/sci/ - Science & Math

>>15437848
Not exactly a science question but I guess people will know an answer here.
What is the best version of Ted Kaczynskis manifest? Is there a printed version one should get or are they all expensive cash grabs?

guess what? we looked at the data, and it turns out /sci/ posters have significantly less sex than the general population.

I just learned that there are tons of scientists from places I would consider legit that are saying race is not biological and is only a social construct. Am I missing something? Obviously the traits we use to distinguish between the races are biological phenotypes, so how would it not be biological?

>>15438476
Purely using a phenotype to subdivide humans into categories is not fine-grained enough. Sure, all british may look alike, but those that belong to royalty are still distinctively different to poor people. The economic state does play a role in a humans attitude, character traits and thus is also a predictor for behavior.

Also race-definitions are in general quite arbitrary. Usually people just pick out some phenotypic markers and use them to subdivide living beings. Which works fine unless you completely rely on those markers as a predictor for behavior. A german shepherd is easily distinguished from a mastiff and in general they also show different forms of behavior, e.g. shepherd listening to commands, mastiff being relaxed. But through experience in life their characters are shaped, traumatic episodes could lead to an aggressive mastiff. Though the race of the dog still plays a role in how he acts.

>>15438476
There is no black or asian or white gene, it's very arbitrary. Japanese people look distinctly different from Chinese people but they're both asians etc.
Historically the modern concept of race has mostly been important in colonial provinces like the USA where there is a significant cultural difference between black people and white people whereas in Europe nationality used to be more significant (its changing nowadays for various reasons).

>>15438476
biological phenotypes are real but race is a concept which gets distorted and molded at will by pretty much all scientists. there is no single universally agreed standard of exactly how different two individuals have to be for us to consider them as belonging to different races. among animals, it might be really small, among humans, no such thing as race is accepted thanks to left-wingers completely dominating anthropology and the social sciences

>>15438532
>Purely using a phenotype to subdivide humans into categories is not fine-grained enough
What would make it fine grained enough? The only thing I would imagine is neccessary is the ability to detect a multi modal distribution.
>Also race-definitions are in general quite arbitrary.
Isn't it just as arbitrary as defining when somone has a testosterone deficiency? We can change the threshold for when someone is positive or negative for a disease based on our ethical preferences. We will get false positives and false negatives either way, but we except the arbitrariness and lack of accuracy. Why is race different? I can predict better than random guessing someone's behavior based on how I classify their race, even if I don't always find it moral to use race data to make decisions.

From plotting this in MATLAB, I'm pretty sure the only bifurcation value is a=0 since the origin (which is a fixed point) goes from stable for a <= 0 to unstable for a > 0. How would I go about showing this analytically in order to show I didn't miss any other bifurcation values or fixed points?

>>15438848
Easiest would be to just chug it into xppaut and let it do the bifurcation analysis for you. That is what most people in the field do. There the steady states are computed numerically for a given bifurcation parameter which is then varied to find all organizations of the system.

Solving those equations analytically is quite a mess. You would need to find steady states by finding x and y that satisfy x'=0 and y'=0, then compute the stability of steady states via the Eigenvalue of the Jacobian matrix. From there you can find for which a the stability changes. What book is this from?

is there any way I can undo the brain damage from alcohol?

>>15438848
First convert the system into polar coordinates

[eqn]x = r \cos(\varphi) \\
x' = \cos(\varphi) r' - r \sin(\varphi) \varphi' \\
y = r \sin(\varphi) \\
y' = \sin(\varphi) r' + r \cos(\varphi) \varphi' [/eqn]
You plug this into the system.
[eqn]
\cos(\varphi) r' - r \sin(\varphi) \varphi' = -r \cos(\varphi) - r \sin(\varphi) + a \cos(\varphi) \\
\sin(\varphi) r' + r \cos(\varphi) \varphi' = r \cos(\varphi) - r \sin(\varphi) + a \sin(\varphi)

[/eqn]
Multiply the first equation with cos(phi) and the second with sin(phi) and add them
[eqn]
r' = - r + a
[/eqn]
Now instead multiply the first equation with -sin(phi) and the second with cos(phi) and add them
[eqn] r \varphi' = r[/eqn]


Now it becomes trivial.

Im trying to solve this simple physics problem: a rod of length A, which is nailed to the wall (assume no friction) in point A, is in in its unstable position of equilibrium. After letting it fall (initial speed = 0), calculate the angular velocity of the rod when it passes through its stable equilibrium position (see drawing)

This could very easily be solved with the conservation of mechanical energy, but im trying to do it by calculating the angular acceleration and with that the time it takes to reach the equilibrium point and with that the angular speed. Here is what I've done:
[math]Mg\frac{L}{2}=I_A\alpha[/math]
where I_A is the moment of inertia of the rod, and alpha the angular acceleration. Thus [math] \alpha =\frac{3g}{2L}rad/s [/math]
which is constant, thus from the uniformly accelerated circular motion equation,
[math]\theta(t)=\frac{1}{2}\alpha t^2[/math]
from which i can get the time needed to go through an angle of pi, which is the equilibrium position
[math]t=2\sqrt\frac{\pi L}{3g}[/math]
and since the angular velocity [math]\omega=\alpha t =\sqrt\frac{3g\pi}{L}[/math]
while the correct answer is [math]\sqrt\frac{6g}{L}[/math]
I dont understand where is my mistake. i get the right answer using the conservation of energy.

>>15439367
of length L*

>>15439367
The angular acceleration depends on the angle

>>15439369
isnt it a constant of 3g/2L? or is that wrong? and if it isnt wrong than how does it depend on the angle?

>>15439369
>>15439373
nevermind i got what you meant, i forgot to add the sine of the angle between the two vectors in the first vector product to get alpha, it should work now

Does anyone have any experience with hunting bugs for Open Source Vulnerability programmes?

>>15439287
Wean yourself alcohol completely and do mental exercises to strengthen what's left of your brain, encourage new connections and work on memory. There is no undo, there is only salvage what's left.
Good news is that you can live well with substantial damage.

Does anyone understand what the fuck is this question asking?

>>15439853
there are n stairs on the staircase
frog moves from bottom to top in k steps, where k is no greater than n
What's the total number of ways for the frog to do this?
e.g. if n=2, then we have k=1 (only possible solution being that the frog jumps directly up) and k=2 (only possible solution being that the frog jumps to the first step, then to the second step)

>>15439853
Yes.
Essentially the questions asks for the cardinality of the set
[eqn]\left\{ (x_1, x_2, ,\ldots, x_k) \in \{1,\ldots,n\}^k \middle| \sum_{i=1}^k x_i = n \right \}[/eqn]

At least that's what I'm assuming.

>In a family of two children at least one is a boy what is the probability that the second is a boy.
Why is BG considered different to GB?

>>15439890
Meant to be:
>In a family of two children at least one is a boy what is the probability that both are boys.

hi im not science but these last 2 google searches (pic rel) returned the same result

what happens if oxygen has more than 6 or 8 electrons in the "excited state"

>>15438476
Here is an article in the new york times by David Reich who is probably the most respected population geneticist out there and he's trying to convince a liberal audience that yes in fact race has a biological basis:
https://www.nytimes.com/2018/03/23/opinion/sunday/genetics-race.html

'Scientists' like anthropologists who claim it does not are just parroting some politically convenient groupthink

>>15439890
Because they mean different things? One means first is a boy, second is a girl. The other means first is a girl, second is a boy.

>>15439893
In that question you have three possible scenarios:

1st born child is a boy, 2nd is a girl
1st born child is a girl, 2nd is a boy
1st born child is a boy, 2nd is a boy

So the odds of two boys is 1 in 3

>>15440424
What if they are twins? Then there is no first and second.

/med/ still dead so can any smart boy tell me which kind of tapeworm has big wide proglottods with hooks/spiky bits and deep dark red colour assuming it isn't just from internal micro bleeds as prior ones I saw where white or sickly yellow? Going off bits I see each time I go to toilet to shit, I have hookworms (very thin thread like worms in varying lengths unless I confused them for other ones) and I had pinworms (3 rounds of pyrantel seemed to do the job)

What is in the other side of a black hole? Would there even be such a thing?

>>15440539
We can draw space time diagrams of what happens past the event horizon but all paths eventually end up in a singularity. However no physicist thinks a singularity actually exists, it's just the limit where our math breakdown. So all we can really say is currently we do not know. We have no way to see inside, or to see what state of matter all the mass inside becomes.

>>15440481
>Then there is no first and second.
Yes there is, there is oldest and youngest.

>>15441253
What if they are siamese twins?

>>15441290
Still makes no difference.

>>15440424
why does 1st born, 2nd born matter? if i have been told that one is a boy, why would i care about when they were born?

>>15441609
It's not so much about the order that they were born as that saying "at least one is a boy" does not actually tell you which one the boy in question is, so we can't say either is definitively a boy.

>>15441609
> why would i care about when they were born?
You wouldn't. It is simply a way to highlight the fact that BG is not the same thing as GB, the order matters.

>>15437848
My question is:
>How are we sure that physical propertys of a particle is inherent property rather then relational, having in mind that observing particle in isolation is not posible?

>>15441736
>having in mind that observing particle in isolation is not posible?
what gave you that idea?

>>15441753
The mear observation of a particle is act of interaction is it not?
You allways measure things in relatiom to something and with something.

>>15441762
You can measure properties like charge, mass or spin without a direct interaction.

>>15441765
How?

>>15441767
Because they generate fields that can be measured at a large distance from the source.

>>15441773
Okay, but what makes you sure they wouls generate fields if there was only that particular particle and nothing else?
As i said, how can you be sure it is not caused because of relation to other systems?

>>15441777
Because if that was the case the field would not be constant. It would (randomly) vary as it interacted with the background even if only by a small amount.

I might be interpreting this wrong, but does this mean that if i apply any pressure to an amount of Hydrogen or Helium gas, it will expand due to repulsion forces?

>>15441736
The properties of particles follow from theories of quantum fields (of which they are excitations) and in these theories you can consider the case of there being a single particle alone in the universe. Given that these theories agree with experiment in the many situations they have been tested this is evidence that you can consider things like mass, spin, charge to be inherent to the individual particle.

>>15441787
The graph is ambiguous because it is not clear which quantities are being held constant. It is clear pressure is changing and the number of moles is held constant, but then there is still a free parameter in the equation of state. Is this constant volume, constant temperature, constant entropy (adiabatic)? etc

Anyway since P is increasing having PV/RT increase certainly does imply volume is increasing, it just means V or T aren't changing fast enough to compensate the P increase.

>>15441808
>certainly does imply volume
Sorry I mean "does *not* imply"

>>15441736
>>15441765
>>15441773
>Because they generate fields that can be measured at a large distance from the source.
NTA, but what is the device that allows this called?

Because while I can buy a double slit experiment setup, that'll produce a wave interference pattern, for about $200, I can't find one that includes a "photon detector" at the slits for the double line effect.

Further, I can't find any explanation as to how this is achieved in any paper. There's "which-way" experiments in a Wikipedia article (with varying results), but they all seem to involve blocking or otherwise restricting the particle involved.

Leaves me wondering if the whole thing really is just a thought experiment between Einstein and Bohr, and it hasn't really been confirmed that light acts as anything other than a wave at various levels of restriction.

