

Is there a way to know whether a molecule's bond is polar, nonpolar, or ionic without memorizing every atom's electronegativity value?

>>11349899
Nah.
>>11350731
Why don't you ask them?
>>11351299
http://docs.mathjax.org/en/latest/input/tex/eqnumbers.html
>>11351417
That's the standard intuition, yes.
>>11352117
Applied math.
>>11352133
Even if it was tallied as a stupid question, it's not necessarily stupid, and we don't necessarily believe it to be stupid.
"Stupid questions" is basically "the remainder" or "others".
>then why do you call them stupid questions
Because I felt like it, and judyposter went with it.

Probably a pretty stupid one but bear with me: With any given formula, is it possible to add another operation that reduces the rate of change as it closes a given number, never reaching it?
For example, f(x)=x^2 would make it grow infinitely, but what can I do if I want it to never reach 10?

How can I get better at understanding geometric arguments in physics? I feel like the derivations both in class and in Purcell go right over my head. Is this something worth asking my professor about for some help on? I don't wanna waste his time either.

>>11352830
multiply by the logistics function and set the carrying capacity to 10

So let me get this straight: Making a group of matter "stable" involves making its entropy much higher than the entropy of all of its possible decay states at zero pressure, correct?

So where does the strange matter hypothesis come from? The hypothetical strange matter needs to somehow lose a bunch of potential energy without decaying in the process. If it's not shedding its potential energy through decay, what else can it supposedly use instead? Is there some gap in the understanding of the strong nuclear force where it's uncertain how much potential energy a baryon can shed in its own formation?

>>11352830
Not stupid. All you have to do is find a function whose output never exceeds a certain value (i.e., a global maximum). Then you compose your original function into that function, and you can scale it for the maximum that you want to have.
I'm using your example to make a function that modifies the original f=x^2 to never exceed 10.
I know at least one function that has a global maximum, 1-e^(-kx) for any particular positive number k. (I know this function as something called a cumulative distribution function, which has the property that its results always lie between 0 and 1). I'm gonna call this bigF(x).
If the results of bigF(x) are never greater than 1, so the results of bigF(f) = bigF(x^2) won't be either.
If you want something that never exceeds 10 instead of 1, we don't have to choose a different bigF. We can multiply whatever we got from bigF by 10 so instead of ranging from 0 to 1, it ranges from 0*10 to 1*10, or 0 to 10.
On the picture, you can see that f_original = x^1 doesn't have a global maximum. bigF(x), or 1-e^(-kx), does. (I chose k = 0.001; you can choose a different one if you want to change how fast your result starts nearing its maximum.)
Giving the results of f_original (which is x^2) to bigF as input, we get something that also never exceeds 1. Then, we can multiply that by 10 to get something that never exceeds 10.

>>11352859
You study geometry, obviously.
Euclid's Elements are absolutely excellent and the best recommendation I can give.
You can complement them with a more recent text, like "Affine Maps, Euclidean Motions and Quadrics", which is decent at best, but I'll mention nonetheless because I don't know anything better.

Read them while commuting or something.

>>11352859
Example of a derivation you didn't get?

>>11352743
Any bond between two nonmetals is going to be covalent. You can check for polarity when you draw the Lewis diagram and consider the bond angles. Any metal and nonmetal is ionic. Other than that, chem is largely memorization. I dunno ochem lol.
>>11352778
>we don't necessarily believe it to be stupid.
This is true. All of these questions are technical stupid considering the thread they are, but those listed explicitly as stupid are more miscellaneous. That or you are asking for book reccs or career advice.

how are you supposed to read this
what is the relationship between the part inside max and the part inside min, are they alternatives?
why does it say otherwise on the very right
and what does the if relate to
sorry i'm retarded

>>11352964
nvm figured it out

>>11352863
I didn't know about this function, it seems quite handy. Thanks!
>>11352901
Thanks for the step by step explanation. I really appreciate the extra mile, it tells me the right though process to follow in the future.

Is the uncertainty principle a confounding factor in the stability of atomic isotopes? Like, is a small nucleus more uncertain with its energy due to the position of the nucleons being more well defined, whereas the nucleons in a larger nucleus have less definite positions?

that Hidden Figures film/diversity propaganda was on tonight, the one about how three ladies of dark complexion are the sole drivers of the entire space project without whom nothing could have even made it into space

I remember /sci/ shredding it but what did the three actually do?

>>11352911
Thank you based Touhou anon. What translation of the Elements do you recommend?
>>11352921
I looked back at my own post and realized I worded it badly. I meant to say that a lot of the derivations I could not do on my own, yet when my professor covers them or I give in and look at the solutions for Purcell, it's clear as day. That being said, the electric field of a ring of charge, proving that the electric field within a hollow shell is 0, and picrel 1.69 all fucked me up.

>>11353065
Forgot picrel

>>11352892
>correct
No. Stability is determined by the local minima of the energy functional [math]S:\mathcal{V}\rightarrow\mathbb{R}[/math], where [math]\mathcal{V} = \mathcal{I}\times \mathcal{E}[/math] is the space of configurations. If we write [math]S_E:\mathcal{I}\rightarrow \mathbb{R}[/math] for each [math]E\in\mathcal{E}[/math], decays [math]A\rightarrow B[/math] between critical points [math]A,B\in\mathcal{I}[/math] of [math]S_E[/math], at a [math]fixed~E\in\mathcal{E}[/math], can be understood as the meta/instability of [math]A[/math] with a unstable manifold of dim [math]> 0[/math] into [math]B[/math]. Over a certain range [math]E\in D\subset\mathcal{E}[/math] this may be the case, until a phase transition occurs at [math]E_0 \in \partial D[/math] where the unstable manifold from [math]A[/math] into [math]B[/math] becomes stable. This means that across this phase transition, the configuration [math]A[/math] is more stable than [math]B[/math]. If [math]S_\mathcal{E}[/math] is Cerf then these phase transitions are characterized by the Floer cohomology groups [math]\hat{H}^\ast_E(\mathcal{I}),\check{ H}^\ast_E(\mathcal{I}),\overline{H}^\ast_E(\mathcal{I})[/math] as [math]E\in\mathcal{E}[/math] varies.
Strange matter is entirely reasonable, as the formation of stars (exotic or otherwise) constitute such a phase transition characterized by these cohomology groups, hence there is no reason to expect top/down quarks to still be the lowest energy. It is on you to prove that, in the case of SM where [math]\mathcal{I}[/math] is an infinite dimensional [math]SU(3)\times SU(2)\times U(1)[/math]-equivariant jet bundle, that these Floer groups are insensitive to these star-formation phase transitions.

>>11353065
Fitzpatrick.
There honestly aren't that many options in libgen, and I originally read it in my own language.
Also, a word of warning:
Do not algebraize the explanations or results. If you do that, you're basically just reviewing Euclidean geometry from school.
For example, proposition 35 says >Parallelograms which are on the same base and between the same parallels are equal
to one another.
Where equal stands for having the same measure. You emphatically shouldn't degeometrize this kind of thing by just associating it to "the area of a parallelogram is the base times the height."

>>11353060
Never mind, I checked Warosu
>>/sci/thread/S8531771#p8531853
So they were just backroom number crunchers I guess being a calculator was pretty respectable work back in the day, especially for women.
pic related is the real Katherine Johnson

Reminds me a little of the women that contributed to the Bruteforce modeling the positional variation of the poles
24:15
https://www.youtube.com/watch?v=Yze1YAz_LYM

>>11353095
>having the same measure
Bad habit from analysis, I meant the same area.

Is this really the difference between matrix and rank 2 contravariant tensor? Is it the general notion of this kind of tensor analogue via isomorphism or does it just works for the cartesian tensor?

Let [math]\mathbb{K}^n[/math] the finite dimensional coordinate vector space and [math]\mathbf{\beta}= \{ v_{i} \}[/math] a basis for this space.

If [math]\mathbf{\beta}'= \{ v_{i}' \}[/math] is new basis for which [math]x \in V[/math], will transform to the new basis as [math]x'=Ax[/math].

