/sci/ - Science & Math

Prove germ theory isn't true

>>15310646
ok, what am I missing
>2-tailed hypothesis test, normal distribution
>sample mean = 624.16, hypothesised mean = 650, standard deviation = 409.42 and number of trials = 51
>z-value = -0.45072
easy BUT

>calculator gives a p-value of 0.6522
>z-table gives a p-value of 0.65272
tried various online calculators and they give either of the above. i dont know why they dont all say the same as the z-table. am i an idiot? which is it?

Consider the polynomial [math]f(x)=x^3-x+1 \in \mathbb Z_3[x][/math]. Consider [math]R=\mathbb Z_3 [x] / (f(x))[/math], a finite field of order 27. Find an element [math]r \in R^{\times }[/math] which is a generator. The hint provided is along the lines of "Let [math]\alpha \in R[/math] denote the element [math]x+(f(x))[/math], so that [math]f(\alpha)=0[/math]." Don't see how thats relevant or what the path to solving is tho. I guess I want to find some explicit r with order 27?

>>15310802
You assume the data follows a normal distribution (the null hypothesis [math]H_0[/math]).
Your 'z-value' is the test statistic you are using to test this hypothesis. The 'p-value' is the probability, given that [math]H_0[/math] is true, of observing data at least as extreme as you did (in either direction since 2-tailed). From your assumption you know that [math]z|H_0~\text{Normal}(0,1)[/math], so you can calculate that this p-value is about 0.65.

>>15310828
I understand this but I need to calculate the exact p-value, the issue I'm having is that I am getting two slightly different p-values depending on the method used

>>15310831
Probably because of rounding. In R this is
2*pnorm((624.16 - 650) / (409.42 / sqrt(51)))
which yields 0.6521901 (rounded of course)

f(Divide[π,2]) = cos(Divide[π,2]) * sec(Divide[π,2]) + sin(2*Divide[π,2])

My calc teacher want us to find the period of the function, its value and the graph of it, but isn't it undefined?

Could someone help me understand the response I was given here? >>>15285875

I think I understand the first part, of how we justify that it has degree 3, but how do we explicitly compute f and solve for the constants c1, c2, c3?

>>15310988
correction: >>>15308792

Can anyone find an example of two matrices A,B that don't have a common eigenvector such that eig(A+B) subseteq eig(A) + eig(B)?

How do you determine the strength of various dessicants? Some function of water vapor pressure and the Keq of the hydrate?

How would you compare CaSO4, H2SO4, P4O10, 3A sieves, and so on?

Consider[eqn]V_{\mathbb{C}} \ni (x,\, y) \mapsto \overline{(x,\, y)} = (x,\, -y) \in V_{\mathbb{C}}[/eqn]where [math]V_{\mathbb{C}}[/math] is the complexification of a real vector space [math]V[/math]. I have to show that this is a real-linear transformation. What do they mean by "real-linear"?

>>15312508
You have to prove that it's an endomorphism if you view V_C as a vector space over R.

>>15312568
okay thanks

>>15310646
Suppose I have one function f(x, y) infinitely differentiable on y, but only continuous on x. If f^n(x, y) is the n-th partial derivative with respect to y, is f^n(x, y) continuous on x for all n?

For example,

f(x, y + h) = f(x, y) + f'(x, y)h + R_1

By the integral form of the remainder R_1, f'(x, y) should be continuous in x, no? Then by induction it should be true.

>>15310820
Do you know why that quotient is a field? Do you know that any finite multiplicative subgroup of a field is cyclic? What is the order of the cyclic group in this case?

26 = 13 * 2
x^2 is not the identity
Is x^13 the identity?

In a coastal town, the city government wants to build a bridge that
track the movement of the tides. The variation of the tides during a day can be
modeled as a sinusoidal function of the form y(t) = A sin(B(t − C)) + D, where
A, B, C, D ∈ R are constant parameters to be adjusted. If a high tide occurs
at 6:00 am and a low tide occurs at noon, and the amplitude of the change in
water level is 4 meters, find the equation that represents this variation along
of the day. At what time will the water level be 2 meters above average?
Help

Is 5 days enough time to study for freshman mid terms and get an A? I have the mid terms for two modules due in 5 days. AAAAAAAAAAAhhh

>>15313363
5 minutes is enough dumbshit, pay attention in class

Could you break down salt into chloride and sodium?

Had a theoretical idea of mixing finely crushed salt into flour then dusting the air (like creating a cloud of flour). Then you ignite the flour turning it into a small explosive. But would that explosion have enough energy to break the bonds in salt and create chlorine gas?

>>15313641
if you want the chlorine, you could just... do electrolysis? instead of your weird flour explosive?

>>15313647
Of course that would be the best way but I'm curious if my shitty idea would still work. I know it's not practical

>>15312796
Sounds good to me.
Alternatively use continuity of f'(x,y) in y to get uniform convergence and exchange limit orders with Moore-Osgood

>>15313641
>enough energy to break the bonds in salt and create chlorine gas?
You mean a bunch of sodium and chloride ions which would immediately bond back together once they lost some of that momentum?

>>15310802
What is variance?

>>15310802
You are getting different values because z tables are only precise up to 2 decimals. How can you be so retarded? You can clearly see the answers are very similar.

>In this figure, a piston P is attached to a cylinder. The piston is connected to a wheel with two shafts, AB and BC. The shaft AB is attached to circumference of the wheel. The wheel is being rotated with an uniform velocity of ω = 100 rad/s. Determine the velocity of the piston at the moment when A, C and center of the wheel are linear and θ = 60° and dθ/dt = 500 rad/s.

Please help me with my homework.

>>15313911
You don't define theta. Is it H on the figure?

>>15313926
Yes. I'm having trouble because there are two angular velocities in action.

>>15313974
define three vectors v_B, v_A, v_C corresponding to the points in the figure. There are 6 unknown time derivatives of the components, and the goal is to find the time derivative of the x-component of v_C.

-The time derivative of the y component of v_C is zero by the problem set up.
-You know both components of the time derivative of v_A since it is just given by rotation plus center of mass motion, both of which depend on omega (and the radius).
-You know both components of the time derivative of v_B-v_A since you are also gien the angular velocity for H (or theta).
-You know that the time derivative of the magnitude |v_B-v_C|^2 vanishes since the length of the piston is constant.

This is enough equations to solve for what you want.

>>15314012
Could the problem also be solved in this manner?

>>15314036
Sure, the only difference between that solution and my suggestion is the last point about using the time derivative of |v_B-v_C|^2 vanishing. Also note that your problem is slightly more complicated than that (since point A is on a wheel).

I've started an engineering degree and am doing a calculus 1 course with a lot of applications.
I'm managing with the stuff basically, I think. I can do most of the simple problems alright. I mean it's usually just a case of following a rule or formula. But the worded questions (of which there are a lot) are messing me up. I take a look at them and I'm like "I have no idea what this question is asking me to do"

And I can follow the examples we're given but that doesn't seem to help much when every question is different.
I dunno. It's frustrating me. Am I retarded or what? Maybe I shouldn't be in an engineering degree.

>>15314045
I see. But what of dθ/dt? Does that have no effect on the velocity of piston?>>15313911
>at the moment when A, C and center of the wheel are linear
Couldn't that point be the instantaneous center when the velocity is zero?

>>15314067
>I see. But what of dθ/dt? Does that have no effect on the velocity of piston?
Yes it affects the time derivative of v_B-v_A like I said.

>Couldn't that point be the instantaneous center when the velocity is zero?
I don't know. I don't think so but it's not obvious to me without actually doing the problem. I know the strategy I presented to you will work though. Tough problem for a first year physics class (I assume), good luck

>>15312822
Its a field since f is irreducible in Z_3 and the multiplicative subgroup removes 0 so it has order of 27-1=26. i think we have:
x^2=x^2
and
x^13=2=/=1
therefore
x^26=id and order(x)=26 and so α is a generator

is this the right approach? does it make sense to talk about x as an element of R? im a bit confused about what the elements of R actually are still

>>15314281
like i see that x^3-x+1 implies x^3=x+2, from which u can derive x^13=2 and x^26=id, but then what is the actual generator of the group? is it just x itself? im sort of deducing this stuff bsaed on previous computational examples ive seen, but i still dont get what the actual generator is

>>15313641
You might reduce some trace sodium to Na(0), but it'd instantly reoxidize in air. Cl- wouldn't be touched, you either need a strong oxidizer (stronger than O2) or an electrolysis to get it into Cl2. Wheat flour, if anything, would act as a reducing agent only. Heat generally doesn't reverse reactions.

>>15314281
It's exactly what you have there, the elements of R are equivalence classes of the polynomials of degree less than 3, hence the quotient has 27 elements (by counting). The multiplicative group is just the 26 nonzero equivalence classes. The equivalence class of x (let's say [x]) is a generator because powers of [x] greater than 2 must be some other polynomial [p(x)] = p([x]) of degree less than 3. By Lagrange, its order must be either 2, 13, or 26. Turns out it's 26. In fact, a cyclic group has exactly one subgroup of order d for each divisor d, so R^x has only two non-trivial proper subgroups (the cyclic group of order 2 and the cyclic group of order 13). This means there are at least 13 polynomials (those outside the cyclic subgroup of order 13) [p(x)] = p([x]) that generate the whole group. It's not always the case that [x] is a generator. For example, it doesn't work when you consider F_2/(p(x)) for p(x) = x^4 + x^3 + x^2 + x + 1, since [x] is of order 5.

