/sci/ - Science & Math

Unanswered questions:

Math questions:
>>14943494
>>14939857
>>14939615 ([math] \{a\} - \{\{a\}\}=\{a\}[/math] and [math]\bigcup \{a\} = a [/math], the latter notation is for the union of sets inside that set so [math]\bigcup \{a,b,c, \ldots \} = a \cup b \cup c \cup \ldots [/math])
>>14939561
>>14938936
>>14937255
>>14936535
>>14935322 (provide some context to your question)

Physics questions:
>>14940222 (use Newton's second law a=F/m and Newton's universal law of gravitation to figure out the acceleration of each object)
>>14938501
>>14936566
>>14936328

Chemistry question:
>>14940814

Biology questions:
>>14940282 (two bivalents make up a tetrad)
>>14938540
>>14937018
>>14935759

Engineering questions:
>>14945862 (try asking >>>/3/)
>>14944912

Stupid questions:
>>14945939
>>14943461
>>14943119 (someone give him the pasta)
>>14942701
>>14940139
>>14939213 (it's probably called a normal person)
>>14938114
>>14935276
>>14935160 (It's ok as long as you keep away from his talks about real numbers)
>>14934618
>>14934601

Don't know where to put this question, probably stupid:
>>14938434

How do I master the mean inequalities?

>>14945693
>anon, that's *exactly* the reason
Anon, if someone makes fun of you and then actually helps you and you turn up your nose and impolitely ignore them you're a little bitch.

>>14946072
No, it means you're normal.

>>14946053
We've all seen mathfag and even physics book paths, but where's the path for chemistry?

>>14946113
Look at the /sci/ wiki

>>14939615
Just to add to what >>14946057 said, the key fact you will need to prove is that
[math] \{\{a\}\} \not \in \{a\}[/math], or equivalently [math] \{a\} \neq a [/math]. This follows mainly from the Axiom of Foundation (a.k.a. the Axiom of Regularity).
>>14938936
Strange question. Should it be x>1/2?
>>14936535
The Golden Ratio is the positive solution to the equation [math]x^2 = 1+x[/math]. You get the [math]\sqrt{5}[/math] when you apply the quadratic theorem. As far as I know, the equation comes from the question: what is the size x>1 for which we can take a 1-by-x rectangle, remove a 1-by-1 square from it, and still get a rectangle of the same shape?
>>14943119
Here you go, friend

>>14946113
for starters, choose a subfield

>>14946053
It's so fucking hard to find good fucking university level physics lectures online. Does anyone have a recommendation for optics (physics lectures, not watered down engineering lectures pls [good engineering lectures are also no problem though])?

>>14946053
>master1200
Die

>>14946119
Now that I read this again. What the fuck did he mean by mathematics being phased out? It doesn’t even seem remotely close now and I’m pretty sure this letter dates like a decade ago. He’s not even talking about baby tier math, he’s taking about real mathematics. Am I missing something or do we already have the technology?

How

>>14946240
What is your problem?

>>14946057
>sample
>>14946312
Some people just don't like it when others post lower res versions and samples.

>>14946345
Didn't pay attention to that one, here's the higher res one.
The OP pic is from pixiv which doesn't allow full res downloads without an account and i'm not creating an account just for that.

>>14946432
bugmenot usually solves that.

>>14936566
the whole idea behind particle entanglement is conservation of momentum, not necessarily energy. i dont believe theres anything in QM that would imply energy isnt locally conserved.
>>14936328
you could do something like that, assume every point looks like [math]T(t)=(T_0 - T_f)e^{-\alpha t} + T_f[/math] where [math]\alpha[/math] is some simple function of position, like the distance to the nearest source/sink or whatever. it wont be exact. if youre using computers, i recommend reading a bit on numerical methods, which will give you pretty much exact solutions without having to actually solve a PDE.
>>14944912
headphones, like speakers, are just two wires. buy some dollar store earbuds, cut them, and solder them into the back of the board.
>>14934618
wumi!

>>14946053
This outfit is really borderline. I am not sure if she is a maid or not. She lacks an apron and the hat is wrong, but everything else is maid-like?

Why does Touhou have so many vampire maids and why isn't it a show or a book or some form not a videogame?

I don't play videogames but it seems like it has a lot of maids and that makes it interesting.

I guess for math my question is if people think Wolfy"s triangles would be prettier if they had colors instead of just black and white? Maybe I should put symbols on it? I was writing in a book about Computer Experiments and I decided to replicate Wolfy's triangles because her experiment is famous and now I am trying to figure out how do I make them nicer?

Thank you for reading my post.

>>14946514
>you could do something like that
Wait what?
I thought the whole reason heat transfer was an actual research subject was because you couldn't do shit like that and get reasonable results.
>>14946519
It's a normal dress.

>>14946527
>Wait what?
if you read somewhere that you couldnt do that then youre probably right, i was just throwing stuff out there. although im dubious that heat transfer is an "actual research subject", considering the heat equation is the most basic bitch PDE that you solve day 1 in a course. maybe youre thinking of heat transfer combined with navier-stokes.

>>14946527
It has the same sleeves and belt as a maid dress and it has a ribbon like this maid.

It almost looks like a recolor of a maid dress, but no apron.

This question regards Sum of Products (Disjunctive normal form) and Product of sums (Conjunctive normal form).

Is there a way to know whether SoP or PoS has the fewest literals and gates without having to draw out the kmap for both? (or even one kmap?)

Just wondering if there are any tricks or patterns that you can use as a general guide for knowing.

>>14946532
>considering the heat equation is the most basic bitch PDE that you solve day 1 in a course.
The heat equation is such a stupid fucking PDE that you can know the system's state at infinity without putting any brain into it.
It's such a stupid equation that the inverse problem is a nightmare.
It's such a stupid equation that it has an evolution kernel.

But I don't think you can disregard convection and non-homogeneous media and just use it for everything.

>>14946481
Thank you for letting me know about that although the site is down right now.

>>14946543
if youre not writing out the SoP or the PoS, and youre not drawing either kmap or logic gates, then what do you have access to? how are you representing the logic?

Why is food science as a discipline never really discussed? I get that it's a small group compared to engineers/chemists/microbiology but it covers some pretty broad topics

>>14946712
maybe just writing out the kmap for one of them like SoP. is it possible to easily tell if the PoS form will have fewer literals just from looking at the SoP?

>>14946793
>>14946712
or having it in a kmap optimized SoP form already, and then looking at that and knowing if PoS will have fewer literals

>>14946053
What in the fuck are these things pol keeps posting?

>>14946053
I was thinking about AI again. The library maid came from a place that used to have a big space computer called Award Computer. Award Computer's job was that when a maid is nice to a human, Award Computer activates and sends her a Maid Candy. The maids used to spend a lot of time arguing about how to get more Maid Candy faster and collaborated on projects to get more candy.

How is a big space computer different from a normal computer? Is it required that the Award Computer candy machine be in space? Do things get computed different in a big space computer? Is it made of the same things as an Earth computer? Is there any benefit to being a space computer? Why is she shaped like a starfish spiral?

I realized something just now also. I am wasting images by posting these maids and I should stop! I need something that runs the latest stable diffusion and I can make a computer draw this stuff so other people can see it! Probably my idea would be clearer if I had drawings and the computer can do really nice drawings!

I am going to stop posting anime maids and start posting images of what she is showing me!

Does anybody know can stable diffusion run on Science Computer or do I need a special machine? I will ask Dra/g/ons!

Thank you for reading my post.

>>14946818
just read the /sdg/ OP
you can run it on your own PC but you'll need a very good GPU

>>14946818
>Does anybody know can stable diffusion run on Science Computer or do I need a special machine?
It supposedly runs on any GPU that has 4gb of memory and float16 support or ~6gb of memory (the last one is from personal experience).

easy calc question (first year of uni), i want to know if what I did in calculating the limit of this succession is correct

[math]\lim_{n \to \infty }(\frac{n^2-n^\frac{3}{2}+1}{n^2-5n})^{n(3+sin(n))}[/math]

I know that the exponent can be at most 4n. So I know that this whole succession is always less or equal to this other succession:

[math] \lim_{n \to \infty }(1+\frac{1}{\frac{n^2-5n}{5n-n^\frac{3}{2}+1}})^{(\frac{n^2-5n}{5n-n^\frac{3}{2}+1})(\frac{5n-n^\frac{3}{2}+1}{n^2-5n})4n}[/math]

which for n -> infinity is asymptotic to:

[math]\sim e^{-4\sqrt n}[/math]

which goes to 0
therefore since:
1.the second succession which is bigger than the first one goes to zero
2. the first succession is bigger than 0 for every n
=> the first succession goes to zero by the squeeze theorem

In the definition of graded ring, what is the symbol • denoting in R_n • R_m here? Direct product?