[math]\mathcal{T}[/math] is a tensor if the matrix of its componets [math]\mathbf{T}[/math] transforms with basis change like [math]\mathbf{T}'= \mathbf{A} \mathbf{T} \mathbf{A^{T}}[/math]

While the matrix [math]\mathbf{L}[/math] of a linear operator [math]\mathcal{L}[/math] tranforms as: [math]\mathbf{L'}= \mathbf{A} \mathbf{L} \mathbf{A^{-1}}[/math].

Is just that?

Why have we evolved to have poopie on our butts?

Is the many-worlds interpretation intrinsically untestable, or does it have variants that are theoretically testable?

Can somebody explain using DNA as storage to me?
Does that mean that I can have the entire SNES library on a patch of my skin or something?

>>11353152
First of all, in a framed vector space a basis change is orthogonal, so [math]A^T = A^{-1}[/math].
A rank-[math](p,q)[/math] tensor is an element of [math]V^p \otimes W^q[/math], while a linear map is an element of [math]\operatorname{Hom}(V,W)[/math]. Suppose [math]W=V[/math], in order to make the identification we need a few things:
1. [math]\operatorname{Hom}(V,V) \cong \operatorname{Hom}({\bf 1},V^*\otimes V)\cong V^*\otimes V[/math], which identifies a linear map [math]T:V\rightarrow V[/math] as a "[math]V[/math]-valued" vector in [math]V[/math], and
2. [math]V^* \cong V[/math] whence [math]V^* \otimes V \cong V \otimes V[/math], so the linear map [math]T[/math] is a bona fide rank-[math](1,1)[/math] tensor, and
3. an orientation on [math]V[/math] so that we can perform "musical isomorphism" [math]V\otimes V \xrightarrow{\sim} V^2[/math] by [math]T_j^i \mapsto T_{ij}[/math], so [math]T[/math] can also be identified with a rank-[math](0,2)[/math] tensor.
In finite dimensional vector spaces all of these are true, but they may fail when [math]V[/math] is a non-commutative ring module or when it's infinite dimensional. In particular Riesz [math]V\cong V^*[/math] need not hold on general Banach spaces.

>>11353152
A rank 2 tensor is a matrix generated by two vectors, one describing a location in space and the other describing orientation.
Literally just a square of numbers with a slick transformation rule lol

>>11352499
whisky guy again, please help me understand why I can't just treat differentials as infinitesimals. answering with a link or two is fine, I don't mind reading for myself.

>>11348081
Just wait.
Everyone dies eventually.

I'm experimenting with plotting lines on log-linear charts for trading-related reasons. I found a formula for doing exactly this at: https://en.wikipedia.org/wiki/Semi-log_plot#log-linear_plot

It works perfectly, but I'm baffled by how they generalized the function from [math]F_1[/math] to [math]F(x)[/math]. If anyone could shed some light on this, it would satisfy my curiosity.

Thanks in advance.

Assume you have two have two balls of different materials on vacuum, and the two balls eventually collide inelastically.
Is it correct to assume that the ball made of the less dense material will be the one to break (assuming the system has enough energy for it)

>>11350482
Thanks for your comment. The math is on a high level than I understand, but I can look up the terms and you clearly relate them with each other in any case. I was hoping for some simpler (or at least alternative) way to relate [math]|m\rangle[/math] and [math]w(j)[/math] other than through "Fourier transform [math]|m\rangle[/math] into [math]\k\rangle[/math] which is part of the Bloch wavefunction [math]|\Psi\rangle = |u(k)\rangle \otimes |k\rangle[/math] and then Fourier transform this whole thing and you get [math]|w(j)[/math]". Intuition tells me these kinds of schemes can be completed like a commutative diagram. But maybe that's just not the case, however I fail to see the elegance.

But the point is that,
If I may ask, how is this related to Wannier states?

>>11353950
Pardon the errors
>But the point is that,
I wanted to greentext this

>>11353152
matrix is literally just a bunch of numbers.
2-rank tensor is something which is represented by a matrix once a basis has been chosen. transformation laws is what determines the type (contravariant, covariant or (1,1)).
you have theorems on how to rewrite a matrix of linear transformation or bilinear form when a basis is changed, but tensors already have these transformation laws built-in. nothing more, but also nothing less.
if you want, rank 2 tensors are "coordinate-free" definitions of matrices

>>11353222
No interpretation of QM is testable. Most physicist don't even agree on which is best, in fact, most agree that there is no good interpretation.

So dream interpretation is pure pseudoscience right?

>>11354179
If it claims to be a science, then yes. I try to interpret my dreams regularly but it's in no way pseudoscience lol. Of what use is science when you're dealing with subjective phenomena that literally no one else experiences or can experience?

>>11354193
I do it too but I mean stuff like how they say nightmares of being stabbed = feelings of being wounded by comments or actions IRL or peoples faces melting during a lucid dream = feeling inadequacy IRL

>>11353529

Woah, i love you. Thank you <3
But can you tell me something about the difference between [math]T_{i}^{j}[/math], [math]T_{ij}[/math] and [math]T^{ij}[/math], is there something else i should know about beside they being different simply by the way they transform under basis change?

>>11353152
>>11353607
>>11353956
Thanks.

>>11353898
You can treat them as infinitesimals, if you are careful. *In real life*, so long as you are dealing with first order, ordinary differentials, you can treat them as very small quantities awaiting a limiting process to zero and it doesn't matter. This is justified by how the first, total derivative is defined. Suppose you have a function [math]
x:t\to x(t) [/math]. Since we are in real life, you can safely assume that this function is infinitely differentiable and that it is continuous and all of its derivatives are continuous (this is where mathniggers want to lynch you). By definition
[eqn] \frac{\text{d} x}{\text{d} t}\equiv\lim_{\Delta t\to0}\frac{x(t+\Delta t)-x}{\Delta t}=\lim_{\Delta t\to0}\frac{\Delta x}{\Delta t} [/eqn]
You can see that the derivative of x with resp. to t is dx/dt, which is defined as the limit of the ratio of two quantities. It is a property of limits that the limit of a ratio is the ratio of the limits and so it is totally legal to maneuver [math] \text{d}x\approx\Delta x [/math] and [math] \text{d}t\approx\Delta t [/math] around like algebraic quantities, cancelling and multiplying them and whatnot, so long as you keep in the back of your mind that this is all just shorthand. It makes math majors seethe, do it often. However, shit gets a lot more complicated when dealing with second derivatives or higher or partial derivatives, and the same logic definitely doesn't apply.
>>11353937
No, it isn't. Things like ductility, toughness, and hardness are also there to consider. What would happen if a ball of glass collided with a ball of lead?

>>11353898
You can treat them as infinitesimals, if you are careful. *In real life*, so long as you are dealing with first order, ordinary differentials, you can treat them as very small quantities awaiting a limiting process to zero and it doesn't matter. This is justified by how the first, total derivative is defined. Suppose you have a function [math]
x:t\to x(t) [/math]. Since we are in real life, you can safely assume that this function is infinitely differentiable and that it is continuous and all of its derivatives are continuous (this is where mathniggers want to lynch you). By definition
[eqn] \frac{\text{d} x}{\text{d} t}\equiv\lim_{\Delta t\to0}\frac{x(t+\Delta t)-x}{\Delta t}=\lim_{\Delta t\to0}\frac{\Delta x}{\Delta t} [/eqn]
You can see that the derivative of x with resp. to t is dx/dt, which is defined as the limit of the ratio of two quantities. It is a property of limits that the limit of a ratio is the ratio of the limits and so it is totally legal to maneuver [math] \text{d}x\approx\Delta x [/math] and [math] \text{d}t\approx\Delta t [/math] around like algebraic quantities, cancelling and multiplying them and whatnot, so long as you keep in the back of your mind that this is all just shorthand. It makes math majors seethe, do it often. However, shit gets a lot more complicated when dealing with second derivatives or higher or partial derivatives, and the same logic definitely doesn't apply.
>>11353937
No, it isn't. Ductility, toughness, and hardness are also properties to consider. What would happen if a ball of glass collided with a ball of lead?