They "define" another element a as simply [x] because the quotient k[x]/(p(x)) when p(x) is irreducible is isomorphic to a field extension k[a], where a is "some" root of p(x) in k[x] outside the field k. It's pointless to speak in these terms when one is working in arbitrary fields though, because the root a is precisely defined as the equivalence class [x]. You can always imagine the complex numbers as the prototypical example. Q isn't algebraically closed, so some roots of some polynomials in Q[x] will lie outside Q, and we know how those look like (the square root of 3, i, whatever). But you can use a similar construction to simply "define" a root a = [x] in the quotient Q[x]/(p(x)).

Can somebody please tell me what the fuck is going on here?

[eqn]f \in k([0,\tau] \times \mathbb{R})[/eqn]

if f is a function f(x(t)) and t is on the interval [0,[math]\tau[/math]], and f has asymptotes for 2 and -2. How am I supposed to find an inequality with k in it? I don't even understand the notation

https://en.wikipedia.org/wiki/Outline_of_statistics
There are a lot of topics and theories here. Which ones are useful to know?

>>15315271
Please ignore this question: I will find a book for a general understanding instead.

>>15315302
Don't know anything about statistics but wikipedia is pretty good for math topics desu. Sometimes you don't need a whole book.

>>15315302
One thing you have to realise is statistics pedagogy is yet to catch up with modern statistics. Pretty much every undergraduate book still puts most emphasis on Classical Frequentist statistics. Even Machine Learning is not given much priority. So you won't be learning REAL statistics until quite a few books.

>>15314653
thank you so much for this explanation, this explained lots of the intuition behind what was happening here <3

We can create antiparticles right?
We don't have the technology yet, but theoretically we could create antiwater.
Imagine there's antiwater in a perfect vacuum and I go slurp it all up and drink it.
What happens? Would it just annihilate the second it touches my tongue, or does it specifically have to touch normal water in order to annihilate?

>>15315396
it doesn't care about the specific atoms at play. a particle and its antiparticle meeting are sufficient, so as soon as electron and positron (or any other pairing) touch they annihilate.

what does this symbol mean

>>15315423
skip to next video

>>15315423
If your book doesn't have an index of symbols, it's not worth reading. If you're using lecture notes, you should have paid attention during the lecture.

>>15315532
Neither, tt was from a conference paper presentation. I found it in their paper after asking. I thought maybe it was a common symbol I didn't know but I think they just made something up

>>15315423
normal (not necessarily proper) subgroup. at least that's the only thing I've seen it used for

>>15315589
they were using it to say the set has element wise greater than for the whole set

emission / absorption spectrum questions
>how much of the matching frequency will the atom absorb? (if there are like a million photons and 1 electron)
>will it always emit the same frequency it absorbed, or can it emit other frequencies that atom has?
>does it emit in all directions, 1 direction, the same or opposite direction, a random direction?
>photons can only carry 1 frequency right, because they ARE the frequency?
>do the frequencies not absorbed pass through or bounce off the atom? (or both)

>>15316068
>how much of the matching frequency will the atom absorb?
depends on the other net forces acting on it, how much it can actually move or whatever under those tiny pulls. How much resistance. Same reason some bulbs draw 100W and some 50 from the same wire.

I have this idea for a double jump apparatus. Okay, so imagine you jump, but an AI or whatever is tracking your movements and the floor rises with you. In the air, you crouch down like your getting ready to jump again, and then the moment your feet touch the platform, it locks into place so you can push off of it for a second jump. And then it unlocks and drops back where it was. You're with me so far? That makes sense? Would that work?

Now imagine if, instead of a floor, it was like a little platform attached to your shoe with hydraulic pistons running up along your calves. The computer would track your legs and adjust the piston to keep the platform at the same level while your legs moved from the jumping motion. Now these platforms may have to get detached and turn this into something you can only use once before reloading, but, like, would that work at all? Is there anything there at all? I'm not asking if it's possible to go out and make this right now, just if it sounds theoretically plausible.

>>15310646
Wouldn't the "benefit of the doubt" be that he's clinically retarded and not trolling surely?

People can't help if they're clinically retarded, who would put a troll above that?

>>15316171
You're basically talking about this, but with a solenoid or other linac in the springy part (and a battery, or compressed air tank).

>>15316208
and btw these are super fucking dangerous and cripple plenty of kids and adults every year. Your body doesn't have a control system for these dynamics like it does your muscles and masses.

>>15316068
> how much of the matching frequency will the atom absorb? (if there are like a million photons and 1 electron)
If a photon with some frequency is absorbed then the atom can't absorb another because it's already at a new energy level. So one until it emits that energy.

> will it always emit the same frequency it absorbed, or can it emit other frequencies that atom has?
if allowed it can fall down to the lowest energy state in multiple hops but that is far, far less likely than a single jump.

> does it emit in all directions, 1 direction, the same or opposite direction, a random direction?
random direction

> do the frequencies not absorbed pass through or bounce off the atom? (or both)
(potentially) both, that is entirely what scattering cross-section calculations are about.

>>15316213
>another because it's already at a new energy level.
Yes it can. It can jump up more than one energy level without emitting anything.

>>15316210
Oh yeah, I'm not gonna try and make it or anything, it's just a thought I had a while back and could never decide if it was just a troll logic thing I was too stupid to find the fault in.

>>15316215
Not a photon of the same original frequency.

>>15316213
>random direction
I don't think this is truly right either. It's just a noisy function of the photon and atom's relative geometry, initial conditions or whatever, which is pointlessly small and unprofitable to matter for most applications so it's hardly ever calculated or estimated.

>>15316221
>Not a photon of the same original frequency.
oh yeah sorry, I missed this part of OP. It'd be a different amount of energy and momentum obviously my b.

could there have been life on venus in the distant past

>>15316215
>>15316221
is it still the same atom at that point? I thought they used spectrographs to identify atoms, but if it can go up a level then absorb an entirely new frequency, how can you even identify what you're observing?

>>15316226
how would one calculate it?

>>15316226
That's scattering, not absorption followed by emission at a later time. An excited atom emits a photon in a random direction. It's irrelevant the path of the particle that initially put it into that state.

>>15316240
>a random direction
Do you mean random as in "totally quantum unpredictable," or random as in "depends on the electron's exact angle in orbit when it happens, which is virtually unmeasurable"

>>15316235
>it can go up a level then absorb an entirely new frequency, how can you even identify what you're observing?
it is the same nucleus obviously, otherwise it would absorb the original frequency instead.

>>15316248
since when do electrons have angles? I thought they were dimensionless and orientationless?

>>15316248
both

>>15316235
> thought they used spectrographs to identify atoms
They do, you are literally describing Spectroscopy. Shine light a gas, you point a camera at it. Certain frequencies are absorbed and then re-emitted, that light no longer hits the camera so you see dark lines.

>>15316270
maybe I'm thinking about it wrong, but here's my confusion
>atom that absorbs green
>shine white light at it
>you don't get green back
>energy level increases
>atom no longer absorbs green, is now absorbing blue
how can you identify that this is a green absorbing atom and not a blue absorbing atom if its state changed?

>>15316264
the electron is orbiting at a certain plane and at any given time its "located" at a (fuzzy) angle as well. That's why magnetism. Wouldn't magnetically aligned atoms all emit their electrons in plane?

>>15316280
>identify t
that's why you use the entire spectrum and not just one line in it. Some spikes do overlap unmeasurably close but never all. So the overall thing is different.

>>15316280
That (almost) never happens because the difference between the lowest energy state and the next is far larger than the next jump. The difference in energy levels decreases with each jump. You'd have to absorb green and then say far, far infra-red to jump to the 2nd level.

>>15316281
spin orientation has absolutely nothing to do with the orientation of an atom.

>>15316352
Okay, does that mean it doesn't matter at all though??

>>15316281
>orbiting at a certain plane
>>15316352
>spin
sorry, I mean axial orientation, not its plane w.r.t. the nucleus.

>>15316377
Atoms at rest have no meaningful orientation, all you can assign is spin.

>>15316171
> but, like, would that work at all?
No. This is just a fancy version of the idea of "pulling yourself up by your own bootstraps".

You can't push off from something that's attached to you. If the platforms detach just before you push off from them, they'd have to have significant mass, otherwise you'd just push them away without gaining any momentum yourself. And that mass would just weigh you down on the first jump.

If they were heavy enough, it would "work" in the sense of doing a double jump; it's just that the overall jump height would be no better than you'd get from a normal jump and probably worse due to it being less intuitive.

>>15316414
Okay, you seem to still be missing my question. Does an electron's spin or pole axis truly have ZERO effect on its emission geometry? Is this relationship literally uncorrelated, or just very hard to measure and sample.

>>15316442
> Does an electron's spin or pole axis truly have ZERO effect on its emission geometry?
Yes. Emission is completely random.

>>15316465
interesting, thanks.

>>15316465
>Emission is completely random.
how do lasers emit all in one direction, in phase?