>>14947114
The set {a*b : a in Rn and b in Rm}

Can someone give me a bijection from (0 1) to [a b]?

>>14947139
Have you tried googling for bijection from an open interval to a closed interval? You will find hundreds of pages showing one.

>>14947214
yes but I was unable to figure out how to make one for [a b]

Could anyone knows how implement BFGS , Nonlinear conjugate gradient , and other algorithms of optimization with Rosen Brock function in python or MATLAB?

>>14947023
>which for n -> infinity is asymptotic to:
>[math] \displaystyle ∼e^{−4\sqrt{n}}[/math]

[math] \displaystyle \frac{n^2-5n}{n^2-n^{\frac{3}{2}}+1} [/math] has to go to infinity for that trick to work. also you changed the [math]n^2[/math] in the numerator to [math]5n[/math] in the second limit, is that a typo?

>>14947023
Try [math](1 + x)^y \approx 1 + xy[/math] in a first order Taylor approximation in terms of [math]x[/math].

>>14947394
Those algorithms are implemented in scipy.optimize. You might have to code the Rosenbrock function, but that's it. If you want to do it by hand, I'd start with wikipedia.

What does it mean that "the source and drain capacitances *dominate* over the gate capacitance"? Does it simply mean that they are bigger? Why would one sum capacitances?

How is interrupting people an autism symptom?
Normies interrupt each other all the time. Sometimes I'm in a group conversation and Alex tries to start speaking, but Bob immediately cuts him off by talking louder, so I wait until Bob finishes and ask Alex what was he trying to say, because hyperfocusing on their own thoughts and not noticing other people are trying to speak is just a very normal normie thing.

>>14946053
How the FUCK do I solve this? Can someone at least give me a hint? I don't even see how angular momentum could be happening here??

>>14947645
Forgot picture

>>14947645
>>14947646
Just use the angular momentum formula [math]\sum_i \vec{r_i} \times \vec{p_i}[/math], where the sum goes through all particles, [math]\vec{r}[/math] is position and [math]\vec{p}[/math] is momentum.
>inb4 where does the third dimension come from
It points outside of the paper.

When writing notes in LaTeX, is there a good program to create images with? Primarily sketches, like the ones you'd make on paper.

>>14947250
If f(x) maps (0,1)->[0,1], then g(x)=af(x)+(b-a) maps (0,1)->[a,b].

>>14947646
Just write "Read my paper".

>>14947646
>>14947749
kek

>>14946287
Let A be the original matrix; let C be the square submatrix you end up with; and let B be the intermediate rectangular matrix that is as wide as C but as tall as A.

It's pretty obvious that [math]\mathrm{rk}A = \mathrm{rk}B[/math]. And since every row of A is a linear combination of basis rows, you can take these same linear combinations of the corresponding rows of B to see that [math]\mathrm{rk}B \leq \mathrm{rk}C[/math]. But we also know [math]\mathrm{rk}B \geq \mathrm{rk}C[/math] since C embeds into B, and hence they are equal.
>>14947682
>sketches, like the ones you'd make on paper
Are you asking for: a latex GUI that also near-seamlessly lets you draw with some kind of touchscreen or tablet? If so, I would also like to know.

Right now I just use MS-paint on windon'ts and Tuxpaint on GNU/Linux

>>14947619
It is an indicator that someone may have difficulty interpreting social cues or appreciating other people's needs. On it's own it isn't a diagnosis of anything. It can be a symptom of other disorders, too. ADHD is probably the most common one.

Dumb heat question.

Due to space issues, I have to put some paper on top of the radiators, would the paper burn if the radiator gets top hot?

I currently have some wood on top of the radiator and has the stacks of paper on top of that.

I am extremely intelligent how do I use this to defeat all of my enemies all at once?

>>14947831
I was looking for something that uses mouse and keyboard mostly, since I don't have a tablet.
It seems like Inkscape is relatively easy to start but LaTeX's Tikz package would be better long-time, although a headache to start.

>>14947919
>LaTeX's Tikz package would be better long-time, although a headache to start.
Tikz never stops being a headache lol

Why is internet speed limited? Doesn't the same speed consume the roughly same electricity? What exactly is the resource that is being used?

>>14947880
If you were actually intelligent you could work this out yourself.

>>14947943
By speed do you mean bitrate?
Transmission consume both electricity and bandwidth. The reason it's finite is because bandwidth is a finite resource, and because the medium/infrastructure caps the data rate (you can't transmit too much data at once in all cases so easily otherwise it's just errors galore on the receiver's end).

>>14947943
>Why is internet speed limited?
Think about HOW the information is transmitted through a wire. How would you do it? The first modems were devices that you would literally plug your telephone receiver, pic related. Two of these would be connected by a regular phone call, and they would communicate by a series of noises. How fast can a computer distinguish different noises? Pic related was a pretty bad way of doing it, and according to the filename achieved a rate of 300 bits per second. That's due to a number of things, but a slow computer (limiting how fast it can recognize symbols) and the receiver picking up random sounds (causing errors in the information received) are both part of it. Similarly for the computer that sent the signal.

Think about it a while, and you will probably find some ways to mitigate the limitations... but also some other sources of noise error, and other bottlenecks to speed. And with the modern internet bouncing your signal from router to ISP to who-knows-where and back, those possible bottlenecks only multiply. Fascinating stuff though.
>>14948017
Just to elaborate on this anon's answer: "Bandwidth" here just means the maximum bitrate. The name comes by way of Information Theory, and is supposed to be an analogy to "bandwidth" in radio communications, where it refers to a range of radio frequencies -- not a really similar thing, but I guess it made sense to nerds in the 1950s-1980s.

>>14947433
oh yeah i made a typo. so,
[math]\lim_{n \to \infty }(1+\frac{1}{\frac{n^2-5n}{n^2-n^\frac{3}{2}+1}})^{(\frac{n^2-5n}{n^2-n^\frac{3}{2}+1})}[/math]
doesn't converge to e because it's not in the form of 1^infinity. Right?

Assuming that my typo was the actual succession (so in the starting succession we have 5n instead of n^2 at the numerator. Is everything else right or are there still mistakes?

>>14947867
If the radiator is working properly, the paper won't catch fire. The pictured sort has a single combined input/return. Steam flows upwards into the radiator, gives up some heat, condenses into liquid, which flows downwards and returns to the boiler to be heated again. The water in the system needs to become steam to travel from the boiler to the radiator (without a pump), but it doesn't need to be much hotter than boiling point. In fact, as soon as the water becomes steam, it rises away from the heat source. A higher operating pressure would elevate the boiling point, but those systems work at low pressures typically, with safety relief valves built in.

do you think that there is a limit to how much time you can study in a row? I don't struggle with motivation or focus but after a while I start making stupid mistakes and forgetting things I just read

>>14947867
All the houses I've stayed in with this style of radiator had a fitted, lacquered wood shelf that would sit on top.A couple bookends and you've got yourself a handy little shelf.

>>14946053
Why is a set countable if there is an injection from it to N? Shouldn't it be a bijection?

>>14948100
Yes, mix studying with 5minute breaks, and break at lunch for 60 minutes language practice + 60 minutes vigorous exercise. You will be a Man my son

>>14948102
There are two common definitions of countable both with their annoying drawbacks:
>countable means injection into N
Now you have to say "countably infinite" a lot of the time
>countable means bijection with N
Now you have to say "finite or countable" a lot of the time

Just be aware of the standard your book is using. Hope this helps

>>14948102
Some authors consider finite sets to be countable too.

>>14948104
>Now you have to say "finite or countable" a lot of the time
Actually
Now you have to say "finite or countable" sometimes

I ended up making a thread on the dra/g/on maid board.

>>>/g/89487959

>>14946846
I tried /sdg/ but it was full of people generating things I didn't want to look at. Discussions were about prompts and not how to use the tech and the install directions are gated in someone's Google drive and I don't really want to use Google if I can avoid it, because they are evil and non-privacy respecting.