>>11354291
not yukari.
technically a tensor of type (p,q) is a map which takes q vectors and p covectors and gives a number. this can be interpreted as a map which takes q vectors and gives back a single p-vector (whatever that is). in particular, a (1,1) tensor is a linear transformation and (0,2) is a bilinear form.
if you look at tensors only as matrices containing some information, then transformation laws are really all there is to it. if you want tensors to actually do something (e.g. evaluate a bilinear form on a pair of vectors, or apply a linear transformation to a vector), then tensors of different types are fundamentally different.

>>11353934
To generalize they simply replaced x1 with x.
But at the same time they also replaced m with the formula they gave for m at the start, which is probably what caused your confusion.

One of the tricks I learnt during undergrad while learning differential calculus basics was this one, generally used to evaluate summations.
If [math] S = x_{1} x{2} x{3} … x_{n} [/math]
Take natural log to both sides (assuming all domain conditions are satisfied) and differentiate both sides
[math] \frac{S'}{S} = \frac{x_1'}{x_1} + \frac{x_2'}{x_2} + \frac{x_3'}{x_3} … \frac{x_n'}{x_n} [/math]
Generally used if you can spot a series with the derivative of terms in the denominator in the numerator.
[math] Eg: used to simplify \sum_{r=1}^{n} r tanr by writing terms as \frac{rsinrx}{cosrx} [/math]

I am able to solve the questions on this trick but I never understood WHY exactly this works.
Is there some way we can relate the standard examples shown to demonstrate the sums on this trick to the fundamental definition of the derivative.
Also theoretically, is there some way to solve questions which would otherwise be solved using differentiation without using this trick? I just feel that we shouldn't need this trick theoretically and there has to be some kind of an alternate way for each question. In some of the examples there is no relation to anything like differentiation.
Sorry if this is a dumb question, I may be a brainlet.

One of the tricks I learnt during undergrad while learning differential calculus basics was this one, generally used to evaluate summations.
If [math]S=x_1 x_2 x_3…x_n[/math]
Take natural log to both sides (assuming all domain conditions are satisfied) and differentiate both sides
[math]\frac{S}{S}=\frac{x_1'}{x_1}+\frac{x_2'}{x_2}+\frac{x_3'}{x_3}…+\frac{x_n'}{x_n}[/math]
Generally used if you can spot a series with the derivative of terms in the denominator in the numerator.
Eg:used to simplify [math]\sum_{r=1}^{n}rtanr [/math] by writing terms as [math] \frac{rsinrx}{cosrx}[/math]

I am able to solve the questions on this trick but I never understood WHY exactly this works.
Is there some way we can relate the standard examples shown to demonstrate the sums on this trick to the fundamental definition of the derivative.
Also theoretically, is there some way to solve questions which would otherwise be solved using differentiation without using this trick? I just feel that we shouldn't need this trick theoretically and there has to be some kind of an alternate way for each question. In some of the examples there is no relation to anything like differentiation.
Sorry if this is a dumb question, I may be a brainlet.

>>11354310
>ductility, toughness, and hardness are also properties to consider
Thanks, friend

why does (-x)^2 generate a positive answer and -x^2 a negative one?

>>11354487
Theorem One: Let [math]S = \Pi x_i[/math]. Then [math]\log S = \log \Pi x_i = \Sigma \log x_ i[/math].
Proof: basic properties of the logarithm.
Theorem two: [math]D \log [f(x)] = \frac{f'(x)}{f(x)}[/math].
Proof: basic application of the chain rule.
>>11354569
The second one is read as [math]-(x^2)[/math] because of PEMDAS.

>>11354569
(-x)^2=(-x)(-x)=(-1)x(-1)x=(-1*-1)(x*x)=x^2
-x^2=(-1)x^1
>>11354565
yw friend

>>11354572
>>11354574
thanks guys

>>11354349
Thank you. I have been reading A Gentle Introduction To Tensors by Boaz Porat and i find it very good. This was the explanation i was looking for, but without reading this notes and other articles and posts all around the web no one can understand what is being said here. This is to all curious anons who stumbled upon the replies to my post: If you really know linear algebra go read the notes by Porat. But if you don't, go back to proof books and don't ask what a tensor is because you will not understand. You've been warned.

>>11354310
>It is a property of limits that the limit of a ratio is the ratio of the limits and so it is totally legal to maneuver dx≈Δxdx≈Δx and dt≈Δtdt≈Δt around like algebraic quantities, cancelling and multiplying them and whatnot, so long as you keep in the back of your mind that this is all just shorthand.
the conclusion is correct, but it doesn't follow from the premise ("the limit of a ratio is the ratio of the limits") at all. you shouldn't have written that.

Hey bros, retard here. My question pertains to mathematics in general. "When you come to a question that immediately stumps you, what do you do?"
When I'm hit with these integration problems, I'm just 'stuck' and immediately go to slader or some solution site, I know this is a HORRIBLE habit and I'd like other anons input on how they approach problems when they're at a loss of what to do.

>>11354383
Thanks. If they just left m there, I would have been way less confused.

how to go from line 2 to line 3?

>>11353950
Wannier states are by definition the filled states in real space, and they form irreps of the symmetries on each fundamental domain [math]\Gamma[/math]. They in general define a Hilbert bundle [math]\simeq \mathcal{F}^{-1}P_\text{occ}\mathcal{H}[/math] in real space, the phase of which as you move about [math]\Gamma[/math], or equivalently the Berry phase of the Bloch states within the first Brillouin zone, will determine exactly the Chern number in your TI/TSC. Specifically, it is [math]c_1 = -\operatorname{arg}\operatorname{det}P_\text{occ} \mod 2\pi \in\mathbb{Z}[/math] where [math]P_\text{occ}[/math] is the Fredholm projector on [math]\mathcal{H}[/math] onto the occupied states.
The unfilled states will also form irreps, but as I've said before these irreps will in fact trivialize the topology of the system. This is why we don't call them Wannier states. Since, as you know, the topology determine the presence of gapless edge states, we essentially only want to count the chirality on the occupied bands, otherwise [math]c_1[/math] will always be zero since gapless states always connect occupied and unoccupied bands, acquiring chiralities [math]\pm 1[/math] resp. for each gapless state, whence they cancel.

>>11354310
>>11352148

Okay, I think I've got it now, someone just check me on this:

I understood the gist of PDEs, because you've got a relationship of derivatives of multiple variables and you look for antiderivative(?) solutions with respect to each variable. When it came to ODEs though, I kept seeing simple examples like f'(x) = x^2, for which the solution is (x^3)/3. When I saw these examples I wondered what the difference was from a function, the term ODE and process of finding a solution is redundant to getting the antiderivative of a derivative.

But that's an absolutely trivial ODE that exists only for learning purposes. A more common/practical situation would be one where I have two functions f(x) and g(x), for which I have not observed any relationship, but I do have the relationship of their derivatives, say f'(x) = 5 sin g'(x). This is an ODE from which I would search for the equation (antiderivative?) describing a function z(x) (which may be one of many or may not exist in entirety) which is the relation of f(x) and g(x).

Am I hitting the mark here?

I know It probably seems like I'm overshooting my level of understanding, but I tend to digest subjects best from an overview of everything in a book at once to see how stuff links up from above before grinding on exercises to get an actual working knowledge of how things behave.

>>11354743
I reread the previous chapter and look through the examples for tricks.
>>11354829
[math](\frac{1+ \sqrt{5} }{2})^2 = \frac{1}{4} + \frac{5}{4} + \frac{ \sqrt{5}}{2} = 1 + \frac{1+\sqrt{5}}{2}[/math]
Similarly for the other one.

Does A ∪ (B ∩ C) = (A ∩ B) ∪ (A ∩ C)? Wolfram just says it might or might not.

>>11354861
Let [math]x \in A[/math] but [math]x \notin B, ~ C[/math].
Then x is clearly in the left side, but it's not in the right side.

>>11354879
Oh wow that makes so much more sense. We've been told to draw venn diagrams for our set theory but I'm not very visual.

>>11354846
ODEs look more like [math]f'(x) = f(x)[/math]. try to find all functions which satisfy the relation.

