/sci/ - Science & Math

The official, temporary, memevirus season /sci/sqt/ discord: https://discord.gg/gbjGVu
Feel free to shitpost, join or not. It doesn't matter.

~UNANSWERED~

Math+CS
>>11477131
>>11479831
>>11484181
>>11484561
>>11486380

Physics+Engi
>>11477686
>>11483271 (are they?)
>>11484639 (phase is the measure of angular displacement between two signals at a given time. sine is separated from cosine by 90 degress; if cosine has phase zero, then sine has a phase of -90 degrees.)

Bio+Physiology+Memevirus
>>11479007
>>11481895 (talking about tomatoes)
>>11483586 (have sex)
>>11483785
>>11486544

Stupid
>>11478399 (it's not, it's cute)
>>11478970
>>11479609
>>11480187
>>11481752
>>11483174 (wonderful contribution)
>>11483341
>>11483315
>>11484192

>>11486380
Expand everything to first order.

working on a project that involves which in concept is a "genuinely non-euclidean first person shooter"

you've seen things like Duke 3D and Antichamber which use sort of "smoke and mirrors" re: generating non-euclidean geometry, but at least on the majority of continuous parts of these games the geometry is still fundamentally Euclidean.

Anyway, I'm working on a Wolf-3D style renderer specifically with the intent of making something genuinely non-Euclidean, to that extent, I'm replacing the draw checks associated with the camera with a hyperbolic distance metric, which works okay in generating distinct images. The issue is in quantizing movement of the player, you can generate hyperbolic geometries but being able to move through it is another thing. Does anyone have any good reference for this kind of thing?

tl;dr, how do I model physics in a non-Euclidean space, i.e., hyperbolic or elliptic? What points are "close" to a given point in non-Euclidean geometry? And any advice for metrics/distance function on hyperbolic geometry?

Anybody have any good reference material for this stuff?

>>11487281
I think you might have to full on learn Riemannian geometry.
Basically, your player has a velocity vector and an acceleration vector. His movement goes along the geodesic defined by the velocity vector, and his velocity gets updated by the parallel transport of the previous velocity vector plus the changes brought in by acceleration, and the direction he's facing also changes according to parallel transport.
The easiest route is probably picking up a General Relativity text and reading the appendix.

Dumb frogposter here
Please help me understanding the linear transformation properties anons:

A transformation is linear when
1) T(v + w) = T(v) + T(w)
2) aT(v) = T(av)

As far as I know, 1) preserves additivity of vectors and 2) associativity. But why is this "linear"? What does linear even mean in this context? I'm so clueless

>>11487328
>What does linear even mean in this context?
I means exactly 1) and 2). What don't you understand?

>>11487328
>2) associativity
Anon the [math]a[/math] is a scalar not a vector lol.
>>11487298
Here
http://www.supermath.info/InfiniteSeriesandtheResidueTheorem.pdf

>>11487177
When I fart do the farts become clouds in the sky?

>>11487328
>What does linear even mean in this context?
It means its graph is a line.

>>11487336
>a is a scalar not a vector
I know, I meant associativity of scalar multiplication. My bad though, it was clearly poorly worded.

>>11487335
I don't really see how this preserves lines being lines, the points in space being equaly spaced and so on.

>>11487328
An affine transformation sends lines to lines (technically, an affinity, but muh FTA and muh R has no automorphisms).
A linear transformation is an affine transformation which sends zero to zero.

ALTERNATIVELY: in a vector space, you can sum vectors, and multiply by a scalar. For a linear transformation, it essentially doesn't matter whether you sum before or afterwards, or multiply by a scalar before or afterwards.

>>11487348
> how this preserves lines being lines
In a vector space, a line is given by [math]a+ \lambda b[/math], for fixed [math]a, b \in V[/math] and [math]\lambda[/math] going through all the scalars.
For an affine transformation, we have [math]T(a+ \lambda b) = T(a)+ \lambda T(b)[/math], which is naturally also a line.

>>11487374
>affine
Linear.

how do we feel about network science?

>>11487376
>Linear
affine

>>11488624
>we
think for yourself retard

>>11487177
Could someone explain electrophiles and nucleophiles to me please? I'm confused about them.

Hey all, so I have a quick probability theory question for something I'm working on (not homework). Let's say I have a sequence of iid random variables and some continuous function g(), could be bounded or unbounded. The RVs are nice, no Cauchy distributions or other weird shit.

Now the sample mean of the iid random variables converges in probability to the population mean by the law of large numbers. If I want to say g(sample mean) converges to g(population mean) that's just continuous mapping theorem. But is there anything equivalent that says that the sample mean of g(random variables) converges to g(population mean)? Surely there has to be a name for this.

>>11489250
If g() is nonlinear I wouldn't expect the sample mean of g(x) to equal g(population mean of x).
For example, suppose the population of x consists half of +1, and half of -1, and suppose g(x) = x^2. Then the population mean of x is 0, and g(population mean) is still 0. g(x) == 1 at any sample point, so the sample mean of g(x) is still 1.

>>11489460
Hmm, that's a good counterexample. I suspect I'll have to specify some conditions for when this is true. For what it's worth, my RVs are all exponential family distributions. I might have to figure out some conditions for g() as well. I might dig into exponential family theory, maybe Morris/Efron worked on this back in the 70s or 80s.

so glycerol is supposedly not considered toxic and is regularly added to food
yet this datasheet says that it can be toxic and has warnings like irritating to skin and to "wash off immediately"
lethal dose to rats is 2g/kg
would this mean I would be likely to die if I orally ingested around 100g ?
are some MSDSes full of shit?

>>11489928
the first image sats 12g while this one says 2g
I must be dumb or something because I do not get it

>>11487328
first tenet of algebra: Geometry is just pictures we tell ourselves to let ourselves sleep at night

Now, linear means that each input to the function T is weighted, valued equally. So the effect of u in T(u) is the same thing as the effect of u in T(v+u). as in... T(v+u) - T(u) = T(v). Think of it like a linear function, y= ax. Each unit x matters equally and grows y corresponding to a, regardless where it is. But in any other function, a sine, a parabola, a quintic etc, where you add in the x matters. Parabola(u) != parabola (u+v) because inputting one unit of growth to input will vary the output differently depending on where the input is grown from ; Delta x^2 from 1 to 0 is different than delta x^2 from 100 to 99

>>11489937
>>11489928
Not all sources will concur. Lots of things normies say are safe are not safe. Some things are dangerous for rats and only slightly dangerous for humans. irritant warnings can come from very mild/potential irritation

>>11490241
it makes some sense I suppose
I looked up plain sugar which has the same irritant warnings though an LD50 in rats of lik 30000mg

How the fuck am I supposed to synthesize this using only methane, cyclohexane, and styrene for carbon? This is early into an OChem II class, so nothing that advanced. I'm pretty sure it's just a bunch of successive Wittig reactions, but I don't know how to get the regioselectivity when adding to the central cyclohexane.

>got a degree in aerospace engineering
>can't find a job here in Italy
time to move to Germany I guess

>>11490501
Wilkommen, Herr Corona!

>>11490408
two things. your reaction can have multiple products, and you just need to isolate pic related. the other thing is, USE the methane as a bufferswapper

>>11487177
what is the difference between Grassman numbers and differential forms?

>>11490553
one is a number, the other is a form.

>>11487177
Could solanine be useful in anyway against coronvirus

>>11490553
A differential form is a section of the exterior bundle, a grassman number is an element of the exterior algebra.

I posted this but nobody responded :(, can't delete the thread.
What should I get into to spend my time getting paid for designing/studying Pathogens and developing a treatment or vaccines for the named above?
I was thinking that the best one for that field would be Biochemistry but in my country, they have no job opportunities at all.
I could go for Chemical Engineering to "play it safe" and do a masters/Ph.D. specific to the area I like.
What do y'all think?
Sorry if I can't express myself that well but English is my 3rd language.
No, I'm not a pajeet btw.

>>11490590
so differential forms are Grassman numbers?

>>11490695
No.

>>11490695
That's not what I said, is it?

>Igor, this is Grichka.
>It's worse than we thought. It's happening this time.
>You could call it a catastrophe, a pandemic, the apocalypse. Yes.
>Here is what I need you to do. Take the truck.
>Head to one of the large stores, I mean one of the wholesale stores. Sam's Club or Costco.
>Walk directly to the toilet paper section. I've calculated the capacity of the truck. It will hold 68 of the 80 packs.
>You need to fill the truck to capacity and get the toilet paper back to the compound. Now.

question: explain this phenomenon.

>>11490700
>>11490710
excuse me i'm just trying to grasp this from a physics perspective. I know that differential forms on a manifold are viewed as smooth sections of wedge products of the cotangent bundle of a manifold [math]\mathcal{M}[/math]. This is [math]\Omega(\mathcal{M})[/math]. Isn't this an exterior algebra?

>>11490727
No.
At each point [math]p \in M[/math] we have a tangent vector space [math]T_pM[/math], and this vector space has an exterior algebra [math]\wedge (T_pM)[/math]. By gluing all these exterior algebras we obtain the exterior bundle [math]\wedge TM[/math]. Sections of [math]\wedge TM[/math] are differential forms.

There might be some variations in notation, tho. So say, a diferential form might need to be a homogenous element (entirely in one of the [math]\wedge ^k (TM)[/math] ) or whatever.

>>11490749
Small mistake, it's [math]\wedge ( T^*_p M)[/math] and [math]\wedge T^*M[/math]

if i eat a bunch of cans of tuna right now because im poor and under quarentine is it ok?

i usually dont eat tuna or fish in general so i probably have low mercury levels although i have merucry fillings..... so can i use up all my saved up mercury allowances??

>>11490749
Yeah i get it now thanks. The confusion was due to the way I thought differential form fields are defined..mainly that we dont take the “wedge of T*Ms” rather than the union of the exterior algebras of the T*Ms.

>>11490930
youll be fine, just avoid canned tuna after

>>11487177
Can someone pls help with this proof im trying to do?
Using the definitions found in chapter 2 of Rudin's principles of mathematical analysis, im trying to prove that every neighborhood is an open set, with the added twist that im trying to do it by proving that the complement is open (this doesn't result in circularity as far as I know).