>>14946851
Thank you maids from touhou. I will try to figure out what GPU the science computer has. This is the first time in ny life that GPUs have done anything interesting so I am playing catch-up.

>>14948100
Study until you get bored or tired. If you're just bored, study something more interesting until previous topic interests you again.

If you are tired but not bored, go take a nap to recover energy and continue studies after you wake up and eat a snack.

I don't know if you are American or not, or your snack preference so idk exactly what to recommend. Trail mix is usually nice. A pack of nuts or dried fruit is nice too. Pretzels are pretty good too. Drink selection should be made according to how you handle caffeine. I don't like a lot of caffeine, so I avoid energy drinks, but I have heard people say nice things about Monster.

>>14946053
Can anyone tell me why
(where x = 4)
5x +8 -2x +5 = -39
but
5x +8 - 2x +5 = 25
Which one's right?

>>14948305
Surely one small gap between the minus and the number in question, shouldn't be able to change the result? How am I supposed to git gud at math if I can't even get over this? It's fucking bullshit.

>>14948307
I have no idea what you're trying to ask, that polynomial evaluated at X=4 is 25

Is it appropriate to interpret Taylor series as characterizing any differentiable function?
In other words, do all of the derivatives evaluated at a single point of a function completely describe the function itself?

>>14948311
I'm asking why,
1. if you keep the same expressions but change "... - 2x..." to -(2x),
2. where the brackets are 'imagined,'
3. the answer changes from 25, to -39?
4 (extra) isn't that notation extremely fragile?

x = np.array([-1.2, 1])
def f(x):
d = len(x)
if d == 2:
x_store = np.zeros((1,2)) # storing x values
print(x_store)
why print cannot work?

>>14948342
first polynomial: 5×4+8-2×4+5
which is 20+8-8+5=25
second "modified" polynomial: 5×4+8-(2x4)+5
which is once again 20+8-(8)+5
No idea where the -39 comes from, but watch basic algebra on khan academy to get a better grasp on this

How do I prove that if a set of vectors in [math] F^n [/math] span a subspace of dimension [math] p [/math], then resultant vectors in [math] F^{n-1} [/math] obtained by deleting the element in the [math] i[/math]-th position from every vector, span a subspace of dimension [math] p-1 [/math].

>>14948163
Probably I am going to be banned soon for maidposting. Hopefully the information I need is collected before that so I can use it while banned. If this is the last thing I get to say in this thread, thank you for helping me.

I am going to see if I can build a closet supercomputer to make stable diffusion drawings.

>>14948080
>doesn't converge to e because it's not in the form of 1^infinity. Right?
right, in fact its not hard to see that it just converges to 2.
>Is everything else right or are there still mistakes?
maybe?
[math] \displaystyle
x = \frac{-n^{\frac{3}{2}}+5n+1}{n^2-5n} \\
\lim_{n \to \infty} \left( 1 + \frac{1}{x^{-1}} \right) ^{xx^{-1}4n} \; \; \; \; \; \; \; x^{-1} \to \infty \\
\lim_{n \to \infty} e ^ {x4n} \\
\lim_{n \to \infty} e ^ {4 \frac{-n^{\frac{5}{2}}+5n^2+n}{n^2-5n}} \to e^{-4\sqrt{n}}
[/math]
im not sure if the jump from line 2 to 3 is justified. you cant just pull exponents out of limits like that.
did you try what wemi suggested? >>14947507
>>14948349
put "global x_store" at the top of the function under def f(x), otherwise it gets deleted once the function is done.

>>14948450
Nevermind.

>>14948317
It works like that for functions [math]\mathbb{C} \to \mathbb{C} [/math] but not for functions [math]\mathbb{R} \to \mathbb{R} [/math].
https://en.wikipedia.org/wiki/Non-analytic_smooth_function

>>14948317
the more derivatives you take the wider it gets. but there is some agreement between the nth and n+1th derivative that prevents divergence. it can unfurl to infinity, from the extra data on one point. rather strange, because it is not one point, but infinite operations within the topological density of point.

>>14948342
There is no conceivable way of misinterpreting that expression that ends up with -39.

>>14948342
i really really want to know how you get -39

>>14948305
>>14948840
>>14948852
[math]5x + 8(-2)x + 5 = 20 - 64 + 5= -39[/math]. What program are you using that actually interprets it this way?

How do you factor: [math]49n^{2}-3[/math]?
Should I just leave it as it is?

>>14949000
(7n - sqrt(3))((7n + sqrt(3))

>>14949005
thanks fren.

I’m learning alg geo from hartshorne and he doesn’t really treat the case of varieties over non algebraically closed fields.
For algebraically closed fields, he gives a functor from varieties over k (as defined in chapter 1), to “abstract” vatieties over k (a scheme over k with certain conditions). And this functor is fully faithful.
Does the same functor essentially work for non algebraically closed fields? My understanding is that A_k^1 (as a scheme) has closed points which don’t correspond to points of k. For example, over R, there’s the closed point corresponding to the ideal generated by X^2+1.
I guess you can also look at the residue fields of points to determine if they’re actually in R or not, but it still seems like the scheme version of a variety over k carries more info than the more concrete version you get by looking at zeroes of polynomials.
I guess I don’t have any particular question, I’m just wondering exactly what the relation here is

Can anyone write this pseudocode in MATLAB or python?

Is there any scientifically backed way to stop or slow a receding hairline without risking your testosterone?

>>14947845
>ADHD is probably the most common one.
That solves the puzzle, thanks.

>>14946053
What are the practical applications of snap crackle pop lock and drop? Joun e i can sort of understand, and i get absement, but all the others?

I've heard that ADHD is often comorbid with Autism. Is there a neurological reason for this?

What is Flanders Scarlet's favorite blood type?

>>14949462
Yes, the neural circuits involved in both overlap.

>>14949463
If we go by japanese blood type horoscope, Flandre probably prefers B type and Remilia A type.

>>14949509
Remilia canonically prefers B-

>>14949512
Unfortunately.

>>14949284
I really wish this was a show. It would be a better vampire show than Castlevania was. Castlevania didn't have any maids or fanservice in it.

>>14949509
>>14949512
My theory is that flan prefers B+ because she is very much remis opposite. Remi is polite and formal while flan is informal and sometimes sassy. Remi is outgoing and likes parties while flan is a shut in. Remi is a tactician while flan can rely simply on brute force. Ect.

>>14949546
How is B+ B-'s opposite?

>>14949553
Because it is + instead of - also I'm completely rationalizing this shit because I'm B+

>>14949556
>also I'm completely rationalizing this shit because I'm B+
Same but I'm A+.

>>14949526
There already is a touhou anime.

Back to science questions.
Why is it that the mitochondrial DNA of animals and fungi deviate more from the standard genetic code than the plastids and mitochondria of green plants and algae. Green plants and algae are so much more ancient and diverse so I would think there should be more deviation not less if it was completely random.

>>14949567
Yeah and its boring as shit

>>14949594
>>14949567
Is it a maid show?

Also maybe the mushroom answer is part of why they taste so good? Mushroom tastes nice on everything.

>>14949594
Fair enough.
>>14949609
No.

Next science question:

What are some effective nootropics that don't require a prescription?

Where does maid anon live? I wanna dress her up in cute girl clothes.

>>14949630
https://www.gwern.net/Nootropics

how to prove that [math]13^n - 7[/math] is divisible by 6? I think I'm supposed to use induction but I get stuck on it

>>14950037
[math]13^n=(12+1)13^{n-1}[/math] so you can gradually decompose it to 13 - 7 which is divisible by 6.

Why does the anti-derivative give you the area under the curve. In simple terms, like for multiplication and division it can go x*z = y & y/z = x. That's an intuitive opposite. You always know it right away.
But I don't understand why I can use x^2 for the function 2x, or -cos(x) for sin(x) to find the entirety of the area. When I graph and visualise these functions the idea makes sense, x^2 growing much faster than 2x because of the ever growing accumulation for the area, sin(x) and -cos(x) continuously adding and subtracting from the area.
Am I thinking about it wrong referencing */ and +- opposites?
Do I think of each anti-derivative as standing on its own for each of its corresponding derivative function, that is they all need their own proof which can be done in different ways? Which is why you usually just memorise them in calculus classes?

>>14950155
because the instataneous change of the function

F(x) = the area under the graph of f between points a and x

is an infinitely thin rectangle with height equal to f(x)

>>14950182
but what I don't get is how integral(b)-integral(a) gives you the area under a continuous, changing line between those two points.
How does it sum all of those infinitely thin rectangles from subtracting 1 substitutions into the function with another. Does that simply depend on the specific anti-derivative and how it was found?

