>>16465871
i was the one with the question about suramin

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What's the book on linear algebra that everyone here recommends? There's a particular one that's a borderline meme, I think the name started with an 's' or had a prominent 's' sound in it.

>>16466124
Serge Lang

fuck, the plasma physics general is gone, and i finally had a question to ask in it...

does anyone here do RIE for semiconductor device fabrication? i need two recipes for InAs and AlGaSb

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So I just want to make sure I'm with the book so far:

A discrete probability function is when:

a) 0<=p(s) for each s in S.
b) the sum of all p(s) is equal to 1.

Now with pic related, p(s) would be a discrete probability function, but p(A) wouldn't since it doesn't add up to 1 and fails (b)? But we could still use it to say "Well, the odds of flipping a heads on an odd-numbered flip are 2/3rds? Or does it failing (b) make it useless for us?

>>16466223
P(A) is not a distribution, it is the probability of an event.

>>16466223
p(A) isnt a function like p(s) is. p(s) is "probability that first heads appears on the sth toss", which is a function of s, and p(A) is "probability that A occurs", with A being "first heads is on an odd toss", so its just a constant (equal to 2/3, as shown).

>>16466236
>>16466247
jfc

thank you, that's confusing

How hot is free floating radiation? If you refined some nuclear material could you use it for a more direct process like melting glass or heating up an oven?

>>16466258
What do you mean by free floating radiation? Also note that temperature isn't a property of single particles (such as alpha/beta/gamma radiation), it is a macroscopic property of a collection of atoms (their average kinetic energy).

>>16466124
LADR by Sheldon Axler. The new edition is free, open access

>>16466294
>What do you mean by free floating radiation
How I worded it was probably better in my head. I meant the alpha, beta, and gamma particles that are emitted. The idea was trying to grasp at was if they can use nuclear energy to power steam engines via bolling water, could you just take multiple rods or material and just use the heat to warm up material directly.

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If you had two weeks to learn Analysis up until the Riemann Integral, how would you do it?

Pi equals 3

>>16466746
Read Tao's book on Analysis 1 I guess, or any other book, idk. Not a math guy

I think I'm getting filtered by linear logic exponentials. I get that ! restores structural rules for terms "on the left" and ? for terms "on the right", and I understand their duality, but what about a ? on the left or a ! on the right? Why not just have one symbol that is it's own dual?
Maybe the actual filter is Sequence Calculus. I only ever use the "classical" notation (don't know what it's called) of classical logic in undergrad and this shit is confusing me.

Anyone know of any good popsci YouTube stuff I can show my girlfriend, especially videos about plants?
She's interested in science but she's not the most academically-inclined and needs visuals to learn, which is why I'm looking for popsci

>>16466851
I don't really watch plant videos so I can't give a rec.
But for popsci with high production value and visuals, look into stuff like Veritasium, Kurgesagt, AsapScience, Mark Rober, or PBS channels like Be Smart (the PBS channels are good). Im not saying I myself love em, but they are popular to most people.

Maybe be more specific for other people to answer, like does she like gardening, or plant biology, or wtv?

>>16466860
Thank you, PBS is a good rec. She's enjoyed PBS eons in the past with me
I'd say I'm looking for videos related to plant biology, ecology, and maybe plant evolution. She's also expressed interest in learning how to identify flowers.

>>16466851
Crash Course Botany maybe?

>>16466675
> could you just take multiple rods or material and just use the heat to warm up material directly.
You would die from radiation sickness since they would be highly radioactive and since nothing would be cooling them you might die from the heat first. But sure, you *could* do that.

hello sqt. im a neet studying math on their own and really interested in number theory. i would like a book recommendation for cool multiplicative number theory like modern stuff sieves and shit. here are the books ive read:

steins fourier analysis and complex analysis
apostols analytic number theory
apostol modular functions and dirichlet series in number theory
chandrasekharans elliptic functions
a good chunk of whittaker watsons special functions section (havent read the classical analysis part in a meaningful way)
derek lawdens elliptic functions and applications
berndt heckes theory of modular forms and dirichlet series

so im extremely good at dealing with special functions and their applications to most things especially number theory, which probs can be seen by list above. however im kinda weak about the real analysis stuff. i struggle at non trivial estimates of contour pieces or stuff like using phragmen lindelof. im familiar with the basic stuff and the standard rigour of analysis but this might be because i havent just gotten up and taken a real analysis book yet. im worried this is gonna hurt me when studying number theory. i also wanna get into algebraic nt but my algebra background is just groups rings and modules from dummit&foote and langs undergrad algebra so i have a lot to learn.
what do i do /sci/bros?

Is there a group [math]G[/math] that contains finite-index subgroups [math]A,B \le G[/math] with [math]A \cong \mathbb{Z}, B \cong \mathbb{Z}^2[/math]?

>>16467140
If both the indexes were finite than so would be their ratios, which is index of A in B. Contradiction

>>16467153
>index of A in B
A doesn't have to be a subgroup of B.

>>16467140
OP here, I think I figured it out. Assume towards contradiction there is such a group. The intersection [math]A \cap B[/math] is a subgroup of [math]A \cong \mathbb{Z}[/math] and therefore cyclic. But [math][B : A \cap B] \le [G : A] < \infty[/math], contradicting the fact that [math]B \cong \mathbb{Z}^2[/math] has no finite index cyclic subgroups.

What is the simplest way to prove that any fraction can be written as the sum of distinct egyptian fractions?

>>16467440
I'm not sure a "simple" method exists since it involves a fair amount of number theory. I think the oldest was by Fibonacci - https://en.wikipedia.org/wiki/Greedy_algorithm_for_Egyptian_fractions

>>16467440
fundamental theorem of arithmetic
the prime factorization of a number is unique

Had to take a 2 year break for school due to family reasons, will I be okay to take linear algebra? I'm pretty good algebra side but my calculus is a bit rusty

>>16467473
How does that help? Egyptian fractions are the sum of distinct unit fractions. So 2/5 = 1/5 + 1/5 would not be correct, but 2/5 = 1/3 + 1/15 would be.