So far ive got:

Let [math]x \in \mathbb{X}[/math] with [math](\mathbb{X},d)[/math] a metric space.
Then the neighborhood with radius [math]r > 0[/math] centered at [math]x[/math] is the set:

[eqn]\mathcal{N}_r (x) = \{ y \in \mathbb{X} : d(x,y) < r \}[/eqn]

and this its complement is:

[eqn](\mathcal{N}_r (x))^c = \{ y \in \mathbb{X} : d(x,y) \geq r \}[/eqn]

I want to see that if I take [math]y \in [(\mathcal{N}_r (x))^c]'[/math] (the set of all limit points of [math](\mathcal{N}_r (x))^c[/math]) then it follows that [math]y \in (\mathcal{N}_r (x))^c[/math] thus proving that [math](\mathcal{N}_r (x))^c[/math] is closed and in consequence, its complement, the original neighborhood, is open.

Let [math]y \in [(\mathcal{N}_r (x))^c]'[/math], then, as [math]y[/math] is a limit point of [math](\mathcal{N}_r (x))^c[/math] it follows that [math]\forall \epsilon > 0[/math] there exists [math]\mathcal{N}_{\epsilon} (y) : \mathcal{N}_{\epsilon} (y) \ni y' \neq y , y' \in \mathcal{N}_r (x))^c[/math]

What im having trouble with is what comes next, im having trouble finding a suitable radius [math]\epsilon[/math such that I can set up a triangle inequality and conclude that [math]d(x,y) \geq r[/math] , I had tried with [math]\epsilon = d(x,y) - r[/math] but that fails if [math]d(x,y) = r[/math]

>>11491251
Correct me if im wrong, but isn't the complement of every open set a closed set. Therefore, the complement of the neighborhood would have to be closed as the neighborhood is open (look at theorem 2.23). This is intuitive because if all points are interior points, then there is a distinctive boundary of "outer points" such that the neighborhood is not in the set. This would be in the complement and would cause this set to be closed.

>>11487177
https://www.youtube.com/watch?v=8HWpiXyI8yo
what is actually going on here?

>>11491251
Take a sequence [math](y_n)_n \subset N_r(y')^c[/math] converging to [math]y \in (N_r(y)^c)'[/math] and enclose each of them in a neighborhood [math]N_{\epsilon_n}(y_n)[/math]. Since metric spaces are Hausdorff the fact that [math]y_n \neq x'[/math] for all [math]x'\in N_r(x)[/math] and [math]n[/math] means that you can find [math]\delta_n,\epsilon_n > 0[/math] such that [math]N_{\epsilon_n}(y_n)\cap N_{\delta_n}(x') = \emptyset[/math]. What does this tell you?

>>11491279
Im trying to prove that a neighborhood is open using the same exact fact you're mentioning and not the other way around.

>>11491288
I think I kinda understand what youre trying to say but I dont know what a haussdorf space is and I havrnt learned about sequences yet
You dont think a triangle inequality can be set up? It looks very intuitive its just I cajt find thr right combination of distances

>>11487177
What kind of dog is that?

>>11491319
>You dont think a triangle inequality can be set up?
I'm not sure if the triangle inequality is useful here because right at the boundary the distance to the original neighborhood becomes infinitely small.
Hausdorff means that you can separate distinct points with their neighborhoods, but you really need to know how sequences work here. In essence, the argument is that you are able to bound every term in the sequence away from [math]every[/math] single point in [math]N_r(x)[/math] by some [math]\epsilon >0[/math]; in particular, they're bounded away from [math]x[/math] by [math]\epsilon + r[/math]. This should tell you that the limit of [math]y_n\rightarrow y[/math] will never "leak" into [math]N_r(x)[/math].

If I have the homogeneous 1D wave equation, what regularity do I need on the initial conditions to apply the d'Alembert formula?
Do I need any regularity to obtain non-classical solutions from it?

If I have a sequence [math] (a_n)_{n \in \mathbb{N}} [/math] which is in [math] l^2 [/math], can I prove that [math] \left (\frac{a_n}{\cos(n)} \right )_{n \in \mathbb{N}} [/math] is also in [math] l^2 [/math]?

>>11491251
Use the fact that [math]f(y) = d(x, y)[/math] is a continuous function. It then follows trivially.
>how do I prove it's continuous
You use triangle.
>>11491749
Pretty sure you can't. [math]\cos n[/math] gets arbitrarily next to zero, so you take a sequence like 1/2^n and just keep pushing the individual elements further back to that they fall somewhere where [math](1/2)^n / \cos j >1[/math]. Then the sum explodes.

what is the metal material in cheap jewellery? i'm talking like 25 cent gumball machine prizes, where you get a little emblem necklace or whatever and you can snap it right half in two. it's almost got the internal appearance of chocolate, structure-wise.

>>11491800
plastic

>orientation is simply which side of the car the wheel is

>>11491806
is it actually plastic? what with a metallized coating or something, like mylar?

>>11491841
>is it actually plastic?
yes

>what with a metallized coating or something
coating or paint yes

>like mylar?
mylar is plastic yes and some of the more popular mylar products are mylar coated with aluminium

>>11491811
>going for a ride on my friend's Klein surface
>suddenly notice that the wheel is now on the left side of the car

>>11491345
Shit I get you, the problem is that I havent learned about sequences yet.
Is there no way to get an expression that proves that the lower bound of [math]d(x,y)[/math] is r ?

can you separate sugar from a liquid or at least test for sugar?
i did an experiment with the enzyme lactase and milk, left it for a few days in a a warm place and came back to see that milk has changed.

>>11492157
Jesus Christ anon, I already told you. It's just [math]f(y)[/math] being continuous.
For [math]y' \in [ \mathcal{N}_r(x)^c]'[/math], we have that, for any [math]\delta > 0[/math], there's an [math]\epsilon > 0[/math] such that [math]y \in \mathcal{N}_{ \epsilon}(y') [/math] implies that [math]|f(y')-f(y)| < \delta [/math], and in particular [math]f(y') > f(y) - \delta [/math]. But there's always some [math]a_{ \epsilon} \in \mathcal{N}_{r}(x)^c [/math] such that [math]a_{ \epsilon} \in \mathcal{N}_{ \epsilon}(y') [/math] by definition, so [math]f(y') > f(a) - \delta \geq r - \delta[/math] for any [math]\delta > 0[/math].
It shortens to half a line if you can use sequences.

>>11492230
In retrospective, there's actually no real reason to define [math]f[/math] if we can't just abridge the whole thing with sequences.
So we get that, for [math]y' \in [ \mathcal{N}_r(x)^c]'[/math] and [math]\epsilon > 0[/math], there's [math] a_{ \epsilon} \in \mathcal{N}_{r}(x)^c [/math] such that [math]a_{ \epsilon} \in \mathcal{N}_{ \epsilon}(y')[/math]. Then [math]||y'-x|| \geq ||x - a_{ \epsilon } || - ||a_{\epsilon} - y'|| \geq r - \epsilon [/math] for any [math]\epsilon > 0[/math].

So, as far as we currently know, baryon number of the universe has to be perfectly conserved, whilst lepton number of the universe can slightly fluctuate due to neutrinos turning into anti-neutrinos. Is that right?

>>11492400
There is no internal symmetry that leads to the conservation of baryon number via Noether; in fact anomalies such as chiral-ABJ in SM can violate it.

Why do we describe the complex plane as [math]\mathbb {I\cup R}[/math] (I think that notation's correct), rather than replacing it with some form of [math]\mathbb I^2[/math] that more accurately reflects its modular nature?

(I am uncomfortable with saying that the "real" axis of a complex plane always equals the real axis of an xy plot, since the complex numbers projected onto a real xy domain are not surjective)

>>11492458
>(I am uncomfortable with saying that the "real" axis of a complex plane always equals the real axis of an xy plot, since the complex numbers projected onto a real xy domain are not surjective)
what

>>11492458
>I∪R
nobody uses that. complex numbers are denoted [math]\mathbb{C}[/math].

>>11492458
This is actually painful to read

I meant *modulo nature
And the I^2 doesn't make perfect sense, I know; but my question is, why don't we express the complex plane purely in terms of imaginary axes?

>>11492487
Feel free to define how that would work, specifically.

>>11492487
so what do you suggest exactly ? that we write something else than [math]a+bi[/math] ?

>>11492487
More like modulo any understanding how vector spaces work, and it's the structure that misses the point of a complex structure

>>11492475
plz no bulli
>>11492491 >>11492501
I don't know how. What is required to define a vector space?
According to pg.4 of this: http://web.math.ucsb.edu/~padraic/ucsb_2014_15/ccs_proofs_f2014/ccs_proofs_f2014_lecture9.pdf
you need to define closure, identity, commutativity, and associativity and such, right?

>>11492514
Do you think just taking the union of the imaginary and real lines gets you their span?

>>11492487
>why don't we express the complex plane purely in terms of imaginary axes?
because [math]\mathbb{C}[/math] naturally contains [math]\mathbb{R}[/math] as a subset and this is actually one of [math]\mathbb{C}[/math]'s most important properties. why would you want to get rid of it?

>>11487253
link expired pls remake

>>11492518
no but the sum does!, at least sum of any vecties blabla

>>11492533
https://discord.gg/fzJ5vp

>>11492523
It's more a question of semantics, [math]\mathbb C[/math] would still be [math]\mathbb C[/math].
It doesn't feel like [math]\mathbb R[/math] fully captures the modulo nature of [math]\mathbb C[/math], where i=i, i^2=-1, i^3=-i, i^4=1, i^5=i etc.
Since the modulo function destroys information, it feels like it should require special treatment. And the fact that we limit the angles of phasors from [math]0\leq \theta \lt 2\pi[/math] also feels like a red flag

>>11492541
-1 = -1, (-1)^2 = 1, (-1)^3 = -1, (-1)^4 = 1, (-1)^5 = -1, etc.

>>11492543
Yes, that logicaly follows.

>>11492545
The point is that your gibberish about the "modulo nature of C" applies just as well to R itself. Do you think R doesn't fully capture the nature of R?

>>11492541
>Since the modulo function destroys information,
dude what ? no, it doesn't. in real numbers you have (-1)^2 = 1, is this "loss of information" ?

>>11492553
>applies just as well to R itself.
That's fair, but it doesn't seem to be a problem in [math]\mathbb R[/math].
When you get into the infinitesimal rotations within [math]\mathbb C[/math], it feels like a lot of information gets lost when you constrain theta of a phasor between 0 and 2pi. The way I see it, theta should go from [math]-\infty\lt \theta \lt \infty[/math]; If you want to project it onto a 2D plane where [math]0\leq \theta\lt 2\pi[/math], shouldn't a distinction be made that you're no longer just in the Reals, but now you're in a modulo system?