([x] - floor function)
[math]a > 1 \\ x_n := \frac{1}{n^2 + 1} + \frac{1}{n^2 + 2} + ... + \frac{1}{n^2 + [na]}[/math]
[math] \lim_{n \to \infty } x_n [/math]
Any idea what could this be? I've got a bunch of these fucking sums and I can't figure it out.

What's the difference between the chain rule and the general power rule, they look the same and does the same thing.

>>14950623
General power rule is a special case of the chain rule

>>14950485
Isn't it 0? since the limit of each term is 0?

>>14950657
the harmonic series diverges even though it's a sum of terms that converge towards 0

>>14950659
I meant that each term gets smaller as n gets bigger, in the harmonic series the second term is 1/2 and as n gets bigger it stays 1/2 and doesn't change, this is not true for your sum.

Anyway, I'm pretty sure you can show its 0 with some inequality like this
[eqn] x_n = \frac{1}{n^2+1} + \frac{1}{n^2+2} + \dotsb + \frac{1}{n^2+ \lfloor na \rfloor} \leq \frac{1}{n^2+1} + \frac{1}{n^2+1} + \dotsb + \frac{1}{n^2+1} = \frac{n}{n^2+1} \\ \lim_{n \rightarrow \infty} \frac{n}{n^2+1} = 0 \geq \lim_{n \rightarrow \infty} x_n [/eqn]
Since the this is a sum of non-negative terms then the original sum must be 0.

>>14950657
No, you can calculate the lower bound using the Am > Hm inequality (though it's useless because I'm looking for the upper bound).

bros, I'm getting good at maths. Problems that seemed impossible are easy now. Finals are near and I feel like I have a good grasp on the material. Thanks for answering my questions. Feels good to not feel like a retard.

>>14950673
Hmm... that does make sense actually. Now that I look at it the limit of the lower bound I calculated >>14950674 is also zero
Though I think it only holds up if the series is actually convergent which I don't think is necessarily true

>>14950673
Just a slight note, i got confused on the number of terms, i thought there were [math]n[/math] where you just multiply by [math]a[/math], but it's actually [math] \lfloor na \rfloor [/math], anyway this doesn't change anything because
[eqn] x_n \leq \frac{\lfloor an \rfloor}{n^2+1} \leq \frac{an}{n^2+1} [/eqn]And [math]a[/math] is a constant so the limit is still 0 for the last one.
>>14950692
>Though I think it only holds up if the series is actually convergent which I don't think is necessarily true
Well each [math]x_n[/math] is a finite series, you're looking for the limit of the sequence [math]x_1, x_2, x_3, \ldots[/math] you can check that [math]x_{n+1} < x_n[/math] so this is a decreasing sequence that is bounded below so it must converge.

>>14950037
this is pretty easy to see at a glance:
-7 divided by 6 has a remainder of 5 (think about it)
13 has a remainder of 1, which means that [math]13^n[/math] has a remainder of [math]1^n[/math], which is just 1

>>14950037
Just use modular arithmetic [eqn] 13 \equiv 1 \mod 6 \\ 13^n \equiv 1^n \mod 6 \\ 13^n \equiv 1 \mod 6 [/eqn]
Adding this to [eqn] -7 \equiv -1 \mod 6 [/eqn]We get [eqn] 13^n - 7 \equiv 0 \mod 6 [/eqn]

>>14946053
>nickel atoms like pearls into a string of of molecular beads made of carbon and sulfur, and began testing.
so the "beads" are made out of carbon and sulfur AND nickel?
they took smaller pieces of nickel (atoms) and added it to a mixture of carbon and sulfur?
help me out here

Can someone confirm if I've done this integral correctly? It's been a loooong time since I've done anything like this

>>14950771
>you can check that [math]x_n+1<x_n[/math]
The problem is that this is not straightforward to show for every [math]x_n[/math] because for each [math]n+1[/math], each term gets smaller but you get a larger amount of terms (multiplied by the constant [math]a[/math]), so in effect for big [math]a[/math] it may not be true for all [math]n[/math]
[eqn]\sum^{[na + a]} _{k = 1}\frac{1}{(n+1)^2 + k} < \sum^{[na]} _{k = 1}\frac{1}{n^2 + k}[/eqn]
I'm not actually sure whether it's actually decreasing for all [math]a[/math].. there may be big enough values where the first few terms of the [math]x_n[/math] series are increasing, although you can probably find a sufficiently large [math]n[/math] such that the series is decreasing (but I'm not sure how to prove that).

>>14951036
ah that sine function cant be simplified to just 1, I forgot -1 was a thing

>>14951036
It looks like you used big pipe at the end but "bigg" pipe would have been better
[math] \int_a^b f'(x)\,dx= f(x)\bigg|_a^b [/math]

>>14951165
idk how to get the bigg pipe in microsoft word. Im using unicode because I dont know latex

In the diagram I made, if object A is moving at a certain speed in a certain direction and I know that object B has a fast enough speed that it can collide with object A, how do I determine the angle to give object B?

I've been trying to figure this out for hours and I'm not making any progress.

>>14951184
let [math]\alpha[/math] and [math]\beta[/math] be the angles inside the triangle at points [math]A[/math] and [math]B[/math], respectively, and let [math]a[/math] and [math]b[/math] be, confusingly, the paths that [math]B[/math] and [math]A[/math] travel, respectively (its traditional to name the sides of a triangle after the angles opposite to them). then by the law of cosines:
[math] \displaystyle
\frac{\cos(\alpha)}{a} = \frac{\cos(\beta)}{b}
[/math]
you want to equate the time it takes the objects to travel their paths, i.e.
[math] \displaystyle
\frac{b}{v_1}=\frac{a}{v_2} \\
bv_2 = av_1
[/math]
multiply the first equation by some stuff and cancel out the denominators (since theyre equal)
[math] \displaystyle
\frac{\cos(\alpha)}{a} \left( \frac{v_1}{v_1} \right) = \frac{\cos(\beta)}{b} \left( \frac{v_2}{v_2} \right) \\
\frac{v_1 \cos(\alpha)}{av_1} = \frac{v_2 \cos(\beta)}{bv_2} \\
v_1 \cos(\alpha) = v_2 \cos(\beta) \\
\beta = \cos ^{-1} \left( \frac{v_1 \cos(\alpha)}{v_2} \right)
[/math]

Whats the formal name of adding something to both sides of an equation?

I dont even know what to search because i dont know the name

like:
a = b
a -x = b -x

I don't understand this. Where does that factor come from?

Explain "if and only if" to [me] a third grader.

>>14951345
I would just call it a property of equality. This is the first thing that shows up when I google that.


>>14951363
It means two things are both true or both false.
Let's say x if and only if y
Then these are possible
-x is true and y is true
-x is false and y is false

These are not possible
-x is true and y is false
-x is false and y is true

>>14951376
So they are dependent on one another to be true (In a general circumstance)?

Can someone solve this question or atleast tell me how it is solved? I missed the classes where sir taught this and I don't have any friends in that class to take the notes from.
All I remember is that sir said to draw 4 graphs and use the one that yield a straight line. But, like I said, I have no idea what he's talking about so I can't do anything. This assignment is due in a few hours and idk what to do.

>>14951361
its the area of the trapezoid that you clearly thought you were too good to draw.
[eqn]
\int_0^5 \frac{1}{50} (10 - x ) dx = \frac{1}{50} \bigg |10x - \frac{x^2}{2} \bigg |_0^5 = \frac{1}{50} \bigg |\frac{20x - x^2}{2} \bigg |_0^5 = \frac{1}{50} \bigg |\frac{(20 - x)x}{2} \bigg |_0^5 = \frac{1}{50} \cdot \frac{(20-5) \cdot 5}{2} = \frac{1}{50} \cdot \frac{(10+5) \cdot 5}{2}
[/eqn]

give me one reason why automating the arts was a good idea

I suck at word questions, so ples help.
The answer to questions 3 and 5 are the same, right? I'll post the answer I got in a reply to this post.

>>14951690

>>14951690
>>14951696
I think you forgot to include the students who want spanish, french and latin in v)

>>14951701
I don't understand. They're asking of the 16 students who take french, how many take Spanish. Isn't it 7 (4+3)?