>>15441847
You are looking at particle experiments, not how to measure the strength of a field. Also the double-slit experiment measures neither.

>>15441784
Sorry for naging, but why wouldnt it be constant?
I researched a bit, observing mass, inherent mass and mass trough interaction. Observing and interaction mass are clear to me. Idk why i have hard time acccepting that quarks for example are just setted that way without any other reasson... just like you know.. here is a quark and here is his inherent mass and this is the way things will be no matter anything. I dont know.. maybe i am overthinking it.

>>15438476
>tons of scientists from places I would consider legit that are saying race is not biological and is only a social construct
No scientist that I can find is actually saying this. They are simply saying the biological factors that make up race are amorphous and not causally related to what the laymen would use the differentiate between races, particularly in terms of genetics. There are genetic markers that are more common in certain races (and in some places, these may even affect your insurance rates), but all of them are found in other populations to some degree or another. This is possibly due to that genetic bottleneck we had some 70K years ago, that left us so inbred that we could nearly share cancers with one another.

>>15439513
Try >>>/g/

>>15441870
The anon that asked the question was asking about relational values for the fundamental properties of a particle. Those relations would change depend on its neighbours. It's not a stupid question as many physicists in the past have suggested properties like position aren't intrinsic (like having x,y,z coordinates) but are instead relative.

>>15441877
Thats me yes. Thank you for summary friend.
Altho, its my fault, what i wanted to imply is not that the mass is determined by its relation to enviorment.it is in. Implication that my question has is: How can we be certain there would be such a thing as mass if there was nothing towards what it could be manifested. So what i am really asking is, how are we sure that mass would exist as a property without there being interaction of anykind. Maybe i am making mistake by thinking of propertys as interactions between entitysy...

>>15441900
Delving into philosophy, but you can't say much of anything about anything without anything to compare it to. If the entire universe was a single particle, you couldn't say much of anything about said particle (what kind it was, what its velocity or vector is, what sort of energy it has, etc.) If it was even slightly more complicated, say a proton, you could infer mass from the Higgs mechanism, but a fundamental particle on its own as the entirety of existence with absolutely zero interactions has no measurable properties, as there's nothing to measure against.

Granted, at the same time, we wouldn't be around to measure or talk about the thing, so we can just chalk up that dead end to anthropic principle.

>>15441900
Generally speaking interactions are effects caused by fundamental properties. You can't exactly have an electric interaction if there isn't any charges.

However mass specifically is a lot more complicated because of mass-energy equivalence and quantum field renormalization. Essentially there is a particle's intrinsic mass and the mass we actually measure (which is due to interactions).

Someone please explain to me how sewing machine’s works. Why does it matter at all that the hole is at the front of the needle. If the needle isn’t being pulled all the way through it can’t possibly hold together. Having a second thread doesn’t make sense, how does it loop through the first in order to hold the cloth? None of this makes sense. Even looking at images doesn’t make sense.

Please help. It’s witchcraft

>>15441981
Agreed its hard philosophical speculation.
ChatGpt told me there are experiment suggesting particles do have inherent propertys, so relational perspective kinda falls off.
But this.. this fact that we will ever get to know what thing is only via how it interact with other stuff... ahhh
For all we know it could be ghost potatoe.
>>15441996
Yes, agreed.

Anyhow thank you for responses and clarification anons.

how do i convince administration that adding a third semester of linear algebra, but still trying to cram everything into a single semester of abstract algebra is NOT a good idea?

>>15443291
Probably make the comparison to literally every other university program and ask them why they think this choice is better since 1-2 courses in linear algebra seem to be enough everywhere else.

I want to do a fun project and measure the temperature of food using sonic waves. How can I do this?

>>15443325
You can't.

>>15437848
Where can I learn high school-level math fast? Khan Academy?

>>15443593
Yes.

>>15443745
thanks!

>>15442030
A sewing machine effectively pushes a loop of thread through the fabric with the needle, then pushes a second loop (from the bobbin) through that loop, then the next loop through the fabric goes through that loop, and so on. Each of the loops gets pulled tight after the other loop has passed through it, and the friction holds everything together (topologically, there's nothing stopping it from unravelling).

Making a machine that sews like by hand would be hard. It would need to to continuously release and re-grab the needle.

>>15443360
Why?

I'm OP from >>/sci/thread/S15430306
Anon in that thread requested I post an update here.

I heard back from the teaching professor at my own institution. That professor didn't seem to fully grasp what I suggested, but I did get feedback that will help me to better explain it. I also got a reading list, specifics of who to send this idea to, and a suggestion to consider doing a PhD in physics.

Not really interested in the latter, they couldn't afford me and I don't want to move, but it's pretty flattering I guess.

>>15443922
What mechanism do you think allows you to measure temperate using sound waves? And how do you think you can do that in a "fun project"?

>>15443962
>What mechanism do you think allows you to measure temperate using sound waves?
NTA but sonar, if you could get it working at a molecular level.

>>15443962
Well I imagine if I know the general density of something and I can figure out the speed by getting the time it took to traverse through it I might be able to determine the objects internal temperature?

>>15443980
Oh I know how to do it but I'm asking that anon to see if he knows, and why he thinks it can be done as a cheap hobby experiment. My point isn't that it's not possible, but that it's not possible for him.

>>15443991
well I don't know and I am trying to learn.

>>15443991
>Oh I know how to do it
Cool because I was just spitballing.

>>15443995
Each material (food) has a different profile for Temperature vs Speed of Sound inside that material. So first you would need to know what that is, since a database for that information doesn't exist you would have to create it yourself. Then once you have that you need to somehow measure the speed of your sound wave only through that food sample. All of that is technically possible though would require a lot of time and equipment. Then there is the fact the speed of sound doesn't vary much on the scales you are talking about. In air the difference between room temperate and freezing air is only 4%, the sensitivity of your temperate measurement based on speed isn't going to be very accurate.

Is there ever going to be medical technology that can allow you to change your body into any form you wish like a video game avatar? Imagine going from being a buff 6'2 guy to looking like 9S from NieR in the span of an hour.

>>15444126
probably not, no

>>15443893
So instead of thinking of it as a pair of threads I look at it as a series of loops that hold together like tying your shoe.

The dude who thought of this is smarter than me

>>15444189
sad. it'd be cool to be able to change your body on a whim, but i suppose that's what VR is for.

The fundamental postulate of statistical mechanics is a talmudic pilpol

>>15444358
Statistical mechanics agrees with experiment so the fundamental postulate is valid. The question of why it works is actually a trendy recent topic in physics called the eigenstate thermalization hypothesis.

>>15444374
The Eigenstate thermalization hypothesis is a quantized talmudic pilpul

>body requires a 12km run daily just to get my metabolism working
Think I got some kind of stomach bug or illness. I'm 22 but feel like I'm in my fifties if I don't get any exercise. Any suggestions or recommendations? Not sure if talking to my doctor about this would help since my symptoms are largely anecdotal.

Given 1 < a < b < e , where e is the base of thenatural log, then prove that:

[ln(2b) - ln(2a)]/2 < (b^2) - (a*b)

>>15444546
Talk to your doctor. You may have asthma or allergy induced asthma. If you're around any smokers, weed smokers, etc, allergies can also trigger asthma. Alternatively, could be psychiatric. There's no way to know and the list of possibilities are miles long. I only mention the asthma as a lot of people these days are in apartments smoking marijuana rather than cigarettes, and both can induce allergic reaction related asthma symptoms even if you can't smell it. Realistically though it could be anything, other allergens, a circulatory issue, depression, other drug use, etc. Point is, you have to talk to your doctor. "Hey doc I feel like a grandpa at home and only feel better after exercising outside for a long time", first idea would be indoor allergen but all of them require testing.

>>15445146
[eqn]
\frac{\ln(2b) - \ln(2a)}{2} = \int_{2a}^{2b} \frac{1}{2x} dx < \int_{2a}^{2b} \frac{b}{2} dx = b^2 - ab
[/eqn]

any math studs?

in proving sqrt(2) is irrational, we get to tthe point where we can express sqrt(2)=p/q as sqrt(2)= k/m where k,m is any integer. The proof relies on the assumption that p/q have no common divisors as that is the standard form of a rational number. but k/m can reach a form where it doesnt have any common divisors for example if k=2 and m=4

wtf how does "no common divisors" serve as a logical contradiction point when for any value of k and m we can eventually reach a form with no common divisors

>>15445593
>the assumption that p/q have no common divisors
All pairs of integers have common divisors as 1 and -1 always divisors.
>doesnt have any common divisors for example if k=2 and m=4
There you have four common divisors (1,-1,2,-2).


If x = a/b for integers a,b then it can also be written as x = c/d with c = a/gcd(a,b) and d=b/gcd(a,b).
c and d will be coprime integers that means gcd(c,d) = 1 that implies that every rational number can be written a quotient of two coprime integers.

>>15445593
It feels like you're misunderstanding the proof.
We say that p and q have no common divisors other than 1 (i.e. are in lowest terms), but by squaring both sides and multiplying by 2, we find that p^2=2q^2, i.e. p^2 is even, so p is even as well. Hence p=2k for some k, so p^2=(2k)^2=4k^2. So 4k^2=2q^2, which implies that 2k^2=q^2, and so q must be even as well.
The problem, then, lies in the fact that p and q are both necessarily even, and thus share a common factor of 2, meaning that p/q can't actually be in lowest terms. We can keep iterating this process infinitely, so no such rational number exists.
This isn't a problem you run into with roots that ARE rational. For example, if we try sqrt(4)=x/y, we very quickly find that x^2=4y^2=(2y)^2, so we just find that x=2y and so x/y=2.

>>15445593
It feels like you're misunderstanding the proof.
We say that p and q have no common divisors other than 1 (i.e. are in lowest terms), but by squaring both sides and multiplying by 2, we find that p^2=2q^2, i.e. p^2 is even, so p is even as well. Hence p=2k for some k, so p^2=(2k)^2=4k^2. So 4k^2=2q^2, which implies that 2k^2=q^2, and so q must be even as well.
The problem, then, lies in the fact that p and q are both necessarily even, and thus share a common factor of 2, meaning that p/q can't actually be in lowest terms. We can keep iterating this process infinitely, so no such rational number exists.
This isn't a problem you run into with roots that ARE rational. For example, if we try sqrt(4)=x/y, we very quickly find that x^2=4y^2=(2y)^2, so we just find that x=2y and so x/y=2. (Or -2, but I digress. The point is, it doesn't devolve into infinite descent.)

>>15445621
>>15445632
this is what i have cooked up

Theorem: There exists an x such that if y=sqrt(x) then y is irational

Proof: suppose y rational and let x=2. then sqrt(2)=a/b. This implies both a and b are even and furthermore contradicts the definition of rational numbers. Thus we can conclude for x=2 y is irrational.

>>15445803
>Theorem: There exists an x such that if y=sqrt(x) then y is irational
This "theorem" is wrong. There exists an x such that 1 = sqrt(x) but 1 is not irrational.

>>15445803
> This implies both a and b are even
How?