>>16467497
You need more logic, sets, functions, capital-sigma "[math]\Sigma[/math]" notation (summation) and proofs for linear algebra. Very little calculus

>>16467112
Try exploring the MAA Book Review Repository. For example:
>Multiplicative functions
https://old.maa.org/press/maa-reviews/famous-functions-in-number-theory
>A summary of the elementary number theory everyone should know
https://old.maa.org/press/maa-reviews/number-theory-an-introduction-to-mathematics
https://old.maa.org/press/maa-reviews/a-guide-to-elementary-number-theory
>The reach of number theory into other areas of math
https://old.maa.org/press/maa-reviews/number-theory-an-introduction-to-mathematics
>If you are feeling lucky
https://old.maa.org/press/maa-reviews/unsolved-problems-in-number-theory
>A popularizer of number theory
John Stilwell (not really pop-sci, just very friendly but rigorous texts)
>Three classic treatises , not for the faint of heart
https://link.springer.com/book/10.1007/978-1-4612-9957-8
https://link.springer.com/book/10.1007/978-3-642-61945-8
https://link.springer.com/book/10.1007/978-1-4684-9884-4

>>16467112
any suggestions for this?

>>16467590
>>16467589
holy shit sorry for the timing

So universe expands, but locally galaxies, objects, etc. don't expand because of gravity, electromagnetic, weak and strong forces, they bind them together. Particles themselves don't expand because they're point objects.
But the question is, why fundamental forces also don't "expand" with space? F=ma, a=v/t, v=s/t. s should expand with space, hence F also should expand.

>>16468247
s is displacement, that is a fixed distance irrespective of the expansion of space. what you are talking about is relative velocity = distance/time + speed of expansion.

>>16468247
>Particles themselves don't expand because they're point objects.
Sorta. As of now, force that expands universe isn't too powerful, but it's getting stronger and at one point it might start ripping apart galaxies, celestial bodies, molecules, atoms and even hadrons.

I'm trying to minimize the variance of the d0-d11 by controlling the v0-v11. v is constrained to positive values. This is what I've got so far, but I'm not sure what else I should be doing to it to get closer to solution.
Apologies for smoothbraining. I haven't done have-to-actually-think-about-it math in a long fucking time. Your guidance will be heavily appreciated.
d0 = (14 * (v0 + 85))
d1 = (13 * (v1 + 85)) - ((v0 + 85)
d2 = (12 * (v2 + 85)) - ((v1 + 85)
d3 = (11 * (v3 + 85)) - ((v2 + 85)
d4 = (10 * (v4 + 85)) - ((v3 + 85)
d5 = (9 * (v5 + 85)) - ((v4 + 85)
d6 = (8 * (v6 + 85)) - ((v5 + 85)
d7 = (7 * (v7 + 85)) - ((v6 + 85)
d8 = (6 * (v8 + 85)) - ((v7 + 85)
d9 = (5 * (v9 + 85)) - ((v8 + 85)
d10 = (4 * (v10 + 85)) - ((v9 + 85)
d11 = (3 * (v11 + 85)) - ((v10 + 85)

>>16468247
Empty space will expand. Non empty space will not expand. So areas inside galaxies will not, but the empty space between galaxies will, far as I know

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Dumb scifi writer here zero scientific background

Would it be possible to use large amounts of light, or just light in general, to accelerate particles or matter in order to generate energy? I'm making a power plant in my story that uses a "cascading light accelerator" that accelerates matter to generate energy through the use of a geographic phenomenon in a planet, and I want to know more on how that could work. I don't plan to explain it in depth and the plant itself will be mostly mumbo jumbo but if there's anything interesting to know about this, I'd like to hear it

>>16468489
Light does have momentum, it can move things. It's the basis for how solar sails work, and solar radiation pressure has to be taken into account for satellite orbits. But scientifically speaking it's an incredibly weak force that needs the power of a sun (or supernova) to have any "oomf".

>>16468465
Occupied space also expands, objects just collapse on themselves until they're their previous size.

>>16468538
that sounds good enough. Thanks

>>16468489
yes, the photoelectric effect is basically this.You can use it to generate electric current too but it sucks compared to using like a normal battery. Someone already mentioned solar sails. Its the whole E^2=(mc^2)^2 + (pc)^2 thing, even if a particle has no mass (photon), it can still carry momentum

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>>16468464
I think you have a quadratic program. I can't get the solution because maxima's lapack looks broken.

>>16468865
Continued: the smallest eigenvalue is 4e-14 the rest are around 100, so it seems like you'll have a solution.

>>16468874
v =[0.000000 6.538462 11.169872 16.469988 22.896999 30.877444 41.047181 54.435312 72.822552 99.564510 141.766128 217.255376]
var = 559636.5
But with the (near) zero eigenvalue maybe you can add something to the minimum without changing the result.

>be starting chemical engineer
>senior asks me to help him
>cost reduction project where we try to find the optimum for a specific stream
>initial model was simple
>stream A enters system 1 which costs money, then enters system 2 which nets us money
>now system 2 has been split into system 2a which costs us money
>and system 2b which nets us money
>however the fraction that goes to 2a vs 2b is dependent on stream A
>more of A means more goes to 2a
so what do I do now? because now this fraction depends on A but the costs of A depends on this fraction. Feels like I'm running into an issue here

>>16469675
The best explanation of dynamic programming is in Peters and Timmerhaus plant design book.

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There's a quarter of a unit circle. Then an infinite sequence of semi-circles are put inside, each having always four contact points. The sequence is made by putting the next semi-circle in the upper right gap like shown in picrelated.

What is the diameter of the n:th semi-circle?

>>16469934
The radius of the first semicircle, which I will denote [math]R[/math], is [math]\sin\theta[/math] where [math]\theta[/math] is the angle of the unit circle (from the positive horizontal, increasing downwards) whose point on the unit circle is the bottom right corner of the semicircle.

It's also obvious that [math]\cos\theta=2\sin\theta[/math], which yields [math]\theta=\arctan(1/2)[/math]. Therefore, [math]R=\frac{1}{\sqrt{5}}[/math].

Next, denote [math]\varphi[/math] to be the angle from the first semicircle's positive horizontal (increasing upward) where the second semicircle's bottom left corner is. Additionally, similarly to [math]\theta[/math], denote the angle to the bottom left corner of the second semicircle from the positive horizontal of the unit circle as [math]\theta'[/math]. The radius of the second semicircle, which I will denote [math]r[/math], is [math]\sin\theta'[/math], and also [math]R-R\sin\varphi[/math]. Additionally, we have the relation

[eqn]R+R\cos\varphi+2r=\cos\theta'[/eqn]

WolframAlpha gives me a really complicated exact solution (and a numerical answer) for r, so it suffices to say that this can be repeated over and over to find the radius of the nth semicircle.