>>11492588
>blah blah blah covering space

How can an antenna transmit a wave lenght larger than it itself is?

>>11492588
>That's fair, but it doesn't seem to be a problem in R.
It's literally EXACTLY the same problem

>>11492819
no shit. I meant in a practical sense

>>11492458
The complex field is usually described as [math]\mathbb{C} = (\mathbb{R}^2 , +, \times)[/math]
With [math](a,b) + (c,d) = (a+b,c+d)[/math] and [math](a,b) \times (c,d) = (ac-bd,ad+bc)[/math]
The fact that there's an isomorphism between [math]\mathbb{R}^2[/math] and [math]\mathbb{C}[/math] is a completely different matter from constructing [math]\mathbb{C}[/math] , and most definitely no one constructs [math]\mathbb{C}[/math] as the union of the imaginary numbers with the reals, that wouldn't even make sense bro.

Is this the place to ask for help with my homework? Just for clarification, I've already solved the question, but I think it may have been a fluke and I want to be sure I did it properly. It's really basic finance, but please don't point me to /biz/, they are all memeing retards.

I'm calculating current yield on a bond. The problem is that I don't know whether I am supposed to calculate with the bid price or the ask price, and they both give the 'correct' answer because I am only supposed to round to two decimal places.

>>11492919
Yes.
But if it`s bond stuff, you`ll have better luck in >>>/wsr/

Redpill me on the query complexity model of Hamiltonian simulation. It feels like one big meme to me, because as you approach lower bounds on query complexity, the actual oracle you're querying gets more and more convoluted to implement (in both quantum and classical resources).

"Consider a nonempty set Z, and a field K, such that [math] K^Z = { f : Z \rightarrow K / f \ is \ a \ function}[/math] and let's define:
[math] + : K^Z × K^Z \rightarrow K^Z, \ (f+g)(x) =f(x) +g(x) \ \forall x \in Z,[/math]
[math] \cdot : K × K^Z \rightarrow K^Z, \ (\lambda \cdot f)(x) = \lambda \cdot f(x) \forall x \in Z.[/math]
Then K^Z is a K-vector space."
I don't fully get this. As far as I get it this means that Z^K are all the functions which goes from Z to K and that fulfill those two conditions, but I can't see what that actually means. I would really appreciate if someone could give me an example and a counterexample of two functions that fulfill those conditions.

>>11493788
I was a bit confused at your notation at first and thought you meant sections of a graph. Now, [math]K^Z[/math] is basically the function space of functions [math]f:Z \rightarrow K[/math]. Recall the definition of a vector space [math]V[/\math] over a field [math]K[/math], this just means that there is an operation of vector addition and scalar multiplication which are both binary, and satisfy a bunch of conditions (associativity, commutativity, you get the drill).

In your example, we have defined vector addition and scalar multiplication. Now, endowed with these operations, show that [math]K^Z[/math] satisfies all the axioms (you can find them on Wikipedia). If you want to be pompous, let [math]F[/math] and [math]G[/math] be vector addition and scalar multiplication respectively. Then, we have defined:

[math]F(f,g)=f+g[/math]
[math]F(\lambda,f)=\lambda f[/math]

Where then we define (pointwise!):

[math](f+g)(x)=f(x)+g(x)[/math]
[math](\lambda f)(x)=\lambda f(x)[/math]

Is this a bit more clear anon?

>>11493820
Some wonky formatting, should be a vector space [math]V[/math] over a field [math]K[/math], and instead [math]G(\lambda,f)=\lambda f[/math]. A final remark, regarding examples/counterexamples, the binary operations you have given are definitions as I've mentioned in my previous post.

>>11493820
Sorry there was supposed to be brackets around K^z definition but I don't know why they weren't displayed. I think I got that, but I can't "see" it that's why I asked for examples. I just found an example here https://en.wikibooks.org/wiki/Linear_Algebra/Definition_and_Examples_of_Vector_Spaces#ex:FcnsNToRIsVecSp but I would really want to see a counterexample to actually know if I understood it, thanks for your help anyway anon.

>>11493887
I see. It's just a definition, it's not about seeing whether there's something there or not. We define our binary operations and then we define the result of these binary functions pointwise so that they're indeed a function in [math]K^Z[/math].

Simple example, let both the set and the field be the real numbers. Then, suppose that we have [math]\lambda=\pi[\math], [math]f, f(x)=x[/math] and [math]g, g(x)=2x[/math]. By our definitions:

[math](f+g)(x)=x+2x=3x[/math]
[math](\lambda f)(x)=\pi x[/math]

Is this a bit more clear? There aren't any counterexamples since it's just a definition, not a claim or statement.

>>11493918
Then ALL functions I worked with during HS were "vectors" of the set of functions that goes fro R to R? And same goes for the set of functions that goes from R^n to R? Wew
Thanks again anon

>>11493972
Anytime fren. A vector is just an element of the underlying set in the vector space, in which this case it would be [math]K^Z[/math] (or for our example [math]\mathbb{R}^{\mathbb{R}}[/math], and an element is just a function as you have described it. You can later define a normed space which is a vector space with an appropriate norm function, and allows you to deduce a bunch of properties of functions.

Friendly reminder that asking whether "thing" is "a vector" is useless. Vector spaces is all there is; it's only sometimes that we feel like talking about some particular elements of the space.

Is derivative of a (continuous) function always a linear function?
If so, what is the double derivative? How do you take derivative of a linear function? It's the same isn't it?

>>11494274
The derivative operator by itself is a linear map, but the derivative of a function is not necessarily linear. Consider [math]f(x)=x^3[/math]. Then, its derivative is [math]x^2[/math] which is clearly not linear. The double derivative is just the derivative of the first derivative, an easy way to compute is to just let your derivative function be [math]g[/math] and then compute the derivative normally to obtain the second derivative.

>>11494323
Then why did 3b1b animate a derivative of polynomial (similar to [math]x^3 [/math]) as just a tangent slope?

>>11494331
The little nitpick that's messing you up here is that the derivative EVALUATED AT A PARTICULAR POINT is the slope of the tangent line AT THAT POINT. But it doesn't make any sense to call the derivative a tangent slope without specifying WHERE on the curve it's tangent.
The derivative is a function; you give it a point, and it tells you what the tangent slope is there. That function doesn't have to be a linear one.

>>11494359
That's what I thought, but wasn't sure with terminology. Thanks

>take-home exam because you know what
>posted online on two separate pages, one has the exam file, the other is the submission page
>only ever look at the file page because why the fuck would I look at the submission page before I finish
>only thing on that page is "enjoy"
>do it by myself because lockdown, and I also have no friends
>80% of the questions are straight up from the textbook, without even having to look at it I remembered how to solve them from doing the problem sets
>look up the singular hard question, don't really find a single answer but piece something together
>due today, look at submission page
>"THIS IS A CLOSED NOTE TEST TO BE TAKEN BY YOURSELF BLAH BLAH BLAH"
>feel massively guilty because I obviously did not do it closed note
Did I cheat?

Why do endogenous and exogenous hormones cause a hypertrophy of the endometrium in post menopausal women, but steroids cause testicles in men to atrophy?

How do you factorize the polynomial in the denominator?

>>11494200
Correctamundo.
>>11494401
>did I cheat
No, anon, you did God's work.
>>11494958
Set [math]z = x^2[/math] and solve normally for z, then plug the identity back in.
By the by, does anyone have a good reference book for this sort of bullshit? I feel like it might come in handy sometimes.

>>11494964
>Set z=x2z=x2 and solve normally for z, then plug the identity back in.

That's not how you do it, if you plug in z=x^2 and solve you get z=(-1+-3i)/2 and that's for sure not the result I'm supposed to get. I'm supposed to get (x^2−x+1)(x^2+x+1)

If time is a scalar magnitude, does that mean that time represented in the X axis is always positive in both the right and the left side of 0?

>>11494990
no

>>11494997
why

Could I die from a perforated colon?

>>11494982
Don't mind the touhou posters, they're actually all pretty dim. Just do this

[math] x^4 + x^2 + 1 = (x^4 + 2x^2 + 1) - x^2 = (x^2 + 1)^2 - x^2 = (x^2 + 1 - x)(x^2 + 1 + x) [/math]

>>11494200
based retard

>>11495112
How do you go from
[math](x^2+1)^2−x^2[/math]
to [math](x^2+1−x)(x^2+1+x)[/math]

>>11495056
because that makes no fucking sense, does it?

>>11495129
>[math] a^2 - b^2 = (a - b)(a + b) [/math]

Anon, I...

>>11494990
Time is not a scalar, it's a component of a 4-vector.

>>11494200
This was such a conceptual hurdle for me. Being a physicsfag, I got so confused by the inconsistent ways the term "vector" was used. Is it an arrow, a quantity with direction and magnitude, an n-tuple, a thing that transforms like a vector?
Fucking hell.

I'm an undergrad student and one of professors thought I had potential and took me into their research lab. Now that I'm working in it I'm getting overwhelmed and frustrated by my incompetence. Do I a.) magically get my shit together, b.) tell my PI to reduce my responsibility, or c.) accept frustration?

Why does [math]\sum_{n=1}^∞\dfrac{1}{n}[/math] diverge and
[math]\sum_{n=1}^∞\dfrac{1}{n^m} \leftrightarrow m\geq 2[/math] converges?

Why
[math]\sum_{n=1}^\infty\dfrac{1}{n}[/math] diverges, and
[math]\sum_{n=1}^\infty\dfrac{1}{n^m}\leftrightarrow m\geq2[/math] converge?

>>11495348
The sum converges the moment [math]m>1[/math]. This is because the Riemann zeta is holomorphic to the right of 1.

>>11495348
Jackass intuition is that [math]\int x^m ~ dx= \frac{ x^{m+1}}{m+1}[/math] when [math]m \neq -1[/math] , which always gives you a function which converges at infinity when [math]m < -1[/math] and bounds the sum above, while naively plugging in [math]m=0[/math] into the formula gives you infinity.

>>11495383
*naively plugging in [math]m = -1[/math].