>>15445593
i too have often found the step in that proof where we assume no common factors to be a bit handwavy. even the vsauce video glossed over it without explanation. why are we allowed to assume this about the fraction? why does it result in a contradiction? here, in my opinion, is a better worded proof:
>suppose a/b = sqrt(2)
>…
>then a and b must have a common factor of 2
>but this implies that a/b can never be brought into lowest terms, for there will always be a factor of 2 you can factor out
>however, all finite integer ratios can be brought into lowest terms
>therefore, a and b cannot be integers
basically, you cant just factor out twos forever, which is what a/b = sqrt(2) would imply. in fact you cant factor any number forever, so its safe to assume that if a ratio exists, a simplified version of it must exists as well. if you can prove a simplified version does not exist, then the ratio itself does not exist.

>>15446114
>why are we allowed to assume that about the fraction?
>proceeds to explain exactly why we're allowed to assume that about the fraction
It's a proof by contradiction. We assume that there exists a lowest-terms representation, and show that this representation can't actually be lowest terms.

>why are we allowed to assume that about the fraction?
>proceeds to explain exactly why we're allowed to assume that about the fraction
It's a proof by contradiction. We assume that there exists a lowest-terms representation, and show that this representation can't actually be lowest terms, using literally the very argument you've provided here. Hence no such representation exists.

>>15446127
>>rhetorically asks question
>>proceeds to answer it
i didnt say it wasnt valid, i said it was confusing.
>We assume that there exists a lowest-terms representation
that is not how most novice readers interpret it. the line is usually written as "let a/b be equal to sqrt(2) in lowest terms." then when the common factor of two is shown, a lot of people think that only contradicts the second part of that statement, but not the first. see: >>15445593

in terms of long term (centuries) structural integrity is it better for structures to be above or below ground? underground structures essentially don't oscillate, but are subject to stronger vibrations during earthquakes. or how would this even work?

Does this happen to (other) animals too?

>>15438476
The ones saying that would often come up with another term that functions more or less the same way in biology, but without the political baggage.

>>15446221
of course
https://www.youtube.com/watch?v=lveKtu1BSIw

>>15446221
Dude, untwist your balls

Let x be a rational number. Prove that if xy is irrational, then y is irrational.

Suppose x,y are both rational numbers. The product of two rational numbers will always yield another rational number, thus it cant be the case
that xy is irrational. If y is rational then xy is rational. Therefore if xy is irrational then y is irrational.

is it insufficient to just say the product of two rational numbers is also rational?

>>15447062
yes, thats sufficient. its a basic proof by contradiction
>assume y is rational
>then xy is rational, a contradiction
>therefore, y is not rational

>PhD advisor's favorite phrase is unhelpful "good job"
>PhD students and seminar speakers basically say "interesting" and change topic
How do I get feedback on my research? No one around me seems to give a shit.

>>15447206
you don't. presentations are a farce. the entirety of academia is a circus and most of you probably don't even do real science.

How do I convert the top format to the bottom format? What are the steps?

>>15447404
[eqn]
m = \frac{y-b}{x-a} \implies m(x-a) = y-b \implies y = m(x-a)+b
[/eqn]

>>15446153
Depends on what you're making them out of and where in the world they are. Above ground is subject to the sun, wind, rain, heat, cold. Underground is subject to more vibration from earthquakes, but also, corrosion, bugs, mold, water tables. I'm probably forgetting some.

>>15447408
Oh, multiply both sides by (x-a), duh. I don't know why that didn't occur to me. I've been looking at this stuff for too long. Thank you, anon.

you know how there are space rocks in our solar system with diamonds? and if someone mind those diamonds they would make the value of diamonds on earth worthless? are there any other places with resources like gold, copper, oil, lithium, etc... how feasible would it be to mine on other things in outer space??

why doesnt like spacex or nasa do that then everyone will have bling and electronics and slave children wont have to mine in cobalt farms

>>15447420
>if someone mind those diamonds they would make the value of diamonds on earth worthless?
Diamonds are already basically worthless. They are just controlled by diamond cartels that artificially inflate the prices.

>>15447427
okay dude i was just using that as an example of how things wouldn't be scarce

>>15447420
a big problem is (solar) radiation which is especially important for gold and metals in general due to their use in electronics which are otherwise sensitive to radiation. furthermore dumping huge amounts of any material, be it ground or space, into the public would make you get promoted into a missing person and all of the materials mysteriously disappear. on the other hand slowly selling it would make profits long term as long as you leave out the radiation part which can have consequences elsewhere. also what other anon said.

>>15447441
can you say that in layman?

so its not possible to bring back gold or metals because they'll evaporate from radiation?? or will radiation damage the gold/metal or something
but then how do we send rovers and space probes out?

then you say that you'll get murdered by the resources industry for making their resource worthless well okay any industry can be disrupted

>>15447483
>can you say that in layman?
space=filled with radiation=bad

>so its not possible to bring back gold or metals because they'll evaporate from radiation?? or will radiation damage the gold/metal or something
>but then how do we send rovers and space probes out?
radioactive metal=bad. radiation can't be removed. you can get anything back to earth as it's been done with moon rocks. also asteroids cuz gravity.

>then you say that you'll get murdered by the resources industry for making their resource worthless well okay any industry can be disrupted
yeah. as such it would be a terrible idea to go in front of a camera and scream "I GOT TONS OF GOLDS AND DIAMONS FROM SPACE! YOU JEWS ARE WORTHLESS NOW! SUCK MY DIIIIII- ACK!". being subtle is key the same way if you were to find a fuckton of gold in the wild.

>>15447065
My teacher told me it was insufficent. What can i argue to change her mind?>

>>15447504
tell her to google what a contrapositive is

>>15447504
also, it may be the case that you proof is technically correct but shes saying it wasnt verbose enough.

>>15447501
so is space mining a feasible investment or not?

so all of the resources on earth are all we got? is we all just gonna starve to death? humans are going to eventually go extinct and then what? the universe will go on without us? that can't happen.

>>15447523
mine asteroids? no. mine other planets? yes.

Could someone show me how to do b and c for picrel?
I solved the ABCD matrix but my prof didn't give me the formulas to do b and c?

>>15447508
lmao
>>15447513
This is what is stated on my book regarding proofs
" a mathematician may include some discussion of the strategy behind the proof to make the proof easier to read. Usually, however, it is up to readers to figure this out for themselves."

I was also told that the explicitness of the proof depends on the audience that is reading. But the only person reading it is my teacher. Other professors have agreed that my proof is right. I dont know how to argue my position as to why she should mark it right.

>>15447565
dont know what to tell ya, ask your department head what your options are.
>>15447552
b just requires knowing what a decibel is (and remembering that voltage decibels are actually duodecibels).
for c, set up two equations, one for the condition that each port (the network is symmetrical so) has an input impedence of 50, and one for the condition that the attenuation is -10db.

>>15447583
>duodecibels
actually "duodeci" would be twelve. i looked it up and twenty is "viginti", so it'd be "vigintibels." im gonna start using that to piss people off.

>>15447583
i pay 3k to get cucked by the subjective thought of a "mathematician" awesome

>>15447583
I know how to convert to decibels.

So for (b) alpha would just be 10^(-10/20) = 0.316?

I saw two different formulas on google and one said to do a double negative (i.e. 10^(-(-10)/20)) and another just said to do 10^(-10/20) = 0.316.

I'm just not sure which of the two is correct here.

>>15447591
>I'm just not sure which of the two is correct here.
try plugging in -20db for both formulas and see which one gives you a reasonable answer.

>>15447590
it happens, anon, dont cry over it

>>15447599
ill try

>>15447593
Would reasonable be the one that gives me the smaller number, since we're dealing with an attenuation factor here?

My professor didn't really dive that deep into the calculation for; nor did he give the explicit formula for it.

>>15447621
>Would reasonable be the one that gives me the smaller number, since we're dealing with an attenuation factor here?
bingo

>>15447622
arigato, anon-kun.

I'm GONNA FUCKING MAKE IT

>>15447632
np, fren, ygmi. just a heads up, i dont really know what it means by "all ports are matched to 50ohms", i just kinda guessed when i made my first answer.

Is there a relation between the euler characteristic of a particle and it's spin?

>>15446221
>Does this happen to (other) animals too?
over the last 3 weeks i have been obsessed with this and as soon as i have a little harm in the testicles i fear it's torsion. this is due to reading the account of some guy who had one and said it's insanely painful and had to rush the hospital because it leads to gangrene.
Just 1h ago i had a little feeling in the left one and now i cant stop thinking about this

>>15447206
that's why you need to do research in the same lab as the people who love the field

>>15447206
Depends on your relationship with your advisor. I found that it took me asking him in a 1-on-1 meeting for feedback on how my project was going.
Only about half of bosses (advisors included) are good at giving negative feedback, and probably one in twenty is good at giving positive feedback

>>15447728
>euler characteristic of a particle
what is this?

the spin or helicity stems from the representation of the poincare-lorentz group
ie mass and helicity are casimirs of this symmetry group

this is why spin is not a quantum phenomena, but a relativistic one, and nobody has a clue why quantized spin stems from the symmetry group of the universe.
https://en.wikipedia.org/wiki/Representation_theory_of_the_Poincaré_group?useskin=vector
https://en.wikipedia.org/wiki/Representation_theory_of_the_Lorentz_group?useskin=vector

>>15447755
The Euler Characteristic is an geometric property of an object that remains invariant under homeomorphic transformations.

How do I solve picrel?
I got the ABCD matrix for this/individual ABCD matrices for each part of the circuit and I'm a bit stuck here.

Is it just
(V1 I1) = (Matrix for L3 and R2)*(V2 R2)

I2 = V2/ZL

Then use PL = (|IL|)^2/R2?

I don't know if I'm even doing it right. Not sure how to approach it desu.

Let V be a vector space of finite dimension over some field. We say a linear operator T: VV is a projection if T^2 = T. Suppose P,Q are projections such that PQ = 0. Must QP = 0 as well?

Which plugin does 4chan use to render TeX code?

>>15447949
No, you can already find counterexample in 2 dimensions.

>>15447998
I figured it out eventually. thanks anon

>>15447964
> MathJax.
Right click any TeX and it tells you.

>>15448149
Thanks

What's the likely cause of this?
https://www.theguardian.com/commentisfree/2023/may/18/british-workers-sick-economy-cure-tories-ill-nhs-strikers
>Here’s how things currently stand. The number of people who say they are not looking for work because of long-term ill health rose by 86,000 in the first three months of 2023 to 2.55m and is now 438,000 higher than it was before the start of the pandemic three years ago.

I don't get this. Doesn't the last equation only have a single point as solution?

>>15448408
Only if B is real.
For example, consider B=i and C=0 (for simplicity)

>>15447755
Garbage tier post

>>15448431
Reads like a GPT reply.

>>15448161
>What's the likely cause of this?
Aging demographics and covid infections causing a lot of long term damage, especially where brain damage is concerned.

>>15448161
Probably long covid. 86k on 2.55m is a 3.4% rise in 3 months, which is much higher than you'd see just from an ageing population.

I'm a mental midget doing orgo 1 and the amount of exceptions makes me think all this stuff was just made up.
To what extend is it actually made up; just figments of the imaginations of a schizophrenic, middle-aged academic groupmind?

Is there evidence for this?

Let [math]p(x) = \prod_{i \in I} p_i (x)[/math] be a product of monic linear polynomials (not necessarily distinct). I need to show that if [math]q[/math] is monic and divides [math]p[/math] then [math]q(x) = \prod_{i \in J} p_i (x)[/math] for some subset [math]J[/math] of the index set.