I'm trying to write an equation that relates concentrations of one neurotransmitter to GABA's concentrations, ignoring the rest. I have some graphs gathered showing mM of each neurotransmitter over time (of a neuron firing, shows lifetime of neurotransmitter in extracellular space before re-uptaken by a neuron), and am assuming that since a neurotransmitter transporter functions to uptake by incidentally binding to Cl- and funneling both the Cl- ion and the neurotransmitter into the neuron, and since GABA transporters also funnel Cl- ions into a neuron to make an action potential harder to fire making any uptake of GABA hamper release of dopamine, I can write something...what do I try?

how can I understand why the Euclidean algorithm is true intuitively, preferably without any equations?
the wikipedia article has graphs about it which I don't really understand, I understand the procedure but I don't understand why it's true
I don't understand why gcd(a,b) = gcd(b, a mod b), pls no algebraic proof

>>11354894
Okay well that seems like it puts me back to square one. How am I not just finding a general antiderivative/primitive?

>>11355022
Because you could have something like [math] y''+(y')^2+y=\cos x [/math] which definitely cannot be solved by simply integrating

What keeps you going to that next question?

How to enter TeX mode on /sci/?

Name all words in
>pic related

>>11354969
>gcd(a,b) = gcd(b, a mod b)
I'll give an informal proof.
We have that [math]d|a[/math] and [math]d|b[/math]. Because of this, we know that [math]d|a-b[/math]. To prove it, we just use [math]a=dr_a[/math] and [math]b=dr_b[/math] for some [math]r_a, ~ r_b[/math], and then we have [math]a-b = d(r_a - r_b)[/math].
So for whatever [math]a[/math] we start with, we can just keep subtracting [math]b[/math] from it and that won't change [math]a[/math] and [math]b[/math]'s common divisors.
So we could, for example, keep subtracting [math]b[/math] until [math]a[/math] is as small as possible, which gives us gcd(a,b) = gcd(b, a mod b).
>>11355182
See https://i.imgur.com/vPAp2YD.png
Also, see the TeX button on the posting box.

>>11355203
any know a website that will help me understand how to solve that?
my answer is {(0,0,0) (1,0,0) (2,0,0) (0,0,1) (1,1,1) (2,1,0) (0,0,2) (1,2,1) (2,2,2)}

>>11355209
Thanks. I was particularly looking for the keyword.
It's "math", thanks! [math]\checkmark[/math]
That's mentioned nowhere on 4chan.

Is there a method of analyzing geometry in the way i'm about to describe?

say you reduce a circle to 100 nodules, or even pixels. Many of these square pixels will vary in how many neighboring pixels they have. so say the average each has is like, 3.3/4. Is there a way to calculate this? and for various shapes? Say, various flat shapes which are also composed of 100 pixels.

I was thinking of some mesh like analysis on some CADs or something, there's gotta be a better way though. I would like to quantify the relationship between wall contact and shape for a project, I'm not sure where to start though with this

>>11355212
3*3*3=27
Aren't you missing at least 18 words there?
Assuming [math]\mathbb{F}^3_3 = {\mathbb{Z}_3}^3[/math] of course, but your answer looks like you are trying to write that (and failing at that).

>>11354835
Your knowledge is unparallelled, Anon. I'll look further into it, even without my idea it there is plenty of theory behind this that I don't know of.

>>11355263
Define the pixel touching rate per derivative. You have to do this any way as the number is algorithm and coordinate system dependent. For instance a horizontal lines has value 1. y=x has and a vertical line one. To do this for the coordinate system you describe you paramtrize your line in. The pixel overlap=f(dx/ds) -> Average overlape=integral(f(dx/ds)ds)/
Please cite this post in your work/homework assignment

How do I stop being dumb?

>>11355314
You flatter me sweetie, I just happen to also be doing a PhD in TQFT/topological ordering. For the general theory behind the classification problem, I personally recommend Bernevig's text on TI/TSCs and his set of notes https://arxiv.org/abs/1506.05805..

>be me
>freshman
>retard
>don't get fucking math (calculus (limit, integral), matrices) at all
>can't read
What do?
I need a cheatsheet, that has minimal amount of words (they distract and annoy me, since I can't read).
Basically solution of your typical exercises.
Do you have such thing?

>>11352778
>>11349899
what else is there then?

I need to make a solution of a certain chemical in water.
Will the same amount of substance be distributed evenly in both 10 ml and 100 ml of water?

>>11355058
realizing im smarter than people i hate who will butcher the good things in life if i don’t dominate them preemptively

>>11355285
idk what i;m missing, that's why i;m asking for a website that will teach me to do these things

>>11356130
I doubt there is a better way to teach that than to say:
"Simply count in Base 3:
000, 001, 002, 010, 011, 012, 020, 021, 022, 100, ..."

>>11350711
sorry about the delay. What does k represent here? just any number greater than N?

I hate to ask, but I've got 2 bases of A,B and a function f A->B
What is meant with
B[f]A?

>>11350711
sorry im a little confused here. so k is just any integer 1 or greater?

assuming yes, this is what I can say about the comparison

[math]|X_{N+k}-X|\geq|X_{n+1}-X|>|X_{N}-X|>R|X_{n}-X|[/math]

I'm not really sure where to go from here.

>>11350711
sorry for the delay. im a bit confused. what is k? just any integer 1 or greater?

[math]|X_{n+1}-X|<R|X_{n}-X|<|X_{N}-X|[/math]

not really sure where to go from here

also not sure what to do about the [math]|X_{N+k}-X|[/math]

Scenario:
>spiral corridor
>light source arbitrarily placed in said corridor
>walls allow for partial dispersion of light
How can I determine the intensity of light anywhere within the spiral? What do I need to study to figure this out?

Why were early scientists often priests?

>>11356581
Never mind I got it. Apparently it was just a way to write the transformation matrix

>>11356744
>What do I need to study to figure this out?
Physics would be my best guess.
What kind of light source are you talking about? That seems like a critical point to consider.

I have a drawing tablet and I'm tired of using so much paper working practice problems.
I've tried using Paint, Gimp, and Krita for digital "whiteboard" software but they're all photo editing/drawing programs, not optimized for writing/working math problems.

for those of you that do the same, what software do you use?

>>11354969
An ideal of two numbers a,b, denoted by (a,b), is just the set of all possible numbers you can get by adding a and b in different ways. For example a+b, 4a+16b or -23a+104b. As it turns out, in integers, every such ideal is actually generated by one element, which is called the greatest common divisor of a and b. gcd(a,b) = d such that (d)=(a,b). But the ideal generated by (a,b) is the same as ideal generated by (a,b+ka) for any k, since b = (b+ka) - ka. So the euclidean algorithm works by repeatedly reducing one of the terms until it's smaller than the other, and then going the other way around.

>>11356747
science was the way to explore god's creation and learn its function to better divine his purpose. That, and experiments are fucking expensive. The church foot the bill for the priests.

Uh, really simple. I have a homogenically charged sphere, charge Q, radius R. What's the energy it takes to add the charge dq (same charge density ρ) to it? The description also said assume you're adding a spherical shell with dq and dr to the sphere.
Methinks you'd just use the regular old formula for potential energy outside the sphere from with the negative integral from ∞ to r and use dq for q, but that sounds a bit too simple and I assume I'm not supposed to have the dq in the solution.
What am I missing?

>>11357088
For [math] r>R [/math] we have [eqn] E(r)=\frac{Q}{4\pi\epsilon_0r^2}=\frac{\rho R^3}{3\epsilon_0r^2} [/eqn] Because [math] E(r)=-\frac{\partial V}{\partial r} [/math] (definition), we get [eqn] V=-\int_\infty^RE\text{ d}r=\frac{\rho R^2}{3\epsilon_0} [/eqn]
For a small charge [math] \text{d}q [/math] being added to the sphere, we have
[eqn]\text{d}U=V\text{ d}q=\frac{\rho R^2}{3\epsilon_0}\text{ d}q=\frac{Q}{4\pi\epsilon_0R}\text{ d}q[/eqn]
which is a formula you can find in many texts.
>>11356747
The same reason lots of 17th and 18th century mathematicians were aristocrats. Has nothing to do with god and all to do with class.

Can there be group isomorphisms between a field [eqn]\mathbb{F}_{p^n}[/eqn] and a Group G that is NOT a field?

More Particular between [eqn]\mathbb{F}_{p^n}[/eqn] and [eqn]\mathbb{Z} / n]\mathbb{Z}[/eqn] where n is NOT a prime.