>>15316475
When such an electron decays without external influence, emitting a photon, that is called spontaneous emission. The phase and direction associated with the photon that is emitted is random.

Lasers use a combination of stimulated emission, which requires an external electromagnetic field, and mirrors.

[eqn][\exists\ J \in End(V): J^2 = -1] \implies [\dim_{\mathbb{R}}V = 2n \text{ where } n \in \mathbb{N}][/eqn]How can I show this? Things I showed so far or already know
>the inversion is true
>if the dimension of V is 1, then there is no such J
>J is an isomorphism

>>15317758
You can't show this because it's not a true statement.
Consider [math]V = \mathbb{C}[X][/math]. which is an infinite dimensional vector space over [math]\mathbb{R}[/math]. but
[math]J(p(X)) = i \cdot p(X)[/math] is an endomorphism on it with [math]J^2 = -1[/math].

>>15317758
>>15317770
With the further restriction that V is finite dimensional try applying the determinant. Remember that the determinant of the negative identity is -1 if the dimension is odd and 1 if the dimension is even.

>>15317770
I should have said that V has to be finitely dimensional. Excuse me.

>>15317758
Odd-degree polynomials have at least one real root, hence matrices on odd-dimensional vector spaces have at least one real eigenvalue.

>>15317785
>>15317788
thanks

Say I have some curve in [math]\mathbb{R}^3[/math] which is parametrized by trigonometric polyonials.
So something like {(cos(t), sin(2t), cos(3t) + sin(t)) : [math] t \in \mathbb{R}[/math]}
Under what circumstances is the Zariski closure of this curve strictly greater than the curve?
>>15317788
I haven't been here in a few years and have to say it's reassuring how the same old touhoufags are still going strong.

>>15317849
I think it's not hard to figure out anon, good luck, read the definition

>>15317862
Are you sure this is as easy as you think it is?
Since in some cases (such as a simple unit circle in the xy plane), the closure is just the curve itself, you seem to imply that it's always the case.
So for any point not on the curve I have to find some polynomial vanishing on the curve but not on that point.
It's not at all obvious to me how to do that.

>>15317912
Shut the fuck up worthless talentless trash and solve it yourself

So why does it diverge instead of converging at 2/5

>>15318074
sry

Let t be transcendent over a field K. What does it mean that the set of polynomials P(t) is "algebraically equivalent" to the set P(x)?

>>15318082
You are adding infinitely many numbers all between 1/3 and 2/5. Of course it diverges.

Oh yeah I forgot we add them up I keep on thinking its the same rules as finding a limit

>>15318114
>>15318105

Liouville-Ostrowski Theorem for the integrability of elementary functions has no wiki article. Wtf! Does anyone know of a yt vid where someone goes through the proof? I didn't find anything.

>>15318147
Do you mean this?
https://en.wikipedia.org/wiki/Liouville%27s_theorem_(differential_algebra)

>>15318162
yep, that's it

>>15310646
How can I make my page get nicer?

>>>/g/92490328<div class="xa23b"><span class="xa23t"></span><span class="xa23i"></span></div>

>>15318087
isomorphic as rings

>>15310646
i have been recently absorbed by age reversal and life extension research. Last time i took that topic up i was 11, not a lot of info was avilable and now a lot of things changed.
During researching i encountered 2 big names Aubrey de Gray - prophesizing endless life and David Sinclair - machinism theorist who sells NMN.

I am highly skeptical of anything new (for no reason, nothing bad ever happened to me when trying out things) so i don't fully trust any of their talk.

here's the question fellow /sci/ssors, do you think aging is preventable? Is anyone here updated on the aging science news and can enlighten me?

can someone explain the april fools joke to me?

>>15318713
>powered by GPT-4chan ▲
that's it, that's the joke, it implies that 4shids is inflating it's userbase with bots and allowing AI research on hands on site

For school physics tutoring you can dm me in discord aslan#4650, first lesson free

im taking a course for networks but its extremely low quality and shallow, they dont explain in any depth at all, they just want you to memorize certain catchphrases.

Is the Tanenbaum book on computer networks any good for an introduction? or is there any prerequisite knowledge?

Let's say the probability that a particular person dies at any instant of time it 5%. If I try to compute the probability that that person survives the next 5 seconds, I would do (1 - p(death at instant))^(number of instants in 5 seconds). There are an infinite number of instants in 5 seconds. So this quantity becomes 0, so there's a 0% chance I survive the next 5 seconds. What's wrong with this logic?

>>15319165
you need to assign probability to a time frame to calculate how long it takes to have supposed 100% probability cause because let's say every second the probability rises by 5% (100>105>105.25 etc). Even then 5% probability can repeat every second effectively making the probability calculation ineffective or at least not exponential because 5% every second is still 5% three hours later just like a coin toss is 50% every time you toss a coin no matter how many times you get tails or heads.

>>15318711
bump

You guys like stats right?
Pls solve these for me, I’m at my wits end and this homework platform is unforgiving

>>15319263
if you don't want to learn that, just use ChatGPT , you'll get a B and a pass

>>15319165
Instants are not a real unit of time measurement, so you can't get a meaningful survival rate out of that.
On the other hand, something like "0.05 deaths per second" is reasonable for describing a population (e.g. based on average life expectancy data), and you can normalize this figure to be per capita if you like.
Mathematically speaking, the thing you're really after is known as a "Poisson process".

>>15319165
Nothing is wrong with that. That's the right answer given the assumptions you started with.

>>15319263
looks like a confidence interval problem.
c & d use CLT should be easy to solve via google.

First I am hearing of chebyshev's theorem.

When mass warps spacetime it creates a gradually shaped well, right? But say something was spinning near the speed of light. Would spacetime warp gradually like it was a center of high mass?

Is there a way to compress data connections (say followers, who follows whom) as polynomials? Like you draw a line/graph that represents a person, then if they follow someone they intersect with that person's line. I know this would just get insane lines like picrel, but if one line can repersent hundreds of potential connections surely this is a way to compress the size of the data that expresses these relationships? Like picrel

Also, how would you express this, is it even possible? I'm guessing it would have to not be random for this to happen.

>>15319784
Correct.

https://en.wikipedia.org/wiki/ADM_formalism#ADM_energy_and_mass
https://en.wikipedia.org/wiki/Kerr_metric

How do I determine the value of the Lipschitz constant L (with ||f(a) - f(b)|| < L||a - b||) if f is a function from R^2 -> R^2?

I've never taken any functional analysis or anything like that, I just need to find a value for it with my given function f

So I'm dealing with the set-theoretical identity between a vector space V and its complex conjugate vector space [math]\overline{V}[/math]. So the map is this
[eqn]id: V \ni v \mapsto v \in \overline{V}?[/eqn]Do I understand this correctly?

>>15319874
you have to adjust for the different norm. You probably have to use the Euklidean norm for two coordinates. That's the alteration you have to somehow factor in. In practice, I would first graph the function with a 3d plotter and then work out the proof later on.

>>15319876
Additional question: What is an antilinear isomorphism? Is it just an antilinear, bijective map or is there more to it?

>>15319876
The identity will not be a linear map between those vector spaces.
Let [math]\circ[/math] be the scalar multiplication on [math]V[/math] and [math]\bullet[/math] be the scalar multiplication on [math]\bar V[/math].
For [math]L:V \to \bar V[/math] to be a linear map you need that for all [math]x,y \in V[/math] and [math]\alpha \in \mathbb{C}[/math] that

[eqn]L(\alpha \circ x + y) = \alpha \bullet L(x) + L(y) = \bar \alpha \circ L(x) + L(y)[/eqn]

>>15319886
Well the norm isn't defined, but thanks. I'll go and graph it

>>15319929
So did I understand the definition of the identity correctly? That it is
[eqn]id: V \ni v \mapsto \overline{\overline{v}} \in \overline{V}?[/eqn]I know that this is not linear (my task is to prove that the identity is antilinear; which is true for the such-defined mapping). I was confused because the exercise didn't provide me with a formal definition.

>>15319932
In that case, they most certainly want you to use the euclidean norm.

When dealing with a complex conjugated vector space [math]\overline{V}[/math] can you simply assume that canonical identities of the conjugation, like e.g. [math]\overline{v + w} = \overline{v} + \overline{w}[/math] , are true?

>>15319970
nvm this question, it's solved

Something has a 30% chance of happening and it happens 4 out of 5 times. What are the overall chances of this happening?

I believe this is a binomial distribution, so using that I got ~.028
Why is this different from (.3)^4 * .7? Is it because there are additional possibilities of when the failure occurs (i.e. first trial, second trial...)?
If I asked the probability of one specific scenario happening- happens the first four times and then doesn't happen- is .3^4 * .7 valid?

>>15320414
>the probability of one specific scenario happening- happens the first four times and then doesn't happen- is .3^4 * .7 valid?
yes. But in a sense, there are five ways this can happen to satisfy your original question; any position of the single failure. {AAAAB AAABA AABAA ABAAA BAAAA} are all okay, right? So the binomial distribution accounts for that; multiply your formula by five.

Let's say we have a pendulum system with the standard equation [math]\ddot{\theta} = - \frac{g}{l}\sin\theta[/math]. Now imagine I want to push the pendulum bob at a regular interval in order to perturb the system with a random direction and force. How do I 'accurately' model this?