I tried induction on the degree of [math]q[/math] but it didn't get my anywhere. I'd appreciate any help.
Also, does the statement remain true if we relax the condition that the [math]p_i[/math] are linear polynomials and replace it with the assumption that the [math]p_i[/math] are irreducible?

>>15437848
Hey anon, I fail my math midterm on series and series of function. How do i cope?

Do you think I could get through:
>Limits
>Average rates of change
>First principles differentiation
>Linear approximation
>Differentiation rules
>Implicit differentiation
>Derivates and graphs
>Concavity, the second derivative, local and global extrema
>Optimization
>Integration, Riemann sums and the Definite Integral
>Fundamental theorem of calculus
>Areas between curves and accumulation of change
>Volumes of revolution
To the point where I'd answer ~50% of questions correctly in 18 days?
I procrastinated like a retard and have a midwit IQ

I'm trying to evaluate a double integral over a cylindrical shell with radius [math]R[/math]. How do I parametrize it?
I know that going from [math]\mathrm{d}x\mathrm{d}y[/math] to [math]\mathrm{d}\theta\mathrm{d}z[/math] I get a jacobian of [math] R [/math] but I have no idea why

why doesn't the temperature of an ideal gas in a Joule/free expansion change?
Or is it just an experimental result without a why? (assuming you cant talk about the microscopical/stastical mechanics level)

>>15450876
>Joule/free expansion
well this
https://en.wikipedia.org/wiki/Joule_expansion?useskin=vector#Ideal_gases

>>15450918
>Because the internal energy does not change and the internal energy of an ideal gas is solely a function of temperature, the temperature of the gas does not change
Okay, but my teacher started with the Joule expansion to give a mathematical proof that the internal energy of an ideal gas is only a function of temperature, so it would be like saying
>since the temperature of a Joule expansion doesn't change, you can prove that the internal energy is only in function of temperature, and since it's only a function of temperature and the internal energy of an ideal gas in a Joule expansion doesn't change then its temperature doesn't change
so I'm assuming that there is another way to prove that it's only a function of temperature, otherwise this would be circular reasoning

>>15450938
It's almost a definition. For an ideal gas the only energy the particles can have is kinetic energy, so the total energy in the system can only depend on the total number of particles and the temperature.

>>15450757
Wasn't the trick to this just looking at the polynomials as functions?
If [math]p(x) = q(x)d(x)[/math], and, say, [math]q(x)[/math] has degree one, then clearly [math]q(x) = p_i(x)[/math] for some [math]i \in J[/math], since otherwise [math]q(x) = x - a[/math] and [math]0 = q(a)d(a) = p(a) = \prod_{i \in J} p_i(a) \neq 0[/math]

>>15450757
This statement is not true at all. Consider the following polynomials with coefficients in [math]\mathbb{Z}/4\mathbb{Z}[/math].

[eqn]p(x) = x^2 \\
p_1 (x) = x \\
p_2 (x) = x \\
q (x) = x + 2[/eqn]

Then [math]p(x) = p_1(x) p_2(x)[/math] and [math]q(x)[/math] divides [math]p(x)[/math] since [math]p(x) = (x+2)q(x)[/math] but [math]q(x)[/math] can't be written as a product of the [math]p_i(x)[/math].

what are the implications of being the major of your star system?

>>15451307
>major of your star system
what is that supposed to mean?

The rod is uniformly charged. I need to find [math]v_0[/math] such that the velocity of the particle at [math]A[/math] is 0. The particle has charge [math]q[/math] and mass [math]m[/math].

How the fuck do I do that? I'm stuck on this problem for more than 2 hours now, I tried approaching it in two different ways but with no success.

I tried:

1- Using conservation of energy [math]U_P+\frac{1}{2}mv_0^2 = U_A[/math];

2- Finding [math]E[/math] from [math]V[/math] and using it on [math]m\ddot x = qE[/math]. Then I used the equations for [math]v[/math] and [math]x[/math] to eliminate [math]t[/math], and then I applied the given conditions.

Both methods didn't work. Any help is appreciated.

>>15451612
fuck, forgot the picture, my bad

>>15451612
>>15451615
use [math]W=Fd[/math]. [math]W[/math] is the initial energy of the particle, and [math]Fd[/math] is an integral that looks something like
[math] \displaystyle
\int_{\frac{L}{3}}^{L} F(x)dx
[/math]
note that [math]F(x)[/math] is another integral, unless your professor gave you a formula for an equivalent point charge for a coaxial rod or something. let me know if you get stuck.

>>15451637
By [math]Fd[/math] you mean force x distance, right? Also, based on these limits of integration, [math]A[/math] should be the origin?

>>15451646
>By [math]Fd[/math] you mean force x distance, right?
yea
>Also, based on these limits of integration, A should be the origin?
no, [math]x=0[/math] would be the right end of the rod for the bounds i posted.

>>15451649
I wrote [math]F[/math] as [math]qE[/math] and calculated the electric field.
Then I plugged back on the integral of [math]F[/math] and integrated. But the problem is that I end up with a [math]\ln(0)[/math] in one of the terms.
btw, the answer is supposed to be [math]v_0 =\sqrt{\frac{q\lambda \ln(2)}{2\pi\epsilon_0 m}}[/math]

>>15451649
>>15451682
nvm, I did it. thanks for the help anon!

>>15439871

The N^th coefficient of the taylor series of 1/(1-x)^k around x=0

>>15451682
>>15451710
shit sorry 4chanx decided not to ping me.
anytime fren <3

>>15451236
Well the claim is obvious if q has degree 1, but desu I don't see how this argument generalizes to arbitrary degrees..

>>15451283
Thanks but my question is about polynomials over a field, sorry I should have mentioned that explicitly

I have a very rusty fuel tank that needs cleaning, I want to try using electrolysis.
Just to make sure I'm doing it right:
1) Is a PC PSU (the 12 volt rail, specifically) going to get the job done?
2) What electrolyte do I need to use? Will regular old salt do, or do I need something more specific, and at what percentage of concentration?
3) What polarity do I need to use to remove the rust from the tank and have it deposit on some iron rods?

>>15452747
There are literally hundreds of video guides on YouTube for this *exact* topic. Watch one.

Question for math/physics people:
What is the shape of this mirror?
S=2F
https://en.wikipedia.org/wiki/Curved_mirror

It is not parabola and it is not spherical. In these two shapes, the rays do not focus in single point. What is it then?
Does this shape even exist?

>>15451905
In polynomial rings over a field are integral domains. That has some very useful consequences.
The algebraic multiplicities of the roots in a product of polynomials are the sums of the algebraic multiplicites of the roots in the factors.
Also the sum of algebraic multiplicities of all roots of a non-zero polynomial can't be higher then it's degree.

Let [math]p(x) = q(x) r(x)[/math], let [math]n[/math] be the sum of algebraic multiplicities of all roots of [math]p(x)[/math], let [math]k[/math] be the sum of algebraic multiplicities of all roots of [math]q(x)[/math] and [math]l[/math] be the sum of algebraic multiplicities of all roots of [math]r(x)[/math].

Since [math]p(x)[/math] is a product of linear factors [math]\deg(p(x)) = n[/math] with the things mentioned above you get

[math]\deg(q(x)) \geq k = n - l \geq \deg(p(x)) - \deg(r(x)) = \deg(q(x))[/math]
so [math]\deg(q(x)) = k [/math] which implies that [math]q(x)[/math] can be written as a product of linear factors. No linear factor can appear more often than the algebraic multiplicity of the corresponding root of [math]p(x)[/math].

>>15453031

>>15453107
This does not help me. I am asking about mirrors, not lenses.
And specifically about that type of mirror that is on the picture.
What is the name of the curvature of the mirror or equation of the curvature, with that kind of properties ?

how do I order these compounds in order of growing acidity?

>>15453031
>What is the shape of this mirror?
concave. any concave mirror will do that.
>It is not parabola and it is not spherical. In these two shapes, the rays do not focus in single point.
i think you misread the wiki page. the page said that parallel rays hitting a spherical concave mirror do not converge. the rays in your picture are not parallel, and that stipulation only applied to spherical mirrors anyway, not parabolic ones.

>>15453420
It says for that image
>Real image
>Inverted (vertically)
>Same size
>Image formed at centre of curvature

My understanding is, that if I put some object of some size (not infinitely small!) at 2F above principal axis, that I will get a projection of that below of principal axis at 2F. Some shape and size as the object, just inverted.
At least that is what I want to achieve.
A projection of an object that is not distorted (can be inverted). What shape of curve should I use?
parabola and circular arc doesn't work.

Let say an example: If I put phone screen at some distance from curved mirror, and then I put a piece of paper at some other place, I want to get a projection of the phone screen on that piece of paper. (or projection on a wall). Not distorted, but can be inverted.

>>15453414
by odrering from highest pka( least acidic) to lowest pka (most acidic).

what do i do if i'm the main character of this space colony?

>>15453457
>parabola and circular arc doesn't work.
i dont see why they wouldnt
>If I put phone screen at some distance from curved mirror, and then I put a piece of paper at some other place, I want to get a projection of the phone screen on that piece of paper.
if you want to project an image like that onto a screen, the screen needs to be at the exact location of the image. in the case of the original pic, the piece of paper would need to be directly below the phone.

>>15453031
Well...why not a hyperbola? They have a focus after all.

>>15453459
i see but assuming im doing an exam and dont have the pka tables with me. Or do i have to memorize them

Anyone can explain Gateaux/Frechet differentials like I'm a retard? I know it's a generalization of directional differentials but I want more intuition. Any good material on the topic?

>>15453031
It's presumed to be parabolic, i.e. a circular paraboloid. An exact paraboloid results in a flat object perpendicular to the axis generating an image which is an exact scaled copy. Any other shape results in aberration.

Mirrors made for telescopes (and similar applications) are within a fraction of fraction of a wavelength of a paraboloid. Less demanding applications may use other shapes; e.g. the mirrors placed at blind junctions to view approaching traffic are typically spherical, as that's an easier shape to make.

>>15453648
Right, first things first.
When you think about unbounded linear operators you probably think of something like the differential on [math]L^2[/math], and that's one kind of unbounded linear operator, but another very important kind is "choose an uncountable Hamel basis for the Banach space and go sicko mode by fucking permuting it around or something". It's defined everywhere, it's obviously linear but it's probably not continuous.
A Gateaux derivative is just a regular directional derivative, except it's not necessarily a continuous linear map, it can also be some wacky monstrosity. If it's not a wacky monstrosity, it's a Frechet derivative.

do i thank my family or supervisor first in the acknowledgments section

How certain are you to die if you blend the seeds of 50 apples to a paste and then eat it? Why can I not find any information on this?

>>15454495
An average apple seed contains 0.6 milligrams of cyanide, an adult weighing 70 kilograms (154 pounds) would need to consume around 150-300 apple seeds to reach a lethal dose of cyanide.

>>15454561
Alright, make it the seeds of 300 apples for a 70kg individual. How certain is death? What about puking?

give me a purely philosophical (no physics allowed) argument for why spacetime exists or it doesn't

>>15455004
does anything exist?

>>15455004
>give me a purely philosophical (no physics allowed) argument for why spacetime exists or it doesn't
For stuff to change it has to change in some space, and since that space also changes how things change in it will do so differently or at different rates.

I don't understand what watts are. If I have a 100 watts light bulb does that mean it takes up 100 watts every second?