>>11495375
I would like to ask why that is too, but I feel like it will be some difficult math.
But it's pretty counter-intuitive that 1+(1/2)+(1/3)+...+(1/1000000)+... doesn't converge to something, and 1+(1/4)+(1/9)+(1/16)+... does

>>11495348
Try the integral test. You can notice that for the first sum, it will be larger than the integral of [math]1/x[/math] from 1 until infinity, yet the latter quantity diverges, so the sum must diverge as well. For the other sums, try comparing them to the integral [math]1/x^{m}[/math] from 1 to infinity where m is larger than 2. This integral converges, and after doing some manipulations you can bound the sum above by this integral, since the integral converges, the sum must converge as well.

>>11495515
The integral for m larger or equal than 2 will be larger than the second sum in your question but where n starts from 2 (which is basically equivalent as adding finite terms to a convergent sum still makes it convergent).

Could you follow my reasoning and provide feedback? 1/3

If for a cellular automaton:

The upper bifurcation value of [math]\displaystyle\frac{1}{{2}}+\frac{1}{{{2}\sqrt{{{2}}}}}[/math] is equal to [math]\displaystyle{{\cos}^{2}{\left(\frac{\pi}{{8}}\right)}}[/math]. Its conjugate, the lower bifurcation value is [math]\displaystyle\frac{1}{{2}}-\frac{1}{{{2}\sqrt{{{2}}}}}[/math] and by naturally [math]\displaystyle{{\sin}^{2}{\left(\frac{\pi}{{8}}\right)}}[/math], both adding up to 1.
[math]\displaystyle{{\cos}^{2}{\left(\frac{\pi}{{8}}\right)}}-{{\sin}^{2}{\left(\frac{\pi}{{8}}\right)}}=\frac{1}{\sqrt{{{2}}}}[/math]

When broken into factors:

[math]\displaystyle \cos{{\left(\frac{\pi}{{8}}\right)}}- \sin{{\left(\frac{\pi}{{8}}\right)}}[/math] is a very good approximation to [math]\displaystyle \cos{{\left({1}{r}{a}{d}\right)}}[/math]

[math]\displaystyle \cos{{\left(\frac{\pi}{{8}}\right)}}+ \sin{{\left(\frac{\pi}{{8}}\right)}}\approx \cos{{\left({1}\right)}}+ \sin{{\left({1}\right)}}[/math]

[math]\displaystyle \sin{{\left({1}\right)}}\approx{{\cos}^{2}{\left(\frac{\pi}{{8}}\right)}}[/math]

So the model is approximating its live values to cosine and sine of 1 radians as far as I can tell.

[math]\displaystyle \cos{{\left({1}\right)}}{\left( \cos{{\left({1}\right)}}+ \sin{{\left({1}\right)}}\right)}\approx{{\cos}^{2}{\left({1}\right)}}+ \cos{{\left({1}\right)}} \sin{{\left({1}\right)}}[/math]

[math]\displaystyle=\frac{1}{{2}}{\left( \cos{{\left({2}\right)}}+ \sin{{\left({2}\right)}}+{1}\right)}\approx\frac{1}{\sqrt{{{2}}}}[/math]

[math]\displaystyle= \cos{{\left({2}\right)}}+ \sin{{\left({2}\right)}}\approx\sqrt{{{2}}}-{1}[/math]

[math]\displaystyle={1}+ \sin{{\left({4}\right)}}\approx{3}-{2}\sqrt{{{2}}}[/math]

>>11495716
2/3

[math]\displaystyle= \sin{{\left({4}\right)}}\approx{2}-{2}\sqrt{{{2}}}[/math]

[math]\displaystyle={{\sin}^{2}{\left(\frac{\pi}{{8}}\right)}}=\frac{1}{{2}}-\frac{1}{{{2}\sqrt{{{2}}}}}[/math]

[math]\displaystyle={\left(-\frac{ \sin{{\left({4}\right)}}}{{2}}\right)}^{2}\approx{{\sin}^{2}{\left(\frac{\pi}{{8}}\right)}}[/math]

[math]\displaystyle=\frac{1}{{{4}-{2}\sqrt{{{2}}}}}=\frac{1}{{2}}+\frac{1}{{{2}\sqrt{{2}}}}={{\cos}^{2}{\left(\frac{\pi}{{8}}\right)}}[/math]

[math]\displaystyle=\frac{1}{{{2}+ \sin{{\left({4}\right)}}}}\approx{{\cos}^{2}{\left(\frac{\pi}{{8}}\right)}}[/math]

[math]\displaystyle={1}-\frac{{{\sin}^{2}{\left({4}\right)}}}{{4}}\approx{{\cos}^{2}{\left(\frac{\pi}{{8}}\right)}}[/math]

[math]\displaystyle=\frac{3}{{4}}+\frac{{{\cos}^{2}{\left({4}\right)}}}{{4}}~{{\cos}^{2}{\left(\frac{\pi}{{8}}\right)}}=\frac{1}{{2}}+\frac{1}{{2}}\sqrt{{2}}[/math]

[math]\displaystyle={{\cos}^{2}{\left({4}\right)}}~\sqrt{{{2}}}-{1}[/math]

This is an even better result.

Solve for x (closest approximation, in rad):

[math]\displaystyle{{\sin}^{2}{\left({x}\right)}}\approx{2}-\sqrt{{{2}}}[/math]

[math]\displaystyle{{\cos}^{2}{\left({x}\right)}}\approx\sqrt{{{2}}}-{1}[/math]

[math]\displaystyle{\left|{ \sin{{\left({x}\right)}}}\right|}=\sqrt{{{2}-\sqrt{{{2}}}}}[/math]

[math]\displaystyle{\left|{ \cos{{\left({x}\right)}}}\right|}=\sqrt{{\sqrt{{{2}}}-{1}}}[/math]

[math]\displaystyle \arcsin{{\left(\sqrt{{{2}-\sqrt{{{2}}}}}\right)}}={0.871611162}{r}{a}{d}[/math]

[math]\displaystyle \arccos{{\left(\sqrt{{\sqrt{{{2}}}-{1}}}\right)}}={0},{871611162}{r}{a}{d}[/math]

is the root angle of my cellular automaton.

where sin(x) + cos (x) ˜ sqrt(2)

[math]\displaystyle{{\sin}^{2}{\left({x}\right)}}-{{\cos}^{2}{\left({x}\right)}}=- \cos{{\left({2}{x}\right)}}={3}-{2}\sqrt{{{2}}}[/math]

[math]\displaystyle- \cos{{\left({2}{x}\right)}}={0.17157287525}[/math]

>>11495718
3/3

[math]\displaystyle- \sec{{\left({2}{x}\right)}}={5.82842712475}={{\cot}^{2}{\left(\frac{\pi}{{8}}\right)}}[/math]

This concludes my steps of reasoning. Being the mathlet that I am, I need /sqt/ to tell me if these steps are valid.

Afternotes:

5.82842712475 = square of silver ratio = [math]\displaystyle{\left({1}+\sqrt{{{2}}}\right)}^{2}[/math]

Once hyperbolic transformations are applied for:

[math]\displaystyle{e}^{u}=\frac{ \cos{{x}}}{ \sin{{x}}}[/math]

[math]\displaystyle \sec{{\left({2}{x}\right)}}= \coth{{u}}={{\cot}^{2}{\left(\frac{\pi}{{8}}\right)}}[/math]

The critical temperature of a square-lattice 2D Ising model is:

$\displaystyle{k}_{{{B}}}\frac{{T}_{{{c}}}}{{J}}=\frac{2}{ \ln{{\left({1}+\sqrt{{{2}}}\right)}}}$

from where I've derived automaton's critical point (highest live cell count) as the Golden Ratio from here.

>>11495718
3/3

[math]\displaystyle- \sec{{\left({2}{x}\right)}}={5.82842712475}={{\cot}^{2}{\left(\frac{\pi}{{8}}\right)}}[/math]

This concludes my steps of reasoning. Being the mathlet that I am, I need /sqt/ to tell me if these steps are valid.

Afternotes:

5.82842712475 = square of silver ratio = [math]\displaystyle{\left({1}+\sqrt{{{2}}}\right)}^{2}[/math]

Once hyperbolic transformations are applied for:

[math]\displaystyle{e}^{u}=\frac{ \cos{{x}}}{ \sin{{x}}}[/math]

[math]\displaystyle \sec{{\left({2}{x}\right)}}= \coth{{u}}={{\cot}^{2}{\left(\frac{\pi}{{8}}\right)}}[/math]

The critical temperature of a square-lattice 2D Ising model is:

[math]\displaystyle{k}_{{{B}}}\frac{{T}_{{{c}}}}{{J}}=\frac{2}{ \ln{{\left({1}+\sqrt{{{2}}}\right)}}}[/math]

from where I've derived automaton's critical point (highest live cell count) as the Golden Ratio from here.

name a page where I can outsource shit to shitskins

>>11495883
Listen very carefully.
Call your ISP's tech support, and pretend your router doesn't work. Ranjesh will introduce himself and ask you to unplug the router and wait fifteen seconds. Take advantage of the time window to ask Ranjesh if he has any cousins or nephews who are in university and need money. Ranjesh will tell you about his nephew Rajaan. Arrange with Ranjesh for Rajaan to do your homework, and make sure to carefully look over his work, because while the lad may be smart, he doesn't speak the best english. Then, send a pack of frozen Meatballs and Mozzarella Hot Pockets to Fort Knox on the mail to thank me for helping you.

Is there a good 3d anatomy program that is either free or that can be pirated easily?

>>11496062
>Fort Knox on the mail
in*
9.89/10

Is there a library for representing mathematical equations as objects that can do simple integrals and derivatives? Preferable in java

>>11494990
negative numbers are scalars

>>11496272
What about complex numbers?

I'm studying civil engineering and have to choose a construction project for a project management class i'm taking. Just wondering if you guys know of any websites/resources which detail aspects of past construction projects from a project management perspective. E.g. websites that provide detailed case studies for construction projects, would be ideal!

Hi, sci, when I calculate equations with several solutions with SOLVE my casio only gives me one of the results, I know this can be done in another way, this is just an easy example, but I need to get both solutions for a more complex equation. How do I do it using Solve?

Btw, the equation I need the answer to is: 15*tan(x) - 4.9*(15/(cos(x)*44.3)^2 = 30

>>11496364
Assuming that your Casio:
> sets [math]x_0=0[/math]
>applies Newton's method
You can try to cheat the calculator by writing x+1 or x+2 to punt the initial guess somewhere else so it hopefully reaches another answer. You might also be able to manually input an initial guess.
Overall, I posted that because I'm bored and you should just read the manual.