My first thought was adding a Gaussian perturbation term [math]\ddot{\theta}_p = \sum_i \alpha_i \exp(-\beta(t-t_i)^2)[/math] but it's not working too well

I'm trying to learn Lean 4 and whenever I uses the `cases` tactic it gives the resulting terms weird names like †h which are a pain to use in proofs. How do I rename them? If I do `Cases with x y` it throws an error.

and ignore my shitty proofs thanks

>>15320634
Fourier Transform and do it in the algebraic space I’d think.

>>15320634
Oh and now that I think of it, trapezoid numerical method gets you spectral accuracy with periodicals like that

[eqn]\lim_{x \to 0^+} \frac{1 + x\ln{x}}{x} = \frac{1 + \lim_{x \to 0^+} x\ln{x}}{\lim_{x \to 0^+} x} = \lim_{x \to 0^+} \frac{1 + 0}{x} = \infty[/eqn]Is this legitimate?

I Want someone to convince me to not commit sodoku tonight.

I've always been the textbook case of being smart enough to do menial work but dumb enough where I won't be synthesizing something new. I lack common sense and most times people will correct me but in a very condescending way or they won't correct me because they would rather keep me down. Though I did somehow fumble my way into an engineering program. I don't really learn quickly enough so my grades have been falling(no I'm not killing myself just because of school but it is a contributing factor). Though I believe this might be a result of the education being catered to being perfect the first time rather than being centered around letting people make mistakes. I also don't believe in the idea that hardwork will let a person achieve success. It's all circumstances/scenarios and who you know. The idea of hardwork is a slave mentality to a certain extent.

I don't really see any point in most things these days. Yeah video games /tv shows/ movies are nice distractions. But nothing really interests me or motivates me. I see most relationships (friendships included) though I don't have any real friends) as a chain around my neck. But that's a personal view.

>>15320725
why over complicate and not just = lim 1/x + lim x/xlnx = 0 + lim lnx

>>15320738
Stop worrying and enjoy your life.

>>15320740
1/x with x to 0 is infinity. So you have a infinity - infinity situation in this case. I think you overlooked the limes

>>15310646
What causes cancer /sci/?

>>15320745
There is nothing to enjoy tho

>>15320725
For me I'd go[eqn]\lim_{x \to 0^+} \frac{1 + x \ln{x}}{x} = \frac{x + x^2 \ln{x}}{x^2} = \lim_{x \to 0^+} \frac{1 + 2x \ln{x} + x}{2x} = \lim_{x \to 0^+} \frac{1}{0} = \infty[/eqn]

>>15320758
Fuck, missed out the lim for the second one.

>gravity can bend light
>gravity does not SLOW light
>but if gravity is super powerful enough (black hole) it can completely stop light
Any ideas why there's a discontinuity ? Goes from not slowing it at all to completely pulling it in?

>>15320758
I don't see how the extra step makes it any clearer. In the end you're doing the exact same thing, you could've done in the beginning.

>>15320753
dude think about the awesome Marvel and Star Wars movies that you're gonna miss :-)

>>15320763
black holes arent infinitely dense.

that's just the limit of the object, which can never be realized.

>>15320772
I don't like marvel or Star wars movies.

>>15320769
Oh, never mind. Since the numerator goes to 1 and the denominator goes to 0, you can do your steps no problem.

>>15320738
well you could always critically think about what you like and what you'd like to work towards, but i suspect you don't want to (think or work, that is) so..
just stick it out fag, you don't need good grades, go get a comfy job in hvac or whatever the fuck and travel the world in yearly vacations impregnating whores all over.
more importantly fix your mentality around
>it's all about who you know
>relationships are a chain around my neck
incompatible views, i recc dropping the latter. civilization is built by midwits and who they know/ drink with
>>15320753
nah there's a lot, if you don't enjoy what you're doing, go do something else
there's a lot to see in life.
GET OUT OF DOING THE SAME OLD SHIT IF IT'S CLEARLY NOT WORKING FOR YOU

>>15320785
Imagine how happy yu vill be when you eat ze bugs and live in ze pod and yu vill own nozing after ze forz industrial revolution.

>>15320791
I don't believe in a hedonistic life style. Only insecure fools attempt to live that kind of life.

The problem with HVAC and other technical jobs like electricians/masonry etc is that it will take a physical toll on your body by the time you're 40-50 years old. You dont really get to enjoy your retirement to the fullest extent as a result. Also there is always a chance that robots could take those kinds of jobs.

I have never really enjoyed talking with other people. It was always a means to an end for me.

>>15320841
i'm not a believer in hedonism myself, but i'm sure you'll enjoy it for a lil bit while you wait to get perspective - not that it matters tho. there's many many other things you could do, par example the supreme normalfag route of enjoying your uni life and then starting a family (don't bother giving your argument for why you don't want a family, i don't care). i'm not going to prescribe something specific
just stop trying so hard to find reasons to not enjoy your life and GET OUT OF DOING THE SAME OLD SHIT IF IT'S CLEARLY NOT WORKING FOR YOU
especially while you're still young

how do you all write decimals?
for example, would you write 0.123 or .123?
I was taught that the zero is superfluous, but most people I have encountered write a zero before the decimal.

>>15320884
>just stop trying so hard to find reasons to not enjoy your life

I see what you're saying. But I'm not really trying hard to find reasons for not living out of the blue. I'm looking at every little thing that has happened in my life, good and bad, and deciding based on that. It's not about doing the same old shit.

What you don't know is how much my brain is fucked when I'm in around people(even those that are familiar with me). I can articulate my thoughts just fine through writing but for some reason my brain can't let me articulate what I'm thinking through speech properly. There comes a point where sentences don't come out grammatically correct through speech or I'm reaching for a word but it's not coming to me. So people don't take me seriously as a result. That's another aspect.

>>15320922
I'll skip the zero if it's in my own notes, but if I intend to show it to anybody I throw it in. Makes it easier to recognise it's 0.123 and not 123 with a little mark right by it

>>15320922
decimals are harmful, use fractions

>>15320946
>brain is fucked
this is nothing that can't be practiced and improved upon - i had/have similar problems.
i don't know what more to tell you though. life is worth living, i can promise you that. please stay alive and try a lot of different things, i'm sure you'll find something sooner or later that you'd want to live for, something that gives you great satisfaction
my final note is just that you should try to not get too caught up in your own mind/thoughts

>>15320758
let x lnx = y. the lim x- > 0+ is the same as lim y -> 0+ and your new function becomes (1+y)/exp(W(y)) which trends toward unity.

>>15320763
Probably the only actualy way to understand what's happening is studying Lorentzian geometry.

>>15320738
First off, it sounds like you might be depressed. I know the meme of people not trusting doctors but you should try to get professional help: depression is a real mental affliction and you will only realize that once you have gotten over it. Second, your school problems are probably not just due to your intelligence or lack thereof. I take it you never study together with anyone, well that is making your time a hell of a lot less enjoyable and challenging. Third, video games and TV shows should not be a reason to live. I won't deny that there's never any artistic merit to them, but when it comes to entertainment, what makes it worthwhile is enjoying it with other people. Otherwise you are nothing but a consumer.

Before you're going to throw away your life, try to make yourself sure it's really as hollow as you make it sound, because it probably isn't. Search for value in things: skills, stories, people, music, places, etc. If after having genuinely tried something you find you don't like it, explain to yourself why and move to the next. If you've already decided on committing suicide, what do you have to lose? In the best case you find a reason to live, and in the worst case you've proven yourself right.

>>15310646
>new thread
>I was never notified
Man, this sucks. Anyway.

Unanswered questions:

Previous questions:
>>15293182

Maths questions:
>>15286044
>>15288722
>>15290099
>>15290834
>>15298482
>>15298677
>>15299353
>>15300877
>>15305775
>>15307308
>>15308007
>>15308370

Physics questions:
>>15309963

Chemistry questions:
>>15305583 (RbCl and BaF2)

Stupid questions:
>>15286231
>>15287191
>>15288104
>>15290489
>>15291792
>>15291850
>>15292340
>>15298869
>>15301482
>>15301495
>>15305185

>>15321105
Thank you for your response.

A couple of things.
I trust doctors. I don't trust pharmaceuticals. I think psychology has a long way to go and I don't think centering treatment around medicine is the way to go for depression/ anxiety (especially now that the research shows that gut bacteria may have huge influence on these issues).

I have never found studying with other people productive. I prefer to learn at my own pace. I don't like slowing anyone else down. Besides everyone ends up not liking me anyway since I ask dumb obvious questions or because I'm incapable of communicating my thoughts properly, so as a result of that I'm of no use to them since they are trying to learn too.

>>15321390
NTA. Consider the probability of various life outcomes compared to the attitude you are currently taking. Is life trajectory better or worse after having medical and psychiatric intervention? On the whole, usually better. Can individual ineptitude make things worse? Absolutely. Is that cause to write off the potential benefit? Nope.
>I trust doctors. I don't trust pharmaceuticals. I think psychology has a long way to go and I don't think centering treatment around medicine is the way to go for depression/ anxiety (especially now that the research shows that gut bacteria may have huge influence on these issues).
Yeah sure and somehow obesity is not the ultimate cause of eating excess calories. You, like those people, are seeking excuses rather than treatment.