Side note, thanks for the problem anon. This was interesting to solve.

>>16470070
That eqn didn't render. In math form:

[math]R+R\cos\varphi+2r=\cos\theta'[/math]

Let [math](G, \mu)[/math] be a probability space and set [math](\Omega, \mathbb{P}) = (G^\mathbb{N}, \mu^\mathbb{N})[/math]. Let [math]X_i: \Omega \to G[/math] the projection on the ith coordinate, written [math]X_i (\omega) = \omega_i[/math], and let [math]\mathcal{A}_n[/math] be the sigma algebra generated by [math]X_1 \dots X_n[/math]. I want to prove that for any (say bounded) [math]F: G^{n+1} \to \mathbb{R}[/math] we have [math]\mathbb{E} [F(X_1, \dots, X_n, X_{n+1}) | \mathcal{A}_n] = \int_G F(X_1, \dots, X_n, t) d\mu (t)[/math].

It boils down to proving that if [math]A_n \in \mathcal{A}_n[/math] then [math]\int_{A_n} F(\omega_1 , \dots, \omega_n, \omega_{n+1}) d\mathbb{P} (\omega) = \int_{A_n} \int_G F(\omega_1 , \dots, \omega_n, t) d\mu(t) d\mathbb{P}(\omega)[/math]. How does this follow from independence of [math]X_{n+1}[/math] from [math]A_n[/math]?

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Dumb question, but why does only this term survive in QFT? if <0|psi+=0 and psi|0>=0 shouldn't this term die due to normal ordering like all the others?

This probably doesn't deserve it's own thread, unless the premise is expanded to a /sci/ck/ deal. What is the optimal amount of times to flip potatoes to get as many sides fried and crunchy in the least time possible? It's an optimization problem, which I haven't done in years, and it involves probabilities, which I've never been great at. Here's some assumptions:

>The potatoes are cooked and only the sides need to be fried.
If you don't know already, baking your potatoes before you fry them gets you better results than cooking them in the pan at a low heat before you fry them.

>The potatoes have been cubed perfectly
Obviously they haven't and there will be a lot of strange shapes because you're dicing an irregular shape.

>One side on each potato is fried before the first turn

>Flipping the potatoes turns each cube to a random side

At this point I'm stuck because I've always been weak on probability. There's a 1/6 chance of each side being fried each time they're turned, but one side has already been fried, so to my mind 1/6th of the sides have been fried and there's a 5/6 chance of frying a new side on the first turning. And that's as far as I can take it with my half remembered and half-assed probability ability. What's the next step? What formula am I missing? Is this a combinatorics problem?

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Is it normal for Boston fern (Nephrolepis exaltata) leaves to turn yellowish during fall and winter? I heard overwatering can cause fronds to turn yellow, but I can rule that out, since the soil has a good drainage and I'm careful with watering.

>>16470999
The psi operator isn't acting on the vacuum |0>, it is acting on the two particle state |i>. Each psi field kills off one of the b^\dagger operators on the right, and the \psi^\dagger operators kill off the b operators on the left in <f|.

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>>16471058
pbinom(0, flips, 1/6) is part way there:

1 0.83333333
2 0.69444444
3 0.57870370
4 0.48225309
5 0.40187757
6 0.33489798
7 0.27908165
8 0.23256804
9 0.19380670
10 0.16150558
11 0.13458799
12 0.11215665
13 0.09346388
14 0.07788657
15 0.06490547
16 0.05408789
17 0.04507324
18 0.03756104
19 0.03130086
20 0.02608405

If you pick a face, flip it 20 times, 2.6% of the cubes will not cook that face. But if you flip it too often it won't become as crispy, because the potato surface temperature will be lower.

>>16471780
That's not quite what I'm after. The average number of cooked faces per cube is more germane because then you can set it up as an optimization problem to maximize the cooked sides per flip.

>But if you flip it too often it won't become as crispy
I should have listed the assumption that the amount of time spent on each face was sufficient to crisp it, but you have a point. What if the flips are only sufficient to cook it halfway to crispy so that each side needs to be fried twice to actually crisp up. You could flip it nearly twice as often and you'd have less of a chance of burning a face.

Do books get checked for accuracy? I lost points for paraphrasing a book answer

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>>16471994
8 flips is ideal. If you keep flipping the burned side will get more burned faster than the average number of cooked sides increases 6*pgeom(flips, 1/6). But your burned-raw preference may be different, and I might be off by 1 since the first side starts out cooked already.

>>16471058
Fry them in oil for the quickest results. In general, the closer to sphere shape the more even the cook, but gradation actually adds complexity to texture. Any conceivable goal could min-maxed, such as shoe string fries to maximize crunchyness, and steak fries for nice body.

>>16472205
No. What gave you the idea they would be?

>>16470521
Stripping away some details of your question, I think you essentially want to prove that [math]\mathbb E[f(X,Y)|X]=g(X)[/math], where [math]g(x)=\mathbb E[f(x,Y)][/math].
I'm not really sure how your intermediate step plays into this (not a fan of the notation tbdesu), but I would prove the above first for indicator functions and then use the ``standard machinery'' or a monotone class argument.

You have, for example, for [math]A,B \in\mathcal B(\mathbb R)[/math], [math]\mathbb E[1_A(X)1_B(Y)|X]=1_A(X)\mathbb E[1_B(Y)][/math] by measurability and independence, while [math]g(x)=\mathbb E[1_A(x)1_B(Y)|X]=1_A(x)\mathbb E[1_B(Y)][/math] (also by independence), implying the above holds for [math]f(x,y)=1_{A\times B}(x,y)[/math].

>>16472218
It is the class's textbook

>>16472299
Of course textbooks can be wrong, why do you think errata or new editions exist? But was the textbook wrong or was your paraphrasing of it incorrect?

How much time elapses for a person who attempts partial hanging to actually decease? Help me write my thesis.

>>16472208
Thanks, anon. I'll take your word for it and brush up on probability so I can do this myself next time.