>>11355781
Qualitative analysis, game theory, etc.
>>11357336
>a Group G that is NOT a field?
I was going to write the full construction of a field structure on G by pulling multiplication back and forward, but then I realized that doing that is much stupider than I first thought.
>where n is NOT a prime.
No, because characteristic.

Can anyone remind me why units/rates are described with an inverse power? I can't remember why the units of a rate for example is mol/m^-2*s^-1m or just m/s^-1, from my notes.

t. grad student after a looong hiatus.

>>11357628
You use inverse powers so all the units can be multiplied together without division, like m*s^-1 for speed or kg*m^2*s^-2 for energy

>>11355446
Thanks for the reference. I'm doing my master's thesis on topological insulators, but to my shame I lack a lot of mathematical insight. Category theory, fiber bundles, and even stuff like Pontryagin duality is relatively unknown to me.

Has anyone tested if potential energy has any effect on the outcome of measurements of quantum systems?

Using euler's, I get
120cos(wt+pi/3) + j*120sin(wt+pi/3)
So where's the answer?

>>11357882
If [math] \mathbf{V}_{rms}=A\exp({j\phi}) [/math], then [math] V(t)=A\sqrt{2}\cos(\omega t+\phi) [/math]. When moving from phasor domain to time domain, you're only going to care about the real part of the phasor. Electrical engineering is just incredibly gay like that. Also, remember that the root-mean-square of the amplitude V of a sinusoidal signal is V/sqrt2.
The correct answer is a).

>>11357643
Thank you!

How do I get a 14 inch dick?

If I have a device that randomly generates either a 1 or a 0 with a 50/50 chance, then I hook the device up to a rig that will drop a bowling ball whenever the device outputs 1, will the device's output stop being 50/50 and instead be weighted slightly more towards 1?

>>11358078
buy a strapon

>>11358116
no
>>11358078
be black~

In the last thread I asked if there was a transfinite hyper-real number w in *R that satisfies 10^(-w)=0 and someone said no, there's no REAL number. Is that supposed to mean "real" as in any transfinite extension to the reals, or did they just not know the difference between R and *R?

>>11357800
That is to be expected if it's a physics masters, since physicists don't really concern themselves with the mathematical formalisms. Some even consider the classification problem to be solved in the 80's. Bernevig himself is doing qauntum chemistry stuff now.
I'd say you don't need much of the math if you're mainly focused on the physics, but it'd be good to at least go through Altland-Zirnbauer and Kitaev's papers on the classification of non-interacting topological orders.
>>11357880
I don't know if there's a lower bound on how stupid a question can be on /sqt/ but energy quite literally determines the ground state, which is what almost all measurements [math]\opreatorname{tr}\rho A[/math] are made on. Of course changing the potential energy is going to change your measurements.

Is mathematics so logically pure that the conclusions we derive from its study must necessarily also come to the intelligent beings of any universe that could possibly exist? In other words if physics is the study of the most objective aspects our universe, is mathematics the study of what must necessarily be true in any state of existence, otherwise it would be too contradictory to exist at all?

How to get better at mentally aligning orthogonal planes/lines/vectors in 3-space? My ability to rotate single objects and to translate surfaces is getting better, but when I have a level surface or something like that and need to find tangent lines or normals or when I have planes I just cannot follow the curvature/geometry of the object I want to relate things with and it fucks everything up for me. The algebra is not a problem but I feel like I'm cheating myself by reducing everything to lin alg and calculus.

Say cigarette cancer deaths are X% per age group Y.
If I want to adjust for age in the population, I just divide the X by the ratio of the population in that age group?

Say 9% of cancer deaths is in the age group 60-65. Say that age group is 3% of the population. The age-adjusted risk in that age group is then 9/3=3x ?

>>11358834
>The age-adjusted risk in that age group
Is the number you started with. The 9%.

With multiplication you get the percentage of this age group dying due to smoking with respect to the entire population.
With division you get a ratio that has its uses, but no immediate meaning.

math/chemistry brainlet here. My question regards pedagogy more than a problem or equation. Why are educators surprised, or feign concern, that x number of people fail their classes, when in every single math/chem class students are exposed to practice problems that a child could do, but in homeworks and exams there are problems of varying degrees, often with steps never addressed in class.
I know time limits what one can cover in class, but you'd think people would manage their time wisely and try to at least incorporate a problem or two that's more difficult than the braindead shit they teach students and then expect them to understand or apply to more complicated shit.

>>11358626
>did he really not know the difference between R and *R
My bad, I assumed it was some sort of meme notation and you meant the reals.
>>11358960
If I gave you a list of sixth grade math/physics problems, could you honestly tell me which ones a sixth grader would find hard?

>>11359119
Let me rephrase: the example problems done in every math/chem class are too, too simple, and never involve the number of steps, nor do they match the complexity, of what is asked on homeworks or exams.

>>11359124
What do you mean simple and complicated?
Some exercises are short, and some exercises are long. Clearly, doing a bunch of short exercises is preferable.

Hi again. My study language isn't English and even then some of these things I only know by abbreviations since their full name wasn't on the slides, but:
On dielectric strength, is the max. field strength referring to what I assume is the "vacuum" field strength or the "net" (Enet = Evak/εr, in our notation) field strength? If I have some material with a dielectric strength of 69V/m is my base field 69V/m or is that the field reduced by the dielectric field?
Sorry for my bad expression, thanks for your help.

>>11358960
Dangerously redpilled.

Is it normal to be completely lost your first couple abstract algebra lectures? I feel like I missed something even though I took the suggested class path my school recommended. This is the first time I feel stumped even in lecture.

Okay, i'm working through a sequence problem sheet and i stumbled with this one

Let [math]a_{n}[/math] be a sequence of real numbers such that [math]a_{n} \in \mathbb{Z}, \forall n \in \mathbb{N}[/math]. Prove that if [math]lim_{n \to \infty} a_{n}=l[/math] then there exist [math]n_{0}\in\mathbb{N}[/math] such that [math]a_{n}=l,~ \forall n \geqslant n_{0}[/math]

Is this asking me to prove that after a certain point of the sequence, every member of the sequence is equal to its limit?
As if the sequence stopped at whatever [math]l[/math] and then just kept repeating the same number?
If so, i'm confused on how to approach this.
If i'm mistaken with the first assumption please enlighten me.

>>11359129
What do you misunderstand? I mean what I said. Regardless of problems differing in length, the simple reality of it is that to save time, professors/teachers feed students easy shit to practice, and yet they expect them to understand and apply concepts in ways they have never seen before, because again, they never bother to teach those things; you're better off skipping lectures, reading the book on your own, and going to the tutoring lab for any questions or to practice. There is always concern about students being shit at math and chem, and it seems that no educational body really wants to tackle the issue at hand: time. They think they can solve it by making tutoring sessions mandatory, instead of simply extending class time so educators can fucking teach problems of varying difficulties, so that students can grasp concepts better. It is astounding to me that in shit like math and science, where precision and step by step understanding is imperative not only to succeed, but also as a foundation of knowledge, is treated with such utter disregard. And they have the fucking gall to act surprised about this.

How does one increase geometric intuition and also specifically the ability to embed and rotate geometric objects mentally? I can essentially rotate and translate any simple polygon and place markers on the object, color faces, label vertices etc. but when I have to decompose the object and/or embed it within another object or space, analyze its surface, relate its angles with something else outside of just the representation in an ambient space I lose pretty much all visualizing ability. The struggle gets severe enough that I have to physically manipulate analogous objects irl just to correct my wrong mental image of the transformation.

>>11359365
Depends how badly you're lost. If you're completely floundering to the point where you can't fake your way through simple questions and are en-route to failing the class, that's not a normal hiccup that will resolve itself, and you should find some help.
If you're pushing through the course but don't feel you're grasping anything on a level higher than arbitrary symbol-shuffling, then this happens to most students and you're doing pretty well if you legitimately get past it by the end of your first semester.

>>11359398
If you are doing a limit of a series in the integers ([math]a_n \in \mathbb{Z}[/math]), then the only way for there to be a limit is exactly what you are asked to show.
In the reals you could do shit like [math]a_n = 1 - 0.1^n[/math]. Not possible here.