Let's even assume there IS a causal mechanism in some % of cases anyway. You have no way to know that would be your case. Especially given inconsistent results or null findings https://www.ncbi.nlm.nih.gov/pmc/articles/PMC9157303/

You don't understand the research you read, so relying on said misunderstanding of research is only going to fuck you up further.

>>15321442
Thanks for the link and educating me
But the obesity argument seems like a false equivalency and I would never argue against that.
I still think CBT(cognitive behavioral therapy) is the best method except for extreme cases.

>>15321495
>But the obesity argument seems like a false equivalency and I would never argue against that.
Not equivalent, but analogous. Perhaps loosely, at least with regard to people making inferences from misunderstanding research.
>I still think CBT(cognitive behavioral therapy) is the best method except for extreme cases.
Nothing stops you from trying that and finding out the hard way, for good or ill.

>>15321442
>as a whole
There is no body of research and no total weight of all outcomes. You are arguing in completely bad faith without any knowledge on the topic.

>>15321552
>There is no body of research and no total weight of all outcomes. You are arguing in completely bad faith without any knowledge on the topic.
OR you could've interpreted what "as a whole" meant with respect to the whole of the factors considered, rather than assume I would make inferences from unknowns. Who is arguing in bad faith here?

What exactly is my teacher doing I get doing the ratio test and replacing everything with n+1

I get the terms canceling out and then Supescript 4 sqrt(n) & Supescript 4 sqrt(n) +1 cancelling out because 1 becomes insignificant.

Why does -2x in the denominator turn into 2x, is it because of the absolute value signs, why are they not removed then

But why doesn't she find c the center

where does the one come from in 2x < 1

>>15321567
> is it because of the absolute value sign
yes.
> why are they not removed then
because that would imply a complete difference range of values for x: |2x| < 1 means -1 < 2x < 1, but 2x < 1 would mean that -infinity < 2x < 1 which is not correct for the convergence test.
> where does the one come from
because that's the radius when using the ratio test

Photons always move at the speed of light. Do they warp spacetime around them?

>>15321636
They have energy and all energy warps spacetime.

>>15321442
I'm not really interested in this debate, especially since this is under the post of a guy contemplating suicide.

Why is QM mostly a probabilistic theory? Is it because that's the random nature of particles on a sub-atomic scale or because they're too small to measure with modern technology?

Why is it called Statistical Mechanics and not Probabilistic Mechanics. There is no statistics.

>>15322406
Statistics is the study of probability distributions. You don't care what individual particles are doing, but what they do instead as a whole.

>>15322400
As far as we know, particles have inherently random behaviour. Non-believers tried to rule this out using hidden variable theories. Bell's theorem crushed them.

https://en.wikipedia.org/wiki/Hidden-variable_theory
https://en.wikipedia.org/wiki/Bell%27s_theorem

>>15322400
we don't know

>>15322400
>Why is QM mostly a probabilistic theory? Is it because that's the random nature of particles on a sub-atomic scale or because they're too small to measure with modern technology?
You can't measure something at that scale without changing it by definition. To measure some energy, you must have it interact with some other energy. That changes what you've just measured because you've altered it in some way as a result of that exchange. Random, but conditional.

Anyway, yes, it is random but it is conditionally random. A common example is the three-polarizer paradox https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Quantum_Tutorials_(Rioux)/07%3A_Quantum_Optics/7.13%3A_The_Three-Polarizer_Paradox
So two filters at 0 degrees and 90 degrees = 0% passthrough. Adding a third filter at 45 degrees, 0->45 = 50% passthrough, 45->90 = 50% of the remainder (12.5% of the inital light) passes, even though 0->90 = 0%.
Each interaction creates a new probability as the remaining photons are now oriented in a way that let them pass through the new 2nd filter. As a result, 50% of those photons now pass through the 90 degree filter instead of 0%. Rather than what you'd expect filtering macro objects, where the remainder would still be 0%. This demonstrates very easily what the randomness is. The conditional "weird part" is the fact adding more interactions, or "measurements", change the results unlike with macro objects.
https://www.youtube.com/watch?v=MzRCDLre1b4

There are many ways one could interpret experiments beyond what the experiments show. I simply leave it as what the experiment shows, so a kind of instrumentalism. It will likely remain this way until further experiment or experimentally validated theory offers a better demonstration of some alternative.

>>15322467
>>15322400
I am very tired though so I maybe fucked up something the video is probably better at explaining it.

>>15322467
>Rather than what you'd expect filtering macro objects, where the remainder would still be 0%
Why would you expect that for macro objects?

>>15322473
>Why would you expect that for macro objects?
Because macro objects are not being changed by the new 2nd filter. If the 0 filter and 90 filter remove 100% of something, for macro objects adding a third filter in-between still results in 100% of things being filtered.

>>15322483
But using the filters clearly changes the objects because if you don't have the 0 degree filter and just the 90 degree filter some objects will still pass, so it's not unnatural to expect the 45 degree filter to change the objects too

>>15322499
I am not sure what is being lost in the communication.

Say you have 2 shapes in the macro world. Circles and squares.
Filter 1 removes all circles
Filter 2 removes all squares
Now add a third filter
Filter 1 removes all circles
Filter 2 removes all squares or lets all squares through
Filter 3 is pointless, or the added filter 2 does nothing and filter 3 still removes all squares.
In the quantum world filter 2 instead changes some of the squares to triangles and some remainder get through the square-shaped holes.

That is not a great analogy but I am at a loss trying to figure out where in the macro world you think you get the same result, except to be cheeky and make up some contrived scenario that is clearly not relevant.

>>15322507
I see what you're trying to say but you're assuming some specific mechanism for how the filter works which I don't think has to be necessarily assumed

>>15322507
>In the quantum world filter 2 instead changes some of the squares to triangles and some remainder get through the square-shaped holes.
I am too tired. I meant to get through the final square-removing holes.

>>15322510
>I see what you're trying to say but you're assuming some specific mechanism for how the filter works which I don't think has to be necessarily assumed
what in the hell are you even talking about

They need to make coffee stronger for this shit I put myself through

>>15322519
Ok, let me be direct. The polarizer thing just follows from classical electromagnetism so I don't think there's anything quantum about it

>>15322527
>Ok, let me be direct. The polarizer thing just follows from classical electromagnetism so I don't think there's anything quantum about it
Go ahead and show me your formulas and work. I am all ears. Enlighten me.

>>15322527
Hang on, I had a thought on this problem. Did you not watch the video or read through my link? Because my explanation relies on your having done so, somewhat, to understand what part of this is not classical. Is that where the problem is? Is that why you claimed I am "assuming a mechanism not observed"?

There are experiments demonstrating things not shown in classical EM. If you are unaware of those, if you think this is somehow not based on experiment in reality validating quantum field theory, then maybe I can understand why you are saying these things. You need to explain further what, exactly, you are talking about.

>>15322540
https://www.feynmanlectures.caltech.edu/I_33.html#Ch33-S4

>>15322552
Yeah that just reaffirms what I am thinking here >>15322547

So you're trying to say the mechanism from quantum field theory and the eigenstates is what you do not believe in? Is that why you claimed I was "assuming a mechanism"? Because nothing in what you just linked describes or would explain quanta. That is simply a description of what the macro world observes, and to that extent and degree it is right. That stands to reason as they are both describing what was observed in this case.

If you specifically want experiments or demonstration of quanta, that is why you go from there to examples such as stern-gerlach https://en.wikipedia.org/wiki/Stern%E2%80%93Gerlach_experiment

I use the polarizer example notwithstanding because it demonstrates something classical EM does not. Namely, that probability with quanta does not work like macro objects in that interactions change them. If you are claiming the quanta is "not in evidence" you have been trying to mislead people by asking questions you are not genuinely interested in hearing answer to.

>>15322568
I'm just saying that the polarization law can be explained classically like in that feynman lecture

>>15319165
Nothing is wrong with the logic. What you are (probably) meaning to use is a hazard rate, which is a measure of the `force of mortality' in continuous time.

I'm gonna fucking explode if I keep getting this report returned, it's feels so fucking simple but no matter what I fucking do I always get it fucking returned, please help me. It's the last thing I have to do for my mechanics class and the supervisor is just fucking retarded giving horrible advice. Thank you.

"Two riders of different masses collide horizontally with equal
great speeds. Derive an expression for the lighter rider's speed after
the collision. The expression should be simplified as far as possible."

>>15322578
>I'm just saying that the polarization law can be explained classically like in that feynman lecture
No you're not.
>>15322510
>I see what you're trying to say but you're assuming some specific mechanism for how the filter works which I don't think has to be necessarily assumed
You wanted an explanation of quantum mechanics so I gave you an example and how it relates to quantum mechanics. The equations are, also, in quantum field theory. That it can be explained in other ways is of no importance nor is it relevant. It is simply very easy to see and understand, and is something useful to map explanations with probabilities onto with quanta, and can mathematically be modeled with quantum field theory.

You are not "just saying" the polarization can be modeled in classical EM, which is true, you are claiming some assumption exists as if it is not demonstrated experimentally. Obviously if I am assuming to explain something about quantum mechanics I ASSUME you want the explanation from quantum field theory. Unless you are being dishonest. So what, exactly, did you mean "assuming some specific mechanism"? The mechanism is not assumed, it is experimentally demonstrated in other ways.