>>15455087
100 joules (of energy) per second

>>15454488
supervisor and with stronger words than you use for your family

A hollow cylinder of mass M, radius R has one of its bases on a plane, it has negligible height (think of it as a ring) and it's rotating around its axis with a certain constant angular velocity w0. A mass is placed in contact with the internal walls of the cilinder while its rotating (so by conservation of angular momentum it now has another angular velocity, still constant), assume no friction between the plane and the cilinder and the plane and the mass but there is friction between the mass and the cylinder.
What's the minimum static friction coefficient such that the mass rotates together with the cylinder?

The answer is coefficient>g/(w2/R), where w is the angular velocity after the mass is placed

>>15437848
Hypothetically if I literally had to sleep 16 hours a day my whole life, does my age of adulthood remain similar assuming my life expectancy is the same as usual, despite experiencing "less life", and can't be as heavily in education, training, or employment?

Why is the expression "people of colour" an actual thing? It sounds like you hate black people so much you can't even say "black" without retching, so you euphemistically call them "people of colour".

>>15455683
>you
well that wasn't me

>>15455719
https://en.wikipedia.org/wiki/Generic_you

>>15455683
i say “colored people” and no ones called me out on it yet.

>>15455683
it's an expression you can use to exclude white people without making it blatantly obvious that you're racist

>>15455592
This is just a trivial centripetal acceleration problem. The normal force must be mw^2/R for it to move in a circle, and the coefficient of static friction times this equals the weight mg.

I was boiling water and I noticed that it boils as a sine wave, I mean the bubbles stop and then they come back and it grows in intensity and then it goes down until there are no more bubbles and this repeats in cycles of about 8 seconds
I strongly suspect this is because I'm using an induction stove (I have no idea on how it works but I'm guessing the way current flows is creating this pattern), or is it something else?

>>15456095
probably just nucleation causing chain reactions of bubbling

>>15456095
pwm with a period so long you can see it.
https://en.m.wikipedia.org/wiki/Pulse-width_modulation
there’s an electroboom video showing his stove doing this but im at work and cant go looking for it.

>>15455880
Based.
>>15455947
Even if you use POC like that nobody considers kikes to be POCs tho.

>>15456141
well i cant find the video, i give up.
>>15456407
modern leftists consider jews to be in the same oppressor class as whites. hating israel is mostly a leftist thing now. i predict in a decade or so antisemitism will widely be considered a left wing standpoint. i also predict liberal arts degrees will become a right wing thing; itll be a sign of wealth, as only the rich can afford to get a useless degree.
this is all in regards to the states, of course.

How can I impart visible color to a 60% w/v sucrose solution that won't diffuse into a 50% sucrose layer on top of it?

bump

If [math]w\propto u[/math] and [math]w\propto v[/math] how do we get [math]w\propto uv[/math]?
I thought [math]w\propto u\therefore w=ku[/math] and [math]w\propto v\therefore w=lv[/math] which means that, if [math]u\neq v[/math], either [math]k[/math] or [math]l[/math] are not constants which means either [math]w\propto u[/math] or [math]w\propto v[/math] is false.
Now how do you get [math]w\propto uv[/math]?

>>15458649
Okay, by multiplying the two statements we get
[math] w^2=kluv [/math] which seems the best you
can get for [math] w^2 \propto uv [/math].

>>15458676
>>15458676
Yeah, but I saw a statement that was like [math]w\propto uv[/math] and it seems like to me that a function can't be proportional to two functions at once not equal to each other.

>>15458685
>>15458676
Well, it could be a typo--easy to overlook--or that
what you noticed is correct--one function can't be
proportional to two separate functions at once.

From [math] w^2 \propto uv [/math], you can't
square root the w without the u and v being
involved. And if u and v are the same, w^2 is
proportional to itself and [math] w \propto u [/math], for example.

>>15458720
No, I saw this sort of thing watching a video on population models. It seemed pretty good but this part seemed suspect to me:
> https://youtu.be/_wWpZYL1zU8?t=522
I just looked something up and it said that you have to hold a variable constant whilst working with the other variable but then, if you're holding one variable constant how do you get that the function is proportional to the product of the variables? and how are you "allowed" to make that step in the first place (i.e., in what contexts would one variable be constant whilst the other varies; the example used in the video doesn't seem to be one because if P varies so does M-P)?

>>15458746
>>15458720
Now I understand you. In the video, P and M-P are
related expressions as you have figured it.
Note that P is a function of time that changes
and M is a constant or number.
So, how does the [math] dP \over dt [/math]
remain the same with the product of the two
expressions?

Two equations: [math] {dP \over dt}=k_1 P [/math] and [math] {dP \over dt}=k_2 (M-P) [/math]
Note that when P goes up in value, M-P goes down.
To combine the effects of growth and decay in
one equation, they must be multiplied to give the
logistic equation with proportion constant.
This goes completely against how proportional
functions combine in a logical way, but we want
the "effect" to show itself. Pic related is the result
of a graph of the equation--fast increase and tapering
to the limit. The actual basis of creation I'm not sure about.

>>15447846
I would think using an equivalent circuit for the mutual inductor, then using frequency domain to simplify should work.

Any quantum mechanics books with a nice and proper definition of pure states and mixed states? Insofar they basically only vaguely define them in relation to the density matrix.

>>15459775
What does "proper" mean to you? A pure state is a vector in the Hilbert space. It defines a probability distribution for whatever observable you want to calculate via the density matrix. A mixed state is not a vector in the Hilbert space but is a weighted sum of pure density matrices. Intuitively the probabilities involved in the weighted sum in the mixed states reflect our ignorance of what the true pure state is, whereas the probabilities associated to the pure state density matrix are fundamental as far as we know.

>>15453462
lead us to the next kind of safe planet or expand the colony so we may prosper

Idk if this belongs in this board but:

Ive liked astronomy since I was young and thought rockets and astronauts were cool as shit

I still do, but a question has been bothering me

When you have one container with high pressure air (like a balloon or a gas tank) in an area of lower pressure, they try to equalise, but the strength of the material (rubber or metal) stops it from exploding up to a certain pressure, where the material cannot hold back the force of the compressed gas. This makes sense to me.

When space however, its completely different. Sure, stronger metals and stuff are used, but shouldnt the pressure from the air compared to outside the rocket or astronaut suit be something like infinitely stronger compared to outside it, considering space is a vacuum? Or does the pressure outside the container not play a role in its strength?

>>15459898
As in, defining pure states (other than just saying that a state is pure when the density matrix is rho=1 for one element and null for the rest/trace of squared rho is 1), shows some basic properties and such, and does the analog thing for mixed states.

Complete n00b here, how do I get Mathematica to simplify this integral? I know it must have a simple solution (see https://physics.stackexchange.com/a/176805/329254)) but I'm just getting this monstrosity from Mathematica.

The integral in question:
[eqn]
\int _{-\frac{L}{2}}^{\frac{L}{2}}\int _{-\frac{L}{2}}^{\frac{L}{2}}\frac{1}{\left(h^2+x^2+y^2\right)^{3/2}}dxdy
[/eqn]

>>15460702
MATHEMATICA CHADS... I KNEEL

>>15460292
If you want a textbook that tells you what you already seem to know, try pretty much any quantum information textbook. For instance the von Neumann entropy of a pure state density matrix is zero, and the converse holds too as far as I know.

Can someone help me with pic related? It's the last task I have left for next weeks assignment and I am stuck on all of them

>>15460778
First define a probability space [math](\Omega, \mathscr{A},P)[/math].
The sample space is just all possible die results. Since the game takes at most 100 turns you can just use a finite sample space for example
[eqn]\Omega = \{1,2,3\}^{100} \times \{1,2,3,4,5,6\}^{100} \times \{1,2,3\}^{100} \times \{1,2,3,4,5,6\}^{100} \times \{1,2,3,4,5,6\}^{100} \times \{1,2,3,4,5,6\}^{100}[/eqn]
The event space can be taken as the power set since you have a finite sample space.
[eqn]\mathscr{A} = 2^\Omega[/eqn]
The probability measure is also easy since none of the 6 dice are biases
[eqn]P(A) = \frac{|A|}{|\Omega|}[/eqn]

Now you can define integer valued random variables on the probability space that count how many steps the players move each turn. So for the first player take the random variables
[math]X_i (\omega) = \omega_i [/math] for [math]i=1,...,100[/math]
For the second player
[math]Y_i (\omega) = \omega_{100+i} [/math] for [math]i=1,...,100[/math]
For the third player
[math]Z_i (\omega) = \omega_{200+i} + \omega_{300+i}[/math] for [math]i=1,...,100[/math]
and for the last player
[math]W_i (\omega) = \omega_{400+i} + \omega_{500+i}[/math] for [math]i=1,...,100[/math]

Now with everyone defined you can start to calculate for (a)
[eqn]E[X_i] = 1 \cdot P(X_i = 1) + 2 \cdot P(X_i = 2) + 3 \cdot P(X_i = 3) = 2[/eqn]
do the same for the other players too.
For (b)
[eqn]V[X_i] = E[X_i^2] - E[X_i]^2 = \frac{2}{3} [/eqn]
The standard derivation is the square root of that and similiar for the other players.
For (c) just divide the number of fields by the expected values from (a). (d) and (e) should be obvious from the numbers given by the other problems.

Our professor defined [math]T=h\nu-\phi[/math], where [math]\phi[/math] is the work function (also defined as the energy needed to remove an electron of a solid surface). But in wikipedia the work function is defined as [math] W=-e\phi-E_f[/math], or [math]\phi=V-\frac W e[/math], where W is the work function meanwhile [math]\phi[/math],es the electrostatic potential. Are both definitions equivalent and I just can't see it or are they defining different things?

>>15460873
Your confusion is because the [math]\phi[/math] in both equations are not representing the same thing.

In T [math]\phi[/math] is energy (in electronvolts). In W [math]\phi[/math] is the electrostatic potential which is why [math]e\phi[/math] gives you an energy.

can I do integration by parts on a integral where the integrand are vectors and there's a dot product?
take this integral for example
[eqn]
\int_{t_1}^{t_2}\frac{\mathrm{d}^2\mathbf{r}}{\mathrm{d}t^2}\cdot\frac{\mathrm{d}\mathbf{r}}{\mathrm{d}t}\mathrm{d}t
[/eqn]
can I do integration by parts here like I would normaly do?

>>15461160
No, you have to do the dot product first (which means you are integrating some scalar rather than vectors)

"The data of the stopping potential vs. wavelength in an experience of illumination of a sodium plate are
λ (Å) 2000 3000 4000 5000 6000
V0(Volts) 4.20 2.06 1.05 0.41 0.03

Obtain graphically the work function φ"
I don't really get what I have to do. I know that
[math]T=h\nu-\phi[/math], so [math]q*V_0=\frac{hc}{\lambda} - \phi[/math]. So I got [math]V_0[/math] and [math]\lambda[/math], but still not having q and [math]\phi[/math]. I asked a classmate what I was supposed to do and he said it was just making a fitting in python but I'm not really sure how to do this in this exercise, any advice in how to proceed?
>>15460953
thanks bro, makes sense

>>15460843
Ahh now it makes sense, thanks anon.

How do I read and understand scientific papers? In order to keep up with the advances in AI and robotics, I was gonna start reading new papers that come out, but I have 0 idea what anything in them means. The whole thing looks Greek to me. I can't understand so many jargon let alone know what they're actually doing.