>>11487177
[math]E^2=p^2c^2+m^2c^4[/math] if [math]v=c[/math] then [math]E^2=2m^2c^4[/math] is this false simply because division by zero or something more physically meaningful?

>force is derivative of momentum against time
>(classical) momentum is mass times velocity
>derivative of a product is the first factor times the derivative of the second plus the derivative of the first factor times the second
Which one of these is untrue so that force on a body gaining/losing mass isn't just (dm/dt)v + ma. Why does it depend on the velocity at which mass is added/removed

>>11487177
A quick math question:
What terminology do I use to ask Wolfram Alpha how fast an exponential curve is doubling?

I'm CS so a bit soft on math and I've been applying half-remembered stuff on WA to make predictions on when the pandemic reaches numbers in particular places and to check for how well different outbreaks map to exponential curves and where the residuals deviate. It's interesting stuff but I'm stretching my knowledge a bit.

>>11495883
>name a page where I can outsource shit to shitskins
Fiverr would probably do desu

>>11497233
nvm, I'll answer my own question.
Take the fit equation from the exponential fit, refactor it as [math](x+n)/x=2[/math] and solve for n.
WA cleans up my input a bit too and gives me the below.


>[math]exp fit {1,1,1,1,1,2,4,4,5,5,5,5,5,5,5,6,8,8,12,20,28,39,52,66,102,155}[/math]
>[math]0.0143746 e^(0.356006 x)[/math]
>[math]Solve[E^(-0.356006 x + 0.356006 (n + x)) == 2, {n}][/math]

The answer is n≈1.94701 which if you're curious, is the time to double for New Zealand's total confirmed cases of coronavirus.

How to solve?

I got (3ab/2)*1/(b-a) not a/(b-a). Can anyone explain where the a in the answer came from?

Recently (before the weather grew warmer) my snot has been drying unusually. Normally it dries in pebble-like formations, globs. But it has since started drying in slate-like slabs or flakes, with a very watery underlayer.
Does anyone have any clue why this might be? It's not concerning to me but I don't see why it would have happened if there were no changes in allergens in my environment and no dietary changes

>>11497653
Use the first row to substitute for one of w, x or y in the other rows. Then you have three equations with three unknowns and can use Gaussian substitution.

>>11497653
Copy and paste it in.
https://www.symbolab.com/solver/linear-system-of-equations-calculator
>>11497654
Typo, I think.

>>11497653
The first three equations define a hyperplane. The last one asks for the intersection of the hyperplane with the unit sphere.
Naturally, if the solution is unique, the hyperplane is tangent to the sphere, and thus the solution is the hyperplane's normal appropriately scaled and oriented (norm one for the sphere, then make sure it's in the hyperplane by choosing the sign).
>>11497654
Looks like a typo to me.

How would I go about proving this?

>>11497786
There's some [math]\epsilon > 0[/math] such that [math]f(x_0) - 2 \epsilon > g(x_0)[/math]. There are thus [math]\delta _1 > 0[/math] and [math] \delta _2 > 0[/math] such that [math]|f(x)- f(x_0)| < \epsilon [/math] when [math] x \in N_{ \delta_1} (x_0)[/math] and [math]|g(x)-g(x_0)| < \epsilon [/math] when [math] x \in N_{ \delta _2} (x_0)[/math]. Take the neighborhood as [math]N = N_{ \delta_1} (x_0) \cap N_{ \delta _2} (x_0) [/math].
Finishing is left as an exercise to the reader.

>>11497802
thx bb

>>11497653
>>11497780
>The first three equations define a hyperplane.
nope, they define a line. that is, if they're in a general position, I'm not gonna check. if that's the case, the solution will be either a pair of points, a single point, or an empty set, depending on the relative position of the line and the sphere.

>>11497814
Oh shit, I'm going insane.
Thanks.

>>11492248
BtW I did manage the proof using only triangle inequalities, quite ugly tho.

>>11487177
G'day,

How do you work out the density form the mass flow rate formula? I initially took it as the density of water though appear to have gotten a completely different answer (98kg/s as opposed to the 75kg/s that is the correct amount).

>>11497786
>>11497802
Or just use the fact that f-g is continuous and that the inverse image under f-g of (0, infinity) is open.

>>11487177
A religious friend of mine does not believe that humans have evolved from apes. Other than goosebumps, the appendix, and the coccyx, what other pieces of evidence can I use to show her how retarded it is to refute evolution?

>>11497856
Seems about right looking at the units, what book?

>>11497863
don't bother using logic. When people like that go into denial, it's more effective to lead them into self-doubt than trying to convince them of anything.

>>11496474
v cannot be c though. I don't see any division by zero either.
>>11496476
The basic form of newton's second law is for a point mass only. Mass is assumed to be constant so [math] \dot{p}=m\dot{v} [/math]. Expressions like [math] F=\dot{m}v [/math] follow from applying the conservation of mass and Reynolds transport theorem to a control volume, and not just from Newton's law alone.
>>11497856
Use a table or some other reference (it's probably in the appendix of your textbook. If not, usually just assume 999 kg/m^3) to look up the density of water [math] \rho [/math] at that temperature. You are given the volumetric flow rate [math] Q=...\text{m}^3\text{/s}. [/math] Mass flow rate is [math]
\dot{m}=\rho Q [/math] and weight flow rate is [math] \dot{w}=\gamma Q=\rho g Q [/math] where g is gravitational acceleration.
>>11497863
The fossil record.

>>11497860
I don't think you can, because [math]f[/math] and [math]g[/math] are only specified to be continuous at [math]x_0[/math] .
I'm not sure if there was some version of that equivalence for functions only continuous at one point that I don't remember, tho.

/med/ question here.

I’m a fucking idioy and gave in to peer pressure night like a cuck. I snorted cocaine. Made my mood swin rapidly into an uncontrollable rage. I got some sleep and feel calm now, but my breathing is labored. Did I fuck anything up permanently by snorting this shit once?

>>11497900
No. Cocaine has a pretty short half life. You're going to be okay.

Can any math chad advice me here? I am in my late 20s and I have come to realise me being a brainlet when it comes to math is a major reason for my inability to do well in many of the fields I want to get into. So I am re-learning math hopefully master certain topics to a bachelor's degree level.(I am not deluded enough to think I can self teach every topic a math undergrad learns).

With that being said math will to me be a tool to get better at a different field(business/finance or programming) many of which will become far easier to master with a strong math background. So my questions are this.

1) What are some topics that are valuable in other fields like business, finance and CS?

2) Should I re-learn some physics as well? I remember having a conversation with an anon here who said physics phds are sought after from big name consultant companies to even big soccer clubs for sports science.

Sorry if this is vague. Just trying to lay out a path for myself.

Sleep related question here. Due to anxiety and depression, I have not been sleeping properly for 2 years. My sleep is partial like half sleep half awake which makes my eyes and body hurt when I wake up. What are some long term damages I have done to myself?

>>11487177
What is sum of 1+1 , i dunno the answer for that ?

>>11497922
basic college math and physics is just:
thomas' calculus
fundamentals of physics hallyday & resnick

>>11497922
Just do the khan academy math IMO, that runs though calculus. Pretty good, I learned more from that than my actual classes lol. That's pretty much a sufficient baseline for whatever you might need for business or programming or physics. I think Business is kind of the safest go-to degree program really, even if that just leaves you as overlord of a retail store or whatever that's still a perfectly good living though not very glamorous.

Why is any number inside a [math]\sin[/math] function multiplied by [math]\pi[/math] -> [math]\sin (x * \pi)[/math] always returns zero?

Say that I have some set [math]X[/math] and some set [math]Y[/math]. If I know that two functions [math]f, g[/math] satisfy

[math]f(x) \leq g(y) \; \forall \; x \in X, y \in Y[/math]

then does it follow that

[math]\mathrm{sup}_{x \in X} f(x) \leq \mathrm{inf}_{y \in Y} g(y)[/math] ?

Note that the inf/sup are not necessarily achieved.

>>11497932
Thanks but my question is physics useful to the fields I mentioned ?

>>11497958
it doesn't. sin(pi/2)=1

>>11497966
i said multiplied, not divided

>>11497972
Right. So did I. pi/2 is 0.5 multiplied by pi.

>>11497956
Thanks bro. Also is physics relevant?

>>11497958
Because [math] \sin\theta [/math] represents the vertical distance of a point on the unit circle from the x-axis, at an angle of [math] \theta [/math] from the x-axis (with counter-clockwise being a positive angle). [math] \pi [/math] is one half rotation, [math] 2\pi [/math] is two halves or one whole rotation, [math] 3\pi [/math] is one and a half rotations, etc. Every half rotation means there is no distance between the point and the x-axis. So sine is 0.

Suppose I have [math]f : X \to Y[/math], [math]g : Y \to Z[/math] such that [math]g[/math] is continuous and their composition [math]f \circ : X \to Z[/math] is continuous. Can I conclude that [math]f[/math] is also continuous?

>>11497981
I am asking because physics isn't also an end goal for me.

I was just wondering if any of it will boost my proficiency in business/finance/econ/cs.

>>11497988
Meant [math]f \circ g : X \to Z[/math] but you get what I mean.

>>11497978
oh, i forgot the condition [math]x \in \mathbb{N}[/math]

>>11497900
Well at least you learned your lesson.

>>11497988
what if g constant

>>11498003
Crap, I guess that's a no.

>>11498068
exactly

>>11497927
2.
>>11497993
Because [math]\sin 0=0[/math], [math]\sin \pi = 0 [/math], and we have the periodicity relation [math]\sin ( x + 2 \pi )= \sin x [/math], which implies [math] \sin (x + n 2 \pi ) = \sin x [/math] for [math]n \in \mathbb{Z}[/math] by induction.

>>11497924
>Due to anxiety and depression, I have not been sleeping properly for 2 years
Have you seen a doctor about this? That's really bad.
>What are some long term damages I have done to myself?

Why would you want to know that? Just correct your sleep pattern and you should be fine. I should think your memory of the last two years will have suffered as a result.

i dont know what this is called so i dont know how to google it. i tried probability but that wasnt it. say i'm throwing a dice 10 times. how do i calculate the odds of getting a six atleast once?