>>15322321
I (responding to my own post) am retarded and completely misread Anon's post. Please disregard.

>>15322593
No problem happens to everyone.

>>15319263
These are not difficult, you really should just open your textbook.
750, 840, 95%, 662 and 786 is what I get.

>>15322586
>You wanted an explanation of quantum mechanics
I wasn't the one who asked the question
>So what, exactly, did you mean "assuming some specific mechanism"?
I meant that you were assuming that the macroscopic object interacted in a certain way with the polarizer and hence the experiment seemed paradoxical.

>>15322600
>I wasn't the one who asked the question
Thank you for clarifying.
>I meant that you were assuming that the macroscopic object interacted in a certain way with the polarizer and hence the experiment seemed paradoxical.
In terms of quanta that is absolutely the case that it is paradoxical if viewed from the ordinary sense of macroscopic objects. Classical EM is not describing it the same way for the same thing, even if it arrives at the same results by modeling continuous waves at macro scales.

Are you happy now? Do you understand why I said it does not matter to what I was explaining that it can be explained some other way, which would not at all answer anything about quantum mechanics?

>>15322608
>Are you happy now?
Yes, but you should probably give a more quantum example next time

>>15322612
>Yes, but you should probably give a more quantum example next time
As a matter of demonstrating how probabilities work at quantum scales this example works perfectly fine. It is something people can see and visualize while also understanding that, as individual units, the probabilities do not obey what macro objects would. I won't change it because someone nitpicked about something irrelevant and completely missed the point as a result. My examples don't change to prevent your embarrassment.

Other than that sure go ahead suggest what you'd prefer if it isn't on that basis.

>>15320922
with 0 that's the standard in my country

Consider the bases B_1 = [math]\{b_1,\, b_2,\, b_3,\, b_4\}[/math] and B_2 = [math]\{2ib_1 + b_2 + 4b_4,\, 3b_2 + 7b_3 + ib_4,\, 0.5b_3 + ib_4,\, -ib_3 - ib_4\}[/math]. Is
[eqn]\begin{pmatrix} 2i & 0 & 0 & 0 \\ 1 & 3 & 0 & 0 \\ 0 & 7 & 0.5 & -i \\ 4 & i & i & -i\end{pmatrix}[/eqn] the change of basis matrix [math]_{B_1}[id]_{B_2}[/math]? I mean is this the matrix that transforms a corrodinate vector represented by B_2 to a coordinate vector represented by B_1 or is it vice versa?

>>15321166
>>I was never notified
How do you envision this? I think this thread was made after the last one archived.

>>15288722
That book is more theoretical (which is good if you can spare the time), and indeed less focused on engineers.
>>15288104
Take differences and/or ratios between consecutive terms.
>>15292340
*shrug* SICP is fun, go for it if you don't know how to program.

>>15322584
That question is so badly written you don't know if they meant an elastic or inelastic collision.

I presume they want elastic but it's a terrible example. I mean if two riders going fast crashed into each other head on I'm fairly sure the result would be zero velocity and a lot of screaming from the wreckage.

>>15323042
Yea and I keep getting it wrong, it sounds so simple but it's always wrong

>>15323062
You can just google the fucking answer: https://www.khanacademy.org/science/physics/linear-momentum/elastic-and-inelastic-collisions/a/what-are-elastic-and-inelastic-collisions

It's even simpler than the general case as both the initial velocities are identical.

Not really getting this whole harmonic series convergence stuff

wouldn't 1/n converge since as n goes to infinity it converges on 0

[math]0^0=0/0[/math]
?

>>15323076
I have googled it, and results show exactly what I calculated, it's fucked

>>15323084
>>15323082
is that for me I don't get it how does it become 0/0

>>15323082
>wouldn't 1/n converge since as n goes to infinity it converges on 0
No, the individual terms converging does not necessarily mean the series itself converge.
To put it in a more intuitive way, the idea is that 1/n does not converge towards 0 fast enough for its sum to converge.

>>15323086
And what did you calculate?

are these dipole arrows correct? and where would the lone pair's moment point to?

>>15323084
[math]0^n = 0,\ n^0 = 1[/math]. Both can't be true when [math]n = 0[/math] hence [math]0^0[/math] is undefined.

>>15323082
The convergence of the sequence to 0 is only a necessary not a sufficient condition for the convergence of the series.

>>15323106
Alright I can get that since at 1 you start already 1 itself I guess.

But what about 1 / n^2, that converges right, why is that line of fast enough at n^x where x>1, it seems random and arbitrary

>>15323156
Look, the area under the function 1/(2x) is lesser than the sum of 1/x, right? Because you have at least 1 + 1/2 + 1/6 + 1/8 + ... in the first case and you have 1 + 1/2 + 1/3 + ... in the second. Now ask yourself how can I determine the area of the function 1/(2x)? Well you simply take the integral of the 1/(2x) and then you look if the interval diverges; so you end up with 1/2 ln(x) as the integral of 1/(2x). Obviously the ln(x) is not bounded, it diverges. We know that the area of 1/(2x) is smaller than whatever the sum of 1/x is; so naturally the sum of 1/x also has to converge, don't you agree?

>>15323185
*diverge not converge in the last sentence, sorry

>>15323123
there is no unless statement in math?

>>15323156
>But what about 1 / n^2
That famously converges to pi^2/6 yes.

>>15323192
>>15323185
I guess that makes sense, I have a test tomorrow and this is making my head hurt so I'll just except the definitions given to me from here on

>>15323118
good question. Lone pairs do apparently (from googling) create a dipole moment, which would be pointing away from the N atom.
Keep in mind though, that the Cl and O also have 3 and 2 lp's respectively, so the overall moment would still be pointing towards the Cl and O.

>>15321166
couldn't you write a script of some sort that notifies you when a new sqt appears?

>>15323289
Perfect, thank you for the response

how do astronomers differentiate between red shift and extinction?

>>15323334
extinction leaves absorption lines unchanged since it only attenuates higher frequency photons, while Doppler/red shifting causes the lines to be shifted as well

>>15323202
There is but how would that help?

Struggling a lot with basic stuff while solving complex equation, forgetting a lot of rules (exponential, factoring, taking common elements) while doing calc and trigno

Makes me kinda depressed and while learning new things i just play with identities a lot and not able to click that idea in mind makes me feel retarded, i cant do double digits multiplication in my head and feeling a lot depressed. Pls help anons, i just can't give up on engineering

>>15323412
you forget what you don't use
you won't remember the derivative power rule in 3 years from now

>>15323460
Is it normal to forget the most basic stuff ?

>>15323460
but that's the only one worth remembering

>>15323412
Statistically, it's the same for the majority of the people in your program.

[eqn]\left(1 + \frac{a}{x}\right)^{bx} = e^{ab}[/eqn]I came across this very helpful identity. Anyone knows a quick proof this?

>>15323578
It trivially follows from the definition of e.

[eqn] e = \lim_{n \to \infty} \left(1 + \frac{1}{n} \right)^n = \lim_{x \to \infty} \left(1 + \frac{1}{\frac{x}{a}} \right)^{\frac{x}{a}} = \lim_{x \to \infty} \left(1 + \frac{a}{x} \right)^{\frac{x}{a}}\\

e^{ab} = \lim_{x \to \infty} \left(1 + \frac{a}{x} \right)^{bx} [/eqn]

>>15323578
That is only true in the limit x -> infinity

>>15323578
probably something involving the taylor series

>>15323608
great thank you

>>15323578

[math](a^b)^c = a^(bc)[/math]

?

>>15323608

[math] (a^b)^c = a^{bc} [/math]

another useful identity for you

Do analytic bump functions exist?
i.e. a smooth function with compact support that has a taylor series everywhere

>>15323692
Smooth functions with compact support exist like
[eqn]f(x) = \begin{cases} \exp \left(\frac{1}{x^2 - 1} \right) & |x| \leq 1 \\ 0 & |x| > 1 \end{cases}[/eqn]
A smooth function has a taylor series everywhere but the taylor series does not have to be equal to the function in more than one point.

Analytic functions with compact support don't exist except for the constant zero function. To see why simply try to compute the taylor series in a point outside of the support. Clearly the series will be constant zero.

>>15323716
welp that sucks

Okay so for limit comparison tests I know you have to get a b series, but how does 1/2n=3 become 1/n

Also is it right that it diverges if a is greater than 0

Also if b is 1/n and I already know that diverges why do I have to continue doesn't that mean the whole thing is divergent

Forgive me for not knowing latex.
Let p be the canonical mapping from X to X/R, q the canonical mapping from Y to Y/S, where R and S are relations. Furthermore, let f be a mapping from X to Y. The first part of the exercise I'm stuck on has me proving that the following two conditions are equivalent
(i) There exists a function f* from X/R to Y/S such that f* composed with p = q composed with f.
(ii) For all x', x'' in X, R{x',x''} => S{f(x'),f(x'')}
I substituted the functions into a theorem from a previous chapter to get the equivalence, but the second part has me stuck. It asks to derive the uniqueness of the map f* using condition (ii). Now it's easy if you use condition (i), since if g, h are functions such that g comp p = q comp f and h comp p = q comp f, then for all x in X, g(f(x)) = h(f(x)), so on their domains (X/R) they would be equal, hence f* is unique.
But using condition (ii) I can't get anywhere because the implication isn't helpful in establishing uniqueness (I guess). It's certainly possible that I'm overlooking an argument in the text, but if someone has an idea I would certainly appreciate it. Thanks!