>>14951720
Oh, you answered 7. My bad.
But the answer to 3 would be 4, only spanish and french, not latin.

>>14951725
3) Asks how many students want French and Spanish. It doesn't say French and Spanish only. can't you say 7 for that as well?
Again, this is why I hate word questions.

When i see a Reimman sum in a class without specifying that it is from positive to negative infinity, that is still what we assume, correct?

>>14951738
Yes it's bullshit.
But the way I interpret it 3) is the same as Spanish and French and not Latin.
But feel free to interpret it another way. It's ambiguous.

>>14951759
Thank you, fren. You can have this waifu I made in sd and haven't shared with anyone yet.

>>14951779
Elfuuuuuu

>>14951291
Thank you for your answer!

>>14951383
bump. I need the answer

>>14951090
Apologies i got confused again and thought there were only n terms. You're right, for a large value of [math] a [/math] the sum on the left can be bigger than the sum on the right, however you can find that each [math] x_n [/math] has a lower and upper bound using an inequality similar to the one in >>14950673
[eqn] \frac{an-1}{n^{2}+an}\le\frac{\left\lfloor an\right\rfloor }{n^{2}+\left\lfloor an\right\rfloor }\leq x_{n}\leq\frac{\left\lfloor an\right\rfloor }{n^{2}+1}\le\frac{an}{n^{2}+1} [/eqn]
And since [eqn] \lim_{n\rightarrow\infty}\frac{an-1}{n^{2}+an}=\lim_{n\rightarrow\infty}\frac{an}{n^{2}+1}=0 [/eqn]Then sequence of [math] x_n [/math] must converge to 0 by the squeeze theorem.

>>14951383
>>14951901
The rate of the reaction can be described as the change of the concentration of the reactant [A] with time[eqn] \frac{d[A]}{dt}=-k[A]^{n} [/eqn]Where [math]n[/math] is the order of the reaction. This is a separable differential equation which you can solve easily to end up with
[eqn] \begin{matrix} \frac{1}{[A]^{n-1}}=\frac{1}{[A]_{0}^{n-1}}+\frac{kt}{n-1} & \text{For $n\neq1$} \\ \ln[A]=\ln[A]_{0}-kt & \text{For $n=1$} \end{matrix} [/eqn]
Where [math] [A]_0 [/math] is the initial concentration. Therefore you should plot [math] [A], \ln[A], [A]^{-1}, [A]^{-2} [/math] against time, these plots correspond to zero, first, second and third order kinetics respectively. You should see which one of these plots gives a line to find the order of the reaction.

>>14952192
I didn't think of applying the squeeze theorem here at all. Thanks a lot!

>>14952192
I plotted the graphs. One of the other guys found a Chegg answer that says the correct one is the 2nd order due to high R^2 value. But, in my graphs, the 3rd order one is the one with a straight line.

>>14952307
Determine the rate constant and the initial concentration from your second order kinetic line equation (the slope of the line is [math] k/2 [/math] and the y intercept is [math] 1/[A]_0^2 [/math])
Then after determining the rate constant and the initial concentration graph
[eqn] [A] = \sqrt{\frac{1}{(1/[A]_0^{2}) + (k/2)t}} [/eqn]
And see if it fits the original data then compare to the answer of the other guy.

>>14952342
Thanks. I found out it's the 3rd order.

stupid question
simple proof for the fact that [math]b \in \mathbb{Q} \land b>0, \lim_{n \rightarrow \infty} \frac{\lfloor nb \rfloor}{n} = b [/math]?
i feel like there has to be a nice elegant proof of this property

>>14952729
[math]\displaystyle \lim_{n \to \infty} \dfrac{nb - 1}{n} = \lim_{n \to \infty} \dfrac{nb}{n} + \lim_{n \to \infty} \dfrac{1}{n} = b[/math]
[math]\displaystyle b = \lim_{n \to \infty} \dfrac{nb - 1}{n} \leq \lim_{n \to \infty} \dfrac{\lfloor nb \rfloor }{b} \leq \lim_{n \to \infty} \dfrac{nb}{b} = b[/math]

Bros, how do I even do this? I have no clue how to derive the answer it's asking for.

Also, do I have to put the parenthesis I've indicated with the black pen? Because not doing that correctly will mess with the order of operations.

>>14952745
Goofed the fraction on the second half lmao.
[math]\displaystyle b = \lim_{n \to \infty} \dfrac{nb - 1}{n} \leq \lim_{n \to \infty} \dfrac{\lfloor nb \rfloor }{n} \leq \lim_{n \to \infty} \dfrac{nb}{n} = b[/math]

>>14952729
Use the fact that
[math] nb-1 < \lfloor nb \rfloor \leq nb [/math]
Which follows from the definition of the floor function, the limits of these two bounds are the same
[eqn] \lim_{n \rightarrow \infty} \frac{nb-1}{n} = \lim_{n \rightarrow \infty} \frac{nb}{n} = b [/eqn]
So now you can just say the limit is b with the squeeze theorem.

>>14952747
Better pic

Is there a series where [math]\lim a_n = +\infty \land \forall n \in \mathbb{N} a_n > 0 \land \frac{a_{n+1}}{a_n} < 1[/math]?

>>14952756
bro wtf are you sure you got the question written down correctly?

>>14952790
The harmonic series

>>14952790
>>14952796
Actually no wait, ignore what i said, you can't have such a series because [math] \frac{a_{n+1}}{a_n}<1 \Longrightarrow a_{n+1} < a_n [/math] which means the series is decreasing so it can't diverge to positive infinity.

>>14952796
Nope. [math]\frac{\frac{1}{n}}{\frac{1}{n+1}} = \frac{n+1}{n} > 1[/math]

>>14952802
I phrased that wrong, I meant a series where [math]\frac{a_{2n+1}}{a_{2n}} < 1[/math], i.e. every odd term is smaller than the previous even term. I think it could be overall divergent to infinity if also every odd term is smaller than the next odd term.

>>14951877
thank you for your (You) <3

>>14952811
so [math]a_n[/math] is the n-th term and not the n-th partial sum? you really need to specify that.
If the terms are all positive they're getting larger then the series is divergent.

Would the series 0+1+0+1+0+1+... fit your criteria?

>>14952811
Series or sequence?

>>14952833
I forgot this is a thing in english (we call series and sequences by the same thing in my native language)
I meant sequence

>>14952838
consider
1,0,2,1,3,2,4,3,5,4,6,5,7,6,8,7,...

>>14952846
Nice.

How do I go around proving pic related? working on a neighbourhood where the partial derivatives of the metric tensor are 0, btw.

>>14952871
What's the [math]\delta[/math] for?

>>14952899
variation, delta of f(x) is differentiating f(x+th) in t and evaluating the result at t=0. It commutates with normal derivation and basically has the same product/chain rules as it.

>>14952924
I'm not curious about what's a variation, I'm curious about what's being variated.

You have an infinitesimal anti-symmetric variation of [math]g^{\mu \nu}[/math] which induces a variation of [math]\Gamma ^{\lambda} _{\mu \lambda, \nu}[/math]?

>>14952932
It's just being used as a method in a larger process to prove Einstein's equation

what is ramification theory from a homotopical point of view?

Are there any youtube videos that are good for sets and relations? I'm drowning in all the symbols and terms.

Like a "collection of equivalence classes of a relation on a set"

I was maybe 50% on sets, and after cartesian products, ordered pairs, and now relations its too much

Where the fuck does y-x comes from?

>>14953174
x < y iff y - x > 0

>>14953191

>>14952792
Is the question wrong, anon?

>>14953255
>question says y is basically linear in terms of x but the dfferential of y is a seventh degree polynomial in terms of x
I dunno sounds retarded to me.

> a+b=x
> 2(a+b) =2x

what is the term for the relationship between these two formulas?

>>14953547
Scalar multiple

>>14953255

How do I prove that [math]f(x)\geq ln(f(x))[/math]?

>>14953754
What if f(x) < 0

>>14953761
damn

>>14953547
linearly dependent

>>14953754
If [math]f(x)[/math] is strictly positive consider [math]y := f(x)[/math].

For [math]y\geq 1[/math]:
[eqn] \ln(y) = \int_1^y \frac{dt}{t} \leq \int_1^y dt = y - 1 < y[/eqn]
For [math]0 < y < 1[/math]:
[eqn] \ln(y) = - \int_y^1 \frac{dt}{t} < 0 < y[/eqn]

>>14953822
thanks!