>>11498176
the probability of NOT getting six atleast once is the probability of getting six zero times.

the prob of not getting six in a single throw is 5/6 and you have 10 throws which are independent. therefore the prob of not getting six in 10 throws is (5/6)^10.

what you want is probability of the opposite which is therefore 1 - (5/6)^10

I hope I'm right, I fucking hate probability

If M is an invertible nxn real matrix with positive determinant, what would be a nice path in GL_n(R) from M to the identity?
I can think of one using the QR decomposition but is there a simpler path?

>>11498206
using QR or polar decomposition is a pretty patrician way because it almost proves that O(n) is a deformation retract of GL(n)

I guess you must go through an orthogonal matrix anyway. so you think of the columns in the matrix as of a basis and you apply gram-schmidt, which can clearly be done in a "movie" version. that gets you to an orthogonal matrix and now you apply some kind of inductive argument. first you rotate the basis to bring the first column to (1,0,..,0). now you rotate to bring second column to (0,1,0,..,0) but through rotations which keep (1,0,..,0) fixed. so you're actually doing the first step but one dimension lower. the (n-1)-th step necessarily brings also the n-th vector to (0,.0,+-1), the sign depending on the determinant.

I can't think of anything simpler.

>>11498206
Suggestion as follows:
Construct it explicitly for [math]GL(2, \mathbb{R}[/math] through muh angles.
For [math]GL(n, \mathbb{R}[/math], correct it piecewise. That is, take [math]v \in R^n[/math] and it's preimage [math]w \in R^n[/math], and extend to a full base. Apply a trick like the one for [math]GL(2, \mathbb{R}[/math] on the span to rotate [math]v[/math] to its image while keeping the rest of the base with fixed images, and then keep on repeating. Eventually halts.
>why wouldn't it work for -1 determinant
You have one dimension left and it's on the other side, so to correct you need to pass through zero.

Who's the best 2hu girl?

whats physical chemistry like in college?

>>11498480
Jo'on

>>11498480
Choosing Remilia is your fate.
https://www.strawpoll.me/19610286

>>11498564
>they deactivated the bot that votes 100 times for literally every poll made on the website specifically when I come up with a joke I want to do with it
FUCK

I've had this stomach pain in my upper abdomen for 2 days now. I'm pretty sure it's purely psychological and fear-based because yesterday afternoon I convinced myself I was fine and I felt great. As soon as I thought about COVID-19 though my stomach took a plunge, I lost my appetite and for the shivers.

I don't have any breathing problems I am aware of, but my chest does feel right at times. Pretty sure that's a placebo to. I'm just afraid of SUDDEN acute respiratory syndrome and dying.

How can I quell my stomach? It's really the only thing bothering me at this point, but whenever I think of COVID or getting it I go back to unbearable upper abdomen stomach area pain and a little bit of the chest.

>>11498615
Hm interesting. Can you even imagine pain? Are you sure it doesn't actually hurt you?

And if you're really sure it's psychological, then you have to calm yourself down yourself lol.
I can motivate you by telling you that you won't die from this pain if it's imagined, so if you're not able to suppress your fear, you'll be in pain but you'll survive. Now realize how unnecessary that anxiety is.
Also, statistically you will not get corona

>>11498626
I'm fascinated by your claim that statistically I will not get Corona. Do you mind telling me more about that?

I think I've driven myself into an anxiety based fear cooped up self quarantined here for weeks. I read the media too much and I took one look at the overwhelmed hospitals and all the international and government actions and it horrified me.

If your claim is true though, that statistically I will not get Corona assuming I just practice proper social distancing and PPE, then there's no reason to worry. This is just a fear train run amok and I hopped onto it.

>>11498643
Actually, it's difficult to measure if you'll statistically get Corona, and it depends on your location, etc.
But so far there's pretty low number of infected people, considering it's a pandemic and it hasn't yet been 2 weeks since we took action globally.
Also even if you get it, you have less than 8% (Italy example) chance of dying, so that's unlikely too.
Do what is advised and you'll minimize the chances even more, but don't overthink it, because it will turn into obsession. You're unlikely to die, unless our governments lie, which is unlikely too.
Just chill and see what God has prepared for you

>>11498663
Thanks bro. I'm from New England not too close to NYC but still.

Idk how I can escape my prison of fear and anxiety. It disables me and I know that in the unfortunate event that I do get Corona and I do show symptoms, my anxiety will shoot up so high I'll probably die from it before I could ever die of corona. I am deathly afraid of the sudden part of not being able to breathe and suffocating to death. It's unlikely but the chance is still there. I have to conquer my fear though. Fear is the enemy. Fear is the mind-killer. Fear is a little enemy that brings total obliteration.

>>11498206
It's trivial to write sign-preserving row operations as paths. Just compose those.

Anyone likes me?
>Too much pressure, close deadlines
>Be anxious
>Can't do shit and procrastinate all day
>No pressure, no concrete deadlines
>Too relaxed and procrastinate all day
I hate myself for this but this is me for the past 10 years. How do I fix this?

>>11498925
>you can reobtain the identity matrix from a matrix with positive determinant through sign preserving elementary row operations
This is literal hell to prove, anon.
> for [math]a[/math] and [math]b[/math] collumns, we have [math](a, b) \rightarrow (a+b, b) \rightarrow (a+b, -a) \rightarrow (b, -a)[/math], and by repeating the exact same thing once more we obtain that [math](-a, -b)[/math] can be obtained from [math](a, b)[/math] through sign preserving elementary collumn operations
>need to do an entire rodeo to show that Gaussian elimination can be deformed into something that gives you a diagonal matrix whose signs can then be corrected 2 by 2 with the above proposition
Why would you do this?

So I have devised a plan to curb my crippling jewtube addiction, at least until the next set of exams.
I will block access to it from my router, then change the router's access password to something I won't know until the next exams are over.
My question is if there's some way to do that cryptographically or something, have an encrypted password that can only be decrypted after a certain date.
I'm a quasiNEET so asking someone to hold onto the password is not really an option.

Any actuaries here? How hard was it to find a job?

So surely this is a frequently asked stupid question, but where can i get a run down on (non-rigorous) calculus up to and including multivariable/vector? I want to speed review but don’t have the time to slog through a 1,000 page tomb like Stewart’s Calculus.

>>11499059
Realize that to be happy, you have to also be working hard or you will despise yourself for being a slacker. Realize that to not get paralyzed from deadlines, etc, you have to be working hard and proactively, or you will find yourself cornered again.

>>11498176
You want the binomial distribution.

>>11495389
is it counter-intuitive that log(x) doesn't converge?

>>11497992
you mean [math] g \circ f : X \to Z[/math]

Brainlet medfag here trying to learn MRI physics.

What the fuck is a magnetic 'moment'? Explain to me like I'm literally retarded.

I'm an embalmer, and I'm about to run out of gloves. What are some good alternatives? This counts as science right? Science adjacent at the very least.

any way to remove scars at home? or reduce them? money doesn't matter much but going outside does they're about 5 years old now

What's the motivation behind why Cartan subalgebras are useful/interesting? Why should I care?

help with this gay shit pls

Exercise 8
Lengths of the sides of a right-angled triangle are three consecutive terms of an
arithmetic sequence. Calculate the length of the sides
a) perimeter of the triangle is 72 cm
b) area of the triangle is 54 cm2

>>11500341
The smallest side is [math]a[/math], the second smallest is [math]a+b[/math], and the third is [math]a+2b[/math], because they're in arithmetic sequence.
Then, because the perimeter is 72, we have [math]a+(a+b)+(a+2b)=3(a+b)=72[/math], and [math](a+b) = 24[/math].
Since the area is equal to the two cathetus multiplied divided by two, we also have [math]\frac{ a(a+b) }{2}=54[/math], and [math]a(a+b)=108[/math]. Swapping in the value of [math]a+b[/math] rom earlier, we have [math]a \times 24 = 108[/math] and [math]a=4.5[/math]. Then [math]b = 24 - a = 19.5[/math].

>>11500366
>>11500341
lol I thought that a) and b) were two independent questions and not properties of one triangle

>>11500381
FUCK YOU'RE RIGHT
Shit, this problem is ass.

>>11500398
Actually I'm not sure, I'm not the guy who asked this (>>11500341) question.

>>11500402
No, no, I checked if Pythagoras worked for the numbers I gave.
Anyhow: we call the smallest side [math]a-b[/math], the middle one [math]a[/math], and the hypothenuse is [math]a+b[/math]. Then Pythagoras gives [math](a-b)^2+a^2 = (a+b)^2[/math], which expands to [math]a^2 + a^2 + b^2 - 2ab = a^2 + b^2 + 2ab[/math], which simplifes to [math]a = 4b[/math], which you just swap into the perimeter/area fomula for both cases.

>>11500409
elegant

>>11499819
If you were a chemist in the 30's you could be discovering LSD right now

Can somebody explain to me what exacly is the [math]f \rightarrow g[/math] in functions? but not just [math]f[/math] and [math]g[/math] but anything that's to both sides of the arrow, i never really got that, if someone can provide articles about it even great

>>11500771
f:AB
means f is a function with domain A and codomain B. That means the arguments f takes are all the elements of A and the values of f lie in B.
>https://en.wikipedia.org/wiki/Function_(mathematics)

>>11500792
>f:AB
meant [math]f:A\to B [/math] (filter deletes the unicode arrow symbol)

>>11499936
You can get info about the irreps of Lie algebras from them. Specifically since characters are class functions, we can get the irrep characters from the Cartans, which can tell you the rank and module properties of the irrep spaces themselves. By studying the roots and their Weyl groups, you can then get the full ladder algebra structure on the irreps.

"The adsorption in
the micropore should not be considered as that of molecules
onto a solid surface but as the filling of molecules into a
nanospace where a deep potential field is generated by the
overlapping of all the wall potentials"

Please explain

>>accepted applicants will be notified by March 2020

>less than 1 week left of March
>committee probably can't even meet to finalize their decision because of corona-chan

reeeeee just tell me whether I've been accepted or not

Stupid question

Will this crisis actually collapse the American government/global economy and render it to Brazil status? Or will this disease go away and economy will recover sooner than all the doomers anticipate?


Trying to gauge whether this is an all out mass death global catastrophe American empire dying in less and a year scenario or a deadly pandemic for one year the return to normal.

I was asked to do a drawing of a small pipe system, and the pipes are supposed to be welded. How do I represent that in the drawing? I just put a line there (red circle) but I don't know if there is some sort of symbol for it.

Keep in mind that I have 0 knowledge of piping stuff or mechanical engineering for that matter, all I know how to use autocad.