Suppose you have two discrete subsets in a compact subspace of R2. For each point in the first you want to find the closest point in the second set. What would the order of that function be?
N2 probably right?

>>15461770
Just like researchers there's a need to read much literature that sometimes it's a lot of informations, you just look it up, in universities you used to have professors and libraries and other research papers hence why research was a derivative of universities, now it's much more open to everyone. (internet)

>>15460191
https://en.wikipedia.org/w/index.php?title=Effect_of_spaceflight_on_the_human_body&useskin=vector#Vacuum

>>15438476
I think it's just leftists exaggerating the fact that all categories are just made up by humans to help organize the world in the mind.

>>15438848
What practical application is this math for?

Let's say you were in a space suit and were suddenly teleported in the middle of a super void, would you still be able to see your space suit? Assuming you had no helmet lights on. Or how far away from a star would you need to be for light to no longer be within the human visible spectrum?

If you dissolve a salt in water, do the resulting ions distribute themselves equally over the entire body of water or do they stay in the same spot?

I want to get into Galois theory, what are the prerequisites? I have basically no math background right now (I'm a first year EE student so I only took the basic maths exams like calculus and linear algebra)

>>15462552
Read Dummit and Foote.

>>15462552
pick up any undergrad algebra textbook for a reasonably soft introduction to everything you need or >>15462613

is raw chicken/salmonella dangerous to open wounds/cuts or only from ingesting?

Ok so super poor college student here. My phone started to spark last night and after a bit of extreme surgery I managed to get the battery pack out before it exploded/caught fire.
It seems to be...punctured. I have it stored in a glass jar in side a metal box for now, but I am worried.

I cannot afford to give this to an e-waste dump. They charge too much. How do I safely disable this thing from being a genuine hazard?

Yes I tried googling before asking, but all the answers are AI generated slop telling me to the right thing for the enviroment and recycle

>>15462901
Wow that sounds bad. I hope someone answers this. Sadly this is out of my scope of knowledge.
>>15462678
Theoretically possible but unlikely.
However, blood always a much more terrible place for an infection to begin. The places it can spread from entering your bloodstream...
yeah no thanks
>>15462346
What kind of water? Assuming suspended I will answer as follows
Per Brownian motion the ions would not stay in the same spot
>>15462298
>Let's say you were in a space suit and were suddenly teleported in the middle of a super void, would you still be able to see your space suit?
No
>Assuming you had no helmet lights on. Or how far away from a star would you need to be for light to no longer be within the human visible spectrum?
Distance doesn't matter per se, two equidistant stars from an individual are visible based on their brightness. The human visible spectrum is dominated by visible light, not "near" or "far" light.
So essentially however far until it falls out of the visible human light spectrum, so about 6.5 apparent magnitude
>>15460191
Space is not a perfect vacuum.We call it a vacuum because it can for most purposes be considered one.
It's also not really important which pressure is more extreme, what matters is the difference between the two pressures (inside and outside).
It also helps that the ISS is maintained at a pressure far above what most people consider "normal" but is still liveable.

All that said, it is a fact our current science/materials can handle the pressure differential.

>>15461160
Yes. You can prove it by writing out the sum for the dot product and then integrate-by-parts each term.

>>15462901
>How do I safely disable this thing from being a genuine hazard?
throw it in a large metal bucket of water outside and wait a couple days

Did a puzzle problem with my kid. Figure out the value of all the coins in a piggy bank given some constraints. Let's say: p for pennies, n for nickels, d for dimes, q for quarters.

n, d, q are each prime and distinct
p = d(n + d)
p = q + 24

Straightforward enough to determine n=2, which yields the relation between dimes and quarters:
d (d + 2) - 24 = q

So we plugged in primes for d until we hit a prime for q. This happened pretty early (d=5, q=11) so we're done. Still, we tried up to 37 dimes and no further prime quantities of quarters.

I'm wondering this, though: Did the author write the problem knowing that there are many possible solutions and that the alternate solutions were too large to be reasonable answers? Or, was the author able to anticipate that there would only be one prime solution somehow? If so, how did he do this?

tl;dr:
given:
d (d + 2) - 24 = q
where d and q must be positive and prime
is it possible to anticipate whether or not there are solutions other than d=5, q=11

>>15463062
You can write it as a quadratic

[math]d(d + 2) - 24 = q \implies d^2 + 2d - (24 + q) = 0[/math]

Given we only want the positive root (d can't be negative),

[math]d = \dfrac{-2 \pm \sqrt{4 + 4(24 + q)}}{2} \\
d = \dfrac{-2 + \sqrt{100 + 4q}}{2} = -1 + \sqrt{25 + q}[/math]

Since d must be an integer then 25 + q must be a square number. So we can write,

[math]25 + q = a^2 \implies q = a^2 - 25 = (a + 5)(a - 5)[/math] where a is some integer > 5.

So the only time q can be prime is when a = 6 giving q = 11, otherwise q has at least two factors so can never be prime.

>>15463211
Your explanation is terrific. I wasn't expecting something I could follow so easily. Thank you very much!

If anyone is curious, this was question 10 in a book is called:
Challenging Mathematical Teasers
by J.A.H. Hunter
Dover, 1980

I have a very basic physics question about energy.
I was told the following definition of a System:
>"A system is a set of two or more bodies exerting internal forces on each other. Forces exerted from bodies outside of the system are called external forces"
Then I was told the following definition of an Energy:
>"A system possesses energy if it has the ability to do work. You can do work on a system with an external force to transfer energy into the system"
So as I understand it, Energy is a property of a system of bodies, and not a body itself.
So if I have a road, a car, and a passenger, the passenger will have different kinetic energy in a passenger-road system and a different kinetic energy in a passenger-car system.
But let's consider another example: suppose, we have two bodies of mass m,M, where m<M, that are moving relative to each other with a constant velocity v. In the frame of reference of each object, the other one is moving with the same speed v. So the kinetic energy of body M in the frame of reference of body m is larger, than the kinetic energy of body m in the frame of reference of body M.
So, does that mean that the energy of the M-m system is larger than m-M system? But shouldn't it be the same if it's the same system? Or does it work in such a way, that it isn't symmetric, and so that M-m system is considered a different system than m-M system?

Any good resources to learn undergrad mechanics ? (Video lectures particular, I tried resnick halliday but my knowledge of calc is a little poor cause calculus 1 is a co-requisite to it)

>>15463859
The numerical value of the energy depends on the chosen reference frame

>>15464044
yea, I know that, but what does it mean when we say a "system of bodies has energy" then?

>>15464081
It means what it says

Why is -1 considered "larger" than -2? Or why is any smaller negative considered a larger number than a larger negative?
Please don't give me a response with the number like because that doesn't explain it.

>>15464106
I assume your confusion comes from a perceived ambiguity between "larger" and "greater than" as phrases.
"Greater than" refers to overall value unambiguously; in this case, -1 is greater than -2, and we can very easily show this is intuitive by, say, adding 3 to both of them. This reduces to 2 being greater than 1, which I don't believe you should have any problem with.
"Larger" can refer, depending upon your mindset, to either value, or absolute value, which is distance from 0 on the number line. If we run with the latter interpretation, then you are right to be confused, because -2 would indeed be "larger" than -1; however, this is not usually the sense people mean it in.
This, of course, generalises to any pair of negatives.

>>15463908
Bump

y=x2+1 ...(1)
x=(-y±√y2-4)/2

now, x belongs to c if y>2

from (1) i can see that if y=1.64 then x=0.8 now inserting y=1.64 will not yield a real value. what's wrong here?

>>15464255
retarded, b=0 ,c= 1-y

>>15463859
>>15464081
You can define kinetic energy for individual bodies within the system. Your book is introducing the notion of system because there is potential energy associated with interaction between the bodies in the system.

Let [math]\mu[/math] be Lebesgue measure on the unit interval [math][0,1][/math] equipped with the Borel sigma algebra. Suppose [math][0,1] = \bigcup_{i=1}^n A_i = \bigcup_{i=1}^n B_i[/math] where both unions are disjoint, and also [math]\mu(A_i) = \mu(B_i)[/math] for all [math]i[/math]. Is it obvious that [math]\exists[/math] a measure-preserving isomorphism [math]\phi: [0,1] \to [0,1][/math] sending each [math]A_i[/math] to [math]B_i[/math]? How to define such a map pointwise, or even setwise?

can someone explain how R2-cR2 isnt [0 1-c c-c^2] ?

>>15464548
It's a typo. They mean R_2-cR_3. It happens

>>15464551
okay that what I was assuming. thanks

is this guy a credible source of info?

https://www.youtube.com/watch?v=gXvuJu1kt48

>>15464146
Thanks. Great answer.
I guess you could say that -1 is relatively larger than -2 but -2 but -2 is absolutely larger than -1?

>>15463908
Bump

My book says that a Carnot cycle is the only possible revertible cycle that operates between two sources.
But, for example, between how many sources is a revertible Stirling cycle operating? I would say infinite but where are all the infinite heat sources? I don't get it

I'm going through a high school math problems book and now I'm on quadratics. One of the problems is to describe and graph function [math]y=|x-x^2|-x[/math] which I did successfully, except that I didn't detect this weird curvature when [math]0<x<1[/math] (pic related). So my question is: how was I supposed to detect it?

>>15466305
1. Write the function in piecewise form based on the absolute value
2. Compute its first and second derivatives

>>15466305
[eqn]y = |x - x^2| - x = \begin{cases} -x^2 & \text{ for } x - x^2 \geq 0 \\
(x - 1)^2 - 1 & \text{ for } x - x^2 < 0 \end{cases}[/eqn]

To solve those inequalites you factor them and note that a product is
>zero if at least one factor is zero
>positive if no factor is zero and an even number of factors is negative
>negative if no factor is zero and an odd number of factors is negative

So
[eqn]y = \begin{cases} -x^2 & \text{ for } 0 \leq x \leq 1 \\
(x - 1)^2 - 1 & \text{ for } 0<x \lor 1 < x \end{cases}[/eqn]

>>15466323
[math]x < 0[/math] in the last line.

is this right?
[math]yx(y^{-1}) = xy(y^{-1}) = x(yy^{-1}) = x1 = x[/math]

using commutativity, associativity, multiplicative inverse and identity?

I'm stuck on C, I'd appreciate help

>>15466475
Compute the eigendecomposition of [math]A[/math].

>>15466479
I've never heard of that before, I don't think it's been taught in our class, this is the solution, is this what you're talking about?

>>15466481
Yeah pretty much. You can write [math]A=V\Lambda V^{-1}[/math], where [math]V=[\bar{v}_1,\bar{v}_2][/math] and [math]\Lambda[/math] is a matrix having [math]\lambda_1,\lambda_2[/math] on its diagonal.
Then notice that [math]A^{2023}=V\Lambda^{2023}V^{-1}=V\Lambda V^{-1}=A[/math] again.

>>15466475

>uses special, school-defined algebra
>good

Consider a rod on a plane rotating about a fixed point [math]\mathcal{O}[/math]. One end of the rod is at a distance [math]a[/math] of [math]\mathcal{O}[/math] and the other is at a distance [math]b[/math].
The linear mass density of the rod is given by [math]\mu = \alpha x[/math] where [math]x[/math] is the distance of an arbitrary point on the rod to [math]\mathcal{O}[/math] and [math]\alpha[/math] is a positive constant. Given that the angular velocity of the rod is [math]\omega[/math], find the total kinetic energy of the rod.