What are the values of the limits [math]\lim_{x\rightarrow 1} \left(\frac{2-x}{x}\right)^{\frac{1+\sqrt x}{1-x}}[\math] and [math]\lim_{x\rightarrow 0}\frac{1-\cos x\cos 2x}{x^2}[\math]

>>16472471
Let me fix that for you:

What are the values of the limits [math]\lim_{x\rightarrow 1} \left(\dfrac{2-x}{x}\right)^{\dfrac{1+\sqrt{x}}{1-x}}[/math]
and
[math]\lim_{x\rightarrow 0}\dfrac{1-\cos{x}\cos{2x}}{x^2}[/math]

>>16472479
I blame moot

>>16472471
>>16472479
nta, here's the second one:

[math]\lim_{x\to 0}{\frac{1-\cos{x}\cdot\cos{2x}}{x^2}}[/math]

>>16472700
Use L'Hospital's rule twice, that should give you 5/2

If I have a son and put in effort to ensure he always has his highest possible natural testosterone growing up (age appropriate exercise, good diet, blood tests to check for possible lack of micronutrients, etc.) will that make him have a larger dick? What about trying to maximize his testosterone while he's still in his mom's placenta? Would it be about increasing the mother's testosterone?

>>16472749
Teach him to take care of his hair because hair loss is on the rise due to lifestyle/enviroment. You will want a high level of testosterone only up to that age which is associated with the male baldness pattern

>>16472943
don't care, baldness is non-existent in my family + being very successful in your youth counts way more than being bitter about hair when you're 45

What body has ever been observed to move in a way that is continuous? Regular algebra is what holds when limits aren't going to zero, so velocity is only the derivative of position in theory, as a simplified ideal model.

>>16472967
This is exactly what differential equations are for; the Navier-Stokes equations are perhaps the most infamous example.

>>16472974
I don't understand. If no bodies move continuously how can dif eq be a solution? Is there some kind of dark math correction that can make it differentiable?

>>16472967
Infinities do not exist in nature. Our physical laws are tools to *describe* reality, that doesn't mean they are reality.

>>16472471
>>16472479
The fist limit you can write in the form [math]\lim_{x\rightarrow 1} f(x) = \lim_{x\rightarrow 1} e^{\ln{f(x)}}[/math]. You can then write the logarithm section as:

[math]
\dfrac{1 + \sqrt{x}}{1 - x} \ln{\dfrac{2 - x}{x}} \\
= \dfrac{1 + \sqrt{x}}{1 - x} \left(\ln(2 - x) - \ln(x)\right) \\
= \dfrac{1 + \sqrt{x}}{1 - x} \left(\ln(1 + (1 - x)) - \ln(1 - (1 - x))\right) \\
= \dfrac{1 + \sqrt{x}}{1 - x} \left( ((1 - x) - (1 - x)^2/2 + (1 - x)^3/3 - \ldots) - (-(1 - x) - (1 - x)^2/2 - (1 - x)^3/3 - \ldots) \right) \\
= \dfrac{1 + \sqrt{x}}{1 - x} ( 2(1 - x) + 2(1 - x)^3/3\ + \ldots) \\
= 2(1 + \sqrt{x}) ( 1 + (1 - x)^2/3\ + ...) \\
[/math]

And so in the limit [math]{x \rightarrow 1}[/math] this simplifies to 4. So the final answer is [math]e^4[/math].

The second limit is just L'Hospital until the denominator no longer contains x.

>>16472334
It said shells become minerals in the ground

>>16473411
Limestone can be formed from shells over geological time periods but that is a rock not a mineral.

What is a proof in mathematics?

I don't get the idea, is it about coming to the same place from two different sides? Like, if we are on corner and I tell you about a store in the opposite corner of that same block, then either walking left or right around the block without crossing streets should get you to the same place, proving that the store is indeed on the opposite corner.

I want to understand the philosophy, the logic behind it. Any references? Like, how do people actually get to the level of math to go on problems like "prove 2+2=4"? All I can think is:
>2+2=4
>Because
>2=1+1
>4=1+1+1+1
>So
>(1+1)+(1+1) = 1+1+1+1
>Therefore
>2+2=4
But then again, what the fuck am I doing? Going around in circles.

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>>16473431
You have to start from somewhere, assumptions that are so basic they cannot be proven - they are simply taken to be true. Those are the axioms of mathematics.

> how do people actually get to the level of math to go on problems like "prove 2+2=4"
You can do such things using fields like symbolic logic and set-theory (ZFC) but again they each have axioms. Even more basic that 1 + 1 = 2 (which they can then later on prove).

>>16473431
If you really want to know pick up the text Journey Into Mathematics: An Introduction to Proofs by Joseph J. Rotman (2006), you dont need ZFC at the beginning like the other anon said.
>2+2=4
A proof of this fact involves defining what "+" is. This definition is non-trivial but once you prove (an argument that most likely will go over the beginner's head) that it has the desired properties, that fact is easy to get, somewhat like your argument.
Look at this too:
https://www.ma.imperial.ac.uk/~buzzard/xena/natural_number_game/

>>16473431
Real any mathematician's first analysis book. Terrence Tao's Analysis I is very accessible.

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Chemistry chads, will bleach have any sort of reaction with petroleum jelly (Vaseline)?
I am going to bleach some warts on my foot, and I plan on protecting the surrounding skin with Vaseline.

Tl;Dr what I want to learn about is the intersection of electricity and chemistry.
Is electrochemistry a thing?
Would it be better to investigate QED and chemistry seperately or is there literature specific to electricity doing things to molecules?

when taking the std deviation of a series of measurements of a physical quantity >=0 (say, a time duration), I'm getting that the result is x+-y with y > x, implying that there's a chance for the value of being negative. is there a better way to model random errors taking into account that my quantity can't be negative?

>>16474360
>electrochemistry
Sure, it's a thing. Do you have any portable device? Is there a battery in that device? Electrochemistry is a science behind that battery.

>>16474499
I would try the Box-cox transformation where you may end up depending on log(length+1) being normally distributed.