>>11500771
Sometimes you'll see functions defined by something that looks like:
[math]f: A \rightarrow B[/math]
[math]f: a \rightarrow b[/math]

Where [math]A[/math] is the domain of [math]f[/math], [math]B[/math] is the codomain, [math]a \in A[/math] and [math]b[/math] is some expression given in terms of a.

hey so relativity

is ftl travel actually a conceptual possibility?

say one mass of objects is moved to another region of space, but travels faster than light to do so

say the travel takes 1 minute. they stay for a few hours, then go back to their destination. will their destination have only changed a few hours? or will some interstellar shit have happened?

probably been asked a million time desu

>>11501088
Does your boss have any knowledge? You need to know what type of weld it is and add the correct symbol.

>>11501242
[math]f : a \mapsto b = f(a)[/math] is the pedantically correct way to notate the mapping

Are there any actual studies that prove cortisol levels from stress affect skin and acne or is it just a meme?

>A (multiplicative) group G can be considered as a category with one object, G. Let hom(G,G) be the set of elements of G; composition of morphisms a,b is simply the composition ab given by the binary operation in G.

I'm having trouble seeing how a group element could be considered a morphism. The fact that group operations are closed seems to account with the requirement that hom(G,G) x hom(G,G) -> hom(G,G), but that does not explain how a group element can be thought of as a morphism f: G -> G.
I apologize for the stupidity, this is the first time I have tried to learn anything about categories.

>>11501305
Well, an element does act on the group itself by left multiplication, doesn't it?

>>11501305
The group multiplication is a map [math]G\times \rightarrow G[/math] from [math](a,b)\mapsto ab[/math]. Fix [math]a\in G[/math] then we can define a homorphism [math]M_a \in \operatorname{Hom}(\ast,\ast)[/math] sending [math]M_a(\ast) = a\cdot \ast[/math], and composition satisfies monoidality [math]M_a \circ M_b (\ast) = M_a(b\cdot \ast) = ab \cdot \ast = M_{ab}\cdot \ast[/math], hence as [math]a\in G[/math] varies we get an isomorphism [math]\operatorname{Hom}(\ast,\ast) \cong G[/math].

>>11501305
Basically, imagine I have an abstract object [math]A[/math] and a structure [math]a[/math]. So, say, [math]A[/math] could be a metric space and [math]a[/math] could be a metric, or [math]A[/math] could be a set and [math]a[/math] isn't anything. The duo [math](A, a)[/math] usually has a group of structure preserving symmetries, which form [math]A[/math]'s group of symmetries.
In categoryshittery, you can just take any group [math]G[/math] and pretend it's the group of symmetries of an object in a category, by literally saying "I have a category with one object, and the morphisms are this group."

>>11501311
Yes I see that now.
>>11501315
Thank you anon this cleared it up nicely.
>>11501319
This went over my head but thank you for responding.

>>11501259
All he told me was that it is stainless steel, so I guess not even he knows the weld type.

I unironically function the best with 25-27 hours circadian rhythm. Like, I periodically wake up in the mornings, then after a few days in the afternoons, then a few days later in the evenings, then early morning, and repeat...
I can't go to sleep after 24hours because I'm not tired yet. Is this unhealthy?

Hey bros, how exactly do I check if my answer is correct or not for the topic Infinite Series? I had wolframalpha help me with Integrals with it's step-by-step but I don't think wolframalpha does that (or I'm messing up with it), anyone else got a diff website that can show me the steps?

>>11501653
it's called "non-24 sleep wake disorder." It's a bitch. Keeping a 9-5 will kill you, and you won't have energy for friends and family.
But if you're a NEET recluse, it's otherwise perfectly healthy.
There is no cure.

>>11501315
>>11501305
>>11501311
You are missing the point. In a category, the arrows can be literally anything, not necessarily functions from G to G. So we just define a category with one objects and one morphism for every group element. Since group operations are associative and there is an identity element which corresponds to the identity morphism, we see that G can be thought of as a category with one object. No need to involve any actual from G to G. Because in general arrows are not maps.

>>11502301
that's not really true if I spend enough time in red light and deny myself caffeine, turn off electronics I can brute force a 2-4 AM bedtime even though my body largely follows this pattern you're describing.

>>11502783
Yeah adrenaline, good habits, and life goals really goes a long way. But slip up for a moment and you're fucked

>>11487177
What do you guys know about the National Geospatial Intelligence Agency?

>>11502779
You are right, but it's the same exact representation that you are suggesting and it's easier to understand it through maps.

I have a unit sphere S^2 and I want to put N points on it so that the minimum distance between 2 points is as large as possible. What is the asymptotic behavior of the minimum distance between two points of such a distribution of N points, d(N)? The distance could be euclidean or geodesic, the result will be the same as N->infinity.

>>11502892
The categories are isomorphic, sure, but the point of a category is to abstract away the irrelevant details, and in this case imo the maps are irrelevant. The only thing that matters here is how arrows behave, and we can simply define them to behave like group elements. Only the structure is relevant: the composition of arrows.
I think this is similar to thinking of groups axiomatically as having elements or using Cauchy's theorem to consider them as transformations of the group G->G.
As a matter of fact, this is exactly what you do: I suggest the arrows be just the group elements and you insist to change them to be elements of Sym(G): transformations G->G. Why?
Do you also find groups easier to understand as subgroups of Sym(G)?

>>11502892
Plus there are many ways of picking maps. For example instead of g-> (x-> gx) you could have picked g-> (x->x(g^-1)). I just said that in a category it's irrelevant what the underlying maps are or what they mean, all that matter is how they behave, so just make a take for every element of G and define arrow composition to correspond to group multiplication. Simple, easy, no need to jump through hoops and arbitrary choices.

>>11502779
to be absolutely fair, >>11501305 is about realizing G as a category where G is the object. then some representation G -> Hom(G,G) is necessary.

however I agree with you 100% that if the sentence read "a group is precisely a category with single object and isomorphisms" without the assumptions that G must the object, then the point would be as you've described it

>>11501256
Even going at lightspeed is impossible for anything with mass since it would take infinite energy to push something with mass to go that fast. But time will stop for you if you somehow go that fast.
Some people like to think that going faster than light would make time go backwards. The thing is that even if time is going backwards for the person traveling faster than light they arent actually traveling back in time since its only their own time that is going backwards, everything else is still moving forward in time. Honestly I dont know what would happen to you if time was going backwards for you.
It is true thought that going near the speed of light will make your time slow down.
My question would be if you were traveling in a space ship that has a constant force from the engines pushing you at 1m/s/s and you traveled 10 light years. Assuming the world operated in newtonian physics and you calculated how long it would take. Is that time you calculated the same amount of time you would experience in a world with relativity while inside the spaceship?
I find this interesting because it would mean that you can travel 10 light years in less than 10 years from your perspective.

>>11503161
It does not specify that the category must be subcategory of sets and functions or any other. Look at what the original poster quoted in greentext. G is just the label of the one object. He doesn't specify that the morphisms are represented by some actual maps because you don't need to do that! What is written in greentext is entirely correct.
The point where OP is confused is that he thinks that morphisms necessarily represent functions, and I explained that they don't. Morphisms are just some objects with a rule for composition. It can mean anything you want. In this example, the quoted author makes the arrows to correspond with group elements and composition with group multiplication. There does not have to be anything "behind" the arrows/morphisms, they're just what you want them to be.

>>11503067
>so that the minimum distance is as large as possible
First, we use the geodesic distance. We call the minimum distance [math]r[/math], and for any of the N points, we add in the entire unit disk around it with radius r on the surface of the sphere. All these disks have empty intersection, and thus, we actually have the classical circle packing in a sphere problem.
I dunno anything about circle packings in a sphere, tho, good luck finding and reading articles.

I'm trying to follow the theorem that states that fourier transformation of a discrete real valued even function is also real valued even function. I understand the even part and I am able to prove it. What I don't get is the real part. How can I show that the fourier transform is also real valued? What causes the coefficient at the sine summand to go to zero?

>>11503203
Sine is odd, your function is even. Odd*even=odd. (for functions) so your integral is 0.

>>11503208
I'm an idiot. Thank you very much, I understand now.

>>11503187
>radius r
*radius [math]\frac{r}{2}[/math]

>>11503169
>It does not specify that the category must be subcategory of sets and functions or any other.
okay, well, I guess it really doesn't. nevertheless I think that labeling the single element "G" is unfortunate.

Any rec ebook for learning highway/road construction?

can anyone tell me what's wrong with this elaboration?

[math]\int \csc^3 x dx \newline

\int \csc^2 x \csc x dx\newline

u = \csc x \implies du = - \csc x \cot x dx\newline
- dv = \csc^2x \implies - v = \cot x + C\newline

- \cot x \csc x - (\int(- \cot x)(- \csc x \cot x)dx)\newline
- \cot x \csc x - \int \cot^2 x \csc x dx\newline
- \cot x \csc x - \int (\csc^2 x - 1) \csc x dx\newline
- \cot x \csc x - \int \csc^3 x \csc x dx\newline
- \cot x \csc x - \int \csc^3 x dx - \sin\newline

\int \csc^3 x dx = - \cot x \csc x + \sin x - \csc^3 x dx\newline
2 \int \csc^3 x dx = - \cot x \csc x - \sin x\newline
\int \csc^3 x = (- \cot x \csc x - \sin x)/2
[/math]

I'm quite tired from solving multiple integrals so it's very likely that there are multiple mis-signaling here and there, but I'm afraid my main mistakes are wrong presumptions and substitutions.

>>11503396
so I broke latex, guess I'm bad at a variety of things. pic related is my somewhat formatted process

I like to play with graphs of functions, especially the graphs of solutions of differential equations. If you modulate the coefficients of the solutions, intuitively the shape of the graph can change quite drastically. I notice that when you smoothly vary the coefficients, the graph itself smoothly varies.

In my mind, it's almost as if I'm watching a 2d projection of a 3d surface, where I'm moving along the Z axis.Is there some term for this? Interpreting the family of solutions of a differential equation as a 3d surface? Is there any utility or insight that it can provide?

Is critical point really that strict? I'd think that water would be liquid under 5,000,000 atm.

>>11503396
Wide thread is wide. The dx thing is calculus, eww.