>>11359586
similar question to his: is there an underlying intuition to be gained from the algebraic manipulations and inequality juggling with intro analysis proofs or is the intuition nearly always latent within the theorems and definitions? I have never really gained insight from doing a proof involving limits or sequences, its always from a long period of mulling over a theorem I didn’t fully understand the implications of in my free time.

>>11359398
Think about what a limit point is. A limit point has to have infinitely many sequences contained in any of its neighborhoods. Since each integer is spaced at least by 1, if the sequence doesn't stabilize then you can certainly shrink your neighborhood such that it doesn't include any points in the sequence at all. This contradiction means that either you don't have a limit point at all or the sequence has to stabilize.

I'm studying calc and trying to understand limits. I don't understand their definition,
>Let f(x) be a function defined on an interval that contains x=a, except possibly at x=a.
How does an interval have x = a, but not at x = a?

How do I solve [math]\sin(y)\sqrt{1-\cos(x)}+\sin(x)\sqrt{1-\cos(y)}=0[/math] for all [math]0\leq x < y \leq 2\pi[/math]? I feel like I miss a trig identity.

>>11359743
[math] f(x)=\frac{1}{x} [/math] is defined at all points on the interval [math] (-1,1)
[/math] except at the point [math] x=0 [/math] which is contained within that interval.

>>11359752
[math]\sin(y)\sqrt{1-\cos(x)}+\sin(x)\sqrt{1-\cos(y)}=0 \\
\iff (sin(y)=0 \vee cos(x)=1) \wedge (sin(x)=0 \vee cos(y)=1)[/math]

>>11347903
I'm looking at a polyhedron that's probably unsuitable for making dice out of because it'll always land with the pole of 3 faces up instead of a single face (pentagonal Icositetrahedron). I tried looking up a simple mathematical term for fair dice > 4 sides that don't do this, and the only thing I could find was "isohedral zonohedron." This is retarded and there's another term for "polyhedron made of uniform polygons with parallel sides only," right?
I haven't been in a class in years, this is for my own interest not schoolwork. Pic related, the guy who designed it said they're unsuitable as dice.

I am trying to create a keras neural network to predict distance on roads between two points in city. I am using Google Maps to get travel distance and then train neural network to do that.
Neural network architecture:

from keras.optimizers import SGD
sgd = SGD(lr=0.00000001)
from keras.models import Sequential
from keras.layers import Dense, Activation
model = Sequential()
model.add(Dense(100, input_dim=4 , activation='relu'))
model.add(Dense(100, activation='relu'))
model.add(Dense(1,activation='sigmoid'))
model.compile(loss='mse', optimizer='sgd', metrics=['mse'])


Then i divide sets to test/train

Xtrain=train[['p1Lat','p1Lon','p2Lat','p2Lon']]/100
Ytrain=train[['distnaceInMeters']]/100000
Xtest=test[['p1Lat','p1Lon','p2Lat','p2Lon']]/100
Ytest=test[['distnaceInMeters']]/100000

Then i fit data into the model, but loss stays the same:

history = model.fit(Xtrain, Ytrain,
batch_size=1,
epochs=1000,
# We pass some validation for
# monitoring validation loss and metrics
# at the end of each epoch
validation_data=(Xtest, Ytest))

I later print the data:

prediction = model.predict(Xtest)
print(prediction)
print (Ytest)


But result is the same for all the inputs:


[[0.26150784]
[0.26171574]
[0.2617755 ]
[0.2615582 ]
[0.26173398]
[0.26166356]
[0.26185763]
[0.26188275]
[0.2614446 ]
[0.2616575 ]
[0.26175532]
[0.2615183 ]
[0.2618127 ]]
distnaceInMeters
2 0.13595
6 0.27998
7 0.48849
16 0.36553
21 0.37910
22 0.40176
33 0.09173
39 0.24542
53 0.04216
55 0.38212
62 0.39972
64 0.29153
87 0.08788

I can not find the problem. What is it? I am new to machine learning.

>>11359761
it's a matter of odd vs even number of sides, right? In theory you could create a prism where the an even sided prism uses the faces as their numbers and an odd sided prism uses the edges. I don't know the math behind "fair dice" but if you're literally just trying to make dice, there are other ways than tiling a sphere.

>>11359571
Play with platonic solids and puzzles as a child
Dabble in transcendental meditation, sensory deprivation, and other hippie crap as a teen
Try not to let the alienation and derealization get to you
Remember that schizophrenia often doesn't hit until your late 20's

>>11359759
I don't think this is true, x in [0,2*pi] and y= 2*pi - x seems to be a solution.

>>11359772
Not isohedral because the faces aren't uniform and a prism only imposes one pair of parallel faces. If the top and bottom were triangles they'd be the *only* parallel faces.
"Fair dice" are defined as polyhedra with an equal chance of landing on any given face (i.e. no polyhedra made up of differing polygon faces where smaller ones would be less likely to land). Fair numeric distribution is achieved by assigning each opposing face a pair of numbers that sum to the number of faces + 1.

>>11359772
Also, that die does have an even number of sides (Icositetrahedron = d24). When you make a deltoid icositetrahedron i.e. kite-shaped sides the resulting polyhedron has the desired properties (uniform faces only in parallel). Even icositetrahedra made from triangles do the same thing. Not surprisingly those have polygons where all edges are the same length unlike the pentagonal one above -- which is also chiral, a rarity for isohedra.

>>11359794
You are right, I'm a retard.
Didn't consider sin(x) can get negative.

>>11347903
>2 dimensions: tangent line to the curve
ok
>3 dimensions: tangent plane to the surface
ok
>n dimensions: tangent space to the manifold
i guess it checks out, BUT
>3 dimensions
consider a curve in R^3. tangent line is defined often in the direction of its normal vector. but could you define a tangent plane such that it contains the line in the direction of the normal vector and a vector perpendicular to the normal vecor? And then would it not make sense to think of all lines making up that tangent plane as tangent lines to the curve in R^3 at the given point a?

>>11360095
>tangent line is defined often in the direction of its normal vector
What? Tangent is the opposite of normal.

>>11360095
A curve in R^3 is a 1D manifold. Hence its tangent space is a 1D vector space.
You're confusing the embedding of an n-dim manifold into (n+k)-dim Euclidean space for the actual manifold itself. For instance, the curve and the line are 1D objects. The surface and plane are 2D.
Extrinsic topology/geometry is useful for visualization but gives misleading intuition, IMO.

>>11360095
>tangent line is defined often in the direction of its normal vector
It's not.
>but could you define a tangent plane such that it contains the line in the direction of the normal vector and a vector perpendicular to the normal vecor?
You can take a plane that contains the tangent vector and the binormal, yes.
>And then would it not make sense to think of all lines making up that tangent plane as tangent lines to the curve in R^3 at the given point a?
No.

>>11360118
My post was not correct in explaining what i meant but yours still makes sense, thanks

>>11359218
bls resbond :DD:dDDD:D

Has anyone tested if quantum particles "favor" outcomes that produce higher entropy?

is the hilbert transform just my original signal phase shifted +-90 degrees?

Is 23 too old to start a mathematics degree?

>>11360316
yes

>>11360316
no

>>11360323
>>11360326
samefag

Where the fuck are the janitors

>>11359218
>>11360178
Dielectric strength is the maximum electric field a material can be in before it breaks down. Not D=E/ε or Enet=E/εr, but E, or Evak as you call it.

>>11360349
What threads seem to be the problem? Are you using the report function?

>>11360364
Are you blind or the idiot making the retarded and schizo threads

take a look on the catalog

>>11360367
I don't appreciate the attitude, anon. I use filters, hide and report off topic threads. I recommend you do the same.

>>11359365
Yeah.
Did you have a good linear algebra course beforehand? Linear algebra has this thing where the abstract objects all have concrete realizations, so the student gradually learns to bridge both sides of the subject to gain a grasp of doing algebra.
>>11359398
Set [math]\delta = 0.4[/math].
>>11359763
>>>/g/sqt
>>11360367
The catalog always looks like that.
Are you new here?

Does this board have a Discord?

>>11360178
>>11359218
Dielectric strength is the maximum [math] \mathbf{E} [/math] a material can be subjected to. In an isotropic material the convention is that is measured against the true electric field [math] \mathbf{E} [/math], not the "net" electric field or electric displacement.