>>15310646
Good Night /Sci/entists!

I have been working more on typesetting the Maid Space book. I figured out how to make the spacing bigger so the Maid Dance is more readable, but it looks off-center and I don't know why. It appears the center is being determined by the upper left corner of the Maid Space somehow?

How can I make the centering look nicer and is it possible to make the Maid Decoration longer or should I try to find a different symbol?

Thank you /sci/entists for reading my post.

slow thread

Could someone explain this simplification in more detail to me? It's not clicking. The source is from Apostol Vol. I if that's needed.

>>15324045
The essential part of the comparison test that the
limit comparison test doesn't have is the inequality
between a_n and b_n series.

If you let b_n become 1/2n, this series is less
than 1/(2n-3) at each step from n=2. We know
that half of a harmonic series still diverges and
since the limit is positive, a_n diverges.

It's possible to, say, have a divergent series
but the limit be negative or something else
mainly because limit comparison test has
no comparison between a_n and b_n built
in.

>>15324045
> how does 1/2n=3 become 1/n
Huh? That doesn't happen anywhere in that example. I think you might be confusing the Limit Comparison Test with the Ratio Test.

> is it right that it diverges if a is greater than 0
[math]c = lim a_n/b_n[/math] if c > 0 then either both series converge or both series diverge.

> why do I have to continue
see above, since you know b (1/n) diverges and c > 0 then a_n must also be divergent.

really stupid poster here

like really really dumb

here is my homework:

>Select a process or phenomenon from everyday life that you are interested in analyzing.
>Use a histogram to plot the frequency at which the observations occur. Does the histogram resemble any probability distribution that you know of?
>Apply a distribution fitting tool to determine which distribution model is most appropriate.
>Identify which parameters are most appropriate in describing the selected distribution.

pretty simple right?
however im retarded

The phenomenon I want to observe is the probability of getting at least one copy of character of rarity five at least one time using certain amount of in game currency, the game is a gacha.

here is an example of the data the data was obtained from one column here:
>https://docs.google.com/spreadsheets/d/1FeF2xC15EL_NudUV5R3xqltwHANbZf9hk_5UH7G6O4U/edit?usp=sharing

>0.008
>0.015936
>0.023808512
>0.031618044
>0.0393651
>0.047050179
>0.054673777
>0.062236387
>0.069738496

for my histogram I used 253 of all the data so I fit exponential line an be done with it
I made the histogram and got pic related I assume its all right but feel free to correct me

here are my 3 questions:
1.what are the parameters the homework is asking about, how do I get them?
2. I assume that the parameters were the 31.614 and the power but I'm not sure, I know that function in the histogram is suposed to describe the line that was draw there, is that function what they are asking me about?
3. This is the most important one, I know all of this is used to try and predict a value right?
but I want to predict the probability to get the character if I use certain amout of in game currency but when I use the function I get some weird values, like for example I use x=40 in this formula:
=31.614*EXP(0.1573*40)
I get 17078.91704 and that number really doesnt make sense I would expect it to be something like 0.9 or something not such a big number

I hope you guys can help me

What are some good resources for learning or preparing for calculus? I heard Khan academy is good.

about extension of scalars:
given a k-algebra A and a field extension L/k. Is it true that
[math]M_n(A)\otimes_k L \simeq M_n(A\otimes_k L)[/math]
as L-algebras?

>>15324721
Mr. Tarrou on YouTube was a big help

pic rel shows what certain operations on the ordinal numbers look like formally.
Can these formulas be applied to any set in the sense that, for example, if [math]X[/math] is some ring, then division is possible in [math]X[/math] if and only if [math]\forall a,b\in X\;\forall a>0\;\exists! c,d\in X\;\big[ a\cdot c+d=b\big]\land\big[ d<a\big][/math]?

Any recommendations for a digital microscope? The stuff I like to inspect is mostly inorganic like rocks and coins and shit but it would be nice if I could also look at things like my hands or even prepared slides. Currently have one of these pieces of shit and for $12 it works OK.

This problem has been driving me fucking mental for hours now, I tried [math]\int_{0}^{20} 9800 * x * 2 * (6 - \frac{3x}{10})dx but apparently that doesn't work

>>15325091
Hey I have one those, it's really crap.
How much budget do you have? I lucked out and got a second-hand Zeiss optical microscope from my workplace.
Good ones like that are going to be well over a grand but with 200-500 dollars you can get bench-top chinkshit fully digital ones

Of what use is it that the dual space of the vector space of polynomials is isomorphic to the vector space of formal power series? What can you actually do with this result?

If h(x) = [(x+9)^1/2 - 3]/x
What is the limit of h(x) = 1/6 when x tends to 0 and when the real number 0 is not a part of the h domain

>>15324441
I realized I don't have to fix this. The cross section of people who like math and people who like maids and people who read 4Chan has a high amount of autistim. I can just publish a first edition with a lot of typesetting wrong and someone who can't handle badly centered columns will tell me how to fix it because they want the misaligned columns to go away.

Ignoring this for now to work more in written content. If anyone knows how to fix the centering, please tell me. If not I will just keep writing for now.

>>15325246
>What can you actually do with this result?
Share a secret
https://dl.acm.org/doi/10.1145/359168.359176

anyone know where i can find a copy of pic related? scihub doesn't seem to have the article and my two main private trackers for written content (myanonamouse and iptorrents) don't seem to have it either.
alternatively a link to a site with similar big journals would be fine too, if one exits

---

Cite this article
Tirole, R., Vezzoli, S., Galiffi, E. et al. Double-slit time diffraction at optical frequencies. Nat. Phys. (2023).

DOI
https://doi.org/10.1038/sNIGGER41567-NIGGER023-NIGGER01993-w

>>15325268
Clearly
[eqn] \lim_{x \to 0} h(x) = f'(9)[/eqn]
where
[eqn]f(x) = \sqrt{x}[/eqn]

So caclulate the derivative with the power rule [math]f'(x) = \frac{1}{2\sqrt{x}}[/math] and then plug in [math]x=9[/math]. So [math]f'(9) = \frac{1}{6}[/math].

>>15325198
The pressure is function of density, gravity and depth; the triangle is irrelevant

>>15325091
I suggest you go analog and put a camera on top. Nothing digital will ever beat a good lense or two. So get a stereo microscope.
If you want to get clear images while also being able to look manually, i suggest a trinocular setup, but if you only want the digital picture a mononocular arrengement will work too.
It also makes upgrading easyer, stick a 100$ shit camera on top if you like, or pop a 2000$ pro camera up there and take professional HD pics.

>>15325291
I meant to say force

>>15324516
I think I was confused because she turned the 1 in the numerator into n for b_n. I guess it makes sense n=> 1 but all the examples I had before were done by removing +5 or + something else from the denominator

Consider V = W + W* where W* is the dual space of W and + is the direct sum of the two vector spaces; all vector spaces are finitely dimensional. I want to find an as simple as possible isomorphism Phi between V and V*. My first intuition was[eqn]\Phi: V \ni (w,\, w^*) \mapsto (w^*,\, w) \in V^*.[/eqn]But that didn't work out. Do you think they only want me to use the evaluation mapping here or is there a simpler isomorphism?

>>15325372
>>15324516
The n is on the numerator because of reciprocals.
Makes the ratio easier to calculate the limit.

Since 1/(2n-3) is a rational function, usually
dropping everything except the highest degree
term does the trick. Hence, why I used 1/2n.

For a finitely dimensional vector space W, is this[eqn](W \oplus W^*)^* = W^* \oplus W^{**}[/eqn]correct? V* denotes the dual space of a vector space V.

>>15310646
>stupid questions thread
>assuming other threads are pretty smart
>on /sci/

Do electron spins spontaneously flip? Like, if we separate a beam of electrons into +1/2 and -1/2 beams (Zeeman effect), could the +1/2 spin beam be further split into two more beams?

>>15325421
They are isomorphic at least but then both sides are isomorphic to [math]W^2[/math].
I don't think they are equal as sets if you use the strict set theoretic definitions of all terms.

How do I find the dot product for a word problem, I've only ever been giving dot product questions in numerical form a dot b

>>15325395
Ah I see so b_n was 1/n because the denominator is half of a_n and then she multiplied ok thx

>>15325459
What would you say is an easy isomorphism between W + W* and (W + W*)*? It is + the direct sum.

>>15325468
you just need to multiply the lengths of the two vectors and the cosine of the angle inbetween

>>15324511
The sums telescope. For example [math]\sum_{k=1}^n f(k) - f(k-1) = f(1) - f(0) + f(2) - f(1) + \cdots = f(n)-f(0)[/math].