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I just want someone to check my Nernst equation here. So I'm wanting to electrochemically reverse the reaction of ferric chloride with copper metal:
[math]Fe^{3+}_{(aq)} + Cu_{(s)} \longrightarrow Fe^{2+}_{(aq)} +Cu^{1+}_{(aq)}[/math]
The standard potential for this is 0.77-0.52 = 0.25V, polarity be damned.
The Nernst equation simplifies to this:
[math]E_{cell} = E^0_{cell} - 0.026 * ln(Q_r)[/math]
Doing some simple math, Q can never be greater than 15000, because that would make E cross 0V, and my 5V supply will easily be able to push Q beyond reasonable concentrations (153E-69). But actually calculating Q is a bit stranger. If I understand:
[math]Q_r = \frac{[Fe^{2+}][Cu^{1+}]}{[Fe^{3+}]}[/math]
But I think I can assume both that [math][Fe^{2+}] = [Cu^{1+}][/math], and that [math][Fe^{3+}] = k - [Fe^{2+}][/math], where k is the maximum/final concentration of unconsumed ferric chloride, which I'll assume has a maximum value of about 7.86mol/L.
That gives:
[math]Q_r = \frac{[Fe^{2+}]^2}{7.86-[Fe^{2+}]}[/math]
And I can feed this into the Nernst equation:
[math]E_{cell} = E^0_{cell} - 0.026 * ln \left( \frac{[Fe^{2+}]^2}{7.86-[Fe^{2+}]} \right) [/math]
Which is something I can plot a graph of.

Pic related is the graph with a logarithmic X axis representing the magnitude of the concentration of Fe2+ ions, and the Y axis is the cell potential. If chemistry class has taught me anything it's that concentrations below 10^-7 don't exist (lol), so I can assume I'll need no more than 1.2V for my cell.

Is this correct? As a physicist, having something that isn't unitless inside a natural log gives me the heebie jeebies.

>>16475229
The units are in the E0. 100 year old chemistry journals pleasantly leave out most units: they had a convention. The unit police haven't taken my Nernst equation. Assuming you did the half-cell calculation correctly, you need to apply more than 1.2V because there's an anode reaction, ohmic loss in the electrolyte, and electrode kinetics.

>>16475251
>The units are in the E0
But it's a measure of the voltage?
I think in reality, the concentration of each species is being divided by its standard concentration (i.e. 1mol/L), so the units all cancel out inside the logarithm. Because concentrations have to be relative to something. Maybe that's what the square brackets implied all along.

>Assuming you did the half-cell calculation correctly
That's mainly what I'm posting for. Does the [Fe3+] = k-[Fe2+] make sense? I'm getting second thoughts since concentrations are generally multiplicative and not additive.

>you need to apply more than 1.2V
If I understand, the E_cell value from the Nernst equation for my given reactant concentrations is what it takes to bring the cell to a steady-state, with no reaction progressing either way. 1.2V is the highest that will ever go, give or take another 5% from temperature variations. Since I'm going for a trickle-charging reactor, I figure I can just run my reaction from a ≥1.5V regulator with some series resistance and call it good. The important part is having enough voltage to prevent my copper cathode from corroding in any circumstance, while being able to push the reaction through to completion.

But I'm pretty sure some O2 and maybe CO2 have dissolved into my spent etchant, so I've no clue what the hell is going to happen when I put it all together. So long as it doesn't eat up my chinese "platinum" electrode I guess.

>>16474041
Use duck tape, Mongoloid.

Hey guys, I need help finding an identity I lost some time ago. Unfortunately, I can only be vague about this as otherwise I would have found it again myself.

It involved finite series, where each term was the product of a^i and b^j, with i and j being the parts indexed.

The identity showed a way to refactor the equation, which could be useful for further proofs/simplifications.

The identity was first proven by a famous marhematician/physicist.

I can't remember more than that and it is really killing me. Please. Help me /sci/, you are my only hope.

>>16475809
Should add, this identity doesn't show up in any general lists on common infinite/finite series equations.

>>16475809
>>16475812
Your description is incredibly vague and could match any number of series.. If its not on this list then good luck remembering.

https://en.wikipedia.org/wiki/Series_expansion

>>16475849
Yeah, I know. Thankstho.

>>16474041
Depends on the presence of unsaturated hydrocarbons. Aromatics especially can form some nasty compounds in the presence of bleach. All this information should be available through the company's SDS, or possibly PubMed.

A quick test to see if anything is reacting is to simply mix them, and feel if the vessel gets warm. A good demonstration of this phenomenon is to drip bleach on white vs brown paper towels, in case you don't trust your judgement.

>>16473431
>What is a proof in mathematics?
It is a game where you use certain agreed upon rules to go from one set of statements ('axioms') to another ('theorems')

Why is IQ so incredibly resistant to positive change?

>>16476182
IQ is Jewish cess

>>16476191
Got any proof? Ashkenazis meming their way to the top is funny but not indicative that the psychometrics are bunk

>>16476156
Thanks, I'll mix a little bit first then.

>>16475809
Does this look right?
https://www.wolframalpha.com/input?i2d=true&i=Sum%5BSum%5BPower%5Bx%2Ci%5DPower%5By%2Cj%5D%2C%7Bj%2C1%2Cm%7D%5D%2C%7Bi%2C1%2Cn%7D%5D&lang=es
Simply the product of the closed form for each geometric series, a=1.
https://en.wikipedia.org/wiki/Geometric_series

>>16476429
Yes thank you very much

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If you have a container with liquid in it and you tilt the container and the water level always stays the same during the tilt, is it possible that the container walls are not circular in shape?

Can someone tell me if this proof is correct?

[math]
\begin{aligned}
\lim_{x \to 5} \frac{1}{x-3} & = 1/2\\
0 < |x-5| & < \delta\\
|\frac{1}{x-3} - \frac{1}{2}| & < \epsilon\\
|\frac{2}{2(x-3)} - \frac{x-3}{2(x-3)}| & < \epsilon\\
|\frac{5-x}{2(x-3)}| & < \epsilon\\
\frac{1}{2}*|x-5|*\frac{1}{|x-3|} & < \epsilon\\
-\delta < x-5 & < \delta \\
5 - \delta < x & < 5 + \delta\\
|x - 3|\ & \text{has an upper bound of}\ |5 + \delta - 3|\\
|x - 3| \in & (2 - \delta, 2 + \delta) \ \text{Now assume \(\delta < 1\)}\\
|x - 3| \in & (1, 3) \ \text{Use 1 since it has the greater reciprocal}\\
\frac{1}{2}*|x-5|*1 & < \epsilon\\
|x-5| & < 2 \epsilon\\
\delta = 2 \epsilon
\end{aligned}
[/math]

>>16477354
With your answer, if epsilon is 1000, will delta be less than 1? Your answer for delta has to work for every positive epsilon. Instead, multiply everything by |x-3|, then use the triangle inequality on |x-3| to compare to delta.

When you maneuver around to get to a point with f(delta) < epsilon, solve for delta, but be wary of dividing by a negative number in an inequality.