Does pair production happen without the involvement of W bosons? If so, is this the only identity change that doesn't involve W bosons?

>>11360420
Yeah, of course there is, but you'll need to pass a little test to get in.
Prove that the assignation [math]X \rightarrow ~ i_X \omega[/math] defines a surjective linear map [math]\mathcal{H} _{loc} (W) \rightarrow H^1 (W; \mathbb{R})[/math]. Deduce an exact sequence of vector spaces [eqn] 0 \rightarrow \mathcal{H} (W) \rightarrow \mathcal{H} _{loc} (W) \rightarrow H^1 (W; \mathbb{R} ) \rightarrow 0[/eqn]
Where [math](W, \omega)[/math] is a symplectic manifold, [math]\mathcal{H} (W)[/math] is the vector space of Hamiltonian vector fields, [math]\mathcal{H} _{loc} (W)[/math] are locally Hamiltonian vector fields, and the cohomology is de Rham.
Shouldn't take you more than a minute, it is basically just applying the definitions.

Can science study itself recursively?

Do massive objects such as black holes exhibit enough directionality, that something observing within the "aperture" of the object could use it as a telescope?

Do I have any hope of passing Calculus 3 if I struggle with Calculus 2?

>>11359398
the limit exists so there is an [math] n_0\in\mathbf{N} [/math] such that [math] |a_{n}-l|<1/2 [/math] whenever [math] n\geq n_0 [/math]. now [math] a_n-l\in\mathbf{Z} [/math] so...

>>11360821
It should be not-too-hard, provided you eventually do figure out calc 2. Calc 3 is just calc 2 with some linear algebra mixed in

>>11360543
For each [math]X\in \mathcal{H}(W)[/math], let [math]f_\alpha \in C^\infty(U_\alpha)[/math] be the associated Hamiltonian function on the local patch [math]U_\alpha \subset W[/math]. By definition we have [math](\iota_X\omega - df)_{U_\alpha} = 0[/math]; indeed, this is Poincare's lemma stating that, since the Lie derivative [math]L_X\omega = 0 = d\iota_X \omega + \iota_X d\omega = d\iota_X \omega[/math] and [math]\iota_X d\omega[/math] is closed, [math]\iota_X\omega \in C^1(W)[/math] is locally exact. Hence we know that [math]\operatorname{ker}(\mathcal{H}_\text{loc}(W)\rightarrow H^1(W)) \cong \mathcal{H}(W)[/math] is the space of [math]X[/math] with globally defined Hamiltonians; namely the [math]f_\alpha[/math]'s can be patched together.
What is the obstruction to this patching? It has to do with coordinate transitions [math]g_{\alpha\beta}[/math] on overlaps [math]U_\alpha\cap U_\beta[/math], or alternatively [math]g_{\alpha\beta} = f_\alpha^{-1} df_\beta[/math], which can be trivialized if the log function is well-defined [math]g=d\ln f[/math] for [math]f = f_\alpha|_{U_\alpha \cap U_\beta} = f_\beta|_{U_\alpha\cap U_\beta}[/math]. Of course, as we all know, classes of these obstructions is the first Cech cohomology [math]\check{H}^1(W)[/math], which over [math]\mathbb{R}[/math] is de Rham [math]\check{H}^1(W) \cong H_\text{dR}^1(W)[/math].
Alternatively one can notice that, since both Cech and de Rham satisfies the dimensionality axiom (in addition to the EM axioms) of cohomology theory, the fact that the coefficients lie in a field of characteristic 0 means that, on finite CW spaces, there is only one cohomology theory: the singular cohomology [math]H^1(W,\mathbb{R})[/math]; hence [math]\check{H}^1(W) \cong H^1(W) \cong H^1_\text{dR}[/math].

>>11360835
I see

My issue with Calc 2 is every math class I've taken up to this point was basically "heres a problem, use the steps you already know to solve it", whereas calc 2 seems very "here's a problem, try and figure out which of the many methods you know to go about doing something like this works" and my lizard brain can't do

If you had a magical orb that just sprayed out high energy neutrons constantly, what elements would the environment around it slowly converge towards?

Massive brainlet here. I'm struggling with the following math exercise:
Find a∈R a>0 / the minimum and maximum distance from (4,2) to the curve x^2+y^2=a are √5 and 3√5 respectively.
I tried to do it using Lagrange multipliers. The thing is that I'm struggling with an stupid system of equations, it seems easy af but I don't know if I'm to tired of what but I can't fuckin solve it.
This is the system of equations:
•(x-4)/x=(y-2)/y
•x^2+y^2=a
•5=(x-4)^2+(y-2)^2

My idea is to find some solutions to that system of equations and compare then to this system of equations:
•(x-4)/x=(y-2)/y
•x^2+y^2=a
•45=(x-4)^2+(y-2)^2
And then choose the "a" that appears in both systems. I know my English is broken af sorry for that.

>>11360199
What makes you think QM violates principle of least action? The classical bit is always the biggest contribution to the partition function by Atiyah-Bott localization.
>>11360511
No. Go learn what vertices contribute to pair production in Peskin-Schroeder. And don't come back until you've read from cover to cover.
>>11360891
>If you had a magical orb that just sprayed out high energy neutrons
I'd shove it up your ass so you reach critical mass and die.
What's with all these stupid questions lately? This isn't a daycare.

>>11360900
First [math]x^2 + y^2 = a[/math] is a circle [math]S^1_r[/math] with radius [math]r=\sqrt{a}[/math], hence the Lagrangian reads [math]L[x] = \int_S dV_y d_g(x,y)[/math], where [math]d_g(x,y) = \sqrt{g_{ij}(x-y)^i(x-y)^j}[/math] is the metric induced by the metric tensor [math]g[/math]. By using polar coordinates you don't even need Lagrange multipliers.

>>11360917
Professor doesn't really like that we use polar coordinates in this kind of exercises. So we should try to avoid them.
I used f(x,y)=(x-4)^2+(y-2)^2 and g(x,y)=x^2+y^2+a. Since ∇g(x,y)=0 just when (x,y)=(0,0) and that point doesn't belong to the circumference I can apply Lagrange multipliers.
So I have:
•2(x-4)=λ 2x
•2(y-2)=λ 2y
•x^2+y^2=a
Then I resolved and got the systems I already posted. They doesn't seem that hard to resolve but I'm too brainlet.

>>11348740
I use mendeley to organize the PDFs i use for my citations

>>11349062
Ingest a large dose of cannabis while in a relaxing environment. Makes you so retarded and inept that the only thing you can do is relax (or have a panic attack). If it works right you feel like youve been on vacation for a month and you are ready to get back to work

>>11360961
>(((Elsevier)))

>>11361068
Wait its owned by elsevier? Also i get all my papers from scihub cuz my uni is too cheap to provide access to most journals

>>11361078
>Wait its owned by elsevier?
They bought it a few years back.

Based on the core outline of these two CS degrees, what one sounds better from a learning and future prospects (not research) angle
>ignore the electives
>include electives

Thanks /sci/

>>11347903
why did we EVER wake up one day and think 'yup gunna add a point at infinity boi'.

Just got off a phone interview with a company for a research chemist position. They're a small company that contracts to manufactures to produce their successful research.
They said one thing they sometimes do with new hires is hire them through a temp agency for 6 months, then hire them outright.
Sounded a bit sketchy, but I've also never applied to a real job before (just internships/summer work).
Is this something that happens, or should I be concerned?
Thanks in advance.

>>11361859
I'd definitely go with #2 if I wanted a professional dev job after uni.