How do I apply gauss law when there are more than 1 dielectrics?

$1 million prize for solving one of the 6 major math problems?

>chemical potential
>gibbs free energy
whats the difference

5+5*5^5 tetrated to 5 pentated to 5 hexated to 5 ... =?

>>14954283
5 + (5 * (5 ^ 5) )

>>14946053

>>14954580
meant to ask how one determines the outer bounds, If someone can walk me through it

>>14953268
>>14953608
Thanks, anons

>>14954587
You solve
2 cos(theta) >= 1

I saw that a lot of people were getting an iPad for their maths at uni, at first I couldn't really understand why but I made the plunge to get one and I have to say, its helped massively. I think a small part that no ones thinks of is having to sift through the scraps of paper to find a little equation. I got the iPad air with the apple pencil and a paper style screen protector and it has made learning a lot more enjoyable this year (which I think was needed) and one else have an iPad for learning?

>>14955048
How much did they pay you for this post

Artin algebra

Does anyone have solutions or even answers to exercises in Chapter 1 Matrices?

Why is an ordered pair defined like this?

{x | x = {a} v x = {a,b}}

why is {b,b} not included? and why is {a} included on its own? why

The exam was supposed to be multiple choice, three hours - now its multiple choice with the whole day to do it (as in they release it at midnight and we have until the following midnight to submit). Is this a good sgin?

>>14955075
Multiple elements in the set are the same
{b,b} = {b}
If you include it then an ordered pair becomes non-ordered because
(a,b) = {{a},{a,b},{b}} = {{b},{a,b},{a}} = (b,a)
So basically you failed to define an ordered pair

Remember that you're trying to define it in such a way that (a,b) is not equal to (b,a)

>>14955083
ty for explaining anon i will keep working to try and understand

Just had a E&M exam and I cannot for the life of me figure out what my prof was asking from me.
Question posed as follows:
Suppose we have an infinite conducting checkerboard where each black/white cell is insulated from adjacent cells. We charge each "black" cell with potential +V and each "white" cell with potential -V. If we move far away from the infinite checkerboard, the potential tends towards zero.
Is the following statement true or false? If we made the cells smaller, the potential would tend towards zero faster as we moved away from the checkerboard.

I suspect that this question has to do with Green functions and the "magic rule" but I am stumped as to how I should be interpreting either in the context of this problem.

>>14955222
i dont have a clue but what if you made the cells larger, or infinitely larger. Would that do anything?

>>14949462
Anon, don't entangle your mind with these pop sci "diagnoses".
It's fucking aggravating and disgusting that big pharma has so much power to dole out this shit like candy because people were butthurt over the fact someone didn't give a fuck about their meager existence.
Not only that, liberal judicial systems are in on it, and will weaponize it to further strip you of your humanity - invalidating/discounting and/or discrediting any accomplishments, in addition to, jeopardizing potential opportunities until death.

>>14955225
This was literally my line of thinking. If we made the cells infinitely large, the electric field produced by either a black or white cell would be constant (opposite in direction, though) regardless of distance from the board, which obviously corresponds to a non-zero potential at infinity. So, if we did the opposite, the reverse should be true.
Typing this just now made me think a little harder in the limit that each cell's size tends towards zero. In that case, adjacent cells would act like point dipoles, which means the checkerboard can be (in this limit) thought of as an infinite planar dielectric with polarization field vector P.

>>14955239
>If we made the cells infinitely large, the electric field produced by either a black or white cell would be constant (opposite in direction, though) regardless of distance from the board, which obviously corresponds to a non-zero potential at infinity.
uh, no. making the cells infinitely large would be like having an infinite plane of potential, which still goes to zero at infinity.
heres a hint: you can think of a non-ionized atom as an electric dipole.

>>14955280
oh shit nvm, potential would be infinite. ive brought shame on my wife but im not a bitch so im not deleting the post.

>>14955048
I have an iPad regular that I bought for learning, yes. Then I bought an Pro for drawing.

I've been using it to take notes for like 3 years now, my biggest advice is to switch to paper occasionally. Once you've gotten used to taking notes on the iPad (6 months+), you'd start to get lazy and take bad notes.

It's definitely good for organizing as it organizes everything by default. You can also use it to annotate pdfs.

>>14955076
Sure.

>>14955280
I think I have a bit of an intuition of what the question was asking from me in the second part of my post. When shrink the cell sizes down to zero, the plane becomes an infinite sheet of dipoles that is electrically neutral but essentially acts as an infinite dielectric sheet under the influence of (what I can only assume to be) a fictitious. My stance then tends towards true, but I'm still confused on why a uniformly polarized sheet of dielectric material would produce an electric field that tends faster towards zero as you move away from it versus the checkerboard scenario with non-zero and non-finite cell sizes.

>>14955292
Daddy Griffiths is here to help!

>>14955308
a fictitious constant electric field.*

>>14955308
i literally had Griffiths pulled up to check the section on dipoles when i found that bit about the infinite plane.
at any rate, i think this would be a pretty hard question to answer rigorously. did you find pic rel yet?

>>14955064
Just post the questions that confuse you.
>>14955222
Sounds intuitively true.
>>14955308
>I'm still confused on why a uniformly polarized sheet of dielectric material would produce an electric field that tends faster towards zero as you move away from it versus the checkerboard scenario with non-zero and non-finite cell sizes.
For points really close to say, the middle of a black square in the checkerboard, the field from the white squares doesn't really cancel out the field from the black square, because muh distance.
>>14955326
>pretty hard question to answer rigorously
It's not even stated rigorously.

>>14955332
>It's not even stated rigorously.
what parts are ambiguous?

>>14955335
>it's not rigorous hence it's ambiguous
Not really how this works.

But the last bit is unspecific.
>the potential would tend towards zero faster as we moved away from the checkerboard

>>14955326
Not trying to answer this question rigorously, this is a purely conceptual question.

>>14955337
>But the last bit is unspecific.
Say the potential due to a checkerboard of cell size [math] d \times d [/math] is of the form [math]V\sim \frac{1}{r^l}[/math], then a checker board of cell size [math]b \times b[/math] would have a potential of the form [math]V\sim \frac{1}{r^m}[/math] where [math] m > l [/math]. Is this true?

>>14955352
Shit, forgot to add that [math] b < d [/math]. Also, [math]r[/math] is just the distance from the checkerboard.

How do I solve a linear system of equations over a commutative ring (you can assume Z/nZ if it's easier) which is not a field?
Either I was never taught this in algebra or I've forgotten and I'm brainfarting on how I'm supposed to go about it

>>14955487
https://en.m.wikipedia.org/wiki/Diophantine_equation

What are the conditions on the side called where it says

[math]a,b \neq{ 0} and~2a \neq{b}[/math]

And how do you put them into Wolfram?

What are some classics or recommended books for control theory or feedback control ? Is this even the right place to ask this ? pls no bully

I'm taking my first course in linear algebra, and was looking for (REDACTED). After looking at a bunch of answers, I've noticed that there's a tendency to mention sets, or (topic we haven't got to yet).

Maybe I just need some more humility and to study more, but it also feels like there's a psychological tendency to give "correct" (general?) answers that end up more complicated because now we have two or three interacting parts.

Is it just me?

>>14955729
What?

The product of subgroups HK in an abelian group is also a subgroup, since trivially HK = KH. Right?

Explain.
[math] \nu [/math] = nullity
[math] \rho [/math] = rank

>>14955777
Nevermind.
>>14955729
What?

Help me with this notation, please.
I know what attaching spaces via the gluing map means. But what does the gluing via the Cartesian product of the map and {1} mean?

>>14955810
[math]\phi: S^p \times D^q \to M[/math]
[math]\phi \times \{ 1 \}: S^p \times D^q \to M \times I[/math], where the second coordinate always takes the value 1.

>>14955831
Thank you, that makes sense

>prove that if [math](a_n)[/math] is a sequence of positive numbers which diverges to infinity, then
[math]\displaystyle \lim_{n \to \infty}\frac{1}{2^n} (1 + \frac{a_2}{a_1})(1 + \frac{a_3}{a_2})...(1 + \frac{a_{n+1}}{a_n}) = +\infty[/math]
i think this is easy to show for an increasing sequence but I'm not sure how to approach this for any sequence that diverges to infinity. any clues?