>>16477489
or dividing by 0 too

>>16477354
x can't be 3 since you'd be doing 1/0, so you have the options of 3 < x < 5 or 5 < x < 7. Just do inequalities for both, solve for [math] delta < 2 [/math], and choose the minimum.

>>16476182
why is liver so resistant to positive change? what about the stomach?
brain develops based on the genes it has, everything is extremely precise. it accepts plenty of plasticity but if it allowed just about any changes, it would end up extremely prone to fuckups and actual loss of IQ.

not a math fag so i may be vague with what i need, but perhaps some anon here has a solution
(not really important context: I work in a textile factory. One stage of production is to take a bobbin and spin it onto a... circular arm thing, to create a loose loop of yarn that can later be washed)
there are 15 slots on the machine. depending on the type of material involved, there are different numbers of spins, that deplete the bobbins to different degree. At some point the color needs to be changed and then it comes to the math. the question is:
>how to count the lowest possible number of spins needed to deplete all bobbins, on the condition that all but the last material loops must be 99%+ full?

hope it makes sense, ask more questions if something needs to be clarified

>>16474499
Maybe it's actually negative and your assumption is weong

>>16477530
It's incredibly easy to lose IQ though. Trauma, exposure to toxic substances like lead/mercury, head injuries, even insufficient stimulation. It's just not possible to get it to work better in general like a muscle for some reason lol

>>16477489
If epsilon is 1000 you would use a delta of 2000 and still get an answer for the limit that is within 1000. [eqn]|\frac{1}{5-2000-3} - \frac{1}{2}| < 1000[/eqn]
>>16477512
|x-3| is 3, so x would be 6 in that case.

My answer is the same as this guy gets but he works it out slightly different:
https://youtube.com/watch?v=AfrnYS5S8VE&t=482

Slightly revised the end:
[math]
\begin{aligned}
|x - 3| \in & (2 - \delta, 2 + \delta) \ \text{Now assume \(\delta \leq 1\)}\\
|x - 3| \in & (1, 3) \ \text{Use 1 since it has the greater reciprocal}\\
\frac{1}{2}*|x-5|*\frac{1}{|x-3|} < \frac{1}{2}*\delta*1 & = \epsilon\\
\delta & = 2 \epsilon
\end{aligned}
[/math]

>>16477708
Guy, I didnt say delta had to be less than 1, you did. Where did you think I pulled that out of?

>>16478053
I learned enough at this point to know that the answer is [eqn]\delta = min(1, 2 \epsilon)[/eqn]

The point is to prove that the limit exists, and that is a proof. The reply I made was mistaken. If you could pick an value for delta, then you might pick 2 or something very close to it and get an undefined value, of a huge number.
I'm not sure what you mean by 'multiply everything by |x-3|' but ultimately the result is you will just get a smaller value for delta.

>>16477683
How the fuck is severe head trauma 'extremely easy'?
it's just the same easy to lose function in any other internal organ.
I also recommend you to read what IQ actually is, since you don't seem to be grasping the idea here.
and fuck yes, it is possible to make brain work better

>>16478150
>severe
Not even lol, a couple mild bumps a week is enough for severe degeneration
And youre kinda going against consensus here....

>>16478150
>just the same easy

>>16478198
you were clearly given one too many mild bump in your career

>>16478150
>fuck yes it is possible to make brain work better
https://www.youtube.com/watch?v=Cycon01RT18
from a respected Univeristy of Toronto lecturer

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What's his fucking problem?

I was actually diagnosed with ADHD as a child but my ADHD has only gotten worse as an adult. I have fucked myself by avoiding medication since I was 13? What medication I was prescribed either made me feel horrible or caused heart problems.

The only possible other thing off the top of my head affecting my brain could have been being struck with a hollow metal pipe on the right side of my head near where the temple and jaw meet 6 years ago (had a circular cut on my head for awhile). Had there been a little bit more force I imagine my skull would have been gibbed.

I am doing well in MIT right now as a CS undergrad but I feel like a complete idiot compared to my peers and professors. My knowledge and understanding of topics to me feels superficial and only sufficient to pass a course. Around a couple weeks after a course its like my knowledge dematerializes unless I actively try to maintain it but doing this for all the subjects is just unfeasible for me. I seem to keep speaking before I think making me look like an utter fool. I will catch myself saying things and force myself to stop because they are absurd. I am spending obscene hours studying on courses labeled to only be for 15-20 hours typically. I believe I am having alot of difficulty focusing and I don't recall it being anywhere as bad as it is now when I was a child.

should I try meds again?

>>16478490
Instead of hoarding knowledge by reviewing flash cards or your cheat sheets you have to find a use for it. If you can't find a use then there's no point in remembering.

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Any estimate on how long it would take the plant to cover the whole planet giving the information in the pic?
It's not simple exponential growth but rather something much much faster since it's the growth rate that is doubling each time interval rather than the size itself.

Why do scientists so often use the phrase "small mammals" while referring to Mesozoic mamnals?

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What is this thing equal to?

>>16478957
[math]a\ln{2}[/math]

>>16479444
cool how the euler's number just pops up in math. I just created that series out of my own imagination and didn't expect it to be related to the e.

>>16479538
You take the geometric series formula
[eqn]\sum_{k=0}^\infty a x^k = \frac{1}{1 - x}[/eqn]
you integrate it
[eqn]\sum_{k=0}^\infty \frac{a}{k+1} x^{k+1} = - \log(1 - x)[/eqn]
multiply by -1 and then take the limit as x goes to -1.

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What went wrong with this calculation? I tried to calculate the radius of the small circle and I got this answer as in picrel. But when I made the sme diagram on Desmos, it turns out there's a gap between the circles meaning that the small radius should actually be a bit larger. But I can't figure out what I did wrong.

>>16479656
oh, never mind. I think I realized it. I assumed that the line there goes through the circle center and the left intersection point simultaneously which is wrong.

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Why is a flying kite generally in the shape of a kite, instead of say a square or diamond?
What makes this better for flying?
I found this
>The quadrilateral with the greatest ratio of perimeter to diameter is a kite, with 60°, 75°, and 150° angles.
But not sure if related.
NASA’s page says you can use the same formulas we use for airplanes, I.e. the lift and drag.
https://www.grc.nasa.gov/www/k-12/airplane/kiteaero.html
And says it’s basically surface area for a kite, but that implies you could use a circle for a kite, but no one does.
Has anyone used NASA’s kite modeler?
https://www.grc.nasa.gov/www/k-12/airplane/kiteprog.html

>>16479701
Look up power kites of you want something with more pull.