I first wrote the expression for the kinetic energy of an infinitesimal portion of the rod [math]\mathrm{d}K = \frac{1}{2}\omega^2x^2\mathrm{d}m[/math], then I wrote [math]\mathrm{d}m[/math] as [math]\mathrm{d}m = \alpha x \mathrm{d}x[/math] and then I rewrote [math]\mathrm{d}K[/math] as [math]\mathrm{d}K = \frac{1}{2}\omega^2\alpha x^3\mathrm{d}x[/math].
Now, my question is, should I do two integrals, one going from 0 to [math]a[/math] and the other going from 0 to [math]b[/math], or should I assume that, let's say, [math]a[/math] is the smallest portion of the rod and integrate from [math]-a[/math] to [math]b[/math]?

>>15466538
1. It's a rod
2. Who gives a fuck
3. Who gives a fuck if you measure things about it
4. What does it all mean?

>>15437848
Hello, I am trying to understand Cauchy's argument principle and I need some help.
https://en.wikipedia.org/wiki/Argument_principle

Basically, I am using a test function f(z):
\begin{equation}
f(z) = \frac{2z+1}{(z^2-z-5)^2}
\end{equation}
with the following derivative:
\begin{equation}
f'(z) = \frac{6z^2+2z+8}{(z^2-z-5)^3}
\end{equation}
My contour is from 2 to 2i to -2 to -2i and back to 2. Basically just a little diagonal box around the origin.
My understanding is that my function has one zero and two poles. But one pole does not lie within the bounds of the contour. So with one zero and one pole within the bounds of the contour, by the Cauchy argument principle, the following integral should yield 0:
\begin{equation}
\frac{1}{2 \pi i} \oint \frac{f'(z)}{f(z)} dz
\end{equation}
However, when I actually calculate I get something much more messy, with logs and an arctangent. Am I misapplying the Cauchy argument principle here, and if so, what am I missing?

would it be possible to create uranium through fusion?

would it be feasible to create uranium235 through fusion to make nuclear weapons?

hello, this isnt really a stupid question but i dont know where else to ask

there's a discussion on /pol/ as to whether the answer is 17 or 21, and most of us are getting 21 but the image and some users say 17

>>15467123
oops i forgot here is the link to the discussion >>428556050

>>15467123
> most of us are getting 21
that just confirms /pol/ is the lowest iq board.

>>15467123
How is that not -13???

>>15467152
Ops my dyslexia is showin. I flipped the 8 and 5

>>15437848
Stooped question but I want to learn topology. Can someone give me a breakdown what I need to know before studying it and a list of resources for studying it.

>>15437848
Reposting because I messed up the Latex, sorry folks.


Hello, I am trying to understand Cauchy's argument principle and I need some help.
https://en.wikipedia.org/wiki/Argument_principle

Basically, I am using a test function f(z):
[eqn]
f(z) = \frac{2z+1}{(z^2-z-5)^2}
[/eqn]
with the following derivative:
[eqn]
f'(z) = \frac{6z^2+2z+8}{(z^2-z-5)^3}
[/eqn]
My contour is from 2 to 2i to -2 to -2i and back to 2. Basically just a little diagonal box around the origin.
My understanding is that my function has one zero and two poles. But one pole does not lie within the bounds of the contour. So with one zero and one pole within the bounds of the contour, by the Cauchy argument principle, the following integral should yield 0:
[eqn]
\frac{1}{2 \pi i} \oint \frac{f'(z)}{f(z)} dz
[/eqn]
However, when I actually calculate I get something much more messy, with logs and an arctangent. Am I misapplying the Cauchy argument principle here, and if so, what am I missing?

>>15467167
If I recall correctly, you have to account for the
order of the poles. So, the denominator of f(z) has
4 poles--2 poles for each root. One of the roots is
in the contour, so that's two poles total. The zero
is order one--so one zero. Altogether that should
be 1-2 = -1. The resulting answer of the integral
is -2*pi*i.

Trying to figure out the math on this basket weave tile pattern, what would the measurement of the small squares be? Please excuse the crude drawing didn’t have a straight edge around

>>15467421
Is it 10”x10”? Please tell me it was not that fucking simple

>>15467425
It was that fucking simple.

>>15467435
Sometimes I wonder how I put my pants on in the morning. Thanks

>>15467024
possible? yes. feasible? no. you don't perform fusion on ultraheavy elements, it's too energetically unfavorable. the best plan would be to do what we currently do, which is extract the naturally-existing 235 from the dominant 238, or perform neutron bombardment on 238 to turn it into plutonium-239 which decays to u-235 (and is itself fissile)

>>15467123
>and most of us are getting 21
and the best part is you seriously believe you have an intelligent opinion on political matters

>>15467421
it's 5"x5". if the horizontal rectangles have a width of 30" and the vertical ones 20", it's obvious the small squares have a width and height of 5", since the width of two small squares + the vertical rectangle equate to the width of the horizontal rectangle.

An analog op-amp integrator has a gain of -20dB per decade in its pracitcal region, and is identical to a 1-pole lowpass filter.
The integral of [math]sin(x)[/math] is [math]sin(x+\phi)[/math], but a sinusoid with 10x the frequency, [math]sin(10x)[/math], has an integral of [math]0.1sin(x+\phi)[/math]. In effect, a mathematical integrator has a "gain" of -10dB per decade,

Doesn't this mean analog integrators should be tailored to -10dB/decade instead of -20dB/decade? How would a "half-pole" LPF even work? Would a low-Q LPF even retain its integrator ability?

where should i talk about foundations of logic, /sci/ or /his/
it's philosophy, but it's still applicable to maths isn't it. so where should it be discussed.

>>15466486
>>15466511
I fell asleep but I managed to figure it out now, thank you lads

>>15467123
the masterrace everybody

Testing something real quick.

>>15467262
Hi, thanks for responding.
The trouble is that when I do this integral (in Maple) and divide by 2*pi*i, what I get is actually not an integer at all. I actually get the following:
[eqn]
\frac{1}{2 \pi i} \oint \frac{f'(z)}{f(z)} dz
= \frac{1}{2 \pi i} \left( \frac{3\pi}{4} +\arctan\left(\frac{3}{29}\right) -\frac{3\ln(3)}{2} +\frac{\ln(5)}{2} \right)
\approx -0.257 i
[/eqn]

>>15467999
checked

>>15468031
Maple gives me -1.

(Integrating over a circle rather than a diagonal box for simplicity but it should make no difference.

>>15467999
Test checked!

>>15437848
I've just started electromagnetics and we're in Coulomb's Law. I'd like to test it, but how would I apply a "known charge" to an object? I was thinking of testing it using balloons.

>>15468068
Alright, I fixed it and now I get -1 as well. I still don't 100% understand the fix. I had a parametrized path p(t) as a function of t. Apparently, when you do this, you need to include not only f'(p(t))/f(p(t)) in the integrand, but also you have to multiply by p'(t). I had neglected to do this. Once I included this, I got -1 like you.

>>15468125
>I still don't 100% understand the fix.
>I had a parametrized path p(t) as a function of t. Apparently, when you do this, you need to include not only f'(p(t))/f(p(t)) in the integrand, but also you have to multiply by p'(t).
Chain rule.

>>15467909
You should make note of the second method. It can help quite a lot in quickly calculating powers of matrix with nice patterns.

>>15468031
>>15468068
Even though you went through the integration to
obtain the answer, the nice thing about the
argument principle is you didn't have to touch the
integral AT ALL. Only note the poles and zeros
of f(z) and there's your -1.

I feel like I can easily find two paths which will invalidate this limit but my head is completely empty, any ideas? I've tried x=y and such, as well as x=y^n or x=x^n as well as x/y as a constant. I'm just not seeing it

>>15468271
I figured something out, is this valid?
x^4 = y^12 which gives
y^22/y^24+7y^22
simplifies to
1/y^2+7
y -> 0 = 1/7

>>15468271
Try going polar.

Let [math] y=r\sin(\theta) [/math] and [math] x=r\cos(\theta) [/math].

Factoring for r in both numerator and denominator gives: [math] {r^{14} \cos^{4}(\theta) \sin^{10}(\theta)\over r^8 (\cos^8(\theta)+7r^{14}\sin^{14}(\theta))} [/math]

Allow the limit as r goes to zero and you'll see the whole function
goes to zero. Thus, the limit exists.

>>15468356
It doesn't exist, it's indeterminate.

>>15468305
You can also do y^10 = x^4 which will give you a different limit (1)

>>15468356
>>15468359
No...you can fix theta to be whatever angle you like
and it'll still work out. Did you account for the division
of r^14 and r^8?

>>15468356
>Allow the limit as r goes to zero and you'll see the whole function goes to zero.
it very obviously does not
factoring out r^8 over r^8 you end up with r^6 in the numerator and r^14 in the sine term of the denominator, which on its own should be enough to indicate that it absolutely does not tend to 0 as r does

>>15468356
>>15468376
That doesn't work since x and y do not both go to zero when theta goes to zero.

>>15468383
>>15468376
I know x and y do not both go to zero at the same
time, that's how sine and cosine function. You have
to work completely in polar, radius and theta.

The theta only cycles you around the origin but
the action is done by the radius going to zero.
That's when it all zeroes out.

>>15468406
It's still indeterminate, you can use a different path and get a different limit.

>>15468489
>>15468406
Okay, tell me how you managed to get indeterminate
and what path(s) are you using?

>>15468515
enter the origin along the curve cos=r sin^2 (the tangent to the y axis as theta goes to pi/2)

An object is moving with a speed of 6.25m/s, is deacelerated at a rate given by dv/DT = -2.5√v. where v is the instantaneous speed. The time taken by the object to come to rest is ?

How to get started on this questions ? Not able to solve :(

>>15468554
Also, how any tips to solve these type of variable speed/acceleration questions ?

>>15468554
>>15468567
rearrange so you get something like dt = f(v) dv then integrate both sides with the appropriate limits for the velocity (the initial value for v is 6.25 and the final is 0)

>>15468527
>>15468515
I see...you actually cannot do that.

In rectangular (x,y) coordinates we need
an equation usually that brings us to the
origin since x and y go to zero independently.
Polar coordinates doesn't have that issue
because r can go to the origin but theta can't.

Thus, we cannot create a curve based in polar
(even if it mimics rectangular equations) for
this problem since theta can be fixed for anything.
This is the exact reason why we switch to polar
for some problems because it's as easy as letting
a variable (r) go to zero than building a curve for it.

Here is what I found on Symbolab:
https://www.symbolab.com/solver?or=gms&query=%5Clim_%7B%5Cleft(x%2Cy%5Cright)%5Crightarrow%5Cleft(0%2C0%5Cright)%7D%5Cfrac%7Bx%5E%7B4%7Dy%5E%7B10%7D%7D%7B%5Cleft(x%5E%7B8%7D%2B7y%5E%7B22%7D%5Cright)%7D

On Google, I can graph its surface by: x^4*y^10/(x^8+7y^22)
and if you zoom in close enough the origin is on a
flat surface between two very high peaks. This is
because the function is discontinuous at the origin only
and the "mountains" form two parabola-looking curves
from the denominator surrounding but not touching zero.