>>15325445
It's a common trick to keep out schizos

To prove an increasing sequence is bounded, all I have to do is prove that no term xn of the sequence exceeds an integer L right?

what's up with these fake wikipedia websites where low-life annoying losers scrap wikipedia and copy + paste the articles into their own webpages. Why does google not filter this trash? Wtf

>>15324441
Off center how? These lines (pic) are the same length. Keep in mind the matrix element includes the numbers to either side, which are of differing lengths. If you want them to be the same you could add some phantom characters to the top left corner number, for example.

>>15325540
correct, although L doesn't have to be an integer (even if it's usually more convenient to pick one).

>>15325540
if there exists an N such that |x_n| < N for all x_n of (x_n), then it's bounded. You need the absolute value for the lower-bound

>>15325549
>>15325553
This is a bit confusing to me since if I try to think about it logically, an increasing sequence should have a limit of infinity since it increases infinitely, I know this line of thinking is wrong but I can't understand why.

>>15325558
consider
0.1
0.11
0.111
etc.

>>15325559
Doesn't the sum of this sequence eventually go to infinity with a large enough n?

>>15324528
Your picture makes no sense at all. A probability is a number between 0 and 1, not something like 97. You should also probably flip the axes. I'm not looking through your 2000 line document but what you should want as a parameter is the probability of obtaining an item. Then your independent variable is the number of rolls and the dependent variable is the probability of obtaining your item.
Assuming the rolls are independent you will have a geometric distribution.

Also it looks like your hw wants you to _estimate_ a distribution's parameters rather than just modelling it and calculating a probability.

>>15325562
>sum
You are correct... but the sum of a sequence is a series. A sequence itself does not involve summation at all.

>>15325562
The sum of all increasing sequences goes to infinity. It's a necessary condition for the convergence of series that the sequence of the elements of the series converges to 0.

>>15325566
Oh I think I'm conflating the terms in my head. So finding the limit of a series is finding the value which xn eventually converges to?

>>15325569
*I meant sequence not series

>>15325282
There appears to be a preprint on arxiv: https://arxiv.org/abs/2206.04362
Stuff usually gets added to scihub in batches and this nature paper was published only yesterday.

>>15325569
>>15325574
Correct. In fact, that's how convergence of a sequence is defined.

>>15325585
Thank you for the help

>>15310646
>tfw moderately smart/slightly above average but not very smart.

People often think that I am arrogant and that I think I am smarter than I am because I take an interest in things like science but my intellectual ability is not stellar or amazing.

Has anyone else experienced this?

>>15325602
Yes, many of us have gone outside before and met other people and found that they have a plethora of different opinions and personalities.

>>15325620
>plethora

I need some help with an Epsilon Delta proof:
Prove: lim x->∞ (6x-5/3x-12)=2.
I know the first step should be |xn-2|. I did that and simplified to |19/(3x-12)| but I don't know where to go from there. Usually there is no denominator when I've done these proofs.

>>15325683
*When I say no denominator, I mean there isn't a x-c in the denominator.

>>15325685
Use the fact that lim a/b = lim a / lim b

I have no Idea how to do this and don't even know what I'd type in to find a lesson online

>>15325688
I don't think I can use that when doing it via an epsilon delta proof.

>>15325695
Why not? Delta-Epsilon is just a method to prove a limit.

>>15325691
This should be the Dirichlet eta function for x = 4, so you might be able to calculate eta(4). If you have the solution calculate the 0.999 times eta(4). Then set up a script to determine the n, maybe a computer can determine it by brute-force. This is how I would do it.

>>15325736
Addition: Wolfram alpha and the formula I found the wiki-page for the eta()-function say eta(4) = 7pi^4/720, so now you have an exact number.

>>15325691
>>15325736
?? just find K such that [math]|\sum_{n=K}^\infty\frac{(-1)^n}{n^4}|<0.001[/math], K=6.

Would i just take the derivatives of the given y functions and implement them into the original equation just to see if everything cancels out and it equals zero?

Consider three sets M, N and U. Set U is disjoint from M and N. Can I presume [math]|M| \neq |N| \implies |M \cup U| \neq |N \cup U|[/math] or do I have to prove this? M, N and U can be either finite or infinite, there are restrictions on this.

>>15325773
oh wait this is wrong, nvm

>>15325772
Yea

>>15325753
okay but why does my teacher say the answer is 5 isn't that greater than 0.01 as she points out in her work

So when getting the cross product do I remove any negative signs since here -8 is written as 8

>>15325778
thanks

my teacher's solution to these problems wont load so i cant check myself or get myself unstuck.

i was wondering what i would need to do for part b. the wronskian i got was -te^t but i dont think thats right

>>15325827
forgot image my b

>>15325775
Wtf when is this wrong?

>>15325834
Oh right infinite sets.

>>15324196
Shameless samefag here, does anyone have any ideas?

>>15325683
Your first step should be to assume an epsilon > 0 and start from the statement [math] | f(x) - 2 | < \epsilon [/math], then from that create a chain of if [math] ( \impliedby ) [/math] statements that ends up to something like [math] x > M( \epsilon) [/math] where [math] M( \epsilon ) [/math] is any function of epsilon. In this case, it would be:
[math] \impliedby x > \frac{19}{3 \epsilon} + 4 [/math]

what's an example for a bilinear transformation in R^2?

My prof ends up finding the parametric hamiltonian in pic related, by solving the 1 dimensional Schrodinger equation in a periodic potential using periodic boundary conditions.
The problem is I can't understand his derivation. Anyone knows a book where I can find something like this?

How to accomplish pic related as indicated by the red arrow? Sorry this isn't real analysis or something actually interesting and thanks in advance.

I need some help with the intergral test,

I get urning e^-n into 1/e^n because you move down a number with a negative exponent

I get that its continuous because we don't divide by 0 at any value of n

I get that its postitive because at no value of n is it negative

I get that its decreasing since e > 1 and as you increase the value of the denominator you get closer to 0

I don't get what's happening afterwards and youtube videos aren't heling me. As far as I can tell I can only tell that shes adding 0 and 1/e so relaces the n exponent with 1

Is the anti derivative of 1/e^n =1/t ?
Does she use t for a reason or did she pick a variable at random?

>>15326010
add up the terms in red

>>15326047

>>15326053
How have I never seen this method before? The American education system fucking failed me lol.

are there any good videos that can teach me to solve this sort of problem?

>>15325790
This shows that the sum of the terms _starting at 6_ are less than 0.01, so you need the first 5 in your partial sum.

>>15325793
-(-15-(-7))=8
>>15325891
[math](x,y)\mapsto xy[/math]
>>15326057
Looks like a mistake when plugging in the upper bound of t.

Why is this statement false:
If {xn} is a sequence, and for any e > 10^(-10000), there is a big N s.t. n > N -> |xn - 1| implies lim n->∞ xn = 1.

>>15326456
I don't follow your statement but I assume you're asking why we require [math]\varepsilon>0[/math] instead of [math]\varepsilon>\text{something small but positive}[/math]? If so the constant sequence [math]x_n=1+10^{-10001}[/math] should help you understand.

>>15325285
Thank you. Is there any other way to reach this result without using the power rule?

>>15326207
>>15326238
Well thanks everyone but I'm pretty sure I just failed, We we're supposed to make our formula sheet for the test and I forgot mine at home

Out of curiosity though
7n
/
29^(n-1)
n =1, n goes to infinte sum symbol

What method was the best to use for this one to find if it converges

>>15326888
sorry 7n should be 7^n

>>15326888
>>15326904

[eqn] \sum_{n=1}^\infty {7^n \over 29^{n-1}} [/eqn]

Clearly, this is a geometric series with powers
of "n" on the numerator and denominator. Split
the inside sum like this:

[eqn] \sum_{n=1}^\infty {1 \over 29}{\left({7 \over {29}}\right)}^n [/eqn]

The common ratio is less than one, so the series
converges and has an actual sum, namely:

[eqn] {{1 \over 29} \over {1-{7 \over 29}}}-{1 \over 29} [/eqn]

>>15326948
Does that work regardless of what 29^(n-x) x is

>>15326954
>>15326948
Yes...whatever "x" is has to be accounted
for as the first term...1/29^2, 1/29^3, etc...

>>15326971
Wait are you saying the answer is same whatever x is or x determines the first term.
I think its the first but I'm not sure

>>15326984
>>15326971
The sum changes because the first term changes

>>15326991
>>15326984
>>15326971
hmm

okay whats
7^n
/
29^n-2

>>15327007
and the whole sum thing
n = 1, n goes to infinite

>>15327007
>>15326991
The sum will be:
[eqn] {{1 \over 29^2} \over {1-{7 \over 29}}}-{1 \over 29^2} [/eqn]

>>15327021
okay I think I'm getting it
is 7/29 = r

>>15327071
>>15327021
Yes, it's the common ratio.

>>15325583
perfect, thanks

>>15327074
cool thx, I'm having trouble knowing which series test to do which test is
7n^2
/
29^2

1
/
sqrt(n)

e/n -e/n+1


I don't need the answers necessarily just which test

>>15327084
shit sry first was meant to be
7^n2
/
29^n2

>>15325583
thank you

>>15327084
First one is actually a subsequence of the
geometric series. The "n^2" skips terms that "n"
normally has, but use the ratio test.

The second is a p-series test with exponent
equalling 1/2.

The third looks like telescoping series and
the sum needs expansion to see it happen.