But yeah you can kite a Circle. It's just harder to build. But look at the Asians flying dragon kites and whatnot

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>>16479721
Sure you can, it’s just not steerable

>>16478957
This is a generalisation of the alternating harmonic series (which you get when [math]a=1[/math]), and it's very important to note: As written, this series converges to [math]a\ln{2}[/math], but it's not really accurate to say that it is "equal" to that value. It doesn't have a well-defined sum at all, actually.
This is because it's a conditionally-convergent series; that is, it converges, but if you take the absolute value of every term, you get a series which diverges instead (in this case, the harmonic series times [math]a[/math]). Conditionally-convergent series have the property that you can rearrange the terms without changing any of them, and the sum changes as a result.

For example, if we follow each positive term with two negatives instead of one, we get [math]a -\frac{a}{2} -\frac{a}{4} + \frac{a}{3} - \frac{a}{6} -\frac{a}{8}... = \frac{a}{2} - \frac{a}{4} + \frac{a}{6} - \frac{a}{8}...[/math], and you'll notice that just by rearranging, we've gotten the same series as originally, but halved - so now it instead converges to [math]\frac{a\ln{2}}{2}[/math].

In fact, we can go a step further: A conditionally-convergent series can be rearranged to converge to anything you want, or to diverge entirely
https://en.wikipedia.org/wiki/Riemann_series_theorem

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above: textbook definition
under: translation by chatgpt

why do they do this? just to make us suffer?

>>16479974
At least the first one sort of defines "linear combination". A "combination" of vectors must allow more operations than "linear combination".

>>16478714
I'd assume growth rate is volumetric speed, i.e. how many cubic metres it increases its volume by every second, but it might also be area speed (square metres per second) or linear (metres per second) since it seems to be a long snaky plant. Either way, you'd just write an equation like [math] dK/dt = Kr_0 \times 2^{1,000,000t} [/math] where t is time in seconds, K is the aforementioned volume/area/length, and Kr0 is the initial value of its speed. Then you integrate it and give it a K0 constant for the initial value of its size. Integrating an exponential just results in another exponential.

You can estimate the size of the city and plug 1 hour into the resulting equation to solve for one constant, but you'll still have another constant, there aren't enough definite degrees of freedom since you can't easily estimate its size in the t=20s panel, or the time passed in the 2nd panel. But you can make guesses for that second one that will give you a finite range.

>>16465871
Okay, I'll admit I'm dumb, and I would like some help.
I figured out this spreadsheet formula to find the distance (d) to the horizon on a sphere of radius (r) from an eye height of (h), all in the same units.
d = r * acos ( r / ( r + h ))

Now I'd like to solve for (h) instead, but can't wrap my head around the algebra because of the acos function.
Can/would anyone help with this?

>>16480988
Your question is flawed since your distance to the horizon equation is wrong. Basic Pythagoras should tell you [math]r^2 + d^2 = (r + h)^2[/math].

>>16481006
>Basic Pythagoras
Your unhelpful snark is wrong. Over a sphere pythagoras is only an approximation over short distances.

>>16467440
CS fag here. Came up with this computationally reasonable algorithm, which universality can serve as a proof. Just for fun.

Corollary 1: Any rational number, represented in any base, either terminates or ends in repeating digits.
Corollary 2: Any rational number less than 1, if it has repeating digits in a given base, consists of only repeating digits and no non-repeating component.

1. Express the fraction in binary, accounting for repeating digits. For example, 11/17 becomes 0.(10100101)*.
2. Express the first iteration of the repeating component, or the non-repeating digits if there is no repetition, as a sum of fractional powers of 2. So 0.10100101 becomes 1/2 + 1/8 + 1/64 + 1/256. If there are no repeating digits, you're done.
3. Express the rest of the repetition by shifting the initial fraction right by one repetition. In other words, divide the initial fraction by 2 to the power of the length of the repeating sequence. In this case, 2^8 = 256, so (11/17)/256 = 11/4352, or 0.00000000(10100101)*.
4. Express the numerator as a sum of binary terms, and simplify. 11/4352 becomes 8/4352 + 2/4352 + 1/4352, which simplifies to 1/544 + 1/2176 + 1/4352. The fractions will always simplify to have a numerator of 1, because all numerators are powers of 2 and the denominator is a larger power of 2.
6. Add the non-repeating and repeating terms together. 11/17 = 1/2 + 1/8 + 1/64 + 1/256 + 1/544 + 1/2176 + 1/4352.

Note 1: Shifting the repeating component right and expressing the first iteration separately is the key to this approach, and is required. This is what allows all fractions to have unit numerators, simply by reducing their magnitude arbitrarily in a consistent way.
Note 2: While the overall method of converting numbers to sums of integer fractions would work in any base, it's important to do this in binary, because binary is the only base where all nonzero fractions have unit numerators.

>>16479789
>and it's very important to note [fussy mathematical nitpicking]
yeah, not really though

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>>16480988
[math] \cos ( \arccos ( x ) ) = x [/math]

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>>16480988
Kek... sometimes ChatGPT is the better way to go.

>>16481087
But we do see the knives behind their backs.

>>16481023
What?? It's exact for a perfect sphere.

Thought of a strange probability question. Assume we have a shuffled deck of 52 playing cards. We draw a card from this deck and don't look at it. We then add this card to a second identical deck and shuffle. We then draw a card from the second deck. What is the probability of drawing an ace?
My train of thought is that the probability would be [math]\frac{1}{13}[/math]. Since we gain no information, this selection should be identical to just shuffling the decks together and drawing a card. I just wanted to see if this made sense.

>>16481276
Yes it makes sense. 1/13 times the card you add will be an ace, so effectively your 53 card deck contains 53/13 aces.

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>>16481235
Tell me you're a flat-Earther without telling me you're a flat-Earther.

>>16481151
>sometimes
aren't LLMs scoring high in math olympiads?
I must say it feels very good to see mathfags get shit on by computers
so much for
>MUH MATH CREATIVITY
>MUH IQ

when in reality all you need to be good at math is good memory to absorb textbooks and then apply the rules in the correct order like an autist

>>16481398
Not really a fair comparison. Math olympiads always feature solved problems, which both the competitors and the AI can know about in advance, and are therefore BS. The intention was to have competitors demonstrate creativity by coming up with the solution on their own, which some do, but the only way to actually ensure that is do things that haven't been done before. In that sense I don't see any LLMs publishing novel insights, and in general they are quite bad at extrapolation and exploration.