>>14946072
wemi

>>14955863
wouldn't a sequence of positive numbers diverging to infinity be necessarily increasing though

Please tell me because I am uninformed:

If, hypothetically, say we had four people who each built for themselves a perfect, ideal, most precise atom clock; would we have four completely synchronous time measurements?

Can four such hypothetical-clocks, when each measures time potentially quantum-level dimensions - and we know of basic improbability - ever display the exact same time when measured from independent channels?

Additionally, time in this way is relatively and physically tied to the biases of the observer, can individuals actually experience time with actual, measurable differences, and does the "speed of time" fluctuate or is it constant throughout?

>>14955931
Retard, google harmonic series.

>>14955956
good thing the topic here is sequences, not series

>>14955863
Use AM-GM

[eqn] \lim_{n \to \infty} \prod_{k=1}^n \frac{1 + \frac{a_{k+1}}{a_k}}{2} \geq \lim_{n \to \infty} \prod_{k=1}^n \sqrt{\frac{a_{k+1}}{a_k}} = \lim_{n \to \infty} \sqrt{\frac{a_{n+1}}{a_1}} = \infty [/eqn]

>>14955962
>>14955972
wait, nevermind I misread what you wrote.

>>14955949
No

You would have four different clocks each with a different location in space, reading 4 separate times, subject to variables. The readings would be measured relatively using a protocol such as network time (ntp) and arranged by stratum. Stratum 0 being a caesium atomic clocksource, then 1 and 2 as distributors of these time sources, with the remaining 14 stratum levels serving as a type of consensus network.

>>14956010

I love you, thank you

In real polynomial functions,
I have three roots and are asked to find all possible grade 4 polynomials that go through those.
I can't find anything on how to solve it.
The guide has the answers, but I don't know how to go about it. Using the factored form I get a grade 3 solution.
The textbook simply squares each factor, giving three possible solutions. But can't really see why or how.
Roots in the problem were 0, -3 and 5. (Grade 4)

Any help would be gladly appreciated.

>>14956381
You've found the third degree polynomial [math]p(x) = ax(x + 3)(x - 5)[/math]. If a fourth degree polynomial zeroes on 0, -3 and 5 it can be written as [math]p(x)q(x)[/math], where [math]q(x)[/math] has degree one. Now, [math]q(x)[/math] can't have any roots other than [math]0, -3, 5[/math], so it needs to be one of [math]x, x + 3, x - 5[/math], hence the possible solutions are the ones you've listed.

>>14956399
Thanks a lot for replying!
Well, that does make sense. Though there's still something I can't make sense out of.
A root being a value for x such that its corresponding y=0 in a given function, can possibly be repeated within a function without being the same one (and hence, not repeated)? How so?
Thanks again.

Let V be a vector space. dim(V) = 8. Let H, K be subspaces of V. dim(H) = 5, dim(K) = 4.
Then;
a. K [math]\subseteq[/math] H
b. dim(H[math]\cap [/math]K) = 1
c. H + K = V
d. H [math] \cap [/math] K != (the empty set)

by logic my answer would be d, but im not sure why. Can you tell me why the others are weong?

Also, while trying to use Grassmann formula I got stuck on this; if I know dim(H1) and dim(H2), where H1 and H2 are two subspaces, can i also know dim(H1+H2)? without actually knowing H1 and H2, only their dimension. I wonder if the fact that any n-dimensional subspace is isomorph to K^n (where K is a generic field) can help with this.

>>14956430
>A root being a value for x such that its corresponding y=0 in a given function, can possibly be repeated within a function without being the same one (and hence, not repeated)? How so?
If [math]f, g[/math] are functions and [math]f(a) = 0[/math], then trivially [math]fg(a) = f(a) g(a) = 0[/math], since anything times 0 is zero.

Also, if [math]f[/math] and [math]g[/math] are both differentiable, and [math]f(a) = g(a) = 0[/math], then [math]fg'(a) = f'(a)g(a) + f(a)g'(a) = 0[/math]. In the particular case of [math]f[/math] and [math]g[/math] being polynomials, we use this to say that [math]fg[/math] has a zero of degree two at 0.

>>14956456
>.a zero of degree two at 0.a zero of degree two at 0.
At a, I mean.

>>14956440
H+K is a subspace of V so its dimension is at most the dimension of V.


dim(H) + dim(K) = dim(H∩K) + dim(H+K) <= dim(H∩K) + dim(V)
9 <= dim(H∩K) + 8
1 <= dim(H∩K)

>>14956477
alright i think i understand why it's D. so does this result mean that given the dimension of two subspaces and the dimension of the vector space that they belong in, I can't know exactly what the dimension of their intersection is? Because it would depend on the choice of the two actual subspaces, right? thank you

trying to get a performance boost in solving Ax = b hundreds of thousands of times
A and b change, but A is always a minor of a larger matrix M
can A^-1 be computed any quicker by precomputing the major inverse M^-1?
(also i'm holding off on matrix decomposition because this is over a finite field as well)

>>14956635
> can A^-1 be computed any quicker by precomputing the major inverse M^-1?
In general, no. However there are much better methods that do not require computing the inverse. Go read up on LU decomposition.

>>14956663
pivoting makes the later code messy, but for my current use case RREF has yet to fail even once
also L_minor and U_minor can probably be computed directly from M since every element is a power of a primitive

Is it accurate to say Direct Current has a frequency of 0 Hz, or does frequency simply not apply as a magnitude in DC?

How do you prove Bezout's Identity without sets?

how do i convert the boxed part into polar?

>>14957051
[math]x = r \cos(\theta)[/math]
[math]y = r \sin(\theta)[/math]

"There is no significant physical distinction between interference and diffraction. It has, however, become somewhat customary, if not always appropriate, to speak of interference when considering the superposition of only a few waves and diffraction when treating a large number of waves. Even so, one refers to multiple-beam interference in one context and diffraction from a grating in another"
Just read that from Hetch's Optics. If so, why do we study interference and diffraction as two different phenomena? Why don't we just study diffraction as a multiple light beam interference? Are they really the same phenomena?

>>14957051
substitute the polar expressions for x and y and solve
you will get [math]r = 4\cos\theta[/math]
>>14956966
https://proofwiki.org/wiki/B%C3%A9zout%27s_Identity#Proof_3
>>14956952
DC doesn't have a frequency in the usual sense but when taking the Fourier transform of an AC signal people will refer to the 0 Hz value as the "DC component", this is ultimately a terminology question, both ways of thinking are kind of correct

>>14957117
And what did he mean by " if not always appropriate", when is it not appropriate?

>>14957119
im sorry but could you walk me through the steps to find 4 cos theta, i just dont understand how you arrived at that by replacing x and y with r cos, r sin respectively

>>14957117
if n is the number of sources, then the behavior is qualitatively different for small n (interference) and [math]n\rightarrow\infty[/math] (diffraction)
in addition, you should find in the textbook that the limit to infinity allows diffraction to use a different (simpler) mathematical derivation
it's analogous to the difference between mechanics and thermodynamics, thermodynamics ultimately emerges from the mechanics of very many rigid atoms, but instead of calculating all the equations of motion, you abstract and look at the overall outcome without caring about the exact value of n as long as it's very large

>>14957142
[math]\sin^2\theta+\cos^2\theta=\ldots?[/math]

>>14957179
but there is a -2 by the cos
so im reading it as (cos -2)^2 +sin^2 =4

>>14957224
So expand the bracket.

>>14956456
I will copy this reply and keep it for the time being. I have an (a different one) exam tomorrow, and in the afternoon I felt kind of sick. I couldn't yet completely parse it, but if I'm able, I'm going back to it, and if needed, back here for questions, or at least a thanks.
Thanks for replying, anon.

what's the best way to stop a chlorine + muriatic acid reaction?
>be pool guy
>carry jugs of chlorine and muriatic acid in my truck and in my dolly cart
>shitty cap on a jug of acid let some leak onto a stack of chlorine tabs at the bottom of the cart
>hear sizzling
>catch a good whiff of something
>look down and see the chlorine puck smoking
>push cart near edge of pool and hold my breath while I separate out all the chemicals
>mfw

https://healthfully.com/clicking-problems-in-the-wrists-8596152.html
>Injuries or congenital malformations of the wrist may cause the wrist bones to fail to line up correctly, which can lead to rubbing and clicking.
How common is this? My wrists have clicked since forever and it has had absolutely no effects whatsoever on me.