Can anyone help me with this?
Z = Z1*Z2/(Z1+Z2);
V = sqrt(P*conj(Z))
P1 = V^2/Z1
P2 = V^2/Z2
but this is wrong apparently

>>11360839
That's correct, but you didn't post an account, so I can't add you.
>>11360891
I have a more interesting question.
If I had a gun that shoots electros made of lava, and my arch-nemesis Klaus had a gun that shoots electrons made of ice, who would win? Specifically, the battle is occuring on the U.S. - Mexico border, and on my side is a luchador named Roberto, while on his side is a black man dressed as a cowboy named Clyde.
I am six feet tall and handsome, while Klaus is slightly taller but also slightly less handsome. I am also wrecked with guilt over my past crimes against the mexican people and trying to redeem myself, while Klaus is a merciless killer.
Roberto is a protector of the mexican spirit and he studied TQFT in his free time. He wields a chainsaw which is actually a really small black hole, and around the black hole a bunch of steel electrons spin really fast (they all have spin 1/2 because this makes them spin harder) and this does a lot of damage when he cuts things.
Clyde is a tall, strong man, and has the kindest smile you'll ever see. He's also a navy seal and my childhood friend. He has a gun that shoots dark matter, and I don't know what it does because I don't know how dark matter works, but I'm sure it kills a lot of people!
>>11361943
Are you talking about projective spaces or compactifications?

>>11362162
>Z = Z1*Z2/(Z1+Z2)
yes
>V = sqrt(P*conj(Z))
what lmao
You've got [math] \mathbf{V} [/math]. You've got [math] Z=\frac{Z_1Z_2}{Z_1+Z_2} [/math]. Power over [math] Z_1 [/math] is [math] \mathbf{I}^2Z_1=\frac{\mathbf{V}^2}{Z^2}Z_1 [/math]. Active power is the real part of this complex number. You figure out the rest.
>>11362187
No LARPing in here.
>>11360420
Antimony#5965

>>11362214
>>11362162
Whoops, just noticed the source isn't given. Anyway, you know the total active power dissipated. You know all impedances. You know how to calculate voltage from this. Hint: use the current divider rule to see the proportion of current that flows into Z1 as opposed to Z2.

>>11362187
>Are you talking about projective spaces or compactifications?
projective spaces, yes

>>11362248
Well, the idea is to add points where parallel lines meet. This lets you say that any two lines meet, and simplifies proofs.

>inb4 QM and GR are contradictory
Suppose for a moment we ignore the contradictions and just play things straight. Under the Copenhagen interpretation, what would spacetime curvature produced by probability clouds be like? Would it be a tiny bit curved everywhere in the cloud, with amounts weighted by the probability of finding the particle in a given location? What about the aggregate effect of all the overlapping clouds from different particles?

>>11362280
>if angels rode unicorns, how many could dance on the head of a needle?

>>11361859
Both seem weak on the theoretic end.

Wouldn't want to have missed computation theory and automata theory.
I seriously hope those are incorporated in some of the practical stuff I see there.

>>11362280
Many Worlds seems like the more reasonable interpretation, if you ask me.
Everything falls into place without introducing arbitrary new shit.

why is sin^2 (x) the same as (sin x)^2 but sin^(-1) x isn't the same as (1/sin x)

This is from Spivak, chpt 2, problem 21, using problems 1-18 and 1-19 (a proof of the Schwarz inequality). I don't see how x= or y= what they do here.

>>11362913
hold off on answering this, I have misapplied portions of the question.

>>11362910
convention. really sin^2(x) should stand for sin(sin(x))

>>11362932
Yeah.
>>11362910
Use sin(x)^(-1)

if I have velocity as a function of height
something like v(y) = (whatever function) m/s
and I take the derivative of that function with respect to height (dv/dy), what is the unit?
is it 1/s or is it m/s^2?? or something else

[math] \frac{\partial v}{\partial y} [/math] is something called strain-rate and is a quantity that you come across often in fluid and continuum mechanics. For example, a Newtonian fluid has the property that the shearing force on a parcel of fluid is proportional to strain-rate. Units would be 1/time. You can even generalize this and take the vector gradient of velocity to get the strain-rate tensor: [math] \nabla\mathbf{v} [/math].

yikes
>>11363083 is for >>11363033

>>11363083
>>11363085
thanks
I was so confused as to why I wasnt getting pascals from 1/s then I realized multiplying everything by m/m afterwards got me the right units

>>11347903
Why does my brain insist on playing music in my head all the time, it never stops not even meditating.

>>11363105
You need to work on your meditation and mindfulness, then.

>>11362122
Thanks anon, yes that's the general consensus with local forums etc
1# is a 'third rate uni' in this region but not really bad because I finished a unrelated degree there already. Whereas #2 is a Tech uni and a level above #1.

>>11362437
Thanks anon
#2 is very math weak based on forum reports even though this one seems better for straight to development work. I suspect I will miss theory like you mentioned and more.. I plan to catch up on complex components anyway because its interesting.

A graduate on a local forum said he was doing masters and he had to 'catch up' on core cs components that weren't taught, I guess that's not too much of an issue if I am being proactive in self learning.

Thanks for your replies anons, it helps

OK, I'm hung up on this. Why is the lambda absent from the solution? I don't seem to understand how 18a is applicable..

I totally get how to go from
[math]
0 < \lambda^2(y_1^2 + y_2^2) - 2\lambda(x_1y_1 + x_2y_2) + (x_1^2 + x_2^2) [/math]
to
[math]
\frac{2(x_1y_1 + x_2y_2)}{\lambda(y_1^2 + y_2^2)} - \frac{(x_1^2 + x_2^2)}{\lambda^2(y_1^2 + y_2^2} < 0 [/math]

But there are still some lingering lambdas, and I'm still not sure how this relates to 18a.

>>11364359
>>11364326
is this what I'm missing?
[math]
\frac{2(x_1y_1 + x_2y_2)}{(y_1^2 + y_2^2)} - \frac{4(x_1^2 + x_2^2)}{(y_1^2 + y_2^2)} < \frac{2(x_1y_1 + x_2y_2)}{\lambda(y_1^2 + y_2^2)} - \frac{(x_1^2 + x_2^2)}{\lambda^2(y_1^2 + y_2^2} < 0
[/math]

I still don't see where squaring the first term fits in here:
[math]
[\frac{2(x_1y_1 + x_2y_2)}{(y_1^2 + y_2^2)}]^2 - \frac{4(x_1^2 + x_2^2)}{(y_1^2 + y_2^2}
[/math]

>>11364326
>>11364359
>>11364390
nevermind, this link outlined my issue: https://math.stackexchange.com/questions/147515/help-understanding-proof-of-schwarz-inequality

ok still stuck, I don't get how the discriminant of the quadratic was derived in the link above. It's of the form [math][\frac{b}{a}]^2 - 4\frac{c}{a}[/math], and that doesn't make sense to me currently.

>>11364442
>>11364407
pic related

So, still stuck.
[math]
0 < \lambda^2(y_1^2 + y_2^2) - 2\lambda(x_1y_1 + x_2y_2) + (x_1^2 + x_2^2)
[/math]
so it's a quadratic with
[math]
a=(y_1^2+y_2^2), b=(-2(x_1y_1 + x_2y_2), c=(x_1^2+x_2^2), x=\lambda
[/math]
so then
[math]
\lambda = \frac{-[-2(x_1y_1 + x_2y_2)] \pm \sqrt{[-2(x_1y_1 + x_2y_2)]^2 - 4(y_1^2+y_2^2)(x_1^2+x_2^2)}}{2(y_1^2+y_2^2)}
[/math]
and it has no solution iff the discriminant is less than 0, but how do I prove the discriminant is less than 0? And if I did, how do I proceed to wrap up the problem?

>>11364750
I guess really I'm just looking to prove the statement "Since the equation cannot equal 0, it has no solution and the discriminant is negative."

>>11364786
>>11364750
ok I got this bitch all wrapped up, but I still feel like I'm jumping to conclusions in regards to assuring myself that [math]b^2 - 4c < 0[/math]. I know it's true, my proof just sucks. Here it goes:

Prove that if [math]ax^2 + bx + c > 0 [/math] for all x, then the discriminant is negative (aka, [math]b^2 - 4ac < 0[/math].

First complete the square
[math]
a(x + \frac{b}{2a}) > \frac{b^2}{4a^2} - \frac{c}{a}
[/math]

Since this applies to all x, say [math] x = \frac{b}{2a} [/math], then the left hand side is zero. Now multiply both sides by 4a^2 if you really want, and wa-la.
[math] 0 > b^2 - 4ac \rightarrow \[/math]

Is this bad math? It feels like bad math. If so can you provide an alternate proof?

why sqt so slow today