>>16481398
>good memory
Which is a component of what psychometric?
Hint: it starts with an I

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>>16481305
The first formula involving acos is the distance is you have to "walk" across the surface to reach the horizon, it is not a straight direct line. Pythagoras is the distance you see.

I've tried to do the research myself but I'm getting conflicting information.

My maternal grandfather had male pattern baldness.

What are the chances that I will also bald? I have two brothers and both of them are balding, does that make a difference to the chance that I will bald?

Somehow, my mother has a sister, who has 3 sons, and none of them have balded at all, does this mean that aunty can only really be my mothers half sister and grandmother was a whore?

>>16481080
>casually "proves" [math]\ln{2} = \frac{\ln{2}}{2}[/math]
>nah bro that's just nitpicking

>>16479789
The Riemann series theorem doesn't mean that conditionally convergent series don't converge, it just means that infinite series don't have the same commutativity that finite series do.

>>16481523
Thank you for the clarification and distinction.
Walking distance was indeed what I was looking for (hence the reference to the trig functions in my original request).
It's useful to know line of sight, but that one was easy.
I'll somehow be more specific when I use either.

>>16481798
>The Riemann series theorem doesn't mean that conditionally convergent series don't converge
good thing that I explicitly said that it still converged and that the problem sat in the usage of the term "equals"

>>16481803
nta
do you think that absolutely convergent series "equal" their converged values? whats your reason for drawing a line between the two?

>>16482630
>do you think that absolutely convergent series "equal" their converged values?
up to an arbitrary point of precision, yes
>whats your reason for drawing a line between the two?
you can rearrange the terms in 1+1/2+1/4+1/8+... however you please - the limit of the partial sums is still 2, and so for all intents and purposes the entire series can be said to equal 2 exactly
this idea falls apart with conditionally-convergent series since they can be made to converge to any value desired, or to not converge at all

>>16482685
Commutativity only applies to finite sums.

>>16482699
...and absolutely convergent infinite ones.

>>16482685
>up to an arbitrary point of precision, yes
>for all intents and purposes the entire series can be said to equal 2 exactly
you dont think absolutely convergent series equal their converged values. if i write
[math] \displaystyle
2 + 2 = 4 \\
\Sigma_{n = 0}^{\infty} \frac{1}{2^n} = 1
[/math]
you dont consider those two equal signs to mean the same thing. you think one of them is a "real" equal sign, and the other is a "fake" equal sign.

>>16482720
>[math] \displaystyle \Sigma_{n = 0}^{\infty} \frac{1}{2^n} = 1 [/math]
oops mb

>>16482720
>you dont consider those two equal signs to mean the same thing
Why shouldn't I?
The whole point of a limit is that "equal to arbitrary precision" just means outright equal, isn't it?

>>16476211
Got any proof that Ashkenazi exist?
>>16477544
Could you use this?

Linear Feet = π × (Outer Diameter - Inner Diameter) × Number of Revolutions.

Idk how to account for thickness, which would constantly decrease the difference between ID and OD. Nobody’s going to answer you anyway but maybe I can increase your chances by being retarded.

>>16482741
>The whole point of a limit is that "equal to arbitrary precision" just means outright equal, isn't it?
and yet you felt the need to point it out when i asked if they were equal. you pointed it out because you feel that it makes the equality materially different than a "normal" equality.
>>16482719
multiplication is commutative under scalars, but not matrices. does that mean its incorrect to say that the product of two matrices "equals" anything?

>>16481455
>a component
yes, the one that every modern computer has by default
math sissies are unironically going to get filtered faster than social science sissies
in reality, they already have, but the gatekeeping is strong in math departments

>>16482785
>does that mean its incorrect to say that the product of two matrices "equals" anything?
No, because matrix multiplication is a different operation than scalar multiplication.
Meanwhile, scalar addition is scalar addition, no matter how many times you do it.

I don't get why in induction in the induction step you assume that the statement holds for any n and then you have to prove that it holds for n+1. If you assume that the statement holds for any n, aren't you basically assuming the thing you're supposed to prove?

>>16483087
>you assume that the statement holds for any n
no, you PROVE that the statement holds for one particular n, and youre allowed to pick any n. generally you pick an n thats easy to prove, say n_0, and then every other n piggybacks off of that proof.
first you prove the statement S holds for n_0 (to be verbose, you prove that S(n) holds for n = n_0). then you prove that S(n) implies S(n+1). once youve proved both of those, youve proven that S(n) holds for n >= n_0, because S(n_0) was explicitly proven, which then implies S(n_0 + 1), which implies S(n_0 + 2), and so on...

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If I skip these problems, what will happen? I hate fractional exponents like you wouldn't believe

Does uni get better after all the retarded children filter out after the first year?

>>16485350
The book picked them for a reason. You're doing arc length integrals, which means most of them will clean themselves up when you actually do the work. It'll also prepare you for doing line integrals in the future.

Do I have this right?

The Laplace transform [math]\mathcal{L}{f}[/math] is a ring homomorphism from the set of appropriate functions (exponential or less growth, finite discontinuities) where + is our usual addition of functions and [math]\cdot[/math] is [math]\ast[/math], the convolution of two functions, to the set of rational functions where + and [math]\cdot[/math] are our usual addition and multiplication. The multiplicative identity in the first set is [math]\delta[/math], the Dirac delta function.

Am I missing something? From what I'm reading convolution is associative and distributes over addition. Is there some pair of functions in our set whose convolution isn't?

>>16485836
No.

>>16485864
the image isnt correct. laplace of shifted unit step (which has expo or less growth and finite discontinuity) is not rational

I'm feeling extra dumb today but how would I draw a graph with 6 vertices with a degree of three?

>>16486827
The easiest way would be to draw a graph with 9 vertices total. 6 with degree of 3 and 3 with with degree of 6 where each degree 3 vertex is connected will all degree 6 vertices.

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What if you have two tangent unit circles and a third circle with radius R, being tangent to the unit circles. Then you start putting circles in the space surrounded by those three circles, putting the next one on top of the previous one like shown in picrelated.

Given R, what is the radius of the n:th circle in the sequence?

>>16486866
Use Descartes' theorem