^I mean it would be liquid water at 1500 degrees C at 5,500,000 atm, right?

https://en.wikipedia.org/wiki/Superheated_water
https://en.wikipedia.org/wiki/Critical_point_(thermodynamics)

>>11487177

>>11503471
yeah and I can't delete the post now, sorry for the havoc

I showed that it commutes with the boundary. How do I show that it induces an isomorphism in reduced homology?

what kind of effect does watching porn constantly have if any? I'm not masturbating to it, when I beat off I don't watch porn but I use it as background noise and video constantly on my tv, you can easily watch it for a couple minutes then get back to what you're doing, like how some people have tv shows for "background" my gf recently found out about it and said it's really weird and probably unhealthy, is she right?

I'm going through Tao's Analysis book and I'm double guessing one exercise of his. Can /sci/ help in verifying my proof?

Lemma 2.2.10. Let a be a positive number. Then there exists exactly one natural number b such that b + + = a.

Definition 2.2.1 (Addition of natural numbers). Let m be a natural
number. To add zero to m, we define 0 + m := m. Now suppose
inductively that we have defined how to add n to m. Then we can add
n++ to m by defining (n++) + m := (n + m)++.

Axiom 2.4. Different natural numbers must have different successors;
i.e., if n, m are natural numbers and n != m, then n++ != m++. Equivalently , if n++ = m++, then we must have n = m.

Proof. We induct on b. For the base case, 0 + + = a, since a is positive, then it must not equal
0, so let us define a as 1. Then 0 + + = 1, by definition 1 = 0 + +, and so 0 + + = 0 + +, this
shows the existence. Now, by Axiom 2.4, 0 = 0, which shows the uniqueness. Inducting on b,
(b + +) + + = a, this is just the definition of addition and so this closes our induction.

>>11503574
Never mind figured it out :)

>>11503637
The proof of uniqueness seems correct.
The existence proof is weird. Specifically, I don't really know whatever it is you did.
My suggestion is as follows: set the property [math]P[/math] as [math]x \in \mathbb{N}[/math] is either zero or the successor of some number, and use induction to show every natural number satisfies [math]P[/math].
I know it feels like some sort of cheating to trivialize the base case like that, but it works fine.

What's exacly [math]\mathbb{R}^2[/math]?

>>11503728
The set {(a,b) where a and b are real numbers}.

Sanity check:
A singular simplex is homologic to a singular simplex with two vertices replaced?
It seems to be true by homotopy invariance (homotopic maps induce chain homotopic homomorphisms of homology). Is this true?

>>11503396
>>11503401
anyone? i think i can go further and make it more eligible than that, for what's worth

>>11503734
That's it? what's the square for?

If acceleration ([math]\overrightarrow{a}[/math]) is the change of velocity, then why is gravity (as an acceleration) constant? shouldn't that be a velocity?

>>11503738
>replaced
What the fuck do you mean replaced?
Consider the unit interval [math][0, 1][/math] and the boundary of the triangle just in case.
>chain homotopic homomorphisms of homology
Pretty sure there are five or six typos in there.
>>11503728
[math]R \times R[/math], the cartesian product.
>>11503746
It's there by analogy with multiplication.

>>11500993
Hard to say without context, but it sounds like an overintellectualization.
They probably want to emphasize that adsorption is controlled by potential fields, and potential fields can be linearly combined, so that the chemists making the particle know what to focus on.

>>11503757
>What the fuck do you mean replaced?
Meant to say interchanged (switched).

>>11503779
If I'm understanding you correctly, flipping vertices is a homeomorphism from the simplex to itself, so yes.
Then again, you might have some orientation reversal stuff.

>>11487177
I want to understand genetics, but I've never learned how to study well, is there any book or website I could read? Also, is anki any good?

>>11503793
What I meant to ask is if for any singular simplex f: D^n ->X , if f' is f precomposed with the homeomorphism that switches two vertices of the simplex D^n, then f + f' is the boundary of some chain. I've proven this in 2-dim case. Is this also true in higher dimensions?

>>11503857
Ah. It's true, because swapping two of the vertices swaps the orientation and thus [math]f[/math] and [math]f'[/math] correspond to opposite homology classes.

>>11503756
Things don't fall at a constant speed.

>>11503869
Yes, that's what I want to prove. How do you prove this?

>>11503067
Consider minimization of the (occupation of) Coulomb potential [math]\exp\left[-\sum\limits_{i<j}^N q_{ij}\ln|z_i-z_j| \right][/math] in the 2D plane [math]\mathbb{C}[/math]; for uniform charges [math]q_{ij} = 1[/math], the minimal configuration gives us an arrangement of points such that the inter-particle distance [math]|z_i-z_j|[/math] is maximized. Hence let us consider [math]Z_N[\{z_i\}_{i},g] = e^{-\sum\limits_{i<j}^N\ln |z_i-z_j|_g}[/math], where [math]|z|_g = \sqrt{g_{z\overline{z}}|z|^2}[/math] is the norm on the Riemann sphere [math]\overline{\mathbb{C}}[/math] with respect to a metric [math]g_{z\overline{z}}[/math] of positive scalar curvature. The goal is then to minimize the average [math]\overline{Z}_N[g] = \sum_{\{z_i\}_i\in Q_N(\overline{\mathbb{C}})}Z_N[\{z_i\}_i,g][/math] as a functional on the moduli space [math]\mathscr{M}_0[/math] of conformal classes of metrics, where [math]Q_N(\overline{\mathbb{C}}) = (\overline{\mathbb{C}}^N\setminus\Delta)/S_N[/math] is the configuration space, and find the asymptotics [math]\overline{Z}^*[/math] of [math]\inf \overline{Z}_N[/math]. Suppose a minimizer [math]g^*\in\mathscr{M}_0[/math] exists such that [math]\inf \overline{Z}_N = \overline{Z}_N[g^*][/math] is achieved with some [math]g^*[/math] such that the inter-particle distance [math]|z_i-z_j|_{g^*}\sim r_*[/math] is approximately uniform (it'd be interesting to prove this), then each summand reads [math]Z_N[\{z_i\}_i,g^*] = \frac{1}{N!}\prod_{i<j}^N\frac{1}{|z_i^*-z_j^*|_{g^*}} \sim \frac{r_{*}^{-N}}{N!}[/math], hence [math]\overline{Z}_N[g^*] \sim \sum_{k\leq N} \frac{r_*^{-k}}{k!} \rightarrow e^{-r_*}[/math], and the asymptotic dependence is suppressed exponentially.

>>11503871
They do fall at [math]9.8\rfrac{m}{s^2}[/math]

>>11503894
that isn't a speed

>>11503896
you still get what i'm saying, gravity is always the same here

>>11503902
Only approximately though. Gravity isn't actually constant, and it decreases with elevation. What even is your question?

What's a good short course on trig. before I continue with Calc 1? I am done watching professor Leonard's intermediate algebra playlist, but he doesn't have a completed Precalc playlist yet. Suggestions?

>>11503923
khan academy

This isn't right, is it? The Dijkstra's Alg does give 5.
I can't get this question to save myself. Help please.

Why is half a circle pi radians? Pi is defined as circumference over diameter, so I don't understand why one full rotation is 2pi instead of 1pi

>>11504085
Nevermind now I see it. In radians you take radius, not diameter

>>11504082
What's an MDST? Minimum distance what?

>>11503746
>>11503728
It's notation, because it's easier than writing (and reading) [math]\mathbb{R} \times \mathbb{R}[/math], where [math]\times[/math] is the Cartesian product on sets. Furthermore it readily generalizes to [math]n[/math]-fold Cartesian products, i.e. instead of writing [eqn]\underbrace{\mathbb{R} \times \cdots \times \mathbb{R}}_{n \text{ times}}[/eqn] we can just write [math]\mathbb{R}^n[/math]. This also nicely generalizes further to different types of products, for example [math]n[/math]-fold tensor products can be written conveniently as [math]\mathbb{R}^{\otimes n}[/math].

I should also note that most authors will assume [math]\mathbb{R}^2[/math] not as a mere set, but as a vector space over the reals with its elementwise addition and scalar multiplication. Again, because who the fuck unironically writes [math](\mathbb{R}^2, +, \cdot)[/math] outside of pedagogical texts?

>>11504094
Spanning tree. Sorry.

>>11504085
Apparently Euler used pi flexibly, allowing it to be 1/4 circumference, 1/2 circumference, 1/1 circumference etc. To him, it originally just meant "perimeter"
If you don't want to screw around with the 2, there's "tau" which is just short for 2pi.
But IMO, pi should just be redefined to ~6.28

>>11504130
>If you don't want to screw around with the 2, there's "tau"
No there isn't. Use pi.

>>11504120
>can't even fucking find a spanning tree with diameter 6
I think they just fucked it up.

>>11504135
Ya I thought so too. Do you have any idea what a solution might be though? It's kind of killing me.

>>11504136
Can you gimme the text so I can check specifically how they phrased Djikstra?

>>11504138
The text is West's Intro to Graph Theory. This is how they phrased it in the book.

If acceleration is by definition the variation of speed, how can it ever be constant?

Or does it mean that speed is changing at a constant rate (at a constant acceleration)?

>>11504138
This is the specific question, part b.

>>11504163
You have a ball that has a speed v=3t meters per second. So at time t=1, it has speed 3 m/s; at t=4 it has speed 3*4=12 m/s, etc. The speed increases linearly. Acceleration is the derivative of time. a=dv/dt=3 m/s^2. This is constant acceleration.

>>11504185
oh, now i get it, thanks man

>>11504155
>>11504173
FUCK. I can't comeup with anything. Sorry lad.

>>11504196
Thanks for easing my mind about the other part. I think I found one, but the assignment is due soon lol so I'll just have to submit my guess.

Electrophiles are molecules who are attracted to negative charged sites in other molecules, for example the oxygen double bonded to a carbon atom, while nucleophiles are attracted to positive charged sites in molecules, for example Hydrogen atoms bonded to a more electronegative atom

How does n + m being positive prove that a + b is negative? Either I'm grossly misunderstanding something or this is a typo. Anyone?

>>11503396
>>11503401
bampu. pls i'm not lazy, i just don't see what's wrong about this

>>11504373
What's the definition of negative integer being used? [math]a[/math] is negative if [math]-a[/math] is a nonzero natural number?
If so, it's immediate. n+m is natural, [math]-[-(n+m)]=n+m[/math].

So soap is supposed to be able to destroy viruses by washing your hand for at least 20 seconds.
How about for alcohol? How long does it take for alcohol to destroy viruses?