>>14956795
oops, i'm silly, M isn't even square
there's really no good way to speed this up short of chaining common blocks of the minors

How can I count the size of the similarity class (of matrices in particular)? By dividing the order of the group by the order of centralizer of representative of this class or are there any other possibilities?
Group, for example, is GL(2, k), where k is a field.

I get why x = 21 but how do you work that out?

Is P(A-B) = P(A) - P(B)?

I think yes because it's true for intersection and complement is A intersection ~B

>>14957896
no, empty set is in LHS but not RHS

>>14956795
>pivoting
Speaking of pivoting:

>thank you for calling [math]\mathtt{lu}(A)[/math], here's your decomposition [math]A=PLU[/math], with complimentary row reorderings for numerical stability
>so [math]P^TA=LU[/math], thanks. this means that i can call [math]\mathtt{lu}(P^TA)[/math] and get a proper triangular decomposition with identity permutation, right?
>...
>...right?

Scipy, at least, doesn't guarantee this.
I can see that the usual partial-pivoting approach doesn't let you control the final [math]P[/math], but is it possible to implement a pivoting strategy that can ensure consistency (returns [math]P^TA=ILU[/math]) while still retaining good stability properties?

>>14957896
Only true if B is a subset of A

Proof:
Assume [math] B \subseteq A [/math] then A can be written as the union of two disjoint sets [math] A = (A-B) \cup B [/math], So [math] P(A) = P \left( (A-B) \cup B \right) = P(A-B) + P(B) [/math] and after rearrangement you get [math] P(A-B) = P(A) - P(B) [/math]

>>14958177
You showed that it's true if B is a subset of A. Not that it's true ONLY in that case

>>14957896
Let A be heads, and B be tails.

>>14958177
Right, it's bad wording on my part, i tend to forget "if" and "only if" have a precise meaning and use them sloppily.
You're right that there are cases in which this statement is true when B is not a subset if A.
To turn this into an "if and only if statement" you impose the condition [math] P(B-A) = 0 [/math] so the statement reads "[math] \text{$P(A-B) = P(A) - P(B)$ if and only if $P(B-A) = 0$} [/math]"

Proof:
Starting with [math] P(A-B) = P(A) - P(B) [/math] we have
[math] P(A) = P(A-B) + P(B) \\ = P((A-B) \cup B) \\ = P(A \cup B) \\ = P( A \cup (B-A) ) \\ = P(A) + P(B-A) \\ \Rightarrow P(B-A) = 0[/math]
The proof for the converse (starting with [math] P(B-A) = 0 [/math] ) is exactly the same but with the steps in reverse.

>>14958267
meant for >>14958185

>>14958185
It's an equivalence proof.

The linear combination of two vectors in R2 always generates the whole plane as long as those two vectors aren't proportional right? Does this hold true for R^N? That is, if I have n linearly independent vectors in an n-dimensional space, do they always have to generate the whole space?

>>14958447
If dimension of a finite dimensional vector space is [math] n[/math], then a set of [math] n[/math] vectors being linearly independent is equivalent to them being a basis. So, yes.

[math]a > 1 \\ \displaystyle \lim_{n \to \infty} n \left( \frac{1}{n^2} + \frac{1}{(n+1)^2} + \cdots + \frac{1}{\lfloor na \rfloor^2} \right)[/math]
how would you solve this? I can't figure it out

>>14957490
The article is talking about spontaneous clicking from simple movements (crepitus from inflammation) which not the same as the clicking you get when stretching a joint (joint crepitus or joint cracking), the former is associated with pain and the latter is not.
See https://en.wikipedia.org/wiki/Crepitus

>>14958534
>The article is talking about spontaneous clicking from simple movements
That's what I have. Both of my wrists click whenever I move them backwards fast enough and to a far enough angle. And I can repeatedly perform the motion and they will repeatedly click. The joints aren't cracking.

>>14957918
>>14958177
>>14958185
>>14958266
>>14958267
Tyvm anons, I realized I messed up my proof by taking complement of B not complement of power set of B

>>14958585
Are there any signs of inflammation? i.e. is there pain, redness, swelling, hotness or loss of function?
If not then you probably shouldn't worry about it too much, otherwise you might want to see a doctor.

>>14958652
>Are there any signs of inflammation? i.e. is there pain, redness, swelling, hotness or loss of function?
There's absolutely nothing.

Can you answer my original question for how common is this?

Someone explain this to me

sqrt(3)x - 1 = 0

first answer
sqrt(3)x = 1
x = 1/sqrt(3)x

second answer
sqrt(3)x =1
square both sides
3x=1
x=1/3

Is it some pemdas shit, why is pemdas even true like why does multiplication come before addition they just told me it does but never said why

>>14958669
>Can you answer my original question for how common is this?
I don't know, sorry.

>>14958652
Not him, but if I move my hand downwards and draw a circle to the right it crackles heavily. No pain normally, but it feels that it isn't so good a thing.
Also I can do this crackling sound with my fingers when closing a fist.
Is any of those a matter that would perhaps need attention?

>>14958671
Could you be precise in your notations? Here it's undecipherable or totally wrong.

>>14958671
>why is pemdas even true like why does multiplication come before addition
Because we say it is. It's just a notation.
The rest of your post is incoherent.

>>14958671
>first answer
>sqrt(3)x = 1
>x = 1/sqrt(3)x
That should be x = 1/sqrt(3) at the end, which is correct.

>second answer
>sqrt(3)x =1
>square both sides
>3x=1
When squaring both sides, you need to square the x as well:
>sqrt(3)x =1
>square both sides correctly:
>sqrt(3)^2 x^2 = 1^1
>3 x^2 = 1
>x^2 = 1/3
>x = sqrt(1/3) = 1/sqrt(3)

>>14958681
But it (a)?effects the answer what do you mean it's just notation, in my mind I think of it as just a style thing for convience, but doesn't change the answer??

>>14958687
oh yeah forgot about that, thanks

>>14956456
What would f(x) and g(x) refer to, as being different functions? (In our case we're talking about a single function so I'm guessing... well, I think I'm having trouble parsing that.
Also, what would be g' and/or f'?

Once again, thanks for replying.

>>14958678
It shouldn't be a source of worry unless it's associated with pain and/or inflammation.

>>14958511
What the fuck is the formula?

>>14958511
figure out how to write the question correctly first because that is meaningless gibberish.

>>14958719
>>14958725
What's wrong with the question?

>>14958729
The denominator has no logical progression

>>14958745
? It does. [math] n, n + 1, n + 2, \dots, \lfloor na \rfloor [/math].

>>14958719
>>14958725
>>14958745
NTA but i think he meant this
[eqn] \lim_{n\rightarrow\infty}n\left(\sum_{k=n}^{\lfloor na\rfloor}\frac{1}{k^{2}}\right) [/eqn]

>>14958751
Wrong. As written it states (n + i)^2, that does not relate to (the floor of) na

>>14958511
>The number of terms increase with n
>Each term decreases with n
It sounds like a Riemann sum so rewrite it into one

[eqn] \lim_{n \to \infty} n \left( \frac{1}{n^2} + \frac{1}{(n+1)^2} + \ldots + \frac{1}{\lfloor na \rfloor^2} \right) = \lim_{n \to \infty} n \sum_{k=0}^{an} \frac{1}{(n+k)^2} = \lim_{n \to \infty} \sum_{k=0}^{an} \frac{1}{\left(1+\frac{k}{n} \right)^2} \frac{1}{n} = \int_0^a \frac{1}{(1+x)^2} dx = \frac{a}{1+a}

[/eqn]

>>14958719
>>14958725
I think you would write it out like this using sigma notation:
[math]\displaystyle n \sum^{\lfloor na \rfloor}_{k = n}\frac{1}{k^2}[/math]
I just transcribed how my professor wrote it.

>>14958777
I think you might just have autism desu.

>>14958779
The upper limit of the integral should be a - 1 instead of a, but otherwise correct

>>14958780
That would make more sense.

>>14958782
Mathematicians call it rigour.

>Thread reached bump limit
I'm going to make a new thread, which 2hu should i make the OP?

>>14958858
nvm someone already made a new one

>>14958813
>>14958813
>>14958813
>>14958813

>>14958177
>P((A−B)∪B)=P(A−B)+P(B)
what does + mean here?

>>14958972
It means addition on the real numbers

If A and B are disjoint then
P(A ∪ B) = P(A) + P(B)

I believe this property above is called finite additivity of the probability measure