Can anyone help me with this Lean problem? The question is trivial but I have no idea how to tell lean what I want.

Getting hard filtered by spherical and cylindrical coordinates, you're supposed to write E in spherical and cylindrical coordinates and then compute the equation for cylindrical coordinates, but I'm stuck

Do I need to know any linear algebra to study ODEs?
t. got through Calculus I-III without knowing anything about linear algebra except for determinants and matrix multiplication

>>15467810
>>>/x/
>>15468956
I think you're either using some unconventional notation or you must have misinterpreted something.
rho = sqrt(r^2+z^2)
r = rho*sin(phi) = sqrt(x^2+y^2)
z = rho*cos(phi)
x = r*cos(theta)
y = r*sin(theta)
Cartesian coordinates are of the form (x, y, z), with x, y, and z being the distances parallel to each respective axis.
Cylindrical coordinates are of the form (r, theta, z), r being the length of a vector going along the x and y axes and perpendicular to the z-axis, and theta being an angle measured between the x and y axis.
Spherical coordinates are of the form (rho, theta, phi). Rho is the length of a 3-dimensional vector. Theta is the same angle as its cylindrical equivalent. Phi is an angle measured between the r-vector and the z-axis.
In some cases, you might see people use r instead of rho for the vector used in spherical coordinates, but I think that would just make converting from cylindrical to spherical coordinates and vice-versa much more confusing.

Otherwise, I think the formulas you're using and the procedures you're carrying out to be OK.

>>15468968
If you need to solve, for example, a system of
ODEs putting them in a matrix form is quite helpful.
There are also other linear algebra ideas brought
up in context for solutions to ODEs, but starting
off you might see just single variable equations
which are easier.

>>15469002
I'm using the same notation as our books, I noticed I made an error where I factored out r^2 which lead to 2r^2, which is incorrect, I've got an exam tomorrow and everything seems to be fine except for spherical & cylindrical coordinates so I'm up late which will probably ruin my performance and well... The closest I can get to is r^2 >= sqrt(rcos(phi)-1/4), which is incorrect. Here are my teacher's solution to a past exam which is where I got this from

>>15468176
>>15468068
Alright, now I'm trying something similar in Maple, but with the Zeta function. I'm integrating in a box from 13i to 1+13i to 1+15i to 15i and then back to the starting point of 13i. I believe I should get a result of 1 because there is one zero of the Zeta function in that range, and no poles.
[eqn]
\frac{1}{2 \pi i} \oint \frac{\zeta'(z)}{\zeta(z)} dz = 1
[/eqn]

However, the result I actually get is 0. (see pic related).

Can someone explain why the highlighted part is true?

>>15469200
(force of H+ on whole sphere) = (force of H+ on H+) + (force of H+ on H-)
so
(force of H+ on whole sphere) = (0) + (F)

>>15469196
>>15468176
That's interesting. Since the box is not touching
the only pole the zeta function has (where s=1)
and has a documented zero inside the contour,
this means that the pole is well away from the
box and the integral is zero by Cauchy's integral
theorem (Cauchy-Goursat), not argument principle.

Also, the argument principle calls for meromorphic
functions where the zeta function is holomorphic
at the box region.

>>15469219
I thought that would be the case, but what throw me off was that the integral would be calculated with the contributions of H+. I was trying to convince myself that the force contributed by the H+ would be equal of the force that was being neglected on the H-. But I don't think it's that trivial to see, at least for me, and using the argument that you gave makes the problem very easy.

Thank you, anon.

>>15469283
Okay, I think this is getting to my confusion. I was kind of thinking of holomorphic as a stricter version of meromorphic. For meromorphic, wikipedia says
>In the mathematical field of complex analysis, a meromorphic function on an open subset D of the complex plane is a function that is holomorphic on all of D except for a set of isolated points, which are poles of the function.
So I just figured holomorphic functions were meromorphic, and the set of isolated points was just the empty set. Apparently this is wrong, but I'm not sure why.

Anyway, my goal here is to use some kind of contour integral to "detect" zeros of the zeta function, scanning along the critical strip, getting a different result when the contour surrounds a zero. If this only works for meromorphic functions, and not holomorphic functions like the zeta function, is there any way to save this project, or is it doomed to fail?

>>15469283
>>15469317
Also I just realized, even if zeta is holomorphic, zeta'/zeta isn't. That should have poles at all the zeroes of the zeta function, right? So Cauchy's integral theorem shouldn't apply.

>>15469324
No. The only pole is where s=1, which becomes
the harmonic series. I'd figure that if you make the
box big which includes that pole, the zeta is
meromorphic in that box and so argument principle.

>>15469324
>>15469283
You got it!

>>15469352
...No, I don't think I did. You referenced Cauchy's integral theorem for why my integral should come out to be zero. But the integrand here is not holomorphic, so I am suggesting that Cauchy's integral theorem should not apply. Instead, the argument principle should apply, right?

>>15469356
>>15469352
Yes, the argument principle applies and 1-1 = 0 as expected.

>>15469356
Also, I just realized the Maple extract you had
has zeta'/zeta and not zeta. Silly me.

>>15469361
But the contour doesn't have the pole of zeta contained in it. Note that the argument principle DOES have f'/f as the integrand, not just f. So actually, if the argument principle applies, then it should be 1-0 = 1.

>>15469371
I know, I said as much here: >>15469366.
zeta'/zeta is the one that works for you

>>15469377
Well then I guess I still don't know why when I run it in Maple I get 0.

>>15469378
I saw that you have zeta'/zeta in Maple. The
argument principle applies because there is a
zero and a pole (from the zeros of the zeta function)
inside the contour. That's the only way you can
have zero. It is definitely not the integral theorem.

>>15469390
I feel like we're talking in circles a little bit, didn't we just agree (here >>15469371 and here >>15469377) that the argument principle should be giving me an answer of 1, not 0? Since my contour surrounds a zero of the zeta function, and doesn't surround any poles.

>>15469401
Okay. I agree with >>15469371 completely.
The argument principle has f'/f as the integrand
and not f. Just like how you had it on Maple, no?

What did Maple give as the answer? Zero.
The reason: zeta'/zeta has poles inside the
contour, just one. There has to be a zero that's
in the contour as well (not sure what's the value).
So, 1-1 = 0.

I was mixed up before when I was talking about
f = zeta alone and not f'/f.

>>15469430
But I've said here >>15469196 that my contour is from 13i to 1+13i to 1+15i to 15i and then back to the starting point of 13i. This contour does not contain the only pole at 1. It does however contain the first non-trivial zero of the zeta function, 0.5 +14.134... which you can see at this link:
https://mathworld.wolfram.com/RiemannZetaFunctionZeros.html

>>15469442
Yes, I agree the contour contains the nontrivial
zero 0.5+14.134i . The only pole for zeta is far
away from the contours. Again, this is about zeta.

Now, for zeta'/zeta. Division by zeta turns all the
zeros into poles. So, I know that 0.5+14.134i is
now a pole. But Maple gave the answer of zero
for the whole integral. Why? There has to be a
zero in the contour, specifically the zero coming
out of zeta' . If you find that zero from zeta' and
it resides in that box, you'll see that we have it.

>>15469465
I think you may be a little confused about what the argument principle is. I will quote it from wikipedia here:
https://en.wikipedia.org/wiki/Argument_principle
>Specifically, if f(z) is a meromorphic function inside and on some closed contour C, and f has no zeros or poles on C, then
[eqn]
\frac{1}{2 \pi i} \oint \frac{f'(z)}{f(z)} dz = Z - P
[/eqn]
>where Z and P denote respectively the number of zeros and poles of f(z) inside the contour C.

So here, I am replacing f(z) with zeta(z). Note that, for the argument principle, we care only about the zeroes and poles of zeta(z), NOT the zeroes and poles of zeta'(z)/zeta(z). So, for my given contour,
Z = 1
P = 0
right?

>>15469516
I see it now. I agree I was confused.
It's f(z) = zeta we're concerned with and
not zeta'/zeta. With the definition you provided
I have to disagree. Here's the reason:

The function f(z) has to be meromorphic inside
and on the closed contour. However, zeta is
holomorphic inside and on the contour--
we have a zero in the contour and not the pole.
On this distinction, we cannot do 1-0 at all.

Therefore, the integral is zero by integral theorem.

>>15469565
The integral theorem definitely does not apply, because the integrand here is not holomorphic. (Note that the integral theorem requires the ENTIRE integrand to be holomorphic, which in this case is zeta'/zeta).

For the note that zeta(z) is holomorphic and therefore not meromorphic, I would point you back to my comment here >>15469317 where I explained how I view holomorphic as a stricter version of meromorphic. I may be incorrect in that understanding, but I still don't understand how it is incorrect.

>>15469580
It's the other way around actually. Meromorphic
basically means holomorphic except at the poles.

I agree with zeta'/zeta being meromorphic, but
ultimately we can regard it as holomorphic to
an extent.

Now that you mention this, I'm going to look up
other theorems.

>>15469621
>Meromorphic basically means holomorphic except at the poles

Yes, and zeta(z) is holomorphic except at all of its poles (of which, it has none). So is zeta(z) meromorphic?

Would you say that the argument principle cannot be applied to any region with P=0?

>>15469628
meant to say it has none within the region enclosed in the contour.

>>15469628
In a vacuous sense...yes. You can say zeta is
meromorphic with zero poles in the contour.
In fact I saw a video where something like this
is being used for the argument principle 43 sec.
in: https://www.youtube.com/watch?v=CdonFbLduk4

>>15469672
So then would you agree that something has gone wrong if Maple is giving me 0 for the result, when the argument principle predicts I should get 1? Like maybe I am entering it wrong into Maple or something, but the problem as formulated should give me an answer of 1, right?

>>15469679
Yes...something happened for sure. I might not
have an immediate answer for this, but you can
play around a bit to see if anything is entered
wrong or change the contours.

I'm still looking for relevant theorems.

>>15469688
>Good to know that we are in agreement at least. I've attached a pic of my Maple code if you or someone else who has Maple wants to reproduce what I've done, or find a mistake in it.

p1, p2, p3, and p4 represent the corners of the box.
q1 represents the parametrized path from p1 to p2, q2 represents the parametrized path from p2 to p3, etc.
And then just below that I perform the contour integral.

>>15469697
You can carry this over to the next /mg/ or /sqt/
if you want.

>>15469697
This is equivalent to parametrizating the whole box as
[eqn] \gamma(t) = \begin{cases} q_1(t) & t\in[0,1] \\
q_2(t-1) & t \in ]1,2] \\
q_3(t-2) & t \in ]2,3] \\
q_4(t-3) & t \in ]3,4] \end{cases}[/eqn]
What I don't like about this parametrization is that this is not [math]C^1[/math] as the derivatives are discontinuous in the corners.
Taking a cubic spline instead of piecewise linear functions will make [math]\gamma[/math] continuous differentiable on the whole box and Maple will give 1.

If [math]\psi \in \mathbb{C}[/math] then, afaik, [math]|\psi|^2=\psi^* \psi[/math] (the * is the conjugate). If [math]\psi[/math] has the form [math]ce^{(a-bi)},\ a,b,c \in \mathbb{R}[/math], then [math]|\psi|^2=c^2e^{2a}[/math] for any [math]a,b,c \in \mathbb{R}[/math], right?